Commun. Math. Phys. 304, 1–48 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1214-0
Communications in
Mathematical Physics
From Open Quantum Systems to Open Quantum Maps Stéphane Nonnenmacher1 , Johannes Sjöstrand2 , Maciej Zworski3 1 Institut de Physique Théorique, CEA/DSM/PhT, Unité de Recherche Associée au CNRS, CEA-Saclay,
91191 Gif-sur-Yvette, France. E-mail:
[email protected]
2 Institut de Mathématiques de Bourgogne, UFR Science et Techniques, 9 Avenue Alain Savary, B.P. 47870,
21078 Dijon Cedex, France. E-mail:
[email protected]
3 Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, USA.
E-mail:
[email protected] Received: 30 April 2010 / Accepted: 25 October 2010 Published online: 9 March 2011 – © The Author(s) 2011. This article is published with open access at Springerlink.com
Abstract: For a class of quantized open chaotic systems satisfying a natural dynamical assumption we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories. 1. Introduction and Statement of the Results In this paper we show that for a class of open quantum systems satisfying a natural dynamical assumption (see §2.2) the study of the resolvent, and hence of scattering, and of resonances, can be reduced, in the semiclassical limit, to the study of open quantum maps, that is of finite dimensional quantizations of canonical relations obtained by truncation of symplectomorphisms derived from the classical Hamiltonian flow (Poincaré return maps). We first explain the result in a simplified setting. For that consider the Schrödinger operator P(h) = −h 2 + V (x) − 1, V ∈ Cc∞ (Rn ),
(1.1)
and let t be the corresponding classical flow on T ∗ Rn (x, ξ ) (Fig 1): def
t (x, ξ ) = (x(t), ξ(t)), x (t) = 2ξ(t), ξ (t) = −d V (x(t)), x(0) = x, ξ(0) = ξ. Equivalently, this flow is generated by the Hamilton vector field H p (x, ξ ) =
n ∂p ∂ ∂p ∂ − ∂ξ j ∂ x j ∂ x j ∂ξ j j=1
(1.2)
2
S. Nonnenmacher, J. Sjöstrand, M. Zworski
Fig. 1. An example of a potential, V ∈ Cc∞ (R2 ), to which the results apply: the Hamiltonian flow is hyperbolic on the trapped set in a range of energies – see [38, App. C]. In this example each energy surface p −1 (E) is three dimensional, so the Poincaré section is two dimensional as shown in Fig. 2
associated with the classical Hamiltonian p(x, ξ ) = |ξ |2 + V (x) − 1.
(1.3)
The energy shift by −1 allows us to focus on the quantum and classical dynamics near the energy E = 0, which will make our notations easier.1 We assume that the Hamiltonian flow has no fixed point at this energy: dp p−1 (0) = 0. The trapped set at any energy E is defined as K E = {(x, ξ ) ∈ T ∗ Rn : p(x, ξ ) = E, t (x, ξ ) remains bounded for all t ∈ R}. (1.4) def
The information about spectral and scattering properties of P = P(h) in (1.1) can be obtained by analyzing the resolvent of P, R(z) = (P − z)−1 ,
Im z > 0,
and its meromorphic continuation – see for instance [33] and references given there. More recently semiclassical properties of the resolvent have been used to obtain local smoothing and Strichartz estimates, leading to applications to nonlinear evolution equations – see [14] for a recent result and for pointers to the literature. In the physics literature the Schwartz kernel of R(z) is referred to as the Green’s function of the potential V . The operator P has absolutely continuous spectrum on the interval [−1, ∞); nevertheless, its resolvent R(z) continues meromorphically from Im z > 0 to the disk D(0, 1), in the sense that χ R(z)χ , χ ∈ Cc∞ (Rn ), is a meromorphic family of operators, with poles independent of the choice of χ ≡ 0 (see for instance [41, Sect. 3] and [39, Sect. 5]). The multiplicity of the pole z ∈ D(0, 1) is given by def m R (z) = rank χ R(w)χ dw, z
where the integral runs over a sufficiently small circle around z. We now assume that at energy E = 0, the flow t is hyperbolic on the trapped set K 0 and that this set is topologically one dimensional. Hyperbolicity means [24, Def. 17.4.1] that at any point ρ = (x, ξ ) ∈ K 0 the tangent space to the energy surface splits into the neutral (RH p (ρ)), stable (E ρ− ), and unstable (E ρ+ ) directions: Tρ p −1 (0) = RH p (ρ) ⊕ E ρ− ⊕ E ρ+ ,
(1.5)
1 There is no loss of generality in this choice: the dynamics of the Hamiltonian ξ 2 + V˜ (x) at some energy √ E > 0 is equivalent with that of ξ 2 + V˜ /E − 1 at energy 0, up to a time reparametrization by a factor E. The same rescaling holds at the quantum level.
From Open Quantum Systems to Open Quantum Maps
3
Fig. 2. A schematic view of a Poincaré section = j j for K E inside p −1 (E). The flow near K E can be described by an ensemble of symplectomorphisms between different components j – see §2.2 for abstract assumptions and a discussion why they are satisfied when the flow is hyperbolic on K E and K E has topological dimension one. The latter condition simply means that the intersections of K E with j ’s are totally disconnected
this decomposition is preserved through the flow, and is characterized by the following properties: ∃ C > 0, ∃ λ > 0, d exp t H p (ρ)v ≤ C e−λ|t| v , ∀ v ∈ E ρ∓ , ±t > 0.
(1.6)
When K 0 is topologically one dimensional we can find a Poincaré section which reduces the flow near K 0 to a combination of symplectic transformations, called the Poincaré map F: see Fig. 2 for a schematic illustration and §2.2 for a precise mathematical formulation. The structural stability of hyperbolic flows [24, Thm. 18.2.3] implies that the above properties will also hold for any energy E in a sufficiently short interval [−δ, δ] around E = 0, in particular the flow near K E can be described through a Poincaré map FE . Under these assumptions, we are interested in semiclassically locating the resonances of the operator P(h) in a neighbourhood of this energy interval: def
R(δ, M0 , h) = [−δ, δ] + i[−M0 h log(1/ h), M0 h log(1/ h)], where δ, M0 are independent of h ∈ (0, 1]. Here the h log(1/ h)-size neighbourhood is natural in view of results on resonance free regions in case of no trapping – see [26]. To characterize the resonances in R(δ, M0 , h) we introduce a family of “quantum propagators” quantizing the Poincaré maps FE . Theorem 1. Suppose that t is hyperbolic on K 0 and that K 0 is topologically one dimensional. More generally, suppose that P(h) and t satisfy the assumptions of §2.1–§2.2. Then, for any δ > 0 small enough and any M0 > 0, there exists h 0 > 0 such that there exists a family of matrices, {M(z, h), z ∈ R(δ, M0 , h), h ∈ (0, h 0 ]}, holomorphic in the variable z, and satisfying h −n+1 /C0 ≤ rank M(z, h) ≤ C0 h −n+1 , C0 > 1,
4
S. Nonnenmacher, J. Sjöstrand, M. Zworski
such that for any h ∈ (0, h 0 ], the zeros of def
ζ (z, h) = det(I − M(z, h)), give the resonances of P(h) in R(δ, M0 , h), with correct multiplicities. The matrices M(z, h) are open quantum maps associated with the Poincaré maps FRe z described above: for any L > 0, there exist a family of h-Fourier integral operators, {M(z, h)}, quantizing the Poincaré maps FRe z (see §2.3.2 and §3.3), and projections h (see §5.2.2) of ranks h −n+1 /C0 ≤ rank h ≤ C0 h −n+1 , such that M(z, h) = h M(z, h) h + O(h L ). The statement about the multiplicities in the theorem says that ζ (w) 1 dw m R (z) = 2πi z ζ (w) 1 tr (I − M(w))−1 M (w)dw. =− 2πi z
(1.7)
(1.8)
A more precise version of Theorem 1, involving complex scaling and microlocally deformed spaces (see §3.4 and §3.5 respectively), will be given in Theorem 2 in §5.4. In particular Theorem 2 gives us a full control over both the cutoff resolvent of P, χ R(z)χ , and the full resolvent (Pθ,R − z)−1 of the complex scaled operator Pθ,R , in terms of the family of matrices M(z, h); for this reason, the latter is often called an effective Hamiltonian for P. The mathematical applications of Theorem 1 and its refined version below include simpler proofs of fractal Weyl laws [43] and of the existence of resonance free strips [30]. The advantage lies in eliminating flows and reducing the dynamical analysis to that of maps. That provides an implicit second microlocalization without any technical complication (see [43, §5]). The key is a detailed understanding of the operators M(z, h) stated in the theorem. Relation to semiclassical trace formulæ. The notation ζ (z, h) in the above theorem hints at the resemblance between this determinant and a semiclassical zeta function. Various such functions have been introduced in the physics literature, to provide approximate ways of computing eigenvalues and resonances of quantum chaotic systems – see [10,20,47]. These semiclassical zeta functions are defined through formal manipulations starting from the Gutzwiller trace formula – see [42] for a mathematical treatment and references. They are given by sums, or Euler products, over periodic orbits where each term, or factor is an asymptotic series in powers of h. Most studies have concentrated on the zeta function defined by the principal term, without h-corrections, which strongly resembles the Selberg zeta function defined for surfaces of constant negative curvature. However, unlike the case of the Selberg zeta function, there is no known rigorous connection between the zeroes of the semiclassical zeta function and the exact eigenvalues or resonances of the quantum system, even in the semiclassical limit. Nevertheless, numerical studies have indicated that the semiclassical zeta function admits a strip of holomorphy
From Open Quantum Systems to Open Quantum Maps
5
O2
ϕ s
O1
O3
Fig. 3. This figure, taken from [34], shows the case of symmetric three disc scattering problem (left), and the associated Poincaré section (right). The section is the union of the three coball bundles of circle arcs (in red) parametrized by s (the length parameter on the circle, horizontal axis), and cos ϕ (vertical axis), where ϕ is the angle between the velocity after impact and the tangent to the circle. Green, blue, red strips correspond to different regions of forward escape; they are bounded by components of the stable manifold. The trapped set, T , shown in yellow, is the intersection of the latter with the unstable manifold
beyond the axis of absolute convergence, and that its zeroes there are close to actual resonances [10,48]. The traces of M(z, h)k , k ∈ N admit semiclassical expressions as sums over periodic points, which lead to a formal representation of ζ (z, h) = exp −
∞ tr M(z, h)k k=1
k
as a product over periodic points. That gives it the same form as the semiclassical zeta functions in the physics literature. In this sense, the function ζ (z, h) is a resummation of these formal expressions. As will become clear from its construction below, the operator M(z, h) is not unique: it depends on many choices which affect the remainder term O(h L ) in (1.7). However, the zeroes of ζ (z, h) in R(δ, M0 , h) are the exact resonances of the quantum Hamiltonian. Comments on quantum maps in the physics literature. Similar methods of analysis have been introduced in the theoretical physics literature devoted to quantum chaos. The classical case involves a reduction to the boundary for obstacle problems: when the obstacle consists of several strictly convex bodies, none of which intersects a convex hull of any other two bodies, the flow on the trapped set is hyperbolic. The reduction can then be made to boundaries of the convex bodies, resulting with operators quantization Poincaré maps – see Gaspard and Rice [17], and for a mathematical treatment Gérard [18], in the case of two convex bodies, and [31, §5.1], for the general case. Figure 3 illustrates the trapped set in the case of three discs. The semiclassical analogue of the two convex obstacle, a system with one closed hyperbolic orbit, was treated by Gérard and the second author in [19]. The approach of that paper was also based on the quantization of the Poincaré map near this orbit.
6
S. Nonnenmacher, J. Sjöstrand, M. Zworski
A reduction of a more complicated quantum system to a quantized Poincaré map was proposed in the physics literature. Bogomolny [4] studied a Schrödinger operator P(h) with discrete spectrum, and constructed a family of energy dependent quantum transfer operators T (E, h), which are integral operators acting on a hypersurface in the configuration space. These transfer operators are asymptotically unitary as h → 0. The eigenvalues of P(h) are then obtained, in the semiclassical limit, as the roots of the equation det(1 − T (E)) = 0. Smilansky and co-workers derived a similar equation in the case of closed Euclidean 2-dimensional billiards [13], replacing T (E) by a (unitary) scattering matrix S(E) associated with the dual scattering problem. Prosen [35] generalized Bogomolny’s approach to a nonsemiclassical setting. Bogomolny’s method was also extended to study quantum scattering situations [16,32]. Open quantum maps have first been defined in the quantum chaos literature as toy models for open quantized chaotic systems (see [28, §2.2], [29, §4.3] and references given there). They generalized the unitary quantum maps used to mimic bound chaotic systems [11]. Some examples of open quantum maps on the 2-dimensional torus or the cylinder, have been used as models in various physical settings: Chirikov’s quantum standard map (or quantum kicked rotator) was first defined in the context of plasma physics, but then used as well to study ionization of atoms or molecules [9], as well as transport properties in mesoscopic quantum dots [46]. Other maps, like the open baker’s map, were introduced as clean model systems, for which the classical dynamics is well understood [29,36]. The popularity of quantum maps mostly stems from the much simplified numerical study they offer, both at the quantum and classical levels, compared with the case of Hamiltonian flows or the corresponding Schrödinger operators. For instance, the distribution of resonances and resonant modes has proven to be much easier to study numerically for open quantum maps, than for realistic flows [7,25,27,28,37]. Precise mathematical definitions of quantum maps on the torus phase space are given in [28, §4.3–4.5]. Organization of the paper. In the remainder of this section we give assumptions on the operator P and on the corresponding classical dynamical system, in particular we introduce a Poincaré section and map associated with the classical flow. We refer to results of Bowen and Walters [8] to show that these assumptions are satisfied if the trapped set supports a hyperbolic flow, and is topologically one dimensional, which is the case considered in Theorem 1. In §3 we recall various tools needed in our proof: pseudodifferential calculus, the concept of semiclassical microlocalization, local h-Fourier integral operators associated to canonical tranformation (these appear in Theorem 1), complex scaling (used to define resonances as eigenvalues of nonselfadjoint Fredholm operators), microlocally deformed spaces, and Grushin problems used to define the effective Hamiltonians. In §4 we follow a modified strategy of [42] and construct a microlocal Grushin problem associated with the Poincaré map on . No knowledge of that paper is a prerequisite but the self-contained discussion of the problem for the explicit case of S 1 given in [42, §2] can illuminate the complicated procedure presented here. In [42, §2] one finds the proof of the classical Poisson formula using a Grushin problem approach used here. Because of the hyperbolic nature of the flow the microlocal Grushin problem cannot directly be made into a globally well-posed problem – see the remark at the end of §4. This serious difficulty is overcome in §5 by adding microlocal weights adapted to the flow. This and suitably chosen finite dimensional projections lead to a well posed Grushin problem, with an effective Hamiltonian essentially given by a quantization of the Poincaré map: this fact is summarized in Theorem 2, from which Theorem 1 is a simple corollary.
From Open Quantum Systems to Open Quantum Maps
7
2. Assumptions on the Operator and on Classical Dynamics Here we carefully state the needed assumptions on quantum and classical levels.
2.1. Assumptions on the quantum Hamiltonian P(h). Our results apply to operators P(h) satisfying general assumptions given in [30, §3.2] and [43, (1.5), (1.6)]. In particular, they apply to certain elliptic differential operators on manifolds X of the form X = XR
J Rn \BRn (0, R) , j=1
where R > 0 is large and X R is a compact subset of X . The reader interested in this higher generality should consult those papers. Here we will recall these assumptions only in the (physical) case of differential operators on X = Rn . We assume that P(h) = aα (x, h)(h Dx )α , (2.1) |α|≤2
where aα (x, h) are bounded in C ∞ (Rn ), aα (x, h) = aα0 (x)+O(h) in C ∞ , and aα (x, h) = aα (x) is independent of h for |α| = 2. Furthermore, for some C0 > 0 the functions aα (x, h) have holomorphic extensions to {x ∈ Cn : | Re x| > C0 , | Im x| < | Re x|/C0 },
(2.2)
they are bounded uniformly with respect to h, and aα (x, h) = aα0 (x) + O(h) on that set. Let P(x, ξ ) denote the (full) Weyl symbol of the operator P, so that P = P w (x; h D; h), and assume P(x, ξ ; h) → ξ 2 − 1
(2.3)
when x → ∞ in the set (2.2), uniformly with respect to (ξ, h) ∈ K ×]0, 1] for any compact set K Rn (here, and below, means that the set on the left is a pre-compact subset of the set on the right). We also assume that P is classically elliptic: def aα (x)ξ α = 0 on T ∗ Rn \{0}, (2.4) p2 (x, ξ ) = |α|=2
and that P is self-adjoint on L 2 (Rn ) with domain H 2 (Rn ). The Schrödinger operator (1.1) corresponds to the choices |α|=2 aα ξ α = |ξ |2 , aα ≡ 0 for |α| = 1, and a0 (x) = V (x) − 1. The assumption (2.3) shows that we can also consider a slowly decaying potential, as long as it admits a holomorphic extension in (2.2).
2.2. Dynamical assumptions. The dynamical assumptions we need roughly mean that the flow t on the energy shell p −1 (0) ⊂ T ∗ X can be encoded by a Poincaré section, the boundary of which does not intersect the trapped set K 0 . The abstract assumptions below are satisfied when the flow is hyperbolic on the trapped set which is assumed to be topologically one dimensional – see Proposition 2.1.
8
S. Nonnenmacher, J. Sjöstrand, M. Zworski
To state the assumption precisely, we notice that aα0 (x)ξ α p(x, ξ ) =
(2.5)
|α|≤2
is the semi-classical principal symbol of the operator P(x, h D; h). We assume that the characteristic set of p (that is, the energy surface p −1 (0)) is a simple hypersurface: dp = 0
on p −1 (0).
(2.6)
Like in the Introduction, we denote by t = exp(t H p ) : T ∗ X → T ∗ X def
the flow generated by the Hamilton vector field H p (see (1.2)). Our assumptions on p(x, ξ ) ensure that, for E close to 0, we still have no fixed point in p −1 (E), and the trapped set K E (defined in (1.4)) is a compact subset of p −1 (E). We now assume that there exists a “nice” Poincaré section for the flow near K 0 , namely finitely many compact contractible smooth hypersurfaces k ⊂ p −1 (0), k = 1, 2, . . . , N with smooth boundaries, such that ∂k ∩ K 0 = ∅, k ∩ k = ∅, k = k , H p is transversal to k uniformly up to the boundary,
(2.7) (2.8)
For every ρ ∈ K 0 , there exist ρ− ∈ j− (ρ) , ρ+ ∈ j+ (ρ) of the form ρ± = ±t± (ρ) (ρ), with 0 < t± (ρ) ≤ tmax < ∞, such that {t (ρ); −t− (ρ) < t < t+ (ρ), t = 0} ∩ k = ∅, ∀ k. (2.9) We call a Poincaré section the disjoint union def
N k . = k=1
The functions ρ → ρ± (ρ), ρ → t± (ρ) are uniquely defined (ρ± (ρ) will be called respectively the successor and predecessor of ρ). They remain well-defined for ρ in some neighbourhood of K 0 in p −1 (0)) and, in such a neighbourhood, depend smoothly on ρ away from . In order to simplify the presentation we also assume the successor of a point ρ ∈ k belongs to a different component: If ρ ∈ k ∩ K 0 for some k, then ρ+ (ρ) ∈ ∩ K 0 for some = k.
(2.10)
The section can always be enlarged to guarantee that this condition is satisfied. For instance, for K 0 consisting of one closed orbit we only need one transversal component to have (2.7)–(2.8); to fulfill (2.10) a second component has to be added. We recall that hypersurfaces in p −1 (0) that are transversal to H p are symplectic. In fact, a local application of Darboux’s theorem (see for instance [23, §21.1]) shows that we can make a symplectic change of variables in which p = ξn and H p = ∂xn . If ⊂ {ξn = 0} is transversal to ∂xn , then (x1 , · · · xn−1 ; ξ1 , · · · , ξn−1 ) can be chosen as coordinates on . Since ω p−1 (0) = n−1 j=1 dξ j ∧ d x j , that means that ω is nondegenerate. The local normal form p = ξn will be used further in the paper (in its quantized form).
From Open Quantum Systems to Open Quantum Maps
9
The final assumption guarantees the absence of topological or symplectic peculiarities:
k T ∗ Rn−1 with smooth boundary, and a symplectic There exists a set
k → k which is smooth up the boundary together (2.11) diffeomorphism κk :
k in T ∗ Rn . with its inverse. We assume that κk extends to a neighbourhood of
k . In other words, there exist symplectic coordinate charts on k , taking values in The following result, due to Bowen and Walters [8], shows that our assumptions are realized in the case of 1-dimensional hyperbolic trapped sets. Proposition 2.1. Suppose that the assumptions of §2.1 hold, and that the flow t K 0 is hyperbolic in the standard sense of Eqs. (1.5,1.6). Then the existence of satisfying (2.7)–(2.11) is equivalent with K 0 being topologically one dimensional. Remark. Bowen and Walters [8] show more, namely the fact that the sets {k } can be chosen of small diameter, and constructed such that ∩ K 0 forms a Markov partition for the Poincaré map. Small diameters ensure that (2.11) holds, while, as mentioned before, (2.10) can always be realized by adding some more components. Proposition 2.1 shows that the assumptions of Theorem 1 imply the dynamical assumptions made in this section. The proof of [38, App. C] shows that the following example of “three-bumps potential”, P = −h 2 + V (x) − 1, x ∈ R2 ,
V (x) = 2
3
exp(−R(x − xk )2 ),
k=1
xk = (cos(2π k/3), sin(2π k/3)), satisfies our assumptions as long as R > 1 is large enough (see Fig. 1). 2.3. The Poincaré map. Here we will analyze the Poincaré map associated with the Poincaré section discussed in §2.2, and its semiclassical quantization. 2.3.1. Classical analysis. The assumptions in §2.2 imply the existence of a symplectic relation, the so-called Poincaré map on .
k using κk given in (2.11), so that the More precisely, let us identify k ’s with Poincaré section =
N
k
k=1
Let us call def
T = K0 ∩ =
N k=1
k ⊂
N
T ∗ Rn−1 .
k=1
Tk the reduced trapped set.
k
The map def
f : T −→ T , ρ −→ f (ρ) = ρ+ (ρ)
10
S. Nonnenmacher, J. Sjöstrand, M. Zworski
Σ1
Σ2
Σ3
F
13
D42
A13
D12 F12
D13
D53
A12
Fig. 4. Schematic representation of the components Fik of the Poincaré map between the sets Dik and Aik (horizontal/vertical ellipses). The reduced trapped set Ti is represented by the black squares. The unstable/ stable directions of the map are the horizontal/vertical dashed lines
t (see the notation of (2.9)) is the Poincaré map for K 0 . It is a Lipschitz bijection. The decomposition T = k Tk allows us to define the arrival and departure subsets of T :
Dik = {ρ ∈ Tk ⊂ k : ρ+ (ρ) ∈ Ti } = Tk ∩ f −1 (Ti ), def
def
Aik = {ρ ∈ Ti ⊂ i : ρ− (ρ) ∈ Tk } = Ti ∩ f (Tk ) = f (Dik ). For each k we call J+ (k) ⊂ {1, . . . , N } the set of indices i such that Dik is not empty (that is, for which Ti is a successor of Tk ). Conversely, the set J− (i) refers to the predecessors of Ti . Using this notation, the map f obviously decomposes into a family of Lipschitz bijections f ik : Dik → Aik . Similarly to the maps ρ± , each f ik can be extended to a neighbourhood of Dik , to form a family of local smooth symplectomorphisms def
Fik : Dik −→ Fik (Dik ) = Aik , where Dik (resp. Aik ) is a neighbourhood of Dik in k (resp. a neighbourhood of Aik in i ). Since our assumption on K 0 is equivalent with the fact that the reduced trapped set T is totally disconnected, we may assume that the sets {Dik }i∈J+ (k) (resp. the sets {Aik }k∈J− (i) ) are mutually disjoint. We will call def
Dk = i∈J+ (k) Dik ,
def
Ai = k∈J− (k) Aik .
Notice that, for any index i, the sets Di , Ai both contain the set Ti , so they are not disjoint. We will also define the tubes Tik ⊂ T ∗ X containing the trajectories between Dik and Aik : def
Tik = {t (ρ), : ρ ∈ Dik , 0 ≤ t ≤ t+ (ρ)}.
(2.12)
See Fig. 4 for a sketch of these definitions, and Fig. 5 for an artistic view of Tik The maps Fik will be grouped into the symplectic bijection F between k Dk and k Ak . We will also call F the Poincaré map, which can be viewed as a symplectic relation on . We will sometimes identify the map Fik with its action on subsets of T ∗ Rn−1 :
ik −→ A
ik ,
ik = κ −1 ◦ Fik ◦ κk : D F i
ik def D = κk−1 (Dik ),
ik def A = κi−1 (Aik ).
From Open Quantum Systems to Open Quantum Maps
11
Fig. 5. Trajectories linking the boundaries of the departure set Dik ⊂ k and the arrival set Aik ⊂ i . Note the stretching and contraction implied by hyperbolicity. These trajectories and Dik ∪ Aik form the boundary of the tube Tik defined by (2.12)
Using the continuity of the flow t , we will show in §4.1.1 that the above structure can be continuously extended to a small energy interval z ∈ [−δ, δ]. The Poincaré map for the flow in p −1 (z) will be denoted by Fz = (Fik,z )1≤i,k≤N (see §4.1.1 for details). In the case of K 0 supporting a hyperbolic flow, a structural stability of K z holds in a stronger sense: the flows t K z and t K 0 are actually orbit-conjugate (that is, conjugate up to time reparametrization) by a homeomorphism close to the identity. [24, Thm. 18.2.3]. 2.3.2. Quantization of the Poincaré map In this section we make more explicit the operator M(z, h) used in Theorem 1. The semiclassical tools we are using will be recalled in §3. Let us first focus on a single component Fik : Dik → Aik of the Poincaré map. A
ik ) is a quantization of the symplectomorphism Fik (more precisely, of its pullback F semiclassical (or h-) Fourier integral operator, that is a family of operators Mik (h) : L 2 (Rn−1 ) → L 2 (Rn−1 ), h ∈ (0, 1], whose semiclassical wavefront set satisfies
ik × D
ik , WFh (Mik ) A
(2.13)
ik . (h-FIOs are defined in §3.3, and which is associated with the symplectomorphism F and WFh is defined in (3.9) below).
ik means the following thing: for any a ∈ Being associated to the symplectic map F
ik ), the quantum operator Opw (a) transforms as follows when conjugated by Cc∞ ( A h Mik (h) w 1−2δ
∗ Opw Mik (h)∗ Opw h (a)Mik (h) = Oph (αik Fik a) + h h (b),
(2.14)
where the symbol αik ∈ Sδ (T ∗ Rn−1 ) is independent of a, αik = 1 on some neighbourhood of Tk in k , and b ∈ Sδ (T ∗ Rn−1 ), for every δ > 0. Here Opw h denotes the semiclassical Weyl quantization on R2(n−1) (see Eq. (3.1), and Sδ (T ∗ Rn−1 ) is the symbol class defined in §3.1. The necessity to have δ > 0 in (2.14) comes from the slightly exotic nature of our Fourier integral operator, due to the presence of some mild exponential weights – see §3.5 below. The property (2.14), which is a form of Egorov’s Theorem, characterizes Mik (h) as
ik (see [42, Lemma 2] and a semiclassical Fourier integral operator associated with F [15, §10.2] for that characterization).
12
S. Nonnenmacher, J. Sjöstrand, M. Zworski
We can then group together the Mik (h) into a single operator-valued matrix (setting Mik (h) = 0 when i ∈ J+ (k)): M(h) : L 2 (Rn−1 ) N −→ L 2 (Rn−1 ) N , M(h) = (Mik (h))1≤i,k≤N . We call this M(h) a quantization of the Poincaré map F. The operators M(z, h) in Theorem 1 will also holomorphically depend on z ∈ R(δ, M0 , h), such that for each z ∈ R(δ, M0 , h) ∩ R the family (M(z, h))h∈(0,1] is an h-Fourier integral operator of the above sense. Comment on notation. Most of the estimates in this paper include error terms of the type O(h ∞ ), which is natural in all microlocal statements. To simplify the notation we adopt the following convention (except in places where it could lead to confusion): u ≡ v ⇐⇒ u − v = O(h ∞ ) u ,
Su T u + v ⇐⇒ Su ≤ O(1)( T u + v ) + O(h ∞ ) u ,
(2.15)
with norms appropriate to context. Since most estimates involve functions u microlocalized to compact sets, in the sense that, u − χ (x, h D)u ∈ h ∞ S (Rn ), for some χ ∈ Cc∞ (T ∗ Rn ), the norms are almost exclusively L 2 norms, possibly with microlocal weights described in §3.5. The notation u = OV ( f ) means that u V = O( f ), and the notation T = OV →W ( f ) means that T u W = O( f ) u V . Also, the notation neigh(A, B)
forA ⊂ B,
means an open neighbourhood of the set A inside the set B. Starting with §4, we denote the Weyl quantization of a symbol a by the same letter a = a w (x, h D). This makes the notation less cumbersome and should be clear from the context. Finally, we warn the reader that from §4 onwards the original operator P is replaced by the complex scaled operator Pθ,R , whose construction is recalled in §3.4. Because of the formula (3.13), that does not affect the results formulated in this section. 3. Preliminaries In this section we present background material and references needed for the proofs of the theorems. 3.1. Semiclassical pseudodifferential calculus. We start by defining a rather general class of symbols (that is, h-dependent functions) on the phase space T ∗ Rd . For any δ ∈ [0, 1/2] and m, k ∈ R, let Sδm,k (T ∗ Rd ) = a ∈ C ∞ (T ∗ Rd × (0, 1]) : ∀ α ∈ Nd , β ∈ Nd , ∃ Cαβ > 0,
β |∂xα ∂ξ a(x, ξ ; h)| ≤ Cαβ h −k−δ(|α|+|β|) ξ m−|β| , def
1
where ξ = (1 + |ξ |2 ) 2 . Most of the time we will use the class with δ = 0 in which case we drop the subscript. When m = k = 0, we simply write S(T ∗ Rd ) or S for the class of symbols. In the paper
From Open Quantum Systems to Open Quantum Maps
13
d = n (the dimension of the physical space) or d = n − 1 (half the dimension of the Poincaré section), and occasionally (as in (2.13) d = 2n − 2, depending on the context. The quantization map, in its different notational guises, is defined as follows w a w u = Opw h (a)u(x) = a (x, h D)u(x) 1 x + y i x−y,ξ / h def = u(y)dydξ, a ,ξ e (2π h)d 2
(3.1)
and we refer to [12, Chap. 7] for a detailed discussion of semiclassical quantization (see also [40, App.]), and to [15, App. D.2] for the semiclassical calculus for the symbol classes given above. We denote by δm,k (Rd ) or m,k (Rd ) the corresponding classes of pseudodifferential operators. The quantization formula (3.1) is bijective: each operator A ∈ δm,k (Rd ) is exactly represented by a unique (full) symbol a(x, ξ ; h). It is useful to consider only certain equivalence classes of this full symbol, thus defining a principal symbol map – see [15, Chap. 8]: σh : δm,k (Rd ) −→ Sδm,k (T ∗ Rd )/Sδm−1,k−1+2δ (T ∗ Rd ). m,k onto Sδm,k /Sδm−1,k−1+2δ . The combination σh ◦ Opw h is the natural projection from Sδ The main property of this principal symbol map is to “restore commutativity”:
σh (A ◦ B) = σh (A)σh (B). Certain symbols in S m,0 (T ∗ Rd ) admit an asymptotic expansion in powers of h, a(x, ξ ; h) ∼ h j a j (x, ξ ), a j ∈ S m− j,0 independent of h,
(3.2)
j≥0
such symbols (or the corresponding operator) are called classical, and make up the subclass Sclm,0 (T ∗ Rd ) (the corresponding operator class is denoted by clm,0 (Rd )). For any operator A ∈ clm,0 (Rd ), its principal symbol σh (A) admits as representative the h-independent function a0 (x, ξ ), first term in (3.2). The latter is also usually called the principal symbol of a. In §3.5 we will introduce a slightly different notion of leading symbol, adapted to a subclass of symbols in S(T ∗ R) larger than Scl (T ∗ Rd ). The semiclassical Sobolev spaces, Hhs (Rd ) are defined using the semiclassical Fourier transform, Fh : 1 def def
u 2H s = ξ 2s |Fh u(ξ )|2 dξ, Fh u(ξ ) = u(x)e−i x,ξ / h d x. (3.3) h (2π h)d/2 Rd Rd Unless otherwise stated all norms in this paper, • , are L 2 norms. We recall that the operators in (Rd ) are bounded on L 2 uniformly in h, and that they can be characterized using commutators by Beals’s Lemma (see [12, Chap. 8] and [43, Lemma 3.5] for the Sδ case):
ad N · · · ad1 A L 2 →L 2 = O(h (1−δ)N ) A ∈ δ (X ) ⇐⇒ (3.4) for linear functions j (x, ξ ) on Rd × Rd , where ad B A = [B, A].
14
S. Nonnenmacher, J. Sjöstrand, M. Zworski
For a given symbol a ∈ S(T ∗ Rd ) we follow [42] and say that the essential support is contained in a given compact set K T ∗ Rd , ess-supph a ⊂ K T ∗ Rd , if and only if ∀ χ ∈ S(T ∗ Rd ), supp χ ∩ K = ∅ !⇒ χ a ∈ h ∞ S (T ∗ Rd ). The essential support is then the intersection of all such K ’s. Here S denotes the Schwartz space. For A ∈ (Rd ), A = Opw h (a), we call WFh (A) = ess-supph a,
(3.5)
the semiclassical wavefront set of A. (In this paper we are concerned with a purely semiclassical theory and will only need to deal with compact subsets of T ∗ Rd . Hence, we won’t need to define noncompact essential supports). 3.2. Microlocalization. We will also consider spaces of L 2 functions (strictly speaking, of h-dependent families of functions) which are microlocally concentrated in an open set V T ∗ Rd : def
H (V ) = {u = (u(h) ∈ L 2 (Rd ))h∈(0,1] , such that ∃ Cu > 0, ∀ h ∈ (0, 1], u(h) L 2 (Rd ) ≤ Cu ,
∃ χ ∈ Cc∞ (V ), χ w (x, h Dx ) u(h) = u(h) + OS (h ∞ )}.
(3.6)
The semiclassical wave front set of u ∈ H (V ) is defined as: WFh (u) = T ∗ Rd \{(x, ξ ) ∈ T ∗ Rd : ∃ a ∈ S(T ∗ Rd ), a(x, ξ ) = 1, a w u L 2 = O(h ∞ )}. (3.7) The condition (3.7) can be equivalently replaced with a w u = OS (h ∞ ), since we may always take a ∈ S (T ∗ Rd ). This set obviously satisfies WFh (u) V . Notice that the condition does not characterize the individual functions u(h), but the full sequence as h → 0. We will say that an h-dependent family of operators T = (T (h))h∈(0,1] : S (Rd ) → S (Rk ) is semiclassically tempered if there exists L ≥ 0 such that
x −L T (h)u H −L ≤ C h −L x L u H L , h ∈ (0, 1), x = (1 + x 2 )1/2 . def
h
h
Such a family of operators is microlocally defined on V if one only specifies (or considers) its action on states u ∈ H (V ), modulo OS →S (h ∞ ). For instance, T is said to be asymptotically uniformly bounded on H (V ) if ∃ C T > 0 ∀ u ∈ H (V ) ∃ h T,u > 0, ∀ h ∈ (0, h T,u ), T (h)u(h) L 2 (Rk ) ≤ C T Cu . (3.8) Two tempered operators T, T are said to be microlocally equivalent on V , iff for any u ∈ H (V ) they satisfy (T − T )u L 2 (Rk ) = O(h ∞ ); equivalently, for any χ ∈ Cc∞ (V ), (T − T )χ w L 2 →L 2 = O(h ∞ ).
From Open Quantum Systems to Open Quantum Maps
15
If there exists an open subset W T ∗ Rk and L ∈ R such that T maps any u ∈ H (V ) into a state T u ∈ h −L H (W ), then we will write T = T (h) : H (V ) −→ H (W ), and we say that T is defined microlocally in W × V . For such operators, we may define only the part of the (twisted) wavefront set which is inside W × V : WFh (T ) ∩ (W × V ) = (W × V )\{(ρ , ρ) ∈ W × V : ∃ a ∈ S(T ∗ Rd ), b ∈ S(T ∗ Rk ), a(ρ) = 1, b(ρ ) = 1, bw T a w = O L 2 →L 2 (h ∞ )}. (3.9) def
(h) : If WFh (T ) ∩ (W × V ) W × V , there exists a family of tempered operators T 2 2
are microlocally equivalent on V , while T
is OS →S (h ∞ ) L → L , such that T and T outside V , that is
◦ a w = O(h ∞ ) : S (Rd ) → S (Rk ), T for all a ∈ S(T ∗ Rd ) such that supp a ∩ V = ∅. This family, which is unique modulo OS →S (h ∞ ), is an extension of the microlocally defined T (h), see [15, Chap. 10]. 3.3. Local h-Fourier integral operators.. We first present a class of globally defined h-Fourier integral operators following [42] and [15, Chap. 10]. This global definition will then be used to define Fourier integral operators microlocally. Let (A(t))t∈[−1,1] be a smooth family of selfadjoint pseudodifferential operators, ∀t ∈ [−1, 1],
∗ d A(t) = Opw h (a(t)), a(t) ∈ Scl (T R ; R),
where the dependence on t is smooth, and WFh (A(t)) ⊂ T ∗ Rd , in the sense of (3.5). We then define a family of operators U (t) : L 2 (Rd ) → L 2 (Rd ), h Dt U (t) + U (t)A(t) = 0. U (0) = I d. (3.10) An example is given by A(t) = A = a w , independent of t, in which case U (t) = exp(−it A/ h). The family (U (t))t∈[−1,1] is an example of a family of unitary h-Fourier integral operators, associated to the family of canonical transformations κ(t) generated by the (time-dependent) Hamilton vector fields Ha0 (t) . Here the real valued function a0 (t) is the principal symbol of A(t) (see (3.2)), and the canonical transformations κ(t) are defined through d κ(t)(ρ) = (κ(t))∗ (Ha0 (t) (ρ)), κ(0)(ρ) = ρ, ρ ∈ T ∗ Rd . dt If U = U (1), say, and the graph of κ(1) is denoted by C, we conform to the usual notation and write U ∈ Ih0 (Rd × Rd ; C ), where C = {(x, ξ ; y, −η) : (x, ξ ) = κ(y, η)}. Here the twisted graph C is a Lagrangian submanifold of T ∗ (Rd × Rd ). In words, U is a unitary h-Fourier integral operator associated to the canonical graph C (or the symplectomorphism κ(1) defined by this graph). Locally all unitary h-Fourier
16
S. Nonnenmacher, J. Sjöstrand, M. Zworski
integral operators associated to canonical graphs are of the form U (1), since each local canonical transformation with a fixed point can be deformed to the identity, see [42, Lemma 3.2]. For any χ ∈ S(T ∗ Rd ), the operator U (1) χ w , with χ ∈ S(T ∗ Rd ) is still a (nonunitary) h-Fourier integral operator associated with C. The class formed by these operators, which are said to “quantize” the symplectomorphism κ = κ(1), depends only on κ, and not on the deformation path from the identity to κ. This can be seen from the Egorov characterization of Fourier integral operators – see [42, Lemma 2] or [15, §10.2]. Let us assume that a symplectomorphism κ is defined only near the origin, which is a fixed point. It is always possible to locally deform κ to the identity, that is construct a family of symplectomorphisms κ(t) on T ∗ Rd , such that κ(1) coincides with κ in some neighbourhood V of the origin [42, Lemma 3.2]. If we apply the above construction to get the unitary operator U (1), and use a cutoff χ ∈ S(T ∗ Rd ), supp χ V , then the operator U (1)χ w is an h-Fourier integral operator associated with the local symplectomorphism κ V . Furthermore, if there exists a neighbourhood V V such that χ V ≡ 1, then U (1)χ w is microlocally unitary inside V .
For an open set V Rd and κ a symplectomorphism defined in a neighbourhood V of V , we say that a tempered operator T satisfying
) −→ H (κ(V
)), T : H (V is a microlocally defined unitary h-Fourier integral operator in V , if any point ρ ∈ V has a neighbourhood Vρ ⊂ V such that T : H (Vρ ) −→ H (κ(Vρ )) is equivalent to a unitary h-Fourier integral operator associated with κ Vρ , as defined by the above procedure. The microlocally defined operators can also be obtained by oscillatory integral constructions — see for instance [30, §4.1] for a brief self-contained presentation. An example which will be used in §4.1 is given by the standard conjugation result, see [42, Prop. 3.5] or [15, Chap. 10] for self-contained proofs. Suppose that P ∈ clm,0 (Rd ) is a semi-classical real principal type operator, namely its principal symbol p = σh (P) is real, independent of h, and the Hamilton flow it generates has no fixed point at energy zero: p = 0 !⇒ dp = 0. Then for any ρ0 ∈ p −1 (0), there exists a canonical transformation, κ, mapping V = neigh((0, 0), T ∗ Rd ) to κ(V ) = neigh(ρ0 , T ∗ Rd ), with κ(0, 0) = ρ0 and p ◦ κ(ρ) = ξn (ρ) ρ ∈ V, and a unitary microlocal h-Fourier integral operator U : H (V ) → H (κ(V )) associated to κ, such that U ∗ PU ≡ h Dxn : H (V ) → H (V ). While ξn is the (classical) normal form for the Hamiltonian p in V , the operator h Dxn is the quantum normal form for P, microlocally in V . The definition of h-Fourier integral operators can be generalized to graphs C associated with certain relations between phase spaces of possibly different dimensions. Namely, if a relation C ⊂ T ∗ Rd × T ∗ Rk is such that its twist C = {(x, ξ ; y, −η) ; (x, ξ ; y, −η ) ∈ C}
From Open Quantum Systems to Open Quantum Maps
17
is a Lagrangian submanifold of T ∗ (Rd × Rk ), then one can associate with this relation (microlocally in some neighbourhood) a family of h-Fourier integral operators T : L 2 (Rk ) → L 2 (Rd ) [2, Def. 4.2]. This class of operators is denoted by Ihr (Rd × Rk ; C ), with r ∈ R. The important property of these operators is that their composition is still a Fourier integral operator associated with the composed relations. 3.4. Complex scaling. We briefly recall the complex scaling method of Aguilar-Combes [1] – see [39,41], and references given there. In most of this section, this scaling is independent of h, and allows to obtain the resonances (in a certain sector) for all operators P(h), h ∈ (0, 1], where P(h) satisfies the assumptions of §2.1. For any 0 ≤ θ ≤ θ0 and R > 0, we define θ,R ⊂ Cn to be a totally real deformation of Rn , with the following properties: θ ∩ BCn (0, R) = BRn (0, R), θ ∩ Cn \BCn (0, 2R) = eiθ Rn ∩ Cn \BCn (0, 2R), (3.11) n α θ = {x + i f θ,R (x) : x ∈ R }, ∂x f θ,R (x) = Oα (θ ). If R is large enough, the coefficients of P continue analytically outside of B(0, R), and we can define a dilated operator: def
Pθ,R = P θ,R ,
u) Pθ,R u = P( ˜ θ,R ,
is the holomorphic continuation of the operator P, and u˜ is an almost analytic where P extension of u ∈ Cc∞ (θ,R ) from the totally real submanifold θ,R to neigh(θ,R , Cn ). The operator Pθ,R − z is a Fredholm operator for 2θ > arg(z+1) > −2θ . That means that the resolvent, (Pθ,R − z)−1 , is meromorphic in that region, the spectrum of Pθ,R in that region is independent of θ and R, and consists of the quantum resonances of P. To simplify notations we identify θ,R with Rn using the map, Sθ,R : θ,R → Rn , θ,R x −→ Re x ∈ Rn ,
(3.12)
and using this identification, consider Pθ,R as an operator on Rn , defined by −1 ∗ ∗ (here S ∗ means the pullback through S). We note that this identificaton (Sθ,R ) Pθ,R Sθ,R satisfies ∗ C −1 u L 2 (Rn ) ≤ Sθ,R u L 2 (θ,R ) ≤ C u L 2 (Rn ) ,
with C independent of θ if 0 ≤ θ ≤ θ0 . The identification of the eigenvalues of Pθ,R with the poles of the meromorphic continuation of (P − z)−1 : Cc∞ (Rn ) −→ C ∞ (Rn ) from {Im z > 0} to D(0, sin(2θ )), and in fact, the existence of such a continuation, follows from the following formula (implicit in [39], and discussed in [45]): if χ ∈ Cc∞ (Rn ), supp χ B(0, R), then χ (Pθ,R − z)−1 χ = χ (P − z)−1 χ .
(3.13)
This is initially valid for Im z > 0 so that the right-hand side is well defined, and then by analytic continuation in the region where the left-hand side is meromorphic. The reason
18
S. Nonnenmacher, J. Sjöstrand, M. Zworski
2 2
Fig. 6. The complex scaling in the z-plane used in this paper
for the Fredholm property of (Pθ,R − z) in D(0, sin(2θ )) comes from the properties of the principal symbol of Pθ,R – see Fig. 6. Here for convenience, and for applications to our setting, we consider Pθ,R as an operator on L 2 (Rn ) using the identification Sθ,R above. Its principal symbol is given by pθ,R (x, ξ ) = p(x + i f θ,R (x), [(1 + id f θ,R (x))t ]−1 ξ ), (x, ξ ) ∈ T ∗ Rd , (3.14) where the complex arguments are allowed due to the analyticity of p(x, ξ ) outside of a compact set — see §2.1. We have the following properties Re pθ,R (x, ξ ) = p(x, ξ ) + O(θ 2 ) ξ 2 , Im pθ,R (x, ξ ) = −dξ p(x, ξ )[d f θ,R (x)t ξ ] + dx p(x, ξ )[ f θ,R (x)] + O(θ 2 ) ξ 2 .
(3.15)
This implies, for R large enough, | p(x, ξ )| ≤ δ, |x| ≥ 2 R !⇒ Im pθ,R (x, ξ ) ≤ −Cθ.
(3.16)
For our future aims, it will prove convenient to actually let the angle θ explicitly depend on h: as long as θ > ch log(1/ h), the estimates above guarantee the Fredholm property of (Pθ,R − z) for z ∈ D(0, θ/C), by providing approximate inverses near infinity. We will indeed take θ of the order of h log(1/ h), see (3.31).
3.5. Microlocally deformed spaces. Microlocal deformations using exponential weights have played an important role in the theory of resonances since [21]. Here we take an intermediate point of view [26,43] by combining compactly supported weights with the complex scaling described above. We should stress however that the full power of [21] would allow more general behaviours of p(x, ξ ) at infinity, for instance potentials growing in some directions at infinity. Let us consider an h-independent real valued function G 0 ∈ Cc∞ (T ∗ Rd ; R), and rescale it in an h-dependent way: G(x, ξ ) = Mh log(1/ h)G 0 (x, ξ ),
M > 0 fixed.
(3.17)
From Open Quantum Systems to Open Quantum Maps
19
For A ∈ m,0 (Rd ), we consider the conjugated operator e−G
w (x,h D)/ h
Ae G
w (x,h D)/ h
= e− adG w (x,h D) / h A L−1 (−1) 1 ad G w (x,h D) A + R L , = ! h
(3.18)
=0
where RL =
(−1) L L!
1
e−t G
w (x,h D)
0
1 ad G w (x,h D) h
L Aet G
w (x,h D)
dt.
The semiclassical calculus of pseudodifferential operators [12, Chap. 7],[15, Chap. 4, App. D.2] and (3.17) show that
1 ad G w (x,h D) h
A = (M log(1/ h)) (ad G w0 (x,h D) ) A ∈ (Mh log(1/ h)) h−∞,0 (Rd ), ∀ > 0.
Since G w 0 L 2 →L 2 ≤ C 0 , functional calculus of bounded self-adjoint operators shows that
exp(±t G w (x, h D)) ≤ h −tC0 M , so we obtain the bound, R L = O L 2 →L 2 (log(1/ h) L h L−2tC0 M ) = O L 2 →L 2 (h L−2tC0 M−Lδ ), with δ > 0 arbitrary small. Applying this bound, we may write (3.18) as e−G
w (x,h D)/ h
Ae G
w (x,h D)/ h
∼
∞ (−1) 1 ad G w (x,h D) A ∈ m,0 (Rd ). ! h
(3.19)
=0
In turn, this expansion, combined with Beals’s characterization of pseudodifferential operators (3.4), implies that the exponentiated weight is a pseudodifferential operator: exp(G w (x, h D)/ h) ∈ δ0,C0 M (Rd ), ∀δ > 0.
(3.20)
Using the weight function G, we can now define our weighted spaces. Let Hhk (Rd ) be the semiclassical Sobolev spaces defined in (3.3). We put HGk (Rd ) = e G
w (x,h D)/ h
Hhk (Rd ), u H k = e−G def
w (x,h D)/ h
G
u H k , h
(3.21)
and u, v H k = e−G G
w (x,h D)/ h
u, e−G
w (x,h D)/ h
v H k . G
As a vector space, HGk (Rd ) is identical with Hhk (Rd ), but the Hilbert norms are different. In the case of L 2 , that is of k = 0, we simply put HG0 = HG .
20
S. Nonnenmacher, J. Sjöstrand, M. Zworski
The mapping properties of P = p w (x, h D) on HG (Rd ) are equivalent with those of def
w
w
PG = e−G / h P e G / h on L 2 (Rd ), which are governed by the properties of the (full) symbol pG of PG : formula (3.19) shows that pG = p − i H p G + O(h 2 log2 (1/ h)).
(3.22)
At this moment it is convenient to introduce a notion of leading symbol, which is adapted to the study of conjugated operators such as PG . For a given Q ∈ S(T ∗ Rd ), we say that q ∈ S(T ∗ Rd ) is a leading symbol of Q w (x, h D), if β
∀α, β ∈ N d , h −γ ∂xα ∂<ξ > (Q − q) = Oα,β (< ξ >−|β| ),
(3.23)
that is, (Q − q) ∈ S 0,−γ (T ∗ Rd ) for any γ ∈ (0, 1). This property is obviously an equivalence relation inside S(T ∗ Rd ), which is weaker than the equivalence relation defining the principal symbol map on h (see §3.1). In particular, terms of the size h log(1/ h) are “invisible” to the leading symbol. For example, the leading symbols of pG and p are the same. If we can find q independent of h, then it is unique. For future use we record the following: Lemma 3.1. Suppose Q w (x, h D) : HG (Rd ) −→ HG (Rd ), Q ∈ S(T ∗ Rd ), is self-adjoint (with respect to the Hilbert norm on HG ). Then this operator admits a real leading symbol. Conversely, if q ∈ S(T ∗ Rd ) is real, then there exists Q ∈ S(T ∗ Rd ) with leading symbol q, such that Q w (x, h D) is self-adjoint on HG (Rd ). Proof. This follows from noting that −G Qw G = e def
w/h
Q w (x, h D)e G
w/h
has the same leading symbol as Q w (x, h D), and that self-adjointness of Q w on HG is 2 equivalent to self-adjointness of Q w G on L : the definition of HG in (3.21) (the case of k = 0) gives Q w u, v HG = e−G
w/h
Q w u, e−G
w/h
−G v L 2 = Q w G (e
w/h
u), e−G
w/h
v L 2 . #
The weighted spaces can also be microlocalized in the sense of §3.2: for V T ∗ Rd , we define the space def
HG (V ) = {u = u(h) ∈ HG (Rd ), : ∃ Cu > 0, ∀h ∈ (0, 1], u(h) HG (Rd ) ≤ Cu
∃ χ ∈ Cc∞ (V ), χ w u = u + OS (h ∞ )}. (3.24)
w
In other words, HG (V ) = e G (x,h D)/ h H (V ). This definition depends only on the values of the weight G in the open set V .
From Open Quantum Systems to Open Quantum Maps
21
For future reference we state the following Lemma 3.2. Suppose T : H (V ) → H (κ(V )) is an h-Fourier integral operator associated to a symplectomorphism κ (in the sense of §3.3), and is asymptotically uniformly bounded (in the sense of (3.8). Take G 0 ∈ Cc∞ (neigh(κ(V ))), G = Mh log(1/ h)G 0 . Then the operator: T : Hκ ∗ G (V ) → HG (κ(V ))
(3.25)
is also asymptotically uniformly bounded with respect to the deformed norms. Proof. Since the statement is microlocal we can assume that V is small enough, so that T ≡ T0 A in V , where T0 is unitary on L 2 (Rd ) and A ∈ h . As in the proof of Lemma 3.1 the boundedness of (3.25) is equivalent to considering the boundedness of e−G
w (x,h D)
T0 e(κ
∗ G)w (x,h D)/ h
Aκ ∗ G : L 2 (Rd ) → L 2 (Rd ),
where Aκ ∗ G = e−(κ def
∗ G)w (x,h D)/ h
Ae(κ
∗ G)w (x,h D)/ h
.
Because of (3.19), we have uniform boundedness of Aκ ∗ G on L 2 . Unitarity of T0 means that it is sufficient to show the uniform boundedness of T0−1 e−G
w (x,h D)/ h
T0 e(κ −1
= e−M log(1/ h)(T0
∗ G)w (x,h D)/ h
∗ w Gw 0 (x,h D)T0 ) M log(1/ h)(κ G 0 ) (x,h D)
e
on L 2 . Egorov’s theorem (see [15, §10.2]) shows that −∞,−1 ∗ T0−1 G w (Rd ). 0 (x, h D)T0 = G κ (x, h D), G κ − κ G 0 ∈ h −∞,0 ∗ 2 2 (Rd ), the Baker-Campbell-Hausdorff formula Since [G w κ , κ G 0 ] = h B, B ∈ h 2 for bounded operators shows that
T0−1 e−G
w (x,h D)/ h
T0 e(κ w
∗ G)w (x,h D)/ h ∗
w
= e−M log(1/ h)G κ (x,h D) e M log(1/ h)(κ G 0 ) (x,h D) w ∗ w 2 2 = e M log(1/ h)(−G κ (x,h D)+κ G 0 ) (x,h D))+O L 2 →L 2 (log(1/ h) h ) = exp O L 2 →L 2 (h log(1/ h)) = Id + O L 2 →L 2 (h log(1/ h)). This proves uniform boundedness of globally defined operators T0 A, and the asymptotic uniformly boundedness in the sense of (3.8) of T on spaces of microlocally localized functions. # 2 Alternatively, we can compare exp(M log(1/ h)G w ) with (exp(M log(1/ h)G ))w and use product forκ κ mulæ for pseudodifferential operators – see [43, App.] or [15, Sect. 8.2].
22
S. Nonnenmacher, J. Sjöstrand, M. Zworski
3.6. Escape function away from the trapped set. In this section we recall the construction of the specific weight function G which, up to some further small modifications, will be used to prove Theorems 1 and 2. Let K E ⊂ p −1 (E) be the trapped set on the E-energy surface, see (1.4), and define = K δ def = KE. (3.26) K |E|≤δ
The construction of the weight function is based on the following result of [19, App.]: , U ⊂ V , there exists G 1 ∈ C ∞ (T ∗ X ), such for any open neighbourhoods U, V of K that G 1U ≡ 0, H p G 1 ≥ 0,
H p G 1 p−1 ([−2δ,2δ]) ≤ C, H p G 1 p−1 ([−δ,δ])\V ≥ 1. (3.27)
These properties mean that G 1 is an escape function: it increases along the flow, and (as specified by the strictly increases along the flow on p −1 ([−δ, δ]) away from K neighbourhood V ). Furthermore, H p G is bounded in a neighbourhood of p −1 (0). Since such a function G 1 is necessarily of unbounded support, we need to modify it to be able to use HG -norms defined in §3.5 (otherwise methods of [21] could be used and that alternative would allow more general behaviours at infinity, for instance a wide class of polynomial potentials). For that we follow [43, §§4.1,4.2,7.3] and [30, §6.1]: G 1 is modified to a compactly supported G 2 in a way allowing complex scaling estimates (3.16) to compensate for the wrong sign of H p G 2 . Specifically, [30, Lemma 6.1] states that for any large R > 0 and δ0 ∈ (0, 1/2) we can construct G 2 with the following properties: G 2 ∈ Cc∞ (T ∗ X ) and ∗ Hp G2 ≥ 0 on TB(0,3R) X, ∗ Hp G2 ≥ 1 on TB(0,3R) X ∩ ( p −1 ([−δ, δ])\V ), H p G 2 ≥ −δ0 on T ∗ X.
(3.28)
Let def
G = Mh log(1/ h)G 2 , with M > 0 a fixed constant. Then, in the notations of §3.5, we will be interested in the complex-scaled operator Pθ,R : HG2 (Rn ) −→ HG (Rn ), for a scaling angle depending on h: θ = θ (h) = M1 h log(1/ h),
M1 > 0 fixed.
(3.29)
Inserting the above estimates in (3.22), we get | Re pθ,R,G (ρ)| < δ/2, Re ρ ∈ / V, !⇒ Im pθ,R,G (ρ) ≤ −θ/C1 ,
(3.30)
provided that we choose [30, §6.1] M δ0 M ≥ M1 ≥ , for some C > 0. (3.31) C C Assuming that the constant M0 appearing in the statement of Theorem 1 satisfies 0 < M0 ≤ M1 for δ > 0 and h > 0 small enough, the rectangle R(δ, M0 , h) is contained in the uncovered region in Fig. 6, hence the scaling by the angle (3.29) gives us access to the resonance spectrum in the rectangle R(δ, M0 , h). In §5.3 we will need to further adjust M0 with respect to M1 .
From Open Quantum Systems to Open Quantum Maps
23
3.7. Grushin problems. In this section we recall some linear algebra facts related to the Schur complement formula, which are at the origin of the Grushin method we will use to analyze the operator Pθ,R . For any invertible square matrix decomposed into 4 blocks, we have −1 p11 p12 q11 q12 −1 −1 !⇒ p11 = = q11 − q12 q22 q21 , p21 p22 q21 q22 −1 exists (which implies that q22 , and hence p11 , are square matrices). provided that q22 −1 : We have the analogous formula for q22 −1 −1 q22 = p22 − p21 p11 p12 .
One way to see these simple facts is to apply gaussian elimination to p11 p12 P= p21 p22 so that, if p11 is invertible, we have an upper-lower triangular factorization: −1 p12 1 p11 p11 0 . P= −1 p21 1 0 p22 − p21 p11 p12
(3.32)
The formula for the inverse of p11 leads to the construction of effective Hamiltonians for operators (quantum Hamiltonians) P : H1 → H2 . We first search for auxiliary spaces H± and operators R± for which the matrix of operators P − z R− : H1 ⊕ H− −→ H2 ⊕ H+ , R+ 0 is invertible for z running in some domain of C. Such a matrix is called a Grushin problem, and when invertible the problem is said to be well posed. When successful this procedure reduces the spectral problem for P to a nonlinear spectral problem of lower dimension. Indeed, if dim H− = dim H+ < ∞, we write −1 P − z R− E(z) E + (z) , = E − (z) E −+ (z) R+ 0 and the invertibility of (P − z) : H1 → H2 is equivalent to the invertibility of the finite dimensional matrix E −+ (z). The zeros of det E −+ (z) coincide with the eigenvalues of P (even when P is not self-adjoint) because of the following formula: tr (P − w)−1 dw = − tr E −+ (w)−1 E −+ (w) dw, (3.33) z
z
valid when the integral on the left hand sideis of trace class – see [44, Prop. 4.1] or verify it using the factorization (3.32). Here z denotes an integral over a small circle centered at z. The above formula shows that dim ker(P − z) = dim ker E −+ (z). The matrix E −+ (z) is often called an effective Hamiltonian for the original Hamiltonian P – see [44] for a review of this formalism and many examples. In the physics literature, this reduction is usually called the Feshbach method. We illustrate the use of Grushin problems with a simple lemma which will be useful later in §5.3.
24
S. Nonnenmacher, J. Sjöstrand, M. Zworski
Lemma 3.3. Suppose that def
P =
P R− R+ 0
: H1 ⊕ H− −→ H2 ⊕ H+ ,
where H j and H± are Banach spaces. If P −1 : H2 → H1 exists then P is a Fredholm operator ⇐⇒ R+ P −1 R− : H− → H+ is a Fredholm operator, and ind P = ind R+ P −1 R− . Proof. We apply the factorization (3.32) with p11 = P, p12 = R− , p21 = R+ , p22 = 0. Since the first factor is invertible we only need to check the Fredhold property and the index of the second factor: 1 P −1 R− , 0 −R+ P −1 R− and the lemma is immediate.
#
4. A Microlocal Grushin Problem In this section we recall and extend the analysis of [42] to treat a Poincaré section ⊂ p −1 (0) for a flow satisfying the assumptions in §2.2. In [42] a Poincaré section associated to a single closed orbit was considered. The results presented here are purely microlocal in the sense of §3.2, first near a given component k of the section, then near the trapped set K 0 . In this section P is the original differential operator, but it could be replaced by its complex scaled version Pθ,R , since the complex deformation described in §3.4 takes place far away from K 0 . Also, when no confusion is likely to occur, we will often denote the Weyl quantization χ w of a symbol χ ∈ S(T ∗ Rd ) by the same letter: χ = χ w. 4.1. Microlocal study near k . First we focus on a single component k of the Poincaré section, for some arbitrary k ∈ {1, . . . , N }. Most of the time we will then drop the subscript k. Our aim is to construct a microlocal Grushin problem for the operator i (P − z), h near = k , where | Re z| ≤ δ, | Im z| ≤ M0 h log(1/ h), and δ will be chosen small enough so that the flow on t K Re z is a small perturbation of t K 0 . 4.1.1. A normal form near k . Using the assumption (2.11) and a version of Darboux’s theorem (see for instance [23, Theorem 21.2.3]), we may extend the map κk = κ :
k → k to a canonical transformation
k in T ∗ Rn , κk defined in a neighbourhood of
From Open Quantum Systems to Open Quantum Maps
25
k def
k , |xn | ≤ , |ξn | ≤ δ}, = {(x, ξ ) ∈ T ∗ Rn ; (x , ξ ) ∈ such that
κk (x , 0, ξ , 0) = κk (x , ξ ) ∈ k ,
p ◦
κk = ξn .
(4.1)
k ) the neighbourhood of k in T ∗ X in the range of
We call k =
κ k ( κk . The “width along the flow” > 0 is taken small enough, so that the sets {k , k = 1, . . . , N } are mutually disjoint, and it takes at least a time 20 for a point to travel between any k and its successors. The symplectic maps
κk allow us to extend the Poincaré section to the neighbouring energy layers p −1 (z), z ∈ [−δ, δ]. Let us call
k ∩ {ξn = z}). κk,z =
κ k ( def
Then, if δ > 0 is taken small enough, for z ∈ [−δ, δ] the hypersurfaces
k ) = {
k } k (z) = κk,z ( κ (x , 0; ξ , z), (x , ξ ) ∈ are still transversal to the flow in p −1 (z). Using this extension we may continuously def
jk ) ⊂ k (z), and by consequence deform the departure sets D jk into D jk (z) = κk,z ( D −1 the tubes T jk into tubes T jk (z) ⊂ p (z) through a direct generalization of (2.12). The tube T jk (z) intersects j (z) on the arrival set A jk (z) ⊂ j (z); notice that for z = 0,
jk ) (equivalently A
jk (z) = κ −1 (A jk (z)) is the latter is in general different from κ j,z ( A j,z
jk (0)). These tubes induce a Poincaré map F jk,z bijectively generally different from A relating D jk (z) with A jk (z). The following lemma, announced at the end of §2.3.1, shows that for |z| small enough the interesting dynamics still takes place inside these tubes: the trapped set is stable with respect to variations of the energy. Lemma 4.1. Provided δ > 0 is small enough, for any z ∈ [−δ, δ] the trapped set K z jk T jk (z). As a consequence, in this energy range the Poincaré map associated with (z) fully describes the dynamics on K z . Proof. From our assumption in §2.1, there exists a ball B(0, R) (the “interaction region”) such that, for any E ∈ [−1/2, 1/2], the trapped set K E must be contained inside ∗ ∗ TB(0,R) X . If R is large enough, any point ρ ∈ p −1 (z)\TB(0,R) X, z ≈ 0, will “escape fast” in the past or in the future, because the Hamilton vector field is close to the one corresponding to free motion, 2 j ξ j ∂x j . Hence we only need to study the behaviour ∗ Rn . of points in p −1 (z) ∩ TB(0,R) ∗ Let us define the escape time from the interaction region TB(0,R) X : for any ρ ∈ ∗ TB(0,R) X , tesc (ρ) = inf{t > 0, max(|πx t (ρ)|, |πx −t (ρ)|) ≥ R}. def
For any E ∈ [−1/2, 1/2], the trapped set K E can be defined as the set of points in p −1 (E) for which tesc (ρ) = ∞. Let us consider the neighbourhood of K 0 formed by the interior of the union of tubes, ( Tik )◦ . By compactness, the escape time is bounded ∗ from above outside this neighbourhood, that is in p −1 (0) ∩ TB(0,R) X \( Tik )◦ , by some
26
S. Nonnenmacher, J. Sjöstrand, M. Zworski
finite t1 > 0. By continuity of the flow t , for δ > 0 small enough, the escape time in ∗ the deformed neighbourhood p −1 (z) ∩ TB(0,R X \( Tik (z))◦ will still be bounded from above by 2t1 : this proves that K z Tik (z). # def
A direct consequence is that the reduced trapped sets T j (z) = (z) ∩ K z are contained inside D j (z). For any set S(z) depending on the energy in the interval z ∈ [−δ, δ], we use the notation def S(z). (4.2) S = |z|≤δ
We will extend the notation to complex values of the parameter z ∈ R(δ, M0 , h), identifying S(z) with S(Re z). 4.1.2. Microlocal solutions near . Let us now restrict ourselves to the neighbourhood of k , and drop the index k. The canonical transformation
κ can be locally quantized using the procedure reviewed in §3.3, resulting in a microlocally defined unitary Fourier integral operator
) −→ H (), U ∗ P U ≡ h Dxn , microlocally in
. U : H (
(4.3)
For z ∈ R(δ, M0 , h), we consider the microlocal Poisson operator K(z) : L 2 (Rn−1 ) → L 2loc (Rn ), [K(z) v+ ](x , xn ) = ei xn z/ h v+ (x ),
(4.4)
which obviously satisfies the equation (h Dxn − z) K(z) v+ = 0. For v+ microlocally concentrated in a compact set, the wavefront set of K(z) v+ is not localized in the flow direction. On the other hand, the Fourier integral operator U is
to . Therefore, we use a smooth cutoff function well-defined and unitary only from χ , χ = 1 in , χ = 0 outside a small open neighbourhood of (say, such that
), and define the Poisson operator |xn | ≤ 2 inside
) → H ( ). K (z) = χw U K(z) : H ( def
) ⊂ L 2 (Rn−1 ), to a microlocal solution of the This operator maps any state v+ ∈ H ( equation (P − z)u = 0 in , with u ∈ H ( ). As we will see below, the converse holds:
). each microlocal solution in is parametrized by a function v+ ∈ H ( In a sense, the solution u = K (z)v+ is an extension along the flow of the transverse data v+ . More precisely, K (z) is a microlocally defined Fourier integral operator associated with the graph
, |xn | ≤ } ⊂ T ∗ (X × Rn−1 ). C− = {(
κ (x , xn , ξ , Re z); x , ξ ), (x , ξ ) ∈
(4.5)
a short trajectory segEquivalently, this relation associates to each point (x , ξ ) ∈ ment through the point
κ (x , 0; ξ , Re z) ∈ (Re z). We use the notation C− since this relation is associated with the operator R− defined in (4.13) below. Back to the normal form h Dxn , let us consider a smoothed out step function, χ0 ∈ C ∞ (Rxn ), χ0 (xn ) = 0
for xn ≤ −/2, χ0 (xn ) = 1
for xn ≥ /2.
From Open Quantum Systems to Open Quantum Maps
27
We notice that the commutator (i/ h)[h Dxn , χ0 ] = χ0 (xn ) is localized in the region of the step and integrates to 1: this implies the normalization property (i/ h)[h Dxn , χ0 ]K(z)v+ , K(¯z )v+ = v+ 2L 2 (Rn−1 ) ,
(4.6)
where •, • is the usual Hermitian inner product on L 2 (Rn ). Notice that the right hand side is independent of the precise choice of χ0 . We now bring this expression to the neighbourhood of through the Fourier integral operator χw U . This implies that the Poisson operator K (z) satisfies: (i/ h)[P, χ w ]K (z)v+ , K (¯z )v+ ≡ v+ 2
). for any v+ ∈ H (
(4.7)
Here the symbol χ is such that χ w ≡ U χ0w U ∗ inside , so χ is equal to 0 before − () and equal to 1 after () (in the following we will often use this time-like terminology referring to the flow t ). In (4.7), we are only concerned with [P, χ w ]
. Hence, at microlocally near , since the operator χw U is microlocalized in × this stage we can ignore the properties of the symbol χ outside . The expression (4.7) can be written
) → H (
). K (¯z )∗ [(i/ h)P, χ w ]K (z) = I d : H (
(4.8)
Fixing a function χ with properties described after (4.7) and writing χ = χ f (where f is for forward), we define the operator R+ (z) = K (¯z )∗ [(i/ h)P, χ f ] = K(¯z )∗ U ∗ χw [(i/ h)P, χ f ] def
(4.9)
(from here on we denote χ = χ w in similar expressions). This operator “projects” any
). But it is important to notice that u ∈ H () to a certain transversal function v+ ∈ H ( R+ (z) is also well-defined on states u microlocalized in a small neighbourhood of the : the operator χ w [(i/ h)P, χ f ] cuts off the components of u outside full trapped set K . Hence, we may write )) → H (
). R+ (z) : H (neigh( K Equation (4.8) shows that this projection is compatible with the above extension of the transversal function:
) → H (
). R+ (z) K (z) = I d : H (
(4.10)
) and microlocal solutions to This shows that transversal functions v+ ∈ H ( (P − z)u = 0 are bijectively related. Since | Im z| ≤ M0 h log(1/ h) and |xn | ≤ 2
), we have the bounds
(resp. |xn | ≤ inside inside
K (z) L 2 →L 2 = O(h −2 M0 ), R+ (z) L 2 →L 2 = O(h − M0 ). Just as K (¯z )∗ , R+ (z) is a microlocally defined Fourier integral operator associated with the relation
} ⊂ T ∗ (Rn−1 × X ), (4.11) C+ = {x , ξ ; (
κ (x , xn , ξ , Re z)), (x , xn , ξ , Re z) ∈ namely the inverse of C− given in (4.5). In words, this relation consists of taking any ρ ∈ ∩ p −1 (Re z) and projecting it along the flow on the section (z).
28
S. Nonnenmacher, J. Sjöstrand, M. Zworski
We now select a second cutoff function χb with properties similar with χ f , and satisfying also the nesting property χb = 1 in a neighbourhood of supp χ f .
(4.12)
With this new cutoff, we define the operator
) → H (). R− (z)u − = [(i/ h)P, χb ] K (z) : H (
(4.13)
), this operator creates a microlocal solution Starting from a transversal data u − ∈ H ( in and truncates by applying a pseudodifferential operator with symbol H p χb . Like K (z), it is a microlocally defined Fourier integral operator associated with the graph C− . Its norm is bounded by R− (z) L 2 →L 2 = O(h − M0 ). 4.1.3. Solving a Grushin problem. We are now equipped to define our microlocal
), we want to solve the system Grushin problem in . Given v ∈ H (), v+ ∈ H ( (i/ h)(P − z)u + R− (z)u − = v, (4.14) R+ (z)u = v+ ,
). with u ∈ L 2 (X ) a forward solution, and u − ∈ H ( Let us show how to solve this problem. First let
u be the forward solution of (i/ h)(P − z)
u = v, microlocally in . That solution can be obtained using the Fourier integral operator U in (4.3) and the easy solution for h Dxn . We can also proceed using the propagator to define a forward parametrix: T def def
u = E(z) v, E(z) = e−it (P−z)/ h dt. (4.15) 0
The time T is such that T () ∩ = ∅ (from the above assumption on the separation between the k we may take T = 5). By using the model operator h Dxn , one checks that the parametrix E(z) transports the wavefront set of v as follows: t (WFh (v) ∩ p −1 (Re z)). (4.16) WFh (E(z)v) ⊂ WFh (v) ∪ T (WFh (v)) ∪ 0≤t≤T
In general,
u does not satisfy R+ (z)
u = v+ , so we need to correct it. For this aim, we solve the system (i/ h)(P − z) u + R− (z)u − ≡ 0, (4.17) R+ (z) u ≡ v+ − R+ (z)
u through the Ansatz
u − = −v+ + R+ (z)
u, u = −χb K (z) u − .
(4.18)
Indeed, the property (P − z) K (z) ≡ 0 ensures that (i/ h)(P − z) u = −R− (z)u − . We then obtain the identities u = −K (¯z )∗ [(i/ h)P, χ f ] χb K (z) u − R+ (z) ≡ −K (¯z )∗ [(i/ h)P, χ f ] K (z) u − ≡ −u − .
From Open Quantum Systems to Open Quantum Maps
29
The second identity uses the nesting assumption (H p χ f )χb = H p χ f , and the last one results from (4.8). This shows that the Ansatz (4.18) solves the system (4.17). Finally,
, for v ∈ H () and v+ ∈ H (
) (u =
u + u , u − ) solves (4.14) microlocally in × respectively. Furthermore, these solutions satisfy the norm estimate
u + u − h −5M0 ( v + v+ ).
(4.19)
The form of the microlocal construction in this section is an important preparation for the construction of our Grushin problem in the next section. In itself, it only states that, for v microlocalized near , (i/ h)(P − z)u = v can be solved microlocally near in the forward direction. . We will now extend the construction of the Grushin 4.2. Microlocal solution near K problem near each k , described in §4.1, to obtain a microlocal Grushin problem near . This will be achieved by relating the construction near k to the the full trapped set K one near the successor sections j . We now need to restore all indices k ∈ {1, . . . , N } in our notations.
k ) ⊂ L 2 (Rn−1 ) is the space 4.2.1. Setting up the Grushin problem. We recall that H (
k (see (3.6). For u ∈ L 2 (X ) microlocally of functions microlocally concentrated in ∗ concentrated in neigh( K , T X ), we define
1 ) × · · · × H (
N ), R+ (z)u = (R+1 (z)u, . . . , R+N (z)u) ∈ H (
(4.20)
)) → H (
k ) was defined in §4.1 using a cutoff χ k ∈ where each R+k (z) : H (neigh( K f Cc∞ (T ∗ X ) realizing a smoothed-out step from 0 to 1 along the flow near k . Similarly, we define N
1 ) × · · · × H (
N ) → H (∪k=1 R− (z) : H ( k ),
R− (z)u − =
N
j
j
R− (z)u − ,
N u − = (u 1− , . . . , u − ).
(4.21)
1 k (z) was defined in (4.13) in terms of a cutoff function χ k ∈ C ∞ (T ∗ X ) which Each R− c b also changes from 0 to 1 along the flow near k , and does so before χ kf . Below we will impose more restrictions on the cutoffs χbk . With these choices, we now consider the microlocal Grushin problem (i/ h)(P − z)u + R− (z)u − ≡ v, (4.22) R+ (z)u ≡ v+ .
The aim of this section is to construct a solution (u, u − ) microlocally concentrated in a small neighbourhood of K 0 × κ1−1 (T1 ) × · · · × κ N−1 (T N ), provided (v, v+ ) is concentrated in a sufficiently small neighbourhood of the same set. To achieve this aim we need to put more constraints on the cutoffs χbk . We assume that each χbk ∈ Cc∞ (T ∗ X ) is supported near the direct outflow of Tk . To give a precise
30
S. Nonnenmacher, J. Sjöstrand, M. Zworski
A14
V1 2 b
V0
b
2
T12
3 f
A12
3
A13
3 b
1
Fig. 7. Schematic representation of (part of) the neighbourhoods V1 ⊂ V0 of K 0 (resp. green shade and green dashed contour), some sections k (thick black) and arrival sets Ak j ⊂ k (red). We also show the tubes ±± −− T12 connecting 2 with A12 (the dashed lines indicate the boundaries of T12 ), the supports of the cutoffs k 3 χb and χ f (dot-dashed line), and two trajectories in K 0 (full lines inside V1 ). (color on-line only)
jk (see (2.12), (4.2) by removing condition, let us slightly modify the energy-thick tubes T or adding some parts near their ends: jk , −s2 2 < t < t+ (ρ) + s1 2}, s1 s2 def = {t (ρ) : ρ ∈ D T jk
si = ±.
−− do not intersect the neighbourhoods k , j , With this definition, the short tubes T jk ++ intersect both (see Fig. 7). while the long tubes T jk We then assume that −− , χbk (ρ) = 1 for ρ ∈ (4.23) T jk j∈J+ (k)
and supp χbk is contained in a small neighbourhood of that set. Furthermore, we want the cutoffs {χbk }k=1,...,N to form a microlocal partition of unity near K 0 : there exists a containing all long tubes: neighbourhood V0 of K ++ jk , (4.24) T V0 ⊃ k, j
and such that N k=1
χbk (ρ) ≡ 1 for ρ ∈ V0 .
(4.25)
From Open Quantum Systems to Open Quantum Maps
31
These conditions on χbk can be fulfilled thanks to the assumption (2.10) on the section . A schematic representation of these sets and cutoffs is shown in Fig. 7. 4.2.2. Solving the homogeneous Grushin problem. Let us first solve (4.22) when v ≡ 0.
k is mapped through κk,z to a subset of k (z). The The wavefront set WFh (v+k ) ⊂ microlocal solution K k (z)v+k , initially concentrated inside the neighbourhood k , can be extended along the flow to a larger set +k , which intersects the successors j (z) of k (z) ++ (we recall that j = k according to assumpand contains the union of tubes j∈J+ (k) T jk tion (2.10). This can be done by extending the symplectomorphism
κk , the associated unitary Fourier integral operator Uk , and replace the cutoff function χk by a function χ+k supported in the set +k ; we can then define the extended Poisson operator as:
) → H (+k ). K k+ (z) = χw+ Uk K(z) : H ( k
Assuming κk,z (WFh (v+k )) is contained in the departure set Dk (z) ⊂ k (z), the extended ++ (z). In microlocal solution K k+ (z)v+k is concentrated in the union of tubes ∪ j∈J+ (k) T jk that case, we take as our Ansatz def
u k = χbk K k+ (z) v+k .
(4.26)
tmax = max{t+ (ρ), ρ ∈ k Dk (z), | Re z| ≤ δ},
(4.27)
Set def
the maximal return time for our Poincaré map. Then the above Ansatz satisfies the estimate
u k L 2 h −M0 (tmax +) v+ H ( D
k ) .
(4.28)
Due to the assumption (4.23), the cutoff χbk effectively truncates the solution only near the sections k (z) and j (z), j ∈ J+ (k), but not on the “sides” of supp χbk . Hence, the expression (i/ h)(P − z)u k ≡ [(i/ h)P, χbk ] K k+ (z) v+k
(4.29)
k (z)v k supported near D (z), and other comcan be decomposed into one component R− k + ponents supported near the arrival sets A jk (z) ⊂ j , due to the “step down” of χbk near A jk (z). The assumption (4.25) ensures that j
[(i/ h)P, χbk ] ≡ −[(i/ h)P, χb ] microlocally near A jk (z),
(4.30)
so the expression in (4.29) reads k (i/ h)(P − z)u k ≡ R− (z)v+k −
j
[(i/ h)P, χb ] K k+ (z) v+k .
(4.31)
j∈J+ (k)
Now, for each j ∈ J+ (k) we notice that K k+ (z) v+k is a solution of (P − z)u = 0 near A jk (z), so this solution can also be parametrized by some transversal data “living” on
32
S. Nonnenmacher, J. Sjöstrand, M. Zworski
the section j (z) (see the discussion before (4.5). This data obviously depends linearly on v+k , which defines the monodromy operator M jk (z): K k+ (z)v+k ≡ K j (z) M jk (z)v+k , microlocally near A jk (z).
(4.32)
k ⊂
jk (z) ⊂
k to A
j , they The operators M jk (z) are microlocally defined from D
k ) for = j. The identity (4.8) provides an explicit formula: are zero on H ( D M jk (z) = K j (¯z )∗ [(i/ h)P, χ f ]K k+ (z) = R+ (z)K k+ (z). j
j
(4.33)
Before further describing these operators, let us complete the solution of our Grushin problem. Combining (4.31) with (4.32), we obtain j k (i/ h)(P − z)u k ≡ R− (z)v+k − R− (z)M jk (z)v+k . (4.34) j∈J+ (k)
k This shows that the problem (4.22) in the case v = 0 and a single vk+ , WFh (v+k ) ⊂ D is solved by j
u ≡ χbk K k+ (z) v+k , u k− = −v+k , u − = M jk (z)v+k , j ∈ J+ (k). We now consider the Grushin problem with v = 0, v+ = (v+1 , . . . , v+N ) with each v+k
k . By linearity, this problem is solved by microlocalized in D u≡ χbk K k+ (z) v+k , k j u−
j
≡ −v+ +
M jk (z)v+k .
(4.35)
k∈J− ( j)
, while u −j From the above discussion, u is microlocalized in the neighbourhood V0 of K
j ∪ A
j (z). is microlocalized in D Let us now come back to the monodromy operators. The expression (4.33) shows that M jk (z) is a microlocal Fourier integral operator. Since we have extended the solution
jk ) is K k (z) v+k beyond k , the relation associated with the restriction of K k+ (z) on H ( D a modification of (4.5), of the form jk
jk , − ≤ t ≤ tmax + }, C− = {(t (
κk,z (ρ)); ρ), ρ ∈ D
such that the trajectories cross j . On the other hand, the relation C+ associated with j R+ (z) is identical with (4.11). By the composition rules, the relation associated with M jk (z) is
jk , ρ = κ −1 ◦ F jk,z ◦ κk,z (ρ) = F
jk,z (ρ)}. C jk = {(ρ , ρ), ρ ∈ D j,z This is exactly the graph of the Poincaré map F jk,z : D jk (z) → A jk (z), seen through the coordinates charts κk,z , κ j,z .
jk ) → H ( A
jk (z)) is When z is real, the identity (4.8) implies that M jk (z) : H ( D microlocally unitary. Also, the definition (4.33) shows that this operator depends holomorphically on z in the rectangle R(δ, M0 , h). To lowest order, the z-dependence takes the form M jk (z) = M jk (0) Opw h (exp(i z t˜+ / h)) (1 + O(h log(1/ h))),
From Open Quantum Systems to Open Quantum Maps
33
where t˜+ ∈ Cc∞ (Rn−1 ; R+ ) is an extension of the return time associated with the map
jk,z on D
jk . For z ∈ R(δ, M0 , h), this operator satisfies the asymptotic bound F −M0 tmax
M jk (z) H ( D ).
k )→H ( A
j (z)) = O(h
(4.36)
4.2.3. Solving the inhomogeneous Grushin problem. It remains to discuss the inhomogeneous problem (i/ h)(P − z)u + R− u − ≡ v,
(4.37)
, which satisfies for v microlocalized in a neighbourhood V1 of K −+ V1 ⊂ T jk
(4.38)
j,k
−+ intersects k only near D k , see Fig. 7). (each tube T jk −− . We Let us first assume that v is microlocally concentrated inside a short tube T jk use the forward parametrix E(z) of (i/ h)(P − z) given in (4.15) with the time T = tmax + 5,
(4.39)
and consider the Ansatz def
u = χbk E(z) v.
(4.40)
According to the transport property (4.16), E(z)v is microlocalized in the outflow of −− , so the cutoff χ k effectively truncates E(z)v only near A jk (z) ⊂ j . The partition T b jk of unity (4.25) then implies that j
(i/ h)(P − z)u ≡ v + [(i/ h)P, χbk ] E(z) v ≡ v − [(i/ h)P, χb ] E(z) v. Also, the choice of the time T ensures that E(z)v is a microlocal solution of (P −z)u = 0 near A jk (z), so j
E(z)v ≡ K j (z)R+ (z)E(z)v microlocally near A jk (z). Thus, we can solve (4.37) by taking u − ≡ R+ (z)E(z)v, u − = 0, = j. j
j
j
jk (z)), and that The propagation of wavefront sets given in (4.16) shows that u − ∈ H ( A +− WFh (u) ⊂ T jk does not intersect the “step up” region of the forward cutoffs χ f , so that R+ (z)u ≡ 0 for all = 1, . . . , N . k ), we can replace the cutoff χ k If v is microlocally concentrated in V1 ∩ ∪|t|≤ t ( D b in (4.40) by χb , χbk + ∈J− (k)
and apply the same construction. The only notable difference is the fact that R+k (z)u may k . be a nontrivial state concentrated in ∪|t|≤ D
34
S. Nonnenmacher, J. Sjöstrand, M. Zworski
In both cases, we see that u + u − h −M0 (tmax +2) v . By linearity, the above procedure allows to solve (4.37) for any v microlocalized inside the neighbourhood V0 . This solution produces a term R+ u, which can be solved away using the procedure of §4.2.2. Notice that R+ u h −M0 (tmax +) v . We summarize the construction of our microlocal Grushin problem in the following = K δ in Proposition 4.2. For δ > 0 small enough, there exist neighbourhoods of K j ) in
j , V+j , and V−j , j = 1, · · · N , κ −1 ( T T ∗ X , V+ and V− , and neighbourhoods of
j such that for any (v, v+ ) ∈ H (V+ ) × H (V+1 ) × · · · H (V+N ), we can find (u, u − ) ∈ H (V− ) × H (V−1 ) × · · · H (V−N ), satisfying i (P − z)u + R− (z)u − ≡ v, h
R+ (z)u ≡ v+ microlocally in V+ × V+1 × · · · VN+ .
Here R± (z) are given by (4.20) and (4.21). Furthermore, the solutions satisfy the norm estimates
u + u − h −M0 (2tmax +2) ( v + v+ ), where tmax is the maximal return time defined in (4.27). One possible choice for the above sets is def
k , V−k = D
k ∪ V+ = V1 , V− = V0 , V+k = D
k (z). A
| Re z|≤δ
u,
u − ) the solution for the inhomogeneous problem Proof. Take v ∈ H (V1 ), and call (
(4.37). Then the propagation estimate (4.16) implies that
u is concentrated inside the j ++
j (z)). larger neighbourhood V0 ⊂ ∪ j,k T jk (see (4.24)), while
u− ∈ H ( A
k ) so, provided the data satisfies v+k ∈ D
k , the computations We have R+k (z)
u ∈ H (D of §4.2.2 show how to solve the homogeneous problem with data (v+ − R+ (z)
u ). That solves the full problem. The expressions (4.35) show that the solutions to the homoge k ∪ A
k (z). neous problem ( u, u k− ) are microlocalized, respectively, in V0 and in D # Remark. The proof of the proposition shows that the neighbourhoods V+k and V−k are different. For given data (v, v + ), the solutions (u, u − ) will not in general be concentrated in the same small set as the initial data. This, of course, reflects the fact that a neighbourhood V of K 0 is not invariant under the forward flow, but escapes along the unstable direction. In order to transform the microlocal Grushin problem described in this proposition into a well-posed problem, we need to take care of this escape phenomenon. This will be done using escape functions in order to deform the norms on the spaces L 2 (X ) (as described in §3.5), but also on the auxiliary spaces L 2 (Rn−1 ).
From Open Quantum Systems to Open Quantum Maps
35
5. A Well Posed Grushin Problem The difficulty described in the remark at the end of §4 will be resolved by modifying the norms on the space L 2 (X ) × L 2 (Rn−1 ) N , through the use of exponential weight functions as described in §3.5. These weight functions will be based on the construction described in §3.6. In most of this section we will consider the scaled operator Pθ,R globally, so we cannot replace it by P any longer. To alleviate notation, we will write this operator P = Pθ,R , θ = M1 h log(1/ h),
R & C0 ,
(5.1)
where C0 is the constant appearing in (2.2), and M1 > 0 is a constant (it will be required to satisfy (3.31) once we fix the weight G, and is larger than M0 appearing in Theorem 1). We will first discuss the local construction near each k and then, as in the previous section, adapt it to construct a global Grushin problem. Our first task is still microlocal: we explain how a deformation of the norm on L 2 (X ) by a suitable weight function G can be used to deform the norms on the N auxiliary
k . spaces L 2 (Rn−1 ), microlocally near 5.1. Exponential weights near k . As in §4.1, in this subsection we work microlocally in the neighbourhood k of one component k (k is the neighbourhood described in §4.1); we drop the index k in our notations. Notice that the complex scaling has no effect in this region, so P ≡ Pθ,R . We will impose a constraint on the weight function
. The construction of the local solution G near , and construct weight functions g on performed in §4.1 will then be studied in these deformed spaces.
0 ∈ C ∞ (T ∗ Rn ), so that Take a function g 0 ∈ Cc∞ (Rn−1 ), and use it to define G
0 (x , xn , ξ , ξn ) = g 0 (x , ξ ) in
. G Then, using the Fourier integral operator U given in (4.3), one can construct a weight function G 0 ∈ S(T ∗ X ) such that
w ∗ microlocally near . Gw 0 ≡ U (G 0 ) U Notice that G 0 now depends on h through an asymptotic expansion G 0 (h) ∼ h j G 0, j , G 0, j ∈ Cc∞ (T ∗ X ) independent of h.
(5.2)
j≥0
0 ◦
This weight satisfies G 0,0 = G κ −1 in , and the invariance property [P(h), G w 0 (x, h D)] ≡ 0
microlocally in .
(5.3)
As in §3.5, we rescale these weight functions by def
G = Mh log(1/ h) G 0 ,
def
g = Mh log(1/ h) g 0 .
(5.4)
Still using the model h Dxn , one can easily check the intertwining property
) → H ( ), G w (x, h Dx ; h) K (z) ≡ K (z) g w (x , h Dx ; h) : H ( e−G
w (x,h D
x ;h)/ h
K (z) ≡ K (z) e−g
w (x ,h D
x ;h)/ h
) → H ( ). : H (
(5.5)
) Using the weights G and g we define the microlocal Hilbert spaces HG ( ) and Hg ( by the method of §3.5. We need to check that the construction of a microlocal solution performed in §4.1.2 remains under control with respect to these new norms.
36
S. Nonnenmacher, J. Sjöstrand, M. Zworski
Lemma 5.1. The operators
) → HG ( ), z ∈ R(δ, M0 , h) K (z) : Hg ( satisfy the analogue of (4.7). Namely, taking a cutoff χ jumping from 0 to 1 near as
) will satisfy in §4.1.2, then any v+ ∈ Hg ( [(i/ h)P, χ w ] K (z) v+ , K (¯z ) v+ HG ≡ v 2Hg .
(5.6)
Proof. From the cutoff χ we define the deformed symbol χG through χGw (x, h D) = e−G def
w (x,h D)/ h
χ w (x, h D) e G
w (x,h D)/ h
.
The symbol calculus of §3.5 shows that χG also jumps from 0 to 1 near , so that (returning to the convention of using χ for χ w ) [(i/ h)P, χ ]K (z)v+ , K (¯z )v+ HG ≡ e−G/ h [(i/ h)P, χ ]K (z)v+ , e−G/ h K (¯z )v+ L 2 ≡ K (¯z )∗ [(i/ h)PG , χG ] K (z) e−g/ h v+ , e−g/ h v+ L 2 ≡ K (¯z )∗ [(i/ h)P, χG ] K (z) e−g/ h v+ , e−g/ h v+ L 2 ≡ e−g/ h v+ 2 ≡ v+ 2Hg . In the second line we used (5.5), the third line results from P ≡ PG , due to (5.3), and the last one from (4.7) applied to χG . # Equation (5.5) shows that, for z ∈ R(δ, M0 , h), the operators K (z) and R± (z) defined respectively in (4.9) and (4.13), satisfy the same norm estimates with respect to the new norms as for the L 2 norms: −M0
K (z) Hg ( ),
)→HG () = O(h −M0 ),
R+ (z) HG ()→Hg (
) = O(h
−M0
R− (z) Hg ( ).
)→HG () = O(h
(5.7) (5.8)
The arguments presented in §4.1 carry over to the weighted spaces, and the microlocal solution to the problem (4.14) constructed in §4.1.3 satisfies the norm estimates (5.9)
u HG + u − Hg h −5M0 v HG + v+ Hg . Given a function G 0,0 (x, ξ ) satisfying H p G 0,0 = 0 in , one can iteratively construct a full symbol G 0 of the form (5.2), such that (5.3) holds. Now, the lower order terms in G 0 may change the norms only by factors (1 + O(Mh log(1/ h))), so the same norm estimates hold if we replace G 0 by its principal symbol G 0,0 in the definition of the new norms. As a result, we get the following
0 (x , xn , ξ , ξn ) = g 0 (x , ξ ), G 0 ∈ Cc∞ (X ) satisfying G 0 = Proposition 5.2. Take G −1
G0 ◦
κ in , and G = Mh log(1/ h) G 0 ,
0 . g = Mh log(1/ h) G
). Then, the estimates (5.7)–(5.9) hold in the spaces HG (), Hg (
From Open Quantum Systems to Open Quantum Maps
37
5.2. Globally defined operators and finite rank weighted spaces. In this section we transform our microlocal Grushin problem into a globally defined one. This will require transforming all the microlocally defined operators (R± (z), M jk (z)) into globally defined operators acting on L 2 (X ) or L 2 (Rn−1 ). Because our analysis took place near the trapped set K 0 , we will need to restrict our auxiliary operators to some subspaces of L 2 (Rn−1 ) obtained as images of some finite rank projectors. These subspaces are composed of functions microlocalized near K 0 . To show that the resulting Grushin problem is well-posed (invertible), the above construction must be performed using appropriately deformed norms on the spaces L 2 (X ), L 2 (Rn−1 ), obtained by using globally defined weight functions G, g j . Our first task is thus to complete the constructions of these global weights, building on §3.6 and §5.1. 5.2.1. Global weight functions. We will now construct global weight functions G ∈ Cc∞ (X ), g j ∈ Cc∞ (T ∗ Rn−1 ) (one for each section j ). For this, we will use the construction of an escape function away from K 0 presented in §3.6, and modify it near the Poincaré section so that it takes the form required in Proposition 5.2, and allows us to define auxiliary escape functions g j . These weight functions will allow us to to define finite rank realizations of the microlocally defined operators R± (z) and M(z). Our escape function G 0 ∈ S(T ∗ X ) is obtained through a slight modification of the weight G 2 (x, ξ ) described in (3.28). The modification only takes place near the trapped , and in particular near the sections j . The following lemma is easy to verify. set K Lemma 5.3. Let { j , } j=1,...,K be the neighbourhoods of j described in §4.1.1, j and j be small neighbourhoods of j , j j j , and let V be a small neighδ (see (3.26)). Then there exists G 0 ∈ Cc∞ (T ∗ X ) such that bourhood of K Hp G0 ≥ 1
∗ on TB(0,3R) X ∩ p −1 ([−δ, δ])\W,
Hp G0 = 0
on j ,
def
W = V∪
N
j ,
j=1
Hp G0 ≥ 0 H p G 0 ≥ −δ0
on
(5.10)
∗ TB(0,3R) X, ∗
on T X.
→ (see §4.1.1), we can construct G 0 Besides, using the coordinate charts
κj : j j
is independent of the energy variable ξn ∈ [−δ, δ]. such that G 0 ◦
κ j j The last assumption (local independence on ξn ) is not strictly necessary, but it simplifies our construction below, making the auxiliary functions g j independent of z — see Proposition 5.2. For the set V we assume that V V1 , where V1 is the set defined in (4.38). As a consequence, there exists a set V1 , with V V1 V1 with the following property. Consider the parametrix E(z) (4.15) with the time T = tmax + 5. Then there exists t1 > 0 such that, for any ρ ∈ p −1 ([−δ, δ])\V1 , the trajectory segment {t (ρ), 0 ≤ t ≤ T } spends a time t ≥ t1 outside of W . The main consequence of this property is a strict increase of the weight along the flow outside V1 : ∗ X ∩ p −1 ([−δ, δ])\V1 , ∀ρ ∈ TB(0,2R)
G 0 (T (ρ)) − G 0 (ρ) ≥ t1 .
(5.11)
38
S. Nonnenmacher, J. Sjöstrand, M. Zworski
(Here we use the fact that T is small enough, so that a particle of energy z ≈ 0 starting ∗ ∗ inside TB(0,2R) at t = 0 will remain inside TB(0,3R) up to t = T .) The set V will be further characterized in the next subsection. From now on, we will take for weight function G = Mh log h G 0 with such a function G 0 , and use it to define a global Hilbert norm • H k (X ) as in (3.21). As in Proposition G 5.2, we define, for each j = 1, . . . , N , the auxiliary weight
j, g j (x , ξ ) = Mh log(1/ h) G 0 ◦
κ j (x , 0, ξ , 0), (x , ξ ) ∈ def
(5.12)
and extend it to an element of Cc∞ (T ∗ R(n−1) ), so that the deformed Hilbert norm w
v Hg j = e−g j (x ,h Dx )/ h v L 2 (Rn−1 ) is globally well-defined. Proposition 5.2 shows that our microlocal construction near j satisfies nice norm estimates with respect to the spaces HG (X ), Hg j . To see the advantages of having weights which are escape functions we state the following lemma which results from applying Lemma 3.2 to the Fourier integral operator exp(−it P/ h): Lemma 5.4. Suppose that ρ1 = t (ρ0 ) for some t > 0, and that def
= G 0 (ρ1 ) − G 0 (ρ0 ) > 0. Suppose also that χ j ∈ Cc∞ (T ∗ X ), j = 0, 1, have their supports in small neighbourhoods of ρ j ’s. Then for h small enough we have
e−it P/ h χ0w HG →HG ≤ h M/2 ,
χ1w e−it P/ h HG →HG ≤ h M/2 .
(5.13)
5.2.2. Finite dimensional projections. We want to construct a finite dimensional subj space of the Hilbert space Hg j (Rn−1 ), such that the microlocal spaces Hg j (V± ) are both ∞ approximated by it modulo O(h ). For each j = 1, . . . , N , let S j , S j be two families of open sets with smooth boundaries in T ∗ Rn−1 , satisfying
κ −1 j (T j ) S j S j ⊂ D j ,
j = 1, . . . , N .
(5.14)
k j . In particular, each S j , S j splits into disjoint components Sk j Sk j ⊂ D Once these sets are chosen, we need to adjust the set V in Lemma 5.3, making it thinner if necessary: δ , V1 ) and t0 > 0 such Lemma 5.5. For δ > 0 small enough, there exists V = neigh( K that the following property holds:
k j ∩S j For any indices j = 1, . . . , N , k ∈ J+ ( j), any z ∈ [−δ, δ] and any point ρ ∈ D
such that its successor Fk j,z (ρ) does not belong to Sk , then the trajectory between κ j,z (ρ) and Fk j,z (κ j,z (ρ)) spends a time t ≥ t0 outside of W = V ∪ Nj=1 j .
From Open Quantum Systems to Open Quantum Maps
39
Σk
Sk
Dkj
V Σj
Akj
Sj
Fig. 8. Schematic representation (inside some energy layer p −1 (z)) of the neighbourhood V and the sets Sk , S j . The departure/arrival sets Dk j , Ak j are similar to the ones appearing in Fig. 4. The sets Sk , S j are represented through their images in k , j through κk,z , κ j,z . We show 3 trajectories staying inside V all the time, and one ending outside Sk
The time t0 is necessarily smaller than the maximal return time tmax of (4.27); on the other hand, t0 increases if we decrease the width ∼ of the sets j . See Fig. 8 for a sketch. Now, let Q j = Q j (x , ξ ; h) ∈ S(T ∗ Rn−1 ), with leading symbol q j independent of h (the leading symbol is meant in the sense of (3.23)). We choose that leading symbol to be real and have the following properties: q j (ρ) < 0,
ρ ∈ Sj,
q j (ρ) > 0,
ρ ∈ T ∗ Rn−1 \S j , lim inf q j (ρ) > 0.
(5.15)
ρ→∞
Lemma 3.1 shows that one can choose Q j so that Q wj (x , h Dx ) : Hg j (Rn−1 ) −→ Hg j (Rn−1 ) is self-adjoint. Under the assumptions (5.15), we know that Q j has discrete spectrum in a fixed neighbourhood of R− when h > 0 is small enough. Let def def (5.16) H j = j Hg j (Rn−1 ) , where j = 1R− Q wj (x , h Dx ) , that is, j is the spectral projection corresponding to the negative spectrum of Q wj . In particular,
j Hg j →Hg j = 1,
dim(H j ) ∼ c j h 1−n , c j > 0.
(5.17)
40
S. Nonnenmacher, J. Sjöstrand, M. Zworski def
We group together these projectors in a diagonal matrix h = diag( 1 , . . . , N ) def
projecting Hg1 (Rn−1 ) × · · · Hg N (Rn−1 ) onto H = H1 × · · · H N . The space H j will be equipped with the norm • Hg j . For future reference we record the following lemma based on functional calculus of pseudodifferential operators (see for instance [12, Chap. 7]): Lemma 5.6. For any uniformly bounded family of states u = (u(h) ∈ L 2 (Rn−1 ))h→0 , WFh (u) S j !⇒ u − j u Hg j = O(h ∞ ) u Hg j . In §5.1 we used the microlocally defined operators j
j ). R+ (z) : HG ( j ) → Hg j ( j
Renaming them R+,m (z) (where m stands for microlocal) we now define def
j
j
R+ (z) = j R+,m : HG (X ) → H j .
(5.18)
The estimate (5.8) together with the above lemma shows that
R+ (z) HG (X )→H j = O(h −M0 ), j
z ∈ R(δ, M0 , h).
(5.19)
j
The operators R+ (z) are globally well-defined once we choose a specific realization of j R+,m (z), which gives a unique definition mod O(h ∞ ). We have thus obtained a family of operators def
R+ (z) = (R+1 , . . . , R+N ) : HG (X ) −→ H1 × · · · H N . j
In turn, the operators R− (z) are obtained by selecting a realization of the microlocally j
j ), and restricting that realization to H j : defined operator R−,m (z) on Hg j ( j
j
R− (z) = R−,m (z) j : H j −→ HG (X ).
(5.20)
Again, these operators are well defined mod O(h ∞ ). Putting together (5.8) with (5.17) ensures that
R− (z) H j →HG = O(h −M0 ). j
We group these operators into R− (z) : H1 × · · · H N −→ HG (X ), R− (z)u − =
N j=1
(5.21) j j R− (z) u − ,
u− =
N (u 1− , . . . , u − ).
From Open Quantum Systems to Open Quantum Maps
41
5.3. A well posed Grushin problem. With these definitions we consider the following Grushin problem: def
P(z) : HG2 × H → HG × H, H = H1 × · · · H N , (i/ h)(Pθ,R (h) − z) R− (z) def , z ∈ R(δ, M0 , h). P(z) = R+ (z) 0
(5.22)
Since Pθ,R (h) − z (which we will denote by P − z for short) is a Fredholm operator, so is P(z), as we have only added finite dimensional spaces. For Im z > 0 the operator (P − z) is invertible, so Lemma 3.3 shows that the index of P(z) is 0. Hence, in order to prove that P(z) is bijective it suffices to construct an approximate right inverse and then use a Neumann series. The rest of this section will be devoted to the proof of this (approximate) right invertibility of P(z). 5.3.1. A well-posed homogeneous problem. As before we first consider the homogeneous problem (i/ h)(P − z)u + R− (z)u − = 0,
R+ (z)u = v+ ,
(5.23)
where only one component v+k is nonzero (we may assume that v+k H1 = 1). For that we adapt the methods of §4.2.2. We construct an approximate solution using the extended
k ), so its action Poisson operator K k+ (z) (that operator acts on the microlocal space Hgk ( on Hk is well-defined modulo O(h ∞ )), and take u = χbk K k+ (z) v+k , where χbk is the backwards cutoff function with properties given in (4.12),(4.23) and (4.25). The fact that G increases along the trajectories implies that u satisfies the same norm bound as with the “old norms” (see (4.28):
u HG (X ) h −M0 (tmax +) v+k Hk . The microlocally defined operator satisfies k (z) u ≡ v+k + O Hgk (h ∞ ), R+,m
R+,m (z) u = O Hg j (h ∞ ), j
j = k.
As a result, projecting the left-hand side onto Hk has a negligible effect: R+k (z) u = k (v+k + O(h ∞ )) = v+k + OHk (h ∞ ). Following (4.29) we write (i/ h)(P − z)u ≡ [(i/ h)P, χbk ] K k+ (z)v+ ∈ HG (X ).
(5.24)
As noticed in §4.2.2, the transport properties of K k+ (z) show that u is microlocalized ++ (z), so the right-hand side in (5.24) splits into a inside the union of tubes ∪ j∈J+ (k) T jk
k , and other components concentrated near the arrival component concentrated near D sets A jk (z), j ∈ J+ (k). We rewrite (4.34) for the present data: j k (z)v+k − R−,m (z)M jk (z)v+k . (5.25) (i/ h)(P − z)u ≡ R−,m j∈J+ (k)
42
S. Nonnenmacher, J. Sjöstrand, M. Zworski
jk (z) ⊂
j , which is not Each state M jk (z)v+k is microlocalized inside the arrival set A contained in S j in general – see the remark at the end of §4 and Fig. 8. j Consequently one could fear that replacing the operators R−,m (z) by the truncated j operators R− (z) would drastically modify the above right hand side. The microlocally weighted spaces HG , Hg j have been constructed precisely to avoid this problem. The mechanism is a direct consequence of the relative properties of the sets S j and V explained in Lemma 5.5. Namely, a point ρk ∈ S jk is either “good”, if its image ρ j = F jk,z (ρk ) ∈ S j , or “bad”, in which case G 0 (ρ j ) − G 0 (ρk ) ≥ t0 .
(5.26)
χ j ∈ Cc∞ (S j ), χ j = 1 on S j , χ j = 0 outside neigh (S j , S j ).
(5.27)
Let us choose a cutoff
k ) → H ( A
jk (z)) is uniformly Since the Fourier integral operator M jk (z) : H ( D bounded, (5.26) implies the norm estimate (see Lemma 5.4) ∀v+k ∈ Hk ,
k Mt0 −M0 tmax
(1 − χ w
v+k Hk , j ) M jk (z) v+ Hg j h
z ∈ R(δ, M0 , h).
For this estimate to be small when h → 0, we require the ratio M0 /M to be small enough to ensure the condition t0 −
M0 tmax ≥ t0 /2 > 0. M
(The bounds (3.31) and M0 ≤ M1 show that the ratio M0 /M can indeed be chosen arbitrary small.) k On the other hand, χ w j M jk (z) v+ is microlocalized inside neigh(S j , S j ), so w k ∞ Lemma 5.6 implies that ( j − 1)χ j M jk (z) v+ = O Hg j (h ). Putting these estimates altogether, we find that ∀v+k ∈ Hk ,
M jk (z) v+k = j M jk (z) v+k + O(h Mt0 /2 ) v+k .
(5.28)
This crucial estimate shows that the projection of M jk (z) v+k on H j has a negligible effect. We now define the finite rank operators j M jk (z) k : Hk → H j , j ∈ J+ (k),
jk (z) def
= h M(z) h . M = in short M(z) 0 otherwise, (5.29) These operators satisfy the same norm bounds (4.36) as their infinite rank counterparts. j Using these operators, and remembering that the operators R− : H j → HG (X ) are −M bounded by O(h 0 ), we rewrite (5.25) as j k
jk (z) v+k + O(h Mt0 /3 ) v+k . (i/ h)(P − z)u ≡ R− (z)v+k − R− (z) M j∈J+ (k)
Generalizing the initial data to arbitrary v+ ∈ H1 × · · · × H N , we obtain
From Open Quantum Systems to Open Quantum Maps
43
Proposition 5.7. Assume z ∈ R(δ, M0 , h). Let v+ ∈ H. Then there exists (u, u − ) ∈ HG2 (X ) × H such that (i/ h)(P − z)u + R− (z) u − = O(h Mt0 /3 ) v+ H in HG (X ), (5.30) ∞ R+ (z)u = v+ + O(h ) v+ H in H, (5.31)
u HG (X ) h −M0 (tmax +) v+ H , u − H h −M0 tmax v+ H . (5.32) The second part of the solution, u − , is of the form −M0 tmax
− I d)v+ , M(z)
, u − = ( M(z) H →H h
jk (z)) j,k=1,...,N is the matrix of operators defined in (5.29). where M(z) = (M
jk (z), j ∈ J+ (k), for z ∈ [−δ, δ]: We collect some properties of the operators M
jk (z)) ⊂ S j × S k .
jk (z) is uniformly bounded, and WF ( M • M h ˜ • take ρk ∈ S k , ρ j = F jk,z (ρk ) ∈ S j : (1) if the trajectory segment connecting the points κk,z (ρk ), κ j,z (ρ j ) is contained in
jk (z) is an h-Fourier integral operator of W , then microlocally near (ρ j , ρk ), M order zero with associated canonical transformation F˜ jk,z = κ −1 j,z ◦ F jk,z ◦ κk,z ;
jk (z) (2) if furthermore the above segment is disjoint from the support of G, then M is microlocally unitary near (ρ j , ρk ); (3) if, on the opposite, this segment contains a part outside W , then there exist χ j ∈ Cc∞ (neigh(ρ j )), χk ∈ Cc∞ (neigh(ρk )), equal to 1 near ρ j and ρk respectively, and a time t (ρk ) > 0 independent of the exponent M, such that w M t (ρk )
χw ) : Hgk → Hg j . j M jk (z)χk = O(h
For z ∈ R(δ, M0 , h) similar statements hold, modulo the fact that the symbol of the Fourier integral operator is multiplied by exp(−i zt+ / h), which modifies the order of the operator. 5.3.2. A well-posed inhomogeneous problem. Let us now consider the inhomogeneous problem (i/ h)(Pθ,R − z)u + R− (z)u − = v
v ∈ HG (X ).
(5.33)
We will use a partition of unity to decompose v into several components. Take ψδ ∈ S(T ∗ X ), ψδ = 1 near p −1 ([−δ/2, δ/2]), and ψδ = 0 outside −1
δ p ([−δ, δ]). The operator (Pθ,R − z) is elliptic outside p −1 [−δ/2, δ/2]. Taking ψ
δ ⊂ p −1 ([−δ/2, δ/2]), the operator similar with ψδ but with supp ψ def
δw ) : HG2 → HG L = (Pθ,R − z − i ψ
is invertible, with uniformly bounded inverse L −1 ∈ h0 . Hence, by taking u = (h/i)L −1 (1 − ψδw ) v, we find
δw )u + O(h ∞ ) u = (1−ψδw ) v + O(h ∞ ) v , (i/ h)(Pθ,R −z)u = (i/ h)(Pθ,R −z −i ψ
44
S. Nonnenmacher, J. Sjöstrand, M. Zworski
which solves our problem for the data (1 − ψδw ) v. The first equality uses pseudodiffer δ : ential calculus and the fact that ψδ ≡ 1 on the support of ψ
δw L −1 (1 − ψδw ) = OS →S (h ∞ ). ψ Let us now consider the data (ψδw v) microlocalized in p −1 ([−δ, δ]). We split this state using a spatial cutoff ψ R ∈ Cc∞ (X ), such that ψ R = 1 in B(0, R), ψ R = 0 outside B(0, 2R). To solve the equation (i/ h)(Pθ,R − z)u =
v,
v = (1 − ψ R ) ψδw v,
(5.34)
we take the Ansatz u = E(z)
v,
(5.35)
with E(z) the parametrix of (4.15) (with P replaced by Pθ,R ), for the same time T = tmax + as in (4.39). It satisfies v − e−i T (Pθ,R −z)/ h
v. (i/ h)(Pθ,R − z)u =
(5.36)
The time T is small enough, so that ∗ ∗ t p −1 ([−δ, δ])\TB(0,R) X ∩ TB(0,R/2) X = ∅, 0 ≤ t ≤ T. Hence, the states
v (t) = e−it (Pθ,R −z)/ h
v def
∗ X for t ∈ [0, T ]. The estimate (3.30) (adapted are all microlocalized outside TB(0,R/2) to the weight G 0 ) then implies that [30, Lemma 6.4]
2 Im (Pθ,R − z)
v (t),
v (t) HG h ≤ (−M1 /C1 + 2M0 ) log(1/ h), ∀t ∈ [0, T ],
∂t
v (t) 2HG =
where C1 > 0 is independent of the choice of M1 . Once more, we assume M0 /M1 is small enough so that −M1 /C1 + 2M0 ≤ −M1 /2C1 , and hence
e−i T (Pθ,R −z)/ h
v HG ≤ C h M1 T /2C1
v HG , so the problem (5.34) is solved modulo a remainder O(h M1 T /2C1 ). ∗ We now consider the component (ψ R ψδw v) microlocalized in TB(0,2R) ∩ p −1 ([−δ, δ]). We split it again using a cutoff ψV1 ∈ Cc∞ (V1 ), ψV1 = 1 in the set V1 V1 (see the discussion after Lemma 5.3). To solve the problem for the inhomogeneous data
v = (1 − ψVw1 )ψ R ψδw v, we use the Ansatz (5.35), resulting in the estimate (5.36). The microlocalization of
v outside of V1 , together with the assumption (5.11), implies the norm estimate (see Lemma 5.4)
e−i T (Pθ,R −z)/ h
v HG ≤ C h Mt1 /2−M0 T
v HG .
From Open Quantum Systems to Open Quantum Maps
45
Again, we assume M0 /M small enough, so that Mt1 /2 − M0 T ≥ Mt1 /3. We have solved the problem for
v up to a remainder O(h Mt1 /3 )
v HG . We finally consider the data
v = ψVw1 ψ R ψδw v microlocalized inside V1 . For this data, −− , we can use the microlocal analysis of §4.2.3. If WFh (
v ) is contained inside V1 ∩ T jk v ) (see the Ansatz (4.40)) will intersect j inside the arrival set then WFh (χbk E(z)
jk (z), but not necessarily inside S j . However, the same phenomenon as in Lemma 5.5 A −− (z), occurs: there exists a time t3 > 0 such that, for any z ∈ [−δ, δ] and any ρ ∈ V1 ∩T jk ρ+ (ρ) ∈ j (z)\κ j,z (S j ) !⇒ G 0 (ρ+ (ρ)) − G 0 (ρ) ≥ t3 .
(5.37)
j
v using the cutoff χ j of (5.27), the property (5.37) implies If we decompose R+,m (z)E(z)
that v Hg j = O(h Mt3 /2−M0 T )
v HG .
(1 − χ w j ) R+,m (z)E(z)
j
Again we assume M0 /M small enough, so that Mt3 /2 − M0 T ≥ Mt3 /3. Hence, if we set v u − = R+ (z)χ w j E(z)
j
j
v + O(h ∞ )
v HG = R+,m (z)χ w j E(z)
j
= R+,m (z)E(z)
v + O(h Mt3 /3 )
v HG , j
v HG . we end up with a solution of (5.33) modulo a remainder O(h Mt3 /3 )
We recall that M1 /M is bounded by (3.31), so all the above error estimates can be put in the form O(h cM )
v HG , with c > 0 independent of M: we have thus shown that the problem (5.33) admits a solution for any v ∈ HG , up to this remainder. We may then apply Proposition 5.7 to solve the resulting homogeneous problem, and get an approximate solution for the full problem (5.22). We summarize this solution in the following Proposition 5.8. Assume z ∈ R(δ, M0 , h). Let (v, v+ ) ∈ HG × H. Then there exists (u, u − ) ∈ HG2 × H such that (i/ h)(P − z)u + R− (z)u − = v + O(h cM )( v HG + v+ H ) in HG (X ), (5.38) R+ (z)u = v+ + O(h ∞ ) ( v HG + v+ H ) in H,
u H 2 + u − H h −M0 (2tmax +2) v HG + v+ H . (5.39) G
5.4. Invertibility of the Grushin problem. We can transform this approximate solution into an exact one. The system (5.38) can be expressed as an approximate inverse of P(z): u v
= E(z) , u− v+ (5.40) cM
P(z) E(z) = I + R(h) : HG × H −→ HG × H, R(h) = O(h ). For h small enough the operator I + R(h) can be inverted by a Neumann series, so we obtain an exact right inverse of P(z),
(I + R(z))−1 . E(z) = E(z)
46
S. Nonnenmacher, J. Sjöstrand, M. Zworski
Since P(z) is of index zero, E(z) is also a left inverse, which proves the well-posedness of our Grushin problem (5.22). Theorem 2. We consider h > 0 small enough, and z ∈ R(δ, M0 , h). For every (v, v+ ) ∈ HG × H, there exists a unique (u, u − ) ∈ HG2 × H such that (i/ h)(Pθ,R − z)u + R− (z)u − = v, (5.41) R+ (z)u = v+ , where R± (z) are defined by (5.18) and (5.20). The estimates (5.39) hold, so if we write v E E+ u = E(z) , E(z) = , E − E −+ v− u− then the following operator norms (between the appropriate Hilbert spaces) are bounded by:
E , E + , E − , E −+ = O(h −M0 (2tmax +2) ).
(5.42)
Moreover, we have a precise expression for the effective Hamiltonian:
+ OH→H (h c M ) = −I + M(z, h), E −+ (z) = −I + M(z) def
(5.43)
is the matrix of “open quantum maps” defined in (5.29) and described after where M(z) Proposition 5.7. Remark. If we restrict the parameter z to a rectangle of height | Im z| ≤ Ch instead of | Im z| ≤ M0 h log(1/ h), the bounds (5.43) become E ∗ (z) = O(1). Theorem 1 and formula (1.8) follow from this more precise result. In fact, the equality (3.13) shows that 1 tr Rθ,R (w) dw, (5.44) rank χ R(w)χ dw = rank χ Rθ,R (w)χ dw = − 2πi z z z see [41, Prop. 3.6] for the proof of the last identity in the simpler case of compactly supported perturbations, and [39, Sect. 5] for the general case. The well-posedness of our Grushin problem means that we can apply formula (3.33) recalled in §3.7. It shows that the right-hand side in (5.44) is equal to 1 tr E −+ (w)−1 E −+ (w) dw, 2πi z def
which in view of (5.43) gives (1.8). The exponent L = c M in the remainder of (5.43) depends on the integer M > 0 used in the scaling of the weight function G, which can be chosen arbitrary large, independently of c > 0. Acknowledgements. We would like to thank the National Science Foundation for partial support under the grant DMS-0654436. This article was completed while the first author was visiting the Institute of Advanced Study in Princeton, supported by the National Science Foundation under agreement No. DMS-0635607. The first and second authors were also partially supported by the Agence Nationale de la Recherche under the grant ANR -09-JCJC-0099-01. Thanks also to Edward Ott for his permission to include Fig. 3 in our paper. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
From Open Quantum Systems to Open Quantum Maps
47
References 1. Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schröfdinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971) 2. Alexandrova, I.: Semi-Classical Wavefront Set and Fourier Integral Operators. Can. J. Math 60, 241–263 (2008) 3. Balazs, N.L., Voros, A.: The quantized baker’s transformation. Ann. Phys. (NY) 190, 1–31 (1989) 4. Bogomolny, E.B.: Semiclassical quantization of multidimensional systems. Nonlinearity 5, 805–866 (1992) 5. Borgonovi, F., Guarneri, I., Shepelyansky, D.L.: Statistics of quantum lifetimes in a classically chaotic system. Phys. Rev. A 43, 4517–4520 (1991) 6. Bony, J.-M., Chemin, J.-Y.: Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. France 122, 77–118 (1994) 7. Borgonovi, F., Guarneri, I., Shepelyansky, D.L.: Statistics of quantum lifetimes in a classically chaotic system. Phys. Rev. A 43, 4517–4520 (1991) 8. Bowen, R., Walters, P.: Expansive One-parameter Flows. J. Diff. Eq. 12, 180–193 (1972) 9. Chirikov, B.V., Izrailev, F.M., Shepelyansky, D.L.: Dynamical stochasticity in classical and quantum mechanics. Math. Phys. Rev., 2, 209–267 (1981); Soviet Sci. Rev. Sect. 2 C: Math. Phys. Rev. 2, Chur: Harwood Academic, 1981 10. Cvitanovi´c, P., Rosenquist, P., Vattay, G., Rugh, H.H.: A Fredholm determinant for semiclassical quantization. CHAOS 3, 619–636 (1993) 11. Degli Esposti, M., Graffi, S. (eds): The mathematical aspects of quantum maps. Heidelberg: Springer, 2003 12. Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the semi-classical limit. Cambridge: Cambridge University Press, 1999 13. Doron, E., Smilansky, U.: Semiclassical quantization of chaotic billiards: a scattering theory approach. Nonlinearity 5, 1055–1084 (1992); Rouvinez, C., Smilansky, U.: A scattering approach to the quantization of Hamiltonians in two dimensions – application to the wedge billiard. J. Phys. A 28, 77–104 (1995) 14. Burq, N., Guillarmou, C., Hassell, A.: Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics. GAFA 20, 627–656 (2010) 15. Evans, L.C., Zworski, M.: Lectures on Semiclassical Analysis, http://math.berkeley.edu/~zworski/ semiclassical.pdf 16. Georgeot, B., Prange, R.E.: Fredholm theory for quasiclassical scattering. Phys. Rev. Lett. 74, 4110–4113 (1995) 17. Gaspard, P., Rice, S.A.: Semiclassical quantization of the scattering from a classically chaotic repellor. J. Chem. Phys. 90, 2242–2254 (1989) 18. Gérard, C.: Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes. Mémoires de la Société Mathématique de France Sér. 2(31), 1–146 (1988) 19. Gérard, C., Sjöstrand, J.: Semiclassical resonances generated by a closed trajectory of hyperbolic type. Commun. Math. Phys. 108, 391–421 (1987) 20. Gutzwiller, M.: Chaos in classical and quantum mechanics. New York: Springer, 1990 21. Helffer, B., Sjöstrand, J.: Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.) 24, 1–228 (1986) 22. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. I, II, Berlin: Springer-Verlag, 1983 23. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. III, IV, Berlin: Springer-Verlag, 1985 24. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press, 1997 25. Keating, J.P., Novaes, M., Prado, S.D., Sieber, M.: Semiclassical structure of quantum fractal eigenstates. Phys. Rev. Lett. 97, 150406 (2006) 26. Martinez, A.: Resonance free domains for non globally analytic potentials. Ann. Henri Poincaré 3(4), 739–756 (2002). Erratum: Ann. Henri Poincaré 8(7), 1425–1431 (2007) 27. Nonnenmacher, S., Rubin, M.: Resonant eigenstates for a quantized chaotic system. Nonlinearity 20, 1387–1420 (2007) 28. Nonnenmacher, S., Zworski, M.: Distribution of resonances for open quantum maps. Commun. Math. Phys. 269, 311–365 (2007) 29. Nonnenmacher, S., Zworski, M.: Fractal Weyl laws in discrete models of chaotic scattering. J. Phys. A 38, 10683–10702 (2005) 30. Nonnenmacher, S., Zworski, M.: Quantum decay rates in chaotic scattering. Acta Math. 203, 149– 233 (2009) 31. Nonnenmacher, S., Sjöstrand, J., Zworski, M.: Fractal Weyl law for open quantum chaotic maps. In preparation
48
S. Nonnenmacher, J. Sjöstrand, M. Zworski
32. Ozorio de Almeida, A.M., Vallejos, R.O.: Decomposition of Resonant Scatterers by Surfaces of Section. Ann. Phys. (NY) 278, 86–108 (1999) 33. Petkov, V., Zworski, M.: Semi-classical estimates on the scattering determinant. Ann. H. Poincaré 2, 675–711 (2001) 34. Poon, L., Campos, J., Ott, E., Grebogi, C.: Wada basin boundaries in chaotic scattering. Int. J. Bifurcation and Chaos 6, 251–266 (1996) 35. Prosen, T.: General quantum surface-of-section method. J. Phys. A28, 4133–4155 (1995) 36. Saraceno, M., Vallejos, R.O.: The quantized D-transformation. Chaos 6, 193–199 (1996) 37. Schomerus, H., Tworzydlo, J.: Quantum-to-classical crossover of quasi-bound states in open quantum systems. Phys. Rev. Lett. 93, 154102 (2004) 38. Sjöstrand, J.: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60, 1–57 (1990) 39. Sjöstrand, J.: A trace formula and review of some estimates for resonances. In: Microlocal analysis and spectral theory (Lucca, 1996), 377–437, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 490, Dordrecht: Kluwer Acad. Publ., 1997, pp. 377–437 40. Sjöstrand, J.: Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations. http://arxiv.org/abs/0802.3584v3 [math.sp], 2009 41. Sjöstrand, J., Zworski, M.: Complex scaling and the distribution of scattering poles. J. AMS 4, 729– 769 (1991) 42. Sjöstrand, J., Zworski, M.: Quantum monodromy and semiclassical trace formulae. J. Math. Pure Appl. 81, 1–33 (2002) 43. Sjöstrand, J., Zworski, M.: Fractal upper bounds on the density of semiclassical resonances. Duke Math. J. 137, 381–459 (2007) 44. Sjöstrand, J., Zworski, M.: Elementary linear algebra for advanced spectral problems. Ann. l’Inst. Fourier 57, 2095–2141 (2007) 45. Tang, S.H., Zworski, M.: From quasimodes to resonances. Math. Res. Lett. 5, 261–272 (1998) 46. Tworzydlo, J., Tajic, A., Schomerus, H., Beenakker, C.W.: Dynamical model for the quantum-to-classical crossover of shot noise. Phys. Rev. B 68, 115313 (2003) 47. Voros, A.: Unstable periodic orbits and semiclassical quantisation. J. Phys. A 21, 685–692 (1988) 48. Wirzba, A.: Quantum Mechanics and Semiclassics of Hyperbolic n-Disk Scattering Systems. Phys. Rep. 309, 1–116 (1999) Communicated by S. Zelditch
Commun. Math. Phys. 304, 49–68 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1213-1
Communications in
Mathematical Physics
Generalized Pseudo-Kähler Structures Johann Davidov1,2, , Gueo Grantcharov3 , Oleg Mushkarov1, , Miroslav Yotov3 1 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.
E-mail:
[email protected];
[email protected]
2 “L. Karavelov” Civil Engineering Higher School, 1373 Sofia, Bulgaria 3 Department of Mathematics, Florida International University, Miami, FL 33199, USA.
E-mail:
[email protected];
[email protected] Received: 4 May 2010 / Accepted: 18 October 2010 Published online: 11 March 2011 – © Springer-Verlag 2011
Abstract: In this paper we consider pseudo-bihermitian structures – pairs of complex structures compatible with a pseudo-Riemannian metric. We establish relations of these structures with generalized (pseudo-) Kähler geometry and holomorphic Poisson structures similar to that in the positive definite case. We provide a list of compact complex surfaces which could admit pseudo-bihermitian structures and give examples of such structures on some of them. We also consider a naturally defined null plane distribution on a generalized pseudo-Kähler 4-manifold and show that under a mild restriction it determines an Engel structure. 1. Introduction Bihermitian structures have recently received serious attention due to their relations to supersymmetric sigma models in theoretical physics and generalized geometry. However one of the reasons they were introduced in [3] was the observation that the selfdual component of the Weyl tensor of an oriented Riemannian 4-manifold determines a restriction on the number of (local) complex structures compatible with the metric and the orientation. The possibilities are 0, 1, 2, or ∞, if we do not distinguish structures differing by sign. The bihermitian structures thus arise naturally on 4-manifolds with 2 different (up to sign) compatible complex structures. About 15 years earlier than the paper [3], these structures appeared in the physics literature [12], where the target spaces of the sigma-models with (2, 2)-sypersymmetry were identified with Riemannian manifolds admitting 2 compatible complex structures satisfying additional differential restrictions. An impulse for development of this topic in geometry and string theory was the interpretation of bihermitian structures in terms of the so-called generalized Kähler Partially supported by “L. Karavelov” Civil Engineering Higher School, Sofia, Bulgaria under contract No 10/2009. Partially supported by CNRS-BAS joint research project Invariant metrics and complex geometry, 2008– 2009.
50
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
structures [16,20], the latter being equivalent to the geometry induced on the target of a N = (2, 2) supersymmetric sigma model [12,18]. This interpretation brought an important new viewpoint for studying deformations of such structures and led to a number of new examples [14,17]. On a pseudo-Riemannian 4-manifold of neutral signature (+, +, −, −) there are analogs for most of the notions in the Riemannian case. In particular, compatible complex structures and self-duality are well defined, unlike the Lorentzian case. Many results in the neutral setting are similar to results in the Riemannian case but there are also important differences. In this note we develop the notion of a pseudo-bihermitian structure which was considered also in the physics literature [13]. We show that, in the same way as in the Riemannian case, it can be related to (twisted) generalized pseudo-Kähler structures (Sect. 3) as well as to holomorphic Poisson structures (Sect. 4). In Sect. 5 we show that the 3-dimensional complex flag manifold Fl carries a generalized Kähler structure. We also prove that any holomorphic line bundle on Fl is a holomorphic Poisson module with respect to a Poisson structure of a special type. In Sect. 6 we provide a list of all compact complex surfaces which might carry pseudo-bihermitian structures. It contains the list of bihermitian surfaces obtained in [3]. In Sect. 7 we adapt a construction of [16,21] to find examples of pseudo-bihermitian structures, which are collected in Proposition 10. Note that no Kodaira surface admits generalized Kähler structures [4,5], but it admits a generalized pseudo-Kähler structure. We consider also some other differences between the Riemannian and the neutral setting. The first one is related to the basic observation that on a 4-dimensional vector space two complex structures J+ and J− inducing the same orientation are compatible with a positive-definite inner product iff J+ J− + J− J+ = 2 p I d for a constant p with | p| < 1. The same holds for structures compatible with a split-signature inner product, but this time | p| > 1. The difference appears when the above identities are considered globally on a 4-manifold. If p is a function with | p| < 1 at each point, then there always exists a unique conformal class of positive-definite metrics compatible with J+ and J− . However we show in Sect. 7, Example 3 that there are compact 4-manifolds admitting two such structures J+ and J− with | p| > 1 at every point which are not compatible with a global pseudo-Riemannian metric, despite the fact that locally such a metric always exists. Another difference comes from the fact that there is a naturally defined null-plane distribution on any pseudo-bihermitian manifold, which is totally real with respect to both complex structures. We show in Sect. 8 that, under a mild restriction this distribution is an Engel structure, which is a good analog of a contact structure in dimension four [29]. 2. Pseudo-Bihermitian Structures In this section we consider the indefinite analog of bihermitian structures on 4-manifolds. An almost para-hypercomplex structure on a smooth 4-manifold M (also called an almost complex product [1] or a neutral almost hypercomplex structure [11]) consists of three endomorphisms J1 , J2 , J3 of T M satisfying the relations J12 = −J22 = −J32 = −I d, J1 J2 = −J2 J1 = J3
(1)
of the imaginary units of the paraquaternionic algebra (split quaternions). A metric g on M is called compatible with the structure {J1 , J2 , J3 } if g(J1 X, J1 Y ) = −g(J2 X, J2 Y ) = −g(J3 X, J3 Y ) = g(X, Y )
(2)
Generalized Pseudo-Kähler Structures
51
(such a metric is necessarily of neutral signature (+, +, −, −)). In this case we say that {g, J1 , J2 , J3 } is an almost para-hyperhermitian structure. For any such a structure we define three 2-forms i setting i (X, Y ) = g(Ji X, Y ), i = 1, 2, 3. If the Nijenhuis tensors of J1 , J2 , J3 vanish, the structure {g, J1 , J2 , J3 } is called parahyperhermitian and (J1 , J2 , J3 ) is called para-hypercomplex. When additionally the 2-forms i (X, Y ) = g(Ji X, Y ) are closed, the para-hyperhermitian structure is called para-hyperkähler (also called hypersymplectic [19] and neutral hyperkähler [11]). Hypercomplex or para-hypercomplex structures can be obtained in the following way. Consider a 4-manifold with two complex structures J+ and J− such that J+ J− + J− J+ = 2 p I d
(3)
for a function p. Suppose that | p|<1 at each point. Then J+ , K = √ 1 2
1− p 2
[J+ , J− ], S= − √ 1
1− p 2
(J− +
p J+ ) form an almost hypercomplex structure (cf. e.g.[13]). Thus the complex structures J+ and J− are compatible with a positive definite metric. If | p| > 1 at every point, then 1 1 J+ , K = [J+ , J− ], S = − (J− + p J+ ) 2 p2 − 1 p2 − 1 form an almost para-hypercomplex structure [13]. Hence by [9] there is a locally defined metric compatible with the structure {J+ , K , S}. It is clear that the structure J− is also compatible with this metric. Conversely, if the structures J+ and J− are compatible with a pseudo-Riemannian metric g, so will be K and S, hence g is of neutral signature. Note that, unlike the positive definite case, given J+ and J− , such a metric may not exist globally (see Example 3 in Sect. 7). It follows from the above discussion that if | p| = 1 at every point, then J+ and J− yield the same orientation. This is a consequence from the well-known fact that two non-collinear (almost) complex structures on a 4-manifold both compatible with a pseudo-Riemannian metric determine opposite orientations exactly when they commute. Definition 1. If J+ = ±J− are complex structures on a 4-manifold compatible with a pseudo-Riemannian metric g and if they yield the same orientation, then (g, J+ , J− ) is said to be a pseudo-bihermitian structure. Such a structure is called strict if J+ = ±J− at every point. Note that if (g, J+ , J− ) is a pseudo-bihermitian structure, then J+ and J− satisfy identity (3) with p = − 21 g(J+ , J− ). The following lemma is well-known in the positive definite case. For the neutral case it is stated in [13] and proved in [25] for generalized Kähler structures. For the sake of completeness we provide a new proof, which works both in the positive and neutral-signature cases. Lemma 1. Let J+ and J− be complex structures on a 4-manifold such that J+ J− + J− J+ = 2 p I d for p = const and | p| > 1. Then {J+ , K , S} is a para-hypercomplex structure. Proof. We have to prove that the almost product structures K and S are integrable. To do this we shall use a local neutral metric g compatible with the structure {J+ , K , S}.
52
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
Then J− is also compatible with g and p = − 21 g(J+ , J− ). Denote by F ± the Kähler 2-form of (g, J± ). Then a standard formula for the Hermitian structure (g, J± ) gives: g((∇ X J± )(Y ), Z ) = (∇ X F ± )(Y, Z ) 1 = (d F ± (J± X, Y, J± Z ) + d F ± (J± X, J± Y, Z )), 2
(4)
where ∇ is the Levi-Civita connection of g. Since the dimension of the manifold is four, there is a unique 1-form θ± (the Lee form) such that d F ± = θ± ∧ F ± . Then g((∇ X J± )(Y ), Z ) = g(X, Z )θ± (J± Y ) − g(J± X, Z )θ± (Y ) −g(X, Y )θ± (J± Z ) − g(J± X, Y )θ± (Z ). It follows that 2X (g(J+ , J− )) = 2g(∇ X J+ , J− ) + 2g(J+ , ∇ X J− ) = −θ+ ([J+ , J− ]X ) + θ− ([J+ , J− ]X ). Thus 2d(g(J+ , J− )) = −(θ+ − θ− ) ◦ [J+ , J− ].
(5)
In view of the identity 2 p = − g(J+ , J− ), the condition p = const leads to θ+ = θ− since [J+ , J− ] = 2 p 2 − 1K = 0 at every point. Then, using the identity S = − √ 12 (J− + p J+ ), we see that the fundamental 2-form F S of S is a linear comp −1
bination of F − and F + with constant coefficients. Hence d F S = θ+ ∧ F S , so the Lee form of (g, S) is θ+ . Let F K be the fundamental 2-form of (g, K ) and denote its Lee form by θ K . Take a g-orthogonal basis of tangent vectors {E 1 , E 2 , E 3 , E 4 } with ||E 1 ||2 = ||E 2 ||2 = 1, ||E 3 ||2 = ||E 4 ||2 = − 1. Set εi = ||E i ||2 , i = 1, 2, 3, 4. Then the identities d F K = θ K ∧ F K and 4
εi d F K (E i , K E i , Z ) = 2
i=1
4
εi g((∇ Ei K )(K E i ), Z )
i=1
give θ K (Z ) = −
4
εi [g((∇ Ei K )(E i ), K Z )
i=1
for any tangent vector Z . Since K = −J+ S, we have θ K (Z ) = −
4 i=1
εi [g((∇ Ei J+ )(S E i ), J+ S Z ) −
4
εi [g((∇ Ei S)(E i ), S Z ).
i=1
Using (4) and the fact that d F + = θ+ ∧ F + one can easily see that the first term on the right-hand side vanishes. The second term is θ S (Z ). Thus θ K = θ S = θ+ , therefore the structures K and S are integrable [24].
Generalized Pseudo-Kähler Structures
53
3. Generalized Pseudo-Kähler Structures Recall that a H -twisted generalized complex structure on a smooth manifold M is an endomorphism I of the bundle T M ⊕ T ∗ M satisfying the following conditions: (a) I 2 = −I d, (b) I preserves the natural metric X + ξ, Y + η =
1 (ξ(Y ) + η(X )), 2
X, Y ∈ T M, ξ, η ∈ T ∗ M,
(c) the +i-eigensubbundle of I in (T M ⊕ T ∗ M) ⊗ C is involutive with respect to the H -twisted Courant bracket defined by 1 [X + ξ, Y + η] H = [X, Y ] + L X η − L Y ξ − (dı X η − dı Y ξ ) + ı Y ı X H, 2 where H is a closed 3-form. The integrability condition (c) is equivalent to the vanishing of the Nijenhuis tensor N H (A, B) = [A, B] H − [I A, I B] H + I [I A, B] H + I [A, I B] H ,
A, B ∈ T M ⊕ T ∗ M.
The space of 2-forms 2 (M) acts on T M ⊕ T ∗ M as eb (X + ξ ) = X + ξ + ı X b for any b ∈ 2 (M). Then the Courant bracket satisfies [eb (A), eb (B)] H = [A, B] H +db . In particular if I is a generalized complex structure, integrable with respect to the H -twisted Courant bracket, then J = e−b I eb is a generalized complex structure, integrable with respect to the (H − db)-twisted Courant bracket. So whenever H is exact, H = db for some 2-form b, the structure I is called untwisted since the structure J is integrable with respect to the Courant bracket with vanishing 3-form. Following M.Gualtieri [16,18] we introduce the following: Definition 2. A (twisted) generalized pseudo-Kähler structure is a pair of commuting (twisted) generalized complex structures I1 , I2 : T M ⊕ T ∗ M → T M ⊕ T ∗ M, such that the ±1-eigenspaces L ± of G = I1 I2 are transversal to T M and the canonical inner product on T M ⊕ T ∗ M is non-degenerate on L ± . Using the same proof as in [16,18], we have Theorem 2. A H -twisted generalized pseudo-Kähler structure on a manifold M is equivalent to a quadruple (g, J+ , J− , b), where g is a pseudo-Riemannian metric, J+ and J− are g-Hermitian complex structures, and b is a 2-form such that d + F + = −d − F − = H + db, where F ± is the Kähler form of (g, J± ) and d ± is the imaginary part of the ∂-operator of J± . 4. Holomorphic Poisson Structures In this section we prove an indefinite analog of the well-known result [21] that a generalized Kähler manifold carries a holomorphic Poisson structure. In fact, we have the following slightly more general result. Theorem 3. Let (M, g) be a pseudo-Riemannian manifold and let J+ , J− be two complex structures on M compatible with g and such that d + F + = −d − F − . Then M admits a J+ -holomorphic Poisson structure which vanishes iff [J + , J − ] = 0.
54
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
Proof. Let be the bivector field on M determined by the endomorphism [J+ , J− ] − i J+ [J+ , J− ] of T C M and the complex bilinear extension of g. We shall prove that is a holomorphic Poisson field. To show that is holomorphic we shall use the Chern connection D + of the pseudo-Hermitian structure (g, J+ ). It is defined by the identity g(D +X Y, Z ) = g(∇ X Y, Z ) − 21 d F + (J± X, Y, Z ), where ∇ is the Levi-Civita connection of g. As in the positive case, D + is a Hermitian connection such that the restriction of its (0, 1) part on the holomorphic tangent bundle is the ∂-operator of J+ . In view of (4) and the identity d ± F ± (X, Y, Z ) = −d F ± (J± X, J± Y, J± Z ), we have 2g((D +X J− )(Y ), Z ) = 2g(D +X J− Y, Z ) + 2g(D +X Y, J− Z ) = 2g((∇ X J− )(Y ), Z ) − d F + (J+ X, J− Y, Z )−d F + (J+ X, Y, J− Z ) = d F − (J− X, Y, J− Z ) + d F − (J− X, J− Y, Z ) − d F + (J+ X, J− Y, Z ) − d F + (J+ X, Y, J− Z ) = d − F − (X, J− Y, Z ) + d − F − (X, Y, J− Z ) + d + F + (X, J+ J− Y, J+ Z ) + d + F + (X, J+ Y, J+ J− Z ). Thus 2g((D +X J− )(Y ), Z ) = −d + F + (X, J− Y, Z ) − d + F + (X, Y, J− Z ) + d + F + (X, J+ J− Y, J+ Z ) + d + F + (X, J+ Y, J+ J− Z ).
(6)
The 3-form d + F + has no (3, 0) and (0, 3)-components, so d + F + (A, B, C) = d + F + (J+ A, J+ B, C) + d + F + (J+ A, B, J+ C) + d + F + (A, J+ B, J+ C). Applying this identity to the last two terms in (6) we get 2g((D +X J− )(Y ), Z ) = −d + F + (J+ X, J− Y, J+ Z ) − d + F + (J+ X, J+ J− Y, Z ) − d + F + (J+ X, Y, J+ J− Z ) − d + F + (J+ X, J+ Y, J− Z ). (7) Set Q = [J+ , J− ]. Then, since D + J+ = 0, we have 2g((D +X Q)(Y ), Z ) − g((D +J+ X Q)(Y ), J+ Z ) = −g((D +X J− )(Y ), J+ Z ) − g((D +X J− )(J+ Y ), Z ) −g((D +J+ X J− )(Y ), Z ) + g((D +J+ X J− )(J+ Y ), J+ Z ). Applying (6) to the first and the second term, and (7) to the third and the fourth term, we easily get g((D +X Q)(Y ), Z ) − g((D +J+ X Q)(Y ), J+ Z ) = 0.
(8)
As in [3] and [21], consider the form (X, Y ) = g(Q X, Y ). The (1, 1)-part of this form with respect to J+ vanishes since (J+ X, J+ Y ) = −g(J+2 J− J+ X, Y ) − g(J− J+2 X, J+ Y ) = g(J− J+ X, Y ) + g(J− X, J+ Y ) = −(X, Y ).
Generalized Pseudo-Kähler Structures
55
Then the (0, 2)-component of is (0,2) (X, Y ) = (0,2) (X (0,1) , Y (0,1) ) = (X (0,1) , Y (0,1) ) 1 1 = [(X, Y ) − (J+ X, J+ Y )] + i[(J+ X, Y ) + (X, J+ Y )] 4 4 1 1 = [(X, Y ) + i(X, J+ Y )] = [g(Q X, Y ) + ig(Q X, J+ Y )] 2 2 1 = g( , X ∧ Y ). (9) 2 It follows that is of type (2, 0) with respect to J+ . Moreover, we have g(D +X +i J+ X , Y ∧ Z ) = 2(D +X +i J+ X (0,2) )(Y, Z ) = [g((D +X Q)(Y ), Z ) + ig((D +X Q)(Y ), J+ Z )] + i[g((D +J+ X Q)(Y ), Z ) + ig((D +J+ X Q)(Y ), J+ Z )] = [g((D +X Q)(Y ), Z ) − g((D +J+ X Q)(Y ), J+ Z )] + i[g((D +X Q)(Y ), J+ Z ) + g((D +J+ X Q)(Y ), Z )]. Hence, by (8), g(D +X +i J+ X , Y ∧ Z ) = 0 for every X, Y, Z ∈ T M. This shows that D +X +i J+ X = 0, therefore is a holomorphic section of the anti-canonical bundle
2 T (1,0) M of (M, J+ ). To prove that the Schouten-Nijenhuis bracket [ , ] vanishes, we note first that it is enough to show that [Re , Re ] = 0. Indeed, since is holomorphic, it is easy to see in local holomorphic coordinates that [ , ] = 0. Note also that [Re , I m ] = [I m , Re ] since Re and I m are of degree 2. Thus, we have 0 = [ , ] = [Re , Re ] + [I m , I m ]. Suppose that [Re , Re ] = 0. Then we get [I m , I m ] = 0, hence [ , ] = 2i[Re , I m ]. Because [ , ] is of type (3,0) and purely imaginary, we conclude that [ , ] = 0. According to (9), the endomorphism Q of T M corresponds to the bivector field Re via the metric g. Then, in view of [28, Prop. 1.9], the equality [Re , Re ] = 0 is equivalent to Gg((∇ Q X Q)(Y ), Z ) = 0, where G means the cyclic sum over X, Y, Z and ∇ is the Levi-Civita connection of g. To prove the latter identity we use the fact that the Levi-Civita connection ∇ and the Chern connection D + of (g, J+ ) are related by 1 g(∇ X Y, Z ) = g(D +X Y, Z ) − d + F + (X, J+ Y, J+ Z ). 2 Set P = J+ J− + J− J+ . Then, by (6) we have 2g((∇ X Q)(Y ), Z ) = 2g((∇ X QY, Z ) + 2g(∇ X Y, Q Z ) = 2g((D +X Q)(Y ), Z ) − d + F + (X, J+ QY, J+ Z ) − d + F + (X, J+ Y, J+ Q Z ) = d + F + (X, PY, Z ) + d + F + (X, Y, P Z ) + 2d + F + (X, J− Y, J+ Z ) + 2d + F + (X, J+ Y, J− Z ).
(10)
56
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
Therefore 2Gg((∇ Q X Q)(Y ), Z ) = G[d + F + (Q X, PY, Z ) + d + F + (Q X, Y, P Z ) +2d + F + (Q X, J− Y, J+ Z ) + d + F + (Q X, J+ Y, J− Z )]. Using the skew-symmetry of d + F + , it is easy to see that G[d + F + (Q X, PY, Z ) + d + F + (Q X, Y, P Z )] = 2G[d + F + (J+ J− X, J+ J− Y, Z ) − d + F + (J− J+ X, J− J+ Y, Z )]. We have d + F + = −d − F − , so d + F + is of type (2, 1) + (1, 2) for both J+ and J− . Therefore d + F + (A, B, C) = Gd + F + (J+ A, J+ B, C) = Gd + F + (J− A, J− B, C). It follows that Gg((∇ Q X Q)(Y ), Z ) = G[d + F + (J+ J− X, J+ J− Y, Z ) − d + F + (J− J+ X, J− J+ Y, Z ) +d + F + (J+ J− X, J− Y, J+ Z ) − d + F + (J− J+ X, J− Y, J+ Z ) +d + F + (J+ J− X, J+ Y, J− Z ) − d + F + (J− J+ X, J+ Y, J− Z )] = G[d + F + (J− X, J− Y, Z ) − d + F + (J+ X, J+ Y, Z ) +d + F + (J− X, J− Y, Z ) − d + F + (J+ X, Y, J+ Z ) +d + F + (J− X, Y, J− Z ) − d + F + (J+ X, J+ Y, Z )] = 3[d + F + (X, Y, Z ) − d + F + (X, Y, Z )] = 0. This proves that [Re , Re ] = 0 which implies, as we have mentioned, that [ , ] = 0, i.e. is a Poisson field. One can also prove that the field is Poisson using the fact that the 2-vector corresponding to the endomorphism J+ + J− is Poisson [26] and its (2, 0)-part is a constant multiple of . A holomorphic Poisson structure on a complex surface is merely a holomorphic section of its anti-canonical bundle. Using this fact N. Hitchin [21] proposed a simple way for constructing generalized Kähler structures on Del Pezzo surfaces. A different approach by M. Gualtieri [17] based on the notion of generalized complex branes extends this construction to higher-dimensional Fano manifolds. Here, we state a modification of his result which can be proved in the same way as [17, Theorem 7.1]. Theorem 4. Let L be a holomorphic line bundle on an n-dimensional compact complex manifold M with holomorphic Poisson structure σ such that c1 (L)n = 0. Let (g0 , J0 ) be a pseudo-Kähler structure with Kähler form F0 ∈ c1 (L). Consider σ and F0 as homomorphisms σ : (T C M)∗ → T C M and F0 : T C M → (T C M)∗ , and suppose that the following conditions are satisfied: (i) σ ◦ F0 = ∂ X 1,0 for some (1, 0) vector field X 1,0 ; (ii) [Re X 1,0 , I m σ ] = 0 for the Schouten-Nijenhuis bracket. Then the choice of a Hermitian structure on L with curvature F0 determines a family of generalized pseudo-Kähler structures (gt , Jt , J0 ) with Jt = φt∗ (J0 ) for a 1-parameter group of diffeomorphisms φt such that Jt = J0 for t = 0 only at the points of M where σ = 0.
Generalized Pseudo-Kähler Structures
57
Remark 1. Using Theorem 4 or the construction in [22], one expects to produce examples of generalized pseudo-Kähler structures on ruled surfaces over a Riemann surface of genus greater than one. For example, consider a ruled surface M over a curve C of genus g > 1 obtained as a projectivization of a vector bundle V of degree deg(V ) < 2 − 3g. Its anti-canonical bundle has a nowhere-vanishing holomorphic section s and the choice of a Hermitian metric on it will produce a curvature 2-form F0 = dd c log|s|2 . Suppose that F0 is non-degenerate at each point. Then Theorem 4 and [22] produce generalized pseudo-Kähler structure with non-trivial canonical bundle. Note that when V = O⊕L is decomposable, the admissible metrics on M considered in [2] define Hermitian metrics on the anti-canonical bundle of M which are candidates to provide such F0 . However one can check that none of these metrics has a non-degenerate Ricci tensor. In case deg(V ) > 2 − 2g, there are metrics with this property but there is no holomorphic Poisson structure. So, it is an open question whether any ruled surface admits a generalized pseudo-Kähler structure. Note that R. Goto [15] has recently constructed positive definite generalized Kähler structures on some of these surfaces using more general deformations of Kähler-Poisson structures [14] than that considered in [17]. However his approach is based on elliptic methods and can not be adapted directly to the pseudoRiemannian case. 5. Generalized Pseudo-Kähler Structures on 3-Dimensional Flag Manifold Consider the complex flag manifold Fl = {(L , V )| 0 ∈ L ⊂ V ⊂ C3 , dim L = 1, dim V = 2}. It can be embedded into CP2 ×CP2 as the quadric Fl = {(x0 , x1 , x2 ; y0 , y1 , y2 ) ∈ CP2 × CP2 | x0 y0 + x1 y1 + x2 y2 = 0}. Let ω be the Kähler form of the standard Kähler structure on CP2 normalized so that ω is integral. Denote by p1 and p2 the projections of CP2 × CP2 onto the first and the second factor. Set ω1 = p1∗ ω and ω2 = p2∗ ω. The restrictions of these forms to Fl will be denoted by the same symbols. Lemma 5. For any integers a and b with ab < 0 and a + b = 0, the form F = aω1 + bω2 is non-degenerate on Fl. Proof. Suppose that for such a and b the 2-form F = aω1 + bω2 is degenerate at some point of Fl. The group U (3), embedded diagonally in U (3) ×U (3), acts transitively and holomorphically on Fl and F is invariant under this action. It follows that F degenerates at every point of Fl. This implies that the top degree F 3 vanishes since deg F = 2. We have ωi3 = ( pi∗ ω3 )|Fl = 0 for i = 1, 2. Therefore F 3 = 3ab(a ω12 ∧ ω2 + b ω1 ∧ ω22 ). Let ψ : Fl → Fl be the holomorphic map induced by the map ψ([x], [y]) = ([y], [x]) on CP2 × CP2 . It is clear that ψ ∗ ω1 = ω2 and ψ ∗ ω2 = ω1 . Therefore 0 = ψ ∗ (aω12 ∧ ω2 + bω1 ∧ ω22 ) = aω1 ∧ ω22 + bω12 ∧ ω2 . Then (a + b)(ω12 ∧ ω2 + ω1 ∧ ω22 ) = 0 and we get the identity (a + b)(ω1 + ω2 )3 = 3(a + b)(ω12 ∧ ω2 + ω1 ∧ ω22 ) = 0. But the latter identity does not hold since ω1 + ω2 is the Kähler form of Fl induced by the product of the Fubini-Studi forms on each factor of CP2 , a contradiction. Later in the paper we’ll need the following: Lemma 6. Let U and V be commuting holomorphic vector fields on a complex manifold and ϕ a smooth function on the manifold. Then (U ∧ V ) ◦ dd c ϕ = i∂((U ϕ)V − (V ϕ)U ).
58
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
Proof. We use the identity d c = 21 (i∂ − i∂) and the fact that for any (0, 1)-vector field Z , [U, Z ](1,0) = 0. Then we have 2(dd c ϕ)(U, Z ) = iU (∂ϕ(Z )) + i Z (∂ϕ(U )) − i∂ϕ([U, Z ]) = iU Z ϕ + i ZU ϕ − [U, Z ]ϕ = 2i ZU ϕ, so ıU dd c ϕ = i∂(U ϕ). From here we get (U ∧ V ) ◦ dd c ϕ = ıU dd c ϕ ⊗ V − ı V dd c ϕ ⊗ U = i∂(U ϕ) ⊗ V − i∂(V ϕ) ⊗ U = i∂((U ϕ)V − (V ϕ)U ). Now we are ready to prove the following: Proposition 7. The flag manifold Fl admits a generalized pseudo-Kähler structure. Proof. Take arbitrary integers a and b with ab < 0, a + b = 0. Then F0 = aω1 + bω2 is non-degenerate by Lemma 5, so it determines a pseudo-Kähler metric on Fl. Since the form F0 is integral, it determines a Hermitian holomorphic line bundle L on Fl with curvature F0 . We have c1 (L)3 = 0 since c1 (L)3 is represented by the invariant form F03 on Fl and the 2- form F0 is non-degenerate. Now we want to define a holomorphic Poisson structure on Fl as σ = Z 1 ∧ Z 2 for two commuting holomorphic vector fields Z 1 and Z 2 . Let Z 1 and Z 2 be the fields on CP2 × CP2 generated by the complex 1-parameter groups (x0 , x1 , x2 ; y0 , y1 , y2 ) → (et x0 , e−t x1 , x2 ; e−t y0 , et y1 , y2 ) and (x0 , x1 , x2 ; y0 , y1 , y2 ) → (et x0 , x1 , e−t x2 ; e−t y0 , y1 , et y2 ), respectively. Clearly Z 1 and Z 2 are commuting holomorphic vector fields tangent to Fl. Then Z 1 ∧ Z 2 is a holomorphic Poisson structure on Fl. To show that Fl admits a generalized pseudo-Kähler structure it remains only to check conditions (i) and (ii) in Theorem 4. Denote by X the holomorphic vector field on CP2 generated by the group (x0 , x1 , x2 ) → (et x0 , e−t x1 , x2 ). Then Z 1 = (X ◦ p1 , −X ◦ p2 ). Similarly Z 2 = (Y ◦ p1 , −Y ◦ p2 ), where Y is the vector field on CP2 generated by the group (x0 , x1 , x2 ) → (et x0 , x1 , e−t x2 ) The bi-vector field τ = X ∧ Y is a holomorphic section of the anti-canonical bundle of CP2 . Set f = ln ||τ ||2 , where the norm is taken with respect to metric yielded by the normalized Fubini-Study metric g of CP2 . We claim that, although f is defined only outside of the zero set of τ , the functions X f and Y f are globally defined and smooth. To check this we use the standard coordinates of CP 2 . For the coordinates z 1 = xx01 , z 2 = xx20 , set gαβ = g(
∂ ∂ , ) and G (z) = g11 g22 − |g12 |2 . ∂z α ∂z β
Then ||τ ||2 = 4|z 1 z 2 |2 G (z) and we have ∂ ∂ ∂ ∂ ∂ − z2 , Y = −z 2 , τ = 2z 1 z 2 ∧ , ∂z 1 ∂z 2 ∂z 2 ∂z 1 ∂z 2 ln G (z) ln G (z) ln G (z) X f = −3 − 2z 1 − z2 , Y f = −1 − z 2 . ∂z 1 ∂z 2 ∂z 2 X = −2z 1
(11)
Generalized Pseudo-Kähler Structures
In the coordinates u 1 =
59
x0 x1 , u 2
=
x2 x1
we have
∂ ∂ ∂ ∂ ∂ + u2 , Y = −u 2 , τ = −2u 1 u 2 ∧ , ∂u 1 ∂u 2 ∂u 2 ∂z 1 ∂z 2 ln G (u) ln G (u) ln G (u) X f = 3 + 2u 1 + u2 , Y f = −1 − u 2 . ∂u 1 ∂u 2 ∂u 2 X = 2u 1
Finally, in the coordinates v1 =
x0 x2 , v2
=
x1 x2
(12)
we have
∂ ∂ ∂ ∂ ∂ ∂ − v2 , Y = v1 + v2 , τ = 2v1 v2 ∧ , ∂v1 ∂v2 ∂v1 ∂v2 ∂v1 ∂v2 ln G (v) ln G (v) ln G (v) ln G (v) X f = v1 − v2 , Y f = 2 + v1 − v2 . ∂v1 ∂v2 ∂v1 ∂v2 X = v1
(13)
It follows from (11), (12), (13) that τ vanishes on the analytic set C = {[x] ∈ CP2 : x0 x1 x2 = 0} and that X f, Y f can be extended to smooth functions on a neighborhood of every point of C. Since CP2 \C is dense, we see that X f, Y f can be extended to unique smooth functions on the whole space CP2 . We shall denote the extensions by the same symbols. Identities (11), (12), (13) imply also that if ζ = (ζ1 , ζ2 ) is a standard coordinate system of CP2 , we have dd c ln ||τ ||2 = dd c ln G (ζ ) on CP2 \C. Therefore dd c ln ||τ ||2 on CP2 \C is the Ricci from of the standard Kähler structure on CP2 . As it is well-known, the Ricci form of this structure is equal to 3 times the Kähler form. Thus, since we are working with the normalized Kähler form, we have dd c ln ||τ ||2 = 3λω, where λ > 0 is a constant. Hence, for k = 1, 2, dd c (ln ||τ ||2 ◦ pk ) = 3λpk∗ ω = 3λ ωk on the set M = {(x0 , x1 , x2 ; y0 , y1 , y2 ) ∈ CP2 × CP2 | x0 x1 x2 y0 y1 y2 = 0}. Thus on M we have 1 (Z 1 ∧ Z 2 )(a dd c (ln ||τ ||2 ◦ p1 ) + b dd c (ln ||τ ||2 ◦ p2 )). (14) (Z 1 ∧ Z 2 ) ◦ F0 = 3λ It follows from (14) and Lemma 6 that if we set X 1,0 =
i {[a(X f ) ◦ p1 − b(X f ) ◦ p2 ]Z 2 − [a(Y f ) ◦ p1 − b(Y f ) ◦ p2 ]Z 1 }, 3λ
where f = ln ||τ ||2 as above, we have (Z 1 ∧ Z 2 ) ◦ F0 = ∂ X 1,0 on the open set M. This identity holds everywhere since the vector field X 1,0 is smooth on CP2 × CP2 and M is dense. Thus condition (i) of Theorem 4 is satisfied for σ = Z 1 ∧ Z 2 . To show that condition (ii) also holds, we note that [X 1,0 , Z 1 ∧ Z 2 ] = −
i {a([X, Y ] f ) ◦ p1 + b([X, Y ] f ) ◦ p2 }Z 1 ∧ Z 2 = 0 3λ
since [X, Y ] = 0. The function f is real-valued, so X f = X f, Y f = Y f , and we have [X 1,0 , Z 1 ∧ Z 2 ] =
i {a([X, Y ] f ) ◦ p1 + b([X, Y ] f ) ◦ p2 }Z 1 ∧ Z 2 = 0. 3λ
Using the identities [X, X ] = [Y, Y ] = [X, Y ] = [X , Y ] = 0, it is easy to see that [X 1,0 , Z 1 ∧ Z 2 ] − [X 1,0 , Z 1 ∧ Z 2 ] = 0. It follows that [Re X 1,0 , I m(Z 1 ∧ Z 2 )] = 0. Then, by Theorem 4, the flag manifold Fl admits a generalized pseudo-Kähler structure.
60
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
Note that Fl admits also a usual generalized Kähler structure [14]. Corollary 8. Any holomorphic line bundle on the 3-dimensional flag manifold Fl carries a structure of a holomorphic Poisson module with respect to the holomorphic Poisson structure U1 ∧ U2 defined by commuting holomorphic vector fields U1 and U2 . Proof. First we notice that any two commuting vector fields on Fl span a maximal torus in the algebra sl(3, C) of the holomorphic vector fields on Fl and all such tori are conjugate in the group of biholomorphisms. So, we may assume that the vector fields U1 and U2 in the corollary coincide with Z 1 and Z 2 defined in the proof of Proposition 7. Denote by K the canonical bundle of CP2 . It is well known that every holomorphic line bundle over Fl is of the form L mn = m3 p1∗ K + n3 p2∗ K , where m, n ∈ Z. If we consider K with the metric induced by the normalized Fubini-Study metric of CP2 , the curvature form of K with respect to its canonical connection is equal to the Kähler form ω. Therefore the form m3 p1∗ ω + n3 p2∗ ω = m3 ω1 + n3 ω2 represents the first Chern class of L mn . Denote this form by F and set σ = Z 1 ∧ Z 2 . We have seen above that there is a (1, 0)-vector field X 1,0 such that σ ◦ X 1,0 = ∂ X 1,0 and [X 1,0 , σ ] = 0. Now the corollary follows from [17, Prop. 10] since the first Chern class of L mn coincides with its Atiyah class. 6. The Four-Dimensional Case In dimension four, a pseudo-hermitian metric is either positive (negative) definite or of signature (2,2). Using the results in Sect. 4 we shall prove the following: Theorem 9. Let (M, g, J+ , J− ) be a compact pseudo-bihermitian 4-manifold. (i) If d + F + = −d − F − , then (M, J+ ) (and (M, J− )) is one of the following complex surfaces: a complex torus, a K3 surface, a primary Kodaira surface, a blow-up of a surface of class V I I0 , a ruled surface described in [7] with χ ±τ divisible by 4, where χ and τ are the Euler characteristic and the signature of M. (ii) If the bihermitian structure is strict, then (M, J+ ) (and (M, J− )) is one of the following: a complex torus, a K3 surface, a primary Kodaira surface, a properly elliptic surface of odd first Betti number, a Hopf surface, a minimal Inoue surface without curves. Proof. According to Theorem 3, under assumption (i) there is a non-zero holomorphic section of the anti-canonical bundle of (M, J+ ). Such surfaces with even first Betti number are described in [7] and they exhaust the first four cases in (i). The restriction on χ ±τ in the last case comes from Matsushita’s topological condition for existence of a split-signature metric [27]. For the case of surfaces with odd first Betti number, we notice that the proof of Proposition 2.3 in [8] shows that either the Kodaira dimension of (M, J+ ) (and (M, J− )) is −∞ or its canonical bundle is holomorphically trivial. Then the Kodaira classification of minimal compact complex surfaces [6] leads to the list in (i). Part (ii) follows from the fact that the canonical bundle is topologically trivial in the case of strictly pseudo-bihermitian surfaces, since the 2-form (0,2) given by (9) provides a non-vanishing section. So one can use the well-known list of the surfaces with vanishing first Chern class [30] Remark 2. Notice that, by [7, Lemma 2.1], if a compact complex surface is not minimal and has a nowhere-vanishing holomorphic section of the anti-canonical bundle, then its minimal model also admits such a section. Moreover, the dimension of the space
Generalized Pseudo-Kähler Structures
61
of holomorphic sections decreases by at most one after a blow-up. It keeps the same dimension only if the blow-up is at a base point of the anti-canonical linear system. This leads to additional restrictions on the possible blow-ups of surfaces in case (i), but we shall not discuss this question here. Remark 3. There are generalized pseudo-Kähler manifolds (M, g, J+ , J− ) so that J+ and J− induce opposite orientations. In the four dimensional case such structures commute. In any dimension, for a generalized pseudo-Kähler manifold with commuting J+ and J− , the same reasoning as in [5] shows that the holomorphic tangent bundle of (M, J+ ) splits into a sum of two holomorphic subbundles. Conversely, if the holomorphic tangent bundle of a compact complex surface (M, J ) splits, then by [5] there is a generalized (pseudo) Kähler structure (g, J+ , J− ) such that J+ = J and [J+ , J− ] = 0. 7. Generalized Pseudo-Kähler Structures Via Deformations of Para-Hyperkähler Structures It has been observed in [3,16,21] that one can explicitly define a generalized Kähler structure by means of a hyperkähler structure. Given a para-hyperkähler structure, a similar construction can be applied to obtain a generalized pseudo-Kähler structure. Let {g, J1 , J2 , J3 } be a para-hyperkähler structure on a 4-manifold M with J12 = −J22 = −J32 = −I d and J3 = J1 J2 . We would like to construct two commuting generalized almost complex structures I1 and I2 following [21]. To do this we need complex valued 2-forms β1 and β2 on M which satisfy (β1 − β2 )2 = (β1 − β2 )2 = 0, β1 = β2 , β1 = β2
(15)
at every point. We set exp(βk ) = 1+βk + 21 βk2 , k = 1, 2, and (X +ξ ). exp(βk ) = ı X exp(βk )+ ξ ∧ exp(βk ) for X + ξ ∈ T M ⊕ T ∗ M (the Clifford action of T M ⊕ T ∗ M on the forms). Then E k = {A ∈ (T M ⊕ T ∗ M)C | A. exp(βk ) = 0} is the +i-eigenspace of a generalized almost complex structure Ik . If βk is closed, Ik is Courant integrable [16,21]. It is shown in [21, Lemma 1] that I1 and I2 commute. Moreover, E 1 ∩ E 2 ⊕ E 1 ∩ E 2 is the (−1)-eigenspace of I1 I2 and E 1 ∩ E 2 ⊕ E 1 ∩ E 2 is the (+1)-eigenspace. Note also that E 1 ∩ E 2 = {U − ıU β1 | U ∈ T C M, ıU β1 = ıU β2 } ([21]). Thus, for A = U − ıU β1 ∈ E 1 ∩ E 2 , B = V − ı V β1 ∈ E 1 ∩ E 2 , we have A + A, B + B = −Re{(β1 − β1 )(U, V )} = −Re{(β2 − β2 )(U, V )} = −Re{(β1 − β2 )(U, V )}.
(16)
Now, given a para-hyperkähler structure {g, J1 , J2 , J3 } on a 4-manifold M, set J+ = J1 and J− = a J1 +b J2 +c J3 , where a, b, c are fixed numbers such that a 2 −b2 −c2 = 1 and a = 1. Then J+ and J− are complex structures compatible with the metric g satisfying the identity J+ J− + J− J+ = −2a I d. As in Sect. 2, set 1 1 K = √ [J+ , J− ], S+ = − √ (J− − a J+ ). 2 a2 − 1 a2 − 1
(17)
62
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
Then {g, J+ , K , S+ } is a para-hyperhermitian structure with S+ = J+ K . Let F + (X, Y ) = g(J+ X, Y ), F K (X, Y ) = g(K X, Y ) and ω (X, Y ) = g(S+ X, Y ) be the corresponding fundamental 2-forms. Similarly, if 1 S− = √ (J+ − a J− ), 2 a −1 then {g, J− , K , S− } is a para-hyperhermitian structure with S− = J− K . We denote the fundamental 2-forms of J− and S− by F − and ω , respectively. Set ω+ = ω + ω , ω− = ω − ω . Then
ω+ (X, Y ) = ω− (X, Y ) =
a+1 g(J+ X − J− X, Y ) = a−1 a−1 g(J+ X + J− X, Y ) = a+1
a+1 + (F (X, Y ) − F − (X, Y )), a−1
a−1 + (F (X, Y ) + F − (X, Y )). a+1
(18)
In particular, the forms ω+ and ω− are closed since F + and F − are so. Identity (10) implies that ∇[J+ , J− ] = 0, thus ∇ K = 0. Therefore the form F K is also closed. Now, similar to [16] we set β1 = F K + iω+ , β2 = −F K + iω− . Conditions (15) for these forms are equivalent to 2 − 4(F K )2 = 0. F K ω+ = F K ω− = ω+ ω− = ω+2 + ω−
(19)
Let X be a tangent vector with g(X, X ) = 1. Then {X, J+ X, K X, S+ X } is a g-orthonormal basis of tangent vectors. Using (17), (18) and the paraquaternionic identities, it is easy to see that (ω+ ∧ ω+ )(X, J+ X, K X, S+ X ) = 4(a + 1), (ω− ∧ ω− )(X, J+ X, K X, S+ X ) = −4(a − 1), (ω+ ∧ ω− )(X, J+ X, K X, S+ X ) = 0, (F K ∧ ω± )(X, J+ X, K X, S+ X ) = 0. We also have (F K ∧ F K )(X, J+ X, K X, S+ X ) = 2. It follows that identities (19) are satisfied. The identity β1 − β2 = 2F K + i(ω+ − ω− ) implies that a vector U ∈ T C M satisfies ıU (β1 − β2 ) = 0 if and only if a 2 − 1K U + i J+ U − ia J− U = 0. (20) Thus E 1 ∩ E 2 = {U − ıU β1 | U ∈ T C M, U satisfies (20)}. Let L − be the (−1)-eigenspace of I1 I2 acting on T M ⊕ T ∗ M. Any X + ξ ∈ L − can be written as X + ξ = U + U , where U = 21 (X + iY ) ∈ E 1 ∩ E 2 , Y ∈ T M, and ξ = ıU β1 − ıU β1 . In this notation, (20) is equivalent to a 2 − 1K X − J+ Y + a J− Y = 0, (21) a 2 − 1K Y + J+ X − a J− X = 0.
Generalized Pseudo-Kähler Structures
63
In fact, either of these identities is a consequence of the other one. For every V = 1 2 (Z + i T ) ∈ E 1 ∩ E 2 , we have a+1 Re(β1 − β 1 )(U, V ) = [g(J+ X − J− X, T ) − g(J+ Y − J− Y, Z )] a−1 a+1 =− [g(X, J+ T − J− T ) + g(J+ Y − J− Y, Z )]. a−1 √ Applying K to the second identity of (21) we get a 2 − 1Y = S+ X − aS− X . This gives a a2 a 2 − 1J+ Y = −K X + √ X+√ J+ J− X, a2 − 1 a2 − 1 1 a a 2 − 1J− Y = a K X + √ X+√ J− J+ X. 2 2 a −1 a −1 It follows that a−1 a 2 − 1(J+ Y − J− Y ) = (a − 1)(a + 2)K X + √ X. a2 − 1 Similarly, a−1 a 2 − 1(J+ T − J− T ) = (a − 1)(a + 2)K Z + √ Z. a2 − 1 Then
(a − 1)Re(β1 − β 1 )(U, V ) = −
a−1 g(X, Z ). a+1
(22)
Suppose that X + ξ, A = 0 for every A ∈ L − . Take any Z ∈ T M and set T = (a 2 − 1)−1/2 [S+ Z − aS− Z ]. Then V = 21 (Z + i T ) satisfies (20). Indeed we have a 2 − 1K Z − J+ T + a J− T 1 a = a 2 − 1K Z − √ (−K Z − a J+ S− Z ) + √ (J− S+ Z + a K Z ) 2 2 a −1 a −1 [J+ , J− ]Z 1 1 =√ (2a 2 K Z + a(J+ S− Z + J− S+ Z )) = √ (2a 2 K Z − a 2 √ ) 2 2 a −1 a −1 a2 − 1 1 (2a 2 K Z − 2a 2 K Z ) = 0. =√ a2 − 1 Moreover, a 2 − 1K T + J+ Z − a J− Z = K S+ Z − a K S− Z + J+ Z − a J− Z = 0. Thus V ∈ E 1 ∩ E 2 and, by our assumption, (16) and (22), we have g(X, Z ) = 0. Since the latter identity holds for every Z , we conclude that X = 0. Then Y = (a 2 − 1)−1/2 [S+ X − aS− X ] = 0, hence U = 0, thus ξ = ıU β1 −ıU β1 = 0. This proves that the canonical inner
64
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
product on T M ⊕ T ∗ M is non-degenerate on L − . Moreover, the inclusion T M ∩ L − ⊂ E 1 ∩ E 2 and identity (20) imply that T M ∩ L − = {0}. Similar arguments show that the metric . , . is non-degenerate on the (+1)-eigenspace L + of I1 I2 and T M ∩ L + = {0}. Thus I1 , I2 is a generalized pseudo-Kähler structure on M. We can deform this structure using the arbitrary smooth function f on M. Let Ht be the flow of the F K -Hamiltonian vector field ı d f F K , so Ht∗ (F K ) = F K . Define γ1 = F K + i(ω + Ht∗ ω ), γ2 = −F K + i(ω − Ht∗ ω ). Then γ1 − γ2 = 2F K + 2i Ht∗ ω = Ht∗ (2F K + 2iω ) = Ht∗ (β1 − β2 ) and γ1 − γ 2 = β1 −β 2 . It follows that for small t, the forms γ1 and γ2 define a generalized pseudo-Kähler structure. Finally, let us note that a generalized pseudo-Kähler structure can be explicitly defined by means of the pseudo-Kähler structures (g, J+ ), (g, J− ) and [16, (6.14)]. Example 1. The construction above can be applied to 4-tori and primary Kodaira surfaces since each of these surfaces admits a para-hyperkähler structure (see, for example, [23,24]). Recall that the Kodaira surfaces do not admit any (positive) generalized Kähler structure [4,5]. Example 2. Any para-hyperhermitian structure which is locally conformally para-hyperkähler can be deformed as in [3] to obtain a strictly pseudo-bihermitian structure. The universal cover of the locally conformally para-hyperkähler manifold M is globally conformally para-hyperkähler. The deformation is performed on its para-hyperkähler structure such that Ht is invariant with respect to the fundamental group of M. Then one obtains a generalized pseudo-Kähler structure which after a (global) conformal change descends to a pseudo-bihermitian structure on the quotient. In particular, there are pseudo-bihermitian metrics on properly elliptic surfaces of odd first Betti number and the Inoue surfaces of type S + [10]. These surfaces do not admit any (positive) bihermitian structure [4]. On the other hand the quaternionc Hopf surfaces admit both bihermitian and pseudo-bihermitian structures since they have both hyperhermitian and para-hyperhermitian metrics [10]. They also have bihermitian metrics arising from twisted generalized Kähler structures [5], however it is not clear whether these surfaces admit twisted generalized pseudo-Kähler structures. The same question is open for K3 surfaces too. Notice that the above constructions produce “complementary” examples of bihermitian and pseudo-bihermitian structures on the surfaces in the lists in Theorem 9. We summarize the examples obtained so far in: Proposition 10. Generalized pseudo-Kähler structures exist on complex 2-tori and primary Kodaira surfaces. Pseudo- bihermitian structures exist also on the quaternionic Hopf surfaces, properly elliptic surfaces with odd first Betti number and Inoue surfaces of type S + . Example 3. Here we provide examples of complex structures J+ and J− satisfying the relation (3) J+ J− + J− J+ = 2 p I d for a nonconstant function p with | p| > 1, which are not compatible with any global neutral metric. Consider Example 1 above in the case of a complex torus which is a
Generalized Pseudo-Kähler Structures
65
product of 2 elliptic curves. It admits a holomorphic involution φ without fixed points, such that the quotient is a smooth complex surface. This surface is called a hyperelliptic surface of type Ia . One can check that the natural para-hypercomplex structure of the torus descends to a para-hypercomplex structure on the quotient, but it admits no compatible para-hyperhermitian metrics [10]. In particular, one can fix a para-hyperkähler family of φ-invariant complex structures on the torus and can deform any two structures of this family via the procedure described in Example 2. The Hamiltonian deformations Ht are defined by a single function and if one chooses this function to be φ-invariant, then both (J+ )t = J+ and (J− )t are φ-invariant for all t. Since they satisfy the relation (3) for small t, they descend to structures which satisfy the same identity on the quotient hyperelliptic surface. Since | p| > 1 at any point for fixed t, K = 0 everywhere. If there were a compatible metric, then the fundamental forms F K + i F J+ K obtained as in the consideration above would provide a trivialization of the canonical bundle, which is absurd because the canonical bundle of a hyperelliptic surface is not topologically trivial. 8. Null-Planes of 4-Dimensional Pseudo-Bihermitian Metrics In this section we show that, under a mild restriction, a naturally defined null-plane distribution on a pseudo-bihermitian 4-manifold M determines a local Engel structure. Recall that an Engel structure is by definition a 2-dimensional distribution D on a 4-manifold M such that rank[D, D] = 3 and rank[D, [D, D]] = 4 at each point of M. These structures have been actively investigated recently (see the Introduction in [29] for an overview). They admit canonical coordinates and are preserved by small C 2 -deformations. The global existence of an oriented Engel structure on an oriented compact manifold leads to triviality of its tangent bundle. Moreover, Vogel [29] showed that the converse also holds - any paralellizable 4-manifold admits such a structure. Let (M, g, J+ , J− ) be a pseudo-bihermitian 4-manifold with J+ J− + J− J+ = 2 p I d, where | p| > 1. Let F ± and θ± be the Käher and the Lee form of (g, J± ), respectively. Suppose that the pseudo-bihermitian structure is defined by a (twisted) generalized pseudo-Kähler one. Then d + F + + d − F − = 0 by Theorem 2 and taking the Hodge-dual 1-forms we get θ+ + θ− = 0. If we set K = [J+ , J− ]/2 p 2 − 1 as above, then K 2 = I d and K = ±I d. Moreover, g(K X, Y ) = −g(X, K Y ), in particular the eigenspaces of K consists of isotropic vectors. Lemma 11. For the endomorphism N± = J+ + ( p ± p 2 − 1)J− of T M, we have K er N± = I m N± = ∓1 − eigenspace o f K . Proof. It is easy to see that N±2 = 0 and K er N+ ∩ K er N− = {0}. Moreover, − K J+ = J+ K = −K J− = J− K =
p p2
−1
1 p2 − 1
J+ − J+ −
1 p2
−1
p p2 − 1
J− , J− .
(23)
It follows that if K± is the ±1-eigenspace of K , then K− ⊂ K er N+ and K+ ⊂ K er N− . Hence dim K er N± ≥ 2. This implies the lemma since the kernels of N+ and N− are transversal and the dimension of the ambient space is 4.
66
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
Denote the vector field dual to θ± w.r.t. g by the same letter. Set X = (J+ + f J− )θ+ , where f = p − p 2 − 1 and Y = θ+ + K θ+ . Clearly X, Y ∈ K er N− . One can easily see that X and Y are isotropic. Assume that |θ+ |x = 0 at some point x ∈ M. Then the vector fields X and Y are linearly independent at x. Indeed, suppose that λX + μY = 0 at x for some constants λ and μ. Applying N− to both sides of this identity, we get μ(N− + N− K )θ+ = 0 at x. Then, using (23), we compute easily that μ(J+ θ+ − f J− θ+ )x = 0. If J+ θ+ = f J− θ+ at x, we would have |θ+ |x = f (x)|θ+ |x , hence |θ+ |x = 0, a contradiction. Therefore μ = 0, thus λ(J+ θ+ + f J− θ+ )x = 0. This implies λ = 0 since |θ+ |x = 0. Now define a 2-plane in Tx M setting Dx = Span(X, Y )x .
(24)
Theorem 12. Let (M, g, J+ , J− ) be a (twisted) generalized pseudo-Kähler 4-manifold with nowhere vanishing Lee forms θ+ = −θ− and such that J+ J− + J− J+ = 2 p I d with | p| > 1. Then the null distribution D defined by (24) is an Engel structure on an open subset of M or the flow of Y consists of null-geodesics. Proof. Set N = N− . Then D = K er N = I m N . We are going to calculate N [X, Y ] and show that it is proportional to N J+ Y . This will imply that [X, Y ] ∈ Span(X, Y, J+ Y ), so rank[D, D] = 3. Then we will show that [Y, J+ Y ] has vanishing J+ X component iff ∇Y Y = FY for some smooth function F. This proves that either the flow of Y is geodesic or rank[D, [D, D]] = 4 on an open subset of M. For the Levi-Civita connection we have [3]: 2(∇ X J± )Y = g(X, Y )J± θ± + g(J± X, Y )θ± + θ± (J± Y )X − θ± (Y )J± X, and therefore 2(∇ X N )Y = g(X, Y )(J+ − f J− )θ+ + g((J+ − f J− )X, Y )θ+ + θ+ ((J+ − f J− )Y )X − θ+ (Y )(J+ − f J− )X + 2X ( f )J− Y, (25) since θ− = −θ+ . Also p = −1/2g(J+ , J− ) = 1/4tr (J+ ◦ J− ) and we get by (5) that dp =
1 θ+ ◦ [J+ , J− ] = 2
p 2 − 1θ+ ◦ K .
(26)
We have X, Y ∈ K er N and g(X, X ) = g(Y, Y ) = 0, g(X, Y ) = g(X, J+ X ) = g(Y, J+ Y ) = 0, g(J+ X, Y ) = ( f 2 − 1)|θ+ |2 = 2( f p − 1)|θ+ |2 .
(27)
Then the vector fields X, Y, J+ X, J+ Y form a basis of the tangent space at each point of M. We have also that g(θ+ , J+ J− θ+ ) = g(θ+ , J− J+ θ+ ) = −g(J+ θ+ , J− θ+ ) = p|θ+ |2 , Y = θ+ +
J+ J− − 2 p I d + J+ J− − f θ+ + J+ J− θ+ θ+ = . 2 2 p −1 p2 − 1
(28)
Generalized Pseudo-Kähler Structures
67
Now (∇ X N )Y − (∇Y N )X = −N ∇ X Y + N ∇Y X = −N [X, Y ], since N X = N Y = 0. To compute N [X, Y ] we use the fact that J+ X = − f J− X, J+ Y = − f J− Y . Hence by (25) and (27) we have 2|θ+ |−2 (∇ X N )Y = 2( f 2 − 1)θ+ − 2J+ X = −2 f p 2 − 1Y, since θ+ (J+ Y ) = g(J+ Y, θ+ ) = 0 by (28), θ+ (Y ) = |θ+ |2 and, in view of (26) and Lemma 11, X ( f ) = f θ+ (K X ) = f θ+ (X ) = f g(X, θ+ ) = 0. Similarly 2|θ+ |−2 (∇Y N )X = −2( f 2 − 1)θ+ + 2( f p − 1)Y − 2 f J− X = 2( f p + f p 2 − 1 − 1)Y = 0, θ+ , θ+ ) = − |θ+ |2 + f p|θ+ |2 by (28) and since θ+ (X ) = 0, θ+ (J+ X ) = g(−θ+ + f J+ J− 2 Y ( f ) = f g(Y, θ+ ) = f |θ+ | . So N [X, Y ] = f p 2 − 1|θ+ |2 Y . We can easily check that N J+ + J+ N = 2( p f − 1)I d. Then N J+ Y = 2( p f − 1)Y so [X, Y ] ∈ Span(X, Y, J+ Y ). It follows from (27) that X, Y, J+ Y are linearly independent at every point, hence rank[D, D] = 3. If [Y, J+ Y ] has nowhere-vanishing J+ X -component, then rank[[D, D], D] = 4, so D is an Engel structure. To find the J+ X -component of [Y, J+ Y ] we use that [Y, J+ Y ] = ∇Y J+ Y − ∇ J+ Y Y . First observe that (∇Y N )Y = 0, so ∇Y Y ∈ Span{X, Y }. We have also that 2(∇Y J+ )Y = − θ+ (Y )J+ Y = − |θ+ |2 J+ Y . Since ∇Y J+ Y = (∇Y J+ )Y + J+ ∇Y Y , then ∇Y J+ Y ∈ Span{J+ X, J+ Y }. Moreover, the J+ X component of ∇Y J+ Y is equal to the J+ X -component of J+ ∇Y Y which is also the X -component of ∇Y Y . On the other hand we have 2(∇ J+ Y N )Y = −θ+ (Y )(J+ − f J− )(J+ Y ) = 2 p f |θ+ |2 Y, since (J+ − f J− )J+ Y = J+ (J+ + f J− )Y − 2 p f Y = −2 p f Y . So N ∇ J+ Y Y = − p f |θ+ |2 Y and ∇ J+ Y Y ∈ Span{X, Y, J+ Y } does not have a J+ X -component. Then [Y, J+ Y ] = ∇Y J+ Y − ∇ J+ Y Y has nowhere-vanishing J+ X -component iff ∇Y Y has nowhere-vanishing X -component. To finish the proof notice that ∇Y Y ∈ Span{X, Y } and if its X -component vanishes locally, ∇Y Y = FY which in turn means that the flow of Y is geodesic. Note finally that if p = const, then θ+ = θ− and the distribution D is integrable. Acknowledgements. The authors express their gratitude to V. Apostolov for helpful discussions and comments on a preliminary version of this paper. Part of this work was done during the visit of the first and the third-named authors at the Abdus Salam School of Mathematical Sciences, GC University Lahore, Pakistan and the second named author’s visit to the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences. The authors thank the two institutions for their hospitality.
References 1. Andrada, A., Salamon, S.: Complex product structures on Lie algebras. Forum Math. 17(2), 261–295 (2005) 2. Apostolov, V., Calderbank, D., Gauduchon, P., Tonnesen-Friedmann, C.: Hamiltonian 2-forms in Kähler geometry. IV. Weakly Bochner-flat Kähler manifolds. Comm. Anal. Geom. 16(1), 91–126 (2008) 3. Apostolov, V., Gauduchon, P., Grantcharov, G.: Bihermitian structures on complex surfaces. Proc. Lond. Math. Soc. 79(2), 414–428 (1999), Corrigendum 92, 200–202 (2005) 4. Apostolov, V.: Bihermitian surfaces with odd first Betti number. Math. Z. 238(3), 555–568 (2001)
68
J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov
5. Apostolov, V., Gualtieri, M.: Generalized Kaehler manifolds, commuting complex structures, and split tangent bundles. Commun. Math. Phys. 271(2), 561–575 (2007) 6. Barth, W., Hulek, K., Peters, C., Van de Ven, A.: Compact complex surfaces. Heidelberg: Springer, Second Edition, 2004 7. Bartocci, C., Macri, E.: Classification of Poisson surfaces. Commun. Contemp. Math. 7(1), 89–95 (2005) 8. Bottacin, F.: Poisson structures on moduli spaces of sheaves over Poisson surfaces. Invent. Math. 121(2), 421–436 (1995) 9. Davidov, J., Grantcharov, G., Muskarov, O., Yotov, M.: Parahyperhermitian surfaces. Bull. Math. Soc. Sci. Math. Roumanie 52(100), No 3, 281–289 (2009) 10. Davidov, J., Grantcharov, G., Muskarov, O., Yotov, M.: Work in progress 11. Dunajsky, M., West, S.: Anti-self-dual conformal structures in neutral signature. In: Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Zurich: Eur. Math. Soc. 2008, pp. 113–148 12. Gates, S., Hull, C., Rocek, M.: Twisted multiplets and new supersymmetric nonlinear sigma models. Nucl. Phys. B248, 157–196 (1984) 13. Goteman, M., Lindstrom, U.: Pseudo-hyperkahler Geometry and Generalized Kähler Geometry. Lett. Math. Phys., doi:10.1007/s11005-010-0456-7, Dec. 2010 14. Goto, R.: Poisson structures and generalized Kähler submanifolds. J. Math. Soc. Japan 61(1), 107–132 (2009) 15. Goto, R.: Unobstructed K-deformations of generalized complex structures and bihermitian structures. http://arxiv.org/abs/0911.2958v2 [math.DG], 2010 16. Gualtieri, M.: Generalized complex geometry. Oxford University DPhil Thesis. http://arxiv.org/abs/math/ 040122v1 [math.DG], 2004 17. Gualtieri, M.: Branes on Poisson varieties. http://arxiv.org/abs/0710.2719v2 [math.DG], 2010 18. M. Gualtieri: Generalized Kähler geometry. http://arxiv.org/abs/1007.34852v1 [math.DG], 2010 19. Hitchin, N.: Hypersymplectic quotients. Acta Acad. Sci. Tauriensis 124, (suppl), 169–180 (1990) 20. Hitchin, N.: Generalized Calabi-Yau manifolds. Q. J. Math. 54(3), 281–308 (2003) 21. Hitchin, N.: Instantons, Poisson structures and generalized Kähler geometry. Commun. Math. Phys. 265, 131–164 (2006) 22. Hitchin, N.: Bihermitiqan structures on Del Pezzo surfaces. J. Symplectic Geom. 5(1), 1–8 (2007) 23. Kamada, H.: Neutral hyperkähler structures on primary Kodaira surfaces. Tsukuba J. Math. 23, 321–332 (1999) 24. Kamada, H.: Self-dual Kähler metrics of neutral signature on complex surfaces. PhD thesis, Tohoku University, 2002 25. Lindstrom, U., Rocek, M., von Unge, R., Zabzine, M.: Generalized Kahler manifolds and off-shell supersymmetry. Commun. Math. Phys. 269, 833–849 (2007) 26. Lyakovich, S., Zabzine, M.: Poisson geometry of sigma models with extended supersymmetry. Phys. Lett. B 548(3–4), 243–251 (2002) 27. Matsushita, Y.: Fields of 2-planes and two kinds of almost complex structures on compact 4-dimensional manifolds. Math. Z. 207(2), 281–291 (1991) 28. Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Progress in Mathematics, Vol. 118, BaselBoston: Birkhäuser, 1994 29. Vogel, T.: Existence of Engel structures. Ann. Math. 169, 79–137 (2009) 30. Wall, C.T.C.: Geometric structures and complex surfaces. Topology 25, 119–153 (1986) Communicated by A. Kapustin
Commun. Math. Phys. 304, 69–93 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1226-9
Communications in
Mathematical Physics
Colored Group Field Theory Razvan Gurau Perimeter Institute for Theoretical Physics Waterloo, Waterloo, ON N2L 2Y5, Canada. E-mail:
[email protected] Received: 14 May 2010 / Accepted: 15 December 2010 Published online: 8 March 2011 – © Springer-Verlag 2011
Abstract: Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to higher dimensional topological spaces. The perturbative development of the usual GFT’s is rather involved combinatorially and plagued by topological singularities (which we discuss in great detail in this paper), thus very difficult to control and unsatisfactory. Both these problems simplify greatly for the “colored” GFT (CGFT) model we introduce in this paper. Not only this model is combinatorially simpler but also it is free from the worst topological singularities. We establish that the Feynman graphs of our model are combinatorial cellular complexes dual to manifolds or pseudomanifolds, and study their cellular homology. We also relate the amplitude of CGFT graphs to their fundamental group.
1. Introduction: Group Field Theory Group Field Theories (GFT) [1–5] are quantum field theories over group manifolds. They generalize random matrix models and random tensor models [6–10]. GFT’s arise naturally in several discrete approaches to quantum gravity, like Regge calculus [11], dynamical triangulations [12,13] or spin foam models [14,15] (see also [16] for further details). To define a GFT we choose a group G, and the quatum field φ, a real scalar function over n copies of the group φ : G ⊗n → R. Furthermore we require that φ is invariant under arbitrary permutations σ and simultaneous multiplication to the left of all its arguments, φ(hgα1 , . . . , hgαn ) = φ(gα1 , . . . , gαn ), ∀h ∈ G, φ(gασ (1) , . . . , gασ (n) ) = φ(gα1 , . . . , gαn ), ∀σ ∈ Sn .
(1)
70
R. Gurau
As a quantum field theory, a GFT in n dimensions is defined by the action functional [17] 1 S= )φα0 ...αn−1 [dg] φα0 ...αn−1 K −1 (gα0 , . . . , gαn−1 ; gα0 , . . . gαn−1 2 λ + (2) [dg] V (gα 0 , . . . , gα n+1 )φα 0 ...α 0 . . . φα n+1 ...α n+1 , 0 0 n−1 n−1 0 n−1 n+1 where we used the shorthand notations φ(gα0 , . . . gαn−1 ) = φα0 ...αn−1 , and [dg] for the integral over the group manifold with Haar measure of all group elements appearing in the arguments of the integrand. The vertex operator V encodes the connectivity dual to a n simplex, whereas the propagator K encodes the connectivity dual to the gluing of two n simplices along (n − 1) simplices1 of their boundary. A Feynman graph with vertices V and propagators K is dual to a “gluing of simplices”. From a mathematical standpoint, GFT is a tool to investigate the properties of this gluing of simplices. For instance, if the gluing represents a manifold one can define topological invariants [18] using GFT’s. In discrete approaches to quantum gravity the gluing of simplices is interpreted as a space-time background, making GFT a combinatorial, background independent theory whose perturbative development generates space-times. This is further supported as, for the simplest choice of V and K , the Feynman amplitude of a graph reproduces the partition function of a BF theory discretized on the gluing of simplices. BF theory becomes Einstein gravity after imposing the Plebanski constraints, hence it is natural to suppose that a choice of operators K and V exists which reproduces the partition function of the latter. This line of research has been explored in [19–23], to define more realistic GFT’s whose semiclassical limit [24,25] exhibits the expected behavior. In this paper we are interested in the combinatorial and topological aspects of the Feynman diagrams of the action (2). As these do not depend on the specifics of K and V we will opportunistically make the simplest choices. Our study is motivated by the following analogy with matrix models. Even though only identically distributed matrix models are topological in some scaling limit ([6]), it turns out that the power counting of more involved models (like for instance the Gross-Wulkenhaar model [26,27]) is again governed by topological data. In fact, from a quantum field theory perspective one can argue that the only matrix models for which some renormalization procedure can be defined are those with topological power counting. Pursuing this analogy one step further, the non trivial fixed point of the Gross-Wulkenhaar model [28,29] opens up the intriguing possibility that GFTs are UV complete quantum field theories. However a perturbative development of the usual GFT runs almost immediately into a very subtle and extremely serious problem: some gluings of simplices dual to GFT graphs present “wrapping singularities”. Such gluings are not manifolds or pseudomanifolds! This is extremely surprising, especially as it appears to contradict the most basic results of [30] concerning n-dimensional rectilinear gluings of simplices! This apparent contradiction is explained by the subtle differences between arbitrary gluings of simplices dual to GFT graphs and rectilinear gluings used in [30]. In Sect. 2 we give a very detailed discussion of these singularities, explain the precise relation between GFT graphs and the classical results of [30] and provide abundant examples. The pathological “wrapping singularities” plague the perturbative development of the usual GFT models and virtually any result about the topology of gluings of simplices dual to GFT graphs can be established only under some restrictions (obviously the 1 That is the field φ is associated to (n − 1) simplices, and its arguments g to (n − 2) simplices.
Colored Group Field Theory
71
1
A3
10 13 12
12
01
A0
13
0
01 02 03
21 20 23
2
A2
02
23
03 30 31 32
3
A1
Fig. 1. GFT vertex in three dimensions and dual tetrahedron
most common is to require that the gluing is rectilinear). However up to this work none of these restrictions could be promoted at the level of the action to ensure that all GFT graphs respect it. In Sect. 3 we introduce the new “colored” GFT (CGFT) model. In Sect. 4 we prove that all CGFT graphs are free of the infamous wrapping singularities and that they always correspond to pseudo-manifolds. We then undertake the first steps in a systematic study of the topology of CGFT graphs by means of the graph combinatorial cellular complex and cellular homology. The amplitudes of CGFT graphs are subsequently related to the first homotopy group of the graph complex in Sect. 6, and a necessary condition for a graph to be homotopically trivial is derived. Section 7 draws the conclusion of our work and Appendix A presents explicit examples of homology groups for several CGFT graphs. 2. GFT Graphs in Three Dimensions In this section2 we will discuss in detail the Feynman graphs of the familiar GFT’s in three dimensions and describe precisely the “wrapping singularities” mentioned in the Introduction. The usual GFT action in three dimensions is 1 S= [dg] φα0 α1 α2 φα0 α1 α2 + Sint , 2 (3) λ Sint = [dg] φα03 α02 α01 φα10 α13 α12 φα21 α20 α23 φα32 α31 α30 , 4 where φα0 α1 α2 ≡ φ(gα0 , gα1 , gα2 ), and gαi j = gα ji in Sint . The GFT vertex generated by Sint , is represented on the left in Fig. 1. Each field φ in Sint is associated to a halfline of the GFT vertex. Every two fields in Sint share a group element, consequently every two halflines of the GFT vertex share a strand (depicted as a solid line in Fig. 1). We label the halflines of the GFT vertex 0, 1, 2 and 3 and each strand by the (unordered) couple of labels of the two halflines which share it. The halflines 0 and 1 share the strand 01 (or 10), the halflines 0 and 2 share the strand 02 (or 20), etc. The GFT vertex is dual to a tetrahedron represented on the right in Fig. 1. The half lines of the vertex are dual to the triangles bounding the tetrahedron. The edge shared by two triangles is dual to the strand shared by the corresponding halflines. For simplicity, the labels of the triangles have been omitted in Fig. 1. Instead, we labeled the vertex 2 We would like to thank the anonymous referee for suggesting the addition of this section.
72
R. Gurau
Fig. 2. GFT lines in three dimensions
Fig. 3. The two tadpoles in three dimensional GFT
Fig. 4. Second order graphs in three dimensional GFT
of the tetrahedron opposite to the triangle 0 by A0 , the one opposite the triangle 1 by A1 , etc. In the sequel the GFT vertex will be called a stranded vertex, to emphasize its internal strand structure. The quadratic part of the action 3 connects two GFT vertices via an arbitrary permutation of the strands, yielding the six possible choices for the lines, presented in Fig. 2. The GFT lines are dual to the identification of two triangles and each permutation of the strands encodes one of the six possible ways to identify two triangles. The GFT lines will sometimes be referred to as stranded lines to emphasize their internal strand structure. The perturbative development of GFT is indexed by the Feynman graphs of the action 3, dual to arbitrary gluings of tetrahedra, built with stranded vertices and stranded lines. They will be called stranded graphs. Stranded graphs are the higher dimensional generalization of the ribbon graphs of matrix models.
2.1. Combinatorics of GFT graphs. Due to the strand structure of the GFT lines and vertices the combinatorics of GFT graphs is somewhat involved. One needs to track carefully which halflines are contracted into the GFT lines, as well as count every line with a factor six. To illustrate this consider the double tadpole graph made of one vertex and two lines. Choose a halfline (say 0 in Fig. 1). It has three choices of a partner to contract into a line (1, 2 and 3 respectively). The remaining halflines necessarily connect into a line. Thus a first rough count gives 3 × 62 different graphs. This is of course a gross overestimate. For instance choosing 1 or 3 yields the same graph (on the left in Fig. 3) while choosing 2 yields the graph on the right in Fig. 3. The number of distinct graphs drops to 2 × 62 . This is still a large overestimate as, for specific choices of the permutations of the strands some of these tadpoles are still related by symmetry (we will see an example later on). 2 At second order the very naive counting of contraction yields [3! + 2 24 ] × 64 (corresponding to the two cases schematically drawn in Fig. 4). This number is again an extreme overestimate and can be reduced drastically. Fixing precisely the symmetry factors of the graphs requires tracking carefully the structure of the strands and consequently is rather involved. In fact, to our knowledge this has not yet been done even for the first and second order graphs.
Colored Group Field Theory
73 l1
L1
f1 l1
F1 F2 L2
l2 l1
l2
f2
l2
Fig. 5. A GFT graph G 1 dual to a gluing with negative Euler characteristic
2.2. Wrapping singularities in GFT graphs. We now come to the second part of our analysis, namely the topology of the gluing of tetrahedra dual to a GFT graph. As mentioned in the Introduction, the results of Sect. 3.2 in [30]3 are proved for rectilinear gluings of simplices. A rectilinear gluing (Definition 3.2.1. of [30]) is, crucially, non-branching, namely any two triangles identified by a gluing belong to exactly two distinct tetrahedra. The gluings of tetrahedra dual to all the GFT graphs at first order (Fig. 3) and some GFT graphs at second order (on the right in Fig. 4) are not rectilinear: there exist triangles belonging to the same tetrahedron which are identified. One can not apply directly the results of [30]. When reconsidering carefully the topology of arbitrary gluings we will discover that some of them present wrapping singularities and are not manifolds or pseudo manifolds. Of utmost importance in the sequel is the notion of link of a vertex (see Sect. 3.2 in [30]). Any vertex v in a gluing of tetrahedra belongs to several tetrahedra τi . For each tetrahedron τi , we denote σi the triangle of the tetrahedron τi opposite to v (that is not containing v). The link of v, denoted lkv is the gluing of the σi . A neighborhood of a vertex is the topological cone over its link, C|lkv| = lkv×[0,1] lkv×{1} . The main result in Sect. 3.2 of [30] is Proposition 3.2.8. The Euler characteristic χ (X ) of any closed connected three dimensional rectilinear gluing X , with k vertices v1 . . . vk and links lk(v1 ) obeys 1 χ (lkvi ), (4) χ (X ) = k − 2 i
where χ (lkvi ) is the Euler characteristic of the link lkvi . In particular, as the links are two dimensional surfaces, their Euler characteristic is at most 2 and one recovers the classical result that the Euler characteristic of a three dimensional closed connected pseudo-manifold is positive χ (X ) ≥ 0. Consider now the GFT graph represented on the left in Fig. 5. It is dual to the nonrectilinear gluing (using the labeling in Fig. 1): 3 We thank the anonymous referee for pointing us to this excellent reference which we will use extensively in the sequel.
74
• •
R. Gurau
the gluing of the triangles A1 A2 A3 ≡ A0 A3 A2 , by identifying the vertices A1 ≡ A0 , A2 ≡ A3 , A3 ≡ A2 and the edges 01 ≡ 10, 02 ≡ 13, 03 ≡ 12. the gluing of the triangles A1 A0 A3 ≡ A0 A1 A2 by identifying A1 ≡ A0 , A0 ≡ A1 , A3 ≡ A2 and the edges 21 ≡ 30, 20 ≡ 31, 23 ≡ 32.
This gluing has 2 vertices ( A0 = A1 and A2 = A3 ), 4 edges (01, 02 = 13, 03 = 12 and 23), 2 triangles ( A1 A2 A3 = A0 A3 A2 and A1 A0 A3 = A0 A1 A2 ) and one tetrahedron (A0 A1 A2 A3 ), hence an Euler characteristic χ (X ) = −1 < 0. Therefore this gluing is not a pseudo-manifold. Even worse, this gluing does corespond to an abstract simplicial complex (Definition 2.1 in [31]) and not even a trisp (Definition 2.44 in [31]). In Appendix B we show that Eq. (4) is a necessary condition that a gluing of tetrahedra actually is a pseudo-manifold (we do not claim that it is sufficient). In our quest to find a GFT such that all its graphs are pseudo-manifolds, Eq. (4) will be our guide. We will first try to understand the profound origin of its failure for general GFT graphs, and then we will try to build a model such that Eq. (4) holds for all its graphs. The proof of Lemma 3.2.8 in [30] relies on the following detailed balance valid for a rectilinear gluing • • •
each edge of the gluing accounts for two vertices of the links, each triangle of the gluing accounts for three edges of the links, each tetrahedron of the gluing accounts for four triangles of the links.
Denoting V, E, T and the number of vertices, edges, triangles and tetrahedra of the gluing X , and v, e, and t the number of vertices, edges and triangles of the links, the total Euler characteristic of the links is written: χ (lkvi ) = v − e + t = 2E − 3T + 4, (5) i
but, as every tetrahedron is bounded by four triangles, 4 = 2T , hence 1 χ (lkvi ) = E − T + = V − χ (X ). 2
(6)
i
The links defined for rectilinear gluings generalize to arbitrary gluings dual to GFT graphs. A link is a gluing of triangles, hence it is dual to a ribbon graph, called the link graph. The link graphs can be accessed directly starting from the GFT graph by the following algorithm (see [32]): •
•
for all GFT vertices erase all strands belonging to a halfline. For instance, erasing the strands belonging to the halfline 0 in Fig. 1, we obtain the ribbon vertex with strands 12, 13 and 23. Then repeat this for all halflines of the GFT vertex to obtain four “descendent” ribbon vertices. connect the strands of the ribbon vertices as dictated by the GFT lines.
An example is presented in Fig. 6, where each connected ribbon graph is dual to the link of a vertex in the gluing of tetrahedra. The detailed balance would translate for GFT graphs as • • •
each face (closed strand) of a GFT graph accounts for two faces in the link graphs. each line in the GFT graph accounts for three lines in the link graphs. each GFT vertex accounts for four vertices in the link graphs.
Colored Group Field Theory
75
Fig. 6. Link graphs in GFT
Fig. 7. A GFT graph G 2 related by symmetry to G 1
Fig. 8. A third singular graph G 3
The first item above does not hold in arbitrary GFT graphs. For the graph G 1 in Fig. 5 we drawn the two link graphs on the right. The faces F1 and F2 of the GFT graph account each for only one face in the link graphs, f 1 and f 2 (that is an edge of the tetrahedron contributes only one vertex on a link, and not two). Thus although the link graphs have four vertices and six lines, they only have a total of six faces, and not eight. The faces of the links, f 1 and f 2 , consist each of two lines l1 , respectively l2 , while the faces F1 and F2 consist of only one line L 1 respectively L 2 . That is f 1 and f 2 wrap twice around F1 and F2 , hence the name “wrapping singularity”. The graph of Fig. 7 presents the same phenomenon. This illustrates the difficulty to accurately compute symmetry factors: surprisingly G 1 and G 2 are related by symmetry. A somewhat different example of a graph for which Eq. 4 fails is given in Fig. 8. In this case one of the links is planar, thus its Euler characteristic is 2, while the second is 2 non-orientable, and its Euler characteristic is 1 (the link is homeomorphic with RP ). Hence k − 21 i χ (lkvi ) = 1/2 which is not even an integer. The wrapping singularities are generic in the usual GFT: any graph having a tadpole insertion like in Fig. 9 will have a wrapping singularity. Lemma 1. The gluing dual to a GFT graph with a wrapping singularity will not obey Eq. (4) hence, by Appendix B, is not a pseudo-manifold.
76
R. Gurau l1 L1
f1 l1
F1
l1
Fig. 9. Wrapping singularities in arbitrary GFT graphs
Proof. Every GFT vertex always gives four ribbon vertices hence (in the notation of Eq. (5)) t = 4 always holds. Any halfline of the GFT vertex will contribute to three of the four descendent ribbon vertices, thus e = 3T always holds also. A strand in the GFT vertex will belong to two different ribbon vertices (for instance the strand 12 in Fig. 1 belongs to the ribbon vertices 12, 13, 23 and 12, 01, 02). A face in the link corresponding to a strand either passes through only one of these two vertices (in which case there exists a second, distinct face in some link graph which passes through the second vertex) or it passes through both and generates a wrapping singularity. Hence v ≤ 2E and v = 2E if and only if a graph has no wrapping singularity, hence 1 χ (X ) ≤ k − χ (lkvi ), (7) 2 i
and the equality holds only if the graph is free of wrapping singularities.
It is therefore a very important question whether there exists a GFT action such that all its graphs are free of wrapping singularities. The central point of this paper is that indeed such an action exists: it is the colored GFT model we define in Sect. 3. The CGFT model brings also many other advantages. In generates fewer graphs than the usual GFT model (no graph at first order and only one graph at second order) simplifying the combinatorics. Not only are its graphs free of wrapping singularities but also we will prove that all graphs are dual to pseudo-manifolds. It is, to this day, the only example of a GFT action whose graphs are all dual to pseudo-manifolds. The faces and link graphs are easily identified. All links are orientable surfaces (of arbitrary genus). The homology groups and fundamental group of graphs can easily be defined related to the Feynman amplitudes of the graphs. 3. The Colored GFT Model We will define a fermionic colored GFT model invariant under a global color transformation. One can alternatively consider a bosonic CGFT model, but the latter does not exhibit this tantalizing invariance. Instead of an unique bosonic field, consider n + 1 Grassmann fields, ψ 0 , . . . , ψ n : G ⊗n → G, {ψ i , ψ j } = 0,
(8)
ψ → ψ¯ such that ψ 1 ψ 2 = −ψ¯2 ψ¯1 , ψ¯¯ = −ψ.
(9)
with hermitian conjugation
Colored Group Field Theory
77
The fields ψ have no symmetry properties under permutations of the arguments, but are all invariant under simultaneous left action of the group on all their arguments. The upper index p denotes the color of the field ψ p . A (global) color transformation is an internal rotation U ∈ SU (n + 1) on the grassmann fields (ψ i ) = U i j ψ j , (ψ¯ i ) = ψ¯ j (U i j )∗ = ψ¯ j (U −1 ) ji .
(10)
The only quadratic form invariant under color transformation is
ψ¯ p ψ p .
(11)
p
The interaction is a monomial in the fields. The only monomial in ψ invariant under color rotation is ψ0 . . . ψn,
(12)
(ψ 0 ) . . . (ψ n ) = U 0i0 . . . U nin ψ i0 . . . ψ in = det(U )ψ 0 . . . ψ n .
(13)
as
The hermitian GFT action of minimal degree, invariant under (global) color rotation is S= (14) [dg] ψ¯ p ψ p + [dg] ψ 0 ψ 1 . . . ψ p + [dg] ψ¯0 ψ¯1 . . . ψ¯ p , p
where the arguments of ψ p and ψ¯ p in the interaction terms are chosen to reproduce the combinatorics of the GFT vertex (see [17]), that is, denoting g pq = gq p the group element associated to the strand connecting the halflines p and q, ψ p (g p−1 p , g p−2 p , . . . , g0 p , g pn , . . . g pp+1 ).
(15)
The interaction part of the action of Eq. (14) has two terms. We call the vertex involving only ψ’s the positive vertex and represent it like in Fig. 10. The second term, involving only ψ¯ is similar, but the colors turn anticlockwise around it. The vertices have a detailed internal structure encoding the connectivity of the arguments g From Eq. (14) we conclude that the propagator of the model is formed of n parallel strands and always connects two halflines of the same color, one on a positive vertex and one on a negative vertex. We orient all lines from positive to negative vertices. The strand structure of the vertex and propagator is rigid. A stranded colored GFT graph admits therefore a simplified representation as a colored graph. The colored graph is obtained by collapsing all the strands of the lines in “thin” lines, and all the strands of the vertices in point vertices. Conversely, given a colored graph with thin lines and point vertices one can reconstruct the stranded graph associated to it. Figure 11 depicts a CGFT graph either as a stranded graph (on the right) or as colored graph (on the left).
78
R. Gurau
(0,1,...,n) 1
2
g12 g 01
0
g13 g23
g 0n g34
3 Fig. 10. nD vertex 0
0 1 v
1
3
1
v
2
2
2 3 Fig. 11. The second order CGFT graph
4. Bubble Homology in Colored GFT The strand structure of the vertices and propagators render the Feynman graphs of our model topologically very rich. This should come as no surprise, as in three dimensions for instance, gluings dual to CGFT graphs include not only all orientable piecewise linear three dimensional manifolds (see [33] and references therein), but also pseudomanifolds. The topology of the dual gluing of simplices is encoded in the topology of CGFT graphs and in the rest of this paper we will study the latter. An important notion associated to CGFT graphs is that of p-bubble. Represent the graph G as colored graph with thin lines and point vertices. Let C be an ordered set of colors C ⊂ {0, 1, . . . , n} of cardinality p. A p-bubble of colors C, denoted BVC , is the connected subgraph of G made only of lines of colors C and with vertex set V. The graph itself is not considered a (n + 1)-bubble. The graph in Fig. 11 has 3-bubbles Bv012 , Bv013 , Bv023 and Bv123 , 2-bubbles Bv011 v2 , 1 v2 1 v2 1 v2 1 v2 02 03 12 13 23 0 1 2 3 Bv1 v2 , Bv1 v2 , Bv1 v2 , Bv1 v2 , Bv1 v2 , 1-bubbles Bv1 v2 , Bv1 v2 , Bv1 v2 , Bv1 v2 , and finally 0-bubbles Bv1 , Bv2 . The bubbles are the combinatorial encoding of the vertices, lines, faces and link graphs discussed in Sect. 2 for CGFT graphs. Trivially Remark 1. The 0-bubbles of G are its vertices and the 1-bubbles are its lines. The following remarks are a bit more interesting. Remark 2. The 2-bubbles of the colored CGFT graph G are the faces (closed strands) of the associated stranded graph.
Colored Group Field Theory
79
3
vp+2
32 31 30
2
21 20 23 30 31 32
3
0
12 13 10
10 13 12
0
03 02 01
1
1
01 02 03
23 20 21
2
01 02 03
0
32 31 30
3
vp
vp+1
Fig. 12. Stranded and colored graph
Proof. Consider three successive vertices v p , v p+1 and v p+2 along a strand, as depicted in Fig. 12. Their halflines have a color index 0, 1, 2 or 3. The strands are identified, in each vertex, by the couple of colors of the halflines they belong to. The lines have only parallel strands and connect (respecting the colorings of the halflines) two vertices of opposite orientation. This drawing essentially proves the result. The strand 12, common to the halflines of colors 1 and 2 on v p necessarily connects with the strand 12 on v p+1 which in turn connects to the strand 12 on v p+2 . Thus the labels (12) are conserved all along the strand. This is the fundamental difference between the usual GFT graphs and the CGFT graphs and render the latter much better behaved. The 2-bubble 12 is obtained by deleting all the lines of colors 0 and 3 of the colored graph. In the stranded graph representation one deletes all the lines of color 0 and 3 together with all their strands (as these strands have at least a label 0 or 3). Therefore the 2-bubble 12 is the closed strand (face) 12 in Fig. 12.
Remark 3. The 3-bubbles of the colored CGFT graph G are the link graphs of the associated stranded graph. Proof. Again Fig. 12 essentially proves the result. Consider the vertex v p+1 in Fig. 12 and the ribbon vertex (of a link graph) obtained by deleting all the strands belonging to the halfline 0. This vertex is made of strands of colors 12, 13 and 23 of v p+1 . We will call it a 123 ribbon vertex. The labels of the strands are conserved along the lines, thus the vertex 123 coming from v p+1 connects, via the line of color 2 with the ribbon vertex (coming from v p ) containing both strands 12 and 23, hence necessarily the 123 ribbon vertex coming from v p . Consequently, all the ribbon vertices of the link graph are 123 vertices coming from various GFT vertices. The link graph is then a connected component obtained from the stranded graph by deleting all lines of color 0 together with all their strands, hence a 3-bubble.
80
R. Gurau
Note that this algorithm to identify link graphs in a colored GFT graph is a drastic simplification with respect to the one valid for arbitrary GFT graphs presented in Sect. 1.
Taking a careful look at Fig. 12 and the algorithm giving the link graphs we notice that the latter are necessarily ribbon graphs with no twists along the ribbons, hence Remark 4. The link graphs of a colored GFT graph are dual to gluings representing orientable, closed surfaces. A face, say 12, will always contribute two different faces, belonging to two different link graphs, one with colors 123 and the other with colors 012. Remark 5. A CGFT graph has no wrapping singularities, hence the dual gluing respects Eq. 4. Most importantly, the main topological property of CGFT graphs is Theorem 1. Any CGFT is dual to a simplicial pseudo-manifold. Proof. Step 1. In this proof we use [31]. We first prove that the dual of any CGFT graphs is a finite abstract simplicial complex. A finite abstract simplicial complex (Definition 2.1 [31]) is a finite set A together with a collection of subsets such that if X ∈ and Y ⊆ X then Y ∈ . Consider the dual gluing of tetrahedra. Its vertices are dual to the 3-bubbles of the i jk CGFT graph, hence its set of vertices is written A = {BV }. The edges of the gluing are dual to the 2-bubbles of the graph. Every 2-bubble is shared by exactly two ij i jk i jl 3-bubbles, and we denote it by the couple of the two 3-bubbles e = BV = {BV1 , BV2 } and V ⊂ V1 ∩ V2 every triangle is dual to a line, and every line belongs to exactly three i jk i jl 3-bubbles, l = Bvi 1 v2 = {BV1 , BV2 , BVikl3 } and {v1 v2 } ⊂ V1 ∩ V2 ∩ V3 . Finally, every tetrahedron is dual to a vertex shared by exactly four 3-bubbles v1 = Bv1 = i jk i jl jkl {BV1 , BV2 , BVikl3 , BV4 } and v1 ∈ V1 ∩ V2 ∩ V3 ∩ V4 . is the set of all vertices edges triangles and tetrahedra (supplemented by the empty set). i jk i jl i jk i jl jkl Consider the subset {BV1 , BV2 , BVikl3 } ⊂ {BV1 , BV2 , BVikl3 , BV4 }, and v1 ∈ V1 ∩ V2 ∩ V3 ∩ V4 . Then v1 ∈ V1 ∩ V2 ∩ V3 , and necessarily the line of color i touching v1 , Bvi 1 v2 , i jk
i jl
is a line in all subgraphs BV1 , BV2 and BVikl3 . Hence {v1 , v2 } ⊂ V1 ∩ V2 ∩ V3 , and i jk
i jl
Bvi 1 v2 = {BV1 , BV2 , BVikl3 } ⊂ .
i jk
i jl
i jk
i jl
jkl
Consider now the subsets {BV1 , BV2 } ⊂ {BV1 , BV2 , BVikl3 , BV4 }, v1 ∈ V1 ∩ V2 ∩ V3 ∩ ij
V4 . Then v1 ∈ V1 ∩ V2 , and necessarily the entire face BV , with v1 ∈ V is a face in i jk i jl ij i jk i jl both subgraphs BV1 and BV2 . Hence V ⊂ V1 ∩ V2 and BV = {BV1 , BV2 } ⊂ . The i jk
i jl
i jk
i jl
same reasoning applies for the subsets {BV1 , BV2 } ⊂ {BV1 , BV2 , BVikl3 } with {v1 v2 } ⊂ V1 ∩ V2 ∩ V3 . The subsets with one element are in the vertex set A hence always in consequently
an abstract simplicial complex. On a more abstract note, the CGFT graph itself is an abstract complex ([34], p. 125). An abstract complex is a partially ordered set , to each of whose elements θ is associated a nonnegative integer dθ (the dimension of the element θ ) such that θ1 < θ2 ⇒ dθ1 < dθ2 . The dimension zero elements are the 0-bubbles, {Bv }. The
Colored Group Field Theory
81
dimension one elements are the 1-bubbles seen as sets of vertices, Bvi 1 v2 = {Bv1 , Bv2 }. The dimension two elements are the 2-bubbles seen as sets of lines (thus sets of sets of ij vertices) BV = {Bvi 1 v2 , . . . } such that {v1 , v2 }, · · · ⊂ V. The dimension three elements are the 3-bubbles, seen as sets of faces BV = {BV , . . . } such that V ∈ V . The complex is the set of all elements partially ordered by ∈. i jk
ij
Step 2. One needs just to check the definition. A 3-dimensional simplicial pseudo-manifold is a finite abstract simplicial complex with the following properties: •
•
•
it is non-branching: Each 2-dimensional simplex is a face of precisely two 3-dimensional simplices. This is obviously respected as every GFT line connects exactly two distinct GFT vertices in a graph (note that this condition is broken by generic GFT graphs). it is strongly connected: Any two 3-dimensional simplices can be joined by a “chain” of 3-dimensional simplices in which each pair of neighboring simplices have a common 2-dimensional simplex. Again trivial: in any connected CGFT graph any path connecting two vertices is dual to a “chain”. it has dimensional homogeneity: Each simplex is a face of some 3-dimensional simplex. This is easy to check by following the proof that is an abstract simplicial complex.
The p-bubbles are the building blocks of a combinatorial graph complex and generate an associated homology. Denote the set of p-bubbles by B p , and define the p th chain group as the finitely generated group C p (G) = {α p },
αp =
C ∈Bp BV
C C C cV BV , cV ∈ Z.
(16)
The chain groups define a homology groups via a boundary operator, Definition 1. The p th boundary operator d p acting on a p-bubble BVC with colors C = {i 1 , . . . i p } is • for p ≥ 2, d p (BVC ) =
(−)q+1 q
B C ∈B p−1 V V ⊂V C =C \i
C
B V ,
(17)
q
which associates to a p-bubble the alternating sum of all ( p − 1)-bubbles formed by subsets of its vertices. • for p = 1, as the lines Bvi 1 v2 connect a positive vertex (say v1 ) to a negative one, say v2 d1 Bvi 1 v2 = Bv1 − Bv2 . • for p = 0, d0 Bv = 0. These boundary operators give a well defined homology as
(18)
82
R. Gurau
Lemma 2. The boundary operators respect d p−1 ◦ d p = 0. Proof. To check this consider the application of two consecutive boundary operators on a p-bubble d p−1 d p (BVC ) = (−)q+1 d p−1 BVC (19) B C ∈B p−1 V
q
=
(−)q+1
+
B C ∈B p−1 V V ⊂V C =C \i
q
V ⊂V C =C \i q
(−)
r
r >q
r
q
B C ∈B p−2 V V ⊂V C =C \i
(−)r +1
BV ,
B C ∈B p−2 V V ⊂V C =C \i
BVC
r
C
(20)
r
where the sign of the second term changes, as ir is the r − 1th color of C \ i q if q < r . The two terms cancel if we exchange q and r in the second term.
The operators d p provide direct access to the attaching maps of p cells to p−1 cells. This allows one to define inductively a cellular complex associated to every CGFT graph. We present below the construction for colored GFT graphs in three dimensions. Theorem 2. A colored GFT graph G with four colors is a three dimensional cellular complex X 3 , with cells identified by the p-bubbles. Proof. Although straightforward this proof is somewhat abstract and technical. We will follow the classical reference [35] and employ the notations therein. The vertices of the 0 ) and the lines are associated to graph are associated to 0 dimensional cells (denoted eB v 1 1 dimensional cells (denoted eB ) attached to the vertices to form the 0-skeleton X 0 i v1 v2
and the 1-skeleton X 1 of our complex (see Example 0.1 in [35]). ij Consider now a 2 bubble BV . It is a closed circuit of lines with alternating colors. This circuit is isomorphic with a circle S1 , hence there exists a bijection ϕBi j : S1 → X 1 . The V
two cell associated to the 2-bubble is a topological cone over the circle e2 i j = CS1 = D 2 , BV
hence a two dimensional disk. We attach the two cells e2 i j via the maps ϕBi j and obtain BV
V
X 2 . As a set, X 2 is the disjoint union e2 . Up to this stage our complex
the 2-skeleton of our complex and the two cells X 2 = X 1 Bi j
V
ij
BV
of the one skeleton is just the classical
CW complexes of ribbon graphs. i jk A 3-bubble BV is itself a colored graph with three colors and by the previous construction has an associated two dimensional cell complex Y 2 i jk , with zero, one and two BV
cells denoted f B0v , f B1i and f 2i j . By Remarks 3 and 4, Y 2 i jk is the cell complex of a link BV BV v1 v2 i jk homeomorphic with an oriented closed connected surface of genus g, Mg (BV ).4 The 4 That is M is the two dimensional sphere S2 , M is the torus S1 × S1 , etc. 0 1
Colored Group Field Theory
83 i jk
3 three cells of our complex are topological cones eB i jk = C M g (BV ) over the surfaces i jk
Mg (BV ). The cells of Y 2 i jk are in one to one correspondence with some 0, 1 or 2 bubble BV
ij
i jk
i jk
ij
Bv , Bvi 1 v2 , BV , that is subgraphs of BV . But, as BV is itself a subgraph G, Bv , Bvi 1 v2 , BV 0 , e1 , e2 in the 2 are also subgraphs of G, hence one to one to some 0, 1, and 2 cell eB i ij v BV
BV
skeleton of the graph X 2 . Each f 2i j can then be mapped by some φ f 2 onto the correBV
sponding
e2
i jk
ij
BV
. We attach the three cell BV to
defined as ϕBi jk | f 2 V
ij BV
ij BV
X2
= φf2 .
i jk
by the map ϕBi jk : Mg (BV ) → X 2 V
ij BV
As a last remark note that, as the three cells are cones over arbitrary oriented surfaces, the complex X 3 is not a CW complex. It becomes one if and only if all the two complexes Y 2 i jk are spheres S2 (as the cone over a spheres S2 is a three disk D3 ). In BV
turn by Remark 1 and 5 this implies that the Euler characteristic χ (X 3 ) = 0, hence X 3 a three dimensional closed connected manifold. 5. Minimal and Maximal Homology Groups Before analyzing the Feynman amplitudes of CGFT graphs we will give some generic properties of the bubble homology induced by the boundary operator (17). Detailed examples of homology computations for graphs are presented in Appendix A. First, by definition d0 acting on 0-bubbles is zero. Thus, for any graph,
ker(d0 ) = Z, (21) N
where N = |B0 | is the number of vertices (0 bubbles) of the graph. Our first result concerns the minimal homology group of a colored graph. Lemma 3. For connected closed graphs H0 = Z. Proof. The operators d1 acting on a one chain is d1 α1 = cvi 1 v2 d1 Bvi 1 v2 .
(22)
Bvi 1 v2 ∈B1
The matrix of d1 is then an incidence matrix of the oriented graph, where a line enters its positive end vertex and exists from its negative end vertex, ⎧ if Bvi 1 v2 enters Bv ⎪ ⎨ 1,
Bv Bvi v = −1, if Bvi 1 v2 exits Bv . (23) 1 2 ⎪ ⎩ 0, else We will compute the Im(d1 ) using a contraction procedure. We start by choosing a line Bvi 1 v p connecting the two distinct vertices v1 and v p . We collect all terms containing
84
R. Gurau
either Bv1 or Bv p in Eq. (22) to get d1 α1 = Bv1 Bv Bvi v cvi 1 v p + 1 1 p + Bv p Bv p Bvi
1v p
+
v=v1 ,v p
vq =v p
cvi 1 v p +
Bv1 Bv Bvi v cvi 1 vq 1 1 q
vr =v1
j Bv B B j cvv v vv
Bv p Bv p Bvi
p vr
cvi p vr
.
(24)
Using Bv p Bvi
= − Bv Bvi v = ±1, we rewrite Eq. (24) as 1 1 p d1 α1 = (Bv1 − Bv p ) Bv Bvi v cvi 1 v p + (Bv1 − Bv p ) Bv Bvi v cvi 1 vq 1 1 1 p 1 q 1v p
+
vq =v p
vq =v p
+
Bv p Bv Bvi v cvi 1 vq 1 1 q
v=v1 ,v p
+
vr =v1
Bv p Bv p Bvi
p vr
cvi p vr
j Bv B B j cvv , v vv
(25)
and perform the change of basis in C0 (G), Bv 1 = Bv1 − Bv p , Bv q = Bvq , vq = v1 ,
(26)
under which Equation (24) becomes d1 α1 = Bv 1 Bv Bvi v cvi 1 v p + 1 1 p +
vq =v p
+
vq =v p
Bv p Bv Bvi v cvi 1 vq 1 1 q
v=v1 ,v p
Bv 1 Bv Bvi v cvi 1 vq 1 1 q +
vr =v1
j Bv B B j cvv . v vv
Bv p Bv p Bvi
p vr
cvi p vr (27)
The first line of Eq. (27) is the only one involving Bv 1 , which is linearly independent from all other Bv , therefore it spans a direction in Im(d1 ). As cvi 1 v p ∈ Z and
Bv p Bvi v = ±1, we have 1 p
Im(d1 ) = Z ⊕ . . .
(28)
The second and third lines of Eq. (27) corespond to the incidence matrix of a graph in which all vertices Bvq who were connected by lines to Bv1 are now connected by lines with Bv p , that is the graph obtained from G by contracting the line Bvi 1 v p and gluing the two vertices Bv1 and Bv p into an unique vertex. We iterate this contraction procedure for a spanning tree, that is N − 1 times. Once such a tree is contracted, the final graph has only one vertex (rosette) v and all remaining j lines are loop lines. For any remaining loop line, the coefficient in this final sum of cvv will be B B j + B B j = 0. v
vv
v
vv
Colored Group Field Theory
85
Therefore the image of d1 is precisely Im(d1 ) =
Z ⇒ H0 =
N −1
and the kernel of d1 is ker(d1 ) =
ker(d0 ) = Z, Im(d1 )
(29)
Z.
(30)
L−(N −1)
For the maximal homology group of a graph G a similar result holds. In fact a contraction procedure for a spanning tree in the graph implied that the first homotopy group is H0 = Z, and a contraction procedure for a spanning tree in the gluing of simplices dual to a CGFT graph will in turn imply that Hn = Z. Note first that Im(dn+1 ) = 0. We have the lemma Lemma 4. For a connected closed nD graph G, ker(dn ) = Z, Im(dn ) =
Z.
(31)
|Bn |−1
Proof. Denote the set of all colors C = {0, 1, . . . , n} and consider the boundary of an arbitrary n chain C C C dn cV BV = cV dn BVC C ∈Bn BV
i
C ∈B n BV
C =C\i
=
i
C cV
C ∈B n BV
j
C =C\i
=
i, j; j
BC ∈B n−1 V C =C\i\ j
BC ∈B n−1 V
(−) j+1 BVC | j
i
C =C\i\ j;V ⊂V
C C BVC (−) j+1 cV | V⊃V + (−)i cV | C =C\i
V⊃V
C =C\ j
C = (−)i cC if C = C \ i and Eq. (32) becomes Renaming the cV V C C BVC (−)i+ j+1 cV | V⊃V − cV | V⊃V . i, j; j
BC ∈B n−1 V C =C\i\ j
C =C\i
C =C\ j
.
(32)
(33)
C are equal, hence First, note that Eq. (33) is zero if and only if all cV
ker(dn ) = Z ⇒ Hn = Z.
(34)
To determine the image of dn , consider the gluing dual to the GFT graph G. It has vertices dual to the n-bubbles of G and edges to the (n −1)-bubbles. We orient edges dual to BVC ∈ Bn−1 with colors C = C \ i \ j and j < i from the vertex dual to BVC ∈ Bn
86
R. Gurau
with V ⊂ V and colors C = C \ i to the vertex dual to BVC ∈ Bn with V ⊂ V and colors C = C \ j. The matrix of the operator dn is then the transposed of the incidence matrix of the : ⎧ ⎪ 1, if the edge dual to BVC enters the vertex dual to BVC ⎪ ⎨ (35) BC BC = −1, if the edge dual to B C exits the vertex dual to B C , V V ⎪ B V ⎪ ⎩ 0, else
and redefining BVC = BVC (−)i+ j+1 Eq. (33) is rewritten: C dn αn = BVC BC BC cV . C BV
V
(36)
V
C ’s are associated to vertices and B C to lines. Therefore In the dual gluing , the cV V C and B C swapped! Eq. (36) has the same form as (24), with the cV V ˜
C Call T˜ a rooted tree in the dual gluing , and for all dual vertices BVC call B˜ V˜ the dual tree line touching this dual vertex and going towards the root. Under the change of C , the quadratic form (36) becomes variables parallel to (27), performed this time on the cV C˜ C C C˜ C B˜ V˜ + (37) cV BC BC BB dn αn = . C ∈Bn BV
B˜ V˜ BV
C ∈ ˜ BV /T
B
V
C˜
The vectors in brackets are linearly independent as each of them contains a B˜ V˜ correC ∈Z sponding to a different tree line in T˜ . As C˜ C = ±1, |T˜ | = |Bn | − 1 and cV B˜ V˜ BV
we conclude that Im(dn ) =
Z.
(38)
|Bn |−1
The other homology groups H p , 0 < p < n depend of the particular CGFT graph one analyzes. Note that in three dimensions, in order to compute H1 and H2 one needs only to determine the kernel and image of the operator d2 . Several examples are presented in Appendix A. 6. Amplitudes and Homotopy After the study of the homology groups of a GFT graph, the next natural step is to study its homotopy groups. We will only define here a homotopy equivalence for curves which will enable us to define the fundamental group of a GFT graph. In strict parallel to triangulated polyhedra, we define an edge path as an ordered sequence of vertices [vn , . . . v1 ] connected by lines (that is, ∀i, ∃Bvl i+1 vi ). An edge loop is a closed edge path, v n = v 1 . The paths [v i+k , v i+k−1 , . . . , v i+1 , v i ] and [v i+k v i+k+1 , . . . v n , v 1 , . . . v i−1 , v i ], are homotopically equivalent if the set of vertices
vn , . . . v1
span a 2 bubble.
(39)
Colored Group Field Theory
87
The edge path group of a CGFT graph as follows. Start by associating group elements g ∈ G to all lines of the graph. For all faces BVab of the graph, the set of vertices V = vn , . . . v1 is a closed path. The relations defining the fundamental group are associated to the face of the graph and write RBab = V
j ab Bvi+1 vi ∈BV
σ (Bvi
i+1 vi j Bvi+1 vi
g
)
= e,
(40)
where e is the unit element of G, and σ (Bvi i+1 vi ) is 1 (or −1) if the vertex vi is positive (or negative). Finally, we set the group elements associated to a spanning tree T in the graph to e. The fundamental group of the graph is the group of words with generators j gB j , Bvi+1 vi ∈ / T and relations (40). vi+1 vi
On the other hand, as proved in [32] for a more general case, the amplitude of a colored GFT graph is AG = [dg] δ(RBab ) = [dg] δ(RBab ), (41) ab BV
V
g∈ /T
ab BV
V
for any tree T ∈ G. Therefore the Feynman amplitude of a graph is the volume of the relations defining the fundamental group π1 of the graph over the base GFT group G!5 In this light the main result of [32] can be translated as follows (see [32] for the appropriate definitions): “type 1 graphs dual to manifold space-times are homotopically trivial”. It is however difficult to give a complete characterization of homotopically trivial graphs. A first step in this direction is given by Lemma 5 below. Lemma 5. If a closed three dimensional graph is homotopically trivial, then all its 3-bubbles are planar. Proof. If a graph is homotopically trivial then it is homologically trivial H1 = 0. Equation (30) then implies
Im(d2 ) = Z. (42) L−(N −1)
As d2 is defined on B2 , with |B2 | = F, and ker(d2 ) ⊃ Im(d3 ) we conclude that F − [L − (N + 1)] ≥ B − 1, where B = |B3 |. On the other hand, Eq. 4, implies for the dual CGFT graph N−L+F−B=− gBC , C ∈B3 BV
V
(43)
(44)
where gBC is the genus of the 3-bubble BVC . Equations (44) and (43) imply that V
gBC = 0, ∀BVC ∈ B3 , therefore all the 3-bubbles are planar.
V
5 In mathematical terms the volume of the representation variety π → G, [36]. 1
88
R. Gurau
This lemma shows that a homotopically trivial graph is dual to a manifold gluing whose associated complex is a CW complex. The reciprocal of Lemma 5 is not true. The first example in Appendix A is a non-homotopically trivial graph whose bubbles are all planar. The amplitude of a CGFT graph is governed by the fundamental group π1 . As the first homology group H1 we introduced in Sect. 4 is just the abelianization of π1 , it is reasonable to suppose that a rough power counting estimate should depend on H1 , whose evaluation for specific examples is much simpler than the one of π1 .
7. Conclusion We started out work by an indepth study of singularities in the usual GFT models. We concluded that a certain type of topological singularities (baptized wrapping singularities) is generic for such theories always associated to graphs whose dual gluings are not pseudo-manifolds. We then introduced a new (fermionic) group field theory whose combinatorics and topology is much more controlled. We were able to establish that all its graphs are dual to simplicial pseudo-manifolds and encode a combinatorial cellular complex. Using an appropriate boundary operator we detailed the homology of the CGFT graphs, introduced a homotopy transformation for paths on graphs and related the Feynman amplitude of graphs with the fundamental group. A large amount of work should now be carried out in several directions. In a more quantum field theoretical approach one should not only continue the preliminary studies on the power counting of GFT’s [5,32] but also look for nontrivial fermionic instanton solutions corresponding to [37] and their possible relation with matter fields. On the other hand our results could be used as a purely mathematical tool to further the understanding of three dimensional topological spaces. The encoding of the bubble complex in colored graphs provides a bridge between topological and combinatorial notions opening up the possibility to obtain, using the latter, new results on the former. Acknowledgments. The author would like to express his deepest thanks to an anonymous referee whose detailed report and many remarks led to a substantial rewriting and significant improvement of this paper. We are also greatly indebted to Matteo Smerlak. Not only did he point out to us the body of literature on manifold crystallization, but also, our numerous discussions on the topology of GFT graphs have been hugely beneficial for this work. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Mi-nistry of Research and Innovation.
A. Homology Computations In this appendix we compute the homology groups for three examples of four colored graphs.
A.1. First example. Consider the four colored graph of Fig. 13. The reader can check that all lines connect a positive and a negative vertex. The 3-bubbles of this graph are Bv012 , Bv013 , Bv023 and Bv123 . The 2-bubbles 1 ...v8 1 ...v8 1 ...v8 1 ...v8 (faces) are
Colored Group Field Theory
89 v4
v3
1 0
2
2 1
v8
3
v7
0
3
3
3
v6
0
v5
1 2
2 0 v1
v2
1
Fig. 13. A first example of 3D graph
Bv011 v2 v8 v7 , Bv013 v4 v6 v5 , Bv021 v7 v3 v5 , Bv024 v6 v2 v8 , Bv031 v7 v6 v4 , Bv032 v8 v5 v3 , Bv121 v2 v6 v5 , Bv123 v4 v8 v7 , Bv131 v2 v3 v4 , Bv135 v6 v7 v8 , Bv231 v5 v8 v4 , Bv232 v3 v7 v6 .
(45)
The 1-bubbles are Bv01 v7 , Bv02 v8 , Bv03 v5 , Bv04 v6 , Bv11 v2 , Bv13 v4 , Bv15 v6 , Bv17 v8 , Bv21 v5 , Bv22 v6 , Bv23 v7 , Bv24 v8 , Bv31 v4 , Bv32 v3 , Bv35 v8 , Bv36 v7 ,
(46)
while the 0-bubbles are vertices. We need to analyze the kernel and the image of the operator d2 . Acting on a two chain d2 , we write 01 01 d2 α2 = c1287 [Bv11 v2 + Bv17 v8 − Bv01 v7 − Bv02 v8 ] + c3465 [Bv13 v4 + Bv15 v6 − Bv03 v5 − Bv04 v6 ] 02 02 +c1735 [Bv21 v5 + Bv23 v7 − Bv01 v7 − Bv03 v5 ] + c4628 [Bv22 v6 + Bv24 v8 − Bv04 v6 − Bv02 v8 ] 03 03 +c1764 [Bv31 v4 + Bv36 v7 − Bv01 v7 − Bv04 v6 ] + c2853 [Bv32 v3 + Bv35 v8 − Bv02 v8 − Bv03 v5 ] 12 12 [Bv21 v5 + Bv22 v6 − Bv11 v2 − Bv15 v6 ] + c3487 [Bv23 v7 + Bv24 v8 − Bv13 v4 − Bv17 v8 ] +c1265 13 13 +c1234 [Bv31 v4 + Bv32 v3 − Bv11 v2 − Bv13 v4 ] + c5678 [Bv35 v8 + Bv36 v7 − Bv15 v6 − Bv17 v8 ] 23 23 +c1584 [Bv35 v8 + Bv31 v4 − Bv21 v5 − Bv24 v8 ] + c2376 [Bv32 v3 + Bv36 v7 − Bv22 v6 − Bv23 v7 ].
(47) A lengthy but straightforward computation shows that
Im(d2 ) = Z ⊕ 2Z, kerd2 = Z. 8
Using Lemma 3we conclude that ker(d1 ) = that Im(d3 ) = 3−1 Z. Therefore
(48)
3
16−7 Z and using Lemma 4 we conclude
H0 = Z, Z ker(d1 ) = 9 = Z2 , H1 = Im(d2 ) Z ⊕ 2Z 8 Z ker(d2 ) = 3 = 0, H2 = Im(d3 ) 3Z H3 = Z.
(49)
90
R. Gurau
0
v4
v3
1
1
2
2 0
3
v8
v7
v5
v6
3
3
3 0
2
1
1
2
0
v2
v1
Fig. 14. A second example of 3D graph
Note first that these homology groups match those of RP 3 . Second, direct inspection shows that all the bubbles of this graph are planar. Indeed one can prove that the dual gluing is RP 3 , but this requires more effort than merely computing homology groups.
A.2. Second example. Consider now the graph of Fig. 14. , Bv012 , Bv123 , Bv123 , Bv013 and The 3-bubbles of this graph are Bv012 1 v2 v6 v5 3 v4 v8 v7 1 v5 v8 v4 2 v3 v7 v6 1 ...v8 023 Bv1 ...v8 . The 2-bubbles are Bv011 v2 v5 v6 , Bv013 v4 v8 v7 , Bv021 v2 v5 v6 , Bv023 v4 v8 v7 , Bv031 v2 v3 v4 , Bv035 v6 v7 v8 , Bv121 v5 , Bv122 v6 , Bv124 v8 , Bv123 v7 , Bv131 v5 v8 v4 , Bv132 v3 v7 v6 , Bv231 v5 v8 v4 , Bv232 v3 v7 v6 .
(50)
The 1-bubbles are Bv01 v2 , Bv05 v6 , Bv03 v4 , Bv07 v8 , Bv11 v5 , Bv12 v6 , Bv13 v7 , Bv14 v8 , Bv21 v5 , Bv22 v6 , Bv23 v7 , Bv24 v8 , Bv31 v4 , Bv32 v3 , Bv35 v8 , Bv36 v7 .
(51)
Lemmas 3 and 4 give H0 = Z, ker(d1 ) =
Z,
H3 = Z, Im(d3 ) =
Z.
(52)
Z → H2 = 0.
(53)
9
5
Manipulations similar to the ones for the first example give Im(d2 ) =
9
Z → H1 = 0, ker(d2 ) =
5
These homology groups are consistent with those of S3 . A direct computation of the Feynman amplitude shows that indeed this graph is homotopically trivial and, unsurprisingly, the dual gluing is indeed S3 .
Colored Group Field Theory
91 v2
3 2
0
v3
1
2
1
1
3
1
v1
v4 0 v6
2
0 v5
3
Fig. 15. A third example of a graph
A.3. Third example. For the final example take the graph of Fig. 15. The 3-bubbles of this graph are Bv012 , Bv013 , Bv023 , Bv123 . 1 v2 v3 v4 v5 v6 1 v2 v3 v4 v5 v6 1 v2 v3 v4 v5 v6 1 v2 v3 v4 v5 v6
(54)
Its 2-bubbles are Bv011 v2 v3 v4 v5 v6 , Bv021 v2 v3 v4 v5 v6 , Bv031 v2 v3 v4 v5 v6 , Bv121 v2 v3 v4 v5 v6 , Bv131 v2 v3 v4 v5 v6 , Bv231 v2 , Bv233 v4 , Bv235 v6 ,
(55)
and its 1-bubbles are Bv01 v6 , Bv02 v3 , Bv04 v5 , Bv11 v4 , Bv12 v5 , Bv13 v6 , Bv21 v2 , Bv23 v4 , Bv25 v6 , Bv31 v2 , Bv33 v4 , Bv35 v6 .
(56)
Again from 3 and 4 we get H0 = Z, ker(d1 ) =
Z,
H3 = Z, Im(d3 ) =
7
Z,
(57)
3
and a direct computation gives
Im(d2 ) = Z → H1 = Z ⊕ Z; ker(d2 ) = Z → H2 = 0.
(58)
3
5
This graph represents a pseudo-manifold with two isolated singularities with links isomorphic with the torus. B. Euler Characteristic of Pseudo-manifolds in Three Dimensions This result is classical but, for completeness, we present here a proof. Lemma 6. If a 3 dimensional gluing of tetrahedra with k vertices is a pseudo-manifold then 1 χ (X ) = k − χ (lkvi ). (59) 2 i
Proof. A pseudo-manifold X can be transformed into a manifold with boundary, M X by removing its singular points by surgery. That is, as a set, X admits a decomposi(i) (i) tion X = M X ∪i C[∂ M X ], with M X a manifold with boundary ∂ M X = ∪i ∂ M X and (i) (i) C[∂ M X ] is the topological cone over ∂ M X .
92
R. Gurau (1)
The first singular point has some neighborhood whose boundary is ∂ M X , a closed (1) connected surface, and eliminating it by surgery from X we obtain M X , a pseudomanifold with one less singular point but with a boundary having a connected com(1) (1) ponent. The initial pseudo-manifold is written X = M X ∪ C[∂ M X ]. Also note that (1) (1) (1) M X ∩ C[∂ M X ] = ∂ M X , and, as X is locally compact, the Euler characteristic obeys an inclusion-exclusion relation (1)
(1)
(1)
χ (M X ) + χ (C[∂ M X ]) = χ (X ) + χ (∂ M X ).
(60)
The connected components ∂ M X(i) are two dimensional surfaces. The Euler characteristic of a cone over a surface is always 1. To prove this, let a simplicial complex on (1) ∂ M X with vertices vi , lines l j and triangles tk . It lifts to a simplicial complex of the cone C[∂ M X(1) ] obtained by coning all simplices to an unique vertex a. The simplicial complex of the cone has one extra vertex, a itself, one extra edge avi for each vertex vi , one triangle al j for each line l j and one tetrahedron atk for each triangle tk . The Euler characteristic of the cone is written (1)
χ (C[∂ M X ]) = 1 + |vi | − (|l j | + |vi |) + (|t j | + |l j |) − |t j | = 1.
(61)
Hence (1)
(1)
χ (M X ) = χ (X ) − 1 + χ (∂ M X ),
(62)
and iterating for all isolated singularities we conclude χ (M X ) = χ (X ) − C + χ (∂ M X ),
(63)
where C is the number of connected components of the boundary of M X (equal to the number of singular points of X ). But for any manifold with boundary M X in three dimensions, χ (M X ) = 21 χ (∂ M X ), hence 1 χ (X ) = C − χ (∂ M X ). 2
(64)
One can continue to eliminate by surgery also regular points of M X . Applying this procedure to all the internal points (singular or not) of a gluing of simplices proves the lemma.
References 1. Boulatov, D.V.: A Model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629 (1992) [arXiv:hep-th/9202074] 2. Freidel, L.: Group field theory: An overview. Int. J. Theor. Phys. 44, 1769 (2005) [arXiv:hep-th/0505016] 3. Oriti, D.: The group field theory approach to quantum gravity, in [15]. [arXiv:gr-qc/0607032] 4. Oriti, D.: Quantum gravity as a quantum field theory of simplicial geometry. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds.) Quantum Gravity. Birkhaeuser, Basel (2007). [arXiv:gr-qc/0512103] 5. Magnen, J., Noui, K., Rivasseau, V., Smerlak, M.: Scaling behaviour of three-dimensional group field theory. Class. Quant. Grav. 26, 185012 (2009) [arXiv:0906.5477[hep-th]] 6. David, F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985) 7. Ginsparg, P.H.: Matrix models of 2-d gravity. [arXiv:hep-th/9112013] 8. Gross, M.: Tensor models and simplicial quantum gravity in > 2-D. Nucl. Phys. Proc. Suppl. 25A, 144 (1992)
Colored Group Field Theory
93
9. Ambjorn, J., Durhuus, B., Jonsson, T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6, 1133 (1991) 10. Sasakura, N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991) 11. Williams, R.: Quantum Regge calculus, in Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter. Cambridge University Press, Cambridge (2009) 12. Ambjorn, J., Jurkiewicz, J., Loll, R.: Reconstructing the universe. Phys. Rev. D 72, 064014 (2005) [arXiv:hep-th/0505154] 13. Ambjorn, J., Jurkiewicz, J., Loll, R.: The universe from scratch. Contemp. Phys. 47, 103–117 (2006) [arXiv:hep-th/0509010] 14. Oriti, D.: Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity. Rept. Prog. Phys. 64, 1703 (2001) [arXiv:gr-qc/0106091] 15. Perez, A.: Spin foam models for quantum gravity. Class. Quant. Grav. 20, R43 (2003) [arXiv:grqc/0301113] 16. Oriti, D. (ed.): Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter. Cambridge University Press, Cambridge (2009) 17. De Pietri, R., Petronio, C.: Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4. J. Math. Phys. 41, 6671 (2000) [arXiv:gr-qc/0004045] 18. Turaev, V.G., Viro, O.Y.: State sum invariants of 3 manifolds and quantum 6j symbols. Topology 31, 865 (1992) 19. Barrett, J.W., Naish-Guzman, I.: The Ponzano-Regge model. Class. Quant. Grav. 26, 155014 (2009) [arXiv:0803.3319 [gr-qc]] 20. Engle, J., Pereira, R., Rovelli, C.: The loop-quantum-gravity vertex-amplitude. Phys. Rev. Lett. 99, 161301 (2007) [arXiv:0705.2388 [gr-qc]] 21. Engle, J., Pereira, R., Rovelli, C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008) [arXiv:0708.1236 [gr-qc]] 22. Livine, E.R., Speziale, S.: A new spinfoam vertex for quantum gravity. Phys. Rev. D 76, 084028 (2007) [arXiv:0705.0674 [gr-qc]] 23. Freidel, L., Krasnov, K.: A new spin foam model for 4d gravity. class. quant. grav. 25, 125018 (2008) [arXiv:0708.1595 [gr-qc]] 24. Conrady, F., Freidel, L.: On the semiclassical limit of 4d spin foam models. Phys. Rev. D 78, 104023 (2008) [arXiv:0809.2280 [gr-qc]] 25. Bonzom, V., Livine, E.R., Smerlak, M., Speziale, S.: Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model. Nucl. Phys. B 804, 507 (2008) [arXiv:0802.3983 [gr-qc]] 26. Grosse, H., Wulkenhaar, R.: Renormalisation of phi**4 theory on non-commutative R**4 in the matrix base. Commun. Math. Phys. 256, 305 (2005) [arXiv:hep-th/0401128] 27. Gurau, R., Magnen, J., Rivasseau, V., Vignes-Tourneret, F.: Renormalization of non-commutative phi**4(4) field theory in x space. Commun. Math. Phys. 267, 515 (2006) [arXiv:hep-th/0512271] 28. Disertori, M., Gurau, R., Magnen, J., Rivasseau, V.: Vanishing of beta function of non commutative phi(4)**4 theory to all orders. Phys. Lett. B 649, 95 (2007) [arXiv:hep-th/0612251] 29. Geloun, J.B., Gurau, R., Rivasseau, V.: Vanishing beta function for Grosse-Wulkenhaar model in a magnetic field. Phys. Lett. B 671, 284 (2009) [arXiv:0805.4362 [hep-th]] 30. Thurston, W.P.: Three-Dimensional Geometry and Topology: Vol 1. University Presses of California, Columbia and Princeton, ISBN 13: 9780691083049, ISBN 10: 0691083045 31. Kozlov, D.: Combinatorial Algebraic Topology. Springer, Heidelberg, ISBN-10: 354071961X, ISBN-13: 978-3540719618 32. Freidel, L., Gurau, R., Oriti, D.: Group field theory renormalization–the 3d case: power counting of divergences. Phys. Rev. D 80, 044007 (2009) [arXiv:0905.3772 [hep-th]] 33. Mulazzani, M.: Lins-Mandel graphs representing 3-manifolds. Discr. Math. 140, 107 (1995) 34. Alexandrov, P.S.: Combinatorial Topology. Dover Publications, ISBN-10: 0486401790, ISBN-13: 9780486401799 35. Hatcher, A.: Algebraic Topology. Cambridge University Press, ISBN: 0-521-79160-X, ISBN:0-52179540-0 36. Mulase, M., Penkava, M.: Volume of representation varieties. [arXiv:math/0212012] 37. Fairbairn, W.J., Livine, E.R.: 3d spinfoam quantum gravity: matter as a phase of the group field theory. Class. Quant. Grav. 24, 5277 (2007) [arXiv:gr-qc/0702125] Communicated by A. Connes
Commun. Math. Phys. 304, 95–123 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1137-1
Communications in
Mathematical Physics
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories Detlev Buchholz1,∗ , Gandalf Lechner2 , Stephen J. Summers3,∗∗ 1 Institut für Theoretische Physik and Courant Centre “Higher Order Structures in Mathematics”,
Universität Göttingen, 37077 Göttingen, Germany. E-mail: [email protected]
2 Fakultät für Physik, Universität Wien, 1090 Vienna, Austria. E-mail: [email protected] 3 Department of Mathematics, University of Florida, Gainesville, FL 32611, USA. E-mail: [email protected]
Received: 15 May 2010 / Accepted: 24 May 2010 Published online: 9 October 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com
Dedicated to Sergio Doplicher on the occasion of his seventieth birthday Abstract: Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel’s strict deformations of C ∗ – dynamical systems with automorphic actions of Rn , whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita–Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations. 1. Introduction Recent advances in algebraic quantum field theory have led to purely algebraic constructions of quantum field models on Minkowski space, both classical and noncommutative [5,9,11–13,17,18,22–24,26,29], many of which cannot be achieved by the standard methods of constructive quantum field theory. Some of these models are local and free, some are local and have nontrivial S–matrices, and yet others manifest only certain remnants of locality, though these remnants suffice to enable the computation of nontrivial S–matrix elements. In order to construct a quantum field model on noncommutative Minkowski space, Grosse and one of us [17] have deformed the free quantum field in a certain manner to find a family of theories which are Poincaré covariant and comply with a slightly weakened version of the principle of Einstein causality (“wedge locality”). As pointed out in [17], a completely analogous deformation can be carried out on a free field on ∗ Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen. ∗∗ Research supported by the NSF Grant DMS-0901370.
96
D. Buchholz, G. Lechner, S. J. Summers
classical Minkowski space. In [13] two of us presented a generalization (called a warped convolution) of that deformation which can be applied to any Minkowski space quantum field model in any number of dimensions. This deformation results in a family of distinct theories which are wedge–local and covariant under the representation of the Poincaré group associated with the initial, undeformed theory. It turns out that also the S–matrix changes under this deformation, and the scattering is nontrivial even if the scattering of the initial theory is trivial. When taking the free quantum field as the initial model, this deformation coincides with that of Grosse and Lechner. It provides the first fully consistent examples of relativistic quantum field theories on four–dimensional Minkowski space describing nontrivial elastic scattering processes [13,17]. Warped convolution was subsequently studied in the language of Wightman quantum field theory in [18], where it was shown that the deformation of the field operators can be understood as resulting in a certain deformation of the canonical product on the Borchers–Uhlmann algebra – the algebra of test functions canonically associated with a Wightman theory. This was the first indication that the deformation of operators resulting from warped convolution may be equivalent to a deformation of the operator product. A well known example of this latter type is the strict deformation theory of C ∗ – dynamical systems with an action of Rn developed by Rieffel in [28]. It was originally introduced for the quantization of classical models. We shall show in this paper that the warped convolution applied to any C ∗ –dynamical system provides a covariant representation of the corresponding deformed Rieffel algebra. In particular, all states in a covariant representation of the initial system can be lifted to states on the deformed algebra (compare [21, Cor. 4.4]). In spite of this tight relation between the two deformation procedures, the concept of warped convolution appears to be more appropriate in applications to quantum field theory. For there one has to deal simultaneously with a multitude of different deformations and to establish relations between the resulting operators. This can be done most conveniently in a common representation space of the various Rieffel algebras, and such a space is provided by the warped convolution procedure. Suitably adjusting the deformation parameters to the geometry of Minkowski space, we apply the warped convolutions to quantum field theories, as outlined in [13], and prove as well as extend the results given there. As we shall further explain, any quantum field theory on Minkowski space can be constructed from a (causal) Borchers triple consisting of a von Neumann algebra, a representation of the Poincaré group and a vector representing the vacuum state. The physical constraints of causality and covariance can conveniently be expressed in terms of a few conditions on these triples. We shall show that these properties are preserved under a distinguished group of warped convolutions, thereby giving rise to interesting new theories. The article is organized as follows. In Sect. 2 we prove and extend the results about warped convolution given in [13]. These extensions allow us to establish the relation with Rieffel deformed dynamical systems. In Sect. 3 a restricted family of warped convolutions is applied to Borchers triples to construct quantum field theories in two spacetime dimensions. We show, in particular, that the Tomita–Takesaki modular objects associated with such triples remain fixed under these deformations. The application of the warped convolutions to general relativistic quantum field theories in higher dimensions is discussed in Sect. 4. We present there the salient results given in [13] in the framework of causal Borchers triples and also establish further physically relevant properties of the deformed theories, not addressed in [13]. Finally, we briefly discuss prospects for further development of this approach in Sect. 5.
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
97
2. Warped Convolutions and Rieffel Deformations We clarify here the relation between the notion of warped convolution, recently introduced in [13], and the strict deformation of C ∗ –algebras established in [28] by Rieffel. In either case one proceeds from a C ∗ –dynamical system (A, Rn ), cf. [27]. It consists of a C ∗ –algebra A equipped with a strongly continuous automorphic action of the group Rn which will be denoted by α. In order to relate the two settings, it will be convenient to consider the system (A, Rn ) in a covariant representation. That is, we regard A as a concrete C ∗ –algebra on a Hilbert space H on which the automorphisms α are implemented by the adjoint action of a weakly continuous unitary representation U of Rn , αx (A) = U (x)AU (x)−1 , x ∈ Rn . As a matter of fact, this assumption imposes no significant restriction of generality. For if the abstract algebra A can be represented faithfully on some separable Hilbert space, then there also exists a faithful covariant representation of (A, Rn ), cf. [27, Lemma 7.4.9 and Prop. 7.4.7]. Furthermore, since the adjoint action α of the unitary representation U can be extended to the algebra B(H) of all bounded operators on H, no generality will be lost when we proceed to the C ∗ –dynamical system (C, Rn ), where C ⊂ B(H) is the C ∗ –algebra of all operators on which α acts strongly continuously. We shall then be able to restrict to suitable subalgebras A of C as necessary. 2.1. Rieffel deformations. We begin by considering the C ∗ –algebra C of all uniformly continuous bounded functions A : Rn → B(H). The algebraic structure of C is the natural one inherited from B(H), i.e. the algebraic operations in C are pointwise defined, ( A + B)(x) = A(x) + B(x), ( A B)(x) = A(x) B(x), A∗ (x) = A(x)∗ x ∈ Rn , and the norm is given by1 || A|| = sup || A(x)||. x∈Rn
Following Rieffel [28], we consider the subalgebra C ∞ ⊂ C of smooth (in the norm topology) elements A, i.e. ||∂ μ A|| < ∞ for all multi-indices μ.2 Note that the elements C ∈ B(H) act as multipliers on C ∞ , if one identifies C with the corresponding constant function in C ∞ (denoted by the same symbol). Clearly, the maps (C, A) → C A and (C, A) → AC are norm continuous in both variables, ||C A|| ≤ ||C|| || A|| ≥ || AC||, and the multiplication by C commutes with the operations of differentiation and integration on C ∞ . In the subsequent analysis we shall find it necessary to integrate the functions x → A(x). In order to handle the fact that these functions are in general not absolutely integrable with respect to Lebesgue measure due to a lack of suitable decay properties, we introduce mollifiers L n : Rn → C. A convenient choice is given by (i + xk )−1 , x ∈ Rn . L n (x) = (i + x1 + · · · + xn )−1 k=1,...,n 1 Risking some confusion, we use the same symbol for the norms on C and B(H). 2 We use the notation μ = (μ , . . . , μ ), ∂ μ = ∂ μ1 · · · ∂ μn and (∂ μ A)(x) = ∂ μ A(x), where x are n 1 k x x x1 xn the components of x with respect to a fixed orthonormal basis in Rn and ∂xk are the corresponding partial
derivatives, k = 1, . . . n.
98
D. Buchholz, G. Lechner, S. J. Summers
Because of this simple form one easily verifies that (∂ μ L n )(x) = Nn,μ (x)L n (x), where Nn,μ is smooth and bounded for any multi-index μ; moreover L n ∈ L 1 (Rn ). We therefore choose L n as a universal mollifier on C ∞ . It follows from the preceding remarks that, for any multi-index μ, the functions μ x → ∂x (L n (x) A(x)) are Bochner integrable in B(H) with respect to the Lebesgue measure. Moreover, applying Leibniz’s rule, one gets d x ||∂xμ (L n (x) A(x)) || ≤ cn,μ || A|||μ| , A ∈ C ∞ , where cn,μ does not depend on A, and we have introduced the norms, m ∈ N0 , ||∂ μ A||. || A||m = μ, |μ|≤m
The following technical lemma is a basic ingredient in the subsequent discussion. In its proof, we make use of arguments furnished by Rieffel [28]. Lemma 2.1. Let A, B ∈ C be n + 1 times continuously differentiable and let f ∈ S(Rn × Rn ) with f (0, 0) = 1. (i) The norm limit of Bochner integrals in B(H), . d xd y f (εx, εy) e−i x y A(x)B(y) = A × B , lim (2π )−n ε→0
exists and does not depend on f . Here x y, x, y ∈ Rn is any symmetric bilinear form on Rn with determinant 1 or −1. (ii) With L n as above, there exists a polynomial u, v → Pn (u, v) on Rn ×Rn of degree n + 1 in the components of u and v, respectively, such that −n A × B = (2π ) d xd y e−i x y Pn (∂x , ∂ y ) L n (x) A(x) L n (y)B(y) , where the integral is defined as a Bochner integral in B(H). (iii) || A × B|| ≤ cn || A||n+1 ||B||n+1 , for a universal constant cn . (iv) Let C ∈ B(H). Then (C A × B) = C( A × B), ( A × BC) = ( A × B)C, ( AC × B) = ( A × C B) , and the linear map C → A × C B is continuous on the unit sphere of B(H) in the strong operator topology. Proof. (i) Crucial for the result are certain properties of the function x, y → e−i x y . Namely, for each polynomial x, y → Q n (x, y) of degree n + 1 in the variables x and y, respectively, there is a corresponding polynomial Pn in the same family such that Q n (x, y) e−i x y = Pn (−∂x , −∂ y ) e−i x y ,
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
99
and vice versa. The step from the right-hand side to the left-hand side is easily accomplished by differentiation; the opposite direction can likewise be established, noticing that the Fourier transform of x, y → e−i x y is again of this form. Choosing Q n (x, y) = L n (x)−1 L n (y)−1 , x, y ∈ Rn , with L n specified above, one observes that d xd y f (εx, εy) e−i x y A(x)B(y) = d xd y Q n (x, y) e−i x y f (εx, εy) L n (x) A(x) L n (y)B(y) = d xd y Pn (−∂x , −∂ y )) e−i x y f (εx, εy) L n (x) A(x) L n (y)B(y) = d xd y e−i x y Pn (∂x , ∂ y ) f (εx, εy) L n (x) A(x) L n (y)B(y) , where Pn is the polynomial corresponding to the chosen Q n . In view of the smoothness and rapid decay properties of the integrands, the integrals are defined as Bochner integrals in B(H), and the last equality is obtained by partial integration. Decomposing μ Pn (∂x , ∂ y ) into a sum of monomials of the form ∂x ∂ yν with |μ|, |ν| ≤ n + 1, performing the differentiations and taking into account the properties of L n , one gets by an application of the dominated convergence theorem, lim d xd y e−i x y Pn (∂x , ∂ y ) f (εx, εy) L n (x) A(x) L n (y)B(y) ε→0 (2.1) = d xd y e−i x y Pn (∂x , ∂ y ) L n (x) A(x) L n (y)B(y). Note that the derivatives of x, y → f (εx, εy) contain powers of ε as factors and therefore disappear in the limit. In particular, the limit does not depend on the choice of f . Assertion (ii) has been established in (2.1). From this relation one also obtains the estimate (iii), because of the properties of L n established above. The proof of the equalities in (iv) is another straightforward consequence of (ii) and is therefore omitted. It remains to establish the continuity of the map. In view of the continuity and decay properties of the functions appearing in the representation (ii) of the product ×, the integrals underlying the definition of A × C B can be approximated in norm, uniformly for C ∈ B(H), ||C|| ≤ 1, by finite sums of the form cμ,ν,i,k (∂ μ A)(xi ) C (∂ ν B)(yk ) , μ,ν,i,k
where cμ.ν,i,k are constants which do not depend on C. Since the operators (∂ μ A)(xi ), (∂ ν B)(yk ) are bounded, the stated continuity properties of the map with respect to C then follow.
Within this general setting, the Rieffel deformations of the C ∗ –dynamical system (C, Rn ) [28] can be presented as follows. Let C ∞ ⊂ C be the ∗ –algebra of smooth elements with respect to the action of α and let Q be a real skew symmetric matrix
100
D. Buchholz, G. Lechner, S. J. Summers
relative to the chosen bilinear form on Rn , i.e. x Qy = −y Qx, x, y ∈ Rn . One then considers for A, B ∈ C ∞ the functions in C ∞ given by . . x → Aα Q (x) = α Qx (A) , y → Bα (y) = α y (B) and sets . A × Q B = Aα Q × Bα ,
A, B ∈ C ∞ .
It has been shown by Rieffel [28] that × Q defines an associative product on C ∞ , the Rieffel product, which is compatible with the ∗ –operation. In view of the normalization of the bilinear form chosen in Lemma 2.1 (i), the original identity operator 1 still acts as the identity with respect to the new product × Q . Moreover, there exists a C ∗ – norm on the deformed algebra (C ∞ , × Q ). It is of interest here that, by Lemma 2.1, the Rieffel product extends to more general functions in C in a natural manner. We shall take advantage of this fact in the following subsection. 2.2. Warped convolution. We turn now to the discussion of the warped convolution on (C, Rn ) introduced in [13]. Many of the results below were stated there and provided with sketches of proofs; in addition to supplying complete proofs of those assertions, here we also prove results which strengthen and complement those discussed in [13]. The weakly continuous unitary representation U of Rn implementing α enters into the definition of the warped convolution. In a first step we want to give proper meaning to the formal expressions B α Qx (A) d E(x) and B d E(x) α Qx (A), A ∈ C ∞ , where Q is any real n × n matrix, E is the spectral resolution of U and B ⊂ Rn is any bounded Borel set. If F is a finite–dimensional projection, it follows from the spectral calculus that the integrals B α Qx (A) Fd E(x) and B d E(x)F α Qx (A) are well–defined in the strong operator topology. Moreover, since U (y) = ei x y d E(x), y ∈ Rn , one obtains for any test function f as in Lemma 2.1, α Qx (A) Fd E(x) = lim (2π )−n d xd y f (εx, εy)e−i x y α Qx (A) F E(B)U (y) , ε→0 B −n d E(x) Fα Qx (A) = lim (2π ) d xd y f (εx, εy)e−i x y E(B)U (x)F α Qy (A). ε→0
B
Hence, adopting the notation in Lemma 2.1, α Qx (A) Fd E(x) = Aα Q × F U B , d E(x) Fα Qx (A) = U B F × Aα Q , B
B
. where we have introduced the function x → U B (x) = E(B) U (x), which is an ele∞ ment of C since the set B is bounded. Choosing any net of projections F converging monotonically to the identity operator 1, it follows from part (iv) of Lemma 2.1 that the right-hand side of these equalities converges in the strong operator topology. Hence the corresponding limits of the integrals on the left-hand side exist in B(H) and can be used to define the warped convolution integrals as . α Qx (A) d E(x) = lim α Qx (A) F d E(x) = Aα Q × U B , F 1 B B (2.2) . d E(x) α Qx (A) = lim d E(x) Fα Qx (A) = U B × Aα Q . B
F 1 B
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
101
If the spectrum sp U of U is compact, the integrals do not depend on B if B ⊃ sp U , so one can proceed to the limit B Rn in B(H). In the general case, however, we have a priori no control on the continuity properties of the resulting operators. In order to cope with this problem, we consider the dense domain D ⊂ H of vectors which are smooth with respect to the action of U . Let P = (P1 , . . . , Pn ) be the generators of U and let . x → URn (1 + P 2 )−n−1 (x) = U (x)(1 + P 2 )−n−1 . This function is an element of C which is n +1 times continuously differentiable. Making use of the first half of part (iv) of Lemma 2.1, we thus get for ∈ D, Aα Q × U B = Aα Q × U B (1 + P 2 )−n−1 (1 + P 2 )n+1 = Aα Q × E(B) URn (1 + P 2 )−n−1 (1 + P 2 )n+1 . In the latter expression we can proceed to the limit B Rn according to the second half of part (iv) of Lemma 2.1, since the projections E(B) converge strongly to 1 in this limit. In view of relation (2.2), this proves the existence of the first type of integrals, . α Qx (A) d E(x) = lim α Qx (A) d E(x) B Rn B (2.3) = Aα Q × URn (1 + P 2 )−n−1 (1 + P 2 )n+1 . Part (iii) and the first half of part (iv) of Lemma 2.1 imply that the functions x → αx Aα Q × URn (1 + P 2 )−n−1 = U (x) Aα Q U (x)−1 × URn (1 + P 2 )−n−1 are elements of C ∞ , since x, y → (U (x) Aα Q U (x)−1 )(y) = αx+Qy (A) is smooth in both variables. So it is also clear that the domain D is stable under the action of the integrals. For the proof of existence we make use of the fact of the second type of integrals, that the functions x → αx (1 + P 2 )n+1 A(1 + P 2 )−n−1 , A ∈ C ∞ are elements of C ∞ .3 Thus, by the first half of part (iv) of Lemma 2.1, we get for ∈ D, U B × Aα Q = U B × Aα Q (1 + P 2 )−n−1 (1 + P 2 )n+1 = U B × (1+ P 2 )−n−1 (1+ P 2 )n+1 Aα Q (1+ P 2 )−n−1 (1+ P 2 )n+1 = URn E(B)(1 + P 2 )−n−1 × (1 + P 2 )n+1 Aα Q (1 + P 2 )−n−1 ×(1 + P 2 )n+1 , where in the latter expression we can proceed again to the strong limit E(B) 1 if B Rn . In view of relation (2.2), this proves existence of the strong limits . d E(x) α Qx (A) = lim d E(x) α Qx (A) B Rn B = URn (1 + P 2 )−n−1 × (1 + P 2 )n+1 Aα Q (1 + P 2 )−n−1 ×(1 + P 2 )n+1 . 3 This can be seen by pulling the components of P through from the left to the right of A, noticing that their commutators with A yield derivatives of A.
102
D. Buchholz, G. Lechner, S. J. Summers
By a similar argument as before, one finds that the domain D is stable under the action of the second type of integrals, as well. It follows at once from the preceding results and Lemma 2.1 that one can conveniently present the two types of integrals on the domain D in terms of the strong limits −n α Qx (A) d E(x) = lim (2π ) d xd y f (εx, εy) e−i x y α Qx (A) U (y) , ε→0 (2.4) d E(x) α Qx (A) = lim (2π )−n d xd y f (εx, εy) e−i x y U (x) α Qy (A) . ε→0
We shall make use of these relations throughout the subsequent analysis. In the following, we limit ourselves to the particularly interesting case where the matrix Q entering into the definition of the integrals is skew symmetric relative to the chosen bilinear form on Rn . It is understood without further mention that the integrals are defined on the common stable domain D. Lemma 2.2. Let Q be any real skew symmetric matrix on Rn and let A ∈ C ∞ . Then (i) d E(x) α Qx (A) = α Qx (A) d E(x) and ∗ (ii) α Qx (A) d E(x) ⊃ α Qx (A∗ ) d E(x). Proof. (i) As pointed out above, one has for ∈ D, −n d xd y f (εx, εy) e−i x y α Qx (A) U (y) . α Qx (A) d E(x) = lim (2π ) ε→0
The integration can be restricted to the submanifold (ker Q)⊥ × (ker Q)⊥ , since the remaining integrals merely produce factors of 2π . Substituting x → x + Q −1 y and taking into account that Q −1 is skew symmetric, one gets d xd y f (εx, εy) e−i x y α Qx (A) U (y) = d xd y g(εx, εy) e−i x y U (y)α Qx (A) = d xd y g(εy, εx) e−i x y U (x) α Qy (A) , . where g(x, y) = f (x + Q −1 y, y). In the limit ε → 0 one obtains lim (2π )−n d xd y g(εy, εx) e−i x y U (x) α Qy (A) = d E(x) α Qx (A) , ε→0
proving assertion (i). (ii) For the proof of the second assertion, note that
∗ d xd y f (εx, εy) e−i x y α Qx (A) U (y) = d xd y f (εx, εy) ei x y U (−y) α Qx (A∗ ) = d xd y f (εy, −εx) e−i x y U (x) α Qy (A∗ ).
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
103
Hence, for , ∈ D, , α Qx (A) d E(x) = lim (2π )−n , dx dy f (εx, εy) e−i x y α Qx (A) U (y) ε→0 −n = lim (2π ) dx dy f (εy,−εx) e−i x y U (x) α Qy (A∗ ), ε→0 = d E(x) α Qx (A∗ ) , . The assertion now follows from the preceding step.
We may therefore meaningfully declare the following definition as in [13]. Definition 2.3. Let Q be a real skew symmetric matrix on Rn and let A ∈ C ∞ . The corresponding warped convolution A Q of A is defined on the domain D by means of the preceding results according to . A Q = d E(x) α Qx (A) = α Qx (A) d E(x). In particular, 1 Q = 1. We shall next show that the warped convolution provides a representation of the . algebra (C ∞ , × Q ) defined by A → π Q (A) = A Q . The argument proceeds through a number of steps. It is apparent from the definition that the map π Q is linear and, by the second part of the preceding lemma, we have π Q (A)∗ ⊃ π Q (A∗ ). The proof that π Q is also multiplicative requires more work. Lemma 2.4. Let Q be a real skew symmetric matrix on Rn and let A, B ∈ C ∞ . Then (understood as an equality on D) A Q B Q = (A × Q B) Q , where × Q denotes the Rieffel product on C ∞ . In other words, π Q (A)π Q (B) = π Q (A × Q B). Proof. Let f, g be test functions as in Lemma 2.1. Recalling that A × Q B ∈ C ∞ can be approximated in norm according to A × Q B = lim (2π )−n dvdw f (δv, δw) e−ivw α Qv (A)αw (B) , δ→0
one finds for ∈ D,
(2π )2n (A × Q B) Q = lim
ε,δ→0 −ivw−i x y
×e
dvdwd xd y f (δv, δw)g(εx, εy) α Qv+Qx (A)αw+Qx (B) U (y) ,
in the sense of strong convergence. Similarly, one has 2n (2π ) A Q B Q = lim dvdwd xd y g(εv, εw) f (δx, δy) ε,δ→0 −ivw−i x y
×e
α Qv (A) U (w) α Qx (B) U (y).
104
D. Buchholz, G. Lechner, S. J. Summers
In both cases the limits are to be performed in the given order. In order to see that these limits coincide, one rewrites the two integrals. For the first one, by substituting (v, w) → (v − x, w − Qx) and bearing in mind that Q is skew symmetric, one obtains dvdwd xd y f (δv, δw)g(εx, εy) e−ivw−i x y α Qv+Qx (A)αw+Qx (B) U (y) = dvdwd xd y h δ,ε (v, w, x, y) e−ivw−i x(y+Qv−w) α Qv (A) αw (B) U (y) , . where h δ,ε (v, w, x, y) = f (δ(v − x), δ(w − Qx)) g(εx, εy) . For the second integral, by substituting (w, y) → (w − Qx, y + Qx − w) and making use again of the fact that Q is skew symmetric, one finds dvdwd xd y g(εv, εw) f (δx, δy) e−ivw−i x y α Qv (A) U (w) α Qx (B) U (y) = dvdwd xd y kδ,ε (v, w, x, y) e−ivw−i x(y+Qv−w) α Qv (A) αw (B) U (y) , where kδ,ε (v, w, x, y) = f (δx, δ(y+ Qx −w))g(εv, ε(w− Qx)). Thus the two integrals coincide apart from the mollifying test functions h δ,ε and kδ,ε , respectively. In order to show that the two integrals have the same limits, one proceeds as in the proof of part (i) of Lemma 2.1. Again one finds by Fourier transformation that for any given polynomial L on R4n there is a corresponding polynomial P such that L(v, w, x, y) e−ivw−i x(y+Qv−w) = P(−∂v , −∂w , −∂x , −∂ y ) e−ivw−i x(y+Qv−w) . A convenient choice for L is given by L(v, w, x, y) = (L n (v)L n (w)L n (x)L n (y))−1 ,
(2.5)
where L n are the mollifiers introduced before. With this choice one gets by partial . . integration, setting u = (v, w, x, y), du = dvdwd xd y and ∂ = (∂v , ∂w , ∂x , ∂ y ), du h δ,ε (u) e−ivw−i x(y+Qv−w) α Qv (A) αw (B) U (y) = du e−ivw−i x(y+Qv−w) P(∂) h δ,ε (u) L(u)−1 α Qv (A) αw (B) U (y) . The derivatives in the second line are well–defined, since A, B ∈ C ∞ and ∈ D. Moreover, all derivatives of u → L(u)−1 are absolutely integrable, and the derivatives of u → h δ,ε (u) produce factors of δ and ε, respectively. Thus in the limit of small δ and ε one can replace in the above integral the test function h δ,ε by its value at the origin, i.e. 1. The same argument applies if one replaces h δ,ε by kδ,ε , proving equality of the limits of the respective integrals.
At this point, the operators π Q (A) = A Q , A ∈ C ∞ , are well–defined only on the dense, invariant domain D, defining there a ∗ –algebra. We next show that they may be extended to bounded operators on H, in contradiction to an assertion made in [13]. In the proof we make use of the fact that the algebra (C ∞ , × Q ) admits a C ∗ –norm || · || Q , cf. [28, Ch. 4]. It thus can be completed to a C ∗ –algebra, denoted by (C Q , × Q ), to which the group of automorphisms αx , x ∈ Rn , extends in a strongly continuous manner [28, Prop. 5.11]. As is well known, every positive element of a C ∗ –algebra has a positive square root in the algebra. We need here the following more detailed information.
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
105
Lemma 2.5. Let A ∈ C ∞ be strictly positive in (C Q , × Q ), i.e. √ A − δ 1 = B ∗ × Q B for some δ > 0 and B ∈ (C Q , × Q ). Then its positive square root A ∈ (C Q , × Q ) is also an element of C ∞ . Proof. The form of A implies that its spectrum is contained in the interval [δ, A Q ]. √ As the square root z → z is holomorphic in a complex neighborhood of this√region and C ∞ is closed under the holomorphic calculus [28, Cor. 7.6], it follows that A ∈ C ∞ .
With the help of this lemma we can show now that the operators π Q (A) = A Q , . A ∈ C ∞ are bounded. For the operators (δ + a)2 1 − A∗ × Q A, a = A Q , are elements ∞ of C and strictly positive in (C Q , × Q ) for every δ > 0. Thus their positive square roots . B = (δ + a)2 1 − A∗ × Q A ∈ (C Q , × Q ) are elements of C ∞ according to the preceding lemma. Bearing in mind the properties of π Q established thus far, we therefore have for any ∈ D, (δ + a)2 2 − π Q (A)2 = , π Q (δ + a)2 1 − A∗ × Q A = , π Q (B ∗ × Q B) = π Q (B)2 ≥ 0, where we made use of the fact that B ∗ = B since B is positive. Hence we obtain π Q (A) ≤ (A Q + δ) , ∈ D. Since D is dense in H and δ > 0 was arbitrary, we conclude that the operators π Q (A) can be extended to the whole Hilbert space with operator norms satisfying the bound π Q (A) ≤ A Q ,
A ∈ C∞.
(2.6)
Moreover, it follows from this estimate and the preceding results that the representation π Q : C ∞ → B(H) can be continuously extended to a representation of the C ∗ –algebra (C Q , × Q ) on H. We summarize these findings. Theorem 2.6. The map . π Q (A) = A Q ,
A ∈ C∞ ,
extends to a representation of the Rieffel–deformed C ∗ –algebra (C Q , × Q ) on H. In particular, one has the bound π Q (A) ≤ A Q , A ∈ (C Q , × Q ). Example. Of particular interest in physics are the cases where the spectrum of U contains an atomic part. Without loss of generality one may then assume that {0} is part of the atomic spectrum with corresponding invariant vector .4 Since the algebra C ∞ is weakly dense in B(H), it is clear that is cyclic for C ∞ ; moreover, because of the invariance of under the action of U , one also has C ∞ ⊂ D. Within this setting the 4 Note that proceeding from the group U (x) to the group U (x) = eiq x U (x), x ∈ Rn , merely amounts to q a translation A → α Qq (A) of the operators A in the original warped convolution.
106
D. Buchholz, G. Lechner, S. J. Summers
relation between the warped convolutions and the Rieffel deformations can be exhibited quite easily. For, as a consequence of the invariance of , one obtains for A, B ∈ C ∞ , A Q B = lim (2π )−n d xd y e−i x y f (εx, εy) α Qx (A) U (y) B
ε→0 −n = lim (2π ) d xd y e−i x y f (εx, εy) α Qx (A) α y (B)
ε→0
= (A × Q B) . In particular, A Q = A ,
A ∈ C∞.
(2.7)
Making use of the associativity of the product × Q on C ∞ , it is therefore clear that for A, B, C ∈ C ∞ , A Q B Q C = A Q (B × Q C) = (A × Q B × Q C) = (A × Q B) Q C . We return now to the discussion of the general case and exhibit further interesting properties of the representations π Q introduced above. Proposition 2.7. Let π Q be the representation of the C ∗ –algebra (C Q , × Q ) established by the preceding theorem. (i) π Q is α–covariant, i.e. for any A ∈ (C Q , × Q ) π Q (αx (A)) = U (x)π Q (A)U (x)−1 , x ∈ Rn . (ii) π Q induces a bijective map of C ∞ onto itself. (iii) π Q is faithful, i.e. π Q (A) = A Q , A ∈ (C Q , × Q ). (iv) π Q is irreducible. Proof. (i) Let A ∈ C ∞ . Since the domain D is stable under the action of the unitaries U (x), relation (2.3) and Lemma 2.1 imply U (x)A Q U (x)−1 = (αx (A)) Q , proving the assertion for A ∈ C ∞ . The continuity properties of π Q and the automorphic action of αx on (C Q , × Q ) then yield assertion (i). (ii) According to [28, Thm. 7.1], the smooth elements of (C Q , × Q ) are exactly the elements of C ∞ . It therefore follows from the continuity of the map π Q that the functions x → π Q (αx (A)), A ∈ C ∞ , are smooth; hence π Q (A) = A Q ∈ C ∞ for A ∈ C ∞ . The proof that π Q C ∞ is bijective requires a computation: In view of the preceding observation, one may apply the warping procedure with underlying matrix −Q to the operator A Q , giving (A Q )−Q . Now according to relation (2.4) one has on the domain D, (2π )2n (A Q )−Q = lim dvdwd xd y f (εv, εw) f (δx, δy) e−ivw−i x y α Qx−Qv (A)U (y)U (w) , ε,δ→0
where the limits are to be performed in the given order. Substituting (v, w) → (x −v, w − y), the integral can be transformed into dvdwd xd y f (ε(x − v), ε(w − y)) f (δx, δy) eivw−i xw−i yv α Qv (A)U (w).
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
107
As the x, y–integration in the latter integral involves only ordinary functions, it is straightforward to compute its limit for δ → 0, giving n dvdw (1/ε)2n f (w/ε, −v/ε) e−ivw α Qv (A)U (w), (2π ) where f denotes the Fourier transform of f . It is also apparent that the latter expression converges to (2π )2n A as ε → 0. Hence (A Q )−Q = A for A ∈ C ∞ . Now if π Q (A) = A Q = 0, it follows that A = (A Q )−Q = 0, so π Q C ∞ is injective; similarly, interchanging the role of Q and −Q, one has π Q (A−Q ) = (A−Q ) Q = A, so π Q C ∞ is also surjective. (iii) Since π Q is α–covariant, its kernel ker π Q is α–invariant. Hence, in view of the strongly continuous action of α on (C ∞ , × Q ), the space ker π Q C ∞ is dense in ker π Q . But this space coincides with {0}, since π Q C ∞ is injective according to the preceding result. Consequently, π Q ( · ) defines a C ∗ –norm on (C Q , × Q ), which must coincide with · Q because of the uniqueness of such norms. (iv) The final assertion follows from the fact that π Q C ∞ is surjective. So its range contains C ∞ , which is weakly dense in B(H).
Let us turn now to the case of an abstractly given C ∗ –dynamical system (A, Rn ) equipped with some strongly continuous representation α : Rn → Aut A. Denoting by A∞ the smooth elements of A, one obtains by arguments given by Rieffel [28] and sketched at the end of Sect. 2.1 a deformed ∗ –algebra (A∞ , × Q ) with C ∗ –norm · Q for any given skew symmetric matrix Q. Its C ∗ –completion will be denoted (A Q , × Q ). Here we have used Q as an upper index in order to distinguish the abstract setting from the concrete one used thus far. Let (π, H) be an α–covariant representation of A on a Hilbert space H, i.e. on H there exists a weakly continuous unitary representation U of Rn such that U (x)π(A)U (x)−1 = π(αx (A)) ,
A ∈ A.
(2.8)
Consequently π(A∞ ) ⊂ C ∞ , so one can define for any A, B ∈ A∞ the operators π(A) Q ∈ C ∞ and the product π(A)× Q π(B); moreover, π(A)× Q π(B) = π(A × Q B). After having established the properties of the warping procedure on C ∞ , it is almost evident that the covariant representation (π, H) of A induces a covariant representation (π Q , H) of (A Q , × Q ). It is fixed by setting . π Q (A) = π(A) Q ,
A ∈ A∞ .
(2.9)
By Theorem 2.6, the operators π Q (A) are bounded. Moreover, it follows from Lemma 2.2 that π Q (A)∗ = (π(A) Q )∗ = (π(A)∗ ) Q = π(A∗ ) Q = π Q (A∗ ). Similarly, Lemma 2.4 implies π Q (A)π Q (B) = π(A) Q π(B) Q = (π(A) × Q π(B)) Q = π(A × Q B) Q = π Q (A × Q B). Finally, one may employ the analogue of Lemma 2.5 in the abstract setting and the reasoning thereafter to obtain π Q (A) ≤ A Q , A ∈ A∞ . Hence the homomorphism
108
D. Buchholz, G. Lechner, S. J. Summers
π Q : A∞ → B(H) can be extended by continuity to a representation of (A Q , × Q ), as claimed. From the first part of Proposition 2.7 it follows that U (x)π Q (A)U (x)−1 = U (x)π(A) Q U (x)−1 = (U (x)π(A)U (x)−1 ) Q = π(αx (A)) Q = π Q (αx (A)) , for all A ∈ A∞ . So the representation π Q is also covariant, hence π Q (A∞ ) ⊂ C ∞ . Depending on the properties of the chosen representation (π, H) of A, the map π Q : A∞ → C ∞ may not be injective or surjective. But according to part (ii) of the preceding proposition one has π(A) Q = 0 if and only if π(A) = 0, A ∈ A∞ . Furthermore, in view of the continuity of the action α on A and A Q , the inclusions ker π A∞ ⊂ ker π and ker π Q A∞ ⊂ ker π Q are dense in the norms · and · Q , respectively. Thus it follows that π Q is faithful if and only if π is faithful. Theorem 2.8. Let (π, H) be an α–covariant representation of the C ∗ –algebra A. The homomorphism π Q : A∞ → B(H), fixed by the relation . π Q (A) = π(A) Q , A ∈ A∞ , extends continuously to an α–covariant representation of the C ∗ –algebra (A Q , × Q ). Moreover, π Q is faithful if and only if π is faithful. So the warping method provides a representation of the deformed algebras in the same Hilbert space as the undeformed algebra, enabling the direct comparison of deformed operators corresponding to different Q. This point will prove to be useful in the physical context treated below.
2.3. Further properties of warped convolutions. Even though the warped convolutions may be viewed as merely generating certain specific representations of Rieffel algebras, it will be advantageous to base the subsequent discussion directly on them without referring to the Rieffel setting. The reasons for this are threefold: (a) It will be necessary to deal with subalgebras of the algebra of smooth operators which are not invariant under the automorphic action of the translations. So there is no corresponding Rieffel algebra, but the warping procedure is still meaningful. (b) It will be necessary to consider warped operators A Q , AQ and their sums and products for different matrices Q, Q . Such operations can be carried out in the framework of warped convolutions more easily than in the Rieffel setting, where one has to use Hilbert modules instead of Hilbert spaces. (c) We shall need to establish algebraic properties of the warped operators arising from spectral properties of the unitary representation U , which are not available in the Rieffel setting. Returning to the Hilbert space framework, we first exhibit some general covariance properties of the warped convolutions, cf. [13]. To this end we consider (anti)unitary operators V whose adjoint actions on the translations U induce linear transformations of Rn . It follows at once that for any such V the algebra C ∞ is stable under the corresponding adjoint action, V C ∞ V −1 = C ∞ , and V D = D. The following result is the first instance where we must deal with warped convolutions for different choices of the underlying matrix Q.
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
109
Proposition 2.9. Let V be a unitary or antiunitary operator on H such that V U (x)V −1 = U (M x), x ∈ Rn , for some invertible matrix M. Then, for A ∈ C ∞ , V A Q V −1 = (V AV −1 ) σ M Q M T , where M T is the transpose of M with respect to the chosen bilinear form, σ = 1 if V is unitary and σ = −1 if V is antiunitary. Proof. Making use of relation (2.4) for real f , one commences from the equalities of strong integrals V d xd y e−i x y f (εx, εy) α Qx (A) U (y) V −1 = d xd y e−iσ x y f (εx, εy) α M Qx (V AV −1 ) U (M y) = d xd y e−i x y f (εσ M T x, εM −1 y) α σ M Q M T x (V AV −1 ) U (y) , where the last equality is obtained by substituting (x, y) → (σ M T x, M −1 y). Applying these relations to any vector ∈ D and taking into account V −1 D = D, the assertion follows in the limit of small ε.
Next, we establish a result which is fundamental for the applications to physics. We shall show that the warped convolutions preserve certain specific commutation properties of the operators in C ∞ for appropriate choices of the underlying skew symmetric matrices depending on the spectrum of the representation U [13]. Proposition 2.10. Let A, B ∈ C ∞ be operators such that [α Qx (A), α−Qy (B)] = 0 for all x, y ∈ sp U . Then [A Q , B−Q ] = 0. Proof. Returning to the definition of the warped convolutions by the spectral calculus and making use of Lemma 2.2, one finds for vectors , with compact spectral support
, A Q B−Q = lim , (A) (B)F d E(y) , d E(x)Fα α Qx −Qy F,F 1
where F, F are finite–dimensional projections. Now
, α−Qy (B)F d E(y) d E(x)Fα Qx (A) = , d E(x)Fα Qx (A) α−Qy (B)F d E(y) = , d E(x)Fα−Qy (B) α Qx (A)F d E(y) , where the step from the first to the second line is justified by the fact that the given expression can be decomposed into a finite sum of product measures multiplied with smooth functions. The second step is a consequence of the commutativity properties of A and B. Introducing the notation u = (v, w, x, y) ∈ R4n and picking any test function
110
D. Buchholz, G. Lechner, S. J. Summers
u → h(u) which is equal to 1 at 0, it follows from the spectral representation of U that the latter integral is equal to du h(εu) e−ivx−i yw , U (v)Fα−Qy (B) α Qx (A)F U (w). lim (2π )−2n ε→0
Adopting now the arguments and notation in the final part of the proof of Lemma 2.4, one finds that for the polynomial L (2.5) there exists a corresponding polynomial P such that du h(εu) e−ivx−i yw , U (v)Fα−Qy (B) α Qx (A)F U (w) = du e−ivx−i yw P(∂) h(εu) L(u)−1 , U (v)Fα−Qy (B) α Qx (A)F U (w). After having performed the differentiations in the last integral, one sees by an application of the dominated convergence theorem that the composite limit ε → 0, F, F 1 is independent of the order in which the individual limits are carried out and also does not depend on the choice of h. Thus one has, in particular, , A Q B−Q = lim (2π )−2n
ε→0
du h(εu) e−ivx−i yw , U (v)α−Qy (B) α Qx (A)U (w).
As before, one takes advantage of the fact that the integration may be restricted to the submanifold (ker Q)⊥ × · · · × (ker Q)⊥ ⊂ R4n , since the remaining integrals merely produce factors of 2π . So the preceding integral can be recast as du e−ivx−i yw h(εu) , U (v)α−Qy (B) α Qx (A)U (w) = du e−ivx−i yw h(εu) , U (w)α−Qy+v−w (B) α Qx+v−w (A)U (v) = du e−ivx−i yw k(εu) , U (w)α−Qy (B) α Qx (A)U (v), where k(v, w, x, y) = h(v, w, x − Q −1 (v − w), y + Q −1 (v − w)). The last equality is the result of the substitution (x, y) → (x − Q −1 (v − w), y + Q −1 (v − w), under which e−ivx−i yw does not change because of the skew symmetry of Q. Proceeding to the limit of small ε, one obtains by relation (2.4) and Lemma 2.2, du e−ivx−i yw k(εu) , U (w)α−Qy (B) α Qx (A)U (v) lim (2π )−2n ε→0
= , B−Q A Q . This shows that , A Q B−Q = , B−Q A Q . Since , were arbitrary elements of a dense set of vectors, the assertion now follows.
We finally discuss the structure of the family of maps given by the warped convolutions. According to Proposition 2.7 (ii), these maps act bijectively on C ∞ and therefore can be composed and have inverses. In fact, they form a group which is homomorphic to Rn(n−1)/2 , as can be seen from the next proposition.
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
111
Proposition 2.11. Let Q 1 , Q 2 be skew symmetric matrices. Then (A Q 1 ) Q 2 = A Q 1 +Q 2 , A ∈ C ∞ . Proof. To begin, note that for any continuous bounded function f of the generator P of U one has A Q f (P) = (A f (P)) Q , as a consequence of relation (2.3) and part (iv) of Lemma 2.1. Let ∈ D be any vector with compact spectral support and let f be a test function such that f (P) = . It follows that (A Q 1 ) Q 2 = (A Q 1 f (P)) Q 2 f (P). Picking nets of finite–dimensional projections F, F converging to 1, making use of the spectral calculus, which implies f (P) d E(z) = f (z) d E(z), z ∈ Rn , and recalling the definition of the warped convolutions, one obtains in the sense of weak convergence (A Q 1 ) Q 2 = lim f (x) f (y) α Q 1 x+Q 2 y (A) Fd E(x)F d E(y). F 1,F 1
Here the limits are taken in the given order and the (strong) limit F 1 has been interchanged with the y–integration by an application of the dominated convergence theorem. Since the function x, y → f (x) f (y)α Q 1 x+Q 2 y (A) is smooth and rapidly decreasing in norm, one can interchange the limits. The product measure d E(x)F d E(y) converges weakly in the sense of distributions to δ(x − y) d xd E(y) as F 1, where δ(x − y) d x is the Dirac measure at y; hence one obtains (A Q 1 ) Q 2 = lim α(Q 1 +Q 2 )x (A) Fd E(x) f (P)2 = A Q 1 +Q 2 . F 1
The desired conclusion then follows, because the space of vectors with compact spectral support is dense in H.
Note that this result does not entail a composition law of the representations π Q of the Rieffel algebras, since their ranges do not, in general, fit with their respective domains. Further Results. Most of the preceding results can be established in a setting of unbounded operators. One proceeds again from a continuous unitary representation U of Rn and considers the ∗ –algebra F of all operators F for which there is some n F ∈ N such that the functions x → (1 + P 2 )−n F αx (F)(1 + P 2 )−n F are arbitrarily often differentiable in norm. The operators F ∈ F are defined on the domain D and leave it invariant. Making use of the fact that there is a version of Lemma 2.1 in this setting, one can define the Rieffel product × Q on F; the warped convolutions of the elements of F can be defined as well and are elements of F. Moreover, Lemmas 2.2 and 2.4 hold without changes, so the warped convolutions define an (unbounded) ∗ –representation of (F, × Q ), and Propositions 2.9, 2.10 and 2.11 hold as well. We refrain from giving the proofs here. 3. Warped Convolutions and Borchers Triples We consider now warped convolutions in the context of Borchers triples, invented by Borchers [3] for the construction and analysis of relativistic quantum field theories. This setting is, on the one hand, more restrictive than the preceding one, since one deals with unitary representations U of the translations Rn , n ≥ 2, with certain specific spectral properties. On the other hand, one considers subalgebras of B(H) on which the adjoint action α of U merely induces endomorphisms for semigroups of translations in the set . W = {x = (x0 , x1 , . . . , xn−1 ) ∈ Rn : x1 ≥ |x0 |}.
112
D. Buchholz, G. Lechner, S. J. Summers
Definition 3.1. A Borchers triple (R, U, ) (relative to W) consists of (a) a von Neumann algebra R ⊂ B(H), (b) a weakly continuous unitary representation U of Rn on H whose spectrum is containedin the closed forward light cone V+ = { p = ( p0 , p1 , . . . , pn−1 ) ∈ Rn : 2 } and which satisfies α (R) ⊂ R, x ∈ W, p0 ≥ p12 + · · · + pn−1 x (c) and a unit vector ∈ H which is invariant under the action of U and is cyclic and separating for R.
By condition (c), Tomita–Takesaki theory [30,31] is applicable to the pair (R, ), and we shall denote by , J the associated modular operator and involution. In this context Borchers [3] proved the following remarkable theorem (see [15] for a simpler proof). Theorem 3.2. Let (R, U, ) be a Borchers triple relative to W. Denoting by ϑ(t), t ∈ R, and j the transformations acting on x = (x0 , x1 , . . . xn−1 ) ∈ Rn by . ϑ(t) x = (cosh(2π t)x0 + sinh(2π t)x1 , sinh(2π t)x0 + cosh(2π t)x1 , x2 , . . . , xn−1 ) , . j x = (−x0 , −x1 , x2 , . . . xn−1 ) , one has (i) it U (x) −it = U (ϑ(t)x) for x ∈ Rn and t ∈ R, (ii) J U (x)J = U ( j x) for x ∈ Rn . Moreover, (R , U, ) is a Borchers triple relative to −W, where R = J RJ is the commutant of R. Proof. The assertion for n = 2 is proven in [3]. Setting x⊥ = (0, 0, x2 , . . . , xn−1 ), conditions (b), (c) in Definition 3.1 imply U (x⊥ )RU (x⊥ )−1 = R and U (x⊥ ) = . The uniqueness of the modular objects then entails that and J both commute with all U (x⊥ ), completing the proof in the general case.
We shall show now that the family of Borchers triples is stable under the deformations induced by warped convolutions corresponding to certain specific choices of the skew symmetric matrix5 Q. Moreover, the modular objects of the deformed triples coincide with those of the original one. This observation is of relevance in quantum field theory, which will be discussed at the end of this section. We begin with some technical remarks. Let C ∞ be, as above, the ∗ –algebra of all ∞ smooth elements in B(H) under the adjoint action of the translations and let R = R C ∞ . In view of condition (b) in Definition 3.1, one obtains elements of R∞ by smoothing any element R ∈ R with Schwartz test functions f having support in W, . R( f ) = d x f (x) αx (R). (3.1) These weak integrals are elements of R∞ since, by construction, they are smooth and contained in the von Neumann algebra R. Choosing sequences f n of test functions with support in W which approximate the Dirac measure at 0, one sees that R∞ is dense in R in the strong operator topology, and consequently is cyclic for R∞ . By the same . ∞ reasoning one finds that is also cyclic for R = R C∞. 5 Having in mind applications to quantum field theory, we choose henceforth the Lorentz product x y = n n x0 y0 − n−1 m=1 x m ym , x, y ∈ R , as the bilinear form on R .
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
113
Now let Q be any real skew symmetric matrix on Rn which is W–compatible in the sense that QV+ ⊂ W. This constraint on Q will become important in the following. The corresponding warped operators A Q , A ∈ R∞ , are defined as in the previous section. Since they are bounded and satisfy A Q ∗ = A∗ Q , they generate a von Neumann algebra, called a warped algebra for short. With a slight abuse of notation, we write . R Q = {A Q : A ∈ R∞ } . For the proof that the warped triple (R Q , U, ) is again a Borchers triple, we note that, as a consequence of Proposition 2.9, one has αx (R Q ) = αx (R) Q ⊂ R Q for x ∈ W. So condition (b) in Definition 3.1 is satisfied. Furthermore, since is cyclic for R∞ , it is also cyclic for R Q as a consequence of Eq. (2.7). In order to see that is separating for R Q , let A ∈ R∞ , A ∈ R ∞ . Then [αx (A), α y (A )] = 0 for x ∈ W, y ∈ −W, and taking into account that Q sp U ⊂ Q V+ ⊂ W, it follows from Proposition 2.10 that [A Q , A −Q ] = 0. Thus (R )−Q ⊂ (R Q ) . But Eq. (2.7) implies that is cyclic for (R )−Q and thus a fortiori for (R Q ) . Hence is separating for R Q , and condition (c) in Definition 3.1 holds as well. Theorem 3.3. Let (R, U, ) be a Borchers triple relative to W and let Q be W–compatible. Then the resulting warped triple (R Q , U, ) is also a Borchers triple relative to W. In view of this theorem, we may apply modular theory to the warped triple. We shall show next that the corresponding modular objects coincide with the original ones. To this end we need the following technical lemma. Lemma 3.4. Let (R, U, ) be a Borchers triple relative to W and let S = J 1/2 be the corresponding Tomita conjugation given by the closure of the map S A = A∗ , A ∈ R. Then the subdomain R∞ is a core for S. Proof. Let R ∈ R and let f n be a sequence of real test functions with support in W such that in the sense of strong convergence limn R( f n ) = R , cf. relation (3.1) and the remarks thereafter. Since R( f n ) ∈ R∞ ⊂ R and lim S R( f n ) = lim R( f n )∗ = lim R ∗ ( f n ) = R ∗ = S R , n
n
n
the conclusion follows, because R is a core for S by definition.
We are now in a position to establish the invariance of the modular objects of Borchers triples under the warping procedure. Theorem 3.5. Let (R, U, ) be a Borchers triple relative to W with modular objects , J , and let Q be a W–compatible matrix. Then the modular objects Q , J Q associated with the warped triple (R Q , U, ) coincide with those of the original triple, i.e. Q = , J Q = J.
114
D. Buchholz, G. Lechner, S. J. Summers
Proof. Let S Q be the Tomita conjugation associated with the warped triple (R Q , U, ) and let S be the Tomita conjugation associated with (R, U, ). Since A Q ∈ R Q for A ∈ R∞ , Eq. (2.7) and Lemma 2.2 imply S Q A = S Q A Q = (A Q )∗ = (A∗ ) Q = A∗ = S A . According to the preceding lemma, R∞ is a core for S, hence S Q ⊃ S. By the Tomita–Takesaki theory [30,31], the adjoint S Q ∗ of S Q is theS Tomita conjugation associated with ((R Q ) , U, ), and similarly S ∗ is the Tomita conjugation associated with (R , U, ). It was shown in the proof of Theorem 3.3 that (R )−Q ⊂ (R Q ) . Thus, as A −Q ∈ R −Q for A ∈ R ∞ , one obtains by another application of Eq. (2.7) and Lemma 2.2, S Q ∗ A = S Q ∗ A −Q = (A −Q )∗ = (A ∗ )−Q = A ∗ = S ∗ A . By the preceding lemma R ∞ is a core for S ∗ , hence S Q ∗ ⊃ S ∗ and consequently S ⊃ S Q , since both conjugations are closed operators. Thus S Q = S and, by the uniqueness of the polar decomposition, the desired conclusion follows.
An immediate consequence of this theorem is the observation that R Q = R −Q .
(3.2)
Indeed, Theorem 3.2 and Proposition 2.9 imply J R Q J = (J RJ )− j Q j and it is also straightforward to verify that j Q j = Q for any W–admissible matrix Q. Since J = J Q , the asserted equation then follows from Tomita–Takesaki theory. Let us discuss now the physical significance of these findings. As was pointed out in [3], Theorem 3.2 allows one to use the Borchers triple (R, U, ) as a building block for the construction of a quantum field theory in two spacetime dimensions. Identifying the cone W ⊂ R2 defined above with the corresponding wedge shaped region in two– . dimensional Minkowski space, one interprets A(W) = R as the algebra generated by observables which are localized in W. Moreover, noticing that the transformations ϑ(t), t ∈ R, and j introduced in Theorem 3.2 have the geometrical meaning of Lorentz boosts and spacetime reflection, respectively, one can consistently extend the representation U of the translations R2 to a continuous (anti)unitary representation of the proper Poincaré group P+ . It is given by . U (λ) = U (x)J σ it , λ = (x, j σ ϑ(t)) ∈ P+ , where x ∈ R2 , t ∈ R and σ ∈ {0, 1}. Thus J represents the PCT–operator. With the help of this representation one can define the algebras generated by observables in the transformed wedge regions λW, λ ∈ P+ by setting . A(λW) = U (λ)RU (λ)−1 , λ ∈ P+ . This definition is consistent, since the stability group of the wedge W in P+ consists of the boosts ϑ(t), t ∈ R, whose corresponding automorphic action leaves the algebra R invariant according to Tomita–Takesaki theory. The resulting assignment W. → A(W. ) of wedge regions to algebras defines a net (pre–cosheaf) on R2 . It is Poincaré covariant by construction and causal. In fact, since j maps the wedge W onto its spacelike complement W = −W, one has A(W ) = U ( j)A(W)U ( j)−1 = J RJ = R = A(W) ,
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
115
where the third equality follows from Tomita–Takesaki theory. So the observables in spacelike separated wedges commute, in accordance with the principle of Einstein causality. In this way any Borchers triple defines a relativistic quantum field theory in two spacetime dimensions, cf. [3] for more details. The upshot of these considerations is the insight that, as a consequence of the preceding three theorems, the warped triples (R Q , U, ) generate in the same manner another causal and covariant net W. → A Q (W. ) by setting . A Q (λW) = U (λ)R Q U (λ)−1 , λ ∈ P+ . Thus the warping procedure provides a tool for the consistent deformation of two– dimensional quantum field theories without changing the underlying representation of the Poincaré group. We shall further elaborate on this observation in the next section. 4. Warped Convolutions in Quantum Field Theory In this section we examine applications of the warping procedure to relativistic quantum field theories in more than two spacetime dimensions. Thus we interpret Rn , n > 2, as Minkowski space equipped with the standard metric fixed by the Lorentz product, cf. footnote 5. The identity component of its isometry group, the Poincaré group, is the ↑ ↑ semidirect product P+ = Rn L+ of the spacetime translations Rn and the proper ↑ orthochronous Lorentz transformations L+ . In a manner similar to the preceding section, we describe the theories in the algebraic setting of local quantum physics [20] by a qualified version of the concept of Borchers triple. Additional constraints arise since, on the one hand, the group generated by the translations, along with the boosts and reflection emerging from the modular structure of the triple, does not act transitively on the set of wedge regions in Rn if n > 2. ↑ The smallest subgroup of the Poincaré group which fulfills this condition is P+ . So one needs from the outset an action of this group on the underlying algebra R, which one interprets again as the algebra of observables localized in the given wedge region . W = {x = (x0 , x1 , . . . , xn−1 ) ∈ Rn : x1 ≥ |x0 |}. On the other hand, one must ensure that this action is consistent with the principle of Einstein causality, according to which observables in spacelike separated regions must commute. The resulting consistency conditions can be expressed in terms of the triple in an evident manner, cf. [2, Prop. 7.3.22]. They lead to the concept of a causal Borchers triple. Definition 4.1. A causal Borchers triple (R, U, ) relative to W consists of (a) a von Neumann algebra R ⊂ B(H), ↑ ↑ (b) a weakly continuous unitary representation U of P+ such that, λ ∈ P+ , U (λ)RU (λ)−1 ⊂ R if λW ⊂ W , U (λ)RU (λ)−1 ⊂ R if λW ⊂ W , and the spectrum of the abelian subgroup U Rn of the spacetime translations is contained in the closed forward lightcone V+ , (c) and a unit vector ∈ H, describing the vacuum, which is invariant under the action of U and is cyclic and separating for R.
116
D. Buchholz, G. Lechner, S. J. Summers
Remark. In two spacetime dimensions any Borchers triple determines a causal Borchers triple by the modular construction in the preceding section. As there is no element in ↑ P+ which maps the wedge W into its spacelike (causal) complement W , the second constraint in condition (b) is trivially satisfied in this case. In order to flip the wedge one ↑ needs the spacetime reflection j, which is an element of P+ ⊃ P+ . As we have seen, its corresponding action on R is consistent with Einstein causality as a consequence of modular theory. In higher dimensions one either has to posit causality from the outset, as we do, or one has to impose additional constraints on the modular structure of the triple which imply it, cf. [4,5,7,10,19]. With the above input one can define the algebras corresponding to arbitrary regions in Rn in a straightforward manner, which we briefly recall. Making use of the fact that ↑ P+ acts transitively on the wedge regions, one begins with the wedge algebras by setting . ↑ A(λW) = U (λ)RU (λ)−1 , λ ∈ P+ .
(4.1)
This definition is consistent, since λ1 W = λ2 W implies that the transformation λ−1 2 λ1 is an element of the stability group of W, and R is stable under the adjoint action of the corresponding unitary operators according to the first part of condition (b). Simi−1 −1 ⊂ R, hence A(λ W) ⊂ larly, if λ1 W ⊂ λ2 W, it follows that U (λ−1 1 2 λ1 )RU (λ2 λ1 ) A(λ2 W). Thus the family of wedge algebras complies with the condition of isotony. The wedge algebras also transform covariantly under the adjoint action of the representation −1 −1 U by their very definition. Moreover, if λ1 W ⊂ (λ2 W) , then U (λ−1 2 λ1 )RU (λ2 λ1 ) ⊂ R according to the second part of condition (b). Hence A(λ1 W) ⊂ A(λ2 W) in accordance with Einstein causality. The algebras corresponding to arbitrary causally closed convex regions O ⊂ Rn are determined from the wedge algebras A(W· ) by setting A(O) = W· ⊃O A(W· ). It is apparent that the resulting assignment O → A(O) inherits the structure of a causal and covariant net on Rn , i.e. of a local quantum theory [20]. It should be noted, however, that within the present general framework the algebras corresponding to bounded regions may happen to be trivial. We shall comment on the physical significance of this possibility at the end of this section. We now want to use our warping procedure to deform causal Borchers triples. Additional constraints on the underlying skew symmetric matrices arise due to the extra conditions imposed on such triples. In fact, Q must have the following form with respect to the coordinates chosen in the specification of the wedge W ⊂ Rn : ⎛
⎞ ··· 0 ··· 0⎟ ··· 0⎟ ⎟ . . .. ⎟ . .⎠ 0 0 0 ··· 0
0 ⎜ζ . ⎜0 Q=⎜ ⎜. ⎝ ..
ζ 0 0 .. .
0 0 0 .. .
(4.2)
for fixed ζ ≥ 0. In the special but physically most interesting case of n = 4 dimensions, one can admit matrices of the more general form ⎛
0 . ⎜ζ Q=⎝ 0 0
ζ 0 0 0 0 0 0 −η
⎞ 0 0⎟ η⎠ 0
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
117
for fixed ζ ≥ 0, η ∈ R. Note that these matrices are skew symmetric with respect to the Lorentz product. The following facts pointed out in [17] are crucial for the consistent deformation of the triples and, in turn, determine the choice of the admissible matrices Q [17, Lemma 2]. (i) Q V+ ⊂ W. ↑ (ii) Let λ = (x, ) ∈ P+ be such that λW ⊂ W. Then QT = Q. ↑ (iii) Let λ = (x, ) ∈ P+ be such that λ W ⊂ W . Then QT = −Q. Any matrix Q with these properties is said to be W–admissible (qualifying the notion of W–compatibility introduced in the preceding section). Given a causal Borchers triple (R, U, ) relative to W, we proceed as in the preceding section and define for fixed W–admissible matrix Q the warped von Neumann algebra . R Q = {A Q : A ∈ R∞ } . The corresponding warped triple (R Q , U, ) is again a causal Borchers triple. For the proof of this fact we make use of Proposition 2.9, according to which ↑
U (λ)A Q U (λ)−1 = (U (λ)AU (λ)−1 )QT , λ = (x, ) ∈ P+ , for all A ∈ C ∞ . Taking into account properties (ii) and (iii) of Q given above, we conclude that U (λ) R Q U (λ)−1 = (U (λ) R U (λ)−1 ) Q ⊂ R Q if λW ⊂ W , U (λ) R Q U (λ)−1 = (U (λ) R U (λ)−1 )−Q ⊂ (R )−Q if λW ⊂ W . But from Eq. (3.2) one has (R )−Q = (R Q ) ; hence the warped triple satisfies condition (b) in Definition 4.1. In the proof of Theorem 3.3, it was shown that is cyclic and separating for R Q , so the triple also complies with condition (c). Theorem 4.2. Let (R, U, ) be a causal Borchers triple relative to W and let Q be a W–admissible matrix. The corresponding warped triple (R Q , U, ) is again a causal Borchers triple relative to W. So the deformations induced by the warped convolutions are consistent with the basic principles of local quantum physics. It is noteworthy that also certain more specific features persist under these deformations, such as the physically significant property of wedge duality. This property can be encoded into a maximality condition on the Borchers triple, which implies that the underlying algebra cannot be enlarged without coming into conflict with causality. Definition 4.3. Let (R, U, ) be a causal Borchers triple relative to W. The triple is said ↑ to be maximally causal if U (λ)RU (λ)−1 = R for any λ ∈ P+ such that λW = W . It immediately follows from the definition of the wedge algebras that, under these circumstances, A(W· ) = A(W· ) for all wedges W· , i.e. wedge duality obtains. Proposition 4.4. Let (R, U, ) be a maximally causal Borchers triple relative to W and let Q be a W–admissible matrix. Then the corresponding warped Borchers triple (R Q , U, ) is also maximally causal.
118
D. Buchholz, G. Lechner, S. J. Summers ↑
Proof. Let λ ∈ P+ with λW = W . Then property (iii) of the W–admissible matrix Q, Proposition 2.9 and the maximality condition imply that U (λ) R Q U (λ)−1 = (U (λ) R U (λ)−1 )−Q = R −Q . Equation (3.2) completes the proof.
Let us turn now to the question whether the deformed Borchers triples generate new theories. It is apparent that equivalent triples, as defined below, give rise to isomorphic nets of observable algebras and therefore must be identified. Definition 4.5. Let (R1 , U1 , 1 ) and (R2 , U2 , 2 ) be two causal Borchers triples. The triples are equivalent if there exists an isometry V : H1 → H2 between the underly↑ ing Hilbert spaces such that V R1 = R2 V , V U1 (λ) = U2 (λ)V for all λ ∈ P+ , and V 1 = 2 . Note that the algebras encountered in Borchers triples are generically isomorphic to the unique hyperfinite factor of type III1 and hence to each other. Thus the nontrivial requirement in the definition is the condition that the isometry V intertwines, besides the algebras, the respective representations of the Poincaré group. Although one may expect that the warped Borchers triples are generally inequivalent to the original ones, there does not yet exist an argument to that effect. It has been shown in [13,17] that in theories describing massive particles the elastic scattering matrix changes under these deformations, thereby providing a rather indirect proof that the respective Borchers triples must be inequivalent. We present here an alternative argument, covering a large family of theories in more than two spacetime dimensions. It is based on the following lemma, whose proof is given in the Appendix. There we also comment on the additional physically meaningful spectral constraint on the translations made in the hypothesis. Lemma 4.6. Let (R, U, ) be a causal Borchers triple relative to W ⊂ Rn , n ≥ 3, such that sp U Rn contains some point in the interior of V+ and let Q = 0 be a W–admissible generic form (4.2). Then is cyclic for at most one of the matrix of the algebras λ∈N αλ (R) and λ∈N αλ (R Q ), where N is any given neighborhood of the ↑ identity in P+ . The following observation about the relation between Borchers triples and their warped descendants is an immediate consequence of this result. Proposition 4.7. Let (R, U, ) be a causal Borchers triple relative to W ⊂ Rn , n ≥ 3, such that sp U Rn contains some point in the interior of V+ and let be cyclic for ↑ λ∈N αλ (R) for some neighborhood N of the identity in P+ . Then (R Q , U, ) and (R, U, ) are inequivalent for any W–admissible matrix Q = 0 of the generic form (4.2). Proof. Let V be some unitary operator which intertwines the two triples. Then V λ∈N αλ (R) V −1 = λ∈N αλ (V R V −1 ) = λ∈N αλ (R Q ). Hence = V is cyclic for the latter algebra as well, in conflict with the preceding lemma.
In the familiar examples of quantum field theories which have been rigorously constructed so far, such as (generalized) free field theories in physical spacetime and interacting field theories in lower dimensions [16], the vacuum is known to be cyclic for
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
119
the algebras affiliated with compact spacetime regions (Reeh–Schlieder property). Thus, applying the warping procedure to the corresponding Borchers triples, one ends up with inequivalent, i.e. new theories. However, the local algebras in the deformed theories no longer have the Reeh–Schlieder property, according to the preceding lemma. In fact, even for the algebras affiliated with pointed spacelike cones, which are of relevance in gauge theory [8], is not cyclic. Thus in more than two spacetime dimensions the warped algebras can, in general, not be interpreted in terms of some underlying point fields. Yet, as was pointed out in [13], the warped theories admit a meaningful physical interpretation with respect to noncommutative Minkowski space (Moyal space). In fact, the first examples of such theories appeared in that setting [17]. We recall that noncommutative Minkowski space is described by coordinate operators X μ satisfying the commutation relations [X μ , X ν ] = i θμν 1, where θμν = −θνμ are real constants, μ, ν = 0, 1, . . . , n − 1. It is straightforward to verify that in more than two dimensions there always exist certain lightlike coordinates X ± which commute and thus can be simultaneously diagonalized. Hence it should be possible to localize fields and observables with respect to these coordinates, thereby dislocalizing them in the remaining ones. In particular, the wedges W considered here are possible localization regions in noncommutative Minkowski space, whereas bounded regions and pointed spacelike cones are not. On the basis of this interpretation, the algebras corresponding to the latter regions are expected to be trivial, in line with the preceding lemma. Now, apart from the wedges, there are other cylindrical regions (such as the intersections of opposite wedges) which are possible localization regions. It is therefore an intriguing question whether the corresponding algebras in the warped theories are nontrivial. An affirmative answer would support their interpretation in terms of noncommutative Minkowski space. We hope to return to this problem elsewhere. 5. Conclusions In this investigation we have clarified the relation between the warped convolution of C ∗ –dynamical systems, proposed in [13], and the strict deformation of such systems, established by Rieffel [28]. It turned out that, for fixed deformation matrix Q, the warped convolution induces a faithful covariant representation of the corresponding Rieffel algebra, if the original dynamical system is given in a faithful covariant representation. Thus, from this point of view, the warped convolution provides little new information. Yet, whereas the Rieffel deformations were introduced for the purpose of quantizing classical systems with Poisson bracket given by a fixed Q, warped convolutions were conceived for the deformation of quantum field theories. Within the latter framework one must deal simultaneously with a multitude of different deformation matrices Q and establish relations between the resulting operators. The warping procedure is more appropriate in this context, since all warped deformations of a given dynamical system are concretely presented in a single Hilbert space, irrespective of the choice of Q. For the discussion of the field theoretic aspects it has proven to be convenient to make use of the concept of causal Borchers triples (R, U, ). The algebras of observables attached to arbitrary regions in Minkowski space can be reconstructed from any such triple, thereby specifying a covariant and causal quantum theory. Within this setting the problem of constructing a theory thus presents itself as follows. One first has to devise a continuous unitary representation U of the Poincaré group on some Hilbert
120
D. Buchholz, G. Lechner, S. J. Summers
space which satisfies the relativistic spectrum condition with vacuum vector . This task can be accomplished, e.g., by specifying the stable particle content of the theory and performing the standard Fock space construction. In a second step one must exhibit a von Neumann algebra R on this space satisfying certain specific compatibility conditions with respect to the action of U , which allow one to interpret R as an algebra of observables localized in a given wedge region of Minkowski space. It should be noted that the nets of local observable algebras appearing in any quantum field theory can be realized in this way. Disregarding systems with an unreasonably large number of local degrees of freedom, the algebraic structure of R is known to be model independent, i.e. the algebras corresponding to different theories are isomorphic [6]. One may thus take as prototype the von Neumann algebra R0 generated by free (non–interacting) fields on Fock space which are smeared with test functions having support in the given wedge region. Despite this concrete setting, the problem of identifying other proper examples of such algebras R is notoriously difficult. The strategy pursued in the present investigation is based on the general idea of deforming a given causal Borchers triple, such as (R0 , U, ), without changing the representation U . The warping procedure provides a consistent method to that effect. It leads to a family of new examples of causal Borchers triples in any number of spacetime dimensions. However, the deformations of Borchers triples obtained by the warping procedure are rather special and of limited physical interest. It therefore seems worthwhile to fathom the potential of the general idea underlying this construction. Since the representation ↑ U of P+ induces the pertinent constraints on the admissible algebras R, one may try to generalize the formula for the warped deformations by the ansatz . A= dλ dλ K (λ, λ ) αλλ (A0 ) L(λ, λ ) , A0 ∈ R0 , ↑
where dλ denotes the Haar measure on P+ (or a subgroup thereof) and K , L are suitable operator valued kernels. The consistency conditions on the algebra R generated by the deformed operators can then be re-expressed in terms of transformation properties of these kernels under the adjoint action of the representation U . In two spacetime dimensions these constraints simplify considerably. There it suffices if the kernels K , L transform covariantly under the adjoint action of the unitary ↑ representation U of P+ and is cyclic and separating for the resulting deformed von Neumann algebra R. One may then proceed as in Sect. 3 and extend the representation U to a representation of P+ by adding to it the modular conjugation associated with (R, ) which can be interpreted as PCT–operator. The algebras corresponding to arbitrary wedges can be obtained from R by the adjoint action of the resulting (anti)unitary representation of P+ . Indeed, there is evidence that a large family of integrable models on two–dimensional Minkowski space, considered by one of us, can be subsumed in this manner [24,25]. The prospect of finding other interesting deformations of this kind also in higher spacetime dimensions seems promising. Moreover, the method can also be transferred to quantum field theories on curved spacetimes having a sufficiently large isometry group [14]. Thus the algebraic methods presented here shed new light on the yet unsolved constructive problems in relativistic quantum field theory. Acknowledgements. GL wishes to thank S. Waldmann for interesting discussions about Rieffel deformations.
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
121
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendix We give here the proof of Lemma 4.6, which concludes that, given any neighborhood ↑ N of the identity in P+ , is cyclic for at most one of the algebras λ∈N αλ (R) and λ∈N αλ (R Q ). Moreover, we comment on the significance of the spectral constraint made in the hypothesis of the lemma. We begin by noting that it suffices to establish the assertion for arbitrarily small ↑ neighborhoods N of the identity in P+ ; for it then holds for all bigger neighborhoods as ↑ P+ is a rotation by well. In particular, one may assume that λ0 N λ−1 0 = N , where λ0. ∈ π which maps W onto W . Assume now that is cyclic for S = λ∈N αλ (R) ⊂ R and let A ∈ S C ∞ . Then for any λ ∈ N one has αλ−1 (A) ∈ R∞ , so the warped operators αλ−1 (A) Q are well–defined and αλ (αλ−1 (A) Q ) ∈ αλ (R Q ). By Proposition 2.9 αλ (αλ−1 (A) Q ) = AQT , where is the image of λ under the canonical homomor↑
↑
phism mapping P+ onto L+ . Hence AQT ∈ αλ (R Q ), λ ∈ N . Assume now, for a reductio ad absurdum, that is also cyclic for λ∈N αλ (R Q ). Then, since
λ∈N
αλ (R Q )
=
λ∈N
αλ (R Q ) ⊃
⎛ αλ (αλ0 (R Q )) = αλ0 ⎝
λ∈N
⎞ αλ (R Q )⎠ ,
λ∈N0
where the inclusion obtains because (R Q , U, ) is a causal Borchers triple, one con cludes that is separating for λ∈N αλ (R Q ). But Eq. (2.7) entails AQT = A Q , and consequently AQT = A Q , λ ∈ N . Proposition 2.11 then yields AQT −Q = A = A Q−QT , λ ∈ N . By explicit computation one finds that the sums of matrices of the form QT − Q, λ ∈ N , include all multiples of Q. Hence Am Q = A, m ∈ Z, by another application of Proposition 2.11. The same is true for the smooth operators . A ∈ T = αλ0 (S) ⊂ αλ0 (R) ⊂ R , as one sees by applying again Proposition 2.9. Pick now an arbitrary compact subset in the interior of the forward lightcone V+ , so Q is a compact subset in the interior of W. Hence for any given x, y ∈ Rn and sufficiently large m ∈ N, the wedges W + x + m Qu and W + y − m Qv lie spacelike to each other for all u ∈ and v ∈ V+ . As explained in Sect. 4, one therefore has for any A ∈ R, A ∈ R , the equality [αx+m Qu (A), α y−m Qv (A )] = 0. Now let A ∈ S C ∞ , A ∈ T C ∞ , let be any vector with spectral support with respect to U Rn contained in , and let be any other vector with compact spectral support. According to the preceding step and Proposition 2.9 one has αx (A) = αx (Am Q ) = (αx (A))m Q and similarly α y (A ) = (α y (A ))−m Q . So one obtains by the same line of arguments as in the proof of Proposition 2.10, , αx (A) α y (A ) = lim , (αx (A))m Q (α y (A ))−m Q m→∞
= lim , (α y (A ))−m Q (αx (A))m Q m→∞
= , α y (A ) αx (A) ,
122
D. Buchholz, G. Lechner, S. J. Summers
where in the second equality the support properties of and the above commutation properties of A, A have been used. Thus, varying , within the above limitations, one arrives at E() [αx (A), α y (A )] = 0 for x, y ∈ Rn , where E( · ) denotes the spectral resolution of U Rn . ∞ This equality has been established for A ∈ S C ∞ and A ∈ T C . But if N is sufficientlysmall, the algebra S is mapped into itself by all translations in the open convex cone λ∈NW; appealing to the discussion following relation ∞ (3.1) allows one to conclude that S C ∞ is weakly dense in S and, similarly, T C is weakly dense ∈ T . Moreover, for any u, v ∈ Rn in T . So the equality holds for all A ∈ S and A there is a w ∈ Rn such that αw (T ) ⊃ αu (T ) αv (T ). (This follows from the the Poincaré covariance discussed in Sect. 4 and the geometry of wedge regions). Hence E() [A, T ] = 0 for A ∈ S and T ∈ y∈Rn α y (T ). Since is cyclic for S it is also cyclic for T = αλ0 (S). The spectral condition on U Rn therefore implies that the elements of y∈Rn α y (T ) are invariant under translations. In particular U (x) ∈ y∈Rn α y (T ), x ∈ Rn , cf. [1, Theorem 4.6]. Thus E() [A, U (x)] = 0, x ∈ Rn , and consequently E() A = 0, A ∈ S. It is then clear that E() = 0 for any compact subset in the interior of V+ . So the spectrum of U Rn is confined to the boundary of the lightcone V+ , i.e. there is no spectral point in its interior, contradicting the hypothesis of the lemma. This completes the proof of the lemma. Finally, let us discuss the significance of the assumption that the spectrum of U Rn intersects the interior of V+ . As a matter of fact, disregarding the trivial case sp U = {0}, this input is a consequence of the additivity of the energy–momentum spectrum, which can be established in the present setting if is (apart from a phase) the only unit vector in the underlying Hilbert space which is invariant under translations [20, Chap. II.5.4]. The possibility that sp U consists of the boundary of V+ (and thus is not additive) can only be realized in theories where the Lorentz symmetry is spontaneously broken. With the help of one–dimensional chiral fields which one assigns to lightrays, one can manufacture such examples, and these are stable under the warping procedure. Since these examples seem to be merely of academic interest, we do not present them here. References 1. Araki, H.: Mathematical Theory of Quantum Fields. Oxford: Oxford University Press, 1999 2. Baumgärtel, H., Wollenberg, M.: Causal Nets of Operator Algebras. Berlin: Akademie Verlag, 1992 3. Borchers, H.-J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 15–332 (1992) 4. Borchers, H.-J.: On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41, 3604–3673 (2000) 5. Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759– 785 (2002) 6. Buchholz, D., D’Antoni, C., Fredenhagen, K.: The universal structure of local algebras. Commun. Math. Phys. 111, 123 (1987) 7. Buchholz, D., Dreyer, O., Florig, M., Summers, S.J.: Geometric modular action and spacetime symmetry groups. Rev. Math. Phys. 12, 475–560 (2000) 8. Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982) 9. Buchholz, D., Lechner, G.: Modular nuclearity and localization. Ann. Henri Poincaré 5, 1065–1080 (2004) 10. Buchholz, D., Summers, S.J.: An Algebraic characterization of vacuum states in Minkowski space. 3. Reflection maps. Commun. Math. Phys. 246, 625–641 (2004)
Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories
123
11. Buchholz, D., Summers, S.J.: Stable quantum systems in Anti-de Sitter space: Causality, independence and spectral properties. J. Math. Phys. 45, 4810–4831 (2004) 12. Buchholz, D., Summers, S.J.: String– and brane–localized causal fields in a strongly nonlocal model. J. Phys. A 40, 2147–2163 (2007) 13. Buchholz, D., Summers, S.J.: Warped convolutions: A novel tool in the construction of quantum field theories. In: Quantum Field Theory and Beyond, edited by Seiler, E., Sibold, K. Singapore: World Scientific, 2008, pp. 107–121 14. Dappiaggi, C., Lechner, G., Morfa-Morales, E.: Deformations of quantum field theories on spacetimes with Killing vector fields. Commun. Math. Phys. (2010). arXiv:1006.3548 (to appear) 15. Florig, M.: On Borchers’ theorem. Lett. Math. Phys. 46, 289–293 (1998) 16. Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, Berlin-Heidelberg-New York: Springer Verlag, 1987 17. Grosse, H., Lechner, G.: Wedge–local quantum fields and noncommutative Minkowski space. JHEP 0711, 012 (2007) 18. Grosse, H., Lechner, G.: Noncommutative deformations of Wightman quantum field theories. JHEP 0809, 131 (2008) 19. Guido, D.: Modular covariance, PCT, Spin and Statistics. Ann. Inst. Henri Poincaré 63, 383–398 (1995) 20. Haag, R.: Local Quantum Physics. Berlin, Heidelberg and New York: Springer Verlag, 1992 21. Kaschek, D., Neumaier, N., Waldmann, S.: Complete positivity of Rieffel’s quantization by actions of Rd . J. Noncommut. Geom. 3, 361–375 (2009) 22. Lechner, G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003) 23. Lechner, G.: On the existence of local observables in theories with a factorizing S-matrix. J. Phys. A 38, 3045–3056 (2005) 24. Lechner, G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008) 25. Lechner, G.: Article in preparation 26. Mund, J., Schroer, B., Yngvason, J.: String–localized quantum fields and modular localization. Commun. Math. Phys. 268, 621–672 (2006) 27. Pedersen, G.K.: C∗ –Algebras and Their Automorphism Groups. London-New York-San Francisco: Academic Press, 1979 28. Rieffel, M.A.: Deformation quantization for actions of Rd . Memoirs A.M.S. 506, 1–96 (1993) 29. Schroer, B.: Modular localization and the bootstrap–formfactor program. Nucl. Phys. B 499, 547– 568 (1997) 30. Takesaki, M.: Tomita’s Theory of Modular Hilbert Algebras and Its Applications. Berlin-Heidelberg-New York: Springer Verlag, 1970 31. Takesaki, M.: Theory of Operator Algebras. Volume II, Berlin-Heidelberg-New York: Springer Verlag, 2003 Communicated by Y. Kawahigashi
Commun. Math. Phys. 304, 125–174 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1211-3
Communications in
Mathematical Physics
The Spectral Action and Cosmic Topology Matilde Marcolli1 , Elena Pierpaoli2 , Kevin Teh1 1 Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA.
E-mail: [email protected]; [email protected]
2 Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA.
E-mail: [email protected] Received: 20 May 2010 / Accepted: 21 October 2010 Published online: 22 February 2011 – © Springer-Verlag 2011
In memory of Andrew Lange Abstract: The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincaré dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincaré homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces. Contents 1. 2. 3. 4.
5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . The Problem of Cosmic Topology . . . . . . . . . . . . 2.1 Laplace spectrum and cosmic topology . . . . . . . The Spectral Action and Cosmic Topology . . . . . . . 3.1 The spectral action functional . . . . . . . . . . . . Recalling the Case of S 3 . . . . . . . . . . . . . . . . . 4.1 Euclidean model . . . . . . . . . . . . . . . . . . . 4.2 The Poisson summation formula . . . . . . . . . . 4.3 The spectral action in 4-dimensions . . . . . . . . Slow-Roll Potential from the Spectral Action . . . . . . 5.1 Nonperturbative corrections and slow-roll potential 5.2 Slow-roll parameters . . . . . . . . . . . . . . . . 5.3 Slow-roll parameters for the S 3 -topology . . . . . . The Quaternionic Cosmology and the Spectral Action . 6.1 The Dirac spectra for SU (2)/Q8 . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
126 127 129 129 130 131 131 132 132 133 133 134 135 137 137
126
M. Marcolli, E. Pierpaoli, K. Teh
6.2 Trivial spin structure: nonperturbative spectral action . . . . 6.3 Nontrivial spin structures: nonperturbative spectral action . . 6.4 Slow-roll potential and parameters for the quaternionic space 7. Poincaré Homology Sphere: Dodecahedral Cosmology . . . . . . 7.1 Generating functions for spectral multiplicities . . . . . . . . 7.2 The Dirac spectrum of the Poincaré sphere . . . . . . . . . . 7.3 The double cover Spin(4) → S O(4) . . . . . . . . . . . . . 7.4 The spectral multiplicities . . . . . . . . . . . . . . . . . . . 7.5 The spectral action for the Poincaré sphere . . . . . . . . . . 7.6 Slow-roll potential in dodecahedral cosmologies . . . . . . . 8. Flat Cosmologies . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The spectral action on the flat tori . . . . . . . . . . . . . . . 9. Geometric Engineering of Inflation Scenarios via Dirac Spectra . 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Appendix: Lens Spaces, a False Positive . . . . . . . . . . . . . 11.1The trouble with the Dirac spectrum on lens spaces . . . . . 11.2Multiplicities, first case . . . . . . . . . . . . . . . . . . . . 11.3The spectral action and the Poisson formula . . . . . . . . . 11.4The other spectrum . . . . . . . . . . . . . . . . . . . . . . 11.5The spectral action for |D| . . . . . . . . . . . . . . . . . . 11.6The slow-roll potential: a false positive . . . . . . . . . . . . 11.7Lens spaces: a discrepancy . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
138 139 140 142 142 143 144 144 145 147 148 149 152 156 156 157 157 158 162 166 168 172 173
1. Introduction Noncommutative geometry provides models of particle physics, with matter Lagrangians coupled to gravity, based on an underlying geometry which is a product of an ordinary 4-dimensional (commutative) spacetime manifold by a small noncommutative space which determines the matter content of the model. The spectral action functional, which is defined for metric noncommutative spaces (spectral triples) is obtained as the trace of a cutoff of the Dirac operator of the spectral triple by a test function. The asymptotic expansion of the spectral action delivers a classical Lagrangian, which contains gravitational terms (Einstein–Hilbert, conformal gravity, cosmological term) and a coupled matter Lagrangian. For a suitable choice of the noncommutative space the latter recovers the Standard Model Lagrangian and some extensions with right-handed neutrinos, and more recently with supersymmetric QCD, see [6,10]. In trying to understand the cosmological implications of this model, one can work as in [26] with the asymptotic expansion of the spectral action functional, but this only delivers models of the very early universe, near the unification epoch. These can be potentially interesting, as the model contains different possible mechanisms of inflation, related to the presence of effective gravitational and cosmological constants. However, one cannot extrapolate that form of the model towards the modern universe, due to the possible presence of non-pertubative effects in the spectral action at lower energies. The spectral action in its non-perturbative form is typically very difficult to compute exactly. However, the recent result of [9] shows that, for a spacetime whose spatial sections are 3-spheres S 3 , Wick rotated and compactified to a Euclidean model S 3 × S 1 , the spectral action can be computed explicitly in non-perturbative form, through a careful use of the Poisson summation formula.
The Spectral Action and Cosmic Topology
127
In particular, we show here that the non-perturbative correction observed in [9] for perturbations D 2 → D 2 +φ 2 of the Dirac operator gives rise to a slow-roll potential V (φ) for the field φ, which can be used as a model for inflation. We compute the corresponding slow-roll parameters. These are independent of the artifact of the Euclidean compactification to S 3 × S 1 . In particular the dependence on the β parameter coming from the size of the S 1 factor disappears. Moreover, while in the Euclidean compactification, the energy scale and the sphere radius a are independent quantities, for a Lorentzian geometry with the Friedmann form of the metric, both the scale factor a(t) and the energy scale (t) become time-dependent quantities, related by = 1/a(t). Since in the explicit nonperturbative form of the spectral action only the product a appears, the resulting slow-roll potential continues to make sense in the Lorentzian signature, and the dependence on the scale factor a(t) disappears through being matched with the = 1/a(t) scale. Thus, the slow-roll mechanism obtained from the non-perturbative form of the spectral action can be Wick rotated back to the Lorentzian Friedmann form of the geometry and used as a model from which to derive estimates on cosmological parameters such as the spectral index n s and the tensor-to-scalar ratio. Thus, we obtain a slow-roll potential for inflation from the non-perturbative corrections to the spectral action, which is potentially sensitive to the geometry and topology of the underlying 3-dimensional sections of spacetime. This is interesting, in the perspective of deriving cosmological signatures of possibly non-simply connected topologies. This is known as the problem of cosmic topology and it has been widely studied by cosmologists in recent years. We review briefly the current state of understanding of this problem and the list of those that are currently considered to be the most likely candidates for non-simply connected cosmic topologies. Cosmological constraints show that flat or nearly flat, very sightly positively curved, geometries are preferred over negatively curved ones. Combined with requirements of homogeneity on the geometry, this selects as the most likely candidates the flat tori and quotients (Bieberbach manifolds) or the sphere and quotient spherical forms. We compare here the behavior of two among the most promising spherical candidates, the quaternionic space and the Poincaré homology sphere or dodecahedral space, and we show that, in the gravity model based on the spectral action functional, they both behave like the 3-sphere in terms of the resulting model of inflation with slow-roll potential. We then analyze the case of flat tori, and we show that these instead show a distinctly different behavior in terms of possible models of inflation. Finally, in an appendix we discuss the case of lens spaces, where a discrepancy in the existing mathematical literature on the Dirac spectrum gives rise to a “false positive” in terms of the relation between cosmic topology and inflation. Our method consists of extending the non-perturbative calculation of the spectral action of the sphere given in [9] to all these other cases, by subdividing the Dirac spectrum into a union of arithmetic progressions with multiplicities that can be interpolated via values of polynomials at points of the spectrum. Then one can apply the Poisson summation formula to each of these progressions and obtain a complete explicit computation of the spectral action non-pertubatively.
2. The Problem of Cosmic Topology The problem of cosmic topology is the question of whether the spatial topology of the universe can be constrained on the basis of available cosmological data, especially
128
M. Marcolli, E. Pierpaoli, K. Teh
coming from the cosmic microwave background (CMB). A general introduction to cosmic topology is given in [23]. It was known since the mid ’90s that the CMB anisotropies may produce constraints on the geometry of the universe [22]. In fact, the constraints on the 0 parameter favor a spatial geometry that is either flat or nearly flat, slightly positively curved (see [5]). However, even with the curvature severely constrained by cosmological data, there are still different possible multiconnected topologies that support a homogeneous metric with given nearly flat constant curvature. The curvature constraints thus suggest that spherical space forms S 3 / , flat tori T 3 and Bieberbach manifolds T 3 / are all good possible candidates for cosmic topologies [38]. Since the first year of WMAP data [36], the problem of cosmic topology became especially interesting for the main reason that a multiconnected topology may be able to account for some of the anomalies observed in the CMB anisotropies. In fact, the WMAP data suggested possible violations of statistical isotropy in the angular correlation function of the temperature fluctuations. The main anomalies are the quadrupole suppression, the small value of the two-point temperature correlation function at angles above 60 degrees, and the anomalous alignment of the quadrupole and octupole [37]. These anomalies could be an indication of the presence of interesting (non-simply connected) cosmic topologies. As discussed for instance in [35], there are at present three main approaches to investigating the question of cosmic topology. • A search for multiple imaging in the CMB sky would reveal the periodicities caused by the matching of sides of a fundamental domain for a manifold that is a compact quotient of a model geometry (flat or spherical). This type of search is known also as “circles in the sky method”. • A non-trivial cosmic topology is expected to violate the statistical isotropy of the angular power spectrum of CMB anisotropies, that is, the rotational invariance of n-point correlations. • Different cosmic topologies may also be detectable through correlation patterns of the CMB anisotropy field which may be detectable through the coefficients of the expansion of the field in spherical harmonics. At present there are no conclusive results that either prove or disprove the presence of a non-simply connected spatial topology in the universe. Although encouraging initial results [25] suggested that one of the most widely studied candidates for cosmic topology, the Poincaré homology sphere or dodecahedral space, could account for the missing large angle correlations of the two-point angular correlation function of the temperature spectrum of the CMB, attempts to account for the quadrupole-octupole alignment in this topology have failed [40]. A “circle in the skies” search based on the first year of WMAP data [12] also failed to identify any non-simply connected topologies. An explicit description of all the different candidate spherical spaces was given in [17], with an analysis of how they may be detectable through “crystallographic method” through the presence of spikes in the pair separation histogram for three dimensional catalogs of cosmic objects. In addition to the three approaches mentioned above, there has been recently also an analysis of the problem of cosmic topology from the point of view of residual gravity acceleration, [34]. This predicts that, in a non-simply connected topology which is a quotient of either the flat 3-dimensional space or the sphere by a discrete group of isometries, a test particle of negligible mass that feels the gravitational influence of a nearby massive object should also feel a gravitational effect from the translates of the same massive objects in nearby fundamental domains of the group action. This gives rise to a
The Spectral Action and Cosmic Topology
129
gravitational effect qualitatively similar to dark energy. Due to symmetries, it is shown in [34] that this effect vanishes at first order, but has nontrivial third order effects. In the particular case of the Poincaré homology sphere, it vanishes also at third order and only has non-trivial fifth order effects. It is also shown that, in cases like tori T 3 with three different translation lengths, the residual gravity acceleration effect tends to pull the space back to its most symmetric form with three equal translation lengths. Thus, cosmology dynamically prefers the most symmetric forms for a given topology. Recently, predictions of possible cosmic topologies have also been obtained within brane-world scenarios in string theory [27]. 2.1. Laplace spectrum and cosmic topology. What is especially interesting from our point of view is the fact that the way possible non-simply connected topologies manifest themselves in the CMB is mostly through properties of the Laplace spectrum of these manifolds. More precisely, in cosmology it is customary to express the temperature fluctuations of the CMB as a series in the spherical harmonics Ym , of the form ∞ T = am Ym . T
(2.1)
=0 m=−
One then looks at the correlation for the parameters am . In the case of S 3 , the off diagonal terms vanish, am a∗ m = C δ δmm ,
(2.2)
while the diagonal terms C give the temperature anisotropy power spectrum. In the case of a spherical space form S 3 / , with a finite group of isometries, it is well known that these correlation functions in general no longer have vanishing off-diagonal components. The information on the different topologies is therefore encoded in the eigenfunctions of the Laplacian, which replace the usual spherical harmonics of S 3 in the computation of these correlation functions. Thus, an explicit computation of the spectrum and eigenfunctions of the Laplacian on the candidate 3-manifolds, as in [24,32], can be used to produce simulated CMB skies for the different candidate topologies, which are then compared to the WMAP data for the observed CMB. Various statistical tests have been developed to compare different candidate topologies and search for a best fit with observational data. In particular, in the case of the simplest spherical geometries, a comparison based on Bayesian analysis was given recently in [29], where simulated CMB maps for these different topologies are also exhibited. The work [29] compares the cosmological predictions of suppression of power at low for five different spherical manifolds: the simply connected case S 3 , the quaternionic space, and the three exceptional geometries, octahedral, truncated cubic, and dodecahedral. 3. The Spectral Action and Cosmic Topology In this paper we follow a very different point of view on the problem of cosmic topology. We work within a particular theoretical model of gravity coupled to matter, which arises from the noncommutative geometry models of particle physics developed in [11] and
130
M. Marcolli, E. Pierpaoli, K. Teh
more recently [10]. These models are based on extending ordinary spacetime to a product by small extra dimensions, which, unlike in string theory models, are not manifolds but noncommutative spaces. Within these models, one has a natural choice of an action functional, which is the spectral action of [8]. This is essentially an action functional for gravity on the product space X × F, with X the ordinary 4-dimensional spacetime manifold and F the fiber noncommutative space. The large energy asymptotic expansion of the spectral action functional delivers a Lagrangian with gravity terms including the usual Einstein–Hilbert action with a cosmological term, and additional conformal gravity terms. Moreover, in the asymptotic expansion one also finds a coupled matter Lagrangian, which depends on the choice of the non-commutative space F. It is shown in [10] that, for a suitable choice of F, the resulting Lagrangian recovers the full Lagrangian of an extension of the minimal Standard Model with right-handed neutrinos and Majorana mass terms. More recent work [6] shows that a modified choice of the space F leads to a further extension of the Standard Model that incorporates supersymmetric QCD. In all these cases, the main feature of these noncommutative geometry models is that one has a non-perturbative action functional on X × F, whose asymptotic expansion delivers a Lagrangian for particle physics coupled to gravitational terms. In other words, gravity on the noncommutative product space X × F manifests itself as gravity coupled to matter on the ordinary (commutative) spacetime manifold X . 3.1. The spectral action functional. The generalization of Riemannian geometry in the world of noncommutative geometry is provided by the notion of spectral triples. These are data (A, H, D), with A the algebra of functions on the (possibly noncommutative) space, H the Hilbert space of square integrable spinors, and D the Dirac operator. The information corresponding to the metric tensor is encoded in the Dirac operator. The spectral action functional of [8] is defined as Tr( f (D/)), where is the energy scale and f is a test function, usually a smooth approximation of a cutoff function. This is regarded as a spectral formulation of gravity in noncommutative geometry. This action functional has an asymptotic expansion at high energies Tr( f (D/)) ∼ f k k −|D|−k + f (0)ζ D (0) + o(1), (3.1) k∈DimSp
∞
k! where f k = 0 f (v)v k−1 dv, f 0 = f (0), f −2k = (−1)k (2k)! f (2k) (0), and integration is given by residues of zeta function ζ D (s) = Tr(|D|−s ) at the points k in the Dimension Spectrum, that is, at poles of the zeta function. It suffices for our purposes to concentrate only on the gravitational sector of the noncommutative geometry model, because that is where we expect to see a signature of cosmic topology to appear. This means that, instead of computing the spectral action on the product space X × F, with a noncommutative space F that accounts for the matter terms in the Lagrangian arising from the asymptotic expansion, we only compute it on the underlying commutative spacetime manifold X . More precisely, in Sect. 4 we recall the computation for the sphere case done in [9]. In the following sections we obtain explicit non-perturbative computations of the spectral actions on various 3-manifolds that are interesting candidates for cosmic topologies. We then show in §5 that perturbations of the Dirac operator of the form D 2 → D 2 +φ 2 , as considered in [9] for the sphere case, provide a slow-roll potential for a field conformally coupled to gravity, which provides a mechanism for inflation.
The Spectral Action and Cosmic Topology
131
We then compute the spectral action and the corresponding slow-roll parameters explicitly for the various different topologies, two spherical ones and a flat case, and we show that in the spherical cases the slow-roll parameters agree with those for the simply connected topology while in the flat case they are different. We discuss separately the case of lens spaces in the Appendix. Since the spectral action is computed non-perturbatively, the results are not confined to the very early universe near unification energy, but extend to lower energies, so that predictions about slow-roll parameters and cosmological properties like tensor-to-scalar ratio and spectral index can, in principle, be compared with observational data. To obtain a more precise model that can be directly compared with data one should also include further corrections to the slow-roll parameters coming from the additional matter sector (the noncommutative space F), which will not be considered in this paper. The main conclusion is that, in models of gravity coupled to matter based on noncommutative geometry and the spectral action functional, there is a coupling of topology and inflation: different spatial topologies can have measurable effects on the tensor-to-scalar ratio and spectral index, through their effects on the slow-roll parameters of a slow-roll potential coming from perturbations of the Dirac operator and non-perturbative effects in the spectral action functional. 4. Recalling the Case of S3 In this section we recall briefly the results of Chamseddine–Connes [9] on the nonperturbative calculation of the spectral action for the 3-sphere, which we need later, when we compare their result to the analogous computation in the cases of the other candidate cosmic topologies. 4.1. Euclidean model. We are interested in investigating cosmological signatures of the spatial topology of the universe. This means the topology of a spatial 3-dimensional section of the 4-dimensional Lorentzian spacetime describing the universe. It is customary, in working with the spectral action functional of noncommutative geometry, to Wick rotate to a Euclidean model of gravity on a compact manifold. Thus, instead of working with a non-compact Lorentzian 4-manifold which is topologically a cylinder S × R, with S a compact 3-dimensional Riemannian manifold, we Wick rotate and compactify to a Riemannian 4-manifold X = S × S 1 , where the size of S is unaltered and the size of the compactified S 1 acquires the meaning of a thermodynamic parameter, an inverse temperature β, as in [9]. The spectral action of a 3-dimensional manifold S is related in [9] to the 4-dimensional geometry X = S × S 1 by showing that the transform ∞ (u − x)−1/2 h(u) du (4.1) k(x) = x
relates the spectral action functionals for the 3-dimensional and 4-dimensional geometries by Tr(h(D 2X /2 )) ∼ 2β Tr(k(D S2 /2 )), and the Dirac operator D X of the form with β the size of the circle 0 DS ⊗ 1 + i ⊗ DS1 , DX = DS ⊗ 1 − i ⊗ DS1 0
(4.2)
S1
where D S 1 has spectrum β −1 (Z + 1/2).
(4.3)
132
M. Marcolli, E. Pierpaoli, K. Teh
4.2. The Poisson summation formula. The Poisson summation formula states that, for a test function h in Schwartz space h ∈ S(R), one has h(n) = h(n), (4.4) n∈Z
n∈Z
or the more general form
h(x + λn) =
n∈Z
1 2πinx n e λ h( ), λ λ
(4.5)
n∈Z
with λ ∈ R∗+ and x ∈ R, where h is the Fourier transform h(x) = h(u) e−2πiux du. R
(4.6)
In [9] the Poisson summation formula is applied to a test function of the form h(u) = P(u) f (u/), where P(u) is a polynomial function that gives a smooth interpolation for the multiplicities of the Dirac eigenvalues on the 3-sphere S 3 and f is a smooth approximation to a cutoff function, used in the spectral action functional. This allows for an explicit nonperturbative computation of the spectral action functional in the case of the 3-sphere. As shown in §2.2 of [9], for a sphere S 3 with radius a the Dirac spectrum is given by ±a −1 ( 21 + n) for n ∈ Z, with multiplicity n(n + 1). The Poisson summation formula as above gives a spectral action of the form 1 Tr( f (D/)) = (a)3 f (0) + O((a)−k ) f (2) (0) − (a) 4 1 = (a)3 v 2 f (v) dv − (a) f (v) dv + O((a)−k ), (4.7) 4 R R where f (2) denotes the Fourier transform of v 2 f (v). 4.3. The spectral action in 4-dimensions. The corresponding computation of the spectral action for S 3 × S 1 is done in [9] using Poisson summation
1 1 g(n + , m + ) = 2 2 2
(n,m)∈Z
(−1)n+m g (n, m),
(4.8)
(n,m)∈Z2
for a function of two variables g(u, v) = 2P(u) h(u 2 (a)−2 + v 2 (β)−2 ),
(4.9)
where P(u) is the polynomial that interpolates the multiplicities of the spectrum on the sphere S 3 of radius a and h(D 2 /2 ) is the Schwartz function in the spectral action, and β is the size of the circle S 1 , which has Dirac spectrum β −1 (Z + 1/2). One obtains then the spectral action on S 3 × S 1 using Poisson summation on Z2 . This gives Tr(h(D 2 /2 )) = g (0, 0) + O(−k )
(4.10)
The Spectral Action and Cosmic Topology
133
for any k > 0, where g (n, m) =
R2
g(u, v)e−2πi(xu+yv) du dv,
(4.11)
and the error term (n,m) =(0,0) (−1)n+m g (n, m) is estimated to be smaller than −k . One obtains from (4.10)
∞
Tr(h(D 2 /2 )) = π 4 a 3 β 0
1 u h(u) du − π aβ 2
∞
h(u) du + O(−k ). (4.12)
0
One can consider particular classes of test functions h which approximate well enough an even cutoff function on the Dirac spectrum. The class of functions used in [9] is test functions of the form h(x) = P(π x)e−π x with P a polynomial, and in particular, among these, a good approximation to a cutoff function given by the test functions h n (x 2 ) with h n (x) =
n (π x)k k=0
k!
e−π x .
(4.13)
5. Slow-Roll Potential from the Spectral Action We show here that the perturbations of the Dirac operator considered in [9], of the form D 2 → D 2 + φ 2 , give rise to a slow-roll potential for inflation. We compute the corresponding slow-roll parameters in the case of S 3 . We then compute the potential and slow-roll parameters in the case of the other candidate cosmic topologies and we compare them with the case of S 3 .
5.1. Nonperturbative corrections and slow-roll potential. One of the most interesting aspects of the results of [9] is that, under the perturbation D 2 → D 2 + φ 2 of the Dirac operator, one finds a potential V (φ) for a scalar field φ conformally coupled to gravity, which at low energies behaves like a quartic Higgs potential, but which has additional nonperturbative corrections, which have the effect, at higher energies of flattening out the form of the potential so that it is asymptotic to a constant. This gives it the typical form of the slow-roll potentials used in models of inflation in cosmology. The replacement D 2 → D 2 + φ 2 , corresponding to a shift h(u) → h(u + φ 2 /2 ) in the test function (assumed of the form h(x) = P(π x)e−π x as above) produces a potential for the field φ, which, for sufficiently small values of the parameter x = φ 2 /2 , recovers the usual quartic potential for the field φ, conformally coupled to gravity, which on S 3 × S 1 is of the form ∞ 1 1 − π 2 βa 3 h(v)dv φ 2 + πβah(0) φ 2 + πβa 3 h(0) φ 4 . (5.1) 2 2 0 These correspond to a term of the form X R φ 2 dvol, giving the conformal coupling to gravity, from the 4th Seeley-de Witt coefficient, together with a quadratic mass term and a quartic potential, respectively from the second and 4th coefficient.
134
M. Marcolli, E. Pierpaoli, K. Teh
However, for larger values of the parameter x = φ 2 /2 , the potential obtained from the nonperturbative calculation of [9] levels out. Theorem 7 of [9] shows that one has on S 3 × S 1 , ∞ ∞ Tr(h((D 2 + φ 2 )/2 ))) = 2π 4 βa 3 h(ρ 2 )ρ 3 dρ − π 2 βa h(ρ 2 )ρdρ 0
0
π +π βa V(φ / ) + 2 βa W(φ 2 /2 ) + (), 2 (5.2) 4
3
2
2
where the error term () is exponentially small in and the funcions V and W are given by ∞ x V(x) = u(h(u + x) − h(u))du, W(x) = h(u)du. (5.3) 0
0
This is the typical behavior expected from a slow-roll potential used in scenarios for inflationary cosmology based on the Standard Model of elementary particles, as in the recent paper [15]. In particular, we are interested in deriving the associated slow-roll parameters. 5.2. Slow-roll parameters. A way to obtain models of inflation with slow roll potential is to have a theory with a non-minimal coupling of a scalar field to gravity via the curvature R. For a version of a Higgs based inflation see [15]. We show here that the nonperturbative corrections to the Higgs potential in the spectral action obtained in [9] present a similar scenario. Notice that, in the derivation of the slow-roll potential from the spectral action, we have replaced the Minkowskian spacetime geometry with a compactified Euclidean model in order to compute the spectral action nonperturbatively and then derive the slow-roll potential from the perturbation of the Dirac operator. However, once we have obtained a Lagrangian for gravity coupled to a scalar field φ that will be responsible for inflation, we can continue the same Lagrangian back to Minkowskian signature and consider the effects of the slow-roll potential over a Minkowskian spacetime given by a Friedmann metric with assigned topology on the spatial sections. Consider a Minkowskian space-time metric of the form ds 2 = a(t)2 ds S2 − dt 2 ,
(5.4)
where ds S2 is the assigned Riemannian homogeneous metric on the 3-manifold S (the candidate cosmic topology) and a(t) is the scale factor. In models of inflations based on a scalar field with a slow-roll potential V (φ), the accelerated expansion phase a/a ¨ > 0 is governed by the equation a¨ = H 2 (1 − ), a where the Hubble parameter H 2 (φ) is related to the slow roll potential V (φ) by 8π 1 2 H (φ) 1 − (φ) = V (φ), 3 3m 2Pl
The Spectral Action and Cosmic Topology
135
where m Pl is the Planck mass and (φ) is the first slow-roll parameter satisfying the equation of state m2 (φ) = Pl 16π
V (φ) V (φ)
2 .
(5.5)
The inflationary phase is characterized by (φ) < 1. The second slow-roll parameter has the form η(φ) =
m 2Pl 8π
V (φ) V (φ)
−
m 2Pl 16π
V (φ) V (φ)
2 .
(5.6)
These parameters enter in two important measurable quantities: the spectral index and the tensor-to-scalar ratio, which are given respectively by n s = 1 − 6 + 2η, r = 16.
(5.7)
We remark that, from the cosmological viewpoint, the model we consider here will only be a toy model, in the sense that, as in [9] we only look at the purely gravitational part of the spectral action, and we do not consider the effect of the presence of matter coming from the presence of the additional noncommutative space as extra dimensions. This simplification has the advantage that it allows us to focus only on the nonperturbative effects on the Higgs potential, without having to carry around additional terms that are not directly affected by the 3-dimensional spatial topology. However, one should keep in mind that the resulting slow-roll parameters will also be affected by the matter contributions, as described in §4 of [26]. So, in particular, the values we obtain here for these parameters, in the simplifying assumption that drops the matter part, need not meet the observational constraints. The main point for us is to show that there is a contribution to these slow-roll parameters that can be different from that of the sphere in certain candidate cosmic topologies such as the flat tori or equal to that of the sphere in other candidate topologies such as quaternionic or dodecahedral spaces. For this reason we drop the matter terms that would not differ in the various cases. 5.3. Slow-roll parameters for the S 3 -topology. We now compute the slow-roll parameters resulting from the nonperturbative corrections to the Higgs potential of [9], in the case where the underlying spatial topology is the 3-sphere. Theorem 5.1. The slow-roll potential V (x) = π 4 βa 3 V(φ 2 /2 ) +
π 2 βaW(φ 2 /2 ), 2
(5.8)
with V and W as in (5.3), and x = φ 2 /2 , has slow-roll parameters m2 (x) = Pl 16π
2 ∞ h(x) − 2(a)2 x h(u)du x 2 ∞ 0 h(u)du + 2(a) 0 u(h(u + x) − h(u))du
(5.9)
136
M. Marcolli, E. Pierpaoli, K. Teh
and η(x) =
m 2Pl h (x) + 2(a)2 h(x) x 8π 0 h(u)du + 2(a)2 0∞ u(h(u + x) − h(u))du
2 ∞ h(x) − 2(a)2 x h(u)du m 2Pl x − , 2 ∞ 16π 0 h(u)du + 2(a) 0 u(h(u + x) − h(u))du
(5.10)
written in the variable x = φ 2 /2 . Proof. We have, as in Lemma 8 of [9], ∞ V (x) = − h(u)du
and
V (x) = h(x),
x
while W (x) = h(x) and W (x) = h (x). So, if we write V (φ) 1 V (φ) 2 and B = A= , 2 V (φ) V (φ) so that m2 m2 = Pl A and η = Pl (B − A), 8π 8π we find
2 ∞ h(x) − 2(a)2 x h(u)du 1 x A= 2 ∞ 2 0 h(u)du + 2(a) 0 u(h(u + x) − h(u))du and B = x 0
h (x) + 2(a)2 h(x) ∞ . h(u)du + 2(a)2 0 u(h(u + x) − h(u))du
Remark 5.2. The slow-roll parameters obtained in this way are independent of the scale β, as one should expect since that was an artifact introduced by our passing to a Euclidean model to perform calculations with the spectral action, while they depend on both the energy scale and the scale factor a, but only through their product a. This again fits in well with cosmological models, since we know that, for a cosmology described by a Friedmann metric (5.4), the time dependence of the energy scale factor is related through (t) ∼ 1/a(t), so that their product is a constant C independent of time. Thus, we can rewrite (5.9) and (5.10) for the 3-sphere in the form
2 ∞ h(x) − 2C x h(u)du m 2Pl x ∞ (x) = , 16π 0 h(u)du + 2C 0 u(h(u + x) − h(u))du η(x) =
m 2Pl h (x) + 2πC h(x) x 8π 0 h(u)du + 2C 0∞ u(h(u + x) − h(u))du
2 ∞ h(x) − 2C x h(u)du m 2Pl x ∞ − . 16π 0 h(u)du + 2πC 0 u(h(u + x) − h(u))du
(5.11)
We now compare this inflation model derived from the nonperturbative spectral action on the sphere with the case of other nontrivial topologies.
The Spectral Action and Cosmic Topology
137
6. The Quaternionic Cosmology and the Spectral Action Let Q8 denote the group of quaternion units {±1, ±i, ± j, ±k}. It acts on the 3-sphere, with the latter identified with the group SU (2). The resulting quotient manifold SU (2)/Q8 plays an interesting role as a possible cosmic topology candidate, in view of the recent results of [29] on the statistical comparison of various spherical space forms in terms of the best fit for either the power spectrum C or the off-diagonal part of the correlation matrices. As shown in the study of correlation matrices, as exhibited in Fig. 3 of [29], the quaternionic space, unlike the other nontrivial topologies considered in their study, shows no additional structure in the off-diagonal correlations with respect to the spherical case. Thus, the analysis of off-diagonal terms in the correlation functions does not suppress the Bayesian factor of the quaternionic space, while it suppresses those of all the other nontrivial topologies. While the other model comparison carried out in [29] using the power spectrum C does not favor this topology, the particular behavior of the off-diagonal terms seems sufficiently interesting to develop additional possible tests for comparing the quaternionic geometry SU (2)/Q8 to the ordinary spherical geometry. 6.1. The Dirac spectra for SU (2)/Q8. As we show here, the main reason why the case of SU (2)/Q8 can be treated with the same technique used in [9] for the sphere S 3 is because the Dirac spectrum is given in terms of arithmetic progressions indexed over the integers, so that one can again apply the same type of Poisson summation formula. This is not immediately the case for other spherical geometries. More precisely, we recall from [18] that one can endow the 3-manifold SU (2)/Q8 with a 3-parameter family of homogeneous metrics, depending on the parameters ai ∈ R∗ , i = 1, 2, 3. The different possible spin structures j on SU (2)/Q8 correspond to the four group homomorphisms Q8 → Z/2Z with 0 ≡ 1 and Ker( j ) = {±1, ±σ j }, with σ j the Pauli matrices. The Dirac operator for each of these spin structures and its spectrum are computed explicitly in [18]. The case we are interested in here is only the one where the metric has parameters a1 = a2 = a3 = 1, for which SU (2)/Q8 is a spherical space form. For this case the Dirac spectrum was also computed in [2]. In this case, see Corollary 3.2 of [18], the Dirac spectrum for SU (2)/Q8 with the spherical metric a1 = a2 = a3 = 1, is given in the case of the spin structure 0 by ⎧3 with multiplicity 2(k + 1)(2k + 1) ⎪ 2 + 4k ⎪ ⎪ ⎨ 3 + 4k + 2 with multiplicity 4k(k + 1) 2 (6.1) 3 ⎪ − − 4k − 1 with multiplicity 2k(2k + 1) ⎪ ⎪ 2 ⎩ − 23 − 4k − 3 with multiplicity 4(k + 1)(k + 2), where k runs over N. For all the other three spin structures j , j = 1, 2, 3, the spectrum is given by ⎧3 with multiplicity 2k(2k + 1) ⎪ 2 + 4k ⎪ ⎪ ⎨ 3 + 4k + 2 with multiplicity 4(k + 1)2 2 (6.2) 3 ⎪ ⎪ − 2 − 4k − 1 with multiplicity 2(k + 1)(2k + 1) ⎪ ⎩ 3 − 2 − 4k − 3 with multiplicity 4(k + 1)2 , again with k ∈ N.
138
M. Marcolli, E. Pierpaoli, K. Teh
6.2. Trivial spin structure: nonperturbative spectral action. By replacing k with −k − 1 in the third row and k with −k − 2 in the fourth row, we rewrite the spectrum (6.1) in the form 3 with multiplicity 2(k + 1)(2k + 1) 2 + 4k (6.3) 3 2 + 4k + 2 with multiplicity 4k(k + 1), where now k runs over the integers Z. This expresses the spectrum in terms of two arithmetic progressions indexed over the integers. Now the condition that allows us to apply the Poisson summation formula as in [9] is the fact that the multiplicities can be expressed in terms of a smooth function of k. This is the case, since the multiplicities in (6.3) for an eigenvalue λ are given, respectively, by the functions P1 (λ) and P2 (λ) with 5 1 2 3 u + u+ , 4 4 16 7 1 2 3 P2 (u) = u − u − . 4 4 16 P1 (u) =
(6.4)
We then obtain an explicit nonperturbative calculation of the spectral action for SU (2)/Q8 as follows. Theorem 6.1. The spectral action on the 3-manifold S = SU (2)/Q8, with the trivial spin structure, is given by Tr( f (D/)) =
1 1 (a)3 f (0) + (), f (2) (0) − (a) 8 32
(6.5)
with a the radius of the 3-sphere SU (2) = S 3 , with the error term satisfying |()| = O(−k ) for all k > 0, and with f (k) denoting the Fourier transform of v k f (v) as above. Namely, the spectral action for SU (2)/Q8 is 1/8 of the spectral action for S 3 . Proof. Consider a test function for the Poisson summation formula which is of the form s h(u) = g(4u + ), 2
for some s ∈ Z.
Then (4.4) gives n∈Z
1 iπ sn n s exp( ) g ( ), g(4n + ) = 2 4 4 4
(6.6)
n∈Z
which we apply to gi (u) = Pi (u) f (u/), with Pi as in (6.4) and f the Schwartz function in the spectral action approximating a cutoff. This gives an expression for the spectral action on S = SU (2)/Q8 with the trivial spin structure, and with the sphere S 3 = SU (2) of radius one, which is of the form Tr( f (D/)) =
Z
=
3 7 g1 (4n + ) + g2 (4n + ) 2 2
Z
Z
1 3πin 7πin n n exp( ) g1 ( ) + exp( ) g2 ( ). 4 4 4 4 4 4
1
Z
(6.7)
The Spectral Action and Cosmic Topology
139
Assuming that f is a Schwartz function, then gi is also Schwartz, hence so is gi . Therefore, for each k ∈ N, we get an estimate of the form 1 n =0
4
n | gi ( )| ≤ Ck −k . 4
This shows that we can write the right hand side of (6.7) as the terms involving gi (0) plus an error term that is of order O(−k ). One then computes g1 (0) =
1 3 (2) 3 5 f (0) + 2 f (0). f (1) (0) + 4 4 16
(6.8)
1 3 (2) 3 7 f (0) − 2 f (0), f (1) (0) − 4 4 16
(6.9)
Similarly, one has g2 (0) =
so that one obtains for the spectral action in (6.7), 1 g2 (0)) + O(−k ) g1 (0) + ( 4 1 1 = 3 f (0) + O(−k ). f (2) (0) − 8 32
Tr( f (D/)) =
(6.10)
The case with the 3-sphere SU (2) = S 3 of radius a is then analogous, with the spectrum scaled by a factor of a −1 , which is like changing to a in the expressions above, so that one obtains (6.5). 6.3. Nontrivial spin structures: nonperturbative spectral action. The computation of the spectral action on SU (2)/Q8 in the case of the non-trivial spin structures j with j = 1, 2, 3 is analogous. One starts with the Dirac spectrum (6.2) and writes it in the form of two arithmetic progressions indexed over the integers 3 with multiplicity 2k(2k + 1) 2 + 4k (6.11) 3 + 4k + 2 with multiplicity 4(k + 1)2 . 2 In this case one again has polynomials interpolating the values of the multiplicities. They are of the form 3 1 2 1 u − u− , 4 4 16 1 2 1 1 P2 (u) = u + u + . 4 4 16 P1 (u) =
(6.12)
We then obtain the following result. Theorem 6.2. The spectral action on the 3-manifold S = SU (2)/Q8, for any of the non-trivial spin structures j , j = 1, 2, 3, is given by the same expression (6.5) as in the case of the trivial spin structure 0 .
140
M. Marcolli, E. Pierpaoli, K. Teh
Proof. It is enough to observe that the sum of the two polynomials (6.12) that interpolate the spectral multiplicities, P1 (u) + P2 (u) =
1 2 1 u − 2 8
is the same as in the case (6.4) of the trivial spin structure. One then has the same value of 1 1 1 g1 (0) + g2 (0) = (P1 (u) + P2 (u)) f (u/) du, 4 4 4 R which gives the spectral action up to an error term of the order of O(−k ).
6.4. Slow-roll potential and parameters for the quaternionic space. We compute now the slow-roll potential and slow-roll parameters for the case of the quaternionic cosmic topology S = SU (2)/Q8. We first compute the spectral action in the Euclidean 4-dimensional model S × S 1 , from which we obtain the slow-roll potential by a perturbation of the Dirac operator as in the case of S 3 . Theorem 6.3. The spectral action on the 4-manifold S × S 1 with S = SU (2)/Q8 is given by π Tr(h(D / )) = 4 a 3 β 8 2
2
0
∞
π u h(u) du − 2 aβ 16
∞
h(u) du + O(−k ),
0
(6.13) namely 1/8 of the spectral action for S 3 × S 1 . Proof. The eigenvalues for the operator D 2 /2 on S × S 1 are 1 s (4k + )2 (a)−2 + (m + )2 (β)−2 , 2 2 for SU (2) = S 3 of radius a and S 1 of radius β, with multiplicities 2Pi (u), where Pi (u) are the polynomials (6.4) and (6.12) that interpolate the spectral densities for S = SU (2)/Q8 and the integer s also varies according to the arithmetic progressions in the spectrum (6.3) or (6.11). For a given integer s, the Poisson summation formula over Z2 gives
1 s g(4n + , m + ) = 2 2 2
(n,m)∈Z
(n,m)∈Z2
πins 1 n exp( ) (−1)m g ( , m), 4 4 4
(6.14)
where g is a Schwartz function of the form (4.9). We have two functions gi (u, v) = 2Pi (u) h(u 2 (a)−2 + v 2 (β)−2 ),
(6.15)
with Pi as in (6.4) and (6.12), respectively, for the trivial and non-trivial spin structure.
The Spectral Action and Cosmic Topology
141
In the case of the trivial spin structure one writes the spectral action as Tr(h(D 2 /2 )) =
Z2
=
1 1 3 7 g1 (4n + , m + ) + g2 (4n + , m + ) 2 2 2 2 2 Z
1 Z2
3πin n exp( )(−1)m g1 ( , m) 4 4 4
1 n 7πin + g2 ( , m). exp( )(−1)m 4 4 4 2
(6.16)
Z
The main term that contributes to (6.16) is g2 (0, 0) g1 (0, 0) + 2 = 2 aβ (P1 (ax) + P2 (ax)) h(x 2 + y 2 ) d x d y R2 1 2 2 2 = aβ (a) x − h(x 2 + y 2 ) d x d y 4 R2 ∞ ∞ π h(ρ 2 ) ρ 3 dρ − 2 aβ h(ρ 2 ) ρ dρ. = π 4 a 3 β 2 0 0
(6.17)
The error term (n,m) =(0,0)
1 n gi ( , m) 4 4
can be estimated as in [9] and is of the order of O(−k ), for all k > 0. The result for the non-trivial spin structures is the same, since we have seen that the sum P1 (u) + P2 (u) is the same. This gives (6.13) after the change of variables ∞ ∞ 1 ∞ 1 ∞ h(ρ 2 ) ρ 3 dρ = u h(u) du and h(ρ 2 ) ρ dρ = h(u) du. 2 0 2 0 0 0
Since the spectral action for S × S 1 in this case only differs from the one of S 3 × S 1 by a multiplicative factor 1/8, we obtain the following for the slow-roll potential and the slow-roll parameters. Proposition 6.4. The slow-roll potential for S = SU (2)/Q8 is V (φ) =
1 1 π 4 βa 3 V(φ 2 /2 ) + π 2 βaW(φ 2 /2 ), 8 16
with V and W as in (5.3), and the slow-roll parameters are the same (5.11) as for the 3-sphere. Thus, in this case the slow-roll inflation model derived from the spectral action does not distinguish the standard cosmic topology S 3 from the quaternionic case SU (2)/Q8.
142
M. Marcolli, E. Pierpaoli, K. Teh
7. Poincaré Homology Sphere: Dodecahedral Cosmology The Poincaré homology sphere, which is the quotient of the 3-sphere S 3 by the binary icosahedral group , is also commonly referred to as the dodecahedral space, due to the fact that the action of on S 3 has a fundamental domain that is a dodecahedron. The dodecahedral space is obtained by gluing together opposite faces of a dodecahedron with the shortest clockwise twist that matches the faces. This space has been regarded as a likely candidate for the cosmic topology problem and extensively studied for testable cosmological signatures with all the methods presently available, [7,17,23,25,29,32, 34,40,41]. In particular, the three-year WMAP results confirmed the main anomalies: quadrupole suppression, small value of the two-point temperature correlation function at large angles, and quadrupole–octupole alignment. A recent analysis [7] of the Poincaré dodecahedral space based on the explicit computation of the Laplace spectrum and the construction of the resulting simulated CMB sky with more precise estimates of higher modes up to ∼ 30 finds a good match to the WMAP data regarding the two-point temperature correlation function. Thus, the dodecahedral space remains at present one of the most likely candidates, although it fails to account for other anomalies like the quadrupole–octupole alignment [40]. We give here the explicit computation of the spectral action functional for the dodecahedral space, and we show then in §7.6 that, in our model, from the point of view of the resulting inflation slow-roll parameters, the dodecahedral space behaves like the sphere, so that it cannot be ruled out as a candidate cosmic topology in a gravity model based on the spectral action. 7.1. Generating functions for spectral multiplicities. To compute explicitly the Dirac spectrum of the Poincaré homology sphere, we use a general result of Bär [2], which gives a formula for the generating function of the spectral multiplicities of the Dirac spectrum on space forms of positive curvature. In the generality of [2], one considers a manifold M that is a quotient M = S n / of an n-dimensional sphere, n ≥ 2, with the standard metric of curvature one, with ⊂ S O(n + 1) a finite group acting without fixed points. It is shown in [2] that the classical Dirac operator on S n has spectrum n [n/2] k + n − 1 ± . (7.1) + k , k ≥ 0, with multiplicities 2 k 2 The eigenvalues of M are the same as the eigenvalues of S n , but with smaller multiplicities. The spin structures of M are in 1-1 corrrespondence with homomorphisms : → Spin(n + 1), such that ◦ = id , where is simply the double cover map from Spin(n + 1) to S O(n + 1). If D is the Dirac operator on M, then to specify the spectrum of M, for one of these spin structures, one just needs to know the multiplicities, m(±(n/2 + k)), k ≥ 0. These are encoded in two generating functions F+ (z) = F− (z) =
∞ k=0 ∞ k=0
m(
n + k, D)z k , 2
m(−(
n + k), D)z k . 2
(7.2)
(7.3)
The Spectral Action and Cosmic Topology
143
It is elementary to show that these power series have radius of convergence at least 1 about z = 0. Now denote the irreducible half spin representations of Spin(2m) by + ρ + : Spin(2m) → Aut (2m ),
− ρ − : Spin(2m) → Aut (2m ), ± are the positive and negative spinor spaces. Let χ ± : Spin(2m) → C where 2m be the character of ρ ± . It is shown in [2] that the generating functions of the spectral multiplicities have the form
F+ (z) =
1 χ − ((γ )) − z · χ + ((γ )) , || det (12m − z · γ )
(7.4)
1 χ + ((γ )) − z · χ − ((γ )) . || det (12m − z · γ )
(7.5)
γ ∈
F− (z) =
γ ∈
7.2. The Dirac spectrum of the Poincaré sphere. In order to compute explicitly the Dirac spectrum of the Poincaré homology sphere, it suffices then to compute the multiplicities by computing explicitly the generating functions (7.4) and (7.5). Let be the binary icosahedral group. To carry out our computations, we regard S 3 as being the set of unit quaternions, and is the following set of 120 unit quaternions: • 24 elements are as follows, where the signs in the last group are chosen independently of one another: 1 {±1, ±i, ± j, ±k, (±1 ± i ± j ± k)}. 2
(7.6)
• 96 elements are either of the following form, or obtained by an even permutation of coordinates of the following form: 1/2(0 ± i ± φ −1 j ± φk),
(7.7)
where φ is the golden ratio. Then acts on S 3 by left multiplication. Similarly, if S 3 is regarded as the unit sphere in R4 , then S O(4) acts on S 3 by left multiplication. In this way, we may identify a + bi + cj + dk ∈ , with the following matrix in S O(4): ⎛
⎞ a −b −c −d ⎜ b a −d c ⎟ ⎝c d a −b ⎠ d −c b a
144
M. Marcolli, E. Pierpaoli, K. Teh
7.3. The double cover Spin(4) → S O(4). Let us recall some facts about the double cover Spin(4) → S O(4). Let SL3 SU (2) be the group of left isoclinic rotations: ⎞ ⎛ a −b −c −d ⎜ b a −d c ⎟ , ⎝c d a −b ⎠ d −c b a where a 2 + b2 + c2 + d 2 = 1. Similarly, let S 3R rotations: ⎛ p −q −r s ⎜q p ⎝ r −s p s r −q
SU (2) be the group of right isoclinic ⎞ −s −r ⎟ , q ⎠ p
where p 2 + q 2 + r 2 + s 2 = 1. Then Spin(4) SL3 × S 3R , and the double cover : Spin(4) → S O(4) is given by (A, B) → A · B, where A ∈ SL3 , and B ∈ S 3R . The complex half-spin representation ρ − is just the projection onto SL3 , where we identify SL3 with SU (2) via ⎛ ⎞ a −b −c −d a − bi d + ci ⎜ b a −d c ⎟ → . ⎝c d a −b ⎠ −d + ci a + bi d −c b a The other complex half-spin representation ρ + is the projection onto S 3R , where we identify S 3R with SU (2) via ⎛
⎞ p −q −r −s t p − qi s + ri s −r ⎟ ⎜q p ⎝ r −s p q ⎠ → −s + ri p + qi . s r −q p 7.4. The spectral multiplicities. We define our spin structure : → Spin(4) to simply be A → (A, I4 ). It is obvious that this map satisfies ◦ = id . Therefore, given γ = a + bi + cj + dk ∈ , we see that χ − ((γ )) = 2a, χ + ((γ )) = 2. We then obtain the following result by direct computation of the expressions (7.4) and (7.5), substituting the explicit expressions for all the group elements. This can be done using Mathematica. Theorem 7.1. Let S = S 3 / be the Poincaré sphere, with the spin structure described here above. The generating functions for the spectral multiplicities of the Dirac operator are √ 16(710647 + 317811 5)G + (z) F+ (z) = − , (7.8) √ √ (7 + 3 5)3 (2207 + 987 5)H + (z)
The Spectral Action and Cosmic Topology
145
where G + (z) = 6z 11 + 18z 13 + 24z 15 + 12z 17 − 2z 19 −6z 21 − 2z 23 + 2z 25 + 4z 27 + 3z 29 + z 31 and H + (z) = −1 − 3z 2 − 4z 4 − 2z 6 + 2z 8 + 6z 10 + 9z 12 + 9z 14 + 4z 16 −4z 18 − 9z 20 − 9z 22 − 6z 24 − 2z 26 + 2z 28 + 4z 30 + 3z 32 + z 34 , and
√ 1024(5374978561 + 2403763488 5)G − (z) F− (z) = − , √ √ (7 + 3 5)8 (2207 + 987 5)H − (z)
(7.9)
where G − (z) = 1 + 3z 2 + 4z 4 + 2z 6 − 2z 8 − 6z 10 −2z 12 + 12z 14 + 24z 16 + 18z 18 + 6z 20 , and H − (z) = −1 − 3z 2 − 4z 4 − 2z 6 + 2z 8 + 6z 10 + 9z 12 + 9z 14 + 4z 16 −4z 18 − 9z 20 − 9z 22 − 6z 24 − 2z 26 + 2z 28 + 4z 30 + 3z 32 + z 34 . We can then obtain explicitly the spectral multiplicities from the Taylor coefficients of F+ (z) and F− (z), as in 7.2 and 7.3. 7.5. The spectral action for the Poincaré sphere. In order to compute the spectral action, we proceed as in the previous cases by identifying polynomials whose values at the points of the spectrum give the values of the spectral multiplicities. We obtain the following result. Proposition 7.2. There are polynomials Pk (u), for k = 0, . . . , 59, so that Pk (3/2 + k + 60 j) = m(3/2 + k + 60 j, D) for all j ∈ Z. The Pk (u) are given as follows: Pk = 0, whenever k is even. 1 1 1 P1 (u) = − u + u2. 48 20 60 1 1 3 − u + u2. P3 (u) = 80 12 60 7 1 13 − u + u2. P5 (u) = 240 60 60 17 3 1 P7 (u) = − u + u2. 240 20 60 11 1 2 7 − u+ u . P9 (u) = 80 60 60
146
M. Marcolli, E. Pierpaoli, K. Teh
19 47 1 + u + u2. 48 60 60 29 1 1 2 − u+ u . 240 4 60 1 11 17 − u + u2. 80 60 60 37 19 1 − u + u2. 240 60 60 13 1 79 + u + u2. − 240 20 60
P11 (u) = − P13 (u) = P15 (u) = P17 (u) = P19 (u) = P21 (u) = P23 (u) = P25 (u) = P27 (u) = P29 (u) =
23 1 3 − u + u2. 16 60 60 7 1 71 + u + u2. − 240 12 60 9 1 2 53 − u+ u . 240 20 60 1 19 29 − u + u2. 80 60 60 29 1 59 + u + u2. − 240 60 60 9 1 11 + u + u2. 48 20 60 7 1 23 − u + u2. 80 12 60 23 1 47 + u + u2. − 240 60 60 13 1 2 77 − u+ u . 240 20 60 1 13 19 − + u + u2. 80 60 60
P31 (u) = − P33 (u) = P35 (u) = P37 (u) = P39 (u) =
17 1 7 + u + u2. 48 60 60 1 1 31 + u + u2. − 240 4 60 1 31 47 − u + u2. 80 60 60 11 1 23 + u + u2. − 240 60 60 19 3 1 − + u + u2. 240 20 60
P41 (u) = − P43 (u) = P45 (u) = P47 (u) = P49 (u) =
The Spectral Action and Cosmic Topology
147
1 7 1 + u + u2. 16 60 60 11 1 1 =− + u + u2. 240 12 60 1 1 7 + u + u2. =− 240 20 60 39 59 1 = − u + u2. 80 60 60 1 119 59 + u + u2. =− 240 60 60
P51 (u) = − P53 (u) P55 (u) P57 (u) P59 (u)
Proof. These are computed directly from the Taylor coefficients of the generating functions of the spectral multiplicities (7.8) and (7.9). We then obtain the nonperturbative spectral action for the Poincaré sphere. Theorem 7.3. Let D be the Dirac operator on the Poincaré homology sphere S = S 3 / , with the spin structure : → Spin(4) with A → (A, I4 ). Then, for f a Schwartz function, the spectral action is given by 1 1 3 (2) 1 f (0) − f (0) , (7.10) Tr( f (D/)) = 60 2 8 which is precisely 1/120 of the spectral action on the sphere. Proof. The result follows by applying Poisson summation again, to the functions g j (u) = P j (u) f (u/). This gives, up to an error term which is of the order of O(−k ) for any k > 0, the spectral action in the form Tr( f (D/)) =
59 1 1 g j (0) = P j (u) f (u/)du. 60 60 R j=0 j
It suffices then to notice that 59 j=0
P j (u) =
1 2 1 u − . 2 8
The result then follows as in the sphere case.
7.6. Slow-roll potential in dodecahedral cosmologies. The dodecahedral space S = S 3 / , with the binary icosahedral group, also behaves in the same way as the quaternionic space SU (2)/Q8 with respect to the properties of the slow-roll potential and slow-roll parameters. Namely, the slow-roll potential is a multiple of the potential for the sphere S 3 and the slow-roll parameters are therefore equal to those of the sphere. Theorem 7.4. The spectral action for the manifold S × S 1 , with S = S 3 / the Poincaré dodecahedral space, is given by 1/120 of the spectral action of S 3 × S 1 (4.12), ∞ ∞ π 4 3 π 2 a β aβ uh(u)du − h(u)du, (7.11) Tr(h(D 2 /2 )) ∼ 120 240 0 0
148
M. Marcolli, E. Pierpaoli, K. Teh
up to an error term of the order of O(−k ). The slow-roll potential V (φ) obtained by replacing D 2 → D 2 + φ 2 is also 1/120 of the potential for the 3-sphere, V (φ) =
π 4 3 π 2 βa V(φ 2 /2 ) + βaW(φ 2 /2 ), 120 240
(7.12)
with V and W as in (5.3). The slow-roll parameters are the same (5.9), (5.10) as for the sphere S 3 . Proof. We showed in Theorem 7.3 that the spectral action for the Poincaré dodecahedral space S = S 3 / is 1/120 of the spectral action of the 3-sphere of radius one. Changing the radius a of the 3-sphere has the effect of changing → (a) in the expression (7.10) of the spectral action, as in the case of the sphere. We then obtain the spectral action Tr(h(D 2 /2 )) for the product S × S 1 , as in Theorem 6.3 using Poisson summation applied to the functions gi (u, v) = 2Pi (u)h(u 2 (a)−2 + v 2 (β)−2 ), with Pi (u) as in Proposition 7.2, namely Tr(h(D 2 /2 )) =
59
(n,m)∈Z2 i=0
3 1 gi (60n + i + , m + ). 2 2
Then, the Poisson summation formula applied to these functions shows that the spectral action on the product of the dodecahedral space by the circle is given by Tr(h(D 2 /2 )) =
1 gi (0, 0) + O(−k ). 60 i
We compute this as in the sphere case, using the fact that i Pi (u) = u 2 /2 − 1/8. This gives (7.11), as in the case of the sphere and of the quaternionic space. The slow-roll potential is then obtained exactly as in the previous cases. 8. Flat Cosmologies Another very promising candidate for possible non-simply-connected cosmic topologies is given by the flat manifolds: flat 3-dimensional tori and their quotients, the Bieberbach manifolds. Simulated CMB skies have been computed for tori and for all the Bieberbach manifolds in [33]. The method is the same as in the analysis of simulated CMB skies for spherical space forms of [24,32], namely through the explicit computation of the spectrum and eigenforms of the Laplacian. In the case of flat tori, the basis given by planar waves is more directly adapted to the topology, while the basis in spherical waves is better suited for comparison between simulated and observed CMB sky. So the analysis of [33] of the Laplace spectra and eigenfunctions uses the transition between these two bases. The resulting simulated CMB skies are suitable for an investigation for flat cosmic topologies with the “circles in the sky” method. A statistical analysis of distance correlations between cosmic sources, aimed at identifying possible signatures of cosmic topologies given by flat tori with the method of “cosmic crystallography” was performed in [20].
The Spectral Action and Cosmic Topology
149
While an early analysis of the anomalies of the anisotropy spectrum of the CMB (the quadrupole suppression, the small value of the two-point temperature correlation function at large angles, and the quadrupole-octupole alignment) suggested that flat tori would account for all of these anomalies, if one of the sides of the fundamental domain is of the order of half the horizon scale, the more detailed analysis of [30] excludes this possibility on the basis of the “circles in the sky method” and of the S-statistic test, measuring reflection symmetry. Nonetheless, the flat tori remain at present one of the most promising possible candidates for multiconnected cosmic topologies. An analysis of how to produce a quadrupole-octupole alignment for a flat torus with cubic fundamental domain, depending on the size of the torus, was given in [1]. However, the alignment obtained in this way is not strong enough to account for the observed anomaly. Comparison with candidates such as dodecahedral and octahedral cosmologies shows that in these spherical topologies one has either no alignment or an anti-alignment, which appears to favor the flat tori. We show here that, from the point of view of our model of gravity based on the spectral action functional, a cosmic topology given by a flat torus generates an inflation potential and slow-roll parameters that are different from those of the spherical topologies considered in the previous sections. 8.1. The spectral action on the flat tori. Let T 3 be the flat torus R3 /Z3 . The spectrum of the Dirac operator, denoted D3 , is given in Theorem 4.1 of [4] as ± 2π (m, n, p) + (m 0 , n 0 , p0 ) ,
(8.1)
where (m, n, p) runs through Z3 . Each value of (m, n, p) contributes multiplicity 1. The constant vector (m 0 , n 0 , p0 ) depends on the choice of spin structure. Theorem 8.1. The spectral action Tr( f (D32 /2 )) for the torus T 3 = R3 /Z3 is independent of the spin structure on T 3 and given by 3 2 2 Tr( f (D3 / )) = f (u 2 + v 2 + w 2 )du dv dw + O(−k ), (8.2) 4π 3 R3 for arbitrary k > 0. Proof. By (8.1), we know the spectrum of D32 is given by 4π 2 (m, n, p) + (m 0 , n 0 , p0 ) 2 , where (m, n, p) runs through Z3 , and each value of (m, n, p) contributes multiplicity 2. Given a test function in Schwartz space, f ∈ S(R), the spectral action is then given by 2 4π ((m + m 0 )2 + (n + n 0 )2 + ( p + p0 )2 ) . Tr( f (D32 /2 )) = 2f 2 3 (m,n, p)∈Z
In three dimensions, the Poisson summation formula is given by g(m, n, p) = g (m, n, p), Z3
Z3
150
M. Marcolli, E. Pierpaoli, K. Teh
where the Fourier transform is defined by g(u, v, w)e−2πi(mu+nv+ pw) dudvdw. g (m, n, p) = R3
If we define
g(m, n, p) = f
4π 2 ((m + m 0 )2 + (n + n 0 )2 + ( p + p0 )2 ) , 2
(8.3)
and apply the Poisson summation formula, we obtain the following expression for the spectral action: Tr( f (D32 /2 )) = 2 g (m, n, p) (m,n, p)∈Z3
= 2 g (0, 0, 0) + O(−k ) 2 4π ((u + m 0 )2 + (v + n 0 )2 + (w + p0 )2 ) du dv dw =2 f 2 R3
+O(−k ) 3 = f (u 2 + v 2 + w 2 )du dv dw + O(−k ). 4π 3 R3 g (m, n, p) = O(−k ) for arbitrary k > 0 is elementary, The estimate (m,n, p) =0 using the fact that f ∈ S(R). We observe that the nonperturbative spectral action is independent of the choice of spin structure. Now let X = T 3 × Sβ1 . We then compute the spectral action for the operator D 2X as a direct consequence of the previous result. Theorem 8.2. On the 4-manifold X = T 3 × Sβ1 , with the flat torus of size and with the product Dirac operator D X as in (4.3), the spectral action is given by 4 β3 ∞ uh(u)du + O(−k ) (8.4) Tr(h(D 2X /2 )) = 4π 0 for arbitrary k > 0. Proof. For the operator D 2X , with D X as in (4.3) the spectral action Tr(h(D 2X /2 )) is given by 1 1 2 4π 2 2 2 2 . ) 2h ((m + m ) + (n + n ) + ( p + p ) ) + (r + 0 0 0 ()2 (β)2 2 4 (m,n, p,r )∈Z
We set
g(u, v, w, y) = 2 h
4π 2 2 y2 2 2 (u + v + w ) + 2 (β)2
The Poisson summation formula then gives 1 g(m + m 0 , n + n 0 , p + p0 , r + ) = 2 4 (m,n, p,r )∈Z
(m,n, p,r )∈Z4
.
(−1)r g (m, n, p, r ).
The Spectral Action and Cosmic Topology
151
Since we have h ∈ S(R), we can estimate that the error term g (m, n, p, r ) (m,n, p,r ) =(0,0,0,0)
is bounded by O(−k ) for arbitrary k > 0. We then obtain Tr(h(D 2X /2 )) = g (0, 0, 0, 0) + O(−k ). We have
g (0, 0, 0, 0) =
R4
2h
4π 2 y2 2 2 2 (u + v + w ) + ()2 (β)2
du dv dw dy.
This gives 4 β3 4 β3 V ol(S 3 ) ∞ 2 2 2 2 h(u + v + w + y ) du dv dw dy = h(ρ 2 )ρ 3 dρ 4π 3 R4 4π 3 0 which gives Tr(h(D 2X /2 )) from which we obtain (8.4).
4 β3 = 2π
∞
ρ 3 h(ρ 2 )dρ + O(−k ),
0
We now consider the effect of introducing the perturbation D 2 → D 2 + φ 2 in the spectral action. We write as above ∞ V(x) = u (h(u + x) − h(u)) du. 0
We then have the following. Theorem 8.3. The perturbed spectral action on the flat tori is of the form Tr(h((D 2X + φ 2 )/2 )) = Tr(h(D 2X /2 )) +
4 β3 V(φ 2 /2 ). 4π
(8.5)
The corresponding slow-roll potential is of the form V (φ) =
4 β3 V(φ 2 /2 ), 4π
and the slow-roll parameters are given by
2 ∞ m 2Pl x h(u)du ∞ = 16π 0 u(h(u + x) − h(u))du ⎛
2 ⎞ ∞ 2 h(u)du m h(x) 1 x ⎠. ∞ η = Pl ⎝ ∞ − 8π 2 u(h(u + x) − h(u))du u(h(u + x) − h(u))du 0 0
152
M. Marcolli, E. Pierpaoli, K. Teh
Proof. The result follows directly from (8.4) upon writing 4 β3 ∞ 2 2 2 uh(u)du + O(−k ) Tr(h((D X + φ )/ )) = 4π 0 4 β3 ∞ = u(h(u + x) − h(u))du 4π 0 4 β3 ∞ + uh(u)du + O(−k ), 4π 0 and computing the slow-roll parameters as in (5.5) and (5.6).
Notice how the absence of the W(φ 2 /2 ) term in the slow-roll potential for the case of flat tori gives rise to slow-roll parameters that are genuinely different from those we computed for the spherical geometries. This shows that, in noncommutative geometry models of gravity based on the spectral action functional, there is a nontrivial relation between cosmic topology (or at least the underlying curvature geometry) and the shape of the induced inflation slow-roll potential and parameters. The case of the other flat geometries, the Bieberbach manifolds, can be handled with similar techniques, based on the explicit computation of their Dirac spectra given in [31].
9. Geometric Engineering of Inflation Scenarios via Dirac Spectra If one renounces the assumption of homogeneity and constant curvature, which reduces the candidate topologies to spherical and flat space forms, one finds that it is possible to engineer different inflation scenarios, by changing the slow-roll potential and the resulting slow roll parameters by modifying the metric on a fixed topology and change accordingly the Dirac spectrum and the resulting spectral action. In the spherical examples we computed explicitly in the previous sections the Dirac spectra tend to have non-trivial multiplicities. These reflect the very symmetric form of the geometry. On the contrary, it is shown in [14] that, for a generic Riemannian metric on a given smooth compact 3-dimensional manifold M, all the Dirac eigenvalues are simple. Moreover, the result of [13] shows that, for a given L > 0 and an assigned sequence of non-zero real numbers − L < λ 1 < λ2 < λ3 < · · · < λ N < L ,
(9.1)
it is possible to construct, on an arbitrary smooth compact spin 3-manifold M, a Riemannian metric g such that the non-zero spectrum of corresponding Dirac operator D M in the interval (−L , L) consists of the simple eigenvalues Spec(D M ) ∩ ((−L , L)\{0}) = {λ j } j=1,...,N .
(9.2)
The way to obtain a Dirac spectrum with these properties is to start with a metric on M for which the dimension of the kernel of the Dirac operator is minimal, compatibly with the constraint given by the index theorem. By rescaling this metric one ensures that no other eigenvalue occurs in an interval (−3L , 3L). One then performs a connected sum with N copies of S 3 , so that the resulting manifold is still topologically the same as M. One endows each of the 3-spheres with a Berger metric as in [21], scaled so that the
The Spectral Action and Cosmic Topology
153
interval (−2L , 2L) contains only one eigenvalue of the Dirac operator. Then applying a surgery formula one obtains the desired eigenvalues for the Dirac spectrum on the connected sum manifold, [13]. For simplicity, to avoid handling separately a possible kernel, let us consider here a variant of the spectral action where one only sums over the non-zero spectrum of D. We write this as Tr ( f (D/)) := f (λ/). (9.3) λ∈Spec(D)\{0}
We then have the following result, which allows us to construct, on a given 3-manifold a metric with prescribed spectral action. Lemma 9.1. Let M be a compact smooth 3-manifold, with a given spin structure. Let f be a smooth function, compactly supported inside an interval [−L/, L/]. Then, for any given λ > 0, there is a metric gλ,L on M such that the spectral action for the resulting Dirac operator is Tr ( f (D/)) =
ˆ f (0) + O(−k ), λ
(9.4)
for arbitrary k > 0. Proof. Let λn = η + nλ be a progression indexed by the integers n ∈ Z, with λ > 0 and η = 0. Let {λn 0 + j } j=1,...,N be the points of this sequence that lie in the interval (−L , L). We assume that λn = ±L for all n. Using the method of [13] we construct, by taking connected sums with Berger spheres, a metric on M for which the Dirac operator D has Spec(D) ∩ ((−L , L) \ {0}) given by the simple eigenvalues λn 0 + j , with j = 1, . . . , N . For a test function supported in [−L/, L/] we then have Tr ( f (D/)) =
f (λn /).
n∈Z
We can then use the Poisson summation formula
g(η + nλ) =
n∈Z
1 2πinη n exp( ) g( ) λ λ λ
n∈Z
to g(u) = f (u/). We estimate as in the previous cases that 1 2πinη n exp( ) g ( ) ≤ O(−k ), λ λ λ n =0
for arbitrary k > 0, so that we are left with the term 1 g (0) = f (0). λ λ
154
M. Marcolli, E. Pierpaoli, K. Teh
Notice, for example, that we can apply this result starting from any one of the spherical topologies we analyzed in the previous sections. In such cases one starts with the round metric, so one does not have a kernel to worry about. One then scales it so as not to have any other eigenvalue in the interval (−3L , 3L) and proceeds to modify the metric by taking connected sums with the Berger spheres to insert the desired eigenvalues in the interval (−L , L). Thus, on a given underlying topology one can significantly alter the form of the spectral action by this method, at the cost of no longer having a homogeneous metric. We now show the effect this operation has on the inflation slow-roll potential, even though such non-homogeneous metrics are clearly less interesting in terms of candidate cosmologies. We now see how this procedure can be used to construct different possible slow-roll potentials. Let (λn , P) denote the following data: • A progression λn = η + nλ, for n ∈ Z, with λ > 0 and η = 0. • A polynomial P(u) = αu 2 + γ with the property that P(λn ) = m n is a non-negative integer, for all n. In the following, as above, we assume that either we start from a manifold M with a metric for which D M has trivial kernel, or else we modify the spectral action on M × S 1 to count only the non-zero part of the spectrum of D M . Proposition 9.2. Let M be a compact smooth 3-dimensional manifold endowed with a spin structure. Given a smooth compactly supported test function h such that h ≡ 1 on an interval [−T, T ] and decays rapidly to zero outside of this interval, and given a choice of data (λn , P) as above, there exists a Riemannian metric g on M such that the resulting Dirac operator D of the form (4.3) on M × Sβ1 has spectral action π 4 βα ∞ 2π 2 βγ ∞ 2 2 u h(u) du + h(u) du + O(−k ), (9.5) Tr(h(D / )) = λ λ 0 0 for arbitrary k > 0. Proof. Consider the sequence of non-negative integers m n = P(λn ). Let ⊂ Z2 be the set of pairs (n, m) such that xn,m (, β) :=
λ2n (m + 1/2)2 + ∈ (0, T ). 2 (β)2
We can assume that all other points of the form xn,m , for (n, m) ∈ / lie outside of the support of h. Let (−L , L) be an interval that contains all the points λn for which the set of m ∈ Z with (n, m) ∈ is non-empty. For all n ∈ Z such that the set of (n, m) ∈ is nonempty, choose sufficiently small, non-intersecting open intervals Un, = (λn − , λn + ) around the value λn , such that all λ ∈ Un, have the property (m+1/2) λ that ∈ (0, T ), for all m such that (n, m) ∈ . Then choose m n points 2 + (β)2 λn,1 < λn,2 < · · · < λn,m n inside Un, . By the construction of [13], by taking connected sums with suitable Berger spheres, we can obtain on M a metric for which the Dirac operator D M has spectrum satisfying Spec(D M ) ∩ ((−L , L)\{0}) = {λn, j } with n = n 0 , . . . , n 0 + N and j = 1, . . . , m n . Let D be the corresponding Dirac operator (4.3) on M × Sβ1 . The spectral action Tr(h(D 2 /2 )) is computed by 2 h(λ2n, j −2 + (m + 1/2)2 (β)−2 ). 2
2
n, j
The Spectral Action and Cosmic Topology
155
Given the construction of the λn, j above, this is also equal to 2P(λn )h(λ2n −2 + (m + 1/2)2 (β)−2 ). (n,m)∈Z2
We let g(u, v) = 2P(u)h(u 2 −2 +v 2 (β)−2 ) and we obtain, by the Poisson summation formula, 1 2πinη 1 n ) g ( , m). g(nλ + η, m + ) = (−1)m exp( 2 λ λ λ 2 2 (n,m)∈Z
(n,m)∈Z
Estimating as before the sum of terms with (n, m) = (0, 0) to be bounded by O(−k ) for arbitrary k > 0, this gives Tr(h(D 2 /2 )) =
1 g (0, 0) + O(−k ), λ
where we then have g (0, 0) = 2(αu 2 + γ ) h(u 2 −2 + v 2 (β)−2 ) du dv R2 4 2 2 2 2 2 = βα (u + v ) h(u + v ) du dv + 2 βγ h(u 2 + v 2 ) du dv R2 R2 ∞ ∞ ρ 3 h(ρ 2 ) dρ + 2 βγ 4π h(ρ 2 ) dρ = 4 βα2π 0 ∞ ∞0 4 2 u h(u) du + βγ 2π h(u) du. = βαπ 0
0
We then have the following result, which shows that altering the spatial metric on suitable bubbles (the Berger spheres with which one performs a connected sum) consequently alters the form of the inflation potential and slow-roll parameters induced by the spectral action. Corollary 9.3. For a 3-manifold M with a metric constructed as in Proposition 9.2 above, the induced slow-roll potential has the form V (φ) =
π 4 αβ 2π 2 γβ V(φ 2 /2 ) − W(φ 2 /2 ), λ λ
with V and W as in (5.3). The corresponding slow-roll parameters are given by m2 (x) = Pl 16π m2 η(x) = Pl 8π
α2 V (x) − 2γ W (x) α2 V(x) − 2γ W(x)
2
α2 V (x) − 2γ W (x) 1 − α2 V(x) − 2γ W(x) 2
α2 V (x) − 2γ W (x) α2 V(x) − 2γ W(x)
2
.
Proof. This follows directly from the previous result, by computing Tr(h(D 2 + φ 2 )/2 ) in the same way as in the previous cases.
156
M. Marcolli, E. Pierpaoli, K. Teh
Notice that, after rotating back to Lorentzian signature with a metric of the Friedmann form (5.4), the factor in the slow-roll parameter appears in fact multiplied by the scale factor a(t) of the Friedmann metric, which gives a constant term by the relation (t) ∼ 1/a(t), as in the previous cases. One is left with the freedom of modifying the slow-roll parameters by changing the modified metric on the Berger spheres and correspondingly affecting the values of the parameters α and γ . One can also obtain potentials of a more general form, if one constructs, via the same method, spectra that are only partially given by arithmetic progressions. An example of this sort is computed explicitly in the Appendix: it gives rise to a genuinely different shape of the potential V (φ). 10. Conclusions In models of high-energy physics based on noncommutative geometry, the spectral action functional of [8] is proposed as an action functional for gravity, or for gravity coupled to matter when additional noncomumutative extra dimensions are introduced in the geometry of the model. We concentate here on the purely gravitational part of the model, without noncomumutative extra dimensions, and we compute the explicit nonperturbative form of the spectral action functional for three among the more likely candidates for the problem of cosmic topology: the quaternionic space SU (2)/Q8 and the Poincaré dodecahedral space S 3 / , with the binary icosahedral group, and the flat tori. We show that when one computes the spectral action for the 4-dimensional manifold obtained by Wick rotating and compactifying the corresponding space-time to a product of the given 3-manifold by a circle, one obtains as non-perturbative effect a slow-roll potential for a field φ coming from perturbations D 2 → D 2 + φ of the Dirac operator of the 4-dimensional geometry. We compute the slow-roll parameters for the resulting slow-roll potential V (φ) and show that they make sense when rotating back to the original Minkowskian spacetime. We see that, in the case of the quaternionic and the dodecahedral space, the slow-roll parameters are the same as for the ordinary case of the sphere S 3 , while in the case of the flat tori the potential one obtains in this way behaves significantly differently from the spherical cases. This shows that cosmological models based on noncommutative geometry predict that different candidate cosmic topologies may give rise to different inflation scenarios, and different values for testable cosmological parameters. 11. Appendix: Lens Spaces, a False Positive Lens spaces are quotients of the sphere S 3 by the action of a finite cyclic group Z/N Z. They have been considered among the candidate cosmic topologies, especially in [39], which shows simulated CMB maps for lens spaces and computes the expected CMB anisotropies for some of these topologies. The surprising result of the analysis of [39] is that instead of finding an increasingly suppressed quadrupole with increasing N , the low multipoles are enhanced instead of being suppressed for large N . Thus, the simulated power spectra of [39] suggest that to maintain consistency with the WMAP data, one cannot exceed the range N ≤ 15. On the other hand, in the same work [39] the lens space case is analyzed from the point of view of the “circles in the sky” method and it is shown that potentially detectable periodicities (matching circles) would appear only in the range N > 7.
The Spectral Action and Cosmic Topology
157
11.1. The trouble with the Dirac spectrum on lens spaces. Consider in particular lens spaces L N = SU (2)/Z N , with N ≥ 3, which are quotients of the sphere SU (2) = S 3 by the action of the finite cyclic group Z N = Z/N Z acting on SU (2) ⊂ C2 by ω 0 , with ω N = 1. (11.1) 0 ω−1 For these lens spaces, Bär gave an explicit computation of the Dirac spectrum in [3]. The result states that, for the canonical spin structure, the spectrum is of the form (i) (ii)
−i N − 21 , with multiplicity 2i N , i = 0, 1, 2, . . . , − 21 ± m, with multiplicity m, m = 2, 3, . . . , −(m + 1) < i N ≤ (m − 2),
(11.2)
where we have taken on the sphere S 3 the round metric of radius one. In the even case N = 2N , there is also a second spin structure for which the Dirac spectrum is given in [3] as (i) −(N + i N ) − 21 with multiplicity 2(N + i N ), i = 0, 1, 2, . . . , (ii) − 21 ± m with multiplicity m, m = 2, 3, . . . 1 − m < i N + N2 ≤ m − 2.
(11.3)
Unfortunately, this result of [3] appears to be incorrect, as we discuss in §11.7 below. However, we still show here what the spectral action and slow-roll potential would be for a manifold with Dirac spectrum as above, because the computation itself exhibits some interesting features that we have not encountered in the other spherical and flat examples and that may be useful in different contexts, for manifolds whose Dirac spectrum only partially decomposes as a union of arithmetic progressions. We show that the incorrect calculation of the Dirac spectrum of L N of [3] leads to a “false positive” result of a spherical cosmic topology which gives rise to an inflation scenario different from the simply connected case. However, as we show in §11.7 below, with the correct calculation of the Dirac spectrum for the lens spaces, the inflation potential is in fact again the same as for the case of the sphere, just as in the other spherical cases we computed in this paper. 11.2. Multiplicities, first case. We consider here the problem of computing the explicit nonperturbative form of the spectral action for an operator D with spectrum of the form (11.2). We start by writing the multiplicities in a more convenient form. The multiplicity in row (i) is already in a nice form. To handle row (ii), we need to break it up into the subsets corresponding to each equivalence class m ≡ j (mod N ), where j ∈ {0, 1, . . . , N − 1}. In order to determine the multiplicity of −1/2 ± m, it is convenient to replace the upper and lower bounds of −(m + 1) < i N ≤ (m − 2) with the smallest and largest values of i N which satisfy the inequality. Lemma 11.1. The multiplicity of −1/2 ± m is given by 2m(m − j) N 2 2m − 2m j + m N N
for m ≡ j
mod N , with j = 0, 1,
for m ≡ j
mod N , with j = 2, 3, . . . , N − 1.
(11.4)
158
M. Marcolli, E. Pierpaoli, K. Teh
Proof. We first look at the case where m ≡ j (mod N ), j = 0, 1. In this case, the bound −m − 1 < i N ≤ m − 2 can be replaced by j − m ≤ i N ≤ m − j − N, by adding j + 1 to the left-hand side and subtracting N + j − 2 from the right-hand side. What matters is that 0 < j + 1 ≤ N , and 0 ≤ N + j − 2 < N . If m = k N + j, where k = 0, 1, 2, . . ., then we see that there are 2k values of i which satisfy the inequality, and hence −1/2 ± m has multiplicity 2 km = (2m(m − j))/N . Notice that when m is 0 or 1, this formula gives us a multiplicity of zero, which is good, since in row (ii), the index m begins at m = 2. We then consider the case m ≡ j (mod N ), j = 2, 3, . . . , N − 1. One can replace the upper and lower bounds −m − 1 < i N ≤ m − 2 by j − m ≤ i N ≤ m − j, by adding j + 1 to the left-hand side, and subtracting j − 2 from the right-hand side. One has 0 < j + 1 ≤ N , and 0 ≤ j − 2 < N . If m = k N + j, k = 0, 1, 2, . . ., then we see that there are 2k + 1 values of i which satisfy the inequality, and hence −1/2 ± m has multiplicity (2k + 1)m = (2m 2 − 2m j + m N )/N . 11.3. The spectral action and the Poisson formula. One sees then that, unlike the cases of the spherical topologies we analyzed before, one cannot simply write the whole spectrum (11.2) as a union of arithmetic progressions indexed over the integers. However, it is still possible to extend the positive and the negative part of the spectrum, separately, to unions of such arithmetic progressions. This provides us with a different method, still based on the Poisson summation formula, to compute the spectral action, which may turn out to be useful in other cases. This is our main reason for including the full calculation here, despite the fact that it does not give the correct answer for lens spaces. One finds that the multiplicities, for the positive and the negative parts of the spectrum, can be interpolated by polynomials, in the following way. Lemma 11.2. For m > 0, the multiplicity of −1/2 + m, when m ≡ i mod N , is given by Pi+ (−1/2 + m), with Pi+ the polynomials 1 2 2 2 u + u+ , N N 2N 2 1 (11.5) P1+ (u) = u 2 − , N 2N 1 − 2j + N 2 2 − 2j + N P j+ (u) = u 2 + u+ , j = 2, 3, . . . , N − 1. N N 2N For m ≥ 0, the multiplicity of −1/2−m, when m ≡ mod N is given by P− (−1/2−m), with P− the polynomial P0+ (u) =
2 2 u + N 2 P1− (u) = u 2 + N 2 P j− (u) = u 2 + N
P0− (u) =
1 2 u+ , N 2N 3 4 u+ , N 2N 1 + 2j − N 2 + 2j − N u+ , N 2N
(11.6) j = 2, 3, . . . , N − 1.
The Spectral Action and Cosmic Topology
159
The multiplicity of −i N − 1/2, for i ≥ 0, is given by P − (−i N − 1/2), with P − (u) = −2u − 1.
(11.7)
Proof. This follows directly from the expressions for the multiplicities given in Lemma 11.1 above. One can then make the following observation on computing the spectral action. Lemma 11.3. Let D be an operator with spectrum (11.2). Given a Schwartz function f , there are Schwartz functions f + and f − , respectively supported on the positive and negative reals, with the property that f = f + + f − on (−∞, −α]∪[α, ∞), with Iα = (−α, α) an interval with Spec(D) ∩ Iα = ∅. The spectral action for D is then computed by Tr( f (D/)) = Tr( f + (D/)) + Tr( f − (D/)).
(11.8)
Proof. One observes from (11.2) that there is a gap in the spectrum of D around zero. Thus, it is possible to replace the function f with a pair of Schwartz functions f + and f − , which are, respectively, equal to f on the positive and negative parts of the spectrum and that have support contained only in the positive or negative reals. Since the values of f and f + + f − on an open neighborhood of the spectrum are the same, the value of the spectral action is unchanged. We can now compute the two terms on the right-hand side of (11.8). Theorem 11.4. Let f + be a Schwartz function supported on the positive reals, chosen as in Lemma 11.3. Then, for an operator D with spectrum of the form (11.2) one has 1 3 (2) (1) 2 f + (0) + 2 (11.9) Tr( f + (D/)) = f + (0) + + (), N (k) where the error term is of order + () = O(−k ), for any k > 0, and where f + is the k Fourier transform of v f + (v), as above.
Proof. We define g +j (u) = P j+ (u) f + (u/), with P j+ as in (11.5), for j = 0, . . . , N − 1, so that we can write the spectral action with test function f + in the form Tr( f + (D/)) =
−1 N
g +j (−1/2 + k N + j).
(11.10)
k∈Z j=0
In fact, extending the sum to m ∈ Z does not change anything, since all terms with m ≤ 0 fall outside of the support of f + . We can then apply the Poisson summation formula to compute this expression. The analog of (6.6) now gives 1 (2 j − 1)πik k exp g +j ( ). (11.11) g +j (k N + (2 j − 1)/2) = N N N k∈Z
k∈Z
The same argument used in [9] to estimate the remainder term applies here to give, for any k > 0, 1 k | g + ( )| ≤ O(−k ), N j N k =0
160
M. Marcolli, E. Pierpaoli, K. Teh
so that (11.10) can then be written as Tr( f + (D/)) =
N −1 1 + g j (0) + O(−k ). N
(11.12)
j=0
We then obtain the following values for g +j (0), using the form (11.5) of the polynomials P j+ : 2 3 (2) 2 1 f + (0) + 2 f + (0), f +(1) (0) + N N 2N 2 1 (2) f + (0), g1+ (0) = 3 f + (0) − (11.13) N 2N 2 2 − 2 j + N 1 − 2 j + N (2) (1) 2 f + (0), g +j (0) = 3 f + (0) + f + (0) + N N 2N j = 2, 3, . . . , N − 1
g0+ (0) =
This then gives N −1 1 + 1 3 (2) (1) 2 f + (0) + 2 g j (0) = f + (0) , N N j=0
while the terms with f + (0) in this case add up to zero, since N −1 2 2 − 2j + N + = 1, N N j=2
and
N −1 j=2
1 − 2j + N = 0. 2N
The argument for the term with f − is similar. We have the following result. Theorem 11.5. Let f − be a Schwartz function supported on the negative reals, chosen as in Lemma 11.3. Then, for D an operator with spectrum (11.2) one has 1 3 (2) (1) Tr( f − (D/)) = 2 f − (0) + 2 (11.14) f − (0) + − () N with the error term − () = O(−k ), for any k > 0. − − − Proof. We set g − j (u) = P j (u) f − (u/), with P j as in (11.6), and g (u) = − − P (u) f − (u/), with P as in (11.7. Then we can write the spectral action on L N , with the Schwartz function f − , in the form ⎞ ⎛ N −1 ⎠ ⎝g − (k N − 1/2) + Tr( f − (D/)) = g− j (−1/2 + k N − j) . (11.15) k∈Z
j=0
Then one can again use the Poisson summation formula 1 −(2 j + 1)πik k − g j (k N − (2 j + 1)/2) = exp g− j ( ) N N N k∈Z
k∈Z
(11.16)
The Spectral Action and Cosmic Topology
and
g − (k N − 1/2) =
k∈Z
161
1 −πik k exp g − ( ), N N N
(11.17)
k∈Z
and an estimate of the error terms 1 1 k k −k | g− | g − ( )| ≤ O(−k ), j ( )| ≤ O( ) and N N N N k =0
k =0
as in [9] to write (11.15) as ⎞ ⎛ N −1 1 ⎝ − −k ⎠ Tr( f − (D/)) = g− g (0) + j (0) + O( ). N
(11.18)
j=0
One can then compute these values using the explicit form of the polynomials P j− and P − of (11.6) and (11.7) and one obtains 2 3 (2) 2 1 (1) f − (0) + 2 f − (0) f − (0) + N N 2N 2 4 3 (2) (1) f − (0) g1− (0) = 3 f − (0) + 2 f − (0) + N N 2N (11.19) 2 3 (2) 2 + 2 j − N 2 (1) 1 + 2j − N f − (0) + f − (0) + f − (0), g− j (0) = N N 2N j = 2, 3, . . . , N − 1
g0− (0) =
(1) g − (0) = −22 f − (0). f − (0) −
Thus, since N −1 N −1 2 4 2 + 2j − N 1 3 1 + 2j − N + + − 2 = 1 and + + − 1 = 0, N N N 2N 2N 2N j=2
j=2
one then has g − (0) +
N −1
(2)
(1)
3 2 g− j (0) = 2 f − (0) + f − (0).
(11.20)
j=0
Thus, one obtains (11.14).
This gives a complete nonperturbative calculation of the spectral action as follows. Theorem 11.6. The spectral action for an operator D with spectrum (11.2) is given by 1 3 (2) (2) (1) (1) Tr( f (D/)) ∼ 2 ( f + (0) + f − (0)) + 2 ( f + (0) + f − (0)) , (11.21) N up to an error term of the order of O(−k ). Proof. This follows directly from Lemma 11.3 and Theorems 11.4 and 11.5.
162
M. Marcolli, E. Pierpaoli, K. Teh
In particular, one is especially interested in the case where the function f is a Schwartz function that approximates a cutoff function on an interval [−α, α]. In this case, f is an even function and one can assume that the two functions f + and f − can be chosen to be mirror images, so that f + (x) = f − (−x). Corollary 11.7. Let f be an even Schwartz function such that the f + and f − of Lemma 11.3 satisfy f + (x) = f − (−x). Then the spectral action of Theorem 11.6 is given by Tr( f (D/)) =
4 3 (2) f + (0) + O(−k ). N
(11.22)
(k) Proof. The function f ± is the Fourier transform of v k f ± (v), so that (k) v k f ± (v) dv = v k f ± (v) dv. f ± (0) =
R
R±
Using f + (v) = f − (−v) one sees that ∞ v 2 f + (v)dv =
−∞
0
so that
(2) f + (0)
=
(2) f − (0),
0
v 2 f − (v)dv
while
∞
v f + (v)dv = −
0
0 −∞
v f − (v)dv,
f −(1) (0). The 2 -terms then cancel. so that f +(1) (0) = −
11.4. The other spectrum. We also show, in a similar way, how one can compute the spectral action for an operator D whose spectrum is of the form given in (11.3). As before, the multiplicity in row (i) is already in a nice form, while for row (ii) we obtain the following. Lemma 11.8. The multiplicity of − 21 ± m in (11.3) is given by 2m(m − j) for m ≡ j N 2m(m − j + N ) for m ≡ j N
mod N ,
j = 0, 1, . . . ,
mod N ,
j=
N + 1, 2
N + 2, . . . , N − 1. 2
(11.23)
Proof. To handle row (ii), we need to break it up into the pieces m ≡ j (mod N ), where j ∈ {0, 1, . . . , N − 1}. Similar to the previous spectrum, to find nice expressions for the multiplicities, it will be convenient to replace the upper and lower bounds with the highest and lowest values of i N + N2 that satisfy the inequality. We first consider the case with m ≡ j (mod N ), j = 0, 1, . . . , N /2 + 1. The bound 1 − m < i N + N2 ≤ m − 2 can be replaced by N N N + j − m ≤ iN + ≤m− j− 2 2 2 by adding N2 + j − 1 to the left hand side and subtracting j − 2 + N2 from the right-hand side. We check that 0 < N2 + j − 1 ≤ N , and that 0 ≤ j − 2 + N2 < N . If m = k N + j,
The Spectral Action and Cosmic Topology
163
where k = 0, 1, 2, . . ., then we see that there are 2k values of i which satisfy the inequality, and hence −1/2 ± m has multiplicity 2km = (2m(m − j))/N . Once again, when m is 0 or 1, this formula gives us a multiplicity of zero, as is necessary, since in row (ii), the index m begins at m = 2. We then look at the case with m ≡ j (mod N ), j = N /2 + 2, . . . , N − 1. Here the range 1 − m < i N + N2 ≤ m − 2 becomes j −m−
N N N ≤ iN + ≤m− j+ 2 2 2
by adding j − 1 − N2 to the left hand side and subtracting j − 2 − N2 from the right hand side. Again, we check that 0 < j − 1 − N2 ≤ N , and that 0 ≤ j − 2 − N2 < N . If m = k N + j, where k = 0, 1, 2, . . ., then we see that there are 2k + 2 values of i which satisfy the inequality, hence −1/2±m has multiplicity (2k +2)m = (2m(m − j + N ))/N .
We can then compute the polynomials that interpolate the spectral multiplicities in the following way. Lemma 11.9. For m > 0, the multiplicity of −1/2 + m, for m ≡ j mod N , is given by the values P j+ (−1/2 + m) of the polynomials 2 2 2 − 2j 1 − 2j N u + u+ for j = 0, . . . , + 1, N N 2N 2 (11.24) 2 2 − 2 j + 2N 1 − 2 j + 2N N + 2 P j (u) = u + u+ for j = + 2, . . . , N − 1. N N 2N 2
P j+ (u) =
For m ≥ 0, the multiplicity of −1/2 − m, for m ≡ mod N , is given by the values P− (−1/2 − m) of the polynomials 1 + 2j N 2 2 2 + 2j u + u+ for j = 0, . . . , + 1, N N 2N 2 (11.25) 1 + 2 j − 2N N 2 2 + 2 j − 2N − 2 P j (u) = u + u+ , for j = + 2, . . . , N − 1. N N 2N 2
P j− (u) =
The multiplicity of −(N +i N )− 21 , for i ≥ 0, is given by the value P − (−(N +i N )−1/2) of the polynomial P − (u) = −2u − 1.
(11.26)
Proof. This follows directly from the expressions for the multiplicities given in Lemma 11.8 above. We then have the following result. Theorem 11.10. Let f + be a Schwartz function supported on the positive reals, chosen as in Lemma 11.3, and let D be an operator with spectrum given by (11.3). The spectral action is of the form 1 3 (2) (1) 2 f + (0) − 2 Tr( f + (D/)) = f + (0) + O(−k ). (11.27) f + (0) − N
164
M. Marcolli, E. Pierpaoli, K. Teh
Proof. We set g +j (u) = P j+ (u) f + (u/), with P j+ as in (11.24), for j = 0, . . . , N − 1. Then, arguing as in Theorem 11.4, we see that the spectral action is computed by N −1 1 + Tr( f (D/)) = g j (0) + O(−k ). N j=0
We can compute each term explicitly using (11.24), and we obtain 2 3 (2) 2 − 2 j 2 (1) 1 − 2j f + (0) + f + (0) + f + (0) N N 2N N for j = 0, . . . , + 1, 2 (11.28) 2 2 − 2 j + 2N 2 (1) 1 − 2 j + 2N (2) + 3 f + (0) + f + (0), g j (0) = f + (0) + N N 2N N for j = + 2 . . . , N − 1. 2 g +j (0) =
We have N /2+1 j=0 N /2+1 j=0
2 − 2j + N 1 − 2j + 2N
N −1 j=N /2+2 N −1 j=N /2+2
2 − 2 j + 2N = −1, N 1 − 2 j + 2N = −1. 2N
This then gives N −1
g +j (0) = 23 f + (0). f +(2) (0) − 2 f +(1) (0) −
j=0
We then proceed in a way similar to Theorem 11.5 for the case of a test function supported on the negative reals. Theorem 11.11. Let f − be a Schwartz function supported on the negative reals, chosen as in Lemma 11.3. The spectral action for an operator D with spectrum given by (11.3) is given by 1 3 (2) (1) Tr( f − (D/)) = 2 f − (0) + 32 f − (0) + O(−k ). (11.29) f − (0) + N − − − Proof. We set g − j (u) = P j (u) f − (u/), with P j as in (11.25), and g (u) = − − P (u) f − (u/), for P as in (11.26). By the same reasoning of Theorem 11.5 we see that, up to an error term of the order of O(−k ), the spectral action Tr( f − (D/)) is given by ⎞ ⎛ n−1 1 ⎝ − ⎠ g− g (0) + j (0) . N j=0
The Spectral Action and Cosmic Topology
165
We then find 2 3 (2) 2 + 2 j 2 (1) 1 + 2j f − (0) + f − (0) + f − (0), N N 2N N for j = 0, . . . , + 1 2 (11.30) 2 3 (2) 2 + 2 j − 2N 2 (1) 1 + 2 j − 2N − f − (0) + f − (0) g j (0) = f − (0) + N N 2N N + 2, . . . , N − 1 for j = 2
g− j (0) =
and (1) g − (0) = −22 f − (0). f − (0) −
We have N /2+1 j=0 N /2+1 j=0
2 + 2j + N 1 + 2j + 2N
N −1 j=N /2+2 N −1 j=N /2+2
2 + 2 j − 2N − 1 = 3, N 1 + 2 j − 2N − 1 = 1, 2N
so we obtain g − (0) +
n−1
3 (2) 2 (1) g− j (0) = 2 f − (0) + 3 f − (0) + f − (0).
j=0
We then assemble these two cases together and we obtain the following expression for the spectral action. Theorem 11.12. Let f be a Schwartz function on the real line, and let D be an operator with spectrum given by (11.3). For f + and f − chosen as in Lemma 11.3, with f = f + + f − on an open neighborhood of the spectrum of D, the spectral action is given by 1 3 (2) (2) 2 ( f + (0) + Tr( f (D/)) = f − (0)) N (1) (1) + 2 (3 f − (0) − f + (0)) + ( f − (0) − f + (0)) (11.31) up to an error term of the order O(−k ) for arbitary k > 0. (2) (2) f − (0) and In particular, if the function f is an even function, then f + (0) = (1) (1) f + (0), while f − (0) = − f + (0), so one obtains f − (0) = (2)
(1)
f + (0) − 42 f + (0). Tr( f (D/)) = 43 We see then that the resulting spectral action for the two spectra (11.2) and (11.3) of [3] is different, unlike what we have seen in all the other explicit cases of Dirac spectra
166
M. Marcolli, E. Pierpaoli, K. Teh
on manifolds for which we explicitly computed the spectral action, where the spectral action is independent of the spin structure, even though the Dirac spectrum itself may be different for different spin structures. This is in clear contrast with the lens spaces calculation with the correct Dirac spectrum that we describe in Sect. 11.7, below. However, it is interesting to notice that the two spectra (11.2) and (11.3) have the property that the spectral action computed for the operator |D| instead of D restores the symmetry, namely it gives the same result for the two spectra. 11.5. The spectral action for |D|. We consider again an operator D that has as spectrum either (11.2) or (11.3). We replace D by |D| and we proceed to the same calculation of the spectral action as before. Theorem 11.13. Let D = F|D| be an operator with spectrum (11.2). Let f be an even Schwartz function and f + and f − be as in Lemma 11.3, with f = f + + f − on an open neighborhood of the spectrum of D, and with f − (−x) = f + (x). Then the spectral action Tr( f (|D|/)) is given by 1 3 (2) 4 f + (0) + 22 (11.32) Tr( f (|D|/)) = f +(1) (0) + O(−k ). N Proof. Let λ±j,m = ±m − 1/2 and λi− = −i N − 1/2 be the arithmetic progressions of the Dirac spectrum (11.2) on L N with the canonical spin structure. We have obtained in Lemma 11.2 polynomials P j+ (u), P j− (u) and P − (u) such that, for m > 0, P j+ (λ+j,m ) is the spectral multiplicity of λ+j,m , while for m ≤ 0, P j− (λ−j,m ) is the spectral multiplicity of λ−j,m , and P − (λi− ) is the spectral multiplicity of λi− . When we replace D by |D|, we want new polynomials P¯ j− (u) and P¯ − (u), with the property that, for m ≤ 0, P¯ j− (−λ−j,m ) is the spectral multiplicity of λ−j,m and P¯ − (−λi− ) is the spectral multiplicity of λi− . It suffices to choose P¯ j− (−u) = P j− (u) and P¯ − (−u) = P − (u). All the polynomials P j+ , P j− and P − of Lemma 11.2 are of the form c2 u 2 +c1 u+c0 for suitable coefficients ck . Thus, while the P j+ remain the same, the corresponding P¯ j− and P¯ − will be of the form c2 u 2 − c1 u + c0 . More precisely, we obtain 2 P¯0− (u) = u 2 − N 2 − P¯1 (u) = u 2 − N 2 − P¯ j (u) = u 2 − N P¯ − (u) = 2u − 1.
1 2 u+ , N 2N 3 4 u+ , N 2N 1 + 2j − N 2 + 2j − N u+ , N 2N
(11.33) j = 2, 3, . . . , N − 1, (11.34)
Similarly, for f even with f = f + + f − on an open neighborhood of the spectrum, as before, and with f + (−u) = f − (u), we have f + (−λ−j,m ) = f − (λ−j,m ) and f + (−λ−j ) = f − (λ−j ). Correspondingly, we now set u u u ¯− g +j (u) = P j+ (u) f + ( ), g¯ − g¯ − (u) = P¯ − (u) f − ( ). j (u) = P j (u) f − ( ), (11.35)
The Spectral Action and Cosmic Topology
167
We see then that Tr( f (|D|/)) is given by ⎞ ⎛ −1 N −1 N 2 j − 1 2 j + 1 1 ⎝ )+ ) + g¯ − (k N + )⎠ . g +j (k N + g¯ − j (k N + 2 2 2 k∈Z
j=0
(11.36)
j=0
We then use Poisson summation as before and we find ⎞ ⎛ N −1 N −1 1 ⎝ + −k − ⎠ Tr( f (|D|/)) = g ¯− g j (0) + j (0) + g¯ (0) + O( ). (11.37) N j=0
j=0
We then see that N −1
(2)
(1)
g +j (0) = 23 f + (0) + 2 f + (0)
j=0
as before, while N −1
3 (2) 2 (1) 3 (2) 2 (1) − g ¯− j (0) + g¯ (0) = 2 f − (0) − f − (0) = 2 f + (0) + f + (0),
j=0
so that (11.32) holds.
We now see that this is the same result obtained from the second spectrum (11.3). Theorem 11.14. Let D = F|D| be an operator with spectrum (11.3). Let f be an even Schwartz function and f + and f − be as in Lemma 11.3, with f = f + + f − on an open neighborhood of the spectrum of D, and with f − (−x) = f + (x). Then the spectral action Tr( f (|D|/)) is given by 1 3 (2) (1) Tr( f (|D|/)) = 4 f + (0) + 22 (11.38) f + (0) + O(−k ). N Proof. The argument is the same as in the previous case, but applied to the eigenvalues and multiplicities (11.3) and the polynomials P j+ , P j− and P − of Lemma 11.9. We then compute Tr( f (|D|/)) as in the case of the canonical spin structure, using the corre− sponding polynomials P¯ j− and P¯ − and the functions g¯ − j and g¯ as above. We obtain again the expression (11.37), where in this case N −1
(2) (1) g +j (0) = 23 f + (0) f + (0) − 2 f + (0) −
j=0
and N −1
3 (2) 2 (1) − g ¯− j (0) + g¯ (0) = 2 f − (0) − 3 f − (0) + f − (0)
j=0 (2)
(1)
= 23 f + (0), f + (0) + 32 f + (0) + so that Tr( f (|D|/)) is given by 1 3 (2) (1) (2) (1) 2 f + (0) − 2 f + (0) + 23 f + (0) , f + (0) − f + (0) + 32 f + (0) + N up to an error term of the order of O(−k ). This gives again the same (11.38) as for the canonical spin structure.
168
M. Marcolli, E. Pierpaoli, K. Teh
11.6. The slow-roll potential: a false positive. Now we use this result to compute the spectral action for an operator D2 with D=
0 D ⊗ 1 + i ⊗ DS1 D ⊗ 1 − i ⊗ DS1 0
,
(11.39)
where D S 1 has spectrum β −1 (Z + 1/2), and D is an operator with spectrum either (11.2) or (11.3). The spectrum of the operator D2 will be contained in the set of values of the form ± 2 (λ j,m ) (a)−2 + λ2n (β)−2 and (λi− )2 (a)−2 + λ2n (β)−2 , where λ±j,m and λi− are the arithmetic progressions associated to the spectrum of D and λn = n + 1/2 are the eigenvalues on a circle of radius one. We see that the pairs of points (u, v) in R2 which are of the form (λ±j,m , λn ) or (λi− , λn ) all lie outside of a vertical strip around u = 0. We fix a value α < 1 such that the strip u ∈ (−α, α) contains one such pair. We also fix a k > 0, which will determine the order O(−k ) of error in the spectral action computation. Let then (u k ) be a smooth function, which is equal to zero for u ≤ 0 and is equal to one for u ≥ α. Lemma 11.15. Suppose given a polynomial P(x) = c2 x 2 + c1 x + c0 , and let h be a Schwartz function on R. The difference between the integrals I1 =
R2
P(x) (x k (a)k ) h(x 2 + y 2 ) d x d y
(11.40)
and
∞ π/2
I2 = 0
−π/2
(
c2 2 ρ + c1 ρ cos θ + c2 ) h(ρ 2 ) ρ dρ dθ 2
(11.41)
is bounded by |I1 − I2 | = O(−k ).
(11.42)
Proof. The difference I1 − I2 is computed by R
α
P(x) (x (a) ) h(x + y ) d x k
k
2
2
0
αk dy ≤ C . (a)k
We can then proceed to compute the spectral action. Theorem 11.16. The spectral action for the operator D of (11.39), where D has spectrum (11.2) is of the form Tr(h(D2 /2 )) = 2π 4 a 3 β 0
∞
∞
u h(u) du + 23 a 2 β
u 1/2 h(u) du + O(−k ).
0
(11.43)
The Spectral Action and Cosmic Topology
169
Proof. For a given k > 0, we choose a cutoff (u k ) as in Lemma 11.15. We then set + (u) = (u) and − (u) = + (−u). We then consider, for j = 0, . . . N − 1, functions of the form g +j (u, v) = 2P j+ (u) + (u k ) h(u 2 (a)−2 + v 2 (β)−2 ),
(11.44)
where the polynomials P j+ (u) are as in (11.5). We also set k 2 −2 ¯− + v 2 (β)−2 ), g− j (u, v) = 2 P j (u) − (u ) h(u (a)
(11.45)
with the polynomials P¯ j− (u) of (11.33) and g − (u, v) = 2 P¯ − (u) − (u k ) h(u 2 (a)−2 + v 2 (β)−2 ),
(11.46)
with P¯ − (u) as in (11.34). The spectral action Tr(h(D2 /2 )), for D with spectrum (11.2), is then given by N −1
Tr(h(D2 /2 )) =
g +j (n N +
j=0
+
N −1
1 (2 j − 1) ,m + ) 2 2
g− j (n N −
j=0
1 (2 j + 1) ,m + ) 2 2
1 1 +g − (n N − , m + ). 2 2
(11.47)
We compute it by applying the Poisson summation formula to the functions (11.44), (11.45), and (11.46). We obtain, as in the previous cases, ⎞ ⎛ N −1 N −1 1 ⎝ Tr(h(D2 /2 )) = g +j (0, 0) + g− g − (0, 0)⎠ + O(−k ). (11.48) j (0, 0) + N j=0
j=0
We then use Lemma 11.15 to estimate the integrals, up to an error term of order O(−k ), to be of the form ∞ π/2 N −1 + 2 ρ 2 (a)2 + ρ cos θ (a) h(ρ 2 ) ρ dρ dθ g j (0, 0) = 2 aβ 0
j=0
−π/2
∞
= 2π 4 a 3 β
∞
ρ 3 h(ρ 2 )dρ + 23 a 2 β
0
ρ 2 h(ρ 2 )dρ,
0
where we used the fact that N −1
P j+ (u) = 2u 2 + u.
j=0
After a change of variables, we write the above as N −1 j=0
g +j (0, 0)
= π a β 4 3
0
∞
∞
uh(u) du + a β 3 2
0
u 1/2 h(u) du.
(11.49)
170
M. Marcolli, E. Pierpaoli, K. Teh
Similarly, using the approximation of Lemma 11.15 we obtain, up to an error term of the order of O(−k ), N −1
g− g − (0, 0) = 2π 4 a 3 β j (0, 0) +
j=0
∞
ρ 3 h(ρ 2 )dρ + 23 a 2 β
0
∞
= π a β 4 3
ρ 2 h(ρ 2 )dρ
0
∞
uh(u) du + a β 3 2
0
∞
u 1/2 h(u) du,
0
(11.50) where we used the fact that N −1
P¯ j− (u) + P¯ − (u) = 2u 2 − u
j=0
and that − (u) = + (−u). This then gives (11.43).
The case where D has spectrum (11.3) is analogous and yields the same result. Theorem 11.17. Consider the operator D of (11.39), where D has spectrum (11.3). The spectral action is of the form ∞ ∞ u h(u) du + 23 a 2 β u 1/2 h(u) du + O(−k ). Tr(h(D2 /2 )) = 2π 4 a 3 β 0
0
(11.51) Proof. One proceeds exactly as in Theorem 11.16, but using the expressions for P j+ , P¯ j− and P¯ − as in Theorem 11.14. One then has N −1
N −1
P j+ (u) = 2u 2 − u − 1,
j=0
P¯ j− (u) + P − (u) = 2u 2 − 3u + 1,
j=0
so that, using − (u) = + (−u), one correspondingly obtains ∞ N −1 + 4 3 3 2 3 2 g j (0, 0) = 2π a β ρ h(ρ )dρ − 2 a β 0
j=0
∞
ρ 2 h(ρ 2 )dρ
0
∞
−2π 2 aβ h(ρ 2 )ρdρ 0 ∞ ∞ 4 3 3 2 u h(u) du − a β u 1/2 h(u) du = π a β 0 0 ∞ h(u)du. −π 2 aβ
(11.52)
0
Similarly, one obtains N −1
g− g − (0, 0) = π 4 a 3 β j (0, 0) +
j=0
∞
u h(u) du + 33 a 2 β
0
∞
u 1/2 h(u)du
0 ∞
+π 2 aβ
h(u)du. 0
(11.53)
The Spectral Action and Cosmic Topology
171
Thus, adding these contributions one obtains then the same (11.51) as in the previous case. We then see that the form of the associated potential V (φ) coming from the perturbations D2 → D2 + φ 2 is very different from the 3-sphere and the other spherical manifolds computed in this paper, quaternionic and dodecahedral space. We set ∞ ∞ V(x) = u (h(u + x) − h(u)) du, Z(x) = u 1/2 (h(u + x) − h(u)) du, 0
0
(11.54) in the variable x = φ 2 /2 . Proposition 11.18. Let D be the operator of (11.39). We have Tr(h((D2 + φ 2 )/2 )) = Tr(h(D2 /2 )) + 2π 4 a 3 βV(φ 2 /2 ) + 23 a 2 βZ(φ 2 /2 ). (11.55) Thus, the potential V (x), for x = φ 2 /2 is of the form V (x) = 2π 4 a 3 β V(φ 2 /2 ) + 23 a 2 β Z(φ 2 /2 ). Proof. This is an immediate consequence of Theorems 11.16 and 11.17.
(11.56)
We then see that the form of the slow-roll parameters is also different in this case. Proposition 11.19. The slow-roll parameters for the potential V (x) are given by (x) =
m 2Pl A, 8π
and
η(x) =
m 2Pl (B − A), 8π
(11.57)
where 1 πCV (x) + Z (x) 2 , 2 πCV(x) + Z(x) πCV (x) + Z (x) B= . πCV(x) + Z(x) A=
(11.58) (11.59)
Proof. This follows directly from the definition of the slow-roll parameters, having imposed the condition (t) ∼ 1/a(t), so that a = C, on the Friedmann form of the spacetime back in Lorentzian signature. This calculation creates a “false positive” which gives the impression that there are spherical manifolds for which the inflation potential and slow-roll parameters are genuinely different from those of the sphere. This would make for a much stronger correlation between inflation and cosmic topology than what we have observed in the previous section, with different inflation scenarios not only between spherical and flat cases, but even between different topologies with the same underlying spherical geometry. However, this turns out not to be the case. The true story of the lens spaces, described in the coming section, shows that in fact, with the correct calculation of the Dirac spectrum, they behave exactly as the other spherical topology, with the same slow-roll parameters as in the simply connected case.
172
M. Marcolli, E. Pierpaoli, K. Teh
11.7. Lens spaces: a discrepancy. In this section we compute the Dirac spectrum for lens space using the same generating function technique due to Bär [2], that we used to compute the Dirac spectrum for the Poincaré homology sphere, and compare the results to the calculation in [3]. For simplicity, let us just consider the space L N = SU (2)/Z N in the case N = 4 with the canonical spin structure. By applying Eqs. (7.4), (7.5), one obtains that the generating functions for the spectral multiplicities for L4 , with the canonical spin structure, are given by: 2(z + 5z 3 + z 5 + z 7 ) , (−1 + z 2 )3 (1 + z 2 )2 2(1 + z 2 + 5z 4 + z 6 ) . F− (z) = − (−1 + z 2 )3 (1 + z 2 )2 F+ (z) = −
(11.60) (11.61)
Proceeding in precisely the same manner as in the case of the Poincaré homology sphere, one obtains the following lemma. Lemma 11.20. There are polynomials Pk (u), for k = 0, . . . , 3, so that Pk (3/2+k+4 j) = m(3/2 + k + 4 j, D) for all j ∈ Z. The Pk (u) are given as follows: Pk = 0, whenever k is even, 1 1 1 P1 (u) = − u + u 2 , 8 2 2 1 3 1 P3 (u) = − + u + u 2 . 8 2 2 Before comparing these multiplicities with the ones given in [3], let us first compute the nonperturbative spectral action of the lens space. Theorem 11.21. Let D be the Dirac operator on L4 , with the canonical spin structure. Then, for f a Schwartz function, the spectral action is given by 1 1 3 (2) f (0) − f (0) , (11.62) Tr( f (D/)) = 4 4 which is precisely 1/4 of the spectral action on the sphere. Proof. As usual the result follows by applying Poisson summation to the functions g j (u) = P j (u) f (u/). This gives, up to an error term which is of the order of O(−k ) for any k > 0, the spectral action in the form 3 1 1 Tr( f (D/)) = g j (0) = P j (u) f (u/)du. 4 4 R j=0 j It suffices then to notice that 3 j=0
1 P j (u) = u 2 − . 4
The result then follows as in the sphere case.
The Spectral Action and Cosmic Topology
173
Observe that this time around, the spectral action is a constant multiple of the spectral action of the sphere, and so one obtains the same slow-roll parameters as in the simply connected case, just as with the other spherical space forms. Let us compare the multiplicities obtained using the generation function method in Lemma 11.20 with those obtained using the results of [3] in Lemma 11.2. By setting N = 4 in Lemma 11.2 it is immediately evident that the two sets of multiplicities do not agree. As a side remark, even if we replace −(m + 1) < i N with −(m − 2) ≤ i N in Eq. (11.2) when performing the computation of Lemma 11.2, this just results in altering P0± , and P1± very slightly, while leaving the other P j± unchanged, and the resulting multiplicities still do not agree with the multiplicities of Lemma 11.20. We are inclined to believe that the generating function method of Lemma 11.20 gives the correct answer because of two reasons. First, the generating function method leads to a spectral action of 1/|G| times the spectral action of S 3 , where G is the group acting on S 3 , and this is exactly the result we obtained for the other spherical space forms. Secondly, if one computes the Dirac spectrum of SU (2)/Q8 using the generating function method, one gets the same answer as obtained by Ginoux in [18], where the Dirac spectrum of SU (2)/Q8 is computed using representation theoretic methods. Moreover, the result of our Lemma 11.20 also agrees with the computation for lens spaces in [16], where the effect of cosmic topology on the Casimir energy is computed. References 1. Aurich, R., Lustig, S., Steiner, F., Then, H.: Cosmic microwave background alignment in multi-connected universes. Class. Quantum Grav. 24, 1879–1894 (2007) 2. Bär, C.: The Dirac operator on space forms of positive curvature. J. Math. Soc. Japan 48(1), 69–83 (1996) 3. Bär, C.: The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces. Arch. Math. 59, 65–79 (1992) 4. Bär, C.: Dependence of Dirac Spectrum on the Spin Structure. In: Séminaires & Congrès, 4. Bouoguignon J.P., Bânson, T., Hija-â, O. (eds.) Global Anal. and Harmonic Anal. (Luming, 2000), Paris: French Math. Soc., 2000, pp. 17–33 5. de Bernardis, P., Ade, P.A.R., Bock, J.J., Bond, J.R., Borrill, J., Boscaleri, A., Coble, K., Crill, B.P., De Gasperis, G., Farese, P.C., Ferreira, P.G., Ganga, K., Giacometti, M., Hivon, E., Hristov, V.V., Iacoangeli, A., Jaffe, A.H., Lange, A.E., Martinis, L., Masi, S., Mason, P.V., Mauskopf, P.D., Melchiorri, A., Miglio, L., Montroy, T., Netterfield, C.B., Pascale, E., Piacentini, F., Pogosyan, D., Prunet, S., Rao, S., Romeo, G., Ruhl, J.E., Scaramuzzi, F., Sforna, D., Vittorio, N.: A flat Universe from high-resolution maps of the cosmic microwave background radiation. Nature 404, 955–959 (2000) 6. van den Broek, T., van Suijlekom, W.D.: Supersymmetric QCD and noncommutative geometry. http:// arXiv.org/abs/1003.3788v1 [hepth], 2010 7. Caillerie, S., Lachièze-Rey, M., Luminet, J.P., Lehoucq, R., Riazuelo, A., Weeks, J.: A new analysis of the Poincaré dodecahedral space model. Astron. and Astrophys. 476(2), 691–696 (2007) 8. Chamseddine, A., Connes, A.: The spectral action principle. Commun. Math. Phys. 186(3), 731–750 (1997) 9. Chamseddine, A., Connes, A.: The uncanny precision of the spectral action. Commun. Math. Phys. 293, 867–897 (2010) 10. Chamseddine, A., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11(6), 991–1089 (2007) 11. Connes, A.: Gravity coupled with matter and foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996) 12. Cornish, N.J., Spergel, D.N., Starkman, G.D., Komatsu, E.: Constraining the topology of the universe. Phys. Rev. Lett. 92, 201302 (2004) 13. Dahl, M.: Prescribing eigenvalues of the Dirac operator. Manus. Math. 118, 191–199 (2005) 14. Dahl, M.: Dirac eigenvalues for generic metrics on three-manifolds. Ann. Global Anal. Geom. 24, 95–100 (2003) 15. De Simone, A., Hertzberg, M.P., Wilczek, F.: Running inflation in the Standard Model. Phys. Lett. B 678, 1–8 (2009)
174
M. Marcolli, E. Pierpaoli, K. Teh
16. Dowker, J.S.: Spherical universe topology and the Casimir effect. Class. Quant. Grav. 21, 4247–4271 (2004) 17. Gausmann, E., Lehoucq, R., Luminet, J.P., Uzan, J.P., Weeks, J.: Topological lensing in spherical spaces. Class. Quant. Grav. 18, 5155–5186 (2001) 18. Ginoux, N.: The spectrum of the Dirac operator on SU2 /Q 8 . Manus. Math. 125(3), 383–409 (2008) 19. Gomero, G.I., Reboucas, M.J., Tavakol, R.: Detectability of cosmic topology in almost flat universes. Class. Quant. Grav. 18, 4461–4476 (2001) 20. Gomero, G.I., Reboucas, M.J., Teixeira, A.F.F.: Spikes in cosmic crystallography II: topological signature of compact flat universes. Phys. Lett. A 275, 355–367 (2000) 21. Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974) 22. Kamionkowski, M., Spergel, D.N., Sugiyama, N.: Small-scale cosmic microwave background anisotropies as a probe of the geometry of the universe. Astrophys. J. 426, L57–60 (1994) 23. Lachièze-Rey, M., Luminet, J.P.: Cosmic topology. Phys. Rep. 254, 135–214 (1995) 24. Lehoucq, R., Weeks, J., Uzan, J.P., Gausmann, E., Luminet, J.P.: Eigenmodes of three-dimensional spherical spaces and their applications to cosmology. Class. Quant. Grav. 19, 4683–4708 (2002) 25. Luminet, J.P., Weeks, J., Riazuelo, A., Lehoucq, R.: Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background. Nature 425, 593– 595 (2003) 26. Marcolli, M., Pierpaoli, E.: Early universe models from noncommutative geometry. http://arXiv.org/abs/ 0908.3683v1 [hepth], 2009 27. McInnes, B.: APS instability and the topology of the brane-world. Phys. Lett. B 593(1-4), 10–16 (2004) 28. Nelson, W., Sakellariadou, M.: Natural inflation mechanism in asymptotic noncommutative geometry. Phys. Lett. B 680, 263–266 (2009) 29. Niarchou, A., Jaffe, A.: Imprints of spherical nontrivial topologies on the cosmic microwave background. Phys. Rev. Lett. 99, 081302 (2007) 30. de Oliveira-Costa, A., Tegmark, M., Zaldarriaga, M., Hamilton, A.: Significance of the largest scale CMB fluctuations in WMAP. Phys. Rev. D 69, 063516 (2004) 31. Pfäffle, F.: The Dirac spectrum of Bieberbach manifolds. J. Geom. Phys. 35, 367–385 (2000) 32. Riazuelo, A., Uzan, J.P., Lehoucq, R., Weeks, J.: Simulating Cosmic Microwave Background maps in multi-connected spaces. Phys. Rev. D 69, 103514 (2004) 33. Riazuelo, A., Weeks, J., Uzan, J.P., Lehoucq, R., Luminet, J.P.: Cosmic microwave background anisotropies in multiconnected flat spaces. Phys. Rev. D 69, 103518 (2004) 34. Roukema, B.F., Róza´nski, P.T.: The residual gravity acceleration effect in the Poincaré dodecahedral space. Astron. and Astrophy. 502, 27 (2009) 35. Souradeep, T., Hajian, A.: Statistical isotropy of CMB anisotropy from WMAP. http://arXiv.org/abs/astroph/0502248v1, 2005 36. Spergel, D.N., Verde, L., Peiris, H.V., Komatsu, E., Nolta, M.R., Bennett, C.L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer, S.S., Page, L., Tucker, G.S., Weiland, J.L., Wollack, E., Wright, E.L.: First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: determination of cosmological parameters. Astrophys. J. Suppl. 148, 175–194 (2003) 37. Tegmark, M., de Oliveira-Costa, A., Hamilton, A.: A high resolution foreground cleaned CMB map from WMAP. Phys. Rev. D. 68, 123523 (2003) 38. Uzan, J.P., Kirchner, U., Ellis, G.F.R.: WMAP data and the curvature of space. Mon. Not. Roy. Astron. Soc. 344, L65 (2003) 39. Uzan, J.P., Riazuelo, A., Lehoucq, R., Weeks, J.: Cosmic microwave background constraints on lens spaces. Phys. Rev. D 69, 043003 (2004) 40. Weeks, J., Gundermann, J.: Dodecahedral topology fails to explain quadrupole-octupole alignment. Class. Quant. Grav. 24, 1863–1866 (2007) 41. Weeks, J., Lehoucq, R., Uzan, J.P.: Detecting topology in a nearly flat spherical universe. Class. Quant. Grav. 20, 1529–1542 (2003) Communicated by A. Connes
Commun. Math. Phys. 304, 175–186 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1201-5
Communications in
Mathematical Physics
Spectral Properties of the Renormalization Group at Infinite Temperature Mei Yin Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA. E-mail: [email protected] Received: 21 May 2010 / Accepted: 20 August 2010 Published online: 27 January 2011 – © Springer-Verlag 2011
Abstract: The renormalization group (RG) approach is largely responsible for the considerable success that has been achieved in developing a quantitative theory of phase transitions. Physical properties emerge from spectral properties of the linearization of the RG map at a fixed point. This article considers RG for classical Ising-type lattice systems. The linearization acts on an infinite-dimensional Banach space of interactions. At a trivial fixed point (zero interaction), the spectral properties of the RG linearization can be worked out explicitly, without any approximation. The results are for the RG maps corresponding to decimation and majority rule. They indicate spectrum of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, only residual spectrum. This may serve as a lesson in what one might expect in more general situations. 1. Introduction We consider renormalization group (RG) transformations for Ising-type lattice spin systems on Zd . Our original lattice is denoted by L and our image lattice is denoted by L . The image lattice L indexes a partition of L into cubical blocks, all with the same cardinality bd . Thus for each site y in L , there is a corresponding block yo that is a subset of L, given by yo = {x : byi −
b−1 b−1 ≤ xi ≤ byi + , 1 ≤ i ≤ d} 2 2
(1.1)
b−2 b ≤ xi ≤ byi + , 1 ≤ i ≤ d} 2 2
(1.2)
for odd blocking factor b; and yo = {x : byi −
for even blocking factor b. More generally, for each subset Y of L , there is a corresponding union of blocks Y o that is a subset of L. A spin variable σx = ±1 is assigned
176
M. Yin
to each site x in L, and a block spin variable σy = ±1 is assigned to each site y in L . If X is a finite subset of the original lattice, then σ X denotes the spin variable x∈X σx . Similarly,if Z is a finite subset of the image lattice, then σ Z denotes the block spin variable z∈Z σz . The main physical properties of L are encoded in the Hamiltonian H (σ ) = − X J (X )σ X , where J is the original interaction defined on nonempty finite subsets of L. Likewise, the main physical properties of L are encoded in the Hamiltonian H (σ ) = − Y J (Y )σY , where J is the resulting interaction defined on nonempty finite subsets of L . Here is the formal definition of the RG map: X J (X )σ X e Y J (Y )σY σ y∈L Ty (σ, σy )e = , (1.3) X J (X )σ X Y J (Y )σY σ e σ e where σ and σ (normalized sums) denote the product probability measures on {+1, −1}L and {+1, −1}L , respectively, and Ty (σ, σy ) denotes a specific RG probability kernel, which depends only on σ through yo , and satisfies both a symmetry condition, Ty (σ, σy ) = Ty (−σ, −σy ), and a normalization condition,
(1.4)
Ty (σ, σy ) = 1
(1.5)
σ
for every σ and every y. Notice that because of (1.4) and (1.5), Ty (σ, +1) = Ty (σ, −1) = 1. σ
(1.6)
σ
In the following, we restrict our attention to a special kind of deterministic probability kernel: There is a function φy (σ ) that depends only on σ through yo , and Ty (σ, σy ) = 2δ(φy (σ ), σy ). Our basic assumption is that the original interaction J lies in a Banach space Br , with norm ||J ||r = sup |J (X )|erl(x,X ) , (1.7) x∈L X :x∈X
where the constant r ≥ 0, d is a metric on L, and l(x, X ) = sup{d(x, y) : y ∈ X }, with the convention that l(x, ∅) = 0. Correspondingly, there is a paired Banach space Br∗ . As |
J1 (X )J2 (X )| ≤
X
|J1 (X )|
X
=
x∈L X :x∈X
≤
sup
x∈L x∈X
≤ sup
1 |J2 (X )| |X |
x∈X
1 |J1 (X )||J2 (X )| |X |
1 |J2 (X )|e−rl(x,X ) |J1 (X )|erl(x,X ) |X |
x∈L X :x∈X
X :x∈X
|J1 (X )|erl(x,X ) ·
sup
x∈L x∈X
1 |J2 (X )|e−rl(x,X ) , |X |
Spectral Properties of the Renormalization Group at Infinite Temperature
177
a suitable Br∗ norm is defined by ||J ||r∗ =
sup
x∈L x∈X
1 |J (X )|e−rl(x,X ) . |X |
(1.8)
Notice that here Br is technically not the dual space of Br∗ , and Br∗ is technically not the dual space of Br . The spaces Br and Br∗ are paired in the sense that each one is part of the dual space of the other, or in other words, each one consists of continuous linear functions defined on the other. We study the situation when ||J ||r = 0 (indication of infinite temperature). We consider the spectrum of the linearization L(J ) of two commonly used RG transformations, decimation and deterministic majority rule with odd blocking factor. We show that this spectrum is of an unusual kind: dense point spectrum for which the adjoint operators L∗ (J ) have no point spectrum at all, but only residual spectrum. Remark 1. In this paper, spectrum is crudely divided into 3 types [1]: For a bounded linear operator A acting on a Banach space A : B → B: 1. λ is in the point spectrum ⇐⇒ there exists B u = 0, such that (A − λ)u = 0, i.e., Kernel(A − λI) is nontrivial. 2. λ is in the residual spectrum ⇐⇒ λ is not in the point spectrum, and Range(A − λI) = B. 3. λ is in the continuous spectrum ⇐⇒ λ is not in the point spectrum or the residual spectrum, Range(A − λI) = B, and Range(A − λI) = B. This definition is too simple to fully capture the notion of continuous spectrum, but it will be adequate for our purposes. Israel [2] found the operator bound of L(J ) for decimation in a Banach algebra setting, but did not go into detail about the spectral type of this transformation. He also examined the operator bound of L(J ) for majority rule on the triangular lattice. These results are extended by the present investigation, which includes the spectral type of L(J ) and L∗ (J ) for decimation (Theorems 1 and 2) and majority rule (Theorems 3 and 4). Even though this investigation is focused on the RG transformation acting on a system very close to a trivial interaction, it serves as a test case—after all, if it is reasonably difficult to compute the spectrum of the RG map, then one can get an idea of what to expect by computing in a simple case. If even this case has bizarre spectral properties, then it may serve as a lesson in what to expect in more general situations. 2. Some General Results Proposition 1. The renormalized coupling constants J are given by the expression J (Z ) = σ Z log(W (σ )), (2.1) σ
where W (σ ) is the frozen block spin partition function given by W (σ ) = Ty (σ, σy )e X J (X )σ X . σ y∈L
(2.2)
178
M. Yin
Proof. In order to write down an explicit expression of J , we use Fourier series on the group {+1, −1}L . If H (σ ) = − Y J (Y )σY , then J (Z ) = σ −H (σ )σ Z . We see that ⎛ ⎞ J (Z ) = σ Z log ⎝ Ty (σ, σy )e X J (X )σ X ⎠ σ y∈L
σ
+
σ
σ Z
log
e
Y
J (Y )σY
σ
An important observation here is that log
−
σ
σ Z
log
σ
e
Y
J (Y )σY
e
σ
and log
X
J (X )σ X
σ
e
. (2.3)
X
J (X )σ X
are constants with respect to σ Z ; thus, when summing over all possible image configurations σ , they both vanish. Proposition 2. Suppose the original interaction J is at infinite temperature. Then for every subset W of the original lattice and every subset Z of the image lattice, the partial (Z ) derivative ∂∂ JJ (W ) of the RG transformation is given by the expression ∂ J (Z ) = Ty (σ, σy )σW σ Z . ∂ J (W ) σ σ
(2.4)
y∈L
Proof. We take the derivative of both sides of (2.1) with respect to J (W ): J (X )σ X σ W ∂ J (Z ) σ y∈L Ty (σ, σy )e X = σZ . J (X )σ X X ∂ J (W ) σ y∈L Ty (σ, σy )e σ
(2.5)
When J is at infinite temperature, i.e., ||J ||r = 0, J (X ) = 0 for every subset X of the original lattice. Definition 1. For every subset Z of the image lattice, the linearization L(J ) of the RG transformation for J at infinite temperature is given by a linear function of K (which indicates variation from infinite temperature), L(J )K (Z ) =
∂ J (Z ) W
∂ J (W )
K (W ),
(2.6)
where W ranges over all finite subsets of the original lattice. Definition 2. The adjoint of the linearization L∗ (J ) of the RG transformation for J at infinite temperature is characterized by the usual correspondence between adjoint operators, K 1 (X )L(J )K 2 (X ) = K 2 (Y )L∗ (J )K 1 (Y ), (2.7) X
Y
where X ranges over all finite subsets of the image lattice, and Y ranges over all finite subsets of the original lattice. Definition 3. A constant pure magnetic field is one such that K (X ) = 0 except for one-point sets {x}, where K ({x}) = m, a constant.
Spectral Properties of the Renormalization Group at Infinite Temperature
179
3. Spectrum of the Linearization of Decimation Transformation and its Adjoint at Infinite Temperature Proposition 3. Consider decimation transformation with blocking factor b and a probability kernel defined by φy (σ ) = σby ,
(3.1)
where by = b(y1 , . . . , yd ) = (by1 , . . . , byd ). Suppose the original interaction J is at infinite temperature. Then for every subset Z of the image lattice, the linearization L(J ) of this transformation is given by the expression L(J )K (Z ) = K (bZ ),
(3.2)
where bZ = ∪z∈Z {bz}. Proof. We evaluate (2.4) explicitly: ∂ J (Z ) = δ(W, bZ ) = δ(W, bZ ), ∂ J (W ) σ
(3.3)
where δ is the Kronecker delta function. Proposition 4. Consider the adjoint of decimation transformation with blocking factor b and a probability kernel defined by φy (σ ) = σby .
(3.4)
Suppose the original interaction J is at infinite temperature. Then for every subset Z of the original lattice, the adjoint of the linearization L∗ (J ) of this transformation is given by the expression K (Y ) if Z = bY; ∗ L (J )K (Z ) = (3.5) 0 otherwise. Proof. We notice that in this case, (2.7) becomes K 1 (X )L(J )K 2 (X ) = K 1 (X )K 2 (bX ). X
(3.6)
X
Without loss of generality, we assume L∗ (J )K ({0}) = 0, which amounts to an index shift. Theorem 1 (Israel). Suppose the original interaction J is at infinite temperature. Then in the Banach Space Br , the spectrum of the linearization of the decimation transformation L(J ) is all point spectrum, |λ| ≤ 1. Proof. The proof of this theorem follows from several propositions. Proposition 5. ||L(J )|| = 1.
180
M. Yin
Proof. We check that for each fixed x ∈ L, X :x∈X |L(J )K (X )|erl(x,X ) ≤ ||K ||r , which would imply ||L(J )|| ≤ 1. By (3.2), |L(J )K (X )|erl(x,X ) = |K (bX )|erl(x,X ) X :x∈X
≤
X :x∈X
|K (bX )|e
rl(bx,bX )
≤
X :bx∈bX
|K (X )|erl(bx,X ) ≤ ||K ||r .
(3.7)
X :bx∈X
The claim is verified when we realize that a constant pure magnetic field is an eigenvector with eigenvalue 1. Corollary 1. Every eigenvalue λ of L(J ) satisfies |λ| ≤ 1. Proposition 6. Every |λ| ≤ 1 is an eigenvalue. Proof. For a generic λ, we display one eigenvector here. In fact, with some further thought, it is not hard to show that there are infinitely many eigenvectors for each λ. The eigenvector K is defined by K ({(bn , 0, . . . , 0)}) = λn K ({(1, 0, . . . , 0)}) = λn
(3.8)
for n ≥ 0, and for all the other subsets X, K (X ) is set to zero. Moreover, we have stricter restrictions on the eigenvector K that lies in a Banach space Br : r > 0. Proposition 7. For λ = 0 and for every finite subset |X | > 1, we must have K (X ) = 0 for the eigenvector K . Proof. This follows from the observation that we can always pick a site, say x, in X , such that l(x, X ) > 0. As a result, bn x is a site in bn X , and l(bn x, bn X ) = bn l(x, X ) > 0. Since L(J )K (X ) = K (bX ), we must have K (bn X ) = λn K (X ). Then due to the fact that K is an eigenvector, we need to ensure that |λ|n |K (X )|er b
n l(x,X )
< ∞.
(3.9)
The following statements concern translation-invariant Hamiltonians. In this case, it is believed that the RG map should be almost a contraction near the trivial fixed point. Almost means that except for a few degrees of freedom (maybe just one) it should be a contraction. If one restricts oneself to even interactions, then it should actually be a contraction—reflecting the fact that if we start with an even interaction in the very high temperature phase, then the RG map would drive it to the zero interaction. Proposition 8. Restricted to the translation-invariant even subspace of Br : r = 0, the point spectrum of L is |λ| < 1. Proof. For |λ| < 1, the eigenvector K may be defined by K ({x, y}) = 1
(3.10)
for x, y that are nearest-neighbors, and in general, K (bX ) = λK (X ). However, no such eigenvector would work for |λ| = 1. Suppose the nontrivial eigenvector K (X ) = m = 0 for some finite subset X : |X | > 1. Due to translation-invariance, for arbitrary n, all sets Y with the same shape as bn X will have |K (Y )| = m. In particular, there will be infinitely many subsets Z containing 0 with |K (Z )| = m, which implies ||K ||r = ∞.
Spectral Properties of the Renormalization Group at Infinite Temperature
181
Proposition 9. Restricted to the translation-invariant even subspace of Br : r > 0, the point spectrum of L is λ = 0. Proof. This follows from Propositions 7 and 8. Theorem 2. Suppose the original interaction J is at infinite temperature. Then in the Banach Space Br∗ , the spectrum of the adjoint of the linearization of the decimation transformation L∗ (J ) is all residual spectrum, |λ| ≤ 1. Proof. The proof of this theorem follows from several propositions. Proposition 10. ||L∗ (J )|| ≤ 1. Proof. By (3.5), 1 ∗ 1 |L (J )K (X )|e−rl(x,X ) ≤ |K (X )|e−rl(bx,bX ) sup sup |X | |X | x∈X bx∈bX x∈L
bx∈L
≤
sup
x∈L x∈X
1 |K (X )|e−rl(x,X ) . |X |
(3.11)
Proposition 11. For every λ = 0, there is no nontrivial eigenvector. Proof. Fix an arbitrary finite subset X of the infinite lattice, after a finite number of iterations of L∗ (J ) (say n times), X will not be of the form bY for some Y . Thus λn+1 K (X ) = (L∗ (J ))n+1 K (X ) = 0, which implies K (X ) = 0. Proposition 12. For λ = 0, there is no nontrivial eigenvector. Proof. Suppose the nontrivial eigenvector K (X ) = m = 0 for some finite subset X , then the crucial fact that we can always find Y , with L∗ (J )K (Y ) = K (X ) will do the job. As L∗ (J )K (Y ) = λK (Y ) = 0, we reach a contradiction. Corollary 2. In the Banach Space Br∗ , the point spectrum of L∗ (J ) is empty. Proof of Theorem 2 continued. The only thing left to show now is that for |λ| ≤ 1, Range(λI − L∗ (J )) = Br∗ . Define K ({(1, 0, . . . , 0)}) = 1, and K (X ) = 0 for all other subsets X . We will show that K can not be approximated by any K in Range(λI −L∗ (J )) within distance 1/2. To see this, note that for n ≥ 0, K ({(bn+1 , 0, . . . , 0)}) = λS({(bn+1 , 0, . . . , 0)}) − S({(bn , 0, . . . , 0)})
(3.12)
for some S that lies in Br∗ . Suppose 1 1 ≥ ||K − K ||r∗ = |K (X ) − K (X )|e−rl(x,X ) sup 2 |X | x∈X
≥ ≥
x∈L
sup
x=(bn ,0,...,0) x∈X ∞ n
1 |K (X ) − K (X )|e−rl(x,X ) |X |
|K ({(b , 0, . . . , 0)}) − K ({(bn , 0, . . . , 0)})|
(3.13)
n=0
= |λS({(1, 0, . . . , 0)}) − 1| + |λS({(b, 0, . . . , 0)}) − S({(1, 0, . . . , 0)})| + · · · . (3.14)
182
M. Yin
Then, as |λ| ≤ 1, for any n ≥ 0, 1 ≥ |λn+1 S({(bn , 0, . . . , 0)}) − λn S({(bn−1 , 0, . . . , 0)})| + · · · 2 +|λ2 S({(b, 0, . . . , 0)}) − λS({(1, 0, . . . , 0)})| + |λS({(1, 0, . . . , 0)}) − 1|. (3.15) By the triangle inequality, this implies |λn+1 S({(bn , 0, . . . , 0)}) − 1| ≤
1 , 2
(3.16)
which further implies |λn+1 S({(bn , 0, . . . , 0)})| ≥
1 . 2
(3.17)
Using |λ| ≤ 1 again, we have |S({(bn , 0, . . . , 0)})| ≥
1 . 2
(3.18)
But then, ||S||r∗ =
∞
sup
x∈L x∈X
1 |S({(bn , 0, . . . , 0)})| = ∞. |S(X )|e−rl(x,X ) ≥ |X |
(3.19)
n=0
Remark 2. Notice the similarity between the adjoint operators L(J )/L∗ (J ) in our Banach spaces and left/right translation in l ∞ /l 1 . L(J ) acts like left translation and L∗ (J ) acts like right translation on sequences (X, bX, . . .) for all possible subsets X . Moreover, ignoring multiplicity of the eigenvalues, the spectrum of L(J ) is the same as that of left translation in l ∞ , and the spectrum of L∗ (J ) is the same as that of right translation in l 1 . This might be related to the fact that the norms in our Banach spaces are something like combinations of l ∞ and l 1 norms. 4. Spectrum of the Linearization of Majority Rule Transformation and its Adjoint at Infinite Temperature s−1 For notational convenience, in this section, we set s = bd and ν = s−1 /2s−1 . 2
Proposition 13. Consider majority rule transformation with odd blocking factor b and a probability kernel defined by ⎛ ⎞ φy (σ ) = sign ⎝ σx ⎠. (4.1) x∈yo
Suppose the original interaction J is at infinite temperature. Then for every subset Z of the image lattice, the linearization L(J ) of this transformation is given by the expression χ (W ∩ zo )K (W ), (4.2) L(J )K (Z ) = where χ (W ∩ zo ) =
σ
W :W ⊂Z o z∈Z σ
Tz (σ, σz )σW ∩zo σz .
Spectral Properties of the Renormalization Group at Infinite Temperature
183
Proof. We evaluate (2.4) explicitly: ∂ J (Z ) = σW \Z o Tz (σ, σz ) Tz (σ, σz )σW ∩zo σz . ∂ J (W ) σ σ
Since
z∈Z / σ
σ
(4.3)
σ
z∈Z
σW \Z o = 0 for W not completely contained inside Z o , it follows that W ⊂ Z o .
Proposition 14. Consider majority rule transformation with odd blocking factor b and a probability kernel defined by ⎛ ⎞ φy (σ ) = sign ⎝ σx ⎠. (4.4) x∈yo
Suppose the original interaction J is at infinite temperature. Then for every subset Z of the original lattice, the adjoint of the linearization L∗ (J ) of this transformation is given by the expression L∗ (J )K (Z ) = χ (Wn )K (∪{n}), (4.5) Wn
where Z = ∪Wn and Wn ⊂ no . Proof. We notice that in this case, (2.7) becomes K 1 (X )L(J )K 2 (X ) = K 1 (X ) χ (Y ∩ xo )K 2 (Y ) X
X
=
Y =∪Wn
Y :Y ⊂X o x∈X
K 2 (Y )
χ (Wn )K 1 (∪{n}).
(4.6)
Wn
Theorem 3. Suppose the original interaction J is at infinite temperature. Then in the Banach Space Br , the spectrum of the linearization of the majority rule transformation L(J ) is all point spectrum, |λ| ≤ sν. Proof. The proof of this theorem follows from several propositions. Proposition 15. Consider an Ising-type spin system on an odd polygon A with cardinality |A|. Fix a certain vertex V and a certain subset W of the vertices. If σa ∈ {+1, −1} satisfies σ A σa > 0, then |A|−1 σW σa | ≤ σV σa = |A|−1 /2|A|−1 , (4.7) | n
σ
σ
2
is the binomial coefficient. Proof. We first show that σ σW σa = 0 for any W with even cardinality. This is due to a symmetry argument. If there is a spin configuration with σW σa = 1, then flipping the spins at every vertex, we will have a configuration with σW σa = (−1)|W | (−1) = sum will be zero. (−1)|W |+1 = −1. Vice versa. Thus the total Next we investigate into the special case σ σV σa , where V is any fixed vertex. The explicit calculation is easy to carry out. Due to symmetry, we only consider σV = 1 in the following, and there are |A| − 1 vertices for which the spins are yet to be assigned. where
k
184
M. Yin
1. σa = 1, if there are more 1’s than −1’s in the overall spin configuration, i.e., as long as the number of −1’s does not exceed |A|−1 2 . It is not hard to see that there are |A|−1 |A|−1 |A|−1 + 1 + · · · + |A|−1 of them. 0 2
2. σa = −1, if there are more −1’s than 1’s in the overall spin configuration, i.e., as it is not hard to see that there are long as the number of −1’s exceeds |A|−1 2 . Again, |A|−1 |A|−1 |A|−1 |A|−1 |A|−1 |A|−1 of them. |A|+1 + |A|+3 + · · · + |A|−1 = |A|−3 + |A|−5 + · · · + 0 2
2
2
In conclusion, when σV = 1, there are
|A|−1 |A|−1 2
2
more spin configurations for σa to be 1
rather than to be −1. A similar result holds for σV = −1. Thus considering all possible |A|−1 spin configurations, there are 2 |A|−1 more spin configurations for σV σa to be 1 rather |A|−1 2 than to be −1. It follows that σ σV σa = |A|−1 /2|A|−1 . 2 Finally we consider σ σW σa for any W with odd cardinality. Without loss of generality, suppose V ⊂ W . For a fixed spin configuration, σV σa = σW σa can only occur when there is an odd number of −1’s and an odd number of 1’s in the spin configuration for vertices in W \V . For such a configuration, we notice the following important fact: Suppose it has the extra property that unequal numbers of −1’s and 1’s are assigned for the remaining |A| − 1 vertices of A\V , then if we flip the spins at every vertex other than V, σV σa will change sign. Moreover, at the same time, the sign of σW σa also changes, so sum does not change. Therefore, we see that the difference in σ σW σa and the total σ σV σa can only be caused by the following scenario: Equal numbers of −1’s and 1’s are assigned for the remaining |A| − 1 vertices of A\V , and there is an odd number of −1’s and an odd number of 1’s in the spin configuration for vertices in W \V . It is |A|−1 not hard to see that there are at most 2 |A|−1 of them. Thus σ σW σa varies between 2 |A|−1 |A|−1 − |A|−1 /2|A|−1 and |A|−1 /2|A|−1 , and our claim follows. 2
2
Proposition 16. ||L(J )|| = sν.
Proof. We check that for each fixed x ∈ L, X :x∈X |L(J )K (X )|erl(x,X ) ≤ sν||K ||r , which would imply ||L(J )|| ≤ sν. As x ∈ X, L(J )K (X ) is a linear combination of K (Y )’s, each one with coefficient bounded above by ν by (4.7). Ignoring the coefficients of K (Y )’s, we can then collect terms according to which one of the sites in xo belongs to Y . (When |Y ∩xo | > 1, K (Y ) can be classified into either one of the s groups.) Moreover, each Y has size no smaller than X , the exponential factor changes to a larger quantity after the action of L(J ). We see that each collection is bounded above by ||K ||r by definition. The claim is verified when we realize that a constant pure magnetic field is an eigenvector with eigenvalue sν. Corollary 3. Every eigenvalue |λ| ≤ sν. Proposition 17. Every |λ| ≤ sν is an eigenvalue. Proof. For a generic λ, we display one eigenvector here. In fact, with some further thought, it is not hard to show that there are infinitely many eigenvectors for each λ. The eigenvector K is defined by K ({(
b−1 b−1 ,..., )}) = λ/ν − (s − 1), 2 2
(4.8)
Spectral Properties of the Renormalization Group at Infinite Temperature
185
and K ({x}) = 1
(4.9)
b−1 o for ( b−1 2 , . . . , 2 ) = x ∈ 0 . In general, for n = 0, K is defined by s K ({m}) = o λ/ν K ({n}) for m ∈ n . For all the other subsets X, K (X ) is set to zero. Corollary 4. The spectrum of L(J ) diverges as 2s π as the blocking factor b gets large.
Proof. This follows from an easy application of Stirling’s formula: √ 2s s 2π · (s − 1)(s − 1)s−1 e−(s−1) . ∼ sν ∼ s−1 π 2π s−1 s−1 e−(s−1) 2s−1 2
(4.10)
2
Theorem 4. Suppose the original interaction J is at infinite temperature. Then in the Banach Space Br∗ , the point spectrum of the adjoint of the linearization of the majority rule transformation L∗ (J ) is empty. Moreover, every |λ| ≤ ν is in the residual spectrum of L∗ (J ). Proof. The proof of this theorem follows from several propositions. Proposition 18. For every λ = 0 and λ = ν, there is no nontrivial eigenvector. Proof. Fix an arbitrary finite subset X . For λ = 0, K (X ) is either zero or a nonzero constant multiple of K ({0}) as a result of the action of L∗ (J ). In particular, λK ({0}) = L∗ (J )K ({0}) = ν K ({0}), which implies that K ({0}) = 0. Proposition 19. For λ = 0, there is no nontrivial eigenvector. Proof. Suppose the nontrivial eigenvector K (X ) = m = 0 for some finite subset X , then the crucial fact that we can always find Y , with L∗ (J )K (Y ) a nonzero constant multiple of K (X ) will do the job. As L∗ (J )K (Y ) = λK (Y ) = 0, we reach a contradiction. Proposition 20. For λ = ν, every nontrivial eigenvector has norm infinity. Proof. We must have K ({0}) = m = 0 in order for K to be nontrivial. As ν K ({x}) = L∗ (J )K ({x}) = ν K ({0})
(4.11)
for x ∈ 0o , we see that K ({x}) = m also. Following in similar fashion, K ({n}) = m for arbitrary n. But then, ||K ||r∗ = ∞. Proof of Theorem 4 continued. The only thing left to show now is that for |λ| ≤ ν, Range(λI − L∗ (J )) = Br∗ . Define K ({(0, 0, . . . , 0)}) = 1, and K (X ) = 0 for all other subsets X . We will show that K can not be approximated by any K in Range(λI −L∗ (J )) within distance 1/4. To see this, note that for n ≥ 0, K ({(bn+1 , 0, . . . , 0)}) = λS({(bn+1 , 0, . . . , 0)}) − ν S({(bn , 0, . . . , 0)}) (4.12) for some S that lies in Br∗ . And in particular, K ({0, . . . , 0}) = (λ − ν)S({0, . . . , 0}).
(4.13)
186
M. Yin
Suppose 1 1 sup ≥ ||K − K ||r∗ = |K (X ) − K (X )|e−rl(x,X ) 4 x∈X |X | x∈L
≥ |K ({(0, 0, . . . , 0)}) − K ({(0, 0, . . . , 0)})| ∞ |K ({(bn , 0, . . . , 0)}) − K ({(bn , 0, . . . , 0)})| + n=0
= |(λ − ν)S({(0, 0, . . . , 0)}) − 1| +|λS({(1, 0, . . . , 0)}) − ν S({(0, 0, . . . , 0)})| + · · · . Then, as |λ| ≤ ν ≤
1 2,
(4.14)
for any n ≥ 0,
λ 1 λ ≥ |( )n+1 (λ − ν)S({(bn , 0, . . . , 0)}) − ( )n (λ − ν)S({(bn−1 , 0, . . . , 0)})| + · · · 2 ν ν λ +| (λ − ν)S({(1, 0, . . . , 0)}) − (λ − ν)S({(0, 0, . . . , 0)})| ν +|(λ − ν)S({(0, 0, . . . , 0)}) − 1|. (4.15) By the triangle inequality, this implies λ 1 |( )n+1 (λ − ν)S({(bn , 0, . . . , 0)}) − 1| ≤ , ν 2 which further implies λ 1 |( )n+1 (λ − ν)S({(bn , 0, . . . , 0)})| ≥ . ν 2 Using |λ| ≤ ν ≤
1 2
(4.16)
(4.17)
again, we have |S({(bn , 0, . . . , 0)})| ≥
1 . 2
(4.18)
But then, ||S||r∗ =
∞
sup
x∈L x∈X
1 |S(X )|e−rl(x,X ) ≥ |S({(bn , 0, . . . , 0)})| = ∞. |X |
(4.19)
n=0
Acknowledgements. This work began at the Isaac Newton Institute in Cambridge during the 2008 program in Combinatorics and Statistical Mechanics, organized by Alan Sokal. The author owes deep gratitude to her advisor Bill Faris for his continued help and support. She also thanks Tom Kennedy, Doug Pickrell, and Bob Sims for their kind and helpful suggestions and comments. The author appreciated the opportunity to talk about this work in the 2009 workshop in Renormalization Group and Statistical Mechanics in Vancouver, organized by David Brydges, Joel Feldman, and A.C.D. van Enter, and is grateful for the feedback from many of these people.
References 1. Dunford, N., Schwartz, J.: Linear Operators, Part I. New York: Interscience Publishers, 1958 2. Israel, R.B.: Banach algebras and Kadanoff transformations. In: Fritz, J., Lebowitz, J.L., Szász, D. (eds.) Random Fields, Vol. II, Amsterdam: North-Holland, 1981, pp. 593–608 Communicated by M. Salmhofer
Commun. Math. Phys. 304, 187–228 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1208-y
Communications in
Mathematical Physics
q (2) On a Correspondence between SUq (2), E q (1, 1) and SU Kenny De Commer Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy. E-mail: [email protected] Received: 1 June 2010 / Accepted: 19 September 2010 Published online: 13 February 2011 – © Springer-Verlag 2011
Abstract: In a previous paper, we showed how one can obtain from the action of a locally compact quantum group on a type I -factor a possibly new locally compact quantum group. In another paper, we applied this construction method to the action of quantum SU (2) on the standard Podle´s sphere to obtain Woronowicz’s quantum E(2). In this paper, we will apply this technique to the action of quantum SU (2) on the quantum projective plane (whose associated von Neumann algebra is indeed a type I -factor). The locally compact quantum group which then comes out at the other side turns out to be the extended SU (1, 1) quantum group, as constructed by Koelink and Kustermans. We also show that there exists a (non-trivial) quantum groupoid which has at its corners (the duals of) the three quantum groups mentioned above. 0. Introduction This is part of a series of papers ([2,3]) devoted to an intriguing correspondence between and the quantizations of SU (2), E(2) SU (1, 1), with the latter two groups being respectively the non-trivial two-folded covering E(2) Z2 of the Euclidean transformation group of the plane, and the normalizer of SU (1, 1) inside S L(2, C) (which contains SU (1, 1) as an index 2 normal subgroup). In a sense, their duals form a trinity of ‘Morita equivalent locally compact quantum groups’. There then exists a ‘linking quantum groupoid’ combining these three quantum groups into one global structure, and it is important to understand for example the (co)representation theory of this object. In this paper, we will treat the ‘groupoid von Neumann algebra of the linking quanq (2) and tum groupoid between the duals of SUq (2), E SU q (1, 1)’. This object consists of three ‘corners’, corresponding to linking quantum groupoids of the pairs inside. The q (2) linking quantum groupoid between the pair consisting of the duals of SUq (2) and E was treated in [2]. However, we will give here an alternative description which is more Supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”.
188
K. De Commer
in line with how a second linking quantum groupoid will be presented, namely the one between the duals of SUq (2) and SU q (1, 1). It is this linking quantum groupoid which will be the main object of study in the present article. The third linking quantum groupoid q (2) and between the duals of E SU q (1, 1) can then easily be obtained by a composition procedure, while the global ‘3×3 linking quantum groupoid’ is simply the three separate linking quantum groupoids pasted together. Let us now describe these objects and constructions in more detail, beginning with providing some more information on the quantum groups we mentioned. We note that all q’s which appear in this article are real numbers satisfying 0 < q < 1, and that we denote by N0 the set of natural numbers with 0 excluded. q (2) and SU q (1, 1). Of the above quantum groups, The quantum groups SUq (2), E SUq (2) is the most well-known one, and the easiest to handle. It is an example of a compact quantum group in the sense of Woronowicz ([20,24]), and was introduced by him in [19] as a ‘twisted’ or q-version of the ordinary SU (2)-group. It appears in different guises, depending on what type of functions one considers on this quantum group: polynomial, continuous or measurable. All of these viewpoints can be shown to correspond to the same ‘virtual object’ that is SUq (2), and the passage-way between them is easy to describe. In this paper, we will only need the von Neumann algebraic picture, so we will state the definition in this context, even though this is certainly not the most suitable way to present it. We first introduce some terminology. Definition 0.1. A von Neumann bialgebra (M, M ) consists of a von Neumann alge¯ satisfying the bra M and a faithful normal unital ∗ -homomorphism M : M → M ⊗M coassociativity condition ( M ⊗ ι) M = (ι ⊗ M ) M . The following is a definition of SUq (2) on the von Neumann algebra level. Definition 0.2. Denote I+ = N, and denote by H+ the Hilbert space l 2 (I+ ) ⊗ l 2 (Z). Consider on it the operators a+ = 1 − q 2k ek−1,k ⊗ 1, k∈N0
b+ = (
q k ekk ) ⊗ S,
k∈N
where the ei j denote the standard matrix units, and where S denotes the forward bilateral shift. Then the von Neumann bialgebra (L ∞ (SUq (2)), + ) consists of the von Neumann algebra ¯ (Z) ⊆ B(H+ ), L ∞ (SUq (2)) = B(l 2 (I+ ))⊗L equipped with the unique unital normal ∗ -homomorphism ¯ ∞ (SUq (2)) + : L ∞ (SUq (2)) → L ∞ (SUq (2))⊗L which satisfies
+ (a+ ) = a+ ⊗ a+ − qb+∗ ⊗ b+ + (b+ ) = b+ ⊗ a+ + a+∗ ⊗ b+ .
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
189
This particular von Neumann bialgebra (L ∞ (SUq (2)), + ) will in fact have some extra structure which really qualifies it as ‘the space of bounded measurable functions on a (locally) compact quantum group’, but we will not need this extra structure in this paper. Of course, one has to verify that the above definition is meaningful. There are two ways of establishing this: the first and more natural one is to introduce first SUq (2) in a different way (by considering say its associated Hopf ∗ -algebra), and then to use its extra structure (the existence of an invariant positive state) to pass to the von Neumann algebra level, and to prove the equivalence with the above definition. A second way consists of finding a unitary which implements + on the generators a+ and b+ . The coassociativity condition will then automatically be satisfied, since it is satisfied on the generators a+ and b+ of L ∞ (SUq (2)). This method is a lot more computational, and makes use of some non-trivial q-analytic facts. However, it is this approach which is most suited for the purpose of this article. Before introducing this method, let us first make some remarks on notation. We will use standard notation for all things q (see [6]). More precisely, for n ∈ N ∪ {∞} and a ∈ C, we denote (a; q)n =
n−1
(1 − q k a),
k=0
and (a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n . . . (am ; q)n , while we denote by r ϕs the basic hypergeometric functions. We also borrow the following notation from [11]. a Definition 0.3. The entire function z → | q, z , depending on the parameters b a, b ∈ C, is defined as
a | q, z b
=
∞ (a; q)n (bq n ; q)∞ n=0
(q; q)n
1
(−1)n q 2 n(n−1) z n .
Then if b ∈ C\q −N , we have a a | q, z . | q, z = (b; q)∞ 1 ϕ1 b b We can now state the following proposition. We refer to Appendix A for some information on how it can be deduced from observations in the literature. Proposition 0.4. Writing again I+ = N, we denote by Pq+2 the following function on I+ × I+ × I+ : Pq+2 ( p, v, w) = (−q) p−w q ( p−w)(v−w) ×
q 2v+2 q 2v−2w+2
1/2
(q 2w+2 ; q 2 )∞
1/2
(q 2 ; q 2 )∞ (q 2 p+2 , q 2v+2 ; q 2 )∞ | q 2 , q 2 p−2w+2 ,
(1)
190
K. De Commer
or equivalently, Pq+2 ( p, v, w) = (−q) p q vw+ p(v+w) × 3 ϕ2
q −2w
1/2
(q 2v+2 , q 2 p+2 ; q 2 )∞ 1/2
1/2
(q 2 ; q 2 )∞ (q 2 ; q 2 )w −2v q q −2 p | q 2, q 2 . 0 0
(2)
Then for r, s, t ∈ Z and p ∈ I+ , the vectors + Pq+2 ( p, v, w)ev ⊗ er + p−w ⊗ ew ⊗ es− p+v ξr,s, p,t = v,w∈I+ v−w=t
form an orthonormal basis of H+ ⊗ H+ = (l 2 (I+ ) ⊗ l 2 (Z)) ⊗ (l 2 (I+ ) ⊗ l 2 (Z)). Moreover, denoting by W+ the unitary + W+ : H+ ⊗ H+ → l 2 (Z) ⊗ l 2 (Z) ⊗ H+ : ξr,s, p,t → er ⊗ es ⊗ (e p ⊗ et ),
we have W+∗ (1 ⊗ x)W+ = + (x),
¯ (Z). for all x ∈ L ∞ (SUq (2)) = B(l 2 (I+ ))⊗L
+ Note that there is some freedom in the choice of the ξr,s, p,t if we only want them to implement the comultiplication. However, the above form is a natural one to choose. q (2). Also this object was introduced Let us now move on to the quantum group E by Woronowicz (at least on the operator algebraic level, [21]), and is a q-version of the group of matrices
a 0 | a, b ∈ C, |a| = 1 , b a −1
which has an alternative abstract description as the double cover of the group E(2) of Euclidian transformations of the plane. Also in this case, one has a set of different structures to consider, depending on which function algebra one is interested in. However, the passage between these structures, notably between the algebra of polynomial functions and the algebra of bounded continuous/measurable functions, is now not so straightforward as in the previous case. The main obstacles are the lack of a well-behaved invariant functional on the purely algebraic level, and the necessity to work with unbounded operators in the operator algebraic setting. This prohibits to treat the possible correspondence q (2) however, things are still within a general framework. For the particular case of E well-behaved. q (2) is essentially the one which appears in [21], but The following definition of E lifted to the von Neumann algebraic setting. Definition 0.5. Denote I0 = Z, and denote by H0 the Hilbert space l 2 (I0 ) ⊗ l 2 (Z). Consider on it the unitary operator a0 = S ∗ ⊗ 1, where S ∗ denotes the backward bilateral shift (acting on the first factor), and the unbounded normal operator b0 which has the linear span of basis vectors en ⊗ ek as its core, with b0 en ⊗ ek = q n en ⊗ ek+1 ,
k, n ∈ Z.
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
191
q (2)), 0 ) consists of the von Neumann Then the von Neumann bialgebra (L ∞ ( E algebra q (2)) = B(l 2 (I0 ))⊗L ¯ (Z) ⊆ B(H0 ), L ∞( E equipped with the unique unital normal ∗ -homomorphism q (2)) → L ∞ ( E q (2))⊗L q (2)) ¯ ∞( E 0 : L ∞ ( E which satisfies
0 (a0 ) = a0 ⊗ a0 ˙ 0∗ ⊗ b0 , 0 (b0 ) = b0 ⊗ a0 +a
where +˙ means ‘the closure of the sum of two unbounded operators’. q (2)), 0 ) carries extra structure which makes E q (2) eligible Also in this case, (L ∞ ( E to be called a locally compact quantum group. As for SUq (2), it is of course not obvious on first sight that the above definition makes sense. But one can again find a unitary implementing it: the following proposition could in principle be deduced from the results of [10], but we will give another argument in the main body of the text, based on Proposition 0.4 and the results of [2] (see Proposition 4.2). We remark that such an implementing unitary was also considered in [23]. Proposition 0.6. Writing again I0 = Z, we denote by Pq02 the following function on I0 × I0 × I0 : 1 0 2 2 p−2w+2 . Pq02 ( p, v, w) = (−q) p−w q ( p−w)(v−w) 2 2 | q , q q 2v−2w+2 (q ; q )∞ Then for r, s, t ∈ Z and p ∈ I0 = Z, the vectors 0 ξr,s, Pq02 ( p, v, w)ev ⊗ er + p−w ⊗ ew ⊗ es− p+v p,t = v,w∈I0 v−w=t
form an orthonormal basis of H0 ⊗H0 = (l 2 (I0 )⊗l 2 (Z))⊗(l 2 (I0 )⊗l 2 (Z)). Moreover, denoting by W0 the unitary 0 W0 : H0 ⊗ H0 → l 2 (Z) ⊗ l 2 (Z) ⊗ H0 : ξr,s, p,t → er ⊗ es ⊗ e p ⊗ et ,
we have W0∗ (1 ⊗ x)W0 = 0 (x),
q (2)) = B(l 2 (I0 ))⊗L ¯ (Z). for all x ∈ L ∞ ( E
Finally, we have the locally compact quantum group SU q (1, 1) to discuss. A first ques tion that immediately comes to mind is: why SU q (1, 1) and not SUq (1, 1)? This is because of the ‘no-go theorem’ of Woronowicz (cf. [22]), which says that SUq (1, 1) simply can not exist as a locally compact quantum group. This is not as bad as it sounds: due to a key observation of Korogodsky ([13]), it turns out that a close companion to SU (1, 1) allows a q-deformation into a locally compact quantum group, namely the normalizer SU (1, 1) of SU (1, 1) inside S L(2, C). But one had to wait till [11] for the first rigorous results that this object really existed in the operator algebraic framework (and more particularly, fitted in the setting of [14]).
192
K. De Commer
The presentation in [11] in fact started from a concrete unitary implementing the coalgebra structure, because it turned out that the accompanying Hopf algebra structure was too weak to capture all necessary information. Hence the treatment of this quantum group had a lot more q-analytic machinery running in the background. To state the definition of SU q (1, 1), we first present some auxiliary notation as in the original paper [11]. (+)
(−)
Notation 0.7. We denote I− = Z, I− = N− 0 = {m ∈ Z | m < 0}, and I− = (+) (−) (±) I− I− , the disjoint union. We write p ∈ I− as p± when we interpret it as an element in I− , and we write p for an indeterminate element in { p+ , p− }. We denote c : I− → Z2 : p± → ±, so that p = pc(p) . The way in which we will present the definition is slightly different from the one in [11]. We again refer to Appendix A for more information on the equivalence between the two definitions. Definition 0.8. Denote by Pq−2 the following function on I− × I− × I− : for ρ, ν, ω ∈ {±}, we put (−ρq) p−w v+1 1 ( p+v−w)( p+v−w+1) ν q2 √ 2 1/2 (−ρq −2 p , −νq −2v ; q 2 )∞ −νq 2v+2 2 2 p−2w+2 . · · | q , ρωq 1/2 νωq 2v−2w+2 (q 4 ; q 4 )∞ (−ωq −2w ; q 2 )∞
Pq−2 ( pρ , vν , wω ) =
Denote H− = l 2 (I− ) ⊗ l 2 (Z). Then for p ∈ I− and r, s, t ∈ Z, the vectors − = Pq−2 (p, v, w)ev ⊗ er + p−w ⊗ ew ⊗ es− p+v ξr,s,p,t v,w∈I− v−w=t c(v)c(w)=c(p)
form an orthogonal basis of H− ⊗ H− . If we then define W− as the unitary − → er ⊗ es ⊗ ep ⊗ et , W− : H− ⊗ H− → l 2 (Z) ⊗ l 2 (Z) ⊗ H− : ξr,s,p,t
the application x → − (x) := W−∗ (1 ⊗ x)W− ¯ (Z). defines a von Neumann bi-algebra structure on L ∞ ( SU q (1, 1)) = B(l 2 (I− ))⊗L We will not present here the associated (and incomplete) Hopf algebraic picture. We refer the reader to the original paper [11] for this. However, it should be mentioned that the incompleteness of the Hopf algebraic picture is in some sense related to the fact that in the operator algebraic picture, certain ‘off-diagonal corner operators’ are introduced, (+) (−) namely the ones intertwining the I− and I− -part. In the Hopf-algebraic setting, there is no trace of these. For this reason, we have some doubt that SU q (1, 1) should really be interpreted as a q-deformation of SU (1, 1). Rather, it seems to us that it is a ‘noncommutative blow-up’ of the ordinary SU (1, 1)-group (by changing C into the Morita equivalent M2 (C), in some vague sense). For this reason, it is perhaps better to stick with Woronowicz’s nomenclature ‘extended quantum SU (1, 1)-group’.
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
193
Connecting the quantum groups by means of a linking quantum groupoid. We now come to the notion of a linking weak von Neumann bialgebra between these structures. A general theory of such objects was treated in [4] (see also [2] for some motivation), but we will here only present the essence of the structure for the situation at hand. The observation is quite simple: consider the Hilbert spaces H+ , H0 and H− introduced in the definitions of the previous subsection, and form the direct⎛sum Hilbert space ⎞ H− H = H− ⊕ H0 ⊕ H+ , which we may present in the column form ⎝ H0 ⎠. Then we H+ have a left action on this by the direct sum von Neumann algebra q (2)) ⊕ L ∞ (SUq (2)) L ∞ ( SU q (1, 1)) ⊕ L ∞ ( E ⎛ ∞ ⎞ SU q (1, 1)) 0 0 L ( ⎠. q (2)) =⎝ 0 L ∞( E 0 0 0 L ∞ (SUq (2)) But the definition of the von Neumann algebras of these quantum groups immediately suggests how this pattern can be completed at the non-diagonal entries: simply define ¯ (Z), L (μ, ν) = B(l 2 (Iν ), l 2 (Iμ ))⊗L where μ, ν ∈ {−, 0, +}, and with the Iμ defined in the definitions of the previous subsection. Then the L (μ, μ) coincide with the L ∞ -von Neumann algebras of our respective quantum groups, while we now also have the action of ⎛ ⎞ ⎛ ⎞ L (−, −) L (−, 0) L (−, +) H− ⎝ H0 ⎠ . Q = ⎝ L (0, −) L (0, 0) L (0, +) ⎠ on L (+, −) L (+, 0) L (+, +) H+ (We want to stress however that the L (μ, ν) for μ = ν are not von Neumann algebras, only Hilbert W∗ -bimodules!) The next objective is then to generalize the comultiplications of the diagonal entries L (μ, μ) to the off-diagonal parts. Also this is easy to do given all the structure at hand. To introduce this comultiplication, let us first remark that by the notation ¯ (μ, ν), we mean the σ -weak closure of the algebraic tensor product of L (μ, ν)⊗L these two spaces inside B(Hν ⊗ Hν , Hμ ⊗ Hμ ). We can then collect all these tensor products together in a balanced or ‘C3 -fibred’ tensor product of Q with itself: ⎛ ⎞ ¯ (−, −) L (−, 0)⊗L ¯ (−, 0) L (−, +)⊗L ¯ (−, +) L (−, −)⊗L ¯ (0, −) L (0, 0)⊗L ¯ (0, 0) L (0, +)⊗L ¯ (0, +) ⎠ , Q ∗ Q := ⎝ L (0, −)⊗L ¯ (+, −) L (+, 0)⊗L ¯ (+, 0) L (+, +)⊗L ¯ (+, +) L (+, −)⊗L ⎞ H− ⊗ H− which is a unital von Neumann subalgebra of B ⎝ H0 ⊗ H0 ⎠. The von Neumann H+ ⊗ H+ ¯ algebra Q ∗ Q can also be identified with a corner of the tensor product Q ⊗Q, namely ¯ with h(Q ⊗Q)h, where h is the projection ⎛
h = 1− ⊗ 1− + 10 ⊗ 10 + 1+ ⊗ 1+ ,
194
K. De Commer
the 1μ denoting the units in L (μ, μ). (It is clear how to form then the triple fibred product Q ∗ Q ∗ Q etc.) We can now define the comultiplication maps μν on the off-diagonal parts: they are given by ¯ (μ, ν) : x → Wμ∗ (1 ⊗ x)Wν , μν : L (μ, ν) → L (μ, ν)⊗L where the Wμ were defined in the definitions of the previous subsection. Of course, one must prove that μν has the above range, but this is not so difficult to establish in a direct manner (see e.g. the proof of Proposition 3.8 in [11]). We can further collect these maps together into a map ¯ Q : Q → Q ∗ Q ⊆ Q ⊗Q by the formula ⎛⎛ ⎞⎞ ⎛ ⎞ x−− x−0 x−+ −− (x−− ) −0 (x−0 ) −+ (x−+ ) Q ⎝⎝ x0− x00 x0+ ⎠⎠ = ⎝ 0− (x0− ) 00 (x00 ) 0+ (x0+ ) ⎠ , +− (x+− ) +0 (x+0 ) ++ (x++ ) x+− x+0 x++ where xμν ∈ L (μ, ν). This map is then obviously a faithful normal ∗ -homomorphism by definition of the maps μν . Whether it is unital depends on the precise choice of range: if one takes Q ∗ Q as the range, then the map is unital; on the other hand, if one ¯ as the range, then it is not. This is simply because the unit of Q ∗ Q is chooses Q ⊗Q ¯ which we introduced above. the projection Q (1) = h of Q ⊗Q We can now state one of the main observations in this paper. ¯ is coassociative. Theorem 0.9. The comultiplication Q : Q → Q ⊗Q Using the terminology of [1] in the von Neumann algebraic setting, this will qualify (Q, Q ) as a weak von Neumann bialgebra. Because of its particular structure, we call it a (3×3-)linking weak von Neumann bialgebra (see [4]). It can be interpreted as (the groupoid von Neumann algebra pertaining to) a kind of quantized groupoid with three q (2) and classical objects and the duals of the quantum groups SUq (2), E SU q (1, 1) as its isotropy groups. This is also the reason why we then call these duals ‘Morita equivalent quantum groups’, as the previous description is closely related to how Morita equivalence between (classical) groupoids is defined by means of a linking groupoid. See again [2] for some more intuition behind these concepts.
Projective corepresentations. We now comment on the way we prove this theorem. Our method is not straightforward, and in fact, we must admit that we have not even tried very hard to prove Theorem 0.9 by direct means. This is because we hope that our method, though roundabout, is much better suited for generalization. The main idea to prove Theorem 0.9 is the following. We first make the apparently unrelated and easy observation that for a locally compact group G, there is a close connection between actions on (separable) type I -factors on the one hand (i.e., actions on von Neumann algebras of the form B(H ) for some (separable) Hilbert space H ), and (measurable) unitary 2-cocycle functions on G on the other. Indeed, given such an action, one can choose for each group element a unitary implementing the associated automorphism, and this will then provide one with an -projective representation for
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
195
some unitary 2-cocycle function (which can be taken to be measurable if the unitaries are well-chosen). The philosophy is now that in the quantum setting, the proper generalization of a 2-cocycle function is a (2×2-)linking weak von Neumann bialgebra. Indeed, we showed in [4] that, given any coaction of a von Neumann bialgebra on a type I -factor (which we then called a projective corepresentation, see Sect. 1 of the present article), one can construct from this a linking weak von Neumann bialgebra (uniquely determined up to isomorphism). Observe that we started with one von Neumann bialgebra, but that a linking weak von Neumann bialgebra has two von Neumann bialgebras inside. Indeed, the other von Neumann bialgebra is ‘hidden somewhere’ in the projective corepresentation! This is a generalization of the notion of twisting a von Neumann bialgebra by means of a unitary 2-cocycle. (In fact, our main example will arise from a genuine 2-cocycle twisting, but in a non-natural way. We will therefore not emphasize it in this paper, but refer to Proposition 4.3 of [4] to see the connection.) The main observation then is that for SUq (2), there are two very natural such projective representations, namely by considering the action on either the standard Podle´s sphere, or on a certain Z2 -quotient of the equatorial Podle´s sphere (which can be interpreted as a quantum projective plane, [8]). Indeed, one can show that the von Neumann algebras associated with these quantum homogeneous spaces are both type I -factors. Thus one can consider their associated 2×2-linking weak von Neumann bialgebras, and combine them (by a composition procedure) into a 3×3-linking weak von Neumann bialgebra. This will turn out to be precisely the object described in Theorem 0.9, hence proving the claimed coassociativity property in an indirect way. Of course, with this discussion alone, it is not clear why one should expect the quanq (2) and tum groups E SU q (1, 1) to pop out of these constructions. In fact, we are not sure if one can figure out a priori precisely which quantized Lie group will appear, but one can get some information on its associated quantized Lie algebra. Indeed, in [3], an infinitesimal picture was presented of a dual version of the object (Q, Q ). This is in fact how we discovered the possibility to ‘deform’ or ‘twist’ SUq (2) into the other two quantum groups (see [3] for some more information, and for the link with actions on quantum homogeneous spaces). It is our hope then that this method will allow us to obtain locally compact quantum group versions of q-deformations of some higher-dimensional Lie groups. (We are allowed to use the terminology locally compact quantum groups, which correspond to von Neumann bialgebras with invariant weights, by Proposition 3.7 of [4].) Indeed, the fact that as complicated a quantum group as SU q (1, 1) can be obtained from this procedure, gives good hope. We want to stress that the advantage of this method is the following: actions of a compact quantum group, even on a type I -factor, can be described in a purely algebraic way. Then our general principle gives for free a new locally compact quantum group (and a linking structure), and the difficult analytic computations are relegated to an identification problem, not an existence problem. However, at the moment of writing, such generalizations have not been attempted yet, so it may well be that we are looking at an isolated phenomenon in the setting of quantum Lie groups (though it should be mentioned that on the infinitesimal level, the higher-dimensional analogues are very easily obtained). Contents of the paper. In the first section, we will state the main facts concerning the theory of projective corepresentations of von Neumann bialgebras (taken from [4]). The second section begins with some preliminaries on the action of SUq (2) on the so-called ‘equatorial Podle´s sphere’ and on the quantum projective plane (we take [8] as
196
K. De Commer
the reference here, since both these objects are treated there together, and moreover the same conventions as ours are used). We then present the spectral decomposition of the action of SUq (2) on the quantum projective plane (but relegate the proof to Appendix B), and find in this way a concrete unitary which ‘implements’ this action. In the third section, we apply to this action the ‘projective corepresentation ⇒ linking weak von Neumann bialgebra’ construction we explained in the Introduction, and show that the resulting object coincides with a 2×2-corner of the structure described in Theorem 0.9. In particular, this will show that this 2×2-corner has indeed a coassociative coproduct. In the fourth section, we revisit some of the material of [2] to show that another of the 2×2-corners of the object in Theorem 0.9 has a coassociative coproduct. We then end this section with the proof of Theorem 0.9. q (2) and In Appendix A, we show that the definitions of SUq (2), E SU q (1, 1) we gave in the Introduction are equivalent to the usual ones. In Appendix B, we carry out the computation of the spectral decomposition of SUq (2) on the quantum projective plane. In Appendix C, we prove some summation formulas for basic hypergeometric functions which were used in the article. Conventions and notations. By N, we denote the set of natural numbers with zero included. By N0 , we mean N\{0}. (This is important to mention since another convention is followed in [11]!) We will mostly work with Hilbert spaces of the form l 2 (I ), where I is an index set. We then denote by ei the canonical basis vectors, and by ei j the corresponding canonical matrix units in B(l 2 (I )). We denote by ωi j the normal functional ei , · e j on B(l 2 (I )) (we assume linearity in the second factor). When we write ei j or e j with i or j ∈ / I , the element is interpreted to be zero. ¯ The ordinary The spatial tensor product between von Neumann algebras is denoted ⊗. tensor product between Hilbert spaces is denoted as ⊗. The algebraic tensor product between vector spaces is denoted as . When A ⊆ B(H1 , H2 ) and B ⊆ B(H2 , H 3 ) are linear spaces of maps between n certain Hilbert spaces, we will denote B · A = { i=1 bi ai | n ∈ N0 , bi ∈ B, ai ∈ A}. We use the leg numbering notation for operators on tensor products of Hilbert spaces, as is customary in quantum group theory. E.g., if Z : H ⊗2 → H ⊗2 is a certain operator, we denote by Z 13 the operator H ⊗3 → H ⊗3 acting as Z on the first and third factor, and as the identity on the second factor. In many formulas, we will use the notation ± and ∓. This means that such a formula splits up into two formulas, one in which every ± is replaced by + and ∓ by −, and one in which ± is replaced by − and ∓ by +. 1. Projective Corepresentations We already used the terminology ‘projective corepresentation of a von Neumann bialgebra’ in the Introduction. Let us spell out the definition. Definition 1.1. Let (M, M ) be a von Neumann bialgebra, and H a Hilbert space. By a (unitary left) projective corepresentation of (M, M ) on H , we mean a coaction ¯ α : B(H ) → M ⊗B(H ), that is, a faithful unital normal ∗ -homomorphism satisfying the coaction property (ι ⊗ α)α = ( M ⊗ ι)α.
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
197
For the applications in the subsequent sections, we will always have H = l 2 (N) (in a ‘natural’ way). For the rest of this section, we then fix a von Neumann bialgebra (M, M ) and a left coaction α of (M, M ) on B(l 2 (N)). We further assume that M is represented on a Hilbert space K in a normal, faithful, unit-preserving way, so that we may identify M ⊆ B(K ). The following notion was introduced in [4]. Definition 1.2. Denote I = α(e00 )(K ⊗ l 2 (N)). The unitary G : K ⊗ l 2 (N) → I ⊗ l 2 (N) : ξ → (α(e0i )ξ ) ⊗ ei i∈I
will be called the implementing unitary of α. It is easy to see that the above map G is indeed a well-defined unitary. Its adjoint is given by G ∗ : I ⊗ l 2 (N) → K ⊗ l 2 (N) : ξ ⊗ ei → α(ei0 )ξ.
(3)
For any x ∈ B(l 2 (N)), we then have G ∗ (1 ⊗ x)G = α(x), which follows most easily if one takes x a matrix unit. We also note that the matrix coefficients of G may be interpreted as Clebsch-Gordan coefficients of α. Remark. For the purposes of this section, it will be convenient to keep I the concrete Hilbert space as given above. However, in the later applications, it is more suitable to ‘reparametrize’ I , i.e. to take a unitarily equivalent copy. In this paper, this will not cause any difficulties. However, we remark that taking a different parametrization inside the same Hilbert space has some representation-theoretic consequences (see [4], Prop. 3.5). Notation 1.3. We denote N ⊆ B(K , I ) for the σ -weak closure of the linear span of the set {(ι ⊗ ω0i )(G)m | i ∈ N, m ∈ M} ⊆ B(K , I ), i.e. the σ -weak closure of the right M-module generated by the elements in the first row of G (recall that the functionals ωi j were introduced at the end of the Introduction). We denote O = N ∗ ⊆ B(I , K ) for the space of adjoints of elements in N . Finally, we denote by P the σ -weak closure of the set N · O ⊆ B(I ). By definition, N is a right M-module. It is further easy to compute that (ι ⊗ ω0i )(G)∗ (ι ⊗ ω0k )(G) = (ι ⊗ ωik )(α(e00 )) ∈ M,
for all i, k ∈ N,
so that O · N ⊆ M ⊆ B(K ). We then also have that N · O ⊆ B(I ) is a ∗ -algebra, and hence P is a von Neumann algebra. The following was proven in [4], Prop. 3.6. The proof is not very hard, and follows quite immediately from the two distinguishing properties of G, namely its unitarity and the fact that it implements α.
198
K. De Commer
Proposition 1.4. Write Q for the space P N I ⊆B . O M K Then Q is a unital von Neumann subalgebra of B is norm-dense in I and O · I is norm-dense in K .
I K
. Moreover, we haveN · K
1P 0 , will be a linking von The final properties imply that (Q, f ), with f = 0 0 Neumann algebra (between P and M), in the sense that both f and (1 − f ) are full projections (i.e. O · N and N · O are σ -weakly dense in respectively M and P). We will occasionally write the components P, N , O, M as Q i j , i, j ∈ {1, 2}. We now show that the von Neumann algebra Q of the previous proposition is endowed with more structure, namely, that it carries a coassociative comultiplication. For the proof of the following proposition, we again refer to [4], Prop. 3.6.
Proposition 1.5. Let G be the unitary implementing the projective corepresentation α : P N 2 (N)), and let Q = ¯ be the von Neumann algebra as B(l 2 (N)) → M ⊗B(l O M constructed above. 2 (N)), and, denoting ¯ Then G ∈ N ⊗B(l ¯ ¯ P ⊗P N ⊗N I ⊗I Q∗Q= ⊆B , ¯ ¯ O ⊗O M ⊗M K ⊗K there exists a unique unital, normal, faithful and coassociative ∗ -homomorphism Q : ¯ i j , such that the restriction of Q to M Q → Q ∗ Q such that Q (Q i j ) ⊆ Q i j ⊗Q coincides with M , and such that, denoting by N the restriction of Q to N , we have ( N ⊗ ι)(G) = G13 G23 . The previous proposition thus shows how the coaction α of (M, M ) on B(l 2 (N)) has given rise to a linking weak von Neumann bialgebra (Q, Q ) (see Definition 0.3 of [4]), and in particular to a new von Neumann bialgebra (P, P ) = (Q 11 , 11 ). It is this construction method which we explained in the Introduction. In this paper, the projective corepresentation α under consideration will be the restriction of another coaction α . The following discussion is devoted to the extra structure that will be present in this case. Consider the von Neumann algebra B(l 2 (N)) ⊕ B(l 2 (N)). We can identify it with C(Z2 , B(l 2 (N))), the space of functions from Z2 to B(l 2 (N)), where we will write Z2 = {−, +} for convenience, and where the −-fiber is the left B(l 2 (N))-component. We can then consider the following maps: the projection maps obtained by evaluation, π± : B(l 2 (N)) ⊕ B(l 2 (N)) → B(l 2 (N)) : x → x(±), and the diagonal embedding map d : B(l 2 (N)) → B(l 2 (N)) ⊕ B(l 2 (N)) : x → d(x) = x ⊕ x.
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
199
Then π+ ◦ d = π− ◦ d = ι B(l 2 (N)) , the identity map. We also have the flip map σ : B(l 2 (N)) ⊕ B(l 2 (N)) → B(l 2 (N)) ⊕ B(l 2 (N)) : x ⊕ y → y ⊕ x, which induces an action of Z2 . Then d(B(l 2 (N)) consists precisely of the fixed elements (+) (−) for σ . We will further write ekl = 0 ⊕ ekl and ekl = ekl ⊕ 0, and similarly for the units in these fibers: 1(+) = 0 ⊕ 1 and 1(−) = 1 ⊕ 0. We will now assume that the von Neumann bialgebra (M, M ) has a coaction 2 ¯ (N)) ⊕ B(l 2 (N))), α : B(l 2 (N)) ⊕ B(l 2 (N)) → M ⊗(B(l
which is equivariant with respect to σ : α ◦ σ = (ι ⊗ σ ) ◦ α. Then it is clear that α restricts to a coaction of M on d(B(l 2 (N))). We then further assume that our given coaction α on B(l 2 (N)) and the restriction of α to d(B(l 2 (N))) coincide by the isomorphism d : B(l 2 (N)) → d(B(l 2 (N))): α ◦ d = (ι ⊗ d)α. Notation 1.6. We denote 2 ¯ α : C(Z2 , B(l 2 (N))) → M ⊗B(l (N)). α± = (ι ⊗ π± )
For the following proposition, recall that we use the notation G for the projective unitary corepresentation associated with α (Definition 1.2), the notation I for the space α(e00 )(K ⊗ l 2 (N)), and the notation (P, P ) for the von Neumann bialgebra as constructed in Prop. 1.5. Proposition 1.7. Denote 2 ¯ (N)). e = α+ (1(+) − 1(−) ) ∈ M ⊗B(l
Then there exists a self-adjoint grouplike unitary e in P such that e = G ∗ (e ⊗ 1)G. Recall that the group-like property means that P (e) = e ⊗ e. Proof. Denote (+)
p˘ + = α+ (e00 ), (−) p˘ − = α+ (e00 ),
then p˘ + and p˘ − are orthogonal projections summing to α(e00 ). In particular, we have p˘ ± ≤ α(e00 ), and hence they correspond to projections p± in B(I ) by the formula 2 (N)), we actually have ¯ p˘ ± = G ∗ ( p± ⊗ e00 )G. Since p˘ ± ∈ M ⊗B(l (ι ⊗ ω0k )(G)(ι ⊗ ωkl )( p˘ ± )(ι ⊗ ω0l (G))∗ ∈ P. p± = k,l∈N
Then e := p+ − p− is a self-adjoint unitary in P. We prove that it satisfies the conditions above.
200
K. De Commer
Write q+ = α+ (1(+) ). Then we have G ∗ ( p+ ⊗ 1)G = G ∗ ( p+ ⊗ ei0 e00 e0i )G i
=
G ∗ (1 ⊗ ei0 )G p˘ + G ∗ (1 ⊗ e0i )G
i
=
(+) α(ei0 ) α+ (e00 )α(e0i )
i
=
(+) (−) (+) (+) (−) α+ ((ei0 + ei0 )e00 (e0i + e0i ))
i
= α+ (1(+) ). This proves that e = G ∗ (e ⊗ 1)G. We now prove that e is grouplike. First, we compute that ( P (e) ⊗ 1)G13 G23 = = = =
( P (e) ⊗ 1)( N ⊗ ι)(G) ( N ⊗ ι)((e ⊗ 1)G) ( N ⊗ ι)(G e) G13 G23 ( M ⊗ ι)( e).
On the other hand, (e ⊗ e ⊗ 1)G13 G23 = ((e ⊗ 1)G)13 ((e ⊗ 1)G)23 = (G e)13 (G e)23 = G13 e13 G23 e23 = G13 G23 (ι ⊗ α)( e) e23 . So from the above two computations, we see that it is sufficient to see if e), (ι ⊗ α)( e) e23 = ( M ⊗ ι)( i.e. that e is an α-cocycle. Bringing e23 to the other side, and writing out the expressions with use of the coaction property (ι ⊗ α ) α = ( M ⊗ ι) α , this becomes, writing h = 1(+) − 1(−) , α+ )( α (h)(1 ⊗ h)). (ι ⊗ α)( α+ (h)) = (ι ⊗ Now α+ )((ι ⊗ d) α+ (h)), (ι ⊗ α)( α+ (h)) = (ι ⊗ so it is sufficient to prove that α (h)(1 ⊗ h), (ι ⊗ d) α+ (h) = α− (h). But this follows immediately which is equivalent with the identity α+ (h) = − from the equivariance of α with respect to σ , and the fact that σ (h) = −h. It is convenient to split up G with the aid of the projections constituting the group-like element e above.
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
201
Notation 1.8. Let α be a σ -equivariant coaction of (M, M ) on B(l 2 (N)) ⊕ B(l 2 (N)) which restricts to α as above. Let e ∈ P be the group-like element of Proposition 1.7, and write e = p+ − p− with p+ and p− orthogonal projections in P. Then we write G (±) = ( p± ⊗ 1)G. It is clear that G (±) are then isometries with range ( p± I ) ⊗ l 2 (N). Let us record the following fact. Lemma 1.9. Using the notation from Proposition 1.5 (w.r.t. α) and Notation 1.8, we 2 (N)) and ¯ have that G (±) ∈ N ⊗B(l (+) (±)
(−) (∓)
( N ⊗ ι)G (±) = G13 G23 + G13 G23 . 2 (N)). Moreover, ¯ Proof. As p± ∈ P, it is immediate that G (±) = ( p± ⊗ 1)G ∈ N ⊗B(l as e is a group-like element in P, it follows that
P ( p± ) = p+ ⊗ p± + p− ⊗ p∓ . As N (x y) = P (x) N (y) for x ∈ P and y ∈ N , and ( N ⊗ι)G = G13 G23 , the formula in the statement of the lemma follows. 2. On the Action of SUq (2) on the Quantum Projective Plane 2.1. The equatorial Podle´s sphere and the quantum projective plane. In [2], we showed how one can apply the theory of the previous section to the action of SUq (2) on the standard Podle´s sphere (i.e. the one which arises as the quotient space by the S 1 -action). In this paper, we will need to consider another Podle´s sphere, namely the equatorial one (which is the other extreme point in the moduli space of Podle´s spheres). Let us recall the definition in the version which will be of most use to us. The equivalence of this definition with the ordinary one as a universal object can be found for example in [8]. We also refer to that paper for the notion of the quantum projective plane (note that this ‘projective’ is unrelated to the one of the previous section!). We will keep using the notations σ, d and π± we introduced near the end of the previous section. Definition 2.1. Denote by Y (±) and W (±) the following operators on l 2 (N): Y (±) = ± 1 − q 4k ek−1,k , k∈N0
W (±) = ±
q 2k ekk .
k∈N
Consider then Y, W ∈ C(Z2 ,
B(l 2 (N)))
= B(l 2 (N)) ⊕ B(l 2 (N)), with
Y (μ) := Y (μ) , W (μ) := W (μ) . 2 ) generated by Y and W is called the space of conThen the unital C∗ -algebra C(Sq∞ 2 . tinuous functions on the equatorial Podle´s sphere Sq∞ 2 ), given The Z2 -action σ on C(Z2 , B(l 2 (N))) restricts to an action of Z2 on C(Sq∞ on the generators as σ (Y ) = −Y and σ (W ) = −W . We denote the space of Z2 -fixed elements as C(RPq2 ), and call it the space of continuous functions on the quantum projective plane RPq2 .
202
K. De Commer
2 . Since it comIt is well-known that SUq (2) has a natural ergodic action α on Sq∞ mutes with the Z2 -action (see e.g. Remark 4.2 in [8]), we then also have an ergodic action α of SUq (2) on RPq (2). Now in Lemma 6.4 of [18], it is shown that the ensuing 2 ) is obtained by applying the functional SUq (2)-invariant state ω on C(Sq∞
ω(x) =
1 (ω(x(+)) + ω(x(−))) 2
2 ) ⊆ C(Z , B(l 2 (N))), where ω is Tr( · D) with D the trace class operaon C(Sq∞ 2 2 ) = C(Z , B(l 2 (N))), it tor (1 − q 2 )Diag(q 2k ). From this, and the fact that C(Sq∞ 2 2 ) = π (C(S 2 )) , with π the GNSfollows that the von Neumann algebra L ∞ (Sq∞ ω ω q∞ representation, may be identified with C(Z2 , B(l 2 (N))), in such a way that π ω (Y ) and π ω (W ) coincide with respectively Y and W . 2 It then also follows that L ∞ (RPq2 ) = π ω (C(RPq )) may be identified with 2 2 2 ∼ B(l (N)) = d(B(l (N))) ⊆ C(Z2 , B(l (N))), the space of constant functions from Z2 to B(l 2 (N)). As it is well-known (and easy to show) that any ergodic action of SUq (2) on a C∗ -algebra can be completed to a coaction of L ∞ (SUq (2)) on the von Neumann algebraic completion of the C∗ -algebra in its GNS-representation with respect to the actioninvariant state on it, we then obtain, from the ordinary algebraic definition of the actions of SUq (2) on the equatorial Podle´s sphere (cf. [16,17]) and on the quantum projective plane, the following von Neumann algebraic descriptions. 2 ) ∼ Definition 2.2. We denote by α the unique coaction of L ∞ (SUq (2)) on L ∞ (Sq∞ = 2 C(Z2 , B(l (N))) such that
α (Y ∗ ) = (a+∗ )2 ⊗ Y ∗ − q(1 + q 2 )a+∗ b+ ⊗ W − qb+2 ⊗ Y, α (W ) = a+∗ b+∗ ⊗ Y ∗ + (1 − (1 + q 2 )b+∗ b+ ) ⊗ W + b+ a+ ⊗ Y, α (Y ) = −q(b+∗ )2 ⊗ Y ∗ − q(1 + q 2 )b+∗ a+ ⊗ W + a+2 ⊗ Y. α to a coaction This coaction is then Z2 -equivariant. We denote by α the restriction of on L ∞ (RPq2 ) = B(l 2 (N)) ∼ = d(B(l 2 (N))). Note then in particular that α is a coaction of the form treated in the final part of the previous section. 2.2. Spectral decomposition of the SUq (2)-action on the quantum projective plane. In the previous section, we showed that the action of SUq (2) on the quantum projective plane RPq2 gives rise to a coaction 2 ¯ α : B(l 2 (N)) → L ∞ (SUq (2))⊗B(l (N)),
which is hence a projective corepresentation of L ∞ (SUq (2)) on l 2 (N) in the terminology of Definition 1.1. In this section, we want to find an explicit description of the associated unitary G which we introduced in Definition 1.2. (−) (+) Denote again by e00 and e00 the matrix units at position 00 in resp. the − and + fiber (±) of B(l 2 (N)) ⊕ B(l 2 (N)). Then e00 are precisely the spectral projections at eigenvalue ±1 of W . Hence to determine α(e00 ), we should try to determine the eigenvectors for
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
203
1 and −1 of α+ (W ), using the notation introduced at the end of the previous section (Notation 1.6). To make the enunciation of the following proposition and subsequent ones more succinct, the following notation, extending the one in Notation 0.7, will come in handy. (+)
(−)
Notation 2.3. We denote J (+) = I− for the set Z. We denote J (−) = −I− − 1 for the set N. We denote by J the disjoint union J = J (+) J (−) . We then use the same notational conventions for elements in J as for elements in I− . Proposition 2.4. The spectrum of the operator α+ (W ) equals {±q 2r , 0 | r ∈ N}, with 0 not occurring in the point spectrum. For r ∈ N, an orthonormal basis for the eigenspace (t, p) I±q 2r of ±q 2r is given by the vectors ξr,± with p, t ∈ Z and p + r ∈ J (±) , determined by the formula (t, p)
ξr,± =
∞
Q q 2 ( p± , r, n) en ⊗ et−n ⊗ e p+n ∈ H+ ⊗ l 2 (N),
n=0
with Q q 2 ( p± , r, n) = (∓q)r (±1)n q
n(n−1) 2
1/2
(∓q 2 p+2r +2 ; q 2 )n (q 4 p+4n+4 ; q 4 )∞ 1/2
1/2
1/2
1/2
(∓q 2 p+2r +2 ; q 2 )∞ (q 4 ; q 4 )r (−1; q 2 )∞ (q 2 ; q 2 )n −2n q q −2r ±q −2 p−2n 2, q2 . · 3 ϕ2 | q ∓q −2 p−2n−2r 0
The proof of this proposition will be presented in Appendix B. Now denote by I = I{−1,1} = I1 + I−1 the range of the spectral projection of α+ (W ) associated with the set {−1, 1}. This then equals the range space of α(e00 ). Further recall that I− denotes the set I−(+) I−(−) = Z N− 0 . Then, by the above results, we can define a unitary map (t, p)
(±)
u : I → H− = l 2 (I− ) ⊗ l 2 (Z) : ξ0,± → e− p−1 ⊗ e p+t , (±)
where we recall that en = en ± for n ∈ I− . In the following proposition, we will again use the notation I+ = N. Proposition 2.5. The map (t, p)
(±)
G : H+ ⊗ l 2 (N) → H− ⊗ l 2 (N) : ξr,± → e− p−r −1 ⊗ e p+t−r ⊗ er defines a unitary, and (u ∗ ⊗ 1)G coincides with the unitary constructed in Definition 1.2. Proof. The fact that G is a well-defined unitary is of course immediate by the previous (t, p) proposition. Then, from the way α+ (Y ∗ ) acts on the non-normalized eigenvectors ηr,±
204
K. De Commer
in the proof of Proposition 2.4 (see the identities (12), (13) and (14) in Appendix B), we have, for r ∈ N, t ∈ Z and p ∈ J (±) , that (t, p)
(±)
G ∗ (u ⊗ 1) ξ0,± ⊗ er = G ∗ e− p−1 ⊗ e p+t ⊗ er (t+2r, p−r )
= ξr,± (∓1)r (t+2r, p−r ) = (t+2r, p−r ) ηr,± ηr,± (∓1)r (t, p) (−1)r = α+ (Y ∗ )r η0,± (t, p) ∗ r α+ (Y ) η0,± =
(±1)r (t, p) α+ (Y ∗ )r ξ0,±
(t, p)
α+ (Y ∗ )r ξ0,± .
But (t, p)
(t, p)
α+ (Y ∗ )r ξ0,± 2 = α+ (Y )r α+ (Y ∗ )r ξ0,± , ξ0,± t, p
(t, p)
(t, p)
= α+ ((q 4 W 2 ; q 4 )r )ξ0,± , ξ0,± = (q 4 ; q 4 )r . So (t, p)
−1/2
G ∗ (u ⊗ 1) ξ0,± ⊗ er = (±1)r (q 4 ; q 4 )r −1/2
= (q 4 ; q 4 )r
(t, p)
α+ (Y ∗ )r ξ0,±
(t, p)
α+ (Y ∗ )r α+ (W )r ξ0,± −1/2
= q −r (r +1) (q 4 ; q 4 )r
−1/2
= q −r (r +1) (q 4 ; q 4 )r
(t, p)
α+ ((Y W )∗ )r ξ0,±
(t, p)
α(((Y W )∗ )r e00 )ξ0,±
(t, p)
= α(er 0 )ξ0,± , which, by Eq. (3) after Definition 1.2, proves the proposition.
As mentioned in the first section, we may treat G itself as the implementing unitary of α, as the unitary u only serves to reparametrize the Hilbert space α(e00 )(H+ ⊗ l 2 (N)). Now further denote e : l 2 (I− ) ⊗ l 2 (Z) → l 2 (I− ) ⊗ l 2 (Z) : en(±) ⊗ el → ± en(±) ⊗ el , denote p± = 21 (1 ± e) and G (±) = ( p± ⊗ 1)G. Then e is precisely the self-adjoint unitary which appeared in Proposition 1.7, so our notation is consistent with the one introduced in Notation 1.8. It is then also easy to see that G (±) α+ (W ) = ±(1 ⊗ W (+) )G (±) , G α+ (Y (∗) ) = ±(1 ⊗ (Y (+) )(∗) )G (±) . (±)
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
205
Hence α+ (x) = ±(1 ⊗ x (+) )G (±) , G (±)
2 for all x ∈ L ∞ (Sq∞ ).
Write now G (±) =
∞
(±) Gr,s ⊗ er s
r,s=0 (±)
(±)
as a σ -weakly converging sum, where Gr,s : l 2 (I+ ) ⊗ l 2 (Z) → l 2 (I− ) ⊗ l 2 (Z). Proposition 2.6. For n ∈ N, k ∈ Z, we have n(n−1) (∓q 2s+2r −2n+2 ; q 2 )n (q 2n+2 ; q 2 )∞ 1 (±) en ⊗ ek = √ (∓q)r (±1)n q 2 Gr,s 1/2 1/2 1/2 2 (∓q 2s+2r +2 ; q 2 )∞ (q 4 ; q 4 )r (q 4 ; q 4 )s −2n q −2r ±q −2s q (±) · 3 ϕ2 | q 2 , q 2 en−r −s−1 ⊗ ek−r +s . −2s−2r ∓q 0
1/2
1/2
Remark. Using transformation formula (III.11) of [6], we have −2n −2r −2n q q q −q −2s −q −2r q −2s 2 2 n 2 2 = (−1) . ϕ , q ϕ , q | q | q 3 2 3 2 q −2s−2r 0 q −2s−2r 0 Hence we can also write (±) Gr,s en
n(n−1) (∓q 2s+2r −2n+2 ; q 2 )n (q 2n+2 ; q 2 )∞ 1 ⊗ ek = √ (∓q)r q 2 1/2 1/2 1/2 2 (∓q 2s+2r +2 ; q 2 )∞ (q 4 ; q 4 )r (q 4 ; q 4 )s −2n q ±q −2r q −2s (±) · 3 ϕ2 | q 2 , q 2 en−r −s−1 ⊗ ek−r +s . ∓q −2s−2r 0
1/2
1/2
(±)
Proof. We have that for n ∈ N, k, l ∈ Z and m ∈ I− , (±) (±) (±) em ⊗ el , Gr,s en ⊗ ek = em ⊗ el ⊗ er , G en ⊗ ek ⊗ es (±) = G ∗ em ⊗ el ⊗ er , en ⊗ ek ⊗ es (l+2r +m+1,−m−r −1) , en ⊗ ek ⊗ es = ξr,± (k+n,s−n)
= δm,n−r −s−1 δl,k−r +s ξr,±
, en ⊗ ek ⊗ es . (t, p)
The proposition then follows immediately by the concrete form of the ξr,± given in Proposition 2.4, after rearranging some of the q-shifted factorials. Definition 2.7. We define the following operators: 1/2
L 0+ : l 2 (I+ ) ⊗ l 2 (Z) → l 2 (I0 ) ⊗ l 2 (Z) : en ⊗ ek → (q 2n+2 ; q 2 )∞ en ⊗ ek , a0 : l 2 (I0 ) ⊗ l 2 (Z) → l 2 (I0 ) ⊗ l 2 (Z) : en ⊗ ek → en−1 ⊗ ek , (±)
L −0 : l 2 (I0 ) ⊗ l 2 (Z) → l 2 (I− ) ⊗ l 2 (Z) : en ⊗ ek → q
n(n+1) 2
(∓q −2n ; q 2 )∞ en(±) ⊗ ek , 1/2
f : l 2 (I− ) ⊗ l 2 (Z) → l 2 (I− ) ⊗ l 2 (Z) : en(±) ⊗ ek → (±1)n+1 en(±) ⊗ ek .
206
K. De Commer
Note that the map L 0+ also appeared in [2] and [23]. Also note that the above operators are indeed all bounded. (±)
Definition 2.8. For r, s ∈ N, we define polynomials K r,s as follows: (±) K r,s (x) = 2 ϕ1
q −2 min{r,s} −q −2 min{r,s} 2 2 | q , q x . ±q 2|r −s|+2
Proposition 2.9. For s ≥ r , we have (±1)s 1 (±) = √ q 2 (r −s)(3r +s+1) Gr,s 2 1/2
×
(q 4 ; q 4 )s
(±q 2s−2r +2 ; q 2 )∞ (±) r +s+1 (±) ∗ f L −0 a0 L 0+ K r,s (b+ b+ )b+s−r . 1/2 (q 4 ; q 4 )∞ (q 4 ; q 4 )r
For r ≥ s, we have (±1)r 1 (±) Gr,s = √ q 2 (s−r )(3s+r +1) 2 (±q 2r −2s+2 ; q 2 )∞ (±) r +s+1 (±) ∗ L −0 a0 L 0+ K r,s (b+ b+ )(−qb+∗ )r −s . 1/2 4; q 4) 4 4 (q ∞ (q ; q )s 1/2
×
(q 4 ; q 4 )r
Proof. For s ≥ r , we have, by applying the transformation formula (III.6) of [6] with respect to q −2r as the terminating factor, that 3 ϕ2
q −2n q −2r ±q −2s | q 2, q 2 ∓q −2s−2r 0
= (−q 2r −2n )r
(±q 2s−2r +2 ; q 2 )r (∓q 2s+2 ; q 2 )r
2 ϕ1
q −2r −q −2r 2 2n+2 , | q , q ±q 2s−2r +2
while for r ≥ s, we have, applying transformation formula (III.6) of [6] with respect to q −2s as the terminating factor, 3 ϕ2
q −2n ±q −2r q −2s | q 2, q 2 ∓q −2s−2r 0
= (−q
2s−2n s (±q
)
2r −2s+2 ; q 2 )
(∓q 2r +2 ; q 2 )s
s
2 ϕ1
q −2s −q −2s 2 2n+2 . | q ,q ±q 2r −2s+2
Then with the above transformation formulas at hand, the proposition follows straight(±) forwardly from the formulas for Gr,s in Proposition 2.6 and the remark following it, applying only elementary (although tedious) computations with shifted q-factorials and q-powers. (±)
Remark. It seems odd that in the formula for Gr,s , an extra ‘parity operator’ f appears when switching from r ≥ s to s ≥ r . We have no real conceptual reason to explain this phenomenon.
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
207
3. Identification of the Reflected Quantum Group In the previous section, we found an explicit description of the unitary G implementing the projective corepresentation α of L ∞ (SUq (2)) on B(l 2 (N)) ∼ = L ∞ (RPq2 ). As G was 2 2 2 defined as a unitary from H+ = l (I+ ) ⊗ l (Z) to H− = l (I− ) ⊗ l 2 (Z), we will have that the space N for G, introduced in Notation 1.3, will be a subspace of B(H+ , H− ). Lemma 3.1. The equality N = L (−, +) holds. ¯ (Z) ⊆ Proof. We recall that L (−, +) was just the space B(l 2 (I+ ), l 2 (I− ))⊗L B(H+ , H− ). As it is immediately observed from Proposition 2.6 that all G0,r = (+) (−) + G0,r commute with 1 ⊗ S, with S the bilateral forward shift on l 2 (Z), it follows G0,r ¯ (Z) that G0,r ⊆ L (−, +) for all r ∈ N. As L ∞ (SUq (2)) = L (+, +) = B(l 2 (I+ ))⊗L by definition, and N is generated by the G0,r as a right L (+, +)-module (again by definition), we obtain the inclusion N ⊆ L (−, +). (±) Now, using the notation introduced at the end of the previous section, we have G0,r ∈ (+) (e p0 ⊗ 1) N by Lemma 1.9. Take p ∈ N. It follows then from Proposition 2.6, that G0,0 is a non-zero scalar multiple of the matrix unit e( p−1)+ ,0 ⊗ 1 in B(H+ , H− ), while (±) G0, p (e00 ⊗ S − p ) is a non-zero scalar multiple of the matrix unit e(− p−1)± ,0 ⊗ 1. So N contains all matrix units of the form ep,0 ⊗ 1, for p ∈ I− . As N is by definition closed under right multiplication with elements in L (+, +), it follows that indeed N = L (−, +).
Itthen follows immediately from this lemma that the linking von Neumann alge P N associated with G, using again Notation 1.3, equals precisely the bra O M P N = {−, +}-part of the von Neumann algebra Q in Theorem 0.9, i.e. O M L (−, −) L (−, +) , with in particular M = L ∞ (SUq (2)). We also recall that L (+, −) L (+, +) we denoted by and μν (or M , N , . . .) the comultiplication and its constituents on Q, as obtained by the method explained in Proposition 1.5. Further recall that we defined another collection of maps μν on L (μ, ν) in the discussion preceding Theorem 0.9. Proposition 3.2. The comultiplications μν and μν coincide for μ, ν ∈ {−, +}. From this proposition (and from Proposition 1.5), it will then immediately follow that the μν for μ, ν ∈ {−, +} are coassociative. In particular, as we will not actually need to use that − is coassociative, this will give an alternative, indirect proof for the coassociativity of the comultiplication − on L ∞ ( SU q (1, 1)) (first established in [11]). Proof (of Proposition 3.2). The fact that ++ = ++ , the natural comultiplication on L ∞ (SUq (2)), was part of Proposition 1.5. Further, as L (+, −) = L (−, +)∗ , and L (−, +) · L (+, −) is σ -weakly dense in L (−, −), the equalities −− = −− and +− = +− will immediately follow from the equality −+ = −+ . But −+ (x y) = −+ (x)++ (y) and −+ (x y) = −+ (x)++ (y) for all elements x ∈ L (−, +) and y ∈ L (+, +). So we see that it is sufficient to prove, for p ∈ I− , that N (ep0 ⊗ 1) =
208
K. De Commer
+ N (ep0 ⊗ 1). As both these expressions vanish on ξr,s, p,t for p ∈ N0 , it is already − sufficient to prove that, for p ∈ I− = Z N0 and r, s, t ∈ Z, we have + − N (ep0 ⊗ 1)ξr,s,0,t = ξr,s,p,t .
Using the formulas for the Gr,s in Proposition 2.6, we see that for p ∈ N, (−q −2 p+2 ; q 2 ) p (q 2 p+2 ; q 2 )∞ 1 1 (+) G0,0 (e p0 ⊗ 1) = √ q 2 p( p−1) e( p−1)+ ,0 ⊗ 1 1/2 2 (−q 2 ; q 2 )∞ 1/2
1/2
and 1/2
(q 2 ; q 2 )∞ 1 (±) e ⊗ S p. G0, p (e00 ⊗ 1) = √ 1/2 (− p−1)± ,0 4 4 2 (∓q 2 p+2 ; q 2 )1/2 ∞ (q ; q ) p + Then since, for m, n, p ∈ N and r, s, t, k ∈ Z, we have ++ (emn ⊗ S k )ξr,s, p,t = + δn, p ξr,s,m,t+k by Proposition 0.4, and N (x y) = N (x) M (y) for x ∈ N and y ∈ M = L ∞ (SUq (2)), we see that it is sufficient to prove that for p ∈ N0 , we have − ξr,s,( p−1)+ ,t =
√ − 1 p( p−1) 2q 2
1/2
(−q 2 ; q 2 )∞
(+)
1/2 1/2 (−q −2 p+2 ; q 2 ) p (q 2 p+2 ; q 2 )∞
+ N (G0,0 )ξr,s, p,t , (4)
and that for p ∈ N, we have − ξr,s,(− p−1)± ,t+ p =
4 4 1/2 √ (∓q 2 p+2 ; q 2 )1/2 ∞ (q ; q ) p (±) + 2 N (G0, p )ξr,s,0,t . 1/2 2 2 (q ; q )∞
From Lemma 1.9, it follows that (+) )= N (Gr,s
± j∈N
and (−) )= N (Gr,s
± j∈N
(±)
(5)
(±)
Gr, j ⊗ G j,s
(±) Gr,(∓) j ⊗ G j,s
as σ -weakly converging sums. We will in the following use the alternative expression (2) for the function Pq+2 which (±)
appears in Proposition 0.4. Then using the formula from Proposition 2.6 for the G0, j -
term, and the slightly different formula in the remark following it for the G (±) j,0 -term, we get from the formulas in the previous paragraph, after some easy simplifications, (+)
+ N (G0,0 )ξr,s, p,t 1 1 1 = (−1) p (±1)v (∓q) j q vw+ p(v+w+1) q 2 v(v−1) q 2 w(w−1) 2 ± v,w∈I+ j∈Z v−w=t
1/2
1/2
1/2
(q 2v+2 ; q 2 )∞ (q 2 p+2 ; q 2 )∞ (∓q 2 j−2v+2 ; q 2 )v (∓q 2 j−2w+2 ; q 2 )w (±q 2 j+2 ; q 2 )∞ (q 2 ; q 2 )w (q 4 ; q 4 )∞ −2w −2v −2 p q q q (±) (±) | q 2 , q 2 ev− j−1 ⊗ er + p−w+ j ⊗ ew− j−1 ⊗ es+v− p− j , · 3 ϕ2 0 0 ·
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
209
where we recall that by convention ea = 0 when a ∈ / I− . Now we apply the change of variables j = w + m = v + n with m − n = t. Changing then the order of summation, we get, using still the notation J (+) = Z and J (−) = N, (+) + N (G0,0 )ξr,s, p,t =
×
1 1 2 (−1) p q 2 (2t p+2 p+t −t) 2
1/2 (q 2 p+2 ; q 2 )∞ (±1)t (q 4 ; q 4 )∞ ±
(∓q)m
(−1)w q w(2v+2 p)
v,w≥0 v−w=t
m,n∈J (±) m−n=t 1/2
1/2
(q 2v+2 ; q 2 )∞ (∓q 2n+2 ; q 2 )v (∓q 2m+2 ; q 2 )w (±q 2m+2w+2 ; q 2 )∞ (q 2 ; q 2 )w −2w −2v −2 p q q q (±) (±) 2 2 | q , q e−n−1 ⊗ er + p+m ⊗ e−m−1 ⊗ es−n− p . · 3 ϕ2 0 0 ·
Now with the conditions on m, n, v, w as in this summation, we can write 1/2
2m+2 (∓q 2n+2 ; q 2 )1/2 ; q 2 )1/2 v (∓q w =
(∓q 2n+2 ; q 2 )∞
1/2
(∓q 2m+2 ; q 2 )∞
(∓q 2m+2 ; q 2 )w .
Then we can apply the summation formula in Proposition C.1 with x = ±q 2m and y = ±q 2n to obtain (+) + )ξr,s, p,t N (G0,0
=
1/2 1 (−1; q 2 ) p (q 2 p+2 ; q 2 )∞ 1 2 (−1) p q 2 (2t p+2 p+t −t) (±1)t 2 (q 4 ; q 4 )∞ ±
(∓q)m
m,n∈J (±) m−n=t
−2n ∓q (±) (±) 2 2m+2 p+2 e−n−1 · | q , ±q ⊗er + p+m ⊗e−m−1 ⊗es−n− p . 1/2 q 2t+2 (∓q 2m+2 ; q 2 )∞ 1/2
(∓q 2n+2 ; q 2 )∞
Plugging this into the right-hand side of equality (4), a straightforward comparison of coefficients proves the validity of the identity (4). The other case (5) is entirely similar. We first write Pq+2 again in the 3 ϕ2 -form as in + the previous case, as this then simplifies immediately inside the expression for ξr,s,0,t . (μ)
Next, for μ ∈ {−, +}, write the expansion for N (G0, p ) as (μ)
N (G0, p ) =
± j∈N
(±μ)
G0, j
(±)
⊗ G j, p ,
210
K. De Commer (±μ)
+ and, when evaluating it on ξr,s,0,t , use the formula in Proposition 2.6 for the factor G0, j , and the slightly different expression given in the remark below that proposition for the (±) factor G j, p . We find (μ)
+ N (G0, p )ξr,s,0,t
=
1 1 1 1 (±μ)v (∓1) j q vw+ 2 v(v−1)+ 2 w(w−1)+ j 1/2 1/2 2 (q 4 ; q 4 )∞ (q 2 ; q 2 ) (q 4 ; q 4 ) p + ∞
± v,w∈I i, j∈Z v−w=t i− j= p
1/2
1/2
(∓q 2i−2w+2 ; q 2 )∞ (∓μq 2 j−2v+2 ; q 2 )∞ (q 2v+2 ; q 2 )∞ (q 2w+2 ; q 2 )∞ (±μq 2 j+2 ; q 2 )∞ (∓q 2i+2 ; q 2 )∞ −2w −2 j −2 p q ±q q (±μ) (±) 2 2 × 3 ϕ2 | q , q ev− j−1 ⊗ er −w+ j ⊗ ew−i−1 ⊗ es+v− j+ p . ∓q −2i 0 ×
If we then apply the change of variables j = v + m and i = w + n and change the order of summation, we obtain (μ)
+ N (G0, p )ξr,s,0,t 1 (−μ)t q 2 t (t+1) 1 = 4 4 1/2 2 (q 4 ; q 4 )∞ (q 2 ; q 2 )1/2 ∞ (q ; q ) p ±
×
∞
(−μ)w q 2w q (2n−2m−2 p)w 2
w=0
× 3 ϕ2
q −2w ±q −2n+2 p−2w ∓q −2n−2w
1/2
(∓q)m (∓μq 2m+2 , ∓q 2n+2 ; q 2 )∞
m,n∈J (±) m−n=−t− p
(±μq 2n−2 p+2w+2 , q 2n−2m−2 p+2w+2 , q 2w+2 ; q 2 )∞ (∓q 2n+2w+2 ; q 2 )∞ q −2 p (±μ) (±) | q 2 , q 2 e−m−1 ⊗ er +n− p ⊗ e−n−1 ⊗ es−m+ p . 0
We now remark that the scalar appearing after the summation over m equals precisely the (q 2 ;q 2 )∞ 2n−2 p and y = ±q 2m , where g ( p,μ) expression (∓q w 2n+2 ) gw ( p, μ)(x, y) with x = ±q ∞ is the function appearing in Proposition C.2. So using the summation formula of that proposition, the above sum reduces to (μ)
+ N (G0, p )ξr,s,0,t 1 1/2 1 (−μ)t q 2 t (t+1) (q 2 ; q 2 )∞ = 2 (q 4 ; q 4 )∞ (q 4 ; q 4 )1/2 p ±
×
m,n∈J (±) m−n=−t− p
2m+2 ; q 2 )1/2 ∞ m (∓μq (∓q) 1/2 2n+2 2 (∓q ; q )∞
∓μq −2m (±μ) (±) 2 2n−2 p+2 e−m−1 ⊗ er +n− p ⊗ e−n−1 ⊗ es−m+ p . | q , ±μq μq 2n−2m+2 (μ)
+ Plugging this expression for N (G0, p )ξr,s,0,t into the right hand side of the identity (5), we can again conclude, after a tedious simplification, that both sides are equal.
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
211
q (2) 4. The Linking Weak von Neumann Bialgebra Between SUq (2) and E The main part of this section consists in showing that, in Theorem 0.9, the μν with μ, ν ∈ {+, 0} are coassociative. This will then let us conclude the proof of that theorem in a straightforward manner. For the {0, +}-part of (Q, Q ), we do not have to go through as much trouble as for the SU q (1, 1)-case, as we have already treated a lot of material concerning it in [2]. We do remark however that we have not yet substantiated the claim that 0 , in the form represented in Proposition 0.6, equals the comultiplication in Definition 0.5, which we will momentarily write as L ∞ ( Eq (2)) for distinction. This equality will be proven in Proposition 4.2. But let us observe already that W0 is a well-defined unitary: this follows straightforwardly from the q-Hankel orthogonality relations for the 1 ϕ1 -q-Bessel functions (see [10], Eq. (5.9), and our Appendix A for the relation of our Pq02 -functions with these 1 ϕ1 -q-Bessel functions). As we had observed already, it is then easy to see q (2)) to L ∞ ( E q (2))⊗L q (2)) (c.f. the ¯ ∞( E that 0 is a well-defined map from L ∞ ( E proof of Proposition 3.8 in [11]). Let us state one of the main results from that paper (see [2], Theorem 3.13 and Theorem 4.4). We will use notations as in the Introduction. Recall that the operators a0 and L 0+ we will use were defined in Definition 2.7. Proposition 4.1. There exists a unital, normal, coassociative ∗ -homomorphism ¯ (0, 0) L (0, +)⊗L ¯ (0, +) L (0, 0) L (0, +) L (0, 0)⊗L : → ¯ (+, 0) L (+, +)⊗L ¯ (+, +) L (+, 0) L (+, +) L (+, 0)⊗L which restricts to maps ¯ (μ, ν), μν : L (μ, ν) → L (μ, ν)⊗L and such that 00 = L ∞ ( Eq (2)) , ++ = L ∞ (SUq (2)) and ∞ k k k ∗ k (q 2 ; q 2 )−1 0+ (L 0+ ) = k a0 L 0+ b+ ⊗ a0 L 0+ (−qb+ ) , k=0
this last sum being convergent in norm. The coassociativity of the μν will then follow immediately from the following proposition. Proposition 4.2. For all μ, ν ∈ {0, +}, we have μν = μν . Proof. We observe that the proposition will follow once we prove that 0 (a0 ) = 00 (a0 ) and 0+ (L 0+ ) = 0+ (L 0+ ). Indeed, suppose this is satisfied. Then as the equality (L ∞ (SUq (2)) =)+ = + was part of the previous proposition, we will have that, for x ∈ L ∞ (SUq (2)) and m ∈ Z, 0+ (a0m L 0+ x) = 0 (a0 )m 0+ (L 0+ )+ (x) = 0 (a0 )m 0+ (L 0+ )+ (x) = 0+ (a0m L 0+ x).
212
K. De Commer
But the elements of the form a0m L 0+ x are easily seen to be σ -weakly dense in L (0, +). So then 0+ = 0+ follows from this. As in the beginning of Proposition 3.2, this will allow us to conclude μν = μν for all μ, ν ∈ {0, +}. We now first show that 0 (a0 ) = a0 ⊗ a0 . By the definition of 0 as given in Proposition 0.6, this means that, for r, s, t ∈ Z and p ∈ I0 = Z, we should have 0 ξr,s, p−1,t =
v,w∈I0 v−w=t
Pq02 ( p, v, w)ev−1 ⊗ er + p−w ⊗ ew−1 ⊗ es− p+v .
Comparing coefficients, this reduces to the identity Pq02 ( p − 1, v − 1, w − 1) = Pq02 ( p, v, w) for all p, v, w ∈ I0 . This is in fact immediate from the definition of Pq02 , as the formula only involves pairwise differences of the v, w and p. + Thus the only point left to prove is that 0+ (L 0+ ) = 0+ (L 0+ ). Choose a vector ξr,s, p,t + with r, s, t ∈ Z and p ∈ I+ = N. It is then sufficient to prove that 0+ (L 0+ )ξr,s, p,t = + 0+ (L 0+ )ξr,s, p,t . Using the definition of 0+ , and the formulas for L 0+ and 0+ (L 0+ ) stated in the previous proposition, this becomes the identity ∞
1/2
0 (q 2 p+2 ; q 2 )∞ ξr,s, p,t =
v,w∈I+ k=0 v−w=t
1/2
Pq+2 ( p, v, w)(−q)k q k(v+w)
(q 2v+2 , q 2w+2 ; q 2 )∞ (q 2 ; q 2 )k
· ev−k ⊗ er + p−w+k ⊗ ew−k ⊗ es− p+v−k . −1/2
Multiplying both sides with (−q)− p (q 2 ; q 2 )∞ (q 2 p+2 ; q 2 )∞ , and using the definition of Pq02 (see Proposition 0.6) and expression (2) for the function Pq+2 (see Proposition 0.4), this becomes
(−q)
−w ( p−w)(v−w)
q
v,w∈I0 v−w=t
=
0 q 2v−2w+2
| q ,q 2
2 p−2w+2
ev ⊗ er + p−w ⊗ ew ⊗ es− p+v
∞ (q 2v+2 , q 2w+2 , q 2k+2 ; q 2 )∞ (−q)k q vw+( p+k)(v+w) (q 2 ; q 2 )∞
v,w∈I+ k=0 v−w=t −2w q × 3 ϕ2
q −2v q −2 p 2 2 | q , q ev−k ⊗ er + p−w+k ⊗ ew−k ⊗ es− p+v−k . 0 0
But the coefficients on the right hand side make sense for all v, w ∈ I0 , and are moreover = 0 when v or w is not in I+ . We may hence take the v-w-summation on the right over I0 , and we may also take the k-summation over Z. Then a comparison of coefficients
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
213
shows that we must prove the following summation formula: for all v, w ∈ Z and p ∈ N, 0 2 2 p−2w+2 | q , q (−q)−w q ( p−w)(v−w) q 2v−2w+2 =
∞ k=−∞
· 3 ϕ2
(q 2v+2k+2 , q 2w+2k+2 , q 2k+2 ; q 2 )∞ (q 2 ; q 2 )∞ | q 2, q 2 .
(−q)k q (v+k)(w+k)+( p+k)(v+w+2k) q −2w−2k q −2v−2k q −2 p 0 0
But, with t = v − w, and by making the change of variables k → k + w, the right hand side can be rewritten as (−q)−w q ( p−w)t × 3 ϕ2
q −2k
∞
(−q)k q 3k
k=0 q −2t−2k
0
(q 2k−2w+2 , q 2t+2k+2 ; q 2 )∞ (q 2 ; q 2 )k | q 2, q 2 ,
2 +2(t+ p−w)k
q −2 p 0
and the identity then follows by applying Proposition C.3 with respect to x = q −2w and y = q 2t . This finishes the proof. Combining this result with the work of the previous section, it is now an easy task to finish the proof of Theorem 0.9. Proof (of Theorem 0.9). Using the notation introduced before the theorem, we have shown in the previous sections that all maps μν are coassociative, except for those with μ = ν inside {−, 0}. But take x ∈ L (0, +) and y ∈ L (+, −). Then from the definition of the μν , it follows immediately that 0− (x y) = 0+ (x)+− (y), and hence (0− ⊗ ι)0− (x y) = (0+ ⊗ ι)0+ (x)(−0 ⊗ ι)−0 (y) = (ι ⊗ 0+ )0+ (x)(ι ⊗ +− )+− (y) = (ι ⊗ 0− )0− (x y), where the second identity follows from the results of the previous sections. As the set L (0, +) · L (+, −) ⊆ L (0, −) is σ -weakly dense in L (0, −), we have proven that 0− is coassociative. The coassociativity of −0 then follows immediately by applying the ∗ -operation. This concludes the proof. q (2) Remark. One can also prove an analogue of Theorem 0.9 if we replace SUq (2), E and SU q (1, 1) by the respective Z2 -quotients S Oq (3), E q (2) and SU q (1, 1)/Z2 ∼ = Oq (1, 2) (although this final quantum group may require a different interpretation, see the remark after Definition 0.8). The reason is that the actions of SUq (2) on the Podle´s spheres descend to S Oq (3), so that one may equally well apply our theory of projective corepresentations to S Oq (3) with respect to the standard Podle´s sphere and the quantum projective plane, as to obtain in this way a 3×3-linking weak von Neumann bialgebra between the mentioned quantum groups. In fact, the only reason why we have worked with their double covers is that these have received more attention in the literature, so that it was easier for us to refer to the known results concerning these quantum groups.
214
K. De Commer
Acknowledgements. I would like to express my gratitude to E. Koelink for discussions on the material of this paper and its presentation, and for providing some indispensible references concerning basic hypergeometric functions and their use in quantum group theory.
q (2)) and A. On the Tensor Product Representations of L ∞(SUq (2)), L ∞( E ∞ q (1, 1)) L ( SU Let us comment on the proof of Proposition 0.4, Proposition 0.5, and the equivalence of the Definition 0.8 with the one in [11]. We will use the same notations as in the Introduction. The case of SUq (2). We are not aware of Proposition 0.4 appearing explicitly in the literature, although it is surely known to the experts (see for example the remarks following Proposition 3.3 of [11]). We will give some information on how one can arrive at it. We remark that this was also worked out in [7], which unfortunately remained unpublished (we would like to thank W. Groenevelt for providing us with this manuscript). For p, w ∈ N and t ∈ Z, let us denote pw (q 2 p ; q 2t , 0 | q 2 ) =
∞ (q −2w , q 2 )k (q 2t+2k+2 ; q 2 )∞ k=0
(q 2 ; q 2 )k
q 2k( p+1) .
We remark that for t ≥ 0, we then have pw (q 2 p ; q 2t , 0 | q 2 ) = (q 2t+2 ; q 2 )∞ pw (q 2 p ; q 2t , 0 | q 2 ), where the pw are the ordinary Wall polynomials (in the variable q 2 p ) −2w q 0 2p 2t 2 2 2 p+2 . | q ,q pw (q ; q , 0 | q ) = 2 ϕ1 q 2t+2 +2 the following functions on I+ × I+ × I+ : Denote by P q +2 ( p, v, w) = (−1) p−w q ( p−w)(v−w+1) P q
pw (q 2 p ; q 2v−2w , 0 | q 2 ) 1/2
1/2
1/2
(q 2v+2 ; q 2 )∞ (q 2 ; q 2 ) p (q 2 ; q 2 )w
.
(6)
For t = v − w ≥ 0, the expression on the right coincides with the normalized Wall polynomials as defined for example in [12], Eq. (2.5), except that an extra sign-factor (−1) p appears. We remark that we have the following symmetry relation: with p, w, t ∈ N such that w ≥ t, we have +2 ( p, w, v). +2 ( p, v, w) = P P q q
(7)
This follows from observing that for t = v − w ≤ 0, the summation for pw starts only at k = −t, by making the change of variables k → k + t, and by simplifying this expression. Now applying the transformation which appears at the end of Sect. 2 of [12], we +2 = P +2 as defined by the expression (2) in Proposition 0.4 (initially, this get that P q q transformation should only be used for the case v − w ≥ 0, but using the symmetry relation (7), and the obvious v-w-symmetry of the function Pq+2 defined by (2), it applies
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
215
in all cases). On the other hand, using a limit version of the identity III.(3) of [6] to the +2 , one sees that also P +2 = P +2 as defined by the expression (1) in Proposifunction P q q q tion 0.4 (again, this transformation only applies for the case v − w ≥ 0, but one can use either analyticity in the variable t = v − w or Proposition 6.6.(1) of [11] to see that it applies for all cases). Now by (for example) Eq. (2.7) of [12], and by the symmetry relation (7), we conclude that ∞ v,w∈N v−w=t
+2 ( p , v, w) = δ p, p , +2 ( p, v, w) P P q q
as well as ∞
+2 ( p, v, w)Pq 2 ( p, v , w ) = δw,w δv,v P q
p=0 + whenever v − w = v − w . We then obtain that the ξr,s, p,t are indeed an orthonormal basis of H+ ⊗ H+ , and W+ a unitary transformation. This proves the first part of Proposition 0.4. To prove that W+ implements + , one needs the following q-contiguous relations for the -function: 2v+2 q 2 2 p−2w | q ,q q 2v−2w+2 2v+4 2v+2 q q 2v+2 2 2 p−2w 2v+2 2 2 p−2w+2 = (1 − q +q , ) | q ,q | q ,q q 2v−2w+2 q 2v−2w+2
which was proven in Lemma 6.5 of [11] (see Eq. (6.2)), and 2v+2 q 2 2 p−2w+2 q 2 p−2w | q , q q 2v−2w+2 2v+2 2v q q 2 2 p−2w 2 2 p−2w+2 = − + , | q , q | q , q q 2v−2w q 2v−2w which can be proven in a similar way: namely, expanding the right hand side leads us to ∞ (q 2v−2w+2k+2 ; q 2 )∞ k=0
=−
(q 2 ; q 2 )k
[(−(q 2v+2 ; q 2 )k + q 2k (q 2v ; q 2 )k )](−1)k q k(k−1) q (2 p−2w)k
∞ (q 2v+2 ; q 2 )k−1 (q 2v−2w+2k+2 ; q 2 )∞ k=1
(q 2 ; q 2 )k−1
(−1)k q k(k−1) q (2 p−2w)k .
Then by the change of variables k → k − 1, this simplifies to the left hand side of the above expression. From this, one then easily verifies, using expression (1) of Proposition 0.4, that the operators + (a+ ) and + (b+ ) act as follows on our basis vectors: + + = 1 − q 2 p ξr,s, + (a+ )ξr,s, p,t p−1,t , + p + + (b+ )ξr,s, p,t = q ξr,s, p,t+1 ,
216
K. De Commer
where the right-hand side of the first expression is interpreted as zero when p = 0. Hence W+ satisfies + = + (x) +∗ (1 ⊗ x)W W for x ∈ {a, b}, and hence for all x ∈ L ∞ (SUq (2)). This finishes the proof of Proposition 0.4. We should also remark that an implementing unitary for + (in fact, a concrete description of the regular left corepresentation of SUq (2)) was obtained in [15]. q (2). Proposition 0.6 was proven in the course of Proposition 4.2. HowThe case of E ever, we should remark that in [10], a spectral decomposition of the operator 0 (b0∗ b0 ) was obtained. This gives an eigenvector-basis that, as with [12] in the previous paragraph, is almost the same basis as the one we obtain, but modulo the appearance of some extra minus sign. We should also note that Pq02 ( p, v, w) = (−q) p−w Jv−w (q p−w ; q 2 ), where 1 Jα (z; q ) = z 2 (q ; q 2 )∞ 2
α
0 q 2α+2
| q ,q z 2
2 2
is the 1 ϕ1 -q-Bessel function. The case of SU q (1, 1). Finally, we must comment on the definition of SU q (1, 1) we presented. The equivalence with the one in [11] is straightforward, though there is one small difference which requires some explanation. ¯ ∞ ( We first introduce a certain operator ∈ L ∞ ( SU q (1, 1))⊗L SU q (1, 1)). As e ∞ is a self-adjoint unitary inside L ( SU q (1, 1)), we may identify W ∗ (e) with C(Z2 ), sending e to the function ± → ±1. As e is group-like, this will moreover intertwine − with the natural comultiplication on C(Z2 ) coming from the group structure on Z2 . Denote then by ω ∈ C(Z2 ) ⊗ C(Z2 ) the 2-cocycle function ω(μ, ν) = 1 if μ and ν not both − ω : Z2 × Z2 → S 1 : ω(μ, ν) = −1 if μ = ν = −, ¯ ∞ ( SU q (1, 1))⊗L SU q (1, 1)), using the above idenand denote by its image in L ∞ ( tification. More explicitly, we have that is given on H− ⊗ H− as the operator ev ⊗ er ⊗ ew ⊗ es = ω(c(v), c(w))ev ⊗ er ⊗ ew ⊗ es . The 2-cocycle property of ω will give the following identity for : ( − ⊗ 1)(− ⊗ ι)( − ) = (1 ⊗ − )(ι ⊗ − )( − ), i.e. is a unitary 2-cocycle ([5]) for − . This implies that (L ∞ ( SU q (1, 1)), − ( · ) ∗ ) is again a well-defined von Neumann bi-algebra. Let us now turn to the definition of SU q (1, 1) presented in [11]. We first remark that in [11], one identifies our set I− with a subset of R by the correspondence : p± ↔
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
217
±q − p . Then using the notation a p introduced in Definition 3.1 of [11], which for each p ∈ (I− ) is a function (I− ) × (I− ) → R, we have the following correspondence between these a p and our Pq−2 , by a straightforward comparison: Pq−2 (p, v, w) = ω(c(w), c(v))c(v)v−w c(p) p−w a(p) ((w), (v)). Denote then by Fr,s,±q − p ,t the basis elements introduced in Definition 3.6 of [11] (we remark that in their description the l 2 (I− ) and l 2 (Z)-part or switched with respect to our convention; we will change their ordering of the two factors to have compatibility with our presentation). Then using the second symmetry relation in Proposition 3.5 of [11], we have Fr,s,±q − p ,t =
a(p) ((v), (w))ev ⊗ er + p−w ⊗ ew ⊗ es− p+v
v,w∈I− v−w=t c(v)c(w)=c(p)
=
c(p) p c(v)v c(w)w a(p) ((w), (v))ev ⊗ er + p−w ⊗ ew ⊗ es− p+v
v,w∈I− v−w=t c(v)c(w)=c(p)
=
=
ω(c(w), c(v))Pq−2 (p, v, w)ev ⊗ er + p−w ⊗ ew ⊗ es− p+v
v,w∈I− v−w=t c(v)c(w)=c(p) − ξr,s, p,t .
− the comultiplication as introduced in [11] by This implies that, if we denote by means of the vector basis F, we have the following relation: − (x) = − (x) ∗ ,
for all x ∈ L ∞ ( SU q (1, 1)).
But in fact, (L ∞ ( SU q (1, 1)), − ( · ) ∗ ) is an isomorphic copy of (L ∞ ( SU q (1, 1)), 1 − ( · )). To see this, we remark that, as we have taken ω as an S -valued 2-cocycle on f (μν) Z2 , it is a coboundary: we have ω(μ, ν) = f (μ) f (ν) with f (+) = 1 and f (−) = i. Denote then by u the unitary u : H− → H− : ep ⊗ er → f (c(p)) ep ⊗ er , and denote φ : L ∞ ( SU q (1, 1)) → L ∞ ( SU q (1, 1)) : x → uxu ∗ . Then one gets that = (u ∗ ⊗ u ∗ )− (u), and so − (φ(x)) = (φ ⊗ φ)( − (x) ∗ ),
for all x ∈ L ∞ ( SU q (1, 1)).
− ) are isomorThis thus proves that (L ∞ ( SU q (1, 1)), − ) and (L ∞ ( SU q (1, 1)), phic von Neumann bialgebras, and justifies our use of the basis ξ − .
218
K. De Commer
+ (W ) B. Spectral Decomposition of α In this Appendix, we will determine the spectral decomposition of the action of SUq (2) on the equatorial Podle´s sphere and on the quantum projective plane. We are not aware of this concrete computation having been carried out explicitly in the literature, but the method is quite standard (and could probably be shortened somewhat). We will use the notation as introduced in Sect. 2.2. Proposition B.1. The eigenspace of the eigenvalue ±1 of α+ (W ) has a basis of ortho(t, p) normal vectors ξ0,± , where t ∈ Z and p ∈ J (±) , these vectors being given by the formula (t, p) ξ0,±
∞ 1/2 1/2 (∓q 2 p+2 ; q 2 )n (±q 2 p+2n+2 ; q 2 )∞ n n(n−1) 2 = (±1) q en ⊗ et−n ⊗ e p+n . 1/2 1/2 (−1; q 2 )∞ (q 2 ; q 2 )n n=0
Proof. Denote, for t, p ∈ Z, K
t, p
= {en ⊗ em ⊗ ek | m + n = t, k − n = p} ⊆ l 2 (N) ⊗ l 2 (Z) ⊗ l 2 (N),
where S denotes the closed linear span of a set S. Then, since for n, k ∈ N and m ∈ Z, we have, by the definition of α+ (W ) in Definition 2.2, α+ (W )(en ⊗ em ⊗ ek ) = q n (1 − q 2n+2 )1/2 (1 − q 4k+4 )1/2 en+1 ⊗ em−1 ⊗ ek+1 +(1 − (1 + q 2 )q 2n )q 2k en ⊗ em ⊗ ek + q n−1 (1 − q 2n )1/2 (1 − q 4k )1/2 en−1 ⊗ em+1 ⊗ ek−1 , we see that α+ (W ) restricts to each K t, p , and determines there a Jacobi matrix. We make a distinction in the analysis between the case p ≥ 0 and p < 0. First we consider the case p ≥ 0. Then we have a unitary map l 2 (N) → K t, p such that en → en ⊗ et−n ⊗ e p+n . Under this identification, the restriction of α+ (W ) becomes the Jacobi matrix W ( p) with W ( p) en = q n (1 − q 2n+2 )1/2 (1 − q 4( p+n)+4 )1/2 en+1 +(1 − (1 + q 2 )q 2n )q 2( p+n) en + q n−1 (1 − q 2n )1/2 (1 − q 4( p+n) )1/2 en−1 . Each eigenvalue then arises with multiplicity one, and an eigenvector at eigenvalue x ( p) ( p) ( p) is given by ηx = f n (x)en , where f n is the sequence of functions satisfying ( p) ( p) f −1 = 0, f 0 = 1 and ( p)
( p)
q n (1 − q 2n+2 )1/2 (1 − q 4 p+4n+4 )1/2 f n+1 (x) + (1 − (1 + q 2 )q 2n )q 2 p+2n f n (x) ( p)
( p)
+ q n−1 (1 − q 2n )1/2 (1 − q 4 p+4n )1/2 f n−1 (x) = x f n (x). Denote ( p)
f n (x) = (−1)n q −
n 2 4 p+3 2 − 2 n
1/2
(q 4 ; q 4 ) p+n 1/2 1/2 (q 4 ; q 4 ) p (q 2 ; q 2 )n
( p)
Pn (−q 2 p+2 x),
q (2) and On a Correspondence between SUq (2), E SU q (1, 1) ( p)
( p)
for some other function Pn . Then Pn
219
should itself satisfy the recurrence relation
( p)
( p)
(1 − q 4( p+n+1) )Pn+1 (x) − ((1 − (1 + q 2 )q 2n )q 2n+4 p+2 + 1)Pn (x) ( p)
( p)
+ q 2n+4 p+2 (1 − q 2n )Pn−1 (x) = (x − 1)Pn (x). ( p)
Hence, for example by [9], Sect. 3.11, we see that Pn (x) = Pn (x; q 2 p , −q 2 p ; q 2 ), where Pn (x; a, b; q) denotes the big q-Laguerre polynomial −n 0 x q Pn (x; a, b; q) = 3 ϕ2 | q, q . qa qb So ( p) f n (x)
1/2
(q 4 ; q 4 ) p+n
2
n − n2 − 4 p+3 2 n
= (−1) q
× 3 ϕ2
1/2
1/2
(q 4 ; q 4 ) p (q 2 ; q 2 )n q −2n 0 −q 2 p+2 x 2 2 | q , q . q 2 p+2 −q 2 p+2
(8)
α+ (W ) will be spanned Now we know that the eigenspace H±1 for the eigenvalue ±1 of by the H±1 ∩ K t, p , with p, t ∈ Z. We now show that each W ( p) has eigenvalues 1 and −1, still assuming p ≥ 0. We first observe the trivial transformation formula −2n −2n 0 ∓q 2 p+2 0 q q 2 2 2 2 | q , q = 2 ϕ1 | q ,q . 3 ϕ2 ±q 2 p+2 q 2 p+2 −q 2 p+2 Then, using a limit form of the q-Vandermonde formula ([6], Eq. 1.5.3), we get −2n q 2( p+1)n 0 q 2 2 | q , q = (∓1)n q n(n−1) . 2 ϕ1 2 p+2 ±q (±q 2 p+2 ; q 2 )n Hence, using the formula (a 2 ; q 2 )n = (a; q)n (−a; q)n , we get ( p) f n (±1)
= (±1) q
n(n−1) 2
= (±1)n q
n(n−1) 2
n
1/2
(∓q 2 ; q 2 ) p+n 1/2
1/2
1/2
(q 2 ; q 2 )n (∓q 2 ; q 2 ) p (±q 2 p+2 ; q 2 )n 1/2
(∓q 2 p+2 ; q 2 )n 1/2
1/2
(q 2 ; q 2 )n (±q 2 p+2 ; q 2 )n
.
So, by using a limit form of Heine’s summation formula ([6], Eq. II.5), 2 p+2 ∞ ∓q ( p) 2 2 | f n (±1)| = 1 ϕ1 | q , −1 ±q 2 p+2 n=0
=
(−1; q 2 )∞ , (±q 2 p+2 ; q 2 )∞
and ( p)
η±1 =
1/2
(−1; q 2 )∞
1/2
(±q 2 p+2 ; q 2 )∞
.
(9)
220
K. De Commer ( p)
Hence ±1 appears as an eigenvalue of W ( p) with eigenvector η±1 ∈ K t, p . We then find that the intersection of the eigenspace H±1 with the closed linear span of the K t, p with p ≥ 0 has an orthonormal basis consisting of the vectors (t, p)
ξ0,± =
∞ 2 p+2 ; q 2 )1/2 (±q 2 p+2n+2 ; q 2 )1/2 n(n−1) (∓q n ∞ (±1)n q 2 en ⊗ et−n ⊗ e p+n . 2 )1/2 (q 2 ; q 2 )1/2 (−1; q n ∞ n=0
We move on to the case p < 0. Now we have a unitary map l 2 (N) → K t, p such that ek corresponds to ek− p ⊗ et−k+ p ⊗ ek . Under this identification, the restriction of α+ (W ) becomes the Jacobi matrix W ( p) with W ( p) ek = q k− p (1 − q 2k−2 p+2 )1/2 (1 − q 4k+4 )1/2 ek+1 + (1 − (1 + q 2 )q 2(k− p) )q 2k ek + q k− p−1 (1 − q 2k−2 p )1/2 (1 − q 4k )1/2 ek−1 . Again, each eigenvalue arises with multiplicity one, and an eigenvector at eigenvalue x ( p) ( p) ( p) f k (x)ek , where now f k is the sequence of functions satisfying is given by ηx = ( p) ( p) f −1 = 0, f 0 = 1 and ( p)
( p)
q k− p (1 − q 2k−2 p+2 )1/2 (1 − q 4k+4 )1/2 f k+1 (x) + (1 − (1 + q 2 )q 2(k− p) )q 2k f k (x) ( p)
( p)
+ q k− p−1 (1 − q 2k−2 p )1/2 (1 − q 4k )1/2 f k−1 (x) = x f k (x). Now put k2
( p)
f k (x) = (−1)k q − 2 − ( p)
(−2 p+3)k 2
( p)
for some function Pk . This Pk
(q −2 p+2 ; q 2 )k (−q 2 ; q 2 )k 1/2
1/2
( p)
Pk (−q 2 x),
1/2 (q 2 ; q 2 )k
should then satisfy the recursion formula
( p)
( p)
(1 − q 2k−2 p+2 )(1 + q 2k+2 )Pk+1 (x) − ((1 − (1 + q 2 )q 2k−2 p )q 2k+2 + 1)Pk (x) ( p)
( p)
+ q 2k−2 p+2 (1 − q 2k )Pk−1 (x) = (x − 1)Pk (x). ( p)
( p)
So Pk is again a big q-Laguerre polynomial, namely Pk (x) = Pk (x; q −2 p , −1; q 2 ). We then have ( p)
k2
f k (x) = (−1)k q − 2 −
(−2 p+3)k 2
(q −2 p+2 ; q 2 )k (−q 2 ; q 2 )k 1/2
1/2
1/2
(q 2 ; q 2 )k −2k 0 −q 2 x q 2 2 . × 3 ϕ2 | q , q q −2 p+2 −q 2
(10)
We now show that W ( p) has 1 in its spectrum, but not −1. We first show that 1 is in the spectrum. Using again the q-Vandermonde formula, we have −2k −2k 0 −q 2 0 q q 2 2 2 2 | q , q = 2 ϕ1 | q ,q 3 ϕ2 q −2 p+2 q −2 p+2 −q 2 = (−1)k q k(k−1)
q −2 pk+2k (q −2 p+2 ; q 2 )
k
.
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
221
Then ( p) f k (1)
=q
k(k−1) 2
q
1/2
(−q 2 ; q 2 )k
− pk
1/2
1/2
(q 2 ; q 2 )k (q −2 p+2 ; q 2 )k
.
So, using again the limit form of Heines summation formula, we get ∞ −q 2 ( p) 2 −2 p | f k (1)|2 = 1 ϕ1 | q , −q q −2 p+2 k=0
=
(−q −2 p ; q 2 )∞ . (q −2 p+2 ; q 2 )∞
( p)
Hence the formal sum η1 gives in fact a well-defined element inside K ( p)
η1 =
(−q −2 p ; q 2 )∞
t, p
with
1/2 1/2
(q −2 p+2 ; q 2 )∞
,
(11)
and we find that the intersection of the eigenspace H1 with the closed linear span of the K t, p with p < 0 has an orthonormal basis consisting of the vectors (t, p)
ξ0,+ =
∞
q
k(k−1) 2
q − pk
1/2
1/2
(−q 2 ; q 2 )k (q 2k−2 p+2 ; q 2 )∞ 1/2
ek− p ⊗ et−k+ p ⊗ ek .
1/2
(−q −2 p ; q 2 )∞ (q 2 ; q 2 )k
k=0
Now for n ≥ − p, we have 1/2
1/2
(−q 2 ; q 2 )n+ p (q 2n+2 ; q 2 )∞ 1/2
1/2
(−q −2 p ; q 2 )∞ (q 2 ; q 2 )n+ p
1/2
=
1/2
(−q 2 ; q 2 )∞ 1/2
1/2
(−q 2n+2 p+2 ; q 2 )∞ (−q −2 p ; q 2 )∞
·
(q 2n+2 p+2 ; q 2 )∞ 1/2
(q 2 ; q 2 )n
,
and 1/2
(−q 2 ; q 2 )∞ 1/2
1/2
(−q 2n+2 p+2 ; q 2 )∞ (−q −2 p ; q 2 )∞
=q
(− p)(− p−1) 2
1/2
(−q 2 p+2 ; q 2 )n 1/2
(−1; q 2 )∞
.
Hence we can also write (t, p)
ξ0,+ =
∞ n=0
q
n(n−1) 2
1/2
1/2
(−q 2 p+2 ; q 2 )n (q 2 p+2n+2 ; q 2 )∞ 1/2
1/2
(−1; q 2 )∞ (q 2 ; q 2 )n
en ⊗ et−n ⊗ e p+n ,
which is precisely the same formula as for the p ≥ 0 case. Indeed, we could also have checked directly that this is also an eigenvector of norm 1 for the eigenvalue 1. Now we show that W ( p) has no eigenvector for −1 when p < 0. First remark that, using the transformation formulas (III.5) and (III.1) of [6], we can rewrite, for − p > 0, −2k 0 q2 q 2 2 | q ,q 3 ϕ2 q −2 p+2 −q 2 −2k −2 p q q −2k 2 −1 2 2 | q , −q = (−q ; q )k 2 ϕ1 q −2 p+2 2 (q −2 p ; q 2 )∞ (−q −2k+2 ; q 2 )∞ q −q 2 2 −2 p . = ϕ | q , q 2 1 −q −2k+2 (−q −2k ; q 2 )k (q −2 p+2 ; q 2 )∞ (−q 2 ; q 2 )∞
222
K. De Commer
But clearly 1 ≤ 2 ϕ1
q 2 −q 2 | q 2 , q −2 p −q −2k+2
for all − p > 0. So to see if ∞
q −k
2 −(−2 p+3)k
k=0
≤
∞
( p) 2 k=0 | f k (−1)|
2 ϕ1
q 2 −q 2 | q 2 , q −2 p −q 2
= +∞, we have to check if
(q −2 p+2 ; q 2 )k (−q 2 ; q 2 )k (−q −2k+2 ; q 2 )2∞ = +∞. (q 2 ; q 2 )k (−q −2k ; q 2 )2k
Leaving out more non-essential factors regarding the convergence, the sum simplifies to ∞
q −k
2 −(−2 p+3)k
(1 + q −2k )−2 ,
k=0
which is clearly divergent.
It is now not so hard to find the complete spectrum of α+ (W ) by using the operators α+ (Y ) and its adjoint, as given in Definition 2.2. Proof (of Proposition 2.4). The fact that α+ (W ) will have its spectrum inside {±q 2n , 0 | n ∈ N}, with 0 not in the point-spectrum, is immediate. Using the same notation as in the previous proposition, denote, for r ∈ N and t, p ∈ Z, t, p by Kr,± the eigenspace of the eigenvalue ±q 2r inside K t, p (which could be zerodimensional). Then for r > 0, we have, by using the commutation relations between W, Y and Y ∗ , t, p
t−2, p+1
α+ (Y )Kr,± = Kr −1,± , α+ (Y )∗ Kr,± = Kr +1,± t, p
t+2, p−1
.
(t, p)
Denote, for r ∈ N and t, p ∈ Z, by ηr,± the formal eigenvectors which we denoted as ( p)
η±q 2r ∈ K t, p in the previous proposition. Then by combining the above remark concerning α+ (Y ) and its adjoint with the previous proposition, we see that in the +-case, all values of p and r correspond to actual eigenvectors, while in the −-case, we have the (t, p) restriction r + p ≥ 0. Moreover, the linear span of the ηr,± , where r, ±, t and p range over all admissible values, will then be a dense subspace of l 2 (N) ⊗ l 2 (Z) ⊗ l 2 (N). ( p) Now observe that for any p ∈ Z and x ∈ R, we have f 0 (x) = 1. So for p > 0, we have (t, p)
α+ (Y ∗ )ηr,± = −q(1 − q 4 p )1/2 e0 ⊗ et+2 ⊗ e p+n + · · · , while (t+2, p−1)
ηr +1,±
= e0 ⊗ et+2 ⊗ e p−1 + · · · .
So, on comparing these leading coefficients, we find that (t, p)
(t+2, p−1)
α+ (Y ∗ )ηr,± = −q(1 − q 4 p )1/2 ηr +1,±
.
(12)
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
223
Similarly, by comparing the coefficients of the e− p ⊗ et+ p ⊗ e0 for p ≤ 0, we obtain (t, p)
α+ (Y ∗ )ηr,± = −
(1 ± q 2r +2 ) (t+2, p−1) q − p+1 ηr +1,± . (1 − q −2 p+2 )1/2
(13)
Since Y Y ∗ = 1 − q 4 W 2 by an easy computation, we then have, taking norms of both sides, that, for p > 0, (t, p)
(t+2, p−1) 2
(1 − q 4r +4 )ηr,± 2 = q 2 (1 − q 4 p )ηr +1,±
,
and for p ≤ 0, (t, p)
(1 − q 4r +4 )ηr,± 2 =
(1 ± q 2r +2 )2 −2 p+2 (t+2, p−1) 2 q ηr +1,± . (1 − q −2 p+2 )
Hence for p ≥ 0, we have (t, p)
ηr,± 2 = q −2
(1 − q 4r ) (t−2, p+1) 2 η , 1 − q 4 p+4 r −1,±
while for p < 0, we get (t, p)
ηr,± 2 = q 2 p
(1 − q −2 p )(1 ∓ q 2r ) (t−2, p+1) 2 ηr −1,± . (1 ± q 2r )
By induction, and using the expressions (9) and (11), we then find the following formulas for the norm squared of the η-vectors: if p ≥ 0, we have (t, p)
ηr,± 2 = q −2r
(q 4 ; q 4 )r (−1; q 2 )∞ , (q 4 p+4 ; q 4 )r (±q 2 p+2r +2 ; q 2 )∞
if r + p ≥ 0 > p, we have (t, p)
ηr,± 2 = q − p = q−p
2+ p
(q 2 ; q 2 )− p (±q 2 ; q 2 )r (∓q 2 ; q 2 )r + p (t+2 p,0) 2 ηr + p,± (±q 2 ; q 2 )r + p (∓q 2 ; q 2 )r
2 − p−2r
(q 2 ; q 2 )− p (∓q 2r +2 ; q 2 )∞ (−1; q 2 )∞ , (±q 2r +2 ; q 2 )∞ (∓q 2r +2 p+2 ; q 2 )∞
and for 0 > r + p > 0 we have (t, p)
ηr,+ 2 = q r
2 −r +2r p
(q 2 ; q 2 )r (q 2 ; q 2 )− p (−q −2r −2 p ; q 2 )∞ . (−q 2 ; q 2 )r (q 2 ; q 2 )− p−r (q −2r −2 p+2 ; q 2 )∞
Denote by (t, p)
ξr,± =
(∓1)r (t, p) ηr,±
(t, p)
ηr,±
(14)
224
K. De Commer
the normalized eigenvectors for α+ (W ) (the reason for the extra sign factor (∓1)r is (t, p) that Proposition 2.5 would hold). Then, from the expressions for ηr,± in the previous proposition, namely (t, p)
∞
(t, p)
n=0 ∞
ηr,± = ηr,± =
( p)
f n (±q 2r )en ⊗ et−n ⊗ e p+n , ( p)
f k (±q 2r )ek− p ⊗ et−k+ p ⊗ ek ,
p ≥ 0, p < 0,
k=0 ( p)
( p)
where the f n and f k are defined by the respective formulas (8) and (10), together (t, p) with the above formulas for the norm, we have concrete expressions for the ξr,± in terms of 3 ϕ2 -functions. We next want to write these in a different form. Using, for p ≥ 0, the transformation formula
q −2n 0 ∓q 2 p+2r +2 2 2 | q , q ±q 2 p+2 ∓q 2 p+2 −2n −q −2r q 2 2r +2 = (∓q −2 p−2n ; q 2 )−1 ϕ | q , q n 2 1 ±q 2 p+2
3 ϕ2
(∓q 2 p+2r +2 ; q 2 )n = (−1)n (∓q −2 p−2n ; q 2 )n (±q 2 p+2 ; q 2 )n −2n q −2r ±q −2 p−2n q 2 2 , · 3 ϕ2 , q | q ∓q −2 p−2n−2r 0 which is a combination of the identities (III.5) and (III.6) of [6], we find, after some (t, p) simplifications, that the ξr,± for p ≥ 0 satisfy the formula in the statement of the proposition. In case p < 0, we can use the same identities, together with (III.11) of [6], to get, for k = n + p ≥ 0, the transformation formula
q −2k 0 ∓q 2r +2 2 2 | q ,q 3 ϕ2 q −2 p+2 −q 2 −2k ∓q −2 p−2r q −2k 2 −1 2 2r +2 | q , ±q = (−q ; q )k 2 ϕ1 q −2 p+2 (∓q 2r +2 ; q 2 )k = (−q −2 p )k (−q −2k ; q 2 )k (q −2 p+2 ; q 2 )k −2k q ±q −2r q 2 p−2k 2 2 · 3 ϕ2 | q ,q ∓q −2k−2r 0 (∓q 2r +2 ; q 2 )k = (−q −2 p )k −2k (−q ; q 2 )k (q −2 p+2 ; q 2 )k −2k−(−2 p) q −2k ±q −2r q 2 2 | q · 3 ϕ2 , q ∓q −2k−2r 0
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
225
(∓q 2r +2 ; q 2 )k = (±1)k− p (−q −2 p )k (−q −2k ; q 2 )k (q −2 p+2 ; q 2 )k −2k+2 p q −2r ±q −2k q 2 2 , · 3 ϕ2 , q | q ∓q −2k−2r 0 (t, p)
from which, again after some simplifications, we see that the formula for ξr,± in the proposition is still valid in this case (where we note again that, in the case of negative eigenvalues, the r, p are restricted by the condition r + p ≥ 0). C. Some Summation Formulas for Basic Hypergeometric Functions Proposition C.1. Suppose x, y ∈ C0 and p ∈ N. For w ∈ N, write ( p) (x, gw
w 2w2 +2 pw
y) = (−1) q
x (q ( )w y
Then ∞ w=0
( p) gw (x,
y) 3 ϕ2
= (−1; q ) p 2
2w+2 x ; q 2 ) (xq 2w+2 ; q 2 ) (−q 2 x; q 2 ) ∞ ∞ w y . (q 2 ; q 2 )w
q −2w q −2w xy q −2 p | q 2, q 2 0 0
− 1y | q 2 , q 2 p+2 x . q 2 xy
Proof. As the 3 ϕ2 -term is a polynomial of fixed degree p, it is easy to see that, expanding this term, the resulting double sum on the left-hand side is absolutely convergent, so that we can change the order of summation later on. Moreover, both sides are analytic in each variable x and y, so we can also restrict our attention to the situation x, y ∈ / ±q 2Z . Then the identity can be rewritten in the form ∞
(−1)w q 2w
2 +2 pw
w=0
x ( )w y
−2w −2w y (−q 2 x; q 2 )w q q q −2 p 2 2 x ϕ , q | q 3 2 0 0 (q 2 x; q 2 )w (q 2 xy ; q 2 )w (q 2 ; q 2 )w − 1y (−1; q 2 ) p = 2x 2 | q 2 , q 2 p+2 x . (15) q 2 xy (q y ; q )∞ (q 2 x; q 2 )∞ ×
Now on the left hand side, we can expand the 3 ϕ2 -term as a sum over l ranging from 0 to w. Changing the order of summation, we can write the entire expression as a double sum over l : 0 → ∞ and w : l → ∞. Then changing the variable w to w − l, and reversing the order of summation again, the above left hand side expression can be simplified to ∞
(−1)w q 2w
w=0
×
∞ l=0
2 +2 pw
(−1)l q 2lp
(−q 2 x; q 2 )w x ( )w 2 x 2 y (q y ; q )w (q 2 x; q 2 )w (q 2 ; q 2 )w
(q −2 p ; q 2 )l (−q 2w+2 x; q 2 )l . (q 2w+2 x; q 2 )l (q 2 ; q 2 )l
226
K. De Commer
Now the sum over l is just
2 ϕ1
q −2 p −q 2w+2 x 2 , −q 2 p . By the q-Vandermonde | q q 2w+2 x
formula ([6], (1.5.2)), this equals becomes
(−1;q 2 ) p . (q 2w+2 x;q 2 ) p
So the left hand side expression in (15)
∞ (−1; q 2 ) p (−q 2 x; q 2 )w w 2w2 +2 pw x w ) (−1) q ( . (q 2 x; q 2 ) p y (q 2 xy ; q 2 )w (xq 2 p+2 ; q 2 )w (q 2 ; q 2 )w
(16)
w=0
Now the sum over w can be written as −q 2 x c−1 2 2 p+2 x . lim 2 ϕ2 | q , −cq q 2 p+2 x q 2 xy c→0 y Using (III.4) of [6], this becomes (− 1y ; q 2 )∞ (q 2 xy ; q 2 )∞
2 ϕ1
1 −q 2 x 0 2 , , − | q q 2 p+2 x y
which by a limit version of [6], (III.1) can be transformed into − 1y 1 | q 2 , q 2 p+2 x . q 2 xy (q 2 p+2 x, q 2 xy ; q 2 )∞ (Note that the intermediate step is only valid for restricted values of y and x, but the total transformation is valid in all cases by analyticity.) Then plugging this back into (16), we find the identity (15) we were after. Proposition C.2. Suppose p ∈ N and x, y ∈ C0 with x ∈ / −q −2N−2 p−2 . For w ∈ N, write (q 2 x ( p,±) gw (x, y) = (∓1)w q 2w ( )w y
2w+2 x ; q 2 ) (±q 2w+2 x; q 2 ) (−q 2 p+2 x; q 2 ) ∞ ∞ w y . 2 2 (q ; q )w
Then ∞ w=0
( p,±) gw (x,
y) 3 ϕ2
q −2w q −2w x1 q −2 p | q 2, q 2 −q −2 p−2w x1 0
=
∓ 1y 2 2 | q , ±q x . ±q 2 p+2 xy
Proof. As for the previous proposition, both sides of the identity are analytic in x and y, so we may restrict our attention to the case x, y ∈ / ±q 2Z . We will then only give the proof for the +-case, as the −-case is completely similar. We again expand the 3 ϕ2 -factor on the left hand side as a sum over the variable l : 0 → w, change the order of w and l, replace w by the variable w − l, and again change the order of summation. Then we obtain that the expression on the left hand side can be simplified to w=0
(−q 2 p+2 x; q 2 )w 2 x (−1)w q 2w ( )w 2 x 2 y (q y ; q )w (q 2 x; q 2 )w (q 2 ; q 2 )w
×
∞ l=0
(−1)l q l
2 +l
(q −2 p ; q 2 )l x q 2l(w+ p) ( )l 2w+2 x 2 . 2 2 y (q y ; q )l (q ; q )l
q (2) and On a Correspondence between SUq (2), E SU q (1, 1)
The sum over l thus equals
1 ϕ1
227
q −2 p 2 , q 2w+2 p+2 x , which, by the limit form | q y q 2w+2 xy
of Heines summation formula ([6], (II.5)) can be reduced to sum over w can be rewritten as (q
2 p+2 x
y
; q )∞ lim 2
c→0
2 ϕ2
(q 2w+2 p+2 xy ;q 2 )∞ (q 2w+2 xy ;q 2 )∞
x −q 2 p+2 x c−1 | q 2 , −cq 2 2 2 q x q p+2 xy y
. Then the
,
which, by using again [6], (III.4) and (III.1), reduces to the right hand side of the identity we wanted to prove. Proposition C.3. Suppose x, y ∈ C and p ∈ N. For k ∈ N, write ( p)
fk
= (−q)k q 3k
2 +2kp
(x y)k
(q 2k+2 x, q 2k+2 y; q 2 )∞ . (q 2 ; q 2 )k
Then ∞
( p)
f k (x, y) 3 ϕ2
k=0
q −2k q −2k 1y q −2 p | q 2, q 2 0 0
=
0 q2 y
| q 2 , q 2 p+2 x .
Proof. Both sides of the identity again depend analytically on the variables x and y, so we may suppose that x, y ∈ / {0} ∪ q 2Z . As in the previous propositions, we again expand the 3 ϕ2 -factor on the left hand side as a sum over the variable l : 0 → k, change the order of k and l, replace k by the variable k − l, and again change the order of summation. Then we obtain that the expression on the left hand side can be simplified to −2 p ∞ 2k+2 x, q 2k+2 y; q 2 ) q ∞ k 3k 2 +2 pk k (q 2 2k+2 p+2 (−q) q (x y) x . | q ,q 1 ϕ1 q 2k+2 x (q 2 ; q 2 )k k=0
x;q )∞ But the 1 ϕ1 -expression can be simplified to (q(q 2k+2 x;q by the limit version of Heines 2) ∞ summation formula(Eq. (1.5.1) in [6]). The remaining sum over k can then be shown to 0 equal precisely | q 2 , q 2 p+2 x by a (double) limit version of Jackson’s transq2 y formation formula (Eq. (1.5.4) in [6]). This concludes the proof. 2k+2 p+2
2
References 1. Böhm, G., Nill, F., Szlachányi, K.: Weak Hopf algebras. I. Integral theory and C∗ -structure. J. Algebra 221(2), 385–438 (1999) 2. De Commer, K.: On a Morita equivalence between the duals of quantum SU (2) and quantum E(2). http://arxiv.org/abs/0912.4350v2 [math.QA], 2009 3. De Commer, K.: On the construction of quantum homogeneous spaces from ∗ -Galois objects. Alg. Rep. Theory. http://arXiv.org/abs/1001.2153v2 [math.QA], 2010. doi:10.1007/s10468-011-9265-7 4. De Commer, K.: Comonoidal W∗ -Morita equivalence for von Neumann bialgebras, to be published in J. Noncommut. Geom., preprint available at http://arXiv.org/abs/1004.0824v1 [math.OA], 2010 5. Enock, M., Vainerman, L.: Deformation of a Kac algebra by an Abelian subgroup. Communi. Math. Phys. 178, 571–596 (1996) 6. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge, U.K.: Cambridge University Press, 1990
228
K. De Commer
7. Groenevelt, W.: Tensor products for quantum SU (2). Unpublished manuscript, 2003 8. Hajac, P., Matthes, R., Szyma´nski, W.: Quantum Real Projective Space, Disc and Spheres. Alg. Rep. Theory 6(2), 169–192 (2003) 9. Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Delft University of Technology. Report no. 98-17 (1998) 10. Koelink, H.T.: The quantum group of plane motions and the Hahn-Exton q-Bessel function. Duke Math. J. 76(2), 483–508 (1994) 11. Koelink, E., Kustermans, J.: A locally compact quantum group analogue of the normalizer of SU (1, 1) in S L(2, C). Communi. Math. Phys. 233, 231–296 (2003) 12. Koornwinder, T.H.: The addition formula for little q-Legendre polynomials and the SU (2) quantum group. SIAM J. Math. Anal. 22(1), 295–301 (1991) 13. Korogodsky, L.I.: Quantum Group SU (1, 1)Z2 and super tensor products. Communi. Math. Phys. 163, 433–460 (1994) 14. Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Sci. Ecol. Norm. Sup. 33(6), 837–934 (2000) 15. Lance, E.C.: An Explicit Description of the Fundamental Unitary for SU (2)q . Communi. Math. Phys. 164, 1–15 (1994) 16. Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum sphere of Podle´s I: an algebraic viewpoint. K-Theory 5, 151–175 (1991) 17. Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14(3), 193–202 (1987) 18. Tomatsu, R.: Compact quantum ergodic systems. J. Funct. Anal. 254, 1–83 (2008) 19. Woronowicz, S.L.: Twisted SU (2) group, An example of a non-commutative differential calculus. Publications of RIMS, Kyoto University 23(1), 117–181 (1987) 20. Woronowicz, S.L.: Compact matrix pseudogroups. Communi. Math. Phys. 111, 613–665 (1987) 21. Woronowicz, S.L.: Quantum E(2)-group and its Pontryagin dual. Lett. Math. Phys. 23, 251–263 (1991) 22. Woronowicz, S.L.: Unbounded elements affiliated with C∗ -algebras and noncompact quantum groups. Communi. Math. Phys. 136, 399–432 (1991) 23. Woronowicz, S.L.: Quantum SU (2) and E(2) groups, Contraction procedure. Communi. Math. Phys. 149, 637–652 (1992) 24. Woronowicz, S.L.: Compact quantum groups. In: Symétries quantiques (Les Houches, 1995), Amsterdam: North-Holland, 1998, pp. 845–884 Communicated by A. Connes
Commun. Math. Phys. 304, 229–280 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1207-z
Communications in
Mathematical Physics
Hilbert Expansion from the Boltzmann Equation to Relativistic Fluids Jared Speck1, , Robert M. Strain2, 1 Department of Pure Mathematics & Mathematical Statistics, University of Cambridge, Wilberforce Road,
Cambridge CB3 0WB, United Kingdom. E-mail: [email protected]
2 Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 209 South 33rd Street,
Philadelphia, PA 19104-6395, USA. E-mail: [email protected] Received: 11 June 2010 / Accepted: 4 September 2010 Published online: 22 February 2011 – © Springer-Verlag 2011
Abstract: We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellians. The Maxwellians are constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations. Contents 1.
2. 3.
Introduction and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Lorentzian geometry and the mass shell M . . . . . . . . . . . . . . . . 1.3 Hypotheses on the collision kernel . . . . . . . . . . . . . . . . . . . . 1.4 Maxwellians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The energy-momentum tensor and the particle current for rB . . . . . . 1.6 The relativistic Euler equations and their relationship to the relativistic Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Outline of the structure of the article . . . . . . . . . . . . . . . . . . . Local Existence for the Relativistic Euler Equations . . . . . . . . . . . . . The Relativistic Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . 3.1 Expression for the collision operator . . . . . . . . . . . . . . . . . . .
230 233 235 236 237 238 239 243 247 247 247 250 251
J.S. was supported by the Commission of the European Communities, ERC Grant Agreement No 208007. R.M.S. was supported in part by the NSF grant DMS-0901463.
230
J. Speck, R. M. Strain
3.2 Macroscopic quantities and conservation laws for rB . . . . . ¯ 3.3 Macroscopic quantities for a Maxwellian M = M(n, θ, u; P) 3.4 The invertibility of the maps H(n, z) and P(n, z) . . . . . . . 3.5 Regimes of hyperbolicity for the rE system . . . . . . . . . . . 3.6 Bessel function identities and inequalities . . . . . . . . . . . 3.7 The Hilbert expansion . . . . . . . . . . . . . . . . . . . . . . 3.8 Relativistic Boltzmann estimates . . . . . . . . . . . . . . . . Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
252 254 257 259 262 265 267 278 278
1. Introduction and Main Results The special relativistic Boltzmann (rB from now on) equation provides a statistical description of a gas of relativistic particles that are interacting through binary collisions in Minkowski space, which we denote by M. The dynamic variable is the one-particle empirical measure F ε ≥ 0, which represents the average number of particles of fourmomentum P at each spacetime point x ∈ M. The four-momentum P of a particle of rest mass m 0 is future-directed1 and satisfies the normalization condition Pκ P κ = −m 20 c2 , where the constant c denotes the speed of light. Consequently, we may view F as a def function of time t ∈ R, space x¯ ∈ R3 , and three-momentum P¯ = (P 1 , P 2 , P 3 ) ∈ R3 , with2 3 P 0 = m 20 c2 + (P a )2 . a=1
A more geometric point of view is offered in Sect. 1.2, where it is explained how to view F as function on the mass shell M = {(x, P) ∈ M × Tx M | Pκ P κ = −m 20 c2 , P is future-directed}, def
which is a submanifold of T M, the tangent bundle of M, and which is diffeomorphic to R4 × R3 . The relativistic Boltzmann equation (rB from now on) in the unknown F ε is P κ ∂κ F ε =
1 C(F ε , F ε ), ε
(1.1)
where C(·, ·) is the collision operator (defined in (1.10), and the dimensionless parameter ε is the Knudsen number. It is the ratio of the particle mean free path to a characteristic (physical) length scale. Intuitively, when ε is small, the continuum approximation of fluid mechanics is expected to be valid. In this setting, we anticipate that the system of particles can be faithfully modeled through the use of macroscopic quantities, such as pressure, proper energy density, etc., whose evolution is prescribed by the equations of relativistic fluid mechanics, that is, the relativistic Euler equations. As a first rigorous step in this direction, we show that any sufficiently regular solution ¯ θ (t, x), ¯ u(t, x)) ¯ of the relativistic Euler system (rE from now on) satisfying the (n(t, x), 1 In the inertial coordinate system we use throughout this article, future-directed vectors P satisfy P 0 > 0. 2 The formula for P 0 holds only in a coordinate system in which the spacetime metric g has the components
gμν = diag(−1, 1, 1, 1).
Hilbert Expansion from Relativistic Boltzmann to Euler
231
technical conditions (1.39) can be used to construct a corresponding family of classical solutions F ε of the rB equation. Here, θ denotes the fluid temperature, n denotes the fluid proper number density, and u denotes the fluid four-velocity. Roughly speaking, the technical conditions are the assumption that θ (t, x) ¯ is uniformly positive with only mild fluctuations, that n(t, x) ¯ is uniformly bounded from above and below away from 0, and that the spatial components of the four-velocity, namely u 1 (t, x), ¯ u 2 (t, x), ¯ and 3 u (t, x), ¯ are uniformly small. In Lemma 1.1, we give a simple proof that these conditions are always satisfied near the constant fluid states. Under these assumptions, our estimates show that as ε → 0+ (in the hydrodynamic limit), the rB solution F ε converges to the relativistic Maxwellian3 M associated to the solution of the rE system; see Sect. 1.4 for the definition of a Maxwellian, which is a function in equilibrium with the collision process; i.e., C(M, M) = 0. Thus, our results show that for small ε, there are near-local equilibrium solutions to the rB equation whose underlying dynamics are effectively captured by the rE system. For the non-relativistic Boltzmann equation, these are called “normal solutions” in Grad [24]. The main strategy of our proof is to perform a Hilbert expansion (see Sect. 3.7) for F ε . We write F ε = F0 +
6
εk Fk + ε3 FR;ε .
(1.2)
k=1
Inserting this expansion into (1.1) and equating like powers of ε results in a hierarchy of equations. It turns out that F0 , . . . , F6 , which do not depend on ε, can be solved for: F0 must be Maxwellian, while F1 , . . . , F6 solve linear equations with inhomogeneities. Thus all of the difficult (and ε−dependent) analysis is contained in the analysis of the remainder term FR;ε , which is carried out in Sect. 3.8. Our methods and results can be viewed as an extension of the program initiated by Caflisch [8], who proved analogous results for the non-relativistic Boltzmann equation and Euler equations [8]. We will utilize strategies from Caflisch as well as Guo [31] and Guo-Jang-Jiang [32] to perform the Hilbert expansion. We will also use relativistic Boltzmann estimates from the work of the second author [46,48] in order to control the expansion. Additionally, we develop several necessary tools to study this problem in the setting of special relativity. In particular, we develop a mathematical theory for the kinetic equation of state, which is described just below. A more detailed discussion of the existing literature related to our result is located in Sect. 1.8. Before proving the aforementioned results, we will sketch a proof of local existence (in Sect. 2) for the relativistic Euler equations. This local existence result ensures that there are in fact solutions to the rE system that can be used in the aforementioned construction. However, we mention upfront that during the course of our investigation, we ran into several technical difficulties that, to our surprise, seem to be unresolved in the literature. The first concerns the fundamental question of which fluid variables can be used as state-space variables in the rE system. In addition to the four-velocity u, there are five other fluid variables that play a role in the ensuing discussion: the aforementioned variables n and θ, together with the entropy per particle η, the pressure p, and the proper energy density ρ. In order to close the Euler equations, one must assume relations between the fluid variables. In this article, we assume that the three relations (1.27a)–(1.27c) hold between the five non-negative variables n, θ, η, p, and ρ. As will be discussed below, these choices were not made arbitrarily, but are in fact satisfied by 3 These are also known as Jüttner distributions.
232
J. Speck, R. M. Strain
the macroscopic quantities n[M], θ [M], η[M], p[M], and ρ[M] corresponding to a relativistic Maxwellian M; these quantities are defined in Sect. 3.2. We emphasize that the relations (1.27a)–(1.27c) are required in order for the rE system to arise from the rB equation in the hydrodynamic limit. Now it is commonly assumed that as a consequence of the three relations, any two of n, θ, η, p, ρ uniquely determine the remaining three. In particular, during our construction of the fluid solutions, we need to be able to go back and forth between the variables (n, θ ) and the variables (η, p), i.e., we need to 2 be able to invert the smooth map (n, z) → (H(n, z), P(n, z)), where z = mk B0 cθ , k B > 0 denotes Boltzmann’s constant, and H and P are defined in (1.29a)–(1.29b) below. However, we were unable to find a fully rigorous proof of the invertibility of this map in the literature. Consequently, in Lemma 3.5 below, we use asymptotic expansions for Bessel functions to rigorously verify the local invertibility of this map outside of a compact set. In particular, we show that the map is locally invertible whenever θ is sufficiently large, and whenever θ is sufficiently small and positive. Additionally, the numerical plot in Fig. 1 (see Sect. 3.4), which covers the compact set in question, strongly suggests that the map (n, z) → (H(n, z), P(n, z)) = (η, p), is an auto-diffeomorphism of the region (0, ∞) × (0, ∞). This would imply that we can always smoothly transform back and forth between (n, θ ) and (η, p) in the region of physical relevance, i.e., the region in which all of the quantities are positive; see Conjecture 1 in Sect. 2 below. Similarly, in Lemma 3.6, we rigorously prove that outside of the same compact set of temperature values, there exists a kinetic equation of state p = f kinetic (η, ρ), which gives the fluid pressure p as a function of the entropy per particle η and the proper energy density ρ. 4 A related issue is the fact that in order for the rE system to be well-posed and causal ∂f under a general equation of state p = f (η, ρ), it is sufficient to prove that 0 < ∂ρ < 1. η ∂f We explain why the positivity of ∂ρ is needed for our proof of local existence in Remark η 2.1 below, while the mathematical connection between the inequality ∂∂ρf < 1 and the η
speed of sound propagation being less than the speed of light is explained in e.g. [44]. Now in the case of the kinetic equation of state p = f kinetic (η, ρ), ∂ fkinetic ∂ρ can be η
written as a function of θ alone. It is possible to write down a closed form expression for this latter quantity (see Eq. (3.32), but since the formula is a rather complicated one involving ratios of Bessel functions, we have only analytically verified the inequality ∂ f kinetic 0 < ∂ρ < 1 (again using asymptotic expansions for Bessel functions) outside of η
the same compact set discussed in the previous paragraph; see Lemma 3.6. Therefore, the fully rigorous version of our local existence result is currently limited to initial data whose temperature avoids the compact set in question. However, we have numerically ∂ f kinetic 1 observed that in fact, the stronger inequality 0 < ∂ρ < 3 should hold for all η
θ > 0; see Conjecture 2 in Sect. 2, and Fig. 2 in Sect. 3.5. This stronger inequality would √imply that the speed of sound under the kinetic equation of state is never larger than 1/3 times the speed of light. In view of these complications, when stating the hypotheses for our local existence theorem (Theorem 1 in Sect. 1.7), we make careful assumptions on the fluid initial data that are designed to ensure that they fall within the regime of hyperbolicity, and within a regime in which the aforementioned map (n, θ ) → (η, p) is invertible with 4 By causal, we mean that the speed of sound is less than the speed of light.
Hilbert Expansion from Relativistic Boltzmann to Euler
233
smooth inverse. However, if our two conjectures are in fact correct, then the relations (1.27a)–(1.27c) imply that many of these assumptions are automatically verified whenever the fluid variables are positive. Aside from these complications, our local existence theorem is a standard result. However, there are several additional aspects of it that are worthy of mention. First, we avoid the use of symmetrizing variables in our proof. We instead use the framework of energy currents, which was first applied by Christodoulou to the rE system in [11], and which was later expounded upon by the first author in [44]. We also remark that our local existence result only applies to initial data with proper energy density ρ that is uniformly positive. In particular, we avoid addressing the complicated issue of the free-boundary problem for the relativistic Euler equations. A related comment is that our local existence result produces a spacetime slab [0, T ] × R3 on which the uniform positivity property is preserved. We remark that in view of the assumptions on n, θ mentioned near the beginning of the article, we will only study fluid solutions belonging to compact subsets of the regions of interest to us, that is, regions where the maps H and P are rigorously known to be invertible. Thus, on such compact subsets, the uniform positivity of ρ is an automatic consequence of the continuity of the map (n, z) → ρ, which is implicitly defined by the relations (1.27a)–(1.27b). 1.1. Notation and conventions. We now summarize some notation and conventions that are used throughout the article. M denotes Minkowski space, while M denotes the mass shell. In general, Latin (spatial) indices a, b, j, k, etc., take on the values 1, 2, 3, while Greek indices κ, λ, μ, ν, etc., take on the values 0, 1, 2, 3. Indices are raised and lowered with the Minkowski metric gμν and its inverse (g −1 )μν . For most of the article, we work in a fixed inertial coordinate system on M, in which case gμν = (g −1 )μν = diag(−1, 1, 1, 1),
(1.3)
3 and P κ Q κ = −P 0 Q 0 + a=1 P a Q a . Here and throughout, we use Einstein’s summation convention that repeated indices, with one “up” and one “down” are summed over. When differentiating with respect to state-space variables, we use the notation ∂U |V , to mean partial differentiation with respect to the quantity U while V is held constant. def def def ¯ 2 )1/2 , and | P| ¯ 2= We define x¯ = (x 1 , x 2 , x 3 ), P¯ = (P 1 , P 2 , P 3 ), P 0 = (m 20 c2 + | P| 3 def a 2 0 −1 ¯ ¯ ¯ ¯ ˆ a=1 (P ) . Furthermore Q, P , Q are treated similarly. We also define P = (P ) P, ˆ We use the symbol and similarly for Q. ∂ ∂ ∂ def = (∂1 , ∂2 , ∂3 ), , , ∂x¯ = ∂x1 ∂x2 ∂x3 to denote the spatial coordinate gradient. The Sobolev norm · H N of a Lebesgue measurable function f (x) ¯ on R3 is defined in the usual way: ⎛ f H N = ⎝ def
x¯
| α |≤N
x¯
⎞1/2 ∂α f 2L 2 ⎠ x¯
,
(1.4)
234
J. Speck, R. M. Strain
where ∂α = ∂1n 1 ∂2n 2 ∂3n 3 , α = (n 1 , n 2 , n 3 ) is a spatial coordinate-derivative multi-index, def and | α | = n 1 + n 2 + n 3 . Here and throughout, we use the abbreviation Hx¯N = Hx¯N (R3x¯ ). ¯ G( P) ¯ as follows: We also define the L 2P¯ inner product of two functions F( P), F, G P¯ =
¯ ¯ P. ¯ F( P)G( P)d
def
R3P¯
(1.5)
For brevity, we sometimes write ·, · = ·, · P¯ . The L 2 (R3P¯ ) norm is denoted | · |2 . ¯ G(x, ¯ as inner product of two functions F(x, ¯ P), ¯ P) Similarly, we define the L 2x; ¯ P¯ def
F, G x; ¯ P¯ =
def
R3x¯
R3P¯
¯ ¯ xd ¯ F(x, ¯ P)G( x, ¯ P)d ¯ P.
We denote the corresponding norm by f 2 = f H 0 (R3 ×R3 ) = f L 2 (R3 ×R3 ) . We x¯ x¯ P¯ P¯ furthermore define the norm def
def ¯ h∞ = ess supx∈R3 , p∈R3 |h(x, ¯ P)|.
x¯
P¯
For each ≥ 0, we also define the weight function w as /2 def ¯ = ¯2 1 + | P| . w = w ( P)
(1.6)
We then define a corresponding weighted L ∞ norm by ¯ x, ¯ ¯ P)|. h∞, = ess supx∈R3 , p∈R3 |w ( P)h( def
x¯
P¯
¯ over a measurable We define the Hx¯N norm of a Lebesgue measurable function f (x) subset E ⊂ R3x¯ by ⎛ f H N (E) = ⎝
def
x¯
| α |≤N
⎞1/2 ∂α f 2L 2 (E) ⎠ x¯
,
where f L 2 (E) =
1/2 | f |2 d x¯
def
x¯
,
E
and similarly for the other norms and inner products over a subset. If X is a normed function space, then we use the notation C j ([0, T ], X ) to denote the set of j-times continuously differentiable maps from (0, T ) into X that, together with their derivatives up to order j, extend continuously to [0, T ]. We sometimes use the notation A B to mean that there exists an inessential uniform constant C such that A ≤ C B. Generally C will denote an inessential uniform constant whose value may change from line to line. For essential constants, we always write down their dependence explicitly.
Hilbert Expansion from Relativistic Boltzmann to Euler
235
1.2. Lorentzian geometry and the mass shell M. In this article, we primarily work in a fixed inertial coordinate system on M, which is a global rectangular coordinate system {x μ }μ=0,1,2,3 in which the spacetime metric gμν has the form (1.3). Note the sign convention of (1.3). This is the most common sign convention found in the relativity literature, but it is opposite of the sign convention that is sometimes found in the relativistic Boltzmann literature. Our coordinate system {x μ }μ=0,1,2,3 represents a special choice of a “space-time” splitting. We identify x 0 with ct, where c is the speed of light and t is time, while we def identify (x 1 , x 2 , x 3 ) = x¯ with a “spatial coordinate”: x = (x 0 , x 1 , x 2 , x 3 ) = (ct, x). ¯ We often work with the coordinate t rather than x 0 . Note that ∂0 = 1c ∂t . The mass shell is a subset of T M = ∪x∈M Tx M, def
the tangent bundle of M. In the following, we use an inertial coordinate system {x μ , Y ν }μ,ν=0,1,2,3 on T M, where {x μ }μ=0,1,2,3 is the inertial coordinate system on M, and ∂ Tx M = Y κ κ (Y 1 , Y 2 , Y 3 , Y 4 ) ∈ R4 . ∂x x 4 In the above expression, (Y 1 , Y 2, Y 3 , Y 4 ) are the coordinates of vectors in Tx M R ∂ ∂ ∂ ∂ relative to the basis { ∂ x 0 , ∂ x 1 , ∂ x 2 , ∂ x 3 }. The mass shell M is defined to be x
x
x
x
M = {(x, P) ∈ T M | Pκ P κ = −m 20 c2 and P 0 > 0}. def
Let us also define Mx = M ∩ Tx M, def
and ¯ = P0 = φ( P) def
¯ 2. m 20 c2 + | P|
It follows that the map : R3 → Mx defined by (P 1 , P 2 , P 3 ) = (φ(P 1 , P 2 , P 3 ), P 1 , P 2 , P 3 ), def
is a diffeomorphism, and we can use it to put coordinates on Mx ; i.e., any element of ¯ P) ¯ Mx , viewed as a submanifold of Tx M, has components (P 0 , P 1 , P 2 , P 3 ) = (φ( P), relative to our rectangular coordinate system. It follows that {x μ , P j }μ=0,1,2,3; j=1,2,3 is a global coordinate system on M. If (x, P) is an element of M, then we often slightly ¯ We similarly identify F(x, P) with abuse notation by identifying (x, P) with (t, x, ¯ P). ¯ F(t, x, ¯ P). We recall that there is a canonical measure def ¯ dμg¯ = |det(g)|d ¯ P, (1.7)
236
J. Speck, R. M. Strain
associated to g, ¯ the first fundamental form of Mx . We remark that g¯ is Riemannian since Mx is a spacelike hypersurface in Tx M. This measure will allow us to define (in a geometrically invariant manner) integration over the surface Mx . Recall that since Mx is (relative to the coordinate system {Y ν }ν=0,1,2,3 on Tx M) the level set Mx = {(P 0 , P 1 , P 2 , P 3 ) ∈ Tx M | P 0 − φ(P 1 , P 2 , P 3 ) = 0}, it follows that g¯ = ∗ g, where ∗ g is the pullback of g by . Simple calculations imply that in our inertial coordinate system, we have g¯ jk =
∂κ ∂λ P j Pk g = − + δ jk , ( j, k = 1, 2, 3). κλ ∂ P j ∂ Pk (P 0 )2
(1.8)
Using (1.7) and (1.8), we compute that the canonical measure associated to g¯ can be expressed as follows relative to the coordinate system P¯ on Mx : dμg¯ =
1 ¯ d P, P0
(1.9)
where we have used the fact that |det(g)| ¯ = (P 0 )−2 . We remark that (1.9) is valid only in an inertial coordinate system, and that integrals relative to the measure dμg¯ will play a central role in the definitions and analysis of Sect. 3.2. 1.3. Hypotheses on the collision kernel. In order to state our hypotheses on the collision kernel, we introduce the following expression for the Boltzmann collision operator, which is local in (t, x); ¯ we will elaborate upon it in Sect. 3.1: 0 ¯ ¯ ¯ C(F, G) = P vø σ (, ϑ)[F( P¯ )G( Q¯ ) − F( P)G( Q)]d Qdω, (1.10) R3 ×S2
where we have suppressed the dependence of F and G on (t, x). ¯ Note that the collision ¯ Q), ¯ the operator acts only on the P variables. In the above expression, vø = vø ( P, Møller velocity, is defined by 2 2 √ P¯ Q¯ 1 P¯ Q¯ c s def c ¯ ¯ vø = vø ( P, Q) = − 0 − 2 0 × 0 = , (1.11) 2 P0 Q c P Q 4 P 0 Q0 where × denotes the cross product in R3 . In (1.10), σ is the differential cross-section, ¯ Q) ¯ is defined by or the collision kernel. The relative momentum ( P, def = (P κ − Q κ )(Pκ − Q κ ) = −2(P κ Q κ + m 20 c2 ) ≥ 0, (1.12) ¯ Q, ¯ P¯ , Q¯ ) is defined by while the scattering angle ϑ( P, cos ϑ = (P κ − Q κ )(Pκ − Q κ )/2 . def
(1.13)
¯ Q¯ below in (3.5). Finally, Here the variables P and Q are defined in terms of P, 2 ¯ ¯ s( P, Q), which is defined by c s = the energy in a center-of-momentum frame,5 can be expressed as (1.14) s = −(P κ + Q κ )(Pκ + Q κ ) = 2 −P κ Q κ + m 20 c2 ≥ 0. 5 A center-of-momentum frame is a Lorentz frame in which P μ + Q μ = P μ + Q μ = (√s, 0, 0, 0).
Hilbert Expansion from Relativistic Boltzmann to Euler
237
Notice that s = 2 + 4c2 . We warn the reader that this notation, which is used in [12], may differ from other authors notation by a constant factor. Furthermore, ω is an element of S2 (viewed as a submanifold of R3 ), which can (with the exception of the north pole) be parameterized by the angles (ϑ, ϕ) ∈ (0, π ] × (0, 2π ], where ϑ is from above, and ϕ is an azimuthal angle. Relative to these coordinates, we have that dω = sinϑdϑdϕ. The function σ depends on the chosen model of particle interaction. For the remainder of the article, we assume the following: Hypotheses on the collision kernel. We assume that there are constants C1 , C2 , a, and b such that the differential cross-section σ , which is listed above in (1.10) and below in (3.4), satisfies the inequalities σ (, ϑ) ≤ C1 a + C2 −b σ0 (ϑ), where • C1 and C2 are non-negative, • There exists a number γ > −2 such that 0 ≤ σ0 (ϑ) ≤ sinγ ϑ, • 0 ≤ a < min(2, 2 + γ ), and 0 ≤ b < min(4, 4 + γ ). We also assume that there exist constants C ≥ 1 and β ∈ (−4, 2) such that 1 0 β/2 ¯ ≤ C(P 0 )β/2 . (P ) ≤ ν( P) C
(1.15)
¯ the collision frequency, is defined by Here ν( P), def ¯ ¯ ¯ ν( P) = dQ dω vø σ (, ϑ) J ( Q), R3
S2
¯ is the global relativistic Maxwellian defined by and J ( Q) 0 def ¯ = e−cQ /(k B θ M ) , J ( Q)
(1.16)
and θ M > 0 is a constant. The lower bound in (1.15) is satisfied if there is a suitable lower bound for σ. See [16,18,46,49], and we refer specifically to [46] for more details. The upper bound in (1.15) is sufficient to deduce some of the estimates that we use below, such as Lemma 3.12, Lemma 3.13, Lemma 3.14, and Lemma 3.15. Our hypotheses originate from the general physical assumption introduced in [16]; see also [17] for further discussions. Standard references in relativistic Kinetic theory include [9,12,18,53,45]. 1.4. Maxwellians. We now introduce the Maxwellians, a special class of functions on M that play a fundamental role in connecting the relativistic Boltzmann equation to the relativistic Euler equations. Given any functions n = n(t, x), ¯ θ = θ (t, x), ¯ u μ = u μ (t, x) ¯ on M such that n > 0, θ > 0, u 0 > 0, u κ u κ = −c2 , we define the corresponding ¯ as follows: Maxwellian M = M(n, θ, u; P) z zu κ P κ def ¯ , (1.17) M = M(n, θ, u; P) = n exp m 0 c2 4π m 30 c3 K 2 (z)
238
J. Speck, R. M. Strain
where the dimensionless variable z is defined by z= def
m 0 c2 , kBθ
(1.18)
k B is again Boltzmann’s constant, and K 2 (z) is the Bessel function defined in (3.43). It can be shown (see e.g. [12, Chap. 2]) that C(F, F) ≡ 0
⇐⇒
¯ F is a Maxwellian of the form M(n, θ, u; P).
(1.19)
For this reason, a Maxwellian M is said to be in local equilibrium with the collision process. If n(t, x), ¯ θ (t, x), ¯ and u μ (t, x) ¯ are constant valued, then M is said to be in ¯ appearing in (1.16) is a global global equilibrium. Note also that the function J ( P) Maxwellian. It will play a distinguished role in the analysis of Sect. 3.8. In fact, we will assume that the local Maxwellian corresponding to the fluid solution is uniformly ¯ see (1.39). comparable to powers of J ( P); Remark 1.1. Although C(M, M) = 0, it is not true in general that M is a solution to ¯ = 0. Nevertheless, our the rB equation (1.1); i.e., in general, P κ ∂κ M(n, θ, u; P) main result (Theorem 2) shows that under certain assumptions, including that (n, θ, u) are solutions to the rE system, there is a solution of (1.1) near M.
1.5. The energy-momentum tensor and the particle current for rB. We now define TBolt z [F], which is the energy-momentum-stress-density tensor (energy-momentum tensor for short) for the relativistic Boltzmann equation, and I Bolt z [F], which is the particle ¯ these quantities are defined as follows: current. Given any function F( P), μν
TBolt z [F] = c μ I Bolt z [F]
def
R3P¯
¯ P μ P ν F( P)
d P¯ , (0 ≤ μ, ν ≤ 3), P0
¯ ¯ d P , (0 ≤ μ ≤ 3). =c P F( P) 0 P R3P¯ def
(1.20)
μ
Whenever there is no possibility of confusion, we abbreviate TBolt z = TBolt z [F], and similarly for the additional quantities depending on F that appear below. The conser¯ is a vation laws can be summarized as follows (see Lemma 3.2): whenever F(t, x, ¯ P) classical solution to the relativistic Boltzmann equation, the following conservation laws hold: μκ
∂κ (TBolt z [F]) = 0, (0 ≤ μ ≤ 3), κ ∂κ (I Bolt z [F]) = 0.
(1.21)
¯ and (n, θ, u) are It is explained in the next section that whenever F = M(n, θ, u; P), a solution to the rE system, then the above conservation laws for F hold even though F need not be a solution to the rB equation. This fact will play an important role during our discussion of the relationship of the rE system to the rB equation.
Hilbert Expansion from Relativistic Boltzmann to Euler
239
1.6. The relativistic Euler equations and their relationship to the relativistic Boltzmann equation. In this section, we recall some basic facts about the rE equations in Minkowski space. This is intended to serve as background for Theorem 1, which is stated in Sect. 1.7, and proved in Sect. 2. For a detailed discussion of the rE system, we refer the reader to Christodoulou’s survey article [10]; here we only provide a brief introduction. Our other goal in this section is to illustrate some of the formal correspondences between the rE system and the rB equation. These provide a heuristic basis for the expectation that the rE system should emerge from the rB equation in the hydrodynamic limit. The rE system models the evolution of a perfect fluid evolving in a Lorentzian spacetime. In Minkowski space, the spacetime of special relativity, they are μκ
∂κ T f luid = 0, (0 ≤ μ ≤ 3), ∂κ I κf luid = 0.
(1.22)
Here the energy-momentum-stress-density tensor (energy-momentum tensor for short) for a perfect fluid has components μν
T f luid = c−2 (ρ + p)u μ u ν + p(g −1 )μν , def
(μ, ν = 0, 1, 2, 3),
(1.23)
where ρ ≥ 0 is the proper energy density, p ≥ 0 is the pressure, and u is the fourvelocity. The four-velocity is a future-directed (i.e., u 0 > 0 in our inertial coordinate system) vectorfield that satisfies the normalization condition u κ u κ = −c2 .
(1.24)
μ
The vectorfield I f luid is the particle current. It is proportional to the four-velocity: μ
I f luid = nu μ , def
(μ = 0, 1, 2, 3).
The quantity n ≥ 0 is the proper number density. All of these quantities are functions of (t, x) ¯ ∈ R × R3 . By projecting the first divergence in (1.22) in the direction parallel to u and onto the g−orthogonal complement of u, we can rewrite (1.22) (omitting some standard calculations) in the well-known form ∂κ (nu κ ) = 0, u ∂κ ρ + (ρ + p)∂κ u κ = 0, (ρ + p)u κ ∂κ u μ + μκ ∂κ p = 0, κ
(μ = 0, 1, 2, 3),
(1.25a) (1.25b) (1.25c)
where , the two-tensor that projects onto the g−orthogonal complement of u (i.e. μκ u κ = 0) has the components μν = c−2 u μ u ν + (g −1 )μν , (μ, ν = 0, 1, 2, 3). def
(1.26)
The above equations are redundant in the following sense: if (1.24) and (1.25b) hold, and if (1.25c) holds for μ = 1, 2, 3, then it follows that (1.25c) also holds when μ = 0. Equations (1.22) are not closed because there are more unknowns than equations. In order to close the rE system in a manner compatible with the rB equation, we will make use of the additional fluid variable η, a non-negative quantity known as the entropy per
240
J. Speck, R. M. Strain
particle. We now make the following assumptions, which are of crucial importance: we assume that the fluid variables n, θ, η, p, ρ are bound by the relations n p = k B nθ = m 0 c2 , z 2 K 1 (z) + 3 p, ρ = m0c n K 2 (z) −η K 2 (z) K 1 (z) n = 4π e4 m 30 c3 h −3 exp exp z , kB z K 2 (z)
(1.27a) (1.27b) (1.27c)
where k B > 0 is Boltzmann’s constant, h > 0 is Planck’s constant, and z is defined in (1.18). Furthermore, the K j (·) are modified second order Bessel functions, which are defined in Lemma 3.7. The origin of these relations, which are fundamentally connected to the properties of Maxwellians, is explained later in this section. Using the above relations, we can deduce the local solvability of any one of the variables n, θ, η, p, ρ in terms of any two of the others whenever we know that the necessary partial derivatives are non-zero (knowing this allows us to apply the implicit ∂ρ function theorem). In particular, whenever ∂ p > 0, we can locally solve for p as a η
function f kinetic of η and ρ :
p = f kinetic (η, ρ).
(1.28)
We refer to f kinetic as the kinetic equation of state. Note that this equation of state is discussed in Synge [53]. For future use, we denote by H and P respectively the smooth maps from (n, z) to η, p induced by the above relations: K 1 (z) K 2 (z) exp z , (1.29a) η = H(n, z) = k B ln (4π e4 m 30 c3 h −3 n −1 z K 2 (z) p = P(n, z) = m 0 c2 nz −1 .
(1.29b)
In the next section, we sketch a standard proof, which is based on energy estimates, of Theorem 1, i.e., for local existence for the rE system under the relations (1.27a)–(1.27c). During the proof, we work with the unknowns (η, p, u 1 , u 2 , u 3 ), the reason being that a framework for deriving energy estimates in these variables via the method of energy currents has been developed; this is explained in detail during the proof of the theorem. To derive the energy estimates, we of course need an equivalent (for C 1 solutions) formulation of the rE system (1.22) in terms of (η, p, u 1 , u 2 , u 3 ). To prove the equivalence of the systems, one needs the following identity, which is shown to be a consequence of the relations (1.27a)–(1.27c) in Proposition 3.4 below: ∂ρ ρ+p=n . (1.30a) ∂n η Using (1.25a), (1.27a)–(1.27c), and several applications of the chain rule, we deduce the following well-known version of the rE system: u κ ∂κ η = 0, u κ ∂κ p + q∂κ u κ = 0, κ (ρ + p)u ∂κ u μ + μκ ∂κ p = 0,
(μ = 0, 1, 2, 3),
(1.31a) (1.31b) (1.31c)
Hilbert Expansion from Relativistic Boltzmann to Euler
μν = c−2 u μ u ν + (g −1 )μν , ∂ p def . q = c2 (ρ + p) ∂ρ η def
241
(μ, ν = 0, 1, 2, 3), (1.31d) (1.31e)
Although our local existence theorem is proved using the variables (η, p, u 1 , u 2 , u 3 ), in order to construct the relativistic Maxwellian (1.17), which plays a fundamental role in our analysis of the rB equation, we require the availability of the variables (n, θ, u 1 , u 2 , u 3 ). Thus, we need to be able to solve for (n, z) as a smooth function of (η, p). Remarkably, we could not find such a result in the literature. Thus, in Lemma 3.4, we rigorously show that we can locally solve for (n, z) in terms of (η, p) if 0 < z ≤ 1/10 or z ≥ 70. Furthermore, based on numerical observations in the region 1/10 < z < 70, we make the following conjecture: Conjecture 1. The map (n, z) → (H(n, z), P(n, z)) is an auto-diffeomorphism of the region (0, ∞) × (0, ∞), where the maps H and P are defined in (1.29a)–(1.29b). This conjecture is based on a numerical plot (see Fig. 1), in which z 5 ∂∂zp appears to be η
negative for all z > 0. Examining the proof of Lemma 3.4, it is clear that the negativity ∂p of ∂z would imply that the conjecture is true. We remark that by (3.24), analytically η
verifying the negativity of this quantity is equivalent to demonstrating the following inequality for all z > 0 : K 1 (z) 2 K 1 (z) 4 +z (1.32) 3 − z − < 0. K 2 (z) K 2 (z) z Another fundamental quantity in thesystem (1.31a)–(1.31e) is the speed of sound, the square of which is defined to be c2 ∂∂ρp . It is a fundamental thermodynamic assumpη ∂p tion that 0 < ∂ρ < 1 for physically relevant equations of state. As we alluded to η in the Introduction, the positivity of ∂∂ρp is required in order for the rE system to be η
hyperbolic, while the upper bound of 1 implies that the speed of sound propagation is less than the speed of light. In Remark 2.1, we explain exactly how we use the positivity in our proof of local existence, while the analytic and geometric connections between the upper bound of 1 and the speed of sound propagation being less than the speed of light is explained in e.g. [44]. Remarkably, there seems to be no rigorous proof in the literature that these inequalities hold for the kinetic equation of state (1.28) in every regime. Consequently, in Lemma 3.6, we analytically verify that the equation of state (1.28) exists and satisfies these inequalities in the same regime discussed above, namely for 0 < z ≤ 1/10 and z ≥ 70. Moreover, we conjecture that the following stronger statement is true. Conjecture 2. Under the relations (1.27a)–(1.27c), p can be written as a smooth, positive function of η, ρ on the domain (0, ∞) × (0, ∞), i.e., the equation of state (1.28) is well-defined for all (η, ρ) ∈ (0, ∞) × (0, ∞). Furthermore, on (0, ∞) × (0, ∞), we have that ∂ p def ∂ f kinetic (η, ρ) < 1. 0< (1.33) (η, ρ) = ∂ρ ∂ρ 3 η
η
242
J. Speck, R. M. Strain
Our conjecture is based upon a plot of (∂z |η ρ)/(∂z |η p) (which can be expressed in terms 2
of a function of z = mk B0 cθ alone) that covers the region in question, i.e., the region 1/10 < z < 70. Our plot is labeled as Fig. 2 of Sect. 3.5. Furthermore, we note that according to Eq. (3.32), proving the inequality (1.33) is equivalent to proving that the following inequality holds for z > 0 : def
2 K 1 (z) K 1 (z) 4 + z −z K 2 (z) K 2 (z) K 1 (z) 3<3+z + 2 K 2 (z) K 1 (z) K 1 (z) 3K +z K −z− 2 (z) 2 (z) In these inequalities (∂z |η ρ)/(∂z |η p) =
4 z
< ∞.
(1.34)
∂ f kinetic −1 ∂ρ η
is the middle term expressed as
a function of z. Now that we have provided a thorough discussion of the subtleties, we state our: Running Assumption: For the remainder of the article, we restrict our attention to regimes in which any three of the five macroscopic variables n, θ, η, p, ρ can be written as smooth functions of the remaining two, and in which 0 < ∂∂ρp (η, ρ). η
To avoid burdening the paper, we do not point out this assumption every time it is made. Thus, under these running assumptions, any two of the five variables, together with the spatial components u 1 , u 2 , u 3 of the four-velocity, may be considered to be state-space variables (i.e., they completely determine the state of the system), and the phrase “a solution to the rE system” is a well-defined concept, independent of the choice of state-space variables. The role of Lemmas 3.5 and 3.6, then, is to guarantee that there is a large regime in which these conditions hold. Furthermore if Conjectures 1 and 2 are correct, then due to the relations (1.27a)– (1.27c), these conditions are automatically verified. We now explain the connection between solutions to the rE system and solutions to ¯ as defined in the rB equation. First, given any relativistic Maxwellian M(n, θ, u; P) (1.17), we can define an associated proper number density n[M], temperature θ [M], entropy per particle η[M], pressure p[M], and proper energy density ρ[M]. The first ¯ while precise two quantities are defined to be the first two arguments in M(n, θ, u; P), definitions of the remaining three (as well as equivalent definitions of the first two) are given in Sect. 3.2. Furthermore, it can be shown that μν
μν
TBolt z [M] = T f luid ,
μ
μ
I Bolt z [M] = I f luid .
(1.35)
Equation (1.35) should be interpreted as follows: the tensor obtained by substituting p = p[M], ρ = ρ[M], and u μ = I μ [M]/n[M] into the definition (1.23) agrees with the tensor calculated using (1.20). The key point is the following: as shown in Proposition 3.3, the macroscopic quantities n[M], θ [M], η[M], p[M], ρ[M] satisfy the relations (1.27a)–(1.27c). Therefore we must use the same relations to close the rE equations. Alternatively, if we are given smooth functions (η, p, u 1 , u 2 , u 3 ) of (t, x) ¯ that verify the rE equations (1.31a)–(1.31e) on [0, T ] × R3 , we can construct a local relativistic ¯ (as in (1.17) as long as the maps (1.29a)–(1.29b) are invertMaxwellian M(n, θ, u; P) ible in a neighborhood of (η, p)([0, T ] × R3 ). Now for any solution of rE system, the identity (1.35) implies that (1.21) holds for the Boltzmann energy-momentum tensor
Hilbert Expansion from Relativistic Boltzmann to Euler
243
constructed out of M. Furthermore, as noted in Sect. 1.4, the right-hand side of the rB equation vanishes for M : C(M, M) = 0. At this point one may be tempted to conclude that M itself is a solution to the rB equation. In fact, this is not the case in general; this is exactly the difficulty that Theorem 2 addresses. Its content can roughly be expressed as follows: if ε is small enough and positive, then M is “close to” a solution F ε of the rB equation. We remark that even though M itself is not in general a solution of (1.1), it nevertheless plays a prominent role in our construction of a local solution of (1.1). In particular, the validity of the Hilbert expansion of Sect. 3.7 relies heavily on the fact that ¯ is a Maxwellian and that (n(t, x), M(n(t, x), ¯ θ (t, x), ¯ u(t, x); ¯ P) ¯ θ (t, x), ¯ u(t, x)) ¯ is a solution to the rE system. 1.7. Statement of main results. In this section, we state our two independent theorems. The first concerns the existence of local solutions to the rE system satisfying the technical assumptions that are used in the proof of our main result. The second is our main result, which provides criteria that ensure that solutions to the rE system are the hydrodynamic limit of solutions to the rB equation. For clarity, we first state our assumptions on the initial data in a separate paragraph. ˚ = (n, ˚ θ˚ , u˚ 1 , u˚ 2 , u˚ 3 ) be initial data for the relInitial Data for the rE System. Let V 2 def ativistic Euler equations under the relations (1.27a)–(1.27c), and define z˚ = m 0 c˚ . Let def
kB θ
˚ z˚ )(R3 ) ⊂ (0, ∞)×(0, ∞) be the image of the initial data (n, ˚ z˚ ), and assume n,z = (n, that there exists a compact convex set n,z ⊂ (0, ∞) × (0, ∞) containing n,z in its def
interior. Let η, p = (H, P)(n,z ) ⊂ (0, ∞) × (0, ∞), where the functions H(n, z) and P(n, z) are defined in (1.29a) and (1.29b). Let η, p be a compact convex subset of (0, ∞) × (0, ∞) containing η, p . Note in particular that our assumptions imply that there exist constants n 1 > 0 and θ1 > 0 such that def
˚ x), ¯ n 1 ≤ inf n( x∈ ¯ R3
θ1 ≤ inf θ˚ (x). ¯ x∈ ¯ R3
Assume that the map (n, z) → (H(n, z), P(n, z)) = (η, p) has a well-defined, smooth inverse on η, p that maps η, p into a compact subset of (0, ∞) × (0, ∞) containing K 1 (z) + 3 p be as defined in (1.27b), and view ρ, η, and p as funcn,z . Let ρ = m 0 c2 n K 2 (z) tions of (n, z) on n,z . Assume that on the set (η(n, z), ρ(n, z)) | (n, z) ∈ n,z , p can (η,ρ) be written as a smooth function p = f kinetic (η, ρ) and that 0 < ∂ fkinetic . ∂ρ def
Remark 1.2. As discussed at the end of the Introduction, the careful assumptions on the fluid initial data are designed to ensure that they fall within the regime of hyperbolicity, and to ensure the invertibility of the maps between the solution variables (n, θ ) and (η, p). However, if our two conjectures are correct, then both of these conditions are automatically verified whenever the fluid variables are positive. In any case, Lemmas 3.5 and 3.6 together show that whenever θ˚ is uniformly small and positive, or in whenever θ˚ is uniformly large, these conditions hold. Remark 1.3. The additional convexity assumptions on n,z and η, p are technical conditions that are used in our proof of Theorem 1, e.g. to conclude (2.1); see [42,
244
J. Speck, R. M. Strain
Prop. B.0.4] for a discussion on the role of convexity in this context (roughly speaking, convexity is needed so that one can apply the mean value theorem). Theorem 1 (Local Existence for the rE System). Consider initial data that are subject to the restrictions described above. Assume that N ≥ 3 and that there exist constants 2 n > 0, θ > 0 such that (n, mk B0 cθ ) ∈ n,z (defined above), and such that ˚ − V N < ∞, V H x¯
(1.36)
where V = (n, θ , 0, 0, 0). Then these data launch a unique classical solution V = (n, θ, u 1 , u 2 , u 3 ) to the rE system existing on a nontrivial slab [0, T ] × R3 upon which def
0< 0<
inf
n(t, x), ¯
inf
θ (t, x). ¯
(t,x)∈[0,T ¯ ]×R3
(t,x)∈[0,T ¯ ]×R3
N −2 k V has the following regularity: V − V ∈ C N −2 ([0, T ] × R3 ) k=0 C ([0, T ], H N −k ). If in addition the initial data are such that the inequalities (1.39) below are strictly verified at t = 0, then there exists a spacetime slab [0, T ] × R3 of existence, with 0 < T ≤ T, upon which the condition (1.39) remains verified. ˚ − V N < δ, and δ is sufficiently small, then there exist constants Finally, if V H x¯
θ∗ > 0 and C > 0, and a slab of existence [0, T ] × R3 , with T ≥ C /δ, upon which the following bounds are satisfied: θ∗ < θ (t, x) ¯ < 2θ∗ , −1 j |c u (t, x)| ¯ ≤ Cδ, ( j = 1, 2, 3).
(1.37) (1.38)
As is shown below in Lemma 1.1, if δ is sufficiently small, then the bounds (1.37)–(1.38) will also imply that the technical conditions (1.39) are verified on [0, T ] × R3 . Remark 1.4. The conclusions of this theorem regarding the slab [0, T ] × R3 follow easily from the regularity properties of the solution. Remark 1.5. There are two main obstacles to extending our principal result, which is Theorem 2 below, to a global-in-time existence result for the rB solution. The first is that local solutions to the rE system tend to form shocks in finite time. In fact, in [11], Christodoulou showed that there are data arbitrarily close to that of a uniform, quiet fluid state (i.e., V ≡ V) that launch solutions which form shocks in finite time. Since our construction of a local solution to the rB equation relies on the availability of the solution to the rE system, the breakdown of the fluid solution could in principle allow for a breakdown in the Boltzmann solution. The second obstacle is the possible breakdown of the technical condition (1.39) satisfied by the fluid solution; this breakdown is sometimes avoidable. More specifically, the condition (1.39), which plays a key role in the analysis of Sect. 3.8, may break down in finite time even before the shock happens. However, we are aware of a class of data that launch solutions for which (1.39) holds until the time of shock formation. The details are contained in [11]; we offer a quick summary. One considers initial data for the rE system that satisfy the assumptions of Theorem 1, and the following additional assumptions: the data are irrotational and isentropic (i.e, η ≡ const), and they coincide with the constant state V = (n, θ , 0, 0, 0) outside of the unit sphere centered at the origin of the Cauchy hypersurface {t = 0}. If the departure,6 from constant state is ≤ δ, where δ is sufficiently 6 The notion of smallness is measured by a Sobolev norm of suitably high order.
Hilbert Expansion from Relativistic Boltzmann to Euler
245
small, then the estimates of [11, Theorem 13.1] imply the following fact: on the exterior of an outgoing sound cone C emanating from a sphere of radius 1− contained in {t = 0}, where = (δ) is a sufficiently small positive number satisfying 0 < ≤ 1/2, n and θ can be continuously extended7 to [0, Tmax ] × R3 \ Cint , where Cint denotes the interior of C, and [0, Tmax ) is the maximal time interval of classical existence, i.e., the time of first shock formation. Furthermore, the estimates |n(t, x) − n| ≤ Cδ, |θ (t, x) ¯ − θ| ≤ Cδ, and |c−1 u j (t, x)| ¯ ≤ Cδ, ( j = 1, 2, 3), hold on the region on [0, Tmax ] × R3 \Cint . Christodoulou’s theorem does not prove that the same estimates hold in Cint , but on p. 6, of [11], he remarks that the estimates do hold on the interior, and are in fact easier to prove than the exterior estimates. Under these conditions, we may piece together the conclusions from the two regions [0, Tmax ]×R3 \Cint and Cint to conclude the following: for sufficiently small δ, an inequality of the form (1.39) is satisfied on any slab [0, T ] × R3 , with T < Tmax . Consequently, the conclusions of Theorem 2 hold on such a slab. Now that we have a large class of suitable solutions to the rE system available, we are ready to state our main theorem. Note that our main theorem is independent of Theorem 1; the role of Theorem 1 is to ensure that there are fluid solutions that can be used in the hypotheses of Theorem 2. Theorem 2. Let (n(t, x), ¯ θ (t, x), ¯ u(t, x)) ¯ be a sufficiently regular (see Remark 1.8 below) solution to the relativistic Euler equations (1.22) for (t, x) ¯ ∈ [0, T ] × R3x¯ . ¯ as in (1.17). Assume Construct the local Maxwellian M(n(t, x), ¯ θ (t, x), ¯ u(t, x); ¯ P) that there exist constants C > 0, θ M > 0, and α ∈ (1/2, 1) such that for every 0 def ¯ ∈ [0, T ] × R3 × R3 , the global Maxwellian J ( P) ¯ = (t, x, ¯ P) e−c P /(k B θ M ) from x¯ P¯ (1.16) verifies the inequalities ¯ J ( P) ¯ ≤ C J α ( P). ¯ ≤ M(n(t, x), ¯ θ (t, x), ¯ u(t, x); ¯ P) C
(1.39)
Define initially ¯ = M(0, x, P) ¯ + F ε (0, x, ¯ P)
6
¯ + ε3 FR;ε (0, x, ¯ ≥ 0. εn Fn (0, x, ¯ P) ¯ P)
n=1
Then ∃ε0 > 0 such that for each 0 < ε ≤ ε0 there exists a unique classical solution ¯ ∈ F ε of the relativistic Boltzmann equation (1.1) of the form (1.2) for all (t, x, ¯ P) [0, T ] × R3x¯ × R3P¯ . Furthermore, there exists a constant C T = C T (M, F1 , . . . , F6 ) > 0 such that for all ε ∈ (0, ε0 ) and for any ≥ 9, the following estimates hold: ε3/2 sup FR;ε (t)/ M(t) + sup FR;ε (t)/ M(t) 0≤t≤T
∞,
≤ C T ε3/2 FR;ε (0)/ M(0)
2
0≤t≤T
∞,
+ FR;ε (0)/ M(0) + 1 . 2
Recall that FR;ε is the remainder from (1.2). Moreover, we have that sup F ε (t) − M(t)2 + sup F ε (t) − M(t)∞ ≤ C T ε,
0≤t≤T
0≤t≤T
where the constants C T > 0 are independent of ε. 7 Even though the L ∞ norm of the solution remains bounded, the Sobolev norm of the solution blows up as t ↑ Tmax .
246
J. Speck, R. M. Strain
Remark 1.6. Conditions which would imply (1.39) are standard in the Hilbert expansion literature. It generally seems to be unclear at the moment how to remove them in the context of classical solutions. We choose to make the assumption (1.39) rather than a more stringent assumption of moderate temperature variation in order to make it clear that (1.39) is all that is needed. The condition (1.39) is used to ensure that the local Maxwellian M, and the terms in the Hilbert expansion Fn , have sufficient momentum decay; we use this condition in Sect. 3.8. Remark 1.7. We have not specified precisely the initial conditions for the terms F1 (0), . . . , F6 (0), and FR;ε (0) in Theorem 2. They are constructed via the Hilbert expansion in Sect. 3.7, after one has the fluid initial conditions as in Theorem 1. Remark 1.8. In the hypotheses of Theorem 2, we have not been specific in stating the regularity needed from the relativistic Euler equations that is required to build the terms F1 , . . . , F6 in the Hilbert expansion. This is because we are not attempting to optimize the amount of regularity needed in the Hilbert expansion. In particular, including additional terms in the expansion would require a smoother solution to the rE system. However, for the purposes of this article, it is certainly sufficient to have (n(t, x), ¯ θ (t, x), ¯ u(t, x)) ¯ ∈ C 7 ([0, T ] × R3x¯ ) with suitable L 2x¯ integrability. In the next lemma, we prove that near-constant, non-vacuum fluid states necessarily verify an inequality of the form (1.39). Lemma 1.1. Assume that the initial data for the rE system satisfy the assumptions of ¯ be the local Theorem 1, including the smallness assumption in δ, and let M(n, θ, u; P) Maxwellian (1.17) corresponding to the solution. Then if δ is sufficiently small, there exist a non-trivial slab of the form [0, C /δ] × R3x¯ , and constants C > 0, θ M > 0, α ∈ ¯ ∈ [0, C /δ] × R3 × R3 , the global Maxwellian (1/2, 1), such that for every (t, x, ¯ P) x¯ P¯ 0 def ¯ = ¯ verify the inequalities (1.39). J ( P) e−c P /(k B θ M ) from (1.16) and M(n, θ, u; P) Proof. To prove that (1.39) holds, we first recall that by the conclusions of Theorem 1, there exist a non-trivial slab [0, C /δ] × R3x¯ , and constants n min > 0, n max > 0, θ∗ > 0, and C > 0, such that relative to our inertial coordinate system, the solution satisfies the following inequalities on [0, C /δ] × R3x¯ : ¯ < n max , 0 < n min ≤ n(t, x) 0 < θ∗ < θ (t, x) ¯ < 2θ∗ , |c−1 u j (t, x)| ¯ ≤ Cδ, ( j = 1, 2, 3). Therefore, using the Cauchy-Schwarz inequality, it is easy to show that there exist dimensionless constants α ∈ (1/2, 1), C1 > 0, C2 > 0, depending on the dimen−2 sionless constants m −1 0 c k B θ∗ and δ, such that if δ is sufficiently small, then for all ¯ (t, x, ¯ P) ∈ [0, C /δ] × R3x¯ × R3P¯ we have u Pκ κ −1 0 −1 0 c C P ≤ m −1 ≤ m −1 1 0 0 c C2 P , kBθ (1.40) C2 ≤ α −1 C1 . We also observe that since u, P are both future-directed and timelike, we have that u κ P κ ≤ −|u κ u κ |1/2 |Pλ P λ |1/2 = −m 0 c2 . In particular, u κ P κ < 0. It thus follows from definitions (1.16) and (1.17), the uniform bounds on n(t, x) ¯ and θ (t, x) ¯ (which are strict for θ ), and inequality (1.40), that for any positive constant θ M satisfying
Hilbert Expansion from Relativistic Boltzmann to Euler
247
C2 ≤ m 0 c2 /(k B θ M ) ≤ α −1 C1 , −3 3 there exists a dimensionless constant C > 1, depending on the constants m −3 0 c h n min , −3 −3 3 −1 −2 m 0 c h n max , m 0 c k B θ∗ , and δ, such that inequality (1.39) holds.
1.8. Historical background. There have been numerous important contributions to the subject of fluid dynamic limits of the non-relativistic Boltzmann equation. Due to length constraints, it is impossible to give a comprehensive list. We only point out a brief few works, including the early work of Grad [24,25]. In the context of DiPerna-Lions [14] renormalized weak solutions, we mention the fluid limits shown in [1–5,13,22, 23,34,38]. We refer to the review articles [21,37,54] for a more comprehensive list of references. For fluid limits in the context of strong solutions to the non-relativistic Euler and Boltzmann equations, we mention the work of Nishida [39], Caflisch [8], Bardos-Ukai [6], Guo [28–30], Liu-Yang-Yu [35], Guo-Jang-Jiang [32] and recently Guo-Jang [26]. In this work we will use the Hilbert expansion approach from Caflisch [8] combined with the recent developments in Guo-Jang-Jiang [32] to allow a non-zero initial condition for ¯ the remainder: FR;ε (0, x, ¯ P). Other works which connect to and motivate different elements of our estimates/ results include: [11,27,44,46,48,51,52,50]. They will be discussed in more detail at the appropriate time in the following developments. Much less is known for the relativistic Boltzmann equation. Formal fluid limit calculations are shown in the textbooks [9,12]. Linearized hydrodynamics are also studied in [15]. Our new contribution is to prove the existence of “normal solutions” via a Hilbert expansion to the full non-linear relativistic Boltzmann equation. It would be useful to check if these types of solutions can also be constructed for the relativistic Landau equation, as in for instance [50]. 1.9. Outline of the structure of the article. We now briefly outline the remainder of this article. In Sect. 2, we sketch a proof of local existence for the rE system that is based on well-known techniques. In Sect. 3, we provide additional background for the relativistic Boltzmann equation. In Sect. 3.1, we provide an expression for the Boltzmann collision operator that will be used in the subsequent analysis. In Sect. 3.2, we study macroscopic ¯ and we discuss the corquantities associated to a particle density function F(t, x, ¯ P), responding conservation laws that hold whenever F is a solution to the rB equation. In Sect. 3.3, we discuss the thermodynamic relations that hold between the macroscopic quantities associated to the special case F = M, where M is a Maxwellian. In Sect. 3.4, we discuss the issue of solving for (n, θ ) in terms of (η, p), assuming that the aforementioned thermodynamic relations hold. In Sect. 3.5, we prove that there are regimes of hyperbolicity for the rE system under the same thermodynamic relations. In Sect. 3.7, we carry out the Hilbert expansion for F ε in detail. Finally, in Sect. 3.8, we provide a proof of our main theorem. 2. Local Existence for the Relativistic Euler Equations In this section we sketch a proof of local existence for the rE system; i.e, we sketch a proof of Theorem 1 of Sect. 1.7. During the course of the proof, which is mostly along standard lines, we provide references that indicate where one may find the omitted details.
248
J. Speck, R. M. Strain
Sketch of the proof of Theorem 1. Theorem 1 can be proved using the method of energy currents, a technique which was first applied to the rE system by Christodoulou in [11]. For complete details, one can first look in [44], which describes in detail how to derive energy estimates for the linearized rE system; we sketch this derivation below. Once one has these energy estimates, the proof of local existence for the non-relativistic Euler equations given in [36] can be easily modified to handle the rE system. One may also consult [33,41], or [44] for the essential ideas on how to finish the proof once one has energy estimates for the linearized system. Furthermore, although we do not need it for this article, we remark that [44] contains a proof of continuous dependence on initial data. def In order to apply this method, we use W = (η, p, u 1 , u 2 , u 3 ) as our unknowns for Eq. (1.31a)–(1.31e). The assumptions on the data allow us to use the functions def ˚ = H and P from (1.29a)–(1.29b) to transform the initial data (see Remark 1.3) V def ˚ = (η, ˚ u˚ 1 , u˚ 2 , u˚ 3 ) such that ˚ θ˚ , u˚ 1 , u˚ 2 , u˚ 3 ) to initial data W ˚ p, (n, ˚ − W N < ∞, W (2.1) Hx¯
and such that ˚ x), ¯ 0 < inf η( x∈ ¯ R3
˚ x), ¯ 0 < inf p( x∈ ¯ R3
where W = (η, p, 0, 0, 0). Here, η and p are equal to H(n, θ ) and P(n, θ ) respectively. That (2.1) follows from (1.36) can be shown via Sobolev-Moser type estimates; see e.g. the Appendix of [44]. Typical proofs of local existence are based on either the construction of convergent sequence of iterates, or a contraction mapping argument. Both of these arguments require that one prove energy estimates for the linearized Euler equations, which are the following system: def
u μ ∂μ η˙ = F,
uk q 0 ∂0 u˙ k + q ∂k u˙ k = G, u μ ∂μ p˙ + u˜ μj ∂μ p˙ = H j , ( ρ+ p ) u μ ∂μ u˙ j +
(2.2a) (2.2b) ( j = 1, 2, 3).
(2.2c)
= is the “background” (which can be In the above equations, W def ˙ = (η, ˙ p, ˙ u˙ 1 , u˙ 2 , u˙ 3 ) is the thought of as the previous iterate in an iteration scheme), W variation (which can be thought of as either the next iterate or one of its spatial derivadef terms that arise from tives), and the terms b = (F, G, H1 , H2 , H3 ) are the inhomogeneous 3 def 0 2 μν = the iteration + differentiation procedure. Furthermore, u = c + j=1 ( u j )2 , def
( η, p, u1, u2, u3)
u μ u ν + (g −1 )μν , and q is the function of the state-space variables from (1.31e) evaluated at the background. To deduce energy estimates for the linearized systems, one first defines energy cur˙ which are vectorfields that depend quadratically on the variations W: ˙ rents J, u0 2 u k u˙ k ( u k u˙ k )2 def ˙ W) ˙ = , p˙ + 2 0 p˙ + ( J˙ 0 (W, u 0 η˙ 2 + ρ+ p ) u 0 u˙ k u˙ k − q u ( u 0 )2 (2.3) j k )2 u ˙ u ( u def k j j 2 2 j j k ˙ W) ˙ = , ( j = 1, 2, 3). p˙ + 2u˙ p˙ + ( J˙ (W, u η˙ + ρ+ p ) u u˙ u˙ k − q ( u 0 )2
Hilbert Expansion from Relativistic Boltzmann to Euler
249
The two key properties of the above energy currents, which are discussed in detail in ˙ W) ˙ is a positive definite quadratic form in W, ˙ [44], are (i) it can be shown that J˙ 0 (W, ˙ and (ii) if W is a solution to the linearized rE system (2.2a)–(2.2c), then it can be shown ˙ W)] ˙ does not depend on the derivatives of W. ˙ More specifically, the that ∂κ [J˙ κ (W, following formula holds: μ u uk u˙ k p˙ p˙ 2 + 2∂0 ∂μ J˙ μ = (∂μ u μ )η˙ 2 + ∂μ q u0 μ uj ( u k u˙ k )2 u μ k k − 2 u k u˙ ( ∂μ u˙ j + ∂μ [( ρ+ p ) u ] u˙ u˙ k − ρ+ p) ( u 0 )2 u0 u0 + 2ηF ˙ +2
u j H j u k u˙ k pG ˙ + 2u˙ k Hk − 2 , q ( u 0 )2
(2.4)
where (F, G, H1 , H2 , H3 ) are the inhomogeneous terms in (2.2a)– (2.2c). We therefore can define an energy E(t) ≥ 0 by def ˙ W) ˙ d x, J˙ 0 (W, E2 (t) = ¯ R3
(2.5)
˙ L 2 . More specifically, property (i) can be used to show and E can be used to control W that if q, p, ρ are uniformly bounded from above and away from 0 on [0, t] × R3 , and the | u j | are bounded from above on [0, t] × R3 , then there exists a constant C > 0 ∂t W L ∞ , and ∂x¯ W L ∞ , such that on [0, t], we depending only on the values of W, have ˙ L 2 ≤ E ≤ CW ˙ L2 . C −1 W
(2.6)
Remark 2.1. If q < 0, then J˙ 0 is no longer positive definite, and inequality (2.6) fails. p ∂ p 2 ρ+ p ), it is clear that the non-negativity of ∂∂ Since q = c ∂ ρ ( ρ plays a fundamental η
η
role in the well-posedness of the rE system. This explains the significance of Lemma 3.6 and Conjecture 2. Furthermore, we note that the conditions on the initial data guarantee def ˚ x) ¯ is uniformly positive. that q( ¯ = q(0, x) Furthermore, the Cauchy-Schwarz inequality for integrals, (2.4), and (2.6) imply that ˙ W)] ˙ ˙ ∂κ [J˙ κ (W, L 1 ≤ CW L 2 b L 2 ≤ CEb L 2 .
(2.7)
Using (2.5), the divergence theorem (assuming suitable fall-off conditions at infinity), and (2.7), we have that d d 2 ˙ W)] ˙ d x¯ E = c 0 E2 = c ∂κ [J˙ κ (W, 3 dt dx R ≤ CEb L 2 , which leads to the following energy estimate for the linearized system: d E(t) ≤ Cb L 2 . dt
(2.8)
250
J. Speck, R. M. Strain
The availability of (2.8) is the fundamental reason that the rE system has localin-time solutions belonging to a Sobolev space. This concludes our abbreviated discussion of the existence aspect of Theorem 1 in terms of W; we comment on the variables def V = (n, θ, u 1 , u 2 , u 3 ) near the end of the proof. Let us now assume that we have a local solution W to the nonlinear rE equations near the constant state W; we will make a few remarks about the time of existence. In the nonlinear case, one can define energies (with N ≥ 3 so that various Sobolev-Moser type inequalities are valid) def E2N (t) = J˙ 0 ∂α (W − W), ∂α (W − W) d x. ¯ 0≤| α |≤N
R3
Furthermore, all of the inhomogeneities that arise upon differentiating the rE equations are of quadratic order or higher. Consequently, near a constant state, the inhomogeneous terms analogous to the term b L 2 on the right-hand side of (2.8) can be bounded by CE2N (t), and the resulting energy estimate is d E N (t) ≤ CE2N (t). dt
(2.9)
Consequently, we may apply Gronwall’s inequality to (2.9) and use property (2.6) to deduce an a-priori estimate of the form W − W H N ≤ C x¯
˚ W−W HN x¯
˚ 1−CtW−W HN
.
(2.10)
x¯
This fact is a It follows from (2.10) that the time of existence is at least of size C/δ. consequence of a standard continuation principle that is available for hyperbolic PDEs (consult e.g. [33, Chap. 6] or [43] for the essential ideas), which implies that the solution exists as long as the a-priori energy estimates lead to the conclusion that W − W H N x¯ is sufficiently small. We remark that in the formula (2.10), C is a numerical constant that depends on an a-priori assumption concerning the subset of (0, ∞) × (0, ∞) to which the pair (η, p) belongs, and on B, where B > 0 is a fixed a-priori upper bound for W − W H N . These a-priori assumptions imply that the solution escapes neither x¯ the regime of hyperbolicity, nor a convex subset (see Remark 1.3) of the regime in which the map (n, z) → (H(n, z), P(n, z)) = (η, p) is invertible; on the time inter val [0, C/δ], these a-priori assumptions can be shown to hold through a bootstrap argument. Furthermore, once we have shown (2.10), this inequality can be translated into an inequality for the original variables V; Sobolev-Moser type estimates allow us to estimate (here we again use the convexity assumption discussed in Remark 1.3) V − V H N ≤ CW − W H N . This is possible because V can be written as a smooth x¯ x¯ function of W whenever (η, p) belongs to the domain of the inverse of the aforementioned map. 3. The Relativistic Boltzmann Equation In this section, we provide some additional background material on the relativistic Boltzmann equation. The rB equation is often expressed in the physics literature (see
Hilbert Expansion from Relativistic Boltzmann to Euler
251
e.g. [7,12]) in the Lorentz-invariant form of (1.1) where the collision kernel C is defined by c d Q¯ d Q¯ d P¯ ¯ Q| ¯ P¯ , Q¯ )[F( P¯ )G( Q¯ )− F( P)G( ¯ ¯ C(F, G) = W ( P, Q)]. 2 R3 Q 0 R3 Q 0 R3 P 0 In our analysis below, it will often be convenient to divide both sides of (1.1) by P 0 . We ˆ defined by therefore introduce the normalized velocity P, c def ¯ Pˆ = 0 P, P and Q(·, ·) = def
c C(·, ·). P0
Then (1.1) is equivalent to
1 ∂t F + Pˆ · ∂x¯ F = Q(F, F), (3.1) ε def where Pˆ · ∂x¯ = Pˆ 1 ∂1 + Pˆ 2 ∂2 + Pˆ 3 ∂3 . On the right-hand side of the above expression for C, the variables P and Q represent the pre-collisional four-momenta of a pair of particles, while P and Q represent their post-collisional four-momenta. We assume that the particle collisions are elastic, in which case the conservation of energy8 and 3-momentum can be expressed as (μ = 0, 1, 2, 3). (3.2) P μ + Q μ = P μ + Q μ , ¯ Q| ¯ P¯ , Q¯ ), which is a Lorentz scalar, can be expressed as The transition rate W ( P, ¯ Q| ¯ P¯ , Q¯ ) = sσ (, ϑ)δ (4) (P μ + Q μ − P μ − Q μ ), W ( P,
(3.3)
δ (4)
is a Dirac delta function, and σ (, ϑ) is the differential cross-section or where scattering kernel. The other quantities are defined in (1.12), (1.13), and (1.14). We have several remarks to make. First, the Lorentz invariance of the left-hand side of (1.1) is manifest, while the Lorentz invariance of the right-hand side follows from that of d P¯ ¯ Q¯ ); see (1.9). Next, we remark that ϑ in (1.13) is well-defined (and similarly for Q, P 0 under (3.2), but that it may not be in general [18]; i.e., in general, the right-hand side of (1.13) may be larger than 1 in magnitude. Finally, we point out that both the left and ¯ right-hand sides of (3.1) are functions of (t, x, ¯ P). 3.1. Expression for the collision operator. In this section, we provide an alternate expression (3.4) for the collision operator, which is derived from carrying out certain integrations in a center-of-momentum frame. This is the expression for the collision operator that we will use in our analysis for the remainder of the article. We remark that yet another expression for the collision operator was derived in [20]; see [47,48] for an explanation of the connection between the expression from [20] and the one in (3.4) below. One may use Lorentz transformations as described in [12] and in some detail in [47,49] to reduce the delta functions in (3.3), thereby obtaining ¯ ¯ ¯ Q(F, G) = dQ dω vø σ (, ϑ) [F( P¯ )G( Q¯ ) − F( P)G( Q)], (3.4) R3
S2
¯ Q), ¯ the Møller velocity, is given by (1.11). where vø = vø ( P, 8 In our inertial coordinate system, the “energy” of a four-vector is c times its 0 component, while its 3-momentum comprises its final 3 components.
252
J. Speck, R. M. Strain
The post-collisional 3-momenta in the expression (3.4) can be written as follows: ¯ ¯ P¯ + Q¯ ¯ ( P + Q) · ω , + ω + (γ − 1)( P¯ + Q) P¯ = ¯ 2 2 2 | P¯ + Q| ¯ · ω ( P¯ + Q) P¯ + Q¯ ¯ ¯ ¯ , − ω + (γ − 1)( P + Q) Q = ¯ 2 2 2 | P¯ + Q|
(3.5)
√ where γ = (P 0 + Q 0 )/ s, and · is the ordinary Euclidean dot product in R3 . The energies can be expressed as P 0 + Q0 ¯ + √ ω · ( P¯ + Q), 2 2 s P 0 + Q0 ¯ − √ ω · ( P¯ + Q). Q 0 = 2 2 s P 0 =
It is clear that P, Q, P , Q satisfy (3.2). Additionally, it is explained in [49] that after carrying out the integrations in a center-of-momentum frame, the scattering angle becomes ¯ Q, ¯ and satisfies cos ϑ = k ·ω, where ω ∈ S2 ⊂ R3 , and k = k( P, ¯ Q) ¯ ∈ a function of P, 2 3 ¯ ¯ S ⊂ R . We remark that an expression for k( P, Q) is given in [49, Eq. (5.37)] and [47], but that its precise form is not needed here.
3.2. Macroscopic quantities and conservation laws for rB. In this section, we study ¯ We also dismacroscopic quantities associated to a particle density function F(t, x, ¯ P). ¯ cuss conservation laws that hold whenever F(t, x, ¯ P) is a solution to the relativistic Boltzmann equation (1.1). This material is quite standard [12], but we include it for μν convenience. We begin by recalling that the energy-momentum tensor TBolt z [F] and the μ particle current I Bolt z [F] for the relativistic Boltzmann equation are defined in (1.20). ¯
Note that the term dPP0 on the right-hand side of (1.20) is exactly the canonical measure defined in (1.7). We now define the following macroscopic quantities:
¯ ¯ ln[h 3 F(t, x, ¯ − 1 dP, P μ F(t, x, ¯ P) ¯ P)] P0 R3 2 2 def κ λ −c n = gκλ I Bolt z I Bolt z , μ
S = −k B c def
uμ = def
κ
μ n −1 I Bolt z , 2
u κ u = −c , ρ= def
p
μν
= def
η= def
−2
κλ c u κ u λ TBolt z, μ κλ ν κ TBolt z λ , −c−2 n −1 u κ S κ .
(3.6a) (3.6b) (3.6c) (3.6d) (3.6e) (3.6f) (3.6g)
Here, S is the entropy four flow, n is the proper number density, u is the four-velocity, ρ is the proper energy density, p μν is the pressure tensor, η is the entropy per particle, def and μν = c−2 u μ u ν + (g −1 )μν is as defined in (1.26).
Hilbert Expansion from Relativistic Boltzmann to Euler
253
We also decompose the pressure tensor by defining the pressure p and the viscous pressure tensor v μν : p μν = pμν + v μν , def
where p= def
1 1 κλ κλ p κλ = κλ TBolt z. 3 3
(3.7)
We remark that κλ vκλ = 0. The following lemma will be useful in verifying numerous identities, especially those of Proposition 3.3. μ
Lemma 3.1. Let ν be a matrix (of real numbers) such that κμ λν gκλ = gμν ,
(μ, ν = 0, 1, 2, 3),
and such that det() = 1 (i.e. a proper Lorentz transformation). Consider the change on Tx M from one inertial coordinate system to another induced of coordinates P ← P def μ κ μ by : P = κ P . Then under this change of coordinates, I μ , u μ , and S μ transform μν as the components of a vector, while TBolt z and p μν transform as the components of a two-contravariant tensor. In an arbitrary coordinate system, these statements take the following form: I μ = μκ Iκ, μ μ κ u = κ u , μ μ κ S = κS ,
μν TBolt z μν
p
= =
(μ = 0, 1, 2, 3), (μ = 0, 1, 2, 3),
(3.8a) (3.8b)
(μ = 0, 1, 2, 3),
(3.8c)
κλ Bolt μκ νλ T z, μ ν κλ κ λ p ,
(μ, ν = 0, 1, 2, 3), (μ, ν = 0, 1, 2, 3).
(3.8d) (3.8e)
Proof. We give only the proof of (3.8a); the remaining statements follow similarly. Equation (3.8a) is equivalent to the following statement, where we abbreviate ¯ = F( P): ¯ F(t, x, ¯ P) ¯ ¯ μ 1 2 3 dP κ F(1α P α , 2β P β , 3γ P γ ) d P . (3.9) P F(P , P , P ) 0 = μκ P 0 ¯ R3 P P P∈ R3 Recall that Lorentz transformations preserve the form of the metricg, so that gμν = ¯ 2 and gμν = diag(−1, 1, 1, 1), which implies in particular that P 0 = m 20 c2 + | P| ¯ 2 . It thus follows from the discussion in Sect. 1.2, and in particular 0 = m 2 c2 + | P| P 0 ¯
¯
Eq. (1.9), that dPP0 = dP P0 . Equation (3.9) now follows from the standard change of variables formula for integration. The conservation laws for the relativistic Boltzmann equation are given in the following lemma. ¯ be a C 1 solution to (1.1). Then the conservation Lemma 3.2 [9, Chap. 2]. Let F(t, x, ¯ P) laws (1.21) hold. Note that (1.21) corresponds to the fluid conservation laws (1.22).
254
J. Speck, R. M. Strain
¯ In this section, we 3.3. Macroscopic quantities for a Maxwellian M = M(n, θ, u; P). prove the following proposition: Proposition 3.3. Let n(t, x) ¯ > 0, θ (t, x) ¯ > 0 be positive functions on M, and let u(t, x) ¯ be a future-directed vectorfield on M satisfying (1.24). Consider the corresponding local ¯ defined in (1.17) with (1.18). Then the following relativistic Maxwellian M(n, θ, u; P) relations hold for the quantities defined in (3.6a)–(3.7): I μ [M] = n[M]u μ , (μ = 0, 1, 2, 3), n[M] p[M] = k B n[M]θ = m 0 c2 , z[M] K 1 (z[M]) ρ[M] = 3 p[M] + m 0 c2 n[M] K 2 (z[M])
(3.10a) (3.10b) (3.10c)
p[M]
K 3 (z[M]) ! "# $ = m 0 c n[M] − k B n[M]θ, K 2 (z[M]) μν μν (3.10d) TBolt z [M] = T f luid [M], (μ, ν = 0, 1, 2, 3), K 1 (z[M]) −η K 2 (z[M]) 4 3 3 −3 exp z[M] n[M] = 4π e m 0 c h exp kB z[M] K 2 (z[M]) K K (z[M]) (z[M]) −η 2 3 3 3 −3 exp z[M] . = 4π m 0 c h exp kB z[M] K 2 (z[M]) (3.10e) 2
μν
By T f luid [M], we mean the energy-momentum tensor that results from inserting p[M] and ρ[M] into the expression on the right-hand side of (1.23). Proof. It is well-known that since u(x) is a future-directed vector satisfying (1.24), there exists an inertial coordinate system in which (u 0 , u 1 , u 2 , u 3 ) = (c, 0, 0, 0) (at the spacetime point x); such a frame is known as a “rest frame” for u. By Lemma 3.1, it suffices to check that (3.10a)–(3.10e) hold in such a rest frame. Using definition (1.20), it follows that I μ can be expressed in a rest frame for u as z −z P 0 d P¯ μ P exp . I [M] = cn m0c P0 4π m 30 c3 K 2 (z) R3 μ
(3.11)
By symmetry, it follows that I μ [M] is proportional to (1, 0, 0, 0), and therefore also to u. Let us denote the proportionality constant by A, so that in any inertial coordinate system, we have that I μ [M] = Au μ . We thus have that A = I 0 /u 0 . Furthermore, it follows from (3.11) that in a rest frame for u, we have that −z P 0 z ¯ d P. exp A=n m0c 4π m 30 c3 K 2 (z) R3
(3.12)
Hilbert Expansion from Relativistic Boltzmann to Euler
255
¯ and making We carry out the integration in (3.12) using spherical coordinates for P, ¯ = m 0 cz −1 (λ2 − z 2 )1/2 , the change of variables λ = z P 0 /(m 0 c). This implies that | P| ¯ = m 0 cz −1 λ(λ2 − z 2 )−1/2 , and we obtain d| P| ∞ 1/2 z 3 3 −3 −λ 2 2 λ A=n c z λe − z dλ. (3.13) 4π m 0 4π m 30 c3 K 2 (z) λ=z From the Bessel function identity (3.44) in the case j = 2, it follows that A = n, which completes the proof of (3.10a). ¯ 2, To prove (3.10b), we note that in a rest frame for u, it follows that P κ P λ κλ = | P| where μν is defined in (1.26). Inserting this formula into definition (3.7), using def¯ and using the integration inition (1.20), integrating with spherical coordinates for P, variable λ as above, we have that 1 z −z P 0 d P¯ 2 ¯ p[M] = cn | P| exp 3 4π m 30 c3 K 2 (z) R3 m0c P0 ∞ 3/2 z 1 4 4 −4 −λ 2 2 λ = cn c z e − z dλ. (3.14) 4π m 0 3 4π m 30 c3 K 2 (z) λ=z 2
Referring to definition (3.43), it follows that p[M] = n m 0zc , which proves (3.10b). To prove (3.10c), we note that in a rest frame for u, u κ u λ P κ P λ = c2 (P 0 )2 . Inserting this formula into definition (3.6e), carrying out the integration in spherical P¯ coordinates, using the change of variables λ as above, referring to definition (3.43), and using the recursion formula (3.45) in the case j = 2, we have that z −z P 0 0 d P¯ ρ[M] = cn P exp m0c 4π m 30 c3 K 2 (z) R3 λ=∞ z 4 4 −4 = cn c z e−λ λ2 (λ2 − z 2 )1/2 dλ 4π m 0 4π m 30 c3 K 2 (z) λ=z λ=∞ z 4 4 −4 e−λ (λ2 − z 2 )3/2 dλ = cn 4π m 0 c z 4π m 30 c3 K 2 (z) λ=z λ=∞ z 4 4 −2 c z e−λ (λ2 − z 2 )1/2 dλ +cn 4π m 0 4π m 30 c3 K 2 (z) λ=z 3K (z) (z) z K 2 1 = m 0 c2 n + K 2 (z) z2 z K 1 (z) K 3 (z) (3.15) = 3nk B θ + m 0 c2 n = m 0 c2 n[M] − k B n[M]θ. K 2 (z) K 2 (z) Combining this identity with (3.10b), we deduce (3.10c). To prove (3.10d), we first note that in a rest frame for u, we have that μν
T f luid [M] = diag(ρ[M], p[M], p[M], p[M]), and μν TBolt z [M]
z −z P 0 d P¯ μ ν = cn P P exp . m0c P0 4π m 30 c3 K 2 (z) R3
(3.16)
256
J. Speck, R. M. Strain μν
The fact that μ = ν ⇒ TBolt z [M] = 0 follows from symmetry. The fact that 00 [M] = ρ[M] follows directly from comparing the integral expressions (3.15) TBolt z jj
and (3.16), while the fact that TBolt z [M] = p[M] (there is no summation in j here) follows from comparing the integral expressions (3.14) and (3.16) using symmetry. To prove (3.10e), we notice that in a rest frame for u, we have that u κ P κ = −c P 0 . Inserting this formula into definitions (3.6a) and (3.6g), and evaluating the two integrals that arise as in (3.13) and (3.15), we have that η[M] =
=
=
=
z −z P 0 (−k B n )n exp m0c 4π m 30 c3 K 2 (z) R3 & % ' ( 0 −z P z 3 + ln h n × − 1 d P¯ m0c 4π m 30 c3 K 2 (z) z −z P 0 z P 0 d P¯ kB exp m0c 4π m 30 c3 K 2 (z) m 0 c R3 % & ' ( z z −z P 0 3 d P¯ exp ln h n −1 −k B m0c 4π m 30 c3 K 2 (z) 4π m 30 c3 K 2 (z) R3 z K 1 (z) z 4 4 3K 2 (z) kB + 4π m 0 c z2 z 4π m 30 c3 K 2 (z) m 0 c % & ' ( z K 2 (z) z 3 −k B ln h n − 1 4π m 30 c3 3 3 3 3 z 4π m 0 c K 2 (z) 4π m 0 c K 2 (z) ' ( % & K 1 (z) z − k B ln h 3 n −1 . (3.17) kB 3 + z K 2 (z) 4π m 30 c3 K 2 (z) −1
Equation (3.10e) now follows from (3.17) and simple algebraic manipulation (solve for n). The next lemma was used in Sect. 2 to derive an equivalent version of the rE system; i.e., Eq. (1.31a)–(1.31e). More specifically, only Eq. (3.19) was used. However, as an aside, we also discuss the fundamental thermodynamic relation (3.18) (see [10]), which can, in consideration of the positivity of n and θ, be used to show that η can be written as a smooth function of n, ρ. Proposition 3.4. Assume that the functional relations (3.10b), (3.10c), and (3.10e) hold for the variables p, ρ, n, θ, and η. Then the additional relations also hold: ∂ρ nθ = , (3.18) ∂η n ∂ρ ρ+p=n . (3.19) ∂n η Proof. To ease the notation, we use the notation (1.18) and abbreviate K j = d K j (z), K j = dz K j (z), where K j (z) is the Bessel function defined in (3.43). To begin the proof of (3.18), we first note that by the chain rule, it follows that
Hilbert Expansion from Relativistic Boltzmann to Euler
257
∂ρ ∂ρ ∂z = . ∂η n ∂z n ∂η n
(3.20)
We claim that Eq. (3.20) leads to the following identities: ( % K 3 − K 2−1 K 2 K 3 + z −2 K 2 ∂ρ nm 0 c2 = ∂η n kB z z −1 K 2 − z −2 K 2 + K 3 + z −1 K 3 − K 2−1 K 2 K 3 # $! " = nθ,
=1
(3.21)
which completes the proof of (3.18). To see that (3.21) holds, we note that differentiating the last equality in (3.10c) leads to the relation ) * K K K ∂ρ 1 3 3 = m 0 c2 n − 22 + 2 , (3.22) ∂z n K2 z K2 while differentiating each side of (3.10e) with respect to η (while n is held constant) leads to the following identity: , ∂z + −1 −1 −2 −1 −1 z − k −1 K − z K + K + z K − K K K 0= 2 3 3 2 3 2 2 B z K 2 (z). ∂η n
(3.23) Inserting (3.22) and (3.23) into the left-hand side of (3.21) implies the first equality. The second equality in (3.21) follows from (3.46) in the case j = 2, which implies that the term above the under-braces is equal to 1, and from the definition of z. ∂ρ ∂ρ The proof of (3.19) follows similarly using the chain rule identity ∂n = ∂n + η z ∂ρ ∂z ∂z ∂n η , and we omit the calculations. n
3.4. The invertibility of the maps H(n, z) and P(n, z). In this short section, we state and prove Lemma 3.5, which addresses the issue of solving for (n, θ ) in terms of (η, p). This lemma rigorously shows that Conjecture 1 is true outside of a compact set of θ values. Furthermore, at the end of the section, we provide Fig. 1, which is our numerical evidence for the validity of the conjecture. For the purposes of avoiding repetition, during the proof of Lemma 3.5, it is convenient to use notation that is defined below in the proof of Lemma 3.6. However, logically speaking, the proof of Lemma 3.5 comes before the proof of Lemma 3.6. Lemma 3.5. Consider the smooth maps η = H(n, z) and p = P(n, z) defined in (1.29a) and (1.29b) respectively. Then the map (n, z) → (H(n, z), P(n, z)) is invertible with smooth inverse if 0 < z ≤ 1/10 or z ≥ 70. Proof. Using (1.27a) and (1.27c), it follows that p exp kηB is a smooth function of z alone. Therefore, by the implicit function theorem, since p > 0 holds whenever z > 0, we can locally solve for z in terms of η, p if ∂∂zp = 0. We will show that ∂∂zp < 0 holds for 0 < z ≤ 1/10 and z ≥ 70.
η
η
258
J. Speck, R. M. Strain
Fig. 1. z 5 ∂∂zp plotted as a function of z η
We begin by quoting Eq. (3.31), which states that ∂ z |η p K 1 (z) K 1 (z) 2 4 =3 +z −z− . p K 2 (z) K 2 (z) z
(3.24)
Using (3.24) and the expansions (3.33)–(3.34), it follows that z
∂ z |η p = −4 + z 2 /2 + 3z1 (z) + z 4 /4 + z 2 2 (z). p
Thus, using the bounds (3.35), we obtain that z
∂ z |η p < −3, p
(0 < z ≤ 1/10).
Since p > 0, it follows that ∂z |η p < 0 whenever 0 < z ≤ 1/10 as desired. On the other hand, using the expansions (3.38) and (3.39), it follows that z
∂ z |η p 45 = −5/2 + + 3z 1 (z) + z 2 2 (z). p 8z
Therefore, using the bounds (3.40), it follows that z
∂ z |η p < −1, (z ≥ 70). p
Therefore, by the implicit function theorem, we can solve for z in terms of η, p if −2 pz from (1.27a), the same is true of n. This 0 < z ≤ 1/10 or z ≥ 70. Since n = m −1 0 c completes the proof of Lemma 3.5.
Hilbert Expansion from Relativistic Boltzmann to Euler
259
Remark 3.1. It is clear from the proof of the lemma that Conjecture 1 can be shown by ∂p demonstrating the negativity of ∂z for all z > 0. Thus, the numerical plot in Fig. 1, η
which was created with Maple 11.0, is the motivation for our conjecture.
3.5. Regimes of hyperbolicity for the rE system and the existence of the kinetic equation of state. In this section, we prove that whenever θ is sufficiently small and positive or sufficiently large, there exists an equation of state of the form (1.28), i.e., of the form p = f kinetic (η, ρ). Furthermore, under the same temperature assumptions, we show ∂ f kinetic that the equation of state satisfies 0 < ∂ρ < 1. We remark that ∂ fkinetic ∂ρ can be η
η
expressed as a function of θ alone. As previously discussed, this condition is sufficient to ensure the hyperbolicity of the rE system in these temperature regimes; in particular, ∂ f kinetic as discussed in Remark 2.1, the condition 0 < ∂ρ plays a fundamental role in η
the proof of local existence. Our result rigorously shows that outside of a compact set of θ values, a slightly weaker version of Conjecture 2 holds. Furthermore, at the end of this section, we provide Fig. 2, which is our numerical evidence for the validity of the conjecture. Also see the discussion at the end of Sect. 1.6. For convenience, we use the variable z from (1.18) during the statement and proof of the lemma. Lemma 3.6 (Hyperbolicity of the rE system). Assume that the functional relations (3.10b), (3.10c), and (3.10e) hold for the macroscopic variables n, θ, η, p, and ρ. Then if 0 < z ≤ 1/10 or z ≥ 70, p can be expressed as a smooth function f kinetic of η and ρ : p = f kinetic (η, ρ).
Fig. 2. ∂∂ρp plotted as a function of z η
260
J. Speck, R. M. Strain
Furthermore, the following estimate holds for 0 < z ≤ 1/10 : ∂p − 1 ≤ z2. ∂ρ η 3 Additionally, the following estimate holds for z ≥ 70 : ∂ρ − 3z ≤ 41. ∂p η 5
(3.25)
(3.26)
Remark 3.2. We did not attempt to be optimal in our estimate of the error terms on the right-hand sides of the above inequalities. Proof. It follows from (3.10b), (3.10c), and (3.10e) that −η K 2 (z) K 1 (z) p = 4π e4 m 40 c5 h −3 exp , (3.27) exp z kB z2 K 2 (z) K 1 (z) ρ= p z +3 . (3.28) K 2 (z) ∂ | p We use the following version of the chain rule: ∂∂ρp η = ∂zz |ηη ρ . Using (3.28), we further deduce that ∂ z |η ρ p d K 1 (z) K 1 (z) =3+z + z . (3.29) ∂ z |η p K 2 (z) ∂z |η p dz K 2 (z) We then use the identities z K 1 = K 1 − z K 2 and K 2 = −2z −1 K 2 − K 1 , which follow from (3.45) and (3.46), to compute that d K 1 (z) K 1 (z) K 1 (z) 2 z =4 +z − z, (3.30) dz K 2 (z) K 2 (z) K 2 (z) ∂ z |η p K 1 (z) K 1 (z) 2 4 =3 +z (3.31) −z− . p K 2 (z) K 2 (z) z Combining (3.29), (3.30), and (3.31), we have that 2 K 1 (z) K 1 (z) 4 + z −z ∂ z |η ρ K 2 (z) K 2 (z) K 1 (z) =3+z + . 2 ∂ z |η p K 2 (z) K 1 (z) K 1 (z) 4 3K + z − z − K 2 (z) z 2 (z)
(3.32)
Using Corollary 3.8, we can write z K 1 (z) = + 1 (z), K 2 (z) 2 K 1 (z) 2 z2 = + 2 (z), K 2 (z) 4
(3.33) (3.34)
where for 0 < z ≤ 1/10, we have the estimates |1 (z)| ≤ 2z 2 ,
|2 (z)| ≤ 2z 3 .
(3.35)
Hilbert Expansion from Relativistic Boltzmann to Euler
261
Inserting these expansions into (3.32) and multiplying the numerator and denominator of the fraction by z, we have that ∂ z |η ρ z2 z 2 + 4z1 + z 4 /4 + z 2 2 =3+ + z1 + 2 . ∂ z |η p 2 z /2 + 3z1 + z 4 /4 + z2 − 4
(3.36)
Using the bounds (3.35), it follows that for 0 < z ≤ 1/10, the second term on the right-hand side of (3.36) (i.e. the z 2 /2 term) partially cancels the last term (i.e., the large fraction which is negative), which implies that 7z 2 ∂ | ρ zη − 3 ≤ , ∂ z |η p 10
(0 < z ≤ 1/10).
(3.37)
The facts that p can be expressed as a smooth function of η and ρ whenever 0 < z ≤ 1/10, and that inequality (3.25) is verified both easily follow from (3.37). To prove (3.26), we again use Corollary 3.8 to write K 1 (z) = 1− K 2 (z) K 1 (z) 2 = 1− K 2 (z)
3 15 + 2 + 1 (z), 2z 8z
(3.38)
3 6 + + 2 (z), z z2
(3.39)
40 . z3
(3.40)
where for z ≥ 10, we have that | 1 (z)| ≤
16 , z2
| 2 (z)| ≤
Inserting these expansions into (3.32) and multiplying the numerator and denominator of the fraction by z, it follows that ∂ z |η ρ 3z 3 15 = + + + z 1 + ∂ z |η p 5 2 8z
9 4
+
15 2z
+ 4z 1 + 65 z 2 1 + z 2 2 + 25 z 3 2 −5/2 +
45 8z
+ 3z 1 + z 2 2
.
(3.41)
Using the bounds (3.40) and the expression (3.41), it can be checked that for z ≥ 70, we have that ∂ | ρ 3z zη − ≤ 41. ∂ z |η p 5
(3.42)
The facts that p can be expressed as a smooth function of η and ρ whenever z ≥ 70, and that inequality (3.26) is verified, both easily follow from (3.42). Remark 3.3. Notice that Conjecture 2 is equivalent to the conjecture that the right-hand side of (3.32) is > 3 for all z > 0. In Fig. 2, we present a numerical plot, which was created with Maple 11.0, that covers the set of z values lying outside of the scope of Lemma 3.6, and that suggests that this conjecture is true. Note that the inequalities of the conjecture are stronger than those proved in the lemma.
262
J. Speck, R. M. Strain
3.6. Bessel function identities and inequalities. We now state the technical lemma that contains the Bessel function properties that we have used throughout this article. The expansion (3.47) (including the error terms) and inequality (3.48) can be found in [40]. The remaining identities can be found in [12, Chap. 2]. Lemma 3.7 (Properties of Bessel functions). Let K j (z) be the Bessel function defined by λ=∞ j def (2 ) j! 1 K j (z) = e−λ (λ2 − z 2 ) j−(1/2) dλ, ( j ≥ 0). (3.43) (2 j)! z j λ=z Then the following identities hold: 2 j−1 ( j − 1)! 1 λ=∞ −λ 2 K j (z) = λe (λ − z 2 ) j−(3/2) dλ, ( j > 0), (2 j − 2)! z j λ=z K j (z) K j+1 (z) = 2 j + K j−1 (z), ( j ≥ 1), z also d dz
(3.45)
K j+1 (z) , ( j ≥ 0), (3.46) zj ) * n−1 π −z −n −m e K j (z) = A j,m z γ j,n (z)z + , ( j ≥ 0, n ≥ 1), (3.47) 2z
K j (z) zj
(3.44)
=−
m=0
where the following additional identities and inequalities also hold: A j,0 = 1, A j,m =
(4 j 2 − 1)(4 j 2 − 32 ) · · · (4 j 2 − (2m − 1)2 ) , ( j ≥ 0, m ≥ 1), m m!8
|γ j,n (z)| ≤ 2 exp [ j 2 − 1/4]z −1 |A j,n |, ( j ≥ 0, n ≥ 1),
(3.48)
K j (z) < K j+1 (z), ( j ≥ 0). The following corollary of Lemma 3.7 is used in the proof of Lemma 3.6. Corollary 3.8. For 0 < z ≤ 1/10, the following inequalities hold: K (z) z 1 − ≤ 2z 2 , K 2 (z) 2 K (z) 2 z 2 1 − ≤ 2z 3 . K 2 (z) 4 For z ≥ 10, the following inequalities hold: K (z) 3 15 1 −1+ − 2 ≤ K 2 (z) 2z 8z K (z) 2 6 3 1 −1+ − 2 ≤ K 2 (z) z z
(3.49a) (3.49b)
16 , z3
(3.50a)
40 . z3
(3.50b)
Hilbert Expansion from Relativistic Boltzmann to Euler
263
Proof. We remark that throughout the proof, we make no attempt to be optimal in our . λ=∞ estimates. Using (3.43) in the case j = 1, and the fact that λ=0 λe−λ dλ = 1, it follows that ( % λ=z λ=∞ z 2 −λ −λ z K 1 (z) − 1 = − e λ dλ + λe 1− − 1 dλ. λ λ=0 λ=z The first integral is trivially bounded in magnitude by z 2 /2. Using the fact that 1 − (z/λ)2 − 1 ≤ (z/λ)2 on the domain 0 ≤ z/λ ≤ 1, it follows that the second integral is bounded in magnitude by λ=∞ z e−λ dλ ≤ z. λ=0
We therefore conclude that z 1 K 1 (z) − ≤ 1 + . z 2
(3.51)
Using (3.44) in the case j = 2 and similar arguments, which we leave to the reader, we also conclude that z 2 (3.52) K 2 (z) − 2 ≤ 1 + . z 3 Using (3.51) and (3.52), together with simple algebraic estimates, it follows that for 0 ≤ z ≤ 1/10, we have K (z) z 1 − ≤ 2z 2 . K 2 (z) 2 This proves (3.49a). Inequality (3.49b) follows from similar reasoning; we leave the details to the reader. To prove (3.50a), we first decompose K 1 (z) 1+ A B3 2 = = (1 + A) 1 − B + B − , (3.53) K 2 (z) 1+ B 1+ B where 15 3 γ1,3 − + 3 , 8z 128z 2 z 105 15 γ2,3 + B= + 3 , 8z 128z 2 z A=
(3.54a) (3.54b)
and the γ j,n are from (3.47). For the remainder of the proof, we will now assume that z ≥ 10; all of our estimates will hold on this domain. Now using (3.47), it can be checked that the following inequalities hold: |γ1,3 | ≤
1 , 4
|γ2,3 | ≤ 1.
(3.55)
264
J. Speck, R. M. Strain
Consequently, it is easy to check that the following estimates hold: 1 ≤A 4z 1 ≤B z 15 B − 8z 2 15 2 B − 8z
1 , 2z 2 ≤ , z 1 ≤ 2, z 4 ≤ 3. z ≤
(3.56a) (3.56b) (3.56c) (3.56d)
Using simple algebraic calculations, it follows from the expansions (3.53), (3.54a), and (3.54b) that K 1 (z) 3 15 =1− + + O(z −3 ). K 2 (z) 2z 8z 2
(3.57)
In (3.57), the symbol O(z −3 ) denotes the cubic (in z −1 ) and higher-order terms that arise K 1 (z) in the expansion of K . We now estimate this O(z −3 ) term by using the expansions 2 (z) (3.53)–(3.54b) to split it into the following 3 pieces: 15 γ1,3 B3 2 −B + B − , (3.58a) I = − + 3 128z 2 z 1+ B 3 15 B3 II = − B− + B2 − , (3.58b) 8z 8z 1+ B 2 γ1,3 γ2,3 15 B3 2 III = 3 − 3 + B − . (3.58c) − z z 8z 1+ B 3
B It is easy to see by sign considerations (i.e., using B > 0) that | − B + B 2 − 1+B | ≤ |B|. Using also (3.55) and (3.56b), we conclude that the following inequality holds: 15 15 2 3 γ1,3 1 ≤ 3. |I | ≤ − + 3 |B| ≤ + 2 (3.59) 2 2 128z z 128z 4z z 4z
For the term I I, we use similar sign considerations, together with the estimates (3.56b) and (3.56c) to conclude that 15 3 1 2 4 3 ≤ 3. + (3.60) |I I | ≤ B − + |B|2 ≤ 8z 8z 8z z 2 z 2 z Finally, for the term I I I, we use the fact that B > 0, together with (3.55), (3.56b), and (3.56d) to conclude that % ( 2 γ γ 15 1 4 8 53 1 1,3 2,3 2 3 |I I I | ≤ 3 + 3 + B − + |B| ≤ 3 + 3 + 3 + 3 ≤ 3 . z z 8z 4z z z z 4z (3.61) Adding (3.59), (3.60), and (3.61), we arrive at (3.50a). Inequality (3.50b) can be shown directly from (3.50a); we omit the details.
Hilbert Expansion from Relativistic Boltzmann to Euler
265
3.7. The Hilbert expansion. In this section, we perform a Hilbert expansion for the rB equation (3.1). We decompose the solution F ε as the sum (1.2) where F0 , F1 , . . . , F6 in (1.2) will be independent of ε. Also, FR;ε is called the remainder term; it will depend upon ε. Our main goal in this section is to explain how one can prove Proposition 3.9, which summarizes the behavior of F0 , F1 , . . . , F6 ; the remainder term FR;ε is analyzed in detail in the next section. We begin by inserting the expansion (1.2) into (3.1) to obtain 6
ε (∂t + Pˆ · ∂x¯ )Fk + ε3 (∂t + Pˆ · ∂x¯ )FR;ε = k
k=0
6
εi+ j−1 Q(Fi , F j )
k=0 i+ j=k 0≤i, j≤6
+ε5 Q(FR;ε , FR;ε ) +
6
ε2+k Q(FR;ε , Fk ) + Q(Fk , FR;ε ) + A.
k=0
def i+ j−1 Q(F , F ). Equating like powers of ε on each Above A = 12 i j k=7 i+ j=k,1≤i, j≤6 ε side of the equation, we obtain the following system: 0 = Q(F0 , F0 ),
(3.62)
∂t F0 + Pˆ · ∂x¯ F0 = Q(F0 , F1 ) + Q(F1 , F0 ), ∂t F1 + Pˆ · ∂x¯ F1 = Q(F0 , F2 ) + Q(F2 , F0 ) + Q(F1 , F1 ), .. . ˆ Q(Fi , F j ), ∂t F5 + P · ∂x¯ F5 = Q(F0 , F6 ) + Q(F6 , F0 ) +
(3.63)
i+ j=6 1≤i, j≤6
while the remainder satisfies the equation 1 ∂t FR;ε + Pˆ · ∂x¯ FR;ε − Q F0 , FR;ε + Q FR;ε , F0 = ε2 Q(FR;ε , FR;ε ) ε 6 + εi−1 Q(Fi , FR;ε ) + Q(FR;ε , Fi ) + ε2 A, (3.64) i=1
with
def A = −ε ∂t F6 + Pˆ · ∂x¯ F6 + εi+ j−6 Q(Fi , F j ). i+ j>6 1≤i, j≤6
By (1.19), Eq. (3.62) implies that F0 must be a relativistic local Maxwellian ¯ = M = M(n(t, x), ¯ as in (1.17). Consequently, the F0 (t, x, ¯ P) ¯ θ (t, x), ¯ u(t, x); ¯ P), remaining equations in (3.63) and below involve the linear operator: L(h) = − {Q (h, M) + Q (M, h)} . def
We remark that L is an integral operator involving only the momentum space variables. Furthermore, L is a linear Fredholm operator that can be inverted as long as the inhomogeneity (i.e., the terms in (3.63), and the equations below it, which are not
266
J. Speck, R. M. Strain
of the form L(Fi )) is perpendicular to the five dimensional null space of the adjoint ¯ P 0 }. This null space can operator L † : Null(L † ) = span{φ1 , . . . , φ5 } = span{1, P, be seen easily from the standard pre-post change of variables. In the preceding discussion, the notion of perpendicular and adjoint is the one corresponding to the usual L 2 momentum space inner product defined in (1.5). The operator L has the null space ¯ MP 0 }. Null(L) = span{M, M P, The aforementioned perpendicularity conditions can be checked by direct calculation, so that Eq. (3.63) and the one below it imply that /
φi , ∂t F0 + Pˆ · ∂x¯ F0
0
= 0, (i = 1, . . . , 5), (3.65) F1 = −L −1 ∂t F0 + Pˆ · ∂x¯ F0 + 1 , / 0 φi , ∂t F1 + Pˆ · ∂x¯ F1 − Q(F1 , F1 ) = 0, (i = 1, . . . , 5), (3.66) P¯ F2 = −L −1 ∂t F1 + Pˆ · ∂x¯ F1 − Q(F1 , F1 ) + 2 . P¯
Above 1 and 2 are elements of the null space of L (i.e., L(1 ) = L(2 ) = 0). Applying the operator ∂μ to each side of (1.20), differentiating under the integral, and using (3.10d), it follows that Eq. (3.65) implies that the relativistic Euler equations (1.22) are verified by the macroscopic quantities n[M], θ [M], and u[M] corresponding to the Maxwellian M = F0 . This explains the fact that in order to initiate the Hilbert expansion, we solve (with the help of Theorem 1) for a smooth solution to the rE system using the variables (n, θ, u). As discussed in Cercignani-Kremer [9, Sect. 5.5], the parameters in the expansion of 1 in terms of the basis {M, MP 0 , MP 1 , MP 2 , MP 3 } satisfy a linearized inhomogeneous version of the relativistic Euler equations; with the help of the expression for F1 in (3.65), this enables us to find a solution F1 to Eq. (3.63). Furthermore, the higher order correction terms F2 , F3 , . . . , F6 can be solved for in the same way, where the corresponding inhomogeneous terms in the linearized relativistic Euler equations depend upon the previous terms in the expansion. We refer to Cercignani-Kremer [9, Sect. 5.5] for more details on these terms in the expansion. We remark that a careful treatment of the non-relativistic Hilbert expansion is found in [8,24,25]. In particular, we are using the argument from [8]. These arguments carry over directly (once one identifies the null spaces in the relativistic case, as we have done above). We will use the following results, which are not studied in detail here. First, the terms ¯ F1 , . . . , F6 are smooth in (t, x), ¯ and they also have decay in the momentum variables, P. Consider, for example, F1 . For the non-relativistic version of L, Grad [25] and Caflisch ¯ Their argument carries over [8] argue that L −1 preserves decay in the momentum P. directly to our case of the relativistic Boltzmann equation as follows. We can combine the argument in [8,25] with the relativistic estimates (1.15), Lemma 3.13, Lemma 3.14, and arguments as in [52], to see that indeed L −1 leads preserves momentum decay. This −1 q ˆ ∂t F0 + P · ∂x¯ F0 decays at infinity as fast as M for any to the conclusion that −L 0 < q < 1. Thus, F1 will decay as fast as Mq , and F1 is smooth in (t, x) ¯ since the parameters in the expansion of 1 in terms of M, MP 0 , MP 1 , MP 2 , MP 3 solve linear equations with forcing terms coming from the smooth functions n, θ, u. This argument is similar for the higher order terms in the expansion. The next proposition summarizes the estimates that we use in the next section.
Hilbert Expansion from Relativistic Boltzmann to Euler
267
Proposition 3.9. Let (n(t, x), ¯ θ (t, x), ¯ u(t, x)) ¯ be a smooth solution (see Remark 1.8) of the rE equations (1.22) on a time interval [0, T ] × R3x¯ . Form the relativistic Maxwellian ¯ as in (1.17). Then the terms F1 , . . . , F6 of the Hilbert expansion F0 = M(n, θ, u; P) are smooth in (t, x) ∈ [0, T ] × R3x¯ and for any 0 < q < 1, they have momentum decay given by F j (t, x, ¯ ≤ C(q)Mq (n(t, x), ¯ ¯ P) ¯ θ (t, x), ¯ u(t, x); ¯ P), ( j = 1, 2, . . . , 6). ¯ The constants in this bound are independent of (t, x, ¯ P). 3.8. Relativistic Boltzmann estimates. In this section we prove our main result, Theorem 2. Using Theorem 1, we may assume that there is a sufficiently smooth solution (n, θ, u) to the relativistic Euler equations satisfying all of the desired properties in Theorem 2. ¯ We can then construct the local relativistic Maxwellian M(n(t, x), ¯ θ (t, x), ¯ u(t, x); ¯ P) as in (1.17). After the analysis of Sect. 3.7, the main point left is to estimate solutions to the equation for the remainder (3.64). We will outline the main strategy for these estimates after the statements of Lemma 3.10 and Lemma 3.11 below. It will be useful to express the remainder as √ def f ε = FR;ε / M. (3.67) We use the notation f 2 = f L 2 (R3 ×R3 ) throughout this section. We define the x¯ P¯ linearized relativistic Boltzmann collision operator around M by √ √ def L(h) = −M−1/2 Q Mh, M + Q M, Mh . def
We also define a nonlinear operator by (h, f ) = M−1/2 Q def
√
Mh,
√
Mf .
(3.68)
We recall the notation from Sect. 1.1. We further define the weighed L 2 (R3x¯ × R3P¯ ) “dissipation” norm by 2 def ¯ |h(x, ¯ 2. hν = d x¯ d P¯ ν( P) ¯ P)| R3x¯
R3P¯
def ¯ is given by (1.15). Recall the ¯ = ν(J )( P), Above the the “collision frequency”, ν( P) def def ¯ = ¯ Furthermore, w1 ( P). weight function (1.6). We will sometimes write w = w( P) def ¯ ¯ ¯ P)/ J ( P). (3.69) h ε = FR;ε (t, x,
It will then be sufficient to estimate f ε 2 (t) and h ε ∞, (t) to conclude Theorem 2. We prove the needed estimates in Lemma 3.10 and Lemma 3.11 just below. Let P denote the orthogonal L 2 (R3P¯ ) projection with respect to the null space of the linear operator L, which is √ √ √ √ √ M, P¯ 1 M, P¯ 2 M, P¯ 3 M, P 0 M .
268
J. Speck, R. M. Strain
We know from e.g. [18,20,46] that there exists a number δ0 > 0 such that Lh, h P¯ ≥ δ0 {I − P}h2ν .
(3.70)
We will furthermore use the following L 2 - L ∞ estimates. Lemma 3.10 (L 2 Estimate). We consider a smooth solution (see Remark 1.8) ¯ θ (t, x), ¯ u(t, x)) ¯ to the relativistic Euler equations (1.22) generated by The(n(t, x), ¯ f ε , h ε be defined in (1.17), (3.67), and (3.69) respectively, orem 1. Let M(n, θ, u; P), and let δ0 > 0 be as in the coercivity estimate (3.70). Then there exist constants ε0 > 0 and C = C(M, F0 , F1 , . . . , F6 ) > 0, such that for all ε ∈ (0, ε0 ) we have √ d δ0 f ε 22 (t) + {I − P} f ε 2ν (t) ≤ C{ εε3/2 h ε ∞, (t) + 1} f ε 22 + f ε 2 . dt 2ε Above and below the constant C(M, F0 , F1 , . . . , F6 ) depends upon the L 2 norms and the L ∞ norms of the terms M, F0 , F1 , . . . , F6 as well as their first derivatives. Lemma 3.11 (L ∞ Estimate). Under the assumptions of Lemma 3.10, there exists ε0 > 0 and a positive constant C = C(M, F0 , F1 , . . . , F6 ) > 0, such that for all ε ∈ (0, ε0 ) and for any ≥ 9 we have % ( sup ε3/2 h ε ∞, (s) ≤ C ε3/2 h 0 ∞, + sup f ε 2 (s) + ε7/2 .
0≤s≤T
0≤s≤T
As we will soon explain, these two lemmas together imply Theorem 2. These estimates are motivated by the L 2 − L ∞ framework from [31]. Similar lemmas have also been used to study the non-relativistic Hilbert expansion in [32]. Indeed, the short proof of Theorem 2 is essentially extracted from [32], modulo these lemmas. The main strategy is as follows. We first control the remainder equation (3.64) as in Lemma 3.10 using the L 2 energy estimates from [46] such as Lemma 3.12 below. To finish the proof of Theorem 2, we also need L ∞ estimates such as those in Lemma 3.11. These can be proven using Duhamel’s principle (3.74) and further estimates from [46] such as Lemmas 3.13, 3.14, and 3.15 below. In this framework, the key idea is to control the solution in L ∞ by the L 2 norms of the solution and the L ∞ norm of the initial data. Proof of Theorem 2. We will use the main estimates in Lemma 3.10 and Lemma 3.11. After applying the standard Gronwall inequality to the differential inequality in Lemma 3.10, we obtain f ε 22 (t) + 1 ≤ C f ε 22 (0) + 1 eCta(T,ε) , where a(T, ε) = def
√
ε sup ε3/2 h ε ∞, (t) + 1. 0≤t≤T
By Lemma 3.11, a(T, ε) b(T, ε) with ) * def √ 3/2 ε 7/2 b(T, ε) = ε ε h 0 ∞, + sup f (s)2 + ε + 1. 0≤s≤T
Hilbert Expansion from Relativistic Boltzmann to Euler
269
We have shown f ε 2 (t) ≤ C ( f 0 2 + 1) eCtb(T,ε) . Notice that e x ≤ C(1 + x), if 0 ≤ x ≤ 1. Therefore, for ε sufficiently small, on some short time interval, we have % ) *( √ ε 3/2 ε f 2 (t) ≤ C ( f 0 2 + 1) 1 + ε ε h 0 ∞, + sup f (s)2 . 0≤s≤T
Hence, there exists ε0 > 0 such that for 0 ≤ ε ≤ ε0 we may conclude sup f ε (s)2 ≤ C T 1 + f 0 2 + ε3/2 h 0 ∞, . 0≤s≤T
This procedure works for some short time interval, and then the inequality above follows in general by a continuity argument. This last estimate and Lemma 3.11 together imply the main estimate in Theorem 2. For the remainder of this paper, we will discuss the proofs of Lemmas 3.10 and 3.11. To this end, we will use the following nonlinear estimate. Lemma 3.12. For any ≥ 9, we have the following estimate for the collision operator (3.68): (h 1 , h 2 ), h 3 ¯ ≤ Ch 3 ∞, h 2 2 h 1 2 . (3.71) P Furthermore, if χ is any rapidly decaying function, then we have (h 1 , χ ), h 3 ¯ + (χ , h 1 ), h 3 ¯ ≤ Ch 3 ν h 1 ν . P P
(3.72)
In the proof of this lemma below, we only require that χ satisfies the rapid decay ¯ ≤ C(P 0 )−m , with m > 3. On the other hand, in our applications condition χ ( P) below, we will consider smooth functions χ with exponential decay, which is a property possessed by the relativistic Maxwellians defined in (1.17). Proof of Lemma 3.12. We notice from (3.68) and (1.39) that 0 ¯ | (h 1 , h 2 )| dQ dω vø σ (, ϑ) e−α Q h 1 ( P¯ )h 2 ( Q¯ ) R3 S2 def 0 ¯ ¯ 2 ( Q) ¯ = + dQ dω vø σ (, ϑ) e−α Q h 1 ( P)h I + I I. R3
S2
We estimate first the piece without post-collisional velocities, denoted I I above: 0 I I, h 3 ¯ ¯ 2 ( Q)h ¯ 3 ( P) ¯ d P¯ d Q¯ dω vø σ (, ϑ) e−α Q h 1 ( P)h P
R3
R3
h 3 ∞,
R3
S2
d P¯
R3
d Q¯
S2
dω vø σ (, ϑ)
h 1 ( P)h ¯ 2 ( Q) ¯ (Q 0 ) (P 0 )
.
Above, we made use of the trivial estimate e−α Q (Q 0 )− . Applying the CauchySchwarz inequality, we conclude that ) *1/2 h i ( P) ¯ 2 1 I I, h 3 ¯ h 3 ∞, ¯ ¯ dP dQ dω vø σ (, ϑ) . P (Q 0 ) (P 0 ) R3 R3 S2 0
i=1,2
270
J. Speck, R. M. Strain
¯ Q) ¯ symmetry to interchange the values of P¯ and Q¯ in the integral We use the ( P, involving h 2 above. Since ≥ 9, we observe from our hypothesis above (1.15) that vø σ (, ϑ) 0 − ¯ dQ dω 1. (P ) 3 2 (Q 0 ) R S Estimates of this type are proven for instance in [46, Lemma 3.1]. From here, the first estimate in Lemma 3.12 follows for term I I . Similarly, for term I we have 0 I, h 3 ¯ ¯ ¯ ¯ dP dQ dω vø σ (, ϑ) e−α Q h 1 ( P¯ )h 2 ( Q¯ )h 3 ( P) P R3
R3
S2
) 1
h 3 ∞,
R3
i=1,2
d P¯
R3
d Q¯
S2
dω vø σ (, ϑ)
*1/2 h i ( P¯ )2 (Q 0 ) (P 0 )
.
Above, we used the ( P¯ , Q¯ ) symmetry to interchange the values of P¯ and Q¯ in the integral involving h 2 ( P¯ ) above. By the pre-post collisional change of variables, which 0 0 ¯ Q¯ = P 0 Q 0 d P¯ d Q¯ , we have is d Pd P Q R3
d P¯
R3
=
R3
d P¯
d Q¯
dω vø σ (, ϑ)
h i ( P¯ )2
(Q 0 ) (P 0 ) h i ( P) ¯ 2 ¯ dQ dω vø σ (, ϑ) . (Q 0 ) (P 0 ) R3 S2 S2
Above, we used the fact that the kernel of the integral is invariant with respect to the relativistic pre-post collisional change of variables from [19]. Now it can be seen that P 0 Q 0 P 0 and Q 0 Q 0 P 0 . This is the content of [20, Lemma 2.2]. From here, since ≥ 9 we observe that there exists a small δ > 0 such that vø σ (, ϑ) vø σ (, ϑ) 0 − /2+δ ¯ d Q¯ dω ≤ C(P ) d Q dω 1. 0 0 (Q ) (P ) (Q 0 ) /2+δ R3 S2 R3 S2 This establishes (3.71). ¯ (P 0 )−m for any To prove (3.72), we choose a cut-off function χ satisfying χ ( P) m > 3. We will estimate the I and I I terms from the top of this proof with h 2 = χ in the first case. In particular, as before, we have that 0 I I, h 3 ¯ ≤ C ¯ ¯ ¯ 3 ( P) ¯ dP dQ dω vø σ (, ϑ) e−α Q h 1 ( P)h P R3
R3
S2
h 3 ν h 1 ν . We also have I, h 3 ¯ ≤ C P ≤C
R3
d P¯
1 j∈1,3
R3
R3
d Q¯
d P¯
S2
R3
0 ¯ dω vø σ (, ϑ) e−α Q (Q 0 )−m h 1 ( P¯ )h 3 ( P)
d Q¯
S2
2 1/2 dω vø σ (, ϑ) (Q 0 )−m (Q 0 )−m h j ( P¯ ) ,
Hilbert Expansion from Relativistic Boltzmann to Euler
271
where we have used similar reasoning as we did for terms. After the pre-post the previous I, h 3 ¯ ≤ Ch 3 ν h 1 ν as desired. change of variables, as before, we conclude that P We have thus shown that (h 1 , χ ), h 3 P¯ h 3 ν h 1 ν . The last estimate involving the term (χ , h 1 ) follows in exactly the same way. With this lemma, we are ready for the Proof of Lemma 3.10. Since FR;ε satisfies (3.64), the function f ε from (3.67) satisfies the equation ) √ * ˆ · ∂x¯ } M + P 1 {∂ t ε ε ε ∂t f + Pˆ · ∂x¯ f + L( f ) = f ε + ε2 f ε , f ε √ ε M +
6
√ √ ¯ εi−1 Fi / M, f ε + f ε , Fi / M + ε2 A,
i=1
where A¯ = −εM−1/2 {∂t + Pˆ · ∂x¯ }F6 +
√ √ εi+ j−6 Fi / M, F j / M .
i+ j>6 1≤i, j≤6
We now take the L 2 inner product (in R3x¯ × R3P¯ ) of this equation with f ε . It follows from (3.70) that 2 3 1 1 d δ0 ε ε ε ε ˆ ∂t f + P · ∂x¯ f + L( f ), f f ε 22 + {I − P} f ε 2ν . ≥ ε 2 dt ε x; ¯ P¯ In the remainder of the proof √we will give upper bounds for the other terms. We point ¯ Furthermore, out that M−1/2 {∂t + Pˆ · ∂x¯ } M is a first-order polynomial in P. {∂ + Pˆ · ∂ }√M x¯ t ¯ C(n, u, θ ). √ ≤ w1 ( P) M Here C = C(n, u, θ ) depends upon the L ∞ norm of the first derivatives of the fluid def variables. Choose a˜ = 2/(4 − b), where b is any number subject to the constraints described just above (1.15). With that, for any κ > 0, we obtain 4) 5 √ * {∂t + Pˆ · ∂x¯ } M ε ε ¯ = d x¯ d P + d x¯ d P¯ f ,f √ κ κ ¯ ¯ M w ( P)≥ w ( P)≤ 1 1 a˜ a˜ x; ¯ P¯ ε
ε
ε κ f L 2 ≤ C2 (n, u, θ )w1 f ε L ∞ (w( P)≥ ¯ ) εa˜ √ + C∞ (n, u, θ ) w1 f ε 2L 2 (w( P)≤ κ . ¯ ) εa˜
Above, C2 (n, u, θ ) denotes a constant which depends upon L 2 norms of the first derivatives of the fluid variables and the L ∞ norms of the fluid variables alone. This estimate follows from the Cauchy-Schwarz inequality. The constant C∞ (n, u, θ ) also depends
272
J. Speck, R. M. Strain
only on the L ∞ norms of the fluid variables and their first-order derivatives. Furthermore, from (1.39), (3.69) and (3.67), we observe that 2 ¯ f ε ( P) ¯ ≤ w1− ( P) ¯ w ( P)h ¯ ε ≤ ε w ( P)h ¯ ε , if w1 ( P) ¯ ≥ κ. |w1 ( P)| 2/ a ˜ κ εa˜
The above estimate utilizes − 1 ≥ 8 ≥ (4 − b) = 2/a˜ and J < CM in (1.39). Additionally, since a1˜ = 2 − b2 , we have the following weight estimates: w = w 1/a˜ w b/2−1 ≤
κ 1/a˜ b/2−1 κ 1/a˜ 0 b/2 ¯ ≤ κ. w (P ) , if w( P) ε ε εa˜
With these two computations, we estimate 4) * 5 {∂ + Pˆ · ∂ }√M t x¯ f ε, f ε √ M
¯
x; ¯ P
√ ε2 2/a˜ h ε ∞, f ε 2 + C∞ (n, u, θ ) w f ε 2L 2 (w( P)≤ κ ¯ ) κ εa˜ √ 2 ε ε ε 2 ≤ Cκ ε h ∞, f 2 + C wP f L 2 (w( P)≤ κ ¯ ) εa˜ √ ε 2 + C w{I − P} f L 2 (w( P)≤ κ ¯ ) εa˜
≤ Cκ ε2 h ε ∞, f ε 2 + C f ε 22 + C
κ 1/a˜ {I − P} f ε 2ν .
Choosing κ to be sufficiently small, these are the estimates that we will use for the term involving derivatives of the local Maxwellian M. We use Lemma 3.12 and (3.69) to obtain for ≥ 9 that √ 2 3/2 ε h ∞, f ε 22 . ε ( f ε , f ε ), f ε x; ¯ P¯ εε Above we have also used (1.39) with J ≤ CM. We utilize the second estimate in Lemma 3.12 and Proposition 3.9 to achieve 6 / 0 / √ √ ε0 i−1 ε ε ε Fi / M, f , f ε + f , Fi / M , f x; ¯ P¯ x; ¯ P¯ i=1
6
εi−1 f ε 2ν P f ε 2ν + {I − P} f ε 2ν
i=1
ε f ε 2L 2 + {I − P} f ε 2ν . ε
√ We have used Proposition 3.9, (1.16), and (1.39) to conclude that Fi / M is a rapidly decaying function as in the statement of Lemma 3.12. Similar to the above estimates, ¯ f ε ¯ ε2 f ε L 2 f ε L 2 . Note that ε = 1 was added above so we have ε2 A, P ε that we can obtain the estimate in Lemma 3.10 by absorbing this term into the l.h.s. of the inequality for ε > 0 sufficiently small. In particular, we conclude our estimate by δ0 choosing κ small and then supposing that ε ≤ 2C .
Hilbert Expansion from Relativistic Boltzmann to Euler
273
We are now ready to consider the L ∞ estimate for h ε in Lemma 3.11. We first expand √ √ −J −1/2 {Q(M, J h) + Q( J h, M)} = ν(M)h − K (h), where K = K 2 − K 1 . This will be an important term in the equation for the remainder (3.64) once we plug in the ansatz (3.69). This is observed in the proof of Lemma 3.11. The operators K 1 (h) and K 2 (h) are defined as √ def K 1 (h) = J −1/2 Q− (M, J h), √ √ def K 2 (h) = J −1/2 {Q+ (M, J h) + Q+ ( J h, M)}, while √ def ν(M) = J −1/2 Q− ( J , M) = Q− (1, M). Above, the operators Q± are the usual gain and loss parts of Q from (3.4). More specifically, the operators K i can be expressed as % ( ¯ M( P) def ¯ ¯ , K 1 (h) = dωd Q¯ vø σ (, ϑ) J ( Q) h( Q) ¯ R3 ×S2 J ( P) % ( ¯ ) J ( P def dωd Q¯ vø σ (, ϑ) M( Q¯ ) K 2 (h) = h( P¯ ) ¯ R3 ×S2 J ( P) % ( ) ¯ J ( Q + dωd Q¯ vø σ (, ϑ) M( P¯ ) h( Q¯ ) . ¯ R3 ×S2 J ( P) Given any small number η > 0, we choose a smooth cut-off function χ = χ () satisfying 1 if ≥ 2η, χ () = 0 if ≤ η. This Lorentz-invariant cut-off function was previously used in [46]. We now define J ( P¯ ) ¯ def 1−χ ¯ ¯ K 2 (h) = dωd Q (1 − χ ()) vø σ (, ϑ) M( Q ) h( P ) ¯ R3 ×S2 J ( P) J ( Q¯ ) ¯ ¯ ¯ dωd Q (1 − χ ()) vø σ (, ϑ) M( P ) (3.73) + h( Q ). ¯ R3 ×S2 J ( P) We also define K 1 (h) in the same way. We will use the splitting K = K 1−χ + K χ . We will use the Hilbert-Schmidt form for K χ ; it is given by ¯ Q) ¯ h( Q), ¯ d Q¯ k χ ( P, (i = 1, 2). K χ (h) = 1−χ
R3
This form is computed explicitly in [46, App.]; see also [12]. Since the closed form ¯ Q) ¯ can be quite complicated, we simply state the folexpression for the kernel k χ ( P, lowing useful estimate:
274
J. Speck, R. M. Strain
Lemma 3.13 [46, Lemma 3.2]. There exists a constant ζ > 0 such that the kernel enjoys the estimate −ζ χ ¯ ¯ k ( P, ¯ Q) ¯ P 0 Q0 C1 + (P 0 + Q 0 )−(β)− /2 e−c| P− Q| , where (β)− = max{−β, 0}, β is the parameter, and C1 is the constant from above (1.15). Note that in the notation of this paper we use P 0 to denote the quantity which is called p0 in [46]. The estimate above is proved in [46, Lemma 3.4] for the case of soft potentials, i.e. (β)− = b, a = 0, and C1 = 0 in (1.15). However the generalization to the case above is immediate and follows directly from the proof. In the case above, we have the exact estimate ζ = min{1 − (a + (γ )− )/2, min {2 − |γ |, 4 − b, 2} /4}. We will now quote the estimate of the operator K 1−χ as follows: Lemma 3.14 [46, Lemma 4.6]. Fix ≥ 0. Then given any small η > 0, we have ¯ 1−χ (h( P)) ¯ ≤ ηe−γ P 0 h∞ . w ( P)K Above the constant γ > 0 is independent of η. We will also use the following nonlinear estimate: Lemma 3.15 [46, Lemma 5.2]. For any ≥ 0, we have the following L ∞ estimate for the nonlinear Boltzmann collision operator: √ √ ¯ ¯ −1/2 ( P)Q ¯ ¯ ν( P)h h 1 J , h 2 J ( P) w ( P)J 1 ∞, h 2 ∞, . We are now ready to prove Lemma 3.11. Proof of Lemma 3.11. From (3.64) and (3.69), we obtain √ ν(M) ε 1 ε2 √ ∂t h ε + Pˆ · ∂x¯ h ε + h − K (h ε ) = √ Q h ε J , h ε J ε ε J 6 √ √ 1 ˜ εi−1 √ Q Fi , h ε J + Q h ε J , Fi + ε2 A, + J i=1 with 1 ε def εi+ j−6 √ Q(Fi , F j ). A˜ = − √ {∂t + Pˆ · ∂x¯ }F6 + J J i+ j>6,i≤6, j≤6 We will prove Lemma 3.11 by iterating twice the representation of solutions to this equation in terms of the Duhamel formula. def ˆ − s). Let K = K 1−χ + K χ be For purposes of the proof, we define y1 = x¯ − P(t the splitting from (3.73) and define t def ν(t, s) = dτ ν(M)(τ, x). ¯ s
Hilbert Expansion from Relativistic Boltzmann to Euler
275
¯ ≈ ν(M)( P). ¯ From this and (1.15), we have Furthermore, using (1.39) we have ν(J )( P) that ¯ − s) ≈ (P 0 )β/2 (t − s). ν(t, s) ≈ ν(J )( P)(t Here and below A ≈ B indicates that ∃ C ≥ 1 such that C1 A ≤ B ≤ C A. Now by the Duhamel formula we have ν(t, 0) ¯ = exp − ˆ P) ¯ h ε (t, x, h ε0 (x¯ − Pt, ¯ P) ε 1 t ν(t, s) ¯ K 1−χ (h ε )(s, y1 , P) + ds exp − ε 0 ε 1 t ν(t, s) ¯ K χ (h ε )(s, y1 , P) + ds exp − ε 0 ε t √ 2 √ ν(t,s) ε ¯ + ds e− ε √ Q h ε J , h ε J (s, y1 , P) J 0 6 t √ ν(t, s) i−1 1 ds exp − ε √ Q Fi , h ε J + ε J 0 i=1 √ ¯ +Q h ε J , Fi (s, y1 , P) t ν(t, s) 2 ˜ ¯ ε A(s, y1 , P). + ds exp − (3.74) ε 0 We will now simply write ν(t, s) as ν(t − s). We will use the following basic estimate several times below: t ν(t − s) ν ε. ds exp − ε 0 We will now estimate each of the terms in (3.74). Given η > 0, we recall the splitting K = K 1−χ + K χ as in (3.73). From Lemma 3.14, for any η > 0, the K 1−χ term in (3.74) multiplied by w is bounded as w t ν(t, s) 1−χ ε ¯ K ds exp − (h )(s, y , P) 1 ε ε 0 t η ν(t − s) sup ||h ε (t)||∞, ν η ε sup ||h ε (t)||∞ . ds exp − 2 0≤t≤T ε 0 0≤t≤T Next, we use Lemma 3.15 to conclude that √ w ε √ √ Q h J , h ε J ν( P) ¯ h ε 2∞, . J Then the fourth line in (3.74) is bounded by t ν(t − s) 2 ν h ε (s)2∞, ε3 sup h ε (s)2∞, . ds exp − Cε ε 0≤s≤t 0
(3.75)
276
J. Speck, R. M. Strain
For the following terms in (3.74), from Lemma 3.15 again, we have 6 √ √ i−1 w ε ε ε √ Q Fi , h J + Q h J , Fi J i=1
≤C
√ ¯ h ε ∞, εi−1 ν( P) Fi / J
6 i=1
∞,
¯ h ε ∞, . ≤ Cν( P)
We used Proposition 3.9 and the upper bound in (1.39) to conclude that have √ ≤ C. Thus, the fifth line in (3.74) is bounded by Fi / J ∞,
t 0
ν(t − s) νh ε (s)∞, ≤ Cε sup h ε (t)∞, . ds exp − ε 0≤t≤T
(3.76)
As above, the last line in (3.74) is clearly bounded by Cε3 . We have now estimated all the terms in (3.74) save one, which we denote by Iγ , and which is defined by t ν(t, s) def 1 ¯ Iγ = ds exp − K χ h ε (s, y1 , P). ε 0 ε Collecting the above estimates, we have established sup ε3/2 h ε (t)∞, ≤ Cε sup ε3/2 h ε (t)∞, + Cε3 sup ε3/2 h ε (t)2∞, + Cε3
0≤t≤T
0≤t≤T 3/2
+ Cε
0≤t≤T
¯ 3/2 Iγ . h 0 ∞, + Cw ( P)ε
To bound Iγ , we will input h ε in the form of (3.74) back into Iγ just below. ¯ Q). ¯ With this notation, it follows that We recall that the kernel of K χ (h) is k χ ( P, t ν(t, s) 1 ¯ Q) ¯ h ε (s, y1 , Q). ¯ Iγ = ds exp − d Q¯ k χ ( P, 3 ε ε R 0 We plug (3.74) for h ε into the above to obtain t ν(t, s) 1 ¯ Q) ¯ Iγ = ds exp − d Q¯ k χ ( P, 3 ε ε R 0 ¯ ν ( Q)(s, 0) ˆ − s) − Qs, ˆ Q) ¯ + Iγ ,γ + Hγ . (3.77) × exp − h ε0 (x¯ − P(t ε ˆ − s) − Q(s ˆ − s ). Furthermore, we define We also introduce the notation y2 = x¯ − P(t def
¯ ν( P)(t, s) 1 ¯ Q) ¯ = ds exp − d Q¯ k χ ( P, ε ε R3 0 ¯ s ν ( Q)(s, s ) 1 χ ε ¯ × K h (s , y2 , Q). ds exp − ε ε 0
Iγ ,γ
def
t
Hilbert Expansion from Relativistic Boltzmann to Euler
277
Additionally, the term Hγ can be expressed as ¯ ν( P)(t, s) 1 ¯ Q) ¯ Hγ = ds exp − d Q¯ k χ ( P, ε ε R3 0 ¯ s ν ( Q)(s, s ) 1 1−χ ε ¯ × K h (s , y2 , Q) ds exp − ε ε 0 t ¯ ν( P)(t, s) 1 ¯ Q) ¯ + ds exp − d Q¯ k χ ( P, ε ε R3 0 ¯ s ν ( Q)(s, s ) × ds exp − ε 0 2 √ √ ε ¯ × √ Q h ε J , h ε J (s , y2 , Q) J t ¯ ν( P)(t, s) 1 ds exp − + ε ε 0 ¯ s ν ( Q)(s, s ) ¯ Q) ¯ d Q¯ k χ ( P, ds exp − × ε R3 0 6 √ √ 1 ¯ × εi−1 √ Q Fi , h ε J + Q h ε J , Fi (s , y2 , Q) J i=1 t ¯ ν( P)(t, s) 1 ¯ Q) ¯ ds exp − d Q¯ k χ ( P, + ε ε R3 0 ¯ s ν ( Q)(s, s ) 2 ˜ ¯ × ε A(s , y2 , Q). ds exp − ε 0
t
(3.78)
Above we are using the necessary additional notation s ¯ def ν ( Q)(s, s ) = dτ ν(M)(τ, x). ¯ s
¯ ≈ ν(M)( Q), ¯ which implies We have ν(J )( Q) ¯ ¯ ν ( Q)(s, s ) ≈ ν(J )( Q)(s − s ), which we will now simply write as ν(s − s ) as we did in the previous case of ν. Using similar arguments to (3.75) and (3.76), we can control all the terms in (3.77) and (3.78) except the second term in (3.77) by the following upper bound: % ( C h 0 ∞, + ε3 sup h ε (s)2∞, + ε sup h ε (s)∞, + Cε3 . 0≤s≤T
0≤s≤T
We now concentrate on the second term in (3.77), Iγ ,γ . We claim that for any small η > 0, the following estimate holds: w Iγ ,γ ≤ η sup h ε (s)∞, + Cη sup f ε (s)2 . 0≤s≤T
0≤s≤T
278
J. Speck, R. M. Strain
This is proved in [46, Lemma 4.4]. We note that [46, Lemma 4.4] is explained in detail for the soft potentials and for general momentum weights parametrized by k ≥ 0. Here we only use the k = 0 case from [46, Lemma 4.4]. Furthermore, this estimate can be easily extended to the full range of hard and soft-potentials. The proof for the hard potentials follows in exactly the same way as the estimate for the soft potentials, but several technical simplifications of the proof are possible in the hard potential case. With this last estimate we will now finish the proof. We first collect all of our estimates as follows: sup ε3/2 h ε (s)∞, ≤ C{η + η } sup ε3/2 h ε (s)∞, + ε7/2 C
0≤s≤T
+Cη ε
3/2
h 0 ∞, + ε
3/2
C
0≤s≤T 3/2 ε
sup ε 0≤s≤T
h (s)2∞, + Cη sup f ε (s)2 . 0≤s≤T
We now choose η and η small in such a way that C{η + η } < 21 . Then for sufficiently small ε > 0, we obtain % ( sup ε3/2 h ε (s)∞, ≤ C ε3/2 h 0 ∞, + sup f ε (s)2 + ε7/2 ,
0≤s≤T
0≤s≤T
and we conclude our proof.
Acknowledgements. RMS thanks Princeton University, where this project was initiated, for its generous hospitality. JS echoes the sentiments of RMS, and also thanks Mihalis Dafermos and Willie Wong for useful discussions. RMS and JS both thank the University of Cambridge for support during the completion of this article.
References 1. Arsénio, D.: On the Boltzmann equation: Hydrodynamic limit with long-range interactions and mild solutions. Ph.D. dissertation, Department of Mathematics, New York University, September 2009 2. Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46(5), 667–753 (1993) 3. Bardos, C., Golse, F., Levermore, C.D.: Acoustic and Stokes limits for the Boltzmann equation. C. R. Acad. Sci. Paris Sér. I Math. 327(3), 323–328 (1998) 4. Bardos, C., Golse, F., Levermore, D.: Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles. C. R. Acad. Sci. Paris Sér. I Math. 309(11), 727–732 (1989) 5. Bardos, C., Golse, F., Levermore, D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Statist. Phys. 63(1-2), 323–344 (1991) 6. Bardos, C., Ukai, S.: The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math. Models Methods Appl. Sci. 1(2), 235–257 (1991) 7. Boisseau, B., van Leeuwen, W.A.: Relativistic Boltzmann theory in D + 1 spacetime dimensions. Ann. Physics 195(2), 376–419 (1989) 8. Caflisch, R.E.: The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math. 33(5), 651–666 (1980) 9. Cercignani, C., Medeiros Kremer, G.: The relativistic Boltzmann equation: theory and applications. Progress in Mathematical Physics, Vol. 22. Basel: Birkhäuser Verlag, 2002 10. Christodoulou, D.: The Euler equations of compressible fluid flow. Bull. Amer. Math. Soc. (N.S.) 44(4), 581–602 (2007) (electronic) 11. Christodoulou, D.: The formation of shocks in 3-dimensional fluids. Zürich: European Mathematical Society, 2007 12. de Groot, S.R., van Leeuwen, W.A., van Weert, Ch.G.: Relativistic kinetic theory. Amsterdam, North-Holland Publishing Co., 1980
Hilbert Expansion from Relativistic Boltzmann to Euler
279
13. De Masi, A., Esposito, R., Lebowitz, J.L.: Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Comm. Pure Appl. Math. 42(8), 1189–1214 (1989) 14. DiPerna, R.J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2) 130(2), 321–366 (1989) 15. Dudy´nski, M.: On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics. J. Stat. Phys. 57(1-2), 199–245 (1989) 16. Dudy´nski, M. Ekiel-Je˙zewska M.L.: On the linearized relativistic Boltzmann equation. I. Existence of solutions. Commun. Math. Phys. 115(4), 607–629 (1988) 17. Dudy´nski, M., Ekiel-Je˙zewska, M.L.: The relativistic Boltzmann equation - mathematical and physical aspects. J. Tech. Phys. 48, 39–47 (2007) 18. Glassey, R.T.: The Cauchy problem in kinetic theory. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1996 19. Glassey, R.T., Strauss, W.A.: On the derivatives of the collision map of relativistic particles. Transport Theory Statist. Phys. 20(1), 55–68 (1991) 20. Glassey, R.T., Strauss, W.A.: Asymptotic stability of the relativistic Maxwellian. Publ. Res. Inst. Math. Sci. 29(2), 301–347 (1993) 21. Golse, F.: The Boltzmann equation and its hydrodynamic limits. In: Evolutionary equations. Vol. II, Dafermos, C., Feireisl, E. (eds.) Handbook Diff. Equations, Amsterdam: Elsevier/ North Holland, 2005, pp. 159–301 22. Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155(1), 81–161 (2004) 23. Golse, F., Saint-Raymond, L.: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. (9) 91(5), 508–552 (2009) 24. Grad, H.: Principles of the kinetic theory of gases. In: Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase. Berlin: Springer-Verlag, 1958, pp. 205–294 25. Grad, H.: Asymptotic theory of the Boltzmann equation. II. In: Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962), Vol. I. New York: Academic Press, 1963, pp. 26–59 26. Guo, Y., Jang, J.: Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System. Comm. Math. Phys. (in press). http://arxiv.org/abs/0910.5512v1 [math.AP], 2011 27. Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169(4), 305–353 (2003) 28. Guo, Y.: Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm. Pure Appl. Math. 59(5), 626–687 (2006) 29. Guo, Y.: Erratum: “Boltzmann diffusive limit beyond the Navier-Stokes approximation”. Comm. Pure Appl. Math. 59(5), 626–687 (2006) 30. Guo, Y.: Erratum: “Boltzmann diffusive limit beyond the Navier-Stokes approximation”. Comm. Pure Appl. Math. 60(2), 291–293 (2007) 31. Guo, Y.: Decay and continuity of Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. 197(3), 173–809 (2010) 32. Guo, Y., Jang, J., Jiang, N.: Local Hilbert expansion for the Boltzmann equation. Kinet. Relat. Models 2(1), 205–214 (2009) 33. Hörmander, L.; Lectures on nonlinear hyperbolic differential equations. In: Mathématiques & Applications (Berlin) [Mathematics & Applications], Vol. 26. Berlin: Springer-Verlag, 1997 34. Lions, P.-L., Masmoudi, N.: From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Ration. Mech. Anal. 158(3), 173–193, 195–211, 2001 35. Liu, T.-P., Yang, T., Yu, S.-H.: Energy method for Boltzmann equation. Phys. D 188(3-4), 178–192 (2004) 36. Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. New York: Springer-Verlag, 1984 37. Masmoudi, N.: Some recent developments on the hydrodynamic limit of the Boltzmann equation. In: Mathematics & mathematics education (Bethlehem, 2000). 2002, pp. 167–185 38. Masmoudi, N., Levermore, C.D.: From the Boltzmann equation to an incompressible Navier-StokesFourier system. Arch. Rat. Mech. Anal. in press (2009) 39. Nishida, T.: Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys. 61(2), 119–148 (1978) 40. Olver, F.W.J.: Asymptotics and special functions. AKP Classics, Wellesley, MA: A K Peters Ltd., 1997. Reprint of the 1974 original. New York: Academic Press 41. Shatah J., Struwe, M.: Geometric wave equations, Courant Lecture Notes in Mathematics, Vol. 2. New York: New York University Courant Institute of Mathematical Sciences, 1998 42. Speck, J.: On the questions of local and global well-posedness for the hyperbolic pdes occurring in some relativistic theories of gravity and electromagnetism. PhD dissertation, Piscataway, NJ, 2008 43. Speck, J.: The non-relativistic limit of the Euler-Nordström system with cosmological constant. Rev. Math. Phys. 21(7), 821–876 (2009)
280
J. Speck, R. M. Strain
44. Speck, J.: Well-posedness for the Euler-Nordström system with cosmological constant. J. Hyperbolic Differ. Equ. 6(2), 313–358 (2009) 45. Stewart, J.M.: Non-equilibrium relativistic kinetic theory. Berlin, New York: Springer-Verlag, 1971 (English) 46. Strain, R.: Asymptotic stability of the relativistic Boltzmann equation for the Soft Potentials. Commun. Math. Phys. 300(2), 529–597 (2010) 47. Strain, R.: Coordinates in the relativistic Boltzmann theory. Kinet. Relat. Models 4(1, special issue), 345–359. http://arxiv.org/abs/1011.5093v1 [math.Ap], 2011 48. Strain, R.M.: Global Newtonian limit for the relativistic Boltzmann equation near vacuum. SIAM J. Math. Anal. 42(4), 1568–1601 (2010) 49. Strain, R.M.: An energy method in collisional kinetic theory. Ph.D. dissertation, Division of Applied Mathematics, Brown University, May 2005 50. Strain, R.M., Guo, Y.: Stability of the relativistic Maxwellian in a collisional plasma. Commun. Math. Phys. 251(2), 263–320 (2004) 51. Strain, R.M., Guo, Y.: Almost exponential decay near Maxwellian. Comm. Part. Diff. Eqs. 31(1-3), 417– 429 (2006) 52. Strain, R.M., Guo, Y.: Exponential decay for soft potentials near Maxwellian. Arch. Rat. Mech. Anal. 187(2), 287–339 (2008) 53. Synge, J.L.: The relativistic gas. Amsterdam: North-Holland Publishing Company, 1957 54. Villani, C.: A review of mathematical topics in collisional kinetic theory, In: Handbook of mathematical fluid dynamics, Vol. I. Amsterdam: North-Holland, 2002, pp. 71–305 Communicated by P. Constantin
Commun. Math. Phys. 304, 281–293 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1217-x
Communications in
Mathematical Physics
Comment on “Random Quantum Circuits are Approximate 2-designs” by A.W. Harrow and R.A. Low (Commun. Math. Phys. 291, 257–302 (2009)) Igor Tuche Diniz1,2 , Daniel Jonathan1 1 Instituto de Física, Universidade Federal Fluminense, Niterói, Brazil. E-mail: [email protected] 2 Institut Néel-CNRS, Grenoble, France. E-mail: [email protected]
Received: 21 June 2010 / Accepted: 27 September 2010 Published online: 9 March 2011 – © Springer-Verlag 2011
Abstract: In [A.W. Harrow and R.A. Low, Commun. Math. Phys. 291(1):257–302 (2009)], it was shown that a quantum circuit composed of random 2-qubit gates converges to an approximate quantum 2-design in polynomial time. We point out and correct a flaw in one of the paper’s main arguments. Our alternative argument highlights the role played by transpositions induced by the random gates in achieving convergence. 1. Introduction Quantum k-designs [1] are statistical ensembles over the sets of states or operators of a quantum system that faithfully reproduce the k th moments of the respective uniform distributions. These pseudo-random ensembles are of interest since they can often be efficiently simulated in a physical system. In other words, while physically generating random states or operators of an n-qubit quantum system requires resources that grow exponentially in n, pseudorandom objects may require only polynomial resources [2]. They are thus a practical tool for a wide variety of communication and computation tasks that make use of random quantum objects (e.g., [3–5]). In ref. [6], Harrow and Low (HL) have provided an example of an efficient construction of a quantum 2-design for operators of an n-qubit system, i.e., one that can be physically implemented using resources that scale polynomially with n. Unlike previous constructions with this property [7–9], their scheme appears to be efficient also for higher values of k [10]. The construction is based on a random quantum circuit model [2]: at each step of the circuit, a pair of qubits is chosen at random, and a 2-qubit gate is applied to them, drawn from some ensemble μ over the set of all such gates. The pseudorandom n-qubit operators that result from this procedure have second moments whose evolution can be reduced to a classical Markov chain [11,12]. In particular, the (approximate) convergence of this chain to its stationary state is sufficient to ensure This comment refers to doi:10.1007/s00220-009-0873-6.
282
I. T. Diniz, D. Jonathan
the convergence of the pseudorandom operator ensemble to an approximate quantum 2-design [6]. In this note we wish to point out and correct a flaw in a significant step of this analysis, on which the main results of ref. [6] directly depend. Specifically, the proof of Corollary 5.1 (p. 284), a statement concerning the number of steps required for the convergence of the Markov chain, is incorrect. We give an alternative argument showing that the statement itself is indeed valid. Our proof highlights the role played by transpositions induced by the random gates in achieving convergence. We assume that the reader is familiar with ref. [6]. In Sect. 2 we summarize some of its results, explaining where they are affected by the flawed step. In Sect. 3 we explain the flaw itself, giving an explicit counterexample. In Sect. 4 we give the general idea of our argument, and develop some preliminary results using standard tools from Markov chain theory and group representations. Section 5 contains our main result, with several details left to the Appendix. 2. Summary of Results in [6] Following a strategy introduced in [11,12], the first part of ref. [6] establishes a map from the evolution of second moments of a random quantum circuit to a classical Markov chain P with state space P = {0, 1, 2, 3}n . When the ensemble μ is chosen to be the uniform (Haar) distribution over U (4), P turns out to have a particularly simple form, described by the following algorithm: given a position p = ( p1 , . . . pn ) ∈ P , choose a new position p as follows: − choose randomly and uniformly a pair of indices 1 ≤ i = j ≤ n, − if pi = p j = 0, do nothing, − if ( pi , p j ) = (0, 0), replace the pair with any element of {0, 1, 2, 3}2 \(0, 0), choosing uniformly from the 15 possibilities. (1) 1 The corresponding Markov matrix P( p, p ) has the form P = n(n−1) i= j Pi j , where Pi j affects only the i, j coordinates of p. Apart from an isolated stationary state 0 = (0 . . . 0), this Markov chain is ergodic, with stationary state given by the uniform distribution π( p) = (4n − 1)−1 , ∀ p ∈ P \{0}.
(2)
The key technical problem is then to analyze the convergence time of P, measured for example by its mixing time in the trace norm: (3) tmi x P (ε) := max min t P t ( p, ·) − π T V < ε . p∈ P \0
HL’s approach is to concentrate first on the much smaller Markov chain Z which tracks the number of nonzero coordinates (i.e., the Hamming weight H ( p) = |{i | pi = 0 }|) of states evolving under the P chain. This ‘zero chain’ is ergodic on the state space Z = {1, . . . , n}, with stationary state n H 3 (4) ζπ (H ) = Hn ; H ∈ Z . 4 −1
Comment on “Random Quantum Circuits are Approximate 2-designs”
283
Its only nonvanishing transition probabilities are (Eq. (5.2) in [6]): n −1 3 Z (H, H + 1) = H (n − H ) , 5 2 n −1 1 Z (H, H − 1) = H (H − 1) , (5) 5 2 2H (3n − 2H − 1) . Z (H, H ) = 1 − Z (H, H − 1) − Z (H, H + 1) = 1 − 5n(n − 1) Determining a tight upper bound on the mixing time tmi x Z (ε) of this chain turns out to be quite tricky. The main difficulty is dealing with states with small values of H , which by Eq. (5) only have probability O(1/n) of evolving. Nevertheless, after a laborious calculation, HL are able to show in Theorem 5.1 that tmi x Z (ε) = (n log(n/ε)). The next step in the analysis is the one that concerns us in this Comment. In Corollary 5.1, Harrow and Low state that, once the Z chain has approximately mixed, then O(n ln(n/ε)) further steps suffice to ensure the convergence of the P chain as a whole, so that
Corollary 5.1 [6]. The full (P) chain mixes in time tmi x P (ε) = n log nε . It is important to emphasize that, despite its moniker, this result is in fact an independent theorem that does not follow automatically from other results in [6]. It is also a vital step in the main argument of the paper, as it implies immediately (see Eq. (5.7) and Theorem 4.1) that the spectral gap of the P chain is of order (1/n). This fact is, in turn, necessary for the main conclusions of the paper, viz. Theorem 2.2 giving the polynomial bound for the convergence of a random quantum circuit to a 2-design. Unfortunately, as we now show, the demonstration of Corollary 5.1 given in [6] is flawed. 3. Flaw in the Proof of Corollary 5.1 The argument given in [6] is based on the well-known ‘coupon collector’ scenario [13,14], where one must complete a collection of n different coupons by acquiring them at random. In the present context, each ‘coupon’ corresponds to a coordinate i of p, which is ‘collected’ when it is first chosen in Eq. (1) together with another j such that ( pi , p j ) = (0, 0). HL carefully show that, if the Z chain has already converged, then after O(n ln(n/ε)) circuit steps, the probability that all coordinates have been ‘hit’ in this sense is greater than 1 − ε. The crux of their argument is however the following statement (p. 284): Once each site of the full chain has been hit, (...) the chain has mixed. This is because, after each site has been hit, the probability distribution over the states is uniform. Indeed, if this were true, then standard results, based on the concept of a ‘strong stationary time’ (SST)1 would allow the bound on the ‘album completion’ time τ to be converted into one on the P chain’s mixing time. Unfortunately, however, the quoted 1 An SST [14,15] is an instant τ when the distribution X of the chain conditional on a certain event τ occurring matches the stationary one π . More precisely, X τ must be obtained independently of τ , and of the initial state y of the chain, i.e.: Py {X τ = x, τ = t} = π(x)Py {τ = t}. Under these circumstances a bound on the mixing time can be established (see, e.g., Proposition 6.10 in [14]).
284
I. T. Diniz, D. Jonathan
statement is incorrect: the probability distribution conditioned on all sites being hit is in fact not uniform, and an SST-type argument cannot be used. This is already apparent in Eq. (1): note that, conditioned on a site i having just been hit, its value pi has probability 1/5 of becoming 0 and 4/15 of becoming 1, 2 or 3. In particular, since this is true of the last site to be hit, the overall distribution for p conditioned on all sites being hit cannot be uniform. One can also construct an explicit counterexample. Choose for example n = 3 qubits (the simplest nontrivial case) and initial state y = (0 0 1). Starting from y, consider those evolutions such that all three sites are ‘collected’ after two circuit steps. By exhausting all such cases, it is straightforward to check that the conditional probability of reaching each final state is not uniform. For example: the probabilities of obtaining (1 0 0) or (0 0 1) have a ratio 3:2. 4. Alternative Strategy and Symmetry Analysis While it is conceivable that, with appropriate tweaking, an SST-based argument might still be found for Corollary 5.1, we have been unable to do so. We propose instead a different strategy, based on reducing the analysis of the P chain to that of another well-known problem in Markov chain theory: the repeated random transposition of n objects. Note that other kinds of argument may also be possible, for instance via coupling (A. Harrow, private communication). Much is known about the random transposition chain [14,16,17]; in particular, P. Diaconis and collaborators have shown that it converges to within ε of a random permutation after (n ln(n/ε)) steps.2 In order see how this result applies to the problem at hand, let us define the set of states sharing the same Hamming weight H : G H := { p |H ( p) = H }.
(6)
Since the Z chain mixes after (n ln(n/ε)) circuit steps, then at that point the total probability for each G H is approximately correct. However, the probability distributions within each set may still be uneven, and so it is not yet possible to ensure that the full P chain has mixed to its uniform stationary state. Note now that all elements of G H are equivalent up to permutations of their indexes and/or of the values 1, 2 or 3 of their nonzero coordinates. One can thus expect that applying a random permutation of these variables will result in the mixing of P. Lemmas 1 and 2 below show that this is indeed true. The remaining question is then: how do we ensure that such a permutation is applied? A simple way is to do it ‘by hand’. For example, once the Z chain has mixed, we can apply an efficient permutation-generating algorithm such as the Durstenfeld-Knuth shuffle [18], which requires O(n) transpositions to generate an exactly randomly distributed permutation of the indexes of p. In physical terms, each transposition can be implemented by a SWAP gate on the corresponding qubit pair. Subsequently, all we need is to apply independent permutations of the values 1, 2, 3 on each site. These can all be 3 done in parallel, by applying a random choice from the set of Pauli rotations {σi }i=1 on each qubit (compare e.g. the C1 /P1 -twirl in [9]). The overall number of circuit steps for the entire algorithm is therefore still (n ln(n/ε)). Once this is done, the remainder of the argument in [6] implies that an approximate quantum 2-design will indeed have been generated. 2 In fact, much sharper statements can be made [14,16,17], but these are not necessary here.
Comment on “Random Quantum Circuits are Approximate 2-designs”
285
Of course, following this strategy requires switching mid-way from the ‘pure’ random quantum circuit model described by Harrow and Low to a different algorithm. This is irrelevant if all that is required is an efficient means of generating a 2-design. Our interest here, however, is to show that the same result is also achieved within the original random circuit model. Specifically, in Sec. 5 we will show that, once the Z chain has mixed, the P chain itself performs the role of a random transposition chain. Diaconis et al’s results then ensure that P mixes in (n ln(n/ε)) additional steps, and so the overall number of steps will also be of order (n ln(n/ε)). Before we formalize these ideas, it is useful to exploit the symmetries of P in order to reduce its analysis to that of a simpler chain, which we call Q. This requires some elementary results from the application of group representation theory to Markov chains [17,19]. Markov Chain Projections. Suppose a Markov chain M, with state space M , is invariant under an action of some group G, i.e.: M(g(x), g(y)) = M(x, y), ∀g ∈ G, ∀x, y ∈ M . If G a , G b ⊆ M are orbits induced by the group action, then the rule
M(x, y); x ∈ G a (7) N (G a , G b ) := y∈G b
defines3 a new Markov chain N over the set of all orbits, {G i } ≡ N . This ‘projected’ chain can be seen as a coarse-graining of the original one. Every probability distribution μ(x) over M has a natural projection νμ (a) = x∈G a μ(x) on N . In particular, if μ is a stationary distribution for M, then νμ is a stationary distribution for N . Also, every eigenfunction h of N can be lifted onto a corresponding eigenfunction f of M, with the same eigenvalue, defined by f (x) := h(a), ∀x ∈ G a (see e.g. Lemma 12.8 in [14]). The converse is, in general, not true, since eigenfunctions of M can project to zero. Thus the projected chain can have fewer eigenvalues than the original [14,19], and simpler dynamics. In particular, if both chains are ergodic, N mixes at least as fast as M. The Z chain is an example of a projection of P. By Eq. (1), the transition probabilities P( p, p ) of the P chain are insensitive to whether the nonzero coordinates of p and p are equal to 1,2 or 3 (they only distinguish these values from 0). They are also invariant under permutations of the indexes of p, p . The group subsuming both these symmetries is isomorphic to the wreath product4 S3 Sn . The corresponding orbits in P are precisely the sets G H , and the projected chain resulting from Eq. (7) is the Z chain. Q chain. It is useful to define a less coarsely-grained projection of P, which we call Q, with state space Q ≡ {0, 1}n (the vertices of a unit hypercube). Consider the action on P by the subgroup S3n ⊂ S3 Sn formed by independent permutations of the values 1, 2 and 3 of each coordinate of p. The resulting set of orbits is isomorphic to Q , under the bijection q ↔ G q = { p | pi = 0 ⇔ qi = 0 }. By Eq. (7), the corresponding projected chain is
Q(q, q ) = P( p, p ); ∀ p ∈ G q p ∈G q 3 Note that this sum is independent of the choice of x. 4 This is the semidirect product S n S , where φ is the natural homomorphism of S n induced by elements 3 φ n 3 of Sn .
286
I. T. Diniz, D. Jonathan
Table 1. Transition probabilities Q (i, j) (qi q j , qi q j ) and M (i, j) (qi q j , qi q j ). On the left column we have the
initial values qi q j and on the top line the final values qi q j Q (i, j) 00 01 10 11
00 1 0 0 0
01 0 1/5 1/5 1/5
10 0 1/5 1/5 1/5
M (i, j) 00 01 10 11
11 0 3/5 3/5 3/5
00 1 0 0 0
01 0 1/4 0 1/4
10 0 0 1/4 1/4
11 0 3/4 3/4 1/2
with stationary state on Q \{0} given by the projection of π in Eq. (2): νπ (q) =
1 3 H (q ) . 4n − 1
(8)
1 (i, j) , where each Like P, Q may be written as a convex sum Q = n(n−1) (i= j) Q Q (i, j) (q, q ) vanishes except for pairs q, q that differ only at coordinates i and j. When restricted to these coordinates, the matrix Q (i, j) always has the same form, given in Table 1. The reason for defining Q is that, despite being a projection of P, the two chains have completely equivalent dynamics - we can therefore restrict ourselves to studying the simpler chain.5 As we now show, this happens because the P chain does not distinguish between the elements within each orbit G q . Lemma 1. The mixing times tmi x Q (ε) and tmi x P (ε) are equal for all ε > 0. Proof. Since P is a reversible Markov chain, the t th power of its matrix can be expanded as P t ( p, p ) = π( p )
| P|
f j ( p) f j ( p )λtj ,
j=1
where f j : P → R| P | ∈ l 2 (π ) are the eigenfunctions of P, with corresponding eigenvalues λ j , and which are orthonormal with respect to the stationary measure π (see Lemma 12.2 in [14] ). Similarly,
Q (q, q ) = νπ (q ) t
| Q |
h j (q)h j (q )α tj ,
j=1
where h j ∈ l 2 (νπ ), α j are the eigenfunctions of Q and corresponding eigenvalues. As previously noted, each h j can be lifted to a corresponding f j with the same eigenvalue, given by f j ( p) = h j (q), ∀ p ∈ G q . Note now that, by Eq. (1), P( p1 , p1 ) = P( p2 , p2 ); ∀ p1 , p2 ∈ G q , p1 , p2 ∈ G q 5 A related strategy is used in [11].
(9)
Comment on “Random Quantum Circuits are Approximate 2-designs”
287
In other words, P can be written as a block-constant matrix, with rank equal to that of Q. This implies that the eigenfunctions ‘lifted’ from h j are the only eigenfunctions of P with non-zero eigenvalues. For each p ∈ G q , we have then | P|
t P t ( p, p ) − π( p ) = π( p ) f j ( p) f j ( p )λ j − 1 j=1 p ∈ P p ∈ P | Q |
−H (q ) t = 3 νπ (q ) h j (q)h j (q )λ j − 1 j=1 q ∈ Q p ∈G q
Q t (q, q ) − νπ (q ) = (10) q ∈ Q
since there are 3 H (q ) elements in G q . Finally, since the orbits G q for q = 0 partition P \{0}, we obtain the desired result by substituting in Eq. (3) Note that the Z chain is also a projection of the Q chain under the natural action of Sn on Q , with orbits G H . Thus every probability distribution ν(q) over Q \{0} has a projection ζν (H ) = q ∈G H ν(q) over Z \{0}. We are now ready to formalize the intuitive argument given at the beginning of this section. Given a permutation σ ∈ Sn , let Aσ be its natural representation as a Markov matrix acting on the space of probability distributions over Q :6 [ν Aσ ](q) := ν(σ (q)),
(11)
where σ (q) i = q σ −1 (i) is the natural action of σ on Q . We can extend this representation to any probability distribution over Sn by taking convex combinations of the Aσ .In particular, the uniform distribution is represented by the Markov matrix 1 S = n! σ ∈Sn Aσ . The following lemma shows that applying this random permutation to any distribution ν over Q brings it as close to the stationary state of Q as its projection ζν is to the stationary state of Z . Lemma 2. ν S − νπ T V = ζν − ζπ T V . Proof. By definition the orbits G H are invariant under permutations, so
q ∈G H
[ν S](q) =
1 [ν Aσ ](q) = ν(q) = ζν (H ). n! σ ∈Sn q ∈G H
q ∈G H
Also, since S Aσ = S, ∀σ , then ν S is a constant function on G H : ν S(q) =
ζν (H ) , ∀q ∈ G H . |G H |
(12)
6 Here, as is usual in the Markov chain literature [14], ν is a row vector and the Markov matrix A acts on σ the left.
288
I. T. Diniz, D. Jonathan
By Eq. (8), the same is also true for νπ . Thus, using also Eq. (7): 1 |ν S(q) − νπ (q)| 2 H ∈ Z q ∈G H
1 = ζν − ζπ T V . = ν S(q) − ν (q) π 2 H ∈ Z q ∈G H q ∈G H
ν S − νπ T V =
(13)
5. Proof of Corollary 5.1 in [6] In this section we show how the Q chain itself induces a random permutation of the indexes of q, and how this leads to our desired result, Corollary 5.1 of [6]. Let us begin by introducing the random transposition chain T studied by Diaconis et al [14,16,17]. Consider a set of n different objects occupying n positions, and subject to the following evolution rule: at each step, two values 1 ≤ i, j ≤ n are selected independently at random, and the objects at these positions are swapped. If i = j, nothing happens. Formally, this can be seen as a random walk on Sn , with transition probabilities between permutations σ and ρ given by (14) T (σ, ρ) = τ ρσ −1 , where τ is the probability distribution over Sn defined by ⎧ ⎨ 1/n, α = I τ (α) = 2/n 2 , α is a transposition ⎩ 0, otherwise.
(15)
This chain is ergodic and converges to the uniform distribution. As we have already mentioned, Diaconis et al. showed that this occurs with mixing time tmi x T (ε) = (n ln(n/ε)).
(16)
Returning now to the components Q (i, j) of the Q chain (see Table 1), notice that each can be rewritten as the convex sum Q (i, j) =
1 (i, j) 4 (i, j) T + M , 5 5
(17)
where T (i, j) represents the transposition of coordinates i and j and M (i, j) is still a Markov matrix. Thus, Q can be seen as the combination of two Markov chains 1 4 T p + M, (18) 5 5 1 1 (i, j) and M ≡ (i, j) , respectively. where T p ≡ n(n−1) i= j T i= j M n(n−1) The T p chain represents a random transposition of the components of q. Though similar to T , it is based on a different representation of the permutation group: here the transpositions T (i, j) act on the state space Q , and not Sn itself. As a result, T p is reducible to independent chains on each of the orbits G H . Furthermore, since T p Q=
Comment on “Random Quantum Circuits are Approximate 2-designs”
289
lacks the identity component present in Eq. (15), an even (resp. odd) number of steps will always lead to an even (resp. odd) permutation of q. Thus T p is a non-convergent, periodic chain. The latter difficulty can be easily removed by rewriting Eq. (18) as Q=
1 4 ˜ T + M, 5 5
(19)
1 ˜ where T = n1 I + n−1 n T p is now aperiodic, and M = M + 4n T p − I , is an ergodic Markov chain on Q \{0} for n ≥ 3.7 Alternatively, T may also appear in Eq. (18) if we modify the definition of the two-qubit gate ensemble μ, allowing at each step an extra probability 1/n of applying the identity gate. In this case, the P chain in Eq. (1) becomes P = n1 I + n−1 n P. The corresponding modification of Q leads to 4 1 n−1 1 4 1 I+ M ≡ T + M . (20) Q = T + 5 5 n n 5 5 The T chain is not ergodic, as it is still reducible into independent chains TH on each orbit G H . In particular, T does not have a unique stationary state. Nevertheless, it does converge to the random permutation S over Q , and the mixing time given in Eq. (16) is still valid, in the following generalized sense: Lemma 3. Each initial distribution ν on Q converges under T to its randomized version ν S. In addition, Eq. (16) remains valid under the generalized notion (21) tmi x T (ε) = max min t νT t − ν S T V ≤ ε . ν
Proof. This follows from the fact that each TH is isomorphic to a projection of T in the sense of Eq. (7). See the Appendix for details. Turning now to the M chain, note from its definition that it is symmetric under permutations of the site indexes, and in particular under transpositions. Thus M (or its ˜ M ) commutes with T . variants M, This property gives us an intuitive picture of how the Q chain behaves. According to Eq. (19), each step of Q can be seen as a random choice between moving according ˜ A sequence of t steps will, for large enough t, contain roughly t/5 steps to T or to M. ˜ Note that the latter are the only steps where the Hamming of T and 4t/5 steps of M. weight can change, and thus only they contribute to the convergence of the zero chain Z . Moreover, since the T and M˜ chains commute, we can consider that all these M˜ steps happen first. Once Z has converged, all we need is to wait for the subsequent T steps to build up to a random permutation of site indexes. Lemmas 1 and 2 then ensure that the full P chain will have converged. Let us now formalize this argument Proof of Corollary 5.1 in [6]. Let γ > 0, and let t0 = tmi x Z (γ ), so that the state ζ of the Z chain after t0 steps satisfies ζ − ζπ T V ≤ γ . By Lemma 2, the corresponding state ν of the Q chain at that moment lies within the ball (22) B(γ ) := ν ν S − νπ T V ≤ γ . 7 This is true since it can be shown [20] that i) M is ergodic on \{0} and ii) its eigenvalues are lowerQ 1 . Thus the eigenvalues of M ˜ are all > −1 for n ≥ 3. bounded by − 23 − 3(n−1)
290
I. T. Diniz, D. Jonathan
Define now a mixing time for Q for initial conditions restricted to this ball tmi x Q (ε, γ ) := max min t ν Q t − νπ ≤ ε . TV
ν∈B(γ )
In the Appendix, we show that this time is bounded by the mixing time of T : Lemma 4. 5 4 2
2 −2δ tmi x Q (ε, γ ) < δ + δ + tmi x T ε − γ − e 2 5
(23)
(24)
for all ε > 0 and all γ ≥ 0, δ > 0 satisfying ε > e−2δ + γ . Choosing γ = ε/2, δ 2 = 21 ln(4/ε) gives 4 25 1 ln(4/ε) + tmi x T (ε/4) . tmi x Q (ε, ε/2) < 4 2 5 Taking into account Eq. (16) and the fact that ln(4/ε) ≤ n ln(4n/ε) , ∀n ≥ 1, it follows that there exists an integer K such that 2
tmi x Q (ε, ε/2) < K n ln(4n/ε). Thus the mixing time for the entire Q is tmi x Q (ε) ≤ tmi x Z (ε/2) + tmi x Q (ε, ε/2) = (n ln(n/ε)), where we use the fact (Theorem 5.1 of [6]) that tmi x Z (γ ) = (n ln(n/γ )). Finally, applying Lemma 1 proves our desired result Acknowledgements. The authors acknowledge the support of Brazilian funding agencies CNPq and FAPERJ. This work is part of the Brazilian National Institute of Science and Technology of Quantum Information (INCT-IQ). We thank Aram Harrow for encouraging comments, and Roberto I. Oliveira for helpful discussions.
A. Appendix A.1. Proof of Lemma 3. Since all elements of G H are equivalent under permutations, and transpositions generate all permutations, TH is irreducible; it is also aperiodic due to the identity component in T . Thus, TH is ergodic, and it is easy to see that its stationary state is the uniform distribution over G H . In other words, any initial distribution ν H over G H converges to ν H S (see Eq. (12)). Since T = H TH , the same is true for any initial distribution ν over Q . Let us now link TH with Diaconis’ T chain. By the orbit-stabilizer theorem, G H is isomorphic to the quotient Sn /N H , where N H ⊂ Sn is the stabilizer of some element 0 ∈ G . Explicitly, we identify x ∈ G xH H H ↔ gx N H , where gx is any permutation 0 ) = x. Since T can be described using the probability distribution τ such that gx (x H H in Eq. (15), but with the transpositions acting on G H , it follows (see, e.g. Lemma 3 in Sec. 3F of [17]) that its transition matrix is TH (x, y) = τ (g y N H gx−1 ) = T (gx , g y N H ),
(25)
where we have used Eq. (14). Note now that T is invariant under the action of N H on Sn given by h(g) = gh. The set of orbits of this action is precisely Sn /N H ∼ = GH. Comparing Eq. (25) and Eq. (7), it is clear that TH is (isomorphic to) the projection of T with respect to this action. Thus, as discussed in Sec. 4, the mixing time for TH is at most equal to that of the T , in Eq. (16). Finally, the same is true for T since T = H TH .
Comment on “Random Quantum Circuits are Approximate 2-designs”
291
A.2. Proof of Lemma 4. Let p = 1/5. After t steps of Q, the T V distance to the stationary state νπ is, from Eq. (20): t t t i t−i i ˜ t−i p (1 − p) T M − νπ d(t) := ν Q − νπ T V = ν i i=0 TV t
t ≤ (26) pi (1 − p)t−i νT i − νπ . TV i i=0
In the first equation we have used the fact that T and M˜ commute, and in the second ˜ and that the triangle inequality, and also the facts that νπ is the stationary state for M, applying an ergodic Markov matrix can never increase the T V distance to its stationary state. Let us now split √Eq. (26) into two sums √ d1 (t), d2 (t), containing respectively terms with i ≤ pt − δ t and i > pt − δ t, where δ > 0 is some constant such that t > (δ/ p)2 . In order to bound d1 (t) we can use the fact that T V distances between probability distributions are always ≤ 1, so that √
pt−δ
t t pi (1 − p)t−i . d1 (t) ≤ i i=0
This is a sum of terms in the tail of the binomial distribution, which can again be bound, for any t > (δ/ p)2 using e.g. the Hoeffding inequality [21] d1 (t) ≤ exp(−2δ 2 ). We can also bound d2 (t), as follows: since ν ∈ B(γ ), then for each value of i: i νT − νπ ≤ νT i − ν S + ν S − νπ T V ≤ νT i − ν S + γ . TV
TV
TV
Furthermore, since T is ergodic on each orbit G H , and the initial state ν converges to ν S by Lemma 3, then the T V distance with respect to this state is non-increasing at each step of chain [14]. Thus, in d2 are at most equal to that of the term with all T V√distances the smallest value i = pt − δ t + 1: ⎡ ⎤ t √
t pi (1 − p)t−i ⎦ νT pt−δ t +1 − ν S + γ d2 (t) ≤ ⎣ TV i √ i= pt−δ t +1 √ ≤ νT pt−δ t +1 − ν S + γ , TV
since the sum is over part of the binomial distribution. Combining both bounds: √ d(t) ≤ exp(−2δ 2 ) + γ + νT pt−δ t +1 − ν S . TV
(27)
Given now any ε > 0, choose γ , δ ≥ 0 satisfying e−2δ + γ < ε, and choose also t to be the first instant for which √ 2 max νT pt−δ t +1 − ν S ≤ ε − e−2δ − γ . (28) 2
ν∈B(γ )
TV
292
I. T. Diniz, D. Jonathan
(This instant exists, by Lemma 3). Substituting in eq. (27) and using Eq. (23): tmi x Q (ε, γ ) ≤ t. Define now, in analogy to Eq. (23), tmi x T (ε, γ ) := max min t νT t − ν S T V ≤ ε ≤ tmi x T (ε), ν∈B(γ )
(29)
(30)
with the inequality resulting since tmi x T (ε) maximizes over a larger set. Then we can restate Eq. (28) as √! √ 2 pt − δ t < pt − δ t + 1 = tmi x T ε − γ − e−2δ , γ . √ This inequality, which is quadratic in t, can be inverted to give " √
1 2 2 −2δ δ + δ + 4 p tmi x T ε − γ − e t< ,γ . 2p Using Eqs. (29) and (30) we obtain the relation between the mixing times of Q and T : " 1 2
2 −2δ δ + δ + 4 p tmi x T ε − γ − e tmi x Q (ε, γ ) < 2p References 1. Ambainis, A., Emerson, J.: Quantum t-designs: t-wise independence in the quantum world. In: Proceedings of the 22nd Annual IEEE Conference on Computational Complexity, Los Alanitos, CA: IEEE, 2007, pp. 129–140 2. Emerson, J., Weinstein, Y.S., Saraceno, M., Lloyd, S., Cory, D.G.: Pseudo-random unitary operators for quantum information processing. Science 302, 2098 (2003) 3. Hayden, P., Leung, D., Shor, P.W., Winter, A.: Randomizing quantum states: Constructions and applications. Commun. Math. Phy. 250(2), 371–391 (2004) 4. Bendersky, A., Pastawski, F., Paz, J.P.: Selective and efficient estimation of parameters for quantum process tomography. Phys. Rev. Lett. 100(19), 190403 (2008) 5. Harrow, A., Hayden, P., Leung, D.: Superdense coding of quantum states. Phys. Rev. Lett. 92(18), 187901 (2004) 6. Harrow, A.W., Low, R.A.: Random quantum circuits are approximate 2-designs. Commun. Math. Phy. 291(1), 257–302 (2009) 7. DiVincenzo, D.P., Leung, D.W., Terhal, B.M.: Quantum Data Hiding. IEEE Trans. Inf Theory 48(3), 580–599 (2002) 8. Gross, D., Audenaert, K., Eisert, J.: Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys. 48, 052104 (2007) 9. Dankert, C., Cleve, R., Emerson, J., Livine, E.: Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80(1), 012304 (2009) 10. Brown, W., Viola, L.: Convergence rates for arbitrary statistical moments of random quantum circuits. Phys. Rev. Lett. 104, 250501 (2010) 11. Dahlsten, O.C.O., Oliveira, R., Plenio, M.B.: The emergence of typical entanglement in two-party random processes. J. Phys. A-Math. Theo. 40, 8081–8108 (2007) 12. Oliveira, R., Dahlsten, O.C.O., Plenio, M.B.: Generic entanglement can be generated efficiently. Phys. Rev. Lett. 98, 130502 (2007) 13. Feller, W.: An Introduction to Probability Theory and Its Applications. Volume 1. New York: Wiley, 3rd edition, 1968 14. Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times: With a Chapter on Coupling from the Past by James G. Propp and David B. Wilson. Providence, RI: Amer. Math. Soc., 2008
Comment on “Random Quantum Circuits are Approximate 2-designs”
293
15. Aldous, D., Diaconis, P.: Shuffling cards and stopping times. The American Mathematical Monthly 93(5), 333–348 (1986) 16. Diaconis, P., Shahshahani, M.: Generating a random permutation with random transpositions. Probability Theory and Related Fields 57(2), 159–179 (1981) 17. Diaconis, P.: Group representations in probability and statistics. Hayward, CA: Inst. Math. Stat., 1988 18. Knuth, D.E.: The art of computer programming, Vol. 2: Seminumerical algorithms. Reading, MA: Addison-Wesley, 3rd edition, 1997 19. Boyd, S., Diaconis, P., Parrilo, P., Xiao, L.: Symmetry Analysis of Reversible Markov Chains. Internet Mathematics 2(1), 31 (2005) 20. Diniz, I. Tuche de A.: Algoritmos quânticos para a geração de unitários pseudo-aleatórios. Master’s thesis, Universidade Federal Fluminense, Brazil, 2009 21. Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Amer. Stat. Assoc. 58, 13–30 (1963) Communicated by M.B. Ruskai
Commun. Math. Phys. 304, 295–328 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1228-7
Communications in
Mathematical Physics
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials Genadi Levin Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel. E-mail: [email protected] Received: 18 May 2009 / Accepted: 2 November 2010 Published online: 31 March 2011 – © Springer-Verlag 2011
Abstract: For an infinitely renormalizable quadratic map f c : z → z 2 + c with the sequence of renormalization periods {km } and rotation numbers {tm = pm /qm }, we −1 log | p | > 0, then the Mandelbrot set is locally connected at prove that if lim sup km m c. We prove also that if lim sup |tm+1 |1/qm < 1 and qm → ∞, then the Julia set of f c is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor. 1. Introduction Theorem 1. Suppose that for some quadratic polynomial f (z) = z 2 + c0 there is an increasing sequence n m → ∞ of integers, such that f n m is simply renormalizable, and lim sup m→∞
log | pm | > 0, nm
(1)
where pm /qm ∈ (−1/2, 1/2] denotes the rotation number (written in lowest terms and with qm ≥ 1) of the separating fixed point of the renormalization f n m . Then c0 lies in the boundary of the Mandelbrot set M and M is locally connected at c0 . If, in addition, the renormalizations f n m are non-primitive (satellite), then the same conclusion holds under a weaker condition: lim sup m→∞
log qm > 0. nm
(2)
The sequence {n m } above is a subsequence of the sequence of all renormalization periods of f . If we suppose that {n m } represents all the renormalization periods of f , and Research supported in part by an ISF grant number 799/08.
296
G. Levin
all of them are satellite and, moreover, pm /qm are close enough to zero, then a weaker combinatorial condition is sufficient, see Theorem 7(1)–(1’) as well as Theorem 2(1). The following problems are central in holomorphic dynamics: MLC conjecture: “The Mandelbrot set is locally connected”, and its dynamical counterpart: “For which c is the Julia set Jc of f c locally connected?” The MLC conjecture is equivalent to the following rigidity conjecture: if two quadratic polynomials with connected Julia sets and all periodic points repelling are combinatorially equivalent, then they are affinely conjugate. The MLC implies that hyperbolic dynamics is dense in the space of complex quadratic polynomials [8] (see also [50]). Yoccoz (see [17]) solved the above problems for at most finitely renormalizable quadratic polynomials as follows: for such a non hyperbolic map f c , the Julia set Jc of f c is locally connected (provided f c has no neutral periodic points), and the Mandelbrot set M is locally connected at c. (For the (sub)hyperbolic maps and maps with a parabolic point the problem about the local connectivity of the Julia set had been settled before in the works by Fatou, Douady and Hubbard and others, see e.g. [4].) At the same time, infinitely renormalizable dynamical systems have been studied intensively, see e.g. [14,34,35] and references therein. In his work on renormalization conjectures, Sullivan [53] (see also [37]) introduced and proved so-called “complex bounds” for real infinitely renormalizable maps with bounded combinatorics. Roughly speaking, complex bounds mean that a sequence of renormalizations is precompact. This property became the focus of research. It has played a basic role in recent breakthroughs in the problems of local connectivity of the Julia set and rigidity for real and complex polynomials, see [1,15,19–21,24,25,27,49]. On the other hand, there are maps without complex bounds. Indeed, Douady and Hubbard showed the existence of an infinitely renormalizable f c , such that its small Julia sets do not shrink (and thus f c has no complex bounds) but still such that the Mandelbrot set is locally connected at this c, see [38,52]. Theorem 1 provides a first class of combinatorics of infinitely renormalizable maps f c without (in general) complex bounds, for which the Mandelbrot set is locally connected at c. Obviously, previous methods in proving the local connectivity of M do not work in this case. Our method is based on an extension result for the multiplier of a periodic orbit beyond the domain where it is attracting, see the next section. That the maps with the combinatorics described in Theorem 1 in general do not have complex bounds and can have non locally connected Julia sets follow from the second result, Theorem 2 stated below, which makes the qualitative construction of Douady and Hubbard of an infinitely renormalizable quadratic map with a non-locally connected Julia set, quantitative. It is based on the phenomenon known as cascade of successive bifurcations (see e.g. [26]), and is the following. Let W be a hyperbolic component of the interior of the Mandelbrot set, so that f c has an attracting periodic orbit of period n for c ∈ W . Given a sequence of rational numbers tm = pm /qm = 0 in (−1/2, 1/2] (here and below pm , qm are assumed to be co-primes and with qm ≥ 1), choose a sequence of hyperbolic components W m as follows: W 0 = W , and, for m ≥ 0, the closure of the hyperbolic component W m+1 touches the closure of the hyperbolic component W m at the point cm with the internal argument tm = pm /qm (see the next section). When the parameter c crosses cm from W m to W m+1 , the periodic orbit of period n m = nq0 . . . qm−1 which is attracting for c ∈ W m “gives rise” to another periodic orbit of period n m+1 = n m qm which becomes attracting for c ∈ W m+1 . Thus when the parameter c moves to a limit parameter through the hyperbolic components W m , the dynamics undergoes a sequence of bifurcations precisely at the parameters cm , m = 0, 1, . . .. In the case when n = 1 and tm = 1/2 for all m ≥ 0, the parameters cm are real, and we get the famous period-doubling cascade on the real line known since the 1960’s
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
297
[43]. The corresponding limit parameter lim cm = c F = −1.4 . . .. The Julia set Jc F is locally connected [18,27]. Douady-Hubbard’s construction shows that if the sequence {tm } tends to zero fast enough, then the periodic orbits generated by the cascade of successive bifurcations at the parameters cm , m > 0, stay away from the origin. This implies that the Julia sets of the renormalizations of f c∗ for c∗ = lim cm do not shrink, and Jc∗ is not locally connected at zero. Their construction is by continuity, and it does not give any particular sequence of tm . By the Yoccoz bound for limbs, see [17], {tm } can be chosen inductively so close to zero that M is locally connected at c∗ . Milnor suggests in [39], p. 21, that the convergence of the series: ∞
|tm |1/qm−1 < ∞
(3)
m=1
could be a criterion that the periodic orbits generated by the cascade after all bifurcations stay away from the origin. Theorem 2 shows that the condition (3) is indeed sufficient for this, but not the optimal one (cf. [28]). We give a weaker sufficient condition and thus prove: Theorem 2. Let t0 , t1 , . . . , tm , . . . be a sequence of rational numbers tm = pm /qm ∈ (−1/2, 1/2]\{0}, such that limm→∞ qm = ∞ and lim sup |tm |1/qm−1 < 1.
(4)
m→∞
Let W be a hyperbolic component of period n, and the sequence of hyperbolic components W m , m = 0, 1, 2, . . ., is built as above, so that W m+1 touches W m at the point cm with the internal argument tm . Then: (1) the sequence {cm }∞ m=0 converges to a limit parameter c∗ ∈ ∂ M and M is locally connected at c∗ , (2) the map f c∗ is infinitely renormalizable with non locally connected Julia set. If, for instance, |tm+1 |1/qm ≤ 1/2 for big indexes m, then qm+1 ≥ 2qm → ∞, and Theorem 2 applies. Let us make some further comments. As it is mentioned above, we prove (2) by showing that the conditions of Theorem 2 imply that the cascade of bifurcated periodic orbits of f c∗ stay away from the origin. The conclusion (2) of Theorem 2 breaks down if not all its conditions are valid. Indeed, if tm = 1/2 for all m, then |tm+1 |1/qm = 1/21/2 < 1. At the same time, as it is mentioned above, the limit Julia set Jc F is locally connected. Theorem 2 is a consequence of a more general Theorem 7, see Sect. 4. Note in conclusion that there is a similarity between Theorems 2, 7 and the celebrated Bruno - Yoccoz criterion of the (non-)linearizability of quadratic map near its irrational fixed point.1 In Sect. 2 we state Theorems 3, 5 proved in [28], and Theorems 4, 6 that we prove here, see the last Sect. 5. Theorem 1 is derived from Theorems 5, 6 in Sect. 3. In turn, Theorem 7 is stated and proved in Sect. 4. It implies Theorem 2. Throughout the paper, B(a, r ) = {z : |z − a| < r }. We use both symbols exp(z) and e z to denote the exponential of z ∈ C. 1 This reflects a similarity between the phenomena of non local connectivity of the Julia set in both cases. Recently Xavier Buff and Alexandre Dezotti proposed a far reaching conjecture about this analogy [2].
298
G. Levin
2. Multipliers 2.1. Hyperbolic components. A component W of the interior of M is called an n-hyperbolic if f c , c ∈ W , has an attracting periodic orbit OW (c) of period n. Denote by ρW (c) the multiplier of OW (c). By the Douady-Hubbard-Sullivan theorem [4,7,36], ρW is a analytic isomorphism of W onto the unit disk, and it extends homeomorphically to the boundaries. Given a number t ∈ (−1/2, 1/2], denote by c(W, t) the unique point in ∂ W with the internal argument t, i.e. ρW at this point is equal to exp(2π t). The root of W is the point cW = c(W, 0) with the internal argument zero. If t = p/q is a rational number, we will always assume that p, q are co-primes and q ≥ 1. For any rational t = 0, denote by L(W, t) the connected component of M\{c(W, t)} which is disjoint with W . It is called the t-limb of W . Denote also by W (t) a nq-hyperbolic component with the root point c(W, t); its closure touches W at this point. The limb L(W, t) contains W (t). The hyperbolic component W is called primitive if its root cW is not a point in the closure of other hyperbolic component. Otherwise it is non-primitive. Throughout the paper, we use a well-known notion of the external ray and its angle (or argument), see e.g. [4,7,8,33]. External rays of the Julia set and the Mandelbrot set will be called dynamical rays and parameter rays, respectively. Let W be an n-hyperbolic component. If n > 1, the root cW of W is the landing point of two parameter rays with rational angles 0 < − (W ) < + (W ) < 1. If n = 1, i.e. W is the main cardioid, then cW = 1/4 is the landing point of the only parameter ray of angle zero (and one puts − (W ) = 0, + (W ) = 1 in this case). Period of each angle ± (W ) under the doubling map σ : t → 2t (mod1) is equal to n. The angles ± (W ) can be characterized as follows. The map f = f cW has a parabolic orbit P, and ± (W ) are the nearest arguments among arguments of the external rays of f , that land at each point of P. See also Sect. 3.1. The wake (see [17]) of a hyperbolic component W is the only (open) component W ∗ of the plane containing W cut by the parameter rays of arguments ± (W ) (together with their common landing point cW ). The points of the periodic orbit OW (c) as well as its multiplier ρW extend as analytic functions to the wake W ∗ [17,32]. Moreover, |ρW | > 1 in W ∗ \W . Finally, the following relation will be used. If W is an n-hyperbolic component and p/q = 0 is rational, then + (W ( p/q)) − − (W ( p/q)) =
(s+ − s− )(2n − 1) . 2nq − 1
(5)
Here the integers 0 ≤ s− < s+ ≤ 2n − 1 are the “periods” in the 2n -expansions of the angles ± (W ) of the root of W , i.e. ± (W ) = s± /(2n − 1). This follows from Douady’s tuning algorithm [6], see also [32] (formula (6.1) with d = 2n combined with Theorem 7.1) or Proposition 2.4.3 of [51]. 2.2. Analytic extension of the multiplier. Given C > 1, consider an open set of points in the punctured ρ-plane defined by the inequality |ρ − 1| > C log |ρ|.
(6)
It obviously contains the set D∗ = {ρ : 0 < |ρ| ≤ 1, ρ = 1} and is disjoint with an interval 1 < ρ < 1 + . Denote by (C) the connected component of which
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
299
contains the set D∗ completed by 0. Denote also by log (C) the set of points L = log ρ = x + i y, ρ ∈ (C), |y| ≤ π . (C) is simply-connected. More precisely, the intersection of log (C) with any vertical line with x = x0 > 0 is either empty or equal to two (mirror symmetric) intervals. If C > 4, then x < 2/(C − 2) for all L = x + i y in log (C). If C is large enough, log (C) contains two (mirror symmetric) domains bounded by the lines y = ±(C/2)x (x > 0) and y = ±π . Let W be an n-hyperbolic component of M. The map ρW from W onto the unit disk c → ρW (c) has an inverse, which we denote by c = ψW (ρ). It is defined so far in the unit disk. In [28] we prove the following. Theorem 3. (a) There exists B0 > 0 as follows. Suppose that, for some c, the map f c has a repelling periodic orbit of exact period n, and the multiplier of this orbit is equal to ρ(c). Assume that |ρ(c)| < e. Then |ρ (c)| 4n log |ρ(c)| + (1 + o(1)) (7) |ρ(c) − 1| ≤ B0 n |ρ(c)| as n → ∞. (b) Denote n = (n −1 4n B0 ) and consider its log-projection n = log (n −1 4n B0 ) = {L = x + i y : exp(L) ∈ n , |y| ≤ π } . log
Then the function ψ = ψW extends to a holomorphic function in the domain n . ˜ n of n defined by its log-projection ˜ log (c) The function ψ is univalent in a subset n = log log ˜ n , |y| ≤ π } as follows: ˜ n = n \{L : |L − Rn | < Rn }, {log ρ = x + i y : ρ ∈ where Rn depends on n only and has an asymptotics Rn = (2 + O(2−n ))n log 2 as ˜ n by ψ is contained in the wake W ∗ . n → ∞. Finally, the image of In the present paper we find a bigger extension for the function ψW in the case of non-primitive W : Theorem 4. There exists K˜ > 0 as follows. Let W be a non-primitive n-hyperbolic component, i.e. W = Z (t0 ), for some n 0 -hyperbolic component Z and some t0 = p0 /q0 = 0, and n = n 0 q0 . Then ψW extends to a univalent function in a domain n 0 ,t0 which con˜ n and a neighborhood of the point 1 ∈ ∂n , and is defined by its log-projection sists of as follows: log ˜ log n 0 ,t0 := L = x + i y : exp(L) ∈ n 0 ,t0 , |y| ≤ π = n ∪ B(0, d), log log where d = K˜ min{n| p0 |4−n 0 , n −1 }. If q0 is large enough, n 0 ,t0 coincides with n ∪ B(0, d).
Along with Theorem 3, the proof of Theorem 4 is based on geometric relations between the multiplier maps ρ Z and ρW of the hyperbolic components Z and W = Z (t0 ) respectively near the common point c0 = c(Z , t0 ) = c(W, 0) of their boundaries. We start with the following known relation [13]: |(dρW /dρ Z )(c0 )| = q02 . Then we show in Lemmas 4.1 and 4.2–4.3 that ρW as a function of ρ Z extends to a univalent function in a disk centered at the point ρ Z (c0 ) and of radius proportional to r (n 0 , p0 /q0 ) = min{1/(n 0 q03 ), 2−n 0 /2 | p0 /q0 |} as far as the function ψ Z = ρ Z−1 allows ˜ n 0 . It follows to do such an extension, i.e. univalent. The latter holds in the domain log 2 ˜ that d is roughly q0 min{r (n 0 , p0 /q0 ), dist (2πi p0 /q0 , ∂ n 0 )}. See Sect. 5.1 for the complete proof. Note that the proof of Lemma 4.1 uses some classical tools as well as bounds for multipliers from [47] and [30], see Sect. 4.2 and Appendix. This lemma provides also a step in proving Theorem 7.
300
G. Levin
2.3. Limbs. Let W be an n-hyperbolic component. For every t = p/q = 0, consider the hyperbolic component W (t), the corresponding wake W (t)∗ and the limb L(W, t). Then a branch log ρW is well-defined in the open set W (t)∗ , such that log ρW (c) → 2πit as c → c(W, t), and, for every c ∈ L(W, t), the point log ρW (c) is contained in the following round disk (Yoccoz’s circle): n log 2 n log 2 |< , (8) Yn (t) = L : |L − 2πit + q q see [17,46] and references therein. See also Theorem 10 of Appendix. Comment 1. An important corollary of the Yoccoz bound (8) is that every point in M ∩ W ∗ either belongs to the closure of the hyperbolic component W , or belongs to some limb L(W, p/q) (see [17,40]). Theorem 1 will be a consequence of the following two bounds on the size of limbs. The first one is proved in [28] and the proof is based on Theorem 3(a)–(b) and (8). Theorem 5. There exists A > 0, such that, for every n-hyperbolic component W and every t = p/q ∈ (−1/2, 1/2], the diameter of the limb L(W, t) is bounded by: diam L(W, t) ≤ A
4n . | p|
(9)
The second bound concerns the non-primitive components. Its proof is based on Theorems 3(a), 4, 5, and on (8), and it states the following. Theorem 6. There exists A˜ > 0, such that, if an n-hyperbolic component W is not primitive, then n
8 diam L(W, p/q) ≤ A˜ . q
(10)
See Sect. 5.2 for the proof. 3. Rigidity 3.1. Simple renormalization. We follow some terminology as in [34], see also [40]. For the theory of polynomial-like maps, see [9]. Let f be a quadratic polynomial with connected Julia set. The map f n is called renormalizable if there are open disks U and V such that f n : U → V is a polynomial-like map with a single critical point at 0 and with connected Julia set Jn . The map f n : Jn → Jn has two fixed points counted with multiplicity: β (non-separating) and α. Denote them by βn and αn . The renormalization is simple if any two small Julia sets f i (Jn ), i = 0, 1, . . . , n − 1 cannot cross each other, i.e. they can meet only at some iterate of βn . If these small Julia sets don’t meet, the renormalization f m n is called of disjoint type, or primitive. Otherwise it is called of β-type, non-primitive, or satellite type. Every repelling periodic point z of f of period n has a well-defined rational rotation number p/q ∈ (−1/2, 1/2], which is defined by the order at which f n permutes (locally) q external rays landing at z in the couterclockwise direction. If the fixed point αn of f n is repelling, then it has a non-zero rational rotation number p/q, which can be defined equivalently as follows: f n : U → V is the hybrid equivalent to a quadratic polynomial which lies in the p/q-limb of the main cardioid.
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
301
3.2. Demonstration of Theorem 1. We split the proof into a few steps. A. Let f n : U → V be a simple renormalization of f , βn its β-fixed point, and n−1 the periodic orbit containing βn . We use some notions and results On = { f i (βn )}i=0 from [8,40]. The characteristic arc I (On ) = (τ− (On ), τ+ (On )) of On is the shortest arc (measured in S 1 ) between the external arguments of the rays landing at the points of On . Then τ± (On ) are the arguments of two dynamical rays that land at the point βn = f (βn ) of On , and c0 = f (0) lies in the sector bounded by these rays and disjoint with 0. Furthermore, the two parameter rays of the same arguments τ± (On ) land at a single parameter c(On ). The point c(On ) is the root of a hyperbolic component denoted by Wn , and the above parameter rays completed by c(On ) bound the wake Wn∗ of this component. Denote by L(On ) = Wn∗ ∩ M the corresponding limb. Thus, (A1) c0 ∈ L(On ), (A2) moreover, the rotation number of the α-fixed point αn of the renormalization is p/q if and only if c0 ∈ L(Wn , p/q) - the limb of Wn which is attached at the point of ∂ Wn with the internal argument p/q. B. Let m < m . By [34], since all renormalizations f n m are simple, n m divides n m . Furthermore, by (A2), (B1). τ− (On m )<− (Wn m ( pm /qm ))≤τ− (On m ) < τ+ (On m ) ≤ τ+ (Wn m ( pm /qm )) < τ+ (On m ), thus the sets L(On m ) form a decreasing sequence of compact sets containing c0 . Therefore, for the local connectivity of M at c0 it is enough to prove that the set S = ∩m L(On m ) consists of a single point. (B2). It is easy to see that τ+ (On m ) − τ− (On m ) → 0. Indeed, by (B1) and (5), τ+ (On m ) − τ− (On m ) ≤ (2n m − 1)2 /(2n m qm − 1), and qm > 2. Thus, there exists lim τ± (On m ) = τ0 .
(11)
Note that τ0 is not periodic under the doubling map σ (t) = 2t (mod 1). C. Let c be any point from S. (C1) All periodic points of f c are repelling. Indeed, obviously, f c cannot have an attracting cycle. If f c has an irrational neutral periodic orbit then c lies in the boundary of a hyperbolic component contained in S, a contradiction with (11). If f c has a neutral parabolic periodic orbit, then c is the landing point of precisely two parameter rays with periodic arguments, if c = 1/4, and the only ray landing at it, of zero argument, if c = 1/4, again the same contradiction. Thus, all cycles are repelling. Consider the so-called real lamination λ(c) of f c [22]. It is a minimal closed equivalence relation on S 1 that identifies two points whenever their prime end impressions intersect. For every m, {τ− (On m ), τ+ (On m )} ⊂ λ(c). Since λ(c) is closed, τ0 ∈ λ(c). Moreover, for every m, there is a pair of angles {τm− , τm+ }, such that their images under the doubling map {σ (τm± )} ⊂ {τ± (On m )}, and {τm− , τm+ } is contained in the class of λ(c) corresponding to the point βn m . Passing to the limit, we get that the pair {τ0 /2, τ0 /2 + 1/2} is the critical class of λ(c). The following statements are known after Thurston and Douady and Hubbard and proved in a much more general form in [23], Prop. 4.10: if the critical class {τ0 /2, τ0 /2 + 1/2} is contained in a class of λ(c), then λ(c) is determined by (the itinerary of) τ0 . Since τ0 is the same for all c ∈ S, we conclude that the real laminations of all f c , c ∈ S, coincide. In particular, for every c ∈ S, f cn m is simply renormalizable
302
G. Levin
because by [34] the simple renormalizations are detected by the lamination. The renormalization f cn m is hybrid equivalent to some f T (c,m) [9]. Denote nˆ k = n m+k /n m , k > 0. k Then f Tnˆ (c,m) is simply renormalizable, and the rotation number of its α-fixed point is pm+k /qm+k because it is a topological invariant. (C2) By (A2), T (c, m) is contained in the intersection of a decreasing sequence of limbs L m,k , k = 1, 2, . . ., such that L m,k is attached to a hyperbolic component of period nˆ k at the internal argument pm+k /qm+k . By Theorem 5, the diameter of L m,k , n m+k
diam L m,k
4nˆ k 4 nm ≤A =A . | pm+k | | pm+k |
Since lim sup(log | pm |)/n m > 0, one can find and fix m in such a way, that n m+k
4 nm = 0. lim inf k→∞ | pm+k | It means, that, for the chosen m, the limbs L m,k (k → ∞) shrink to a point cˆ = T (c, m), so that cˆ depends on m but is independent of c ∈ S. Thus f cn m is quasi-conformally conjugate to f cˆ , for all c ∈ S. D. Assume that the compact S has at least two different points. Then S contains a point c1 ∈ ∂ M different from c0 . By (C1)–(C2), the renormalizations f cn0m of f c0 and f cn1m of f c1 are quasi-conformally conjugate, all periodic points of f c0 , f c1 are repelling, and λ(c0 ) = λ(c1 ). We are in a position to apply Sullivan’s pullback argument (see [37]). Using a quasi-conformal conjugacy near small Julia sets (rather than on the postcritical set) and an appropriate puzzle structure, we arrive at a quasi-conformal conjugacy between f c0 , f c1 . Since c1 ∈ ∂ M, then according to [9], c0 = c1 . This finishes the proof of Theorem 1 under the condition (1). E. Now assume that every renormalization f m n is also non-primitive. Then the hyperbolic components Wn m are non-primitive as well [40], and we can apply the bound of Theorem 6 instead of Theorem 5. It gives the same conclusions under the weaker condition (2). The proof stands the same with some obvious changes in (C2). 4. Non locally Connected Julia Sets 4.1. Statement. Let t0 , t1 , . . . , tm , . . . be a sequence of non-zero rational numbers tm = pm /qm ∈ (−1/2, 1/2]. Let us introduce the following conditions: (Y 0)a sup |tm |q0 . . . qm−1 < ∞.
(12)
m≥1
(Y 0)b inf
m≥2
qm−1 | log | pm−1 |tm |
q0 . . . qm−2
> 0.
(13)
(Y 1) There exists β > 0, such that lim sup m→∞
max{(q0 . . . qm−1
qm 2 ) , exp(βq
0 . . . qm−2 )}
> 0.
(14)
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
303
(S) for some k ≥ 0 and γ > 0, ∞
u k,m H (u k,m+1 ) < ∞, qm (1 − u k,m )
(15)
exp(γ qk . . . qm−1 ) 1/qm = tm+1 max qk . . . qm , ,m ≥ k | pm q m |
(16)
m=k
where u k,m
(we set qk . . . qm−1 = 1, if m = k), and H (u) = 16u∞ k=1
(1 + u 2k )8 . (1 − u 2k−1 )8
(17)
In particular, H : [0, 1) → [0, ∞) is strictly increasing from zero to infinity, and extends to a holomorphic function in the unit disk. For more information about H , see Subsect. 4.3. See also a remark on the condition (S) in the beginning of Subsect. 4.5. Let now W be a hyperbolic component of some period n ≥ 1, and the sequence {W m } of hyperbolic component is built as in the Introduction, in other words, W 0 = W , and the closure of the hyperbolic component W m+1 touches the closure of the hyperbolic component W m at the point cm ∈ ∂ W m with internal argument tm . Theorem 7. The following statements hold. 1. If the conditions (Y 0)a –(Y 0)b are satisfied, then the sequence of parameters cm , m = 0, 1, 2, . . . , converges to a limit parameter c∗ . 1’. If, additionally, the condition (Y 1) is satisfied, then the Mandelbrot set is locally connected at the limit parameter c∗ . 2. If the conditions (Y 0)a –(Y 0)b and (S) are satisfied, then the map f c∗ is infinitely renormalizable with non locally connected Julia set. Theorem 2 stated in the Introduction is a simple corollary of Theorem 7: Proposition 1. Assume that lim m→∞ qm = ∞ and, for some a < 1 and all m large enough, |tm+1 | ≤ a qm .
(18)
Then the conditions (Y 0)a−b , (Y 1), and (S) are satisfied. Proof. One can assume that, for every m ≥ m 0 , qm is large enough and (18) holds. Then (18) implies qm+1 > qm3 , for m ≥ m 0 , which, in turn, implies that qm > (qm 0 . . . qm−1 )2 qm 0 ,
(19)
for m > m 0 . Using this, we get, for m ≥ m 0 + 2, qm /(1/a)q0 ...qm−2 ≥ (1/a)qm−1 a q0 ...qm−2 = (1/a)qm−1 −q0 ...qm−2 → ∞. In particular, (Y 1) holds with β = log(1/a). As for (Y 0)a−b , we have (for m ≥ m 0 + 2): 3/2
|tm |q0 . . . qm−1 < a qm−1 q0 . . . qm 0 −1 (qm 0 . . . qm−2 )qm−1 < a qm−1 qm−1 (q0 . . . qm 0 −1 ),
304
G. Levin
and the latter sequence is bounded (it tends to zero). It proves (Y 0)a . In turn, by (18)–(19), log | pm−1 qm−1 /tm | qm−1 log(1/a) ≥ → ∞, q0 . . . qm−2 q0 . . . qm−2 which proves (Y 0)b . Let us verify (S). Using (19) with k > m 0 instead of m 0 , we 3/(2q ) have, for m ≥ k: u k,m < a max{(qm m , exp(γ qk . . . qm−1 /qm )} < a1 , for some a < a1 < 1, provided k ≥ m 0 is large enough and m ≥ k. Fixing such k, we have for m−k m ≥ k, H (u k,m+1 )u k,m /(1−u k,m ) < H (a1 )a1 /(1−a1 ). On the other hand, qm > 23 , m ≥ k. Therefore, (15) holds, too. Note that, with the help of the bound (29), see Subsect. 4.3, one can easily find sequences tm , such that |tm |1/qm−1 → 1 and such that the conditions of Theorem 7 hold. The rest of the section occupies the proof of Theorem 7. 4.2. Bifurcations. Let W be an n-hyperbolic component, and let c0 ∈ ∂ W have an internal argument t0 = p/q = 0. Consider the periodic orbit O(c) = {b j (c)}nj=1 of f c which is attracting when c ∈ W (that is, O is the orbit denoted by OW in Subsect. 2.1). Then all b j (c) as well as the mulptiplier ρW (c) of O(c) are holomorphic in W and extend to holomorphic functions in c in the whole wake W ∗ of W . As we know, the function ρW (c) is injective near c0 . Consider the inverse function ψW . It is well defined and univalent ˜ n , which includes the unit disk and a neighborhood of the point in the domain ρ0 = exp(2πi p/q), so that ψW (ρ0 ) = c0 . It is convenient to use also the composition log
ψW = ψW ◦ exp . ˜ n , which includes the left half-plane It is defined and holomorphic in the domain {L : Re(L) < 0} and a neighborhood of the point 2πi p/q. Recall that W ( p/q) denotes a hyperbolic component touching W at the point c0 . The following well-known picture describes the (local) bifurcation near c0 . For the proof, see e.g. [5] (for n = 1), [28] or [40]. Let us fix a disk B(0, δ), where δ > 0 is so small, that ψW is univalent in B(0, 1)∪ B(ρ0 , δ q ). Given s ∈ B(0, δ), define ρ = ρ0 +s q and c = ψW (ρ). Fix a small neighborhood E of the set O(c0 ). For 1 ≤ k ≤ n, there exists a function Fk , which is defined and holomorphic in B(0, δ), Fk (0) = 0, Fk (0) = 0, p/q such that, for every s ∈ B(0, δ), s = 0, the points bk, j (s) = bk (c0 ) + Fk (s exp(2πi qj )), nq 1 ≤ k ≤ n, 0 ≤ j ≤ q −1, are the only fixed points of f c in the neighborhood E, which n are different from the points of O(c) = {bk (c)}k=1 . They form a periodic orbit O p/q (c) of f c of period nq, which collides with O(c) as c → c0 . Denote Bˆ = ψW (B(ρ0 , δ q )). p/q The multiplier of O p/q (c) is the product 2nq 1≤k≤n,0≤ j≤q−1 bk, j (s), which is invariant under the change s → s exp(2πi/q). Hence, this multiplier is, in fact, a non-constant ˆ which takes the value 1 at c = c0 . As the map f c holomorphic function on c ∈ B, has at most one non-repelling periodic orbit, the cycle O p/q (c) is attracting for c ∈ Bˆ ∩ W ( p/q). Therefore, for such c, O p/q (c) is just the cycle OW ( p/q) (c) of period nq, which exists and attracting throughout W ( p/q). In particular, the multiplier of O p/q (c) is just the multiplier ρW ( p/q) (c) of the attracting periodic orbit of f c , for c ∈ W ( p/q). log
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
305
Let us make a general remark. Assume that, for some m ≥ 1 and for any c in some domain , the map f cm has no fixed points with multiplier 1. (For example, this is the case, for any m, if is a hyperbolic component.) Then, by the Implicit Function Theorem, every fixed point of f cm as well as its multiplier is defined locally as a holomorphic function, which has an analytic contination along every curve in . As for the continuation of the multiplier function, a weaker condition is enough. By the above local bifurcation picture, the multiplier of a periodic orbit of f c of period m extends analytically through a neighborhood of any parameter cˆ unless f cˆ has a periodic orbit of (exact) period m with multiplier 1 (i.e., cˆ is the root of a primitive m-hyperbolic component). Assume now that, for any c ∈ , the map f c has no periodic orbits of period m with multiplier 1. Then we have that the multiplier of any periodic orbit of f c of period m, which is defined locally near c ∈ , has an analytic continuation along every curve in , which starts at c. We will be concerned with the problem of holomorphic (=analytic) extensions (=continuations) of the multiplier functions ρW and ρW ( p/q) from a domain to a bigger domain. As the multiplier ρW ( p/q) (c) is holomorphic in a small neighborhood Bˆ of c0 and, by the above, it extends from Bˆ to holomorphic functions defined in W ( p/q) and in W , ρW ( p/q) extends to a holomorphic function defined in the simply-connected domain Bˆ ∪ W ∪ W ( p/q). Recall also that ρW ( p/q) has an analytic continuation from W ( p/q) to the wake W ( p/q)∗ , and |ρW ( p/q) | > 1 in W ( p/q)∗ \W ( p/q). Thus ρW ( p/q) is holomorphic in the domain W ∪ Bˆ ∪ W ( p/q)∗ , and |ρW ( p/q) | > 1 in (W ∪ W ( p/q)∗ )\W ( p/q). ˆ the function ρW ( p/q) is an implicit function of Now, since ρW is univalent in W ∪ B, ρ = ρW in B(0, 1)∪ B(ρ0 , δ q ). We study ρW ( p/q) as a function of ρW whenever it makes sense. The following relation between ρW ( p/q) and ρW at the bifurcation parameter c0 is proved in [13]: dρW ( p/q) q2 (c0 ) = − . dρW ρ0
(20)
Another important ingredient for us is an inequality connecting the multipliers ρW ( p/q) (c) and ρW (c) when c lies in the hyperbolic component W . For c ∈ W , such that ρW (c) = 0, the following bound takes place: | log ρW ( p/q) (c)|2 | log ρW (c) − 2πi p/q|2 < q2 , log |ρW ( p/q) (c)| − log |ρW (c)|
(21)
for some branch of log ρW ( p/q) (c) and any branch of log ρW (c). For the proof, see Appendix. Let us introduce the function log
W, p/q = ρW ( p/q) ◦ ψW . It is holomorphic in the left half-plane union with a neighborhood of the boundary point
2 2πi p/q, and W, p/q (2πi p/q) = 1, W, p/q (2πi p/q) = −q . (The latter holds by (20).) We want to know how far W, p/q is univalent. The main technical part is contained in the next Lemma 4.1 having an independent interest. Suppose g : B(0, 1) → C is a univalent function, and U a simply-connected domain, such that B(0, 1) ∪ U is also a simply-connected domain. (Below, U will be either a disk or the image of a disk by exponential map.) We say that g has a univalent extension to B(0, 1) ∪ U , if there is a function (denoted by the same letter g), which is holomorphic in B(0, 1) ∪ U , coincides with g in B(0, 1) and is (globally) univalent in B(0, 1) ∪ U .
306
G. Levin
Lemma 4.1. For every X > 0 there exist 0 < 0 < < 1 depending only on X , such that the following properties hold. Assume that, for some 0 < r < 1, the function ψW has a univalent extension to B(0, 1) ∪ U , where U = exp(B(2πi p/q, r )) = {exp(w) : w ∈ B(2πi p/q, r )} . log
Assume further that the topological disk V = ψW (B(2πi p/q, r )) = ψW (U ) containing c0 obeys the following two disjointness properties: (a) V is disjoint with any limb L(W, p /q ) of W other than L(W, p/q) and such that q ≤ q + 1, (b) V is disjoint with the subset of M, which is outside of the wake of W : V ∩ (M\W ∗ ) = ∅. Besides, V is disjoint with the parameter ray of argument zero, i.e. with the set R0 = {c > 1/4}. Then the following conclusions hold: I. The function ρW ( p/q) (c) extends to a holomorphic function defined in the domain V p/q := V ∪W ( p/q)∗ ∪W , and, for each λ ∈ B(0, 1) there is a unique cλ ∈ V p/q , such that ρW ( p/q) (cλ ) = λ. Clearly, cλ ∈ W ( p/q). II. If rq 2 < X , then the function W, p/q is well-defined and univalent in the disk 3 2
B(2πi p/q, r ), and 23 q 2 < |W, p/q (L)| < 2 q , for every L ∈ B(2πi p/q, 0 r ). In particular, the following covering property holds: for every θ ≤ 0 r , the image of B(2πi p/q, θ ) under the map W, p/q covers B(1, 2q 2 θ/3) and is covered by B(1, 3q 2 θ/2). Furthermore, the function ψW ( p/q) (the inverse to ρW ( p/q) : W ( p/q) → B(0, 1)) has a univalent extension to B(0, 1) ∪ B(1, 2q 2 0 r/3). Comment 2. Roughly speaking, the conditions (a) and (b) guarantee that the function ρW ( p/q) extends analytically through every point in V ∩ W ∗ and V \W ∗ respectively. The restriction (a) is “local” (inside of the wake) while (b) is a “global” one (in the rest of M). We give bounds on r from below in terms of n and p/q to satisfy (a) and (b) in Lemma 4.2 and Lemma 4.3 respectively. The following combinatorial fact appears in Lemma 6.1 of [28] (in a slightly different form). For completeness, we reproduce its short proof here: Proposition 2 (cf. [5]). Let W be an n-hyperbolic component. Let also c ∈ L(W, t ) ∪ c(W, t ), for some t = p /q and q > 2. Assume that f cn Q has a fixed point with the multiplier 1. Then Q ≥ q − 1. Proof. Consider the dynamical plane of f c . The critical value c lies in the sector S bounded by the dynamical rays of arguments ± (W (t )) and disjoint with 0. On the nQ other hand, c belongs to a petal at a fixed point a of the map f c with the multinQ plier 1. Hence, a is in the closure of the same sector S, too. Since ( f c ) (a) = 1, nQ every dynamical ray of f c , which lands at a, is fixed by f c (see e.g. [41]). But since c = 1/4, a is a landing point of at least two rays. Therefore, there are two rays Rt1 , Rt2 , nQ 0 < t1 < t2 < 1 (landing at a), which are fixed by f c and which lie in the closure of S. Then + (W (t )) − − (W (t )) ≥ t2 − t1 ≥ (2n Q − 1)−1 . Apply the formula (5). It gives us n Q ≥ nq − 2n + 1, that is, Q ≥ q − 1.
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
307
Now we start the proof of Lemma 4.1. We drop some indices and write ψ = ψW , ρ p/q = ρW ( p/q) , and also = W, p/q , B = B(2πi p/q, r ). First, we show Part I. We prove a more general statement, see Proposition 3 below. Given the hyperbolic component W and the point t0 = p/q = 0, we define a simply-connected domain D(W, p/q) as follows: D(W, p/q) = C\(R0 ∪ M1 ∪ M2 ∪ M3 ).
(22)
Here R0 = {c > 1/4} is the parameter ray to the root c = 1/4 of the main cardioid, and Mi , 1 ≤ i ≤ 3, are the following subsets of M: (i) M1 = M\W ∗ . Note that M1 is a continuum, which contains cW and c = 1/4 (unless W is the main cardioid and M1 = {1/4}). (ii) M2 = ∪L(W, p /q ), over all p /q = p/q, such that q ≤ q + 1. (iii) M3 is the shortest subarc of the simple closed curve ∂ W , which contains all c(W, p /q ) with p /q as in (ii), i.e., p /q = p/q and q ≤ q + 1. In other words, if, for 0 < t1 < t2 < 1, we denote l(t1 , t2 ) = {c(W, t) : t1 < t < t2 } (the open subarc of ∂ W with the end points c(W, t1 ) and c(W, t2 ), which is disjoint with cW ), then M3 = ∂ W \l(t− , t+ ). Here t± ∈ (0, 1) are the closest points to t0 of the form p /q = 0 with q ≤ q + 1 from the left and from the right of t0 (note that t± exist since 1/(q + 1) < t0 < q/(q + 1)). The set R0 ∪ M1 ∪ M2 ∪ M3 is connected, closed, unbounded, and does not separate the plane. So, D(W, p/q) is a simply-connected domain. Let us show that the domain V p/q = W ∪ W ( p/q)∗ ∪ V is contained in D(W, p/q). Indeed, W and W ( p/q)∗ are disjoint with R0 as well as with Mi , i = 1, 2, 3. Also, by the condition (a), V is disjoint with M2 and, by the condition (b), V is disjoint with M1 ∪ R0 . To show that V is disjoint with M3 , it is enough to prove that, for every r , 0 < r < r , the domain V (r ) = log ψW (B(2πi p/q, r )) is disjoint with M3 . Fix such r . Let us use the condition that ψ has a univalent extension to B(0, 1)∪U . It implies that the domain W ∪V = ψ(B(0, 1)∪U ) is simply-connected. We have: V (r ) = ψ(U (r )), where U (r ) = exp(B(2πi p/q, r )). And since r < r , the boundary of W ∪ V (r ) = ψ(B(0, 1) ∪ U (r )) is a simple closed curve. In particular, V (r ) ∩ ∂ W is the closure l0 of a single open arc l0 = ψ(∂ B(0, 1) ∩ U (r )) containing c0 because otherwise there would be a common point in the boundaries of W and V (r )\W outside of the closed arc l 0 , a contradiction with the fact that ∂(W ∪ V (r )) is a simple curve. By the condition (a), l0 ⊂ l(t− , t+ ), and, by the definition, M3 = ∂ W \l(t− , t+ ). Therefore, V (r ) ∩ M3 = ∅. We have shown that V p/q ⊂ D(W, p/q). As we know, ρ p/q is well-defined and holomorphic in the domain D0 = Bˆ ∪ W ( p/q)∗ ∪ W , where Bˆ is a small neighborhood of the common boundary point c0 of W and W ( p/q)∗ . As W ( p/q)∗ , W are subsets of D(W, p/q) and Bˆ is small, one can assume that D0 ⊂ D(W, p/q). Since V p/q ⊂ D(W, p/q), Part I follows immediately from a general Proposition 3. The function ρ p/q extends from D0 to a holomorphic function defined in the domain D(W, p/q), and, for each λ ∈ B(0, 1) there is a unique cλ ∈ D(W, p/q), such that ρ p/q (cλ ) = λ. Besides, cλ ∈ W ( p/q).
308
G. Levin
Proof. The function ρ p/q has an analytic continuation along every curve starting at c0 , which does not contain a parameter c, such that f c has a periodic orbit of period nq with multiplier 1. We will call such parameters c suspicious. Every suspicious point lies in the boundary of M. Denote by I the set of those suspicious points, which are outside of the wake W ( p/q)∗ . Let us prove that I ∩ D(W, p/q) = ∅.
(23)
Assume the contrary, i.e., there is c ∈ D(W, p/q)\W ( p/q)∗ , such that f c has a periodic orbit of period nq with multiplier 1. Since c ∈ ∂ M ∩ D(W, p/q)\W ( p/q)∗ , then either (1) c ∈ l(t− , t+ ), or (2) c ∈ L(W, p /q ), where p /q = p/q and q > q + 1. As f c , for c ∈ ∂ W , has a neutral periodic orbit of period n, the case (1) is excluded. The case (2) is excluded by the above Proposition 2 (with Q = q). Thus (23) holds. Denote the closure of W ( p/q)∗ by K . The function ρ p/q has a holomorphic extension from W ( p/q)∗ to a small neighborhood S of K , because ∂ K ∩ M = {c0 } and ρ p/q is holomorphic in the neighborhood Bˆ of c0 . One can assume that S ⊂ D(W, p/q). Since K is an unbounded continuum not separating the plane, D = D(W, p/q)\K is a simply-connected domain, too. By (23), D does not contain any suspicious point. Hence, the function ρ p/q , which is holomorphic in a subdomain S\K of D , has an analytic continuation along every curve in D . By the Monodromy Theorem, ρ p/q has a well-defined analytic continuation ρ˜ p/q from S\K to D , and, by the Uniqueness Theorem, ρ˜ p/q on D and ρ p/q on K define an analytic continuation of ρ p/q to D(W, p/q) = D ∪ K . Thus we have shown that the function ρ p/q extends from D0 to a holomorphic function defined in the domain D(W, p/q). As ρ p/q : B(0, 1) → W ( p/q) is a homeomorphism and |ρ p/q | > 1 in (W ( p/q)∗ \W ( p/q)) ∪ W , then, for every |λ| ≤ 1 there is one and only one cλ in W ( p/q)∗ ∪ W , such that ρ p/q (cλ ) = λ. Moreover, cλ ∈ W ( p/q). Denote ˜ Observe D˜ = D(W, p/q)\(W ( p/q)∗ ∪ W ). It remains to show that |ρ p/q | > 1 in D. that, for every c ∈ D(W, p/q), except for perhaps finitely many c (for which c is the root of a non-primitive nq-hyperbolic components), ρ p/q is the multiplier of some periodic orbit of f c of period nq. Assume now, by a contradiction, that, for some |λ| ≤ 1 and ˜ ρ p/q (c1 ) = λ. Then c1 lies in the closure of some nq-hyperbolic component c1 ∈ D, W1 . Since c1 ∈ / M1 ∪ L(W, p/q), then W1 lies in a limb of W other than its p/q-limb. Hence, for some p /q = p/q, c1 ∈ W1 ⊂ L(W, p /q ) ∪ {c(W, p /q )}. We have: q > q + 1, because otherwise c1 ∈ M2 . On the other hand, consider the root c˜ of W1 . nq Then the map f c˜ has a fixed point with multiplier 1. By Proposition 2, q ≤ q + 1, which is a contradiction, because q > q + 1. Let us pass to the proof of Part II. It has three main ingredients. The first one is the inequality (21). The second one is Proposition 4 below, which follows, for example, from the theory of quasinormal families due to P. Montel [42]. A family of holomorphic functions in a domain D is called quasinormal in D, if every sequence of maps of the family contains a subsequence, which converges locally uniformly in D except for, possibly, a finite number of points. The points where the convergence is not locally uniform are called irregular points. If the number of irregular points is always at most N , the family is called quasinormal of order at most N (if N = 0, the family is normal). It is clear from the Maximum Principle, that outside of the irregular points the sequence converges locally uniformly to infinity. Montel [42] proves the following main criterion of quasinormality: A family of functions which are holomorphic in a domain, where it takes at most N0 times the value 0 and at most N1 times the value 1 is quasinormal in the domain of order at most the minimum of N0 , N1 . As an immediate corollary, we have:
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
309
Proposition 4. Assume that {g} is a family of functions which are holomorphic in the unit disk B(0, 1), and such that: (1) g(z) = 0 if and only if z = 0, (2) there exists Z = 0, so that each g of the family takes the value Z in at most one point, and (3) there exists z 0 = 0, so that the set {g(z 0 )} is bounded. Then the family {g} is uniformly bounded on any compact in the unit disk. The last ingredient is the following statement, which is Lemma I from [44]: Proposition 5. Let ω(z) = αz + c2 z 2 + · · · be regular and |ω(z)| ≤ |z| for |z| < 1, and let ω(z) further satisfy ω(z) = 0 for 0 < |z| < 1; then ω(z) is univalent inside the circle |z| = R(α) with R(α) = 1 + log(1/|α|) − [(1 + log(1/|α|))2 − 1]1/2 . Let us turn to the proof of II. Consider = ρ p/q ◦ ψ ◦ exp. It is holomorphic in B and such that (2πi p/q) = 1 and (2πi p/q) = −q 2 . Define g(w) =
(r w + 2πi p/q) − 1 . rq 2
(24)
Then g is holomorphic in the unit disk, moreover, g(0) = 0 and g (0) = −1. We are going to show that g satisfies Conditions (1)–(3) of Proposition 4 with Z = −1/(2X ) and with z 0 = −1/2. (1) Assume g(w) = 0. It means that ρ p/q (c) = 1, for some c ∈ V . By Part I, then c = c0 , i.e. w = 0. (2) Assume g(w) = −1/(2X ). Then, for some c1 ∈ V , ρ p/q (c1 ) = 1 − rq 2 /(2X ). Since rq 2 < X , the point λ = 1 − rq 2 /(2X ) ∈ (1/2, 1), i.e., it lies in the unit disk. Hence, by Part I, such c1 is unique, that is, w is the only solution of the equation g(w) = −1/(2X ). (3) Let us fix w = −1/2. Now, apply (21) with c = ψ ◦ exp(−r/2 + 2πi p/q) and log ρW (c) = −r/2 + 2πi p/q: 2 | log(1 + rq 2 g(−1/2))|2 1 2 | − r/2| < q = rq 2 . 2 log |1 + rq g(−1/2))| −(−r/2) 2
Geometrically, it means that the point 1+δg(−1/2) belongs to the set E = {exp(z) : |z − δ/4| < δ/4}, where δ = rq 2 . If δ < δ0 , where δ0 is a small fixed number, then E is contained in a small disk around 1 of radius at most δ. Hence, |g(−1/2)| < 1. On the other hand, if δ ≥ δ0 and δ < X , then |g(−1/2)| < (1 + exp(δ))/δ ≤ (1 + exp(X ))/δ0 . We have checked that the conditions (1)-(3) hold for every g as above. Therefore, by Proposition 4, for every X > 0 there is C, such that |g(w)| < C, for |w| < 9/10. Now we can apply Proposition 5 to the function ω(z) = g(9z/10)/C, |z| < 1. We get that g is univalent in the disk |w| < := (9/10)R(9/(10C)). It means that is univalent in the disk B(2πi p/q, r ). By the classical distortion bounds for univalent maps, there exists 0 , which depends on only, such that 2/3 < | (L)/ (2πi p/q)| < 3/2, for L ∈ B(2πi p/q, 0 r ). Since | (2πi p/q)| = q 2 , this proves the covering property. To complete the proof of Part II, let us show that the function ψW ( p/q) has a univalent extension to B(0, 1) ∪ B(1, 2q 2 0 r/3). Indeed, by what we have just proved, log −1 is well-defined and univalent in B(1, 2q 2 r/3). In the function ρ −1 0 p/q = ψW ◦ other words, ψW ( p/q) = ρ −1 p/q has an analytic continuation from B(0, 1) to the domain B := B(0, 1)∪ B(1, 2q 2 0 r/3) (denote this continuation again by ψW ( p/q) ). In order to
310
G. Levin
show that ψW ( p/q) is univalent in B , observe that ψW ( p/q) (B ) ⊂ W ( p/q)∪ V ⊂ V p/q , and, by Part I, ρ p/q is a holomorphic function in V p/q . Thus, ψW ( p/q) in B has a welldefined inverse function ρ p/q . The proof of the lemma is completed. Comment 3. Obviously, Lemma 4.1 remains valid if the constant 0 is replaced by a 1−|z| 1+|z|
smaller one. Using the classical bounds (1+|z|) 3 ≤ | f (z)| ≤ (1−|z|)3 for any univalent function f (z) = z + . . . in the unit disk (see e.g. [12]), it is easy to check that, by the optimality of the bounds, 0 < /8, and one can take 0 = /16. In the next two lemmas we give some bounds on r from below in terms of n and p/q to satisfy the conditions (a) and (b) of the above Lemma 4.1. Lemma 4.2 (cf. [5]). Let n ≥ 1 and p/q = 0. Given an n-hyperbolic component log log W , consider the corresponding function ψW . Assume that ψW extends to a function, which is defined and univalent on the disk B = B(2πi p/q, 1/(2nq 3 )). Then the domain log V = ψW (B) is disjoint with any limb L(W, p /q ) other than L(W, p/q) and such
that q ≤ q + 1. Proof. The image of any limb L(W, p /q ) by the function log ρW is contained in 2 n log 2 Yoccoz’s circle Yn ( p /q ) = {L : |L − (2πi p /q + n log q )| < q }. Let us estimate
the distance d between the point 2πi p/q and the circle Yn ( p /q ), where p /q = p/q and q ≤ q + 1. Then | p /q − p/q| ≥ 1/(qq ) and, hence,
1/2 2 p
p n log 2 2 n log 2 2π d= − + −
q q q q
1/2 4π 2 1 2 ≥
+ (n log 2) − n log 2 q q2 ≥
4π 2 q 2 (q + 1) 4π 2 q2
1 + (n log 2)2
1/2
≥ + n log 2
1 , 2nq 3
for all n ≥ 1 and q ≥ 2. Lemma 4.3. Let n ≥ 1 and p/q = 0. Set 1 p 2 1 , n/2 rˆ = min n q 2
p . q log
Given an n-hyperbolic component W , consider the corresponding function ψW . log Assume that ψW extends to a function, which is defined and univalent on the disk log B = B(2πi p/q, rˆ ). Then the domain V = ψW (B) is contained in the wake W ∗ . In particular, condition (b) of Lemma 4.1 holds. Proof. Each point of the periodic orbit OW (c) of period n has an analytic continuation from W to the wake W ∗ and, by continuity, to W ∗ \{cW }. (In fact, one can extend it analytically, for example, to C\({c > 1/4} ∪ (M\W ∗ )).) Assume that V is not contained in W ∗ . Then there is c1 ∈ V \M, such that c1 lies on a parameter ray of argument
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
311
tc1 ∈ {t± (W )}. Consider now the dynamical plane of f c1 and its dynamical rays (for the definiton of dynamical rays to the disconnected Julia set, see e.g. [33] and Subsect. 6.2 of the Appendix). Then a point of the periodic orbit OW (c1 ) of f c1 of period n must be the landing point of a (non-smooth) dynamical ray of f c1 of argument tc1 (see e.g. [32]). As tc1 is a periodic point of the doubling map σ of period n, we can employ Corollary 6.1 of the Appendix. Denote by ρ1 the multiplier of the periodic orbit OW (c1 ). By the condition of the present lemma, log ρ1 lies in the disk B. In particular, log |ρ1 | < rˆ . On the other hand, by Corollary 6.1, we obtain that a branch of log ρ1 lies in the circle Y = Yn (0, δ(n, (1/n) log |ρ1 |)). Therefore, the disks B and Y must intersect. Let us show that this is impossible. Indeed, by (62)-(63) of Corollary 6.1, and since log |ρ1 | < rˆ , then Y ⊂ B(R, R), where R=
nπ log 2 . arctan (2nnπ −1)ˆr
We need to check that (2π p/q)2 + R 2 > (ˆr + R)2 , or (2π p/q)2 > rˆ 2 + 2Rrˆ . Denote β = nπ/((2n − 1)ˆr ). Consider two cases. If β ≥ 1, then R < 4n log 2, and 2 4 2 p 1 1 p 2 p p 2 rˆ 2 + 2Rrˆ ≤ + 8n log 2 ≤ . (1 + 8 log 2) < 2π n q n q q q If β < 1, then we use that arctan β > β/2 and, hence, R<
nπ(log 2)2(2n − 1)ˆr = 2ˆr (2n − 1) log 2. nπ
Thus
2 1 p p 2 n rˆ + 2Rrˆ < rˆ (1 + 4(2 − 1) log 2) ≤ 1 + 4(2 − 1) log 2 < 2π . q 2n q 2
2
n
We can now combine Lemmas 4.2-4.3 with Lemma 4.1 and get an effective version of Lemma 4.1. As 1/(2nq 3 ) ≤ (1/n)( p/q)2 , the minimum of radii introduced in Lemmas 4.2-4.3 is the number r (n, p/q) defined by 1 1 p . (25) r (n, p/q) = min , 2nq 3 2n/2 q Put r = r (n, p/q) in Lemma 4.1. Since rq 2 ≤ 1/(2nq) < 1, we can apply Lemma 4.1 with X = 1 and find corresponding 0 < 0 < < 1. Then, by Lemma 4.1 (I)–(II), the function W, p/q is well-defined in B(2πi p/q, r (n, p/q)) and univalent in B(2πi p/q, r (n, p/q)). Moreover, p 2 p p ⊂ W, p/q B 2πi , 0 r n, B 1, q 2 0 r n, 3 q q q 3 2 p . (26) ⊂ B 1, q 0 r n, 2 q 3 As 3q 0 r2(n, p/q) < 4nq < 1, a branch of log is well-defined in B(1, 3q 2 0 r (n, p/q)/2), such that it vanishes at the point 1. Therefore, using this branch of log, the function 2
λW, p/q := log W ( p/q) = log ◦ρW ( p/q) ◦ ψW ◦ exp
(27)
312
G. Levin
is well-defined and univalent in B(2πi p/q, 0 r (n, p/q)) and vanishes only at the point 2πi p/q. Next, we apply the derivative estimate for W ( p/q) , see Lemma 4.1 (II). Using (26) and that 0 < 1/8 (see Comment 3) we easily conclude from the definition of λW, p/q that q 2 /2 < |λ W, p/q | < 2q 2 in B(2πi p/q, 0 r (n, p/q)). Finally, by the last conclusion of Lemma 4.1(II) the function ψW ( p/q) has a univalent extension to B(0, 1)∪ B(1, 2q 2 0 r (n, p/q)/3). We define a constant Q 0 by the condition: for q > Q 0 and 0 < ≤ 1/(4q), 4 (28) exp(B(0, )) ⊂ B 1, . 3 We use this with = q 2 0 r (n, p/q)/2 < 1/(4q), where q > Q 0 . We get part A of the following statement, which will serve as an induction argument in the proof of the Main Lemma 4.5. Recall that ρ0 = exp(2πi p/q) and c0 = ψW (ρ0 ). Lemma 4.4. There exists 0 < 0 < 1 as follows. Assume that ψW has a univalent extension to B(0, 1) ∪ U , where U = exp(B(2πi p/q, r (n, p/q))). Then A-B hold. A. A. The function W, p/q = ρW ( p/q) ◦ ψW ◦ exp is holomorphic in B(2πi p/q, r (n, p/q)), and it is equal to 1 only at the center 2πi p/q. Furthermore, the function λW, p/q introduced above: λW, p/q = log ◦ρW ( p/q) ◦ ψW ◦ exp is well-defined and univalent in B(2πi p/q, 0 r (n, p/q)), and it is equal to zero only at the center 2πi p/q. For every L ∈ B(2πi p/q, 0 r (n, p/q)), we have: −1 1 2
2 2 q < |λW, p/q (L)| < 2q . In particular, the inverse function λW, p/q is defined and univalent in B(0, 0 q 2 r (n, p/q)/2), and 1 2
< |(λ−1 W, p/q ) (L)| < 2 2 2q q there. Finally, if q > Q 0 , the function ψW ( p/q) has a univalent extension to B(0, 1)∪ Uˆ , where Uˆ = exp(B(0, 0 q 2 r (n, p/q)/2)). B. For each k = 1, . . . n, there exists a function Fk , which is defined and holomorphic in the disk 1/q 1 S = |s| < r (n, p/q) , 2 such that Fk (0) = 0, Fk (0) = 0, and the following holds. For every s ∈ S, s = 0, and corresponding ρ = ρ0 + s q , the points bk (c0 ) + Fk (s exp(2πi qj )), k = 1, . . . , n, j = 0, . . . , q − 1, form the periodic orbit O p/q (c) of f c of period nq, where c = ψW (ρ). The multiplier of O p/q (c) is equal to ρW ( p/q) (c). It remains to prove part B. It holds locally, in a neighborhood of the point s = 0 (see the discussion in the beginning of Sect. 4.2, where the functions Fk were introduced). Let us show that the functions Fk don’t have singularities in the disk S. Indeed, otherwise being continued along a curve in S, which starts at s = 0 and ends at some s1 = 0, the multiplier ρW ( p/q) of O p/q becomes equal to 1, for some c1 = c0 . Here c1 = ψW (ρ1 ), where
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
313
|ρ1 −ρ0 | = |s1 |q < r (n, p/q)/2. Besides, ρ1 = ρ0 . It is easy to check that then a branch of log ρ1 is contained in B(2πi p/q, r (n, p/q)), and log ρ1 = log ρ0 = 2πi p/q. On the other hand, by Part A, ρW ( p/q) ◦ ψW ◦ exp takes the value 1 in B(2πi p/q, r (n, p/q)) only at the point 2πi p/q, a contradiction. 4.3. The function H . Definition 4.1. Define a real strictly increasing smooth function H : [0, 1) → [0, ∞), H (0) = 0 as follows. Let G be the set of all holomorphic functions g : B(0, 1) → C\{1}, such that g(w) = 0 if and only if w = 0. Then H (u) = sup{|g(w)| : |w| ≤ u, g ∈ G}. There is an explicit expression for H . It is obtained as follows. Let J (w) be a holomorphic function in B(0, 1), such that J (0) = 0, J (0) > 0, and J : B(0, 1)\{0} → C\{0, 1} is an infinite unbranched cover. Such function is investigated in [44], see also [10,16,45]. By the Schwarz lemma, H (u) = max|w|=u |J (w)|. On the other hand, by [44], J (w) =
(1+w ) 16w∞ k=1 (1+w2k−1 )8 . (Apparently, J is equal to the square of the so-called elliptic mod2k 8
(1+u ) ulus, see e.g. [45].) Thus, H (u) = −J (−u) = 16u∞ k=1 (1−u 2k−1 )8 . Using a bound for |J (w)| proved in [44], we get: 2k 8
H (u) ≤
1 exp (−π 2 / log u). 16
(29)
4.4. Main lemma. Theorem 7 will be a consequence of Lemma 4.5 below and a renormalization argument. It is easy to check the existence of Q 1 , such that, for q > Q 1 and | p/q| ≤ 1/2, p 1 ⊂ {ρ : Re(ρ) < 1} . exp B 2πi , 3 q 2q Recall that r (n, p/q) = min{1/(2nq 3 ), | p/q|2−n/2 }, and the constant Q 0 is defined by the condition (28) before Lemma 4.4. Lemma 4.5. Let 0 be the constant from Lemma 4.4. Set 0 , Q = max{Q 0 , Q 1 }. 4 Let t0 , t1 , . . . , tm , . . . be a sequence of rational numbers tm = pm /qm ∈ (−1/2, 1/2]\{0}. Let W 0 be the main cardioid. Denote W m = W m−1 (tm−1 ), m = 1, 2, . . ., in other words, the closure of the hyperbolic component W m touches the closure of the hyperbolic component W m−1 at the point cm−1 := c(W m−1 , tm−1 ) with the internal argument tm−1 . Denote α=
n 0 = 1, n m = q0 q1 . . . qm−1 , m > 0, i.e., n m is the period of the attracting periodic orbit of f c , for c ∈ W m .
314
G. Levin
(C1) Assume that, for m ≥ 0, we have: qm > Q and |t0 | <
α α α 2 pm−1 1 | pm−1 qm−1 | = , m ≥ 1. , , |tm | < qm−1 r n m−1 , min 8π 4π qm−1 4π 2n m 2n m−1 /2
(30) Then the sequence cm converges to some c∗ ∈ ∂ M. (C1’) If, additionally, qm 2 n /2 m→∞ max{n m , n m−1 2 m−1 }
lim sup
>
8 , α
(31)
then the Mandelbrot set is locally connected at c∗ . (C2) Assume that the conditions of (C1) hold, and assume also that |t0 |H (u 0 ) +
∞ m=0
um 1 H (u m+1 ) < , qm (1 − u m ) 100
(32)
where 2n m /2 |1/qm , m ≥ 0. u m = |32tm+1 max 2n m+1 , | pm q m |
(33)
Then the map f c∗ is infinitely renormalizable with non locally connected Julia set. Proof. Introduce some notations. Let 1 pm 1 pm = min rm = r n m , , | | , m = 0, 1, . . . , qm 2n m qm3 2n m /2 qm 2 B0 := B(0, α), Bm := B(0, αqm−1 rm−1 ), m = 1, 2, . . . , B˜ m := B(2πi pm /qm , rm ), Um = exp( B˜ m ), m = 0, 1, . . . .
As usual, ψW m is the function, which is inverse to the multiplier function ρW m . Using notations of Lemma 4.4, denote log
ψm = ψW m = ψW m ◦ exp, λm = λW m ,tm = log ◦ρW m+1 ◦ ψW m ◦ exp . For m ≥ 0, λm is defined and holomorphic in a neighborhood of the point 2πitm , and a branch of the log is chosen so that λm (2πitm ) = 0. Note that cm = ψm (2πitm ). We have explicitly ψW 0 (ρ) = ρ/2 − (ρ/2)2 , so that ψW 0 is holomorphic in the plane and is univalent in any domain, which does not contain two points ρ1 = ρ2 with ρ1 + ρ2 = 2, particularly, it is univalent in the half-plane {Re(ρ) < 1}. By the choice of Q 1 and Q, ψW 0 is univalent in B(0, 1) ∪ U0 . This will allow us to start applying Lemma 4.4. Now, let us verify that, for m ≥ 0, B˜ m ⊂ Bm \0. Indeed, 2π | pm /qm | 2π | pm /qm | ≥ 16π > 1. ≥ rm 1/(2n m qm3 )
(34)
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
315
Hence, if rˆm denotes the radius of Bm and using the condition (30), 2π
| pm | | pm | + rm < 4π < rˆm . qm qm
This proves (34). Note that ψ0 = ψW 0 ◦ exp is holomorphic in C. Set C = max{|(ψ0 ) (L)| : L ∈ B0 }. Claim 1. For m = 1, 2, 3, . . ., the following holds. (i) The function ψm extends to a univalent map defined in Bm . Moreover, the univalent function ψW m : B(0, 1) → W m has a univalent extension to B(0, 1) ∪ exp(Bm ). (ii) For L ∈ Bm , |ψm (L)| ≤ C
2m . n 2m
(iii) For any k ≥ 0, denote Rk = ψk (Bk ), (so that ck ∈ Rk ). Then Rm ⊂ Rm−1 , and the diameter of the set Rm is less than 2Cα2m /n 3m . In particular, {Rm } shrink to a point c∗ , which is the limit of the sequence cm . (iv) If, for some m > 0, qm 8 > , α max n 2m , n m−1 2n m−1 /2
(35)
then the limb L(W m , tm ) is contained in Rm . We prove (i)–(iii) of Claim 1 by induction in m = 1, 2, 3, . . .. The proof is an almost straightforward application of Lemma 4.4. We have: ψm+1 = ψm ◦ λ−1 m , m ≥ 0,
(36)
whenever the right hand-side is defined. Let us prove (i)–(iii) for m = 1. As we checked before Claim 1, ψW 0 has a univalent extension to B(0, 1) ∪ U0 , that is, the conditions of Lemma 4.4 hold, for W = W 0 , and p0 , q0 instead of p, q. By Lemma 4.4 (A), the inverse λ−1 0 is defined and univa(B ) is contained in B˜ 0 . Also, for L ∈ B1 , lent in B1 = B(0, (0 /4)q02 r0 ), and λ−1 1 0 −1
1 2 < |(λ0 ) (L)| < q 2 . By (36), ψ1 extends to a univalent map defined in B1 , and, by 2q 2 0
0
the Chain Rule, for L ∈ B1 ,
|ψ1 (L)| ≤ C
2 2 = C 2. q02 n1
The property (iii) follows from (i)–(ii): R1 = ψ1 (B1 ) = ψ0 ◦ λ−1 0 (B1 ) ⊂ ψ0 (B0 ) = R0 . Finally, by the last conclusion of Lemma 4.4 (A), ψW 1 has a univalent extension to B(0, 1) ∪ exp(B1 ). Step of induction. This is an obvious modification of the argument for m = 1. So, assume (i)–(ii) hold for m ≥ 1. By (34) and (i), the conditions of Lemma 4.4 hold,
316
G. Levin
for W = W m and pm , qm instead of p, q. Therefore, the inverse λ−1 m is defined and univalent in Bm+1 = B(0, αqm2 rm ), also ˜ λ−1 m (Bm+1 ) ⊂ Bm ⊂ Bm ,
(37)
1 2
< |(λ−1 m ) (L)| < 2 . 2qm2 qm
(38)
and, for L ∈ Bm+1 ,
By (36), ψm+1 extends to a univalent map defined in Bm+1 , and, by the Chain Rule and the induction assumption, for L ∈ Bm+1 ,
|ψm+1 (L)| ≤ C
2m 2 2m+1 = C . n 2m qm2 n 2m+1
In turn, Rm+1 = ψm+1 (Bm+1 ) = ψm ◦ λ−1 m (Bm+1 ) ⊂ ψm (Bm ) = Rm , i.e. (iii) holds as well, for m + 1. Finally, by the last conclusion of Lemma 4.4 (A), ψW m+1 has a univalent extension to B(0, 1) ∪ exp(Bm+1 ). The induction is completed. Let us prove (iv). Assuming the condition of (iv) holds, let us check that Yoccoz’s circle Yn m (tm ) is contained in Bm . Indeed, Yn m (tm ) is contained in the disk centered at 2πi pm /qm of radius 2n m log 2/qm , while, by (35),
1 2n m log 2 2αn m log 2 qm−1 α log 2 α log 2 2 min qm−1 rm−1 . < , ≤ ≤ qm 2 2n m 2n m−1 /2 2 8 max{n 2m , n m−1 2n m−1 /2 }
Then, using (30),
α α log 2 2 | pm | 2n m log 2 2 + qm−1rm−1 < αqm−1 + < rm−1 , 2π qm qm 2 2
which is the radius of Bm . Thus Yn m (tm ) ⊂ Bm , hence, L(W m , tm ) ⊂ ψm (Yn m (tm )) ⊂ ψm (Bm ) = Rm . This finishes the proof of Claim 1. It yields immediately the statements (C1)-(C1’) of the lemma. Now we pass to the proof of the statement (C2). Let us denote by Ok (c) the n k -periodic orbit of f c , which is attracting if c ∈ W k , k = 0, 1, 2, . . .. As we know, Ok (c) extends holomorphically for c in the wake (W k )∗ of W k . Given m ≥ 1, consider any point b(c) of the periodic orbit Om (c), for c ∈ (W m )∗ . By (34) and by Claim 1(i), ψW m−1 has a univalent extension to B(0, 1) ∪ exp( B˜ m−1 ) (for m = 1, this was checked before Claim 1). Thus the conditions of Lemma 4.4 hold, for W = W m−1 and p/q = tm−1 . Then the conclusion (B) of that lemma tells us that the function b extends to a neighborhood of cm−1 in the following sense. There is a holomorphic function Z of a local parameter s in the disk r m−1 1/qm−1 Sm−1 := {|s| < vm−1 }, vm−1 = , 2 such that it matches b(c), i.e., b(c) = Z (s) for c = cm−1 (s) := ψW m−1 (exp(2πitm−1 ) + s qm−1 ). j
Moreover, the points Z (seqm−1 ), where j = 1, . . . , n m−1 −1 and eq = exp(2πi/q), also belong to Om . We denote by b+ the point Z (seqm−1 ) of Om , which is uniquely defined for s ∈ Sm−1 . Let us estimate the distance between b = Z (s) and b+ = Z (seqm−1 ).
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
317
Claim 2. |Z (s)| < 3 for s ∈ Sm−1 , m = 1, 2, . . .. Proof of Claim 2. For |c| < 5, every point z, such that |z| ≥ 3, escapes under the dynamics of f c . As Z (s) ∈ Jcm−1 (s) , it is then enough to check that |cm−1 (s)| < 5. Fix s ∈ Sm−1 and denote c˜ = cm−1 (s). If c˜ ∈ M, then |c| ˜ ≤ 2, so we assume c˜ ∈ / M. We have: 3 |ρW m−1 (c) ˜ − exp(2πitm−1 )| = |s|qm−1 < rm−1 /2, where rm−1 ≤ 1/(2n m−1 qm−1 )≤ 1/16. Hence, rm−1 1 < rm−1 ≤ log |ρW m−1 (c)| ˜ < log 1 + . 2 16 On the other hand, consider the Riemann map R : {|w| > 1} → C\M, where R(w) = w + α0 + O(1/|w|) as w → ∞. If R(w) ˜ = c, ˜ then, in the notations of Subsect. 6.2 of the Appendix, |w| ˜ = |Bc˜ (c)| ˜ = exp(2ac˜ ). In turn, by Theorem 9 of Subsect. 6.2, ac˜ ≤
1 n m−1
log |ρW m−1 (c)| ˜ ≤
1 , 16
that is, |w| ˜ ≤ exp(1/8). By a general property of univalent maps (see e.g. [12], Ch.II), for every r ≥ 1, the complement of the image R({|w| > r }) belongs to the disk |c−α0 | ≤ 2r . Setting here r = 1 and using that −2, 1/4 ∈ M and α0 ∈ R, we get −2 < α0 ≤ 0 (in fact, it is well-known that α0 = −1/2). Then, setting r = exp(1/8), we get finally that |c| ˜ < 2 + 2 exp(1/8) < 5. Claim 2 is proved. Thus |Z (s)| < 3 for s ∈ Sm−1 . Therefore, |s|
2π vm−1 |Z (seqm−1 ) − Z (s)| < 3 . 2 qm−1 1 − |s| 2
(39)
vm−1
Let us detect s for Om , which corresponds to the limit parameter c∗ . Using (36)–(37), we have : −1 ˜ ˜ c∗ ∈ ψm+1 (Bm+1 ) = ψm ◦ λ−1 m (Bm+1 ) ⊂ ψm ( Bm ) = ψm−1 ◦ λm−1 ( Bm ).
As λm−1 (2πitm−1 ) = 0, then (38) gives us that ˜ λ−1 m−1 ( Bm ) ⊂ B(2πitm−1 , C), where C=
2 2 qm−1
(2π |tm | + rm ) ≤
2 2 qm−1
2π |tm | +
(40)
1 2n m qm3
<
16|tm | . 2 qm−1
(41)
Note that, by (30), 2 16|tm |/qm−1
rm−1 /2
< 32
α 2 < < 1. 4π π
(42)
Therefore, if sm−1 ∈ Sm−1 is such that c∗ = cm−1 (sm−1 ), then 1/qm−1 1/qm−1 2 16|tm |/qm−1 |sm−1 | 2n m−1 /2 < = 32|tm | max 2n m , , vm−1 rm−1 /2 | pm−1 qm−1 | (43)
318
G. Levin
and is less than 1. Now we consider two consecutive periodic orbits Om and Om+1 . Let bm+1 (c) be any point of Om+1 (c), for c ∈ (W m+1 )∗ . By the above, there is a holomorphic function Z m+1 of the local parameter s ∈ Sm , such that bm+1 (c) = Z m+1 (s), where c = cm (s), provided cm (s) ∈ (W m+1 )∗ . For s ∈ Sm , we have: ρW m−1 (cm (s)) ∈ ρW m−1 ◦ ψW m (B(e2πitm , rm /2)) ⊂ ρW m−1 ◦ ψW m ◦ exp( B˜ m ). ˜ Consider the set := ρW m−1 ◦ ψW m ◦ exp( B˜ m ), in other words, = exp(λ−1 m−1 ( Bm )). 2πit m−1 }. Now By (40)–(42) along with (34), ⊂ exp(B(2πitm−1 , (2/π )(rm−1 /2)))\{e we use the definition of the constant Q 0 , see (28), with = (2/π )(rm−1 /2). As < 3 1/(4qm−1 ) ≤ 1/(4q) and qm−1 > Q 0 , we therefore have that 2 rm−1 4 ⊂ B exp (2πitm−1 ) , , exp B 2πitm−1 , π 2 3 and since 4/3 < rm−1 /2, we conclude that the simply-connected domain is a subset of B(e2πitm−1 , rm−1 /2)\{e2πitm−1 }. It allows us to fix a branch p of the function v → (v−e2πitm−1 )1/qm−1 , which is defined for v ∈ . We thus get a well defined passage map pˆ : s → p◦ρW m−1 (cm (s)) from the local parameter s ∈ Sm of Om+1 (c) to the corresponding local parameter in Sm−1 of the points of Om (c). Now, let Zˆ m (s) := Z m ( p(s)), ˆ ˆ ˆ s ∈ Sm , be a point of Om , for which Z m (0) = Z m+1 (0). For b = Z m (s) of Om , there is + (s) of O , i.e. Zˆ + (s) = Z ( p(s)e the corresponding point b+ = Zˆ m m m ˆ qm−1 ). Consider the m + functions Z m+1 , Zˆ m , and Zˆ m holomorphic in Sm . Observe that Z m+1 (s) = Zˆ m (s) if and + (s) = Zˆ (s) in S . Introduce a new function only if s = 0, and Zˆ m m m ζm (s) =
Z m+1 (s) − Zˆ m (s) . + (s) − Zˆ (s) Zˆ m m
By the above, ζm (s) obeys the following properties: (i) it is holomorphic in the disc Sm , (ii) ζm (s) = 1 in Sm , (iii) ζm (s) = 0 if and only if s = 0. We conclude that
|ζm (s)| ≤ H
|s| vm
,
(44)
where the function H is defined in Sect. 4.3. As for the limit parameter s = sm , we get |sm | ˆ + |Z m+1 (sm ) − Zˆ m (sm )| ≤ H | Z m (sm ) − Zˆ m (sm )|. vm
(45)
In turn, by (39), |sm−1 |
2π vm−1 + | Zˆ m (sm ) − Zˆ m (sm )| = |Z m (sm−1 eqm−1 ) − Z m (sm−1 )| < 3 . (46) qm−1 1 − |sm−1 |2 2 vm−1
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
319
∗ Here Z m+1 := Z m+1 (sm ) is any point of the periodic orbit Om+1 of the limit map ∗ f c∗ while Z m := Zˆ m (sm ) is a point of the periodic orbits Om of the same map f c∗ , ∗ . Because H and t/(1 − t 2 ) are increasing functions, which is determined by the Z m+1 we conclude from (45), (46), and (43):
u m−1 2π H (u m ), (47) qm−1 1 − u m−1 1/qk n k /2 where u k = 32|tk+1 | min 2n k+1 , |2pk qk | , k = 0, 1, . . .. ∗ determines a point Z ∗ In turn, the point Z m m−1 of the periodic orbit Om−1 , and so on until a point Z 0∗ of O0 . The inequality (47) holds for every m ≥ 1. It remains to estimate |Z 1∗ − Z 0∗ |. Since Z 0∗ is a fixed point of f c∗ , we compare |Z 1∗ − Z 0∗ | with |β ∗ − Z 0∗ |, where β ∗ is the second fixed point of f c∗ . By similar considerations, we can write ∗ ∗ |Z m+1 − Zm |≤3
|Z 1∗ − Z 0∗ | ≤ H (u 0 )|β ∗ − Z 0∗ |. On the other hand, |β ∗ − Z 0∗ | = |1 − ρ ∗ |, where ρ ∗ is the multiplier of Z 0∗ . Hence, |β ∗ − Z 0∗ | ≤ |1 − exp(2πit0 )| + | exp(2πit0 ) − ρ ∗ | ≤ 2π |t0 | + 1/(2q03 ) < 7|t0 |. We have finally, for m = 0, 1, 2, . . .,
∞ 1 u k−1 ∗ |Z m+1 − Z 0∗ | < 6π |t0 |H (u 0 ) + H (u k ) . qk−1 1 − u k−1
(48)
k=1
∗ is any point of the periodic orbit Om+1 of f c∗ , for any m ≥ 0. On the other Here Z m+1 hand, since ρ ∗ = 2Z 0∗ and |ρ ∗ | > 1 − 1/(2q03 ) > 1/2, then |Z 0∗ | > 1/4. Hence, under the condition (32), for every m ≥ 1, the periodic orbit Om of f c∗ lies outside of a fixed neighborhood B(0, δ) of zero, where δ > 1/4 − 6π/100 > 0. It is well known that this implies the non local connectivity of Jc∗ (see [52] for a detailed proof of this fact).
4.5. Proof of Theorem 7. First, let us make a remark on the condition (S) of Theorem 7. We claim that the value u k,m in (S) can be replaced by exp(γ˜ qk . . . qm−1 ) 1/qm u˜ k,m = Atm+1 max Bqk . . . qm , , | pm q m | for any positive A, B, and γ˜ . Indeed, given k, if k1 > k is big enough and m ≥ k1 , then u˜ k1 ,m /u k,m < 1. Therefore, if (15) holds and since H is increasing, then ∞ m=k1
∞ u˜ k1 ,m u k,m H (u˜ k1 ,m+1 ) ≤ H (u k,m+1 ) < ∞. qm (1 − u˜ k1 ,m ) qm (1 − u k,m ) m=k1
Vice versa, given k, if k1 > k is big enough and m ≥ k1 , then u k1 ,m /u˜ k,m < 1, i.e., if (15) holds with u˜ k,m instead of u k,m , then it also holds with u k1 ,m .
320
G. Levin
Now we can restate the conditions (Y0), (Y1), (S) as follows: (Y 0 )a−b There is C > 0, such that, for all m large enough, q ...q C 0 m−2 |tm | < min . , | pm−1 qm−1 | exp − q0 . . . qm−1 C
(49)
(Y1’) There exist β > 0, C1 > 0, such that lim sup m→∞
qm > C1 . max (q0 . . . qm−1 )2 , exp(βq0 . . . qm−2 )
(50)
(S’) For some k ≥ 0, ∞ m=k
u˜ k,m H (u˜ k,m+1 ) < ∞, qm (1 − u˜ k,m )
(51)
where 2qk ...qm−1 /2 1/qm u˜ k,m = |32tm+1 max 2qk . . . qm , | , m ≥ k. | pm |qm
(52)
(We set qk . . . qm−1 = 1, if m = k.) It is enough to find a tail Tk0 = {tm }∞ m=k0 of the sequence T0 = {tm }∞ , which satisfies corresponding conditions of Lemma 4.5 with m=0 n = 1. Namely, let us start with the 1-hyperbolic component W0 (the main cardioid) and the tail Tk0 = {tk0 , tk0 +1 . . .} in place of W and T0 = {t0 , t1 , . . .}, respectively. Then we get a sequence of hyperbolic components W0k0 ,m , m ≥ k0 , where W0k0 ,k0 = W0 and, for m > k0 , the closure of W0k0 ,m touches the closure of W0k0 ,m−1 at the point ck0 ,m−1 with internal argument tm−1 . By a well known straightening procedure, see [9] and the proof of Theorem 1, the following implications hold. If the sequence of parameters ck0 ,m converges, as m → ∞, to some ck0 ,∗ , then the sequence cm also converges, to some c∗ . If, moreover, M is locally connected at ck0 ,∗ , then M is locally connected at c∗ , and if the Julia set Jck0 ,∗ is not locally connected, then Jc∗ is not locally connected, too. Thus to prove the theorem it is enough to find a tail Tk0 , such that: (a’) the condition (Y 0 )a−b for the whole sequence T0 implies the condition (30) for the tail Tk0 , (b’) the condition (Y1’) for T0 implies the condition (31) for Tk0 , and (c’) the conditions (Y 0 )a−b and (S’) for T0 implies the condition (32) for Tk0 . Notice that (a’) and (b’) are obviously true, for any k0 large enough, because q0 . . . qk0 −1 → ∞ as k0 → ∞. Let us show (c’). Assume that the condition (S’) holds, for a fixed large enough k. Since u˜ k,m is decreasing in k, for every k0 large enough, ∞ m=k0
u˜ k0 ,m 1 H (u˜ k0 ,m+1 ) < . qm (1 − u˜ k0 ,m ) 200
Thus to satisfy (32) for some Tk0 instead of T0 , it is enough to check that |tk0 |H (u˜ k0 ,k0 ) tends to zero as k0 tends to ∞. Using again that u˜ k,m decreases as k increases, it follows from (S’), that, for a fixed k and k0 → ∞, u˜ k,k0 −1 u˜ k0 −1,k0 −1 H (u˜ k0 ,k0 ) ≤ H (u˜ k,k0 ) → 0. qk0 −1 (1 − u˜ k0 −1,k0 −1 ) qk0 −1 (1 − u˜ k,k0 −1 )
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
321
On the other hand, using (Y 0 )a−b it is easy to see that, as k0 → ∞, |tk0 |qk0 −1 u˜ k0 −1,k0 −1 < → 0. |tk0 |/ qk0 −1 (1 − u˜ k0 −1,k0 −1 ) u˜ k0 −1,k0 −1 This completes the proof of Theorem 7. 5. Non-primitive Components 5.1. Proof of Theorem 4. Let W = Z (t0 ), Z be an n 0 -hyperbolic component, and t0 = p0 /q0 = 0. By Theorem 3(c), the function ψ Z extends to a univalent map ˜ log defined in B(0, 1) ∪ exp(B(2πit0 , r0 )), where r0 = dist (2πit0 , ∂ n 0 ). We would like to apply Lemma 4.1. The radius r in Lemma 4.1 will be specified as follows: r = min{r0 , r (n 0 , p0 /q0 )}, where r (n 0 , p0 /q0 ) = min{1/(2n 0 q03 ), | p0 /q0 |2−n 0 /2 }. Let us check the conditions of Lemma 4.1 with this specific r , with X = 1, and with Z , p0 /q0 , W in place of W , p/q, W ( p/q) respectively. Indeed, by the above and by Lemmas 4.2-4.3, we have that ψ Z has a univalent extension to B(0, 1) ∪ U , where U = exp(B(2πit0 , r )), and, furthermore, the domain V = ψ Z (U ) satisfies the conditions (a)-(b) of Lemma 4.1. Therefore, we indeed can apply Lemma 4.1 with these data. We conclude, that: (i) the function ρW extends to a holomorphic function defined in the log domain Vˆ = V ∪ W ∗ ∪ Z , (ii) the function Z ,t0 = ρW ◦ ψ Z is defined and univalent in B(2πit0 , 0 r ), and 2q02 /3 < | Z ,t0 | < 3q02 /2 in B(2πit0 , 0 r ), and (iii) the function ψW has a univalent extension to B(0, 1) ∪ B(1, 2q02 0 r/3) (where 0 < 0 < 1 is a universal constant). Now, applying Theorem 3(c) to the n 0 q0 -hyperbolic component W , we ˜ n 0 q0 . This along with (iii) gives us that ψW is holomorphic get that ψW is univalent in in the domain 2 2 ˜ n 0 q0 ∪ B 1, q02 0 r . ˜ n 0 q0 ∪ B(0, 1) ∪ B 1, q02 0 r = := 3 3 ˜ n 0 q0 ) ⊂ W ∗ , Let us show that ψW is univalent in . Indeed, by Theorem 3(c), ψW ( and by (ii)–(iii), 2 2 2 2 log log −1 ⊂ ψ Z (B (2π t0 , 0 r )) ⊂ V, ψW (B(1, q0 0 r )) = ψ Z ◦ Z ,t0 B 1, q0 0 r 3 3 therefore, ψW () ⊂ Vˆ . But (i) tells us that ρW is well-defined in Vˆ . Then ρW is a well-defined inverse function to ψW in , i.e., ψW is univalent in . It remains to check that the domain n 0 ,t0 introduced in Theorem 4 is a subset of . It is easy to check that, for some K > 0 and all n 0 , q0 , K −1 P < r0 = ˜ log dist (2πit0 , ∂ n 0 ) < K P, where n 0 |t0 | t02 P = min , . 4n 0 n 0 Obviously, 1/(2n 0 q03 ) < t02 /n 0 and n 0 |t0 |4−n 0 < |t0 |2−n 0 /2 . Combining this, we get that, for some universal K 1 > 0, n 0 |t0 | 1 2 2 n| p0 | 1 2 q0 0 r > K 1 q0 min . , min , = K 1 3 4n 0 n 0 q03 4n 0 n
322
G. Levin
Taking a bit smaller 0 < K˜ < K 1 and putting d = K˜ min n| p0 |4−n 0 , n −1 , we have that exp(B(0, d)) ⊂ B(1, 2q02 0 r/3). Recall that n 0 ,t0 is defined by its log-pro˜ log jection as n 0 q0 ∪ B(0, d). Hence, indeed, n 0 ,t0 ⊂ , while ψW is univalent in . Finally, it is easy to show that for q0 large enough B(0, d) covers the portion of the disk log {L : |L − Rn 0 q0 | < Rn 0 q0 }, which is deleted from n 0 q0 (see Theorem 3(c)), hence, log log ˜ n 0 q0 above can be replaced by n 0 q0 . 5.2. Proof of Theorem 6. The proof consists of a consideration of several cases using Theorem 4 and Theorem 3. Note that in some cases we get much better bounds for the size of the limb L(W, p/q). Keeping the notations of Subsect. 5.1, W = Z ( p0 /q0 ), where Z is an n 0 -component, and so W is an n-component, where n = n 0 q0 . Denote by D the diameter of L(W, p/q). If q ≤ 8n , the bound holds trivially, with A˜ = 4. So, in what follows, q > 8n . d Assume n|qp| ≥ 1000 , where d = K˜ min{ n|4np00 | , n1 } is taken from Theorem 4. If n |qp| ≥
K˜ 1000n ,
n n n then, by Theorem 5, D ≤ A |4p| ≤ A(1000n 2 K˜ −1 ) 4q ≤ A˜ 8q , for some A˜
K˜ n| p0 | 1000 4n 0 ,
independent on n, q. If n |qp| ≥
D≤A
then similarly
4n 0 +n 8n 4n ≤ 1000 A ≤ A˜ , | p| q K˜ | p0 |q
for some A˜ independent on n, q. Hence, one can assume that n|t| = n
| p| d < . q 1000
(53)
By Theorem 4 and Koebe, for every L ∈ B(0, d/2), |(ψW ) (L)| ≤ 81|(ψW ) (0)|. log
log
(54)
In turn, by Theorem 3(a) and (20), |(ψW ) (0)| = q0−2 |ψ Z (exp(2πit0 ))| ≤ q0−2 log
≤ q0−2
B0 4n 0 (1 + o(1)) n 0 | exp(2πit0 ) − 1|
B1 4n 0 B1 4n 0 < , n 0 | p0 /q0 | n
(55)
for some absolute constant B1 > 0. Let us show that Yoccoz’s circle Yn (t) = {L : 2 n log 2 |L −(2πit + n log q )| < q } is covered by B(0, d/2). First, 2n log 2/q < 2n|t| < d/2, by (53). Hence, it is enough to check that (2π t)2 + (n log 2/q)2 < (d/2 − n log 2/q)2 , or (2π t)2 < (d/2)2 (1 − 4n log 2/(qd)), and this holds by (53). Thus Yn (t) ⊂ B(0, d/2). Therefore, by (54)-(55), D < 2n(log 2/q)81B1
4n 0 8n < 200B1 . n q
Acknowledgements. The author is indebted to Alex Eremenko for discussions, and especially for answering the author’s question about the function H and for finding the reference [44] (see Sect. 4). The author thanks the referee for a careful reading of the paper and many helpful comments leading to several important improvements and corrections.
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
323
6. Appendix: Geometric Bounds for the Multiplier 6.1. Periodic points on the boundary of a basin of attraction. Let us fix an n-hyperbolic component W . For c ∈ W , the map f c has the attracting periodic orbit O(c) of period n, and ρW (c) denotes its multiplier. Given a rational number p/q = 0, consider the point c(W, p/q) of ∂ W with the internal argument p/q. As we know, see the beginning of p/q Sect. 4.2, for c near c(W, p/q), there is a unique periodic orbit Oc of f c of period nq, which collides with O(c) as c → c(W, p/q). Its multiplier ρW ( p/q) (c) extends to a function, which is defined and holomorphic in the union W ∪ Bˆ ∪ W ( p/q), where Bˆ is a small neighborhood of c(W, p/q). Here we prove the inequality (21) of Sect. 4.2: Theorem 8. For c ∈ W , such that ρW (c) = 0, | log ρW ( p/q) (c)|2 | log ρW (c) − 2πi p/q|2 < q2 , log |ρW ( p/q) (c)| − log |ρW (c)|
(56)
for some branch of log ρW ( p/q) (c) and any branch of log ρW (c). Proof. This follows from Theorem A of [47] and Theorem 2 of [30] (see also [46]). Let us fix c ∈ W with ρW (c) = 0, and consider the dynamical plane of f c . Denote by the component of the immediate basin of attraction of O(c), which contains the critical point 0 and a point b ∈ O(c). A Riemann map R : → B(0, 1) normalized by R(b) = 0 and with an appropriate arg R (b) conjugates f cn : → to the Blaschke product Bρ (z) = z(z + ρ)/(1 + ρz), ¯ where ρ = ρW (c). The Julia set of Bρ is the unit circle S 1 . Given a rational number p/q = 0, the map Bρ : S 1 → S 1 has a unique rotation set ( p/q) with the rotation number p/q. (This means that the restriction Bρ : ( p/q) → ( p/q) can be lifted and extended to an increasing continuous ˜ + 1) = B(x) ˜ map B˜ : R → R, such that B(x + 1, which then defines in a usual manner the rotation number p/q, see e.g. [3].) The set ( p/q) is a repelling periodic orbit of Bρ of period q. Let be the multiplier of ( p/q), so that > 1. The following bound is proved in [47], Theorem A. For every branch L of the logarithm log ρ, 0 < log < q 2
|L − 2πi p/q|2 . 2|Re(L)|
(57)
The inverse map R −1 : B(0, 1) → has a radial limit denoted by R −1 (w) at every repelling periodic point w ∈ S 1 of Bρ , and the point R −1 (w) is either repelling or parabolic periodic point of f cn : ∂ → ∂, see [48,46]. (Alternatively, here and below one could use the fact that the boundary ∂ is locally connected, because f c is a hyperbolic polynomial; in turn, this implies that R −1 has a homeomorphic continuation to the boundary ∂ B(0, 1).) The set O˜ c ( p/q) = {R −1 (w) : w ∈ ( p/q)} is a periodic orbit of f cn of period q, which lies in ∂ and is repelling, because f c is hyperbolic. Denote by ρ˜c its multiplier. Let w0 ∈ ( p/q) and z 0 = R −1 (w0 ). Now we proceed as in [30]. Linearizing q Bρ near w0 by a conformal map g, g(0) = w0 , we find a map h = R −1 ◦ g, which is conformal near 0 and such that h maps a small semidisk D() = {w : |w| < , I m(w) > 0} nq into and conjugates w → w with f c : → wherever it makes sense. Let us nq fix also a small r > 0 so that f c is linearizable in B(z 0 , r ) and let μ : B(z 0 , r ) → C, nq μ(z 0 ) = 0, be a conformal map, which conjugates f c with its linear part z → ρ˜c z. As R −1 has the radial limit z 0 at w0 , by Lindelof’ theorem, R −1 (w) → z 0 uniformly in any Stolz angle at w0 . Hence, h(w) → z 0 uniformly in {|w| < , arg(w) ∈ (t, π − t)},
324
G. Levin
for any t ∈ (0, π/2). Thus, for any t ∈ (0, π/2) there is t > 0, such that h(D(t , t)) ⊂ B(z 0 , r ), where D(t , t) = {|w| < t , arg(w) ∈ (t, π − t)}. Set U = ∪t D(t , t) and ˜ ), where h˜ = μ ◦ h. Then U and V are topological disks, such that 0 ∈ ∂U , V = h(U k U ⊂ U := {z : z ∈ U }, ∪∞ k=0 U = {z : I m(z) > 0}, and 0 ∈ ∂ V , V ⊂ ρ˜c V , and h˜ : U → ρ˜c V is a conformal homeomorphism that conjugates the linear maps z → z and z → ρ˜c z on U . This is the framework of Theorem 2 of [30]. It states that under these conditions | log ρ˜c |2 < 2 log , log |ρ˜c |
(58)
for some branch log ρ˜c . Combining (57) and (58), we get | log ρ˜c |2 | log ρW (c) − 2πi p/q|2 < q2 , log |ρ˜c | − log |ρW (c)|
(59)
for some branch log ρ˜c and any branch log ρW (c). Now, let us show that, when c is close to c(W, p/q), then the periodic orbit O˜ c ( p/q) is just the periodic orbit O p/q (c). Indeed, −1 (r e2πi p/q ), 0 < r < 1}. Then let c → c(W, p/q) radially, i.e. along the curve γ = {ρW there is a branch of log ρW (c) converging to 2πi p/q as c → c(W, p/q) and, by (59) we conclude that log ρ˜c → 0 as c → c(W, p/q). Since every periodic orbit of f c(W, p/q) other than O(c(W, p/q)) is repelling, it follows by continuity, that O˜ c ( p/q) collides with O(c) as c → c(W, p/q) along γ . On the other hand, O p/q (c) is the only periodic orbit with this property. Hence, O˜ c ( p/q) and O p/q (c) coincide for c ∈ γ close to c(W, p/q), therefore, they coincide for every c ∈ W close to c(W, p/q). Thus their multipliers are equal, i.e., for every c ∈ W , such that ρW (c) = 0, ρ˜c in (59) can be replaced by ρW ( p/q) (c). 6.2. Disconnected Julia set. Suppose c ∈ C\M, i.e., Jc is totally disconnected. Let Bc be the Bottcher coordinate function for f c at infinity, i.e., Bc is defined and univalent in a neighborhood of infinity, such that Bc (z)/z → 1 as z → ∞ and Bc ( f c (z)) = [Bc (z)]2 . The function G c (z) = log |Bc (z)| = limn→∞ 21n log | f cn (z)| extends to a harmonic function defined in the basin of infinity Ac = C\Jc , such that G c (z) → 0 as z → ∂ Ac . In turn, Bc extends to a univalent map defined in the domain {z : G c (z) > G c (0)}. In particular, the value Bc (c) is well-defined. (By [7], the map c → Bc (c) is a holomorphic isomorphism of the complement of M onto the complement of B(0, 1).) Let us introduce parameters ac > 0 and tc ∈ [0, 1) such that Bc (c) = exp(2ac + 2πitc ). Note that ac = G c (0) and tc is the argument of the parameter ray to M, which passes through c ∈ C\M. Let z be a periodic point of f c of period n, and ρ = ( f cn ) (z) its multiplier. If one puts in [11], Theorem 1.6, d = 2, a = b = ac and k = 0, we get: Theorem 9. 1 log |ρ| ≥ ac . n
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
325
Now, we give a bound for log ρ that involves ac , tc , and a rotation number of z, which is similar to the Yoccoz bound (8) though holds for non-connected Julia sets. We consider dynamical rays to the disconnected set Jc (see e.g. [33]). Each ray is either an unbounded smooth curve, which crosses every level curve {z : G c (z) = r } orthogonally and terminates in Jc , or a one-sided limit of such smooth rays. In the latter case, the ray is called non-smooth, or left (right) if it is a limit of smooth rays from the left (right). A ray is non-smooth if and only if it contains a critical point of the function G c , i.e., a preimage by some f ck , k ≥ 0, of the critical point 0 ∈ A∞ . Every ray has a well-defined angle (or argument) though some angle may correspond to two non-smooth rays, which are left and right limits of smooth ones. Denote by (z) the set of angles of dynamical rays that land at z. The following is proved in [33]. The set (z) is a non-empty closed nowhere dense subset of the unit circle S 1 = R/Z. It is finite if and only if it contains a rational angle t0 . In this case, t0 is a periodic point under the doubling map σ : t → 2t (mod 1) of some period nq, q ≥ 1, and every other point t ∈ (z) is periodic by σ of the same period. The rotation number p/q of z is the order at which f cn permutes (locally) each cycle of q dynamical rays that land at z. Let us assume that the rotation number of z is p/q, and choose t0 ∈ (z). Consider the following periodic orbit of the map σ n : t0 = {σ kn (t0 ) : k = 0, 1, . . . , q − 1}. It is a subset of (z). We associate to t0 an angle α ∈ (0, π ) as follows [32]. First, the map σ has a natural extension to a half-cylinder S˜ = {(x, y) : x ∈ S 1 , y ≥ 0} by σ (x, y) = (2x(mod 1), 2y). Consider the set {Nw } of all vertical segments in S˜ with top points w, over all w such that σ j (w) = wc := (iac /π, tc ), for some 1 ≤ j ≤ n. Each segment Nw is a vertical “needle” with the top w and a base point xw ∈ S 1 . Given Nw , let xl be the point of the set t0 , which is the closest to the base point xw , xw = xl , from the left (measured in S 1 ). Consider the triangle with the vertices w, xw , and xl . Let αl (Nw ) be its angle at the vertex w. Define αl as the minimum of the αl (Nw ), over all Nw . The angle αr is defined similarly using the points from the right of xw . Then the angle α is said to be αl + αr , if the ray of argument t0 is smooth, and otherwise either αl or αr , depending on whether left or right ray of the angle t0 lands at z. We have the following generalization of (8): Theorem 10 (see [32]). A branch of log ρ is contained in the disk p nπ log 2 nπ log 2 < Yn ( p/q, α) = L : L − 2πi + . q qα qα We apply this bound when p/q = 0/1, and t0 (a periodic point of σ of period n) is the argument of a non-smooth, say, left, ray. In particular, tc ∈ t0 . In this case, α = αl ≥ δ(n, ac ), where δ(n, a) = arctan
(2n
π . − 1)a
(60)
Indeed, we check, that, for every “needle” Nw , αl (Nw ) ≥ δ(n, ac ).
(61)
Let σ j (w) = wc , 1 ≤ j ≤ n. Denote by s the distance between the points xw , xl (measured on S 1 = R/Z). Let us show that s ≥ 1/(2 j (2n − 1)). Indeed, otherwise 2 j s < 1, hence, s = s0 /2 j , where s0 is the distance between σ j (xl ) and tc = σ j (xw ). In the considered case, σ j (xl ) and tc are different points of the periodic orbit t0 . Since the
326
G. Levin
distance between any two points of t0 is at least 1/(2n − 1), then s ≥ 1/(2 j (2n − 1)). Thus, in any case, s ≥ 1/(2 j (2n − 1)), and then tan αl (Nw ) =
s 1/(2 j (2n − 1)) ≥ = tan δ(n, ac ). (ac /π )/2 j (ac /π )/2 j
Therefore, (61) is checked. Under this setting, Theorem 9 and Theorem 10 imply immediately: Corollary 6.1. Assume tc is a periodic point of σ : S 1 → S 1 of period n, and a (onesided) dynamical ray of f c of angle tc lands at a periodic point of f c of period n with multiplier ρ. Then, for a branch of log ρ, log ρ ∈ Yn (0, δ(n,
1 log |ρ|)) = {L : |L − Rn,|ρ| | < Rn,|ρ| }, n
(62)
where Rn,|ρ| =
nπ log 2 . nπ arctan (2n −1) log |ρ|
(63)
Note that (62) is proved in [31] as well. References 1. Avila, A., Kahn, J., Lyubich, M., Shen, W.: Combinatorial rigidity for unicritical polynomials. Ann. Math. 170, 783–797 (2009) 2. Buff, X.: Arithmetical conditions for non locally connected Julia sets. Slides of a talk at the conference “Complex structures and dynamics”, September 21–26, 2009, Bedlewo, Poland 3. Bullett, S., Sentenac, P.: Ordered orbits of the shift, square roots, and the devil’s staircase. Math. Proc. Camb. Phil. Soc. 115(3), 451–481 (1994) 4. Carleson, L., Gamelin, T.: Complex Dynamics. Berlin-Heidelberg-NewYork: Springer-Verlag, 1993 5. Cheritat, A.: Estimates on the speed of explosion of the parabolic fixed points of quadratic polynomials and applications. Preprint Nov. 23, 1999, available at http://www.math.univ.toulouse.fr/~cherltat/e_ publi2.php 6. Douady, A.: Algorithms for computing angles in the Mandelbrot set. In: Chaotic Dynamics and Fractals, ed. Barnsley, M., Demko, S. San Diego: Acad. Press, 1986, pp. 155–168 7. Douady, A., Hubbard, J.H.: Iteration des polynomes quadratiques complexes. C.R.A.S., t. 294, 123– 126 (1982) 8. Douady, A., Hubbard, J.H.: Etude dynamique des polynomes complexes. I, II. Pub. Math. d’Orsay 84-02 (1984), 85-04 (1985) 9. Douady, A.J.H. Hubbard: On the dynamics of polynomial-like maps. Ann. Sc. Ec. Norm. Sup. 18, 287– 343 (1985) 10. Eremenko, A.: E-mail communication, August 25–27, 2007 11. Eremenko, A., Levin, G.: Estimating the characteristic exponents of polynomials (in Russian). Theor. Funk., Funkts. Anal. i Prilozhen, 58, 30–40 (1993). Translation in J. Math. Sci. 85(5), 2164–2171 (see also http://www.ma.huji.ac.il/~levin/) 12. Goluzin, G.M.: Geometric theory of functions of complex variable. Moscow: Nauka, 1966 13. Guckenheimer, J., McGehee, R.: A proof of the Mandelbrot N-square conjecture. Institut Mittag-Leffler. Report No. 15, 1984 ( available at http://www.math.cornell.edu/~gucken/publications_10.html) 14. Graczyk, J., Swiatek, G.: The Real Fatou Conjecture. Annals of Math. Studies 144, Princeton, N.J.: Princeton Univ. Press, 1998 15. Graczyk, J., Swiatek, G.: Generic hyperbolicity in the logistic family. Ann. Math. 146(1), 1–52 (1997) 16. Hurwitz, A.: Uber eine Anwendung der elliptischen Modulfunktionen auf einem Satz der allgemainen Funktiontheorie. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich 49, 242–253 (1904) 17. Hubbard, J.H.: Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz. In: “Topological Methods in Modern Mathematics.” Houston, TX: Publish or Perish, 1993
Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials
327
18. Hu, J., Jiang, Y.: The Julia set of the Feigenbaum quadratic polynomial is locally connected. Preprint 1993 19. Kozlovski, O., Shen, W., Van Strien, S.: Rigidity for real polynomials. Ann. Math. 165, 749–841 (2007) 20. Kozlovski, O., Shen, W., Van Strien, S.: Density of hyperbolicity in dimension one. Ann. Math. 166, 145–182 (2007) 21. Kozlovski, O., Van Strien, S.: Local connectivity and quasi-conformal rigidity for non-renormalizable polynomials. Sept. 2006, http://arxiv.org/abs/math/0609710v3 [math.Ds], 2009 22. Kiwi, J.: Real laminations and the topological dynamics of complex polynomials. Adv. in Math. 184, 207–267 (2004) 23. Kiwi, J.: Combinatorial continuity in complex polynomial dynamics. Proc. London Math. Soc. 91(3), 215–248 (2005) 24. Kahn, J., Lyubich, M.: Local connectivity of Julia sets for unicritical polynomials. Ann. Math. 170, 413– 426 (2009) 25. Lyubich, M.: Dynamics of quadratic polynomials, I–II. Acta Math. 178, 185–297 (1997) 26. Levin, G.: Sequences of bifurcations of one-parameter families of mappings. Uspekhi Mat. Nauk, 37, n. 3 (225), 189–190 (1982) (Russian). English trans.: Russ. Math. Surv. 37, n. 3, 211-212 (1982) 27. Levin, G., Van Strien, S.: Local connectivity of the Julia set of real polynomials. Ann. Math. 147, 471–541 (1998) 28. Levin, G.: Multipliers of periodic orbits of quadratic polynomials and the parameter plane. Israel J. Math. 170(1), 285–315 (2009) 29. Levin, G.: On explicit connections between dynamical and parameter spaces. J. d’Analyse Math. 91, 297– 327 (2003) 30. Levin, G.: On Pommerenke’s inequality for the eigenvalue of fixed points. Colloq. Math. LXII(Fasc.1), 168–177 (1991) 31. Levin, G., Sodin, M.: Polynomials with disconnected Julia sets and Green maps. Hebrew Univ. of Jerusalem, Preprint 23/1990-91 (see http://www.ma.huji.ac.il/~levin/levsod.pdf) 32. Levin, G.: Disconnected Julia set and rotation sets. Ann. Scient. Ec. Norm. Sup. 4 series, t. 29, 1–22 (1996) 33. Levin, G., Przytycki, F.: External rays to periodic points. Isr. J. Math. 94, 29–57 (1996) 34. McMullen, C.: Complex Dynamics and Renormalization. Annals of Math. Studies 135, Princeton, N.J.: Princeton Univ. Press, 1994 35. McMullen, C.: Renormalization and 3-manifold which Fiber over the Circle. Annals of Math. Studies 142, Princeton, N.J.: Princeton Univ. Press, 1996 36. McMullen, C., Sullivan, D.: Quasiconformal homeomorphisms and dynamics III. The Teichmuller Space of a Holomorphic Dynamical System. Adv. Math. 135, 351–395 (1998) 37. De Melo, W., Van Strien, S.: One-Dimensional Dynamics. Ergebnisse Series 25, Berlin-Heidelberg-NewYork: Springer Verlag, 1993 38. Milnor, J.: Local connectivity of Julia sets: Expository lectures. Stony Brook IMS preprint 1992/11, available at http://arxiv.org/abs/math/9207220v1 [math.Ds], 1992 39. Milnor, J.: Non locally connected Julia sets constructed by iterated tuning. Birthday lecture for “Douady 70”. Revised May 26, 2006, SUNY, 28 pp. (available at http://www.math.sunysb.edu/~jack/tune-b.pdf) 40. Milnor, J.: Periodic orbits, external ray and the Mandelbrot set: an expository account. “Geometrie Complexe et Systemes Dynamiques”. Colloque en l’honneur d’Adrien Douady (Orsay, 1995). Asterisque 261, 277–331 (2000) 41. Milnor, J.: Dynamics in One Complex Variable. Third Edition. Annals of Math. Studies 160, Princeton, N.J.: Princeton Univ. Press, 2006 42. Montel, P.: Familles Normales de Fonctions Analytiques. Paris, Gauthier-Villars, 1927 43. Myrberg, P.J.: Alteration der reellen polynome zweiten grades. Ann. Acad. Sci. Fenn. A 259, 1–10 (1958) 44. Nehari, Z.: The elliptic modular function and a class of analytic functions first considered by Hurwitz. Amer. J. Math. 69, 70–86 (1947) 45. Nehari, Z.: Conformal Mappings. NewYork: McGraw-Hill Book Company, 1952 46. Petersen, C.L.: On the Pommerenke-Levin-Yoccoz inequality. Erg. Th. Dynam. Sys. 13(4), 785–806 (1993) 47. Petersen, C.L.: No elliptic limits for quadratic maps. Erg. Th. Dynam. Sys. 19, 127–141 (1999) 48. Pommerenke, Ch.: On conformal mappings and iteration of rational functions. Complex Var. Th. Appl. 5(2–4), 117–126 (1986) 49. Shen, W.: On the metric properties of multimodal interval maps and C 2 density of Axiom A. Invent. Math. 156(2), 301–403 (2004) 50. Schleicher, D.: On fibers and local connectivity of Mandelbrot and Multibrot sets. Proc. Symp. Pure Math. 72, 477–507 (2004) 51. Schleicher, D.: Internal addresses in the Mandelbrot set and irreducibility of polynomials. PhD thesis, Cornell University, 1994, available at http://arxiv.org/abs/math/9411238v2 [math.Ds], 1994
328
G. Levin
52. Sorensen, D.E.K.: Infinitely renormalizable quadratic polynomials with non-locally connected Julia set. J. Geom. Anal. 10, 169–206 (2000) 53. Sullivan, D.: Bounds, quadratic differentials and renormalization conjectures. In: Mathematics into the Twenty-First Century, AMS Centennial Publications 2, Providence, RI: Amer. Math. Soc., 1991 Communicated by S. Smirnov
Commun. Math. Phys. 304, 329–368 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1227-8
Communications in
Mathematical Physics
On Vanishing Theorems for Vector Bundle Valued p-Forms and their Applications Yuxin Dong1,2, , Shihshu Walter Wei3, 1 Institute of Mathematics, Fudan University, Shanghai 200433, Peoples’s Republic of China 2 Key Laboratory of Mathematics, for Nonlinear Sciences, Ministry of Education, Fudan University,
Shanghai, People’s Republic of China. E-mail: [email protected]
3 Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-0315, USA.
E-mail: [email protected] Received: 13 June 2009 / Accepted: 30 July 2010 Published online: 17 April 2011 – © Springer-Verlag 2011
Abstract: Let F : [0, ∞) → [0, ∞) be a strictly increasing C 2 function with F(0) = 0. We unify the concepts of F-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce F-Yang-Mills fields, F-degree, F-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or p √ with the minus sign) on manifolds. When F(t) = t, 1p (2t) 2 , 1 + 2t − 1, and 1 − √ 1 − 2t, the F-Yang-Mills field becomes an ordinary Yang-Mills field, p-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce the E F,g −energy functional (resp. F-Yang-Mills functional) and derive the first variational formula of the E F,g −energy functional (resp. F-Yang-Mills functional) with applications. In a more general frame, we use a unified method to study the stressenergy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors are naturally linked to F-conservation laws and yield monotonicity formulae, via the coarea formula and comparison theorems in Riemannian geometry. Whereas a “microscopic” approach to some of these monotonicity formulae leads to celebrated blow-up techniques and regularity theory in geometric measure theory, a “macroscopic” version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems for p−forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems for F−harmonic maps (which include harmonic maps, p-harmonic maps, exponentially harmonic maps, minimal graphs and maximal space-like hypersurfaces, etc.), F−Yang-Mills fields, extended Born-Infeld fields, and generalized YangMills-Born-Infeld fields (with the plus sign and with the minus sign) on manifolds, etc. Supported by NSFC grant No 10971029, and NSFC-NSF grant No 1081112053.
Research was partially supported by NSF Award No DMS-0508661, the OU Presidential International
Travel Fellowship, and the OU Faculty Enrichment Grant.
330
Y. Dong, S. W. Wei
As another consequence, we obtain the unique constant solution of the constant Dirichlet boundary value problems on starlike domains for vector bundle-valued 1-forms satisfying an F-conservation law, generalizing and refining the work of Karcher and Wood on harmonic maps. We also obtain generalized Chern type results for constant mean curvature type equations for p−forms on Rm and on manifolds M with the global doubling property by a different approach. The case p = 0 and M = Rm is due to Chern.
1. Introduction A theorem due to Garber, Ruijsenaars, Seiler and Burns [GRSB] states that every harmonic map u : Rm → S m with finite energy must be constant(m > 2). This result has been generalized by Hildebrandt [Hi] and Sealey [Se1] to harmonic maps into arbitrary Riemannian manifolds from more general domains, for example from an hyperbolic m-space form, or from Rm with certain globally conformal flat metrics, where m > 2. In the context of harmonic maps, the stress-energy tensor was introduced and studied in detail by Baird and Eells [BE]. Following Baird-Eells [BE], Sealey [Se2] introduced the stress-energy tensor for vector bundle valued p−forms and established some vanishing theorems for L 2 harmonic p−forms. Liouville type theorems for vector bundle valued harmonic forms or forms satisfying certain conservation laws have been treated by [KW] and [Xi1]. These follow immediately from monotonicity formulae. A similar technique was also used by [EF1] and [EF2] to show nonexistence of L 2 −eigenforms of the Laplacian (on functions and differential forms) on certain complete noncompact manifolds of nonnegative sectional curvature. On the other hand, in [Ar], M. Ara introduced the F−harmonic map and its associated stress-energy tensor. Let F : [0, ∞) → [0, ∞) be a C 2 function such that F > 0 on [0, ∞) , and F(0) = 0. A smooth map u : M → N between two Riemannian manifolds is said to be an F−harmonic map if it is a critical point of the following F−energy functional E F given by E F (u) =
F( M
|du|2 )dv 2
(1.1)
with respect to any compactly supported variation, where |du| is the Hilbert-Schmidt norm of the differential du of u, and dv is the volume element of M. When F(t) = t, p 1 α t 2 p (2t) , (1 + 2t) (α > 1, dim M = 2) , and e , the F−harmonic map becomes a harmonic map, a p−harmonic map, an α-harmonic map, and an exponentially harmonic map respectively. One of these striking features is that we can use, for example p-harmonic maps to study topics or problems that do not seem to be approachable by ordinary harmonic maps (in which p = 2) (see e.g. [We2,We3,LWe]). In addition to the above examples, F−energy functionals and their critical points arise widely in geometry and physics. Recall that a minimal hypersurface in Rm+1 , given as the graph of the function u on a Euclidean domain satisfies the following differential equation and is a solution of Plateau’s problem (for any closed m − 1-dimensional submanifold in the minimal graph as a given boundary): div
∇u 1 + |∇u|2
= 0.
(1.2)
Vanishing Theorem
331
If a maximal spacelike hypersurface in Minkowski space Rn,1 (with the coordinate n 1 n 2 2 (t, x , · · · , x ) and the metric ds = dt − i=1 (d x i )2 ) is given as the graph of the function v on a Euclidean domain, then the function v satisfies ∇v div = 0. (1.3) 1 − |∇v|2 m Obviously √ the solutions u and v√are F−harmonic maps from a domain in R to R with F = 1 + 2t − 1 and F = 1 − 1 − 2t respectively, with respect to any compactly supported variation. In [Ca], Calabi showed that Eqs. (1.2) and (1.3) are equivalent over any simply connected domain in R2 . Along the lines of Calabi, Yang [Ya] showed that, for m = 3, Eqs. (1.2) and (1.3) over a simply connected domain are, respectively, equivalent instead to the vector equations ∇×A ∇× =0 (1.4) 1 ∓ |∇ × A|2
(where A is a vector field in R3 and ∇ × ( · ) is the curl of ( · ) ) which arise in the nonlinear electromagnetic theory of Born and Infeld [BI]. This observation leads Yang [Ya] to give a generalized treatment of Eqs. of (1.2) and (1.3) expressed in terms of differential forms as follows: dω δ (1.5) = 0, ω ∈ A p (Rm ) 1 + |dω|2 and
δ
dσ
1 − |dσ |2
= 0,
σ ∈ Aq (Rm )
(1.6)
(where d is the exterior differential operator and δ is the codifferential operator), and a reformulation of Calabi’s equivalence theorem in arbitrary n dimensions. Born-Infeld theory is of contemporary interest due to its relevance in string theory ([BN,DG,Ke,LY, Ya,SiSiYa]). It is easy to verify that the solutions of (1.5) and (1.6) are critical points of the following Born-Infeld type energy functionals E +B I (ω) = 1 + |dω|2 − 1 dv, (1.7) Rm
and E− B I (σ ) =
Rm
1−
1 − |dσ |2 dv,
(1.8)
respectively. By choosing a sequence of cutoff functions and integrating by parts, Sibner-Sibner-Yang [SiSiYa] established a Liouville theorem for the L 2 exterior derivative dω of a solution ω of (1.5). They also introduced Yang-Mills-Born-Infeld fields and obtained a Liouville type result for finite-energy solutions of a generalized self-dual equation reduced from the Yang-Mills-Born-Infeld equation on R4 . In this paper, we unify the concepts of F-harmonic maps, minimal hypersurfaces in Euclidean space, maximal spacelike hypersurfaces in Minkowski space, and Yang-Mills
332
Y. Dong, S. W. Wei
Fields, and introduce F-Yang-Mills fields, F-degree, F-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manip √ folds (cf. Definitions 3.2, 4.1, 6.1 and 8.1). When F(t) = t , 1p (2t) 2 , 1 + 2t − 1 , √ and 1 − 1 − 2t , the F-Yang-Mills field becomes an ordinary Yang-Mills field, a p-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce the E F,g −energy functional (resp. F-Yang-Mills functional) and derive the first variational formula of the E F,g −energy functional (resp. F-YangMills functional) (Lemmas 2.5 and 3.1) with applications. In a more general frame, we use a unified method to study the stress-energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors lead to a fundamental integral formula (2.10), and are naturally linked to F-conservation laws. For example, we prove that every F−Yang-Mills field satisfies an F-conservation law. In particular, every p−Yang-Mills field satisfies a p-conservation law (cf. Theorem 3.1 and Corollary 3.1). As an immediate consequence, the simplified integral formula (2.11), from (2.10) holds for vector bundle valued forms satisfying an F−conservation law in general, and holds for the F−YangMills field in particular. This yields monotonicity inequalities, via the coarea formula and comparison theorems in Riemannian geometry (cf. Theorem 4.1 and Proposition 4.1). Whereas a “microscopic” approach to monotonicity formulae leads to celebrated blowup techniques due to E. de-Giorgi [Gi] and W.L. Fleming [Fl], and regularity theory in geometric measure theory (cf. [FF,A,SU,PS,HL,Lu]). For example, the regularity results of Allard [A] depend on the monotonicity formulae for varifolds. The regularity results of Schoen and Uhlenbeck [SU] depend on the monotonicity formulae for harmonic maps which they derived for energy minimizing maps; monotonicity properties are also dealt with by Price and Simon [PS] for Yang-Mills fields, and by Hardt-Lin [HL] and Luckhaus [Lu] for p-harmonic maps. A “macroscopic” version of these monotonicity formulae enable us to derive some Liouville type results and vanishing theorems under suitable growth conditions on Cartan-Hadamard manifolds or manifolds which possess a pole with appropriate curvature assumptions (e.g. Theorems 5.1 and 5.2). In particular, our results are applicable to F−harmonic maps, F−Yang-Mills fields, extended BornInfeld fields, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds, and obtain the first vanishing theorem for p-Yang-Mills fields (cf. Theorems 5.3–5.8). In fact, we introduce the following E F,g −energy functional: |d ∇ σ |2 )dvg E F,g (σ ) = F( (2.12) 2 M for forms σ ∈ A p−1 (ξ ) with values in a Riemannian vector bundle ξ , or study an even more general functional E F,g (ω) for forms ω ∈ A p (ξ ) (see (2.5)), introduced by Lu-Shen-Cai [LSC]. Naturally, the stress-energy tensor associated with E F,g (σ ) or E F,g (ω) plays an important role in establishing Liouville type results for extremals of E F,g or forms satisfying an F−conservation law. Our growth assumptions in Liouville type theorems in the general settings (cf. (5.1), (5.4), Theorems 5.1 and 5.2) are weaker than the assumption of finite energy for harmonic maps due to Garber, Ruijsenaars, Seiler and Burns [GRSB], Sealey [Se1], and others, or finite F-energy for F-harmonic maps due to M. Kassi [Ka], or L p growth for vector bundle valued forms due to J.C. Liu [Li1], or the slowly divergent F−energy condition (e.g. (5.3)) for harmonic maps and Yang-Mills fields that was first introduced
Vanishing Theorem
333
by H.S. Hu in [Hu1,Hu2], for F-harmonic maps due to Liao and Liu [LL2], and for an extremal of E F,g -energy functional treated by M. Lu, W.W. Shen and K.R. Cai [LSC](see Theorem 10.1, Examples 10.1 and 10.2 in the Appendix). Furthermore, our estimates in the monotonicity formulae are sharp in the sense that in special cases, they recapture the monotonicity formulae of harmonic maps [SU] and Yang-Mills field [PS] (cf. Corollary 4.1. and Remark 4.2). In addition to establishing vanishing theorems and Liouville type results, the monotonicity formulae may be used to investigate the constant Dirichlet boundary value problem as well. We obtain the unique constant solution of the constant Dirichlet boundary value problem on starlike domains for vector bundle-valued 1-forms satisfying an F-conservation law (cf. Theorem 6.1), generalizing and refining the work of Karcher and Wood on harmonic maps [KW]. Notice that our constant boundary value result holds for any starlike domain, while the original result in [KW] was stated for a disc domain. For an extended Born-Infeld field ω ∈ A p (Rm ) with the plus sign, we give an upper bound of the Born-Infeld type energy E +B I (ω; G(ρ)) of the p-form ω over its “graph” G(ρ) in Rm+k (cf. Proposition 7.1). This recaptures the volume estimate for the minimal graph of f due to P. Li and J.P. Wang, when ω = f ∈ A0 (Rm ) = C ∞ (Rm ) (cf. [LW]). As further applications, we obtain vanishing theorems for extended Born-Infeld fields (with the plus sign or with the minus sign) on manifolds under an appropriate growth condition on E ± B I -energy, and for generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds under an appropriate growth condition on YM± B I -energy (cf. Theorems 7.1, 8.1, and 8.2, Propositions 7.2 and 8.1). The case M = Rm and dω ∈ L 2 , where ω is a Born-Infeld field (hence ω has finite E B I -energy, √ 2 by the inequality 1 + t 2 − 1 ≤ t2 for any t ∈ R) is due to L. Sibner, R. Sibner and Y.S. Yang (cf. [SiSiYa]). Being motivated by the work in [We1,We2] and [LWW], we consider constant mean curvature type equations for p−forms on Rm and thereby obtain generalized Chern type results for constant mean curvature type equations for p−forms on Rm and on manifolds with the global doubling property by a different approach (cf. Theorems 9.1–9.4). The case p = 0 and M = Rm is due to Chern (cf. Corollary 9.1). This paper is organized as follows. Generalized F-energy functionals and F-conservation laws are given in Sect. 2. In Sect. 3, we introduce F-Yang-Mills fields. In Sect. 4, we derive monotonicity formulae. Liouville type results and vanishing theorems are established in three Subsects. 5.1, 5.2– 5.3 of Sect. 5. In Sect. 6, we treat constant Dirichlet Boundary Value Problems for vector valued 1-forms. Extended Born-Infeld fields and exact forms are presented in Sect. 7. In Sect. 8, we introduce generalized Yang-Mills-Born-Infeld fields (with the plus sign and with the minus sign) on manifolds. Generalized Chern type results on manifolds are investigated in Sect. 9. In the last section, we provide an appendix of a theorem on E F,g -energy growth. Throughout this paper let F : [0, ∞) → [0, ∞) be a strictly increasing C 2 function with F(0) = 0, and let M denote a smooth m−dimensional Riemannian manifold (mostly m > 2); all data will be assumed smooth for simplicity unless otherwise indicated.
2. Generalized F-Energy Functionals and F-Conservation Laws Let (M, g) be a smooth Riemannian manifold. Let ξ : E → M be a smooth Riemannian vector bundle over (M, g), i.e. a vector bundle such that each fiber is equipped with a
334
Y. Dong, S. W. Wei
positive inner product , E . Set A p (ξ ) = ( p T ∗ M ⊗ E) the space of smooth p−forms on M with values in the vector bundle ξ : E → M. The exterior differential operator d ∇ : A p (ξ ) → A p+1 (ξ ) relative to the connection ∇ E is given by d ∇ σ (X 1 , . . . , X p+1 ) =
p+1
(−1)i+1 ∇ XEi (σ (X 1 , . . . , X i , . . . , X p+1 ))
i=1
+
(−1)i+ j σ ([X i , X j ], X 1 , . . . , Xi , . . . , X j , . . . , X p+1 ),
i< j
(2.1) where the symbols covered by are omitted. Since the Levi-Civita connection on T M is torsion-free, we also have (d ∇ σ )(X 1 , . . . , X p+1 ) =
p+1
(−1)i+1 (∇ X i σ )(X 1 , . . . , X i , . . . , X p+1 ).
(2.2)
i=1
For two forms ω, ω ∈ A p (ξ ), the induced inner product is defined as follows:
ω, ω =
ω(ei1 , . . . , ei p ), ω (ei1 , . . . , ei p ) E i 1 <···
=
1
ω(ei1 , . . . , ei p ), ω (ei1 , . . . , ei p ) E , p!
(2.3)
i 1 ,...,i p
where {e1 , · · · em } is a local orthonormal frame field on (M, g). Relative to the Riemannian structures of E and T M, the codifferential operator δ ∇ : A p (ξ ) → A p−1 (ξ ) is characterized as the adjoint of d via the formula
d ∇ σ, ρ dvg =
σ, δ ∇ ρ dvg , M
M
where σ ∈ ∈ one of which has compact support, and dvg is the volume element associated with the metric g on T M. Then (δ ∇ ω)(X 1 , . . . , X p−1 ) = − (∇ei ω)(ei , X 1 , . . . , X p−1 ). (2.4) A p−1 (ξ ), ρ
A p (ξ ),
i
For ω ∈ A p (ξ ), set |ω|2 = ω, ω defined as in (2.3). The authors of [LSC] defined the following E F,g -energy functional given by |ω|2 )dvg , F( (2.5) E F,g (ω) = 2 M where F : [0, +∞) → [0, +∞) is as before. For our purpose, we also allow the domain of F to be [0, c) , where c is a positive number. In fact, we will study the case F : [0, 21 ) → [0, 1) in Sect. 7. The stress-energy associated with the E F,g -energy functional is defined as follows (cf. [BE,Ba,Ar,LSC]): S F,ω (X, Y ) = F(
|ω|2 |ω|2 )g(X, Y ) − F ( )(ω ω)(X, Y ), 2 2
(2.6)
Vanishing Theorem
335
where ω ω denotes a 2−tensor defined by: (ω ω)(X, Y ) = i X ω, i Y ω .
(2.7)
Here , is the induced inner product on A p−1 (ξ ), and i X ω is the interior multiplication by the vector field X given by (i X ω)(Y1 , . . . , Y p−1 ) = ω(X, Y1 , . . . , Y p−1 ) for ω ∈ A p (ξ ) and any vector fields Yl on M, 1 ≤ l ≤ p − 1. When F(t) = t and ω = du for a map u : M → N , S F,ω is just the stress-energy tensor introduced in [BE]. For two 2−tensors T1 , T2 ∈ (⊗2 T ∗ M), their inner product is defined as follows:
T1 , T2 = T1 (ei , e j )T2 (ei , e j ), (2.8) i, j
where {ei } is an orthonormal basis with respect to g. Suppose M is a complete Riemannian manifold. We calculate the rate of change of the E F,g -energy integral E F,g (ω) when the metric g on the domain or base manifold is changed. To this end, we consider a compactly supported smooth one-parameter variation s of the metric g, i.e. a smooth family of metrics gs such that g0 = g. Set δg = ∂g ∂s |s=0 . Then δg is a smooth 2-covariant symmetric tensor field on M with compact support. Lemma 2.1. For ω ∈ A p (ξ )( p ≥ 1), then dE F,gs (ω) 1 = s=0 ds 2
S F,ω , δg dvg , M
where S F,ω is as in (2.6). Proof. From [Ba], we know that d|ω|2gs = − ω ω, δg ds s=0 and d dvgs ds
|s=0
=
1
g, δg dvg . 2
Then
2 d |ω|2gs dE F,gs (ω) |ω|2 d |ω| ) ) dvgs = F ( dv + F( g ds 2 ds 2 2 ds M M s=0 s=0 1 |ω|2 |ω|2 )g − F ( )ω ω, δg dvg =
F( 2 M 2 2 1 =
S F,ω , δg dvg . 2 M
|s=0
Remark 2.1. When F(t) = t, the above result was derived by Sanini in [San] and by Baird in [Ba].
336
Y. Dong, S. W. Wei
For a vector field X , we denote by θ X its dual one form, i.e., θ X (·) = g(X, ·). By definition, the 2-tensor ∇θ X is given by (∇θ X )(Y, Z ) = (∇Y θ X )(Z ) = Y (θ X (Z )) − θ X (∇Y Z ) = g(∇Y X, Z ).
(2.9)
Lemma 2.2 (cf. [Xi1]). |ω|2 ) = i X d ∇ ω + d ∇ i X ω, ω − ω ω, ∇θ X , 2
ω(ei , e j1 , . . . , e j p−1 ), (∇ei ω)(X, e j1 , . . . , e j p−1 )
d ∇ i X ω, ω = ∇X (
j1 <···< j p−1 ;i
+ ω ω, ∇θ X . Next, we have the following result in which F(t) = t is known in [Se2] and [Xi1]: Lemma 2.3. Let ω ∈ A p (ξ )( p ≥ 1) and let S F,ω be the stress-energy tensor defined by (2.6), then for any vector field X on M, we have |ω|2 ∇ |ω|2 ) δ ω, i X ω + F ( ) i X d ∇ ω, ω 2 2 − i 2 ω, i X ω , |ω|
(div S F,ω )(X ) = F (
grad(F (
2
))
where grad ( • ) is the gradient vector field of •. Proof. By using Lemma 2.2 and (2.9), we derive the following (div S F,ω )(X ) =
m
∇ei S F,ω (ei , X ) − S F,ω (ei , ∇ei X )
i=1
=
m i=1
∇ei
F(
|ω|2 |ω|2 ) ei , X − F ( ) i ei ω, i X ω 2 2
|ω|2 |ω|2 ) ei , ∇ei X + F ( ) i ei ω, i ∇ei X ω 2 2 m |ω|2 |ω|2 ) ei , X − ei (F ( )) i ei ω, i X ω ei F( = 2 2 −F(
i=1
|ω|2 |ω|2 )ei i ei ω, i X ω + F ( ) i ei ω, i ∇ei X ω 2 2 m |ω|2 |ω|2 )− )) i ei ω, i X ω ei (F ( = ∇ X F( 2 2 −F (
i=1
|ω|2 |ω|2 )ei i ei ω, i X ω + F ( ) i ei ω, i ∇ei X ω −F ( 2 2 |ω|2 |ω|2 ) i X d ∇ ω + d ∇ i X ω, ω − F ( ) ω ω, ∇θ X = F ( 2 2
Vanishing Theorem
337
− i
grad(F ( |ω|2
−F ( = F (
|ω|2 2
)
2
))
ω, i X ω + F (
|ω|2 ∇ ) δ ω, i X ω 2
ω(ei , e j1 , . . . , e j p−1 ),(∇ei ω)(X, e j1 , . . . , e j p−1 )
j1 <···< j p−1 ;i
|ω|2
) i X d ∇ ω + d ∇ i X ω, ω − i 2 ω, i X ω grad(F ( |ω|2 )) 2 |ω|2 ∇ |ω|2 ∇ +F ( ) δ ω, i X ω − F ( ) d i X ω, ω . 2 2
Definition 2.1. ω ∈ A p (ξ ) ( p ≥ 1) is said to satisfy an F- conservation law if S F,ω is divergence free, i.e. the (0, 1)−type tensor field div S F,ω vanishes identically (div S F,ω ≡ 0). Lemma 2.4 ([Ba]). Let T be a symmetric (0, 2)−type tensor field. Let X be a vector field, and θ X be its dual 1-form, then div(i X T ) = (div T )(X ) + T, ∇θ X . Proof. Let {ei } be a local orthonormal frame field. Then div(i X T ) =
m
∇ei (i X T ) (ei ) i=1
m
∇ei (T (X, ei )) − T (X, ∇ei ei ) = i=1
=
m
(∇ei T )(X, ei ) +
i=1
m
T (∇ei X, ei )
i=1
= (div T )(X ) +
m
T (ei , e j ) g(∇ei X, e j ).
i, j=1
This via (2.9) proves the lemma. Let D be any bounded domain of M with C 1 −boundary. By applying T = S F,ω to Lemma 2.4 and using Stokes’ theorem, we immediately have the following: S F,ω (X, ν)dsg = S F,ω , ∇θ X + (div S F,ω )(X ) dvg , (2.10) ∂D
D
where ν is the unit outward normal vector field along ∂ D with (m − 1)-dimensional volume element dsg . In particular, if ω satisfies an F−conservation law, we have S F,ω (X, ν)dsg = S F,ω , ∇θ X dvg . (2.11) ∂D
D
It should be pointed out that the formulae (2.10) and (2.11) were also derived in [LSC]. We will give some important applications of (2.11) later.
338
Y. Dong, S. W. Wei
Now we introduce a new E F,g -energy functional as follows: For σ ∈ A p−1 (ξ ), E F,g (σ ) =
F( M
|d ∇ σ |2 )dvg . 2
(2.12)
This functional includes the functionals for F−harmonic maps (in which σ is a map between two Riemannian manifolds), and Born-Infeld fields (in which σ is the potential of an electric field or a magnetic field and M = R3 ; cf. [Ya]) as its special cases, etc. Lemma 2.5 (The First Variation Formula for E F,g -energy functional). d E F,g (σt ) |t=0 = − dt
τ F (σ ), η dvg M
for any η ∈ A p−1 (ξ ) with compact support, where σt = σ + tη and τ F (σ ) = ∇ 2 −δ ∇ (F ( |d 2σ | )d ∇ σ ). Furthermore, the Euler-Lagrange equation of E F,g is F (
|d ∇ σ |2 ∇ )τ (σ ) + i 2 d σ = 0, grad(F ( |dσ2 | )) 2
(2.13)
where τ (σ ) = −δ ∇ d ∇ σ . Proof. We compute
|d ∇ σ |2 ∇ ) d σ, d ∇ η dvg 2 M |d ∇ σ |2 ∇ )d σ ), η dvg =
δ ∇ (F ( 2 M = − τ F (σ ), η dvg ,
d E F,g (σ + tη) |t=0 = dt
F (
M
where |d ∇ σ |2 ∇ τ F (σ ) = −δ ∇ (F ( )d σ ) 2
m ∇ 2 |d σ | ∇ )d σ (ei , · · · , ·) ∇ei F ( = 2 i=1
=
m i=1
= F (
ei (F (
|d ∇ σ |2 ∇ |d ∇ σ |2 ))d σ (ei , · · · , ·) + F ( )(∇ei d ∇ σ )(ei , · · · , ·) 2 2
|d ∇ σ |2 ∇ )τ (σ ) + i 2 d σ. grad(F ( |dσ2 | )) 2
From Lemma 2.3 and the above expression (2.13) for τ F (σ ), we immediately have the following
Vanishing Theorem
339
Corollary 2.1. For σ ∈ A p−1 (ξ ), we have (div S F,d ∇ σ )(X ) = − τ F (σ ), i X d ∇ σ + F (
|d ∇ σ |2 ) i X (d ∇ )2 σ, d ∇ σ . 2
In particular, if τ F (σ ) = 0 and (d ∇ )2 σ = 0, then div S F,d ∇ σ = 0. Remark 2.2. In some cases, the condition (d ∇ )2 σ = 0 is satisfied automatically. For example, if σ ∈ A p−1 (M) := ( p−1 T ∗ M) , or σ = dϕ ∈ A1 (ϕ −1 T N ), where ϕ : M → N is a smooth map, then we have (d ∇ )2 σ = 0. Corollary 2.2 ([BE,Ka]). Let ϕ : M → N be an F−harmonic map. Then div S F,dϕ = 0. In particular, if F(t) = t and ϕ : M → N is a harmonic map, we have div SId,dϕ = 0. 3. F-Yang-Mills Fields In this section we introduce F-Yang-Mills functionals and F-Yang-Mills fields. Just as F−harmonic maps play a role in the space of maps between Riemannian manifolds, so do F- Yang-Mills fields in the space of curvature tensors (associated with connections on the adjoint bundles of principal G-bundles) over Riemannian manifolds. Let P be a principal bundle with compact structure group G over a Riemannian manifold M. Let Ad(P) be the adjoint bundle Ad(P) = P × Ad G,
(3.1)
where G is the Lie algebra of G. Every connection ρ on P induces a connection ∇ on Ad(P). We also have the Riemannian connection ∇ M on the tangent bundle T M, and the induced connection on p T ∗ M ⊗ Ad(P). An AdG invariant inner product on G induces a fiber metric on Ad(P) and making Ad(P) and p T ∗ M ⊗ Ad(P) into Riemannian vector bundles. Although ρ is not a section of 1 T ∗ M ⊗ Ad(P) via its induced connection ∇, the associated curvature R ∇ , given by R ∇ X,Y = [∇ X , ∇Y ] − 2 ∇[X,Y ] , is in A (Ad(P)). Let C be the space of connections ∇ on Ad(P). We now introduce Definition 3.1. The F- Yang-Mills functional is the mapping YM F : C → R+ given by 1 YM F (∇) = F( ||R ∇ ||2 )dv, (3.2) 2 M where the norm is defined in terms of the Riemannian metric on M and a fixed AdG invariant inner product on the Lie algebra G of G. That is, at each point x ∈ M , its norm ||R ∇ ||2x = ||Re∇i ,e j ||2x , i< j
where {e1 , · · · , en } is an orthonormal basis of Tx (M) and the norm of Re∇i ,e j is the standard one on Hom(Ad(P), Ad(P))-namely, S, U ≡ trace (S T ◦ U ).
340
Y. Dong, S. W. Wei
Definition 3.2. A connection ∇ on the adjoint bundle Ad(P) is said to be an F-YangMills connection and its associated curvature tensor R ∇ is said to be an F-Yang-Mills field, if ∇ is a critical point of YM F with respect to any compactly supported variation in the space of connections on Ad(P) . A connection ∇ is said to be a p-Yang-Mills connection and its associated curvature tensor R ∇ is said to be a p-Yang-Mills field, if ∇ is a critical point of the p-Yang-Mills functional YM p with respect to any compactly supported variation, where YM p (∇) = 1p M |R ∇ | p dv , and p ≥ 2. Lemma 3.1 (The First Variation Formula for F-Yang-Mills functional YM F ). A ∈ A1 (Ad(P)) and ∇ t = ∇ + t A be a family of connections on Ad(P). Then
Let
d YM F (∇ t )|t=0 = M δ ∇ F ( 21 ||R ∇ ||2 )R ∇ , A dv. dt Furthermore, The Euler-Lagrangian equation for YM F is 1 F ( ||R ∇ ||2 )δ ∇ R ∇ − i 1 ∇ 2 R ∇ = 0 grad F ( 2 ||R || ) 2 or δ
∇
1 F ( ||R ∇ ||2 )R ∇ 2
(3.3)
= 0.
Proof. By assumption, the curvature of ∇ t is given by R ∇ = R ∇ + t (d ∇ A) + t 2 [A, A], t
where [A, A] ∈ A2 (Ad(P)) is given by [A, A] X,Y = [A X , AY ] . Indeed, for any local vector fields X, Y on M, with [X, Y ] = 0 , we have R∇ X,Y = (∇ X + t A X )(∇Y + t AY ) − (∇Y + t AY )(∇ X + t A X ) t
2 = R∇ X,Y + t[∇ X , AY ] − t[∇Y , A X ] + t [A X , AY ] 2 = R∇ X,Y + t∇ X (AY ) − t∇Y (A X ) + t [A, A] X,Y ∇ 2 = R∇ X,Y + t (d A) X,Y + t [A, A] X,Y .
Thus 1 1 t F( ||R ∇ ||2 ) = F( ||R ∇ ||2 + t R ∇ , d ∇ A + ε(t 2 )), 2 2 where ε(t 2 ) = o(t 2 ) as t → 0. Therefore 1 t F( ||R ∇ ||2 + t R ∇ , d ∇ A + ε(t 2 )) dv YM F (∇ ) = 2 M and d YM F (∇ t )|t=0 = dt
1 F ( ||R ∇ ||2 ) R ∇ , d ∇ A dv 2
1
δ ∇ F ( ||R ∇ ||2 )R ∇ , A dv. = 2 M M
Vanishing Theorem
341
This derives the Euler-Lagrange equation for YM F as follows
1 0 = δ ∇ F ( ||R ∇ ||2 )R ∇ 2 m 1 =− (∇ei F ( ||R ∇ ||2 )R ∇ )(ei , ·) 2 i=1
1 = F ( ||R ∇ ||2 )δ ∇ R ∇ − i grad(F ( 1 ||R ∇ ||2 )) R ∇ . 2 2 Example 3.1. The Euler-Lagrangian equation for the p-Yang-Mills functional YM p is δ ∇ (||R ∇ || p−2 R ∇ ) = 0 or ||R ∇ || p−2 δ ∇ R ∇ − i grad(||R ∇ || p−2 ) R ∇ = 0.
(3.4)
Theorem 3.1. Every F−Yang-Mills field R ∇ satisfies an F-conservation law. Proof. It is known that R ∇ satisfies the Bianchi identity d ∇ R ∇ = 0.
(3.5)
Therefore, by Lemma 2.3, Lemma 3.1 and (3.5), we immediately derive the desired div S F,R ∇ = 0. Definition 3.3. ω ∈ Ak (ξ )(k ≥ 1) is said to satisfy a p-conservation law ( p ≥ 2) if p S F,ω is divergence free for F(t) = 1p (2t) 2 , i.e. for any vector field X on M, we have |ω| p−2 δ ∇ ω, i X ω + |ω| p−2 i X d ∇ ω, ω − i grad(|ω| p−2 ) ω, i X ω = 0.
(3.6)
As an immediate consequence, one has Corollary 3.1. Every p−Yang-Mills field R ∇ satisfies a p-conservation law. The F-conservation law√is crucial to our subsequent √ development. F-Yang-Mills fields in the cases F(t) = 1 + 2t − 1 and F(t) = 1 − 1 − 2t will be explored in Sect. 8. 4. Monotonicity Formulae In this section, we will establish monotonicity formulae on Cartan-Hadamard manifolds or more generally on complete manifolds with a pole. We recall a Cartan-Hadamard manifold is a complete simply-connected Riemannian manifold of nonpositive sectional curvature. A pole is a point x0 ∈ M such that the exponential map from the tangent space to M at x0 into M is a diffeomorphism. By the radial curvature K of a manifold with a pole, we mean the restriction of the sectional curvature function to all the planes which contain the unit vector ∂(x) in Tx M tangent to the unique geodesic joining x0 to x and pointing away from x0 . Let the tensor g − dr dr = 0 on the radial direction ∂, and is just the metric tensor g on the orthogonal complement ∂ ⊥ . We’ll use the following comparison theorems in Riemannian geometry:
342
Y. Dong, S. W. Wei
Lemma 4.1 (cf. [GW]). Let (M, g) be a complete Riemannian manifold with a pole x0 . Denote by K r the radial curvature of M. (i) If −α 2 ≤ K r ≤ −β 2 with α > 0, β > 0, then β coth(βr )[g − dr ⊗ dr ] ≤ H ess(r ) ≤ α coth(αr )[g − dr ⊗ dr ]. (ii) If K r = 0, then 1 [g − dr ⊗ dr ] = H ess(r ). r (iii) If − (1+rA2 )1+ ≤ K r ≤ 1− r
B 2
B (1+r 2 )1+
with > 0 , A ≥ 0 , and 0 ≤ B < 2 , then A
[g − dr ⊗ dr ] ≤ H ess(r ) ≤
e 2 [g − dr ⊗ dr ]. r
(iv) If −Ar 2q ≤ K r ≤ −Br 2q with A ≥ B > 0 and q > 0, then √ √ B0 r q [g − dr ⊗ dr ] ≤ H ess(r ) ≤ ( A coth A)r q [g − dr ⊗ dr ] q+1 2 1/2 for r ≥ 1, where B0 = min{1, − q+1 }. 2 + (B + ( 2 ) )
Proof. (i), (ii), and (iv) are treated in Sect. 2 of [GW]. (iii) Since for every > 0 ,
d s 1 2 − = − (1 + s ) , ds 2 (1 + s 2 )1+ we have ∞
∞ A A B B < ∞ and < 1. s ds = s ds = 2 )1+ 2 )1+ (1 + s 2 (1 + s 2 0 0 Now the assertion is an immediate consequence of the quasi-isometry theorem due to A B Greene-Wu [GW, p.57] in which 1 ≤ η ≤ e 2 and 1 − 2 ≤ μ ≤ 1.
Analogous to [Ka], (in which (iv) is employed) for a given function F, we introduce the following Definition 4.1. The F-degree d F is defined to be d F = sup t≥0
t F (t) . F(t)
For most of this paper, d F is assumed to be finite, unless otherwise stated. Lemma 4.2. Let M be a complete manifold with a pole x0 . Assume that there exist two positive functions h 1 (r ) and h 2 (r ) such that h 1 (r )[g − dr ⊗ dr ] ≤ H ess(r ) ≤ h 2 (r )[g − dr ⊗ dr ]
(4.1)
on M\{x0 }. If h 2 (r ) satisfies r h 2 (r ) ≥ 1,
(4.2)
then
S F,ω , ∇θ X ≥ (1 + (m − 1)r h 1 (r ) − 2 pd F r h 2 (r )) F( where X = r ∇r .
|ω|2 ), 2
(4.3)
Vanishing Theorem
343
∂ Proof. Choosing an orthonormal frame {ei , ∂r }i=1,...,m−1 around x ∈ M\{x0 }. Take X = r ∇r . Then
∇∂ X = ∂r
∂ , ∂r
(4.4)
m−1 ∂ r H ess(r )(ei , e j )e j . = ∂r
∇ei X = r ∇ei
(4.5)
j=1
Using (2.6), (2.9), (4.4) and (4.5), we have |ω|2 )(1 + r H ess(r )(ei , ei )) 2 m−1
S F,ω , ∇θ X = F(
i=1
−
m−1
F (
i, j=1
−F (
|ω|2 )(ω ω)(ei , e j )r H ess(r )(ei , e j ) 2
∂ ∂ |ω|2 )(ω ω)( , ). 2 ∂r ∂r
(4.6)
By (4.1), we get |ω|2 ) (1 + (m − 1)r h 1 (r )) 2 m−1 |ω|2 −F ( ) (ω ω)(ei , ei )r h 2 (r ) 2
S F,ω , ∇θ X ≥ F(
i=1
−F (
|ω|2 2
)(ω ω)(
∂ ∂ , ) ∂r ∂r
|ω|2 ) (1 + (m − 1)r h 1 (r ) − 2 pd F r h 2 (r )) 2 |ω|2 )(r h 2 (r ) − 1) i ∂ ω, i ∂ ω . +F ( ∂r ∂r 2
≥ F(
(4.7)
The last step follows from the fact that m−1
(ω ω)(ei , ei ) + (ω ω)(
i=1
=
∂ ∂ , ) ∂r ∂r
m
ω(ei , e j1 , · · · , e j p−1 ), ω(ei , e j1 , · · · , e j p−1 )
1≤ j1 <···< j p−1 ≤m i=1
= p|ω|2 , where em =
∂ ∂r .
Now the lemma follows immediately from (4.2) and (4.7).
Theorem 4.1. Let (M, g) be an m−dimensional complete Riemannian manifold with a pole x0 . Let ξ : E → M be a Riemannian vector bundle on M and ω ∈ A p (ξ ). Assume that the radial curvature K r of M satisfies one of the following three conditions:
344
Y. Dong, S. W. Wei
(i) −α 2 ≤ K r ≤ −β 2 with α > 0, β > 0 and (m − 1)β − 2 pαd F ≥ 0; (ii) K r = 0 with m − 2 pd F > 0; (iii) − (1+rA2 )1+ ≤ K r ≤ (1+rB2 )1+ with > 0 , A ≥ 0 , 0 < B < 2 , and A
B m − (m − 1) 2 − 2 pe 2 d F > 0. If ω satisfies an F−conservation law, then
1 ρ1λ
Bρ1 (x0 )
F(
1 |ω|2 )dv ≤ λ 2 ρ2
Bρ2 (x0 )
F(
|ω|2 )dv 2
for any 0 < ρ1 ≤ ρ2 , where ⎧ m − 2 p βα d F i f K r satis f ies (i) ⎨ m − 2 pd F i f K r satis f ies (ii) λ= A ⎩ B − 2 pe 2 d F i f K r satis f ies (iii). m − (m − 1) 2
(4.8)
(4.9)
Proof. Take a smooth vector field X = r ∇r on M. If K r satisfies (i), then by Lemma 4.1 and the increasing function βr coth(βr ) → 1 as r → 0 , (4.2) holds. Now Lemma 4.2 is applicable and by (4.3), we have on Bρ (x0 )\{x0 } , for every ρ > 0, |ω|2
S F,ω , ∇θ X ≥ (1 + (m − 1)βr coth(βr ) − 2 pd F αr coth(αr )) F( ) 2
|ω|2 αr coth(αr ) ) F( ) = 1 + βr coth(βr )(m − 1 − 2 pd F βr coth(βr ) 2
|ω|2 |ω|2 α > 1 + 1 · (m − 1 − 2 pd F · · 1) F( ) = λF( ), β 2 2 coth(αr ) provided that m−1−2 pd F · βα ≥ 0 , since βr coth(βr ) > 1 for r > 0 , and coth(βr ) < 1, for 0 < β < α , and coth is a decreasing function. Similarly, from Lemma 4.1 and Lemma 4.2, the above inequality holds for the cases (ii), and (iii) on Bρ (x0 )\{x0 }. Thus,
by the continuity of S F,ω , ∇θ X and F( |ω|2 ) , and (2.6), we have for every ρ > 0, 2
S F,ω , ∇θ X ≥ λF(
|ω|2 ) 2
in Bρ (x0 ), (4.10)
|ω|2 ∂ ) ≥ S F,ω (X, ) ρ F( 2 ∂r
on ∂ Bρ (x0 ).
It follows from (2.11) and (4.10) that
|ω|2 )ds ≥ λ ρ F( 2 ∂ Bρ (x0 )
Bρ (x0 )
F(
|ω|2 )dv. 2
(4.11)
Hence we get from (4.11) the following:
F( |ω|2 )ds 2
∂ Bρ (x0 )
|ω|2 Bρ (x0 ) F( 2 )dv
≥
λ . ρ
(4.12)
Vanishing Theorem
345
The coarea formula implies that d |ω|2 |ω|2 )dv = )ds. F( F( dρ Bρ (x0 ) 2 2 ∂ Bρ (x0 ) Thus we have d dρ
F( |ω|2 )dv 2
Bρ (x0 )
|ω|2 Bρ (x0 ) F( 2 )dv
≥
λ ρ
(4.13)
for a.e. ρ > 0. By integration (4.13) over [ρ1 , ρ2 ], we have |ω|2 |ω|2 ln F( F( )dv − ln )dv ≥ ln ρ2λ − ln ρ1λ . 2 2 Bρ2 (x0 ) Bρ1 (x0 ) This proves (4.8). Remark 4.1. (a) The theorem is obviously trivial when λ ≤ 0. (b) A study of Laplacian comparison on Cartan-Hadmard manifolds with Ric M ≤ −β 2 has been made in [Di]. By employing our techniques, as in the proofs of Lemma 4.2 and Theorem 4.1, some monotonicity formulas under appropriate curvature conditions, can be derived. (c) Whereas curvature assumptions (i) to (iii) cannot be exhaustive, our method is unified in the following sense: Regardless how radial curvature varies, as long as we have Hessian comparison estimates (4.1) with bounds satisfying (4.2), and the factor 1 + (m − 1)r h 1 (r ) − 2 pd F r h 2 (r ) ≥ c > 0 in (4.3) for some constant c, and ω satisfies an F− conservation law, then we obtain a monotonicity formula (4.8) for E F,g (ω)−energy, for an appropriate λ > 0. Corollary 4.1. Suppose M has constant sectional curvature −α 2 (α 2 ≥ 0). Let m − 1 − 2 pd F ≥ 0, if α = 0 , and m − 2 pd F > 0 if α = 0. Let ω ∈ A p (ξ ) be a ξ −valued p−form on M m satisfying an F-conservation law. Then 1 |ω|2 |ω|2 1 ) dv ≤ ) dv F( F( m−2 pd F m−2 pd F 2 2 Bρ1 (x0 ) Bρ2 (x0 ) ρ1 ρ2 for any x0 ∈ M and 0 < ρ1 ≤ ρ2 . Proof. In Theorem 4.1, if we take α = β = 0 for the case (i) or a = 0 for the case (ii), this corollary follows from (4.8) immediately. Remark 4.2. When F(t) = t and ω is the differential of a harmonic map or the curvature form of a Yang-Mills connection, then we recover the well-known monotonicity formulae for the harmonic map or Yang-Mills field (cf. [PS]). Proposition 4.1. Let (M, g) be an m−dimensional complete Riemannian manifold whose radial curvature satisfies (iv) − Ar 2q ≤ K r ≤ −Br 2q with A ≥ B > 0 and q > 0. Let√ω ∈ √ A p (ξ ) satisfy an F−conservation law, and δ := (m − 1)B0 − 2 pd F A coth A ≥ 0, where B0 is given in Lemma 4.1. Suppose (4.15) holds. Then |ω|2 |ω|2 1 1 F( F( ) dv ≤ ) dv (4.14) 2 2 ρ11+δ Bρ1 (x0 )−B1 (x0 ) ρ21+δ Bρ2 (x0 )−B1 (x0 ) for any 1 ≤ ρ1 ≤ ρ2 .
346
Y. Dong, S. W. Wei
Proof. Take X = r ∇r . Applying Lemma 4.1, (4.2), and (4.3), we have
2
S F,ω , ∇θ X ≥ F( |ω|2 ) 1 + δr q+1 and |ω|2 ∂ |ω|2 ) = F( ) − F ( ) i ∂ ω, i ∂ ω on ∂ B1 (x0 ), ∂r ∂r ∂r 2 2 |ω|2 ∂ |ω|2 S F,ω (X, ) = ρ F( ) − ρ F ( ) i ∂ ω, i ∂ ω on ∂ Bρ (x0 ). ∂r ∂r ∂r 2 2 It follows from (2.11) that |ω|2 |ω|2 ρ F( ) − F ( ) i ∂ ω, i ∂ ω ds ∂r ∂r 2 2 ∂ Bρ (x0 ) 2 2 |ω| |ω| ) − F ( ) i ∂ ω, i ∂ ω ds F( − ∂r ∂r 2 2 ∂ B1 (x0 ) |ω|2 ). ≥ (1 + δr q+1 )F( 2 Bρ (x0 )−B1 (x0 ) S F,ω (X,
Whence, if
∂ B1 (x0 )
then
F(
|ω|2 |ω|2 ) − F ( ) i ∂ ω, i ∂ ω ds ≥ 0 , ∂r ∂r 2 2
|ω|2 ) ds ≥ (1 + δ) ρ F( 2 ∂ Bρ (x0 )
Bρ (x0 )−B1 (x0 )
F(
(4.15)
|ω|2 ) dv 2
for any ρ > 1. Coarea formula then implies 2 d Bρ (x0 )−B1 (x0 ) F( |ω|2 ) dv 1+δ dρ ≥ |ω|2 ρ Bρ (x0 )−B1 (x0 ) F( 2 ) dv
(4.16)
for a.e. ρ ≥ 1. Integrating (4.16) over [ρ1 , ρ2 ], we get |ω|2 |ω|2 ) dv − ln ) dv ln F( F( 2 2 Bρ2 (x0 )−B1 (x0 ) Bρ1 (x0 )−B1 (x0 ) ≥ (1 + δ) ln ρ2 − (1 + δ) ln ρ1 . Hence we prove the proposition. Corollary 4.2. Let K r and δ be as in Proposition 4.1, and ω satisfy an F−conservation law. Suppose 2 |ω|2 d F i ∂ ω ≤ ∂r 2
(4.17)
on ∂ B1 , or F( |ω|2 ) − F ( |ω|2 )|i ∂ ω|2 ≥ 0 on ∂ B1 . Then (4.14) holds. 2
2
∂r
Proof. The assumption (4.17) implies that (4.15) holds, and the assertion follows from Proposition 4.1.
Vanishing Theorem
347
5. Vanishing Theorems and Liouville Type Results In this section we list some results in the following three subsections, that are immediate applications of the monotonicity formulae in the last section.
5.1. Vanishing theorems for vector bundle valued p-forms. Theorem 5.1. Suppose the radial curvature K r of M satisfies the condition in Theorem 4.1. If ω ∈ A p (ξ ) satisfies an F−conservation law and Bρ (x0 )
F(
|ω|2 ) dv = o(ρ λ ) as ρ → ∞, 2
(5.1)
where λ is given by (4.9), then F( |ω|2 ) ≡ 0 , and hence ω ≡ 0. In particular, if ω has finite E F,g −energy, then ω ≡ 0. 2
Definition 5.1. ω ∈ A p (ξ ) is said to have slowly divergent E F,g -energy, if there exists a positive continuous function ψ(r ) such that
∞ ρ1
dr = +∞ r ψ(r )
(5.2)
for some ρ1 > 0 , and lim
ρ→∞ B (x ) ρ 0
F( |ω|2 ) dv < ∞. ψ(r (x)) 2
(5.3)
Remark 5.1. (1) Hesheng Hu introduced the notion of slowly divergent energy (in which F(t) = t , ω = du , or ω = R ∇ ), and made a pioneering study in [Hu1,Hu2]. (2) In [LL2] and [LSC], the authors established some Liouville results for F−harmonic maps or forms with values in a vector bundle satisfying an F−conservation law under the condition of slowly divergent energy. Obviously Theorem 5.1 improves all these growth conditions, as its special cases of F , and expresses the growth condition more explicitly (cf. Theorem 10.1, Examples 10.1 and 10.2 in Appendix). Theorem 5.2. Suppose M and δ satisfy the condition in Proposition 4.1. If ω ∈ A p (ξ ) satisfies an F−conservation law, (4.15) holds, and Bρ (x0 )
F(
|ω|2 )dv = o(ρ 1+δ ) as ρ → ∞ 2
(5.4)
then ω ≡ 0 on M − B1 (x0 ). In particular, if ω has finite E F,g −energy, then ω ≡ 0 on M − B1 (x0 ). Notice that Theorem 5.2 only asserts that ω vanishes in an open set of M. If ω possesses the unique continuation property, then ω vanishes on M everywhere (cf. Corollaries 5.2 and 5.4).
348
Y. Dong, S. W. Wei
5.2. Liouville theorems for F-harmonic maps. Let u : M → N be an F−harmonic map. Then its differential du can be viewed as a 1-form with values in the induced bundle u −1 T N . Since ω = du satisfies an F−conservation law, we obtain the following Liouville-type Theorem 5.3. Let N be a Riemannian manifold. Suppose the radial curvature K r of M and λ satisfy the condition in Theorem 4.1 in which p = 1. Then every F−harmonic map u : M → N with the following growth condition is a constant. Bρ (x0 )
F(
|du|2 ) dv = o(ρ λ ) as ρ → ∞. 2
(5.5)
In particular, every F−harmonic map u : M → N with finite F−energy is a constant. Proof. This follows at once from Theorem 5.1 in which p = 1 and ω = du. Remark 5.2. This is in contrast to a Liouville Theorem for F-harmonic maps into a domain of strictly convex function by a different approach (cf. Theorem 12.1 in [We2]). Theorem 5.4 (Liouville Theorem for p-harmonic maps). Let N be a Riemannian manifold. Suppose the radial curvature K r of M satisfies one of the following three conditions: (i) −α 2 ≤ K r ≤ −β 2 with α > 0, β > 0 and (m − 1)β − pα ≥ 0; (ii) K r = 0 with m − p > 0; (iii) − (1+rA2 )1+ ≤ K r ≤ (1+rB2 )1+ with > 0 , A ≥ 0 , 0 < B < 2 , and A
B m −(m − 1) 2 − pe 2 > 0. Then every p−harmonic map u : M → N with the following p−energy growth condition (5.6) is a constant: 1 |du| p dv = o(ρ λ ) as ρ → ∞, (5.6) p Bρ (x0 )
where ⎧ ⎨
m − p βα , if K r satisfies (i) m − p, if K r satisfies (ii) λ= A ⎩ B m − (m − 1) 2 − pe 2 , if K r satisfies (iii).
(5.7)
In particular, every p−harmonic map u : M → N with finite p−energy is a constant. Proof. This follows immediately from Theorem 5.3 in which F(t) = d F = 2p .
p 1 2 p (2t)
and
Remark 5.3. The case 1p Bρ (x0 ) |du| p dv = o((ln ρ)q ) as ρ → ∞ for some positive number q is due to Liu-Liao [LL1]. Corollary 5.1. Let M , N , K r , λ and the growth condition (5.6) be as in Theorem 5.4, in which p = 2. Then every harmonic map u : M → N is a constant.
Vanishing Theorem
349
Theorem 5.5. Let M , N , K r , and δ satisfy the condition of Proposition 4.1 in which p = 1. Suppose (4.15) holds for ω = du. Then every F−harmonic map u : M → N with the following growth condition is a constant on M − B1 (x0 ): Bρ (x0 )
F(
|du|2 )dv = o(ρ 1+δ ) as ρ → ∞ 2
(5.8)
on M − B1 (x0 ). In particular, if u has finite F−energy, then u ≡ const on M − B1 (x0 ). Proof. This follows at once from Proposition 4.1. Proposition 5.1. Let (M, g) be an m−dimensional complete Riemannian manifold 2q 2q whose radial curvature √ satisfies √ −Ar ≤ K r ≤ −Br with A ≥ B > 0 and q > 0. If δ := (m − 1)B0 − p A coth A ≥ 0, where B0 is given in Lemma 4.1. Suppose (4.15) holds p−harmonic map u : M → N with the growth condition for ω = pdu. Then every 1 1+δ ) as ρ → ∞ is a constant on M − B (x ) , In particular, |du| dv = o(ρ 1 0 p Bρ (x0 ) if u has finite p−energy, then u ≡ const on M − B1 (x0 ). Corollary 5.2. Let M , N , K r , δ , (4.15), and the growth condition be as in Proposition 5.1, in which p = 2. Then every harmonic map u : M → N is a constant. Proof. This follows immediately from Proposition 5.1 and the unique continuation property of a harmonic map. 5.3. Applications in F-Yang-Mills fields. Let R ∇ be an F−Yang-Mills field, associated with an F−Yang-Mills connection ∇ on the adjoint bundle Ad(P) of a principle G-bundle over a manifold M. Then R ∇ can be viewed as a 2-form with values in the adjoint bundle over M , and by Theorem 3.1, ω = R ∇ satisfies an F−conservation law. Theorem 5.6 (Vanishing Theorem for F-Yang-Mills fields). Let M , K r , and λ satisfy the condition in Theorem 4.1 in which p = 2. Suppose F−Yang-Mills field R ∇ satisfies the following growth condition: Bρ (x0 )
F(
|R ∇ |2 ) dv = o(ρ λ ) as ρ → ∞. 2
(5.9)
Then R ∇ ≡ 0 on M. In particular, every F−Yang-Mills field R ∇ with finite F−Yang-Mills energy vanishes on M. Proof. This follows at once from Theorem 5.1 in which p = 2 and ω = R ∇ .
Theorem 5.7 (Vanishing Theorem for p-Yang-Mills fields). Suppose the radial curvature K r of M satisfies one of the following conditions: (i) −α 2 ≤ K r ≤ −β 2 with α > 0, β > 0 and (m − 1)β − 2 pα ≥ 0; (ii) K r = 0 with m − 2 p > 0; (iii) − (1+rA2 )1+ ≤ K r ≤ (1+rB2 )1+ with > 0 , A ≥ 0 , and 0 < B < 2 , and A
B m − (m − 1) 2 − 2 pe 2 > 0.
350
Y. Dong, S. W. Wei
Then every p−Yang-Mills field R ∇ with the following growth condition vanishes: 1 |R ∇ | p dv = o(ρ λ ) as ρ → ∞, (5.10) p Bρ (x0 ) where
⎧ ⎨
m − 2 p βα , if K r satisfies (i) m − 2 p, if K r satisfies (ii) λ= A ⎩ B − 2 pe 2 > 0, if K r satisfies (iii). m − (m − 1) 2
(5.11)
In particular, every p−Yang-Mills field R ∇ with finite YM p −energy vanishes on M. Corollary 5.3. Let M , N , K r , λ , and the growth condition (5.12) be as in Theorem 5.10, in which p = 2. Then every Yang-Mills field R ∇ ≡ 0 on M. Theorem 5.8. Suppose M , K r , and δ , satisfy the same conditions of Proposition 4.1 in which p = 2 , and (4.15) holds for ω = R ∇ . Then every F−Yang-Mills field R ∇ with the following growth condition vanishes on M − B1 (x0 ): |R ∇ |2 )dv = o(ρ 1+δ ) as ρ → ∞. F( (5.12) 2 Bρ (x0 ) In particular, if R ∇ has finite F−Yang-Mills energy, then R ∇ ≡ 0 on M − B1 (x0 ). Proof. This follows immediately from Proposition 4.1. Proposition 5.2. Let (M, g) be an m−dimensional complete Riemannian manifold 2q 2q whose radial curvature satisfies −Ar √ √ ≤ K r ≤ −Br with A ≥ B > 0 and q > 0. Let δ := (m − 1)B0 − 2 p A coth A ≥ 0, where B0 is given in Lemma 4.1, and let (4.15) ω = R ∇ . Then every p−Yang-Mills field R ∇ with the growth condition hold for 1 ∇ p 1+δ ) as ρ → ∞ vanishes on M − B (x ). In particular, if 1 0 p Bρ (x0 ) |R | dv = o(ρ R ∇ has finite p−Yang-Mills energy, then R ∇ ≡ 0 on M − B1 (x0 ).
Corollary 5.4. Let M , K r , δ , (4.15) , and the growth condition be as in Proposition 5.2, in which p = 2. Then every Yang-Mills field R ∇ ≡ 0 on M. Proof. This follows at once from Proposition 5.2, and the unique continuation property of Yang-Mills field. Further applications will be treated in Sect. 8. 6. Constant Dirichlet Boundary Value Problems To investigate the constant Dirichlet boundary value problems for 1-forms, we begin with Definition 6.1. The F-lower degree l F is given by t F (t) . t≥0 F(t)
l F = inf
Vanishing Theorem
351
Definition 6.2. A bounded domain D ⊂ M with C 1 boundary ∂ D is called starlike if there exists an interior point x0 ∈ D such that ∂ , ν ∂ D ≥ 0, (6.1)
∂r x0 where ν is the unit outer normal to ∂ D , and the vector field ∂ ∂r x0 (x)
∂ ∂r x0
is the unit vector field
such that for any x ∈ D\{x0 } ∪ ∂ D , is the unit vector tangent to the unique geodesic joining x0 to x and pointing away from x0 . It is obvious that any convex domain is starlike. Theorem 6.1. Suppose M satisfies the same condition of Theorem 4.1 and D ⊂ M is a bounded starlike domain with C 1 boundary. Assume that the F-lower degree l F ≥ 1/2. If ω ∈ A1 (ξ ) satisfies an F−conservation law and annihilates any tangent vector η of ∂ D, then ω vanishes on D. Proof. By assumption, there exists a point x0 ∈ D such that the distance function r x0 satisfies (6.1). Take X = r ∇r , where r = r x0 . From the proofs of Theorem 4.1, we know that |ω|2 ), (6.2)
S F,ω , ∇θ X ≥ cF( 2 where c is a positive constant. Since ω ∈ A1 (ξ ) annihilates any tangent vector η of ∂ D, we easily derive the following on ∂ D: ∂ S F,ω (X, ν) = r S F,ω ( , ν) ∂r
2 |ω|2 ∂ ∂ |ω| = r F( ) , ν − F ( ) ω( ), ω(ν) 2 ∂r 2 ∂r
2 ∂ |ω|2 |ω| = r , ν F( ) − F ( )|ω|2 ∂r 2 2 ∂ |ω|2 , ν (1 − 2l F )F( ) ≤ 0. ∂r 2 From (2.11), (6.2) and (6.3), we have |ω|2 0≤ )dv ≤ 0, cF( 2 D which implies that ω ≡ 0. ≤ r
(6.3)
Corollary 6.1. Suppose M and D satisfy the same assumptions of Theorem 6.1. Let u : D → N be a p−harmonic map ( p ≥ 1) into an arbitrary Riemannian manifold N . If u|∂ D is constant, then u| D is constant. p
Proof. For a p−harmonic map u, we have F(t) = 1p (2t) 2 . Obviously d F = l F = Take ω = du. This corollary follows immediately from Theorem 6.1.
p 2.
Remark 6.1. When M = Rm and D = Bρ (x0 ), this result, Corollary 6.1, recaptures the work of Karcher and Wood on the constant Dirichlet boundary value problem for harmonic maps [KW]. The result of Karcher and Wood was also generalized to harmonic maps with potential by Chen [Ch] and p−harmonic maps with potential by Liu [Li2] for disc domains.
352
Y. Dong, S. W. Wei
7. Extended Born-Infeld Fields and Exact Forms In this section, we will establish Liouville type theorems for solutions of the extended Born-Infeld equations (1.5) and (1.6) proposed by [Ya]. Using Hodge star operator ∗ , we can rewrite Eqs. (1.5) and (1.6) as dω d∗ (7.1) = 0, ω ∈ A p (Rm ) 2 1 + |dω| and
d∗
dσ 1 − |dσ |2
= 0,
σ ∈ Aq (Rm )
(7.2)
respectively. As pointed out in the Introduction, the solutions of (7.1) and (7.2) are critical points of the E +B I -energy functional and the E − B I -energy functional respectively. Notice -energy functional are E F,g −energy functhat the E +B I -energy functional and the E − BI √ √ tionals with F(t) = 1 + 2t − 1 (t ∈ [0, +∞)), and F(t) = 1 − 1 − 2t (t ∈ [0, 1/2)) respectively. Definition 7.1. The extended Born-Infeld energy functional with the plus sign on a manifold M is the mapping E +B I : A p (M) → R+ given by 1 + |dω|2 − 1 dv (7.3) E +B I (ω) = M
and the extended Born-Infeld energy functional with the minus sign on a manifold q + M is the mapping E − B I : A (M) → R given by − 1 − 1 − |dσ |2 dv. (7.4) E B I (σ ) = M
E− B I ) with respect to any compactly supported vari-
A critical point ω of E +B I (resp. σ of ation is called an extended Born-Infeld field with the plus sign (resp. with the minus sign) on a manifold.
Obviously Corollary 2.1 implies that the solutions of (7.1) and (7.2) satisfy F−conservation laws. Now we recall the equivalence between (7.1) and (7.2) found by [Ya] as follows: Let ω ∈ A p (Rm ) be a solution of (7.1) with 0 ≤ p ≤ m − 2. Then dω τ =±∗ (7.5) 1 + |dω|2 is a closed (m − p−1)−form. Since the de Rham cohomology group H m− p−1 (Rm ) = 0, there exists an (m − p − 2)−form σ such that τ = dσ . It is easy to derive from (7.5) the following: |dω|2 =
|dσ |2 1 − |dσ |2
(7.6)
Vanishing Theorem
353
and dω = ±(−1) p(m− p) ∗
dσ 1 − |dσ |2
.
(7.7)
The Poincaré Lemma implies that σ satisfies (7.2) with q = m − p − 2. Using (7.6), we get 1 + |dω|2 − 1 2 1 − 1 − |dσ | = 1 + |dω|2 ≤ 1 + |dω|2 − 1. (7.8) Conversely, a solution σ of (7.2) with 0 ≤ q ≤ m − 2 gives us a solution ω ∈ A p (Rm ) of (7.1) with p = m − q − 2, and 1 − 1 − |dσ |2 2 1 + |dω| − 1 = . (7.9) 1 − |dσ |2 Let’s first consider Eq. (7.1) and let ω be a solution of (7.1). Choose an orthonormal
m , basis ω1 , . . . , ωk of A p (Rm ) consisting of constant differential forms, where k = p and for each 1 ≤ α ≤ k , ωα = d x j1 ∧ · · · ∧ d x j p α for some 1 ≤ j1 < · · · < j p ≤ m. Then we may write ω = kα=1 f ωα . So ω may be m . Let M = (x, ω(x)) regarded as a map ω : Rm → p (Rm ) Rk , where k = p be the graph of ω in Rm+k and let G(ρ) be the extrinsic ball of radius ρ of the graph centered at the origin of Rm+k given by G(ρ) = M ∩ B m+k (ρ). Set ωρ =
k α=1
where f ρα (x) =
f ρα ωα ,
⎧ ⎨
ρ if f α > ρ f α (x) if | f α (x)| ≤ ρ . ⎩ −ρ if f α < ρ
For any δ > 0, let φ be a nonnegative cut-off function defined on Rm given by ⎧ 1 on B m (ρ) ⎪ ⎪ ⎨ (x) φ = (1+δ)ρ−r on B m ((1 + δ)ρ)\B m (ρ) . δρ ⎪ ⎪ ⎩ 0 on Rm \B m ((1 + δ)ρ)
(7.10)
(7.11)
354
Y. Dong, S. W. Wei
Proposition 7.1. Let ω ∈ A p (Rm ) be an extended Born-Infeld field with the plus sign on Rm . Then the Born-Infeld type energy of ω over G(ρ) satisfies the upper bound √ E +B I (ω; G(ρ)) ≤ m kωm ρ m ,
m and ωm is the volume of the unit ball in Rm . where k = p Proof. Taking inner product with φωρ , we may get from (1.5) or (7.1) that dω 0=
d ∗ ( ), φωρ d x m 1 + |dω|2 R dω =
, d(φωρ ) d x m 1 + |dω|2 R dω dω =
, dφ ∧ ωρ d x + φ , dωρ d x. 1 + |dω|2 1 + |dω|2 Rm Rm √ 1 Using the fact that |dφ| = |∇φ| ≤ δρ and |ωρ | ≤ kρ, we have
B m ((1+δ)ρ)
φ
dω 1 + |dω|2
, dωρ d x ≤ ≤
So B m (ρ)∩{| f α |≤ρ}
|dω|2 1 + |dω|2
B m ((1+δ)ρ)
√
dx ≤
|dφ||ωρ ||dω| dx 1 + |dω|2
k Vol B m ((1 + δ)ρ) − B m (ρ)) . δ
√
k Vol B m ((1 + δ)ρ) − B m (ρ)) . δ
Because G(ρ) ⊂ M ∩ (B m (ρ) × [−ρ, ρ]), we have + 1 + |dω|2 − 1 d x E B I (ω; G(ρ)) ≤ B m (ρ)∩{| f α |≤ρ} ≤ 1 + |dω|2 d x − V ol(B m (ρ) ∩ {| f α | ≤ ρ})
B m (ρ)∩{| f α |≤ρ}
|dω|2 + 1 d x − V ol(B m (ρ) ∩ {| f α | ≤ ρ}) 2 m α 1 + |dω| B (ρ)∩{| f |≤ρ} √
k V ol B m ((1 + δ)ρ) − B m (ρ)) ≤ √δ
k ωm (1 + δ)m ρ m − ρ m . = δ ≤
Let δ → 0, we have
√ E +B I (ω; G(ρ)) ≤ m kωm ρ m .
Vanishing Theorem
355
Remark 7.1. When ω = f ∈ A0 (Rm ) = C ∞ (Rm ), the above result is the volume estimate for the minimal graph of f (cf. [LW]). √ Lemma 7.1. (i) If√F(t) = 1 + 2t − 1 with t ∈ [0, +∞), then d F = 1 and l F = 1/2. (ii) If F(t) = 1 − 1 − 2t with t ∈ [0, 1/2), then d F = +∞ and l F = 1. √ Proof. (i) For F(t) = 1 + 2t − 1, we have t F (t) t = √ √ F(t) 1 + 2t( 1 + 2t − 1) √ 1 + 2t + 1 = √ for t ∈ (0, +∞). 2 1 + 2t Hence, 1 1 t F (t) = + √ F(t) 2 2 1 + 2t
for t ∈ [0, +∞).
(7.12)
By definition, we√get d F = 1 and l F = 1/2. (ii) For F(t) = 1 − 1 − 2t, we have t t F (t) = √ √ F(t) 1 − 2t(1 − 1 − 2t) √ 1 1 + 1 − 2t for t ∈ (0, ). = √ 2 2 1 − 2t Hence, t F (t) 1 1 = + √ F(t) 2 2 1 − 2t
1 for t ∈ [0, ). 2
(7.13)
By definition, we obtain d F = +∞ and l F = 1. √ By applying Corollary 4.1 to M = Rm and F = 1 + 2t − 1, we immediately get the following: Theorem 7.1. Let ω ∈ A p (Rm ) be an extended Born-Infeld field with the plus sign on Rm . If m > 2 p and ω satisfies the following growth condition: 1 + |dω|2 − 1 d x = o(ρ m−2 p ) as ρ → ∞ Bρ (x0 )
for some point x0 ∈ Rm , then dω = 0 , and ω is exact. In particular, if ω has finite E +B I −energy, then ω is exact. Remark 7.2. In [SiSiYa], the authors proved the following: Let ω be a solution of (7.1). If dω ∈ L 2 (Rm ) (m ≥ 3) or dω ∈ L 2 (R2 ) ∩ H1 on R2 , where H1 is the Hardy space, √ 2 then dω ≡ 0. In view of the inequality 1 + t 2 − 1 ≤ t2 for any t ≥ 0, it is clear that being in L 2 ensures finite E +B I −energy. Using the duality between solutions of (7.1) and (7.2), we have
356
Y. Dong, S. W. Wei
Proposition 7.2. Let σ ∈ Aq (Rm ) be a q−form with m−4 2 < q < m − 2. If σ is an extended Born-Infeld field with the minus sign on Rm , and σ satisfies the following growth: 1 − 1 − |dσ |2 d x = o(ρ 2q−m+4 ) as ρ → ∞, (7.14) 1 − |dσ |2 Bρ (x0 ) then dσ = 0 , and σ is exact. In particular, if σ has finite E − B I −energy, then σ is exact. Proof. By the duality between (7.1) and (7.2), we get a solution ω from the solution σ of (7.2), where ω satisfies (7.1) and (7.9). Since p = m − q − 2, the condition q > m−4 2 is equivalent to m > 2 p. Obviously (7.9) and (7.14) imply 1 + |dω|2 − 1 d x = o(ρ m−2 p ) as ρ → ∞. Bρ (x0 )
Therefore Theorem 7.1 implies that dω = 0 which is equivalent to dσ = 0. Proposition 7.3. Let σ ∈ Aq (Rm ) be a q−form with q < m−2 2 . Suppose that σ is an extended Born-Infeld field with the minus sign on Rm , satisfying |dσ |2 ≤ 1 − Then 1
m q+1
Bρ1 (x0 )
ρ1
1−
for any 0 < ρ1 ≤ ρ2 . Proof. Let F(t) = 1 −
√
(q + 1)2 . (m − q − 1)2
1 − |dσ |2 d x ≤
1 m q+1
ρ2
(7.15)
Bρ2 (x0 )
1−
1 − |dσ |2 d x
(7.16)
1 − 2t. For the distance function r on Rm , we have
1 [g − dr ⊗ dr ], (7.17) r where g is the standard Euclidean metric. Taking X = r ∇r , using (4.6) and (7.17), we have at those points x ∈ Rm , where dσ (x) = 0 , H ess(r ) =
|dσ |2 |dσ |2
S F,dσ , ∇θ X = m F( ) − q F ( )|dσ |2 2 2 2 F ( |dσ2 | )|dσ |2 |dσ |2 ). = m−q F( 2 2 F( |dσ2 | )
(7.18)
From (7.13), it is easy to see that (7.15) is equivalent to, for every x ∈ Rm , F ( |dσ2 | )|dσ |2 2
m −q
2 F( |dσ2 | )
= m − q(1 +
1 1 − |dσ |2
)≥
m q +1
(7.19)
which implies
S F,dσ , ∇θ X ≥
m |dσ |2 F( ) on q +1 2
Bρ (x0 ).
(7.20)
Therefore we can prove this proposition by using (7.20) in the same way as we prove Theorem 4.1, via (4.10).
Vanishing Theorem
357
Corollary 7.1. In addition to the same hypotheses of Proposition 7.3, if σ satisfies m 1 − 1 − |dσ |2 d x = o(ρ q+1 ) as ρ → ∞, Bρ (x0 )
then dσ ≡ 0 , and σ is exact. In particular, if σ has finite E − B I −energy, then σ is exact. 8. Generalized Yang-Mills-Born-Infeld Fields (with the plus sign and with the minus sign) on Manifolds In [SiSiYa], L. Sibner, R. Sibner and Y.S. Yang consider a variational problem which is a generalization of the (scalar valued) Born-Infeld model and at the same time a quasilinear generalization of the Yang-Mills theory. This motivates the study of YangMills-Born-Infeld fields on R4 , and they prove that a generalized self-dual equation whose solutions are Yang-Mills-Born-Infeld fields has no finite-energy solution except the trivial solution on R4 . In this section, we introduce the following Definition 8.1. The generalized Yang-Mills-Born-Infeld energy functional with the plus sign on a manifold M is the mapping YM+B I : C → R+ given by YM+B I (∇) = 1 + ||R ∇ ||2 − 1 dv (8.1) M
and the generalized Yang-Mills-Born-Infeld energy functional with the minus sign + on a manifold M is the mapping YM− B I : C → R given by YM− (∇) = 1 − 1 − ||R ∇ ||2 dv. (8.2) BI M
The associate curvature form R ∇ of a critical connection ∇ of YM+B I (resp. YM− B I ) is called a generalized Yang-Mills-Born-Infeld field with the plus sign (resp. with the minus sign) on a manifold. √ √ By applying F(t) = 1 + 2t − 1 and F(t) = 1 − 1 − 2t to Theorem 3.1, we obtain Corollary 8.1. Every generalized Yang-Mills-Born-Infeld field (with the plus sign or with the minus sign) on a manifold satisfies an F-conservation law. Theorem 8.1. Let the radial curvature K r of M satisfy one of the three conditions (i), (ii), and (iii) in Theorem 4.1 in which p = 2 and d F = 1. Let R ∇ be a generalized Yang-Mills-Born-Infeld field with the plus sign on M. If R ∇ satisfies the following growth condition: 1 + ||R ∇ ||2 − 1 dv = o(ρ λ ) as ρ → ∞, Bρ (x0 )
where
⎧ α if ⎨m − 4β if λ= m−4 A ⎩ B − 4e 2 if m − (m − 1) 2
K r satis f ies(i), K r satis f ies (ii), K r satis f ies (iii),
then its curvature R ∇ ≡ 0. In particular, if R ∇ has finite YM+B I -energy, then R ∇ ≡ 0.
358
Y. Dong, S. W. Wei
√ Proof. By applying Corollary 8.1 and F(t) = 1 + 2t − 1 to Theorem 4.1 in which d F = 1 , by Lemma 7.1(i), and p = 2 , for R ∇ ∈ A2 (Ad P), the result follows immediately. Theorem 8.2. Suppose M has constant sectional curvature −α 2 (α 2 ≥ 0). Let R ∇ be a generalized Yang-Mills-Born-Infeld field with the plus sign on M. If m > 4 and R ∇ satisfies the following growth condition:
Bρ (x0 )
1 + ||R ∇ ||2 − 1 dv = o(ρ m−4 )
as ρ → ∞,
then its curvature R ∇ ≡ 0. In particular, if R ∇ has finite YM+B I -energy, then R ∇ ≡ 0. Proof. This follows at once by applying α = β in conditions (i) and (ii) of Theorem 8.1. Corollary 8.2. Let R ∇ be a Yang-Mills-Born-Infeld field with the plus sign on Rm . If m > 4 and R ∇ satisfies the following growth condition:
Bρ (x0 )
1 + ||R ∇ ||2 − 1 d x = o(ρ m−4 ) as ρ → ∞,
then its curvature R ∇ ≡ 0. In particular, if R ∇ has finite YM+B I -energy, then R ∇ ≡ 0. If we replace dσ with R ∇ and set q = 2 in Proposition 7.3, by a similar argument, we obtain the following Proposition 8.1. Let R ∇ be a Yang-Mills-Born-Infeld field with the minus sign on Rm . Suppose m > 6 , ||R ∇ ||2 ≤
m 2 − 6m m 2 − 6m + 9
and Bρ (x0 )
1−
m
1 − ||R ∇ ||2 d x = o(ρ 2 ) as ρ → ∞.
Then R ∇ = 0. It is well-known that there are no nontrivial Yang-Mills fields in Rm with finite Yang-Mills-energy for m ≥ 5 (in contrast with R4 , where the problem is conformally invariant and one obtains Yang-Mills fields with finite YM-energy by pullback from S 4 (cf. [JT])). In Corollary 8.2, for the case m ≥ 5, we obtain a similar result for Yang-MillsBorn-Infeld field (with the plus sign) on Rm . It’s natural to ask if there exists a nontrivial Yang-Mills-Born-Infeld field (with the plus sign) on R4 with finite YM+B I -energy.
Vanishing Theorem
359
9. Generalized Chern type Results on Manifolds A Theorem of Chern states that every entire graph xm+1 = f (x1 , . . . , xm ) on Rm of constant mean curvature is minimal in Rm+1 . In this section, we view functions as 0-forms and consider the following constant mean curvature type equation for p−forms ω on Rm ( p < m) and on manifolds with the global doubling property by a different approach (being motivated by the work in [We1,We2] and [LWW]): dω δ (9.1) = ω0 , 1 + |dω|2 where ω0 is a constant p−form. (Thus when p = 0, (9.1) is just the equation describing graphic hypersurface with constant mean curvature.) Equivalently, (9.1) may be written as dω d∗ (9.2) = ξ0 , 1 + |dω|2 where ξ0 is a constant (m − p)−form. Theorem 9.1. Suppose ω is a solution of (9.2) on Rm . Then ξ0 = 0. Proof. Obviously, for every (m − p)−plane in Rm , there exists a volume element d of , such that ξ0 | = c d, for some constant c. Let i : → Rm be the inclusion mapping. If follows from (9.2) and Stokes’ theorem that for every ball B(x0 , r ) of radius r centered at x0 in ⊂ Rm , and its boundary ∂ B(x0 , r ) with the surface element d S, we have 0 ≤ |c|ωm− p r m− p = c d B(x ,r ) 0 = i ∗ ξ0 B(x0 ,r ) dω ∗ = di ∗ 1 + |dω|2 B(x0 ,r ) dω = i∗ ∗ 1 + |dω|2 ∂ B(x0 ,r ) dω dS ≤ 1 + |dω|2 ∂ B(x0 ,r ) ≤ (m − p)ωm− p r m− p−1 , where ωm− p is the volume of the unit ball in . Hence we get 0 ≤ |c| ≤
m−p , r
which implies that c = 0 by letting r → ∞.
(9.3)
360
Y. Dong, S. W. Wei
This generalizes the work of Chern: Corollary 9.1 ([Che]). Let p = 0 in Theorem 9.1. Then the graph of ω over R m is a minimal hypersurface in Rm+1 . Proof. As p = 0, we may assume that ω = f for some function f on Rm . Then (9.1) is equivalent to div
∇f
= c,
1 + |∇ f |2
(9.4)
where c is a constant. Now the assertion follows from Theorem 9.1. Corollary 9.2. Let p = 0 and m ≤ 7 in Theorem 9.1. Then the graph of ω over Rm is a hyperplane in Rm+1 . Proof. This follows at once from Corollary 9.1 and Bernstein Theorems for minimal graphs (cf. [Be,Al,Gi,Si]). Corollary 9.3. Let p = 0 and |∇ω|(x) ≤ β (for all x ∈ Rm , where β > 0 is a constant) in Theorem 9.1. Then the graph of ω over Rm is a hyperplane in Rm+1 , for all m ≥ 2. Proof. This follows at once from Corollary 9.1 and Harnack’s theorem due to Moser (cf. [Mo], p.591). In fact, we can give a further generalization. Theorem 9.2. Let ω be a differential form of degree p on Rm , satisfying d∗
dω
1 + |dω|2
= ξ,
(9.5)
where ξ is a differential form of degree m − p on Rm . Suppose there exists an (m − p)−plane in Rm , with the volume element d, such that ξ | = g(x)d, off a bounded set K in , where g is a continuous function on \K with c = inf x∈\K |g(x)|. Then c = 0. Proof. We consider two cases: Case 1. g assumes both positive and negative values: By the intermediate value theorem, g assumes value 0 at some point, and thus c = inf x∈\K |g(x)| = 0. Case 2. g is a nonpositive or nonnegative function: Since K ⊂ is bounded, choose a sufficiently large r0 < r so that K ⊂ B(x0 , r0 ), where B(x0 , r0 ) is the ball of radius r0 centered at x0 in ⊂ Rm . Let 0 ≤ ψ ≤ 1 be the cut off function such that ψ ≡ 1 on B(x0 , r0 ) and ψ ≡ 0 off B(x0 , 2r ) ⊂ , and |∇ψ| ≤ Cr1 (cf. also Lemma 1 in [We1]). Let i : → Rm be the inclusion mapping. Multiplying (9.5) by ψ, and applying the
Vanishing Theorem
361
divergence theorem, we have cωm− p (r
m− p
m− p − r0 )
≤ = = ≤
ψ(x)g(x)d B(x ,r )\B(x0 ,r0 ) 0 ∗ ψi ξ B(x0 ,r )\B(x0 ,r0 ) dω ∗ ψdi ∗( ) 1 + |dω|2 B(x0 ,r )\B(x0 ,r0 ) dω d |∇ψ| 2 1 + |dω| B(x0 ,2r )\B(x0 ,r0 ) dω dS + 1 + |dω|2 ∂ B(x0 ,r0 ) m− p−1
≤ ωm− p C1 2m− p r m− p−1 + (m − p)ωm− p r0
,
where ωm− p is the volume of the unit ball in . Hence m− p
0 ≤ cωm− p (1 −
m− p−1
r0 ωm− p C1 2m− p (m − p)ωm− p r0 + ) ≤ r m− p r r m− p
implies that c = 0 by letting r → ∞. Corollary 9.4. There does not exist a solution of (9.5) such that ξ | = g(x)d, off a bounded set K in some (m − p)−plane in Rm with c > 0, where g is a continuous realvalued (not necessary nonnegative or nonpositive) function, and c = inf x∈\K |g(x)|. Corollary 9.5. Let f be a function satisfying ∇f =c div 1 + |∇ f |2 off a bounded subset K ⊂ Rm , where c ≡const. Then c = 0. In particular, every graph of f of constant mean curvature off a cylinder Rm \(B(x0 , r0 ) × R) is minimal. Proof. This follows at once from Theorem 9.2 in which ω = d f and p = 0. In particular, we choose K = B(x0 , r0 ). Remark 9.1. This result, Corollary 9.4, recaptures Corollary 9.1, a theorem of Chern, in which K is an empty set. Notice that Chern’s result was also generalized to graphs with higher codimension and parallel mean curvature in Euclidean space by Salavessa [Sa1]. Next we consider the following equation: dσ δ = ρ0 , 1 − |dσ |2
(9.6)
which generalizes the constant mean curvature equation for spacelike hypersurfaces.
362
Y. Dong, S. W. Wei
Theorem 9.3. Let σ be a differential form of degree q(≤ m − 1) on Rm , satisfying dσ d∗ (9.7) = τ0 , 1 − |dσ |2 where τ0 is a constant (m − q)−form. If 1 = o(r ), 1 − |dσ |2
(9.8)
where r is the distance from the origin, then τ0 = 0. Proof. Obviously, for every (m − q)−plane in Rm , there exists a volume element d of , such that τ0 | = c d. Let i : → Rm be the inclusion mapping. For every ball B(x0 , r ) of radius r centered at x0 in ⊂ Rm , and its boundary ∂ B(x0 , r ), by using (9.7) and Stokes’ theorem, we have c d |c|ωm−q r m−q = B(x0 ,r ) dσ ∗ = i ∗( ) 1 − |dσ |2 ∂ B(x0 ,r ) dσ dS ≤ 2 1 − |dσ | ∂ B(x0 ,r ) 1 ≤ (m − q) sup { }ωm−q r m−q−1 , 1 − |dσ |2 ∂ B(x0 ,r ) where ωm−q is the volume of the unit ball in . Hence |c| ≤
m −q r
1 sup { } 1 − |dσ |2 ∂ B(x0 ,r )
implies that c = 0 by letting r → ∞. Remark 9.2. (1) When q = 0, (9.6) describes a spacelike graphic hypersurface with constant mean curvature. It is known that √ 1 2 is bounded iff the Gauss image 1−|dσ |
of the hypersurface is bounded (cf. [Xi2,Xi3]). Such kind of Chern type results under growth conditions were obtained in [Do,Sa2] for spacelike graphs as well. (2) A similar generalized Chern type result can be established for the following more general equation:
∇ 2 ∇ |d σ | ∇ F( )d σ = ρ0 . δ 2 Using a different technique or idea (cf. [We1,We2,LWW]), one can extend the above results to a complete noncompact manifold M that has the global doubling property, i.e., ∃D(M) > 0 such that ∀r > 0 , ∀x ∈ M , V ol(B(x, 2r )) ≤ D(M)V ol(B(x, r )).
(9.9)
Examples of complete manifolds with the global doubling property include complete noncompact manifolds of nonnegative Ricci curvature, in particular Euclidean space Rm .
Vanishing Theorem
363
Theorem 9.4. Let ω be a differential form of degree p on M that has the global doubling property, and satisfies dω d∗ = ξ, (9.10) 1 + |dω|2 where ξ is a differential form of degree m − p on M. Suppose there exists an (m − p)− dimensional submanifold in M, with the volume element d, such that ξ | = g(x)d, off a bounded set K in , where g is a continuous function on \K with c = inf x∈\K |g(x)|. Then c = 0. Proof. Proceed as in the proof of Theorem 9.2, it suffices to show the result holds for g ≥ 0 or g ≤ 0. Let K ⊂ B(x0 , r0 ), where B(x0 , r0 ) is the geodesic ball of radius r0 in M , centered at x0 . Let 0 ≤ ψ ≤ 1 be the cut off function such that ψ ≡ 1 on B(x0 , r0 ) and ψ ≡ 0 off B(x0 , 2r ), and |∇ψ| ≤ Cr1 (cf. also Lemma 1 in [We1]). Let i : → M be the inclusion mapping. Then multiplying both sides of Eq. (9.10) by ψ , integrating over the annulus B(x0 , 2r )\B(x0 , r0 )(⊂ M\K ) , and applying Stokes’ theorem, we have c(V ol(B(x0 , r )) − V ol(B(x0 , r0 )) ≤ ψ(x)g(x) d B(x ,r )\B(x0 ,r0 ) 0 ≤ ψg (x) d B(x ,2r )\B(x0 ,r0 ) 0 = ψi ∗ ξ B(x0 ,2r )\B(x0 ,r0 ) dω ∗ = ψdi ∗( ) 1 + |dω|2 B(x0 ,2r )\B(x0 ,r0 ) dω d ≤ |∇ψ| 1 + |dω|2 B(x0 ,2r )\B(x0 ,r0 ) dω dS + 1 + |dω|2 ∂ B(x0 ,r0 ) C1 V ol(B(x0 , 2r )) ≤ V ol(∂ B(x0 , r0 )) + , (9.11) r where d S is the area element of ∂ B(x0 , r ). Hence, dividing (9.11) by V ol(B(x0 , r )) , one has c(1 −
V ol B(x0 , r0 ) V ol(∂ B(x0 , r0 )) C1 D(M) )≤ + →0 V ol(B(x0 , r )) V ol(B(x0 , r )) r
as r → ∞ , since M has infinite volume (by Lemma 5.1 in [LWW]).
10. Appendix: A Theorem on E F,g -Energy Growth In this Appendix, we provide a theorem on E F,g -energy growth, with examples (cf. Examples 10.1 and 10.2). These in particular, imply that our growth assumptions
364
Y. Dong, S. W. Wei
(5.1) and (5.4) in Liouville type results are weaker than the existing growth conditions such as finite E F,g -energy, slowly divergent E F,g -energy (cf. (5.3)), (10.6), and (10.7). f (r ) We say that f (r ) ∼ g(r ) as r → ∞ , if lim supr →∞ g(r ) = 1 , and f (r ) g(r ) as r → ∞ , otherwise. We say that f (r ) g(r ) for large r , if there exist positive constants k1 and k2 such that k1 g(r ) ≤ f (r ) ≤ k2 g(r ) for all large r , and f (r ) g(r ) for large r otherwise. Lemma 10.1. Let ψ(r ) > 0 be a continuous function such that ∞ dr = +∞ ρ0 r ψ(r )
(5.2)
for some ρ0 > 0. Then (i) ψ(r ) can not go to infinity faster than r λ , i.e., limr →∞ (ii) If
) limr →∞ ψ(r rλ
exists for some λ > 0 , then lim
r →∞
ψ(r ) rλ
= ∞ , for any λ > 0.
ψ(r ) = 0, rλ
(10.1)
f (r ) g(r ) , and ψ(r ) r λ . Proof. Suppose on the contrary, i.e. limr →∞ ) limr →∞ ψ(r rλ
ψ(r ) rλ
= c < ∞, where c = 0 (resp.
= ∞). Then there would exist ρ1 > 0 such that if r ≥ ρ1 , ψ(r ) > 2c r λ (resp. ψ(r ) > kr λ , where k > 0 is a constant.) This would lead to
∞ ∞ 2 ∞ dr dr dr r esp. k < ∞, ≤ 1+λ c ρ1 r 1+λ ρ1 r ψ(r ) ρ1 r contradicting (5.2), by the continuity of ψ(r ) if ρ0 < ρ1 . Theorem 10.1. Let ω ∈ A p (ξ ) have slowly divergent E F,g −energy. That is,
F( |ω|2 ) dv < ∞ ψ(r (x)) 2
lim
ρ→∞ B (x ) ρ 0
(5.3)
for some continuous function ψ(r ) > 0 satisfying (5.2). Then (i) For any λ > 0 , limr →∞ (ii) If limr →∞
ψ(r ) rλ
ψ(r ) rλ
= ∞.
exists for some λ > 0 , then Bρ (x0 )
F(
|ω|2 ) dv = o(ρ λ ) as ρ → ∞. 2
(5.1)
Proof. In view of Lemma 10.1 and (10.1), we have for every > 0 , there exists ρ2 > 0 , such that if r > ρ2 , then ψ(r ) <
rλ , 2(L + 1)
(10.2)
Vanishing Theorem
365
2 F( |ω|2 ) where L := limρ→∞ Bρ (x0 ) ψ(r (x)) dv (by assumption 0 ≤ L < ∞ ). Hence by the definition of L , there exists ρ3 > 0 such that if ρ > ρ3 , then
F( |ω|2 ) dv < L + 1. ψ(r (x)) 2
Bρ (x0 )
(10.3)
2 Since limρ→∞ ρ1λ Bρ (x0 ) F( |ω|2 ) dv = 0 , we have for every > 0 , there exists 2 ρ4 > 0 , such that if ρ > ρ4 , then 1 ρλ
Bρ2 (x0 )
F(
|ω|2 ) dv < . 2 2
(10.4)
It follows that for every > 0 , one can choose ρ5 = max{ρ2 , ρ3 , ρ4 } , such that if ρ > ρ5 , then via (10.2) (10.3) and (10.4), we have 1 ρλ
2 F( |ω|2 ) ψ(r (x)) |ω|2 ) dv + F( dv 2 ρλ Bρ2 (x0 ) Bρ (x0 )\Bρ2 (x0 ) ψ(r (x)) 2 F( |ω|2 ) r λ < + dv 2 2(L + 1) Bρ (x0 )\Bρ2 (x2 ) ψ(r (x)) ρ λ 2 F( |ω|2 ) ≤ + dv 2 2(L + 1) Bρ (x0 ) ψ(r (x)) (10.5) < + = . 2 2
1 |ω|2 ) dv = λ F( 2 ρ Bρ (x0 )
That is, (5.1) holds. Example 10.1. Let ω ∈ A p (ξ ) have the growth rate lim
ρ→∞ B (x ) ρ 0
F( |ω|2 ) dv < ∞ (ln r (x))q 2
(10.6)
for some number q ≤ 1. Then ω has slowly divergent E F,g −energy (5.3), as ψ(r ) = (ln r )q satisfies (5.2) for any number q ≤ 1. Furthermore, as an immediate consequence of Theorem 10.1, ω has the growth rate Bρ (x0 )
F(
|ω|2 ) dv = o(ρ λ ) as ρ → ∞ 2
(5.1)
for any λ > 0. The following is an example of ψ(r ) that does not satisfy (5.2), yet ω has the growth rate (5.1):
366
Y. Dong, S. W. Wei
Example 10.2. Let ω ∈ A p (ξ ) have the growth rate
F( |ω|2 )
ρ→∞ B (x ) ρ 0
(ln r (x))q
lim
2
dv < ∞
(10.7)
for some number q > 1. Then ψ(r ) = (ln r )q does not satisfy (5.2) for any number q > 1. Since (ln ρ)q goes to infinity slower than ρ λ for any q , λ > 0 , it is evident that ω has the growth rate (5.1), via (10.3) for any λ > 0. Acknowledgments. The authors wish to thank Professors J.G. Cao and Z.X. Zhou, and Ye Li for their helpful discussions. They also wish to thank the editor, the referee, Communications in Mathematical Physics, and Springer for making the present form of the paper possible.
References [A] [Al]
Allard, W.K.: On the first variation of a varifold. Ann. of Math. 95(2), 417–491 (1972) Almgren, F.J. Jr.: Some interior regularity theorems for minimal surfaces and extension of Bernstein’s theorem. Ann. of Math. 84(2), 277–292 (1966) [Ar] Ara, M.: Geometry of F−harmonic maps. Kodai Math. J. 22, 243–263 (1999) [Ba] Baird, P.: Stress-energy tensors and the Lichnerowicz Laplacian. J. Geom. Phys. 58(10), 1329–1342 (2008) [Be] Bernstein, S.: Sur un theoreme de geometrie at ses application aux equations aux derivees partielles du type elliptique. Comm. Soc. Math. Kharkov 15(2), 38–45 (1915-C1917) [BE] Baird, P., Eells, J.: A conservation law for harmonic maps. In: Geometry Symposium, Utrecht 1980, in: Lecture notes in Mathematics, Vol. 894, Berlin, Heidleberg-NewYork: Springer, 1982, pp. 1–25 [BI] Born, M., Infeld, L.: Foundation of a new field theory. Proc. R. Soc. London Ser. A., 144, 425–451 (1934) [BN] Barbashov, B.M., Nesterenko, V.V.: Introduction to the relativistic string theory. Singapore World Scientific, 1990 [Ca] E. Calabi, Examples of Bernstein problems for some nonlinear equations. In: Global Analysis, (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Providence, RI: Amer. Math. Soc., 1970, pp. 223–230 [Ch] Chen, Q.: Stability and constant boundary-value problems of harmonic maps with potential. J. Aust. Math. Soc. (Series A) 68, 145–154 (2000) [Che] Chern, S.S.: On the curvature of a piece of hypersurface in euclidean space. Abh. Math. Sem. Hamburg 29, 77–91 (1965) [CZ] Chen, Q., Zhou, Z.R.: On gap properties and instabilities of p-yang-mills fields. Canad. J. Math. 59(6), 1245–1259 (2007) [DG] Deser, S., Gibbons, G.W.: Born-infeld actions. Class. Quantun 15, L35-9 (1998) [Di] Ding, Q.: The Dirichlet problem at infinity for manifolds of nonpositive curvature. In: Gu, C. H. (ed.) et al., Differential geometry, Proc. of the Symp. in honour of Prof. Su Buchin on his 90th birthday (Shanghai), Singapore: World Sci., 1993, pp. 48–58 [Do] Dong, Y.X.: Bernstein theorems for spacelike graphs with parallel mean curvature and controlled growth. J. Geom. Phys. 58(3), 324–333 (2008) [EF1] Escobar, J.F., Freire, A.: The spectrum of the Laplacian of manifolds of positive curvature. Duke Math. J. 65, 1–21 (1992) [EF2] Escobar, J.F., Freire, A.: The differential form spectrum of manifolds of positive curvature. Duke Math. J. 69, 1–41 (1993) [EFM] Escobar, J.F., Freire, A., Min-Oo, M.: L 2 vanishing theorems in positive curvature. Indiana Univ. Math. J. 42(4), 1545–1554 (1993) [FF] Federer, H., Fleming, W.H.: Normal and integral currents. Ann. of Math. 72, 458–520 (1960) [Fl] Fleming, W.H.: On the oriented plateau problem, rend. Circ. Mat. Palermo 11(2), 69–90 (1962) [Gi] de Giorgi, E.: Una estensione del theorema di bernstein. Ann. Scuola Norm. Sup. Pisa 19(3), 79– 85 (1965) [GRSB] Garber, W.D., Ruijsenaars, S.N.M., Seiler, E., Burns, D.: On finite action solutions of the nonlinear σ −model. Ann. Phys. 119, 305–325 (1979) [GW] Greene, R.E., Wu, H.: Function theory on manifolds which possess a pole. Lecture Notes in Math. 699, Berlin, Heidleberg-New York:Springer-Verlag, 1979
Vanishing Theorem
367
Hardt, R., Lin, F.H.: Mappings minimizing the L p norm of the gradient. XL Comm. Pure App. Math. 40(5), 555–588 (1987) [Hi] Hildebrandt, S.: Liouville theorems for harmonic mappings, and an approach to Bernstein theorems. Ann. Math. Stud. 102, 107–131 (1982) [Hu1] Hu, H.S.: On the static solutions of massive Yang-Mills equations. Chinese Annals of Math. 3, 519– 526 (1982) [Hu2] Hu, H.S.: A nonexistence theorem for harmonic maps with slowly divergent energy. Chinese Ann. of Math. Ser. B 5(4), 737–740 (1984) [JT] Jaffe, A., Taubes, C.: Vortices and Monopoles: Structures of Static Gauge Theories. Boston: Birkhauser, 1980 [Ka] Kassi, M.: A liouville theorem for F−harmonic maps with finite F−energy. Electronic J. Diff. Eqs. 15, 1–9 (2006) [Ke] Ketov, S.V.: Many faces of Born-Infeld theory. http://arxiv.org/abs/hep-th/0108189v1, 2001 [KW] Karcher, H., Wood, J.C.: Non-existence results and growth properties for harmonic maps and forms. J. Reine Angew. Math. 353, 165–180 (1984) [La] Lawson, H.B. Jr.: The theory of gauge fields in four dimensions. CBMS Regional Conference Series in Mathematics 58, Providence, RI: Amer. Math. Soc., 1985 [LY] Lin, F.H., Yang, Y.S.: Gauged harmonic maps, Born-Infeld electromagnetism, and Magnetic Vortices. Comm. Pure. Appl. Math. LVI, 1631–1665 (2003) [Li1] Liu, J.C.: Vanishing theorem for L p −forms valued on vector bundle (Chinese). J. of Northwest Normal University (Natural Science) 38(4), 20–22 (2004) [Li2] Liu, J.C.: Constant boundary-valued problems for p− harmonic maps with potential. J. Geom. Phys. 57, 2411–2418 (2007) [LL1] Liu, J.C., Liao, C.S.: The energy growth property for p− harmonic maps. J. East China Normal Univ. (Natural Sci.) no. 2, 19–23 (2004) [LL2] Liu, J.C., Liao, C.S.: Nonexistence Theorems for F− harmonic maps (Chinese). J. Math. Research and Exposition 26, 311–316 (2006) [LSC] Lu, M., Shen, X.W., Cai, K.R.: Liouville type Theorem for p−forms valued on vector bundle (Chinese). J. Hangzhou Normal Univ. (Natural Sci.) 7, 96–100 (2008) [Lu] Luckhaus, S.: Partial hölder continuity for minima of certain energies among maps into a riemannian manifold. Indiana Univ. Math. J. 37, 349–367 (1988) [LW] Li, P., Wang, J.P.: Finiteness of disjoint minimal graphs. Math. Res. Lett. 8, 771–777 (2001) [LWe] Li, J.F., Wei, S.W.: A p-harmonic approach to generalized Bernstein problem. Commun. Math. Anal. Conference 1, 35–39 (2008) [LWW] Lee, Y.I., Wang, A.N., Wei, S.W.: On a generalized 1-harmonic equations and the inverse mean curvature flow. J. Geom. Phys. 61(2), 453–461 (2011) [Mo] Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm Pure App. Math. XIV, 577–591 (1961) [RS] Rigoli, M., Setti, A.: Energy estimates and liouville theorems for harmonic maps. Intern. J. Math. 11(3), 413–448 (2000) [PS] Price, P., Simon, L.: Monotonicity formulae for Harmonic maps and Yang-Mills fields. preprint, Canberra 1982. Final verson by P. Price, A monotonicity formula for Yang-Mills fields, Manus. Math. 43, 131–166 (1983) [Sa1] Salavessa, I.M.: Graphs with parallel mean curvature. Proc. A.M.S. 107(2), 449–458 (1989) [Sa2] Salavessa, I.M.: Spacelike graphs with parallel mean curvature. Bull. Bel. Math. Soc. 15, 65–76 (2008) [San] Sanini, A.: Applicazioni tra variet‘a riemanniane con energia critica rispetto a deformazioni di metriche. Rend. Mat. 3, 53–63 (1983) [Se1] Sealey, H.C.J.: Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory. Math. Soc. Camb. Phil. Soc. 91, 441–452 (1982) [Se2] Sealey, H.C.J.: The stress energy tensor and vanishing of L 2 harmonic forms. Preprint, 1983 [SiSiYa] Sibner, L., Sibner, R., Yang, Y.S.: Generalized Bernstein property and gravitational strings in BornInfeld theorey. Nonlinearity 20, 1193–1213 (2007) [Si] Simons, J.: Minimal varieties in Riemannian manifolds. Ann. of Math. 88(2), 62–105 (1968) [SU] Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Diff. Geom. 17, 307–335 (1982) [We1] Wei, S.W.: On 1-harmonic functions, SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 127, (2007) 10 pp. [We2] Wei, S.W.: P-harmonic geometry and related topics. Bull. Transilv. Univ. Brasov Ser. III 1(50), 415–453 (2008) [We3] Wei, S.W.: Representing homotopy groups and spaces of maps by p-harmonic maps. Indiana Univ. Math. J. 47, 625–670 (1998) [HL]
368
[Xi1] [Xi2] [Xi3] [Ya]
Y. Dong, S. W. Wei
Xin, Y.L.: Differential forms, conservation law and monotonicity formula. Scientia Sinica (Ser A) XXIX, 40–50 (1986) Xin, Y.L.: On Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space. Comm. Math. Helv. 66, 590–598 (1991) Xin, Y.L.: A rigidity theorem for a space-like graph of higher codimension. Manus. Math. 103(2), 191–202 (2000) Yang, Y.S.: Classical solutions in the Born-Infeld theory. Proc. R. Soc. Lond. A 456, 615–640 (2000)
Communicated by N.A. Nekrasov
Commun. Math. Phys. 304, 369–393 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1230-0
Communications in
Mathematical Physics
Distortion of the Poisson Bracket by the Noncommutative Planck Constants Artur E. Ruuge, Freddy Van Oystaeyen Department of Mathematics and Computer Science, University of Antwerp, Middelheim Campus Building G, Middelheimlaan 1, B-2020 Antwerp, Belgium. E-mail: [email protected]; [email protected] Received: 15 October 2009 / Accepted: 15 December 2010 Published online: 17 April 2011 – © Springer-Verlag 2011
Abstract: In this paper we introduce a kind of “noncommutative neighbourhood” of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a “distortion” of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical shadow. 1. Introduction In the present paper we describe a mathematical construction which can be perceived as a kind of noncommutative neighbourhood of the parameter → 0 of the semiclassical approximation of quantum theory (the Planck “constant”). This construction can be of interest in noncommutative algebraic geometry, as well as in mathematical physics, and, informally speaking, it is linked to an idea of a “quantization on a noncommutative space”. In quantum mechanics, if we speak about quantization, then this normally implies that we have a linear map Q : A → B between two algebras defined over a ring of formal power series C[[]], where A is commutative (the classical observables), and B is noncommutative (the quantum observables). Since the map Q does not need to be an algebra homomorphism, the multiplication on B induces a noncommutative associative product ∗ on A, f ∗ g = f g + B1 ( f, g) + 2 B2 ( f, g) + . . . , for f, g ∈ A, where B1 , B2 , . . . are bilinear maps. The first map B1 gives rise to a Poisson bracket on A, { f, g} := B1 ( f, g) − B1 (g, f ),
370
A. E. Ruuge, F. Van Oystaeyen
and, in fact, “quantization” is always a quantization in the direction of a given Poisson structure {−, −}cl on A (i.e. one is asked to construct a map Q such that {−, −} = {−, −}cl ). It is natural to consider a more general case where both algebras A and B can be noncommutative. Then, in a certain sense, a quantization Q : A → B consists in making things “more noncommutative”. Naively, this might look like a technical generalization of what already exists, but in reality one runs into a serious conceptual problem here. How do we define a noncommutative analogue of the Poisson bracket? Given an arbitrary noncommutative algebra A, if we impose the antisymmetry, the Leibniz rule, and the Jacobi identity (i.e. the usual axioms of the Poisson bracket), then it can easily turn out that there are no “interesting” Poisson structures on it [22,34,35], and essentially the only Poisson structure is given by the commutator [ f, g] := f g − g f, f, g ∈ A. Therefore, there exist different approaches to define a noncommutative generalization of the Poisson bracket. One of the possibilities is to modify the axioms by introducing twists (twisted antisymmetry, twisted Leibniz rule, twisted Jacobi identity), what is a common practice, for instance, in the theory of coloured Lie algebras and their representations [16,33]. Another possibility is to perceive the problem in terms of homotopy theory and to generalize the Jacobi identity “up to homotopy” [1,2,11,12,31]. The Poisson bracket then gets replaced with an infinite collection of operations {−, −, . . . , −}n : A⊗n → A, of different arity n = 1, 2, . . ., satisfying the L ∞ -algebra kind of axioms. The third possibility, which has recently received some additional attention in the literature [4,5, 9,13,32], is to consider the “double Poisson” bracket, −, − : A⊗2 → A⊗2 , i.e. a bracket with the values in A ⊗ A, rather than in A. The latter seems to be a reasonable approach for the algebras which are “far from commutative”. For example, there exists a canonical double Poisson structure on the free noncommutative algebra Cξ1 , ξ2 , . . . , ξn on a finite number of symbols ξ1 , ξ2 , . . . , ξn . To keep the story short, the idea of a noncommutative Poisson bracket leaves some space for creativity. The main construction of the present paper can be perceived as follows. If we look at an abstract d-dimensional quantum mechanical system with coordinates x1 , x2 , . . . , xd and momenta p1 , p2 , . . . , pd , then we have the canonical commutation relations x j ] = −i, [ pi , p j ] = 0 = [ xi , x j ], [ pi , for i, j ∈ [d] := {1, 2, . . . , d}, where is central. Why do we put the commutator [ pi , x j ] into the centre? If one considers the bracketed expressions over the generators xi , p j , i, j ∈ [d], which have a length of at least three, then this yields just 0, 0, . . .. Essentially, we suggest to go to another extreme: let us declare every next commutator a new variable. In other words, let us replace with an infinite collection of symbols { } , where varies over all finite planar binary trees (i.e. all possible bracketings) equipped with a leaf labelling from the set [2d], −→ i , [i, j] , [[i, j],k] , [[[i, j],k],l] , [[i, j],[k,l]] , . . . , where i, j, k, l, . . . vary over [2d] := {1, 2, . . . , 2d}. The commutation relations are as follows: j] = [i, j] , [ [i, j] , k ] = [[i, j],k] , [ [i, j] , [k,l] ] = [[i, j],[k,l]] , . . . , [ i ,
Noncommutative Planck Constants
371
for i, j, k, l, . . . ∈ [2d]. This yields an infinite dimensional noncommutative algebra, which we denote E2d () and perceive as a “noncommutative neighbourhood” of , having in mind an analogy with the terminology of [18] for the noncommutative manifolds. One can now identify the generators corresponding to the coordinates and momenta with the noncommutative Planck constants i , i ∈ [2d], corresponding to the trees with only one leaf: 1 , p1 = x1 = d+1 ,
p2 = 2 , x2 = d+2 ,
... pd = d , ... xd = 2d . q
In the present paper, we construct a natural q-deformation E2d () of the algebra E2d () corresponding to a 2d × 2d matrix of formal variables, q = qi, j , qi,i = 1, qi, j = q −1 j,i , where i, j ∈ [2d]. In particular, every commutator [i , j ], i, j ∈ [2d], in the defining j − q j,i j i . relations of E2d () gets replaced with a q-commutator [ i , j ]q := i q After that we consider a series of truncations (E2d ())N , N = 1, 2, 3, . . . , obtained q by factoring out of the ideals in E2d (), generated by the symbols with the number of 1 leaves || > N . Note that the commutation relations of (E2d )2 , where 1 is a 2d × 2d matrix with all entries equal to one, appears in [6], but in a different context. We prove, that for every N = 1, 2, 3, . . ., there exists an analogue of the Weyl quantization map (the q-Weyl quantization), which we define as a linear map q q q W N : gr (E2d ())N → (E2d ())N , where gr(−) denotes taking the associated graded with respect to a filtration induced by the degrees of the generators deg( ) := || − 1. q
It turns out, that this map W N has an nontrivial property in comparison to the well known Weyl quantization map in quantum mechanics. For an abstract physical system with two degrees of freedom (d = 2), the latter is a linear map Q Weyl : C[ p, x][] → C p, x []/( p x − x p = −i), where one puts an -adic filtration on the domain of Q Weyl . It is linear not just over C, but over the ring C[], so, in particular, we have: Q Weyl ( f ) = Q Weyl ( f ), for any monomial f in x and p. On the other hand, if we denote the canonical image of in the associated graded, and take, for example, a generic monomial u in the variables k , k ∈ [2d], and a symbol [i, j] , where i, j ∈ [2d], i = j, then we obtain: q q W N ([i, j] u) = [i, j] W N (u), if N > 2. In this sense, the generators corresponding to the trees with the number of leaves || 2 are just as good in the role of “observables” as the generators i , i ∈ [2d]. We suggest to refer to this fact as distortion of quantization by the noncommutative Planck constants. If one considers a representation theory q of (E2d ())N , say, for N = 3, then one can define an “observable” which does not have an analogue in quantum theory. Does this only distort the conventional description of our physical reality, or does it make some physical sense? For the Weyl quantization map Q Weyl , one can extract the Poisson bracket on C[x, p][] as a first semiclassical correction to the multiplication of classical observables: i Q Weyl f g − { f, g} + O(2+| f |+|g| ) = Q Weyl ( f )Q Weyl (g), 2
372
A. E. Ruuge, F. Van Oystaeyen
where O(n ) stands for an element of degree n in the -adic filtration on C[ p, x][], deg() = 1, deg( p) = 0, deg(x) = 0, and | f | and |g| denote the degrees of homogeneous elements f, g ∈ C[ p, x][]. This bracket {−, −} is bilinear not just over C, but q over C[]. In the present paper we perform a similar extraction of a bracket −, − N q from the q-Weyl quantization map W N , for N = 1, 2, 3, . . ., q q q q W N uv + u, v N + O(2 + |u| + |v|) = W N (u)W N (v), q where u, v ∈ gr (E2d ())N are homogeneous elements of degrees |u| and |v|, respectively, and O(n), n = 0, 1, 2, . . ., stands for an element of degree at least n in the filtration induced by the grading. Note that this corresponds to considering “semiclassics” with → 0 and q fixed, which is different from [3,7]. The distortion of quantization has its q counterpart on the properties of the corresponding bracket −, − N : it is linear only over C. We call this fact a distortion of the Poisson bracket by the noncommutative Planck constants and describe its properties as of a natural candidate for a noncommutative Poisson bracket (the “direction” of quantization on a noncommutative geometry). 2. Planar Binary Trees It is convenient to realize the collection of all planar binary trees as follows. Take your favourite singleton {∗} (i.e. ∗ is just a symbol). Set Y0 := ∅ and Y1 := {∗}. For every n = 2, 3, 4, . . ., define Yn recursively as a set of all pairs T = (u, v), where u ∈ Y p , v ∈ Yq , p, q 1, p + q = n. The tree L(T ) := u is termed the left branch of T , the tree R(T ) := v is termed the right branch of T , and |T | := n is termed the number of leaves in T . Denote Y := ∞ n=0 Yn . The planar binary trees Y naturally encode all possible bracketings over a set of symbols. Suppose we have an alphabet . Consider a word of length three: w = x1 x2 x3 , where x1 , x2 , x3 ∈ . The set Y3 contains two elements, ((∗, ∗), ∗) and (∗, (∗, ∗)). The first element corresponds to ((x1 x2 )x3 ), and the second element corresponds to (x1 (x2 x3 )). The set Y4 contains already five elements, and if we take, for example, (∗, ((∗, ∗), ∗)) ∈ Y4 and a word w = x1 x2 x3 x4 over , then the corresponding bracketed expression is (x1 ((x2 x3 )x4 )), etc. The number of elements #Yn = Cn−1 , for n = 1, 2, 3, . . ., is given by the Catalan numbers, Cm :=
(2m)! , (m + 1)!m!
where m = 0, 1, 2, . . .. Fix an alphabet . A leaf-labelled planar binary tree can be perceived as a pair = (T, w), where T ∈ Y, and w is a word of length |T | over . For example, the set Y2 contains only one element (∗, ∗), and if we take x1 , x2 ∈ , then = ((∗, ∗), x1 x2 ) is a leaf-labelled tree, such that the leaf on the left branch is labelled with x1 , and the leaf on the right branch is labelled with x2 . Denote Y() the set of all leaf labelled planar binary trees over the alphabet . The left branch of a leaflabelled tree = (T, w) ∈ Y(), w = x1 x2 . . . xn , xi ∈ , i ∈ [n], is defined as L() := (L(T ), w ), where w = w1 w2 . . . wm , m = |L(T )|, and the right branch is defined as R() := (R(T ), w ), where w = wm+1 wm+2 . . . wn . The set Y of all planar binary trees can be totally ordered as follows. Let T, T ∈ Y, T = T . If |T | < |T |, then set T ≺ T , and if |T | > |T |, then set T T . In case |T | = |T |, both trees must have at least two leaves, since #Y1 = 1, while
Noncommutative Planck Constants
373
T = T . Therefore, one can look at the left and right branches: T = (L(T ), R(T )) and T = (L(T ), R(T )). Set T ≺ T , if L(T ) ≺ L(T ), and T T , if L(T ) L(T ). If L(T ) = L(T ), then compare the right branches and set T ≺ T if and only if R(T ) ≺ R(T ). Since the number of leaves in the branches is always strictly smaller than the number of leaves in the whole tree, this defines recursively a total order ≺ on Y. Assume that there is a total order ≺ on the alphabet (here we overload the notation ≺ and denote the total orders on different sets with the same symbol, assuming the apparent ambiguity is always resolved by the context). It induces a total order on the words: x1 x2 . . . xm ≺ y1 y2 . . . yn if and only if (x1 ≺ y1 ) ∨ (x1 = y1 & x2 . . . xm ≺ y2 . . . yn ), where xi , y j ∈ , i ∈ [m], j ∈ [n]. Therefore the leaf-labelled trees Y() can be totally ordered as well: (T, w) ≺ (T , w )
:⇔
(T ≺ T ) ∨ (T = T & w ≺ w ),
where (T, w), (T , w ) ∈ Y(). If the total number of letters is finite, i.e. # < ∞, then there is a unique bijection Y() Z>0 , which respects the order, ∼
ν : Y() → Z>0 , ≺ ⇔ ν() < ν( ), where , ∈ Y(). We will use this map later to define the standard ordering for the monomials in the noncommutative Planck constants , ∈ Y([2d]). The total order on [2d] is assumed to be <. Note that there is a naturally defined operation on Y corresponding to the concatenation of trees, T ∨ T := T , where T = (T, T ) ∈ Y. Since |T ∨ T | = |T | + |T |, one has also a concatenation on the leaf-labelled trees: (T, w) ∨ (T , w ) := (T ∨ T , ww ), for every (T, w), (T , w ) ∈ Y(). The latter generalizes naturally to a composition of trees as follows. If one takes a collection of leaf-labelled trees i = (Ti , wi ) ∈ Y(), i ∈ [n], and a leaf-labelled tree = (T, σ¯ ) ∈ Y([n]), where σ¯ := (σ (1), σ (2), . . . , σ (n)) is a word over [n] corresponding to a permutation σ ∈ Sn , then deleting, for every i ∈ [n], the i th leave in and inserting in its place the tree i , one obtains a tree T with a labelling w = w1 w2 . . . wn . Denote the result ≡ (T , w ) as Tσ (1 , 2 , . . . , n ). If σ = id, then write just = T (1 , 2 , . . . , n ). In particular, 1 ∨2 = (∗, ∗)(1 , 2 ). 3. The Bracketing Algebra Let q = qi, j be a 2d × 2d matrix of formal variables qi, j (where d is a fixed positive integer) satisfying qi,i = 1 and qi, j = q −1 j,i , i, j ∈ [2d]. Extend this notation as follows: q, :=
m n
qiα , jβ ,
α=1 β=1
for every , ∈ Y([2d]), where (i 1 , i 2 , . . . , i n ) is the leaf-labelling of , n = || (we write the words over [2d] as sequences separated by commas), and ( j1 , j2 , . . . , jm ) is the leaf-labelling of , m = | |. Observe that
374
A. E. Ruuge, F. Van Oystaeyen
q, = 1, q, = q−1 , , for , ∈ Y([2d]). (q) Denote K 2d the ring of polynomials with complex coefficients in the variables q (q) {qi, j }i< j and {qi,−1j }i< j , i, j ∈ [2d]. Consider an algebra E2d () over K 2d generated by an infinite collection of symbols { }∈Y ([2d]) satisfying the relations − q , = [ , ]q := ∨ , where , ∈ Y([2d]). Observe that the notation implies ∨ = 0, ∨ , ∨ = −q ,
(1)
for any , ∈ Y([2d]). Therefore, one may select a collection of independent generators as { }∈Y + ([2d]) , where the set Y + ([2d]) is recursively defined as follows: ∈ Y + ([2d]), ∈ Y + ([2d]),
if || = 1, if L(), R() ∈ Y + ([2d]) & L() ≺ R(). q
(q)
(q)
(q)
Definition 1. The algebra E2d () := K 2d {ξ }∈Y + ([2d]) /I2d , where I2d is the ideal generated by the relations [ξ , ξ ]q = ξ∨ , where and vary over Y + ([2d]), and q ≺ , is termed the bracketing algebra. The canonical images ∈ E2d ([2d]) of + the symbols ξ , ∈ Y ([2d]), are termed the noncommutative Planck constants. q
Remark 1. Looking for a natural name for the algebra E2d ([2d]), we came up with the fact that bracketing is also an important concept in the phenomenological philosophy of Edmund Husserl, which means something like “suspending of judgement” that precedes a phenomenological analysis. It is also termed epoché (επ ωχ η), so perhaps another name q for E2d ([2d]) could be the “epoché algebra”. It is natural to keep the notational convention (1) for the noncommutative Planck constants , ∈ Y + ([2d]), extending the notation to all leaf-labelled trees ∈ Y([2d]). The underlying vector space of the bracketing algebra is naturally graded as deg := || − 1, where || is the number of leaves in ∈ Y + ([2d]). Note that || − 1 coincides with the number of internal vertices in . The induced decreasing filtration respects the mulq q tiplication on E2d (), so one obtains an algebra filtration F • E2d (), q F n E2d () := { | ∈ Y + ([2d]), deg( ) n},
(2)
where n = 0, 1, 2, . . ..
q q Definition 2. The associated graded algebra A2d () := gr F E2d () with respect to filtration (2) is termed the classical shadow of the bracketing algebra. We denote the canonical images of in the classical shadow as ∈ A2d (), ∈ q Y([2d]) (i.e. without the hats). Observe that A2d () is nothing else but an infinite dimensional q-affine space, q
= q , ,
Noncommutative Planck Constants
375 (q)
for , ∈ Y([2d]). A “quantization” should be perceived as a linear over K 2d map q q Q : A2d () → E2d (). One obtains an example of such a map as follows. Denote q B(A2d ()) the monomial basis in the classical shadow formed by the monomials of the shape 1 2 . . . n , where n ∈ Z>0 , 1 , 2 , . . . , n ∈ Y([2d]), such that 1 2 · · · n , the symbol means “ or =”. For an element w = 1 2 . . . n ∈ q B(A2d ()), set 2 . . . n . Q : w → 1 (q)
q
(3) q
q
This extends by K 2d -linearity to the whole A2d (), Q : A2d () → E2d (). For a pair q of basis elements w, w ∈ B(A2d ()), define q(w, w ) by w w = q(w, w )ww . The commutation relations imply that (w, w ) := Q(w )Q(w) − q(w, w )Q(w)Q(w ) ∈ q F |w|+|w |+1 E2d (), where |w| and |w | are the degrees of w and w , respectively. Furq thermore, there exists a unique element w, w ∈ A2d () of degree |w| + |w | + 1, such that (w, w ) = Q(w, w ) + Z ,
(4) (q)
where Z ∈ F |w|+|w |+2 E2d (). Extending the notation −, − by K 2d -bilinearity, one q q q obtains a bracket −, − : A2d () ⊗ A2d () → A2d (), where the tensor product is taken over K q . q
q
q
Definition 3. The linear map Q : A2d () → E2d () defined by (3) is termed the normal q q q-quantization on the classical shadow. The bilinear map −, − : A2d ()⊗A2d () → q A2d () defined by (4) is termed the normal q-Poisson bracket on the classical shadow. Remark 2. The normal q-Poisson bracket is a graded map of degree +1. If we perceive the generators corresponding to the trees with a single leaf, || = 1, as the analogues of the coordinates and momenta of a d-dimensional quantum mechanical system, then intuitively −, − corresponds to the Poisson bracket {−, −} multiplied by the semiclassical parameter → 0. It is worth pointing out that the normal q-quantization map (3) can be described using the calculus of functions of ordered operators [25,27,26,19] (the μ-structures). For a q monomial w = 1 2 . . . n ∈ B(A2d ()), 1 2 · · · n , one has: νn ν1 ν2 1 2 . . . n , Q(w) =
where . . . denotes the autonomous bracket [26], νi := ν(i ), i = 1, 2, . . . , n, and ∼ ν : Y([2d]) → Z>0 is the numbering described in the previous section. Basically, the indices νi indicate the “order of action” of the symbols i , , i ∈ [n], and the symbol with the smallest number is put in the right most position in the product. For every N = 1, 2, 3, . . ., consider the space of smooth complex functions S(R N ), which decay at infinity faster than any power of a polynomial (the Schwartz space). Suppose we have a noncommutative algebra O described in terms of generators ξ1 , ξ2 , . . . , ξm and a finite number of noncommutative polynomial relations R1 , R2 , . . . , Rk ∈ Cξ1 , ξ2 , . . . , ξm , O := Cξ1 , ξ2 , . . . , ξm /(R1 , R2 , . . . , Rk ).
376
A. E. Ruuge, F. Van Oystaeyen
Denote Ai ∈ O the canonical image of ξi , i ∈ [m]. If f (x1 , x2 , . . . , x N ) ∈ S(R N ) is a polynomial function in x1 , x2 , . . . , x N , then there is a well-defined notation σ (1) σ (2)
σ (N )
f = f ( Ai1 , Ai2 , . . . , Ai N ) ∈ O, for every permutation σ ∈ S N , and every collection i 1 , i 2 , . . . , i N ∈ [m], where the indices atop correspond simply to the order of factors in the products. For example, if N = 3, and f (x1 , x2 , x3 ) = x1 x25 x37 , then f corresponding to a permutation σ (1) = 2, σ (2) = 3, σ (3) = 1 is going to be Ai52 Ai1 Ai73 , etc. A natural extension of this notation N to the set S := ∞ N =1 S(R ) is termed a μ-structure (for a complete list of axioms, see [19,26]). In [19] the authors consider a problem of quantization (the “asymptotic” quantization) in a setting where one is given a family of defining relations R1(ε) , R2(ε) , . . . , Rk(ε) ∈ Cξ1 , ξ2 , . . . , ξm containing a “small” commutative parameter ε → 0. This yields a family of noncommutative algebras Oε . It can happen that Oε admits a left regular representation, i.e. for every f (x1 , x2 , . . . , xm ) ∈ S(Rm ) (recall, that m is the number of generators ξ1 , ξ2 , . . . , ξm ), and for every j ∈ [m], there exists a unique gε (x1 , x2 , . . . , xm ) ∈ S(Rm ), such that j (ε)
1 (ε))
2 (ε)
1 (ε)
m
2 (ε)
m
(ε) A j f ( A1 , A2 , . . . , A(ε) m ) = gε ( A1 , A2 , . . . , Am ), (ε)
where Ai is the canonical image of ξi in Oε , i ∈ [m]. If we assume that ε is specialm ized to a particular value ε ∈ [0, 1], then this defines the operators L(ε) j : S(R ) → (ε)
(ε)
S(Rm ), f → gε = L j ( f ), representing the generators A j ∈ Oε , j ∈ [m]. There is a star product ε on S(Rm ) defined by 1 (ε)
2 (ε)
m
f ε g := f (L1 , L2 , . . . , L(ε) m )g, for any f, g ∈ S(Rm ). A series of examples of such products is considered in [25]. According to the general philosophy advocated in [19,25], one should define the generalized quantum Yang-Baxter equation as a system of equations (ε) R (ε) j (ξi → Li , i ∈ [m]) = 0,
j ∈ [k],
where one replaces to symbols ξi with the operators of the left regular representation (ε) (ε) Li , i ∈ [m], in the noncommutative polynomials R j ∈ Cξ1 , ξ2 , . . . , ξm , j ∈ [k]. q Being applied to our case, we can connect the classical shadow A2d () with the bracq keting algebra E2d () by a homotopy, modifying the relations for the noncommutative Planck constants as − q , = ε ∨ , where ε ∈ [0, 1], and , ∈ Y([2d]). If one reinterprets ε and perceives it as a formal q commutative parameter, i.e. as a central generator for a central extension of E2d (), then it becomes quite natural to consider the problem of quantization in terms of truncations q of the Rees ring corresponding to the filtration F • E2d (). In the next section we describe the corresponding star product ε explicitly, and this, as a side-effect, yields a left reguq lar representation of the truncations (E2d ())N , N = 1, 2, . . ., on the q-commutative polynomial rings.
Noncommutative Planck Constants
377
4. Normal Noncommutative Quantization In quantum mechanics of a d-dimensional system with canonical momenta z1 = p1 , z2 = p2 , . . . zd = pd , and coordinates z d+1 = x1 , z d+2 = x2 , . . . z 2d = xd , if we have a pair of monomials j1 j2 i1 i2 jn im g= z i1 zi2 . . . zim , z j1 z j2 . . . z jn , f = where i α , jβ ∈ [2d], α ∈ [m], β ∈ [n], then their product f g can again be expressed as 1 2 2d z1 , z2 , . . . z 2d , f g = ( f ∗ g) where ( f ∗ g)(z 1 , z 2 , . . . , z 2d ) ∈ C[z 1 , z 2 , . . . , z 2d ][], and is the Planck constant, [ z i , z j ] = −i(δi, j−d − δi−d, j ), where i, j ∈ [2d], is central, and δ is the Kronecker symbol. Extending by C[]linearity, one obtains a noncommutative product on C[z 1 , z 2 , . . . , z 2d ][], which is most easily described in terms of the Wick contractions, z i z j := −iδi, j−d ,
(5)
for i, j ∈ [2d]. For the monomials f = z i1 z i2 . . . , z im and g = z j1 z j2 . . . z jn as above, we have: f ∗ g =
n m
z iα z jβ (z i1 . . . zˇ iα . . . z im ) (z j1 . . . zˇ jβ . . . z jn ),
α=1 β=1
where the check mark atop denotes that the corresponding symbol in the product is omitted. In what follows, it is natural to reinterpret the sum over Wick contractions as a sum over leaf-labelled trees of degree two (i.e. the symbol of the contraction can be perceived as an unlabelled planar binary tree with two leaves). Consider the epoché algebra q E2d () around the semiclassical parameter . Recall, that we have defined a numbering ∼ ν : Y([2d]) → Z>0 on the collection of all leaf-labelled trees, but we can still use it to define the ordering of factors in the products of the generators , ∈ Y + ([2d]). q Denote B(E2d ()) a set of all monomials of the shape w 1 ,2 ,...,m := 2 . . . 1 m , where 1 2 · · · m , i ∈ Y + ([2d]), i ∈ [m], m ∈ Z>0 . q
Proposition 1. The set B(E2d ()) is a basis of the underlying vector space of the epoché q algebra E2d (). Proof. The fact claimed is a straightforward consequence of the “diamond lemma” ]q ) is always greater than ν( ) and [28], based on the observation that ν([ , ν( ), where , ∈ Y + ([2d]), and the observation that the length of a monomial ∈ B(E q ()), is smaller than the length of w , where w . w w [ , ]q w , w 2d
378
A. E. Ruuge, F. Van Oystaeyen
Recall now that we have a decreasing filtration F • E2d () induced by the grading deg( ) = || − 1 of the vector space spanned over all noncommutative Planck conq stants , ∈ Y([2d]). The associated graded A2d () (i.e. the classical shadow) has a q basis B(A2d ()) formed by the monomials q
w1 ,2 ,...,m := 1 2 . . . m ,
(6)
where 1 2 · · · m , i ∈ Y + ([2d]), i ∈ [m], m ∈ Z>0 , and is the canonical image of , ∈ Y + ([2d]). This yields a vector space isomorphism ϕq : w1 ,2 ,...,m → w 1 ,2 ,...,m
(7) q
between the underlying vector spaces of the classical shadow A2d () and the epoché q q u , v ∈ B(E2d ()), then the basis property implies that there algebra E2d (). If we take q exists
a unique function cu,v : B(E2d ()) → C with a finite support , such that u v= w w ) w. Therefore, we can induce a noncommutative product on the ∈ cu,v ( q classical shadow A2d () as follows: u ) ϕq−1 ( v) = ϕq−1 (
cu,v ( w )ϕq−1 ( w ),
w ∈ (q)
imposing bilinearity of over K 2d . This star product should be perceived as a natural analogue of the star product ∗ in quantum mechanics. Of course, if we had considered q a central extension of E2d () by a formal central generator ε, [ , ]q = ε ∨ , q we would have obtained a product ε on the extension A2d () ⊗ C[ε]. In this sense, ε corresponds to a semiclassical parameter → 0 in quantum mechanics. q q q To describe the star product : A2d () ⊗ A2d () → A2d () explicitly, we need some notation. In place of the Wick contractions (5), define , if ∈ Y + ([2d]), + := (8) 0, otherwise, for any ∈ Y([2d]). Recall that if we have an unlabelled planar binary tree T ∈ Y with n leaves, |T | = n, then we have a notation T (1 , 2 , . . . , n ) ∈ Y([2d]), for any i ∈ Y([2d]), i ∈ [n]. The number of leaves in = T (1 , 2 , . . . , n ) equals || = |1 | + |2 | + · · · + |n |, and the labelling is inherited from the labellings of the arguments. For a finite sequence = (1 , 2 , . . . , n ) of trees i ∈ Y([2d]), i ∈ [n], and a permutation σ ∈ Sn , set σ := (σ (1) , σ (2) , . . . , σ (n) ). Define a coefficient q( , σ ) by σ (1) σ (2) . . . σ (n) = q( , σ )1 2 . . . n .
(9)
In particular, q( , id) = 1. Note that q( , σ ) is just some product of the entries of the matrix q = qi, j i, j∈[2d] , and observe that q( , σ ◦ τ ) = q( , σ )q( σ , τ ),
(10)
Noncommutative Planck Constants
379
for any σ, τ ∈ Sn . Therefore, q( , σ )−1 = q( σ , σ −1 ). For a vector of positive integers (m 1 , m 2 , . . . , m p ), such that m 1 + m 2 + · · · + m p = n, denote (m 1 ,m 2 ,...,m p )
Sn
where l j :=
:= {σ ∈ Sn | σ (l j + 1) < · · · < σ (l j + m j ), j ∈ [ p]},
j−1
m i , j ∈ [ p]. We need also a notation: 1, if 1 2 · · · m , θ ≡ θ1 ,2 ,...,m := 0, otherwise,
i=1
(11)
where = (1 , 2 , . . . , m ), i ∈ Y([2d]), i ∈ [m]. q
Theorem 1. The normal star product of a pair of elements of B(A2d ()) in the notation (6), (8), (11) is defined by the formula w1 ,2 ,...,m wm+1 ,m+2 ,..., N =
×
(|T |,|T |,...,|T p |) σ ∈S N 1 2
N
p=1
T1 ,T2 ,...,T p ∈Y , |T1 |+|T2 |+···+|T p |=N
+ (q( , σ ))−1 θG( T , ,σ ) G
, ,σ ) 1 (T
+G
, ,σ ) 2 (T
. . . +G
, ,σ ) p (T
, (12)
= (T1 , T2 , . . . , T p ) ∈ Y N , and where = (1 , 2 , . . . , N ) ∈ (Y + ([2d])) N , T , σ )), , T G( , T , σ ) = (G 1 ( , T , σ ), G 2 ( , T , σ ), . . . , G n ( , σ ) := T j ( G j ( , T )+1) , σ (l j (T )+2) , . . . σ (l j (T )+|T j |) ), σ (l j (T ) = where l j (T
j−1 i=1
|Ti |, for j ∈ [ p].
Proof. The formula (12) can be proven by induction. Basically it says that we should consider a collection = (1 , 2 , . . . , N ) of the leaf labelled trees i , i ∈ [N ], (where i i+1 , unless i = m), split it in all possible ways into p ordered subsets of the sizes m 1 , m 2 , . . . , m p > 0, m 1 + m 2 + · · · + m p = N , where p varies over [N ], and then span all possible trees T1 , T2 , . . . , T p over these groups of arguments, |Ti | = m i , i ∈ [ p]. + The notation of the shape θ defined in (11) and the notation from (8) select the terms which appear after the process of reordering of factors using the commutation relations − q , = ∨ , and q( , σ )−1 is the corresponding braiding coefficient. q
Proposition 2. The star product of a pair of elements of the basis B(A2d ()) in the notation (6) can be expressed as follows: w1 ,2 ,...,m wm+1 ,m+2 ,..., N = 1 2 · · · N . q
∼
q
(13)
Proof. One needs to apply the vector space map ϕq : A2d () → E2d () defined by (7) to the left and the right-hand sides of (13), to expand the definitions (6) of the basis elements on the left hand side, and to use the fact that ϕq is an algebra isomorphism ∼ q q (A2d (), ) → E2d (), by construction.
380
A. E. Ruuge, F. Van Oystaeyen
The normal star product respects the decreasing filtration F • (A2d (), ) induced by the grading deg( ) = || − 1, || is the number of leaves in ∈ Y + ([2d]). The q underlying vector space of F n (A2d (), ), n = 0, 1, 2 . . . , is spanned by the products 1 2 · · · p , where p = 1, 2, . . . , and |1 | + |2 | + · · · + | p | − p = n, i ∈ q q Y + ([2d]), i ∈ [ p]. The algebras (A2d (), ) and E2d () are canonically isomorphic as q filtered algebras (by construction), and the classical shadow A2d () can therefore be q • identified with the associated graded algebra of F (A2d (), ). Since taking a q-commutator increases the filtration degree by one, if we look at it in the classical shadow, q this yields a graded bilinear map −, − (a bracket on A2d ()) of degree +1. q
Proposition 3. The q-commutator in F • (A2d (), ) induces a graded bilinear map of q degree +1 on the classical shadow A2d (), q
q
q
q
−, − : A2d () ⊗ A2d () → A2d (), which satisfies the q-Poisson bracket axioms. Proof. Take a pair of basis elements w = w1 ,2 ,...,m ∈ B(A2d ()) and w =
m q |i |−m, w1 ,2 ,...,n ∈ B(A2d ()). Their degrees are as follows: |w| ≡ deg(w) = i=1
n and |w | ≡ deg(w ) = j=1 | j | − n. Recall that this notation implies 1 2 · · · m , and 1 2 · · · n . From the explicit formula (12) for the star product, we have: q
w w = ww +
m n (i, j) −1 + q ∨ , σm,n i ∨ (m . . . i+1 i=1 j=1
j
× i−1 . . . 1 ) (n . . . j+1 j−1 . . . 1 ) + O|w|+|w |+2 , denotes ∨ where O|w|+|w |+2 stands for an element of filtration degree |w| + |w | + 2, the concatenation of the lists = (m , . . . , 2 , 1 ) and = (n , . . . , 2 , 1 ), and (i, j) σm,n ∈ Sm+n denotes a permutation (i, j)
(i, j)
(i, j)
(σm,n (1), σm,n (2), . . . , σm,n (m + n)) := (i, j, 1, . . . , i − 1, i + 1, . . . , m, m + 1, . . . , m + j − 1, m + j + 1, . . . , m + n). The opposite star product w w looks totally similar. Taking the q-commutator [w, w ]q := w w − q(w , w)w w, where q(w , w) is determined from ww = q(w , w)w w, one extracts w, w as the element of degree |w| + |w | + 1 satisfying [w, w ]q = w, w + O|w|+|w |+2 . It is straightforward to check that w, w = −q(w , w)w , w, i.e. the first axiom of a q-Poisson bracket (the q-antisymmetry) is satisfied. If we take q another arbitrary element w = w1 ,2 ,...,k ∈ B(A2d ()), then one can establish the other two axioms, i.e. the q-Leibniz rule w, w w = w, w w + q(w, w )w w, w , and the q-Jacobi identity w, w , w = w, w , w + q(w, w )w , w, w ,
Noncommutative Planck Constants
381
by a straightforward computation. Observe that with these properties, the bracket −, − is determined by its values on the generators , which are just , = ∨ , for , ∈ Y + ([2d]). 5. Distortion of the Weyl Quantization We are now interested in a q-analogue of the Weyl quantization. In quantum mechanics of a d-dimensional system, this is basically a symmetrization map W : A → B, where A = C[z 1 , z 2 , . . . , z 2d ] ⊗ C[h], and B = Cξ1 , ξ2 , . . . , ξ2d [η]/I, where I is the ideal generated by the canonical commutation relations [ξi , ξd+ j ] = −iη, [ξi , ξ j ] = 0, and [ξd+i , ξd+ j ] = 0, where [−, −] denotes a commutator, and i, j ∈ [d]. The canonical images of ξ1 , ξ2 , . . . , ξd in B are denoted as z1 = p1 , z2 = p2 , . . . , zd = pd (the canonical momenta), the canonical images of ξd+1 , ξd+2 , . . . , ξ2d are denoted as z d+1 = x1 , z d+2 = x2 , . . . , z 2d = xd (the coordinates), and the canonical image of η is denoted (the Planck constant). The Weyl quantization map is a linear map W : A → B defined on the monomials in generators as follows: W : h m zi1 zi2 . . . zin →
m z iσ (1) z iσ (2) . . . z iσ (n) , n! σ ∈Sn
where m ∈ Z0 , i α ∈ [2d], α ∈ [n], n ∈ Z>0 . A characteristic property of the Weyl quantization which selects W from the other possible quantizations (like, for example, the normal quantization) is the affine equivariance. If we take an arbitrary 2d ×2d matrix A over C, then we can act with it on the column of the generators (z 1 , z 2 , . . . , z 2d )T , or on the column of their quantized analogues ( z 1 , z 2 , . . . , z 2d )T , where (−)T denotes transposition. One can first act, and then quantize, or first quantize, and then act. The affine equivariance is the property that the two results coincide for any A: W ((Az)i1 (Az)i2 . . . (Az)in ) =
1 (A z)iσ (1) (A z)iσ (2) . . . (A z)iσ (n) , n!
(14)
σ ∈Sn
z)i := 2d z j , i ∈ [2d]. where (Az)i := 2d j=1 Ai, j z j , and (A j=1 Ai, j q We would like to define a q-Weyl quantization map as a linear map W (q) : A2d () → q q q E2d (), where E2d () is the epoché algebra discussed in the previous sections, and A2d () is its classical shadow. Suppose we wish to define it by the formula (q) σ (2) . . . W (q) : 1 2 . . . n → C1 ,2 ,...,n (σ ) σ (1) σ (n) , σ ∈Sn
(q)
where C1 ,2 ,...,n (σ ) are some coefficients, and i ∈ Y + ([2d]), i ∈ [n], n ∈ Z>0 , 1 2 · · · n . Then a natural condition that 2 . . . 1 n ∈ F p+1 E2d (), W (q) (1 2 . . . n ) − q
where p = |1 | + |2 | + · · · + |n | − n, and F • E2d () is the filtration on E2d () described in the previous sections, yields a condition on the coefficients (q) C1 ,2 ,...,n (σ )q( , σ ) = 1, q
σ ∈Sn
q
382
A. E. Ruuge, F. Van Oystaeyen
where = (1 , 2 , . . . , n ), and one uses the notation (9). The problem remains: what (q) is the correct way to define the coefficients C1 ,2 ,...,n satisfying this condition? It is natural to generalize the affine equivariance in the quantum mechanical case (14) into the left and right affine coequivariance. It is convenient to consider the truncations q q (E2d ())N and (A2d ())N , for any N ∈ Z>0 , since the corresponding sums are going to be in this case finite. Formally, these truncations can be perceived as setting all the generators and to zero, if || > N , ∈ Y + ([2d]). Let + + Y N ([2d]) := { ∈ Y ([2d]) | || N }, ) and denote (N the canonical image of , || N , in the truncation (A2d ())N of the ) + ([2d]), and denote (N classical shadow, ∈ Y the canonical image of , || N , N q in the truncation (E2d ())N . q Let M2d be an algebra of the shape
q (15) M2d = C {, }, ∈Y + ([2d]) /I, q
where , denote the generators written in a matrix form, and I is an ideal generated by a countable collection of noncommutative polynomials. Denote A, the canoniq cal image of , in M2d . The left affine coequivariance condition is the following q q equality in M2d ⊗ (E2d ())N : (q,N ) (N ) (N ) A1 ,1 . . . An ,n ⊗ C ,..., (σ ) . . . =
n
1
+ ([2d]), 1 ,...,n ∈Y N σ ∈Sn
σ ∈Sn , + ([2d]) 1 ,...,n ∈Y N
σ (1)
σ (n)
(q,N ) (N ) (N ) C1 ,...,n (σ )Aσ (1) ,1 . . . Aσ (n) ,n ⊗ . . . , 1
n
(16)
+ ([2d]), n ∈ Z . This is a condition on a collection where 1 , 2 , . . . , n ∈ Y >0 N (q,N )
of coefficients C1 ,...,n (σ ). Similarly, the right affine coequivariance condition is an q q equality in (E2d ())N ⊗ M2d : (q,N ) (N ) (N ) C ,..., (σ ) . . . ⊗ A1 ,1 . . . An ,n + ([2d]), 1 ,...,n ∈Y N σ ∈Sn
=
1
σ ∈Sn , + ([2d]) 1 ,...,n ∈Y N
n
σ (1)
(q,N )
σ (n)
(N )
(N )
1
n
C1 ,...,n (σ ) . . . ⊗ A1 ,σ (1) . . . A,n ,σ (n) ,
(17)
+ ([2d]), n ∈ Z . This is another condition on the coefwhere 1 , 2 , . . . , n ∈ Y >0 N (q,N ) (q) ficients C1 ,...,n (σ ). These coefficients are supposed to define a linear map W N : q q (A2d ())N → (E2d ())N , (q) (q) (N ) (N ) (N ) (N ) (N ) (N ) W N : 1 2 . . . n → σ (2) . . . C1 ,2 ,...,n (σ ) σ (1) σ (n) , (18) σ ∈Sn
Noncommutative Planck Constants
383
and to satisfy σ ∈Sn
(q,N ) C1 ,2 ,...,n (σ )q( , σ ) = 1,
(19)
+ ([2d]))n . Let us term the latter condition the exiswhere = (1 , 2 , . . . , n ) ∈ (Y N tence of a classical limit for the quantization map. Observe that the notation q( , σ ) defined in (9) can explicitly be described as q j ,i , q( , σ ) := 1i< j n, σ −1 (i)>σ −1 ( j)
where q, corresponds to = q , , for , ∈ Y([2d]). Theorem 2. The conditions of left and right affine coequivariance (16), (17), in addition to the condition of existence of a classical limit (19), determine the ideal I in (15) and (q,N ) the coefficients C1 ,2 ,..., N (σ ) for the quantization map (18), q( , σ ) , (q( , κ))2 κ∈Sn
(q,N )
C1 ,2 ,...,n (σ ) =
+ ([2d]))n , σ ∈ S , N ∈ Z . The ideal I is where = (1 , 2 , . . . , n ) ∈ (Y n >0 N generated by the quantum matrices type relations:
2 ,1 1 ,2 + q1 ,2 2 ,2 1 ,1 = q1 ,2 1 ,1 2 ,2 + q1 ,2 1 ,2 2 ,1 , and 1 ,2 2 ,1 + q1 ,2 2 ,2 1 ,1 = q1 ,2 1 ,1 2 ,2 + q1 ,2 2 ,1 1 ,2 , where 1 , 2 , 1 , and 2 vary over Y + ([2d]). (q,N )
Proof. Let us first derive another generic fact about the coefficients C1 ,2 ,...,n (σ ). Fix a positive integer N . Take a pair of permutations σ, τ ∈ Sn and look at the coefficient (N ) (N ) (N ) (N ) corresponding to σ ◦τ ∈ Sn . On one hand, since τ (1) . . . τ (n) = q( , τ )1 . . . n , where = (1 , . . . , n ), we must have (q,N ) (q) (N ) (N ) (N ) (N ) W N (τ (1) . . . τ (n) ) = q( , τ ) C1 ,...,n (σ ) σ (1) . . . σ (n) + Z, σ ∈Sn
where Z ∈ F m+1 (E2d ())N , m = |1 | + · · · + |n | − n. On the other hand, a straight(q) forward application of the formula for W N yields (q,N ) (q) (N ) (N ) (N ) (N ) W N (τ (1) . . . τ (n) ) = Cτ (1) ,...,τ (n) (σ ) (τ ◦σ )(1) . . . (τ ◦σ )(n) . q
σ ∈Sn
384
A. E. Ruuge, F. Van Oystaeyen
Changing the summation index to σ = τ ◦ σ and comparing the two expressions, one obtains (q,N ) (q,N ) Cτ (1) ,...,τ (n) (τ −1 ◦ σ ) = q( , τ )C1 ,...,n (σ ).
(20)
+ ([2d]))n , and look at the left affine coequivariFix now = (1 , 2 , . . . , n ) ∈ (Y N ) and the right-hand side R N ( ). ance condition (16). Denote the left hand side L N ( Changing the summation indices in the expression for L N ( ) to i = σ (i) , i ∈ [n], one obtains L N ( ) = A1 , −1 . . . An , −1 σ
+ ([2d]) σ ∈S 1 ,...,n ∈Y n N
(q,N )
(1)
σ
(n)
(N ) (N ) (σ ) . . . .
⊗ C
,..., −1 σ −1 (1) σ (n)
n
1
Observe now that if one has an abstract expression f ( ) depending on = + ([2d]))n and wishes to take a sum over all tuples (1 , . . . , n ) ∈ (Y , then one N can do it as follows: 1 f ( ) = f ( ρ ), w1 ,2 ,...,n + + n ∈(Y N ([2d]))
ρ∈Sn
1 ,2 ,...,n ∈Y N ([2d]), 1 2 ···n
where w1 ,2 ,...,n := #{σ ∈ Sn | σ = }. Hence,
) = L N (
1
+ ([2d]), 1 ,...,n ∈Y N 1 ···n , σ,ρ∈Sn
w1 ,...,n
(q,N )
⊗ C
,..., −1 (ρ◦σ −1 )(1) (ρ◦σ )(n)
A1 ,
(ρ◦σ −1 )(1)
. . . An ,
(ρ◦σ −1 )(n)
(N ) (N ) (σ ) . . . . ρ(1)
ρ(n)
The same trick applied to R N ( ) yields:
) = R N (
1
σ,ρ∈Sn , + ([2d]), 1 ,...,n ∈Y N 1 ···n
w1 ,...,n
Aσ (1) ,
ρ(1)
. . . Aσ (n) ,
(q,N ) ) ) ⊗ C1 ,...,n (σ ) (N . . . (N . ρ(1)
ρ(n)
(21)
ρ(n)
(N )
Now, if we look at the products of in the associated graded algebra and take the ) = R N ( ) leaves us with leading term, then the equality L N ( (q,N ) A1 , −1 . . . An , −1 C (σ )q( , ρ) ,..., σ,ρ∈Sn
=
(ρ◦σ
σ,ρ∈Sn
)(1)
Aσ (1) ,
ρ(1)
(ρ◦σ
)(n)
. . . Aσ (n) ,
ρ(n)
(ρ◦σ −1 )(1)
(ρ◦σ −1 )(n)
(q,N ) C1 ,...,n (σ )q( , ρ),
Noncommutative Planck Constants
385
+ ([2d]))n , and every for every n ∈ Z>0 , every = (1 , 2 , . . . , n ) ∈ (Y = N + ([2d]))n , such that · · · . The left hand side (1 , 2 , . . . , n ) ∈ (Y n 1 2 N of this equality can be simplified, if one takes into account the property (10), which implies q( , ρ) = q( , (ρ ◦ σ −1 ) ◦ σ ) = q( , ρ ◦ σ −1 )q(ρ◦σ −1 , σ ).
Introducing an index of summation κ = ρ ◦σ −1 in place of ρ, and invoking the condition of existence of classical limit (19), one obtains: A1 , . . . An , q( , κ) κ∈Sn
=
κ (1)
σ,ρ∈Sn
κ (n)
Aσ (1) ,
ρ(1)
. . . Aσ (n) ,
(q,N )
ρ(n)
C1 ,...,n (σ )q( , ρ),
(22)
+ ([2d]), such that , Consider now the case n = 2. For every 1 , 2 , 1 , 2 ∈ Y 1 2 N since q( , id) = 1, we have
, (12)) A1 ,1 A2 ,2 + A1 ,2 A2 ,1 q( (q,N ) = C1 ,2 (id) A1 ,1 A2 ,2 + A1 ,2 A2 ,1 q( , (12)) (q,N ) + C1 ,2 ((12)) A2 ,1 A1 ,2 + A2 ,2 A1 ,1 q( , (12)) , where id, (12) ∈ S2 are the two elements of the symmetric group S2 . From the classical limit condition (19), one obtains: (q,N ) (q,N ) , (12)) = 1. C1 ,2 (id) + C1 ,2 ((12))q( (q,N )
(q,N )
Observe that q( , (12)) = q1 ,2 . Express now C1 ,2 (id) via C1 ,2 ((12)) and sub(q,N )
stitute the result into the previous equality. Cancelling out C1 ,2 ((12)), one arrives at A2 ,1 A1 ,2 + A2 ,2 A1 ,1 q1 ,2 − q1 ,2 A1 ,1 A2 ,2 + A1 ,2 A2 ,1 q1 ,2 = 0. (23) Since the level of truncation N is arbitrary, this corresponds precisely to the first half of (q) the defining relations of M2d . The second half stems in a totally similar manner from the right coequivariance condition (17). (q,N ) We still need to define the coefficients C1 ,...,n (σ ). Let n ∈ Z>0 . Denote , ) := A1 , . . . An , q( , κ), L n ( κ∈Sn
κ (1)
κ (n)
+ ([2d]))n . Note that where = (1 , . . . , n ) and = (1 , . . . , n ) are in (Y N , ) is precisely the left-hand side of (22) expressing the left coequivariance L n ( condition (16), so there is also a right analogue Rn ( , ) := Aκ (1) ,1 . . . Aκ (n) ,n q( , κ),
κ∈Sn
386
A. E. Ruuge, F. Van Oystaeyen
which corresponds to the right coequivariance condition (17). We claim that the following properties hold: L n ( σ , ) = q( , σ )L n ( , ),
Rn ( , σ ) = q( , σ )Rn ( , ),
(24)
for any σ ∈ Sn . This can be done by induction in n. Consider L n ( , ), for example. If n = 2, then one arrives at (23). If n > 2, then proceed as follows. Observe, that if the property mentioned holds for some given σ ∈ Sn and τ ∈ Sn , and for any and , then L n ( σ ◦τ , ) = q( σ , τ )L n ( σ , ) = q( σ , τ )q( , σ )L n ( , ), and it holds for σ ◦ τ ∈ Sn as well due to (10), Therefore, it suffices to check it only on the generators of Sn . Rewrite L n ( , ) as follows: L n ( , ) =
n
A1 ,
((mn)◦ρn+ )(1)
m=1 ρ∈Sn−1
× q( , (mn))q( (mn) )=
. . . A1 ,
((mn)◦ ρ )(n−1)
n
An ,m
L n−1 ( n−1 , ( (mn) )n−1 )q( , (mn)),
m=1
where ρn+ denotes the canonical image of ρ ∈ Sn−1 in Sn , such that ρn+ (n) = n and ρn+ (i) = ρ(i), i < n, and the symbol (mn) ∈ Sn denotes the transposition of m and n, and (−)n−1 corresponds to a truncation of a string of symbols, so that n−1 = (mn) . If one now makes an inductive assumption (1 , . . . , n−1 ), and, similarly, for that (24) holds for n − 1, then this implies L n ( λ+n , ) = q( , λ+n )L n ( , ), for all th λ ∈ Sn−1 . In a totally similar way, isolating the first, but not the n factor in the products, one can show, that L n ( μ+1 , ) = q( , λ+1 )L n ( , ), for any μ ∈ Sn−1 , where μ+1 ∈ Sn + + is the permutation, such that μ1 (1) = 1, and μ1 (i + 1) = μ(i), i ∈ [n − 1]. Since the collection of permutations of the shape λ+n and μ+1 , where λ, μ ∈ Sn−1 , generate the whole Sn , the property claimed follows. The second equality in (24) is established in a similar way. Return now to the left coequivariance condition L N ( ) = R N ( ), where = + ([2d]))n . Consider first the case where all = (1 , . . . , n ) ∈ (Y i 0 ∈ N + YN ([2d]), i ∈ [n]. The commutation relations for the ideal I (more precisely, those that stem from the right coequivariance condition), imply an equality A0 , . . . A0 , = λ(1) λ(n) ) = R N ( ) q( , λ)A0 ,1 . . . A0 ,n , for any λ ∈ Sn . Therefore, the requirement L N ( acquires in this case the shape:
+ ([2d]), σ,ρ∈S 1 ,...,n ∈Y n N 1 ···n
A0 ,1 A0 ,2 . . . A0 ,n ⊗
(q,N ) × q( , ρ ◦ σ −1 )C
,..., −1 (ρ◦σ −1 )(1) (ρ◦σ )(n)
ρ(1)
ρ(2)
. . .
ρ(n)
(q,N ) (σ ) − q( , ρ)C0 ,...,0 (σ ) = 0,
where := (1 , 2 , . . . , n ). The existence of the classical limit condition and the
(q,N ) fact q( (0) , σ ) = 1 imply σ ∈Sn C0 ,...,0 (σ ) = 1. Invoking the property (20), one
Noncommutative Planck Constants (q,N )
obtains C
(ρ◦σ −1 )(1)
387
,...,
(ρ◦σ −1 )(n)
(q,N ) (σ ) = q( , ρ ◦ σ −1 )C (1),..., (n) (ρ). Performing the
summation over κ = ρ ◦ σ −1 ∈ Sn yields: A0 ,1 A0 ,2 . . . A0 ,n ⊗ ρ(1)
+ ([2d]), σ,ρ∈S 1 ,...,n ∈Y n N 1 ···n
ρ(2)
. . .
ρ(n)
(q) (q,N ) × Z n ( )C ,..., (σ ) − q( , ρ) = 0, 1
n
where (q) Z n ( ) :=
(q( , σ ))2 ,
σ ∈Sn
for any = (1 , . . . , n ) ∈ (Y([2d]))n . If we look at this expression in the associated graded algebra, i.e. the product . . . becomes . . . = ρ(1) ρ(n) ρ(1) ρ(n) q( , ρ) . . . , then it follows, that the only candidate for the coefficients in n
1
(q,N ) (q) (q) the case 1 · · · n is just C ,..., (σ ) = q( , ρ)/Z n ( ) =: C¯ (σ ). n 1 If we use this expression for all , not just 1 · · · n , then the property (q) (q) , ρ)C¯ (ρ ◦ σ ), just q( , ρ ◦ σ ) = q( , ρ)q(ρ , (σ )) implies that C¯ (σ ) = q( ρ
what is needed to satisfy (20). It remains to check if the equation L N ( ) = R N ( ), + ([2d]))n , is indeed satisfied. Using the property of the where = (1 , . . . , n ) ∈ (Y N (q) coefficients C¯ (σ ) mentioned, one obtains:
) = L N (
1 ···n , ρ∈Sn
1 (q) (q) A1 , . . . An C¯ (ρ) ⊗ H , ρ(1) ρ(n) w
(q) . . . where H := . Now, using a trick of “inserting a κ∈Sn q( , κ)κ (1) κ (n)
(q) ¯ unit” (σ )q( , σ ) = 1, and then invoking the property L n ( σ , ) = C σ ∈Sn q( , σ )L n ( , ), one arrives at
) = L N (
+ ([2d]))n , ∈(Y N 1 ···n
1 (q) (q) A ⊗ H , , w
(25)
(q) ¯ (q) (σ )A , . . . A C¯ (q) where A := (ρ). For the right-hand σ,ρ∈Sn C n ρ(n) 1 ρ(1) , ) in the shape (21), one can first invoke the property Rn ( , ρ ) = side R N ( q( , ρ)Rn ( , ) to obtain ) = R N (
1 ···n , σ ∈Sn
1 ¯ (q) (q) (σ )Aσ (1) ,1 . . . Aσ (n) ,n ⊗ H . C w
388
A. E. Ruuge, F. Van Oystaeyen
(q) Now, inserting the unit κ∈Sn C¯ (κ)q( , κ) = 1, and using again the property ) = q( Rn ( , κ , κ)Rn ( , ), one obtains the same expression (25) as for L N ( ). ) = R N ( ). The Therefore, the left coequivariance requirement (16) is satisfied, L N ( right coequivariance is established in a totally similar way. (q,N )
It is quite remarkable that the coefficients C1 ,..., N (σ ) for the q-Weyl quantization
(q)
q
q
W N : (A2d ())N → (E2d ())N that emerge in the proof do not depend on the level of truncation N . Therefore, we immediately obtain the projective limit W (q) : q q (q) A2d () → E2d () of W N as N → ∞, W (q) : 1 2 . . . n →
1
(q) Z n ( ) σ ∈Sn
σ (2) . . . q( , σ ) σ (1) σ (n) ,
(26)
, σ ) is determined for = (1 , 2 , . . . , n ) ∈ Y + ([2d]), where the coefficient q( by (σ (1)) (σ (2)) . . . (σ (n)) = q( , σ )1 2 . . . n , and the “partition function”
(q) (q) Z n ( ) is just Z n ( ) = σ ∈Sn (q( , σ ))2 . Definition 4. The map W (q) : A2d () → E2d () defined by (26) is termed the q-Weyl quantization map, q = qi, j , i, j ∈ [2d]. q
q
The map W (q) allows to induce another star product on the classical shadow q A2d (), W (q) ( f )W (q) (g) = W (q) ( f g), q
q
for f, g ∈ A2d (). Recall that we already have a product on A2d (), stemming ∼ q q from the normal quantization, and an algebra isomorphism ϕ : (A2d (), ) → E2d (). q q q Denote πm : A2d () → F m E2d ()/F m+1 E2d () the canonical projection in the m th component of the associated graded algebra, m ∈ Z0 , and perceive ϕ as a vector q space map. Since for any homogeneous element f ∈ A2d () of degree | f | we have q | f |+1 E2d (), it follows that W ( f ) − ϕ( f ) ∈ F m−1 (πm ◦ ϕ −1 ) W (q) ( f )W (q) (g) − W (q) πl ( f g) = πm ( f g), l=0
where m ∈ Z0 . Applying this formula recursively, one obtains an explicit expression for every component of f g. q
Proposition 4. For every f, g ∈ A2d () and every m ∈ Z0 , the following holds: πm ( f g) = (πm ◦ ϕ −1 )
m r =0
· · · ◦ (W
(q)
(−1)r
◦ πl2 ◦ ϕ
Proof. Induction by m = 0, 1, 2, . . ..
0l1
(W (q) ◦ πlr ◦ ϕ −1 ) ◦ . . .
) ◦ (W (q) ◦ πl1 ◦ ϕ −1 ) W (q) ( f )W (q) (g) .
Noncommutative Planck Constants
389
Let us extract the “first semiclassical correction” from the product corresponding to the q-Weyl quantization W (q) , f, g(q) := π| f |+|g|+1 ( f g), q
where f, g ∈ A2d () are homogeneous elements of degrees | f | and |g|, respectively. Definition 5. The linear graded map −, −(q) : A2d ()⊗A2d () → A2d () of degree +1 is termed the canonical distortion of the q-Poisson bracket on the classical shadow q A2d (). q
q
q
(q)
Look at the q-antisymmetrization f, g− := f, g(q) − q(g, f )g, f (q) , where q(g, f ) is determined by f g = q(g, f )g f, f and g are homogeneous. Extending q this bilinearly, one obtains another bracket on A2d (). In quantum mechanics of a d-dimensional system with coordinates x = (x1 , x2 , . . . , xd ) and the canonically conjugate momenta p = ( p1 , p2 , . . . , pd ), the first semiclassical correction to the Weyl product of a pair of classical observables F(x, p) and G(x, p) is of the shape (−i/2){F, G}, where {−, −} is the canonical Poisson bracket, while the antisymmetrization corresponds to −i{F, G}. Up to a factor λ = 1/2 independent on F and G, these two results coincide. In the q-deformed case the two brackets are different, i.e. such λ (possibly, depending on q) does not exist. Proposition 5. The canonical distortion −, −(q) of the q-Poisson bracket satisfies the 2-cocycle condition uv, w(q) − uv, w(q) + u, vw(q) − u, v(q) w = 0, q
(q)
for any u, v, w ∈ A2d (). The q-antisymmetrized bracket −, −− yields a q-Poisson q structure on A2d (). Proof. These facts are a straightforward consequence of the associativity of the q-Weyl q product on the q-affine space A2d (). It is important to point out that the bracket −, −(q) stemming from the q-Weyl quantization is not bilinear with respect to the generators of degree || 2. In this sense, these generators “distort” already the classical picture (i.e. the Poisson bracket) and should be perceived as dynamical variables just as with || = 1, which correspond to the classical coordinates x = (x1 , x2 , . . . , xd ) and momenta p = ( p1 , p2 , . . . , pd ). 6. Noncanonical Distortions In this section we would like to discuss briefly some other possibilities to distort the classical Poisson bracket {−, −}. Informally, the basic idea of “distortion” is to add new variables into the picture and to extend the bracket in a nontrivial way. We have q introduced the epoché algebra E2d () as a kind of noncommutative neighbourhood of the semiclassical parameter of a mechanical system with d degrees of freedom, but, of course, for the mathematical construction the nature of this parameter is not important. One encounters the canonical commutation and anticommutation relations, for example, in statistical physics considering the creation and annihilation operators of the particles
390
A. E. Ruuge, F. Van Oystaeyen
constituting a system (different types of bosons and fermions). Extending these relations with a central parameter g (the interaction parameter), the generic shape of these relations is as follows: ψi− ψ +j − ε j,i ψ +j ψi− = gδi, j , − − + + + + ψi− ψ − j − ε j,i ψ j ψi = 0, ψi ψ j − ε j,i ψ j ψi = 0, where i, j ∈ Z0 , ψi+ are the creation operators, and ψi− are the annihilation operators, and εi, j ∈ {−1, +1} depending on the statistics of a pair of particles associated with the indices i and j. The difference is that the number of degrees of freedom d is now infinite, but one can√still introduce the analogues of coordinates xi and momenta xi ∓ i pi )/ 2. In the condensed matter physics, it is a common practice pi as ψi± = ( to consider an asymptotics with respect to g → 0 (the quasiparticle approximation), and one may consider a noncommutative neighbourhood around g (this parameter is sometimes termed the external Planck constant). Look at a second quantized observable B of polynomial type: B=
∞
m=1 i 1 ,i 2 ,...,i m ∈Z0 , α1 ,α2 ,...,αm ∈Z2
(α ,α ,...,αm ) (α1 ) (α2 ) (α ) zi1 z i2 . . . zim m ,
2 Bi1 ,i12 ,...,i m
(α ,α ,...,α )
(α)
m 2 is not zero, and zi = xi , if where only finite number of coefficients Bi1 ,i12 ,...,i m (α) ¯ ¯ ¯ ¯ pi , if α = 1, i ∈ Z0 , Z2 = {0, 1}. The commutation relations α = 0, and zi = between xi and p j , i, j ∈ Z0 , imply that one can assume without loss of generality a (α ,α2 ,...,αm ) certain symmetry or antisymmetry of the coefficients Bi1 ,i12 ,...,i with respect to the m permutations of indices. This leads to the following construction. In the definitions of the q-analogues of the Weyl quantization, of the star product, and q q of the Poisson bracket, one can restrict oneself to some subspaces of A2d () and E2d () described, for example, in terms of q-symmetrizators and antisymmetrizators. In other q q words, fix a projector P : A2d () → A2d (), P 2 = P, and consider a product f P g = q P( f g), for f, g ∈ PA2d (). This modified product does not need to be associative, (q) and therefore the corresponding first “semiclassical” correction f, gP = P( f, g(q) )
(q)
does not need to be a 2-cocycle. Let us write T f 1 , f 2 , . . . , f n P for a bracketing of q ( f 1 , f 2 , . . . , f n ) ∈ (A2d ())n described by a planar binary tree T ∈ Y with n leaves, (q) |T | = n. Basically, one obtains a collection of brackets {T −, −, . . . , −P }T ∈Y , related to each other in a more general way than the q-Jacobi and the q-Leibniz identities. It might be of interest to describe this collection in terms of the homotopy theory. One can consider the construction mentioned not on the whole classical shadow q q A2d (), but taking a truncation (A2d ())N at some level N ∈ Z>0 . As an example to have in mind, take N = 2 (we denote the generators as i and i, j in this case, q q i, j ∈ [2d], i < j), and consider a linear map P : (A2d ())2 → (A2d ())2 , P : i1 , j1 . . . im , jm k1 . . . kn →
σ ∈Sm
m cij11,...,i ,..., jm (σ )
×i1 , jσ (1) . . . im , jσ (m) k1 . . . kn ,
Noncommutative Planck Constants
391
where i μ , jμ , kν ∈ [2d], μ ∈ [m], ν ∈ [n], m, n ∈ Z0 , and the coefficients
i 1 ,...,i m i 1 ,...,i m i 1 ,...,i m m 2 cij11,...,i ρ∈Sm (b j1 ,..., jm (σ )) , where b j1 ,..., jm (σ ) is the “braiding” ,..., jm (σ ) = b j1 ,..., jm (σ )/ factor determined by m ξi1 ξ jσ (1) . . . ξim ξ jσ (m) = bij11,...,i ,..., jm (σ )ξi 1 ξ j1 . . . ξi 1 ξ jm ,
for a collection symbols satisfying ξi ξ j = −q j,i ξ j ξi , like i, j = −q j,i j,i . Another way to distort the classical Poisson bracket {−, −} would be to consider more complicated quadratic commutation relations for the noncommutative Planck constants , ∈ Y([2d]), such as the reflection equation algebra relations [14,15]. One can also think of more general star products in analogy with the twisted products induced by a coquasitriangular structure on a Hopf algebra [17,24,30]. For a collection of leaf-labelled trees = (1 , . . . n ) ∈ (Y + ([2d]))n , and I ⊂ [n], I = {i 1 < i 2 < · · · < ir }, write q ¯ I := i 1 . . . ir , and set I := [n]\I . Consider a bilinear product R on A2d (), R( [n] R := I , ) ¯ , [m]
I ⊂[n] J ⊂[m]
J
I
J¯
where ∈ (Y + ([2d]))n , ∈ (Y + ([2d]))m , and R is a bilinear map R : A2d () ⊗ q q A2d () → A2d (). If R is associative, then R is determined by its restriction R¯ : ( , ) → R( , ) to V × V , where V is the vector space spanned over the generators , ∈ Y + ([2d]). For the product corresponding to the normal q-quantization, q this is a map which factors through V, R¯ : V × V → V ⊂ A2d (), and it is a graded map of degree +1 (the grading on V is given by deg( ) = || − 1). Extending the analogy with the q-Poisson bracket, one may say that this R¯ corresponds to a “coquasitriangular structure multiplied by ”. Denote L (R) the operators of left multiplication on the classical shadow by , L (R) = R −, ∈ Y([2d]). The commutation relations q − q , = ∨ in E2d (), where , ∈ Y + ([2d]), are translated into a system q
L (R)L (R) − q , L (R)L (R) = L∨ (R), where and vary over Y + ([2d]), which can be perceived as a generalized quantum q Yang-Baxter equation for E2d (). 7. Discussion As has already been pointed out in the Introduction, the problem of quantization on a noncommutative space, as well as the concept of the/a noncommutative geometry [8,10,20,21,23,29] itself, admits various approaches. In the most simple case, the philosophy described in the present paper can be expressed as follows. If one starts with a Poisson bracket {−, −}, say, on an algebra C[x1 , . . . , xd , p1 , . . . , pd ] of polynomial observables of a classical mechanical system, then what one should do first of all is to span a vector space V¯ over a collection of symbols x¯1 , x¯2 , . . . , x¯d and p¯ 1 , p¯ 2 , . . . , p¯ d , and to embed it into a Z-graded vector space V as a component of degree zero, V¯ ⊂ V . It is suggested to construct the ambient vector space V as a span over the symbols indexed by the planar binary leaf-labelled trees ∈ Y + ([2d]) (the notation is described in the main text), deg( ) = || − 1. In particular, x¯i and p¯ j , i, j ∈ [d], correspond to with || = 1. Next, one defines a filtered algebra E2d () generated by the symbols , ∈ Y + ([2d]) (the noncommutative Planck constants) satisfying the relations
392
A. E. Ruuge, F. Van Oystaeyen
[ , ] = ∨ , for , ∈ Y + ([2d]), where the filtration is the decreasing filtration induced by the length of polynomials filtration on the tensor algebra T (V ). A quantization is a linear map W : A2d () → E2d (), where A2d () is the associated graded algebra of E2d () (the classical shadow), satisfying the left and right affine coequivariance and the existence of the classical limit conditions. Conceptually, an analogue of a Poisson bracket is the first semiclassical correction to the star product on the classical shadow, W ( f g) = W ( f )W (g), described by −, − : A2d () ⊗ A2d () → A2d (), which is a graded linear map of degree +1 with respect to the grading induced by the Z-grading on V . Intuitively, the algebra E2d () is a kind of noncommutative neighbourhood of the semiclassical parameter ,
[i, j] , [[i, j],k] , [[i, j],[k,l]] , . . . , −→ i ,
where i, j, k, l, . . . ∈ [2d], corresponding to an idea to replace every next commutator with a new variable. In fact, the nature of the parameter is not important, and one can consider the same construction around λ−1 , where λ → ∞ is the scaling parameter for renormalization. The advantage of our approach is that it admits a natural Hopf algebraic q deformation (in this paper we describe the q-deformation E2d (), where q is a 2d × 2d matrix of formal variables). It would be of interest to study this in more detail, but we leave it for another paper. Acknowledgements. This work is supported by FWO (Fonds Wetenschappelijk Onderzoek – Vlaanderen), the research project G.0622.06 (“Deformation quantization methods for algebras and categories with applications to quantum mechanics”).
References 1. Alekseev, A., Kosmann-Schwarzbach, Y.: Manin pairs and moment maps. J. Diff. Geom. 56(1), 133–165 (2000) 2. Alekseev, A., Kosmann-Schwarzbach, Y., Meinrenken, E.: Quasi-Poisson manifolds. Canad. J. Math. 54(1), 3–29 (2002) 3. Beggs, E. J., Majid, S.: Semiclassical differential structures. Pacific J. Math. 224(1), 1–44 (2006) 4. Van den Bergh, M.: Double Poisson algebras. Trans. Amer. Math. Soc. 360(11), 5711–5769 (2008) 5. Bocklandt, R., Le Bruyn, L.: Necklace Lie algebras and noncommutative symplectic geometry. Math. Z. 240(1), 141–167 (2002) 6. Castro, C.: On modified Weyl-Heisenberg algebras, noncommutativity, matrix-valued Planck constant and QM in Clifford spaces. J. Phys. A 39(45), 14205–14229 (2006) 7. Chaichian, M., Demichev, A.P., Kulish, P.P.: Quasi-classical limit in q-deformed systems, noncommutativity and the q-path integral. Phys. Lett. A 233(4–6), 251–260 (1997) 8. Connes, A.: Cyclic cohomology and noncommutative differential geometry. Proceedings of the International Congress of Mathematicians, Vol. 1, (Berkeley, Calif., 1986), Providence, RI: Amer. Math. Soc., 1987, pp. 879–889 9. Crawley-Boevey, W., Etingof, P., Ginzburg, V.: Noncommutative geometry and quiver algebras. Adv. Math. 209(1), 274–336 (2007) 10. Drinfel’d, V.G.: Quasi-Hopf algebras. (Russian) Algebra i Analiz 1(6), 114–148 (1989), translation in Leningrad Math. J. 1(6), 1419–1457 (1990)
Noncommutative Planck Constants
393
11. Farkas, D.R.: A ring-theorist’s description of Fedosov quantization. Lett. Math. Phys. 51(3), 161–177 (2000) 12. Fialowski, A., Penkava, M.: Deformation theory of infinity algebras. J. Algebra 255(1), 59–88 (2002) 13. Ginzburg, V.: Non-commutative symplectic geometry, quiver varieties, and operads. Math. Res. Lett. 8(3), 377–400 (2001) 14. Gurevich, D.I.: Algebraic aspects of the quantum Yang-Baxter equation. (Russian) Algebra i Analiz 2(4), 119–148 (1990), translation in Leningrad Math. J. 2(4), 801–828 (1991) 15. Gurevich, D.I., Pyatov, P.N., Saponov, P.A.: Representation theory of an algebra of a (modified) reflection equation of GL(m|n) type. (Russian) Algebra i Analiz 20(2), 70–133 (2008), translation in St. Petersburg Math. J. 20(2), 213–253 (2009) 16. Hartwig, J.T., Larsson, D., Silvestrov, S.D.: Deformations of Lie algebras using σ -derivations. J. Algebra 295(2), 314–361 (2006) 17. Hirshfeld, A.C., Henselder, P.: Star products and quantum groups in quantum mechanics and field theory. Ann. Physics 308(1), 311–328 (2003) 18. Kapranov, M.: Noncommutative geometry based on commutator expansions. J. Reine Angew. Math. 505, 73–118 (1998) 19. Karasev, M.V., Maslov, V.P.: Nonlinear Poisson brackets. Geometry and quantization. Translated from the Russian by A. Sossinsky and M. Shishkova. Translations of Mathematical Monographs, 119. Providence, RI: Amer. Math. Soc., 1993 20. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66, 157–216 (2003) 21. Kontsevich, M.: Formal (non)commutative symplectic geometry. The Gel’fand Mathematical Seminars, 1990–1992, Boston, MA: Birkhuser Boston, 1993, pp. 173–187 22. Kubo, F.: Finite-dimensional non-commutative Poisson algebras. J. Pure Appl. Algebra 113(3), 307–314 (1996) 23. Le Bruyn, L.: Noncommutative geometry and Cayley-smooth orders. In: Pure and Applied Mathematics (Boca Raton), 290. Boca Raton, FL: Chapman & Hall/CRC, 2008 24. López Peña, J., Panaite, F., Van Oystaeyen, F.: General twisting of algebras. Adv. Math. 212(1), 315–337 (2007) 25. Maslov, V.P.: An application of the method of ordered operators to the problem of obtaining exact solutions. (Russian) Teoret. Mat. Fiz. 33(2), 185–209 (1977) 26. Maslov, V.P.: Operational methods. Translated from the Russian by V. Golo, N. Kulman, G. Voropaeva. Moscow: Mir Publishers, 1976 27. Maslov, V.P.: Nonstandard characteristics in asymptotic problems. (Russian) Usp. Mat. Nauk 38(6) (234), 3–36 (1983) 28. Newmann, M.H.A.: On theories with a combinatorial definition of “equivalence”. Ann. of Math. 43, 223–243 (1942) 29. Van Oystaeyen, F., Verschoren, A.: Noncommutative algebraic geometry. An introduction. Lecture Notes in Mathematics, 887. Berlin: Springer-Verlag, 1981 30. Panaite, F., Van Oystaeyen, F.: Quasi-Hopf algebras and representations of octonions and other quasialgebras. J. Math. Phys. 45(10), 3912–3929 (2004) 31. Penkava, M.: L-infinity algebras and their cohomology. http://arxiv.org/abs/q-alg/9512014v1, 1995 32. Pichereau, A., Van de Weyer, G.: Double Poisson cohomology of path algebras of quivers. J. Algebra 319(5), 2166–2208 (2008) 33. Richard, L., Silvestrov, S.D.: Quasi-Lie structure of σ -derivations of C[t ±1 ]. J. Alg. 319(3), 1285–1304 (2008) 34. Xu, P.: Noncommutative Poisson algebras. Amer. J. Math. 116(1), 101–125 (1994) 35. Yao, Y., Ye, Y., Zhang, P.: Quiver Poisson algebras. J. Algebra 312(2), 570–589 (2007) Communicated by A. Connes
Commun. Math. Phys. 304, 395–409 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1231-z
Communications in
Mathematical Physics
Poincaré Polynomial of Moduli Spaces of Framed Sheaves on (Stacky) Hirzebruch Surfaces Ugo Bruzzo1,2 , Rubik Poghossian3,4 , Alessandro Tanzini1,2 1 Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, I-34136 Trieste, Italia.
E-mail: [email protected]; [email protected]
2 Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italia 3 Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica,
I-00133 Roma, Italia
4 Yerevan Physics Institute, Alikhanian Br. st. 2, 0036 Yerevan, Armenia.
E-mail: [email protected] Received: 18 November 2009 / Accepted: 22 November 2010 Published online: 8 April 2011 – © Springer-Verlag 2011
Abstract: We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincaré polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the mathematical framework for the counting of D4-D2-D0 branes bound states on total spaces of the bundles OP1 (− p). 1. Introduction In this paper we study the moduli space of framed torsion-free sheaves on a Hirzebruch surface F p . We start by studying the geometry of this moduli space within the framework of Huybrechts-Lehn’s theory of framed modules. We then consider a natural toric action on the moduli space, determining its fixed points and the weight decomposition of the tangent spaces at the fixed points. This will in turn allow us to compute the Morse indexes at the fixed points, and therefore the Poincaré polynomial of the moduli space. As a byproduct, we show that the moduli spaces we consider are irreducible. A formula for D-branes counting on the total spaces of the bundles OP1 (− p) (which we denote X p in this paper) was given in [12] and [14] using string theory techniques and two-dimensional reduction, respectively. Hirzebruch surfaces F p are projective compactifications of the spaces X p , and indeed the results of this paper provide a proof of the This research was partly supported by the INFN Research Project PI14 “Nonperturbative dynamics of gauge theory”, by PRIN “Geometria delle varietà algebriche”, by an Institutional Partnership Grant of the Humboldt foundation of Germany and European Commission FP7 Programme Marie Curie Grant Agreement PIIF-GA-2008-221571.
396
U. Bruzzo, R. Poghossian, A. Tanzini
conjectural formulas of [12,14] for integer values of the first Chern class. Actually our computations also make sense for fractional first Chern classes (to be more precise, for Chern classes c1 (E) = mp C, where m is an integer, and C is the “exceptional curve” in F p ). This may be interpreted as a computation related to torsion-free sheaves on a stacky compactification of X p , as we shall show in Sect. 4. This relates to the stacky compactifications of ALE spaces discussed in [24]. The structure of this paper is as follows. In Sect. 2 we provide some physical motivations. In Sect. 3 we give some details about the structure of the moduli spaces we want to study. In Sect. 4 we compute the Poincaré polynomial of these moduli spaces. In Sect. 5 we draw some conclusions and check our formula for the Poincaré polynomial in some particular cases. 2. Physical Motivations In this short section we want to provide some physical motivations and perspectives for the mathematical results we give in this paper. The reader who is not interested in this may safely skip to the next section. The analysis of D-brane bound states has many applications in string theory compactifications and supersymmetric gauge theories. It is a relevant problem to provide a rigorous mathematical framework for these studies. In this paper we will consider the D4-D2-D0 system; in flat space this is a well studied case, known to correspond to the moduli space of framed torsion free sheaves on P2 . The low-energy effective theory of the D4-branes is N = 4 Super Yang-Mills theory, supporting an instanton bundle with first and second Chern characters fixed by the number of D2-branes and D0-branes respectively. Singularities due to point-like instantons require the introduction of new degrees of freedom, the “instanton freckles” [19] in order to partially compactify the instanton moduli space. The resolution obtained via non-commutative instantons in [29] precisely leads to the moduli space of framed torsion free sheaves on P2 . In this paper we are interested in considering the same D-brane system on the orbifold C2 / , ∈ U (2). In the physics literature these systems were mainly analyzed in the context of the Ooguri-Strominger-Vafa conjecture [31], which relates the counting of microstates of supersymmetric black holes to that of bound states of Dp-branes wrapping cycles of the internal Calabi-Yau threefold in string theory compactifications. In [2] the example of a local Calabi-Yau modeled as the total space of a rank two bundle over a Riemann surface g was studied in detail, providing some support for the conjecture. In that example, the D4 branes wrap the total space of an O(− p)-bundle over the Riemann surface. Very little is known about the moduli space of instantons on such spaces; indeed the authors of [2] proceeded by conjecturing a reduction to the base g . An argument clarifying this reduction was provided in [4] via path-integral localization. In this paper we propose that the moduli space relevant for D4-D2-D0 branes counting on X p is the moduli space of framed torsion-free sheaves on the global quotient P2 / with a minimal resolution of the singularity at the origin. This variety is a toric Deligne-Mumford stack X p whose coarse space is the Hirzebruch surface F p ; it corresponds to a “stacky” compactification of X p obtained by adding a divisor C˜ ∞ P1 / . We will show that torsion free sheaves framed on C˜ ∞ provide solutions with fractional first Chern class, which is a distinctive feature of D-brane countings on orbifold spaces. We will compute the Poincaré polynomial of the moduli space of these framed sheaves via localization and show that the resulting generating function of Euler characteristics is in agreement with the D-brane counting formulae of [12] and [14]. This has a nice
Poincaré Polynomial of Moduli Spaces of Framed Sheaves
397
expression in terms of modular forms. The details of this construction are presented in Sect. 4. Let us remark that the low-energy effective theory supported by the D4-branes is a Vafa-Witten twisted N = 4 supersymmetric theory on X p [34]. The results that we obtain are in agreement with the available calculations of the instanton partition function (with non-commutative resolution), namely the case p = 1, where the blowup formulas of [34,35] apply, and p = 2. Indeed, X 2 is the A1 ALE space [18,22] (for details on instanton counting on A p−1 ALE spaces the reader may refer to [11,12,18]). It would be very interesting to have a full comparison with our results, though at the moment this is not possible due to the lack of an ADHM-like construction of instantons on more general quotients C2 / , ∈ U (2). We finally notice that supersymmetric twisted N = 2 theories were studied on arbitrary toric manifolds using localization techniques in [28], obtaining a nice “twodimensional topological vertex” interpretation of the results. It is natural to expect our partition functions to behave in a similar way; indeed, in some low-rank cases we are able to provide factorized formulae for the generating functional of the Euler characteristics of the moduli space of sheaves, see Sect. 5, although further studies are needed to provide an interpretation of our results in terms of equivariant topological vertexes [21]. 3. Moduli of Framed Sheaves on Hirzebruch Surfaces In this section we briefly study the structure of the moduli space of framed sheaves on Hirzebruch surfaces. 3.1. Hirzebruch surfaces. We denote by F p the p th Hirzebruch surface F p = P(OP1 ⊕ OP1 (− p)), which is the projective closure of the total space X p of the line bundle OP1 (− p) on P1 (a useful reference about Hirzebruch surfaces is [3]). This may be explicitly described as the divisor in P2 × P1 , F p = {([z 0 : z 1 : z 2 ], [z : w] ∈ P2 × P1 | z 1 w p = z 2 z p }. Denoting by f : F p → P2 the projection onto P2 , we let C∞ = f −1 (l∞ ), where l∞ is the “line at infinity” z 0 = 0. The Picard group of F p is generated by C∞ and the fibre F of the projection F p → P1 . One has 2 C∞ = p,
C∞ · F = 1,
F 2 = 0.
The canonical divisor K p may be expressed as K p = −2C∞ + ( p − 2)F. We shall consider the moduli space M p (r, k, n) parametrizing isomorphism classes of pairs (E, φ), where • E is a torsion-free coherent sheaf on F p , whose topological invariants are the rank r , the first Chern class c1 (E) = kC, and the discriminant (E) = c2 (E) −
r −1 2 c (E) = n; 2r 1
398
U. Bruzzo, R. Poghossian, A. Tanzini
• φ is a framing on C∞ , i.e., an isomorphism of the restriction of E to C∞ with the trivial rank r sheaf on C∞ : ∼ O⊕r . φ: E → |C∞
C∞
In constructing the moduli space M p (r, k, n) one only considers isomorphisms which preserve the framing, i.e., (E, φ) (E , φ ) if there is an isomorphism ψ : E → E such that φ ◦ ψ = φ. While a point in M p (r, k, n) should always be denoted as the class
of a pair (E, φ), we shall be occasionally a little sloppy in our notation, omitting the framing φ. For r = 1, the value of k is irrelevant, and every moduli space M p (1, k, n) turns out to be isomorphic to the Hilbert scheme X [n] p parametrizing length n 0-dimensional subschemes of X p . The structure of the moduli space for r > 1 can be studied using an ADHM description [32], which shows that the moduli space is a smooth algebraic variety of dimension 2r n. However we are not going to give this description here. 3.2. Structure of the moduli space. A general result about the structure of moduli spaces of framed sheaves on a projective surface X was given by Nevins [30], who showed that, under some mild rigidity condition, there is a fine moduli scheme. However in the specific case at hand much more can be said. The structure of the moduli space may be studied by showing that, under some conditions, and after a suitable choice a polarization in X , and a certain stability parameter, framed sheaves yield stable pairs according to the theory developed by Huybrechts and Lehn [15,16]. The moduli space turns out to be a quasi-projective separated scheme of finite type [7]. The obstruction to smoothness at a point [(E, φ)] of the moduli space lies in the kernel of the trace map Ext2 (E, E(−D)) → H 2 (X, O(−D)), where D is the framing divisor. In our case Ext2 (E, E(−C∞ )) = 0, hence the moduli space is smooth (and then is an algebraic variety). The result one gets in our case is the following. Proposition 3.1. The moduli space M p (r, k, n), when nonempty, is a smooth quasiprojective variety of dimension 2r n. Its tangent space at a point [E] is isomorphic to the vector space Ext 1 (E, E(−C∞ )). 3.3. Toric action. The two-dimensional algebraic torus C∗ × C∗ acts on F p according to G t1 ,t2
p
p
([z 0 : z 1 : z 2 ], [z : w]) −−−−→ ([z 0 : t1 z 1 : t2 z 2 ], [t1 z : t2 w]). The divisors C = f −1 ([1 : 0 : 0]) and C∞ are fixed under this action. Note that one has C = C∞ − p F as divisors modulo linear equivalence. Moreover, this action has four fixed points, i.e., p1 = ([1 : 0 : 0], [0 : 1] and p2 = ([1 : 0 : 0], [1 : 0] lying on the exceptional line C, and two points lying on the line at infinity C∞ . The invariance of C∞ implies that the pullback G ∗t1 ,t2 defines an action on M p (r, k, n). Moreover we have an action of the diagonal maximal torus of Gl(r, C) on the framing. Altogether we have an action of the torus T = (C∗ )r +2 on M p (r, k, n) given by ∗ (t1 , t2 , e1 , . . . , er ) · (E, φ) = (G −1 , (1) ) E, φ t1 ,t2
Poincaré Polynomial of Moduli Spaces of Framed Sheaves
399
where φ is defined as the composition ∗ (G −1 t1 ,t2 ) φ
e1 ,...,er
⊕r ⊕r ∗ ∗ ⊕r −−−−−→ (G −1 (G −1 t1 ,t2 ) E|C∞ − t1 ,t2 ) OC∞ −→ OC∞ −−−−→ OC∞ .
We study now the fixed point sets for the action of T on M p (r, k, n). This is basically the same statement as in [26] (see also [25] and [13]), but for the sake of completeness we give here a sketch of the proof. Proposition 3.2. The fixed points of the action (1) of T on M p (r, k, n) are sheaves of the type E=
r
Iα (kα C),
(2)
α=1
where Iα is the ideal sheaf of a 0-cycle Z α supported on { p1 } ∪ { p2 } and k1 , . . . , kr are integers which sum up to k. The α th factor in this decomposition corresponds, via the framing, to the α th factor in OC⊕r∞ . Moreover, r p p 2 2 n = + kα − k = + (kα − kβ )2 , (3) r 2r 2r α<β
α=1
where is the length of the singularity set of E. Proof. We first check that if E is fixed under the T -action, then E = ⊕rα=1 Eα , where each Eα is a T -invariant rank-one torsion-free sheaf on F p . Let K be the sheaf of rational functions on F p . Then E = E ⊗ K is free as a K-module. Choose a trivialization E ⊕rα=1 Eα which, when restricted to C∞ , provides an eigenspace decomposition for the action of T . Then set Eα = Eα ∩ E, i.e., pick up the holomorphic sections of Eα . This provides the desired decomposition. By taking the double dual of E, and using T -invariance, we obtain E ∗∗ rα=1 OF p (kα C), whence E has the form (2). Since the 0-cycles Z α have to be T -invariant and should not be supported on C∞ , they must be supported as claimed. The numerical equality (3) follows from a straight calculation. The exact identification of the fixed points is obtained by using some Young tableaux combinatorics. The precursor of this technique is Nakajima’s seminal book [23] where the case of the Hilbert scheme of points of C2 is treated (from a gauge-theoretic viewpoint, this is the “rank 1 case” for framed instantons on S 4 ). This was generalized to higher rank in [6,10,27]. A word on notation: if Y is a Young tableau, |Y | will denote the number of boxes in it. (i) In the case at hand, it turns out that to each fixed point one should attach an r -ple {Yα } of pairs of Young tableaux (so i = 1, 2 and α = 1, . . . , r ). Write Z α = Z α(1) ∪ Z α(2) , (i) (i) where Z α is supported at pi . The Young tableau {Yα }, for i and α fixed, is attached to the ideal sheaf I Z (i) as usual: choose local affine coordinates (x, y) around pi and make α
(i)
a correspondence between the boxes of {Yα } and monomials in x, y in the usual way (cf. [23]); then I Z (i) is generated by the monomials lying outside the tableau. Now the α identity (3) may be written as p n= |Yα1 | + |Yα2 | + (kα − kβ )2 . (4) 2r α α<β
400
U. Bruzzo, R. Poghossian, A. Tanzini
Looking for all collections of Young tableaux and strings of integers k1 , . . . , kr satisfying this condition together with rα=1 kα = k, one enumerates all the fixed points. 3.4. Nonemptiness of the moduli space. The moduli spaces M p (r, k, n) may be given an ADHM description, as shown in [32]. The analysis performed in [32] implies the following results. Note that the value of k may be normalized in the range 0 ≤ k < r − 1 upon twisting by OF p (C), and in the sequel we shall assume that this has been done. Proposition 3.3. (i) The moduli space M p (r, k, n) is nonempty if and only if the −1 number n − r2r pk 2 is an integer, and the bound n≥N=
pk (r − k) 2r
(5)
holds. (ii) All sheaves in M p (r, k, N ) are locally free. (iii) The moduli space M1 (r, k, N ) is isomorphic to the Grassmannian variety G k (r ) of k-planes in Cr . For p > 1, the space M p (r, k, N ) is isomorphic to a rank ( p − 1) vector bundle on G k (r ). In particular, for all p ≥ 1 the moduli space M p (r, k, N ) has the homotopy type of a compact space. 4. Poincaré Polynomial In this section we determine the weight decomposition of the toric action on the tangent space to the moduli space at the fixed points, and use this information to compute the Poincaré polynomial of the moduli spaces M p (r, k, n). 4.1. Defining the Poincaré polynomial. Let X be a space whose cohomology with rational coefficients is finite-dimensional. One defines the Poincaré polynomial of X as Pt (X ) = (dim H n (X, Q)) t n . n≥0
4.2. Sheaves on stacky Hirzebruch surfaces. Actually our computations also make sense for c1 (E) = kC with k = m/ p for integer m, and p ≥ 2. This can be justified by considering a “stacky compactification” of X p ; instead of adding the divisor C∞ , we add C˜ ∞ C∞ /Z p . One obtains a Deligne-Mumford stack X p , whose so-called coarse space may be identified with the Hirzebruch surface F p , and one has a a morphism π : X p → F p . Since the push-forward functor π∗ : Coh(X p ) → Coh(F p ) is exact [1], one has an isomorphism H i (X p , F) H i (F p , π∗ F)
(6)
for any coherent sheaf F ∈ Coh(X p ) and all i. Owing to this basic fact, we can reduce all our computations to the coarse space F p , by appropriately taking into account the push-forward of the relevant sheaves. The theory developed in [9] allows one to prove the following lemma.
Poincaré Polynomial of Moduli Spaces of Framed Sheaves
401
Lemma 4.1. The Picard lattice of X p is freely generated over Z by the divisor C˜ ∞ and the inverse image F˜ = π −1 (F). Moreover one has π( p C˜ ∞ ) = C∞ , ˜ = F. π( F)
(7)
p (r, k, n) be the moduli space of torsion-free rank r sheaves E on X p , with Let M ˜ = kC and discriminant n, that are framed on C˜ ∞ to the sheaf O⊕r . Here k = m/ p c1 (E) C˜ ∞ for some integer m. The fixed points under the torus action are as in Proposition 3.2, except that in this case the kα ’s are kα = m α / p, m α ∈ Z.
p (r, k, n) a fully developed theory Remark 4.2. We do not have for the moduli spaces M as we described in the previous section for the spaces M p (r, k, n). On the other hand,
p (r, k, n) may be replaced when k takes integer values, in the following treatment M p by M (r, k, n), so that all results are rigourous. Therefore, the results for non-integer values of k are to be regarded as heuristic, in the sense that they need a more rigourous proof, involving the construction of moduli spaces of framed sheaves on the stacks X p . 4.3. Weight decomposition of the tangent spaces. We compute here the weights of the
p (r, k, n) action of the torus T on the irreducible subspaces of the tangent spaces to M at the fixed points (E, φ) of the action. According to the decomposition (2), the tangent
p (r, k, n) Ext 1 (E, E(−C˜ ∞ )) splits as space T(E,φ) M Ext 1 (E, E(−C∞ )) = Ext 1 (Iα (kα C), Iβ (kβ C − C˜ ∞ )). α,β
The factor Ext 1 (Iα (kα C), Iβ (kβ C − C˜ ∞ )) has weight eβ eα−1 under the maximal torus of Gl(r, C). So we need only to describe the weight decomposition with respect to the remaining action of T 2 = C∗ × C∗ . From the exact sequence 0 → Iα → O → O Z α → 0 we have in K-theoretic terms, Ext • (Iα (kα C), Iβ (kβ C − C˜ ∞ )) = Ext • (O(kα C), O(kβ C − C˜ ∞ )) − Ext • (O(kα C), O Z β (kβ C − C˜ ∞ )) − Ext • (O Z α (kα C), O(kβ C − C˜ ∞ )) + Ext • (O Z α (kα C), O Z β (kβ C − C˜ ∞ )).
(8)
We should note that Ext 0 (O(kα C), O(kβ C − C˜ ∞ )) H 0 (X p , O(−n αβ C − C˜ ∞ )) H 0 F p , O −([n αβ ] + 1)C∞ + pn αβ F = 0, where n αβ = kα − kβ and [n αβ ] is its integer part. In order to show the above vanishing we used the relation C = p(C˜ ∞ − F), the isomorphism (6) and Lemma 7. More precisely, for our purposes we only need to compute the push-forward of line bundles; this is done along the lines of [5]. Analogously Ext 2 (O(kα C), O(kβ C − C˜ ∞ )) H 0 (F p , O(([n αβ ] + 1)C∞ − pn αβ F)) = 0 ,
402
U. Bruzzo, R. Poghossian, A. Tanzini
since the divisors C∞ and F intersect. We are thus left with the computation of the Ext 1 groups Ext1 (O(kα C), O(kβ C − C˜ ∞ )) = H 1 (F p , O(−n αβ C − C˜ ∞ )). We distinguish three cases according to the values of [n αβ ]. In the first case ([n αβ ] = 0) one again easily sees that H 1 (X p , O(−n αβ C − C˜ ∞ )) = H 1 (F p , O( p{n αβ }F −C∞ )) = 0, where {n αβ } is the fractional part, {n αβ } = n αβ − [n αβ ]. In the second case ([n αβ ] > 0) we get [n
αβ H 1 (X p , O(−n αβ C − C˜ ∞ )) ⊕d=0
]−1
H 0 (P1 , OP1 ( pd + p{n αβ })).
(9)
We prove this result by induction on [n αβ ] > 0 using the exact sequence 0 → O(−(n αβ + 1)C − C˜ ∞ ) → O(−n αβ C − C˜ ∞ ) → OC (−n αβ C) → 0. (10) For n αβ = 1 one readily obtains (9). On the other hand from (10) we get 0 → H 0 (P1 , O(n αβ p)) → H 1 (X p , O(−(n αβ + 1)C − C˜ ∞ )) → H 1 (X p , O(−n αβ C − C˜ ∞ )) → H 1 (P1 , O(n αβ p)) = 0, and by the inductive hypothesis 0 → H 0 (P1 , O(n αβ p)) → H 1 (X p , O(−(n αβ + 1)C − C˜ ∞ )) [n
αβ → ⊕d=0
]−1
H 0 (P1 , OP1 ( pd + p{n αβ })) → 0,
so that (9) is proved. Since H 0 (P1 , OP1 ( pd + p{n αβ })) is the space of homogeneous poly pd+ p{n } − pd+ p{n αβ }+i nomials of degree pd + p{n αβ } in two variables, it equals i=0 αβ t1−i t2 in the representation ring of T 2 . Finally, we get the weights of this summand of the tangent space as −j L α,β (t1 , t2 ) = eβ eα−1 t1−i t2 . (11) i, j≥0,i+ j− pn αβ ≡0 mod p, i+ j≤ p(n αβ −1)
For the third case (n αβ < 0) we have analogously L α,β (t1 , t2 ) = eβ eα−1
j+1
t1i+1 t2 .
(12)
i, j≥0, i+ j+2+ pn αβ ≡0 mod p, i+ j≤− pn αβ −2
p (r, k, n) at the fixed These terms contribute to the weight decomposition of T(E,φ) M points together with the remaining terms in (8), which are essentially the same as in the P2 case modulo some rescalings of the arguments (in a sense, we look at X p as the resolution of singularities of C2 /Z p , and rescale the arguments to achieve Z p -equivariance). In this way we get
p (r, k, n) T(E,φ) M r p(k −k ) Y1 p(k −k ) Y2 p p = L α,β (t1 , t2 ) + t1 β α Nα,β (t1 , t2 /t1 ) + t2 β α Nα,β (t1 /t2 , t2 ) , α,β=1
(13)
Poincaré Polynomial of Moduli Spaces of Framed Sheaves
403
where L α,β (t1 , t2 ) is given by (11) and (12), while ⎧ ⎫ ⎬ ⎨ −lY (s) −a (s) Y 1+a (s) 1+l (s) Y t1 β t2 Yα + t1 Yα t2 β Nα,β (t1 , t2 ) = eβ eα−1 × ,(14) ⎩ ⎭ s∈Yα
s∈Yβ
a well known expression for the C2 case (first introduced in [10]). Here Y denotes an r -ple of Young tableaux, while for a given box s in the tableau Yα , the symbols aYα and lYα denote the “arm” and “leg” of s respectively, that is, the number of boxes above and on the right of s. 4.4. Computing the Poincaré Polynomial. We shall closely follow the method presented in [25], Sect. 3.4. We choose a one-parameter subgroup λ(t) of the r + 2 dimensional torus T , λ(t) = (t m 1 , t m 2 , t n 1 , . . . , t nr ), by specifying generic weights such that m 1 = m 2 n 1 > n 2 · · · > nr > 0. We have now a different fixed locus; as far as the action on the surface F p is concerned, the entire “exceptional line” C is pointwise fixed under this action. Accordingly, the fixed points in M(r, k, n) still have the form (2), with Z α fixed under the action of the diagonal subgroup of C∗ × C∗ , which means that the points in the supports of the 0-cycles Z α are arbitrary points in C. Each component of the fixed point set is parametrized by an r -ple of pairs ((k1 , Y1 ), . . . , (kr , Yr )) with n=
r
|Yα | +
α=1
r p (kα − kβ )2 and kα = k. 2r α<β
α=1
Remark 4.3. This action allows one to prove that the space M p (r, k, n) has the homotopy type of a compact space, thus generalizing point (iii) of Proposition 3.3. This is needed to fully justify our computation of the Poincaré polynomial of M p (r, k, n) which uses, albeit indirectly, Morse theory. We may define a set p
M0 (r, k, n) =
0≤ ≤n−N
Mlf (r, k, n − ) × Sym X p p
p
(where Mlf (r, k, n) is the open subscheme of M p (r, k, n) given by the locally free p points) and a map γ : M p (r, k, n) → M0 (r, k, n) by letting γ ((E, φ)) = (E ∗∗ , φ), supp(E ∗∗ /E) ). p
The space M0 (r, k, n) may be given a schematic structure by looking at it as a Uhlenp beck-Donaldson partial compactification of Mlf (r, k, n), and the morphism γ is then p projective (hence proper) [8,17]. There is a natural action of C∗ on M0 (r, k, n), which p is compatible with the action on M p (r, k, n). The “deepest” stratum in M0 (r, k, n) is isomorphic to M p (r, k, N ) × Symn−N X p , and its intersection Z with the fixed locus of the C∗ action is isomorphic to a product of symmetric products of copies of the exceptional curve C (hence, is compact). An analysis analogous to the one developed in [23,26] shows that γ −1 (Z ) has the same homotopy type of M p (r, k, n).
404
U. Bruzzo, R. Poghossian, A. Tanzini
To link with the previous construction, we should notice that each fixed point E¯ of the × T r action corresponding to an r -ple ((k1 , Y1 ), . . . , (kr , Yr )) determines a fixed point ((k1 , ∅, Y1 ), . . . , (kr , ∅, Yr )) of the T -action which lies in the same compo¯ Since the × T r -module structure of the tangent space nent of the fixed locus as E.
p (r, k, n) at a fixed point of × T r does not change if we move the fixed point T(E ,φ) M in its component, we may choose ((k1 , ∅, Y1 ), . . . , (kr , ∅, Yr )). In this case (13) reduces to
p (r, k, n) = T(E,φ) M
r
p(kβ −kα )
L α,β (t1 , t1 ) + t1
α,β=1
p Y Nα,β (1, t1 ) .
(15)
By (14) we have ⎛ p Y Nα,β (1, t1 )
= eβ eα−1 × ⎝
p(1+aYα (s)) t1
s∈Yα
+
⎞ − p aYβ (s) ⎠. t1
s∈Yβ
Our task now is to compute the index of the critical points, that is, the number of terms in (15) for which one of the following possibilities holds: (i) (ii) (iii) (iv)
the weight of t1 is negative, the weight of t1 is zero and the weight of e1 is negative, the weights of t1 , e1 are zero and the weight of e2 is negative, the weights of t1 , e1 , e2 are zero and weight of e3 is negative, …… (r + 1) the weights of t1 , e1 ,…,er −1 are zero and the weight of er is negative. The contribution of the diagonal (α = β) terms of (15) turns out to be r
(| Yα | −l(Yα )).
α=1
The nondiagonal (α = β) terms can be rewritten as ⎛ ⎝ L α,β (t1 , t1 ) + L β,α (t1 , t1 ) α<β
+
eβ s∈Yα
+
eα
eβ s∈Yβ
eα
p(1+aYα (s)+kβ −kα ) t1
eα p (−aYα (s)−kβ +kα ) + t eβ 1
p(−aYβ (s)+kβ −kα ) t1
eα p (1+aYβ (s)+kα −kβ ) t eβ 1
+
⎞ ⎠.
(16)
We compute now the L terms in these expressions. Due to the formulas (11), (12), if kα − kβ ≥ 0 only the term L α,β (t1 , t1 ) contributes to the index. This contribution is easy to compute: i, j≥0,i+ j− pn αβ ≡0 mod p, i+ j≤ p(n αβ −1)
1=
1 [n αβ ] p[n αβ ] + 2 − p + p[n αβ ]{n αβ }. 2
Poincaré Polynomial of Moduli Spaces of Framed Sheaves
405
If, instead, kα − kβ < 0 there is a contribution from the L β,α (t1 , t1 ) term which is equal to
1=
i, j≥0, i+ j>0, i+ j+ pn βα ≡0 mod p, i+ j≤ p(n βα −1)
1 [n βα ] p[n βα ] + 2 − p 2 + p[n βα ]{n βα } − δ p{n βα },0 .
Summarizing, the contribution of the L terms to the index is 1 lα,β
2 [n αβ ] p[n αβ ] + 2 − 1 2 [n βα ] p[n βα ] + 2 −
=
p + p[n αβ ]{n αβ } p + p[n βα ]{n βα } − δ p{n βα },0
if n αβ ≥ 0,
(17)
otherwise.
The only operation to be yet performed is the sum over the boxes of Yα in (16). A careful analysis shows that the contribution is | Yα | + | Yβ | −n α,β , where n α,β
=
of columns of Yα that are longer than kα − kβ if kα − kβ ≥ 0, of columns of Yβ that are longer than kβ − kα − 1 otherwise.
(18)
With this information we may compute the desired Poincaré polynomial.
p (r, k, n) is Theorem 4.4. The Poincaré polynomial of M
p (r, k, n)) = Pt (M
r fixed points
(α)
Here m i
(α)
t 2(|Yα |−l(Yα ))
α=1
∞ 2(m t i
t2
i=1
+|Y |+|Y |−n ) − 1 2(lα,β α β α,β . t −1
+1)
α<β
is the number of columns in Yα that have length i.
Corollary 4.5. The generating function for the Euler characteristics of the moduli spaces
p (r, k, n) is M
p (r, k, n)) q n+ P−1 (M
pk 2 2r
z = k
θ3 ( vp | τp )
k,n
r
η(τ ˆ )2
with q = e2πiτ and z = e2πiv .
p (r, k, n)) and Proof. It just follows by setting t = −1 in the above expression for Pt (M the standard formulas for the (quasi)-modular functions θ3 (v|τ ) = η(τ ˆ )=
1 2
q 2 n e2πivn ,
n∈Z ∞
(1 − q l ).
l=1
406
U. Bruzzo, R. Poghossian, A. Tanzini
5. Some Comparisons and Consequences 5.1. Irreducibility of the moduli space. First, we note that our computation of the Poincaré polynomial of the spaces M p (r, k, n) allows us to conclude that these spaces are irreducible. Theorem 5.1. For all p ≥ 1, and all values of the parameters r, k, n, the moduli space M p (r, k, n) of framed sheaves on the Hirzebruch surface F p is irreducible. Proof. By letting t = 0 in the formula for the Poincaré polynomial we see that M p (r, k, n) is connected, and since it is smooth (Proposition 3.1), it is also irreducibile. 5.2. Hilbert schemes of points. For r = 1 Theorem 4.4 yields the Poincaré polynomial of the Hilbert schemes of points of the spaces X p . The generating function for these is given by ∞
n Pt (X [n] p )q =
n=0
∞
1
j=1
(1 − t 2 j−2 q j )(1 − t 2 j q j )
.
(19)
This is independent of p, according to what was noted in [23, Cor. 7.7]: the Betti numbers of the Hilbert scheme of the total space of a line bundle on a Riemann surface do not depend on the line bundle. Indeed formula (19) is the formula of [23, Cor. 7.7] for = P1 . 5.3. The rank 2 case. One can examine M p (2, k, n) in some detail. In the case p = 1 our result coincides with Corollary 3.19 of [25], where it is also shown (identity (3.20)) that the generating function of the Poincaré polynomial can be further factorized to obtain the elegant product formula
1 (2, 0, n)) q n = Pt (M
n∈Z
∞
1 (1 − t 4d q d )2 (1 − t 4d−2 q d )(1 − t 4d−4 q d ) d=1 2 t 2k(2k+1) q k . ×
(20)
k∈Z
Our computations also agree with some explicit characterizations of the spaces M1 (2, k, n) given in [7]. 5.4. The case p = 2. This is considered in [33]. Besides the usual line bundles O(kα C) with integer kα , the author (without much mathematical justification) considers there also line bundles with half-integer Chern class kα (see the discussion after Eq. (3.35) of [33]). This is what we get in the “stacky” case for p = 2. It is interesting that incorporating the terms corresponding to the half-integer kα again leads to a product formula (see (2.62)–(2.64) of [33])
2 (2, 0, n)) q n Pt (M n∈ Z2
=
∞
1 − (−t 2 q 2 )d t −2
d=1
(1 − t 4d q d )(1 − t 4d−2 q d )2 (1 − t 4d−4 q d )(1 + (−t 2 q 2 )d )
1
1
.
(21)
Poincaré Polynomial of Moduli Spaces of Framed Sheaves
407
This strongly suggests that one should be able to write a general factorized expression for the generating function of the Poincaré polynomial in the case of the stacky Hirzebruch surfaces for all values of p.
5.5. The case of minimal discriminant. We would also like to compare our computation with the result in Proposition 3.3, which implies that whenever n reaches the lower pk bound N = 2r (r − k), the moduli space M p (r, k, N ) is homotopic to the Grassmannian variety G k (r ). Coherently with the fact that all sheaves in M p (r, k, N ) are locally free, in computing the Poincaré polynomial for this case one only considers empty Young tableaux, and takes values kα that obey the condition r
kα =
α=1
r α=1
kα2 .
This implies that each kα is either 0 or 1. Thus the fixed points in the case (5) are in a one-to-one correspondence with the sequences k = (k0 , k1 · · · , kr ) with kα ∈ {0, 1} and 1 = k. Such sequences may be conveniently represented as Young tableaux (below denoted as Yk ) embedded in a rectangle of size (r − k) × k as follows: starting from the left top of the rectangle successively draw a vertical (horizontal) line segment of unit size if you encounter 0 (1). According to Theorem 4.4 the Poincaré polynomial for this case is 2l Pt = t α,β . (22) k α<β = 0, besides the cases with kα = 1 and kβ = 0 for which According to Eq. (17) all lα,β lα,β = 1. But the number of latter cases exactly matches with the number of hooks of the given Young tableau. Since the number of hooks is the same as the number of boxes, we obtain Pt = t 2|Yk | . (23) k
Thus the Poincaré polynomial is nothing but the generating function (with respect to the parameter q = t 2 ) of the partitions with number of parts (i.e. length) ≤ r − k and with largest part ≤ k. This generating function is known to coincide with the “q-binomial coefficient” [20, p. 27] (1 − q r −k+1 ) · · · (1 − q r ) r = , k (1 − q) · · · (1 − q k ) which shows that the Poincaré polynomial (23) coincides with the Poincaré polynomial of the Grassmannian G k (r ). Acknowledgements. We thank Hua-Lian Chang, Rainald Flume, Francesco Fucito, Emanuele Macrı, Dimitri Markushevich, J. Francisco Morales and Claudio Rava for useful suggestions. Special thanks to Fabio Nironi for enlightening discussions on the stacky Hirzebruch surfaces. The second author acknowledges hospitality and support from SISSA.
408
U. Bruzzo, R. Poghossian, A. Tanzini
References 1. Abramovich, D., Vistoli, A.: Compactifying the space of stable maps. J. Amer. Math. Soc. 15, 27–75 (2002) 2. Aganagic, M., Ooguri, H., Saulina, N., Vafa, C.: Black holes, q-deformed 2d Yang-Mills, and nonperturbative topological strings. Nucl. Phys. B 715, 304–348 (2005) 3. Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces. Vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Berlin: Springer-Verlag, second ed., 2004 4. Bonelli, G., Tanzini, A.: Topological gauge theories on local spaces and black hole entropy countings. Adv. Theor. Math. Phys. 12, 1429–1446 (2008) 5. Borne, N.: Fibrés paraboliques et champ des racines. Int. Math. Res. Not. IMRN 2007, no. 16, Art. ID rnm049, 38 pp. (2007) 6. Bruzzo, U., Fucito, F., MoralesJ.F. Tanzini, A.: Multi-instanton calculus and equivariant cohomology. J. High Energy Phys. 0305, 054 (2003) 7. Bruzzo, U., Markushevich, D.: Moduli spaces of framed torsion-free sheaves on projective surfaces. http://arXiv.org/abs/0906.1436v2 [math.AG], 2009 8. Bruzzo, U., Markushevich, D., Tikhomirov, A.: Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces. http://arXiv.org/abs/1009.0856v2 [math.AG], 2010 9. Fantechi, B., Mann, E., Nironi, F.: Smooth toric DM stacks. http://arXiv.org/abs/0708.1254v2 [math.AG], 2009 10. Flume, R., Poghossian, R.: An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential. Int. J. Mod. Phys. A 18, 2541–2563 (2003) 11. Fucito, F., Morales, J.F., Poghossian, R.: Multi instanton calculus on ALE spaces. Nucl. Phys. B 703, 518–536 (2004) 12. Fucito, F., Morales, J.F., Poghossian, R.: Instantons on toric singularities and black hole countings. J. High Energy Phys. 0612, 073 (2006) 13. Gasparim, E., Liu, C.-C.M.: The Nekrasov conjecture for toric surfaces. Commun. Math. Phys. 293, 661–700 (2010) 14. Griguolo, L., Seminara, D., Szabo, R.J., Tanzini, A.: Black holes, instanton counting on toric singularities and q-deformed two-dimensional Yang-Mills theory. Nucl. Phys. B 772, 1–24 (2007) 15. Huybrechts, D., Lehn, M.: Framed modules and their moduli. Int. J. Math. 6, 297–324 (1995) 16. Huybrechts, D., Lehn, M.: Stable pairs on curves and surfaces. J. Alg. Geom. 4, 67–104 (1995) 17. Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31, Braunschweig: Friedr. Vieweg & Sohn, 1997 18. Kronheimer, P.B., Nakajima, H.: Yang-Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990) 19. Losev, A., Nekrasov, N., Shatashvili, S.L.: Freckled instantons in two and four dimensions. Class. Quant. Grav. 17, 1181–1187 (2000) 20. Macdonald, I.G.: Symmetric functions and Hall polynomials. New York: The Clarendon Press/Oxford University Press, 1995 21. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and DonaldsonThomas theory, I. Compos. Math. 142, 1263–1285 (2006) 22. Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76, 365–416 (1994) 23. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. Vol. 18 of University Lecture Series, Providence, RI: Amer. Math. Soc., 1999 24. Nakajima, H.: Sheaves on ALE spaces and quiver varieties. Moscow Math. J. 7, 699–722 (2007) 25. Nakajima, H., Yoshioka, K.: Lectures on instanton counting. In: Algebraic structures and moduli spaces. Vol. 38 of CRM Proc. Lecture Notes, Providence, RI: Amer. Math. Soc., 2004, pp. 31–101 26. Nakajima, H., Yoshioka, K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Invent. Math. 162, 313–355 (2005) 27. Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2003) 28. Nekrasov, N.A.: Localizing gauge theories. In: Proc. of the 14th International Congress on Mathematical Physics (ICMP 2003), Lisbon, Portugal, 28 Jul - 2 Aug 2003, Hackensaer, NJ: World Scientific, 2005, pp. 645–654 29. Nekrasov, N., Schwarz, A.S.: Instantons on noncommutative R4 , and (2,0) superconformal six dimensional theory. Commun. Math. Phys. 198, 689–703 (1998) 30. Nevins, T.A.: Representability for some moduli stacks of framed sheaves. Manus. Math. 109, 85–91 (2002) 31. Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. Phys. Rev. D 70(3), 106007 (2004)
Poincaré Polynomial of Moduli Spaces of Framed Sheaves
409
32. Rava, C.: ADHM data for framed sheaves on Hirzebruch surfaces. PhD thesis, SISSA, Trieste, 2010 2 /Z and Weil conjecture. http://arXiv. 33. Sasaki, T.: O(−2) blow-up formula via instanton calculus on C 2 org/abs/hep-th/0603162v2, 2006 34. Vafa, C., Witten, E.: A strong coupling test of S-duality. Nucl. Phys. B 431, 3–77 (1994) 35. Yoshioka, K.: Betti numbers of moduli of stable sheaves on some surfaces. Nucl. Phys. B Proc. Suppl. 46, 263–268 (1996) Communicated by N.A. Nekrasov
Commun. Math. Phys. 304, 411–432 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1232-y
Communications in
Mathematical Physics
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models Ed Segal Department of Mathematics, Imperial College of London, London SW7 2BW, United Kingdom. E-mail: [email protected]
Received: 14 December 2009 / Accepted: 22 November 2010 Published online: 3 April 2011 – © Springer-Verlag 2011
Abstract: We define the category of B-branes in a (not necessarily affine) LandauGinzburg B-model, incorporating the notion of R-charge. Our definition is a direct generalization of the category of perfect complexes. We then consider pairs of Landau-Ginzburg B-models that arise as different GIT quotients of a vector space by a one-dimensional torus, and show that for each such pair the two categories of B-branes are quasi-equivalent. In fact we produce a whole set of quasi-equivalences indexed by the integers, and show that the resulting auto-equivalences are all spherical twists. Contents 1. 2. 3.
Introduction . . . . . . . . . . . . 1.1 A sketch proof for W = 0 . . . Landau-Ginzburg B-Models . . . . Quotients of a Vector Space by C∗ 3.1 Quasi-equivalences . . . . . . 3.2 Spherical B-branes . . . . . . 3.3 Spherical twists . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
411 414 416 421 422 426 428
1. Introduction The starting point for this paper is the celebrated result of Orlov [13] that the derived category of a Calabi-Yau hypersurface Y in projective space is equivalent to the triangulated category of graded matrix factorizations for the homogeneous polynomial f defining Y . In physicists’ language this is the statement that the categories of topological B-branes are the same in the sigma model with target Y and in the Landau-Ginzburg model with superpotential f . This is just part of a much deeper conjecture which goes back to Witten [17] (based on earlier observations by Vafa and others) and which states
412
E. Segal
that the conformal field theories associated to these two models are different limit points in the same moduli space, the so-called Stringy Kähler Moduli Space (SKMS). The B-twist of a conformal field theory is expected to be independent of your position in the SKMS, so it follows that the B-twisted theories of each model should be equivalent and in particular that their B-brane categories are the same. The basics of Witten’s idea are easy to understand, even for a mathematician. A Landau-Ginzburg model is a Kähler manifold X with a holomorphic function W called the superpotential. From such a thing one can write down a standard supersymmetric Lagrangian, analogous to the Lagrangian in classical mechanics coming from a Riemannian manifold equipped with a real-valued potential function. We consider the LG model, X = Cn+1 x1 ,...,xn , p ,
W = f (x) p,
where f is a homogeneous degree n polynomial in the xi ’s. Now we ‘gauge’ this theory, which means we try and divide by the C∗ symmetry under which each xi has weight 1 and p has weight −n. The resulting theory should be the same, at least in some limit, as the theory coming from the quotient LG model. However we should take the quotient carefully, which means take a GIT quotient, and here we have two choices. One is the total space of the canonical bundle on Pn−1 , X + = K Pn−1 , and the other is the affine orbifold X − = [Cn /Zn ]. The function W descends to either of these, so they are both LG models. Physically, there is a parameter (the complexified Fayet-Iliopoulos parameter) in the Lagrangian of the gauged theory, and the theories on X + and X − are expected to appear at two different limits of this parameter. The important thing about a LG model is the set of critical points for the function W . On X + this will be the hypersurface Y ⊂ Pn−1 . If we assume that this is smooth, then locally around Y the function W is just quadratic in the normal directions, physically these directions are then ‘massive modes’ and can be ‘integrated out’ in the low-energy theory. What this means is that we should expect the theory coming from (X + , f p) to look essentially like a theory based just on Y , with no superpotential. We conclude that the theory on Y is connected, in the moduli space of theories, to the theory on the orbifold [Cn /Zn ] with superpotential W . This is called the Calabi-Yau/Landau-Ginzburg correspondence. On the level of conformal field theories, this story is beyond the reach of current mathematical technology. However, all the spaces here are Calabi-Yau, which means the theories admit ‘B-twists’ which are topological field theories (more precisely they are Topological Conformal Field Theories/Cohomological Field Theories) and these are much more tractable. And as mentioned above, the B-twisted theories should be independent of the FI parameter, and so the result to prove is that the B-model (the B-twisted TCFT) arising from Y is the same as the B-model arising from the Landau-Ginzburg model (X − , f p). The open sector of a B-model is the category of B-branes, these are a type of boundary condition for the CFT. This should be a Calabi-Yau dg-category. For a LG model with W = 0, the category of B-branes is (a dg-enhancement of) the bounded derived category
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
413
of coherent sheaves. When W = 0 we need a generalization of this, the idea (due to Kontsevich) is to use objects that are like chain-complexes of sheaves, except that we now have d 2 = W instead of d 2 = 0. We define this category, denoted Br (X, W ) for a LG model (X, W ), in Sect. 2. In particular, the homotopy category of Br (X − , f p) is the category of matrix factorizations of f on X − = [Cn /Z n ], and the homotopy category of (Y, 0) is D b (Y ). Orlov’s result is thus the statement that H0 (Br (Y, 0)) = H0 (Br (X − , f p)), i.e. the homotopy categories of the open sectors of the two B-models are the same. In fact this goes a long way to proving that the whole of the B-models are the same, to get the full statement one would have to show that the open sectors are equivalent as Calabi-Yau dg-categories, the closed sectors should then follow by Costello’s theorem [3]. However this is not the aim of the current paper. Rather, we wanted to try to re-prove Orlov’s result following more closely the ideas in Witten’s construction, which does not appear in Orlov’s proof. In particular we wanted to see the equivalence as the composition of two equivalences: (X − , f p) ←→ (X + , f p) ←→ Y This both clarifies the result and suggests how to generalise it. For the first equivalence, we assume that the fundamental relationship between X − and X + is that they are birational, being related by a change of GIT quotient is just a special case of this. Hence we conjecture (Conjecture 2.15) that the B-models associated to any two birational CalabiYau LG models are equivalent (to be more precise, we just conjecture that their open sectors are equivalent as dg-categories). This conjecture will be obvious to experts; it is a generalization of a theorem of Bridgeland [2] and lies in the same circle of ideas as Ruan’s Crepant Resolution conjecture [14]. Unfortunately we do not get as far as addressing the second equivalence in this paper. We’ll just remark that although it looks mysterious, it is just a global version of a fairly classical result by Knörrer [9], and a closely related result is proved by Orlov [12]. What we actually manage to prove in this paper is a slight generalisation of the first equivalence, and so a small step towards our general conjecture. We show (Theorem 3.3) that if X + and X − are two different GIT quotients of a vector space V by C∗ , and W is an invariant polynomial on V , then Br (X + , W ) Br (X − , W ) are equivalent dg-categories. Our proof borrows heavily from the work of Hori, Herbst and Page [5], in which they give a detailed physical argument for a generalisation of Orlov’s result. Their key idea is a grade restriction rule. Their reasoning involves A-branes and is mathematically rather mysterious, however the rule itself will be instantly familiar to anyone who knows Beilinson’s Theorem [1]. Our improvement on their result, other than making it mathematically rigourous, is that they work only with the objects of the B-brane category whereas we include the morphisms as well (the massless open strings). We want to explain one last aspect of the physics picture. The SKMS (the space in which the FI parameters live) can be explicitly described for our examples: it is a cylinder with one puncture, and the two GIT quotient LG models live at either end of the cylinder. The category of B-branes is the same for all points in this space, however we cannot trivialise it globally, i.e. there is monodromy. Therefore to get an equivalence between the B-brane categories of X + and X − we must pick a path between the two ends of the cylinder. Up to homotopy there are Z such paths, so we should find Z such
414
E. Segal
equivalences. This is what we, and Orlov, find. By composing equivalences we get autoequivalences of the B-brane category at either end; this is the monodromy around the puncture. The puncture is the limit where the mass of a particular brane goes to zero, and the monodromy should be a Seidel-Thomas spherical twist around this brane. There is also monodromy around each end of the cylinder; this should just be given by tensoring with the O(1) line-bundle. Now we can explain the layout of the paper: In Sect. 1.1 we give a sketch of our method for the special case that W = 0. This means we don’t have to worry about LG models, we just deal with the more familiar case of derived categories of coherent sheaves. We show that we can construct Z many derived equivalences between X + and X − , and that the resulting autoequivalences are spherical twists. In Sect. 2 we explain properly what a LG B-model is, and what the category of B-branes is. In Sect. 3 we describe the class of examples we will consider. These are pairs of LG B-models (X + , W ) and (X − , W ) that are different GIT quotients of a vector space by C∗ . In Sect. 3.1 we prove our main result, that there are Z many quasi-equivalences between the categories of B-branes on (X + , W ) and (X − , W ). In Sect. 3.2 we describe the resulting auto-quasi-equivalences of the category of B-branes on (X + , W ). We show (more-or-less) that they are spherical twists. The technology of LG B-models is in its infancy, so many of the arguments of the last two sections are rather messy and ad-hoc. In particular the ‘more-or-less’ of the previous paragraph is because we do not have a proper theory of Fourier-Mukai transforms. We apologise to the reader for this unsatisfactory state-of-affairs, and hope that later treatments will clean these results up a bit. 1.1. A sketch proof for W = 0. As we will see in Sect. 2, a special case of the category of B-branes in a Landau-Ginzburg B-model is the category Perf(X ) of perfect complexes on a smooth space X , which is a dg-model for the derived category D b (X ). We thought it would be helpful to explain the proof of our results in this special case, as D b (X ) is probably more familiar than Br (X, W ). Also the proof in this case is quite simple and still contains the important points for the more general case, the hard work in generalizing is mostly technicalities. For this sketch, we’ll use the example of the standard three-fold flop. This is of course well understood and we will say nothing particularly original, but we will indicate afterwards how to generalise. Let V = C4 with co-ordinates x1 , x2 , y1 , y2 , and let C∗ act on V with weight 1 on each xi and weight −1 on each yi . There are two possible GIT quotients X + and X − , depending on whether we choose a positive or negative character of C∗ . Both are isomorphic to the total space of the bundle O(−1)⊕2 over P1 . Both are open substacks of the Artin quotient stack X = [V /C∗ ] given by the semi-stable locus for either character. Let ι± : X ± → X denote the inclusions. This stacky point of view makes it clear that there are (exact) restriction functors ι∗± : D b (X ) → D b (X ± ).
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
415
By D b (X ) we mean the derived category of the category of C∗ -equivariant sheaves on V . This contains the obvious equivariant line-bundles O(i) associated to the characters of C∗ . The unstable locus for the negative character is the set {y1 = y2 = 0} ⊂ V . Consider the Koszul resolution of the associated sky-scraper sheaf: (y1 ,y2 )
(y2 ,−y1 )
K − = O(2) −→ O(1)⊕2 −→ O. Then ι− K − is exact, it is the pull-up of the Euler sequence from P1y1 :y2 . On the other hand ι+ K − is a resolution of the sky scraper sheaf OP1x :x along the zero section. Similar 1 2 comments apply for the Koszul resolution K + of the set {x1 = x2 = 0}. Let Gt ⊂ D b (X ) be the triangulated subcategory generated by the line bundles O(t) and O(t + 1). This is the grade restriction rule of [5], we are restricting to characters lying in the ‘window’ [t, t + 1]. Claim 1.1. For any t ∈ Z, both ι∗+ and ι∗− restrict to give equivalences ∼
∼
D b (X + ) ←− Gt −→ D b (X − ). To see that these functors are fully-faithful it suffices to check what they do to the maps between the generating line-bundles, so we just need to check that Ext•X (O(t + k), O(t + l)) = Ext•X ± (O(t + k), O(t + l)) for k, l ∈ [0, 1], i.e. HX• (O(i)) = H X• ± (O(i)) for i ∈ [−1, 1], and this is easily verified. To see that they are essentially surjective we need to know that the two given line bundles generate D b (X ± ). This is essentially a corollary of Beilinson’s Theorem [1]. One way to see it is to first observe that the set {O(i), i ∈ Z} generates D b (X ± ) because X ± is quasi-projective, then use twists of the exact sequence ι± K ± repeatedly to resolve any O(i) by a complex involving only O(t) and O(t + 1). So for any t ∈ Z we have a derived equivalence ∼
t : D b (X + ) −→ D b (X − ) passing through Gt . Composing these, we get auto-equivalences ∼
b b −1 t+1 t : D (X + ) −→ D (X + ).
To see what these do, we only need to check them on the generating set of line-bundles {O(t), O(t + 1)}. Applying t to this set is easy, it just sends them to the same linebundles on X − .1 To apply −1 t+1 however, we first have to resolve O(t) in terms of O(t +1) 1 The easiest sign convention is to keep y and y as degree -1 on both sides, i.e. it’s the O(−1) bundle on 1 2 P1y1 :y2 that has global sections. Otherwise t sends O(t) to O(−t).
416
E. Segal
and O(t + 2). We do this using the exact sequence ι− K − (t). The result is that −1 t+1 t sends (−y2 ,y1 )
O(t) → [O(t + 2) −→ O(t + 1)⊕2 ] O(t + 1) → O(t + 1). Claim 1.2. −1 t+1 t is an inverse spherical twist around OP1x
1 :x2
(t).
A spherical twist is an autoequivalence discovered by [15] associated to any spherical object in the derived category, i.e. an object S such that Ext(S, S) = C ⊕ C[−n] for some n (i.e. the homology of the n-sphere). It sends any object E to the cone on the evaluation map [RHom(S, E) ⊗ S −→ E]. The inverse twist sends E to the cone on the dual evaluation map [E −→ RHom(E, S)∨ ⊗ S]. The object OP1x :x (t) ι+ K − (t) is spherical, and the inverse twist around it sends 1 2 O(t + 1) to itself and O(t) to the cone (−y2 ,y1 )
[O(t) −→ ι+ K − (t)] [O(t + 2) −→ O(t + 1)⊕2 ], which agrees with −1 t+1 t . To complete the proof of the claim we would just need to check that the two functors also agree on the Hom-sets between O(t) and O(t + 1). Now instead let V = C p+q with co-ordinates x1 , ..., x p , y1 , ..., yq . Let C∗ act linearly on V with positive weights on each xi and negative weights on each yi . The two GIT quotients X + and X − are both the total spaces of orbi-vector bundles over weighted projective spaces. We must assume the Calabi-Yau condition that C∗ acts through S L(V ). Let d be the sum of the positive weights, so the sum of the negative weights is −d. The above argument goes through word-for-word, where now Gt = O(t), . . . , O(t + d − 1) . 2. Landau-Ginzburg B-Models A Landau-Ginzburg model is a Kähler manifold X equipped with a holomorphic function W . We are only interested in the B-model on (X, W ), and this doesn’t need the metric, just the complex structure. Also we want to work in the algebraic world, so for us X will be a smooth scheme (or stack) over C. When W = 0, it is a standard slogan that the category of B-branes is the derived category D b (X ) of coherent sheaves on X . However the category of B-branes should really be a dg-category, whose homotopy category is D b (X ) (for background on dg-categories,
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
417
we recommend [16]). A good model is given by Perf(X ), the category of perfect complexes. The objects of Perf(X ) are bounded complexes of finite-rank vector bundles, and the morphisms are given by Hom(E • , F • ) = (Hom(E • , F • ) ⊗ A0,• ) (this is what we might call the ‘Dolbeaut’ version of Perf(X ), other versions are possible as we will discuss below). The differential here is a sum of the Dolbeaut differential ∂¯ and the differential on Hom(E • , F • ), which itself is the commutator with the differentials on E • and F • . The homology of this complex is Ext• (E • , F • ) = Hom D b (X ) (E • , F • ). Futhermore since X is smooth every object in D b (X ) is quasi-isomorphic to a complex in Perf(X ), so H0 (Perf(X ) = D b (X )) as required. We need to generalise this for W = 0. Kontsevich’s idea was to modify the definition of a chain-complex, replacing d 2 = 0 with d 2 = W . This doesn’t make sense on a Z-graded complex, so the usual procedure (at least in the mathematics literature) is to work instead with Z2 -graded complexes. However there is another possibility, standard in the physics literature, which is to replace the ‘homological’ grading with the notion of R-charge (strictly speaking, vector R-charge). This is a geometric action of C∗ on X , under which W must have weight 2. Then we can define a B-brane to be a C∗ equivariant vector bundle E, with an endormorphism d of R-charge 1, and the condition d 2 = W 1 E makes sense. If the C∗ action is trivial then we are forced to take W = 0, and we recover the definition of a perfect complex. Also, the definition of the morphism chain-complexes in Perf(X ) adapts easily, as we shall see. Definition 2.1. A Landau-Ginzburg B-model is the following data: • A smooth n-dimensional scheme (or stack) X over C, • A choice of function W ∈ O X (the ‘superpotential’), • An action of C∗ on X (the ‘vector R-charge’), such that 1. W has weight (‘R-charge’) equal to 2. 2. −1 ∈ C∗ acts trivially. From now on we’ll call the C∗ acting in this definition C∗R to distinguish it from other C∗ actions that will appear later. Remark 2.2. In physics terms, Axiom 2 follows from the fact that the axial R-charge symmetry is acting trivially. It implies that the sheaf of functions O X is supercommutative under the C∗R grading. We could relax it, but keep supercommutativity, by allowing X to be a superspace. Definition 2.3. A B-brane on a Landau-Ginzburg B-model (X, W ) is a finite-rank vector bundle E, equivariant with respect to C∗R , equipped with an endomorphism d E of R-charge 1 such that d E2 = W · 1 E . If we wanted to be more pretentious we could say that X is a space endowed with a sheaf of curved algebras (W is the curvature) and that a B-brane is a locally free sheaf of curved dg-modules over X . We can shift the R-charge on a B-brane E by tensoring with a line bundle associated to a character of C∗R . We denote these shifts by E[n] for n ∈ Z. This agrees with the homological shift functor in the following special case:
418
E. Segal
Example 2.4. Let W = 0 and C∗R act trivally. Then a B-brane is just a bounded complex of vector bundles. Note that since −1 ∈ C∗R acts trivially every B-brane splits as a direct sum E = E ev ⊕ E od of its Z2 -eigen-bundles, and d E exchanges these sub-bundles. There is a weaker definition of Landau-Ginzburg B-model where we keep only the trivial action of Z2 ⊂ C∗R , thus only this Z2 -grading remains. We shall make no use of this weaker definition, except for the following example. Example 2.5. Let X = Cn and W be any polynomial. This defines a LG B-model in the weak (Z2 -graded) sense. Then for a B-brane (E, d E ) both E ev and E od must be trivial bundles, so d E is given by a matrix 0 d E0 dE = d E1 0 whose square is W 1. This is a called a matrix factorization of W . We can’t in general add R-charge to this example. But we can if we orbifold it, as follows. Example 2.6. Let X = [Cnx1 ,...,xn × C∗p /C∗G ], where C∗G (the gauge group) acts with weight 1 on each xi and weight −k on p. This is equivalent as a stack to [Cn /Zk ]. Let C∗R act with weight 0 on each xi and weight 2 on p. If we pick a superpotential W = f (x) p, where f (x) is a homogeneous degree k polynomial in the xi ’s, then this defines a LG B-model (for k = n it is the orbifold phase of the Witten construction described in the Introduction). Every C∗R -equivariant vector bundle on X is the direct sum of C∗R -equivariant line-bundles; these are given by the lattice Z2 /(−k, 2). This bijects with the subset Z × [0, 1] ⊂ Z2 . This means that we can consider a B-brane on (X, W ) to be given by a pair (E 0 , E 1 ) of graded free modules over the ring C[x1 , ..., xn ], where each xi has degree 1, and graded maps d E0 : E 0 → E 1 ,
d E1 : E 1 → E 0
with d E0 d E1 = d E1 d E0 = f . This is called a graded matrix factorization. Now we want to define the morphisms between two B-branes. We will precisely mimic the construction of Perf, by first defining a homomorphism bundle and then taking derived global sections of it. Recall that a B-brane on the LG B-model (X, 0) is a C∗R -equivariant bundle E on X equipped with an endomorphism d E of R-charge 1 whose square is zero. Let dgR Vect(X ) be the category whose objects are B-branes on (X, 0) and whose morphisms are all morphisms of vector bundles. This is a dg-category, and when the C∗R action on X is trivial it is just the category dgVect(X ) of complexes of vector bundles on X . It is also a monoidal category, since we can tensor equivariant bundles and their endomorphisms in the usual way.
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
419
Now let (X, W ) be any LG B-model, and let (E, d E ), (F, d F ) be two B-branes on (X, W ). We have a C∗R -equivariant vector bundle Hom(E, F) = E ∨ ⊗ F, and this carries an endomorphism d E,F = 1∨E ⊗ d F − d E∨ ⊗ 1 F of R-charge 1. One can check that 2 d E,F =0
(the two copies of W that appear cancel each other). This means that the pair (Hom(E, F), d E,F ) is an object of dgR Vect(X ). Furthermore, given a third B-brane (G, dG ), we have composition maps Hom(E, F) ⊗ Hom(F, G) → Hom(E, G) and these are closed and of degree zero. Definition 2.7. Given an LG-model (X, W ) we define a category Br (X, W ) enriched over the category dgR Vect(X ). The objects of Br (X, W ) are the B-branes on (X, W ), and the morphisms between two branes E and F are given by (Hom(E, F), d E,F ). ∗
∗
We need to fix a monoidal functor R : Vect(X)CR → dgVectCR that sends a C∗R equivariant vector bundle to a bounded C∗R -equivariant chain-complex of vector spaces that computes its derived global sections. Since we are working with smooth spaces over C we will use Dolbeaut resolutions, i.e. we define ¯ R(E) = ((E ⊗ A0,• X ), ∂), ˇ but we could also use other models such as Cech resolutions with respect to some C∗R -invariant open cover. Now Hom(E, F) is an object in dgR Vect(X ). This means that R(Hom(E, F)) = (Hom(E, F) ⊗ A0,• X ) is a bi-complex, graded by R-charge and by Dolbeaut degree, with differential ¯ d E,F + ∂. As usual we may collapse this bi-complex to a complex. If we apply this to all pairs of branes simultaneously we get the following: Definition 2.8. Given an LG-model (X, W ) we define the dg-category of B-branes to be Br (X, W ) := R(Br (X, W )). The monoidalness of R ensures that this is indeed a category.
420
E. Segal
Example 2.9. Let W = 0 and C∗R act trivially on X . Then Br (X, 0) = Perf(X ), the category of perfect complexes. Since X is smooth the homotopy category of this is H0 (Br (X, 0)) = D b (X ). Example 2.10. Let X = [Cn /Zn ] as in Example 2.6. Then the functor means ‘take Zn -invariants’, and this is exact, so we may let R X = X . The homotopy category of Br (X, W ) is the category of graded B-branes DGr B(W ) defined by Orlov [13]. Remark 2.11. Br (X, W ) should only depend on a (Zariski) neighbourhood of the critical locus of W . This has been proved (without R-charge and on the level of homotopy categories) by Orlov [11]. Remark 2.12. As far as we are aware this definition is new in the mathematics literature, but it is almost classical in the physics literature, see e.g. [6]. Remark 2.13. We could make the definition of a B-brane more general by allowing the endomorphism d E to be derived, i.e. d E ∈ (End(E) ⊗ A0,• X ) with R-charge plus Dolbeaut degree equal to 2. Similarly we could generalize the definition of the LG B-model by allowing W to be a closed element of A0,• X . The advantage of this more general definition of B-brane is that the resulting category contains mapping cones, i.e. it is pre-triangulated. However notice that in Example 2.9 above Perf(X ) is already pre-triangulated, this leads us to suspect that at least when W ∈ O X our more restricted category of B-branes is in fact pre-triangulated as well. When X is affine this is obvious. Remark 2.14. Since the Hom sets are actually bi-complexes, and the Dolbeaut grading is bounded, we have a spectral sequence converging to the homology of R(Hom(E, F)) whose first page is (H • (Hom(E, F)), d E,F ). A map f : (X, W ) → (X , W ) of LG B-models is just a map from X to X commuting with the R-charges and such that f ∗ W = W . Assuming that the derived global sections functors R X and R X are chosen compatibly we get a dg-functor f ∗ : Br (X , W ) → Br (X, W ). Similarly a birational map between (X, W ) and (X , W ) is a birational map from X to X that commutes with R-charge and sends W to W . Conjecture 2.15. Let (X, W ) and (X , W ) be birational LG B-models, and assume that X and X are Calabi-Yau. Then there is a quasi-equivalence Br (X, W ) Br (X , W ). In the next section we prove a special case of this conjecture. As was explained in the Introduction, this is a conservative version of the real conjecture, which is that the B-models associated to (X, W ) and (X , W ) are equivalent. We state this version since it is not yet proved that the B-model exists.
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
421
3. Quotients of a Vector Space by C∗ Take a vector space V , and equip it with a linear action of C∗ , which we’ll denote by C∗G (the ‘gauge group’). We require that C∗G acts through S L(V ). We have a stack quotient X = [V /C∗G ]. There are also two possible GIT quotients of V by C∗G associated to the characters ±1 of C∗G . From the stacky point of view these are open sub-stacks ι± : X ± = [V±ss /C∗G ] → X consisting of the semi-stable loci given by either character. All of these spaces are Calabi-Yau. Now choose an action of C∗R on V that commutes with the gauge-group action. Note that both GIT quotients are then preserved by C∗R . Let W be a function on V that is invariant with respect to C∗G and has R-charge 2. Then we have three Landau-Ginzburg B-models ι−
ι+
(X + , ι∗+ W ) → (X , W ) ← (X − , ι∗− W ). ι∗+ W
(3.1)
ι∗− W
and just W . From now on we’ll abuse notation and call both Both GIT quotients are the total space of orbi-vector bundles over weighted projective space. To see this, let V = Vx ⊕ Vy ⊕ Vz be the decomposition of V into eigenspaces with positive, negative and zero C∗G weights. Then X + projects down to PVx , and it is the total space of the vector bundle associated to the graded vector space Vy ⊕ Vz . Similarly X − is the total space of Vx ⊕ Vz over PVy . For our sign conventions, it is simplest if we agree that PVy is Proj of a negatively graded ring, so that the O(−1) line bundle on PVy is the one that has global sections. If we don’t adopt this then whenever we restrict to X − we have to flip the signs of all line-bundles. Let d be the sum of the positive eigenvalues of C∗G on V , since C∗G acts through S L(V ) the sum of the negative eigenvalues is −d. We’ll make repeated use of the following fairly classical fact: Lemma 3.1 [4]. p HPVx (O(k))
⎧ ⎨ (OVx )k p = 0, k ≥ 0 = (OVx )d−k p = dimPVx , k ≤ −d ⎩ 0 otherwise,
where (OVx )k is the polynomial on Vx with C∗G -degree k. Corollary 3.2. H X0 + (O(k)) = (OV )k for all k, and p
H X + (O(k)) = 0 for p > 0 and k > −d. Proof. By adjunction and affineness of the projection X + → PVx , we have H X + (O(k)) = HPVx (S • (Vy ⊕ Vz )∨ ⊗ O(k)). p
p
422
E. Segal
3.1. Quasi-equivalences. In this section we will prove Theorem 3.3. There is a natural set of quasi-equivalences Br (X + , W ) ∼ = Br (X − , W ) parametrised by Z. The key idea of the proof of this theorem comes from [5]. Using restriction functors shown in 3.1, we will identify both Br (X + , W ) and Br (X − , W ) with one of a set of full subcategories Gt ⊂ Br (X , W ) parameterized by t ∈ Z. Note that every vector bundle on X is a direct sum of the obvious line bundles O(k), k ∈ Z. Let Gt ⊂ Br (X , W ) be the full subcategory consisting of B-branes (E, d E ) where all the summands of E come from the set O(t), . . . , O(t + d − 1). We will show that the functors ι∗± : Br (X , W ) → Br (X ± , W ) become quasi-equivalences when restricted to any of the subcategories Gt , thus proving Theorem 3.3. Recall that a dg-functor between dg-categories is a quasi-equivalence if the induced map on homotopy categories is an equivalence. This means that it must be a quasiisomorphism on Hom sets (quasi-fully-faithful) and surjective on homotopy-equivalence classes of objects (quasi-essentially-surjective). Lemma 3.4. For any t ∈ Z, both functors ι∗± : Gt → Br (X ± , W ) are quasi-fully-faithful. Proof. Obviously we need only show the proof for ι∗+ . Let (E, d E ) and (F, d F ) be any two B-branes in Gt . We get corresponding B-branes ι∗+ (E, d E ) and ι∗+ (F, d F ) on X + . Then Hom Br (X ,W ) ((E, d E ), (F, d F )) = RX (Hom(E, F)) and Hom Br (X + ,W ) (ι∗+ (E, d E ), ι∗+ (F, d F )) = R X + (ι∗+ Hom(E, F)). We wish to show that the map ι∗+ is a quasi-isomorphism between these two complexes. Recall (Remark 2.14) that the homology of both complexes can be computed by spectral sequences whose first pages are (HX• (Hom(E, F)), d E,F ) and (H X• + (ι∗+ Hom(E, F)), ι∗+ d E,F ).
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
423
On X , taking global sections just means taking C∗G -invariants, which is exact, so for any line-bundle O(k), HX• (O(k)) = HX0 (O(k)) = (OV )k , and by Corollary 3.2 this is also true on X + when k > −d. Since Hom(E, F) is a direct sum of line-bundles from the set O(1 − d), . . . , O(d − 1) the induced map ι∗+ : HX• (Hom(E, F)) → H X• + (ι∗+ Hom(E, F)) is an isomorphism between the first pages of the two spectral sequences. Hence ι∗+ is a quasi-isomorphism. We will deduce quasi-essential-surjectivity from the following lemma, which is essentially Beilinson’s Theorem [1]. Lemma 3.5. For any t ∈ Z, any C∗R -equivariant vector bundle E on X + has a finite C∗R -equivariant resolution by direct sums of shifts of line-bundles from the set O(t), . . . , O(t + d − 1). Proof. Recall that all vector bundles on X are direct sums of the character line bundles. Since X + is quasi-projective, E is a quotient of ι∗+ V for some vector bundle V on X , and we can choose this quotient to be C∗R -equivariant. Then we have a map V → ι+∗ E which is surjective on X + . Since X is smooth, the kernel of this map has a finite resolution by vector bundles, which we again may choose to be C∗R -equivariant. The restriction of this resolution to X + , together with V , give a finite C∗R -equivariant resolution of E by direct sums of character line-bundles. Thus it is sufficient to prove the lemma for the line-bundles O(k). On PVx we have the Euler exact sequence (∧• Vx∨ , ¬x) = [0 → O(−d) → · · · → O → 0] which resolves O(−d) in terms of O(−d + 1), . . . , O, and the C∗R -action on Vx means that it is C∗R -equivariant. Pull this up to X + . By repeatedly using twists of this exact sequence we see that any line-bundle O(k) has a C∗R -equivariant resolution by shifts of line bundles from the set O(t), . . . , O(t + d − 1). Lemma 3.6. For any t, both functors ι∗± : Gt → Br (X ± , W ) are quasi-essentially-surjective. Proof. Again we only show the proof for ι∗+ . Let (E, d E ) be a B-brane on (X + , W ). By Lemma 3.5 we can C∗R -equivariantly resolve E by a complex ∂E
∂E
∂E
q
E −s → · · · → E −1 → E 0 E,
424
E. Segal
where every term is a direct sum of shifts of line bundles O(k) with t ≤ k ≤ t + d − 1. If we let E − p [ p], E= p
then ∂E is an endomorphism of E with R-charge 1. We’re going to show that we can perturb ∂E to an endomorphism dE whose square is W 1E , and that the resulting B-brane (E, dE ) is homotopic to (E, d E ). To see that this proves the lemma, let Eˆ be the vector bundle on X given by the same direct sum of line-bundles as E. Then ˆ H X0 + (End(E)) = HX0 (End(E)) ˆ so we have a (see Corollary 3.2), so dE is the restriction of an endomorphism dEˆ of E, ˆ B-brane (E, dEˆ ) ∈ Gt that restricts to give (E, dE ). So every B-brane is homotopic to a B-brane lying in ι∗+ Gt , which is the statement of the lemma. As well as the R-charge, we will need to keep track of the grading on E that comes from it being a complex, let’s call this the homological grading. Of course ∂E also has homological grade 1. Now consider the complex E and the bundle E as objects in the usual derived category of sheaves on X + , which are quasi-isomorphic under the map q. The line bundles making up E have no higher Ext groups between them (Cor. 3.2 again), so we have quasi-isomorphisms H 0 (End(E)) ∼ = R H om X + (E, E) ∼ = R H om X + (E, E).
(3.2)
Here we are using the homological grading on the LHS and the Dolbeaut grading on the RHS, but the quasi-isomorphims are also equivariant with respect to R-charge. This means we can find an element D0 ∈ H 0 (End(E)) which is closed with respect to ∂E , has R-charge 1, and maps to the endomorphism d E of E, i.e. d E q = q D0 . We can use D0 to perturb the endomorphism ∂E of E. Unfortunately this does not yet make it a B-brane for (X + , W ), rather we have (∂E + D0 )2 = D02 = W 1E − [∂E , D−1 ] for some element D−1 ∈ H 0 (End(E)) which has homological grade -1 and R-charge 1. Here we write [∂E , −] to denote the supercommutator with respect to the R-charge grading; strictly speaking this is the differential on H 0 (End(E)) that comes from considering (E, 0) as a B-brane on (X + , 0) rather than as a complex of sheaves in D b (X + ), but the difference is irrelevant and the signs are more convenient this way. If we perturb further by D−1 we get 2 , (∂E + D0 + D−1 )2 = W 1E + [D0 , D−1 ] + D−1
and notice that now all the unwanted terms have homological degree at most −1. We claim we can iterate this process, and since the homological degree is bounded it will terminate. Indeed, we wish to solve (∂E + D)2 = W 1E ,
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
425
where D = D0 + D−1 + D−2 + · · · is a series of terms of decreasing homological grade and R-charge 1. The piece of this equation in homological grade −k < 0 is [∂E , D−k−1 ] + (D 2 )−k = 0. Assume that we have found D0 , .., D−k such that this equation holds in homological grades > −k. By (3.2), H 0 (End(E)) has no homology in negative degrees, so we can find D−k−1 if (D 2 )−k is closed. But [∂E , (D 2 )−k ] = [∂E , D 2 ]−k+1 =
−k+1
[[∂E , Di−1 ], D−k−i+1 ]
i=0
=
−k+1
[(D 2 )i , D−k−i+1 ]
i=0
= [D 2 , D]−k+1 = 0, so inducting on k our solution D exists. We let dE = ∂E + D so (E, dE ) is a B-brane on (X + , W ). It remains to show that it is homotopic to the brane (E, d E ). To see this we consider the dga End Br (X + ,W ) ((E, d E ) ⊕ (E, dE )) = (End(E ⊕ E) ⊗ A0,• ). This carries its usual grading (the sum of R-charge and Dolbeaut grade) and also the ¯ ∂E homological grading from E. Its differential is a sum of terms induced from d E , ∂, and the D−k , these have homological grading 0, 0, 1 and −k respectively. Thus we can filter this dga by defining F p End Br (X + ,W ) ((E, d E ) ⊕ (E, dE )) to be the sum of the bi-graded pieces that have (usual grade) − (homological grade) ≥ p, then this filtration is compatible with the differential and the algbra structure. Also the filtration is bounded, in the sense that the induced filtration on any (usual) graded subspace is bounded. This is a sufficient condition for the associated spectral sequence of dgas to converge [10]. To get page 1 of this spectral sequence we take the homology of the term of the differential which has bi-degree (1, 1), this is the term induced from ∂E . The diffential on page 1 is induced from d E , ∂¯ and D0 , and D0 was chosen so that it induced d E on ∂E -homology. So page 1 is (End(E ⊕ E) ⊗ A0,• ) = End Br (X + ,W ) ((E, d E ) ⊕ (E, d E )). This is concentrated in homological grade zero, so the spectral sequence collapses at page 2. We deduce that in the homotopy category the objects (E, dE ) and (E, d E ) are isomorphic.
426
E. Segal
3.2. Spherical B-branes. We use the same set-up as in the previous subsection, but from now on we assume that C∗G has no zero eigenvalues in V , so V = Vx ⊕ Vy are the positive and negative C∗G -eigenspaces. The zero section gives an inclusion PVx → X + , and there is an associated sky-scraper sheaf OPVx . This is a spherical object in the derived category D b (X + ). We are going to modify it so as to produce a spherical object in the category of B-branes Br (X + , W ). Under our definition a B-brane is a vector bundle, so it is supported over the whole of X + (it is ‘space-filling’). However a better definition should allow arbitrary coherent sheaves, which in particular can be supported just on subschemes. Then no modification of OPVx would be necessary, we could just equip it with the zero endomorphism, which does indeed square to W because W ≡ 0 along the zero section. We have not attempted to develop such a definition because the presence of local Ext groups makes defining the morphisms between such objects significantly more difficult. Instead we shall resolve OPVx by vector bundles, and deform the resolution. Nevertheless the resulting object does behave as if it was supported just on the zero section (Prop. 3.8).
Let ∂ yi , dy i be dual bases of Vy and Vy∨ , and yi the corresponding co-ordinates. Consider the Koszul resolution of OPVx : ∼
(∧• Vy∨ , ¬i yi ∂ yi ) −→ OPVx . We will deform the differential to make it a B-brane on (X + , W ), and show that it is still spherical. Write W as W = yi f i . (3.3) i
This is possible since W is gauge invariant, and has R-charge 2 so has no constant term. We define a B-brane on (X + , W ) by the C∗R -equivariant vector bundle S := ∧• (Vy∨ [1]) and the endomorphism d S :=
¬yi ∂ yi + ∧ f i dy i .
i
It is easy to check that d S2 = W 1 S . Proposition 3.7. The B-brane (S, d S ) is independent, up to isomorphism, of the choice of splitting (3.3)
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
427
Proof. Let W = i yi fˆi be another choice of splitting, and dˆS the correspondingto prove the lemma in the case that f i = fˆi for i > 2. In that case we have fˆ1 = f 1 + y2 g, fˆ2 = f 2 − y1 g, for some g. We have inverse isomorphisms 1 S + ∧(gdy 1 ∧ dy 2 ) : (S, d S ) → (S, dˆS ), 1 S − ∧(gdy 1 ∧ dy 2 ) : (S, dˆS ) → (S, d S ), and it is easy to check that these are closed. Let ζ : PVx → X + denote the zero section. Proposition 3.8. For any B-brane (E, d E ) on (X + , W ), the homology of Hom Br (X + ,W ) ((E, d E ), (S, d S )) can be computed from a spectral sequence whose first page is HP•Vx (ζ ∗ E ∨ ) with the differential induced from d E . Note that since W = 0 on the zero section, d E does indeed induce a differential on HP•Vx (ζ ∗ E ∨ ). Proof. The bundle S, as well as being C∗R -equivariant, is graded by the powers in the exterior algebra. Let’s call this the exterior grading, and write dS = ∂S + DS for the terms of exterior grade -1 and +1 (∂ S is the usual Koszul differential). Consider Hom Br (X + ,W ) ((E, d E ), (S, d S )) = (Hom(E, S) ⊗ A0,• ). This carries its usual grading which is the sum of R-charge and the Dolbeaut grading, and also an exterior grading induced from the grading on S. The differential has terms induced from ∂ S , D S , d E and ∂¯ having bi-degrees (1, −1), (1, 1), (1, 0) and (1, 0) respectively. We now proceed by a similar argument to the one used at the end of Lemma 3.6. Define a filtration by letting F p Hom Br (X + ,W ) ((E, d E ), (S, d S )) ⊂ Hom Br (X + ,W ) ((E, d E ), (S, d S )) be the direct sum of the bi-graded pieces whose total degree is ≥ p, then the differential preserves this filtration, and is bounded for any fixed total of the Dolbeaut grade and R-charge. Page 1 of the associated spectral sequence is given by taking the homology of the term induced from ∂ S only, so it is (Hom(E, OPVx ) ⊗ A0,• ) ∼ = RPVx (ζ ∗ E ∨ ) ¯ This is concentrated in exterior grade zero, so with differential induced from d E and ∂. this spectral sequence collapses after this page. To compute page 2, we can use a second spectral sequence (essentially the one from Remark 2.14) by remembering that the complex on page 1 is actually a bi-complex under the Dolbeaut grading and R-charge.
428
E. Segal
Corollary 3.9. (S, d S ) is either a spherical object or zero in H0 (Br (X + , W )). Proof. By Corollary 3.2, HP•Vx (ζ ∗ S ∨ ) = HP•Vx (O) ⊕ HP•Vx (O(−d)) = C ⊕ C,
where the second copy of C has some bi-degree depending on the dimensions and R-charges of Vx and Vy . Either the spectral sequence collapses at this point (which it usually will for degree reasons) and (S, d S ) is spherical, or it converges to 0 and (S, d S ) is contractible. Example 3.10 (Flop with superpotential). Let V = C4 with C∗G weights 1, 1, −1, −1, so both GIT quotients are isomorphic to O(−1)⊕2 . Let W = x1 y1 + x2 y2 (and pick any P1 compatible C∗R action). We can take (S, d S ) to be (y2 ,−y1 )
O(2) o
(x2 ,−x1 )
/
O(1)⊕2 o
(y1 ,y2 )
/
(x1 ,x2 )
O
so Hom Br (X + ,W ) ((S, d S ), (S, d S )) ∼ = RP1 (ζ ∗ S ∨ ) ∼ =0 and so (S, d S ) is contractible. In fact one would expect the whole category Br (X + , W ) in this example to be zero by Knörrer periodicity.
3.3. Spherical twists. We continue with the same class of examples as in the previous subsection. We have shown in Theorem 3.3 that for each t ∈ Z we have quasiequivalences ι∗+
ι∗−
Br (X + , W ) ←− Gt −→ Br (X − , W ). On the homotopy categories these can be inverted, so we have Z-many equivalences ∼
t : H0 (Br (X + , W )) −→ H0 (Br (X − , W )) passing through the categories H0 (Gt ), and hence we have autoequivalences −1 t+1 t of Br (X + , W ). The statement that we would like to be able to make is that −1 t+1 t is an inverse spherical twist around the spherical object (S(t), d S ), in the sense of [15]. Unfortunately such a statement would require a proper theory of Fourier-Mukai transforms for Landau-Ginzburg B-models, and we have not developed such a theory. Instead we’re going to settle for a less clean statement, which we prove below (Theorem 3.13). Recall that an inverse spherical twist on a space X is an auto-equivalence of the derived category D b (X ) that sends an object E to the cone on the natural map [E −→ RHom X (E, S)∨ ⊗ S], where S is a fixed spherical object in D b (X ). We have shown (Cor. 3.9) that we have an object (S, d S ) ∈ Br (X + , W ) that is either spherical or zero, we can twist it by O(t)
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
429
to get other B-branes S(t) that are either spherical or zero. What we’re going to do is construct, for any B-brane (E, d E ) ∈ Br (X + , W ), a suitable map
E : E → H∨ ⊗ S(t), where H is a complex such that H Hom Br (X + ,W ) (E, S(t)) and then show that −1 t+1 t sends E to the cone on E . If S(t) is spherical, this is a good approximation to showing that −1 t+1 t is a spherical twist (at least on objects). If S(t) −1 is zero, it shows that t+1 t is the identity (at least on objects). We begin with another Corollary of Proposition 3.8. Lemma 3.11. Let (E, d E ) ∈ ι∗+ Gt . Then Hom Br (X + ,W ) ((E, d E ), (S(t), d S )) ∼ = (HP0Vx (ζ ∗ E ∨ (t), d E∨ ). Proof. Hom Br (X + ,W ) (E, S(t)) = Hom Br (X + ,W ) (E(−t), S) which by Prop. 3.8 can be computed from HP•Vx (ζ ∗ E ∨ (t)). But E is a direct sum of line bundles O(k) with t ≤ k < t + d, so by Lemma 3.1, HP•Vx (ζ ∗ E ∨ (t)) = HP0Vx (ζ ∗ E ∨ (t)) = C⊕m E , where m E is the number of copies of O(t) appearing in E, and the spectral sequence collapses. Pick an (E, d E ) ∈ ι∗+ Gt . For notational convenience let us define H := (HP0Vx (ζ ∗ E ∨ (t)), d E∨ ). If we were in the special case when W = 0 and we had chosen d E = 0 then there would be a canonical map (the unit of the adjunction)
0 : E → H∨ ⊗ S(t). This map just projects E onto its O(t)⊕m E summand and then includes this as the final term of H∨ ⊗ S(t). When d E = 0 the map 0 is not closed, so we cannot take its mapping cone. We can fudge this using the following: Lemma 3.12. There is a closed map of R-charge 0,
E = 0 + 1 + ... : E → H∨ ⊗ S(t), where i has exterior grade i. Recall that the ‘exterior grade’ refers to the grading on S that comes from its underlying vector bundle being an exterior algebra.
430
E. Segal
Proof. We use the iterative technique from Lemma 3.6. Consider the complex Hom X + (E, H∨ ⊗ S(t)). This is bigraded by R-charge and exterior grade, and carries a differential d composed of terms d = d−1 + d0 + d1 of exterior grade −1, 0, and 1. The term d−1 just comes from the Koszul differential ∂ S on S. If we just take d−1 homology, the complex is acyclic except in exterior grade 0, where it is HomPVx (O(t)⊕m E , O(t)⊕m E ). We want to solve d E = 0, which in exterior grade k is d−1 k+1 = −d0 k − d1 k−1 . Suppose we have solved this for all exterior grades ≤ k. Then d−1 (−d0 k − d1 k−1 ) = d0 d−1 k + d1 d−1 k−1 + d02 k−1 = −d0 (d0 k−1 + d1 k−2 ) − d1 (d0 k−2 + d1 k−3 ) + d02 k−1 = 0. If k ≥ 1 then by acyclicity an k+1 exists. To check that an 1 exists we need to check that d0 0 is zero in d−1 -homology, which means calculating the component of it that maps O(t)⊕m E ⊂ E to O(t)⊕m E ⊂ H∨ ⊗ S(t). But this is zero, because the differential on H∨ cancels the component of d E that maps O(t)⊕m E to itself. Write (C E , dC ) for the mapping cone of E . Theorem 3.13. For any (E, d E ) ∈ ι∗+ Gt , −1 t+1 ◦ t ([(E, d E )]) [(C E , dC )] in the homotopy category of Br (X + , W ). Proof. Calculating t ([(E, d E )] is easy since (E, d E ) ∈ ι∗+ Gt , it is given by exactly the same data as (E, d E ) but considered as a brane on X − . To apply −1 t+1 to it we have to replace it with a homotopy equivalent brane that lies in ι∗− Gt+1 , which we know we can do by Lemma 3.6. In fact we can do this fairly explicitly: split E into its factors E = O(t)⊕m E ⊕ E , where E is a direct sum of line bundles from {O(t + 1), . . . , O(t + d − 1)}, then we can resolve E (recall Lemma 3.5) by the complex ∼ ¯ ⊕m E ⊕ E −→ (E, ∂E ) := S(t) E,
where S¯ is the complex ¯ ∂ ¯ ) := (∧≥1 (Vy∨ [1]), ¬i yi ∂ yi ) ( S, S
Equivalences Between GIT Quotients of Landau-Ginzburg B-Models
431
given by truncating (S, ∂ S ). Now we run the algorithm of Lemma 3.6 to get a brane (E, dE ) ∈ ι∗− Gt+1 , which we can then transport back to X + . This brane is graded by the powers of the exterior algebra; as before we call this the exterior grading. Define an exterior grading on C E by putting E in grade zero and shifting the exterior grading on H∨ ⊗ S(t) by 1 (as one usually would for a mapping cone). The differential on C E is then a sum of terms of exterior grade ≥ −1, and the term of exterior grade −1 is just the term induced from the Koszul differential ∂ S on S. Denote this term by ∂C . Then (E, ∂E ) and (C E , ∂C ) are branes on the LG model (X + , 0), and they are clearly homotopy equivalent. Indeed, (C E , ∂C ) is [O(t)⊕m E ⊕ E
( j ⊕m E ,0)
−→ (S(t)⊕m E , ∂ S )],
where j : O → S is the inclusion of O = ∧0 Vy∨ → S, and the cone on j is clearly ¯ ∂ ¯ ). This means we have maps homotopy equivalent to ( S, S f0 h1
4 CE k
*
E
y
hˆ 1
g0
forming a homotopy equivalence (with respect to ∂C and ∂E ), where f 0 and g0 have both R-charge and exterior grade 0 and h 1 and hˆ 1 have R-charge −1 and exterior grade 1. We claim we can use our iterative trick once again to perturb these maps by terms of increasing exterior grade until we get a homotopy equivalence between (C E , dC ) and (E, dE ). The argument is much the same as before: firstly observe that R H om X + ((C E ⊕ E, ∂C ⊕ ∂E ), (C E ⊕ E, ∂C ⊕ ∂E )) has homology only in exterior grade zero, because ∂C and ∂E have homology only in exterior grade zero. Secondly, let h1 0 0 f0 H1 = F0 = g0 0 0 hˆ 1 be the elements of this dga that we want to perturb, and let d = dC ⊕dE and ∂ = ∂C ⊕∂E . The equations we want to solve are [d, F] = 0, F 2 = 1 + [d, H ], which are equivalent to [∂, F] = −[(d − ∂), F], [∂, H ] = F 2 − 1 − [(d − ∂), H ], and it is easy to check that if these equations hold in exterior grade ≤ k, then the righthand-sides are closed with respect to [∂, −] so by acyclicity they can be solved in exterior grade k + 1. Acknowledgements. I’d like to thank Richard Thomas for helpful suggestions, Manfred Herbst for patiently explaining [5] to me, and the geometry department at Imperial College for sitting through some lectures on this material when it was in preliminary form. Some results closely related to those of this paper (although using the ‘derived category of singularities’ description of the category of B-branes) have been been found independently by [7] and [8, Sect. 7].
432
E. Segal
References 1. Be˘ılinson, A. A: Coherent sheaves on Pn and problems in linear algebra. Funktsional. Anal. i Prilozhen 12(3), 68–69 (1978) 2. Bridgeland, T.: Flops and derived categories. Invent. Math. 147(3), 613–632 (2002) 3. Costello, K.: Topological conformal field theories and Calabi-Yau categories. Adv. Math 210(1), 165–214 (2007) 4. Dolgachev, I.: Weighted projective varieties. In: Group actions and vector fields (Vancouver, B.C., 1981), Volume 956 of Lecture Notes in Math. Berlin: Springer, 1982, pp. 34–71 5. Herbst, M., Hori, K., Page, D.: Phases of n = 2 theories in 1+1 dimensions with boundary. http://arXiv. org/abs/0803.2045v1 [hepth], 2008 6. Herbst, M., Lazaroiu, C.-I.: Localization and traces in open-closed topological Landau-Ginzburg models. J. High Energy Phys. 0505, 044 (2005) 7. Herbst, M., Walcher, J.: On the unipotence of autoequivalences of toric complete intersection Calabi-Yau categories, 2009. http://arXiv.org.abs/0911.4595v1 [math.A6], 2009 8. Kapustin, A., Katzarkov, L., Orlov, D., Yotov, M.: Homological mirror symmetry for manifolds of general type. Cent. Eur. J. Math. 7(4), 571–605 (2009) 9. Knörrer, H.: Cohen-Macaulay modules on hypersurface singularities. I. Invent. Math 88(1), 153–164 (1987) 10. McCleary, J.: A user’s guide to spectral sequences. Volume 58 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, Second edition, 2001 11. Orlov, D. O.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.) 240–262 (2004) 12. Orlov, D. O.: Triangulated categories of singularities, and equivalences between Landau-Ginzburg models. Mat. Sb 197(12), 117–132 (2006) 13. Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. In: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Volume 270 of Progr. Math.. Boston, MA: Birkhäuser Boston Inc., 2009, pp. 503–531 14. Ruan, Y.: The cohomology ring of crepant resolutions of orbifolds. In: Gromov-Witten theory of spin curves and orbifolds. Volume 403 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2006, pp. 117–126 15. Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001) 16. Toën, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007) 17. Witten, E.: Phases of N = 2 theories in two dimensions. Nuclear Phys. B. 403(1–2), 159–222 (1993) Communicated by A. Kapustin
Commun. Math. Phys. 304, 433–457 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1233-x
Communications in
Mathematical Physics
Non-Abelian Multiple Vortices in Supersymmetric Field Theory Chang-Shou Lin1 , Yisong Yang2 1 Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan, ROC.
E-mail: [email protected]
2 Department of Mathematics, Polytechnic Institute of New York University, Brooklyn,
New York 11201, USA. E-mail: [email protected] Received: 30 December 2009 / Accepted: 4 December 2010 Published online: 2 April 2011 – © Springer-Verlag 2011
Abstract: In this paper, we consider a system of non-Abelian multiple vortex equations governing coupled SU (N ) and U (1) gauge and Higgs fields which may be embedded in a supersymmetric field theory framework. When the underlying domain is doubly periodic, we prove the existence and uniqueness of an n-vortex solution under a necessary and sufficient condition explicitly relating the domain size to the vortex number n and the Higgs boson masses. When the underlying domain is the full plane, we use a constructive approach to establish the existence and uniqueness of an n-vortex solution. 1. Introduction It is well known that the notion of vortices is important to many fields of theoretical physics. In superconductivity theory, the appearance of magnetically charged vortices marks the onset of mixed states typical in type II superconductors [2,23,24,28]. Besides, dually (i.e., both electrically and magnetically) charged vortices arise in a wide range of areas in condensed-matter physics including high-temperature superconductivity [32,36], optics [11], the Bose–Einstein condensates [27,31], the quantum Hall effect [42], and superfluids [40]. In cosmology, vortices generate topological defects known as cosmic strings [26,30,49] which give rise to useful mechanisms for matter formation in the early universe [1,14,16,22]. In the Weinberg–Salam unified theory of electromagnetic and weak interactions, the W - and Z -boson generated vortices [3–6,12] exist [10,43,44,46,54] which anti-screen magnetic field in contrast to the classical Meissner effect. More recently, non-Abelian vortices are studied in the context of supersymmetric gauge field theory [8,25] for their relevance in some fundamental problems such as quark confinement in QCD [8,9,41,48] and reconnection/collision of cosmic strings [19,29]. In the study of multivortex solutions of non-Abelian gauge field theory, the governing equations are often given as nonlinear systems of second-order elliptic equations and challenge a systematic understanding. On the other hand, these problems provide opportunities for developing new analytical techniques and methods, as witnessed by the
434
C.-S. Lin, Y. Yang
progress in the studies of electroweak vortices [10,43,54] and non-Abelian Chern– Simons vortices [33,35,38,53,54]. The purpose of the present paper is to develop an existence and uniqueness theory for the solutions to the multivortex equations derived in [18,25] in the context of supersymmetric field theory. Although there has been no rigorous construction of solutions in the literature yet due to the difficult structure of the equations, a calculation of the dimensions of the moduli space is carried out in [25] adapting the index theorem method of Weinberg [51,52] and assuming a solution exists at each topological sector. Thus, our work on existence fills this gap. In particular, our existence and uniqueness theorems for non-Abelian multiple vortex solutions provide a concrete and precise realization of the moduli space dimensionality calculation given in [25], as in the Abelian case [28,51,52]. The rest of the paper is outlined as follows. In Sect. 2, we present the formulation in [18] that gives rise to the multiple vortex equations to be studied here. In Sect. 3, we consider the vortex equations over a spatially doubly periodic domain modeling a condensed lattice structure [47]. We then derive some natural constraints as a result of the periodic setting and show how these constraints determine the quantized Abelian and non-Abelian fluxes. We also show that these constraints are natural in the sense that they do not lead to the Lagrange multiplier problem. As a by-product, we obtain a necessary condition for the existence of an n-vortex solution. In Sect. 4, we find n-vortex solutions by a constrained minimization process. We then discuss some interesting special cases. In Sect. 5, we establish our main existence and uniqueness theorem concerning doubly periodic n-vortex solutions by showing that the necessary condition obtained in Sect. 3 is also sufficient. We prove the existence of a solution by deriving suitable a priori estimates and using a degree theory approach. It is remarkable that such a necessary and sufficient condition for the existence of an n-vortex solution is explicitly expressed in terms of various physical parameters. The uniqueness of a solution then follows as a consequence of the convexity of an action functional governing the multiple vortex equations. In Sect. 6, we prove the existence of an n-vortex solution by an iterative sub- and supersolution method. Note that in general there is a lack of maximum principle in systems of equations and sub- and supersolution methods usually cannot be implemented. We then remark that the iterative method may also be applied to the doubly periodic situation studied in Secs. 3–5 if a sufficient condition guaranteeing the existence of a suitable subsolution holds. Moreover, we note the exponential decay properties of the solutions and apply these properties to show that the Abelian and non-Abelian fluxes assume the same quantized values as those in the doubly periodic situation. As in the doubly periodic situation, the uniqueness of the solution constructed follows from the convexity of an action functional. In Sect. 7, we draw our conclusions of the paper. 2. Field-Theoretical Formulation Following [18], let Wμ and wμ be gauge fields given over the groups U (N ) and U (1) respectively. Taking the fundamental representation, the N Higgs fields are given by an N by N complex matrix H . The Lagrangian density is defined by
with
L = K − V,
(2.1)
1 1 K = Tr − 2 (Fμν )2 + Dμ H Dμ H † − 2 ( f μν )2 , 2g 4e
(2.2)
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
V=
g2 e2 Tr(H H † 2 ) + (Tr(H H † − c1 N ))2 , 4 2
435
(2.3)
where X = X − Tr(X )1 N stands for the traceless part of an N by N matrix X , the gauge-covariant derivatives are defined by Dμ H = ∂μ H + iWμ H + iwμ H = ∂μ H + iWμ H + iwμ 1 N H,
(2.4)
the non-Abelian and Abelian gauge field curvatures are given by Fμν = ∂μ Wν − ∂ν Wμ + i[Wμ , Wν ],
f μν = ∂μ wν − ∂ν wμ ,
(2.5)
and g, e, c > 0 are coupling parameters. The non-Abelian and Abelian gauge fields are both massive with the mass spectra √ √ (2.6) m g = g c, m e = e 2N c, respectively, and the mass ratio γ given by mg g = √ , γ = me e 2N
(2.7)
which will be seen to be an important quantity in our variational solution of the problem. We are interested in multiple vortex solutions which depend on the x 1 and x 2 coordinates, and the gauge fields wμ and Wμ are nontrivial in their μ = 1, 2 components only. It is shown in [18] that energy-minimizing solutions of the equations of motion of L are governed by the following Bogomol’nyi–Prasad–Sommerfield (BPS) self-dual [13,39] system of equations D1 H + iD2 H = 0, m 2g H H † , F12 = 2c m2 f 12 = e Tr(H H † − c1 N ). 2c
(2.8) (2.9) (2.10)
(It should be noted that the coupling parameters appearing in the Lagrange density defined by (2.1)–(2.3) are taken in a such a way in order to achieve the above BPS reduction.) After some algebra [18,20], the system (2.8)–(2.10), away from the vortex points, may be recast in terms of two real-valued functions ψe and ψg into the form ψg m 2e −ψe −ψg N −1 e + [N − 1]e (2.11) ψe + e = m 2e , N ψg m 2g (N − 1) −ψ (2.12) e e e−ψg − e N −1 = 0. ψg + N The system is also studied in [25] at the critical coupling m e = m g or γ = 1. Following [18], if we prescribe point vortices with unit vortex number at p1 , . . . , pn ∈ , then N ψg + ln |x − p j |2 N −1 are smooth functions for x ∈ R2 near p j , j = 1, . . . , n. N ψe + ln |x − p j |2 and
(2.13)
436
C.-S. Lin, Y. Yang
For vortices with multiple vortex numbers, an extended representation may straightforwardly be obtained. In view of (2.13), we can set − N ψe = u 1 , −
N ψg = u 2 , N −1
(2.14)
in (2.11) and (2.12) to arrive at the governing equations: n u 1 (N −1) u1 u2 u 1 = −N m 2e + m 2e e N + N u 2 + [N − 1]e N − N + 4π δ p j (x), (2.15) j=1 n u 1 (N −1) u1 u2 u 2 = m 2g e N + N u 2 − e N − N + 4π δ p j (x).
(2.16)
j=1
3. Periodic Vortices and Natural Constraints As in [4,5,10,17,43], we first consider a compact setting in which the vortices are generated over a doubly periodic domain . Physically, this situation amounts to imposing gauge-periodicity for field configurations, initially formulated conceptually by ’t Hooft in [47], and realized concretely in the context of the Ginzburg–Landau vortices in [50]. In the present context, such a formulation renders the periodicity of the functions u 1 and u 2 in Eqs. (2.15) and (2.16), modulo , which gives rise to the situation to be studied below. Within the context of the above described doubly periodic situation, we introduce [7] a source function u 0 satisfying 4π n + 4π δ p j (x), x ∈ ; u 0 ≤ 0, || n
u 0 = −
(3.1)
j=1
and set u 1 = u 0 + v1 , u 2 = u 0 + v2 .
(3.2)
Then (2.15)–(2.16) take the following regularized form: v1 (N −1) v1 v2 4π n + m 2e eu 0 + N + N v2 + [N − 1]e N − N , || v1 (N −1) v1 v2 4π n v2 = + m 2g eu 0 + N + N v2 − e N − N . || v1 = −N m 2e +
(3.3) (3.4)
Formally, (3.3)–(3.4) are the Euler–Lagrange equations of the action functional v1 (N −1) 1 (N − 1) |∇v1 |2 + |∇v2 |2 + N eu 0 + N + N v2 I (v1 , v2 ) = 2 2 2m g 2m e v1 v2 4π n 4π n(N − 1) − v1 + v2 dx. (3.5) +N (N − 1)e N N − N − 2 m e || m 2g ||
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
437
However, there are natural integral constraints resulting from integrating Eqs. (3.3)– (3.4): v1 (N −1) v1 v2 4π n eu 0 + N + N v2 + [N − 1]e N − N dx = N || − 2 , (3.6) J1 (v1 , v2 ) ≡ me v1 (N −1) v1 v2 4π n eu 0 + N + N v2 − e N − N dx = − 2 . (3.7) J2 (v1 , v2 ) ≡ mg From (3.6)–(3.7), we have v1 (N −1) 4π n 4π n(N − 1) N eu 0 + N + N v2 dx = N || − 2 − , m m 2g e v1 v2 4π n 4π n −N N N e dx = N || − 2 + 2 . m mg e
(3.8) (3.9)
In view of (3.6), (3.8), and (3.9), we arrive at the necessary conditions for the existence of an n-vortex solution as 4π n (3.10) N || > 2 , me 4π n 4π n(N − 1) , (3.11) N || > 2 + me m 2g 4π n 4π n N || > 2 − 2 . (3.12) me mg It is obvious to note that (3.11) implies (3.10) and (3.12). Therefore, these necessary conditions are actually summarized into the single condition (3.11). We now study whether the constraints (3.6) and (3.7) give rise to the so-called “constraints problem” due to the issue of the Lagrange multipliers. For this purpose, let (v1 , v2 ) be a critical point of I subject to the constraints (3.6) and (3.7). Then there are real numbers σ1 and σ2 such that (d1 I + σ1 d1 J1 + σ2 d1 J2 )(v1 , v2 ) = 0, (d2 I + σ1 d2 J1 + σ2 d2 J2 )(v1 , v2 ) = 0, (3.13) where d1 and d2 denote the Fréchet differentiation with respect to the first and second arguments, respectively. Concretely, (3.13) gives us the expressions
v v1 v2 1 4π n u 0 + N1 + (NN−1) v2 −N N w1 dx w1 − N − 2 ∇v1 · ∇w1 + e + [N − 1]e 2 m e || me v1 (N −1) v1 v2 σ1 + eu 0 + N + N v2 + [N − 1]e N − N w1 dx N v1 (N −1) v1 v2 σ2 + eu 0 + N + N v2 − e N − N w1 dx = 0, (3.14) N v v1 v2 1 4π n u 0 + N1 + (NN−1) v2 −N N w2 dx w2 + 2 ∇v2 · ∇w2 + e −e 2 m g || mg v1 (N −1) v1 v2 σ1 eu 0 + N + N v2 − e N − N w2 dx + N v1 (N −1) v1 v2 1 σ2 e u 0 + N + N v2 + e N − N w2 dx = 0, (3.15) + N N −1
438
C.-S. Lin, Y. Yang
where w1 , w2 are arbitrary (doubly-periodic) trial functions. Setting w1 = w2 ≡ 1 in (3.14) and (3.15) and using the constraints (3.6) and (3.7), we get v1 (N −1) v1 v2 eu 0 + N + N v2 + [N − 1]e N − N dx σ1 v1 (N −1) v1 v2 + σ2 eu 0 + N + N v2 − e N − N dx = 0, (3.16) v1 (N −1) v1 v2 σ1 eu 0 + N + N v2 − e N − N dx v1 (N −1) v1 v2 1 e u 0 + N + N v2 + (3.17) e N − N dx = 0. + σ2 N −1 Multiplying (3.17) by (N − 1) and adding the result into (3.16), we obtain σ1 + σ2 = 0. On the other hand, inserting (3.6) and (3.7) into (3.16), we have 4π n 4π n σ1 N || − 2 = σ2 2 , (3.18) me mg which, by virtue of (3.10), implies that σ1 and σ2 cannot have opposite signs. Consequently, we must have σ1 = σ2 = 0, which indicates that all the terms in (3.13) or (3.14) and (3.15) arising from the Lagrange multipliers are automatically absent. We can now compute the fluxes generated from the Abelian and non-Abelian gauge field curvatures. Following [18] and using (2.14) and (3.2), we see that the Abelian and non-Abelian gauge field curvatures f 12 and F12 are given respectively by ψg m 2e −ψe −ψg m2 + [N − 1]e N −1 − e e e 2N 2 2 v1 (N −1) v1 v2 m2 m (3.19) = e eu 0 + N + N v2 + [N − 1]e N − N − e , 2N 2 ψg (N − 1) 2 −ψe −ψg F12 = e mge − e N −1 T 2N v1 v2 (N − 1) 2 u 0 + v1 + (N −1) v2 = mg e N N (3.20) − e N − N T, 2N where T = diag 1, − N 1−1 , . . . , − N 1−1 . Therefore, in view of the right-hand sides of (3.6) and (3.7), we can integrate (3.19) and (3.20) to obtain the Abelian and non-Abelian fluxes, respectively, of an n-vortex solution to be 1 (3.21) Abelian = f 12 dx = −2π n , N (N − 1) non-Abelian = T. (3.22) F12 dx = −2π n N f 12 =
Thus, roughly speaking, the Abelian and non-Abelian fluxes of an n-vortex solution are 1/N and (N − 1)/N of the flux of the Abrikosov–Nielsen–Olesen n-vortex solution [2,28,37,50], as stated in [18].
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
439
4. Solution by Constrained Minimization We assume throughout this section that the condition (3.11) holds. In Sect. 3, we have seen that a solution of (3.3)–(3.4) may be obtained by solving the constrained minimization problem η = inf {I (v1 , v2 ) | (v1 , v2 ) satisfies the constraints (3.6) and (3.7)} .
(4.1)
On the other hand, using (3.8) and (3.9), we can simplify (3.5) into the form
1 (N − 1) 2 2 I (v1 , v2 ) = |∇v1 | + |∇v2 | dx 2 2m 2g 2m e 4π n(N − 1) 4π n 4π n v 2 + N N || − 2 , − N || − 2 v 1 + me m 2g me
(4.2)
where and in the sequel we use f to denote the average value of a function f over , f =
1 ||
f dx.
(4.3)
We will carry out our minimization procedure over the standard Sobolev space W 1,2 (). We recall the Trudinger–Moser inequality [7,21]
ev dx ≤ C exp
1 16π
|∇v|2 dx , v ∈ W 1,2 (), v dx = 0.
(4.4)
For v1 , v2 satisfying (3.6) and (3.7), we make the decomposition v1 = v 1 + v1 , v2 = v 2 + v2 so that v 1 , v 2 ∈ R and
v1 dx =
v2 dx = 0.
(4.5)
We rewrite (3.8) and (3.9) as e
v 1 (N −1) N + N v2
v1
(N −1)
eu 0 + N + N v2 dx = C1 > 0, v v v1 v2 1 2 eN−N e N − N dx = C2 > 0.
(4.6) (4.7)
From these we obtain v v v1 (N −1) 1 2 v 1 = ln C1 C2N −1 − ln eu 0 + N + N v2 dx − (N − 1) ln e N − N dx , (4.8) v v v1 (N −1) C1 1 2 − ln eu 0 + N + N v2 dx + ln e N − N dx . (4.9) v 2 = ln C2
440
C.-S. Lin, Y. Yang
Inserting (4.8) and (4.9) into (4.2), we have 1 (N − 1) 2 2 I (v1 , v2 ) = |∇v1 | + |∇v2 | dx 2 2m 2g 2m e v v v 4π n 1− 2 u 0 + N1 + (NN−1) v2 N N ln e dx + (N − 1) ln e dx + N || − 2 me v v v 4π n(N − 1) 1 2 u 0 + N1 + (NN−1) v2 N − N dx ln e − ln e dx + m 2g 4π n(N − 1) C1 4π n 4π n . (4.10) ln + N N || − − N || − 2 ln C1 C2N −1 + me m 2g C2 m 2e Applying Jensen’s inequality of the form 1 1 f e dx ≥ exp f dx , || ||
(4.11)
we obtain from (4.5) the lower bounds v v v 1 1 2 u 0 + N1 + (NN−1) v2 e dx ≥ || exp u 0 dx , e N − N dx ≥ ||. (4.12) || Hence, in view of (3.10), (4.10), and (4.12), we arrive at the inequality (N − 1) 1 I (v1 , v2 ) ≥ |∇v1 |2 + |∇v2 |2 dx 2 2m 2g 2m e v 4π n(N − 1) u 0 + N1 + (NN−1) v2 ln e dx − C3 , − m 2g
(4.13)
where C3 > 0 is a constant independent of v1 and v2 . On the other hand, using the assumption u 0 ≤ 0 (see (3.1)) and (4.4), we have v v 1 (N −1) u 0 + N1 + (NN−1) v2 e dx ≤ e N + N v2 dx
2
∇v 1 (N − 1) 1
+ ∇v2
dx ≤ C exp
16π N N (1 + ε) 1 (N − 1)2 2 −1 2 dx , (4.14) ≤ C exp |∇v | + (1 + ε )|∇v | 1 2 16π N2 N2 where ε > 0 is an arbitrary interpolation parameter. Applying (4.14) in (4.13), we obtain 1 1 n(N − 1)(1 + ε) I (v1 , v2 ) ≥ − |∇v1 |2 dx 2 m 2e 2N 2 m 2g n(N − 1)2 (1 + ε−1 ) (N − 1) 1 − + |∇v2 |2 dx − C4 2m 2g 2N 2 ≡ δ1 (ε) |∇v1 |2 dx + δ2 (ε) |∇v2 |2 dx − C4 , (4.15)
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
441
where C4 > 0 is an irrelevant constant. To ensure the two quantities δ1 (ε) and δ2 (ε) to stay positive, we require n(1 + ε) < n(1 + ε−1 ) <
2N 2 m 2g (N − 1)m 2e 2N 2 (N − 1)2
=
2N 2 γ 2, (N − 1)
.
(4.16) (4.17)
The conditions (4.16) and (4.17) are sufficient for the existence of an n-vortex solution which may be obtained by solving the minimization problem (4.1) as follows. Applying (4.16) and (4.17) in (4.15), we see that the action I is bounded from below over the field configurations taken from the Sobolev space W 1,2 () (say). Thus η in (4.1) is well defined. Let {vk,1 , vk,2 } be a minimizing sequence of (4.1). In view of (4.15) , v } is a bounded sequence in W 1,2 (). and the Poincaré inequality, we see that {vk,1 k,2 , v } is weakly convergent to an Without loss of generality, we may assume that {vk,1 k,2 1,2 element (V1 , V2 ) (say) in W (). Using the compact embedding W 1,2 () → L p () ( p ≥ 1) and (4.4), we see that the functionals defined by the left-hand sides of (3.8) and (3.9) are weakly continuous over W 1,2 (). Therefore, inserting v1 = vk,1 , v2 = vk,2 in (4.8) and (4.9) and letting k → ∞, we see that V1 , V2 and V 1 = lim v 1 , V 2 = lim v 2 k→∞
k→∞
(4.18)
satisfy Eqs. (3.10) and (3.11) as well. Consequently, (V1 , V2 ), where V1 = V1 +V 1 , V2 = V2 + V 2 , is a solution to (4.1) in the natural function space W 1,2 (). Below are two interesting special cases concerning an n-vortex solution realized as a minimizer of (4.1). (a) N = 2 (this is the important U (2) non-Abelian gauge group situation). Then the conditions (4.16) and (4.17) become n(1 + ε) <
8m 2g m 2e
n(1 + ε−1 ) < 8.
,
(4.19) (4.20)
We are interested in getting the largest possible number of vortices under a suitable mass ratio γ = m g /m e . From (4.20), we see that such largest number is n = 7 which imposes the condition ε > 7. In view of (4.19), we have n < m 2g /m 2e = γ 2 when ε is close to 7. Hence we are led to the condition 2 mg γ2 = > 7. (4.21) me In other words, when (4.21) holds, there are vortices with vortex number n up to 7. (b) n = 1 and N ≥ 2. That is, we are interested in the existence of, at least, a single vortex solution in the non-Abelian model. In this situation, (4.17) can always be satisfied with a suitable choice of ε > 0, ε>
(N − 1)2 . N 2 + 2N − 1
(4.22)
442
C.-S. Lin, Y. Yang
In view of (4.22) and (4.16) (with n = 1), we arrive at 2 mg N −1 γ2 = ≡ r (N ). > 2 me (N + 2N − 1)
(4.23)
The function r (N ) defined by the right-hand side of (4.23) decreases when N ≥ 3 with r (2) = 1/7 and r (3) = 2/15. Thus the maximum of r (N ) (N ≥ 2) is r (2) = 1/7 and (4.23) is guaranteed for arbitrary N ≥ 2 if γ =
mg 1 >√ . me 7
(4.24)
√ In other words, when 7m g > m e , for any U (N ) (N ≥ 2) gauge group situation, an action I -minimizing n = 1 vortex solution exists. More generally, for an n-vortex situation, we may arrive at the condition γ2 =
m 2g m 2e
>
n(N − 1) . (2 − n)N 2 + 2n N − n
(4.25)
We note that, unlike (3.11), the conditions derived above are all independent of the underlying periodic spatial domain size ||. 5. Existence Theorem for Spatially Periodic Vortices In this section, we develop an existence theory for multiple vortices over the periodic domain discussed in the previous two sections. Our main result shows that (3.11) is also a sufficient condition for the existence of solutions. More precisely, we state Theorem 5.1. Equations (2.15)–(2.16) possess a solution if and only if (3.11) holds, or || >
4π n 4π n(N − 1) + . N m 2e N m 2g
Moreover, when a solution of Eqs. (2.15)–(2.16) exists, it must be unique. Technically, the existence of doubly periodic vortices in field-theoretical models [10,17,34,38,43,45,46,54] is a much more difficult issue than the existence of planar vortices due to the presence of the integral constraints and often challenges a complete understanding. Our existence and uniqueness theorem here may be regarded as a rare case as in the classical Abelian Higgs model [50] when a complete and explicit characterization for the existence of an n-vortex solution can be spelled out. For convenience, we set v = v1 , w = v2 , λ1 = m 2e , λ2 = m 2g in (3.3)–(3.4). Then v, w satisfy 4π n v N −1 v−w , (5.1) v = λ1 eu 0 e N + N w + (N − 1)e N − N + || 4π n v N −1 v−w w = λ2 eu 0 e N + N w − e N + . (5.2) || We have the following a priori bounds for a solution pair of (5.1)–(5.2).
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
443
Lemma 5.2. Let (v, w) be a solution of (5.1)–(5.2). Then we have the bounds u 0 + v ≤ 0, u 0 + w ≤ 0, v − w ≤
N . N −1
Proof. For a proof of the fact that u 0 + v ≤ 0, u 0 + w ≤ 0, see the proof of Lemma 6.1. Furthermore, the proof of Lemma 6.2 establishes N N ≤ . v − w ≤ N ln N −1 N −1 To prove Theorem 5.1, we need the following lemma concerning the case when blow-up occurs. Lemma 5.3. Let {(vk , wk )} be a sequence of solutions to Eqs. (5.1)–(5.2) with λ1 = λ1,k and λ2 = λ2,k . Suppose that λ1,k → λ1 , λ2,k → λ2 , and sup{|vk (x)| + |wk (x)| | x ∈ } → ∞, as k → ∞. Then λ1 and λ2 satisfy || =
4π n 4π(N − 1) + . N λ1 N λ2
(5.3)
Proof. We use the notation (4.3) and denote the right-hand of (5.1)–(5.2) by f k and gk , respectively, v (x) vk −wk 4π n u 0 kN + NN−1 wk (x) , vk = f k ≡ λ1,k e e + (N − 1)e N − N + || vk N −1 vk −wk 4π n wk = gk ≡ λ2,k eu 0 e N + N wk − e N . + || By Lemma 5.2 and theory of linear elliptic partial differential equations, there are positive constants Mα and Mα such that
vk − v k C 1,α () ≤ Mα f k L ∞ () ≤ Mα ,
(5.4)
wk − w k C 1,α () ≤ Mα gk L ∞ () ≤ Mα ,
(5.5)
and
for α ∈ (0, 1). In particular,
∇vk L ∞ () + ∇wk L ∞ () ≤ Mα . Now suppose sup{|vk (x)| | x ∈ } → ∞. Since vk (x) + u 0 (x) ≤ 0, we have u 0 (x) dx. vk ≤ −
By (5.4), v k → −∞ and vk (x) → −∞ uniformly on as k → ∞. By Lemma 5.2, vk (x) − wk (x) ≤ N /(N − 1). Hence, v k − wk ≤
N . N −1
444
C.-S. Lin, Y. Yang
Suppose lim inf k→∞ (v k − w k ) = −∞. Passing to a subsequence if necessary, we may assume limk→∞ (v k − w k ) = −∞. Then, by (5.4) and (5.5), we have vk (x) − wk (x) → −∞ uniformly on as k → ∞. Thus, f k (x) → −λ1 N +
4π n uniformly on as k → ∞. ||
By (5.4), a subsequence of {vk − v k } (still denoted by {vk − v k }) converges to V (say) by the estimates of linear elliptic equations, and V satisfies V = −λ1 N +
4π n on . ||
(5.6)
Integrating (5.6) leads to λ1 N || = 4π n.
(5.7)
On the other hand, by (3.11), we have || >
4π n 4π(N − 1) + . N λ1,k N λ2,k
Thus || > λ4π1 Nn , which contradicts (5.7). Therefore, we conclude that {v k − wk } is a bounded sequence. Thus, wk → −∞ as k → ∞, which implies wk (x) → −∞ uniformly on as k → ∞. By passing to a subsequence, we may assume vk − v k → V and wk − wk → W in C 2 (), v k − w k → σ in R, as k → ∞. Then (V, W, σ ) satisfies 4π n V −W +σ , V = λ1 (N − 1)e N − N + || V −W +σ 4π n − λ2 e N . W = || By integrating both equations, it yields V −W +σ 4π n N || − , e N dx = N − 1 λ1 (N − 1) V −W +σ 4π n e N dx = . λ2 Combining the above two quantities, we have || =
4π n 4π n(N − 1) + , λ1 N λ2 N
which is (5.3). Hence, the proof of Lemma 5.3 is finished.
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
445
We will apply Lemma 5.3 and the degree theory to prove Theorem 5.1. To start with, we consider the case λ1 = λ2 . Lemma 5.4. Suppose λ1 = λ2 . Then Eqs. (5.1)–(5.2) possess a unique solution if and n only if || > 4π λ1 , which is exactly the condition (3.11) when the g-mass and e-mass are identical. Furthermore, the topological degree for Eqs. (5.1)–(5.2) is equal to 1. Proof. Since λ1 = λ2 , we obtain from (5.1)–(5.2) the equation (v − w) = λ1 N (e
v−w N
− 1).
By the maximum principle, it is easy to get v ≡ w. Thus, v satisfies v = λ1 (eu 0 +v − 1) +
4π n . ||
(5.8)
n It is well-known [50] that (5.8) possesses a solution if and only if || > 4π λ1 . Furthermore, the solution is unique with the Morse index 0. Thus, the topological degree = (−1)Morse index = 1.
Proof of Theorem 5.1. By Lemma 5.3, under the condition || >
4π n 4π n(N − 1) + , N λ1 N λ2
solutions of (5.1) and (5.2) are uniformly bounded on . Thus, the Leray–Schauder topological degree can be defined. In the following, we explain how to apply the topological theory to our equations. Define C˙ α () = {u ∈ C α () | u = 0}, 0 < α < 1, and set X = C˙ α () × C˙ α (). Choose a, b ∈ R such that f (x, v(x) + a, w(x) + b) dx = 0, (5.9) g(x, v(x) + a, w(x) + b) dx = 0, (5.10)
where 4π n v N −1 v−w , f (x, v, w) = λ1 eu 0 (x) e N + N w + (N − 1)e N − N + || 4π n v N −1 v−w g(x, v, w) = λ2 eu 0 (x) e N + N w − e N + . || It is easy to see a and b can be uniquely determined whenever v and w are given. For (v, w) ∈ X , define T (v, w) = (V, W ) ∈ X , where (V, W ) ∈ X is the unique solution of the system V = f (x, v + a, w + b), W = g(x, v + a, w + b),
446
C.-S. Lin, Y. Yang
so that a fixed point of T is a solution of Eqs. (5.1) and (5.2). Besides, elliptic theory may be used to show that T is completely continuous. Moreover, by Lemma 5.3, there is a constant M > 0 such that for any fixed point (v, w) of T in X we have
v C α () + w C α () ≤ M. Therefore, we can define the Leray-Schauder degree d(λ1 , λ2 ) for T . Clearly, by Lemma 5.4, we have d(λ, λ) = 1 for λ large. Thus, by the homotopy invariant of the 4π n(N −1) n d(λ1 , λ2 ), and the fact that the set {(λ1 , λ2 )||| > 4π } is path-connected, N λ1 + N λ2 we have d(λ1 , λ2 ) = d(λ, λ) = 1. The existence of solutions to Eqs. (5.1)–(5.2) follows from the non-vanishing of the Leray–Schauder topological degree. Furthermore, the convexity of the functional (3.5) indicates that (3.5) can at most have one critical point. Thus the uniqueness of a solution of Eqs. (2.15)–(2.16) follows. This completes the proof of Theorem 5.1. In fact, to establish the existence proof of Theorem 5.1 here, it suffices to apply a fixed-point theorem argument directly and it is not necessary to use degree theory. However, since a degree theory description provides more information about the solution, we give it above. 6. Construction of Planar Vortices We now consider the vortex equations (2.15)–(2.16) over the full plane R2 subject to the topological boundary value condition u 1 (x) → 0, u 2 (x) → 0 as |x| → ∞.
(6.1)
First, we state Lemma 6.1. Suppose (u 1 , u 2 ) is a solution of (2.15)–(2.16) over R2 subject to the boundary condition (6.1). Then u 1 ≤ 0 and u 2 ≤ 0. Proof. We first prove u 2 ≤ 0 in R2 . Assume u 2 > 0 at some point in R2 . Due to (6.1) and u 2 (x) → −∞ as x → p j , u 2 must attain its maximum at some (say) x2 ∈ R2 , say, at x2 ∈ R2 , such that u 2 (x2 ) = max{u 2 (x) | x ∈ R2 } > 0. Applying the maximum principle, we obtain by (2.16) e
N −1 N u 2 (x 2 )
1
≤ e− N u 2 (x2 ) .
Obviously, the inequality implies u 2 (x2 ) ≤ 0, which yields a contradiction. So, we have proved u 2 (x) ≤ 0 in R2 . The case u 1 (x) ≤ 0 in R2 can be proved similarly. Suppose otherwise that there is some x1 ∈ R2 such that u 1 (x1 ) = max{u 1 (x) | x ∈ R2 } > 0. Then the maximum principle applied to (2.15) yields e
u 1 (x1 ) N −1 N + N u 2 (x 1 )
+ (N − 1)e
u 1 (x1 ) u 2 (x1 ) N − N
≤ N.
(6.2)
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
447
An elementary calculation shows that e
N −1 N t
t
+ (N − 1)e− N ≥ N if t ≤ 0.
Thus we have in view of (6.2) that u 1 (x1 ) u 1 (x1 ) N −1 u 2 (x1 ) Ne N ≤ e N e N u 2 (x1 ) + (N − 1)e− N ≤ N, which implies u 1 (x1 ) ≤ 0, and it yields a contradiction. Lemma 6.2. Suppose (u 1 , u 2 ) is a solution of Eqs. (2.15)–(2.16) over R2 subject to (6.1). Then u 1 N −1 u1 u2 w(x) ≡ e N + N u 2 + (N − 1)e N − N (x) ≤ N , x ∈ R2 . Proof. Note that lim|x|→+∞ w(x) = N . Assume w(x0 ) = max{w(x) | x ∈ R2 } > N .
(6.3)
By the maximum principle, we have at x0 the inequality 0 ≥ w
2
u1 N − 1 u1 N − 1
+ u 2 + ∇ + u 2
=e N N N N
u1 u2 u1 u1 u2 u 2
2
−N N +(N − 1)e − + ∇ −
N N N N u u 1 N −1 u1 u2 u1 N − 1 u2 1 . ≥ e N + N u2 + u 2 + (N − 1)e N − N − N N N N u 1 N −1 N + N u2
(6.4)
We consider x0 ∈ { p1 , . . . , pn } first. By using Eqs. (2.15) and (2.16), we have 2 u 1 N −1 m e (N − 1)m 2g u1 N − 1 + u2 = + e N + N u2 N N N N u1 u2 N −1 2 N −1 2 me − m g e N − N − m 2e , + N N and u2 − = N N u
1
m 2g m 2e − N N
e
u 1 N −1 N + N u2
+
N − 1 2 m 2g me + N N
u1
u2
e N − N − m 2e . (6.5)
The equation above and (6.4) lead us at x0 to the inequality 0 ≥ w u 1 u 2 u 1 N −1 u 1 u 2 2 m 2 u 1 N −1 ≥ e e N + N u 2 + (N − 1)e N − N e N + N u 2 + (N − 1)e N − N N u 1 N −1 u 1 u 2 2 u1 u2 N − 1 2 u 1 + N −1 u 2 mg e N N + −eN−N − m 2e e N + N u 2 + (N − 1)e N − N N m 2e w(w − N ) ≥ N > 0, which is a contradiction. Therefore x0 = p j for some j = 1, 2, . . . , n.
448
C.-S. Lin, Y. Yang
We note that u 1 − u 2 has no singularity at p j . For x = p j , u 1 N −1 u1 u2 w(x) = e N + N u 2 + (N − 1)e N − N (x) ≤ w( p j ) 1
= (N − 1)e N (u 1 −u 2 )( p j ) . Thus u 1 (x) − u 2 (x) ≤ (u 1 − u 2 )( p j ), i.e., p j is also a maximum point of u 1 − u 2 . By (6.5) and by the maximum principle, we see that u u2 1 − (pj) 0≥ N N u1 ( p j ) u2 ( p j ) N − 1 2 m 2g = me + e N − N − m 2e N N u1 ( p j ) u2 ( p j ) m 2e − N N (N − 1)e > −N . N Hence w( p j ) = (N − 1)e
u1 ( p j ) u2 ( p j ) N − N
< N,
which again yields a contradiction to (6.3). Therefore, Lemma 6.2 is proved. By Lemma 6.1, u 1 = 0 and u 2 = 0 is a supersolution to Eqs. (2.15) and (2.16) over R2 . Thus we may be able to apply a monotone iterative scheme to solve Eqs. (2.15)– (2.16) subject to (6.1). We remark that in general, such a scheme cannot work for a system of equations even when there is a supersolution. For Eqs. (2.15) and (2.16), we note that both the nonlinearities in Eqs. (2.15) and (2.16) satisfy u1 u2 u2 N − 1 u 1 N −1 u 2 ∂ u 1 + N −1 u 2 eN e N eN N + (N − 1)e N − N = − e− N < 0, ∂u 2 N
and u1 u2 u2 ∂ u 1 + N −1 u 2 1 u 1 N −1 eN N − e N − N = e N e N u 2 − e− N < 0, ∂u 1 N
whenever u 1 < 0 and u 2 < 0. Lemma 6.3. Let w be the unique solution of w
w = λ(e − 1) + 4π
n
δ p j in R2 , w(x) → 0 as |x| → ∞,
(6.6)
j=1
where λ = min(N m 2e , m 2g ). Then (w, w) is a subsolution to Eqs. (2.15) and (2.16).
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
449
Proof. The existence and uniqueness of the solution w to (6.6) has been established in [28]. Let w1 = w2 = w. Since w j (x) < 0 in R2 , j = 1, 2, we have w1
w1 = λ(e
− 1) + 4π
n
δpj
j=1
≥
m 2e
n w1 N −1 w1 w2 + N w2 −N N N e + (N − 1)e − N + 4π δpj , j=1
and w2 = λ(ew2 − 1) + 4π
n
δpj
j=1 n w1 N −1 w1 w2 2 + N w2 −N N N + 4π ≥ mg e −e δpj . j=1
Thus (w1 , w2 ) is a subsolution to Eqs. (2.15) and (2.16). Now we can state our main theorem regarding planar vortices. Theorem 6.4. There exists a unique solution (u 1 , u 2 ) to Eqs. (2.15)–(2.16) over R2 subject to the boundary condition (6.1). Proof. Let U0 be the solution of U0 − K U0 = 4π
n
δ p j (x), x ∈ R2 ; U0 (x) → 0
as |x| → ∞,
(6.7)
j=1
where K = 2(m 2g +m 2e )ec0 for some positive constant c0 to be chosen later. Let G K (x; p) be the Green function of − K with singularity at p. Clearly, U0 = 4π
n
G K (x; p j ).
j=1
√ By scaling, G K (x; 0) = G 1 ( K x; 0), where G 1 is the Green function of − 1. Thus,
U0 L ∞ (R2 ) ≤ c1 , where c1 is a constant independent of K . By induction, let (u l,1 , u l,2 ) be the solution of u l,1 − K u l,1 = f 1 (u l−1,1 , u l−1,2 ) − K u l−1,1 + 4π
n
δpj ,
j=1
u l,2 − K u l,2 = f 2 (u l−1,1 , u l−1,2 ) − K u l−1,2 + 4π
n j=1
(6.8) δpj ,
450
C.-S. Lin, Y. Yang
for l = 1, 2, . . . , where u 0,1 = u 0,2 = U0 and u N −1 u v f 1 (u, v) = m 2e e N + N v + (N − 1)e N − N − N , u N −1 u v f 2 (u, v) = m 2g e N + N v − e N − N . We claim w ≤ u l,1 ≤ 0, w ≤ u l,2 ≤ 0,
(6.9)
and u l+1,1 ≤ u l,1 , u l+1,2 ≤ u l,2 for l = 0, 1, 2, . . . ,
(6.10)
provided that c0 is large enough. By the maximum principle, u 0,1 = u 0,2 ≤ 0 in R2 . By (6.8), (u 1,1 − u 0,1 ) − K (u 1,1 − u 0,1 ) = m 2e eu 0,1 − 1 − K u 0,1 ≥ 0, provided that K ≥ m 2e .
(6.11)
Then by the maximum principle, u 1,1 (x) − u 0,1 (x) ≤ 0. Similarly, (u 1,2 − u 0,2 ) − K (u 1,2 − u 0,2 ) = m 2g eu 0,2 − 1 − K u 0,2 ≥ 0, and the maximum principle implies u 1,2 ≤ u 0,2 , provided that K ≥ m 2g .
(6.12)
Before we proceed with the proof of the case l = 2, we have to estimate u 1,1 (x) − u 1,2 (x). We may assume K ≥ 1. By (6.8), we have −(u 1,1 (x) − u 1,2 (x)) + K (u 1,1 (x) − u 1,2 (x)) = (m 2e − m 2g ) 1 − eu 0,1 (x) . By the maximum principle, the above equation gives max{u 1,1 (x) − u 1,2 (x) | x ∈ R2 } ≤ m 2e + m 2g .
(6.13)
By induction, we may assume u l,1 (x) ≤ u l−1,1 (x) ≤ · · · ≤ u 0,1 (x), u l,2 (x) ≤ u l−1,2 (x) ≤ · · · ≤ u 0,2 (x), u l,1 (x) − u l,2 (x) ≤ c0 ,
(6.14)
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
451
where K = (m 2g + m 2e )ec0 and c0 is chosen later. To prove (6.14) for the case l + 1, we note that the system (6.8) yields (u l+1,1 − u l,1 ) − K (u l+1,1 − u l,1 ) = g1 (u l,1 ; u l,2 ) − g1 (u l−1,1 ; u l−1,2 ), where
s N −1 s t g1 (s; t) = m 2e e N + N t + (N − 1)e N − N − N − K s,
and g1 is defined in the convex set = {(s, t) | s ≤ 0, t < 0, s − t ≤ c0 }. By the choice of K , we have 1 s + N −1 t N − 1 s − t ∂g1 (s; t) = m 2e eN N + eN N − K ∂s N N ≤ m 2e ec0 − K < 0, and ∂g1 = m 2e ∂t
N − 1 s + N −1 t N −1 s−t eN N − eN N N N
<0
for (s, t) ∈ . Thus, g1 (s; t) is decreasing in both s and t. By induction, u l,1 ≤ u l−1,1 and u l,2 ≤ u l−1,2 . Hence, (u l+1,1 − u l,1 ) − K (u l+1,1 − u l,1 ) ≥ 0 in R2 , which implies u l+1,1 − u l,1 ≤ 0 in R2 . For u l+1,2 (x), we also have (u l+1,2 − u l,2 ) − K (u l+1,2 − u l,2 ) = g2 (u l,1 ; u l,2 ) − g2 (u l−1,1 ; u l,2 ), where
Again, we have
s N −1 s t g2 (s; t) = m 2g e N + N t − e N − N − K t in . ∂g2 ∂s (s; t)
≤ 0,
∂g2 ∂t (s; t)
≤ 0 for (s, t) ∈ . Thus,
(u l+1,2 − u l,2 ) − K (u l+1,2 − u l,2 ) ≤ 0, which yields u l+1,2 − u l,2 ≤ 0 in R2 . To prove the last inequality of (6.14), we have −(u l+1,1 − u l+1,2 ) + K (u l+1,1 − u l+1,2 ) ul,1 N −1 u l,1 u l,2 = K (u l,1 − u l,2 ) + (m 2g − m 2e ) e N + N ul,2 − (m 2e (N − 1) + m 2g )e N − N + m 2e N ≤ K (u l,1 − u l,2 ) + m 2e N − m 2e (N − 1)e
u l,1 −u l,2 N
.
452
C.-S. Lin, Y. Yang
Let u l+1,1 (x0 ) − u l+1,2 (x0 ) = max{u l+1,1 (x) − u l+1,2 (x) | x ∈ R2 }. Then the maximum principle yields u l+1,1 (x0 ) − u l+1,2 (x0 ) ≤ u l,1 (x0 ) − u l,2 (x0 ) u l,1 (x0 )−u l,2 (x0 ) 1 N . m 2g N − m 2e (N − 1)e + K
(6.15)
Let c1 be the root of c1
m 2g N = m 2e (N − 1)e N , and set c0 to be c0 ≥ c1 + m 2g N .
(6.16)
If u l,1 (x0 ) − u l,2 (x0 ) ≥ c1 , then m 2g N − m 2e (N − 1)e
u l,1 (x0 )−u l,2 (x0 ) N
≤ 0, and then
u l+1,1 (x0 ) − u l+1,2 (x0 ) ≤ u l,1 (x0 ) − u l,2 (x0 ) ≤ c0 . If u l,1 (x0 ) − u l,2 (x0 ) < c1 , then u l+1,1 (x0 ) − u l+1,2 (x0 ) ≤ c1 +
m 2g N K
≤ c0 .
This finishes the proof of (6.14). It remains to prove w defined in Lemma 6.3 is a lower bound of u l,1 and u l,2 for all l. For l = 0, we have (w − u 0,1 ) − K (w − u 0,1 ) = λ(ew − 1) − K w ≥ 0,
K ≥ λ.
R2 .
Similarly, we have w ≤ u 0,2 . By induction, we assume Thus, w − u 0,1 ≤ 0 in w ≤ u l,1 and w ≤ u l,2 . Then, by using m 2g ≥ λ, we have (w − u l+1,1 ) − K (w − u l+1,1 ) ≥ g1 (w; w) − g1 (u l,1 ; u l,2 ) ≥ 0, since u l,1 ≥ w and u l,2 ≥ w. Hence, w ≤ u l+1,1 by the maximum principle. The same argument works for the inequality w ≤ u l+1,2 . Therefore, (6.9) is proved. After (6.9) is established, it is obvious to see that (u l+1,1 , u l+1,2 ) converges to (U1 , U2 ) uniformly in any compact set in R2 \{ p1 , . . . , pn }, and (U1 , U2 ) is a solution to Eqs. (2.15)–(2.16), which satisfies w ≤ U j ≤ 0 in R2 ,
j = 1, 2.
Thus, lim|x|→∞ U j (x) = 0 for j = 1, 2. Now suppose (u 1 , u 2 ) is any solution of Eqs. (2.15)–(2.16). By Lemma 6.1, u 1 (x) ≤ 0 and u 2 (x) ≤ 0 for x ∈ R2 . We note that by Lemma 6.2, u1
u2
(N − 1)e N − N ≤ N . c0
Hence, (u 1 (x), u 2 (x)) ∈ if c0 satisfies (N − 1)e N ≥ N . Applying the monotone arguments, we have u 1 (x) ≤ u l,1 (x), u 2 (x) ≤ u l,2 (x) for all x ∈ R2 . By taking the limit l → ∞, we obtain u 1 ≤ U1 and u 2 ≤ U2 in R2 , which implies (U1 , U2 ) is the maximal solution.
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
453
Corollary 6.5. Let (U1 , U2 ) and (U1 , U2 ) be the maximal solutions of Eqs. (2.15)–(2.16) 2 2 2 2 2 with m 2e , m 2g and m 2 e , m g , respectively. If m e ≥ m e and m g ≥ m g , then U j (x) ≥ U j (x), j = 1, 2. Corollary 6.5 is an immediate consequence of Lemma 6.2. We remark that our iterative method in this section may be carried out also for the problem over a periodic spatial domain (studied in Sects. 3–5) provided that, in addition to (3.11), the equation w = λ(ew − 1) + 4π
n
δ p j (x), x ∈ ,
(6.17)
j=1
has a solution for λ = min(N m 2g , m 2e ), which requires that the condition λ|| = || min(N m 2g , m 2e ) > 4π n
(6.18)
be fulfilled, which may be viewed as a sufficient condition for the validity of an iterative construction. It is interesting to note that, unlike the sufficient conditions derived in Sect. 4 for the minimization construction, the condition (6.18) depends on the domain size ||. Staying away from the vortex points p1 , . . . , pn and linearizing Eqs. (2.15) and (2.16) around u 1 = 0, u 2 = 0, we have the linearized equations U1 = m 2e U1 , U2 = m 2g U2 .
(6.19)
In view of (6.19) and the asymptotic condition (6.1), it is standard to show that a solution (u 1 , u 2 ) of (2.15)–(2.16) subject to (6.1) enjoys the exponential decay estimates u 1 (x) = O(e−(1−ε)m e |x| ), u 2 (x) = O(e−(1−ε)|m g |x| ) as |x| → ∞,
(6.20)
where 0 < ε < 1 can be made arbitrarily small. As a consequence of (6.20), (2.14), (3.19), and (3.20), we see that the Abelian and non-Abelian curvatures, f 12 and F12 , decay to zero following the same exponential laws (6.20) obeyed by u 1 and u 2 , respectively. In particular, the Abelian and non-Abelian fluxes over R2 , R2 f 12 dx and R2 F12 dx, are both well defined. Below, we show that they are actually the same quantized quantities given in (3.21) and (3.22). To proceed, we need to elaborate on the singular source terms in (2.15) and (2.16). For this purpose, define ρ(t) to be a smooth monotone increasing function over t > 0 so that ⎧ ⎨ ln t, t ≤ 21 , ρ(t) = (6.21) 0, t ≥ 1, ⎩ ≤ 0, for all t > 0. Let δ > 0 be such that Bδ ( p j ) = {x | |x − p j | < δ}, j = 1, 2, . . . , n, are disjoint. Namely, Bδ ( p j ) ∩ Bδ ( pk ) = ∅, p j = pk , j, k = 1, . . . , n. Consider the functions |x − p j | , j = 1, . . . , n. u 0j (x) = 2ρ δ
454
C.-S. Lin, Y. Yang
Then u 0j ≤ 0, u 0j = 0 for |x − p j | ≥ δ, and u 0j = 4π δ p j (x) − g j , where, of course,
δ supp(g j ) ⊂ x
≤
x − p j | ≤ δ , 2 and g j is smooth, j = 1, . . . , n. Besides, we have
R2
g j dx =
(−u 0j ) dx δ
|x− p j |≥ 3
Define the background function u 0 =
n
u 0 = 4π
=
0 j=1 u j . N
|x− p j |= 3δ
∂u 0j ∂n
ds = 4π.
Then
δ p j − g0 ,
j=1
where g0 =
n j=1
gj,
R2
g0 dx = 4π n.
(6.22)
The function u 0 has the useful property that u 0 ≤ 0, u 0 = 0on R2 \δ , where δ = ∪nj=1 Bδ ( p j ). Thus, as in Sect. 3, the substitution (3.2) transforms (2.15) and (2.16) into the equations v1 (N −1) v1 v2 v1 = g0 − N m 2e + m 2e eu 0 + N + N v2 + [N − 1]e N − N , (6.23) v1 (N −1) v1 v2 v2 = g0 + m 2g eu 0 + N + N v2 − e N − N . (6.24) On the other hand, using (6.20) and standard elliptic estimates, we see that u 1 , u 2 ∈ W 2, p (R2 \δ ) for any p > 2 (say). Consequently, all the first derivatives of u 1 and u 2 decay to zero as |x| → ∞ (cf. [28]). Differentiating (2.15) and (2.16) in R2 \δ and considering the linearized equations again, we conclude that all first derivatives of u 1 and u 2 vanish at infinity exponentially fast. Since u 0 is of compact support, we see that all first derivatives of v1 and v2 vanish at infinity exponentially fast. As a consequence of this observation and the divergence theorem, we have v1 dx = v2 dx = 0. (6.25) R2
R2
Integrating (6.23) and (6.24) and applying (6.22) and (6.25), we have v1 (N −1) v1 v2 4π n eu 0 + N + N v2 + [N − 1]e N − N − N dx = − 2 , 2 me R v1 (N −1) v1 v2 4π n eu 0 + N + N v2 − e N − N dx = − 2 , 2 mg R
(6.26) (6.27)
replacing (3.6) and (3.7). Hence, proceeding analogously as before, we can show that the Abelian and non-Abelian fluxes over R2 are given by the same quantized quantities
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
455
as described in (3.21) and (3.22) over , respectively, which verifies again the statement made in [18] in the full plane setting. Finally, to establish the uniqueness of a solution of the Eqs. (6.23) and (6.24) which vanishes at infinity of R2 , we note that such a solution is a critical point of the action functional 1 (N − 1) I (v1 , v2 ) = |∇v1 |2 + |∇v2 |2 2 2 2m e 2m 2g R v v1 (N − 1) u 0 + N1 + (NN−1) v2 u0 +N e −e − + v2 N N v1 v2 v v1 v2 v2 1 − − + g0 + N (N − 1) e N N − 1 − + (N − 1) 2 dx, N N m 2e mg (6.28) defined over W 1,2 (R2 ). Since this functional is convex in W 1,2 (R2 ), its critical point in W 1,2 (R2 ) is unique. 7. Conclusions In this paper, we have presented two sharp existence and uniqueness theorems for the non-Abelian multiple vortices arising in a coupled SU (N ) and U (1) gauge Higgs model embedded in a supersymmetric field theory framework studied in the recent work [18]. Specifically, our results are concerned about vortices over a doubly periodic spatial domain and over the full plane R2 . In the situation over a periodic domain , we have obtained a necessary and sufficient condition for the existence of a unique n-vortex solution realizing an arbitrarily prescribed distribution of n vortices which indicates that larger domain size || or larger Abelian and non-Abelian Higgs boson masses, m e and m g , permit a higher vortex number n. Furthermore, we have found sufficient conditions under which the n-vortex solution may be obtained through either a constrained minimization process or a monotone iterative construction. The condition derived for the validity of the minimization method is independent of the domain size but depends on the mass ratio γ = m g /m e . On the other hand, however, the condition derived for the validity of the iteration method depends on the domain size but is independent of the mass ratio. In the situation over the full plane R2 , we have proved the existence and uniqueness of a solution realizing an arbitrarily prescribed distribution of finitely many vortices so that the Abelian and non-Abelian Higgs and gauge fields are shown to approach their limiting values exponentially fast at infinity. In both doubly periodic and full plane situations, the total Abelian and non-Abelian gauge field fluxes of an n-vortex solution are quantized n-multiples of those of 1-vortex solutions. Acknowledgements. We would like to thank Norisuke Sakai for helpful communications regarding some issues of the vortex equations studied here. We would also like to thank the referee for several constructive suggestions.
References 1. Abel, T., Stebbins, A., Anninos, P., Norman, M.L.: First structure formation. II. Cosmic strings plus hot dark matter models. Astrophy. J. 508, 530–534 (1998) 2. Abrikosov, A.A.: On the magnetic properties of superconductors of the second group. Sov. Phys. JETP 5, 1174–1182 (1957)
456
C.-S. Lin, Y. Yang
3. Ambjorn, J., Olesen, P.: Anti-screening of large magnetic fields by vector bosons. Phys. Lett. B 214, 565– 569 (1988) 4. Ambjorn, J., Olesen, P.: On electroweak magnetism. Nucl. Phys. B 315, 606–614 (1989) 5. Ambjorn, J., Olesen, P.: A magnetic condensate solution of the classical electroweak theory. Phys. Lett. B 218, 67–71 (1989) 6. Ambjorn, J., Olesen, P.: A condensate solution of the classical electroweak theory which interpolates between the broken and the symmetric phase. Nucl. Phys. B 330, 193–204 (1990) 7. Aubin, T.: Nonlinear Analysis on Manifolds: Monge–Ampére Equations. Berlin-New York: Springer, 1982 8. Auzzi, R., Bolognesi, S., Evslin, J., Konishi, K., Yung, A.: Nonabelian superconductors: vortices and confinement in N = 2 SQCD. Nucl. Phys. B 673, 187–216 (2003) 9. Auzzi, R., Bolognesi, S., Evslin, J., Konishi, K.: Nonabelian monopoles and the vortices that confine them. Nucl. Phys. B 686, 119–134 (2004) 10. Bartolucci, D., Tarantello, G.: Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Commun. Math. Phys. 229, 3–47 (2002) 11. Bezryadina, A., Eugenieva, E., Chen, Z.: Self-trapping and flipping of double-charged vortices in optically induced photonic lattices. Optics Lett. 31, 2456–2458 (2006) 12. Bimonte, G., Lozano, G.: Z flux-line lattices and self-dual equations in the standard model. Phys. Rev. D 50, 6046–6050 (1994) 13. Bogomol’nyi, E.B.: The stability of classical solitons. Sov. J. Nucl. Phys. 24, 449–454 (1976) 14. Brandenberger, R., Kaiser, N., Schramm, D., Turok, N.: Galaxy and structure formation with hot dark matter and cosmic strings. Phys. Rev. Lett. 59, 2371–2374 (1987) 15. Chern, J.L., Chen, Z.Y., Lin, C.S.: Uniqueness of topological solutions and the structure of solutions for the Chern–Simons system with two Higgs particles. Commun. Math. Phys. 296, 323–351 (2010) 16. Coulson, D., Ferreira, P., Graham, P., Turok, N.: Microwave anisotropies from cosmic defects. Nature 368, 27–31 (1994) 17. Caffarelli, L., Yang, Y.: Vortex condensation in the Chern–Simons Higgs model: an existence theorem. Commun. Math. Phys. 168, 321–336 (1995) 18. Eto, M., Fujimori, T., Nagashima, T., Nitta, M., Ohashi, K., Sakai, N.: Multiple layer structure of nonAbelian vortex. Phys. Lett. B 678, 254–258 (2009) 19. Eto, M., Hashimoto, K., Marmorini, G., Nitta, M., Ohashi, K., Vinci, W.: Universal reconnection of non-Abelian cosmic strings. Phys. Rev. Lett. 98, 091602 (2007) 20. Eto, M., Isozumi, Y., Nitta, M., Ohashi, K., Sakai, N.: Solitons in the Higgs phase – the moduli matrix approach. J. Phys. A 39, R315–R392 (2006) 21. Fontana, L.: Sharp borderline Sobolev inequalities on compact Riemannin manifolds. Comment. Math. Helv. 68, 415–454 (1993) 22. Gibbons, G., Hawking, S., Vachaspati, T (eds.): The Formation and Evolution of Cosmic Strings. Cambridge: Cambridge Univ. Press, 1990 23. Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity, In: Collected Papers of L. D. Landau (edited by D. Ter Haar), New York: Pergamon, 1965, pp. 546–568 24. Gustafson, S., Sigal, I.M., Tzaneteas, T.: Statics and dynamics of magnetic vortices and of Nielsen–Olesen (Nambu) strings. J. Math. Phys. 51, 015217 (2010) 25. Hanany, A., Tong, D.: Vortices, instantons and branes. J. High Energy Phys. 0307, 037 (2003) 26. Hindmarsh, M.B., Kibble, T.W.B.: Cosmic strings. Rep. Prog. Phys. 58, 477–562 (1995) 27. Inouye, S., Gupta, S., Rosenband, T., Chikkatur, A.P., Grlitz, A., Gustavson, T.L., Leanhardt, A.E., Pritchard, D.E., Ketterle, W.: Observation of vortex phase singularities in Bose-Einstein condensates. Phys. Rev. Lett. 87, 080402 (2001) 28. Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Boston: Birkhäuser, 1980 29. Jeannerot, R., Rocher, J., Sakellariadou, M.: How generic is cosmic string formation in SUSY GUTs. Phys. Rev. D 68, 103514 (2003) 30. Kibble, T.W.B.: Some implications of a cosmological phase transition. Phys. Rep. 67, 183–199 (1980) 31. Kawaguchi, Y., Ohmi, T.: Splitting instability of a multiply charged vortex in a Bose–Einstein condensate. Phys. Rev. A 70, 043610 (2004) 32. Khomskii, D.I., Freimuth, A.: Charged vortices in high temperature superconductors. Phys. Rev. Lett. 75, 1384–1386 (1995) 33. Lin, C.S., Ponce, A.C., Yang, Y.: A system of elliptic equations arising in Chern–Simons field theory. J. Funct. Anal. 247, 289–350 (2007) 34. Lin, C.S., Prajapat, J.V.: Vortex condensates for relativistic Abelian Chern–Simons model with two Higgs scalar fields and two Gauge fields on a torus. Commun. Math. Phys. 288, 311–347 (2009) 35. Lucia, M., Nolascob, M.: SU (3) Chern–Simons vortex theory and Toda systems. J. Diff. Eqs. 184, 443– 474 (2002)
Non-Abelian Multiple Vortices in Supersymmetric Field Theory
457
36. Matsuda, Y., Nozakib, K., Kumagaib, K.: Charged vortices in high temperature superconductors probed by nuclear magnetic resonance. J. Phys. Chem. Solids 63, 1061–1063 (2002) 37. Nielsen, H., Olesen, P.: Vortex-line models for dual strings. Nucl. Phys. B 61, 45–61 (1973) 38. Nolasco, M., Tarantello, G.: Vortex condensates for the SU (3) Chern–Simons theory. Commun. Math. Phys. 213, 599–639 (2000) 39. Prasad, M.K., Sommerfield, C.M.: Exact classical solutions for the ’t Hooft monopole and the Julia–Zee dyon. Phys. Rev. Lett. 35, 760–762 (1975) 40. Shevchenko, S.I.: Charged vortices in superfluid systems with pairing of spatially separated carriers. Phys. Rev. B 67, 214515 (2003) 41. Shifman, M., Yung, A.: Non-Abelian string junctions as confined monopoles. Phys. Rev. D 70, 045004 (2004) 42. Sokoloff, J.B.: Charged vortex excitations in quantum Hall systems. Phys. Rev. B 31, 1924–1928 (1985) 43. Spruck, J., Yang, Y.: On multivortices in the electroweak theory I: existence of periodic solutions. Commun. Math. Phys. 144, 1–16 (1992) 44. Spruck, J., Yang, Y.: On multivortices in the electroweak theory II: Existence of Bogomol’nyi solutions in R2 . Commun. Math. Phys. 144, 215–234 (1992) 45. Tarantello, G.: Multiple condensate solutions for the Chern–Simons–Higgs theory. J. Math. Phys. 37, 3769–3796 (1996) 46. Tarantello, G.: Self-Dual Gauge Field Vortices. Progress in Nonlinear Differential Equations and Their Applications 72, Boston-Basel-Berlin: Birkäuser, 2008 47. ’t Hooft, G.: A property of electric and magnetic flux in nonabelian gauge theories. Nucl. Phys. B 153, 141–160 (1979) 48. Tong, D.: Monopoles in the Higgs phase. Phys. Rev. D 69, 065003 (2004) 49. Vilenkin, A., Shellard, E.P.S.: Cosmic Strings and Other Topological Defects. Cambridge: Cambridge U. Press, 1994 50. Wang, S., Yang, Y.: Abrikosov’s vortices in the critical coupling. SIAM J. Math. Anal. 23, 1125–1140 (1992) 51. Weinberg, E.J.: Multivortex solutions of the Ginzburg-Landau equations. Phys. Rev. D 19, 3008–3012 (1979) 52. Weinberg, E.J.: Index calculations for the Fermion-vortex system. Phys. Rev. D 24, 2669–2673 (1981) 53. Yang, Y.: The relativistic non-Abelian Chern–Simons equations. Commun. Math. Phys. 186, 199–218 (1997) 54. Yang Y.: Solitons in Field Theory and Nonlinear Analysis. New York-Berlin: Springer, 2001 Communicated by I.M. Sigal
Commun. Math. Phys. 304, 459–498 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1235-8
Communications in
Mathematical Physics
Local Causal Structures, Hadamard States and the Principle of Local Covariance in Quantum Field Theory Claudio Dappiaggi1,2 , Nicola Pinamonti2 , Martin Porrmann3 1 Erwin Schrödinger Institut für Mathematische Physik, Boltzmanngasse 9, 1090 Wien, Austria.
E-mail: [email protected]
2 II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149,
22761 Hamburg, Germany. E-mail: [email protected]; [email protected]
3 Quantum Research Group, School of Physics, University of KwaZulu-Natal and National
Institute for Theoretical Physics, Private Bag X54001, Durban 4001, South Africa. E-mail: [email protected] Received: 15 January 2010 / Accepted: 10 December 2010 Published online: 3 April 2011 – © Springer-Verlag 2011
Abstract: In the framework of the algebraic formulation, we discuss and analyse some new features of the local structure of a real scalar quantum field theory in a strongly causal spacetime. In particular, we use the properties of the exponential map to set up a local version of a bulk-to-boundary correspondence. The bulk is a suitable subset of a geodesic neighbourhood of an arbitrary but fixed point p of the underlying background, while the boundary is a part of the future light cone having p as its own tip. In this regime, we provide a novel notion for the extended ∗ -algebra of Wick polynomials on the aforesaid cone and, on the one hand, we prove that it contains the information of the bulk counterpart via an injective ∗ -homomorphism while, on the other hand, we associate to it a distinguished state whose pull-back in the bulk is of Hadamard form. The main advantage of this point of view arises if one uses the universal properties of the exponential map and of the light cone in order to show that, for any two given backgrounds M and M and for any two subsets of geodesic neighbourhoods of two arbitrary points, it is possible to engineer the above procedure such that the boundary extended algebras are related via a restriction homomorphism. This allows for the pull-back of boundary states in both spacetimes and, thus, to set up a machinery which permits the comparison of expectation values of local field observables in M and M . 1. Introduction In the framework of quantum field theory over curved backgrounds, we witnessed a considerable series of leaps forward due to a novel use of advanced mathematical techniques combined with new physical insights leading to an improved understanding of the underlying foundations of the theory. It is far from our intention to give a recollection of all of them, but we would like to draw attention at least to some of them. On the one hand, in [9], the principle of general local covariance was formulated leading to the realisation of a quantum field theory as a covariant functor between the category of globally hyperbolic (four-dimensional) Lorentzian manifolds with isometric embeddings as
460
C. Dappiaggi, N. Pinamonti, M. Porrmann
morphisms and the category of C ∗ -algebras with unit-preserving monomorphisms as morphisms and also to the new interpretation of local fields as natural transformations from compactly supported smooth function to suitable operators. On the other hand, the presence of a nontrivial background comes with the grievous problem of the a priori absence of a sufficiently large symmetry group to identify a natural ground state as in Minkowski spacetime where Poincaré invariance enables this. Nonetheless, it is still possible to identify a class of physically relevant states as those fulfilling the so-called Hadamard condition. This guarantees that the ultraviolet behaviour of the chosen state mimics that of the Minkowski vacuum at short distances as well as that the quantum fluctuations of observables such as the smeared components of the stress-energy tensor are bounded. From a practical point of view, the original characterisation of the Hadamard form was realised by means of the local structure of the integral kernel of the two-point function of the selected quasi-free state in a suitably small neighbourhood of a background point. Unfortunately, such a criterion is rather difficult to check in a concrete example and a real step forward has been achieved in [38,39] in which the connection between the Hadamard condition and the microlocal properties of the two-point function is proven and fully characterised. This result has prompted a series of interesting developments in the analysis of physically relevant states in a curved background, but we focus mainly on a few recent advances (cf. [13–15]) where it has been shown that, either in asymptotically flat or in cosmological spacetimes, it is possible to exploit the conformal structure of the manifold to identify a preferred null submanifold of codimension one, the conformal boundary. On the latter it is possible to coherently encode the information of the bulk algebra of observables and to identify a state fulfilling suitable uniqueness properties whose pullback in the bulk satisfies the Hadamard condition, being at the same time invariant under all spacetime isometries. The main problem in the above construction is the need to find a rigid and global geometric structure which acts as an auxiliary background out of which the bulk state is constructed. Hence the local applicability of a similar scheme seems rather limited; yet one of the main goals of the present paper is to show that such a procedure can indeed be set up at a local level and for all spacetimes of physical interest. In particular, this statement is established on the basis of a careful use of some rather well-known geometrical objects. To be more precise, the point of view taken is the following: if one considers an arbitrary but fixed point p in a strongly causal four-dimensional spacetime, it is always possible to single out a geodesic neighbourhood where the exponential map is a local diffeomorphism. Within this set we can also always select a second point q such that . the double cone D ≡ D( p, q) = I + ( p) ∩ I − (q) is a globally hyperbolic spacetime. This line of reasoning has a twofold advantage: on the one hand, one can single out a local natural null submanifold of codimension one, C p+ , as the portion of J + ( p) contained in the closure of D, while, on the other hand, we are free to repeat the very same construction for a second point p with associated double cone D in another spacetime M . Since the exponential map is invertible and the tangent spaces T p (M) and T p (M ) are isomorphic, it turns out that it is possible to engineer all the geometric data in such a way that the two boundaries C p+ and C p+ can be related by a suitable restriction map, the only freedom being the choice of a frame at p and at p . These two advantages can be used to draw some important conclusions on the structure of local quantum field theories. More precisely, we shall focus on a real scalar field theory in D, with generic mass m and with generic curvature coupling ξ . The associated
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
461
quantum observables are described by the Borchers-Uhlmann algebra or rather by the extended algebra of fields. In particular, we shall show that it is possible to construct a scalar field theory also on C p and, as a novel result, that also, in the boundary, there exists a natural notion of extended algebra which is made precise here. Apart from the check of mathematical consistency of our definition, we reinforce our proposal by showing that there exists an injective ∗ -homomorphism between the bulk and the boundary counterparts. The relevance of this result is emphasised by the identification of a natural state on the boundary, whose pull-back in D via turns out to be of Hadamard form. This result provides a potential candidate for a local vacuum in the large class of backgrounds to be considered here. Yet we still have not made profitable use of the second advantage outlined before. As a matter of fact, we can now consider two arbitrary strongly causal spacetimes M and M as well as two points therein so that the relevant portions of the two boundaries, C p and C p , say, associated with the double cones, can be related by a suitable restriction map. The construction of the boundary field theory shows that such a map becomes an injective homomorphism between the boundary extended algebras, hence allowing for the construction of a local Hadamard state in two different backgrounds starting from the same building block on the boundary. The results presented in the present work have some antecedents in the concept of relative Cauchy evolution developed in [9]. Knowing a theory (and its corresponding Hadamard function) in the neighbourhood of a Cauchy surface, such a method permits one to reconstruct the theory in the neighbourhood of any other Cauchy surface of the same spacetime. The deformation arguments, see [18], play a crucial role in obtaining the Hadamard property for the deformed state in particular. Another related key result is also the one presented in [45] about the local quasi-equivalence of quasifree Hadamard states. In the present paper, using the null cones as hypersurfaces on which to encode the quantum information, we succeed in giving an extended algebra of observables without knowing the state in a neighbourhood of such a surface. Hence, based on the new method presented here, it is possible to determine quantum states out of their form on null surfaces alone and thus in a spacetime-independent way. We are now in a position to have a reference state with respect to which we can compare the expectation values of the same field observables in two different spacetimes. In particular, if one of these is (a portion of) Minkowski spacetime, it is obvious that the result of the comparison will be related to the geometric data of the second background which can now be assessed with a crystal clear procedure. Furthermore, we shall show that this method admits an interpretation within the language of category theory, so that it becomes manifest that our proposal is not in contrast with the principle of general local covariance, but can actually be seen as a generalisation. As a matter of fact, it reduces to the latter whenever isometric embeddings are involved, in which case the fields recover their interpretation as natural transformations as in [9], i.e., they transform in a covariant manner under local isometries. To reinforce the above procedure we also provide an explicit example of this “comparison” strategy considering a massless real scalar field minimally coupled to scalar curvature both in Minkowski and in a Friedman-Robertson-Walker spacetime with flat spatial sections. We demonstrate how the difference of the expectation values of the regularised squared scalar fields in these two spacetimes can be expanded into a power series of a suitable local coordinate system (null-advanced) yielding, at first order, a contribution dependent on the structure of the so-called scale factor of the curved background.
462
C. Dappiaggi, N. Pinamonti, M. Porrmann
Since we have already extensively discussed the plan of action, we only briefly sketch the synopsis of the paper. In Sect. 2 we shall analyse all the geometric structure needed. Although most of the material, devoted to the construction of frames and of the exponential map, is rather well-known in the literature, we try nonetheless to recollect it here to provide guidance through the construction of the main geometric objects required, the boundary in particular. In Sect. 3 we shall tackle the problem of constructing a quantum scalar field theory on a null cone; in particular, in Subsects. 3.1 and 3.2 we discuss the structure of the bulk and boundary algebras therein while, in Subsect. 3.3, we identify the distinguished boundary state. The novel construction of the extended algebra on the boundary is presented in Subsect. 3.4 and all these results are connected to the bulk counterpart in Subsect. 3.5. Eventually, in Sect. 4, we discuss, by means of the language of categories, the scheme which leads to the possibility to compare field theories on different spacetimes. The concrete example mentioned above is in Subsect. 4.2. Section 5 summarises the paper and sets out a few conclusions as well as possible future investigations. 2. Frames and Cones As outlined in the Introduction, the keyword of this paper is “comparison,” i.e. our ultimate goal will be to correlate quantum field theories in different backgrounds both at the level of algebras and of states and, moreover, to try in the process also to extract information on the local geometry. To this avail one needs crystal clear control both of the underlying background and of its properties. Therefore, we cannot consider arbitrary manifolds, but need to focus only on those which are of physical relevance insofar as they can carry a full-fledged quantum field theory. If we keep in mind this perspective, we shall henceforth call spacetime a four-dimensional, Hausdorff, connected smooth manifold M endowed with a Lorentzian metric whose signature is (−, +, +, +). Then, consequently, M is also second countable and paracompact [19,20]. Customarily one also requires that M be globally hyperbolic (see for example [1] or [9]) in order to have a well-defined Cauchy problem for the equations of motion ruling the dynamics of standard free field theories. The next natural step is the identification of further local geometric structures which could serve as a useful tool in the comparison of two different field theories on two different spacetimes, M and M . It is known that, for a real scalar field theory, it suffices to require that the two spacetimes are either isometrically embedded into each other or conformally related [9,34]. The drawback of this approach is that only a few pairs M and M fulfill such criteria and potentially interesting cases, such as when M coincides with Minkowski spacetime and M with de Sitter, are excluded. A natural alternative would be to consider pairs of spacetimes M and M related by a global diffeomorphism, but, unfortunately, these maps do not preserve the geometric structures at the heart of the quantum or even of the classical field theory. A typical example of such a problem arises in connection with the equations of motion of a dynamical system whenever these are constructed out of the spacetime metric. The action of a generic diffeomorphism preserves their form only in special cases, viz. when they are related to isometries. Hence we would return to the original scenario. Apart from these remarks we should also keep in mind the idea, briefly sketched in the Introduction, to exploit a bulk-to-boundary reconstruction procedure along the lines of [13,14]. At a global level, this requires the existence of a conformal boundary structure, a feature shared only by a certain class of manifolds. Since we want to consider a scenario
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
463
as general as possible, a viable alternative is to focus only on the local structures of the underlying spacetimes. In the remainder of this section, we show how to substantiate this heuristic idea if one carefully uses certain properties of the exponential map. 2.1. Frames and the exponential map. The aim of this subsection is to introduce the basic geometric tools to be used. Most of the concepts are certainly well-known in the literature and the reader might refer either to [26] for a full-fledged analysis of those related to bundles and their properties or to [28,33] for a discussion focused on the differential geometric aspects. Nonetheless, it is worthwhile to recapitulate part of them since they will play a pivotal role in this paper and we can, at the same time, fix the notation. Consider an arbitrary four-dimensional differentiable manifold M. To any point p ∈ M, we can associate • a linear frame F p of the tangent space, i.e., a non-singular linear mapping e : R4 → T p (M), or, equivalently, an assignment of an ordered basis e1 , …, e4 of T p (M). It is straightforward to infer that the set of all such linear frames F M at an arbitrary but fixed p ∈ M naturally comes with a right and free action of the group G L(4, R) which is tantamount to the possible changes of basis in R4 , i.e., (A, e) → e A, where e A denotes the ordered basis Aij ei for all A ∈ G L(4, R). Thus F M can be endowed with the following additional structure: • Given a four-dimensional differentiable manifold M, a frame bundle is the principal bundle F M = F[G L(4, R), π , M] built from the disjoint union p F p M, where F M → M is the projecF p M is identified with the typical fibre G L(4, R) and π : tion map. Furthermore, the tangent bundle T M can be constructed as the associated bundle T M = F M ×G L(4,R) R4 . We emphasise the well-known fact that the structure introduced last guarantees that the typical fibre of the tangent bundle at any point p is R4 regardless of the chosen manifold, a fact we shall use in the forthcoming discussion. Following [28,33], recall that • for any p ∈ M, if D p is the set of all vectors v in T p (M) such that the geodesic γv : [0, 1] → M admits v as a tangent vector in 0, then the exponential map at p is exp p : D p → M with exp p (v) = γv (1); of the 0-vector in T p (M) • for any point p ∈ M there always exists a neighbourhood O such that the exponential map is a diffeomorphism onto an open subset O ⊂ M. Fur is star-shaped, O is called a normal neighbourhood, and the thermore, whenever O inverse map therein will be denoted exp−1 p : O → O. Although the existence of open sets where the exponential map is a diffeomorphism suggests a way to compare local quantum field theories on different manifolds, we also need to single out a preferred structure of codimension 1, since we wish to implement a bulk-to-boundary procedure. To this avail all the manifolds are henceforth endowed with a smooth Lorentzian metric, which entails the following additional features: • Since a linear frame at a point p ∈ M can be seen as the assignment of an ordered basis of R4 , one can endow this latter vector space with the standard Minkowski metric η, which, by construction, is invariant under the Lorentz group S O(3, 1).
464
C. Dappiaggi, N. Pinamonti, M. Porrmann
In this case the frame bundle becomes F M = F[S O(3, 1), π , M] which is also referred to as the bundle of orthonormal frames over M. Furthermore, if the spacetime is oriented and time-oriented, we can further reduce the group to S O0 (3, 1), the component of S O(3, 1) connected to the identity. • Every point in a Lorentzian manifold admits a normal neighbourhood (see Proposition 7 and also Definition 5 in Chap. 5 of [33]). • There is always a choice of coordinates, called normal coordinates, such that, in these coordinates, the pull-back of the metric g under the inverse of the exponential map equals η (the Minkowski metric in standard coordinates) on the inverse image of the point p. • Since we shall ultimately need to single out a sort of preferred codimension 1 structure, it is rather important that, in a Lorentzian manifold, the so-called Gauss lemma holds true (Lemma 1 in Chap. 5 of [33]). In particular, this entails that, given any ⊂ T p (M) having p as its own tip, then the p ∈ M, if we consider the null cone C subset C ∩ O is mapped into a local null cone in O ⊂ M which consists of initial segments of all null geodesics starting at p. We are now in a position to outline the building blocks of our geometric construction. Let us consider two spacetimes (M, g) and (M , g ) and two generic points p ∈ M and p ∈ M , together with their normal neighbourhoods O p and O p . If we equip each tangent space with an orthonormal basis via a frame, e : R4 → T p (M) and e : R4 → T p (M ), we are also free to introduce a map i e,e : T p (M) → T p (M ) which is constructed simply by identifying the elements of the two ordered bases. The strategy is now to exploit the fact that the exponential map is a diffeomorphism (hence invertible) in a geodesic neighbourhood to introduce a map ı e,e : O p → O p such that . ı e,e = exp p ◦ i e,e ◦ exp−1 (2.1) p . It is important to stress a few further aspects of this last definition: • The map ı e,e is well defined only when exp−1 p can be inverted on the image of −1 (O ) ⊂ O p . Therefore, for the sake of nota , that is when i ◦ exp i e,e ◦ exp−1 p e,e p p tional simplicity, when we write ı e,e it is always assumed that such a requirement is satisfied. Furthermore, for every point p we can always consider a sufficiently smaller subset of O p , retaining all its properties, where the above inclusion holds true. • The map ı e,e , which maps a sufficiently small O to O , is not unique, in the sense that it depends on the chosen orthonormal frames e and e . We have always the freedom to act with an element of the structure group of the fibre (be it S O(3, 1) or S O0 (3, 1) depending on the scenario considered) which maps an orthonormal basis into a second one, and this either on T p (M) or T p (M ). Such arbitrariness cannot be lifted and, for this reason, we have explicitly indicated the two frames in the mapping ı e,e . • Despite the freedom mentioned above, the map ı e,e is invariant under the action of a single element of the structure group of the fibre on both e and e , i.e., there exists an equivalence relation: We say that ı e,e ∼ ı e, ˜ e˜
(2.2)
if and only if there exists an element ∈ S O0 (3, 1) such that e˜ = e and e˜ = e . This equivalence relation shall actually play a relevant role in the discussion of Sect. 4.
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
465
As a related point, notice that, if the spacetime M is isometrically embedded into M , a scenario close to the hypotheses in [9], each isometry φ : M → M induces an isomorphism between the orthonormal frame bundles F M and F M since the metric structure is preserved. In this case every local character of the manifold M is preserved under φ (see for example Chap. 3 of [33]) and, hence, one can consider a sufficiently small subset of the normal neighbourhood of any p ∈ M as well as of φ( p) ∈ M so that our construction yields the following commutative diagram: Op ⏐ ⏐ φ
exp−1 p
−−−−→
T p (M) ⏐ ⏐i e,(φ∗ ◦ e)
Oφ( p) ←−−−− Tφ( p) (M ) expφ( p)
.
Notice that the presence of φ∗ ◦ e in place of a generic e can be justified as follows: If we call ( , ) p the inner product between vectors in T p (M), then for any v, w ∈ T p (M), one has (v, w) p = (φ∗ (v), φ∗ (w))φ( p) , which, upon introduction of a local frame e : R4 → T p (M), yields (v, w) p = (e(vi ), e(wi )) p = (φ∗ ◦ e(vi ), φ∗ ◦ e(wi ))φ( p) , where vi , wi ∈ R4 . Moreover, if a generic e is used in place of φ∗ ◦ e there is no guarantee that the previous diagram commutes. A counterexample can actually be constructed considering two isometrically related spacetimes, which are not rotationally invariant and taking for e , φ∗ ◦ e rotated by some generic angle. 2.2. Double cones and their past boundary. The analysis of the previous subsection is a first step towards the setup of a full-fledged procedure which allows for the local comparison of quantum field theories on different spacetimes. We shall now single out a preferred submanifold of codimension 1 on which to apply a bulk-to-boundary reconstruction. To this avail we have to ensure in the first place that one can consistently assign to the background M a well-defined quantum field theory. Since we are only interested in local quantities, the usual hypothesis of global hyperbolicity of the spacetime can be moderately relaxed and, henceforth, we shall assume M to be strongly causal [2], i.e., for every point p ∈ M, there exists an arbitrarily small convex, causally convex neighbourhood Op , which means that no non-spacelike curve intersects Op in a disconnected set. In other words, Op itself is globally hyperbolic. From a physical point of view, this simply forces us to require that, ultimately, the theory coincides with the usual quantisation procedure on each of these subsets, while, from a geometrical perspective, the discussion of the preceding section still holds true since we are entitled to select Op ⊆ O p , the normal neighbourhood of p, in such a way that the exponential map is a local diffeomorphism also on Op . Furthermore, a rather useful class of sets is constructed out of the so-called double cones, D( p , q) = I + ( p ) ∩ I − (q) ⊂ M, where I ± stand, respectively, for the chronological future and past while q ∈ Op . Notice that both p and q can be arbitrary but, for our construction, we shall always suppose that at least one of them coincides with p, henceforth p ≡ p. It is also interesting that D( p, q) is an open and still globally hyperbolic subset of Op . In the forthcoming
466
C. Dappiaggi, N. Pinamonti, M. Porrmann
discussion the boundary of this region will also be relevant and we point out that the closure D( p, q) is a compact set (see for example Chap. 8 in [46]) which coincides with J + ( p) ∩ J − (q). Furthermore, it is also important to recall both that the set of (the closures of) double cones can be used as a base of the topology of Op and that, under the previous assumptions, we can also freely consider the image of D( p, q) under the inverse exponential map exp−1 p , denoted by U ( p, q). The reader should bear in mind that U ( p, q) is not necessarily the closure of a double cone in T p (M) ∼ R4 with respect to the flat metric since only (portions of) cones in T p (M), having p as their tip, are mapped in (portions of) those in O p and vice versa. Nonetheless, this construction allows for the identification of the main geometrical structure needed, since the very existence of D( p, q) and the properties of this set as well . as of J + ( p) under the exponential map suggest to consider C p+ = ∂ J + ( p) ∩ D( p, q) as the natural boundary on which to encode data from a field theory in the bulk. The bulk here means D( p, q) which is a genuine globally hyperbolic submanifold of M on which a full-fledged quantum field theory can indeed be defined. From a geometrical point of view, a few interesting intrinsic properties of C p+ can readily be inferred, namely, to start with, C p+ is generated by future directed null geodesics in particular originating from p. Notice that the latter are not complete since the set we are interested in is constrained to D( p, q) ⊂ Op and, therefore, its image under exp−1 p in + T p (M) identifies a portion of a future directed null cone C constructed with respect to the flat metric η, where this portion is topologically equivalent to I × S2 , I ⊆ R. Yet all these properties are universal, thus they do not depend on the choice of a specific frame e at p. This is not the case for the form of the image of C p+ in C + under exp−1 p or the pull-back of the metric in normal coordinates under exp∗p . These clearly depend upon the coordinate system considered (individuated by e) and, hence, the possible choices of e and of coordinates on C p+ deserve a more detailed discussion. If one starts from the observation that the double cones of interest all lie in a normal neighbourhood, a first natural guess is to select the standard normal coordinates constructed out of the frame e. In this setting the metric can be expanded as 1 Rμανβ ( p)σ α (q, p)σ β (q, p) + O(3), 3 where σ (q, p) is the so-called Synge’s world function, i.e. half of the square of the geodesic distance between p and q (see Sect. 2.1 of [35]). Here σ α and σ β denote the covariant derivatives of σ performed at p, whereas O(3) is a shortcut to stress that the metric is approximated up to cubic quantities in the normal coordinates. Unfortunately, both the coordinate system and the expansion are not well suited to be used in the analysis of the geometry of the null cone C p+ , since one would like to have a local chart where it is manifest that C p+ is a null hypersurface. Furthermore, for later purposes, we also need to discuss some properties of the metric in the vicinity of C p+ as a whole and not only in a neighbourhood of p. To this end, it is useful to employ the so called retarded coordinates as introduced in [36,37]. We also refer to the review [35], which has the advantage to clearly discuss the explicit relation between these new coordinates and the normal ones (or, also, the Fermi-Walker ones). Let us briefly recall the construction of these retarded coordinates. Consider a timelike geodesic γ˜ through p with unit tangent vector u. In this setting one can define a coordinate r as the field . r (q) = −σα (q, p ) u α ( p ), gμν (q) = ημν −
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
467
where q ∈ O and p ∈ γ˜ are connected by a light-like geodesic originating from p and pointing towards the future. With σα (q, p ) we mean the covariant derivative at p of the geodesic distance. The net advantage of r is that, on C p+ , it can be read as an affine parameter of the null geodesics emanating from p. In other words, once an orthonormal frame e is chosen in such a way that e0 ( p ) = u, the scalar field r on C +p is unambiguously fixed. We can now define the full retarded coordinates as (u, r, x A ), where u labels the family of forward null cones with tips lying on γ˜ (see Eq. (154) and the preceding discussion in [35]), and C p+ = p ∈ D( p, q) | u( p ) = 0 , while x A are local coordinates on S2 . Notice that one could alternatively switch to the more common local chart (θ, ϕ) of S2 at p and we shall do so whenever needed. Moreover, in this coordinate system, the most generic form of the metric reads [12] ds 2 = −α du 2 + 2υ A du d x A − 2e2β du dr + g AB d x A d x B ,
(2.3)
where α, υ A , β and g AB are smooth functions depending on the coordinates. Notice that here r ∈ (0, ∞), while u ranges over an open set I ⊆ R which contains 0. Moreover, the x-coordinates on the sphere give rise to a volume element with respect to (2.3) of the form |g AB | d x A ∧ d x B = |g AB | | sin θ | dθ ∧ dϕ, (2.4) where the symbol | · | under the square root is kept to recall that we are actually referring to the determinant of the matrices involved. Notice also that, depending on the chosen coordinates x A on S2 , the switch to (θ, ϕ) yields a harmless additional contribution to the metric coefficients; this justifies the two symbols g AB and g AB , although, henceforth, we shall mostly stick to the last one. It is also remarkable that, whenever Rμν γ˙ μ γ˙ ν = 0, γ a generator of C p+ , one can prove that, on C p+ , (2.3) simplifies (see formula (2.36) in [12]) to ds 2 = −α du 2 − 2 du dr + r 2 (dθ 2 + sin2 θ dϕ 2 ),
(2.5)
where the standard coordinates (θ, ϕ) on the 2-sphere are used in place of x A . Apart from being much simpler, this form is more closely related to the standard Bondi one which is canonically used in the implementation of bulk-to-boundary techniques as devised in [13,14] for a large class of asymptotically flat and of cosmological spacetimes. Unfortunately, contrary to these papers, here the scenario is much more complicated and, furthermore, the cone does not seem to display any particular symmetry group to be exploited, such as for example the BMS in [13]. Yet, the situation is not as desperate as one might think, since, ultimately, for our purposes it will only be relevant that the metric at p becomes the Minkowski one, our coordinates being √ constructed out of an orthonormal frame at p. In particular, this means that, at p, |g AB | will become proportional to r , which, in this special scenario, can be seen both as the affine null parameter introduced above, or, equivalently, as the standard radial coordinate in Minkowski spacetime constructed out of the orthonormal frame in T p (M) ∼ R4 . Before concluding this section, we briefly compare (2.5) with the corresponding expression in Minkowski spacetime, where the flat metric can be written as ds 2 = −dU 2 + 2 dU dr + r 2 (dθ 2 + sin2 θ dϕ 2 ),
(2.6)
468
C. Dappiaggi, N. Pinamonti, M. Porrmann
. U = t + r denoting the light coordinate constructed out of the time and spherical coordinates. The cone with tip at 0 is characterised by U = 0 and, also in this case, r is an affine parameter along the null geodesics emanating from 0. It is important to stress that the pull-back of (2.3) under exp∗p tends to (2.6) when approaching the point exp p (0) = p. Finally, we comment on the behaviour of D( p, q) under (2.1). As mentioned before, exp−1 p does not map a double cone in M into one in T p (M), but, nonetheless, we can still adapt the choice of q in such a way that ı e,e (D( p, q)) is properly contained in a sufficiently large double cone D( p, q ) ⊂ Op . 3. Algebras of Observables on the Bulk and on the Boundary In the previous section, we focused on the introduction and analysis of the main geometric tools needed. In particular, we recall once more that the main geometrical objects are the double cones D( p, q) which are globally hyperbolic spacetimes in their own right. Since, in the forthcoming discussion, neither p nor q will play a distinguished role, we shall omit them, hence using D in place of D( p, q). More importantly, we are now entitled to introduce a well-defined classical field theory and, for the sake of simplicity, we shall henceforth only deal with a free real scalar field with generic mass m and generic curvature coupling ξ . Let us recollect some standard properties of such a physical system along the lines, e.g., of [47]. Consider ϕ : D → R which fulfills the following equation of motion:
. Pϕ = g + ξ R + m 2 ϕ = 0, m 2 > 0 and ξ ∈ R, (3.1) where g = −∇ μ ∇μ is the d’Alembert wave operator constructed out of the metric g while R is the scalar curvature. Since this is a second-order hyperbolic partial differential equation, each solution with smooth and compactly supported initial data on a Cauchy surface can be constructed as the image of the following map: : C0∞ (D) → C ∞ (D),
(3.2)
where is the causal propagator defined as the difference of the advanced and the . retarded fundamental solutions. Furthermore, each ϕ f = ( f ) satisfies the following support property: supp(ϕ f ) ⊆ J + (supp( f )) ∪ J − (supp( f )), and, if S(D) denotes the set of solutions of (3.2) with smooth compactly supported initial data on any Cauchy surface of D, then this turns out to be a symplectic space when endowed with the weakly non-degenerate symplectic form,
σ (ϕ f , ϕh ) = dμ() ϕ f ∇n ϕh − ϕh ∇n ϕ f = dμ(D)( f h), ∀ f , h ∈ C0∞ (D).
D
(3.3) Here the integral is independent of the choice of Cauchy surface , as can be noticed from the last equation, while dμ(), dμ(D) and n are, respectively, the metric-induced measures and the vector normal to . As a last ingredient, these properties can be exploited in combination with the fact that, by construction, D is contained in a larger globally hyperbolic open set (O in the notation of the previous section), in order to conclude that ϕ f can be unambiguously
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
469
extended to a solution of the very same equation throughout O . This can be proved by recalling that both D and O are globally hyperbolic and by invoking the uniqueness of the causal propagator. As a consequence we are entitled to consider the restriction of ϕ f on C p+ which yields ϕ f C p+ ∈ C ∞ (C p+ ). (3.4) 3.1. Quantum algebras on D. After the setup of a classical field theory, we consider a suitable quantisation scheme to be described as a two-fold process: in a first step we shall select a suitable algebra of fields which fulfills the necessary commutation relations and, second, we choose a quantum state as a functional on this algebra in order to compute the expectation values of the relevant observables. Thus let us proceed in logical sequence starting from D, the bulk spacetime, where we can introduce Fb (D) as the subset of sequences with a finite number of elements lying in ⊗ns C0∞ (D), n≥0
where n = 0 yields C by definition while ⊗ns denotes the n-fold symmetric tensor product. According to this definition it is customary to denote a generic F ∈ Fb (D) as a finite sequence {Fn }n , where each Fn ∈ ⊗ns C0∞ (D). We can now promote Fb (D) to a topological ∗ -algebra equipping it with • a tensor product · S such that S(F p ⊗ G q ), (F · S G)n = p+q=n
where S is the operator which realises total symmetrisation; • a ∗ -operation via complex conjugation, i.e., {Fn }∗n = {F n }n for all F ∈ Fb (D); • the topology induced by the natural one of ⊗ns C0∞ (D). The above more traditional realisation of Fb (D) can be replaced by a novel point of view, thoroughly developed in [7,5]. To be specific, consider Fb (D) as a suitable subset of the functionals over C ∞ (D), the smooth field configurations. Explicitly, F ∈ Fb (D) yields a functional F : C ∞ (D) → R out of the standard pairing between ⊗n C ∞ (D) and ⊗n C0∞ (D), denoted by , , via ∞
. 1 Fn , ϕ n . F(ϕ) = n!
(3.5)
n=0
In order to grasp the connection between the two perspectives, it is useful to introduce a Gâteaux derivative, dn (n) ⊗n F(ϕ + λh) , ∀h ∈ C ∞ (D), F (ϕ)h = dλn λ=0
F (n) (0).
so that Fn ≡ We shall use alternatively both pictures in the forthcoming analysis. The key point in the subsequent quantisation scheme consists in the modification of the algebraic product · S to yield a new one, , which is constructed out of the causal propagator , unambiguously defined according to (3.2),
470
C. Dappiaggi, N. Pinamonti, M. Porrmann
(F G)(ϕ) =
∞ i n (n) ⊗n (n) F (ϕ), G (ϕ) , 2n n!
∀F, G ∈ Fb (D).
(3.6)
n=0
By direct inspection one realises that F G still lies in Fb (D) and, more importantly, that (Fb (D), ) gets the structure of a ∗ -algebra under the operation of complex conjugation. It is important to notice that, up to now, we have not used the existence of the equation of motion (3.1) and, therefore, we can refer to Fb (D) as an off-shell algebra. On the other hand, if one wants to encompass also the dynamics of the classical system, one needs only to divide Fb (D) by the ideal I which is the set of elements in Fb (D) generated by those of the form P j Fn (x1 , . . . , xn ), where P j is the operator in (3.1) applied to the . j th variable in Fn ∈ ⊗ns C0∞ (D). The outcome is the on-shell algebra Fbo (D) = FbI(D ) which inherits the ∗ -operation from Fb (D) and is nothing but the more commonly used field algebra. At this stage, it is important to remark that neither Fb (D) nor its on-shell version Fbo (D) contain all the elements needed to fully analyse the underlying quantum field theory. As a matter of fact, objects such as the components of the stress-energy tensor involve products of fields evaluated at the same spacetime point, an operation which is a priori not well-defined due to the distributional nature of the fields themselves. To circumvent this obstruction, a standard procedure calls for the regularisation of these illdefined objects by means of a suitable scheme which goes under the name of Hadamard regularisation. We shall not dwell here on the technical details but only highlight some aspects in the Appendix. The interested reader is referred to [23,24] for a full account. In the functional language used before, the problem mentioned above translates into the impossibility to include in Fb (D) objects of the form dμ(g) f (x)ϕ 2 (x), F(ϕ) = D
where dμ(g) is the metric-induced volume form, while f is a test function in C0∞ (D) and ϕ ∈ C ∞ (D). Actually, the star product (3.6) applied to a couple of such fields involves the ill-defined pointwise product of with itself. To solve this problem, we shall follow the line of reasoning of [7]. Namely, we introduce a new class of functionals, Fe (D), which have a finite number of non-vanishing derivatives, the n th of which has to be a symmetric element of the space of compactly supported distributions E (D n ), whose wave front sets, moreover, satisfy the following restriction:
+ n − n = ∅, (3.7) ∪ D×V WF(Fn ) ∩ D × V ±
where V corresponds to the forward and to the backward causal cone in the tangent space, respectively. The closure symbol indicates that we also include the tip of the cone in the set of future or past directed causal vectors. We can make Fe (D) a ∗ -algebra if we extend naturally the ∗ -operation of Fb (D) and endow it with a new product, H , whose well-posedness was first proved in [6,8,23,24]. The explicit form is realised as (F H G)(ϕ) =
∞ 1 (n) F (ϕ), H ⊗n G (n) (ϕ) , n! n=0
(3.8)
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
471
where H ∈ D (D 2 ) is the so-called Hadamard bi-distribution. We shall briefly introduce and discuss it in the Appendix, but, for our purposes, it is important to recall that, on the one hand, it satisfies the microlocal spectrum condition, hence yielding a natural substitute for the notion of positivity of energy out of its wave front set, while on the other hand it suffers from an ambiguity. At the level of the integral kernel, only the antisymmetric part of the Hadamard bi-distribution is fixed to i/2 times the causal propagator . Also the singular structure is unambiguously determined by the choice of the background. Otherwise there always exists the freedom to add a smooth symmetric function which, in our scenario, means that, if H, H are two Hadamard distributions, then the integral kernel of H − H is a symmetric element of C ∞ (D 2 ). Yet, as far as the algebra is concerned, this freedom boils down to an algebraic isomorphism i H ,H : (Fe (D), H ) → (Fe (D), H ), namely (cf. [5,23]) i H ,H = α H ◦ α −1 H , ∞ . 1 H ⊗n , F (2n) . α H (F) = n!
(3.9)
n=0
As for the algebra generated by compactly supported smooth functions, also the extended one, Fe , has its on-shell counterpart, Feo , constructed from the quotient with the ideal generated by the equation of motion applied to the elements of Fe . One of the advantages of the formalism employed is the possibility to easily transcribe the overall construction in terms of categories, hence yielding a crystal-clear mathematical picture of the relevant structures and their relations. This was first advocated and utilised in the seminal paper [9], where the principle of general local covariance was formulated in this language to which we shall also stick. In particular, we shall now recast the above discussion in this different perspective, while the actual relation with [9] will only later be outlined in Sect. 4. Hence, we shall use the following categories: DoCo: The objects are defined as follows: For every spacetime M, as in the previous section, we consider the oriented and time-oriented double cones D( p, q) with the property that there exists a normal neighbourhood O p ⊂ M centred in p that contains D( p, q). Hence, an object is a triple D ≡ [D( p, q), O p , e]. Recall that since both O p and the double cones are globally hyperbolic, the choice of time and space orientation is always possible. The morphisms are the maps ı e,e : O p → O p introduced in (2.1) such that ı e,e D (D( p, q)) ⊂ D ( p , q ), furthermore they maps e to e . Although the subscript which refers to e and e is a small abuse of notation, we feel that its presence might help the reader to focus on the ingredients here at play. The composition rule for the morphisms is defined in terms of the composition of the maps ı e,e , and hence the associativity derives from the associativity of the composition. Notice that, as per (2.1), an arrow between two objects exists only if the source double cone is sufficiently small so that its image under exp−1 p is contained in the domain of the definition of exp p . This caveat does not spoil the associativity property of the composition of arrows. DoCoiso : This is the subcategory of DoCo obtained by keeping the same objects but restricting the possible morphisms of DoCo to those which are isometric embeddings. Algi : The objects are unital, topological, ∗ -algebras Fi (D) with i = b, bo, constructed as above, while for i = e, eo, we further restrict the objects to the equivalence classes generated by identifying extended algebras which are isomorphic under the map (3.9). The morphisms are equivalence classes of injective, unit-preserving, ∗ -homomorphisms.
472
C. Dappiaggi, N. Pinamonti, M. Porrmann
Since the key ingredients to construct both Fb (D) and Fbo (D) are just the causal propagator from (3.2) and the operator (3.1) realising the equations of motion, their uniqueness in any D suggests that the association of a suitable algebra, Fi : D → Fi (D),
i = b, bo,
(3.10)
with each double cone D can be promoted to a functor between DoCoiso and Algi . To this end, in order to define the action of Fi on morphisms of DoCoiso , notice that Fi (D) is an algebra generated by smooth and compactly supported functions on D, hence Fi (ı e,e ) is just the injective ∗ -homomorphism which associates with every element in Fi (D) its image under ı e,e . Furthermore, Fi defined in that way enjoys the covariance property and maps idD , the identity of D in DoCoiso , to idFi (D ) , the identity of Fi (D) in Algi , i.e., Fi (ı e,e ) ◦ Fi (˜ı e ,e˜ ) = Fi (ı e,e ◦ ı˜e ,e˜ )
Fi (idD ) = idFi (D ) .
and
Notice that, in the case of the on-shell algebra, the equation of motion is left unchanged by any morphism in DoCoiso . It is also important to note that singling out extended algebras which are related via (3.9) ensures that the ambiguity in the choice of the Hadamard bi-distribution does not spoil the well-posedness of (3.10) when i = e. All these assertions can be proved noticing that, due to the discussion presented after (2.1), the category DoCoiso is just a subcategory of the category of local manifolds introduced in [9] where similar results are discussed. It would be desirable to extend the functor (3.10) to the category DoCo that has a larger group of morphisms. Unfortunately, this is not straightforward and, actually, not even possible. If we consider two generic globally hyperbolic regions D and D in DoCo, related by ı e,e as in (2.1), we can draw the following diagram: F
D −−−−→ Fi (D) ⏐ ı e,e ⏐ D −−−−→ Fi (D ) F
(3.11) .
To have a functor between DoCo and some Algi requires existence of a well-defined morphism F (ı e,e ) between Fi (D) and Fi (D ). But this is not possible since, in general, ı e,e is not an isometry and, thus, it does neither map solutions of (3.1) in D into those of the same equation (but out of a different metric) in D , nor does it preserve the causal propagator. Hence it spoils the -operation and does not preserve the singular structure of the Hadamard bi-distribution which only depends upon the underlying geometry. As an aside, a positive answer to the present question will only be possible considering the off-shell classical ∗ -algebras (Fb , · S ), but, as soon as quantum algebras are employed, the situation looks grim. Yet it is possible to circumvent this obstruction making profitable use of the geometric properties of the portion of the future directed light cone in any D in order both to set up a bulk-to-boundary correspondence and to later compare the outcome on the boundary of different spacetimes.
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
473
3.2. Quantum field theory on the boundary. In order to fulfill our goals, it is mandatory as a first step to understand how to construct a full-fledged quantum field theory on the light cone and this will be the main aim of this subsection. Our procedure derives from the experience gathered in similar scenarios where a field theory on a null surface was constructed such as, e.g., in [13–15,31] (see also [11,21,43] for further analyses in similar contexts). Therefore, following the same philosophy as in these papers, we shall first show that it is possible to assign to the boundary a natural field algebra and that there is a “natural” choice for the relevant quantum state. To this avail, in this subsection, we shall consider the cone as an abstract manifold on its own, not regarding it as a particular portion of the boundary of a specific globally hyperbolic double cone D, since the connection with the bulk theory will be presented only later. Nevertheless, we have to keep in mind that the algebra to be constructed has to be large enough in order to contain the images of some suitable projections of all the elements of the algebra in the bulk D. This will be the most delicate point of the whole construction because it is not sufficient to consider an algebra generated by compactly supported data on the cone; we have to extend this set to more generic elements. Let us start with the introduction of three distinct sets in R × S2 ⊂ R4 relevant for the following construction, namely, employing the standard coordinates, we define C p+ = (V, θ, ϕ) ∈ R × S2 | V ∈ I ⊂ R, (θ, ϕ) ∈ S2 , (3.12) where I is the open interval (0, V0 (θ, ϕ)) with a positive, bounded smooth function V0 (θ, ϕ) on the sphere. The other two regions will be denoted C p and C , respectively, where the coordinate V is allowed to extend over (0, ∞) or the full real line, so that C p+ ⊂ C p ⊂ C . We stress that, with a slight abuse of notation, we employ the symbol C p+ as in the previous sections although we are not referring to an actual cone since, ultimately, (3.12) will indeed coincide with J p+ ∩ ∂D, if we employ the same conventions and nomenclatures as in the preceding analysis. Furthermore notice that C is not completely independent of p which still is required to be a point in this set. Yet, to avoid worsening an already cumbersome notation, we leave such a dependence implicit. As a natural next step, we need to identify a suitable space of functions on the boundary and, to this avail, viewing C p immersed in R4 , we define . S (C p ) = ψ ∈ C ∞ (C p ) | ψ = h f C p , f ∈ C0∞ (R4 ) and h ∈ C ∞ (C p ) , (3.13) where h vanishes uniformly on S2 , as V → 0, while each derivative along V tends to a constant uniformly on S2 . As far as this subsection is concerned we can safely choose h to be equal to V . Furthermore, S (C p ) turns out to be a symplectic space when endowed with the following strongly non-degenerate symplectic form: dψ dψ . ψ − ψ d V ∧ dS2 , ∀ψ, ψ ∈ S (C p ), (3.14) σC (ψ, ψ ) = dV dV Cp where dS2 is the standard measure on the unit 2-sphere. The reason for such an apparently strange choice of S (C p ) is the later need to relate the theory on these sets to the one in the bulk of a double cone. Hence the most natural choice of compactly supported smooth functions on C p would not fit into the overall picture since a general solution of the Klein-Gordon equation (3.1) with smooth compactly
474
C. Dappiaggi, N. Pinamonti, M. Porrmann
supported initial data on some D would propagate on the light cone C p to a function which is also supported on the tip. This point corresponds to V = 0 in the above picture and, thus, it does not lie in C p . On the contrary, we shall show in the next subsection that (3.13) is indeed the natural counterpart on the boundary which arises from the set of solutions of (3.1). In order to introduce the relevant algebra of observables, we follow the same philosophy as in Subsect. 3.1, introducing Ab (C p ), whose generic element F is a sequence {Fn }n with a finite number of elements in ⊗ns S (C p ), (3.15) n≥0
where ⊗ns again denotes the symmetrised n-fold tensor product and the first term in the sum is C. Notice that the -superscript is introduced in this subsection in order to avoid a potential confusion with the similar symbols used for the counterpart in the bulk. In order to promote (3.15) to a full topological ∗ -algebra, we have to endow it with • a ∗ -operation which is the complex conjugation, i.e., {Fn }∗n = {Fn }n for all F ∈ Ab (C p ); • multiplication of elements such that for any F , G ∈ Ab (C p ), S(F p ⊗ G q ); (F · S G )n = p+q=n
• the topology induced by the natural topology of S (C p ), namely the topology of smooth functions on C p . Although well-defined, this algebra is not suited to be put in relation to data in the bulk and, thus, the above product has to be deformed once more. To this avail, we employ . the functional point of view as in (3.5), i.e., if X = ∈ C ∞ (C p ) | V −1 ∈ C ∞ (C ) , then F ∈ Ab (C p ) yields a functional F : X → R out of , , the pairing between ⊗n X and ⊗n S (C p ), F () =
∞ 1 n F , , ∀ ∈ X. n! n n=0
By direct inspection of the definition of S (C p ) in (3.13), one notices that , is nothing but the standard inner product on (0, ∞) × S2 between compactly supported functions and smooth ones, taken with respect to the measure d V ∧ dS2 . Although the theory on C p has no equation of motion built in and, hence, no causal propagator such as (3.2), we can nonetheless introduce a new B -product on Ab (C p ), namely ∞
. in Fn (), nσC G n () , ∀ ∈ X, (F B G )() = n 2 n!
(3.16)
n=0
where σC is the integral kernel of (3.14), i.e.,
σC (V, θ, ϕ), (V , θ , ϕ ) = −
∂2 sign(V − V ) δ(θ, θ ), ∂V ∂V
(3.17)
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
475
where δ(θ, θ ) stands for δ(θ − θ )δ(ϕ − ϕ ) and the derivatives have to be taken in the weak sense. Notice that σC is defined as a distribution on C0∞ (C p2 ) which, by direct inspection, can be extended also to S 2 (C p2 ). Finally, B is well-defined because only a finite number of elements appear in the sum on the right-hand side of (3.17) and, thus, convergence is not an issue here. We can hence conclude this subsection with a proposition whose proof follows from the preceding discussion.
Proposition 3.1. The pair Ab (C p ), B equipped with the ∗ -operation introduced above is a well defined ∗ -algebra. In the next section we shall discuss the form of a certain class of quantum states on this algebra to eventually use them to extend the boundary algebra of observables analogous to the procedure on the bulk. 3.3. Natural boundary states. The next step in our construction is the introduction of an extended algebra on the boundary, but, in the present scenario, there is no standard definition of an Hadamard state or of a bi-distribution; and this lack of a class of a priori physically relevant states hinders an imitative repetition of the use of the function H as in (3.8). Therefore, a natural bi-distribution on C p+ is needed and, for this purpose, our choice is the following weak limit: .
1 1 δ(θ, θ ), (3.18) ω (V, θ, ϕ), (V , θ , ϕ ) = − lim+ π →0 (V − V − i)2 which has the advantage of being at the same time a well-defined element of D (C 2 ), where C ∼ R × S2 and D denotes the space of distributions over the test functions in C0∞ (C ). Such an expression already appeared in different, albeit related, scenarios where a scalar quantum field theory was studied [13,15,16,27,31,43,44]. It is important to remark that in the first two of these papers, (3.18) was actually used as the building block to construct a quasi-free pure state for a scalar field theory built on a three-dimensional null cone which represented the conformal boundary of a suitable class of spacetimes. In all these cases, the particular geometric structure as well as the presence of a particular symmetry group entails that (3.18) fulfills suitable uniqueness properties and, furthermore, gives rise to a full-fledged Hadamard state in the bulk. Therefore we shall employ the above expression as the natural candidate bi-distribution on the boundary, proving in the next sections that, when we realise C p+ as part of the boundary of a double cone, we can also construct a physically meaningful state on the bulk D out of (3.18). The above bi-distribution can be read as a functional ω : S (C p ) × S (C p ) → R, but it is actually much more convenient to recall that C p ⊂ C . Within this perspective, since C is topologically R × S2 and each element in S (C p ), together with the V -derivative, also lies in L 2 (C , d V ∧ dS2 ), the following expression for the 2-point function is meaningful: 1 ψ(V, θ, ϕ)ψ (V , θ, ϕ) d V d V dS2 (θ, ϕ) , (3.19) ω(ψ, ψ ) = − lim+ π →0 R2 ×S2 (V − V − i)2 where the delta function over the angular coordinates is already integrated out. The distribution (3.19) satisfies a suitable continuity condition, as for example shown in [30],
|ω(ψ, ψ )| C ψ L 2 ∂V ψ L 2 + ψ L 2 + ∂V ψ L 2 < ∞, (3.20)
476
C. Dappiaggi, N. Pinamonti, M. Porrmann
which allows for the extension of ω to the space of square-integrable functions whose derivative along the V -coordinate is also an element of L 2 . Furthermore, this also entails the possibility to perform a Fourier-Plancherel transform along V (cf. Appendix C in [30]) resulting in a much more manageable form for (3.19), (k, θ, ϕ) ψ (k, θ, ϕ) , ω(ψ, ψ ) = dk dS2 (θ, ϕ) 2k (k) ψ (3.21) R×S2
where (k) is the step function equal to 1 if k 0 and 0 otherwise. It should be stressed that the presence of (k) corresponds to the physical intuition of taking only positive frequencies, also because, on the cone, the only causal directions are the lines at constant angular variables. Under special circumstances this idea has a clear connection with the geometrical bulk data as well as with the Hadamard property of a bulk state constructed out of (3.18) (cf. for example [15,29,30]). As a last step in this subsection, we underline that the above analysis entails two relevant remarks. The first one concerns the wave front set of the bi-distribution ω on C 2 ∼ (R × S2 )2 . This was already studied in Lemma 4.4. of [30] yielding WF(ω) ⊆ A ∪ B, where A=
(V, θ, ϕ, ζV , ζθ , ζϕ ), (V , θ , ϕ , ζV , ζθ , ζϕ ) ∈ (T ∗ C )2 \{0} | V = V , θ = θ , ϕ = ϕ , 0 < ζV = −ζV , ζθ = −ζθ , ζϕ = −ζϕ
and B=
(3.22)
(3.23)
(V, θ, ϕ, ζV , ζθ , ζϕ ), (V , θ , ϕ , ζV , ζθ , ζϕ ) ∈ (T ∗ C )2 \{0} | θ = θ , ϕ = ϕ , ζV = ζV = 0, ζθ = −ζθ , ζϕ = −ζϕ . (3.24)
Although, at this stage, this result is only an aside, it will play a pivotal role in the discussion of Subsecs. 3.4 and 3.5. In particular, if we recall that C p ⊂ C , it turns out that the wave front set of ω on C0∞ (C p2 ) can only be smaller or equal to A ∪ B, and actually it corresponds to A ∪ B restricted to (T ∗ C p )2 . As a second remark, note that it is possible to construct a new algebra, say Ab (C ) on the full C starting from (3.13) and considering the set L 2 (C , d V ∧ dS2 ) in place of S (C p ), while keeping the same ∗ -operation and composition rule. On the one hand, it is straightforward, that Ab (C p ) is a ∗ -subalgebra of Ab (C ) while, on the other hand, we can see that the two-point function ω as in (3.18) can be used as a building block of a quasi-free state for Ab (C ). Hence the same conclusion can be drawn for Ab (C p ) since, by construction, the antisymmetric part of ω is equal to 2i σ . The only possible issue is positivity, but this is solved by direct inspection of (3.21) whose right-hand side is manifestly greater than 0 once ψ = ψ . It is important to point out, for the sake of completeness, that an almost identical analysis appears in [13,15], though performed at the level of Weyl algebras. In summary, we get Proposition 3.2. The Gaussian (quasi-free) state constructed out of the distribution ω enjoys the following properties: 1. It is a well-defined algebraic state on Ab (C p ) and on Ab (C ).
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
477
2. It is a vacuum with respect to ∂V , in the sense given in [40]. 3. It is invariant under the change of the local frame, hence invariant under the action of S O0 (1, 3). Proof. The first point can be analysed by checking linearity, positivity and normalisability of the state on Ab (C p ). Since the state is quasi-free, it is enough to examine these properties for the functional ω on S (C p ) × S (C p ), where they follow from the previous discussion. The proof of the second point derives from the observation that ω, as a state on Ab (C ), is invariant under translations and from the fact that the Fourier-Plancherel transform of the integral kernel of ω along the V -direction only contains positive frequencies, as is clear from (3.21). The third point can be proved recalling a result in [13], namely, ω on Ab (C ) is invariant under the action of an infinite-dimensional group, the so-called Bondi-MetznerSachs group (BMS). In short, if one switches from the coordinates (V, θ, ϕ) to (V, z, z¯ ) obtained out of a stereographic projection, the BMS maps . z → z = (z) = az+b cz+d , ad − bc = 1, a, b, c, d ∈ C, V → V = K (z, z¯ )(V + α(z, z¯ )),
where z¯ transforms as the complex conjugate of z, α(z, z¯ ) ∈ C ∞ (S2 ) and . K (z, z¯ ) =
1 + |z|2 . |az + d|2 − |bz + c|2
Hence, by direct inspection of the above formulae, the BMS group is seen to be the regular semidirect product S L(2, C) C ∞ (S2 ). Most notably one observes that there exists a proper subgroup which is a homomorphism to S O(3, 1) and thus the state ω turns out to be invariant under the group sought-for. 3.4. Extended algebra on the boundary. In the previous subsection, we introduced the boundary algebra together with a suitable notion of -product, but this is still not sufficient to intertwine the boundary data with those on the bulk because we lack a counterpart for the extended algebra of observables on C p . Yet, due to the results of the last subsection, we have all the ingredients to construct it. As a starting point, we define the building block of the extended algebra as follows. Definition 3.3. We call An the set of elements Fn ∈ D (C pn ) that fulfill the following properties: 1. Compactness: The Fn are compact towards the future, i.e., the support of Fn is contained in a compact subset of C n ∼ (R × S2 )n . 2. Causal non-monotonic singular directions: The wave front set of Fn contains only causal non-monotonic directions which means that + − . WF(Fn ) ⊆ Wn = (x, ζ ) ∈ (T ∗ C p )n \{0} | (x, ζ ) ∈ V n ∪ V n , (x, ζ ) ∈ Sn , (3.25) +
where (x, ζ ) ≡ (x1 , . . . , xn , ζ1 , . . . , ζn ) ∈ V n if, employing the standard coordinates on C p , for all i = 1, …, n, (ζi )V > 0 or ζi vanishes. The subscript V here refers
478
C. Dappiaggi, N. Pinamonti, M. Porrmann −
to the component along the V -direction on C p . Analogously, we say (x, ζ ) ∈ V n if every (ζi )V < 0 or ζi vanishes. Furthermore, (x, ζ ) ∈ Sn if there exists an index i such that, simultaneously, ζi = 0 and (ζi )V = 0. 3. Smoothness Condition: The distribution Fn can be factorised into the tensor product of a smooth function and an element of An−1 when localised in a neighbourhood of V = 0, i.e., there exists a compact set O ⊂ C p such that, if ∈ C0∞ (C p ) so that it . is equal to 1 on O and = 1 − , then for every multi-index P in {1, . . . , n} and for every i n, . f = F˜n (u x Pi+1 ,...,x Pn ) x P · · · x P ∈ C ∞ (C Pi ), 1
i
(3.26)
where F˜n : C0∞ (C pn−i−1 ) → D (C pi ) is the unique map from C0∞ (C pn−i−1 ) to D (C pi ) determined by Fn using the Schwartz kernel theorem. Furthermore, u x Pi+1 ,...,x Pn ∈ C0∞ (C pn−i ), and we have specified the integrated variables x Pi+1 , …, x Pn . For every j i, ∂V1 · · · ∂V j f lies in C ∞ (C pi )∩L 2 (C pi , d V P1 ∧dS2P1 · · · d V Pi ∧dS2Pi )∩L ∞ (C pi ), while the limit of f as V j tends uniformly to 0 vanishes uniformly in the other coordinates. Remark 3.4. a) In Property 2 of Definition 3.3 we required that WF(Fn ) ∩ Sn = ∅, viz., no spatial directions are present in the wave front set of Fn . Even if such an extra condition is not stipulated in the definition of the elements on Fe , here we have to add it because, later on, we want to multiply elements of A with ω and in the wave front set of ω there are spatial directions, to be specific, the intersection WF(ω) ∩ S2 is not empty. b) Thanks to the smoothness condition, the last requirement in Definition 3.3, the distributions in An can be extended over ⊗n S (C p ) and such an extension is unique. Notice that S (C p ) is strictly contained in A and that the candidate to play the role of the extended algebra on C p thus is Ae (C p ) = Ans , n 0
where Ans is the subset of totally symmetric elements in An defined in Definition 3.3. Moreover, the first space in the previous direct sum is C and only sequences with a finite number of elements are considered. We can now endow this set with the structure . of ∗ -algebra by introducing the ∗ -operation {Fn }∗ = {Fn } for all F ∈ Ae . The composition law arises from a modification of B by means of the state constructed in the sequel of (3.18). It is a priori clear that such a procedure intrinsically depends on the particular ω considered. Nonetheless, our choice will later be justified both through its connection with the bulk data and by well-posedness of the new structure. If we stick to the functional representation, we can thus introduce ω : Ae (C p ) × Ae (C p ) → Ae (C p ), ∞ 1 (n) F (), ωn G (n) () , (F ω G )() = n! n=0
for all F , G ∈ Ae (C p ) and for all ∈ C ∞ (C p ).
(3.27)
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
479
Proposition 3.5. The operation (3.27) is a well-defined product in Ae . Proof. As a starting point, notice that (3.27) is bilinear by construction and that, by definition of Ae (C p ), there are only a finite number of non-vanishing elements F (n) and G (n) . Accordingly, (3.27) consists of finite linear combinations of terms that formally look like S
C p2k
F j (x1 , . . . , x j ) ω(x1 , y1 ) · · · ω(xk , yk ) G l (y1 , . . . , yl )
k
dμ(xi ) dμ(yi )
i=1
(3.28) with k j and k l, while S realises symmetrisation in the non-integrated variables (xk+1 , . . . , x j , yk+1 , . . . , yl ) and dμ is the measure d V ∧ dS2 (θ, ϕ) on C p , written here in the usual coordinates. Therefore, the proof amounts to showing that it is possible to give a rigorous meaning to expressions like (3.28) and that the result of such an operation is an element of A j+l−2k . First, we follow the proof of [23], and, to this avail, examine if (3.28) can be seen as the target of 1 ∈ C ∞ (C p2k ) under the linear map determined, with the help of the Schwartz kernel theorem, by the distribution resulting from multiplication of the two distributions ω⊗k ⊗ I ⊗( j+l−2k) ∈ D (C j+l ) and F j ⊗ G l ∈ A j+l , where I denotes the identity operator on A1 . Let us start by discussing the well-posedness of the multiplication of distributions presented above. This can be checked by examining the structure of the wave front set of the single objects to verify that their composition never contains the zero section. The key ingredients for this can be readily inferred using Theorem 8.2.9 in [25],
WF(F j ⊗ G l ) ⊂ W j ∪ {0} × (Wl ∪ {0}) \{0}, (3.29) and WF(ω⊗k ⊗ I j+l−2k ) ⊂ ( A ∪ B ∪ {0})k × {0}\{0},
(3.30)
where, as usual, we have not specified the dimension of the zero section in the cotangent space. Furthermore, A, B and W j are defined in (3.23), (3.24) and (3.25), respectively. It is now possible to apply Theorem 8.2.10 in [25] since the above wave front sets never sum up to the zero section. This is tantamount to realising that for every n and m, since + − ± An ⊂ V n × V n and V n ∩ Wn = ∅, B n × {R3 }m ∩ W2n+m = ∅ and An ∩ (Wn × Wn ) = ∅. j+l The outcome is that, in (3.28), the pointwise product of F j ⊗ G l ∈ D (C p ) with ω⊗k ⊗ I j+l−2k ∈ D (C p ) is still a well-defined element of D (C p ), whose wave front set satisfies the inclusion
WF F j ⊗ G l · ω⊗k ⊗ I j+l−2k ⊂ WF(F j ⊗ G l ) ∪ {0} + WF ω⊗k ⊗ I j+l−2k ∪{0}, j+l
j+l
(3.31) where, as usual, the sum of two wave front sets is defined as the sum on the fibres of the cotangent spaces. Unfortunately, this does not suffice to show the well-posedness j of (3.28). Since F j and G l are not compactly supported on C p and C pl , respectively,
j+l their product does not lie in E C p . Hence we cannot directly test the linear map
480
C. Dappiaggi, N. Pinamonti, M. Porrmann
stemming from (F j ⊗ G l ) · ω⊗k ⊗ I j+l−2k on the unit constant function on C p2k in order to infer that (3.28) is an element in A j+l−2k . Let us hence proceed by in the case k = 1 we can test the linear map
showing that arising from (F j ⊗ G l ) · ω ⊗ I j+l−2 on 1 and that the result of this operation is an element of A j+l−2 . The case for a generic k arises from a recursive application of the very same procedure and, eventually, the application of an operator realising the total symmetrisation. Thus we are interested in
(F j ⊗ G l ) · ω ⊗ I j+l−2 dμ(x1 ) dμ(y1 ), C p2
where F j ∈ A j and G l ∈ Al . We exploit Property 3 of Definition 3.3 and notice that, if the smoothness condition holds for a compact set O, it also holds for every larger compact set O1 containing O ∈ C p . We can thus find a common set O1 for which the smoothness condition property is true at the same time for F j = ( + ) F j and G l = ( + ) G l with respect to a common compactly supported function equal to 1 on O1 . Effectively, the above integral is divided into the sum of four different ones, which we now analyse separately. Part I) The first term is
((x1 )F j ⊗ (y1 )G l ) · ω ⊗ I j+l−2 dμ(x1 ) dμ(y1 ). C p2
(3.32)
j+l In this case the integral can be considered as the smearing of a distribution in D C p with a test function in C0∞ (C p2 ). Hence, Theorem 8.2.12 of [25] ensures that, using the notation introduced there, the result of (3.32) is a distribution whose wave front set is contained in W j+l−2 ∪ (W j × Wl ) ◦ (A × {0}) ⊂ W j+l−2 as given in (3.25). Notice that, in the proof of the last inclusion, we have used (3.29) and (3.30). Hence Property 2 in Definition 3.3 holds. That said, Property 1 is automatically satisfied since, by hypothesis, F j ∈ A j and G l ∈ Al , while Property 3 holds true for the resulting distribution as (3.26) is valid a priori in all variables and, thus, left untouched for those which have not been integrated out in (3.32). Part II) The second term is
( (x1 )F j ⊗ (y1 )G l ) · ω ⊗ I j+l−2 dμ(x1 ) dμ(y1 ). C p2
(3.33)
The analysis is rather simple if we make profitable use of (3.26) in the integrated vari. ables and interpret the previous integral in the weak sense. Namely, let f = (x1 )F j (u)
. j−1 and f = (x j+1 )G l (u ) for some u ∈ C0∞ C pl−1 , then for some u ∈ C0∞ C p
we have that the operation ω( f, f ) is well defined due to the continuity property (3.20) satisfied by ω. Hence, Properties 1, 2 and 3 in Definition 3.3 hold true because they are satisfied by (x1 )F j ⊗ (y1 )G l . Parts III & IV) The remaining two terms are substantially identical and we treat only one of them. Hence consider
(F j ⊗ G l ) · ω ⊗ I j+l−2 dμ(x1 ) dμ(y1 ). (3.34) C p2
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
481
In order to cope with this integral we introduce a new larger factorisation η + η = 1 with η ∈ C0∞ (C p ) such that η = 1 on a large compact set properly containing the closure of supp() so that both supp(η ) ∩ supp() = ∅ and η = η . If now G l is substituted with (η + η )G l we obtain another splitting. On the one hand, since η ∈ C0∞ (C p ), the analysis of F j ⊗ ηG l boils down to that of Case I, while, on the other hand, (V )[ η ](V ) (V −V )2
turns out to be smooth on C p , since by construction (V − V )2 > 0 for V on the support of and V on that of η . Hence, if we write the smoothness
condition ∞ (3.26) by means of η as η (x j+1 )G l (tl−1 ) = f (x j+1 ), where tl−1 ∈ C0 C pl−1 , we . obtain that u = ω( f ) is a compactly supported smooth function on C , thus yielding that F j (u) is a well-posed operation as u is compactly supported. Furthermore, in order to conclude the analysis of the present case, due to Theorem 8.2.12 in [25], we notice that the wave front set of (3.34) is contained in W j+l−2 given in (3.25) and that Property 3 in Definition 3.3 holds true just by applying (3.26) before smearing it.
The result of this subsection is that Ae (C p ), ω is a full-fledged topological ∗ -algebra. Furthermore,
due to the compactness property stated in Definition 3.3, the subalgebra + (Ae C p ), ω defined by restriction of the domain of the test distributions to C p+ is a well-defined topological ∗ -algebra and, thus, we are ready to discuss the intertwining relations between bulk and boundary data. 3.5. Interplay between the algebras and the states on D and on C p+ . We are now in the position to discuss a connection between the field theories described above, hence setting up a bulk-to-boundary correspondence and identifying an Hadamard state in the bulk. The whole subsection is devoted to this issue, but, as a starting point, we need to recapitulate the geometric structure in order to clearly relate Subsects. 2.2 and 3.2. Recall that we consider the globally hyperbolic subset O contained in a geodesic neighbourhood of an arbitrary but fixed point p in a strongly causal spacetime M. In O we single out a double cone D ≡ D( p, q), which plays the role of the bulk spacetime, while the set ∂ J + ( p) ∩ D is our selected boundary. Up to the choice of an orthonormal
frame inA p, the latter can be seen as the locus u = 0 in the natural coordinate system u, r, x introduced in Subsect. 2.2 which is furthermore endowed with the metric (2.3). In terms of the structure of Subsect. 3.2, we can identify the boundary as C p+ with a small caveat with respect to the coordinates used. While it is always possible to switch from x A (A = 1, 2) to the standard (θ, ϕ), the role of V as a coordinate is played by r , the affine parameter on the null geodesics √ of the cone. As a last point, the role of the function h in (3.13) is taken in general by 4 |g AB |, where g AB are the metric components appearing in (2.3) evaluated at u = 0 and | · | is kept to indicate the determinant. It is interesting to notice that whenever the conditions for the reduction of (2.3) to (2.5) are fulfilled, h can be set to V ≡ r , (see also the relation between the volume elements of the sphere in different coordinates (2.4)). Furthermore, in the retarded coordinates used, the exponential map becomes an identity. Hence, if not strictly necessary, we shall not indicate it anymore. Now we proceed in two steps. The first one consists in proving the possibility of introducing a well-defined map from the extended algebra in D to the one on C p+ ⊂ C p , while, in the second, we prove that this map is also well-behaved with respect to the algebra structures.
482
C. Dappiaggi, N. Pinamonti, M. Porrmann
Theorem 3.6. Let D be a double cone and regard the portion C p+ of the boundary as part of a cone C p . Let us introduce the linear map : Fe (D) → Ae (C p+ ) by setting . n (Fn ) = 4 |g AB |1 · · · 4 |g AB |n ⊗n (Fn ) + n , (3.35) (C p )
where is the causal propagator (3.2), |C p+ denotes the restriction on C p+ and the subscripts 1, . . . , n entail dependence of the root on the coordinates of the i th cone. Then, the following properties hold true:
ˆ 1 and is an element of D (C p+ × D)n . ˆ n , the integral kernel of n , is equal to ⊗n 1) ˆ n satisfies The wave front set of ˆ n ) ⊂ (WF( ˆ 1 ) ∪ {0})n \{0}. WF(
(3.36)
ˆ 1 ), then: Furthermore, if (x, ζx ; y, ζ y ) ∈ WF( (a) neither ζx nor ζ y vanish. (b) (ζx )r = 0. (c) (ζx )r 0 if and only if −ζ y is future directed. 2) The image of Fe (D) under lies in Ae (C p ). Proof. We prove the above properties in two separate steps.
√ ⊗n I Construction of 4 |g AB | C and the wave front set of its integral kernel. In the normal neighborhood O p ⊂ M which contains D we can select a subset O ⊂ (J − (D)\J − ( p)) ⊂ O p which is a globally hyperbolic open set that extends D over C p+ , but neither over p nor over the future of D. The existence of a similar set results from the global hyperbolicity of M which contains O p and thus also D. Let us indicate ˆ ∈ D (O ×D) the integral kernel of : C ∞ (D) → C ∞ (O ) defined by restricting by 0 the map in (3.2). It holds true that ˆ = (x1 , ζ1 ; x2 , ζ2 ) ∈ T ∗ O × T ∗ D\{0} | (x1 , ζ1 ) ∼ (x2 , −ζ2 ) , WF() (3.37) where the equivalence relation (x1 , ζ1 ) ∼ (x2 , ζ2 ) means that there exists a null geodesic γ with respect to the metric g in D which contains both x and y. Furthermore, g μν (ζ1 )ν and g μν (ζ2 )ν are the tangent vectors of the affinely parametrised geodesic γ in x and y, respectively [38,39]. We proceed by restricting one entry of the causal propagator1 on C p+ , while leaving the other localised in D. To this end, let us define the embedding χ of C p+ × D to O × D, whose action, in retarded coordinates on O , is defined as χ (r, θ, ϕ; x2 ) = ˆ (0, r, θ, ϕ; x2 ). According to Theorem 8.2.4 of [25], the restriction of the first entry of ˆ = ∅, on C p+ by means of the pullback under χ is well defined provided that Mχ ∩WF() where Mχ is the set of normals of χ . In order to verify this statement about the empty intersection, we notice that, using the definition employed in such a theorem, Mχ ⊂ Nχ = (x1 , ζ1 ; x2 , ζ2 ) ∈ T ∗ O × T ∗ D x1 ∈ C p+ ⊂ O , ζ1 = (ζ1 )u du, (ζ1 )u ∈ R . 1 The same procedure was employed in the proof of Proposition 4.3 in [30] or in the work [22].
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
483
ˆ = ∅; consider (x1 , ζ1 ; x2 , ζ2 ) ∈ Nχ and the null geodesic We prove that Nχ ∩ WF() γ originating from x1 whose tangent vector in x1 is equal to g −1 (ζ1 ). Notice that, in the retarded coordinates, the only non-vanishing component of g −1 (ζ1 ) is (g −1 (ζ1 ))r which implies that γ is contained in C p+ and, in particular, does not enter D. For this ˆ is the empty set and hence the hypotheses reason the intersection of Nχ with WF() ˆ C , the pullback of ˆ under χ , can be of Theorem 8.2.4 of [25] are fulfilled. Thus ∗ ˆ C ) ⊂ χ WF(). ˆ In particular, this entails that, if defined unambiguously and WF( ˆ C ) with x ∈ C p+ and y ∈ D, it enjoys properties (a), (b) and (c) (x, ζx ; y, ζ y ) ∈ WF( stated above. In order to verify (b), suppose this were not true and consider (x, ζx ; y, ζ y ) ∈ ˆ C ), where (ζx )r = 0. Thus there should exist an element (x, ζx ; y, ζ y ) such WF( that χ ∗ (x, ζx ; y, ζ y ) = (x, ζx ; y, ζ y ), where ζx is a null covector whose components are (ζx )r = 0, (ζx )θ = (ζx )θ and (ζx )ϕ = (ζx )ϕ , while (ζx )u is a fixed number in R. Since g −1 (ζx ) has to be null and since (g −1 (ζx ))u = 0, the only possibility is (ζx )θ = (ζx )ϕ = 0. Hence the only non-vanishing component of g −1 (ζx ) is the r -component, which implies that ζx = (ζx )u du, thus (x, ζx ; y, ζ y ) ∈ Nχ . At this point we have ˆ ∩ Nχ = ∅ so that (ζx )r has to be different from reached a contradiction because WF() zero. Notice that (a) and (c) result from the constraint imposed by the equivalence relaˆ in (3.37) and from the observation that the projection of tion ∼ in the wave front set of ζx on C p+ under χ ∗ does not change the causal direction (past/future). In order to accom√ ˆ C with 4 |g AB | ⊗ 1 which plish this part of the proof, we only need to multiply every is smooth because it is the 4th order root of a smooth positive function. Hence the wave ˆ n as the tensor front set of the resulting distribution is left unchanged. Thus we define √ . ⊗n + n ˆ ˆ product of distributions n = (h C ) ∈ D ((C p × D) ), where h = 4 |g AB | ⊗ 1, ˆ n ) enjoys the inclusion (3.36). By the Schwartz and, due to Theorem 8.2.9 in [25], WF( √ kernel theorem we obtain the linear map n = ( 4 |g AB | C )⊗n whose integral kernel ˆ n. is II On the image of . Notice that, since every F ∈ Fe (D) is composed of a finite number of Fn , it is sufficient to prove that the generic Fn is mapped to an element of Ans by ˆ n , the integral kernel of n , and or, rather, by n . Moreover, the pointwise product of n I ⊗ Fn , I the unit constant function in D (C p ), is well-defined because their wave front sets do not sum up to the zero section, as one can infer from (3.7), from Theorem 8.2.9 in [25] and from (3.36) along with the discussion presented above. More precisely, we have that WF(I n ⊗ Fn ) ⊂ {0} × WF(Fn ), where {0} is the zero section in T ∗ C p+ n while ˆ n ) has non-vanishing components on that cotangent space, thus every element in WF( n ˆ WF(I ⊗ Fn )+WF(n ) cannot contain the zero section. Hence the Hörmander criterion for the multiplication of distributions, Theorem 8.2.10 in [25], is fulfilled and, moreover, ˆ n ) · (I n ⊗ Fn ) can be tested on any compactly supported the resulting distribution ( smooth characteristic function η on the support of Fn yielding
the n (Fn ) sought-after. Theorem 8.2.12 in [25] guarantees that n (Fn ) ∈ D C p+ n and that WF(n (Fn )) ⊂
ˆ n ) · (I n ⊗ Fn ) ⊂ Wn as in (3.25). (x1 , ζx1 ; . . . ; xn , ζxn ; y1 , 0; . . . ; yn , 0) ∈ WF ( ˆ 1 ) ∩ (S1 × T ∗ D) = ∅ and that, In order to verify the last inclusion, we notice that WF( ˆ n )·(I n ⊗ Fn )∩(V ± making once more use of Theorem 8.2.10 in [25], WF( n ×{0}) = ∅,
484
C. Dappiaggi, N. Pinamonti, M. Porrmann
where {0} is the zero section in T ∗ D n . Thus the wave front set of n (Fn ) is contained in Wn , and this is tantamount to the second condition in Definition 3.3. As for the first one, this can be shown to hold true since, by construction, supp(Fn ) ⊂ K n ⊂ D n , K a compact set, and hence, due to the support property of , there exists another compact set K constructed as the closure of J − (K ) ∩ C p+ in C such that K n contains the support of n (Fn ). The third and last requirement can also be established by recalling that the singular support of the causal propagator (3.2) is contained in the set of the null geodesics. Furthermore, those emanating from the support of any Fn (recall that supp(Fn ) ⊂ K ) intersect C p on a compact set that is disjoint from p in particular. Hence the causal propagator is a smooth function whenever one entry is smoothly localised2 on the support of Fn and the other one on a neighbourhood of p. Furthermore, even after multiplication √ both by the function as in Definition 3.3 and by 4 |g AB |, such smooth function is square-integrable and bounded, together with its V -derivative, in a suitable open set of C p ∼ R+ × S2 such that V ∈ (0, V0 ), because it is a restriction to C p+ of a smooth func√ tion defined in a neighbourhood of p multiplied by 4 |g AB |. For the same reason, the √ ˆ C tends to zero whenever one entry of the causal propagator limit V → 0 of 4 |g AB | is localised on some compact set in D. Finally, notice that n (Fn ) is totally symmetric whenever Fn has this property, and this completes the proof that n (Fn ) ∈ Ans . As an intermediate step, we proceed by discussing the effect of the map on the symplectic form. Proposition 3.7. The projection : Fb (D) → Ae (C p+ ) is a symplectomorphism, i.e., for every f, h ∈ C0∞ (D), σ (ϕ f , ϕh ) = σC (1 f, 1 h),
(3.38)
with σ taken as in (3.3). Proof. Let ϕ f = f and ϕh = h, where is as in (3.2), and consider both a Cauchy . surface of D and the portion of O1 = D ∩ I − () whose boundary is formed by the + null surface C p and by . Then the current . Jμ = ϕ f ∂μ ϕh − ϕh ∂μ ϕ f
satisfies dμ() n μ Jμ = σ (ϕ f , ϕh ) with n μ the unit future directed vector normal to . Hence we can apply the divergence theorem to Jμ in O1 considered as a subregion of a larger globally hyperbolic spacetime, O , that contains D. The result is that, since ∇ μ Jμ = 0 in O1 in particular, the following identity holds, σ (ϕ f , ϕh ) = dμ(C p+ ) n μ Jμ . C p+
Furthermore, the right-hand side of the preceding equation can be rewritten in terms of the retarded coordinates on C p+ and, if one uses the relation between the volume elements on the sphere (2.4) in spherical and local coordinates, it becomes ∂ ∂ |g AB | ϕ f ϕh − ϕh ϕ f dr ∧ dS2 , (3.39) ∂r ∂r R+×S2 2 The localisation is realised by pointwise multiplication with smooth functions of suitable support.
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
485
where both ϕ f and ϕh are evaluated on C p+ , a legitimate operation as explained at the beginning of this section, and, furthermore, they vanish on the complement of C p+ in C of the preceding expression, we can consider p . Finally, due to the antisymmetry √ √ 4 |g AB | ϕ f = 1 f as well as 4 |g AB | ϕh = 1 h, and a direct inspection shows that (3.39) equals σC (1 f, 1 g) as given in (3.14) setting r = V . Remark 3.8. Notice that the ideal I generated by the equations of motion (3.1) is mapped by to 0 ∈ Ae (C ), because is a weak solution of (3.1). Hence the image of both Fb (D) and Fbo (D) under lie in Ae (C ); actually they coincide. On the basis of this remark, we stress another important property of . Proposition 3.9. The map is injective when acting on the on-shell extended algebra Feo (D). Proof. Let us recall that the action of on F = {Fn }n ∈ Fe (D) is determined by the actions of n = ⊗n 1 on its components Fn . We shall hence analyse the kernel of 1 seen as a map from T (D) to some functionals on S (C p+ ), where the elements of T (D) are the compactly supported symmetric distributions whose wave front sets enjoy (3.7). . We shall prove that ker(1 ) = K = Pu u ∈ T (D) . Given u ∈ E (D) there exists a sequence of u j ∈ C0∞ (D) whose support is contained in K , a proper compact subset of D, such that u j → u weakly for j → ∞. Consider then the following chain of equalities: − ( f ) u j = ( f ) P A (u j ) = σ (( f ), (u j )) = σC (1 ( f ), 1 (u j )), M
M
where P is the operator realising the equation of motion and A is its advanced fundamental solution. Furthermore, does intersect neither K nor J + (K ), and hence, on , A (u j ) is equal to R (u j ). Notice that in order to obtain the second equality, we have to choose two elements of a family of Cauchy hypersurfaces, and , which do not intersect K and such that J + (K ) ∩ = ∅ and J − (K ) ∩ = ∅, and integrate by parts twice. The resulting integral on vanishes due to the support property of A , while the integral on M is zero since P(( f )) = 0. Thus the remaining integral is precisely the symplectic form computed on . Furthermore, the last equality derives from Proposition 3.7. We proceed by writing ∂ 4 4 ( f ) u j = −2 |g AB | (u j ) |g AB | ( f ) d V dS2 . ∂V M Cp Passing now to the weak limit yields 1 (u)(−2∂V 1 f ) = u(( f )). From the previous . discussion we obtain that, letting S = −2∂V 1 (C0∞ (D)), the condition 1 (u)(S) = 0 is equivalent to u((D(D))) = 0, and the latter implies u = Pu for some u ∈ E (D). Since, as a functional, 1 (u) are defined on a set larger than S we have that ker(1 ) ⊂ K . In order to obtain the opposite inclusion, let us define by √ R the operator that realises the restriction on C p+ and the subsequent multiplication by 4 |g AB |. Notice that 1 is defined as the composition R ◦, where now is the map from T (D) to D (O ), where O is the normal neighbourhood containing D. Hence ker(1 ) ⊃ ker() = K . Note that K is contained in the ideal I divided out of Fe (D) in order to obtain Feo (D). The proof can then be concluded by applying a similar procedure to n to verify that also ker(n ) is contained in the ideal I .
486
C. Dappiaggi, N. Pinamonti, M. Porrmann
Before continuing the analysis of the map acting on the extended algebra Fe (D), we show that the pull-backs both of the symplectic form σC and of the boundary state ω have a nice interplay with the symplectic form in the bulk and with the Hadamard states in general. The next proposition deals with the singular structure of the state ω when pulled back in the bulk. Proposition 3.10. Under the assumptions of Theorem 3.6, . Hω = ∗ ω
(3.40)
is an Hadamard bi-distribution constructed as the pull-back of ω as in (3.18) under as in (3.35). Proof. The proof of this proposition can be performed by restricting attention to the compactly supported smooth functions on D. Let us start by showing that Hω is a good distribution on C0∞ (D 2 ), hence continuous in the topology of compactly supported smooth functions. To this end, notice that Hω ( f, g) = ω(1 ( f ), 1 (g)), moreover, ω(1 ( f ), 1 (g)) enjoys the continuity stated in (3.20). Furthermore, the L 2 norms present in (3.20) are controlled by the supremum norms of 1 ( f ), 1 (g) and their r -derivatives on some compact set in C (here taken as the entire cone). The proof of the continuity of Hω can be concluded employing the continuity of : C0∞ (M) → C ∞ (M), √ which is spoilt neither by the restriction on C p+ nor by the multiplication by 4 |g AB |. Notice that the antisymmetric part of Hω equals the symplectic form (3.3) which is preserved by the action of (Prop. 3.7), and Hω satisfies the equation weakly because so does which is used in the definition of . Furthermore, Hω ( f, f ) is positive for every f , because ω is a state for the boundary algebra. We thus conclude that Hω is the two-point function of a quasi-free state for Fb (D). Hence, in order to prove the Hadamard property, due to the work of Radzikowksi [38], it is only necessary to check that the wave front set of Hω satisfies the microlocal spectrum condition. This can be verified following the procedure envisaged in [22,30]. For completeness, we shall summarise here the main steps of such a proof. 1) It suffices to show that the microlocal spectrum condition holds locally in D, namely, when Hω is restricted on a generic compact set K 2 with K ⊂ D. We hence have to show that WF(Hω ) = (x1 , ζ1 ; x2 , −ζ2 ) ∈ T ∗ K 2 \{0} |(x1 , ζ1 ) ∼ (x2 , ζ2 ), ζ1 " 0 , (3.41) where ∼ is the equivalence relation of (3.37), while ζ1 " 0 indicates that ζ1 is a future directed vector. According to the preceding discussion, to show the inclusion ⊃, we make use of Theorem 5.8 of [41] which can be applied once ⊂ holds and yields the other relation as thesis. 2) In order to show that ⊂ holds in (3.41), notice that the past directed null geodesics originating from K in O p intersect C p+ on a region contained in a compact set N ⊂ C p+ . We stress that p ∈ C p+ , and hence p ∈ N . Thus, if we smoothly localise ˆ 1 (the integral kernel of 1 ) on N × K , where N is the complement of N in C p+ , the resulting object is described by a smooth function which is square-integrable on ˆ 1 is kept fixed in K . C p+ and so is its V -derivative, also when an entry of
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
487
3) We shall hence introduce a partition of unity on C p+ , N + N = 1, such that N ∈ C0∞ (C p+ ) is equal to 1 on the compact set N . Hence it vanishes on the intersection of a sufficiently small neighbourhood of p with C p+ . Inserting two such partitions of unity in ∗ ω and employing multilinearity, Hω becomes the sum of
ˆ 1 ⊗ ( N + ) ˆ1 . four terms, ω ( N + N ) N ˆ 1 ⊗ N ˆ 1 ). In this case, we 4) The only one which contributes to WF(Hω ) is ω( N notice that, due to the form of WF(ω) given in (3.22) and to the constraint enjoyed ˆ 1 ) as discussed in 1) of Theorem 3.6, we can apply Theorem 8.2.13 of by WF( [25] in order to obtain that the inclusion sought holds for the wave front set of this term. 5) All the other three terms have vanishing wave front sets. Let us briefly discuss them ˆ 1 when restricted separately. In particular, due to the regularity shown by N 2 + ˆ 1 ⊗ ˆ to K , the composition of (N N 1 ) with ω on C p can be computed 2 and yields a smooth function on K . The remaining two terms can be addressed ˆ 1 ⊗ N ˆ 1 ). At this point, similarly. To this end, let us concentrate on ω(N notice that the supports of N and of N have non-vanishing intersection; however, such an intersection is contained in a further compact set R. We can thus insert another partition of unity, R + R , in order
to divide such a term in two
ˆ 1 . Hence, ω ( ) ˆ 1 has a ˆ 1 ⊗ N ˆ 1 ⊗ N parts ω ( R + ) N
R
N
R
vanishing wave front set because the supports of N R and N are disjoint and ω is represented by a smooth function on their Cartesian product. Furthermore, since both N R and N are in C0∞ (C p+ ), we can estimate the wave front set of
ˆ 1 ⊗ N ˆ 1 employing once more Theorem 8.2.13 of [25] to obtain ω R N
that it is equal to the empty set.
We can now prove a second theorem which focuses on the effect of the map on the algebraic structures and on the boundary state. In particular, we shall individuate an Hadamard state in the bulk. Theorem 3.11. Under the assumptions of Theorem 3.6, one has:
∗ -homomorphism between the algebras F (D), 1) induces a unit-preserving e H ω
and Ae (C p+ ), ω .
2) is an injective ∗ -homomorphism when acting “on shell” on Feo (D), Hω . Proof. We only prove the first statement. The second arises by direct inspection from this one and from Proposition 3.9. First notice that automatically preserves the ∗ -operation because n = n , hence n (Fn )∗ = n (Fn∗ ). Thus we only need to verify the statement on the -products. In particular, we look for Hω : Fe (D) × Fe (D) → Fe (D) such that, (F Hω G) = (F) ω (G), ∀F, G ∈ Fe (D),
and, at the same time Fe , Hω is isomorphic to (Fe , H ).
(3.42)
488
C. Dappiaggi, N. Pinamonti, M. Porrmann
The natural candidate arises from the analysis performed in Proposition 3.10 and, in particular, from the distribution Hω introduced in (3.40) to be plugged in (3.8) in place of H . This is a well-defined procedure since Hω is of Hadamard form as proved in Proposition 3.10, and, hence, (Fe , H ) turns out to be isomorphic to (Fe , Hω ), the isomorphism being realised as in (3.9). We are thus left with the task to verify (3.42) for every F and G in Fe (D). If we exploit both the definitions and the bilinearity of all the -products involved, this reduces to the requirement to show that
l+m−2k (Fl ⊗ G m )(Hω ⊗k ) = (l Fl ⊗ m G m )(ω⊗k ) for l + m − 2k 0. The last relation directly results from bilinearity, (3.40) and from the fact that k = k1 ⊗ k2 for all k1 , k2 > 0 such that k1 + k2 = k. Remark 3.12. Notice that it is possible to turn the injective homomorphism into a bijection if we restrict attention to the local von Neumann algebras defined as the double commutant in the GNS representation of a quasifree Hadamard state of the C ∗ -algebra generated by the local Weyl operators constructed out of the symplectic forms (3.3) and (3.14) in D and on the boundary C p+ , respectively. This last claim is based on the invertibility of 2 on the weak solutions of the Klein-Gordon equation, where we recall that the Goursat problem with compact initial data on C p , in general, yields a solution of (3.1) whose restriction on any Cauchy surface of D is not compact. Alas, the von Neumann algebra mentioned does not contain relevant physical observables such as the components of the regularised stress-energy tensor or the regularised squared fields, objects we would like to use in order to extract information about the local geometric data such as the scalar curvature.
4. Interplay with General Covariance and Comparison between Spacetimes We are now in the position to collect all our results in a comprehensive framework which will exhibit both a nice interplay with the principle of local covariance, as devised in [9], and the possibility to compare quantum field theories on different backgrounds both at the level of algebras and of states. To this avail, the construction in Subsect. 3.5 will play a pivotal role, and the natural language we adopt is that of categories, which was already introduced in Subsect. 2.1. In particular, notice that the construction of an extended algebra of observables on a double cone D can be realised as a suitable functor between the categories DoCoiso and Alg, although it is not possible to extend such a functor to DoCo. Notwithstanding this obstruction, the additional structure which arises from both the boundary and the field theory defined thereon allows us to circumvent the above problem in a way that also puts us in a position to compare field theories in different spacetimes. This requires the introduction of a further category, namely, BAlg: The objects of this category are the extended boundary (topological ∗ -)algebras presented in Subsect. 3.4, constructed on all possible C p+ , while the morphisms are the unit preserving ∗ -homomorphisms among them. The key point consists in making a profitable use of the ∗ -homomorphisms introduced in Theorem 3.6, in order to establish that ◦ F indeed defines a functor between the two categories DoCo and BAlg, which admits the following pictorial, but inspiring
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
489
diagrammatic representation, F
D −−−−→ Fe (D) −−−−→ Ae (C p+ ) ⏐ ⏐ ⏐αı ı e,e ⏐ e,e D −−−−→ Fe (D ) −−−− → F
Ae (C p+ )
(4.1) .
The arrow αıe,e traces back to the analysis in Subsect. 3.2, where it was shown that the boundary theory can be constructed and analysed independently of the specific bulk. Hence, αıe,e is the ∗ -homomorphism αıe,e : Ae (C p+ ) → Ae (C p+ ) whose action on the F ∈ Ae (C p+ ) is defined as follows: By means of the push-forward, it is αıe,e (Fn ) = ı e,e ∗ Fn , n
n
on ı e,e (C p+ ) ⊂ (C p+ )n , while αıe,e (Fn ) = 0 on the complement of ı e,e (C p+ ) in (C p+ )n . Such an operation is well-defined because Fn has compact support towards the future and, when extended on the closure of C p+ , ı e,e maps p to p . We hence have the following
Proposition 4.1. Consider Ae : DoCo → BAlg, whose action on the objects and morphisms is seen as follows: • the action of Ae on the objects of DoCo is such that Ae (D) = ◦Fe (D) = Ae (C p+ ); • the action of Ae on the morphisms ı e,e is such that Ae (ı e,e ) = αıe,e . Then Ae is a functor between the two categories. Proof. In order to show that Ae : DoCo → BAlg is a functor, notice that the covariance property holds and the identity is preserved, i.e., Ae (ı e,e ) ◦ Ae ( ιe ,e ) = Ae (ı e,e ◦ ιe ,e ),
Ae (idD ) = idAe (C p+ ) ,
as can be seen by direct inspection of the definition of Ae (ı e,e ).
As for the connection with the principle of general local covariance, we recall that, in the most general case, it is not possible to find a direct relation between F (D) and F (D ), unless the embedding : F (D ) → F (C ) can be inverted on the image of composed with αıe,e . This is indeed what happens whenever, e.g, ı e,e is an isometry (or, at worst, a conformal isometry [34]) which preserves the base point p. Hence we are working in DoCoiso . Under this assumption, the discussion about causality, usually an integral part of the reasoning as in [7,9], does not have to be performed directly, since its essence is already encoded in the analysis of the properties of the map . A similar statement holds true also for the time-slice axiom (see [10], in particular). Especially, since the theory on the boundary is, to a certain extent, non-dynamical, there is no such axiom in our boundary framework and the one in the bulk is automatically assured by and its properties.
490
C. Dappiaggi, N. Pinamonti, M. Porrmann
4.1. Comparison of expectation values in different spacetimes. The aim of this section is to clarify in which sense one can compare two field theories on two different backgrounds. We shall first explain the procedure abstractly and then give a concrete example. Bearing in mind (4.1), it is straightforward to realise that, whenever we assign a state ω on Ae (C p ), we can pull it back either on Fe (D ) via or on Fe (D) via αıe,e ◦, and the information about the bulk geometry is indeed restored by and . This is a rather general feature which holds true regardless of the global structure of the spacetimes in which D and D are embedded. Yet, on practical grounds, it is natural to choose one of the two double cones embedded in the four-dimensional Minkowski spacetime, where our capability of performing explicit computations of physical quantities is enhanced due to the large symmetry of the background. To be more specific on this point, consider a double cone D as a subset of (R4 , η) while D lies in a generic strongly causal spacetime M , chosen in such a way that there exist two frames e and e in T p (R4 ) and T p (M ), respectively, yielding a well-defined ı e,e : D → D . At this point we can apply the construction discussed for the local fields and algebras, related on the boundary via the map αıe,e . As a next step, following also the general philosophy of [9], we consider observables constructed out of the same local fields either on D or D . Here we suppose that there exists ı e,e : D → D and hence, from the same field, we form ( f ) ∈ Fe (D) and (ı e,e ∗ f ) ∈ Fe (D ), where is a local field in the sense of [9]. We compute their expectation values on the pull-back of a suitable boundary state yielding an Hadamard counterpart in the bulk. In general, the difference between the results depends on the geometric data of both D and D , and thus we are ultimately comparing quantum field theories on different backgrounds. Nonetheless, from a computational point of view, this procedure is still too involved, since one has to cope with the singular structure of the chosen state(s). Even if we restrict attention to those fulfilling the Hadamard condition, one would still need to take care of the regularisation procedure of the observables, a hassle which can be avoided. The idea is to consider on each double cone two bulk Hadamard states constructed out of the pull-back of different boundary counterparts and to work with their difference. In this case the integral kernel of such a difference is known to be smooth, an advantage which strongly reduces the computational efforts. Although the price to pay is the introduction of two states on the light cone, a natural candidate as one of them is the distinguished reference state ω which arises from (3.40) in Proposition 3.10. Hence we are left with the need to assign only one extra datum. Before we discuss an explicit example of this procedure, we stress the most important properties of both ω and of its pull-back in the bulk, say ∗ ω. These can be inferred from both Proposition 3.2 and Theorem 3.11: 1. Local Lorentz invariance: According to the third item of Proposition 3.2, ω turns out to be invariant under a large set of geometric transformations on the full cone C which contains the boundary. In particular, ω is invariant under the natural action of the subgroup of the Lorentz group which corresponds to isometries of the neighbourhood where the state is defined. Since the map is constructed substantially out of the causal propagator (3.2) in a geodesic neighbourhood, its action on ω via pull-back does not spoil the above property, i.e., ∗ ω is invariant under the above assumptions. 2. Microlocal structure: The wave front set of ω is contained in the union of (3.23) and (3.24) and, most notably, it does not contain directions to the past. In particular,
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
491
this allows for the proof of Proposition 3.10 according to which ∗ ω satisfies the Hadamard property in the bulk double cone. 3. Behaviour as a “vacuum”: The boundary state ω turns out to be invariant under rigid translations of the V -coordinate, or, in other words, it is a vacuum with respect to the transformation generated by the vector ∂V . This statement can be proved exactly as in [15,29] for the counterpart on the conformal boundary of an asymptotically flat or of a cosmological spacetime, viz., from the explicit form of the two-point function (3.21). This also entails that the energy computed on the cone with respect to ∂V is minimised. Unfortunately, this property has not a strong counterpart in the bulk, but, if the bulk can be realised as an open set in (R4 , η), then ∗ ω is seen to coincide with the Minkowski vacuum for massless fields. 4.2. An application: Extracting the curvature. In this subsection we shall present an explicit application which follows the guidelines given above. For simplicity consider on the one hand a double cone D realised as an open subset of Minkowski spacetime (R4 , η), where the metric η has the standard diagonal form with respect to the Cartesian coordinates (t, x, y, z) induced by the standard orthonormal frame e of R4 . As D we consider a double cone which can be embedded in a homogeneous and isotropic solution of Einstein’s equation with flat spatial section. This is a Friedman-Robertson-Walker spacetime (M , g), where g = a 2 (t) η and a(t) ∈ C ∞ (I, R+ ) with I ⊆ R and a(0) = 1. Here t refers to the so-called conformal time and thus we still consider the coordinates (t, x, y, z) induced by the standard frame of R4 , indicated as e to distinguish it from e the previous one. Furthermore, notice that, in view of the special form of g, e = a(t) . Since the underlying spacetimes are conformally related, their causal structures and, in particular, the double cones coincide. Consider two points p = (0, 0, 0, 0) and q = (t , 0, 0, 0) and the corresponding double cones D( p, q) ⊂ R4 and D ( p, q) ⊂ M . In this framework the map ı e,e : D(x0 , x1 ) → D (x0 , x1 ) turns out to be trivial. Next, we choose a minimally coupled real scalar field theory, viz., φ : D → R, φ = 0, where is the d’Alembert wave operator constructed out of the metric in D. We stress once more that we consider the very same equation also in D . If we follow the guidelines of Subsect. 3.1, we can construct Fe (D) and Fe (D ) and their counterparts on the boundaries C p and C p . As outlined in the previous subsection, we now consider two algebraic states on Ae (C p ), one, ω, is the reference state, while the other can be arbitrary, provided that the pull-back to the bulk via still fulfills the Hadamard condition. This requirement is not too restrictive since, e.g., any state which differs from ω by a smooth function on the boundary and vanishes in a neighbourhood of the tip is an admissible choice. In particular, consider another Gaussian state ω : Ae (C p ) → C, whose two-point function has the following form: 1 ω (ψ1 , ψ2 ) = ω(ψ1 , ψ2 ) + dr dr dS2 (θ, ϕ) ψ1 (r, θ, ϕ)ψ2 (r , θ, ϕ). 4π R+ ×R+ ×S2 (4.2) Notice that the integral on the right-hand side entails that the integral kernel of ω − ω is not smooth because it contains a δ-like singularity in the angular coordinates. Despite this fact ω can be pulled back to every spacetime still yielding an Hadamard bi-distribution.
492
C. Dappiaggi, N. Pinamonti, M. Porrmann
The last statement can be proved operating in the same way as in the proof
of Proposition 3.10 with ω replaced by ω − ω. The key new feature, yielding WF Hω −ω = ∅, is the fact that, while (ζx )r is never equal to zero for every (x, ζx ; y, ζ y ) ∈ WF(1 ), every (x, ζx ; y, ζ y ) ∈ WF(ω − ω) is such that (ζx )r = (ζ y )r = 0. We are now ready to consider the expectation values of suitable observables from the point of view of both D and D . The most natural one is the expectation value of φ 2 ( f ), where f ∈ C0∞ (D) (and hence also in C0∞ (D )), with respect to the bulk states constructed out of ω and ω. Notice that φ 2 ( f ) is a shortcut for saying that we are actually considering u f = f (x)δ(x − y) ∈ A2s (D) ⊂ Fe (D). One of the advantages of this construction is that, in this case, we are allowed to keep a more general stance, namely, we can substitute f (x) by a Dirac function peaked at the point xt = (t, 0, 0, 0) with t ∈ (0, t ). This is tantamount to consider : φ 2 : (xt ), where u = δ(x − xt )δ(x − y) ∈ As2 (D). Now (3.35) can be used to evaluate 2 u in both (R4 , η) and (M , g) by means of the explicit form of the causal propagator. In Minkowski spacetime it looks like (see [17] or [35]) . δ(t − t − |x − x |) δ(t − t + |x − x |) (x, x ) = − + , 4π |x − x | 4π |x − x |
(4.3)
where t is the time coordinate and x the three-dimensional spatial vector in Euclidean coordinates. The counterpart of in D can be directly evaluated exploiting the conformal transformation between (M , g) and (R4 , η). The d’Alembert wave equation in the first spacetime (M , g) corresponds in the flat one to φ −
a φ = 0, a
(4.4)
where stands for time derivation and = −∇ μ ∇μ . Furthermore, the causal propagator of this partial differential equation is related to the one in M via M (x, y) =
1 (x, y). a(tx )a(t y )
(4.5)
If we follow the procedure discussed in Chap. 4 of [35], we get (x, x ) = (x, x ) +
V (x, x ) (t − t − |x − x |) − (−t + t − |x − x |) , 4π
where V (x, x ) is a smooth function whose explicit form is derived from the Hadamard a (x) recursive relations for (4.4). In particular, it also holds that V (x, x) = − 2a(x) . We are now ready to compare the expectation values in (ω − ω). The Minkowski side of this operation yields, by direct computation, (ω − ω)(R 2 u) = 4
1 , 4
while in the cosmological setting
(ω − ω) 2M u = 4π
0
∞
(xt , r∗ ) r∗ a 3 (r∗ )dr∗ a(t)a(r∗ )
2 ,
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
493
where we have rewritten the integral in the r -variable in terms of r∗ , the affine parameter of the null cone in Minkowski spacetime. The defining relation between these two variables is dr = a 2 (r∗ ) dr∗ . The above integral can be rewritten by means of (4.5) as
(ω − ω) 2M u = 4π
∞ 0
(xt , r∗ )
a 2 (r∗ ) dr∗ + a(t)
t/2
0
V (xt , r∗ )
a 2 (r∗ ) dr∗ a(t)
2 ,
in which the first integral, via (4.3), yields
(ω − ω) 2M u =
0
∞
δ(t − 2r∗ )
a 2 (r∗ ) dr∗ + a(t)
t/2
0
V (xt , r∗ )
a 2 (r∗ ) r∗ dr∗ a(t)
2 .
Let us now expand a(t) in a power series around the point x0 ,
(ω − ω)
2M u
1 a 2 (t/2) t2 − a (0) + O(t 3 ) = 2 a(t) 4
2 ,
where, in the derivative, we have exploited that, at first order in t, V (xt , r∗ ) =
a (0) + O(t), a(0)
also due to the rotational symmetry of M . Notice that, in the above formula, also r∗ is of order O(t). If we now expand both a(t) and a(t/2) in a Taylor series, we obtain
(ω − ω) 2M u = a2 +
3a 4
2 a a
2 t 2 + O(t 3 )
,
where all the functions a together with their derivatives are evaluated at t = 0. We can 4 summarise the discussion, finally calculating the difference between (ω − ω) R u 2
M and (ω − ω) 2 u ,
4
3 M 2 2 3 (ω − ω) R 2 u − (ω − ω) 2 u = a (0) t + O(t ), 4 with a(0) = 1. The interpretation of this upshot is that, exactly as expected, the above comparison yields a result which, at first order, allows us to extract precise information on the a priori unknown geometric data via a measurement, in this case, the first derivative of the scale factor at the point x0 in a Friedman-Robertson-Walker universe.
494
C. Dappiaggi, N. Pinamonti, M. Porrmann
5. Summary and Outlook In this paper we have achieved a twofold goal: on the one hand we propose a novel way to look at the properties of a local quantum field theory in a suitable curved background, while, on the other hand, the very same construction yields a mechanism which allows for the comparison of expectation values of field observables in different spacetimes. More specifically, starting from a careful analysis of the underlying geometry, we realise that only moderate assumptions are needed to reach our goals, viz., our general setting consists of an arbitrary strongly causal manifold M in which we identify an arbitrary but fixed double cone D ≡ D( p, q) = I + ( p) ∩ I − (q) strictly contained in a normal neighbourhood of p. Since D is globally hyperbolic, we can consider therein a real scalar field theory along the lines of (3.1) and therefore follow the general quantisation scheme which particularly calls for the association of a Borchers-Uhlmann algebra of observables with the chosen system. This algebra can be extended, both enlarging the set of its elements and the defining product, in order to encompass also a priori more singular objects, such as the Wick polynomials, which constitute the so-called extended algebra. The very deep reason for choosing D ⊂ M lies in its boundary and, more properly, on the portion of J + ( p) which it contains. This is a differentiable submanifold of codimension 1 on which it is possible to construct a genuine free scalar field theory, following exactly the same procedure successfully employed for the causal boundary of an asymptotically flat or cosmological spacetime in [13,14]. The main novel result in this framework arises from the construction of an extended algebra also for the boundary theory—Ae (C p ) in the main body—whose well-posedness is justified both by its mathematical properties and by its relation to the bulk counterpart. Hence, the latter is embedded in Ae (C p ) by means of , an injective ∗ -homomorphism. The advantage of this picture is the possibility to make use of a long tradition, originating from [27], which allows us to exploit the geometrical properties of the boundary to identify for the algebra thereon a natural state which can be pulled-back to the bulk via , yielding a counterpart which satisfies the microlocal spectrum condition; hence it is of Hadamard form. This guarantees that we can identify a local state in D which is physically well-behaved. In physical terms it means that this state is the same for all inertial observers at p. In other words, the bulk state as well as the one on the boundary are invariant under a natural action of S O0 (3, 1). Thus it can be identified as a sort of local vacuum on a curved spacetime independent of the frame. The second goal of comparing expectation values on different backgrounds is based on the above construction. More precisely, we consider not just one but actually two regions as above in two a priori different spacetimes M and M . The construction of the field theories proceeds as usual, but now we make use of the invertibility of the exponential map in geodesic neighbourhoods in order to engineer the double cones D ⊂ M and D ⊂ M so that we can map the boundary in D to that in D via a local diffeomorphism. This procedure can be brought to the level of boundary extended algebras which thus can be related by means of a suitable restriction homomorphism. The advantage is the previously unknown possibility to carefully use the distinguished state identified on each C p in order to compare the bulk expectation values of field observables constructed for the theories in D and D . The important point is that this new perspective is completely compatible with the standard principle of general local covariance when applicable as devised in [9], and, actually, it complements it corroborating its significance. Furthermore, since nothing prevents us from choosing one of the spacetimes as the Minkowski one, one can concretely check how the proposed machinery allows for the
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
495
comparison of the expectation values of the field observables, making manifest the role and the magnitude of the geometric quantities. We stress this point by means of a simple example involving a massless minimally coupled field in the flat and in a cosmological spacetime. It seems safe to claim that there are several possibilities to apply our procedure to many other cases of physical interest. These are certainly not the only roads left open, and actually even the identified bulk Hadamard state should be studied in more detail. As a matter of fact, it is interesting to understand whether it is connected in any way with the states of minimum energy that appear in Friedman-Robertson-Walker spacetimes [32]. We leave this as well as the myriad of other questions for future investigations. Acknowledgements. The work of C.D. is supported by a Junior Research Fellowship from the Erwin Schrödinger Institute and he gratefully acknowledges it, while that of N.P. is supported by the German DFG Research Program SFB 676. We would like to thank Bruno Bertotti, Romeo Brunetti, Klaus Fredenhagen, Thomas-Paul Hack, Valter Moretti and Jochen Zahn for profitable discussions on the topics of the present paper. C.D. also gratefully acknowledges the support of the National Institute for Theoretical Physics of South Africa for his stay at the University of KwaZulu-Natal.
A. Hadamard States This appendix briefly recollects some properties of Hadamard states which are used throughout the main text. Since most of the material has already been proved in several different alternative ways in the literature, we limit ourselves to giving the main statements and the necessary references. Let us stress that, from a physical perspective, Hadamard states are the natural candidates for physical ground states of a quantum field theory on a curved background, since their ultraviolet behaviour mimics that of the Minkowski vacuum at short distances and, furthermore, they guarantee that the quantum fluctuations of the expectation values of observables, such as the smeared components of the stress-energy tensor, are finite. In the subsequent discussion we always assume that we are dealing with a quasi-free state on a suitable field algebra constructed on a globally hyperbolic spacetime (M, g) from a field satisfying an equation of motion such as (3.1). We stick to this assumption because it is consistent with the main body of the paper, but the reader should keep in mind that such an hypothesis could be relaxed (see for example [42]). As a starting point we state a global criterion characterising Hadamard states [38,39]. Definition A.1. A state ω satisfies the Hadamard condition and is thus called an Hadamard state if and only if WF(ω) = (x, k x ; y, −k y ) ∈ T ∗ M 2 \{0} (x, k x ) ∼ (y, k y ), k x " 0 , where, in this expression, ω actually stands for the integral kernel of the two-point function associated with ω. The relation (x, k x ) ∼ (y, k y ) indicates that there exists a null geodesic γ connecting x to y such that k x is coparallel and cotangent to γ at x and k y is the parallel transport of k x from x to y along γ . The requirement k x " 0 means that the covector k x is future directed. The above condition on the wave front set is rather useful and often employed on practical grounds to check whether a given state really is Hadamard or not. Nonetheless, it is possible to provide another definition via the so-called Hadamard form, which has been rigorously introduced in [27].
496
C. Dappiaggi, N. Pinamonti, M. Porrmann
Definition A.2. A state ω is said to be of the (local) Hadamard form if and only if in any convex normal neighbourhood the integral kernel of the associated two-point function can be written as ω(x, y) = H (x, y) + W (x, y), where H (x, y) = lim+ →0
σ (x, y) U (x, y) + V (x, y) ln , σ (x, y) λ2
(A.1)
and the limit is to be understood in the weak sense. Here, U, V , as well as W are smooth functions, while λ is a reference length; furthermore, . σ (x, y) = σ (x, y) ± 2i (T (x) − T (y)) + 2 with > 0. In the above formula, T is a time function, such that ∇T is timelike and future directed on the full spacetime (M, g). In addition, if we apply (3.1) either to the x- or to the y-variable, the result has to be a smooth function. The existence of a time function T is guaranteed on any globally hyperbolic manifold [3,4] as these can be decomposed as × R, where is a smooth Cauchy surface and R is the range of the time function T . A completely satisfactory definition of the Hadamard form requires some more work to rule out spacelike singularities, to circumvent convergence problems of the series V , which is only asymptotic, and, finally, to assure that the definition depends neither on a special choice of the temporal function T nor on the convex normal neighbourhood employed. In strict terms, we have only defined the local Hadamard form here. A stronger and more satisfactory definition, the so-called global Hadamard form, has been introduced in [27]. It reinforces the local form extending it from the convex normal neighbourhoods to certain “causally-shaped” neighbourhoods of a Cauchy surface, thereby ruling out spacelike singularities. However, in [39], it has been shown that the local Hadamard form already implies the global Hadamard form. Another important fact is that the singular structure (A.1) is completely determined by the geometry of the background and the equation of motion. This of course does not hold for W which encodes the full state dependence. References 1. Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society Publishing House, 2007 2. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry, Vol. 202 of Monographs and Textbooks in Pure and Applied Mathematics. New York: Marcel Dekker, 2nd ed., 1996 3. Bernal, A.N., Sánchez, M.: On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem. Commun. Math. Phys. 243(3), 461–470 (2003) 4. Bernal, A.N., Sánchez, M.: Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes. Commun. Math. Phys. 257(1), 43–50 (2005) 5. Brunetti, R., Dütsch, M., Fredenhagen, K.: Perturbative Algebraic Quantum Field Theory and the Renormalization Groups. http://arxiv.org/abs/0901.2038v2 [math-ph], 2009, to appear Adv. Theor. Math. Phys. 6. Brunetti, R., Fredenhagen, K.: Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds. Commun. Math. Phys. 208(3), 623–661 (2000)
Local Causal Structures, Hadamard States and the Principle of Local Covariance in QFT
497
7. Brunetti, R., Fredenhagen, K.: Quantum Field Theory on Curved Backgrounds. In: Bär, C., Fredenhagen, K. (eds), Quantum Field Theory on Curved Spacetimes—Concepts and Mathematical Foundations, Vol. 786 of Lecture Notes in Physics, chap. 5, Berlin-Heidelberg: Springer-Verlag, 2009, pp. 129–155 8. Brunetti, R., Fredenhagen, K., Köhler, M.: The Microlocal Spectrum Condition and Wick Polynomials of Free Fields on Curved Spacetimes. Commun. Math. Phys. 180(3), 633–652 (1996) 9. Brunetti, R., Fredenhagen, K., Verch, R.: The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory. Commun. Math. Phys. 237(1-2), 31–68 (2003) 10. Chilian, B., Fredenhagen, K.: The Time Slice Axiom in Perturbative Quantum Field Theory on Globally Hyperbolic Spacetimes. Commun. Math. Phys. 287(2), 513–522 (2009) 11. Chmielowiec, W., Kijowski, J.: Hamiltonian description of radiation phenomena: Trautman–Bondi energy and corner conditions. Rep. Math. Phys. 64(1-2), 223–240 (2009) 12. Choquet-Bruhat, Y., Chru´sciel, P.T., Martín-García, J.M.: The light-cone theorem. Class. Quantum Grav. 26(13), 135011 (2009) 13. Dappiaggi, C., Moretti, V., Pinamonti, N.: Rigorous Steps Towards Holography in Asymptotically Flat Spacetimes. Rev. Math. Phys. 18(4), 349–415 (2006) 14. Dappiaggi, C., Moretti, V., Pinamonti, N.: Cosmological Horizons and Reconstruction of Quantum Field Theories. Commun. Math. Phys. 285(3), 1129–1163 (2009) 15. Dappiaggi, C., Moretti, V., Pinamonti, N.: Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property. J. Math. Phys. 50(6), 062304 (2009) 16. Dappiaggi, C., Moretti, V., Pinamonti, N.: Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime. http://arxiv.org/abs/0907.1034v1 [gr-qc], 2009 17. Friedlander, F.G.: The Wave Equation on a Curved Space-Time. Cambridge-London-New York-Melbourne: Cambridge University Press, 1975 18. Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity Structure of the Two-Point Function in Quantum Field Theory in Curved Spacetime, II. Ann. Phys. (N.Y.) 136(2), 243–272 (1981) 19. Geroch, R.: Spinor Structure of Space–Times in General Relativity. I. J. Math. Phys. 9(11), 1739–1744 (1986) 20. Geroch, R.: Spinor Structure of Space–Times in General Relativity. II. J. Math. Phys. 11(1), 343–348 (1970) 21. Herdegen, A.: Infrared Problem and Spatially Local Observables in Electrodynamics. Ann. Henri Poincaré 9(2), 373–401 (2008) 22. Hollands, S.: Aspects of Quantum Field Theory in Curved Spacetime. Ph.D. thesis, University of York, 2000. Advisor: B. S. Kay 23. Hollands, S., Wald, R.M.: Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime. Commun. Math. Phys. 223(2), 289–326 (2001) 24. Hollands, S., Wald, R.M.: Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime. Commun. Math. Phys. 231(2), 309–345 (2002) 25. Hörmander, L.: The Analysis of Linear Partial Differential Operators I – Distribution Theory and Fourier Analysis, Vol. 256 of Grundlehren der mathematischen Wissenschaften. Berlin-Heidelberg-New York: Springer-Verlag, 2nd ed., 1990 26. Husemoller, D.: Fibre Bundles, Vol. 20 of Graduate Texts in Mathematics. New York-Berlin-Heidelberg: Springer-Verlag, 3rd ed., 1994 27. Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep. 207(2), 49–136 (1991) 28. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry – Volume I. New York-London: Interscience Publishers, 1963 29. Moretti, V.: Uniqueness Theorem for BMS-Invariant States of Scalar QFT on the Null Boundary of Asymptotically Flat Spacetimes and Bulk-Boundary Observable Algebra Correspondence. Commun. Math. Phys. 268(3), 727–756 (2006) 30. Moretti, V.: Quantum Out-States Holographically Induced by Asymptotic Flatness: Invariance under Spacetime Symmetries, Energy Positivity and Hadamard Property. Commun. Math. Phys. 279(1), 31–75 (2008) 31. Moretti, V., Pinamonti, N.: Holography and SL(2, R) symmetry in 2D Rindler space–time. J. Math. Phys. 45(1), 230–256 (2004) 32. Olbermann, H.: States of low energy on Robertson–Walker spacetimes. Class. Quantum Grav. 24(20), 5011–5030 (2007) 33. O’Neill, B.: Semi-Riemannian Geometry: With Applications to Relativity, Vol. 103 of Pure and Applied Mathematics. New York-London-Paris: Academic Press, 1983 34. Pinamonti, N.: Conformal Generally Covariant Quantum Field Theory: The Scalar Field and its Wick Products. Commun. Math. Phys. 288(3), 1117–1135 (2009) 35. Poisson, E.: The Motion of Point Particles in Curved Spacetime. Living Rev. Relativity 7(6), 2004, available at http://www.livingreviews.org/lrr-2004-6, 2004
498
C. Dappiaggi, N. Pinamonti, M. Porrmann
36. Poisson, E.: Retarded coordinates based at a world line and the motion of a small black hole in an external universe. Phys. Rev. D 69(8), 084007 (2004) 37. Preston, B., Poisson, E.: Light-cone coordinates based at a geodesic world line. Phys. Rev. D 74(6), 064009 (2006) 38. Radzikowski, M.J.: Micro-Local Approach to the Hadamard Condition in Quantum Field Theory on Curved Space-Time. Commun. Math. Phys. 179(3), 529–553 (1996) 39. Radzikowski, M.J.: A Local-to-Global Singularity Theorem for Quantum Field Theory on Curved SpaceTime. Commun. Math. Phys. 180(1), 1–22 (1996) 40. Sahlmann, H., Verch, R.: Passivity and Microlocal Spectrum Condition. Commun. Math. Phys. 214(3), 705–731 (2000) 41. Sahlmann, H., Verch, R.: Microlocal Spectrum Condition and Hadamard Form for Vector-Valued Quantum Fields in Curved Spacetime. Rev. Math. Phys. 13(10), 1203–1246 (2001) 42. Sanders, K.: Equivalence of the (Generalised) Hadamard and Microlocal Spectrum Condition for (Generalised) Free Fields in Curved Spacetime. Commun. Math. Phys. 295(2), 485–501 (2010) 43. Schroer, B.: Bondi-Metzner-Sachs symmetry, holography on null-surfaces and area proportionality of “light-slice” entropy. http://arxiv.org/abs/0905.4435v4 [hep-th], 2010 44. Sewell, G.L.: Quantum Fields on Manifolds: PCT and Gravitationally Induced Thermal States. Ann. Phys. (N.Y.) 141(2), 201–224 (1982) 45. Verch, R.: Local Definiteness, Primarity and Quasiequivalence of Quasifree Hadamard Quantum States in Curved Spacetime. Commun. Math. Phys. 160(3), 507–536 (1994) 46. Wald, R.M.: General Relativity. Chicago-London: The University of Chicago Press, 1984 47. Wald, R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. Chicago-London: The University of Chicago Press, 1994 Communicated by Y. Kawahigashi
Commun. Math. Phys. 304, 499–511 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1245-6
Communications in
Mathematical Physics
Twistor Theory on a Finite Graph Paul Baird, Mohammad Wehbe Département de Mathématiques, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu, CS 93837, 29238 Brest Cedex, France. E-mail: [email protected]; [email protected] Received: 1 February 2010 / Accepted: 15 December 2010 Published online: 22 April 2011 – © Springer-Verlag 2011
Abstract: We show how the description of a shear-free ray congruence in Minkowski space as an evolving family of semi-conformal mappings can naturally be formulated on a finite graph. For this, we introduce the notion of holomorphic function on a graph. On a regular coloured graph of degree three, we recover the space-time picture. In the spirit of twistor theory, where a light ray is the more fundamental object from which spacetime points should be derived, the line graph, whose points are the edges of the original graph, should be considered as the basic object. The Penrose twistor correspondence is discussed in this context. 1. Introduction Two appealing ideas, both due to R. Penrose, provide a different perspective to our understanding of physical fields. The first of these is to try to build up space-time and quantum mechanics from combinatorial principles. One way to attempt this is from socalled spin networks, and Penrose argues how 3-dimensional space arises from systems with large angular momentum. A spin network is a graph whose vertices have degree 3 (the number of edges incident with each vertex is 3) and whose edges are labeled by an integer which represents twice the angular momentum [19]. The second idea is to consider twistor space, the space of null geodesics, as the more basic object from which space-time points should be derived [20]. Twistor diagrams can be considered as a natural adaptation of the combinatorial perspective to the twistor program [20]. Our principal aim in this article is to give an alternative way in which combinatorial structures arise from the twistorial construction of fields. One of the basic objects of twistor theory, a shear-free ray congruence, can be viewed alternatively as a semi-conformal complex valued mapping which evolves in time [4–6]. This latter object, which we shall consider as a physical field, is perfectly suited to be defined on a finite graph (or network). In this context, we shall refer to the function as holomorphic, since in the plane, a semi-conformal mapping is either holomorphic or
500
P. Baird, M. Wehbe
anti-holomorphic. What is fascinating is that only certain graphs support a holomorphic function. If the order of the graph is sufficiently small, computer programs can be used to generate such functions. To a graph endowed with a holomorphic function ϕ : V () → C, where V () is the set of vertices of , we can associate its twistor dual L , whose vertices are the edges of the original graph, sometimes called the line-graph, as well as a function ψ : V (L ) → C. In the spirit of twistor theory, where light rays are considered to be the fundamental objects, we consider the graph L as the basic object from which physical fields and space-time points should be2 deduced. Indeed, a vertex of arises as a complete subgraph in L upon which ψ vanishes. An outline of the paper is as follows. We first of all explain how a shear-free ray congruence on 4-dimensional Minkowski space can be viewed as an evolving family of complex-valued semi-conformal mappings on 3-dimensional space-like slices. This is the basis of our generalization to graphs. In Sect. 3, we discuss finite graphs. In particular, we recall the notion of holomorphic mapping between graphs and introduce the concept of holomorphic function on a graph. Holomorphic mappings are then characterized by the property that they preserve holomorphic functions (Proposition 3.3). The properties of holomorphic functions are discussed in relation to quantum graphs, spin networks and orthographic projection. A holomorphic function on a graph is equivalent to an isotropic 1-form which vanishes around closed cycles. On a regular graph of degree 3 oriented by colour, we show how, from an isotropic 1-form, we can recover a spinor field defined on the vertices, which corresponds to the spinor field defining a shear-free ray congruence on spacetime. Finally, in Sect. 4, we discuss the twistor correspondence between a graph and its line-graph.
2. Shear-Free Ray Congruences on Minkowski Space The Penrose twistor correspondance associates to a light ray in Minkowsi space M4 , a point in a 5-dimensional CR-submanifold N 5 of CP 3 [17]. We can obtain N 5 as follows. We first of all compactify M4 by adding a light cone at infinity to obtain the manifold 4 M diffeomorphic to S 1 × S 3 . The Hopf fibration π : CP 3 → S 4 is the map given by π([z 1 , z 2 , z 3 , z 4 ]) = [z 1 + z 2 j, z 3 + z 4 j] ∈ HP 1 , where we use homogeneous coordinates [z 1 , z 2 , z 3 , z 4 ] for points of CP 3 and where HP 1 is the quaternionic projective space. On identifying HP 1 with S 4 and letting S 3 be the equatorial 3-sphere given by Re ((z 1 + z 2 j)/(z 3 + z 4 zj)) = 0 for z 3 + z 4 i = 0, together with the point at ∞, we see that π([z 1 , z 2 , z 3 , z 4 ]) ∈ S 3 if and only if z 1 z 3 + z 2 z 4 + z 1 z 3 + z 2 z 4 = 0. We then define N 5 = π −1 (S 3 ). There is a natural identification of N 5 with the unit tangent bundle T 1 S 3 to S 3 after which it follows, since S 3 is parallelizable, that N 5 is 4 diffeomorphic to S 3 × S 2 . If we consider S 3 as the compactified slice t = 0 in M , then 5 3 2 a point (x, v) of N ∼ = S × S gives the light ray passing through x with direction v.
Twistor Theory on a Finite Graph
501
The picture can be unified by introducing the flag manifold F12 of pairs (, ) consisting of resp. 1- and 2-dimensional subspaces of C4 with ⊂ and considering the double fibration:
CP 3 ∪ N5
F12
G 2 (C4 ) ∪ 4 M
where G 2 (C4 ) is the Grassmannian of complex 2-dimensional subspaces of C4 and where the left projection is given by (, ) → and the right by (, ) → . A point of CP 3 determines a plane in G 2 (C4 ), called an α-plane, given by all the containing 4 . This plane may or may not intersect M ; if it does it does so in a null geodesic. The points of CP 3 which give light rays are precisely the points of N 5 . In order to describe a shear-free ray congruence (SFR), it is useful to have the notion of conformal foliation. We formulate this in terms of a semi-conformal mapping, which will be the fundamental object we discuss later in the context of graphs. A Lipschitz map ϕ : (M m , g) → (N n , h) between Riemannian manifolds is said to be semi-conformal if, at each point x ∈ M where ϕ is differentiable (dense by Radmacher’s Theorem), the derivative dϕx : Tx M → Tϕ(x) N is either the zero map or is conformal and surjective on the orthogonal complement of ker dϕx (called the horizontal distribution). Thus, there exists a number λ(x) (defined almost everywhere), called the dilation, such that λ(x)2 g(X, Y ) = ϕ ∗ h(X, Y ), for all X, Y ∈ (ker dϕx )⊥ . If ϕ is of class C 1 , then we have a useful characterisation in local coordinates, given by β
g i j ϕiα ϕ j = λ2 h αβ , where (x i ), (y α ) are coordinates on M, N , respectively and ϕiα = ∂(y α ◦ ϕ)/∂ x i . The fibres of a smooth submersive semi-conformal map determine a conformal foliation, see [26] and conversely, with respect to a local foliated chart, we may put a conformal structure on the leaf space with respect to which the projection is a semi-conformal map. We then have the identity: (LU g) (X, Y ) = −2U (ln λ) g(X, Y ), for U tangent and X, Y orthogonal to the foliation. This latter equation can be taken to be the characterisation of a conformal foliation. Specifically, a foliation is called conformal if there is a function a = a(U ) which depends only on U , such that (LU g) (X, Y ) = a(U ) g(X, Y ), for U tangent and X, Y orthogonal to the foliation. The relation between a and the dilation λ can now be deduced by calculating the mean curvature of the horizontal distribution; see, for example [2]. A shear-free ray congruence on a region A ⊂ M4 is a foliation by null-geodesics which is without shear. That is, if W represents the tangent vector field to the congruence, then at a point x ∈ A, the metric complement W ⊥ is 3 dimensional and contains W itself; if we take a 2-dimensional spacelike complement S in W ⊥ , then for the congruence to be shear-free, we require Lie transport of vectors in S along W to be conformal. This property is independent of the choice of S. By the Kerr Theorem, locally an analytic
502
P. Baird, M. Wehbe
shear-free ray congruence is defined by the intersection of N 5 with a complex analytic surface S [16,22]. In general the congruence of light rays defined by N 5 ∩ S will be multivalued with singularities. Solutions of the zero rest-mass field equations
∇ A A ϕ AB...L = 0 are then given by considering a function f (z 1 , z 2 , z 3 , z 4 ) homogeneous of degree −n −2 and taking a contour integral in an appropriate way. This is the basis of the Penrose transform, which is an integral transform from sheaf cohomology in the twistor space into the space of massless fields; see [12,20,28] for details. In [4], the equations for an SFR are reformulated in such a way that will enable us to adapt them to the context of graphs. Specifically, if W is tangent to a future pointing congruence of null curves on a region A ⊂ M4 , then at each point (t, x) ∈ M4 , we can decompose W into its timelike and spacelike components: W = ∂t + U , where U is a unit tangent to the slice Rt3 = {(t, x1 , x2 , x3 ) ∈ M4 : t const }. Then W is tangent to an SFR if and only if Rt3 (i) ∂U ∂t = −∇U U (1) (ii) 0 = (LU g)(X + iY, X + iY ) , where {X, Y, U } is an orthonormal basis tangent to Rt3 at each point and g is the standard Euclidean metric on Rt3 . Indeed, the unit direction field U can be represented by a spinor field [μ A ] ∈ CP 1 and then (1) is equivalent to the usual spinor representation of an SFR: μ A μ B ∇ A A μ B = 0. 4
M W = 0, whereas (1)(ii) is Note that (1)(i) is equivalent to the geodesic condition ∇W equivalent to the property that U be tangent to a conformal foliation on each slice Rt3 . Furthermore, one can show that if (1)(i) is satisfied everywhere and (1)(ii) on an initial slice R03 , then (1)(ii) is satisfied for all t [4]. The Riemannian analogue of this extension, with SFR replaced by integrable Hermitian structure, is discussed in [3]. If we locally integrate the vector field U , so that for each t it is tangent to the fibres of a semi-conformal mapping ϕ = ϕt : Bt → C (Bt open in Rt3 ), the above equations are equivalent to the pair [4]: (i) d ∂ϕ ∂t (U ) = −τ (ϕ) (2) (ii) 0 = g(grad ϕ, grad ϕ) ,
where grad ϕ is the (complex) gradient with respect to the metric g on Rt3 . In fact one can easily check that (2) is invariant under the replacement of ϕt by ψt = ζt ◦ ϕt , where ζt is an arbitrary conformal transformation of a domain of the complex plane; this is precisely the gauge freedom one requires in the choice of ϕt . 3. Holomorphic Functions on a Graph A finite graph of order n is a set V of cardinality n endowed with a binary relation ∼. For x, y ∈ V , if x ∼ y we will say that x and y are neighbours, or are joined by an edge and we will represent this diagrammatically by drawing a line segement between
Twistor Theory on a Finite Graph
503
x and y. We suppose in what follows that the relation ∼ is symmetric, so that edges are not directed, although most of our discussion also applies to directed graphs. We prefer to use the term directed, rather than the more usual oriented; the latter term being reserved for a notion of orientation of a (undirected) graph, rather akin to orientation of a manifold, which we will define later. We do not allow the relation ∼ to be reflexive, so that the graph does not contain loops, neither do we allow multiple edges, although once more, the discussion can be adapted to this more general situation. We can represent the edges as a subset E of the formal symmetric product V V and so express the graph as the pair = (V, E). It will often be convenient to represent an edge (x, y) ∈ E using the notation x y, or, if we impose a direction on the edge, by xy. We say that the edge x y is incident with the vertex x (and also with y). It is our aim to represent fields purely in terms of the combinatorial properties of graphs and as far as possible to dispense with notions of (semi-) Riemannian geometry. However, a natural generalisation of our theory is to endow each edge with a real number, called its length and to consider what are called metric graphs. One can even go further, and suppose that an angle is defined between edges incident with a given vertex; however, this now becomes an approximation of (semi-)Riemannian geometry and would defeat our purpose of developing a purely combinatorial theory. Many notions of Riemannian geometry translate into combinatorial properties of graphs. A useful reference is the book by Chung [11], which uses slightly different conventions. We outline below those notions which are essential to our development. Given a graph = (V, E), to each x ∈ V , we define its degree m(x) to be the number of edges incident with x. A graph is called regular if m(x) = m is constant for each vertex. We define the tangent space at x ∈ V , to be the set Tx := {xy : x y ∈ E}. That is, each element of Tx is a directed edge, with base point x and end point y ∼ x. Given a function ϕ : V → R N with values in a Euclidean space and a vector X = xy ∈ Tx , we define its directional derivative in the direction X to be the number dϕx (X ) = ϕ(y) − ϕ(x). Note that we could extend the notion of tangent space to include all linear combinations of edges xy, y ∼ x, to obtain a vector space, but we prefer to use a discrete concept for the tangent space. If ω : Tx → R N , then we define its co-derivative at x to be the quantity d∗ ω(x) = −
1 ω(xy). m(x) y∼x
If for each x ∈ V we have given a map ω = ωx : Tx → R N , then provided ω(xy) = −ω( yx), we will refer to ω as an R N -valued 1-form. In particular, if f : V → R is a function, then d f is a 1-form and we have d∗ d f (x) = −
1 ( f (y) − f (x)) m(x) y∼x
= f (x) − = f (x) ,
1 f (y) m(x) y∼x
504
P. Baird, M. Wehbe
where we define the Laplacian of f to be the quantity f (x) := f (x) −
1 f (y). m(x) y∼x
Define the L 2 inner product of two functions f, g : V → R to be the sum m(x) f (x)g(x) , f, g = x∈V
and the inner product of their derivatives to be (d f, dg) =
d f (xy)dg(xy) =
1 ( f (y) − f (x))(g(y) − g(x)), 2 y∼x x∈V
x y∈E
the factor of one half appearing in the last term, since here each edge is counted twice. Then f, g = (d f, dg). so that the Laplacian is a self-adjoint operator on L 2 (V, R). It follows that its eigenvalues are real and non-negative [11]. The notion of semi-conformal mapping between graphs was introduced by H. Urakawa in 2000 [24,25]. More recently, these have been called holomorphic mappings by M. Baker and S. Norine in their development of Riemann surface theory in the context of finite graphs [8,9]. Motivated by our Proposition 3.3 below, we shall also refer to these as holomorphic mappings between graphs. Let 1 = (V1 , E 1 ) and 2 = (V2 , E 2 ) be two (not necessarily finite) graphs. Then a mapping ϕ : V1 → V2 between the vertices is defined to be a mapping of graphs, if, whenever x ∼ y(x, y ∈ V1 ) we have, either ϕ(x) = ϕ(y), or ϕ(x) ∼ ϕ(y). In this case we will write: ϕ : 1 → 2 . Definition 3.1. Let ϕ : 1 = (V1 , E 1 ) → 2 = (V2 , E 2 ) be a mapping of graphs. Then we say that ϕ is holomorphic at x ∈ V1 if, on setting z = ϕ(x), for all z ∼ z, the number λ(x, z ) := {x ∼ x : ϕ(x ) = z } , is well-defined and depends only on x (i.e. it is independent of the choice of z ) in which case we write λ(x) = λ(x, z ). We say that ϕ is holomorphic if it is holomorphic at every point. In this case, if x ∈ V1 is such that ϕ(y) = ϕ(x) for all y ∼ x, we set λ(x) = 0 and so obtain a well-defined function λ : V1 → N, called the dilation of ϕ. The above definition can easily be extended to mappings of metric graphs, where now 1 and 2 are endowed length functions 1 , 2 defined on the edges E 1 , E 2 , respectively [1]. The dilation is then replaced by the function λ(x, z ) =
2 (ϕ(x)z ) x ∼x ϕ(x )=z
1 (x x )
.
Twistor Theory on a Finite Graph
505
Fig. 1. Example of a finite graph endowed with a holomorphic function
An automorphism of a graph = (V, E) is a bijective mapping ϕ : V → V such that x ∼ y if and only if ϕ(x) ∼ ϕ(y). It follows that an automorphism is holomorphic with dilation identically equal to 1. We interpret such a mapping as the analogue of an isometry in the setting of smooth manifolds. Thus a holomorphic map generalizes this notion. Given a graph = (V, E) and a vertex x ∈ V , then a function f : V → R is harmonic at x if f (x) = 0 – we will call such a function a local harmonic function. In [24,25] it is shown that a mapping between graphs pulls back local harmonic functions to local harmonic functions if and only if it is holomorphic. Mappings which preserve local harmonic functions in a wider context are called harmonic morphisms [7]. We now introduce one of the fundamental objects of our study, namely a holomorphic function on a graph (Fig. 1) Definition 3.2. Let = (V, E) be a (not necessarily finite) graph, then a function ϕ : V → C is called holomorphic at x ∈ V if (dϕ(x y))2 = (ϕ(y) − ϕ(x))2 = 0. y∼x
y∼x
We say that ϕ : → C is holomorphic if it is holomorphic at every vertex x ∈ V . The notion is a natural adaptation of that of a semi-conformal mapping ϕ : M m → C from a Riemannian m-manifold into the complex plane, as discussed in Sect. 2. For, ϕ : U ⊂ R2 → C is semi-conformal if and only if 2 2 ∂ϕ ∂ϕ ∂ϕ ∂ϕ = 0. + =4 ∂x ∂y ∂z ∂z That is, if and only if ϕ is holomorphic or anti-holomorphic. But on a graph, we do not a priori have a notion of orientation, which in the plane is precisely what distinguishes holomorphic from anti-holomorphic, which justifies the above definition. However, we do sacrifice linearity in the equation for holomorphicity, which is an essential ingredient in the study by Baker and Norine who develop their theory using harmonic functions. We now prove an analogue in the context of holomorphic functions, of a theorem of Urakawa [24,25], that holomorphic mappings between graphs are characterized as those mappings which preserve harmonic functions. Proposition 3.3. Let ϕ : 1 = (V1 , E 1 ) → (V2 , E 2 ) be a mapping between graphs. Then ϕ is holomorphic if and only if it preserves local holomorphic functions, that is, if f : V2 → C is holomorphic at ϕ(x)(x ∈ V1 ), then f ◦ ϕ is holomorphic at x. In particular, if ϕ : 1 → 2 is holomorphic, then f ◦ ϕ is also holomorphic for every holomorphic function f : V2 → C.
506
P. Baird, M. Wehbe
Proof. Suppose that ϕ : 1 → 2 is holomorphic and let f : V2 → C be holomorphic at y ∈ V2 . Consider the function f ◦ ϕ. We show that it is holomorphic at each point x with ϕ(x) = y. Now
2 ( f ◦ ϕ)(x ) − ( f ◦ ϕ)(x) = ( f (ϕ(x )) − f (y))2 x ∼x
x ∼x
= λ(x)
( f (y ) − f (y))2 = 0 ,
y ∼y
by the holomorphicity of ϕ. Conversely, suppose that ϕ : 1 → 2 preserves local holomorphic functions. Let y ∈ V2 and let x ∈ ϕ −1 (y) ∈ V1 . If there is only one vertex y1 ∼ y, then the condition of holomorphicity at x is trivially satisfied, so we may suppose there are at least two distinct vertices joined by an edge to y. Let y1 , y2 ∼ y. We want to show that λ(x, y1 ) = λ(x, y2 ). Consider the function f holomorphic at y given by f (y) = 0, f (y1 ) = i, f (y2 ) = 1 and f (y ) = 0 for all y ∼ y with y = y1 , y2 . By hypothesis, f ◦ ϕ is holomorphic at x, so that, if x1 , . . . , xr ∼ x satisfy ϕ(x1 ) = · · · = ϕ(xr ) = y1 and xr +1 , . . . , xr +s ∼ x satisfy ϕ(xr +1 ) = · · · = ϕ(xr +s ) = y2 , then
2 ( f ◦ ϕ)(x ) − ( f ◦ ϕ)(x) = −r + s , x ∼x
which must vanish, so that r = s and λ(x, y1 ) = λ(x, y2 ). Since y1 , y2 ∼ y are arbitrarily chosen, we conclude that ϕ is holomorphic. We will consider a pair (, ϕ), of a graph together with a holomorphic function ϕ : → C, as a (static) field. Later on, we will consider how to introduce a dynamic into the field. It may be appropriate in the context of quantum field theory to view ϕ as a probability amplitude defined at each vertex. Note that if ϕ : → C is a holomorphic function, then so is cϕ + a for any complex constants a, c ∈ C. A holomorphic function can be viewed as a special case of a more general object, which we refer to as an isotropic 1-form. Definition 3.4. Let ω be a 1-form defined on a graph = (V, E). Then we call ω isotropic if (ω(x y))2 = 0 , y∼x
at each vertex x ∈ V . Then the derivative dϕ of a holomorphic function is an isotropic 1-form. Conversely, we require an integrability condition on ω in order that it be the derivative of a function. This amounts to the requirement that k ω(ek ) should vanish around any cycle {ek }k (a cycle being a sequence of directed edges {e1 , e2 , . . . , er } such that the point of arrival of ek is the start point of ek+1 with er +1 then being identified with e1 ). For if this is the case, then we define ϕ at a fixed vertex x0 , say to take the value ϕ0 and then set ϕ(y) = ϕ0 + ω(xy) for y ∼ x. Continuation of this process to all vertices is well-defined on account of the cycle condition. A quantum graph is a metric graph, such that each edge supports a solution to the 1-dimensional Schrödinger equation with a compatibility condition at each vertex; see
Twistor Theory on a Finite Graph
507
[15] and the references cited therein. We can view the pair (, ϕ) of a graph endowed with a holomorphic function as a similar structure, where we replace a solution to the 1-dimensional Schrödinger equation on an edge xy by the amplitude ϕ(y) − ϕ(x). The compatibility condition at each vertex becomes y∼x (ϕ(y) − ϕ(x))2 = 0. A spin network, in its more recent formulation, consists of a graph where each edge has a label which corresponds to a representation of a particular group. To each vertex is associated an intertwiner which relates these different representations. The original spin networks of Penrose consist of regular graphs with each vertex having degree 3 and with the associated group SU(2) [23]. Note that the character of an irreducible representation is an algebraic integer, that is, it is the root of some monic equation. We do not know if there may be a deeper connection between spin networks and pairs (, ω), where ω is an isotropic 1-form with the different values ω(xy) corresponding to characters of representations satisfying polynomial identities at each vertex. Another interesting construction is the following. Given n complex numbers z 1 , z 2 , . . . , z n satisfying nk=1 z k 2 = 0, then one can construct an n-dimensional cube in Rn such that there exists an orthogonal projection from Rn onto C which maps one vertex v0 ∈ Rn to 0 ∈ C and its neighbouring vertices v1 , v2 , . . . , vn ∈ Rn to the points z 1 , z 2 , . . . , z n . Conversely, given any orthogonal projection π : Rn → C, then 2 the complex numbers z k = π(vk − v0 ) satisfy z k = 0. This property is known under the name of Gauss’ fundamental theorem of axonometry [14] and when n = 3, the projection of the vertices is known as orthographic projection. For example, the three dimensional cube supports the holomorphic function indicated in Fig. 2. The projection of the vertices of other regular polyhedra satisfy other polynomial equations. For example, the equation (z 1 + · · · z n )2 − (n + 1)(z 1 2 + · · · + z n 2 ) = 0 is satisfied by the orthogonal projections z 1 , . . . , z n of the vertices of a regular tetrahedron [13]. The case of the cube shows how we can see an n-dimensional space arising from a regular graph (, ϕ) with common vertex degree n endowed with a holomorphic function. Specifically, at each vertex x, the complex numbers ϕ(y) − ϕ(x)(y ∼ x) generate a cube in Rn . On some infinite graphs, the construction of a holomorphic function can be easily achieved. For example, let be the integer lattice in R N , with edges joining vertices whose components differ by 1 in a single entry. Then given any complex valued function g0 defined on the set {(x1 , x2 , . . . , x N −1 , 0) ∈ Z N } and another one g1 defined on {(x1 , x2 , . . . , x N −1 , 1) ∈ Z N }, we can now construct a holomorphic function ϕ by
Fig. 2. The 1-skeleton of the cube endowed with a holomorphic function
508
P. Baird, M. Wehbe
Fig. 3. The 1-skeleton of the cube with an orientation giving colouring
extension. Explicitly, ϕ(x1 , x2 , . . . , x N −1 , 2) is obtained by solving the equation N −1
(g1 (x1 , . . . xk − 1, . . . , x N −1 , 1) − g1 (x1 , . . . xk , . . . , x N −1 , 1))2
k=1
+ (g0 (x1 , . . . , x N −1 , 0) − (g1 (x1 , . . . , x N −1 , 1))2 + (ϕ(x1 , . . . , x N −1 , 2) − (g1 (x1 , . . . , x N −1 , 1))2 = 0 for ϕ(x1 , . . . , x N −1 , 2), and so on. In general, at each step there will be two solutions and so infinitely many branches will be defined on R N . We can view such holomorphic functions as solving an intitial value problem: given a function g and its normal derivative on a hypersurface S, find a holomorphic function ϕ which coincides with g and has the same normal derivative on S. However, finding finite graphs which support a holomorphic function seems much harder and at present, using a computer, we can only test examples with a small number of vertices. For example, MAPLE fails to find a holomorphic function on the 1-skeleton of the dodecahedron in a reasonable time, however, it does show the existence of isotropic 1-forms. We now wish to show how, given a graph endowed with a holomorphic function, we can recover a spinor field on the graph. Let = (V, E) be a regular graph with common vertex degree m. An orientation on is a colouring of the edges of the graph with the numbers 1, 2, . . . , m. By a colouring, we mean an assignment of a number k ∈ {1, 2, . . . , m} to each edge so that no two edges incident with the same vertex have the same colour. For example, the 1-skeleton of the cube, above, is coloured as in Fig. 3. Let = (V, E) be a regular graph of degree 3 which is oriented by the colours {1, 2, 3}. Suppose further, that is endowed with an isotropic 1-form ω. Then, given a vertex x ∈ V , we can associate to x a triple of complex numbers ξ(x) = (ξ1 , ξ2 , ξ3 ), where x ∼ y1 , y2 , y3 , ξk = ω(x yk ) and we suppose the edge x yk has colour k(k = 1, 2, 3). Since ξ1 2 + ξ2 2 + ξ3 2 = 0, the symmetric matrix ξ3 −ξ2 − ξ3 i (A, B ∈ {0, 1}) ( AB ) := ξ3 ξ2 − ξ3 i has determinant zero and so can be written in the form AB = μ A μ B , for some spinor (μ A ) ∈ C2 (defined up to sign). We therefore have a spinor field μ A on that provides the analogue of the spinor field on R3 which generates an SFR in Minkowski space, as described in Sect. 2.
Twistor Theory on a Finite Graph
509
We can proceed further and construct the analogue of the vector field U (tangent to the associated conformal foliation in the smooth case) at each vertex. In fact, μ = μ0 /μ1 = −(ξ2 +iξ3 )/ξ1 represents the direction of U in the chart given by stereographic projection, so that 1 2 2 |ξ + iξ | − |ξ | , −ξ (ξ + iξ ) . U= 2 3 1 1 2 3 |ξ1 |2 + |ξ2 + iξ3 |2 It is now possible to consider the discrete analogue of Eq. (2): ∂ϕn (U ) = −ϕn , d ∂n for a family of complex-valued functions {ϕn } parametrized by the natural numbers, equivalently: dϕn+1 (U ) = −ϕn .
(3)
However, care needs to be taken in the choice of sign of U when applying this equation, since our construction has essentially only found a non-oriented direction U at each vertex. In the case when ϕn is a given holomorphic function, we can ask whether (3) determines successive functions ϕn+1 which are also holomorphic. We do not have a general result to this effect, but it does turn out to be the case for the graph consisting of the 1-skeleton of the cube. The following table constructs the successive holomorphic function, which is unique up to addition of a constant. Vertex 1 2 3 4 5 6 7 8
ξ
U
dϕn+1 (U )
(1,
√1 (1, 0, −1) 2 √1 (1, 0, −1) 2 √1 (1, 0, −1) 2 − √1 (1, 0, −1) 2 − √1 (1, 0, −1) 2 − √1 (1, 0, −1) 2 − √1 (1, 0, −1) 2 − √1 (1, 0, −1) 2
√1 (ϕn+1 (2) − ϕn+1 (7)) 2 √1 (ϕn+1 (1) − ϕn+1 (8)) 2 √1 (ϕn+1 (5) − ϕn+1 (4)) 2 √1 (ϕn+1 (3) − ϕn+1 (6)) 2 √1 (ϕn+1 (3) − ϕn+1 (6)) 2 √1 (ϕn+1 (5) − ϕn+1 (4)) 2 √1 (ϕn+1 (1) − ϕn+1 (8)) 2 √1 (ϕn+1 (2) − ϕn+1 (7)) 2
√ 2i, 1) √ (−1, 2i, −1) √ (1, − 2i, 1) √ −(1, 2i, 1) √ −(1, 2i, 1) √ (1, − 2i, 1) √ (−1, 2i, −1) √ (1, 2i, 1)
ϕ√n √ − 32 ( 2 + i) √ √ 2 3 ( 2 − i) √ √ − 32 ( 2 − i) √ √ 2 ( 2 + i) √3 √ 2 3√( 2 + i) √ − 32 ( 2 − i) √ √ 2 3√( 2 − i) √ − 32 ( 2 + i)
ϕn+1 √
2 2 3√ 2( 2+i) √3 2( 2+i) √3 2 2 3 2i 3
0 0 2i 3
4. The Twistor Correspondence Between Graphs Twistor space, as first introduced by R. Penrose [17], is the space whose points correspond to light rays in Minkowski space. More precisely, there is a 5-real dimensional CR-submanifold of CP 3 whose points are the light rays. In order to complete the picture it is necessary to compactify and to complexify M4 to the complex Grassmannian G 2 (C4 ) of complex 2-planes through the origin in C4 . Via the twistor double fibration, a point of CP 3 now determines an α-plane in G 2 (C4 ), which, if it intersects the real space M4 , does so in a null geodesic (see, for example [28]). On the other hand, associated to the three-dimensional space forms is their minitwistor space: the space of all geodesics. For example, the mini-twistor space of R3 is
510
P. Baird, M. Wehbe
Fig. 4. The line-graph of the graph of Fig. 1
the complex surface given by the tangent bundle to the 2-sphere: T S 2 ; each line in R3 being defined by its direction u ∈ S 2 and its displacement from the origin c ∈ Tu S 2 (c is the unique vector starting at the origin which hits the line at right angles) (see, [7]). In view of these correspondences, it is very natural to define the twistor dual of a graph to be the graph whose vertices are the edges of the original graph, where two vertices are connected if and only if the corresponding edges in the original graph are incident. This dual graph is a well-known classical concept called the line-graph. Precisely, given a graph = (V, E), then the line-graph or twistor dual of is the graph L = (E, T ), where, for X, Y ∈ E, we have X ∼ Y if and only if X and Y are incident in . The only connected graph that is isomorphic to its line-graph is a cyclic graph and H. Whitney showed that, with the exception of the graphs K 3 (the complete graph on three vertices) and K 1,3 (the bipartite graph with edges joining one vertex to three other unconnected vertices)), any two connected graphs with isomorphic line graphs are isomorphic [30]. Not every graph arises as the line-graph of a graph, specifically, there are nine classified graphs, such that provided a given graph L doesn’t contain one of them as a subgraph, then L = L is the line-graph of some graph [27,10]. As an example, Fig. 4 shows the line-graph of the graph of Fig. 1. We can now pursue the twistor correspondence, so that a vertex of a graph corresponds to a complete subgraph of the line graph L . This latter object is then the discrete analogue of the complex projective line corresponding to all the light rays passing through a given point. If now is endowed with an isotropic 1-form ω : T → C, then, on giving each edge a direction, we can define a corresponding dual function ψ : V (L ) → C, by ψ(X ) = ω(xy), where X = x y has direction xy. It follows that if x ∈ V () and C x is the complete subgraph of L corresponding to x, then ψ(X )2 = 0. (4) X ∈C x
Note that this latter condition is independent of the choice of direction given to each edge in . Conversely, given a graph L which is the line graph of a graph , and a function ψ : V (L) → C satisfying (4) for each complete subgraph C x corresponding to a vertex x ∈ V (), then on giving each edge in a direction, we can define an isotropic 1-form on . If further is regular of degree three and oriented by colour, as described in the previous section, we then have a spinor field μ A on giving the analogue of an SFR. This provides a discrete analogue of the Kerr Theorem, which associates to a complex analytic surface in CP 3 , a shear-free ray congruence in Minkowski space.
Twistor Theory on a Finite Graph
511
References 1. Anand, C.K.: Harmonic morphisms of metric graphs. In: Harmonic morphisms, harmonic maps and related topics. Research Notes in Mathematics, Vol. 413. (ed. C. Anand, P. Baird, E. Loubeau, J. C. Wood), Boca Raton, FL: Chapman and Hall/CRC, 2000, pp. 97–108 2. Baird, P.: Harmonic maps with symmetry, harmonic morphisms and deformations of metrics. Research Notes in Mathematics, Vol. 87. Boston: Pitman, 1983 3. Baird, P., Eastwood, M.G.: CR geometry and conformal foliations. http://arxiv.org/abs/1011.4717v1 [math.Ds], 2010 4. Baird, P., Wehbe, M.: Shear-free ray congruences on curved space-times. http://arXiv.org/abs/0909. 0241v1 [math-ph], 2009 5. Baird, P., Wood, J.C.: Harmonic morphisms, conformal foliations and shear-free ray congruences. Bull. Belg. Math. Soc. 5, 549–564 (1998) 6. Baird, P., Wood, J.C.: Harmonic morphisms and shear-free ray congruences. http://arXiv.org/abs/math/ 0306390v1 [math.Ds], 2003 7. Baird, P., Wood, J.C.: Harmonic Morphisms between Riemannian Manifolds. London Math. Soc. Monograph, 39. Oxford: Oxford University Press, 2003 8. Baker, M., Norine, S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Advances in Math. 215(2), 766–788 (2007) 9. Baker, M., Norine, S.: Harmonic morphisms and hyperelliptic curves. Int. Math. Res. Not. 2914–2955 (2009) 10. Beineke, L.W.: Derived graphs and digraphs. In: Beiträge sur Graphentheorie. (ed. H. Sachs, H. Voss and H. Walther). Leipzig: Teubner, 1968, pp. 17–33 11. Chung, F.R.K.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, Vol. 92. Washington, DC, 1997 12. Eastwood, M.G.: Introduction to Penrose transform. In: The Penrose Transform and Analytic Cohomology in Representation Theory, Proceedings of an AMS-IMS-SIAM Conference Held at Mount Holyoke College, Massachusetts, Cont. Math. Vol. 154. Providence, RI: Amer. Math. Soc. 1993, pp. 71–75 13. Eastwood, M.G., Penrose, R.: Drawing with complex numbers. Math. Intelligencer 22, 8–13 (2000) 14. Gauss, C.F.: Werke, Zweiter Band. Königlichen Gesellschaft der Wissenschaften, Göttingen, 1876 15. Gnutzman, S., Smilansky, U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006) 16. Huggett, S., Todd, P.: Introduction to Twistor Theory. London Math. Soc. Lecture Notes. Cambridge: Cambridge University Press, 1985 17. Penrose, R.: Twistor algebra. J. Math. Phys. 8, 66–345 (1967) 18. Penrose, R.: Solutions of the zero-rest-mass equations. J. Math. Phys. 10(1), 38–39 (1969) 19. Penrose, R.: Angular momentum: an approach to combinatorial space-time. In: Quantum Theory and Beyond. ed. T. Bastin, Cambridge: Cambridge Univ. Press, 1971, pp. 151–180 20. Penrose, R.: Twistor Theory, its aims and achievements. In: Quantum Gravity, C.J. Isham, R. Penrose, D.W. Sciama. eds., Oxford: Clarendon Press, 1975, 268–407 21. Penrose, R., Rindler, W.: Spinors and space-time. Vol. 1. Two-spinor calculus and relativistic fields, Cambridge Monographs on Mathematical Physics, 2nd edn, 1987 (1st edition, 1984) 22. Penrose, R., Rindler, W.: Spinors and space-time. Vol. 2. Spinor and twistor methods in space-time geometry. Cambridge Monographs on Mathematical Physics, 2nd edn. 1988 (1st edition, 1986). Cambridge and New York: Cambridge University Press, 1988 23. Rovelli, C., Smolin, L.: Spin networks and quantum gravity. Phys. Rev. D 52, 59–5743 (1995) 24. Urakawa, H.: A discrete analogue of the harmonic morphism. In: Harmonic morphisms, harmonic maps and related topics. Research Notes in Mathematics, Vol. 413 (ed. C. Anand, P. Baird, E. Loubeau, J.C. Wood), 97–108. Boca Raton, FL: Chapman and Hall/CRC, 2000 25. Urakawa, H.: A discrete analogue of the harmonic morphism and Green kernel comparison theorems. Glasgow Math. J. 42, 319–334 (2000) 26. Vaisman, I.: Conformal foliations. Kodai Math. J. 2, 26–37 (1979) 27. van Rooij, A.C.M., Wilf, H.S.: The interchange graph of a finite graph. Acta Math. Acad. Sci. Hungar. 16, 263–269 (1965) 28. Ward R.S., Wells R.O. Jr.: Twistor geometry and field theory. Cambridge Monographs on Mathematical Physics. Cambridge and New York: Cambridge University Press, 1990 29. Wehbe, M.: Aspects twistoriels des applications semi-conformes. Thesis, Université de Bretagne Occidentale, 2009 30. Whitney, H.: Congruent graphs and the connectivity of graphs. Amer. J. Math. 54, 150–168 (1932) Communicated by P.T. Chru´sciel
Commun. Math. Phys. 304, 513–581 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1242-9
Communications in
Mathematical Physics
Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff R. Alexandre1,2 , Y. Morimoto3 , S. Ukai4 , C.-J. Xu5,6 , T. Yang7 1 IRENAV Research Institute, French Naval Academy, Brest-Lanvéoc 29290, France.
E-mail: [email protected]
2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China 3 Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan.
E-mail: [email protected]
4 17-26 Iwasaki-cho, Hodogaya-ku, Yokohama 240-0015, Japan.
E-mail: [email protected]
5 School of Mathematics, Wuhan University, Wuhan 430072, P. R. China 6 Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12,
76801 Saint Etienne du Rouvray, France. E-mail: [email protected]
7 Department of mathematics, City University of Hong Kong, Hong Kong, P. R. China.
E-mail: [email protected] Received: 14 February 2010 / Accepted: 13 November 2010 Published online: 26 April 2011 – © Springer-Verlag 2011
Abstract: We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem. Contents 1. 2.
3. 4.
5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . Non-isotropic Norms . . . . . . . . . . . . . . . . . . . 2.1 Coercivity and upper bound estimates . . . . . . . 2.2 Definition and properties of the non-isotropic norm 2.3 Upper bound estimates . . . . . . . . . . . . . . . Commutator Estimates . . . . . . . . . . . . . . . . . . 3.1 Non-isotropic norm in R6x,v . . . . . . . . . . . . . 3.2 Weighted estimates on commutators . . . . . . . . Local Existence . . . . . . . . . . . . . . . . . . . . . 4.1 Energy estimates for a linear equation . . . . . . . 4.2 Existence for the linear equation . . . . . . . . . . 4.3 Convergence of approximate solutions . . . . . . . Qualitative Study on the Solutions . . . . . . . . . . . . 5.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . 5.2 Non-negativity . . . . . . . . . . . . . . . . . . . Full Regularity . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
514 519 519 521 528 530 530 531 538 538 541 543 545 546 551 554
514
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
6.1 Formulation of the problem . . . . . . 6.2 Gain of regularity in velocity variable 6.3 Gain of regularity in space variable . . 6.4 Higher order regularity . . . . . . . . 7. Global Existence . . . . . . . . . . . . . . 7.1 Macroscopic energy estimate . . . . . 7.2 Microscopic energy estimate . . . . . 7.3 A Priori estimate . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
555 556 558 564 565 566 570 578 579
1. Introduction We consider the Cauchy problem for the inhomogeneous Boltzmann equation f t + v · ∇x f = Q( f, f ),
f |t=0 = f 0 ,
(1.1)
where f = f (t, x, v) is the density distribution function of particles, having position x ∈ R3 and velocity v ∈ R3 at time t. Here, the right hand side of (1.1) is given by the Boltzmann bilinear collision operator, which is given in the classical σ −representation by Q(g, f ) =
R3 S2
B (v − v∗ , σ ) g∗ f − g∗ f dσ dv∗ ,
where f ∗ = f (t, x, v∗ ), f = f (t, x, v ), f ∗ = f (t, x, v∗ ), f = f (t, x, v), and for σ ∈ S2 , v =
|v − v∗ | v + v∗ |v − v∗ | v + v∗ + σ, v∗ = − σ, 2 2 2 2
which gives the relation between the post and pre collisional velocities. Recall that we have conservation of momentum and kinetic energy, that is, v + v∗ = v + v∗ and |v|2 + |v∗ |2 = |v |2 + |v∗ |2 . The kernel B is the cross-section which can be computed in different physical settings. In particular, the non-negative cross section B(z, σ ) depends only on |z| and the z scalar product |z| , σ . In most cases, the kernel B cannot be expressed explicitly, but to capture its main properties, one may assume that it takes the form π v − v∗ ,σ , 0≤θ ≤ . B(|v − v∗ |, cos θ ) = (|v − v∗ |)b(cos θ ), cos θ = |v − v∗ | 2
An important example is the inverse power law potential ρ −r with r > 1, ρ being the distance between two particles, in which the cross section has a kinetic factor given by (|v − v∗ |) ≈ |v − v∗ |γ ,
4 γ =1− , r
The Boltzmann Equation Without Angular Cutoff
515
and a factor related to the collision angle containing a singularity, b(cos θ ) ≈ K θ −2−2s when θ → 0+, for some constants K > 0 and 0 < s = r1 < 1. The cases with 1 < r < 4, r = 4 and r > 4 correspond to so-called soft, Maxwellian molecule and hard potentials respectively. In the following discussion, this type of cross sections, with the parameters γ and s given above, will be kept in mind. As a fundamental equation in kinetic theory and a keystone in statistical physics, the Boltzmann equation has attracted, and is still attracting, a lot of research investigations since its derivation in 1872. A large number of mathematical works have been performed under Grad’s cutoff assumption, avoiding the non-integrable angular singularity of the cross-sections, see for example [22,23,34,39,40,46,48,49,67,68] to cite only a few, further references being given in the review [70,73]. However, except for the hard sphere model, for most of the other molecule interaction potentials, such as the inverse power laws recalled above, the cross section B(v − v∗ , σ ) is a non-integrable function in angular variable and the collision operator Q( f, f ) is a nonlinear singular integral operator velocity variable. By no means to be complete, let us now review some previous works related to the Boltzmann equation in the context of such singular (or non-cutoff) cross-sections. For other references and comments, readers are referred to [5,73]. The mathematical study for the Boltzmann equation, without assuming Grad’s cutoff assumption, can be traced back at least to the work by Pao in the 1970s [64] which is about the spectrum of the linearized operator. In the 1980s, the existence of weak solutions to the spatially homogeneous case was proved by Arkeryd in [17] for the mild singular case, that is, when 0 < s < 21 , and by using an abstract Cauchy-Kovalevskaya theorem, Ukai in [69] proved the local existence of solutions to both the spatially homogeneous and inhomogeneous equations, in the space of functions which are analytic in x and Gevrey in v. For a long time, the mathematical study of singular cross-sections was limited to these results and a few others, most of them related to the spatially homogeneous case concerning only the existence. An important step was initiated by the works of Desvillettes and his collaborators in the 1990s, showing partial regularization results for some simplified kinetic models, cf. [26–29,31,33,72]. After the well known result of DiPerna and Lions [34] for the cutoff case, Lions was able to show the gain of regularity of solutions in the Landau case [50], which is a model arising as a grazing limit of the Boltzmann equation. It was then expected that this kind of singular cross sections should lead to smoothing effect on solutions, that is, the solutions have higher regularity than the initial data. For example, it should be similar to the case when one replaces the collision operator in the Boltzmann equation by a fractional Laplacian in the velocity variable, that is, a fractional Kolmogorov-type equation [61]. Certainly, the results of Lions [51] and Desvillettes [27–29] have influenced the research in this direction. It is therefore not surprising that a systematic approach, using the entropy dissipation and/or the smoothing property of the gain part of the collision operator, was initiated and has been developed to an almost optimal stage through the efforts of many researchers, such as Alexandre, Bouchut, Desvillettes, Golse, Lions, Villani and Wennberg. The underlying tools have proved to be very useful for the study on the mathematical theory regarding the regularizing effect for the spatially homogeneous problems for which the theory can now be considered as quite satisfactory,
516
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
cf. [6–16,24,29,30,32,47,59,60,71], and the references therein, see also for a much more detailed discussion [5]. Compared to the spatially homogeneous problems, the original spatially inhomogeneous Boltzmann equation is of course physically more interesting and mathematically more challenging. For the existence of weak solutions, we mention two results regarding the Cauchy problem. One is about the local existence between two moving Maxwellians proved in [3] by constructing the upper and lower solutions, another is the global existence of renormalized solutions with defect measures shown in [16] where the solutions become weak solutions if the defect measures vanish. On the other hand, the local existence of classical solutions was proved in [12] in some weighted Sobolev spaces. However, in view of the above available results, the mathematical theory for noncutoff cross-sections is so far not satisfactory. This is in sharp contrast to the cutoff case, for which the theories have been well developed, see [19–21,34,36,46,52,53,67,68,70] and the references therein. For the study of the regularizing effect, one of the main difficulties comes from the coupling of the transport operator with the collision operator, which is similar to the Landau equation studied in [25]. To overcome this difficulty, a generalized uncertainty principle à la Fefferman [38] (see also [56–58]) was introduced in [8,9] for the study of smoothing effects of the linearized and spatially inhomogeneous Boltzmann equation with non-cutoff cross-sections. In order to complete the full regularization process, recently, in [12], by using suitable pseudo-differential operators and harmonic analysis, we have developed sharp coercivity and upper bounds of the collision operators in Sobolev space, together with the estimation on the commutators with these pseudo-differential operators. More precisely, in [10–12], for classical solutions, we established the hypo-ellipticity of the Boltzmann operator, using a generalized version of the uncertainty principle. The present work is a continuation of our collaborative program since 2006 [9–12]. Comparing to the cutoff case, we aim to settle a mathematical framework similar to the studies first proved by Ukai, see [67,68], and fitted into an energy method by Liu and collaborators [52,53] and Guo [46] which has led to a clean theory for the Cauchy problem in the cutoff case, for solutions close to a global equilibrium. In this paper, we will establish the global existence of non-negative solutions in some Sobolev space for the Boltzmann equation near a global equilibrium and prove the full regularity in all variables for any positive time. As mentioned in the abstract, one of the main ingredients in the proof is the introduction of a new non-isotropic norm which captures the main feature of the singularity in the cross-section. This new norm is in fact the counterpart of the coercive norm which was introduced by Guo [45] as an essential step for Landau equation. It is not known if there is any equivalence of this norm to some Sobolev norm, in contrast to the case of the Landau equation. However, since it is designed to be equivalent to, and to have much simpler expression than, the Dirichlet form of the linearized collision operator, this norm not only works extremely well for the description of the dissipative effect of the linearized collision operator through the H-theorem, but also well fits for the upper bound estimation on the nonlinear collision operator. Here, we would like to mention the work by Mouhot and Strain [62,63] about the gain of moment in a linearized context due to the singularity in the cross-section. Such a gain of moment which is well described by the new non-isotropic norm is in fact crucial for the proof of the global existence.
The Boltzmann Equation Without Angular Cutoff
517
We now come back to the problem considered in this paper. To make the presentation as simple as possible, and to concentrate on the singularity of the grazing effect, we shall study the Maxwellian molecule type cross-sections with mild singularity, that is, the case when π v − v∗ B(|v − v∗ |, cos θ ) = b(cos θ ), ,σ , 0≤θ ≤ , cos θ = |v − v∗ | 2 and b(cos θ ) ≈ K θ −2−2s ,
θ → 0+ ,
(1.2)
with 0 < s < 21 . The general case will be left to our future work. In order to prove the global existence, we need to use the complete dissipative effect of the collision operator. Similar to the angular cutoff case, such dissipative effect can be fully represented by the dissipation of the linearized collision operator on the microscopic component of the solution through the H-theorem. Thus, as usual, we consider the Boltzmann equation around a normalized Maxwellian distribution 3
μ(v) = (2π )− 2 e−
|v|2 2
.
√ Since μ is the global equilibrium state satisfying Q(μ, μ) = 0, by setting f = μ+ μg, we have Q(μ +
√
μg, μ +
√ √ √ √ √ μ g) = Q(μ, μ g) + Q( μ g, μ) + Q( μ g, μ g).
Denote √ √ (g, h) = μ−1/2 Q( μ g, μ h). Then the linearized Boltzmann operator takes the form √ √ Lg = L1 g + L2 g = −( μ , g) − (g, μ ). And the original problem (1.1) is now reduced to the Cauchy problem for the perturbation g, gt + v · ∇x g + Lg = (g, g), t > 0 ; (1.3) g|t=0 = g0 . This problem will be considered in the following weighted Sobolev spaces. For k, ∈ R, set H k (R6x,v ) = f ∈ S (R6x,v ) ; W f ∈ H k (R6x,v ) , where R6x,v = R3x × R3v and W (v) = v = (1 + |v|2 ) /2 is the weight with respect to the velocity variable v ∈ R3v . The main theorem can be stated as follows.
518
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Theorem 1.1. Assume that the cross-section satisfies (1.2) with 0 < s < 1/2. Let g0 ∈ H k (R6 ) for some k ≥ 3, ≥ 3 and √ f 0 (x, v) = μ + μ g0 (x, v) ≥ 0. Then there exists ε0 > 0, such that if admits a unique global solution
g0 H k (R6 ) ≤ ε0 , the Cauchy problem (1.3)
g ∈ L ∞ ([0, +∞[ ; H k (R6 )). √ Moreover, f (t, x, v) = μ + μ g(t, x, v) ≥ 0 and g ∈ C ∞ (]0, +∞[ ×R6 ). Actually, for the uniqueness, we can prove the following stronger result, which might be of independent interest. Note that here we do not need to assume that f is a small perturbation of μ. Theorem 1.2. Under the same condition on the cross-section, for 0 < T ≤ +∞ 2s (R3 )). Suppose that f , f ∈ and l > 2s + 7/2, let f 0 ≥ 0, f 0 ∈ L ∞ (R3x ; Hl+2 1 2 v 2s ∞ 3 3 L (]0, T [×Rx ; Hl+2 (Rv )) are two solutions to the Cauchy problem (1.1). If one solution is non-negative, then f 1 ≡ f 2 . Throughout this paper, we assume that the cross-section satisfies the condition (1.2) with 0 < s < 1/2 except otherwise stated. The rest of the paper will be organized as follows. In the next section, we will introduce a new non-isotropic norm and prove some essential coercivity and upper bound estimates on the collision operators with respect to this new norm. In order to study the gain of regularity of the solution, we need to apply some pseudo-differential operators on the Boltzmann equation. For this purpose, in Sect. 3, we study the commutators of the collision operators with the pseudo-differential operators. In Sect. 4, we will apply the energy method for the Boltzmann equation and obtain the local existence theorem. In Sect. 5, we will study the uniqueness and the non-negativity of the solutions. This new method for proving non-negativity can be applied to the case with angular cutoff. For more detailed discussion on the non-negativity problem, refer to [15]. In Sect. 6, the full regularity is proved along the approach of [12]. Finally, the global existence of the solution will be given in the last section. For this, the macro-micro decomposition introduced by Guo [45] will be used for the estimation on the macroscopic component. Note. After finishing this paper, we were informed by R. Strain of his recent paper in collaboration with P. Gressmann [41], showing also the existence of global solutions to the Cauchy problem by using a different approach. Notice that their solution is in different function space which does not lead to full regularity because of the weak regularity in the velocity variable. Note added in September, 2010. Several new results have been announced along the same line of development since the submission of the current paper. For the reader’s references we mention [13–15,42,41–44]. The main difference of the results is the range of admissible values of γ : γ > −1 − 2s in the first 3 papers and γ > max(−3, −3/2 − 2s) in the latter 4 papers.
The Boltzmann Equation Without Angular Cutoff
519
2. Non-isotropic Norms In this section, we study the bilinear collision operator given by Q(g, f ) = b(cos θ ) g∗ f − g∗ f dσ dv∗ , R3 S2
through harmonic analysis. Since the collision operator acts only with respect to the velocity variable v ∈ R3 , (t, x) is regarded as a parameter in this section.
2.1. Coercivity and upper bound estimates. Let g ≥ 0, g ≡ / 0, g ∈ L 12 L log L(R3v ). It was shown in [6] that there exists a constant cg > 0 depending only on the values of g L 1 and g L log L such that for any smooth function f ∈ H s (R3v ), we have 2
cg f 2H s (R3 ) ≤ (−Q(g, f ), f ) L 2 (R3v ) + Cg L 1 (R3v ) f 2L 2 (R3 ) . v
v
(2.1.1)
Besides this, we still need some functional estimates on the Boltzmann collision operators. The first one, given below, is about the boundedness of the collision operator in weighted Sobolev spaces, see [1,2,4,5,12,47]. Theorem 2.1. Assume that the cross-section satisfies (1.2) with 0 < s < 1. Then for any m ∈ R and any α ∈ R, there exists C > 0 such that Q( f, g) Hαm (R3v ) ≤ C f L 1 +
α +2s
m+2s (R3 ) (R3v ) g H(α+2s) + v
(2.1.2)
m+2s (R3 ). for all f ∈ L 1α + +2s (R3v ) and g ∈ H(α+2s) + v
We now turn to the linearized operator. First of all, by using the conservation of energy |v∗ |2 + |v |2 = |v∗ |2 + |v|2 , we have μ(v∗ ) = μ−1 (v) μ(v∗ ) μ(v ). Thus,
√ √ ( f, g)(v) = μ−1/2 μ∗ f ∗ μ g − μ∗ f ∗ μ g dv∗ dσ b(cos θ ) √ = b(cos θ ) μ∗ f ∗ g − f ∗ g dv∗ dσ. (2.1.3) It is well-known that L (acting with respect to the velocity variable) is an unbounded symmetric operator on L 2 (R3v ). Moreover, its Dirichlet form satisfies
√ √ (Lg, g) L 2 (R3 ) = − ( μ , g) + (g, μ ), g L 2 (R3 ) v v
1/2 1/2 1/2 = b(cos θ ) (μ∗ ) g − (μ∗ ) g + g∗ (μ) − g∗ (μ )1/2 (μ∗ )1/2 gdv∗ dσ dv
= b(cos θ ) (μ∗ )1/2 g − (μ∗ )1/2 g + g∗ (μ )1/2 − g∗ (μ)1/2 (μ∗ )1/2 g dv∗ dσ dv
= b(cos θ ) (μ)1/2 g∗ − (μ )1/2 g∗ + g(μ∗ )1/2 − g (μ∗ )1/2 (μ)1/2 g∗ dv∗ dσ dv
= b(cos θ ) (μ )1/2 g∗ − (μ)1/2 g∗ + g (μ∗ )1/2 − g(μ∗ )1/2 (μ )1/2 g∗ dv∗ dσ dv
2 1 = dv∗ dσ dv b(cos θ ) (μ∗ )1/2 g − (μ∗ )1/2 g + (μ)1/2 g∗ − (μ )1/2 g∗ 4 ≥ 0. (2.1.4)
520
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
The third line in the above equation is obtained by using the change of variables (v, v∗ ) → (v , v∗ ). The fourth line follows from the change of variables (v, v∗ ) → (v∗ , v) and then the fifth line follows from the fourth one by using the change of variables (v, v∗ ) → (v , v∗ ). And the second to last line is just the summation of the previous four lines. Note that the Jacobians of the above coordinate transformations are equal to 1. Moreover, it follows from the above formula that (Lg, g) L 2 (R3v ) = 0 if and only if Pg = g, where
√ Pg = a + b · v + c|v|2 μ, with a, c ∈ R, b ∈ R3 . Here, P is the L 2 -orthogonal projection onto the null space N = Span
√ √ √ √ √ μ , v1 μ , v2 μ , v3 μ , |v|2 μ .
The following result on the gain of moment of order s in the linearized framework is essential in the sequent analysis. Theorem 2.2 (Theorem 1.1 of [63]). Assume that the cross-section satisfies (1.2) with 0 < s < 1. Then there exists a constant C > 0 such that (Lg, g) L 2 (R3v ) ≥ C (I − P)g2L 2 (R3 ) . s
v
For the bilinear operator ( ·, · ), we need the following two formulas. For suitable functions f, g, the first formula coming from (2.1.3) is √ ( f, g)(v) = Q( μ f, g) +
b(cos θ )
√ μ∗ − μ∗ f ∗ g dv∗ dσ.
(2.1.5)
On the other hand, applying the change of variables (v, v∗ ) → (v , v∗ ) in (2.1.3) gives (( f, g), h) L 2 (R3v ) = =
√ b(cos θ ) μ∗ f ∗ g − f ∗ g h b(cos θ ) μ∗ f ∗ g − f ∗ g h .
By adding these two lines, the second formula is 1 (( f, g), h) L 2 (R3v ) = 2
√ μ∗ h − μ∗ h . b(cos θ ) f ∗ g − f ∗ g
(2.1.6)
The following lemma shows that L1 controls L. Lemma 2.3. Under the condition (1.2) on the cross-section with 0 < s < 1, we have (L1 g, g) L 2 (R3v ) ≥
1 (Lg, g) L 2 (R3v ) . 2
(2.1.7)
The Boltzmann Equation Without Angular Cutoff
521
Proof. From (2.1.3) and similar changes of variables, we have √ (L1 g, g) L 2 (R3v ) = − ( μ , g), g L 2 (R3 ) v
2 1 = b(cos θ ) (μ∗ )1/2 g − (μ∗ )1/2 g dv∗ dσ dv 2
2 1 = b(cos θ ) (μ )1/2 g∗ − (μ)1/2 g∗ dv∗ dσ dv 2
2 1 = b(cos θ ) (μ∗ )1/2 g − (μ∗ )1/2 g 4
2 1/2 1/2 dv∗ dσ dv. + (μ ) g∗ − (μ) g∗ Therefore, (2.1.7) follows from (A + B)2 ≤ 2(A2 + B 2 ) and (2.1.4).
2.2. Definition and properties of the non-isotropic norm. The non-isotropic norm associated with the cross-section b(cos θ ) is defined by 2 √ 2 b(cos θ )μ∗ g − g + b(cos θ )g∗2 μ − μ , |||g|||2 = (2.2.1) where the integration is over R3v × R3v∗ × S2σ . Thus, it is a norm with respect to the velocity variable v ∈ R3 only. As we will see later, the reason that this norm is called non-isotropic is because it combines both derivative and weight of order s due to the singularity of cross-section b(cos θ ). The following lemma gives an upper bound of this non-isotropic norm by some weighted Sobolev norm. Lemma 2.4. Assume that the cross-section satisfies (1.2) with 0 < s < 1. Then there exists C > 0 such that |||g|||2 ≤ C||g||2Hss
(2.2.2)
for any g ∈ Hss (R3v ). Proof. Applying (2.1.2) with α = −s and m = −s gives
≤ C|| f 2 || 1 ||g|| H s ||g|| H s ≤ C|| f ||2 2 ||g||2 s . (2.2.3) Q( f 2 , g), g Hs L s s L 2 3 L (R v )
2s
s
On the other hand,
Q( f , g), g 2
2 2 b(cos θ ) f g − f g g ∗ ∗ L 2 (R3v )
= b(cos θ ) f ∗2 g − g g+ g 2 b(cos θ ) f ∗2 − f ∗2 . =
522
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
For the first term in the last equation, using b(a − b) = 21 (a 2 − b2 ) − 21 (a − b)2 yields
1 2 Q( f , g), g 2 3 = b(cos θ ) f ∗2 g 2 − g 2 L (R v ) 2
2 1 − b(cos θ ) f ∗2 − f ∗2 . b(cos θ ) f ∗2 g − g + g 2 2 2 2 By the change of variables (v∗ , v ) → (v∗ , v), the first term above is also 21 bg ( f ∗ − 2 f ∗ ). Thus, it follows that
2 1 1 Q( f 2 , g), g 2 3 = − b f ∗2 − f ∗2 , b f ∗2 g − g + g2 L (R v ) 2 2 and then b f ∗2
g −g
2 2 2 2 bg f ∗ − f ∗ . ≤ 2 Q( f , g), g +
2
By using (2.2.3) and the cancellation lemma from [6], we get 2 b f ∗2 g − g ≤ C|| f ||2L 2 ||g||2Hss s
+ C||g||2L 2 || f ||2L 2 ≤ C|| f ||2L 2 ||g||2Hss . s
(2.2.4)
√ μ gives √ √ |||g|||2 ≤ C( μ 2L 2 ||g||2Hss + ||g||2L 2 μ 2Hss ) ≤ C||g||2Hss .
Thus, choosing f =
s
This completes the proof of the lemma.
s
In the context of usual weighted Sobolev spaces, this last result is likely to be optimal. Next we will show that this non-isotropic norm is controlled by the linearized operator. First of all, we shall need the following preliminary computation. Lemma 2.5. For any φ ∈ Cb1 , we have b(cos θ )|φ(v∗ ) − φ(v∗ )|dσ ≤ Cφ |v − v∗ |2s ≤ Cv2s v∗ 2s , σ
where Cφ depends on φC 1 = φ L ∞ + φ L ∞ . b
Proof. It follows from Taylor’s formula that |φ(v∗ ) − φ(v∗ )| ≤ Cφ |v∗ − v∗ | ≤ Cφ sin
θ |v − v∗ |, 2
and |φ(v∗ ) − φ(v∗ )| ≤ Cφ . Then for any δ ∈ (0, π/2), δ π/2 sin(θ/2) 1 b(cos θ )|φ(v∗ ) − φ(v∗ )|dσ ≤ Cφ |v − v∗ | dθ + dθ θ 1+2s θ 1+2s 0 δ σ ≤ Cφ |v − v∗ |δ −2s+1 + δ −2s .
The Boltzmann Equation Without Angular Cutoff
If |v − v∗ |−1 ≤ σ
If |v − v∗ | ≤ σ
π 2,
523
by choosing δ = |v − v∗ |−1 , we get
b(cos θ )|φ(v∗ ) − φ(v∗ )|dσ ≤ Cφ |v − v∗ |2s ≤ Cv2s v∗ 2s . 2 π,
we have
b(cos θ )|φ(v∗ ) − φ(v∗ )|dσ ≤ Cφ |v − v∗ | ≤ Cφ
And this completes the proof of the lemma.
2 ≤ Cv2s v∗ 2s . π
Up to the kernel of L, the following lemma gives the equivalence between the nonisotropic norm and the Dirichlet form of L. Lemma 2.6. For g ∈ N ⊥ , we have (Lg, g) L 2 (R3v ) ∼ |||g|||2 .
(2.2.5)
Here A ∼ B means that there exists two generic constants C1 , C2 > 0 such that C1 A ≤ B ≤ C2 A. Proof. We first deal with the lower bound estimate starting with the terms linked to L2 . Since √ − (L2 g, g) L 2 (R3v ) = (g, μ ), g L 2 (R3 ) , we get from (2.1.5) that √ √ − (L2 g, g) L 2 (R3v ) = Q( μg, μ), g L 2 (R3 ) v √ + b(cos θ ) μ∗ − μ∗ g∗ μ g.
(2.2.6)
Using (2.1.2) with α = 0, m = 0, the first term on the right hand side of (2.2.6) can be estimated by √ √ √ √ Q( μg, μ), g L 2 (R3 ) ≤ ||Q( μg, μ)|| L 2 g L 2 v √ √ ≤ C|| μg|| L 1 || μ|| H 2s g L 2 ≤ C||g||2L 2 . 2s
2s
For the second term on the right hand side of (2.2.6), we have √ μ∗ − μ∗ g∗ μ g dvdv∗ dσ b(cos θ )
√ μ∗ − μ∗ g∗ (μ )1/4 (μ )1/4 − (μ)1/4 g = b(cos θ ) √ + b(cos θ ) μ∗ − μ∗ g∗ (μ )1/4 (μ)1/4 g.
524
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Thus, √ b(cos θ ) μ∗ − μ∗ g∗ μ g 1/2 √
2 ≤ b(cos θ ) μ∗ − μ∗ |g|2 (μ )1/4
1/2
2 1/4 1/4 2 1/4 − (μ) |g∗ | (μ ) b(cos θ ) (μ )
1/2 √ b(cos θ ) μ∗ − μ∗ |g|2 (μ )1/4 (μ)1/4
1/2 √ 2 1/4 1/4 b(cos θ ) μ∗ − μ∗ |g∗ | (μ ) (μ)
× + × 1/2
≤ I1
1/2
× I2
1/2
+ I3
1/2
× I4 .
Using Lemma 2.5 with φ = μ1/4 gives σ
1/4 b(cos θ ) μ∗ − (μ∗ )1/4 dσ ≤ C|v − v∗ |2s ≤ C < v >2s < v∗ >2s .
1/4 1/2 Since μ∗ (μ ) = (μ∗ )1/4 (μ )1/4 (μ )1/4 = (μ∗ )1/4 μ1/4 (μ )1/4 , we get
b(cos θ )|(μ∗ )1/2 − (μ∗ )1/2 | |g|2 (μ )1/2 dvdv∗ dσ
≤C b(cos θ ) (μ∗ )1/4 − (μ∗ )1/4 (μ∗ )1/4 + (μ∗ )1/4 |g|2 (μ )1/2 ≤C b(cos θ ) (μ∗ )1/4 − (μ∗ )1/4 (μ∗ )1/4 (μ )1/2 |g|2 +C b(cos θ ) (μ∗ )1/4 − (μ∗ )1/4 (μ∗ )1/4 (μ )1/2 |g|2
≤C v∗ 2s (μ∗ )1/4 v2s |g|2 + v∗ 2s (μ∗ )1/4 v2s μ1/4 |g|2 dvdv∗
I1 + I3 ≤ C
≤ C(g2L 2 (R3 ) + g2L 2 (R3 ) ). s
For I2 , by using the change of variables (v, v∗ ) → (v∗ , v) and then (v , v∗ ) → (v, v∗ ), one has
2 b(cos θ ) (μ )1/4 − (μ)1/4 |g∗ |2 (μ )1/4
2 = b(cos θ ) (μ∗ )1/4 − (μ∗ )1/4 |g|2 (μ∗ )1/4
≤C v∗ 2s (μ∗ )1/4 v2s |g|2 dvdv∗ ≤ Cg2L 2 (R3 ) . s
The Boltzmann Equation Without Angular Cutoff
525
For I4 , using the change of variables (v, v∗ ) → (v , v∗ ) implies that √ b(cos θ ) μ∗ − μ∗ |g∗ |2 (μ )1/4 (μ)1/4 √ = b(cos θ ) μ∗ − μ∗ |g∗ |2 (μ )1/4 (μ)1/4
≤C v2s (μ)1/4 v∗ 2s |g∗ |2 dvdv∗ ≤ Cg2L 2 (R3 ) . s
In summary, we obtain |(L2 g, g)| ≤ Cg2L 2 .
(2.2.7)
s
For the term involving L1 , using (2.1.6) yields √ (L1 g, g) L 2 (R3v ) = − ( μ, g), g L 2 (R3 ) v
2 1 1/2 1/2 = b(cos θ ) (μ∗ ) g − (μ∗ ) g 2
2 1 = b(cos θ ) (μ∗ )1/2 (g − g) + g((μ∗ )1/2 − (μ∗ )1/2 ) 2
2 1 1 ≥ b(cos θ )μ∗ (g − g)2 − b(cos θ )g 2 (μ∗ )1/2 − (μ∗ )1/2 , 4 2 where we used the inequality (a + b)2 ≥ 21 a 2 − b2 . Then 1 b(cos θ )μ∗ (g − g)2 (L1 g, g) L 2 (R3v ) ≥ 4
2 + b(cos θ )g 2 (μ∗ )1/2 − (μ∗ )1/2
2 3 − b(cos θ )g 2 (μ∗ )1/2 − (μ∗ )1/2 . 4 We now apply (2.2.4) and the change of variables (v, v∗ ) → (v∗ , v) to get
2 b(cos θ )g 2 (μ∗ )1/2 − (μ∗ )1/2 ≤ C||g||2L 2 ||μ1/2 ||2Hss ≤ C||g||2L 2 . s
Therefore, (L1 g, g) L 2 ≥
1 |||g|||2 − Cg2L 2 . s 4
Thus, we have from (2.2.7), (Lg, g) L 2 = (L1 g, g) L 2 + (L2 g, g) L 2 1 ≥ |||g|||2 − C||g||2L 2 . s 4 By Theorem 2.2, we have from the assumption g ∈ N ⊥ that
s
526
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
|||g|||2 ≤ 4 (Lg, g) L 2 + C||g||2L 2 ≤ C˜ (Lg, g) L 2 , s
which gives the lower bound estimation. For the upper bound estimate, we have
2 1 b(cos θ ) (μ∗ )1/2 g − (μ∗ )1/2 g (L1 g, g) L 2 (R3v ) = 2
2 1 = b(cos θ ) (μ∗ )1/2 (g − g) + g((μ∗ )1/2 − (μ∗ )1/2 ) 2
2 ≤ b(cos θ )μ∗ (g − g)2 + b(cos θ )g 2 (μ∗ )1/2 − (μ∗ )1/2 ≤ |||g|||2 . By (2.1.7), we have (Lg, g) L 2 (R3v ) ≤ 2|||g|||2 . The proof of Lemma 2.6 is then completed.
The next result shows that the non-isotropic norm controls the Sobolev norm of both derivative and weight of order s. Lemma 2.7. There exists C > 0 such that
|||g|||2 ≥ C ||g||2H s + ||g||2L 2 .
(2.2.8)
s
Proof. Write |||g|||2 =
R6 S2
+
2 b(cos θ )μ∗ g(v) − g(v ) dσ dv∗ dv
R6 S2
2 b(cos θ )g∗2 μ1/2 (v) − μ1/2 (v ) dσ dv∗ dv ≡ A + B.
According to the calculation of Propositions 1 and 2 in [6], we have ξ −3 A = (2π ) b ˆ g(ξ ˆ + )|2 ·σ μ(0)| ˆ g(ξ ˆ )|2 + μ(0)| |ξ | R3 S2 ¯ˆ ) dσ dξ −2Re μ(ξ ˆ − )g(ξ ˆ + )g(ξ 1 ξ 2 − ≥ · σ ( μ(0) ˆ − | μ(ξ ˆ | g(ξ ˆ )| b )|)dσ dξ 2(2π )3 R3 |ξ | S2 ≥ C1 |ξ |2s |g(ξ ˆ )|2 dξ ≥ C1 2−2s (1 + |ξ |2 )s |g(ξ ˆ )|2 dξ ≥ C1 2
|ξ |≥1 −2s
g2H s (R3 ) v
|ξ |≥1
− C1 g2L 2 (R3 ) , v
where we have used Lemma 3 in [6] that ξ · σ (μ(0) ˆ − |μ(ξ ˆ − )|)dσ ≥ C1 |ξ |2s , b |ξ | S2
∀|ξ | ≥ 1.
(2.2.9)
The Boltzmann Equation Without Angular Cutoff
Similarly,
527
ξ 2 (0)|μ 1/2 (ξ )|2 + g 1/2 (ξ + )|2 g2 (0)|μ b B = (2π ) ·σ |ξ | R3 S2 1/2 (ξ + ) μ 1/2 (ξ ) dσ dξ −2Re g2 (ξ − )μ 2 1 ξ 2 (0) μ 1/2 (ξ + ) − μ 1/2 (ξ ) dσ dξ = b · σ g 2(2π )3 R3 S2 |ξ |
ξ 1 1/2 (ξ ) μ 1/2 (ξ + )dσ dξ · σ g2 (0) − Re g2 (ξ − ) μ b + 3 (2π ) R3 S2 |ξ | = B1 + B2 . −3
For B1 , one has B1 =
2 ξ 1/2 + 1/2 (ξ ) dσ dξ · σ g2 (0) μ (ξ ) − μ |ξ | R3 S2 2 ξ 1/2 − · σ μ μ(2ξ ) b (ξ ) − 1 dσ dξ = C1 g2L 2 (R3 ) v |ξ | R3ξ S2
b
≥ C2 g2L 2 (R3 ) , v
where
R3ξ
C2 = C1
μ(2ξ )
S2
b
2 ξ 1/2 − · σ μ (ξ ) − 1 dσ dξ > 0. |ξ |
For the second term on the right hand side, by using 1/2 (ξ ) μ 1/2 (ξ + ) ≥ C μ μ(2ξ ), for some positive constant C, we have
ξ 1/2 (ξ ) μ 1/2 (ξ + )dσ dξ ·σ g2 (0) − Re g2 (ξ − ) μ b B2 = |ξ | R3 S2
ξ ·σ g2 (0) − Re g2 (ξ − ) b μ(2ξ )dσ dξ. ≥C |ξ | R3 S2 ξ ·σ b g 2 (v) 1 − cos(ξ − · v) dv μ(2ξ )dσ dξ. =C 3 2 3 |ξ | R S Rv We now use Bobylev’s technique [18] to have ξ v · σ ψ(ξ − · v)dσ = · σ ψ(ξ · v − )dσ, b b |ξ | |v| S2 S2 so that
v · σ 1 − cos(ξ · v − ) μ(2ξ )dσ dξ dv |v| R3v R3 S2 − v v 2 =C ·σ μ(0) − μ dσ dv g (v) b 3 2 |v| 2 R S v ≥C g 2 (v)|v|2s dv ≥ C2−2s g2L 2 (R3 ) − Cg2L 2 (R3 ) .
B2 ≥ C
g 2 (v)
|v|≥1
b
s
v
v
528
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
where we have used (2.2.9) and the change of variables in σ by exchanging ξ/|ξ | and v/|v|. Finally, by choosing a suitably small constant 0 < λ < 1, |||g|||2 = A + B1 + B2 ≥ λA + B1 + λB2 ≥ C(g2H s (R3 ) + g2L 2 (R3 ) ), v
and this concludes the proof of the lemma.
v
2.3. Upper bound estimates. To apply the energy method, we need some upper bound estimate on the collision operator in terms of the non-isotropic norm which will be given in the following proposition. For this, we first prove Lemma 2.8. There exists C > 0 such that b(cos θ ) f ∗2 (g − g)2 ≤ C || f ||2L 2 |||g|||2 . s
(2.3.1)
Proof. Different from Lemma 2.4, we apply the Bobylev formula [18] to have b(cos θ )μ∗ (g − g)2 dv∗ dσ dv 1 ξ · σ μ(0)(| ˆ g(ξ ˆ )|2 + |g(ξ = ˆ + )|2 ) b (2π )3 |ξ |
−2Re μ(ξ ˆ − )g(ξ ˆ + )g(ξ ˆ ) dξ dσ 1 ξ = ·σ μ(0)| ˆ g(ξ ˆ ) − g(ξ ˆ + )|2 b (2π )3 |ξ |
+2Re μ(0) ˆ − μ(ξ ˆ − ) g(ξ ˆ + )g(ξ ˆ ) dξ dσ, and
b(cos θ ) f ∗2 (g − g)2 dv∗ dσ dv 1 ξ · σ f 2 (0)|g(ξ = ˆ ) − g(ξ ˆ + )|2 b (2π )3 |ξ |
+2Re f 2 (0) − f 2 (ξ − ) g(ξ ˆ + )g(ξ ˆ ) dξ dσ.
Since μ(0) ˆ = 1, f 2 (0) = f 2L 2 , we obtain b(cos θ ) f ∗2 (g − g)2 dv∗ dσ dv = f 2L 2 b(cos θ )μ∗ (g − g)2 dv∗ dσ dv
2 ξ 2 − + · σ Re μ(0) ˆ − μ(ξ ˆ − f ) g(ξ ˆ ) g(ξ ˆ ) dξ dσ b L2 (2π )3 |ξ |
ξ 2 + 2 (0) − 2 (ξ − ) g(ξ · σ Re f f ˆ ) g(ξ ˆ ) dξ dσ. b + (2π )3 |ξ |
The Boltzmann Equation Without Angular Cutoff
529
For the last term, we note that ξ ξ − 2 −iv·ξ − 2 2 b f (v) 1 − e · σ | f (0) − f (ξ )|dσ ≤ b ·σ dv dσ. |ξ | |ξ | v Now consider
If |v||ξ | ≥
2 π,
ξ − b · σ 1 − e−iv·ξ dσ. |ξ | S2
we choose δ =
1 |ξ ||v|
−
≤ π/2 to have |1 − e−iv.ξ | ≤ |v||ξ | sin θ for any −
0 ≤ θ ≤ δ. And if π2 ≥ θ ≥ δ, we have |1 − e−iv.ξ | ≤ 2. Hence, δ π/2 1 1 ξ − · σ 1 − e−iv·ξ dσ ≤ C|v||ξ | b sin θ dθ + C dθ 1+2s 1+2s |ξ | θ S2 0 θ δ ˜ 2s |ξ |2s . ≤ C|v||ξ |δ −2s+1 + C δ −2s ≤ C|v| On the other hand, if |v||ξ | ≤ π2 , we have directly π/2 1 ξ − · σ 1 − e−iv·ξ dσ ≤ C b |v| |ξ | sin θ dθ 1+2s 2 |ξ | θ S 0 ˜ ˜ 2s |ξ |2s . ≤ C|v||ξ | ≤ C|v| Thus, we have ξ 2 2 ˆ (ξ )dξ dσ ≤ C f 2L 2 g2H s . f (0) − f 2 (ξ − ) |g| b · σ s |ξ | By using the regular change of variables ξ → ξ + , and by noticing that ξ − = φ(ξ + , σ ) = ξ + −
|ξ + |
σ,
θ |ξ − | = |ξ + | tan , 2
cos θ2 ∂(ξ + ) 1 2 ∂(ξ ) = 4 cos θ/2,
cos
θ ξ+ = + · σ, 2 |ξ |
we have ξ 2 2 + f (0) − ˆ (ξ )dξ dσ b · σ f 2 (ξ − ) |g| |ξ | ξ+ 1 2 2 2 (φ(ξ + , σ )) |g| f ˆ 2 (ξ + )dξ + dσ b 2( · σ ) − 1 (0) − f = cos2 θ/2 |ξ + | ≤ C f 2L 2 g2H s . s
Hence,
ξ − + 2 2 b · σ Re f (0) − f (ξ ) g(ξ ˆ )g(ξ ˆ )dξ dσ ≤ C f 2L 2 g2H s . s |ξ | Similarly, we have ξ − + ≤ C√μ 2 2 g2 s . b · σ Re μ(0) ˆ − μ(ξ ˆ ) g(ξ ˆ ) g(ξ ˆ )dξ dσ H Ls |ξ | Therefore, we have proved (2.3.1) by using (2.2.8).
530
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
In view of a future application of the energy method, the scalar product of the collision operator with a test function is given by Proposition 2.9. There exists C > 0 such that
(( f, g), h) L 2 (R3 ) ≤ C || f || L 2 |||g||| + ||g|| L 2 ||| f ||| |||h|||. s s Proof. Note that
(( f, g), h) L 2 (R3 ) = μ−1/2 Q(μ1/2 f, μ1/2 g), h 2 3 L (R ) 1/2 f ∗ g − f ∗ g h = b(cos θ )μ∗
1/2 1 1/2 = b(cos θ ) f ∗ g − f ∗ g μ∗ h − μ∗ h 2 2 1/2 1 ≤ b(cos θ ) f ∗ g − f ∗ g 2
2 1/2 × b(cos θ ) (μ∗ )1/2 h − (μ∗ )1/2 h ≤
1 1/2 A × B 1/2 . 2
For B, we have
2 B≤2 b(cos θ )μ∗ (h − h)2 + 2 b(cos θ )h 2∗ (μ )1/2 − μ1/2 = 2|||h|||2 , where we have used the change of variables (v, v∗ ) → (v , v∗ ) for the first term and (v, v∗ ) → (v∗ , v) for the second term. Similarly, b(cos θ )g∗ 2 ( f − f )2 . A≤2 b(cos θ ) f ∗ 2 (g − g)2 + 2 Then (2.3.1) implies that
A ≤ C || f ||2L 2 |||g|||2 + ||g||2L 2 ||| f |||2 , s
s
which completes the proof of the proposition.
3. Commutator Estimates 3.1. Non-isotropic norm in R6x,v . We now define the norm associated with the collision operator on the space of (x, v). For m ∈ N, ∈ R, set ⎧ ⎨
⎫ ⎬ α g(x, · )|||2 d x < +∞ , B m (R6x,v ) = g ∈ S (R6x,v ); |||g|||2Bm (R6 ) = |||W ∂x,v ⎩ ⎭ 3 |α|≤m Rx
where ||| · ||| is the non-isotropic norm defined in (2.2.1).
The Boltzmann Equation Without Angular Cutoff
531
First of all, one has Lemma 3.1. For any ≥ 0, γ , β ∈ N3 , γ
γ
γ
|||W ∂x ∂vβ Pg|||B0 (R6 ) + |||P(W ∂x ∂vβ g)|||B0 (R6 ) ≤ C ,β ||∂x g||2L 2 (R6 ) , (3.1.1) 0 0 C0 |||g|||2B0 (R6 ) − C2 ||g||2L 2 (R6 ) ≤ (Lg, g) L 2 (R6x,v ) ≤ C3 |||g|||2B0 (R6 ) ,
(3.1.2)
||g||2L 2 (R6 ) + ||g||2L 2 (R3 ;H s (R3 )) ≤ C|||g|||2B0 (R6 ) ≤ C||g||2L 2 (R3 ;H s (R3 )) . x v x v l l+s l+s l
(3.1.3)
0
0
and
Proof. By definition of the orthogonal projection operator P, we have Pg = ag (t, x)μ1/2 +
3
bg, j (t, x) v j μ1/2 + cg (t, x)|v|2 μ1/2 ,
j=1
with
ag (t, x) =
R3v
g(t, x, v)μ1/2 (v)dv, cg (t, x) =
and
1 g(t, x, v) √ (v 2 − 3)μ1/2 (v)dv, 6 v
bg, j (t, x) =
v
g(t, x, v) v j μ1/2 (v)dv,
j = 1, 2, 3.
Thus (3.1.1) can be obtained by integration by parts. To get (3.1.2), we use (2.2.2) and (2.2.5) to obtain |||g|||2B0 (R6 ) ≥ C (Lg, g) L 2 (R6x,v ) ≥ C0 |||(I − P)g|||2B0 (R6 ) 0
0
C0 |||g|||2B0 (R6 ) − C0 |||Pg|||2B0 (R6 ) ≥ 0 0 2 C0 2 2 |||g|||B0 (R6 ) − C2 ||g|| L 2 (R6 ) . ≥ 0 2 Finally, (3.1.3) follows directly from (2.2.2) and (2.2.8). 3.2. Weighted estimates on commutators. We will use the following notation, for γ ∈ N3 , T (F, G, μγ ) = Q(μγ F, G) + b(cos θ ) (μγ )∗ − (μγ )∗ F∗ G dv∗ dσ ,
(3.2.1)
√ √ where μγ = pγ (v) μ(v) = ∂ γ ( μ ) is a Maxwellian type function of variable v. In this notation, (2.1.5) is equivalent to √ ( f, g) = T ( f, g, μ ). And the Leibniz formula gives ∂xα ∂vβ ( f, g) =
α1 +α2 =α, β1 +β2 +β3 =β
,α2 Cβα11,β T (∂xα1 ∂vβ1 f, ∂xα2 ∂vβ2 g, μβ3 ). (3.2.2) 2 ,β3
First of all, let us recall the following lemma from [12].
532
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Lemma 3.2. Let ≥ 0, 0 < s < 1/2. There exists C > 0 such that
≤ C f 1 3 g 2 3 h L 2 (R3 ) . W Q( f, g) − Q( f, W g) , h L (R ) L (R ) 2 3 L (R )
v
Using this result, we shall show that Proposition 3.3. For any ≥ 0,
W T (F, G, μγ ) − T (F, W G, μγ ), h ≤ C||F|| L 2 ||G|| L 2 ||h|| L 2s .
L 2 (R3v )
(3.2.3)
Proof. From (3.2.1), it follows that
W T (F, G, μγ ) − T (F, W G, μγ ), h 2 3 L (R v )
= W Q(μγ F, G) − Q(μγ F, W G), h 2 3 L (R v )
+ b(cos θ )(μγ ∗ − μγ ∗ )F∗ G W − W h = B1 + B2 . Lemma 3.2 implies that B1 ≤ C||μγ F|| L 1 ||G|| L 2 ||h|| L 2 ≤ C||F|| L 2 ||G|| L 2 ||h|| L 2 .
For B2 , since we have assumed that 0 < s < 1/2, we get 1/2 2 2 2 |W − W | b(cos θ )|F∗ | |G | B2 ≤ sin θ 1/2
2 2 × . b(cos θ ) sin θ μγ ∗ − μγ ∗ |h| Equation (2.2.4) implies that
2 b(cos θ ) μγ ∗ − μγ ∗ |h|2 ≤ C||μγ ||2Hss ||h||2L 2 , s
while, using
|W − W |2 ≤ sin2 θ (W∗ )2 + (W )2 ≤ sin2 θ (W∗ )2 (W )2 , we get
b(cos θ )|F∗ |2 |G |2
|W − W |2 2 ≤ b(cos θ ) sin θ (W F)2 ∗ (W G) sin θ ≤ C||F||2L 2 ||G||2L 2 ,
which leads to completion of the proof of the proposition.
The Boltzmann Equation Without Angular Cutoff
533
Similarly, we have also Proposition 3.4. There exists a constant C > 0 such that
T (F, G, μγ ), h L 2 (R3 ) ≤ C ||F|| L 2s |||G||| + ||G|| L 2s |||F||| |||h|||. v
(3.2.4)
Proof. By the Cauchy-Schwarz inequality, we have
b(cos θ )(μγ ∗ )1/2 F∗ G − F∗ G h T (F, G, μγ ), h L 2 (R3 ) = v
1 = b(cos θ ) F∗ G − F∗ G (μγ ∗ )1/2 h − (μγ ∗ )1/2 h 2 2 1/2 1 ≤ (cos θ ) F∗ G − F∗ G 2
2 1/2 1/2 1/2 × b(cos θ ) (μγ ∗ ) h − (μγ ∗ ) h
≤
1 ˜ 1/2 A × B˜ 1/2 . 2
By using the estimation of the term A in the proof of Proposition 2.9, it follows that
A˜ ≤ C ||F||2L 2 |||G|||2 + ||G||2L 2 |||F|||2 s
s
and
B˜ ≤ C ||μγ ||2L 2 |||h|||2 + ||h||2L 2 |||μγ |||2 ≤ C|||h|||2 . s
s
We are now ready to prove the following estimate with differentiation and weight. Proposition 3.5. For any ≥ 3, and N ≥ 3, we have, for all β ∈ N6 , |β| ≤ N ,
W ∂ β ( f, g ), h x,v L 2 (R 6
x,v
≤ C|| f || N 6 |||g||| N 6 |||h||| 0 6 . B0 (R ) H (R ) B (R ) )
Remark 3.6. In fact, this proposition holds even when > 23 + 2s, and N > 23 + 2s. Here, we consider the case when ≥ 3, N ≥ 3 with 0 < s < 1/2 for the simplicity of the notations. Proof. Using the Leibniz formula (3.2.2) gives
β β W ∂x,v ( f, g), h 2 6 = Cβ21,β3 T (∂ β1 f, W ∂ β2 g, μβ3 ), h 2 6 L (Rx,v ) L (Rx,v )
β + Cβ21,β3 W T (∂ β1 f, ∂ β2 g, μβ3 ) − T (∂ β1 f, W ∂ β2 g, μβ3 ), h 2 6 . L (Rx,v )
534
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Then from (3.2.3), we get
W T (∂ β1 f, ∂ β2 g, μβ ) − T (∂ β1 f, W ∂ β2 g, μβ ), h 3 3 2 6 L (Rx,v ) 1/2 ≤C ! ≤C
R3x
∂ β1 f 2L 2 (R3 ) ∂ β2 g2L 2 (R3 ) d x
v
h L 2s (R6x,v )
v
∂ β1 f L ∞ (R3 ; L 2 (R3 )) ∂ β2 g L 2 (R6 ) h L 2s (R6x,v ) , x v x,v ∂ β1 f L 2 (R6 ) ∂ β2 g L ∞ (R3 ;L 2 (R3 )) h L 2s (R6x,v ) ,
x,v
x
if |β1 | ≤ 1 ; if |β1 | ≥ 2.
v
Since |β1 | ≤ 1 implies |β1 | + 3/2 < 3 ≤ N and |β1 | ≥ 2 implies |β2 | + 3/2 < |β|, it follows that
W T (∂ β1 f, ∂ β2 g, μβ ) − T (∂ β1 f, W ∂ β2 g, μβ ), h 3 3 2 6 L (R ) x,v
≤ C f H N (R6 ) g H |β | (R6 ) |||h|||B0 (R6 ) .
(3.2.5)
x,v
0
On the other hand, if |β1 | ≤ 1 so that |β1 | + 23 + s < 3 ≤ N , we get from (3.2.4)
T (∂ β1 f, W ∂ β2 g, μβ ), h 3 2 6 L (Rx,v )
≤C ∂ β1 f 2L 2 (R3 ) |||W ∂ β2 g|||2 + ||W ∂ β2 g||2H s (R3 ) d x R3x
s
+
β2
R3x
W ∂
v
g2L 2 (R3 ) s v
v
|||∂ β1 f |||2 + ||∂ β1 f ||2H s (R3 ) d x
1/2
|||h|||B0 (R6 )
v
0
≤ C ∂ β1 f L ∞ (R3x ; L 2s (R3v )) + ||∂ β1 f ||2L ∞ (R3 ; H s (R3 )) |||g|||B|β2 | (R6 ) |||h|||B0 (R6 ) x
s
v
0
s
0
≤ C f H |β1 |+3/2+s+ (R6 ) |||g|||B|β2 | (R6 ) |||h|||B0 (R6 ) , Hence, for |β1 | ≤ 1, we have
T (∂ β1 f, W ∂ β2 g, μβ ), h 3 L 2 (R 6
x,v
≤ C f H 3 (R6 ) |||g||| |β2 | |||h|||B0 (R6 ) . s B (R6 ) 0 )
We now consider the case when |β1 | ≥ 2. First of all, assume 2 ≤ |β1 | ≤ |β| − 1 so that |β2 | = |β| − |β1 | − |β3 | ≤ |β| − 2. Then, we get
T (∂ β1 f, W ∂ β2 g, μβ ), h 3 2 6 L (Rx,v ) ≤ C f H |β1 | (R6 ) ||W ∂ β2 g|| L ∞ (R3x ; Hss (R3v )) s
β2 +W ∂ g L ∞ (R3x ; L 2s (R3v )) ||∂ β1 f || L 2 (R3x ; Hss (R3v )) |||h|||B0 (R6 ) 0
≤ C f H |β1 |+s (R6 ) ||W
3/2+ s β2 x v ∂ g||2L 2 (R6 ) |||h|||B0 (R6 ) 0
s
s
≤ C f H |β|−1+s (R6 ) |||g|||B|β|−2+3/2+s+ (R6 ) |||h|||B0 (R6 )
s
0
≤ C f H |β| (R6 ) |||g|||B|β| (R6 ) |||h|||B0 (R6 ) . s
0
The Boltzmann Equation Without Angular Cutoff
535
We turn next to the case when β1 = β, for which we have
√ T (∂ β f, W g, μ ), h 2 6 = (∂ β f, W g ), h L (Rx,v )
L 2 (R6x,v )
.
Since we want to avoid using the non-isotropic norm of f on the right hand side, we cannot not use the estimate (2.2.3) to complete the proof. So we proceed in a different way: Use firstly (2.1.5) to get
√
(∂ β f, W g), h 2 6 = Q( μ ∂ β f, W g), h 2 6 L (Rx,v ) L (Rx,v ) √ β + b(cos θ ) μ∗ − μ∗ (∂ f )∗ (W g) hdv∗ dσ dvd x. On one hand, (2.1.2) with m = 0, α = −s, implies that
Q(√μ ∂ β f, W g), h 2 6 L (Rx,v )
√ ≤ Ch L 2s (R6 ) μ ∂ β f L 2 (R3 ; L 1 (R3 )) W g L ∞ (R3x ; Hs2s (R3v )) x v 2s
≤ C f H |β| (R6 ) ||W g||| H 3/2+2s+ (R6 ) |||h|||B0 (R6 ) . s
0
On the other hand, we can write √ b(cos θ ) μ∗ − μ∗ (∂ β f )∗ (W g) hdv∗ dσ dvd x √ = b(cos θ ) μ∗ − μ∗ (∂ β f )∗ (W g) h − h dv∗ dσ dvd x √ + b(cos θ ) μ∗ − μ∗ (∂ β f )∗ (W g) h dv∗ dσ dvd x = D1 + D2 . By the Cauchy-Schwarz inequality, one has 1/2
2 |D1 | ≤ b(cos θ )|(∂ β f )∗ |2 |(W g) |2 (μ∗ )1/4 − (μ∗ )1/4 dv∗ dσ dvd x ×
1/2
2 1/4 1/4 2 (h − h ) dv∗ dσ dvd x . b(cos θ ) μ∗ + (μ∗ )
Lemma 2.5 yields
2 b(cos θ )|(∂ β f )∗ |2 |(W g) |2 (μ∗ )1/4 − (μ∗ )1/4 dv∗ dσ dvd x ≤C |(∂ β f )∗ |2 |(W g)|2 v2s v∗ 2s dv∗ ddvd x R3x
≤C ≤
R3x
R6v,v∗
||∂ β f ||2L 2 (R3 ) ||W g||2L 2 (R3 ) d x ≤ C||∂ β f ||2L 2 (R6 ) ||W g||2L ∞ (R3 ; L 2 (R3 )) s
v
s
v
3/2+ C|| f ||2H N (R6 ) ||W x g||2L 2 (R6 ) s
s
s
≤ C|| f ||2
|β|
Hs (R6 )
x
|||g|||2B2 (R6 ) ,
s
v
536
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
while from Lemma 2.8, we get
2 1/4 b(cos θ ) μ∗ + (μ∗ )1/4 (h − h )2 dv∗ dσ dvd x 1/2 ≤4 b(cos θ )μ∗ (h − h )2 dv∗ dσ dvd x ≤ C|||h|||2B0 (R6 ) . 0
Therefore, we obtain |D1 | ≤ C|| f || H |β| (R6 ) |||g|||B2 (R6 ) |||h|||B0 (R6 ) .
s
0
For the term D2 , we have √ β b(cos θ ) (∂ μ − μ f ) (W g) h dv dσ dvd x ∗ ∗ ∗ ∗
√ b(cos θ ) = μ∗ − μ∗ (∂ β f )∗ (W g)hdv∗ dσ dvd x β (∂ f )∗ W g |h|v2s v∗ 2s dv∗ dvd x ≤C R3x
≤C
R3x
R6v,v∗
∂ β f L 1 (R3 ) W g L 2s (R3v ) h L 2s (R3v ) d x v 2s
≤ C∂ β f L 2 (R3 ; L 2 x
3 3/2+2s+ (Rv ))
W g L ∞ (R3x ; L 2s (R3v )) h L 2s (R6 ) ,
so that |D2 | ≤ C|| f || H |β|
3/2+2s+ (R
6)
|||g|||B2 (R6 ) |||h|||B0 (R6 ) .
Therefore, it follows that
(∂ β f, W g), h L 2 (R 6
x,v
≤ C|| f || H |β|
3/2+2s+
(R 6 )
0
)
|||g|||B2 (R6 ) |||h|||B0 (R6 ) .
(3.2.6)
0
Finally, for the case |β1 | ≥ 2, since 3/2 + 2s < 3 ≤ N , we have also
≤ C f N 6 |||g||| N 6 |||h||| 0 6 . T (∂ β1 f, W ∂ β2 g, μβ ), h 3 B0 (R ) H (R ) B (R ) L 2 (R 6 ) x,v
The proof of the proposition is then completed.
By using the argument in the proof of the above proposition, the next proposition follows from the Sobolev imbedding theorems. Proposition 3.7. For any ≥ 3, we have, for all β ∈ N6 , |β| ≤ 2,
W ∂ β ( f, g ), h ≤ C|| f || 3 6 |||g||| 3 6 |||h||| 0 6 . (3.2.7) x,v H (R ) B (R ) B (R ) 2 6 L (Rx,v )
Finally, the linear operators can be also estimated as follows.
0
The Boltzmann Equation Without Angular Cutoff
537
Proposition 3.8. For ≥ 3, we have for any β ∈ N6 ,
≤ C|β|, || f || |β| 6 |||h||| 0 6 . W ∂ β L2 ( f ), h x,v B0 (R ) H (R ) 2 6 L (R ) If |β| ≥ 1, we have
L1 (W ∂ β g) − W ∂ β L1 (g), h x,v x,v 2 6 L (R )
≤ C|β|, ||g|| H |β| (R6 ) + |||g|||B|β |−1 (R6 ) |||h|||B0 (R6 ) , 0 and for |β| = 0,
L1 (W g) − W L1 (g), h
≤ C||g|| 2 6 |||h||| 0 6 . L (R ) B0 (R ) 2 6 L (R )
(3.2.8)
(3.2.9)
(3.2.10)
Remark 3.9. On the right hand side of (3.2.7), the term |||g|||B3 (R6 ) comes from the Sobolev imbedding L ∞ (R3x ; Hs2s (R3v )) ⊃ Hs
3/2+2s+
(R6 ) ⊃ B03 (R6 ),
where is any small positive number. Thus the order of differentiation is equal to 3. Note that this is due to the nonlinearity in the operator (·, ·). For the linear operators, the estimates given in (3.2.8–3.2.10) do not involve this term. Proof. For the proof of (3.2.8), by using the Leibniz formula (3.2.2), we have
β − W ∂x,v L2 ( f ), h
L 2 (R6x,v )
√ β = W ∂x,v ( f, μ ), h
β √ = Cβ 1,β T (∂ β1 f, W ∂ β2 μ , μβ3 ), h 2
3
L 2 (R6x,v )
L 2 (R6x,v )
β √ √ + Cβ 1,β W T (∂ β1 f , ∂ β2 μ , μβ3 ) − T (∂ β1 f , W ∂ β2 μ , μβ3 ), h 2
3
L 2 (R6x,v )
= E1 + E2 .
Then (3.2.5) implies
β1 √ √ |E 2 | = Cβ ,β W T (∂ β1 f , ∂ β2 μ , μβ3 ) − T (∂ β1 f , W ∂ β2 μ , μβ3 ), h 2 6 2 3 L (Rx,v ) √ ≤ C f |β| 6 μ |β| 3 |||h|||B0 (R6 ) ≤ C f |β| 6 |||h|||B0 (R6 ) , H (R )
H (R )
0
H (R )
and (3.2.4) implies also,
β1 √ Cβ2 ,β3 T (∂ β1 f, W ∂ β2 μ , μβ3 ), h |E 1 | = ≤ C f H |β| (R6 ) |||h|||B0 (R6 ) ,
0
where for the case when β1 = β, we have used (3.2.6).
0
L 2 (R6x,v )
538
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
√ For (3.2.9), since −L1 (g) = ( μ, g), by using again the Leibniz formula (3.2.2), we have
β β − W ∂x,v L1 (g) − L1 (W ∂x,v g), h 2 6 L (R )
√ √ β β = W ∂x,v ( μ , g) − ( μ , W ∂x,v , g), h 2 6 L (R )
√ β1 β β 1 2 = Cβ ,β T (∂ μ , W ∂ g, μβ3 ), h 2 2
|β2 |≤|β|−1
+
β Cβ 1,β 2
L (R6x,v )
3
3
√ √ W T (∂ β1 μ , ∂ β2 g, μβ3 ) − T (∂ β1 μ , W ∂ β2 g, μβ3 ), h
L 2 (R6x,v )
= F1 + F2 .
Then (3.2.5) implies
√ √ β Cβ 1,β W T (∂ β1 μ , ∂ β2 g, μβ3 ) − T (∂ β1 μ , W ∂ β2 g, μβ3 ), h 2 6 |F2 | = 2 3 L (Rx,v ) √ ≤ C μ |β| 3 g |β | 6 |||h|||B0 (R6 ) ≤ Cg |β | 6 |||h|||B0 (R6 ) , H (R ) H (R ) H (R ) x,v x,v 0 0
which also gives (3.2.10). On the other hand, for F1 , (3.2.4) implies that, when |β2 | ≤ |β| − 1,
T (∂ β1 √μ , W ∂ β2 g, μβ ), h 3 2 6 L (Rx,v )
√ ≤C ∂ β1 μ 2L 2 (R3 ) |||W ∂ β2 g|||2 + ||W ∂ β2 g||2H s (R3 ) d x R3x
+
R3x
s
β2
W ∂
g2L 2 (R3 ) s v
v
v
|||∂
β1 √
μ ||| + ||∂ 2
β1 √
√ ≤ C μ H |β1 |+s (R3 ) |||g|||B|β2 | (R6 ) |||h|||B0 (R6 ) v
s
μ ||2H s (R3 ) v
1/2
dx
|||h|||B0 (R6 ) 0
0
≤ C|||g|||B|β|−1 (R6 ) |||h|||B0 (R6 ) .
0
Then the proof of the proposition is completed.
4. Local Existence 4.1. Energy estimates for a linear equation. We now consider the following Cauchy problem for a linear Boltzmann equation with a given function f , ∂t g + v · ∇x g + L1 g = ( f, g) − L2 f,
g|t=0 = g0 ,
which is equivalent to the problem: ∂t G + v · ∇x G = Q(F, G), with F = μ +
√
μ f and G = μ +
√
μ g.
G|t=0 = G 0 ,
(4.1.1)
The Boltzmann Equation Without Angular Cutoff
539
We shall now study the energy estimates on (4.1.1) in the function space H N . For N ≥ 3, ≥ 3 and β ∈ N6 , |β| ≤ N , taking
β β W 2 ∂x,v g (t, x, v), ϕ(t, x, v) = (−1)|β| ∂x,v as a test function on R3x × R3v , we get
" β # 1 d β W ∂ β g2L 2 (R6 ) + W ∂x,v , v · ∇x g, W ∂x,v g 2 6 L (R ) 2 dt
β β + W ∂x,v L1 (g), W ∂x,v g 2 6 L (R )
β β β β = W ∂x,v ( f, g), W ∂x,v g 2 6 − W ∂x,v L2 ( f ), W ∂x,v g L (R )
where we have used the fact that
β β v · ∇x W ∂x,v g , W ∂x,v g
L 2 (R 6 )
L 2 (R 6 )
,
= 0.
Applying now Propositions 3.5 and 3.8, we get for any 3 ≤ k ≤ N and |β| ≤ k,
1 d β 2 β β g , W ∂x,v g 2 6 ∂ g L 2 (R6 ) + L1 W ∂x,v L (R ) 2 dt 2 2 ≤ C f H k (R6 ) |||g|||Bk (R6 ) + g H k (R6 ) + f H k (R6 ) |||g|||Bk (R6 )
+ g H k (R6 ) + |||g|||Bk−1 (R6 ) |||g|||Bk (R6 ) .
By taking summation over |β| ≤ k, Lemma 2.3 together with (3.1.2) and the CauchySchwarz inequality imply that d C0 g2H k (R6 ) + |||g|||2Bk (R6 ) ≤ Ck, f H k (R6 ) |||g|||2Bk (R6 ) dt 2 + Ck, g2H k (R6 ) + f 2H k (R6 ) + |||g|||2 k−1
B
(R 6 )
,
3 ≤ k ≤ N . (4.1.2)
For k = 1, 2, Proposition 3.7 is used to get d C0 g2H k (R6 ) + |||g|||2Bk (R6 ) ≤ Ck, f H 3 (R6 ) |||g|||2B3 (R6 ) dt 2 + Ck, g2H k (R6 ) + f 2H 3 (R6 ) + |||g|||2 k−1
B
(R 6 )
,
(4.1.3)
while for k = 0, d C0 g2L 2 (R6 ) + |||g|||2B0 (R6 ) ≤ C0, f H 3 (R6 ) |||g|||2B0 (R6 ) dt 2
2 2 + C0, g L 2 (R6 ) + f H 3 (R6 ) ,
where C0 is the constant in (3.1.2), which is independent on k, and N .
(4.1.4)
540
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Take N ≥ 3, when k ≥ 2, by taking a linear combination of (4.1.2) and (4.1.3), we have C2 d C0 2 2 g k−1 6 + g H k (R6 ) + 2 0 |||g|||2Bk (R6 ) H (R ) dt 2Ck, 2 Ck,
C0 f H N (R6 ) |||g|||2B N (R6 ) + g2H N (R6 ) + f 2H N (R6 ) ≤ 2 C d 0 |||g|||2 k−1 6 + g2 k−1 6 + H (R ) B (R ) dt 2
C0 f H N (R6 ) |||g|||2B N (R6 ) + g2H N (R6 ) + f 2H N (R6 ) ≤ 2 +Ck−1, f H N (R6 ) |||g|||2B N (R6 ) + g2H N (R6 ) + f 2H N (R6 ) + |||g|||2 k−2
B
By induction and by using (4.1.4), we have the following estimate ⎞ ⎛ d ⎝ (0 |||g|||2 N 6 ck,l g2H k (R6 ) ⎠ + C B (R ) dt 0≤k≤N
)N , f N 6 |||g|||2 N 6 + g2 N 6 + f 2 N 6 , ≤C H (R ) H (R ) H (R ) B (R )
(0 < C0 , ck, and C )N , . Notice that for some positive constants C g2H N (R6 ) ∼ ck,l g2H k (R6 ) .
0≤k≤N
(R 6 )
.
(4.1.5)
(4.1.6)
We are now ready to prove the following theorem. Theorem 4.1. Let N ≥ 3, ≥ 3. Assume that g0 ∈ H N (R6 ) and f ∈
L ∞ ([0, T ]; H N (R6 )). If g ∈ L ∞ ([0, T ]; H N (R6 )) L 2 ([0, T ]; B N (R6 )) is a solution of Cauchy problem (4.1.1), then there exists 0 > 0 such that if f L ∞ ([0,T ]; H N (R6 )) ≤ 0 ,
we have g2L ∞ ([0,T ]; H N (R6 )) + ||g||2L 2 ([0,T ]; B N (R6 )) ≤ CeC T (g0 2H N (R6 ) + 02 T ), (4.1.7)
for a constant C > 0 depending only on N , . Proof. Choosing 0 = ⎛
(0 C )N , , 2C
we have, from (4.1.5), ⎞
(0 C |||g|||2B N (R6 ) 2 0≤k≤N )N , (g2 N 6 + 02 ) ≤ C( ≤ 2C ck,l g2H k (R6 ) + 02 ), H (R )
d ⎝ dt
ck,l g2H k (R6 ) ⎠ +
0≤k≤N
The Boltzmann Equation Without Angular Cutoff
and
541
⎞ ⎛ (0 −C t C d ⎝ −Ct 2 ck,l g H k (R6 ) ⎠ + |||g|||2B N (R6 ) ≤ C02 e−Ct . e e dt 2 0≤k≤N
Thus we get (4.1.7) for some constant C > 0 and this completes the proof of the theorem. 4.2. Existence for the linear equation. With the energy estimate given in the above subsection, we can now prove the following local existence theorem by using the Hahn-Banach theorem. Theorem 4.2. Let ≥ 3, N ≥ 3 and g0 ∈ H N (R6 ). There exists 0 > 0 such that if f L ∞ ([0,T ]; H N (R6 )) ≤ 0 ,
then the Cauchy problem (4.1.1) admits a unique solution g ∈ L ∞ ([0, T ]; H N (R6 )) ∩ L 2 ([0, T ]; B N (R6 )). Proof. We consider the following Cauchy problem : P g ≡ ∂t g + v · ∇x g + L1 g − ( f, g) = H, g(0) = g0 .
(4.2.1)
For h ∈ C ∞ ([0, T ]; S(R6x,v )) with h(T ) = 0, we define g, P N∗ , h L 2 ([0, T ];H N (R6 )) = (P g, h) L 2 ([0, T ];H N (R6 )) ,
so that P N∗ , is the adjoint of the linear operator P in the Hilbert space L 2 ([0, T ]; H N (R6 )). Set W = w = P N∗ , h; h ∈ C ∞ ([0, T ]; S(R6x,v )) with h(T ) = 0 , which is a dense subspace of L 2 ([0, T ]; H N (R6 )). And we also have P N∗ , (h) = −∂t h + (v · ∇x )∗ h + L∗1 h + ∗ ( f, h). Then
h, P N∗ , h
H N (R6 )
1 d ||h(t)||2H N (R6 ) + (v · ∇x h, h) H N (R6 ) 2 dt + (L1 (h), h) H N (R6 ) − (( f, h), h) H N (R6 ) .
=−
Same as Theorem 4.1, for || f || L ∞ ([0,T ]; H N (R6 )) ≤ 0 , we have
e2C(s−t) h, P N∗ , h H N (R6 ) dt t T ≥ ||h(t)||2H N (R6 ) + Ce2C(s−t) |||h(s)|||2B N (R6 ) ds. T
t
542
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Thus, for all 0 < t < T , ||h(t)||2H N (R6 ) + C||h||2L 2 ([t,T ]; B N (R6 )) ≤ C h, P N∗ , h L 2 ([t, T ]; H N (R6 ))
≤ C||P N∗ , (h)|| L 2 ([t, T ];H N (R6 )) ||h|| L 2 ([t, T ];H N (R6 )) .
Hence, we get ||h|| L 2 ([0, T ]; H N (R6 )) + ||h|| L ∞ ([0, T ]; H N (R6 ))
≤ C T ||P N∗ , (h)|| L 2 ([0, T ]; H N (R6 )) .
(4.2.2)
Since ||h|| L 2 ([t, T ];H N (R6 )) ≤ C||h|| L 2 ([t,T ]; B N (R6 )) ,
we also have ||h|| L 2 ([0,T ]; B N (R6 )) ≤ C||P N∗ , (h)|| L 2 ([0, T ]; H N (R6 )) .
(4.2.3)
Next, we define a functional G on W as follows G(w) = (H, h) L 2 ([0, T ]; H N (R6 )) + (g0 , h(0)) H N (R6 ) .
N (R6 )), (3.1.3) gives Then, if H ∈ L 2 ([0, T ]; H −s
|G(w)| ≤ H L 2 ([0, T ]; H N (R6 )) h L 2 ([0, T ]; H N (R6 )) + g0 H N (R6 ) h(0) H N (R6 ) −s +s ≤ CH L 2 ([0, T ]; H N (R6 )) ||h|| L 2 ([0,T ]; B N (R6 )) + g0 H N (R6 ) h(0) H N (R6 ) −s
≤ C||P N∗ , (h)|| L 2 ([0, T ]; H N (R6 )) ≤ C||w|| L 2 ([0, T ]; H N (R6 )) ,
where we have used (4.2.2) and (4.2.3).
Thus, G is a continuous linear functional on W; · L 2 ([0, T ]; H N (R6 )) . Now, there exists g ∈ L 2 ([0, T ]; H N (R6 )) such that for any w ∈ W,
G(w) = (g, w) L 2 ([0, T ]; H N (R6 )) ,
by the Hahn-Banach Theorem. For any h ∈ C ∞ ([0, T ]; S(R6x,v )) with h(T ) = 0, we have g, P N∗ , h L 2 ([0, T ];H N (R6 )) = (H, h) L 2 ([0, T ];H N (R6 )) + (g0 , h(0)) H N (R6 ) ,
and by the definition of the operator P N∗ , , we have also
P g, h˜
L 2 ([0, T ];L 2 (R6 ))
= H, h˜
L 2 ([0, T ];L 2 (R6 ))
˜ + g0 , h(0)
L 2 (R 6 )
where ˜ ) = 0, h˜ = N W 2 N h ∈ C ∞ ([0, T ]; S(R6 )) with h(T
,
The Boltzmann Equation Without Angular Cutoff
543
1 where = (1 − x,v ) 2 . Since N W 2 N is an isomorphism on h : h ∈ C ∞ ([0, T ]; N (R6 )), then g ∈ S(R6 )) with h(T ) = 0 , we have shown that if H ∈ L 2 ([0, T ];H −s N 2 6 L ([0, T ];H (R )) is a solution of the Cauchy problem (4.2.1). It remains to take H = −L2 ( f ) = ( f,
√
μ),
to get (H, h) L 2 ([0, T ]; H N (R6 )) ≤ C f L 2 ([0, T ]; H N (R6 )) ||h|| L 2 ([0,T ]; B N (R6 )) .
Then G is also continuous on W. And this completes the proof of Theorem 4.2.
4.3. Convergence of approximate solutions. In this subsection, we prove the local existence theorem. Theorem 4.3. Let N ≥ 3, ≥ 3. There exist 1 , T > 0 such that if g0 ∈ H N (R6 ) and g0 H N (R6 ) ≤ 1 ,
then the Cauchy problem (1.3) admits a solution g ∈ L ∞ ([0, T ]; H N (R6 )) ∩ L 2 ([0, T ]; B N (R6 )). Remark 4.4. By the equation in (1.3), we have, for 0 < s < 1/2, By using the equation (1.3), we have, for 0 < s < 1/2, N −1 ∂t g, v · ∇x g ∈ L 2 ([0, T ]; H −1 (R6 )).
Moreover, if we go back to Eq. (1.1), we have that f = μ + μ1/2 g ∈ H N ([0, T ] × × R3 )), for any ∈ N and any bounded domain ⊂ R3x , and thus the Sobolev embedding implies that f is a classical solution of Eq. (1.1) if N > 7/2 + 1. We will use this properties for the smoothing effect of Theorem 1.1. For the proof of Theorem 4.3, we consider the sequence of approximate solutions defined by the following Cauchy problem, n ∈ N, ∂t f n+1 + v · ∇x f n+1 = Q( f n , f n+1 ) ,
f n+1 |t=0 = f 0 ,
where f n = μ + μ1/2 g n and f 0 = f 0 . Note that it is also equivalent to ∂t g n+1 + v · ∇x g n+1 + L1 g n+1 − (g n , g n+1 ) = −L2 g n ,
g n+1 |t=0 = g0 . (4.3.1)
544
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Proposition 4.5. Let N ≥ 3, ≥ 3. There exist 1 , T > 0 such that if g0 ∈ H N (R6 ) and g0 H N (R6 ) ≤ 1 ,
the Cauchy problem (4.3.1) admits a sequence of solutions {g n , n ∈ N} ⊂ L ∞ ([0, T ]; H N (R6 )) ∩ L 2 ([0, T ]; B N (R6 )). Moreover, for all n ∈ N, g n L ∞ ([0,T ]; H N (R6 )) + g n L 2 ([0,T ]; B N (R6 )) ≤ 0 ,
(4.3.2)
where 0 is the constant in Theorem 4.1. Proof. Equation (4.3.2) will be proved by induction on n. Firstly, consider the equation ∂t g 1 + v · ∇x g 1 + L1 g 1 − (g0 , g 1 ) = −L2 g0 ,
g 1 |t=0 = g0 .
When 1 < 0 , the existence of g 1 is given by Theorem 4.2 satisfying g 1 ∈ L ∞ ([0, T ]; H N (R6 )) ∩ L 2 ([0, T ]; B N (R6 )). From Theorem 4.1, we can deduce g 1 L ∞ ([0,T ]; H N (R6 )) + g 1 L 2 ([0,T ]; B N (R6 )) ≤ CeC T g0 H N (R6 ) ,
Thus (4.3.2) holds when 1 is chosen to be small compared to 0 . For n ≥ 1, under the assumption that g n L ∞ ([0,T ]; H N (R6 )) + g n L 2 ([0,T ]; B N (R6 )) ≤ 0 ,
Theorem 4.2 yields the existence of g n+1 ∈ L ∞ ([0, T ]; H N (R6 )) ∩ L 2 ([0, T ]; B N (R6 )). From Theorem 4.1, we can deduce g n+1 2L ∞ ([0,T ]; H N (R6 )) + g n+1 2L 2 ([0,T ]; B N (R6 )) ≤ CeC T (g0 2H N (R6 ) + 02 T ).
and this gives g n+1 L ∞ ([0,T ]; H N (R6 )) + g n+1 L 2 ([0,T ]; B N (R6 )) ≤ 0 ,
when T > 0 is sufficiently small, Thus we prove (4.3.2) for all n ∈ N, and this completes the proof of the proposition.
The Boltzmann Equation Without Angular Cutoff
545
It remains to prove the convergence. Set w n = g n+1 − g n and deduce from (4.3.1) that ∂t w n + v · ∇x w n + L1 w n − (g n , w n ) = (w n−1 , g n ) − L2 w n−1 ,
w n |t=0 = 0.
Similar to the computation for (4.1.4), we obtain d C0 w n 2L 2 (R6 ) + |||w n ||2B0 (R6 ) ≤ C0, g n H 3 (R6 ) |||w n |||2B0 (R6 ) dt 2 + C0, w n−1 L 2 (R6 ) (|||g n |||B3 (R3 ) + 1)|||w n |||B0 (R6 ) .
If 0 is sufficiently small, this yields, d w n 2L 2 (R6 ) + C1 |||w n |||2B0 (R6 ) ≤ C2 w n−1 L 2 (R3 ) , dt which, in turn, gives, if T is sufficiently small, w n L ∞ ([0,T ];L 2 (R6 )) ≤ λw n−1 L ∞ ([0,T ];L 2 (R6 )) ,
{g n }
is a Cauchy sequence in for some λ ∈ (0, 1). Thus we conclude that the sequence L ∞ ([0, T ]; L 2 (R6 )). Let g be the limit function. By interpolation with the uniform estimates (4.3.2), we see that the sequence is strongly convergent in L ∞ ([0, T ]; H N −δ (R6 )) ∩ L 2 ([0, T ]; B N −δ (R6 )) for any δ > 0. Furthermore, by using Eq. (4.3.1) and Proposition 4.5, we see that {∂t g n } N −1 is uniformly bounded in L ∞ ([0, T ]; H −1 ), so that it is a compact set in the function space N −1−2δ C 1−δ (]0, T∗ [ ; Hl−1 ( × R3v ))
for any bounded domain ⊂ R3x . Now we can take the limit in Eq. (4.3.1) and thus g is a solution of Cauchy problem (1.3). Finally, by a standard weak compactness argument, we can extract a subsequence of approximate solutions such that g n → g ∈ L ∞ ([0, T ]; H N (R6 )) weakly*, g n → g ∈ L 2 ([0, T ]; B N (R6 )) weakly, which shows that g ∈ L ∞ ([0, T ]; H N (R6 )) ∩ L 2 ([0, T ]; B N (R6 )). Now the proof of Theorem 4.3 is complete. 5. Qualitative Study on the Solutions In this section, we will prove two main qualitative properties of the solutions to the problem considered in this paper, that is, the uniqueness and non-negativity.
546
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
5.1. Uniqueness. The uniqueness of solutions can be proved in a larger function space as stated in Theorem 1.2. To obtain this theorem, we now first prove two preliminary results in the following lemmas. Set ϕ(v, x) = v, x2 = 1 + |v|2 + |x|2 and Wϕ,l =
vl (1 + |v|2 )l/2 . = ϕ(v, x) 1 + |v|2 + |x|2
Lemma 5.1. For l ≥ 4, we have |Wϕ,l
W Wl + Wl−3 θ 3,∗ l−3 θ + sin − Wϕ,l | ≤ C sin Wϕ,l,∗ 2 ϕ(v∗ , x) 2
≤ C θ Wl Wϕ,3,∗ , + θ l−2 Wϕ,l,∗
where Wϕ,l,∗ =
Wl,∗
ϕ(v∗ , x)
(5.1.1)
, and also for l ≥ 1,
|Wϕ,l − Wϕ,l | ≤ C sin
Wl Wl,∗ θ Wl + Wl,∗ ≤ Cθ . 2 ϕ(v , x) ϕ(v , x)
(5.1.2)
Proof. For k ≥ 0, a ≥ 0, set Fk (λ) =
λk . λ+a 2
d d Then for λ ∈ [1, ∞[, we have dλ Fk (λ) ≥ 0 if k ≥ 1 and dλ 2 Fk (λ) ≥ 0 if k ≥ 2. Since d dλ Fk (λ) is positive and increasing on [1, ∞[ if k ≥ 2, it follows from the mean value theorem that for λ, λ ≥ 1,
|Fk (λ) − Fk (λ )| ≤
d Fk (λ + |λ − λ |)|λ − λ |. dλ
Setting λ = v2 , λ = v 2 , we have d Fk (2(v2 + |v − v |2 )) 2|v| + |v − v | |v − v | dλ ≤ 2k Fk−1/2 (2(v2 + |v − v |2 ))|v − v |, √ d because |λ − λ | ≤ 2|v − v ||v| + |v − v |2 ≤ |v|2 + 2|v − v |2 and λ dλ Fk (λ) ≤ k Fk−1/2 (λ). Therefore, we obtain, choosing a = |x|2 , vl−1 |v − v | |v − v |l |Wϕ,l − Wϕ,l | ≤ Cl + v2 + |v − v |2 + a v2 + |v − v |2 + a |Fk (v2 ) − Fk (v 2 )| ≤
≤ Cl |v − v |vl−3 F1 (v2 ) + Cl = I + I I.
|v − v |l v2 + |v − v |2 + a (5.1.3)
The Boltzmann Equation Without Angular Cutoff
547
Note that v2 ≤ 2v∗ 2 + 2|v − v∗ |2 . Since F1 is increasing, we have v∗ 2 + |v − v∗ |2 I ≤ Cl |v − v |vl−3 v∗ 2 + |v − v∗ |2 + |x|2 θ Wl + Wl−3 W3,∗ . ≤ Cl sin 2 ϕ(v∗ , x) On the other hand
|v − v∗ |l sinl θ2 I I ≤ Cl 1 + 1 − sin2 θ2 |v|2 + 21 sin2 θ2 |v∗ |2 + |x|2 θ Wl + Wl,∗ . ≤ Cl sinl−2 2 ϕ(v∗ , x)
Since v and v are symmetric, we get the first conclusion. The second one is a direct consequence of the first inequality of (5.1.3). Lemma 5.2. Let l ∈ N. If 0 < s < 1/2, there exists C > 0 such that Wϕ,l Q( f, g) − Q( f, Wϕ,l g) , h L 2 (R6 ) ≤ C f L ∞ (R3 ;L 1 (R3 )) Wϕ,l g L 2 (R6 ) h L 2 (R6 ) . x
l
(5.1.4)
v
Moreover, if l ≥ 5 then Wϕ,l Q( f, g) − Q( f, Wϕ,l g) , h L 2 (R6 ) ≤ CWϕ,l f L 2 (R6 ) g L ∞ (R3 ;L 2 (R3 )) h L 2 (R6 ) . x
l
(5.1.5)
v
Proof. It follows from (5.1.2) that Wϕ,l Q( f, g) − Q( f, Wϕ,l g) , h L 2 (R6 ) b f ∗ g (Wϕ,l − Wϕ,l ) h dvdv∗ dσ d x = ≤C b |θ | |(Wl f )∗ | |(Wϕ,l g) | |h| dvdv∗ dσ d x =C b |θ ||(Wl f )∗ | |(Wϕ,l g)| |h | dvdv∗ dσ d x ≤C
1/2 b |θ | |(Wl f )∗ | |(Wϕ,l g)| dvdv∗ dσ d x 2
×
b |θ | |(Wl f )∗ | |h |2 dvdv∗ dσ d x
1/2
= C J1 × J2 . Clearly, one has J12
≤
C f L ∞ (R3 ;L 1 (R3 )) Wϕ,l g2L 2 (R6 ) x l v
≤ C f L ∞ (R3 ;L 1 (R3 )) Wϕ,l g2L 2 (R6 ) . x
l
v
S2
b(cos θ ) |θ | dσ
548
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Next, by the regular change of variables v → v , cf. [6,16], we have J22 =
D0 (v∗ , v )|(Wl f )∗ ||h |2 dv∗ dv d x,
where
θ (v∗ , v , σ ) b(cos θ (v∗ , v , σ ))dσ 2 S2 cos (θ (v∗ , v , σ )/2) π/4 ψ −1−2s sin ψ dψ, ≤C
D0 (v, v ) = 2
0
and cos ψ =
v − v∗ · σ, ψ = θ/2, |v − v∗ |
dσ = sin ψdψdφ.
Thus, J22 ≤ C f L ∞ (R3 ;L 1 (R3 )) h2L 2 (R6 ) , x
l
v
and this together with the estimate on J1 give (5.1.4). We now prove (5.1.5) by using (5.1.1) instead of (5.1.2). For this purpose, when l ≥ 5, we write Wϕ,l Q( f, g) − Q( f, Wϕ,l g) , h L 2 (R6 ) ≤C b |θ | |(Wϕ,3 f )∗ | |(Wl g) | |h| dvdv∗ dσ d x l−2 |(Wϕ,l f )∗ | |g | |h| dvdv∗ dσ d x + b |θ | = M1 + M2 . The estimation on M1 can be obtained following the proof of (5.1.4) except for the x variable. Indeed, M1 ≤ C
1/2 b |θ | |(Wϕ,3 f )∗ | |(Wl g)|2 dvdv∗ dσ
1/2
2
×
dx b |θ | |(Wϕ,3 f )∗ | |h | dvdv∗ dσ ≤ Cg L ∞ (R3 ;L 2 (R3 )) Wϕ,3 f L 1 (R3v ) h L 2 (R3v ) d x x
l
v
≤ Cg L ∞ (R3 ;L 2 (R3 )) x
l
v
1/2
Wϕ,5 f 2L 2 (R3 ) d x v
≤ CWϕ,5 f L 2 (R6 ) g L ∞ (R3 ;L 2 (R3 )) h L 2 (R6 ) . x
l
v
1/2 h L 2 (R3v ) d x
The Boltzmann Equation Without Angular Cutoff
549
M2 can be estimated as follows. Firstly, we have 2 2 2 l−2 M2 = C b |θ | |(Wϕ,l f )∗ ||g| |h | dvdv∗ dσ d x 3 ≤ C2 b |θ | l−2− 2 |g||(Wϕ,l f )∗ |2 dvdv∗ dσ d x 3 × b |θ | l−2+ 2 |g||h |2 dvdv∗ dσ d x = M2,1 × M2,2 . Then, if l − 2 −
3 2
− 2s − 1 > −1, that is, l > 2s +
3 2
+ 2, we have
M2,1 ≤ Cg L ∞ (R3x ;L 1 (R3v )) Wϕ,l f 2L 2 (R6 ) . On the other hand, for M2,2 we need to apply the singular change of variables v∗ → v . The Jacobian of this transform is ∂v∗ 8 4 8 −2 ∂v = |I − k ⊗ σ | = |1 − k · σ | = sin2 (θ/2) ≤ 16θ , θ ∈ [0, π/2]. Notice that this gives rise to an additional singularity in the angle θ around 0. Actually, the situation is even worse in the following sense. Recall that θ is no longer a legitimate polar angle. In this case, the best choice of the pole is k = (v − v)/|v − v| for which polar angle ψ defined by cos ψ = k · σ satisfies (cf. [6, Fig. 1]) ψ=
π −θ , 2
π π ψ ∈ [ , ]. 4 2
dσ = sin ψdψdφ,
This measure does not cancel any of the singularity of b(cos θ ), unlike the case in the usual polar coordinates. Nevertheless, this singular change of variables yields 3 M2,2 = C b(cos θ ) |θ | −2+ 2 |g| |h |2 dvdv∗ dσ d x ≤C D1 (v, v )|g| |h |2 dvdv d x, when l − 2 > D1 (v, v ) =
3 2
+ 2s because
S2
3
θ l−2+ 2 −2 b(cos θ )dσ ≤ C
π/2 π/4
(
3 π − ψ)−2−2s+l−2+ 2 −2 dψ ≤ C. 2
Therefore, M2,2 ≤ Cg L ∞ (R3x ;L 1 (R3v )) h2L 2 (R6 ) . Now the proof of (5.1.1) is completed by using the imbedding estimate for l > 23 , g L 1 (R3v ) ≤ Cg L 2 (R3 ) . l
And this completes the proof of the lemma.
v
We are now ready to conclude the proof of the uniqueness theorem.
550
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Proof of Theorem 1.2. Set F = f 1 − f 2 . Then we have Ft + v · ∇x F = Q( f 1 , F) + Q(F, f 2 ) , F|t=0 = 0.
(5.1.6)
Let S(τ ) ∈ C0∞ (R) satisfy 0 ≤ S ≤ 1 and S(τ ) = 1, |τ | ≤ 1; S(τ ) = 0, |τ | ≥ 2. Set S N (Dx ) = S(2−2N |Dx |2 ) and multiply Wϕ,l S N (Dx )2 Wϕ,l F to (5.1.6). Integrating and letting N → ∞, we have 1 d Wϕ,l F(t)2L 2 (R6 ) = Wϕ,l Q( f 1 , F) + Wϕ,l Q(F, f 2 ) , Wϕ,l F L 2 (R6 ) 2 dt −(v · ∇x (ϕ −1 )Wl F, Wϕ,l F) L 2 (R6 ) , because (v · ∇x S N (Dx )Wϕ,l F, S N (Dx )Wϕ,l F) L 2 (R6 ) = 0. The second term on the right hand side is estimated by Wϕ,l F2L 2 (R6 ) . Since f 1 ≥ 0, from the coercivity of −(Q( f 1 , g), g) it follows that Q( f 1 , Wϕ,l F) , Wϕ,l F L 2 (R6 ) ≤ C f 1 (t) L ∞ (R3x ; L 1 (R3v )) Wϕ,l F(t)2L 2 (R6 ) . x, v
By Lemma 5.2, we have Wϕ,l Q( f 1 , F) − Q( f 1 , Wϕ,l F) , Wϕ,l F L 2 (R6 ) ≤ C f 1 L ∞ (R3 ;L 1 (R3 )) Wϕ,l F2L 2 (R6 ) , x
and
v
l
Wϕ,l Q(F, f 2 ) − Q(F, Wϕ,l f 2 ) , Wϕ,l F L 2 (R6 ) ≤ C f 2 L ∞ (R3 ;L 2 (R3 )) Wϕ,l F2L 2 (R6 ) . x
l
v
Finally, for l > 7/2 + 2s, we have Q(F, Wϕ,l f 2 ) , Wϕ,l F L 2 (R6 ) ≤ CQ(F, Wϕ,l f 2 ) L 2 (R6 ) Wϕ,l F(t) L 2 (R6 ) F(t, x, · ) 2 x2 f 2 (t, x, · )2H 2s (R3 ) d x ≤ CWϕ,l F(t) L 2 (R6 ) 1 3 2 L 2s (Rv ) v 3 ϕ l+2s x Rx
1/2
≤ CWϕ,l F(t)2L 2 (R6 ) f 2 (t) L ∞ (R3 ; H 2s (R3 )) , x v l+2s because x−2 ≤ Wϕ,2 and x2 /ϕ is a bounded operator on H 2s uniformly with respect to x. Thus, we have, for any 0 < t < T , d Wϕ,l F(t)2L 2 (R6 ) dt l
≤ C f 1 L ∞ (]0,T [×R3 ; L 1 (R3 )) + f 2 L ∞ (]0,T [×R3 ; H 2s (R6 )) Wϕ,l F(t)2L 2 (R6 ) . x v x v l l+2s l
Therefore, Wϕ,l F(0) L 2 (R6 ) = 0 which implies Wϕ,l F(t) L 2 (R6 ) = 0 for all t ∈ [0, T [. And this gives f 1 = f 2 , and thus completes the proof of Theorem 1.2.
The Boltzmann Equation Without Angular Cutoff
551
Remark 5.3. For the function space considered in Theorem 1.1, the uniqueness of solutions is a direct consequence of Theorem 1.2 if there exists a non-negative solution. It is because g0 ∈ H k and g ∈ L ∞ (]0, T [×H k ) with k, ≥ 3 imply f 0 = √ √ μ + μg0 ∈ L ∞ (R3x ; Hm2s ) and f = μ + μg ∈ L ∞ (]0, T [×R3x ; Hm2s ) for any m, respectively, and k > 3/2 + 2s. 5.2. Non-negativity. In this subsection, we will prove the non-negativity of the solution obtained in Theorem 1.1. Theorem 5.4. Let N ≥ 3, ≥ 3. There exists 1 > 0 such that if g0 ∈ H N (R6 ) with μ + μ1/2 g0 ≥ 0 and g0 H N (R6 ) ≤ 1 , and g ∈ L ∞ ([0, T ]; H N (R6 )) is a solution of
the Cauchy problem (1.3), then we have μ + μ1/2 g ≥ 0 on [0, T ] × R6 . Proof. By applying Remark 5.3 on the uniqueness to the Cauchy problem (1.3), it is enough to prove the non-negativity of the approximate solutions given by Proposition 4.5, that is, f n = μ + μ1/2 g n ≥ 0 ,
n ∈ N.
(5.2.1)
Again, this will be proved by induction. It is clearly true for n = 0 by taking g 0 = g0 , and so we now assume that it is true for some n and will prove that (5.2.1) is true for n + 1. From (4.3.1), f n+1 = μ + μ1/2 g n+1 is the solution of the following Cauchy problem: ! ∂t f n+1 + v · ∇x f n+1 = Q( f n , f n+1 ), (5.2.2) f n+1 |t=0 = f 0 = μ + μ1/2 g0 ≥ 0. Take the convex function β(s) =
1 − 2 1 (s ) = s (s − ) 2 2
with s − = min{s, 0}, and notice that βs (s) =
dβ(s) = s−. ds
Setting φ(x, v) = (1 + |x|2 + |v|2 )−2 and noticing that βs ( f n+1 )φ(x, v) = min{ f n+1 , 0}φ(x, v) ∈ L ∞ ([0, T ]; L 1 (R3x ; L 2 (R3v )), we have by (5.2.2), d β( f n+1 )φd xdv = Q( f n , f n+1 ) βs ( f n+1 )φ d xdv 6 dt R6 R n+1 − v · ∇x (β( f )φ)d xdv − (φ −1 v · ∇x φ) β( f n+1 )φd xdv, R6
R6
where the first term on the right hand side is well defined by Theorem 2.1, because 2s (R3 )). Since the second term vanishes and f n belongs to L ∞ ([0, T ] × R3x ; L 12s ∩ H2s v |v · ∇x φ| ≤ Cφ, we obtain d β( f n+1 )φd xdv ≤ Q( f n , f n+1 )βs ( f n+1 )φd xdv + C β( f n+1 )φd xdv. dt R6 R6 R6
552
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
For the first term on the right hand side, we have Q( f n , f n+1 ) βs ( f n+1 )φd xdv R6
= b(cos θ ) f ∗n f n+1 − f ∗n f n+1 βs ( f n+1 )φ R6x,v
=
R6
x,v +
R3v∗ ×S2σ
R6x,v
R3v∗ ×S2σ
b(cos θ ) f ∗n
f n+1 − f n+1 βs ( f n+1 )φ
βs ( f n+1 ) f n+1 φ
R3v∗ ×S2σ
= I + I I.
b(cos θ ) f ∗n − f ∗n
From (4.3.2), we have, for any n ∈ N, √ f n L ∞ ([0,T ]×R3x ; L 1 (R3v )) ≤ 1 + μ g n L ∞ ([0,T ]×R3x ; L 1 (R3v )) ≤ 1 + Cg n L ∞ ([0,T ]×R3x ; L 2 (R3v )) ≤ 1 + C0 , so that the cancellation lemma from [6] implies that
b(cos θ ) f ∗n − f ∗n = C f n (t, x, v)dv R3v∗ ×S2σ
R3v n
≤ C|| f || L ∞ ([0,T ]×R3x ; L 1 (R3v )) ≤ C ,
while βs (s)s = 2β(s) implies that |I I | ≤ C
R6
β( f n+1 )φd xdv.
On the other hand, by the convexity of β, that is, βs (a)(b − a) ≤ β(b) − β(a) , and the assumption that f ∗n ≥ 0, we get
b(cos θ ) f ∗n f n+1 − f n+1 βs ( f n+1 )φ I = R6x,v
≤
R6x,v
≤
R6x,v
− ≤
R3v∗ ×S2σ
R3v∗ ×S2σ
R3v∗ ×S2σ
R3v∗ ×S2σ
R3v∗ ×S2σ
= I1 + I2 .
β( f n+1 ) − β( f n+1 ) φ
b(cos θ ) f ∗n β( f n+1 ) − f ∗n β( f n+1 ) φ
R6x,v
R6x,v
b(cos θ ) f ∗n
b(cos θ )β( f n+1 ) f ∗n − f ∗n φ
b(cos θ ) f ∗n β( f n+1 )
φ −φ +C
R6
β( f n+1 )φd xdv
The Boltzmann Equation Without Angular Cutoff
553
Applying Taylor’s formula to the first term gives 1 I1 = dτ b(cos θ ) f ∗n β( f n+1 ) v − v · ∇v φ v + τ (v − v)) . R6x,v
0
R3v∗ ×S2σ
Since |v − v| = |v − v∗ | sin
θ θ ≤ v v∗ sin , 2 2
(5.2.3)
by setting vτ = v + τ (v − v), 0 < τ < 1, 0 ≤ θ ≤ π/2, we have √ θ 2 |v| ≤ |vτ | + |v − v| ≤ |vτ | + sin |v| + |v∗ |. (|v| + |v∗ |) ≤ |vτ | + 2 2 Then (1 + |x|2 + |v|2 ) ≤ C(1 + |x|2 + |vτ |2 )(1 + |v∗ |2 ), which implies |∇v φ(x, vτ )| ≤ (1 + |x|2 + |vτ |2 )−5/2 ≤ Cφ(x, v) So we obtain
v∗ 5 . v
|I1 | ≤ C|| f n || L ∞ ([0, T ]×R3 ; L 1 (R3 )) x
6
v
R6
β( f n+1 )φd xdv.
Again from (4.3.2), we have, for any n ∈ N, f n L ∞ ([0,T ]×R3 ; L 1 (R3 )) ≤ μ1/2 L 1 (R3 ) + μ1/2 g n L ∞ ([0,T ]×R3 ; L 1 (R3 )) ≤ C(1 + 0 ). x
6
v
6
Finally, we have obtained, for 0 < t < T , d n+1 β( f )φ d x dv ≤ C β( f n+1 )φ d x dv, dt R6 R6
x
6
v
β( f n+1 )|t=0 = 0.
Therefore, for 0 < t < T , and by continuity, β( f n+1 (t))φ d x dv = 0, R6
which implies that f n+1 (t, x, v) ≥ 0 for (t, x, v) ∈ [0, T ] × R6x,v . Therefore, the proof of Theorem 5.4 is completed. Remark 5.5. Note that the above analysis can be extended to the strong singularity case. Indeed, by writing I1 = b(cos θ ) f ∗n β( f n+1 ) v − v · ∇v φ(v) R6x,v
1 + 2
R3v∗ ×S2σ 1
dτ 0
= I11 + I12 ,
R6x,v
R3v∗ ×S2σ
2 b(cos θ ) f ∗n β( f n+1 ) v − v ∇v2 φ v + τ (v − v))
554
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
since we have |∇v2 φ(x, vτ )| ≤ (1 + |x|2 + |vτ |2 )−3 ≤ Cφ(x, v)
v∗ 6 , v2
it follows from (5.2.3) that |I12 | ≤ C|| f n || L ∞ ([0, T ]×R3 ; L 1 (R3 )) x
On the other hand, setting k = v − v =
v−v∗ |v−v∗ |
v
8
R6
β( f n+1 )φd xdv.
and writing
1 1 |v − v∗ | (σ − (σ · k)k) + ((σ · k) − 1)(v − v∗ ), 2 2
we have I11 =
1 b(cos θ ) f ∗n β( f n+1 ) (cos θ − 1) (v − v∗ ) · ∇v φ(v), 2 R6x,v R3v∗ ×S2σ
where we have used the symmetry that S2 b(σ · k) (σ − (σ · k)k) dσ = 0. Therefore, we have β( f n+1 )φd xdv, |I11 | ≤ C|| f n || L ∞ ([0, T ]×R3 ; L 1 (R3 )) x
v
6
R6
and the same estimation holds also in the strong singularity case. 6. Full Regularity We now prove the smoothing effect of the Cauchy problem for the non-cutoff Boltzmann equation. More precisely, the main result of this section is given by Theorem 6.1. Assume that 0 < s < 1/2. There exists 1 > 0 such that if g0 ∈ H33 (R6 ) with μ + μ1/2 g0 ≥ 0, g0 H 3 (R6 ) ≤ 1 , and g ∈ L ∞ ([0, T ]; H33 (R6 )) is the solution 3
of Cauchy problem (1.3), then we have g ∈ C ∞ (]0, T [×R6 ).
Let us recall that H k (R7t,x, ), H k (R6x, v ) and H k (R3v ) denote some weighted Sobolev spaces with the weight defined in the variable v. Since the regularity result to be proved is local in space and time, for convenience, we define the corresponding local version of weighted Sobolev spaces. For 0 ≤ T1 < T2 < +∞, and any given open domain ⊂ R3x , define Hlm (]T1 , T2 [× × R3v ) =
f ∈ D (]T1 , T2 [× × R3v );
ϕ(t)ψ(x) f ∈ Hlm (R7 ) , ∀ ϕ ∈ C0∞ (]T1 , T2 [), ψ ∈ C0∞ () .
The proof of Theorem 6.1 will be divided into several steps.
The Boltzmann Equation Without Angular Cutoff
555
6.1. Formulation of the problem. First of all, we recall the main result in [12] given below. Theorem 6.2. Assume that 0 < s < 1, 0 ≤ T1 < T2 < +∞, ⊂ R3x is an open domain. Let f be a non-negative solution of the Boltzmann equation (1.1) satisfying f ∈ H 5 (]T1 , T2 [× × R3v ) for all ∈ N. Moreover, assume that f satisfies the nonvacuum condition f (t, x, ·) L 1 (R3v ) > 0,
(6.1.1)
for all (t, x) ∈ ]T1 , T2 [×. Then we have f ∈ H +∞ (]T1 , T2 [× × R3v ) ⊂ C +∞ (]T1 , T2 [× × R3v ), for any ∈ N. To apply this result, let us firstly note that, by Theorem 4.3 and Theorem 5.4, under the assumption of Theorem 6.1, the unique solution of the Cauchy problem (1.3) satisfies g L ∞ ([0, T ]; H 3 (R6 )) ≤ 0 , 3
μ+
√
μ g ≥ 0.
√ Therefore, f = μ + μ g ≥ 0 is a solution of the Boltzmann equation (1.1). On the other hand, we can choose 0 > 0 small enough such that √ || μ g|| L ∞ ([0,T ]×R3x ; L 1 (R3v )) ≤ Cg L ∞ ([0, T ]; H 2 (R6 )) ≤ C0 < 1, where C is the Sobolev constant of the imbedding H 2 (R3x ) ⊂ L ∞ (R3x ). Thus, for any (t, x) ∈]0, T [×R3x , √ f (t, x, v)dv = 1 + μ g(t, x, v)dv R3v
R3v
√ ≥ 1 − || μ g|| L ∞ ([0,T ]×R3x ; L 1 (R3v )) > 0 ,
(6.1.2)
√ so that f = μ + μ g satisfies condition (6.1.1). By using Eq. (1.1) and Remark 4.4, we have also, for any ∈ N, 0 < T1 < T2 < T and bounded open domain ⊂ R3x , f =μ+
√
μ g ∈ H 3 (]T1 , T2 [× × R3v ).
Note that we can not apply directly Theorem 6.2 because we now only know that f has regularity just in H 3 (]T1 , T2 [× × R3v ). To overcome this, we prove the following theorem. Theorem 6.3. Under the assumptions of Theorem 6.1, we have, for any 0 < T1 < T2 < T and bounded open domain ⊂ R3x , f =μ+ holds for all ∈ N.
√
μ g ∈ Hl5 (]T1 , T2 [× × R3v ),
556
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
The proof of this theorem is similar but easier than that of Theorem 6.2 which was proved in [12]. In fact, since we have g n → g, by mollifying the initial data and using the uniqueness of the solution, we do not need at all to mollify the solution as in [12]. It follows that to obtain the above regularization result, we only need to prove the à priori estimate on the smooth solution, which can be deduced from [12]. To make the paper self-contained, we give a proof here. Let us recall that here we consider the Maxwellian molecule type cross-sections with the mild singularity. Here and below, φ denotes a cutoff function satisfying φ ∈ C0∞ and 0 ≤ φ ≤ 1. Notation φ1 ⊂⊂ φ2 stands for two cutoff functions such that φ2 = 1 on the support of φ1 . Take the smooth cutoff functions ϕ, ϕ2 , ϕ3 ∈ C0∞ (]T1 , T2 [) and ψ, ψ2 , ψ3 ∈ C0∞ () such that ϕ ⊂⊂ ϕ2 ⊂⊂ ϕ3 and ψ ⊂⊂ ψ2 ⊂⊂ ψ3 . Set f 1 = ϕ(t)ψ(x) f, f 2 = ϕ2 (t)ψ2 (x) f and f 3 = ϕ3 (t)ψ3 (x) f . For α ∈ N6 , |α| ≤ 3, define α g = ∂x,v (ϕ(t)ψ(x) f ) ∈ L ∞ (]T1 , T2 [; L 2 (R6 )).
Then the Leibniz formula yields the following equation: gt + v · ∂x g = Q( f 2 , g) + G, (t, x, v) ∈ R7 ,
(6.1.3)
where G=
α1 +α2 =α, 1≤|α1 | + ∂ α (ϕt ψ(x) f
Cαα21 Q ∂ α1 f 2 , ∂ α2 f 1 + v · ψx (x)ϕ(t) f ) + [∂ α , v · ∂x ](ϕ(t)ψ(x) f )
≡ A + B + C.
(6.1.4)
Then we can take W 2 g as a test function for Eq. (6.1.3). It follows by integration by parts on R7 = R1t × R3x × R3v that
0 = W Q( f 2 , g), W g 2 7 + G, W 2 g 2 7 , L (R )
L (R )
which is sufficient for obtaining the required initial regularity.
6.2. Gain of regularity in velocity variable. The next step is to show the gain of regularity in the velocity variable by using the coercivity of the collision operator. Proposition 6.4. Under the assumption of Theorem 6.1, for any 0 < T1 < T2 < T and bounded open domain ⊂ R3x , one has sv f 1 ∈ L 2 (Rt ; H 3 (R6 )), 1
for any ∈ N, where v = (1 − v ) 2 , f 1 = ϕ(t)ψ(x) f with ϕ ∈ C0∞ (]T1 , T2 [), ψ ∈ C ∞ ().
The Boltzmann Equation Without Angular Cutoff
557
Proof. Firstly, the local positive lower bound (6.1.2) implies that inf
(t,x)∈Supp ϕ×Supp ψ1
f 2 (t, x, ·) L 1 (R3v ) = c0 > 0.
Thus, the coercivity estimate (2.1.1) gives
− Q( f 2 , W g), W g 2 7 L (R )
Q( f 2 , W g), W g 2 3 d xdt =− L (R v ) t∈Supp ϕ x∈Supp ψ1 C0 W g(t, x, ·)2H s (R3 ) ≥ Rt
v
R3x
− C f 2 (t, x, ·) L 1 (R3v ) W g(t, x, ·)2L 2 (R3 ) d xdt v
≥
C0 sv W
g2L 2 (R7 )
− C T f 2 L ∞ (R 4
t,x ;
L 1 (R3v )) W
g2L ∞ ([0,T ]; L 2 (R6 )) .
Since f 2 L ∞ (R 4
t,x ;
L 1 (R3v ))
≤ C f 2 L ∞ ([0,T ];
3/2+
H3/2+ (R6 ))
,
and W g2L ∞ ([0,T ]; L 2 (R6 )) ≤ C f 1 2L ∞ ([0,T ]; H 3 (R6 )) ,
for l > 3/2, we have sv W
g2L 2 (R7 )
2 ≤ + G, W g 2 7 L (R )
+ W Q( f 2 , g) − Q( f 2 , W g), W g 2 7 C f 2 2L ∞ ([0,T ]; H 3 (R6 ))
L
. (6.2.1) (R )
By applying Lemma 3.2, the third term on the right hand side of (6.2.1) can be estimated as follows:
W Q( f 2 , g) − Q( f 2 , W g) , W g 2 7 L (R ) ≤ C T f 2 L ∞ ([0,T ]; H 3/2+ (R6 )) g L ∞ ([0,T ]; L 2 (R6 )) g L 2 (R7 )
+3
≤ C T 2 f 2 L ∞ ([0,T ]; H 3/2+ (R6 )) f 1 2L ∞ ([0,T ]; H 3 (R6 )) . +3
For the second term in (6.2.1), we shall prove the following claim: For 0 < s < 1/2, one has
G, W 2 g ≤ C f3 ∞ L ([0,T ]; H 3 (R6 )) L 2 (R 7 )
+ f 2 2L ∞ ([0,T ]; H 3 (R6 )) sv W g L 2 (R7
t,x,v )
In fact, recalling the expression (6.1.4), it is easy to get B2L 2 (R7 ) + C2L 2 (R7 ) ≤ C T f 3 2L ∞ ([0,T ];
3 (R6 )) H +1
.
. (6.2.2)
558
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
For the term A, firstly recall that α1 + α2 = α, |α| ≤ 3 and |α2 | ≤ 2. In the following, we will apply Theorem 2.1 with m = −s. We separate the discussion on A into two cases. Case 1. If |α1 | = 1, we have Q(∂ α1 f 2 , ∂ α2 f 1 )(t, x, ·)2H s (R3 ) d xdt Rt
R3x
≤C
Rt
R3x
v
∂ α1 f 2 (t, x, ·)2L 1
3 +2s (Rv )
≤ C∂ α1 f 2 2L ∞ (R4
1 3 t,x ; L +2s (Rv ))
≤ C T f 2 2L ∞ ([0,T ];
3 (R6 )) H +3
∂ α2 f 1 (t, x, ·)2H s
Rt
R3x
3 +2s (Rv )
∂ α2 f 1 (t, x, ·)2H s
3 +2s (Rv )
f 1 2L ∞ ([0,T ];
3 (R6 )) H +2s
Case 2. If |α1 | ≥ 2, then |α2 | ≤ 1, it follows that ∂ α1 f 2 (t, x, ·)2L 1 (R3 ) ∂ α2 f 1 (t, x, ·)2H s Rt
R3x
+2s
≤ C∂ α2 f 1 2L ∞ (R4
v
3 l+2s (Rv )
d xdt
.
d xdt
∂ α1 f 2 (t, x, ·)2L 2 3 d xdt l+3/2+δ+2s (Rv ) R3x f 2 2L ∞ ([0,T ]; H 3 (R6 )) . C T f 1 2 ∞ 1+3/2+ (R6 )) L ([0,T ]; H +2s +3 s 3 t,x ; Hl+2s (Rv ))
≤
d xdt
Rt
By combining these two cases, we have
≤ C T f 2 2 ∞ G, W 2 g L ([0,T ]; 2 7 L (R )
+ f 2 2L ∞ ([0,T ];
3 (R6 )) H +1
3 (R6 )) H +3
sv W g L 2 (R7
t,x,v )
.
Therefore, we obtain sv W g2L 2 (R7 ) ≤ C 1 + f 3 L ∞ ([0,T ]; The proof of the proposition is then completed.
4 H 3 (R6 ))
.
6.3. Gain of regularity in space variable. With the gain of regularity in the variable v given in the above subsection, we now want to prove the gain of regularity in the variable x. Here, the hypo-elliptic nature of the equation will be used. For this purpose, we introduce a more general framework. First of all, let us consider a transport equation in the form of f t + v · ∇x f = g ∈ D (R7 ),
(6.3.1)
where (t, x, v) ∈ R7 . In [9], by using a generalized uncertainty principle, we proved the following hypoelliptic estimate.
The Boltzmann Equation Without Angular Cutoff
559
Lemma 6.5. Assume that g ∈ H −s (R7 ), for some 0 ≤ s < 1. Let f ∈ L 2 (R7 ) be a weak solution of the transport equation (6.3.1), such that sv f ∈ L 2 (R7 ) for some 0 < s ≤ 1. Then it follows that s(1−s )/(s+1)
x
f ∈ L2
ss − s+1
s(1−s )/(s+1)
(R7 ), t
f ∈ L 2−
s s+1
(R7 ),
where • = (1 − • )1/2 . Of course g is typically linked with the Boltzmann collision operator. Through the above uncertainty principle, we have the following result on the gain of regularity in the variable x Proposition 6.6. Under the hypothesis of Theorem 6.1, one has sx0 f 1 ∈ L 2 (Rt ; H 3 (R6 )), for any ∈ N and 0 < s0 =
(6.3.2)
s(1−s) (s+1) .
Proof. For any ∈ N, it follows from Proposition 6.4 that sv W g ∈ L 2 (R7 ), while using the upper bound estimation given by Theorem 2.1 with m = −s, we get W Q( f 2 , g) ∈ L 2 (R4t,x ; H −s (R3v )). On the other hand, (6.2.2) gives W G ∈ L 2 (R4t,x ; H −s (R3v )). By using Eq. (6.1.3), it follows that ∂t (Wl g) + v · ∂x (Wl g) = Wl Q( f 2 , g) + Wl G ∈ H −s (R7 ). Finally, by using Lemma 6.5 with s = s, we can conclude (6.3.2) and this completes the proof of the proposition. Therefore, under the hypothesis f ∈ L ∞ ([0, T ]; H 3 (R6 )) for all ∈ N, it follows that for any ∈ N we have sv (ϕ(t)ψ(x) f ) ∈ L 2 (Rt ; H 3 (R6 )), sx0 (ϕ(t)ψ(x) f ) ∈ L 2 (Rt ; H 3 (R6 )).
(6.3.3)
We now improve this partial regularity in the variable x. Since fractional derivatives will be involved, it is not surprising that a Leibniz type formula for fractional derivatives in the variable x is needed. We shall use the following one, taken from [12]. Let 0 < λ < 1. Then there exists a positive constant Cλ = 0 such that |Dx |λ Q ( f, g) = Q |Dx |λ f, g + Q f, |Dx |λ g |h|−3−λ Q ( f h , gh ) dh , + Cλ (6.3.4) R3
with f h (t, x, v) = f (t, x, v) − f (t, x + h, v),
h ∈ R3x .
560
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
With this preparation, we need a preliminary step, given by Proposition 6.7. Let 0 < λ < 1 and f ∈ L ∞ ([0, T ]; H 3 (R6 )) be a solution of (1.1). Assume that, for all ∈ N, we have sv f 1 ∈ H 3 (R7 ), λx f 1 ∈ H 3 (R7 ). Then, one has for any ∈ N, sv λx f 1 L 2 (Rt ; H 3 (R6 )) ≤ C f 2 L ∞ ([0,T ]; H 3 (R6 )) +3 s × v f 1 H 3 (R7 ) + λx f 1 H 3 +2s
+2s
(R 7 )
.
α f and α ∈ N6 , |α| ≤ 3. We choose W |D |λ ψ 2 (x) |D |λ W g as Proof. Set g = ∂x,v 1 x x 2 a test function for Eq. (6.1.3). Then
(v · ∂x ψ2 (x)) |Dx |λ W g, ψ2 (x) |Dx |λ W g 2 7 L (R )
λ λ = ψ2 (x) |Dx | W Q( f 2 , g), ψ2 (x) |Dx | W g 2 7 L (R )
λ λ + ψ2 (x) |Dx | W G, ψ2 (x) |Dx | W g 2 7 . L (R )
One has
≤ C λ ∂ α f 1 2 (v · ∂x ψ2 (x)) |Dx |λ W g, ψ2 (x) |Dx |λ W g x L +1 (R7 ) , 2 7 L (R ) and the same estimation for the linear term of G in (6.1.4),
ψ2 (x)|Dx |λ W (B + C) , ψ2 (x) |Dx |λ W g ≤ C λ ∂ α f 1 2 x L +1 (R7 ) . L 2 (R 7 ) For the nonlinear terms of G in (6.1.4), we shall use the Leibniz formula (6.3.4). First of all, the coercivity estimate (2.1.1) gives, as in (6.2.1),
− Q( f 2 , ψ2 (x)|Dx |λ W g), ψ2 (x)|Dx |λ W g 2 7 L (R )
≥
C0 sv ψ2 (x)|Dx |λ W −C T f 2 L ∞ (R4
t,x ;
g2L 2 (R7 )
L 1 (R3v )) ψ2 (x)|D x |
λ
W g2L ∞ ([0,T ]; L 2 (R6 )) .
On the other hand, the upper bound estimate of Theorem 2.1 with m = −s gives,
Q(|Dx |λ f 2 , ψ1 (x)W g), ψ1 (x)|Dx |λ W g 2 7 L (R )
≤ C|Dx |
λ
f 2 L ∞ (R4 , L 1 (R3 )) ψ2 (x)sv W t,x v 2s
≤ C f 2 L ∞ ([0,T ];
3/2+λ+
H3/2+2s+ (R6 ))
g L 2 (R7 ) ψ2 (x)|Dx |λ sv W g L 2 (R7 ) 2s
ψ2 (x)sv W g L 2 (R7 ) ψ2 (x)|Dx |λ sv W g L 2 (R7 ) 2s
≤ δψ2 (x)|Dx |λ sv W g2L 2 (R7 ) + Cδ f 2 2L ∞ ([0,T ];
H33 (R6 ))
sv g2L 2
+2s (R
7)
.
The Boltzmann Equation Without Angular Cutoff
561
For the cross term coming from the decomposition (6.3.4), by using again Theorem 2.1 with m = −s, we get R3
|h|−3−λ Q(( f 2 )h , (W g)h ), ψ22 (x)|Dx |λ W g
dh L 2 (R 7 )
≤ |Cλ |ψ2 (x)|Dx |λ sv W g L 2 (R7 ) × |h|−3−λ ( f 2 )h L ∞ (R4 , L 1 (R3 )) sv (W g)h L 2 (R7 ) dh t,x v 2s 2s 3 R
≤ δψ1 (x)|Dx |λ sv W g2L 2 (R7 ) + Cδ f 2 2L ∞ ([0,T ];
H33 (R6 ))
sv g2L 2
+2s (R
7)
.
Hence, we have
|Dx |λ Q( f 2 , ψ2 (x)W g) − Q(|Dx |λ f 2 , ψ2 (x)W g), ψ2 (x)|Dx |λ W g 2 7 L (R ) ≤ δψ2 (x)|Dx |λ sv W g2L 2 (R7 ) + Cδ f 2 2L ∞ ([0,T ];
H33 (R6 ))
sv g2L 2
+2s (R
7)
.
In conclusion, we get ψ2 (x)|Dx |λ sv W g2L 2 (R7 )
≤ C f 2 2L ∞ ([0,T ]; H 3 (R6 )) |Dx |λ g2L 2 (R7 ) + sv g2L 2 (R7 ) 3 +2s +2s
λ 2 + |Dx | W Q ( f 2 , g) − Q f 2 , W g , ψ2 (x)|Dx |λ W g 2 7 L (R )
+ |Dx |λ W A, ψ22 (x)|Dx |λ W g 2 7 L (R )
= I + II + III. For the term II, again, formula (6.3.4) yields,
|Dx |λ W Q ( f 2 , g) − Q f 2 , W g , ψ22 (x)|Dx |λ W g 2 7 L (R )
λ λ 2 = W Q |Dx | f 2 , g − Q |Dx | f 2 , W g , ψ2 (x)|Dx |λ W g 2 7 L (R )
λ λ 2 λ + W Q f 2 , |Dt,x | g − Q f 2 , W |Dx | g , ψ2 (x)|Dx | W g 2 7 L (R )
+ Cλ |h|−3−λ W Q (( f 2 )h , gh ) − Q ( f 2 )h , W gh , ψ22 (x)|Dx |λ W g R3
L 2 (R 7 )
Since 0 < s < 1/2 , Lemma 3.2 implies
W Q |Dx |λ f 2 , g − Q |Dx |λ f 2 , W g , ψ 2 (x)|Dx |λ W g 2 2 7 L (R ) ≤ C |Dx |λ f 2 L ∞ (R4
t,x
, L 1 (R3v )) g L 2 (R4t,x , L 2 (R3v )) |D x | λ
λ
≤ C f 2 L ∞ ([0,T ]; H 3/2+λ+ (R6 )) g L 2 (R3 ) |Dx | g L 2 (R7 ) , +3/2+
g L 2 (R7 )
dh.
562
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
and
W Q f 2 , |Dx |λ g − Q f 2 , W |Dx |λ g , ψ 2 (x)|Dx |λ W g 2 2 7 L (R ) ≤ C f 2 L ∞ (R4
t,x
, L 1 (R3v )) |D x |
λ
g L 2 (R4
t,x
≤ C f 2 L ∞ ([0,T ]; H 1+3/2+ (R6 )) |Dx |
λ
+3/2+
, L 2 (R3v )) |D x |
λ
g L 2 (R7 )
g2L 2 (R3 ) .
For the cross term,
R3
|h|−3−λ
W Q (( f 2 )h , gh ) − Q ( f 2 )h , W gh , ψ22 (x)|Dx |λ W g
L 2 (R 7 )
dh
≤ C f 2 ∞ g L 2 (R3 ) |Dx |λ g L 2 (R7 ) . 1+3/2+ L ([0,T ]; H (R6 )) +3/2+
Thus, we have II ≤ C f 2 L ∞ ([0,T ]; H 3 (R6 )) |Dx |λ g2L 2 (R3 ) . +2 We now consider the last term I I I . Recall that A stands for the nonlinear terms from G, A= Cαα21 Q ∂ α1 f 2 , ∂ α2 f 1 . α1 +α2 =α α1 =0
We have
|Dx |λ Q ∂ α1 f 2 , ∂ α2 f 1 , W ψ 2 (x)|Dx |λ W g 2 L 2 (R 7 ) ! * * s λ ≤ Cv ψ1 (x)|Dx | W g L 2 (R7 ) × * Q |Dt,x |λ ∂ α1 f 2 , ∂ α2 f 1 * L 2 (R4
−s 3 t,x ;H (Rv ))
* * + * Q ∂ α1 f 2 , |Dx |λ ∂ α2 f 1 * L 2 (R4 ;H −s (R3 )) t,x v * * * * −3−λ +* Q ∂ α1 ( f 2 )h , ∂ α2 ( f 1 )h dh * * * h
m 3 L 2 (R4t,x ;Hl+|γ |/2 (Rv ))
+ .
We divide the discussion into two cases. Case 1. |α1 | = 1 (then |α2 | ≤ 2). Applying (2.1.2) with m = −s. We have * * * Q |Dx |λ ∂ α1 f 2 , ∂ α2 f 1 * 2 4 −s 3 L (Rt,x ;H (Rv ))
≤
C λx ∂ α1 f 2 L 2 (Rt ; (L ∞ (R3 ;L 1 (R3 ))) sv ∂ α2 f 1 L ∞ (Rt ; L 2 (R6 )) x +2s v +2s
≤ C λx f 2 H 1+3/2+ (R7 ) sv f 1 H 2+1/2+ (R7 ) , +3/2++2s +2s * α * * Q ∂ 1 f 2 , |Dx |λ ∂ α2 f 1 * 2 4 −s 3 ≤ C∂
α1
L (Rt,x ;H (Rv ))
f 2 L ∞ (R 4
1 3 t,x ;L +2s (Rv ))
≤ C f 2 L ∞ ([0,T ]; H 1+3/2+
+3/2++2s
sv λx ∂ α2 f 1 L 2
7 +2s (R ) s v f 1 H 3 (R7 ) , (R6 )) +2s
The Boltzmann Equation Without Angular Cutoff
and
* * * *
R3
563
* * h −3−λ Q ∂ α1 ( f 2 )h , ∂ α2 ( f 1 )h dh * *
≤C
L 2 (R4t,x ;H −s (R3v ))
|h|−3−λ ∂ α1 ( f 2 )h L ∞ (R4
1 3 t,x ; L +2s (Rv ))
≤ C ∂ α1 f 2 L ∞ (R4
1 3 t,x ;L +2s (Rv ))
≤ C f 2 L ∞ ([0,T ]; H 1+3/2+
+3/2++2s
sv ∂ α2 ( f 1 )h L 2
+2s (R
sv ∂ α2 ∇x f 1 L 2
+2s (R
7)
dh
7)
sv f 1 H 3 (R7 ) . (R6 )) +2s
Case 2. |α1 | ≥ 2. By the same argument as above, one has * * * Q |Dx |λ ∂ α1 f 2 , ∂ α2 f 1 * 2 4 −s 3 L (Rt,x ;H (Rv )) * α * λ α2 1 * + Q ∂ f 2 , |Dx | ∂ f 1 * L 2 (R4 ;H −s (R3 )) t,x v ≤ C f 2 L ∞ ([0,T ]; H 1+3/2+ (R6 )) sv f 1 H 3 (R7 ) + λx f 1 H 3 +2s
+3/2++2s
7 +2s (R )
.
When |α1 | = 2, we have * * * * α −3−λ 1 ( f ) , ∂ α2 ( f ) * * dh h Q ∂ 2 h 1 h * 3 * 2 4 R L (Rt,x ; H −s (R3v )) ≤ C |h|−3−λ ∂ α1 ( f 2 )h L ∞ (Rt ; (L 2 (R3 ; L 1 (R3 ))) v
+2s
x
× sv ∂ α2 ( f 1 )h L 2 (Rt ; (L ∞ (R3 ; L 2 (R3 ))) dh x v +2s ≤ C ∇x ∂ α1 f 2 L ∞ (Rt ; (L 2 (R3 ; L 1 (R3 ))) sv ∂ α2 f 1 L 2 (Rt ; (L ∞ (R3 ; L 2 (R3 ))) x v x v +2s +2s s ≤ C f 2 L ∞ ([0,T ]; H 3 6 v f 1 H 3 (R7 ) , +3/2++2s (R )) +2s while when |α1 | = |α| = 3, we have * * * * α −3−λ * Q ∂ ( f 2 )h , ( f 1 )h dh * * 3h * 2 4 −s 3 R L (Rt,x ;H (Rv )) ≤ C |h|−3−λ ∂ α ( f 2 )h L 2 (R4 ;L 1 (R3 )) sv ( f 1 )h L ∞ (R4 t,x
≤ C ∂ α f 2 L 2 (R4
1 3 t,x ; L +2s (Rv ))
≤
+2s
2 3 t,x ; L +2s (Rv ))
v
dh
sv ∇x f 1 L ∞ (R4
2 3 t,x ; L +2s (Rv ))
C f 2 2L ∞ ([0,T ]; H 3 6 . +3/2++2s (R ))
Thus, by the Cauchy-Schwarz inequality, we obtain III ≤ δsv ψ1 (x)|Dx |λ W g2L 2 (R7 ) + Cδ f 2 2L ∞ ([0,T ]; H 3 (R6 )) sv f 1 2H 3 +3
+2s
(R 7 )
+ λx f 1 2H 3
+2s
(R 7 )
Finally, we get sv ψ1 (x)|Dx |λ W g2L 2 (R7 )
≤ Cδ f 2 2L ∞ ([0,T ]; H 3 (R6 )) sv f 1 2H 3 (R7 ) + λx f 1 2H 3 (R7 ) , +3 +2s +2s
and the proof of the proposition is completed.
.
564
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
We are now ready to show the gain of at least order 1 regularity in the variable x. Proposition 6.8. Under the hypothesis of Theorem 6.1, one has 3 7 1+ε x (ϕ(t)ψ(x) f ) ∈ H (R ),
(6.3.5)
for any ∈ N and some ε > 0. Proof. By fixing s0 =
s(1−s) (s+1) ,
then (6.3.3) and Proposition 6.7 with λ = s0 imply sv sx0 g ∈ H 3 (R7 ).
It follows that (sx0 g)t + v · ∂x (sx0 g) = sx0 Q( f 2 , g) + sx0 G ∈ H −s (R7 ). By applying Lemma 6.5 with s = s, we can deduce that sx0 +s0 (ϕ(t)ψ(x) f ) ∈ H 3 (R7 ), for any ∈ N. If 2s0 < 1, by using again Proposition 6.7 with λ = 2s0 and Lemma 6.5 with s = s, we have 3 7 3s0 3 7 0 sv (ϕ(t)ψ(x) f ), 2s x (ϕ(t)ψ(x) f ) ∈ H (R ) ⇒ x (ϕ(t)ψ(x) f ) ∈ H (R ).
Choose k0 ∈ N such that k0 s0 < 1,
(k0 + 1)s0 = 1 + ε > 1.
Finally, (6.3.5) follows from Proposition 6.7 with λ = k0 s0 by induction. And this completes the proof of the proposition. 6.4. Higher order regularity. The proof of Theorem 6.3 will now be concluded with the above preparation. From Proposition 6.4, Proposition 6.8 and Eq. (1.1), it follows that for any ∈ N, sv (ϕ(t)ψ(x) f ),
∇x (ϕ(t)ψ(x) f ) ∈ H 3 (R7 ).
This fact will be used to show higher order regularity in the variable v by using the following Proposition 6.9. Let 0 < λ < 1. Assume that, for any cutoff functions ϕ ∈ C0∞ (]T1 , T2 [), ψ ∈ C0∞ () and all ∈ N, λv (ϕ(t)ψ(x) f ),
∇x (ϕ(t)ψ(x) f ) ∈ H 3 (R7 ).
Then, for any cutoff function and any ∈ N, 3 7 λ+s v (ϕ(t)ψ(x) f ) ∈ H (R ).
The Boltzmann Equation Without Angular Cutoff
565
For the proof, we choose W 2v λ W g as a test function for (6.1.3), and then proceed as in the proof of Proposition 6.7. The only difference is in the estimation on the commutator with the convection:
" λ # , v · ∂x W g, λ W g ≤ Cλ g 2 7 ∇x g 2 7 , v v v L (R ) L (R ) L 2 (R 7 ) since
" λ # v , v · ∂x = λλ−2 ∂v · ∂ x , v
and λ−2 ∂v are bounded operators in L 2 . This is the reason why we need in the first v step the gain of the regularity of order 1 in the variable x. To complete the proof of the full regularization result, firstly, exactly as Proposition 6.3.5, we can get 3 7 1+ε v (ϕ(t)ψ(x) f ) ∈ H (R ),
for any ∈ N and some ε > 0. Therefore, we obtain that there exists > 0 such that for any ∈ N and any cutoff functions ϕ(t) and ψ(x), ϕ(t)ψ(x) f ∈ Hl4+ (R7 ). Notice that the estimate in Proposition 6.7 can be modified as follows. In fact, we can obtain sv λx f 1 L 2 (Rt ; H 4 (R6 )) ≤ C f 2 H 4+ (R7 ) +3
s × v f 1 H 4 (R7 ) + λx f 1 H 4 (R7 ) . +2s +2s By using this, the proof of f ∈ H 4+ (]T1 , T2 [× × R3v ), ∀ ∈ N ⇒ f ∈ H 5 (]T1 , T2 [× × R3v ), ∀ ∈ N, is direct and this completes the proof of Theorem 6.3 by the bootstrapping argument. 7. Global Existence We shall establish a global energy estimate for the Cauchy problem (1.3). For ease of exposition, and unless necessary, generic constants will be dropped out from the estimates in this section. Finally, all in all, we shall follow and adapt the method initiated by Guo [46] on the estimation on the macroscopic components. Here we point out that his method works also for the non-cutoff case but only when the estimations of the nonlinear and related terms are carried out in terms of the non-isotorpic norms (2.2.1). We also note that his method has been generalized to various directions. Among them, a few are the external force case [35,37], the Vlassov-Maxwell-Boltzmann equation [65], the soft potential case [66,74] and the Landau equation [45,74]. Independently of Guo’s method [46] which is based on the macro-micro decomposition near a global Maxwellian, the energy method based on the macro-micro decomposition around a local Maxwellian has also been developed with application to the classical fluid dynamical equations [52–54]. Further references are found in these papers.
566
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Introduce the macro-micro decomposition near the absolute Maxwellian μ:
g = Pg + (I − P)g = g1 + g2 , Pg = g1 = a + b · v + c|v|2 μ1/2 . In this section, the following result on the energy estimate will be proved. Theorem 7.1. For N , ≥ 3, let T > 0 and suppose that g is a classical solution to the Cauchy problem (1.3) on [0, T ]. There exist constants M0 , M1 > 0 such that if max E(t) ≤ M0 ,
0≤t≤T
then g enjoys the estimate
t
E(t) +
D(τ )dτ ≤ M1 E(0),
0
for any t ∈ [0, T ], where E = g2H N (R6 ) ∼ (a, b, c)2H N (R3 ) + g2 2H N (R6 ) ,
x
is the instant energy functional, and D = ∇x (a, b, c)2H N −1 (R3 ) + |||g2 |||2B N (R6 )
x
the total dissipation rate. The proof of this theorem is divided into two parts, that is, the estimation on the macroscopic component and the microscopic component respectively. 7.1. Macroscopic energy estimate. By the macro-micro decomposition, the equation in (1.3) is reduced to ∂t a + b · v + c|v|2 μ1/2 + v · ∇x a + b · v + |v|2 c μ1/2 = −(∂t + v · ∇x )g2 + Lg2 + (g, g). Using v · ∇x b · v =
vi v j ∂i b j =
i, j
we deduce ⎧ (i) ⎪ ⎪ ⎪ ⎪ ⎨ (ii) (iii) ⎪ ⎪ (iv) ⎪ ⎪ ⎩ (v)
vi |v|2 μ1/2 vi2 μ1/2 vi v j μ1/2 vi μ1/2 μ1/2
vi2 ∂i bi +
i
: : : : :
∇x c ∂t c + ∂i bi ∂i b j + ∂ j bi ∂t bi + ∂i a ∂t a
vi v j (∂i b j + ∂ j bi ),
i> j
= −∂t rc + lc + h c , = −∂t ri + li + h i , = −∂t ri j + li j + h i j , i = j, = −∂t rbi + lbi + h bi , = −∂t ra + la + h a ,
(7.1.1)
where r = (g2 , e) L 2v ,
l = −(v · ∇x g2 + Lg2 , e) L 2v ,
h = ((g, g), e) L 2 (R3v ) ,
stand for rc , · · · , h a , while e ∈ span{vi |v|2 μ1/2 , vi2 μ1/2 , vi v j μ1/2 , vi μ1/2 , μ1/2 }.
The Boltzmann Equation Without Angular Cutoff
567
Lemma 7.2. Let ∂ α = ∂xα , α ∈ N3 , |α| ≤ m, m ≥ 3. Then, ∂ α (a, b, c)2 L 2 (R3x ) ≤ ∇x (a, b, c) H m−1 (R3x ) (a, b, c) H m−1 (R3x ) . Proof. Let k = |α|. Then, one has for k = 0 that (a, b, c)2 L 2 (R3x ) ≤ (a, b, c) L 6 (R3x ) (a, b, c) L 3 (R3x ) ≤ ∇x (a, b, c) L 2 (R3x ) (a, b, c) H 2 (R3x ) , since 1/3
2/3
ab L 2 (R3x ) ≤ a L 6 (R3x ) b L 3 (R3x ) ≤ ∇x a L 2 (R3x ) b L ∞ (R3 ) b L 2 (R3 ) x
x
≤ ∇x a L 2 (R3x ) b H 2 (R3x ) . Also for k = 1, we have ∂(a, b, c)2 L 2 (R3x ) ≤ (a, b, c)∂(a, b, c) L 2 (R3x ) ≤ (a, b, c) L ∞ (R3x ) ∇x (a, b, c) L 2 (R3x ) ≤ (a, b, c) H m−1 (R3x ) ∇x (a, b, c) L 2 (R3x ) , and for 2 ≤ k ≤ m, ∂ α (a, b, c)2 L 2 (R3x ) ≤
∂xk (a, b, c)∂xk−k (a, b, c) L 2 (R3x )
k ≤ k2
≤
∂xk (a, b, c) L ∞ (R3x ) ∂xk−k (a, b, c) L 2 (R3x )
≤ (a, b, c) H m−1 (R3x ) ∇x (a, b, c) H m−1 (R3x ) . And this completes the proof of the lemma.
Lemma 7.3 (Estimate of r, l, h). Let ∂ α = ∂xα , ∂i = ∂xi , |α| ≤ N − 1, N ≥ 3. Then, one has ∂i ∂ α r L 2 (R3x ) + ∂ α l L 2x ≤ g2 H N (R3x , L 2 (R3v )) ≡ A1 ,
(7.1.2)
∂ α h L 2 (R3x ) ≤ ∇x (a, b, c) H N −2 (R3x ) (a, b, c) H N −1 (R3x ) +(a, b, c) H N −1 (R3x ) g2 H N −1 (R3x , L 2 (R3v )) + g2 2H N −1 (R3 , L 2 (R3 )) ≡ A2 . x
v
(7.1.3)
Proof. Equation (7.1.2) follows from ∂i ∂ α r L 2 (R3x ) ≤ (∂i ∂ α g2 , e) L 2 (R3v ) L 2 (R3x ) ≤ ∂i ∂ α g2 L 2 (R6x,v ) , and ∂ α l L 2x ≤ (∇x ∂ α g2 , ve) L 2 (R3v ) + (∂ α g2 , L∗ e) L 2 (R3v ) L 2 (R3x ) ≤ ∂xα g2 H 1 (R3x , L 2 (R3v )) .
568
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
We prove (7.1.3) as follows: 1/2 h= b(cos θ )μ∗ (g∗ g − g∗ g)edvdv∗ dσ
1/2 = b(cos θ )gg∗ (μ1/2 )∗ e − μ∗ e dvdv∗ dσ = b(cos θ )(μ1/2 g)(μ1/2 g)∗ q(v ) − q(v) dvdv∗ dσ ≡ (g, g) =
2
2
(gi , g j ) =
i, j=1
(i j) ,
i, j=1
where q = q(v) is some polynomial of v. Firstly, we have (11) = η j ηk (ψ j , ψk ), η j ,ηk ∈{a,b,c}
where ψ j (v) ∈ N . Clearly, |(ψ j , ψk )| < ∞, so that by virtue of Lemma 7.2, ∂ α (11) L 2 (R3x ) ≤ ∂ α (a, b, c)2 L 2 (R3x ) ≤ ∇x (a, b, c) H N −2 (R3x ) (a, b, c) H N −1 (R3x ) . On the other hand, (g, f ) L 2 (R3x ) ≤ μ1/2 g L 2 (R3 ;L 1 (R3 )) μ1/2 f | L 2 (R3 ;L 1 (R3 )) ≤ g L 2 (R6x,v ) f L 2 (R6x,v ) x
v
3
x
3
v
which yields for |α| ≤ N − 1, ∂ α (12) L 2 (R3x ) ≤ ∂ α (a, b, c) L 2 (R3x ) ∂ α g2 L 2 (R6x,v ) α
∂
(21)
≤ (a, b, c) H N −1 (R3x ) g2 H N −1 (R3x ,L 2 (R3v )) ,
L 2 (R3x ) ≤ ∂ α g2 L 2 (R6x,v ) ∂ α (a, b, c) L 2 (R3x )
≤ g2 H N −1 (R3x ,L 2 (R3v )) (a, b, c) H N −1 (R3x ) ,
∂ α (22) L 2 (R3x ) ≤ ∂ α g2 2L 2 (R6
x,v )
Now the proof of (7.1.3) is completed.
≤ g2 2H N −1 (R3 ,L 2 (R3 )) . v
x
Lemma 7.4 (Macro-energy estimate). Let |α| ≤ N − 1. ∇x ∂ α (a, b, c)2L 2 (R3 ) x d α (∂ r, ∇x ∂ α (a, −b, c)) L 2 (R3x ) + (∂ α b, ∇x ∂ α a) L 2 (R3x ) ≤− dt +g2 2H N (R3 ,L 2 (R3 )) + D1 E1 ,
(7.1.4)
v
x
where D1 = ∇x (a, b, c)2H N −1 (R3 ) + g2 2H N (R3 ,L 2 (R3 )) x
x
v
is a dissipation rate and E1 = (a, b, c)2H N −1 (R3 ) + g2 2H N −1 (R3 ,L 2 (R3 )) = g2H N −1 (R3 ,L 2 (R3 )) x
is an instant energy functional.
x
v
x
v
The Boltzmann Equation Without Angular Cutoff
569
Proof. (a) Estimate of ∇x ∂ α a. Let A1 , A2 be as in Lemma 7.3. From (7.1.1) (iv), ∇x ∂ α a2L 2 (R3 ) = (∇x ∂ α a, ∇x ∂ α a) L 2 (R3x ) x
= (∂ α (−∂t b − ∂t r + l + h), ∇x ∂ α a) L 2 (R3x ) ≤ R1 + Cη (A21 + A22 ) + η∇x ∂ α a2L 2 (R3 ) .
R1 = −(∂ α ∂t b + ∂ α ∂t r, ∇x ∂ α a) L 2 (R3x )
x
d α (∂ (b + r ), ∇x ∂ α a) L 2 (R3x ) + (∇x ∂ α (b + r ), ∂t ∂ α a) L 2 (R3x ) dt d ≤ − (∂ α (b + r ), ∇x ∂ α a) L 2 (R3x ) dt +Cη (∇x ∂ α b2L 2 (R3 ) + A21 ) + η∂t ∂ α a2L 2 (R3 ) ,
=−
x
x
(b) Estimate of ∇x ∂ α b. From (7.1.1) (iii) and (ii), ∂ j ∂ α (∂ j bi +∂i b j )+∂i ∂ α (2∂i bi − ∂i b j ) x ∂ α bi + ∂i2 ∂ α bi = j=i
j=i α
∇x ∂
α
b2L 2 (R3 ) x
+ ∂i ∂
α
b2L 2 (R3 ) x
= ∂i ∂ (−∂t r + l + h), = −(x ∂ α bi + ∂i2 ∂ α bi , ∂ α b) L 2 (R3x ) = R2 + R3 + R4 ,
where R2 = (∂t ∂ α r, ∂i ∂ α b) L 2 (R3x ) =
d α (∂ r, ∂i ∂ α b) L 2 (R3x ) + (∂i ∂ α r, ∂t ∂ α b) L 2 (R3x ) dt
d α (∂ r, ∂i ∂ α b) L 2 (R3x ) + Cη A21 + η∂t ∂ α b2L 2 (R3 ) , x dt α α 2 α 2 R3 = −(∂ l, ∂i ∂ b) L 2 (R3x ) ≤ Cη A1 + η∂i ∂ b L 2 (R3 ) , ≤
x
R4 = −(∂ α h, ∂i ∂ α b) L 2 (R3x ) ≤ Cη A22 + η∂i ∂ α b2L 2 (R3 ) . x
(c) Estimate of ∇x ∂ α c. From (7.1.1) (i), ∇x ∂ α c2L 2 (R3 ) = (∇x ∂ α c, ∇x ∂ α c) L 2 (R3x ) = (∂ α (−∂t r + l + h), ∇x ∂ α c) L 2 (R3x ) x
≤ R5 + Cη (A21 + A22 ) + η∇x ∂ α c2L 2 (R3 ) , x
where R5 = −(∂ α ∂t r, ∇x ∂ α c) L 2 (R3x ) = −
d α (∂ r, ∇x ∂ α c) L 2 (R3x ) + (∇x ∂ α r, ∂t ∂ α c) L 2 (R3x ) dt
d α (∂ r, ∇x ∂ α c) L 2 (R3x ) + Cη A21 + η∂t ∂ α c2L 2 (R3 ) . x dt α (d) Estimate of ∂t ∂ (a, b, c). ≤−
∂t ∂ α (a, b, c) L 2 (R3x ) = ∂ α ∂t Pg L 2 (R6x,v )
= ∂ α P (−v · ∇x g − Lg + (g, g)) L 2 (R6x,v ) = ∂ α P(v · ∇x g) L 2 (R6x,v )
≤ ∇x ∂ α (a, b, c) L 2 (R3x ) + ∇x ∂ α g2 L 2 (R6x,v ) .
(7.1.5)
570
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Combining all the above estimates and taking η > 0 sufficiently small, we deduce ∇x ∂ α (a, b, c)2L 2 (R3 ) x d α (∂ r, ∇x ∂ α (a, −b, c)) L 2 (R3x ) + (∂ α b, ∇x ∂ α a) L 2 (R3x ) ≤− dt +A21 + A22 + η∇x ∂ α g2 2L 2 (R6 ) . x,v
Finally, choosing |α| ≤ N − 1, and using Lemma 7.3, we obtain A21 + A22 + η∇x ∂ α g2 2L 2 (R6
x,v )
≤ D1 E1 + ηg2 2H N (L 2 (R3 )) ,
which completes the proof of Lemma 7.4.
x
v
7.2. Microscopic energy estimate. In this subsection, We shall use Lemma 2.6 and Proposition 3.5 to estimate the microscopic component. Step 1. Let α ∈ N3 , |α| ≤ N , and apply ∂ α = ∂xα to (1.3) to have, ∂t (∂ α g) + v · ∇x (∂ α g) + L(∂ α g) = ∂ α (g, g), and take the L 2 (R6x,v ) inner product with ∂ α g. By Lemma 2.6, we have 1 d α 2 ∂ g L 2 (R6 ) + D1 ≤ J, x,v 2 dt
(7.2.1)
where D1 is a dissipation rate |||∂ α g2 |||2 d x = |||∂ α g2 |||2B0 (R6 ) , D1 = R3
0
and J is given by J = (∂ α (g, g), ∂ α g) L 2 (R6 ) =
2
(∂ α (gi , g j ), ∂ α g2 ) L 2 (R6 )
i, j=1
=
2
J (i j) .
i, j=1
First, consider J (11) . We have, with ψ j ∈ N , |J (11) | ≤ ∂ α (g1 , g1 ) L 2 (R6 ) ∂ α g2 L 2 (R6 )
≤ ∂ α (g1 , g1 ) L 2 (R3v ) L 2 (R3x ) ∂ α g2 L 2 (R6 ) , ∂ α (g1 , g1 ) L 2 (R3v ) ≤ |∂ α (η j ηk )|(ψ j , ψk ) L 2 (R3v ) , η j ,ηk ∈{a,b,c}
The Boltzmann Equation Without Angular Cutoff
571
(ψ j , ψk )2L 2 (R3 ) v 2 1/2 = b(cos θ )μ∗ {(ψ j )∗ ψk − (ψ j )∗ ψk }dv∗ dσ dv
=
μ
b(cos θ )μ∗ {( p j )∗ pk
2 − ( p j )∗ pk }dv∗ dσ
dv < ∞,
(7.2.2)
where p j ∈ {1, v, |v|2 }. Consequently, by virtue of Lemma 7.2, |J (11) | ≤ ∂ α (a, b, c)2 L 2 (R3x ) ∂ α g2 L 2 (R6 ) ≤ ∇x (a, b, c) H N −2 (R3x ) (a, b, c) H N −1 (R3x ) g2 H N (R3x ,L 2 (R3v ))
≤ (a, b, c) H N −1 (R3x ) ∇x (a, b, c)2H N −2 (R3 ) + |||g2 |||2B N (R6 ) . x
0
On the other hand, using Proposition 3.5 gives |J (12) | ≤ g1 H N (R6 ) |||g2 |||B N (R6 ) |||g2 |||B N (R6 ) 0
0
0
≤ ||(a, b, c)|| H N (R3x ) |||g2 |||2B N (R6 ) , 0
|J (21) | ≤ g2 H N (R6 ) |||g1 |||B N (R6 ) |||g2 |||B N (R6 ) 0
0
0
≤ ||g2 || H N (R6 ) ||(a, b, c)|| H N (R3x ) |||g2 |||B N (R6 ) , 0
0
|J (22) | ≤ g2 H N (R6 ) |||g2 |||B N (R6 ) |||g2 |||B N (R6 ) 0
0
0
≤ ||g2 || H N (R6 ) |||g2 |||2B N (R6 ) . 0
0
Taking the summation of (7.2.1) for |α| ≤ N , N ≥ 3, we have Lemma 7.5. Let N ≥ 3. Then, d 1/2 g2H N (R3 ,L 2 (R3 )) + |||g2 |||2B N (R6 ) ≤ D2 E2 , x v dt 0
(7.2.3)
where D2 = ∇x (a, b, c)2H N −1 (R3 ) + |||g2 |||2B N (R6 ) , x
E2 =
g2H N (R3 ,L 2 (R3 )) x v
0
∼ (a, b, c)2H N (R3 ) + g2 2H N (R3 ,L 2 (R3 )) . x
x
v
Step 2. Let ∂ α = ∂xα , 1 ≤ |α| ≤ N , N ≥ 3, and apply W ∂ α to (1.3). We have ∂t (W ∂ α g) + v · ∇x (W ∂ α g) + L(W ∂ α g) = S1 + S2 , where S1 = W ∂ α (g, g),
S2 = [L, W ](∂ α g).
Take the L 2 (R6x,v ) inner product with W ∂ α g to deduce 1 d W ∂ α g2L 2 (R6 ) + D2 ≤ G, 2 dt
(7.2.4)
572
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
where D2 is a dissipation rate |||(I − P)W ∂ α g|||2 d x D2 = R3
1 ≥ |||∂ α g|||2B0 (R6 ) − (∇x (a, b, c) H N −1 (R3x ) + ||g2 ||2H N (R3 ,L 2 (R3 )) ). x v 2 Here we have used for 1 ≤ |α| ≤ N , |||PW ∂ α g|||2 d x ≤ ||∂ α g||2L 2 (R6
x,v )
R3
≤ ∇x (a, b, c)2H N −1 (R3 ) + ||g2 ||2H N (R3 ,L 2 (R3 )) . x
x
v
On the other hand, G is defined by G = G 1 + G 2 , G i = (Si , W ∂ α g) L 2 (R6 ) ,
i = 1, 2.
The estimation on G 1 and G 2 will be given in the following lemmas. Lemma 7.6. Let N ≥ 3, ≥ 3. Then, for E and D defined in Theorem 7.1, we have G 1 ≤ E 1/2 D. Proof. First, write
G1 =
(W ∂ α (gi , g j ), W ∂ α g) L 2 (R6 ) =
i, j=1,2
(i j)
G1 .
i, j=1,2
Proceeding as in (7.2.2), (11)
|G 1 | ≤ W 2 ∂ α (g1 , g1 ) L 2 (R6 ) ∂ α g L 2 (R6 )
≤ W 2 ∂ α (g1 , g1 ) L 2 (R3v ) L 2 (R3x ) ∂ α g L 2 (R6 ) , ∼ ∂ α {(a, b, c)2 } L 2 (R3x ) W 2 (ψ j , ψk ) L 2 (R3v ) ∂ α g L 2 (R6 ) , W 2 (ψ j , ψk )2L 2 (R3 ) v 2 1/2 W 2 = b(cos θ )μ∗ {(ψ j )∗ ψk − (ψ j )∗ ψk }dv∗ dσ dv
=
μW
4
b(cos θ )μ∗ {( p j )∗ pk
2 − ( p j )∗ pk }dv∗ dσ
dv < ∞.
Since 1 ≤ |α| ≤ N , (11)
|G 1 | ≤ ∇x (a, b, c) H N −1 (R3x ) (a, b, c) H N −1 (R3x ) ×(∇x (a, b, c) H N −1 (R3x ) + ||g2 || H N (R3x ,L 2 (R3v )) ). On the other hand, we have 2 |||g1 |||B N (R6 ) =
3 |β|≤N Rx
β |||W ∂x,v g1 (x, · )|||2 d x ≤ (a, b, c)2H N (R3 ) . x
(7.2.5)
The Boltzmann Equation Without Angular Cutoff
573
This fact and Proposition 3.5 yield α |G (12) 1 | ≤ ||g1 || H N (R6 ) |||g2 |||B N (R6 ) |||W ∂ g|||B0 (R6 )
0
≤ ||(a, b, c)|| H N (R3x ) |||g2 |||B N (R6 ) |||∂ α g|||B0 (R6 ) ,
(21)
|G 1 | ≤ ||g2 || H N (R6 ) |||g1 |||B N (R6 ) |||W ∂ α g|||B0 (R6 )
0
≤ g2 H N (R6 ) ||(a, b, c)|| H N (R3x ) |||∂ α g|||B0 (R6 ) , (22) |G 1 |
α
≤ ||g2 || H N (R6 ) |||g2 |||B N (R6 ) |||W ∂ g|||B0 (R6 )
≤ g2 H N (R6 ) |||g2 |||B N (R6 ) |||∂ α g|||B0 (R6 ) .
Noticing that for 1 ≤ |α| ≤ N , |||∂ α g|||2B0 (R6 ) ≤ |||W ∂ α g1 (x, · )|||2 d x +
0
R3x
R3x
|||W ∂ α g2 (x, · )|||2 d x
≤ ∇x (a, b, c)2H N −1 (R3 ) + |||g2 |||B N (R6 ) ,
x
we now conclude the proof of the lemma.
We shall evaluate G 2 . In view of Proposition 3.8,
α α |G 2 | ≤ [L1 , W ]∂ g, W ∂ g 2 6 L (R )
α α + W L2 (∂ g), W ∂ g 2 6 + L2 (W ∂ α g), W ∂ α g α
L (R )
α
L 2 (R 6 )
≤ ||∂ g|| L 2 (R6 ) |||∂ g|||B0 (R6 ) ≤ (∇x (a, b, c) H N −1 (R3x ) + ∂ α g2 L 2 (R6 ) )|||∂ α g|||B0 (R6 )
≤ ∇x (a, b, c)2H N −1 (R3 ) + Cη ||∂ α g2 ||2L 2 (R6 ) + η|||∂ α g|||2B0 (R6 ) , (η > 0). x
Thus, we have established Lemma 7.7. Let 1 ≤ |α| ≤ N , N ≥ 3. Then, d α 2 ∂ g L 2 (R6 ) + |||∂ α g|||2B0 (R6 ) dt ≤ E 1/2 D + ||∂ α g2 ||2L 2 (R6 ) + ∇x (a, b, c)2H N −1 (R3 ) . x
(7.2.6)
Step 3. We need also to estimate W g2 . Apply W (I − P) to the equation in (1.3) to have ∂t (W g2 ) + v · ∇x (W g2 ) + L(W g2 ) = W (g, g) + W [v · ∇x , P]g + [L, W ]g2 . And then take the inner product with W g2 to get d W g2 2L 2 (R6 ) + D3 ≤ H, dt
574
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
where
D3 = ≥
R3
|||(I − P)W g2 |||2 d x
1 |||g2 |||2B0 (R6 ) − Cg2 L 2 (R3 ) , 2
while H = H1 + H2 + H3 , H1 = (W (g, g), W g2 ) L 2 (R6 ) , H2 = (W [v · ∇x , P]g, W g2 ) L 2 (R6 ) , H3 = ([L, W ]g2 , W g2 ) L 2 (R6 ) . (i j) H1 = (W (gi , g j ), W g2 ) L 2 (R6 ) = H1 . i, j=2
i, j=2
Proceeding as in (7.2.5), (11)
|H1
| ≤ W 2 (g1 , g1 ) L 2 (R6 ) g2 L 2 (R6 )
≤ W 2 (g1 , g1 ) L 2 (R3v ) L 2 (R3x ) g2 L 2 (R6 ) ∼ (a, b, c)2 L 2 (R3x ) W 2 (ψ j , ψk ) L 2 (R3v ) g2 L 2 (R6 ) ,
W 2 (ψ j , ψk )2L 2 (R3 ) v 2 1/2 2 W = b(cos θ )μ∗ {(ψ j )∗ ψk − (ψ j )∗ ψk }dv∗ dσ dv
=
μW 4
b(cos θ )μ∗ {( p j )∗ pk − ( p j )∗ pk }dv∗ dσ
2 dv < ∞.
Then we have, by using Lemma 7.2, (11)
|H1
| ≤ ∇x (a, b, c) L 2 (R3x ) (a, b, c) H 1 (R3x ) g2 L 2 (R6 ) .
On the other hand, we have |||W g1 (x, · )|||2 d x ≤ (a, b, c)2L 2 (R3 ) . |||g1 |||2B0 (R6 ) = R3x
x
This fact and Proposition 3.5 yield (12)
|H1
| ≤ ||g1 || H N (R6 ) |||g2 |||B N (R6 ) |||W g2 |||B0 (R6 )
0
≤ ||(a, b, c)|| H N (R3x ) |||g2 |||2B N (R6 ) , (21) |H1 |
≤ ||g2 || H N (R6 ) |||g1 |||B N (R6 ) |||W g2 |||B0 (R6 )
0
≤ g2 H N (R6 ) ||(a, b, c)|| H N (R3x ) |||g2 |||B0 (R6 ) , (22) |H1 |
≤ ||g2 || H N (R6 ) |||g2 |||B N (R6 ) |||W g2 |||B0 (R6 )
≤ g2 H N (R6 ) |||g2 |||B N (R6 ) . 2
0
The Boltzmann Equation Without Angular Cutoff
575
And H2 is evaluated as follows: |H2 | ≤ |(W 2 [v · ∇x , P]g, g2 ) L 2 (R6 ) | ≤ ∇x g L 2 (R6 ) g2 L 2 (R6 ) ≤ ∇x (a, b, c)2L 2 (R3 ) + g2 2H 1 (R3 ;L 2 (R3 )) . x
v
Finally, in view of Proposition 3.8 ,
|H3 | ≤ [L1 , W ]g2 , W g2 2 6 L (R )
+ W L2 (g2 ), W g2 2 6 L
+ L2 (W g2 ), W g2 2 6 (R ) L (R )
≤ g2 L 2 (R6 ) |||g2 |||B0 (R6 ) .
Since it holds by interpolation inequality that g2 L 2 (R6 ) ≤ ηg2 L 2 (R6 ) + Cη g2 L 2 (R6 ) +s ≤ η|||g2 |||B0 (R6 ) + Cη g2 L 2 (R6 ) ,
(7.2.7)
for any small enough η > 0, we have established Lemma 7.8. d g2 2L 2 (R6 ) + |||g2 |||2B0 (R6 ) dt ≤ E 1/2 D + ||g2 ||2L 2 (R6 ) + ∇x (a, b, c)2L 2 (R3 ) . x
Step 4. Let β ∂ β = ∂x,v = ∂ α ∂ γ , ∂ α = ∂xα , ∂ γ = ∂vγ , |β| = |α| + |γ | ≤ N , γ = 0, N ≥ 3,
and apply W ∂ β (I − P) to (7.2.4) to have ∂t (W ∂ β g2 ) + v · ∇x (W ∂ β g2 ) + L(W ∂ β g2 ) = W ∂ β (g, g) + [v · ∇x , W ∂ β ]g2 −W ∂ β [P, v · ∇]g + [L, W ∂ β ]g2 + W ∂ β (∂t + v · ∇x )g1 . And then take the L 2 (R6x,v ) inner product with W ∂ β g2 to get 1 d β ∂ g2 2L 2 (R6 ) + D4 ≤ K . 2 dt Here D4 is a dissipation rate given by |||(I − P)W ∂ β g2 |||2 d x D4 = R3
1 ≥ |||∂ β g2 |||2B0 (R6 ) − C||∂ α g2 ||2L 2 (R6 ) . 2
(7.2.8)
576
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
γ where we used, with ψ ∈ N and ψ˜ = (−1)|γ | ∂v (W ψ) where β = α + γ and γ β α ∂ = ∂ x ∂v ,
|||PW ∂ β g2 |||2 = |||∂ α (ψ, W ∂ γ g2 ) L 2 (R3v ) ψ|||2
˜ ∂ α g2 ) L 2 2 2 3 |||ψ|||2 ≤ ||∂ α g2 ||2 2 6 . = (ψ, L (R ) v L (R ) x
x,v
On the other hand, K is given by K = (W ∂ β (g, g), W ∂ β g2 ) L 2 (R6 ) +([v · ∇x , W ∂ β ]g2 , W ∂ β g2 ) L 2 (R6 ) −(W ∂ β [P, v · ∇]g, W ∂ β g2 ) L 2 (R6 ) +([L, W ∂ β ]g2 , W ∂ β g2 ) L 2 (R6 ) +(W ∂ β (∂t + v · ∇x )g1 , W ∂ β g2 ) L 2 (R6 ) = K1 + K2 + K3 + K4 + K5. Lemma 7.9. Let N ≥ 3. Then |K 1 | ≤ E 1/2 D. Proof. First, write K1 =
(W ∂ β (gi , g j ), W ∂ β g2 ) L 2 (R6 ) =
i, j=2
(i j)
K1 .
i, j=2
In view of Lemma 7.2, (11) |K 1 | = (W ∂ β (g1 , g1 ), W ∂ β g2 ) L 2 (R6 ) ≤ B2 ∂ α (a, b, c)2 L 2 (R3x ) W ∂ β g2 L 2 (R6 ) ≤ ∇x (a, b, c) HxN −1 (a, b, c) HxN −1 W ∂ β g2 L 2 (R6 ) , where B22 ∼ W ∂vγ (ψ j , ψk )2L 2 (R3 ) v 1/2 W = b(cos θ )(∂vγ1 μ)∗ 2 × (∂vγ2 ψ j )∗ (∂vγ3 ψk ) − (∂vγ2 ψ j )∗ (∂vγ3 ψk ) dv∗ dσ dv 2 2 = μW b(cos θ )μ∗ q∗ (q j )∗ qk − (q j )∗ qk dv∗ dσ dv < ∞. Here, q, q j , qk are polynomials of v. On the other hand, we have 2 |||W ∂ β g1 (x, · )|||2 d x ≤ (a, b, c)2H N (R3 ) . |||g1 |||B N (R6 ) =
|β|≤N
R3x
x
The Boltzmann Equation Without Angular Cutoff
577
This point and Proposition 3.5 yield (12)
|K 1
| ≤ ||g1 || H N (R6 ) |||g2 |||B N (R6 ) |||W ∂ β g2 |||B0 (R6 )
0
≤ ||(a, b, c)|| H N (R3x ) |||g2 |||B N (R6 ) , 2
(21) |K 1 |
= ||g2 || H N (R6 ) |||g1 |||B N (R6 ) |||W ∂ β g2 |||B0 (R6 )
0
≤ g2 H N (R6 ) ||(a, b, c)|| H N (R3x ) |||g2 |||B N (R6 ) , (22)
|K 1
| = ||g2 || H N (R6 ) |||g2 |||B N (R6 ) |||W ∂ β g2 |||B0 (R6 )
0
≤ g2 H N (R6 ) |||g2 |||2B N (R6 ) .
Now the proof of the lemma is completed.
K 2 , K 3 , K 4 , K 5 are estimated as follows. We have, for |β| = |α| + |γ | ≤ N , γ = 0, |K 2 | = ([v · ∇x , W ∂ β ]g2 , W ∂ β g2 ) L 2 (R6 ) ≤ W ∂xα+1 ∂vγ −1 g2 L 2 (R6 ) W ∂ β g2 L 2 (R6 ) ≤ Cη W ∂xα+1 ∂vγ −1 g2 2L 2 (R6 ) + ηW ∂ β g2 2L 2 (R6 ) . Note that |K 3 | = |(W ∂ β [P, v · ∇]g, W ∂ β g2 ) L 2 (R6 ) , W ∂ β g2 ) L 2 (R6 ) | ≤ |(∂ γ W 2 ∂ β [P, v · ∇]g , ∂ α g2 ) L 2 (R6 ) |
≤ ∇x ∂ α (a, b, c) L 2 (R3x ) + ∇x ∂ α g2 L 2 (R6x,v ) ∂ α g2 L 2 (R6x,v ) ≤ ∇x (a, b, c)2H N −1 (R3 ) + g2 2H N (R3 ,L 2 (R3 )) . x
x
v
In view of Proposition 3.8, |K 4 | = ([L, W ∂ β ]g2 , W ∂ β g2 ) L 2 (R6 )
β ≤ [L1 , W ∂x,v ]g2 , W ∂ β g2 2 6 L (R )
β β + W ∂x,v L2 (g2 ), W ∂ β g2 2 6 + L2 (W ∂x,v g2 ), W ∂ β g2 2 6 L (R ) L (R )
≤ ||g2 || H |β| (R6 ) + |||g2 |||B|β|−1 (R6 ) |||W ∂ β g2 |||B0 (R6 )
0
Hence |K 4 | ≤ Cη ||g2 ||2
|β|
H (R6 )
+ |||g2 |||2 |β|−1 B
(R 6 )
+ η|||W ∂ β g2 |||2B0 (R6 ) . 0
578
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
Finally, recalling (7.1.5), |K 5 | = (W ∂ β (∂t + v · ∇x )g1 , W ∂ β g2 ) L 2 (R6 ) ≤ |(∂ γ W 2 ∂ β (∂t + v · ∇x )g1 , ∂ α g2 ) L 2 (R6 ) | ≤ (∂ α (∂t + ∇x )(a, b, c) L 2 (R3x ) ∂ α g2 L 2 (R6 ) , (|α| ≤ N − 1, |γ | ≥ 1)
≤ ∇x ∂ α (a, b, c) L 2 (R3x ) + ∇x ∂ α g2 L 2 (R6x,v ) ∂ α g2 L 2 (R6x,v ) ≤ ∇x (a, b, c)2
HxN −1
+ g2 2H N (R3 ,L 2 (R3 )) . v
x
Now using (7.2.7) we conclude the Lemma 7.10. Let |β| = |α + γ | ≤ N , |α| ≤ N − 1, |γ | ≥ 1, N ≥ 3. Then, d β ∂ g2 2L 2 (R6 ) + |||∂ β g2 |||2B0 (R6 ) dt ≤ E 1/2 D + ||g2 ||2 |β| 6 H (R )
+|||g2 |||
2
|α|+|γ |−1
B
(R 6 )
+ ∇x (a, b, c)2H N −1 (R3 ) + g2 2H N (R3 ,L 2 (R3 )) . (7.2.9) x
7.3. A Priori estimate. We take the linear combination Cα(1) (7.1.4)α + Cα(2) (7.2.3)α + |α|≤N −1
+C
|α|≤N
(4)
(7.2.8) +
x
v
Cα(3) (7.2.6)α
1≤|α|≤N
|β|=|α+γ |≤N ,|α|≤N −1,|γ |≥1 (1)
(2)
(3) Cα,γ ((7.2.9)α,γ . (3)
(5)
With a suitable choice of the coefficients Cα , Cα , Cα , C (4) , Cα,γ , we get d ˜ ˜ E + D ≤ H, dt where E˜ = −
|α|≤N −1
+
|α|≤N
. Cα(1) (∂ α r, ∇x ∂ α (a, −b, c)) L 2 (R3x ) + (∂ α b, ∇x ∂ α a) L 2 (R3x )
1≤|α|≤N
+C (4) g2 2L 2 (R6 ) +
D˜ =
|α|≤N −1
+
1≤|α|≤N
1/2
|β|=|α+γ |≤N ,|α|≤N −1,|γ |≥1
+
|α|≤N
(5) β Cα,γ W ∂x,v g2 2L 2 (R6 ) ,
Cα(2) |||∂xα g2 |||2B0 (R6 ) 0
Cα(3) |||∂xα g2 |||2B0 (R6 ) + C (4) |||g2 |||2B0 (R6 )
|β|=|α+γ |≤N ,|α|≤N −1,|γ |≥1
H = D1 E1
Cα(3) ∂xα g2L 2 (R6 )
Cα(1) ∇x ∂xα (a, b, c)2L 2 (R3 ) x
+
Cα(2) ∂xα g2L 2 (R6 ) +
(7.3.1)
1/2
+ D2 E2
+ DE 1/2 .
(3) β Cα,γ |||∂x,v g2 |||2B0 (R6 ) ,
The Boltzmann Equation Without Angular Cutoff
579
Clearly, it holds that ˜ D ∼ D, ˜ E 1/2 ∼ E, and H ≤ DE. Now (7.3.1) gives /
0
E 1/2 (t) + 1 − C sup E(τ ) 0≤τ ≤t
t
D(τ )dτ ≤ CE(0),
0
which leads to the closure of à priori estimate and then completes the proof of Theorem 7.1. Now, the proof of Theorem 1.1 can be completed by the usual continuation argument based on Theorem 4.3 and Theorem 7.1. Acknowledgements. The last author’s research was supported by the General Research Fund of Hong Kong, CityU#103109. Finally the authors would like to thank the City University of Hong Kong, Kyoto University and Wuhan University for financial support during each of their stays, mainly starting from 2006. These supports have enabled the final conclusion through our previous papers.
References 1. Alexandre, R.: Remarks on 3D Boltzmann linear operator without cutoff. Transp. Th. Stat. Phys. 28-5, 433–473 (1999) 2. Alexandre, R.: Around 3D Boltzmann operator without cutoff. A New formulation. Math. Mod. Num. Anal. 343, 575–590 (2000) 3. Alexandre, R.: Some solutions of the Boltzmann equation without angular cutoff. J. Stat. Phys. 104, 327–358 (2001) 4. Alexandre, R.: Integral kernel estimates for a linear singular operator linked with Boltzmann equation. Part I: Small singularities 0 < ν < 1. Indiana Univ. Math. J. 55(6), 1975–2021 (2006) 5. Alexandre, R.: A review of Boltzmann equation with singular kernels. Kin. Rel. Mod. 2(4), 551–646 (2009) 6. Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Rat. Mech. Anal. 152, 327–355 (2000) 7. Alexandre, R., ElSafadi, M.: Littlewood-Paley decomposition and regularity issues in Boltzmann homogeneous equations.I. Non cutoff and Maxwell cases. Math. Mod. Meth. Appl. Sci. 15, 907–920 (2005) 8. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Uncertainty principle and regularity for Boltzmann type equations. C. R. Acad. Sci. Paris, Ser. I 345, 673–677 (2007) 9. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Uncertainty principle and kinetic equations. J. Funct. Anal. 255, 2013–2066 (2008) 10. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Regularity of solutions for the Boltzmann equation without angular cutoff. C. R. Math. Acad. Sci. Paris, Ser. I 347(13–14), 747–752 (2009) 11. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Local existence for non-cutoff Boltzmann equation. C. R. Math. Acad. Sci. Paris, Ser. I 347(21–22), 1237–1242 (2009) 12. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Regularizing effect and local existence for non-cutoff Boltzmann equation. Arch. Rat. Mech. Anal. 198(Issue 1), 39–123 (2010) 13. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. J. Funct. Anal. http://hal.archives-ouvertes.fr/ hal-00496950/fr/ (To appear) 14. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential. Anal. Appl. http://hal.archives-ouvertes.fr/hal00510633/fr/, 2010 (To appear)
580
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang
15. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Qualitative properties of solutions to the Boltzmann equation without angular cutof. Arch. Rational Mech. Anal. (To appear) 16. Alexandre, R., Villani, C.: On the Boltzmann equation for long-range interaction. Comm. Pure Appl. Math. 55, 30–70 (2002) 17. Arkeryd, L.: Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rat. Mech. Anal. 77(1), 11–21 (1981) 18. Bobylev, A.: The theory of nonlinear, spatially uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev. C. Math. Phys. 7, 111–233 (1988) 19. Bouchut, F., Desvillettes, L.: A proof of smoothing properties of the positive part of Boltzmann’s kernel. Rev. Mat. Iberoam. 14, 47–61 (1998) 20. Bouchut, F., Golse, F., Pulvirenti, M.: Kinetic equations and asymptotic theory. Series in Appl. Math., Paris: Gauthiers-Villars, 2000 21. Boudin, L., Desvillettes, L.: On the singularities of the global small solutions of the full Boltzmann equation. Monat. für Math. 131, 91–108 (2000) 22. Cercignani, C.: The Boltzmann equation and its applications. Applied Mathematical Sciences, 67, Berlin-Heidelberg-NewYork: Springer-Verlag, 1988 23. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical theory of Dilute gases. Applied Mathematical Sciences, 106. New York: Springer-Verlag, 1994 24. Chen, H., Li, W.-X., Xu, C.-J.: The Gevrey hypoellipticity for a class of kinetic equations Comm. PDE. 36, 693–728 (2011), doi:10.1080/03605302.2010.507689 25. Chen, Y., Desvillettes, L., He, L.: Smoothing effects for classical solutions of the full Landau equation. Arch. Rat. Mech. Anal. 193, 21–55 (2009) 26. Desvillettes, L.: About the regularization properties of the non cut-off Kac equation. Commun. Math. Phys. 168, 417–440 (1995) 27. Desvillettes, L.: Regularization for the non Cutoff 2D Radially Symmetric Boltzmann Equation with a Velocity Dependant Cross Section. Trans. Th. Stat. Phy. 25(3–5), 383–394 (1996) 28. Desvillettes, L.: Regularization Properties of the 2-Dimensional Non Radially Symmetric Non Cutoff Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules. Trans. Th. Stat. Phys. 26(3), 341–357 (1997) 29. Desvillettes, L.: About the use of the Fourier transform for the Boltzmann equation. Summer School on “Methods and Models of Kinetic Theory” (M& MKT 2002). Riv. Mat. Univ. Parma 7(2), 1–99 (2003) 30. Desvillettes, L., Furioli, G., Terraneo, E.: Propagation of Gevrey regularity for solutions of Boltzmann equation for Maxwellian molecules. Trans. Amer. Math. Soc. 361, 1731–1747 (2009) 31. Desvillettes, L., Villani, C.: On the spatially homogeneous Landau equation for hard potentials. Part I: existence, uniqueness and smoothness. Comm. Part. Diff. Eqs. 25-1-2, 179–259 (2000) 32. Desvillettes, L., Wennberg, B.: Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm. Part. Diff. Eqs. 29-1-2, 133–155 (2004) 33. Desvillettes, L., Golse, F.: On a model Boltzmann equation without angular cutoff. Diff. Int. Eqs. 13 (4-6), 567–594 (2000) 34. DiPerna, R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130, 321–366 (1989) 35. Duan, R.-J.: Stability of the Boltzmann equation with potential forces on torus. Physica D 238, 1808–1820 (2009) 36. Duan, R.-J., Li, M.-R., Yang, T.: Propagation of singularities in the solutions to the Boltzmann equation near equilibrium. Math. Mod. Meth. Appl. Sci. 18, 1093–1114 (2008) 37. Duan, R.-J., Ukai, S., Yang, T.: A combination of energy method and spectral analysis for study of equations of gas motion. Front. Math. China 4(2), 253–282 (2009) 38. Fefferman, C.: The uncertainty principle. Bull. Amer. Math. Soc. 9, 129–206 (1983) 39. Grad, H.: Asymptotic Theory of the Boltzmann Equation II. In: Rarefied Gas Dynamics, J. A. Laurmann, ed, vol. 1, New York: Academic Press, 1963, pp. 26–59 40. Grad, H.: Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Symp. Appl. Math. Vol. 17, Providence, RI: Amer. Math. Soc., 1965, editor R. Finn, pp. 154-183 41. Gressman, P.-T., Strain, R.-M.: Global strong solutions of the Boltzmann equation without cutoff. http://www.math.upenn.edu/~strain/preprints/gsNoncut1.pdf, 2009 42. Gressman, P.-T., Strain, R.-M.: Global classical solutions of the Boltzmann equation with long range potential. Proc. Nat. Acad. Sci. U.S.A 107(13), 5744–5749 (2010) 43. Gressman, P.-T., Strain, R.-M.: Global classical solutions of the Boltzmann equation with long range interations and soft potentials. Preprint 2010, avaliable at http://arXiv.org/abs/1002.3639v2 [math.AP], 2010 44. Gressman, P.-T., Strain, R.-M.: Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Preprint 2010, http://arXiv.org/abs/1007.1276v2 [math.AP], 2010 45. Guo, Y.: The Landau equation in a periodic box. Comm. Math. Phys. 231, 391–434 (2002)
The Boltzmann Equation Without Angular Cutoff
581
46. Guo, Y.: The Boltzmann equation in the whole space. Indiana Univ. Maths. J. 53(4), 1081–1094 (2004) 47. Huo, Z.H., Morimoto, Y., Ukai, S., Yang, T.: Regularity of solutions for spatially homogeneous Boltzmann equation without Angular cutoff. Kin. Rel. Mods. 1, 453–489 (2008) 48. Kaniel, S., Shinbrot, M.: The Boltzmann equation. I. Uniqueness and local existence. Commun. Math. Phys. 59, 65–84 (1978) 49. Lions, P.L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications, I, II, and III. J. Math. Kyoto Univ., 34, 391–427, 429–461, 539–584 (1994) 50. Lions, P.L.: On Boltzmann and Landau equations. Philos. Trans. Roy. Soc. London Ser. A 346(1679), 191–204 (1994) 51. Lions, P.L.: Regularity and compactness for Boltzmann collision operator without angular cut-off. C. R. Acad. Sci. Paris Séries I 326, 37–41 (1998) 52. Liu, T.-P., Yu, S.-H.: Micro-macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1), 133–179 (2004) 53. Liu, T.-P., Yang, T., Yu, S.-H.: Energy method for Boltzmann equation. Phys. D 188, 178–192 (2004) 54. Liu, T.-P., Yang, T., Yu, S.-H., Zhao, H.-J.: Nonlinear stability of rarefaction waves for the Boltzmann equation. Arch. Ration. Mech. Anal. 181, 333–371 (2006) 55. Lu, X.A.: Direct method for the regularity of the gain term in the Boltzmann equation. J. Math. Anal. Appl. 228, 409–435 (1998) 56. Morimoto, Y.: The uncertainty principle and hypoelliptic operators. Publ. RIMS Kyoto Univ. 23, 955–964 (1987) 57. Morimoto, Y.: Estimates for degenerate Schrödinger operators and hypoellipticity for infinitely degenerate elliptic operators. J. Math. Kyoto Univ. 32, 333–372 (1992) 58. Morimoto, Y., Morioka, T.: The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators. Bull. Sc. Math. 121, 507–547 (1997) 59. Morimoto, Y., Ukai, S.: Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff. J. Pseudo-Dier. Oper. Appl. 1, 139–159 (2010) 60. Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Regularity of solutions to the spatially homogeneous Boltzmann equation without Angular cutoff. Disc. Cont. Dyn. Sys. Series A 24(1), 187–212 (2009) 61. Morimoto, Y., Xu, C.-J.: Hypoelliticity for a class of kinetic equations. J. Math. Kyoto Univ. 47, 129–152 (2007) 62. Mouhot, C.: Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Commun. Part. Diff. Eqs. 31, 1321–1348 (2006) 63. Mouhot, C., Strain, R.M.: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J. Math. Pures Appl. (9) 87(5), 515–535 (2007) 64. Pao, Y.P.: Boltzmann collision operator with inverse power intermolecular potential, I, II. Commun. Pure Appl. Math. 27, 407–428, 559–581 (1974) 65. Strain, R.M.: The Vlasov-Maxwell-Boltzmann system in the whole space. Commun. Math. Phys. 268(2), 543–567 (2006) 66. Strain, R., Guo, Y.: Exponential decay for soft potentials near Maxwellian. Arch. Rat. Mech. Anal. 187(2), 287–339 (2008) 67. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974) 68. Ukai, S.: Les solutions globales de l’equation de Boltzmann dans l’espace tout entier et dans le demiespace. C. R. Acad. Sci. Paris Ser. A-B 282 (1976), no. 6, Ai, A317–A320 69. Ukai, S.: Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff. Japan J. Appl. Math. 1-1, 141–156 (1984) 70. Ukai, S.: Solutions of the Boltzmann equation In: Pattern and Waves – Qualitative Analysis of Nonlinear Differential Equations eds. M. Mimura, T. Nishida, Studies of Mathematics and Its Applications 18, Tokyo: Kinokuniya-North-Holland, 1986, pp. 37–96 71. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rat. Mech. Anal. 143, 273–307 (1998) 72. Villani, C.: Regularity estimates via entropy dissipation for the spatially homogeneous Boltzmann equation. Rev. Mat. Iberoam. 15(2), 335–352 (1999) 73. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds) Handbook of Mathematical Fluid Mechanics. Amsterdem: NorthHolland, 2002 74. Yu, H.: Convergence rate for the Boltzmann and Landau equations with soft potentials. Proc. Royal Soc. Edinburgh 139A, 393–416 (2009) Communicated by P. Constantin
Commun. Math. Phys. 304, 583–584 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1241-x
Communications in
Mathematical Physics
Erratum
Erratum to: Equilibrium States for Interval Maps: Potentials with sup ϕ − inf ϕ < h t op ( f ) Henk Bruin1 , Mike Todd2 1 Department of Mathematics, University of Surrey, Guildford, Surrey 602 7XH, UK.
E-mail: [email protected]
2 School of Mathematics and Statistics, University of St. Andrews, North Haugh, St. Andrews,
Fife, KY16 9SS, Scotland. E-mail: [email protected] Received: 21 December 2010 / Accepted: 15 February 2011 Published online: 14 April 2011 – © Springer-Verlag 2011
Commun. Math. Phys. 283, 579–611 (2008)
This is a correction to the proof of Lemma 4: Lemma 4. Assume that f is a C 3 multimodal map with non-flat critical points, take α ∈ (0, 1] and let max := max{c : c ∈ Crit}. Then there exists K = K (#Crit, max , α) such that if lim inf |D f n ( f (c))| K n
for all c ∈ Crit,
then formula (8) of the original paper, i.e., sup i
τ i −1
| f k (X i )|α < ∞
(8)
k=0
holds for every inducing scheme obtained as in Proposition 1 on a sufficiently small neighbourhood of Crit. Clearly, if lim inf n |D f n ( f (c))| → ∞ then (8) holds for all α > 0 simultaneously. Proof. This proof is a correction after Rivera-Letelier and Shen drew our attention to a dubious part in the earlier proof. We will use Theorem A in [RS] to fix the gap. This theorem, translated to our notation, says that for every β > 0, there is K = K (#Crit, max , β) and ρ > 0 such that if minc∈Crit lim inf n |D f n ( f (c))| ≥ K , then for every interval J of length |J | < ρ each component of f −n (J ) has length ≤ n −β . To give one intermediate step, the condition on the derivatives along critical order implies that f satisfies a backward contraction property with constant r , see [BRSS, Theorem 3], which is then The online version of the original article can be found under doi:10.1007/s00220-008-0596-0.
584
H. Bruin, M. Todd
used in [RS, Theorem A], assuming r = r (β) > 1 is sufficiently large, to show that the components of f −n (J ) have length ≤ n −β . To apply this for our case, we take β > 1/α and induce on a set X of length < ρ and then for each partition element X i , the image f k (X i ) is a component of the τi − k th τi −1 k preimage of X , and therefore has length ≤ (τi − k)−β . This gives k=0 | f (X i )|α ≤ τi −αβ < ∞ uniformly over all X i . Hence formula (8) holds. k=1 k References [BRSS] [RS]
Bruin, H., Rivera-Letelier, J., Shen, W., van Strien, S.: Large derivatives, backward contraction and invariant densities for interval maps. Invent. Math. 172, 509–533 (2008) Rivera-Letelier, J., Shen, W.: Statistical properties of one-dimensional maps under weak hyperbolicity assumptions. http://arxiv.org/abs/1004.0230v1 [math.Ds], 2010
Communicated by G. Gallavotti
Commun. Math. Phys. 304, 585–635 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1243-8
Communications in
Mathematical Physics
Boundary Value Problems for the Stationary Axisymmetric Einstein Equations: A Disk Rotating Around a Black Hole Jonatan Lenells Department of Mathematics, Baylor University, One Bear Place # 97328, Waco, TX 76798, USA. E-mail: [email protected] Received: 7 March 2010 / Accepted: 5 November 2010 Published online: 22 April 2011 – © Springer-Verlag 2011
Abstract: We solve a class of boundary value problems for the stationary axisymmetric Einstein equations involving a disk rotating around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. In the absence of a disk, they reduce to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk. Contents 1. 2.
3. 4.
5. 6.
Introduction . . . . . . . . . . . . . . . . . . Disk/Black-Hole Systems . . . . . . . . . . . 2.1 The Ernst equation . . . . . . . . . . . . 2.2 The boundary value problem . . . . . . . 2.3 The solution . . . . . . . . . . . . . . . . 2.4 Axis and horizon values . . . . . . . . . . Example . . . . . . . . . . . . . . . . . . . . 3.1 Numerical data . . . . . . . . . . . . . . Spectral Theory . . . . . . . . . . . . . . . . 4.1 A bounded and analytic eigenfunction . . 4.2 The main Riemann-Hilbert problem . . . 4.3 The global relation . . . . . . . . . . . . 4.4 Linearizable boundary conditions . . . . . 4.5 The auxiliary Riemann-Hilbert problem . A Scalar Riemann-Hilbert Problem . . . . . . 5.1 Formulation of the scalar RH problem . . 5.2 Solution of the scalar RH problem . . . . Theta Functions . . . . . . . . . . . . . . . . 6.1 Explicit expression for the Ernst potential 6.2 The metric functions e2U and a . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
586 588 588 589 590 594 597 598 600 601 602 605 605 605 608 611 615 616 616 618
586
7.
Axis and Horizon Values . . . . . . . . 7.1 Values near the regular axis . . . . 7.2 Values of a0 and K 0 . . . . . . . . 7.3 Values near the black hole horizon 8. Parameter Ranges . . . . . . . . . . . 8.1 Singularity structure . . . . . . . . 8.2 Dependence on w4 . . . . . . . . Appendix A. Condensation of Branch Points A. 1. The first degeneration . . . . . . A. 2. The second degeneration . . . . . References . . . . . . . . . . . . . . . . . .
J. Lenells
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
619 620 622 624 626 626 627 628 630 632 634
1. Introduction Two of the most famous solutions of the stationary axisymmetric Einstein equations are the Kerr black hole and the Neugebauer-Meinel disk. The former was discovered by Kerr in 1963 [13] and the latter by Neugebauer and Meinel in the 1990s [21–23]. In this paper, we construct analytic solutions of a class of boundary value problems (BVPs) for the stationary axisymmetric Einstein equations which combine the Kerr and Neugebauer-Meinel spacetimes. Thus, the BVPs considered involve a finite disk of dust rotating uniformly around a central black hole. In the limit of a vanishing disk, the solutions tend to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk. The constructed disk/black-hole systems are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. Given the importance of the Kerr and Neugebauer-Meinel solutions, we believe that the class of solutions presented here could also be of interest. We emphasize, however, that the new solutions involve a disk whose inner rim starts right at the event horizon of the black hole, whereas a physically correct BVP would allow for a small gap between the disk and the horizon. Thus, we expect the presented solutions to involve some dust particles traveling at superluminal speeds near the horizon. The general analysis of rotating relativistic bodies is exceedingly complicated because it involves the study of free BVPs for the Einstein equations, which are nonlinear partial differential equations in four dimensions. However, in cases where the surface of the body is known and the motion is stationary and axisymmetric (a reasonable assumption in many astrophysical situations), the physical problem gives rise to a BVP for a single integrable equation in two dimensions—the celebrated Ernst equation. The integrability of the Ernst equation implies that powerful solution-generating techniques are at hand. Thus, through the application of suitable nonlinear transformations, new stationary axisymmetric spacetimes can be generated from already known ones. Furthermore, a large class of solutions of the Ernst equation can be given explicitly in terms of theta functions on Riemann surfaces [18]. In this way, it is possible to write down a large number of exact analytic solutions to the stationary axisymmetric Einstein equations and to study them using the methods of algebraic geometry cf. [17]. Nevertheless, for the solution of a concrete BVP, the power of this approach is often limited. Indeed, although a large class of exact solutions can be produced, the problem of determining the particular solution within this class that satisfies the given BVP is in general a highly nonlinear problem. It is therefore remarkable that Neugebauer and Meinel in the 1990s were able to solve explicitly, using constructive methods, the BVP corresponding to the physically relevant situation of a rotating dust disk. The structure of
Boundary Value Problems for the Einstein Equations
587
an infinitesimally thin, relativistic disk of dust particles which rotate uniformly around a common center was first explored numerically by Bardeen and Wagoner 40 years ago [3], who also pointed out that “there may be some hope of finding an analytic solution” [2]. Biˇcák notes in the comprehensive review [4] that the subsequent construction of such an analytic solution by Neugebauer and Meinel represents “the first example of solving the BVP for a rotating object in Einstein’s theory by analytic methods.” Let us point out that since the Neugebauer-Meinel solution can be written in terms of theta functions on Riemann surfaces of genus 2, it belongs to the general class of solutions introduced in [18] and can therefore be analyzed by means of algebro-geometric methods. On the other hand, the Kerr black hole is the most famous example of a stationary axisymmetric spacetime and has had an immense impact on the development of general relativity and astrophysics (see e.g. [5]). The approach in this paper is primarily inspired by the work of Neugebauer, Meinel, and collaborators [24], but also by a novel method for the analysis of BVPs for integrable PDEs which has been developed by Fokas and his collaborators within the past 15 years. A central development in the theory of nonlinear PDEs in the second half of the 20th century, and continuing to the present, has been the introduction of the Inverse Scattering Transform (IST). This technique was put forward in the famous 1967 paper [11] in connection with the Korteweg-de Vries (KdV) equation and the range of its applicability began to unfold with the investigation of the nonlinear Schrödinger (NLS) equation [ZS]. One of the most important later developments in this area has been the generalization of the IST formalism from initial-value to initial-boundary value problems introduced by Fokas [8,9] and subsequently developed further by several authors cf. [10]. The Fokas method consists of two steps: (a) Construct an integral representation of the solution characterized via a matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane, where k denotes the spectral parameter of the associated Lax pair. Since this representation involves, in general, some unknown boundary values, the solution formula is not yet effective. (b) Characterize the unknown boundary values by analyzing a certain equation called the global relation. In general, this characterization involves the solution of a nonlinear problem; however, for certain so-called linearizable boundary conditions, step (b) can be solved in closed form. In a recent work [20] steps (a) and (b) were implemented for the class of BVPs of the Ernst equation corresponding to a thin rotating disk of finite radius. In particular, it was found that the boundary conditions of the particular BVP corresponding to the uniformly rotating Neugebauer-Meinel disk are linearizable. The present paper is to some extent a continuation of [20]: The main novel observation is that the BVP which combines the Kerr black hole boundary conditions with those of a uniformly rotating disk is also linearizable. Physically, disk/black-hole systems are important as models for black hole accretion disks in the context of active galactic nuclei and X-ray binaries [1]. Accretion disks are flattened astronomical objects made of rapidly rotating gas which slowly spirals onto a central gravitating body. Accretion onto a black hole is generally assumed to be thin and axisymmetric (see e.g. [25]) and many of the most energetic phenomena in the universe have been attributed to the accretion of matter onto black holes. In particular, active galactic nuclei and quasars are believed to be the accretion disks of supermassive black holes. We refer to [1] and references therein for more information on the physical aspects of black hole accretion disks. Our approach here is mathematical and we do not investigate any possible physical relevance of the presented solutions. In particular, we will refer to the disk as a ‘disk of dust’ although the assumption that the disk stretches all the way to the horizon presumably forces some of the dust particles to travel at
588
J. Lenells
superluminal velocities. We mention in this regard that the BVP corresponding to a dust ring with finite inner radius (i.e. the BVP obtained by inserting a small gap between the horizon and the innermost dust particles) is believed to not be exactly solvable. In fact, the calculations in [14] suggest that it is impossible, even in the Newtonian limit, to obtain explicit solutions describing a self-gravitating dust ring around a central point mass. The manuscript is organized as follows. In Sect. 2, we formulate our disk/black-hole BVP and write down its full solution in terms of theta functions. In Sect. 3, we consider a particular example. In Sects. 4–7, the derivation of the solution is presented in several steps. In Sect. 8, we consider the singularity structure of the solution and its dependence on various parameters. In the Appendix, we consider the relationship between the solution derived here and the general class of solutions of the Ernst equation studied in [19,15]. 2. Disk/Black-Hole Systems In this section we introduce the Ernst equation, formulate the BVP corresponding to a dust disk rotating uniformly around a central black hole, and present its solution in terms of theta functions. 2.1. The Ernst equation. The metric of a stationary axisymmetric vacuum spacetime can be written in the Weyl-Lewis-Papapetrou form ds 2 = e−2U e2κ (dρ 2 + dζ 2 ) + ρ 2 dϕ 2 − e2U (dt + adϕ)2 , (2.1) where ρ ≥ 0 and ζ ∈ R are Weyl’s canonical coordinates and t, ϕ are chosen so that ∂t and ∂ϕ are the two commuting asymptotically timelike and spacelike Killing vectors, respectively, cf. [26]. The metric functions e2U , a, and e2κ are functions of ρ and ζ alone. Following standard practice, we introduce a real-valued potential b by aρ = ρe−4U bζ ,
aζ = −ρe−4U bρ .
(2.2)
It can be shown that the Einstein field equations for the metric (2.1) reduce to the following single nonlinear PDE in two dimensions for the complex-valued Ernst potential f = e2U + ib: f + f¯ 1 f ρρ + f ζ ζ + f ρ = f ρ2 + f ζ2 , ρ > 0, ζ ∈ R. (2.3) 2 ρ The real part of f will be denoted by e2U despite the fact that it may take on negative values. We also need the concept of a corotating frame. Given ∈ R, we define the coordinates (ρ , ζ , ϕ , t ) corotating with the angular velocity by ρ = ρ,
ζ = ζ,
ϕ = ϕ − t,
t = t.
In these new coordinates, the metric (2.1) retains its form and the corotating metric functions U , a , κ are related to U, a, κ via (2.4a) e2U = e2U (1 + a)2 − 2 ρ 2 e−4U , (1 − a )e2U = (1 + a)e2U ,
κ − U = κ − U.
(2.4b)
Boundary Value Problems for the Einstein Equations
589
Fig. 1. The exterior domain D of a disk of radius ρ0 with the black hole horizon stretching along the imaginary axis from −ir1 to ir1
The Ernst equation retains its form in the corotating system and we denote the corotating Ernst potential by f := e2U + ib .
2.2. The boundary value problem. We now formulate the BVP for the Ernst equation (2.3) which corresponds to a finite dust disk rotating uniformly around a central black hole. The formulation involves five parameters: the radius ρ0 > 0 and angular velocity ∈ R of the disk; the ‘radius’ r1 > 0 and angular velocity h ∈ R of the black hole horizon; and the (necessarily constant) value of the corotating metric function e2U on the disk. We will henceforth work with a complex variable z = ρ + iζ and write f (z) for f (ρ, ζ ).1 Let D denote the exterior of a finite disk of radius ρ0 > 0, i.e. D consists of all z ∈ C with strictly positive real part which do not belong to the interval [0, ρ0 ], see Fig. 1. The horizon of the black hole stretches in the z-plane along the imaginary axis from −ir1 to ir1 . We consider the problem of finding a function f such that: • f satisfies (2.3) in D. • f (z) = f (¯z ) (equatorial symmetry).
(2.6a) (2.6b)
• f (z) → 1 as |z|2 → ∞ (asymptotic flatness).
(2.6c)
•
∂f ∂ρ (iζ )
= 0 for all|ζ | > r1 (regularity on the rotation axis).
• f (ρ ± i0) = e2U (+i0) for 0 < ρ < ρ0 (boundary condition on the disk). •e
2Uh (iζ )
= 0 for 0 < |ζ | < r1 (boundary condition on the horizon).
(2.6d) (2.6e)
The boundary conditions (2.6d) and (2.6e) are the boundary conditions corresponding physically to a uniformly rotating dust disk and a rotating black hole, respectively, cf. [24]. If one sets r1 = 0 in (2.6) (i.e. one removes the black hole), then the solution of the obtained BVP is the Neugebauer-Meinel disk rotating with angular velocity . If one sets ρ0 = 0 in (2.6) (i.e. one removes the disk), then the solution of the obtained BVP is the Kerr black hole rotating with angular velocity h . 1 In general, given a function h(ρ, ζ ), we will suppress the dependence on z¯ and write h(z) for h(ρ, ζ ) even when h is not analytic.
590
J. Lenells
Fig. 2. The Riemann surface z presented as a two-sheeted cover of the complex k-plane with five branch cuts when Re k4 < ζ . The contours γ and + are also shown
2.3. The solution. The formulation of the BVP (2.6) involves the five independent parameters ρ0 , , r1 , h , and the constant value of e2U on the disk. However, it turns out that the condition that the solution be nonsingular at the rim of the disk imposes one relation among these parameters, so that the class of solutions is parametrized by only four parameters. It is convenient to adopt a parametrization in terms of the four parameters ρ0 , r1 , w2 , and w4 , where w2 and w4 are two real quantities related to the other parameters via Eqs. (5.9) below. For a given choice of these parameters, the corresponding solution f of the BVP (2.6) can be written in terms of theta functions on the family of Riemann surfaces {z }z∈D defined as follows: Let k1 , k¯1 , . . . , k4 , k¯4 ∈ C denote the eight zeros of w 2 (k) + 1, where w(k) =
w4 k 4 + w2 k 2 + ρ02 (w2 − w4 ρ02 ) (k 2 − r12 )
,
(2.7)
ordered so that k j , j = 1, . . . , 4, have negative imaginary parts and so that Re k1 ≤ Re k2 ≤ Re k3 ≤ Re k4 . Since these eight zeros are symmetrically distributed with respect to the origin, we have −k1 = k¯4 and −k2 = k¯3 . For each z = ρ +iζ , z is defined as the hyperelliptic Riemann surface of genus 4 consisting of all points (k, y) ∈ C2 such that y 2 = (k + i z)(k − i z¯ )
4
(k − k j )(k − k¯ j ),
(2.8)
j=1
together with two points at infinity required to make the surface compact. We introduce branch cuts in the complex k-plane from k j to k¯ j , j = 1, . . . , 4, and from −i z to i z¯ , see ˆ = C ∪ {∞}, we let k + and k − denote the points which project onto k Fig. 2. For k ∈ C and which lie in the upper and lower sheet of z , respectively. By definition, the upper (lower) sheet is characterized by y/k 5 → 1 (y/k 5 → −1) as k → ∞. In view of the assumption of equatorial symmetry (2.6b), we will in the sequel always assume that ζ ≥ 0. We will also, for the sake of definiteness, assume that 0 < Re k3 < r1 < Re k4 .
(2.9)
Boundary Value Problems for the Einstein Equations
591
Moreover, we assume that ζ = Re k3 and ζ = Re k4 , so that no branch cuts overlap (the solution for ζ = Re k j , j = 3, 4, can be obtained by continuity). For n complex numbers {a j }n1 , we let [a1 , . . . , an ] denote the directed contour n−1 ∪ j=1 [a j , a j+1 ]. We let γ denote the contour on z which projects to the contour [r1 , Re k3 + , k3 + ] ∪ [k¯3 − , Re k3 − , Re k2 + , k2 + ] ∪ [k¯2 − , Re k2 − , −r1 ] (2.10) in the complex k-plane, where > 0 is an infinitesimally small positive number, and which lies in the upper sheet for Re k < ζ and in the lower sheet for Re k > ζ . We define
+ as the contour in the upper sheet of z which lies above the segment = [−iρ0 , iρ0 ]. The contours + and γ are shown in Fig. 2 in the case when Re k4 < ζ . In order to define theta functions associated with z , we need to introduce a basis of the first homology group H1 (z , Z) of z . Since the Riemann surface z depends on z, there are three qualitatively different cases to consider: (1) The cut [−i z, i z¯ ] lies to the right of [k4 , k¯4 ] (i.e. Re k4 < ζ ). (2) The cut [−i z, i z¯ ] lies between [k3 , k¯3 ] and [k4 , k¯4 ] (i.e. Re k3 < ζ < Re k4 ). (3) The cut [−i z, i z¯ ] lies to the left of [k3 , k¯3 ] (i.e. 0 ≤ ζ < Re k3 ). For these three cases, we let {a j , b j }4j=1 be the canonical basis of H1 (z , Z) shown in (1), (2), and (3) of Figs. 3, 4 respectively. Thus, for j = 1, . . . , 4, a j surrounds the cut [k j , k¯ j ], whereas b j enters the upper sheet on the right side of [−i z, i z¯ ] and exits again on the right side of [k j , k¯ j ]. We define {ω j }41 as the canonical basis of the space of holomorphic one-forms on z dual to {a j , b j }. Then
ωi = δi j ,
ωi = Bi j ,
aj
i, j = 1, . . . , 4,
bj
where B is the period matrix associated with the cut system {a j , b j }. We let ω = (ω1 , ω2 , ω3 , ω4 )T . The 4 × 4 matrix B is symmetric and has a positively definite imaginary part. The associated theta function (v|B) is defined by (v|B) =
e
2πi
1 T 2N
B N +N T v
,
v ∈ C4 .
(2.11)
N ∈Z4
We let ω P Q denote the Abelian differential of the third kind on z , which has two simple poles at the points P and Q with residues +1 and −1, respectively, and whose a-periods vanish, i.e. a j ω P Q = 0 for j = 1, . . . , 4. We can now state our main result. Theorem 2.1. Solution of the disk/black-hole BVP Let ρ0 , r1 , w2 , w4 be strictly positive numbers such that (2.9) holds. [The requirement that the solution be singularity-free imposes further restrictions on these parameters, see Sect. 8.] Let the function h(k) be defined by h(k) =
1 ln w(k)2 + 1 − w(k) , πi
k ∈ = [−iρ0 , iρ0 ].
(2.12)
592
J. Lenells
Fig. 3. The homology basis {a j , b j }41 on the hyperelliptic Riemann surface z of genus g = 4 in the case of (1) Re k4 < ζ , (2) Re k3 < ζ < Re k4 , and (3) 0 ≤ ζ < Re k3
Fig. 4. Three-dimensional picture of the hyperelliptic Riemann surface z in the case when Re k4 < ζ
Boundary Value Problems for the Einstein Equations
593
Define the z-dependent quantities u ∈ C4 and I ∈ R by u= hω + ω, I = hω∞+ ∞− + ω∞+ ∞− .
+
γ
Then the function
+
γ
∞− u − −i z ω|B
eI , f (z) = ∞− u + −i z ω|B
(2.13)
(2.14)
satisfies the BVP (2.6) with the prescribed values of ρ0 and r1 , and with the values of h , , and e2U (+i0) given by
w4 h e2U0 + −2w4 4h e2U0 2 1 2U (+i0) 2U0 1 − , = , e = e , h = − ahor h w4 e2U0 + 22h (2.15) where ahor ∈ R denotes the (necessarily constant) value of the metric function a on the horizon and e2U0 ∈ R denotes the real part of f (+i0). Explicit expressions for the constants ahor and f (+i0) are presented in Propositions 2.4 and 2.5 below. Moreover, define the z-dependent quantity L by dh 1 + dκ1 (κ1 ) h(κ2 )ωκ + κ − (κ2 )+ hω−r + ,−r − −sgn(ζ −r1 ) hωr + r − L=− 1 1 1 1 1 1 + 2
dk
+
1 + lim ω + − − sgn(ζ − r1 ) ωr + r − − 2 ln , (2.16) 1 1 2 →0 γ1 ( ) −r1 ,−r1 γ2 ( ) where γ1 ( ) denotes the contour γ with the segment covering [−r1 , −r1 + ] removed, γ2 ( ) denotes the contour γ with the segment covering [r1 − , r1 ] removed, and the prime on the integral along indicates that the integration contour should be deformed slightly before evaluation so that the pole at κ2 = κ1 is avoided.2 Then the metric functions e2U , a, e2κ of the line element (2.1) corresponding to the Ernst potential (2.14) are given for z ∈ D by ⎞ ⎛ ∞− ∞− ω + ω|B) (u + Q(0) ρ −i z i z ¯ ⎝ e2U (z) = e I , a(z) = a0 − − Q(u)⎠ e−I , i z¯ Q(u) Q(0) Q(0)(u + −i z ω|B) (2.17a)
i z¯
e2κ(z) = K 0
(u|B)(u + −i z ω|B) L e , i z¯ (0|B)( −i z ω|B)
(2.17b)
where Q(v) is defined by Q(v) =
∞− ω|B)(v + i z¯ ω|B) , i z¯ (v|B)(v + −i z ω|B)
(v +
∞− −i z
v ∈ C4 ,
(2.18)
and the two constants a0 , K 0 ∈ R are given explicitly by Eqs. (2.26) and (2.27) below. 2 The result is indepedent of whether the contour is deformed to the right or to the left of the pole.
594
J. Lenells
Remark 2.2. 1. The function h(k),√k ∈ , is well-defined by the right-hand side of (2.12), because w 2 + 1 ≥ 0 and w 2 + 1 − w > 0 for k ∈ . 2. Unless stated otherwise, all integrals in this paper along paths on Riemann surfaces for which only the endpoints are specified are assumed to lie within the fundamental polygon obtained by cutting the Riemann surface along the given cut basis. This implies that some integrals will depend on the particular choice of the a j ’s and b j ’s within their respective homology classes. It is convenient to fix the a j ’s and b j ’s so that they are invariant under the involution k ± → k ∓ . For ζ > Re k4 , this is accomplished by fixing3 b j = b j+1 ∪ [k¯ j+1 , k j ]+ ∪ [k j , k¯ j+1 ]− ,
b4 = [i z¯ , k4 ]+ ∪ [k4 , i z¯ ]− ;
j = 1, 2, 3,
and by letting a j , j = 1, . . . , 4, be the path in the homology class specified by Fig. 3 which as a point set consists of the points of z lying directly above [k j , k¯ j ]. This implies the important symmetry ω(k + ) = −ω(k − ). For z with 0 ≤ ζ < Re k4 and for other Riemann surfaces below, we will assume that an analogous fixing of the cut basis which assures ω(k + ) = −ω(k − ) has been made. 3. The limit as → 0 of the expression within brackets on the right-hand side of (2.16) always exists and is finite because of the pole structure of ω−r + ,−r − and ωr + r − . 1 1 1 1 4. The assumption that r1 satisfies (2.9) is made for simplicity and is not essential. The relevant formulas in the case when (2.9) does not hold can be obtained from those presented here by analytic continuation.
2.4. Axis and horizon values. Of particular interest are the values of the Ernst potential and of the metric functions on the regular axis {iζ | ζ > r1 } and on the black hole horizon {iζ | 0 < ζ < r1 }. In the limit ρ → 0, the Riemann surface z degenerates since the branch cut [−i z, i z¯ ] shrinks to a point. Thus, the values of f on the ζ -axis are given in terms of quantities defined on the Riemann surface defined by the equation y 2 =
4
(k − k j )(k − k¯ j ),
(2.19)
j=1
i.e., is the Riemann surface z with the cut [−i z, i z¯ ] removed. We introduce a canonical cut basis {a j , bj }31 on according to Fig. 5 and let ω = (ω1 , ω2 , ω3 )T denote the dual basis of holomorphic one-forms. We let B denote the associated period matrix and introduce the short-hand notation (v) := (v|B ), v ∈ C3 , for the associated theta function. Let γ denote the ζ -dependent contour on which projects to the contour (2.10) in the complex k-plane and which lies in the upper sheet for Re k < ζ and in the lower sheet for Re k > ζ . Define the ζ -dependent quantities u ∈ C3 , I ∈ R, and K ∈ C by
u =
+
hω +
γ
ω,
I =
+
hω∞ + ∞−
+
γ
ω∞ + ∞− ,
K =
∞− k4
ωζ + ζ − . (2.20)
3 For two complex numbers z and z , [z , z ]+ and [z , z ]− denote the covers of [z , z ] in the upper 1 2 1 2 1 2 1 2 and lower sheets of z , respectively.
Boundary Value Problems for the Einstein Equations
595
Fig. 5. The cut system {a j , bj }3j=1 on the degenerated Riemann surface of genus g = 3
Let γ + denote the contour on with the same projection onto the complex k-plane as γ , but which lies entirely in the upper sheet, and define J ∈ R by ⎧ ⎨ + hωζ + ζ − + γ ωζ + ζ − , ζ > r1 , + r1 J = (2.21) ⎩ + hωζ + ζ − + + γ + ωζ + ζ − , 0 < ζ < r1 . r− 1
γ+
The prime on the integral along indicates that the path should be deformed slightly before evaluation, so that it avoids the pole of the integrand at k = ζ + . It is irrelevant for the formulas below if this deformation is performed so that the pole lies to the right or to the left of γ + , since these two choices yield values of J which differ by a multiple of 2πi and J only appears exponentiated. In the same way, the value of J changes by irrelevant multiples of 2πi if loops surrounding ζ ± are added to the contour from r1− to r1+ . Proposition 2.3 (Solution on the regular axis). The behavior of the solution (2.14) near the regular axis {iζ | ζ > r1 } is given by f (ρ + iζ ) = f (iζ ) + O(ρ 2 ), where
ρ → 0, ζ > r1 ,
(2.22)
∞− ∞− (u − ζ − ω ) − (u − ζ + ω )e J −K I −J f (iζ ) = e , ζ > r1 . (2.23) ∞− ∞− (u + ζ − ω ) − (u + ζ + ω )e−J −K
The behavior of the metric functions e2U , a, and e2κ in (2.17) near the regular axis is given by e2U (ρ+iζ ) = e2U (iζ ) + O(ρ 2 ), a(ρ + iζ ) = O(ρ 2 ), e2κ(ρ+iζ ) = 1 + O(ρ 2 ), ρ → 0, ζ > r1 , where e2U (iζ )
− ∞− ∞ (u )2 ( ζ − ω )2 − ( ζ + ω )2 e−2K e I −J , ζ > r1 . = ∞− ∞− (0)2 (u + ζ − ω )2 − (u + ζ + ω )2 e−2J −2K (2.24)
596
J. Lenells
Define the constant L 0 by dh 1 + L0 = − dκ1 (κ1 ) h(κ2 )ωκ + κ − (κ2 ) + hω−r + ,−r − − hωr + r − 1 1 1 1 1 1 2
dk
+
+
1 + lim ω + − − ωr + r − − 2 ln , (2.25) 1 1 2 →0 γ1+ ( ) −r1 ,−r1 γ2+ ( ) where γ1+ ( ) denotes the contour γ + with the segment covering [−r1 , −r1 + ] removed, and γ2+ ( ) denotes the contour γ + with the segment covering [r1 − , r1 ] removed. Define the constants a0 , K 0 ∈ R by ∞− R+ (u + 2 k4 ω ) −I k4 ω∞+ ∞− Re , (2.26) e a0 = −2i lim R→∞ (u ) K0 =
(0)2 −L 0 e , (u )2
(2.27)
where the right-hand sides are understood to be evaluated at some ζ > r1 .4 Define the ζ -dependent quantities L and M by dh 1 L = − dκ1 (κ1 ) h(κ2 )ωκ + κ − (κ2+ ) + hω−r + hωr + r − − + 1 1 1 ,−r1 1 1 2 + dk
+
+
+ 1 + lim ω + − + ωr + r − −2 ln − hωζ + ζ − , 0 < ζ < r1 , 1 1 2 →0 γ1 ( ) −r1 ,−r1 γ2 ( )
+ (2.28) and 1 M = lim 2 →0
(ζ − )+
(ζ − )−
ωζ + ζ −
− 2 ln − ln 4 − πi ,
0 < ζ < r1 ,
(2.29)
where γ1 ( ) denotes the contour γ with the segment covering [−r1 , −r1 + ] removed, and γ2 ( ) denotes the contour γ with the segment covering [r1 − , r1 ] removed. Proposition 2.4 (Solution on the horizon). The behavior of the solution (2.14) near the black hole horizon {iζ | 0 < ζ < r1 } is given by f (ρ + iζ ) = f (iζ ) + O(ρ 2 ),
ρ → 0, 0 < ζ < r1 ,
(2.30)
where f (iζ ) = −
∞− ζ + J −K ζ + ω ) − (u − ζ + ω + ζ − ω )e e I −K , ∞− ∞− ζ+ +K J (u + ζ − ω )− (u + ζ − ω + ζ − ω )e
(u −
∞−
0 < ζ < r1 . (2.31)
The behavior of the metric functions e2U , a, and e2κ in (2.17) near the black hole horizon is given by 4 Their values are independent of the choice of ζ > r . 1
Boundary Value Problems for the Einstein Equations
597
e2U (ρ+iζ ) = e2U (iζ ) + O(ρ 2 ), a(ρ + iζ ) = ahor + O(ρ 2 ), e2κ(ρ+iζ ) = e2κhor + O(ρ 2 ), where e
2U (iζ )
=−
(u +
ζ+
ζ− (0)2
(2.32)
ω )2
∞− ω )2 − ( ζ + ω )2 e−2K × e I +J , (2.33) ∞− ∞− ζ+ (u + ζ − ω )2 − (u + ζ − ω + ζ − ω )2 e2J +2K ∞− (u + 2 ζ − ω ) (0)4 −I −M = a0 + ,
2 e − ζ+ − ∞ ∞ (u + ζ − ω ) ( ζ − ω )2 − ( ζ + ω )2 e−2K (
ahor
ρ → 0, 0 < ζ < r1 ,
e2κhor = −K 0
(u
+
∞− ζ−
ζ+ ζ−
(2.34) ω )2
(0)2
eJ
+L
.
(2.35)
Moreover, ahor and e2κhor are constants independent of 0 < ζ < r1 . By taking limits in the above formulas we can find the values of f at the origin z = +i0 and at the point z = ir1 , where the regular axis meets the horizon. Proposition 2.5. Solution at z = +i0 and z = ir1 The value of the solution (2.14) at z = ir1 is given by ∞− (u − r + ω ) 1 (2.36) e I −K ζ =r1 . f (ir1 ) = − ∞− (u + r − ω ) 1
Define the value of
J
at ζ = 0 by hω0+ 0− + J |ζ =0 =
+
r1+
r1−
+
γ+
ω0 + 0− ,
where the primes on the integrals indicate that the contours should be deformed slightly before evaluation so that they pass to the left of the pole at k = 0+ . Then the value f (+i0) of the solution (2.14) at z = +i0 is given by the right-hand side of (2.31) evaluated at ζ = 0. 3. Example Before presenting the derivation of the solution presented in the previous section, we wish to consider a concrete example. In this regard we note that all the quantities appearing in Sect. 2 can easily be computed explicitly using standard results for Riemann surfaces. In order to find the canonical basis {ωi }41 and the period matrix B, we start with the holomorphic one-forms {ηi }41 defined by ηi =
k i−1 dk , y
i = 1, . . . , 4.
(3.1)
598
J. Lenells
The ηi ’s form a noncanonical basis of holomorphic differentials (see e.g. [6]). Defining the two 4 × 4 matrices A and Z by (A−1 )i j = ηi , Zi j = ηi , i, j = 1, . . . , 4, (3.2) aj
bj
we find ω = Aη,
B = AZ .
(3.3)
The one-form ω∞+ ∞− on z is given explicitly by 4 k 4 dk k 4 dk + ω∞+ ∞− = − ωj, y y aj
(3.4)
j=1
whereas the one-form ωζ + ζ − on is given by ωζ + ζ −
y (ζ + )dk − = (k − ζ )y (k) 3
j=1
a j
y (ζ + )dk ωj . (k − ζ )y (k)
(3.5)
For k’s which lie some distance away from the branch cuts, the value of y in (2.8) can be evaluated according to 4 k − k¯ j k − i z ¯ ˆ (k − k j ) , k ∈ C, y(z, k + ) = (k + i z) k + iz k − kj j=1
where the branches with strictly positive real part are chosen for the square roots. For k ∈ [−i z, i z¯ ] and > 0 infinitesimally small, we have 4 k − k¯ j k − i z ¯ ˆ y(z, (k + )+ ) = (k + i z) −i (k − k j ) , k ∈ C; k + iz k − kj j=1
similar expressions are valid when k ∈ [k j , k¯ j ]. Using formulas of this type, it is straightforward to numerically evaluate all the expressions presented in Sect. 2. In fact, the theta functions are particularly suitable for numerical evaluation, because the strictly positive imaginary part of B implies that only a small number of terms in the sum (2.11) have to be included. In this way, we have verified for several examples all formulas of Sect. 2 to high precision.
3.1. Numerical data. Here we consider the particular example for which ρ0 = 1,
r1 =
1 , 2
w2 = 3,
w4 =
1 . 10
Numerically, we find k1 ≈ −0.95 − 5.48i,
k2 ≈ −0.21 − 0.95i.
(3.6)
Boundary Value Problems for the Einstein Equations
599
Fig. 6. The real and imaginary parts of the solution (2.14) for the choice of parameters specified in (3.6)
Fig. 7. The values of f in the equatorial plane ζ = 0+ for the choice of parameters specified in (3.6)
Fig. 8. The axis and horizon values of f for the choice of parameters specified in (3.6)
Thus, (2.9) is satisfied and we compute ≈ 0.055,
h ≈ 0.14,
f (+i0) ≈ −0.17,
f (ir1 ) ≈ −0.94i,
and a0 ≈ −18.17,
K 0 ≈ 9.43,
e2κhor ≈ −93.46.
The real and imaginary parts of f are shown in Fig. 6. Note that the real part of f is continuous but not smooth at the endpoints ±ir1 of the horizon. Moreover, as expected, the imaginary part of f has a jump across the disk. The values of f in the equatorial plane ζ = 0+ are shown in Figs. 7, 8. The metric functions ae2U and e2κ are shown in Fig. 9. Note that ae2U = 0 on the regular part of the axis. Moreover, e2κ = 1 on the regular axis and e2κ = e2κhor on the horizon. Figure 10 shows the real parts of the corotating potentials f and f h in the equatorial plane and along the ζ -axis, respectively. In accordance with the boundary conditions (2.6d) and (2.6e), e2U is constant along the disk and e2Uh vanishes along
600
J. Lenells
Fig. 9. The metric functions ae2U and e2κ for the choice of parameters specified in (3.6)
Fig. 10. The real parts of the corotating potentials f and f h in the equatorial plane and along the ζ -axis, respectively
the horizon. It can also be verified to high accuracy that the metric functions a and κ defined by (2.17) satisfy the appropriate equations, i.e. (cf. [17]) az =
iρ bz e4U
and
κz =
ρ f z f¯z , 2e4U
(3.7)
and that da z=ρ+i0 = 0, dζ
0 < ρ < ρ0 .
(3.8)
Equation (3.8) implies that the imaginary part of f is constant along the disk in accordance with (2.6d). 4. Spectral Theory We now turn to the proof of the results of Sect. 2. The proof will proceed through four main steps, presented in Sects. 4, 5, 6, and 7, respectively. The first step consists of analyzing the Lax pair for Eq. (2.3) and formulating two matrix Riemann-Hilbert (RH) problems: one main RH problem (which can be formulated for any choice of boundary conditions) and one auxiliary RH problem (which can be formulated because the boundary conditions of the BVP (2.6) are linearizable). In the second step, we show that these two matrix RH problems can be combined into a single scalar RH problem. The third step consists of solving this scalar RH problem explicitly in terms of theta functions. In the final fourth step, we prove Propositions 2.3–2.5 concerning the behavior of the solution near the ζ -axis; this step will follow from a study of the theta function formulas of Theorem 2.1 in the limit ρ → 0.
Boundary Value Problems for the Einstein Equations
601
4.1. A bounded and analytic eigenfunction. The elliptic Ernst equation (2.3) admits the Lax pair z (z, k) = U (z, k)(z, k), (4.1) z¯ (z, k) = V (z, k)(z, k), where the 2 × 2-matrix valued function (z, k) is an eigenfunction, k is a spectral parameter, and the 2 × 2-matrix valued functions U and V are defined by 1 ¯ 1 1 k − i z¯ f¯z¯ f z¯ f¯z λ f¯z λ U= , V = , λ(z, k) = . 1 ¯ ¯ λ f f f f k + iz z z f + f f + f λ z¯ z¯ We write the Lax pair (4.1) in the form d = W Let
0 σ1 = 1
1 , 0
where
0 σ2 = i
W := U dz + V d z¯ . −i , 0
1 σ3 = 0
(4.2) 0 . −1
Suppose f is a solution of the BVP (2.6). Following the same procedure as in [20], we define a solution (z, k) of (4.2) with the following properties: • For each z, (z, ·) is a map from the Riemann surface Sz to the space of 2 × 2 matrices, where Sz is defined by the equation λ2 =
k − i z¯ . k + iz
We view Sz as a two-sheeted covering of the Riemann k-sphere endowed with a branch cut from −i z to i z¯ ; the upper (lower) sheet is characterized by λ → 1 (λ → −1) as k → ∞. • satisfies the initial conditions 1 1 − + ˆ (4.3) , lim [(z, k )]2 = , k ∈ C, lim [(z, k )]1 = 1 −1 z→i∞ z→i∞ where, for a 2 × 2-matrix A, [A]1 and [A]2 denote the first and second columns of A, respectively. • obeys the symmetries (z, k + ) = σ3 (z, k − )σ1 ,
(z, k + ) = σ1 (z, k¯ + )σ3 ,
ˆ k ∈ C.
(4.4)
• is analytic for k ∈ Sz away from the set + ∪ − ∪ {−r1± , r1± }, where + and
− denote the coverings of = [−iρ0 , iρ0 ] in the upper and lower sheets of Sz , respectively. We emphasize that , in general, has singularities (simple poles) at the points k ± for k = −r1 and k = r1 . These poles arise since the Lax pair (4.1) is singular at points where e2U = 0. Physically, the points at which e2U vanishes make up the boundary of the ergospheres of the spacetime (within these surfaces there can be no static observer with respect to infinity). To see that e2U = 0 at z = ir1 and z = −ir1 , we note that the metric function a vanishes identically on the regular axis (cf. [26]). Thus, evaluating
602
J. Lenells
(2.4a) at ρ = 0, we find U (iζ ) = U (iζ ) for ζ > r1 . The boundary condition (2.6e) together with the continuity of f imply that e2U (iζ ) → 0 as ζ ↓ r1 . In addition to the eigenfunction (z, k), we will also need its corotating analog (z, k). This eigenfunction satisfies the Lax pair equations (4.1) with f replaced by f and the initial conditions (4.3) with replaced by . The eigenfunctions and are related by [24] (z, k) = (z, k)(z, k),
k ∈ Sz ,
(4.5)
where (z, k) = (1 + a)I − ρe−2U σ3 + i(k + i z)e−2U (λ(z, k)σ1 − I)σ3
(4.6)
and I denotes the 2×2 identity matrix. The corotating eigenfunction h (z, k) is defined analogously. 4.2. The main Riemann-Hilbert problem. Evaluation at ρ = 0 of Eq. (2.4a) with replaced by h yields e2Uh (iζ ) = e2U (iζ ) (1 + h a(iζ ))2 ,
0 < ζ < r1 .
(4.7)
Note that e2U < 0 along the horizon, which lies inside the ergosphere. Thus, in view of the boundary condition (2.6e), we find that a ≡ ahor on the black hole horizon, where ahor is a constant given by ahor = −1/ h .
(4.8)
The next proposition expresses the values of on the ζ -axis in terms of two spectral functions F(k) and G(k). We let f 1 denote the value of f at z = ir1 . Proposition 4.1. The values of on the ζ -axis can be expressed in terms of two spectral functions F(k) and G(k) as f (iζ )1 ˆ A(k), ζ > r1 , k ∈ C, (iζ, k + ) = (4.9a) f (iζ ) − 1 f (iζ )1 ˆ T (k)A(k), 0 < ζ < r1 , k ∈ C, (4.9b) (iζ, k + ) = f (iζ ) − 1 1 f (iζ )1 ˆ (4.9c) T (k)σ1 A(k)σ1 , −r1 < ζ < 0, k ∈ C, (iζ, k + ) = f (iζ ) − 1 2 f (iζ )1 ˆ σ A(k)σ1 , ζ < −r1 , k ∈ C, (4.9d) (iζ, k + ) = f (iζ ) − 1 1 where the 2 × 2-matrix valued functions A(k), T1 (k), and T2 (k) are defined by F(k) 0 ˆ , k ∈ C, (4.10) A(k) = G(k) 1 1 i 2(k − r1 )h − i f 1 ˆ (4.11) T1 (k) = , k ∈ C, 2 −i f 2(k − r ) 2(k − r1 )h 1 h + i f1 1 ¯ T2 (k) = T1 (−k),
ˆ k ∈ C.
Boundary Value Problems for the Einstein Equations
603
The functions F(k) and G(k) have the following properties: ˆ i.e. viewed as functions on Sz they satisfy • F and G are unique functions of k ∈ C, F(k + ) = F(k − ),
G(k + ) = G(k − ),
ˆ k ∈ C.
(4.12)
ˆ • F(k) and G(k) are analytic for k ∈ C\(
∪ {r1 , −r1 }). ¯ F and G obey the symmetries • Under the conjugation k → k, ¯ F(k) = F(k),
¯ G(k) = −G(k),
ˆ k ∈ C.
(4.13)
• In the limit k → ∞, F(k) = 1 + O(1/k),
G(k) = O(1/k),
k → ∞.
(4.14)
Proof. For z = iζ , λ = 1 for all k on the upper sheet of Sz . The axis values of are thus determined by integration of the equation d = W (iζ, k + ),
(4.15)
where W (iζ, k + ) =
1 d f¯ f + f¯ d f
d f¯ . df
(4.16)
Since the real part of f vanishes at z = ±ir1 , Eq. (4.15) breaks down at ζ = ±r1 . Integration of (4.15) for ζ in each of the four intervals (−∞, −r1 ), (−r1 , 0), (0, r1 ), and (r1 , ∞) yields f (iζ ) 1 + ˆ (iζ, k ) = U1 (k), ζ > r1 , k ∈ C, f (iζ ) −1 f (iζ ) 1 + ˆ U2 (k), 0 < ζ < r1 , k ∈ C, (iζ, k ) = f (iζ ) −1 f (iζ ) 1 + ˆ U3 (k), −r1 < ζ < 0, k ∈ C, (iζ, k ) = f (iζ ) −1 f (iζ ) 1 + ˆ U4 (k), ζ < −r1 , k ∈ C, (iζ, k ) = f (iζ ) −1 where the matrices U j (k), j = 1, . . . , 4, are independent of ζ . The initial conditions (4.3) imply that U1 = A for some functions F(k) and G(k). This establishes (4.9a). The value of at z = −i∞ is obtained from the value at z = i∞ by integrating W along a large semicircle at infinity. During this integration k changes sheets. Thus, using (4.9a) and the fact that W vanishes for large z, we compute 1 G(k) 1 1 + − + . lim (z, k ) = lim (z, k ) = lim σ3 (z, k )σ1 = 0 F(k) 1 −1 z→−i∞ z→i∞ z→i∞ (4.17) This shows that U4 = σ1 A(k)σ1 and proves (4.9d).
604
J. Lenells
We now use continuity of the matrices and h at the points z = ±ir1 to find U2 and U3 . Let f 2 denote the value of f at z = −ir1 . The conditions that (iζ, k + ) be continuous at ζ = r1 and ζ = −r1 are 1 1 f¯1 f¯2 (A(k) − U2 (k)) = 0 (σ1 A(k)σ1 − U3 (k)) = 0, and f 1 −1 f 2 −1 (4.18a) respectively. In view of Eqs. (4.5), (4.6), and (4.8), the conditions that h (iζ, k + ) be continuous at ζ = r1 and ζ = −r1 are −1 0 −1 0 1 f¯1 + 2ih (k − r1 ) A(k) = 2ih (k − r1 ) U2 (k), 1 0 1 0 f 1 −1 (4.18b) and
f¯2 f2
−1 1 + 2ih (k + r1 ) 1 −1
0 0
−1 σ1 A(k)σ1 = 2ih (k + r1 ) 1
0 U3 (k), 0 (4.18c)
respectively. The top and bottom rows of each of the four matrix equations in (4.18) are linearly dependent since f 1 and f 2 are purely imaginary. Combining the four bottom rows into two matrix equations, we find f1 f1 −1 −1 A(k) = U2 (k), f 1 + 2ih (k − r1 ) −1 2i(k − r1 ) 0 f2 f2 −1 −1 σ1 A(k)σ1 = U3 (k). f 2 + 2ih (k + r1 ) −1 2ih (k + r1 ) 0 Using that f 1 = − f 2 in view of the equatorial symmetry, we deduce from these equations that U2 (k) = T1 (k)A(k) and U3 (k) = T2 (k)σ1 A(k)σ1 , where T1 and T2 are given by (4.11). This proves (4.9b) and (4.9c). The properties of F and G are proved as in [20]. The functions F(k) and G(k) jump across = [−iρ0 , iρ0 ]. Let F + , G + and F − , G − denote the values of F and G for k to the right and left of , respectively. It follows as in [24] (see also [20]) that − (z, k) = + (z, k)D(k), k ∈ + ;
− (z, k) = + (z, k)σ1 D(k)σ1 , k ∈ − , (4.19)
where the jump matrix D is given in terms of F ± and G ± by −1 − + F (k) 0 F (k) 0 . D(k) = G + (k) 1 G − (k) 1
(4.20)
For a given z, Eq. (4.19) provides the jump condition for a matrix RH problem on the Riemann surface Sz satisfied by (z, k). We will refer to this as the main RH problem.5 In general, given both the Dirichlet and Neumann boundary values for a BVP for 5 A complete formulation of this problem also involves specifying residue conditions at the four points ±r1± as well as a normalization condition.
Boundary Value Problems for the Einstein Equations
605
the Ernst equation, it is possible to determine the spectral functions F and G, compute the jump matrix D, and then obtain the Ernst potential f from the asymptotics of the solution of the main RH problem. However, for a well-posed problem only one of these boundary values is specified. In our analysis of (2.6) we will therefore instead use the global relation and the symmetry of the boundary conditions to formulate an auxiliary RH problem from which F and G can be determined.
4.3. The global relation. The equatorial symmetry of the solution f of (2.6) implies that the spectral functions F(k) and G(k) satisfy an important relation. Recalling the axis values (4.9) of , the following proposition is proved in the same way as Proposition 4.3 in [20]. Proposition 4.2. The spectral functions F(k) and G(k) defined in Proposition 4.1 satisfy −1 −1 −1 T1 A+ (k)σ1 A−1 + (k)T1 = T2 σ1 A− (k)σ1 A− (k)σ1 T2 ,
k ∈ ,
(4.21)
where A(k) is defined in terms of F(k) and G(k) by Eq. (4.10) and A± denote the values of A to the right and left of , respectively. Equation (4.21) is referred to as the global relation.
4.4. Linearizable boundary conditions. In general, the global relation alone is not sufficient for determining F and G. However, for boundary conditions satisfying sufficient symmetry, so-called linearizable boundary conditions, there exist another algebraic relation satisfied by F and G. The boundary conditions specified in (2.6) turn out to be linearizable. Indeed, recalling the axis values (4.9) of , the following proposition is proved in the same way as Proposition 5.3 in [20]. Proposition 4.3. The spectral functions F(k) and G(k) satisfy the relation −1 (B −1 −1 σ1 σ3 B)(T1 A+ σ1 A−1 + T1 ) −1 −1 −1 = −(T1 A+ σ1 A−1 + T1 )(B σ1 σ3 B), k ∈ ,
(4.22)
where we use the short-hand notation B and for f (+i0) 1 B := , := (+i0, k + ) = 1 − I + ike−2U0 (σ1 − I)σ3 . f (+i0) −1 h (4.23)
4.5. The auxiliary Riemann-Hilbert problem. Combining the relations (4.21) and (4.22), we can formulate a RH problem for the 2 × 2-matrix valued function M(k) defined by −G(k) F(k) −1 ˆ M(k) = A(k)σ1 A (k) = 1−G(k)2 , k ∈ C. (4.24) G(k) F(k)
606
J. Lenells
Proposition 4.4. Suppose f is a solution of the BVP (2.6). Let f 1 := f (ir1 ) ∈ iR denote the value of f at z = ir1 . Then the spectral functions F(k) and G(k) are given by F(k) = M12 (k),
G(k) = M22 (k),
ˆ k ∈ C,
where M is the unique solution of the following RH problem: ˆ • M(k) is analytic for k ∈ C\(
∪ {−r1 , r1 }). • Across , M(k) satisfies the jump condition S(k)M− (k) = −M+ (k)S(k),
k ∈ ,
(4.25)
where M+ and M− denote the values of M to the right and left of , respectively, and S(k) is defined by S(k) = T1−1 B −1 −1 σ1 σ3 BT2 σ1 ,
k ∈ .
(4.26)
• M has the asymptotic behavior M(k) = σ1 + O(1/k),
k → ∞.
(4.27)
• The entries of M have simple poles at k = r1 and k = −r1 . The associated residues are given by 1 − f1 1 1 f1 −| f 1 |2 , Res , (4.28) Res M(k) = M(k) = r1 −r1 − f1 α − f 12 f 1 α f 1 / f¯1 where α=
d + ζ =r1 e2U (iζ ) dζ
(4.29)
and d + /dζ denotes the right-sided derivative. Proof. We deduce from (4.21) and (4.22) that the function M defined in (4.24) satisfies the jump condition (4.25). The asymptotic behavior (4.27) follows from the properties of F and G. By evaluating the first symmetry in (4.4) at the branch point i z¯ and taking the limit as z approaches the regular axis, we find (cf. Eqs. (2.63)–(2.64) in [24]) iIm f (iζ ) , ζ > r1 , Re f (iζ ) −iIm f (iζ ) G(ζ ) = , ζ < −r1 . Re f (iζ )
1 , Re f (iζ ) | f (iζ )|2 , F(ζ ) = Re f (iζ )
G(ζ ) =
F(ζ ) =
(4.30a) (4.30b)
The poles of F and G arise since Re f (iζ ) = 0 at ζ = ±r1 . Equations (4.30) together with the equatorial symmetry of f yield Res F(k) = r1
1 , α
Res G(k) = r1
f1 , α
Res F(k) = − −r1
| f 1 |2 , α
Res G(k) = − −r1
f1 , α
where α is given by (4.29). The residue conditions (4.28) follow immediately from these relations.
Boundary Value Problems for the Einstein Equations
607
In the last step of the proof, we show that M21 (k) does not have poles at the possible zeros of F, despite the form of (4.24). We first extend the definition (4.26) of S(k) to ˆ by all k ∈ C ¯ 2 σ1 . S(k) = T1−1 B −1 (+i0, k + )−1 σ1 σ3 (−i0, k + ) BT
(4.31)
This definition is consistent with (4.26). Indeed, since f is equatorially symmetric, (¯z , k + ) = (z, −k¯ + ),
(4.32)
¯ = (+i0, k + ) = (−i0, k + ) for k ∈ . We claim that the matrices S and so that M satisfy tr(SM) = 0,
ˆ k ∈ C.
(4.33)
Indeed, the same type of argument used to prove Proposition 4.3 shows that the function R = −1 (ρ + i0, k)σ1 σ3 (ρ − i0, k) is independent of ρ. Evaluation at ρ = 0 using the axis values yields + + ¯ R(k + ) = A−1 T1−1 B −1 −1 (+i0, k )σ1 σ3 (−i0, k ) BT2 σ1 Aσ1 .
Evaluation at ρ = ρ0 yields Tr R = 0. The preceding two equations imply (4.33). It follows from (4.33) that G 2 −1 must vanish whenever F has a zero. The auxiliary RH problem presented in Proposition 4.4 can be used to determine the spectral functions F and G. These spectral functions can then be used to compute the jump matrix and to set up the main RH problem. However, in analogy with linearizable BVPs for other integrable PDEs, we expect that the jump condition of the auxiliary RH problem can also be substituted directly into the main RH problem with the result that the unknown quantities in the main RH problem disappear. In fact, an example of this mechanism was observed by Neugebauer and Meinel in the case of a rigidly rotating disk. They discovered that the analogs of the main and auxiliary RH problems can be combined into a single scalar RH problem from which the Ernst potential f can be directly recovered, see [24]. It turns out that a similar approach can be adopted in the present case—in the next section, we will combine the main and auxiliary RH problems with respective jump conditions (4.19) and (4.25) into a scalar RH problem on the Riemann surface z introduced in Sect. 2.
608
J. Lenells
5. A Scalar Riemann-Hilbert Problem We let the scalar-valued function w(k) be defined by 1 ˆ k ∈ C, w(k) = − tr(S(k)), 2 and define two 2 × 2-matrix valued functions L and Q by L(z, k) = (z, k)σ1 −1 (z, k),
(5.1)
k ∈ Sz ,
(5.2)
and Q(z, k) = −(z, k)A(k)−1 S(k)A(k)(z, k)−1 − w(k)I,
k ∈ Sz .
(5.3)
Lemma 5.1. The functions L, Q, and w have the following properties: • The traces and determinants of Q and L satisfy tr Q = 0,
tr L = 0,
det L = −1,
det Q = −1 − w2 .
(5.4)
• Q can be alternatively written as Q(z, k) = (z, k)σ1 A(k)−1 S(k)A(k)σ1 (z, k)−1 + w(k)I.
(5.5)
• Q and L admit the symmetries Q(z, k − ) = −σ3 Q(z, k + )σ3 , • Q has no jump across
±:
±,
L(z, k − ) = σ3 L(z, k + )σ3 .
(5.6)
whereas L satisfies the following jump conditions across
(Q + wI)L− = −L+ (Q + wI),
k ∈ +,
(Q − wI)L− = −L+ (Q − wI),
k ∈ −.
• QL = −LQ; in particular, tr(QL) = 0. • Let Lˆ 22 = L21 Q11 + L22 Q21 . Then Lˆ 222 − L221 (1 + w 2 ) = Q221 .
(5.7)
(5.8)
• w has the form w(k) =
w4 k 4 + w 2 k 2 + w 0 , k 2 − r12
where w4 , w2 , w0 are real coefficients explicitly given by 22 2h , (5.9a) e2U0 ( − h )2 | f 0 |2 ( − h )2 + f 12 ( − 2h ) + 4i f 1r1 2 h + 2h 4r12 2 − 1 w2 = , 2e2U0 ( − h )2 (5.9b)
1 w0 = − 2U 2 −2ib0 f 13 − 2i f 12 r1 h + f 1 + 2ir1 h 8e 0 h
−| f 0 |2 f 12 − 4i f 1 r1 h − 4r12 2h + 1 + f 14 + f 12 + 4i f 1r1 h − 4r12 2h , w4 = −
(5.9c) where f 0 and f 1 denote the values of f at z = +i0 and z = ir1 , respectively.
Boundary Value Problems for the Einstein Equations
609
Proof. The first three properties in (5.4) are immediate from Eqs. (5.1)–(5.3). The fourth property follows since det(S) = −1 and det Q = det(−A−1 (S + wI)A−1 ) = det(S + wI) = det S − w2 . Using (4.33) and the definitions of A and M, a computation shows that A−1 (S + wI)A = −σ1 A−1 (S + wI)Aσ1 . Using this identity in the definition (5.3) of Q, we find (5.5). The symmetries (5.6) follow from (5.2), (5.3), and (5.5) together with the first symmetry in (4.4). By (4.19) and (4.20), A−1 and σ1 A−1 do not jump across + and − , respectively. It follows from the expressions (5.3) and (5.5) that Q does not jump across ± . For k ∈ + , the definitions (5.2) and (5.3) of L and Q show that −1 −1 −1 (Q− + wI)L− = −− A−1 − S A− σ1 − = −− A− SM− A− −
(5.10)
−1 −1 −1 − L+ (Q+ + wI) = + σ1 A−1 + S A+ + = + A+ M+ S A+ + .
(5.11)
and
Using that A−1 does not jump across + together with the jump condition (4.25), we see that the right-hand sides of (5.10) and (5.11) are equal. Similarly, using (5.5), the fact that σ1 A−1 has no jump across − , and (4.25), we find the jump across − . This proves (5.7). Since M is tracefree and tr(SM) = 0, we deduce that M S + S M + 2wM = 0. In view of the definitions of Q and L, this implies QL = −LQ. Equation (5.8) follows by direct computation using the identity tr(QL) = 0 and the four properties in (5.4). The last statement concerning the form of w follows from (4.31) and (5.1) by direct computation. The condition that M does not jump at the endpoints of implies that tr S(±iρ0 ) = 0, i.e. w0 = ρ02 (w2 − w4 ρ02 ).
(5.12)
In particular, the function h(k) defined in (2.12) vanishes at the endpoints of . Lemma 5.2. There exist points {m j }41 ⊂ C such that 8 f 2 2h Q21 (k) = ( f + f¯)( f 0 + f¯0 )( − h )2
4
j=1 (k − m j ) . (k 2 − r12 )
(5.13)
610
J. Lenells
Proof. By (5.6), Q21 is a unique function of k, i.e. Q21 (k + ) = Q21 (k − ). Thus (k 2 − r12 )Q21 is an entire function of k ∈ C. The existence of {m j }41 satisfying (5.13) therefore follows if we can show that (k 2 − r12 )Q21 (k) = As k → ∞, we have f (z) + (z, k ) = f (z)
8 f 2 2h k 4 + O(k 3 ), ( f + f¯)( f 0 + f¯0 )( − h )2
k → ∞.
1 + O(1/k), −1
k → ∞. (5.15)
A(k) = I + O(1/k),
(5.14)
Thus, by (5.3) and (4.31), (k 2 − r12 )Q(z, k + ) = − =
0 1 −1 0
f¯ f
0 8k 4 2 2h
( f 0 + f¯0 )(−h )2
42 2h
( f + f¯)( f 0 + f¯0 )( − h )2
f¯ f
−1 1 − w4 k 4 I + O(k 3 ) −1 2 f¯ k 4 + O(k 3 ), k → ∞. f − f¯
f¯ − f 2f
(5.16) The (21)-entry of this equation yields (5.14). ˆ by Define four points {k j }4j=1 ⊂ C w +1= 2
w42
4
j=1 (k − k j )(k (k 2 − r12 )2
− k¯ j )
.
We assume that the k j ’s are ordered as in Subsect. 2.3. Let Sˆ z denote the double cover of the Riemann surface Sz defined by adding cuts [k j , k¯ j ], j = 1, . . . , 4, both on the upper and lower sheets of Sz . Thus a point (k, ±λ, ±μ) of Sˆ z is specified by giving a ˆ together with a choice of sign of λ and of point k ∈ C ! 4 ! μ = " (k − k j )(k − k¯ j ). j=1
We specify the sheets so that λ → 1 (λ → −1) as k → ∞ on sheets 1 and 2 (sheets 3 and 4), and μ ∼ k 4 (μ ∼ −k 4 ) as k → ∞ on sheets 1 and 3 (sheets 2 and 4). As k crosses the cut [−i z, i z¯ ], λ changes sign whereas the sign of μ remains unchanged. As k crosses any of the other cuts, μ changes sign whereas the sign of λ remains unchanged. Consider the function H defined by √ Lˆ 22 − L21 w 2 + 1 H (z, k) = , k ∈ Sˆ z , (5.17) √ Lˆ 22 + L21 w 2 + 1 √ where Lˆ 22 = √ L21 Q11 + L22 Q21 . We fix the sign of the root w 2 + 1 in (5.17) by 2 + 2 requiring √ that w + 1 = −w4 k + O(k) as k → ∞ . Since w4 > 0, this implies that w 2 + 1 ≥ 0 for k ∈ + . The eigenfunction of the Lax pair (4.1) satisfies det = −2e2U F(k) (see Eq. (2.65) in [24]), so that the entries of L may have poles at
Boundary Value Problems for the Einstein Equations
611
the points which project to the zeros of F(k) in the Riemann k-sphere. However, H has no singularities at these points. Therefore, in view of (5.8), the possible zeros and poles of H belong to the set in Sˆ z which projects to {±r1 } ∪ {m j }41 , and if H has a double pole at (m j , λ, μ), then it has a double zero at (m j , λ, −μ), j = 1, . . . , 4. By the symmetries (5.6), we have Lˆ 22 (k, λ, μ) = Lˆ 22 (k, −λ, μ),
L21 (k, λ, μ) = −L21 (k, −λ, μ).
Therefore H (k, λ, μ) =
1 . H (k, −λ, μ)
Similarly, we have H (k, λ, μ) = 1/H (k, λ, −μ). Consequently, H → 1/H whenever k crosses one of the cuts of the two-sheeted Riemann surface z defined by (2.8). We can therefore view H as a single-valued function on z with the values on the upper sheet given by the values of H on sheet 1 of Sˆ z , and the values on the lower sheet given by the inverses of these values. 5.1. Formulation of the scalar RH problem. We want to formulate a scalar RH problem in terms of the complex-valued function ψ(z, k) defined by ψ(z, k) =
log H (z, k) , y
k ∈ z .
(5.18)
However, since log H is a multi-valued function on z , this definition of ψ needs to be supplemented by a choice of branches for the logarithm. We will fix a single-valued representative of ψ on z by introducing cuts which connect the zeros and poles of H . Across these cuts ψ will jump by multiples of 2πi/y. The problem is that even though (5.8) implies that all zeros and poles of H lie in the cover of the set {±r1 } ∪ {m j }41 , the exact distribution of these zeros and poles is not known. It is therefore not clear at this stage how to make a consistent choice of branches. We address this problem by considering the limit in which the solution f approaches the Kerr solution. For a solution near the Kerr solution, we can utilize the Kerr expressions for F and G to compute H explicitly to first order. This will give us the correct choice of branches in the Kerr limit and by continuity this choice extends also to more general solutions. The Ernst potential for the Kerr black hole rotating with angular velocity h and with a horizon stretching from−ir1 to ir1 is given by f kerr = where R± are defined by R± =
R+ e−iδ + R− eiδ − 2r1 , R+ e−iδ + R− eiδ + 2r1
(±r1 − ζ )2 + ρ 2 ,
and the parameter δ ∈ (−π/2, 0) is related to h by h =
i f 1kerr (1 + ( f 1kerr )2 ) 2r1 (1 − ( f 1kerr )2 )
,
f 1kerr = i tan(δ/2).
612
J. Lenells Table 1. The change arg H in arg H as k traverses each of the cycles in the cut basis {a j , b j }4j=1
Cycle a1 a2 a3 a4
arg H −4π −4π 4π 4π
arg H −8π −8π −12π −4π
Cycle b1 b2 b3 b4
The value at the origin is given by f 0kerr =
cos(δ) − 1 . cos(δ) + 1
We consider adding a slowly rotating disk to the Kerr solution. Using the Kerr values for f 0 and f 1 , we compute w4 , w2 , w0 according to (5.9) with 1. The branch points {k j }41 are found by solving the equation w 2 +1 = 0. As → 0, k1 and k4 tend to infinity, whereas k2 and k3 approach finite values. The spectral functions F kerr (k) and G kerr (k) are given explicitly by (cf. Sect. 2.4 in [24]) F kerr (k) = G kerr (k) =
22h (k 2 − r12 ) + 2ih f 1kerr k − ( f 1kerr )2 22h (k 2 − r12 ) (2ih r1 − f 1kerr )( f 1kerr )2 22h (k 2 − r12 )
,
.
For definiteness, we consider the example of r1 = 1/2 and δ = −1/2. Assuming that z = ρ + iζ with ρ 1 and ζ 1, we compute Q and L to first order by substituting the axis values (4.9a) for together with the values of the Kerr solution into the right-hand sides of (5.2) and (5.3). We find that H has double poles and double zeros at the points in the sets − − − − − + + + + + + {m − 1 , m 2 , −r1 , m 3 , m 4 , r1 } and {m 1 , m 2 , −r1 , m 3 , m 4 , r1 },
respectively. As → 0, m 1 → k1 and m 4 → k4 , whereas m 2 and m 3 converge to values close (but not equal) to k2 and k3 , respectively. Let {a j , b j }41 be the particular cycles on z specified in Remark 2.2. As k traverses each of these cycles, the argument of H changes by the amount arg H according to Table 1. A choice of branches for log H consistent with the above properties is obtained by introducing cuts on z according to Fig. 11. The introduced cuts run from the poles of H to the zeros of H . Letting (log H )+ and (log H )− denote the values of log H for z just to the right and to the left of a cut, respectively, we have (log H )+ + 4πi = (log H )− . An overall choice of branch is made by requiring that log H → 2 log f as k → ∞+ , where an appropriate branch is chosen for log f , see Eq. (5.20) below. ˆ ConseFor this choice of branches, we have log H (k + ) = − log H (k − ), k ∈ C. quently, ψ(z, k) is a unique function of k, i.e. ψ(z, k + ) = ψ(z, k − ),
ˆ k ∈ C.
ˆ → C. ˆ We therefore view ψ(z, ·) as a function C Proposition 5.3. Let ψ be defined by (5.18) with the choice of branches for log H specˆ →C ˆ has the following properties: ified above. Then the function ψ(z, ·) : C
Boundary Value Problems for the Einstein Equations
613
Fig. 11. The additional cuts introduced on z in order to make log H a single-valued function
ˆ ∪ ∪4 [m j , k j ] ∪ {±r1 } . • ψ(z, k) is analytic for k ∈ C\ j=1 • Across , ψ(z, k) satisfies the jump condition 2 ln ψ − (z, k) = ψ + (z, k) + y(z, k + )
√
1 + w2 − w √ 1 + w2 + w
,
k ∈ ,
(5.19)
where ψ + and ψ − denote the values of ψ to the right and left of the cut, respectively. • Across the directed intervals [k j , m j ], j = 1, 2; [m j , k j ], j = 3, 4; [r1 , k3 ], [k¯3 , k2 ], and [k¯2 , −r1 ], ψ(z, k) satisfies the jump condition ψ − (z, k) = ψ + (z, k) +
4πi . y(z, k + )
• As k → m j , j = 1, 2, ψ(z, k) satisfies ψ(z, k) =
2 log(k − m j ), y(z, m +j )
k → m j,
j = 1, 2.
k → m j,
j = 3, 4.
• As k → m j , j = 3, 4, ψ(z, k) satisfies ψ(z, k) =
−2 log(k − m j ), y(z, m +j )
• As k → −r1 , ψ(z, k) satisfies ψ(z, k) =
2 log(k + r1 ), y(z, −r1+ )
k → −r1 .
• As k → r1 , ψ(z, k) satisfies ψ(z, k) =
−2 log(k − r1 ), y(z, r1+ )
k → r1 .
614
J. Lenells
• As k → k j , j = 1, . . . , 4, ψ(z, k) satisfies ψ(z, k) =
2πi , y
k → kj,
j = 1, . . . , 4,
where y = y(z, k + ) for k just to the left of the cut [k j , m j ] for j = 1, 2, and just to the left of the cut [m j , k j ] for j = 3, 4 and is analytically continued around the endpoint k j so that y = y(z, k − ) to the right of the cut. • As k → ∞, ψ→
2 log( f ) + O(1/k 6 ), k5
k → ∞.
(5.20)
Proof. We first show that ψ satisfies the jump condition (5.19). Algebraic manipulation of (5.7) using the identity tr(QL) = 0 and the properties in (5.4) shows that the functions L21 and Lˆ 22 = L21 Q11 + L22 Q21 satisfy L21+ = −2w Lˆ 22− + (1 + 2w 2 )L21− , k ∈ +, (5.21) Lˆ 22+ = (1 + 2w 2 )Lˆ 22− − 2w(1 + w2 )L21− , L21+ = 2w Lˆ 22− + (1 + 2w 2 )L21− , k ∈ −. (5.22) Lˆ 22+ = (1 + 2w 2 )Lˆ 22− + 2w(1 + w 2 )L21− , Thus ±2 √ 1 + w2 − w H+ (k), H− (k) = √ 1 + w2 + w
k ∈ ±.
√ Equation (5.19) follows from here since 1 + w2 ≥ 0 and 1 + w 2 ± w > 0 for k ∈ . The behavior as k → ±r1 and k → m j follows since H has double zeros at −r1 , m +j , j = 1, 2, and double poles at r1 , m +j , j = 3, 4. In order to find the behavior of ψ as k → ∞, we note that, by (5.15), L(z, k + ) =
f¯ f
f¯ 1 σ −1 1 f
1 −1
−1 + O(1/k),
k → ∞.
Thus, by (5.16), 4 1 + f 2 2 2h Lˆ 22 (z, k ) = − k 2 + O(k), ( f + f¯)( f 0 + f¯0 )( − h )2 +
Since
√
k → ∞.
w 2 + 1 = −w4 k 2 + O(k) as k → ∞+ , we find H (z, k + ) = f (z)2 + O(1/k),
k → ∞.
log H 2 log f + O(1/k 6 ), → y k5
k → ∞.
Therefore
Boundary Value Problems for the Einstein Equations
615
5.2. Solution of the scalar RH problem. The solution of the scalar RH problem presented in Proposition 5.3 is
2 dk )2 − w(k ) ψ(z, k) = 1 + w(k ln πi y(z, k + )(k − k) 2 4 dk dk − 2 +2 y(z, k + )(k − k) y(z, k + )(k − k) j=1 [k j ,m j ] j=3 [k j ,m j ] dk +2 , k ∈ z , + + + [r1 ,k3 ] [k¯3 ,k2 ] [k¯2 ,−r1 ] y(z, k )(k − k) (5.23) where we used that
√
√ 1 + w 2 + w = − ln 1 + w2 − w , ln
k ∈ .
By deforming contours, we can replace the last three integrals on the right-hand side of (5.23) with an integral along γ , where γ is the contour on z defined in Sect. 2. We define two divisors K and M on z . K is defined by K=
4
kj,
(5.24)
j=1
whereas M is defined as the sum of the points in z which lie above the set {m j }41 and which are double poles of H , i.e. − + + M = m− 1 + m2 + m3 + m3 .
(5.25)
We can then write (5.23) as
dk 2 )2 − w(k ) ln ψ(z, k) = 1 + w(k πi + y(z, k )(k − k) M dk dk + 2 , −2 y(z, k )(k − k) K γ y(z, k )(k − k) where the integrals are contour integrals on z and the prime on the integral from K to M indicates that the paths of integration do not necessarily lie in the complement of the cut basis {a j , b j }. In view of (5.20), this leads to 4 M 4 k dk k dk k 4 dk − − , (5.26a) h(k) log f = + y y y K
γ n−1 M n−1 k dk k dk k n−1 dk = + , n = 1, . . . , 4, (5.26b) h(k) + y y y K
γ where h(k) is defined by (2.12). Remark 5.4. Although Eqs. (5.26) were derived under the assumption that the solution is a small perturbation of the Kerr solution, they are valid more generally. Indeed, the crucial facts used in the derivation are that H has a double pole at r1+ and a double zero at −r1+ , and these properties are preserved under a continuous deformation. It is conceivable that the double poles of H that make up M will change sheets under such a deformation so that (5.25) has to be modified, but the resulting Eqs. (5.26) remain unchanged.
616
J. Lenells
6. Theta Functions In this section we derive explicit expressions for the Ernst potential f and the metric functions e2U and a in terms of theta functions. 6.1. Explicit expression for the Ernst potential. We will show that the right-hand side of (5.26a) can be expressed in terms of the theta function on z . We will first assume that the integration paths from K to M in (5.26) lie in the fundamental polygon determined by the cut basis; later we will see that the result is the same also when this is not the case. Let {η j }4j=1 denote the noncanonical basis of holomorphic one-forms on z defined in (3.1) and let A be the matrix defined in (3.2). Then the canonical basis is ω = Aη. Let u, I ∈ C4 be defined by (2.13). Applying A to (5.26b), we find M u= ω. (6.1) K
Using that ω∞ + ∞ − = −
k 4 dk + γ T η, y
for some vector γ ∈ C4 , Eq. (5.26a) yields f = e−
M K
ω∞+ ∞− +I
,
(6.2)
where the terms involving γ cancelled because of (5.26b). Formula (2.14) for f will follow if we can prove that ∞−
u − −i z ω M e− K ω∞+ ∞− = (6.3) ∞− , u + −i z ω where (v) := (v|B). Let e( j) and π ( j) denote the j th columns of the 4 × 4 identity matrix I and the period matrix B, respectively. Then (v + e( j) |B) = (v|B),
1
(v + π ( j) |B) = e−2πi(v j + 2 B j j ) (v|B),
v ∈ C4 . (6.4)
The Jacobian J ac(z ) of z is defined as the complex torus C4 /L, where L is the discrete lattice generated by the e( j) ’s and the π ( j) ’s. We define the map ϕ : z → C4 by k ϕ(k) = ω, k ∈ z , −i z
with the contour fixed to lie within the fundamental polygon. Then ϕ composed with the projection C4 → J ac(z ) is the Abel map with base point −i z. We write M = #4 − + 4 j=1 M j , where, for j = 1, . . . , 4, M j = m j or M j = m j . Let K = ϕ(K) ∈ C . An
Boundary Value Problems for the Einstein Equations
617
argument following pp. 322–325 in [6] shows that K projects to the vector of Riemann constants in J ac(z ). Thus the functions (ϕ(P) − ϕ(K) + K) , (ϕ(P) − ϕ(M) + K)
P ∈ z ,
(6.5)
and e
−
P j=1 ∞−
#4
ωM j k j
,
P ∈ z ,
(6.6)
both have simple poles at the points of M and simple zeros at the points of K. Moreover, the general identity ([6], p. 67) R ω R S = 2πi ωj, j = 1, . . . , 4, R, S ∈ z , (6.7) bj
S
implies that as a j is traversed the functions (6.5) and (6.6) both get multiplied by 1 and as b j is traversed they both get multiplied by e−2πi constant and we deduce that e
−
P j=1 ∞−
#4
ωM j k j
=
M K
ωj .
Hence their quotient is a
(ϕ(P) − ϕ(K) + K)(ϕ(∞− ) − ϕ(M) + K) . (ϕ(P) − ϕ(M) + K)(ϕ(∞− ) − ϕ(K) + K)
(6.8)
Using the identity
R S
ω∞+ ∞− =
∞+ ∞−
ωRS ,
R, S ∈ z ,
and the fact that K = ϕ(K), evaluation of (6.8) at P = ∞+ yields M
(ϕ(∞+ )) ϕ(∞− ) − K ω M . e− K ω∞+ ∞− = M
ϕ(∞+ ) − K ω (ϕ(∞− ))
(6.9)
(6.10)
Our choice of the cut basis {a j , b j } implies that ϕ(∞+ ) = −ϕ(∞− ) modulo a-periods.
(6.11)
Thus, since (v) is an even function, we arrive at (6.3). Now suppose the integration paths from K to M in (5.26) do not lie within the fundamental polygon. Then there exist integer vectors p, q ∈ Z4 such that equations (6.1) and (6.2) get replaced by M M # − ω + − − 4j=1 p j b ω∞+ ∞− +I j u= ω + Bp + q and f =e K ∞ ∞ , K
respectively. However, a computation using (6.4), (6.7), and (6.10) shows that the terms involving p and q cancel, so that f is still given by (2.14). We can now complete most of the proof of Theorem 2.1; the derivation of the formula for e2κ will be postponed to the Appendix. We first establish the formulas in (2.15): The expression for h follows from (4.8); the expression for follows by solving (5.9a) for recalling that e2U0 < 0 and w4 > 0; and the expression for e2U (+i0) follows by evaluating (2.4a) at z = +i0.
618
J. Lenells
6.2. The metric functions e2U and a. Using formula (2.14), which was established in the previous subsection, the expression for the metric function e2U in (2.17) can be derived as follows cf. [15]. Since the entries of u are purely imaginary, I ∈ R, and (v) ¯ = (v) for v ∈ C4 ([17], p. 203), Eq. (2.14) yields ⎛ ∞− ∞− ⎞ (u − −i z ω) (u − i z¯ ω) ⎠ eI . (6.12) + f + f¯ = ⎝ ∞− ∞− (u + −i z ω) (u + i z¯ ω) Let E(P, Q) denote the prime form on z . Applying Fay’s identity ([17], p. 205) P3 P4 E(P3 , P1 )E(P2 , P4 ) v+ ω v+ ω E(P3 , P4 )E(P2 , P1 ) P2 P1 P3 P4 E(P3 , P2 )E(P1 , P4 ) v+ ω v+ ω + E(P3 , P4 )E(P1 , P2 ) P1 P2 P4 P3 ω+ ω , v ∈ C4 , = (v) v + P2
P1
(i z¯ , ∞+ , −i z, ∞− )
with (P1 , P2 , P3 , P4 ) = −i z 2Q(0) v + ω v+
∞+
= (v) v +
−i z
∞+
ω+
∞−
i z¯
∞−
i z¯
to (6.12), we find −i z ω − v+ ω v+
i z¯
ω ,
∞−
∞+
ω (6.13)
where Q(0) =
1 E(−i z, i z¯ )E(∞+ , ∞− ) , 2 E(−i z, ∞− )E(∞+ , i z¯ )
and we used that (Lemma 3.12 in [17]) E(−i z, ∞+ )E(i z¯ , ∞− ) = −1. E(−i z, ∞− )E(i z¯ , ∞+ )
∞+ By Proposition 3.11 in [17], Q(0) can be written as in (2.18). Letting v = u + −i z ω = ∞− ∞− ∞− u − −i z ω and dividing by (u + −i z )(u + i z¯ ω), Eq. (6.13) yields ∞+ ∞+ −i z (u + −i z ω) (u + i z¯ ω) (u)(u + i z¯ ω) − = 2Q(0) ∞− ∞− ∞− ∞− . (6.14) (u + −i z ω)(u + i z¯ ω) (u + i z¯ ω) (u + −i z ω) Equations (6.12) and (6.14) lead to the expression for e2U in (2.17). By (5.7) in [16], we have ⎞ ⎛ ∞− ∞− (u + ω + ω) −i z i z¯ − 1⎠ , (a − a0 )e2U = −ρ ⎝ i z¯ Q(0)Q(u)(u + −i z ω)
(6.15)
where a0 ∈ R is a constant determined by the condition that a = 0 on the regular axis. In view of the expression for e2U , this yields the expression for the metric function a given in (2.17a). An alternative derivation of Eq. (6.15) is presented in the Appendix.
Boundary Value Problems for the Einstein Equations
619
Fig. 12. The axis-adapted homology basis {a˜ j , b˜ j }4j=1 on z
7. Axis and Horizon Values In this section we consider the limits of the formulas in Theorem 2.1 as z approaches a point on the ζ -axis. As ρ ↓ 0, the Riemann surface z degenerates since the branch cut [−i z, i z¯ ] shrinks to a point. This type of degeneration of a Riemann surface is analyzed in Chap. III of [7]. In order to utilize the results of [7], we introduce an axis-adapted cut basis {a˜ j , b˜ j }4j=1 on z by T a˜ A = 0 b˜
a 0 b A
⎛
1 ⎜0 A=⎝ 0 0
where
0 1 0 0
0 0 1 0
⎞ −1 −1⎟ . −1⎠ −1
The axis-adapted cut basis is displayed in Fig. 12 in the case when ζ > Re k4 . Note that a˜ 4 surrounds the collapsing cut [−i z, i z¯ ]. According to the transformation formula for theta functions (Eq. (12) in [7]), there exists a constant c0 independent of v and B such that ˜ = c0 (v|B) (v| ˜ B)
(7.1)
whenever v˜ = A−1 v = Av. We define u˜ and I˜ as the analogs of u and I in the axis-adapted basis, i.e. u˜ =
+
h ω˜ +
γ
ω, ˜
I˜ =
+
h ω˜ ∞+ ∞− +
γ
ω˜ ∞+ ∞− .
Since ω˜ = Aω, we have u˜ = Au. Since ω∞+ ∞− = ω˜ ∞+ ∞− − 2πi ω˜ 4 , ˜ ˜ := (v| B), we have I = I˜ − 2πi u˜ 4 . Thus, introducing the shorthand notation (v) the Ernst potential (2.14) can be expressed in terms of the axis-adapted basis as f (z) =
˜ u˜ − ( ˜ u˜ + (
∞− −i z
ω) ˜
−i z
ω) ˜
∞−
˜
e−2πi u˜ 4 + I .
(7.2)
620
J. Lenells
Let denote the degenerated Riemann surface defined in (2.19). According to [7], we have the following expansions as ρ ↓ 0: ω˜ j = ωj + O(ρ 2 ),
1 ω + − + O(ρ 2 ), ω˜ 4 → (7.3a) 2πi ζ ζ ζ+ B˜ i4 → ωi + O(ρ 2 ), i = 1, 2, 3,
j = 1, 2, 3;
B˜ i j → Bi j + O(ρ 2 ), i, j = 1, 2, 3;
ζ−
(7.3b) B˜ 44
M 1 ln ρ + + O(ρ 2 ), = πi πi
(7.3c)
where M ∈ C is a constant. The path of integration from ζ − to ζ + in (7.3b) must be chosen as the limit of the cycle b˜4 . For example, for Re k3 < ζ < Re k4 , this path is [ζ, k¯4 ]− ∪[k¯4 , ζ ]+ . Whether this path lies within the fundamental polygon on depends on the particular representatives of the homology cycles a j and bj . 7.1. Values near the regular axis. We first consider the case when z approaches a point on the regular axis. We define c ∈ C4 and c ∈ C3 by ∞− ∞− ω, ˜ c = ω . (7.4) c= k4
k4
Lemma 7.1. The following limits hold as z approaches a point on the regular axis (i.e. as ρ → 0 with r1 < ζ ):
u˜ = ˜ u˜ − ˜ u˜ + ˜ u˜ +
−i z
˜ u˜ +
∞−
−i z
i z¯
−i z
ω˜ +
∞−
i z¯
˜ u˜ +
∞−
−i z
∞−
∞−
i z¯
J 2πi
+ O(ρ 2 ),
= u −
ω˜
= u +
ω˜
= u +
ω˜ ω˜ ω˜
u
∞− ζ−
∞−
ζ−
∞−
ζ−
c=
c
+ O(ρ 2 ),
K 2πi
ω − u −
ω − u +
ω + u +
∞−
ζ+
∞−
ζ+
∞−
ζ+
(7.5a)
ω e J
−K
+ O(ρ 2 ),
(7.5b)
ω e−J
−K
(7.5c)
ω e−J
+ O(ρ 2 ),
−K
+ O(ρ 2 ),
(7.5d) = (u ) + βρ + O(ρ 2 ),
˜ u) ( ˜ = (u ) − βρ + O(ρ 2 ),
(7.5e) 1 = − (u + 2c )e−J −2K −M + O(1), ρ
(7.5f)
where β ∈ C is a constant and u , K , J , and M are defined in (2.20), (2.21), and (2.29). The equations obtained from (7.5) by replacing u, ˜ u , and J by 0 everywhere are also valid.
Boundary Value Problems for the Einstein Equations
621
Proof. The expansions in (7.5a) follow immediately from (7.3a). We will also show (7.5b); the proofs of the other expansions are similar. We first assume that ζ > Re k4 . In order for the argument of the theta function to have a finite limit, we shift the integration limit from −i z to k4 . Our choice of the cut system {a˜ j , b˜ j } implies that ∞− ˜ + s, ω˜ = c + Br (7.6) −i z
where r, s ∈
R4
are defined by r = (0, 0, 0, 1/2)T ,
Therefore, ˜ u˜ −
∞− −i z
s = (0, 0, 0, −1/2)T .
˜ − s) = ˜ u˜ − c − Br ω˜ = (
1
e2πi( 2 N
T
(7.7)
˜ −s)) ˜ Br B˜ N +N T (u−c−
. (7.8)
N ∈Z4
Using (7.7), we can write the right-hand side of (7.8) as
e
2πi
# 3 1 2
i, j=1
#3 #3 Ni B˜ i j N j + 12 B˜ 44 N4 (N4 −1)+ i=1 Ni (u˜ i −ci )+N4 (u˜ 4 −c4 )+ 12 N4 B˜ i4 Ni (N4 − 21 )+ i=1
.
N ∈Z4
In view of (7.3c), only the terms with N4 = 0 and N4 = 1 give nonzero contributions in the limit ρ → 0. Equations (7.3) and (7.5a) imply that the subleading terms, which also receive contributions from the terms with N4 = −1 and N4 = 2, are of O(ρ 2 ). We find ∞− 2πi 1 N T B N − 1 N T ζ + ω +N T (u −c )
2 2 ζ− ˜ u˜ − ω˜ = e −i z
N ∈Z3
+
e
2πi
1 T 2N
N ∈Z3
=
1 u −c − 2
B N + 21 N T
ζ+
ζ−
ω+N T (u −c ) +J −K +πi
ζ+ ζ−
−
1 u −c + 2
ζ+
+ O(ρ 2 )
eJ
ζ−
−K
+ O(ρ 2 ). (7.9)
Since c +
1 2
ζ+ ζ−
ω =
∞−
ζ−
ω ,
c −
1 2
ζ+ ζ−
ω =
∞−
ζ+
ω ,
(7.10)
this proves (7.5b) in the case when ζ > Re k4 . Similar arguments apply when ζ < Re k3 or Re k3 < ζ < Re k4 . For example, if Re k3 < ζ < Re k4 , then Eq. (7.6) gets replaced by ∞− ˜ + t, ω˜ = c + Br −i z
where r = (0, 0, 0, 1/2)T ,
1 1 1 T 1 t = − ,− ,− ,− . 2 2 2 2
(7.11)
622
J. Lenells
Letting t = (−1/2, −1/2, −1/2), this leads to the following analog of Eq. (7.9): ∞− + 1 ζ ˜ u˜ − ω˜ = u − c − ω −t 2 ζ− −i z + 1 ζ ω − t e J −K + O(ρ 2 ). (7.12) − u − c + 2 ζ− Taking into account that, for Re k3 < ζ < Re k4 , ∞− ∞− + + 1 ζ 1 ζ ω +t = ω , c − ω +t = ω , c + 2 ζ− 2 ζ− ζ− ζ+
(7.13)
we again arrive at (7.5b). Lemma 7.2. Let Q be given by (2.18). As ρ → 0 with r1 < ζ , ⎡ ⎤ ∞− 2 ∞− 2 1 Q(u) = 2 ⎣ u + ω − u + ω e−2J −2K ⎦ + O(ρ 2 ). − + (u ) ζ ζ (7.14) The behavior of Q(0) as ρ → 0 is given by the expression obtained by replacing u and J with zero in the right-hand side of (7.14). Proof. In view of (7.1), the expression for Q(u) is invariant under the change of cut basis from{a j , b j } to {a˜ j , b˜ j }, i.e. Q(u) =
∞−
− ˜ u˜ + ∞ ω) ω) ˜ ( ˜ i z¯ . i z¯ ˜ u) ˜ u˜ + ( ˜ ( ˜ −i z ω)
˜ u˜ + (
−i z
Utilizing the limits of Lemma 7.1, we find (7.14).
(7.15)
˜
By applying the results of Lemma 7.1 to formula (7.2) and using that e I = e I −2πi u˜ 4 = + O(ρ 2 ) as ρ → 0, we find that f is given by (2.22) near the regular axis. The e expression (2.24) for e2U on the regular axis follows by applying the results of Lemma 7.2 Q(0) I to the equation e2U = Q(u) e . The limiting behavior a = O(ρ 2 ) follows from (6.15); the fact that the terms of O(ρ) vanish in the expansion of a is most easily seen from (2.2). The behavior e2κ = 1 + O(ρ 2 ) near the regular axis follows from (2.17b) and the condition that κ = 0 on the regular axis; the fact that the terms of O(ρ) vanish in the expansion of e2κ is most easily seen from (3.7). This completes the proof of Proposition 2.3. I −J
7.2. Values of a0 and K 0 . The constant a0 is determined by (6.15) and the condition that a = 0 on the regular axis. We find ∞− ∞− ρe−2U (u + −i z ω + i z¯ ω) , ζ > r1 . a0 = lim i z¯ ρ→0 Q(0) Q(u)(u + −i z ω)
Boundary Value Problems for the Einstein Equations
623
Substituting into this equation the expression (2.17a) for e2U and passing to the axisadapted basis, we find − − ˜ u˜ + ∞ ω˜ + ∞ ω) ˜ 2πi u˜ − I˜ ρ ( −i z i z¯ 4 e . a0 = lim i z¯ 2 ρ→0 Q(0) ˜ u˜ + ( ω) ˜ −i z
By Lemmas 7.1 and 7.2, this yields a0 = −
(u + 2c ) −M −I −2K (0)4 e . (7.16)
2 ∞− ∞− (u ) ( ζ − ω )2 − ( ζ + ω )2 e−2K
Expression (2.26) for a0 is obtained by letting ζ → ∞ in (7.16). Indeed, since lim
ζ →∞
(0)4
2 = 1 ∞− ∞− ( ζ − ω )2 − ( ζ + ω )2 e−2K
and u , c , I are independent of ζ > r1 , we find (u + 2c ) −I −M −2K e e a0 = − lim . ζ →∞ (u )
(7.17)
The constant M is given by6 (ζ −x)+ 1 M = lim ω + − − 2 ln x − ln 4 − πi . 2 x→0 (ζ −x)− ζ ζ The combination M + 2K remains finite in the limit ζ → ∞ and we find + * + R 1 lim − lim (M + 2K ) = ω∞+ ∞− − 2 ln R − ln 4 − πi . ζ →∞ 2 R→∞ R−
(7.18)
Equations (7.17) and (7.18) imply (2.26). The constant K 0 in (2.17b) is determined by the condition that e2κ = 1 on the regular part of the axis. In order to compute K 0 we first rewrite (2.17b) in terms of the axis-adapted cut system. Note that ω−r + ,−r − = ω˜ −r + ,−r − − 2πi ω˜ 4 , ωr + r − = ω˜ r + r − −2πi ω˜ 4 , ωκ + κ − = ω˜ κ + κ − − 2πi ω˜ 4 . 1
1 1
1
1 1
1 1
1 1
(7.19)
dκ1 dh dk (κ1 ) = 0, we find 1 dh + ˜ u) ˜ u˜ + i z¯ ω) ( ˜ ( −i z ˜ − 2 + dκ1 dk (κ1 ) + h(κ2 )ω˜ κ1+ κ1− (κ2 )+ + h ω˜ −r1+ ,−r1− − + h ω˜ r1+ r1− = K0 e ˜ ( ˜ ξ¯ ω) (0) ξ ˜
Since e2κ
1
1
+
×e
1 2
lim →0
˜ −r + ,−r − − γ ( ) ω˜ r + r − −2 ln γ1 ( ) ω 2 1 1 1 1
,
ζ > r1 .
Taking the limit as ρ → 0 of this expression, we find (2.27). 6 Expressions of this type are considered in [28].
624
J. Lenells
7.3. Values near the black hole horizon. The limits as z approaches the black hole horizon have a slightly different flavor than those considered in the previous subsection, because u˜ 4 diverges as ρ ↓ 0 with 0 < ζ < r1 . In fact, u˜ 4 = −
P 1 ln ρ + + O(ρ 2 ), πi πi
ρ ↓ 0, 0 < ζ < r1 ,
(7.20)
where 1 P = 2
+
hωζ + ζ −
1 + lim 2 →0
γ ( )
ωζ + ζ −
+ 2 ln + ln 4 ,
and γ ( ) denotes the contour γ with the segments which lie above the interval [ζ − , ζ + ] removed.7 Note that
e2P +2M = e J ,
0 < ζ < r1 .
(7.21)
Lemma 7.3. The following limits hold as z approaches a point on the black hole horizon (i.e. as ρ → 0 with 0 < ζ < r1 ): ˜ u˜ −
* ∞− ∞− 1 ω˜ = 2 − u − ω e J −2M −K + ρ −i z ζ − + +
˜ u˜ + ˜ u˜ +
u −
∞
ζ+
ω +
ζ
ω
ζ−
+
e2J −2M −2K
+ O(1),
(7.22)
∞− ∞− ∞− ζ+ ω˜ = u + ω − u + ω + ω e J +K + O(ρ 2 ), −i z
∞−
i z¯
=
ω˜
ζ−
u +
∞− ζ−
ω
ζ−
+
u +
∞− ζ−
ζ−
ω +
ζ+ ζ−
ζ+ 1 = − u + ω e J −M + δ + O(ρ), ρ ζ− −i z ζ+ 1 ˜ u) ( ˜ = u + ω e J −M + δ + O(ρ), ρ ζ− ∞− ∞− ∞− ˜ u˜ + ω˜ + ω˜ = u + 2 ω + O(ρ),
˜ u˜ +
i z¯
ω
e J +K + O(ρ 2 ),
ω˜
−i z
(7.23)
ζ−
i z¯
where δ is a constant. Proof. We prove (7.22) in the case when Re k3 < ζ < r1 ; the proofs of the other identities are similar. For Re k3 < ζ < r1 , ∞− ˜ − t), ˜ u˜ − ˜ u˜ − c − Br ω˜ = ( −i z
7 For 0 < ζ < r , the contour γ contains the covering in the upper sheet of [ζ − , ζ ] and the covering in 1 the lower sheet of [ζ, ζ + ].
Boundary Value Problems for the Einstein Equations
625
where r and t are defined in (7.11). We can write the right-hand side as
e
2πi
N ∈Z4
×e
2πi
#
3 i, j=1
#
3 i=1
Ni B˜ i j N j + 21 B˜ 44 N4 (N4 −1)
#3 #3 Ni (u˜ i −ci )+N4 (u˜ 4 −c4 )− i=1 Ni ti + 21 N4 B˜ i4 Ni (N4 − 21 )+ i=1
.
The factor involving the divergent quantities B˜ 44 and u˜ 4 is
e
2πi
1 ˜ 2 B44 N4 (N4 −1)+N4 u˜ 4
= ρ N4 (N4 −3) e N4 (N4 −1)M +2N4 P (1 + O(ρ 2 )).
Thus the diverging terms are of O(ρ −2 ) and arise when N4 = 1, 2. We find
∞− + 1 2P −K 2πi 21 N T B N + 12 N T ζζ− ω +N T (u −c −t ) ˜ u˜ − ω˜ = − 2 e e ρ −i z 3 +
1 2M +4P −2K e ρ2
e
2πi
N ∈Z
1 T 2N
B N + 32 N T
ζ+ ζ−
ω +N T (u −c −t )
+ O(1).
N ∈Z3
Using (7.13) and (7.21), we find (7.22).
Lemma 7.4. Let Q be given by (2.18). As ρ → 0 with 0 < ζ < r1 , ∞− ∞− ζ+ (u + ζ − ω )2 − (u + ζ − ω + ζ − ω )2 e2J +2K 2 Q(u) = −ρ + O(ρ 4 ), ζ+ −2M 2 2J (u + ζ − ω ) e
Q(0) =
∞− ( ζ −
ω )2
∞− − ( ζ + (0)2
(7.24) ω )2 e−2K
+ O(ρ 2 ).
(7.25)
Proof. By applying the limits in Lemma 7.3 to (7.15), we find the statement for Q(u). The limit of Q(0) is obtained as in the case of the regular axis, since the diverging factors involving u˜ 4 are not present. By applying the limits of Lemma 7.3 to Eq. (7.2), we find that f is given by (2.30) near the horizon. Similarly, the behavior of e2U near the horizon follows by applyQ(0) −2πi u˜ 4 + I˜ e . The ing Lemma 7.4 and the expansion (7.20) to the equation e2U = Q(u) expression for ahor is established by applying the results of Lemmas 7.3 and 7.4 to the axis-adapted version of (6.15). We next show formula (2.32) for the behavior of e2κ near the horizon. Equation (3.7) implies that the terms of O(ρ) in the expansion of e2κ vanish, so we only have to determine the leading term. Suppose 0 < ζ < r1 . Then, recalling (7.19) and using that + dκ1 dh dk (κ1 ) = 0, we can write L as defined in (2.16) in terms of axis-adapted quantities as 1 dh L=− dκ1 (κ1 ) h(κ2 )ω˜ κ + κ − (κ2+ ) + h ω˜ −r + ,−r − + h ω˜ r + r − 1 1 1 1 1 1 + 2
dk
+
1 + lim ω˜ + − + ω˜ r + r − − 2 ln − 2πi h ω˜ 4 − 2πi u˜ 4 . 1 1 2 →0 γ1 ( ) −r1 ,−r1 γ2 ( )
+
626
J. Lenells
In view of (7.20) and (7.21), we find
e L = ρ 2 e L +2M −J + O(ρ),
ρ → 0, 0 < ζ < r1 ,
where L is defined by (2.28). Applying this expansion together with Lemma 7.3 to the expression for e2κ in Theorem 2.1, we arrive at (2.32). This completes the proof of Proposition 2.4. We conclude this section by proving Proposition 2.5. As ζ ↓ r1 , we have e−J → 0 whereas e K tends to a bounded constant. The expression (2.36) for f 1 := f (ir1 ) therefore follows immediately from (2.23). Similarly, the statement in Proposition 2.5 regarding f 0 := f (+i0) follows by taking the limit ρ ↓ 0 in (2.31). 8. Parameter Ranges In this section we consider the singularity structure of the solution (2.14) and its dependence on the four parameters ρ0 , r1 , w2 , and w4 . 8.1. Singularity structure. The solution f presented in (2.14) is continuous but not smooth at the point z = ±ir1 , where the regular axis meets the horizon. Moreover, Im f has a jump across the disk. Away from these points f is smooth except possibly at points in the set where the denominator of (2.14) vanishes. Physically, we are interested in solutions which are singularity-free away from the disk and the horizon. A complete characterization of the singularity-free solutions involves , - determining for which choices ∞− of ρ0 , r1 , w2 , w4 the set z ∈ D | (u + −i z ω) = 0 is empty. We will not complete this analysis here, but we will indicate how a large class of singularity-free solutions can be constructed starting with parameters corresponding to a Kerr background. In Subsec. 5.1, the Kerr solutions were parametrized in terms of the parameters r1 > 0 and −π/2 < δ < 0. However, since the map π
2 : − , 0 → R>0 δ → w2 = (8.1) tan δ sin δ 2 is one-to-one, we may also adopt a parametrization in terms of r1 > 0 and w2 = 2/(tan δ sin δ) > 0. Let frkerr denote the unique Kerr solution corresponding to the 1 w2 parameters r1 > 0 and w2 > 0. Moreover, let f denote the solution in (2.14) corresponding to some strictly positive parameters ρ0 , r1 , w2 , w4 . Then f → frkerr as ρ0 ↓ 0 1 w2 and w4 ↓ 0 with r1 , w2 held fixed. Indeed, consider perturbing a Kerr background solution f kerr by adding a small disk of radius ρ0 rotating with angular velocity . In the limit ρ0 ↓ 0 and ↓ 0, the jump contour + in the RH problem disappears and reduces to the identity matrix. Thus, the BVP (2.6) reduces to the Kerr black hole BVP and the perturbed solution f approaches f kerr in this limit. Substituting the Kerr values of f 0 and f 1 and letting ↓ 0 in (5.9), we find that w0 → 0,
w2 →
( f 0kerr )2 − 1 2 f 0kerr
=
2 , tan δ sin δ
and
w4 → 0,
(8.2)
as the Kerr background is approached. This leads to the relation (8.1) between δ and w2 . In view of (5.12), the vanishing limiting value of w0 is achieved by letting ρ0 ↓ 0. Thus, for small values of w4 > 0 and ρ0 > 0, the solution f corresponding to {ρ0 , r1 , w2 , w4 } is a small perturbation of the Kerr background solution frkerr . In par1 w2 ticular, f is singularity-free for sufficiently small perturbations. By increasing w4 and
Boundary Value Problems for the Einstein Equations
627
Fig. 13. The dependence on w4 of the parameters h , , e2U0 = Re f (+i0), and e2U0 = Re f (+i0) for the example specified by (8.3)
ρ0 , larger perturbations of the background are obtained until the construction eventually breaks down and the solutions become singular. In this way, a large class of singularity-free solutions can be constructed. Numerical data suggest that given strictly positive values of the parameters ρ0 , r1 , and w2 , there exists an interval [0, w4max ], w4max > 0, such that all solutions f corresponding to {ρ0 , r1 , w2 , w4 } with w4 ∈ (0, w4max ) are free of singularities. In the following subsection, we illustrate the general situation by considering a typical example. 8.2. Dependence on w4 . We let
ρ0 = 1,
r1 =
1 , 2
w2 = 3,
(8.3)
and consider the solution f given in (2.14) corresponding to {ρ0 , r1 , w2 , w4 } as w4 > 0 varies. We find that the solution is free of singularities for 0 < w4 < w4max , where w4max ≈ 0.27051. The example presented in Sect. 3 corresponds to taking w4 = 1/10. The dependence on w4 of several parameters is displayed in Figs. 13 and 14. The parameter w4 is analogous to the variable μ used in [24] to parametrize the Neugebauer-Meinel solutions and Figs. 13 and 14 are the analogs of Fig. 2.9 in [24]. To see how the solution f becomes singular as w4 increases beyond w4max , we note that as w4 < w4max increases, the ergosphere of the solution f grows larger and larger until it eventually, in the limit w4 ↑ w4max , envelops all of spacetime. As w4 increases beyond w4max a singularity of f enters the domain D at z = +∞ and moves inward along the positive real axis. The graph of the singular function Re f for w4 = 1/2 > w4max is shown in Fig. 15.
628
J. Lenells
Fig. 14. The dependence on w4 of the parameters b0 = Im f (+i0) and f 1 = f (ir1 ) for the example specified by (8.3)
Fig. 15. The graph of Re f for w4 = 1/2 > w4max . The disk and the black hole are too small to be visible
Acknowledgement. The author is grateful to M. Ehrnström and A. S. Fokas for helpful remarks on a first version of the manuscript and to the two referees for several valuable suggestions.
Appendix A. Condensation of Branch Points In this Appendix we show that the Ernst potential (2.14) is related via a certain limiting procedure to the class of solutions of the Ernst equation studied in [19,15]. By applying this limiting procedure to the formula for the metric function e2κ given in [15], we will also establish the expression (2.17b) for e2κ and so complete the proof of Theorem 2.1. The limiting operation involves partially degenerating a Riemann surface by letting branch points coalesce along the curve + and at the points ±r1 . The construction of new solutions of the Ernst equation through this type of ‘condensation’ of branch points along curves was first described in [19]. ˆ z be a Riemann surface of genus g > 4 obtained by adding g − 4 branch cuts Let g−4 ˆ z is defined by the equation {[E j , F j ]} j=1 to z . Let ξ = −i z. Then yˆ 2 = (k − ξ )(k − ξ¯ )
g−4 4 (k − ki )(k − k¯i ) (k − E i )(k − Fi ). i=1
i=1
Boundary Value Problems for the Einstein Equations
629
ˆ z which is the natural generalization of the basis Let {aˆ j , bˆ j } j=1 be the cut basis on 4 {a j , b j }1 on z , i.e. for j = 1, . . . , 4, aˆ j surrounds the cut [k j , k¯ j ]; for j = 5, . . . g, aˆ j surrounds the cut [E j−4 , F j−4 ]; the cycle bˆ j enters the upper sheet on the right side of [−i z, i z¯ ] and exits again on the right side of [k j , k¯ j ] for j = 1, . . . , 4 and on the right side of [E j−4 , F j−4 ] for j = 5, . . . , g. For simplicity, we will assume that ζ > Re k4 . ˆ be ˆ w) ˆ := (w| ˆ B) Let ωˆ = (ωˆ 1 , . . . , ωˆ g )T denote the canonical dual basis and let ( the associated theta function. Let p, q ∈ Cg be vectors which are indepedent of z and which satisfy the reality condition Bˆ p + q ∈ Rg . The theta function with characteristics p, q ∈ Rg is defined by
1 T ˆ T ˆ p ˆ = (vˆ + Bˆ p + q| B)e ˆ 2πi 2 p B p+ p (v+q) (v| ˆ B) , vˆ ∈ Cg . q g
Then p ∞+ ˆ ˆ ( ξ ω) q , fˆ = − ˆ p ( ∞ ω) ˆ ξ q
(A.1)
is a solution of the Ernst equation (2.3) and the corresponding metric function e2κˆ is given by p ξ¯ p ˆ ˆ ( ξ ω) (0) ˆ q q e2κˆ = Kˆ 0 , (A.2) ˆ ( ˆ ξ¯ ω) (0) ˆ ξ where Kˆ 0 ∈ C is a constant determined by the condition that e2κˆ = 1 on the regular axis [19,15]. We choose E 1 = −r1 − i ,
F1 = E¯ 1 ,
E 2 = r1 − i ,
F2 = E¯ 2 ,
ˇ z of genus 6 by where > 0 is a small number, and define the Riemann surface yˇ 2 = (k − ξ )(k − ξ¯ )
4 2 (k − ki )(k − k¯i ) (k − E i )(k − Fi ). i=1
i=1
ˇ z is obtained from z by adding two short vertical cuts centered at −r1 In other words, and r1 , respectively. The cut basis {aˇ j , bˇ j }6j=1 is shown in Fig. 16. The condensation of branch points will now proceed in two steps: In the first step, we let the branch points E j+2 , F j+2 , j = 1, . . . , g − 6 condense along the curve . In doing ˇ z and the Ernst potential fˆ approaches ˆ z degenerates to this, the Riemann surface ˇ ˇ z . Intuitively fˇ has a disk, but no a solution f defined in terms of theta functions on ˇ z degenerates to z and we will black hole. In the second step, we let → 0. Then find that fˇ approaches the solution f in (2.14).
630
J. Lenells
ˇz Fig. 16. The homology basis {aˇ j , bˇ j }6j=1 on the Riemann surface
A. 1. The first degeneration. Let p = ( p, ˇ m) ∈ C6 × Rg−6 ,
q = 0,
where the components of the vector m ∈ Rg−6 satisfy 0 < m j < 1/2, j = 1, . . . , g − 6. We consider the limit E j+2 , F j+2 → κ j , j = 1, . . . , g − 6, in which the branch cut [E j+2 , F j+2 ] shrinks to a point κ j ∈ . In this limit, (cf. Eq. (7.3)) 1 1 ; ωˆ 1 , . . . ωˆ g → ωˇ 1 , . . . ωˇ 6 , ωˇ κ + κ − , . . . , ωˇ κ + κ − 2πi 1 1 2πi g−6 g−6 κ+ j ˆ ˇ ˆ Bi j → Bi j , i, j = 1, . . . , 6; Bi, j+6 → ωˇ i , i = 1, . . . , 6, j = 1, . . . , g − 6; 1 Bˆ i+6, j+6 → 2πi
κ− j
κ +j
κ− j
ωˇ κ + κ − , i
i
i, j = 1, . . . , g − 6, i = j;
1 ln |E j+2 − F j+2 | + O(1), j = 1, . . . , g − 6. Bˆ j+6, j+6 = πi ˆ z , we have For two points P, Q ∈
Q Q
2πi 1 N T Bˆ N +N T ( Q ω+Bp) ˆ 2πi 21 p T Bˆ p+ p T P ωˆ p P 2 ˆ ωˆ = e . (A.4) e 0 P N ∈Zg
Letting N = ( Nˇ , n) ∈ Z6 ×Zg−6 and using that p = ( p, ˇ m), we find that the factor in the sum on the right-hand side involving the diverging quantities Bˆ j+6, j+6 , j = 1, . . . , g −6, is eπi
#g−6 j=1
n j (n j +2m j ) Bˆ j+6, j+6
.
Consequently, since 0 < m j < 1/2 by assumption, all terms in the sum in (A.4) approach zero except the ones with n ≡ 0. We infer that the sum on the right-hand side of (A.4) converges to ⎛ ⎞ Q κ+ g−6 j ˇ ⎝ ωˇ + mj ωˇ + Bˇ pˇ ⎠ . (A.5) P
j=1
κ− j
Boundary Value Problems for the Einstein Equations
631
We let the κ j ’s condense onto the curve with a density determined by the measure dm(κ) defined by dm(κ) = −
1 dh (κ)dκ, 2 dκ
κ ∈ ,
where h is the function defined in (2.12). Then, integrating by parts and using that h vanishes at the endpoints of , we find
g−6
mj
κ +j
κ− j
j=1
ωˇ →
dm(κ)
κ+
κ−
ωˇ = u, ˇ
(A.6)
where uˇ ∈ C6 is defined by uˇ =
+
h ω. ˇ
Combining (A.4)-(A.6), we find Q Q Q p pˇ ˇ ˆ ˇ uˇ + ωˆ → ωˇ e L/2+ dm(κ) P ωˇ κ + κ − , 0 0 P P
(A.7)
where Lˇ is defined by 1 Lˇ = − 2
dh (κ1 ) dk
dκ1
h(κ2 )ωκ + κ − (κ2+ ), 1 1
and the prime on the integral along indicates that the integration contour should be deformed slightly before evaluation so that the pole at κ2 = κ1 is avoided.8 Applying this formula to (A.1), we arrive at the following limit of fˆ: ∞+ pˇ ˇ (uˇ + ξ ω) ˇ 0 ˇ eI , (A.8) fˆ → fˇ = ∞− p ˇ ˇ (uˇ + ξ ω) ˇ 0 where Iˇ ∈ R is defined by Iˇ =
dm(κ)
∞+
∞−
ωˇ κ + κ − =
+
h ωˇ ∞+ ∞− .
Moreover, applying Eq. (A.7) to the expression for e2κˆ in (A.2), we find
e
2κˆ
→e
2κˇ
= Kˇ 0
ˇ u) ˇ uˇ + ξ¯ ω) ξ¯ ( ˇ ( ˇ ξ ˇ e L e dm(κ) ξ ωˇ κ + κ − , ξ¯ ˇ ( ˇ (0) ˇ ξ ω)
8 The result is indepedent of whether the contour is deformed to the right or to the left of the pole.
(A.9)
632
J. Lenells
where Kˇ 0 is a constant independent of z. For some constant C, we have P P − ξ¯ − P ωξ ξ¯ ˇ z, e 0 =C , P∈ P −ξ because both sides have simple poles at ξ , simple zeros at ξ¯ , and are analytic elsewhere ˇ z . Hence, on ξ¯ κ+ P − ξ¯ κ + ωˇ κ + κ − = − ωξ ξ¯ = log − ∈ πi + 2πiZ . P − ξ P=κ κ− ξ It follows that the last exponential factor in (A.9) is independent of z and can be absorbed into Kˇ 0 . Thus, ˇ u) ˇ uˇ + ξ¯ ω) ( ˇ ( ˇ ξ ˇ 2κˇ ˇ (A.10) e = K0 eL . ξ¯ ˇ ˇ (0)( ξ ω) ˇ ˇ z as the cuts A. 2. The second degeneration. We now consider the degeneration of centered at ±r1 collapse. In the limit → 0, 1 1 ω + −, ω+ − ; ωˇ 1 , . . . ωˇ 6 → ω1 , . . . ω4 , 2πi −r1 ,−r1 2πi r1 r1 B B1 + O( 2 ), Bˇ = B1T B2 where B is the period matrix on z , the 4 × 2 matrix B1 is defined by + + −r1 r1 B1 = ω ω , −r1−
r1−
and the 2 × 2 matrix B2 is given by ⎛ 1 πi (ln + c− ) B2 = ⎝ −r1+ 1 2πi −r − ωr + r − 1
1 2πi
r1+ r1−
ω−r + ,−r − 1
1 πi (ln
1 1
where c+ , c− ∈ C are constants. Moreover, ⎛
+
+ c+ )
1
⎞ ⎠,
⎞
hω
⎟ ⎜ 1 + hω−r + ,−r − ⎟ . uˇ → ⎜
2πi ⎝ 1 1 ⎠ 1 2πi + hωr + r − 1 1
We have ⎞ ⎛
Q Q Q
2πi 1 Nˇ T Bˇ Nˇ + Nˇ T (u+ ˇ p) ˇ ω ˇ + B ˇ 2πi 21 pˇ T Bˇ pˇ + pˇ T (u+ ˇ P ω) ˇ p P 2 ⎠ ⎝ ˇ uˇ + e ωˆ = e . 0 P Nˇ ∈Z6
(A.11)
Boundary Value Problems for the Einstein Equations
633
Letting Nˇ = (N , n) ∈ Z4 × Z2 and choosing, for some 0 < α < 1/2, pˇ = (0, 0, 0, 0, α, −α), the same type of argument that led to (A.5) shows that all terms in the sum approach 0 as → 0 except those with n ≡ 0. It follows that the sum in (A.11) converges to + + Q −r1 r1 u+ ω where u= hω + α − ω. −r1−
+
P
r1−
On the other hand, the rightmost exponential factor in (A.11) can be written as
e
2πi
1 2
Q
pˇ T Bˇ p+ ˇ P ω) ˇ ˇ pˇ T (u+
= eπiα ×e
2(B ˇ 55 −2 Bˇ 56 + Bˇ 66 )
α
+
hω−r + ,−r − −α 1
1
hωr + r − +α
+
Q
1 1
P
ω−r + ,−r − −α 1
Q P
1
ωr + r − 1 1
.
Applying these formulas to (A.8), we find fˇ → f =
(u + (u +
where
I =
+
hω∞+ ∞− + α
The contours in the integrals
−r1+ −r1−
and
r1+ r1−
∞+ ξ
ω)
ξ
ω)
∞−
−r1+ −r1−
−
r1−
r1+
(A.12)
ω∞+ ∞− .
r1−
are the limits of the cycles bˇ5 and bˇ6 , respec-
tively. By deforming these contours, we find that + + −r1 r1 − ω∞+ ∞− , ω∞+ ∞− = −r1−
eI ,
−r1+
−r1−
γ
ω−
r1+
ω=
r1−
γ
ω,
where γ is the contour on z defined in Sect. 2. Therefore, in the limit α → 1/2, u and I become exactly the u and I of Theorem 2.1 and the solution f in (A.12) becomes the Ernst potential in (2.14). This provides the promised link between the solutions in [19] and the solution presented in this paper. By applying the same limiting procedure to Eq. (A.10) we will determine the corresponding metric function e2κ . We have e
πiα 2 ( Bˇ 55 −2 Bˇ 56 + Bˇ 66 )
=
2α 2
e
−r + α 2 c+ +c− − −1 ωr + r − −r1
1 1
(1 + O( )),
→ 0.
Hence, for the quotient in (A.10) we find ξ¯ −r + ˇ u) ˇ uˇ + ξ¯ ω) (u)(u + ξ ω) 4α 2 2α 2 (c+ +c− )−2α 2 −r −1 ωr + r − ˇ ( ˇ ( ξ 1 1 1 Kˇ 0 = e ξ¯ ˇ ( ˇ ξ¯ ω) (0) ˇ (0)( ω) ξ ξ
×e
+
hω−r + ,−r −− + hωr + r − 1
1
1 1
e
1 ξ¯ 2 ξ
ω−r + ,−r −−21 1
1
ξ¯ ξ
ωr + r − 1 1
(1 + O( )). (A.13)
634
J. Lenells
This expression vanishes in the limit → 0. However, this behavior is compensated by the fact that the constant Kˇ 0 diverges as → 0, so that the limit of e2κˇ is finite and non-zero. The last exponential factor on the right-hand side of (A.13) can be absorbed into K 0 . Indeed, the same type of argument that we used to find (A.10) shows that this factor is independent of z. The constants c+ and c− are given by the expressions obtained by replacing ζ in the right-hand side of (2.29) with r1 and −r1 , respectively. Letting α → 1/2 and using that (r1 −δ)+ r+ (−r1 −δ)+ 1 lim ω−r + ,−r − + ωr + r − − 4 ln δ − 2 ω−r + ,−r − δ→0
(−r1 −δ)−
1
(r1 −δ)−
1
= 2 lim
δ→0
1 1
γ1 (δ)
ω−r + ,−r − − 1
1
γ2 (δ)
r1−
1
1
ωr + r − − 2 ln δ , 1 1
we infer that the limit of e2κˇ is given by (2.17b). We finally point out that formula (6.15) for the metric function a can be derived in a similar way. Indeed, the metric function aˆ corresponding to the solution fˆ in (A.1) is given by [15] ⎞ ⎛ − − ˆ p (0) ˆ p ( ∞ ωˆ + ¯∞ ω) ˆ ξ ξ ⎟ ⎜ q q ˆ − 1⎟ (aˆ − aˆ 0 )e2U = −ρ ⎜ ⎠, ⎝ − − ˆ ˆ p ( ¯∞ ω) ˆ p ( ∞ ω) ˆ ˆ Q(0) ξ ξ q q where ˆ Q(0) =
ˆ (
∞−
− ˆ ¯∞ ω) ω) ˆ ( ˆ ξ . ξ¯ ˆ ( ˆ (0) ω) ˆ ξ
ξ
An application of the above limiting procedure to this expression yields (6.15). References [1] Abramowicz, A., et al: Theory of black hole accretion discs. Edited by M. A. Abramowicz, G. Björnsson, J. E. Pringle. Cambridge: Cambridge University Press, 1999 [2] Bardeen, J.M., Wagoner, R.V.: Uniformly rotating disks in general relativity. Astrophys. J. 158, L65–L69 (1969) [3] Bardeen, J.M., Wagoner, R.V.: Relativistic disks. I. Uniform rotation. Astrophys. J. 167, 359–423 (1971) [4] Biˇcák, J.: Selected solutions of Einstein’s field equations: their role in general relativity and astrophysics. In: Einstein’s field equations and their physical implications, Edited by B. G. Schmidt, Lecture Notes in Physics, Vol. 540, Berlin: Springer-Verlag, 2000, pp. 1–126 [5] Chandrasekhar, S.: The mathematical theory of black holes. Reprint of the 1992 edition, Oxford Classic Texts in the Physical Sciences, New york: The Clarendon Press Oxford University Press, 1998 [6] Farkas, H. M., Kra, I.: Riemann surfaces. 2nd edition, Graduate Texts in Mathematics 71, New York: Springer-Verlag, 1992 [7] Fay, J. D.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics 352, Berlin-New York: Springer-Verlag, 1973 [8] Fokas, A.S.: A unified transform method for solving linear and certain nonlinear PDEs. Proc. Roy. Soc. Lond. A 453, 1411–1443 (1997) [9] Fokas, A.S.: Integrable nonlinear evolution equations on the half-line. Com. Math. Phys. 230, 1–39 (2002) [10] Fokas, A. S.: A unified approach to boundary value problems. CBMS-NSF regional conference series in applied mathematics, Philadelphia: SIAM, 2008
Boundary Value Problems for the Einstein Equations
635
[11] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett 19, 1095–1097 (1967) [12] Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley-Interscience [John Wiley & Sons], 1978 [13] Kerr, R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237–238 (1963) [14] Klein, C.: Counter-rotating dust rings around a static black hole. Class. Quantum Grav 14, 2267– 2280 (1997) [15] Klein, C., Korotkin, D., Shramchenko, V.: Ernst equation, Fay identities and variational formulas on hyperelliptic curves. Math. Res. Lett. 9, 27–45 (2002) [16] Klein, C., Richter, O.: Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces. Phys. Rev. D 58, 124018 (1998) [17] Klein, C., Richter, O.: Ernst equation and Riemann surfaces. Analytical and numerical methods. Lecture Notes in Physics, 685. Berlin; Springer-Verlag, 2005 [18] Korotkin D., A.: Finite-gap solutions of stationary axisymmetric Einstein equations in vacuum, Theoret. and Math. Phys. 77, 1018–1031 (1989) [19] Korotkin, D. A., Matveev, V. B.: On theta-function solutions of the Schlesinger system and the Ernst equation. (Russian) Funkt. Anal. i Pril. 34(4), 18–34, 96 (2000); translation in Funct. Anal. Appl. 34(4), 252–264 (2000) [20] Lenells, J., Fokas, A.S.: Boundary value problems for the stationary axisymmetric Einstein equations: a rotating disk. Nonlinearity 24, 177 (2011) [21] Neugebauer, G., Meinel, R.: The Einsteinian gravitational field of the rigidly rotating disk of dust. Astroph. J 414, L97–L99 (1993) [22] Neugebauer, G., Meinel, R.: General relativistic gravitational field of a rigidly rotating disk of dust: Axis potential, disk metric, and surface mass density. Phys. Rev. Lett. 73, 2166–2168 (1994) [23] Neugebauer, G., Meinel, R.: General relativistic gravitational field of a rigidly rotating disk of dust: Solution in terms of ultraelliptic functions. Phys. Rev. Lett. 75, 3046–3047 (1995) [24] Meinel, R., Ansorg, M., Kleinwächter, A., Neugebauer, G., Petroff, D.: Relativistic figures of equilibrium. Cambridge: Cambridge University Press, 2008 [25] Pringle, J.E.: Accretion discs in astrophysics. Ann. Rev. Astron. Astrophys. 19, 137–162 (1981) [26] Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact solutions of Einstein’s field equations. Second Edition. Cambridge: Cambridge University Press, 2003 [27] Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focussing and one- dimensional self-modulation in nonlinear media. Soviet Physics-JETP 34, 62–69 (1972) [28] Yamada, A.: Precise variational formulas for abelian differentials. Kodai Math. J. 3, 114–143 (1980) Communicated by P.T. Chru´sciel
Commun. Math. Phys. 304, 637–647 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1244-7
Communications in
Mathematical Physics
Construction of N -Body Initial Data Sets in General Relativity Piotr T. Chru´sciel1 , Justin Corvino2 , James Isenberg3 1 Gravitationsphysik, University of Vienna, A1190, Vienna, Austria.
E-mail: [email protected]
2 Department of Mathematics, Lafayette College, Easton, PA 18042, USA.
E-mail: [email protected]
3 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA.
E-mail: [email protected] Received: 16 March 2010 / Accepted: 28 October 2010 Published online: 24 April 2011 – © Springer-Verlag 2011
Abstract: Given a collection of N solutions of the (3 + 1) Einstein constraint equations which are asymptotically Euclidean and vacuum near infinity, we show how to construct a new solution of the constraints which is itself asymptotically Euclidean, and which contains specified sub-regions of each of the N given solutions. This generalizes earlier work which handled the time-symmetric case, thus providing a construction of large classes of initial data for the many body problem in general relativity. 1. Introduction An important problem in any theory of gravitation is the description of the motion of many-body systems. Various approximation schemes have been proposed to analyze this in general relativity, but no rigorous treatment has been provided thus far. A first step towards a solution of this question is to provide wide classes of initial data which solve the general relativistic constraint equations, and which are relevant to the problem at hand. There exists a rich family of initial data sets modeling isolated gravitational systems, but the non-linear nature of the constraints must be addressed when attempting to incorporate several such systems into a single one. In recent work [4] we have shown how this can be done by a gluing construction, under the restrictive hypothesis of time-symmetry. The aim of this work is to remove this restriction. Given a Riemannian metric g and a symmetric (0, 2)-tensor K on three-manifold M, the Einstein constraints map can be written in the form1 −2(g jk K i j;k − (K j j );i ) . Φ(g, K ) = R(g) − K i j K i j + (K i i )2 The condition then for (g, K ) to be the first and second fundamental forms of M embedded in a Ricci-flat space-time is that the vacuum constraint equations Φ(g, K ) = (0, 0) be satisfied. 1 Here and throughout this paper, we use the Einstein summation convention.
638
P. T. Chru´sciel, J. Corvino, J. Isenberg
We recall that (g, K ) on the exterior E of a ball in R3 constitutes an asymptotically Euclidean end (to order ) provided there are coordinates in which, for multi-indices |α| ≤ + 1, |β| ≤ , |∂ α (gi j − δi j )(x)| = O(|x|−|α|−1 ),
|∂ β K i j (x)| = O(|x|−|β|−2 ),
(1)
where ∂ denotes the partial derivative operator. Note that throughout the rest of this work, we require that ≥ 2. We say that (M, g, K ) is asymptotically Euclidean if M is the union of a compact set and a finite number of ends, all of which are asymptotically Euclidean for (g, K ) in the sense defined above. One readily verifies that every asymptotically Euclidean (AE) end possesses a well-defined energy-momentum vector (m, p). The starting point for constructing initial data for an N body system is the choice of the bodies. Each body is separately designated by the choice of an AE initial data set, and by the specification of a fixed interior region in each such AE solution. To perform this construction, we also specify a vacuum AE end disjoint from the interior region in each of the N data sets. We choose a collection of points which roughly locate the N bodies on a fiducial flat background. We then construct a new initial data set on a manifold obtained by excising a neighborhood of infinity in each chosen end, and then gluing these into a manifold Mext which is R3 with N disjoint balls, centered around the chosen points, removed. In this way, the bodies are made to interact, by gluing the various ends into a fixed end. We now construct initial data on the resulting manifold which: i) is identical to the initial data from the N bodies away from Mext , hence containing N chosen regions isometric to the specified interior regions of the bodies; ii) solves the vacuum constraints on Mext ; iii) is identical to a space-like slice of a Kerr space-time sufficiently far from the bodies; and iv) has the centers of the bodies in a configuration which is a scaled version of the chosen configuration, where the scale factor can be chosen arbitrarily above a certain threshold. We emphasize that we preserve the original solutions away from a neighborhood of the gluing region. If the bodies are vacuum to begin with, we produce a solution to the vacuum constraints everywhere, but we can allow the initial data sets to be non-vacuum away from the chosen AE end. The construction actually produces a family of solutions depending on a parameter ; roughly speaking, the bodies cannot be arbitrarily close together, but must be separated by a distance above a certain threshold. For each , the distance between the bodies is on the order O( −1 ); the smaller is taken, the further apart the bodies will be, and thus the weaker will be their initial interaction. We note that the energy-momentum four-vector of Mext in the resulting solution tends to the sum of the four-vectors of the bodies as → 0+ . We now state our main theorem. For clarity of exposition, we assume that the initial data sets are C ∞ -smooth, and refer the interested reader to [6,8] to formulate the case of a finite degree of regularity. Theorem 1. For each k = 1, . . . , N , let (E k , g k , K k ) be a three-dimensional AE end which solves the vacuum constraints Φ(g k , K k ) = 0, with time-like energy-momentum four-vector (m k , pk ). Let Uk ⊂ E k be a pre-compact neighborhood of the boundary N ∂ E k , and let Mext = R3 \ k=1 Bk , where B 1 , . . . , B N are pairwise disjoint closed balls in R3 . There is an 0 > 0 so that for 0 < < 0 , there is a solution (Mext , g , K ) of the vacuum constraint equations, with one AE end, containing the disjoint union N k k −1 k=1 (Uk , g , K ), so that the distances between distinct Uk are O( ). Near infinity
Construction of N -Body Initial Data Sets in General Relativity
639
(Mext , g , K ) is isometric to a space-like slice of a Kerr ADM energy metric, with the N N momentum (m(g ), p ) satisfying m(g ) − k=1 m k < and p − k=1 pk < . The construction described here allows us to glue together any finite number of asymptotically Euclidean ends which solve the vacuum constraint equations, and the construction is local near infinity in each end; i.e., any given compact subset of the end can be realized isometrically in the final metric g . The local nature of the construction implies that we can allow the N original solutions to have multiple ends, and we can also allow nonzero matter fields supported outside a neighborhood of infinity in the chosen ends, as indicated below. Corollary 1. Let (Mk , g k , K k ), k = 1, . . . , N , be three-dimensional initial data sets with vacuum AE ends E k ⊂ Mk of respective time-like ADM energy-momentum (m k , pk ). Let Uk ⊃ Mk \E k be chosen subdomains with E k ∩ Uk precompact. There is an 0 > 0 so that for 0 < < 0 , there is an initial data set (M, g , K ) which contains N a region U isometric to k=1 (Uk , g k ), for which (M\U, g , K ) has one AE end, with the same properties as those of (Mext , g , K ) as in Theorem 1. 2. Preliminaries 2.1. Kerr-Schild coordinates. In order to establish certain estimates needed for the gluing carried out below, it will be convenient to use the explicit Kerr-Schild form of the Kerr metric (cf., e.g., [2]): 2m r˜ 3 m , gμν = ημν + 4 θ θ = η + O μ ν μν r˜ + a 2 z 2 |x| where η is the Minkowski metric (we are using the signature (−, +, +, +)), with θμ d x μ = dt −
r˜ 2
z 1 r˜ (xd x + ydy) + a(xdy − yd x) − dz , 2 +a r˜
where (x 0 , x 1 , x 2 , x 3 ) = (t, x, y, z), and where r˜ is defined implicitly as the solution of the equation r˜ 4 − r˜ 2 (x 2 + y 2 + z 2 − a 2 ) − a 2 z 2 = 0 . We will continue to use the term Kerr-Schild coordinates for a coordinate system which has been obtained by a Lorentz transformation from the above. 2.2. Global charges. In establishing the main results, we use the fact that the Kerr space-times yield a family of initial data sets which admit coordinates (such as KerrSchild) with sufficient approximate parity symmetry that allows the definition of angular momentum J and centre of mass c, in addition to the four-momentum (m, p). Indeed the Kerr initial data sets we use satisfy the Regge-Teitelboim asymptotic conditions, which say that in suitable AE coordinates the following estimates also hold: α
∂ gi j (x) − gi j (−x) = O(|x|−|α|−2 ), ∂ β K i j (x) + K i j (−x) = O(|x|−|β|−3 ). (2)
640
P. T. Chru´sciel, J. Corvino, J. Isenberg
The space of initial data satisfying (2) is known to be dense in the space of vacuum AE data [8]. Using such a coordinate system, we can compute the energy and linear and angular momenta using flux integrals at infinity (dσe is Euclidean surface measure, ν is the Euclidean outward normal, and r = |x|):
1 lim m= gi j,i − gii, j ν j dσe , 16π R→∞ {r =R} i, j 1 lim pi = (K i j − K gi j )ν j dσe , 8π R→∞ {r =R} j 1 j lim Ji = (K jk − K g jk )Yi ν k dσe , 8π R→∞ {r =R} j,k ⎤ ⎡
1 ⎣ lim gik δ k ν i − gii ν ⎦ dσe . x gi j,i − gii, j ν j − mc = 16π R→∞ {r =R} i, j
i
Note that in the last term, we can replace g in the center integrand by (g − gEucl ). Taken together, these give a set of ten Poincaré charges associated to the end. We emphasize that we do not impose condition (2) on the initial data for the bodies; rather, we show in Proposition 1 that we can modify the given vacuum end of each body to a vacuum end in Kerr, preserving the data set away from the end. We recall the relation between the charge integrals and the constraints. Indeed, these charges arise from integrating the constraints against elements of the cokernel of the linearized constraint operator. By linearizing at the Minkowski data, we have Φ(gEucl + h, K ) = DΦ(h, K ) + Q(h, K ), where Q(h, K ) = h ∗ ∂ 2 h + ∂h ∗ ∂h + ∂h ∗ K +h∗∂ K +K ∗ K , where “∗” denotes a linear combination (with smooth bounded coefficients) of some metric contractions Euclidean coordinates of the tensor product. In at the Minkowski data, DΦ(h, K ) = j (−2(K i j, j − K j j,i )), i, j (h i j,i j − h ii, j j ) . Thus ∗ for any vector and scalar pair (Y, N ) which satisfies DΦ (Y, N ) = (0, 0), we have as a consequence of integration by parts that (Y, N ) · Φ(gEucl + h, K )dμe {R0 ≤r ≤R} = B(R) − B(R0 ) + (Y, N ) · Q(h, K )dμe , (3) {R0 ≤r ≤R}
where dμe is the Euclidean volume measure, and where B(R) = (h i j,i − h ii, j ) (−2)Y i (K i j − K δi j ) + N {r =R}
−
i
(N,i h i j − N, j h ii ) ν j dσe .
i
By letting Y be a Euclidean Killing vector field, or letting N be a constant or a coordinate function x , we can easily relate B(R) to one of the above surface integrals defining the ADM energy-momenta.
Construction of N -Body Initial Data Sets in General Relativity
641
2.3. Hamiltonian formulation of the Poincaré charges. It will be convenient to use the Hamiltonian formulation of the Poincaré charges, as we review now (see Appendix E of [6] and references therein). Let S be a three-dimensional spacelike hypersurface in a four-dimensional Lorentzian space-time (M , g). ¯ Suppose that M contains an open set U with a time coordinate t (with range not necessarily equal to R), as well as a “radial” coordinate r ∈ [R, ∞), leading to local coordinate systems (t, r, v A ), with (v A ) providing local coordinates on a two-dimensional sphere. We further require that S ∩ U = {t = 0}. Assume that the metric g¯ μν approaches the Minkowski metric ημν as r tends to infinity. Set Ω = d x 0 ∧ d x 1 ∧ d x 2 ∧ d x 3 , and d Sαβ = Ω( ∂ ∂x α , ∂ ∂x β , ·, ·). The Hamiltonian analysis of vacuum general relativity in [3] leads to the formula 1 H (S , g, ¯ X) = 2
∂∞ S
Uαβ d Sαβ
for the Hamiltonian H (S , g, ¯ X ) associated to the flow of a vector field X , assumed to be a Killing vector field for the Minkowski metric, where 8π Uνλ = √
1 g¯ βγ ∇˚ κ (| det g| ¯ g¯ γ [ν g¯ λ]κ )X β + | det g| ¯ g¯ α[ν ∇˚ X λ] α , | det g| ¯
with ∇˚ the Levi-Civita connection for the Minkowski metric, and det g¯ = det(g¯ ρσ ). The integration over ∂∞ S is taken, as usual, as a limit of integrals over spheres tending to infinity. We let Greek indices run from 0 to 3, with x 0 = t, and A[μν] = 21 (Aμν − Aνμ ). We note that under enough approximate parity (such as holds for the Kerr data we consider), the above integrals converge [6]. If S , viewed as a hypersurface in (M , g), ¯ has first and second fundamental forms (g, K ) satisfying (1) and (2), then for the appropriate choice of X , the Hamiltonian yields the energy and momenta defined earlier. Indeed, if we let X = ∂ x∂ μ , we get the energymomentum one-form: H (S , g, ¯ ∂ x∂ μ ) = pμ , where p0 = m. With X = x 0 ∂∂x i + x i ∂ ∂x 0 , i ¯ X) = Ji. we get H (S , g, ¯ X ) = mc , and with X = i j x j ∂ ∂x , we have H (S , g, We note that these Killing vectors correspond to Killing Initial Data, or KIDs, for the Minkowski metric [1]. In Gaussian coordinates about S , the future-pointing unit normal is ∂t∂ , and for Y tangent to S , N ∂t∂ + Y is Killing for the Minkowski metric if and only if (DΦ)∗ (Y, N ) = (0, 0), where DΦ is the linearization of the constraints operator at the Minkowski data. We let Uαβ (Y, N ) correspond to X = N ∂t∂ + Y . In fact the surface integrals for the Hamiltonian at finite radii can be related to the constraints operator Φ(g, K ) = (τi , ρ) by the following identity, where X = N ∂t∂ + Y i ∂∂x i corresponds to the KID (Y, N ), and q is a quadratic form in (gi j − δi j , coefficients uniformly bounded in terms of bounds on gi j and g i j :
αβ
{x 0 =0,r =R}
U (Y, N )d Sαβ =
{x 0 =0,r =R0 }
1 + 8π
∂gi j ∂xk
, K i j ) with
Uαβ (Y, N )d Sαβ
{x 0 =0,R0 ≤r ≤R}
Y i τi + Nρ + q dμg .
(4)
642
P. T. Chru´sciel, J. Corvino, J. Isenberg
3. Constructing N-Body Initial Data by Gluing 3.1. Many-Kerr initial data sets. In this section we show how to attach a collection of N Kerr ends into one AE end while still solving the vacuum constraints. A special case of Theorem 2 below, for very special configurations (e.g., identical bodies placed symmetrically about a center) has earlier been established in [6, Sect. 8.9]. For 0 ≤ a < b, let Γ ( y, a, b) = {x ∈ R3 : a < |x − y| < b} be a coordinate annulus of inner radius a and outer radius b centred at y. Theorem 2 (Many-Kerr initial data sets). For each k = 1, . . . , N , let (E k , g k , K k ) be a Kerr asymptotically Euclidean three dimensional end, with Poincaré charges Q k := (m k , pk , m k ck , J k ) . Here the first entry is the ADM mass, the second is the ADM momentum, ck is the centre of mass, and J k is the total angular momentum. Suppose that the closures of the annuli N Γ (ck , 1, 3) are pairwise disjoint, and that k=1 (m k , pk ) is time-like. There exists 0 > 0 such that for each 0 < < 0 there is a vacuum asymptotically flat initial data set (E, g , K ) containing isometrically N
Γ ( −1 ck , −1 , 2 −1 ), g k , K k
.
k=1
N Proof. Let m T = k=1 m k . Since m T is non-zero, we can translate the origin of coorN dinates so that the center of mass k=1 m k ck vanishes. Let B(r0 ) ⊂ R3 be a Euclidean ball of radius r0 = 5 + max{|c1 |, . . . , |c N |}, centred at the origin. N Γ (c , 0, 1) . Let χ be a We now construct a family of data on Ω0 := B(r0 )\ k k=1 smooth nondecreasing function so that χ (t) = 0 for t < 9/4 and χ (t) = 1 for t > 11/4. Define (g˜ , K˜ ) on Ω0 as follows: • On each Γ (ck , 1, 2), we let (g˜ , K˜ ) be equal to (g,k , K ,k ), where (g,k , K ,k ) are initial data for a Kerr metric in Kerr-Schild coordinates with global charges Q ,k = (m k , pk , m k ck , 2 J k ) . • On each Γ (ck , 2, 3), (g˜ (x), K˜ (x)) = (1 − χ (|x − ck |))(g,k , K ,k ) + χ (|x − ck |)(gEucl , 0) . • On Γ (0, r0 − 1, r0 ), we let (g˜ (x), K˜ (x)) = (1−χ (|x|−r0 +3))(gEucl , 0)+χ (|x|−r0 +3) × (g,ext (x), K ,ext (x)), where (g,ext (x), K ,ext (x)) is a Kerr metric with global charge N N N m k + δm , pk + δ p , m ,ext δc, 2 J k + δ J , Q ,ext =
k=1
=:m ,ext
k=1
and where θ := (δm, δ p, δc, δ J) ∈ Θ ⊂ R10 . Θ is a compact convex set which will be specified below.
k=1
Construction of N -Body Initial Data Sets in General Relativity
• On Γ (0, 0, r0 − 1)\
N
k=1 Γ (ck , 1, 3),
643
let (g˜ , K˜ ) = (gEucl , 0).
In what follows we will use the fact, which follows directly from the above definition and from the Kerr-Schild form of the Kerr metric, that for any k ∈ N, we have (g˜ , K˜ ) − (gEucl , 0)C k (Ω0 ) ≤ C ,
(5)
for some constant C = C(k). We thus have Φ(g˜ , K˜ ) = O(), and Φ(g˜ , K˜ ) = 0 in N a neighborhood of ∂Ω, where Ω = B(r0 )\ k=1 Γ (ck , 0, 2). The goal is to modify (g˜ , K˜ ) by adding a smooth deformation (δg , δ K ) supported in Ω so that Φ(g˜ + δg , K˜ + δ K ) = 0. The proof proceeds in two stages, as we now describe. Recall that the linearized constraints operator DΦ at the Minkowski data has a ten-dimensional cokernel K0 = ker(DΦ ∗ ) spanned by the KIDs mentioned earlier (Sect. 2.3); this cokernel is an obstruction to solving the full system Φ(g˜ + δg , K˜ + δ K ) = 0 for δg and δ K . However, (g˜ , K˜ ) is close to the Minkowski data, and the constraints Φ(g˜ , K˜ ) vanish on a neighborhood of ∂Ω, and so in particular Φ(g˜ , K˜ ) belongs to the appropriate weighted spaces used in [6,8]. Therefore, by applying Theorem 2 of [8] or Corollary 5.11 of [6], we can at least solve the equation up to cokernel: we can find, for each θ ∈ Θ as above, a smooth deformation (δgθ , δθ ) which is supported in Ω, satisfies (δgθ , δ K θ )C 3 (Ω) ≤ C (C is independent of θ and ), and also satisfies Φ(g˜ + δgθ , K˜ + δ K θ ) ∈ ζ K0 , where ζ is a smooth weight function that vanishes on ∂Ω. For instance, in the notation of [6], ζ = ψ 2 , for ψ a smooth function which near ∂Ω takes the form ψ = e−s/d , where d is a defining function for the boundary ∂Ω, and where s > 0. The collection of L 2 (dμe )-projections of Φ(g˜ + δgθ , K˜ + δ K θ ) onto a basis for K0 defines a map Θ θ → R10 which is continuous in θ . We show that this map vanishes for some θ , and that will complete the proof; a rescaling then yields Theorem 2. We now focus on computing these projections, which can be thought of as balance equations as we shall see. The primary tool for these computations is integration by parts together with the flux integrals (4) and (3). Note that the difference |dμe − dμg˜ | is O(), and that the Taylor expansion around the Minkowski data (gEucl , 0) yields Φ(g˜ + δgθ , K˜ + δ K θ ) = DΦ(g˜ + δgθ − gEucl , K˜ + δ K θ ) + O( 2 ). As in Sect. 2.3, integrating this quantity against the basis of KIDs we get the boundary integrands of (3), integrated over ∂Ω, plus O( 2 ) terms. Alternatively we can use an analogue of the integration-by-parts formula (4) to derive the balance equations in a form analogous to (8.7) of [6]. In either case, recall that on ∂Ω our solution up to cokernel agrees with Kerr data. For such data, the differences between the limiting surface integrals (which give the charges) and surface integrals at finite radius are given by the divergence theorem as in (4). On the other hand, the Kerr data solves the vacuum constraints, so that the integrand over the annulus in (4) reduces to q. In the case of the data (g,ext , K ,ext ) under consideration, this results in a O( 2 ) term; note that we use Kerr-Schild coordinates, in which the metric components take the form (ημν +O(m ,ext )). Keeping all this in mind, we can now compute the balance equations. Working first with e(1) = (0, 1), the KID corresponding to the Minkowskian Killing vector ∂t∂ (cf. Sect. 2.3), we obtain 1 θ ˜ θ e(1) , Φ(g˜ + δg , K + δ K ) L 2 (Ω) = Uαβ (e(1) )d Sαβ + O( 2 ) 16π ∂Ω = δm + O( 2 ).
(6)
644
P. T. Chru´sciel, J. Corvino, J. Isenberg
Next choosing e(1+i) = (∂i , 0), for each i = 1, 2, 3, to be the KID corresponding to the Minkowskian Killing vector ∂∂x i , we calculate the balance equation to be 1 e(1+i) , Φ(g˜ + δgθ , K˜ + δ K θ ) L 2 (Ω) = δpi + O( 2 ) . 16π
(7)
Now let e(4+i) , for each i = 1, 2, 3, be the KID corresponding to the Minkowskian Killing vector t ∂∂x i + x i ∂t∂ ; at t = 0 this corresponds to (Y, N ) = (0, x i ). The boundary integral around each ck is thus αβ i U (0, x )d Sαβ = Uαβ (0, x i − cki + cki )d Sαβ ∂ B( ck ,1) ∂ B( ck ,1) αβ i i = U (0, x − ck )d Sαβ + Uαβ (0, cki )d Sαβ ∂ B( ck ,1) ∂ B( ck ,1) 0+O( 2 )
= cki
∂ B( ck ,1)
Uαβ (0, 1)d Sαβ +O( 2 )
m k +O( 2 )
=
m k cki
+ O( 2 ) ,
where the first integral in the second line vanishes, up to quadratic terms in the metric, by definition of the centre of mass; we have also used linearity of the global charges with respect to the KIDs, and the fact that the ADM mass is the Poincaré charge associated to the KID (0, 1). It follows that the associated balance equation reads 1 e(4+i) , Φ(g˜ + δgθ , K˜ + δ K θ ) L 2 (Ω) = m ,ext δci − m k cki +O( 2 ) . (8) 16π k=1 N
0
Finally, let ∂ = ∂ ∂x , and let e(7+i) , i = 1, 2, 3, be the KIDs corresponding to the Minkowskian Killing vector i j x j ∂ ; thus (Y, N ) = (i j x j ∂ , 0). We then have j j αβ j U (i j x ∂ , 0)d Sαβ = Uαβ (i j (x j − ck + ck )∂ , 0)d Sαβ ∂ B( ck ,1) ∂ B( ck ,1) j = Uαβ (i j (x j − ck )∂ , 0)d Sαβ ∂ B( ck ,1) +
2 Jki +O( 2 )
Uαβ (i j ck ∂ , 0)d Sαβ j
∂ B( ck ,1) j
= i j ck
∂ B( ck ,1)
Uαβ (∂ , 0)d Sαβ +O( 2 )
pk +O( 2 )
= i j ck pk + O( 2 ) . j
Construction of N -Body Initial Data Sets in General Relativity
645
We conclude that (recall that Q ,ext has angular momentum 2
N k=1
J k + δ J)
1 e(7+i) , Φ(g˜ + δgθ , K˜ + δ K θ ) L 2 (Ω) = δ J i − (ck × pk )i + O( 2 ). 16π N
(9)
k=1
N
We let Θ = (0, 0, 0, k=1 ck × pk ) + B0 , where B0 is a closed ball around the origin chosen so that Q ,ext has time-like four-momentum. We can invoke now the Brouwer fixed point theorem, in a way similar to the proof of Theorem 8.1 2 of [6], to conclude that there exist 0 small enough so that the right-hand-sides of the balance equations (6)-(9) can be all made to vanish for all 0 < < 0 , which is the desired result.
3.2. Reduction to Kerr asymptotics. We recall the well-known result of [6,8] which states that any AE vacuum end satisfying the Regge-Teitelboim condition (2) (with time-like ADM four-momentum) can be deformed, outside of a compact set, to a new vacuum initial data set such that the data agrees with that of a suitably chosen space-like slice of a Kerr space-time, outside of a compact set. We emphasize that this deformation can be performed to preserve any given pre-compact subset of the end; therefore we can apply this deformation to the data sets for each of our N bodies, preserving as large a compact subset of the original data as we like. We note for our main theorem that the four-momentum of the resulting Kerr can be made as close to the original fourmomentum of the end as we like by performing the deformation at larger coordinate radii. Here we give a significant generalization of the above result, which removes the Regge-Teitelboim assumption (2) in the above gluing: Proposition 1. Let (g, K ) be AE vacuum initial data on the exterior E of a ball in R3 which has time-like ADM four-momentum (m, p). Let > 0. For sufficiently large R, there is a vacuum initial data set (g, ¯ K¯ ) on E so that on E ∩ {|x| ≤ R} we have (g, ¯ K¯ ) = (g, K ), and so that on {|x| ≥ 2R}, (g, ¯ K¯ ) is identical to data from a space-like slice of a suitably chosen Kerr space-time. If (m + δm, p + δ p) is the four-momentum of (g, ¯ K¯ ), then |δm| < and |δ p| < . If, moreover, (2) holds, then also |δc| < and |δ J| < . Proof. The proof is a minor modification of that used in [6,8] to prove the result for the case in which condition (2) holds. In fact the primary modifications which are needed have been introduced in the proof of Theorem 2. Given a vacuum AE end, we let φ R : A1 → A R be the scaling φ R (x) = Rx, and let (g R , K R ) = (R −2 φ ∗R g, R −1 φ ∗R K ). We note that since (2) may not hold, the decay might not be good enough to make the centre of mass and angular momentum well-defined. However, we have j j R R (K jk − (K R ) g Rjk )Yi ν k dσe = R −1 (K jk − K g jk )Yi ν k dσe {r =1} j,k
=O
2 Compare [7] for smoothing arguments.
{r =R} j,k
log R R
.
(10)
646
P. T. Chru´sciel, J. Corvino, J. Isenberg
Similarly, R
{r =1}
⎡
⎣
x
gi j,i − gii, j ν j −
i, j
i
⎤ log R k i ⎦ . gik δ ν − gii ν dσe = O R (11)
These two estimates follow from (3), together with the fact that (g, K ) solve the vacuum constraints. Indeed the estimate Q(h, K ) = O(|x|−4 ) implies that (Y, N ) · Q(h, K ) = O(|x|−3 ) for (Y, N ) = (Yi , x ) = O(|x|). Now applying (3) and the constraints, we obtain B(R) = B(R0 ) + O(log R). We let = R −1 , so that we can use some of the notation from the proof of Theorem 2. As above, we use a cutoff function to glue (g R , K R ) to Kerr data with charges Q ,ext = ((m + δm), ( p + δ p), (m + δm)δc, δ J) on the annulus A1 , so that in a neighborhood of the inner boundary of A1 the data is identically (g R , K R ), and in a neighborhood of the outer boundary the data is identical to the Kerr data. Again, we parametrize a family of such data with θ := (δm, δ p, δc, δ J) ∈ Θ = B0 , where B0 is a closed ball around the origin, chosen so that the four-momentum of Q ,ext is time-like. Let (g˜ , K˜ ) be the resulting glued data. We note that g˜ − gEucl C +1 (A1 ) + K˜ C (A1 ) = O(), that Φ(g˜ , K˜ ) = O(), and that Φ(g˜ , K˜ ) vanishes in a neighborhood of ∂ A1 . As above, we solve the vacuum constraints up to cokernel, so that Φ(g˜ + δgθ , K˜ + δ K θ ) ∈ ζ K0 . We again analyze the projection of Φ(g˜ +δgθ , K˜ +δ K θ ) onto K0 , using the notation from the proof in the last section, along with (10) and (11): 1 e(1) , Φ(g˜ + δgθ , K˜ + δ K θ ) L 2 (A1 ) = δm + O( 2 ), 16π 1 e(1+i) , Φ(g˜ + δgθ , K˜ + δ K θ ) L 2 (A1 ) = δpi + O( 2 ), 16π 1 e(4+i) , Φ(g˜ + δgθ , K˜ + δ K θ ) L 2 (A1 ) = (m + δm)δci + o(1) , 16π 1 e(7+i) , Φ(g˜ + δgθ , K˜ + δ K θ ) L 2 (A1 ) = δ J i + o(1) . 16π The proof now follows from the Brouwer fixed point theorem as before, followed by scaling back to the annulus A R . 3.3. Proof of Theorem 1. Proof. We first apply Proposition 1 to deform the data on each end E k to a new vacuum initial data set which agrees with the data on each Uk , and outside a compact set agrees with data from a suitably chosen space-like slice of a Kerr space-time. A rescaling of all the metrics then reduces the problem to one in which all of the initial data sets are Kerrian outside of a Kerr-Schild coordinate-ball of radius one. The result follows now by applying Theorem 2. Acknowledgements. The authors are grateful to Institut Mittag-Leffler (Djursholm, Sweden), for hospitality and financial support during the initiation of this paper. PTC was supported in part by the Polish Ministry of Science and Higher Education grant Nr N N201 372736. JC was partially supported by NSF grant DMS0707317 and the Fulbright Foundation. JI was partially supported by NSF grants PHY-0652903 and PHY0968612.
Construction of N -Body Initial Data Sets in General Relativity
647
References 1. Beig, R., Chru´sciel, P.T.: Killing Initial Data. Class. Quant. Grav. 14, A83–A92 (1996) 2. Chandrasekhar, S.: The mathematical theory of black holes, Oxford: Oxford UP, 1984 3. Chru´sciel, P.T.: On the relation between the Einstein and Komar expressions for the energy of the gravitational field. Ann. Inst. H. Poincaré 42, 267–282 (1985) 4. Chru´sciel, P.T., Corvino, J., Isenberg, J.: Construction of N -body time-symmetric initial data sets in general relativity. Cont. Math. http://xxx.lanl.gov/abs/0909.1101 (2009, to appear) 5. Chru´sciel, P.T., Delay, E.: Existence of non-trivial asymptotically simple vacuum space-times. Class. Quant. Grav. 19, L71–L79, erratum-ibid, 3389. http://xxx.lanl.gov/abs/gr-qc/0203053 2002 6. Chru´sciel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. de France. 94. http://xxx.lanl.gov/abs/gr-qc/0301073 2003 7. Chru´sciel, P.T., Delay, E.: Manifold structures for sets of solutions of the general relativistic constraint equations. J. Geom. Phys. 51, 442–472. http://xxx.lanl.gov/abs/gr-qc/0309001 2004 8. Corvino, J., Schoen, R.M.: On the asymptotics for the vacuum Einstein constraint equations. J. Diff. Geom, 73, 185–217 (2006) Communicated by M. Aizenman
Commun. Math. Phys. 304, 649–664 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1237-6
Communications in
Mathematical Physics
The Aggregation Equation with Power-Law Kernels: Ill-Posedness, Mass Concentration and Similarity Solutions Hongjie Dong Division of Applied Mathematics, Brown University, 182 George Street, Box F, Providence, RI 02912, USA. E-mail: [email protected] Received: 26 April 2010 / Accepted: 22 November 2010 Published online: 13 April 2011 – © Springer-Verlag 2011
Abstract: We study the multidimensional equation u t + div(uv) = 0, v = aggregation −∇ K ∗ u with initial data in P2 Rd ∩ L p Rd . We prove that with biological relevant potential K (x) = |x|, the equation is ill-posed in the critical Lebesgue space L d/(d−1) Rd in the sense that there exists initial data in P2 Rd ∩ L d/(d−1) Rd d such that the unique measure-valued solution leaves L d/(d−1) R immediately. We also extend this result to more general power-law kernels K (x) = |x|α , 0 < α < 2 for p = ps := d/(d + α − 2), and prove a conjecture in Bertozzi et al. (Comm Pure Appl Math 64(1):45–83, 2010) about instantaneous mass concentration for initial data in P2 Rd ∩ L p Rd with p < ps . Finally, we characterize all the “first kind” radially symmetric similarity solutions in dimension greater than two. 1. Introduction In this paper we consider the multidimensional aggregation equation u t + div(uv) = 0, v = −∇ K ∗ u
(1.1)
for x ∈ Rd and t > 0 with the initial data u(0, x) = u 0 (x), x ∈ Rd . Here d ≥ 2, u ≥ 0, K is the interaction potential, and ∗ denotes the spatial convolution. This equation arises in various models for biological aggregation and problems in granular media; see, for instance, [8,12,13]. The problems of the well-posedness in different spaces, finite-time blowups, asymptotic behaviors of solutions of this equation, as well as the equation with an additional dissipation term, have been studied extensively by a Hongjie Dong was partially supported by the National Science Foundation under agreement No. DMS-0800129.
650
H. Dong
number of authors; see [2,3,5–7,9–12] and references therein. We refer the reader to [4] for a nice review about recent progress on the aggregation equation. In [5] Bertozzi, Laurent and Rosado studied comprehensively the L p theory for the aggregation equation (1.1). Among some other results, they considered radially symα metric kernels where the singularity at the origin is of order |x| for some α > 2 − d, d and proved the local well-posedness of (1.1) in P2 R ∩ L p Rd for any p > ps , where ps = d/(d + α − 2) (see below for the definition of the space P2 Rd ). In the biological relevant case K (x) = |x|, they showed that solutions can concentrate mass instantaneously for initial data in P2 Rd ∩ L p Rd for any p < ps . It remains unknown if (1.1) is well-posed in the critical space P2 Rd ∩ L ps Rd . Another interesting open question is whether one can show a similar instantaneous mass concentration phenomenon for the equation with general power-law potential |x|α . The authors conjectured in [5] that the answer to the second question is positive. The aim of the current paper is to answer these questions. we For the first question, shall construct radially symmetric initial data in P2 Rd ∩ L d/(d−1) Rd , such that the unique measured-valued solution leaves L d/(d−1) Rd immediately for t > 0; see Theorem 2.1. This result implies that (1.1) is ill-posed in P2 Rd ∩ L d/(d−1) Rd , and the well-posedness result for p > d/(d − 1) obtained in [5] is sharp. For the second question, we show that, for any α ∈ (0,2) and any p < ps , there exists radially symmetric initial data in P2 Rd ∩ L p Rd such that the solution concentrates mass at the origin instantaneously. In other words, a Dirac delta appears immediately in the solution. Therefore, we settle down the aforementioned conjecture in [5]. We also prove that, for any α ∈ (0, 2), (1.1) is ill-posed in P2 Rd ∩ L ps Rd by constructing initial data in P2 Rd ∩ L ps Rd such that any weakly continuous measured-valued solution, if it exists, leaves L ps Rd immediately for t > 0. The proofs use some ideas in [5] by considering the flow map driven by the velocity field v. Roughly speaking, there are two steps in the proofs. In the first step, we find a suitable representation of the velocity field v in the polar coordinates, and prove the monotonicity, positivity and asymptotics of the corresponding kernel. For K (x) = |x|, these have already been established in [5] (Lemma 2.3). More delicate analysis is needed for general power-law potential K (x) = |x|α (see Lemma 4.4). In the second step, we deduce certain positive lower bounds for the velocity (Lemmas 2.5, 4.6, and 4.7). Combined with the monotonicity of the velocity in time, we then reduce the problems to study the dynamics of solutions to some ordinary differential equations. In the case p < ps , it is shown that the flow map reaches the origin in a short time, which generates a Dirac delta. While in the critical case p = ps , a Dirac delta may not develop shortly, but the flow map makes the mass concentrate quickly enough near the origin such that the solution u leaves L ps Rd immediately. We also consider profiles of similarity solutions to (1.1) at the blowup time with the potential K (x) = |x|, which conserve mass. This type of solution is an example of “first-kind” similarity solutions; see [1]. In [2], Bertozzi, Carrillo and Laurent constructed radially symmetric first-kind similarity solutions in dimension one and two, and proved that in any odd dimension d ≥ 3 such solutions cannot exist with support on open sets. By observing a certain concavity property of the kernel in the polar coordinates, in Sect. 3 we characterize all the radially symmetric first-kind similarity solutions in dimension d ≥ 3.
The Aggregation Equation with Power-Law Kernels
651
We finish the Introduction by fixing some notation. Most notations in this paper are chosen to be compatible with those in [5]. For r > 0, let Br = x ∈ Rd : |x| < r , Sr = x ∈ Rd : |x| = r . By ωd we mean the surface area of the unit sphere S1 in Rd . We denote P Rd to be the set of all probability measures on Rd , and P2 Rd to be the set of all probability measures on Rd with bounded second moment: P2 (Rd ) := μ ∈ P(Rd ) : |x|2 dμ(x) < ∞ . Rd
2. Ill-Posedness when K (x) = |x| In this section, we prove the following result, which reads that with potential function K (x) = |x|, the aggregation equation (1.1) is ill-posed in P2 ∩ L d/(d−1) Rd . Theorem 2.1. Let K (x) = |x|, k ∈ ((d − 1)/d, 1) and u 0 (x) =
L
1|x|≤1/2 |x|d−1 (− log |x|)k
where
L=
|x|≤1/2
∈ P2 Rd ∩ L d/(d−1) Rd ,
(2.1)
|x|−d+1 (− log |x|)−k d x
is a normalization constant. Let (μt )t∈(0,∞) be the unique measure-valued solution to the aggregation equation (1.1). Then for any t > 0 the density of μt , if it exists, is not in L d/(d−1) Rd . We note that for K (x) = |x|α , α ≥ 1 the global existence and uniqueness of a weakly continuous measure-valued solution was proved in [7]. Moreover, for α < 2, any measure-valued solution will eventually collapse to a Dirac delta at the center of mass in finite time. In [5], it was also proved that if u 0 ∈ P2 then the measure-valued solution stays in P2 . Definition 2.2. Let μ ∈ P Rd be a radially symmetric probability measure. We define μˆ ∈ P([0, +∞)) by
μ(I ˆ ) = μ x ∈ Rd : |x| ∈ I for all Borel sets I in [0, +∞)). We reformulate some results in [5, Sect. 4] as the following lemmas. Lemma 2.3. Let K (x) = |x|, μ ∈ P Rd be a radially symmetric measure. Then for any x = 0, we have ∞ x , (∇ K ∗ μ)(x) = ψ(ρ/|x|) d μ(ρ) ˆ |x| 0
652
H. Dong
where ψ : [0, ∞) → R is a function defined by 1 − ρy1 dσ (y). ψ(ρ) = – S1 |e1 − ρy| Moreover, ψ is continuous, positive, non-increasing on [0, ∞), and ψ(0) = 1,
lim ψ(ρ)ρ =
ρ→∞
d −1 . d
Lemma 2.4. Let K (x) = |x|, and (μt )t∈[0,∞) be a radially symmetric weakly continuous measured-valued solution of the aggregation equation (1.1). Then the vector field v(t, x) = −(∇ K ∗ μt )1x =0 is continuous on [0, ∞) × (Rd \ {0}). For any t ≥ 0, we have μt = X t# μ0 , where, for each x ∈ Rd , X t = X t (x) is an absolutely continuous function on [0, ∞) satisfying d dt X t (x) = v(t, X t (x)) for a.e. t ∈ (0, ∞); X 0 (x) = x, and X t# means the push-forward of a measure by the map X t . Moreover, for any x = 0, X t (x) = Rt (|x|)x/|x|, where Rt is an absolutely continuous, non-negative and non-increasing function in t ∈ [0, ∞), and μˆ t = Rt# μˆ 0 . Consequently, for any x = 0, (∇ K ∗ μt )(x) =
∞
ψ(Rt (ρ)/|x|) d μˆ 0 (ρ)
0
x , |x|
and |(∇ K ∗ μt )(x)| is non-decreasing in t. Next we establish a point-wise lower bound of the velocity v at t = 0. Lemma 2.5. Let K (x) = |x| and u 0 be the initial data defined in (2.1) with k ∈ ((d − 1)/d, 1): u 0 (x) =
L
1|x|≤1/2 . |x|d−1 (− log |x|)k
Then there exists a constant δ1 > 0 such that |(∇ K ∗ u 0 )(x)| ≥ δ1 |x|(− log |x|)1−k for any x ∈ Rd satisfying 0 < |x| < 1/2.
(2.2)
The Aggregation Equation with Power-Law Kernels
653
Proof. Clearly, we have uˆ 0 (ρ) = Lωd (− log ρ)−k 1ρ≤1/2 . By the positivity and continuity of |v(0, ·)| in Rd \{0}, it suffices to prove (2.2) for x ∈ Rd satisfying 0 < |x| < 1/4. It follows from Lemma 2.3 that for any ρ ∈ (|x|, 1/2), ψ(ρ/|x|) ≥ δ0 |x|/ρ for a constant δ0 > 0 independent of |x|. Thus, for any x ∈ Rd satisfying 0 < |x| < 1/4, 1/2 |(∇ K ∗ u 0 )(x)| ≥ ψ(ρ/|x|)uˆ 0 (ρ) dρ |x|
≥ δ0 Lωd |x|
1/2 |x|
(− log ρ)−k
dρ ρ
δ0 Lωd |x| (− log |x|)1−k − (log 2)1−k = 1−k ≥ δ1 |x|(− log |x|)1−k , since 0 < |x| < 1/4. The lower bound (2.2) is proved. We are now ready to prove Theorem 2.1. Proof of Theorem 2.1. By Lemma 2.4, for each x ∈ Rd , |v(t, x)| is non-decreasing in t. Therefore, from the lower bound of |v(x, 0)| (2.2) we infer that for any r ∈ (0, 1/2), d Rt (r ) ≤ −δ1 Rt (r )(− log Rt (r ))1−k . dt Solving the ordinary differential inequality (2.3) gives
(2.3)
(− log Rt (r ))k ≥ (− log r )k + kδ1 t.
(2.4)
From (2.4), we obtain kδ1 t (− log Rt (r ))k ≥1+ , k (− log r ) (− log r )k and by Taylor’s formula, for t sufficiently small, δ2 t (− log Rt (r )) ≥1+ , (− log r ) (− log r )k where δ2 = δ1 /2. We thus obtain Rt (r ) ≤ r e−δ2 t (− log r )
1−k
.
(2.5)
Now we suppose that, for some t > 0, μt has a density function u(t, x) ∈ L d/(d−1) Rd . By Hölder’s inequality, for any r ∈ (0, 1/2), μt (B Rt (r ) ) = u(t, x) d x B Rt (r )
≤
d−1 u
d d−1
d
(t, x) d x
B Rt (r )
≤ u(t, ·) L d/(d−1) (Rd ) Rt (r ).
1
|B Rt (r ) | d
(2.6)
654
H. Dong
On the other hand, by the definitions of μt and Rt , r (− log ρ)−k dρ. μt (B Rt (r ) ) ≥ μ0 (Br ) = Lωd
(2.7)
0
Note that the above inequality is strict only when there is a mass concentration before time t. We combine (2.6), (2.7), and (2.5) to get μt (B Rt (r ) ) Rt (r )
r (− log ρ)−k dρ ≥ Lωd 0 . 1−k r e−δ2 t (− log r )
u(t, ·) L d/(d−1) (Rd ) ≥
(2.8)
However, by L’Hospital’s rule,
r (− log ρ)−k dρ lim 0 r 0 r e−δ2 t (− log r )1−k = lim
r 0 e−δ2
= ∞.
t (− log r )1−k
(− log r )−k 1 + δ2 t (1 − k)(− log r )−k
This gives a contradiction to (2.8) since we assume u(t, ·) ∈ L d/(d−1) Rd . The theorem is proved. 3. Similarity Solutions In this section, we consider the problem of similarity solutions to the aggregation equation. This problem is closely related to the blowup profile for (1.1) at the blowup time. Let us consider mass-conserving similarity solutions 1 x u(t, x) = (3.1) u 0 R(t)d R(t) to the aggregation equation (1.1) with interaction kernel K (x) = |x|. These solutions are “first-kind” similarity solutions, while “second-kind” similarity solutions do not conserve mass. If u is a radially symmetric first-kind similarity solution given by (3.1), then by the homogeneity it is easily seen that x v(t, x) = v0 , v0 = −∇|x| ∗ u 0 . R(t) Moreover, it was proved in [2] that R(t) must be a linear function and on the support of u 0 , v0 = −λx
(3.2)
for some constant λ > 0. In one space dimension, the authors of [2] constructed a first-kind similarity solution x 1 U , u(t, x) = ∗ T −t T∗ − t
The Aggregation Equation with Power-Law Kernels
655
where U is the uniform distribution on [−1, 1]. In any dimension, there is a first-kind similarity measure-valued solution which is a single delta on a sphere with radius shrinking linearly in time [2, Remark 3.8]. Such solution is called a single delta-ring solution. In two space dimensions, a two delta-ring solution was constructed in the same paper, i.e., μˆ 0 = m 1 δρ1 + m 2 δρ2 for some 0 < ρ1 < ρ2 < ∞ and m 1 , m 2 > 0. On the other hand, for any odd dimension d ≥ 3, by using a relation between K (x) and the Newtonian potential they proved the non-existence of radially symmetric first-kind similarity solutions with support on open sets. We characterize all the radially symmetric first-kind similarity measure-valued solutions in the following theorem, which in particular implies that in dimension three and higher there cannot exist first-kind similarity solutions with support on open sets or multi delta rings. Theorem 3.1 (Characterization of similarity solutions). Let d ≥ 3 and K (x) = |x|. Then any radially symmetric first-kind similarity measure-valued solution is of the form x 1 μt (x) = , μ 0 R(t)d R(t) where μˆ 0 = m 0 δ0 + m 1 δρ1
(3.3)
for some constants m 0 , m 1 ≥ 0 and ρ1 > 0. For the proof, first we recall that by Lemma 2.3 for any radially symmetric measurevalued solution μt and t > 0, ∞ x v(t, x) = − (3.4) φ(|x|/ρ) d μˆ t (ρ) , |x| 0 where φ : [0, ∞) → R is a function defined by r − y1 φ(r ) = – dσ (y). S1 |r e1 − y|
(3.5)
The proof of Theorem 3.1 relies on the following observation. Lemma 3.2. i) Let d ≥ 4 and K (x) = |x|. Then the function φ defined by (3.5) is C 2 on [0, ∞) and satisfies φ(0) = 0,
lim φ(r ) = 1, φ (r ) > 0 on [0, ∞),
r →∞
(3.6)
φ (r ) < 0 on (0, ∞).
ii) Let d = 3 and K (x) = |x|. Then the function φ defined by (3.5) is C 1 on [0, ∞) and satisfies (3.6). Moreover, it is concave on [0, ∞), linear on [0, 1] and strictly concave on (1, ∞). Suppose for the moment that Lemma 3.2 is verified. We prove Theorem 3.1 by a contradiction argument. Suppose (μt ) is a radially symmetric first-kind similarity measure-valued solution such that there are two numbers 0 < ρ1 < ρ2 , ρ1 , ρ2 ∈ supp μˆ 0 .
656
H. Dong
Denote w(ρ) := |v0 (ρe1 )|. By (3.4), ∞ w(ρk ) = φ(ρk /ρ) d μˆ 0 (ρ), k = 1, 2. 0
It follows from Lemma 3.2 that φ(ρ1 /ρ) ≥
ρ1 φ(ρ2 /ρ), ρ2
and the inequality is strict for all ρ in a small neighborhood of ρ1 because ρ2 /ρ1 > 1. Since ρ1 ∈ supp μˆ 0 , we get w(ρ1 ) >
ρ1 w(ρ2 ). ρ2
On the other hand, from (3.2) w is a linear function on supp μˆ 0 : w(ρ1 ) =
ρ1 w(ρ2 ). ρ2
Therefore, we reach a contradiction. We finish this section by proving Lemma 3.2. Proof of Lemma 3.2. We rewrite (3.5) as ωd−1 π (r − cos θ )(sin θ )d−2 φ(r ) = dθ, ωd 0 A(r, θ )
(3.7)
where A(r, θ ) = (1 + r 2 − 2r cos θ )1/2 . For r ∈ [0, 1) ∪ (1, ∞), a direct computation gives (r − cos θ )2 ωd−1 π d−2 1 dθ − (sin θ ) φ (r ) = ωd 0 A A3 π ωd−1 (sin θ )d = dθ, ωd 0 A3 and φ (r ) = −
3ωd−1 ωd
0
π
(sin θ )d (r − cos θ ) dθ. A5
Integration by parts yields (sin θ )d+1 3ωd−1 π r (sin θ )d φ (r ) = − − dθ ωd A5 (d + 1)A5 0 3ωd−1 π r (sin θ )d 5(sin θ )d+1r sin θ =− − dθ ωd (d + 1)A7 A5 0 3ωd−1 π r (sin θ )d 5 2 2 r sin =− + 1 − 2r cos θ − θ dθ. ωd A7 d +1 0
(3.8)
(3.9)
The Aggregation Equation with Power-Law Kernels
657
Case 1: d ≥ 4. In this case, we have r 2 + 1 − 2r cos θ −
5 sin2 θ ≥ (r − cos θ )2 . d +1
Therefore, we get φ < 0 on (0, 1) ∪ (1, ∞). Note that because |r − cos θ | ≤ A, r | sin θ | ≤ A, the integrals on the right-hand sides of (3.8) and (3.9) are absolutely convergent and continuous at r = 1 by the dominated convergence theorem. Thus (3.8) and (3.9) hold on the whole region [0, ∞). So we conclude φ ∈ C 2 ([0, ∞)) and φ < 0 on (0, ∞). Moreover, since φ(0) = 0, φ → 1 as r → ∞ and φ > 0 by (3.8), φ is bounded and non-negative on [0, ∞). Case 2: d = 3. As before, the integral on the right-hand side of (3.8) is absolutely convergent and continuous at r = 1 by the dominated convergence theorem. Thus (3.8) holds on the whole region [0, ∞), and we conclude φ ∈ C 1 ([0, ∞)) and φ > 0 on [0, ∞). For d = 3, one can explicitly compute φ. From (3.7), we have 1 π (r − cos θ ) sin θ φ(r ) = dθ 2 0 A(r, θ ) π 1 cos θ sin θ = dθ ∂θ A − 2 0 A 1 = (|1 + r | − |1 − r |) − I, (3.10) 2 where, I :=
1 2
π 0
cos θ sin θ dθ. A
Integrating by parts and using (3.8), we get 1 π (sin2 θ ) 1 π r sin3 θ r I = dθ = dθ = φ (r ). 4 0 A 4 0 A3 2 Thus, φ satisfies φ(r ) =
r − r2 φ (r ) for r ∈ [0, 1]; 1 − r2 φ (r ) for r ∈ (1, ∞),
and φ(1) = 2/3 by (3.7). Solving this ordinary differential equation, we obtain 2r /3 for r ∈ [0, 1]; φ(r ) = 1 1 − 3r 2 for r ∈ (1, ∞), which immediately shows all the remaining claims in ii). The lemma is proved.
Remark 3.3. In the case d = 2, it is easy to check that φ is convex in (0, 1), concave in (1, ∞), and φ → +∞ as r → 1.
658
H. Dong
4. Ill-Posedness and Instantaneous Concentration when K (x) = |x|α In this section, we study the ill-posedness and the instantaneous mass concentration phenomenon for general power-law kernel K (x) = |x|α , 0 < α < 2. In [5], the authors established the instantaneous mass concentration for certain initial data in L p , p < ps in the special case α = 1, and conjectured that it remains true for general α. We prove this conjecture in this section. Theorem 4.1 (Instantaneous mass concentration). Let K (x) = |x|α , α ∈ (0, 2), ps = d/(d + α − 2) and u 0 (x) =
L 1|x|≤1/2 , |x|d+α−2+ε
where
ε ∈ (0, 1), L =
|x|≤1/2
(4.1)
|x|−d−α+2−ε d x
is a normalization constant. Note that u 0 ∈ P2 Rd ∩ L p Rd for any p ∈ [1, d/(d + α − 2 + ε)). Let (μt )t∈(0,∞) be a radially symmetric weakly continuous measure-valued solution to the aggregation equation (1.1). Then for any t > 0 we have μt ({0}) > 0. Therefore mass is concentrated at the origin instantaneously and the solution is singular with respect to the Lebesgue measure. We shall also prove the following theorem which is a generalization of Theorem 2.1. It reads that the aggregation equation (1.1) is ill-posed in the critical space P2 Rd ∩ d L ps R . Theorem 4.2. Let K (x) = |x|α , α ∈ (0, 2), ps = d/(d + α − 2), k ∈ (1/ ps , 1) and u 0 (x) =
d d L R ∩ L R , 1 ∈ P |x|≤1/2 2 p s |x|d+α−2 (− log |x|)k
where
L=
|x|≤1/2
(4.2)
|x|−d−α+2 (− log |x|)−k d x
is a normalization constant. Let (μt )t∈(0,∞) be a radially symmetric weakly continuous measure-valued solution to the aggregation equation (1.1). Then for any t > 0 the density of μt , if it exists, is not in L ps Rd . Remark 4.3. In the case that α ∈ (2 − d, 0), d ≥ 3, we infer from Lemma 4.4 below that the velocity field v is repulsive in the sense that, for a radially symmetric solution, v(t, x) is in the x direction. So one cannot expect the instantaneous mass concentration phenomenon in this case. In fact, we believe that in the case α ∈ (2 − d, 0), the aggregation equation (1.1) is well-posed in L p for p ∈ (1, ∞). Formally, integrating by parts, we obtain for any p ∈ (1, ∞), d u(t, x) p d x = −( p − 1) u(t, x) p div v(t, x) d x, t > 0. dt Rd Rd
The Aggregation Equation with Power-Law Kernels
659
By the definition, div v = −K ∗ u = −α(α + d − 2)|x|α−2 ∗ u ≥ 0.
Therefore, Rd u(t, x) p d x is non-increasing in t. On the other hand, if instead v is defined by v = ∇ K ∗ u, then the velocity field is attractive, and we can see from the proofs below that the results of Theorems 4.1 and 4.2 remain valid in the case α ∈ (2 − d, 0). 4.1. Proof of Theorem 4.1. For fixed α ∈ (2 − d, 2), we define a function ψ : [0, ∞) → R by 1 − ρy1 ψ(ρ) = – dσ (y). (4.3) 2−α |e S1 1 − ρy| The following lemma plays a key role in our proof.
Lemma 4.4. Let K (x) = |x|α , α ∈ (2 − d, 2), μ ∈ P Rd be a radially symmetric probability measure. Then for any x = 0, we have ∞ x (∇ K ∗ μ)(x) = α|x|α−1 ψ(ρ/|x|) d μ(ρ) ˆ . (4.4) |x| 0 Moreover, ψ is continuous, positive, non-increasing on [0, ∞), and ψ(0) = 1,
lim ψ(ρ)ρ 2−α =
ρ→∞
d +α−2 . d
Proof. Equality (4.4) follows from a direct computation. We only prove the second part of the lemma. Since α > 2 − d and |1 − ρy1 | 1 dσ (y) ≤ – dσ (y), – 2−α 1−α S1 |e1 − ρy| S1 |e1 − ρy| the integral on the right-hand side of (4.3) is convergent for any ρ ∈ [0, ∞), and by the dominated convergence theorem, it is easy to see that ψ is continuous on [0, ∞) and ψ(0) = 1. We rewrite (4.3) as ωd−1 π (1 − ρ cos θ )(sin θ )d−2 ψ(ρ) = dθ, (4.5) ωd 0 A(ρ, θ )2−α where A(ρ, θ ) = (1 + ρ 2 − 2ρ cos θ )1/2 . For ρ ∈ [0, 1) ∪ (1, ∞), a direct computation yields ψ (ρ) =
ωd−1 (I1 + I2 ), ωd
(4.6)
660
H. Dong
where
π
(− cos θ )(sin θ )d−2 dθ, A(ρ, θ )2−α 0 π (1 − ρ cos θ )(sin θ )d−2 (ρ − cos θ ) I2 := (α − 2) dθ. A(ρ, θ )4−α 0 Integrating by parts gives us π (sin θ )d−1 1 α − 2 π (sin θ )d ρ I1 = − dθ = dθ. (4.7) d − 1 0 A(ρ, θ )2−α d − 1 0 A(ρ, θ )4−α From (4.6) and (4.7), we get ωd−1 (α − 2) π (sin θ )d−2 ρ(sin θ )2 + (1 − ρ cos θ )(ρ − cos θ ) dθ, ψ (ρ) = 4−α ωd d −1 0 A(ρ, θ ) ωd−1 (α − 2) π (sin θ )d−2 ρ(sin θ )2 d 2 = − A(ρ, θ ) cos θ dθ 4−α ωd d −1 0 A(ρ, θ ) π ωd−1 (α − 2) ρ(sin θ )d d (4.8) dθ + I1 . = 4−α d − 1 ωd 0 A(ρ, θ ) I1 :=
Thanks to (4.8) and (4.7), we have ψ (ρ) =
ωd−1 (α − 2)(d + α − 2) ωd (d − 1)
π 0
ρ(sin θ )d dθ. A(ρ, θ )4−α
Therefore, ψ (ρ) < 0 for ρ ∈ [0, 1) ∪ (1, ∞) and, by the continuity of ψ, we conclude that ψ is non-increasing on [0, ∞). Finally, we consider the asymptotic behavior of ψ(ρ) as ρ → ∞. It follows from (4.5) and (4.7) that π ωd−1 (sin θ )d−2 ψ(ρ) = dθ + ρ I1 2−α ωd 0 A(ρ, θ ) α − 2 (sin θ )d ρ 2 ωd−1 π (sin θ )d−2 + dθ. = ωd 0 A(ρ, θ )2−α d − 1 A(ρ, θ )4−α Therefore, ψ(ρ) goes to zero as ρ → ∞, since α < 2. Moreover, ωd−1 π α−2 lim ψ(ρ)ρ 2−α = (sin θ )d−2 + (sin θ )d dθ ρ→∞ ωd 0 d −1 α−2 ωd−1 π (sin θ )d−2 dθ (sin θ )d−2 + = ωd 0 d d +α−2 > 0. = d Here we used two elementary identities π d −1 π (sin θ )d dθ = (sin θ )d−2 dθ, d 0 0 π ωd−1 (sin θ )d−2 dθ = ωd .
(4.9)
0
Note that since ψ is non-increasing, (4.9) also implies that ψ is positive. The lemma is proved.
The Aggregation Equation with Power-Law Kernels
661
The following lemma can be deduced from Lemma 4.4 in the same way as Lemma 2.4 is deduced from (2.3). We omit the details and refer the reader to [5, Sect. 4]. Lemma 4.5. Let K (x) = |x|α , α ∈ (0, 2), and (μt )t∈[0,∞) be a radially symmetric weakly continuous measured-valued solution of the aggregation equation (1.1). Then the vector field v(t, x) = −(∇ K ∗ μt )1x =0 is continuous on [0, ∞) × (Rd \ {0}). For any t ≥ 0, we have μt = X t# μ0 , where, for each x ∈ Rd , X t = X t (x) is an absolutely continuous function on [0, ∞) satisfying
d dt
X t (x) = v(t, X t (x))
for a.e. t ∈ (0, ∞);
X 0 (x) = x. Moreover, for any x = 0, X t (x) = Rt (|x|)x/|x|, where Rt is an absolutely continuous, non-negative and non-increasing function in t ∈ [0, ∞), and μˆ t = Rt# μˆ 0 . Consequently, for any x = 0, (∇ K ∗ μt )(x) = α|x|α−1
∞
ψ(Rt (ρ)/|x|) d μˆ 0 (ρ)
0
x , |x|
and |(∇ K ∗ μt )(x)| is non-decreasing in t. Next we prove a point-wise lower bound of the velocity v at t = 0. Lemma 4.6. Let K (x) = |x|α , α ∈ (0, 2), and u 0 be the initial data defined in (4.1) with ε ∈ (0, 1): u 0 (x) =
L 1|x|≤1/2 . |x|d+α−2+ε
Then there exists a constant δ1 > 0 such that |(∇ K ∗ u 0 )(x)| ≥ δ1 |x|1−ε
(4.10)
for any x ∈ Rd satisfying 0 < |x| < 1/2. Proof. Clearly, we have uˆ 0 (ρ) = Lωd ρ 1−α−ε 1ρ<1/2 . By the positivity and continuity of v(0, ·) in Rd \{0}, it suffices to prove (4.10) for x ∈ Rd satisfying 0 < |x| < 1/4. It follows from Lemma 4.4 that for any ρ ∈ (|x|, 1/2),
662
H. Dong
ψ(ρ/|x|) ≥ δ0 (|x|/ρ)2−α
(4.11)
for a constant δ0 > 0 independent of |x|. Thus, for any x ∈ Rd satisfying 0 < |x| < 1/4, |(∇ K ∗ u 0 )(x)| ≥ α|x|α−1
1/2
|x|
≥ αδ0 Lωd |x|
ψ(ρ/|x|)uˆ 0 (ρ) dρ 1/2
|x|
ρ −1−ε dρ
αδ0 Lωd |x| |x|−ε − (1/2)−ε ε ≥ δ1 |x|1−ε ,
=
since 0 < |x| < 1/4. The lower bound (4.10) is proved.
Now we are ready to prove Theorem 4.1. Proof of Theorem 4.1. With Lemma 4.6, the proof of Theorem 4.2 is essentially the same as that of Theorem 4.2 [2]. We present it here for completeness. By the definition of the push forward, for any t > 0, we have μt ({0}) = (X t# μ0 )({0}) = μ0 (X t−1 ({0}). Note the solution of the ordinary differential equation r = −δ1 r 1−ε with initial data r0 > 0 reaches zero at a finite time t = r0ε /(εδ1 ). By Lemmas 4.5 and 4.6, for any t > 0, we can find δ > 0 such that X t (Bδ ) = {0}. In other words, Bδ ⊂ X t−1 ({0}). Since μ0 (Bδ ) > 0, we conclude μt ({0}) > 0 for any t > 0. The theorem is proved. 4.2. Proof of Theorem 4.2. The following lemma is an analogy of Lemma 2.5. Lemma 4.7. Let K (x) = |x| and u 0 be the initial data defined in (4.2) with k ∈ (1/ ps , 1): u 0 (x) =
L 1|x|≤1/2 . |x|d+α−2 (− log |x|)k
Then there exists a constant δ1 > 0 such that |(∇ K ∗ u 0 )(x)| ≥ δ1 |x|(− log |x|)1−k
(4.12)
for any x ∈ Rd satisfying 0 < |x| < 1/2. Proof. Clearly, we have uˆ 0 (ρ) = Lωd ρ 1−α (− log ρ)−k 1ρ≤1/2 . As before, it suffices to prove (4.12) for x ∈ Rd satisfying 0 < |x| < 1/4. It follows from Lemma 4.4 that, for any ρ ∈ (|x|, 1/2), (4.11) holds for a constant δ0 > 0 independent
The Aggregation Equation with Power-Law Kernels
663
of |x|. Thus, for any x ∈ Rd satisfying 0 < |x| < 1/4, |(∇ K ∗ u 0 )(x)| ≥ α|x|α−1
1/2 |x|
≥ αδ0 Lωd |x|
ψ(ρ/|x|)uˆ 0 (ρ) dρ 1/2
|x|
(− log ρ)−k
dρ ρ
δ0 Lωd |x| (− log |x|)1−k − (log 2)1−k =α 1−k ≥ δ1 |x|(− log |x|)1−k , since 0 < |x| < 1/4. The lemma is proved.
Proof of Theorem 4.2. We follow the proof of Theorem 2.1 with some modifications. Thanks to Lemmas 4.7 and 4.5, (2.5) remains true for any r ∈ (0, 1/2). Recall that ps = d/(d + α − 2). Now we suppose that, for some t > 0, μt has a density function u(t, x) ∈ L ps Rd . By Hölder’s inequality, for any r ∈ (0, 1/2), μt (B Rt (r ) ) = u(t, x) d x B Rt (r )
u (t, x) d x
≤
ps
B Rt (r )
1 ps
|B Rt (r ) |
1− p1s
≤ u(t, ·) L ps (Rd ) (Rt (r ))2−α . On the other hand, by the definitions of μt and Rt , r μt (B Rt (r ) ) ≥ μ0 (Br ) = Lωd ρ 1−α (− log ρ)−k dρ.
(4.13)
(4.14)
0
We combine (4.13), (4.14), and (2.5) to get μt (B Rt (r ) ) (Rt (r ))2−α
r 1−α ρ (− log ρ)−k dρ ≥ Lωd 0 . 1−k r 2−α e−(2−α)δ2 t (− log r )
u(t, ·) L ps (Rd ) ≥
(4.15)
However, by L’Hospital’s rule,
r 1−α ρ (− log ρ)−k dρ lim 0 r 0 r 2−α e−(2−α)δ2 t (− log r )1−k = lim
r 0 e−(2−α)δ2
= ∞.
t (− log r )1−k
r 1−α (− log r )−k r 1−α (2 − α) 1 + δ2 t (1 − k)(− log r )−k
This gives a contradiction to (4.15) since we assume u(t, ·) ∈ L ps Rd . The theorem is proved.
664
H. Dong
References 1. Barenblatt, G.: Scaling, self-similarity, and intermediate asymptotics. With a foreword by Ya. B. Zeldovich. Cambridge Texts in Applied Mathematics, 14, Cambridge: Cambridge University Press, 1996 2. Bertozzi, A., Carrillo, J., Laurent, T.: Blowup in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22(3), 683–710 (2009) 3. Bertozzi, A., Laurent, T.: Finite-time blow-up of solutions of an aggregation equation in Rn . Commun. Math. Phys. 274(3), 717–735 (2007) 4. Bertozzi, A., Laurent, T.: The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels. Chin. Ann. Math. Ser. B 30(5), 463–482 (2009) 5. Bertozzi, A., Laurent, T., Rosado, J.: L p theory for the multidimensional aggregation equation. Comm. Pure Appl. Math. 64(1), 45–83 (2010) 6. Bodnar, M., Velázquez, J.: An integro-differential equation arising as a limit of individual cell-based models. J. Diff. Eqs. 222(3), 341–380 (2006) 7. Carrillo, J., Di Francesco, M., Figalli, A., Laurent, T., Slepˇcev, D.: Global-in-time weak measure solutions, finite-time aggregation and confinement for nonlocal interaction equations. Preprint 8. Carrillo, J., McCann, R., Villani, C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal. 179(2), 217–263 (2006) 9. Laurent, T.: Local and Global Existence for an Aggregation Equation. Comm. Part. Diff. Eqs. 32(10–12), 1941–1964 (2007) 10. Li, D., Rodrigo, J.: Finite-time singularities of an aggregation equation in Rn with fractional dissipation. Commun. Math. Phys. 287(2), 687–703 (2009) 11. Li, D., Rodrigo, J.: Refined blowup criteria and nonsymmetric blowup of an aggregation equation. Adv. Math. 220(1), 1717–1738 (2009) 12. Li, H., Toscani, G.: Long-time asymptotics of kinetic models of granular flows. Arch. Rat. Mech. Anal. 172(3), 407–428 (2004) 13. Mogilner, A., Edelstein-Keshet, L.: A non-local model for a swarm. J. Math. Bio. 38(6), 534–570 (1999) Communicated by P. Constantin
Commun. Math. Phys. 304, 665–688 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1238-5
Communications in
Mathematical Physics
Ergodicity of Some Open Systems with Particle-Disk Interactions Tatiana Yarmola Department of Mathematics, Mathematics Building, University of Maryland, College Park, MD 20742, USA. E-mail: [email protected] Received: 6 May 2010 / Accepted: 15 November 2010 Published online: 12 April 2011 – © Springer-Verlag 2011
Abstract: We consider steady states for a class of mechanical systems with particledisk interactions coupled to two, possibly unequal, heat baths. We show that any steady state that satisfies some natural assumptions is ergodic and absolutely continuous with respect to a Lebesgue-type reference measure and conclude that there exists at most one absolutely continuous steady state. 0. Introduction Explaining macroscopic phenomena of open systems from the microscopic dynamics is an intriguing subject in statistical mechanics. By an open system we mean a deterministic and energy conserving system that exchanges energy and matter with multiple heat baths. Some examples of such systems have been studied [1–11], but a good understanding of non-equilibrium behavior, i.e. when the heat baths are not equal, has not been reached yet. In this paper we consider a mechanical system that attempts to reproduce the phenomenological laws of thermodynamic transport [9]. It consists of a rectangular domain containing a chain of N identical pinned down disk scatterers that are allowed to rotate freely. See Fig. 1. The particles in the system bounce elastically from the walls and exchange energy with the disks through “perfectly rough” collisions [8,9]. The system is coupled to two possibly unequal heat baths through the openings, which correspond to the “left” and the “right” sides of the rectangle. Once a particle reaches an opening, it leaves the system forever. New particles can be introduced to the system by the heat baths; they are emitted at random times according to some probability distributions that describe injection positions and velocities. Neither energy nor the number of particles is conserved, so the system can be loosely referred to as a “grand-canonical ensemble”. The detailed settings are available in Sect. 1. This paper demonstrates rigorous results regarding invariant densities and ergodicity of the steady states, which we refer to as invariant measures. Theorem 1 provides
666
T. Yarmola
Fig. 1. 1D array of disks in a rectangle
conditions under which an invariant measure is guaranteed to be ergodic and absolutely continuous with respect to a natural reference measure. From Theorem 1 we immediately obtain uniqueness of the absolutely continuous measure as well as the ergodic decomposition. In the equilibrium situation, an absolutely continuous invariant measure can be written down explicitly [7]. By Theorem 1 we conclude that this invariant measure is ergodic. The question of existence and uniqueness of the steady states for other types of open systems that attempt to model thermodynamic transport has been studied to some extent. The best studied examples are chains of anharmonic oscillators coupled at both ends to heat reservoirs [11]. The existence and uniqueness results for such systems have been obtained in [3,5,6] under rather restrictive assumptions on the potential in the chain and coupling to the reservoirs. For open particle systems with indirect particle interactions, existence of the stationary states has only been demonstrated in equilibrium situations by providing them explicitly [1,7]. For non-equilibrium states, existence turns out to be a nontrivial mathematical problem which requires a detailed understanding of the dynamics and we leave it for future work. Uniqueness results for open particle systems are scarce. Ergodicity and thus uniqueness of natural invariant measures for a 1-dimensional particle model was obtained in [1]; we borrow some of the ideas developed there in our proof. For planar geometries, an important first step towards showing ergodicity we rely on was done in [4], where the authors demonstrated that the action of the baths can drive the system from “almost” any state to “almost” any other state in a finite time. We chose simpler geometry in our model in order to focus on the essential properties of the system that lead to absolute continuity and ergodicity of the invariant measures leaving geometric complications aside. However, combined with the results in [4], only minor modifications are needed for our argument to carry through for the class of systems described there. Our argument might potentially be generalized to a wider class of geometries, e.g. presented in [7,10]. The organization of our paper is the following. Section 1 contains a detailed description of the model. We state the main results in Sect. 2. The remaining part of the paper is devoted to the proof of Theorem 1. We start with an outline of proof in Sect. 3, where we state three propositions and show how they imply Theorem 1. The propositions are proven in Sects. 4, 5, 6, and 7. 1. Model Description 1.1. Dynamics of closed systems. Let 0 be a rectangle bounded by y = ±1, x = 0 and x = 2N , where N is an arbitrary positive integer. In the interior of 0 lie N disks D j , 1 ≤ j ≤ N , of equal radii R < 1 centered at (2 j − 1, 0) ∈ 0 , 1 ≤ j ≤ N . The centers of the disks are fixed and the disks are allowed to rotate freely around the center,
Ergodicity of Some Open Systems with Particle-Disk Interactions
667
each carrying a finite amount of kinetic energy derived from its angular velocity. Denote the states of the disks by (ϕ j , ω j ), 1 ≤ j ≤ N , where ϕ j is the disk’s angular position relative to a marked reference point and ω j is the disk’s angular velocity. A number of particles move around in the playground = 0 \ ∪ Nj=1 D j , with particle positions qi ∈ and velocities vi ∈ R2 . Apart from collisions with the boundary of the playground ∂ = ∂0 ∪ (∪ Nj=1 ∂ D j ) the particles are assumed to move freely with constant velocities; particles do not interact with each other. The collisions with ∂0 are specular, and upon collision of a particle with a disk, a certain energy exchange occurs [8,9]. More precisely: The phase space of such system with k particles is ˜ k = ( k × ∂ D1 × · · · × ∂ D N × R2k+N )/ ∼, where q = (q1 , . . . , qk ) ∈ k denotes the positions of k particles, ϕ = (ϕ1 , . . . , ϕ N ) ∈ ∂ D1 × · · · × ∂ D N denotes the angular positions of the disks, v = (v1 , . . . , vk ) ∈ R2k denotes the velocities of k particles, ω = (ω1 , . . . , ω N ) ∈ R N denotes the angular velocities of the N disks, and ∼ is the relation identifying pairs of points on the collision manifold: Mk = {(q, ϕ, v, ω) : qi ∈ ∂ for some i} with the rules of identification as follows: Let vi = ((vi )t , (vi )⊥ ) be the tangential and the normal components of vi . If a particle collides with the boundary of the playground, qi ∈ 0 , then the angle of reflection is equal to the angle of incidence, i.e. (vi )⊥ = −(vi )⊥ ,
(vi )t = (vi )t .
If collision with a disk occurs, qi ∈ ∂ D j for some j, then (vi )⊥ = −(vi )⊥ , (vi )t = (vi )t − Rωj = Rω j +
2η ((vi )t − Rω j ), 1+η
2 ((vi )t − Rω j ), 1+η
where η = mR 2 is a dimensionless parameter relating the moment of inertia of the disc , the mass of the particle m, and the radius of the disc R [8,9]. Throughout the paper we assume that 0 < η < ∞. ˜ k is as follows: X ∼ X , X, X ∈ Mk , if The identification ∼ in the definition of the coordinates of X and X are equal except for vi ’s such that qi ∈ ∂. If qi ∈ 0 , we replace vi in X by vi in X . If qi ∈ ∂ D j , we replace vi and ω j in X by vi and ωj in X . Note that simultaneous collisions of several particles with the same disk are not defined, while there is no problem with simultaneous collisions with different disks and/or ∂0 . ˜ τ on ˜ k by Define the discontinuous flow ˜ τ (q, ϕ, v, ω) = (q + vτ, ϕ + ωτ, v, ω) if no collisions are involved. When collisions occur, the rules of identification are given ˜ τ on ˜ k , where above. Then m k is an invariant measure for m˜ k = (λ2 | )k × ρ|∂ D1 × · · · × ρ|∂ D N × λ2k+N . Here λd is d-dimensional Lebesgue measure and ρ|∂ D j is the uniform measure on the circle ∂ D j [7].
668
T. Yarmola
1.2. Dynamics of open systems. 1.2.1. Coupling to heat baths. Suppose the rectangle 0 has two openings, γ L = {0} × [−1, 1] ∈ ∂0 and γ R = {2N } × [−1, 1] ∈ ∂0 , each connected to a heat bath that absorbs and emits particles. A particle absorbed by one of the baths leaves the system forever. The injection process for each bath is characterized by the following parameters: L and R - the injection rates of the baths. The injection processes are Poisson with rates L and R respectively. ϒ L (υ) and ϒ R (υ) - the distributions of the positions of injection with values in γ L and γR . L (ζ ) and L (ζ ) - the distributions of the angles of injection with values in (− π2 , π2 ). SL (ς ) and S R (ς ) - the distributions of the injected particle speeds with values in (0, ∞). I.e. a particle is injected from the left bath at a random time τ ∈ (0, ∞) given by exponential distribution of rate L , with random position ξ ∈ γ L drawn from ϒ L (υ), at random angle δ drawn from L (ζ ) and with random speed s drawn from SL (ς ). Similarly for the right bath. We assume that each of the distributions ϒ L (υ), ϒ R (υ), L (ζ ), R (ζ ), SL (ς ), and S R (ς ) has positive density on the specified domains. 1.2.2. The phase space. Since we are interested in the invariance properties, we would ˜ k be as like to treat all particles in the system as identical and indistinguishable. Let in Subsect. 1.1. Then the phase space of the system coupled to heat bath(s) is a disjoint union = ∞ k=0 k , ˜ k obtained by identifying the permutations of k particles. where k is the quotient of If we denote unordered sets by {. . .}, under phase space, points in are denoted by X = ({q1 , . . . , qk }, (ϕ1 , . . . , ϕ N ), {v1 , . . . , vk }, (ω1 , . . . , ω N )), with v j understood to be attached to q j . Denote the quotient of the measure m˜ k by m k . Let τ be a continuous-time Markov process on defined as follows: – – –
˜ τ identifying the permutations while no particles enter or exit τ is the quotient of the system; if τ ∈ k for all 0 ≤ τ ≤ τ0 , and a particle exits the system at time τ0 , then τ0 (z) jumps to k−1 ; if τ ∈ k for all 0 ≤ τ ≤ τ0 , and a particle is injected from one of the baths or point sources at time τ0 , then τ0 jumps to k+1 .
It is hard to write down the transition probabilities of τ explicitly because particles may enter at any time. Denote the time-τ transition probability starting at state X for τ by PXτ , if defined. PXτ if not defined for all τ and X : if, for example, X is a state such that two particles will have their first collision at the same time t with the same disk D j , then PXτ is not defined for all τ ≥ t. Note that PXτ is defined if the dynamics is defined with probability 1.
Ergodicity of Some Open Systems with Particle-Disk Interactions
669
Definition 1. We will say that a measure μ is absolutely continuous with respect to a measure ν if for any measurable set A, ν(A) = 0 ⇒ μ(A) = 0. We will say that two measures μ and ν defined on a set C are singular if there exist sets A and B, A ∩ B = ∅, A ∪ B = C such that ν(A) = 0 and μ(B) = 0. When a measure μ is absolutely continuous with respect to ν, we will denote it by μ ν; we will also say that μ has a density with respect to ν. If μ and ν are singular, we will denote it μ ⊥ ν. Definition 2. Let m be a measure on such that for each nonnegative integer k, the conditional measure on k is m k . For our purposes, m is a natural reference measure: if μ is any measure on , we are interested whether μ is absolutely continuous with respect to m. In the rest of this paper, we will refer to m as Lebesgue measure and when we will say that μ is absolutely continuous without mentioning any reference measure, we would mean that μ is absolutely continuous with respect to m. 1.3. Problems of interest. Given a measure ν, the push forward of ν under τ , if defined, is given by ( τ )∗ ν = (PXτ )(A)dν(X ). Note that PXτ = ( τ )∗ δ X when defined. Definition 3. A Borel probability measure μ on is called invariant under τ if its push forward under τ is defined for all τ and μ(A) = (( τ )∗ μ)(A) = (PXτ )(A)dμ(X ).
A classical way of analyzing flow τ is studying the properties of its invariant measures such as existence, uniqueness, absolute continuity with respect to the Lebesgue measure m, and ergodicity. Proving the existence of invariant measures would require very technical arguments that involve dealing with tightness and discontinuities; we leave these problems for future work. In this paper we are going to focus on the issues of absolute continuity with respect to m and ergodicity of the invariant measures provided they exist. The main result of this paper claims that, for the class of systems in consideration, if there exists an invariant measure μ such that the measure of the set of all states with “trapped” particles is zero, then μ is both absolutely continuous and ergodic. 2. Results Definition 4. A state X ∈ is said to contain a trapped particle if either – the velocity of the particle is zero, v = 0, or – the x-component of the velocity of the particle is zero, vx = 0, and the position of the particle has its x-coordinate “between the disks”, i.e. q ∈ ([0, 1 − R] ∪ (∪ Nj=1 [2 j − (1 − R), 2 j + (1 − R)]) ∪[2N − (1 − R), 2N ]) × [−1, 1]. Note that if we evolve the system starting from an initial state containing a trapped particle along any sample path, the system is going to contain a trapped particle at all times.
670
T. Yarmola
Let ST ⊂ be the set of all states with trapped particles. Theorem 1. Suppose there exists a probability measure μ invariant under τ with μ(ST ) = 0. Then μ is absolutely continuous with respect to the Lebesgue measure m and ergodic. Remark 1. Since obviously μ(k ) = 0 for all k, our assertion that μ is absolutely continuous with respect to m means that on each k , it has a density with respect to m k . Corollary 1. If ν is an ergodic measure for τ , then for some k ≥ 0, ν is supported on ∪∞ j=k j and can be represented as a direct product of two measures, ν = μ × π , such that μ is the unique absolutely continuous ergodic measure with μ(ST ) = 0 as in Theorem 1 and π is a singular measure supported on the states in k traced by the trajectories of k trapped particles. Equilibrium Case. Suppose the system is coupled to two equal heat baths, characterized by temperature T and injection rate , i.e. the injection process at each bath is Poisson with rate , the distributions for the positions of injections are uniform on γ L and γ R , |γ L | = |γ R | = |γ |, and upon injection, a particle is assigned a random velocity v sampled from the distribution ce−mβ|v| |v|cos(ϕ)dv, 2
c=
2(mβ)3/2 , √ π
where β = 1/T, m is a particle’s mass, and ϕ ∈ (− π2 , π2 ). Theorem 2. The invariant probability measure μ characterized by the properties below is ergodic. (i) The number of particles in the cell is a Poisson random variable with mean √ √ ar ea() m · √ , λ=2 π |γ | T k −λ
e i.e. μ(k ) = λ n! . (ii) μ has conditional densities ck σk dm k on k , where ck is a normalizing constant and for X ∈ k ,
σk (X ) = e−β(
ω2j j=1 2
N
+m
k i=1
|vi |2 2 )
.
Proof. For η = mR 2 = 1, the invariance of μ is shown in [7]. The argument can be generalized for any 0 < η < ∞. Since μ m and μ(ST ) = 0, ergodicity of μ follows from Theorem 1. The remaining part of the paper is devoted to the proof of Theorem 1. In order to simplify the exposition, we chose to present the proofs of all the technical lemmas for the situation η = 1 only; all proofs could be generalized for any 0 < η < ∞. Aside from technical lemmas, the argument deals with any η, 0 < η < ∞.
Ergodicity of Some Open Systems with Particle-Disk Interactions
671
3. Outline of Proof of Theorem 1 In this section we state three propositions and show that they imply Theorem 1. Our proof uses some ideas from [1]. Idea of proof: Proposition 1 states that absolutely continuous measures stay absolutely continuous under τ . Propositions 2 and 3 imply that any singular measure eventually acquires an absolutely continuous component when evolved under τ . If follows that the singular component of any invariant probability measure with μ(ST ) = 0 must be zero. Propositions 2 and 3 also imply that m-almost all initial states must belong to the same ergodic component; ergodicity follows by the absolute continuity of μ. Denote by –
–
SSC the set of all states in such that a simultaneous collision with the same disk occurs under the evolution of the system with no particle injections, i.e. for some t > 0, qit ∈ ∂ Dk and q tj ∈ ∂ Dk for i = j. Note that the evolution of the system is not defined after such a collision. ST C the set of all states in such that a particle stops under the evolution of the system with no particle injections, i.e. ∃t such that ∀τ > t, vit = 0. When η = 1, this situation occurs when a particle hits a stopped disk tangentially; when η = 1, it i )t occurs when a particle hits a disk with angular velocity ω = (η−1)(v (η+1)R tangentially, where (vi )t is a tangential component of the particle’s velocity upon collision. Let S = ST ∪ SSC ∪ ST C .
Definition 5. A state X ∈ is called admissible if X ∈ S. Given t > 0, the probability that no particles are injected on time interval [0, t] is positive. If μ is an invariant probability measure with μ(ST ) = 0, then it cannot give positive measure to either SSC or ST C , i.e. μ(S) = μ(ST ∪ SSC ∪ ST C ) = 0. It follows that if we start with any measure ν μ, the push forward of ν under the Markov process τ , ( τ )∗ ν, is well defined for all τ > 0. For any measure ν, denote by ν and ν⊥ the absolutely continuous and singular components of ν with respect to the Lebesgue measure m, i.e. ν = ν + ν⊥ , ν m and ν⊥ ⊥ m. Proposition 1. If ν m, then ( t )∗ ν is well defined for any t > 0 and ( t )∗ ν m. In particular, ( t )∗ (μ ) m for any t > 0. Consider a sequence of particle injections c = (c1 , . . . , cn ) at times 0 < t1 < t2 < · · · < tn < T such that at time ti , a particle enters the system at location ξi ∈ γ j , at angle δi ∈ (− π2 , π2 ), and with speed si ∈ (0, ∞). Then, assuming no simultaneous collisions with the same disks occur, one can generate a sample path σ defined on [0, T ] in which one starts from state X ∈ and injects particles into the system according to c. We call C a canonical neighborhood of c if there are disjoint open neighborhoods Ti of ti contained in [0, T ], i of ξi , i of δi , and Si of si such that for each sequence of injections c ∈ C, exactly one particle is injected in each Ti , with position in Q i , angle in i , and speed in Si . No other injections occur in the time interval [0, T ]. We call a canonical neighborhood of σ if there exist an open neighborhood U of X and a canonical neighborhood C of c such that each sample path in starts with an initial condition in U , is generated by a sequence of injections from C, and no simultaneous collisions with same disks occur. Note that for any Y ∈ U , if Y is a set of all sample paths on [0, T ] starting at Y , then the probability that a sample path from Y belongs to is positive.
672
T. Yarmola
Propositions 2 and 3 split the problem of acquiring density for singular measures in the following way: Proposition 2 deals with the simplest situation of acquiring density for a point measure supported on an particle-less initial state Y0 ∈ 0 ; Proposition 3 provides a sample path from any admissible initial state to a particle-less state, from which a density can be acquired using Proposition 2. Proposition 2. Given a state Y0 ∈ 0 with a condition that if η = 1, all disks have nonzero angular velocities, there exist an open neighborhood U0 of Y0 , time T0 , and a set A0 ⊂ 0 with m 0 (A0 ) > 0, such that for any Y ∈ U0 , [( T0 )∗ δY ] has strictly positive density on A0 . In particular, [( T0 )∗ δY ] () = 0. Proposition 3. Given an admissible state X ∈ , a state Y0 ∈ 0 , and a neighborhood U0 ⊂ 0 of Y0 , there exist time T , a sample path σ on [0, T ] that starts at X and ends at Y0 , and a canonical neighborhood of σ , such that each sample path in ends in U0 . Proof of Theorem 1 assuming Propositions 1, 3, and 2. Propositions 2 and 3 imply that for any admissible state X ∈, ∃ a neighborhood U of X , such that ∀Y ∈U, [( T +T0 )∗ δY ] has strictly positive density on A0 and, in particular, [( T +T0 )∗ δY ] () = 0. Assume μ⊥ () = 0. Since μ is invariant with μ(ST ) = 0, μ⊥ (S) = μ(S) = 0. Therefore [( T +T0 )∗ μ⊥ ] () = 0. Applying Proposition 1 we conclude that ∀t > T + T0 , [( t )∗ μ⊥ ] () = 0. Clearly ( t )∗ μ = [( t )∗ (μ )] +[( t )∗ (μ )]⊥ +[( t )∗ (μ⊥ )] +[( t )∗ (μ⊥ )]⊥ . By Proposition 1, ∀t > 0, [( t )∗ (μ )]⊥ () = 0. Therefore, for t > T + T0 , [( t )∗ μ] () > μ (), which contradicts the invariance of μ. This proves the absolute continuity of μ with respect to the Lebesgue measure m. Assume μ1 and μ2 are ergodic measures with μ1 (ST ) = μ2 (ST ) = 0. Then μ i m, i = 1, 2. Suppose there exists a Borel function ϕ such that c1 = ϕdμ1 = ϕdμ2 = c2 . Then by the Random Ergodic Theorem, there exist Ai ∈ , i = 1, 2, with μi (Ai ) = 1 such that for every X ∈ Ai the ergodic averages are equal to ci for a.e. sample path starting from X ; and m(Ai ) > 0, since μi m. Since for any admissible state X ∈ , [( T +T0 )∗ δ X ] has strictly positive density on A0 , the random ergodic averages for all the admissible states are equal for measure 1 sets of sample paths. Since m(S) = 0, either m(A1 ) = 0 or m(A2 ) = 0, a contradiction. Therefore μ1 = μ2 = μ. We prove Proposition 1 in Sect. 4, Proposition 2 in Sect. 5, and Proposition 3 in Sects. 6 and 7. 4. Proof of Proposition 1 In order to show that, given ν m and t > 0, ( t )∗ ν is defined and ( t )∗ ν m, we would like to consider a countable number of subcases depending on how many particles entered on time interval (0, t] and show that for each subcase the time-t push forward of ν is well defined and absolutely continuous. Suppose n particles enter on time interval (0, t]. For n ≥ 1, let Cn = [(0, t] × (γ L ∪ γ R ) × (− π2 , π2 ) × (0, ∞)]n / ∼ be the set of all possible injection parameters for these n particles modulo permutations and let ρn be the natural probability measure
Ergodicity of Some Open Systems with Particle-Disk Interactions
673
on Cn , i.e. the product of appropriate injection distributions modulo permutations (see Subsect. 1.2). Given t1 , . . . , tn−1 with 0 = t0 < t1 < · · · < tn = t, denote by Ct1 ,...,tn−1 the subset of Cn such that j th particle is injected on (t j−1 , t j ]. For any τ > 0, denote by ( nτ )∗ ν the time-τ push forward of ν under the random dynamics assuming that n particles entered on time interval (0, τ ], if defined. Given a subset C ⊂ Cn of injection parameters, denote by ( n,C τ )∗ ν the time-τ push forward of ν under the random dynamics assuming that n particles entered on time interval (0, τ ] with injection parameters from C, if defined. Lemma 1. If ν m, ( 0t )∗ ν is defined and absolutely continuous with respect to m. Moreover, for any n ≥ 1 and a choice of t1 , . . . , tn−1 with 0 = t0 < t1 < · · · < tn = n,Ct1 ,...,tn−1
t, ( t
)∗ ν is defined and absolutely continuous with respect to m.
Proof of Proposition 1 assuming Lemma 1. Let B = {X ∈ : PXt is not defined }. Then for each X ∈ B, there exists a positive measure set C X of injections such that, if we start at X and follow any sequence of injections c ∈ C X , the dynamics is not defined up to time t, i.e. a simultaneous collision with the same disk occurs before time t for each such injection. Assume m(B) > 0. Define A = {(X, c) : X ∈ B; c ∈ C X }. Then by Fubini’s theorem (m × ρn )(A) > 0. Let (X, c) be a Lebesgue density point of A with X admissible and c having no simultaneous injections ((X, c) exists since m(S) = 0 and probability of simultaneous injections is zero). If tc1 , . . . , tcn are the injection times for c, choose any t1 , . . . , tn−1 such that 0 = t0 < tc1 < t1 < · · · < tn−1 < tcn < tn = t (c cannot be an empty sequence of injections since ( 0t )∗ ν is defined by Lemma 1). Let U be any neighborhood of X such that any Y ∈ U is admissible (the set of admissible states is open and dense in ). Then (m × ρn )[A ∩ (U × Ct1 ,...,tn−1 )] > 0, while by Lemma 1 it must be zero. A contradiction. Therefore m(B) = 0 and ( nt )∗ ν is defined. Suppose ( nt )∗ ν has a nonzero singular component supported on a set F ⊂ . Then m(F) = 0 and m{X ∈ : PXt (F) > 0} > 0. By an argument similar to the above we get a contradiction to Lemma 1. Therefore ( nt )∗ ν is defined and absolutely continuous with respect to m, which completes the proof of Proposition 1. Proof of Lemma 1. Since Ct1 ,...,tn−1 ⊂ Cn is such that one particle is injected on each (t j−1 , t j ] and the probability of two particles entering at the same time is zero, to prove Lemma 1 it is enough to treat two situations: 0 particles are injected on (0, t] and 1 particle is injected on (0, t]. Case 1 (0 particles injected). Since m(S) = 0, where S is a set of non-admissible states, and ν m, the time-τ push forward of ν is well defined in this situation. The system behaves as a closed one until some particles exit. We would like to split the possible situations according to the number of particles that exit: 0: no particles exit. Let A0 ⊂ be the set of states such that no particles exit on time interval (0, t]. Decompose A0 = ∪Ak0 such that Ak0 = A0 ∩ k . Then ν| Ak m k 0
and must stay so at all times (0, t], i.e. ( 0τ )∗ (ν| Ak ) m k ∀τ ∈ (0, t]. Indeed, if 0
for some τ ≤ t and a Borel set B, [( 0τ )∗ (ν| Ak )]⊥ (B) = 0 but m k (B) = 0, then 0
(ν| Ak )( −τ B) ≥ [( 0τ )∗ (ν| Ak )]⊥ (B) = 0, while m k ( −τ B) = m k (B) = 0, a 0
0
contradiction to the absolute continuity of ν| Ak . Thus ( 0t )∗ (ν| A0 ) m. 0
674
T. Yarmola
1: 1 particle exits or several particles exit at the same time. Let Aτ1 be the set of states such that 1 particle exits on the time interval (τ, t] and does not collide with ∂\(γ L ∪ γ R ) on that time interval. If several particles exit at the same time, let Aτ1 be the set of states such that neither of these particles collides with ∂\(γ L ∪ γ R ) on the time interval (τ, t]. Then until time τ , the dynamics is equivalent to the dynamics of the closed system and ( 0τ )∗ (ν| Aτ1 ) m. Since after the time τ the particle(s) do not collide with the disk, the particle(s) coordinates are independent from the rest of the system and ( 0t )∗ (ν| Aτ1 ) = ( 0t−τ )∗ (P[( 0τ )∗ (ν| Aτ1 )]) m, where P denotes the projection of the measure on the remaining coordinates in the system. Since the statement is true for any τ ∈ [0, t), ( 0t )∗ (ν| A1 ) m, where A1 = ∪τ ∈[0,t) Aτ1 . n: particles exit at n different times with possibly several exiting at each time. Let An be the set of states such that n particle exits occur on the time interval (0, t], with possibly several particles exiting at the same time. Then An = ∪0=t0 0 half plane of x y-plane bounded by the vertical wall along the y-axis with an opening (−γ /2, γ /2) along the y-axis. Lemma 2 (Particles are injected with 4-dimensional uncertainty). Let ˜ be as above. Assume a particle is injected through the opening at a random entrance time τ ∈ (0, ∞), with a random position ξ ∈ (γ /2, γ /2), at a random angle δ ∈ (−π/2, π/2) and with a random speed s ∈ (0, ∞). Assume also that the distributions for entrance time, position, angle and speed are finite and positive on all the given intervals. Then the measure that describes the probability of finding the particle at a certain position (x t , y t ) with a certain velocity (vxt , v ty ) at any time t > 0 is absolutely continuous with respect to the Lebesgue measure on ˜ × R2 and has positive density everywhere. Proof of Lemma 2. The lemma holds if the mapping f t : (0, ∞) × (γ /2, γ /2) × (−π/2, π/2) × (0, ∞) → (0, ∞) × (−∞, ∞) × (0, ∞) × (−∞, ∞) such that (τ, ξ, δ, s) → (x t , y t , vxt , v ty ) is a diffeomorphism. This boils down to verifying that the Jacobian determinant is nonzero everywhere and that the map is surjective. Simple computations yield that: x t = (t − τ )s cos(δ), y t = −(t − τ )s sin(δ) + ξ, vxt = s cos(δ), and v ty = −s sin(δ) and thus the Jacobian matrix is: ⎛
−s cos(δ) ⎜ s sin(δ) ⎝ 0 0
⎞ 0 −(t − τ )s sin(δ) (t − τ ) cos(δ) 1 −(t − τ )s cos(δ) −(t − τ ) sin(δ) ⎟ ⎠ 0 −s sin(δ) cos(δ) 0 −s cos(δ) − sin(δ)
and its determinant is −s 2 cos(δ), which is nonzero on the domain of definition of f t .
Ergodicity of Some Open Systems with Particle-Disk Interactions
675
Given (x t , y t , vxt , v ty ), let s = (vxt )2 + (v ty )2 , δ = arctan(−v ty /vxt ), ξ=
x t sin(δ) + y t cos(δ) y t − ξ + ts sin(δ) , τ= . cos(δ) s sin(δ)
Such (τ, ξ, δ, s) clearly maps to (x t , y t , vxt , v ty ).
In Lemma 2 the domain ˜ is such that an injected particle cannot collide with the boundary of ˜ at any time. In the real situation, when a particle is injected into the playground , it might collide with ∂ in an arbitrarily short time, depending on the injection parameters. Let C1κ be the set of all particle injection parameters (τ, ξ, δ, s) such that one particle enters on time interval (0, κ], 0 < κ ≤ t, and does not collide with ∂ on time interval (0, κ]; no other particles are injected on time interval (0, t]. Since we assumed that all injection distributions have positive density everywhere and C1κ clearly has a nonempty interior, by Lemma 2 the measure that describes the particle location and velocity at time κ is absolutely continuous with respect to the Lebesgue measure on × R2 ; denote this measure by πC1κ . Suppose we start with a measure ν m and assume that one particle is injected on (0, t] with injection parameters drawn from Cκ1 . Then until time κ the coordinates associated with the injected particle are uncoupled from the coordinates of the rest of the system. Thus ν evolves as in Case 1, ( 0κ )∗ ν is defined, and ( 0κ )∗ ν m. Therefore 1,C1κ
( κ
)∗ ν = [πC1κ ] × [( 0κ )∗ ν] m.
The injected particles have entered by time κ and no other particles enter on time interval (0, t]. Thus on time interval (κ, t] the system behaves as in Case 1, implying that 1,C κ ( t 1 )∗ ν m. Since the above result is true for any κ ∈ (0, t], ( 1t )∗ ν m. 5. Proof of Proposition 2 We are going to “acquire density” starting with an initial state Y0 ∈ 0 by hitting each disk once with a particle possessing a “4-dimensional uncertainty”. Upon each such collision, each disk would acquire a “2-dimensional uncertainty.” That is, we will show that there exists an open set of injections C of N particles, such that the j th particle hits disk D j once and exits the system with no additional collisions by some uniformly selected time T0 . Moreover, ( TN0,C )∗ δY0 m. It will turn out that the same statement is true for nearby states Y ∈ 0 with ( TN0,C )∗ δY varying continuously with Y . In order to prove Proposition 2 we need the following lemmas:
Lemma 3 (Continuity). Let B ⊂ × R2 be an open ball of positions and velocities such that for any (x, y, vx , v y ) ∈ B a particle with position (x, y) and velocity (vx , v y ) at time t = 0 it is going to hit disk D j within time τ0 while not meeting ∂ again on the time interval [0, τ1 ], τ1 > τ0 . Note that the collision with D j must be non-tangential since B is open. Let F j be an open ball of angular positions and velocities of disk D j at time 0. For any t ∈ (τ0 , τ1 ), define f t : B × F j → × R2 × ∂ D j × R such that f t (x, y, vx , v y , ϕ, ω) = (x t , y t , vxt , v ty , ϕ t , ωt ) gives the positions and the velocities of the particle and disk D j at time t. Then f t is continuous for any t ∈ (τ0 , τ1 ).
676
T. Yarmola
Lemma 4 (Acquiring density for a disk). Let B ⊂ × R2 be as in Lemma 3. Assume ϕ,ω at time 0 disk D j has position ϕ and angular velocity ω. Define t : B → S1 × t t R, (x, y, vx , v y ) → (ϕ , ω ) to be the mapping of the particle position and velocity at time zero to the disk position and velocity at time t ∈ (τ0 , τ1 ). Let ν be a measure on ϕ,ω B equivalent to the Lebesgue. Then the push forward measure ( t )∗ ν is absolutely 1 continuous with respect to the Lebesgue measure on S × R and has positive density on some open set. Proof of Proposition 2 assuming Lemmas 3 and 4. Denote the angular positions and angular velocities of the disks in state Y0 by (ϕ1 , ω1 ), . . . , (ϕ N , ω N ). By assumption, if η = 1, ω j = 0, 1 ≤ j ≤ N . Suppose we inject a particle at time τ > 0 with some initial position ξ ∈ γ L , initial angle ϕ ∈ (− π2 , π2 ), and speed s ∈ (0, ∞) arranged in such a way that it first collides non-tangentially with D j at the top, (2 j − 1, R), and exits the system with no additional collisions; this is possible because ω j = 0, 1 ≤ j ≤ N , if η = 1. Since the collision with disk is non-tangential, by Lemmas 2 and 3 there exist open neighborhoods I j of ω j and C j of (τ, ξ, ϕ, s) such that for each ω ∈ I j and (τ, ξ, ϕ, s) ∈ C j , the injected particle follows a nearby path in , hits D j once, and exits the system with no additional collisions within some time T j . Suppose we subsequently hit each disk with a particle as described above. Define a neighborhood U of Y0 by U = ∂ D1 × · · · × ∂ D N × I1 × · · · × I N . N Let C = Nj=1 C j and T = j=1 T j . Then by Lemmas 2 and 4, for any Y ∈
U, ( TN ,C )∗ δY m 0 . Here ( TN ,C )∗ δY denotes the time-T push forward of δY , provided that exactly N particles enter on time interval (0, T ] allowing only injections with parameters in C. Let AY be the set of states on which ( TN ,C )∗ δY has strictly positive density. By Lemma 4, each AY contains an open set and by Lemma 3, AY vary continuously with Y . Therefore there exists an open neighborhood U0 of Y0 , U0 ⊂ U , such that A0 = ∩Y ∈U0 AY contains an open set; clearly m 0 (A0 ) > 0. This completes the proof of Proposition 2. Proof of Lemmas 3 and 4 (η = 1). Assume the disk is of radius R and is centered at the (0, 0) coordinate of the x y-plane and the particle and the disk have initial coordinates (x, y, vx , v y ) and (ϕ, ω) respectively. Denote by τ the collision time with the disk, by θ the angular position of the collision point on the disk measured counterclockwise from the positive direction of the x-axis, and by vt and v⊥ the tangential and the normal velocities of the particle upon collision, with v⊥ representing the velocity after the collision, i.e. pointing outwards. Then, vt = −v y cos(θ ) + vx sin(θ ), v⊥ = −v y sin(θ ) − vx cos(θ ), R cos(θ ) − x R sin(θ ) − y τ = = , vx vy and 2 vt2 + v⊥ = vx2 + v 2y .
One can rewrite Rvt = vx y − v y x, Rv⊥ = −τ (vx2 + v 2y ) − v y y − vx x =
vx2 + v 2y − vt2 .
Ergodicity of Some Open Systems with Particle-Disk Interactions
677
Following the collision, at time t ∈ (τ0 , τ1 ), the coordinates for position and velocity of the particle are: vxt = v⊥ (τ vx + x)/R + ω(τ v y + y), v ty = v⊥ (τ v y + y)/R − ω(τ vx + x), x t = τ vx + x + vx (t − τ ), y t = τ v y + y + v y (t − τ ). The angular velocity of the disk at time t is wt = vt /R and the angular position is ϕ t = [ϕ + ωτ + vt (t − τ )/R] mod 2π R. Clearly f t is continuous, which completes the proof of Lemma 3. In order to prove Lemma 4, we would like to study the matrix of the derivatives of ϕ,ω t : B → S 1 × R, (x, y, vx , v y ) → (ϕ t , ωt ), t ∈ (τ0 , τ1 ). It is: ⎞T ⎛ v v (− vt y −vx )(ω− vRt ) ⎞T ⎛ t vy vy ⊥ ∂ϕ ∂ωt − R 2 (t − τ ) − R2 ⎟ vx2 +v 2y ⎜ ⎟ ⎜ ⎜ ∂x ∂x ⎟ v v v x ⎟ ⎜ ( vt −v y )(ω− Rt ) ⎟ ⎜ ∂ϕ t vx vx ⎟ ∂ωt ⎟ ⎜ ⎜ ⊥ + R 2 (t − τ ) ⎜ ⎜ ∂y ∂y ⎟ vx2 +v 2y R2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ , ⎜ ∂ϕ t ∂ωt ⎟ = ⎜ ( vt y −2vx τ −x)(ω− vt ) ⎜ v⊥ ⎟ ⎜ R y y ⎟ + (t − τ ) ⎜ ⎜ ∂vx ∂vx ⎟ 2 2 2 2 vx +v y R R ⎟ ⎟ ⎜ ⎟ ⎜ t ⎟ ⎜ vt x ⎝ ∂ϕ ∂ωt ⎠ vt (− v −2v y τ −y)(ω− R ) ⎝ x x ⎠ ∂v y ∂v y ⊥ − (t − τ ) − v 2 +v 2 R2 R2 x
y
where T denotes the transpose. Suppose the two rows of the derivative matrix (i.e. columns of the non-transposed matrix) are linearly dependent, then there exists a constant c such that: R 2 (− c=− vy =
vt v y v⊥
− vx )(ω −
vt R)
vx2 + v 2y
vt y R 2 ( v⊥ − 2vx τ − x)(ω − y vx2 + v 2y
=−
+ (t − τ ) =
vt R)
vt x R 2 (− v⊥ − 2v y τ − y)(ω − x vx2 + v 2y
R2 ( vx
vt v x v⊥
− v y )(ω −
vt R)
vx2 + v 2y
+ (t − τ )
+ (t − τ ) vt R)
+ (t − τ ).
Thus either (Rω − vt ) = 0 or vy 2v y τ y x vt vt vx vt vt 2vx τ − = + + = − = − + v⊥ v y v⊥ vx v⊥ y y v⊥ x x ⇔ x = y = vx = v y = 0. From our assumptions it follows that x, y, R, vx , v y = 0; so the derivative matrix has rank 2 unless Rω = vt . Let Aω = {(x, y, vx , v y ) : Rω = vt = R1 (vx y − v y x)}. Clearly ν(Aω ) = 0. For each point (x, y, vx , v y ) ∈ B\Aω , the derivative matrix has rank 2. Therefore the push ϕ,ω forward of ν under t must be absolutely continuous with respect to the Lebesgue ϕ,ω 1 measure on S ×R and has positive density on an open set t (B\Aω ). This completes the proof of Lemma 4.
678
T. Yarmola
6. Flushing Particles Out This section is the 1st step in the proof of Proposition 3. Here we will show the following: Proposition 4. For any admissible state X , there exists a sample path σ X that starts at X and ends at some X 0 ∈ 0 . In order to drive the system from state X with possibly many particles to some particle-less state X 0 , we have to ensure that each particle in X traces a path in from its initial position to one of the exits. Following the ideas in [4], we will describe a class of projected particle paths traced in and show that each can be followed provided that disks have appropriate angular velocities upon collisions. Then we will establish that by injecting particles with appropriate initial conditions we can change the angular velocity of any disk to any given value in an arbitrarily short time. That will enable us to force a particle along a projected particle path by setting the angular velocities of the disks to appropriate values before collisions. Definition 6. A proper projected particle path is a continuous curve γ : [0, 1] → , s → γ (s), such that 1. γ consists of a finite sequence of straight segments meeting at ∂. 2. The incoming and outgoing angles of two consecutive segments of γ meeting ∂0 are equal. 3. Only γ (0) and γ (1) can be in the openings γ L and γ R . 4. γ is nowhere tangent to boundaries of the disks ∪ Nj=1 ∂ D j . Remark 2. Note that a proper projected particle path is allowed to have any non-tangential ‘reflections’ off the boundaries of the disks ∪ Nj=1 ∂ D j . An example of a proper projected particle path is shown in Fig 1. Lemma 5 (Existence of a proper projected particle path). There exits a proper projected particle path from any point (x, y) ∈ ∂ D j to one of the exits γ L or γ R . The statement in Lemma 5 is rather obvious for the geometry we consider. We include a proof for completeness purposes. √
1−R Proof of Lemma 5. Let δ be such that 0 < δ < 2R2−R . If y-coordinate of (x, y) is not in [−δ, δ], i.e. |y| > δ, then by a simple geometric argument there exists a proper projected particle path from (x, y) to the top of the disk D j , (2 j − 1, R) (or the bottom (2 j − 1, −R)), which makes several collisions with the upper wall [0, N ] × {1} (or the lower wall [0, N ] × {−1}). By appending to the proper projected particle path above a segment in that connects (2 j − 1, ±k) to either γ L or γ R , we prove Lemma 5 for the situation when |y| > δ. If y ∈ [−δ, δ], then there exits a segment in that connects (x,√y) to the point on the appropriate nearby disk or exit with y-coordinate between δ and 2R 1−R 2−R . Indeed, by the system’s geometry, a tangent line from (2 j − 1 ± R, 0) to the √
1−R appropriate nearby disk (if applicable) intersects the disk with y-coordinate ± 2R2−R .
In order to force a particle to follow a proper projected particle path, we have to ensure that upon each (non-tangential) collision with a disk, the disk has appropriate angular velocity: the total velocity of the particle after collision is the vector sum of
Ergodicity of Some Open Systems with Particle-Disk Interactions
679
2η v ⊥ = −v ⊥ and v t = v t − 1+η (v t − Rω) and must be parallel to the next segment of the proper projected particle path. We would like to establish that by injecting particles with appropriate initial conditions we can change the angular velocity of any disk to any given value in an arbitrarily short time.
Lemma 6 (Controlling angular velocities of disks in arbitrarily short times). Suppose disk D j rotates with angular velocity ω and none of the particles inside the system will collide with any disk before time τ > 0. Given any ω , there exists a sequence of particle injections on time interval (0, τ ) from the left bath such that: – at time τ the disk D j has angular velocity ω , – at time τ all the injected particles have left the system, and – on time interval (0, τ ) the injected particles follow admissible paths and only hit disks D1 , . . . , D j−1 with the exception of one collision of one particle with disk D j . The same holds for the right bath with appropriate disk renumbering. We prove Lemma 6 in Subsect. 6.3.
6.1. Proof of Proposition 4: no tangential collisions. By definition of an admissible state, one of the following holds for each particle in X under the evolution of the system with no particle injections: 1.
2.
3.
there exists a finite time t > 0 such that the particle exits the system at time t and does not collide with any disks on time interval [0, t], i.e. q t ∈ γ L ∪ γ R and q τ ∈ ∪ Nj=1 ∂ D j ∀τ ∈ [0, t]; there exists a finite time t > 0 such that at time t the particle collides with a disk non-tangentially and no other disk collisions occur on time interval [0, t], i.e. t = 0 and q τ ∈ ∪ N ∂ D q t ∈ ∪ Nj=1 ∂ D j with v⊥ j ∀τ ∈ [0, t]; j=1 there exists a finite time t > 0 such that at time t the particle collides with a disk tangentially with vx = 0 and no other disk collisions occur on time interval [0, t], t = 0 and v = 0 and q τ ∈ ∪ N ∂ D i.e. q t ∈ ∪ Nj=1 ∂ D j with v⊥ x j ∀τ ∈ [0, t]. j=1
In this subsection we are going to assume that for each particle in X either 1 or 2 holds, i.e. under the evolution of the system with no particle injections each particle in X either exits the system or collides with a disk non-tangentially. We will treat the situation with tangential collisions in Subsect. 6.2. Suppose X contains k particles and the above assumption is satisfied. Then using Lemma 5 for the j th particle in X, 1 ≤ j ≤ k, we can assign a proper projected particle path γ j from the particle’s initial position (q 0j , v 0j ) to one of the exits. If we force the particle to follow this path, the times of all collisions are fixed. Indeed, the unique angular velocity of the disk keeps the particle on the path for each (non-tangential) collision, implying that the speeds with which the particle traces segments of the path and therefore the collision times are uniquely determined from the path. If the j th particle j j follows γ j , let τ1 , . . . , τn( j) be the times of collisions with disks, Dk( j,1) , . . . , Dk( j,n( j)) j
j
and let ω1 , . . . , ωn( j) be the required angular velocities. j
Assume first that all τi (for all particles in X and all collisions) are different. Then direct application of Lemma 6 between collisions guarantees the existence of a sample path σ X from state X to some state X 0 ∈ 0 .
680
T. Yarmola j
If some τi happen to coincide, simultaneous collisions with the same disks might occur, making us unable to construct σ X with our choice of paths. We would like to show that we can always choose a collection of nearby paths from the particles’ initial positions to the exits such that no simultaneous collisions with the same disks occur. Note that we cannot, in general, avoid all simultaneous disk collisions since we have no control over the times of the first disk collisions of the particles in X . First collisions, however, cannot occur simultaneously with the same disks since X is admissible. We treat the possibility of simultaneous collisions with different disks as follows: If several, say n, collisions are about to occur in time τ after the previous collision time, all with different disks, we can set the angular velocities of the disks by Lemma 6 in the order of decreasing disk index, in a fraction of time τ, τn , each: Lemma 6 guarantees that the disks with indexes larger than the one we set the angular velocity for are left untouched. In the remaining part of the proof we show that we can choose a collection of paths for particles in X from their initial positions to the exits such that no simultaneous collisions occur. We start with a description of a set of paths to choose from for each particle in X . In the following lemma, we assume that at time 0 the j th particle is the only particle in the system in order to ensure that the system is defined at all times. Lemma 7. Assume that under the evolution of the system with no particle injections the j th particle collides with a disk non-tangentially in some finite time t > 0. Then j j j j there exist open neighborhoods I1 of ω1 , . . . , In( j) of ωn( j) such that for any choice of angular velocities (ω1 ) ∈ I1 , . . . , (ωn( j) ) ∈ In( j) , if a particle starts with its initial j
j
j
j
position (q0 , v0 ), it collides with Dk( j,1) , . . . , Dk( j,n( j)) set to (ω1 ) , . . . , (ωn( j) ) , and exits the system. j
j
j
j
j
j
Lemma 7 follows from Lemma 3 and the continuity of the billiard flow. Lemma 7 guarantees that the j th particle exits the system for any choice of angular j j j j velocities from I1 , . . . , In( j) . If a particle starts at (q0 , v0 ), for different choices of anguj
lar velocities in I1 for disk Dk( j,1) , the particle collides with disk Dk( j,2) at different times. In fact, one gets an open set of possible collision times since the set of possible positions of the particle at any time τ from the collision with disk Dk( j,1) forms a broken line: before any collisions, the particle’s positions are 2η j (v t − R(ω1 ) )]) 1+η 2η 2η Rτ j j j (v t − Rω1 )]) + (ω − (ω1 ) ), = τ (v ⊥ + [v t − 1+η 1+η 1
τ v = τ (v ⊥ + v t ) = τ (v ⊥ + [v t −
where (ω1 ) varies through I1 ; and upon reflections from straight walls the straight line j j of positions becomes a broken line. If we fix a specific (ω1 ) ∈ I1 , a similar argument j j shows that different (ω2 ) ∈ I2 for disk Dk( j,2) yield a range of collision times with j Dk( j,3) ; and so on. j
j
Therefore, we can always pick a collection of (ωi ) ∈ Ii such that no simultaneous collisions occur for all particles in the system. Given such a choice, each particle in X follows a path from its initial position to an exit. Setting the angular velocities of j the disks to (ωi ) at appropriate times by following appropriate sequence of injections j
j
Ergodicity of Some Open Systems with Particle-Disk Interactions
681
provided by Lemma 6, we construct a sample path σ X from state X to some state X 0 in 0 . This completes the proof of Proposition 4 for the initial states X in which particles either hit a disk non-tangentially or exit the system.
6.2. Tangential collisions. In this section we will prove Proposition 4 for the general case, i.e. when for some particles in X the first collision with a disk might be tangential. Since X is admissible, such collisions must occur with vx = 0 and without simultaneous collisions with the same disks. Suppose the first collision of the j th particle is tangential with vx = 0. By Lemma 6 we can ensure that, upon collision, R times the angular velocity of the disk is equal to the velocity of the j th particle, Rω = v = vt . After such a collision the particle continues along the straight line with the same velocity as if no collision has occurred. From that point, the particle will either exit the system, hit a disk non-tangentially, or hit a disk tangentially again. In the third situation we would like to set the angular velocity of the disk to be equal to the velocity of the particle again and continue the process. Since vx = 0 and |v| are kept constant under subsequent iterations of the third situation and upon collisions with straight walls, the third situation can occur at most a finite number of times; then either the particle will exit the system or will collide with a disk non-tangentially. In the later case, by Lemma 5, there exists a proper projected particle path from the non-tangential collision point to an exit. Denote the path traced by the j th particle in this construction by γ j . j
j
Again, the times of all collisions of the j th particle along γ j are fixed; let τ1 , . . . , τn( j) j
j
be the times of collisions with disks, Dk( j,1) , . . . , Dk( j,n( j)) and let ω1 , . . . , ωn( j) be the j
required angular velocities (with Rωi = v = vt at tangential collisions). In the following lemma, we again assume that at time 0, the j th particle is the only particle in the system in order to ensure that the system is defined at all times. Lemma 8. Assume the j th particle has m( j) ≥ 0 tangential collisions before it has a non-tangential collision or exits the system. Then there exist open neighborhoods j j j j j I1 of ω1 , . . . , In( j) of ωn( j) such that for any choice of angular velocities (ω1 ) ∈ I1 , . . . , (ωn( j) ) ∈ In( j) , if a particle starts with its initial position and velocity (q 0j , v 0j ), j
j
j
it possibly collides with Dk( j,1) , . . . , Dk( j,m( j)) set to (ω1 ) , . . . , (ωm( j) ) , collides with j
j
Dk( j,m( j)+1) , . . . , Dk( j,n( j)) set to (ωm( j)+1 ) , . . . , (ωn( j) ) , and exits the system. j
j
j
j
j
j
j
Proof. By Lemma 7 there exist open neighborhoods Im( j)+1,0 of ωm( j)+1 , . . . , In( j),0 of ωn( j) such that for any choice of angular velocities (ωm( j)+1 ) ∈ Im( j)+1 , . . . , (ωn( j) ) ∈ j
j
j
j
j
In( j) if a particle starts with (q 0j , v 0j ), it possibly collides with Dk( j,1) , . . . , Dk( j,m( j)) set to ω1 , . . . , ωm( j) , collides with Dk( j,m( j)+1) , . . . , Dk( j,n( j)) set to (ωm( j)+1 ) ∈ j
j
j
Im( j)+1,0 , . . . , (ωn( j) ) ∈ In( j),0 , and exits the system. At tangential collisions, we chose to set R times the angular velocities of the disks to be equal to the velocities of the colliding particles, Rω = v = vt . If we vary the angular velocities of the disks at tangential collisions keeping sign the same, the particle still follows the same path and exits the system. j
j
j
682
T. Yarmola
Fig. 2. ω = 0 case: the incoming trajectory L is marked by a solid line; outgoing L - by a dashed line; velocities are labeled after collisions j
j
If a particle starts at (q0 , v0 ) and the collision with disk Dk( j,1) is tangential, letting j the angular velocity of Dk( j,1) vary through I1 , we get an open set of possible collision times with Dk( j,2) since the particle follows the same path, only with different speeds. The remaining tangential collisions are treated analogously and non-tangential collisions as in Subsect. 6.1. j j Therefore we can always pick a collection of (ωi ) ∈ Ii such that no simultaneous collisions with the same disks occur for all particles in the system. Given such a choice, each particle in X follows a path from its initial position to an exit. Setting the anguj lar velocities of the disks to (ωi ) at appropriate times using Lemma 6, we construct a sample path σ X from state X to some state X 0 in 0 . This completes the proof of Proposition 4. 6.3. Proof of the Lemma 6 (η = 1). The proof is by induction on j, 1 ≤ j ≤ N . j=1. There are many ways to treat this case, but we choose a very specific one that will be a useful step in treating the induction step. Our method here is identical to the one in the proof of Lemma 4.3 in [4]. Assume first w = 0, i.e. we want to hit D1 radially. Send a particle parallel to the x-axis to hit the disk D1 at (1, 0). If the initial velocity v = (vx , 0) is big enough compared to Rω and the distance from γ L to D1 , the particle will be able to exit the playground without hitting ∂\(γ L ∪ γ R ) again. Note that the larger vx is, the smaller is the angle of reflection; we can introduce any bound on the angle by choosing vx large enough. When w = 0, we send a particle such that it also hits the disk at (1, 0). We can introduce any bound on the angle of incidence in a similar way. Clearly, by making vx sufficiently large and large enough compared to Rw , we can complete the above procedure in an arbitrarily short time. Induction step. Assume the lemma holds for all j ≤ k. We would like to show that it also holds for j = k + 1. We want to send a particle with velocity v = (vx , v y ) such that it first hits disk Dk at (2k − 1, −R) without hitting ∂ along the way. The velocity after collision is (Rωk , −v y ), where ωk is the angular velocity of disk Dk , which we can set to any value in arbitrarily short time by the induction assumption. Consider first the case ω = 0 illustrated in Fig. 2. Then we want to hit disk Dk+1 radially by following the unique trajectory that reflects from the lower boundary of the playground , [0, 2N ] × {−1}, once before hitting Dk+1 ; call this trajectory L. If θ is the angular position of the collision counting counterclockwise from the x-axis and v⊥
Ergodicity of Some Open Systems with Particle-Disk Interactions
683
denotes the normal component of the velocity pointing outwards, then Rω = −v y cos(θ ) + Rωk sin(θ ) = 0
⇒
tan(θ ) =
vy . Rωk
Also v⊥ = −v y sin(θ ) − Rωk cos(θ ) = − tan(θ )Rωk sin(θ ) − Rωk cos(θ ) = −
Rωk , cos(θ )
implying that vx = v⊥ cos(θ ) + Rω sin(θ ) = −Rωk + Rω sin(θ ), v y = v⊥ sin(θ ) − Rω cos(θ ) = −v y − Rω cos(θ ), where vx and v y are the components of the velocity after the collision with disk Dk+1 . If we choose v y and Rωk large compared to Rω, the particle will follow a trajectory L very close to the reversed L on the way from disk D k+1 to disk Dk . By bounding the angle of reflection from disk Dk+1 (choosing v y and Rωk as large as we need), we can ensure that L hits disk Dk in a small neighborhood of (2k − 1, −R). During the flight of the particle to and from the disk Dk+1 , we can reset the angular velocity of disk Dk to a new value Rωk with |Rωk | large enough so that after the second collision with Dk , the particle leaves the system with no additional collisions. By choosing vx , v y , Rωk and Rωk sufficiently large and large enough compared to w, we can ensure that this procedure can be done in an arbitrarily short time. Suppose now that ω = 0. Then we have to hit the disk Dk+1 at a slightly different angular position θ due to the playground geometry. Then Rω = −v y cos(θ ) + Rωk sin(θ )
⇒
vy =
Rωk sin(θ ) − Rω cos(θ )
v⊥ = −v y sin(θ ) − Rωk cos(θ ) Rωk sin(θ ) − Rω Rωk − Rω sin(θ ) ) sin(θ ) − Rωk cos(θ ) = − , = −( cos(θ ) cos(θ ) implying that vx = v⊥ cos(θ ) + Rω sin(θ ) = −Rωk + (Rω + Rω) sin(θ ), v y = v⊥ sin(θ ) − Rω cos(θ ) = −v y + (Rω − Rω) cos(θ ). If we choose θ sufficiently close to θ, v y and Rωk large compared to Rω and Rω , we can ensure that trajectories from and to disk Dk are very close to L. Therefore, on the way back, the particle hits disk Dk in a small neighborhood of (2k − 1, −R) and, by choosing |Rωk | large enough, we can send the particle out of the system without additional collisions. Again, by choosing vx , v y , Rωk , and |Rωk | sufficiently large and large enough compared to Rω and Rω , we can do this procedure in an arbitrarily short time.
684
T. Yarmola
7. Proof of Proposition 3 In this section we complete the proof of Proposition 3. In Sect. 6 we presented a construction of a sample path σ X from any admissible state X to an arbitrary particle-less state X 0 ∈ 0 . In Subsect. 7.1 we would extend this sample to a given state Y0 ∈ 0 , obtaining a sample path σ from X to Y0 defined on some time interval [0, T ]. To finish the proof of Proposition 3, we are also required to show that there exists a canonical neighborhood of σ such that each sample path in ends in U0 , a neighborhood of Y0 . In Subsect. 7.2 we treat the situation when the particles in the initial state X either collide a disk non-tangentially or exit the system. We finish the proof in Subsect. 7.3 by treating the remaining situation with tangential collisions. 7.1. From any state in 0 to any state in 0 . Lemma 9. Given X 0 , Y0 ∈ 0 and T > 0, there exists a sample path σ : X 0 → Y0 on [0, T ]. Proof. Denote the angular positions and velocities of the disks in X 0 and Y0 by (ϕ1 , ω1 ), . . . , (ϕ N , ω N ) and (ϕ1 , ω1 ), . . . , (ϕ N , ωN ) respectively. Divide the time interval [0, T ] into N equal subintervals of length NT . Suppose we can set the angular position and velocity of disk D N to (ωN , ϕ˜ N ) on time interval (0, NT ), ωN . Then if we guarantee that no collisions occur with disk where ϕ N = ϕ˜ N + (N −1)T N T D N on time interval [ N , T ], D N will have angular position and velocity (ϕ N , ωN ) at time T . Similarly we can proceed with setting the angular position and velocity of disk (N −2)T D N −1 to (ωN −1 , ϕ˜ N −1 ) on time interval ( NT , 2T ω N −1 , N ), where ϕ N −1 = ϕ˜ N −1 + N 2T ensuring that if no collisions happen with disk D N −1 on time interval [ N , T ], its angular position and velocity would be (ϕ N −1 , ωN −1 ) at time T . And so on. In order for this procedure to work, we need the following lemma: Lemma 10. Suppose disk D j rotates with angular velocity ω and there are no particles present in the system. Given time t > 0, angular position ϕ , and angular velocity ω , there exists a sequence of particle injections on time interval (0, t) from the left bath such that: – at time t the disk D j has the angular position and velocity (ϕ , ω ), – at time t all the injected particles have left the system, – on time interval (0, t) the injected particles only hit disks D1 , . . . , D j . Proof. The result of Lemma 10 is achieved by the application of Lemma 6 twice: Fix some ω1 > 3+tω t . Apply Lemma 6 to set the angular velocity of D j to w1 in time t t 3 . Let τ1 < 3 be the time of the unique collision with disk D j . Suppose we wait for some time τ < 3t (to be defined later) after 3t and then apply Lemma 6 again to set the angular velocity of D j to ω in time 3t . Let τ2 < 3t be the time of the unique collision with disk D j counted from the time 3t + τ . Then at time t, the angular position and velocity of D j are t 2t ([ϕ + τ1 ω + ( − τ1 + τ + τ2 )ω1 + ( − τ − τ2 )ω ] 3 3
mod 1, ω ).
Ergodicity of Some Open Systems with Particle-Disk Interactions
685
Let ϕ˜ = [ϕ + τ1 ω + ( 3t − τ1 + τ2 )ω1 + ( 2t3 − τ2 )ω ] mod 1; this is a fixed number since all the variables in the expression are fixed. Then we want to pick τ < 3t such that 1 t ϕ = (ϕ˜ + τ (ω1 − ω)) mod 1. This is not a problem since ω1 −ω < 3 by the choice of t w1 , i.e. in time 3 , a disk rotating with angular velocity ω1 − ω makes full revolution and thus, starting at ϕ, ˜ passes through the angular position ϕ at some time τ < 3t . This completes the proof of Lemma 9.
7.2. Proof of Proposition 3: no tangential collisions. In the situation when all particles in X either collide with a disk non-tangentially or exit the system under the evolution of the system with no particle injections, Proposition 3 follows from the lemma below: Lemma 11. Let σ be a sample path from a state X to a state Y on time interval [0, T ] such that each particle present in the system at any time subinterval of [0, T ] follows a proper projected particle path. Then for any neighborhood U of Y , there exists a canonical neighborhood of σ such that each sample path in ends in U . Proof of Lemma 11. Denote by c the sequence of injections that generates σ . Let be any canonical neighborhood of σ ; denote by U X the neighborhood of X and by C the canonical neighborhood of c such that each sample path in starts with an initial condition in U X and follows a sequence of injections from C . Define f : U X × C → as follows: if σ ∈ is a sample path that starts at state X ∈ U X , is generated by a sequence of injections c ∈ C , and ends at state Y , then f (X , ) = Y . To prove Lemma 11, it is enough to show that f is continuous at (X, c). We assumed that along the sample path σ each particle follows a proper projected particle path, i.e. it is only allowed to collide with disks non-tangentially. The continuity of f follows from the following facts: – –
–
– –
If a particle does not collide with ∂ on time interval [τ1 , τ2 ], its position and velocity change continuously. If a particle collides with the wall on time interval [τ1 , τ2 ] and it is not involved in any other collisions with ∂ on time interval [τ1 , τ2 ], then its final position and velocity depend continuously of its initial position and velocity. [This fact follows from the continuity of the billiard flow at collisions]. If a particle collides with a disk non-tangentially on [τ1 , τ2 ] and neither the particle nor the disk is involved in other collisions on time interval [τ1 , τ2 ], then the particle’s position and velocity as well as the disk’s position and angular velocity depend continuously on their initial positions and velocities [follows from Lemma 3]. If a particle exits through γ L or γ R on time interval [τ1 , τ2 ] and does not collide with ∂ on time interval [τ1 , τ2 ], then the coordinates of the other particles and disks are independent from the coordinates of the exiting particle on time interval [τ1 , ∞). The position and velocity of an injected particle depend continuously on the injected parameters [follows from Lemma 2].
7.3. Proof of Proposition 3: tangential collisions. When we constructed a sample path σ X from X to some particle-less state X 0 ∈ 0 in Sect. 6, we first assigned a path in to each particle in X from its initial position to an exit. In order for a particle to follow such a path, the disks had to be set to unique angular velocities at collisions (with Rω = v = vt at tangential collisions). Then we showed that setting the disks to nearby
686
T. Yarmola
angular velocities at collisions makes particles follow nearby paths with collisions happening at nearby times; this was crucial for choosing appropriate particle paths such that no simultaneous collisions with the same disks occur. When choosing nearby paths in order to avoid simultaneous collisions with the same disks, we might have to require that at some tangential collisions, R times the angular velocities of the disks are not equal to the velocities of the colliding particles. And near a tangential collision with Rω = v = vt , final position and velocity of a particle does not depend continuously on initial position and velocity unlike the situation with Rω = v = vt . That prevents us from direct extension of Lemma 11 to the situation when tangential collisions might occur. However, Proposition 3 only requires us to choose a sample path σ and a canonical neighborhood of σ such that each sample path in ends in the given neighborhood U0 . By making the size of the discontinuity small enough, we will still be able to ensure that every sample path in ends in U0 . The following two lemmas imply Proposition 3: Lemma 12. Given > 0, there exists time TX > 0, a sample path σ X on [0, TX ] from X to some particle-less state X 0 and a canonical neighborhood X of σ X , such that each sample path in X ends in an -neighborhood U (X 0 ) of X 0 . Remark 3. Note that state X 0 depends on the choice of σ X , while the size neighborhood U (X 0 ) around X 0 does not.
of the
Lemma 13. Given Y0 ∈ 0 and a neighborhood U0 of Y0 , there exists > 0 such that for any state X 0 ∈ 0 , there exists time T0 > 0, a sample path σ0 : X 0 → Y0 on [0, T0 ], and a canonical neighborhood 0 of σ0 in which each sample path starts in U (X 0 ) and ends in U0 and for any point Y ∈ U (X 0 ), there exists a sample path in 0 that starts at Y . Lemma 13 follows directly from Lemmas 9 and 11. Proof of Lemma 12. As in Sect. 6, denote the initially assigned path in traced j j by the j th particle by γ j . Let τ1 , . . . , τn( j) be the times of collisions with disks, j
j
Dk( j,1) , . . . , Dk( j,n( j)) and let ω1 , . . . , ωn( j) be the required angular velocities (with j
Rωi = v = vt at tangential collisions). In the following lemma, we assume that at time 0 the j th particle is the only particle in the system in order to ensure that the system is defined at all times. Lemma 14. Assume the j th particle has m( j) ≥ 0 tangential collisions before it has a non-tangential collision or exits the system. Then there exist open neighborhoods V j j j j j of (q 0j , v 0j ), I1 of ω1 , . . . , In( j) of ωn( j) such that for any choice of angular velocities (ω1 ) ∈ I1 , . . . , (ωn( j) ) ∈ In( j) , if a particle starts with a position and a velocity from j
j
j
j
V j , it possibly collides with Dk( j,1) , . . . , Dk( j,m( j)) set to (ω1 ) , . . . , (ωm( j) ) , collides j
j
with Dk( j,m( j)+1) , . . . , Dk( j,n( j)) set to (ωm( j)+1 ) , . . . , (ωn( j) ) , and exits the system. j
j
j
j
We will prove Lemma 14 after finishing the proof of Lemma 12. j Let TX be an upper bound on the time it takes for the j th particle to exit along all posj j sible paths described in Lemma 14; and let TX = max1≤ j≤k {TX }. Define g j : V j × I1 × j · · ·×In( j) → 0 as follows: if a particle starts with a position and a velocity from V j , and
Ergodicity of Some Open Systems with Particle-Disk Interactions
687
Dk( j,1) , . . . , Dk( j,n( j)) are set to (ω1 ) ∈ I1 , . . . , (ωn( j) ) ∈ In( j) at potential collision j
j
j
j
times, let g j (q, v, (ω1 ) , . . . , (ωn( j) ) ) be the state of the system at time TX . g j is conj
j
j
j
j
j
j
j
tinuous at (q0 , v0 , ω1 , . . . , ωn( j) ); so there exist sub-neighborhoods V of V j , I1, of I1 . . . , In( j), of In( j) such that for any (q j , v j ) ∈ V , (ω1 ) ∈ I , . . . , (ωn( j) ) ∈ I1, , j
j
j
j
j
j
j
j
|g j (q j , v j , (ω1 ) , . . . , (ωn( j) ) ) − g j (q 0j , v 0j , ω1 , . . . , ωn( j) )| < /2. j
j
j
j
Now we are ready to deal with the k-particle system. As in Sect. 6, we can choose j j a path for each particle in X such that upon each disk collision, (ωi ) ∈ Ii, and no simultaneous collisions with the same disks occur. That defines σ X on [0, TX ]; let X 0 j be the state where σ X ends. To define X choose further sub-neighborhoods of Ii, ’s to ensure that each sample path in X is defined up to time TX . Then each sample path in X ends in an -neighborhood U (X 0 ) of X 0 by the above inequality. Proof of Lemma 14. By Lemma 7 and the fact that near a tangential collision with Rω = v = vt , the particle’s final position and velocity depend continuously on its j j initial position and velocity, there exist open neighborhoods V0 of (q 0j , v 0j ), Im( j)+1,0 of ωm( j)+1 , . . . , In( j),0 of ωn( j) such that for any choice of angular velocities (ωm( j)+1 ) ∈ j
j
j
j
Im( j)+1 , . . . , (ωn( j) ) ∈ In( j) if a particle starts with a position and a velocity from j
j
j
j
j
j
V0 , it possibly collides with Dk( j,1) , . . . , Dk( j,m( j)) set to ω1 , . . . , ωm( j) , collides with Dk( j,m( j)+1) , · · · , Dk( j,n( j)) set to (ωm( j)+1 ) , · · · , (ωn( j) ) , and exits the system. Now we would like to allow an open neighborhood of angular velocities around j j the m th tangential collision. The neighborhood V0 can be split into two parts: (V0 )c j j j (V0 )nc = V0 , where (V0 )c denotes the set of initial positions and velocities such that j j a particle with a position and a velocity from (V0 )c will collide with Dk( j,m( j)) , proj
j
j
j
j
j
vided that Dk( j,1) , . . . , Dk( j,m( j)−1) are set to angular velocities ω1 , . . . , ωm( j)−1 before potential collisions. Then the position and velocity of a particle after collision Dk( j,m( j)) depend continuj ously on its initial position and velocity in (V0 )c even if the angular velocity of Dk( j,m( j)) j j is not equal to ωm( j) . Also, since particles in (V0 )nc do not collide with Dk( j,m( j)) , they exit the system provided the angular velocities of the disks Dk( j,1) , . . . , Dk( j,m( j)−1) j j are set to ω1 , . . . , ωm( j)−1 at appropriate times and angular velocities of the disks j
j
Dk( j,m( j)+1) , . . . , Dk( j,n( j)) are set to values from Im( j)+1,0 , . . . , In( j),0 before collisions. j
j
Therefore there exists an open neighborhood Im( j) of ωm( j) and open subj
j
j
j
j
j
neighborhoods Vm( j) of V0 , Im( j)+1,m( j) of Im( j)+1 , . . . , In( j),m( j) of In( j) such that j
if a particle starts with a position and a velocity from Vm( j) , it possibly colj
j
j
j
lides with Dk( j,1) , . . . , Dk( j,(m( j)−1)) with angular velocities ω1 , . . . , ωm( j)−1 , possibly collides with Dk( j,m( j)) with angular velocity (ωm( j) ) ∈ Im( j) , collides with j
j
j
Dk( j,m( j)+1) , . . . , Dk( j,n( j)) with angular velocities (ωm( j)+1 ) ∈ Im( j)+1,m( j) , . . . , j
j
(ωn( j) ) ∈ In( j),m( j) , and exits the system. The remaining tangential collisions are treated similarly. j
j
688
T. Yarmola
Acknowledgements. I would like to thank my Ph.D. thesis advisor Lai-Sang Young for proposing the problem, fruitful discussions, effective criticism, and useful comments on many drafts of this paper. This work was partially supported by the National Science Postdoctoral Research Fellowship.
References 1. Balint, P., Lin, K.K., Young, L.-S.: Ergodicity and energy distributions for some boundary driven integrable Hamiltonian chains. To appear in Comm. Math. Phys. To Appear 2. Collet, P., Eckmann, J.-P.: A model of heat conduction. Commun. Math. Phys. 287, 1015–1038 (2009) 3. Eckmann, J.-P., Hairer, M.: Non-equillibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212, 105–164 (2000) 4. Eckmann, J.-P., Jacquet, P.: Controllability for chains of dynamical scatterers. Nonlinearity 20(1), 1601– 1617 (2007) 5. Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Entropy production in non-linear, thermally driven Hamiltonian systems. J. Stat. Phys. 95, 305–331 (1999) 6. Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Non-equillibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatues. Commun. Math. Phys. 201, 57–697 (1999) 7. Eckmann, J.-P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1-D models. Commun. Math. Phys. 262(1), 237–267 (2006) 8. Klages, R., Nicolis, G., Rateitschak, K.: Thermostating by deterministic scattering: the periodic Lorentz gas. J. Stat. Phys. 99, 1339–1364 (2000) 9. Larralde, H., Leyvraz, F., Mejía-Monasterio, C.: Transport properties in a modified Lorentz gas. J. Stat. Phys. 113, 197–231 (2003) 10. Lin, K.K., Young, L.-S.: Nonequillibrium Steady States for Certain Hamiltonian Models. To appear in J. Stat. Phys., available at http://arXiv.org/labs/1004.041zv1 [cond-mat.stat-mech], 2010 11. Rey-Bellet, L.: Nonequilibrium statistical mechanics of open classical systems. XIVth International Congress on Mathematical Physics, Hackensack, NJ: World Sci. Publ., 2005, pp. 447–454 Communicated by G. Gallavotti
Commun. Math. Phys. 304, 689–709 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1239-4
Communications in
Mathematical Physics
Comparison Between the Cramer-Rao and the Mini-max Approaches in Quantum Channel Estimation Masahito Hayashi1,2 1 Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan.
E-mail: [email protected]
2 Centre for Quantum Technologies, National University of Singapore,
3 Science Drive 2, Singapore 117542, Singapore Received: 17 May 2010 / Accepted: 10 November 2010 Published online: 20 April 2011 – © Springer-Verlag 2011
Abstract: In a unified viewpoint in quantum channel estimation, we compare the Cramér-Rao and the mini-max approaches, which gives the Bayesian bound in the group covariant model. For this purpose, we introduce the local asymptotic mini-max bound, whose maximum is shown to be equal to the asymptotic limit of the mini-max bound. It is shown that the local asymptotic mini-max bound is strictly larger than the Cramér-Rao bound in the phase estimation case while both bounds coincide when the minimum mean square error decreases with the order O( n1 ). We also derive a sufficient condition so that the minimum mean square error decreases with the order O( n1 ). 1. Introduction In quantum information technology, it is usual to use the quantum channel for sending the quantum state. Since a quantum channel has noise, it is important to identify the quantum channel. In this paper, we consider theoretical optimal performance of quantum channel estimation when we can apply the same unknown channel several times. In order to treat this problem, we employ quantum state estimation theory. In our setting, we can optimize our input state and our measurement [1–14]. As is illustrated in Fig. 1 with n = 4, it is assumed to be possible to use entanglement with the reference system in the measurement process when the channel θ with the unknown parameter θ is applied n times. This setting is mathematically equivalent with the setting given in Fig. 2 with n = 4, which has a single input state in the large input system and a single measurement in the large output system. In this paper, we consider quantum channel estimation with the formulation given by Fig. 2. In the state estimation, when the number n of prepared states goes to infinity, the mean square error (MSE) behaves as the order O( n1 ) as in the estimation of probability distribution. However, in the estimation of the quantum channel, two different analyses were reported concerning asymptotic behavior of MSE. As the first case, in the estimations of depolarizing channels and Pauli channels, the optimal MSE behaves as O( n1 )[1,3].
690
M. Hayashi
Fig. 1. Estimation scheme of the quantum channel
Fig. 2. Simpler estimation scheme of the quantum channel
As the second case, in the estimation of the unitary, the optimal MSE behaves as O( n12 ) [4–11]. In the second case, two different types of results were reported: One is based on the Cramér-Rao approach[4,5]. The other is based on the mini-max approach [6–11]. The Cramér-Rao approach is based on the notion of a locally unbiased estimator, and allows one to give a simple lower bound (the Cramér-Rao bound) to the mean square error (MSE) at a given point. The mini-max approach aims to minimize the maximum of the MSE over all possible values of the parameter. The mini-max approach is more meaningful than the Cramér-Rao approach because the true value of the parameter is unknown. So, the Cramér-Rao bound is just a lower bound in general, while it can be asymptotically achieved in the case of quantum state estimation with n copies of the unknown state. However, the Cramér-Rao bound has been considered so many times in the literature [1,3–5]. The reason seems to be that computing the mini-max bound
Comparison Between Cramer-Rao and Mini-max Approaches
691
is a much harder problem. Indeed, there are only very few examples of calculations of the mini-max bound, and most of them are in the compact group covariant setting, in which, as was shown by Holevo [15], the mini-max bound coincides with the Bayesian average of the MSE over the normalized invariant measure. So, many researchers [6–11] applied this approach to quantum channel estimation under group covariance. As the simplest case for unitary estimation, phase estimation has been treated with the mini-max approach by several papers [9–11], and it has been shown that the minimum MSE that behaves as O( n12 ). On the other hand, the Cramér-Rao approach suggests the noon state as the optimal input [36,37] in phase estimation. The later estimation scheme was experimentally demonstrated in the case of n = 4 [32,33] and n = 10 [34]. Also, another group [31] experimentally demonstrated an estimation protocol concerning the group covariant approach proposed by [35]. In phase estimation with group covariant framework, the asymptotic minimum MSE behaves as O( n12 ) [9,10]. When we focus on the difference θˆ − θ between the true parameter θ and the estimate θˆ , the limiting distribution concerning the random variable n(θˆ − θ ) can be obtained through Fourier transform of a function with a finite domain [11]. However, the Cramér-Rao bound is different from the asymptotic limit of the mini-max bound. So, these results seem to contradict each other. No existing research compares both approaches in a unified viewpoint. The manuscript consists of two parts. In the first part, we discuss the Cramér-Rao approach for channel estimation, and give a simple formula to compute a bound based on the right logarithmic derivative introduced by Holevo [20]. This formula is based on the Choi-Jamiolkowski representation matrix [39,40] of the quantum channel, and holds under the condition that, at the given point, the support of the Choi-Jamiolkowski representation matrix contains the support of its derivative. Under this condition, we prove the additivity of the RLD Fisher information for quantum channels. The second part is about the local asymptotic mini-max bound [38], in which the maximum of the MSE is taken over an interval, the width of which is sent to zero after computing the asymptotic limits. The obtained results are summarized as follows: (1) The local asymptotic mini-max bound is strictly larger than the Cramér-Rao bound in the case of phase estimation. (2) Both bounds coincide with each other when the asymptotic minimum MSE behaves as O( n1 ). (3) The local asymptotic mini-max bound is achievable (Proposition 3), using a variation of the two-step strategy [23,22]. That is, there is an optimal sequence of estimators that achieves the local asymptotic mini-max bound at any point. As a consequence, the maximum of the local asymptotic mini-max bound is shown to be equal to the asymptotic limit of the mini-max bound. However, the Cramér-Rao bound has a different type of achievability. That is, there is an optimal sequence of estimators that achieves the Cramér-Rao bound at a given point. This characteristic is a curious quantum analogue of superefficiency in classical statistics. In estimation of probability distribution, if we assume a weaker condition for our estimator, there exists an estimator that surpasses the Cramér-Rao bound only in measure zero points. Such an estimator is called a superefficient estimator [29]. In the estimation of phase action, we point out that the Cramér-Rao bound can be attained only by a quantum channel version of a superefficient estimator that works at specific points. Since the Cramér-Rao approach is based on the asymptotically locally unbiased condition, we can conclude that the asymptotically locally unbiased condition is too weak for deriving the local asymptotic mini-max bound, which can be attained in all points. Indeed, a similar phenomenon happens in quantum state estimation when we use the large deviation criterion [18].
692
M. Hayashi
This paper is organized as follows. Some of obtained results are based on quantum state estimation with the Cramér-Rao approach. Section 2 is devoted to a review of the Cramér-Rao approach in quantum state estimation. In this section, the symmetric logarithmic derivative (SLD) Fisher information and the right logarithmic derivative (RLD) Fisher information are explained. However, the Cramér-Rao bound is obtained only by the locally unbiased condition. So, it is needed to discuss its relation with the estimator that works globally. In Sect. 3, we treat SLD Fisher information and RLD Fisher information in the quantum channel estimation, and discuss the increasing order of SLD Fisher information. In Sect. 4, we give several examples where the maximum SLD Fisher information increases with O(n 2 ). In Sect. 5, we compare the local asymptotic mini-max bound and the Cramér-Rao bound. We also show the global attainability of the local asymptotic mini-max bound in the channel estimation. It is also shown that the Cramér-Rao bound is closely related to a quantum channel version of superefficiency in the phase estimation. 2. Cramér-Rao Bound in Quantum State Estimation In quantum state estimation, we estimate the true state through the quantum measurement under the assumption that the true state of the given quantum system H belongs to a certain parametric state family {ρθ |θ ∈ ⊂ Rd }. In the following, we consider the case when the number d of parameters is one. Usually, we assume that n quantum systems are prepared in the state ρθ . Hence, the total system is described by the tensor product space H⊗n , and the state of the total system is given by ρθ⊗n . In this case, when we choose a suitable measurement, the MSE decreases in proportion to n −1 as in the estimation of probability distribution. So, we focus on the first order coefficient of the MSE concerning n1 . In the most general setting, any positive operator valued measure (POVM) M n on the total system H⊗n is allowed as an estimator when it takes values in the parameter space ⊂ R. The MSE is given as MSEθ (M n ) := (θˆ − θ )2 Tr ρθ⊗n M n (d θˆ ). In the quantum case, there are several quantum extensions of Fisher information θ when the state ρθ is differentiable at θ and (I − P) dρ dθ (I − P) = 0, where P is the projection to the support of ρθ . The largest one is the right logarithmic derivative (RLD) Fisher information JθR , and the smallest one is symmetric logarithmic derivative (SLD) Fisher information JθS . For these definitions, we define the RLD L θR and the SLD L θS as the operators satisfying 1 S dρθ dρθ = ρθ L θR , = L θ ρθ + ρθ L θS . dθ dθ 2 Then, the RLD and SLD Fisher informations are given by [19–21] JθR := Tr ρθ L θR (L θR )† ,
JθS := Tr ρθ (L θS )2 .
θ 2 When the range of ρθ contains the range of ( dρ dθ ) , the RLD Fisher information has another expression:
JθR = Tr(
dρθ 2 −1 ) ρθ . dθ
(1)
Comparison Between Cramer-Rao and Mini-max Approaches
693
When the state family {ρθ } is given by ρθ := eiθ H |uu|e−iθ H , the condition (1) does not hold, where H is an Hermitian matrix. In this case, the SLD Fisher information is calculated as follows [17]. 4(u|H 2 |u − u|H |u2 ).
(2)
Now, we introduce the unbiased condition by θˆ Tr ρθ⊗n M n (d θˆ ) = θ, ∀θ ∈ . However, this condition is sometimes too restrictive in the asymptotic setting. So, we consider the Taylor expansion at a point θ0 and focus on the first order. Then, we obtain the locally unbiased condition at θ0 : d n ˆ ) = θ0 , M (d θ θˆ Tr ρθ⊗n θˆ Tr ρθ⊗n M n (d θˆ )|θ=θ0 = 1. 0 dθ Under the locally unbiased condition at θ0 , an application of the Schwarz inequality similar to the classical case yields the quantum Cramér-Rao inequalities for both quantum Fisher information. 1 R −1 (J ) , n θ0 1 MSEθ0 (M n ) ≥ (JθS0 )−1 . n
MSEθ0 (M n ) ≥
(3) (4)
Since JθR is greater than JθS , the inequality (4) is more informative than the inequality (3). When the estimator M n is the spectral decomposition of the operator θ0 I + 1S (L θS,(1) + 0 n Jθ
S,(n) · · · + L θ0 ), the equality in (4) holds, where
X ( j)
0
is given as
I ⊗ j−1 ⊗ X ⊗ I ⊗n− j . Then,
we obtain the following inequality:
JθR ≥ JθS . This inequality seems to imply that JθR is not as meaningful as JθS in the one-parametric case. However, as will be explained later, JθR provides a meaningful bound for MSE in the case of channel estimation. In fact, in the asymptotic setting, a suitable estimator usually satisfies the asymptotic locally unbiased condition: d ⊗n ⊗n n n θˆ Tr ρθ0 M (d θˆ ) = θ0 , lim θˆ Tr ρθ M (d θˆ ) lim =1 n→∞ n→∞ dθ θ=θ0 for all points θ0 . Under the above condition, using (4), we obtain the inequality lim sup n MSEθ (M n ) ≥ (JθS )−1 .
(5)
n→∞
Further, by using the two-step method, the bound (JθS )−1 can be universally attained for any true parameter θ [22,23]. So, defining the Cramér-Rao bound: n n {M } satisfies the asymptotic Cθ := infn lim sup n MSEθ (M ) , {M } locally unbiased condition. n→∞
694
M. Hayashi
we obtain Cθ = (JθS )−1 . On the other hand, its multi-parameter case is much more complicated even in the asymptotic setting [24–28]. So, this paper does not treat the multi-parameter case. 3. Maximum SLD and RLD Fisher Informations in Quantum Channel Estimation In this section, we apply the Cramér-Rao approach to estimation of channel. In the quantum system, the channel is given by a trace preserving completely positive (TP-CP) map from the set of densities on the input system H := Cd to the set of densities on the output system K := Cd . By using dd linear maps Fi from S(H) to S(K), any TP-CP dd map can be described by (ρ) = i=1 Fi ρ Fi† . Hence, our task is to estimate the true TP-CP map under the assumption that the true TP-CP map belongs to a certain family of TP-CP maps {θ }. In order to characterize a TP-CP map θ , we formulate the notation concerning states on the tensor product system H ⊗ R, where R is a system of the same dimensionality as H and is called the reference system. Using a linear map A from R to H, we define an element |A of H ⊗ R as follows: A j,k | j H ⊗ |k R , |A := j,k
where {| j H } j=1,...,d and {|k R }k=1,...,d are complete orthonormal systems (CONSs) of H and R. Hence, the relation B ⊗ C|A = |B AC T holds. This notation is applied to the cases of K ⊗ H and K ⊗ R. Now, we focus on the matrix ρ[θ ] := (θ ⊗ id)(|I I |), which is called the Choi-Jamiolkowski representation matrix [39,40]. Then, when the input state is the maximally entangled state | √1 I = dj=1 √1 | j ⊗ | j, the output state is d1 ρ[θ ] = d
d
(θ ⊗ id)(| √1 I √1 I |). When the matrix A A T is a density matrix on H, |AA| is d d a pure state on the product system H ⊗ R. Thus, the output state is given as (θ ⊗ id)(|AA|) = (I ⊗ A T )(θ ⊗ id)(|I I |)(I ⊗ A) = (I ⊗ A T )ρ[θ ](I ⊗ A). θ) In the one-parameter case, we express the derivative dρ( by D[θ ]. dθ When the input state is the product state |vv|⊗|uu|, the output state is θ (|vv|)⊗ |uu| = (I ⊗ u · v T )ρ[θ ](I ⊗ v · u † ). Since
v| Tr K ρ[θ ]|v|uu| = u · v T Tr K ρ[θ ]v · u † = Tr K (I ⊗ u · v T )ρ[θ ](I ⊗ v · u † ) = |uu|, we have v|(Tr K ρ[θ ])|v = 1.
Comparison Between Cramer-Rao and Mini-max Approaches
695
Thus, we obtain Tr K ρ[θ ] = I.
(6)
Taking the derivative in (6), we obtain Tr K D[θ ] = 0.
(7)
Now, we back to our estimation problem. In this problem, our choice is given by a pair of the input state ρ and the quantum measurement M. When we fix the input state, our estimation problem can be reduced to the state estimation with the state family {θ (ρ)|θ ∈ }. In the one-parameter case, we focus on the suprema J R [θ ] := sup J R [θ , ρ], ρ
J S [θ ] := sup J S [θ , ρ], ρ
where J S [θ , ρ] and J R [θ , ρ] are the SLD and RLD Fisher informations when the input state is ρ. In particular, it is important to calculate the supremum J S [θ ] which is smaller than J R [θ ]. When n applications of the unknown channel θ are available, the input state ρn and the measurement M n are given as a state on (H ⊗ R)⊗n and a POVM on (K ⊗ R)⊗n . For a sequence of estimators {(ρn , M n )}, we consider the asymptotic locally unbiased condition: d θˆ Tr θ0 (ρn )M n (d θˆ ) = θ0 , lim θˆ Tr θ (ρn )M n (d θˆ ) lim =1 n→∞ n→∞ dθ θ=θ0 for all points θ0 , and denotes the MSE of (ρn , M n ) by MSEθ (ρn , M n ). Assume that α n J S [⊗n θ ] behaves as O(n ) when n goes to infinity. When {(ρn , M )} satisfies the asymptotic locally unbiased condition, the inequality (5) yields that lim sup n α MSEθ (ρn , M n ) ≥ lim sup n→∞
n→∞
nα . J S [⊗n θ ]
We define the Cramér-Rao bound: n α n {(ρn , M )} satisfies the asymptotic ˜ . Cα [θ ] := inf n lim sup n MSEθ (ρn , M ) {ρn ,M } locally unbiased condition. n→∞ Thus, we obtain C˜ α [θ ] = lim sup n→∞
nα . J S [⊗n θ ]
(8)
Since ⊗m S J S [⊗n+m ] ≥ J S [⊗n θ θ ] + J [θ ], ⊗n
(9)
J [θ ] the limit limn→∞ exists. (For example, see Lemma A.1 of [21].) Thus, C˜ 1 [θ ] n nα can be defined by limn→∞ J S [ ⊗n . ] S
θ
In order to treat the above values, we consider the following condition: (C) The range of ρ[θ ] contains the range of D[θ ]2 .
696
M. Hayashi
Assume that the condition (C) does not hold. When the input state is the maximally entangled state | √1 I , the RLD Fisher information diverges. So, J R [θ ] is infinity. d
Theorem 1. When the condition (C) holds, J R [θ ] = Tr K D[θ ]ρ[θ ]−1 D[θ ] . Proof. Assume that the input state is given by |AA| and A is an invertible matrix. Then, the range of (I ⊗ A T )ρ[θ ](I ⊗ A) contains the range of ((I ⊗ A T )D[θ ](I ⊗ A))2 . Assume that ρ[θ ] is invertible. Using formula (1), we obtain J R [θ , |AA|] = Tr((I ⊗ A T )D[θ ](I ⊗ A))2 ((I ⊗ A T )ρ[θ ](I ⊗ A))−1 = Tr(I ⊗ A T )D[θ ](I ⊗ A)(I ⊗ A T )D[θ ](I ⊗ A)(I ⊗ A)−1 ρ[θ ]−1 (I ⊗ A T )−1 = Tr(I ⊗ A A T )D[θ ]ρ[θ ]−1 D[θ ] = Tr A A T (Tr K D[θ ]ρ[θ ]−1 D[θ ]), where ((I ⊗ A T )ρ[θ ](I ⊗ A))−1 is the inverse of (I ⊗ A T )ρ[θ ](I ⊗ A) on its range. So the supremum of Tr A A T (Tr K D[θ ]ρ[θ ]−1 D[θ ]) with the condition rank A A T = dim H equals Tr K D[θ ]ρ[θ ]−1 D[θ ] . Assume that ρ[θ ] is non-invertible. We choose an arbitrary small real number > 0. Similar calculations and the operator monotonicity of x → −x −1 yield that Tr((I ⊗ A T )D[θ ](I ⊗ A))2 ((I ⊗ A T )(ρ[θ ] + I )(I ⊗ A))−1 ≤ Tr A A T (Tr K D[θ ](ρ[θ ] + I )−1 D[θ ]) ≤ Tr A A T (Tr K D[θ ]ρ[θ ]−1 D[θ ]),
(10)
which is a bounded value due to Condition (C). Now, we focus on two matrixes on the range of (I ⊗ A T )ρ[θ ](I ⊗ A), (I ⊗ A T )ρ[θ ](I ⊗ A) and ((I ⊗ A T )D[θ ](I ⊗ A))2 . Since the right hand side of (10) is independent of , the range of (I ⊗ A T )ρ[θ ](I ⊗ A) contains that of ((I ⊗ A T )D[θ ](I ⊗ A))2 . So, taking the limit → 0, we obtain J R [θ , |AA|] = Tr((I ⊗ A T )D[θ ](I ⊗ A))2 ((I ⊗ A T )ρ[θ ](I ⊗ A))−1 = lim Tr((I ⊗ A T )D[θ ](I ⊗ A))2 ((I ⊗ A T )(ρ[θ ] + I )(I ⊗ A))−1 →+0
≤ Tr A A T (Tr K D[θ ]ρ[θ ]−1 D[θ ]). The remaining task is to show the inequality J R [θ , |AA|] ≤ Tr K D[θ ]ρ[θ ]−1 D[θ ]
(11)
for a non-invertible matrix A. Define (I ⊗ A T )−1 and (I ⊗ A)−1 as the inverses of I ⊗ A T and I ⊗ A whose domains are the ranges of the matrixes I ⊗ A T and I ⊗ A. Letting P and P be the projections to the ranges of the matrixes I ⊗ A T and I ⊗ A, we have (I ⊗ A T ) = P(I ⊗ A T ) = (I ⊗ A T )P = P(I ⊗ A T )P , and (I ⊗ A) = P (I ⊗ A) = (I ⊗ A)P = P (I ⊗ A)P. Then, (I ⊗ A T )ρ[θ ](I ⊗ A) = P(I ⊗ A T )ρ[θ ](I ⊗ A)P = P(I ⊗ A T )P ρ[θ ]P (I ⊗ A)P.
Comparison Between Cramer-Rao and Mini-max Approaches
697
In the following, we consider the case when ρ[θ ] is invertible. The matrix (P ρ[θ ]P )−1 is defined as the inverse of P ρ[θ ]P whose domain and range are the range of P . Thus, J R [θ , |AA|] = Tr P((I ⊗ A T )D[θ ](I ⊗ A))2 P(P(I ⊗ A T )ρ[θ ](I ⊗ A)P)−1 = Tr(I ⊗ A T )D[θ ]P (I ⊗ A)P(I ⊗ A)−1 (P ρ[θ ]P )−1 ×(I ⊗ A T )−1 P(I ⊗ A T )P D[θ ](I ⊗ A) = Tr(I ⊗ A T )D[θ ]P (P ρ[θ ]P )−1 P D[θ ](I ⊗ A) = Tr(P ρ[θ ]P )−1 P D[θ ](I ⊗ A A T )D[θ ]P = Tr(P ρ[θ ]P )−1 D[θ ](I ⊗ A A T )D[θ ] ≤ Tr ρ[θ ]−1 D[θ ](I ⊗ A A T )D[θ ] = Tr A A T (Tr K D[θ ]ρ[θ ]−1 D[θ ]). (12) Therefore, the inequality (11) holds. Next, we consider the case when ρ[θ ] is non-invertible. We choose an arbitrary small real number > 0. Similar calculations and the operator monotonicity of x → −x −1 yield that Tr P((I ⊗ A T )D[θ ](I ⊗ A))2 P(P(I ⊗ A T )(ρ[θ ] + I )(I ⊗ A)P)−1 ≤ Tr A A T (Tr K D[θ ](ρ[θ ] + I )−1 D[θ ]) ≤ Tr A A T (Tr K D[θ ]ρ[θ ]−1 D[θ ]), (13) which is a bounded value due to Condition (C). Now, we focus on two matrixes on the range of P, P(I ⊗ A T )ρ[θ ](I ⊗ A)P and P((I ⊗ A T )D[θ ](I ⊗ A))2 P. Since the right hand side of (13) is independent of , the range of P(I ⊗ A T )ρ[θ ](I ⊗ A)P contains that of P((I ⊗ A T )D[θ ](I ⊗ A))2 P. So, taking the limit → 0, we obtain J R [θ , |AA|] = Tr P((I ⊗ A T )D[θ ](I ⊗ A))2 P(P(I ⊗ A T )ρ[θ ](I ⊗ A)P)−1 = lim Tr P((I ⊗ A T )D[θ ](I ⊗ A))2 P(P(I ⊗ A T )(ρ[θ ] + I )(I ⊗ A)P)−1 →+0
≤ Tr A A T (Tr K D[θ ]ρ[θ ]−1 D[θ ]). Therefore, the inequality (11) holds.
˜ θ } satisfy the condition (C), then Theorem 2. When two channel families {θ } and { the additivity ˜ θ ] = J R [θ ] + J R [ ˜ θ] J R [θ ⊗ holds. ˜ θ . Then, the relation Proof. Let K and K˜ be output systems of the channels θ and (7) guarantees ˜ θ ] = Tr K D[θ ] Tr ˜ D[ ˜ θ ] = 0. Tr K⊗K˜ D[θ ] ⊗ D[ K
(14)
698
M. Hayashi
˜ θ ] = D[θ ] ⊗ ρ[ ˜ θ ] + ρ[θ ] ⊗ D[ ˜ θ ], Theorem 1 and (14) yield Since D[θ ⊗ ˜ θ] J R [θ ⊗ ˜ θ] = Tr K⊗K˜ (D[θ ] ⊗ ρ[
˜ θ ])(ρ[θ ]−1 ⊗ ρ[ ˜ θ ]−1 )(D[θ ] ⊗ ρ[ ˜ θ ] + ρ[θ ] ⊗ D[ ˜ θ ]) +ρ[θ ] ⊗ D[ −1 −1 ˜ ˜ ˜ ˜ = Tr K⊗K˜ (D[θ ]ρ[θ ] D[θ ] ⊗ ρ[θ ] + ρ[θ ] ⊗ D[θ ]ρ[θ ] D[θ ]
˜ θ ]) +2D[θ ] ⊗ D[ ˜ θ ]ρ[ ˜ θ ]−1 D[ ˜ θ ]) = (Tr K D[θ ]ρ[θ ]−1 D[θ ]) ⊗ I + I ⊗ (Tr K˜ D[ ˜ θ ]ρ[ ˜ θ ]−1 D[ ˜ θ ] = Tr K D[θ ]ρ[θ ]−1 D[θ ] + Tr K˜ D[
˜ θ ], = J R [θ ] + J R [
(15)
where Eq. (15) follows from the additivity property concerning the matrix norm:
X ⊗ I + I ⊗ Y = X + Y for any two Hermitian matrixes X and Y .
Corollary 1. When a channel family {θ } satisfies Condition (C), then R J R [⊗n θ ] = n J [θ ]. ⊗n ⊗n R S Since n J S [θ ] ≤ J S [⊗n θ ] ≤ J [θ ], J [θ ] increases in order n under the ⊗n S assumption of Theorem 1, i.e., J [θ ] = O(n). When the rank of ρ[θ ] is the maximum, i.e., dd , this condition holds and J S [⊗n θ ] = O(n). However, there is an example that does not satisfy the above condition but satisfies Condition (C) as follows. So, Condition (C) is weaker than the condition that ρ[θ ] has maximum rank. A channel is called a phase damping channel when the output system K equals the input system H and there exist complex numbers dk,l such that dk,l ρk,l |kl|, {dk,l } (ρ) = k,l
where ρ =
k,l
ρk,l |kl|. In this case, the state ρ[] is written as the following form: ρ[{dk,l } ] =
dk,l |k|kl|l|.
k,l
That is, the range of ρ[] is included by the space spanned by {|k K |k R }. When a channel family {θ } is given as a one-parameter subfamily of {{dk,l } |dk,l is strictly positive.}, Condition (C) holds. Therefore, there exists a channel family that satisfies Condition (C) but consists of non-full-rank channels. Further, we have the following observation. Corollary 2. Assume that Condition (C) holds and there exists a normalized vector u in the input system H such that Tr K D[θ ]ρ[θ ]−1 D[θ ] = u| Tr K D[θ ]ρ[θ ]−1 D[θ ]|u, [u|ρ[θ ]|u, u|D[θ ]|u] = 0, and [IK ⊗ |uu|, ρ[θ ]] = 0. Then, J S [θ ] = J R [θ ], and this bound can be attained by the input pure state |uu| on H. That is, it can be attained without use of the reference system.
Comparison Between Cramer-Rao and Mini-max Approaches
699
Proof. Since [u|ρ[θ ]|u, u|D[θ ]|u] = 0, J S [θ , |uu|] = J R [θ , |uu|]. So, it is sufficient to show that J R [θ , |uu|] = u| Tr K D[θ ]ρ[θ ]−1 D[θ ]|u. We consider the case when the input state is |uu| ⊗ |vv|. Remember that J R [θ , |uu|⊗|vv|] = J R [θ , |uu|]. Since [IK ⊗|uu|, ρ[θ ]] = 0, the equality in (12) holds for A = u · v T . Then, J R [θ , |uu| ⊗ |vv|] = Tr (u · v T )(u · v T )T Tr K D[θ ]ρ[θ ]−1 D[θ ] = u| Tr K D[θ ]ρ[θ ]−1 D[θ ]|u. 4. Examples We consider the case where Condition (C) does not hold. As the simplest example, we consider the one-parameter unitary case, i.e., the case when θ (ρ) = eiθ H ρe−iθ H with an Hermitian matrix H . Using (2), we obtain J S [θ ] = (λmax (H ) − λmin (H ))2 , where λmax (H ) and λmin (H ) are the maximum and minimum of eigenvalues of H . So, we obtain 2 2 J S [⊗n θ ] = n (λmax (H ) − λmin (H )) .
1 0 2 In particular, in the two-dimensional case, when H = , the optimal input is 0 − 21 √1 (|0⊗n + |1⊗n ), which is called the noon state. This unitary estimation is called a 2 phase estimation and this estimation with the noon state was experimentally realized with n = 4 [32,33] and n = 10 [34]. Next, we consider the d-dimensional system Cd spanned by {| j}d−1 j=0 and the unitary
matrix X defined as X | j := | j + 1 mod d. Using a distribution { p j }d−1 j=0 and a real diagonal matrix H with diagonal elements {h j }d−1 j=0 , we define the TP-CP map θ by θ (ρ) :=
d−1
p j X j eiθ H ρe−iθ H X − j .
j=0
This TP-CP map can be regarded as the stochastic application of the unitary X j after the application of the unitary eiθ H . Let h a and h b be the maximum and the minimum eigenvalues of H . Using twodimensional reference system spanned by |0 R and |1 R , we choose the following input state: 1 | n := √ (|a⊗n |0 R + |b⊗n |1 R ). 2 In this case, as the first step, we apply the following measurement {M j }: M j := | j + (a, . . . , a) j + (a, . . . , a)| ⊗ |0 R R 0| +| j + (b, . . . , b) j + (b, . . . , b)| ⊗ |1 R R 1|.
700
M. Hayashi
Fig. 3. Phase estimation with noon state n = 20
When the outcome of this measurement is j , the resulting state is the pure state 1 √ (einh a | j + (a, . . . , a)|0 R + einh b | j + (b, . . . , b)|1 R ). 2
(16)
The SLD Fisher information of the above family is n 2 (h a − h b )2 . Therefore, since the maximum SLD Fisher information behaves as O(n 2 ) at most, J S [⊗n 0 ] behaves as O(n 2 ). 5. Local Asymptotic Mini-max Bound In this section, we consider the relation between the discussion in the previous section and estimating protocols in a different viewpoint. Consider the phase estimation with inputing the noon state √1 (|0⊗n + |1⊗n ). Then, the output state is √1 (einθ/2 |0⊗n + 2
2
e−inθ/2 |1⊗n ). In this case, we cannot distinguish the parameters θ and θ + 2π n . For 1 ⊗n ⊗n √ example, when we apply measurement { (|0 ± |1 )}, the probability with the
2 √1 (|0⊗n + |1⊗n ) equals cos2 nθ/2, as is shown in Fig 3, and the Fisher infor2 equals n 2 . Even if the parameter θ is assumed to belong to (0, π/n], we cannot
outcome
mation distinguish the two parameters θ = π/3n and θ = 2π/3n with such high probability because we have only two outcomes. In order to distinguish two parameters θ = π/3n and θ = 2π/3n in this measurement, we need to repeat this measurement several times, e.g., k. Since the number of application of the unknown unitary is N := kn, the error behaves as k1 N1 , which is different from O( N12 ). So, we cannot conclude that the above method attains the order O( N12 ) concerning MSE. Therefore, we need to discuss what bound can be attained globally, more carefully. For this purpose, we focus on an -neighborhood Uθ, of θ and define the local asymptotic mini-max risk [38]: Cα [θ0 , {(ρn , M n )}] := lim lim sup sup n α MSEθ (ρn , M n ), →0 n→∞ θ∈Uθ , 0
Comparison Between Cramer-Rao and Mini-max Approaches
701
and the local asymptotic mini-max bound: Cα [θ0 ] :=
inf
{(ρn ,M n )}
Cα [θ0 , {(ρn , M n )}].
Concerning the local asymptotic mini-max bound, we have the following two propositions. Proposition 1. When C˜ α [θ0 ] is continuous and the convergence (8) is compactly uniform, Cα [θ0 ] ≥ C˜ α [θ0 ].
(17)
Proof. For any δ > 0, we choose an integer N and an -neighborhood Uθ0 , satisfying Cα [θ0 ] + δ ≥ n α MSEθ (ρn , M n ), ∀n ≥ N , ∀θ ∈ Uθ0 , . We introduce two quantities ηn (θ ) := vn (θ ) :=
(18)
θˆ Tr θ (ρn )M n (d θˆ ), (θˆ − ηn (θ ))2 Tr θ (ρn )M n (d θˆ ).
Then we obtain MSEθ (ρn , M n ) = vn (θ ) + (ηn (θ ) − θ )2 . Let Jθ,n be the Fisher information of the distribution family {Tr θ (ρn )M n (d θˆ )|θ ∈ }. This quantity is smaller than J S [⊗n θ ]. Deforming the classical Cramér-Rao inequality, we obtain vn (θ ) ≥
n (θ) 2 ) ( dηdθ . Jθ,n
Thus, we obtain MSEθ (ρn , M n ) ≥
n (θ) 2 ) ( dηdθ + (ηn (θ ) − θ )2 . Jθ,n
(19)
As is shown later, for any δ > 0, there exists a sufficiently large integer N satisfying the following. For any n ≥ N , there exists θn ∈ Uθ0 , such that dηn (θn ) ≥ 1 − δ. dθ
(20)
Using (19), we obtain MSEθn (ρn , M n ) ≥
n (θ) 2 ) ( dηdθ . Jθn ,n
(21)
Take the limit n → ∞. Then, the continuity of C˜ α [θ0 ], the compact uniformity of the convergence (8), (18), and (21) imply that Cα [θ0 ] + δ ≥ (1 − δ)2 C˜ α [θ0 ]. Taking the limit δ → 0, we obtain (17).
702
M. Hayashi
Finally, we show the existence of the integer N satisfying the above condition given by (20) by using reduction to absurdity. Assume that for any δ > 0, there exists a dηn (θ) < 1 − δ for any θ ∈ Uθ0 , . Thus, subsequence n k such that dθk [ηn k (θ + /2) − (θ + /2)] − [ηn k (θ − /2) − (θ − /2)] = ηn k (θ + /2) − ηn k (θ − /2) − < −δ. Then, max{|ηn k (θ + /2) − (θ + /2)|, |ηn k (θ − /2) − (θ − /2)|} >
δ . 2
That is, max{(ηn k (θ + /2) − (θ + /2))2 , (ηn k (θ − /2) − (θ − /2))2 } >
2 δ2 . 4
Using (19), we obtain max{MSEθ+/2 (ρn k , M n k ), MSEθ−/2 (ρn k , M n k )} ≥
2 δ2 . 4
Since MSEθ+/2 (ρn , M n ) behaves as O( n1α ), we obtain a contradiction.
Proposition 2. Assume that E := supθ∈ |θ | < ∞. When the order parameter α equals 1 and C˜ α [θ0 ] is continuous, C1 [θ0 ] = C˜ 1 [θ0 ] = lim
n
. n→∞ J S [⊗n ] θ
Proof. As is mentioned in Sect. 3, (9) guarantees the convergence of limn→∞
n . J S [⊗n θ ]
It is enough to show the inequality C˜ 1 [θ0 ] ≥ C1 [θ0 ]. For an arbitrary real number δ > 0 and an arbitrary integer m, let {ρm , M m } be a locally unbiased estimator at θ0 such that 1 J S [⊗m θ0 ]
+ δ > MSEθ0 (ρm , M m ).
(22)
We define another coordinate η(θ ) by η(θ ) := θˆ Tr θ (ρn )M n (d θˆ ), and denote the MSE concerning the parameter η of an estimator (ρn , M n ) by MSEη(θ) (ρn , M n ). For any δ > 0, we choose an integer m, a sufficiently small neighborhood Uθ0 , such that 1 C˜ 1 [θ0 ] + δ ≥ S ⊗m m J [θ0 ] 1−δ ≤ for ∀θ, θ ∈ Uθ0 , .
η(θ ) − η(θ ) ≤1+δ θ − θ
(23) (24)
Comparison Between Cramer-Rao and Mini-max Approaches
703
, M nm ) be the estimator given as the average value of n times appliNext, let (ρnm cations of the estimator (ρm , M m ) concerning the original parameter θ . Then, we can choose a sufficiently large number n satisfying the following: When the true parameter , M nm ) belongs to U is θ0 , the estimate of (ρnm θ0 , with probability higher than 1 − δ. The second inequality in (24) guarantees that
1 MSEθ (ρm , M m )+δ E n = MSEθ (ρnm , M nm )+δ E ≥
1 MSEη(θ) (ρnm , M nm ). (1 + δ)2
(25)
Thus, (22), (23), and (25) imply that (1 + δ)2 (
C˜ 1 [θ0 ] 2δ + + δ E) ≥ MSEη(θ0 ) (ρnm , M nm ). nm n
, M nm ) is continuous and (1 + δ)2 ( Since MSEη(θ0 ) (ρnm
C˜ [ ] δ)2 ( 1nmθ0
+
2δ n
C˜ 1 [θ0 ] nm
+ δ(E + 2)) > (1 +
+ δ E), we can choose a sufficiently small number 0 < < such that
(1 + δ)2 (
C˜ 1 [θ0 ] + δ(E + 2)) ≥ MSEη(θ) (ρnm , M nm ) nm
(26)
for θ ∈ Uθ0 , . , M lnm ) be the estimator given as the average value of l times applications Let (ρlnm , M nm ) concerning η. Since the estimator (ρ , M nm ) is unbiased of the estimator (ρnm nm concerning the parameter η,
MSEη(θ) (ρlnm , M lnm ) =
, M nm ) MSEη(θ) (ρnm . l
(27)
We choose a sufficiently large number l satisfying the following: When the true parame , M lnm ) belongs to U ter is θ ∈ Uθ0 , /2 , the estimate η of (ρlnm θ0 , with the probability 1 − pl , where the probability pl exponentially goes to 0 as l → ∞. The first inequality in (24) guarantees that 1 MSEη(θ) (ρlnm , M lnm ) + E pl ≥ MSEθ (ρlnm , M lnm ), ∀θ ∈ Uθ0 , /2 . (1 − δ)2 (28) Thus, the relations (26), (27), and (28) imply (1 + δ)2 C˜ 1 [θ0 ] + δ(E + 2)) + E pl ≥ MSEθ (ρlnm ( , M lnm ), ∀θ ∈ Uθ0 , /2 . l(1 − δ)2 nm (29) Taking the limit l → ∞, we obtain (1 + δ)2 ˜ (C1 [θ0 ] + nmδ(E + 2)) ≥ lim lnm MSEθ (ρlnm , M lnm ), ∀θ ∈ Uθ0 , /2 . l→∞ (1 − δ)2
704
M. Hayashi
Finally, using the above n and m, we define a sequence of estimators {(ρk , M k )} in the following way. For given k, we choose maximum l such that lnm ≤ k. Then, the , M lnm ). In this definition, we only use lnm estimator (ρk , M k ) defined as (ρlnm applications, and the remaining k − lnm applications are discarded. So, we obtain (1 + δ)2 ˜ (C1 [θ0 ] + nmδ(E + 2)) ≥ lim sup k MSEθ (ρk , M k ), ∀θ ∈ Uθ0 , /2 , (1 − δ)2 k→∞ which implies that (1 + δ)2 ˜ (C1 [θ0 ] + nmδ(E + 2)) ≥ C1 [θ0 , {(ρk , M k )}∞ k=1 ] ≥ C 1 [θ0 ]. (1 − δ)2 Since δ > 0 is arbitrary, C˜ 1 [θ0 ] ≥ C1 [θ0 ].
Now, remember that the Cramér-Rao bound can be attained by using the two-step method in the case of state estimation. By using the two-step method [22,23], the local asymptotic mini-max bound Cα [θ0 ] can be attained at all points θ as follows. Proposition 3. Assume that E := supθ∈ |θ | < ∞ and C1 [θ0 ] is continuous. For any δ > 0, there exists a sequence of estimators {(ρn , M n )} such that Cα [θ , {(ρn , M n )}] ≤ Cα [θ ] + δ
(30)
for all points θ . Further, when the parameter space is compact, lim n α minn max MSEθ (ρn , M n ) = max Cα [θ ].
n→∞
(ρn ,M ) θ∈
(31)
θ∈
Proof. We use a two-step method slightly different from [23]. Before applying the unknown channel θ , for any real number δ > 0, we choose an i -neighborhood Uθi ,i satisfying the following three conditions: (1) ∪i Uθi ,i = . (2) For any θi , there exists a sequence of estimators {(ρn (θi ), M n (θi ))} such that Cα [θi ] + δ/2 ≥ lim→0 lim supn→∞ n α supθ∈Uθ , MSEθ (ρn (θi ), M n (θi )). (3) supi:θ∈Uθ , Cα [θi ] ≤ i i i Cα [θ ] + δ/2. We divide √n applications of the unknown channel θ into two groups: √ The first group consists of n applications and the second group consists of n − n applications. In the first step, we apply a POVM M to the first group. This POVM M is a POVM on the single√ system H and is required to satisfy that Jθ is non-degenerate at all points θ . Based on n obtained data, we estimate which i -neighborhood Uθi ,i contains the true parameter, and obtain the first step estimate θiˆ . The error probability Pθ,n of this step √ goes to 0 exponentially, i.e., Pθ,n behaves as e−c n , where c depends on√ θ . In the second step, we apply the estimator (ρn−√n (θi ), M n− n (θi )) to the second group, and obtain our final estimate from the outcome of the estimator √ (ρn−√n (θi ), M n− n (θi )). We express this estimator by (ρn , M n ). Its MSE is evaluated as MSEθ (ρn , M n ) ≤ E Pθ,n + (1 − Pθ,n )
√ n
sup i:θ∈Uθi ,i
MSEθ (ρn−√n (θi ), M n−
(θi )). (32)
Since n α E Pθ,n goes to 0, we obtain lim sup n α MSEθ (ρn , M n ) ≤ n→∞
sup i:θ∈Uθi ,i
Cα [θi ] + δ/2 ≤ Cα [θ ] + δ.
Comparison Between Cramer-Rao and Mini-max Approaches
705
Thus, we obtain (30). Further, the relation (32) yields that sup MSEθ (ρn , M n )
θ∈
≤ sup E Pθ,n + sup (1 − Pθ,n ) θ∈
θ∈
sup i:θ∈Uθi ,i
√ n
MSEθ (ρn−√n (θi ), M n−
(θi )).
The compactness of guarantees that supθ∈ n α E Pθ,n → 0. Thus, lim sup n α sup MSEθ (ρn , M n ) ≤ sup n→∞
θ∈
sup
θ∈ i:θ∈Uθi ,i
Cα [θi ] + δ/2 ≤ sup Cα [θ ] + δ.
Since the part ≥ of (31) is trivial, we obtain (31).
θ∈
Proposition 3 holds even when we replace the MSE by a general error function R(θ, θˆ ) for one-parametric family satisfying the following conditions: (1) The relation R(θ, θˆ ) ∼ = (θˆ − θ )2 holds with a local coordinate when θˆ is close to θ . (2) The maximum of R(θ, θˆ ) exists. Therefore, we can apply Proposition 3 to the following case: Assume that the one-parameter channel family {θ } has a compact group covariant structure, that is, its parameter space is given as an interval [a, b) and there is a unitary representation Uθ of R such that Uθ θ (ρ)Uθ† = θ+θ (ρ). The error is given by mink∈Z (θˆ + k(b − a) − θ )2 instead of the square error (θˆ − θ )2 . In this case, due to the group covariance, Cα [θ ] does not depend on the true parameter θ . Application of Proposition 3 implies that the global min-max error behaves Cα [θ ] n1α . In the phase estimation case, the unknown parameter θ belongs to [0, 2π ), and the minimum of the worst value of the average error maxθ MSEθ (ρn , M n ) behaves 2 as πn 2 [9–11]. That is, the leading decreasing order is O(1/n 2 ) and the leading decreasing coefficient is π 2 when we apply the optimal estimator. Proposition 3 implies that maxθ C2 [θ ] = π 2 . Since C2 [θ ] does not depend on θ due to the homogenous structure, we can conclude that C2 [θ ] = π 2 . 2 ˜ So, the equation J S [⊗n θ ] = n implies the equation C 2 [θ ] = 1. Hence, the ˜ Cramér-Rao bound C2 [θ ] cannot be attained globally in this model. However, it can be attained in a specific point in the following sense. Proposition 4. Assume that E := supθ∈ |θ | < ∞ and C1 [θ0 ] is continuous. For any δ > 0 and any θ0 ∈ , there exists a sequence of estimators {(ρn,θ0 , Mθn0 )} satisfying the asymptotically locally unbiased condition and the relations: lim sup n α MSEθ (ρn,θ0 , Mθn0 ) ≤ Cα [θ ] + δ, ∀θ = θ0 , n→∞
lim sup n α MSEθ0 (ρn,θ0 , Mθn0 ) ≤ C˜ α [θ0 ] + δ. n→∞
In estimation of probability distribution, there exists a superefficient estimator that has smaller error at a discrete set than the Cramér-Rao bound [29]. Since such a superefficient estimator cannot be useful, many statisticians think that it is better to impose a condition for our estimators for removing superefficient estimators. In this classical case, if we assume the asymptotic locally unbiased condition, we have no superefficient estimator. Proposition 4 means that even if the asymptotic locally unbiased condition is assumed, there exists an estimator that behaves in the similar way to a superefficient estimator in the case of unitary estimation. So, we call such an estimator a q-channel-superefficient
706
M. Hayashi
estimator. That is, a sequence of estimators {(ρn , M n )} is called q-channel-superefficient at θ with the order n1α when lim supn→∞ n α MSEθ (ρn , M n ) < Cα [θ ]. Hence, in order to remove the q-channel-superefficiency problem, it is better to adopt the bound Cα [θ ] as the criterion instead of C˜ α [θ ]. Proof. We choose i -neighborhoods Uθi ,i in the same way, and define the neighborhood Uθ0 , 1 . We apply the same first step as Proposition 3 to neighborhoods {Uθi ,i }i ∪ {Uθ0 ,
n 1/4
1 n 1/4
}, and obtain the first step estimate θiˆ . When the first step estimate θiˆ is not
θ0 , we apply the same method as Proposition 3 in the second step. When the first step estimate θiˆ is θ0 , we apply the asymptotically locally unbiased estimator whose MSE behaves as (C˜ α [θ ] + δ)/n α asymptotically. Further, since there exists an asymptotically locally unbiased estimator that surpasses the bound Cα [θ ], the asymptotically locally unbiased condition is too weak for deriving the local asymptotic mini-max bound, which is more meaningful. In order to avoid this problem, it is sufficient to impose the following condition: (CU) The limit limn→∞ n α MSEθ (ρn , M n ) exists for all θ and this convergence is compactly uniform concerning θ . Under Condition (CU), limn→∞ n α MSEθ (ρn , M n ) is continuous concerning θ , and lim n α sup MSEθ (ρn , M n ) = sup
n→∞
lim n α MSEθ (ρn , M n ).
θ∈Uθ0 , n→∞
θ∈Uθ0 ,
Thus, lim n α MSEθ0 (ρn , M n ) = lim sup →0 θ∈Uθ
n→∞
0 ,
lim n α MSEθ (ρn , M n )
n→∞
= lim lim n α sup MSEθ (ρn , M n ) →0 n→∞
θ∈Uθ0 ,
= Cα [θ0 , {(ρn , M n )}]. Therefore, we obtain the following corollary. Corollary 3. When a sequence of estimators {(ρn , M n )} satisfies Condition (CU), lim n α MSEθ (ρn , M n ) ≥ Cα [θ ].
n→∞
Therefore, Condition (CU) is better in estimation of the quantum channel than the asymptotically locally unbiased condition. Finally, we consider the relation with the adaptive method proposed by Nagaoka [16]. In this method, we apply our POVM to each single system H, and we decide the k th POVM based on the knowledge of previous k − 1 outcomes. In this case, Fujiwara [30] analyzed the asymptotic behavior of the MSE of this estimator. Now, we consider the case of nm applications of the unknown channel θ . In this case, we divide nm applications into n groups consisting of m applications. When we apply the adaptive method mentioned in Fujiwara [30] to these groups, the MSE of this estimator behaves C˜ α [θ ] 1 as n J S [ ⊗m , which is close to nm α . So, when α > 1, this method cannot realize the ] θ
1 optimal order O( (nm) α ).
Comparison Between Cramer-Rao and Mini-max Approaches
707
6. Discussion We have compared the Cramér-Rao bound C˜ α [θ ] and the local asymptotic mini-max bound Cα [θ ] in quantum channel estimation, which contains quantum state estimation. When the model has group covariant structure, the local asymptotic mini-max bound Cα [θ ] coincides with the limit of the global mini-max bound. We have also shown that both bounds C˜ α [θ ] and Cα [θ ] coincide in quantum channel estimation when the maximum of SLD Fisher information J S [⊗n θ ] behaves as O(n). The case of state estimation can be regarded as a special case of this case. That is, the conventional state estimation has no difference between both bounds. However, we have shown that the Cramér-Rao bound C˜ α [θ ] is different from the local asymptotic mini-max bound Cα [θ ] in the phase estimation. So, we can conclude that the local asymptotic mini-max bound Cα [θ ] is more meaningful and does not necessarily coincide with the Cramér-Rao bound C˜ α [θ ]. In order to clarify the asymptotic leading order of J S [⊗n θ ], we have derived ⊗n S Condition (C) as a sufficient condition for J [θ ] = O(n). That is, the condition
Tr K D[θ ]ρ[θ ]−1 D[θ ] = ∞ is a necessary condition for square speedup. This condition has been derived from the following two facts. One is the supremum of the RLD Fisher information satisfies the additive property. The other is the RLD Fisher information is an upper bound of the SLD Fisher information. This, the supremum of the RLD Fisher information is the upper bound of the regularized supremum of the SLD Fisher information, which equals the inverse of the Cramér-Rao bound. However, it is an open problem to clarify whether this upper bound can be attained by the regularized SLD Fisher information. Further, Fujiwara and Imai [2] and Matsumoto [14] also obtained another sufficient condition. Since the relation with their conditions is not clear, its clarification is an open problem. Our Condition (C) trivially contains the case when the state ρ[θ ] is a full rank state on the tensor product system while it is not so easy to derive the above full rank condition from Fujiwara and Imai’s condition. Further, we have also obtained another 2 example for J S [⊗n θ ] = O(n ) under Condition (C). This example is a larger class than 2 the unitary model. So, we can expect that J S [⊗n θ ] behaves as O(n ) if Condition (C) does not hold. This is a challenging open problem. Further, in order to treat the phase estimation in the photonic system, it is suitable to restrict the photon number of the input state instead of the number of the unknown application. This formulation is discussed in the next paper [41]. Acknowledgements. The author was partially supported by a Grant-in-Aid for Scientific Research in the Priority Area ‘Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI)’, No. 18079014 and a MEXT Grant-in-Aid for Young Scientists (A) No. 20686026. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme. The author thanks Mr. Wataru Kumagai for helpful comments. He also thanks the referees for helpful comments concerning this manuscript. In particular, the first referee’s report was much help to improve the presentation in Sect. 1.
Lemmas Needed for Theorem 1 Lemma 1. Any strictly positive definite matrix A and any projection P satisfy the inequality A−1 ≥ (P A P)−1 , where
(P A P)−1
is the inverse matrix with the domain P.
(33)
708
M. Hayashi
Proof. Let R be the operator norm of the matrix P A(I − P). Lemma 2 guarantees the inequality P A(I − P) + (I − P)A P ≤ P +
R2 (I − P)
(34)
for any > 0. Thus, A ≤ P A P + (I − P)A(I − P) + P + = P(A + I )P + (I − P)(A +
R2 (I − P)
R2 I )(I − P).
Since the function x → −x −1 is operator monotone, A−1 ≥ (P(A + I )P)−1 + ((I − P)(A + Taking the limit → 0, we obtain (33).
R2 I )(I − P))−1 ≥ (P(A + I )P)−1 .
Lemma 2. Let A be a positive semi-definite matrix and P be a projection. Then, the inequality P A(I − P) + (I − P)A P ≤ P +
R2 (I − P)
(35)
holds for any > 0, where R is the operator norm of the matrix P A(I − P). Proof. Choose an arbitrary normalized vector u. Let t be Pu 2 . Then, √√ u|P A(I − P) + (I − P)A P|u ≤ 2 t 1 − t R R2 R2 = u| P + (I − P)|u, ≤ t + (1 − t) which implies (35).
References 1. Fujiwara, A.: Quantum channel identification problem. Phys. Rev. A 63, 042304 (2001) 2. Fujiwara, A., Imai, H.: A fibre bundle over manifolds of quantum channels and its application to quantum statistics. J. Phys. A: Math. Theor. 41, 255304 (2008) 3. Fujiwara, A., Imai, H.: Quantum parameter estimation of a generalized Pauli channel. J. Phys. A: Math. Gen. 36, 8093 (2003) 4. Fujiwara, A.: Estimation of SU(2) operation and dense coding: an information geometric approach. Phys. Rev. A 65, 012316 (2002) 5. Imai, H., Fujiwara, F.: Geometry of optimal estimation scheme for SU(D) channels. J. Phys. A: Math. Theor. 40, 4391 (2007) 6. Hayashi, M.: Parallel treatment of estimation of SU(2) and phase estimation. Phys. Lett. A 354, 183 (2006) 7. Chiribella, G., D’Ariano, G.M., Perinotti, P., Sacchi, M.F.: Efficient use of quantum resources for the transmission of a reference frame. Phys. Rev. Lett. 93, 180503 (2004) 8. Bagan, E., Baig, M., Munoz-Tapia, R.: Quantum reverse-engineering and reference frame alignment without non-local correlations. Phys. Rev. A 70, 030301 (2004) 9. Luis, A., Perina, J.: Optimum phase-shift estimation and the quantum description of the phase difference. Phys. Rev. A 54, 4564 (1996) 10. Buzek, V., Derka, R., Massar, S.: Optimal quantum clocks. Phys. Rev. Lett. 82, 2207 (1999)
Comparison Between Cramer-Rao and Mini-max Approaches
709
11. Imai, H., Hayashi, M.: Fourier Analytic Approach to Phase Estimation in Quantum Systems. New J. Phys. 11, 043034 (2009) 12. Kahn, J.: Fast rate estimation of an unitary operation in SU(d). Phys. Rev. A 75, 022326 (2007) 13. Hotta, M., Karasawa, T., Ozawa, M.: N-body-extended channel estimation for low-noise parameters. J. Phys. A: Math. Gen. 39, 14465 (2006) 14. Matsumoto, K.: On metric of quantum channel spaces. http://arXiv.org/abs/1005.4759v1 [quant-ph], 2010 15. Holevo, A.S.: Covariant measurements and uncertainty relations. Rep. Math. Phys. 16, 385–400 (1979) 16. Nagaoka, H.: On the parameter estimation problem for quantum statistical models. In: Proc. 12th Symp. on Inform. Theory and its Appl. p 577 (1989); Reprinted in Nagaoka H.: Asymptotic Theory of Quantum Statistical Inference, ed. M Hayashi, Singapore: World Scientific, p. 125, 2005 17. Fujiwara, A., Nagaoka, H.: Quantum Fisher metric and estimation for pure state models. Phys. Lett. A 201, 119 (1995) 18. Hayashi, M.: Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation. J. Phys. A: Math. Gen. 35, 7689 (2002) 19. Helstrom, C.W.: Quantum Detection and Estimation Theory. New York: Academic Press, 1976 20. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. Amsterdam: North-Holland, 1982, (originally published in Russian, 1980) 21. Hayashi, M: Quantum Information: An Introduction. Berlin: Springer, 2006 22. Hayashi, M., Matsumoto, K.: Statistical model with measurement degree of freedom and quantum physics. RIMS koukyuroku No 1055 (Kyoto: Kyoto University) p 96 (1998) (In Japanese); Hayashi, M., Matsumoto, K.: Asymptotic Theory of Quantum Statistical Inference. ed M Hayashi, Singapore: World Scientific, 2005, p. 162 (reprinted, English translation) 23. Gill, R., Massar, S.: State estimation for large ensembles. Phys. Rev. A 61, 042312 (2000) 24. Hayashi, M.: Quantum estimation and the quantum central limit theorem. In: Selected Papers on Probability and Statistics American Mathematical Society Translations Series 2, Vol. 277, Providence RI: Amer. Math. Soc., 2009, pp. 95–123. (It was originally published in Japanese in Bulletin of Mathematical Society of Japan, Sugaku, Vol. 55, No. 4, 368–391 (2003)) 25. Gu¸ta˘ , M., Kahn, J.: Local asymptotic normality for qubit states. Phys. Rev. A 73, 052108 (2006) 26. Gu¸ta˘ , M., Jencova, A.: Local asymptotic normality in quantum statistics. Commun. Math. Phys. 276, 341–379 (2007) 27. Gu¸ta˘ , M., Janssens, B., Kahn, J.: Optimal estimation of qubit states with continuous time measurements. Commun. Math. Phys. 277, 127–160 (2008) 28. Hayashi, M., Matsumoto, K.: Asymptotic performance of optimal state estimation in qubit system. J. Math. Physc. 49, 102101 (2008) 29. LeCam, L.: Asymptotic Methods in Statistical Decision Theory. New York: Springer, 1986 30. Fujiwara, A.: Strong consistency and asymptotic efficiency for adaptive quantum estimation problems. J. Phys. A: Math. Gen. 39, 12489 (2006) 31. Higgins, B.L., Berry, D.W., Bartlett, S.D., Wiseman, H.M., Pryde, G.J.: Entanglement-free Heisenberglimited phase estimation. Nature 450, 393–396 (2007) 32. Nagata, T., Okamoto, R., O’Brien, J.L., Sasaki, K., Takeuchi, S.: Beating the Standard Quantum Limit with Four-Entangled Photons. Science 316(5825), 726 (2007) 33. Okamoto, R., Hofmann, H.F., Nagata, T., O’Brien, J.L., Sasaki, K., Takeuchi, S.: Beating the standard quantum limit: phase super-sensitivity of N-photon interferometers. New J. Phys. 10, 073033 (2008) 34. Jones, J.A., Karlen, S.D., Fitzsimons, J., Ardavan, A., Benjamin, S.C., Briggs, G.A.D., Morton, J.J.L.: Magnetic Field Sensing Beyond the Standard Quantum Limit Using 10-Spin NOON States. Science 324, 1166–1168 (2009) 35. Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation, Graduate Studies in Mathematics 47. Providence. RI: Amer. Math. Soc., 2002 36. Giovannetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004) 37. Giovannetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced “Quantum metrology”. Phys. Rev. Lett. 96, 010401 (2006) 38. Hajek, J.: Local asymptotic minimax and admissibility in estimation. In: Proc. Sixth Berkeley Symp. on Math. Statist. and Prob., Vol. 1, Berkley, CA: Univ. of Calif. Press, 1972, pp. 175–194 39. Choi, M.-D.: Completely Positive Linear Maps on Complex Matrices. Lin. Alg. Appl. 10, 285–290 (1975) 40. Jamiolkowski, A.: Rep. Math. Phys. 3, 275 (1972) 41. Hayashi, M.: Phase estimation with photon number constraint. Prog. Inform. 8, 81–87 (2011) Communicated by M.B. Ruskai
Commun. Math. Phys. 304, 711–722 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1250-9
Communications in
Mathematical Physics
Phase Transitions with Four-Spin Interactions Joel L. Lebowitz1,2,3 , David Ruelle1,2 1 Math. Dept., Rutgers University, Piscataway, NJ 08844, USA. E-mail: [email protected] 2 IHES, 91440 Bures sur Yvette, France. E-mail: [email protected] 3 Physics Dept., Rutgers University, Piscataway, NJ 08844, USA
Received: 20 May 2010 / Accepted: 22 October 2010 Published online: 21 April 2011 – © Springer-Verlag 2011
In memory of Julius Borcea Abstract: Using an extended Lee-Yang theorem and GKS correlation inequalities, we prove, for a class of ferromagnetic multi-spin interactions, that they will have a phase transition (and spontaneous magnetization) if, and only if, the external field h = 0 (and the temperature is low enough). We also show the absence of phase transitions for some nonferromagnetic interactions. The FKG inequalities are shown to hold for a larger class of multi-spin interactions. 1. Introduction The mathematically best understood systems exhibiting phase transitions are Ising models with ferromagnetic interactions on ⊂ Zd . In addition to the special exactly solvable examples, e.g., the nearest neighbor Ising model on Z2 , “almost” everything is known qualitatively about the phase diagram and correlation functions of such systems. This is due to the Lee-Yang theorem [LY], the Asano-Ruelle lemma, correlation inequalities, and low-temperature analysis based on the (broken) symmetry of the ground state.1 In this note we extend some of these results to a class of systems with both pair and four spin interactions. In particular we show that the thermodynamic and correlation functions are analytic if the external magnetic field h does not vanish. Conversely, they will have, when the interactions are ferromagnetic, a phase transition when h = 0 and the temperature is low enough. There is also a class of interactions for which there are no phase transitions at any temperature. Our results are obtained by the use of an extended Lee-Yang theorem [R3] in combination with GKS inequalities. We also obtain FKG inequalities for systems with multispin 1 For a general introduction to rigorous equilibrium statistical mechanics, in particular lattice spin systems, see [R1,I,Sin,Sim]. The results needed here about Lee-Yang and Asano-Ruelle are contained in [R3] and summarized in Appendix A of the present paper, but there is a vast literature on the subject (C.M. Newman, E.H. Lieb, A.D. Sokal, J. Chayes, etc.); the reader may consult [BB1] and [BB2] for a different approach and a large list of references. For correlation inequalities see Sect. 8. For low-temperature analysis see [Sin,Sl] (Pirogov-Sinai theory) and also [HS].
712
J. L. Lebowitz, D. Ruelle
interactions. Finally, we prove Lee-Yang type ferromagnetic behavior for a large class of systems with multispin interactions at sufficiently low temperatures.
2. The model In this note we study classical spin systems such that a configuration σ = (σx )x∈ of spins σx = ±1 in a finite region has energy U (σ )− x h x σx , where h x is the magnetic field at x. It is convenient to use instead of the configuration σ the set X = {x : σx = +1} of “occupied” sites and to consider the partition function Z (z) =
X ⊂
e−βU X
x∈X
e2βh x =
EX zX,
(2.1)
X ⊂
where we have written U (σ ) = U X , E X = e−βU X , z x = e2βh x , z = (z x )x∈ , z X = x∈X z x . The spin-flip symmetry U (σ ) = U (−σ ) is now expressed by E X = E \X . It is known [R3] that in the 2||−1 -dimensional space of partition functions Z satisfying the spin-flip symmetry there is a nonempty open set such that the Lee-Yang property is satisfied: Z (z) = 0 if |z x | < 1 for all x. 2 Furthermore if the Lee-Yang property is satisfied at high temperature, it is satisfied at all temperatures, and the energy is defined by a ferromagnetic pair interaction between spins (see [R3] Theorem 9). In what follows we shall discuss a specific class of models with 4-spin interactions, for which the Lee-Yang property is violated at high temperatures, but such that, at any temperature for which the zeros of Z (z, . . . , z) come close to the positive real axis, the Lee-Yang property holds. In the thermodynamic limit therefore, a phase transition can only occur at zero magnetic field, and examples exist where spontaneous magnetization does indeed occur. The methods used for the proofs will be generalized Lee-Yang theory (with the Asano-Ruelle lemma), and correlation inequalities. Let us expand on what is meant here by absence of phase transition. Consider a bounded finite-range perturbation λV X of the energy U X in (2.2) [i.e., V X = 0 unless diameter X < A and |V X | < B for all X , translation invariance is not required]. We write λ Z (z) = e−β(U X +λVX ) z X . X ⊂
λ (z) [see in Then, for small λ, the Asano-Ruelle lemma gives regions free of zeros for Z particular [LR] in this respect]. Thus, if suitable translation invariance or periodicity is assumed for V , the free energy is analytic in λ, except in the ferromagnetic case, close to h = 0, at low temperature. A concrete version of the models to be discussed is obtained from a planar square Ising model by adding, on alternate distinguished squares, diagonal interactions of the same strength as the interactions on the sides, and a 4-spin interaction (marked by a circle in the figure below). Note that only two distinguished squares meet at each lattice vertex. This will be important for the proof of (i) in Sect. 7. The proof would fail if there were 4-spin interactions on each square, not just alternate squares. 2 The spin-flip symmetry implies that Z (z) = 0 also if |z | > 1 for all x, and therefore we have the x Circle Theorem: all zeros of Z (z, . . . , z) are on the unit circle {z : |z| = 1}.
Phase Transitions with Four-Spin Interactions
713
The energy of a spin configuration σ is thus a sum over distinguished sqares α with four spins σ1α , σ2α , σ3α , σ4α , of contributions of the form σiα σ jα − J4 σ1α σ2α σ3α σ4α . (2.2) U α = −J2 i< j
The partition function Z of a union of distinguished squares is obtained from the product of the partition functions Z α (z 1α , . . . , z 4α ) of the component distinguished squares α by a succession of steps = x0 → · · · → x → · · · → Z . Each step involves a vertex x of as follows: 1) If x coincides with the vertex i of a single distinguished square α, replace in x the corresponding z iα by z x . β 2) If x belongs to the vertices i, j of two squares α and β, write x = A + Bz iα + C z j + β
Dz iα z j , and replace x by A + Dz x (Asano contraction). It is easy to see that, when all variables z iα have been eliminated in favor of the z x (the order in which this is done is unimportant) one obtains Z . The interest of Asano contractions is that if one knows that A + Bu + Cv + Duv = 0 when u ∈ / K, v ∈ / L, where K , L are closed subsets of C disjoint from 0, then A+ Dz = 0 when z ∈ / −K L = {−uv : u ∈ K , v ∈ L} (see Lemma 1 of Appendix). Also, because the partition function Z α of a distinguished square is symmetric, one knows that Z (u 1 , u 2 , u 3 , u 4 ) = 0 if u 1 , u 2 , u 3 , u 4 ∈ / K , where K is any closed circular region containing the 4 zeros of Z α (u, u, u, u) (see Lemma 2 of Appendix: Grace’s theorem). 3. Main Results For the model described above, where J2 ∈ R is arbitrary and J4 ≥ 0, either or both of the following properties hold: (i) there is a neighborhood (independent of ) of the positive real axis {z ∈ C : Imz = 0 and Rez ≥ 0} which is free of zeros of Z (z, . . . , z), (ii) Z satisfies the Lee-Yang property.
714
J. L. Lebowitz, D. Ruelle
Therefore there can be no phase transition except at zero magnetic field. Furthermore, if J2 > 0, J4 ≥ 0, a phase transition actually occurs at low temperature. If J4 > 0, and contains at least one distinguished square, the Lee-Yang property for Z is actually violated at sufficiently high temperature. We also find that there is no phase transition when β and J4 ≥ 0 satisfy β J2 ≤ (log 3)/8; this condition always holds if J2 ≤ 0. 4. Remark: Other Models In order to make Z α (z 1α , . . . , z 4α ) a symmetric function, we have made the contrived assumption that the diagonal interactions in a distinguished square have the same strength as the other interactions. One can actually still obtain the same results as above under more natural conditions3 but the proof is more acrobatic, and left to Appendix B. One can also, instead of a planar system, consider a 3-dimensional model where the spins sit at the middle of the bonds of a diamond lattice. A distinguished square is now replaced by a regular tetrahedron, the four spins at the vertices are in a completely symmetric situation, and the above results remain precisely the same in this new situation. Our general strategy, which could certainly be used for other multiple spin interactions is as follows: (i) using the properties of Asano contractions show that, at high temperature, the partition function Z has no zeros close to the real axis, (ii) show that at low temperature Z satisfies the Lee-Yang property, (iii) use correlation inequalities to prove that a ferromagnetic phase transition is actually present at low temperature.
5. Partition Function of a Distinguished Square Let the energy of a set of 4 spins be given by the symmetric expression
U (σ1 , . . . , σ4 ) = −K 0 − K 2
1≤i< j≤4
[
1 + σi 1 + σ j 1 − σi 1 − σ j · + · ] 2 2 2 2
1 + σ1 1 + σ2 1 + σ3 1 + σ4 1 − σ1 1 − σ2 1 − σ3 1 − σ4 −K 4 [ · · · + · · · ] 2 2 2 2 2 2 2 2 1 1 = −K 0 − K 2 [1 + σi σ j ] − K 4 [1 + σi σ j + σ1 σ2 σ3 σ4 ] 2 8 i< j i< j σi σ j − J4 σ1 σ2 σ3 σ4 (5.1) = −J0 − J2 i< j
with J4 = K 4 /8, J2 = K 4 /8 + K 2 /2, J0 = K 4 /8 + K 2 /2 + K 0 . (K 0 , J0 are unimportant additive constants.) It will be convenient to take K 0 = −3K 2 . We may write Z (z 1 , z 2 , z 3 , z 4 ) =
σ1 =±1
···
σ4 =±1
exp[−β(U (σ1 , . . . , σ4 ) +
h i (1 + σi ))]
i
= bz 1 z 2 z 3 z 4 + (z 2 z 3 z 4 + z 1 z 3 z 4 + z 1 z 2 z 4 + z 1 z 2 z 3 ) + a(z 1 z 2 +z 1 z 3 +z 1 z 4 + z 2 z 3 + z 2 z 4 + z 3 z 4 ) + (z 1 + z 2 + z 3 + z 4 ) + b, 3 Using the notation of Sect. 5 below, we replace the K diagonal interactions by 0 (this leaves J = K /8), 2 2 4 and we assume K 2 , K 4 ≥ 0.
Phase Transitions with Four-Spin Interactions
715
where z i = e2βh i , a = e−β K 2 , b = eβ(K 4 +3K 2 ) . The symmetric polynomial Z is entirely determined by Pab (z) = Z (z, . . . , z) = bz 4 + 4z 3 + 6az 2 + 4z + b, and one can apply Grace’s theorem to obtain properties of the zeros of Z (z 1 , . . . , z 4 ) if one knows the zeros of Pab . 6. Proposition. (the zeros of the partition function Pab ). Let a, b be real > 0. • Condition (i): the zeros of Pab are in the open left-hand plane {z : Rez < 0} is implied by 3a − b > 0 (i.e., e8β J2 < 3). • Condition (ii): the zeros of Pab are on the unit circle {z : |z| = 1} is implied by (1 ≤ b, and 3a ≤ b + 2/b, and 4 ≤ 3a + b). • The condition 4 ≤ 3a + b implies (i) or (ii). This condition, i.e., 4e2β(J2 −J4 ) ≤ 3 + e8β J2 is satisfied in particular by J4 ≥ 0, all J2 . We have P11 (z) = (z+1)4 , so that (i) holds for a = b = 1. Since b > 0, the zeros of Pab will remain in the left-hand plane until they cross the imaginary axis {z : Rez = 0}. This will happen when Pab (i y) = 0 with real y, i.e., when by 4 −4i y 3 −6ay 2 +4i y +b = 0, or by 4 − 6ay 2 + b = 0
and
− y 3 + y = 0.
(6.1)
The second condition (6.1) is satisfied for y = 0, or ±1, so that the first condition gives b = 0, or 2b − 6a = 0. Since b > 0, the crossing occurs at b = 3a, so that the zeros of Pab remain in {z : Rez < 0} for b < 3a. Since we have z −2 Pab (z) = b(z 2 + 1/z 2 ) + 4(z + 1/z) + 6a = b(z + 1/z)2 + 4(z + 1/z) + 6a − 2b, we may write z −2 Pab (z) = 4Rab (ζ ), with 2ζ = z + 1/z and Rab (ζ ) = bζ 2 + 2ζ + (3a − b)/2. √ Since Rab (ζ ) = 0 corresponds to ζ = (−1± 1 − b(3a − b)/2 )/b, assuming that b ≥ 1, and 2−b(3a −b) ≥ 0, and Rab (−1) ≥ 0, implies that the roots of Rab are real ∈ [−1, 1). Therefore the roots of Pab are on the unit circle provided b ≥ 1, 3a ≤ b + 2/b and 4 ≤ 3a + b. Assume 4 ≤ 3a + b. If 3a − b > 0, then (i) holds. Otherwise 3a ≤ b which, together with 4 ≤ 3a + b, implies 2 ≤ b, hence 1 ≤ b. Also 3a ≤ b + 2/b is implied by 3a ≤ b, so that (ii) follows. We have thus shown that 4 ≤ 3a + b implies (i) or (ii) (or both (i) and (ii)). Let g(J2 , J4 ) = 3 − 4e2β(J2 −J4 ) + e8β J2 . Then g(0, 0) = 0, and ∂ J2 g(J2 , 0) = 8β(e8β J2 − e2β J2 ) is > 0 if J2 > 0, and < 0 if J2 < 0. Hence g(J2 , 0) ≥ 0 for all J2 . Since g(J2 , J4 ) ≥ g(J2 , 0) when J4 ≥ 0, we have g(J2 , J4 ) ≥ 0 if J4 ≥ 0, for all J2 .
716
J. L. Lebowitz, D. Ruelle
7. The Zeros of the General Partition Function Z Let be a union of distinguished squares α. As noted in Sect. 2, Z is obtained from the product of the partition functions Z α by some relabelings z iα → z x , and by β Asano contractions (z iα , z j ) → z x . According to Proposition 6 we have = 0 when α zi ∈ / K ⊂ C with different choices (depending on a, b) for the closed set K 0. (i) If 3a − b > 0 we may choose K = {z : Rez < − } for some > 0, and the complement of −K K is a neighborhood of the positive real axis (the inside of a parabola). Therefore Z = 0 when all z x are close to the positive real axis. (ii) If 1 ≤ b, 3a ≤ b + 2/b, and 4 ≤ 3a + b, we may choose K = {z : |z| ≥ 1}, so that −K K is again {z : |z| ≥ 1}. Therefore Z = 0 when all z x are in {z : |z| < 1}, and also, by spin-flip symmetry, when all z x are in {z : |z| > 1}. 8. GKS and FKG Inequalities for Multispin Interactions Consider the energy function H (σ ) = −
JAσ A,
A⊂
where σ A = x∈A σx . We denote by · · · the expectation value which gives the spin −1 configuration σ the probability μ(σ ) = Z exp(−β H (σ )). Assume J X ≥ 0 for all X ⊂ (in particular h x = J{x} ≥ 0 for all x ∈ ). The GKS inequalities then hold:
σ A ≥ 0,
β −1 ∂ σ A /∂ J B = σ A σ B − σ A σ B ≥ 0
for all A, B ⊂ . [See for instance [G] for a proof]. Let now H (σ ) = U (σ ) −
h x σx
x∈
with U = U + U : U (σ ) = −
A⊂
K A
1 + σx 1 − σx , U (σ ) = − . K A 2 2
x∈A
A⊂
x∈A
(We may take K A , K A = 0 if |A| = 1.) We write μ(σ ) = μ X for the probability associated with the spin configuration σ corresponding to the set X = {x : σx = +1} of “occupied” sites. Note that the spin-flip symmetric situation occurs when K A = K A for all A with |A| > 2. The |A| = 2 case is always symmetric, aside from the one-body terms which can be absorbed in the {h x }. Assume that K A , K A ≥ 0 for all A ⊂ . The FKG inequalities then hold, i.e., if σ → f (σ ), g(σ ) are nondecreasing functions, then μ( f g) ≥ μ( f )μ(g) [FKJ].
Phase Transitions with Four-Spin Interactions
717
The proof is based on checking Holley’s criteria for FKG [H]: μ X ∪Y · μ X ∩Y ≥ μ X · μY for all X, Y ⊂ . The h x drop out of the formula. Writing U X instead of U (σ ) we have to check that −U X ∪Y − U X ∩Y ≥ −U X − UY for U = U +U , where U X = − A⊂X K A , and U X = − A⊂\X K A . Define 1 X (A) = 1 if A ⊂ X, = 0 otherwise. It suffices to prove
K A (1 X ∪Y (A) + 1 X ∩Y (A)) ≥
A
K A (1 X (A) + 1Y (A))
A
and similarly with X, Y replaced by \X, \Y , and K A by K A . Since K A , K A ≥ 0, it suffices to show that 1 X ∪Y (A) + 1 X ∩Y (A) ≥ 1 X (A) + 1Y (A) or 1 X ∪Y (A) − 1 X (A) ≥ 1Y (A) − 1 X ∩Y (A) This is true because the left-hand side can vanish only if A ⊂ X ∪ Y or if A ⊂ X , and this implies that the right-hand side also vanishes. The combination of GKS and FKG inequalities, which both hold in the ferromagnetic examples considered, yield much information about the equilibrium properties of the system (see [L1,L2]). It shows in particular that the equilibrium states are linear combinations of those obtained from all-plus and all-minus boundary conditions when the free energy is differentiable in β.
9. Proof of the Main Results Most of the main results stated in Sect. 3 have already been verified in Sect. 7. We now have to check that a phase transition actually occurs at low temperature if J2 > 0, J4 ≥ 0. Putting J4 and the diagonal interactions on distinguished squares = 0, we recover, if J2 > 0, the standard ferromagnetic Ising model: this has spontaneous magnetization at high β. The GKS inequality (Sect. 8) shows that the spontaneous magnetization persists when diagonal interactions on distinguished squares are introduced, and a 4-spin interaction J4 ≥ 0 is added. Finally, if the 4-spin interaction is actually present, i.e., contains at least one distinguished square, and J4 = 0, the Lee-Yang property is actually violated at sufficiently high temperature, as proved in [R3] Theorem 9. We shall now give a sufficient condition such that the Lee-Yang property holds at low temperature for multispin interactions.
718
J. L. Lebowitz, D. Ruelle
10. Proposition. Define the partition function Z (z) associated with a finite set by [exp(−βU X )]z X , Z (z) = X ⊂
where U X = U (σ ) = −
A⊂
K A(
1 + σx 1 − σx KA + K A) + ) = −( 2 2
x∈A
x∈A
A⊂X
A⊂\X
so that we have the spin-flip symmetry U X = U\X . Suppose that for each A ⊂ either K A = 0 or β K A ≥ (|A| − 1) log 2, then the Lee-Yang property is satisfied: Z (z) = 0 if |z x | < 1 for all x ∈ . [For |A| = 2, 3 it suffices to assume β K A ≥ 0, and for |A| = 4 it suffices to assume β K A ≥ log 2.] [Note that this is quite general, and applies in particular to a version of the models in Sects. 2 and 4 having 4-spin interactions on all squares.] The Lee-Yang property is preserved by the product of multiaffine polynomials in disjoint sets of variables, and by Asano contraction (see Lemma 9 in the Appendix below). Therefore, it suffices to prove the Lee-Yang property for the energy function 1 + σx 1 − σx U (A) = −K A ( + ) 2 2 x∈A
x∈A
defined on spin configurations in A. We have thus to study the zeros of (A) Z (A) = [exp(−βU X )]z X = (1 + z x ) + (eβ K A − 1)(1 + zx ) X ⊂A
x∈A
x∈A
which is a symmetric function of the z x , x ∈ A. Using the characterization of Lee-Yang polynomials given in [R3] Theorem 3, and Grace’s Theorem (Theorem 9 in the Appendix below), one finds (see [R3] Remark 4(a)) that Z (A) satisfies the Lee-Yang property provided (1 + z)|A|−1 + (eβ K A − 1) does not vanish when |z| < 1. For |A| > 2, the largest negative number of the form (1 + z)|A|−1 , with |z| ≤ 1, is obtained when arg(1 + z) = π/(|A| − 1), or z = exp(2iπ/ (|A| − 1)). Thus, for |z| ≤ 1, (1 + z)|A|−1 ≥ (1 + exp(
2iπ ))|A|−1 > 1 − 2|A|−1 . |A| − 1
Therefore the Lee-Yang property of Z (A) is ensured by exp(β K A ) − 1 ≥ 2|A|−1 − 1, i.e., β K A ≥ (|A| − 1) log 2. This was our assertion for general |A|. If |A| = 2, it suffices to take K A ≥ 0, because ferromagnetic pair interactions imply the Lee-Yang property. The case |A| = 3 reduces to the case |A| = 2 because 1 1 + σ1 1 + σ2 1 + σ3 1 − σ1 1 − σ2 1 − σ3 · · + · · = [σ1 σ2 + σ2 σ3 + σ3 σ1 + 1] 2 2 2 2 2 2 4 1 1 + σ1 1 + σ2 1 − σ1 1 − σ2 = [( · + · ) + permutations − 1]. 2 2 2 2 2 The case |A| = 4 results from our analysis in Proposition 6.
Phase Transitions with Four-Spin Interactions
719
Acknowledgements. JLL thanks the IHES for its hospitality. We are indebted to Joseph Slawny for helpful comments, and DR thanks Jürg Fröhlich for a useful conversation. The work of JLL was supported in part by NSF grant DMR-0802120 and AFOSR grant AF-FA 9550-07.
A. Appendix: Basic General Results on Multiaffine Polynomials4 A1. Lemma (Asano-Ruelle) [A,R2]. Let K 1 , K 2 be closed subsets of C, with K 1 , K 2 0. If is separately linear in z 1 and z 2 , and if (z 1 , z 2 ) ≡ A + Bz 1 + C z 2 + Dz 1 z 2 = 0 whenever z 1 ∈ / K 1 and z 2 ∈ / K 2 , then ˜ (z) ≡ A + Dz = 0 whenever z ∈ / −K 1 · K 2 . [We have written −K 1 · K 2 = {−uv : u ∈ K 1 , v ∈ K 2 }.] ˜ is called Asano contraction. The map → A2. Theorem (Grace’s theorem). Let P be a complex polynomial of degree n in one variable and be the only polynomial symmetric in its n arguments, separately linear in each, such that (z, . . . , z) = P(z). If the n roots of P are contained in a closed circular region K and z 1 ∈ / K , . . . , zk ∈ / K, then (z 1 , . . . , z n ) = 0. A closed circular region is a closed subset K of C bounded by a circle or a straight line. We allow the coefficients of z n , z n−1 , . . . in P to vanish: we then say that some of the roots of P are at ∞, and we take K noncompact. For a proof see Polya and Szegö [PSz] V, Exercise 145. B. Appendix: Model Without Diagonal Interactions The present model corresponds to the energy (5.1) in which the “diagonal” pairs 1 < 3, 2 < 4 have been removed from the sum over i < j. The partition function of the system of 4 spins on a distinguished square, with vertices labeled consecutively 1,2,3,4, thus has the form Z (z 1 , z 2 , z 3 , z 4 ) = e2β K 2 [eβ(K 4 +2K 2 ) + (z 1 + z 2 + z 3 + z 4 ) + (z 1 z 2 + z 2 z 3 + z 3 z 4 + z 4 z 1 ) + e−2β K 2 (z 1 z 3 + z 2 z 4 ) + (z 1 z 2 z 3 + z 2 z 3 z 4 + z 3 z 4 z 1 + z 4 z 1 z 2 ) + eβ(K 4 +2K 2 ) z 1 z 2 z 3 z 4 ]. For this model we shall prove that if K 2 , K 4 ≥ 0, either or both of the following properties hold: (i) there is a neighborhood (independent of ) of the positive real axis which is free of zeros of Z (z . . . , z), (ii) Z satisfies the Lee-Yang property. (Case (i) occurs at high temperatures.) 4 See the Appendix of [R3] for more details, and [BB1,BB2] for a much more general approach, with many references.
720
J. L. Lebowitz, D. Ruelle
Note that Z (z 1 , z 2 , z 3 , z 4 ) is separately symmetric in z 1 , z 3 , and z 2 , z 4 . Using Grace’s Theorem, we can analyze the zeros of Z in terms of the zeros of P(u, v) = b + 2(u + v) + 4uv + a(u 2 + v 2 ) + 2(u + v)uv + bu 2 v 2 = b(1 + u 2 v 2 ) + 2(u + v)(1 + uv) + 4uv + a(u 2 + v 2 ), where a = e−2β K 2 , b = eβ(K 4 +2K 2 ) . Note that the case K 2 = 0 has already been covered in Sect. 6, so that we can take K 2 > 0. Also, since K 4 = 0 corresponds to the classical Ising model, we can take K 4 > 0. We have thus to consider only the situations where K 2 , K 4 > 0, i.e., 0 < a < 1, and ab > 1, which implies a + b > a + 1/a ≥ 2. In fact, the only assumptions we shall use in what follows are 0 < a < 1 and 0 < a + b − 2. (i) We prove that if |b − a| < 2, then P(u, v) = 0 when Re u, Re v ≥ 0 (this requires only a, b > 0). For a = b = 1, we have P(u, v) = (1 + u)2 (1 + v)2 . One can see that for the zeros (u, v) of P to reach the region Re u, Re v ≥ 0, they must intersect Re u = Re v = 0. We have (with real x, y): P(i x, i y) = b(1 + x 2 y 2 ) + 2i(x + y)(1 − x y) − 4x y − a(x 2 + y 2 ) and P(i x, i y) = 0 yields either (a) or (b): 2 (a): x + y = 0 and b(1 + x 4 ) + (4 − 2a)x 2 = 0, or x 4 − 2( a−2 b )x + 1 = 0, hence a−2 b ≥ 1, i.e., a − b ≥ 2, b−2 2 (b): x y = 1 and 2b − 4 − a(x 2 + 1/x 2 ) = 0, or x 4 − 2( b−2 a )x + 1 = 0, hence a ≥ 1, i.e., b − a ≥ 2.
Therefore, if |b − a| < 2 and P(u, v) = 0, we cannot have Re u ≥ 0 and Re v ≥ 0. There is thus > 0 such that, writing L = {z : Re z ≤ − }, we have z 1 , z 2 , z 3 , z 4 ∈ L
Z (z 1 , z 2 , z 3 , z 4 ) = 0.
implies
(ii) The Lee-Yang property for Z is equivalent to b+(z 1 +z 2 +z 3 )+(z 1 z 2 +z 2 z 3 )+az 1 z 3 +z 1 z 2 z 3 = 0 if |z 1 |, |z 2 |, |z 3 | < 1 by [R3] Theorem 3, or to u(1 + v)2 + b + 2v + av 2 = 0
if |u|, |v| < 1
(B.1)
by Grace’s Theorem. Since a + b − 2 > 0, we may define λ = (ab − 1)/(a + b − 2), so that (1 − λ)2 = (b − λ)(a − λ) and a − λ = (1 − a)2 /(a + b − 2) > 0. Since 1 − a > 0, the above condition (B.1) may now be rewritten as (u + λ)(1 + v)2 + (a − λ)(v 2 + 2
1−λ 2 1−λ v+( ) ) = 0 a−λ a−λ
or v +
u + λ + (a − λ)
1−λ a−λ
v+1
2
= 0
for |u|, |v| < 1
which is equivalent to u + λ + (a − λ)w2 = 0
for |u| < 1, Re w >
1 2
(1 +
1 − λ 2 ) a−λ
(B.2)
Phase Transitions with Four-Spin Interactions
721
1−λ because v → (v + a−λ )/(v + 1) maps the disk {v : |v| < 1} to the half-plane {w : 1 1−λ Re w > 2 (1 + a−λ )}. 1−λ 1−λ The boundary of {w 2 : Re w > 21 (1 + a−λ )} is a parabola {[ 21 (1 + a−λ ) + it]2 : t ∈ R}. Therefore we want
1 1−λ u + λ + (a − λ)[ (1 + ) + it]2 = 0 2 a−λ or 1 1 1 u + λ + (a − λ) + (1 − λ) + (b − λ) + (a − λ + 1 − λ)it − (a − λ)t 2 = 0 4 2 4 or 1 u + (a + b + 2) + (1 + a − 2λ)it − (a − λ)t 2 = 0 4
for |u| < 1, t ∈ R.
(B.3)
Since a + b + 2 > 0, one checks that (B.2) is equivalent to (B.3). We rewrite (B.3) as u + C + Bit − At 2 = 0
if
|u| < 1, t ∈ R
(B.4)
with (1 − a)2 , a+b−2 (1 − a)(b − a) B = 1 + a − 2λ = , a+b−2 1 C = (a + b + 2). 4 A = a−λ=
Equation (B.4) means that the distance of 0 to the parabola C + Bit − At 2 is ≥ 1. For the closest point to 0 of the parabola, we have −2 At (C − At 2 ) + B 2 t = 0, i.e., t = 0, or t 2 = C/A − B 2 /2 A2 if this quantity is > 0. This gives either dist = C > 1, because a + b > 2, or if t 2 = C/A − B 2 /2 A2 > 0: B2 2 B2 C 2 C − − )) + B ( ) A 2 A2 A 2 A2 B2 B2 B4 C C = + B2( − ) = B2( − ). 2 2 4A A 2A A 4 A2
dist 2 = (C − A(
In this last case dist 2 ≥ 1 is equivalent to B4 C + 1 ≤ B2, 2 4A A but since we have C/A − B 2 /2 A2 > 0, hence (C/A)B 2 > B 4 /2 A2 , it suffices to check that B 4 /4 A2 + 1 ≤ B 4 /2 A4 , i.e., 1 ≤ B 4 /4 A2 , or 1≤
(b − a)2 B2 = 2A 2(a + b − 2)
or (b − a)2 ≥ 2(a + b − 2), or (b − a − 1)2 ≥ 4a − 3.
722
J. L. Lebowitz, D. Ruelle
We have thus proved that the Lee-Yang property holds under the assumptions that 0 < a < 1, 0 < a + b − 2, and (b − a − 1)2 ≥ 4a − 3, but either (i) holds, or b − a ≥ 2 , i.e., b − a − 1 ≥ 1, hence (b − a − 1)2 ≥ 4a − 3. We have thus shown that, under the assumptions K 2 > 0, K 4 > 0, which imply 0 < a < 1, 0 < a + b − 2, we have either (i) or (ii) or both, as announced. References [A] [BB1] [BB2] [FKG] [G] [H] [HS] [I] [L1] [L2] [LR] [LY] [PSz] [R1] [R2] [R3] [Sim] [Sin] [Sl]
Asano, T.: Theorems on the partition functions of the heisenberg ferromagnets. J. Phys. Soc. Jap. 29, 350–359 (1970) Borcea, J., Brändén, P.: The lee-yang and polya-schur programs. i. linear operators preserving stability. Invent. Math. 177, 541–569 (2009) Borcea, J., Brändén, P.: The lee-yang and polya-schur programs. ii. theory of stable polynomials and applications. Commun. Pure Appl. Math. 62, 1595–1631 (2009) Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103 (1971) Ginibre, J.: General formulation of the griffiths’ inequalities. Commun. Math. Phys. 16, 310–328 (1970) Holley, R.: Remarks on the fkg inequalities. Commun. Math. Phys. 36, 227–231 (1974) Holsztynski, W., Slawny, W.: Phase transitions in ferromagnetic spin systems at low temperatures. Commun. Math. Phys. 66, 147–166 (1979) Israel, R.B.: Convexity in the theory of lattice gases. Princeton, NJ: Princeton U.P., 1979 Lebowitz, J.L.: GHS and other inequalities. Commun. Math. Phys. 35, 87–92 (1974) Lebowitz, J.L.: Coexistence of phases in ising ferromagnets. J. Stat. Phys. 16, 463–476 (1977) Lieb, E.H., Ruelle, D.: A property of zeros of the partition function for ising spin systems. J. Math. Phys. 13, 781–784 (1972) Lee, T.D., Yang, C.N.: Statistical theory of equations of state and phase relations. ii. lattice gas and ising model. Phys. Rev. 87, 410–419 (1952) Polya, G., Szegö, G.: Problems and theorems in analysis II. Berlin: Springer, 1976 Ruelle, D.: Statistical Mechanics. Rigorous Results. New York: Benjamin, 1969. (Reprint: London: Imperial College Press/Singapore: World Scientific 1999) Ruelle, D.: Extension of the lee-yang circle theorem. Phys. Rev. Lett. 26, 303–304 (1971) Ruelle, D.: Characterization of lee-yang polynomials. Ann. of Math. 171, 589–603 (2010) Simon, B.: The Statistical Mechanics of Lattice Gases. Vol. I, Princeton, NJ: Princeton U.P., 1993 Sinai, Ya.G.: Theory of Phase Transitions: Rigorous Results. Oxford: Pergamon, 1982 Slawny, J.: “Low-temperature properties of classical lattice systems: phase transitions and phase diagrams”. In: Phase Transitions and Critical Phenomena. Vol. 11, C. Domb, J.L. Lebowitz, eds., London: Academic Press, 1987
Communicated by H. Spohn
Commun. Math. Phys. 304, 723–763 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1240-y
Communications in
Mathematical Physics
Local Statistics of Realizable Vertex Models Zhongyang Li Department of Mathematics, Brown University, 151 Thayer St., Providence, RI 02912-1917, USA. E-mail: [email protected] Received: 4 June 2010 / Accepted: 26 November 2010 Published online: 24 April 2011 – © Springer-Verlag 2011
Abstract: We study planar “vertex” models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including the dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an n × n torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics.
1. Introduction A vertex model is a graph G = (V, E), where we associate to each vertex v ∈ V a signature rv . A local configuration at a vertex v is a subset of incident edges of v. A configuration of the graph G is an edge subset of G. The signature rv at a vertex v is a function which assigns a nonnegative real number (weight) to each local configuration at v. The partition function of the vertex model is the weighted sum of configurations X ∈ {0, 1} E , where the weight of a configuration is the product of weights of local configurations, obtained by restricting the configuration at each vertex. Dimers, loop models, and random tiling models are some special examples of vertex models. Direct computations of the partition function of a general vertex model usually require exponential time. On the other hand, using the Fisher-Kasteleyn-Temperley method [4,5], we can efficiently count the number of perfect matchings (dimer configurations) of a finite planar graph. The idea of generalized holographic reduction is to reduce a vertex model on a planar graph to a dimer model on another planar graph, essentially by a linear base change, see Sect. 3. For an original version of the holographic reduction (Valiant’s Algorithm), see [16].
724
Z. Li
However, not all satisfying assignment problems can be reduced to a perfect matching problem (“realized”) using the holographic algorithm. We study the realizability problem of the generalized algorithm for vertex models on the hexagonal lattice and prove that the signature of realizable models form a submanifold of positive codimension of the manifold of all signatures, see Theorem 3.6. An example of realizable models is the 1-2 model, which is a signature on the honeycomb lattice, only one or two edges allowed to be present in each local configuration, see Fig. 13, 14. The realizability problem of Valiant’s algorithm is studied by Cai [2], and the realizability problem of the uniform 1-2 model (not-all-equal relation), a special 1-2 model which assigns all the configurations weight 1, under Valiant’s algorithm is studied by Schwartz and Bruck [14]. Realizable vertex models may be reduced to dimer models in more than one way, that is, using different bases. However, all the dimer models corresponding to the same vertex model are shown to be gauge equivalent, i.e. obtained from one another by a trivial reweighting. One of the simplest vertex configuration models is a graph with the same signature at all vertices. Using the singular value decomposition, we prove that such models on a hexagonal lattice are realizable if and only if they are realizable under orthogonal base change. Moreover, the orthogonal realizability condition takes a very nice form; see Sect. 3.2. We compute the local statistics of realizable vertex models on a hexagonal lattice with the help of the generalized holographic reduction, i.e. for the natural probability measure, we compute the probabilities of given configurations at finitely many fixed vertices, which are proved to be computable by sums of finitely many Pfaffians, see Theorem 10 and Theorem 11. The weak limit of probability measures of the vertex model on finite graphs are of considerable interest. Using an n ×n torus to approximate the infinite periodic graph, we give an explicit integral formula for the probability of a specific local configuration at a fixed vertex, see Sect. 6. These results follow from a study of the zeros of the characteristic polynomial, or the spectral curve, on the unit torus T2 . For a more general result about the intersection of the spectral curve with T2 , see [13]. For example, using our method, we compute the probability that a {001} dimer occurs and for the uniform 1-2 model, and the probability that a {011} configuration occurs at a vertex for the critical 1-2 model, see Examples 4 and 5. The main result of this paper can be stated in the following theorems Theorem 1. For a periodic, realizable, positive-weight vertex model on a hexagonal lattice G with period 1 × 1, assume the corresponding Fisher graph has positive edge weights, then the free energy of G is 1 1 dz dw F := lim 2 log S(G n ) = , log P(z, w) n→∞ n 8π 2 i z iw |z|=1,|w|=1 where G n is the quotient graph G/(nZ × nZ), P(z, w) is the characteristic polynomial. Theorem 2. Assume the periodic vertex model on a hexagonal lattice is realizable to the dimer model on a Fisher graph with positive, periodic edge weights, and assume the entries of the corresponding base change matrices are nonzero. Let λn be the probability measure defined for configurations on a toroidal hexagonal lattice G n . Then for a configuration c at vertices v, p −1 [ wd j ]|Pf(K ∞ )V (d j ) |, lim λn (c, v) =
n→∞
dj
i=1
Local Statistics of Realizable Vertex Models
725
Fig. 1. Relation at a Vertex
where −1 ((u, x1 , y1 ), (v, x2 , y2 )) = K∞
1 4π 2
T2
z x1 −x2 w y1 −y2
Cof(K (z, w))u,v dz dw , P(z, w) i z iw
d j are local dimer configurations on the gadget of the Fisher graph corresponding to v. V (d j ) is the set of vertices involved in the configuration d j . 2. Background 2.1. Vertex models. Let {0, 1}k denote the set of all binary sequences of length k. A vertex model is a graph G = (V, E), where we associate to each vertex v ∈ V a function rv : {0, 1}deg(v) → R+ . rv is called the signature of the vertex model at vertex v. We give a linear ordering on the edges adjacent to v, and we fix such an ordering around each vertex once and for all. This way the binary sequences of length deg(v) are in one-to-one correspondence with the local configurations at v. Each edge corresponds to a digit; if that edge is included in the configuration, the corresponding digit is 1, otherwise the corresponding digit is 0. Hence we can also consider rv as a column vector indexed by local configurations at v: ⎛
⎞ rv (0...00) ⎜ rv (0...01) ⎟ ⎜ ⎟ rv = ⎜ rv (0...10) ⎟. ⎝ ⎠ ... rv (1...11) Example 1 (Signature of the vertex model at a vertex). Assume we have a degree-2 vertex with signature ⎛
⎞ ⎛ ⎞ rv (00) α ⎜ rv (01) ⎟ ⎜ β ⎟ rv = ⎝ = rv (10) ⎠ ⎝ γ ⎠ rv (11) δ that means we give weights to the four different local configurations as in Fig. 1: Assume G is a finite graph. We define a probability measure with sample space the set of all configurations, = {0, 1}|E| . The probability of a specific configuration R is λ(R) =
1 rv (R). S v∈V
(1)
726
Z. Li
The product is over all vertices. rv (R) is the weight of the local configuration obtained by restricting R to the vertex v, and S is a normalizing constant called the partition function for vertex models, defined to be S= rv (R). R∈ v∈V
The sum is over all possible configurations of G. Now we consider a vertex model on a Z2 -periodic planar graph G. By this we mean that G is embedded in the plane so that translations in Z2 act by signature-preserving isomorphisms of G. Examples of such graphs are the square and Fisher lattices, as shown in Fig. 7. Let G n be the quotient of G by the action of nZ2 . It is a finite graph embedded into a torus. Let Sn be the partition function of the vertex model on G n . The free energy of the infinite periodic vertex model G is defined to be F := lim
n→∞
1 log Sn . n2
2.2. Perfect matching. For more information, see [6]. A perfect matching, or a dimer cover, of a graph is a collection of edges with the property that each vertex is incident to exactly one edge. A graph is bipartite if the vertices can be 2-colored, that is, colored black and white so that black vertices are adjacent only to white vertices and vice versa. To a weighted finite graph G = (V, E, W ), the weight W : E → R+ is a function from the set of edges to positive real numbers. We define a probability measure, called the Boltzmann measure μ with sample space the set of dimer covers. Namely, for a dimer cover D, 1 μ(D) = W (e), Z e∈D
where the product is over all edges present in D, and Z is a normalizing constant called the partition function for dimer models, defined to be Z= W (e), D e∈D
the sum over all dimer configurations of G. If we change the weight function W by multiplying the edge weights of all edges incident to a single vertex v by the same constant, the probability measure defined above does not change. So we define two weight functions W, W to be gauge equivalent if one can be obtained from the other by a sequence of such multiplications. The key objects used to obtain explicit expressions for the dimer model are Kasteleyn matrices. They are weighted, oriented adjacency matrices of the graph G defined as follows. A clockwise-odd orientation of G is an orientation of the edges such that for each face (except the infinite face) an odd number of edges point along it when traversed clockwise. For a planar graph, such an orientation always exists [5]. The Kasteleyn matrix corresponding to such a graph is a |V (G)| × |V (G)| skew-symmetric matrix K defined by ⎧ ⎨ W (uv) if u ∼ v and u → v K u,v = −W (uv) if u ∼ v and u ← v ⎩ 0 else.
Local Statistics of Realizable Vertex Models
727
It is known [4,5,8,15] that for a planar graph with a clock-wise odd orientation, the partition function of dimers satisfies Z=
√ det K .
Now let G be a Z2 -periodic planar graph. Let G n be a quotient graph of G, as defined before. Let γx,n (γ y,n ) be a path in the dual graph of G n winding once around the torus horizontally(vertically). Let E H (E V ) be the set of edges crossed by γx (γ y ). We give a crossing orientation for the toroidal graph G n as follows. We orient all the edges of G n except for those in E H ∪ E V . This is possible since no other edges are crossing. Then we orient the edges of E H as if E V did not exist. Again this is possible since G − E V is planar. To complete the orientation, we also orient the edges of E V as if E H did not exist. For θ, τ ∈ {0, 1}, let K nθ,τ be the Kasteleyn matrix K n in which the weights of edges in E H are multiplied by (−1)θ , and those in E V are multiplied by (−1)τ . It is proved in [15] that the partition function Z n of the graph G n is Zn =
1 |Pf(K n00 ) + Pf(K n10 ) + Pf(K n01 ) − Pf(K n11 )|. 2
Let E m = {e1 = u 1 v1 , . . . , em = u m vm } be a subset of edges of G n . Kenyon [7] proved that the probability of these edges occurring in a dimer configuration of G n with respect to the Boltzmann measure Pn is m Pn (e1 , . . . , em ) =
W (u i vi ) |Pf(K n00 )cE + Pf(K n10 )cE + Pf(K n01 )cE − Pf(K n11 )cE |, 2Z n
i=1
where E mc = V (G n )\{u 1 , v1 , . . . , u m , vm }, and (K nθτ ) E mc is the submatrix of K nθτ whose lines and columns are indexed by E mc . The asymptotic behavior of Z n when n is large is an interesting subject. One important concept is the partition function per fundamental domain, which is defined to be lim
n→∞
1 log Z n . n2
Let K 1 be a Kasteleyn matrix for the graph G 1 . Given any parameters z, w, we construct a matrix K (z, w) as follows. Let γx,1 , γ y,1 be the paths introduced as above. Multiply K u,v by z if Pfaffian orientation on that edge is from u to v, otherwise multiply K u,v by 1z , and similarly for w on γ y . Define the characteristic polynomial P(z, w) = det K (z, w). The spectral curve is defined to be the locus P(z, w) = 0. Gauge equivalent dimer weights give the same spectral curve. That is because after Gauge transformation, the determinant multiplies by a nonzero constant, and the locus of P(z, w) does not change. A formula for enlarging the fundamental domain is proved in [3,8]. Let Pn (z, w) be the characteristic polynomial of G n with period 1 × 1, and P1 (z, w) be the characteristic polynomial of G 1 , then Pn (z, w) =
u n =z v n =w
P1 (u, v).
728
Z. Li
Fig. 2. Matchgate
2.3. Matchgates, matchgrids. A matchgate is a planar local graph including a set X of external vertices, i.e. vertices located along the boundary of the local graph. The external vertices are ordered anti-clockwise on the boundary. is called an odd(even) matchgate if it has an odd(even) number of vertices in total(counting both internal and external vertices). For example, the matchgate in Example 2 is an even matchgate. The signature of the matchgate is a vector indexed by subsets of external vertices, {0, 1}|X | . For a subset X 0 ⊂ X , the entry of the signature at X 0 is the partition function of dimer configurations on a subgraph of the matchgate. The subgraph is obtained from the matchgate by removing all the external vertices in X 0 . Example 2 (Signature of a matchgate). Assume we have a matchgate with external vertices 1, 2, 3, and edge weights as illustrated in the Fig. 2: then the signature of the matchgate is ⎞ ⎞ ⎛ ax + by + cz 000 ⎟ ⎜ 001 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ 010 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎜ x ⎟ ⎜ 011 ⎟ ⎜ =⎜ m⎜ ⎟. ⎟ 0 ⎟ ⎜ 100 ⎟ ⎜ ⎟ ⎜ 101 ⎟ ⎜ y ⎟ ⎟ ⎜ ⎜ ⎠ ⎝ 110 ⎠ ⎝ z 0 111 ⎛
A matchgrid M is a weighted planar graph consisting of a collection of matchgates and connecting edges. Each connecting edge has weight 1 and joins an external vertex of a matchgate with an external vertex of another matchgate, so that every external vertex is incident to exactly one connecting edge. 3. Generalized Holographic Reduction In this section we introduce a generalized holographic algorithm to compute the partition function of the vertex model on a finite planar graph in terms of the partition function for perfect matchings on a matchgrid. The idea is using a matchgate to replace each vertex, and perform a change of basis, such that after the base change process, the signature of a vertex becomes the signature of the corresponding matchgate. We describe the algorithm in detail as follows. For a finite graph G, we associate to each oriented edge e a 2-dimensional vector space Ve . To the edge with the reversed orientation, the associated vector space is the dual space, i.e. V−e = Ve∗ . Give a set basis { f e0 , f e1 } for each Ve , satisfying j
f −e ( f ei ) = δi j .
Local Statistics of Realizable Vertex Models
729
Fig. 3. Degree-3 vertex and matchgate
Fig. 4. Degree-4 vertex and matchgate
Let v be a vertex with incident edges el1 , . . . , elk , oriented away from v. The signature of the vertex model at a vertex v, rv , can be considered as an element in Wv = Vel1 ⊗ ... ⊗ Velk . Hence rv has representations under bases F = { f e0 , f e1 }e∈E and B = {be0 , be1 }e∈E as follows: cl cl rv = rv (cl1 · · · clk )bel11 ⊗ · · · ⊗ belkk (2) cl1 ,...,clk
=
cl1 ,...,clk
cl
cl
rv, f (cl1 · · · clk ) f el11 ⊗ · · · ⊗ f elkk ,
(3)
where B are the set of standard bases for each Ve 1 0 0 1 , be = . be = 0 1 cli ∈ {0, 1}. cl1 · · · clk are binary sequences of length k. From the definition of the signature of vertex models, rv (cl1 · · · clk ) is the weight of the configuration cl1 · · · clk at vertex v. We construct a matchgrid M as follows. We replace each vertex v by a matchgate Dv , such that the number of external vertices of Dv is the same as the degree of v, and the edges of the vertex model graph G become connecting edges joining different matchgates in the matchgrid M. Examples of such replacements are illustrated in the Figs. 3, 4. If the signature m v of the matchgate Dv satisfies cl cl rv, f (cl1 · · · clk )bel11 ⊗ · · · ⊗ belkk , (4) mv = cl1 ···clk
that is, the representation of m v under bases B is the same as the representation of rv under bases F, then we have the following theorem:
730
Z. Li
Fig. 5. Figure for Example 3: dimers and loops
Theorem 3. Under the above base change process, the partition function of the vertex model of G is equal to the partition function of the dimer model of M. Proof. There is a natural mapping from ⊗v∈V Wv to C induced by ⊗e∈E φe , where φe is the natural pairing from Ve ⊗ Ve∗ to C. Note that in ⊗v∈V Wv , each Ve and V−e appear exactly once. Since the representation of m v under bases B is the same as the representation of rv under bases F, we have (⊗v∈V m v ) = (⊗v∈V rv ).
(5)
Equation (5) follows from the fact that each φe : Ve ⊗ Ve∗ → C is independent of bases as long as we choose the dual basis for the dual vector space. However, the left side of (5) is exactly the partition function of the dimer model of M, while the right side of (5) is exactly the partition function of the vertex model of G. 0 Define the base change matrix at edge e, Te = f e f e1 . The base change matrix at vertex v, Tv , acting on Wv by multiplication, is defined to be Tv = ⊗{e|e is incident to v, and oriented away from v} Te . In order for a vertex model problem to be reduced to a dimer model problem, one sufficient condition is that at each vertex, the signature of the vertex model rv under the bases F is the same as the signature of the matchgate m v under the standard bases. Namely, Tv m v = rv .
(6)
Equation (6) follows directly from (3) and (4). Example 3. Consider the graph in Fig. 5 with standard dimer signature t rw = rb = r000 r001 r010 r011 r100 r101 r110 r111 t = 01101000 . Define the base change matrix on edges a, b, c to be 0 1 . Ta = Tb = Tc = 1 0 By definition, Tw = Ta ⊗ Tb ⊗ Tc . Note that Tw = (Twt )−1 = Tb . After the base change, we have the standard loop signature r˜w = r˜b = Tw · rw = Tb · rb t = 00010110 .
Local Statistics of Realizable Vertex Models
731
Fig. 6. Periodic Honeycomb Lattice
For instance, after the base change, the dimer configuration 001 with only the c-edge occupied becomes Tw · (b0 ⊗ b0 ⊗ b1 ) = 00000010 , which is the configuration 110, the loop configuration with a-edge and b-edge occupied. However, since the number of vertices in a matchgate is either even or odd, at a vertex d of degree d, the signature of a matchgate must be a 2d−1 dimensional subspace of C2 , d−1 with those 2 entries being 0. These are the entries that correspond to the partition function of dimer configurations on a subgraph of the matchgate with an odd number of vertices, see Example 2.1. This is the parity constraint. As a result, by dimension count we can see that it is not possible to use a holographic algorithm to reduce all vertex models into dimer models. To characterize the special class of vertex models applicable to holographic reduction, we introduce the following definition. Definition 1. A network of relations on a finite graph is realizable, if there exists a system of bases reducing the model to the set of perfect matchings of a matchgrid. Definition 2. A network of relations is bipartite realizable, if it is realizable and the corresponding matchgrid is a bipartite graph. Remark. The above generalizes Valiant’s algorithm [16] in the sense that our basis can be different from edge to edge. As a consequence, our approach results in an enlargement of the dimension of realizable submanifold, which will be shown in the next section.
3.1. Realizability. We are interested in periodic vertex models on the honeycomb lattice with period n × n. The quotient graph can be embedded on a torus T2 = S 1 × S 1 . We classify all the edges into a-type, b-type and c-type according to their direction, and assume b-type and c-type edges have the same direction as the two basic homology cycles (1,0) and (0,1) of torus, respectively, see Fig. 6. Assume the vertex model is realizable, then each corresponding matchgate may have either an even or an odd number of vertices. By enlarging the fundamental domain, we
732
Z. Li
Fig. 7. Permutation of Basis Vectors 1
Fig. 8. Permutation of Basis Vectors 2
can always assume that there are an even number of matchgates with an even number of vertices. Then by permuting rows of matrices on a finite number of edges, we can always have all the matchgates having an odd number of vertices. For example, assume we have a pair of adjacent matchgates with base change matrix on the connecting edge e, Te = f 0 f 1 , as illustrated in Fig. 7. Then if we assume T˜e = f 1 f 0 obtained by permutating two basis vectors of Te , we actually interchange the roles of 0 and 1 at the corresponding digit of the binary sequences as indices of signatures of the matchgate. Namely, for given vertex signature rv , assume Tv m v = rv , and T˜v m˜ v = rv , where T˜v = T˜a ⊗ Tb ⊗ Tc . Then for any binary sequence c1 c2 c3 , the entry m v {c1 c2 c3 } is the same as m˜v {(1−c1 )c2 c3 } , according to Eq. (6). If originally we have an even matchgate at v, by parity constraint, the 001,010,100,111 entries of m v are zero. After the permutation of basis vectors, m˜ v will have 101,110,000,011 entries to be zero. Hence m˜ v has to be an odd matchgate, as illustrate in Fig. 8. By finitely many times of such permutations, we can move two even matchgates to adjacent positions. Then we permutate the basis on the connecting edge, and we can decrease the number of even matchgates by 2. Since we are considering a vertex model with periodic signatures, one period of which can be embedded into a torus; if for one period, we have an odd number of vertices whose signatures are realizable to an odd matchgate, then after enlarging the fundamental domain by 2, we can always have an even number of even matchgates in the bigger torus. Hence if we repeat the process described above, in the end, all matchgates will be odd. Assumption 4. From now on, we make the following assumptions: – All entries of base change matrices are nonzero. – All entries of the matchgate signature are nonzero. Since the honeycomb lattice is a bipartite graph, we can color all the vertices in black and white such that black vertices are adjacent only to white vertices and vice versa.
Local Statistics of Realizable Vertex Models
733
Assume vertex signatures at black and white vertices are as follows: t ij ij ij ij ij ij ij ij rwi j = x000 x001 x010 x011 x100 x101 x110 x111 , t ij ij ij ij ij ij ij ij ij rb = y000 y001 y010 y011 y100 y101 y110 y111 ,
(7) (8)
where (i, j), 1 ≤ i ≤ n, 1 ≤ j ≤ n is the row and column index of white and black vertices. Assume 000 = 1, then entries of signatures are indexed from 1 to 8. Associate to an a-edge, b-edge and c-edge incident to white or black vertices, a basis (i, j,k) (i, j,k) n 0a p0a (i, j,k) (i, j,k) (i, j,k) Ta , = = p n a a (i, j,k) (i, j,k) n 1a p1a (i, j,k) (i, j,k) n 0b p0b (i, j,k) (i, j,k) (i, j,k) Tb , = = p n b b (i, j,k) (i, j,k) n 1b p1b (i, j,k) (i, j,k) n 0c p0c (i, j,k) j,k) Tc = = nc(i, j,k) p(i, c (i, j,k) (i, j,k) n 1c p1c where k = 1 for white vertices and k = 0 for black vertices. By the realizability equation (6), and the fact that all matchgates are odd, the 1st , 4th , 6th , 7th entries of the matchgate signatures m v are 0, so we have a system of 4 algebraic equations at each vertex v. For example, at each black vertex, we have i, j,0
m i, j,0 = Ta and the fact that the
1st
i, j,0
⊗ Tb
i, j,0
⊗ Tc
ij
· rb ,
(9)
entry of m i, j,0 is 0 gives the following equation:
n 0a n 0b n 0c y1 + n 0a n 0b p0c y2 + n 0a p0b n 0c y3 + n 0a p0b p0c y4 + p0a n 0b n 0c y5 + p0a n 0b p0c y6 + p0a p0b n 0c y7 + p0a p0b p0c y8 = 0. That the 4th , 6th , 7th entries of m i, j,0 are 0 give three other similar equations. Similarly at each white vertex, we have i, j,1
m i, j,1 = [(Ta
i, j,1
⊗ Tb
i, j,1 t −1
⊗ Tc
)]
· rwi j .
(10)
The same process gives a system of 4 equations at the white vertex. ij
Let αlm =
(i, j,1)
nlm
(i, j,1) plm
(i, j,0)
nlm
ij
, βlm =
get are linear with respect to
(i, j,0)
, l ∈ {0, 1}, m ∈ {a, b, c}. Then the equations we
plm ij ij ij ij α0a , α1a , β0a , β1a ,
ij
r i j · ui j q i j · ui j = ij ij , i j i j r ·v q ·v i j i j s ·u pi j · u i j = ij ij = ij ij , s ·v p ·v i j i j ξ ·t κi j · ti j = − ij = − , ξ · wi j κ i j · wi j σ i j · ti j λi j · t i j = − ij = − , σ · wi j λi j · wi j
α0a = ij
α1a ij
β0a ij
β1a
which can be solved explicitly. (11) (12) (13) (14)
734
Z. Li
where ⎧ ij ij ij ij ij ij ij ij ⎪ pi j = α0b q i j = α0b , α α α 1 α α α 1 ⎪ 0c 0c 1c 1c 0b 0b ⎪ ⎪ ⎪ ij ij ij ij ij ij ij ij ⎪ ⎪ s i j = α1b r i j = α1b α0c α1b α0c 1 α1c α1b α1c 1 , ⎪ ⎪ ⎪ ⎪ ij ⎪ ⎨ ξ = βi j βi j βi j βi j 1 σ i j = βi j βi j βi j βi j 1 , 0b 0c 0b 0c 0b 1c 0b 1c ⎪ λi j = β i j β i j β i j β i j 1 κ i j = β i j β i j β i j β i j 1 , ⎪ ⎪ ⎪ 1b 0c 1b 0c 1b 1c 1b 1c ⎪ ⎪ i j i j i j i j ij ij ij ij ⎪ i j i j ⎪ v , = = u −x −x x −x −x x x x ⎪ 3 2 7 6 5 ⎪ 1 4 8 ⎪ ⎪ ⎩ wi j = ij ij ij ij t i j = y5i j y6i j y7i j y8i j . y1 y2 y3 y4
(15)
Since α0m = α1m and β0m = β1m as a consequence of the invertibility of the base change matrices, after clearing denominators and some reducing, for any (i, j), Eqs. (11)-(14) are equivalent to 2y2 y8 − 2y6 y4 + (β1c + β0c )(y1 y8 + y2 y7 − y5 y4 − y6 y3 ) +2(y1 y7 − y5 y3 )β1c β0c = 0, −2x1 x7 + 2x5 x3 + (α1c + α0c )(x2 x7 + x1 x8 − x4 x5 − x6 x3 ) +2(x6 x4 − x2 x8 )α1c α0c = 0, 2y8 y3 − 2y4 y7 + (β1b + β0b )(y1 y8 + y3 y6 − y4 y5 − y7 y2 ) +2β1b β0b (y6 y1 − y2 y5 ) = 0, 2x5 x2 − 2x1 x6 + (x1 x8 − x2 x7 − x5 x4 + x3 x6 )(α1b + α0b ) +2(x7 x4 − x3 x8 )α0b α1b = 0.
(16) (17) (18) (19)
If we solve α0b , α1b , β0b , β1b explicitly, a similar process yields 2x3 x2 − 2x1 x4 + (x5 x4 + x1 x8 − x2 x7 − x3 x6 )(α0a + α1a ) +2(x6 x7 − x5 x8 )α0a α1a = 0, 2y6 y7 − 2y5 y8 + (y3 y6 + y2 y7 − y1 y8 − y4 y5 )(β0a + β1a ) +2(y2 y3 − y1 y4 )β0a β1a = 0.
(20) (21)
We get two equations per edge, one involving the a-variables, the other involving the b-variables. But a-variables and b-variables are actually the same thing for each single i, j i, j−1 edge. From Eqs. (16), (17), we can solve basis entries αlc = βlc ; from Eqs. (18), i, j i−1, j (19), we can solve basis entries αlb = β1b . Finally from (11)–(14), we can solve i, j i, j αla and βla , the only constraint left will be αla = βla , which is two polynomial equations with respect to relation signature at each a-edge. Together with Theorem 2 in the Appendix, we have the following theorem: Theorem 5. Under Assumption 4, the realizable signatures on the n × n periodic honeycomb lattice form a 14n 2 dimensional submanifold of the 16n 2 dimensional manifold of all positive signatures; the bipartite realizable signatures on the n × n periodic honeycomb lattice form a 12n 2 dimensional submanifold of the 16n 2 dimensional manifold of all positive signatures.
Local Statistics of Realizable Vertex Models
735
Fig. 9. Matchgrid with 3× 3 Fundamental Domain
For the exact realizability condition, see the Appendix. Under Assumption 4, the weight {111} is non-vanishing at each matchgate. Since the probability measure will not change if all entries of the signature at one vertex are multiplied by a constant, we can assume at each matchgate, the weight of {111} is 1. Therefore, we have Theorem 6. A realizable vertex model on a finite hexagonal lattice G can always be transformed to dimers on M, a Fisher graph as shown in Fig. 9, with the partition function of the vertex model of G equal to the partition function of the dimer model of M, up to multiplication of a constant. It is possible to construct a matchgrid with different weights to produce the same vertex model. Since the holographic reduction is an invertible process, by which we mean that because the base change matrices are nonsingular, we can always achieve the matchgate signature from the vertex signature and vice versa, there is an equivalence relation on dimer models producing the same vertex model. Definition 3. A vertex model is holographically equivalent to a dimer model on a matchgrid, if it can be reduced to the dimer model on the matchgrid using the holographic algorithm, in such a way that the partition function of the vertex model corresponds to perfect matchings of the matchgrid. Two matchgrids are holographically equivalent, if they are holographically equivalent to the same vertex model. Proposition 1. Holographically equivalent matchgrids give rise to gauge equivalent dimer models. Therefore, they have the same probability measure. Proof. See the Appendix.
3.2. Orthogonal realizability. Definition 4. A vertex model is orthogonally realizable if it is realizable by an orthonormal base change matrix on each edge.
736
Z. Li
Consider a single vertex. To the incident edges of the vertex, associate matrices U1 , U2 , U3 . Without loss of generality, we can assume U1 , U2 , U3 ∈ S O(2). In fact, if det Ui = −1, we multiply the first row of Ui by −1. The new signature of the matchgate will be multiplied by −1. This change does not violate the parity constraint. cos αi sin αi 3 U , then U ∈ S O(8), each . Assume U = ⊗i=1 Assume Ui = i − sin αi cos αi term of which is the product of three of cos αi , sin αi . Moreover, the eigenvalues are √ (±α ±α ±α ) −1 1 2 3 . Using trigonometric identities, each entry is a linear combination of e cos(±α1 ± α2 ± α3 ) and sin(±α1 ± α2 ± α3 ). If we further define g = (g1 g2 g3 g4 g5 g6 g7 g8 ) , h = (0 h 2 h 3 0 h 5 0 0 h 8 ) , γ = α1 + α2 − α3 ; ψ = α1 + α3 − α2 ; ϕ = α2 + α3 − α1 , j1 = g4 + g6 + g7 − g1 ; j2 = g1 + g6 + g7 − g4 ; j3 = g1 + g4 + g7 − g6 ; j4 = g1 + g4 + g6 − g7 ; j5 = g3 + g5 + g8 − g2 ; j6 = g2 + g5 + g8 − g3 ; j7 = g2 + g3 + g8 − g5 ; j8 = g2 + g3 + g5 − g8 , P = j2 cos(ϕ) + j7 sin(ϕ); Q = j3 cos(ψ) + j6 sin(ψ); R = j4 cos(γ ) + j5 sin(γ ), K = j1 cos(ϕ + ψ + γ ) − j8 sin(ϕ + ψ + γ ), H = j7 cos(ϕ) − j2 sin(ϕ); L = j6 cos(ψ)− j3 sin(ψ); M = j5 cos(γ ) − j4 sin(γ ), N = j8 cos(ϕ + ψ + γ ) + j1 sin(ϕ + ψ + γ ), then if U g = h, we have 0 4h 2 4h 3 4h 5 4h 8
= = = = =
P = Q = R = K, H + L − M + N, H − L + M + N, N + M + L − H, H + L + M − N.
(22) (23) (24) (25) (26)
By (22), we have tan ϕ = − jj27 , tan ψ = − jj36 , tan γ = − jj45 , tan(ϕ + ψ + γ ) = we define
j1 j8 .
If
u+v , 1 − uv a + b + c + d − abc − abd − acd − bcd . tt (a, b, c, d) = 1 − ab − ac − ad − bc − bd − cd + abcd t (u, v) =
Then the following theorem holds: Theorem 7. A vertex model on a periodic honeycomb lattice with period n ×n is orthogonally realizable if and only if its signatures satisfy the following system of equations tt −
i jk
z7
i jk
z2
i jk
,−
z6
i jk
z3
i jk
,−
z4
i jk
z5
i jk
,−
z1
i jk
z8
=0
∀i, j, k,
Local Statistics of Realizable Vertex Models
737
i j1 ⎫ i j0 i j0 i j1 z z z z ⎪ ⎪ t − 6i j0 , − 4i j0 = t − 6i j1 , − 4i j1 ⎪ ⎪ z3 z5 z3 z5 ⎪ ⎪ i j−1,0 ⎬ i j−1,0 i j1 i j,1 z3 z2 z3 z2 → ∀i, j, t − i j−1,0 , − i j−1,0 = t − i j1 , − i j1 z z7 z6 z7 ⎪ ⎪ 6i−1, j0 ⎪ ⎪ i−1, j0 i j1 i j1 ⎪ z z z z ⎭ t − 7i−1, j0 , − 4i−1, j0 = t − 7i j1 , − 4i j1 ⎪ z2
i j1
z5
z2
z5
i j1
i j1
i j1
where z 1 = x4 + x6 + x7 − x1 , z 2 = x3 + x5 + x8 − x2 , z 3 = x2 + x5 + x8 − x3 , z 4 = i j1 i j1 i j1 x1 + x6 + x7 − x4 , z 5 = x2 + x3 + x8 − x5 , z 6 = x1 + x4 + x7 − x6 , z 7 = x1 + x4 + i j1 i j0 x6 − x7 , z 8 = x2 + x3 + x5 − x8 , and the same relation for zl and y1 , . . . , y8 . x s and y s are defined by (7) and (8). It is trivial to verify these equations in any given situation. Definition 5. A vertex model on a periodic honeycomb lattice with period n × n is positively orthogonally realizable if it is orthogonal realizable and for each vertex (i, j, k), there exists angles ϕ i jk , ψ i jk , γ i jk , such that i jk
sin ϕ =
z4 i jk
i jk
(z 4 )2 + (z 5 )2
,
i jk
sin ψ =
sin γ =
z6
i jk i jk (z 3 )2 + (z 6 )2 i jk z7 i jk
i jk
(z 7 )2 + (z 2 )2
,
,
i jk
sin(−γ − ϕ − ψ) =
z1 i jk
i jk
(z 1 )2 + (z 8 )2
.
Proposition 2. If a vertex model on a bi-periodic hexagonal lattice is orthogonally realizable, then the corresponding dimer configuration has positive edge weights. Proof. Under the assumption that the vertex model have nonnegative signature, we have z1 ≤ z4 + z6 + z7; z4 ≤ z1 + z6 + z7; z6 ≤ z1 + z4 + z7; z7 ≤ z4 + z6 + z7, z2 ≤ z3 + z5 + z8; z3 ≤ z2 + z5 + z8; z5 ≤ z2 + z3 + z8; z8 ≤ z2 + z3 + z5 at all vertices. Since at least three of z 1 , z 4 , z 6 , z 7 are nonnegative, similarly for z 2 , z 3 , z 5 , z 8 , if we take the absolute value for all z i , the above inequalities also hold. Therefore z 42 + z 52 + z 62 + z 32 + z 22 + z 72 − z 12 + z 82 ≥ 0, z 42 + z 52 + z 62 + z 32 + z 12 + z 82 − z 22 + z 72 ≥ 0, 2 2 2 2 2 2 z 4 + z 5 + z 1 + z 8 + z 2 + z 7 − z 32 + z 62 ≥ 0, z 12 + z 82 + z 62 + z 32 + z 22 + z 72 − z 42 + z 52 ≥ 0.
738
Z. Li
Under the assumption that both the vertex models are positively orthogonal realizable, the left side of the above inequalities are exactly edge weights of corresponding matchgates. Theorem 8. If all matchgates have the same signature, for a generic choice of signature, a vertex model on a hexagonal lattice is realizable if and only if it is orthogonal realizable. Proof. Obviously orthogonal realizability implies realizability, we only need to show that realizability implies orthogonal realizability. t Assume r = x1 x2 x3 x4 x5 x6 x7 x8 is a realizable signature for the vertex model. By Lemma 4.1, we can assume the corresponding bases on all edges have real entries. Consider the singular value decomposition for the base change matrix on each edge. Since T−e = (Te−1 )t , multiplying the matrices T by a scalar does not change anything; we can assume that of the entries of the diagonal matrix is 1. Namely, 1 0 Vi i = 1, 2, 3, Ti = Ui 0 λi where Ui , Vi ∈ O(2), and λi is a nonnegative real number. Assume 3 Vi )r = (v1 v2 v3 v4 v5 v6 v7 v8 )t . v = (⊗i=1
1 0 1 0 3 )v and (⊗i=1 )v to be signatures of black 0 λ1i 0 λi and white vertices that are orthogonal realizable. Then by the conditions of orthogonal realizability, we have
3 Then we can consider (⊗i=1
v1 + λ11λ3 v6 + λ11λ2 v7 − λ21λ3 v4 v1 + λ1 λ3 v6 + λ1 λ2 v7 − λ2 λ3 v4 = 1 , 1 1 1 λ3 v2 + λ2 v3 + λ1 λ2 λ3 v8 − λ1 v5 λ3 v2 + λ2 v3 + λ1 λ2 λ3 v8 − λ1 v5
(27)
v1 + λ11λ3 v6 − λ11λ2 v7 + λ21λ3 v4 v1 + λ1 λ3 v6 − λ1 λ2 v7 + λ2 λ3 v4 = , −λ3 v2 + λ2 v3 + λ1 λ2 λ3 v8 + λ1 v5 − λ13 v2 + λ12 v3 + λ1 λ12 λ3 v8 + λ11 v5
(28)
v1 − λ11λ3 v6 + λ11λ2 v7 + λ21λ3 v4 v1 − λ1 λ3 v6 + λ1 λ2 v7 + λ2 λ3 v4 = 1 , 1 1 1 λ3 v2 − λ2 v3 + λ1 λ2 λ3 v8 + λ1 v5 λ3 v2 − λ2 v3 + λ1 λ2 λ3 v8 + λ1 v5
(29)
−v1 + λ11λ3 v6 + λ11λ2 v7 + λ21λ3 v4 −v1 + λ1 λ3 v6 + λ1 λ2 v7 + λ2 λ3 v4 = 1 . 1 1 1 λ3 v2 + λ2 v3 − λ1 λ2 λ3 v8 + λ1 v5 λ3 v2 + λ2 v3 − λ1 λ2 λ3 v8 + λ1 v5
(30)
If we transform fractions into polynomials, and consider (27)+(28)−(29)−(30), (27)+(29)−(28)−(30), and (27)+(30)−(28)−(29), then (λ1 + 1)(λ1 − 1)(v4 v8 + v1 v5 + v3 v7 + v2 v6 ) = 0, (λ2 + 1)(λ2 − 1)(v3 v4 + v5 v6 + v1 v2 + v7 v8 ) = 0, (λ3 + 1)(λ3 − 1)(v4 v2 + v7 v5 + v3 v1 + v8 v6 ) = 0. Then for a generic choice of signature, we must have λ1 = λ2 = λ3 = 1.
Local Statistics of Realizable Vertex Models
739
4. Characteristic Polynomial Assume all entries of the vertex signatures are strictly positive. In this section we prove some interesting properties of the characteristic polynomial. Lemma 1. For a realizable vertex model with positive signature and period 1 × 1, there exists a realization of base change over GL2 (R), (i. e. one can take base change matrices n 0e n 1e to be real), with the property that, at each edge e, α0e α1e < 0 p0e p1e < 0 . Proof. For 1 × 1 fundamental domain, alk = blk , ∀ l, k. By (11)-(14), we have s·t r ·u q ·u p·t =− = = , p·w s·w r ·v q ·v r ·t s·u p·u q ·t =− = = , =− q ·w r ·w s·v p·v
α0a = − α1a from which we derive
( p · t)(r · v) + ( p · w)(r · u) − (r · t)( p · v) − (r · w)( p · u) = 0, (s · t)(q · v) + (s · w)(q · u) − (q · t)(s · v) − (q · w)(s · u) = 0, ( p · t)(q · v) + ( p · w)(q · u) − (q · t)( p · v) − (q · w)( p · u) = 0, (s · t)(r · v) + (s · w)(r · u) − (r · t)(s · v) − (r · w)(s · u) = 0.
(31) (32) (33) (34)
Since each base change matrix is invertible, we have α0i = α1i . Plugging (15) into (31)-(34), and factoring out (α0b − α1b ), or (α0c − α1c ), we obtain that α0c , α1c are two roots of the quadratic polynomial (x6 y5 + y7 x8 + y1 x2 + y3 x4 )z 2 +(−y7 x7 − y3 x3 + y2 x2 + y4 x4 + x6 y6 − x5 y5 + y8 x8 − y1 x1 )z −y6 x5 − x3 y4 − y2 x1 − x7 y8 = 0, and α0b , α1b are two roots of the quadratic polynomial (x3 y1 + y6 x8 + y2 x4 + y5 x7 )z 2 +(y8 x8 − x6 y6 − x5 y5 + y7 x7 − y2 x2 + y3 x3 + y4 x4 − y1 x1 )z −y8 x6 − y7 x5 − x1 y3 − y4 x2 = 0. Under the assumption that xi ,y j are positive, these polynomials have real roots, since the products of two roots are always negative. Then the lemma follows. By definition, the characteristic polynomial P(z, w) is the determinant of an 6 × 6 matrix K (z, w) whose rows and columns are indexed by the 6 vertices in a 1 × 1 fundamental domain, see Fig. 8: ⎛ 0 c −b −1 0 0 ⎞ 1
1
a1 ⎜ −c1 0 ⎜ ⎜ b1 −a1 0 P(z, w) = det ⎜ ⎜ 1 0 0 ⎝ 0 z 0 0 0 w 1 = (z + )(ab − c) + (w + z
0 − 1z 0 ⎟ ⎟ 0 0 − w1 ⎟ ⎟ 0 c2 −b2 ⎟ ⎠ −c2 0 a2 b2 −a2 0 1 z w )(ac − b) + ( + )(bc − a) + a 2 + b2 + 1 + c2 , w w z
740
Z. Li
where a = a1 a2
b = b1 b2
c = c1 c2 ,
(35)
and we consider the non-degenerate case, which means ac − b = 0, bc − a = 0, ab − c = 0. We have the following lemma: Lemma 2. For a realizable vertex model with period 1 × 1, let a, b, c, d be the product of dimer weights (35) after holographic reduction; under Assumption 3.4, we have the following inequalities: (a + b − c − d)(a + c − b − d)(a + d − b − c) > 0 a + b + c + 1 > 0. Here d = d1 d2 . d1 and d2 are original edge weights in the {111} position of the white and the black matchgates obtained from Eq. (6). When we only care about the underlying probability measure, we can multiply all the edge weights by a constant and assume d = 1. Proof. For generic choice of vertex signature, we can assume quotients of basis entries n αi j = pii jj are finite. Apply the realizability equation (6) to a 1 × 1 quotient graph; we have 8 equations for each black vertex, and 8 equations for each white vertex. At each black vertex, 4 equations have 0 on the right side, and 4 equations with a1 , b1 , c1 , d1 on the right side. We divide the 8 equations into 4 groups, each of which has 1 equation with 0 on the right, 1 equation with a non-vanishing edge weight on the right. We take the difference of the two equations in each group, and we get 4 new equations with a non-vanishing edge weight on the right. We perform the same procedure for each white vertex. Then we multiply and add those equations correspondingly; we obtain a + d − b − c = (−α0c + α1c )(y3 x4 + x8 y7 + x6 y5 + y1 x2 ), a + c − b − d = (α0b − α1b )(y1 x3 + y2 x4 + y6 x8 + y5 x7 ), a + b − c − d = (α0a − α1a )(y1 x5 + y4 x8 + y2 x6 + y3 x7 ). Moreover, since holographic reduction leaves the partition function invariant, a + b + c + d = x1 y1 + x2 y2 + x3 y3 + x4 y4 + x5 y5 + x6 y6 + x7 y7 + x8 y8 , where the left side is the partition function of dimers of the 1 × 1 quotient graph, and the right side is the partition function of the vertex model, see Fig. 10. According to Lemma 5.1, α0k and α1k are two real numbers with opposite sign. Among the three terms α0a , α0b , α0c , two of them have the same sign, that can be made positive by properly changing the signs of T . Without loss of generality, we can assume α0b and α0c are both positive. At the black vertices, the {000} entry of matchgate signature corresponds to partition functions of perfect matchings of the generator with all output vertices kept. Since there are odd number of vertices, no perfect matching exists, therefore α0b α0c y5 + α0b y6 + α0c y7 + y8 α0a = − < 0, α0b α0c y1 + α0b y2 + α0c y3 + y4 hence we have −(α0a − α1a )(α0b − α1b )(α0c − α1c ) > 0. The lemma follows from the assumption that entries of relation signatures are positive.
Local Statistics of Realizable Vertex Models
741
Fig. 10. Weighted 1 × 1 Fundamental Domain
The expression of the characteristic polynomial shows that P(z, w) is a smooth function on T2 = {(z, w)||z| = 1, |w| = 1}. We are interested in the intersection of the spectral curve P(z, w) = 0 with unit torus T2 , because it has implications on the convergence rate of correlations. Theorem 4.4 describes the behavior of P(z, w) on T2 . Before stating the theorem, we mention an elementary lemma: Lemma 3. Let f (φ) = A sin φ + B cos φ + C, where A, B, C are real. If C ≥ 0 and A2 + B 2 − C 2 ≤ 0, then f (φ) ≥ 0 for any φ ∈ R. Theorem 9. Under Assumption 4, either the spectral curve after holographic reduction is disjoint from T2 , or intersects T2 at a single real node, that is, for some (z 0 , w0 ) = (±1, ±1), P(z, w) = α(z − z 0 )2 + β(z − z 0 )(w − w0 ) + γ (w − w0 )2 + · · · , where β 2 − 4αγ ≤ 0. Proof. Let Q(θ, φ) = P(eiθ , wiφ ) = [2(ac − b) + 2(bc − a) cos θ ] cos φ + 2(bc − a) sin θ sin φ + 2(ab − c) cos θ + a 2 + b2 + c2 + 1. Consider Q(θ, φ) as a trigonometric polynomial with respect to φ, Q(θ, φ) = A sin φ + B cos φ + C, where A = 2(ac − b) + 2(bc − a) cos θ, B = 2(bc − a) sin θ, C = a 2 + b2 + c2 + 1 + 2(ab − c) cos θ ≥ 0.
742
Z. Li
Define g(cos θ ) = C 2 − A2 − B 2 = 4(ab − c)2 cos2 θ + 4(ab + c)(a 2 + b2 − c2 − 1) cos θ −4(bc − a)2 − 4(ac − b)2 + (a 2 + b2 + c2 + 1)2 , then g is a quadratic polynomial with respect to cos θ with discriminant = 64abc(a + b − c − 1)(a + b + c + 1)(a + 1 − b − c)(a + c − b − 1). The minimal value of g(t) is attained at t0 = −
(ab + c)(a 2 + b2 − c2 − 1) , 2(ab − c)2
where g(t0 ) = −
16(ab − c)2 .
If abc > 0, g(cos θ ) = [2(ab + c) cos θ + a 2 + b2 − c2 − 1]2 + 16abc sin2 θ ≥ 0, therefore Q(θ, φ) ≥ 0, and Q(θ, φ) = 0 only if sin θ = sin φ = 0. If abc < 0, Lemma 4.2 implies that < 0, then g(cos θ ) > 0, therefore Q(θ, φ) > 0. ab+c If abc = 0, we have ab−c = 1. Moreover, (a 2 + b2 − c2 − 1)2 − 4(ab − c)2 = (a + 1 + b + c)(a − 1 − b + c)(a + 1 − b − c)(a − 1 + b − c) > 0, which implies ab + c a 2 + b2 − c2 − 1 > 1, |t0 | = ab − c 2(ab − c) then g(cos θ ) > 0, and Q(θ, φ) > 0. Therefore the only possible spectral curve with T2 is a real point. intersection of the = 0, ∂∂wP (±1,±1) = 0, and H essian[P(z, w)] is It is trivial to check that ∂∂zP (±1,±1)
positive definite wherever a real intersection of P(z, w) = 0 and T2 exists, then the theorem follows.
Local Statistics of Realizable Vertex Models
743
Fig. 11. Configuration 011 at a Black Vertex
5. Local Statistics 5.1. Configuration at single vertex. We are interested in the probability of a specified configuration at a single vertex. Consider a realizable vertex model on a planar finite hexagonal lattice, whose partition function is S. Assume both the vertex with specified signature and its neighbors lie in the interior of the graph. We work out an explicit example here; other cases are very similar. If only the configuration {011} is allowed at a black vertex v, we get a new vertex model, and assume the partition function is S011 . Then the probability that the local configuration {011} appear at v is Pr ({011}, v) =
S011 . S
Hence the local statistics problem reduces to the problem of how to compute S011 efficiently. We will show that S011 can also be computed by the dimer technique with the help of the holographic algorithm. Let us denote the adjacent vertices of v by va , vb , vc , according to the direction of the connecting edges, as illustrated in Fig. 11. Let rv , ra , rb , rc denote the signatures of relations at vertices v, va , vb , vc , then we have ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ a⎞ ra {000} 0 x1 rv {000} ⎜ ra {001} ⎟ ⎜ x a ⎟ ⎜ rv {001} ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 2⎟ ⎜ ra ⎜ rv ⎟ ⎜0⎟ ⎟ ⎜ xa ⎟ {010} {010} ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 3⎟ ⎜r ⎜r ⎟ ⎜y ⎟ ⎟ ⎜ a⎟ ⎜ a {011} ⎟ ⎜ x4 ⎟ ⎜ v {011} ⎟ ⎜ 4 ⎟ ⎜ ⎜ ⎟=⎜ ⎟ ⎟=⎜ ⎟ ⎜ ra {100} ⎟ ⎜ ⎟ ⎜ rv {100} ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ra {101} ⎟ ⎜ ⎟ ⎜ rv {101} ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎝ ra {110} ⎠ ⎝ ⎠ ⎝ rv {110} ⎠ ⎝ 0 ⎠ ⎛
rv {111}
⎛
0
⎞
⎞ rb{000} ⎜ rb{001} ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ rb{010} ⎟ ⎜ x b ⎟ 3 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜r ⎟ x4b ⎟ ⎜ b{011} ⎟ ⎜ ⎜ ⎟ ⎜ ⎟= ⎟ ⎜ rb{100} ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ rb{101} ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ b ⎝ rb{110} ⎠ ⎝ x ⎟ 7⎠ rb{111} x8b
⎛
ra {111}
rc{111}
x8c
⎞ ⎛ ⎞ rc{000} ⎜ rc{001} ⎟ ⎜ x c ⎟ ⎜ ⎟ ⎜ 2⎟ ⎜ rc{010} ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜r ⎟ ⎜ c⎟ ⎜ c{011} ⎟ ⎜ x4 ⎟ ⎜ ⎟=⎜ ⎟ ⎜ rc{100} ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ rc{101} ⎟ ⎜ x6c ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ rc{110} ⎠ ⎝ ⎠
744
Z. Li
Fig. 12. Weighted Matchgate
Here means the entry at the position can be arbitrary. This is because we only allow the configuration {011} at v, therefore the only configurations which actually affect the partition function of the vertex model will be those who do not occupy the a-edge of va , and occupy both the b-edge for vb and the c-edge for vc . We will split each of ra , rb , rc into 2 parts, namely, rl = rl (0) + rl (1), l = a, b, c, below. By definition, the partition function of the vertex model with signature r, ra , rb , rc is a sum of 8 terms, each of which is the partition function of a vertex model with signature rv , ra (i), rb ( j), rc (k) i, j, k ∈ {0, 1}. That is, S{rv ,ra (i),rb ( j),rc (k)}. (36) S011 = S{rv ,ra ,rb ,rc } = i, j,k∈{0,1}
For each S{rv ,ra (i),rb ( j),rc (k)} , we give new bases on edges adjacent to v, such that S{rv ,ra (i),rb ( j),rc (k)} become the partition function of certain local dimer configurations. Namely S{rv ,ra (i),rb ( j),rc (k)}
base change
=
Z {m v (i, j,k),m a (i),m b ( j),m c (k)} ,
(37)
where Z {m v (i, j,k),m a (i),m b ( j),m c (k)} is the partition function of the dimer model on a matchgrid with signature m v (i, j, k), m a (i), m b ( j), m c (k). Assume the original matchgates u a , u b , u c have weights t m l = 0 a2l b2l 0 c2l 0 0 d2l , l = a, b, c, see Fig. 12. We require t m a (1) = 0 0 0 0 c2a 0 0 d2a , t m a (0) = 0 a2a b2a 0 0 0 0 0 , t m b (1) = 0 0 b2b 0 0 0 0 d2b , t m b (0) = 0 a2b 0 0 c2b 0 0 0 , t m c (1) = 0 a2c 0 0 0 0 0 d2c , t m c (0) = 0 0 b2c 0 c2c 0 0 0 .
(38) (39) (40) (41) (42) (43)
Local Statistics of Realizable Vertex Models
745
Notice that for any l ∈ {a, b, c}, m l (0) corresponds to configurations with an unoccupied l-edge; and m l (1) corresponds to configurations with an occupied l-edge. We are trying to determine the new base change matrices on incident edges of v, for each rv , ra (i), rb ( j), rc (k), such that (36), (37), (38)–(43) are satisfied simultaneously. Assume on the adjacent edges of va , vb , vc we have original base change matrix j j n 0k n 1k j Tk = , k = 1, 2, 3; j = a, b, c. j j p0k p1k On an a-type edge adjacent to v, we consider 2 possible base change matrices m 001 0 0 m 111 0 1 , S1 = . S1 = 0 0 1 1 q01 q11 q01 q11 On a b-type edge adjacent to v, we consider 2 possible base change matrices m 002 m 012 m 102 m 112 0 1 S2 = . , S2 = 0 1 q02 0 0 q12 On a c-type edge adjacent to v, we consider 2 possible base change matrices 0 0 1 1 m m m m 03 13 03 13 S30 = . , S31 = 0 1 q03 0 0 q13 j
For signatures rv , ra (i), rb ( j), rc (k), we choose basis S1i , S2 , S3k on edges a, b, c. By the realizability condition at vertex v, we have j
m v (i, j, k) = (S1i ⊗ S2 ⊗ S3k )t · rv = e{i jk} wi jk , where j
k y4 , wi jk = m ii1 q j2 qk3
and e{i jk} is the 8 dimensional vector with entry 1 at the position labeled by the binary sequence {i jk}, and 0 elsewhere. Obviously the choice of new base change matrices, namely, the position of zeros in the new base change matrices, results in the single configuration {i jk} at the matchgate corresponding to v. Now the problem is to determine the entries of the base change matrices satisfying the equations. According to realizability conditions at vertices va , vb , vc , 2 2 ra = i=1 ra (i) = i=1 S1i ⊗ T2a ⊗ T3a · m a (i),
rb = rc =
j 2 2j=1ra ( j) = i=1 T1b ⊗ S2 2 2 k=1 rc (k) = k=1 T1c ⊗ T2c
⊗ ⊗
T3b · m b ( j), S3k · m c (k).
(44) (45) (46)
For the original basis, we have the realizability condition as follows: rl = T1l ⊗ T2l ⊗ T3l · m l .
(47)
746
Z. Li
Substituting ra , rb , rc by (47) in (46) we have the following system of linear equations with respect to entries of S, ⎛ 1 a⎞ ⎛ ⎞ ⎛ a a⎞ m 11 c2 x1a n 11 c2 ⎜ 2 a⎟ ⎜ a⎟ ⎜ na aa ⎟ ⎜ ⎟ m a x ⎜ 2⎟ ⎜ 2⎟ 01 2 ⎟ T2a ⊗ T3a ⎜ = T2a ⊗ T3a ⎜ 01 , ⎜ m 2 ba ⎟ = ⎜ a⎟ a ba ⎟ ⎝ ⎠ ⎝ x n ⎝ 01 2 ⎠ 3 01 2 ⎠ x4a n a11 d2a m 111 d2a ⎛ 1 b⎞ ⎛ b⎞ ⎛ b b⎞ x3 q12 b2 p12 b2 ⎜ 2 b⎟ ⎜ b⎟ ⎜ b b⎟ ⎜ q02 a2 ⎟ ⎜ x4 ⎟ ⎜ ⎟ b b ⎜ p02 a2 ⎟ ⎟ ⎜ ⎟ T1b ⊗ T3b ⎜ ⎜ q 2 cb ⎟ = ⎜ x b ⎟ = T1 ⊗ T3 ⎜ p b cb ⎟ , ⎝ 02 2 ⎠ ⎝ 7 ⎠ ⎝ 02 2 ⎠ b b b db 1 q12 d2 x8 p12 2 ⎛ 1 c⎞ ⎛ ⎞ ⎛ c c⎞ q13 a2 p13 a2 x2c ⎜ 2 c⎟ ⎜ c⎟ ⎜ c c⎟ ⎜ q03 b2 ⎟ ⎜ x4 ⎟ ⎜p b ⎟ c c ⎜ 03 2 ⎟ ⎟ T1c ⊗ T2c ⎜ = T ⊗ T ⎟ 1 2 ⎜ q 2 cb ⎟ = ⎜ ⎜ p c cb ⎟ , c ⎝ 03 2 ⎠ ⎝ x6 ⎠ ⎝ 03 2 ⎠ c c db b 1 x p13 q13 d2 8 2 Under the assumption that original base change matrices are invertible, we have m 111 = n a11 ,
m 201 = n a01 ,
1 b q12 = p12 ,
2 b q02 = p02 ,
1 c = p13 , q13
2 c q03 = p03 .
Therefore, we only need to choose the nonzero entries of the new base change matrices to be the same as the original ones. For the other seven configurations at vertex v, we can use the same method to achieve a similar result. By splitting each one of ra , rb , rc into 2 parts, we express the partition function of the satisfying assignments as the sum of 8 terms, each one of which is the partition function of the relations with signature rv , ra (i), rb ( j), rc (k), i, j, k ∈ {0, 1}. We apply the holographic reduction to each part separately, and we derive that the partition function of the relations with signature rv , ra (i), rb ( j), rc (k), i, j, k ∈ {0, 1} is equal to the partition function of the dimer configurations on the corresponding matchgrids with signature m v (i, j, k), m a (i), m b ( j), m c (k), which corresponds to the local configuration {i jk}. This may not be a dimer configuration; to make it a dimer configuration, let us divide those 8 terms into two groups: one consists of all {i, j, k} satisfying i + j + k ≡ 0 ( mod 2), namely {000}, {011}, {101}, {110}; the other consists of all {i, j, k} satisfying i + j + k ≡ 1( mod 2). The sum of partition functions in the first group is equal to the partition function of dimer configurations in a matchgrid with weights illustrated w000 in the left graph of Fig. 13, where t = w011 +w ; the sum of partition functions 101 +w110 corresponding to {001}, {010}, {100}, {111} is equal to the partition function of dimer configurations in a matchgrid with weights as illustrated in the right graph of Fig. 13, up to a multiplication constant w111 . However, in the left graph we change the parity of the total number of vertices, as a result, there is no dimer. Therefore we have the following theorem
Local Statistics of Realizable Vertex Models
747
Fig. 13. Even and Odd Matchgates
Theorem 10. Assume we have a vertex model H0 on a finite hexagonal lattice, which is holographic equivalent to the dimer model on a Fisher graph F0 . If at a fixed interior vertex v of H0 , only one local configuration is allowed, then the partition function of the new vertex model H1 can be computed by the partition function of the dimer model on a Fisher graph F1 , multiplied by a non-zero constant. F0 and F1 have the same edge weights except for the matchgate corresponding to v. 5.2. Configuration at finitely many vertices. We are interested in the probability that one specified configuration is allowed at each of the given finitely many vertices for a realizable vertex model on a hexagonal lattice. For simplicity, assume all of them are black. Let V0b be the set of all black vertices with specified configuration. To compute the probability, we first give a criterion to construct the new base change matrices on incident edges of V0b according to whether the edge in the dimer configuration is present n0 n1 is the original base change and in the vertex model configuration. Assume p0 p1 matrix on the edge, Base Change Matrices 0 n1 p0 p1 n0 0 p0 p1 n0 n1 0 p1 n0 n1 p0 0
Presence in Vertex Configuration
Presence in Dimer
No
Yes
No
No
Yes
Yes
Yes
No
whether the edges incident to vertices in V0b are present in the configuration of the vertex model is known, given all the specified configurations at V0b . As before, we are going to split the signature of each adjacent white vertex into several parts, and we apply the holographic reduction to each part separately. Our expectation is that after the reduction process, each part will be equivalent to the partition function of a single local dimer
748
Z. Li
configuration. The new base change matrix on each edge is chosen according to whether we want the edge to be present in the dimer configuration or not after the reduction. As before, all the non-vanishing entries of the new bases change matrices are equal to the original ones; we will see how it works below. Consider an arbitrary white vertex w ∈ (V0b ), the neighbors of V0b . Assume rw is the vertex signature at w, and m w is the original dimer signature at the corresponding matchgate. Assume m w = (0 aw bw 0 cw 0 0 dw )t . Let D(w) denote the number of adjacent vertices of w with specified configuration. We classify (V0b ) according to D. If D(w) = 1, we split rw = rw (1) + rw (2). Without loss of generality, assume the left-digit edge(a-type edge, horizontal edge) connects w with b ∈ V0b , and the edge wb is present in the configuration of the vertex model. Assume n 01 n 11 n 02 n 12 n 03 n 13 , T2 = , T3 = , T1 = p01 p11 p02 p12 p03 p13 are original base change matrices on the edges adjacent to w. According to the table, the presence of wb in the vertex model configuration implies two possible choices of the new basis on the horizontal edge, namely, n 01 n 11 n 01 n 11 , S12 = , S11 = p01 0 0 p11 while on the two other incident edges of w we have the original bases T2 , T3 . Assume m w (1) = (0 aw bw 0 0 0 0 0) , m w (2) = (0 0 0 0 cw 0 0 dw ) . Notice that the nonzero entries m w (1) have indices {001} and {010}, which correspond to configurations without the a-edge present; the non-vanishing entries of m w (2) have indices {100} and {111}, which correspond to configurations with the a-edge present. Choose rw (1) = S11 ⊗ T2 ⊗ T3 · m w (1), rw (2) = S12 ⊗ T2 ⊗ T3 · m w (2). Then we can check 3 Ti · m w = rw . rw (1) + rw (2) = ⊗i=1
4 If D(w) = 2, we split rw = i=1 rw (i). Without loss of generality, assume the middle-digit(b-type edge) and right-digit(c-type edge) edges connects w to b2 , b3 ∈ V0b , wb2 is present while wb3 is not present in the configuration of the vertex model. According to our criteria listed in the table, on wb2 we have two different base change matrices: n 02 n 12 n 02 n 12 , S22 = . S21 = p02 0 0 p12
Local Statistics of Realizable Vertex Models
749
On wb3 we have two different base change matrices: n 03 0 0 n 13 , S32 = , S31 = p03 p13 p03 p13 while on the left-digit edge, we keep the original basis T1 , the original basis on the middle and right digit edge are T2 , T3 , as in the D(w) = 1 case. Assume m w (1) = m w (2) = m w (3) = m w (4) =
(0 aw 0 0 0 0 0 0) , (0 0 bw 0 0 0 0 0) , (0 0 0 0 cw 0 0 0) , (0 0 0 0 0 0 0 dw ) .
Obviously, m w (4) corresponds to both edges being present, m w (3) corresponds to neither being present, m w (2) corresponds to the b-edge being present and the c-edge not, and m w (1) corresponds to the b-edge not present and the c-edge present. Again according to the table, choose rw (1) = T1 ⊗ S21 ⊗ S32 · m w (1), rw (2) = T1 ⊗ S22 ⊗ S31 · m w (2), rw (3) = T1 ⊗ S21 ⊗ S31 · m w (3), rw (4) = T1 ⊗ S22 ⊗ S32 · m w (4), we can check 4
rw (i) = ⊗31 Ti · m w = rw .
i=1
4 If D(w) = 3, we split rw = i=1 rw (i), and assume m w (i), i = 1, . . . , 4 as in D(w) = 2. Without loss of generality, assume S11 , S12 as in D(w) = 1, S21 , S22 , S31 , S32 as in D(w) = 2; that is, we have vertex-model signature {110} at w. Choose rw (1) = S11 ⊗ S21 ⊗ S32 · m w (1), rw (2) = S11 ⊗ S22 ⊗ S31 · m w (2), rw (3) = S12 ⊗ S21 ⊗ S31 · m w (3), rw (4) = S12 ⊗ S22 ⊗ S32 · m w (4); 4 again we can check rw = i=1 rw (i) as in D(w) = 2. For all the other local configurations, the same technique works, and we will have a similar result. Assume V0b = {v1 , . . . , v p }, (V0b ) = {w1 , . . . , wk }, then p
S(˜rv1 , . . . , r˜v p ) = [⊗kj=1rw j ⊗ ⊗w∈ (V b r w ] · [⊗q=1 r˜vq ⊗ ⊗b∈V / / 0b rb ] 0) p [⊗kj=1rw j (i j ) ⊗ ⊗w∈ (V = b ) r w ] · [⊗q=1 r˜vq ⊗ ⊗b∈V / / b rb ], i 1 ,...,i k
0
0
750
Z. Li
where r˜vq is the specified configuration at the vertex vq . The first equality follows from the definition of the partition function of satisfying assignments. When we compute the tensor product of relation signatures of all black (white) vertices, we get a vector of dimension 2|E(G)| , where |E(G)| is the number of edges in the planar finite graph G. This vector can be indexed by binary sequences of length |E(G)|. Each binary sequence corresponds to a configuration, and the entry there is the product of weights of the local configurations obtained from the configuration restricted to each black (white) vertices. Obviously the inner product of the vector at black vertices and the vector at white vertices are exactly the partition function of the satisfying assignments. The second equality follows from the multi-linearity of the tensor product. For 1 ≤ j ≤ k, if D(w j ) = 1, i j ∈ {1, 2}; if D(w j ) = 2, 3, i j ∈ {1, 2, 3, 4}. For each summand, we choose a basis on incident edges of vertices in V0b according to whether the edge is present in the dimer configuration and relation configuration, and keep the original bases on all the other edges, as described above. This is realizable because for each b ∈ V0b , whether its incident edges are occupied by dimers are completely determined in each part of the sum. In other words, each term on the right side of the second equality corresponds to a configuration on all the edges incident to vertices in V0b , which is a local dimer configuration at each odd matchgate corresponding to white vertices w’s with D(w) = 3, and can be extended to local dimer configurations at each odd matchgate corresponding to white vertices w’s with D(w) = 1 and D(w) = 2. To make them be dimer configurations at each black matchgate, we divide those configurations into groups according to the following criterion: two configurations are in the same group if and only if the parity of the number of occupied incident edges at each black vertex in V0b is the same. If zero or two incident edges are occupied, we construct an even matchgate with modified weights at the black vertex; if one or three incident edges are occupied, we construct an odd matchgate with modified weights. Notice that the modified weights depend only on the local configuration on the incident edges of the vertex. Since |V0b | = p, we have 2 p different constructions in total, but only 2 p−1 of them admit a dimer cover, each of which has an even number of even matchgates. Therefore we have Theorem 11. Assume we have a realizable vertex model H0 on a finite hexagonal lattice, which is holographic equivalent to the dimer model on a Fisher graph F0 . Let V0b be a subset of black vertices. If for each v ∈ V0b , we only allow one local configuration, this way we obtain a new vertex model H1 . The partition function of H1 is equal to the sum of partition functions of 2 p−1 dimer models on matchgrids F1 , . . . , F2 p−1 , each of which has an even number of even matchgates. Fi (1 ≤ i ≤ 2 p−1 ) are the same F0 , except for the matchgates corresponding to vertices in V0b .
5.3. Ising model and vertex model. Consider the Ising model on a finite Kagome lattice, embedded into a torus, as illustrated in the Fig. 14. The associated honeycomb lattice is illustrated in the figure by the dashed line. Each vertex of the Kagome lattice corresponds to an edge of the honeycomb lattice; hence each Ising spin configuration on the Kagome lattice corresponds to an edge subset of the honeycomb lattice. If the spin is “+”, then the corresponding edge is included in the subset; otherwise the edge is not included. Assume the bonds of the Kagome lattice have interactions J1 , J2 , J3 , as illustrated in Fig. 14. We define a vertex model on the honeycomb lattice with signature at all vertices as follows.
Local Statistics of Realizable Vertex Models
751
Fig. 14. Kagome Lattice and Honeycomb Lattice
⎞ ⎛ e2(J1 +J2 +J3 ) ⎞ rv,000 ⎟ e2J3 ⎜ rv,001 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2J2 ⎟ e ⎜ rv,010 ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ 2J1 ⎟ e ⎜ rv,011 ⎟ ⎜ ⎜ ⎟. ⎜r ⎟= 2J1 ⎟ e ⎜ v,100 ⎟ ⎜ ⎜ ⎟ ⎜r ⎟ 2J2 ⎟ e ⎜ v,101 ⎟ ⎜ ⎜ ⎟ ⎝ rv,110 ⎠ ⎝ ⎠ e2J3 rv,111 e2(J1 +J2 +J3 ) ⎛
This way the probability measure of the Ising model on the Kagome lattice is equivalent to the probability measure of the vertex model on the honeycomb lattice. It is trivial to check that this vertex model is orthogonally realizable. 6. Asymptotic Behavior In this section, we prove the main theorems concerning the asymptotic behavior of realizable vertex models on the periodic hexagonal lattice, as stated in the Introduction. Consider an infinite periodic graph G, with period 1 × 1, see Fig. 9. Our technique to deal with such a graph is to consider a graph G n with n 2 1 × 1 fundamental domains, embed G n into a torus, and consider the limit when n → ∞, see Subsect. 2.2. Our first theorem is about the free energy of the infinite periodic hexagonal lattice. Proof of Theorem 1. Assume Mn is the corresponding matchgrid with respect to G n , and Pn (z, w) is the characteristic polynomial. Obviously Mn is also a quotient graph of a periodic infinite graph modulo a subgraph of Z2 generated by (n, 0) and (0, n). The corresponding Kasteleyn matrices here are defined given the orientation of Fig. 9. For even n, the crossing orientation can be obtained from the orientation of Fig. 9 by reversing all the z-edges and w-edges. By Theorem 3.6, Mn is a Fisher graph, and the partition of the vertex model can be expressed as follows according to the principle of holographic reduction: 1 S(G n ) = Z (Mn ) = | − Pf K n (1, 1) + Pf K n (1, −1) + Pf K n (−1, 1) + Pf K n (−1, −1)|. 2
752
Z. Li
Thus max
u,v∈{−1,1}
|Pf Pn (u, v)| ≤ Z (Mn ) ≤ 2
max
u,v∈{−1,1}
|Pf Pn (u, v)|.
On the other hand, according to the formula of enlarging fundamental domains, 1 1 log max |Pf K n (u, v)| = max log |P(z, w)|. 2 u,v∈{−1,1} u,v∈{−1,1} 2n 2 n n n z =u w =v
By Theorem 9, either P(z, w) has no zeros on T2 , or it has a single real node, in 1 which case any sample point in maxu,v∈{−1,1} 2n 2 z n =u wn =v log |P(z, w)| is at least C n from the real node, for some constant C > 0. Therefore 1 1 S(G n ) = lim max log |P(z, w)| 2 n→∞ n n→∞ u,v∈{−1,1} 2n 2 z n =u wn =v 1 dz dw . = log P(z, w) 2 8π i z iw |z|=1,|w|=1 lim
For each G n , a measure λn is defined as in (1). Assume at a fixed vertices v, we only allow configuration cv . Let μn be the Boltzmann measure of dimer configurations on Mn . Let M˜ n be the matchgrid corresponding to the vertex model which only allows cv at v, as described in Theorem 5.1. Let m v be the matchgate of M˜ n , corresponding to v, and d j be a local dimer configuration at m v , wd j be product of weights of matchgate edges included in d j , and Vd j be the set of external vertices of matchgates m v occupied by dimer configuration d j . In our graph, every matchgate has 3 external vertices. Our second theorem is about the asymptotic behavior of the measure λn . Proof of Theorem 2. According to Theorem 10, λn (c1 , . . . , c p ) =
Z ( M˜ n ) Z ( M˜ n (d j )) = Z (Mn ) Z (Mn ) dj
=
dj
wd j
Z (Mn \V (d j )) , Z (Mn )
where the sum is over all local dimer configurations d j . Since the number of local conZ (Mn \V (d j )) . Note that M˜ n differs figurations is finite, it suffices to consider limn→∞ Z (Mn ) from Mn only on edge weights of m v , hence Mn \V (d j ) and M˜ n \V (d j ) are the same. Given d j , let Wn,d j = Mn \V (d j ) be the subgraph of Mn by removing all vertices occupied by the configuration (d j ), as well as their incident edges. Then Z (Wn,d j ) Z (Mn )
=
1 | −Pf(K n11 (Wn,d j )) + Pf(K n−1,1 (Wn,d j )) 2Z n + Pf(K n1,−1 (Wn,d j )) + Pf(K n−1,−1 (Wn,d j ))|.
Local Statistics of Realizable Vertex Models
753
First of all, let us assume P(z, w) has no zeros on T2 . According to the formula of enlarging fundamental domains, for any (θ, τ ) ∈ {−1, 1}, P f (K nθ,τ ) = 0. Then θ,τ |Pf(K nθ,τ ) E c | = |Pf(K nθ,τ )−1 E ||Pf(K n )|.
In [3,8], it was proved that given two vertices (u, x1 , y1 ) and (v, x2 , y2 ), (K nθ,τ )−1 ((u, x1 , y1 ), (v, x2 , y2 )) =
1 x1 −x2 y1 −y2 Cof(K (z, w))u,v z w . n2 n P(z, w) n z =θ w =τ
Since P(z, w) has no zeros on T2 , we have lim (K nθ,τ )−1 ((u, x1 , y1 ), (v, x2 , y2 )) Cof(K (z, w))u,v dz dw 1 z x1 −x2 w y1 −y2 = 2 4π 2 P(z, w) i z iw T
n→∞
−1 = K∞ ((u, x1 , y1 ), (v, x2 , y2 )).
As n → ∞, each entry of (K nθτ )−1 is convergent, so is the Pfaffian of a finite order sub-matrix (K nθτ )−1 V (d j ) , and we have lim
n→∞
Z (Mn \V (d j )) −1 = |Pf(K ∞ )V (d j ) |. Z (Mn )
If P(z, w) has a real node on T2 , without loss of generality, we can assume the real node is (1, 1). It was proved in [1] that if P(z, w) = 0 has a node at (1, 1), then for any fixed finite subset E, lim
n→∞
1 −1 [Pf(K n−1,1 ) E c + Pf(K n1,−1 ) E c + Pf(K n−1,−1 ) E c ] = Pf(K ∞ )E 2Z n
and Pf(K n1,1 ) E c = 0. n→∞ 2Z n lim
If we take E = V (di, j , . . . , d p, j ), our theorem follows.
7. Simulation of 1-2 Model Assume we have 1-2 model with signature rs = (0 c b a a b c 0)t at all vertices. In other words, at each vertex, either one or two edges are allowed to be present in a local configuration. Sect. 3.2, the By 3π signature is orthogonally realizable with base change cos 4 sin 3π 4 matrix T = on all edges. sin 3π cos 3π 4 4 We define a discrete-time, time-homogeneous Markov chain Mt with state space the set of all configurations of the 1-2 model. For an n × n honeycomb lattice embedded into K . Let (i ) be the set of cona torus, the state space is finite, and let us denote it by {i k }k=1 k figurations that can be obtained from configuration i k by adding or deleting a single edge. Assume p, q, r are edge variables, namely p, q, r ∈ {a, b, c}, and { p, q, r } = {a, b, c}. Define p,+q (i k ) to be the set of configurations of 1-2 model which can be obtained from
754
Z. Li
i k by adding a single q-edge uv; before adding uv, only a p-edge is present at both u and v. Define p,−q (i k ) to be the set of configurations of 1-2 model which can be obtained from i k by deleting a single q-edge uv; after deleting uv, only a p-edge is present at both u and v. Define 0 (i k ) = (i k )\{∪ p,q∈a,b,c, p=q p,+q (i k ) ∪ p,q∈a,b,c, p=q p,−q (i k )}. Define entries of transition matrix for Mt as follows: ⎧ 1 if il ∈ 0 (i k ) ⎪ 3n 2 ⎪ ⎪ ⎪ 1 ⎪ if il ∈ p,+q (i k ) and r ≥ p ⎪ ⎪ 3n 2 ⎪ ⎪ 2 ⎪ 1 r ⎪ if il ∈ p,+q (i k ) and r < p ⎪ ⎨ 3n 2 p 2 P(il |i k ) =
1 if il ∈ p,−q (i k ) and r ≤ p . 3n 2 ⎪ ⎪ ⎪ 2 ⎪ 1 p ⎪ if il ∈ p,−q (i k ) and r > p ⎪ 3n 2 r 2 ⎪ ⎪ ⎪ ⎪ 1 − i ∈ (i ) P(i j |i k ) if il = i k ⎪ ⎪ j k ⎩ 0 else
Obviously, Mt is aperiodic. For more information on Markov chain, see [11,12]. Moreover, we have Proposition 3. Mt is irreducible. Proof. By definition, we only need to prove that any two configurations communicate to each other. We claim that any two dimer configurations can be obtained from each other by finite steps. In fact, the symmetric difference of any two dimer configurations is a union of finitely many loops. Obviously one dimer configuration can be obtained from any other dimer configuration by first adding finitely many edges to achieve their union, then deleting alternating edges of loops. Notice that we have a 1-2 configuration at each step. Hence we only need to prove that any configuration of 1-2 model can reach a dimer by finite steps, each of which is adding or deleting one single edge. Let us start with an arbitrary configuration of 1-2 model. There are 3 types of connected local configurations: loops with even number of edges; zigzag paths with odd number of edges; zigzag paths with even number of edges. For the first and second types, we can always achieve dimers by deleting alternating edges. Hence we can assume that all the zigzag paths with even number of edges are of length 2, and all the other connected local configurations are dimers. There are two types of length-2 paths. One has vertices black-white-black (BWB), and the other has vertices white-black-white (WBW). For each fixed configuration, the number of BWB paths is the same as the number of WBW paths; otherwise the complement graph cannot be covered by dimers. Consider an arbitrary WBW path in a fixed configuration, as illustrated in Fig. 15. One can easily check that no matter what the configuration is, it communicates with a configuration satisfying one of the following two conditions: 1. the number of length-2 edges is equal to the current number of length-2 edges minus 2; 2. it can be moved to each of a WBW configurations containing Bi , for all 1 ≤ i ≤ 6, where Bi is the nearest black vertices to B, as in Fig. 15. Hence if the number of length-2 paths does not decrease, one can transverse all black vertices without meeting a BWB path, because our graph is finite. However, this is impossible because a BWB path always exists as long as a WBW path exists. For an irreducible, aperiodic Markov chain Mt with transition matrix P and stationary distribution π , let x0 be an arbitrary initial distribution. Then lim x0 P n = π.
n→∞
Local Statistics of Realizable Vertex Models
755
Fig. 15. WBW Path
Therefore, in order to sample a configuration, we can approximately sample according to the distribution x0 P N , with large N . To that end, first we choose a fixed dimer configuration with probability 1 as the initial distribution x0 , then we randomly change the configuration by adding or deleting a single edge according to the conditional probability specified by the transition matrix P. If neither adding nor deleting u 1 u 2 ends up with a satisfying configuration, we just keep the previous configuration; else we get a new configuration by adding or deleting u 1 u 2 . Then we repeat the process for N steps. This way we get a sample for distribution x0 P N , if N is sufficiently large, this is approximately a sample for distribution π , which is exactly the distribution given by 1-2 model. Example 4 (Uniform 1-2 Model). Consider the 1-2 model with a = b = c = 1. After the holographic reduction, the signature of each matchgate is √ √ √ √ 3 2 2 2 2 0 00 − . 0 2 2 2 2 It is gauge equivalent to a positive-weight dimer model on a Fisher graph, whose spectral curve does not intersect T2 . We are interested in the probability that a {001} dimer occurs, that is, at a pair of adjacent vertices v1 , v2 connected by an a-edge, only the configuration {001} is allowed. By the technique described in Sect. 5 and Sect. 6, the partition function of the configurations in which a dimer is present at v1 v2 is equal to the sum of two partition functions, one corresponds to both v1 and v2 are replaced by an odd matchgate with signatures √ √ √ √ 2 2 2 2 0 00 − ; 0 − 4 4 4 4 the other corresponds to both v1 and v2 are replaced by an even matchgate with signatures √ √ √ √ 2 2 2 2 00 − 0 − 0 . 4 4 4 4 Both of them have the same partition as a graph with positive weights. Moreover, function 0 1 if we give a base change matrix on v1 v2 edge, and apply holographic reduction, 1 0 we see that the two graphs with positive weights are holographic equivalent. Hence it suffices to consider the one with a pair of odd matchgates with modified weights. Then Pr (a dimer is present at v1 v2 ) =
Z 001 , 18Z
756
Z. Li
where Z 001 is the partition function of dimer configurations with weight 1 on triangles corresponding to v1 and v2 and weight 13 on all the other triangles, and Z is the partition function with weight 13 on all the triangles. Moreover Z 001 = Z 001001 + Z 001111 + Z 010010 + Z 010100 + Z 100100 + Z 100010 + Z 111001 + Z 111111 = Z 001001 + 2Z 001111 + 2Z 010010 + 2Z 010100 + Z 111111 , ˜˜
where Z i jk,i jk is the partition function of dimer configurations with fixed configura˜ on triangles corresponding to v1 and v2 . The second equality follows from tion i jk, i˜ jk symmetry. Meanwhile 1 001001 2 001111 2 010010 2 010100 + Z + Z + Z + Z 111111 = Z , Z 9 3 9 9 then Pr (a dimer is present at v1 v2 ) 8Z 001001 4Z 001111 16Z 010010 16Z 010100 1 1+ + + + = 18 9Z 3Z 9Z 9Z 4 1 4 16 −1 −1 −1 1 + |Pf(K ∞ = )23 | + |Pf(K ∞ )2356 | + |Pf(K ∞ )13 | , 18 3 9 3 where w(zw + 1 + z − 9w) dz dw 3 −1 = (K ∞ )56 , 2 + w 2 ) + 2(z + w) + 2zw(z + w) − 21zw i z iw 2 16π 2 2(z T 3 w(w + z + z 2 − 9zw) dz dw , =− 2 2 2 16π T2 2(z + w ) + 2(z + w) + 2zw(z + w) − 21zw i z iw 1 8z + 8zw − 82w + 9w2 + 9 dz dw −1 = (K ∞ = )36 , 2 2 2 16π T2 2(z + w ) + 2(z + w) + 2zw(z + w) − 21zw i z iw 1 −z − zw − w + 9 dz dw −1 = (K ∞ = )35 . 2 + w 2 ) + 2(z + w) + 2zw(z + w) − 21zw i z iw 2 16π 2 2(z T
−1 (K ∞ )23 = −1 (K ∞ )13 −1 (K ∞ )25 −1 (K ∞ )26
The entries of the inverse matrix can be expressed as elliptic functions. The probability that a {001} dimer occurs is approximately 6%. A sample of the uniform 1-2 model is illustrated in Fig. 16. Example 5 (Critical 1-2 Model). Consider the 1-2 model with a = 4, b = c = 1. After the holographic reduction, it has the same partition function as a positive-weight dimer model on a Fisher graph, whose spectral curve has a single real node on T2 . The probability of the configuration {011}, Pr ({011}) =
1 Z 011 −1 −1 −1 −1 = (|(K ∞ )12 | + |(K ∞ )23 | + |(K ∞ )13 |) + |Pf(K ∞ )123456 |). 3Z 3
Local Statistics of Realizable Vertex Models
Fig. 16. Sample of Uniform 1-2 Model
757
758
Z. Li
Fig. 17. Sample of Critical 1-2 Model
Local Statistics of Realizable Vertex Models
759
Z 011 is the partition function on a Fisher graph with weights 1, 1, 1 on one triangle, and weights 23 , 23 , 13 on all the other triangles. Z is the partition function with weights 23 , 23 , 13 −1 ) | = |(K −1 ) |. Then on all the triangles. By symmetry |(K ∞ 12 ∞ 23 1 2 2 −1 )23 | + |(K −1 )12 | + |(K ∞ 3 9 ∞ 9 dz dw w(4wz + 4 + z − 9w) 1 1 = + 3 6π 2 T2 −7(w 2 + z 2 ) + 32wz(w + z) + 32(w + z) − 114wz i z iw dz dw z(9wz − 4z − w 2 − 4w) 1 + 2 2 2 3π T2 −7(w + z ) + 32wz(w + z) + 32(w + z) − 114wz i z iw 2 1 8 1 18 16z − 85z + 29 = + − + dz √ 3 12π |z|=1 32z − 7 z(32z − 7) z(32z − 7) 4z 2 − 17z + 4 1 9 4 22w 2 − 35w + 8 + − + dw √ 2 3π |w|=1 (32w − 7) w(32w − 7) w(32w − 7) 4w − 17w + 4 23 25 4 65 44 = − arctan − arctan ≈ 39% 48 112π 3 336π 117
Pr ({011}) =
√ For · we choose a branch with positive real part. This integral can be evaluated explicitly because the graph has critical edge weights. A sample of the critical 1-2 model is illustrated in Fig. 17. Question. How large should N be as a function of the size of the graph? Acknowledgements. The author would like to thank Richard Kenyon for many stimulating discussions. The author is also grateful to David Wilson, Béatrice de Tilière, Sunil Chhita for comments, and the anonymous referee for suggestions which helped improve the readability of the present manuscript.
A. Realizability Conditions In this section we give explicit realizability conditions for a periodic honeycomb lattice embedded on a n ×n domain. Let us consider an arbitrary vertex x with adjacent vertices y, z, w. Assume relations have following signatures: r x = (x1 . . . x8 )t r y = (y1 . . . y8 )t , r z = (z 1 . . . z 8 )t rw = (w1 . . . w8 )t . According to the realizability equation (1), after a lengthy computation, we have Theorem 12. A periodic network of relation on n × n honeycomb lattice is realizable if and only if at each vertex x with adjacent vertices y, z, w connected by a, b, c edge, respectively, the signatures satisfy the following equation: X i jk Yi Z j Wk = 0, i, j,k∈{0,1,2}
where Y0 = y1 y4 − y2 y3 , Y1 = y1 y8 + y4 y5 − y2 y7 − y3 y6 , Y2 = y5 y8 − y6 y7 , Z 0 = z1 z6 − z2 z5, Z 1 = z1 z8 + z3 z6 − z4 z5 − z2 z7, Z 2 = z3 z8 − z4 z7, W 0 = w 1 w7 − w3 w5 , W 1 = w1 w8 + w 2 w7 − w5 w4 − w3 w6 , W 2 = w2 w8 − w6 w4 ,
760
Z. Li
and X 000 = x12 (x3 x6 + x2 x7 + x4 x5 − x1 x8 ) − 2x1 x2 x3 x5 , X 001 = x12 (x6 x4 − x2 x8 ) + x22 (x1 x7 − x3 x5 ), X 002 = x22 (x2 x7 − x1 x8 − x3 x6 − x4 x5 ) + 2x1 x2 x4 x6 , X 010 = x32 (x1 x6 − x2 x5 ) + x12 (x7 x4 − x3 x8 ), X 011 = x1 x2 (x4 x7 − x3 x8 ) + x3 x4 (x1 x6 − x2 x5 ), X 012 = x22 (x4 x7 − x3 x8 ) + x42 (x1 x6 − x2 x5 ), X 020 = x32 (x3 x6 − x2 x7 − x4 x5 − x1 x8 ) + 2x1 x4 x3 x7 , X 021 = x42 (x1 x7 − x3 x5 ) + x32 (x6 x4 − x2 x8 ), X 022 = x42 (x2 x7 + x1 x8 + x3 x6 − x4 x5 ) − 2x2 x3 x4 x8 . X 100 X 101 X 102 X 110 X 111 X 112 X 120 X 121 X 122
= = = = = = = = =
x52 (x1 x4 − x2 x3 ) + x12 (x6 x7 − x5 x8 ), x1 x2 (x6 x7 − x5 x8 ) + x5 x6 (x1 x4 − x2 x3 ), x62 (x1 x4 − x2 x3 ) + x22 (x6 x7 − x5 x8 ), x1 x3 (x7 x6 − x5 x8 ) + x5 x7 (x1 x4 − x2 x3 ), x1 x4 x6 x7 − x2 x3 x5 x8 , x4 x8 (x1 x6 − x2 x5 ) + x2 x6 (x4 x7 − x3 x8 ), x72 (x1 x4 − x2 x3 ) + x32 (x6 x7 − x5 x8 ), x3 x7 (x4 x6 − x2 x8 ) + x4 x8 (x1 x7 − x3 x5 ), x82 (x1 x4 − x2 x3 ) + x42 (x6 x7 − x5 x8 ),
X 200 = x52 (x4 x5 − x1 x8 − x3 x6 − x2 x7 ) + 2x1 x5 x6 x7 , X 201 = x52 (x6 x4 − x2 x8 ) + x62 (x1 x7 − x3 x5 ), X 202 = x62 (x1 x8 + x2 x7 + x4 x5 − x3 x6 ) − 2x5 x6 x2 x8 , X 210 = x52 (x4 x7 − x3 x8 ) + x72 (x1 x6 − x2 x5 ), X 211 = x6 x8 (x1 x7 − x3 x5 ) + x5 x7 (x4 x6 − x2 x8 ), X 212 = x82 (x1 x6 − x2 x5 ) + x62 (x4 x7 − x3 x8 ), X 220 = x72 (x1 x8 + x3 x6 + x4 x5 − x2 x7 ) − 2x3 x8 x5 x7 , X 221 = x82 (x1 x7 − x3 x5 ) + x72 (x6 x4 − x2 x8 ), X 222 = x82 (x1 x8 − x3 x6 − x2 x7 − x4 x5 ) + 2x7 x8 x6 x4 . Theorem 13. A periodic network of relation with n × n fundamental domain is bipartite realizable if and only if it is realizable and at any vertex v, the signature satisfies v12 v82 + v22 v72 + v32 v62 + v42 v52 − 2v1 v8 v2 v7 − 2v1 v8 v3 v6 − 2v1 v8 v4 v5 −2v2 v7 v3 v6 − 2v2 v7 v4 v5 − 2v3 v6 v4 v5 + 4v1 v4 v6 v7 + 4v2 v3 v5 v8 = 0 Proof. Without loss of generality, we assume at each matchgate, the signature {111} is 0, and assume v is a black vertex. The {111} entry of the signature of a black vertex
Local Statistics of Realizable Vertex Models
761
is 0 gives us α1a α1b α1c v1 + α1a α1b v2 + α1a α1c v3 + α1a v4 + α1b α1c v5 + α1b v6 + α1c v7 + v8 = 0. (48) The parity constraint implies that the {110} entry is also 0, namely α1a α1b α0c v1 + α1a α1b v2 + α1a α0c v3 + α1a v4 + α1b α0c v5 + α1b v6 + α0c v7 + v8 = 0. (49) From (48)(49), we can solve α1 a, and solution has following form: α1a =
N1 N2 = . D1 D2
Then N1 D2 − N2 D1 = 0. Under the assumption that α0c = α1c , we have 2 (−v2 v5 + v6 v1 )α1b + (v8 v1 + v6 v3 − v4 v5 − v2 v7 )α1b + v8 v3 − v4 v7 = 0.
(50)
From (12), we have 2(v1 v6 − v2 v5 )α0b α1b +(v1 v8 + v3 v6 − v4 v5 − v2 v7 )(α0b +α1b )+2(v3 v8 − v4 v7 ) = 0. (51) 2×(50)−(51), under the assumption that α0b = α1b , we have α1b = −
v1 v8 + v3 v6 − v4 v5 − v2 v7 , 2(v1 v6 − v2 v5 )
which implies that Eq. (50) has double real roots, and its discriminant is 0. That is exactly the statement of the theorem. Proof of Proposition 1. Since holographic reduction is an invertible process, two matchgrids M and Mˆ are holographically equivalent if and only if there exists a basis for each edge, such that one can be transformed to the other using holographic reduction. Obviously holographic equivalent matchgrids have the same dimer partition function. Assume weights m b , m w of M and mˆb , mˆw of Mˆ are as follows: ij m iwj = 0 c1 ij ij m b = 0 c2 ij mˆ iwj = 0 cˆ1 ij ij mˆ b = 0 cˆ2
ij
ij
ij t
ij
ij
ij t
b1 0 a1 0 0 d1 b2 0 a2 0 0 d2
ij ij ij bˆ1 0 aˆ 1 0 0 dˆ1 ij ij ij bˆ2 0 aˆ 2 0 0 dˆ2
t t
, , , .
762
Z. Li
Then by Eqs. (16)-(21), and the uniqueness of the basis on each edge, we have ij ij
a1 b1
ij ij d1 c1
=
ij ij
a1 c1
ij ij
b1 d1
ij ij a1 d1
= a03 a13 = b03
i−1, j i−1, j b2 i−1, j i−1, j a2 c5
= a02 a12 = b02
=
d2
=
d2 a2
ij ij
b1 c1
i, j−1 i, j−1 c2 i, j−1 i, j−1 b2 a2
d2
ij
ij
i, j−1 i, j−1 b13 ,
(52)
ij
ij
i−1, j i−1, j b12 ,
(53)
ij ij
ij ij b2 c2
ij
ij
ij
ij
= a01 a11 = b01 b11 .
(54)
Plugging in (52)-(54) to (9)-(10), we have ⎧ ij i j i, j,0 i, j,0 i, j,0 ij ⎪ cˆ2 = c2 n 01 n 02 p13 · C1 , ⎪ ⎪ ⎪ ⎪ ⎨ bˆ i j = bi j n i, j,0 pi, j,0 n i, j,0 · C i j , 2 2 01 12 03 1 ij i j i, j,0 i, j,0 i, j,0 ⎪ ⎪ aˆ 2 = a2 p11 n 02 n 03 · C1i j , ⎪ ⎪ ⎪ ⎩ ˆi j i j i, j,0 i, j,0 i, j,0 ij d2 = d2 p11 p12 p13 · C1 , ⎧ ij c ij ij ⎪ ⎪ cˆ1 = i, j,1 i,1j,1 i, j,1 · C2 , ⎪ ⎪ n n p ⎪ 01 02 13 ⎪ ⎪ ij ⎪ b1 ij ij ⎪ ˆ ⎪ ⎨ b1 = i, j,1 i, j,1 i, j,1 · C2 , ⎪ ij ⎪ ⎪ aˆ 1 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dˆ1i j =
n 01 p12 n 03
ij a1 i, j,1 i, j,1 i, j,1 p11 n 02 n 03 ij d1 i, j,1 i, j,1 i, j,1 p11 p12 p13
ij
· C2 , ij
· C2 .
To prove that probability measures of M and Mˆ are identical, we only need to prove that for any dimer configuration, the products of weights differ by the same constant factor. Each dimer configuration corresponds to a binary sequence of length N , where N = 3n 2 is the number of connecting edges. Choose an arbitrary edge with basis n0 n1 ˆ adjacent generator weight is . If the edge is occupied, then from M to M, p0 p1 divided by p1 , while adjacent recognizer weight is multiplied by p1 . If the edge is unocˆ adjacent generator weight is divided by n 0 , while the adjacent cupied, then from M to M, recognizer weight is multiplied by n 0 . Therefore, the total effect is for any particular dimer configuration , ij ij ˆ Partition( on M) = C1 C2 , Partition( on M) i, j
which is a constant independent of configuration .
Local Statistics of Realizable Vertex Models
763
References 1. Boutillier, C., Tilière, B.: The Critical Z-invariant Ising Model via Dimers: the Periodic Case. Phys. Rev. C 79, 015502 (2009) 2. Cai, J.: Holographic Algorithms. Current Developments in Mathematics 2005, 111–150 (2007) 3. Cohn, H., Kenyon, R., Propp, J.: A Variational Principle for Domino Tilings. J. Amer. Math. Soc. 14(2), 297–346 (2001) 4. Kasteleyn, P.W.: The Statistics of Dimers on a Lattice. Physica 27, 1209–1225 (1961) 5. Kasteleyn, P.W.: Graph Theory and Crystal Physics. In: Graph Theory and Theoretical Physics, London: Academic Press, 1967 6. Kenyon, R.: An Introduction to the Dimer Model. http://arxiv.org/abs/math/0310326v1 [math.co], 2003, Lecture notes given as a minicourse at ICTP, May, 2002 7. Kenyon, R.: Local Statistics on Lattice Dimers. Ann. Inst. H. Poicare. ´ Probabilités 33, 591–618 (1997) 8. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and Amoebae. Ann. Math. 163(3), 1019–1056 (2006) 9. Kenyon, R., Okounkov, A.: Planar Dimers and Harnack Curve. Duke. Math. J. 131(3), 499–524 (2006) 10. Kirillov, A. Jr.: Introduction of Lie Groups and Lie Algebra. Cambridge Studies in Advanced Mathematics, no. 113, Cambridge: Cambridge univ press, 2008 11. Lawler, G.F.: Introduction to Stochastic Processes. 2nd Edition, Boca Raton, FL: Chapman Hall/CRC, 2006 12. Levin, D., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. Providence, RI: Amer. Math. Soc. 2008 13. Li, Z.: Spectral Curve of a Periodic Fisher Graph. In preparation, available at http://arxiv.org/abs/1008. 3936v2 [math.cv], 2010 14. Schwartz, M., Bruck, J.: Constrained Codes as Network of Relations, Information Theory. IEEE Transactions 54(5), 2179–2195 (2008) 15. Tesler, G.: Matchings in Graphs on Non-orientable Surfaces. J. Combin. Theory Ser. B 78(2), 198–231 (2000) 16. Valiant, L.G.: Holographic Algorithms (Extended Abstract). In: Proc. 45th IEEE Symposium on Foundations of Computer Science, 2004, pp. 306–315 Communicated by S. Smirnov
Commun. Math. Phys. 304, 765–796 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1138-0
Communications in
Mathematical Physics
Categorical Formulation of Finite-Dimensional Quantum Algebras Jamie Vicary Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom. E-mail: [email protected] Received: 13 June 2010 / Accepted: 22 June 2010 Published online: 3 November 2010 – © Springer-Verlag 2010
Abstract: We describe how †-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional ‘quantum algebras’. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of †-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory. 1. Introduction The main purpose of this paper is to describe how †-Frobenius monoids are the correct tool for formulating various kinds of finite-dimensional ‘quantum algebras’. Since †-Frobenius monoids have entirely geometrical axioms, this gives a new way to look at these traditionally algebraic objects. This difference in perspective can be thought of as moving from an ‘internal’ to an ‘external’ viewpoint. Traditionally, we formulate a C*-algebra as the set of elements of a vector space, along with extra structure that specifies how to multiply elements, find a unit element, apply an involution and take norms. This is an ‘internal’ view, since we are dealing directly with the elements of the set. The ‘external’ alternative is to ‘zoom out’ in perspective: we can no longer discern the individual elements of the C*-algebra, but we can see more clearly how it relates to other vector spaces, and these relationships give an alternative way to completely define the C*-algebra. This metaphor is made completely precise by category theory, and the passage between these two types of viewpoint is familiar in categorical approaches to algebra. We proceed in Sect. 2 by introducing our categorical setting, monoidal †-categories with duals, and defining an involution monoid, a categorical axiomatization of an
766
J. Vicary
involutive algebra. Section 3 introduces †-Frobenius monoids, and explores some useful properties of them. We specialize to the category of Hilbert spaces in Sect. 4, and make the connection between †-Frobenius monoids and finite-dimensional C*-algebras precise. An important aspect of the conventional study of C*-algebras is the spectral theorems, for commutative C*-algebras and for normal operators. The †-Frobenius perspective on C*-algebras allows these theorems to be presented categorically in the finite-dimensional case, and we explore this in Sect. 5. We also use the †-Frobenius monoid formalism to explore the construction of alternative quantum theories. This work is relevant to the study of two-dimensional open-closed topological quantum field theories (TQFTs), which model the quantum dynamics of string-like topological structures which can merge together and split apart. It was shown by Lauda and Pfeiffer [22] that such a theory is defined by a symmetric Frobenius monoid equipped with extra structure. If we also add the physical requirement that the theory should be unitary [7] then these become symmetric †-Frobenius monoids, and thus finite-dimensional C*-algebras by Lemma 3.11 and the results of Sect. 4. These are precisely the correct kinds of algebras with which to construct a state-sum triangulation model for the TQFT [16,21], and so we can deduce the following: the two-dimensional open-closed TQFTs which arise from a state sum on a triangulation are precisely the unitary such TQFTs, up to multiplication by a scalar factor. The results presented here are closely tied to finite-dimensional algebras. The author is aware of some work in progress on infinite-dimensional generalizations [5], which requires significant changes to the underlying algebraic structures. However, the importance of the finite-dimensional case should not be underestimated. In the study of topological quantum field theory, in particular, it is often necessary to restrict to finite-dimensional algebras for the constructions to be well-defined, as a consequence of compactness of the topological category. The construction described here can be generalized far beyond the scope of the current paper. In future work, we will describe how higher-dimensional ‘quantum algebras’ can be described as †-Frobenius pseudoalgebras, ‘weakened’ forms of Frobenius algebras which live in a monoidal 2-category. This extends results of Day, McCrudden and Street [13,31]. These higher-dimensional quantum algebras include the fusion C*-categories of considerable importance in the representation theory of quantum groups [19] and in topological quantum field theory [8]. Why †-Frobenius monoids? The key property of †-Frobenius monoids which makes them so useful is contained in the following observation, due to Coecke, Pavlovic and the author [12]. Let (V, m, u) be an associative, unital algebra on a complex vector space V , with multiplication map m : V ⊗ V - V and unit map u : C - V . We can map any element α ∈ V into the algebra of operators on V by constructing its right action, a linear map Rα := m ◦ (id A ⊗ α) : V - V . We draw this right action in the following way:
α
Categorical Formulation of Finite-Dimensional Quantum Algebras
767
The diagram is read from bottom to top. This is a direct representation of our definition of Rα : vertical lines represent the vector space V , the dot represents preparation of the state α, and the merging of the two lines represents the multiplication operation m : V ⊗ V - V . If V is in fact a Hilbert space we can then construct the adjoint map Rα † : V - V . Will this adjoint also be the right action of some element of V ? In the case that (V, m, u) is in fact a †-Frobenius monoid, the answer is yes. We draw the adjoint Rα † by flipping the diagram on a horizonal axis, but keeping the arrows pointing in their original direction: α†
The splitting of the line into two represents the adjoint to the multiplication, and the dot represents the linear map α † : V - C. The multiplication and unit morphisms of the †-Frobenius monoid, along with their adjoints, must obey the following equations (see Definition 3.3):
=
=
=
=
On the left are the Frobenius equations, and on the right are the unit equations. The short horizontal bar in the unit equations represents the unit for the monoid, and the straight vertical line represents the identity homomorphism on the monoid. In fact, we also have two extra equations, since we can take the adjoint of the unit equations. We can use a unit equation and a Frobenius equation to redraw the graphical representation of Rα † in the following way:
α†
α† =
α† =
=
α†
We therefore see that the adjoint of Rα is indeed a right-action of some element: Rα † = Rα , for α = (id A ⊗ α † ) ◦ m † ◦ u.
768
J. Vicary
To better understand this transformation α - α we apply it twice to evaluate (α ) , using the Frobenius and unit equations and the fact that the †-functor is an involution:
(α )† = (α )
=
= (α † )†
= α
α
We see that (α ) = α, and so the operation α - α is an involution. Since taking the adjoint Rα - Rα† is also clearly an involution, the mapping of elements of the monoid into the ring of operators on V is therefore involution-preserving, as it maps one involution into another. We shall see that the mapping is injective and preserves the multiplication and unit of (V, m, u), so in fact we have a fully-fledged involution-preserving monoid embedding as described by Lemmas 3.19 and 3.20. This observation is one reason why †-Frobenius monoids are such powerful tools. In fact, given that the algebra of operators on V is a C*-algebra with ∗-involution given by operator adjoint, and since any involution-closed subalgebra of a C*-algebra is also a C*-algebra, we have already shown that every †-Frobenius monoid in Hilb can be given a C*-algebra norm. Overview of paper. We begin with a description of the categorical structure that we will use to express our results. The categories we will be working with are monoidal †-categories with duals, with nontrivial coherence requirements between the monoidal structure, †-structure and duality structure. These can be seen as not-necessarily-symmetric versions of the strongly compact-closed categories of Abramsky and Coecke [2,3]. We then describe the concept of an involution monoid, a categorical version of the traditional concept of a ∗-algebra, which replaces the antilinear involution with a linear ‘involution’ from an object to its dual. We prove some general results on involution monoids, †-Frobenius monoids and the relationships between them, and give a definition of a unitary †-Frobenius monoid. In Hilb, the category of finite-dimensional complex Hilbert spaces and continuous linear maps, these monoids have particularly good properties, which we explore. We then use these properties to demonstrate in Theorem 4.6 that unitary †-Frobenius monoids in Hilb are the same as finite-dimensional C*-algebras. The spectral theorem for finite-dimensional commutative C*-algebras is an important classical result, and we develop a way to express it using the †-Frobenius toolkit. We first summarize a result from [12], that the category of commutative †-Frobenius monoids in Hilb is equivalent to the opposite of FinSet, the category of finite sets and functions. We generalize this by defining a monoidal †-category to be spectral if its category of commutative †-Frobenius monoids is a finitary topos. We also consider the spectral theorem for normal operators, and give a way to phrase it in an abstract categorical way using the concept of internal diagonalization. Nontrivial examples of spectral categories are provided by categories of unitary representations of finite groupoids HilbG , where G a finite groupoid. In such a category,
Categorical Formulation of Finite-Dimensional Quantum Algebras
769
the spectrum of a commutative generalized C*-algebra — that is, the spectrum of a commutative †-Frobenius monoid internal to the category — is not a set, but an object in a finitary Boolean topos FinSetG . Categories of the form HilbG can therefore be thought of as providing alternative settings for quantum theory, in which the logic of measurement outcomes — while still Boolean — has a richer structure. On a technical level, we also note that this gives a new way to extract a finite groupoid from its representation category, as it is well-known that the groupoid G can be identified in FinSetG as the smallest full generating subcategory. 2. Structures in †-categories The †-functor. Of all the categorical structures that we will make use of, the most fundamental is the †-functor. It is an axiomatization of the operation of taking the adjoint of a linear map between two Hilbert spaces, and since knowing the adjoints of all maps C - H is equivalent to knowing the inner product on H , it also serves as an axiomatization of the inner product. Definition 2.1. A †-functor on a category C is a contravariant endofunctor † : C - C, which is the identity on objects and which satisfies † ◦ † = idC . Definition 2.2. A †-category is a category equipped with a particular choice of †-functor. These †-categories have a long history, sometimes going by the name ∗-categories. In particular, they have been well-used in representation theory, especially by Roberts and collaborators [14,23] under the framework of C*-categories, and by others in the study of invariants of topological manifolds [32]. They have also been used to study the properties of generalizations of quantum mechanics [10,33], where it is not assumed that the underlying categories are C-linear. A useful physical intuition is that the †-functor models the time-reversal of processes, and considering it as a fundamental structure gives an interesting new perspective on the development of physical theories [7]. Given a †-category, we denote the action of a †-functor on a morphism f : A - B as f † : B - A, and by convention we refer to the morphism f † as the adjoint of f . We can now make the following straightforward definitions: Definition 2.3. In a †-category, a morphism f : A - B is an isometry if f † ◦ f = id A ; in other words, if f † is a retraction of f . Definition 2.4. In a †-category, a morphism f : A - B is unitary if f † ◦ f = id A and f ◦ f † = id B ; in other words, if f is an isomorphism and f −1 = f † . Definition 2.5. In a †-category, a morphism f : A - A is self-adjoint if f = f † . Definition 2.6. In a †-category, a morphism f : A - A is normal if f ◦ f † = f † ◦ f . Monoidal categories with duals. We will work in monoidal categories throughout this paper, and we will require that each object in our monoidal categories has a left and a right dual. In the presence of a †-functor there are then some compatibility equations which we can impose, which we will describe in this section. There is an important graphical notation for the objects and morphisms in these categories [18] which we will rely on heavily. We have already made use of it in the
770
J. Vicary
Introduction. Objects in a monoidal category are drawn as wires, and the tensor product of two objects is drawn as those objects side-by-side; for consistency with the equation A ⊗ I A I ⊗ A, we therefore ‘represent’ the tensor unit object I as a blank space. Morphisms are represented by ‘junction-boxes’ with input wires coming in underneath and output wires coming out at the top, and composition of morphisms is represented by the joining-up of input and output wires. For visual consistency, the identity morphism on an object is also not drawn. These principles are demonstrated by the following pictures: B
A
C h
g
f
B g
A
A
Object A or morphism id A
Morphism f :I - A
A
A
Morphism id A ⊗ g
A
Morphism h ◦ (id A ⊗ g)
We will often omit the labels on the wires when it is obvious from the context which object they represent. We now give the definition of duals, and describe their graphical representation. Definition 2.7. An object A in a monoidal category has a left dual if there exists an object A∗L and left-duality morphisms LA : I - A∗L ⊗ A and ηLA : A ⊗ A∗L - I satisfying the triangle equations: A∗
A id A ⊗ LA
? A ⊗ A∗ ⊗ A
id A
- A
ηLA ⊗id A
id A∗
LA ⊗id A∗
? A∗ ⊗ A ⊗ A∗
- A∗ L
(1)
id A∗ ⊗η A
Analogously, an object A has a right dual if there exists an object A∗R and right-duality morphisms RA : I - A ⊗ A∗R and ηRA : A∗R ⊗ A - I satisfying similar equations to those given above. It follows that any two left (or right) duals for an object are canonically isomorphic. To distinguish between the objects A and A∗L , we add arrows to our wires, usually drawing an object A with an upward-pointing arrow and drawing A∗L with a downward-pointing one. We use the same notation for A∗R , which will not lead to confusion since we will soon choose our duals such that A∗L = A∗R for all objects A. We represent the duality morphisms by a ‘cup’ and a ‘cap’ in the following way: A∗L
A A
LA : I - A∗L ⊗ A
A∗L
ηLA : A ⊗ A∗L - I
Categorical Formulation of Finite-Dimensional Quantum Algebras
771
The reason for this is made clear by the representation it leads to for the duality equations: A∗L
A A∗L
=
A
A A∗L
A
=
A∗L
We can therefore ‘pull kinks straight’ in the wires whenever we find them. This is one reason that the graphical representation is so powerful: the eye can easily spot these simplifications, which would be much harder to find in an algebraic representation. Definition 2.8. A monoidal category has left duals (or has right duals) if every object A has an assigned left dual A∗L (or a right dual A∗R ), along with assigned duality morphisms, such that I ∗L = I and (A ⊗ B)∗L = B ∗L ⊗ A∗L (or the equivalent with L replaced with R.) The order-reversing property of the (−)∗L and (−)∗R operations for the monoidal tensor product is important: it allows us to choose a dual for A ⊗ B given duals of A and B independently. In the presence of a braiding isomorphism A ⊗ B B ⊗ A we can suppress this distinction, but this will not be available to us in general. Definition 2.9. In a monoidal category with left or right duals, with an assigned left dual for each object or a chosen right dual for each object, the left duality functor (−)∗L and right duality functor (−)∗R are contravariant endofunctors that take objects to their assigned duals, and act on morphisms f : A - B in the following way: f ∗L := (id A∗ ⊗ ηLB ) ◦ (id A∗ ⊗ f ⊗ id B ∗ ) ◦ ( LA ⊗ id B ∗ ), f
∗R
:=
(ηRB ∗
⊗ id A ) ◦ (id B ∗ ⊗ f
(2)
⊗ id A∗ ) ◦ (id A∗ ⊗ RA∗ ).
(3)
These definitions can be understood more easily by their pictorial representation:
f ∗L
:=
f
f ∗R
:=
f
(4)
Monoidal †-categories with duals. We now investigate appropriate compatibility conditions in the case that our monoidal category has both duals and a †-functor. Definition 2.10. A monoidal †-category is a monoidal category equipped with a †-functor, such that the associativity and unit natural isomorphisms are unitary. If the monoidal category is equipped with natural braiding isomorphisms, then these must also be unitary. We will not assume that our monoidal categories are strict. A good reference for the essentials of monoidal category theory is [24]. In a monoidal †-category we can give abstract definitions of some important terminology normally associated with Hilbert spaces.
772
J. Vicary
Definition 2.11. In a monoidal category, the scalars are the monoid Hom(I, I ). In a monoidal †-category, the scalars form a monoid with involution. Definition 2.12. In a monoidal †-category, a state of an object A is a morphism φ : I - A. Definition 2.13. In a monoidal †-category, the squared norm of a state φ : I - A is the scalar φ † ◦ φ : I - I . If our †-category also has a zero object, we note that it is quite possible for the squared norm of a non-zero state to be zero. For this reason, as it stands, Definition 2.13 seems a poor abstraction of the notion of the squared norm on a vector space. In [33] we describe a way to overcome this problem, but it will not affect us here. Monoidal †-categories have a simpler duality structure than many monoidal categories, as the following lemma shows. Lemma 2.14. In a monoidal †-category, left-dual objects are also right-dual objects. Proof. Given an object A with a left dual A∗L witnessed by left-duality morphisms LA : I - A∗L ⊗ A and ηLA : A ⊗ A∗L - I , we can define RA := ηLA † and ηLA := LA † which witness that A∗L is a right dual for A. Since left or right duals are always unique up to isomorphism, left duals must be isomorphic to right duals in a monoidal †-category. We will exploit this isomorphism to write A∗ instead of A∗L or A∗R , and it follows that A∗∗ A. However, this is not enough to imply that the functors (−)∗L and (−)∗R given in Definition 2.9 are naturally isomorphic; for this we will require extra compatibility conditions. Definition 2.15. A monoidal †-category with duals is a monoidal †-category such that each object A has an assigned dual object A∗ (either left or right by Lemma 2.14) with this assignment satisfying (A∗ )∗ = A, and assigned left and right duality morphisms for each object, such that these assignments are compatible with the †-functor in the following way: LA = ηRA † = ηLA∗† = RA∗ , ηLA = RA † = LA∗† = ηRA∗ , ((−)∗L )† = ((−)† )∗L .
(5)
Since the left and right duality morphisms can be obtained from each other using the †-functor, from now on we will only refer directly to the left-duality morphisms, defining A := LA and η A := ηLA . We note that there does not yet exist a precise theorem governing the soundness of the graphical calculus for this precise type of monoidal category with duals, although we fully expect that one could be proved. The graphical calculus used in this paper should therefore be thought of as a shorthand for the underlying morphisms in the category, rather than a calculational method in its own right. The compatibility condition ((−)∗L )† = ((−)† )∗L looks asymmetrical, as it does not refer to the right-duality functor (−)∗R . We show that it is equivalent to two different compatibility conditions. Lemma 2.16. As a part of the definition of a monoidal †-category with duals, the following compatibility conditions would be equivalent: 1. ((−)∗L )† = ((−)† )∗L , 2. ((−)∗R )† = ((−)† )∗R , 3. (−)∗L = (−)∗R .
Categorical Formulation of Finite-Dimensional Quantum Algebras
773
Proof. From the first two sets of equations between the duality morphisms given in Definition 2.15, it follows directly that ((−)∗L )† = ((−)† )∗R . We combine this with Condition 2 above to show that ((−)∗L )† = ((−)∗R )† , and since the †-functor is an involution, it then follows that (−)∗L = (−)∗R . Since this argument is reversible we have shown that 2 ⇔ 3, and an analogous argument demonstrates that 1 ⇔ 3. In a monoidal †-category the three given conditions will therefore all hold, and in particular the functors (−)∗L and (−)∗R will coincide. We denote this unique duality functor as (−)∗ . We use Condition 1 for Definition 2.15 rather than the more symmetrical Condition, since it follows from a general ‘philosophy’ of †-categories: wherever sensible, require that structures be compatible with the †-functor. We can use this result to demonstrate a useful property of the duality functor (−)∗ . Lemma 2.17. In a monoidal †-category with duals, the duality functor (−)∗ is an involution. Proof. The involution equation is ((−)∗ )∗ = id, and we rewrite this using Lemma 2.16 as ((−)∗L )∗R = id. Writing this out in full, it is easy to demonstrate using the duality equations and the compatibility equations of Definition 2.15. Since the †-functor is also strictly involutive and commutes with the duality functor, their composite is also an involutive functor. Definition 2.18. In a monoidal †-category with duals, the conjugation functor (−)∗ is defined on all morphisms f by f ∗ = ( f ∗ )† = ( f † )∗ . Since the †-functor is the identity on objects, we have A∗ = A∗ for all objects A. To make this equality clear we will write A∗ exclusively, and the A∗ form will not be used. For any morphism f : A - B we can use these functors to construct f ∗ : A∗ - B ∗ , ∗ f : B ∗ - A∗ and f † : B - A, and it will be important to be able to easily distinguish between these graphically. We will use an approach originally due to Selinger [30], in the form adopted by Coecke and Pavlovic [11]. Given the graphical representation of the duality functor (−)∗ given in (4), we could ‘pull the kink straight’ on the right-hand side of the equation. This would result in a rotation of the junction-box for f by half a turn. To make this rotation visible we draw our junction-boxes as wedges, rather than rectangles, breaking their symmetry. The duality (−)∗ is given by composing the conjugation functor (−)∗ and the †-functor, and since geometrically a half-turn can be built from two successive reflections, this gives us a complete geometrical scheme for describing the actions of our functors: B
A f†
f A
B
A∗ f∗ B∗
B∗ f∗ A∗
Our monoidal †-categories with duals are very similar to other structures considered in the literature, such as C*-categories with conjugates [14,34] and stronglycompact-closed categories [2,3]. In these contexts the functors (−)∗ and (−)∗ also play an important role.
774
J. Vicary
Involution monoids. An important tool in functional analysis is the ∗-algebra: a complex, associative, unital algebra equipped with an antilinear involutive homomorphism from the algebra to itself which reverses the order of multiplication. Category-theoretically, such a homomorphism is not very convenient to work with, since morphisms in a category of vector spaces are usually chosen to be the linear maps. However, if the vector space has an inner product, this induces a canonical antilinear isomorphism from the vector space to its dual. Composing this with the antilinear selfinvolution, we obtain a linear isomorphism from the vector space to its dual. This style of isomorphism is much more useful from a categorical perspective, and we use it to define the concept of an involution monoid. We will demonstrate that this is equivalent to a conventional ∗-algebra when applied in a category of complex Hilbert spaces. The natural setting for the study of these categorical objects is a category with a conjugation functor, as defined above. Definition 2.19. In a monoidal category, a monoid is an ordered triple (A, m, u) consisting of an object A, a multiplication morphism m : A ⊗ A - A and a unit morphism u : I - A, which satisfy associativity and unit equations:
=
=
=
(6)
Definition 2.20. In a monoidal †-category with duals, an involution monoid (A, m, u; s) is a monoid (A, m, u) equipped with a morphism s : A - A∗ called the linear involution, which is a morphism of monoids with respect to the monoid structure (A∗ , m ∗ , u ∗ ) on A∗ , and which satisfies the involution condition s∗ ◦ s = id A .
(7)
It follows from this definition that s and s∗ are mutually inverse morphisms, since applying the conjugation functor to the involution condition gives s ◦ s∗ = id A∗ . We also note that for any such involution monoid s : A - A∗ and s ∗ : A - A∗ are parallel morphisms, but they are not necessarily the same. Definition 2.21. In a monoidal †-category with duals, given involution monoids (A, m, u; s A ) and (B, n, v; s B ), a morphism f : A - B is a homomorphism of involution monoids if it is a morphism of monoids, and if it satisfies the involutionpreservation condition s B ◦ f = f∗ ◦ s A .
(8)
If an object B is self-dual, it is possible for the involution s B : B - B to be the identity. Let (B, n, v; id B ) be such an involution monoid. In this case, it is sometimes possible to find an embedding f : (A, m, u; s A ) ⊂- (B, n, v; id B ) of involution monoids even when the linear involution s A is not trivial! We will see an example of this in the next section. The following lemma establishes that the traditional concept of ∗-algebra and the categorical concept of an involution monoid are the same, in an appropriate context. We
Categorical Formulation of Finite-Dimensional Quantum Algebras
775
demonstrate the equivalence for finite-dimensional algebras, since the category of finitedimensional complex vector spaces forms a category with duals. However, involution monoids are useful far more generally, and with a careful choice of conjugation functor could be used just as well to describe infinite-dimensional algebras with an involution. Lemma 2.22. For a unital, associative algebra on a finite-dimensional complex Hilbert space V , there is a correspondence between the following structures: 1. antilinear maps t : V - V which are involutions, and which are order-reversing algebra homomorphisms; 2. linear maps s : V - V ∗ , where V ∗ is the dual space of V , satisfying s∗ ◦ s = id V , and which are algebra homomorphisms to the conjugate algebra on V ∗ . Furthermore, the natural notions of homomorphism for these structures are also equivalent. Proof. We first deal with the implication 1 ⇒ 2. We construct the linear isomorphism s by defining s ◦ φ := (t (φ))∗ for an arbitrary morphism φ : C - V . This is linear, because both t and (−)∗ are antilinear. It is a map V - V ∗ since t (φ) is an element of V , and the complex conjugation functor (−)∗ takes V to V ∗ . Checking the identity s∗ ◦ s = id V , we have s∗ ◦ s ◦ φ = s∗ ◦ (t (φ))∗ = (s ◦ t (φ))∗ = (tt (φ))∗∗ = φ. The monoid homomorphism condition is demonstrated similarly, for arbitrary states φ and ψ of V : s ◦ m ◦ (φ ⊗ ψ) = (t (m ◦ (φ ⊗ ψ)))∗ = (m ◦ (tψ ⊗ tφ))∗ = m ∗ ◦ ((tφ)∗ ⊗ (tψ)∗) = m ∗ ◦ (sφ ⊗ sψ) s ◦ u = (t (u))∗ = u ∗
definition of s t is order-reversing homomorphism order-reversing functoriality of (−)∗ definition of s definition of s, t is homomorphism.
For the implication 2 ⇒ 1, we define t (φ) := (s ◦ φ)∗ for all elements φ of V . The proof that t has the required properties is similar to the proof involved in the implication 1 ⇒ 2. The constructions of s and t in terms of each other are clearly inverse, and so the equivalence has been demonstrated. We now check that homomorphisms between these structures are the same. Our notion of homomorphism between structures of type 2 is given by that in Definition 2.21, and there is a natural notion of homomorphism between monoids equipped with an antilinear self-involution. Consider algebras (A, m, u) and (B, n, v) equipped with antilinear involutive order-reversing homomorphisms t A : A - A and t B : B - B respectively, and let f : A - B be any continuous linear map. It will be compatible with the involutions if t B ◦ f = f ◦ t A . Acting on some state φ of A, and constructing linear maps s A : A - A∗ and s B : B - B ∗ in the manner defined above, we obtain t B ◦ f ◦ φ = s B ∗ ◦ ( f ◦ φ)∗ = s B ∗ ◦ f ∗ ◦ φ∗ and f ◦ t A ◦ φ = f ◦ s A∗ ◦ φ∗ . Equating these and complex-conjugating we have s B ◦ f = f ∗ ◦ s A as required. Conversely, let (A, m, u; s A ) and (B, n, v; s B ) be involution monoids in Hilb, and let f : A - B again be any linear map. If the involution-preservation condition s B ◦ f = f ∗ ◦ s A holds, then applying an arbitrary state φ we obtain s B ◦ f ◦ φ = (t ( f ◦ φ))∗ and f ∗ ◦ s A ◦ φ = f ∗ ◦ (tφ)∗ respectively for the left and right sides of the equation. Equating these and complex-conjugating, we obtain t ( f ◦ φ) = f ◦ (tφ) as required.
776
J. Vicary
3. Results on †-Frobenius Monoids Introducing †-Frobenius monoids. We begin with definitions of the important concepts. Definition 3.1. In a monoidal category, a comonoid is the dual concept to a monoid; that is, it is an ordered triple (A, n, v)× consisting of an object A, a comultiplication n : A - A ⊗ A and a counit v : A - I , which satisfy coassociativity and counit equations:
=
=
=
(9)
If an object has both a chosen monoid structure and a chosen comonoid structure, then there is an important way in which these might be compatible with each other. Definition 3.2. In a monoidal category, a Frobenius structure is a choice of monoid (A, m, u) and comonoid (A, n, v)× for some object A, such that the multiplication m and the comultiplication n satisfy the following equations:
=
=
(10)
Reading these diagrams from bottom to top, the splitting of a line represents the comultiplication n, and merging of two lines represents the multiplication m. This geometrical definition of a Frobenius structure, although well-known, is superficially quite different to the ‘classical’ definition in terms of an exact pairing. The equivalence of these two definitions was first observed by Abrams [1], and an accessible discussion of the different possible ways to define a Frobenius algebra is given in the book by Kock [20]. This geometrical definition was first suggested by Lawvere, and was subsequently popularized in the lecture notes of Quinn [29]. An important property of a Frobenius structure is that it can be used to demonstrate that the underlying object is self-dual. If we are working in a †-category, from any monoid (A, m, u) we can canonically obtain an ‘adjoint’ comonoid (A, m † , u † )× , and it is then natural to make the following definition. Definition 3.3. In a monoidal †-category, a monoid (A, m, u) is a †-Frobenius monoid if it forms a Frobenius structure with its adjoint (A, m † , u † )× . This construction is similar to an abstract Qsystems [23]. Given a †-Frobenius monoid (A, m, u), we refer to m † as its comultiplication and to u † as its counit.
Categorical Formulation of Finite-Dimensional Quantum Algebras
777
Involutions on †-Frobenius monoids. We now look at the relationship between †-Frobenius monoids and the involution monoids of Sect. 2. We will see that a †-Frobenius monoid can be given the structure of an involution monoid in two canonical ways, which in general will be different. Definition 3.4. In a monoidal †-category with duals, a †-Frobenius monoid (A, m, u) has a left involution sL : A - A∗ and right involution sR : A - A∗ defined as follows:
=
=
(11)
sL := ((u † ◦ m) ⊗ id A∗) ◦ (id A ⊗ A∗) sR := id A∗ ⊗ (u † ◦ m) ◦ A ⊗ id A In each case the second picture is just a convenient shorthand, which should literally be interpreted as the first picture. These involutions interact with the conjugation and transposition functors in interesting ways, as we explore in the next lemma. Lemma 3.5. In a monoidal †-category with duals, the left and right involutions of a †-Frobenius monoid satisfy the following equations: sL ∗ = s R , s R ∗ = s L , sL∗ = sL−1 , sR∗ = sR−1 ,
(12) (13)
sL−1 = sR † , sR−1 = sL † .
(14)
Proof. Equations (12) follow from the definitions of the involutions and the graphical representation of the functor (−)∗ , which rotates a diagram half a turn about an axis perpendicular to the page. Equations (13) follow from the †-Frobenius and unit equations; taking the right-involution case, we show this by establishing that sR∗ ◦ sR = id A with the following graphical proof:
=
=
=
Applying the functor (−)∗ to this equation gives sR ◦ sR∗ = id A∗ , establishing that sR and sR∗ are inverse; applying the functor (−)∗ to this argument establishes that sL and sL∗ are inverse. Equations (14) follow from Eqs. (12) and (13) and the properties of the functors (−)∗ , (−)∗ and †. We note that left and right involutions could be defined for arbitrary monoids in a monoidal †-category with duals, but they would not satisfy Eqs. (13) and (14) above. We now combine these results on involutions of †-Frobenius monoids with the concept of an involution monoid from Sect. 2.
778
J. Vicary
Lemma 3.6. In a monoidal †-category with duals, given a †-Frobenius monoid (A, m, u) we can canonically obtain two involution monoids (A, m, u; sL ) and (A, m, u; sR ), where sL and sR are respectively the left and right involutions associated to the monoid. Proof. We deal with the right-involution case; the left-involution case is analogous. We must show that s R : A - A∗ is a morphism of monoids, and that it satisfies the involution condition. We first show that it preserves multiplication, employing the Frobenius, unit and associativity laws:
=
=
=
=
=
We omit the proof that s R preserves the unit, as it is straightforward. The involution condition s R ∗ ◦ s R = id A follows from one of Eqs. (13) in Lemma 3.5. This leads us to the following definition. Definition 3.7. In a monoidal †-category with duals, a †-Frobenius left- (or right-) involution monoid is an involution monoid (A, m, u; s) such that the monoid (A, m, u) is †-Frobenius, and such that the involution s is the left (or right) involution of the †-Frobenius monoid in the manner described by Definition 3.4. A homomorphism of †-Frobenius left- or right-involution monoids would therefore be required to preserve the involution as well as the multiplication and unit, as per Definition 2.21. A useful property of †-Frobenius right-involution monoids is described by the following lemma, which gives a necessary and sufficient algebraic condition for a monoid homomorphism to be an isometry. Lemma 3.8. In a monoidal †-category with duals, a homomorphism of †-Frobenius right-involution monoids is an isometry if and only if it preserves the counit. Proof. Let j : (A, m, u) - (B, n, v) be a homomorphism between †-Frobenius rightinvolution monoids. Assuming that j preserves the counit, we show that it is an isometry by the following graphical argument. The third step uses the fact that j preserves the involution, the fifth that it is a homomorphism of monoids, and the sixth that it preserves the counit.
Categorical Formulation of Finite-Dimensional Quantum Algebras
j† j
=
j∗ j
=
j∗
=
j
779
j j
j =
j
=
j
=
=
Now instead assume that j is an isometry. It is a homomorphism, so we have the unitpreservation equation j ◦ u = v, and therefore j † ◦ j ◦ u = u = j † ◦ v. Applying the †-functor to this we obtain u † = v † ◦ j, which is the counit-preservation condition. Special unitary †-Frobenius monoids. We will mostly be interested in the case when the two involutions are the same, and we now explore under what conditions this holds. Definition 3.9. In a monoidal †-category with duals, a †-Frobenius monoid is unitary if the left involution, or equivalently the right involution, is unitary. That these two conditions are equivalent follows from Lemma 3.5. Definition 3.10. In a braided monoidal †-category with duals, a †-Frobenius monoid is balanced-symmetric if the following equation is satisfied:
=
(15)
The term symmetric is standard (for example, see [20, Sect. 2.2.9]), and describes a similar property that lacks the ‘balancing loop’ on one of the legs of the right-hand side of the equation. In Hilb this loop is the identity and so the concepts are the same, but this may not be the case in other categories of interest. Lemma 3.11. In a monoidal †-category with duals, the following properties of a †-Frobenius monoid are equivalent: 1. it is unitary; 2. it is balanced-symmetric; 3. the left and right involutions are the same; where Property 2 only applies if the monoidal structure has a braiding.
780
J. Vicary
Proof. We first give a graphical proof that 3 ⇒ 2, using Property 3 to transform the second expression into the third:
=
=
=
=
A similar argument shows that 2 ⇒ 3. From Eqs. (14) of Lemma 3.5 it follows that 1 ⇔ 3, and so all three properties are equivalent. We will mostly use the term ‘unitary’ to refer to these equivalent properties, since it is more obviously in keeping with the general philosophy of †-categories, that all structural isomorphisms should be unitary. We also note that if a †-Frobenius left- or rightinvolution monoid is unitary then we can simply refer to it as a ‘†-Frobenius involution monoid’, as the left and right involutions coincide in that case. One particularly nice feature of unitary †-Frobenius monoids is that we can canonically obtain an abstract ‘dimension’ of their underlying space from the multiplication, unit, comultiplication and counit, as the following lemma shows. In a category of vector spaces and linear maps, this dimension will correspond to the dimension of the vector space. Definition 3.12. In a monoidal †-category with duals, the dimension of an object A is given by the scalar A † ◦ A : I - I , and is denoted dim(A). Lemma 3.13. In a monoidal †-category with duals, given a unitary †-Frobenius monoid (A, m, u), dim(A) = u † ◦ m ◦ m † ◦ u; that is, the dimension of A is equal to the squared norm of m † ◦ u. Also, dim(A) = dim(A)∗ . Proof. We demonstrate this with the following series of pictures:
dim(A) =
=
=
=
=
=
=
= dim(A)∗ .
The central diagram is u † ◦ m ◦ m † ◦ u, so this proves the lemma. The notion of the dimension of an object is a crucial one in the theory of monoidal categories with duals, and is studied in depth throughout the literature [9,14,23]. However, we do not rely on it heavily in this paper, and more axioms would be required for our category than those assumed here for the dimension to have good properties, such as being independent of the choice of duality morphisms, or being an element of the integers. We now introduce one final property of a †-Frobenius monoid.
Categorical Formulation of Finite-Dimensional Quantum Algebras
781
Definition 3.14. In a monoidal †-category, a †-Frobenius monoid (A, m, u) is special if m ◦ m † = id A ; that is, if the comultiplication is an isometry. The term special goes back to Quinn [29]. A special †-Frobenius monoid is the same as an abstract Q-system [23], and a useful lemma proved in that reference is that if a monoid (A, m, u) satisfies m ◦ m † = id A , then it is necessarily a special †-Frobenius monoid. It simplifies the expression for the dimension of the underlying space, as demonstrated by this Lemma. Lemma 3.15. In a monoidal †-category with duals, a special unitary †-Frobenius monoid (A, m, u) has dim(A) = u † ◦ u; that is, the dimension of A is equal to the squared norm of u. Proof. Straightforward from Lemma 3.13. Endomorphism monoids. Given any Hilbert space H , it is often useful to consider the algebra of bounded linear operators on H . These give the prototypical examples of C*-algebras, with the ∗-involution given by taking the operator adjoint. In a monoidal category with duals we can construct endomorphism monoids, which are categorical analogues of these algebras of bounded linear operators. These well-known constructions, which go back at least to Müger [27], form an important class of †-Frobenius monoids, and that they have particularly nice properties. Definition 3.16. In a monoidal category, for an object A with a left dual A∗L , the endomorphism monoid End(A) is defined by End(A) := A∗L ⊗ A, id A∗L ⊗ ηLA ⊗ id A , LA . (16) The following lemma describes a well-known connection between categorical duality and Frobenius structures. Lemma 3.17. In a monoidal †-category with duals, an endomorphism monoid is a †-Frobenius monoid. Proof. That the †-Frobenius property holds for an endomorphism monoid End(A) is clear from its graphical representation, which we give here:
=
=
They are examples of the unitary monoids discussed in the previous section. Lemma 3.18. In a monoidal †-category with duals, endomorphism monoids are unitary.
782
J. Vicary
Proof. Following Eq. (11) for the left involution associated to a †-Frobenius monoid, we obtain the following:
This is clearly the identity on A∗ ⊗ A. The right involution is also the identity, by the conjugate of this picture. By Lemma 3.11 the †-Frobenius monoid must therefore unitary. We note that the order-reversing property of the duality functor (−)∗ is crucial here, as the only canonical choice of ‘identity’ morphism A∗ ⊗ A - A ⊗ A∗ would be the braiding isomorphism, but such a braiding is not necessarily present. Also, although the linear involution associated with an endomorphism monoid is the identity, the induced order-reversing antilinear involution on A∗ ⊗ A is certainly not the identity: it is given by taking the name of an operator to the name of the adjoint to that operator, as can be checked by going through the correspondence described in Lemma 2.22. The following lemma is a formal description of the intuitive notion that an algebra should have a homomorphism into the algebra of operators on the underlying space, given by taking the right action of each element. Lemma 3.19. Let (A, m, u) be a monoid in a monoidal category in which the object A has a left dual. Then (A, m, u) has a monic homomorphism into the endomorphism monoid of A. Proof. The embedding morphism h : (A, m, u) ⊂- End(A) is defined by h := (id A∗ ⊗ m) ◦ ( LA ⊗ id A ),
(17)
which has the following graphical representation: A∗ ⊗ A 6 =
h A
We show that it is monic by postcomposing with u ∗ ⊗ id A , which acts as a retraction:
=
=
Categorical Formulation of Finite-Dimensional Quantum Algebras
783
Next we show that h preserves the multiplication operation, employing a duality equation and the associative law:
=
=
Finally, we show that the embedding preserves the unit, employing the unit law:
=
However, as we saw in the Introduction, for the case of †-Frobenius monoids this embedding has a special property: it preserves an involution. We establish this formally in the following lemma. Lemma 3.20. Let (A, m, u; s R ) be a †-Frobenius right-involution monoid. Then the canonical embedding of (A, m, u; s R ) into the †-Frobenius involution monoid End(A) is a morphism of involution monoids. Proof. By Lemma 3.19 the embedding must be a morphism of monoids. Note that we do not need to specify whether we are using the left or right involution of End(A), since by Lemma 3.18 they are both the identity. We must show that this embedding morphism k : A ⊂- A∗ ⊗ A satisfies the involution condition k = k∗ ◦ s R given in Definition 2.21. The proof uses the Frobenius law and the unit law.
=
=
=
It is worth noting that a symmetry has been broken; this lemma would not hold with ‘right-involution’ replaced with ‘left-involution’. This is a consequence of defining the underlying object of our endomorphism monoid to be A∗ ⊗ A rather than A ⊗ A∗ . In a braided monoidal category there would be no essential difference, but we are working at a higher level of generality.
784
J. Vicary
An embedding lemma. We finish this section by demonstrating another general property of †-Frobenius involution algebras. Just as every involution-closed subalgebra of a finitedimensional C*-algebra is also a C*-algebra, we will show that every involution-closed submonoid of a †-Frobenius involution monoid is also †-Frobenius. The next section makes this analogy with C*-algebras precise, but we can prove it here as a general result about †-Frobenius algebras. Lemma 3.21. In a monoidal †-category with duals, let (A, m, u; s) be an involution monoid with an involution-preserving †-embedding into a †-Frobenius left- (or right-) involution monoid. Then (A, m, u; s) is itself a †-Frobenius left- (or right-) involution monoid. Proof. We will deal with the left-involution case; the right-involution case is analogous. Let p : (A, m, u; s) ⊂- (B, n, v; t) be a †-embedding of an involution monoid into a †-Frobenius left-involution monoid. The †-embedding property means that p † ◦ p = id A . In our graphical representation we will use a thin line for A and a thick line for B, and a transition between these types of line for the embedding morphism p. The involutionpreservation condition t ◦ p = p∗ ◦ s is then represented by the following picture:
= s
Applying complex conjugation to p † ◦ p = id A we obtain p ∗ ◦ p∗ = id A∗ , and applying this to the equation pictured above we obtain s = p ∗ ◦ t ◦ p. Also, from the monoid homomorphism equation p ◦ u = v we obtain u = p † ◦ v, and therefore u † = v † ◦ p by applying the †-functor. Using these equations, along with the multiplication compatibility equation p ◦ m = n ◦ ( p ⊗ p), we obtain the following:
s
=
=
=
=
=
The involution is therefore the left involution associated to the monoid. We now show that the monoid is in fact a †-Frobenius monoid. To start with we use the fact that p is an isometry and that it preserves multiplication, along with the unit law
Categorical Formulation of Finite-Dimensional Quantum Algebras
785
of the monoid and the Frobenius law:
=
=
=
=
=
We now employ the fact that p preserves the involution, and then essentially perform the previous few steps in reverse order:
=
=
=
=
The proof for the other Frobenius law is exactly analogous. We have demonstrated that the monoid (A, m, u) is †-Frobenius, and since we have shown that the involution s is the left involution associated to the monoid, it follows that (A, m, u; s) is a †-Frobenius left-involution monoid. 4. Special Unitary †-Frobenius Monoids in Hilb From now on we will mainly work in Hilb, the category of finite-dimensional complex Hilbert spaces and linear maps, which is a symmetric monoidal †-category with duals. Special unitary †-Frobenius monoids have particularly good properties in this setting. The following lemma contains the important insight due to Coecke, Pavlovic and the author, as described in the Introduction and in [12]. Lemma 4.1. In Hilb, a †-Frobenius right-involution monoid admits a norm making it into a C*-algebra. Proof. By Lemma 3.20 a †-Frobenius right-involution monoid (A, m, u) has an involution-preserving embedding into End(A), which is a C*-algebra when equipped with the operator norm. The involution monoid (A, m, u) therefore admits a C*-algebra norm, taken from the norm on End(A) under the embedding. Since the algebra is finitedimensional, the completeness requirement is trivial. We will also require the following important result, which demonstrates a crucial abstract property of the category Hilb. Lemma 4.2. In Hilb, isomorphisms of special unitary †-Frobenius involution monoids preserve the counit.
786
J. Vicary
Proof. Any special unitary †-Frobenius involution monoid is in particular a †-Frobenius right-involution monoid, and so admits a norm with which it becomes a C*-algebra by Lemma 4.1. Finite-dimensional C*-algebras are semisimple, and are therefore isomorphic to finite direct sums of matrix algebras in a canonical way; an isomorphism between two finite-dimensional C*-algebras is then given by a direct sum of pairwise isomorphisms of matrix algebras. We therefore need only show that the lemma is true for special unitary †-Frobenius involution monoids which are matrix algebras, with involution given by matrix adjoint. Let (A, m, u; s) and (B, n, v; t) be special unitary †-Frobenius involution monoids which are both isomorphic to some matrix algebra End(Cn ). Any isomorphism between them must have some decomposition into isomorphisms f : (A, m, u; s) - End(Cn ) and g : End(Cn ) - (B, n, v; t). The statement that g ◦ f preserves the counit is equivalent to the statement that the outside diamond of the following diagram commutes: u†
C - 6
(A, m, u; s)
v† (B, n, v; t) -
Tr f
-
(18)
g
End(Cn ) We will show that each triangle separately commutes, and therefore that the entire diagram commutes. We focus on the triangle involving the isomorphism g; the treatment of the other triangle is analogous. Our strategy is to show that ρg := n1 · v † ◦ g is a tracial state of End(Cn ). It takes the unit to 1, since n1 · v † ◦ g ◦ LB = n1 · v † ◦ v = n1 · dim(B) = 1 n · n = 1, where we used the fact that g is a homomorphism and Lemma 3.15; this is the reason that we require the †-Frobenius monoid to be special. We can simplify the action of ρg on positive elements in the following way, where φ : I - Cn ∗ ⊗ Cn is an arbitrary nonzero state of End(Cn ), and φ is the result of applying the involution to this state:
nρg
φ†
g =
φ
φ
= g φ
φ
φ
g φ
= g φ
g† =
φ†
φ†
g†
g† =
g
g
φ
φ
The expression on the right-hand side is the squared norm of g ◦ φ, which is positive because the inner product in Hilb is nondegenerate and φ is nonzero; this shows that ρg takes positive elements to nonnegative real numbers, and so is a state of End(Cn ). By Lemma 3.11 the involution monoid End(A) is balanced-symmetric, and since we are in Hilb, the balancing loop can be neglected; this means that ρg ◦ (a ⊗ b) = ρg ◦ (b ⊗ a) for all a, b ∈ End(A), and so ρg is tracial. Altogether ρg is a tracial state of a matrix algebra. However, it is a standard result that the matrix algebra on a complex n-dimensional vector space has a unique tracial state given by n1 Tr (for example, see [28, Example 6.2.1]). It follows that ρg = n1 Tr, and so the triangle commutes as required.
Categorical Formulation of Finite-Dimensional Quantum Algebras
787
We can combine this with an earlier Lemma to obtain a very useful result. Lemma 4.3. In Hilb, isomorphisms of special unitary †-Frobenius involution monoids are unitary. Proof. Straightforward from Lemmas 3.8 and 4.2. Given a †-Frobenius monoid in Hilb, we will show that scaling the inner product on the underlying complex vector space produces a family of new †-Frobenius monoids. We first note the following relationship between scaling inner products and adjoints to linear maps. Lemma 4.4. Let V be a complex vector space with inner product (−, −)V and let f : V ⊗n - V ⊗m be a linear map, with the adjoint f † under this inner product. If the inner product is scaled to α · (−, −)V for α a positive real number, the adjoint to f becomes α m−n f † . Proof. Writing the scaled inner product as ((−, −))V and denoting the adjoint to f under this scaled inner product as f ‡ , we must have (( f ◦ x, y))V ⊗m = ((x, f ‡ ◦ y))V ⊗n . Using ((−, −))V ⊗n = α n · (−, −)V ⊗n and making the substitution f ‡ = α m−n f † , we obtain ( f ◦ x, y)V ⊗m = (x, f † ◦ y)V ⊗n which holds by the definition of f † , and so f ‡ is a valid adjoint to f under the new inner product. Lemma 4.5. For a †-Frobenius monoid (A, m, u), scaling the inner product on A by any positive real number gives rise to a new †-Frobenius monoid. Moreover, this scaling preserves unitarity. Proof. This is easy to show using the previous lemma. The †-Frobenius equations will all be scaled by the same factor since they are all composed from a single m and m † , so they will still hold. The unitarity property is an equation involving an m and a u † on each side, and so both sides of this equation will also scale by the same factor. We are now ready to prove our main correspondence theorem between finitedimensional C*-algebras and symmetric unitary †-Frobenius monoids. Theorem 4.6. In Hilb, the following properties of an involution monoid are equivalent: 1. it admits a norm making it a C*-algebra; 2. it admits an inner product making it a special unitary †-Frobenius involution monoid; 3. it admits an inner product making it a †-Frobenius right-involution monoid. Furthermore, if these properties hold, then the structures in 1 and 2 are admitted uniquely. Proof. First, we point out that the norm of Property 1 is not directly related to the inner products of Properties 2 or 3, in the usual way by which a norm can be obtained from an inner product, and sometimes vice-versa. In fact, the norm of a C*-algebra will usually not satisfy the parallelogram identity, and so cannot arise directly from any inner product. We begin by showing 1 ⇒ 2. We first decompose our finite-dimensional C*-algebra into a finite direct sum of matrix algebras. For any such matrix algebra, an inner product is given by (a, b) := Tr(a † b), which is normalized such that Tr(id) = n for a matrix algebra acting on Cn . This gives an endomorphism monoid End(Cn ) in Hilb for each n, which is a unitary †-Frobenius monoid as described by Lemmas 3.17 and 3.18. Such a monoid is not special unless it is one-dimensional; we have m ◦ m † = n · id A∗ ⊗A , where
788
J. Vicary
m is the multiplication for the endomorphism monoid. We rescale the inner product, replacing it with ((a, b)) := n Tr(a † b). As described by Lemma 4.4, writing the adjoint of m under this new inner product as m ‡ , we will have m ‡ = n1 m † , and m ◦m ‡ = id A∗ ⊗A . By Lemma 4.5 this preserves the involution and the unitarity of the monoid, and so we obtain a special unitary †-Frobenius monoid with the same underlying algebra and involution as the original matrix algebra. Taking the direct sum of these for each matrix algebra in the decomposition gives a special unitary †-Frobenius involution monoid, with the same underlying algebra and involution as the original C*-algebra. The implication 2 ⇒ 3 is trivial, and the implication 3 ⇒ 1 is contained in Lemma 4.1, so the three properties are therefore equivalent. We now show that, if these properties hold, the norm and inner product in Properties 1 and 2 are admitted uniquely. It is well-known that a C*-algebra admits a unique norm. Now assume that a finite-dimensional complex ∗-algebra has two distinct inner products, which give rise to two special unitary †-Frobenius involution monoids. Since these monoids have the same underlying set of elements and the same involution, there is an obvious involution-preserving isomorphism between them given by the identity on the set of elements. But by Lemma 4.3 any isomorphism of special unitary †-Frobenius involution monoids in Hilb is necessarily an isometry, and therefore unitary, and so the inner products on the two monoids are in fact the same. As a result, we can demonstrate some equalities and equivalences of categories. Theorem 4.7. The category of finite-dimensional C*-algebras is 1. equal to the category of special unitary †-Frobenius involution monoids in Hilb; 2. equivalent to the category of unitary †-Frobenius involution monoids in Hilb; and 3. equivalent to the category of †-Frobenius right-involution monoids in Hilb; where all of these categories have involution-preserving monoid homomorphisms as morphisms. Proof. We prove 1 by noting that the objects of the category of finite-dimensional C*-algebras are the same as the objects in the category of special unitary †-Frobenius involution monoids in Hilb, since in both cases they are involution monoids satisfying one of the first two equivalent properties of Theorem 4.6, which can only be satisfied uniquely. The morphisms are also the same, and so the categories are equal. For 2 and 3, we note that both of these types of structure admit C*-algebra norms by Lemma 4.1. This gives rise to functors from the categories of 2 and 3 to the category of finite-dimensional C*-algebras. These functors are full and faithful on hom-sets, since the hom-sets have precisely the same definition in both categories, consisting of all involution-preserving algebra homomorphisms. These functors are also surjective on objects, since given a finite-dimensional C*-algebra, by Theorem 4.6 we can find an inner product on the underlying vector space such that the ∗-algebra is in fact a special unitary †-Frobenius involution monoid. Recall that the latter are the objects in the categories of 2 and 3. Since the two functors are full, faithful and surjective, they are therefore equivalences. Our use of the adjective ‘equal’ here perhaps deserves some explanation. It is only appropriate given the way that we have defined the categories of C*-algebras and of special unitary †-Frobenius monoids, with objects being ∗-algebras that have the property of admitting an appropriate norm or inner product. Had we instead defined the objects as being ∗-algebras equipped with their norm or inner product, then the categories would not be equal but isomorphic.
Categorical Formulation of Finite-Dimensional Quantum Algebras
789
Having demonstrated the equivalence between finite-dimensional C*-algebras and †-Frobenius monoids, it becomes clear that Lemmas 3.19 and 3.20 are precisely the finite-dimensional noncommutative Gelfand-Naimark theorem, that any abstract finitedimensional C*-algebra has an involution-preserving embedding into the algebra of bounded linear operators on a Hilbert space. It is striking that these lemmas are quite easy to prove from the †-Frobenius monoid point of view, compared to the traditional C*-algebra perspective. However, to prove Theorem 4.6 we used the decomposition theorem for finite-dimensional C*-algebras from which the finite-dimensional noncommutative Gelfand-Naimark theorem trivially follows, so this does not constitute a new proof; for this, we would need a more direct way to establish the link between finitedimensional C*-algebras and †-Frobenius monoids. In contrast, some properties of C*-algebras are harder to demonstrate from the perspective of †-Frobenius monoids, as demonstrated by Lemma 3.20. The proof of that lemma required 14 applications of identities, while the corresponding property of finitedimensional C*-algebras, that any involution-closed subalgebra is also a C*-algebra, is trivial. 5. Generalizing the Spectral Theorem Classical structures and spectral categories. As a consequence of being able to define finite-dimensional C*-algebras internally to a category, we are also able to state the finite-dimensional spectral theorem categorically. As an introduction to this, we first give a brief summary of some of the main ideas of [12]. We start by introducing an important connection between commutative †-Frobenius monoids and finite sets. Definition 5.1. In a braided monoidal category, a monoid is commutative if the braiding and the multiplication satisfy the commutativity equation:
=
(19)
Theorem 5.2. The category of commutative †-Frobenius monoids in Hilb with involution-preserving1 monoid homomorphisms as morphisms is equivalent to the opposite of FinSet, the category of finite sets. Proof. A commutative †-Frobenius monoid in Hilb is balanced-symmetric, since the balancing is the identity in that category, and is therefore unitary by Lemma 3.11. By Theorem 4.6, the category being constructed is therefore isomorphic to the category of finite-dimensional commutative C*-algebras with algebra homomorphisms as morphisms. We apply the spectral theorem for commutative C*-algebras to obtain the desired result. 1 In fact, this involution-preservation condition is not required: as demonstrated in [12], every homomorphism of finite-dimensional commutative C*-algebras is involution-preserving.
790
J. Vicary
Put more straightforwardly, a choice of commutative †-Frobenius monoid on a Hilbert space defines a basis for that Hilbert space. In fact, the bases for each space are in precise correspondence to the special commutative †-Frobenius monoids, as might be expected from our Theorem 4.6; the same basis will be determined by many different †-Frobenius monoids. Theorem 5.2 motivates the following definition: Definition 5.3. In a braided monoidal †-category, a classical structure is a commutative †-Frobenius monoid. If the underlying object is A, then we say that it is a classical structure on A. Classical structures were first described by Coecke and Pavlovic in [11], and the philosophy of that paper — that a classical structure represents the possible outcomes of a measurement — is embraced here. Definition 5.4. Given a braided monoidal †-category Q, its category of classical structures C(Q) is the category with classical structures in Q for objects, and involutionpreserving monoid homomorphisms as morphisms. Using this notation, the result in Theorem 5.2 can be written as C(Hilb) FinSet op .
(20)
These results give a new perspective on the relationship between finitedimensional Hilbert spaces and finite sets. We can construct a covariant forgetful functor Forget : C(Hilb) - Hilb which takes a classical structure to its underlying Hilbert space. We can also construct a covariant functor Free : FinSet op - Hilb, which takes a set to a Hilbert space freely generated by taking that set as an orthonormal basis, and a function between sets to the adjoint of the linear map that has the same action on the chosen basis. Using the equivalence C(Hilb) FinSetop implied by Theorem 5.2, we see that the functors Forget and Free are naturally isomorphic. We have two quite different points of view, which are both equally valid: a set is a Hilbert space with the extra structure of a special commutative †-Frobenius monoid, and a Hilbert space is a set with the extra structure of a complex vector space. One possible point of view is that a classical structure represents a measurement performed on the underlying Hilbert space, or rather, on the physical system which has that Hilbert space as its space of states. To say ‘the possible results of a measurement form a finite set’ can then be directly interpreted by the formalism: if we are doing our quantum theory in a braided monoidal †-category Q, it is simply the statement that C(Q) FinSet. The emergent ‘classical logic’ with which we reason about these measurement results is then more ‘powerful’ when the category C(Q) has more interesting properties; for example, it could be a fully-fledged elementary topos, as for the case of Hilb. With this in mind, we make the following definition: Definition 5.5. A braided monoidal †-category Q is spectral if C(Q) is an elementary topos. Spectral categories can be thought of as generalized settings for quantum theory which admit a particularly good ‘generalized spectral theorem’, or in which measurement outcomes admit a particularly good logic. We describe a class of spectral categories in Theorem 5.11, which have finite Boolean topoi as their categories of classical objects. We briefly mention a connection to other work. Döring and Isham [15] have developed a topos-theoretic approach to analyzing the logical structure of theories of physics,
Categorical Formulation of Finite-Dimensional Quantum Algebras
791
in which a quantum system is explored through the presheaves on the partially-ordered set of commutative subalgebras of a von Neumann algebra. In finite dimensions von Neumann algebras coincide with C*-algebras, and therefore also with special unitary †-Frobenius monoids in Hilb by Theorem 4.6. Given a †-Frobenius monoid of this type, the partially-ordered set of special commutative sub-†-Frobenius monoids can be constructed categorically, and so Döring-Isham toposes can be constructed directly from any special unitary †-Frobenius monoid in any braided monoidal †-category. The techniques of that research program can then be employed; in particular, we can test whether a generalized Kochen-Specker theorem holds. In fact, we suggest that this approach could be used quite generally to connect the ideas of Döring and Isham to other work on monoidal categories in the foundations of quantum physics, such as that of Abramsky, Coecke and others [4,10]. The spectral theorem for normal operators. We now turn to the spectral theorem for normal operators, which says that a normal operator on a complex Hilbert space can be diagonalized. For complex Hilbert spaces this follows from the spectral theorem for commutative C*-algebras, since any normal operator generates a commutative C*-algebra and the spectrum of this algebra performs the diagonalization. This will not necessarily be the case in an arbitrary monoidal †-category, with C*-algebras replaced by special unitary †-Frobenius monoids. However, we can nonetheless give a direct categorical description of diagonalization. We proceed by introducing two different categorical properties which capture the geometrical essence of the spectral theorem for normal operators, and then showing that they are equivalent. Definition 5.6. In a monoidal category, an endomorphism f : A - A is compatible with a monoid (A, m, u) if the following equations hold:
f =
=
f
f
(21)
m ◦ ( f ⊗ id A ) = f ◦ m = m ◦ (id A ⊗ f ) Definition 5.7. In a braided monoidal †-category, an endomorphism f : A - A is internally diagonalizable if it can be written as an action of an element of a commutative †-Frobenius algebra on A; that is, if it can be written as
f
=
(22) φf
f
= m ◦ (φ f ⊗ id A ),
where m : A ⊗ A - A is the multiplication of a commutative †-Frobenius algebra and φ f : I - A is a state of A.
792
J. Vicary
Lemma 5.8. An endomorphism f : A - A is internally diagonalizable if and only if it is compatible with a commutative †-Frobenius monoid. Proof. Assume that f is internally diagonalizable by the action of an element φ f : I - A of a commutative †-Frobenius monoid (A, m, u), so that f = m ◦(φ f ⊗id A ). The following pictures must be equal by the associativity and commutativity laws, where the multiplication is the morphism m:
φf
=
= φf
= φf
φf
The first picture is f ◦ m, the second is m ◦ ( f ⊗ id A ) and the fourth is m ◦ (id A ⊗ f ), and so f is compatible with the commutative †-Frobenius monoid (A, m, u). Conversely, assuming compatibility of f with a commutative †-Frobenius monoid (A, m, u) and defining φ f = f ◦ u, we have m ◦ (φ f ⊗ id A ) = m ◦ (( f ◦ u) ⊗ id A) = f ◦ m ◦ (u ⊗ id A ) = f, and so f is internally diagonalizable. We now show that any internally-diagonalizable endomorphism must be normal, by the properties of commutative †-Frobenius monoids. Lemma 5.9. If an endomorphism f : A - A is internally diagonalizable, then it is normal. Proof. The statement that f is internally diagonalizable is equivalent to the statement that f can be written as the left-action of a commutative †-Frobenius monoid. By commutativity this is the same as a right action, and using the notation of the Introduction we write this as Rα for an element α ∈ A. We then have f ◦ f † = Rα ◦ Rα † = Rα ◦ Rα , where α is defined as in the Introduction. By commutativity we have Rα ◦ Rα = Rα ◦ Rα , and so f ◦ f † = f † ◦ f . Every internally diagonalizable endomorphism is normal, but is every normal endomorphism internally diagonalizable? This is precisely the content of the conventional spectral theorem for normal operators, and so in Hilb the answer is yes. Lemma 5.10. In Hilb, every normal endomorphism f : A - A is internally diagonalizable. Proof. This follows from the conventional spectral theorem for normal operators. We choose an orthonormal basis set ai : C - A, for 1 ≤ i ≤ dim(A), such that each vector ai is an eigenvector for f . The orthonormal property can be expressed as ai† ◦ a j = δi j idC .
Categorical Formulation of Finite-Dimensional Quantum Algebras
793
This basis set is uniquely determined if and only if f is nondegenerate. We use the morphisms ai to construct a monoid (A, m, u) on A as follows: m :=
dim(A)
ai ◦ (ai† ⊗ ai† ),
i=1
u :=
dim(A)
ai .
i=1
It is straightforward to show that this monoid is in fact a †-Frobenius monoid, which copies the chosen basis for A. Since this monoid only copies eigenvectors of f it follows that it is compatible with f in the sense of Definition 5.6, and so by Lemma 5.8, the morphism f is internally diagonalizable. Classical structures in categories of unitary finite-group representations. An important class of ‘generalizations’ of FinSet is given by the finitary toposes. A topos [26] is a category where the operations familiar from traditional constructive logic can all be defined; in particular, unions, products, function sets and powersets are all available. Technically, a topos2 is a category with all finite limits, in which every object has a power object; the other constructions just mentioned can then be derived. An example is the category of finite G-sets, for a finite group G: objects are finite sets equipped with a G-action, and morphisms are functions between the underlying sets which are compatible with the group actions. That such a category is in fact a topos is far from obvious, and relies on powerful general theorems [25]. Given the explicit connection between FinSet and Hilb established by the equivalence FinSet op C(Hilb), it is natural to ask whether there exist generalizations of Hilb which have other finitary topoi as their categories of classical structures. A topos obtained in this way could be interpreted as giving the classical counterpart to a quantum theory, in contrast to the Döring-Isham toposes discussed after Definition 5.5 which give a direct topos-theoretical view of the quantum structure itself. A heuristic argument puts a stumbling block in front of any such attempt.3 A striking feature of many toposes is that the law of excluded middle can fail, and as a consequence, given a subobject of an object in the topos, the union of the subobject and its complement can fail to give the original object. For a given Hilbert space, a good way to characterize its subobjects is by the projectors on the space. Two projectors P and Q on a Hilbert space represent disjoint subobjects if P Q = 0, and in that case their union as subobjects is represented by the projector P + Q. We now work in a category intended as a generalization of Hilb, assuming only that it is a †-category with hom-sets which are complex vector spaces. Projectors can be defined in this setting as endomorphisms P satisfying P † = P 2 = P, and we can describe disjointness and union using our categorical structure in the manner just described. Given any projector P we will be able to use the complex vector space structure of the hom-sets to construct a new projector (1 − P), where 1 is the identity on the space. This new projector is disjoint with P, and gives the identity under union with P, using the general definitions of these terms given above. In a sense, it therefore seems that the law of excluded middle holds. To avoid this conclusion either the †-functor must go so that 2 Experts will notice that this is the definition of an elementary topos, the most basic type of topos. 3 I am grateful to Christopher Isham for this argument.
794
J. Vicary
projectors cannot be straightforwardly defined, or the complex numbers must go so that we cannot ask that the hom-sets be vector spaces over them, but both are core parts of the mathematical formalism of quantum mechanics which cannot be lightly abandoned. We will skirt around this argument by focusing on those toposes for which the excluded middle does hold: the Boolean toposes, or at least a finitary subclass of these. We will focus on the following types of category: Definition 5.11. A finite quantum Boolean topos is a symmetric monoidal †-category which has a strong symmetric monoidal †-equivalence to a category HilbG of finite-dimensional unitary representations of some finite groupoid G, where Hilb is the category of finite-dimensional complex Hilbert spaces and continuous linear maps. Definition 5.12. A finite Boolean topos is a category equivalent to a topos of the form FinSetG for some finite groupoid G, where FinSet is the topos of finite sets and functions. Theorem 5.13. The category of classical structures in a finite quantum Boolean topos is equivalent to a finite Boolean topos, and every finite Boolean topos arises in this way. Proof. Let Q be a finite quantum Boolean topos, for which by definition there exists a strong symmetric monoidal †-equivalence Q HilbG for a finite groupoid G. There is a canonical forgetful †-preserving functor F : HilbG - Hilb that takes a unitary Grepresentation to the Hilbert space on which G is acting. By abuse of notation we will also write F : Q - Hilb, suppressing the equivalence Q HilbG . A commutative †-Frobenius monoid (A, m, u) in Q gives a commutative †-Frobenius monoid ( F(A), F(m), F(u)) in Hilb, and therefore defines a basis for the Hilbert space F(A) by Theorem 5.2. Each object A of Q, via the equivalence with HilbG , is actually a †-functor A : G - Hilb, and for each g ∈ G the morphism A(g) : F(A) - F(A) is a unitary linear map in Hilb. The morphisms F(m) and F(u) are intertwiners, which can be expressed by the following commuting diagram that holds for all g ∈ G: F(A) ⊗ F(A)
A(g) ⊗ A(g) F(A) ⊗ F(A)
F(m) ? F(A) 6 F(u)
A(g)
F(m) ? - F(A) 6 F(u)
F(I ) ====================== F(I ) Read differently, this diagram is also precisely the condition for A(g) to be a monoid homomorphism for the commutative †-Frobenius monoid ( F(A), F(m), F(u)) in Hilb. Since the morphism A(g) is invertible, it must act as a permutation of the basis of F(A) defined by the monoid, and the commutative †-Frobenius monoid (A, m, u) therefore corresponds to an action of the groupoid G on this basis. Every finite G-action must arise in this way, since any G-action on a finite set gives rise to a linear G-representation on the complex Hilbert space with basis given by elements of the set. Morphisms between commutative †-Frobenius monoids have adjoints which act as set-functions for the induced bases, and these adjoints are compatible with the induced G-actions on the basis elements. It follows that the category of commutative †-Frobenius monoids in Q HilbG is equivalent to the opposite of the category FinSetG .
Categorical Formulation of Finite-Dimensional Quantum Algebras
795
Another way to phrase this result is that the process of taking G-presheaves — either of sets, or of Hilbert spaces — commutes with the process of forming the category of classical objects: C(HilbG ) C(Hilb)G FinSetG . (23) For the functor category HilbG we take only unitary representations, or equivalently †-preserving functors where the †-functor on G takes a morphism to its inverse. It is this result which motivates the term ‘finite quantum Boolean topos’. We also note that we can use this to recover the finite groupoid G from its unitary representation category HilbG , since FinSetG yields G as its smallest full generating subcategory (see [25, Chap. 6]). Given the similarity between the presheaf-style Definitions 5.11 and 5.12, the lemma perhaps seems artificial. In fact, it is known that finite quantum Boolean toposes can be described axiomatically; it follows from the Doplicher-Roberts theorem [14] that, using the terminology of Baez [6], they are precisely the finite-dimensional even symmetric 2-H*-algebras. We also expect that finite Boolean toposes would admit a direct axiomatization, although we do not attempt to give one here. Given the result described here it is interesting to consider a generalization to arbitrary finite-dimensional symmetric 2-H*-algebras. By a generalization of the Doplicher-Roberts theorem [6,17] these are known to be the representation categories of finite supergroupoids. However, we are not aware of any extensions of our results that can be proved along these lines. Acknowledgements. The comments of the anonymous referees have greatly improved this article, and I am also grateful to Samson Abramsky, Bruce Bartlett, Bob Coecke, Chris Heunen, Chris Isham and Dusko Pavlovic for useful discussions. I am also grateful for financial support from EPSRC and QNET. Commutative diagrams are rendered using Paul Taylor’s diagrams package.
References 1. Abrams, L.: Two-dimensional topological quantum field theories and Frobenius algebras. J. Knot Theory and Ram. 5, 569–587 (1996) 2. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, Washington, DC. IEEE Computer Science Press, 2004, pp. 415–425 3. Abramsky, S., Coecke, B.: Abstract physical traces. Theory and Appl. Cat. 14(6), 111–124 (2005) 4. Abramsky, S., Coecke, B.: Handbook of Quantum Logic and Quantum Structures. Volume 2, Chapter Categorical Quantum Mechanics. Maryland Heights, MO: Elsevier, 2008 5. Abramsky, S., Heunen, C.: Classical structures in infinite-dimensional categorical quantum mechanics. In preparation, 2010 6. Baez, J.C.: Higher-dimensional algebra II: 2-Hilbert spaces. Adv. Math. 127, 125–189 (1997) 7. Baez, J.C.: Structural Foundations of Quantum Gravity. Chapter Quantum Quandaries: A CategoryTheoretic Perspective. Oxford: Oxford University Press, 2006, pp. 240–265 8. Bakalov, B., Kirillov, A.: Lectures on Tensor Categories and Modular Functors. Providence, RI: Amer. Math. Soc., 2001 9. Barrett, J.W., Westbury, B.W.: Spherical categories. Adv. in Math. 143(2), 357–375 (1999) 10. Coecke, B., Edwards, B.: Toy quantum categories. In: Quantum Physics and Logic 2008, Electronic Notes in Theoretical Computer Science, 2008. To appear 11. Coecke, B., Pavlovic, D.: The Mathematics of Quantum Computation and Technology, Chapter Quantum Measurements Without Sums. Oxford-NewYork: Taylor and Francis, 2006 12. Coecke, B., Pavlovic, D., Vicary, J.: Dagger-Frobenius algebras in FdHilb are orthogonal bases. Technical Report, RR-08-03, Oxford Univ. Computing Lab., 2008 13. Day, B., McCrudden, P., Street, R.: Dualizations and antipodes. Appl. Categ. Struct. 11, 229–260 (2003) 14. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. Math. 98, 157–218 (1989) 15. Döring, A., Isham, C.: New Structures in Physics, Chapter ‘What is a Thing?’: Topos Theory in the Foundations of Physics. 2008, available at http://arxiv.org/abs/0803.0417 [quant-ph], 2008
796
J. Vicary
16. Fukuma, M., Hosono, S., Kawai, H.: Lattice topological field theory in two dimensions. Commun. Math. Phys. 161(1), 157–175 (1994) 17. Halvorson, H., Müger, M.: Handbook of the Philosophy of Physics, Chapter Algebraic quantum field theory. Amsterdam: North Holland, 2006 18. Joyal, A., Street, R.: The geometry of tensor calculus I. Adv. in Math. 88, 55–112 (1991) 19. Kassel, C.: Quantum Groups. Volume 155 of Graduate Texts in Mathematics. Berlin-HeidelbergNewYork: Springer-Verlag, 1995 20. Kock, J.: Frobenius Algebras and 2D Topological Quantum Field Theories. Cambridge: Cambridge University Press, 2004 21. Lauda, A.D., Pfeiffer, H.: State sum construction of two-dimensional open-closed topological quantum field theories. J. Knot Theory and Ram. 16, 1121–1163 (2007) 22. Lauda, A.D., Pfeiffer, H.: Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras. Topology and Appl. 155(7), 623–666 (2008) 23. Longo, R., Roberts, J.E.: A theory of dimension. K-Theory 11, 103–159 (1997) 24. Mac Lane, S.: Categories for the Working Mathematician. 2nd ed., Berlin-Heidelberg-NewYork: Springer, 1997 25. Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic. Berlin-Heidelberg-NewYork: Springer, 1992 26. McLarty, C.: Elementary Categories, Elementary Toposes. Oxford: Oxford University Press, 1995 27. Müger, M.: From subfactors to categories and topology I. Frobenius algebras in and morita equivalence of tensor categories. J. Pure and Appl. Alg. 180, 81–157 (2003) 28. Murphy, G.J.: C*-Algebras and Operator Theory. London-NewYork: Academic Press, 1990 29. Quinn, F.: Geometry and Quantum Field Theory (Park City, UT, 1991), Chapter Lectures on Axiomatic Topological Quantum Field Theory, Providence, RI: Amer. Math. Soc., 1995, pp. 323–453 30. Selinger, P.: Dagger compact closed categories and completely positive maps. In: Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), Lect. Notes in Theor. Comp. Sci. 170, 139–163 (2007) 31. Street, R.: Frobenius monads and pseudomonoids. J. Math. Phys. 45, 3930–3948 (2004) 32. Turaev, V.: Quantum Invariants of Knots and 3-Manifolds, Volume 18 of de Gruyter Studies in Mathematics. Berlin: Walter de Gruyter, 1994 33. Vicary, J.: Completeness of dagger-categories and the complex numbers. J. Math. Phys., 2010. To appear 34. Zito, P.: 2-C*-categories with non-simple units. Adv. in Math. 210, 122–164 (2007) Communicated by Y. Kawahigashi
Commun. Math. Phys. 304, 797–874 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1218-9
Communications in
Mathematical Physics
Elliptic Hypergeometry of Supersymmetric Dualities V. P. Spiridonov1,2 , G. S. Vartanov3 1 Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia 2 Theory Division, INR RAS, Moscow, Russia. E-mail: [email protected] 3 Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany.
E-mail: [email protected] Received: 25 June 2010 / Accepted: 11 October 2010 Published online: 16 March 2011 – © Springer-Verlag 2011
Abstract: We give a full list of known N = 1 supersymmetric quantum field theories related by the Seiberg electric-magnetic duality conjectures for SU (N ), S P(2N ) and G 2 gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related Römelsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them were proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. General Structure of the Elliptic Hypergeometric Integrals . . . . 3. Superconformal Index . . . . . . . . . . . . . . . . . . . . . . . 3.1 N = 1 superconformal algebra . . . . . . . . . . . . . . . . 3.2 The index . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Calculation of the index . . . . . . . . . . . . . . . . . . . . 4. Seiberg Duality for Unitary Gauge Groups . . . . . . . . . . . . . 5. Intriligator-Pouliot Duality for Symplectic Gauge Groups . . . . . 6. Multiple Duality for S P(2N ) Gauge Group . . . . . . . . . . . . 7. A New S P(2N ) ↔ S P(2M) Groups Duality . . . . . . . . . . . 8. Multiple Duality for SU (2N ) Gauge Group . . . . . . . . . . . . 9. Kutasov-Schwimmer Type Dualities for the Unitary Gauge Group 9.1 SU (N ) gauge group with the adjoint matter field . . . . . . 9.2 Two adjoint matter fields case . . . . . . . . . . . . . . . . . 9.3 Generalized KS type dualities . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
798 800 808 808 811 813 815 817 818 821 822 825 825 826 828
798
V. P. Spiridonov, G. S. Vartanov
9.4 Adjoint, symmetric and conjugate symmetric tensor matter fields . . . . 832 9.5 Adjoint, anti-symmetric and conjugate anti-symmetric tensor matter fields833 9.6 Adjoint, anti-symmetric and conjugate symmetric tensor matter fields . 835 10. KS Type Dualities for Symplectic Gauge Groups . . . . . . . . . . . . . . . 837 10.1 The anti-symmetric tensor matter field . . . . . . . . . . . . . . . . . . 837 10.2 Symmetric tensor matter field . . . . . . . . . . . . . . . . . . . . . . . 838 10.3 Two anti-symmetric tensor matter fields . . . . . . . . . . . . . . . . . 838 10.4 Symmetric and anti-symmetric tensor matter fields . . . . . . . . . . . 840 11. Some Other New Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . 841 11.1 SU ↔ S P groups mixing duality . . . . . . . . . . . . . . . . . . . . . 841 11.2 SU ↔ SU groups mixing duality . . . . . . . . . . . . . . . . . . . . 842 12. S-Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 12.1 SU (N ) gauge group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 12.2 Exceptional cases for unitary gauge groups . . . . . . . . . . . . . . . . 852 12.3 Symplectic gauge group . . . . . . . . . . . . . . . . . . . . . . . . . . 855 13. Exceptional G 2 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 14. ’t Hooft Anomaly Matching Conditions . . . . . . . . . . . . . . . . . . . . 860 15. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 Appendix A. Characters of Representations of Classical Groups . . . . . . . . . 865 Appendix B. Invariant Matrix Group Measures . . . . . . . . . . . . . . . . . . 867 Appendix C. Relevant Casimir Operators . . . . . . . . . . . . . . . . . . . . . 868 Appendix D. Total Ellipticity for the KS Duality Indices . . . . . . . . . . . . . 869 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871 1. Introduction The main goal of this work consists in merging two fields of recent active research in mathematical physics—the Seiberg duality in supersymmetric field theories [75,76] and the theory of elliptic hypergeometric functions [85]. Seiberg duality is an electricmagnetic duality of certain four dimensional quantum field theories with the symmetry group G st × G × F, where the superconformal group G st = SU (2, 2|1) describes properties of the space-time, G is a local gauge invariance group, and F is a global symmetry flavor group. Conjecturally, such theories are equivalent to each other at their infrared fixed points, existence of which follows from a deeply nontrivial nonperturbative dynamics [47,79]. The simplest topological characteristics of supersymmetric theories is the Witten index [99]. Its highly nontrivial superconformal generalization was proposed recently by Römelsberger [72,73] (for N = 1 theories) and Kinney et al [49] (for extended supersymmetric theories). These superconformal indices describe the structure of BPS states protected by one supercharge and its conjugate. They can be considered as a kind of partition functions in the corresponding Hilbert space. Starting from early work [80,95], it is known that such partition functions are described by matrix integrals over the classical groups. The central conjecture of Römelsberger [73] claims the equality of superconformal indices in the Seiberg dual theories. In an interesting work [26], Dolan and Osborn have found an explicit form of these indices for a number of theories and discovered that they coincide with particular examples of the elliptic hypergeometric integrals [89]. This identification allowed them to prove Römelsberger’s conjecture for several dualities either on the basis of known exact computability of these integrals or from the existence of non-trivial symmetry transformations for them.
Elliptic Hypergeometry of Supersymmetric Dualities
799
The general notion of elliptic hypergeometric integrals was introduced by the first author in [81,83]. First example of such integrals discovered in [81] has formed a new class of exactly computable integrals of hypergeometric type called elliptic beta integrals. Such a name was chosen because these integrals can be considered as a top level generalization of the well-known Euler beta integral [1]:
1 0
x α−1 (1 − x)β−1 d x =
(α)(β) , (α + β)
Re α, Re β > 0,
(1.1)
where (x) is the Euler gamma function. Elliptic hypergeometric functions generalize known plain hypergeometric functions and their q-analogues [1]. Moreover, their properties have clarified the origins of many old notions of the hypergeometric world [82]. Limits of the elliptic hypergeometric integrals (or of the elliptic hypergeometric series hidden behind them) matched with the elliptic curve degenerations brought to light new types of q-hypergeometric functions as well [66,67] (see also [10]). In the present work (which was initiated in August 2008 after the first author has known [26]), we extend systematically the Römelsberger and Dolan-Osborn results. More precisely, we present a full list of known N = 1 superconformal field theories related by the duality conjecture for simple gauge groups G = SU (N ), S P(2N ), G 2 . For all of them we express superconformal indices in terms of the elliptic hypergeometric integrals. Using Seiberg dualities established earlier in the literature (see references below) we come to a large number of identities for elliptic hypergeometric integrals. About half of them were proven earlier, which yields a justification of the corresponding dualities. A part of the emerging relations for indices was described in [26], and we prove equalities of superconformal indices for many other dualities. Another half of the constructed identities represents new challenging conjectures requiring rigorous mathematical proof. We give indications how some of them can be proved with the help of hypergeometric techniques. Remarkably, from known relations for elliptic hypergeometric integrals we find many new dualities not considered earlier in the literature. Thus we describe both new elliptic hypergeometric identities and new N = 1 supersymmetric theories obeying an electricmagnetic duality. In particular, we conjecture more than ten new elliptic beta integrals on root systems, which extend the univariate elliptic beta integral of [81]. Analyzing the general structure of all relations for integrals in this paper, we formulate two universal conjectures. Namely, we argue that for the existence of a non-trivial identity for an elliptic hypergeometric integral it is necessary to have a so-called totally elliptic hypergeometric term [82,86,90]. The second conjecture claims that the same total ellipticity (and related modular invariance) is responsible for the validity of ’t Hooft anomaly matching conditions [40], which are fulfilled for all our dualities (the old and new ones). A detailed consideration of the multiple duality phenomenon for the G = S P(2N ) gauge group case and a brief announcement of other results of this work were given in paper [91]. Our results were reported also at the IVth Sakharov Conference on Physics (Lebedev Institute, Moscow, May 2009), Conformal Field Theory Workshop (Landau Institute, Chernogolovka, June 2009), XVIth International Congress on Mathematical Physics (Prague, August 2009), and about ten seminars at different institutes. We thank the organizers of these meetings and seminars for invitations and kind hospitality.
800
V. P. Spiridonov, G. S. Vartanov
2. General Structure of the Elliptic Hypergeometric Integrals We start our consideration from the description of the general structure of elliptic hypergeometric integrals. For any x ∈ C and a base p ∈ C, | p| < 1, we define the infinite product (x; p)∞ =
∞
(1 − x p j ).
j=0
Then the theta function is defined as θ (x; p) = (x; p)∞ ( px −1 ; p)∞ , where x ∈ C∗ . This function obeys the symmetry properties θ (x −1 ; p) = θ ( px; p) = −x −1 θ (x; p) and the addition law θ (xw ±1 , yz ±1 ; p) − θ (x z ±1 , yw ±1 ; p) = yw −1 θ (x y ±1 , wz ±1 ; p), where x, y, w, z ∈ C∗ and we use the convention θ (t x ±1 ; p) := θ (t x, t x −1 ; p).
θ (x1 , . . . , xk ; p) := θ (x1 ; p) . . . θ (xk ; p),
The Jacobi triple product identity for the standard theta series yields θ (x; p) =
1 p n(n−1)/2 (−x)n . ( p; p)∞ n∈Z
For arbitrary q ∈ C and n ∈ Z, we introduce the elliptic shifted factorials n−1 j for n > 0, j=0 θ (xq ; p), θ (x; p; q)n := −n − j −1 j=1 θ (xq ; p) , for n < 0, with the normalization θ (x; p; q)0 = 1. For p = 0 we have θ (x; 0) = 1 − x and θ (x; 0; q)n = (x; q)n = (1 − x)(1 − q x) · · · (1 − q n−1 x), the standard q-Pochhammer symbol [1]. For arbitrary m ∈ Z, we have the quasiperiodicity relations θ ( p m x; p) = (−x)−m p − θ ( p m x; p; q)k = (−x)−mk q θ (x; p; p m q)k = (−x)−
m(m−1) 2
θ (x; p),
− mk(k−1) 2
mk(k−1) 2
q−
p−
km(m−1) 2
mk(k−1)(2k−1) 6
θ (x; p; q)k ,
p−
mk(k−1) m(2k−1) ( 3 −1) 4
θ (x; p; q)k .
We relate bases p, q and r with three complex numbers ω1,2,3 ∈ C in the following way: q=e
ω 2
2π i ω1
,
p=e
ω 2
2π i ω3
, r =e
ω 1
2π i ω3
.
Elliptic Hypergeometry of Supersymmetric Dualities
801
Their “τ → −1/τ ” modular transformed partners are q˜ = e
ω
−2π i ω2
1
,
p˜ = e
ω
−2π i ω2
3
, r˜ = e
ω 3
−2π i ω1
.
Modular parameters τ1 = ω1 /ω2 , τ2 = ω3 /ω2 , τ3 = ω3 /ω1 define three elliptic curves constrained by the condition τ3 = τ2 /τ1 . Elliptic gamma functions are defined as appropriate meromorphic solutions of the following finite difference equation: f (u + ω1 ) = θ (e2π iu/ω2 ; p) f (u), u ∈ C.
(2.1)
Its particular solution, called the standard elliptic gamma function, has the form f (u) = (e2π iu/ω2 ; p, q),
(z; p, q) =
∞ 1 − z −1 p j+1 q k+1 , 1 − zp j q k
(2.2)
j,k=0
where |q|, | p| < 1, z ∈ C∗ (note that the equation itself does not demand |q| < 1). For incommensurate ω1,2,3 , it can be defined uniquely as the meromorphic solution of (2.1) satisfying simultaneously two more equations: f (u + ω2 ) = f (u),
f (u + ω3 ) = θ (e2π iu/ω2 ; q) f (u)
and the normalization condition f ( 3k=1 ωk /2) = 1. The modified elliptic gamma function has the form G(u; ω) = (e
2π i ωu
2
; p, q)(r e
−2π i ωu
1
; q, ˜ r ).
(2.3)
It defines the unique simultaneous solution of Eq. (2.1) and two other equations: f (u + ω2 ) = θ (e
2π iu/ω1
; r ) f (u),
f (u + ω3 ) =
θ (e θ (e
2π i ωu
2
−2π i ωu
; q)
1
; q) ˜
f (u)
with the same normalization condition f ( 3k=1 ωk /2) = 1. Here the third equation can be simplified using the modular transformation for theta functions θ (e
−2π i ωu
1
; q) ˜ = eπ iB2,2 (u|ω1 ,ω2 ) θ (e
where 1 B2,2 (u|ω1 , ω2 ) = ω1 ω2
2π i ωu
2
; q),
ω2 + ω22 ω1 ω2 + u − (ω1 + ω2 )u + 1 6 2
(2.4)
2
is the second Bernoulli polynomial. These statements are based on the Jacobi theorem stating that if a meromorphic ϕ(u) satisfies the system of equations ϕ(u + ω1 ) = ϕ(u + ω2 ) = ϕ(u + ω3 ) = ϕ(u) for ω1,2,3 ∈ C linearly independent over Z, then ϕ(u) = const. The restricted values of bases p n = q m , n, m ∈ Z (or, equivalently, r n = q˜ m or r˜ n = p˜ m ) may be called the torsion points, since the Jacobi theorem fails for them.
802
V. P. Spiridonov, G. S. Vartanov
The function πi
G(u; ω) = e− 3
B3,3 (u|ω)
(e
−2π i ωu
3
; r˜ , p), ˜
(2.5)
where | p|, ˜ |˜r | < 1 and
⎛ 3 2 1 ⎝u 3 − 3u B3,3 (u|ω1 , ω2 , ω3 ) = ωk ω1 ω2 ω3 2 k=1 ⎞ ⎛ 3 3 u ⎝ 2 1 ⎠ + ωk + 3 ω j ωk − ωk 2 4 k=1
1≤ j
k=1
⎞ ω j ωk ⎠
1≤ j
is the third Bernoulli polynomial, satisfies the same three equations and normalization as function (2.3). Hence they coincide, and this fact yields one of the S L(3; Z)-group modular transformation laws for the elliptic gamma function. From the expression (2.5) it is easy to see that G(u; ω) is a meromorphic function of u for ω1 /ω2 > 0, i.e. when |q| = 1. The region |q| > 1 is similar to |q| < 1, it can be reached by a symmetry transformation. The theory of generalized gamma functions was built by Barnes [2]. Implicitly, the function (z; p, q) appeared in the free energy per site of Baxter’s eight vertex model [3] (see also [96] and [28]) – exactly in the form which will be used below in the superconformal indices context. A systematic investigation of its properties was launched by Ruijsenaars in [74]. Its S L(3, Z)-group transformation properties were described in [28]. The modified (“unit circle") elliptic gamma function G(u; ω) was introduced in [83] (see also [21]). Both elliptic gamma functions are directly related to the Barnes multiple gamma function of the third order [31,83]. In terms of the (z; p, q)-function one can write θ (x; p; q)n =
(xq n ; p, q) . (x; p, q)
The short-hand conventions (t1 , . . . , tk ; p, q) := (t1 ; p, q) · · · (tk ; p, q), (t z ±1 ; p, q) := (t z; p, q)(t z −1 ; p, q), (z ±2 ; p, q) := (z 2 ; p, q)(z −2 ; p, q) are used below. The simplest properties of (z; p, q) are: • the symmetry (z; p, q) = (z; q, p), • the finite difference equations of the first order (qz; p, q) = θ (z; p)(z; p, q), ( pz; p, q) = θ (z; q)(z; p, q), • the reflection equation (z; p, q)( pq/z; p, q) = 1, • the duplication formula (z 2 ; p, q) = (z, −z, q 1/2 z, −q 1/2 z, p 1/2 z, − p 1/2 z, ( pq)1/2 z, −( pq)1/2 z; p, q),
Elliptic Hypergeometry of Supersymmetric Dualities
803
• the limiting relations lim (z; p, q) =
p→0
1 , (z; q)∞
lim (1 − z)(z; p, q) =
z→1
1 . ( p; p)∞ (q; q)∞
Definition 1 [82,90]. A meromorphic function f (x1 , . . . , xn ; p) of n variables x j ∈ C∗ , which together with p ∈ C compose all indeterminates of this function, is called totally p-elliptic if f ( px1 , . . . , xn ; p) = · · · = f (x1 , . . . , pxn ; p) = f (x1 , . . . , xn ; p), and if its divisor forms a nontrivial manifold of the maximal possible dimension. Note that here positions of zeros and poles of elliptic functions are considered as indeterminates (i.e., they are not fixed in advance). Consider n-dimensional integrals n dx j I (y1 , . . . , ym ) =
(x1 , . . . , xn ; y1 , . . . , ym ) , xj x∈D j=1
Cn
where D ⊂ is some domain of integration and (x1 , . . . , xn ; y1 , . . . , ym ) is a meromorphic function of x j and yk , where yk denote the “external” parameters. Definition 2 [83]. The integral I (y1 , . . . , ym ; p, q) is called the elliptic hypergeometric integral if there are two distinguished complex parameters p and q such that I ’s kernel (x1 , . . . , xn ; y1 , . . . , ym ; p, q) satisfies the following system of linear first order q-difference equations in the integration variables x j :
(. . . q x j . . . ; y1 , . . . , ym ; p, q) = h j (x1 , . . . , xn ; y1 , . . . , ym ; q; p),
(x1 , . . . , xn ; y1 , . . . , ym ; p, q) where h j are some p-elliptic functions of the variables x j , h j (. . . pxi . . . ; y1 , . . . , ym ; q; p) = h j (x1 , . . . , xn ; y1 , . . . , ym ; q; p). The kernel is called then the elliptic hypergeometric term, and the functions h j (x1 , . . . , xn ; y1 , . . . , ym ; q; p)—the q-certificates. This definition is not the most general possible one of such kind, but it is sufficient for the purposes of the present paper. The elliptic hypergeometric series can be introduced as sums of residues of particular sequences of poles of the elliptic hypergeometric integral kernels [19] and, because of the convergence difficulties, they are less general than the integrals. In the one-dimensional case, n = 1, the structure of admissible elliptic hypergeometric terms can be described explicitly. Indeed, any meromorphic p-elliptic function f ( px) = f (x) can be written in the form f p (x) = z
N θ (tk x; p) , θ (wk x; p)
k=1
N k=1
tk =
N
wk ,
k=1
where z, t1 , . . . , t N , w1 , . . . , w N are arbitrary complex parameters. The positive integer N is called the order of the elliptic function, and the linear constraint on parameters – the balancing condition. From the identity z=
θ (zx, px; p) θ ( pzx, x; p)
804
V. P. Spiridonov, G. S. Vartanov
we see that z is not a distinguished parameter – it can be obtained from tk and wk by appropriate reduction without spoiling the balancing condition. Therefore we set z = 1. Now, for |q| < 1, the general solution of the equation (q x) = f p (x) (x) is
(x) = ϕ(x)
N (tk x; p, q) , (wk x; p, q)
ϕ(x) =
k=1
M M M θ (ak x; q) , ak = bk , θ (bk x; q)
k=1
k=1
k=1
where ϕ(q x) = ϕ(x) is an arbitrary q-elliptic function. However, since ϕ(x) =
M ( pak x, bk x; p, q) , (ak x, pbk x; p, q)
k=1
one can obtain ϕ(x) from ratios of -functions after replacing N by N + 2M and appropriate specification of the original parameters tk and wk with the balancing condition preserved. Therefore we can drop the ϕ(x) function and find that the general elliptic hypergeometric term for n = 1 has the form:
(x; t1 , . . . , t N , w1 , . . . , w N ; p, q) =
N N (tk x; p, q) tk , = 1. (wk x; p, q) wk
k=1
k=1
This function is symmetric in p and q, i.e. we can repeat the above considerations with these parameters permuted. Then, for incommensurate p and q (i.e., when p j = q k , j, k ∈ Z), the equations
(q x) = f p (x) (x),
( px) = f q (x) (x)
determine (x) uniquely up to a multiplicative constant. For |q| > 1, N N (q −1 wk x; p, q −1 ) tk , = 1.
(x; t1 , . . . , t N , w1 , . . . , w N ; p, q) = (q −1 tk x; p, q −1 ) wk k=1
k=1
For |q| = 1, the requirement of meromorphicity in x is too strong. To define integrals in this case one has to use the modified elliptic gamma function G(u; ω), or modular transformations, which we skip for brevity. In analogy with the series case considered in [82], it is natural to extend the notion of total ellipticity to elliptic hypergeometric terms entering integrals [83]. Definition 3. An elliptic hypergeometric integral I (y1 , . . . , ym ; p, q) =
(x1 , . . . , xn ; y1 , . . . , ym ; p, q) x∈D
n dx j xj j=1
is called totally elliptic if all its kernel’s q-certificates h j (x1 , . . . , xn ; y1 , . . . , ym ; q; p), j = 1, . . . , n + m, are totally elliptic functions, i.e. they are p-elliptic in all variables x1 , . . . , xn , y1 , . . . , ym and q. In particular, h j (x1 , . . . , xn ; y1 , . . . , ym ; pq; p) = h j (x1 , . . . , xn ; y1 , . . . , ym ; q; p).
Elliptic Hypergeometry of Supersymmetric Dualities
805 (a)
Theorem 1 (Rains, Spiridonov, 2004). Given Zn → Z maps (m (a) ) = (m 1 , . . . , (a) m n ), a = 1, . . . , M, with finite support, define the meromorphic function
(x1 , . . . , xn ; p, q) =
M
m
(a)
m
(a)
(a)
(x1 1 x2 2 . . . xnm n ; p, q) (m
(a) )
.
(2.6)
a=1
Suppose is a totally elliptic hypergeometric term, i.e. its q-certificates are p-elliptic functions of q and x1 , . . . , xn . Then these certificates are also modular invariant. The proof is elementary. The q-certificates have the explicit form M
(. . . q xi . . . ; p, q) (a) (m (a) ) = θ (x m ; p; q) (a) . h i (x; q; p) = mi
(x1 , . . . , xn ; p, q) a=1
The conditions for h i to be elliptic in x j yield the constraints M
(a)
(a)
(a)
(m (a) )m i m j m k = 0,
(2.7)
a=1 M
(m (a) )m i(a) m (a) j =0
(2.8)
a=1
for 1 ≤ i, j, k ≤ n. The conditions of ellipticity in q add one more constraint M
(a)
(m (a) )m i
= 0.
(2.9)
a=1
The latter equation guarantees that h i has an equal number of theta functions in its numerator and denominator. The modular invariance of h i follows then automatically from the transformation property (2.4). Such a direct relation between total ellipticity and modularity was conjectured to be true in general in [82]. The simplest known nontrivial totally elliptic hypergeometric term corresponds to n = 6, M = 29 and has the form [86] 5 ±1 −1 j=1 (t j x , t j i=1 ti ; p, q) 5 (x ±2 , i=1 ti x ±1 ; p, q) 1≤i< j≤5 (ti t j ; 5
(x; t1 , . . . , t5 ; p, q) =
p, q)
.
Theorem 2 [81]. Elliptic beta integral. For | p|, |q|, |t j | < 1, | 5j=1 t j | < | pq|, one has ( p; p)∞ (q; q)∞ dx = 1, (2.10)
(x; t1 , . . . , t5 ; p, q) 4π i x T where T is the unit circle with positive orientation.
806
V. P. Spiridonov, G. S. Vartanov
The Euler beta integral evaluation formula (1.1) lies at the bottom of this identity. On the corresponding degeneration road one finds many interesting integrals, including the Rahman and Askey-Wilson q-beta integrals [1]. Formula (2.10) served as an entry ticket to the large class of new exactly computable integrals discussed in [19–21,65,83,93], which is essentially extended by the conjectures presented in this paper. In [83,85,87] the elliptic beta integral was generalized to an elliptic analogue of the Gauss hypergeometric function obeying many classical properties. For a survey of this function and its generalizations to higher order elliptic hypergeometric functions and multiple integrals on root systems, see [89]. Two totally elliptic hypergeometric terms associated with the elliptic beta integrals of type I on root systems BCn [20] and An [83] were constructed in [86]. One more similar example for the root system An was built in [93]. Some time ago, using the combination of techniques introduced in [86] and [69], the first author has further generalized the former two terms to an arbitrary number of parameters [90]. For instance, define the kernel n 2n+2m+4 (t z ±1 ; p, q) i j 1 i=1
n (z, t; p, q) = ±1 ±1 ±2 (z i z j ; p, q) j=1 (z j ; p, q) 1≤i< j≤n and the type I BCn -elliptic hypergeometric integral: In(m) (t1 , . . . , t2n+2m+4 ) = where |t j | < 1 and
2n+2m+4 j=1
n dz j ( p; p)n∞ (q; q)n∞
(z, t; p, q) , n n n 2 n!(2π i) zj Tn j=1
t j = ( pq)m+1 .
Theorem 3 [65]. For | pq|1/2 < |t j | < 1, the integrals In(m) satisfy the relation √ √ pq pq (m) (n) . (tr ts ; p, q) Im ,..., In (t1 , . . . , t2n+2m+4 ) = t1 t2n+2m+4 1≤r <s≤2n+2m+4
(2.11) This is an elliptic analogue of the symmetry transformation for some plain hypergeometric integrals established by Dixon in [23]. Theorem 4 [90]. The ratio ρ(z, y; t; p, q) =
1≤r <s≤2n+2m+4
(tr ts ; p, q)−1
n (z; t; p, q) √ √
m (y/ pq; pq/t; p, q)
is the totally elliptic hypergeometric term. That is all ratios ρ(. . . ,qv,. . . )/ρ(. . . , v, . . . ) for v ∈ {z 1 , . . . , z n , y1 , . . . , ym , t1 , . . . , t2n+2m+4 } are p-elliptic functions of all variables z i , yk , tl , and q. The term ρ(z, y; t; p, q) contains elliptic gamma functions with non-removable integer powers of pq in the argument. Therefore the ansatz (2.6) does not cover all interesting totally elliptic hypergeometric terms. As we shall show below, there are also examples of terms depending on fractional powers of pq. For them the total ellipticity condition is slightly modified: it is necessary to consider dilations of the parameter q by appropriate
Elliptic Hypergeometry of Supersymmetric Dualities
807
powers of p. Introducing the variable x0 = ( pq)1/K , K = 1, 2, . . . and adding to the m
(a)
arguments of elliptic gamma functions in (2.6) the multipliers x0 0 , it is not difficult (a) to find the general form of constraints on integers m j and (m (a) ) guaranteeing total ellipticity (with special p K -ellipticity condition for the variable q). However, these constraints look essentially less beautiful than the Diophantine equations described above. Moreover, at the moment it is not clear which part of the modular transformation group survives because of the presence of fractional parts of modular variables in the arguments of respective elliptic functions-certificates. In the present work, we have checked that all nontrivial relations for elliptic hypergeometric integrals described below define totally elliptic hypergeometric terms through the ratios of the corresponding integral kernels. Namely, this property was verified for the equalities of superconformal indices in • the initial Seiberg duality (4.6) and (4.7); S P-groups duality (5.1) and (5.2); [90] • multiple dualities for S P(2N ) gauge group (6.1), (6.2), (6.3) and (6.4); • new duality for S P(2N ) group (7.1) and (7.2); • multiple duality for SU (2N ) gauge group (8.1), (8.2), (8.3) and (8.4); • KS type dualities for unitary groups (9.2) and (9.3) (see Appendix D for a detailed consideration of this case); (9.5) and (9.6); (9.8) and (9.9); (9.10) and (9.11); (9.13) and (9.14); (9.16) and (9.17); (9.19) and (9.20); (9.22) and (9.23); • KS type dualities for symplectic groups (10.2) and (10.3); (10.5) and (10.6); (10.8) and (10.9); (10.11) and (10.12); • confinement for SU (N ) group theories (12.1) and (12.2); (12.3) and (12.4); (12.5) and (12.6); (12.7) and (12.8); (12.9) and (12.10); (12.19) and (12.20); (12.21) and (12.22); (12.25) and (12.26); (12.27) and (12.28); (12.29) and (12.30); (12.31) and (12.32); • confinement for S P(2N ) group theories (12.33) and (12.34); (12.35) and (12.36); (12.37) and (12.38); • dualities for the G 2 gauge group (13.1) and (13.2); (13.3) and (13.4). Our auxiliary file with the details of these verifications is bigger than the present paper. During this work we have found a number of mistakes in the description of hypercharges of the fields in some original papers. On the basis of this large amount of computations, we put forward the following conjecture. Conjecture. The condition of total ellipticity for the elliptic hypergeometric terms is necessary for the existence of the exact integration formulas for elliptic beta integrals or of the nontrivial Weyl group symmetry transformations for the elliptic hypergeometric integrals. It is known that behind each elliptic hypergeometric integral there is a terminating elliptic hypergeometric series appearing from the residue calculus for restricted values of parameters [19]. The above conjecture has a natural meaning in terms of such series—it simply demands that the summation or transformation identities for them involve ratios of Jacobi forms with appropriate quasiperiodicity and modularity properties in the sense of Eichler and Zagier [27]. Already this fact is sufficient (when there are no fractional powers of pq) for the confirmation of the series identities to rather high powers of small log q expansions [19]. For a given elliptic hypergeometric integral there may exist more than one totally elliptic hypergeometric term. For the terms associated with elliptic beta integrals discussed in [86,93] there existed a complementary difference equation with the totally
808
V. P. Spiridonov, G. S. Vartanov
elliptic function coefficients. During our work we have found examples of fake terms which do not lead to identities (or fake anomaly matching conditions without real duality). Therefore analysis of the sufficiency condition for existence of nontrivial identities looks much more neat – it should address the non-uniqueness questions and the list of additional admissible technical tools. Sometimes the ratio of a given elliptic hypergeometric integral kernel to itself with different integration variables yields the totally elliptic hypergeometric term. It may happen that for a fixed set of parameters, it is sufficient to have totally elliptic hypergeometric terms of a more complicated nature than the latter one, and then at least one of them will lead to a nontrivial relation between integrals. 3. Superconformal Index 3.1. N = 1 superconformal algebra. In 4 dimensional space-time the conformal algebra S O(4, 2) is formed by the generators of translations Pa , special conformal transformations K a , S O(3, 1) Lorentz group rotations, Mab = −Mba , and the dilations H . The commutation relations have the form [25] [Mab , Pc ] = i(ηac Pb − ηbc Pa ), [Mab , K c ] = i(ηac K b − ηbc K a ), [Mab , Mcd ] = i(ηac Mbd − ηbc Mad − ηad Mbc + ηbd Mac ), [H, Pa ] = Pa , [H, K a ] = −K a , [K a , Pb ] = −2i Mab − 2ηab H,
(3.1)
where ηab = diag(−1, 1, 1, 1) and all indices take values a = 0, 1, 2, 3. In terms of the matrix ⎞ ⎛ − 21 (Pa − K a ) − 21 (Pa + K a ) Mab ⎟ ⎜ M AB = ⎝ 21 (Pb − K b ) (3.2) 0 iH ⎠, 1 2 (Pb
+ Kb)
−i H
0
where A, B = 0, . . . , 5, relations (3.1) are rewritten in the simpler form [25] [M AB , MC D ] = i(η AC M B D − η BC M AD − η AD M BC + η B D M AC )
(3.3)
with η AB = diag(−1, 1, 1, 1, 1, −1). In the spinorial basis one defines Pα α˙ = (σ a )α α˙ Pa , i Mαβ = − (σ a σ b )βα Mab , 4 where α, α, ˙ β, β˙ = 1, 2, σ a = (I, σ i ),
˙ ˙ K αα = (σ a )αα Ka , i α˙ M β˙ = − (σ a σ b )αβ˙˙ Mab , 4
(3.4)
σ a = (I, −σ i ),
and σ i are the usual Pauli matrices 0 1 0 −i 1 0 1 2 3 , σ = , σ = . σ = 1 0 i 0 0 −1 Using the standard angular momentum generators, we set α˙ J3 J+ J3 J+ Mαβ = , M β˙ = , J− −J3 J − −J 3
(3.5)
Elliptic Hypergeometry of Supersymmetric Dualities
809
with [J3 , J± ] = ±J± , [J+ , J− ] = 2J3 and similar relations for J ± , J 3 . Then the tensor Mab is expressed through these operators as ⎛
Mab
0 ⎜ i ⎜ − 2 (J + + J − − J+ − J− ) =⎜ ⎜ − 1 (J + J − J − J ) − + − ⎝ 2 + −i(J 3 − J3 )
i 2 (J +
+ J − − J+ − J− )
1 2 (J+
+ J − − J + − J− ) −(J3 + J 3 )
0 (J3 + J 3 )
0
− 2i (J+ + J + − J− − J − )
1 2 (J+
⎞
i(J 3 − J3 ) ⎟ i (J + J + − J− − J − ) ⎟ + 2 ⎟. 1 − 2 (J+ + J− + J + + J − ) ⎟ ⎠
+ J− + J + + J − )
0
The conformal algebra (3.1) can be rewritten now as β
β
α˙
β
γ˙
γ˙
α˙
γ˙
[M β˙ , M δ˙ ] = δβ˙ M δ˙ − δδα˙˙ M β˙ , 1 α˙ [M β˙ , Pγ δ˙ ] = −δδα˙˙ Pγ β˙ + δβα˙˙ Pγ δ˙ , 2 1 α˙ γ˙ ˙ [M β˙ , K γ˙ δ ] = δβ˙ K αδ − δβα˙˙ K γ˙ δ , 2 α˙ [M β˙ , H ] = 0,
[Mα , Mγδ ] = δγ Mαδ − δαδ Mγ , 1 β β β [Mα , Pγ δ˙ ] = δγ Pα δ˙ − δα Pγ δ˙ , 2 1 β β [Mα , K γ˙ δ ] = −δαδ K γ˙ β + δα K γ˙ δ , 2 β [Mα , H ] = 0,
(3.6)
˙ ] = −K αβ ˙ , [H, K αβ
[H, Pα β˙ ] = Pα β˙ ,
γ˙ γ˙ γ˙ [Pα β˙ , K γ˙ δ ] = 4 δβ˙ Mαδ − δαδ M β˙ + δβ˙ δαδ H . S O(4, 2) (or SU (2, 2)) algebra can be extended by adding supercharges Q α , Q α˙ and α˙ their superconformal partners S α , S . Supercharges satisfy the anticommutator relations [25,97] {Q α , Q α˙ } = 2Pα α˙ ,
{Q α , Q β } = {Q α˙ , Q β˙ } = 0,
(3.7)
while their superconformal partners obey α˙
˙ {S , S α } = 2K αα ,
α˙
β˙
{S , S } = {S α , S β } = 0.
(3.8)
The cross-anti-commutators of Q α and Sα have the form α˙
{Q α , S } = 0,
{S α , Q α˙ } = 0,
(3.9)
while 1 3 {Q α , S β } = 4 Mαβ + δαβ H + δαβ R , 2 4 1 3 α˙ α˙ {S , Q β˙ } = 4 M β˙ − δβα˙˙ H + δβα˙˙ R , 2 4 where R is the R-charge generating U (1) R -symmetry group.
(3.10)
810
V. P. Spiridonov, G. S. Vartanov
The bosonic and fermionic generators cross-commute as β
β
β
β
[Mα , Q γ ] = δγ Q α − 21 δα Q γ , γ
β
[Mα , Q γ˙ ] = 0,
β
β
[6 pt][Mα , S γ ] = −δα S β + 21 δα S γ ,
γ˙
[Mα , S ] = 0,
α˙
α˙
[M β˙ , Q γ˙ ] = −δγα˙˙ Q β˙ + 21 δβα˙˙ Q γ˙ ,
[M β , Q γ ] = 0, α˙
α˙
γ˙
γ˙ α˙
γ˙
γ˙
γ˙
[M β˙ , S ] = δβ˙ S − 21 δβα˙˙ S ,
[M β , S γ ] = 0, γ
[Pα β˙ , S γ ] = δα Q β˙ ,
(3.11)
[Pα β˙ , S ] = δβ˙ Q α ,
β α˙
˙ ,Q ]=δ S , [K αβ γ γ
˙ , Q ] = δ α˙ S β , [K αβ γ˙ γ˙
[H, Q α ] = 21 Q α ,
[H, Q α˙ ] = 21 Q α˙ ,
[H, S α ] = − 21 S α ,
[H, S ] = − 21 S .
α˙
α˙
The R-charge commutes with all bosonic generators and has non-trivial commutators only with the supercharges and their superconformal partners [R, Q α ] = −Q α ,
[R, Q α˙ ] = Q α˙ ,
[R, S α ] = S α ,
[R, S ] = −S .
α˙
(3.12)
α˙
To simplify the shape of the N = 1 superconformal algebra one introduces the notations ⎛ MAB = ⎝
β
β
Mα + 21 δα H 1 αβ ˙ 2K
1 2 Pα β˙ α˙ M β˙ − 21 δβα˙˙ H
⎞ ⎠ , QA =
Qα α˙ S
B
, Q =
S β Q β˙ . (3.13)
Then the (anti)commutators (3.6), (3.7), (3.8), (3.9), (3.11), (3.12) combine to [26] D MB [MAB , MCD ] = δCB MAD − δA C, 1 B [MAB , QC ] = δCB QA − δA QC , 4
[R, QA ] = −Q A ,
1 B C C C B [MAB , Q ] = −δA Q + δA Q , 4 (3.14)
B
B
[R, Q ] = Q ,
B
B {QA , Q } = 4MB A + 3δA R,
{QA , QB } = 0,
where B δA
=
β
δα 0 0 δβα˙˙
.
A
B
{Q , Q } = 0,
Elliptic Hypergeometry of Supersymmetric Dualities
811
3.2. The index. Suppose an operator Q and its Hermitian conjugate Q † satisfy the relations {Q, Q} = 0,
{Q † , Q † } = 0,
{Q, Q † } = 2H,
(3.15)
where H is the Hamiltonian (= P0 ) of a taken system. This is a universal situation valid down to the non-relativistic quantum mechanics. The Witten index [99] defined as T r (−1)F tells (under certain conditions) whether the supersymmetry is broken spontaneously or not. By definition the operator (−1)F is (−1)F = exp(2π iJz ),
{Q, (−1)F } = 0,
(3.16)
where in the spinorial basis Jz = −J3 − J 3 . It distinguishes bosonic states |b from the fermionic ones | f , (−1)F |b = |b ,
(−1)F | f = −| f .
Because of the cancellation of contributions of states with positive energies to T r (−1)F , this trace formally can be evaluated using the zero-energy states T r (−1)F = n bE=0 − n E=0 , f
(3.17)
are the numbers of bosonic and fermionic ground states. Therewhere n bE=0 and n E=0 f F fore, if T r (−1) = 0, supersymmetry is not broken. However, because of the presence of infinitely many states, one needs a regulator commuting with Q (to save cancellations). Then the regularized Witten index is defined as IW = T r ((−1)F e−β H ),
(3.18)
and formally it does not depend on the parameter β. As to N = 1 superconformal theories, there are different possibilities to realize relaα˙ tion (3.15), due to the superconformal operators S α , S . Namely, one picks a generator † Q with its adjoint Q , such that {Q, Q† } = 2H,
(3.19)
where H does not coincide with the Hamiltonian. Then one can consider the subspace of the Hilbert space composed of the BPS states |ψ annihilated by H, H|ψ = 0, and define the Witten index IW = T r ((−1)F e−β H ). However, the space of such states |ψ is infinite dimensional and one has to introduce other regulators, which leads to a nontrivial generalization of the index itself. For SU (2, 2|1) group, there are four non-trivial choices for supercharges Q, Q† , which can be used for constructing the superconformal index: 3 3 {Q 2 , S 2 } = 2 H − 2J3 + R ; {Q 1 , S 1 } = 2 H + 2J3 + R ; 2 2 3 3 1 2 {Q 2 , −S } = 2 H + 2J 3 − R . {Q 1 , −S } = 2 H − 2J 3 − R ; 2 2 (3.20)
812
V. P. Spiridonov, G. S. Vartanov
The generators commuting with the corresponding pairs of supercharges are 1 ˙ R, P2α˙ , K α2 ; 2 1 Mαβ , H − R, Pα2 , K 2α ; 2
1 ˙ R, P1α˙ , K α1 ; 2 1 Mαβ , H − R, Pα1 , K 1α , 2
α˙
α˙
M β˙ , H +
M β˙ , H +
respectively, see (3.14). Let us stick to the choice 1
Q = Q 1 , Q† = −S , H = H − 2J 3 −
3 R. 2
Composing the matrix [26] M AB
=
β
β
Mα + 21 δα R P
β
Pα −R + 21 H
,
(3.21)
β
where Pα = 21 Pα2 , P = 21 K 2β , and R=H−
1 R 2
we come to the SU (2, 1) Lie algebra with the relations [M AB , MCD ] = δCB M AD − δ AD MCB .
(3.22)
To regularize the trace over the infinite dimensional space of zero modes of H, we use all operators commuting between themselves and with the distinguished supercharges Q and Q† . In our case one additional regulator is t R for some arbitrary complex variable 1 β t restricted as |t| < 1 to ensure damping. Since Mα commute with Q 1 and S , there β is one more regulator x 2J3 , |x| < 1, resolving the degeneracy ensured by Mα . Finally, one defines [72,73] ind(t, x) = T r (−1)F x 2J3 t R e−β H .
(3.23)
This index explicitly depends on the chemical potentials x and t, different from the variable β. In the presence of internal symmetries, one can introduce more regulators to resolve the degeneracies. For U (1)k global symmetry group, one introduces chemical potentials μ j , j = 1, . . . , k, and extends the superconformal index as ind(t, x, μ j ) = T r (−1)F x 2J3 t R e
k
j=1 μ j q j
,
(3.24)
where q j is the generator of j th U (1)-group. For a non-abelian local gauge invariance group G with the maximal torus generators G a , a = 1, . . . , rank G, and a flavor group F with the maximal torus generators F j , j = 1, . . . , rank F, the index reads rank F rank G j a (3.25) ind(t, x, z, y) = T r (−1)F x 2J3 t R e a=1 ga G e j=1 f j F , where ga and f j are the chemical potentials for groups G and F respectively. We assume that the global abelian groups enter the flavor group contributions in (3.25). From the
Elliptic Hypergeometry of Supersymmetric Dualities
813
G representation theory it is known that T r exp( rank gi G i ) = χG (z) is the character i=1 of the corresponding representation of the gauge group G, where z is the set of complex eigenvalues of matrices realizing G. The same is valid for the flavor group F: M T r exp( rank f j F j ) = χ F (y) is the character of the representations forming the j=1 space of free field states, and y is the set of complex eigenvalues of matrices realizing F. Since all physical observables are gauge invariant, one is interested in the index for gauge singlet operators. Therefore formula (3.25) is averaged over the gauge group, which yields the matrix integral rank F rank G j a I (t, x, y) = dμ(g) T r (−1)F x 2J3 t R e a=1 ga G e j=1 f j F , (3.26) G
where dμ(g) is the G-invariant matrix group measure. This is the superconformal index – the key object for our purposes. By construction, it has the meaning of a particular combination of SU (2, 2|1) × G × F group characters naturally restricted to the space of BPS states and integrated over the gauge group. 3.3. Calculation of the index. Explicit computation of the superconformal index for N = 1 theories was performed by Römelsberger [73]. According to his prescription one should first compute the trace in index (3.25) over the single particle states, which yields the formula i(t, x, z, y) =
2t 2 − t (x + x −1 ) χad j (z) (1 − t x)(1 − t x −1 ) t 2r j χ R F , j (y)χ RG , j (z) − t 2−2r j χ R¯ , j (y)χ R¯ , j (z) F G , + (1 − t x)(1 − t x −1 )
(3.27)
j
where the first term represents contribution of the gauge fields belonging to the adjoint representation of the group G, and the sum over j corresponds to the chiral matter superfields ϕ j transforming as the gauge group representations RG, j and non-abelian flavor symmetry group representations R F, j . The functions χad j (z), χ R F , j (y) and χ RG , j (z) are the corresponding characters – their explicit forms for major classical groups are described in Appendix A. In the original Römelsberger formula the denominators are written as 1 − tχ SU (2), f (γ ) + t 2 , where χ SU (2), f (γ ) is the character for the fundamental representation of the SU (2) subgroup in (3.22). Parametrizing it by the eigenvalue x, one comes to (3.27). The U (1) R -group contribution to (3.27) is described by the terms t 2R j and t 2−2R j resulting from a chiral scalar field with the R-charge 2R j and the fermion partner of the conjugate anti-chiral fields whose R-charge is −2R j . In the presence of additional global U (1)-groups the variables r j have the form rj = Rj +
k
q jl μl ,
l=1
j th
where 2R j is the R-charge of the chiral superfield, q jl are the normalized hypercharges of the j th matter superfield for l th U (1)-group and 2μl is the chemical potential for the latter group.
814
V. P. Spiridonov, G. S. Vartanov
To obtain the full superconformal index, this single particle states index is inserted into the “plethystic” exponential with the subsequent averaging over the gauge group: ∞ 1 n n n n i t ,x ,z ,y . (3.28) dμ(g) exp I (t, x, y) = n G n=1
Similar objects appeared in computation of partition functions of different statistical mechanics models and quantum field theories, see, e.g., [3,4,24,29,49,57,58,80,95]. Clearly there are two qualitatively different contributions to superconformal indices – from the matter fields and the gauge fields. The generic form of a matter field single particle states contribution to i(t, x, z, y) in the presence of some global U (1) symmetry group is i S (t, x, y) =
t 2R y − t 2−2R y −1 , (1 − t x)(1 − t x −1 )
(3.29)
where t, x are the same variables as in (3.27) and y = t 2μ is the chemical potential for the U (1) group. It is convenient to introduce new parametrization p = t x, q = t x −1 , w = t 2R y,
(3.30)
where p and q are (in general, complex) parameters satisfying the constraints |q|, | p| < 1. As a result, we can write i S ( p, q, w) =
w − pqw −1 . (1 − p)(1 − q)
Then the described index building algorithm yields (cf. [3]) ∞ ∞ 1 1 − w −1 p j+1 q k+1 n n n iS( p , q , w ) = exp =: (w; p, q). (3.31) n 1 − w p j qk n=1
j,k=0
This result corresponds to formula (69) in [73] after the identifications w := t q u, p := t y, q := t y −1 . However, the fact that this index coincides with the elliptic gamma function was recognized only by Dolan and Osborn in [26]. For the gauge field part one can set q 2t 2 − t (x + x −1 ) p χ − χad j (z). i V ( p, q, z) = (z) = − ad j (1 − t x)(1 − t x −1 ) 1− p 1−q (3.32) For the SU (2) group one has χad j (z) = z 2 + z −2 + 1. Substituting pieces of this expression in the corresponding places of the index, we obtain the following characteristic building blocks ∞ 1 pn qn θ (z 2 ; p)θ (z 2 ; q) 2n −2n (z + z ) = − exp − n n n 1− p 1−q (1 − z 2 )2 n=1
=
1 (1 −
z 2 )(1 − z −2 )(z ±2 ;
p, q)
Elliptic Hypergeometry of Supersymmetric Dualities
and
815
∞ 1 qn pn exp − = ( p; p)∞ (q; q)∞ . − n 1 − pn 1 − qn n=1
Similar expressions are found for field contributions for the higher rank gauge groups. 4. Seiberg Duality for Unitary Gauge Groups First we consider the usual N = 1 supersymmetric quantum chromodynamics (SQCD) as an electric theory with the internal symmetry groups [76] G = SU (N ),
F = SU (N f ) × SU (N f ) × U (1) B ,
where U (1) B is generated by the baryon number charge (the U (1) R group enters the superconformal group). This supersymmetric version of QCD has two chiral scalar multiplets Q and Q˜ belonging to the fundamental f and anti-fundamental f¯ representations of SU (N ) respectively, each carrying a baryon number, and the vector multiplet V in the adjoint representation of G. The field content of the electric theory is collected in the following table: Field
SU (N )
SU (N f )
SU (N f )
Q Q˜
f f¯
1 f¯
V
adj
f 1 1
1
U (1) B
U (1) R
q B = 1 2R Q = N˜ /N f q B = −1 2R Q = N˜ /N f 0 2R V = 1
Here q B , q B denote the baryonic charge and R Q , R Q, R V are half R-charges of the fields. The dual magnetic theory has the symmetry groups ), G = SU ( N
F = SU (N f ) × SU (N f ) × U (1) B ,
= N f − N . Its field content is fixed in the table below where N Field q q˜ M V˜
SU ( N˜ )
SU (N f )
f f¯ 1 adj
f¯ 1 f 1
SU (N f ) 1 f f¯ 1
U (1) B = N / N˜ qB = −N / N˜ qB
0 0
U (1) R 2Rq = N /N f 2R q = N /N f 2R M = 2 N˜ /N f 2R V = 1
This duality is supposed to work only in the conformal window 3N /2 < N f < 3N , following from the demand that both dual theories are asymptotically free in the one-loop approximation. The one-loop beta function for the gauge coupling is given by 11 2 1 g3 T (adj) − T (F) − T (S) , βg = − 16π 2 3 3 3 where T (F) is the sum of Casimir coefficients T (r) (see Appendix C for more details) over all fermions, T (S) is the similar sum over all scalars and T (adj) is T (r) for the adjoint representation. For a summary of this and two loop renormalization group results, see [56].
816
V. P. Spiridonov, G. S. Vartanov
The r j -charges of fields coming from U (1) R and U (1) B currents in the electric theory are r Q = R Q + q B x,
r Q = R Q + q B x,
where x is the U (1) B -group chemical potential. In the magnetic theory we set rq = Rq + q B x,
r q B x, q = R q +
rM = RM .
Then the single particle states index for the electric theory has the form i E ( p, q, z, s, t) = −
p q + 1− p 1−q
χ SU (N ),ad j (z)
1 ( pq)r Q χ SU (N f ), f (s)χ SU (N ), f (z)−( pq)1−r Q χ SU (N ), f (s)χ SU (N ), f (z) f (1 − p)(1 − q) rQ 1−r Q χ SU (N f ), f (t)χ SU (N ), f (z) . (4.1) +( pq) χ SU (N ), f (t)χ SU (N ), f (z) − ( pq)
+
f
For the magnetic theory we have i M ( p, q, z, s, t) = −
q p + 1− p 1−q
χ SU ( N ),ad j (z)
1 1−rq χ ( pq)rq χ SU (N ), f (s)χ SU ( N + ), f (z) − ( pq) SU (N f ), f (s)χ SU ( N ), f (z) f (1 − p)(1 − q) 1−r qχ +( pq)rq χ SU (N f ), f (t)χ SU ( N ), f (z) ), f (z) − ( pq) SU (N f ), f (t)χ SU ( N
+( pq)r M χ SU (N f ), f (s)χ SU (N ), f (t) − ( pq)1−r M χ SU (N ), f (s)χ SU (N f ), f (t) . f f
(4.2)
The superconformal indices take the form (see the invariant measures in Appendix B) IE =
where
N −1 (q; q) N −1 ( p; p)∞ ∞ N! N f N N −1 rQ r Q −1 −1 j=1 (( pq) si z j , ( pq) ti z j ; p, q) dz j i=1 , (4.3) × −1 −1 2π iz j T N −1 1≤i< j≤N (z i z j , z i z j ; p, q) j=1
N j=1
zj =
N f
i=1 si
IM =
N f
N −1 (q; q) N −1 ( p; p)∞ ∞ ! N
N f N
×
where
=
N j=1
i=1
T N−1
i=1 ti
= 1, and
(( pq)r M si t −1 j ; p, q)
1≤i, j≤N f
−1 rq −1 r q j=1 (( pq) si z j , ( pq) ti z j ; −1 −1 (z i z j , z i z j ; p, q) 1≤i< j≤ N
−1 p, q) N dz j , 2π iz j
(4.4)
j=1
z j = 1. Let us renormalize the variables si → ( pq)−r Q si , ti−1 → ( pq)−r Q ti−1 , i = 1, . . . , N f .
(4.5)
Elliptic Hypergeometry of Supersymmetric Dualities
817
Then the superconformal indices are rewritten as the following elliptic hypergeometric integrals: IE =
N −1 (q; q) N −1 ( p; p)∞ ∞ N! N f N
×
−1 −1 N −1 j=1 (si z j , ti z j ; p, q) −1 −1 1≤i< j≤N (z i z j , z i z j ; p, q) j=1
dz j 2π iz j
(4.6)
−1 p, q) N dz j , 2π iz j
(4.7)
i=1
T N −1
and
IM =
N −1 (q; q) N −1 ( p; p)∞ ∞ ! N
N f N
×
i=1
T N−1
(si t −1 j ; p, q)
1≤i, j≤N f
−1 1/ N si z j , T −1/ N ti z −1 j ; −1 −1 (z i z j , z i z j ; p, q) 1≤i< j≤ N j=1 (S
j=1
N f N f −1 = where S = i=1 si , T = i=1 ti , and the balancing condition reads ST ( pq) N f −N . As shown by Dolan and Osborn [26], the equality I E = I M coincides with the An ↔ =2 Am root systems symmetry transformation established by Rains [65]. For N = N this identity is a simple consequence of the symmetry transformation for an elliptic analogue of the Gauss hypergeometric function discovered earlier by the first author in [83]. Note that this equality of integrals is valid for any N f , while the Seiberg duality is expected to exist only in the conformal window, where we have appropriate R−charges yielding an anomaly free theory. One cannot extrapolate the duality outside this window except of the boundary points N f = 3N /2 and N f = 3N (we thank A. Schwimmer and S. Theisen for a discussion on this point). However, this does not mean that for the electric theory outside the conformal window there cannot be different magnetic duals. We present a number of such examples in a separate paper [92]. The needed equality between elliptic hypergeometric integrals is rigorous only under certain constraints on the parameters. The kernels of the integrals are meromorphic functions of integration variables z j ∈ C∗ . There are two qualitatively different geometric sequences of poles of these kernels—some of them converge to zero z j = 0 and others go to infinity. So, the equality I E = I M with the integration contours T on both sides is true provided T separates these two types of pole sequences. In the present situation ˜ ˜ this is guaranteed for |S|1/ N < |si | < 1 and 1 < |ti | < |T |1/ N . All the relations for superconformal indices described below have similar constraints on the parameters, but we shall not describe them for brevity, assuming that the separability conditions for pole sequences are satisfied by the contour T. 5. Intriligator-Pouliot Duality for Symplectic Gauge Groups The electric theory has the overall symmetry group G = S P(2N ), and the following matter field content:
F = SU (2N f ),
818
V. P. Spiridonov, G. S. Vartanov
S P(2N )
SU (2N f )
f
f
Q
U (1) R 1 − (N + 1)/N f
, except for In this and all other tables below we drop the vector superfields V (or V the confining theories where this field is absent), since they are always described by the adjoint representation of G and singlets of F. The dual magnetic theory constructed by Intriligator and Pouliot [43] has the same ), where N = N f − N − 2, with the flavor group and the gauge group G = S P(2 N field content described in the table below: ) S P(2 N
SU (2N f )
U (1) R
f 1
f¯ TA
(N + 1)/N f + 1)/N f 2( N
q M
The conformal window for this duality is 3(N + 1)/2 < N f < 3(N + 1). For these theories we have the following indices (in the renormalized variables) [26]: 2N f N ±1 N N (q; q) N dz j ( p; p)∞ j=1 (ti z j ; p, q) i=1 ∞ IE = , N ±1 ±1 ±2 N 2N N ! 2π iz j T j=1 (z j ; p, q) 1≤i< j≤N (z i z j ; p, q) j=1
(5.1) and
IM =
N (q; q) N ( p; p)∞ ∞ N 2 N!
(ti t j ; p, q)
1≤i< j≤2N f
2N f N
×
N 1/2 t −1 z ±1 ; p, q) i j N ±1 ±1 ±2 (z i z j ; p, q) j=1 (z j ; p, q) j=1 1≤i< j≤ N i=1
T N
j=1 (( pq)
dz j , 2π iz j
(5.2)
2N = 1, the equality with the balancing condition i=1f ti = ( pq) N f −N −1 . For N = N I E = I M is a consequence of the symmetry transformation established in [83]. For , the needed identity (2.11) was proven by Rains in [65]. After arbitrary ranks N , N the degeneration to the rational integrals level, it reduces to the Dixon transformation formula [23]. 6. Multiple Duality for S P(2N) Gauge Group There exists a multiple duality phenomenon, when one electric theory has many magnetic duals. In this section we describe theories with S P(2N ) gauge group, where multiple duality is ensured by W (E 7 ), the Weyl group for the exceptional root system E 7 . However, we skip the description of this group referring for details to [91]. We take N = 1 SQCD electric theory with the symmetry groups G = S P(2N ) and F = SU (8) × U (1). This model has one chiral scalar multiplet Q belonging to the fundamental representations of G and F, a vector multiplet V in the adjoint representation, and the antisymmetric S P(2N )-tensor field X , see the table below: Q X
S P(2N ) f TA
SU (8) f 1
U (1) − N 4−1 1
U (1) R 1 2
0
Elliptic Hypergeometry of Supersymmetric Dualities
819
For N = 1, the field X is absent and the U (1)-group is completely decoupled. In [91] we were giving in tables halves of the R-charges. This electric theory and its particular magnetic dual (with N > 1) were considered in [18]. However, as described in [91], there are other dual theories. In a special section below we show that the ’t Hooft anomaly matching conditions are fulfilled for all these new dualities. The electric superconformal index is IE =
N (q; q) N (( pq)s z i±1 z ±1 ( p; p)∞ j ; p, q) ∞ s N −1 ; p, q) (( pq) ±1 ±1 N 2 N! (z i z j ; p, q) T N 1≤i< j≤N N 8 (( pq)r Q y z ±1 ; p, q) i j dz j i=1 , (6.1) × ±2 2π iz j (z j ; p, q) j=1
where r Q = R Q + e Q s, r X = e X s, and 2R Q = 1/2 is the R-charge of the Q-field, e Q = −(N − 1)/4 and e X = 1 are the U (1)-group hypercharges with s being its chemical potential. The first (new) class of magnetic theories has the symmetry groups G = S P(2N ),
F = SU (4) × SU (4) × U (1) B × U (1).
Its field content is fixed in the table below: q q Y MJ J M
S P(2N ) f f TA 1 1
SU (4) f 1 1 TA 1
SU (4) 1 f 1 1 TA
U (1) B −1 1 0 2 −2
U (1) − N 4−1 − N 4−1 1
2J −N +1 2 2J −N +1 2
U (1) R 1 2 1 2
0 1 1
In the tables of this section the capital index J takes values 0, . . . , N − 1, which is not indicated for brevity. The superconformal index in this magnetic theory is (1)
IM =
N −1
(( pq)r M J yi y j ; p, q)
J =0 1≤i< j≤4
rM
(( pq)
J
yi y j ; p, q)
5≤i< j≤8
N (q; q) N (( pq)s z i±1 z ±1 ( p; p)∞ j ; p, q) ∞ ±1 ±1 2N N ! (z i z j ; p, q) T N 1≤i< j≤N 4 8 ±1 ±1 N rq −2 r q 2 i=1 (( pq) v yi z j ; p, q) i=5 (( pq) v yi z j ; p, q) dz j
×(( pq)s ; p, q) N −1 ×
(z ±2 j ; p, q)
j=1
2π iz j
,
(6.2) where v =
√ 4
y1 y2 y3 y4 and N −1 N −1 s, r s, rY = s, q = R q − 4 4 1 1 − (N − 1 − 2J )s, r M J = R M J − (N − 1 − 2J )s. 2 2
r q = Rq − rMJ = RMJ
820
V. P. Spiridonov, G. S. Vartanov
The second (new) class of dual magnetic theories has the same symmetries as in the previous case, but different representation content as described in the following table: q
S P(2N ) f
SU (4) f
SU (4) 1
U (1) B 1
U (1) − N 4−1
q
f
1
f
−1
− N 4−1
Y MJ
TA 1
1 f
1 f
0 0
2J −N +1 2
1
U (1) R 1 2 1 2
0 1
The index for this magnetic theory is given by (2)
I M = (( pq)s ; p, q) N −1 ×
N −1 4 8 J =0 i=1 j=5
( p; p) N (q; q) N ∞ 2N N !
(( pq)r M J yi y j ; p, q) (( pq)s z i±1 z ±1 j ; p, q)
∞
T N 1≤i< j≤N (z i±1 z ±1 j ; p, q) 4 8 r rq 2 −1 ±1 N q −2 −1 ±1 i=1 (( pq) v yi z j ; p, q) i=5 (( pq) v yi z j ; p, q) dz j , × 2π iz j (z ±2 j ; p, q) j=1
(6.3) where rq = r q =
N −1 1 1 1 − s, rY = s, r M J = − (N − 1 − 2J )s. 4 4 2 2
Finally, the third type of magnetic theories, which was constructed originally by Csáki, Skiba and Schmaltz in [18], has the symmetry groups G = S P(2N ) and F = SU (8) × U (1), and its fields content is S P(2N ) f
SU (8) f
U (1) − N 4−1
U (1) R
q Y MJ
TA 1
1 TA
2J −N +1 2
1
0 1
1 2
Corresponding magnetic superconformal index has the form (3)
I M = (( pq)rY ; p, q) N −1 ×
T N 1≤i< j≤N
N −1
(( pq)r M J yi y j ; p, q)
J =0 1≤i< j≤8
N (( pq)rY z i±1 z ±1 j ; p, q)
(z i±1 z ±1 j ; p, q)
j=1
N (q; q) N ( p; p)∞ ∞ 2N N !
8
rq −1 ±1 i=1 (( pq) yi z j ; p, q) dz j , 2π iz j (z ±2 j ; p, q)
(6.4) where rq =
1 − s(N − 1) 1 − s(N − 1) , rY = s, r M J = s J + . 4 2
The S P(2) gauge group case can be obtained from the tables above by substituting N = 1 and deleting fields X in the electric theory and Y in the magnetic theories, which
Elliptic Hypergeometry of Supersymmetric Dualities
821
decouple completely. The number of mesons in dual theories is reduced as well. Equality of superconformal indices for N = 1 follows from the results of [83], and the needed identities for elliptic hypergeometric integrals for N > 1 were established in [65]. As argued in [91], there should be in total 72 theories dual to each other – this number equals to the dimension of the coset group W (E 7 )/S8 responsible for the dualities (in this respect, see also [55]). 7. A New S P(2N) ↔ S P(2M) Groups Duality We take as the electric theory SQCD based on the symmetry groups G = S P(2M),
F = SU (4) × S P(2l1 ) × S P(2l2 ) × · · · × S P(2l K ) × U (1)
with the fields content fixed in the table below: S P(2M)
SU (4)
S P(2l1 )
S P(2l2 )
...
S P(2l K )
W1
f
f
1
1
...
1
Q1
f
1
f
1
...
1
Q1
f
1
1
f
...
1
... QK
f
1
1
1
...
f
X
TA
1
1
1
...
1
U (1)
U (1) R
−2 − M−N 4 n1 −2 n − 22 n
− 2K 1
0 1 1 1 0
K where n 1 = n 2 = · · · = n K and i=1 li n i = M + N . The dual magnetic theory has G = S P(2N ) and the same flavor group; the fields content is described below: S P(2N )
SU (4)
S P(2l1 )
S P(2l2 )
...
S P(2l K )
w1
f
f
1
1
...
1
q1
f
1
f
1
...
1
q1 ... qK
f
1
1
f
...
1
M−N +2 4 n − 21 n2 − 2
U (1)
U (1) R
f
1
1
1
...
f
n − 2K
Nj
1
TA
1
1
...
1
−2 j − M−N 2
0
M1,k1
1
f
f
1
...
1
−2 − 1 + k − M−N 1 4 2
1 1
0 1 1 1 n
1
f
1
f
...
1
−2 − n 2 + k − M−N 2 4 2
M K ,k K
1
f
1
1
...
f
−2 − K + k − M−N K 4 2
1
Y
TA
1
1
1
...
1
1
0
M2,k2 ...
n
where j = 0, . . . , M − N − 1 and ki = 0, . . . , n i − 1 for any i = 1, . . . , K . Here we assume that M ≥ N (for M = N the fields N j are absent). The superconformal indices have the form IE =
M (q; q) M (t z i±1 z ±1 ( p; p)∞ j ; p, q) ∞ M−1 (t; p, q) ±1 ±1 M 2 M! T M 1≤i< j≤M (z i z j ; p, q) K lr −1 ±1 ±1 M 4 k=1 (ttk z j ; p, q) r =1 j=1 (sr, j z j ; p, q) dz j × 2π iz j (z ±2 ; p, q) rK=1 lr (t nr sr, j z ±1 ; p, q) j=1
j
j=1
j
(7.1)
822
V. P. Spiridonov, G. S. Vartanov
and IM =
N (q; q) N ( p; p)∞ ∞ (t; p, q) N −1 2N N !
M−N −1
i=0
1≤k
(t i+2 tk−1 tr−1 ; p, q)
(t z i±1 z ±1 (t km +1 tr−1 sm,i ; p, q) j ; p, q) ±1 ±1 k (t m tr sm,i ; p, q) T N 1≤i< j≤N (z i z j ; p, q) r =1 m=1 i=1 km =0 K lr ±1 ±1 N 4 k=1 (tk z j ; p, q) r =1 j=1 (sr, j z j ; p, q) dz j , (7.2) × K lr ±1 nr 2π iz j (z ±2 j ; p, q) r =1 j=1 (t sr, j z j ; p, q) j=1 ×
lm n 4 K m −1
with the balancing condition r4=1 tr = t 2+M−N . We have checked that the anomalies of these two theories match (see below), which is a very strong indication that the theories are dual to each other. This is another new duality that we have found. It has rather complicated structure with the flavor group composed of an arbitrary number of simple group components. The renormalization group analysis shows that the asymptotic freedom is present on the electric side for K K M > i=1 li /2 − 1 and on the magnetic side for N > i=1 li /2 − 1. The equality of elliptic hypergeometric integrals I E = I M , which gives another argument supporting this duality, coincides with the Rains Conjecture 1 from [68] (it was used as a starting point for the derivation of the described duality). As we have known after the completion of this work, this conjecture is proven recently by van de Bult [9]. 8. Multiple Duality for SU(2N) Gauge Group We describe now the multiple duality phenomenon for SU (2N ) gauge group. The overall flavor symmetry group of the theories is rather unusual. For N = 1, this multiple duality coincides with that for S P(2) group, see [91]. For N > 2, one has F = SU (4) × SU (4) × U (1)1 × U (1)2 × U (1) B . For N = 2, the flavor subgroup U (1)1 is replaced by SU (2). The field content of the electric theory for N > 2 is shown in the table below: Q Q
SU (2N ) f
A A
SU (4) f
SU (4) 1
U (1)1 0
U (1)2 2N − 2
U (1) B 1
U (1) R
f
1
f
0
2N − 2
−1
1 2 1 2
TA TA
1 1
1 1
1 −1
−4 −4
0 0
0 0
Corresponding superconformal index has the form −1 2N −1 (q; q)2N −1 (U z j z k , V z −1 ( p; p)∞ j z k ; p, q) ∞ IE = −1 −1 (2N )! T2N −1 1≤ j
×
2N 4 j=1 k=1
(sk z j , tk z −1 j ; p, q)
2N −1 j=1
dz j , 2π iz j
(8.1)
where 2N z j = 1 and the balancing condition reads (U V )2N −2 ST = ( pq)2 with 4 j=1 S = k=1 sk and T = 4k=1 tk . This is the two-parameter (higher order) extension of
Elliptic Hypergeometry of Supersymmetric Dualities
823
the type II elliptic beta integral for the root system A2N −1 introduced by Spiridonov in [83]. For N ≥ 2, magnetic dual theories have the same gauge and global symmetry groups. The first (new) dual theory has the field content described for N > 2 in the table below: q
SU (2N ) f
SU (4) f
SU (4) 1
U (1)1 0
U (1)2 2N − 2
U (1) B −1
U (1) R
q
f
1
f
0
2N − 2
1
1 2 1 2
a a Hm G Hm G
TA TA 1 1 1 1
1 1 TA TA 1 1
1 1 1 1 TA TA
1 −1 −1 N −1 1 −N + 1
−4 −4 4N − 8 − 8m 0 4N − 8 − 8m 0
0 0 2 2 −2 −2
0 0 1 1 1 1
where m = 0, . . . , N − 2. This leads to the magnetic index
(1) IM =
N −2
(U N −1 si s j ,V N −1 ti t j ; p, q)
×
( p;
2N −1 (q; q)2N −1 p)∞ ∞
(2N )!
2N 4 j=1 k=1
(V (U V )m si s j , U (U V )m ti t j ; p, q)
m=0
1≤i< j≤4
×
T2N −1 1≤ j
−1 (V z j z k , U z −1 j z k ; p, q) −1 (z −1 j z k , z j z k ; p, q)
2N −1 dz j ( 4 T /Ssk z j , 4 S/T tk z −1 ; p, q) . j 2π iz j
(8.2)
j=1
Our second dual theory was found by Csáki et al in [17]. Its field content for N > 2 is described in the table below: SU (2N ) f
SU (4) f
SU (4) 1
U (1)1 0
U (1)2 2N − 2
U (1) B 1
U (1) R
q
f
1
f
0
2N − 2
−1
1 2 1 2
a a Mk
TA TA 1
1 1 f
1 1 f
1 −1 0
−4 −4 4N − 4 − 8k
0 0 0
0 0 1
q
where k = 0, . . . , N − 1. Its superconformal index has the form (2)
IM =
−1 4 2N −1 (q; q)2N −1 N ( p; p)∞ ∞ ((U V )m sk tl ; p, q) (2N )! m=0 k,l=1
× ×
T2N −1 1≤ j
2N 4 j=1 k=1
−1 (U z j z k , V z −1 j z k ; p, q) −1 (z −1 j z k , z j z k ; p, q)
2N −1 √ √ dz j ( Ssk−1 z j , T tk−1 z −1 ; p, q) . j 2π iz j j=1
(8.3)
824
V. P. Spiridonov, G. S. Vartanov
Our third, again new, duality corresponds to the theory described below for N > 2, SU (2N ) f
SU (4) f
SU (4) 1
U (1)1 0
U (1)2 2N − 2
U (1) B −1
U (1) R
q
f
1
f
0
2N − 2
1
1 2 1 2
a a Mk Hm G Hm G
TA TA 1 1 1 1 1
1 1 f TA TA 1 1
1 1 f 1 1 TA TA
1 −1 0 −1 N −1 1 −N + 1
−4 −4 4N − 4 − 8k 4N − 8 − 8m 0 4N − 8 − 8m 0
0 0 0 2 2 −2 −2
0 0 1 1 1 1 1
q
where k = 0, . . . , N − 1, m = 0, . . . , N − 2. Its superconformal index reads (3)
IM =
4 −1 −1 −1 N (q; q)2N ( p; p)2N ∞ ∞ ((U V )m sk tl ; p, q) (2N )! m=0 k,l=1 ⎤ ⎡ N −2 N −1 N −1 m m ⎣ × si s j , V ti t j ; p, q) (V (U V ) si s j , U (U V ) ti t j ; p, q)⎦ (U 1≤i< j≤4
m=0
−1 2N 4 (V z j z k , U z −1 √ √ j z k ; p, q) 4 4 ( ST sk−1 z j , ST tk−1 z −1 × j ; p, q) −1 −1 2N −1 T 1≤ j
×
2N −1 j=1
dz j . 2π iz j
(8.4)
(1) = From the duality arguments for these field theories, we conjecture that I E = I M (2) (3) I M = I M under certain constraints on the integral parameters, which yield new powerful elliptic hypergeometric integral identities. Instead of the W (E 7 ) Weyl group symmetry in parameters, existing for N = 1, only its subgroup of reflection transformations consistent with the permutational S4 × S4 symmetry group survives. Nevertheless, preliminary considerations indicate that these relations should be provable by an appropriate analog of the method used in [65] for proving W (E 7 )-identities for BC N -integrals of type II. We have checked that the reduction from N f = 4 to N f = 3 realized by the constraint s4 t4 = pq reduces superconformal indices to Spiridonov’s A2N −1 -elliptic beta integral [83], i.e. equality of indices in this case is proven rigorously. For N = 2, superconformal indices are given by the same integrals. However, in this case 1≤ j
Elliptic Hypergeometry of Supersymmetric Dualities
825
have shown, that our multiple SU (2N )-dualities contain S P(2N ) dualities as special subcases. The first mathematical observation that the type II hypergeometric identities for the BC N -root system can be obtained from type II relations for both A2N −1 and A2N root systems has been done in [94], where various new multiple 6 ψ6 summation formulas on root systems have been suggested. Here we extend this observation to the (expected) relations between type II elliptic hypergeometric integrals. On the physical ground, such a relation between the particular SU (4) and S P(4) gauge group dualities was observed in [17]. Note that the further limit U = 1 leads to the SU (2) N -gauge group theories whose indices are given by N -th power of the indices of S P(2)-group models with N f = 4 constructed in [91]. The attempts in [17] to construct an analogous duality for even rank gauge groups SU (2N + 1) have failed. We have succeeded in solving this problem; corresponding results together with the residue calculus details will be presented in a separate paper.
9. Kutasov-Schwimmer Type Dualities for the Unitary Gauge Group Now we pass to generalizations of the Seiberg dualities for unitary and symplectic gauge groups G discovered by Kutasov and Schwimmer (KS) [51,52] and studied in detail in [53] and other papers. For brevity, we skip separate global symmetry group descriptions since they can be read off easily from the field contents of the theories given in the tables. The first column in the tables describes gauge group representations for fields, while other columns, except of the very last one, describe representations and hypercharges for subgroups of the flavor group F. Also, we skip the detailed description of single particle state indices and write out directly the integrals for the superconformal indices together with the balancing condition, if there is any. In this section we describe such dualities for G = SU (N ). 9.1. SU (N ) gauge group with the adjoint matter field. The following electric-magnetic duality is described in [52]. The field content of the electric theory is
Q Q
SU (N ) f
SU (N f ) f
SU (N f ) 1
U (1) B 1
f
1
f
-1
2 r =1 −
ad j
1
1
0
2s
X
U (1) R 2N 2r = 1 − (K +1)N f 2N (K +1)N f = K2+1
The magnetic theory ingredients are collected in the following table:
q
) SU ( N f
SU (N f ) f
SU (N f ) 1
U (1) B N /N
q
f
1
f
−N / N
Y
ad j
1
1
0
Mj
1
f
f
0
U (1) R 2N 2r = 1 − (K +1)N f
2N (K +1)N f 2s = K2+1 4N 2r M j =2 − (K +1)N + 2(Kj−1) +1 f
2 r = 1 −
826
V. P. Spiridonov, G. S. Vartanov
Here j = 1, . . . , K and the dual gauge group dimension is = K N f − N, N
K = 1, 2, . . . ,
(9.1)
with the constraint N f > N /K . 1 Defining U = ( pq)s = ( pq) K +1 , we find the following indices for these theories: IE =
−1 N −1 (q; q) N −1 (U z i z −1 ( p; p)∞ j , U z i z j ; p, q) ∞ (U ; p, q) N −1 −1 −1 N! T N −1 1≤i< j≤N (z i z j , z i z j ; p, q) ×
Nf N
(si z j , ti−1 z −1 j ; p, q)
i=1 j=1
N −1 j=1
dz j , 2π iz j
(9.2)
where Nj=1 z j = 1, the balancing condition reads U 2N ST −1 = ( pq) N f with N f N f S = i=1 si , T = i=1 ti , and IM =
K −1 −1 N N (q; q)∞ ( p; p)∞ (U l−1 si t −1 (U ; p, q) N −1 j ; p, q) N! l=1 1≤i, j≤N f
×
×
T N−1
−1 (U z i z −1 j , U z i z j ; p, q) −1 (z i z −1 j , z i z j ; p, q) 1≤i< j≤ N
Nf N
(U (ST )
K 2N
K
− si−1 z j , U (ST ) 2 N ti z −1 j ;
p, q)
i=1 j=1
−1 N j=1
dz j , 2π iz j
(9.3)
where Nj=1 z j = 1. An important fact is that these theories contain matter fields in the adjoint representation of the gauge group. The conjecture that I E = I M (under appropriate contour separability constraints mentioned earlier) represents a new type of elliptic hypergeometric identities, which was not met earlier [89]. Therefore we describe in Appendix D the total ellipticity property hidden behind this identity. In the large N , N f limit (with fixed N /N f ) the equality of I E and I M was confirmed up to a few terms of the corresponding expansion in [26] using the method of [24].
9.2. Two adjoint matter fields case. This duality was considered by Brodie and Strassler in [7,8]. The electric theory is SU (N ) f
SU (N f ) f
SU (N f ) 1
U (1) 0
f
1
f
0
1 N − N1
X
ad j
1
1
0
0
Y Y
TS
1
1
1
TS
1
1
-1
2 N − N2
Q Q
U (1) B
U (1) R −2 1 − N fN(K +1) 1−
N −2 N f (K +1) 2 K +1 K K +1 K K +1
Elliptic Hypergeometry of Supersymmetric Dualities
827
The magnetic theory has the following matter field content
q
) SU ( N f
SU (N f ) f
SU (N f ) 1
q
f
1
f
X
ad j
1
1
0
Y
ad j
1
1
0
ML J
1
f
f
0
U (1) B
U (1) R N 1 − N f (K +1)
N N N −N
N N f (K +1) 2 K +1 K K +1 2N 2L+K J N f (K +1) + K +1
1−
2−
Here K is odd, 0 ≤ L ≤ K − 1, J = 0, 1, 2, and = 3K N f − N . N
(9.4)
Corresponding electric superconformal index has the form
IE =
N −1 (q; q) N −1 K ( p; p)∞ ∞ (U, U 2 ; p, q) N −1 N! −1 K /2 z z −1 , U K /2 z −1 z ; p, q) (U z i z −1 i j j j , U zi z j , U i × −1 −1 N −1 (z i z j , z i z j ; p, q) T 1≤i< j≤N
×
Nf N
(si z j , ti−1 z −1 j ;
p, q)
i=1 j=1
where
j=1
N
dz j , 2π iz j
(9.5)
1
z j = 1, U = ( pq) K +1 , and the balancing condition reads U N ST −1 = N f N f with S = i=1 si , T = i=1 ti . The magnetic index looks like j=1
( pq) N f
IM =
N −1
Nf 2 K −1 N −1 (q; q) N −1 K ( p; p)∞ ∞ (U, U 2 ; p, q) N −1 (U L+K J/2 si t −1 j ; p, q) N! L=0 J =0 i, j=1
×
×
−1 (U z i z −1 j , U zi z j ;
T N−1
−1 (z i z −1 j , z i z j ; p, q)
1≤i< j≤ N
Nf N i=1 j=1
(U
2−K 2
K /2 z −1 z ; p, q) p, q)(U K /2 z i z −1 j j ,U i
(ST )
3K 2N
si−1 z j , U
2−K 2
(ST )
− 3K 2N
ti z −1 j ;
p, q)
−1 N j=1
dz j , 2π iz j
(9.6)
where Nj=1 z j = 1. Again, the conjectured equality I E = I M is a new type of identities requiring a rigorous proof.
828
V. P. Spiridonov, G. S. Vartanov
9.3. Generalized KS type dualities. These dualities were considered in [42].
9.3.1. First pair of dual theories. Electric theory:
Q Q
SU (N ) f
SU (N f ) f
SU (N f ) 1
U (1) 0
f
1
f
0
TA
1
1
1
TA
1
1
-1
X X
U (1) B 1 N − N1 2 N − N2
U (1) R +2K 2r = 1 − (KN+1)N f 2r = 1 − 2s = 2s =
N +2K (K +1)N f 1 K +1 1 K +1
Magnetic theory: ) SU ( N
SU (N f )
SU (N f )
U (1)
K (N f −2) N K (N f −2) − N N −N f N N −N − f N
U (1) B 1 N 1 − N
q
f
f
1
q
f
1
f
Y
TA
1
1
Y
TA
1
1
Mj
1
f
f
0
0
Pr
1
TA
1
-1
0
r P
1
1
TA
1
0
2 N 2 − N
U (1) R +2K 2r = 1 − (KN+1)N f +2K 2r = 1 − (KN+1)N f 2s = K1+1 2s = K1+1 −N +(2 j+1)N f N N f (K +1) −N +(2r +2)N f N N f (K +1) −N +(2r +2)N f N N f (K +1)
Here j = 0, . . . , K , r = 0, . . . , K − 1, and = (2K + 1)N f − 4K − N , N
K = 0, 1, 2, . . . .
(9.7)
The electric index is 1 N −1 (q; q) N −1 (U z i z j , U −1 ( pq) K +1 z i−1 z −1 ( p; p)∞ j ; p, q) ∞ IE = −1 −1 N! (z i z j , z i z j ; p, q) T N −1 1≤i< j≤N
×
Nf N j=1 k=1
where
N j=1
(sk z j , tk z −1 j ; p, q)
N −1 j=1
dz j , 2π iz j
z j = 1 and U is an arbitrary parameter. The magnetic index is
(9.8)
Elliptic Hypergeometry of Supersymmetric Dualities
Nf K
IM =
K −1
j
(( pq) K +1 sk tl ; p, q)
×
−1 −1 N N (q; q)∞ ( p; p)∞
! N
Nf N
r +1
r
(U −1 ( pq) K +1 sk sl , U ( pq) K +1 tk tl ; p, q)
r =01≤k
j=0 k,l=1
×
829
T N−1
z i z j , U −1 ( pq) K +1 z −1 z −1 ; p, q) (U i j
1≤i< j≤ N
(z i−1 z j , z i z −1 j ; p, q)
1
) 2 s −1 z j , (U U )− 2 ( pq) K +1 t −1 z −1 ; p, q) ((U U k k j 1
1
1
j=1 k=1
−1 N j=1
dz j , 2π iz j
N
z j = 1, the balancing condition looks as ST = ( pq) N f − −N +N N N −N f f 1 N f (K +1) (ST −1 ) N ( pq) 2 N N s , T = t , and U = U . j=1 j j=1 j
where N f
j=1
N +2K K +1
with S =
9.3.2. Second pair of dual theories. Electric theory:
Q Q X X
SU (N ) f
SU (N f ) f
SU (N f ) 1
U (1) 0
f
1
f
0
TS
1
1
1
TS
1
1
−1
U (1) B 1 N − N1 2 N − N2
U (1) R −2K 2r = 1 − (KN+1)N f 2r = 1 − 2s = 2s =
N −2K (K +1)N f 1 K +1 1 K +1
Magnetic theory: ) SU ( N
SU (N f )
SU (N f )
q
f
f
1
q
f
1
f
Y
TS
1
1
Y
TS
1
1
Mj
1
f
f
0
0
Pr
1
TS
1
−1
0
r P
1
1
TS
1
0
U (1)
K (N f +2) N K (N f +2) − N N −N f N N −N f − N
U (1) B 1 N 1 − N 2 N 2 − N
U (1) R −2K 2r = 1 − (KN+1)N −2K 2r = 1 − (KN+1)N
f f
2s = K1+1 2s = K1+1 −N +(2 j+1)N f N N f (K +1) −N +(2r +2)N f N N f (K +1) −N +(2r +2)N f N N f (K +1)
Here j = 0, . . . , K , r = 0, . . . , K − 1, and = (2K + 1)N f + 4K − N , N
K = 0, 1, 2, . . . .
(9.9)
830
V. P. Spiridonov, G. S. Vartanov
The electric index is given by the integral 1 N −1 (q; q) N −1 (U z i z j , U −1 ( pq) K +1 z i−1 z −1 ( p; p)∞ j ; p, q) ∞ IE = N! (z i−1 z j , z i z −1 T N −1 1≤i< j≤N j ; p, q)
N
×
(U z 2j , U −1 ( pq) K +1 z −2 j ; p, q) 1
j=1
Nf
(sk z j , tk z −1 j ; p, q)
k=1
N −1 j=1
dz j , 2π iz j (9.10)
where
N j=1
IM =
z j = 1. The magnetic index is
K
Nf
j=0
k,l=1
U ( pq) ×
r K +1
Nf K −1
(( pq)
j K +1
sk tl ; p, q)
K −1
r =0 1≤k
tk tl ; p, q)
(U
−1
( pq)
r +1 K +1
r sk2 , U ( pq) K +1 tk2 ;
r =0 k=1
×
×
T N−1 N
r +1
(U −1 ( pq) K +1 sk sl ,
N −1 (q; q) N −1 ( p; p)∞ ∞ p, q) ! N
z i z j , U −1 ( pq) K1+1 z −1 z −1 ; p, q) (U i j
1≤i< j≤ N
(z i−1 z j , z i z −1 j ; p, q)
z 2j , U −1 ( pq) K +1 z −2 ; p, q) (U j 1
j=1
×
Nf N
) 2 s −1 z j , (U U )− 2 ( pq) K +1 t −1 z −1 ; p, q) ((U U k k j 1
1
1
j=1 k=1
−1 N j=1
N
z j = 1, the balancing condition reads ST = ( pq) N f − −N +N N N −N f f 1 N f = U N (ST −1 ) N ( pq) 2 N(K +1) . s , T = t , and U j j j=1 j=1
where N f
j=1
dz j , (9.11) 2πi z j
N −2K K +1
with S =
9.3.3. Third pair of dual theories. In comparison with the dualities described in previous two subsections, this case involves non-abelian flavor subgroups of different ranks. The electric theory: Q
SU (N ) f
SU (N f ) f
SU (N f − 8) 1
U (1) +3) −(2K + 1) + 2(4K Nf
Q
f
1
f
X
TA
1
1
1
X
TS
1
1
−1
2K + 1 +
2(4K +3) N f −8
U (1) B 1 N − N1 2 N − N2
U (1) R +2(4K +3) 2r = 1 − N2(K +1)N f N −2(4K +3) 2(K +1)(N f −8) = 2(K1+1) = 2(K1+1)
2 r =1− 2s 2s
Elliptic Hypergeometry of Supersymmetric Dualities
831
The magnetic theory: ) SU ( N
SU (N f )
SU (N f − 8)
q
f
f
1
U (1) +3) 2K + 1 − 2(4K N
U (1) B 1 N
U (1) R +2(4K +3) 2r = 1 − N2(K +1)N
q
f
1
f
+3) −2K − 1 − 2(4K N −8
1 −
N −2(4K +3) 2 r = 1 − 2(K +1)(N f −8)
Y
TA
1
1
−1
Y
TS
1
1
MJ
1
f
f
P2L
1
TS
1
P2M+1
1
TA
1
2L P
1
1
TA
2M+1 P
1
1
TS
f
N 2 N 2 − N
f
1 2(4K +3)(2N f −8) N f (N f −8) +3) −4K − 3 + 4(4K Nf +3) −4K − 3 + 4(4K Nf 2(4K +3) 4K + 3 + N −8 f +3) 4K + 3 + 2(4K N f −8
f
2s = 2(K1+1) 2s = 2(K1+1)
0
2(r + r ) + KJ+1
0
4L+1 4r + 2(K +1)
0
4M+3 4r + 2(K +1)
0
4L+1 4 r + 2(K +1)
0
4M+3 4 r + 2(K +1)
Here J = 0, . . . , 2K + 1, L = 0, . . . , K , M = 0, . . . , K − 1, and = (4K + 3)(N f − 4) − N , N
K = 0, 1, 2, . . . .
(9.12)
The electric index is 1 N −1 (q; q) N −1 (U z i z j , U −1 ( pq) 2(K +1) z i−1 z −1 ( p; p)∞ j ; p, q) ∞ IE = −1 −1 N! (z i z j , z i z j ; p, q) T N −1 1≤i< j≤N
×
N
(U
−1
( pq)
1 2(K +1)
z −2 j ;
p, q)
j=1
×
N −1 j=1
with S=
Nf
N f −8
(sk z j ; p, q)
k=1
(tl z −1 j ; p, q)
l=1
dz j , 2π iz j
(9.13)
N
j=1 z j = 1 and the balancing condition ST U N f N f −8 j=1 t j . The magnetic index is j=1 s j , T =
IM =
−4
N +2
= ( pq) N f −4− 2(K +1) , where
N f N f −8 +1 −1 −1 2K N N J (q; q)∞ ( p; p)∞ (( pq) 2(K +1) si t j ; p, q) N! J =0 i=1 j=1
×
2K
l+1
(( pq) 2(K +1) U −1 si s j ; p, q)
l=0 1≤i< j≤N f
×
2K
×
T N−1
1≤i< j≤ N
2l+1
(( pq) 2(K +1) U −1 si2 ; p, q)
l=0 i=1 m
(( pq) 2(K +1) U ti t j ; p, q)
m=0 1≤i< j≤N f −8
Nf K
f −8 K −1 N
m=0 i=1
z i z j , U −1 ( pq) (U
1 2(K +1)
z i−1 z −1 j ;
(z i−1 z j , z i z −1 j ; p, q)
2m+1
(( pq) 2(K +1) U ti2 ; p, q)
p, q)
832
V. P. Spiridonov, G. S. Vartanov
×
N
⎡ 1
−1 ( pq) 2(K +1) z −2 ; p, q) × ⎣(U
Nf
j
j=1
1
k=1
N f −8
×
) 2 s −1 z j ; p, q) ((U U k
) ((U U
− 21
1 2(K +1)
( pq)
⎤
⎦ tl−1 z −1 j ; p, q)
−1 N
l=1
where
N j=1
dz j , 2πiz j
j=1
(9.14)
1 = S 2 U N −N f N . z j = 1 and U
9.4. Adjoint, symmetric and conjugate symmetric tensor matter fields. This duality was constructed by Brodie and Strassler [8]. The electric theory is SU (N ) f
SU (N f ) f
SU (N f ) 1
U (1) 0
f
1
f
0
1 N − N1
X
ad j
1
1
0
0
Y Y
TS
1
1
1
TS
1
1
-1
2 N − N2
Q Q
U (1) B
U (1) R −2 1 − N fN(K +1) 1−
N −2 N f (K +1) 2 K +1 K K +1 K K +1
The magnetic theory is
q
) SU ( N f
SU (N f ) f
SU (N f ) 1
q
f
1
f
X Y
ad j TS
1 1
1 1
Y NI
TS 1
1 f
1 f
MI
1
f
P2J +1
1
P2J 2J +1 P
1
2J P
U (1)
U (1) B
K N f +2 N K N +2 − Nf
1 N 1 − N
0
0
N −K N f N N −K N − N f
2 N 2 − N
0
0
f
0
0
TA
1
-1
0
TS
1
-1
0
1
1
TA
1
0
1
1
TS
1
0
Here K is odd, I = 0, 1, . . . , K − 1, J = 0, 1, . . . , PK , P˜K , and = 3K N f + 4 − N . N
U (1) R −2 1 − N fN(K +1) 1−
2I K +1
+
2K K +1
−2 N N f (K +1) 2 K +1 K K +1 K K +1
−2 + 2 − 2 N fN(K +1)
2I N −2 K +1 + 2 − 2 N f (K +1) 2J +1 K −2 2 K +1 + K +1 + 2 − 2 N fN(K +1) 2J K N −2 2 K +1 + K +1 + 2 − 2 N f (K +1) +1 K N −2 2 2J K +1 + K +1 + 2 − 2 N f (K +1) 2J K N −2 2 K +1 + K +1 + 2 − 2 N f (K +1)
K −1 2 ,
but there are no fields
(9.15)
Elliptic Hypergeometry of Supersymmetric Dualities
833
The indices are IE =
−1 N −1 (q; q) N −1 (U z i z −1 ( p; p)∞ j , U z i z j ; p, q) ∞ (U ; p, q) N −1 −1 N! (z i z −1 T N −1 1≤i< j≤N j , z i z j ; p, q) × (U K /2 X Y z i z j , U K /2 (X Y )−1 z i−1 z −1 j ; p, q) 1≤i< j≤N
×
N
⎡
⎣(U K /2 X Y z 2j , U K /2 (X Y )−1 z −2 ; p, q) j
j=1
Nf
⎤ ⎦ (si z j , ti−1 z −1 j ; p, q)
i=1
N −1 j=1
dz j , 2π iz j
(9.16) 1
where U = ( pq) K +1 , IM =
N j=1
z j = 1, and
Nf K −1 −1 −1 N N (q; q)∞ ( p; p)∞ −1 L (U L+K si t −1 (U ; p, q) N −1 j , U si t j ; p, q) ! N L=0 i, j=1
×
K −1
((X Y )−1 U J +K /2 si s j , X Y U J +K /2 ti−1 t −1 j ; p, q)
J =0 1≤i< j≤N f K −1
×
Nf 2
((X Y )−1 U 2J + 2 si2 , X Y U 2J + 2 ti−2 ; p, q) K
K
J =0 i=1
× ×
T N−1
−1 (U z i z −1 j , U z i z j ; p, q)
1≤i< j≤ N
−1 (z i z −1 j , z i z j ; p, q)
(U K /2 X
N −K N f N
N
Y N z i z j , U K /2 (X
N −K N f N
N
Y N )−1 z i−1 z −1 j ; p, q)
1≤i< j≤ N
×
N
⎡
⎣(U K /2 X
N −K N f N
N
Y N z 2j , U K /2 (X
N −K N f N
N
Y N )−1 z −2 j ; p, q)
j=1
×
Nf
⎤ (U
2−K 2
X
K N f +2 N
Y
3K N f +4 2N
si−1 z j , U
2−K 2
X
K N +2 − f N
Y
3K N f +4 − 2N
⎦ ti z −1 j ; p, q)
i=1
×
−1 N j=1
dz j , 2π iz j
(9.17)
N f N f where Y = (ST )1/N f , S = i=1 si , T = i=1 ti , X is an arbitrary chemical potential associated with the U (1)-group, and the balancing condition reads U N −2 ST −1 = ( pq) N f . 9.5. Adjoint, anti-symmetric and conjugate anti-symmetric tensor matter fields. This duality was considered in [8]. The electric theory is
834
V. P. Spiridonov, G. S. Vartanov
SU (N ) f
SU (N f ) f
SU (N f ) 1
U (1) 0
f
1
f
0
1 N − N1
X
ad j
1
1
0
0
Y Y
TA
1
1
1
TA
1
1
−1
2 N − N2
Q Q
U (1) B
U (1) R 1 − N fN(K+2+1) 1−
N +2 N f (K +1) 2 K +1 K K +1 K K +1
The magnetic theory is
q
) SU ( N f
SU (N f ) f
SU (N f ) 1
q
f
1
f
X
ad j
1
1
U (1)
U (1) B
K N f −2 N K N −2 − Nf
U (1) R 1 − N fN(K+2+1)
1 N 1 − N
1−
+2 N N f (K +1) 2 K +1
0
0
N −K N f N N −K N f − N
2 N 2 − N
K K +1
+ K2K+1 + 2 − 2 N fN(K+2+1) 2I N +2 K +1 + 2 − 2 N f (K +1) +1 K N +2 2 2J K +1 + K +1 + 2 − 2 N f (K +1) 2 K2J+1 + KK+1 + 2 − 2 N fN(K+2+1) +1 K N +2 2 2J K +1 + K +1 + 2 − 2 N f (K +1) 2J K N +2 2 K +1 + K +1 + 2 − 2 N f (K +1)
Y
TA
1
1
Y
TA
1
1
NI
1
f
f
0
0
MI
1
f
f
0
0
P2J +1
1
TS
1
−1
0
P2J
1
TA
1
−1
0
2J +1 P
1
1
TS
1
0
2J P
1
1
TA
1
0
Here K is odd, I = 0, . . . , K − 1, J = 0, . . . , and
K K +1 2I K +1
K −1 2 , but there are no fields
= 3K N f − 4 − N . N
PK , P˜K ,
(9.18)
The superconformal indices are −1 N −1 (q; q) N −1 (U z i z −1 ( p; p)∞ j , U z i z j ; p, q) ∞ N −1 (U ; p, q) IE = −1 −1 N! T N −1 1≤i< j≤N (z i z j , z i z j ; p, q) (U K /2 X Y z i z j , U K /2 (X Y )−1 z i−1 z −1 × j ; p, q) 1≤i< j≤N
×
Nf N i=1 j=1
(si z j , ti−1 z −1 j ; p, q)
N −1 j=1
dz j , 2π iz j
(9.19)
Elliptic Hypergeometry of Supersymmetric Dualities 1
for U = ( pq) K +1 ,
IM =
N j=1
835
z j = 1, and
Nf K −1 −1 −1 N N (q; q)∞ ( p; p)∞ −1 L (U L+K si t −1 (U ; p, q) N −1 j , U si t j ; p, q) ! N L=0 i, j=1
×
K −1
((X Y )−1 U J +K /2 si s j , X Y U J +K /2 ti−1 t −1 j ; p, q)
J =0 1≤i< j≤N f K −3
×
Nf 2
((X Y )−1 U 2J +1+ 2 si2 , X Y U 2J +1+ 2 ti−2 ; p, q) K
K
J =0 i=1
×
T N−1
−1 (U z i z −1 j , U z i z j ; p, q)
1≤i< j≤ N
−1 (z i z −1 j , z i z j ; p, q)
×
(U K /2 X
N −K N f N
N
Y N z i z j , U K /2 (X
N −K N f N
N
Y N )−1 z i−1 z −1 j ; p, q)
1≤i< j≤ N
×
Nf N
(U
2−K 2
X
K N f −2 N
Y
3K N f −4 2N
si−1 z j , U
2−K 2
X
−
K N f −2 N
Y
−
3K N f −4 2N
ti z −1 j ; p, q)
i=1 j=1
×
−1 N j=1
dz j , 2π iz j
(9.20)
N f N f si , T = i=1 ti , X is an arbitrary where Nj=1 z j = 1, Y = (ST )1/N f , S = i=1 N +2 −1 N f ST = ( pq) . parameter and the balancing condition reads U
9.6. Adjoint, anti-symmetric and conjugate symmetric tensor matter fields. This duality was discussed by Brodie in [8]. The electric theory is
Q
SU (N ) f
SU (N f ) f
SU (N f − 8) 1
U (1) B
f
U (1) x1 = N6 − 1 f x2 = N 6−8 + 1 f
1 N − N1
Q
f
1
X
ad j
1
1
0
0
Y
TA
1
1
1
TS
1
1
-1
2 N − N2
Y
U (1) R +6K 2r1 = 1 − NN (K +1) f
−6K 2r2 = 1 − (N N−8)(K +1) f 2 K +1 K K +1 K K +1
In the original paper [8] there were misprints for the values of U (1)-group hypercharges which were corrected in [50]. The magnetic theory is
836
V. P. Spiridonov, G. S. Vartanov
q
) SU ( N f
SU (N f ) f
SU (N f − 8) 1
q
f
1
f
U (1) 1 − N6f −1 −
U (1) B
U (1) R +6K 1 − NNf (K +1)
1 N 1 − N
6 N f −8
X
ad j
1
1
0
0
Y
TA
1
1
−1
Y
TS
1
1
1
2 N 2 − N
NJ
1
f
f
x1 + x2
0
MJ
1
f
f
x1 + x2
0
PJ
1
TS
1
2x1 − 1
0
J P
1
1
TA
2x2 + 1
0
1−
2J K +1
+
−6K N (N f −8)(K +1) 2 K +1 K K +1 K K +1 2K K +1 + 2r1 + 2r2
2J K +1 2J K +1 2J K +1
+
+
+ 2r1 + 2r2
K K +1
K K +1
+6K + 2 − 2 NNf (K +1)
−6K + 2 − 2 (N f N−8)(K +1)
Here J = 0, 1, . . . , K − 1 and = 3K (N f − 4) − N . N
(9.21)
The superconformal indices are IE =
−1 N −1 N −1 (U z i z −1 (q; q)∞ ( p; p)∞ j , U z i z j ; p, q) (U ; p, q) N −1 −1 N! T N −1 1≤i< j≤N (z i z −1 j , z i z j ; p, q) (U K /2 X Y z i z j , U K /2 (X Y )−1 z i−1 z −1 × j ; p, q) 1≤i< j≤N
×
N
(U K /2 (X Y )−1 z i−2 ; p, q)
i=1
Nf N
N f −8
(si z j ; p, q)
j=1 i=1
(tk z −1 j ; p, q)
k=1
N −1 j=1
dz j 2π iz j
(9.22) 1
for U = ( pq) K +1 , IM =
N j=1
z j = 1, and
N f N f −8 −1 −1 K −1 N N (q; q)∞ ( p; p)∞ (U L+K si t j , U L si t j ; p, q) (U ; p, q) N −1 N! L=0 i=1 j=1
×
K −1
J =0 1≤i< j≤N f
×
Nf K −1 J =0 i=1
×
(U K /2 X −1 Y
T N−1
−1 (U z i z −1 j , U z i z j ; p, q)
1≤i< j≤ N
−1 (z i z −1 j , z i z j ; p, q)
N N z z , U K /2 X Y − N z −1 z −1 ; N i j
i
1≤i< j≤ N
×
N
N
− (U K /2 X Y N z i−2 ; p, q)
i=1
×
N f −8 N j=1 k=1
(X Y U J +K /2 ti t j ; p, q)
1≤i< j≤N f −8
((X Y )−1 U J +K /2 si2 ; p, q)
((X Y )−1 U J +K /2 si s j ; p, q)
Nf N
(U
2−K 2
Y
j
p, q)
3K (N f −4) 2N s −1 z
i
j=1 i=1
(U
2−K 2
Y
−
3K (N f −4) 2N t −1 z −1 ;
k
j
p, q)
−1 N j=1
dz j , 2π iz j
j ; p, q)
Elliptic Hypergeometry of Supersymmetric Dualities
837
1 2(K −2) N −4 f z j = 1, Y = ST −1 X 2N f −8 ( pq) K +1 , and the balancing condiN f N f −8 N −4 −4 N −4 ti . tion reads U X Y ST = ( pq) f with S = i=1 si , T = i=1 The equalities I E = I M for all the dualities described in this section require a rigorous mathematical confirmation. For the moment we have only one justifying argument coming from the total ellipticity condition associated with the kernels of the corresponding pairs of integrals.
where
N
j=1
10. KS Type Dualities for Symplectic Gauge Groups 10.1. The anti-symmetric tensor matter field. For S P(2N ) group the following electric-magnetic duality was discovered by Intriligator in [41]. The electric theory:
Q
S P(2N ) f
SU (2N f ) f
X
TA
1
U (1) R 2(N +K ) 2r = 1 − (K +1)N f 2s =
2 K +1
The magnetic theory: ) S P(2 N
SU (2N f )
q
f
f
Y
TA
1
Mj
1
TA
U (1) R +K ) 2( N 2 r = 1 − (K +1)N f 2s = K2+1
+j N +K 2r j = 2 K K +1 − 4 (K +1)N f
where j = 1, . . . , K , and = K (N f − 2) − N , N
K = 1, 2, . . . .
(10.1)
1
Defining U = ( pq)s = ( pq) K +1 , we find the following indices for these theories: N (q; q) N (U z i±1 z ±1 ( p; p)∞ j ; p, q) ∞ N −1 (U ; p, q) IE = ±1 ±1 N N 2 N! T 1≤i< j≤N (z i z j ; p, q) 2N f ±1 N N dz j i=1 (si z j ; p, q) (10.2) × ±2 2π iz j (z j ; p, q) j=1
j=1
and
IM =
N (q; q) N ( p; p)∞ ∞ (U ; p, q) N −1 N 2 N! K
(U l−1 si s j ; p, q)
l=1 1≤i< j≤2N f
×
T N
1≤i< j≤ N
N (U z i±1 z ±1 j ; p, q)
(z i±1 z ±1 j ; p, q)
2N f i=1
j=1
N (U si−1 z ±1 dz j j ; p, q) , ±2 2π iz j (z j ; p, q) j=1
(10.3) where the balancing condition reads U
2N f 2(N +K ) i=1
si = ( pq) N f .
838
V. P. Spiridonov, G. S. Vartanov
10.2. Symmetric tensor matter field. Another electric-magnetic duality is described by Leigh and Strassler in [54]. The electric theory:
Q
S P(2N ) f
SU (2N f ) f
X
ad j = TS
1
U (1) R +1 2r = 1 − (KN+1)N f 2s =
1 K +1
The magnetic theory: ) S P(2 N
SU (2N f )
U (1) R
q
f
f
+1 2 r = 1 − (K N +1)N f 1 2s = K +1
Y
ad j
1
M2 j , j = 0, . . . , K
1
TA
M2 j+1 , j = 0, . . . , K − 1
1
TS
2(N +1)−2 j N f (K +1)N f 2(N +1)−(2 j+1)N f 2r2 j+1 = 2 − (K +1)N f
2r2 j = 2 −
Here = (2K + 1)N f − N − 2, N
K = 0, 1, 2, . . . .
(10.4)
1
Defining U = ( pq)s = ( pq) 2(K +1) , we find the following superconformal indices: IE =
N (q; q) N ( p; p)∞ ∞ N (U ; p, q) 2N N ! TN ×
N (U z ±2 ; p, q) 2N N f j j=1
(z ±2 j ; p, q)
(U z i±1 z ±1 j ; p, q)
1≤i< j≤N
(z i±1 z ±1 j ; p, q)
(si z ±1 j ; p, q)
i=1 j=1
N dz j 2π iz j
(10.5)
j=1
and
IM
N (q; q) N ( p; p)∞ ∞ N = (U ; p, q) ! 2 N N 2K
(U l si s j ; p, q)
l=0 1≤i< j≤2N f
×
K −1 2N f
(U 2l+1 si2 ;
p, q)
T N
l=0 i=1
×
2N f N N (U z ±2 j ; p, q) j=1
(z ±2 j ; p, q)
(U z i±1 z ±1 j ; p, q)
1≤i< j≤ N
(z i±1 z ±1 j ; p, q)
(U si−1 z ±1 j ;
i=1 j=1
where the balancing condition reads U 2(N +1)
N dz j p, q) , 2π iz j
(10.6)
j=1
2N f i=1
si = ( pq) N f .
10.3. Two anti-symmetric tensor matter fields. This duality was investigated by Brodie and Strassler in [8]. The electric theory:
Elliptic Hypergeometry of Supersymmetric Dualities
839
Q
S P(2N ) f
SU (2N f ) f
U (1) R N +2K +1 1 − (K +1)N f
X
TA
1
Y
TA
1
2 K +1 K K +1
The magnetic theory: ) S P(2 N
SU (2N f )
q
f
f
X
TA
1
Y
TA
1
M J 0 , J = 0, . . . , K − 1
1
TA
M2J 1 , J = 0, . . . , K 2−1
1
TA
M2J +1 1 , J = 0, . . . , K 2−3
1
TS
M J 2 , J = 0, . . . , K − 1
1
TA
U (1) R
+2K +1 N 1 − (K +1)N f
2 K +1 K K +1 N +2K 2 − (K +1)N+1 + K2J+1 f N +2K +1 + 2(2J ) + K 2 − (K K +1 K +1 +1)N f N +2K +1 + 2(2J +1) + K 2 − (K K +1 K +1 +1)N f N +2K +1 + 2J + 2K 2 − (K K +1 K +1 +1)N f
Here K is odd and = 3K N f − 4K − 2 − N . N
(10.7)
For these theories we have the following superconformal indices: IE =
N (q; q) N K ( p; p)∞ ∞ (U, U 2 ; p, q) N −1 N 2 N! K ±1 ±1 2N f N N 2 (U z i±1 z ±1 (si z ±1 dz j j , U z i z j ; p, q) j ; p, q) × , ±1 ±1 ±2 N 2πiz (z z ; p, q) (z ; p, q) j T 1≤i< j≤N i j j i=1 j=1 j=1
(10.8) 1
where U = ( pq) K +1 , the balancing condition reads U N +2K +1
IM =
2N f i=1
si = ( pq) N f , and
N (q; q) N K ( p; p)∞ ∞ (U, U 2 ; p, q) N −1 N 2 N!
×
K −1 2
K −3
(U
J + K2L
si s j ; p, q)
J =0 L=0 1≤i< j≤2N f
×
×
T N
1≤i< j≤ N
N dz j . 2π iz j j=1
2 2N f
K
(U 2J +1+ 2 s 2j ; p, q)
J =0 j=1
2N f N ±1 ±1 (U z i±1 z ±1 (U 1− 2 si−1 z ±1 j , U z i z j ; p, q) j ; p, q) K 2
(z i±1 z ±1 j ; p, q)
K
i=1 j=1
(z ±2 j ; p, q) (10.9)
840
V. P. Spiridonov, G. S. Vartanov
10.4. Symmetric and anti-symmetric tensor matter fields. This duality was found in [8]. The electric theory:
Q
S P(2N ) f
SU (2N f ) f
U (1) R N +2K −1 1 − (K +1)N f
X
TA
1
Y
TS
1
2 K +1 K K +1
The magnetic theory: ) S P(2 N
SU (2N f )
q
f
f
X
TA
1
Y
TS
1
M J 0 , J = 0, . . . , K − 1
1
TA
M2J 1 , J = 0, . . . , K 2−1 M2J +1 1 , J = 0, . . . , K 2−3 M J 2 , J = 0, . . . , K − 1
1
TS
1
TA
1
TA
U (1) R +2K −1 N 1 − (K +1)N f 2 K +1 K K +1 N +2K +1 + 2J 2 − (K K +1 +1)N f N +2K +1 + 2(2J ) + K 2 − (K K +1 K +1 +1)N f N +2K +1 + 2(2J +1) + K 2 − (K K +1 K +1 +1)N f N +2K +1 + 2J + 2K 2 − (K K +1 K +1 +1)N f
Here K is odd and = 3K N f − 4K + 2 − N . N
(10.10)
For these theories we have the following superconformal indices: IE =
N (q; q) N K ( p; p)∞ ∞ (U ; p, q) N −1 (U 2 ; p, q) N N 2 N! K ±1 ±1 2 (U z i±1 z ±1 j , U z i z j ; p, q) × (z i±1 z ±1 T N 1≤i< j≤N j ; p, q) K ±2 N (U 2 z ; p, q) 2N f (s z ±1 ; p, q) N i j dz j j i=1 , × ±2 2π iz j (z ; p, q) j j=1 j=1 1
where U = ( pq) K +1 and the balancing condition reads U N +2K −1 and
IM =
2N f i=1
(10.11)
si = ( pq) N f ,
N (q; q) N K ( p; p)∞ ∞ (U ; p, q) N −1 (U 2 ; p, q) N ! 2N N
×
K −1 2
(U J +
KL 2
si s j ; p, q)
J =0 L=0 1≤i< j≤2N f K −1
×
2 2N f
K (U 2J + 2 s 2j ;
J =0 j=1 N (U 2 z ±2 j ; p, q) K
×
j=1
p, q) 2N f
T N
±1 ±1 2 (U z i±1 z ±1 j , U z i z j ; p, q)
1≤i< j≤ N
(z i±1 z ±1 j ; p, q)
1− K2 −1 ±1 si z j ; i=1 (U ±2 (z j ; p, q)
K
N p, q) dz j . 2π iz j j=1
(10.12)
Elliptic Hypergeometry of Supersymmetric Dualities
841
The equalities I E = I M for all the dualities described in this section represent new elliptic hypergeometric identities requiring a rigorous mathematical confirmation. 11. Some Other New Dualities Let us denote
N +1 N +3 −1 N (q; q) N ( p; p)∞ i=1 r =1 (tr z i , u r z i ; p, q) ∞ I A N (t, u; p, q) = −1 −1 (N + 1)! TN 1≤i< j≤N +1 (z i z j , z i z j ; p, q) ×
N dz j 2π iz j
(11.1)
j=1
N +1
N +3 z j = 1 and the balancing condition i=1 ti u i = ( pq)2 , and N 2N +6 ±1 N (q; q) N ( p; p)∞ i=1 r =1 (tr z i ; p, q) ∞ I BC N (t; p, q) = N ±1 ±1 ±2 2N N ! TN 1≤i< j≤N (z i z j ; p, q) j=1 (z j ; p, q)
with
j=1
×
N dz j 2π iz j j=1
with the balancing condition
(11.2) 2N +6 r =1
tr = ( pq)2 .
11.1. SU ↔ S P groups mixing duality. The first case electric gauge group is G = SU (N + 1), but the dual gauge group is of a different type G = S P(2N ). The flavor symmetry group in both cases is F = SU (N + 3) × SU (N + 3) × U (1) B . The field content of dual theories is described in the tables below: SU (N + 1) SU (N + 3) SU (N + 3) U (1) B U (1) R 2 Q1 f f 1 2 N +3 2 Q2 f 1 f −2 N +3 q1
S P(2N ) f
SU (N + 3) f
SU (N + 3) 1
U (1) B −(N + 1)
q2
f
1
f
N +1
X1
1
TA
1
2(N + 1)
X2
1
1
TA
−2(N + 1)
U (1) R 2 N +3 2 N +3 +1 2N N +3 +1 2N N +3
The superconformal indices are I E = I A N (t1 , . . . , t N +3 , u 1 , . . . , u N +3 ; p, q), (T /ti t j , U/u i u j ; p, q)I BC N (. . . (U/T )1/4 ti . . . , IM = 1≤i< j≤N +3
. . . (T /U )1/4 u i . . . ; p, q), (11.3) where T = 1≤i≤N +3 ti and U = 1≤i≤N +3 u i . The equality I E = I M represents the mixed elliptic hypergeometric integrals transformation proven in [65]. We used this identity as a starting point for finding the described new Seiberg-type pair of field theories.
842
V. P. Spiridonov, G. S. Vartanov
11.2. SU ↔ SU groups mixing duality. Again, we use consequences of the mixed transformations derived in [65]. Corresponding dualities have the flavor symmetry groups F = SU (K ) × SU (N + 2 − K ) × U (1)1 × SU (K ) × SU (N + 2 − K ) × U (1)2 ×U (1) B , for arbitrary 0 < K < N + 2. The matter field content of the initial electric field theory is given in the table Q1
SU (N ) f
SU (N + 2) f
SU (N + 2) 1
U (1) B 1
Q2
f
1
f
−1
U (1) R 2 N +2 2 N +2
In order to verify the ’t Hooft anomalies matching conditions for relevant flavor symmetry subgroups, it is useful to rewrite the latter table as q1
SU (N ) f
SU (K ) f
SU (M) 1
U (1)1 M
SU (K ) 1
SU (M) 1
q2
f
q3
f
q4
f
U (1)2 0
U (1) B 1
1
f
−K
1
1
0
1
1
1
0
f
1
M
−1
1
1
0
1
f
−K
−1
U (1) R 2 N +2 2 N +2 2 N +2 2 N +2
where M = N + 2 − K . The dual theory content is described in the following table: SU (N ) SU (K ) SU (M) f f 1
q1 q2
f
1
f
q3
f
1
1
q4
f
1
1
U (1)1 K (K −2) −K+ N K (K −2) − N MK N −M K N
M
SU (K ) SU (M) 1 1 1
1
f
1
1
f
U (1)2 MK N −M K N K (K −2) −K+ N − K (KN−2)
U (1) B U (1) R 2 1−M N +2 1−K M M −1 K −1
X1
1
f
1
M
1
f
−K
0
X2
1
1
f
−K
f
1
M
0
Y1
1
f
f
K−M
1
1
0
N
Y2
1
1
1
0
f
f
K−M
−N
2 N +2 2 N +2 2 N +2 4 N +2 4 N +2 2N N +2 2N N +2
The superconformal indices have the form I E = I A N −1 (t1 , . . . , t N +2 , u 1 , . . . , u N +2 ; p, q), IM = (tr u s , ts u r , T /ts tr , U/u r u s )I A N −1 (t1 , . . . , t N +2 , 1≤r
where T =
N +2
r =1 tr , U
=
N +2
tr = (T /U ) tr u r u r
= (U/T ) = (U/T ) = (T /U )
r =1
N −K 2N K 2N
K 2N
u r , TK =
r =1 tr , U K
(TK /U K )1/N u r ,
(TK /U K )
N −K 2N
(11.4) K
1/N
tr ,
(U K /TK )1/N tr ,
(U K /TK )
1/N
ur ,
=
K
r =1 u r ,
and
1 ≤ r < K + 1, K + 1 ≤ r ≤ N + 2, 1 ≤ r < K + 1, K + 1 ≤ r ≤ N + 2.
The equality I E = I M for K = 1 was suggested in [83] and the general relation with the complete proof for arbitrary K is given in [65].
Elliptic Hypergeometry of Supersymmetric Dualities
843
12. S-Confinement Following [15,16,76], by s-confinement we mean smooth confinement without chiral symmetry breaking and with a non-vanishing confining superpotential. The theory is confined when its infrared physics can be described completely in terms of gauge invariant composite fields and their interactions. This description has to be valid everywhere in the moduli space of vacua. s-confinement requires also that the theory dynamically generates a confining superpotential. Furthermore, the phase without chiral symmetry breaking implies that the origin of the classical moduli space serves also as a vacuum in the quantum theory. In this vacuum all the global symmetries present in the ultraviolet regime remain unbroken. Finally, the confining superpotential is a holomorphic function of the confined degrees of freedom and couplings, which describe all interactions in the extreme infrared. From the point of view of elliptic hypergeometric functions the s-confinement means that the dual theory gauge group is trivial G = 1 (i.e., there is no ) and the integrals describing superconformal indices are computable vector superfield V exactly, defining highly non-trivial elliptic beta integrals [81]. 12.1. SU (N ) gauge group. In this section we present known examples of the confining theories with the unitary gauge group. For brevity we combine the electric and magnetic theories in a single table separating them by the double line. The magnetic theory fields are denoted using the conventions of [15]. 12.1.1. SU (N ) with (N + 1)( f + f ). [76]: Q
SU (N ) f
SU (N + 1) f
SU (N + 1) 1
U (1) 1
Q
f
1
f
−1
QQ
f
f
0
QN
f
1
N
N Q
1
f
−N
U (1) R 1 N +1 1 N +1 2 N +1 N N +1 N N +1
The superconformal indices for these theories are equal to (after appropriate renormalization of the parameters) N −1 (q; q) N −1 1 ( p; p)∞ ∞ IE = −1 −1 N −1 N! (z i z j , z i z j ; p, q) T 1≤ j
N N +1
(sm z j , tm z −1 j ; p, q)
j=1 m=1
where
N j=1
N −1 j=1
dz j , 2π iz j
(12.1)
z j = 1, and IM =
N +1 m=1
(Ssm−1 , T tm−1 ; p, q)
N +1 k,m=1
(sk tm ; p, q),
(12.2)
N +1 N +1 where S = m=1 sm , T = m=1 tm , with the balancing condition ST = pq. The exact evaluation formula for the integral I E = I M was conjectured and partially confirmed in [83]. Its complete proofs are given in [65,86]. In the simplest p → 0 limit it is reduced to one of the Gustafson integrals [35].
844
V. P. Spiridonov, G. S. Vartanov
12.1.2. SU (2N ) with T A + 2N f + 4 f . The theory with G = SU (2N ) gauge group and flavor group F = SU (2N ) × SU (4) × U (1)1 × U (1)2 was found to be confining in [62,64]. The field content of both theories is described in the table below: Q Q A QQ 2 AQ AN A N −1 Q 2 A N −1 Q 4 2N Q
SU (2N ) f f TA
SU (2N ) 1 f 1
SU (4) f 1 1
U (1)1 −2N 4 0
U (1)2 −2N + 2 −2N + 2 2N + 4
f TA 1 1 1 1
f 1 1 TA 1 1
4 − 2N 8 0 −4N −8N 8N
−4N + 4 −2N + 8 2N 2 + 4N 2N 2 − 2N 2N 2 − 8N −4N 2 + 4N
U (1) R 1 2
0 0 1 2
0 0 1 2 0
We come to the following integrals describing the superconformal indices: IE =
2N −1 (q; q)2N −1 (t z i z j ; p, q) ( p; p)∞ ∞ −1 2N −1 (2N )! (z i z −1 T j , z i z j ; p, q) 1≤ j
×
2N 2N
(tk z −1 j ; p, q)
j=1 k=1
with
2N j=1
4
(si z j ; p, q)
i=1
2N −1 j=1
dz j , 2π iz j
(12.3)
z j = 1, and
IM =
(tt j tk ; p, q)
1≤ j
×
2N 4
(tk si ; p, q)
k=1 i=1
(t
2N −2 2
(t N , T ; p, q) (t N T ; p, q)
si sm ; p, q),
(12.4)
1≤i<m≤4
4 with the balancing condition t 2N −2 ST = pq, where S = i=1 si , T = 2N j=1 t j . Equality I E = I M defines the elliptic beta integral introduced in [83]. It represents an elliptic extension of the Gustafson-Rakha q-beta integral for odd number of integration variables [38]. 12.1.3. SU (2N + 1) with T A + (2N + 1) f + 4 f . These dual models were considered in [62,64]: Q Q A QQ 2 AQ AN Q A N −1 Q 3 2N +1 Q
SU (2N + 1) f f TA
SU (2N + 1) 1 f 1
SU (4) f 1 1
U (1)1 −2N − 1 4 0
U (1)2 −2N + 1 −2N + 1 2N + 5
U (1) R
f TA 1 1 1
f 1 f f 1
3 − 2N 8 −2N − 1 −6N − 3 8N + 4
−4N + 2 −2N + 7 2N 2 + 3N + 1 2N 2 − 3N − 2 −4N 2 + 1
1 2
1 2
0 0 0 1 2 3 2
0
Elliptic Hypergeometry of Supersymmetric Dualities
845
The indices have the form 2N ( p; p)2N ∞ (q; q)∞ (2N + 1)! T2N
IE =
×
2N +1 2N +1
(tk z −1 j ;
(t z i z j ; p, q)
1≤ j
−1 (z i z −1 j , z i z j ; p, q)
p, q)
j=1 k=1
with
2N +1 j=1
IM =
4 i=1
2N dz j (si z j ; p, q) , 2π iz j
(12.5)
j=1
z j = 1, and
(tt j tk ; p, q)
1≤ j
2N +1 4
(tk si ; p, q)(T ; p, q)
k=1 i=1
4 (t N si ; p, q) , (t N T si ; p, q) i=1
(12.6) +1 4 where the balancing condition reads t 2N −1 ST = pq and T = 2N k=1 tk , S = k=1 sk . The equality I E = I M was also suggested in [83] as an elliptic extension of the Gustafson-Rakha q-beta integral with an even number of integrations [38]. 12.1.4. SU (2N + 1) with T A + T A + 3 f + 3 f . Models [16]: Q Q A A kQ Q(A A) A(A A)k Q 2 kQ 2 A(A A)
SU (2N + 1) f f TA TA
AN Q N Q A A N −1 Q 3 N −1 Q 3 A m (A A)
SU (3) f 1 1 1
SU (3) 1 f 1 1
U (1)1 0 0 1 −1
U (1)2 2N − 1 2N − 1 −3 −3
U (1) B 1 −1 0 0
U (1) R
f TA 1 f 1 1 1 1
f 1 TA 1 f 1 1 1
0 −1 1 N −N N −1 −N + 1 0
4N − 2 − 6k 4N − 5 − 6k 4N − 5 − 6k −N − 1 −N − 1 3N 3N −6m
0 2 −2 1 −1 3 −3 0
2 3 2 3 2 3 1 3 1 3
1 3 1 3
0 0
1 1 0
where k = 0, . . . , N − 1 and m = 1, . . . , N . The superconformal indices are written as
IE =
2N ( p; p)2N ∞ (q; q)∞ (2N + 1)! T2N
×
3 2N +1 i=1 j=1
(U z i z j , V z i−1 z −1 j ; p, q)
1≤i< j≤2N +1
−1 (z i z −1 j , z i z j ; p, q)
(si z j , ti z −1 j ; p, q)
2N dz j , 2π iz j j=1
(12.7)
846
V. P. Spiridonov, G. S. Vartanov
where
IM =
2N +1 j=1
3
(U N si , V N ti ; p, q)(U N −1 s1 s2 s3 , V N −1 t1 t2 t3 ; p, q)
i=1
×
z j = 1, and
N −1
⎡ ⎣
j=0
N
((U V ) j ; p, q)
j=1 3
((U V ) j si tk ; p, q)
i,k=1
⎤
(V (U V ) j si sk , U (U V ) j ti tk ; p, q)⎦ ,
1≤i
(12.8) 3 si ti = pq. where the balancing condition reads (U V )2N −1 i=1 The equality I E = I M was derived by Spiridonov in [83] by purely algebraic means as a consequence of other elliptic beta integrals. In the simplest p → 0 limit it reduces to Gustafson’s q-beta integral for the root system A2N [37]. 12.1.5. SU (2N ) with T A + T A + 3 f + 3 f . For N > 2 the models have the form [16]: SU (2N ) f
SU (3) f
Q
f
A A kQ Q(A A)
TA TA
Q
SU (3) 1
U (1)1 0
U (1)2 2N − 2
U (1) B 1
U (1) R
1
f
0
2N − 2
−1
1 3 1 3
1 1
1 1
1 −1
−3 −3
0 0
0 0
f
f
0
4N − 4 − 6k
0
A) m Q2 A(A
TA
1
−1
4N − 7 − 6m
2
mQ 2 A(A A)
1
TA
1
4N − 7 − 6m
−2
2 3 2 3 2 3
AN N A
1 1
1 1
N −N
−3N −3N
0 0
0 0
A N −1 Q 2
TA
1
N −1
N −1
2
N −1 Q 2 A (A A)n
1
TA
−N + 1
N −1
-2
2 3 2 3
1
1
0
−6n
0
0
where k = 0, . . . , N − 1, m = 0, . . . , N − 2 and n = 1, . . . , N − 1. For N = 2 the flavor group is enlarged to F = SU (3) × SU (3) × SU (2) × U (1)2 × U (1) B , and the fields A, A˜ unify to the SU (2)-group doublet. The expressions for the superconformal indices are
IE =
2N −1 (q; q)2N −1 (U z i z j , V z i−1 z −1 ( p; p)∞ j ; p, q) ∞ −1 −1 (2N )! T2N −1 1≤i< j≤2N (z i z j , z i z j ; p, q)
×
2N 3 j=1 i=1
(si z j , ti z −1 j ; p, q)
2N −1 j=1
dz j , 2π iz j
(12.9)
Elliptic Hypergeometry of Supersymmetric Dualities
with
2N j=1
847
z j = 1, and I M = (U N , V N ; p, q)
(U N −1 si sk , V N −1 ti tk ; p, q)
1≤i
×
N
3
((U V ) j−1 si tk ; p, q)
j=1 i,k=1
×
N −2
N −1
((U V ) j ; p, q)
j=1
(V (U V ) j si sk , U (U V ) j ti tk ; p, q),
(12.10)
j=0 1≤i
3 where the balancing condition reads (U V )2N −2 i=1 si ti = pq. The equality I E = I M was also derived in [83] as a consequence of some other elliptic beta integrals. In the simplest p → 0 limit, it reduces to one of Gustafson’s integrals for the root system A2N −1 [37]. Similar to the case of non-confining N f = 4 dualities described earlier, a careful examination of the limit V → 1 (or U → 1) shows that the equality of superconformal indices in this case reduces to the equality of S P(2N )-group confining duality indices discussed in [91]. This means that the elliptic Selberg integral introduced in [19] (see integral (12.35) and its evaluation (12.36) below) is a limiting case of Spiridonov’s An -elliptic beta integral. This result could have been expected since the computation of the latter integral in [83] used the elliptic Selberg integral. 12.1.6. SU (K N f − 1) with N f f + N f f + 1ad j. Taking N = K N f − 1 in (9.1) = 1), we find the s-confining dual theory discussed in [14]. The field content of (or, N these theories is easily found from the tables given in Sec. 9.1. Namely, in the electric theory one should fix N as described; on the magnetic side one should keep all the = 1 in the gauge group part. Therefore for this case mesons and baryons and set N the superconformal index for the electric theory is given by (9.2), and the magnetic superconformal index takes the form
IM =
K
l=1 1≤i, j≤N f
(U
l−1
si t −1 j ;
p, q)
Nf
(U (ST ) 2 si−1 , U (ST )− 2 ti ; p, q), K
K
i=1
(12.11) N f N f 1 where U = ( pq) K +1 , S = j=1 s j , T = j=1 t j , and the balancing condition reads 2K N −2 −1 N f f U ST = ( pq) . For K = 1 one obtains the known A N -root systems integral of type I from Sect. 12.1.1. The conjecture I E = I M for K > 1 represents a new elliptic beta integral requiring rigorous mathematical justification. 12.1.7. SU (3K N f − 1) with N f f + N f f + 2ad j. If we set N = 3K N f − 1 in (9.4), then we obtain the s-confinement discussed in [50]. The superconformal index for the
848
V. P. Spiridonov, G. S. Vartanov
electric theory is given by (9.5), and the magnetic superconformal index takes the form IM =
K −1 2
(U L+K J/2 si t −1 j ; p, q)
L=0 J =0
×
Nf
2−K 2
(U
3K
(ST ) 2 N si−1 , U
2−K 2
(ST )
− 3K 2N
ti ; p, q),
(12.12)
i=1
N f N f 1 where U = ( pq) K +1 , S = j=1 s j , T = j=1 t j , and the balancing condition reads N −1 N f U ST = ( pq) . The equality I E = I M is a new conjectural elliptic beta integral. 12.1.8. SU ((2K + 1)N f − 4K − 1) with N f f + N f f + 2T A . If we set N = (2K + 1)N f − 4K − 1 in (9.7), we obtain the s-confinement discussed by Klein in [50]. The electric superconformal index is given by (9.8), and the magnetic superconformal index takes the form IM =
Nf K
j
(( pq) K +1 sk tl ; p, q)
j=0 k,l=1
×
K −1
Nf
) 2 s −1 , (U U )− 2 ( pq) K +1 t −1 ; p, q) ((U U k k 1
1
1
k=1
r +1
r
(U −1 ( pq) K +1 sk sl , U ( pq) K +1 tk tl ; p, q),
(12.13)
r =0 1≤k
= U 2K N f −4K −1 ST −1 ( pq) K +1 , S = where U is an arbitrary parameter, U N f N f N f − NK+2K +1 . j=1 s j , T = j=1 t j , and the balancing condition reads ST = ( pq) For K = 0 the parameter U drops out, and one obtains the integral discussed in Sect. 12.1.1. The general K > 0 conjecture I E = I M represents another new elliptic beta integral. 12.1.9. SU ((2K + 1)N f + 4K − 1) with N f f + N f f + 2TS . If we set N = (2K + 1)N f + 4K − 1 in (9.9), we obtain again the s-confinement [50]. Corresponding electric superconformal index is given by (9.10), and the magnetic superconformal index takes the form IM
, U −1 ( pq) K +1 ; p, q) = (U 1
×
×
K −1
r =0
1≤k
N f K −1 k=1
) (U U
Nf
j=0
k,l=1
j
(( pq) K +1 sk tl ; p, q)
r +1
r
(U −1 ( pq) K +1 sk sl , U ( pq) K +1 tk tl ; p, q)
) 2 s −1 , (U −1 ( pq) K +1 sk2 , U ( pq) K +1 tk2 ; p, q)((U U k r +1
r =0 − 21
K
( pq)
r
1
1 K +1
tk−1 ;
p, q) ,
(12.14)
Elliptic Hypergeometry of Supersymmetric Dualities 1−K N f −2K K +1
= U 2K N f +4K −1 ST −1 ( pq) where U
849
,S =
N f
j=1 s j , T
N f − NK−2K +1
=
N f
j=1 t j ,
and the
balancing condition reads ST = ( pq) . Presently the conjecture I E = I M is confirmed only for K = 0, which reduces again to the integral of Sect. 12.1.1. 12.1.10. SU ((4K + 3)(N f − 4) − 1) with N f f + (N f − 8) f + T A + TS . If we take N = (4K + 3)(N f − 4) − 1 in (9.12), we obtain the s-confinement [50]. Corresponding electric superconformal index is given by (9.13), and the magnetic superconformal index takes the form IM =
N f N f −8 2K +1
J
(( pq) 2(K +1) si t j ; p, q)
J =0 i=1 j=1
×
2K
(( pq)
l+1 2(K +1)
U
−1
si s j ; p, q)
l=0 1≤i< j≤N f
×
2K
Nf K
2l+1
(( pq) 2(K +1) U −1 si2 ; p, q)
l=0 i=1
(( pq)
m 2(K +1)
U ti t j ; p, q)
m=0 1≤i< j≤N f −8
f −8 K −1 N
2m+1
(( pq) 2(K +1) U ti2 ; p, q)
m=0 i=1 Nf
1
−1 ( pq) 2(K +1) ; p, q) ×(U
) 2 s −1 ; p, q) ((U U k 1
k=1 N f −8
×
1
)− 2 ( pq) 2(K +1) t −1 ; p, q), ((U U l 1
(12.15)
l=1
1 = S 2 U N −N f N and the balancing condition reads where U
U
−4
Nf
s j t j = ( pq) N f −4−
(4K +3)(N f −4)+1 2(K +1)
.
j=1
The equality I E = I M represents another conjectural new elliptic beta integral. 12.1.11. SU (3K N f + 3) with N f f + N f f + ad j + TS + T S . If we take N = 3K N f + 3 for K -odd in (9.15), we obtain again the s-confinement [50]. The corresponding electric superconformal index is given by (9.16), and the magnetic superconformal index is IM =
Nf K −1
−1 L (U L+K si t −1 j , U si t j ; p, q)(U
K 2
X N −K N f Y N ,
L=0 i, j=1 K
U 2 (X N −K N f Y N )−1 ; p, q) ×
K −1
J =0 1≤i< j≤N f
((X Y )−1 U J +K /2 si s j , X Y U J +K /2 ti−1 t −1 j ; p, q)
850
V. P. Spiridonov, G. S. Vartanov K −1
×
Nf 2
((X Y )−1 U 2J +K /2 si2 , X Y U 2J +K /2 ti−2 ; p, q)
J =0 i=1
×
Nf
(U
2−K 2
X K N f +2 Y
3K N f +4 2
si−1 , U
2−K 2
X −(K N f +2) Y −
3K N f +4 2
ti ; p, q),
i=1
(12.16) N f
1 K +1
N f
where U = ( pq) , Y = (ST )1/N f , S = i=1 si , T = i=1 ti , X is an arbitrary parameter, and the balancing condition reads U N −2 ST −1 = ( pq) N f . Again, the proof of the general equality I E = I M is absent. 12.1.12. SU (3K N f −5) with N f f + N f f +ad j + T A + T A . If we take N = 3K N f −5 for K -odd in (9.18), we obtain the s-confinement [50]. The corresponding electric superconformal index is given by (9.19), and the magnetic superconformal index takes the form IM =
Nf K −1
−1 L (U L+K si t −1 j , U si t j ; p, q)
L=0 i, j=1
×
K −1
((X Y )−1 U J +K /2 si s j , X Y U J +K /2 ti−1 t −1 j ; p, q)
J =0 1≤i< j≤N f K −3
×
Nf 2
((X Y )−1 U 2J +1+K /2 si2 , X Y U 2J +1+K /2 ti−2 ; p, q)
J =0 i=1
×
Nf
2−K 2
(U
X K N f −2 Y
3K N f −4 2
si−1 , U
2−K 2
X −(K N f −2) Y −
3K N f −4 2
ti ; p, q),
i=1
(12.17) N f
1 K +1
N f
where U = ( pq) , Y = (ST )1/N f , S = i=1 si , T = i=1 ti , X is an arbitrary parameter, and the balancing condition reads U N −2 ST −1 = ( pq) N f . No proof of the equality I E = I M is known at present. 12.1.13. SU (N ) with N f f +(N f −8) f +ad j + T A + T S . If we set N = 3K (N f −4)−1 in (9.21), we obtain the s-confinement [50]. The corresponding electric superconformal index is given by (9.22), and IM =
Nf K −1
⎡ ⎣
L=0 i=1
×
K −1 J =0
×
Nf
⎤
N f −8
⎡
(U
L+K
si t j , U si t j ; p, q)((X Y ) L
−1
U L+K /2 si2 ;
p, q)⎦
j=1
⎣
((X Y )−1 U J +K /2 si s j ; p, q)
2−K 2
Y
3K (N f −4) 2
⎤ (X Y U J +K /2 ti t j ; p, q)⎦
1≤i< j≤N f −8
1≤i< j≤N f
(U
si−1 ; p, q)
i=1
×(U K /2 (X Y N )−1 ; p, q),
N f −8
(U
2−K 2
Y−
3K (N f −4) 2
tk−1 ; p, q)
k=1
(12.18)
Elliptic Hypergeometry of Supersymmetric Dualities
851
1
where U = ( pq) K +1 , the balancing condition reads U N X −4 Y −4 ST = ( pq) N f −4 with N f −8 N f si , T = i=1 ti , and S = i=1 1 2(K −2) N −4 f Y = X 2 ST −1 ( pq) K +1 . Equality of indices defines another unproven elliptic beta integral evaluation. 12.1.14. New confining duality. Let us take the electric and magnetic N = 1 superconformal field theories described by the tables below: SU (N + 1) f f TA
Q1 Q2 X q1 = Q 1N +1 q2 = Q 1 Q 2
S P(2N ) 1 f 1
SU (N + 3) f 1 1
U (1) 1 − N2+3 N +3
U (1) R 0 1 0
1 f
TA f
N +1 − N2+1
0 1
The dynamically generated superpotential in this case is Wdyn ∝ Q 1N +1 (Q 1 Q 2 )2 . The indices read N (q; q) N ( p; p)∞ ∞ IE = (N + 1)! TN
×
N +1 j=1
where
N +1 j=1
(Sz i−1 z −1 j ; p, q)
1≤i< j≤N +1
−1 (z i z −1 j , z i z j ; p, q)
N
N +3 −1 k=1 (tk z j ; p, q) m=1 (sm z j ; N −1 k=1 (Stk z j ; p, q)
N p, q) dz j , 2π iz j
(12.19)
j=1
= 1, and
IM =
N N +3 (tk sm ; p, q)
(Stk sm−1 ; p, q) 1≤l<m≤N +3 k=1 m=1
(Ssl−1 sm−1 ; p, q)
(12.20)
N +3 sm . with the balancing condition S = m=1 The elliptic beta integral described by the equality I E = I M was discovered by the first author and Warnaar in [93]. Here it defines a new pair of N = 1 supersymmetric quantum field theories dual to each other, which was not considered earlier in the literature. Moreover, it gives a counterexample to the classification of s-confining theories in [15]. Conjecturally, there exists a symmetry transformation for a higher order generalization of I E depending on the bigger number of parameters. Correspondingly, there should exist a more complicated Seiberg duality as well.
852
V. P. Spiridonov, G. S. Vartanov
12.2. Exceptional cases for unitary gauge groups. 12.2.1. SU (6) with 4 f + 4 f . The following pair of models was constructed in [16]: SU (6) f f T3A
Q Q A
SU (4) f 1 1
SU (4) 1 f 1
U (1)1 1 −1 0
U (1)2 3 3 −4
U (1) R 1 1 −1
f f f 1 f 1 1
f f 1 f 1 f 1
0 0 3 −3 3 −3 0
6 −2 5 5 −3 −3 −16
2 0 2 2 0 0 −4
M0 = Q Q M2 = Q A 2 Q B1 = AQ 3 3 B1 = A Q 3 B3 = A Q 3 3 B3 = A3 Q 4 T =A
Their superconformal indices read
( p; p)5∞ (q; q)5∞ 1≤i< j
6 4
(sk z j , tk z −1 j ; p, q)
j=1 k=1
where
6 j=1
5 dz j , 2π iz j
(12.21)
j=1
z j = 1, and I M = (U 4 ; p, q)
4
(sk tl , U 2 sk tl ; p, q)
k,l=1
×
4
(SU sk−1 , SU 3 sk−1 , T U tk−1 , T U 3 tk−1 ; p, q)
(12.22)
k=1
with S = 4k=1 sk , T = 4k=1 tk , and the balancing condition ST U 6 = pq. There is actually a lift of this duality to interacting magnetic theories found in [17]. The theory is self-dual and is based on the SU (6) gauge group and the flavor symmetry group is F = SU (6) × SU (6) × U (1)1 × U (1)1 . The matter content of the dual theories is given in the following tables: the electric theory Q Q A
SU (6) f f T3A
SU (6) f 1 1
SU (6) 1 f 1
U (1)1 1 −1 0
U (1)2 1 1 −2
U (1) R
SU (6) f 1 1 f f
SU (6) 1 f 1 f f
U (1)1 1 −1 0 0 0
U (1)2 1 1 −2 2 −2
U (1) R
1 2 1 2
0
and the magnetic theory q q a M0 M2
SU (6) f f T3A 1 1
1 2 1 2
0 1 1
Elliptic Hypergeometry of Supersymmetric Dualities
853
The electric superconformal index is ( p; p)5∞ (q; q)5∞ 1≤i< j
6 6
(si z j , ti z −1 j ; p, q)
i=1 j=1
5 dz j , 2π iz j
(12.23)
j=1
and the magnetic index is IM =
6 ( p; p)5∞ (q; q)5∞ (si t j , U 2 si t j ; p, q) 6! i, j=1 √ √ 6 6 3 3 S T −1 1≤i< j
×
5 dz j , 2π iz j
(12.24)
j=1
where S =
6
i=1 ,
6
T =
j=1 t j ,
and the balancing condition reads ST U 6 = ( pq)3 .
12.2.2. SU (5) with 3T A + 3 f Models [16]: SU (5) f
SU (3) 1
SU (3) f
U (1) −3
U (1) R
Q A
TA
2 3
f
1
1
0
AQ 2
f
f
−5
A3 Q
T AS
f
0
4 3 2 3
A5
TS
1
5
0
Indices: IE =
3 5 3 ( p; p)4∞ (q; q)4∞ k=1 (sk z i z j ; p, q) (tk z −1 j ; p, q) −1 −1 5! T4 1≤i< j≤5 (z i z j , z i z j ; p, q) j=1 k=1 ×
4 dz j , 2π iz j
(12.25)
j=1
with
5 j=1
z j = 1, and IM =
3 k,l=1
×
3
3
3
(Ss 2j ; p, q)
k=1 sk ,
k, j,l=1;k = j
T =
(Ss j sk ; p, q)
1≤ j
j=1
where S =
(T sk tl−1 ; p, q)
3
k=1 tk ,
(sk2 s j tl ; p, q)
3
(Stl ; p, q)2 ,
(12.26)
l=1
and the balancing condition reads S 3 T = pq.
854
V. P. Spiridonov, G. S. Vartanov
12.2.3. SU (5) with 2T A + 4 f + 2 f . Models [16]: SU (5) f f TA
Q Q A QQ 2 AQ
SU (2) 1 1 f
SU (4) 1 f 1
SU (2) f 1 1
U (1)1 −2 1 0
U (1)2 1 1 −1
U (1) R
1 f TS f 1
f TA 1 f f
f 1 f 1 1
−1 2 −2 1 −3
2 1 −1 −2 1
2 3 2 3 1 3 1 3
A2 Q A3 Q A2 Q 2 Q
Indices:
5 4
(tk z −1 j , u k z j ; p, q)
j=1 k=1
5 j=1
0
1
2 ( p; p)4∞ (q; q)4∞ k=1 (sk z i z j ; p, q) IE = −1 −1 4 5! T 1≤i< j≤5 (z i z j , z i z j ; p, q) ×
with
1 3 1 3
4 dz j , 2π iz j
(12.27)
j=1
z j = 1, and
IM =
4
(SU tk ; p, q)
k=1
×
2
4 2 k=1 l=1 2
(sl2 u k ; p, q)
k,l=1
(tk u l , Stk sl ; p, q)
2
(Su k ; p, q)
k=1
(sk tl tm ; p, q),
(12.28)
k=1 1≤l<m≤4
2
where the balancing condition reads S 3 T U = pq and S = 4 k=1 tk , U = u 1 u 2 .
k=1 sk ,
T =
12.2.4. SU (6) with 2T A + f + 5 f . Models [16]: Q Q A QQ 2 AQ
SU (6) f f TA
A3
A3 Q Q 2 A4 Q
Indices: IE =
SU (2) 1 1 f
SU (5) 1 f 1
U (1)1 −5 1 0
U (1)2 −4 −4 3
U (1) R 0 0
1 f T3S f 1
f TA 1 f TA
−4 2 0 −4 2
−8 −5 9 1 4
0
1 4 1 4 3 4 3 4
1
(U z i z j ; p, q) ( p; p)5∞ (q; q)5∞ −1 −1 5 6! (z T 1≤i< j≤6 i z j , z i z j ; p, q) ×
2
l=1 1≤ j
(sl z j z k ; p, q)
6 5 j=1 k=1
(tk z −1 j ; p, q)
5 dz j , (12.29) 2π iz j j=1
Elliptic Hypergeometry of Supersymmetric Dualities
with
6
i=1 z i
IM =
5
855
= 1, and (U tk ; p, q)
k
2 5
(SU sk t j ; p, q)
k=1 j=1
×
(S 2 t j tk ; p, q)
2
(sk t j tl ; p, q) (12.30)
k=1 1≤ j
(s 3j , Ss j ; p, q),
j=1
1≤ j
where the balancing condition reads S 4 T U = pq and S =
2
k=1 sk ,
T =
5
k=1 tk .
12.2.5. SU (7) with 2T A + 6 f . Models [16]: SU (7) f TA
Q A
SU (2) 1 f
SU (6) f 1
U (1) −5 3
U (1) R
f TS
TA f
−7 7
2 3 1 3
AQ 2 A4 Q
1 3
0
Indices: IE =
1 ( p; p)6∞ (q; q)6∞ −1 −1 7! T6 1≤i< j≤7 (z i z j , z i z j ; p, q) 2
×
(sk z i z j ; p, q)
k=1 1≤i< j≤7
with
7
i=1 z i
IM =
6 7
(tk z −1 j ;
k=1 j=1
6 dz j p, q) , 2π iz j
(12.31)
j=1
= 1, and
6
(S 2 tk ; p, q)
k=1
6 2
(Ssl2 tk ; p, q)
k=1 l=1
2
(sk tl tm ; p, q),
k=1 1≤l<m≤6
(12.32) where the balancing condition reads S 5 T = pq and S = 2k=1 sk , T = 6k=1 tk . All the equalities of superconformal indices of dual theories, I E = I M , described in this section represent new elliptic beta integrals requiring a rigorous proof (the parameter values are assumed to guarantee that only sequences of poles of the integrands converging to zero are located inside the contour T).
12.3. Symplectic gauge group. 12.3.1. S P(2N ) with (2N + 4) f . Models [43]: Q Q2
S P(2N ) f
SU (2N + 4) f
U (1) R 2r = N1+2
TA
2r = N2+2
856
V. P. Spiridonov, G. S. Vartanov
Indices: N (q; q) N 1 ( p; p)∞ ∞ IE = ±1 ±1 N 2N N ! T 1≤i< j≤N (z i z j ; p, q) 2N +4 ±1 N m=1 (tm z j ; p, q) dz j × 2π iz j (z ±2 j ; p, q) j=1
and
IM =
(tm ts ; p, q),
(12.33)
(12.34)
1≤m<s≤2N +4
+4 where the balancing condition reads 2N m=1 tm = pq. The equality I E = I M was introduced and partially justified by van Diejen and the first author in [19] and completely proven in [65] and [83]. Its simplest p → 0 limit yields one of the Gustafson q-beta integrals [35]. 12.3.2. S P(2N ) with 6 f and T A . This duality was considered in [12,18]. The flavor symmetry group is F = SU (6) × U (1) and the field content is S P(2N ) f TA
Q A
SU (6) f 1
U (1) N −1 −3
U (1) R 2r = 13 0
1 TA
−3k 2(N − 1) − 3m
0
Ak Q Am Q
2 3
where k = 2, . . . , N and m = 0, . . . , N − 1. The electric superconformal index is given by the integral IE =
N (q; q) N (t z i±1 z ±1 ( p; p)∞ j ; p, q) ∞ N −1 (t; p, q) ±1 ±1 N 2 N! T N 1≤i< j≤N (z i z j ; p, q) ±1 N 6 N m=1 (tm z j ; p, q) dz j , (12.35) × 2π iz j (z ±2 j ; p, q) j=1
j=1
and the magnetic index is IM =
N j=2
(t ; p, q) j
N
(t j−1 tm ts ; p, q),
(12.36)
j=1 1≤m<s≤6
where the balancing condition reads t 2N −2 6m=1 tm = pq. The equality I E = I M coincides with the elliptic Selberg integral suggested by van Diejen and the first author in [19] and proven in [20] as a consequence of the BCn -elliptic beta integral of type I (its direct proof is given also in [65]). The Selberg integral plays a fundamental role in mathematics and mathematical physics because of a large number of applications [30]. Note that this exactly computable integral gives a confirmation of the KS duality for the special values of parameters N f = 3, K = N .
Elliptic Hypergeometry of Supersymmetric Dualities
857
12.3.3. S P(2M) + 4 f + 2M f + T A . This new confining duality is obtained from the results of Sect. 7 by formal setting N = 0. The models are described in the table: W1 Q1 Q1 ... QK X
S P(2M) f f f
SU (4) f 1 1
S P(2l1 ) 1 f 1
S P(2l2 ) 1 1 f
... ... ... ...
S P(2l K ) 1 1 1
U (1) − M−2 n4 − 21 n − 22
U (1) R 0 1 1
f TA
1 1
1 1
1 1
... ...
f 1
− 2K 1
n
1 0
TA f f
1 f 1
1 1 f
... ... ...
1 1 1
j − M−2 2 n M−2 − 4 − 21 + k1 n M−2 − 4 − 22 + k2
0 1 1
f
1
1
...
f
K − M−2 4 − 2 + kK
W12 X j W1 Q 1 X k 1 W1 Q 2 X k 2 ... W1 Q K X k K
n
1
where j = 0, . . . , M −1, ki = 0, . . . , n i −1 for any i = 1, . . . , K , n 1 = n 2 = . . . = n K K and i=1 li n i = M. The superconformal indices have the form M (q; q) M (t z i±1 z ±1 ( p; p)∞ j ; p, q) ∞ M−1 IE = (t; p, q) ±1 ±1 2 M M! T M 1≤i< j≤M (z i z j ; p, q) K lr −1 ±1 ±1 M 4 k=1 (ttk z j ; p, q) r =1 m=1 (sr,m z j ; p, q) dz j × K lr ±1 nr 2π iz j (z ±2 m=1 (t sr,m z j ; p, q) j ; p, q) r =1 j=1 (12.37) and IM =
M−1
(t i+2 tk−1 tr−1 ; p, q)
lr n 4 K r −1 (t km +1 tk−1 sr,i ; p, q) , (t km tk sr,i ; p, q)
k=1 r =1 i=1 km =0
i=0 1≤k
(12.38) 4
where the balancing condition is r =1 tr = t 2+M . The equality I E = I M was conjectured in [68] and proven in [9]. This duality gives another example of s-confining theories missed in [15]. 12.3.4. S P(2K (N f −2)) with N f f + T A . This duality was considered in [14,50]. From = 0, and the theory is s-con(10.1) we see that the choice N = K (N f − 2) yields N fining. The field content of the electric and magnetic theories is easily found from the tables given in Sect. 10.1. For brevity we skip the electric superconformal index given by (10.2), and present directly the magnetic index I M = (U ; p, q)
−1
K
(U l−1 si s j ; p, q),
(12.39)
l=1 1≤i< j≤2N f
2N 1 where U = ( pq) K +1 and the balancing condition reads U 2K N f −2K i=1f si = ( pq) N f . The conjecture I E = I M represents a new elliptic beta integral. For K = 1 it reduces to the proven relation of Sect. 12.3.1.
858
V. P. Spiridonov, G. S. Vartanov
12.3.5. S P(2(N f − 2 + 2K N f )) with N f f + TS . Looking at (10.4) and fixing N = N f − 2 + 2K N f , we obtain the s-confining theory which was considered in [14,50]. The corresponding electric superconformal index is given by (10.5), and the magnetic index takes the form IM =
K
(U l si s j ; p, q)
l=0 1≤i< j≤2N f
×
K −1
2N f
(U (2l+1)/2 si s j ; p, q)
l=0 1≤i< j≤2N f
(U (2l+1)/2 si±2 ; p, q),
i=1
(12.40) 1
where U = ( pq) K +1 and the balancing condition reads U 2N f −2+4K N f The conjecture I E = I M represents a new elliptic beta integral.
2N f i=1
si =( pq) N f .
12.3.6. S P(2(3K N f − 4K − 2)) with N f f + 2T A . Looking at (10.7) and fixing N = 3K N f − 4K − 2 for odd K , we obtain the s-confining theory which was considered in [50]. The corresponding electric superconformal index is given by (10.8), and the magnetic index takes the form K
I M = (U, U 2 ; p, q)−1
2 K −1
(U J +
KL 2
si s j ; p, q)
J =0 L=0 1≤i< j≤2N f K −1
×
2 2N f
K
(U 2J +1+ 2 s 2j ; p, q),
(12.41)
J =0 j=1
2N 1 where U = ( pq) K +1 and the balancing condition reads U 3K N f −2K −1 i=1f si = ( pq) N f . Rigorous justification of the expected equality I E = I M is absent at the moment. 12.3.7. S P(2(3K N f − 4K + 2)) with N f f + TS + T A . Looking at (10.10) and fixing N = 3K N f −4K + 2 for K odd, we obtain the s-confining theory which was considered in [50]. The corresponding electric superconformal index is given by (10.11), and the magnetic index has the form I M = (U ; p, q)
−1
2 K −1
J =0 L=0 1≤i< j≤2N f
K −1
(U
J + K2L
si s j ; p, q)
2
J =0
2N f
×
K
(U 2J + 2 s 2j ; p, q),
(12.42)
j=1
2N 1 where U = ( pq) K +1 and the balancing condition reads U 3K N f −2K +1 i=1f si = ( pq) N f . The conjectural equality I E = I M is our last new elliptic beta integral for classical root systems.
Elliptic Hypergeometry of Supersymmetric Dualities
859
13. Exceptional G 2 Group G 2 with 5 flavors. This s-confining duality was discussed in [33,60]. The electric theory with the gauge group G 2 and its magnetic dual are described in the table below Q
G2 7
SU (5) f
U (1) R 2r = 15
TS TA f
2 5 3 5 4 5
Q2 Q3 Q4
The superconformal indices are 3 5 5 2 ±1 dz j ( p; p)2∞ (q; q)2∞ k=1 m=1 (tm z k ; p, q) (t ; p, q) , IE = m ±1 ±1 2 2 2 3 2π iz j T 1≤ j
j=1
(13.1) where z 1 z 2 z 3 = 1, |tm | < 1, and IM =
5 m=1
(tm2 ; p, q) (( pq)1/2 tm ; p, q)
1≤l<m≤5
(tl tm ; p, q) (( pq)1/2 tl tm ; p, q)
(13.2)
with the balancing condition 5m=1 tm = ( pq)1/2 . The conjecture I E = I M describes the first elliptic beta integral for exceptional root systems (it was mentioned in [91] and proposed also earlier by M. Ito). Substituting t5 = ( pq)1/2 /(t1 t2 t3 t4 ) in (13.1) and (13.2), and taking the limit p → 0, one obtains the four parameter q-beta integral on the G 2 root system derived in [36]. G 2 with 5 < N f < 12 flavors. This duality was discovered by Pouliot in [61]. The electric theory has gauge group G 2 , but its magnetic dual has SU (N f − 3) gauge group. Their field content is presented in the tables below. The electric theory (the vector superfield V is omitted): G2 7
Q
SU (N f ) f
U (1) R 2r = 1 − N4
f
is omitted): and the magnetic theory (the vector superfield V q
SU (N f − 3) f
SU (N f ) f
q0
f
1
s
TS
1
M
1
TS
U (1) R 2rq = N3 (1 − N 1−3 ) f f 2rq0 = 1 − N 1−3 f
2rs = N 2−3 f 2r M = 2 − N8
f
Corresponding superconformal indices are described by the integrals Nf 3 N f 2 ±1 dz k ( p; p)2∞ (q; q)2∞ k=1 m=1 (tm z k ; p, q) IE = (t ; p, q) , m ±1 ±1 2 2 3 2π iz k T2 1≤ j
k=1
(13.3)
860
V. P. Spiridonov, G. S. Vartanov
where z 1 z 2 z 3 = 1, and N −4
IM =
N −4
( p; p)∞f (q; q)∞f (N f − 3)! ×
T
N f −4
N f −3
×
(t j tk ; p, q)
1≤ j
Nf
(t 2j ; p, q)
j=1 N f −3
(( pq)rs z j z k ; p, q)
−1 (z −1 j zk , z j zk ; 1≤ j
p, q)
(( pq)rs z 2j ; p, q)
j=1
Nf
(( pq)(1−rs )/2 z −1 j ; p, q)
j=1
(( pq)(1−rs )/2 tk−1 z −1 j ; p, q)
k=1
N f −4
j=1
dz j , 2π iz j (13.4)
N f −3 N f where j=1 z j = 1, and the balancing condition reads m=1 tm = ( pq)(N f −4)/2 . The equality I E = I M represents a new symmetry transformation formula for general elliptic hypergeometric integrals on the G 2 root system. Independently, it was also considered earlier by F. A. Dolan. For N f = 5 the integral I M takes the form IM =
( p; p)∞ (q; q)∞ 2
(t j tk ; p, q)
1≤ j
5
(t 2j ; p, q)
j=1
(( pq)1/4 z ±1 ; p, q) 5 (( pq)1/4 t −1 z ±1 ; p, q) dz k=1 j k j . × ±2 ; p, q) (z 2π iz T
(13.5)
Using the univariate elliptic beta integral, one can compute this I M and find the index coinciding with (13.2). As to the general G 2 -transformation I E = I M , it should be a consequence of the original SU (3)-gauge group Seiberg duality. Indeed, let us take N = 3 and set ti−1 = si in the electric index (4.6). Then, if we impose the constraint s N f = pq, we obtain the G 2 -group electric index (13.3) with N f and ti replaced by N f − 1 and si , respectively. Therefore it is expected that the G 2 -magnetic index can be obtained after appropriate restrictions on I M in (4.7). A difficulty lies in computing the limit s N f → pq, since it leads to a diverging integral multiplied by a vanishing coefficient. This limit is currently under investigation.
14. ’t Hooft Anomaly Matching Conditions In this section we describe the standard ’t Hooft anomaly matching conditions [40,97] for some of the new dualities. Needed Casimir operators for unitary and symplectic groups can be found in Appendix C. There exist also the discrete anomalies matching conditions [14], but we skipped their consideration in the present work. Multiple S P(2N ) duality. Let us begin with the multiple duality for S P(2N ) gauge group found in [91] and discussed in Sect. 6. Coincidence of the anomalies is checked for the smaller flavor groups of dual theories. The subgroup SU (4) × SU (4) × U (1) B ×
Elliptic Hypergeometry of Supersymmetric Dualities
861
U (1) × U (1) R of the electric theory has the following triangle anomaly coefficients: SU (4)3L 2N , 3N 2 − 2N − 1 , 2 U (1)2B × U (1) −4N (N − 1),
SU (4)2L × U (1)
U (1)2 × U (1) B 0,
SU (4)2L × U (1) R
− 2N 2 + 1
SU (4)2L × U (1) B
2N
U (1)2B × U (1) R
0
−1 2 − (2N 2 + N + 1).
U (1)2 × U (1) R
U (1) R −(2N 2 + 7N + 1),
(14.1)
−
U (1)3R
N3
We have verified that all three dual magnetic theories have the same anomaly coefficients. Also it is easy to check that the real anomaly is equal to zero in the electric and magnetic theories. Explicitly, for the electric theory one has: 2N +2− 21 8−(2N −2) = 0. S P ↔ S P groups duality. Here we discuss the duality of Sec. 7. In the electric theory, anomaly coefficients for SU (4) × S P(2l1 ) × S P(2l2 ) × . . . × S P(2l K ) ×U (1) ×U (1) R global symmetry group have the values SU (4)3 −2M,
1 M(M − N − 2) 2 SU (4)2 × U (1) R − 2M U (1) R 1 − 6M SU (4)2 × U (1)
−
S P(2li )2 × U (1) −Mn i , S P(2li )2 × U (1) R 0, U (1)3R 1 − 6M 1 −M 3 + 2N M 2 − M N 2 − 4M N − 2M + 2 U (1)2 × U (1) R 2
(14.2)
coinciding with the coefficients in the magnetic theory. Computation of the real gauge anomaly coefficient yields: −4 − (2M − 2) + 2M + 2 = 0. Multiple SU (2N ) duality. The electric theory of Sec. 8 has the following anomaly coefficients for SU (4) × SU (4) × U (1)1 × U (1)2 × U (1) B × U (1) R global symmetry group: SU (4)3 SU (4) × U (1) B SU (4)2 × U (1) R 2
U (1)21 × U (1)2 U (1)2B × U (1)1 U (1)2B × U (1) R U (1)22 × U (1) B U (1) R
2N , 2N , −N ,
SU (4)2 × U (1)1 SU (4)2 × U (1)2 U (1)21 × U (1) B
0 4N (N − 1) 0
−8N (2N − 1),
U (1)21 × U (1) R
− 2N (2N − 1)
0,
U (1)2B × U (1)2 U (1)22 × U (1)1 U (1)22 × U (1) R U (1)3R
32N (2N − 1)
−8N , 0, −8N + 4N 2 − 1,
0 − 32N (N − 1)2 − 2N + 4N 2 − 1 (14.3)
862
V. P. Spiridonov, G. S. Vartanov
which coincide with the anomaly coefficients in all three dual magnetic theories. Calculation of the real gauge anomaly yields: − 21 8 − 2(2N − 2) + 4N = 0. New confining duality. The electric theory of Sec. 12.1.14 has the following anomaly coefficients for S P(2N ) × SU (N + 3) × U (1)1 × U (1)2 × U (1) R global symmetry group SU (N + 3)3 SU (N + 3)2 × U (1) R
N + 1,
U (1)3R
−(N + 1),
S P(2N )2× U (1)
1 − (N + 2)(N + 3) 2 (N + 1)(N + 3) − 2
SU (N + 3)2 × U (1) N + 1, 1 U (1)2 × U (1) R − (N + 1)2 (N + 2)(N + 3) 2 1 U (1) R − (N + 2)(N + 3), S P(2N )2 × U (1) R 0 2
(14.4)
and the same picture holds for the magnetic partner. Calculation of the real gauge anomaly yields: −(N + 3) − (N − 1) + 2(N + 1) = 0. SU ↔ S P groups duality. The anomaly matching for the common global group SU (N + 3) × SU (N + 3) × U (1) B × U (1) R of the duality described in Sect. 11.1 is checked and yields: SU (N + 3)3L
SU (N + 3)2L × U (1) R
N + 1,
SU (N + 3)2L × U (1) B 2(N + 1),
U (1)2B × U (1) R
U (1) R −(N 2 + 2N + 2),
(N + 1)2 N +3 − 8(N + 1)2 −
N 4 − 9N 2 − 10N + 2 . (N + 3)2 (14.5)
U (1)3R
SU ↔ SU groups duality. Here we consider the dualities of Sect. 11.2. The anomaly matching is checked for the global group SU (K ) L × SU (N + 3 − K ) L × U (1)1 × SU (K ) R × SU (N + 3 − K ) R × U (1)2 × U (1) B × U (1) R yielding SU (K )3L
N + 1,
SU (K )2L × U (1) B (N + 1), U (1)2B × U (1)1 0, U (1)21 U (1)21
SU (K )2L × U (1) R SU (K )2L × U (1)1 U (1)2B × U (1) R
(N + 1)2 N +3 (N + 1)(N + 3 − K ) −
− 2(N + 1)2
× U (1) B (N + 1)(N + 3)K (N + 3 − K ) × U (1) R −(N + 1)2 K (N + 3 − K ) U (1) R −(N 2 + 2N + 2)
U (1)3R
(14.6) −
N4
− 9N 2
− 10N + 2 . (N + 3)2
Computation of the real gauge anomaly yields: −2(N + 1) + 2(N + 1) = 0. Comparing the ’t Hooft anomaly matching conditions for all dualities described in this paper and the analysis of total ellipticity of the elliptic hypergeometric terms lying behind the equalities of superconformal indices, we come to the following conjecture.
Elliptic Hypergeometry of Supersymmetric Dualities
863
Conjecture. The condition of total ellipticity for an elliptic hypergeometric term is necessary and sufficient for validity of the ’t Hooft anomaly matching conditions for dual superconformal field theories whose superconformal indices are determined by this term. For proving this hypothesis it is necessary to use the formal mathematical definition of anomalies as cocycles of the gauge groups (see, e.g., [71]). For dual theories we have two, in general different, gauge groups. Therefore the anomaly matching condition looks like an equality of Chern classes of dual theories, and the conditions of total ellipticity—as a condition of vanishing of the combined Chern classes. This problem deserves a separate detailed discussion. 15. Conclusion To summarize our results, on the mathematical side we have conjectured many new symmetry transformations for elliptic hypergeometric integrals or exact evaluation formulas. On the physical side, we have found many new Seiberg dualities. Sections 6, 7, 8, and 11 contain new electric-magnetic dualities for N = 1 SYM theories based on unitary and symplectic gauge groups with specific matter content. Sections 12.1.14 and 12.3.3 contain new examples of S-confining theories derived from known identities for elliptic hypergeometric integrals. It should be clear that this paper does not contain a description of all known dual superconformal field theories. We have limited ourselves only to simple gauge groups G = SU (N ), S P(2N ), and G 2 . First, there are other simple groups G = S O(N ), F4 , E 6 , E 7 , E 8 consideration of which we have skipped. The situation with the dualities for the exceptional groups [22,48,70] is not clear in general (except of the G 2 -cases described above) due to the complexity of the invariants of these groups [11,63]. There are very many dualities involving orthogonal groups S O(N ). Originally we hoped to tackle them as well, but their amount is very big, and it was decided to consider them separately. It is known that many group-theoretical objects for the S O(N ) groups can be obtained as reductions of the S P(2N )-group constructions. Some of such reductions were considered by Dolan and Osborn at the level of superconformal indices [26]. However, there are many dualities that they did not analyze. Many elliptic hypergeometric integrals for the B N (i.e., S O(2N + 1) groups) and D N (i.e., S O(2N ) groups) root systems can be obtained by special restriction of the BC N -integrals (cf. the forms of the corresponding invariant measures given in Appendix B). However, it is not clear at the moment whether superconformal indices of all known S O(N )-group theories and their duals can be obtained in this way. There are also other types of reduction of the indices and dualities, e.g., those leading to dualities outside the conformal windows [92]. Second, we have deliberately skipped consideration of the superconformal indices for extended N > 1 supersymmetric field theories [6,49]. The best known examples correspond to the Seiberg-Witten N = 2 theories [77,78]. Consider the following electric and magnetic theories
Q
S O(3) f
SU (3) f
U (1) R 2 3
q M
S O(4) f 1
SU (3) f TS
U (1) R 1 3 4 3
As discussed by Intriligator and Seiberg [44–46] (see also [32]), the S O(3) Seiberg duality electric model becomes the SU (2) group N = 4 super-Yang-Mills theory in the
864
V. P. Spiridonov, G. S. Vartanov
infrared region after introducing the tree level superpotential Wtr ee ∝ det Q. Superconformal indices have the form 3 3 1/3 s z ±1 ; p, q) j dz ( p; p)∞ (q; q)∞ j=1 (( pq) 1/3 (( pq) s j ; p, q) IE = , ±1 ; p, q) 2 (z 2π iz T j=1
where
(15.1)
3
IM =
j=1 s j
= 1, and
( p; p)2∞ (q; q)2∞ 4 2 3 ×
j=1
T2
2
(( pq) 3 si s j ; p, q)
1≤i< j≤3
1/6 s −1 z ±1 ; i=1 (( pq) i j (z 1±1 z 2±1 ; p, q)
3
2
(( pq) 3 si2 ; p, q)
i=1
p, q)
2 j=1
dz j . 2π iz j
(15.2)
√ √ By a change of integration variables y1 = z 1 z 2 , y2 = z 1 /z 2 in I M , one of the integrations can be taken explicitly with the help of univariate elliptic beta integral, which shows that (15.2) is equal to (15.1). This equality can be obtained as a reduction of the BC N -relations as well [26]. We suppose therefore that it is necessary to consider first all possible S P(2N )-group identities for integrals and then try to reduce them to the relations for superconformal indices of extended supersymmetric dual theories with S O(N ) gauge groups. Third, we skipped the quiver gauge group cases, when there is more than one simple gauge group (which corresponds also to the deconfinement phenomenon [5]). Is is expected that equalities of the superconformal indices for them are mere consequences of the so-called Bailey-type chains (forming a tree) of symmetry transformations for integrals discovered by the first author in [84] and extended in [93] to root systems. Within this context, the duality transformation acquires a simple meaning of the integral transform whose properties resemble the classical Fourier transformation, see [93]. Let us list some other possible applications of our results. Counting of the gauge invariant operators for a number of supersymmetric gauge theories was considered in detail in [34,39]. It is not difficult to see that the corresponding generating functions are obtained from our superconformal indices by taking the limits p, q → 0. To take the simplest possible limit p → 0 one needs first to get rid of the balancing conditions by multiplying a number of parameters by integer powers of p and applying the reflection formula for the elliptic gamma function, see [89]. However, in the present work we have a much larger list of theories where this gauge invariant operators counting technique is applicable (in particular, this concerns the theories described in Sects. 7, 8, 9.2–9.6, 10.2–10.4, 11–13). The limit p → 0 in all these theories leads to q-hypergeometric functions, the meaning of which is not clarified yet from the superconformal index point of view. The subsequent limit q → 0 can be replaced by q → 1 yielding the plain hypergeometric functions, which also should have thus some topological meaning in gauge field theories. Similar clarification is needed for the situations when the elliptic hypergeometric integrals are reduced to terminating elliptic hypergeometric series by some special choices of the parameters, or for the relations between integrals with different powers of p and q. In [83,88], the first author has constructed univariate biorthognal functions associated with the elliptic beta integral. Naturally, it was conjectured there that some multivariable biorthogonal functions exist for all known elliptic beta integrals (which serve as the
Elliptic Hypergeometry of Supersymmetric Dualities
865
orthogonality measures). The first family of such functions was constructed by Rains in [65,66]. As a consequence of our work, the expected number of similar families of multivariable biorthogonal functions has now increased essentially. In [85,87], it was shown that some of the BC N elliptic hypergeometric integrals can be associated with the relativistic Calogero-Sutherland type models. It was conjectured there that other models of such type can be built out of all other existing elliptic beta integrals and their appropriate generalizations. Because we have now the interpretation of the elliptic hypergeometric integrals as superconformal indices of supersymmetric field theories, we come to the natural conjecture that behind each N = 1 superconformal field theory there is a Calogero-Sutherland type model for which these integrals serve either as the topological indices or the wave functions normalizations, respectively. We would like to mention in this context the known appearance of the usual elliptic Calogero-Sutherland models within the N = 2 Seiberg-Witten theories [59]. The group-theoretical interpretation of the elliptic hypergeometric integrals discussed in [73,26,91] and the present paper opens possibilities for general structural theorems on the integrals themselves. It may play a key role in the classification of such integrals on root systems. In particular, it naturally leads to the conjecture that there exist infinitely (countably) many dualities and related elliptic hypergeometric integral identities. All the problems mentioned above deserve detailed investigation either in relation to supersymmetric dualities or on plain mathematical grounds. As to the proofs of many new hypergeometric identities conjectured in this paper, we refer to known methods described in [19,20,65,69,81,83,84,93] (or indicated above in some cases) which are available for their treatment. We plan to consider them case by case depending on their tractability. Acknowledgements. We would like to thank A. A. Belavin, F. A. Dolan, D. I. Kazakov, A. Khmelnitsky, H. Osborn, A. F. Oskin, V. A. Rubakov, A. Schwimmer, M. A. Shifman and S. Theisen for valuable discussions, comments and remarks. The first author is indebted to L. D. Faddeev for discussions on elliptic hypergeometric functions and their applications, which inspired the choice of present paper’s title (it matches in spirit with [98], and it is expected that there exists a noncommutative extension of the elliptic hypergeometry, formulation of which is a rather difficult task). He is also grateful to A. M. Vershik for persistent support and encouragement, which was the main driving force for writing the survey [89]. The second author would like to thank BLTP JINR for its creative atmosphere due to which he was able to join this project. The first author was partially supported by RFBR grants no. 09-01-00271, 09-01-93107-NCNIL-a and, at the final stages of work, by CERN TH (Geneva) and MPIM (Bonn). The second author was partially supported by the Dynasty foundation, RFBR grant no. 08-02-00856-a and grant of the Ministry of Education and Science of the Russian Federation no. 1027.2008.2.
Appendix A. Characters of Representations of Classical Groups Here we present general results for characters of the Lie group representations used in the paper. For the SU (N ) group representations, the characters, depending on x = N xi = 1, are the well known Schur polyno(x1 , . . . , x N ) subject to the constraint i=1 mials λ +N − j det xi j , sλ (x) = s(λ1 ,...,λ N ) (x) = (A.1) N− j det xi where λ is the partition ordered so that λ1 ≥ λ2 ≥ · · · ≥ λ N . They obey the property s(λ1 ,...,λ N ) (x) = s(λ1 +c,...,λ N +c) (x), where c ∈ Z. Therefore one can assume that λ N = 0 without loss of generality.
866
V. P. Spiridonov, G. S. Vartanov
Let us list explicitly the simplest characters. The fundamental and antifundamental representations: χ SU (N ), f (x) = s(1,0,...,0) (x) =
N
xi , χ SU (N ), f = s(1,...,1,0) (x) = χ SU (N ), f (x −1 ).
i=1
The adjoint representation:
χ SU (N ),ad j (x) = s(2,1,...,1,0) (x) =
xi x −1 j − 1.
1≤i, j≤N
The absolutely anti-symmetric tensor representation of rank two: χ SU (N ),T A (x) = s(1,1,0,...,0) (x) = xi x j , χ SU (N ),T A = χ SU (N ),T A (x −1 ). 1≤i< j≤N
The symmetric representation: χ SU (N ),TS (x) = s(2,0,...,0) (x) =
xi x j +
1≤i< j≤N
N
xi2 , χ SU (N ),T S (x)
i=1
= χ SU (N ),TS (x
−1
).
The absolutely anti-symmetric tensor representation of rank three: χ SU (N ),T3A (x) = s(1,1,1,0,...,0) (x) = xi x j xk . 1≤i< j
The absolutely symmetric tensor representation of rank three: χ SU (N ),T3S (x) = s(3,0,...,0) (x) =
N
xi x j xk +
1≤i< j
xi2 x j +
i, j=1,i = j
N
xi3 .
i=1
In the mixed case, we have χ SU (N ),T AS (x) = s(2,1,0,...,0) (x) = 2
xi x j xk +
N
xi2 x j .
i, j=1;i = j
1≤i< j
The Weyl characters for S P(2N ) group are given by the determinant λ +N − j+1 −λ −N + j−1 det xi j − xi j , s(λ1 ,...,λ N ) (x) = N − j+1 −N + j−1 det xi − xi
(A.2)
with λ1 ≥ λ2 ≥ · · · ≥ λ N ≥ 0. For the fundamental and antifundamental representations χ S P(2N ), f (x) = χ S P(2N ), f (x) = s(1,0,...,0) (x) =
N (xi + xi−1 ), i=1
Elliptic Hypergeometry of Supersymmetric Dualities
867
For the adjoint representation χ S P(2N ),ad j (x) = s(2,0,...,0) (x) =
−1 −1 −1 (xi x j + xi x −1 j + xi x j + xi x j ) +
1≤i< j≤N
N (xi2 + xi−2 ) + N . i=1
For the absolutely anti-symmetric representation −1 −1 −1 χ S P(2N ),T A (x) = s(1,1,0,...,0) (x) = (xi x j + xi x −1 j + x i x j + x i x j ) + N − 1. 1≤i< j≤N
As to the exceptional group G 2 , its fundamental representation has the character χ (z 1 , z 2 , z 3 ) = 1 +
3
z i + z i−1 ,
i=1
where z 1 z 2 z 3 = 1. The character for the adjoint representation of G 2 group is −1 −1 z i z j + z i−1 z j + z i z −1 . χ (z 1 , z 2 , z 3 ) = 2 + j + zi z j 1≤i< j≤3
Appendix B. Invariant Matrix Group Measures Here we describe the invariant measures for integrals over classical Lie groups and the exceptional group G 2 . Such a measure for the unitary group SU (N ) with any symmetric function f (z), where z = (z 1 , . . . , z N ), Nj=1 z j = 1, has the form SU (N )
f (z)dμ(z) =
N −1 dz j 1
(z) (z −1 ) f (z) , N ! T N −1 2π iz j j=1
(B.1)
where (z) is the Vandermonde determinant (z i − z j ).
(z) = 1≤i< j≤N
The invariant measure for the symplectic group S P(2N ) with any symmetric function f (z), z = (z 1 , . . . , z N ), has the form S P(2N )
f (z)dμ(z) =
N N dz j (−1) N −1 2 −1 2 (z − z )
(z + z ) f (z) . j j N 2 N ! TN 2π iz j j=1 j=1 (B.2)
For the invariant measures over the orthogonal group S O(N ) and any symmetric function f (z), z = (z 1 , . . . , z N ), one has to distinguish the cases of odd and even N : S O(2N )
f (z)dμ(z) =
1
2 N −1 N ! T N
(z + z −1 )2 f (z)
N dz j , 2π iz j j=1
(B.3)
868
V. P. Spiridonov, G. S. Vartanov
and S O(2N +1)
2 N 1 N dz j (−1) N − 21 −1 2 2 zj − zj f (z)dμ(z) = N
(z + z ) f (z) . 2 N ! TN 2π iz j j=1 j=1 (B.4)
The invariant measure for the exceptional group G 2 and any symmetric function f (z), z = (z 1 , z 2 , z 3 ), where z 1 z 2 z 3 = 1, has the form 2 dz j 1 f (z)dμ(z) = 2
(z + z −1 )2 f (z) . (B.5) 2 3 T2 2π iz j G2 j=1
Appendix C. Relevant Casimir Operators Commutators of the generators T a , a = 1, . . . , dim G, of some classical Lie group G are defined with the help of structure constants f abc T a , T b = i f abc T c . (C.1) It is straightforward to obtain the Casimir operators [97] b n ab (Tra )lm (Tra )ln = C2 (r)δnm , (Tra )m n (Tr )m = T (r)δ ,
(C.2)
n,m
a,l
where r is some irreducible representation. These Casimir operators and the dimension of the representation d(r) are connected through the adjoint representation adj, d(r)C2 (r) = d(adj)T (r).
(C.3)
For checking the ’t Hooft anomaly matching conditions we need the triple Casimir operator which comes from the trace Aabc = T r [T a {T b , T c }].
(C.4)
Then it is convenient to define the operator A(r) relative to the fundamental representation Aabc (r) = A(r)Aabc ,
(C.5)
where Aabc = T r [TFa {TFb , TFc }] and TFa are the generators in the fundamental representation. In the tables below we give the dimensions d(r), the Casimir operators 2T (r), and A(r) for the unitary group and the dimensions d(r), the Casimir operators T (r) for symplectic and G 2 groups. Note that in the verification of the anomaly matchings for unitary groups we use 2T (r).
SU (N ) group:
Irrep r f adj TA TS T3A T3S T AS
d(r) N N2 − 1 1 2 N (N − 1) 1 2 N (N + 1) 1 N (N − 1)(N − 2) 6 1 N (N + 1)(N + 2) 6 1 N (N − 1)(N + 1) 3
2T (r) 1 2N N −2 N +2 1 2 (N − 3)(N − 2) 1 2 (N + 3)(N + 2) N2 − 3
A(r) 1 0 N −4 N +4 1 2 (N − 6)(N − 3) 1 2 (N + 6)(N + 3) N2 − 9
Elliptic Hypergeometry of Supersymmetric Dualities
869
Irrep r f adj = TS TA
S P(2N ) group:
G 2 group:
d(r) 2N N (2N + 1) N (2N − 1) − 1
Irrep r f adj
d(r) 7 14
T(r) 1 2N + 2 2N − 2
T(r) 2 8
Appendix D. Total Ellipticity for the KS Duality Indices In order to illustrate the work hidden behind our conjectures, we briefly describe in this Appendix verification of the total ellipticity for the transformation identity for elliptic hypergeometric integrals associated with the Kutasov-Schwimmer duality from Sect. 9.1. − K First, we change the integration variables z in (9.3) to z = U −1 (ST ) 2 N y and assume that the contours of integration in y-variables can be deformed back to T without crossing the poles. Then the equality of integrals (9.2) and (9.3) is rewritten in the following form: κN
T N −1
E (z, t, s)
N −1 j=1
−1 N dy j dz j = κ N
M (y, t, s) , 2π iz j 2π iy j T N−1 j=1
(D.1)
= K N f − N and where N N −1 (q; q) N −1 ( p; p)∞ ∞ (U ; p, q) N −1 N!
κN = 1
with U = ( pq) K +1 . The kernels of the elliptic hypergeometric integrals are −1 N f (U z i z −1 j , U z i z j ; p, q) N
E (z, t, s) =
−1 (z i z −1 j , z i z j ; p, q)
1≤i< j≤N
M (y, t, s) =
Nf K
(U
l−1
si t −1 j ;
×
Nf N
i=1 j=1
−1 (U yi y −1 j , U yi y j ; p, q)
1≤i< j≤ N
−1 (yi y −1 j , yi y j ; p, q)
p, q)
l=1 i, j=1
(si z j , ti−1 z −1 j ; p, q),
(si−1 y j , U 2 ti y −1 j ; p, q),
(D.2)
i=1 j=1
where
N
i=1 z i
= 1,
N
i=1 yi
1
= U N +N K ( pq)− 2 N f K S K , and U 2N ST −1 = ( pq) N f .
Theorem 5. The function ρ(z, y, t, s) = is a totally elliptic hypergeometric term.
E (z, t, s)
M (y, t, s)
870
V. P. Spiridonov, G. S. Vartanov
Ellipticity of the z-variables q-certificates. As described in Sect. 2, we should consider the ratios h az (z, y, t, s, q) =
ρ(z, y, t, s)|za →qza ,z N →q −1 z N ρ(z, y, t, s) N −1
−1 −1 −1 −1 −1 θ (U z a z −1 j , U z j z N , q z a z j , q z j z N ; p)
j=1, j =a
−1 −1 θ (U q −1 z a−1 z j , U q −1 z −1 j z N , z a z j , z j z N ; p)
=
× ×
−1 −2 −1 −1 −1 θ (U qz a z −1 N , U z a z N , q z a z N , q z a z N ; p)
−1 θ (U q −1 z a−1 z N , U q −2 z a−1 z N , qz a z −1 N , z a z N ; p) Nf i=1
θ (ti−1 z −1 θ (si z a ; p) N ; p) , −1 −1 θ (q si z N ; p) θ (q −1 ti z a−1 ; p)
(D.3)
where a = 1, . . . , N − 1, and check that these are totally elliptic functions. Indeed, h az (z, y, t, s, q) functions are automatically invariant under the transformations 1) sb → psb , s N f → p −1 s N f , 2) tb → ptb , t N f → p −1 t N f , 3) yb → pyb , y N → p −1 y N . Whereas the invariance with respect to the substitutions 4) z c → pz c , z N → p −1 z N for c = a or 5) z a → pz a , z N → p −1 z N uses the balancing condition. Similarly, one checks the invariance with respect to the mixed transformations sb → psb , tc → ptc and yd → p K yd . The most complicated part of the work consists in establishing ellipticity in the variable q. The difficulty comes from the presence of fractional powers of q entering (D.3) through the variable U . Because of that one should scale q by such a power of p that there will be a match with the period of the elliptic functions h az (z, y, t, s, q). Simultaneously, we should preserve the balancing condition and all other constraints on the parameters we have. This is reached by the following transformation of the parameters 6) q → p K +1 q, t N−1f → p (K +1)N f −2N t N−1f ,
y N → p N +N K −N f K (K +1)/2 y N , (D.4)
which leads to U → pU , as required. It is a matter of a neat computation (at the intermediate steps there appears a very cumbersome expression) to show that h az (z, y, t, s, q) do not change under these substitutions. Ellipticity of the y-variables q-certificates. Now we consider the ratios y
h a (z, y, t, s, q) = =
ρ(z, y, t, s)| ya →qya ,y N →q −1 y N ρ(z, y, t, s) −1 N j=1, j =a
× ×
−1 −1 θ (U q −1 ya−1 y j , U q −1 y −1 , ya y j , y j y N ; p) j yN −1 −1 −1 θ (U ya y −1 ya y j , q −1 y −1 ; p) ,q j , U y j yN j yN
−1 θ (U q −1 ya−1 y N , U q −2 ya−1 y N , qya y N−1 , ya y N ; p) −1 −2 −1 θ (U qya y N−1 ya y N , q −1 ya−1 y N ; p) , U ya y N ,q Nf θ (q −1 s −1 y N ; p) θ (U 2 q −1 ti y −1 ; p) i
i=1
θ (si−1 ya ; p)
a
θ (U 2 ti y N−1 ; p)
,
(D.5)
Elliptic Hypergeometry of Supersymmetric Dualities
871
− 1. Again, these are the totally elliptic functions. They are autowhere a = 1, . . . , N matically invariant under the transformations 1) sb → psb , s N f → p −1 s N f , 2) tb → ptb , t N f → p −1 t N f , 3) z b → pz b , z N → p −1 z N . The invariance with respect to the substitutions 4) yb → pyb , y N → p −1 y N , b = a, or 5) ya → pya , y N → p −1 y N uses the balancing condition. The most difficult part is the verification of the invariance with respect to the transformations 6)
q → p K +1 q, U → pU, t N−1f → p (K +1)N f −2N t N−1f ,
y N → p N +N K −N f K (K +1)/2 y N . Ellipticity of the t-parameters q-certificates. Now we need to investigate the functions h at (z, y, t, s, q) = =
ρ(z, y, t, s)|ta →qta ,t N
f
→q −1 t N f
ρ(z, y, t, s) Nf K N θ (U l−1 q −1 si t −1 ; p) l=1 i=1
×
a θ (U l−1 si t N−1f ;
p)
N θ (U 2 q −1 t N f y −1 j ; p) i=1
θ (U 2 ta y −1 j ; p)
j=1
θ (t N−1f z −1 j ; p) θ (q −1 ta−1 z −1 j ; p)
,
(D.6)
where a = 1, . . . , N f −1, and show that they are totally elliptic. Again, invariance under 1) yb → pyb , y N → p −1 y N , 2) sb → psb , s N f → p −1 s N f , and 3) z b → pz b , z N → p −1 z N transformations is automatic. The balancing condition is needed for symmetries 4) tc → ptc , t N f → p −1 t N f , c = a, and 5) ta → pta , t N f → p −1 t N f Computations during the verification of invariance under the transformations 6) q → p K +1 q, U → pU, t N−1f −1 → p (K +1)N f −2N t N−1f −1 ,
y N → p N +N K −N f K (K +1)/2 y N are very lengthy and require a lot of attention to reach the needed statement. Consideration of the s-parameters certificates is equivalent to the t-variables case because of the symmetries of the initial integral kernels. References 1. Andrews, G. E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Math. Appl. 71, Cambridge: Cambridge Univ. Press, 1999 2. Barnes, E.W.: On the theory of the multiple gamma function. Cambr. Trans. 19, 374–425 (1904) 3. Baxter, R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. (NY) 70, 193–228 (1972) 4. Benvenuti, S., Feng, B., Hanany, A., He, Y.H.: Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics. JHEP 0711, 050 (2007) 5. Berkooz, M.: The dual of supersymmetric SU (2k) with an antisymmetric tensor and composite dualities. Nucl. Phys. B 452, 513–525 (1995) 6. Bianchi, M., Dolan, F. A., Heslop, P. J., Osborn, H.: N = 4 superconformal characters and partition functions. Nucl. Phys. B 767, 163–226 (2007) 7. Brodie, J.H.: Duality in supersymmetric SU (Nc ) gauge theory with two adjoint chiral superfields. Nucl. Phys. B 478, 123–140 (1996) 8. Brodie, J.H., Strassler, M.J.: Patterns of duality in N = 1 SUSY gauge theories, or: Seating preferences of theater going nonAbelian dualities. Nucl. Phys. B 524, 224–250 (1998)
872
V. P. Spiridonov, G. S. Vartanov
9. van de Bult, F. J.: An elliptic hypergeometric beta integral transformation. http://arXiv.org/abs/0912. 3812v1 [math.CA], 2009 10. van de Bult, F.J., Rains, E.M.: Basic hypergeometric functions as limits of elliptic hypergeometric functions. SIGMA 5, 59 (2009) 11. Cho, P. L.: Moduli in exceptional SUSY gauge theories. Phys. Rev. D 57, 5214–5223 (1998) 12. Cho, P.L., Kraus, P.: Symplectic SUSY gauge theories with antisymmetric matter. Phys. Rev. D 54, 7640–7649 (1996) 13. Csáki, C., Murayama, H.: Discrete anomaly matching. Nucl. Phys. B 515, 114–162 (1998) 14. Csáki, C., Murayama, H.: New confining N = 1 supersymmetric gauge theories. Phys. Rev. D 59, 065001 (1999) 15. Csáki, C., Schmaltz, M., Skiba, W.: A systematic approach to confinement in N = 1 supersymmetric gauge theories. Phys. Rev. Lett. 78, 799–802 (1997) 16. Csáki, C., Schmaltz, M., Skiba, W.: Confinement in N = 1 SUSY gauge theories and model building tools. Phys. Rev. D 55, 7840–7858 (1997) 17. Csáki, C., Schmaltz, M., Skiba, W., Terning, J.: Selfdual N = 1 SUSY gauge theories. Phys. Rev. D 56, 1228–1238 (1997) 18. Csáki, C., Skiba, W., Schmaltz, M.: Exact results and duality for S P(2N ) SUSY gauge theories with an antisymmetric tensor. Nucl. Phys. B 487, 128–140 (1997) 19. van Diejen, J.F., Spiridonov, V.P.: An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums. Math. Res. Lett. 7, 729–746 (2000) 20. van Diejen, J. F., Spiridonov, V. P.: Elliptic Selberg integrals, Internat. Math. Res. Notices, no. 20, 1083–1110 (2001) 21. van Diejen, J.F., Spiridonov, V.P.: Unit circle elliptic beta integrals. Ramanujan J 10, 187–204 (2005) 22. Distler, J., Karch, A.: N = 1 dualities for exceptional gauge groups and quantum global symmetries. Fortsch. Phys. 45, 517–533 (1997) 23. Dixon, A.L.: On a generalisation of Legendre’s formula K E − (K − E)K = 21 π . Proc. London Math. Soc. 2(1), 206–224 (1905) 24. Dolan, F.A.: Counting BPS operators in N = 4 SYM. Nucl. Phys. B 790, 432–464 (2008) 25. Dolan, F. A., Osborn, H.: On short and semi-short representations for four dimensional superconformal symmetry. Ann. Phys. (NY) 307, 41–89 (2003) 26. Dolan, F.A., Osborn, H.: Applications of the superconformal index for protected operators and q-hypergeometric identities to N = 1 dual theories. Nucl. Phys. B 818, 137–178 (2009) 27. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Math. 55, Boston: Birkhäuser, 1985 28. Felder, G., Varchenko, A.: The elliptic gamma function and S L(3, Z)Z3 . Adv. Math. 156, 44–76 (2000) 29. Feng, B., Hanany, A., He, Y.H.: Counting gauge invariants: the plethystic program. JHEP 0703, 090 (2007) 30. Forrester, P.J., Warnaar, S.O.: The importance of the Selberg integral. Bull. Amer. Math. Soc. (N.S.) 45, 489–534 (2008) 31. Friedman, E., Ruijsenaars, S.: Shintani-Barnes zeta and gamma functions. Adv. Math. 187, 362–395 (2004) 32. Gadde, A., Pomoni, E., Rastelli, L., Razamat, S.S.: S-duality and 2d Topological QFT. JHEP 03, 032 (2010) 33. Giddings, S.B., Pierre, J.M.: Some exact results in supersymmetric theories based on exceptional groups. Phys. Rev. D 52, 6065–6073 (1995) 34. Gray, J., Hanany, A., He, Y. H., Jejjala, V., Mekareeya, N.: SQCD: A geometric apercu. JHEP 0805, 099 (2008) 35. Gustafson, R.A.: Some q-beta and Mellin-Barnes integrals with many parameters associated to the classical groups. SIAM J. Math. Anal. 23, 525–551 (1992) 36. Gustafson, R.A.: Some q-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras. Trans. AMS 341(1), 69–119 (1994) 37. Gustafson, R.A.: Some q-beta integrals on SU (n) and Sp(n) that generalize the Askey–Wilson and Nassrallah-Rahman integrals. SIAM J. Math. Anal. 25, 441–449 (1994) 38. Gustafson, R.A., Rakha, M.A.: q-Beta integrals and multivariate basic hypergeometric series associated to root systems of type Am . Ann. Comb. 4, 347–373 (2000) 39. Hanany, A., Mekareeya, N.: Counting gauge invariant operators in SQCD with classical gauge groups. JHEP 0810, 012 (2008) 40. ’t Hooft, G.: Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking. In: Recent Developments in Gauge Theories (eds. G. ’t Hooft et. al.), New York: Plenum Press, 1980, pp. 135–157 41. Intriligator, K.: New RG fixed points and duality in supersymmetric S P(Nc ) and S O(Nc ) gauge theories. Nucl. Phys. B 448, 187–198 (1995) 42. Intriligator, K.A., Leigh, R.G., Strassler, M.J.: New examples of duality in chiral and nonchiral supersymmetric gauge theories. Nucl. Phys. B 456, 567–621 (1995)
Elliptic Hypergeometry of Supersymmetric Dualities
873
43. Intriligator, K.A., Pouliot, P.: Exact superpotentials, quantum vacua and duality in supersymmetric S P(Nc ) gauge theories. Phys. Lett. B 353, 471–476 (1995) 44. Intriligator, K.A., Seiberg, N.: Phases of N = 1 supersymmetric gauge theories in four-dimensions. Nucl. Phys. B 431, 551–568 (1994) 45. Intriligator, K.A., Seiberg, N.: Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric S O(Nc ) gauge theories. Nucl. Phys. B 444, 125–160 (1995) 46. Intriligator, K. A., Seiberg, N.: Phases of N = 1 supersymmetric gauge theories and electric - magnetic triality. In: Future perspectives in string theory, (Los Angeles 1995), Singapore: World Scientific, 1996 pp. 270–282 47. Intriligator, K.A., Seiberg, N.: Lectures on supersymmetric gauge theories and electric - magnetic duality. Nucl. Phys. Proc. Suppl. 45BC, 1–28 (1996) 48. Karch, A.: More on N = 1 selfdualities and exceptional gauge groups. Phys. Lett. B 405, 280–286 (1997) 49. Kinney, J., Maldacena, J.M., Minwalla, S., Raju, S.: An index for 4 dimensional super conformal theories. Commun. Math. Phys. 275, 209–254 (2007) 50. Klein, M.: More confining N =1 SUSY gauge theories from nonAbelian duality. Nucl. Phys. B 553, 155–204 (1999) 51. Kutasov, D.: A comment on duality in N = 1 supersymmetric non-Abelian gauge theories. Phys. Lett. B 351, 230–234 (1995) 52. Kutasov, D., Schwimmer, A.: On duality in supersymmetric Yang-Mills theory. Phys. Lett. B 354, 315–321 (1995) 53. Kutasov, D., Schwimmer, A., Seiberg, N.: Chiral rings, singularity theory and electric-magnetic duality. Nucl. Phys. B 459, 455–496 (1996) 54. Leigh, R.G., Strassler, M.J.: Duality of Sp(2Nc ) and S O(Nc ) supersymmetric gauge theories with adjoint matter. Phys. Lett. B 356, 492–499 (1995) 55. Leigh, R. G., Strassler, M. J.: Accidental symmetries and N = 1 duality in supersymmetric gauge theory. Nucl. Phys. B 496, 132–148 (1997) 56. Machacek, M. E., Vaughn, M. T.: Two loop renormalization group equations in a general quantum field theory. 1. Wave function renormalization. Nucl. Phys. B 222 83–103(1983); 2. Yukawa couplings, ibid. B 236 221–232 (1984); 3. Scalar quartic couplings, ibid. B 249 70–92(1985) 57. Nakayama, Y.: Index for orbifold quiver gauge theories. Phys. Lett. B 636, 132–136 (2006) 58. Nekrasov, N. A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831– 864 (2003) 59. Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. In: The Unity of Mathematics Eds. P. Etingof, V. Retakh, I. M. Singer, Progr. Math., 244, Boston, MA: Birkhauser, 2006, pp. 525–596 60. Pesando, I.: Exact results for the supersymmetric G 2 gauge theories. Mod. Phys. Lett. A 10, 1871–1886 (1995) 61. Pouliot, P.: Chiral duals of nonchiral SUSY gauge theories. Phys. Lett. B 359, 108–113 (1995) 62. Pouliot, P.: Duality in SUSY SU (N ) with an antisymmetric tensor. Phys. Lett. B 367, 151–156 (1996) 63. Pouliot, P.: Spectroscopy of gauge theories based on exceptional Lie groups. J. Phys. A 34, 8631–8658 (2001) 64. Poppitz, E., Trivedi, S.P.: Some examples of chiral moduli spaces and dynamical supersymmetry breaking. Phys. Lett. B 365, 125–131 (1996) 65. Rains, E.M.: Transformations of elliptic hypergeometric integrals. Ann. of Math. 171, 169–243 (2010) 66. Rains, E.M.: BCn -symmetric abelian functions. Duke Math. J. 135(1), 99–180 (2006) 67. Rains, E.M.: Limits of elliptic hypergeometric integrals. Ramanujan J. 18(3), 257–306 (2009) 68. Rains, E. M.: Elliptic Littlewood identities. http://arXiv.org/abs/0806.0871v1 [math.CO], 2008 69. Rains, E. M., Spiridonov, V. P.: Determinants of elliptic hypergeometric integrals. Funkt. Analiz i ego Pril. 43 (4), 67–86 (2009) (Funct. Anal. Appl. 43 (4), 297–311 (2009)) 70. Ramond, P.: Superalgebras in N = 1 gauge theories. Phys. Lett. B 390, 179–184 (1997) 71. Reiman, A. G., Semenov-Tian-Shansky, M. A., Faddeev, L. D.: Quantum anomalies and cocycles on gauge groups. Funkt. Analiz i ego Pril. 18 (4), 64–72 (1984) (Funct. Anal. Appl. 18 (4), 319–326 (1984)) 72. Römelsberger, C.: Counting chiral primaries in N = 1, d = 4 superconformal field theories. Nucl. Phys. B 747, 329–353 (2006) 73. Römelsberger, C.: Calculating the superconformal index and Seiberg duality. http://arXiv.org/abs/ 0707.3702v1 [hep-th], 2007 74. Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997) 75. Seiberg, N.: Exact results on the space of vacua of four-dimensional SUSY gauge theories. Phys. Rev. D 49, 6857–6863 (1994) 76. Seiberg, N.: Electric–magnetic duality in supersymmetric non-Abelian gauge theories. Nucl. Phys. B 435, 129–146 (1995)
874
V. P. Spiridonov, G. S. Vartanov
77. Seiberg, N., Witten, E.: Electric - magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994), Erratum-ibid. B 430, 485–486(1994) 78. Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in N =2 supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994) 79. Shifman, M. A.: Nonperturbative dynamics in supersymmetric gauge theories. Prog. Part. Nucl. Phys. 39, 1–116 (1997) 80. Skagerstam, B.S.: On the large Nc limit of the SU (Nc ) colour quark-gluon partition function. Z. Phys. C 24, 97–101 (1984) 81. Spiridonov, V. P.: On the elliptic beta function. Usp. Mat. Nauk 56 (1), 181–182 (2001) (Russ. Math. Surv 56 (1), 185–186 (2001)) 82. Spiridonov, V. P.: Theta hypergeometric series. In: Proceedings of the NATO ASI Asymptotic Combinatorics with Applications to Mathematical Physics (St. Petersburg, Russia, July 2001), Eds. Malyshev, V. A., Vershik, A. M., Amsterdam: Kluwer, 2002, pp. 307–327 83. Spiridonov, V. P.: Theta hypergeometric integrals. Algebra i Analiz 15 (6), 161–215 (2003) (St. Petersburg Math. J. 15 (6), 929–967(2003)) 84. Spiridonov, V. P.: A Bailey tree for integrals. Teor. Mat. Fiz. 139, 104–111 (2004) (Theor. Math. Phys. 139, 536–541 (2004)) 85. Spiridonov, V. P.: Elliptic hypergeometric functions. Habilitation thesis, Bogoliubov Laboratory of Theoretical Physics, JINR, September 2004, 218 pp. (Russian) 86. Spiridonov, V.P.: Short proofs of the elliptic beta integrals. Ramanujan J 13, 265–283 (2007) 87. Spiridonov, V. P.: Elliptic hypergeometric functions and Calogero-Sutherland type models. Teor. Mat. Fiz, 150 (2), 311–324 (2007) (Theor. Math. Phys. 150 (2), 266–277 (2007)) 88. Spiridonov, V.P.: Continuous biorthogonality of the elliptic hypergeometric function. Algebra i Analiz (St. Petersburg Math. J.) 20(5), 155–185 (2008) 89. Spiridonov, V. P.: Essays on the theory of elliptic hypergeometric functions. Usp. Mat. Nauk 63(3), 3–72 (2008) (Russ. Math. Surv. 63(3), 405–472 (2008)) 90. Spiridonov, V. P.: Elliptic hypergeometric terms. In: Proc. of the Workshop “Théories galoisiennes et arithmétiques des équations différentielles” (September 2009, CIRM, Luminy, France), to appear, http://arXiv.orb/abs/1003.4491v2 [math.CA], 2010 91. Spiridonov, V.P., Vartanov, G.S.: Superconformal indices for N = 1 theories with multiple duals. Nucl. Phys. B 824, 192–216 (2010) 92. Spiridonov, V.P., Vartanov, G.S.: Supersymmetric dualities beyond the conformal window. Phys. Rev. Lett. 105, 061603 (2010) 93. Spiridonov, V.P., Warnaar, S.O.: Inversions of integral operators and elliptic beta integrals on root systems. Adv. Math. 207, 91–132 (2006) 94. Spiridonov, V. P., Warnaar, S. O.: New multiple 6 ψ6 summation formulas and related conjectures. Preprint August (2009) 95. Sundborg, B.: The Hagedorn transition, deconfinement and N = 4 SYM theory. Nucl. Phys. B 573, 349–363 (2000) 96. Takhtadzhyan, L. A., Faddeev, L. D.: The quantum method of the inverse problem and the Heisenberg X Y Z model. Usp. Mat. Nauk 34(5), 13–63 (1979) (Russ. Math. Surv. 34(5), 11–68 (1979)) 97. Terning, J.: Modern Supersymmetry: Dynamics and Duality. Oxford, UK: Clarendon, 2006 98. Volkov, A.: Noncommutative hypergeometry. Commun. Math. Phys. 258, 257–273 (2005) 99. Witten, E.: Constraints on supersymmetry breaking. Nucl. Phys. B 202, 253–316 (1982) Communicated by A. Kapustin
Commun. Math. Phys. 304, 875–878 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1249-2
Communications in
Mathematical Physics
Erratum
Erratum to: Integral Formulas for the Asymmetric Simple Exclusion Process Craig A. Tracy1 , Harold Widom2 1 Department of Mathematics, University of California, Davis, CA 95616, USA.
E-mail: [email protected]
2 Department of Mathematics, University of California, Santa Cruz, CA 95064, USA.
E-mail: [email protected] Received: 19 March 2011 / Accepted: 23 March 2011 Published online: 14 April 2011 – © Springer-Verlag 2011
Commun. Math. Phys. 279, 815–844 (2008)
This is a correction to the proof of Theorem 2.1 of [1]. An error occurred in the “proof” of Lemma 2.2, which is false. The following should replace the proof of (c) (Sect. II, p. 821ff.), which will correct the proof of the theorem: The initial condition is satisfied by the summand in (2.3) coming from the identity permutation id. So what we have to show is xi −yσ (i) −1 ··· Aσ ξσ (i) dξ1 · · · dξ N = 0, σ =id Cr
Cr
i
or equivalently σ =id Cr
···
Cr
Aσ
x
ξi σ
−1 (i) −yi −1
dξ1 · · · dξ N = 0,
i
when x1 < · · · < x N . We write I (σ ) for the integral corresponding to σ , so that the above becomes I (σ ) = 0. (1) σ =id
For 1 ≤ n < N fix n − 1 distinct numbers i 1 , . . . , i n−1 ∈ [1, N − 1]. Define A = {i 1 , . . . , i n−1 }, The online version of the original article can be found under doi:10.1007/s00220-008-0443-3.
876
C. A. Tracy, H. Widom
and then S N (A) = {σ ∈ S N : σ (1) = i 1 , . . . , σ (n − 1) = i n−1 , σ (n) = N }. When n = 1 this consists of all permutations with N in position 1, and when n = N − 1 it consists of a single permutation. If B is the complement of A ∪ {N } in [1, N ], then σ ∈ S N (A) is determined by the restriction σ |[n+1, N ] : [n + 1, N ] → B. Lemma 2.1. For each A,
I (σ ) = 0.
σ ∈S N (A)
Start of the Proof. When σ ∈ S N (A) the inversions involving N are the (N , i) with i ∈ B. Therefore the integrands in I (σ ) involving these σ may be written i∈B
S(ξ N , ξi ) ×
x
ξi σ
−1 (i) −yi −1
×
{S(ξk , ξ ) : N > k > , σ −1 (k) < σ −1 ()}.
i≤N
The integrals are taken over Cr with r so small that all the denominators in the S-factors are nonzero on and inside the contour. In these integrals we make the substitution ξN →
η
i
,
so that η runs over a circle of radius r N . The integrand becomes p + qη =i, N ξ−1 − η = N ξ−1 N −n (−1) p + qη =i, N ξ−1 − ξi i∈B xσ −1 (i) −xn +y N −yi −1 ×η xn −y N −1 ξi
(2) (3)
i
×
{S(ξk , ξ ) : N > k > , σ −1 (k) < σ −1 ()}.
(The reason that we still have −1 in the exponents in (3) is that dξ N =
(4)
−1 i
Sublemma 2.1. When n = N − 1 we have I (σ ) = 0. Proof. There is a single i ∈ B and (2) is analytic inside the ξi -contour except for a simple pole at ξi = 0. The power of ξi in (3) is x N −x N −1 +y N −yi −1
ξi
,
and since x N > x N −1 and y N > yi , the exponent is positive. Therefore the integrand is analytic inside the ξi -contour, and so the integral is zero. Sublemma 2.2. When n < N − 1 all I (σ ) with σ ∈ S N (A) are sums of lower-order integrals in each of which (2) is replaced by a factor depending on A. The other factors remain the same. In each integral some ξi with i ∈ B is equal to another ξ j with j ∈ B.
Erratum to: Integral Formulas for the Asymmetric Simple Exclusion Process
877
Proof. We may assume that q = 0. This case follows by a limiting argument. We are going to shrink some of the ξi -contours with i ∈ B. Due to the defining property of r , the only poles we pass will come from the product (2). In fact, to avoid double poles later we take ξi ∈ Cri with the ri all slightly different. Take j = max B and shrink the ξ j -contour. The product (2) has a simple pole at ξ j = 0 (the j-factor has the pole and the i-factors with i = j are analytic there) and the power of ξ j in (3) is positive as before, so the integrand is analytic at ξ j = 0. For each k ∈ B with k = j we pass the pole at ξj =
qη
−1 = j,k,N ξ
ξk − p
,
(5)
coming from the k-factor in (2). (Our assumption on the ri assures that there are no double poles.) For the residue we replace the k-factor by −
p + qη
−1 −1 =k, N ξ − η = N ξ , −1 qηξ −2 = j,k,N ξ j
(6)
where in this and the j-factor we replace ξ j by the right side of (5). When i = j, k the i-factor becomes p + qη =i, N ξ−1 − η = N ξ−1 , p (1 − ξi ξk−1 ) and we replace ξ j in the numerator by the right side of (5). We now shrink the ξk -contour. There is a pole of order 2 at ξk = 0 coming from (6) and the j-factor in (2). Since k < j = max B < N , we have y N − yk ≥ 2, so the exponent of ξk in (3) is at least 2. Therefore the integrand is analytic at ξk = 0. The factor (6) has no other poles inside Crk . An i-factor with i = j, k will have a pole at ξk = ξi if ri < rk . There is also the pole at ξk =
qη
−1 = j,k,N ξ
ξj − p
coming from the j-factor. But this relation and (5) imply ξ j = ξk . Thus when we shrink the ξ j -contour and the ξk -contours with k = j we obtain (N − 2)-dimensional integrals in each of which two of the ξ -variables corresponding to indices in B are equal. This proves the sublemma. Sublemma 2.3. For each integral of Sublemma 2.2 there is a partition of S N (A) into pairs σ, σ such that I (σ ) + I (σ ) = 0 for each pair. Proof. Consider an integral in which ξi = ξ j . We pair σ and σ if σ −1 (i) = σ −1 ( j) and σ −1 ( j) = σ −1 (i), and σ −1 (k) = σ −1 (k) when k = i, j. The factor (3) is clearly the same for both when ξi = ξ j , and we shall show that the σ - and σ -factors in (4) are negatives of each other when ξi = ξ j .
878
C. A. Tracy, H. Widom
Assume for definiteness that i < j and σ −1 (i) < σ −1 ( j).
(7)
(Otherwise we reverse the roles of σ and σ .) Then the factor S(ξ j , ξi ) does not appear for σ in (4) but it does appear for σ . This factor equals −1 when ξi = ξ j . To complete the proof it is enough to show that for any k = i, j the product of S-factors involving k and either i or j is the same for σ and σ when ξi = ξ j . There are nine cases, depending on the position of k relative to i and j and the position of σ −1 (k) relative to σ −1 (i) and σ −1 ( j). If k is outside the interval [i, j] and σ −1 (k) is outside the interval [σ −1 (i), σ −1 ( j)] then the products of S-factors for σ and σ are clearly the same. There are five remaining cases, with the results displayed in the table below. The first column gives the position of k relative to i and j, the second column gives the position of σ −1 (k) relative to σ −1 (i) and σ −1 ( j), the third column gives the product of S-factors involving k and either i or j for σ , and the fourth column gives the corresponding product for σ . Keep (7) in mind. i < k < j σ −1 (k) < σ −1 (i) i < k < j σ −1 (i) < σ −1 (k) < σ −1 ( j) i < k < j σ −1 ( j) < σ −1 (k) k j σ −1 (i) < σ −1 (k) < σ −1 ( j)
S(ξk , ξi ) 1 S(ξ j , ξk ) S(ξi , ξk ) S(ξk , ξ j )
S(ξk , ξ j ) S(ξk , ξ j ) S(ξi , ξk ) S(ξi , ξk ) S(ξ j , ξk ) S(ξk , ξi )
In all cases but the second the S-factors are exactly the same for σ and σ when ξi = ξ j . For the second we use S(ξk , ξ ) S(ξ, ξk ) = 1. Sublemmas 2.1– 2.3 give Lemma 2.1. To prove (1) we use induction on N . When N = 2 it folows from Sublemma 2.1. Assume N > 2 and that the result holds for N − 1. For those permutations for which σ (N ) = N , we use the fact that N appears in no involution, so we may integrate with respect to ξ1 , . . . , ξ N −1 and use the induction hypothesis. The set of permutations with σ (N ) < N is the disjoint union of the various S N (A), and for these we apply Lemma 2.1. Acknowledgement. This work was supported by the National Science Foundation through grants DMS0906387 (first author) and DMS-0854934 (second author).
Reference 1. Tracy, C.A., Widom, H.: Integral Formulas for the Asymmetric Simple Exclusion Process. Commun. Math. Phys. 279, 815–844 (2008) Communicated by H. Spohn
