Commun. Math. Phys. 276, 1–21 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0338-8
Communications in
Mathematical Physics
Constraining the Kähler Moduli in the Heterotic Standard Model Tomás L. Gómez1,3 , Sergio Lukic2 , Ignacio Sols3 1 CSIC Instituto de Matemáticas y Fisica Fundamental, Madrid 28006, Spain.
E-mail:
[email protected]
2 Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA.
E-mail:
[email protected]
3 Departamento de Álgebra, Facultad de Matemáticas UCM, Madrid 28040, Spain.
E-mail:
[email protected] Received: 24 December 2005 / Accepted: 11 June 2007 Published online: 19 September 2007 – © Springer-Verlag 2007
Abstract: Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kähler moduli which give rise to realistic compactifications of the strongly coupled heterotic string. When vector bundles are constructed using extensions, we provide simple rules to determine lower and upper bounds to the region of the Kähler moduli space where such compactifications can exist. We show how small these regions can be, working out in full detail the case of the recently proposed Heterotic Standard Model. More explicitly, we exhibit Kähler classes in these regions for which the visible vector bundle is stable. On the other hand, there is no polarization for which the hidden bundle is stable. 1. Introduction Our understanding of Calabi-Yau compactifications of string/M-theory has been increased considerably during the last years. On the one hand, distributions of vacua for type IIB, IIA and type I string theory are much better understood. On the other hand, promising compactifications of the heterotic string have been found at special points of the moduli space. Although a systematic study of distributions of vacua for compactifications of the heterotic string is much harder, because our primitive understanding of their moduli stabilization and the huge amount of vector bundle moduli, we can still find systematic criteria to constrain the regions of the moduli space where realistic vacua should be located. Recently, phenomenologically interesting Calabi-Yau compactifications of the heterotic string have appeared in the literature [2, 5]. Using certain elliptically fibered threefold with fundamental group ZZ3 × ZZ3 , and an SU (4) × ZZ3 × ZZ3 instanton living on the visible E 8 -bundle, give rise to an effective field theory on IR4 which has the particle spectrum of the Minimal Supersymmetric Standard Model (MSSM), with no exotic
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T. L. Gómez, S. Lukic, I. Sols
matter but an additional pair of Higgs-Higgs conjugate superfields. In these models, vector bundles are constructed using vector bundle extensions, which correspond to Hermitian Yang-Mills connections when they are slope-stable. We use this specific construction to exemplify how a systematic selection of realistic Kähler moduli can be done.1 The organization of the paper is as follows: Sect. 2 contains an outline of the natural criteria for selecting Kähler moduli in realistic Calabi-Yau compactifications of the heterotic string. In Sect. 3, we analyze the case of the Heterotic Standard Model, describe the geometry of the elliptic Calabi-Yau and construct its Kähler cone. Section 4 provides lower and upper bounds to the region of the Kähler cone that makes stable the observable vector bundle of the HSM. In such construction we find a destabilizing sub-line bundle for the hidden vector bundle, and exhibit Kähler classes that make stable the visible one. 2. Picking Kähler Moduli The spacetime in a Calabi-Yau compactification of the strongly coupled heterotic string [15], defined through the direct product eleven-dimensional manifold Y = IR4 × X × [0, 1], with X a Calabi-Yau threefold. N = 1 supersymmetry on the four dimensional Effective Field Theory, requires to fix a G 2 -holonomy metric on X × [0, 1] plus gauge connections at the hidden and visible vector bundles, which satisfy the Hermitian Yang-Mills equations. In order to define a barely G 2 -holonomy metric on X × [0, 1] we introduce a calibration 3-form, according to D. Joyce [16], = (At + B)ω ∧ dt + Re(),
(2.1)
which depends on the differential dt along the interval and the holomorphic 3-form and Kähler class ω of the threefold. Such a calibration defines a barely G 2 -holonomy metric on X × [0, 1], where the Kähler class is linearly dilated along the interval; therefore at the visible and hidden boundaries the Kähler classes are ω0 = Bω and ω1 = (A + B)ω (i = 1 stands for the ‘hidden’ boundary and i = 0 for the ‘visible’ one). The set of Kähler classes on X is usually known as the Kähler cone and denoted by K(X ) ⊂ H 2 (X, ZZ). One approach to model building is to attach a SU (n) × G Hermitian Yang-Mills gauge connection at the boundary, to obtain an Effective Field Theory with the commutant of SU (n) × G ⊂ E 8 as gauge group while the N = 1 supersymmetry of the EFT is preserved. Here G is the non-trivial holonomy group associated to a certain flat line bundle. By the theorem of Donaldson and Uhlenbeck-Yau [9], we know that SU (n)-connections that satisfy the Hermitian Yang-Mills equations and slope-stable rank-n holomorphic vector bundles with vanishing first Chern class are in one-to-one correspondence.
2.1. Constraining angular degrees of freedom. Thus, the holomorphic vector bundles Vi → X that we fix at the hidden and visible sectors, have to be slope stable in order to get a sensible vacuum. Slope stability can impose severe constraints on the Kähler moduli. 1 Recently, Donagi and Bouchard [8] have also proposed an independent CY compactification of the heterotic string with the spectrum of the MSSM and no exotic matter, using a different Calabi-Yau with an explicitly slope-stable vector bundle in the observable sector. It would be also interesting to study in detail these questions with the vector bundle which has just appeared in [4], on the same CY [5].
Constraining the Kähler Moduli in the Heterotic Standard Model
3
If Wi → Vi is a rank-m (with m < n) holomorphic torsion free subsheaf2 , then only the ωi ∈ K(X ) that verify 1 1 ωi2 ∧ c1 (Wi ) < ω2 ∧ c1 (Vi ) = 0, (2.2) m X n X i can make Vi stable. At this point we realize that if ωi is a stablemaker for Vi , then N ωi with N ∈ ZZ+ is also a stablemaker. The stablemakers form a subcone Kis (X ) ⊆ K(X ) within the Kähler moduli, [20]. The physical importance of slope stability is clear, [9]: Non-stablemaker classes at the boundary of Kis (X ) make the vector bundle Vi semistable, i.e. we can only find correspondences to connections with reduced gauge group H ⊂ SU (n), thus the gauge dynamics of the Effective Field Theory would be governed by the commutant of H × G ⊂ E 8 instead of SU (n) × G. Usually, a detailed computation of Kis (X ) is difficult because we need to identify every subsheaf Wi of Vi . Note that if h 0 (Wi∨ ⊗ Vi ) = 0, then Wi cannot be a subsheaf of Vi , but the converse is not necessarily true. If the vector bundle Vi is constructed through a non-trivial extension, defined by a short exact sequence 0 −→ VL −→ Vi −→ V R −→ 0,
(2.3)
with Ext1 (V
s R , VL ) = 0, we can give upper and lower bounds to Ki (X ) in a simple way, looking at subsheaves of VL and V R . On the one hand, the set ULi of subsheaves of VL is a subset of the set of subsheaves of Vi , since VL → Vi is injective. This provides an upper bound for cone Kis (X ) of Kähler classes for which Vi is stable:
Kis (X )>
2 = ωi ∈ K(X ) : ωi ∧ c1 (L i ) < 0, ∀ L i ∈ ULi .
(2.4)
X
On the other hand, a subsheaf of Vi gives an element of ULi × URi , where URi is the subset of subsheaves of V R . Indeed, if Wi is a subsheaf of Vi , there is a commutative diagram 0 −→ VL −→ Vi −→ V R −→ 0 ↑ ↑ ↑ 0 −→ W L −→ Wi −→ W R −→ 0
(2.5)
where the vertical arrows are injective, hence we obtain subsheaves W L and W R of VL and V R . This gives a lower bound s < 2 Ki (X ) = ωi ∈ K(X ) : ωi ∧ (c1 (W L )+c1 (W R )) < 0, ∀ W L ∈ ULi , W R ∈ URi . X
(2.6) Note that the ones belonging to ULi are true subsheaves of Vi , and the ones in URi are possible subsheaves of Vi . Therefore, we can construct two bounds to the stablemaker Kähler subcone Kis (X ), Kis (X )< ⊆ Kis (X ) ⊆ Kis (X )> .
(2.7)
2 It is enough to consider reflexive sheaves, i.e., sheaves with W = W ∨∨ . Furthermore, we can assume i i that Wi is semistable.
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T. L. Gómez, S. Lukic, I. Sols
Sometimes we can use further information to discard some pairs (W L , W R ) which do not come from subsheaves Wi of Vi , hence obtaining a better lower bound. For instance, the pair (0, V R ) can be discarded, because it would give a splitting of the defining exact sequence (2.3), but we have assumed that the extension is not trivial, hence has no splitting. Other cases that can be discarded are pairs of the form (0, W R ) when h 0 (W R∨ ⊗ Vi ) = 0. We shall apply these ideas in the next sections to the vector bundles constructed in the Heterotic Standard Model, [2]. 2.2. Constraining radial degrees of freedom. In the last subsection we have seen how to choose rays in the Kähler cone that preserve the slope stability of a given vector bundle, and thus define a consistent gauge group in the effective field theory. On the other hand, radial degrees of freedom in Kis (X ) are related with variations of the volume of X , [11]. We are not free to choose arbitrary volumes for the threefolds at the hidden and observable sector, if we want to preserve sensible values for Newton’s constant and the E 8 gauge coupling, [22]. Using Liouville’s measure, we can estimate the volume of the Calabi-Yau threefold at the point ωi ∈ K(X ) as 3 1 Vol(X )i = ω3 , (2.8) 3! X i thus radial dilations in the Kähler cone ωi → N ωi with N ∈ ZZ+ , map the volume as Vol(X )i → N 3 Vol(X )i . The volume of the threefolds at the boundaries of Y , are related through Witten’s formula [22] ρ 1 Vol(X )1 = Vol(X )0 + 2π ω0 ∧ c2 (V0 ) − c2 (T X ) + O(ρ 2 ), (2.9) P X 2 with P the eleven dimensional Planck length and ρ the length of the M-theory interval. This formula (2.9) holds at first order in ρ, which is the limit where we work, as in (2.1). A more accurate relation between the volumes of the CYs at the boundaries, taking into account the non-linear corrections in ρ, was derived in [6] and [7]. Newton’s constant in the effective supergravity theory on the observable IR4 of Y goes as GN ∼
9P , ρVol(X )0
(2.10)
6P . Vol(X )0
(2.11)
and the E 8 gauge coupling as αGU T ∼
Witten observed in [22], that in order to find realistic values for these physical quantities, the volume of the threefold in the visible sector has to be very large. As the integral in the right-hand side of (2.9) is negative due to Chern-Weil theory, and the identity 2 Tr F ∧ ω = − |F|2 ω3 , (2.12) X
X
3 Being rigorous, we should work with the dimensionfull measure (α ω)3 , although this will be irrelevant for our purposes because α factorizes out in the formulae that we use. In the small volume limit this
approximation can fail, and we should use conformal field theory to give a more accurate estimation.
Constraining the Kähler Moduli in the Heterotic Standard Model
5
he deduced that sensible values for G N and αGU T are only possible for very small Vol(X )1 . Summarizing: Let K0s (X ) and K1s (X ) be the set of Kähler classes that make stable V0 → X and V1 → X , respectively. Physically interesting vacua should be located in rays of the Kähler cone lying in the intersection K0s (X ) ∩ K1s (X ) ⊂ K(X ), such that the relative dilating factor ω0 /ω1 is very large, and the Witten’s correlation 1 1 ρ 1 3 3 ω ∼ ω + 2π ω0 ∧ c2 (V0 ) − c2 (T X ) (2.13) 3! X 1 3! X 0 P X 2 is satisfied. Remark 1. Although the study of distributions of vacua for these models is not as developed as for Calabi-Yau compactifications of the type II string theory, the presence of vacua in these regions of the Kähler moduli space should be statistically favorable along the lines of [10], once the vector bundle, dilaton and complex moduli are stabilized. We have shown how to identify these regions explicitly. In the rest of the paper we determine them for the recently proposed Heterotic Standard Model. 3. The Elliptic Calabi-Yau and its Kähler Cone First, we briefly recall the construction of the Calabi-Yau threefold used the Heterotic Standard Model, following the reference [5]. Let X be the fiber product over IP1 of two rational elliptic surfaces X = B1 ×IP1 B2 , as in the diagram: X π1 π2 B1 ↓ π B2 β1 β2 IP1
(3.1)
This kind of Calabi-Yau threefolds was already studied by C. Schoen in S. The geometry of X is basically encoded in the geometry of the rational elliptic surfaces B1 and B2 . Due to the phenomenological interest in finding threefolds which admit certain Wilson lines4 , the aim of [5] was to look for threefolds X such that ZZ3 × ZZ3 ⊆ Aut( X ). This search was achieved thanks to the existence of certain elliptic surfaces that admit an action of ZZ3 × ZZ3 which can be characterized explicitly through a proper understanding of the Mordell-Weil group of B. Following Kodaira’s classification of singular fibers, our elliptic surfaces B1 and B2 are characterized by three I1 and three I3 singular fibers. Such rational elliptic surfaces are described by one-dimensional families that allow us to build fiber products X , corresponding to smooth Calabi-Yau threefolds. Furthermore, X admits a free action of G = ZZ3 × ZZ3 and the quotient X = X /G is also a smooth Calabi-Yau threefold with fundamental group π1 (X ) = ZZ3 × ZZ3 . Let g1 and g2 be generators of this group, g2 acting as translation tη by a section η of the fibration β : B → IP1 , and g1 acting nontrivially on the base. Let ξ = g1 (σ ) and α B = t−ξ ◦ g1 . The threefold used in the description of the Heterotic Standard Model is X = X /G, although we will work with G-equivariant objects on X . In the rest of this section 4 I.e. flat line bundles with non-trivial holonomy.
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we describe the G-invariant homology rings of B and X , and their corresponding G-invariant Kähler cones (i.e. their ample cones, or spaces of polarizations). For the homology of a surface B, we choose a set of generators: the 0-section σ , the generic fiber F, the 6 irreducible components of the three I3 singular fibers that do not intersect the 0-section 1,1 , 1,2 , . . . , 3,1 , 3,2 , and the two sections generating the free part of the Mordell-Weil group5 ξ and α B ξ . These generators are a basis for H2 (B, ZZ) ⊗ Q, but adding the torsion generator of the Mordell-Weil group η=σ+F−
1 2
1,1 + 2,1 + 3,1 −
1,2 + 2,2 + 3,2 , 3 3
(3.2)
we generate all H2 (B, ZZ). The intersection matrix of the homology generators is as follows: ⎛ σ ⎞T ⎜ F ⎟ ⎟ ⎜ ⎜ 1,1 ⎟ ⎟ ⎜ ⎜ 2,1 ⎟ ⎟ ⎜ ⎜ 3,1 ⎟ ⎟ ⎜ ⎜ 1,2 ⎟ ⎟ ⎜ ⎜ 2,2 ⎟ ⎟ ⎜ ⎜ 3,2 ⎟ ⎟ ⎜ ⎜ ξ ⎟ ⎠ ⎝ αB ξ η
⎛ σ ⎞ ⎛ −1 ⎜ F ⎟ ⎜ 1 ⎟ ⎜ ⎜ ⎜ 1,1 ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎜ 2,1 ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎜ 3,1 ⎟ ⎜ 0 ⎟ ⎜ ⎜ · ⎜ 1,2 ⎟ = ⎜ 0 ⎟ ⎜ ⎜ ⎜ 2,2 ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎜ 3,2 ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎜ ξ ⎟ ⎜ 0 ⎠ ⎝ ⎝ 0 αB ξ 0 η
1 0 0 0 0 0 0 0 0 0 ⎞ 0 0 0 0 0 0 0 1 1 1 ⎟ ⎟ 0 −2 0 0 1 0 0 0 0 1 ⎟ ⎟ 0 0 −2 0 0 1 0 1 0 1 ⎟ ⎟ 0 0 0 −2 0 0 1 0 1 1 ⎟ ⎟ 0 1 0 0 −2 0 0 0 1 0 ⎟ . (3.3) ⎟ 0 0 1 0 0 −2 0 0 0 0 ⎟ ⎟ 0 0 0 1 0 0 −2 1 0 0 ⎟ ⎟ 1 0 1 0 0 0 1 −1 1 0 ⎟ ⎠ 1 0 0 1 1 0 0 1 −1 0 1 1 1 1 0 0 0 0 0 −1
The invariant homology under the action of G = ZZ3 × ZZ3 , is generated by H2 (B, ZZ)G = spanZZ F, t = −σ + 2,1 + 3,1 + 3,2 + 2ξ + α B ξ + η − F , (3.4) where t can be also expressed as the homological sum of three sections, i.e. t = ξ + αB ξ + η ξ . The cohomology ring of X can be expressed as H ∗ (X, Q) = H ∗ ( X , Q)G ,
(3.5)
using the G-invariant cohomology of X . Hence G 2 H (B1 , Q) ⊕ H 2 (B2 , Q) H 2 (B1 , Q)G ⊕ H 2 (B2 , Q)G G = H ( X , Q) = , (3.6) H 2 (IP1 , Q) H 2 (IP1 , Q) 2
that due to (3.4), is the same as X , Q)G H 2 (X, Q) = H 2 ( = spanQ τ1 = π1∗ (t1 ), τ2 = π2∗ (t2 ), φ = π1∗ (F1 ) = π2∗ (F1 ) ,
(3.7)
where t1 and t2 (respectively, F1 and F2 ) are the t-classes (respectively, F-classes) defined in (3.4), corresponding to each surface B1 and B2 . Using Poincaré duality, we know that H 4 (X, Q) is isomorphic to H 2 (X, Q), also H1 (X, ZZ) π1 (X ) = ZZ3 × ZZ3 because the Hurewicz theorem, thus H 1 (X, Q) = H 1 (X, ZZ) ⊗ Q = 0. 5 See Appendix A, for a complete description of the Mordell-Weil group of the elliptic surface.
Constraining the Kähler Moduli in the Heterotic Standard Model
7
The ring H ev ( X , Q)G is generated through the cup product of the generators (3.7), and is isomorphic to H ev ( X , Q)G = Q[τ1 , τ2 , φ]/φ 2 , φτ1 = 3τ12 , φτ2 = 3τ22 ,
(3.8)
with the top cohomology element being τ12 τ2 = τ1 τ22 = 3{pt.}. 3.1. The ample cone of the elliptic surface. As first step to determine the Kähler cone on the threefold, we build the G-invariant ample cone of the rational elliptic surface through Nakai’s criterion. The set of ample classes is by definition the integral cohomology part of the Kähler moduli. Using Looijenga’s classification of the effective curves in a rational elliptic surface [17], we know that the cone of effective classes in H2 (B, ZZ) is generated by the following classes e ∈ H2 (B, ZZ): 1) The exceptional curves e2 := −1, i.e. every section of the elliptic fibration. 2) The nodal curves e2 := −2, i.e. the irreducible components of the singular fibers. 3) The positive classes, i.e. the classes that live in the “future” side of the cone of e2 > 0. Nakai’s criterion for surfaces says that a class s is ample if and only if s · s > 0 and e · s > 0 for every effective curve e. We will apply this criterion to the invariant classes s = a F + bt. • Intersection of s with the exceptional curves. Although there is an infinite amount of exceptional curves or sections in the elliptic surface, we can characterize them completely thanks to our understanding of the Mordell-Weil group. As it is explained in Appendix A, the representation of the Mordell-Weil group E(K ) ZZ ⊕ ZZ ⊕ ZZ3 in End(H2 (B, ZZ)) has as generators: (tξ )∗ , (tα B ξ )∗ and (tη )∗ . Thus, the homology of an arbitrary section can be expressed as y (3.9) xξ yα B ξ zη = (tξ )∗x (tα B ξ )∗ (tη )∗z σ, where xξ (respectively yα B ξ and zη) means xξ = ξ ξ . . . ξ . x
Finding the Jordan canonical forms associated to (tξ )∗ , (tα B ξ )∗ and (tη )∗ , allows us to expand (3.9), explicitly. We exhibit the list of homology classes associated to the sections in Appendix A. Hence, the intersections of the exceptional curves with the generators of the invariant homology are F · [xξ yα B ξ zη] = 1
(3.10)
t · [xξ yα B ξ zη] = x 2 + y 2 − x y − x.
(3.11)
and
It is easy to check that x 2 + y 2 − x y − x as a function ZZ ⊕ ZZ → ZZ is non-negative and becomes zero for (x = 0, y = 0), (x = 1, y = 0) and (x = 1, y = 1). Therefore a G-invariant ample class s = a F + bt has to verify s · [0ξ 0α B ξ zη] = a > 0,
(3.12)
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T. L. Gómez, S. Lukic, I. Sols
and s · [∞ξ ∞α B ξ zη] = a + ∞b > 0, ⇒ b > 0.
(3.13)
• Intersection of s with the nodal curves. The nodal curves are identified with the irreducible components i, j of the singular fibers, thus their intersections with the invariant class s = a F + bt give us s · i, j = b > 0,
(3.14)
an identical result to the inequality (3.13), derived above. • Intersection of s with the positive classes. Let K+ (B) be the cone of positive classes in B, i.e. K+ (B) = {e ∈ H2 (B, ZZ)| e · e > 0}. As K+ (B) is a convex set and we have to take intersections of elements in K+ (B) with invariant classes in H2 (B, ZZ)G , only the intersection K+ (B) ∩ H2 (B, ZZ)G matters. From the intersection matrix of the homology generators, we know that the intersection matrix of the invariant homology H2 (B, ZZ)G is
F t
T F 0 3 · = t 3 1
(3.15)
hence, we find K+ (B) ∩ H2 (B, ZZ)G := e = x F + yt| 6x y + y 2 > 0 ,
(3.16)
being the edges of such a “future” cone F and 6t − F. Furthermore, their intersections with our ample candidate s = a F + bt give us the conditions s · F = (a F + bt) · F = 3b > 0, s · (6t − F) = 18a + 6b − 3b = 18a + 3b > 0,
(3.17)
that do not constrain the inequalities (3.12), and (3.13). Finally, as the cone generated by F and t is within K+ (B) ∩ H2 (B, ZZ)G , the last Nakai condition s · s > 0 or positivity of the Liouville measure is verified. Therefore, the G-invariant ample cone associated to the elliptic surface B is simply K(B)G = spanZZ+ {F, t} .
(3.18)
3.2. Ampleness in the threefold. Once we have characterized the G-invariant ample cone on the rational surface, we can construct G-invariant ample classes on the threefold X as a product of ample classes on the surfaces B1 and B2 . In fact, the following proposition shows that the ample classes on X constructed in this way determine explicitly its G-invariant ample cone K( X )G = K(X ). Proposition 3.1. The G-invariant ample cone of X is K( X )G = spanZZ+ {τ1 , τ2 , φ} .
(3.19)
Constraining the Kähler Moduli in the Heterotic Standard Model
9
Proof. If L i is an ample class in Bi , then π1∗ L 1 ⊗ π2∗ L 2 is an ample class in X , hence K( X )G contains the positive linear span of τ1 , τ2 and φ. To show the opposite inclusion, we apply Nakai’s criterion to some effective classes. Let H = aτ1 + bτ2 + cφ be an ample class. If C1 is the class of a fiber of π1 , 0 < H · C1 = 0a + 3b + 0c = 3b. Analogously, if C2 is the class of a fiber of π2 , we obtain a > 0. Let i : B1 ×IP1 B2 → B1 × B2 . Let C be the class of σ1 ×IP1 σ2 , let c1 , c2 be two integers with c = c1 + c2 , and denote [Bi ] (respectively, [pt]) the class of Bi in H 0 (Bi , ZZ) (respectively, of a point in H 4 (Bi , ZZ)), 0 < H · C = i ∗ ((at1 + c1 f 1 ) ⊗ [B2 ] + [B1 ] ⊗ (bt2 + c2 f 2 )) · i ∗ [σ1 ⊗ σ2 ] = i ∗ ((at1 + c1 f 1 )σ1 [pt] ⊗ σ2 + σ1 ⊗ [pt] (bt2 + c2 f 2 )σ2 ) = i ∗ (c1 [pt] ⊗ σ2 + c2 σ1 ⊗ [pt]) (3.20) = c1 + c2 = c. 4. Slope Stability of the Vector Bundles The concept of (slope) stability of a vector bundle depends on the choice of a polarization H ∈ K(X ) ⊂ H 2 (X, ZZ), i.e., we say that a holomorphic vector bundle E → X is stable iff H 2 · det(·) , (4.1) µ(F) < µ(E) ; with µ(·) = rank (·) for every reflexive subsheaf F → E. By det(E) and det(F) we mean the determinant line bundles associated to E and F. There is a natural bijection between vector bundles on X and G-equivariant vector bundles on X . We will recall a few general remarks on G-invariance and G-equivariance, which will be useful in the rest of this section. Let X be a complex projective variety and G a complex algebraic group acting on it. A subvariety X of X is said to be invariant if g X = X for all g in G. A divisor D is said to be invariant if gD = D for all g in G. A divisor class is said invariant to be if for any divisor D in the class and g and in G, the divisor gD is linearly equivalent to D. An equivariant structure on a vector bundle E on X is a lifting by linear maps E(x) −→ E(gx) (for all g ∈ G) between fibers, of the action of G on X . We will widely use this notion, and sometimes also the notion of equivariant coherent sheaf (we will talk about some equivariant ideal sheaf) so it is convenient to generalize it defining an equivariant structure on a coherent sheaf F on X as a family of isomorphisms ϕgF : F∼ = g ∗ F , for each g ∈ G, so that ϕgF g = ϕgF ϕgF . Equivariant morphisms f : F −→ F , between equivariant sheaves are those such that ϕgF F −→ g ∗ F f ↓ g∗ f ↓
F −→ g ∗ F
ϕgF for all g ∈ G.
(4.2)
(4.3)
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If two vector bundles have an equivariant structure, obviously their tensor products inherit an equivariant structure. If a vector bundle E has an equivariant structure, all of its exterior powers, and in particular its determinant line bundle, det (E), inherit an equivariant structure, and also its dual E ∗ (pointwise, take the inverse of the transposed action). The trivial bundle L = X × C, or O X as an associated sheaf, admits a trivial equivariant structure. A vector bundle with equivariant structure is always invariant, which means, by definition, that g ∗ E is isomorphic to E for any g in G . In the case E is a vector bundle L of rank 1, this definition means that both g ∗ L and L define the same point of Pic(X ), i.e. that the point corresponding to E in Pic(X ) is fixed by the action of the group, or still, in terms of associated divisors, that the corresponding divisor class is invariant. A vector subbundle E ⊂ E of an equivariantly structured bundle E is called an equivariant vector subbundle when g(E (x)) ⊂ E (x) for all x in X and g in G. This is equivalent to say that, for all g ∈ G, the isomorphism E ∼ = g ∗ E given by the equivariant structure applies E into g ∗ E , so this notion still has a meaning when E is just a coherent subsheaf. An equivariant coherent subsheaf E obviously inherits a structure of the equivariant coherent sheaf, as well as its quotient E
= E/E , and we just say that the extension 0 −→ E −→ E −→ E
−→ 0,
(4.4)
is equivariant. An equivariant vector bundle is said to be equivariantly stable if all its equivariant coherent sheaves (enough to check with reflexive) have smaller slope. A section s of equivariantly structured E is called equivariant when, for all x in X and g in G, it is g(s(x)) = s(gx) . When viewing the section, as usual, as a subbundle O X → E, this amounts to say that the subbundle is equivariant and the inherited equivariant structure on the trivial bundle is the trivial equivariant structure. Clearly, the vanishing locus V (s) of an equivariant section is invariant. If the vector bundle E is a line bundle L of rank one, and s is a meromorphic equivariant section of it, i.e. equivariant section defined on a Zariski open set, the divisor it defines is an invariant divisor (not only a divisor of invariant class). We say L is equivariantly effective if it has a nonzero equivariant global (i.e. holomorphic) section. In a surface X , a line bundle L = O X (D) is equivariantly ample when it is equivariant and has positive selfinterseccion, and its itersection number with all equivariantly effective equivariant line bundles is positive. Therefore, ample and equivariant implies equivariantly ample. 4.1. Conditions on the effective divisors. This is an analysis previous to the solution of both problems. We show now that if there exists an effective divisor in the invariant class O B (at + bF) on the elliptic surface, then a ≥ −3b. We start with the following: Remark 2. Denote a the defect quotient a =
a 3
.
(4.5)
Recall that t is the homology sum of three sections, namely ξ , α B ξ and η ξ , which we denote, respectively, s1 , s2 and s3 . The 3a summands in at = as1 + as2 + as3
(4.6)
Constraining the Kähler Moduli in the Heterotic Standard Model
11
can be ordered
, at = s1 + · · · + s3a
(4.7)
so to fullfill the following three conditions: • For all index i such that si = s1 , {s j | j ≤ i and s j = s2 } − { j | j ≤ i and s j = s1 } ≤ a .
(4.8)
• For all index i such that si = s3 , {s j | j ≤ i and s j = s2 } − { j | j ≤ i and s j = s3 } ≤ a .
(4.9)
• For all index i such that si = s2 , {s j | j ≤ i and s j = s1 or s3 } − { j | j ≤ i and s j = s2 } ≤ a . (4.10) Indeed, the following ordering of the 3a summands satisfies the three conditions: take its first 3a summands to be (s1 + s2 + s3 ) + · · · + (s1 + s2 + s3 ).
(4.11)
Next, add summands of the alternating form (s1 + s2 ) + (s3 + s2 ) + (s1 + s2 ) + (s3 + s2 ) + · · ·
(4.12)
(so s has already ocurred a times) and add finally summands s1 , s3 , in no matter which order, until completing a ocurrences of each. The consequence of this remark is the following Lemma 1. For any direct factor OIP1 (l) occurring in the splitting of β∗ O B (at) it is l ≤ a := a3 , i.e. h 0 (β∗ O B (at)(−a − 1)) = 0. Proof. Recall β∗ O B = OIP1 . Order the 3a summands in
, at = s1 + · · · + s3a
(4.13)
as in the former remark. For some index 1 ≤ i < 3a, assume it is already proved that
)(−a − 1)) = 0. h 0 (β∗ O B (s1 + · · · + si−1
(4.14)
It is then enough to prove that h 0 (β∗ O B (s1 + · · · + si )(−a − 1)) = 0.
(4.15)
From
) −→ O B (s1 +· · · + si ) −→ Os1 (s1 +· · · + si ) −→ 0, (4.16) 0 −→ O B (s1 +· · ·+si−1
we obtain
) −→ β∗ O B (s1 +· · ·+si ) −→ OIP1 ((s1 + · · · + si )s1 ) −→ 0. 0 −→ β∗ O B (s1 + · · · + si−1 (4.17)
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T. L. Gómez, S. Lukic, I. Sols
Assume first that si = s1 . Recalling that s12 = −1, s1 s3 = 0, s1 s2 = 1, we have (s1 + · · · + si )s1 = { j | j ≤ i and s j = s2 } − { j | j ≤ i and s j = s1 } ≤ a , (4.18) proving, by consulting the former exact sequence, the wanted vanishing. The vanishing is analogously proved in the case si = s3 . Assume now that si = s2 . Recalling that s22 = −1, s2 s1 = s2 s3 = 1, we have (s1 +· · · + si )s2 = { j | j ≤ i and s j = s1 or s3 }−{ j | j ≤ i and s j = s2 } ≤ a , (4.19)
thus proving also the wanted vanishing.
Corollary. If H 0 (B, O B (at + bF)) = 0,
(4.20)
0 = H 0 (B, O B (at + bF)) = H 0 (IP1 , β∗ O B (at + bF)) = H 0 (IP1 , OIP1 (b) ⊗ β∗ O B (at)),
(4.21)
then a ≥ −3b. Proof. We assume
where β∗ O B (at) is a direct sum of factors OIP1 (l) with l ≤ a/3, by the lemma. Therefore, for some of these factors, we obtain a (4.22) 0 ≤ b+l ≤ b+ . 3 Let be the inverse image of Z under the second projection. 1 Lemma 2. a) If H 0 ( X , O X (a1 τ1 + a2 τ2 + bφ)) = 0, then a1 , a2 ≥ 0 and b ≥ − 3 (a1 + a2 ). b) If H 0 ( X , I (a1 τ1 + bφ)) = 0, then a1 ≥ 0, b ≥ − 13 a1 + 3.
Proof. a) If ai were negative, then the restriction of this section to any elliptic fibre E i of π2 would be O Ei −→ O Ei (a1 ( p1 + p2 + p3 )),
(4.23)
and this is impossible. On the other hand, 0 ∗ H 0 O X (a1 τ1 + a2 τ2 + bφ) = H O B1 (a1 t1 + bF1 ) ⊗ π1∗ π2 O B2 (a2 t2 ) = H 0 (O B1 (a1 t1 + bF1 ) ⊗ β1∗ β2∗ O B2 (a2 t2 )) = H 0 (β1∗ O B1 (a1 t1 ) ⊗ OIP1 (b) ⊗ β2∗ O B2 (a2 t2 )) = H 0( OIP1 (l1i ) ⊗ OIP1 (b) ⊗ OIP1 (l2 j )). i
a1
(4.24)
j
In these sums l1i ≤ a1 := 3 and l2 j ≤ a2 := a32 , because of the former lemma, so if this is nonzero, then for some direct factors OIP1 (l1 ) and OIP1 (l2 ) appearing in the decomposition it is a2 a1 0 ≤ l1 + b + l2 ≤ +b+ (4.25) 3 3
Constraining the Kähler Moduli in the Heterotic Standard Model
13
b) Remark that π1∗ π2∗ I = β1∗ β2∗ I = β1∗ OIP1 (−3) = O X (−3φ).
(4.26)
since β2∗ I = OIP1 (− p1 − p2 − p3 ) ∼ = OIP1 (−3) (because Z lies in the fibers of β2 at three different points p1 , p2 , p3 ∈ IP1 ). Therefore X , I (a1 τ1 + bφ)) = H 0 (B1 , π1∗ π2∗ I ⊗ O B1 (a1 τ1 + bφ)) 0 = H 0 ( = H 0 (B1 , O B1 (a1 τ1 + (b − 3)φ)), and we conclude using the previous corollary.
(4.27)
4.2. The hidden bundle. Let H be a rank-2 subbundle of the vector bundle Eh → X , adjoint representation of the hidden E 8 gauge group, defined through the short exact sequence 0 −→ O X (2τ1 + τ2 − φ) −→ H −→ O X (−2τ1 − τ2 + φ) −→ 0.
(4.28)
By construction of the extension, the determinant line bundle associated to H is trivial, thus the slope of the rank-2 vector bundle is µ(H) = 0. On the other hand, O X (2τ1 + τ2 − φ) admits a morphism to H as it is shown in the diagram (4.28), therefore given a polarization H = O Z+ , we have X (xτ1 + yτ2 + zφ) with x, y, z ∈ Z 2 2 2 µ O X (2τ1 +τ2 − φ) = H · O X (2τ1 +τ2 −φ) = 3(x +2y + 6x z +12yz) > 0, (4.29) that is positive for all H ∈ K(X ), thus µ O X (2τ1 + τ2 − φ) > µ(H), which means that O X (2τ1 + τ2 − φ) is a destabilizing line bundle for H. As H is not stable, we cannot integrate the hermitian Yang-Mills equations in order to construct an SU (2)-instanton on H. We must substitute H in order to find a sensible vacuum for the heterotic string. 4.3. The visible bundle. Here we recall the construction of the visible bundle, [2]. First it is defined as an equivariant rank 2 vector bundle V2 on B of trivial determinant given as a nontrivial extension 0 −→ O B (−2F) −→ V2 −→ I Z (2F) −→ 0,
(4.30)
with Z the scheme of 9 points, together with an equivariant structure on V2 so that this extension is equivariant ∗ 0 −→ O X (−2φ) −→ π2 V2 −→ I (2φ) −→ 0.
(4.31)
Recall that is the lifting to X of Z by the second projection. Then the visible rank 4 vector bundle V4 of the trivial determinant is defined through the extension 0 −→ O(−τ1 + τ2 ) ⊕ O(−τ1 + τ2 ) −→ V4 −→ V2 (τ1 − τ2 ) −→ 0,
(4.32)
together with an equivariant structure making this extension equivariant, and general among such extensions. We will show there exists some equivariant line bundle O X (x 1 τ1 + x 2 τ2 + yφ), thus of corresponding class of divisors H being invariant, i.e. H = x1 τ1 + x2 τ2 + yφ, such
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T. L. Gómez, S. Lukic, I. Sols
that the integers x, y, z are positive (thus O X (x 1 τ1 + x 2 τ2 + yφ) equivariantly ample) and making the equivariant bundle V4 equivariantly stable. The degree of a line bundle O X (a1 τ1 + a2 τ2 + bφ), with respect to the polarization H is H 2 (a1 τ1 + a2 τ2 + b f ) = 3(x1 + x2 + 6y)(a1 x2 + a2 x1 ) + x1 x2 (3a1 + 3a2 + 18b)
(4.33) = 3x2 (2x1 + x2 + 6y)a1 + 3x1 (x1 + 2x2 + 6y)a2 + 6x1 x2 b.
Clearly, this degree function is strictly monotonous with respect to the obvious partial ordering among these line bundles or triples of integers (a1 , a2 , b). Now we will list all possible subsheaves. 1). Possible line subbundles. For this we see the first necessary conditions on a1 , a2 , b for π2∗ V2 to admit O X (a1 τ1 + a2 τ2 + bφ) as an equivariant line subbundle 0 −→ O X (−2φ) −→
π2∗ V2 −→ I (2φ) −→ 0. (4.34) ↑ O (a τ + a τ + bφ) 2 2 X 1 1
If a1 ≤ 0 and a2 ≤ 0 and b ≤ −2 − 13 (a1 + a2 ) is not fulfilled, then the intersection of this subbundle with the one on the left must be null, so O X (a1 τ1 +a2 τ2 + bφ) becomes an equivariant subsheaf of the one on the right, thus giving an equivariant nonzero section of O X (−a1 τ1 − a2 τ2 + bφ) vanishing at . We thus get possibilities 1 i) a1 ≤ 0 and a2 ≤ −1 and b ≤ 2 − (a1 + a2 ), 3 1 ii) a1 ≤ 0 and a2 = 0 and b ≤ −1 − a1 , 3 1 iii) a1 ≤ 0 and a2 ≤ 0 and b ≤ −2 − (a1 + a2 ). 3
(4.35)
For ii) we have used Lemma 2 b). Let us find now necessary conditions for the existence of an equivariant rank 1 reflexive sheaf, i.e. equivariant subbundle O X (a1 τ1 +a2 τ2 +bφ), of V4 : −→ π2∗ V2 (τ1 − τ2 ) −→ 0. 0 −→ O X (−τ1 + τ2 ) ⊕ O X (−τ1 + τ2 ) −→ V4 (4.36) ↑ O X (a1 τ1 + a2 τ2 + bφ) By the same argument as above, combined with our former discusion on equivariant line subbundles of π2∗ V2 , we obtain these possibilities: 1 i.1) a1 ≤ −1 and a2 ≤ 1 and b ≤ − (a1 + a2 ), 3 1 i.2) a1 ≤ 1 and a2 ≤ −2 and b ≤ 2 − (a1 + a2 ), 3 2 1 i.3) a1 ≤ 1 and a2 = −1 and b ≤ − − a1 , 3 3 1 i.4) a1 ≤ 1 and a2 ≤ −1 and b ≤ −2 − (a1 + a2 ). 3 2). Possible reflexive sheaves of rank 2.
(4.37)
Constraining the Kähler Moduli in the Heterotic Standard Model
15
Let us consider now an equivariant reflexive subsheaf of rank 2 ∗ 0 −→ O X (−τ1 + τ2 ) ⊕ O X (−τ1 + τ2 ) −→ V4 −→ π2 V2 (τ1 − τ2 ) −→ 0 (4.38) ↑ R2
having nonnegative degree. Since all of its equivariant line subbundles, as equivariant subbundles of V4 , must have, as seen, negative degree, the reflexive sheaf R2 is equivariantly semistable. If its intersection with the subbundle of V4 in its above presentation were not zero, then there would be a nonzero equivariant morphism R2 −→ O X (−τ1 + τ2 ) ⊕ O X (−τ1 + τ2 ) ,
(4.39)
between both equivariantly semistable sheaves, so that the first should have slope not bigger than the slope of the second, i.e. R2 should have degree not bigger than the degree of the direct sum, which is negative (as seen in the former step). We thus obtain an injection 0 −→ R2 −→ V2 (τ1 − τ2 ) −→ Q −→ 0,
(4.40)
between these equivariant reflexive sheaves of rank 2, thus its quotient Q is a torsion sheaf. We thus obtain a nonzero equivariant morphism O X (a1 τ1 + a2 τ2 + bφ) =
2
R2 −→
2
V2 (τ1 − τ2 ) = O X (2τ1 − 2τ2 ). (4.41)
Therefore, necessarily 1 ii.1) a1 ≤ 2 and a2 ≤ −2 and b ≤ − (a1 + a2 ). (4.42) 3 The top case a1 = 2 and a2 = −2 and b = 0, would give a contradiction to what we want to prove, if it occurred, since no polarization of X giving negative degree to O(−τ1 + τ2 ) ⊕ O(−τ1 + τ2 ) would give negative degree to O X (2τ1 − 2τ2 ), but fortunately it does not occur. Indeed, if this were the case, then the quotient Q would be supported in codimension at least two, but this is incompatible with the kernel R2 of such a quotient being reflexive, unless Q = 0, i.e. R2 ∼ = V2 (τ1 − τ2 ), thus splitting the sequence presenting V4 . This would contradict the genericity of the extension taken in its presentation. Therefore, we get three subcases: 1 ii.1.a) a1 ≤ 1 and a2 ≤ −2 and b ≤ − (a1 + a2 ), 3 1 ii.1.b) a1 ≤ 2 and a2 ≤ −3 and b ≤ − (a1 + a2 ), 3 1 ii.1.c) a1 ≤ 2 and a2 ≤ −2 and b ≤ −1 − (a1 + a2 ). 3
(4.43)
3). Possible rank 3 equivariant reflexive sheaves. We can consider these equivariant subsheaves saturated, i.e. having as a quotient a rank 1 torsion free sheaf, with a line bundle O X (a1 τ1 +a2 τ2 +bφ) as dual. In other words, giving such a subsheaf is equivalent to giving an equivariant line subbundle as in the diagram −→ O 0 −→ π2∗ V2 (−τ1 + τ2 ) −→ V4∨ X (τ1 − τ2 ) ⊕ O X (τ1 − τ2 ) −→ 0. (4.44) ↑ O (a τ + a τ + bφ) 1 1 2 2 X
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T. L. Gómez, S. Lukic, I. Sols
Here we have used that V2∨ ∼ = V2 , since it is a rank two bundle of trivial determinant. Since V4∨ has zero degree for any polarization, all we must show is that the equivariant line subbundle O X (a1 τ1 + a2 τ2 + bφ) has negative degree for the polarization we are considering. If the compositions O X (a1 τ1 + a2 τ2 + bφ) −→ O X (τ1 − τ2 ),
(4.45)
with each of the two direct factors on the right-hand side were both null, then we would have a nonzero equivariant morphism ∗ O X (a1 τ1 + a2 τ2 + bφ) −→ π2 V2 (−τ1 + τ2 ),
(4.46)
and these morphisms have already been analyzed in step one. Therefore, in our situation we are necessarily in one of the following cases: 1 iii.1) a1 ≤ 1 and a2 ≤ −1 and b ≤ − (a1 + a2 ), 3 1 iii.2) a1 ≤ −1 and a2 ≤ 0 and b ≤ 2 − (a1 + a2 ), 3 (4.47) 4 1 iii.3) a1 ≤ −1 and a2 = 1 and b ≤ − − a1 , 3 3 1 iii.4) a1 ≤ −1 and a2 ≤ 1 and b ≤ −2 − (a1 + a2 ). 3 In case iii.1), the top instance (a1 = 1 and a2 = −1 and b = 0) would provide an essential contradiction to what we want, if it ocurred, since no polarization giving O X (τ1 − τ2 ) negative degree could also give negative degree to the bundle O X (−τ1 + τ2 ) ⊕ O (−τ + τ ) in the presentation of V . Fortunately, this instance does not occur. 1 2 4 X Indeed, in such a case the above morphism O (a τ + a τ + bφ) → O (τ − τ ) would 1 1 2 2 1 2 X X be isomorphic, thus splitting the bottom sequence presenting V3 in the diagram 0 −→ O X (−τ1 + τ2 ) ⊕ O X (−τ1 + τ2 ) −→ V4 −→ V2 (τ1 − τ2 ) −→ 0, (4.48) inclusion ↑ of one summand ↑ ↑ id. 0 −→ O (−τ + τ ) −→ V −→ V (τ − τ ) −→ 0, 1 2 3 2 1 2 X in contradiction with the fact that the extension presenting V4 has been taken to be general, with both of its components in the decomposition Ext 1 (V2 (τ1 − τ2 ), O X (−τ1 + τ2 ) ⊕ O X (−τ1 + τ2 )) 1 1 = Ext (V2 (τ1 − τ2 ), O X (−τ1 + τ2 )) ⊕ Ext (V2 (τ1 − τ2 ), O X (−τ1 + τ2 ))
(4.49)
being nonzero. Therefore, the first case splits into three subcases: 1 iii.1.a) a1 ≤ 0 and a2 ≤ −1 and b ≤ − (a1 + a2 ), 3 1 iii.1.b) a1 ≤ 1 and a2 ≤ −2 and b ≤ − (a1 + a2 ), (4.50) 3 1 iii.1.c) a1 ≤ 1 and a2 ≤ −1 and b ≤ −1 − (a1 + a2 ). 3 Summing up, the vector bundle V4 will then be stable if all the subsheaves that we have listed have negative degree. Recall that the degree d(x1 , x2 , y, a1 , a2 , b) is monotonous in a1 , a2 and b, so in each case it is enough to check that it is negative when these numbers take the maximum possible value. Therefore, we get the following sufficient conditions for a polarization to make V4 stable:
Constraining the Kähler Moduli in the Heterotic Standard Model
17
Fig. 1. Polarizations which make V4 stable
Proposition 1. The vector bundle V4 is equivariantly stable for any polarization O X (x 1 , x 2 , y) admitting equivariant structure (for instance, x 1 , x 2 multiple of 3) and making the number d(x1 , x2 , y, a1 , a2 , b) := 3(x1 + x2 +6y)(a1 x2 + a2 x1 )+x1 x2 (3a1 + 3a2 +18b) (4.51) negative for the following triples (a1 , a2 , b) of integers i.1) (−1, 1, 0), i.2) (1, −2, 7/3), i.3) (1, −1, −1), ii.1.b) (2, −3, 0), ii.1.c) (2, −2, −1), iii.1.a) (0, −1, 0), iii.2) (−1, 0, 5/3).
(4.52)
Remark. We have removed some cases which are redundant. For instance, case i.4) corresponds to the point (1, −1, −2), but this case is automatic once case i.3), corresponding to (1, −1, −1), has been checked, since the degree function is monotonous in a1 , a2 and b. Using the proposition, it is easy to find examples of ample sheaves which make V4 stable. For instance, O X (18τ1 + 21τ2 + 49φ). In Figure 1 we have ploted the region of ample bundles which satisfy the conditions of Proposition 1, and hence make stable the vector bundle V4 . Acknowledgements. We would like to thank E-D. Diaconescu, R. Donagi, A. Krause, G. Moore and G. Torroba for useful discussions and correspondence, specially to M. E. Alolnso for help with computer algebra programs (Maple and Singular) and V. Braun for pointing out a computational mistake in a previous version of this paper; fixing the mistake led us to a more explicit form for Proposition 1. S.L. thanks the UCM Department of Algebra for hospitality during his visit.
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T. L. Gómez, S. Lukic, I. Sols
Appendix A. Action of the Mordell-Weil Group on the Homology The Mordell-Weil group E(K ), is defined adding sections fiberwise thanks to the group structure of an elliptic curve, once the zero section is fixed. More rigorously, we define E(K ) in terms of the short exact sequence 0 −→ T −→ H2 (B, ZZ) −→ E(K ) −→ 0
(A.1)
for a certain subgroup T in H2 (B, ZZ). For our elliptic surface, we know that the Mordell-Weil group is isomorphic to ZZ ⊕ ZZ ⊕ ZZ3 and is generated by the sections ξ , α B ξ and η, thus we can express every section as xξ yα B ξ zη for x, y ∈ ZZ and z ∈ ZZ3
(A.2)
with xξ (respectively yα B ξ and zη) meaning xξ = ξ ξ . . . ξ . x
Therefore, if ta : B → B is the Mordell-Weil action of translating by the section a, we have to determine the push forwards (tξ )∗ , (tα B ξ )∗ , (tη )∗ as maps H2 (B) → H2 (B), in order to express the homology class of an arbitrary section as y
[xξ yα B ξ zη] = (tξ )∗x · (tα B ξ )∗ · (tη )∗z σ
(A.3)
with σ the zero section. The push forwards (tξ )∗ , (tη )∗ and (α B )∗ were already determined in [5], using the quotient structure of the Mordell-Weil group on H2 (B, ZZ) and computing intersection numbers with sections. Here, we state their result, and derive (tα B ξ )∗ as (α B )∗ (tξ )∗ (α B )−1 ∗ , hence we have ⎛ σ ⎞ ⎛ 0 0 0 0 0 0 0 0 −1 −1 ⎞ ⎛ σ ⎞ ⎜ F ⎟ ⎜0 1 0 0 1 0 1 0 ⎜ ⎟ ⎜ ⎜ 1,1 ⎟ ⎜ 0 0 1 0 0 0 0 0 ⎜ ⎟ ⎜ ⎜ 2,1 ⎟ ⎜ 0 0 0 0 0 0 −1 0 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ 0 0 0 0 −1 0 0 1 (tξ )∗ · ⎜ 3,1 ⎟ = ⎜ ⎜ 1,2 ⎟ ⎜ 0 0 0 0 0 1 0 0 ⎜ ⎟ ⎜ ⎜ 2,2 ⎟ ⎜ 0 0 0 1 0 0 −1 0 ⎜ ⎟ ⎜ ⎜ 3,2 ⎟ ⎜ 0 0 0 0 −1 0 0 0 ⎝ ⎠ ⎝ 1 0 0 0 0 0 0 0 ξ 0 0 0 0 0 0 0 0 αB ξ
0 0 1 0 0 0 1 2 0
−1 ⎟ ⎜ F ⎟ ⎟ ⎜ ⎟ 0 ⎟ ⎜ 1,1 ⎟ ⎟ ⎜ ⎟ 0 ⎟ ⎜ 2,1 ⎟ ⎟ ⎜ ⎟ 1 ⎟ ⎜ 3,1 ⎟ ⎟·⎜ ⎟, 0 ⎟ ⎜ 1,2 ⎟ ⎟ ⎜ ⎟ 0 ⎟ ⎜ 2,2 ⎟ ⎟ ⎜ ⎟ 1 ⎟ ⎜ 3,2 ⎟ ⎠ ⎝ ⎠ 1 ξ 1 αB ξ
⎛ σ ⎞ ⎛ 0 0 0 0 0 0 0 0 −1 −1 ⎞ ⎛ σ ⎞ ⎜ F ⎟ ⎜ 0 1 1 0 0 0 0 1 −1 0 ⎟ ⎜ F ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ⎜ 1,1 ⎟ ⎜ 1,1 ⎟ ⎜ 0 0 −1 0 0 1 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ⎜ 2,1 ⎟ ⎜ 2,1 ⎟ ⎜ 0 0 0 1 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ 3,1 ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 0 0 −1 1 (tα B ξ )∗ · ⎜ 3,1 ⎟ = ⎜ ⎟·⎜ ⎟, 0 1 ⎟ ⎜ 1,2 ⎟ ⎜ 1,2 ⎟ ⎜ 0 0 −1 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ⎜ 2,2 ⎟ ⎜ 2,2 ⎟ ⎜ 0 0 0 0 0 0 1 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 3,2 ⎟ ⎜ 3,2 ⎟ ⎜ 0 0 0 0 1 0 0 −1 1 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 0 0 0 0 0 0 0 0 1 0 ξ ξ 1 0 0 0 0 0 0 0 1 2 αB ξ αB ξ
(A.4)
(A.5)
Constraining the Kähler Moduli in the Heterotic Standard Model
⎛ σ ⎞ ⎛ 1 0 0 0 1 0 0 ⎜ F ⎟ ⎜ 1 ⎜ ⎟ ⎜ ⎜ 1,1 ⎟ ⎜ −2/3 0 0 0 ⎜ ⎟ ⎜ ⎜ 2,1 ⎟ ⎜ −2/3 0 0 0 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ −2/3 0 0 0 (tη )∗ · ⎜ 3,1 ⎟ = ⎜ ⎜ 1,2 ⎟ ⎜ −1/3 0 1 0 ⎜ ⎟ ⎜ ⎜ 2,2 ⎟ ⎜ −1/3 0 0 1 ⎜ ⎟ ⎜ ⎜ 3,2 ⎟ ⎜ −1/3 0 0 0 ⎝ ⎠ ⎝ 0 0 0 0 ξ 0 0 0 0 αB ξ
19
0 0 0 0 0 0 ⎞ ⎛ σ ⎞ 0 1 1 1 0 0 ⎟ ⎜ F ⎟ ⎟ ⎜ ⎟ 0 −1 0 0 −2/3 1/3 ⎟ ⎜ 1,1 ⎟ ⎟ ⎜ ⎟ 0 0 −1 0 1/3 −2/3 ⎟ ⎜ 2,1 ⎟ ⎟ ⎜ ⎟ 0 0 0 −1 1/3 1/3 ⎟ ⎜ 3,1 ⎟ ⎟·⎜ ⎟. 0 −1 0 0 −1/3 2/3 ⎟ ⎜ 1,2 ⎟ ⎟ ⎜ ⎟ 0 0 −1 0 −1/3 −1/3 ⎟ ⎜ 2,2 ⎟ ⎟ ⎜ ⎟ 1 0 0 −1 2/3 −1/3 ⎟ ⎜ 3,2 ⎟ ⎠ ⎝ ⎠ 0 0 0 0 1 0 ξ 0 0 0 0 0 1 αB ξ (A.6)
Another way of looking at these is as generators of the representation of the three matrices Mordell-Weil group in End H2 (B, ZZ) . The commutation relations [(tξ )∗ , (tα B ξ )∗ ] = 0, [(tξ )∗ , (tη )∗ ] = 0, [(tη )∗ , (tα B ξ )∗ ] = 0 are obeyed and the torsion generator (tη )∗ , verifies (tη )3∗ = 1 as expected. Thus, expanding the equation y
[xξ yα B ξ zη] = (tξ )∗x · (tα B ξ )∗ · (tη )∗z σ for the homology classes of the sections, gives us the following list6 : If x 0 2 1 y ≡ 0 (mod 3) or 1 (mod 3) or 2 (mod 3), z 0 1 2
(A.7)
(A.8)
then xξ yα B ξ zη = (1 − x − y)σ + (1/3x 2 + 1/3y 2 − 1/3x y − x − y)F + 1/3y 1,1 + 2/3x 2,1 +(1/3x +2/3y) 3,1 +2/3y 1,2 +1/3x 2,2 +(2/3x +1/3y) 3,2 + xξ + yα B ξ. If x 1 0 2 (A.9) y ≡ 2 (mod 3) or 0 (mod 3) or 1 (mod 3) z 0 1 2 then xξ yα B ξ zη = (1 − x − y)σ + (1/3x 2 + 1/3y 2 − 1/3x y − x − y + 1)F + (1/3y − 2/3) 1,1 + (2/3x − 2/3) 2,1 + (1/3x + 2/3y − 2/3) 3,1 + (2/3y − 1/3) 1,2 + (1/3x − 1/3) 2,2 + (2/3x + 1/3y − 1/3) 3,2 + xξ + yα B ξ. If x 2 1 0 (A.10) y ≡ 1 (mod 3) or 2 (mod 3) or 0 (mod 3) z 0 1 2 then xξ yα B ξ zη = (1 − x − y)σ + (1/3x 2 + 1/3y 2 − 1/3x y − x − y + 1)F + (1/3y − 1/3) 1,1 + (2/3x − 1/3) 2,1 + (1/3x + 2/3y − 1/3) 3,1 + (2/3y − 2/3) 1,2 + (1/3x − 2/3) 2,2 + (2/3x + 1/3y − 2/3) 3,2 + xξ + yα B ξ. If x 0 2 1 (A.11) y ≡ 1 (mod 3) or 2 (mod 3) or 0 (mod 3) z 0 1 2 6 It can be proven to hold by using induction.
20
T. L. Gómez, S. Lukic, I. Sols
then xξ yα B ξ zη = (1−x − y)σ +(1/3x 2 +1/3y 2 −1/3x y −x − y +2/3)F +(1/3y − 1/3) 1,1 + 2/3x 2,1 + (1/3x + 2/3y − 2/3) 3,1 + (2/3y − 2/3) 1,2 + 1/3x 2,2 + (2/3x + 1/3y − 1/3) 3,2 + xξ + yα B ξ.If
x y z
≡
1 0 0
(mod 3) or
0 1 1
(mod 3) or
2 2 2
(mod 3)
(A.12)
then xξ yα B ξ zη = (1 − x − y)σ + (1/3x 2 + 1/3y 2 − 1/3x y − x − y + 2/3)F + 1/3y 1,1 +(2/3x −2/3) 2,1 +(1/3x +2/3y−1/3) 3,1 +2/3y 1,2 +(1/3x −1/3) 2,2 + (2/3x + 1/3y − 2/3) 3,2 + xξ + yα B ξ. If
x y z
≡
2 2 0
(mod 3) or
1 0 1
(mod 3) or
0 1 2
(mod 3)
(A.13)
then xξ yα B ξ zη = (1 − x − y)σ + (1/3x 2 + 1/3y 2 − 1/3x y − x − y + 2/3)F + (1/3y −2/3) 1,1 +(2/3x −1/3) 2,1 +(1/3x +2/3y) 3,1 +(2/3y −1/3) 1,2 +(1/3x − 2/3) 2,2 + (2/3x + 1/3y) 3,2 + xξ + yα B ξ. If
x y z
≡
0 2 0
(mod 3) or
2 0 1
(mod 3) or
1 1 2
(mod 3)
(A.14)
then xξ yα B ξ zη = (1− x − y)σ +(1/3x 2 +1/3y 2 −1/3x y − x − y +2/3)F +(1/3y − 2/3) 1,1 + 2/3x 2,1 + (1/3x + 2/3y − 1/3) 3,1 + (2/3y − 1/3) 1,2 + 1/3x 2,2 + (2/3x + 1/3y − 2/3) 3,2 + xξ + yα B ξ. If
x y z
≡
1 1 0
(mod 3) or
0 2 1
(mod 3) or
2 0 2
(mod 3)
(A.15)
then xξ yα B ξ zη = (1 − x − y)σ + (1/3x 2 + 1/3y 2 − 1/3x y − x − y + 2/3)F + (1/3y −1/3) 1,1 +(2/3x −2/3) 2,1 +(1/3x +2/3y) 3,1 +(2/3y −2/3) 1,2 +(1/3x − 1/3) 2,2 + (2/3x + 1/3y) 3,2 + xξ + yα B ξ. If
x y z
≡
2 0 0
(mod 3) or
1 1 1
(mod 3) or
0 2 2
(mod 3)
(A.16)
then xξ yα B ξ zη = (1 − x − y)σ + (1/3x 2 + 1/3y 2 − 1/3x y − x − y + 2/3)F + 1/3y 1,1 +(2/3x −1/3) 2,1 +(1/3x +2/3y−2/3) 3,1 +2/3y 1,2 +(1/3x −2/3) 2,2 + (2/3x + 1/3y − 1/3) 3,2 + xξ + yα B ξ.
Constraining the Kähler Moduli in the Heterotic Standard Model
21
References 1. Banks, T., Dine, M.: Couplings and Scales in Strongly Coupled Heterotic String Theory. Nucl. Phys. B 479, 173–196 (1996) 2. Braun, V., He, Y-H., Ovrut, B.A., Pantev, T.: Vector Bundle Extensions, Sheaf Cohomology, and the Heterotic Standard Model. Adv. Theor. Math. Phys. 10, 4 (2006) 3. Braun, V., He, Y-H., Ovrut, B.A., Pantev, T.: Heterotic Standard Model Moduli. JHEP 0601, 025 (2006) 4. Braun, V., He, Y-H., Ovrut, B.A., Pantev, T.: The Exact MSSM Spectrum from String Theory. JHEP 0605, 043 (2006) 5. Braun, V., Ovrut, B.A., Pantev, T., Reinbacher, R.: Elliptic Calabi-Yau Threefolds with Z Z3 × ZZ3 Wilson Lines. JHEP 0412, 062 (2004) 6. Curio, G., Krause, A.: Nucl.Phys. B 602, 172–200 (2001) 7. Curio, G., Krause, A.: Nucl.Phys. B 693, 195–222 (2004) 8. Donagi, R., Bouchard, V.: An SU(5) Heterotic Standard Model. Phys. Lett. B 633, 483–791 (2006) 9. Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford: Oxford University Press, 1990 10. Douglas, M.R.: The Statistics of String/M-Theory Vacua. JHEP 0305, 046 (2003) 11. Douglas, M.R., Fiol, B., Römelsberger, C.: Stability and BPS branes. JHEP 0509, 006 (2005) 12. Grassi, A., Morrison, D.R.: Automorphisms and the Kähler cone of certain Calabi-Yau manifolds. Duke Math. J. 71, 831–838 (1993) 13. Gukov, S., Kachru, S., Liu, X., McAllister, L.: Heterotic Moduli Stabilization with Fractional ChernSimons Invariants. Phys. Rev. D 69, 086008 (2004) 14. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52, New York: Springer-Verlag, 1977 15. Hoˇrava, P., Witten, E.: Eleven-dimensional supergravity on a manifold with boundary. Nucl. Phys. B 475, 94–114 (1996) 16. Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford: Oxford University Press, 2000 17. Looijenga, E.: Rational surfaces with an anti-canonical cycle. Ann. of Math. (2) 114, 267–322 (1981) 18. Namikawa, Yo.: On the birational structure of certain Calabi-Yau threefolds. J. Math. Kyoto Univ. 31, 151–164 (1991) 19. Schoen, C.: On the fiber products of rational elliptic surfaces with sections. Math. Ann. 197, 177–199 (1988) 20. Sharpe, E.: Kähler Cone Substructure. Adv. Theor. Math. Phys. 2, 1441 (1998) 21. Wilson, P.M.H.: The Kähler cone on Calabi-Yau threefolds. Invent. Math. 107, 561–583 (1992) 22. Witten, E.: Strong coupling expansion of Calabi-Yau compactification. Nucl. Phys. B 471, 135–158 (1996) Communicated by M.R. Douglas
Commun. Math. Phys. 276, 23–49 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0339-7
Communications in
Mathematical Physics
Large Time Dynamics of a Classical System Subject to a Fast Varying Force F. Castella1 , P. Degond2 , Th. Goudon3 1 IRISA & IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France.
E-mail:
[email protected]
2 MIP, UMR 5640 (CNRS-UPS-INSA), Université Paul Sabatier - 118, route de Narbonne,
31062 Toulouse Cedex, France. E-mail:
[email protected]
3 Team SIMPAF - INRIA Futurs & Labo. Paul Painlevé, UMR 8524, Université des Sciences et Technologies
Lille 1, F-59655 Villeneuve d’Ascq Cedex, France. E-mail:
[email protected] Received: 15 February 2006 / Accepted: 9 March 2007 Published online: 15 September 2007 – © Springer-Verlag 2007
Abstract: We investigate the asymptotic behavior of solutions to a kinetic equation describing the evolution of particles subject to the sum of a fixed, confining, Hamiltonian, and a small, time-oscillating, perturbation. The equation also involves an interaction operator which acts as a relaxation in the energy variable. This paper aims at providing a classical counterpart to the derivation of rate equations from the atomic Bloch equations. In the present classical setting, the homogenization procedure leads to a diffusion equation in the energy variable, rather than a rate equation, and the presence of the relaxation operator regularizes the limit process, leading to finite diffusion coefficients. The key assumption is that the time-oscillatory perturbation should have well-defined long time averages: our procedure includes general “ergodic” behaviors, amongst which periodic, or quasi-periodic potentials only are a particular case. 1. Setting of the Problem We consider the asymptotic behavior as ε goes to 0 of the solutions f ε (t, x, v) ≥ 0 to the following kinetic equation with relaxation term: t 2 ε ε ε = γ Q( f ε )(t, x, v), (1) ε ∂t f (t, x, v) + H0 (x, v), f + ε V ,x , f ε2 where Q( f ε )(t, x, v) := P( f ε )(t, x, v) − f ε . (2) Here, the Poisson bracket {·, ·} stands as usual for { f, g} = ∇v f · ∇x g − ∇x f · ∇v g. The position, resp. velocity, variables x, resp. v, both belong to the whole space Rd (d ≥ 1), and we shall often make use of the phase-space variable X = (x, v) ∈ R2d . Throughout this text, the Hamiltonian H0 (X ) ∈ C ∞ (R2d ) is assumed given, and confining, i.e. lim H0 (X ) = +∞.
|X |→∞
(3)
24
F. Castella, P. Degond, Th. Goudon
The right-hand-side of (1)-(2) involves a projection operator P, whose value we define as P f ε (t, X ) := f ε (t, H0 (X )), (4) where the quantity f ε (t, E) is the mean value of f ε over the energy shell S E := {X ∈ R2d s.t. H0 (X ) = E}, namely,
1 f (t, E) := f ε (t, X ) δ(H0 (X ) − E), h 0 (E) S E
δ(H0 (X ) − E) S E = H0 −1 (E) . where h 0 (E) := ε
(5) (6)
SE
Here, the measure δ(H0 (X ) − E) over S E is defined as δ(H0 (X ) − E) :=
dσ E (X ) , |∇ X H0 (X )|
(7)
where dσ E (X ) denotes the induced euclidean surface measure over the energy shell S E . The measure δ(H0 (X ) − E) is the standard micro-canonical (or Liouville) measure of statistical physics. The fact that the above objects P f ε (t, X ) and f ε (t, E) are well defined is proved later, under the main assumption that the measure δ(H0 (X ) − E) δ(H0 (X ) − E) < +∞ for almost every E ∈ H0 (R2d ) (see satisfies h 0 (E) = SE
Hypothesis 1), a requirement which somehow reinforces the fact that H0 is assumed confining. Note that throughout this text, the prototype where H0 is the harmonic oscillator H0 = Hharm = (x 2 + v 2 )/2 is relevant. The kinetic equation (1) is written in dimensionless form. The important dimensionless parameters are ε > 0, which goes to zero, and the relaxation parameter γ > 0, considered fixed. We refer to [CDG] for a thorough discussion and motivation of the scaling. The dynamics induced by Eq. (1) may be described as follows: (a)
At leading order, the evolution is driven by transport along the Hamiltonian flow of H0 , i.e. along the solutions to the Hamiltonian ODE: ∂t x(t, x, v) = ∇v H0 (t, x(t, x, v), v(t, x, v)) , ∂t v(t, x, v) = −∇x H0 (t, x(t, x, v), v(t, x, v)) ,
(b)
x(0, x, v) = x, v(0, x, v) = v.
(8)
We shall often use the phase-space notation X (t, X ) = (x(t, x, v), v(t, x, v)). Due to the scaling in (1), transport occurs at the fast time scale t/ε2 and, since H0 is confining, transport roughly induces “oscillatory” trajectories at the fast scale t/ε2 . The reference Hamiltonian H0 is perturbed by the small and oscillatory potential ε V (t/ε2 ). On the one hand, this perturbation is of size ε > 0 when compared to H0 , and ε perturbations of a Hamiltonian flow are known to modify the dynamics by an O(ε2 ) quantity, on time scales of the order 1. Time being rescaled by a factor 1/ε2 in our case, the perturbing term εV is expected to modify the dynamics by a quantity of the order 1. This is usually called a weak-coupling regime.
Large Time Dynamics
(c)
25
On the other hand, the oscillations carried by the potential V (t/ε2 ) at the fast scale t/ε2 may interact with those induced by the transport term at the same scale. Hence only the average effect – in time – of these combined oscillations is expected to influence the dynamics at dominant order. The relaxation term Q( f ε ) = P( f ε )− f ε , discussed below, models complex interaction phenomena. Since the projection operator P( f ε ) projects f ε onto functions of the energy H0 (X ) only, it is clear that Q( f ε ) redistributes the energy uniformly on each energy shell S E , or, in other words, it relaxes f ε to a solution of P f ε = f ε . Yet the fluctuations P f ε − f ε , small but definitely non-zero in (1), are transported along the Hamiltonian flow, which eventually give rise to diffusion in the energy variable. This is typically what already happens for standard diffusion limits in kinetic theory.
In a previous text [CDG], we studied the asymptotic behavior of (1) under the main assumption that the perturbing potential V (t/ε2 , x) is periodic or quasi-periodic in the fast time variable. Assuming also that the Hamiltonian flow of H0 has some stability property (an assumption that we shall need here as well, see Hypothesis 2 below), we proved that f ε goes, in some weak topology, to a function of the energy only, say F(t, H0 (X )), and that the limiting profile F(t, E) (E ∈ R) satisfies a diffusion equation in energy, of the form ∂t [h 0 (E) F(t, E)] − ∂ E [b(E) h 0 (E) ∂ E F(t, E)] = 0,
(9)
where h 0 (E) is defined in (5). The effective coefficient b(E) obtained in [CDG] is nonnegative. It takes into account the average effect, in time, of the resonant interactions between X (t/ε2 ) and the perturbation V (t/ε2 ). It also is given by an almost explicit formula involving auxiliary profile equations. For several reasons, the periodic or quasi-periodic case is unsatisfactory. Technically speaking, the limit equation is easily guessed in the (quasi-)periodic case, upon simply performing a double scale expansion in the spirit of classical homogenization theory [BLP]. On top of that, the actual proofs given in [CDG] rely on the user-friendly framework of double-scale convergence introduced in [Ng] and [A]. In essence, (quasi-)periodicity turns out to be a strong, and quantitative version of ergodicity, for which the analysis eventually reduces to conveniently adapting the tools of double-scale convergence. (Quasi-)periodicity is also restrictive from the physical viewpoint. As discussed below, Eq. (1) may describe the classical evolution, in phase space, of a collection of atoms undergoing the influence of the atomic Hamiltonian H0 , and weakly coupled to an external laser field through εV (t/ε2 ). The potential V then is roughly the electric field. Hence restricting to (quasi-)periodic V ’s means restricting to (quasi-)periodic fields. Last, the quantum equation analogous to (1) has been previously studied in [BCD, BCDG], and other, more general situations than the mere (quasi-)periodic setting have been analyzed there. The analysis performed in [BCD, BCDG] actually shows the key point is that the perturbing potential V should possess well defined long time averages 1 T of the form lim V (s) ds or so. The existence of such long time averages is T →∞ T 0 certainly implied by the much stronger (quasi-)periodicity assumption, but it definitely includes much more general “ergodic” behaviors. Adopting this point of view, our goal in the present paper is to fill the gap, i.e. to describe the asymptotic ε → 0 for more general oscillating potentials V that are not
26
F. Castella, P. Degond, Th. Goudon
(quasi-)periodic in time. The potentials we are interested in are actually of KBM type (after: Krylov, Bogolioubov, Mitropolski): they are required to possess specific long-time averages (Hypothesis 3 below). We stress that the present extension is by no means trivial. First, the mere formulation of the assumption we need on the potential, though of KBM type, is not immediate. It is an original requirement. Second, the (quasi-)periodic setting essentially requires an adaptation of tools from double-scale analysis, and allows to pass to the limit in the standard two-scale topology directly on f ε . This strategy cannot be extended in any way here. We need a variance to split f ε as P f ε + (Id − P) f ε and to pass to the limit in P f ε , exploiting a specific compactness property inherited from the structure of the equation. From this point of view, our proof shares a lot of features with the derivation of the Kubo-like formula in [GP3]. Last, an important difficulty in the analysis of (1) is created by the fact that the operator f → P f does not preserve the smoothness of f , and P f is not even once differentiable in general. This difficulty is linked with the general lack of smoothness of the energy levels S E as E varies. In the (quasi-)periodic setting, the use of a double-scale analysis allows to circumvent this difficulty, with the mere drawback that our homogenization procedure does not provide a corrector. In the present framework, the lack of smoothness of P f is more problematic, and we do need to deal with this difficulty in order to recover the necessary compactness. Our main result is the following. Theorem 1. Let f 0ε ≥ 0 be the initial data for (1). We assume that f 0ε is bounded in L 2 (R2d ). We also suppose the Hamiltonian has well-defined energy levels with finite measure (Hypothesis 1 below), the Hamiltonian flow of H0 is polynomially stable (Hypothesis 2), and the potential V has some well-defined long-time averages (Hypothesis 3). Then, the following holds: (i) The solution f ε (t, X ) to (1) admits the decomposition f ε (t, X ) = P f ε (t, X ) + εgε (t, X ), where gε is bounded in L 2 ((0, T )×R2d ) and P f ε (t, X ) is bounded in C 0 ([0, T ]; L 2 (R2d )). (ii) Up to subsequences, P f ε (t, X ) converges in C 0 ([0, T ]; L 2 (R2d )−weak) towards a function F(t, H0 (X )). (iii) The limiting function F : R+ × R → R+ satisfies the following diffusion equation in D ([0, ∞) × R): ∂t [h 0 (E) F(t, E)] − ∂ E [h 0 (E) b(E) ∂ E F(t, E)] = 0, F(0, E) = lim f 0ε (E) (the limit is in L 2 (R)-weak). ε→0
Equivalently, F satisfies 2 ∂t [h 0 (E) F(t, E)] − h 0 (E) a(E) ∂ E F(t, E) − h 0 (E) b(E) ∂ E,E F(t, E) = 0,
F(0, E) = lim f 0ε (E). ε→0
Here the coefficient a and b are defined from (20) below through a(E) = A (E), b(E) = B (E). They satisfy h 0 (E) a(E) = ∂ E [h 0 (E) b(E)] , b(E) ≥ 0.
Large Time Dynamics
27
Naturally, the above theorem extends the result obtained in [CDG] to non-periodic V ’s. More importantly, it is worth remarking that the formal structure of the formula which defines the diffusion coefficients a and b coincides with the one of eddy diffusivity in turbulence theory, see [Ta]. Before coming to the formulation of Hypothesis 1, 2, and 3, and to the proof of Theorem 1, we wish to make two comments. Equation (1) is the standard setting for the description of an atom in interaction with a light field, say a laser. This is the prototype situation we have in mind: the unperturbed Hamiltonian H0 is the atomic Hamiltonian, while the perturbation εV (t/ε2 ) is the potential energy induced by light in the vicinity of the atom. Equation (1) adopts a classical mechanics description of such an interaction. If a quantum mechanical setting is retained, the kinetic equation (1) becomes a quantum Liouville equation, also known as an atomic Bloch equation. It reads in our case
t 2 ε ε ε iε ∂t ρ (t) = H0 , ρ (t) + ε V , ρ (t) + γ Q(ρ ε (t)). (10) ε2 The unknown ρ ε (t) now is a time dependent trace class operator, called “density matrix” of the atom, all Poisson brackets in (1) have become commutators between operators in (10), and Q(ρ ε ) is a relaxation operator that plays the same role as Q( f ε ). The factor Q(ρ ε ) describes at a heuristic level the observed trend of various atomic systems to relax towards equilibria of the unperturbed Hamiltonian H0 . The relaxation term Q is well documented in the physics literature (see e.g. [Lo]), while the operator Q we introduced in [CDG] comes up in mere analogy with Q, as a classical counterpart of the quantum operator Q. The quantum setting (10) has been completely analyzed in [BCD], for potentials that are either (quasi-)periodic in time, or more generally of KBM type (with optimal convergence rates in the first situation, and no better error estimate than o(1) in the second). The main convergence result obtained in [BCD] is similar in spirit to what we prove here and in [CDG]: the atom tends to be well described by a function of the energy only, i.e. by a function N (t, n) which describes the occupation probability of the atom’s nth eigenstate at time t. The function N (t, n) plays the role of F(t, E). Yet the main difference with the classical case is that energy levels now are discrete, so that the diffusion process obtained classically for F rather becomes a discrete jump process for N . Our second comment concerns the role of the relaxation term Q. From the modeling point of view, a mathematically rigorous derivation of this term goes far beyond the scope of this paper. We simply use it as a way to take into account observed relaxation phenomena. On the other hand, the asymptotic analysis of (1) or (10) is dominated by the resonant interaction between the oscillations of V (t/ε2 ), and those induced by the transport operator ε2 ∂t + {H0 , .}. Technically, the relaxation operators Q or Q somewhat regularize the situation in this respect: they prevent the possibility of too strong resonances (small denominators), through the introduction of some damping in the model. One may then wonder what happens along the asymptotic process if the relaxation term is set to zero in the original Eqs. (1) or (10). In the quantum case, and for (quasi-)periodic potentials V , it turns out smaller damping rates of order O(εµ ) with µ < 1/2 may be considered (see [BCD, BCDG]). The usual (undamped) formulae for the Einstein rate equations [Lo] have been recovered in [BCD, BCDG]. The analysis heavily relies on small denominator estimates, perturbed Diophantine estimates, and other arguments in the same vein, in the spirit of averaging techniques for ODE’s. Yet the condition µ < 1/2
28
F. Castella, P. Degond, Th. Goudon
still means that damping should not be too small with respect to the other perturbations. In this perspective, a deep gap actually separates the case “with damping” from the case “without damping” (in this article as well as in [CDG] and [BCD, BCDG]). The mathematical and physical situation, as well as the limiting process itself, are completely different when the (possibly small) relaxation term is set to zero from the onset: indeed, the limiting equation (9) is time-irreversible, while the associated scaled Eq. (1) only becomes irreversible through the dissipation term. The route we choose here uses a heuristic, and deterministic, relaxation term. The reader may find in [CD, Ca1, Ca2] a similar model “with relaxations” used to rigorously derive the Pauli master equation from the quantum Liouville equation in a deterministic framework. A second, probably more standard approach is the introduction of stochastic averaging in the model, which gives the necessary “loss of memory” in the analysis: to some extent, deterministic relaxation terms play a similar role as the stochastic averaging process. The deep role played by stochastic averaging in the derivation of irreversible equations is very well explained in [CIP] (see also [Sp]). More recently, we may mention [EY1, EY2, PV, LV], or also [KPR]. There are actually several other examples of such an alternative: homogenization of convection(-diffusion) equations (see [GP1, GP2] and the references therein), Lorentz gas involving a billiard (see [BDG] and [BSC]), quantum scattering limit of the Schrödinger equation ([BPR, EY2, PR, PV]...). For the (space-)homogenization of kinetic equations without dissipative term, we refer e.g. to [Al, FH]. We now state the assumptions we need on the Hamiltonian H0 , and the potential V . We begin with the assumptions on H0 , which are essentially the same as in [CDG]. Hypothesis 1. (Energy levels with finite measure). We assume that the confining Hamiltonian H0 satisfies: (i) For almost all E ∈ H0 (R2d ), the set S E = {X = (x, v) ∈ R2d |H0 (X ) = E}, is a smooth orientable 2d − 1 submanifold of R2d . For any such E, we denote the induced euclidean surface measure by dσ E (X ). We also define the (microcanonical or Liouville) measure δ(H0 (X ) − E) as δ(H0 (X ) − E) =
dσ E (X ) . |∇ X H0 (X )|
(ii) For any E as in (i), the set S E has finite measure with respect to δ(H0 (X ) − E), namely h 0 (E) = δ(H0 (X ) − E) < +∞, a.e. E ∈ H0 (R2d ). SE
Remark. As noticed in [CDG], the Sard theorem asserts that Hypothesis 1-(i) is generically satisfied (see e.g. [Mi]). Hence the truly important assumption is point (ii). Hypothesis 2. (Stability of the Hamiltonian flow). Let X : (s, X ) ∈ R × R2d −→ X (s, X ) ∈ R2d be the Hamiltonian flow of H0 , namely d +∇v H0 X (s, X ) = X (s, X ) , X (0, X ) = X = (x, v). (11) −∇ H x 0 ds
Large Time Dynamics
29
We assume that the linearized flow satisfies, for any s ≥ 0 and X ∈ R2d , DX (s, X ) ≤ C(X ) (1 + s) p , DX D2 X (s, X ) ≤ C(X ) (1 + s) p , 2 DX
(12) (13)
for some C(X ) ≥ 0 which is locally bounded in X , and some exponent p ≥ 0, independently of s. Remark. The analysis provided in [CDG] for (quasi-)periodic potentials only requires the first stability estimate (12). In the present text, estimate (13) is required as well. It may be somewhat relaxed into a locally Lipschitz bound of the form |D X /D X (s, X + Y ) − D X /D X (s, X )| ≤ C(X ) |Y | (1 + s) p . We do not detail this unnecessary technical point. Remark. The crucial assumptions on H0 are Hypothesis 1-(ii) and 2. The former allows to define the relaxation operator Q (see Lemma 1 below), while the latter is a strong stability assumption on the Hamiltonian flow of H0 : according to (12), any two trajectories starting with nearby initial data should diverge at most polynomially with time. For the standard Hamiltonian H0 (X ) = v 2 /2 + V0 (x), with V0 any ‘reasonable’ potential, we recall that the generic divergence of two nearby trajectories is more likely exponential in time. Naturally, Hypotheses 1 and 2 are fulfilled in the prototype case of the harmonic oscillator Hharm (X ) = (x 2 + v 2 )/2. We mention that the stability Hypothesis 2 may be somewhat relaxed, so as to include the case of exponential divergence, i.e. the case when (1+s) p is replaced by exp(C1 s) for some C1 > 0. However, this can only be done at the (unreasonable) price of considering large enough values of the relaxation parameter γ , namely γ > C1 . We do not dwell on this aspect of the analysis. As proved in [CDG], Hypothesis 1 allows to properly define the operators and P, as in (4) and (5). This gives a well-defined relaxation operator Q = Id − P in (1). The basic observation is that the co-area formula reads, with the above notations 1 2d f (X ) d X = f (E) h 0 (E) d E. (14) ∀ f ∈ L (R ), R2d
R
Armed with (14), one may indeed deduce (see [CDG]) the following Lemma 1. The operator P defined in (4) satisfies the following properties: (i)
P is a continuous projection operator on L p spaces: P(P f ) = P f,
P f L p (R2d ) ≤ f L p (R2d ) 1 ≤ p ≤ ∞.
Besides P is conservative: for any integrable function, we have Pf dX = f d X. R2d
R2d
30
(ii)
F. Castella, P. Degond, Th. Goudon
P is self-adjoint with respect to the inner product of L 2 (R2d ) (denoted ·, · throughout the paper): for any function f ∈ L 2 (R2d ) and ϕ : R → R such that ϕ(H0 (X )) ∈ L 2 (R2d ), we have ϕ(H0 (X )), (Id − P) f = 0.
(iii)
P is non negative: if f ≥ 0 a.e. X , then P f ≥ 0 a.e. X as well. Moreover, the stronger relation holds: f ≥ 0 a.e. X, and P f = 0 a.e. X =⇒ f = 0 a.e. X.
(iv) The operators f −→ P f and f −→ {H0 , f } are orthogonal, i.e. the relation P {H0 , f } = 0, holds for any f ∈ L 2 (R2d ) such that {H0 , f ∈ L 2 (R2d ). Consequently, for any f, g ∈ L 2 (R2d ) such that {H0 , f and {H0 , g in L 2 (R2d ), we have P({H0 , f } g) = −P( f {H0 , g}). (v) The operator Q = Id − P is a bounded operator on L 2 (R2d ) and the relation |P f − f |2 d X ≥ 0 − Q( f ) f d X = R2d
R2d
holds for any f ∈ L 2 (R2d ). (vi) Let V : R × R2d → R be a C 1 function. Let ϕ ∈ Cc∞ (R). We have P {V(t), ϕ(H0 )} = 0. Remark. Point (v) asserts that Q relaxes towards solutions to P f = f , and the rate of convergence is unity. The proof of point (vi) relies on the equality P {V, ϕ(H0 )} (X ) = P ({V, H0 }) (X ) ∂ E ϕ(H0 (X )) = 0. There remains to specify the behavior of the perturbing potential V (s, x) in the fast time variable s. This is the main new point in the present paper. To motivate our approach, let us describe the situation in the quantum case, i.e. when analyzing (10) instead of (1). The key point in [BCD, BCDG] is the following: one may transform the original PDE (10) into an infinite dimensional ODE, which in turn very much behaves like a finite dimensional ODE with oscillatory coefficients (up to small remainder terms), of the form d t yε (t) = ψ , yε (t) . (15) dt ε2 As a consequence, the quantum problem (10) reduces, in essence, to performing an averaging procedure on the ODE (15). Now, the basic averaging theorem for ODE’s (see e.g. [SV]) asserts that the solution yε (t) to any oscillatory ODE of the form (15), converges in some topology towards the solution of the averaged ODE d 1 T = lim y(t) = ψ (y(t)) ψ (s, y(t)) ds , (16) T →∞ T 0 dt
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31
provided the function ψ(s, z) has KBM dependence in the time variable s. We recall that a function ψ(s, z) is called a KBM function whenever the limit ψ (z) = lim
T →∞
1 T
T
ψ(s, z) ds
(17)
0
exists, for every z. We refer to e.g. [SV] for the precise statements concerning averaging of ODE’s. Periodic, quasi-periodic, or even almost-periodic functions of time, are obvious examples of KBM functions. In the present paper, the analysis shows that an argument similar to the one used in the quantum context [BCD, BCDG] allows to transform the original PDE (1) into a PDE with oscillatory coefficients, of the form (roughly) ∂ t t 2 yε (t, E) = α 2 , E ∂ E yε (t, E) + β , E ∂ E,E yε (t, E), (18) ∂t ε ε2 for some coefficients α and β. This result is very much in the spirit of observation (15) above. The quantitative statement is Proposition 5 below, where the relevant coefficients are the two functions A(t/ε2 , X ) and B(t/ε2 , X ) entering Hypothesis 3. Keeping in mind the paradigm (15)–(16), the natural assumption is that the time dependent coefficients α(s, E), β(s, E) (or A(s, E), B(s, E)) should have well defined long time averages in s. When V is periodic or quasi-periodic in time, this requirement is easily met, since the relevant coefficients then are (quasi-)periodic in time as well [CDG]. The natural extension, which is the purpose of this paper, is the case when the coefficients α, β (or A, B) are merely KBM. All these considerations motivate our main assumption on V : Hypothesis 3. (Existence of long-time averages of V ). (i) We assume that V ∈ Cb3 (R+ × Rd ) is bounded with bounded derivatives up to third order. (ii) Introduce the family of transport operators associated to the Hamiltonian flow of H0 , Su (ϕ)(X ) := ϕ X (u, X ) . For any t ≥ 0, define the functions
t
A(t, X ) :=
e−γ u {V (t) , Su {V (t + u) , H0 }} du,
0
t
B(t, X ) :=
e−γ u {V (t), H0 } × Su {V (t + u) , H0 } du.
0
We assume that A and B admit long-time averages, in that the following limits 2d exist in L ∞ loc (R )-weak- : A (X ) := lim
T →+∞
1 T
0
T
A(s, X ) ds, B (X ) := lim
T →+∞
1 T
T
B(s, X ) ds.
0
(19)
32
F. Castella, P. Degond, Th. Goudon
(iii) Furthermore, we assume that the following limits hold true: t t A 2 , X A (X ), B , X B (X ), ε ε2 + 2d in L ∞ loc (R × R ) − weak − .
(20)
Remark. Assumption (i) may be somewhat relaxed in that V may be assumed Cb2 , with a Lipschitz second derivative, globally in time, locally in X . We do not detail this technical aspect. The above assumption requires some comments. An important hypothesis is point (i). Indeed, we show below (Proposition 2) that the assumed regularity on V and H0 implies the coefficients A and B are bounded, T globally in time t ≥ 0, and locally in X . Hence the sequences 1/T 0 · · · are bounded 2d in L ∞ loc (R ), and they possess (up to subsequences) weak- limits as T → ∞, limits that are automatically independent of time. This ensures the validity of statement (ii), of KBM type. The same argument shows the sequences A(t/ε2 , X ) and B(t/ε2 , X ) + 2d also are bounded in L ∞ loc (R × R ), hence possess weak- limits as ε → 0. The point is, assertion (iii) requires these limits are independent of t. This in turn implies that A(t/ε2 , X ) automatically goes to A (X ), and similarly for B. In summary, assumption (iii) is considerably stronger than (ii). It should also be noted that, in the context of T ODE’s, the averaging of (15) only requires the long time averages 1/T 0 ψ(s, z)ds converge, while in the present PDE context, we do need the reinforced assumption (iii). The remainder of this paper is devoted to the proof of Theorem 1. 2. Proof of Theorem 1 2.1. Two uniform bounds. Our analysis starts with the Proposition 1. Suppose the initial data f 0ε is bounded in L 2 (R2d ). Then, ∞ (R+ ; L 2 (R2d )). (i) The family ( f ε )ε>0 is bounded in L ε ε f − Pf (ii) The family g ε )ε>0 := is bounded in L 2 (R+ × R2d ). ε ε>0
Proof of Proposition 1. We readily observe γ 1 d ε 2 ε ε | f | dX = γ Q( f ) f d X = − 2 |P f ε − f ε |2 d X ≤ 0, 2 dt R2d ε R2d R2d where Lemma 1-(vi) has been used. For later convenience, we also state and prove the following easy, yet crucial, uniform bound on the coefficients A(t/ε2 , X ) and B(t/ε2 , X ) entering Hypothesis 3. Proposition 2. Suppose Hypotheses 2 and 3 are fulfilled. Let 0 < R < ∞ and set E(R) := {X ∈ R2d , |H0 (X )| ≤ R},
B(R) := {X ∈ R2d , |X | ≤ R}.
(i) There exists 0 < ρ(R) < ∞ such that any X ∈ E(R) belongs to B(ρ(R)) as well. Moreover, for any X ∈ E(R) and u ∈ R, we have X (u, X ) ∈ B(0, ρ(R)) too.
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33
(ii) There exists a function C(R), bounded for bounded values of R > 0, such that sup sup sup V (t/ε2 − u) , H0 Su V (t/ε2 ) , H0 ≤ C(R), ε>0 t≥0 X ∈E (R)
sup sup
sup
ε>0 t≥0 X ∈E (R)
V (t/ε2 − u) , Su V (t/ε2 ) , H0 ≤ C(R) (1 + u) p ,
whenever u ≥ 0. The exponent p is as in Hypothesis 3. (iii) There exists a function C(R), bounded for bounded values of R > 0, such that1 C(R) t sup sup sup A 2 , X ≤ p+1 , ε γ ε>0 t≥0 X ∈E (R) C(R) t sup sup sup B . , X ≤ 2 ε γ ε>0 t≥0 X ∈E (R)
Here, functions A and B are as in Hypothesis 3. Proof of Proposition 2. The fact that H0 is confining (3) readily implies point (i). Next, point (ii) comes as an immediate consequence of the regularity assumptions we have made on H0 and V . Indeed, we may write, whenever ε > 0, t ≥ 0, and X ∈ E(R), V (t/ε2 − u) , H0 Su V (t/ε2 ) , H0 ≤ ∇x V 2L ∞ (R×B(ρ(R))) ∇v H0 2L ∞ (B(ρ(R)) ≤ C(R), where C(R) is a locally bounded function of R. Similarly, we have V (t/ε2 − u) , Su V (t/ε2 ) , H0 D X (u, X ) ≤ ∇x V L ∞ (R×B(ρ(R))) × ∞ DX L (B(ρ(R)))
2 × Dx V L ∞ (R×B(ρ(R))) ∇v H0 L ∞ (B(ρ(R))) 2 + ∇x V L ∞ (R×B(ρ(R))) Dx,v H0 L ∞ (B(ρ(R))) ≤ C(R) × (1 + u) p . The last line uses Hypothesis 3. Point (iii) is easily deduced, since, for any ε > 0, t ≥ 0, and X ∈ E(R), we have the two upper bounds +∞ 2 e−γ u du ≤ C(R) γ −1 , |B(t/ε , X )| ≤ C(R) 0 +∞ 2 |A(t/ε , X )| ≤ C(R) e−γ u (1 + u) p du ≤ C(R) γ − p−1 . 0
1 Note that we implicitly assume here γ < 1 – this is no loss of generality.
34
F. Castella, P. Degond, Th. Goudon
2.2. Obtaining an equation for P f ε . The following result completes the previous Proposition 1, according to which f ε = P f ε + O(ε): we derive here an equation for P f ε . We mention that the splitting f ε = P f ε + ε g ε into a (dominant) function of the energy, and a (smaller) fluctuation term, is partly motivated by a similar decomposition occurring in the quantum context, see [BCD, BCDG]. The crucial point is that the dominant function of the energy turns out to satisfy a closed equation (up to small remainder terms). Note that [CDG] follows a completely different approach, based on a double-scale convergence analysis for f ε itself. Proposition 3. The function P f ε = P f ε (t, X ) satisfies the equation ∂t P f ε (t, X ) = L ε P f ε (t, X ) + ε R ε (t, X ) + I ε (t, X ) .
(1)
The leading term L ε (P f ε ) is a second order, linear operator, with memory term and oscillatory coefficients. It has the value L
ε
Pf
ε
t/ε2
e−γ u P
(t, X ) = 0
V (t/ε2 ) , S−u
V (t/ε2 − u) , P f ε (t − ε2 u)
du.
The remainder and initial terms R ε and I ε are R ε (t, X ) =
t/ε2
e−γ u P
0 ε
I (t, X ) = −ε−1 e−γ t/ε
2
V (t/ε2 ) , S−u V (t/ε2 − u) , g ε (t − ε2 u) du, P V (t/ε2 ) , S−t/ε2 g0ε .
The picture given in Proposition 3 is made complete through the next technical statement, according to which operators L ε , R ε , and I ε are bounded in some weak topology. Proposition 4. Let ϕ : R → R be a C ∞ compactly supported function. Then, there exists a constant C > 0, depending on γ , supp(ϕ), H0 and V , such the following estimates hold true: ε ε sup L P f (t, X ) ϕ(H0 (X )) d X ≤ C ϕ W 2,∞ (R) , (2) t≥0
R2d
2 ∞ ε dt ≤ C ϕ 2 2,∞ , R (t, X ) ϕ(H (X )) d X 0 2d W (R ) 0 R ∞ 2 ε dt ≤ C ϕ 2 1,∞ . I (t, X ) ϕ(H (X )) d X 0 2d W (R )
0
R
(3) (4)
Proof of Proposition 3. Proposition 1 naturally leads to splitting the original kinetic equation (1) on f ε into an equation for P f ε , and an equation on g ε = ( f ε − P f ε )/ε. We have on the one hand 1 1 ∂t P f ε = − P V (t/ε2 ), f ε = − P V (t/ε2 ), P f ε − V (t/ε2 ), g ε ε ε = −P V (t/ε2 ), g ε ,
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35
(Lemma 1 has been used, which allows to cancel the O(1/ε) contribution), while, on the other hand ε2 ∂t g ε + H0 , g ε + γ g ε = − V (t/ε2 ), P f ε − ε (Id − P) V (t/ε2 ), g ε . Hence we arrive at a system with a particular triangular structure, namely, ∂t P f ε = −P V (t/ε2 ), g ε , ε2 ∂t g ε + H0 , g ε + γ g ε = − V (t/ε2 ), P f ε − ε (Id − P) V (t/ε2 ), g ε .
(5) (6)
The similar (and crucial) structure is involved in the quantum case (see [BCD, BCDG]). Exploiting this observation, one may first express g ε as a function of P f ε : the method of characteristics readily gives
t/ε2 2 g ε (t, X ) = e−γ t/ε g0ε X (−t/ε2 , X ) − e−γ u V (·/ε2 ) , P f ε 0
× t − ε2 u, X (−u, X ) du t/ε2
−ε e−γ u (Id − P) V (·/ε2 ) , g ε t − ε2 u, X (−u, X ) du.
(7)
0
Now, inserting (7) into (5) roughly gives a closed equation on P f ε , namely ∂t P f ε = L ε P f ε + ε Rε + ε I ε , t/ε2 with Rε = e−γ u P V (t/ε2 ) , (Id − P) S−u V (t/ε2 −u) , g ε (t − ε2 u) du. 0
There remains to observe that, for any u, we have P
V (t/ε2 ) , P S−u
V (t/ε2 − u) , g ε (t − ε2 u)
= 0,
by virtue of Lemma 1. Hence R ε = R ε , and the proposition is proved.
Proof of Proposition 4. Let us pick some R > 0 such that supp ϕ ⊂ (−R, +R). Proposition 2-(i) implies ϕ(H0 (X )) = 0 whenever |X | ≥ ρ(R). Actually, using the energy conservation, the function Su ϕ(H0 ) ≡ ϕ(H0 ) is supported in B(ρ(R)) for any u ≥ 0. This observation is used repeatedly below. First step: Proof of estimate (2) on L ε . Upon using the definition of L ε , performing the obvious integration by parts, and systematically exploiting the identity {V, ϕ(H0 )} = {V, H0 } ∂ E ϕ(H0 ) and so on, we obtain,
36
F. Castella, P. Degond, Th. Goudon
R2d
L ε (P f ε )(t, X ) ϕ(H0 (X )) d X
=
R2d
=
R2d
=
R2d
=
t/ε2 0
e−γ u P V (t/ε2 ), S−u V (t/ε2 − u), P f ε (t − ε2 u) ϕ(H0 ) du d X
t/ε2 −γ u
P f ε (t − ε2 u) V (t/ε2 − u) , Su V (t/ε2 ) , ϕ(H0 ) du d X
t/ε2 −γ u
P f ε (t − ε2 u) V (t/ε2 − u) , Su V (t/ε2 ) , H0 ∂ E ϕ(H0 ) du d X
e
0
e
0
V (t/ε2 − u) , H0 Su V (t/ε2 ) , H0 ∂ E2 E ϕ(H0 ) ∂ E ϕ(H0 ) du d X. + V (t/ε2 − u) , Su V (t/ε2 ) , H0
t/ε2 −γ u
e
R2d
0
P f ε (t − ε2 u)
This together with Proposition 2-(ii) allows to estimate ε ε 2d L (P f )(t, X ) ϕ(H0 ) d X R t/ε2 ≤ C(R) ϕ W 2,∞ (R) e−γ u P f ε (t − ε2 u) 1 B(ρ(R)) (1 + u p ) du d X 2d R 0 ∞ ε e−γ u (1 + u p ) du ≤ C(R) ϕ W 2,∞ (R) f L ∞ (R+ ;L 2 (R2d )) |B(ρ(R))|1/2 ≤ C(R) γ
−p
0
ϕ W 2,∞ (R) ,
where the indicator function 1 B(ρ(R)) is used to keep track of the compact support of ϕ, while the last estimate uses Proposition 1. Second step: Proof of estimate (3) on R ε . The proof follows the same lines as before. Multiplying R ε by the trial function ϕ(H0 ) and performing the obvious integration by parts, we obtain R ε (t, X ) ϕ(H0 (X )) d X R2d
=
=
R2d
g ε (t − ε2 u) e−γ u
V (t/ε2 − u) , Su V (t/ε2 ) , ϕ(H0 ) du d X.
0
R2d
t/ε2
g ε (t − ε2 u) e−γ u V (t/ε2 − u) , H0 } Su V (t/ε2 ) , H0 ∂ E2 E ϕ(H0 ) 0 ∂ E ϕ(H0 ) du d X. + V (t/ε2 − u) , Su V (t/ε2 ) , H0 t/ε2
As a consequence, using Proposition 2-(ii), we recover 2 ∞ ε R (t, X ) ϕ(H0 ) d X dt 0
R2d
≤ C(R) ϕ 2W 2,∞ (R) ≤ C(R)
ϕ 2W 2,∞ (R)
0
2 t/ε2 |g ε (t − ε2 u)| e−γ u(1 + u p ) 1 B(ρ(R)) du d X dt R2d 0
∞
g ε 2L 2 (R×R2d ) γ −2 p ≤ C(R) γ −2 p ϕ 2W 2,∞ (R) .
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37
The last estimate uses Proposition 1. Third step: Proof of estimate (4) on I ε . We write ε 2d I (t, X ) ϕ(H0 ) d X R −1 −γ t/ε2 ε 2 = ε e g0 (X ) St/ε2 V (t/ε ), H0 ∂ E ϕ(H0 (X )) d X R2d 2 ≤ C(R) ϕ W 1,∞ (R) ε−1 e−γ t/ε |g0ε (X )|1 B(ρ(R)) d X ≤ C(R) ϕ W 1,∞ (R) ε Hence +∞ 0
R2d −1 −γ t/ε2
e
f 0ε L 2 (R2d ) .
2 I (t, X ) ϕ(H0 ) d X dt ≤ C(R) ϕ W 1,∞ (R) ε
R2d
+∞
ε−2 e−2γ t/ε dt 2
0
≤ C(R) γ −1 ϕ 2W 1,∞ (R) .
This ends the proof of Proposition 4.
2.3. Analysis of the memory effect. Proposition 3 establishes the equation ∂t (P f ε ) = L ε (P f ε ) + O(ε), where the operator L ε is a second order differential operator involving both a memory effect (this is the term P f ε (t −ε2 u)) and coefficients that oscillate in time (these are the terms V (t/ε2 ) etc.). The next proposition allows to get rid of the memory effect, and to somewhat put in evidence the true difficulty of the present analysis, namely the presence of time oscillatory terms. Proposition 5. For any smooth test functions ζ (t) ∈ Cc∞ (R+ ) and ϕ(E) ∈ Cc∞ (R), we have ∞ P f 0ε (X ) ϕ(H0 (X )) ζ (0) d X + P f ε (t, X ) ϕ(H0 (X )) ζ (t) dt d X 0 R2d R2d
∞ t t ε 2 P f (t, X ) A 2 , X ∂ E ϕ(H0 ) + B , X ∂ E E ϕ(H0 ) + ε ε2 R2d 0 ×ζ (t) d X dt = O(ε), (8) where O(ε) is estimated by Cε for some C, independent of ε, which depends on the test functions ζ and ϕ. Here, the coefficients A(t/ε2 , X ) and B(t/ε2 , X ) are defined in Hypothesis 3. Proof of Proposition 5. Propositions 3 and 4 readily give, upon testing Eq. (1) against ζ (t) ϕ(H0 (X )), ∞ (P f ε )(0, X ) ϕ(H0 (X )) ζ (0) + (P f ε )(t, X ) ϕ(H0 (X )) ζ (t) 2d 2d R R 0 ∞ ε ε L (P f )(t, X ) ϕ(H0 (X )) ζ (t) dt d X = O(ε). + 0
R2d
38
F. Castella, P. Degond, Th. Goudon
Now, the proof of Proposition 4 (estimate (2)) provides the weak form of the operator L ε , namely ∞ L ε (P f ε )(t, X ) ϕ(H0 (X )) ζ (t) dt d X 0
=
R2d ∞
0
t/ε2
e−γ u P f ε (t, X ) × V (t/ε2 ) , H0 Su V (t/ε2 + u) , H0 ∂ E2 E ϕ(H0 ) ∂ E ϕ(H0 ) ζ (t + ε2 u) du d X dt. (9) + V (t/ε2 ) , Su V (t/ε2 + u) , H0 R2d
0
In order to replace ζ (t + ε2 u) by ζ (t) in (9), we evaluate the difference (here, we do not rewrite the exact value of the term between brackets), 2 ∞ t/ε e−γ u P f ε (t, X ) [· · · ] ζ (t + ε2 u) − ζ (t) du d X dt 0 R2d 0 t/ε2 ≤ ε2 × T ζ L ∞ × u e−γ u P f ε L ∞ (R+ ;L 2 (R2d )) [· · · ] L 2 (R2d ) du 0
≤ ε × T ζ L ∞ × C(R) ϕ W 2,∞ × 2
t/ε2
u (1 + u) p e−γ u du
0
≤ C ε2 , where we used Propositions 1 and 2, and the last constant C depends on T , R, γ , ϕ and ζ . The proposition is proved. 2.4. Compactness properties. According to Proposition 5, the limiting dynamics of P f ε may be obtained upon averaging out the coefficients A(t/ε2 , X ) and B(t/ε2 , X ) in Eq. (8). The averaging procedure naturally needs bounds together with compactness properties. These elements are essentially gathered in Proposition 6. Yet the compactness at hand is not enough to conclude at once, since Eq. (8) involves products of weakly convergent sequences P f ε (t, X )× A(t/ε2 , X ) and P f ε (t, X )× B(t/ε2 , X ). The crucial point, which allows to deal with such products, is that the function P f ε (t, X ) possesses some compactness in the time variable t, while A(t/ε2 , X ) and B(t/ε2 , X ) have some compactness in the phase-space variable X . Based on this observation Proposition 7 shows one can pass to the weak limit in the above products. The idea of exploiting this particular structure in products of weakly convergent sequences is borrowed from a similar observation made in [Li] in a different context. It may also be seen as a version of the compensated-compactness principle. Let us come to the details. We first state Proposition 6. Take any time T > 0. −2,1 (i) The sequence (∂t [h 0 (E) (f ε )(t, E)])ε>0 is bounded in L 2 (0, T ; Wloc (R)). ε As a corollary, the sequence (h 0 (E) (f )(t, E))ε>0 is relatively compact in C 0 ([0, T ]; L 2loc (R) − weak), and the sequence ((P f ε )(t, X ))ε>0 is relatively compact in C 0 ([0, T ]; L 2loc (R2d ) − weak).
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39
(ii) There exists a function F(t, E) ∈ C 0 (R+ ; L 2 (R)) such that, up to extracting subsequences, the family h 0 (E) (f ε )(t, E) converges to h 0 (E) F(t, E) in C 0 ([0, T ]; L 2loc (R) − weak), while the sequence (P f ε )(t, X ) converges to F(t, H0 (X )) in C 0 ([0, T ]; L 2loc (R2d ) − weak). Remark. We draw the reader’s attention to the following important difficulty. Proposition 6 asserts the sequence ((P f ε )(t, X ))ε>0 has some compactness in time. This is due to the fact that, when going to the energy variable E, the associated sequence (h 0 (E) (f ε )(t, E))ε>0 is once differentiable in time, with values lying in the negative Sobolev space W −2,1 (R), uniformly in t and ε. Note however that, though both sequences ((P f ε )(t, X ))ε>0 and (h 0 (E) (f ε )(t, E))ε>0 are roughly the same object, yet the time derivative of the sequence ((P f ε )(t, X ))ε>0 does not belong to a negative Sobolev space in any obvious way. From Proposition 6, we are able to deduce Proposition 7. Take two arbitrary test functions ζ (t) ∈ Cc∞ (R+ ) and ϕ(E) ∈ Cc∞ (R). The following convergence result holds: ∞ t (P f ε )(t, X )A 2 , X ζ (t)ϕ(H0 (X )) dtd X ε R2d 0 ∞ F(t, H0 (X ))A (X )ζ (t)ϕ(H0 (X )) dtd X. → ε→0 0
R2d
Similarly, we have, ∞ 0
t (P f ε )(t, X )B , X ζ (t)ϕ(H0 (X )) dtd X ε2 R2d ∞ F(t, H0 (X ))B (X )ζ (t)ϕ(H0 (X )) dtd X. −→ ε→0
0
R2d
Proof of Proposition 6. The whole proposition is essentially a consequence of Proposition 3 combined with the co-area formula (14). We write, taking a compactly supported test function ϕ ∈ W 2,∞ (R), ∂t h 0 (E) (f ε )(t, E) ϕ(E) d E R = ∂t P f ε (t, X ) ϕ(H0 (X )) d X (using the co-area formula) 2d R ε = L P f ε + ε R ε + ε I ε (t, X ) ϕ(H0 (X )) d X (using Proposition 3). R2d
Hence, using Proposition 4, we may upper-bound, 0
T
R
ε
∂t h 0 (E) (f )(t, E) ϕ(E) d E
2 dt ≤ C ϕ 2W 2,∞ (R) ,
−2,1 and the sequence (∂t [h 0 (E) (f ε )(t, E)])ε>0 is bounded in L 2 (0, T ; Wloc (R)).
40
F. Castella, P. Degond, Th. Goudon
On the other hand, Proposition 1 and the co-area formula allow to similarly establish that the sequence (h 0 (E) (f ε )(t, E))ε>0 is bounded in L ∞ (R+ ; L 2loc (R)). Indeed, taking a compactly supported test function ϕ ∈ L 2 (R), we may write ε h 0 (E) (f ε )(t, E) ϕ(E) d E = P f (t, X ) ϕ(H (X )) d X 0 2d R
R
≤ C ϕ(H0 (X )) L 2 (R2d ) ≤ C ϕ L 2 (R) ,
where the first inequality uses Proposition 1 and the second estimate uses once again that H0 is confining. Therefore, standard compactness results give that the sequence (h 0 (E) (f ε ) (t, E))ε>0 is relatively compact in C 0 ([0, T ]; L 2loc (R) − weak). Hence the existence of F(t, E) ∈ C 0 ([0, T ]; L 2loc (R)) such that h 0 (E) (f ε )(t, E) goes to h 0 (E) F(t, E) in C 0 ([0, T ]; L 2loc (R) − weak), as ε → 0. Besides, using the co-area formula again gives the similar compactness property for the sequence ((P f ε )(t, X ))ε>0 . Indeed, taking a compactly supported test function ψ(X ) ∈ L 2 (R2d ), we may write ε (P f )(t, X ) ψ(X ) d X = h 0 (E) (f ε )(t, E) (ψ)(E) d E R2d R −→ h 0 (E) F(t, E) (ψ)(E)d E = F(t, H0 (X )) ψ(X ) d X, R
R2d
where the convergence is uniform as t varies in compact subsets of R+ (thanks to the previous result). The proposition is now proved. Proof of Proposition 7. We only prove the first convergence result, the other one being completely similar. Our proof closely follows that of a similar statement, given in [Li] (Lemma 5.1, p. 12). We choose two cutoff functions in time, resp. in space, (X ) ∈ Cc∞ (R2d ) and χ (t) ∈ Cc∞ (R+ ), such that ≥ 0, χ ≥ 0, = 1, χ = 1. Associated with these regularizing function, we take a small parameter δ > 0, and set t 1 1 X , δ (X ) := 2d , χδ (t) := χ δ δ δ δ together with (P f ε )δ (t, X ) :=
R+
(P f ε )(t + s, X ) χδ (s) ds,
Fδ (t, X ) := F(t + s, X ) χδ (s) ds, + R t t , X := A 2 , X − Y δ (Y ) dY, Aδ 2d ε2 ε R A δ (X ) := A (X − Y ) δ (Y ) dY. R2d
With these notations, the argument, borrowed from [Li] and developed in the subsequent steps below, is the following. First, standard functional analysis shows that Proposition
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41
7 is true when (P f ε ), A(t/ε2 , X ), F, A are replaced by (P f ε )δ , Aδ (t/ε2 , X ), Fδ and A δ , respectively. Second, the compactness of f ε in time (Proposition 6) turns out to imply some compactness of P f ε in time as well. Similarly, the coefficients A(t/ε2 , X ), B(t/ε2 , X ) are smooth in X , uniformly in t/ε2 , and for this reason they possess some compactness in X . From this it follows that the regularized function (P f ε )δ , Aδ (t/ε2 , X ), Bδ (t/ε2 , X ) go to P f ε , A(t/ε2 , X ), B(t/ε2 , X ) in some strong topology as δ goes to zero. The conclusion follows. First step: Proof of a regularized version of the proposition. We write, for any fixed value of δ > 0, ∞ t (P f ε )δ (t, X ) Aδ , X ζ (t) ϕ(H0 (X )) dt d X ε2 R2d 0 ∞ (P f ε )(s, X ) δ (X − Y ) ϕ(H0 (X )) d X = 2d R2d 0 R ×
∞
A
0
t ,Y ε2
=: pδ,ε (s,Y )
χδ (s − t) ζ (t) dt ds dY,
=:aδ,ε (s,Y )
and all integrals actually carry over compact compact sets, say t ∈ [0, T0 ], s ∈ [0, T0 ], |X | ≤ R0 , |Y | ≤ R0 . Now, given δ > 0, and given arbitrary values of the variables s, Y , we have, using the known weak convergence of P f ε (Proposition 6), F(s, H0 (X )) δ (X − Y ) ϕ(H0 (X )) d X. pδ,ε (s, Y ) −→ pδ (s, Y ) := ε→0
R2d
(We used here that the convergence of P f ε is weak in space but pointwise in time). Besides, we have the uniform bound pδ,ε (s, Y ) ≤ C δ (X − Y ) 2 2d ≤ C δ −d , L (R ) Y
where C is independent of ε and δ, but it does depend on ϕ. As a consequence, the dominated convergence theorem gives F(s, H0 (X )) δ (X − Y ) ∀δ > 0, pδ,ε (s, Y ) −→ ε→0
R2d
×ϕ(H0 (X )) d X strongly in L 2loc (R+ × R2d ). On the other hand, the function aδ,ε (s, Y ) satisfies the uniform bound aδ,ε (s, Y ) ≤ C(R) γ − p−1 χδ (s − t) 1 2d ≤ C, L (R ) t
where C is independent of ε and δ (we used Proposition 2). All this information allows to deduce T0 pδ,ε (s, Y ) − pδ (s, Y ) aδ,ε (s, Y ) ds dY 0
B(R0 )
≤ C pδ,ε − pδ L 2 ([0,T0 ]×B(R0 )) = oδ (1),
42
F. Castella, P. Degond, Th. Goudon
where C is independent of ε and δ, and oδ (1) denotes a term which goes to zero with ε, for any fixed δ > 0. Using these notations, we arrive at ∞ t (P f ε )δ (t, X ) Aδ , X ζ (t) ϕ(H0 (X )) dt d X ε2 R2d 0 T0 = pδ (s, Y ) aδ,ε (s, Y ) ds dY + oδ (1)
0
T0
=
B(R0 )
B(R0 )
0
A
t ,Y ε2
× χδ (s − t) ζ (t) ds d X
T0
= 0
B(R0 )
0
B(R0 )
0
F(s, H0 (X )) δ (X − Y ) ϕ(H0 (X ))
dt dY + oδ (1)
T0
A (Y )
T0
× χδ (s − t) ζ (t) ds d X
B(R0 )
F(s, H0 (X )) δ (X − Y ) ϕ(H0 (X ))
dt dY + oδ (1),
+ where the last equality uses the assumed weak convergence of A(t/ε2 , Y ) in L ∞ loc (R × 2d R )-weak- (Hypothesis 3). Hence, we may conclude ∞ t (P f ε )δ (t, X ) Aδ , X ζ (t) ϕ(H0 (X )) dt d X ε2 R2d 0 ∞ Fδ (t, X ) A δ (X ) ζ (t) ϕ(H0 (X )) dt d X + oδ (1). = 0
R2d
Second step: Estimating the effect of the regularization. The previous step allows to write
∞ t ε (P f )(t, X ) A 2 , X − F(t, H0 (X )) A (X ) ζ (t) ϕ(H0 (X )) dt d X ε R2d 0 ∞ t t ε ε (P f )(t, X ) A 2 , X − (P f )δ (t, X ) Aδ ,X = oδ (1) + ε ε2 0 R2d ∞ ×ζ (t) ϕ(H0 (X )) dt d X + [Fδ (t, H0 (X )) A δ (X ) 0
R2d
−F(t, H0 (X )) A (X ) ] ζ (t) ϕ(H0 (X )) dt d X =: oδ (1) + Iδ,ε + IIδ .
(10)
This serves as a definition of the two terms Iδ,ε and IIδ . We now prove that Iδ,ε and IIδ go to zero with δ, uniformly in ε. We begin with the most difficult term Iδ,ε . We first split in the obvious way ∞ t t (P f ε )δ (t, X ) A 2 , X − Aδ , X ζ (t) ϕ(H0 (X )) dt d X Iδ,ε = ε ε2 0 R2d ∞ t (P f ε )(t, X ) − (P f ε )δ (t, X ) A 2 , X ζ (t) ϕ(H0 (X )) dt d X. + ε 0 R2d
Large Time Dynamics
43
Next, going to the energy variable in order to treat the second term later, we split further ∞ t t ε Iδ,ε = (P f )δ (t, X ) A 2 , X − Aδ ,X ζ (t) ϕ(H0 (X )) dt d X ε ε2 R2d 0 ∞ t ε ε h 0 (E) (f )(t, E)−( f )δ (t, E) (A) 2 , E ζ (t) ϕ(E) dt d E + ε R 0 ∞ t t (P f ε )δ (t, X ) A 2 , X − Aδ ,X ζ (t) ϕ(H0 (X )) dt d X = 2 2d ε ε 0 R ∞ t ε ε h 0 (E) (f )(t, E)−( f )δ (t, E) (A)δ , E ζ (t) ϕ(E) dt d E + ε2 R 0 ∞ h 0 (E) (f ε )(t, E) − (f ε )δ (t, E) + R 0 t t , E ζ (t) ϕ(E) dt d E. × (A) 2 , E − (A)δ ε ε2 Here, we have used a regularization of (A)(t, E) in the energy variable E, namely t t (A)δ , E := (A) 2 , E − E χδ s (E ) d E ε2 ε R t s = (A) 2 , E − δ E χ (E ) d E , (11) ε R and s ∈]0, 1/2[ is a parameter that can be chosen arbitrarily (the need for s ∈]0, 1/2[ becomes clear later). At this point, we have split Iδ,ε into (1) (2) (3) + Iδ,ε + Iδ,ε , Iδ,ε = Iδ,ε (i) with the obvious notations. We prove that each term Iδ,ε , i = 1, 2, 3, goes to zero with δ, independently of ε. (1) The first term Iδ,ε is easily bounded by, say, t t (1) , Iδ,ε ≤ C sup sup sup A 2 , X + Y − A 2 , X ∞ ε ε |Y |≤δ ε>0 0≤t≤T0
L (B(R0 ))
where C is independent of ε and δ. Here, we implicitly assumed that supp ⊂ B(1). We also used the uniform boundedness of P f ε in L 2loc and the fact that ζ and ϕ have compact supports. Now, our assumptions on the potential V , the Hamiltonian H0 , and the flow X , readily imply the following uniform Lipschitz bounds, valid for any compact set K ⊂ R+ × R2d , 2 D V t t D2 V ,X +Y − , X ≤ C δ, sup sup sup 2 2 2 2 Dx ε Dx ε |Y |≤δ ε>0 (t,X )∈K DV t t DV ≤ C δ, sup sup sup , X + Y − , X 2 2 Dx ε Dx ε |Y |≤δ ε>0 (t,X )∈K 2 D H0 D 2 H0 sup sup (X + Y ) − (X ) ≤ C δ, 2 2 DX |Y |≤δ (t,X )∈K D X DX DX sup sup sup (u, X + Y ) − (u, X ) ≤ C δ (1 + u) p , D X D X |Y |≤δ ε>0 (t,X )∈K
44
F. Castella, P. Degond, Th. Goudon
where C is independent of ε, δ, and u ≥ 0, but it does depend on the compact set K . An easy adaptation of the proof of Proposition 2 then allows to deduce from these bounds that the function A(t/ε2 , X ) is Lipschitz, locally in X but uniformly in the first variable, i.e. t t sup sup sup A 2 , X + Y − A 2 , X ≤ C δ, (12) ∞ ε ε |Y |≤δ ε>0 0≤t≤T0 L (B(R0 )) where C is independent of ε and δ (but it does depend on R0 , T0 ). The conclusion is (1) Iδ,ε ≤ C δ, where C is independent of ε and δ. (2) The analysis of Iδ,ε follows roughly the same idea: we write (2) Iδ,ε =
∞ 0
ε
ε
ζ (t) h 0 (E) (f )(t, E)−h 0 (E) (f )δ (t, E) ,(A)δ
t , E ϕ(E) dt, ε2
where . , . denotes the duality bracket in the energy variable E. Hence we may safely estimate t (2) ε , Iδ,ε ≤ C δ sup sup ∂t h 0 (E) (f )(t +θ δ, E) ,(A)δ 2 , E ϕ(E) 2 ε ε>0 0≤θ≤1
L ([0,T0 ])
where C is independent of ε and δ. Using now the uniform boundedness of ∂t [h 0 (E) −2,1 (f ε )] in L 2loc (Wloc ), we recover, for some E 0 > 0 that depends on the support of ϕ, t (2) , E ϕ(E) Iδ,ε ≤ C δ sup (A)δ 2,∞ 2 ε ε>0 W (R ) 0≤t≤T0 t ≤ C δ sup (A)δ ε2 , E 2,∞ ε>0 W (]0,E 0 [) 0≤t≤T0 t ≤ C δ 1−2s sup (A) ε2 , E ∞ ε>0 L (]0,E 0 +1[) 0≤t≤T0 t ≤ C δ 1−2s sup . A ε2 , X ∞ ε>0 L (B(R0 )) 0≤t≤T0
Note that the need for using a regularized version of (A)(t, E), namely (A)δ )(t, E), (1) (2) (3) in the splitting Iδ,ε = Iδ,ε + Iδ,ε + Iδ,ε is clear in this estimate: it is enforced by the low ε regularity of ∂t [h 0 (E) (f )] (t, E) in the E variable. The choice 0 < s < 1/2 also appears clearly now. The conclusion is, using Proposition 2-(iii) (2) Iδ,ε ≤ C δ 1−2s , with C independent of δ and ε.
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45
(3)
Last the term Iδ,ε is obviously bounded by, say, t t sup (A) 2 , E + E −(A) 2 , E 2 ε ε s |E |≤δ ε>0 0≤t≤T0 L ([0,E 0 ],h 0 (E) d E) t t (P A) 2 , E + E − (P A) 2 , E ≤ C sup sup sup , 2 ε ε |E |≤δ s ε>0 0≤t≤T0 L (B(R0 )
(3) Iδ,ε ≤ C sup sup
for some given E 0 > 0 (that depends on R0 ), and C is independent of ε and δ. Here we used the co-area formula. Now, estimate (12) establishes that the functions A(t/ε2 , X ) ∈ L 2 (B(R0 )), parametrized by t ∈ [0, T0 ] and ε > 0, satisfy a uniform equi-integrability criterion, hence belong to a relatively compact set of L 2 (B(R0 )). The continuity of the projection operator P readily implies that the functions (P A)(t/ε2 , X ) ∈ L 2 (B(R0 )) also belong to a relatively compact set of L 2 (B(R0 )). This in turn implies that the functions (P A)(t/ε2 , X ), parametrized by t and ε, satisfy a uniform equi-integrability criterion, namely t t (P A) sup sup sup − (P A) , E + E , E −→ 0. (13) 2 2 2 ε ε |E |≤δ s ε>0 0≤t≤T0 L (B(R0 ) δ→0 The conclusion is (3)
Iδ,ε −→ 0, uniformly with respect to ε. δ→0
(i)
Summarizing, the above estimates on Iδ,ε (i = 1, 2, 3) give Iδ,ε −→ 0, uniformly with respect to ε. δ→0
Let us now come to the easier term IIδ . We write as before ∞ Fδ (t, H0 (X )) [A δ (X ) − A (X )] ζ (t) ϕ(H0 (X )) dt d X IIδ = 0 R2d ∞ h 0 (E) [F(t, E) − Fδ (t, E)] (A )δ (E) ζ (t) ϕ(H0 (X )) dt d E + 0 ∞ R h 0 (E) [F(t, E) − Fδ (t, E)] [(A )(E) − (A )δ (E)] + 0
R
×ζ (t) ϕ(E) dt d E, where we defined the regularization of (A )(E) in the energy variable E, (A )δ (E) := (A )(E − E ) χδ s (E ) d E = (A )(E −δ s E ) χ (E ) d E , R
R
(14) and s ∈]0, 1/2[ is a parameter that can be chosen arbitrarily. At this point, we have split IIδ into (1)
(2)
(3)
IIδ = IIδ + IIδ + IIδ ,
46
F. Castella, P. Degond, Th. Goudon
with the obvious notations. A fairly easy analyzing the term Iδ,ε gives successively (1) IIδ ≤ C δ,
adaptation of the estimates produced while (2) IIδ ≤ C δ,
with C independent of δ, and (3)
IIδ −→ 0. δ→0
The adaptation is obtained upon observing that the various functions A(t/ε2 , X ), etc. entering the analysis of Iδ,ε , and their various weak limits (in the relevant spaces) A (X ), etc. that enter the analysis of IIδ , admit the same boundedness properties (typically, if u n u in L 2 , then u L 2 ≤ inf n u n L 2 ). This is enough to reproduce the arguments given for Iδ,ε . Third step: Conclusion. Summarizing, we have proved up to now that
∞ t (P f ε )(t, X ) A 2 , X − F(t, H0 (X )) A (X ) ε R2d 0 ×ζ (t) ϕ(H0 (X )) dt d X = oδ (1) + O(δ), where oδ (1) goes to zero as ε → 0 for any fixed value of δ > 0, and O(δ) is estimated by C δ with C independent of ε and δ. This proves the proposition.
2.5. The homogenization procedure: Proof of the theorem. We are now in position to end the proof of the main theorem. First step: Homogenization procedure. Take two smooth test functions ζ (t) ∈ Cc∞ (R+ ) and ϕ(E) ∈ Cc∞ (R). Proposition 5 asserts
∞ P f 0ε (t, X ) ϕ(H0 (X )) ζ (0) d X + P f ε (t, X ) ϕ(H0 (X )) ζ (t) dt d X R2d 0 ∞ t ε P f (t, X ) A 2 , X ∂ E ϕ(H0 ) + 2d ε 0
R t , X ∂ E2 E ϕ(H0 ) ζ (t) d X dt = O(ε). +B ε2
Hence, Propositions 6 and 7 allow to pass to the limit and eventually obtain F(0, H0 (X )) ϕ(H0 (X )) ζ (0) d X R2d ∞ F(t, H0 (X )) ϕ(H0 (X )) ζ (t) dt d X + 2d 0 ∞ R F(t, H0 (X )) A (X ) ∂ E ϕ(H0 )+B (X ) ∂ E2 E ϕ(H0 ) ζ (t) d X dt = 0. + 0
R2d
Large Time Dynamics
47
The co-area formula next provides
R
∞
h 0 (E) F(0, E) ϕ(E) ζ (0) d E + h 0 (E) F(t, E) ϕ(E) ζ (t) dt d E 0 R ∞ h 0 (E) F(t, E) a(E)∂ E ϕ(E) + b(E)∂ E2 E ϕ(E) ζ (t) d E dt = 0, (15) + 0
R
where a(E) := A (E), b(E) := B (E). Note that both coefficients a and b are well-defined and belong to L ∞ (R) since the two functions A and B belong to L ∞ (R2d ). Note also that Eq. (15) is the weak formulation of the second order equation 2 ∂t [h 0 (E) F(t, E)] + ∂ E [a(E) h 0 (E) F(t, E)] − ∂ E,E [b(E) h 0 (E) F(t, E)] = 0. (16)
There remains to establish the properties of the coefficients a(E) and b(E). This is done in the next step. Second step: Properties of the effective coefficients. The proof we give now establishes the claimed formula h 0 a = ∂ E (h 0 b). It also sheds some light on the connection between the present setting and the case when the potential V is assumed (quasi-)periodic in time. 2d As a preliminary, let h ∈ L ∞ loc (R × R ). We associate to any such function the quantity
∞
χh (t, X ) :=
h t + s, X (s, X ) e−γ s ds,
0 2d which obviously also belongs to L ∞ loc (R × R ). An immediate computation shows χh satisfies the backward transport equation (or: adjoint equation) −∂t χh − {H0 , χh } + γ χh = h,with vanishing initial datum. With this notation, we may introduce the profile χ = χ (t, X ) given by
χ := χ{V , H0 } . The profile χ allows to express the coefficients a and b in the following way:
∞ a(E) = − {V (τ ) , χ (τ )} dτ (E),
0 ∞ b(E) = − {V (τ ) , H0 } χ (τ ) dτ (E), 0
∞ where − · · ·
1 T ···. T 0 0 This observation allows to both prove the non-negativity of b and the relation h 0 a = stands for the weak limit in L ∞ loc − as T → ∞ of
48
F. Castella, P. Degond, Th. Goudon
∂ E (h 0 b). Indeed, we may write, on the one hand ∞ − {V (τ ) , H0 } χ (τ ) dτ 0
T 1 − (∂τ χ (τ ) + {H0 , χ (τ )} − γ χ (τ )) χ (τ ) dτ = lim T →∞ T 0 2 χ (0) − χ 2 (T ) 1 T γ T 2 − = lim H0 , χ 2 /2 dτ + χ (τ ) dτ T →∞ 2T T 0 T 0 T 1 γ T 2 H0 , χ 2 /2 dτ + χ (τ ) dτ , = lim − T →∞ T 0 T 0
from which it follows, using Lemma 1, that ∞ γ T 2 χ (τ ) dτ ≥ 0. P − {V (τ ) , H0 } χ (τ ) dτ = P lim T →∞ T 0 0 Hence b(E) ≥ 0. On the other hand, using the co-area formula and integration by parts, we have, for any trial function ψ ∈ Cc∞ (R), ∞ h0a ψ d E = − {V, χ } ψ(H0 (X )) d X dτ R R2d 0 ∞ − χ {V, ψ(H0 (X ))} d X dτ =− R2d 0 ∞ − χ {V, H0 (X )} (∂ E ψ)(H0 (X )) d X dτ =− R2d 0 = − h 0 b ∂ E ψ d E, R
which proves h 0 a = ∂ E (h 0 b). This ends the proof of our main theorem.
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Communicated by J.L. Lebowitz
Commun. Math. Phys. 276, 51–91 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0329-9
Communications in
Mathematical Physics
The Spectral Shift Function and Spectral Flow N. A. Azamov1 , A. L. Carey2 , F. A. Sukochev1 1 School of Informatics and Engineering, Flinders University of South Australia, Bedford Park,
5042, SA, Australia. E-mail: azam0001;
[email protected]
2 Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia.
E-mail:
[email protected]
Received: 5 May 2006 / Accepted: 7 May 2007 Published online: 28 August 2007 – © Springer-Verlag 2007
Abstract: At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general semifinite von Neumann algebra N and the perturbation V ∈ N . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples where D has compact resolvent belonging to N and V is any bounded self-adjoint operator in N . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].
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Introduction Overview. In [3] we gave an analytic proof that the spectral shift function of M. G. Krein computes the spectral flow under certain restricted circumstances. Spectral flow stems from the work of Atiyah-Patodi-Singer [2] where it is introduced primarily in a topological sense. Subsequently, starting with a suggestion of I. M. Singer at the 1974 Vancouver ICM, the idea that spectral flow could be expressed as the integral of a one form has been extensively studied in the framework of unbounded Fredholm modules (or spectral triples) beginning with [24] and continuing in [15, 16]. The current state of knowledge using an analytic approach due to Phillips [30, 31] is described in detail in [5]. On the other hand the Krein spectral shift function has an extensive history in perturbation theory which may be partly traced from [26, 6, 11, 32]. The spectral shift theory was developed by both physicists and mathematicians in the context of perturbation theory of Schrödinger operators. By the Birman-Krein formula, it is related to the phase of the scattering matrix in the latter’s spectral representation. Krein’s formula for the spectral shift function, which is the motivation for this paper, is restricted to the case where one perturbs an arbitrary unbounded self adjoint operator D0 by a trace class operator. The spectral shift function compares in a sense the relation between the spectrum of D0 and that of its perturbation. In this paper we will extend the theory of the spectral shift function to the situation where we perturb D0 by an arbitrary bounded self adjoint operator but adopt the assumption that D0 has compact resolvent motivated by the notion of spectral triple. One surprising conclusion that follows is that the Atiyah-Patodi-Singer (or APS) index theorem [2] and its generalizations are, in certain instances, computing the spectral shift function. The generalizations of the APS index theorem we are referring to in the previous paragraph have to do with situations where von Neumann algebras other than all the bounded operators on a Hilbert space arise. The standard theory of spectral flow deals with unbounded self adjoint Fredholm operators. Using an approach due to Phillips op cit spectral flow may be defined analytically (but as yet not topologically) for certain operators affiliated to semifinite von Neumann algebras, the so-called Breuer-Fredholm theory, which is reviewed in [5]. In addition the connection between spectral flow in semifinite spectral triples and generalizations of the APS index theorem is explained in [5] with reference to the extensive previous history of the matter. Spectral flow is related to odd degree K -theory and the odd local index theorem in noncommutative geometry [20, 17]. For the purposes of using spectral flow to obtain information about K -theory it is enough to consider unbounded operators D with compact resolvent. However, previous studies of spectral flow formulae have restrictions 2 on D such as theta summability (requiring the heat operator e−t D to be trace class for all t > 0). In this paper we will relax this condition to capture the more general situation where only compact resolvent is needed. By contrast the theory of the spectral shift function was formulated for Schrödinger operators associated to Euclidean spaces and in that setting the assumption of compact resolvent does not generally hold. Thus it is not surprising that the relationship between the Krein spectral shift function and spectral flow should have remained somewhat unexplored until recently. In [3] we noticed that under hypotheses that guaranteed that both the Krein spectral shift function and spectral flow exist then they are essentially the same notion. This led us to the current investigation where we borrow some ideas from spectral flow theory as formulated in a spectral triple to extend the range of situations for which the Krein spectral shift function may be defined. We are then able to obtain new results on the spectral shift function and to prove, by a combination of the methods of [3] and [15, 16]
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that the spectral shift function gives a wide variety of analytic formulae for spectral flow. In fact it is the interaction between these previously disparate theories that makes these advances possible. An additional very important feature of our analysis is its generality; we are able to study self adjoint operators that are affiliated to a general semifinite von Neumann algebra N in contrast to the classical theory which deals with the special case where N is the algebra of all bounded operators on a Hilbert space H. (Note that there are examples of quantum mechanical Hamiltonians whose resolvent lies in a semifinite von Neumann algebra, see e.g. [14].) Summary of results. In order to keep this paper to a reasonable length we have omitted a detailed discussion of spectral flow in von Neumann algebras referring to the paper [5] for this. On the other hand we include in Sect. 1 preliminary results particularly those from [3] which may be less accessible to the reader. This latter paper is very important for the present investigation because it provides a calculus of functions of operators more effective than has been available in the past. The key technical advance that we exploit is the use of double operator integrals (DOI) not in its original form [8] but in the much more effective form discovered in [3]. In particular it is this latter paper that develops a calculus for functions of operators affiliated to general semifinite von Neumann algebras. This calculus replaces the more elementary perturbation theory techniques in [15, 16] and is more user friendly than that of [27]. The main new results on the Krein spectral shift function are in Sect. 2. Here we show that by starting with self adjoint operators D0 with compact resolvent in a semifinite von Neumann algebra we are able to define the Krein spectral shift function for general bounded self adjoint perturbations V. Section 2 also exhibits some properties of the spectral shift function for operators satisfying these hypotheses mostly in the context of preparing the ground for the subsequent discussion of spectral flow. Specifically D0 D0 +V we observe that the spectral projections E (a,b) and E (a,b) of D0 and D0 + V = D1 respectively are finite in N for bounded intervals (a, b). Using a fixed faithful normal semifinite trace τ on N we define (following [7]) the spectral shift measure for the pair D0 , D1 , by 1 Dr D1 ,D0 () = dr, τ V E 0
where is a bounded Borel subset of the real line and Dr = D0 + r V, r ∈ [0, 1]. We consider this Birman-Solomyak formula (formula (16) in the text) for the spectral shift measure as fundamental. It is this formula which in our opinion must be taken as the definition of the generalized spectral shift function of a pair of operators, whenever this expression makes sense for that pair. We then prove that this measure is absolutely continuous with respect to Lebesgue measure and the resulting Radon-Nikodym derivative we define to be the spectral shift function ξ D1 ,D0 (λ). This function can be related to the original spectral shift function of Krein which was introduced in a completely different fashion. Furthermore, under our assumptions, the spectral shift function ξ D1 ,D0 (λ) is in fact a function of a bounded variation and there is a canonical representative which is an everywhere defined function. We remark that in [33] B. Simon outlined (without proof) some similar results on the spectral shift function. In Sect. 3 we present a series of new analytic formulae for spectral flow. These require the use of the spectral shift function and its properties derived in Sect. 2. The formulae
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N. A. Azamov, A. L. Carey, F. A. Sukochev
we obtain subsume those of [16]. We are able to provide a variety of analytic formulae whenever D0 has compact resolvent and show how these formulae can be specialized when there are summability hypotheses imposed. For illustrative purposes we describe two of our main theorems here. Suppose that there is a unitary operator u with D1 = u D0 u ∗ and such that V = u[D0 , u ∗ ] ∈ N . That is, for a dense subalgebra of the C ∗ -algebra generated by u we have a semifinite spectral triple. Then spectral flow from D0 − λ to D1 − λ for any real λ is equal to ξ D1 ,D0 (λ). We relate our results to previous formulae for spectral flow via the following result. For any positive integrable function f on the real line with f (D0 + r V ) trace class for r ∈ [0, 1] and r → f (D0 + r V )1 being 1-summable on [0, 1], we find that spectral flow from D0 − λ to D1 − λ for any real λ is given by 1 sf(λ; D0 , D1 ) = C −1 τ (V f (Dr − λ)) dr, ∞
0
where C = −∞ f (x) d x. In fact in this restricted situation of unitarily equivalent endpoints we can prove that the spectral shift function is constant implying that spectral flow occurs uniformly past any point on the real line not just zero. We also obtain analogues of these results when the endpoint operators are not unitarily equivalent. Then the relation between spectral flow and the spectral shift function is modified by the addition of endpoint correction terms (in a fashion analogous to [16]). The key idea we exploit in this part of the paper is the observation [24] that spectral flow can be written as the integral of an exact one form on the affine space of bounded perturbations of a fixed unbounded operator D0 . It eventuates that there is a way to construct a large class of such one forms that will compute spectral flow when the only constraint on D0 is that it have compact resolvent. This advance is made possible by the understanding of the spectral shift measure afforded by Sect. 2. It then follows that the spectral shift function computes spectral flow with the caveat that there are simple correction terms arising from the zero eigenspace of each endpoint of the path. One motivation for this investigation arises from [17] where the odd local index theorem in noncommutative geometry was deduced from the analytic formula for spectral flow in [16] via an intermediate formula in terms of a cyclic cocycle which was called the ‘resolvent cocycle’. The results and viewpoint of this paper may have other applications to noncommutative geometry besides spectral flow. Specifically we have in mind situations where we would like to avoid ‘summability constraints’ on D0 . 1. Notation and Preliminary Results 1.1. Notation. We write H for a fixed separable complex Hilbert space, and by N we denote a semifinite von Neumann algebra acting on H [34, V. 1. 21]. We use the usual notation DηN for operators D affiliated with N [34, IV. 5, Exercise 3]. We let τ be a faithful normal semifinite trace on N [34, V. 2. 1] and by L 1 (N , τ ) we denote the set of τ -trace class operators affiliated with N [34, V.2, p. 320]. Then we use the notation L1 (N , τ ) = L 1 (N , τ ) ∩ N for the (unitarily) invariant operator ideal [16, Appendix A.2] of all bounded τ -trace class operators with norm ·1,∞ = · + ·1 , where · is the usual operator norm and ·1 := τ (|·|) [34, V.2, p. 320]. By B(R) we denote the T be the spectral σ -algebra of all Borel subsets of R. For a self-adjoint operator T let E projection of T corresponding to ∈ B(R), and let E λT be the spectral projection of T corresponding to (−∞, λ]. Write Cc∞ (R) (respectively, Cc∞ ()) for the set of all
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compactly supported C ∞ -smooth functions on R (respectively, ⊆ R), and B(R) (respectively, Bc (R)) for the set of all bounded Borel functions on R (respectively, compactly supported bounded Borel functions on R). If an operator T affiliated with N is τ -measurable [22, Definition 1.2], then the t th generalized s-number µt (T ) of the operator T is defined by µt (T ) = inf {T E : E is a projection in N with τ (1 − E) t} . If E ∈ N is a τ -finite projection (i.e. τ (E) < ∞), then µt (E) = χ[0,τ (E)) (t), t 0.
(1)
An operator T ∈ N is said to be τ -compact iff lim µt (T ) = 0. The set of all τ -compact t→∞
operators from N forms an ideal in N which is denoted K(N , τ ). The ideal K(N , τ ) coincides with the norm closure of the ideal in N generated by τ -finite projections. Let Rz (D) = (z − D)−1 denote the resolvent of D. If D = D ∗ ηN and Rz (D) ∈ K(N , τ ) for some (and hence for all) z ∈ C \ R, then we say that D has τ -compact resolvent. An operator T ∈ N is called τ -Fredholm [29, Appendix B] if the projections [ker T ], [ker T ∗ ] are τ -finite and there exists a τ -finite projection E ∈ N such that ran(1 − E) ⊆ ran(T ). For a τ -Fredholm operator T one can define its τ -index by τ - ind(T ) = τ ([ker T ]) − τ ([ker T ∗ ]). Definition 1.1. Let N be a semifinite von Neumann algebra and let E, E be two invariant operator ideals over N . We denote the self-adjoint part of E by Esa . Let D0 be a fixed is called self-adjoint operator affiliated with N . A function f : D0 + Esa → f (D0 ) + Esa affinely (E, E )-Fréchet differentiable at D ∈ D0 + Esa , if there exists a (necessarily such that the following equality holds: unique) bounded operator L : Esa → Esa f (D + V ) − f (D) = L(V ) + o(V E ), V ∈ Esa . If E = E then we write L = DE f (D), otherwise we write L = DE ,E f (D). In our case the ideals E and E will be N or L1 (N , τ ). We constantly use some parameters for specific purposes. The parameter r will always be an operator path parameter, i. e. the letter r is used when we consider paths of operators such as Dr = D0 + r V. Very rarely we need another path parameter which we denote by s. We do not use t as a path parameter, since t is used for other purposes later in the paper. The letter λ is always used as a spectral parameter. If we need another spectral parameter we will use µ. We finish this subsection with the following elementary fact which we will use repeatedly. Lemma 1.2. If is an open interval in R and if f ∈ Cck (), then there exist functions √ √ f 1 , f 2 ∈ Cck () such that f 1 , f 2 are non-negative, f = f 1 − f 2 and f 1 , f 2 ∈ Cck (). Proof. Let [a, b] be a closed interval, [a, b] ⊆ and supp( f ) ⊆ (a, b). Take a nonnegative C ∞ -function f 1 f on [a, b] which vanishes at a and b in such a way that √ f 1 is C ∞ -smooth at a and b, and take f 2 = f 1 − f.
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1.2. Self-adjoint operators with τ -compact resolvent. The remaining subsections in this section are technical and dry. The reader may wish to browse this section then move on to Sects. 2 and 3 which contain the main results referring back to the technicalities of this section when needed. In this current subsection we collect some facts about operators with compact resolvent in a semifinite von Neumann algebra. We do not claim any great originality for these. Lemma 1.3. If D = D ∗ ηN has τ -compact resolvent and V = V ∗ ∈ N , then D + V also has τ -compact resolvent. Proof. This follows from the equality Rz (D +V ) = (Rz (D+V )V +1) Rz (D), z ∈ C\R. Lemma 1.4. If D = D ∗ ηN has τ -compact resolvent, then for all compact sets ⊆ R D is τ -finite. the spectral projection E Proof. If D has τ -compact resolvent then the operator (1+D 2 )−1 = (D+i)−1 (D−i)−1 is τ -compact. Since for every finite interval there exists a constant c > 0, not depending D c(1 + D 2 )−1 , the projection E D is also τ -compact, and hence on D such that E τ -finite. Corollary 1.5. If D = D ∗ ηN has τ -compact resolvent, then for all f ∈ Bc (R) the operator f (D) is τ -trace class. Proof. There exists a finite segment ⊆ R such that | f | const χ , so that | f (D)| D. const E Lemma 1.6. [15, Appendix B, Lemma 6] If D0 is an unbounded self-adjoint operator, A is a bounded self-adjoint operator, and D = D0 + A then (1 + D 2 )−1 f (A)(1 + D02 )−1 , √ where f (a) = 1 + 21 a 2 + 21 a a 2 + 4. Lemma 1.7. Let D0 = D0∗ ηN have τ -compact resolvent, and let B R = {V = V ∗ ∈ N : V R} . Then for any compact subset ⊆ R the function D0 +V V ∈ B R → E
is L1 (N , τ )-bounded. D0 +V c0 (1 + (D0 + V )2 )−1 for some constant c0 = c0 () > 0 and Proof. We have E ∗ for every V = V ∈ N . Now, by Lemma 1.6 there exists a constant c1 = c1 (R) > 0, such that for all V ∈ B R ,
(1 + (D0 + V )2 )−1 c1 (1 + D02 )−1 . D0 +V Hence, since D0 has τ -compact resolvent, all projections E , V ∈ B R , are bounded from above by a single τ -compact operator T = c0 c1 (1 + D02 )−1 . This means, that for t > 0, D0 +V µt (E ) µt (T ). D0 +V ) = χ[0,τ (E D0 +V )) (t) and there exists t0 > 0 such that µt0 (T ) 1. Further, by (1) µt (E
D0 +V ) t0 . This implies that for all V ∈ B R , τ (E
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Corollary 1.8. If D0 = D0∗ ηN has τ -compact resolvent, then for any function f ∈ Bc (R) the function V ∈ B R → f (D0 + V )1,∞ is bounded. Corollary 1.9. Let D0 = D0∗ ηN have τ -compact resolvent, r = (r1 , . . . , rm ) ∈ [0, 1]m , V1 , . . . , Vm ∈ Nsa and set Dr = D0 + r1 V1 + · · · + rm Vm . Then Dr (i) for any compact subset ⊆ R the function r ∈ [0, 1]m → E is bounded; 1 (ii) for any function f ∈ Bc (R) the function r ∈ [0, 1] → f (Dr )1 is bounded. An elementary proof of the following lemma can also be found in [15]. Lemma 1.10. If D0 = D0∗ ηN and if V = V ∗ ∈ N , then for any t ∈ R, eit (D0 +V ) converges in ·-norm to eit D0 when V → 0. Proof. Follows directly from Duhamel’s formula t eit (D0 +V ) − eit D0 = ei(t−u)(D0 +V ) i V eiu D0 du. 0
If (S, ν) is a finite measure space then we say that a function f : S → B(H) is measurable, if for any η ∈ H the functions f (·)η, f (·)∗ η : S → H are Bochner measurable. We define the integral of a measurable function by the formula f (σ ) dν(σ )η = f (σ )η dν(σ ), η ∈ H, S
S
and call it an so∗ -integral. If f : S → L1 (N , τ ) is an L1 -bounded function then it is measurable if and only if there exists a sequence of simple (finitely-valued) functions f n : S → L1 (N , τ ) such that f n (σ ) → f (σ ) in the so∗ -topology [3, Sect. 3]. Lemma 1.11. [3, Lemma3.10] If a measurable function f : S → L1 (N , τ ) is uniformly L1 (N , τ )-bounded then S f (σ ) dν(σ ) ∈ L1 (N , τ ) and τ f (σ ) dν(σ ) = τ ( f (σ )) dν(σ ). S
S
1.3. Difference quotients and double operator integrals. Originally, multiple operator integrals of the form ϕ(λ0 , . . . , λn ) d E λD00 V1 d E λD11 V2 . . . Vn d E λDnn , (2) TϕD0 ,...,Dn (V1 , . . . , Vn ) = Rn+1
where D0 , . . . , Dn are self-adjoint operators and V1 , . . . , Vn are bounded operators, were defined as repeated operator integrals [21] or as spectral integrals [8–10] in the case of double operator integrals. It was noted in [3] (see also [28]) that one can give another definition of multiple operator integrals (Definition 1.15). The idea of the new definition is that for functions ϕ(λ0 , . . . , λn ) of the form ϕ(λ0 , . . . , λn ) = α0 (λ0 )α1 (λ1 ) . . . αn (λn ) the expression (2) can be interpreted as α0 (D0 )V1 α1 (D1 )V2 . . . Vn αn (Dn ). Hence, for functions ϕ of the form (7) one can try to define the multiple operator integral by the
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formula (8). Having proved the correctness of this definition (Proposition 1.16), we will be able to work with the usual operator integrals of the form (8), provided that one has a representation of the form of Eq. (7) for the function ϕ. Actually, one has considerable freedom of choice of this representation, and one should try to find that representation of ϕ which is most suitable for the purpose at hand. In our case the function ϕ(λ0 , λ1 ) is the difference quotient f [1] (λ0 , λ1 ) :=
f (λ0 ) − f (λ1 ) λ0 − λ1
(3)
of some function f. We denote by C n,+ (R) the set of functions f ∈ C n (R), such that the j th derivative f ( j) , j = 1, . . . , n, belongs to the space F −1 (L 1 (R)), where F( f ) is the Fourier transform of f. We emphasize that the Fourier transform of a function f ∈ C n,+ (R) need not belong to L 1 (R) (see Lemma 1.12(ii)). Let
:= (s0 , s1 ) ∈ R2 : |s1 | |s0 | , sign(s0 ) = sign(s1 ) , and i dν f (s0 , s1 ) := sgn(s0 ) √ F( f )(s0 ) ds0 ds1 . (4) 2π
If f ∈ C 1,+ (R), then it is not difficult to see that , ν f is a finite measure space [3, Lemma 2.1], so that , ν f can be used for the construction of the double operator integral. The following Birman-Solomyak or BS-representation of f [1] (λ0 , λ1 ) [3, Lemma 2.2] [1] f (λ0 , λ1 ) = α(λ0 , σ )β(λ1 , σ ) dν f (σ ), (5)
where σ = (s0 , s1 ), α(λ0 , σ ) = ei(s0 −s1 )λ0 and β(λ1 , σ ) = eis1 λ1 , is not suitable for our present purposes. Lemma 1.14 provides a modification of this BS-representation for f [1] with which we will constantly work. We include the proof of the following fact for completeness. Lemma 1.12. (i) If f ∈ Cc1 (R), then fˆ ∈ L 1 (R). (ii) the function ϕ(x) = √ x 2 belongs to C 2,+ (R). 1+x
Proof.
(i) Following the proof of [13, Corollary 3.2.33], we have | fˆ|(ξ ) dξ = |ξ + i|−1 |ξ + i|| fˆ|(ξ ) dξ R
R
R
|ξ + i|
−2
1 2
dξ
|ξ + i| | fˆ| (ξ ) dξ 2
R
2
1
2 2
d f (x)
dx
+ f (x) = const < ∞.
dx R (ii) The proof is similar to that of (i).
1 2
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Applying part (i) of this lemma to the first n derivatives of a function f from Ccn+1 (R), we obtain Corollary 1.13. Ccn+1 (R) ⊆ C n,+ (R), n = 1, 2, . . .. The following lemma provides a BS-representation for f [1] , 0 f ∈ Cc2 (R), which will be used throughout this paper. √ Lemma 1.14. Let f ∈ Cc2 (R) be a non-negative function such that g := f ∈ Cc2 (R). If ⊇ supp( f ), then f [1] (λ0 , λ1 ) = (α1 (λ0 , σ )β1 (λ1 , σ ) + α2 (λ0 , σ )β2 (λ1 , σ )) dνg (σ ),
where σ = (s0 , s1 ) and α1 (λ0 , σ ) = ei(s0 −s1 )λ0 g(λ0 ), α2 (λ0 , σ ) = ei(s0 −s1 )λ0 ,
β1 (λ1 , σ ) = eis1 λ1 ,
β2 (λ1 , σ ) = eis1 λ1 g(λ1 ),
(6)
so that α1 (·, σ ), β2 (·, σ ) ∈ Cc2 () for all σ ∈ , and |α1 (·)| , |β2 (·)| g∞ , while α2 (·, σ ), β1 (·, σ ) ∈ C ∞ (R) for all σ ∈ , and |α2 (·)| , |β1 (·)| 1. Proof. The assumption g ∈ Cc2 (R) implies that g ∈ C 1,+ (R) (see Corollary 1.13). Now, g 2 (λ0 ) − g 2 (λ1 ) λ0 − λ1 g(λ0 ) − g(λ1 ) = (g(λ0 ) + g(λ1 )) = g [1] (λ0 , λ1 ) (g(λ0 ) + g(λ1 )) . λ0 − λ1
f [1] (λ0 , λ1 ) =
Hence, using (5), we have f [1] (λ0 , λ1 ) = (α(λ0 , σ )g(λ0 )β(λ1 , σ ) + α(λ0 , σ )g(λ1 )β(λ1 , σ )) dνg (σ ).
If we set α1 (λ, σ ) = α(λ, σ )g(λ), β1 (λ, σ ) = β(λ, σ ), α2 (λ, σ ) = α(λ, σ ) and β2 (λ, σ ) = g(λ)β(λ, σ ), then we see that all the conditions of the lemma are fulfilled. The next step is to recall the definition of the multiple operator integral as it was given in [3]. Let ϕ ∈ B(Rn+1 ) be a bounded Borel function on Rn+1 which admits a representation of the form ϕ(λ0 , λ1 , . . . , λn ) = α0 (λ0 , σ ) . . . αn (λn , σ ) dν(σ ), (7) S
where (S, ν) is a finite measure space and α0 , . . . , αn are bounded Borel functions on R × S. Definition 1.15. For arbitrary self-adjoint operators D0 , . . . , Dn on the separable Hilbert space H, bounded operators V1 , . . . , Vn on H and any function ϕ ∈ B(Rn+1 ) which admits a representation given by (7), the multiple operator integral TϕD0 ,...,Dn (V1 , . . . , Vn ) is defined as (8) TϕD0 ,...,Dn (V1 , . . . , Vn ) := α0 (D0 , σ )V1 . . . Vn αn (Dn , σ ) dν(σ ), S
where the integral is taken in the so∗ -topology.
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Proposition 1.16. [3, Lemma 4.3] The multiple operator integral in Definition 1.15 is well-defined in the sense that it does not depend on the representation (7) of ϕ. The following lemma is a corollary of Lemma 1.14 and the definition of the multiple operator integral. Lemma 1.17. If D0 = D0∗ ηN , if D1 = D0 + V, V = V ∗ ∈ N and if f ∈ Cc2 (R) is a √ non-negative function, such that g := f ∈ Cc2 (R), then T fD[1]1 ,D0 (V ) = (α1 (D1 , σ )Vβ1 (D0 , σ ) + α2 (D1 , σ )Vβ2 (D0 , σ )) dνg (σ ),
where α1 , β1 , α2 , β2 are given by (1.14). We need the following weaker version of [3, Theorem 5.3]. See also [10]. Proposition 1.18. [3, Theorem 5.3] Let N be a von Neumann algebra. Suppose that D0 = D0∗ is affiliated with N , that V ∈ N is self-adjoint and set D1 = D0 + V. (i) If f ∈ C 1,+ (R), then f (D1 ) − f (D0 ) = T fD[1]1 ,D0 (V ). (ii) If f ∈ C 2,+ (R), then the function f : D0 + Nsa → f (D0 ) + Nsa is affinely (N , N )Fréchet differentiable, the equality DN f (D) = T fD,D holds and DN f (D) is [1] ·-continuous, where DN is to be understood in the sense of Definition 1.1. 1.4. Some continuity and differentiability properties of operator functions. We are going to consider spectral flow along ‘continuous’ paths of unbounded Fredholm operators. We will make precise what we mean by continuity in this setting later. However our formulae require more than just continuity. They require us to be able to take derivatives with respect to the path parameter. For this to be feasible we need the full force of the double operator integral formalism. We present the results we will need as a sequence of lemmas. Lemma 1.19. Let (S, ν) be a finite measure space and let f : S → L1 (N , τ ) be a L1 (N , τ )-bounded so∗ -measurable function. Then f (σ ) dν(σ ) f (σ )L1 d |ν| (σ ). L1
Proof. By definition for any η ∈ H the function σ → f (σ )η is Bochner measurable. Hence, the function σ → f (σ ) = supη∈H : η1 f (σ )η is also measurable. Similarly, since the function σ → τ ( f (σ )B) is measurable, the function σ → f (σ )1 = sup B∈N : B1 |τ ( f (σ )B)| is also measurable. Hence, the right-hand side of the last equality is well defined. For η ∈ H with η 1 we have by definition of so∗ -integral (see [3, (2)]) f (σ ) dν(σ )η = f (σ )η dν(σ ) (9) f (σ )η d |ν| (σ ) f (σ ) d |ν| (σ ).
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Hence, the inequality is true for ·-norm. Since A1 = sup B∈N : B1 |τ (AB)| , we have
f (σ ) dν(σ ) =
f (σ ) dν(σ )B
sup
τ 1
B∈N : B1
= sup
τ ( f (σ )B) dν(σ ) B∈N : B1 |τ ( f (σ )B)| dν(σ ) sup B∈N : B1
f (σ )1 d |ν| (σ ),
(10)
where the second equality follows from the definition of so∗ -integral and Lemma 1.11. Combining (9) and (10) completes the proof. In the sequel we will constantly need to take functions of a path of operators. We thus need the following continuity result. For the definition of B R see Lemma 1.7. Proposition 1.20. If D0 = D0∗ ηN has τ -compact resolvent and if f ∈ Cc2 (R) then the operator-valued function A : V ∈ B R → f (D0 + V ) takes values in L1 (N , τ ) and is L1 (N , τ )-continuous. Proof. That A(·) takes values in L1 (N , τ ) follows from Lemma 1.3 and Corollary 1.5. By Lemma 1.2 it is enough to prove continuity for a non-negative function f with √ g = f ∈ Cc2 (R). By Proposition 1.18(i) and Lemma 1.17 we have f (D0 +V )− f (D0 ) = T fD[1]0 +V,D0 (V ) = (α1 (D0 +V, σ )Vβ1 (D0 , σ )+α2 (D0 +V, σ )Vβ2 (D0 , σ )) dνg (σ ).
Hence, by Lemma 1.19, we have f (D0 + V ) − f (D0 )1,∞ α1 (D0 + V, σ )1,∞ V β1 (D0 , σ )
+ α2 (D0 + V, σ ) V β2 (D0 , σ )1,∞ d νg (σ )
g(D0 +V )1,∞ V +V g(D0 )1,∞ d νg (σ )
νg ( ) V (g(D0 + V )1,∞ + g(D0 )1,∞ ). Now, Corollary 1.8 applied to g completes the proof.
Corollary 1.21. If D0 = D0∗ ηN has τ -compact resolvent, r = (r1 , . . . , rm ) ∈ [a, b]m , if V1 , . . . , Vm ∈ Nsa and if Dr = D0 + r V = D0 + r1 V1 + · · · + rm Vm , then for any function f ∈ Cc2 (R) the operator-valued function A : r ∈ [a, b]m → f (D0 + r V ) takes values in L1 (N , τ ) and is L1 (N , τ )-continuous. Next we prove the main lemmas of this section. There are several matters to establish. First we want to be able to differentiate, with respect to the path parameter, certain functions of paths of operators. Then we need to determine formulae for the derivatives and the continuity properties of the derivatives with respect to the path parameter.
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Lemma 1.22. If D1 and D2 are two self-adjoint operators with τ -compact resolvent affiliated with semifinite von Neumann algebra N , if X ∈ Nsa and if f ∈ Cc3 (R) then T fD[1]1 ,D2 (X ) depends L1 -continuously on · perturbations of D1 and D2 . Proof. As usual, we can assume that f is non-negative and its square root g = C 3 -smooth. Let Y1 , Y2 ∈ Nsa . Then by Lemma 1.17,
√
f is
T fD[1]1 +Y1 ,D2 +Y2 (X ) − T fD[1]1 ,D2 (X ) = [α1 (D1 + Y1 , σ )Xβ1 (D2 + Y2 , σ ) + α2 (D1 + Y1 , σ )Xβ2 (D2 + Y2 , σ )
−α1 (D1 , σ )Xβ1 (D2 , σ ) − α2 (D1 , σ )Xβ2 (D2 , σ )] dνg (σ ) = ([α1 (D1 + Y1 , σ ) − α1 (D1 , σ )] Xβ1 (D2 + Y2 , σ )
+α1 (D1 , σ )X [β1 (D2 + Y2 , σ ) − β1 (D2 , σ )] + [α2 (D1 + Y1 , σ ) − α2 (D1 , σ )] Xβ2 (D2 + Y2 , σ ) +α2 (D1 , σ )X [β2 (D2 + Y2 , σ ) − β2 (D2 , σ )]) dνg (σ ). For every fixed σ ∈ by Lemma 1.10 the norms β1 (D2 + Y2 , σ ) − β1 (D2 , σ ) and α2 (D1 + Y1 , σ ) − α2 (D1 , σ ) converge to zero when Y1 , Y2 → 0, and by Corollary 1.8 the L1 -norms of α1 (D1 , σ ) and β2 (D2 + Y2 , σ ) are bounded when Y1 , Y2 → 0. Hence, for every fixed σ ∈ the L1 -norms of the second and third summands in the last integral converge to zero when Y1 , Y2 → 0. Now we are going to show that the same is true for the first and fourth summands. It is enough to prove that for every fixed σ ∈ , for example, α1 (D1 +Y1 , σ )−α1 (D1 , σ )L1 tends to zero. We have α1 (D1 + Y1 , σ ) − α1 (D1 , σ )L1 = ei(s0 −s1 )(D1 +Y1 ) g(D1 + Y1 ) − ei(s0 −s1 )D1 g(D1 ) 1 L i(s0 −s1 )(D1 +Y1 ) i(s0 −s1 )D1 g(D1 + Y1 ) 1 e −e L i(s0 −s1 )D1 + e (g(D1 + Y1 ) − g(D1 )) 1 L i(s0 −s1 )(D1 +Y1 ) i(s0 −s1 )D1 e −e g(D1 + Y1 )L1 + g(D1 + Y1 ) − g(D1 )L1 . It follows from Lemma 1.10 that the first summand converges to zero when s0 , s1 are fixed and Y1 → 0, and it follows from Proposition 1.20 that the second summand also converges to zero. Since by Corollary 1.8 the trace norm of the expression under the last integral is uniformly L1 (N , τ )-bounded with respect to σ ∈ , it follows from Lemma 1.19 that D1 +Y1 ,D2 +Y2 (X ) − T fD[1]1 ,D2 (X ) 1 → 0, T f [1] L
when Y1 , Y2 → 0.
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The following theorem is a version of a well-known Daletskii-S. G. Kre˘ın formula [21]. We would like to give a heuristic argument explaining the formula (11). In the resolvent expansion series 1 1 1 1 1 1 1 = + V + V V + ... z−H −V z−H z−H z−H z−H z−H z−H the second summand is a double operator integral TϕH,H (V ) with ϕ(λ, µ) =
1 1 1 · = z−λ z−µ λ−µ
1 1 − . z−λ z−µ
If H and H + V are bounded operators and f is a function analytic in a neighbourhood of the union of the spectra of H and H + V then the Cauchy integral implies f (H + V ) = f (H ) + T fH,H [1] (V ) + terms of second order. Theorem 1.23. If the von Neumann algebra N is semifinite, D0 = D0∗ ηN has τ -compact resolvent and f ∈ Cc3 (R), then the function f : D ∈ D0 + Nsa → f (D) ∈ Nsa takes values in L1 (N , τ )sa . Moreover, it is affinely (N , L1 )-Fréchet differentiable, the equality DN ,L1 f (D) = T fD,D [1] holds, and DN ,L1 f (D) is (N , L1 )-continuous, so that b T fD[1]r ,Dr (V ) dr, f (Db ) − f (Da ) =
(11)
(12)
a
where V ∈ Nsa , Dr = D0 + r V and the integral converges in L1 (N , τ )-norm. Proof. We have by Proposition 1.18(i) and Lemma 1.17, f (D1 ) − f (D0 ) = T fD[1]1 ,D0 (V ) = (α1 (D1 , σ )Vβ1 (D0 , σ ) + α2 (D1 , σ )Vβ2 (D0 , σ )) dνg (σ ), = (α1 (D0 , σ )Vβ1 (D0 , σ ) + α2 (D0 , σ )Vβ2 (D0 , σ )) dνg (σ )
+ [α1 (D1 , σ ) − α1 (D0 , σ )]Vβ1 (D0 , σ ) dνg (σ ) + [α2 (D1 , σ ) − α2 (D0 , σ )]Vβ2 (D0 , σ ) dνg (σ ) =
D0 ,D0 T f [1] (V ) + (I I ) + (I I I ).
Since α2 is just an exponent and since g ∈ C 2,+ (R), (I I I )L1 = O(V 2 ) can be shown by Duhamel’s formula. The argument is as in the proof of [3, Theorem 5.5]. So,
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N. A. Azamov, A. L. Carey, F. A. Sukochev
it is left to show that (I I )L1 is o(V ). By Lemma 1.19 we have (I I )L1 = [α1 (D1 , σ ) − α1 (D0 , σ )]Vβ1 (D0 , σ ) dνg (σ ) 1 L α1 (D1 , σ ) − α1 (D0 , σ )L1 V β1 (D0 , σ ) dνg (σ )
α1 (D1 , σ ) − α1 (D0 , σ )L1 dνg (σ ). = V
Now, it follows from α1 (·, σ ) ∈ Cc2 (R) (see (1.14)) and Proposition 1.20 that α1 (D1 , σ )− α1 (D0 , σ )L1 → 0, σ ∈ , so that by the Lebesgue dominated convergence theorem we conclude that the last integral converges to 0, and hence (I I )L1 = o(V ). Finally, that DN ,L1 f (D) is (N , L1 )-continuous follows from Lemma 1.22. 1.5. A class F a,b (N , τ ) of τ -Fredholm operators. Our technique for handling spectral flow of paths of unbounded operators is to map them into the space of bounded operators using a particular function. We thus need to discuss some continuity properties of paths of bounded τ -Fredholm operators, analogous to those we described in the unbounded case. Let a < b be two real numbers. Let F a,b (N , τ ) be the set of bounded self-adjoint τ -Fredholm operators F ∈ N such that (F − a)(F − b) ∈ K(N , τ ). For F0 ∈ F a,b (N , τ ) let A F0 = F0 + K(N , τ )sa be the affine space of τ -compact self-adjoint perturbations of F0 . Lemma 1.24. If F0 ∈ F a,b (N , τ ) then A F0 ⊆ F a,b (N , τ ). Proof. If K ∈ K(N , τ )sa then (F0 + K − a)(F0 + K − b) = (F0 − a)(F0 − b) + (F0 − a)K + K (F0 + K − b) ∈ K(N , τ ). Lemma 1.25. If F ∈ F a,b (N , τ ) and h ∈ Bc (a, b) then h(F) ∈ L1 (N , τ ). Proof. The proof is similar to the proof of Lemma 1.4. For any compact subset of (a, b) there exists a constant c0 > 0 such that χ (x) c0 χ[a,b] (x)(b − x)(x − a), so that F F c0 E [a,b] (b − F)(F − a). E
(13)
F ∈ K(N , τ ) and hence E F is Since (b − F)(F − a) ∈ K(N , τ ), it follows that E τ -finite. Now, for any h ∈ Bc (a, b) there exists a compact subset of (a, b) and a F and hence h(F) ∈ L1 (N , τ ). constant c1 such that |h| c1 χ , so that |h(F)| c1 E
Lemma 1.26. If F0 ∈ F a,b (N , τ ), K = K ∗ ∈ K(N , τ ), and if is a compact subset of (a, b), then (i) the function r ∈ [0, 1] L1 (N , τ )-bounded;
→
F0 +r K E takes values in L1 (N , τ ) and is
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65
F0 +K (ii) there exists R > 0 such that the function K ∈ B R ∩ K(N , τ ) → E takes 1 1 values in L (N , τ ) and is L (N , τ )-bounded. F0 +r K Fr Proof. (i) That E = E ∈ L1 (N , τ ) follows from Lemmas 1.25 and 1.26. By Fr Fr c0 E [a,b] (b − Fr )(Fr − a) for all r ∈ [0, 1] and hence by [22, (13) we have E Lemma 2.5]
Fr Fr ) c0 µt E [a,b] (b − Fr )(Fr − a) c0 µt [(b − Fr )(Fr − a)] . µt (E Since (b − Fr )(Fr − a) = (b − F0 )(F0 − a) + r L 1 − r 2 L 2 , where L 1 , L 2 ∈ K(N , τ ), we have by [22, Lemma 2.5(v)] Fr µt (E ) c0 µt/3 [(b − F0 )(F0 − a)] + r µt/3 (L 1 ) + r 2 µt/3 (L 2 )
c0 µt/3 [(b − F0 )(F0 − a)] + µt/3 (L 1 ) + µt/3 (L 2 ) , Fr so that µt (E ) = χ
Fr (t) [0,τ E ]
(14)
is majorized for all r ∈ [0, 1] by a single function
decreasing to 0 when t → ∞, since all three operators (b − F0 )(F0 − a), L 1 and L 2 are τ -compact. The same argument as in Lemma 1.7 now completes the proof. (ii) If F = F0 + K then (b − F)(F − a) = (b − F0 )(F0 − a) + L , where L = (b− F0 )K − K (F0 −a)− K 2 ∈ K(N , τ ). Choose the number R > 0 such that K < R F+K ) = χ
F+K (t) will be implies L < 1. Then by (14) the function t → µt (E ] [0,τ E majorized by a single function decreasing to a number <1, so that the same argument as in Lemma 1.7 again completes the proof. Proposition 1.27. Let F0 ∈ F a,b (N , τ ), K = K ∗ ∈ K(N , τ ), and let h ∈ Cc2 (a, b). Then (i) the function r ∈ R → h(F0 + r K ) takes values in L1 (N , τ ) and is L1 (N , τ )continuous; (ii) there exists R > 0 such that the function K ∈ B R ∩ K(N , τ ) → h(F0 + K ) takes values in L1 (N , τ ) and is L1 (N , τ )-continuous. Proof. The proof of this proposition follows verbatim the proof of Proposition 1.20 with references to Lemmas 1.24, 1.25 and 1.26 instead of Lemmas 1.3, 1.5 and Corollary 1.8. Lemma 1.28. If F1 , F2 ∈ F a,b (N , τ ), X ∈ K(N , τ )sa and h ∈ Cc3 (a, b), then the double operator integral ThF[1]1 ,F2 (X ) takes values in L1 (N , τ ) and is L1 (N , τ )-continuous with respect to norm perturbations of F1 and F2 by τ -compact operators. The proof of this lemma is similar to that of Lemma 1.22 with references to Lemma 1.26(ii) and Proposition 1.27(ii) instead of Corollary 1.8 and Proposition 1.20.
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Theorem 1.29. Let N be a semifinite von Neumann algebra. If F0 ∈ F a,b (N , τ ), h ∈ Cc3 (a, b), then the function h : F ∈ F0 + Ksa (N , τ ) → h(F0 ) + Ksa (N , τ ) takes values in L1 (N , τ )sa . Moreover, it is affinely (K, L1 )-Fréchet differentiable, the equality DK,L1 h(F) = ThF,F [1] holds, and DK,L1 h(F) is (K, L1 ) continuous, so that r0 h(Fr1 ) − h(Fr0 ) = ThF[1]r ,Fr (K ) dr,
r0 , r1 ∈ R,
(15)
r1
where K ∈ Ksa (N , τ ), Fr = F0 + r K and the integral is in L1 (N , τ )-norm. The proof is similar to that of Theorem 1.23 with use of Proposition 1.27(ii) and Lemma 1.28 instead of Proposition 1.20 and Lemma 1.22, and therefore it is omitted. 2. Spectral Shift Function We will take an approach to the notion of spectral shift function suggested by the BirmanSolomyak formula (16). The key point is that once one appreciates that the spectral shift function of M. G. Krein is related to spectral flow in a specific fashion one can reformulate the whole approach to take advantage of what is known about spectral flow as expounded for example in [5]. The theorem in [3] which connects spectral flow and the spectral shift function contains the germ of the idea but one needs the technical machinery of the last section to exploit this. We now explain this different way to approach spectral shift theory which is influenced by ideas from noncommutative geometry. 2.1. The unbounded case. 2.1.1. Spectral shift measure. In order to make our main definition we need to prove a preliminary result which complements [3, Lemma 6.2]. The latter asserts that the Dr function γ (λ, r ) = τ V E λ is measurable for every V ∈ L1 (N , τ ) and D = D ∗ ηN . Lemma 2.1. Let (N , τ ) be a semifinite von Neumann algebra and let D = D ∗ ηN have resolvent. If V = V ∗ ∈ N then the function f : (a, b, r ) ∈ R3 → τ -compact Dr τ V E (a,b) is measurable. Proof. W.l.o.g. we can assume that V 0. It is enough to prove that the function f is measurable with the second variable b and with respect to r. Since √ respect√to Dr Dr τ V E (a,b) = τ V E (a,b) V , we know by [27, Lemma 5.9] that it is enough √ Dr √ to prove that the operator function (r, b) → V E (a,b) V is so∗ -measurable. By [3, 3.2] it is enough to prove that for any ξ, η ∈ H the scalar function √ Proposition Dr √ Dr V E (a,b) V ξ, η = Tr(θ√V ξ,√V η E (a,b) ) is measurable, where θξ,η (ζ ) := ξ, ζ η. √ √ Since the operator θ V ξ, V η is trace class, the measurability of this function follows from [3, Lemma 6.2].
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Definition 2.2. If D0 = D0∗ ηN has τ -compact resolvent and if D1 = D0 + V, V ∈ Nsa , then the spectral shift measure for the pair (D0 , D1 ) is defined to be the following Borel measure on R: 1 Dr dr, ∈ B(R). (16) D1 ,D0 () = τ V E 0
The generalized function ξ D1 ,D0 (λ) =
d D ,D (a, λ) dλ 1 0
(17)
is called the spectral shift distribution for the pair (D0 , D1 ). Evidently, this definition does not depend on a choice of a. By Lemmas 1.3, 1.4, 2.1 and Corollary 1.9(i) the measure exists and is locally finite. Our task now is to show that the spectral shift distribution is in fact a function of locally bounded variation. 2.1.2. Spectral shift function. The main result we wish to establish next is that the spectral shift measure is absolutely continuous with respect to Lebesgue measure. Moreover its density, which we previously referred to as the spectral shift distribution, is in fact a function of locally bounded variation which we will then refer to as the spectral shift function. It is our extension of M. G. Krein’s function to the setting of this paper. Our method of proof is to first establish some trace formulae. Lemma 2.3. (i) Let D = D ∗ ηN have τ -compact resolvent. A function α ∈ B(R) is D ) ( ∈ B(R)), if and only if α(D) ∈ 1-summable with respect to the measure τ (E L1 (N , τ ) and in this case α(λ) τ d E λD . τ (α(D)) = R
Furthermore, for any V = V ∗ ∈ N the function α is 1-summable with respect to the D ), and measure τ (V E α(λ) τ V d E λD . τ (V α(D)) = R
(ii) Let F ∈ F a,b (N , τ ). A function α ∈ B(a, b) is 1-summable with respect to the F ) ( ∈ B(a, b)), if and only if α(F) ∈ L1 (N , τ ) and in this case measure τ (E τ (α(F)) =
b
a
α(λ) τ d E λF .
Furthermore, for any V = V ∗ ∈ N the function α is 1-summable with respect to the F ), and measure τ (V E τ (V α(F)) = a
b
α(λ) τ V d E λF .
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N. A. Azamov, A. L. Carey, F. A. Sukochev
Proof. We give only the proof of (i). W.l.o.g. we can assume that α is a non-negative function. If α is a simple function then the first part of the claim follows from Lemma 1.4. Let αn be an increasing sequence of simple non-negative functions, converging pointwise to α. Then for each of the functions αn the first equality is true. The supremum of the increasing sequence of non-negative operators α n (D) is α(D) of and the
supremum the increasing sequence of numbers R αn (λ) τ d E λD is R α(λ) τ d E λD . Hence,
both non-negative numbers R α(λ) τ d E λD and τ (α(D)) are finite or infinite simultaneously, which proves the first part of the lemma. For the second part we can assume w.l.o.g. that V 0. Then again both parts of the second equality make sense and they are for simple functions. √ equal √
D = τ D V , ∈ B(R), is non-negative and Since the measure τ V E V E √ √ √ √ the supremum of V αn (D) V ∈ L1 (N , τ ) is V α(D) V we have that α(λ) τ V d E λD = lim αn (λ) τ V d E λD n→∞ R R √ √ √ √ V αn (D) V = τ V α(D) V , = lim τ n→∞
D
so that R α(λ) τ V d E λ simultaneously.
√
and τ (V α(D)) = τ
√ V α(D) V are finite or infinite
We need the following version of Fubini’s theorem. Lemma 2.4. (i) For any self-adjoint operator DηN with τ -compact resolvent and V =
D . Let D = D ∗ ηN have τ -compact resolvent and V ∗ ∈ N , let m D,V () = τ V E 0 0 let Dr = D0 + r V. If g ∈ Bc (R), then 1 dr g(λ) m Dr ,V (dλ) = g(λ) D1 ,D0 (dλ). (18) 0
R
R
0
a
a
F , ∈ B(a, b). (ii) For any F ∈ F a,b (N , τ ) and V = V ∗ ∈ N , let m F,V () = τ V E Let F ∈ F a,b (N , τ ) and let Fr = F0 + r V. If g ∈ Bc (a, b), then b 1 b dr g(λ) m Fr ,V (dλ) = g(λ) F1 ,F0 (dλ). Proof. (See also [25, VI.2]). We give only the proof of (i). The measurability of the function r → R g(λ) m Dr ,V (dλ) follows from Lemma 2.1. Note that both integrals are repeated ones. Let ⊇ supp(g) be a finite interval.
By Corollary 1.9 (i) there exists M > 0 such that for all r ∈ [0, 1] we have m Dr ,V () M. If g(λ) = χ (λ), ∈ B(), then 1 1 dr χ (λ) m Dr ,V (dλ) = m Dr ,V () dr = () = χ (λ) (dλ). 0
0
So, (18) is true for simple functions. Let now g be an arbitrary function from Bc (), let ε > 0 and let h be a simple function such that g − h∞ < ε. Then the LHS of (18) is equal to 1 1 dr (g − h)(λ) m Dr ,V (dλ) + dr h(λ) m Dr ,V (dλ) = (I ) + (I I ), 0
0
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and the RHS of (18) is equal to (g − h)(λ) (dλ) + h(λ) (dλ) = (I I I ) + (I V ).
We have (I I ) = (I V ). Further, |(I )| M g − h∞ Mε and |(I I I )| M g − h∞ Mε. The following theorem complements [3, Theorem 6.3]. Theorem 2.5. If D = D ∗ ηN has τ -compact resolvent, if V = V ∗ ∈ N , and if D1 = D0 + V, then the measure D1 ,D0 is absolutely continuous, its density is equal to D0 D1 − τ E (a,λ] + const (19) ξ D1 ,D0 (·) = τ E (a,λ] for almost all λ ∈ R. Moreover, for all f ∈ Cc3 (R) f (D1 ) − f (D0 ) ∈ L1 (N , τ ) and τ ( f (D1 ) − f (D0 )) = f (λ)ξ D1 ,D0 (λ) dλ. (20) R
Proof. By Lemma 1.3 and Corollary 1.5 f (D1 ) − f (D0 ) ∈ L1 (N , τ ). By√Lemma 1.2 we need only consider the case of a non-negative function f with g := f ∈ Cc2 (R). We have by (12), 1 T fD[1]r ,Dr (V ) dr, f (D1 ) − f (D0 ) = 0
where the integral converges in that
L1 (N , τ )-norm.
Hence, it follows from Lemma 1.17
f (D1 ) − f (D0 ) 1 = (α1 (Dr , σ )Vβ1 (Dr , σ ) + α2 (Dr , σ )Vβ2 (Dr , σ )) dνg (σ ) dr. (21)
0
Now, for a fixed σ ∈ , we have τ (α1 (Dr , σ )Vβ1 (Dr , σ ) + α2 (Dr , σ )Vβ2 (Dr , σ )) = τ (V (α1 (Dr , σ )β1 (Dr , σ ) + α2 (Dr , σ )β2 (Dr , σ ))) = (α1 (λ, σ )β1 (λ, σ ) + α2 (λ, σ )β2 (λ, σ )) τ V d E λDr , R
where the last equality uses Lemma 2.3. (That α1 (λ)β1 (λ) + α2 (λ)β2 (λ) belongs to Bc (R) follows from Lemma 1.14.) Hence using (21), and by Lemma 1.11 applied to the finite measure space ([0, 1]× , dr ×νg our previous equality implies that we have: A := τ ( f (D1 ) − f (D0 )) 1 = τ (α1 (Dr , σ )Vβ1 (Dr , σ ) + α2 (Dr , σ )Vβ2 (Dr , σ )) dνg (σ ) dr 0
=
0
1
R
(α1 (λ, σ )β1 (λ, σ ) + α2 (λ, σ )β2 (λ, σ )) τ V d E λDr dνg (σ ) dr.
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Now, by Lemma 2.3, Fubini’s theorem, and Lemma 1.14 we have 1 A= (α1 (λ, σ )β1 (λ, σ ) + α2 (λ, σ )β2 (λ, σ )) dνg (σ ) τ V d E λDr dr
0
= 0
1
R R
f (λ) τ V d E λDr dr.
Finally, by Lemma 2.4 we have 1 A= f (λ) τ V d E λDr dr = R
R
0
f (λ) d D1 ,D0 (λ).
(22)
Let f ∈ Cc1 (R) and take a point a outside of the support of f. Then we have (see [1, Proposition 8.5.5]) A = τ ( f (D1 ) − f (D0 )) = τ ( f (D1 )) − τ ( f (D0 )) D1 D0 − (integrating by parts) = f (λ)dτ E (a,λ] f (λ)dτ E (a,λ] R R D1 D0 =− − τ E (a,λ] dλ. (23) f (λ) τ E (a,λ] R
Comparing (23) and (22) we see that is absolutely continuous with density equal to D0 D1 − τ E (a,λ] + const . (24) ξ D1 ,D0 (λ) = τ E (a,λ] It is worth noting that the formula (20) does not determine the function ξ uniquely, but only up to an additive constant. Remark 1. This theorem is an analogue of [4, Theorem 3.1], in which the existence and absolute continuity of the spectral shift measure were proved for any self-adjoint operator D affiliated with N and τ -trace class operator V ∈ N . As a result of what we have proved to this point we are now in a position to assert that in fact the spectral shift distribution is an everywhere defined function and hence to change our terminology and refer to ξ as a function. Moreover this last lemma enables one to modify ξ so as to make it a function defined everywhere in a natural way. Definition 2.6. If the expression (24) is continuous at a point λ ∈ R, then we define ξ D1 ,D0 (λ) via formula (24). Otherwise, we define the value of the spectral shift function ξ at a discontinuity point to be the half sum of left and right limits. Corollary 2.7. The spectral shift function ξ is a function of locally bounded variation. Proof. This is immediate because ξ is the difference of two increasing functions by the last formula. Lemma 2.8. Let D0 ηN be a self-adjoint operator with τ -compact resolvent, let V ∈ Nsa and let D1 = D0 + V. If f ∈ Bc (R) then ∞ 1 f (λ)ξ D1 ,D0 (λ) dλ = τ (V f (Dr )) dr. −∞
0
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Proof. It follows from Lemma 2.3 and Lemma 2.4 that 1 1 ∞ τ (V f (Dr )) dr = f (λ)τ V d E λDr dr 0
0
=
∞
−∞
−∞
f (λ) 0
1
τ V d E λDr dr.
2.1.3. The spectral shift function for unitarily equivalent operators. The situation where the operators D and D + V, are unitarily equivalent arises naturally in noncommutative geometry in the context of spectral triples. One thinks of the unitary implementing the equivalence as a gauge transformation by analogy with the study of gauge transformations of Dirac type operators. It thus warrants special consideration especially in view of our first result below. Theorem 2.9. Let D be a self-adjoint operator affiliated with N having τ -compact resolvent and let V = V ∗ ∈ N be such that the operators D + V and D are unitarily equivalent. Then the spectral shift function ξ D+V,D of the pair (D + V, D) is constant on R. Proof. The operators f (D + V ) and f (D) are unitarily equivalent and for f ∈ Cc∞ (R) they are τ -trace class by Corollary 1.5. Hence, τ ( f (D + V ) − f (D)) = 0, so that by Theorem 2.5 the equality f (λ)ξ D+V,D (λ) dλ = 0 R
holds for any f ∈ Cc∞ (R). Now, integration by parts shows that ξ (λ) is zero as a generalized function on R, which by [23, Ch. I.2.6] implies that ξ is equal to a constant function. Note that function ξ in this theorem is equal to a constant function everywhere, not just almost everywhere. Our second major result on the spectral shift function in this special context is the following theorem. We shall show in Sect. 3 below that this theorem extends one of the main results of [16]. Theorem 2.10. Let D0 be a self-adjoint operator with τ -compact resolvent, affiliated with N . Let V = V ∗ ∈ N be such that the operators D1 = D0 + V and D0 are unitarily equivalent. If f ∈ Cc2 (R) then 1 ξ D1 ,D0 (µ) = C −1 τ (V f (Dr − µ)) dr, ∀ µ ∈ R, (25) 0 where C = R f (λ) dλ.
Proof. For any fixed µ the operator Dr − µ has τ -compact resolvent by Lemma 1.3 and the function r → τ (V f (Dr − µ)) is continuous by Proposition 1.20, so that the integral on the RHS of (25) exists. By Lemma 2.8 and Theorem 2.9 we have 1 τ (V f (Dr − µ)) dr = f (λ − µ)ξ D1 ,D0 (λ) dλ = ξ(0) f (λ) dλ.
0
R
R
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2.2. The bounded case. As we remarked previously, our technique in the next section for discussing spectral flow in the unbounded case is to map into the space of bounded τ -Fredholm operators. We thus need to develop the theory described in the previous subsections ab initio for the bounded case. Fortunately this is not a difficult task as the proofs are much the same. As we will see, because we are considering bounded perturbations of our unbounded operators, it suffices to consider compact perturbations in the bounded case. Definition 2.11. If F0 ∈ F a,b (N , τ ), K ∈ K(N , τ )sa , F1 = F0 + K and if Fr = F0 + r K , then the spectral shift measure for the pair (F0 , F1 ) is defined to be the following Borel measure on (a, b): 1 Fr F1 ,F0 () = dr, ∈ B(a, b). (26) τ K E 0
The generalized function d F ,F (c, λ), c ∈ (a, b), dλ 1 0 is called the spectral shift distribution for the pair (F0 , F1 ). ξ F1 ,F0 (λ) =
(27)
Evidently, this definition not depend on a choice of c ∈ (a, b). The measurability does Fr may be established following the argument of Lemma of the function r → τ K E 2.1, using Lemma 1.26. It follows that the measure exists and is locally-finite on (a, b). Proposition 2.12. If F0 ∈ F a,b (N , τ ), K ∈ K(N , τ ) and if F1 = F0 + K , then (i) the measure F1 ,F0 is absolutely continuous and its density is equal to F0 F1 + const, λ ∈ (c, b), ξ F1 ,F0 (λ) = τ E (c,λ] − E (c,λ] where c is an arbitrary number from (a, b); (ii) there exists a unique function ξ F1 ,F0 (·) of locally bounded variation on (a, b), such that for any h ∈ Cc2 (a, b) the following equality holds true: b h (λ)ξ F1 ,F0 (λ) dλ. τ (h(F1 ) − h(F0 )) = a
The proof is identical to the proof of Theorem 2.5, with references to 1.24, 1.25, (15) instead of 1.3, 1.5, (12) and hence we omit it. Corollary 2.13. In the setting of Proposition 2.12, if F0 and F1 are unitarily equivalent, then ξ F1 ,F0 is constant on (a, b). The proof is similar to the proof of Theorem 2.9. Definition 2.14. We redefine the function ξ F1 ,F0 at discontinuity points to be half the sum of the left and the right limits of the RHS of the last equality. Thus, the function ξ F1 ,F0 is defined everywhere on (a, b). Lemma 2.15. If F0 ∈ F a,b (N , τ ), K ∈ K(N , τ ), if Fr = F0 + r K , r ∈ [0, 1] and if h ∈ Bc (a, b) then 1 h(λ)ξ F1 ,F0 (λ) dλ = τ (K h(Fr )) dr. R
0
This lemma and its proof are bounded variants of Lemma 2.8, so we omit the details.
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3. Spectral Flow 3.1. The spectral flow function. We wish to avoid a long excursion into the analytic theory of spectral flow preferring the reader to read the early Sections of [5] for the relevant background and definitions. With those prerequisites it is possible to appreciate that the following notions are well defined. We now introduce the spectral flow function on the real line. As usual we have D0 = D0∗ ηN with τ -compact resolvent and D1 = D0 + V with V bounded. Then we define the spectral flow function to be µ → sf(µ, D0 , D1 ), µ ∈ R, where sf(µ, D0 , D1 ) is spectral flow from D0 − µ to D1 − µ along the path Dr − µ = D0 − µ + r V . We remark that the homotopy invariance of spectral flow means that, in the affine space of bounded perturbations of a fixed operator D0 , spectral flow does not depend on the choice of continuous path but only on the endpoints. However one does need to make precise what one means by continuity in a path parameter in the unbounded case. We define the space of self adjoint, unbounded τ -Fredholm operators to be those operators that under the map D → FD = D(1 + D)−1/2 are sent to bounded τ -Fredholm operators in N . This definition is equivalent to the usual definition for unbounded Fredholm operators in a semifinite von Neumann algebra (more details on this can be found in [18].). A path of unbounded τ -Fredholm operators is said to be continuous if its image under this map is continuous in the norm topology on the bounded τ -Fredholm operators. This topology is usually called the Riesz topology and it is different from the graph norm topology used in [12]. A more detailed discussion of topologies on the set of unbounded self-adjoint Fredholm operators and the relevance of these for spectral flow may be found in [36]. We note that the condition of having compact resolvent implies the τ -Fredholm property for the unbounded operator. We now recall, to provide some motivation for the point of view of this section, some ideas from [16] which is couched in the language of noncommutative geometry [19]. In 2 [16] the condition of theta summability (τ (e−t D0 ) < ∞ for all real t > 0) is imposed and then the main result of [16] is an analytic formula for spectral flow from D0 to D1 = D0 + V . The ideas behind this formula go back to [2] and a more complete history may be found in [5]. The formula of [16] is as follows: 1 ε 2 τ V e−ε Dr dr sf(D0 , D1 ) = π 0 1 1 + (ηε (D1 ) − ηε (D0 )) + τ ([ker(D1 )] − [ker(D0 )]) , (28) 2 2 where ηε is a ‘truncated eta invariant’. For an unbounded self-adjoint operator D for 2 which e−t D is τ -trace class for all t > 0, it is defined in [16, Definition 8.1] following [24] by ∞ 1 2 ηε (D) := √ τ De−t D t −1/2 dt. π ε
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When the endpoints are unitarily equivalent the two ηε terms and the kernel correction terms in the formula for spectral flow cancel. In [3] we showed that in the theta summable case and for trace class perturbations the spectral shift function and the spectral flow function differ only by kernel correction terms. Our aim in this paper is to show that this is the case more generally and in fact to go further. We will demonstrate that the spectral shift function provides a way to prove more general analytic formulae for spectral flow than is achieved in [16]. The plan of this section is to first establish a geometric framework that is analogous to that of Getzler [24] and [16]. Then we derive analytic formulae for spectral flow in terms of the spectral shift function in both the case of paths of bounded τ -Fredholm operators and for paths of unbounded τ -Fredholm operators. The starting point is basically the same as that of [16] in that we need a formula for the relative index of two projections. This next result is a strengthening of [16, Theorem 4.1]. Lemma 3.1. Let P and Q be two projections in the semifinite von Neumann algebra N and let a < 0 < b be two real numbers. Let κ be a continuous function such that 2 2 for any s ∈ [0, (b−a) 4 ], κ(s(P − Q) ) is τ -trace class. Then F0 = (b − a)P + a and F1 = (b − a)Q + a are self-adjoint τ -Fredholm operators from F a,b (N , τ ) as is the path Fr = F0 + r (F1 − F0 ), and 1
−1 sf({Fr }) = Ca,b τ F˙r κ [(b − Fr )(Fr − a)] dr, 0
1
where Ca,b = 0 (b − a)κ (b − a)2 (r − r 2 ) dr is a constant, and the derivative F˙r is ·-derivative. Proof. We have F˙r = F1 − F0 = (b − a)(Q − P) and (b − Fr )(Fr − a) = (b − a)2 r (1 − r )(Q − P)2 , so that by assumption κ [(b − Fr )(Fr − a)] is τ -trace class for r ∈ [0, 1]. For each r ∈ (0, 1) define fr (x) = (b − a)xκ (b − a)2 (r − r 2 )x 2 . Then
1
0
τ F˙r κ [(b − Fr )(Fr − a)] dr
1
=
τ (b − a)(Q − P)κ (b − a)2 r (1 − r )(Q − P)2 dr,
(29)
0
and by [16, Theorem 3.1] we have 1
τ F˙r κ [(b− Fr )(Fr −a)] dr = 0
1
τ ( fr (Q − P)) dr
0
1 = fr (1) ind(Q P) dr = Ca,b ind(Q P) = Ca,b sf({Fr }), 0
where the last equality follows from a < 0 < b and the definition of spectral flow.
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Remark 2. In the proof of this lemma we use [16, Theorem 3.1] for functions f without the condition f (1) = 0. But an inspection of the proof of this theorem shows that this condition becomes superfluous, if we rewrite the statement of this theorem as f (1) ind(Q P) = τ [ f (P − Q)], which for functions f with f (1) = 0 becomes just 0 = 0. 3.2. Spectral flow one-forms in the unbounded case. The strategy of [16] is geometric and follows ideas of [24]. The first step in this strategy is summarized in Proposition 3.3 in preparation for which we need an explicit formula for the derivative of a function of a path of operators. The method by which this is achieved in [16] does not apparently generalise sufficiently far to cover the situations considered in this paper. The double operator integral approach of Sect. 2 overcomes this problem. Lemma 3.2. Let D = D ∗ ηN , let X, Y ∈ N and let f ∈ Cc2 (R) be a non-negative √ D,D function such that g := f ∈ Cc2 (R). If Y T fD,D [1] (X ) and X T f [1] (Y ) are both τ -trace class then D,D τ Y T fD,D (X ) = τ X T (Y ) . [1] f [1] Proof. By Lemma 1.17 we have A = τ Y T fD,D [1] (X ) =τ Y (α1 (D, σ )Xβ1 (D, σ ) + α2 (D, σ )Xβ2 (D, σ )) dνg (σ ) i(s1 −s0 )D is1 D i(s1 −s0 )D is1 D e dνg (s0 , s1 ) . g(D)X e +e Xg(D)e =τ Y
Making the change of variables s1 − s0 = t0 , s1 = t1 , and using (4) we have (α1 (D, σ )Xβ1 (D, σ ) + α2 (D, σ )Xβ2 (D, σ )) dνg (σ )
i eit0 D g(D)X eit1 D = sgn(t0 + t1 ) √ 2 2π {(t0 ,t1 )∈R , t0 t1 0} it0 D +e Xg(D)eit1 D F(g)(t0 + t1 ) dt0 dt1 i eit0 D g(D)X eit1 D = sgn(t) √ 2π R t +eit0 D Xg(D)eit1 D dlt F(g)(t) dt, where t = (t0 , t1 ) ∈ R2 : t0 t1 0, t0 + t1 = t and dlt is the Lebesgue measure on t . Thus, by Fubini’s theorem [3, Lemma 3.8] i eit0 D g(D)X eit1 D A = sgn(t) √ τ Y 2π R t it0 D it1 D dlt F(g)(t) dt +e Xg(D)e
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N. A. Azamov, A. L. Carey, F. A. Sukochev
i = sgn(t) √ τ Y eit0 D g(D)X eit1 D 2π R t +Y eit0 D Xg(D)eit1 D dlt F(g)(t) dt i = sgn(t) √ τ X eit1 D Y g(D)eit0 D 2π R t it1 D +X e g(D)Y eit0 D dlt F(g)(t) dt, where the trace and the integral can be interchanged by Lemma 1.11. The integral above D,D coincides with τ X T f [1] (Y ) . The key geometric idea is to regard the analytic formula for spectral flow of [24] and [16] as expressing it as an integral of a one form. As we are dealing with an affine space, the geometry is easy to invoke as we see in the next result. Proposition 3.3. Let D be a self-adjoint operator affiliated with N , having τ -compact resolvent and let f ∈ Cc3 (R). Let α = α f be a 1-form on the affine space D0 + Nsa defined at the point D ∈ D0 + Nsa by the formula f
α D (X ) = τ (X f (D)) , X ∈ Nsa , D ∈ D0 + Nsa .
(30)
Then α is a closed 1-form, and, hence, also exact by the Poincaré lemma. Proof. The proof follows mainly the lines √ of [15], with necessary adjustments. As usual, we can assume that f 0 and g := f ∈ Cc3 (R). We note that the operator X f (D) is τ -trace class, so that the 1-form α is well-defined. Now, by the definition of the exterior differential, for X, Y ∈ N , we have dα D (X, Y ) = £ X α D (Y ) − £Y α D (X ) − α D ([X, Y ]), where £ X is the Lie derivative along the constant vector field X. Since the space D0 +Nsa is flat, we have [X, Y ] = 0. So, we have to prove that £ X α D (Y ) = £Y α D (X ). It follows from Theorem 1.23 that
d
d
A := £ X α D (Y ) = α (Y ) = τ (Y f (D + s X )) D+s X ds s=0 ds s=0
= τ Y DN ,L1 f (D)(X ) = τ Y T fD,D [1] (X ) . Hence, by Lemma 3.2, D,D £ X α D (Y ) = τ Y T fD,D (X ) = τ X T (Y ) = £Y α D (X ), [1] f [1] which implies that α D is a closed 1-form.
Though closedness of a 1-form already should imply its exactness by the Poincaré lemma and contractibility of the domain we follow [16] and give an independent proof of exactness.
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Definition 3.4. Let D0 be a fixed self-adjoint operator with τ -compact resolvent affiliated with N , and let f ∈ Cc (R). We define the function θ f on the affine space D0 + N by the formula 1 f τ (V f (Dr )) dr, θD = 0
where D ∈ D0 + Nsa , V = D − D0 and Dr = D0 + r V. Measurability of the function r → τ (V f (Dr )) follows from Lemma 2.1. Proposition 3.5. Let f ∈ Cc3 (R) and let X ∈ N . Then f
f
dθ D (X ) = α D (X ). Proof. W.l.o.g. we can assume that X is self-adjoint. By definitions f
(A) := dθ D (X )
1 d
d
f θ = τ ((V + s X ) f (Dr + sr X )) dr = ds s=0 D+s X ds s=0 0 1 1 = lim τ ((V + s X ) f (Dr + sr X ) − V f (Dr )) dr s→0 s 0 1 1 1 = lim τ (X f (Dr + sr X )) dr + lim τ (V ( f (Dr + sr X ) − f (Dr ))) dr. s→0 0 s→0 s 0 The first summand of this sum by Proposition 1.20 is equal to 1 τ (X f (Dr )) dr. 0
By Proposition 1.18(i) the second summand is equal to 1 1 1 Dr +sr X,Dr τ V T f [1] (sr X ) dr =lim τ V T fD[1]r +sr X,Dr (r X ) dr lim s→0 s 0 s→0 0 1 1 = τ V T fD[1]r ,Dr (r X ) dr = τ X T fD[1]r ,Dr (V ) r dr, 0
0
where the second equality follows from Lemma 1.22 and the last equality follows from Lemma 3.2. Hence, by Lemma 1.11, 1 τ X f (Dr ) + r T fD[1]r ,Dr (V ) dr (A) = 0 1 Dr ,Dr =τ X f (Dr ) + r T f [1] (V ) dr , 0
where the integral on the RHS is a so∗ -integral. By Proposition 1.20 and Lemma 1.22 the function r ∈ [0, 1] → f (Dr ) + r T fD[1]r ,Dr (V ) ∈ L1 (N , τ ) is L1 (N , τ )-continuous, so that the last integral 1 f (Dr ) + r T fD[1]r ,Dr (V ) dr (B) := 0
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N. A. Azamov, A. L. Carey, F. A. Sukochev
can be considered as Riemann integral. Let 0 = r0 < r1 < · · · < rn = 1 be the partition of [0, 1] into n segments of equal length n1 . By the argument used in the proof of [3, Dr ,Dr j
Theorem 5.8] it can be shown that the L1 (N , τ )-norm of T f [1]j
(V ) − T fD[1]r ,Dr (V ),
r ∈ [r j−1 , r j ], has order n1 . Hence rj n 1 j Dr j ,Dr j Dr ,Dr 1 T L - lim (V ) − j T f [1] (V ) dr n→∞ n n f [1] r j−1 j=1
rj n Dr ,Dr 1 T f [1]j j (V ) − T fD[1]r ,Dr (V ) dr = 0, j n→∞ n r j−1
= L1 - lim
j=1
so that by formula (12) applied to the pair (Dr j−1 , Dr j ) we have n 1 j Dr j ,Dr j 1 f (Dr j−1 ) + T f [1] (B) = L - lim (V ) n→∞ n n j=1
n
1
f (Dr j−1 ) + j f (Dr j ) − f (Dr j−1 ) = L1 - lim n→∞ n j=1 rj n 1 j Dr j ,Dr j Dr ,Dr 1 (V ) − j T f [1] (V ) dr +L - lim T n→∞ n n f [1] r j−1 j=1
n 1
= L1 - lim j f (Dr j ) − ( j − 1) f (Dr j−1 ) = f (D1 ). n→∞ n j=1
The argument before [15, Proposition 1.5] now implies the following corollary. Corollary 3.6. The integral of the 1-form α f along a piecewise continuously differentiable path in D0 + N depends only on the endpoints of the path . Proposition 3.7. If a self-adjoint operator D0 affiliated with N has τ -compact resolvent, D1 , D2 ∈ D0 + Nsa , then for all λ ∈ R, ξ D2 ,D0 (λ) = ξ D2 ,D1 (λ) + ξ D1 ,D0 (λ). Remark. We emphasize that this additivity property is not almost everywhere in the spectral variable but in fact holds everywhere. Proof. It follows from (19) that ξ D2 ,D0 (λ) = ξ D2 ,D1 (λ) + ξ D1 ,D0 (λ) + C, where C is a constant. Multiplying both sides of this equality by a positive Cc2 -function f, and integrating it, by Lemma 2.8 we get αf = αf + αf +C f (λ) dλ, D2 ,D0
D2 ,D1
D1 ,D0
R
where Di ,D j is the straight line path connecting operators Di and D j . The last equality and Corollary 3.6 imply that C = 0.
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3.3. Spectral flow one-forms in the bounded case. Since we obtain our unbounded spectral flow formula from a bounded one we need to study the map D → FD = D(1+ D 2 )−1/2 which takes the space of unbounded self adjoint operators with τ -compact resolvent to the space F −1,1 (N , τ ) of bounded τ -Fredholm operators F satisfying 1 − F 2 ∈ K(N , τ ). Let F0 ∈ F a,b (N , τ ), let h ∈ Cc2 (a, b) and let K = F − F0 , Fr := F0 + r K . We define a 0-form θ and a 1-form α h on the affine space A F0 by the formulae θ Fh =
1
τ (K h(Fr )) dr,
0
and α hF (X ) = τ (X h(F)) , X ∈ K(N , τ ). By Lemmas 1.24 and 1.25, the operators h(Fr ) and h(F) are τ -trace class, so that the forms θ h and α h are well-defined. Proposition 3.8. If F0 ∈ F a,b (N , τ ) and if h ∈ Cc2 (a, b), then dθ Fh (X ) = α hF (X ), where X ∈ K(N , τ ), so that the 1-form α hF is exact. Proof. The proof follows verbatim the proof of Proposition 3.5, with references to Proposition 1.27 and Lemma 1.28 instead of Proposition 1.20 and Lemma 1.22. As in the unbounded case we get the following Corollary 3.9. The integral of the one-form α h depends only on the end-points. Corollary 3.10. Let F j ∈ F a,b (N , τ ), j = 0, 1, 2, such that F2 − F1 , F1 − F0 ∈ K(N , τ ). Then for any λ ∈ (a, b) the following equality holds true ξ F2 ,F0 (λ) = ξ F2 ,F1 (λ) + ξ F1 ,F0 (λ). We omit the proof as it is similar to that of Proposition 3.7. Let −1/2 ϕ(λ) = λ 1 + λ2 . It is easy to see that if D = D ∗ ηN is an operator with τ -compact resolvent, then the operator FD := ϕ(D) belongs to F −1,1 (N , τ ). Proposition 3.11. If D0 = D0∗ ηN is an operator with τ -compact resolvent, and if V = V ∗ ∈ N , D1 = D0 + V, then the following equality holds: ξ D1 ,D0 (λ) = ξ FD1 ,FD0 (ϕ(λ)).
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N. A. Azamov, A. L. Carey, F. A. Sukochev
Proof. Let h ∈ Cc3 (−1, 1) and f (λ) = h(ϕ(λ)). Then by Theorem 2.5, A := τ ( f (D1 ) − f (D0 )) =
f (λ)ξ D1 ,D0 (λ) dλ,
R
and since FD1 − FD0 ∈ K(N , τ ) by [15, Lemma 2.7], we can apply Proposition 2.12 to get
A = τ h(FD1 ) − h(FD0 ) = = =
1
−1 ∞
h (t)ξ FD1 ,FD0 (t) dt
−∞ ∞ −∞
h (ϕ(λ))ϕ (λ)ξ FD1 ,FD0 (ϕ(λ)) dλ f (λ)ξ FD1 ,FD0 (ϕ(λ)) dλ.
Since f is an arbitrary C 2 -function with compact support, comparing the last two formulas we get the equality ξ D1 ,D0 (λ) = ξ FD1 ,FD0 (ϕ(λ)) + C.
(31)
It is left to show that the constant C = 0. Let h be a non-negative function from Cc∞ (−1, 1). By Lemma 2.8 we have
R
h(ϕ(λ))ξ D1 ,D0 (λ) dλ =
1
0
τ V h(FDr ) dr.
(32)
Multiplying the first term of the RHS of (31) by h(ϕ(λ)), integrating it and using Lemma 2.15, we get
A := =
h(ϕ(λ))ξ FD1 ,FD0 (ϕ(λ)) dλ =
R 1
1
−1
τ K h(Fr )(ϕ −1 ) (Fr ) dr,
h(µ)ξ FD1 ,FD0 (µ)(ϕ −1 ) (µ) dµ
0
where K = FD1 − FD0 and Fr is the straight line path connecting FD1 and FD0 . Let g(µ) = h(µ)(ϕ −1 ) (µ). By Corollary 3.9 we have
1
A=
1
τ (K g(Fr )) dr =
0
0
τ F˙ Dr g(FDr ) dr.
By Proposition 1.18(ii) we have F˙ Dr = TϕD[1]r ,Dr (V ). Hence, A= 0
1
τ TϕD[1]r ,Dr (V )g(FDr ) dr.
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Using the BS-representation for ϕ [1] given by (5), it follows from the definition of DOI (8), Lemma 1.12(ii) and Lemma 1.11, that 1 i(s−t)Dr it Dr A= τ e Ve dνϕ (s, t) · g(FDr ) dr
0
= 0
1
τ ei(s−t)Dr V eit Dr g(FDr ) dνϕ (s, t) dr
1 1 = √ τ V eis Dr is ϕ(s)g(F ˆ Dr ) ds dr 2π 0 R 1 1 = √ τ V g(FDr ) eis Dr is ϕ(s) ˆ ds dr 2π 0 R 1 1
= τ V g(FDr )ϕ (Dr ) dr = τ V h(FDr ) dr, 0
(33)
0
since g(ϕ(λ))ϕ (λ) = h(ϕ(λ)). It follows from (31), (32) and (33) that C R h(ϕ(λ)) dλ = 0 and, hence, C = 0. 3.4. The first formula for spectral flow. We establish first a spectral flow formula for bounded τ -Fredholm operators. In this way we avoid a number of difficulties with unbounded operators. Then we make a ‘change of variable’ to get to the unbounded case. First we require some additional notation which is important for establishing a convention for how we handle the situation when the endpoints have a kernel. Let a < 0, b > 0 and let signa,b be the function defined as signa,b (x) = b if x 0, and signa,b (x) = a if x < 0. = signa,b (F), when it is clear from the context what the numbers a We will write F and b are. Definition 3.12. If F ∈ F a,b (N , τ ) and κ is a C 2 -function on [0, ∞) vanishing in a neighbourhood of 0 then for h(λ) = κ ((b − λ)(λ − a)) we define γh (F) as 1 α hFr ( F˙r ) dr, γh (F) = 0
where α h is the closed one-form defined before Proposition 3.8, and {Fr }r ∈[0,1] is the straight line connecting F and F. The following theorem is the analogue in our setting of [16, Theorem 5.7]. It is the fundamental formula that we need as our starting point. Theorem 3.13. Let F0 ∈ F a,b (N , τ ), let K ∈ K(N , τ ) and let F1 = F0 + K . Let κ be a C 2 -function on [0, ∞) vanishing in a neighbourhood of 0, such that the integral of h(λ) = κ ((b − λ)(λ − a)) over (a, b) is equal to 1. Then the spectral flow between F0 and F1 is equal to b sf(F0 , F1 ) = h(λ)ξ F1 ,F0 (λ) dλ + γh (F1 ) − γh (F0 ). a
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Proof. The proof follows ideas of [16, Theorem 5.7]. First of all by Phillips’ definition of spectral flow [5] we have 0 , F 1 ). sf(F0 , F1 ) = sf( F (Note that, a priori we would expect to write 0 ) + sf( F 0 , F 1 ) + sf( F 1 , F1 ), sf(F0 , F1 ) = sf(F0 , F as noted in the proof however there is no spectral flow along the paths joining F and F of [15, Theorem 1.7].). Now, by Lemma 3.1 we have 0 , F 1 ) = sf( F
1
˙ ) dr, α hF ( F r r
0
r }r ∈[0,1] is the straight line path, connecting F 0 and F 1 . By Corollary 3.9 we where { F can replace this path by the (broken) path given on this diagram FO 0 _ _ _/ F 1 γ (F ) −γh (F0 ) h 1 /F 0 1 F Then we get 1 ) = −γh (F0 ) + 0 , F sf( F
1 0
α hFr ( F˙r ) dr + γh (F1 ),
where {Fr }r ∈[0,1] is the straight line path, connecting F0 and F1 . But, setting F1 − F0 = K , we have by Lemma 2.15, 0
1
α hFr ( F˙r ) dr =
0
1
τ (K h(Fr )) dr =
R
h(λ)ξ F1 ,F0 (λ) dλ.
Theorem 3.14. Let F0 ∈ F −1,1 (N , τ ), let K ∈ K(N , τ ) and let F1 = F0 + K . Let κ be a C 2 -function on [0, ∞) vanishing in a neighbourhood of 0, such that the integral of h(λ) = κ(1 − λ2 ) over (−1, 1) is equal to 1. Then the spectral flow function for the pair F0 and F1 is equal to sf(µ; F0 , F1 ) =
1
−1
where h −µ (λ) = h(λ + µ).
h(λ)ξ F1 ,F0 (λ) dλ + γh −µ (F1 − µ) − γh −µ (F0 − µ),
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Proof. By definition we have sf(µ; F0 , F1 ) = sf(F0 − µ, F1 − µ). Since F j − µ ∈ F −1−µ,1−µ (N , τ ), by the previous theorem we have sf(F0 − µ, F1 − µ) 1−µ = h −µ (λ)ξ F1 −µ,F0 −µ (λ) dλ + γh −µ (F1 − µ) − γh −µ (F0 − µ). −1−µ
Since ξ F1 −µ,F0 −µ (λ) = ξ F1 ,F0 (λ + µ), we have sf(F0 − µ, F1 − µ) 1−µ h(λ + µ)ξ F1 ,F0 (λ + µ) dλ + γh −µ (F1 − µ) − γh −µ (F0 − µ) = −1−µ 1
=
−1
h(λ)ξ F1 ,F0 (λ) dλ + γh −µ (F1 − µ) − γh −µ (F0 − µ).
Corollary 3.15. If F0 and F1 are unitarily equivalent, then sf(µ; F0 , F1 ) = ξ F1 ,F0 (µ) = const.
Proof. By Corollary 2.13 the function ξ F1 ,F0 (·) is constant on (−1, 1), so that R h(λ) ξ F1 ,F0 (λ) dλ = ξ F1 ,F0 (0). If F0 and F1 are unitarily equivalent, then γh −µ (F1 − µ) = γh −µ (F0 − µ). Hence, for all µ ∈ (−1, 1), sf(µ; F0 , F1 ) = ξ F1 ,F0 (µ) = ξ F1 ,F0 (0). Lemma 3.16. If F ∈ F −1,1 (N , τ ) and if {h ε }ε>0 is an approximate δ function (by compactly supported even functions) then for all µ ∈ (−1, 1) the limit γµ (F) := lim γh ε (F − µ) ε→0
exists and is equal to ξG,G (0), where G = F − µ. Proof. Since h is an even function we have that
0 −∞
h ε (λ)ξG,G (λ) dλ →
1 ξ (0−) 2 G,G
h ε (λ)ξG,G (λ) dλ →
1 ξ (0+), 2 G,G
and
∞ 0
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then by Lemma as ε → 0. If {G r }r ∈[0,1] is the straight line path connecting G and G, 2.15 we have 1 1
˙ γh ε (G) = αh ε (G r ) dr = τ G˙ r h ε (G r ) dr 0 0 1
= ξG,G h ε (λ)ξG,G (λ) dλ → (0−) + ξG,G (0+) = ξG,G (0) 2 R as ε → 0, by Definition 2.6 of ξ at discontinuity points.
Now we need to handle the situation when the endpoints are not unitarily equivalent. For this we require some additional facts about the ‘end-point correction terms’. The interesting fact which we now establish is that the approach using the spectral shift function differs in a fundamental way from the previous point of view in [16]. The next few results demonstrate this by showing that the spectral shift function absorbs the contribution to the formula due to the spectral asymmetry of the endpoints leaving only kernel correction terms to be handled. Lemma 3.17. If F ∈ F −1,1 (N , τ ) and if µ ∈ (−1, 1), then the following equality holds true 1 γµ (F) = τ [ker(F − µ)]. 2 Proof. Let G = F − µ. We have G G τ [ker G] = τ E (−∞,0] . − E (−∞,0] By Proposition 2.12(ii) and Definition 2.14 the value ξG,G (0) is the half sum of the last expression and G G τ E (−∞,0) = 0. − E (−∞,0) Hence, by Lemma 3.16, γµ (F) = ξG,G (0) =
1 τ ([ker G]) . 2
Theorem 3.18. If F0 , F1 ∈ F −1,1 (N , τ ) such that F1 − F0 ∈ K(N , τ ), then for all µ ∈ (−1, 1), 1 sf(µ; F0 , F1 ) = ξ F1 ,F0 (µ) + (τ [ker(F1 − µ)] − τ [ker(F0 − µ)]). (34) 2 Proof. Replace h in Theorem 3.14 by h ε,µ (thus translate the approximate δ function h ε by µ) and then let ε → 0 using Lemmas 3.16, 3.17. We now see that under hypotheses that guarantee both are defined the spectral flow function and the spectral shift function differ only by kernel corrections terms for the endpoints. We should remark that the occurrence of the correction terms γµ (F j ), j = 1, 2, in the last formula can be explained by the fact that we actually define the spectral flow function and the spectral shift function at discontinuity points in different ways. The spectral shift function is defined as a half-sum of the left and the right limits, while the spectral flow is defined to be left-continuous.
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3.5. Spectral flow in the unbounded case. The formulae for spectral flow in the bounded case may be now used to establish corresponding results in our original setting of unbounded self adjoint operators with compact resolvent. By Proposition 3.11 ξ D1 ,D0 (0) = ξ FD1 ,FD0 (0) and by definition of spectral flow for unbounded operators [5] sf(D0 , D1 ) = sf(FD1 , FD0 ). Hence, it follows from (34) taken at µ = 0 that sf(D0 , D1 ) = ξ D1 ,D0 (0) + γ0 (F1 ) − γ0 (F0 ). Since ker(D) = ker(FD ), we have the following equality: 1 1 sf(D0 , D1 ) = ξ D1 ,D0 (0) + τ [ker(D1 )] − τ [ker(D0 )]. 2 2 If we replace here the operators D0 and D1 by the operators D0 − λ and D1 − λ respectively then we get 1 1 sf(λ; D0 , D1 ) = ξ D1 −λ,D0 −λ (0) + τ [ker(D1 − λ)] − τ [ker(D0 − λ)]. 2 2 Since ξ D1 −λ,D0 −λ (0) = ξ D1 ,D0 (λ) it follows that 1 1 sf(λ; D0 , D1 ) = ξ D1 ,D0 (λ) + τ [ker(D1 − λ)] − τ [ker(D0 − λ)]. 2 2
(35)
3.6. The spectral flow formula using infinitesimal spectral flow. The results on the spectral shift function that we established in Sect. 2 now suggest a new direction for spectral flow theory. Definition 3.19. Let D0 be a self-adjoint operator affiliated with N having τ -compact resolvent. The infinitesimal spectral flow one-form is a distribution-valued one-form D on the affine space D0 + Nsa , defined by formula D (X ), ϕ = τ (X ϕ(D)) ,
X ∈ Nsa , ϕ ∈ Cc∞ (R).
Formally, D (X ) = τ (X δ(D)) , where δ(D) is the δ-function of D. We believe that the notion of infinitesimal spectral flow will have application to the problem of studying spectral flow when the endpoints differ by a D0 relatively bounded perturbation. To this end we establish that spectral flow may be reformulated in terms of it.
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Theorem 3.20. Let D1 ∈ D0 + Nsa . Spectral flow between D0 and D1 is equal to the integral of the infinitesimal spectral flow one-form along any piecewise C 1 -path {Dr }r ∈[0,1] in D0 + N connecting D0 and D1 in the sense that for any ϕ ∈ Cc∞ (R) the following equality holds true: 1 sf(λ; D0 , D1 )ϕ(λ) dλ = Dr ( D˙ r ), ϕ dr. R
0
Formally,
1
sf(D0 , D1 ) = 0
Dr ( D˙ r ) dr,
or
sf(λ; D0 , D1 ) = 0
1
Dr −λ ( D˙ r ) dr.
Proof. By Corollary 3.6 we can choose the path {Dr }r ∈[0,1] to be the straight line path Dr = D0 + r V . It follows from Lemmas 1.3 and 1.4 that the functions λ → τ ([ker(D0 − λ)]) and λ → τ ([ker(D1 − λ)]) can be non-zero only on a countable set. Hence, by (35) and Lemma 2.8 we have sf(λ; D0 , D1 )ϕ(λ) dλ = ξ D1 ,D0 (λ)ϕ(λ) dλ R
R 1
=
1
τ (V ϕ(Dr )) dr =
0
0
Dr ( D˙ r ), ϕ dr.
We remark that the infinitesimal spectral flow one-form is exact in the sense that its value on every test function is exact. 3.7. The spectral flow formulae in the I-summable spectral triple case. The original approach of [16] required summability constraints on the operator D0 . We will now see that if indeed D0 satisfies such conditions then we can weaken conditions on the function f in Theorem 2.10. Lemma 3.21. Let D0 be a self-adjoint operator with a τ -compact resolvent affiliated with
N . Let g be an increasing continuous function on [0, +∞), g(0) 0 and
such that g c(1 + D 2 )−1 ∈ L1 (N , τ ) for all c > 0. Let f (x) = g (1 + x 2 )−1 . Then for any R > 0 and for any V = V ∗ ∈ N the operator f (D + V ) is trace class and the function V ∈ B R → f (D + V )1 is bounded. Proof. By [22, Lemma 2.5(iv)] we have for all t > 0, −1 −1 2 2 = g µt 1 + (D + V ) . µt ( f (D + V )) = µt g 1 + (D + V ) By Lemma 1.7 there exists a constant c = c(R) > 0 such that for any V ∈ B R , −1 −1 1 + (D + V )2 c 1 + D)2 .
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Hence, by [22, Lemma 2.5(iii)] we have −1 −1 2 2 µt ( f (D + V )) g µt c 1 + D = µt g c 1 + D .
Since g c(1 + D 2 )−1 ∈ L1 (N , τ ), the last function belongs to L 1 [0, ∞), which implies that f (D + V ) ∈ L1 (N , τ ). Lemma 3.22. Let D0 , g and f be as in Lemma 3.21. An integral of the one-form f
α D (X ) = τ (X f (D)), X ∈ N , D ∈ D0 + Nsa , along a piecewise smooth path in D0 + Nsa depends only on endpoints of that path. Proof. Let f n be an increasing sequence of compactly supported smooth functions converging pointwise to f and 1 , 2 be two piecewise smooth paths in D0 + Nsa with the same endpoints. Then by Lemma 3.21, Lebesgue dominated convergence theorem and Corollary 3.6 we have αf = lim α fn = lim α fn n→∞ 1 1 n→∞ 1 fn = lim α = lim α fn = αf. n→∞ 2
2 n→∞
2
The condition that g(c(1 + D02 )−1 ) be trace class is a generalised summability constraint. This notion arises naturally for certain ideals I of compact operators (for example for the Schatten ideals I p , p 1, g(x) = x p/2 and we have the notion of p-summability). We have also already remarked on the θ -summable case. Now if there is a unitary u ∈ N with V = u ∗ [D0 , u] bounded then we have, for a dense subalgebra A of the C ∗ -algebra generated by u, a semifinite ‘g-summable’ spectral triple (A, N , D0 ). Moreover D0 + V = u D0 u ∗ so we have unitarily equivalent endpoints. Theorem 3.23. Let f be a non-negative L 1 -function such that f (Dr ) ∈ L1 (N , τ ) for all r ∈ [0, 1], and let r → f (Dr )1 be integrable on [0, 1]. If D0 and D1 are unitarily equivalent then 1 −1 sf(λ; D0 , D1 ) = C τ (V f (Dr − λ)) dr, where C =
∞
−∞
0
f (λ) dλ.
Proof. Unitary equivalence of D0 and D1 implies that two last terms in (35) vanish. In case of f ∈ Bc (R), multiplying (35) by f (λ) and integrating it we get the required equality by Lemma 2.8 and Theorem 2.9. For an arbitrary f ∈ L 1 the claim follows from Lebesgue’s dominated convergence theorem by approximating f by an increasing sequence of step-functions converging a.e. to f. The following corollary recovers two of the main results of [15, 16].
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Corollary 3.24. (i) If D0 is θ -summable with respect to N and if D0 and D1 are unitarily equivalent then 1 ε 2 sf(D0 , D1 ) = τ V e−ε Dr dr. π 0 (ii) If D0 is p-summable (i.e. (1 + D02 )− p/2 ∈ L1 (N , τ )) with respect to N , where p > 1 and if D0 and D1 are unitarily equivalent then 1 − p 2 2 dr, sf(D0 , D1 ) = C −1 τ V 1 + D p r where C p =
∞
−∞
0
p
(1 + λ2 )− 2
dλ. p
Proof. Put f (λ) = e and f (λ) = (1+λ2 )− 2 for (i) and (ii) respectively in Theorem 3.23. The conditions of that theorem are fulfilled by Lemma 3.21. −ελ2
3.8. Recovering η-invariants. To demonstrate that we have indeed generalized previous analytic approaches to spectral flow formulae we still need some refinements. What is missing is the relationship of the ‘end-point correction terms’ to the truncated eta invariants of [24]. In fact Theorem 3.13 combined with some ideas of [16] will now enable us to give a new proof of the original formula (28) for spectral flow with unitarily inequivalent endpoints. Introduce the function ε −3/2 ε(1−λ−1 ) κε (λ) = λ e , π
and let h ε (λ) = κε (1 − λ2 ), f ε (λ) = κε (1 + λ2 )−1 . Lemma 3.25. Let D0 = D0∗ ηN be θ -summable, let V ∈ Nsa and let D1 = D0 + V. Then 1 1 ε 2 h ε (λ)ξ FD1 ,FD0 (λ) dλ = τ V e−ε Dr dr. π 0 −1 Proof. Since h ε (ϕ(µ)) = f ε (µ), by Proposition 3.11 we have ∞ 1 h ε (λ)ξ FD1 ,FD0 (λ) dλ = h ε (ϕ(µ))ξ FD1 ,FD0 (ϕ(µ))ϕ(µ) dµ (A) := −1 −∞ ∞ = f ε (µ)ϕ (µ)ξ D1 ,D0 (µ) dµ. −∞
Further, by Lemmas 2.4 and 2.3, 1 1 ∞
Dr f ε (µ)ϕ (µ) τ V d E µ dr = τ V f ε (Dr )ϕ (Dr ) dr (A) = −∞ 0 0 1 ε 2 3/2 −ε Dr2 2 −3/2 dr (1+ Dr ) e (1+ Dr ) = τ V π 0 1 ε 2 = τ V e−ε Dr dr. π 0
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As we have emphasized previously, the strategy of our proof follows that of [16] in that we deduce the unbounded version of the spectral flow formula for the theta summable case from a bounded version. To this end introduce Fs = D(s + D 2 )−1/2 . Lemma 3.26. [16, Lemma 8.8] We have lim α h ε = 0, δ→0 δ
where δ is the straight line connecting F0 and Fδ . Lemma 3.27. If D = D ∗ ηN is θ -summable, then the following equality holds true 1 (ηε (D) + τ [ker D]) . 2 Proof. We note that 1 − Fs2 = s(s + D 2 )−1 and that F˙s = − 21 D(s + D 2 )−3/2 . The path 1 := {Fs }s∈[0,1] connects sgn(FD ) with FD . If we denote by 2 the straight line path = sign(FD ) then the path −1 + 2 connects FD with F, connecting sgn(FD ) with F so that by Lemma 3.22 applied to f = h ε , and by the argument of [16] and Lemma 3.26 dealing with discontinuity of the path 1 at zero, it follows that γh ε (FD ) = − αhε + αhε . γh ε (FD ) =
1
We have for the first summand 1 hε ˙ hε α = α Fs ( Fs ) ds = 1
0
0
1
2
τ F˙s h ε (Fs ) ds
1 1 τ D(s + D 2 )−3/2 κε 1 − Fs2 ds =− 2 0 1 1 =− τ D(s + D 2 )−3/2 κε s(s + D 2 )−1 ds 2 0 ε 2 1 ε 1 =− τ D(s + D 2 )−3/2 s −3/2 (s + D 2 )3/2 e− s D ds 2 π 0 1 ε 1 − ε D 2 −3/2 =− s τ De s ds 2 π 0 ∞ dt 1 ε 1 2 τ De−εt D √ = − ηε (D). =− 2 π 1 2 t
Let E = [ker D] and let G r = sgn(FD ) + r E be the path 2 . Then the second summand is equal to 1 1
hε ˙ α = τ G r h ε (G r ) dr = τ Eκε (1 − G r2 ) dr. 2
Since 1 − 2
G r2
=
0
0
E(1 − r 2 ),
αhε =
so that γh ε (FD ) =
0
it follows that 1 2 τ Eκε (1 − r ) dr = τ (E)
1 2
0
(ηε (D) + τ [ker D]) .
1
κε (1 − r 2 ) dr =
1 τ (E), 2
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As a direct corollary of these lemmas and Theorem 3.13 we get a new proof of (28). The idea used in the proof of the next theorem, of approximating the exponential function by functions of compact support, has been exploited in the context of the spectral flow formula in [35]. Theorem 3.28. If D0 is θ -summable then the formula (28) holds true. Proof. Let h n be a sequence of smooth non-negative functions, compactly supported on (−1, 1), and converging pointwise to h ε . Recall that Dr = D0 + r V . Then the sequence 1 γh n (FDr ) converges to γh ε (FDr ) and the sequence −1 h n (λ)ξ FD1 ,FD0 (λ) dλ converges 1 to −1 h ε (λ)ξ FD1 ,FD0 (λ) dλ by Lebesgue’s DCT, since θ -summability of D0 implies 1summability of h ε (FDr ). Hence the claim follows from Theorem 3.13, Lemma 3.25 and Lemma 3.27. References 1. Asplund, E., Bungart, L.: A First Course in Integration. New York: Holt, Rinehart and Winston, 1966 2. Atiyah, M., Patodi, V., Singer, I.M.: Spectral Asymmetry and Riemannian Geometry. III. Math. Proc. Camb. Phil. Soc. 79, 71–99 (1976) 3. Azamov, N.A., Carey, A.L., Dodds, P.G., Sukochev, F.A.: Operator integrals, spectral shift and spectral flow. to appear in Canad. J. Math, available at http://arxiv.org/list/math/0703442, 2007 4. Azamov, N.A., Dodds, P.G., Sukochev, F.A.: The Krein spectral shift function in semi-finite von Neumann algebras. Integral Equations Operator Theory 55, 347–362 (2006) 5. Benameur, M.-T., Carey, A.L., Phillips, J., Rennie, A., Sukochev, F.A., Wojciechowski, K.P.: An analytic approach to spectral flow in von Neumann algebras. In: Analysis, geometry and topology of elliptic operators, Hackensack, NJ: World Sci. Publ., 2006, pp. 297–352 6. Birman, M.Sh., Pushnitski, A.B.: Spectral shift function, amazing and multifaceted. Dedicated to the memory of Mark Grigorievich Krein (1907–1989). Integral Equations Operator Theory 30, 191–199 (1998) 7. Birman, M.Sh., Solomyak, M.Z.: Remarks on the spectral shift function. J. Soviet Math. 3, 408–419 (1975) 8. Birman, M.Sh., Solomyak, M.Z.: Double Stieltjes operator integrals. I. In: Problems of Mathematical Physics, No. 1, Spectral Theory and Wave Processes, Leningrad: Izdat. Leningrad. Univ., 1966, pp. 33–67 (Russian) 9. Birman, M.Sh., Solomyak, M.Z.: Double Stieltjes operator integrals. II. In: Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems, Leningrad: Izdat. Leningrad. Univ., 1967, pp. 26–60 (Russian) 10. Birman, M.Sh., Solomyak, M.Z.: Double Stieltjes operator integrals. III, Leningrad: Izdat. Leningrad Univ., 1973, pp. 27–53 (Russian) 11. Birman, M.Sh., Yafaev, D.R.: The spectral shift function, the work of M. G. Krein and its further development. Algebra i Analiz 4, 833–870 (1992) 12. Booβ-Bavnbek, B., Lesch, M., Phillips, J.: Unbounded Fredholm operators and spectral flow. Canad. J. Math. 57, 225–250 (2005) 13. Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics I. Berlin-HeidelbergNewyork: Springer-Verlag, 1979 14. Carey, A.L., Hannabuss, K.C., Mathai, V., McCann, P.: The quantum hall effect on the hyperbolic plane. Commun. Math. Phys 190, 629–673 (1998) 15. Carey, A.L., Phillips, J.: Unbounded Fredholm modules and spectral flow. Canad. J. Math. 50, 673–718 (1998) 16. Carey, A.L., Phillips, J.: Spectral flow in Fredholm modules, eta invariants and the JLO cocycle. K-Theory 31, 135–194 (2004) 17. Carey, A.L., Phillips, J., Rennie, A., Sukochev, F.A.: The local index formula in semifi- nite von Neumann algebras I: spectral flow. Adv. in Math. 202, 451–516 (2006) 18. Carey, A.L., Phillips, J., Rennie, A., Sukochev, F.A.: The local index formula in semifi- nite von Neumann algebras II: even case. Adv. in Math. 202, 517–554 (2006) 19. Connes, A.: Noncommutative Geometry. San Diego: Academic Press, 1994 20. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. GAFA 5, 174–243 (1995) 21. Daletski˘ı, Yu.L., Kre˘ın, S.G.: Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations. Vorone˘z. Gos. Univ., Trudy Sem. Funkcional. Anal. 1, 81–105 (1956) (Russian)
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22. Fack, T., Kosaki, H.: Generalised s-numbers of τ -measurable operators. Pacific J. Math. 123, 269– 300 (1986) 23. Gelfand, I.M., Shilov, G.E.: Generalized functions, Vol. 1, New York: Academic Press, 1964 24. Getzler, E.: The odd Chern character in cyclic homology and spectral flow. Topology 32, 489–507 (1993) 25. Jacobs, K.: Measure and integral. New York-San Francisco-London: Academic Press, 1978 26. Kre˘ın, M.G.: Some new studies in the theory of perturbations of self-adjoint operators. (Russian) First Math. Summer School (Kanev, 1963), Part I, Kiev: Naukova Dumka, 1964, pp. 103.187; English transl. in M.G. Kre˘ın: Topics in differential and integral equations and operator theory, Basel: Birkhäuser, 1983, pp. 107–172 27. de Pagter, B., Sukochev, F.A.: Differentiation of operator functions in noncommutative Lp-spaces. J. Funct. Anal. 212, 28–75 (2004) 28. Peller, V.V.: Multiple operator integrals and higher operator derivatives. http://arxiv.org/list/math.SP/ 0505555, 2005 29. Phillips, J., Raeburn, I.: An index theorem for Toeplitz operators with noncommutative symbol space. J. Funct. Anal. 120, 239–263 (1994) 30. Phillips, J.: Self-adjoint Fredholm operators and spectral flow. Canad. Math. Bull. 39, 460–467 (1996) 31. Phillips, J.: Spectral flow in type I and type II factors - a new approach. Fields Inst. Comm. 17, 137–153 (1997) 32. Simon, B.: Trace ideals and their applications. London Math. Society Lecture Note Series 35, CambridgeLondon: Cambridge University Press, 1979 33. Simon, B.: Spectral averaging and the Krein spectral shift. Proc. Amer. Math. Soc. 126, 1409–1413 (1998) 34. Takesaki, M.: Theory of Operator Algebras. Vol. I, Berlin Heidelberg-Newyork: Springer-Verlag, 2002 35. Wahl, C.: Spectral flow as winding number and integral formulas. http://arxiv.org/list/math/0703287, 2007 36. Wahl, C.: A new topology on the space of unbounded selfadjoint operators and the spectral flow. http://arxiv.org/list/math/0607783, 2006 Communicated by A. Connes
Commun. Math. Phys. 276, 93–115 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0327-y
Communications in
Mathematical Physics
Decay of Weak Solutions to the 2D Dissipative Quasi-Geostrophic Equation César J. Niche, María E. Schonbek Department of Mathematics, UC Santa Cruz, Santa Cruz, CA 95064, USA. E-mail:
[email protected];
[email protected] Received: 15 May 2006 / Accepted: 15 April 2007 Published online: 28 August 2007 – © Springer-Verlag 2007
Abstract: We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ0 is in L 2 only, we prove that the L 2 norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For θ0 in L p ∩ L 2 , with 1 ≤ p < 2, we are able to obtain a uniform decay 2 rate in L 2 . We also prove that when the L 2α−1 norm of θ0 is small enough, the L q norms, 2 for q > 2α−1 , have uniform decay rates. This result allows us to prove decay for the L q norms, for q ≥
2 2α−1 ,
2
when θ0 is in L 2 ∩ L 2α−1 .
1. Introduction and Statement of Results We consider the dissipative 2D quasi-geostrophic equation θt + (u · ∇)θ + (−)α θ = 0, θ (x, 0) = θ0 (x),
(1.1)
where x ∈ R2 , t > 0 and subcritical exponent 21 < α ≤ 1. In this equation, θ = θ (x, t) is a real scalar function (the temperature of the fluid), u is an incompressible vector field (the velocity of the fluid) determined by the scalar function ψ (the stream function/pressure) through u = (u 1 , u 2 ) = (−
∂ψ ∂ψ , ). ∂ x2 ∂ x1
The temperature θ and the stream function ψ are related by ψ = −θ, The second author was partially supported by NSF grant DMS-0600692.
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C. J. Niche, M. E. Schonbek 1
where is the usual operator given by = (−) 2 and defined via the Fourier transform as s 2 f (ξ ) = |ξ |s fˆ(ξ ), s ≥ 0. When α = 21 , “dimensionally, the 2D quasi-geostrophic equation is the analogue of the 3D Navier-Stokes equations” (Constantin and Wu [11]), and the behaviour of solutions to (1.1) is similar to that of the 3D Navier-Stokes equations. For this reason, α = 21 is considered the critical exponent, while α ∈ ( 21 , 1] are the subcritical exponents. Note that when α = 1, (1.1) is the vorticity equation of the 2D Navier-Stokes equations. Besides its intrinsic mathematical interest, the dissipative 2D quasi-geostrophic equation describes models arising in meteorology and oceanography. More specifically, it can be derived from the General Quasi Geostrophic equations by assuming constant potential vorticity and constant buoyancy frequency (see Constantin, Majda and Tabak [10] and Pedlosky [20]). Consider the dissipative quasi-geostrophic equation with subcritical exponent, this is α ∈ ( 21 , 1]. In this article, we address the uniform decay of the L q norm, for q ≥ 2, of weak solutions to (1.1) for the initial data θ0 in different spaces. We first describe results related to the ones obtained here. In his Ph.D. thesis, Resnick [21] proved existence of global solutions to (1.1) for θ0 in L 2 . Moreover, he proved a maximum principle θ (t) L p ≤ θ0 L p , t ≥ 0
(1.2)
for 1 < p ≤ ∞. Constantin and Wu [11] established uniqueness of “strong” solutions (for a precise statement of this and Resnick’s result, see Sect. 2.1) and also showed that for θ0 in L 1 ∩ L 2 , 1
θ (t) L 2 ≤ C(1 + t)− 2α , t ≥ 0.
(1.3)
Their proof relies on an adaptation of the Fourier splitting method developed by Schonbek [22, 23] and on the retarded mollifiers method of Cafarelli, Kohn and Nirenberg [2]. Moreover, they proved that for generic initial data, the decay rate (1.3) is optimal. Using rather general pointwise estimates for the fractional derivative α θ and a positivity lemma, Córdoba and Córdoba [12] gave a new proof of (1.2) and proved decay of solutions when θ0 is in L 1 ∩ L p , for 1 < p < ∞. More specifically, they showed that − p−1 αp
θ (t) L p ≤ C1 (1 + C2 t)
, t ≥ 0,
(1.4)
where C1 and C2 are explicit constants. Working along the same lines, Ju [15] obtained an improved maximum principle of the form θ (t) L p ≤ θ0 L p
1
C p−2 t 1+ p−2
p 2− 2 pα
(1.5)
for θ0 in L 2 ∩ L p , with p ≥ 2 and a constant C = 1. Note that for p = 2, i.e. θ0 in L 2 , this expression reduces to (1.2); this is θ (t) L 2 ≤ θ0 L 2 . We now state the results we prove in this article. As we mentioned in the last paragraph, when θ0 is in L 2 , the decay (1.5) reduces to the maximum principle (1.2) and no decay rate can be deduced. We address this issue in the following theorems, where we prove that the L 2 norm of weak solutions tends to zero but not uniformly, that is, there are solutions with arbitrarily slow decay.
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Theorem 1.1. Let θ be a solution to (1.1) with θ0 ∈ L 2 . Then lim θ (t) L 2 = 0.
t→∞
Theorem 1.2. Let r > 0, > 0, T > 0 be arbitrary. Then, there exists θ0 in L 2 with θ0 L 2 = r such that if θ (t) is the solution with initial data θ0 , then θ (T ) L 2 ≥ 1 − . θ0 L 2 To prove Theorem 1.1 we adapt an argument used in Ogawa, Rajopadhye and Schonbek [19] to prove decay in the context of the Navier-Stokes equations with slowly varying external forces. It consists in finding estimates for the decay of the norm in the frequency space, studying separately low and high frequencies. The decay of the low frequency part is obtained through generalized energy inequalities, while the Fourier splitting method is used to bound the decay of the high frequency part. To construct the slowly decaying solutions of Theorem 1.2 we follow the ideas used by Schonbek [23] to prove a similar result for the Navier-Stokes equations. Namely, we construct initial data θ0λ whose L 2 norm does not change under an appropiate λ-scaling, such that it gives rise to a slowly decaying solution to the linear part of (1.1). We then impose extra conditions on θ0λ to control the term related to the nonlinear part, making it arbitrarily small for small enough values of λ. The next result concerns the decay of the L 2 norm of solutions when the initial data is in L p ∩ L 2 , with 1 ≤ p < 2. Theorem 1.3. Let θ0 ∈ L p ∩ L 2 , where 1 ≤ p < 2. Then, there is a weak solution such that 1 2 − 2α ( p −1)
θ (t) L 2 ≤ C(1 + t)
.
The proof of Theorem 1.3 has similarities with the one for (1.3) in Constantin and Wu [11]. We remark that the decay rate we obtain is of the same type as in (1.4) and (1.5), where θ0 is in L 1 ∩ L p , with 1 < p < ∞ and in L 2 ∩ L p , with p ≥ 2, as proved in Córdoba and Córdoba [12] and Ju [15] respectively. The next theorem is key for establishing decay of the L q norm of solutions, for large enough q. Theorem 1.4. Let θ0
2
L 2α−1 1
tα Moreover, for
2 2α−1
≤ κ. Then, for m =
( m1 − q1 )
2 2α−1
≤ q < ∞,
θ (t) ∈ BC((0, ∞), L q ).
≤ q < ∞, 1
t 2α
+ α1 ( m1 − q1 )
∇θ (t) ∈ BC((0, ∞), L q ).
The proof of Theorem 1.4 is based on ideas used by Kato [17] for proving a similar result for the Navier-Stokes equations. Namely, we construct (in an appropiate space) a solution to (1.1) by successive approximations θn+1 , whose norms are bounded by that of θ1 , θn and ∇θn . This gives rise to a system of recursive inequalities that can be solved if the norm of the data θ0 is small enough. We can then extract a subsequence converging to a solution with a certain decay rate. This preliminary estimate is then used to obtain the decay rate in Theorem 1.4. Note that when α = 1, we recover the rates obtained by Kato [17].
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Remark 1.1. A result related to Theorem 1.4 concerning existence of strong solutions in 2 L p for small data in L q , where 2α−1 < q ≤ p and 1p + αq = α − 21 , was proven by Wu [27]. These solutions exist only in an interval [0, T ], where the size of the initial data 2 tends to zero when T goes to infinity. Notice that the special case of θ0 in L 2α−1 is not covered by the hypothesis. Remark 1.2. It is well known that solutions to the 3D Navier-Stokes equations, i.e. (1.1) with α = 1, are smooth when θ0 is small in L 2 and the solution is in H 1 (see Heywood [13], Kato [17] and Serrin [26]). For the quasi-geostrophic equation with critical expo3 nent α = 21 , Córdoba and Córdoba [12] proved that when θ0 is in H 2 and is small in ∞ L , the solution is in fact classical. These results suggest that the solution obtained in Theorem 1.4 might have better regularity than the one obtained. We now state the result concerning decay of L q norms, for large q. 2
Theorem 1.5. Let θ0 ∈ L 2 ∩ L 2α−1 . Then there exists T = T (θ0 ) such that for t ≥ T 2 and 2α−1 ≤ q < ∞, 1
4a−3
θ (t) L q ≤ C t q α(2α−1) 2
1 −1+ 2α
.
2
By (1.5), when θ0 ∈ L 2 ∩ L 2α−1 , the L 2α−1 norm of the solution tends to zero. Then, after a (possibly long) time T = T (θ0 ), the solution enters the ball of radius κ, where κ is as in Theorem 1.4. Interpolation between the decays in (1.5) and Theorem 1.4, for some q in the appropiate range of values, provides us with a first decay rate. This rate, which is a function of q, can then be maximized, leading us to the result in Theorem 1.5. A similar idea was used by Carpio [3] to obtain analogous results for the Navier-Stokes equations. Remark 1.3. After this work was submitted we received preprints of articles by Carrillo and Ferreira [4–6] in which they prove results directly related to the ones obtained here. The proofs by Carrillo and Ferreira are, in general, rather different from ours, as they work in spaces that are related but not identical to the ones we work in. In [5], they prove Theorem 1.4 in the particular case α = 1 and θ0 ∈ L 2 but with no restriction on the size of the initial data θ0 . Moreover, they obtain estimates for the decay of all derivatives of θ in L 2 ∩ L q , thus showing that the solution is smooth (see Remark 1.2). In the forthcoming preprint [6], Carrillo and Ferreira extend their results to 21 < α ≤ 1 2
and θ0 ∈ L 2α−1 and also obtain decays analogous to those of Theorem 1.5, but in the 2 more restrictive case of initial data θ0 ∈ L 1 ∩ L 2α−1 . Recently, many articles concerning different aspects of the dissipative quasigeostrophic equation have been published. Besides the ones we have already referred to, see Berselli [1], Carrillo and Ferreira [4], Chae [7], Chae and Lee [8], Constantin, Córdoba and Wu [9], Ju [14, 16], Schonbek and Schonbek [24, 25], Wu [27–32] and references contained therein. This article is organized as follows. In Sect. 2, we collect the basic results and estimates we need. In Sect. 3 we prove Theorems 1.1 and 1.2, in Sect. 4 we prove Theorem 1.3 and finally in Sect. 5 we prove Theorems 1.4 and 1.5.
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2. Preliminaries In this section we collect some essential results and estimates concerning solutions to Eq. (1.1).
2.1. Existence and uniqueness of solutions. We first state the existence and uniqueness results we assume througout this article. Theorem 2.1 (Resnick [21]). Let T > 0 be arbitrary. Then, for every θ0 ∈ L 2 and f ∈ L 2 ([0, T ]; H −α ) there exists a weak solution of θt + (u · ∇)θ + (−)α θ = f, θ (x, 0) = θ0 (x), such that θ ∈ L ∞ ([0, T ]; L 2 ) ∩ L 2 ([0, T ]; H α ). Theorem 2.2 (Constantin and Wu [11]). Assume that α ∈ ( 21 , 1], T > 0 and p and q satisfy 1 α 1 + =α− . p q 2
p ≥ 1, q > 0,
Then there is at most one solution θ of (1.1) with initial value θ0 ∈ L 2 such that θ ∈ L ∞ ([0, T ]; L 2 ) ∩ L 2 ([0, T ]; H α ), θ ∈ L q ([0, T ]; L p ). These solutions obey a Maximum Priciple as in (1.2); this is θ (t) L p ≤ θ0 L p , t ≥ 0 for 1 < p ≤ ∞ (see Resnick [21], Córdoba and Córdoba [12] and Ju [15] for proofs). Multiplying (1.1) by θ and integrating in space and time yields s
t
α θ (τ )2L 2 dτ ≤ C, ∀ s, t > 0.
(2.6)
2.2. Estimates. Let θ (x, t) = K α (t, x) ∗ θ0 −
0
t
K α (t − s, x) ∗ u · ∇θ (s) ds
(2.7)
be the integral form of Eq. (1.1), where K α (x, t) is the kernel of the linear part of (1.1), i.e. 1 2α K α (x, t) = ei xξ e−|ξ | t dξ. 2π R2
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Proposition 2.3 (Wu [27]). Let 1 ≤ p ≤ q ≤ ∞. For any t > 0, the operators K α (t) : L p → L q , K α (t) f = K α (t) ∗ f, ∇ K α (t) : L p → L q , ∇ K α (t) f = ∇ K α (t) ∗ f are bounded and K α (t) f L q ≤ Ct
− α1 ( 1p − q1 )
∇ K α (t) f L q ≤ Ct
f L p ,
1 1 1 1 −( 2α + α ( p − q ))
(2.8)
f L p .
(2.9)
The following estimates for the integral term in the right-hand side of (2.7) are an immediate consequence of Proposition 2.3 and they are key in the proof of Theorem 1.4. Lemma 2.4. Let η ≤ µ + ν < 2. Then t t 1 K α (t − s) ∗ u · ∇θ (s) ds η2 ≤ C (t − s)− 2a (µ+ν−η) θ (s) L
0
0
2
Lµ
∇θ (s)
2
Lν
ds
(2.10) and
t
∇ K α (t − s) ∗ u · ∇θ (s) ds η2 L t 1 1 ≤C (t − s)−( 2α + 2a (µ+ν−η)) θ (s)
0
0
Proof. Use (2.8) and (2.9) with q = η2 , p = u · ∇θ
2
L µ+ν
2 µ+ν
≤ Cθ
2
Lµ
∇θ (s)
2
Lν
ds.
(2.11)
and 2
Lµ
∇θ
2
Lµ
which follows from Hölder’s inequality and boundedness of Riesz transform.
Lemma 2.5 (Schonbek and Schonbek [25]). Let β, γ be multi-indices, |γ | < |β| + 2α max( j, 1), j = 0, 1, 2, · · · , 1 ≤ p ≤ ∞. Then x γ Dt D β K α (t)L p = Ct j
|γ |−|β| p−1 2α − j− pα
for some constant C depending only on α, β, γ , j, p. 3. L 2 Decay for Initial Data in L 2 3.1. Proof of Theorem 1.1. Let θ (t) be a solution to (1.1) with θ0 ∈ L 2 . For φ = φ(ξ, t), 2 2 ˆ ˆ . (3.12) + (1 − φ(t))θ(t) θˆ (t)2L 2 ≤ 2 φ(t)θ(t) L2 L2 2 and (1−φ(t))θ(t) 2 the low and high frequency parts ˆ ˆ We call the terms φ(t)θ(t) L2 L2 of the energy respectively. In Propositions 3.1 and 3.3 and Corollary 3.2, we obtain estimates, for an appropiate class of functions φ, that allow us to prove that the low and high frequency parts of the energy tend to zero. These estimates are of similar character to the ones that Ogawa, Rajophadye and Schonbek obtained for the Navier-Stokes equations in [19].
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3.1.1. Energy estimates. We first establish some preliminary estimates which will be needed in the proof of Theorem 1.1. Proposition 3.1. Let ψ ∈ C 1 ((0, ∞), C 1 ∩ L 2 ). Then for 0 < s < t, t ˆ ), ψ θˆ (τ ) − |ξ |α θˆ ψ (τ )2 2 | dτ (s)2 2 + 2 θ(τ (t)2 2 ≤ θˆ ψ |ψ θˆ ψ L
L
t
+2
L
s
2 θˆ (τ )| dτ. |ξ · u θ (τ ), ψ
s
Proof. Let θ (t) be a smooth solution to (1.1). Taking the Fourier transform, multiplying 2 θˆ and integrating by parts we obtain the formal estimate by ψ d αˆ 2 ˆ θˆ 2 2 = 2 ψ θ (t), ψ θ(t) ψ ψ θ (t) − |ξ | 2 L L dt 2 θˆ (t). − 2u · ∇θ (t), ψ Integrating between s and t yields
t 2 2 ˆ ), ψ ˆ ˆ θˆ (τ ) − |ξ |α θˆ ψ (τ )2 2 | dτ θ(τ θ ψ (t) L 2 ≤ θ ψ (s) L 2 + 2 |ψ L s t 2 θˆ (τ )| dτ. |ξ · u θ (τ ), ψ +2 s
The retarded mollifiers method (see Lemma 4.3) allows us to extend this estimate to weak solutions. For full details see Ogawa, Rajopadhye and Schonbek [19].
Corollary 3.2. Let φ ∈ C 1 ((0, ∞), L 2 ). Then for 0 < s < t, t 2α 2 −|ξ |2α (t−s) 2 ˆ ˆ )| dτ. ˆ φ(t) L 2 + 2 |ξ · u θ , e−2|ξ | (t−τ ) φ 2 (t)θ(τ θ φ(t) L 2 ≤ θ (s)e s
η (τ ) = Proof. Take ψ
2α e−|ξ | (t+η−τ ) φ(ξ, t)
for η > 0. Then
ˆ ), ψ η θˆ (τ ) = |ξ |2α ψ η θˆ (τ ) = |ξ |α ψ η θˆ (τ ), ψ η θˆ (τ )2 2 , η θ(τ ψ L so the integrand in the second term in Proposition 3.1 vanishes. Taking the limit as η → 0 (t) = φ(ξ, t) and ψ (s) = e−|ξ |2α (t−s) φ(ξ, t), so we see that ψ t 2α 2α ˆ )| dτ, θˆ φ(t)2L 2 ≤ θˆ (s)e−|ξ | (t−s) φ(t)2L 2 + 2 |ξ · u θ , e−2|ξ | (t−τ ) φ 2 (t)θ(τ s
as we wanted to prove.
Proposition 3.3. Let E ∈ C 1 ((0, ∞), R) and ψ ∈ C 1 ((0, ∞), L ∞ ) such that 1 − ψ 2 ∈ L ∞ ((0, ∞), L ∞ ) and ∇(1 − ψ 2 )ˇ ∈ L ∞ ((0, ∞), L 2 ). Then t ˆ )2 2 dτ E(t)ψ θˆ (t)2 2 ≤ E(s)ψ θˆ (s)2 2 + E (τ )ψ θ(τ L
L
t
+2 s
+2
s
t
s
L
E(τ )|ψ θˆ (τ ), ψ θˆ (τ ) − |ξ |α ψ θˆ (τ )2L 2 | dτ ˆ )| dτ. E(τ )|u · ∇θ (τ ), (1 − ψ 2 (τ ))θ(τ
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Proof. We prove the estimate first for smooth solutions. As in Proposition 3.1, we take the Fourier transform of (1.1) and multiply it by Eψ 2 θˆ . Integrating by parts and then between s and t we obtain the formal estimate t 2 2 ˆ ˆ E(t)ψ θ(t) L 2 ≤ E(s)ψ θ (s) L 2 + E (τ )ψ θˆ (τ )2L 2 dτ s t ˆ )2 2 | dτ E(τ )|ψ θˆ (τ ), ψ θˆ (τ ) − |ξ |α ψ θ(τ +2 L s t E(τ )|u · ∇θ (τ ), (1 − ψ 2 (τ ))θˆ (τ )| dτ. +2 s
Here we used that u · ∇θ, θ = 0. When using the retarded mollifiers method, the conditions 1 − ψ 2 ∈ L ∞ ((0, ∞), L ∞ ) and ∇(1 − ψ 2 )ˇ ∈ L ∞ ((0, ∞), L 2 ) will guarantee the weak convergence of the nonlinear term. For full details see Ogawa, Rajopadhye and Schonbek [19].
3.1.2. Proof of Theorem 1.1. We first prove the following easy estimate. Lemma 3.4. Let m > 0 and
f m (t) =
|ξ |>1
|ξ |2α e−m|ξ |
2α t
dξ.
Then limt→∞ f m (t) = 0. Proof. From the inequality |ξ |2α e−m|ξ |
2α t
e−m|ξ | mt
2α t/2
≤C
it follows that f m (t) =
|ξ |>1
|ξ |2α e−m|ξ |
2α t
dξ ≤ C
Thus, limt→∞ f m (t) = 0.
2α t/2
|ξ |>1 ∞ r e−mtr/2
e−m|ξ |t/2 dξ = C ≤C mt |ξ |>1 1 C 2 e−mtr/2 ≤ 1+ =C . (mt)2 mt (mt)2
e−m|ξ | mt
mt
dξ
dr (3.13)
We choose φ(ξ, t) = e−|ξ | t . Note that φ is the kernel of the solution to the Fourier transform of (1.1). Low frequency energy decay. Using Corollary 3.2 with φ as defined above we obtain 2α
θˆ φ(t)2L 2 ≤ θˆ (s)φ(t − s)φ(t)2L 2 + 2
s
t
ˆ ). |ξ · u θ, φ 2 (t − τ )φ 2 (t)θ(τ
(3.14)
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A standard application of the Dominated Convergence Theorem proves that the first term in the right-hand side of (3.14) tends to zero when t goes to infinity. Now ˆ ) = ξ α · u ˆ ) ξ · u θ (τ ), φ 2 (t − τ )φ 2 (t)θ(τ θ (τ ), |ξ |1−α φ 2 (t − τ )φ 2 (t)θ(τ α 1−α 2 ˆ ) L 1 . ≤ |ξ | u θ (τ ) L ∞ |ξ | φ (t − τ )φ 2 (t)θ(τ (3.15) As u θ (ξ ) = uˆ ∗ θˆ (ξ ), then
α α
ˆ |ξ | u θ (τ ) L ∞ = sup
|ξ | u(ξ ˆ − η)θ (η) dη
ξ ∈R2
≤C
R2
sup
2
R2
ξ ∈R
+ sup
2 ξ ∈R
|ξ − η|α u(ξ ˆ − η)θˆ (η) dη
ˆ |η| u(ξ ˆ − η)θ(η) dη
α
R2
α )u θ ∞ + u ≤ C ( a θ L ∞ ≤ C (α u)θ L 1 + u α θ L 1 L ≤ Cα θ L 2 . Now ˆ ) L 1 ≤ C|ξ |1−2α φ 2 (t − τ )φ 2 (t) L 2 |ξ |α θˆ (τ ) L 2 . |ξ |1−α φ 2 (t − τ )φ 2 (t)θ(τ (3.16) and
|ξ |1−2α φ 2 (t − τ )φ 2 (t)2L 2 =
R
≤
2
|ξ |2−4α e−4|ξ |
|ξ |≤1
2α (2t−τ )
dξ
|ξ |2−4α dξ + f 4 (2t − τ ).
(3.17)
As 21 < α ≤ 1, the first term in the right-hand side of (3.17) is integrable. By Lemma 3.4 and (3.13) we see that f 4 (2t − τ ) ≤
C ≤C 4(2t − τ )2
(3.18)
for t large enough. Then using (3.15), (3.16), (3.17) and (3.18) we obtain that t t ˆ ) dτ ≤ θ (τ ) L ∞ |ξ |1−α φ 2 (t − τ ) |ξ · u θ , φ 2 (t − τ )φ 2 (t)θ(τ |ξ |α u 2 s
s
ˆ ) L 1 dτ × φ 2 (t)θ(τ t θ (τ ) L ∞ |ξ |1−2α φ 2 (t − τ )φ 2 (t) ≤2 |ξ |α u s
× θˆ (τ ) L 2 |ξ |α θˆ (τ ) L 2 dτ t ≤C α θ (τ )2L 2 dτ. s
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Taking limits as s and t go to infinity, (2.6) implies that the low frequency part of the energy goes to zero. High energy frequency decay. Let ψ(ξ, t) = 1 − φ(ξ, t). As 1 − ψ 2 (ξ, t) = 2φ(ξ, t) − φ 2 (ξ, t) = 2e−|ξ |
2α t
− e−2|ξ |
2α t
decays exponentially fast, we can apply Proposition 3.3. After rearranging terms, we obtain 2 2 ˆ ˆ (1 − φ(t))θ(t) = ψ(t)θ(t) ≤ I + I I + I I I + I V, L2 L2
where E(s) 2 ˆ ψ θ(s) , L2 E(t) t 1 E (τ )ψ θˆ (τ )2L 2 − 2E(τ )|ξ |α ψ θˆ (τ )2L 2 dτ, II = E(t) s t 2 III = E(τ )ψ θˆ (τ ), ψ(τ )θˆ (τ ) dτ, E(t) s t 2 E(τ )|u · ∇θ (τ ), (1 − ψ 2 (τ ))θˆ (τ )| dτ. IV = E(t) s I =
We choose E(t) = (1 + t)k , where k > 2. Term I. Since |ψ| ≤ C and θ ∈ L 2 , 1+s k 1+s k 2 ˆ ψ(s)θ(s) ≤ C . I = L2 1+t 1+t Thus, lim I (t) = 0.
t→∞
Term II. We use the Fourier splitting method. Let B(t) = {ξ ∈ R2 : |ξ | ≤ G(t)}, where G is to be determined below. Then E (τ )ψ(τ )θˆ (τ )2L 2 − 2E(τ )|ξ |α ψ(τ )θˆ (τ )2L 2 ˆ )|2 dξ − 2E(τ ) = E (τ ) |ψ θ(τ |ξ |2α |ψ θˆ (τ )|2 dξ R2 B(t) R2 B(t) ˆ )|2 dξ − 2E(τ ) ˆ )|2 dξ + E (τ ) |ψ θ(τ |ξ |2α |ψ θ(τ B(t) B(t) ˆ )|2 dξ ≤ E (τ ) − 2E(τ )G 2α (τ ) |ψ θ(τ R2 B(t) ˆ )|2 dξ − 2E(τ ) +E (τ ) |ψ θ(τ |ξ |2α |ψ θˆ (τ )|2 dξ. B(t)
B(t)
(3.19)
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1 2α k Choosing G(t) = 2(1+t) , we see that E (τ ) − 2E(τ )G 2α (τ ) = 0 so the first term in the right-hand side of (3.19) vanishes. As the last term in (3.19) is negative, it can be dropped, hence t k k−1 ˆ (τ )|2 dξ dτ. II ≤ (1 + τ ) |ψ θ (1 + t)k s B(t) As ψ(ξ, t) = 1 − e−|ξ | t , then |ψ| ≤ |ξ |2α for |ξ | ≤ 1. Then 2 ˆ ˆ )|2 dξ ≤ C G 4α (t) = |ψ θ(τ )| dξ ≤ |ξ |4α |θ(τ 2α
B(t)
B(t)
Then II ≤
k (1 + t)k
t
(1 + τ )k−3 dτ ≤
s
C . (1 + t)2
C , (1 + t)2
so lim I I (t) = 0.
t→∞
2α −|ξ | t = |ξ |2α φ(ξ, t). As Term III. As ψ(ξ, t) = 1 − e−|ξ | t , then ψ = ∂ψ ∂t = |ξ | e E(t) is an increasing function t 2 ˆ ), (1 − φ(τ ))θ(τ ˆ ) dτ III = E(τ )|ξ |2α φ(τ )θ(τ E(t) s t ≤ 2 |ξ |α θˆ (τ ), |ξ |α θˆ (τ ) dτ s t ≤2 α θˆ (τ )2L 2 dτ. (3.20) 2α
2α
s
Taking limits when t and s go to infinity we obtain lim I I I (t) = 0.
t→∞
Term IV. Let ω(ξ, ˆ t) = 1 − ψ 2 (ξ, t). Then ˆ )| ≤ |ξ |α |u |u · ∇θ (τ ), ω(τ ˆ )θ(τ θ (τ )|, |ξ |1−a |ωˆ θˆ (τ )| ˆ ) L 1 . ≤ |ξ |α u θ (τ ) L ∞ |ξ |1−α ωˆ θ(τ
(3.21)
As in (3.16) |ξ |1−α ω(τ ˆ )θˆ (τ ) L 1 ≤ C|ξ |1−2α ω(τ ˆ ) L 2 |ξ |α θˆ (τ ) L 2 . We notice that as 2α − 4 < 0, then |ξ |1−2α ω(τ ˆ )2L 2 = |ξ |2−4α |ω| ˆ 2 dξ R2 2−4α ≤ |ξ | dξ + |ξ |≤1
|ξ |≥1
|ω| ˆ 2 dξ ≤ C.
(3.22)
(3.23)
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C. J. Niche, M. E. Schonbek
Then by (3.21), (3.22) and (3.23),
t
IV ≤ 2
s
s
t
≤2
t
≤2 s
|u · ∇θ (τ ), ω(τ ˆ )θˆ (τ )| dτ θ (τ ) L ∞ |ξ |1−2α ω(τ |ξ |α u ˆ ) L 2 α θ (τ ) L 2 dτ α θ (τ )2L 2 dτ.
As before, letting s and t go to infinity we obtain lim I V (t) = 0.
t→∞
Thus, the high frequency part of the energy goes to zero, which concludes the proof of Theorem 1.1.
3.2. Proof of Theorem 1.2. We briefly describe the idea of the proof. In order to make the decay of a solution to (1.1) arbitrarily slow, we will construct a set of initial data {θ0λ }λ>0 in L 2 such that θ0λ L 2 = θ0 L 2 . The mild solution to (1.1) with initial data θ0λ , θ λ (x, T ) = K α (T ) ∗ θ0λ (x) −
T 0
K α (T − s) ∗ (u λ · ∇)θ λ (s) ds,
(3.24)
has the following property: given T > 0, we can find λ sufficiently close to zero, so that the L 2 norm of the first term of the right-hand side of (3.24) stays arbitrarily close to that of θ0 . For this to hold, θ0λ must be such that: a) the L 2 norm of θ0λ is invariant under the scaling; b) θ0λ gives rise to a self-similar solution to the linear part of (1.1); and c) θ0 is in L p ∩ L q , for appropiate p and q, so that the integral term will be sufficiently small. We remark that as a result of our choice, the L p and L q norms of θ0λ will not be invariant under scaling. We proceed to the proof now. For θ0 in L 2 , it is easy to see that θ0λ (x) = λθ0 (λx) is such that θ0λ L 2 = θ0 L 2 , λ > 0. Then, for these θ0λ , condition a) holds. Now let θ0 be such that θ0λ gives rise to a selfsimilar solution λ to the linear part of (1.1), that is λ (x, t) = λ(λx, λ2α t) is a solution to t + (−)α = 0, λ0 (x) = θ0λ (x).
Decay of Solutions to Quasi-Geostrophic Equation
105
By uniqueness of the solution to the linear part, we have λ (x, t) = K α (t) ∗ λ0 (x) = K α (t) ∗ θ0λ (x), thus λ (t) L 2 = |λ (x, t)|2 d x = λ2 |(λx, λ2α t)|2 d x R2 R2 2α 2α 2α 2 = |(y, λ t)| dy = e−|ξ | λ t |θˆ0 (ξ )| dξ. R2
R2
As a result of this, given T > 0,
λ (T )2 −|ξ |2α λ2α T |θˆ (ξ )|2 dξ 0 L2 R2 e = 1. lim = lim 2 dξ 0 2 2 λ→0 λ→0 | θ (ξ )| R2 0 L
(3.25)
This shows that choosing λ small enough, we can make the ratio of the norms arbitrarily close to 1 for large enough t. We now address the integral term in (3.24). We first notice that K α (t − s) ∗ (u λ · ∇)θ λ (s) L 2 = ∇ K α (t − s) ∗ (u λ θ λ )(s) L 2 ≤ ∇ K α (t − s) L 1 (u λ θ λ )(s) L 2 1
≤ C(t − s)− 2α u λ (s) L p θ λ (s) L q 1
≤ C(t − s)− 2α θ λ (s) L p θ λ (s) L q , where we have used Lemma 2.5 with γ = 0, p = 1, β = 1, j = 0, Hölder’s inequality with 21 = 1p + q1 and boundedness of the Riesz transform. By the Maximum Principle (1.2), θ λ (s) L m ≤ θ0λ L m and as 2
θ0λ L m = λ1− m θ0 L m
(3.26)
then 1
K α (t − s) ∗ (u λ · ∇)θ λ (s) L 2 ≤ C(t − s)− 2α λ ≤ C(t − s)
1 − 2α
2−( 2p + q2 )
θ0 L p θ0 L q
λθ0 L p θ0 L q .
θ0λ
(3.27)
norm of is invariant only when m = 2. Choosing We remark that by (3.26), the θ0 in L p ∩ L q (condition c)) we obtain T 1 K α (T − s) ∗ (u λ · ∇)θ λ (s) L 2 ds ≤ C T 1− 2α λθ0 L p θ0 L q . (3.28) Lm
0
So given > 0 and T > 0, we can choose λ > 0 such that by (3.25), K α (T ) ∗ θ0λ L 2 θ0λ L 2
and by (3.28)
T 0
≥1− , 2
K α (T − s) ∗ (u λ · ∇)θ λ (s) L 2 ds θ0λ L 2
Then θ λ (T ) L 2 ≥ 1 − . θ0λ L 2 This proves our result.
≤
. 2
106
C. J. Niche, M. E. Schonbek
4. L 2 Decay for Initial Data in L p ∩ L 2 , 1 ≤ p < 2 To prove Theorem 1.3, we follow a modified version of the Fourier splitting method, see Constantin and Wu [11]. Similar ideas in the context of the 2D Navier-Stokes equation can be found in Zhang [33]. In order to compute the actual decay rate of the L 2 norm, we need a preliminary estimate, proven in Lemma 4.3, which we then use to establish the right decay. In both proofs we first obtain formal estimates for smooth solutions through the Fourier splitting method and we then use the method of retarded mollifiers of Cafarelli, Kohn and Nirenberg [2] to extend them to weak solutions. The following auxiliary lemmas will be necessary in the sequel. 1
Lemma 4.1. Let h ∈ L p , 1 ≤ p < 2 and let S(t) = {ξ ∈ R2 : |ξ | ≤ g(t)− 2α }, for a continuous function g : R+ → R+ . Then
ˆ 2 dξ ≤ Cg(t) |h|
− α1 ( 2p −1)
.
S(t)
Proof. By Cauchy-Schwarz
ˆ 2 dξ ≤ |h|
S(t)
ˆ 2r dξ |h|
1
1
r
s
dξ
S(t)
,
S(t)
where r1 + 1s = 1. Setting 2r = q, we obtain
1 r
=
2 q
and
1 s
=
2 p
− 1. By the Riesz-Thorin
→ is bounded for p ∈ [1, 2] and Interpolation Theorem, F : p ˆ q p is in L , then h L ≤ h L and as a result of this Lp
ˆ 2 dξ ≤ C |h| S(t)
Lq
2 −1 p
dξ
2
= C (V ol S(t)) p
1 p
+
1 q
= 1. As h
−1
S(t)
= C r (t)
2( 2p −1)
= Cg(t)
− α1 ( 2p −1)
.
Lemma 4.2. Let θ be a solution to (1.1). Then, |u · ∇θ (ξ )| ≤ C|ξ |θ 2L 2 .
Proof. As u · ∇θ (ξ ) = ∇ · uθ (ξ ) = ξ · u θ (ξ ), boundedness of the Fourier transform and of the Riesz transform imply |u · ∇θ (ξ )| = |ξ ||u θ(ξ )| ≤ |ξ |u θ L ∞ ≤ C|ξ |uθ L 1 ≤ C|ξ |θ 2L 2 .
In the next lemma we establish the preliminary decay rate.
Decay of Solutions to Quasi-Geostrophic Equation
107
Lemma 4.3. Let θ be a solution to (1.1) with initial data θ0 in L p ∩ L 2 , 1 ≤ p < 2. Then 1 ˆ 2 dξ ≤ C ln(e + t)−(1+ α ) . |θ| R2
Proof. The first part of the proof consists of a formal argument that proves the expected decay for smooth solutions. At the end of the proof we sketch how to make the argument rigrous. We use the Fourier splitting method, taking 1
B(t) = {ξ : |ξ | ≤ g − 2α (t)}, 1 where g(t) = ( 21 + 2α )[(e+t) ln(e+t)]. From (1.1), after multiplying by θ and integrating
d |θˆ |2 dξ = −2 |ξ |2α |θˆ |2 dξ. 2 dt R2 R Then, as ˆ 2 dξ ≥ 2 2 |ξ |2α |θ| R2
ˆ 2 dξ + (1 + |ξ |2α |θ| B(t)
1 )[(e + t) ln(e + t)]−1 α
(4.29)
B(t)c
|θˆ |2 dξ,
(4.29) becomes d 1 ˆ 2 dξ + (1 + )[(e + t) ln(e + t)]−1 ˆ 2 dξ |θ| |θ| 2 dt R2 α R 1 −1 ≤ (1 + )[(e + t) ln(e + t)] |θˆ |2 dξ. α B(t) 1
Multiplying on both sides by h(t) = [ln(e + t)]1+ α , writing the left-hand side as a derivative and integrating between 0 and t, 1 ˆ 2 dξ ≤ θ0 2 2 [ln(e + t)]1+ α |θ| L R2 t 1 1 ˆ 2 dξ ds. + (1 + )[(e + s)]−1 ln(e + s) α |θ| α B(s) 0 (4.30) Hence, we need to estimate (1.1),
ˆ 2 dξ . From the solution to the Fourier transform of
B(s) |θ|
2α θˆ (ξ, t) = θ0 (ξ )e−|ξ | t +
t
e−|ξ |
2α (t−s)
u · ∇θ (s)ds.
0
We obtain ˆ t)| ≤ |θ0 (ξ )| + |θ(ξ,
0
t
|u · ∇θ | ds
108
C. J. Niche, M. E. Schonbek
which, by Lemma 4.2 leads to ˆ t)| ≤ 2 |θ0 (ξ )| + |θ(ξ, 2
t
2
0
2 |ξ |θ (τ )2L 2
dτ
t ≤ 2 |θ0 (ξ )|2 + t|ξ |2 θ (τ )4L 2 dτ . 0
Then, passing to polar coordinates
ˆ 2 dξ ≤ 2 |θ| B(s)
|θ0 |2 dξ + B(s)
s|ξ |
2 0
B(s)
s
θ (τ )4L 2
dτ
dξ
− 1 ( 2 −1)
≤ C[(e + s) ln(e + s)] α p s π g(s) −2 α 2 3 4 +C r s θ (τ ) L 2 dτ dr dϕ 0 0 0 2 − 1 ( 2 −1) + Cs 2 g α (s) ≤ 2 C[(e + s) ln(e + s)] α p = C[(e + s) ln(e + s)]
− α1 ( 2p −1)
2
+ Cs 2 [(e + s) ln(e + s)]− α ,
(4.31)
where we have used Lemma 4.1 with g(s) = C[(e + s) ln(e + s)] and the Maximum Principle for the L 2 norm of θ . Substituting (4.31) in (4.30) we see that the integral in the right-hand side of (4.30) is finite, so
ˆ 2 dξ ≤ C [ln(e + t)]−(1+ α ) . |θ| 1
R2
The formal part of the proof is now complete. To extend the estimate to weak solutions, we repeat the argument, applying it to the solutions of the approximate equations ∂θn + u n ∇θn + (−)α θn = 0, ∂t where u n = δn (θn ) is defined by δn (θn ) =
t
φ(τ )R⊥ θn (t − δn τ ) dτ.
0
Here the operator R⊥ is defined on scalar functions as R⊥ f = (−∂x2 −1 f, ∂x1 −1 f )
∞ and φ is a smooth function with support in [1, 2] and such that 0 φ(t) dt = 1. For each n, the values of u n depend only on the values of θn in [t − 2δn , t − δn ]. As stated in Constantin and Wu [11], the functions θn converge to a weak solution θ and strongly in L 2 almost everywhere. Since the estimates obtained do not depend on n, they are valid for the limit function θ . The proof is now complete.
Decay of Solutions to Quasi-Geostrophic Equation
109
4.1. Proof of Theorem 1.3. As before, we prove a formal estimate for smooth solutions, which then can be extended to weak solutions by the method of retarded mollifiers. For the formal estimate, we proceed as in Lemma 4.3, employing the Fourier splitting method with 1
B(t) = {ξ : |ξ | ≤ g(t)− 2α } for g(t) = 2α(t + 1). Thus d 1 1 2 2 ˆ ˆ ˆ 2 dξ, |θ| dξ + |θ| dξ ≤ |θ| dt R2 α(t + 1) R2 α(t + 1) B(t) 1
which after using h(t) = (t + 1) α as an integrating factor leads to t 1 1 1 2 2 −1 2 ˆ ˆ (s + 1) α (t + 1) α |θ | dξ ≤ θ0 L 2 + |θ| dξ ds. 0 α R2 B(s)
(4.32)
Working as in (4.31) in Lemma 4.3, using Lemma 4.1 with g(s) = 2(s + 1) and the preliminary estimate from Lemma 4.3 for making 1
θ (τ )4L 2 ≤ θ (τ )2L 2 [ln(e + τ )]−(1+ α ) , we obtain 1 α (t + 1)
t − 1 ( 2 −1)+ α1 −1 2 2 ˆ |θ| dξ ≤ θ0 L 2 + C (1 + s) α p ds 2 R 0 t s 1 −2 1 + θ (τ )2L 2 [ln(e + τ )]−(1+ α ) sg α (s)(1 + s) α −1 dτ ds 0
0
1 2
1
− ( −1)+ α ≤ θ0 2L 2 + C(1 + s) α p t s 1 1 + θ (τ )2L 2 [ln(e + τ )]−(1+ α ) s(1 + s)−( α +1) dτ ds. 0
0
(4.33) Now I (t) =
t 0
s
0 t
(1 + s)
≤C
1
− a1
0
0
t
ds
(1 + τ )
0 t
≤C
1
θ (τ )2L 2 [ln(e + τ )]−(1+ α ) s(1 + s)−( α +1) dτ ds 1 α
[ln(e θ (τ )2L 2
1
+ τ )]−(1+ α ) 1
(1 + τ ) α
dτ
1
1
(1 + τ ) α θ (τ )2L 2
[ln(e + τ )]−(1+ α ) 1
(1 + τ ) α
dτ,
so taking 1
f (t) = (t + 1)
1 α
θ (t)2L 2 ,
− α1 ( 2p −1)+ α1
a(t) = (1 + t)
, b(t) =
[ln(e + τ )]−(1+ α ) 1
(1 + τ ) α
,
110
C. J. Niche, M. E. Schonbek
Eq. (4.33) becomes
t
f (t) ≤ C + a(t) +
f (τ )b(τ )dτ.
0
By Gronwall’s inequality
t
f (t) ≤ f (0) exp
b(τ ) dτ
0
Notice that as
1 2
t
+
a (τ ) exp
0
< α ≤ 1, t b(τ ) dτ = 0
t 0
τ
t
b(s)ds dτ.
(4.34)
1
[ln(e + τ )]−(1+ α ) 1
(1 + τ ) α
< ∞.
Then (4.34) becomes (t + 1) α θˆ (t)2L 2 ≤ C θ0 2L 2 + (1 + t) 1
− α1 ( 2p −1)+ α1
ds,
hence 1 2 ( p −1)
1
θ (t)2L 2 ≤ (t + 1)− α + C(1 + t) α
1 2 ( p −1)
≤ C(1 + t) α
,
which proves the formal estimate. The retarded mollifiers method allows us to extend it to weak solutions.
2 5. L q Decay, for q ≥ 2α− 1 5.1. Proof of Theorem 1.4. We now describe the main ideas behind the proof of 2 Theorem 1.4. For clarity, we let m = 2α−1 . We first prove preliminary estimates of the form 1 1−δ m )
t α ( t
1 2α
θ (t) mδ ≤ C, t > 0,
(5.35)
∇θ (t) L m ≤ C, t > 0
(5.36)
for fixed 0 < δ < 1. To do so, following Katos’s [17] ideas, we construct a solution in m L δ to the integral equation (2.7) by successive approximations θ1 (t) = K α (t) ∗ θ0 ,
θn+1 (t) = K α (t) ∗ θ0 −
0
t
K α (t − s) ∗ (u n · ∇)θn (s) ds, n ≥ 1.
These approximations are such that 1 1−δ m )
t α (
θn+1 (t)
1
m Lδ
≤ K n+1 , t 2α ∇θn+1 (t) L m ≤ K n+1
(5.37)
are bounded by expressions that depend on K 1 , K n and K n only. If θ0 has small L m norm then these recursive relations are uniformly bounded; this is K n ≤ K , K n ≤ K , n ≥ 1 for some K > 0. A standard argument allows us to show that there is a uniformly m converging subsequence θn whose limit is a solution to (2.7) in L δ that obeys (5.35) and (5.36). These preliminary estimates are used to bootstrap a similar argument which proves the results stated in the theorem.
Decay of Solutions to Quasi-Geostrophic Equation
111
Proof. We begin by proving (5.35) and (5.36). Let δ be fixed, 0 < δ < 1. We note first that by (2.8) in Lemma 2.3, θ1 (t)
1 1−δ m )
≤ Ct − α (
m δ
L
θ0 L m ,
and by (2.9) in Lemma 2.3, 1
∇θ1 (t) L m ≤ Ct − 2α θ0 L m . Let K 1 = K 1 = Cθ0 L m . Now assume 1 1−δ m )
t α (
1 2α
t for t > 0. Then θn+1 (t)
∇θn (t) L m ≤ K n
m Lδ
≤ θ1 (t)
m Lδ
θn (t) mδ ≤ K n ,
t
+
1 1−δ m )
≤ K1 t − α (
0
t
+ 0
1 1−δ m )
≤ K1 t − α (
K α (t − s) ∗ (u n · ∇)θn (s)
L
m δ
K α (t − s) ∗ (u n · ∇)θn (s)
L
t
ds m δ
ds
1
(t − s)− αm θn (s) mδ ∇θn (s) L m ds L 0 t 1 1−δ 1 1−δ 1 ≤ K 1 t − α ( m ) + C K n K n (t − s)− αm s − αm − 2α ds +C
0
1 1−δ m )
≤ K1 t − α (
1 1−δ m )
+ C K n K n t − α (
,
(5.38)
where we used boundedness of the Riesz transform and (2.10) in Lemma 2.4 with 2 η = µ = 2δ m and ν = m . By an analogous method
t
∇θn+1 (t) L m ≤ ∇θ1 (t) L m + ∇ K α (t − s) ∗ (u n · ∇)θn (s) L m ds 0 t 1 ≤ K 1 t − 2α + ∇ K α (t − s) ∗ (u n · ∇)θn (s) L m ds 0 t 1 1 δ m ∇θn (s) L m ds ≤ K 1 t − 2α + C (t − s)−( 2α + mα ) θn (s) delta L 0 t 1 1 δ 1−δ 1 − 2α ≤ K1 t + C Kn Kn (t − s)−( 2α + αm ) s − αm − 2α ds ≤ K1 t
1 − 2α
+C
0 1 − 2α Kn Kn t ,
(5.39)
where we used boundedness of the Riesz transform and (2.11) in Lemma 2.4 with η = ν = m2 , µ = 2δ m . We have then that the norms described in (5.37) are respectively bounded by K n+1 ≤ K 1 + C K n K n , K n+1 ≤ K 1 + C K n K n .
112
If K 1 <
C. J. Niche, M. E. Schonbek 1 4c ,
an induction argument allows us to prove that K n ≤ K
Kn ≤ K , 1 . Note that K 0 < for n ≥ 1, where K = 2c of the initial data has to be small. Then 1 1−δ m )
t α (
t
1 2α
1 4c
θn (t)
implies θ0 L m < L
1 , 4c2
thus the L m norm
≤ K,
m δ
∇θn (t) L m ≤ K ,
for n ≥ 1. By a standard argument (see Kato [17] and Kato and Fujita [18] for full details) we can extract a subsequence that converges uniformly in (0, +∞) to a solution θ . Then 1 1−δ m )
t α(
m
θ ∈ BC((0, +∞), L δ ),
1
t 2α ∇θ ∈ BC((0, +∞), L m ). We now use these preliminary estimates to prove the theorem. As before, we construct a solution by successive approximations. Let m ≤ q < ∞. By (2.8) in Lemma 2.3, θ1 (t) L q ≤ Ct
− α1 ( m1 − q1 )
θ0 L m ,
and by (2.9) in Lemma 2.3, ∇θ1 (t) L q ≤ Ct
1 1 1 −( 2α + α ( m − q1 ))
θ0 L m .
Notice that this estimate holds for q ≥ m. Again, set K 1 = K 1 = Cθ0 L m . We want 1
( 1 −1)
1
+1( 1 −1)
to show inductively that the L q norms of t α m q θn (t) and t 2α α m q ∇n+1 θ (t) are uniformly bounded. Then t −1( 1 −1) θn+1 (t) L q ≤ K 1 t α m q + K α (t − s) ∗ (u n · ∇)θn (s) L q ds 0 t −1( 1 −1) − 1 ( 1+δ − 1 ) ≤ K1t α m q + C (t − s) α m q θn (s) mδ ∇θn (s) L m ds L 0 t 2−δ 1 −1( 1 −1) − 1 ( 1+δ − 1 ) ≤ K 1 t α m q + C K n K n (t − s) α m q s − mα − 2α ds 0
≤ K1 t
− α1 ( m1 − q1 )
1
1
1
− ( − ) + C K n K n t α m q ,
(5.40)
2 where we have used (2.10) in Lemma 2.4 with η = q2 , µ = 2δ m and ν = m and we have used the preliminary estimates obtained for θn (t) mδ and ∇θn (t) L m . Proceeding L analogously for the gradient we obtain t ∇θn+1 (t) L q ≤ ∇θ1 (t) L q + ∇ K α (t − s) ∗ (u n · ∇)θn (s) L q ds 0 t −( 1 + 1 ( 1 − 1 )) ≤ K 1 t 2α α m q + ∇ K α (t − s) ∗ (u n · ∇)θn (s) L q ds 0 t 2−δ 1 −( 1 + 1 ( 1 − 1 )) −( 1 + 1 ( 1+δ − 1 )) ≤ K 1 t 2α α m q + C (t − s) 2α α m q s − mα − 2α ds 0
≤ K1t
1 1 1 −( 2α + α ( m − q1 ))
+ K n K n t
1 1 1 −( 2α + α ( m − q1 ))
,
(5.41)
Decay of Solutions to Quasi-Geostrophic Equation
113
where we used (2.11) in Lemma 2.4 with η = q2 , µ = 1
K n+1 = t α K n+1 = t
( m1 − q1 )
2δ m
and ν =
2 m . As before,
setting
θn+1 (t) L q ,
1 1 1 1 2α + α ( m − q )
∇θn+1 (t) L q ,
(5.42)
we obtain K n+1 ≤ K 1 + C K n K n , K n+1 ≤ K 1 + C K n K n . The same arguments that were used for the preliminary estimates apply here, leading to 1
tα
( m1 − q1 )
θ (t) ∈ BC((0, ∞), L q )
and 1
t 2α for
2 2α−1
+ α1 ( m1 − q1 )
∇θ (t) ∈ BC((0, ∞), L q )
≤ q < ∞, which is the desired result.
2 5.2. Proof of Theorem 1.5. Let m = 2α−1 . By (1.5), the L m norm of θ tends to zero, so for times larger than some T = T (θ0 ), θ (t) L m ≤ κ, for κ as in Theorem 1.4. Let m ≤ q < r . Interpolation yields
θ (t) L q ≤ θ (t)aL m θ (t)1−a Lr for a =
m r −q q r −m
and 1 − a =
r q−m q r −m .
θ (t)
Lq
Then
≤ Ct
(1− α1 ) mq
r −q r −m
− α1 ( m1 − q1 )
.
This holds for any r such that q ≤ r < ∞. The optimal decay rate is given by the minimum of the exponent f (r ) = C1
r −q − C2 , r −m
where C1 = (1 − α1 ) mq and C2 = α1 ( m1 − q1 ). As C1 < 0, this is a non-increasing function, so the optimal decay rate is 1 1 1 1 m − − . lim f (r ) = C1 − C2 = 1 − r →∞ α q α m q Then 1
4a−3
θ (t) L q ≤ C t q α(2α−1) which is the desired result.
1 −1+ 2α
, t ≥ T,
Acknowledgements. The authors would like to thank Helena Nussenzveig-Lopes for calling their attention to the articles by Carrillo and Ferreira; José Carrillo and Lucas C.F. Ferreira for providing us with copies of their preprints and for helpful remarks concerning their work and the anonymous referee for his suggestions and comments.
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References 1. Berselli, L.C.: Vanishing viscosity limit and long-time behaviour for 2D quasi-geostrophic equations. Indiana Univ. Math. J. 51, 905–930 (2002) 2. Cafarelli, L., Kohn, H., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982) 3. Carpio, A.: Large-time behaviour in incompressible Navier-Stokes equations. SIAM J. Math. Anal 27, 449–475 (1996) 4. Carrillo, J., Ferreira, L.C.F.: Self-similar solutions and large time asymptotics for the dissipative quasigeostrophic equations. Monatshefte für Math 15, 111–142 (2007) 5. Carrillo, J., Ferreira, L.C.F.: Convergence towards self-similar asymptotic behaviour for the dissipative quasi-geostrophic equations. Banach Center Publ. 74, 95–115 (2006) 6. Carrillo, J., Ferreira, L.C.F.: Asymptotic behaviour for the subcritical dissipative quasi-geostrophic equations. Preprint UAB, 2006 7. Chae, D.: The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16(2), 479–495 (2003) 8. Chae, D., Lee, J.: Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun. Math. Phys. 233(2), 297–311 (2003) 9. Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50, Special Issue, 97–107 (2001) 10. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 11. Constantin, P., Wu, J.: Behaviour of solutions of 2D Quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (electronic) (1999) 12. Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249, 511–528 (2004) 13. Heywood, J.: Open problems in the theory of Navier-Stokes equations of viscous incompressible flow. The Navier-Stokes equations (Oberwolfach, 1988), Lecture Notes in Math. 1431, Berlin: Springer, 1990. pp. 1–22 14. Ju, N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251, 365–376 (2004) 15. Ju, N.: The maximum principle and the global attractor for the 2D dissipative quasi-geostrophic equation. Commun. Math. Phys. 255, 161–182 (2005) 16. Ju, N.: On the two dimensional quasi-geostrophic equations. Indiana Univ. Math. J. 54, 897–926 (2005) 17. Kato, T.: Strong L p solutions of the Navier-Stokes equation in Rm , with applications to weak solutions. Math. Z. 187, 471–480 (1984) 18. Kato, T., Fujita, H.: On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32, 243–260 (1962) 19. Ogawa, T., Rajopadhye, S., Schonbek, M.: Energy decay for a weak solution of the Navier-Stokes equation with slowly varying external forces. J. Funct. Anal. 144, 325–358 (1997) 20. Pedloskym, J.: Geophysical Fluid Dynamics. New York: Springer Verlag, 1987 21. Resnick, S.: Dynamical problems in non-linear advective partial differential equarions. Ph. D. Thesis, University of Chicago, 1995 22. Schonbek, M.: L 2 decay for weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 88, 209–222 (1985) 23. Schonbek, M.: Large time behaviour of solutions to the Navier-Stokes equations. Comm. Partial Diff. Eqs. 11, 733–763 (1986) 24. Schonbek, M., Schonbek, T.: Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J. Math. Anal. 35, 357–375 (electronic) (2003) 25. Schonbek, M., Schonbek, T.: Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Disc. Contin. Dyn. Syst. 13, 1277–1304 (2005) 26. Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Nonlinear problems (Proc. Sympos., Madison, Wis.), Madison, WI: Oniv. ofvisc. Press, pp. 69–98, 1963 27. Wu, J.: Dissipative quasi-geostrophic equations with L p data. Electron. J. Diff. Eq. (2001), No. 56, 13 pp. (electronic) 28. Wu, J.: The 2D dissipative quasi-geostrophic equation. Appl. Math. Letters 15, 925–930 (2002) 29. Wu, J.: The quasi-geostrophic equation and its two regularizations. Comm. Partial Diff. Eqs. 27(5-6), 1161–1181 (2002) 30. Wu, J.: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 36(3), 1014–1030 (2004/05) (electronic) 31. Wu, J.: Solutions of the 2D quasi-geostrophic equation in Hölder spaces. Nonlinear Anal. 62(4), 579–594 (2005)
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32. Wu, J.: The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity 18(1), 139–154 (2005) 33. Zhang, L.: Sharp rates of decay of solutions to 2-dimensional Navier-Stokes equations. Comm. Partial Diff. Eqs. 20, 119–127 (1995) Communicated by P. Constantin
Commun. Math. Phys. 276, 117–130 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0328-x
Communications in
Mathematical Physics
Regge and Okamoto Symmetries Philip P. Boalch École Normale Supérieure et CNRS, 45 rue d’Ulm, 75005 Paris, France. E-mail:
[email protected] Received: 31 May 2006 / Accepted: 17 April 2007 Published online: 13 September 2007 – © Springer-Verlag 2007
Abstract: We will relate the surprising Regge symmetry of the Racah-Wigner 6 j symbols to the surprising Okamoto symmetry of the Painlevé VI differential equation. This then presents the opportunity to give a conceptual derivation of the Regge symmetry, as the representation theoretic analogue of the derivation in [5, 3] of the Okamoto symmetry.
1. Introduction The 6 j-symbols (or Racah coefficients) are real numbers associated to the choice of six irreducible representations Va , . . . , V f of SU(2). They were first published in work of Racah [16] in 1942, and arise in the addition of three angular momenta, which classically can be viewed as adding three vectors in R3 . Apparently ([15]) “there is hardly any branch of physics involving angular momenta where the use of Racah-coefficients is not needed in order to carry out the simplest computation”. (See the two volumes [1, 2] for many more details or the introduction to the tables [20] for a concise summary.) Wigner [23] used a slightly different normalisation so that they have tetrahedral symmetry, and wrote them in the form: ab e . (1) cd f Here a, b, c should be thought of as the lengths of three vectors a, b, c in R3 so the four points 0, a, a + b, a + b + c are the vertices of a (skew) tetrahedron. Then d, e, f should be the lengths of the other three edges of this tetrahedron, i.e. the lengths of a + b + c, a + b, b + c respectively. (Thus each column of (1) contains the lengths of two opposite edges, and the top row abe is a face. See Fig. 1.) Then the 6 j-symbol is invariant under the possible relabellings of this tetrahedron (preserving the relations so one gets 24 = #Sym4 possibilities).
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c
e
b d f a
Fig. 1. Labelling of the edges of a skew tetrahedron
Racah established an explicit formula for the 6 j-symbols as a sum, which has since been equated with the value at 1 of certain 4 F3 hypergeometric functions (and are closely related to the Wilson orthogonal polynomials [24]). Using this explicit Racah formula, in 1959 Regge [18] showed the 6 j-symbols also have the following further symmetry, which is more mysterious: p−a p−b e ab e , = p−c p−d f cd f where p = (a + b + c + d)/2. (Combined with the tetrahedral symmetries this generates a symmetry group isomorphic to Sym3 × Sym4 .) For example classically, recalling that a tetrahedron is determined up to isometry by its edge lengths, one may show (cf. Ponzano–Regge [15] and Roberts [19]) that this Regge action on the set of six edge lengths defines a non-trivial automorphism of the set of (skew) Euclidean tetrahedra, taking a generic tetrahedron to a non-congruent tetrahedron. Earlier Regge [17] found similar extra symmetries of the Clebsch-Gordan 3 j-symbols. The 3 j-symbols are in a sense less canonical, but note that Ponzano–Regge [15] p.7 explain how to obtain the 3 j-symbols as an asymptotic limit of 6 j-symbols and that in this limit the 6 j Regge symmetry becomes the 3 j Regge symmetry. They then wrote: The geometrical and physical content of these [Regge] symmetries is still to be understood and they remain a puzzling feature of the theory of angular momenta. Therefore it is a pleasant result to be able to reduce the problem of their interpretation to the Racah [6j] coefficient only. The basic aim of this article is to give a conceptual explanation of the Regge symmetry of the Racah 6 j coefficient. The key idea is to relate the above 6 j-symbols (for the group SU(2)) to certain three-dimensional 6 j-symbols (i.e. for the group SU(3)). The Regge transformation then arises simply from the natural duality between two dual irreducible representations of SU(3). The layout of the remainder of this article is as follows. First we will give the definition of the 6 j-symbols, then we will relate the Regge symmetry to a symmetry of a completely different object, this time a non-linear differential equation, the Okamoto symmetry of the Painlevé VI equation. Then we will “quantise” (that is, give the representation theoretical analogue of) the derivation of the Okamoto symmetry given in [5, 3], and so give a conceptual derivation of the Regge symmetry (i.e. without using the Racah formula).
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Remark 1. Note that Granovski˘ı and Zhedanov have shown in [8] that the 6 j symbols are determined by a certain noncommutative algebra and so, as they show, finding the Regge symmetry may be reduced to finding the corresponding symmetry of this algebra. 2. Background The 6 j-symbols are real numbers associated to the choice of 6 irreducible representations (irreps) of G := SU(2). We will label irreps Va by positive integers a ≥ 0, so that Va = Syma (V ) is the spin a/2 representation of dimension a + 1, where V is the two-dimensional Hermitian vector space defining G. Given 3 such irreps, with labels a, b, c say, one can form the tensor product Vabc := Va ⊗ Vb ⊗ Vc which again will be a representation of G and will decompose as a direct sum of irreps. Thus given a fourth representation Vd one may consider the multiplicity space Mabcd := HomG (Vd , Vabc ) of G-equivariant maps from Vd into the 3-fold tensor product. Thus Mabcd is a Hermitian vector space with dimension equal to the multiplicity of Vd in Vabc . There are two (almost) canonical unitary bases in Mabcd (‘coupling bases’) and the 6 j-symbols arise as matrix entries of the change of basis matrix between these two bases. The coupling bases arise by decomposing Vabc in the two possible orders: on one hand we may first decompose Vab := Va ⊗ Vb : Ve ⊗ HomG (Ve , Vab ) Vab ∼ = e
which entails the following direct sum decomposition of Mabcd : HomG (Vd , Vec ) ⊗ HomG (Ve , Vab ). Mabcd = HomG (Vd , Vab ⊗ Vc ) ∼ = e
The key point is that each of the terms in this direct sum is either zero or onedimensional (since, for SU(2), any irrep appears at most once in the tensor product of two irreps). Thus choosing a real vector of length one in each one dimensional term yields the 1-2 coupling basis {ve } of Mabcd as e varies, unique up to the sign of each basis vector. (We set ve = 0 if the space HomG (Vd , Vec ) ⊗ HomG (Ve , Vab ) is zero.) Similarly decomposing the 3-fold product in the other order, i.e. first writing V f ⊗ HomG (V f , Vbc ) Vbc ∼ = f
yields a different basis {w f }, the 2-3 coupling basis, adapted to the decomposition HomG (Vd , Va f ) ⊗ HomG (V f , Vbc ). Mabcd = f
Thus given six irreps with labels a, b, c, d, e, f , and a standard sign-convention, one will get two vectors ve , w f in Mabcd and thus a number U (a, b, c, d, e, f ) = ve , w f
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by pairing them using the Hermitian form. (As e and f vary these will be the matrix entries of the unitary change of basis matrix alluded to above—in fact by reality it is real orthogonal.) The 6 j-symbols were defined by Wigner in terms of U by a minor normalisation: U (a, b, c, d, e, f ) ab e , = (−1) p √ cd f (e + 1)( f + 1)
p := (a + b + c + d)/2.
This normalisation is such that the 6 j-symbols admit tetrahedral symmetry, where the coefficients label the six edges of a tetrahedron (containing for example the quadrilateral abcd and faces abe and bc f ). Note that if U is non-zero then p will be an integer. As mentioned in the introduction using the explicit Racah formula for the 6 j-symbols, Regge [18] showed the 6 j-symbols also have the following symmetry: ab e p−a p−b e = , cd f p−c p−d f where p = (a + b + c + d)/2. (Since e, f are fixed, one may just as well view this as a symmetry of the corresponding function U .) More geometrically one may also view this as a symmetry of the set of tetrahedra in R3 (cf. [15, 19]). First note that if Ve appears in Va ⊗ Vb then the triangle inequalities a − b≤ e ≤ a + b hold; i.e. there exists a Euclidean triangle with sides of lengths a, b, e. (One also has the parity condition that a + b + e be even.) Thus if both vectors ve and w f are non-zero then there are four Euclidean triangles with side lengths abe, cde, ad f, bc f respectively. Now these four triangles may or may not fit together to form the faces of a Euclidean tetrahedron; the condition that they do is given by requiring the determinant of the ‘Cayley-Menger matrix’: ⎛
0 a 2 e2 d 2 1
⎜ ⎜ a2 ⎜ ⎜ ⎜ e2 ⎜ ⎜ ⎜ d2 ⎝ 1
⎞
⎟ 0 b2 f 2 1 ⎟ ⎟ ⎟ 2 2 b 0 c 1⎟ ⎟ ⎟ 2 2 f c 0 1⎟ ⎠ 1 1 1 0
to be positive. It is simple to check the Regge transformation preserves the set of all triangle inequalities (although permuting them in a non-trivial way). Moreover a computation will show that the determinant of the Cayley-Menger matrix is preserved too. Thus one may view the Regge transformation as an automorphism of the set of Euclidean tetrahedra, even if we allow real (not necessarily integral) edge lengths (noting that a tetrahedron with non-zero volume is determined by its edge lengths up to isometry, possibly reversing the orientation). Our first step in deriving the Regge symmetry is to note a remarkably similar symmetry of a completely different object, this time of a certain nonlinear differential equation.
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The Painlevé VI differential equation (henceforth PVI ) is the following nonlinear ordinary differential equation for a holomorphic function y(t) with t ∈ C \ {0, 1}: 2 1 1 1 dy 1 1 dy 1 + + + + − y y−1 y−t dt t t − 1 y − t dt t y(y − 1)(y − t) (t − 1) t (t − 1) α + β , + + γ + δ t 2 (t − 1)2 y2 (y − 1)2 (y − t)2
1 d2 y = 2 dt 2
where α, β, γ , δ are four complex constants. PVI is usually thought of as controlling the monodromy preserving (isomonodromic) deformations of rank 2 (traceless) Fuchsian systems with 4 poles on P1 (whose monodromy is a representation of the free group on 3 generators into SL2 (C)). Okamoto [14] proved PVI has a quite nontrivial symmetry: Theorem 1. Choose four complex constants θ = (θ1 , . . . θ4 ) and set α = (θ4 − 1)2 /2,
β = −θ12 /2,
γ = θ32 /2,
δ = (1 − θ22 )/2.
If y(t) is a solution of PVI with parameters θ then, if defined, y + φ/x solves PVI with parameters θ = (θ1 − φ, θ2 − φ, θ3 − φ, θ4 − φ), where φ =
4
1 θi /2
and
2x =
t y + θ3 (t − 1) y − θ1 y − 1 − θ2 + − , y y−t y−1
where y is the derivative of y with respect to t. (Observe the striking similarity with the Regge transformation.) In the next section we will describe exactly how the Okamoto and Regge symmetries are related (in effect showing precisely how the complicated Okamoto action on the pair (y, x) relates to the trivial Regge action on the pair (e, f )). Remark 2. Since the parameters appearing in PVI are now quadratic functions of the θ ’s, if y solves PVI with parameters θ then y will also solve PVI for any parameters obtained from θ by negating any combination of θ1 , θ2 , θ3 and possibly replacing θ4 by 2 − θ4 . Together with the Okamoto transformation these four ‘trivial’ transformations generate a group isomorphic to the affine Weyl group of type D4 (see [14]). Further one may add in transformations corresponding to the Sym4 symmetry group of the affine D4 Dynkin diagram and obtain a symmetry group isomorphic to the affine Weyl group of type F4 . The confusing fact to note is that one still does not obtain symmetries corresponding to all the tetrahedral 6 j symmetries, basically because the PVI flows vary y, x and fix the θ ’s.
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3. Regge and Okamoto We will relate the Regge action on Euclidean tetrahedra to the Okamoto action. Let a1 , a2 , a3 ∈ R3 be three vectors, so that 0, a1 , a1 + a2 , a1 + a2 + a3 are the vertices of a tetrahedron. Denote the other three edge vectors of the tetrahedron by a4 , a5 , a6 so that: a1 + a2 + a3 + a4 = 0, a5 = a1 + a2 ,
a6 = a2 + a3 .
Denote the lengths of these six vectors a1 , . . . , a6 by a, b, c, d, e, f respectively. Now let H be the set of traceless 2 × 2 Hermitian matrices. Thus H is a real threedimensional vector space and we give it a Euclidean inner product by defining A1 , A2 := 2Tr(A1 A2 ). Thus we can view the tetrahedron as living in H by choosing an isometry ϕ : R3 ∼ =H such as ⎛ ⎞ x 1 x y + iz . ϕ ⎝y⎠ = 2 y − i z −x z Then we set A j = ϕ(a j ) ∈ H for j = 1, . . . , 6. Thus the Regge symmetry becomes an action on the set of these 6 Hermitian matrices A j (clearly determined by its action on the first three matrices). Now in the standard isomonodromy interpretation [11] of the Painlevé VI equation the Okamoto symmetry becomes a (birational) action on the set of Fuchsian systems of the form d A1 A2 A3 − A, A := + + , Ai ∈ sl2 (C), (2) dz z z−t z−1 where the coefficients A1 , A2 , A3 are 2×2 traceless complex matrices and t ∈ C\{0, 1} fixed. As above we will set A4 = −(A1 + A2 + A3 ) which is now the residue of Adz at z = ∞. This interpretation comes about by using explicit local coordinates on the space of such systems; up to overall conjugation by SL2 (C), the set of such Fuchsian systems is of complex dimension 6 and local coordinates (near a generic system) are given by θ1 , θ2 , θ3 , θ4 , x, y, where θi is such that Ai has eigenvalues ±θi /2, and where x, y are two explicit algebraic functions of A defined for example in [5] p.199, following [11]. (These are the same x, y appearing in Theorem 1.) Of course one would prefer to view the birational transformation as the intrinsic object, and its explicit coordinate expression as secondary. In particular one might hope for a simpler expression than that given in terms of x, y in Theorem 1. One way to do this, which will be useful here, was observed in [5] Lemma 34. To describe this we should first modify slightly the matrices A1 , A2 , A3 : Let i = Ai + θi /2 A
i = 1, 2, 3
i has eigenvalues 0, θi (i.e. it has rank one and trace θi ). so that A
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Lemma 3. ([5] Lemma 34) Each of the five expressions 1 A 2 ), Tr( A
2 A 3 ), Tr( A
2 A 3 ), 1 A Tr( A
1 A 3 ), Tr( A
3 A 2 A 1 ) Tr( A
is preserved by the birational Okamoto transformation of Theorem 1. This may be proved by a direct coordinate computation; the geometric origin of it is given in [5] (see especially Lemma 34, Remark 30) and may be thought of as the (complexification of the) classical analogue of the ideas we will use in the next section. Now it is straightforward to prove (see [9]) that generically the first two of these expressions (viewed as functions on the set of Fuchsian systems) together with the 1 A 2 ) and four θ ’s make up a system of local coordinates. Let us write λ12 = Tr( A 2 A 3 ). Thus, in these coordinates the Okamoto transformation acts simply λ23 = Tr( A as (θ, λ12 , λ23 ) → (θ1 − φ, θ2 − φ, θ3 − φ, θ4 − φ, λ12 , λ23 )
with φ = 41 θi /2, which looks even more like the Regge transformation, and easily yields the main result of this section: Theorem 2. Suppose A1 , A2 , A3 are 2 × 2 traceless Hermitian matrices with eigenvalues ±θi /2 with θi > 0 and let a, b, c, d, e, f be the edge lengths of the corresponding tetrahedron in R3 (under the isometry ϕ) i.e. the lengths of the vectors corresponding to the six matrices A1 , A2 , A3 , A1 + A2 + A3 , A1 + A2 , A2 + A3 respectively. Then the Okamoto transformation of (A1 , A2 , A3 ) corresponds to the Regge transformation of the tetrahedron with edge lengths a, b, c, d, e, f . Proof. For the first four edge lengths this is easy since (a, b, c, d) = (θ1 , θ2 , θ3 , θ4 ) (note that the triangle inequalities imply φ − θi ≥ 0). We need to also show that the Okamoto transformation preserves the edge lengths e and f . But this is now a simple computation: First note e2 = 2Tr(A1 + A2 )2 = θ12 + θ22 + 4Tr A1 A2 . 1 − θ1 /2)( A 2 − θ2 /2) = 4Tr A 1 A 2 − 2θ1 θ2 so that Then observe 4TrA1 A2 = 4Tr( A 1 A 2 . e2 = (θ1 − θ2 )2 + 4Tr A The first term on the right here is preserved, as is the second term by the lemma above, and so e is preserved since it is positive. Similarly for f . Remark 4. Returning briefly to the complex (not-necessarily Hermitian) picture, the above argument implies that the Okamoto transformation is also characterised as preserving Tr A25 and TrA26 , where A5 = A1 + A2 and A6 = A2 + A3 (and similarly it preserves Tr(A1 + A3 )2 ).
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Remark 5. (Spherical tetrahedra, cf. [21].) Consider three elements M1 , M2 , M3 ∈ SU(2) of the 3-sphere S 3 ∼ = SU(2), and the spherical tetrahedron with vertices I, M1 , M1 M2 , M1 M2 M3 . This has edge lengths li , where TrMi = 2 cos(li ) (i = 1, . . . , 6), where M4 = (M1 M2 M3 )−1 , M5 = M1 M2 , M6 = M2 M3 . One may define a Regge symmetry of the set of such tetrahedra, by acting on the edge lengths exactly as before. This action may be complexified in the obvious way (allow Mi ∈ SL2 (C) and li ∈ C). On the other hand the Okamoto action on the Fuchsian systems (2) induces an action on their monodromy data (i.e. essentially on the space of SL2 (C) representations of the fundamental group of the four-punctured sphere). The fact to be noted is that this action coincides with the above spherical Regge action (taking Mi to be the monodromy around the i th puncture for i = 1, 2, 3, 4); in other words the Okamoto action fixes the functions Tr(M1 M2 ) and Tr(M2 M3 ) of the monodromy data—this was the main result of [10], proved differently in Corollary 35 of [5]. 4. Conceptual Regge Symmetry We will give a conceptual derivation of the fact that the Regge symmetry preserves the 6 j-symbols (i.e. without using the Racah formula). There are two basic steps: • Identify the SU(2) 6 j symbols with certain SU(3) 6 j-symbols, • Use a natural symmetry of these SU(3) 6 j-symbols. These are representation theoretic analogues of the derivation given in [5, 3] of the Okamoto symmetry. (The article [22] helped us to understand this—see also [4] where we first realised that [22] describes a representation theoretic analogue of some things the present author had been thinking about.) First we will set up notation for representations of H := SU(3). Let W be the three-dimensional Hermitian vector space defining H . For any integer a ≥ 0 write Wa := Syma (W ) for the a th symmetric power of W ; an irreducible representation of H . Similarly for integers a ≥ b ≥ 0 write W(a,b) for the irrep of H corresponding to the Young diagram with 3 rows of lengths a, b, 0 resp. (cf. [6]). (Thus in particular Wa = W(a,0) .) One would like to define the SU(3) 6 j-symbols using the same framework as described above for SU(2). This is difficult though since SU(3) is not multiplicity-free and in general one will not obtain a decomposition of the multiplicity spaces into onedimensional pieces, but into pieces of higher dimension. (There are numerous articles discussing this multiplicity problem, and methods to circumvent it.) However things are simpler if we take the three initial representations to be symmetric representations (i.e. of the form Wa ). Then the Pieri rules imply one will again get the desired one-dimensional decomposition and we may proceed as before (and this is the only case we will need here). Thus we choose 3 symmetric irreps, with labels a, b, c say, and form the tensor product Wabc := Wa ⊗ Wb ⊗ Wc . Now, given an arbitrary representation Wλ with λ = ( p, q) one obtains a multiplicity space as before Nabcλ := Hom H (Wλ , Wabc ).
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Similarly to before the two expansions Wa ⊗ Wb ∼ =
W(r,s) ⊗ Hom H (W(r,s) , Wa ⊗ Wb ),
(r,s)
Wb ⊗ Wc ∼ =
W(t,u) ⊗ Hom H (W(t,u) , Wb ⊗ Wc )
(t,u)
yield two decompositions of the multiplicity space: Nabcλ ∼ =
Hom H (Wλ , W(r,s) ⊗ Wc ) ⊗ Hom H (W(r,s) , Wa ⊗ Wb ),
(r,s)
Nabcλ ∼ =
Hom H (Wλ , Wa ⊗ W(t,u) ) ⊗ Hom H (W(t,u) , Wb ⊗ Wc ).
(t,u)
Now the Pieri rules (see [6]) imply that the tensor product of a symmetric representation with any irrep will be multiplicity free (i.e. each irrep that appears in the tensor product will appear exactly once). Thus both of these decompositions of Nabcλ will be into one-dimensional pieces and by choosing real basis vectors of length one vr s , wtu in the corresponding pieces we can define matrix entries as before: U (3) (a, b, c, λ, (r, s), (t, u)) = vr s , wtu . (Again a sign-convention is needed to fix the signs of the basis vectors, but we will not worry about this here—the correct choices will be such that the following result is true.) The basic fact that we will use is that these 3-dimensional 6 j coefficients are equal to 2-dimensional 6 j coefficients: Proposition 6. Choose six integers a, b, c, d, e, f ≥ 0. Then [up to sign] U (a, b, c, d, e, f ) = U (3) (a, b, c, ( p, q), (r, s), (t, u)), where p=
a+b+c+d a+b+e b+c+ f , r= , t= , 2 2 2 q = p − d, s = r − e, u = t − f.
This will be proved below (it is probably quite well-known). First we will describe the desired three-dimensional symmetry and deduce the Regge symmetry: Proposition 7. The three-dimensional 6 j-symbol is symmetric as follows: U (3) (a, b, c, ( p, q), (r, s), (t, u)) = U (3) ( p − a, p − b, p − c, ( p, p − q), ( p − s, p − r ), ( p − u, p − t)).
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Note that this does indeed project back to give the Regge symmetry1 . Thus our task is reduced to justifying the above two propositions. At first sight it may appear that little progress has been made, replacing the Regge symmetry by the symmetry of Proposition 7. But as we shall see, this symmetry arises simply by pairing two dual representations of SU(3). [It is not however simply a matter of dualising all the representations in sight, since the dual of Wa is not a symmetric representation, for a > 0.] Proof (of Proposition 6). The first step is to identify the corresponding weight spaces Mabcd and Nabcλ . This can be done easily using “Howe duality for GLk -GLn ”, as follows (cf. [25, 22])2 . Choose two positive integers k, n and let V, W be complex vector spaces of dimensions k, n respectively. Then their tensor product V ⊗ W is a representation of GL(V ) × GL(W ) and so its d th symmetric power Symd (V ⊗ W ) is also a GL(V ) × GL(W )-module. This decomposes as a direct sum of irreducible GL(V ) × GL(W )-modules in the following way (see [6] Exercise 6.11; apparently this goes back to Cauchy): Symd (V ⊗ W ) ∼ Vλ ⊗ Wλ , = #λ≤k,n |λ|=d
where the sum is over all Young diagrams λ with d boxes and having no more than k or n rows, and Vλ (resp. Wλ ) is the irreducible GL(V )-module (resp. GL(W )-module) corresponding to λ. Choosing bases of V and W allows us to be more explicit. In particular it identifies GL(V ) ∼ = GLk (C) and so picks out a maximal torus (the diagonal subgroup), as well as a Borel subgroup (the upper triangular subgroup) and so allows us to speak of weights and highest weight vectors of GL(V ) modules (similarly for GL(W )). Also we can now view V ⊗ W as the space of linear functions ψ on the set Mk×n of k × n matrices X = (xi j ) with the following action of GLk (C) × GLn (C): (gk , gn )(ψ)(X ) = ψ(gkT Xgn ). (To avoid confusion below when k = n = 3, we will refer to this GLk (C)-action, respectively GLn (C)-action, as the action on the left, resp. right.) Then Symd (V ⊗ W ) = ∗ ) is the set of such functions which are homogeneous polynomials of degree Symd (Mk×n d, with the same action. Thus as GLk (C) × GLn (C)-modules, (k) (n) ∗ Symd (Mk×n )∼ Vλ ⊗ Vλ , (3) = #λ≤k,n |λ|=d (r )
where Vλ is the GLr (C) irrep. with Young diagram λ. ∗ ) can be viewed as the tensor product of the Now, as a GLk module Sym• (Mk×n n ∗ ) ∼ functions on each of the columns of X , i.e. Sym• (Mk×n = 1 Sym• Ck . Moreover if we choose n positive integers µ = (µ1 , . . . , µn ) then we can consider the subspace: ∗ ) S µ Ck := Symµ1 Ck ⊗ · · · ⊗ Symµn Ck ⊂ Sym• (Mk×n 1 Inverting the equations appearing in Proposition 6 yields d = p − q, e = r − s, f = t − u so that Proposition 6 implies U (3) ( p − a, p − b, p − c, ( p, p − q), ( p − s, p − r ), ( p − u, p − t)) = U ( p − a, p − b, p − c, q, r − s, t − u) and so, since q = p − d, r − s = e, t − u = f we do obtain the Regge symmetry U (a, b, c, d, e, f ) = U ( p − a, p − b, p − c, p − d, e, f ) as desired. 2 To proceed explicitly (using similar ideas) see [13, 7].
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of functions which are homogeneous of degree µi in the i th column. This is equivalent ∗ ). to saying they are the vectors of weight µ for the GLn action on Sym• (Mk×n Now what we are really interested in are the GLk multiplicity spaces of the form: µ
Mλ := HomGLk (Vλ(k) , S µ Ck ) whose dimension is the multiplicity of Vλ(k) in this n-fold product of symmetric representations. This multiplicity space may be realised explicitly as the subspace of S µ Ck (k) of vectors of highest weight λ for the GLk action (since each copy of Vλ in S µ Ck has a unique highest weight vector, up to scale). On the other hand by the decomposition ∗ ) of vectors of highest weight λ for the GL action is (3) the subspace of Sym• (Mk×n k (n)
a single copy of Vλ . Intersecting this with the subspace S µ Ck (i.e. the vectors with GLn (C) weight µ) yields the basic result we need (cf. [22] Lemma 3.4):
(n) ∼ M µ between the Lemma 8. The above discussion gives an isomorphism Vλ [µ] = λ (n) weight µ subspace of the GLn (C) representation Vλ and the GLk (C) multiplicity µ space Mλ .
In the 6 j-symbol situation, we are interested in three-fold tensor products of symmetric representations, so n = 3, and our Young diagrams always have at most two non-zero rows, so we may take any k ≥ 2. For k = 2, Lemma 8 implies (3) Mabcd ∼ = Vλ [µ],
where µ = (a, b, c) and where λ has at most 2 rows and is such that Vλ(2) ∼ = Vd as SU(2) representations and the number of boxes in λ equals a +b +c. This implies λ = ( p, q, 0), where p + q = a + b + c and p − q = d, i.e. p = (a + b + c + d)/2 and q = p − d as in the statement of Proposition 6. In turn, for k = 3, Lemma 8 implies (3)
Vλ [µ] ∼ = Nabcλ , and so combining the two gives the desired isomorphism of multiplicity spaces. (Explicitly if we view the 2 × 3 matrices as the first two rows of the 3 × 3 matrices, then these multiplicity spaces are actually equal as spaces of polynomial functions on 3×3 matrices since in both cases they do not depend on the variables in the third row, as λ has at most two non-zero rows.) Remark 9. For the reader familiar with Gelfand-Tsetlin tableaux, we should mention that the weight space Vλ(3) [µ] (and thus the common multiplicity space) admits a basis parameterised by tableaux of the form ⎛ ⎞ p q 0 ⎝ α β ⎠, γ where α, β, γ are integers satisfying interlacing inequalities: p ≥ α ≥ q ≥ β ≥ 0, α ≥ γ ≥ β and should be such that the tableau has ‘weight’ µ = (a, b, c)—the weight of a tableau is the differences of the row sums, i.e. we require γ = a, α + β − γ = b, p+q −(α+β) = c. This gives a simple way to compute the dimension of the multiplicity spaces (i.e. count the tableaux) although we will not need to use this Gelfand-Tsetlin basis (in general it does not coincide with any of the coupling bases).
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To complete the proof of Proposition 6 we need to see the corresponding coupling bases match up under the above isomorphism of multiplicity spaces. (One may use the ∗ ) and so see the Hermitian forms Bargmann-Segal-Fock Hermitian form on Sym• (Mk×n coincide.) One way to do this is to first observe that the 1-2 coupling spaces are the eigenspaces of the GLk (C) quadratic Casimir operator Ck acting on the first two tensor factors of S µ Ck = Syma Ck ⊗ Symb Ck ⊗ Symb Ck , where µ = (a, b, c) and k = 2 or 3. This holds since in general Ck acts (see e.g. [25] p.161) by multiplication by the scalar
m i2 +
mi − m j
(4)
i< j
on the GLk irrep. with Young diagram (m 1 , . . . , m k ). Denote this operation by Ck12 and note it preserves the common multiplicity space Mabcd = Nabcλ ⊂ S µ C2 ⊂ S µ C3 for k = 2, 3. Then we just observe, using (4), that on the multiplicity space the Casimirs differ by a scalar: C312 = C212 + (a + b) and so have the same eigenspaces (and the eigenspace labels match up as stated, that is: r − s = e, r + s = a + b). (Similarly for the 2-3 coupling.) Next we will describe the three-dimensional symmetry which lifts the Regge symmetry: If Vλ(3) is the irrep of GL3 (C) with Young diagram λ = ( p, q, 0) then there is a pairing (3)
(3)
Vλ ⊗ (Vλ )∨ → C, (3)
where (Vλ )∨ is the dual representation. Tensoring this by the p th power D p of the deter(3) (3) minant representation yields a pairing Vλ ⊗ Vλ → D p , where λ = ( p, p − q, 0). (3) (3) This pairs the µ = (a, b, c) weight space of Vλ with the µ weight space of Vλ , where µ = ( p − a, p − b, p − c) and so yields a perfect pairing: (3)
(3)
Vλ [µ] ⊗ Vλ [µ ] → C.
(5)
Now to prove Proposition 7 one just needs to check that the corresponding coupling bases are dual with respect to this pairing. Remark 10. On the level of Gelfand-Tsetlin tableaux the above duality corresponds to negating each tableau element then adding p to each element and finally flipping the tableau about its vertical axis. (Observe that the tableau’s weight has transformed as stated.) Proof (of Proposition 7). We will first describe the above pairing in a different way which will be more convenient. Write G = GL3 (C) and let S a = Syma C3 . By the Pieri rules there is a unique G-equivariant map S a ⊗ S p−a → S p
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since S p appears precisely once in this tensor product3 . (Similarly replacing a by b or c.) Putting these together there is a G-equivariant map S a S b S c ⊗ S p−a S p−b S p−c → S p S p S p pairing the corresponding factors (and omitting to write several ⊗ symbols). Again by the Pieri rules there is a unique (projection) map S p S p S p → D p to the p th power of the determinant representation. Composing with the above map we get a (degenerate) bilinear form: ν : S a S b S c ⊗ S p−a S p−b S p−c → D p . In terms of pairs of polynomials on M3×3 (viewing S α S β S γ as polynomials homogeneous of degrees α, β, γ in the columns 1, 2, 3 resp. as above) this bilinear form ν amounts to multiplication followed by orthogonal projection onto D p (which is just the line spanned by the p th power of the polynomial det : M3×3 → C). (3) (3) Now we wish to relate ν to the natural pairing Vλ [µ] ⊗ Vλ [µ ] → C of (5). For this we view Vλ(3) as a space of polynomials on the 3 × 3 matrices as in Lemma 8 (as the polynomials with highest weight λ for the GL3 -action on the left), but we view Vλ(3) differently as the space of polynomials with lowest weight (0, p − q, p) for the GL3 -action on the left. Then multiplication of functions followed by orthogonal projection onto D p (3) (3) gives a pairing Vλ ⊗ Vλ → D p which one may check is nonzero directly (observing (3) that a highest weight vector of Vλ pairs non-degenerately with a lowest weight vector (3) of Vλ ). Thus by Schur’s lemma this pairing coincides with the natural one up to scale. Restricting to the µ and µ weight spaces respectively and using Lemma 8 thus yields a non-degenerate pairing Nµλ ⊗ Nµ λ → C
(6)
(3) which is a restriction of ν. (To identify Nµ λ ∼ = Vλ [µ ] we use the analogue of Lemma 8 with “highest weight” replaced by “lowest weight” throughout the proof.) Now we need to show that, with respect to the pairing (6), the 1-2 coupling bases on each side are dual (and similarly for the 2-3 bases). For this it is sufficient to prove the non-corresponding coupling basis vectors are orthogonal; the non-degeneracy of the pairing then forces the corresponding coupling basis vectors to pair up. We will show this for the 1-2 coupling (the other coupling being analogous). Write W(x,y) for the irrep. of G with Young diagram (x, y, 0). By the Pieri rules there is a unique map S p S p → W( p, p) and so we have a G-equivariant map,
S a S b S p−a S p−b → S p S p → W( p, p) .
(7)
This enables us to factor ν as follows: S a S b S c S p−a S p−b S p−c → W( p, p) S c S p−c → W( p, p) S p → D p . To see how the coupling subspaces pair up, first expand both S a S b and S p−a S p−b into sums of irreps, so S a S b S p−a S p−b becomes a sum of tensor products of the form 3 More precisely there is a unique subspace of S a S p−a which is G-equivariantly isomorphic to S p , and thus a (unique) orthogonal projection onto this subspace. To lighten the notation we will call this subspace S p here (and similarly below).
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W(r,s) ⊗ W(x,y) , where W(r,s) ⊂ S a S b etc., and (7) maps these to W( p, p) . However using the Littlewood–Richardson rule it is easy to see that there is a nonzero map W(r,s) ⊗ W(x,y) → W( p, p) if and only if x = p − s, y = p − r (and if so it is unique up to scale). This gives the stated correspondence between the 1-2 coupling bases (in the fifth slot of U (3) ). Remark 11. Presumably the argument above extends to the Regge symmetry of the q-deformation of the 6 j-coefficients [12]. References 1. Biedenharn, L.C., Louck, J.D.: Angular momentum in quantum physics. In: Encyclopedia of Mathematics and its Applications, Vol. 8, Reading, MA:Addison-Wesley, 1981 2. Biedenharn, L.C., Louck, J.D.: The Racah-Wigner algebra in quantum theory. In: Encyclopedia of Mathematics and its Applications, Vol. 9, Reading, MA:Addison-Wesley, 1981 3. Boalch, P.P.: Six results on Painlevé VI. SMF, Séminaires et congrès, Vol. 14, to appear, available at http://arxiv.org/abs/math/0503043, 2005 4. Boalch, P.P.: G-bundles, isomonodromy and quantum Weyl groups. Int. Math. Res. Not. no. 22, 1129–1166, (2002) 5. Boalch, P.P.: From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. 90(3), 167–208 (2005) 6. Fulton W., Harris J.: Representation theory. GTM, Vol. 129, Berlin Heidelberg-New York:Springer, 1991 7. Gliske, S., Klink, W.H., Ton-That, T.: Algorithms for computing generalized U(N ) Racah coefficients. Acta Appl. Math. 88(2), 229–249 (2005) 8. Granovski˘ı, Ya.A., Zhedanov, A.S.: Nature of the symmetry group of the 6 j-symbol. Sov. Phys. JETP 67(10), 1982–1985 (1988) 9. Hitchin, N.J.: Geometrical aspects of Schlesinger’s equation. J. Geom. Phys. 23(3–4), 287–300 (1997) 10. Inaba, M., Iwasaki, K., Saito, M.-H.: Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence. I. M. R. N. 2004(1), 1–30 (2004) 11. Jimbo, M., Miwa, T.: Monodromy preserving deformations of linear differential equations with rational coefficients II. Physica 2D, 407–448 (1981) 12. Kirillov, A.N., Yu, N.: Reshetikhin Representations of the algebra Uq (sl(2)), q-orthogonal polynomials and invariants of links. In: Adv. Ser. Math. Phys., Vol. 7, V. Kac, ed., reprinted in: Adv. Ser. Math. Phys. Vol. 11, T. Kohno, ed., Singapore:World Scientific, 1989, pp. 285–339, 202–256 resp. 13. Klink, W.H., Ton-That, T.: Calculation of Clebsch-Gordan and Racah coefficients using symbolic manipulation programs. J. Comput. Phys. 80(2), 453–471 (1989) 14. Okamoto, K.: Studies on the Painlevé equations. I. Sixth Painlevé equation PVI . Ann. Mat. Pura Appl. (4) 146, 337–381 (1987) 15. Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Spectroscopic and group theoretical methods in physics, Bloch, F. ed., NewYork: John Wiley and Sons, 1968, pp. 1–58 16. Racah, G.: Theory of complex spectra II. Phys. Rev. 62, 438–462 (1942) 17. Regge, T.: Symmetry properties of Clebsch–Gordan coefficients. Il Nuovo Cimento X, 544–545 (1958) 18. Regge, T.: Symmetry properties of Racah’s coefficients. Il Nuovo Cimento XI, 116–117 (1959) 19. Roberts, J.: Classical 6 j-symbols and the tetrahedron. Geom. Topol. 3, 21–66 (1999) (electronic) 20. Rotenberg, M., Bivins, R., Metropolis, N., Wooten, J.K.: The 3-j and 6-j symbols. London: Crosby Lockwood and Son Ltd, 1959 21. Taylor, Y., Woodward, C.: 6 j symbols for Uq (sl2 ) and non-Euclidean tetrahedra. Selecta Math. 11, 539–571 (2005) 22. Toledano Laredo, V.: A Kohno-Drinfeld theorem for quantum Weyl groups. Duke Math. J. 112(3), 421–451 (2002) 23. Wigner, E.P.: On the matrices which reduce the Kronecker products of representations of S.R. groups. (1940). Published in: Quantum theory of angular momentum L.C. Biedenharn, ed., New York: Acad. Press, 1965 24. Wilson, J.A.: Some hypergeometric orthgonal polynomials. SIAM J. Math. Anal. 11(4), 690–701 (1980) 25. Želobenko, D.P.: Compact Lie groups and their representations. A.M.S. Trans. Math. Monog. Vol. 40, Providence, RI: Amer. Math. Soc, 1973 Communicated by A. Connes
Commun. Math. Phys. 276, 131–188 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0334-z
Communications in
Mathematical Physics
The Newtonian Limit for Perfect Fluids Todd A. Oliynyk Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, D-14476 Golm, Germany. E-mail:
[email protected] Received: 9 June 2006 / Accepted: 12 June 2007 Published online: 13 September 2007 – © Springer-Verlag 2007
Abstract: We prove that there exists a class of non-stationary solutions to the EinsteinEuler equations which have a Newtonian limit. The proof of this result is based on a symmetric hyperbolic formulation of the Einstein-Euler equations which contains a singular parameter = vT /c, where vT is a characteristic velocity scale associated with the fluid and c is the speed of light. The symmetric hyperbolic formulation allows us to derive independent energy estimates on weighted Sobolev spaces. These estimates are the main tool used to analyze the behavior of solutions in the limit 0. 1. Introduction The Einstein-Euler equations or, in other words, the Einstein equations coupled to a simple perfect fluid are given by the following system of equations: 8π G i j T , c4 = 0,
Gi j = ∇i T i j
(1.1) (1.2)
where the stress-energy tensor for the fluid is given by T i j = (ρ + c−2 p)v i v j + pg i j
(1.3)
with ρ the fluid density, p the fluid pressure, and v the fluid four-velocity normalized by v i vi = −c2 , c the speed of light, and G the Newtonian gravitational constant. The study of the behavior of solutions to these equations in the limit that = vT /c 0, where vT is a characteristic velocity scale associated with the fluid matter is known as Present address: School of Mathematical Sciences, Monash University, Clayton, Vic 3800, Australia.
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the Newtonian limit. By suitably rescaling the gravitational and matter variables (see Sect. 2), the Einstein-Euler equations can be written as G i j = 2κ 4 T i j and ∇i T i j = 0,
(1.4)
where κ = 4π GρT /vT2 , vi v i = − −2 , ρT is a characteristic value for the fluid density, and t = x 4 /vT is a “Newtonian” time coordinate. In the limit 0, one expects that there exists a class of solutions to Einstein-Euler equations (1.4) that approach solutions of the Poisson-Euler equations ∂t ρ + ∂ I (ρw I ) = 0, (I, J = 1, 2, 3) J I J J J ρ(∂t w + w ∂ I w ) = −(ρ∂ + ∂ p), (∂ I = δ I J ∂ J ) = ρ, ( = ∂ I ∂ I )
(1.5) (1.6) (1.7)
of Newtonian gravity in some sense. As above, ρ and p are the fluid density and pressure, respectively, while w I is the fluid (three) velocity. This problem has been studied since the discovery of general relativity by many people and there is a large number of results available in the literature. The majority of results are based on formal expansions in the parameter which are used to calculate the (approximate) values of physical quantities and also to investigate the behavior of the gravitational and matter fields in the limit 0. For some classic and recent results of this type see [2,3,6,9,11–13,20–22,31,41] and references cited therein. The main difficulty with the formal expansions is that they leave completely unanswered the question of convergence. In the absence of a precise notion of convergence, it becomes unclear to what extent the formal expansions actually approximate relativistic solutions. In this paper, we go beyond formal considerations and supply a precise notion of convergence for gravitating perfect fluids as 0. This necessitates introducing suitable variables that are compatible with the limit 0. The metric gi j , which defines the gravitational field, turns out to be singular in this limit. To remedy this problem, we introduce a new gravitational density u¯ i j which is related to the metric via the formula gi j = √ where Qi j =
δI J 0
IJ 4¯u 0 + 2 0 0
Qi j , − det(Q)
0 0 + 4 3 J 4 −1 u¯
(1.8)
0 u¯ I 4 + 4 4 0 0
0 . (1.9) u¯ 44
From this, it not difficult to see that the density u¯ i j is equivalent to the metric gi j for > 0 and is well defined at = 0. For the fluid, we also introduce a new velocity variable wi according to v I = w I and v 4 = 1 + w 4 .
(1.10)
For technical reasons, we only consider isentropic flow where the pressure is related to the density by an equation of state of the form p = f (ρ). Moreover, to formulate a symmetric hyperbolic system for the fluid variables {ρ, v}, we need to deal with the well known problem that the system becomes singular when ρ +c−2 p = 0. This is a particular problem for fluid balls having compact support. To get around this problem, we follow Rendall [34] and use a technique of Makino [24] to regularize the fluid equations so that
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a class of gravitating fluid ball solutions can be constructed. Thus as in [34], we assume an equation of state of the form p = Kρ (n+1)/n ,
(1.11)
where K ∈ R>0 , n ∈ N, and we introduce a new “density” variable α via the formula ρ=
1 α 2n . (4K n(n + 1))n
(1.12)
As discussed by Rendall, the type of fluid solutions obtained by this method have freely falling boundaries and hence do not include static stars of finite radius and so this method is far from ideal. However, in trying to understand the Newtonian limit and post-Newtonian approximations these solutions are almost certainly general enough to obtain a comprehensive understanding of the mathematical issues involved in the Newtonian limit and post-Newtonian approximations. We would also like to remark that the results contained in this article are largely independent of the specific structure of the fluid equations. We therefore expect that the analysis in this paper can be carried over without much difficulty to other matter models whose equations can be formulated as a symmetric hyperbolic system and have a finite propagation speed for the matter density in the limit 0. Our approach to analyze the limit 0 is to use the gravitational and matter variables {¯ui j , wi , α} along with a harmonic gauge to put the Einstein-Euler equations into the following form: b0 (V )∂t V =
1 I 1 c ∂ I V + b I (, V )∂ I V + f (, V )V + g(V )V + h(),
(1.13)
where V comprises both the gravitational and matter variables, and the c I are constant matrices. This system is symmetric hyperbolic and hence by standard theory there exist local solutions. However, the difficulty in analyzing the limit 0 of such solutions is that the equation contains the singular terms −1 c I ∂ I V and −1 g(V )V . Although, singular limits of symmetric hyperbolic equations have been previously analyzed in [5, 19, 37, 38], these results cannot be directly applied to the system (1.13). There are two main difficulties in adapting these results to the Einstein-Euler system. The first is that the Einstein-Euler system (6.1) must be modified by including an elliptic equation, essentially the Newtonian Poisson equation, in order to be of the canonical form required by [5, 19, 37, 38]. This results in a coupled elliptic-hyperbolic system of the form B 0 (W )∂t W =
1 I c ∂ I W + B I (, W )∂ I W + F(, W )W + H (),
(1.14)
where W is related to V via an elliptic equation and F is a non-local functional. The second difficulty is that the initial data which must include a 1/r piece for the metric and hence it cannot lie in the Sobolev space H k . This 1/r type fall-off behavior is crucial for obtaining the correct limit and is intimately tied to the elliptic part of our formulation of the Einstein-Euler system. The standard procedure in general relativity to deal with this type of fall off, at least for elliptic systems, is to replace the spaces H k with the weighted Sobolev spaces Hδk [1, 7]. However, the arguments used in [5, 19, 37, 38] fail for the weighted spaces as the weight used to define the Hδk spaces destroys the integration by parts argument which is used to control the singular term −1 c I ∂ I W in (1.14). Indeed,
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using integration by parts, it follows easily from the definition of the weighted L 2δ innerproduct (see (A.4) with = 1) that 1 − −1 c I ∂ I W |W 2 = − ∂ I (σ −2δ−3 )c I W |W 2 , (1.15) Lδ L 2 where σ (x) = 1 + |x|2 /4. In general, this term will blow up as 0 unless δ = −3/2 which coincides with the standard L 2 norm. However, to include 1/r fall-off, we need to consider −1 < δ < 0 which introduces a singular 1/ term into energy estimates based on the weighted norm Hδk . k (see To overcome this problem, we introduce a sequence of weighted spaces Hδ, Appendix A for a definition) by replacing the weight σ (x) with σ (x) = σ (x). Under this replacement, (1.15) changes to − −1 c I ∂ I W |W 2 ≤ C W |W L 2 , L δ,
δ,
which is no longer singular as 0. This allows us to derive independent energy estimates for solutions to the Einstein-Euler equations. These estimates can then be used to define a precise notion of convergence for gravitating perfect fluids solutions in the limit 0 which is essentially a statement about the validity of the zeroth order expansion in . This is formalized in the following theorem; for a more precise version see Propositions 5.1, 6.1 and 7.8, and Theorems 7.7 and 7.12. k− Theorem 1.1. Suppose −1 < δ < −1/2, k ≥ 3 + s, β j ∈ s =0 C ([0, T ∗ ], Hδ−1 ) is k k+1 I J k I J I a harmonic gauge source function, and α , w ∈ Hδ−1 , z ∈ Hδ , z4 ∈ Hδ−1 is the o
o
free initial data for the Einstein-Euler equations where supp α ⊂ B R∗ for some R ∗ > 0. o
Then for 0 small enough, there exists a T ∈ (0, T ∗ ] independent of ∈ (0, 0 ], and maps u¯ ij (t) − u¯ ij (0), ∂ I u¯ ij (t), ∂t u¯ (t), α (t), wi (t) ∈
s+1
=0
k− C ([0, T ], Hδ−1, ),
∈ C ([0, T ], Hδk+2 ) ∩ C 1 ([0, T ∗ ], Hδk+1 ), k−1 k ) ∩ C 1 ([0, T ∗ ], Hδ−1 ), w I ∈ C 0 ([0, T ∗ ], Hδ−1 k−1 0 ∗ k 1 ∗ ρ ∈ C ([0, T ], Hδ−1 ) ∩ C ([0, T ], Hδ−2 ), 0
∗
such that (i) IJ z wI , wJ φ −∂ K z K I + β I (0) z4I J , (∂t u¯ ij (0)) = −∂ K z K J + β J (0) −∂ K wK + β 4 − g¯ 4J w J − 2 (g¯ 4J w J )2 − g¯ 44 ( 2 g¯ I J w I w J +1) 1 o o o o w4 (0) = − + , g¯ 44 (¯uij (0)) =
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135
wI (0) = w I (0) = w , o
α (0) = α , o
ρ(0) = ρ = (4K n(n + 1))−n α 2n , o
o
where φ = φ(, ρ , w I , z4I J , β j (0), z I J ), and w = w(, ρ , w I , z4I J , β j (0), z I J ) o
(ii)
(iii) (iv) (v)
o
o
o
is the initial data determined by the gravitational constraint equations (see ij Proposition 5.1), and g¯ i j is determined from u¯ (0) by the formulas (1.8) and (3.1), ij I {¯u (x , t), α (x I , t), wi (x I , t)} determines, via the formulas (1.8), (1.9), (1.10), and (1.12), a 1-parameter family (0 < ≤ 0 ) of solutions to the Einstein-Euler 4j Ij equations (1.4) in the harmonic gauge ∂t u¯ + ∂ I u¯ = β j on the common spacetime region (x I , t) ∈ D = R3 × [0, T ], {(x I , t), ρ(x I , t), w I (x I , t)} solves the Euler-Poisson equations (1.5)–(1.7) on the spacetime region D, there exists a constant R ∈ (R ∗ , ∞) independent of ∈ (0, 0 ] such that supp α (t), supp ρ(t) ⊂ B R for all (t, ) ∈ [0, T ] × (0, 0 ], and there exists a constant C > 0 independent of ∈ (0, 0 ] such that j
¯uij (t) − δ4i δ4i (t) L 6 + ∂ I u¯ ij (t) − δ4i δ4 ∂ I (t) H k−1 + v I (t) − w I (t) H k−1 + −1 v 4 (t) − 1 H k−1 + ρ (t) − ρ(t) H k−1 + ∂t ρ (t) − ∂t ρ(t) H k−2 ≤ C
for all (t, ) ∈ [0, T ∗ ] × (0, 0 ]. We remark that the techniques of this paper can also be used to derive convergent expansions in of the type considered in Theorems 2 and 3 of [19] and [38], respectively. These convergent expansions in general differ from the formal post-Newtonian expansions. To get post-Newtonian expansion to a certain order in requires that the initial data must be chosen correctly. In the absence of constraints on the initial data, a general procedure for doing this is discussed in [5]. Due to the fact that there are constraints on the initial data in general relativity, this becomes a non-trivial problem called the initialization problem. See [18] for an extended discussion. The proof of convergence and a discussion of the initialization problem will be presented in a separate paper [27]. We note that similar results for the Vlasov-Einstein system have been derived in [36] using a zero shift maximal slicing gauge. However, unlike [36], our approach is able to handle not only higher order expansions in , but also a wide variety of matter models. We also note that in [16,18], there is another interesting proposal for analyzing the limit as 0 which is based on a gauge for which the Einstein equations are again elliptichyperbolic but distinct from [36]. As in this article, the authors of [16, 18] also propose to use the methods of [5,19,37,38]. However, the required estimates are not proven and it is yet to be verified if this approach would be successful. We remark that the results of this and the companion paper [27] are local in time and therefore address the “near zone” problem. In the special case of spherical symmetry, the situation improves and there are some global results available on the Newtonian limit [26,32]. However, because spherically symmetric systems do not generate gravitational radiation, these results do not shed light on the “far zone” problem for post-Newtonian expansions where radiation plays a crucial role and the 0 limit must be analyzed in the region “close” to future null infinity. We plan to investigate the far zone problem in the near future.
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Our paper is organized as follows: in Sect. 2, we define dimensionless variables for the Einstein-Euler system. Sections 3 and 4 are devoted to introducing variables and a gauge condition that cast the Einstein-Euler equations into a form suitable for analyzing the limit 0. Appropriate initial data which is regular in the limit 0 is constructed in Sect. 5 while in Sect. 6 we prove a local existence theorem for the Einstein-Euler system on the weighted spaces. Finally, in Sect. 7, we show that solutions to the Einstein-Euler system converge as 0 to solutions of the Poisson-Euler system of Newtonian gravity. A precise statement of convergence is contained in Theorem 7.12 which is the main result of this paper. 2. Units Our conventions for units are as follows: L3 M M L i , and [G] = , [ p] = , [v ] = [c] = . L3 LT 2 T MT 2 Note that with these choices the stress-energy tensor has units of an energy density, i.e. [T i j ] = LMT 2 . To introduce dimensionless variables, we define
[x i ] = L , [gi j ] = 1, [ρ] =
ˆ v i = vT vˆ i and ρ = ρT ρ, where vT and ρT are “typical” values for the velocity and the density, respectively. The Einstein-Euler equations then can be written as Gˆ i j = 2κ 4 Tˆ i j and ∇ˆ i Tˆ i j = 0, where =
√ vT 4π GρT , κ= , xˆ i = κ x i , gˆ i j = gi j , 2 c vT
pˆ =
p vT2 ρT
,
and Tˆ i j = (ρˆ + 2 p) ˆ vˆ i vˆ j + pˆ gˆ i j . The normalization vi v i = −c2 , implies that vˆ i vˆ i := gˆ i j vˆ i vˆ j = −
1 . 2
Also, we can introduce a time coordinate t via t = x 4 /vT . With these choices, we have 1 L , [t] = [T ], and [κ] = 2 . T L Thus all our dynamical variables and coordinates are dimensionless and the two constants vT and κ can be used to fix the length and time scales by using units so that
[] = [vˆ i ] = [ρ] ˆ = [ p] ˆ = [g] ˆ = [xˆ i ] = 1, [vT ] =
vT = 1 and κ = 1. In this case we can use t and x 4 interchangeably as long as we remember that they carry different units. To simplify notation, we will drop the “hats” from the hatted variables for the remainder of this article.
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3. Reduced Einstein Equations To aid in deriving the appropriate symmetric hyperbolic system for the gravitational variables, we temporarily introduce a new set of coordinates related to old ones by the simple rescaling x¯ J = x J , x¯ 4 = x 4 / and let ∂i =
∂ ∂ , ∂¯i = i . ∂ xi ∂ x¯
In the new coordinates, the metric g¯ i j and its inverse g¯ i j are given by IJ gI J g I 4 g −1 g I 4 ij ) = (g¯ i j ) = and ( g ¯ . g4J 2 g44 −1 g 4J −2 g 44 Next, consider the metric density ¯ g¯ i j where |g| ¯ = − det(g¯ i j ). g¯ i j = |g|
(3.1)
(3.2)
We note that the metric g¯ i j is related to the density g¯ i j by the following formula: 1 ¯ = − det g¯ i j , g¯ i j = √ g¯ i j where |g| |g| ¯ and hence
IJ 1 g¯ (g ) = √ g¯ 4J |g| ¯ ij
g¯ I 4 . 2 g¯ 44
(3.3)
(3.4)
To obtain a gravitational variable that is regular and non-trivial in the limit 0, we define
1 u¯ i j = 2 g¯ i j − ηi j , (3.5) 4 where
1I 3×3 η = 0 ij
0 −1
is the Minkowski metric density. As stated in the introduction, for > 0, the metric gi j can be recovered from the density u¯ i j via the formulas (1.8)–(1.9). As we shall see, even though the metric gi j is singular in the limit 0, the quantity u¯ i j is well defined at = 0. We note that these variables are closely related to the gravitational variables discovered by Jürgen Ehlers and subsequently used in the papers [17,28,29] to construct stationary/static solutions to the Einstein equations coupled to various matter sources. In the (x¯ i ) coordinate system, the Christofell symbols are given by
¯ ikj = 2 g¯ km (2¯gi g¯ j p − g¯ i j g¯ p )∂¯m u¯ p + 2(¯g p δ(ik ∂¯ j) u¯ p − 2¯g (i ∂¯ j) u¯ k ) . (3.6)
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We note that Christofell symbols in the (x i ) coordinate system are related to the ¯ ikj as follows: A A 4 44 = −2 ¯ 44 , 44 = −1 ¯ 444 , 4A4 = ¯ 4A4 , A A A A = −1 ¯ B4 and BC = ¯ BC . 4AB = 4AB , B4
(3.7) (3.8)
Using (3.6), a straightforward calculation shows that the Einstein tensor G¯ i j is given in terms of the density u¯ i j by
1 2 ij G i j := 2 |g| ¯ G¯ i j = g¯ k ∂¯k (3.9) u¯ + 2 Ai j + B i j + C i j + D i j , 2 where |g| ¯ = − det(¯gi j ),
Ai j = 2 21 g¯ k g¯ mn − g¯ km g¯ n g¯ i p g¯ jq − 21 g¯ i j g¯ pq ∂¯ p u¯ k ∂¯q u¯ mn ,
B i j = 4¯gk 2¯gn(i ∂¯m u¯ j) ∂¯n u¯ km − 21 g¯ i j ∂¯m u¯ kn ∂¯n u¯ m − g¯ mn ∂¯m u¯ ik ∂¯n u¯ j ,
C i j = 4 ∂¯k u¯ i j ∂ u¯ k − ∂¯k u¯ i ∂¯ u¯ jk , 2 k 2 k(i j) D i j := g¯ i j ∂¯k u¯ − 2∂¯k u¯ g¯ .
(3.10) (3.11) (3.12) (3.13) (3.14)
To fix the gauge, we assume that ∂¯i u¯ i j = β j
(3.15)
for prescribed spacetime functions β j = β j (x I , x 4 ). For > 0, ∂¯i u¯ i j = β j implies that ∂¯i g¯ i j = 4 3 β j or equivalently ∂k gk4 = 4 3 β 4 and ∂k gk A = 4 2 β A ,
√ where gi j = − det(gk )g i j is the metric density in the (x k ) coordinates. Thus (3.15) is, for > 0, a generalized harmonic type gauge and is harmonic if the functions β j are chosen to be identically zero. Clearly, if we define E i j := g¯ i j ∂¯k β k − 2∂¯k β (i g¯ j)k , then (3.15) implies that Di j = E i j . Setting
ij 2 ij u¯ + E i j + 2 Ai j + B i j + C i j G R := G i j − D i j + E i j = g¯ k ∂¯k
and T i j := 2 |g| ¯ T¯ i j = |g| ¯
2 IJ T 1 T 4J
1T I 4 T 44
(3.16)
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the Einstein equations G i j = 2 4 T i j in the gauge (3.15) become ij
GR = T i j ,
(3.17)
which we will refer to as the reduced Einstein equations. To write the reduced Einstein equations in first order form, we introduce the variables
if k = I ∂ I u¯ i j ij . u¯ k := ∂¯k u¯ i j = ∂4 u¯ i j if k = 4 The reduced Einstein equations then become
ij ij ij −¯g44 ∂¯4 u¯ 4 = g¯ 4I ∂¯ I u¯ 4 + g¯ I J ∂¯ I u¯ J + E i j + 2 Ai j + B i j + C i j − T i j , g¯ I J ∂¯4 u¯ J = g¯ I J ∂¯ J u¯ 4 , ij
ij
ij ∂¯4 u¯ i j = u¯ 4 ,
or equivalently
1 1 4I 1 ij ij g¯ ∂ I u¯ 4 + g¯ I J ∂ I u¯ J + E i j + Ai j + B i j + C i j − T i j , 1 IJ ij ij IJ g¯ ∂4 u¯ J = g¯ ∂ J u¯ 4 , 1 ij ∂4 u¯ i j = u¯ 4 . Next, define ij
−¯g44 ∂4 u¯ 4 =
ui j := u¯ i j and let
ij
ij
uk := u¯ k ,
(3.18)
V = (r i j ) ∈ M4×4 | det(ηi j + 4r i j ) > 0 .
Then using vector notation
T ij ij ui j := u4 , u J , ui j , we can write the reduced Einstein equations as A4 (u)∂4 ui j = where
1 I 1 C ∂ I ui j + A I (u)∂ I ui j + F¯ i j (, u) − (T i j , 0, 0)T ,
⎛ 0 1 − 4u44 4 ⎝ A (u) = 0 δ I J + 4u I J 0 0 ⎛ ⎞ I J 0 0 δ C I = ⎝δ I J 0 0⎠ , 0 0 0 ⎛ 4I ⎞ I J 4u 4u 0 A I (u) = ⎝4u I J 0 0⎠ , 0 0 0
⎞ 0 0⎠ , 1
(3.19)
(3.20)
(3.21)
(3.22)
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and ij F¯ i j (, u) = (E i j + f¯i j (u, uk ), 0, u4 )T .
(3.23)
The functions f¯i j (u, uk ) are analytic for u ∈ V and moreover are quadratic in uk . Here we are using the notation ij
u = (ui j ) and uk = (uk ). The stress-energy tensor is given in terms of the u variable by IJ IJ 1 δ p0 4u p 0 +√ (T i j ) = ρ(v i v j ) + √ 0 0 0 0 |g| ¯ |g| ¯ I 4 p 0 4u , + 2 p(v i v j ) + √ |g| ¯ 4u4J −1 + 4u44 and hence
where
1 ij 0 (T ) = 0
0
−1 ρ
(3.24)
+ Si j ,
(3.25)
0 |g|v ¯ I v4 ¯ − 1)(v 4 )2 + ((v 4 )2 − 1) |g|v ¯ J v 4 −1 (|g| (ρ + 2 p)v I v J +|g| ¯ −1/2 p(δ I J +4u I J ) pv I v 4 +4|g| ¯ −1/2 pu I 4 +|g| ¯ . pv J v 4 +4|g| ¯ −1/2 pu4J p(v 4 )2 +|g| ¯ −1/2 p(−1+4u44 ) (3.26)
(S i j ) = ρ
We remark that if v 4 − 1 = O(), then S i j is regular in as is easily seen from the above formula and the expansion |g| ¯ = 1 + 4ηi j ui j + f (u),
(3.27)
where f (u) is analytic for u ∈ V and also satisfies f (y) = O(|y|2 ) as y → 0. 4. Regularized Euler Equations There are various approaches to symmetric hyperbolic formulations of the relativistic Euler equations [4, 14, 15, 34, 40]. We use the approach of [4] which is based on fluid projection and the introduction of a Makino variable. In the coordinates (x¯ i ), the Euler equations are given by ∇¯ i T¯ i j = 0,
(4.1)
where T¯ i j = (ρ + 2 p)v¯ i v¯ j + p g¯ i j and the fluid velocity v¯ i is normalized according to 1 . 2
(4.2)
v¯ i ∇¯ j v¯ i = 0
(4.3)
v¯ i v¯ i = − Differentiating (4.2) yields
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which implies v¯ j v¯ i ∇¯ j v¯ i = 0.
(4.4)
Writing out (4.1) explicitly, we have (∂¯i ρ + 2 ∂¯i p)v¯ i v¯ j + (ρ + 2 p)(v¯ j ∇¯ i v¯ i + v¯ i ∇¯ i v¯ j ) + g¯ i j ∂¯i p = 0.
(4.5)
The operator j
j
L i = δi + 2 v¯ j v¯ i j
j
j
projects into subspace orthogonal to the fluid velocity v¯ i , i.e. L i L ik = L k and L i v¯ i = 0. j Using L k to project the Euler equations (4.5) into components parallel and orthogonal to v¯ i yields, after using the relations (4.2)–(4.4), the following system: v¯ i ∂¯i ρ + (ρ + 2 p)L ij ∇¯ i v¯ j = 0,
(4.6)
1 L i ∂¯i p = 0, ρ + 2 p j
(4.7)
Mi j v¯ k ∇¯ k v¯ j + where
Mi j = g¯ i j + 2 2 v¯ i v¯ j . As discussed in the introduction, we introduce a new density variable α via the formula (1.12). Multiplying (4.6) by the square of the function 1 h(α) = 1 + (α)2 , 4n(n + 1) gives dα i ¯ j L ∇ i v¯ = 0, dρ j s 2 dp j ¯ L ∂ j α = 0, Mi j v¯ k ∇¯ k v¯ j + ρ + 2 p dα i
h 2 v¯ i ∂¯i α + h 2 (ρ + 2 p)
where s2 =
1 dp = 2 α2 dρ 4n
is the square of the speed of sound. A simple calculation shows that dα s 2 dp = h 2 (ρ + 2 p) = q, 2 ρ + p dα dρ where q = q(, α) =
1 α. 2nh(α)
(4.8) (4.9)
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This shows that the system (4.8)–(4.9) is symmetric, and moreover at a point where α = 0 and hence p = ρ = 0, it is regular unlike (4.6)–(4.7). This is the point of introducing the Makino variable α. Also note that the pressure is given in terms of the Makino variable by p=
K α 2n+2 . (4K n(n + 1))n+1
(4.10)
Define w I := v¯ I , and w 4 := v¯ 4 −
1
so that v I = w I , and v 4 = 1 + w 4 .
(4.11)
Using vector notation w = (α, wi )T , we can write (4.8) and (4.9) as a 4 ∂4 w = a I ∂ I w + b, where
q L 4j h 2 (1 + w 4 ) , q L i4 Mi j (1 + w 4 ) −h 2 w I −q L Ij I a = , −q L Ij −Mi j w I
a4 =
and
(4.12)
(4.13) (4.14)
j −q L ij ¯ i v¯ b= . j −Mi j ¯ v¯ k v¯
(4.15)
k
From (3.3), (3.5), (3.18), and (3.27), we find that g¯ i j = ηi j + f i j (u),
(4.16)
where the f i j (y) are analytic and satisfy f i j (y) = O(|y|) as y → 0. Also, (3.6) shows that
lp
p ¯ ikj = ηkm 2ηi η j p − ηi j η p um + 2 η p δ(ik u j) − 2η (i uk + f ikj (u, um ) j) (4.17) for functions f ikj (u, um ) that are analytic for u ∈ V, linear in the um , and satisfy f ikj (0, y) = 0. So then Mi j = g¯ i j + 2 2 g¯ ik g¯ j v¯ k v¯ = δi j + m i j (u, w k )
(4.18)
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and j
j
j
j
j
L i = δi + 2 g¯ ik v¯ k v¯ j = δi − δi4 δ4 + i (u, w k ), j
(4.19)
j
where i (u, w k ) and m i j (u, w k ) are analytic for u ∈ V and i (0, 0) = m i j (0, 0) = 0. Using (4.16)–(4.19), the matrices a i and the vector b can be written as 1 0 (4.20) a4 = + aˆ 4 (u, w), 0 δi j α I −w I − 2n δj aI = ˆ w) + α aˆ I (u, w), (4.21) + w I a(u, α I − 2n δi −δi j w I and
0 α bˆ 1 (u, w) · uk
. b= + lp
p −ηim 2η4 η4 p + η p um − 2 η p δ4i u4 − 2η 4 ui bˆ 2 (u, w) · uk 4 (4.22) Note that (i) aˆ 4 , a, ˆ aˆ I , bˆ 1 , and bˆ 2 are analytic in all their variables provided that u ∈ V, 4 I (ii) aˆ , aˆ and aˆ are symmetric, and (iii) aˆ 4 (0, 0) = 0, aˆ I (0, 0) = 0, a(0, ˆ 0) = 0, bˆ 1 (0, 0) = 0, and bˆ 2 (0, 0) = 0. Consequently the system (4.12) is symmetric hyperbolic on a region where (u, w) is small enough to ensure that a 4 is positive definite. This can always be arranged by taking small enough and since we are interested in the limit 0 no generality is lost in assuming this. It is important to realize that the derivation above of (4.12) required that both the Euler equations (4.1) and the fluid velocity normalization (4.2) are satisfied. Alternatively, we can first assume that (4.12) is satisfied and then show that (4.1) and (4.2) are also satisfied. To see this, define N := v¯ i v¯ i + 1/ = g¯ 44 (1/ +w 4 )2 + 1/ +2g¯ 4J (1 + w 4 )w J + g¯ I J w I w J . (4.23) Clearly, N = 0 is equivalent to v¯ i v¯ i = −1/ 2 for > 0. Furthermore, any solution of (4.12) also solves (4.6)–(4.7) for any > 0. So assuming that v¯ is a solution to the system (4.6)–(4.7), contracting (4.7) with v¯ i yields (1 + 2 2 v¯ i v¯ i )v¯ k ∂¯k (v¯ i v¯ i ) = 0. For (2N − 1) = 0, this implies (1 + w 4 )∂4 N = −w I ∂ I N .
(4.24)
Clearly, this is a symmetric hyperbolic equation for N whenever 0 < 1/C ≤ (1+w4 ) ≤ C for some constant C. This can always be arranged at x 4 = 0 by choosing small enough. Therefore, if initially N x 4 =0 = 0, then N = 0 for as long as (1 + w 4 ) stays absolutely bounded and bounded away from zero. Consequently, choosing initial data for the system (4.12) such that N x 4 =0 = 0 will guarantee that the solution will satisfy the full Euler equations (4.5) in an open neighborhood of the hypersurface x 4 = 0. In particular, if {α, wi } is a solution to (4.12) with initial data satisfying N |x 4 =0 , then α is a solution to the equation ∂4 α + X I ∂ I α + Y α = 0,
(4.25)
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T. A. Oliynyk
where X I :=
wI , 1 + w 4
and
Y :=
∇¯ i v¯ i . 2n(1 + w 4 )h 3 (α)
(4.26)
Observe that Y = Y¯ (w 4 , α)(∂t w 4 + ∂ I w I ) + Yˆ (u, w 4 , uk , w I , ), where Y¯ (0, 0) − 1/(2n) = 0, Yˆ (0, . . . , 0) = 0 and Y¯ (w 4 , α 4 ), Yˆ (u, w 4 , uk , w I , α 4 ) are analytic on the region u ∈ V and 1 + w 4 > 0. 5. Newtonian Initial Data Let S0 ∼ = R3 be the hypersurface defined by S0 := {(x I , 0) | (x I ) ∈ R3 }. The covector 4 n i = δi is conormal to S0 implying that constraint equations for the initial data on S0 are given by n i G i j = 2κ 4 n i T i j . Defining C J := −1 (G 4J − κT 4J ) and C 4 := G 44 − κT 44 , we find that C j = 0 is equivalent to n i G i j = 2κ 4 n i T i j for > 0. Also, by defining H j := ∂¯i u¯ i j − β j ,
(5.1)
= 0. the generalized harmonic gauge (3.15) can be written as As will be seen in the proof of the next proposition the equations C j = 0 are regular at = 0. So to find appropriate initial data that is well defined at = 0, we solve the regularized constraint equations C j = 0. Moreover, we must also ensure that the harmonic gauge condition H j = 0 and the fluid normalization N = 0 are satisfied. To solve the constraints C j = 0, H j = 0, and N = 0, we use a implicit function technique based on the work of Lottermoser [23]. We assume that the fluid velocity can be written as (4.10) which is consistent with the expected behavior of the fluid velocity as 0. We will not assume that the density and pressure are related by the equation of state (1.11). Instead, we will consider them as independent prescribed fields for the purpose of finding solutions to the constraint equations. We do this so that the following proposition remains valid for other equations of state. Hj
Proposition 5.1. Suppose −1 < δ < 0, k > 3/2, R > 0 and (ρ, ˜ p, ˜ w˜ I , z˜4I J , β˜ j , z˜ I J ) ∈ k−2 2 k−1 2 k k (Hδ−2 ) × Hδ−1 × (Hδ−1 ) × B R (Hδ ). Then there exists an 0 > 0, an open neighk : borhood U of (ρ, ˜ p, ˜ w˜ I , z˜4I J , β˜ j , z˜ I J ), and analytic maps (−0 , 0 ) × U → Hδ−1 I J k I J I j I J 4 I j (, ρ, p, w , z4 , β , z )→ w , (−0 , 0 )×U → Hδ : (, ρ, p, w , z4 , β , z I J )→ φ, (−0 , 0 ) × U → Hδk : (, ρ, p, w I , z4I J , β j , z I J ) → w I such that for each ij (ρ, p, w I , z4I J , β j , z I J ) ∈ U , (, ρ, p, w I , w 4 , u¯ 4 , β j , ∂¯4 u¯ i j ) is a solution to the three constraints C j = 0, H j = 0, and where (¯ui j ) = (∂t u¯ i j ) =
IJ z w J
N = 0,
w I , φ
z4I J −∂ K z K J + β J
(5.2)
(5.3) −∂ K z K I + β I −∂ K w K + β 4
(t = x 4 ),
(5.4)
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and 1 − g¯ 4J w J − w =− + 4
2 (g¯ 4J w J )2 − g¯ 44 ( 2 g¯ I J w I w J + 1) . g¯ 44
(5.5)
Moreover, if we let φ0 = φ|=0 , w0I = w I |=0 , and w04 = w 4 |=0 , then φ0 , w0I , and w04 satisfy the equations φ0 = κρ, w0I = ∂ I β 4 − ∂ L z4L I + κρw I , and w04 = 0, respectively. Proof. Let u¯ 44 = φ, u¯ I J = z I J , u¯ I 4 = w I , and ∂¯4 u¯ I J = z4I J . Solving H j S = 0 0 yields
and ∂¯4 u¯ 4J = (−∂ I z I J + β J ), ∂¯4 u¯ 44 = −∂ I w I + β 4 (5.6) while solving N S = 0 gives 0
1 − g¯ 4J w J − w =− + 4
2 (g¯ 4J w J )2 − g¯ 44 ( 2 g¯ I J w I w J + 1) . g¯ 44
(5.7)
From (3.3) and (3.5), it is not difficult to verify that w 4 = −1 f (w I , 3 z, 3 w, 2 φ), where f (y) (y = (y1 , . . . , y4 )) is analytic in a neighborhood of (0, 0, 0, 0) and moreover f (y) = O(|y|2 ) as y → 0. Using the relation (5.6) to eliminate ∂¯4 u¯ 44 and ∂¯4 u¯ 4J in favour of w I and z I J , we find that 2 44 g¯ k ∂¯k u¯ + D 44 = φ − ∂ K2 L z K L + 4 2 h 4 ,
2 4J g¯ k ∂¯k u¯ + D 4J = w J − ∂ J β 4 + ∂ L z4L J + 4h J ,
where h 4 = z K L ∂ K L φ + φ∂ K2 L z K L − 2 2 w L ∂ K2 L w K , h J = 2 z K L ∂ K2 L w J + 2 w J ∂ K2 L z K L − 2 w L ∂ K2 L z K J − φ∂k z4K J − 2 z J L ∂ L β 4 . Using this and Eqs. (3.9), (3.10)–(3.14), (3.24)–(3.26), and (4.10)–(4.11), we see that C I = w J + ∂ L z4L J + h J + f I ( 3 z, 3 w, 2 φ, Dz, Dw, Dφ, z4 , (−∂k w I + β 4 ), (−∂ K ζ K L + β L )) − κ S 4I ,
(5.8)
and C 4 = φ − κρ − κ(2w 4 + (w 4 )2 )ρ − ∂ K2 L z K L + 4 2 h 4 + 2 f 4 ( 3 z, 3 w, 2 φ, Dz, Dw, Dφ, z4 , (−∂k w I + β 4 ), (−∂ K ζ K L + β L )) − κ S 44 ,
(5.9)
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T. A. Oliynyk
where the functions f I (y) (y = (y1 , . . . , y9 )) are analytic in a neighborhood of {(0, 0, 0)} × U , where U is any open set and are quadratic in (y4 , . . . , y9 ). Note that S 44 = ρ S14 (, w I , 2 z, 2 w, φ) + pS14 (, w I , 2 z, 2 w, φ) and S 4I = ρw I + ρ S1I (, w I , 2 z, 2 w, φ) + pS2I (, w I , 2 z, 2 w, φ), where the functions S j (y) (y = (y1 , . . . , y7 )) are analytic in a neighborhood of U × {(0, 0, 0)} for any open set U . Using Lemma A.8 and Proposition 3.6 of [17], we see from the above considerations that for any R > 0 there exists an 0 > 0 such that the maps k ) × B R (Hδk )3 −→ Hδk : (, w I , z, w, φ) −→ w4 (−0 , 0 ) × B R (Hδ−1
and k−2 2 k−1 2 k ) × B R (Hδ−1 ) × (Hδ−1 ) × B R (Hδk )3 (−0 , 0 ) × (Hδ−2 k−2 −→ Hδ−2 : (, ρ, p, w I , z4 , β, z, w, φ) −→ C j
are analytic. Since C I |=0 = w I − ∂ I β 4 + ∂ L z4L I − κρw I ,
C 4 |=0 = φ − κρ
(5.10)
k−2 is an isomorphism (see [1], and for −1 < δ < 0 the Laplacian : Hδk → Hδ−2 Proposition 2.2), we can use the analytic version of the implicit function theorem (see [10] Theorem 15.3) to conclude, shrinking 0 if necessary, that there exists an open k−2 2 k ) × (H k−1 )2 × B (H k ) and neighborhood U of any point in (Hδ−2 ) × B R (Hδ−1 R δ δ−1 analytic maps
(−0 , 0 ) × U −→ Hδk : (, ρ, p, w I , z4 , β, z) −→ φ and (−0 , 0 ) × U −→ Hδk : (, ρ, p, w I , z4 , β, z) −→ z such that the constraints are satisfied, i.e. C j (, ρ, p, w I , z4 , β, z, w(, ρ, p, w I , z4 , z), φ(, ρ, p, w I , z4 , z)) = 0 for all (, ρ, p, w I , z4 , β, z) ∈ (−0 , 0 ) × U .
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147
6. Local Existence for the Einstein-Euler System The combined systems (3.19) and (4.12) can be written as b0 (U, V )∂t V =
1 I c ∂ I V + b I (, U, V )∂ I V + f (, U, V )V 1 (t = x 4 ), + g(V )V + h
(6.1)
where U := (0, 0, ui j , 0, 0)T , V :=
ui j o o ij ij ij i T (u4 , u J , δu , α, w ) ,
:= ui j |t=0 = u¯ i j |t=0 ,
δui j := ui j − ui j , o 0 , a 4 (u, wi , α)
(6.3)
A4 (u) 0 I C 0 c I := , 0 0 I 0 A (u) , b I (, U, V ) := 0 a I (, u, wi , α) ⎛ ij ⎞ f¯ (u, uk ) − S i j + 4δui j ∂¯k β k − 8 ∂¯k β (i δu j)k ⎜ ⎟ 0 ⎟, f (, U, V )V := ⎜ ij ⎝ ⎠ u4 i b(, u, uk , w , α) b0 (U, V ) :=
j
g(V )V := (−δ4i δ4 ρ(α), 0, . . . , 0)T , and h :=
4ui j ∂¯k β k − 8 ∂¯k β (i u j)k + ηi j ∂¯k β k − 2∂¯k β (i η j)k o
o
0
(6.2)
(6.4) (6.5) (6.6)
(6.7)
(6.8)
.
(6.9)
For initial data, we will use the following notation: given a function z that depends on time t, we define z := z|t=0 . o
To fix a region on which the system (6.1) is well defined, we note from (3.20), (4.20), and the invertibility of the Lorentz metric (ηi j ) that there exists a constant K 0 > 0 such that − det(ηi j + 4ui j ) > 1/16, 1 + w 4 > 1/16, 1 1 A4 (u) ≥ 1I , a 4 (u, w, α) ≥ 1I 16 16
(6.10) (6.11)
and |A4 (u)| ≤ 16, |a 4 (u, w, α)| ≤ 16
(6.12)
for all |u| ≤ 2K 0 , |wi | ≤ 2K 0 , |α| ≤ 2K 0 . The choice of the bounds 1/16 and 16 is somewhat arbitrary and they can be replaced by any number of the form 1/M and M for
148
T. A. Oliynyk
any M > 1 without changing any of the arguments presented in the following sections. However, since we are interested in the limit 0, we lose nothing by assuming M = 16. k , z I J ∈ H k+1 , z I J ∈ Proposition 6.1. Suppose −1 < δ < 0, k ≥ 3 + s, α , w I ∈ Hδ−1 δ 4 o
o
ij
ij
k , β j ∈ C 1 ([−T, T ], H k ). Let u ¯ , ∂t u¯ and w 4 be the initial data constructed in Hδ−1 δ−1 o
o
o
Proposition 5.1 which, by choosing 0 ≤ 1 small enough, satisfies |wi |, |α |, | u¯ ij | ≤ K 0 for all ∈ (0, 0 ]. o
o
o
Then (i) for each ∈ (0, 0 ], there exists T1 (), T2 () > 0 and a unique solution s+1
V ∈
=0
k− C ((−T1 (), T2 ()), Hδ−1 )
to the system (6.1) with initial data V = (∂t u¯ ij , ∂ J u¯ ij , 0, α , wi ). o
o
o
o
o
(ii) The identities ij
∂t u¯ ij =
u4,
ij
, and u J, = ∂ J u¯ ij
hold where by definition u¯ = −1 u , u = u + δu , and u = u¯ . ij
ij
ij
ij
o
o
o
ij
(iii) The triple {¯u , wi , α } determines, via the formulas (1.12), (3.4), (3.5), and (4.11), a solution to the full Einstein-Euler system (1.1)–(1.2) that satisfies the constraints ∂t u¯ 4 j + ∂ I u¯ I j = β j and v i vi = −
1 . 2
(iv) For some constant C > 0 independent of , the initial data V satisfies the estimate o
V − V 0 H k o
o
δ−1,
≤ C V − V 0 H k o
o
δ−1
≤ C
while
∂t V (0) H k−1 ≤ ∂t V (0) H k−1 ≤ C δ−1,
δ−1
for all ∈ (0, 0 ]. (v) If sup0≤t
T2 () such that the solution V can be continued to the interval (−T1 (), T∗ ). Proof.
(i) Follows directly from Theorem B.5, Proposition B.6, and Corollary B.7, where we use the initial data from Proposition 5.1. (ii) This follows from standard arguments on reductions of 2nd order hyperbolic equations to 1st order symmetric hyperbolic systems. See [39], Sect. 16.3 for details.
Newtonian Limit for Perfect Fluids
149 ij
(iii) By part (ii), the triplet {¯u , wi , α } satisfies the reduced Einstein equations (3.17) ij and the fluid equations (4.12). By construction, {¯u |t=0 , wi |t=0 , α |t=0 } satisfies the constraints N |t=0 = 0, H j |t=0 = 0, and (G 4i − T 4i )|t=0 = 0. The reduced Einstein equations (3.17) can be written in terms of the Einstein density G i j as G i j − g¯ i j ∂¯k Hk + 2∂¯k H(i g¯ j)k = T i j . Using (G 4i − T 4i )|t=0 = 0, we see that
−¯g4 j ∂¯k Hk + 2∂¯k H(4 g¯ j)k
t=0
= 0.
(6.13)
A straightforward calculation then shows that this implies that ∂t H j |t=0 = 0. As discussed in Sects. 4 (see (4.24)), N satisfies a linear symmetric hyperbolic system and hence by uniqueness, it follows that N = 0 for all (x I , t) ∈ R3 × (−T1 (), T2 (). Thus {wi , α } determine a solution, via the formulas (4.11), to the Euler equation which are equivalent to ∇¯ i T i j = 0. So taking the divergence of (6.13) while using ∇¯ i T i j = ∇¯ i G i j = 0 shows that H j satisfies an equation of the form g¯ ik ∂¯ik H j + Q q (¯g, ∂¯k g¯ )∂¯ p Hq = 0, j
jp
jp where the Q q are analytic in g¯ and ∂¯k g¯ . Clearly, this is a linear, 2nd order hyperbolic equation for H j . Since H j |t=0 = ∂t H j |t=0 = 0, we must have H j = 0 for all (x I , t) ∈ R3 × (−T1 (), T2 ()). k (iv) We know from Proposition 5.1 that the map (0, 0 ] → V ∈ Hδ−1, is analytic o
which implies the estimate V − V 0 H k ≤ C for some fixed constant C > 0. δ−1 o o So then
V − V 0 H k o
o
≤ V − V 0 H k
δ−1,
o
o
δ−1
≤ C
by Lemma A.11. Since {¯u , wi , α } solves the reduced Einstein equations (3.17), we have that ¯ ¯ I j + 8 2 ∂ L ∂¯4 u¯ I j + g¯ K L ∂ K2 L u¯ I j + 2 f I j ( 2 u¯ , ∂¯4 u¯ , ∂ L u¯ ) g¯ 44 ∂t ∂ 4 u = 2 S I j ( 2 u¯ , α , wi ), where the f I J are analytic and quadratic in ∂4 u¯ and ∂k u¯ while S I J are also analytic and linear in α and wi . Evaluating this equation at t = 0, and using the following facts from Proposition 5.1, ¯ i j H k −1 ¯uI j H k+1 + ¯u44 H k+1 + ∂t u o
δ
o
δ
δ−1
o
+ α H k o
δ−1
+ wi H k o
δ−1
≤ C, (6.14)
we find upon solving for ∂t ∂¯4 u¯ I j that
∂t ∂¯4 u¯ I j (0) H k−1 ≤ C ∀ ∈ (0, 0 ] δ−1
(6.15)
by the calculus inequalities of Appendix A. But from part (iii), we get that ∂¯4 u¯ 44 + ∂ I u¯ I 4 = 0 and hence differentiating this with respect to t and evaluating at t = 0 yields ¯ I 4 H k−1 ≤ C ∀ ∈ (0, 0 ].
∂t ∂¯4 u¯ 44 (0) H k−1 = ∂ I ∂t u δ−1
o
δ−1
(6.16)
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T. A. Oliynyk
From the estimates (6.14), the fluid equations (4.12) and similar arguments as above show that
∂t α (0) H k−1 ≤ C + ∂t wi (0) H k−1 ≤ C ∀ ∈ (0, 0 ]. δ−1
δ−1
(6.17)
Estimates (6.14)–(6.17) and Lemma A.11 then imply that ∂t V (0) H k−1 ≤ δ−1,
∂t V (0) H k−1 ≤ C for all ∈ (0, 0 ]. δ−1 (v) This is just a statement of the continuation principle of Theorem B.6. 7. The Newtonian Limit Let {V , 0 < ≤ 0 } be the sequence of solutions from Theorem 6.1 where we will always assume that −1 < δ < −1/2 and supp α ⊂ B R for some R > 0. o
If we let Tm () denote the maximal time of existence for the solution V , then V ∈
s+1
=0
k− C ([0, Tm ()), Hδ−1 )⊂
s+1
=0
k− C ([0, Tm ()), Hδ−1, ).
(7.1)
k−
So α ∈ s+1
=0 C ([0, Tm ()), Hδ−1 ) and hence Proposition 3.6 of [17] and Lemma A.8 imply that ρ = ρ(α ) ∈
s+1
=0
k− C ([0, Tm ()), Hδ−2 ).
Using Proposition 2.2 of [1], we can solve the equation = ρ
(7.2)
to find ∈
s+1
=0
C ([0, Tm ()), Hδk+2− ).
To obtain the Newtonian limit, we use to take care of the singular term −1 g(V )V in (6.1) by introducing the new variable ij
ij
ij
ij
j
W := (u4, , u J, , δuij , α , wi ) u J, := u J, − δ4i δ4 ∂ J . Observe that V = W + d , where j
d := (0, δ4i δ4 ∂ J , 0, 0, 0).
(7.3)
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151
Noting that b0 (U , V ) = b0 (U , W ) and b I (, U , V ) = b I (, U , W ),
(7.4)
W satisfies the equation b0 (U , W )∂t W =
1 I c ∂ I W + b I (, U , W )∂ I W + f(, U , W + d )W + H , (7.5)
where H := h − b0 (U , W )∂t d + b I (, U , W )∂ I d + f (, U , W + d )d . k as 0. Therefore by Lemma By construction the initial data V is bounded in Hδ−1 o
A.11, there exists a constant K 1 such that
W |t=0 H k
δ−1,
≤ K 1 for all ∈ (0, 0 ].
(7.6)
Also by definition of W and Lemma A.7, max{ δu L ∞ , α L ∞ , wi L ∞ } ≤ W C 1 ≤ CSob W H k
δ−1,
b
,
(7.7)
where CSob is the constant from Lemma A.7 that is independent. Shrinking 0 if necessary, we can always assume that 20 CSob K 1 < K 0 . Define
(7.8)
τ := min sup τ > 0| sup W (t) H k
δ−1,
0≤t≤τ
≤ 2K 1 and
sup V H k
0≤t≤τ
δ−1,
< ∞ ,1 . (7.9)
From the continuation principle in Theorem 6.1, it is clear that τ satisfies 0 < τ ≤ Tm (). k 7.1. Energy estimates. We will now use energy estimates on the Hδ−1, spaces to show that τ is bounded below by a constant independent of . The strategy we use is that k spaces. All of the results below will be derived under the of [5, 19] adapted to the Hδ, assumption that the 1-parameter family V of solutions has the additional regularity
V ∈
s+1
=0
k+1− C ([0, τ ], Hδ−1 ).
It is then not difficult to use a solution of this type to approximate solutions of the regularity type (7.1) and thereby show that all of the following results also hold for solutions with the regularity (7.1). Since these sort of approximation arguments are standard, we will leave the details to the interested reader. The next lemma contains the basic energy estimate which is the key to deriving estimates independent of . We note that this type of estimate has been derived previously for the standard Sobolev spaces in [5,19]. It also makes clear why we need to introduce the variables W and to put the Einstein-Euler equations into the form (7.5).
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T. A. Oliynyk
Lemma 7.1. Suppose ≥ 0, a 0 ∈ C 1 ([0, τ ], W 1,∞ ), a I ∈ C 0 ([0, τ ], W 1,∞ ), g ∈ 1 ) is a solution to the linear equation C 0 ([0, τ ], L 2λ, ), and that w ∈ C 1 ([0, τ ], Hλ, a 0 ∂t w = a I ∂ I w + g. Then there exists a constant C > 0 independent of such that d w|a 0 w L 2 ≤ C ( div a L ∞ + a L ∞ ) w 2L 2 + g L 2 w L 2 , λ, λ, λ, λ, dt where div a = ∂t a 0 + ∂ I a I and a = (a 1 , a 2 , a 3 ). Proof. Let σ¯ = σ−2λ−3 . Then σ¯ −1 ∂ j σ¯ L ∞ ≤ C for some constant C > 0 that is independent of . Using this, the proof follows by a standard integration by parts argument as in the proof of Lemma B.4. To continue, we estimate, in terms of K 1 , how much the support of α can change as 0. Lemma 7.2. supp α (t) ⊂ B R+32K 1 for all (t, ) ∈ [0, τ ] × (0, 0 ]. Proof. Letting X I , Y¯ and Yˆ be as in Sect. 4 (see (4.26)), we define X I (t) := X I (w4 (t), wJ (t)) and
YI (t) := Y¯ (w4 (t), α (t)) ∂t w4 (t) + ∂ I wI (t) +Yˆ ((u + δu(t)), w 4 (t), uk (t), w J (t), α(t)). o
Using (6.10), (7.7), (7.8), and (7.9), we obtain the bound
X I (t) L ∞ ≤ 32K 1 ∀ (t, ) ∈ [0, τ ] × (0, 0 ].
(7.10)
From Lemmas (A.7) and (A.10), and (7.1), it follows that X I ∈ C 0 ([0, τ ], Cb1 ) and YI ∈ C 0 ([0, τ ], Cb0 ). Therefore the vector field X I can be integrated to get a C 1 flow ψI (t, x) that is well defined for all (t, x) ∈ [0, τ ] × R3 . For each x ∈ R3 , define αx (t) := α (t, ψ (t, x)). Then ∂t ψI (t, x) = X I (t, ψ (t, x)) together with the evolution equation (4.25) implies that d x α (t) + Y (t, ψ (t, x))αx (t) = 0. dt By assumption supp α0 ⊂ B R and hence αx (0) = α (x) = 0 for x ∈ E R := R3 \B R . o Therefore α (t, ψ (t, x)) = 0 all x ∈ E R
(7.11)
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153
by the uniqueness of solutions to ODEs. But τ τ |ψ (t, x) − x| ≤ |∂t ψ (t, x)| = |X (t, ψ (t, x))| ≤ 32K 1 τ ≤ 32K 1 0
0
by (7.10) and 0 < τ ≤ 1. From this, (7.11), and the fact that for each t the map R3 x → ψ (t, x) ∈ R3 defines a C 1 diffeomorphism, it follows that supp α (t) ⊂ B R+32K 1 for all (t, ) ∈ [0, τ ] × (0, 0 ]. Next, we estimate H k+2 in terms of W H k δ
δ−1,
.
Lemma 7.3. Let R¯ = R + 32K 1 and
¯ −(δ−2)−3/2 1 + (1 + R) ¯ 2k . C1 = (1 + R)
Then there exists a constant C > 0 such that
(t) H k+2 ≤ CC1 W (t) nH k δ
δ−1,
for all (t, ) ∈ [0, τ ] × (0, 0 ]. Proof. By Lemma 7.2, the supp α (t) ⊂ B R+32K 1 for all (t, ) ∈ [0, τ ]×(0, 0 ]. Letting R¯ = R + 32K 1 , it follows directly from the definition of the weighted norms that ¯ −η−3/2 u L 2
u L 2 ≤ u L 2η, ≤ (1 + R) for all functions u whose support is contained in B R¯ and for any ∈ (0, 1] and −η−3/2 ≥ 0. Therefore
ρ H k
δ−2
≤ CC1 ρ H k
δ−1,
,
where C > 0 is a constant independent of and ¯ −(δ−2)−3/2 1 + (1 + R) ¯ 2k . C1 = (1 + R) k is an isomorphism and = ρ , we have H k+2 ≤ Since : Hδk+2 → Hδ−2 δ C ρ H k , and hence, by Lemma A.8 (see also (1.12) and (7.3)) and the above estimate δ−2 that
H k+2 ≤ CC1 ρ H k δ
δ−2,
≤ CC1 α nH k
δ−1,
≤ CC1 W nH k
δ−1,
.
We note that for the remainder of this section, all of the constants appearing in the estimates may depend on the fixed constant K 1 . We will often use C to denote constants that depend on K 1 and that may change from line to line. Let Wα = D α W (|α| ≥ 0), b0 = b0 (U , W ), bI = b I (, U , V ) and f = f (, U , W + d )W . The evolution equation (7.5) implies that 1 I c + bI ∂ I W + (b0 )−1 f + (b0 )−1 H . (7.12) ∂t W = (b0 )−1
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T. A. Oliynyk
Differentiating this equation yields 1 I c ∂ I Wα + bI ∂ I Wα + q α |α| ≥ 0,
b0 ∂t Wα =
(7.13)
where q α = b0 [D α , (b0 )−1 ( −1 c I + bI )]∂ I W + b0 D α ((b0 )−1 f ) + b0 D α ((b0 )−1 H ). (7.14) From Lemma A.11, we know, since −1< δ < −1/2, that u¯ H k+1 ≤ |δ+1/2| ¯u H k+1 . δ,
o
o
Since ¯u H k+1 is uniformly bounded in , we get, by Lemmas A.7 and A.11, that
δ
δ
o
U C 1,∞ ≤ CSob U H k+1 ≤ C |δ+1/2|
(7.15)
δ,
b
for some constant C > 0 independent of . So
bi (t) W 1,∞ ≤ C ∀ (t, ) ∈ [0, τ ] × (0, 0 ]
(7.16)
by (7.4), (7.7), (7.9) and (7.15) . Also, note that
d L ∞ + Dd L ∞ ≤ C H k+2 ≤ C and ∂t d L ∞ ≤ C H k+1 δ
δ
by (A.3), (A.24) and Lemmas 7.3 and A.7. The evolution equation (7.12) then implies that
∂t b0 L ∞ = Db0 (U , W ) · ∂t W L ∞ ≤ C(1 + ∂t d H k+1 ). δ
(7.17)
Together (7.16) and (7.17) establish the existence of a constant C > 0 such that
div b (t) L ∞ ≤ C(1 + ∂t (t) H k+1 ) ∀ (t, ) ∈ [0, τ ] × (0, 0 ]. δ
(7.18)
Differentiating (b0 )−1 yields ∂ J (b0 )−1 = −(b0 )−1 (Db0 (U , W ) · (∂ J U , ∂ J W ))(b0 )−1 . This along with (7.15), (7.16), (A.3), (A.24), and Lemmas A.7 and A.9 can be used to control the singular term in (7.14) and results in the following estimate (see also Appendix B.2)
q α (t) L 2
δ−1−|α|,
≤ Pα ( W (t) H k
δ−1,
, (t) H k+2 , ∂t (t) H k+1 ) ∀ t ∈ [0, τ ] δ
δ
(7.19) where Pα (y1 , y2 , y3 ) is a polynomial that is independent of and satisfies P(0) = 0. Note that in deriving this result, we have used the estimate
d H k
δ−1,
+ Dd H k
δ−2,
≤ C H k+2 and ∂t d H k δ
δ−1,
≤ C ∂t H k+1 δ
(7.20) for some C independent of which follows from (A.3), (A.24), and Lemma A.11. Define |||W |||2k,δ−1, := |α|≤k
∂ α W |b0 ∂ α W L 2
δ−|α|,
.
Newtonian Limit for Perfect Fluids
155
Then 1
W (t) H k ≤ |||W (t)|||k,δ−1, ≤ 4 W (t) H k ∀ t ∈ [0, τ ] δ−1, δ−1, 4
(7.21)
by (6.11) and (6.12). Lemma 7.1 combined with the estimates (7.16), (7.18), and (7.19) implies that d |||W |||2k,δ−1, ≤ P(|||W |||k,δ−1, , H k+2 , ∂t H k+1 )|||W |||k,δ−1, δ δ dt or equivalently d |||W (t)|||k,δ−1, ≤ P(|||W (t)|||k,δ−1, , (t) H k+2 , ∂t (t) H k+1 ) ∀ t ∈ [0, τ ] δ δ dt (7.22) for a independent polynomial P(y1 , y2 , y3 ) satisfying P(0) = 0. By Lemma 7.3,
H k+2 can be bounded by a polynomial of W H k that is independent of and δ δ−1, = 0. The differential inequality (7.22) shows that if we can do vanishes for W H k δ−1, the same for ∂t H k+1 then we get an estimate for |||W (t)|||k,δ−1, independent of . δ
Lemma 7.4. There exists a polynomial P(y) with coefficients independent of such that P(0) = 0 and
∂t (t) H k+1 ≤ P( W (t) H k δ
δ−1,
)
for all (t, ) ∈ [0, τ ] × (0, 0 ]. Proof. By (4.12), w := (α , wi )T satisfies an equation of the form a 4 (U , W )∂t w = a I (U , W )∂t w + b1 (U , W )W + b2 (U , W )d and so ∂t w = (a 4 )−1 a I ∂ I w + (a 4 )−1 b1 W + (a 4 )−1 b2 d . Thus
∂t w H k−1 ≤ (a 4 )−1 a I H k−1 DW H k−1 δ−1,
δ−2,
1,
4 −1
+ (a )
b1 H k−1 W H k−1 + (a 4 )−1 b2 H k−1 d H k−1 δ−1,
0,
δ−1,
0,
by Lemma A.8. Also by (7.15), (A.3), (A.24), and Lemmas A.7 and A.9, we have that
(a 4 )−1 a I H k−1 ≤ P( W H k
), DW H k−1 ≤ W H k
(a 4 )−1 b1 H k−1 ≤ P( W H k
), and
δ−1,
1,
δ−1,
0,
δ−1,
δ−2,
,
(a 4 )−1 b2 H k−1 ≤ P( W H k
δ−1,
0,
)
for some polynomial P(y) that is independent of . The above two inequalities along with (7.20) and Lemma 7.3 show that
∂t α H k−1 ≤ ∂t w H k−1 ≤ P( W H k δ−1,
δ−1,
δ−1,
)
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T. A. Oliynyk
for a polynomial P(y) independent of and satisfying P(0) = 0. Using Lemma A.8, the above estimate implies that
,
∂t ρ H k−1 ≤ P W H k δ−1,
δ−2,
where as above P(y) is a polynomial that is independent of . Since ∂t = ∂t ρ , the same arguments used in the proof of Lemma 7.3 can be used to conclude
.
∂t H k+1 ≤ C ∂t ρ H k−1 ≤ P W H k δ
δ−1,
δ−1,
Lemmas 7.3 and 7.4 combined with the estimate (7.22) yield d |||W (t)|||k,δ−1, ≤ P(|||W (t)|||k,δ−1, )|||W (t)|||k,δ−1, ∀ t ∈ [0, τ ] dt
(7.23)
for a polynomial P(y) that is independent of and whose coefficients depend only on K 1 . By Gronwall’s inequality there exists a time T ∗ ∈ (0, 1), independent of , such that if y(t) ≥ 0 is C 1 and satisfies dy/dt ≤ P(y)y, then y(t) ≤ e K 3 t y(0), where K 3 is a constant that depends on K 1 . Therefore |||W (t)|||k,δ−1, ≤ e K 3 t |||W (0)|||k,δ−1, for all (t, ) ∈ [0, min{T ∗ , τ }]×(0, 0 ]. (7.24) Shrinking T ∗ if necessary, we conclude that |||W (t)|||k,δ−1, ≤
3 K1 2
for all (t, ) ∈ [0, min T ∗ , τ ] × (0, 0 ].
(7.25)
Note also that
V (t) H k δ−1, ≤ C for all (t, ) ∈ [0, min{T ∗ , τ }] × (0, 0 ]
(7.26)
by 7.20, 7.21 and Lemma 7.3. Therefore by the definition of τ , we must have 0 < T ∗ < τ for all 0 < ≤ 0 . ˙ := ∂t d Differentiating (7.12) with respect to t, shows that W˙ := ∂t W and d satisfy the equation 1 b0 (U , W )∂t W˙ = c I ∂ I W˙ + b I (, U , W )∂ I W˙ ˙ )W˙ + f¯1 (, U , W , DW , d , Dd , d ˙ , Dd ˙ , ∂t d ˙ ) + ∂t h + f¯2 (, U , W , d , Dd, d for analytic functions f 1 , f 2 with f 2 linear in the last 3 variables. This equation has the same structure (7.5) and it is not difficult to show that the arguments used to derive (7.24) can also be used to obtain the estimate
W˙ (t) H k−1 ≤ C ∀ (, t) ∈ (0, 0 ] × [0, T ∗ ] δ−1,
(7.27)
under the assumption that W˙ (0) H k−1 is bounded as 0. But this is clear from δ−1, Proposition 6.1 and Lemma 7.3 and so the estimate holds. We have proved the following proposition.
Newtonian Limit for Perfect Fluids
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Proposition 7.5. For 0 > 0 small enough, there exists a T ∗ > 0 independent of ∈ (0, 0 ] such that the one parameter family of solutions V exists, for all ∈ (0, 0 ], on a common time interval [0, T ∗ ]. Moreover, there exist constants C > 0, R¯ > 0 such that K0
V (t) H k ≤ C, ∂t V (t) H k−1 ≤ C, δ−1, δ−1, 0 ≤ C, ∂t (t) H k+1 ≤ C,
max{ δu L ∞ , α L ∞ , wi L ∞ } ≤
(t) H k+2 δ
δ
and supp α (t) ⊂ B R¯ for all (, t) ∈ (0, 0 ] × [0, T ∗ ]. 7.2. Properties of the limit equations. To fully understand the limit equations of Sect. 7.3, we first need to consider the following system: ∂t αˆ = −wˆ I ∂ I αˆ −
αˆ ∂ I wˆ I , 2n
(7.28)
αˆ J ˆ ∂ αˆ − wˆ I ∂ I wˆ J − ∂ J , 2n ˆ = ρ, ˆ
∂t wˆ J = −
(7.29) (7.30)
with initial data α(0) ˆ = α and wˆ I (0) = w I , o
o
(7.31)
where α and w I are as defined in Proposition 6.1. This system is precisely the Poissono
o
1 Euler equation written using the Makino variable ρˆ = (4K n(n+1)) ˆ 2n . Indeed, a −n α straightforward calculation shows that (ρ, ˆ wˆ I ) satisfy the Poisson-Euler equations of Newtonian gravity
∂t ρˆ + ∂ I (ρˆ wˆ I ) = 0,
(7.32)
ˆ + ∂ J p), ρ(∂ ˆ t wˆ + wˆ ∂ I wˆ ) = −(ρ∂ ˆ ˆ ˆ = ρ, ˆ J
I
J
J
(7.33) (7.34)
where pˆ = K ρˆ (n+1)/n . Proposition 7.6. There exist a T > 0 and a solution k−1 k ) ∩ C 1 ([0, T ], Hδ−1 ), α, ˆ wˆ I ∈ C 0 ([0, T ], Hδ−1
k+1 ˆ ∈ C 0 ([0, T ], Hδk+2 ) ∩ C 1 ([0, T ], Hδk+1 ), ∂t ˆ ∈ C 0 ([0, T ], Hδ−1 )
to the initial value problem (7.28)–(7.31), where α(t) ˆ has compact support for all t ∈ [0, T ]. Moreover (i) this solution is unique in the class ˜ ∈ C 0 ([0, T ], Hδk+2 )∩C 1 (Rn ×[0, T ]), α, ˜ w˜ ∈ C 0 ([0, T ], H k )∩C 1 (Rn ×[0, T ]) where α(t) ˜ has compact support for all t ∈ [0, T ], and
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T. A. Oliynyk
(ii) the solution also satisfies k−
α, ˆ wˆ I ∈ ∩s+1
=0 C ([0, T ], Hδ−1 ),
k+2−
ˆ ∈ ∩s+1 ˆ ∈ ∩s =0 C ([0, T ], H k+1− ). ), ∂t
=0 C ([0, T ], Hδ δ−1
Proof. Writing the system (7.28)–(7.30) as αˆ I 1 αˆ αˆ 0 −wˆ I − 2n δJ = − ∂t , ∂ I αˆ I J wˆ J wˆ J (4K n(n + 1))n ∂ J ( −1 α 2n ) − 2n δ −wˆ I we see that this system is symmetric hyperbolic with a non-local source term. Since k : Hδk+2 → Hδ−2 is an isomorphism, it is not difficult to adapt the approximation scheme and energy estimates of Appendices B.1 and B.2 to this system. Then as in Appendix B.3, this is enough to produce an existence theorem. Consequently, there exists a T > 0 and a solution k−1 k α, ˆ wˆ I ∈ C 0 ([0, T ], Hδ−1 ) ∩ C 1 ([0, T ], Hδ−1 ).
(7.35)
k−1 k ) ∩ C 1 ([0, T ], Hδ−2 ), ρˆ ∈ C 0 ([0, T ], Hδ−2
(7.36)
Therefore ˜ = −1 ρˆ ∈ C 0 ([0, T ], H k+2 ) ∩ C 1 ([0, T ], H k+1 ). and hence δ δ Differentiating (7.34) with respect to t and using (7.32) yields ˆ = −∂ I (ρˆ wˆ I ). ∂t
(7.37)
k ) and hence −1 (ρˆ w ∈ Hδ−2 ˆ I ) ∈ C 0 ([0, T ], But, (7.35) implies that k+1 ). However, Hδk+2 ). Taking the divergence then gives ∂ I ( −1 ρˆ wˆ I ) ∈ C 0 ([0, T ], Hδ−1 −1 I −1 I ˆ = − ∂ I (ρˆ wˆ ) = −∂ I ( (ρˆ wˆ )) and so ∂t ˆ ∈ C 0 ([0, T ], (7.37) implies that ∂t k+1 Hδ−1 ).
ρˆ wˆ I
C 0 ([0, T ],
The statement about compact support follows from the symmetric hyperbolic equation satisfied by αˆ and the property of finite propagation speed. Uniqueness follows from a slight modification of standard arguments, see [39] Proposition 1.3, Sect. 16.1.
7.3. Convergence as 0. In this section, we identify the limit of the relativistic solutions as 0. To accomplish this, we adapt the arguments of [37], Sect. III. Define ij ij V˜ := (˜u4 , u˜ J , δ u˜ i j , α, ˜ w˜ i )T , α˜ I δj −w˜ I I 2n a˜ := , α˜ I − 2n δi −δi j w I I ij ˜b I := A (δ u˜ ) 0I , 0 a˜ 0 w˜ I ij ˜ S := ρ , w˜ J 4ηi j δ u˜ i j + 2w˜ 4 0
p b˜ := ,
p −ηim 2η4 η4 p + η p u˜ m − 2 η p δ4i u¯ 4 − 2η 4 u¯ i 4 ij ˜ T , f˜(V˜ )V˜ := (−S˜i j , 0, u˜ 4 , b)
Newtonian Limit for Perfect Fluids
159
and h˜ := (ηi j ∂ I β I − 2∂ I β (i η j)I , 0, . . . , 0)T . k+1−r Theorem 7.7. For any r > 0, and V converge in C 0 ([0, T ∗ ], Hloc ) and C 0 k−r k+2 ∗ 1 3 ∗ 0 ∗ ˜ ([0, T ], Hloc ) as 0 to ∈ C (R × [0, T ]) ∩ C ([0, T ], Hδ ) and the unique solution V˜ ∈ C 1 (R3 × [0, T ∗ ]) ∩ C 0 ([0, T ∗ ], H k ) of the system
P ∂t V˜ − b˜ I ∂ I V˜ − f˜(V˜ )V˜ − h˜ = 0,
˜ = 0, c I ∂ I (V˜ − d ) ˜ V (0) = V0 (0), o
˜ = ρ, ˜ where P is the projection onto the L 2 orthogonal complement of { c I ∂ I W = 0 |W ∈ H 1 }. Moreover, (i) there exists a R¯ > 0 such that supp α(t) ˜ ⊂ B R¯ for all t ∈ [0, T ∗ ], k (ii) there exists a ω ∈ C 0 ([0, T ∗ ], Hloc ) such that ∂ I ω ∈ C 0 ([0, T ∗ ], H k−1 ) and ∂t V˜ − b˜ I ∂ I V˜ − f˜(V˜ )V˜ − h˜ − c I ∂ I ω = 0,
(7.38)
(iii) and for δ1 ≥ −1/2, there exists a u˜¯ ∈ C 0 ([0, T ], L 6δ1 ) such that u˜ J = ∂ J u˜¯ i j . ij
Proof. By assumption −1 < δ < −1/2, and so it follows directly from the definition of the weighted norms that for every ≥ 0,
u H ≤ u H
δ−1,
for all u ∈ Hδ−1, .
(7.39)
So by Proposition 7.5, k−1 k V ∈ C 0 ([0, T ∗ ], H k )∩C 1 ([0, T ∗ ], H k−1 ) ⊂ C 0 ([0, T ∗ ], Hδ−1, )∩C 1 ([0, T ∗ ], Hδ−1, )
and ∈ C 0 ([0, T ∗ ], Hδk+2 )∩C 1 ([0, T ∗ ], Hδk+1 ) are uniformly bounded for ∈ (0, 0 ]. Therefore by the Banach-Alaoglu theorem there exists subsequences of and V , ˜ ∈ L 1,∞ ([0, T ∗ ], H k+1 ) ∩ Lip([0, T ∗ ], H k ), which we still denote by and V , and δ δ V˜ ∈ L 1,∞ ([0, T ∗ ], H k ) ∩ Lip([0, T ∗ ], H k−1 ) such that and V converge weakly to ˜ and V˜ , respectively, as 0. By Proposition 7.5, the support of α is uniformly bounded in and hence the support of the weak limit α˜ must also be bounded. From Proposition 6.1, we have that ij ij u J, = ∂ J u¯ . So by Lemmas A.7 and A.11, and (7.15), we find that for δ1 ≥ −1/2 ≥ δ,
ij + uij L 2 ≤ C(1 + V H k )
¯uij L 6 ≤ C ¯uij L 6 ≤ C ¯uij L 6 ≤ C u J, L 2 δ1
δ
δ,
δ−1,
δ,
δ−1,
for a constant C independent of . It follows that u¯ I converges weakly to a u˜¯ i j ∈ L 1,∞ ([0, T ∗ ], L 6δ1 ) for which ∂ J u˜¯ i j = u˜ J .
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T. A. Oliynyk
Now, V satisfies 1 b0 (U , V )∂t V − c I ∂ I (V − d ) + b I (, U , V )∂ I V − f (, U , V )V − h(U ) = 0,
(7.40)
and hence it follows from the boundedness of and V that
c I ∂ I (V − d ) H k−1 ≤ c I ∂ I (V − d ) H k−1 ≤ C. δ−1,
Letting 0 yields ˜ = 0. c I ∂ I (V˜ − d ) Next, applying the projection P (note that V − d ∈ H 1 ) to (7.40) gives P(b0 (U , V )∂t V − b I (, U , V )∂ I V − f (, U , V )V − h(U )) = 0 or equivalently Pb0 P∂t V + Pb0 (1I − P)∂t V − P(bI ∂ I V − f − h ) = 0, where we set b0 = b0 (U , V ), bI = b I (, U ,!V )∂ I V , f = f (, U , V )V , and h = h(U ). Suppose ψ ∈ C0∞ and let u|v = R3 uv d 3 x be the standard L 2 norm. Then ψ|Pb0 (1I − P)∂t V = (1I − P)b0 Pψ|∂t V
(7.41)
as P is a self-adjoint projection operator. Since the imbedding H k (B R ) → H k−r (B R ) k−r (r > 0) is compact for any ball B R , V and converge in C 0 ([0, T ∗ ], Hloc ) and k+2−r 0 ∗ ˜ respectively, as 0. Using this strong convergence C ([0, T ], Hloc ) to V˜ and , 0 2 and we find that " (7.15), # (1I − P)b Pψ → (1I − P)Pψ = 0 in L as 0 and hence 0 ψ|Pb (1I − P)∂t V → 0 by (7.41) and the fact that ∂t V L 2 is uniformly bounded in . Therefore, we have established that Pb0 (1I − P)∂t V −→ 0 weakly in L 2 as 0. The remainder of the proof follows from a straightforward adaptation of the proof of Theorem 2 in [37]. From the block diagonal form of the matrix c I , it is clear that ω can be written as ij
ij
ω = (ω4 , ω I , 0, . . . , 0)T .
Newtonian Limit for Perfect Fluids
161
Using this, we can write the system (7.38) as ∂t α˜ = −w˜ I ∂ I α˜ −
α˜ ∂ I w˜ I , 2n
(7.42)
α˜ KL ˜ ∂t w˜ J = − ∂ J α˜ − w˜ I ∂ I w˜ J − δ I J u˜ 44 , + δ u K L I I 2n
˜ 4I J , ∂t w˜ 4 = −w˜ I ∂ I w˜ 4 − u˜ 44 4 + δI J u ij ∂t u˜ 4 ij ∂t u˜ I ∂t δ u˜ i j ij ∂ J u˜ 4 ij ∂ J u˜ J
= = =
ij ij 4δ u˜ 4I ∂ I u˜ 4 + 4δ u˜ I J ∂ I u˜ J ij ij 4δ u˜ I J ∂ J u˜ 4 + ∂ I ω4 , ij u˜ 4 ,
(7.43) (7.44)
ij + ηi j ∂ I β I − 2∂ I β (i η j)I − S˜i j + ∂ I ω I ,
(7.46) (7.47)
= 0, =
(7.45)
(7.48)
j ˜ δ4i δ4 ,
(7.49)
˜ = ρ, ˜
(7.50)
with initial conditions −1 ˜ u˜ 4 (0) = 0, u˜ iI J (0) = 0, u˜ 44 I = ∂ I φ (φ := ρ(0)), ij
α(0) ˜ = α , w˜ = w , w˜ = 0. I
I
4
(7.52)
o
o
(7.51)
Equation (7.48) immediately implies that ij
u˜ 4 = 0,
(7.53)
and hence, by uniqueness and the fact that δui j (0) = 0, it follows from (7.47) that δ u˜ i j = 0.
(7.54)
˜ But u˜¯ i j ∈ L 6 and Since u˜ J = ∂ J u˜¯ i j , we get from (7.49) that u˜¯ i j = δ4i δ4 . δ1 2 ˜¯ i j ∈ H k ˜ ∈L and so by Theorem 1.2 and Proposition 1.6 of [1], we find that u δ2 δ−2 for 0 > δ2 > δ1 ≥ −1/2 > δ > −1. Since the Laplacian : Hδk2 → Hδk−2 is injective 2 −1 j ˜ and hence for δ2 < 0 (see [1], Proposition 2.2), we must have u˜¯ i j = δ i δ ij
j
4 4
˜ u˜ J = δ4i δ4 ∂ J . ij
j
(7.55)
Substituting (7.53)–(7.55) into (7.42)–(7.49) yields ∂t α˜ = −w˜ I ∂ I α˜ −
α˜ ∂ I w˜ I , 2n
α˜ J ˜ ∂ α˜ − w˜ I ∂ I w˜ J − ∂ J , 2n ˜ = ρ, ˜
∂t w˜ J = −
(7.56) (7.57) (7.58)
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T. A. Oliynyk
and ∂t w˜ 4 = −w˜ I ∂ I w˜ 4 , ij ∂ ωI ∂ I ω4J k I
(7.59) (i
= η ∂ I β − 2∂ I β η ij
I
j)I
+ S˜i j ,
(7.60)
= 0, ˜ ∂t ∂ I = ∂ I ω444 .
(7.61) (7.62)
Since w˜ 4 (0) = 0, uniqueness of solutions to hyperbolic equations implies that w˜ 4 = 0.
(7.63)
˜ w˜ I , α} Proposition 7.6 and (7.56)–(7.58) imply that {, ˜ must satisfy
k−1 k ∩ C 1 [0, T ∗ ], Hδ−1 α, ˜ w˜ I ∈ C 0 [0, T ∗ ], Hδ−1
(7.64)
and
˜ ∈ C 0 [0, T ∗ ], Hδk+2 ∩ C 1 [0, T ∗ ], Hδk+1 ,
k+1 k ˜ ∈ C 0 [0, T ∗ ], Hδ−1 ∂t ∩ C 1 [0, T ∗ ], Hδ−1 .
(7.65) (7.66)
We then get from (7.61) and (7.62) that k ˜ ∈ C 1 ([0, T ∗ ], Hδ−1 ω444 = ∂t )
(7.67)
ω44J = 0.
(7.68)
and
(i j)
Equations (7.54) and (7.63) imply that S˜i j can be written as S˜ i j = 2δ I δ4 w˜ I . We then find from (7.60) that ij
ω I = ∂ I i j ,
(7.69)
where (i j)
i j = −1 (ηi j ∂ I β I − 2∂ I β (i η j)I + 2δ I δ4 w˜ I ).
(7.70)
Note that i j ∈ C 1 ([0, T ∗ ], Hδk+1 ), k−1 k−1 ) and S˜ i j ∈ C 1 ([0, T ∗ ], Hδ−2 ) by (7.64). Therefore since ∂ I β j ∈ C 1 ([0, T ∗ ], Hδ−2 k ω I = ∂ I i j ∈ C 1 ([0, T ∗ ], Hδ−1 ). ij
We collect the above results in the following proposition.
(7.71)
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163
˜ from Theorem 7.7 satisfies Proposition 7.8. The limit solution {V˜ , } ij
δ u˜ i j = u˜ 4 = w˜ 4 = 0, ˜ ∈ C 0 ([0, T ∗ ], Hδk+2 ) ∩ C 1 ([0, T ∗ ], Hδk+1 ),
k+1 k ∂t ∈ C 0 ([0, T ∗ ], Hδ−1 ) ∩ C 1 ([0, T ∗ ], Hδ−1 ),
ij j k+1 k ˜ ∈ C 1 ([0, T ∗ ], Hδ−1 u˜ J = δ4i δ4 ∂ J ) ∩ C 1 ([0, T ∗ ], Hδ−1 ), k−1 k α, ˜ w˜ I ∈ C 0 ([0, T ∗ ], Hδ−1 ) ∩ C 1 ([0, T ∗ ], Hδ−1 ),
˜ α, while {, ˜ w˜ I } solves Eqs. (7.56)–(7.58). Moreover, the ω from Theorem 7.7 is given by ij
ij
ω = (ω4 , ω I , 0, . . . , 0)T , where ij j k ˜ ∈ C 1 ([0, T ∗ ], Hδ−1 ), ω4 = δ4i δ4 ∂t
ij (i j) k ). ω I = ∂ I −1 ηi j ∂ I β I − 2∂ I β (i η j)I + 2δ I δ4 w˜ I ∈ C 1 ([0, T ∗ ], Hδ−1
7.4. Error estimate. To get an error estimate which measures the difference between the relativistic and Newtonian solutions, we adapt the arguments of [37], Sect. IV. Define ˜ − ω and γ := α − α. ˜ Z := V − V˜ + d − d A simple but useful observation is that
γ H k−1 = α − α ˜ Hk
δ−1,
δ−1,
≤ Z H k−1
δ−1,
and
wi − w˜ i H k−1 ≤ Z H k−1 . (7.72) δ−1,
δ−1,
Lemma 7.9. There exists an independent constant C > 0 such that ˜
d (t) − d(t) H k−1 + Dd (t) − Dd(t) H k−1 ≤ (t) − (t) H k+1 δ−1,
δ
δ−2,
≤ C Z (t) H k−1
δ−1,
˜ ˜
∂t d (t) − ∂t d (t) H k−1 ≤ ∂t (t) − ∂t (t) H k ≤ C Z (t) H k−1 + C δ
δ−1,
δ−1,
and
∂t γ H k−2 ≤ C Z (t) H k−1 + C δ−1,
δ−1,
for all (t, ) ∈ [0, T ∗ ] × (0, 0 ]. ˜ are both bounded for all (t, ) ∈ [0, T ∗ ] × Proof. Since the support of α (t) and α(t) (0, 0 ], there exists a independent constant C > 0 such that ˜ H k−1 ≤ ρ − ρ ˜ H k−2 ≤ C ρ − ρ ˜ H k−1 . C −1 ρ − ρ δ−2
˜ = ρ, Also, = ρ , ˜ and :
Hδk+1
→
δ−1,
δ−2
k−1 Hδ−1
is an isomorphism, and therefore
˜ k+1 ≤ ρ − ρ
− ˜ H k−1 ≤ C ρ − ρ ˜ H k−1 ≤ C γ H k−1 ≤ C Z H k−1 (7.73) H δ
δ−2
δ−1,
δ−1,
δ−1,
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T. A. Oliynyk
by Proposition 7.5 and Lemma A.10. From (4.25) and (7.56), it follows that γ satisfies ∂ I w˜ I α, ˜ ∂t γ = −X I ∂ I γ − Y γ + X I − w˜ I ∂ I α˜ + Y − 2n
(7.74)
where X I and Y are given by (4.26). But X I = X I (w4 , wI ) and w˜ I = X I (0, w˜ I ), and hence
X I − w˜ I H k−2 ≤ C wI − w˜ I H k−2 ≤ C Z H k−1 δ−1,
δ−1,
δ−1,
(7.75)
by (7.72), (A.24), Lemma A.10 and Proposition 7.5. Next,
1 ∂ I w˜ I 1 ¯ Y− = Y (V ) − ∂t w4 + ∂ I wI + ∂t w4 2n 2n 2n
1 ∂ I wI − ∂ I w˜ I + Yˆ (U , V ), + 2n where Yˆ (0) = 0 and Y¯ (0) − 1/(2n) = 0. Using (7.15), (A.3), (A.24), Proposition 7.5, and Lemmas A.7–A.10, we can estimate each of the above terms as follows: 1 1 4 I ¯ ¯
Y (V ) − (∂t w + ∂ I w ) H k−2 ≤ Y (V ) −
H k−2 δ, δ−1, 2n 2n
4 I × ∂t w H k−2 + w H k−2 δ−1, δ−1,
∂t V H k−1 + V H k ≤ C, ≤ C V H k δ−1,
δ−1,
δ−1,
1
∂t w4 H k−2 ≤ C ∂t V H k−1 ≤ C, δ, δ−1, 2n 1 I I
(∂ I w − ∂ I w˜ ) H k−2 ≤ C Z H k−1 , δ, δ−1, 2n and
Yˆ (U , V ) H k−2 ≤ C U H k + V H k δ,
δ,
δ−1,
≤ C.
Therefore
Y −
∂ I w˜ I
k−2 ≤ C Z H k−1 + C. δ−1, 2n Hδ,
(7.76)
We can also estimate X I and Y as follows:
X I H k−2 ≤ C V H k δ−1,
δ−1,
≤ C,
(7.77)
Y H k−2 ≤ C( U H k + V H k δ,
δ,
δ−1,
+ ∂t V H k−1 ) ≤ C. δ−1,
(7.78)
The estimates (7.72), (7.75), (7.76), (7.77), (7.78) along with Lemma A.8 imply via Eq. (7.74) that
∂t γ H k−2 ≤ C Z H k−1 + C. δ−1,
δ−1,
(7.79)
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165
˜ = ∂t ρ, Since ∂t = δρ and ∂t ˜ the same arguments used to establish the estimate (7.73) can be used in conjunction with (7.79) to show ˜ k ≤ C Z k−1 + C.
∂t − ∂t H H δ
(7.80)
δ−1,
Finally from (7.73), (7.80), and Lemma A.11, we get the desired estimates ˜ k+1 ≤ C Z k−1
d − d H k−1 + Dd − Dd H k−1 ≤ − H H δ−1,
δ
δ−2,
δ−1,
and ˜ k ≤ C Z k−1 + C ˜ k−1 ≤ ∂t − ∂t
∂t d − ∂t d H H H δ
δ−1,
for some constant C independent of .
δ−1,
Lemma 7.10. There exists a constant C > 0 such that ˜ H k−2 + V (t) − V˜ (t) H k−1 ≤ C for all (t, ) ∈ [0, T ∗ ] × (0, 0 ].
∂t α − ∂t α δ−1,
δ−1,
Proof. From the evolution equation (6.1), we find that Z satisfies the equation b0 ∂t Z =
1 I c ∂ I Z + bI ∂ I Z + F ,
(7.81)
where b0 = b0 (U , V ), bI = b(, U , V ) and ˜ − d )−b0 ∂t ω + bI (∂ I d ˜ − ∂ I d )+bI ∂ I ω F = −b0 ∂t (d ˜ (7.82) −(b0 − 1I )∂t V˜ +(b˜ I − bI )∂ I V˜ + f (, U , V )V − f˜(V˜ )V˜ + h − h. Using (7.15), (A.3), (A.24), Lemmas 7.9, A.7–A.9, and Propositions 7.5 and 7.8, we get the following estimates:
b0 − 1I H k ≤ C( U H k + V H k δ,
δ,
δ−1,
) ≤ C,
(7.83)
˜ − d ) k−1 ≤ (b0 − 1I )∂t (d ˜ − d ) k−1 + ∂t (d −d ˜
b0 ∂t (d ) H k−1 H H δ−1,
δ−1,
δ−1,
˜ − d ) k−1 ≤ C Z k−1 + C, ≤ C( b0 − 1I H k−1 + 1) ∂t (d H H δ,
δ−1,
δ−1,
b0 ∂t ω H k−1 ≤ C( b0 − 1 H k−1 + 1) ∂t ω H k−1 ≤ C, δ,
δ−1,
(7.85)
δ−1,
bI ∂ I ω H k−1 ≤ C bI H k−1 ∂ I ω H k−1 ≤ C bI H k−1 ω H k δ,
δ−1,
δ,
δ−2,
(7.84)
δ−1,
≤ C, (7.86)
˜ k−1 ≤ C.
h − h H
(7.87)
δ−1,
˜ we first note that To estimate the term bI − b, 0 A I (uiJ ) + A I (δuiJ ) I ˜ o b − b = , ˜ 0 a I (u , wi , α , wI , α ) − a I (0, 0, 0, w˜ i , α) where the map a I is analytic. Next, the estimate (7.15) implies that
uij H k−1 , ≤ uij H k−1 + C δuij H k−1 ≤ C + C Z H k−1 . δ
o
δ,
δ−1,
δ−1,
(7.88)
166
T. A. Oliynyk
From Proposition 5.2 and Lemma A.11, we see that the uiJ can be estimated by o
uiJ H k+1 = u¯ iJ H k+1 ≤ |δ+1/2| ¯uiJ H k+1 ≤ C |δ+1/2|+1 . o
δ,
δ,
o
(7.89)
δ
o
Also, from Proposition 7.8 and Lemma 7.9, we obtain ˜ k−1
∂t α − ∂t α ˜ H k−2 + V − V˜ H k−1 ≤ Z H k−1 + d − d H δ−1,
δ−1,
δ−1,
+ ω H k−1 ≤ C Z H k−1 + C. δ−1,
δ−1,
(7.90)
δ−1,
The three estimates (7.88)–(7.90) along with Lemmas A.9 and A.10, and Propositions 7.5 and 7.8, show that
A I (uiJ ) + A I (δuiJ ) H k−1 ≤ C uiJ H k + C δuiJ H k−1 ≤ C + C Z H k−1 , δ,
o
δ,
o
δ−1,
δ−1,
and
a I (u , wi , α , wI , α ) − a I (0, 0, 0, w˜ i , α) ˜ H k−1 δ,
ij i i ≤ C u H k−1 + α − α ˜ H k−1 + w − w˜ H k−1 ≤ C + C Z H k−1 . δ,
δ−1,
δ−1,
δ−1,
Therefore ˜ k−1 ≤ C + C Z k−1 ,
bI − b H H δ−1,
δ−1,
and hence
(b˜ − bI )∂ I V˜ H k−1 ≤ C b˜ − bI H k−1 D V˜ H k−1 ≤ C + Z H k−1 . (7.91) δ,
δ−1,
δ−2,
δ−1,
Next, we notice that f (, U , V )V − f˜(V˜ )V˜ = −ρ F + fˆ(V )V − fˆ(V˜ )V + f¯(, U , V )V , where j
F := −4ρ (δ4i δ4 η pq u¯ pq , 0, . . . , 0)T o
and fˆ and f¯ are analytic. We obtain
fˆ(V )V − fˆ(V¯ )V + f¯(, U , V )V H k−1 ≤ C Z H k−1 + C δ−1,
(7.92)
δ−1,
by the arguments used above. Also, the boundedness of the support of α (t) implies that
F H k−1 ≤ C ρ ηi j u¯ ij H k−1 ≤ C ρ H k−1 ¯u H k−1 ≤ C ρ H k−1 ¯u H k−1 ≤ C. δ−1,
o
δ
δ
o
δ
δ−1,
o
δ
(7.93) So then
f (, U , V )V − f˜(V˜ )V˜ H k−1 ≤ C Z H k−1 + C δ−1,
δ−1,
(7.94)
Newtonian Limit for Perfect Fluids
167
by (7.92) and (7.93). Combining the estimates (7.83)–(7.87), (7.91), (7.92), and (7.94) yields
F H k−1 ≤ C Z H k−1 + C. δ−1,
(7.95)
δ−1,
Letting Z α = D α Z and differentiating Eq. (7.81) yields b0 ∂t Z α =
1 I c ∂ I Z α + bI ∂ I Z α + q α
0 ≤ |α| ≤ k − 1,
where q α = −[D α , b0 ]∂t Z α + [D α , bI ]∂ I Z α + D α F . Using the estimates above along with Propositions 7.5 and 7.8 and the calculus inequalities from Appendix A, we find
∂t Z H k−1 ≤ C Z H k−1 + C, δ−1,
[D
[D
α
α
, b0 ]∂t Z α L 2 δ−1−|α|,
, bI ]∂ I Z α L 2 δ−1−|α|,
δ−1,
≤
C b0
− 1I H k−1 ∂t Z H k−1 ≤ C Z H k−1 + C,
≤
C bI H k−1 D Z H k−2 δ, δ−2,
δ,
δ−1,
δ−1,
≤ C Z H k−1 , δ−1,
and hence
q α H k−1 ≤ C Z H k−1 + C. δ−1,
δ−1,
Combining this estimate with the estimates
∂t b0 + ∂ I bI L ∞ ≤ C, bI L ∞ ≤ C, and Lemma 7.1 shows that d α 0 α Z |b Z ≤ C( Z H k−1 + ) Z H k−1 0 ≤ |α| ≤ k − 1. δ−1, δ−1, dt L 2δ−1−|α|, Summing over α and using Gronwall’s inequality, we get
Z (t) H k−1 ≤ C Z (0) H k−1 + C for all (t, ) ∈ [0, T ∗ ] × (0, 0 ]. δ−1,
δ−1,
This estimate and (7.90) then prove the proposition since Z (0) H k−1 ≤ C by δ−1, Proposition 6.1. We are now ready to prove a precise error estimate for the difference between the relativistic and Newtonian solutions. Proposition 7.11. Suppose −1 < δ < −1/2 and k ≥ 3. Then there exists a constant C > 0 such that j ˜ I ˜ ¯ ij (t) − δ4i δ4 d (t)
¯uij (t) − δ4i δ4i (t) ˜ I (t) H k−1 L 6 + ∂ I u H k−1 + v (t) − w δ,
+
−1
δ−1,
δ−1,
v (t) − 1 H k−1 + ρ (t) − ρ(t) ˜ ˜ H k−1 + ∂t ρ (t) − ∂t ρ(t) H k−2 ≤ C 4
δ−1,
for all (t, ) ∈ [0, T ∗ ] × (0, 0 ].
δ−1,
δ−1,
168
T. A. Oliynyk
Proof. From the evolution equations and Proposition 7.8, we have
j ˜ ij ij = u4 − ω4 , ∂t u¯ ij − δ4i δ4 and hence integrating yields ˜ ¯uij (t) − δ4i δ4 (t) uij − δ4i δ4 φ L 2 + L 2 ≤ ¯ j
j
δ,
δ,
o
t
ij
ij
u4 (s) − ω4 (s) L 2 ds. δ,
0
(7.96) But t 0
ij ij
u4 (s) − ω4 (s) L 2 ds δ,
t
≤ 0
V (s) − V˜ (s) H k−1 + ωi j (s) H k−1 ds (7.97) δ−1,
δ−1,
and j
¯uij − δ4i δ4 φ L 2 ≤ C 3/2
(7.98)
δ,
o
by the calculus inequalities of Appendix A and Proposition 5.1. Also, by Lemma A.4 and u I, = ∂ I u¯ , we have ˜ 6 ≤ C u − δ4i δ d ˜ 2
¯uij − δ4i δ4 L L 4 I, j
ij
j
δ,
δ−1,
˜ + ¯uij (t) − δ4i δ4 (t) L2 j
δ,
˜ + ¯uij (t) − δ4i δ4 (t) L2 . j
(7.99)
δ,
is Recall that ρ = (4K n(n + 1))−n α2n and ρ˜ = (4K n(n + 1))−n α˜ 2n . Since α H k δ−1, bounded as 0, we obtain ˜ H k−1 ≤ C α − α ˜ H k−1 ≤ C V − V˜ H k−1
ρ − ρ δ−1,
δ−1,
δ−1,
(7.100)
by Lemma A.10. We also have that ∂t α H k−2 is bounded as 0, so the formulas δ−1,
∂t ρ =
2n 2n α 2n−1 ∂t α , ∂t ρ˜ = α˜ 2n−1 ∂t α, ˜ (4K n(n + 1))n (4K n(n + 1))n
and the calculus inequalities of Appendix A imply that ˜ H k−2 ≤ C( α − α ˜ H k−2 + ∂t α − ∂t α ˜ H k−2 )
∂t ρ − ∂t ρ δ−1,
δ−1,
δ−1,
≤ C( V − V˜ H k−1 + ∂t α − ∂t α ˜ H k−2 ). δ−1,
δ−1,
(7.101)
Finally, from the definition of V and V˜ , we have I ˜ ˜ I (t) H k−1 + −1 v 4 (t) − 1 H k−1
∂ I u¯ ij (t) − δ4i δ4 d (t) H k−1 + v (t) − w j
δ−1,
≤ C V − V˜ H k−1 . δ−1,
δ−1,
δ−1,
(7.102)
The proof now follows as a direct consequence of Lemma 7.10 and (7.96)–(7.102).
Newtonian Limit for Perfect Fluids
169
In the above error estimate, the norm itself depends on . We now show how to choose norms independent of which are compatible with the error estimate above. First, for any η ∈ R define a norm by
u
, p,η := |α|≤
D α u L ηp .
Recalling that −1 < δ < −1/2, fix η ∈ [δ, −1/2]. Then from (A.24) and Lemma A.11, we get that
u
,2,η−1 ≤ C η+1/2 u H
δ−1,
and u 0,6,η ≤ C η+1/2 u L 6
δ,
(7.103)
for some constant C > 0 independent of . Combining (7.103) with Corollary 7.11 yields the following theorem which is our main result. Theorem 7.12. Suppose −1 < δ < −1/2, −δ ≤ η ≤ −1/2 and k ≥ 3. Then there exists a constant C > 0 such that j ˜ ˜ ¯ ij (t) − δ4i δ4 ∂ I (t)
¯uij (t) − δ4i δ4i (t) 0,6,η + ∂ I u k−1,2,η−1
+ v I (t) − w˜ I (t) k−1,2,η−1 + −1 v 4 (t) − 1 k−1,2,η−1
η+3/2 + ρ (t) − ρ(t) ˜ ˜ k−1,2,η−1 + ∂t ρ (t) − ∂t ρ(t) k−2,2,η−1 ≤ C
for all (t, ) ∈ [0, T ∗ ] × (0, 0 ]. Note that for η = −1/2, we have
u 0,6,−1/2 = u L 6 and u
,2,−3/2 = u H , where u H is the standard Sobolev norm. So the above theorem shows that the difference between the relativistic and Newtonian solutions is of order with respect to the norms · L 6 and · H k−1 . A. Weighted Calculus Inequalities In this and the following sections C will denote a constant that may change value from line to line but whose exact value is not needed. Let V be a finite dimensional vector space with inner product (·|·) and corresponding p norm | · |. For u ∈ L loc (Rn , V ), 1 ≤ p ≤ ∞, δ ∈ R, and ∈ R≥0 , the weighted L p norm of u is defined by
−δ−n/ p u L p if 1 ≤ p < ∞
σ p , (A.1)
u L := δ,
σ−δ u L ∞ if p = ∞ $ 1 where σ (x) := 1 + |x|2 . The weighted Sobolev norms are then defined by 4 ⎧⎛ ⎞1/ p ⎪ ⎪ ⎪ p ⎪⎝ ⎠
D α u L p if 1 ≤ p < ∞ ⎨ δ−|α|, , (A.2)
u W k, p := |α|≤k ⎪ δ, ⎪ α ⎪
D u L ∞ if p = ∞ ⎪ ⎩ δ−|α|, |α|≤k
170
T. A. Oliynyk
where k ∈ N0 , α = (α1 , . . . , αn ) ∈ Nn0 is a multi-index and D α = ∂1α1 . . . ∂nαn . Here ∂i =
∂ , ∂ xi
where (x 1 , . . . , x n ) are the standard Cartesian coordinates on Rn . The weighted Sobolev spaces are then defined as
) k, p k, p n Wδ, = u ∈ Wloc (R , V ) | u W k, p < ∞ . δ,
Directly from this definition, we observe the simple but useful inequality
∂ j u W k ≤ u W k+1 . δ,
(A.3)
δ+1,
k, p
k, p
We note that Wδ,0 are the standard Sobolev spaces and for > 0, the Wδ, are equivalent to the radially weighted Sobolev spaces [1, 7]. For p = 2, we use the alternate notation k := W k,2 . The spaces L 2 and H k are Hilbert spaces with inner products Hδ, δ, δ, δ, u|v L 2 := (u|v)σ−2δ−n d n x, (A.4) δ,
Rn
and
"
u|v H k := δ,
D α u|D α v
#
|α|≤k
L 2δ−|α|,
,
k, p
(A.5) k, p
k . respectively. When = 1, we will also use the notation Wδ = Wδ,1 and Hδk = Hδ,1 Let B R be the open ball of radius R and a R and A R denote the annuli B2R \ B R and B4R \ B R , respectively. Let {φ j }∞ j=0 be a smooth partition of unity satisfying
supp φ0 ⊂ B2 , supp φ j ⊂ A2 j−1 ( j ≥ 1), and φ j (x) := φ1 (21− j x)( j ≥ 1). Scaling gives a one parmeter family of smooth partitions of unity φ j (x) := φ j (x) ( j ≥ 0) which satisfy supp φ0 ⊂ B2/ , supp φ j ⊂ A2 j−1 / ( j ≥ 1), and φ j (x) := φ1 (21− j x)( j ≥ 1). (A.6) Define a scaling operator by S j u(x) := u(2 j−1 x).
(A.7)
This operator satisfies the following simple, but useful identities: S1 = 1I , S j ◦ Sk = Sk ◦ S j = Sk+ j−1 , S j φ j = φ1 ( j ≥ 1),
S j u L p = 2
n(1− j) p
(A.8) (A.9)
u L p ,
(A.10)
S j ◦ D α = 2(1− j)|α| D α ◦ S j .
(A.11)
and
Newtonian Limit for Perfect Fluids
171
Lemma A.1. For 1 ≤ p < ∞, there exists a constant C > 0 independent of ≥ 0 such that 1 p p
u L p ≤ φ0 u L p + C δ,
∞
S j (φ j u) L p ≤ C u L p . p
p
δ,
j=0
Proof. From the identity p
u L p
∞
=
|u| d x + p
|u| p d n x
n
B4/
j=1 a2 j+1 /
and a simple change of variables, it follows that
u L p = σ−δ−n/ p u L p (B4/ ) + p
p
δ,
∞
2n( j−1) S j (σ−δ−n/ p u) L p (a4/ ) . p
(A.12)
j=1
This identity and max σ (x)
−δp−n
min σ (x)
−δp−n
max (S j σ )(x)
−δp−n
min (S j σ )(x)
−δp−n
x∈B 4/
x∈B 4/
=
=
x∈a 4/
x∈a 4/
= =
2 1
−δp−n 2
if −δp − n ≥ 0 , if −δp − n < 0
1 if −δp − n ≥ 0 −δp−n , if −δp − n < 0 2 2 −δp−n
(1 + 22 j ) 2 if −δp − n ≥ 0 , −δp−n 2( j−1) (1 + 2 ) 2 if −δp − n < 0 −δp−n 2
(1 + 22( j−1) ) −δp−n (1 + 22 j ) 2
if −δp − n ≥ 0 , if −δp − n ≤ 0
show that 1 p p p
u L p (B4/ ) ≤ σ−δ−n/ p u L p (B4/ ) ≤ C u L p (B4/ ) C
(A.13)
and 1 − pδ( j−1) p 2
S j u L p (a4/ ) ≤ 2n( j−1) S j (σ−δ−n/ p u) L p (a4/ ) C p ≤ C2− pδ( j−1) S j u L p (a4/ ) (A.14) for some constant C > 0 which is independent of ≥ 0. Using a change of variable, the inequality (A.14) can be written as 1 − pδ( j−1) (1− j)n p 2 2
u L p (a j+1 ) ≤ 2n( j−1) S j (σ−δ−n/ p u) L p (a4/ ) 2 / C p − pδ( j−1) (1− j)n ≤ C2 2
u L p (a j+1 ) . (A.15) 2
/
172
T. A. Oliynyk
From
φk
2 k=0
φ j+k
2
= 1I B4/ and B4/
a2 j+1 /
k=0
and (A.10), we obtain
p
u L p (B4/ )
≤C
p
φ0 u L p
3
+
= 1I a2 j+1 /
(A.16)
p
Sk (φk u) L p
,
(A.17)
k=1
and 2 p
u L p (a
2 j+1 /
)
2n( j+k) S j+k (φ j+k u) L p . p
≤C
(A.18)
k=0
Combining (A.12) with the inequalities (A.13), (A.15), (A.17) and (A.18) yields ⎛ ⎞
u L p ≤ C ⎝ φ0 u L p + p
p
δ,
∞
2− pδ( j−1) S j (φ j u) L p ⎠ p
(A.19)
j=1
for some constant C > 0 independent of ≥ 0. Since supp φ0 ⊂ B4/ and φ0 L ∞ = φ0 L ∞ , we get from (A.13) that
φ0 u L p ≤ φ0 L ∞ u L p (B4/ ) ≤ C σ−δ−n/ p u L p (B4/ ) p
p
p
p
(A.20)
for some constant C > 0 independent of ≥ 0. Next, 2− pδ( j−1) S j (φ j u) L p p
≤ 2− pδ( j−1) 2n(1− j) φ j u L p (A2 j−1 / )
by (A.10) and (A.6),
≤ 2− pδ( j−1) 2n(1− j) φ1 L ∞ φ j u L p (∪1
k=−1 a2 j−k / )
since φ1 L ∞ = φ j L ∞ .
So there exists a constant C > 0 independent of ≥ 0 such that ⎧
p ⎨ C u p p L (B4/ ) + u L p (a8/ ) p − pδ j−1 * 2
S j (φ j u) L p ≤ ⎩ C2− pδ( j−1) 2n(1− j) 2k=0 u Lp p (a
if j = 1
2 j−1+k /
)
if j ≥ 2
.
(A.21) Therefore
φ0 u L p + p
∞
2− pδ( j−1) S j (φ j u) L p p
j=1
⎛
≤ C ⎝ σ−δ−n/ p u L p (B4/ ) + p
∞
⎞ 2n( j−1) S j (σ−δ−n/ p u) L p (a4/ ) ⎠ p
j=1 p
≤ C u L p
δ,
by (A.15), (A.20), and (A.21) by (A.12),
where C > 0 is a constant independent of ≥ 0. The proof then follows from this inequality and (A.19).
Newtonian Limit for Perfect Fluids
173
The above lemma shows that the norm |||u||| L p := φ0 u L p + p
p
δ,
∞
2− pδ( j−1) S j (φ j u) L p p
j=1
is equivalent for 1 ≤ p < ∞, independent of ≥ 0, to the weighted norm u L p . For δ, p = ∞, the appropriate norm is := sup φ0 u L ∞ , 2−δ( j−1) φ j u L ∞ ( j ≥ 1) |||u||| L ∞ δ, and it is easy to see that there exists a constant C > 0 independent of ≥ 0 such that 1 ≤ |||u||| L ∞ ≤ C u L ∞ .
u L ∞ δ, δ, δ, C The same arguments used in proving the previous lemma can be used to establish the following generalization. Lemma A.2. For 1 ≤ p < ∞, let |||u|||
p
:= φ0 u W k, p + p
k, p
Wδ,
∞
2− pδ( j−1) S j (φ j u) W k, p , p
(A.22)
j=1
and for p = ∞ let |||u|||W k,∞ := sup{ φ0 u W k,∞ , 2−δ( j−1) S j (φ j u) W k,∞ ( j ≥ 1)}. δ,
(A.23)
Then there exists a constant C > 0 independent of ≥ 0 such that 1 p p p
u k, p ≤ |||u||| k, p ≤ C u k, p . Wδ, Wδ, Wδ, C For the remainder of this section, we will use the two equivalent norms · W k, p and δ,
||| · |||W k, p interchangeably and refer to both using the notation · W k, p . From (A.22), it δ,
δ,
follows that there exist a constant C > 0 independent of ≥ 0 such that
u W k2 , p ≤ C u W k1 , p whenever k2 ≤ k1 and δ1 ≤ δ2 . δ2 ,
δ1 ,
k ,p
(A.24)
k ,p
Thus we have the inclusion Wδ11, ⊂ Wδ22, for k2 ≤ k1 and δ1 ≤ δ2 . The representation k, p
(A.22) is particularly useful for extending estimates from the usual Sobolev spaces Wδ k, p to the weighted ones Wδ, ( > 0) as the next lemma shows. It also makes clear the philosophy behind deriving weighted Sobolev inequalities which is to derive global estimates from scaling and local Sobolev inequalities [1]. We remark that the norm ||| · |||W k, p , as an alternate representation for the standard δ,1
weighted norms · W k, p , was introduced by Maxwell in [25]. There he used the norm δ,1
to define the weighted Sobolev spaces for non-integral k (see also [4]). Here we will only be interested in integral k.
174
T. A. Oliynyk
Lemma A.3. Suppose 0 > 0 and for all u ∈ C ∞ (Rn , V ), u → F1 (u) is a map that satisfies φ0 F1 (u) = φ0 F1 ((φ0 + φ1 )u), 1 φ j F1 (u) = φ j F1
φ j+k u
k=−1 −( j−1)λ 2 F1 (S j u)
S j F1 (u) =
( j ≥ 1), ( j ≥ 1),
and Fα (α = 2, 3, 4, 5) are linear operators on V . (i) If there is an estimate of the form
F1 (u) W k1 , p1 ≤ C1 F2 (u) W k2 , p2 , where p1 ≥ p2 , then
F1 (u) W k1 , p1 ≤ C F2 (u) W k2 , p2 δ1 ,
δ2 ,
for some constant C > 0 independent of ∈ [0, 0 ] provided δ1 + λ ≥ δ2 . (i i) If there exists an estimate of the form
F1 (u) W k1 , p1 ≤ C1 F2 (u) W k2 , p2 F3 (u) W k3 , p3 +C2 F2 (u) W k4 , p4 F1 (u) W k5 , p5 , 1 1 1 1 1 = + = + (1 ≤ p1 ≤ pα ≤ ∞ α = 2, 3, 4, 5), then p1 p2 p3 p4 p5
F1 (u) W k1 , p1 ≤ C C1 F2 (u) W k2 , p2 F3 (u) W k3 , p3 +C2 F4 (u) W k4 , p4 F5 (u) where
δ1 ,
δ2 ,
δ3 ,
δ4 ,
k ,p
Wδ 5, 5 5
for some constant C > 0 independent of ∈ [0, 0 ] provided δ1 + λ ≥ max {δ2 + δ3 , δ4 + δ5 }. Proof. We only proof part (ii) for 1 ≤ pα < ∞. Part (i) can be proved in a similar manner using the inequality ⎞1/ p ⎛ ⎞1/q ⎛ aj ⎠
⎝
p
aj ⎠
≤⎝
j
q
for a j ≥ 0 and 0 < q ≤ p
(A.25)
j
instead of Hölder’s and Minkowski’s inequalities. See also the proof of Theorem 1.2 in [1]. Recall Hölder’s and Minkowski’s inequalities which state that for 1 ≤ p ≤ q ≤ r < ∞, 1/ p = 1/q + 1/r and any two sequences a j , b j ≥ 0 that the following holds ⎞1/ p ⎛ ⎞1/q ⎛ ⎞1/r ⎛ aj bj ⎠
⎝
p p
j
and
(a j + b j ) p ⎠
⎝ j
q
⎝
j
⎞1/ p
⎛
aj ⎠
≤⎝
⎛
p aj ⎠ j
(A.26)
j
⎞1/ p
≤⎝
brj ⎠
⎛
⎞1/ p
+⎝
p aj ⎠ j
.
(A.27)
Newtonian Limit for Perfect Fluids
175
Next, suppose j ≥ 2. Then
S j (φ j F1 (u)) W1k1 , p1 + 1 + + p1 + p1 1 + + + + + + + −(1− j) p1 λ + = +φ1 S j F1 φ j+k u) + ≤ C2 S j φ j+k u + + F1 + + + + k1 , p1 k=−1 k=−1 W k1 , p1 ⎛ + W + + + 1 1 + + + + + + + + ≤ C2−( j−1) p1 λ ⎝C1 + F2 S j φ j+k u + S j φ j+k u + + F3 + + + + k=−1 k=−1 W k2 , p2 W k3 , p3 ⎞ p1 + 1 + 1 + + + + + + + + + + ⎠ , +C2 + F4 S j φ j+k u + S j φ j+k u + + F5 + + + + p
W k4 , p4
k=−1
W k5 , p5
k=−1
where C > 0 is a constant independent of ≥ 0. Note that in deriving this, we have used the fact that φ1 W k1 ,∞ is bounded for ∈ [0, 0 ]. From the above inequality, we see that 2−δ1 p1 ( j−1) S j (φ j F1 (u)) W1k1 , p1 1 p
≤ C C1 2
−δ2 ( j−1)
k=−1 1
2−δ3 ( j−1)
F3 (S j+k (φ j+k u)) W k3 , p3
× k=−1
+C2 2−δ4 ( j−1)
F2 (S j+k (φ j+k u)) W k2 , p2
1
F4 (S j+k (φ j+k u)) W k4 , p4
k=−1 1
p1
F5 (S j+k (φ j+k u)) W k5 , p5
×
2−δ5 ( j−1)
,
k=−1
where we have used δ1 + λ ≥ max{δ2 + δ3 , δ4 + δ5 }. The above inequality along with (A.26) and (A.27) imply ⎛ ⎝
∞ j=1
⎞1/ p1 p 2−δ1 p1 ( j−1) S j (φ j F1 (u)) W1k1 , p1 ⎠
⎛
⎛
⎜ ≤ C ⎝C1 ⎝
∞
⎞1/ p2 2−δ2 p2 ( j−1) F2 (S j (φ j u)) W2k2 , p2 ⎠ p
j=1
⎛ ×⎝
∞ j=1
⎞1/ p3 p 2−δ3 p3 ( j−1) F3 (S j (φ j u)) W3k3 , p3 ⎠
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T. A. Oliynyk
⎛
∞
+C2 ⎝ ⎛ ×⎝
⎞1/ p4 2−δ4 p4 ( j−1) F4 (S j (φ j u)) W4k4 , p4 ⎠ p
j=1 ∞
⎞1/ p5 ⎞ ⎟ p 2−δ5 p5 ( j−1) F5 (S j (φ j u)) W5k5 , p5 ⎠ ⎠,
j=1
and hence ⎛ ⎝
∞
⎞1/ p1 p 2−δ1 p1 ( j−1) S j (φ j F1 (u)) W1k1 , p1⎠
≤ C C1 F2 (u) W p2 ,k2 F3 (u) W p3 ,k3 δ2 ,
j=1
δ3 ,
+C2 F4 (u) W p4 ,k4 F5 (u) δ4 ,
.
p ,k Wδ 5, 5 5
(A.28) Similar arguments show that
φ
1/ p1
p F1 (u) k1 , p1 + S1 (φ1 F1 (u)) 1k1 , p1 Wδ , Wδ , p1
1
≤ C C1 F2 (u) W p2 ,k2 F3 (u) W p3 ,k3 δ2 ,
1
+C2 F4 (u) W p4 ,k4 F5 (u) δ4 ,
δ3 ,
p ,k
Wδ 5, 5
,
5
(A.29) for some constant C > 0 independent of ∈ [0, 0 ]. The proof now follows from the two inequalities (A.28) and (A.29). The next lemma is a variation of the previous one and can be proved in the same fashion. Lemma A.4. Suppose 0 > 0 and for all u ∈ C ∞ (Rn , V ), u → F1 (u) is a map that satisfies φ0 F1 (u) = φ0 F1 ((φ0 + φ1 )u), 1 φ j F1 (u) = φ j F1 S j F1 (u) =
φ j+k u
k=−1 −( j−1)λ 2 F1 (S j u)
( j ≥ 1), ( j ≥ 1),
and F2 = D P2 , F3 = P3 , F4 = D P4 , and F5 = P5 , where Pα (α = 2, 3, 4, 5) are linear operators on V .
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177
(i) If there exists an estimate of the form
F1 (u) W k1 , p1 ≤ C1 F2 (u) W k2 , p2 , where p1 ≥ p2 , then there exists a constant C > 0 independent of ∈ [0, 0 ] such that
F1 (u) W k1 , p1 ≤ C F2 (u) W k2 , p2 + P2 u W k2 , p2 δ1 ,
δ2 ,
δ2 −1,
provided δ1 + λ ≥ δ2 . (i i) If there exists an estimate of the form
F1 (u) W k1 , p1 ≤ C1 F2 (u) W k2 , p2 F3 (u) W k3 , p3 +C2 F2 (u) W k4 , p4 F1 (u) W k5 , p5 , 1 1 1 1 1 = + = + (1 ≤ p1 ≤ pα ≤ ∞ α = 2, 3, 4, 5), then p1 p2 p3 p4 p5
F1 (u) W k1 , p1 ≤ C C1 F2 (u) W k2 , p2 + P2 u W k2 , p2 F3 (u) W k3 , p3 δ1 , δ2 , δ3 , δ2 −1, +C2 F4 (u) W k4 , p4 + P4 u W k4 , p4 F5 (u) k5 , p5
where
Wδ
δ4 ,
δ4 −1,
5 ,
for some constant C > 0 independent of ∈ [0, 0 ] provided δ1 + λ ≥ max {δ2 + δ3 , δ4 + δ5 }. Remark A.5. By using the generalized Hölder’s inequality, part (ii) of Lemmas A.3 and A.4 can be extended in the obvious fashion if there exist estimates of the form
F1 (u) W k1 , p1 ≤ C F2 (u) W k2 , p2 F3 (u) W k3 , p3 · · · FN (u) W k N , p N , *N 1 where p11 = i=2 pi (1 ≤ p1 ≤ pi ≤ ∞), F1 is as in Lemma A.3, and Fi (i ≥ 2) are of the form Fi = Pi or Fi = D Pi with Pi a linear operator on V . We will now use these two lemmas to extend various inequalities from the standard Sobolev spaces to the weighted ones. All of these inequalities have been derived before by various authors, see for example [1, 4, 7, 8, 25, 30]. The new aspect here is that we show that the constants in the inequalities are independent of ≥ 0 and hence we find inequalities that interpolate between the weighted ( > 0) and the standard ones ( = 0). We begin with a weighted Hölder inequality. Lemma A.6. Suppose 0 > 0, δ1 = δ1 +δ2 and C > 0 independent of ∈ [0, 0 ] such that
1 1 1 = + . Then there is a constant p1 p2 p3
uv L p1 ≤ C u L p2 v L p3 δ1 ,
p
δ2 ,
δ3 ,
p
for all u ∈ L δ22, and v ∈ L δ33, . Proof. Follows directly from Hölder’s inequality and Lemma A.3. Next, we consider weighted versions of the Sobolev inequalities.
178
T. A. Oliynyk
Lemma A.7. (i) For 0 > 0 and k > n/ p there exists a constant C > 0 independent of ∈ [0, 0 ] such that
u L ∞ ≤ C u W k, p δ, δ,
0 and for > 0, u(x) = o(|x|δ ) as |x| → ∞. for all u ∈ Wδ, . Moreover u ∈ Cδ, (ii) For 0 > 0 and 1 ≤ p < n there exists a constant C > 0 independent of ∈ [0, 0 ] such that k, p
u L np/(n− p) ≤ C Du L p
δ−1,
δ,
+ u L p
δ,
1, p
for all u ∈ Wδ, . Proof. (i) The estimate u L ∞ ≤ C u W k, p for some constant C > 0 independent of ≥ 0 δ, δ,
follows from the usual Sobolev inequality u L ∞ ≤ C u W k, p (k > n/ p) and Lemma A.3. Since · W k, p for > 0 is equivalent to · W k, p , the statement δ,
δ,1
u(x) = o(|x|δ ) as |x| → ∞ for > 0 follows from Theorem 1.2 in [1]. (ii) Follows from Lemma A.4 and the Sobolev inequality u L np/(n− p) ≤ C Du L p which holds for all u ∈ W 1, p where 1 ≤ p < n. In addition to the Sobolev inequalities, we will also require weighted versions of the multiplication and Moser inequalities. We first consider the multiplication inequalities. Lemma A.8. Suppose 0 > 0, 1 ≤ p < ∞, k1 , k2 ≥ k3 , k3 < k1 +k2 −n/ p, δ1 +δ2 ≤ δ3 , and V1 × V2 → V3 : (u, v) → uv is a multiplication. Then there exists a constant C > 0 independent of ∈ [0, 0 ] such that
uv W k3 , p ≤ C u W k1 , p v W k2 , p δ3 ,
k ,p
δ1 ,
δ2 ,
k ,p
for all u ∈ Wδ11, and v ∈ Wδ22, . Proof. This proof does not follow directly from Lemma A.3, but can be proved in a simlar fashion. To see this first recall the Sobolev mlutiplication inequality
uv W k3 , p ≤ C u W k1 , p v W k2 , p
(A.30)
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179
which holds for 1 ≤ p < ∞, k1 , k2 ≥ k3 , and k3 < k1 + k2 − n/ p. So ⎛ ⎞1/ p ∞
uv W k3 , p = ⎝ φ0 uv W k3 , p + p
δ3
2− pδ3 ( j−1) S j (φ j uv) W k3 , p ⎠ p
j=1
p ≤ C φ0 u(φ0 + φ1 )v W k1 , p ∞
+
2− pδ3 ( j−1) S j (φ j u)S j
φ j+k v W k1 , p ⎠ p
j=1
k=−1
∞
p/2 ≤ C φ0 u(φ0 + φ1 )v W k3 , p +
⎞2/ p
φ j+k v W k1 , p ⎠ p/2
j=1
+
1
2−( p/2)δ3 ( j−1) S j (φ j u)S j
k=−1
p/2 p/2 ≤ C φ0 u W k1 , p φ0 v W k2 , p ∞
⎞1/ p
1
⎞2/ p
2−( p/2)δ1 ( j−1) S j (φ j u) W k1 , p 2−( p/2)δ2 ( j−1) S j (φ j v) W k2 , p ⎠ p/2
j=1
⎛ ≤C
⎝ φ0 u p k , p W 1
∞
+
⎞1/ p p 2− pδ1 ( j−1) S j (φ j u) W k1 , p ⎠
j=1
⎛ × ⎝ φ0 v W k2 , p + p
∞
⎞1/ p 2− pδ2 ( j−1) S j (φ j v) W k2 , p ⎠ p
j=1
≤ C u W k1 , p v W k2 , p , δ1
δ2
where in deriving the third, fourth, and fifth lines we used (A.25), (A.30), and (A.26), respectively. Lemma A.9. (i) If 0 > 0 and δ1 ≥ max{δ2 + δ3 , δ4 + δ5 }, then there exists a constant C > 0 independent of ∈ [0, 0 ] such that ∞
uv H k ≤ C u H k v L ∞ +
v
u L δ , Hk δ , δ1 ,
δ2 ,
3
δ4 ,
5
k ∞ for all u ∈ Hδk2 , ∩ L ∞ δ5 , and v ∈ Hδ4 , ∩ L δ3 , . (ii) If 0 > 0 and δ1 ≥ max{δ2 + δ3 , δ4 + δ5 }, then there exists a constant C > 0 independent of ∈ [0, 0 ] such that
v L ∞
[D α , u]v L 2 ≤C
Du H k−1 + u L 2 δ3 , δ2 , δ1 −|I |, δ2 −1,
∞
v + Du L ∞ + u k−1 L δ , H δ −1, 4
4
δ5 ,
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T. A. Oliynyk
for all |α| ≤ k, u ∈ Hδk2 , ∩ Wδ1,∞ and v ∈ Hδk−1 ∩ L∞ δ3 , . 4 , 5 ,
m (iii) Suppose 0 > 0, F ∈ C (V, R ) is a map that satisfies D F ∈ Cbk−1 (V, Rm ), and 1 ≤ |α| ≤ k. Then there exists a C > 0 independent of ∈ [0, 0 ] such that
≤ C D F C k−1 u k−1
D α F(u) L 2 L ∞ Du H k−1 + u L 2 δ−|α|,
δ,
δ−1,
b
k ∩ L ∞. for all u ∈ Hδ, (iv) Suppose 0 > 0 and F ∈ Cbk (V, Rm ). Then there exists a C > 0 independent of ∈ [0, 0 ] such that
F(u) H k ≤ C F C k (1 + u k−1 L ∞ ) u H k δ,
δ,
b
k ∩ L ∞. for all u ∈ Hδ,
Proof. Inequalities (i)–(iv) follow directly from (A.24), Lemmas A.3 and A.4, and the following standard Sobolev inequalities: (i) uv H k ≤ C u H k v L ∞ + v H k u L ∞ for all u ∈ H k ∩ L ∞ and v ∈ H k ∩ L ∞. (ii) [D α , u]v L 2 ≤ C Du H k−1 v L ∞ + Du L ∞ v H k−1 for all |α| ≤ k, u ∈ H k ∩ W 1,∞ and v ∈ H k−1 ∩ L ∞ . (iii) Suppose F ∈ C (V, Rm ) is a map that satisfies D F ∈ Cbk−1 (V, Rm ) and 1 ≤ k |α| ≤ k. Then ∂ α F(u) L 2 ≤ C D F C k−1 u k−1 L ∞ Du H k−1 for all u ∈ H ∩ b L ∞. (iv) Suppose F ∈ Cbk (V, Rm ). Then F(u) H k ≤ C F C k (1 + u k−1 L ∞ ) u H k for all b
u ∈ H k ∩ L ∞.
Note that we have used · L ∞ = · L ∞ . 0,
In addition to the Moser inequalities, we also need to know when the map u → F(u) is locally Lipschitz on Hδk . Lemma A.10. Suppose 0 > 0, F ∈ Cb (V, R), F(0) = 0, δ ≤ 0, and k ≤ , and k > n/2. Then for each R > 0 there exists a C > 0 independent of ∈ [0, 0 ] such that
F(u 1 ) − F(u 2 ) H k ≤ C u 1 − u 2 H k δ,
δ,
Proof. See the proof of Lemma B.6 in [30].
k for all u 1 , u 2 ∈ B R (Hδ, ).
We conclude this section with a lemma comparing the norms · L p and · L p . δ
Lemma A.11. (i) If δ ≤ −n/ p, 1 ≤ p ≤ ∞, and 0 ≤ ≤ 1, then −δ−n/ p u L p ≤ u L p ≤ u L p δ
p
for all u ∈ L δ .
δ,
δ
δ,
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181
(ii) If −n/ p < δ, 1 ≤ p < ∞, and 0 < ≤ 1, then
u L p ≤ u L p ≤ −δ−n/ p u L p δ
δ,
δ
p
for all u ∈ L δ . Proof. (i) By assumption 0 ≤ ≤ 1, and so we have σ1 (x) ≤ σ (x) ≤ σ1 (x) −δ−n/ p for all x ∈ Rn . By assumption −δ − n/ p > 0 and so we get −δ−n/ p σ1 ≤ −δ−np −δ−np ≤ σ1 . Therefore, directly from the definition of the weighted norm, we σ find −δ−n/ p u L p ≤ u L p ≤ u L p . Part (ii) is proved in a similar fashion. δ
δ,
δ
B. Quasilinear Symmetric Hyperbolic Systems In this section we establish a local existence and uniqueness theorem for a particular form of the quasilinear symmetric hyperbolic system on the weighted Sobolev spaces Hδk . In [30], we proved a local existence and uniqueness theorem for quasilinear parabolic systems on the Hδk spaces by adapting the approach of Taylor [39] (see Theorem 7.2, p. 330, and Proposition 7.7, p. 334) which is based on using mollifiers to construct a sequence of approximate solutions and then showing that the sequence converges to a true solution. Here, we will again follow the same approach for quasilinear symmetric hyperbolic systems and adapt the local existence and uniqueness theorems of Taylor (see Proposition 2.1, p. 370) to work on the weighted Sobolev spaces. We will only provide a brief sketch of the proof since the proof is very similar to the one in [30] and the details can easily be filled in by the reader. Related existence results have been derived independently in [4] using a different method. The hyperbolic equations that we will consider are of the form b0 (u, v)∂t v = b j (u, v)∂i v + f (u, v)v + h, v|t=0 = v0 ,
(B.1) (B.2)
where (i) the map u = u(t, x) is Rr -valued while the maps v = v(t, x) and h = h(t, x) are Rm -valued, (ii) b0 , b j , f ∈ Cbk (Rr × Rm , Mm×m ) ( j = 1, . . . , n), (iii) b0 and b j ( j = 1, . . . , n) are symmetric, and (iv) there exists a constant ω > 0 such that b0 (ξ1 , ξ2 ) ≥ ω1I m×m for all (ξ1 , ξ2 ) ∈ Rr × Rm .
(B.3)
n B.1. Galerkin method. Let j ∈ C∞ 0 (R ) be any function that satisfies j ≥ 0, j (x) = 0 ! n for |x| ≥ 1, and Rn j (x) d x = 1. Following the standard prescription, we construct from j the mollifier jη (x) := η−n j (x/η) (η > 0) and the smoothing operator
Jη (u)(x) := j ∗ u(x) =
Rn
jη (x − y)u(y) d n y.
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T. A. Oliynyk
Following Taylor ( [39], Ch. 16, Sects. 1 & 2), we first solve the approximating equation b0 (u, Jη vη )∂t vη = Jη b j (u, Jη vη )∂i Jη vη + Jη f (u, Jη vη )Jη vη + Jη h, vη |t=0 = v0 ,
(B.4) (B.5)
and later show that the solutions v converge to a solution of (B.1)–(B.2) as η → 0. Proposition B.1. Suppose T1 , T2 > 0, η > 0, δ ≤ γ ≤ 0, k > n/2, v0 ∈ Hδk , u ∈ C 0 ([−T1 , T2 ], Hγk ), and h ∈ C 0 ([−T1 , T2 ], Hδk ) for some T > 0. Then there exists a T∗ > 0 (T∗ < T1 , T2 ) and a unique vη ∈ C 1 ((−T∗ , T∗ ), Hδk ) that solves the initial value problem (B.4)–(B.5). Moreover if sup0≤t 0 and define F(t, v) := (b0 (u, Jη v))−1 (Jη b j (u, Jη v)∂i Jη v + Jη f (u, Jη v)Jη v + Jη h). Then the approximating equations (B.4)–(B.5) can be written as the first order differential equation v˙ = F(v) ; v(0) = v0 on Hδk . If we can show that F is continuous and is Lipshitz in a neighborhood of v0 in Hδk , then the proof follows immediately from standard existence, uniqueness, and continuation theorems for ODEs on Banach spaces. To prove that F is locally Lipshitz, we first prove the following lemma. Lemma B.2. Suppose δ ≤ γ ≤ 0, > n/2 and that f ∈ Cb (Rm × Rm , Mm×m ). Then for each u ∈ Hγ and R > 0 there exists a constant C > 0 such that
f (u, v1 )v1 − f (u, v2 )v2 H ≤ C v1 − v2 H δ
δ
for all v1 , v2 ∈ B R (Hδ ). Proof. Let f(0,0) = c and g(x, y) = f (x, y) − c so that g(0, 0) = 0. Then f (u, v1 )v1 − f (u, v2 )v2 = c(v1 − v2 ) + (g(u, v1 ) − g(u, v2 ))v1 + g(u, v2 )(v1 − v2 ). Since γ ≤ 0 and > n/2, we get from Lemma A.8 that
f (u, v1 )v1 − f (u, v2 )v2 H ≤ C 1 + g(u, v2 ) Hγ v1 − v2 H δ
+ v1 H g(u, v1 ) − g(u, v2 ) Hγ .
δ
δ
By Lemma A.10, Lemmas A.7 and A.9, and (A.24), we get from the above inequality that
f (u, v1 )v1 − f (u, v2 )v2 H ≤ C( u Hγ , v1 H , v2 H ) v1 − v2 H , δ
δ
where P(y1 , y2 , y3 ) is a polynomial. This proves the lemma.
δ
δ
Using Lemma A.7 of [30], it is not difficult to prove the following variation of the above lemma.
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183
Lemma B.3. Suppose δ ≤ γ ≤ 0, η > 0, > n/2 and that f ∈ Cb (Rm × Rm , Mm×m ). Then for each u ∈ Hγ and R > 0 there exist a constant C > 0 such that
f (u, Jη v1 )D Jη v1 − f (u, Jη v2 )D Jη v2 H ≤ C v1 − v2 H δ
for all v1 , v2 ∈
δ
B R (Hδ ).
The proof now follows easily from the above lemmas, Lemma A.7 of [30], and the estimates of Appendix A, which show that for any R > 0 the map F : ([−T1 , T2 ] × k ) → H k is continuous and moreover there exists a constant C > 0 such that B R (Hδ−1 δ−1
F(t, v1 ) − F(t, v2 ) H k ≤ C v1 − v2 H k for all v1 , v2 ∈ B R (Hδk ). δ
δ
B.2. Energy estimates. Fix k > n/2 + 1. By Proposition B.1, we have a sequence of solutions vη ∈ C 1 ([−T (η), T (η)], Hδk ) (0 < T (η) ≤ T1 , T2 ) to the approximating Eqs. (B.4)–(B.5). The goal is to derive bounds on vη in the Hδk spaces independent of η. To do this, we use energy estimates which we now describe. Lemma B.4. Suppose a 0 ∈ C 1 ([0, τ ], W 1,∞ ), a j ∈ C 0 ([0, τ ], W 1,∞ ), f ∈ C 0 ([0, τ ], L 2λ ) and that w ∈ C 1 ([0, τ ], L 2λ ) satisfies the equation a 0 ∂t w = Jη a j ∂ j Jη w + g. Then there exists a constant C > 0 independent of η > 0 such that d w|a 0 w 2 ≤ C (1 + div a L ∞ + a L ∞ ) w 2L 2 + g L 2 w L 2 , λ λ Lλ λ dt where div a = ∂t a 0 + ∂ j a j and a = (a 1 , . . . , a n ). Proof. First, we have d w|a 0 w 2 = 2 w|a 0 ∂t w 2 + w|∂t a 0 w 2 Lλ Lλ Lλ dt j = 2 w|Jη a ∂ j Jη w 2 + 2 w|g L 2 + w|∂t a 0 w Lλ
λ
L 2λ
.
Letting Jη† denote the adjoint of Jη with respect to the inner-product (A.4), we can write the above expression as d w|a 0 w 2 = 2 Jη† w|a j ∂ j Jη w 2 + 2 w|g L 2 + w|∂t a 0 w 2 . (B.6) λ Lλ Lλ Lλ dt Integration by parts shows that Jη† w|a j ∂ j Jη w 2 = − ∂ j Jη† w|a j Jη w Lλ
L 2λ
− Jη† w|(∂ j a j + a j ρ −1 ∂ j ρ)Jη w
L 2λ
, (B.7)
where ρ = σ1−2λ−n . Since ρ −1 ∂ j ρ L ∞ < ∞, together Lemmas B.7 and B.8 of [30] and (B.7) imply that a L ∞ ) w 2L 2 . (B.8) Jη† w|a j ∂ j Jη w 2 ≤ − ∂ j Jη w|a j Jη w 2 +C(1 + ∂i a i L ∞ + Lλ
Lλ
λ
184
T. A. Oliynyk
Again integrating by parts and using Lemma B.8 of [30], we find that a L ∞ ) w 2L 2 . − ∂ j Jη w|a j Jη w 2 ≤ C(1 + ∂i a i L ∞ + λ
Lλ
(B.9)
The proof now follows from the Cauchy-Schwartz inequality and Eqs. (B.6), (B.8), and (B.9). Let vηα = D α vη , bη0 = b0 (u, Jη vη ), bη = b j (u, Jη vη ) and f η = f (u, Jη vη )Jη vη . The evolution equation (B.4) implies that j
∂t vη = (bη0 )−1 Jη bηj ∂ j Jη vη + (bη0 )−1 f η + (bη0 )−1 h.
(B.10)
Differentiating this equation yields bη0 ∂t vηα = Jη bηj ∂ j Jη vηα + g α ,
(B.11)
where
g α = bη0 [D α , (bη0 )−1 Jη bηj ]∂ j Jη vη + bη0 D α (bη0 )−1 Jη f η +bη0 D α (bη0 )−1 Jη h . (B.12)
To simplify the following estimates, we will assume that b j (0, 0) = 0. It is not difficult to treat the case where b j (0, 0) = 0. Recalling that δ ≤ γ ≤ 0 and k > n/2 + 1, we get from the calculus inequalities of Appendix A and Lemma A.7 of [30] the following estimate
bη0 [D α , (bη0 )−1 Jη bηj ]∂ j Jη vη L 2 ≤ bη0 L ∞ [D α , (bη0 )−1 Jη bηj ]∂ j Jη vη L 2 δ−|α| δ−|α|
0 −1 j 0 −1 j ≤ C (bη ) Jη bη H k ∂ j Jη vη L ∞ + (bη ) Jη bη W 1,∞ ∂ j Jη vη H k−1 δ−1 0 δ−1
≤ C (1 + ( u L ∞ + vη L ∞ )k−1 ) u H k + vη H k vη W 1,∞ 0 0 δ +(1 + u W 1,∞ + vη W 1,∞ ) vη H k , δ
where C is independent of η. By the Sobolev inequality (Lemma A.7) we have
u W 1,∞ ≤ C u Hηk , v W 1,∞ + v W 1,∞ ≤ C vη H k , δ
δ
and hence
bη0 [D α , (bη0 )−1 Jη bηj ]∂ j Jη vη L 2
δ−|α|
≤ P( u Hηk , vη H k ) δ
gα
for a η independent polynomial P(y1 , y2 ) . The other terms in can be estimated in a similar fashion to get
(B.13) ≤ P u Hγk , vη H k , h H k ,
g α L 2 δ−|α|
δ
δ
where as above P(y1 , y2 , y3 ) is an η independent polynomial . It can also be shown using the calculus inequalities and (B.10) that
div b L ∞ ≤ P( u Hγk , vη H k , h H k ). δ
δ
(B.14)
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185
Finally, we note that L ∞ ≤ C.
b Next, if we define
|||vη |||2k,δ := |α|≤k
D α vη |bη0 D α vη
(B.15) L 2δ−|α|
,
then by (B.3) and (B.15) there exists a constant C > 0 independent of η such that C −1 vη H k ≤ |||vη |||k,δ ≤ C vη H k . δ
δ
(B.16)
Since sup0≤t≤T u(t) Hγk < ∞ and sup0≤t≤T ] h(t) H k < ∞, Lemma (B.4) and δ (B.13), (B.14), (B.15), and (B.16) imply that d |||vη |||2k,δ ≤ C(|||vη |||k,δ )|||vη |||k,δ dt
(B.17)
or equivalently d |||vη |||k,δ ≤ P(|||vη |||k,δ ) dt for a polynomial P(y) with positive coefficients that are independent of η > 0. By Gronwall’s inequality, (B.17), and Proposition B.1, this implies that there exists constants T∗ , K > 0, both independent of η > 0, such that T (η) ≥ T∗ and sup vη (t) H k ≤ K .
0≤t≤T∗
δ
(B.18)
Using the time reversed version of the equation (i.e. sending t → −t) we also get, shrinking T∗ if necessary, that sup
−T∗ ≤t≤0
vη (t) H k ≤ K . δ
(B.19)
Finally, from (B.10), (B.18), (B.19), Lemma A.7 of [30], Lemmas A.7 and A.9, and (A.24), we see, increasing K if necessary, that sup
−T∗ ≤t≤T∗
∂t vη (t) H k−1 ≤ K . δ
(B.20)
B.3. Local existence and uniqueness. To get local existence following the approach of Taylor (see Theorem 1.2, p. 362 in [39]), we let η 0 and use the bounds (B.18)–(B.20) obtained from the energy estimates to extract a weakly convergent subsequence of vη which has a limit that solves the initial value problem (B.1)–(B.2). Since the proof is very similar to that of Theorem B.2 in [30], we omit the details. Proposition B.5. Suppose T1 , T2 > 0, δ ≤ γ ≤ 0, k > n/2 + 1, v0 ∈ Hδk , u ∈ C 0 ([−T1 , T2 ], Hγk ) and h ∈ C 0 ([−T1 , T2 ], Hδk ). Then there exists a T∗ > 0 T∗ < min{T1 , T2 } and a v ∈ L ∞ ((−T∗ , T∗ ), Hδk ) ∩ Lip((−T∗ , T∗ ), Hδk−1 ) that solves the initial value problem (B.1)–(B.2).
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Using the estimates of Appendix B of [30] and of Appendix A and B.2 of this paper, it is not difficult to adapt the proofs of Propositions 1.3–1.5, pp. 364–365, in [39] to get the following theorem. k )∩ Theorem B.6. The solution v from Proposition B.5 is unique in L ∞ ((−T1 , T2 ), Hloc k−1 Lip((−T1 , T2 ), Hloc ) and satisfies the additional regularity
v ∈ C 0 ((−T∗ , T∗ ), Hδk ) ∩ C 1 ((−T∗ , T∗ ), Hδk−1 ). Moreover, if T∗ < T2 and sup0≤T 0, δ ≤ γ ≤ 0, k > n/2 + 1, v0 ∈ Hδk , u ∈ C 0 ([−T1 , T2 ], Hγk ) and h ∈ C 0 ([−T1 , T2 ], Hδk ). Then the initial value problem b0 (u)∂t v = b j (u)∂i v + f (u)v + h, v|t=0 = v0
(B.21) (B.22)
has a solution v ∈ C 0 ([−T1 , T2 ], Hδk ) ∩ C 1 ([−T1 , T2 ], Hδk−1 )) k ) ∩ Lip((−T , T ), H k−1 ). that is unique in L ∞ ((−T1 , T2 ), Hloc 1 2 loc
Let [n/2] denote the largest integer with [n/2] ≤ n/2 and k0 = [n/2] + 2. Then differentiating the solution from Theorem B.6, with respect to t, and using Proposition B.7 yields the following result. s
k− Corollary B.8. Suppose k = k0 + s, u ∈
=0 C ([−T1 , T2 ], Hγ ) and h ∈ s k−
). Then the solution from B.5 satisfies the additional regu =0 C ([−T1 , T2 ], Hδ larity v∈
s+1
=0
C ((−T∗ , T∗ ), Hδk− ).
References 1. Bartnik, R.: The Mass of an Asymptotically Flat Manifold. Comm. Pure Appl. Math. 39, 661–693 (1986) 2. Blanchet, L., Damour, T.: Radiative gravitational fields in general relativity. I. General structure of the field outside the source. Phil. Trans. Roy. Soc. Lond. A 320, 379–430 (1986) 3. Blanchet, L., Faye, G., Nissanke, S.: On the structure of the post-Newtonian expansion in general relativity. Phys. Rev. D 72, 44024 (2005) 4. Brauer, U., Karp, L.: Local existence of classical solutions for the Einstein-Euler system using weighted Sobolev spaces of fractional order. Preprint, available at http://www.math.uni-potsdam.de/prof/ a_partdiff/prep/2006_17.pdf, 2006 5. Browning, G., Kreiss, H.O.: Problems with different time scales for nonlinear partial differential equations. SIAM J. Appl. Math. 42, 704–718 (1982)
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38. Schochet, S.: Asymptotics for symmetric hyperbolic systems with a large parameter. J. Differential Equations 75, 1–27 (1988) 39. Taylor, M.E.: Partial differential equations III, nonlinear equations. New York: Springer, 1996 40. Walton, R.A.: A symmetric hyperbolic structure for isentropic relativistic perfect fluids. Houston J. Math. 31, 145–160 (2005) 41. Will, C.M.: Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. III. Radiation reaction for binary systems with spinning bodies. Phys. Rev. D 71, 084027 (2005) Communicated by G.W. Gibbons
Commun. Math. Phys. 276, 189–220 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0333-0
Communications in
Mathematical Physics
Simple Systems with Anomalous Dissipation and Energy Cascade Jonathan C. Mattingly1 , Toufic Suidan2 , Eric Vanden-Eijnden3 1 Department of Mathematics and CNCS, Duke University, Durham, NC 27708, USA.
E-mail: [email protected]
2 Mathematics Department, University of California, Santa Cruz, CA 95064, USA. E-mail: [email protected] 3 Courant Institute, New York University, New York, NY 10012, USA. E-mail: [email protected]
Received: 7 July 2006 / Accepted: 11 May 2007 Published online: 14 September 2007 – © Springer-Verlag 2007
Abstract: We analyze a class of dynamical systems of the type a˙ n (t) = cn−1 an−1 (t) − cn an+1 (t) + f n (t), n ∈ N, a0 = 0, where f n (t) is a forcing term with f n (t) = 0 only for n ≤ n < ∞ and the coupling coef ficients cn satisfy a condition ensuring the formal conservation of energy 21 n |an (t)|2 . Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients cn . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely E|an |2 scales as n −α as n → ∞. Here the exponents α depend on the coupling coefficients cn and E denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable. 1. Introduction and Main Results: Life Starts after Blow-up So little is understood about hydrodynamic turbulence that there is not even consensus on what it is. However, most physicists would agree on the following heuristic picture which has emerged from the works of Kolmogorov, Onsager, Richardson, etc. [Fri95]. In this picture, (fully developed) turbulence refers to the idealized state of an incompressible fluid described by the Navier-Stokes equations in the limit of vanishing molecular viscosity. In this limit, the Navier-Stokes equations formally reduce to the Euler equations,
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and the turbulent solutions should be the most regular solutions of the Euler equations which dissipate energy. This is referred to as anomalous dissipation and is best visualized in the Fourier representation. There it corresponds to a cascade of energy from the small wavenumbers (large spatial scales) where energy is injected (either via the initial condition or by a forcing term in the equation) towards larger and larger wavenumbers (smaller and smaller scales), up to infinity where energy should eventually be dissipated. It is also believed that the cascade of energy implies that the energy spectrum of the turbulent solutions have a power law decay in the wavenumber whose rate can be deduced by dimensional analysis and is 53 in three-dimension of space. Turbulence theory (as we shall refer to the heuristic picture above) also discusses more advanced and more controversial topics such as intermittency. But, without even going into those, most mathematicians would agree that a rigorous confirmation of the basic predictions of turbulence theory is already a tremendous challenge. The best known results on the Navier-Stokes and Euler equations which corroborate the above were obtained in [CET94, Eyi01, DR00, CR06]. These works only indicate that turbulence theory is not blatantly inconsistent. Simpler models, such as the randomly forced Burgers equation or Kraichnan’s model of passive scalar advection (see e.g. [E01, FGV01] for reviews), have also been used to demonstrate that parts of turbulence theory make sense in terms of anomalous dissipation of the weak solutions of the inviscid Burgers equation and the spectrum of energy of the solutions that this implies. Even these simple models remain surprisingly complicated to analyze and a full characterization of the statistical properties of their solutions is still lacking. One of the purposes of the present paper is to illustrate turbulence theory on even simpler models. Many (if not most) of the realistic features have been neglected in our models. Yet, the models possess a rich range of behaviors which depend on the details of the interactions. They provide a simple class of exactly solvable models which can be useful in understanding the inner workings of some energy transfer mechanisms. The solutions of these models are also consistent with much of the claims of turbulence theory. In a way, they offer a setting for the skeptical mathematician to understand the motivation behind these claims, and if this paper succeeds in doing this, we will have achieved our main goal. Next, we introduce the models that we will investigate and we summarize the principal results of the paper. As we will see, the most interesting and meaningful solutions of these models are solutions which have blown-up, such that they have become infinite in some norm. This justifies our claim that “life starts after blow-up”: Disregarding these solutions as nonsensical, as one may be tempted to do at first sight, would, in fact, completely miss the most interesting phenomena displayed by the models, namely that of anomalous dissipation (see Sect. 2 below). Regarding this last point, we note that the concept of anomalous dissipation developed in the present paper has been recently used and nicely generalized in the context of a nonlinear dyadic shell model in Refs. [CFP06a, CFP06b].
1.1. A linear shell model. Consider the equation a˙ n (t) = c (n − 1)an−1 (t) − nan+1 (t) for n ∈ N with the boundary condition a0 (t) = 0 for all t. If c > 0, we can rescale time to fix c = 1; observe also that if an satisfies the equations with the parameter c < 0 then aˆ n (t) = (−1)n+1 an (t) satisfies the equations with parameter |c|.
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In light of these considerations, we set c = 1 and focus our attention on a˙ n (t) = (n − 1)an−1 (t) − nan+1 (t)
(1)
for n ∈ N with the boundary condition a0 (t) = 0 for all t. Although we will see that this calculation is not always correct, on the formal level one has that ∞ ∞ 1d |an (t)|2 = [(n − 1)an−1 (t)an (t) − nan+1 (t)an (t)] 2 dt n=1
=
n=1 ∞
nan (t)an+1 (t) −
n=1
∞
(2) nan (t)an+1 (t) = 0 .
n=1
The second equality is only formal as it assumes that the sum ∞ n=1 nan (t)an+1 (t) is finite and absolutely convergent. To understand this further, consider the evolution of the partial sum n≤N |an (t)|2 . For N ∈ N, 1d |an (t)|2 = −N a N (t)a N +1 (t). 2 dt
(3)
n≤N
The validity of (2) necessitates lim N a N (t)a N +1 (t) = 0.
N →∞
(4)
If this condition is not satisfied, then the formal manipulation in (2) does not hold and the seemingly conservative coupling term in (1) may become a source of anomalous dissipation. We make the concept of anomalous dissipation precise in Sect. 2. But, roughly speaking, it is when seemingly conservative terms have a dissipative effect on the system. In the context of Eq. (1), anomalous dissipation seems to require that the limit as N → ∞ of the right hand side of (3) be negative. In other words, Eq. (1) is dissipative at time t if lim inf nan (t)an+1 (t) > 0. (5) n→∞
If we make the reasonable assumption that limn an+1 /an ∈ (0, ∞), then from (5) the solution of Eq. (1) will be dissipative with a finite dissipation rate provided that 1 an (t) √ n
as n → ∞.1
(6)
At this point some readers may be skeptical since one typically considers equations like (1) with initial data in 2 , the space of square-summable sequences. However, we will see (Theorem 1 in Sect. 3) that Eq. (1) has solutions which exist for all time provided lim sup |an (0)|1/n ≤ 1 . n→∞
1 We say that f (m) g(m) as m → ∞ if there exists an m and c ≥ 1 so that if m > m then 1 1 1 g(m) ≤ f (m) ≤ cg(m). Similarly, we say that f (m) ∼ g(m) as m → m if lim f (m)/g(m) = 1 as 0 c m → m0.
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This condition admits a large class of initial conditions including those which scale as (6). In Sect. 3, we will also see that (1) possesses a wide verity of behavior including conservative, dissipative, and explosive solutions. It might be tempting to dismiss these non-conservative solutions as non-physical solutions arising from pathological data. We now discuss why this is not the case. Consider Eq. (1) with a white-noise forcing in the first coordinate: a˙ n (t) = (n − 1)an−1 (t) − nan+1 (t) + 1n=1 W˙ (t),
(7)
where 1n=m is 1 if n = m and 0 otherwise, and W (t) denotes a standard Brownian motion, i.e. Gaussian process with mean zero and covariance EW (t)W (s) = min(t, s). If one were to accept the formal calculations in (2), showing energy conservation, then E
∞ n=1
|an (t)|2 = E
∞
|an (0)|2 + t
n=1
if the energy is initially finite. Hence, in the forced system energy seems to grow linearly with time and at t = ∞ one expects the system to have infinite energy. These solutions which “blow-up” (in the sense that they have infinite energy) are the most interesting and relevant. In light of the discussion above, one might expect that the energy of the system would grow to be infinite and arrange the an so that the calculation in (2) is not valid since the sum is not rearrangeable. Onsager would then predict that the system would evolve to the state in which the an decayed as fast as possible but still dissipated energy in the sense that (5) holds. √ The reasoning which leads to (6) strongly suggests that the |an | should scale as 1/ n. In fact, if the system is to reach some equilibrium the effect of the dissipation must exactly balance that of the forcing. Specifically, in the stochastic setting when the forcing is W˙ (t)1n=1 , limn n E(an (t)an+1 (t)) → 1 as t → ∞. All of these conclusions turn out to be correct. In particular, in Sect. 6 we prove that if n |an (0)|2 < ∞ then the solutions converge to a unique random variable a ∗∗ = (a1∗∗ , a2∗∗ , · · · ) which is Gaussian with mean zero and whose distribution is the unique stationary measure for the system. Furthermore, this equilibrium state has a √ structure which is consistent with the anticipated 1/ n scaling: ∗∗ = lim E an (t)am (t) = E an∗∗ am
t→∞
1 . n+m−1
(8)
We also show that similar behaviors are observed with a different type of forcing. In particular, if the forcing is constant, a˙ n (t) = (n − 1)an−1 (t) − nan+1 (t) + 1n=1 , then the system evolves to a unique steady state an∗ which scales as √ π Γ ( n2 ) π ∗ . ∼ lim an (t) = an = n−1 t→∞ n nΓ ( 2 )
(9)
(10)
In summary, we see that the forced systems’ energy grows linearly with time if the energy is initially finite. Asymptotically, the system rearranges itself so that it reaches a state which dissipates energy at t = ∞. This state is chosen so that the dissipation rate matches the energy flux√into the system from the forcing. Since the energy flux is finite, this leaves |an | 1/ n as the only choice. A slower decay rate would produce an infinite rate of dissipation and a faster decay rate would produce a system which conserved energy since the limit in (5) would be zero.
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1.2. A second linear shell model. We introduce a second model which also exhibits interesting but different “blow-up” behavior. Consider b˙n = (n − 1)(n − 21 )bn−1 − n(n + 21 )bn+1
(11)
for n ∈ N with the boundary condition b0 (t) = 0 for all t. As in the previous subsection, the unforced equation formally conserves energy since ∞ ∞ 1d 2 bn (t) = [(n − 1)(n − 21 )bn−1 (t)bn (t) − n(n + 21 )bn+1 (t)bn (t)] = 0. (12) 2 dt n=1
n=1
This equality (as in the case of Eq. (1)) is only formal since, in general, the sum cannot be rearranged. As before, to gain insight we consider the partial sums. For N ∈ N, 1d 2 bn (t) = −N (N + 21 )b N (t)b N +1 (t). 2 dt n≤N
With this in mind, (11) will be called conservative if lim N (N + 21 )b N (t)b N +1 (t) = 0,
N →∞
and dissipative if
lim inf N (N + 21 )b N (t)b N +1 (t) > 0. N →∞
Unlike the case of Eq. (1), if one assumes that lim N →∞ b N +1 /b N exists and is in (0, ∞), then the solution of Eq. (11) will be dissipative if bn 1/n as n → ∞. 2 A solution satisfying bn (t) 1/n (if one exists) has finite energy: ∞ n=1 bn (t) < ∞. Thus, this model differs from the first example in that the system can dissipate energy even when the total energy is finite. While we do not prove a general existence result as broad as for (1), we do show (in Theorem 16) that the dynamics for (11) are well defined if ∞ (−1)n bn (0)
(13)
n=1
converges. This is sufficient for our needs: In particular, it covers the case when bn 1/n as n → ∞. The differences between the first and second models are greater than simply the scaling. When started with initial conditions having finite energy the first model conserves energy. In fact the regularity at time t is the same as the regularity of the initial condition. In contrast, if we start (11) with initial data satisfying (13) (and hence with finite energy), the energy decays with time. Furthermore, for almost every t > 0 one has that bn (t) 1/n as n → ∞ and there exists T , depending on the initial data, so that if t > s > T then ∞ ∞ ∞ bn2 (t) < bn2 (s) < bn2 (T ) < ∞ and
n=1
bn2 (t)
n=1
n=1
→ 0 and t → ∞. Turning to the forced setting, consider b˙n = (n − 1)(n − 21 )bn−1 − n(n + 21 )bn+1 + f (t)1n=1 .
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When f (t) = W˙ (t) then ∗∗ Ebn∗∗ bn+m
If f (t) = 1, we have bn∗
1 as n → ∞. n(n + m) 1 as n → ∞. n
1.3. Inviscid limits of the first model. In practice, one is often interested in understanding the limit of equations when the explicit sources of dissipation are removed. To explore this question we investigate Eq. (1) with the addition of an overtly dissipative term and then study the limit as the dissipation is removed. To understand our motivation, recall that we have seen that if the first model is started with finite energy initial data then the formal calculation presented in (2) is valid for all finite times as the energy must be infinity for (2) to fail. Yet, as time tends to infinity, the forced system converges to a steady state with infinite energy for which the calculation presented in (2) fails. In contrast, in the second model the analogous calculation, given in (12), fails at almost every positive time since b(t) 1/n as n → ∞ for almost every t > 0. Since the coupling term produces dissipation at finite times in the second model, it is most interesting to study the effect of extra, explicit dissipation in the first model. To this end, we consider the stochastically forced version of the first model with extra, explicit dissipation sufficient to keep the expected energy of the system finite for all times. In particular, the calculation in (2) is valid in the equilibrium state. We are interested in the structure of this invariant state and how it converges to the steady state without the explicit dissipation (as the dissipation is removed). We will consider two cases, one for which the dissipative term is lower order than the coupling term and one for which it is equal order. Specifically, for p ∈ {0, 1} and any ν > 0, consider the equation α˙ n,ν = −2ν(n − 1) p αn,ν + (n − 1)αn−1,ν − nαn+1,ν + 1n=1 W˙ (t). The case p = 0 corresponds to the uniform damping of all the modes and is analogous to what is called Eckmann damping in the context of fluid mechanics since it is subordinate to the nonlinearity. When p = 1 the intensity of the damping increases with the mode index, and it behaves as a viscous term in the language of fluid mechanics since it is the leading order term. As in the previous2 examples, we force the first coordinate with white noise. Assuming that |αn,ν (0)| < ∞, it is straightforward to see that |αn,ν (t)|2 stays finite and uniformly bounded in time for any ν > 0. Hence, the system remains conservative for all time. Furthermore, as t → ∞ the system converges ∗∗ whose distribution is the unique stationary measure for the to a random variable αn,ν system. Direct calculation in the spirit of (2) shows that E
∞ 1 ∗∗ 2 (n − 1) p |αn,ν | = . ν n=1
Thus, there is no anomalous dissipation in the system: All of the dissipation which balances the forcing comes from the term −2ν(n − 1) p αn (t). In Sect. 8, we will see that for p ∈ {0, 1}, ∗∗ E[αn,ν − an∗∗ ]2 → 0 as ν → 0.
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Both of these steady states are Gaussian with mean zero. The way in which the ∗∗ converges (as ν → 0) to that of the ν = 0 steady state a ∗∗ is different variance of αn,ν n for the two values of p we consider. When p = 0 one has ∗∗ 2 E[αn,ν ] = 21−2ν Γ (1 + 2ν)
1 Γ (2n + 1 − 2ν) 1+2ν . Γ (2n + 2) n
(14)
This shows that variances still decay like a power of n. Notice that for all ν > 0 the total energy is finite. When p = 1 we do not obtain an exact formula but rather that ∗∗ 2 1 1 1 κ2 ≤ E αn,ν , ≤ 2n 2n+2 (κ + ν) 2n + 1 κ 2n + 1
(15)
∗∗ behave as the limiting a ∗∗ for where κ 2 = 1 + ν 2 . Since κ > 1 when ν > 0, the αn,ν n small n but decay exponentially for large n.
1.4. Organization. The remainder of the paper is concerned with proving the statements made in this section. Section 2 contains a precise discussion of the concept of anomalous dissipation. In Sect. 3, we return to the first of the two models introduced in Sect. 1 and illustrate the range of possible dynamics by considering specific initial conditions for which the system can be explicitly solved. In Sects. 4 and 5, we prove the existence of solutions and describe their properties for a wide range of initial data. In Sect. 7, we give the proofs of all of the preceding results. Section 6 discusses the forced setting for the first example and Sect. 8 discusses its inviscid limit. Finally, in Sect. 9 we discuss the second model introduced in Sect. 1: we first describe the qualitative behavior of solutions; then, we prove existence and uniqueness results both in the forced and unforced situations. We note that some of the results about anomalous dissipation and the scaling of the solutions can also be obtained by formally taking the continuous limit in n of (1) and (11). In this limit, these equation formally reduce to hyperbolic conservation laws which were analyzed in [Sri05]. 2. Preliminary: Definition of Anomalous Dissipation The concepts of energy conservation, dissipation, and explosion are straightforward when the total energy of the system is finite. A system is conservative if the energy does not change with time. A system is dissipative (or explosive) if the total energy decreases (or increases) with time. However, as the example in the previous section showed, it is possible to have solutions which one might call dissipative even though the total energy is infinite. We give a definition of the above terms which can be applied to situations where the total energy is infinite. Given a time-dependent sequence, {an (t)}n∈N , define the energy in the block M to N , M < N , by N E M,N (t) = |an (t)|2 . n=M
A given block E M,N is dissipative at time t if E˙M,N (t) < 0. Similarly, we will say it is explosive if E˙M,N (t) > 0. If E˙M,N (t) = 0 then we say the block is conservative at
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time t. If E˙M,N (t) = 0 for all M, N , and t, then the system is at a fixed point. (Note that this is consistent with Example 4 in the next section.) We will say that the system is locally dissipative (locally explosive, or locally conservative) at time t if every finite block is dissipative (explosive or conservative) at time t. In contrast, we will say that a system with E0,∞ = ∞ is dissipative at time t if the limit d lim sup E0,N (t) < 0. dt N →∞ We say it is explosive at time t if lim inf N →∞
d E0,N (t) > 0. dt
When the limit exists we will refer to its absolute value as the rate of energy dissipation or the rate of energy explosion depending on the inequality which is satisfied. We say that the system with E0,∞ = ∞ is conservative at time t if lim
N →∞
d E0,N (t) = 0. dt
If we do not state time explicitly for any of these properties, we mean that the properties hold for all finite times. As the examples of the next section show, it is possible for lim N E0,N = ∞ while lim N E˙0,N = c < 0. Remark 1. It is important to notice that the above categorizations are not exhaustive. It is possible for a system not to fit into any of the categories. This is only an issue when the energy is infinite as we use the definitions at the start when the energy is finite. 3. The Rich Behavior of the First Model The system given by (1) possesses a number of interesting properties beyond those listed in the introductory section. In this section we explore the behavior through a number of examples. A relatively complete theory of the equation will be given in the two sections which follow. We begin with an existence result which covers all of the examples presented. For an infinite vector a = (a1 , a2 , · · · ) with ai ∈ R, define ρ(a) by 1 = lim sup |an |1/n . ρ(a) n→∞
(16)
(ρ(a) is simply the radius of convergence of the power series n an z n ). The following theorem gives an existence and uniqueness result for (1) sufficient for present needs. In particular, it allows initial data with infinite energy ( |an (0)|2 = ∞). A more complete description will be given in Theorem 4 of Sect. 4, where we describe the methodology for solving (1). Theorem 1. If a(0) = (a1 (0), a2 (0), · · · ) is an infinite vector of initial conditions such that ρ(a(0)) > 0, then there exists a unique solution a(t) to (1) with initial conditions a(0) which exist at least up to the time t∗ = arctanh(ρ(a) ∧ 1).
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This existence result, whose proof is given in Sect. 4, covers a wide class of initial data. The dynamical behavior of (1) is quite rich. We list a number of exact solutions which display the range of possible behaviors. Explanations of how these results are obtained will be given in Sect. 4.1. A general discussion of the qualitative properties of solutions of (1) will be given in Sect. 5. Example 1. An energy conserving pulse heading out to infinity: Fixing a1 (0) = 1 and an (0) = 0 for all n = 2, 3, . . . results in the dynamics an (t) =
tanhn−1 (t) . cosh(t)
Even though the solution ∞ decays2 to zero pointwise in n as t → ∞ it conserves energy: ∞ 2 = |a (t)| n n=1 n=1 |an (0)| = 1 for all t ≥ 0. The fact that it conserves energy is consistent with the observations in Eq. (2) and (4) because 1d N tanh N −1 (t) tanh N (t) |an (t)|2 = −N a N (t)a N +1 (t) = − →0 2 dt cosh2 (t) n≤N
as N → ∞. The dynamics of this solution can be understood as pulse moving out to larger and larger N with time while simultaneously spreading out. A simple calculation shows that a N (t) reaches its maximum at a time asymptotic to 21 log(1+4N ) as N → ∞. Hence, as t → ∞ the n for which an which is cresting at time t scales as 41 exp(2t). Example 2. Dissipative solution with finite dissipation rate: If an+1 (0) =
(2n)! 1 ∼√ for n = 0, 1, . . ., 22n (n!)2 πn
then n e−t/2 (2(n − m))!(2m)! tanhn−m (t) for n = 0, 1, . . .. an+1 (t) = √ cosh(t) m=0 22n ((n − m)!m!)2
For each fixed n, we have an (t) ∼
√
2e−t
as t → ∞,
so that the solution decays to zero pointwise in n as t → ∞. On the other hand, since for any fixed time t,
it follows that
∞
e−t/2 1 an (t) ∼ √ √ cosh t π n
n=1 |an (t)|
2
as n → ∞,
= +∞ for all time t ≥ 0 and
e−t 1d <0 |an (t)|2 = −N a N (t)a N +1 (t) → − 2 dt π cosh t n≤N
as N → ∞. For this solution, the calculation in (2) does not hold and the above inequality can be interpreted as a form of anomalous dissipation with finite dissipation rate.
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Example 3. Dissipative solution with infinite dissipation rate: If an (0) = 1 for all n = 1, 2, . . .. then an (t) = e−t for all n = 1, 2, . . . . 2 This solution decays to zero pointwise in n as t → ∞ and ∞ n=1 |an (t)| = +∞ for all t ≥ 0. Notice that 1d |an (t)|2 = −N a N (t)a N +1 (t) = −N e−2t → −∞ 2 dt n≤N
as N → ∞. The formal calculation in (2) does not hold for this solution and, in terms of the definitions of Sect. 2, we view this as a form of anomalous dissipation with infinite dissipation rate. Example 4. A fixed point: If a2n (0) = 0 for n = 1, 2, . . . and a2n+1 (0) =
(2n)! 22n (n!)2
1 ∼√ for n = 0, 1, . . ., πn
then an (t) = an (0) for all n = 1, 2, . . .. This solution is a fixed point of (1). Notice that N 1d |an (t)|2 = −N a N (t)a N +1 (t) = 0 2 dt n=1
for all N ∈ N, but
∞
n=1 |an (t)|
2
√ = +∞ since a2n+1 ∼ 1/ π n as n → ∞.
Example 5. Explosive solution with infinite explosion time: If an (0) = (−1)n+1 for n = 1, 2, . . . , then an (t) = (−1)n+1 et for all n = 1, 2, . . .. In this case,
∞
n=1 |an (t)|
2
= +∞ and
N 1d |an (t)|2 = −N a N (t)a N +1 (t) = N e2t → +∞ 2 dt n=1
as N → ∞. Thus, (2) does not hold for this solution and we see that the above limit is consistent with an infinite explosion rate for the solution. This is consistent with Theorem 1 as ρ(a) = 1 and t∗ = arctanh(1) = ∞. Example 6. Explosive solution with finite explosion time: If an (0) = (−1)n+1 α n with α > 1 for n = 1, 2, . . . , then an (t) =
(−1)n+1 α n for t < t∗ = arctanh(1/α) and all n = 1, 2, . . .. cosh(t) − α sinh(t)
Simple Systems with Anomalous Dissipation and Energy Cascade
This solution blows up at t = t∗ . In this case,
199
∞
n=1 |an (t)|
2
= +∞ for all t < t∗ and
1d N α 2N +1 |an (t)|2 = −N a N (t)a N +1 (t) = → +∞ 2 dt (cosh(t) − α sinh(t))2 n≤N as N → ∞ for all t < t∗ . Notice that this example is consistent with Theorem 1 as ρ(a) = 1/α < 1. The above examples demonstrate the rich range of behavior of the model. In particular, some solutions grow coordinate-wise in time while others decay. The following result gives a criteria for the later. Theorem 2. Let (a1 (0), a2 (0), · · · ) be the infinite vector of initial conditions. If the limit lim +
r →−1
∞
an (0)r n
(17)
n=1
exists and is finite, then for all n ∈ N, |an (t)| → 0 as t → ∞. Looking back at the examples, this result correctly separates those which decay to zero pointwise in n and those which do not. Denote by p the p-summable sequences: For p > 0, := (a1 , a2 , . . .) : p
∞
|an | < ∞ . p
(18)
n=1
If a = (a1 , a2 , . . .) ∈ 1 then (17) exists and is finite. For future reference, we recall the p def p norm · p defined by a p = ∞ n=1 |an | . A complimentary question is to understand for which initial data the system conserves energy. Theorem 3. If a(0) = (a1 (0), a2 (0), · · · ) ∈ 2 , then a(0) 22 = a(t) 22 for all time t ≥ 0. Example 1, 2 and 3 all have nice limits at z = −1 in the sense of (17) and they decay to zero as t → ∞ as dictated by Theorem 2. It is particularly interesting to compare Example 3 and 5. Theorem 2 correctly says that the first decays to zero as time increases while declining to comment on the second. Theorem 3 correctly states that Example 1 conserves energy. However, Theorem 3 is not completely satisfactory in that it only applies to solutions which have finite total energy. A number of our examples have infinite energy. We now turn to understanding in detail the dynamics of (1).
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4. Solution to the Initial Value Problem In this section we show that the initial value problem associated to Eq. (1) is well-posed and admits solutions for a wide class of initial data. Theorem 1 above is an immediate consequence of Theorem 4 below. After giving a general existence and uniqueness theorem, we present specific initial conditions (covered by the existence theorem) for which Eq. (1) admits solutions which conserve energy, dissipate energy, and blow up in finite time. We begin by formally calculating a representation of the solution given by a generating function. We will verify that the representation is valid in the next section. Given initial conditions {an (0) : an (0) ∈ R, n ∈ N}, we assume that a solution an (t) exists and define the generating function G(z, t) =
∞
an+1 (t)z n .
(19)
n=0
Proceeding formally, it is straightforward to verify that G(z, t) would satisfy the following partial differential equation: ∂G ∂G = (z 2 − 1) + zG ∂t ∂z
(20)
with initial condition G 0 (z) = def
∞
an+1 (0)z n .
(21)
n=0 ∂ The first term on the right hand side of (1) comes from z ∂z (zG) and the second from ∂ − ∂z G.
One obtains an ansatz for the form of the solution by solving Eq. (20) by the method of characteristics. By verifying that this ansatz solves the equation, we obtain the following existence and uniqueness result whose proof is postponed until Sect.7. Theorem 4. Consider (1) with the initial condition an (0) such that G 0 (z) is analytic in a neighborhood of the interval (α, 0]. Then the solution of (1) exists and is unique for all t ∈ [0, t∗ ), where t∗ = arctanh(−α ∧ 1). The unique solution is given by G(z, t) an (t) = dz (22) n Γ 2πi z with G(z, t) = def
where ψt (z) = def
ψt (z) (G 0 ◦ φt )(z), cosh(t)
1 z − tanh(t) def and φt (z) = . 1 − z tanh(t) 1 − z tanh(t)
(23)
(24)
Γ is any simple closed contour around the origin within the region of analyticity of G(z, t) (which is non-empty).
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It is worth noting that G(z, t) solves the PDE given in (20) with G(z, 0) = G 0 (z) as initial condition. Notice that finite time existence of solutions only requires that the initial data an (0) have at most exponential growth in n, i.e. there exists C > 0 and γ > 1 such that for all n ∈ N, |an (0)| ≤ Cγ n . If the an (0) decay exponentially then the solution exists for all times (i.e. t∗ = ∞). In addition, (22) implies that an+1 (t) is the n th term of the Taylor series expansion of G(z, t) about z = 0. It is also straightforward to see that (22) defines a semigroup: for any suitable f (z), let 1 ψt (z) f (φt (z)) . cosh(t)
(St f )(z) :=
(25)
Then (23) can be expressed as G(·, t) = St G 0 and it is easy to check that for any t, s > 0, St ◦ Ss f = Ss ◦ St f = St+s f. Since we are particularly interested in knowing the total energy of the solution, it is useful to notice that if G(x, t) is as in (19) then a(t) 22 = def
∞
|an (t)|2 =
n=1
1 2π
2π 0
|G(eiθ , t)|2 dθ = G(eiθ , t) 2L 2 (S 1 ,dθ) . def
We shall give more properties of the solutions of Eq. (1) in Sect. 5 after a brief discussion of the examples given in the previous section.
4.1. Analysis of examples. We use Theorem 4 to calculate exact the solutions given in Sect. 3. The initial data in Example 1 translates into G 0 (z) = 1, so that G(z, t) =
1 . cosh(t) − z sinh(t)
Calculating the Taylor series gives the quoted an (t). In Example 2 one obtains G 0 (z) = √ 1/ 1 − z. Hence, 1 e−t/2 , G(z, t) = √ √ cosh(t) (1 − z)(1 − z tanh(t)) whose Taylor expansion produces the quoted an (t). Example 3 yields G 0 (z)√= 1/(1−z), G(z, t) = e−t√/(1 − z) and the desired an . Example 4 gives G 0 (z) = 1/ 1 − z 2 and et . G(z, t) = 1/ 1 − z 2 ≡ G 0 (z). Example 5 gives G 0 (z) = 1/(1 + z) and G(z, t) = 1+z Example 6 gives G 0 (z) = α/(1 + αz) and G(z, t) =
α 1 cosh(t) − α sinh(t) 1 + αz
it blows up at t = t∗ as stated.
for t < t∗ = arctanh(1/α);
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5. Properties of the Solutions We begin by presenting two results which are more quantitative versions of the results in Theorem 2 and Theorem 3. Together, they highlight the fact that it is possible to have a given coordinate converge to zero while no global energy dissipation is present in the system. This implies that there is a flux of energy out to higher and higher modes. It is also interesting that both of the next two results apply in some situations where the total energy is infinite. Theorem 5. Suppose that G 0 (z) is analytic in a neighborhood of (−1, 0] such that G +0 (−1) = lim + G 0 (x)
(26)
x→−1 x∈R
exists and is finite. Then, an (t) ∼ 2e−t G +0 (−1) as t → ∞. In particular, in a neighborhood of the origin G(z, t) ∼ 2e−t G +0 (−1)/(1−z) as t → ∞. Remark 2. If an = n −α for α < 1, then an (t) ∼ 2e−t Γ (1 − α) as t → ∞. The next result contains Theorem 3 as well as giving control of higher Sobolev-like norms. We recall the Sobolev-like sequence spaces for s ∈ R: def
h s = a = (a1 , a2 , . . .) : a h s < ∞ , where the norm · h s is defined by a 2h s = def
∞
n=1 n
2s |a |2 . n
Theorem 6. If a(0) 22 = def
∞
|an (0)|2 < ∞,
n=1
then a(0) 22 = a(t) 22 for all t ≥ 0. Similarly, for any s ∈ N, if a(0) 2h s = def
∞
n 2s |an (0)|2 < ∞,
n=1
then for any T < ∞ there exists a constant, C(T ), such that sup a(t) 2h s ≤ C(T ).
t∈[0,T ]
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5.1. Finer properties of solutions. We begin by giving conditions guaranteeing that the solution decays exponentially in time. We then turn to the case for which G 0 has singularities on the boundary of the unit circle. We close the section with a result which compares the dynamics obtained by placing a single unit of mass at different locations. Let D(r ) denote the open disk of radius r about the origin. We begin with a simple criteria which guarantees that the an decay exponentially in n. Theorem 7. Suppose G 0 (z) is analytic in the disk D(1 + η) for some η > 0. Then, for each T ≥ 0, there exist constants γ , C > 0 which depend only on η and T such that sup |an (t)| ≤ Cγ n for all n ∈ N.
t∈[0,T ]
In particular, the system conserves energy at all finite times. In order to investigate the behavior of solutions when G 0 (z) is not analytic in D(1+η) for some η > 0, we introduce the following region: ∆(ζ, η, θ ) = {z : |z| ≤ |ζ | + η, | arg(z − ζ ) − arg(ζ )| ≥ θ} . def
(27)
We begin with a careful analysis of the case when there is a single singularity on the unit circle. We contrast the cases for which the singularity is located at ±1 in Theorem 8 and Theorem 9, respectively. The remaining cases are covered by Theorem 10. Theorem 8. Assume that either an (0) ∼ Cn α−1 as n → ∞ for some C = 0 or, more generally, that G 0 (z) satisfies both of the following conditions: i) There exist an A = 0 and α > 0 so that G 0 (z) ∼ A(1 − z)−α
as z → 1.
(28)
ii) G 0 (z) is analytic on ∆(1, η, θ )\{1} for some η > 0 and 0 < θ < π/2. Then, for all time t ≥ 0, an (t) ∼
A (1−2α)t α−1 e n as n → ∞, Γ (α)
(29)
and the solution decays to zero as t → ∞ pointwise in n; more precisely, an (t) ∼ A21−α e−t as t → ∞ . Notice the setting of Theorem 8 the energy of the system is infinite for all that in 2 α > 0: ∞ n=1 an (t) = +∞. However, as a direct consequence of (29) in Theorem 8 and our definition of dissipation in Sect. 2 the following is true: Corollary 1. In the setting of Theorem 8: if 0 < α < 1/2 the system is conservative; if α = 1/2 the system displays a finite dissipation rate; and, if α > 1/2 the system displays an infinite dissipation rate.
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We now consider a singularity at z = −1. The remaining points on the unit circle behave much like the z = 1 in that the solution decays to zero in time. They are discussed in Theorem 10 later in the section. The next result shows that if there is a singularity at z = −1 then the system can explode in time. Theorem 9. Assume that either an (0) ∼ C(−1)n n α−1 as n → ∞ for some C = 0 or, more generally, that G 0 (z) satisfies both of the following conditions: i) There exist an A = 0 and α > 0 so that G 0 (z) ∼ A(z + 1)−α
as z → −1.
(30)
ii) G 0 (z) is analytic on ∆(−1, η, θ )\{−1} for some η > 0 and 0 < θ < π/2. Then, for all time t ≥ 0, an (t) ∼
A (2α−1)t (−1)n n α−1 as n → ∞, e Γ (α)
while for n fixed α as t → ∞, an (t) ∼ 21−α e(2α−1)t ACn−1
where Cnα is the n th coefficient of the Taylor series expansion at z = 0 of the function
1−z α 1 . 1−z 1+z 2 Notice that the energy of the system is again infinite for all α > 0: ∞ n=1 |an (t)| = +∞. However, we have: Corollary 2. In the setting of Theorem 9: If 0 < α < 1/2, the system is conservative; if α = 1/2, it displays a finite explosion rate; and, if α > 1/2, it displays an infinite explosion rate. Remark 3. One must be careful when interpreting the results in Corollary 2 since the system has infinite energy. For example, when α = 1/2 then at each moment of time the system displays a finite explosion rate since energy is pumped in from infinity into any finite collection of modes. However, the rate must slow, falling to zero at t = ∞, since as t → ∞ the system converges to the fixed point given in Example 4. This follows from the fact discussed in Sect. √ 4.1 that the fixed point in Example 4 corresponds to the initial function G(z) = B/ 1 − z 2 for some B = 0. We now give a more general result covering a singularity on the unit circle at any point other than −1. Theorem 8 is a special case of the following result when ζ = 1. Theorem 10. Let ζ be a point on the unit circle not equal to −1. Assume that G 0 (z) behaves as (31) G 0 (z) ∼ A(ζ − z)−α with A = 0 and α > 0 as z → ζ ,
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and that G 0 (z) is analytic on ∆(ζ, η, θ )\{ζ } for some η > 0 and 0 < θ < π/2. Then, for all time t ≥ 0,
1−2α 1+ζ A 1−ζ et + e−t ζtn n α−1 as n → ∞, an (t) ∼ Γ (α) 2 2 where ζt = φt−1 (ζ ). (Notice that φt−1 (1) = 1 and ζt → 1 as t → ∞.) The solution decays to zero as t → ∞ pointwise in n: an (t) ∼ 2e−t
A as t → ∞ . (1 + ζ )α
From the examples above it is natural to conjecture that any singularity on the unit circle dominated by a polynomial-like singularity of degree less than 1/2 will not destroy energy conservation. It can be shown that this intuition is correct for a wide class of initial conditions. We already know that if the initial conditions have finite energy, then energy is conserved. By the reasoning in the section defining anomalous dissipation, it is enough to have 1 lim n 2 |an | = 0 . (32) n→∞
This is possible even if the total energy is infinite. Using the Tauberian theorems in [Hil01] we can show that the dynamics preserves a subset of sequences satisfying (32). These solutions have infinite energy yet conserve energy in the sense of Sect. 2. As these results are tangential and a bit technical we do not give the details. 5.2. The fixed point. By combining the results of the previous section, one can understand a wide range of behavior. We illustrate this by examining the convergence√to a fix point. In Example 4, we saw that the initial data corresponding to G 0 (z) = 1/ 1 − z 2 was invariant under the dynamics. This function is analytic in the open unit disk and has two square root singularities on the unit circle: one at z = 1 and another at z = −1. Up to a technicality, we show that this is the only fixed point that characterizes the initial data that converges to it. If the solution is to exist for all times the function G 0 (z) must be analytic in a neighborhood of (−1, 0]. From Theorem 5, we see that if the limit as z → −1 of G 0 (z) is finite then the solution converges to zero pointwise in n as t → ∞. Hence, if the system converges to a nontrivial steady state, it must have a singularity at z = −1. We assume that the singularity is power-like (i.e. (z + 1)−α ). One can likely deal with other singularities, however, we choose not to pursue this matter here. Theorem 9 implies that the singularity must be order 1/2 if it is a power; otherwise, the system would blow or decay to zero. The question is: Can one make sense of the dynamics when up |an |2 = ∞? The two facts above imply that any initial condition which has only polynomial singularities at z = −1 (if any) and which converges to a fixed point must be of the form: A G 0 (z) = + G˜ 0 (z), (33) 1 (1 + z) 2 where A = 0 and G˜ 0 is analytic in a neighborhood of (−1, 0] and G˜ 0 (z) → G˜ +0 < ∞ as z → −1 from the right along the real axis. By Theorem 5 the dynamics starting from ∞ pointwise in the sequence space. Theorem 8 says that G˜ 0 converge to zero as t → the first term converges to A 2/(1 − z 2 ) as t → ∞. Hence, all data of the form (33) converge to the fix point from Example 4. We have proved the following result:
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Theorem 11. Let (a1 (0), a2 (0), . . .) be an initial condition such that the solutions exist for all time and converge coordinate-wise to a fixed vector (a¯ 1 , a¯ 2 , . . .) which is not the zero vector. Assuming that the G 0 (z) associated to this initial condition has only a power-like singularity at z = −1 then G 0 is of the form given in (33) and ∞ 2 n a¯ n+1 z = A . (34) 1 − z2 n=0
In particular, all fixed points of Eq. (1) are given by the Taylor series of (34). Notice that the fixed points have infinite energy and have the property that all of the an , with n odd, are equal to zero. In Sect. 6, we will see that steady states with less degenerate structure are obtained by forcing the system. Remark 4. If one starts with an initial condition which has a polynomial singularity on the unit circle at ζ = −1 and of an order β + 1/2 with β > 0, then for all finite times the norm a(t) h s will be infinite if s ≥ −β. Yet the dynamics still converges to the fixed point given by Theorem 11. Hence, at t = ∞ all · h s norms are finite for s < 0. 5.3. The effect of shifting the initial condition. In Example 1 we described how a unit of mass placed at a1 spreads out. The following theorem states that the solution obtained by placing a unit of mass in a p behaves in the same way as the solution obtained by placing a mass at a1 except that the picture is shifted p units down the chain. Theorem 12. Let G 0 (z) = z p for some p ∈ N, then tanhn+ p (t) tanhn− p+ξ (t) + βn (t) , cosh(t) cosh(t) where ξ = p − p ∧ n and βn (t) satisfies n p 1 − tanh2 p (t) n≥p . |βn (t)| ≤ p−n 2n p n tanh (t) 1 − tanh (t) n < p an (t) = (−1) p
Proof (Theorem 12). First observe that G(z, t) =
z − tanh(t) p cosh−1 (t) . 1 − z tanh(t) 1 − z tanh(t)
Expanding this we find ∞
G(z, t) =
1 z n (−1) p tanhn+ p (t) + βn tanhn− p+ξ (t) , cosh(t) n=0
where βn (t) = tanh
p∧n
p∧n k n −1 p−k p . tanh (t) − tanh(t) [− tanh(t)] (t) k k k=1
So |βn (t)| ≤ n p
p∧n k p . 1 − tanh2 (t) [tanh(t)] p∧n−k [tanh(t)] p−k k k=1
In both cases the quoted estimate follows by using the binomial theorem.
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6. The Forced System Since (1) may display anomalous dissipation, it is not unreasonable to expect that adding a forcing term to this equation may lead to a (statistical) steady state. We now show that this is indeed the case. Specifically, we study the system a˙ n (t) = (n − 1)an−1 (t) − nan+1 (t) + 1n=m f (t),
(35)
where m ∈ N, and f (t) is either a constant forcing term, f (t) = 1, or a white-noise process, f (t) = W˙ (t) (in the second case (35) has to be properly interpreted as an infinite system of coupled Itô stochastic differential equations). As in the unforced setting, we represent the solution to (35) as in (22) for some G(z, t). an (t) is the n th coefficient in the Taylor series expansion of G(z, t). By Duhamel’s principle one sees that G(z, t) must satisfy the generalization of (20) with the effect of f (t)1n=m included: ∂G ∂G = (z 2 − 1) + zG + F(z, t), ∂t ∂z where F(z, t) = z m−1 f (t) and the initial condition is G(z, 0) = G 0 (z). This equation is valid for both f (t) = 1 and f (t) = W˙ (t) and forcing on any m ∈ N. Using the semigroup representation defined in (25), the solution of the equation above can be represented as t G(z, t) = (St G 0 )(z) + (St−s F)(z, s)ds. (36) 0
We have Theorem 13. Consider (35) with m = 1, f (t) = 1, and initial condition an (0) satisfying the assumptions of Theorem 5. Then, ⎧ π ⎪ ⎪ if n = 1 ⎨ √2 n (37) lim an (t) = an∗ = Γ (2) π t→∞ ⎪ ⎪ · if n ≥ 2. ⎩ n − 1 Γ ( n−1 2 ) Remark 5. Notice that an∗
∼
π n
as n → ∞.
∗ 2 This implies that an∗ has infinite energy: ∞ n=1 |an | = +∞. This is consistent with the fact that the steady state must be dissipative to compensate for the effect of the forcing term since dissipative solutions must have infinite energy. In this simple example we can see how the forcing is balanced explicitly. Mirroring the calculation in (3) for any N : If we start the system {an∗ } at time t = 0 we have 1d ∗ 2 |an (t)| = a1∗ − N a ∗N a ∗N +1 = 0. 2 dt n≤N
Hence, every block conserves energy as must happen at a fixed point.
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Theorem 14. Consider (35) with m = 1, f (t) = W˙ (t), and initial condition an (−T ) satisfying the assumptions of Theorem 5. Then, lim an (t) = an∗∗ (t) ≡
T →∞
t −∞
tanhn−1 (t − s) dW (s) cosh(t − s)
a.s.
(38)
Remark 6. From (38), an∗∗ (t) is a Gaussian process with mean zero and covariance
tanhn+m−2 (t − s) ds cosh2 (t − s) −∞ 1 n, m ∈ N. = n+m−1
∗∗ Ean∗∗ (t)am (t) =
t
(39)
Again, this is consistent with the need for dissipation and implies that the invariant measure for (35) with a white-noise forcing is supported on functions with infinite energy (and, in particular, (39) is not trace-class). In addition, notice that this is consistent with the fact that, at least in expectation, the steady state needs to dissipate precisely the energy pumped into the system. In fact, for any N , 1 d ∗∗ 2 N 1 1 ∗∗ E − =0. |an (t)| = − N E a ∗∗ N (t)a N +1 (t) = 2 dt 2 2 2N n≤N
Proof (Theorem 13). The first term on the right hand-side accounts for the initial condition. Theorem 5 implies that (St G 0 ) → 0 as t → ∞. The second term is given explicitly by t t ψt−s (z) (St−s F)(z, s)ds = (40) (φt−s (z))m−1 f (s)ds. cosh(t − s) 0 0 Letting f (t) = 1 and m = 1, this expression becomes
t
(St−s F)(z, s)ds =
0
t
0
ψt−s (z) ds. cosh(t − s)
It follows that lim
t→∞ 0
t
2
(St−s F)(z, s)ds = √ 1 − z2
z−1 arctan √ 1 − z2
+π .
an∗ is the n th coefficient of the Taylor series expansion at z = 0 of this function. Proof (Theorem 14). Letting f (t) = W˙ (t), m = 1, and considering the initial condition at t = −T , we have
t
−∞
(St−s F)(z, s)ds =
t
−∞
ψt−s (z) dW (s). cosh(t − s)
a ∗∗ (t) is the n th coefficient of the Taylor series expansion at z = 0 of this function.
Simple Systems with Anomalous Dissipation and Energy Cascade
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7. Proofs of the Main Theorems We begin by making a number of observations which will be used in the proofs. ψt and φt each have a single simple pole at z = 1/ tanh(t). Hence, at any finite time both are analytic in an open disk containing the closed unit disk and the Taylor coefficients of their expansions about zero converge to zero exponentially in n. For any fixed t > 0 the map φt is a fractional linear transformation which bijectively maps the open unit disk onto itself and leaves the unit circle invariant. The points z = 1 and z = −1 are the two fixed points. In addition, for every fixed z ∈ D(1)\{1} and fixed neighborhood N of −1, there exists a time T (z, N ) such that φt (z) ∈ N ∩ D(1) for all t > T (z, N ). The behavior of G 0 ◦ φt in a neighborhood of the origin will be important in the analysis which follows. Observe that φt (0) = − tanh(t) and for sufficiently small r > 0, {ρeiθ : θ ∈ [0, 2π ], ρ ∈ [0, r ]} is mapped approximately to {− tanh(t) + ρ(1 − tanh2 (t))eiθ : θ ∈ [0, 2π ], ρ ∈ [0, r ]} and strictly into the closed disk Er = − tanh(t) + ρ(1 − tanh2 (t))eiθ : θ ∈ [0, 2π ], ρ ∈ [0, r/(1 − r )] . For sufficiently small r > 0, Er is strictly contained in the unit disk for all times t. Furthermore, Er is bounded away from the boundary by two lines emanating from −1 of the form {−1 + ρe±iθ : ρ ≥ 0} for some fixed θ ∈ (0, π/2). We recall a basic fact from complex analysis which will be used repeatedly in the arguments that follow. To show that the Taylor coefficients about zero of G 0 ◦ φt converges to those of F as t → ∞ it is sufficient that for all t ≥ 0, F and G 0 ◦ φt are analytic in a fixed, t independent neighborhood of the origin and that G 0 ◦ φt converges uniformly to F on that neighborhood. Proof (Theorem 4). In order that Eq. (22) be well defined, G(z, t) needs to be analytic in a neighborhood of z = 0. Since ψt is analytic for all z ∈ D(1) and each t > 0, the analyticity of G(z, t) about z = 0 is equivalent to the analyticity of G 0 (z) about φt (0) = − tanh(t). As t increases, φt (0) decreases monotonically along the real axis from 0 to −1. We need only show that G 0 is analytic in an open neighborhood of [− tanh(t), 0] in order to complete the proof that Eq. (22) is well defined for all t < t∗ . G 0 is analytic in a neighborhood of the closed interval [− tanh(t), 0] since [− tanh(t), 0] ⊂ (α, 0]. Appealing to the arguments stated at the beginning of this section we see that the image of a small ball about the origin under the mapping φt lies in a thin strip about [− tanh(t), 0]. Hence, the reconstruction formula of Eq. (22) is well defined because Γ can be deformed to lie in a sufficiently small ball about the origin. If t∗ < ∞, then G(z, tanh(t∗ )) fails to be analytic at z = 0 since G 0 (z) in not analytic at α = − tanh(t∗ ); we cannot continue the solution in this case. To see that the an (t) (defined as in the statement of the theorem) do define a solution, observe that by the definition of G(z, t) and integration by parts ∂t G(z, t) (z 2 − 1)∂z G(z, t) + zG(z, t) a˙ n (t) = dz = dz n 2πi z n Γ 2πi z Γ G(z, t) G(z, t) = (n − 1) dz − n dz = (n − 1)an−1 (t) − nan+1 (t). n−1 n+1 Γ 2πi z Γ 2πi z This shows that an (t) given by (22) is indeed a solution of (1) for the initial condition an (0) as long as G(z, t) is analytic around z = 0.
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Finally, to show that the an (t) defined by Eq. (22) is the unique solution of (1) for the initial condition an (0), note that if two different solutions exist for the same initial condition, then their associated G(z, t) must both satisfy (20) for the same initial condition G 0 (z). Since the solution of (20) is unique, this leads to a contradiction. Proof (Theorem 5). Appealing to the discussion at the beginning of the section and the fact that cosh(t) ∼ 21 et as t → ∞, it is enough to show that ψt (z)(G 0 ◦ φt )(z) converges uniformly to G + (−1)/(1 − z) on some neighborhood of the origin. First, observe that ψt (z) converges to 1/(1 − z) as t → ∞ uniformly on any disk contained within the unit disk. From the discussion at the beginning of the section we see that for all t > 0 the disk of radius 1/10 is mapped to a disk contained entirely in the open unit disk and bounded away from the unit circle by lines emanating from −1 of a constant angle. Hence, we have lim
sup |G 0 (φt (z)) − G +0 (−1)| = 0 .
t→∞ |z|<1/10
Proof (Theorem 6). Since (a1 (0), a2 (0), . . .) is square summable, g(r, θ ) := G 0 (r eiθ ) is in L 2 (dθ ) of the unit circle for all r ∈ [0, 1]. Hence, by Plancherel’s theorem, ∞
|an (t)|2 =
n=1
1 2π
2π
|G(eiθ , t)|2 dθ =
0
1 2π
2π 0
|ψt (eiθ )|2 |G 0 (φt (eiθ ))|2 dθ. cosh2 (t)
Introducing the change of variable eiη = φt (eiθ ), we see that ∞
|an (t)|2 =
n=1
=
1 2π 1 2π
2π
0
2π
|ψt (φt−1 (eiη ))|2 1 dη |G 0 (eiη )|2 −1 2 cosh (t) |φt (φt (eiη ))| |G 0 (eiη )|2 dη =
0
∞
|an (0)|2 .
n=1
To obtain the bounds on the weighted norms, we notice that for s ∈ N, ∞ n=1
n 2s |an (t)|2 =
1 2π
2π 0
s ∂ G iθ 2 dθ . (e , t) ∂θ s
Using the same change of variable one easily shows that this term can be bounded in ∂s iθ terms of the L 2 norms of ∂θ s G 0 (e ) for r ≤ s. By assumption, these norms are finite. Proof (Theorem 7). Fix T > 0. By the considerations at the beginning of the section, one sees that for t ∈ [0, T ] the map ψt (z)(G 0 ◦ φt )(z) remains analytic in D(1 + η1 ) for a sufficiently small η1 depending only on T and η. More precisely, η1 is picked to ensure that D(1 + η1 ) maps into D(1 + η) for all t ∈ [0, T ]; the only remaining constraint on η1 is that 1 + η1 < tanh1 T so that ψt (z) is also analytic in D(1 + η1 ). Hence, the power series converges absolutely on D(1 + η1 /2) and |an (t)| ≤ C(1 + η1 /2)n for all n ∈ N for some C > 0.
Simple Systems with Anomalous Dissipation and Energy Cascade
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Proof (Theorem 8, Theorem 10 and Corollary 1). Theorem 8 is a special case of Theorem 10 so we concentrate on the later. By the discussion in the proof of Theorem 5, it is clear that for each moment of time t > 0 there exists a η1 > 0 so that G(z, t) is analytic on ∆(ζt , η1 , θ1 )\{ζt }. η1 may be chosen to be sufficiently small in order to avoid other singularities of G 0 which initially lie outside ∆(ζ, η, θ )\{ζ } and approach D(1) under the dynamics of φt . A similar consideration needs to be taken into account for θ and may result in an increase of θ to a new θ1 . As z → ζt , we see that A [cosh(t) − ζt sinh(t)]α−1 [cosh(t) + ζ sinh(t)]α (ζt − z)α
1−2α A 1+ζ 1−ζ t −t e + e = . 2 2 (ζt − z)α
G(z, t) ∼
The result on the asymptotics in n then follows from Theorem 20 in the Appendix since G(z, t) is analytic on ∆(ζt , η1 , θ1 )\{ζt }. The result for fixed n as t → ∞ is just a restatement of Theorem 5 in this context. The corollary follows directly from the discussion in Section 2. Proof (Theorem 9 and Corollary 2). The proof is similar to that of Theorem 8. Since z = −1 is a fixed point for φt for all t > 0, G(z, t) has a singularity at z = −1 inherited from G 0 (z). Since the circle is invariant under φt , for sufficiently small η1 > 0 and θ1 sufficiently close to π/2 we have that G(z, t) is analytic on ∆(−1, η1 , θ1 ). Direct calculation yields:
1 + tanh(t) α A Ae(2α−1)t 1 = as z → −1 . G(z, t) ∼ α cosh(t) + sinh(t) 1 − tanh(t) (1 + z) (1 + z)α We obtain the quoted result by applying Theorem 20 from the Appendix. The asymptotics in time follow from the fact that Ae(2α−1)t 1 − z α G(z, t) ∼ as t → ∞ 1−z 1+z and direct expression of the right hand side in a Taylor series in z. Corollary 2 follows from the discussion on anomalous dissipation in Sect. 2 and the above results. 8. Inviscid Limits We return to the analysis of the inviscid limits of (1). Fixing p ∈ N and defining Λn =
p
(n − k)
k=1
with the convention that Λn = 1 if p = 0, we consider α˙ n,ν = −2νΛn αn,ν + (n − 1)αn−1,ν − nαn+1,ν + 1n=1 W˙ (t) .
(41)
As mentioned in Sect. 1.1, it is straightforward to see that this system converges to a ∗∗ , α ∗∗ , · · · ). In fact, one has random variable αν∗∗ = (α1,ν 2,ν E
n
∗∗ 2 Λn |αn,ν | =
1 . ν
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Thus, the system does not display anomalous dissipation; the dissipation which balances the energy injection (due to the forcing) comes from the term −2νn p αn (t). Setting ∞ αn+1,ν (t)z n , Gν (z, t) = n=0
one sees that
∂ p Gν ∂Gν ∂Gν + W˙ (t) . = (z 2 − 1) + zGν − 2νz p ∂t ∂z ∂z p
(42)
Using the variation of constants formula we obtain Gν (·, t) = St,ν Gν (·, 0) +
t
(St−s,ν 1) dWs .
0
We will concentrate on the case p ∈ {0, 1}. By the method of characteristics, we find that ⎧ ⎪ e−2νt ⎪ ⎨ for p = 0 ψt (z) f (φt (z)) (St,ν f )(z) = cosh(t) , 1 ⎪ ⎪ ψκt (z − ν) f (ν + κφκt (z − ν)) for p = 1 ⎩ cosh(κt) where κ 2 = 1 + ν 2 . It is interesting to contrast the regularizing effect of the different terms. When p = 0, St,ν simply dissipates energy at a faster rate than St . When p = 1, St,ν has a stronger regularizing effect than St,ν , in that the characteristics are attracted to the circle (ν − κ)eiθ inside of the unit disk and the singularity of ψκt (z − ν) stays uniformly bounded outside of the unit disk for all times. Hence if f has a radius of convergence greater than ν − κ ∼ −1 + ν, then Stν,1 f is analytic on a disk with radius greater than one for all times uniformly. For fixed t, St,ν f converges to St as ν → 0 uniformly on a neighborhood of the origin. Since one also has that St f , St,ν f all go to zero t uniformly on the open disk as t → ∞ for f bounded on the unit disk, we have that −∞ (St−s,ν )1 dWs converges to t −∞ (St−s 1) dWs in mean squared as ν → 0. As before we are primarily interested in these solutions. In this setting they are given by: ⎧ t tanh(t − s)n−1 ⎪ ⎪ dW (s) p=0 e−2ν(t−s) ⎨ cosh(t − s) ∗∗ αn,ν . (43) (t) = −∞ t κ tanh(κ(t − s))n−1 ⎪ ⎪ ⎩ dW (s) p = 1 n cosh(κ(t − s)) −∞ [κ + ν tanh(κ(t − s))] Theorem 15. For all n ∈ N, t ∈ R, and p = 0, 1, one has ∗∗ 2 lim E αn,ν (t) − an∗∗ (t) = 0.
ν→0
∗∗ (t) converges to a ∗∗ (t) as ν → 0 for p = 0, 1. Furthermore, one has the Hence αn,ν n estimates given in (14) and (15).
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213
Proof (Theorem 15). Fix any t. Consider αn (t, ν, 0) and an (t) starting from initial condition zero at time T with T < t. As T → −∞, αn (t, ν, 0) and an (t) converge respectively to αn∗∗ (t, ν, 0) and an∗∗ (t). By the same argument as Theorem 14, one see that (43) holds. Subtracting (43) from (38), one obtains 2 E αn∗∗ (t, ν, 0) − an∗∗ (t) = 2
∞
2 u 2(n−1) (1 + u)−ν − 1 du (u + 2)2n 0 ∞ 2 1 4ν 2 , ≤2 du = (1 + u)−ν − 1 (u + 1)2 (1 + ν)(1 + 2ν) 0
which implies that αn∗∗ (t, ν, 0) → an∗∗ (t) almost surely as ν → 0. The convergence in the other cases is similar. Applying the Itô isometry to (43) proves the quoted value of 2 E αn∗∗ (t, ν, p) for p = 0. The other estimates follow from 1 1 1 ≤ ≤ κ +ν κ + ν tanh(t) κ which holds for t ≥ 0. Remark 7. At first glance, it might seem more natural to consider the system α˙˜ n,ν = −2νn p α˜ n,ν + (n − 1)α˜ n−1,ν − n α˜ n+1 + 1n=1 W˙ (t) . This leads to the following equation for the generating function G˜ν : ∂ G˜ν ∂ p G˜ν ∂ G˜ν = (z 2 − 1) + z G˜ν − 2νz p − 2νD p + W˙ (t) , ∂t ∂z ∂z p ∂ where D p is p applications of the operator defined by (D f )(z) = ∂z (z f (z)) and D0 is the identity operator. Hence, we see that the extra dissipative term contains the derivative of all orders less than or equal to p. Not surprisingly, the result is a mixture of the behavior of (41) for all orders less than or equal to p. In particular, when p = 1 the asymptotic (in time) behavior is given by ∗∗ α˜ n,ν (t)
=
t −∞
κe−ν(t−s) tanh(κ(t − s))n−1 dW (s) [κ + ν tanh(κ(t − s))]n cosh(κ(t − s))
and satisfies the following estimate: ∗∗ 2 κ2 1 ∗∗ ∗∗ E[αn,νκ/2 ]2 ≤ E α˜ n,ν ≤ 2n E[αn,νκ/2 ]2 . 2n+2 (κ + ν) κ 9. A Second Linear Shell Model We begin the analysis of the second model (11) by giving general conditions for the existence of a unique solution of the initial value problem. The technique is similar to that used in Sect. 4.
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J. C. Mattingly, T. Suidan, E. Vanden-Eijnden
Theorem 16. Let {bn (0)} be such that ∞ (−1)n bn (0) < ∞.
(44)
n=1
Then the solution of (11) exists and is unique for all positive times. It can be represented as (−1)n+1 ∂ 2n−1 H bn (t) = (0, t), (45) (2n − 1)! ∂ x 2n−1 where 1 t 2 H (x, t) = Ex H0 (X (t)) exp − X (s)ds , (46) 2 0 and X (t) satisfies the stochastic differential equation 1 d X (t) = −X 3 (t)dt + √ 1 − X 4 (t)dW (t). 2 Ex denotes the expectation conditional on X (0) = x ∈ [−1, 1] and H0 (x) = (−1)n+1 bn (0)x 2n−1 .
(47)
(48)
n∈N
Remark 8. Alternatively, H (x, t) can be expressed as H (x, t) = (−1)n+1 bn (t)x 2n−1 ,
(49)
n∈N
where bn (t) solves (11). Remark 9. If the sequence {bn (0)} is monotone and converges to zero as n → ∞ then the condition in (44) holds. The following theorem summarizes the most interesting properties of solutions of (11). Theorem 17. Suppose that bn (0) satisfies (44). Then, for any positive time t > 0, lim (2n + 1)b2n+1 (t) = C¯ 1 (t) and lim 2nb2n (t) = C¯ 2 (t),
n→∞
n→∞
(50)
where C¯ 1 (t), C¯ 2 (t) ∈ R, C¯ 1 (t), C¯ 2 (t) = 0 for all but finitely many t ∈ [0, ∞). In particular, there exists a T > 0 such that for all t ≥ T , the solution of (11) is dissipative and satisfies ∞ ∞ bn2 (t) < bn2 (T ) < ∞. (51) In fact,
∞
2 n=1 bn (t)
n=1
n=1
→ 0 as t → ∞.
Remark 10. The fact that Eq. (11) dissipates energy at finite times is implicit in the rept resentation (46). As time grows, the factor exp(− 21 0 X 2 (s)ds) converges to zero as exp(−ct) almost surely for some positive deterministic c. (This follows from the law of large numbers and the verifiable assumption that the process is ergodic.) Hence, H (x, t) converges to zero uniformly in x as t → ∞.
Simple Systems with Anomalous Dissipation and Energy Cascade
215
Writing (46) as H (x, t) = (Tt H0 )(x), it is easy to see that Tt defines a (Feller) semigroup with generator L defined by (1 − x 4 ) ∂ 2 f x2 3∂f − x − f 4 ∂x2 ∂x 2 x2 ∂f 1 ∂ (1 − x 4 ) − f = 4 ∂x ∂x 2
(L f )(x) =
for f ∈ C 2 ([−1, 1]). In addition, H (x, t) satisfies x2 ∂H 1 ∂ 4 ∂H (1 − x ) − = H, ∂t 4 ∂x ∂x 2
(52)
(53)
with initial condition H (x, 0) = H0 (x) for x ∈ [−1, 1]. One can check that the boundaries at x = ±1 are entrance boundaries for (53) and H (x, t) satisfies lim (1 − x 4 )
x→±1
∂H = 0. ∂x
(54)
Proof (Theorem 16). Noting that L x 2n−1 = −n(n + 21 )x 2n+1 + (n − 1)(n − 21 )x 2n−3 , we compute ∞
(−1)n+1 b˙n (t)x 2n−1
n=1
=
∞ (−1)n+1 (n − 1)(n − 21 )bn−1 (t) − n(n + 21 )bn+1 (t) x 2n−1 n=1
∞ = (−1)n+1 bn (t) −n(n + 21 )x 2n+1 + (n − 1)(n − 21 )x 2n−3 n=1 ∞ = (−1)n+1 bn (t)L(x 2n−1 ). n=1
Proof (Theorem 17). Associated with (53) we have the eigenvalue problem x2 1 d 4 dφ (1 − x ) − φ, −λφ = 4 dx dx 2
(55)
subject to the boundary conditions lim (1 − x 4 )
x→±1
∂φ = 0. ∂x
It is straightforward to see that the operator in (55) equipped with the boundary condition in (54) is self-adjoint in L 2 [−1, 1]. We now explain why this operator has discrete spectrum. A standard calculation shows that the boundary is an “entrance boundary” in
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J. C. Mattingly, T. Suidan, E. Vanden-Eijnden
the sense of Feller ([Fel54, McK56]), i.e. the diffusion (47), if started from the boundary, enters (−1, 1) and does not return to the boundary. 2 Define L V = 41 ∂x (1 − x 4 )∂x + V (x), where V (x) = − x2 . By standard PDE theory, ∂t u = L V u, subject to the condition limt→0 u(t, x) = δ y (x), has smooth solution in (−1, 1) for any t > 0. We denote this solution by ptV (x, y). For a fixed t > 0, ptV (x, y) is Lipschitz for x ∈ [−1 + , 1 − ] and y ∈ [−1, 1] with a fixed Lipschitz constant Ct and supx,y∈[−1,1] | ptV (x, y)| < Dt . Consider the solution to the following initial value problem: Let f ∈ L 2 ([−1, 1]) and solve ∂t u = L V u with u(0, x) = f (x). The solution is given by
u(t, x) =
ptV (x, y) f (y)dy.
u(t, x) is Lipschitz for x ∈ [−1 + , 1 − ] as the following simple estimate shows: 1 V V |u(t, x) − u(t, x )| = ( pt (x, y) − pt (x , y)) f (y)dy
≤ ≤
−1 1
−1
| ptV (x, y) − ptV (x , y)|| f (y)|dy
Ct |x
−x |
1
−1
| f (y)|dy ≤
2Ct |x
−x |
1
| f (y)| dy 2
−1
21
.
u(t, x) is also bounded in terms of f 2 as follows: |u(t, x)| =
1
−1
ptV (x,
y) f (y)dy ≤ Dt
1
−1
| f (y)|dy ≤ 2Dt
1 −1
| f (y)|
2
21
.
As one can see by a Cantor diagonalization argument in intervals In = [−1 + n1 , 1 − n1 ], V
Tt = et L is a compact self-adjoint operator. Therefore, the spectrum of L V is discrete. Note that the lowest eigenvalue has the following variational representation: 1 1 λ = inf φ
−1 4 (1 −
x 4 )(φ (x))2 + 21 x 2 φ 2 (x) d x , 1 2 −1 φ (x)d x
where the infimum is taken over L 2 equipped with the boundary conditions (54). This shows that the spectrum is strictly positive. Let {φk (x), λk }k∈N be the pair of eigenfunction and eigenvalues such that each φk (x) is odd in x (the even ones do not matter since the initial condition H0 (x) of (53) is odd from (48)). The solution of (53) can be represented as h k e−λk t φk (x), (56) H (x, t) = k∈N
Simple Systems with Anomalous Dissipation and Energy Cascade
where
hk =
1
−1
In turn, (56) implies that bn (t) =
217
H0 (x)φk (x)d x.
h k e−λk t pnk ,
(57)
(−1)n+1 pnk x 2n−1 .
(58)
k∈N
where pnk is defined by φk (x) =
∞ n=1
The pnk satisfy the following recurrence relation inherited from (11): k k −λk pnk = (n − 1)(n − 21 ) pn−1 − n(n + 21 ) pn+1 ,
n ∈ N, p0k = 0.
(59)
The following lemma describes the asymptotic behavior of npn . Lemma 1. For every λk > 0, the recurrence relation in (59) implies that k k lim (2n + 1) p2n+1 = ck1 and lim (2n) p2n = ck2 ,
n→∞
n→∞
(60)
where ck1 and ck2 are nonzero constants whose sign is the same as that of p1k . Proof. Assume p1 > 0 and write (59) as k pn+1 =
λk n(n + 21 )
pnk +
(n − 1)(n − 21 ) n(n + 21 )
k pn−1
n ∈ N, p0k = 0.
We omit the index k in this proof as it plays no role. For sufficiently large n and C depending only on λ, 2 C pn+1 ≤ 1 − + 2 max{ pn , pn−1 }. n n This implies { pn } is bounded. More is true: ! 21 m 2 C 1− + 2 max{ pn , pn−1 }. pm ≤ l l l=n
This implies that
m 1 2 C mpm ≤ exp log m + log 1 − + 2 max{ pn , pn−1 }, 2 l l l=n
which implies further that lim sup mpm < ∞. On the other hand, 2 pn+1 ≥ 1 − pn−1 , n+1
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which implies pm ≥
m l=n
This implies that
2 1− l +1
! 21 pn .
m 1 2 mpm ≥ exp log m − log 1 − pn if n-m=0 mod 2. 2 l +1 l=n
Thus, lim inf mpm > 0. To show that the sequences in the theorem are Cauchy simply compute ! λ 1 1 |(n + 1) pn+1 − (n − 1) pn−1 | = (n + 1) pn+1 + (n − 1) pn−1 − . n(n + 1) 2 n(n + 1 ) 2
Using the fact that {npn } is bounded in n and summing over n we see that the sequence is Cauchy and have proved the lemma. Going back to the proof of Theorem 17, using (60) in (57) implies (50) with C¯ 1 (t) = e−λk t h k ck1 and C¯ 2 (t) = e−λk t h k ck2 . k∈N
k∈N
Note that there is a T > 0 so that C¯ 1 (t)C¯ 2 (t) > 0 for all t ≥ T . Finally, (50) implies that ∞ 1 d 2 bn (t) = − lim N (N + 21 )b N (t)b N +1 (t) = −C¯ 1 (t)C¯ 2 (t) < 0 N →∞ 2 dt n=1
for all t ≥ T which proves (51).
We also consider the system of forced equations b˙n = (n − 1)(n − 21 )bn−1 − n(n + 21 )bn+1 + f (t)1n=m ,
(61)
for n = 1, 2, . . . with boundary condition b0 (t) = 0 for all t and f (t) is either a constant forcing term, f (t) = 1, or a white-noise process, f (t) = W˙ (t). We have Theorem 18. Consider (61) with f (t) = 1, and initial condition bn (0) satisfying (44). Then dk p k n lim bn (t) = bn∗ ≡ , t→∞ λk k∈N
where pkn is defined by (58) and
dk =
1
−1
φk (x)x m d x.
In particular, bn∗ satisfies ∗ ∗ lim (2n + 1)b2n+1 = C1∗ > 0 and lim (2n)b2n = C2∗ > 0.
n→∞
n→∞
Simple Systems with Anomalous Dissipation and Energy Cascade
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Theorem 19. Consider (61) with f (t) = W˙ (t), and initial condition bn (−T ) satisfying (44). Then t lim bn (t) = bn∗∗ (t) ≡ dk pnk e−λk s dW (s), T →∞
−∞
k∈N
where pnk is defined by (58). In particular, b∗∗ (t) is a Gaussian process with mean zero and covariance ∗∗ Ebn∗∗ (t)bm (t) =
dk dk p k p k n m , λ + λk k
k,k ∈N
and we have ∗∗ ∗∗ lim (2n + 1)2 E(b2n+1 (t))2 = C1∗∗ > 0 and lim (2n)2 E(b2n (t))2 = C2∗∗ > 0.
n→∞
n→∞
A. Estimates on Taylor Coefficients For the reader’s convenience, we now state a theorem on the asymptotic of Taylor’s series which can be found in [FO90]. Theorem 20. Let ∆(ζ, η, θ ) be as in (27). Assume that f (z) is analytic in ∆(ζ, η, θ )\{ζ } for some ζ ∈ C, η > 0, and 0 < θ < π/2. If f (z) ∼
K (ζ − z)α
as
z→ζ
for some K > 0 and α ∈ {0, −1, −2, · · · }, then fn ∼
K n α−1 , Γ (α) ζ n+α
where f n is the n th Taylor coefficient of f (z) about z = 0. Acknowledgements. We thank Percy Deift, Charles Fefferman, Stephanos Venakides and Xin Zhou for useful conservations. J. M. is supported in part by the Sloan Foundation and by an NSF CAREER award. T. S is supported in part by NSF grant DMS05-53403. E. V.-E. is supported in part by NSF grants DMS02-09959 and DMS02-39625, and by ONR grant N00014-04-1-0565.
References [CFP06a] [CFP06b] [CET94] [CR06] [DR00]
Cheskidov, A., Friedlander, S., Pavlovi´c, N.: An Inviscid Dyadic Model of Turbulence: The Global Attractor. arXiv:math.AP/06108115 v1, 2006 Cheskidov, A., Friedlander, S., Pavlovi´c, N.: An Inviscid Dyadic Model of Turbulence: The Fixed Point and Onsager’s Conjecture. http://arxiv.org/list/math.AP/0610814, 2006 Constantin, P., E, W., Titi, E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1), 207–209 (1994) Constantin, P., Ramos, F.: Inviscid limit for damped and driven incompressible Navier-Stokes equations in R2 . http://arxiv.org/list/math.AP/0611782 v1, 2006 Duchon, J., Robert, R.: Inertial energy dissipation for weak solutions of incompressible euler and Navier-Stokes equations. Nonlinearity 13(1), 249–255 (2000)
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J. C. Mattingly, T. Suidan, E. Vanden-Eijnden
E, W.: Stochastic hydrodynamics. In: Current developments in mathematics, 2000, Somerville, MA: Int. Press, 2001, pp. 109–147 Eyink, G.L.: Dissipation in turbulent solutions of 2d euler equations. Nonlinearity 14(4), 787–802 (2001) Feller, W.: The general diffusion operator and positivity preserving semi-groups in one dimension. Ann. of Math. (2) 60, 417–436 (1954) Falkovich, G., Gaw¸edzki, K., Vergassola, M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73(4), 913–975 (2001) Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3(2), 216–240 (1990) Frisch, U.: Turbulence. Cambridge: Cambridge University Press, 1995 Hilberdink, T.: A Tauberian theorem for power series. Arch. Math. (Basel) 77(4), 354–359 (2001) McKean, H.P. Jr.: Elementary solutions for certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82, 519–548 (1956) Srinivasan, R.: Simple models with cascade of energy and anomalous dissipation. In: Oliver Buhler, Charles Doering, ed. Fast times and fine scales, Woods Hole Oceanographic Institution Technical Reports. Woods Hole Oceanographic Institution, 2005
Communicated by P. Constantin
Commun. Math. Phys. 276, 221–259 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0332-1
Communications in
Mathematical Physics
On Minimal Affinizations of Representations of Quantum Groups David Hernandez CNRS - UMR 8100, Laboratoire de Mathématiques de Versailles, 45 avenue des Etats-Unis, Bat. Fermat, 78035 Versailles, France. E-mail: [email protected] Received: 21 July 2006 / Accepted: 22 March 2007 Published online: 9 September 2007 – © Springer-Verlag 2007
Abstract: In this paper we study minimal affinizations of representations of quantum groups (generalizations of Kirillov-Reshetikhin modules of quantum affine algebras introduced in [Cha1]). We prove that all minimal affinizations in types A, B, G are special in the sense of monomials. Although this property is not satisfied in general, we also prove an analog property for a large class of minimal affinizations in types C, D, F. As an application, the Frenkel-Mukhin algorithm [FM1] works for these modules. For minimal affinizations of type A, B we prove the thin property (the l-weight spaces are of dimension 1) and a conjecture of [NN1] (already known for type A). The proof of the special property is extended uniformly for more general quantum affinizations of quantum Kac-Moody algebras. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . 2. Background . . . . . . . . . . . . . . . . . . . 3. Minimal Affinizations and Main Results . . . . 4. Preliminary Results . . . . . . . . . . . . . . . 5. Proof of the Main Results . . . . . . . . . . . . 6. Applications and Further Possible Developments References . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction In this paper q ∈ C∗ is fixed and is not a root of unity. Affine Kac-Moody algebras gˆ are infinite dimensional analogs of semi-simple Lie algebras g, and have remarkable applications (see [Ka]). Their quantizations Uq (ˆg), called quantum affine algebras, have a very rich representation theory which has been intensively studied in mathematics and physics (see references in [CP6, DM] and in
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[CP2, FR, Nak1, Nak4] for various approaches). In particular Drinfeld [Dr2] discovered that they can also be realized as quantum affinizations of usual quantum groups Uq (g) ⊂ Uq (ˆg). By using this new realization, Chari-Pressley [CP6] classified their finite dimensional representations. Chari [Cha1] introduced the notion of minimal affinizations of representations of quantum groups: starting from a simple representation V of Uq (ˆg), an affinization of V is a simple representation Vˆ of Uq (ˆg) such that V is the head in the decomposition of Vˆ in simple Uq (g)-representations. Then one can define a partial ordering on the set of affinizations of V and so a notion of minimal affinization for this ordering. For example the minimal affinizations of simple Uq (g)-modules of highest weight a multiple of a fundamental weight are the Kirillov-Reshetikhin modules which have been intensively studied in recent years (for example see [KOS, KNH, Kl, HKOTY, KN, Cha2, Nak4, Nak5, H4, CM3, FL] and references therein). An (almost) complete classification of minimal affinizations was done in [Cha1, CP3, CP4, CP5]. The motivation to study minimal affinizations comes from physics: the affinizations of representations of quantum groups are important objects from the physical point of view, as stressed for example in [FR, Remark 4.2] and in the introduction of [Cha1]. For example in the theory of lattice models in statistical mechanics, they are related to the problem of proving the integrability of the model: the point is to add spectral parameters to a solution of the related quantum Yang-Baxter equation (see [CP6]). A second example is related to the quantum particles of the affine Toda field theory (see [BL, Do]) which correspond to simple finite dimensional representations of quantum affine algebras. In the present paper we prove new results on the structure of minimal affinizations, in particular in the light of recent developments in the representation theory of quantum affine algebras. A particular class of finite dimensional representations, called special modules [Nak4], attracted much attention as Frenkel-Mukhin [FM1] proposed an algorithm which gives their q-character (analog of the usual character adapted to the Drinfeld realization and introduced by Frenkel-Reshetikhin [FR]: they encode a certain decomposition of representations in so called l-weight spaces or pseudo weight spaces). For example the Kirillov-Reshetikhin modules [Nak4, Nak5, H4] are special (this is the crucial point of the proof of the Kirillov-Reshetikhin conjecture). A dual class of modules called antispecial modules is introduced in the present paper (antispecial does not mean the opposite of special), and an analog of the Frenkel-Mukhin algorithm gives their q-character. In the present paper we prove that minimal affinizations in type A, B, G are special and antispecial. We get counterexamples for other types, but we prove in type C, D, F that a large class of minimal affinizations are special or antispecial. In particular the Frenkel-Mukhin algorithm works for these modules. As an application, we prove that minimal affinizations of type A and B are thin (the l-weight spaces are of dimension 1). We also get the special property for analog simple modules of quantum affinizations of some non-necessarily finite quantum Kac-Moody algebras. In the proofs of the present paper, the crucial steps include techniques developed in [H4] to prove the Kirillov-Reshetikhin conjecture and in [H6] to solve the Nakajima’s smallness problem. The general idea is to prove simultaneously the special property and the thin property by induction on the highest weight of the minimal affinizations. This allows to use the elimination theorem [H4] which leads to eliminate some monomials in the q-character of simple modules.
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Nakajima first conjectured the existence of such large classes of special modules for simply-laced cases (see [Nak4]), and the existence of a large class of special minimal affinizations was conjectured by Mukhin in a conversation with the author in the conference “Representations of Kac-Moody Algebras and Combinatorics” at Banff in March 2005. In some situations, the properties are already known or can be proved directly from already known explicit formulas. Indeed, for Kirillov-Reshetikhin modules the special property was proved in [Nak5] (simply-laced case) and in [H4] (non-simply-laced case). So for Kirillov-Reshetikhin modules in classical types, the explicit formulas in [KOS, KNH] are satisfied (the formulas for fundamental representations are given in [KS]) and we can get the properties directly from them. General formulas and the thin properties were proved for irreducible tame modules, which include minimal affinizations, for Yangians of type A [Che1, Che2, NT]. (The author was told by Nakajima that the same results hold for quantum affine algebras of type A by [V].) See also [FM2] for the cases of minimal affinizations, which are evaluation representations in type A. Explicit formulas are also available for twisted yangians in classical types [Mo, Naz1]. But the author did not find in the literature a proof of the correspondence between quantum affine algebras and twisted (or non-simply laced) yangians. In general no explicit formulas for q-characters of quantum affine algebras are available, so our proofs use direct arguments without explicit formulas and are independent of previous results on yangians. In particular this allows to extend uniformly our arguments to previously unknown situations (like type B, C, D, G 2 , F4 ), and to more general quantum affinizations which are not necessarily quantum affine algebras. For quantum affine algebras in classical types, explicit conjectural formulas [NN1, NN2, NN3] are available for a large class of representations including many minimal affinizations (all of them for type A; see [KOS, KNH] for more general formulas). In types A, B, the results proved in the present paper imply [NN1, Conjecture 2.2] for these minimal affinizations. The author did not find in the literature a proof of this result, except for type A as explained above. The main subject of the present paper is minimal affinizations and so we give a proof of [NN1, Conjecture 2.2] in this case. But it is possible to prove [NN1, Conjecture 2.2] for more general representations by using a variation of this proof (this and [NN1, Conjecture 2.2] in types C, D will be discussed in a separate publication). The results of [NT, KS, KOS] and of [NN1, Conjecture 2.2] (and the thin property as their consequence) were explained to the author by Nakajima in an early stage of this research, June 2005. Let us describe the organization of the present paper. In Sect. 2 we give some background on the representation theory of quantum affine algebras. In Sect. 3 we recall the definition of minimal affinizations and state the main results of the paper. In Sect. 4 we give preliminary results, including results from [H6] and discussion about an involution of Uq (ˆg). In Sect. 5 we prove the main result of the paper. In Sect. 6 we explain the proof of [NN1, Conjecture 2.2] for minimal affinizations in types A, B, we state additional results (Theorem 6.6) for more general quantum affinizations, and we discuss possible further developments, in particular on generalized induction systems involving minimal affinizations. 2. Background 2.1. Cartan matrix and quantized Cartan matrix. Let C = (Ci, j )1≤i, j≤n be a Cartan matrix of finite type. We denote I = {1, . . . , n}. C is symmetrizable: there is a matrix
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D = diag(r1 , . . . , rn ) (ri ∈ N∗ ) such that B = DC is symmetric. In particular if C is symmetric then D = In (simply-laced case). We consider a realization (h, , ∨ ) of C (see [B, Ka]): h is a n dimensional Q-vector space, = {α1 , . . . , αn } ⊂ h∗ (set of the simple roots) and ∨ = {α1∨ , . . . , αn∨ } ⊂ h (set of simple coroots) are set such that for 1 ≤ i, j ≤ n, α j (αi∨ ) = Ci, j . Let ∨ 1 , . . . , n ∈ h∗ (resp. ∨ 1 , . . . , n ∈ h) be the fundamental weights (resp. coweights): ∨ ∨ i (α j ) = αi ( j ) = δi, j , where δi, j is 1 if i = j and 0 otherwise. Denote P = {λ ∈ h∗ |∀i ∈ I, λ(αi∨ ) ∈ Z} the set of weights and P + = {λ ∈ P|∀i ∈ I, λ(αi∨ ) ≥ 0} the set of dominant αn ∈ P and weights. For example we have α1 , . . . , 1 , . . . , n ∈ P + . Denote Q = i∈I Zαi ⊂ P the root lattice and Q + = i∈I Nαi ⊂ Q. For λ, µ ∈ h∗ , denote λ ≥ µ if λ − µ ∈ Q + . Let ν : h∗ → h linear such that for all i ∈ I we have ν(αi ) = ri αi∨ . For λ, µ ∈ h∗ , λ(ν(µ)) = µ(ν(λ)). We use the enumeration of vertices of [Ka]. We denote qi = q ri and for l ∈ Z, r ≥ 0, m ≥ m ≥ 0 we define in Z[q ± ]: [m]q ! q l − q −l m . [l]q = , [r ]q ! = [r ]q [r − 1]q · · · [1]q , = −1 m q −q [m − m ]q ![m ]q ! q For a, b ∈ Z, we denote q a+bZ = {q a+br |r ∈ Z} and q a+bN = {q a+br |r ∈ Z, r ≥ 0}. Let C(z) be the quantized Cartan matrix defined by (i = j ∈ I ): Ci,i (z) = z i + z i−1 , Ci, j (z) = [Ci, j ]z . ˜ C(z) is invertible (see [FR]). We denote by C(z) the inverse matrix of C(z) and by D(z) the diagonal matrix such that for i, j ∈ I , Di, j (z) = δi, j [ri ]z . 2.2. Quantum algebras 2.2.1. Quantum groups Definition 2.1. The quantum group Uq (g) is the C-algebra with generators ki±1 , xi± (i ∈ I ) and relations: ±Ci, j ± x j ki ,
ki k j = k j ki , ki x ± j = qi [xi+ , x − j ] = δi, j
ki − ki−1
qi − qi−1
,
1 − Ci, j ± r (−1) (xi± )1−Ci, j −r x ± j (x i ) = 0 (for i = j). r r =0···1−Ci, j q
r
i
This algebra was introduced independently by Drinfeld [Dr1] and Jimbo [J]. It is remarkable that one can define a Hopf algebra structure on Uq (g) by: (ki ) = ki ⊗ ki , (xi+ ) = xi+ ⊗ 1 + ki ⊗ xi+, (xi− ) = xi− ⊗ ki−1 + 1 ⊗ xi− ,
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S(ki ) = ki−1 , S(xi+ ) = −xi+ ki−1, S(xi− ) = −ki xi− , (ki ) = 1, (xi+ ) = (xi− ) = 0. Let Uq (h) be the commutative subalgebra of Uq (g) generated by the ki±1 (i ∈ I ). For V a Uq (h)-module and ω ∈ P we denote by Vω the weight space of weight ω: ω(αi∨ )
Vω = {v ∈ V |∀i ∈ I, ki .v = qi
v}.
In particular we have xi± .Vω ⊂ Vω±αi . We say that V is Uq (h)-diagonalizable if V = ω∈P Vω (in particular V is of type 1). For V a finite dimensional Uq (h)-diagonalizable module we define the usual character dim(Vω )e(ω) ∈ E = Z.e(ω). χ (V ) = ω∈P
ω∈P
2.2.2. Quantum loop algebras. We will use the second realization (Drinfeld realization) of the quantum loop algebra Uq (Lg) (subquotient of the quantum affine algebra Uq (ˆg)): ± (i ∈ I, r ∈ Z), ki±1 (i ∈ I ), Definition 2.2. Uq (Lg) is the algebra with generators xi,r h i,m (i ∈ I, m ∈ Z − {0}) and the following relations (i, j ∈ I, r, r ∈ Z, m, m ∈ Z − {0}):
[ki , k j ] = [ki , h j,m ] = [h i,m , h j,m ] = 0, ±Ci, j ± x j,r ki ,
ki x ± j,r = qi [h i,m , x ± j,r ] = ±
1 [m Bi, j ]q x ± j,m+r , m
+ , x− [xi,r j,r ] = δi, j
− + φi,r +r − φi,r +r
qi − qi−1
,
± ± ± ± ±Bi, j ± ±Bi, j ± ± xi,r x j,r xi,r xi,r x j,r +1 − x ± +1 x j,r − q +1 = q j,r +1 x i,r ,
s ± ± (−1) x± · · · xi,r x± x± · · · xi,r = 0, π(k) j,r i,rπ(k+1) π(s) k q i,rπ(1) k=0···s
π ∈ s
k
i
where the last relation holds for all i = j, s = 1 − Ci, j , all sequences of integers ± ∈ Uq (Lg) r1 , . . . , rs . s is the symmetric group on s letters. For i ∈ I and m ∈ Z, φi,m is determined by the formal power series in Uq (Lg)[[z]] (resp. in Uq (Lg)[[z −1 ]]): ± φi,±m z ±m = ki± exp(±(q − q −1 ) h i,±m z ±m ), m≥0
and
± φi,∓m
= 0 for m > 0.
m ≥1
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Uq (Lg) has a Hopf algebra structure (from the Hopf algebra structure of Uq (ˆg)). ± , For J ⊂ I we denote by Uq (Lg J ) ⊂ Uq (Lg) the subalgebra generated by the xi,m ±1 h i,m , ki for i ∈ J . Uq (Lg J ) is a quantum loop algebra associated to the semi-simple Lie algebra g J of Cartan matrix (Ci, j )i, j∈J . For example for i ∈ I , we denote Uq (Lgi ) = Uq (Lg{i} ) Uqi (Lsl2 ). ± ) is denoted The subalgebra of Uq (Lg) generated by the h i,m , ki±1 (resp. by the xi,r ± by Uq (Lh) (resp. Uq (Lg) ). 2.3. Finite dimensional representations of quantum loop algebras Denote by Rep (Uq (Lg)) the Grothendieck ring of (type 1) finite dimensional representations of Uq (Lg). 2.3.1. Monomials and q-characters. Let V be a representation in Rep(Uq (Lg)). The subalgebra Uq (Lh) ⊂ Uq (Lg) is commutative, so we have: V = Vγ , ± γ =(γi,±m )i∈I,m≥0
± ± where: Vγ = {v ∈ V |∃ p ≥ 0, ∀i ∈ I, m ≥ 0, (φi,±m − γi,±m ) p .v = 0}. ± The γ = (γi,±m )i∈I,m≥0 are called l-weights (or pseudo-weights) and the Vγ = {0} are called l-weight spaces (or pseudo-weight spaces) of V . One can prove [FR] that γ is necessarily of the form:
m≥0
deg(Q i )−deg(Ri )
± γi,±m u ±m = qi
Q i (uqi−1 )Ri (uqi ) Q i (uqi )Ri (uqi−1 )
,
(1)
where Q i , Ri ∈ C(u) satisfy Q i (0) = Ri (0) = 1. The Frenkel-Reshetikhin q-characters morphism χq [FR] encodes the l-weights γ (see also [Kn]). It is an injective ring morphism: ± χq : Rep(Uq (Lg)) → Z[Yi,a ]i∈I,a∈C∗
defined by
χq (V ) = where: mγ = Q i (u) =
a∈C∗
γ
dim(Vγ )m γ ,
q −r Y i,a i,a , i∈I,a∈C∗ i,a
(1 − ua)qi,a , Ri (u) =
a∈C∗
(1 − ua)ri,a .
The m γ are called monomials (they are analogs of weight). We denote by A the set of ± ]i∈I,a∈C∗ . For an l-weight γ , we denote Vγ = Vm γ . We will also monomials of Z[Yi,a p p use the notation ir = Yi,q r for i ∈ I and r, p ∈ Z. For J ⊂ I , χqJ is the morphism of q-characters for Uq (Lg J ) ⊂ Uq (Lg).
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u i,a (m) For a m monomial we denote u i,a (m) ∈ Z such that m = . i∈I,a∈C∗ Yi,a We also denote ω(m) = i∈I,a∈C∗ u i,a (m)i , u i (m) = a∈C∗ u i,a (m) and u(m) = ∗ i∈I u i (m). m is said to be J -dominant if for all j ∈ J, a ∈ C we have u j,a (m) ≥ 0. An I -dominant monomials is said to be dominant. Observe that χq , χqJ can also be defined for finite dimensional Uq (Lh)-modules in the same way. In the following for V a finite dimensional Uq (Lh)-module, we denote by M(V ) the set of monomials occurring in χq (V ). For i ∈ I, a ∈ C∗ we set: −1 Ai,a =Yi,aq −1 Yi,aqi Y j,a i
×
{ j|C j,i =−1}
{ j|C j,i =−2}
−1 −1 Y j,aq −1 Y j,aq
{ j|C j,i =−3}
−1 −1 −1 Y j,aq 2 Y j,a Y j,aq −2 .
(2)
−1 As the Ai,a are algebraically independent [FR] (because C(z) is invertible), for M a −v (m) −1 product of Ai,a we can define vi,a (M) ≥ 0 by M = i∈I,a∈C∗ Ai,a i,a . We put vi (M) = ∨ + a∈C∗ vi,a (M) and v(M) = i∈I vi (M). For λ ∈ −Q we set v(λ) = −λ(1 + · · · + −1 ∨ n ). For M a product of Ai,a , we have v(M) = v(ω(λ)). For m, m two monomials, −1 . we write m ≤ m if m m −1 is product of Ai,a
Definition 2.3. [FM1] A monomial m ∈ A − {1} is said to be right-negative if for all a ∈ C∗ , for L = max{l ∈ Z|∃i ∈ I, u i,aq L (m) = 0} we have ∀ j ∈ I , u j,aq L (m) ≤ 0. Observe that a right-negative monomial is not dominant. We can also define left-negative monomials by replacing max by min in the formula of L in Definition 2.3. Lemma 2.4. [FM1] −1 1) For i ∈ I, a ∈ C∗ , Ai,a is right-negative. 2) A product of right-negative monomials is right-negative. 3) If m is right-negative, then m ≤ m implies that m is right-negative.
We have the same results by replacing right-negative by left-negative. For J ⊂ I and Z ∈ Y, we denote Z →J the element of Y obtained from Z by putting ±1 ± / J . Let β : Z[Y j,b ] j∈I,b∈C∗ → E be the ring morphism such that Y j,a = 1 for j ∈ β(m) = e(ω(m)). Proposition 2.5. [FR, Theorem 3] For V ∈ Rep(Uq (Lg)), let Res(V ) be the restricted Uq (g)-module. We have (β ◦ χq )(V ) = χ (Res(V )). 2.3.2. l-highest weight representations. The irreducible finite dimensional Uq (Lg)-modules have been classified by Chari-Pressley. They are parameterized by dominant monomials: Definition 2.6. A Uq (Lg)-module V is said to be of l-highest weight m ∈ A if there is + .v = 0. v ∈ Vm such that V = Uq (Lg)− .v and ∀i ∈ I, r ∈ Z, xi,r For m ∈ A, there is a unique simple module L(m) of l-highest weight m.
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Theorem 2.7. [CP6, Theorem 12.2.6] The dimension of L(m) is finite if and only if m is dominant. (i) For i ∈ I , a ∈ C∗ , k ≥ 0 we denote X k,a = k ∈{1,...,k} Yi,aq k−2k +1 . i
(i) (i) Definition 2.8. The Kirillov-Reshetikhin modules are the Wk,a = L(X k,a ). (i) We denote by W0,a the trivial representation (it is of dimension 1). For i ∈ I and (i)
a ∈ C∗ , W1,a is called a fundamental representation and is denoted by Vi (a) (in the case g = sl2 we simply write Wk,a and V (a)). For g = sl2 , the monomials m 1 = X k1 ,a1 , m 2 = max(u a (m 1 ),u a (m 2 )) X k2 ,a2 are said to be in special position if the monomial m 3 = a∈C∗ Ya is of the form m 3 = X k3 ,a3 and m 3 = m 1 , m 3 = m 2 . A normal writing of a dominant monomial m is a product decomposition m = i=1,...,L X kl ,al such that for l = l , X kl ,al , X kl ,al are not in special position. Any dominant monomial has a unique normal writing up to permuting the monomials (see [CP6, Sect. 12.2]). It follows from the study of the representations of Uq (Lsl2 ) in [CP1, CP2, FR] that: Proposition 2.9. Suppose that g = sl2 . (1) Wk,a is of dimension k + 1 and: −1 −1 −1 χq (Wk,a ) = X k,a (1 + Aaq k (1 + Aaq k−2 (1 + · · · (1 + Aaq 2−k )) · · · ).
(2) V (aq 1−k ) ⊗ V (aq 3−k ) ⊗ · · · ⊗ V (aq k−1 ) is of q-character: −1 −1 −1 X k,a (1 + Aaq k )(1 + Aaq k−2 ) · · · (1 + Aaq 2−k ).
In particular all l-weight spaces of the tensor product are of dimension 1. (3) For m a dominant monomial and m = X k1 ,a1 · · · X kl ,al a normal writing we have: L(m) Wk1 ,a1 ⊗ · · · ⊗ Wkl ,al . 2.3.3. Special modules and complementary reminders Definition 2.10. For m ∈ A let D(m) be the set of monomials m ∈ A such that there are m 0 = m, m 1 , . . . , m N = m ∈ A satisfying for all j ∈ {1, . . . , N }: (1) m j = m j−1 Ai−1 · · · Ai−1 j ,a1 qi j ,ar j
j qi j
, where i j ∈ I , r j ≥ 1 and a1 , . . . , ar j ∈ C∗ ,
(2) for 1 ≤ r ≤ r j , u i j ,ar (m j−1 ) ≥ |{r ∈ {1, . . . , r j }|ar = ar }|, where r j , i j , ar are as in condition (1). For all m ∈ D(m), m ≤ m. Moreover if m ∈ D(m), then (D(m ) ⊂ D(m)).
Theorem 2.11. [H5, Theorem 5.21] For V ∈ Mod(Uq (ˆg)) a l-highest weight module of highest monomial m, we have M(V ) ⊂ D(m). In particular for all m ∈ M(V ), we have m ≤ m and the vi,a (m m −1 ), v(m m −1 ) ≥ 0 are well-defined. As a direct consequence of Theorem 2.11, we also have: ± Lemma 2.12. For i ∈ I, a ∈ C∗ , we have (χq (Vi (a)) − Yi,a ) ∈ Z[Y j,aq l ] j∈I,l>0 .
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This result was first proved in [FM1, Lemma 6.1, Remark 6.2]. A monomial m is said to be antidominant if for all i ∈ I, a ∈ C∗ , u i,a (m) ≤ 0. Definition 2.13. A Uq (Lg)-module is said to be special (resp. antispecial) if its q-character has a unique dominant (resp. antidominant) monomial. The notion of special module was introduced in [Nak4]. It is of particular importance because an algorithm of Frenkel-Mukhin [FM1] gives the q-character of special modules. It is easy to write a similar algorithm for antispecial modules from the Frenkel-Mukhin algorithm (for example it suffices to use the involution studied in Sect. 4.2). Observe that a special (resp. antispecial) modules is a simple l-highest weight modules. But in general all simple l-highest weight modules are not special. The following result was proved in [Nak4, Nak5] for simply-laced types, and in full generality in [H4] (see [FM1] for previous results). Theorem 2.14. [H4, Theorem 4.1, Lemma 4.4] The Kirillov-Reshetikhin modules are (i) (i) (i) special. Moreover for m ∈ M(Wk,a ), m = X k,a implies m ≤ X k,a A−1 k . i,aqi
Define µ IJ : Z[(A±j,a )→(J ) ] j∈J,a∈C∗ → Z[A±j,a ] j∈J,a∈C∗ , the ring morphism such that µ IJ ((A±j,a )→(J ) ) = A±j,a . For m J -dominant, denote by L J (m →(J ) ) the simple Uq (Lg J )-module of l-highest weight m →(J ) . Define: L J (m) = mµ IJ ((m →(J ) )−1 χqJ (L J (m →(J ) ))). We have: Proposition 2.15. [H6] For a representation V ∈ Rep(Uq (Lg)) and J ⊂ I , there is unique decomposition in a finite sum: χq (V ) = λ J (m )L J (m ). (3) m J -dominant
Moreover for all m J -dominant we have λ J (m ) ≥ 0. Remark 2.16. Let m be a dominant monomial and m ∈ M(L(m)) a J -dominant monomial such that there are no m > m satisfying m ∈ M(m) and m appears in L J (m ). Then from Proposition 2.15 the monomials of L J (m ) are in M(L(m)). It gives inductively from m a set of monomials occurring in χq (L(m)). 2.3.4. Thin modules Definition 2.17. [H6] A Uq (Lg)-module V is said to be thin if its l-weight spaces are of dimension 1. For example for g of type A, B, C, all fundamental representations are thin (it can be established directly from the formulas in [KS]; this thin property was observed and proved with a different method in [H3, Theorem 3.5]; see also [CM2]). Observe that it follows from [H1, Sect. 8.4] that for g of type G 2 , all fundamental representations are thin. For g of type F4 , the fundamental representations corresponding
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to i = 1 and i = 4 are thin, but the fundamental representations corresponding to i = 2 or i = 3 are not thin (see [H3]). For type D, it is known that fundamental representations are not necessarily thin: for example for g of type D4 , the fundamental representations V2 (a) have an l-weight space of dimension 2. Explicit formulas for the q-character of fundamental representation in type D are given in [KS] (the thin fundamental representations of type D are also characterized in [CM2]; see also Remark 2.19 below for a general statement). For m ∈ Z[Yi,a ]i∈I,a∈C∗ a dominant monomial, the standard module M(m) is defined as the tensor product : M(m) =
a∈(C∗ /q Z )
(· · · ⊗ (
Vi (aq)⊗u i,aq (m) ) ⊗ (
i∈I
Vi (aq 2 )
⊗u i,aq 2 (m)
) ⊗ · · · ).
i∈I
It is well-defined as for i, j ∈ I and a ∈ C∗ we have Vi (a) ⊗ V j (a) V j (a) ⊗ Vi (a) and for a ∈ / aq Z , we have Vi (a) ⊗ V j (a ) V j (a ) ⊗ Vi (a). Observe that fundamental representations are particular cases of standard modules. As a direct corollary of a result of Nakajima, there is the following result for simplylaced types: wi ∗ Corollary 2.18. We suppose that g is simply-laced. Let m = i∈I Yi,aq φi where a ∈ C , wi ≥ 0 and φi ∈ Z satisfies (Ci, j < 0 ⇒ |φi − φ j | = 1). Then the standard module M(m) is thin if and only if it is simple as a Uq (g)-module. Proof. It follows from [Nak3, Proposition 3.4] that in this situation the number of monomials inχq (M(m))) is equal to the dimension of the simple Uq (g)-module of highest weight i∈I wi i . Observe that this result is false for not simply-laced g (for example there is a thin fundamental representation for type G 2 which is not simple as a Uq (g)-module, see [H1, Sect. 8.4]). The following remark was communicated to the author by Nakajima: Remark 2.19. In particular for g simply-laced, a fundamental representation is thin if and only if the corresponding coefficient of the highest root is 1 (this point is also a trivial consequence of previously known results, for example the geometric construction [Nak1]). We got also the following example: Proposition 2.20. [H6, Proposition 6.6] Let g = sln+1 and consider a monomial of the form m = Yi1 ,aq l1 Yi2 ,aq l2 · · · Yi R ,aq l R , where R ≥ 0, i 1 , i 2 , . . . , i R ∈ I , l1 , l2 , . . . , l R ∈ Z, satisfying for all 1 ≤ r ≤ R − 1, lr +1 − lr ≥ ir + ir +1 . Then L(m) is thin.
3. Minimal Affinizations and Main Results In this section we recall the definition of minimal affinizations and their classification in regular cases. Then we state the main results which are proved in other sections.
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3.1. Definitions [Cha1] Definition 3.1. For V a simple finite dimensional Uq (g)-module, a simple finite dimensional Uq (Lg)-module L(m) is said to be an affinization of V if ω(m) is the highest weight of V . For V a Uq (g)-module and λ ∈ P + , denote by m λ (V ) the multiplicity in V of the simple Uq (g)-module of highest weight λ. Two affinizations are said to be equivalent if they are isomorphic as Uq (g)-modules. Denote by QV the equivalence classes of affinizations of V and for L an affinization of V denote by [L] ∈ QV its classes. For [L], [L ] ∈ QV , we write [L] ≤ [L ] if and only if for all µ ∈ P + , either (i) m µ (L) ≤ m µ (L ), (ii) ∃ν > µ such that m ν (L) < m µ (L ). Proposition 3.2. ≤ defines a partial ordering on QV . Definition 3.3. A minimal affinization of V is a minimal element of QV for the partial ordering. Remark 3.4. For g = sln+1 , we have evaluation morphisms Uq (Lg) → Uq (g) denoted by eva and ev a (for a ∈ C∗ ) and in particular a minimal affinization L of V is isomorphic to V as a Uq (g)-module. 3.2. Classification The minimal affinizations were classified in [Cha1, CP3, CP4, CP5] for all types, except for type D, E for a weight orthogonal to the special node. For the regular cases (i.e. with a linear Dynkin diagram, that is to say types A, B, C, F4 , G 2 ), the classification is complete: Theorem 3.5. [Cha1, CP3, CP4] Suppose that g is regular and let λ ∈ P + . For i ∈ I let λi = λ(αi∨ ) and for i < n let ci (λ) = q ri λi +ri+1 λi+1 +ri+1 −Ci+1,i −1 and ci (λ) = q ri λi +ri+1 λi+1 +ri −Ci,i+1 −1 . Then a simple Uq (Lg)-module L(m) is a minimal affinization of V (λ) if and only if m (i) is of the form m = i∈I X λi ,ai with (ai )i∈I ∈ (C∗ ) I satisfying one of two conditions: (I) For all i < j ∈ I , ai /a j = i≤s< j cs (λ). (II) For all i < j ∈ I , a j /ai = i≤s< j cs (λ). Observe that we have rewritten the defining formulas of cs , cs [Cha1, CP3, CP4] in a slightly different (more homogeneous) way. Observe that for classical types, minimal affinizations (called generalized Kirillov-Reshetikhin modules) were also studied in [GK]. Remark 3.6. As a consequence of Theorem 3.5, for k ≥ 0 and i ∈ I , the minimal affinizations of V (ki ) are the Kirillov-Reshetikhin modules. For g of type D, and λ ∈ P + , we define with the same formulas ci (λ) for i < n − 1, (i) and we set cn−1 (λ) = q λn−2 +λn +1 . For a monomial m = i∈I X λi ,ai we have analog conditions (I) and (II):
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(I) For all i < j ∈ I , ai /a j = (c j−1 (λ)) j i≤s≤min( j−1,n−3) cs (λ), (II) For all i < j ∈ I , a j /ai = (c j−1 (λ)) j i≤s≤min( j−1,n−3) cs (λ), where j = 0 if j ≤ n − 2 and n−1 = n = 1. It follows from [CP3, Theorem 6.1] that if λn−2 = 0 and m = i∈I X λ(i)i ,ai satisfies (I ) or (I I ), then L(m) is a minimal affinization of V (λ). 3.3. Main results It follows directly from Theorem 2.14 and Remark 3.6 that (see also Proposition 6.8 for an alternative general proof): Corollary 3.7. For i ∈ I and k ≥ 0, the minimal affinizations of V (ki ) are special. In general a minimal affinization is not special. Let us look at some examples. First we consider type C. If m satisfies condition (I I ) of Theorem 3.5, L(m) is not necessarily special. For example consider the case g of type C3 and m = Y2,1 Y2,q 2 Y3,q 7 . L(m) is a minimal affinization of V (22 + 3 ). By the process described in Remark 2.16, −1 −1 −1 −1 −1 −1 2 the monomials 11 13 22−1 24−1 31 33 37 , 1−1 3 15 31 33 37 , 13 15 22 24 35 33 37 , 26 24 33 37 and 33 occur in χq (L(m)) and so L(m) is not special. If m satisfies condition (I ) of Theorem 3.5, L(m) is not necessarily special. For example consider the case g of type C3 and m = Y1,q 3 Y1,q 5 Y1,q 7 Y2,1 . L(m) is a minimal affinization of V (31 + 2 ). By the process described in Remark 2.16, the monomials 11 13 15 17 22−1 31 , 11 13 15 17 24 3−1 5 , −1 2 11 13 15 17 26 , 11 13 15 occur in χq (L(m)) and so L(m) is not special. Eventually, let m = Y1,1 Y1,q 2 Y1,q 4 Y2,q 7 Y2,q 9 Y3,q 14 . We can see as for Y2,1 Y2,q 2 Y3,q 7 that L(m) is not special. Moreover L(m) is antispecial if and only if the module L(Y1,q 14 Y1,q 12 Y1,q 10 Y2,q 7 Y2,q 5 Y3,1 ) is special (see Lemma 4.10 and Corollary 4.11 below). But we can check as for Y1,q 3 Y1,q 5 Y1,q 7 Y2,1 that this module is not special. So L(m) is not special and not antispecial. For types D, there are minimal affinizations which are not special. For example let g of type D4 and m = Y1,q 3 Y1,q 5 Y2,1 . Then L(m) is not special (see [H6, Remark 6.8]). However we prove in this paper: Theorem 3.8. For g of type A, B, G, all minimal affinizations are special and antispecial. Theorem 3.9. For g of type C, F4 and λ ∈ P satisfying λn = 0, all minimal affinizations of V (λ) satisfying (I) (resp. (II)) are antispecial (resp. special). For g of type D and λ ∈ P satisfying λn−1 = λn , all L(m) satisfying (I) (resp. (II)) and an−1 = an are antispecial (resp. special). Note that for type D, the condition an−1 = an is automatically satisfied if λ j = 0 for one j ≤ n − 2. Theorem 3.10. For g of type A, B, all minimal affinizations are thin. Theorems 3.8, 3.9 and 3.10 are proved in Sect. 5. Note for type C, there are minimal affinizations which are not thin: for example consider g of type C4 and m = Y3,1 Y3,q 2 . L(m) is a Kirillov-Reshetikhin and a minimal affinization of V (23 ). By the process described in Remark 2.16, the following −1 −1 −1 −1 monomials occur in χq (L(m)): 31 32 , 21 23 3−1 2 34 41 43 , 21 23 45 43 , 14 21 25 34 45 43 , −1 −1 −1 −1 −1 −1 2 14 21 36 43 , 14 21 34 47 , 16 21 25 34 47 and 16 21 25 36 45 47 . And so by Proposition
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2.15 and Proposition 2.9 the monomial 21 25 27−1 326 45 47−1 occurs in χq (L(m)) with multiplicity larger than 2. For type G 2 , there are minimal affinizations which are not thin: for example let m = Y2,0 Y2,2 . L(m) is a Kirillov-Reshetikhin and a minimal affinization of V (22 ). −1 −1 2 We have 20 22 , 11 13 24−1 22−1 , 1−1 7 19 24 26 28 ∈ M(L(m)), and so Y1,9 Y2,4 Y2,6 occurs in χq (L(m)) with multiplicity larger than 2. 4. Preliminary Results In this section g is an arbitrary semi-simple Lie algebra. We discuss preliminary results which will be used in the proof of Theorem 3.8, 3.9 and 3.10 in the next section. First it is well known that: Lemma 4.1. Let L(m 1 ), L(m 2 ) be two simple modules. Then L(m 1 m 2 ) is a subquotient of L(m 1 ) ⊗ L(m 2 ). In particular M(L(m 1 m 2 )) ⊂ M(L(m 1 ))M(L(m 2 )). 4.1. Results of [H6]
All results of this subsection are preliminary results of [H6].
Lemma 4.2. Let a ∈ C∗ and m be a monomial of Z[Yi,aq r ]i∈I,r ≥0 . Then for m ∈ M(L(m)) and b ∈ C∗ , (vi,b (m m −1 ) = 0 ⇒ b ∈ aq ri +N ). Lemma 4.3. Let V ∈ Rep(Uq (Lg)) be a Uq (Lg)-module and m ∈ M(V ) such that there is i ∈ I satisfying Min{u i,a (m )|a ∈ C∗ } ≤ −2. Then there is m > m in M(V ) i-dominant such that Max{u i,b (m )|b ∈ C∗ } ≥ 2. We recall [H6] that a monomial m is said to be thin if Maxi∈I,a∈C∗ |u i,a (m)| ≤ 1. Proposition 4.4. If V is thin then all m ∈ M(V ) are thin. If V is special and all m ∈ M(V ) are thin, then V is thin. Lemma 4.5. Let m dominant and J ⊂ I . Let v be an highest weight vector of L(m) and L ⊂ L(m) be the Uq (Lg J )-submodule of L(m) generated by v. Then L is an Uq (Lh)-submodule of L(m) and χq (L ) = L J (m). In particular for µ ∈ ω(m) − j∈J Nα j , we have
where µ→(J ) =
dim((L(m))µ ) = dim((L J (m →(J ) ))µ→(J ) ), ∨ j∈J µ(α j )ω j .
Lemma 4.6. Let V = L(m) be a Uq (Lg)-module simple module and consider a monomial m ∈ (M(L(m)) − {m}). Then there is j ∈ I and M ∈ M(V ) j-dominant such that M > m , M ∈ m Z[A j,b ]b∈C∗ and ((Uq (Lg j ).VM ) ∩ (M)m ) = {0}. We have the following elimination theorem: Theorem 4.7. Let V = L(m) be a Uq (Lg)-module simple module. Let m < m and i ∈ I satisfying the following conditions: (i) there is a+ unique i-dominant M ∈ (M(V ) ∩ m Z[Ai,a ]a∈C∗ ) − {m }, (ii) x (V ) = {0}, M r ∈Z i,r
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(iii) m is not a monomial of L i (M), (iv) if m ∈ M(Uq (Lgi ).VM ) is i-dominant, then v(m m −1 ) ≥ v(m m −1 ), (v) for all j = i, {m ∈ M(V )|v(m m −1 ) < v(m m −1 )} ∩ m Z[A±1 j,a ]a∈C∗ = ∅. / M(V ). Then m ∈ Lemma 4.8. Let L(m) be a simple Uq (Lg)-module, and m ∈ M(L(m)) such that all m ∈ M(L(m)) satisfying v(m m −1 ) < v(m m −1 ) is thin. 1) For i ∈ I such that m is not i-dominant, there is a ∈ C∗ such that u i,a (m ) < 0 and m Ai,aq −1 ∈ M(L(m)). i 2) We suppose that g = sln+1 , that there are i ∈ I , a ∈ C∗ satisfying u i,a (m ) = −1 and m Yi,a is dominant. Then there is M ∈ M(L(m)) dominant such that M > m and vn (m M −1 ) ≤ 1, v1 (m M −1 ) ≤ 1. 3) We suppose that g = sln+1 , that there is j ∈ I , such that m is (I − { j})-dominant and if j ≤ n − 1, then for all a ∈ C∗ , (u j,a (m ) < 0 ⇒ u j+1,aq −1 (m ) > 0). Then there is M ∈ M(L(m)) dominant of the form M = m
(A j,aq −1 A j−1,aq −3 · · · Aia ,aq ia − j−1 ),
{a∈C∗ |u j,a (m )<0}
where for a ∈ C∗ , 1 ≤ i a ≤ j. Observe that we can prove in the same way an analogous result where we replace all i ∈ I by i = n − i + 1. 4.2. Involution of Uq (Lg) and simple modules For µ an automorphism of Uq (Lg) and V a Uq (Lg)-module we denote the corresponding twisted module by µ∗ V . The ± ∓ → Yi,a involution of the algebra Y defined by Yi,a −1 is denoted by σ .
± ± → b−m xi,m , For all b ∈ C∗ , let τb be the automorphism of Uq (Lg) defined by xi,m ± ± ∗ −r h i,r → b h i,r , ki → ki . For V a Uq (Lg)-module we have χq (τb V ) = βb (χq (V )), ± ± ) = Yi,ab . So τb∗ L(m) where βb : Y → Y is the ring morphism such that βb (Yi,a L(βb (m)) and χq (τb∗ L(m)) = βb (χq (L(m))).
Lemma 4.9. [Cha1, Proposition 1.6] There is a unique involution σ of the algebra Uq (Lg) such that for all i ∈ I, r ∈ Z, m ∈ Z − {0}: ± ∓ ) = xi,−r , σ (h i,m ) = −h i,−m , σ (ki ) = ki−1 . σ (xi,r ± ∓ Moreover for m ≥ 0, σ (φi,±m ) = φi,∓m . ± ∓ ) = −xi,−r to define an involution of Uq (Lg).) (Observe that we could also use σ (xi,r
Lemma 4.10. We have χq (σ ∗ V ) = σ (χq (V )).
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± ± ∓ Proof. For γ = (γi,±m )i∈I,m≥0 , it follows from the relation σ (φi,±m ) = φi,∓m that ± ∗ q i,a Vγ = (σ V )γ , where γ = (γi,∓m )i∈I,m≥0 . Let Q i (u) = a∈C∗ (1 − ua) and Ri (u) = a∈C∗ (1 − ua)ri,a such that in C[[u ± ]] we have:
deg(Q i )−deg(Ri )
m≥0
± γi,±m u ±m = qi
Q i (uqi−1 )Ri (uqi ) Q i (uqi )Ri (uqi−1 )
.
Then in C[[u ± ]] we have:
deg(Q i )−deg(Ri )
Q i (u −1 qi−1 )Ri (u −1 qi )
deg(Q i )−deg(Ri )
Q i (uqi−1 )Ri (uqi )
∓ γi,∓m u ±m = qi
m≥0
= qi where Q i (u) = identities qi
a∈C∗ (1
Q i (u −1 qi )Ri (u −1 qi−1 ) Q i (uqi )Ri (uqi−1 )
− ua)ri,a−1 and Ri (u) =
a∈C∗ (1
,
− ua)qi,a−1 by using the
−1 −1 1 − au −1 qi−1 1 − a −1 uqi−1 −1 1 − a uqi −1 1 − au qi = q and q = q . i i i 1 − au −1 qi 1 − a −1 uqi 1 − a −1 uqi−1 1 − au −1 qi−1
In particular χ (σ ∗ V ) = σ (χ (V )), where, σ : E → E is defined by σ (e(λ)) = e(−λ). Let w0 be the longest element in the Weyl group of g and i → i be the unique bijection of I such that w0 (αi ) = −αi . Let h ∨ be the dual Coxeter number of g and r ∨ the maximal number of edges connecting two vertices of the Dynkin diagram of g. Corollary 4.11. For m dominant, we have σ ∗ L(m) L(m ), where u i,a (m) m = Y −1 −r ∨ h ∨ . a∈C∗
i,a
q
Proof. A submodule of V is a submodule of σ ∗ V , so V simple implies σ ∗ V simple. As it is proved in [FM1, Corollary 6.9] that the lowest monomial of L(m) is −u i,a (m) , we get the result from Lemma 4.10. i∈I,a∈C∗ Y r ∨ h∨ i,aq
Observe that as a by product we get the following symmetry property: (i) ) is invariant by (τa 2 q r ∨ h ∨ ◦ σ ). Corollary 4.12. If k = k, then χq (Wk,a
For example, this symmetry can be observed on the diagrams of q-characters in [Nak2, Fig. 1] and [H1, Sect. 8]. Let us go back to the main purposes of this paper. First we get a simplification in the proof of Theorem 3.8: Corollary 4.13. In Theorem 3.8, it suffices to prove that all minimal affinizations are special.
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Proof. First suppose that g is of type B or G. Then i = i. If m satisfies condition (I I ) of Theorem 3.5, then m of Corollary 4.11 satisfies condition (I ). Moreover if M is dominant, then σ (M) is antidominant. So we can conclude with Lemma 4.10. If g is of type A, conditions (I) and (II) are the same up to a different numbering. Exactly in the same way we get : Corollary 4.14. In Theorem 3.9, it suffices to prove that the considered simple representations satisfying the condition (II) are special. For V a Uq (Lg)-module, denote by V ∗ the dual module of V . As S(ki ) = ki−1 , we have χ (V ∗ ) = σ (χ (V )). As a direct consequence of [FM1, Corollary 6.9], we have: Lemma 4.15. For m dominant, we have (L(m))∗ L(m ), where m =
u (m) Y i,a ∨ ∨ . i∈I,a∈C∗ i,aq −r h
Note that it was proved in [FM1] that we have the following relation between the ∨ ∨ q-character of (Vi (a))∗ Vi (aq −r h ) and Vi (a): χq ((Vi (a))∗ ) = (τa ◦ σ ◦ τa −1 )(χq (Vi (a))). Proposition 4.16. For m a dominant monomial, we have χ (L(m)) = χ (L((σ (m))−1 )). Proof. From previous results, we have χ (σ ∗ ((L(m))∗ )) = σ (χ ((L(m))∗ )) = χ (L(m)), and σ ∗ ((L(m))∗ ) L(
u i,a (m) i∈I,a∈C∗ Yi,a −1 )
= L((σ (m))−1 ).
u i,a (m) The above proposition can be extended to χ (L(m)) = χ (L( i∈I,a∈C∗ Yi,ba −1 )) for all b ∈ C∗ . Observe that we do not have a direct relation between the monomials of the same weight space: for example for g = sl2 and m = Yq Yq23 , the term of weight in χq (L(m)) 2 −1 is 2Yq Yq 3 Yq−1 5 and the term of weight in χq (L(σ (m))) is Yq −3 + Yq −3 Yq .
4.3. Additional preliminary results (i)
(i)
Lemma 4.17. Let m = X k,a . Let m ∈ M(Wk,a ) and µ ∈ {k, k − 2, . . . , −k + 2}. Then vi,aq µ (m m −1 ) ≥ 1 implies i
vi,aq k (m m −1 ) ≥ 1, vi,aq k−2 (m m −1 ) ≥ 1, . . . , vi,aq µ (m m −1 ) ≥ 1. i
i
i
On Minimal Affinizations of Representations of Quantum Groups
237
Proof. For µ = k the result is clear. We suppose that µ < k and we prove the result by induction on k. For k = 1 the result is clear. For general k ≥ 1 and µ < k, suppose that vi,aq µ (m m −1 ) ≥ 1. So m = m and it follows from Theorem 2.14 that m ≤ m A−1 k . i
By Lemma 4.1,
m
= m 1 m 2 , where m 1 ∈
M(Vi (aqik−1 ))
and m 2 ∈
i,aqi (i) M(W ). k−1,aqi−1
From Lemma 4.2, v j,b (m 1 Y −1 k−1 ) = 0 implies b = aq ri (k−1)+R with R ≥ 1 and so i,aqi µ (i) )−1 ) ≥ 1. So by the induction hypothesis b = aqi . So we have vi,aq µ (m 2 (X k−1,aqi−1 i vi,aq k−2 (m 2 (X i
(i) )−1 ) k−1,aqi−1
≥ 1,vi,aq k−4 (m 2 (X i
(i) )−1 ) k−1,aqi−1
· · · , vi,aq µ (m 2 (X i
≥ 1,
(i) )−1 ) k−1,aqi−1
≥ 1.
We can conclude because it follows from Theorem 2.14 that vi,aq k (m m −1 ) ≥ 1. i
C∗
Lemma 4.18. Let a ∈ and a monomial m ∈ Z[Yi,aq r ]i∈I,r ∈Z . Let m ∈ M(L(m)) and R ∈ Z such that for all i ∈ I , (u i,aq r (m ) < 0 ⇒ r ≤ R). Then there is a dominant monomial M ∈ M(L(m)) ∩ (mZ[Ai,aq r ]{(i,r )|i∈I,r ≤R−ri } ). Proof. From Lemma 2.15 it suffices to prove the result for Uq (slˆ2 ). In this case the result follows from (3) of Proposition 2.9. To conclude this section, let us prove a refined version of Proposition 2.15. For i ∈ I , u i,aq 2r (m) . Define: a ∈ C∗ and m a monomial denote m →(i,a) = r ∈Z Yi,aq 2ri i
L i,a (m) = mµiI ((m →(i,a) )−1 χqi (L i (m →(i,a) ))).
Observe that for a ∈ aqi2Z , m →(i,a) = m →(i,a ) and L i,a (m) = L i,a (m). So the definition can be given for a ∈ (C∗ /qi2Z ). We have: Corollary 4.19. For a representation V ∈ Rep(Uq (Lg)), i ∈ I and a ∈ C∗ , there is a unique decomposition in a finite sum: χq (V ) = λi,a (m )L i,a (m ). →(i,a) {m |(m )
Moreover for all
m
such that
(m )→(i,a)
is dominant}
is dominant, we have λi,a (m ) ≥ 0.
Proof. First we write the decomposition of Lemma 2.15 with J = {i}. Then it follows from Proposition 2.9 that for m an i-dominant monomial we have L i,b ((m )→(i,b) ). L i (m ) = (m )→(I −{i}) b∈(C∗ /qi2Z )
5. Proof of the Main Results In this section we prove Theorems 3.8, 3.9 and 3.10. We study successively the different types.
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5.1. Type A
In this Sect. 5.1, g = sln+1 .
Lemma 5.1. Let λ ∈ P + and L(m) be a minimal affinization of V (λ). Suppose that m satisfies the condition (I I ) (resp. condition (I )) of Theorem 3.5. Let K = max{i ∈ I |λi = 0} (resp. K = min{i ∈ I |λi = 0}). The following properties are satisfied: (1) (2) (3) (4)
For all m ∈ M(L(m)), if v K (m m −1 ) ≥ 1, then v K ,a K q λ K (m m −1 ) ≥ 1. L(m) is special. L(m) is thin. For all m ∈ M(L(m)) we have v j,ak q λk +| j−k| (m m −1 ) = v j,ak q λk +| j−k|−2 (m m −1 ) = · · · = v j,ak q λk +| j−k|−2R (m m −1 ) = 1, where j = max{i|vi (m m −1 ) = 0} (resp. j = min{i|vi (m m −1 ) = 0}), k = max{i ≤ j|λi = 0} (resp. k = min{i ≥ j|λi = 0}), and R = v j (m m −1 ) − 1. Observe that as a consequence of property (4), for b ∈ C∗ , v j,b (m m −1 ) = 0 implies b ∈ {ak q λk +| j−k| , ak q λk +| j−k|−2 , . . . , ak q λk +| j−k|−2R }.
Lemma 5.1 combined with Corollary 4.13 implies Theorem 3.8 and Theorem 3.10 for type A. Proof. We suppose that L(m) satisfies (I I ) (the case (I ) is treated in the same way). We prove by induction on u(m) ≥ 0 simultaneously that (1), (2), (3) and (4) are satisfied. For u(m) = 0 the result is clear. Suppose that u(m) ≥ 1. First we prove (1) by induction on v(m m −1 ) ≥ 0. For v(m m −1 ) = 0 we have m = m and the result is clear. In general suppose that for m such that v(m m −1 ) < v(m m −1 ), the property is satisfied. Suppose that v K (m m −1 ) ≥ 1 and v K ,a K q λ K (m m −1 ) = 0. It suffices to prove that the conditions of Proposition 4.7 with i = K are satisfied. Condition (i) of Proposition 4.7: if M > m and M ∈ M(L(m)), we have v K ,a K q λ K (Mm −1 ) = 0 and so by the induction hypothesis v K (Mm −1 ) = 0. So if we suppose v (m m −1 ) moreover that M ∈ m Z[A K ,a ]a∈C∗ , we have M = m a∈C∗ A KK,a,a , and so we get v (m m −1 ) the uniqueness. For the existence, it suffices to prove that this M = m a∈C∗ A KK,a,a is in M(L(m)). By Lemma 4.6, there is j ∈ I , M ∈ M(L(m)) j-dominant such that M > m and M ∈ m Z[A j,a ]a∈C∗ . By the induction hypothesis on v we have j = K , and so by uniqueness M = M. Condition (ii) of Proposition 4.7: by construction of M we have v K (Mm −1 ) = 0. Condition (iii) of Proposition 4.7: first observe that M ∈ m →(K ) M(L(m(m →(K ) )−1 )). As u(m(m →(K ) )−1 ) < u(m), we have property (4) for L(m(m →(K ) )−1 ) and we get (M)→(K ) = Y K ,a K q λ K −1 Y K ,a K q λ K −3 · · · Y K ,a
Kq
−λ K +1−2R
with R ≥ 0. By Lemma 2.9, m is not a monomial of M(L K (M)).
,
On Minimal Affinizations of Representations of Quantum Groups
239
Condition (iv) of Proposition 4.7: let m ∈ M(Uq (Lg K ).(L(m)) M ) such that →(K ) v(m m −1 ) < v(m m −1 ). Then we have m ∈ M A−1 Z[A−1 K ,b ]b∈C∗ and so (m ) K ,a q λk K
is right negative, so m is not K -dominant. Condition (v) of Proposition 4.7: clear by the induction property on v. Now we prove (2). Let J = {i ∈ I |i < K }. By Lemma 4.1, M(L(m)) ⊂ (m →(J ) M (L(m →(K ) ))) ∪ (M(L(m →(J ) ))m →(K ) ). From Theorem 2.14, all monomials of m →(J ) (χq (L(m →(K ) )) − m →(K ) ) are lower than m A−1 which is right-negative, and K ,a q λ K K
so are not dominant. Let m ∈ (M(L(m →(J ) ))m →(K ) − {m}). If v K (m m −1 ) ≥ 1, it follows from property (1) that m is lower than m A−1 which is right-negative, so m K ,a q λ K K
is not dominant. If v K (m m −1 ) = 0, we have u K ,b (m (m →(K ) )−1 ) ≥ 0 for all b ∈ C∗ . We have m (m →(K ) )−1 ∈ M(L(m →(J ) )) with u(m →(J ) ) < u(m), so by the induction hypothesis on u, m (m →(K ) )−1 is not dominant. So there is i = K , b ∈ C∗ , such that u i,b (m (m →(K ) )−1 ) < 0. As u i,b (m (m →(K ) )−1 ) = u i,b (m ), m is not dominant. So L(m) is special. Now we prove (3). From property (2) and Proposition 4.4, it suffices to prove that all monomials of M(L(m)) are thin. From Lemma 4.3, we can suppose that there is m ∈ M(L(m)) such that there are i ∈ I, a ∈ C∗ satisfying u i,a (m ) = 2 and such that all m satisfying v(m m −1 ) < v(m m −1 ) is thin. Then m is ({1, . . . , i −2}∪{i}∪{i +2, . . . , n})dominant and (u i−1,b (m ) < 0 ⇒ b = aq) and (u i+1,b (m ) < 0 ⇒ b = aq). We can apply (3) of Lemma 4.8 for g{1,...,i−1} and for g{i+1,...,n} . We get M ∈ M(L(m)) dominant satisfying u j1 ,aq j1 −i (M) ≥ 1, u j2 ,aq i− j2 (M) ≥ 1 with j1 < j2 , j1 ≤ i ≤ j2 . From property (2) we have m = M, contradiction with condition (II) of Theorem 3.5. Now we prove (4) by induction on v(m m −1 ) ≥ 0. We can suppose that j = n (Lemma 4.5). So k = K . For v(m m −1 ) = 0 we have m = m and the result is clear. Let be m such that the property is satisfied for m with v(m m −1 ) < v(m m −1 ). Let R ≥ 0 be maximal such that m ≤ m A−1 n,a
λ +n−k kq k
A−1 n,a
λ +n−k−2 kq k
· · · A−1 n,a
λ +n−k−2R+2 kq k
.
We suppose moreover that m ≤ m A−1 n,a
λ +n−k kq k
A−1 n,a
λ +n−k−2 kq k
· · · A−1 n,a
λ +n−k−2R+2 kq k
A−1 n,b
with b = ak q λk +n−k−2R . By the induction property on v, m is (I − {n})-dominant, u n,bq (m ) < 0 and (u n,c (m ) < 0 ⇒ c = bq). By property (3), u n,bq (m ) = −1. m is a monomial of L n (m An,b ). By property (3), we can apply (3) of Lemma 4.8 and we get M ∈ M(L(m)) dominant of the form M = m An,b An−1,bq −1 · · · An−r,bq −r with r ≥ 0. From property (2), we have M = m. So R = 0. So n − r = K , bq −r = a K q λ K, that is to say b = a K q λ K +n−K, contradiction. 5.2. Type B
In this Sect. 5.2, we suppose that g is of type Bn .
5.2.1. Preliminary results for type B. Lemma 5.2. Let a ∈ C∗ , m ∈ Z[Yi,aq r ]i∈I,r ∈Z be a dominant monomial. Consider m ∈ M(L(m)) {1, . . . , n−1}-dominant such that all m ∈ M(L(m)) satisfying v(m m −1 ) <
240
D. Hernandez
v(m m −1 ) is thin. Suppose that m is not dominant and let R = min{r ∈ Z|u n,aq r (m ) < 0}. Then there is M ∈ M(L(m)) {1, . . . , n − 1}-dominant such that m ≥ M > m , m M −1 ∈ Z[Ai,aq R+2(i−n)+4r −1 ]i∈I,r ≤0 , u n,aq R (M) = 0 and for all r ≤ R, u n,aq r (M) ≥ 0 and l≥0 u n,aq R−2−4l (M) > 0. Proof. For shortness of notations, we suppose that m Yn,aq R is dominant (the proof is exactly the same if R = max{r ∈ Z|u n,aq r (m ) < 0}). First there is m 0 = An,aq R−1 An−1,aq R−3 · · · An−α,aq R−1−2α m ∈ M(L(m)), where α ≥ 0 and m 0 is {1, . . . , n − 1}-dominant. If α = 0 we take M = m 0 . Otherwise, u n,aq R−4 (m 0 ) = −1 and u n,b (m 0 ) > 0 implies b = aq R−4 . We continue and we get inductively (at each step the involved monomials are thin by assumption): m r = An,aq R−1−4r An−1,aq R−3−4r · · · An−α+r,aq R−1−2α−2r m r −1 ∈ M(L(m)), where 1 ≤ r ≤ α and m r is {1, . . . , n − 1}-dominant. We take M = m α and the properties are satisfied by construction. Lemma 5.3. Let L(m) be a simple Uq (Lg)-module. Let m ∈ M(L(m)) be such that all m ∈ M(L(m)) satisfying v(m m −1 ) < v(m m −1 ) is thin. Suppose that there are j ∈ (I − {n}) such that u j,b (m ) < 0 and mY j,b is dominant. Moreover we suppose that if j = 1, then u j−1,bq −2 (m ) > 0. Then there is M ∈ M(L(m)) dominant satisfying one of the following conditions: (1) M = m A j,bq −2 A j+1,bq −4 · · · A j+r,bq −2−2r where 0 ≤ r ≤ n − j, (2) M = m (A j,bq −2 A j+1,bq −4 · · · An−1,bq −2n+2 j )An,bq −2n+2 j M where M ∈ Z[Ak,bq −2n+2 j+2(k−n)−4l ]k 0, in case (3), we have u n,bq −3−2n+2 j (M) = u n,bq −1−2n+2 j (M) = 1, in case (4), we have u n−1,bq −2n−4+2 j (M) = 1. Proof. Thanks to the hypothesis u j−1,bq −2 (m ) > 0, we can suppose that j = 1. By using (3) of Lemma 4.8 with g{1,...,n−1} of type An−1 , we get m 1 = m A1,bq −2 A2,bq −4 · · · A1+r,bq −2(r +1) ∈ M(L(m)), {1, . . . , n − 1}-dominant. If m 1 is dominant, then the condition (1) is satisfied, and we set M = m 1 . Otherwise we have r = n − 2, u n−1,bq −2n (m 1 ) = 1, m 1 is not n-dominant and (u n,d (m 1 ) < 0 ⇒ d = bq −2n+3 or d = bq −2n+1 ). If u n,bq −2n+1 (m 1 ) ≥ 0 and u n,bq −2n+3 (m 1 ) = −1, then we can use Lemma 5.2 and so condition (2) is satisfied. If u n,bq −2n+1 (m 1 ) = −1 and u n,bq −2n+3 (m 1 ) ≥ 0, M = m 1 An,bq −2n ∈ M(L(m)) is dominant, so condition (1) is satisfied. If u n,bq −2n+1 (m 1 ) = −1 and u n,bq −2n+3 (m 1 ) = −1, m 2 = m 1 An,bq −2n An,bq −2n+2 ∈ M(L(m)) is n-dominant and (u n−1,d (m 2 ) < 0 ⇒ d = bq −2n+2 ). If m 2 is dominant, condition (3) is satisfied. If u n−1,bq −2n+2 (m 2 ) = −1, M = m 2 An−1,bq −2n ∈ M(L(m)) is dominant as u n,bq −2n+1 (m 2 ) = u n,bq −2n−1 (m 2 ) = 1. So condition (4) is satisfied. The additional properties in the end of the statement are clear by construction of M.
On Minimal Affinizations of Representations of Quantum Groups
241
(n)
5.2.2. Kirillov-Reshetikhin modules Wλ,a . Now we consider the case of a KirillovReshetikhin module in the node n, that is to say a minimal affinization of V (λn ) (observe that in this case condition (I) and condition (II) of Theorem 3.5 are satisfied). (n) . Then Lemma 5.4. Let m = X λ,a
(1) For all m ∈ M(L(m)) and µ ∈ {λ, λ−2, . . . , −λ+2}, vn,aq µ (m m −1 ) ≥ 1 implies vn,aq λ (m m −1 ) ≥ 1, vn,aq λ−2 (m m −1 ) ≥ 1, . . . , vn,aq µ (m m −1 ) ≥ 1. (2) L(m) is special. (3) L(m) is thin. (4) Let m ∈ M(L(m)) satisfying r ∈Z,i
r ∈Z v j,aq λ+2n−2 j+4r (m
m −1 ) − 1.
Proof. (1) follows from Lemma 4.17. (2) follows from Theorem 2.14. Let us prove (3). From property (2) and Proposition 4.4, it suffices to prove that all monomials of M(L(m)) are thin. From Lemma 4.3, we can suppose that there is m ∈ M(L(m)) such that there are l ∈ I, d ∈ C∗ satisfying u l,d (m ) = 2 and such that all m ∈ M(L(m)) satisfying v(m m −1 ) < v(m m −1 ) is thin. We distinguish three cases (α), (β), (γ ). (α) Suppose that there is c ∈ C∗ such that u n,c (m ) ≥ 2. Then one of the two following conditions is satisfied. (α.i): There is b ∈ C∗ such that u n−1,b (m ) = −1, (u n−1,d (m ) < 0 ⇒ d = b), (u n,bq −1 (m ) = 2 or u n,bq −3 (m ) = 2) and (u n,d (m ) = 2 ⇒ (d = bq −1 or d = bq −3 )). (α.ii): There is b ∈ C∗ such that u n−1,b (m ) = u n−1,bq 2 (m ) = −1, u n,bq −1 (m ) = 2, (u n−1,d (m ) < 0 ⇒ (d = b or d = bq 2 )) and (u n,d (m ) = 2 ⇒ d = bq −1 ). Otherwise, by using Proposition 2.15, we would get m ∈ M(L(m)) such that v(m m −1 ) < v(m m −1 ) and m does not satisfy property (3). First suppose that the condition (α.i) is satisfied. We have the following subcases: (α.i.1): u n,bq −1 (m ) ≥ 1 and u n,bq −3 (m ) ≥ 1. Then m An−1,bq −2 ∈ M(L(m)) is (I − {n − 2})-dominant and by (3) of Lemma 4.8 with g{1,...,n−1} we get M ∈ M(L(m)) dominant such that u n−R,bq −2−2R (M) ≥ 1 for an R ≥ 1. By property (2), M = m, contradiction. (α.i.2): u n,bq −3 (m ) = 2 and u n,bq −1 (m ) = 0. Then m = m An−1,bq −2 ∈ M(L(m)) −1 ∈ M(L(m)) such and Yn,bq −3 Yn,bq −1 appears in m . So by Lemma 2.15 there is m that m < m and u n,bq −3 (m ) ≥ 2, contradiction. (α.i.3): u n,bq −1 (m ) = 2 and u n,bq −3 (m ) = 0. Then m = m An−1,bq −2 ∈ M(L(m)) is (I − {n − 2, n})-dominant and ∀ j ∈ I, (u j,bq l (m ) > 0 ⇒ l ≤ −2). So by Lemma 4.18 we get M ∈ M(L(m)) dominant such that vn,bq −4 (m M −1 ) ≥ 1 and m M −1 ∈ Z[A j,bq l ] j∈I,l≤−3 . So u n,bq −1 (M) = u n,bq −1 (m ) = 1. By property (2), M = m. By property (1), we have vn,b (m M −1 ) ≥ 1, contradiction.
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D. Hernandez
Now we suppose that (α.ii) is satisfied. We have the following subcases: (α.ii.1): u n,bq (m ) = 0. Then m = m An−1,bq −2 An−1,b An,b ∈ M(L(m)) and −1 Yn,bq −1 appears in m . So m = m An−1,bq −2 ∈ M(L(m)) Yn−1,bq −4 Yn−1,bq −2 Yn−1,b and u n−1,bq −4 (m ) = 2, contradiction. (α.ii.2): u n,bq (m ) = u n,bq −3 (m ) = 1. Consider m = m An−1,b An−1,bq −2 . Then m ∈ M(L(m)) is (I − {n − 2})-dominant and by (3) of Lemma 4.8 with g{1,...,n−2} we get M ∈ M(L(m)) dominant such that u n−r1 ,bq −2r1 (M) ≥ 1 and u n−r2 ,bq −2−2r2 (M) ≥ 1 with r1 , r2 ≥ 1. By property (2), M = m, contradiction. (α.ii.3): u n,bq (m ) = 1 and u n,bq −3 (m ) = 0. Then m = m An−1,bq −2 An−1,b An,bq −4 ∈ M(L(m)) is (I − {n − 2})-dominant, Yn−1,bq −2 Yn,bq −5 appears in m and ∀ j ∈ I, (u j,d (m ) = −1 ⇒ (( j, d) = (n − 2, bq −2 ) or ( j, d) = (n − 2, b))). So by (3) of Lemma 4.8, there is m ∈ M(L(m)) of the form m = An−2,bq −2 An−3,bq −4 · · · An−R,bq 2−2R m with R ≥ 1 such that ∀ j ∈ I, (u j,d (m ) = −1 ⇒ ( j, d) = (n − 2, bq −2 )). We have u n,bq −5 (m ) = u n−R,bq −2R (m ) = 1. If m is dominant, we have m = m, contradiction. So u n−2,bq −2 (m ) = −1. As moreover u n,bq −5 (m ) = 1, we have a dominant monomial M ∈ M(L(m)) of the form: M = m (An−2,bq −4 An−1,bq −6 An,bq −8 )(An−3,bq −6 An−2,bq −8 An−1,bq −10 An,bq −12 ) · · · (An−r,bq −2r An−r +1,bq −2−2r · · · An,bq −4r ) ×(An−r −1,bq −2−2r An−r,bq −4−2r · · · An−r −1+r ,bq −2−2r −2r ), where r ≥ 1 and r + 1 ≥ r ≥ 0. By property (1), we have M = m. So we have u n−R,bq −2R (m) = u n−R,bq −2R (m ) = 1, contradiction. (β) Suppose that there is b ∈ C∗ such that u n−1,b (m ) ≥ 2. Then we have (u n−2,d (m ) < 0 ⇒ d = bq 2 ) and (u n,d (m ) < 0 ⇒ d = bq). By (3) of Lemma 4.8 with J = {1, . . . , n − 1} and J = {n}, we get a dominant monomial M ∈ M(L(m)) satisfying one of the two following conditions: (β.1) u j1 ,bq 2 j1 −2n+2 (M) = 1, u n,bq −1 (M) = 1 with j1 ≤ n − 1. (β.2) u j1 ,bq 2 j1 −2n+2 (M) = 1, u n−1,b (M) = 1 with j1 ≤ n − 2. From (2) we have m = M, contradiction. (γ ) Suppose that there is i ≤ n − 2 and b ∈ C∗ such that u i,b (m ) ≥ 2. Then m is ({1, . . . , i − 2} ∪ {i} ∪ {i + 2, . . . , n})-dominant. We have (u i−1,d (m ) < 0 ⇒ d = bq 2 ) and (u i+1,d (m ) < 0 ⇒ d = bq 2 ). By applying (3) of Lemma 4.8 and Lemma 5.3, we get M ∈ M(L(m)) dominant such that (M){1,...,n−2} = 1. From property (2) we have m = M, contradiction. Now we prove property (4) by induction on v(m m −1 ) ≥ 0. Let j be as in property (4). For v(m m −1 ) = 0 we have m = m and the result is clear. We suppose that property (4) is satisfied for m satisfying v(m m −1 ) < v(m m −1 ). Let R ≥ 0 maximal such that A−1 · · · A−1 . We suppose moreover that m m −1 ≤ A−1 j,aq λ+2n−2 j j,aq λ+2n−2 j−4 j,aq λ+2n−2 j+4−4R m m −1 ≤ A−1 A−1 · · · A−1 A−1 , j,aq λ+2n−2 j j,aq λ+2n−2 j−4 j,aq λ+2n−2 j+4−4R j,b
On Minimal Affinizations of Representations of Quantum Groups
243
with b = aq λ+2n−2 j−4µ and µ = R. If j ≥ 2, we have u j−1,b (m ) = 1. By the induction hypothesis on v, m is (I − { j})-dominant, u j,bq 2 (m ) = −1 and (u j,d (m ) < 0 ⇒ d = bq 2 ). By property (3), we can apply Lemma 5.3 and we get a dominant monomial M ∈ M(L(m)). From property (2), we have M = m. As u n−1 (m) = 0, we are in the situation (1) or (3) of Lemma 5.3. So m(m )−1 = A j,b A j+1,bq −2 · · · An−1,bq −2(n−1− j) An,bq −2(n− j) (An,bq 2−2(n− j) ), where ∈ {0, 1}. So b = aq λ+2n−2 j , µ = 0 and R = 0, contradiction. 5.2.3. Condition (I). Now we treat the general case of minimal affinization satisfying (n) condition (I) of Theorem 3.5 (except the Kirillov-Reshetikhin modules Wk,a already studied in Lemma 5.4). Lemma 5.5. Let λ ∈ P + and L(m) be a minimal affinization of V (λ) such that m satisfies condition (I) of Theorem 3.5. Let K = min{i ∈ I |λi = 0}. We suppose that K ≤ n − 1. Then the following conditions are satisfied: (1) For all m ∈ M(L(m)) satisfying r ∈Z v K ,a K q 2λ K +4r (m m −1 ) ≥ 1, we have v K ,a K q 2λ K (m m −1 ) ≥ 1. (2) L(m) is special. (3) L(m) is thin. (4) Let m ∈ M(L(m)) satisfying r ∈Z,i
vi,ai q 2i+4r (m m −1 ) ≥ 1},
r ∈Z
k = min{i ≥ j|λi = 0} and R = (
r ∈Z v j,a j q 2λ j +4r (m
(5) Let m ∈ M(L(m)) such that min{i| λn = 0 and
m −1 )) − 1.
r ∈Z vi,ai q 2λi +4r (m
m −1 )
≥ 1} = n. Then
vn,an q λn (m m −1 ) = vn,an q λn −2 (m m −1 ) where R =
r ∈Z vn,an q 2λn +2r (m
= · · · = vn,an q λn −2R (m m −1 ) = 1, m −1 ) − 1.
Proof. We prove by induction on u(m) ≥ 0 simultaneously that (1), (2), (3), (4) and (5) are satisfied. For u(m) = 0 the result is clear. Suppose that u(m) ≥ 1. First we prove (1) by induction on v(m m −1 ) ≥ 0. For v(m m −1 ) = 0 we have m = m −1 m −1 ) and the result is clear. In general suppose that for m such that v(m m ) < v(m −1 −1 the property is satisfied, that v K ,a K q 2λ K (m m ) = 0 and r ∈Z v K ,a K q 2λ K +4r (m m ) ≥ 1. Observe that it follows from Lemma 2.15 and Corollary 4.19 that m is (I − {K })-dom2λ inant and (m )→(K ,a K q K ) is dominant.
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D. Hernandez
If m is not dominant, by Corollary 4.19, there is m ∈ M(L(m)) K -dominant such that m is a monomial of L K ,a K q 2λ K −2 (m ). Moreover from Proposition 2.9, there is b∈ a K q 2λ K +4Z such that A K ,b m ∈ M(L(m)). By the induction property on v, we have −1 →(K ) M(L(m r ∈Z v K ,a K q 2λ K +4r (m A K ,b m ) = 0. So m = m A K ,b . But m ∈ m →(K ) −1 →(K ) −1 ) )). As u(m(m ) ) < u(m), we have property (4) for L(m(m →(K ) )−1 ) (m and we get (m )→(K ) ∈ Y K ,a K q 2λ K −2 Y K ,a K q 2λ K −6 · · · Y K ,a
Kq
−2λ K +2−4R
Z[Y K ,a K q 4r +2λ K ]r ∈Z
with R ≥ 0. By Lemma 2.9, m is not a monomial of M(L K (m )), contradiction. So m is dominant. Let us prove that r ∈Z v K ,a K q 2λ K +4r +2 (m m −1 ) = 0. Observe that m Y K−1,a
Kq
2λ K −2
∈ M(L(mY K−1,a
Kq
2λ K −2
)).
Moreover u j,a (m (m →(K ) )−1 ) < 0 implies j = K and a = a K q 2λ K −2 . As we have u(mY K−1,a q 2λ K −2 ) < u(m), properties (2) and (3) are satisfied by L(mY K−1,a q 2λ K −2 ). So K K we can use (2) of Lemma 4.8 for g{1,...,n−1} of type An−1 and we get a monomial m ∈ M(L(mY K−1,a
Kq
2λ K −2
)) ∩ (m Y K−1,a
Kq
2λ K −2
Z[A j,a K q 2λ K +4r +2(K − j) ] j≤n−1,r ∈Z ),
which is {1, . . . , n − 1}-dominant and satisfying vn−1 (m Y K−1,a vn−1 (m Y K−1,a
Kq
2λ K −2
(m )−1 ) = 0, then mY K−1,a
Kq
2λ K −2
Kq
2λ K −2
(m )−1 ) ≤ 1. If
= m and the result is clear.
Otherwise, consider the unique b ∈ a K q 2λ K +2(n−K )+4Z such that vn−1,b (m Y K−1,a
(m )−1 )
(m )
bq −1 ).
(m )
Kq
2λ K −2
= 1. We have u n,d < 0 ⇒ (d = bq or d = If u n,bq = 0 we use Lemma 5.2 and we get the result. If u n,bq (m ) = −1 and u n,bq −1 (m ) = 0, we use Lemma 5.2, and in particular we get a monomial (mY K−1,a
Kq
2λ K −2
−1 )A−1 j,d ∈ M(L(mY K ,a
Kq
2λ K −2
)),
where d ∈ / a K q 2λ K +4Z , contradiction with condition (II). In the same way if we have u n,bq (m ) = −1 and u n,bq −1 (m ) = −1, then we get a contradiction by using Lemma 5.2 twice. Now it suffices to prove that the conditions of Proposition 4.7 with i = K are satisfied. Condition (i) of Proposition 4.7: if M > m is in M(L(m)), we have necessarily v K ,a K q 2λ K (Mm −1 ) = 0. So by induction hypothesis r ∈Z v K ,a K q 2λ K +4r (Mm −1 ) = 0, and so v K (Mm −1 ) = 0. So if we suppose moreover that M ∈ m Z[A K ,a ]a∈C∗ , we v (m m −1 ) have necessarily M = m a∈C∗ A KK,a,a , and so we get the uniqueness. For the v (m m −1 ) is in M(L(m)). existence, it suffices to prove that this M = m a∈C∗ A KK,a,a By Lemma 4.6, there is j ∈ I , M ∈ M(L(m)) j-dominant such that M > m and M ∈ m Z[A j,a ]a∈C∗ . By induction hypothesis on v we have j = K , and so by uniqueness M = M. Condition (ii) of Proposition 4.7: by construction of M we have v K (Mm −1 ) = 0. Condition (iii) of Proposition 4.7: first observe that M ∈ m →(K ) M(L(m(m →(K ) )−1 )).
On Minimal Affinizations of Representations of Quantum Groups
245
As u(m(m →(K ) )−1 ) < u(m), we have property (4) for L(m(m →(K ) )−1 ) and we get (M)→(K ) ∈ Y K ,a K q 2λ K −2 Y K ,a K q 2λ K −6 · · · Y K ,a
Kq
−2λ K +2−4R
Z[Y K ,a K q 4r +2λ K ]r ∈Z
with R ≥ 0. By Lemma 2.9, m is not a monomial of M(L K (M)). Condition (iv) of Proposition 4.7: consider a monomial m ∈ M(Uq (Lg K ).VM ) Z[A−1 such that v(m m −1 ) < v(m m −1 ). We have m ≤ M A−1 K ,d ]d∈C∗ and so K ,a q 2λk K
(m )→(K ,a K q K ) is right negative, so m is not K -dominant. Condition (v) of Proposition 4.7: clear by the induction property on v. Now we prove (2). Let J = {i ∈ I |K < i}. From Lemma 4.1, 2λ
M(L(m)) ⊂ (m →(J ) M(L(m →(K ) ))) ∪ (M(L(m →(J ) ))m →(K ) ). As all monomials of m →(J ) (χq (L(m →(K ) )) − m →(K ) ) are lower than m A−1 K ,a
λ Kq K
(Theorem 2.14) which is right-negative, they are not dominant. Let m be a mono→(J ) ) < u(m), the induction property mial in (M(L(m →(J ) ))m →(K ) − {m}). As u(m →(K ) −1 implies that m (m ) is not dominant. If l∈Z v K ,a K q 2λ K +4l (m m −1 ) ≥ 1, it folwhich is right-negative, so lows from property (1) that m is lower than m A−1 K ,a K q 2λ K m is not dominant. We suppose that l∈Z v K ,a K q 2λ K +4l (m m −1 ) = 0. We have for all l ∈ Z, u K ,q 2K +4l (m (m →(K ) )−1 ) ≥ 0, and so there is (i, a) ∈ I × C∗ not of the form (K , q 2K +4l ) with l ∈ Z such that u i,a (m (m →(K ) )−1 ) < 0. So u i,a (m ) = u i,a (m (m →(K ) )−1 ) < 0 and m is not dominant. So L(m) is special. Now we prove (3). From property (2) and Proposition 4.4, it suffices to prove that all monomials of M(L(m)) are thin. From Lemma 4.3, we can suppose that there is m ∈ M(L(m)) such that there are l ∈ I, d ∈ C∗ satisfying u l,d (m ) = 2 and such that all m ∈ M(L(m)) satisfying v(m m −1 ) < v(m m −1 ) is thin. We consider subcases as in the proof of Lemma 5.4. If (α.i.1) is satisfied, we get u n−R,bq −2−2R (m) ≥ 1 with R ≥ 1 and (u n,bq −1 (m) ≥ 1 or u n,bq −3 (m) ≥ 1). As −2 − 2R < −3, we get a contradiction with condition (I) of Theorem 3.5. If (α.i.2) is satisfied, we get a contradiction for Lemma 5.4. −1 If (α.i.3) is satisfied, for Lemma 4.2 we get m ∈ M(L(m)) ∩ mZ[Ai,bq l ]i∈I,l≤−3 such that u n,bq −1 (m) = u n,bq −1 (m ) = 1, vn,bq −4 (m m −1 ) ≥ 1. From Lemma 4.2 and Lemma 4.1, we have m ∈ m {1,...,n−1} M(L(m →(n) )), and we get a contradiction for Lemma 5.4. If condition (α.ii.1) is satisfied, we get a contradiction for Lemma 5.4. If condition (α.ii.2) is satisfied, we get as in the proof of Lemma 5.4 that u n−r1 ,bq −2r1 (m) ≥ 1 and u n−r2 ,bq −2−2r2 (m) ≥ 1 with r1 , r2 ≥ 1. Contradiction with condition (I) of Theorem 3.5. If condition (α.ii.3) is satisfied: we follow the proof of Lemma 5.4 and we get m . If m is dominant, we have u n,bq −5 (m ) = u n−R,bq −2R (m ) = 1 with −2R − (−5) ≤ 3 < 2(n − (n − R)) + 4, contradiction with condition (I) of Theorem 3.5. So m is not dominant. Let R, r, r and M be dominant as defined in the proof of Lemma 5.4.
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From the property (2) we have m = M. Observe that r ≤ r + 1. We have u n−R,bq −2R (m) = 1. We study two cases: If n −r − 1 +r = n, we have moreover u n,bq −3−2r −2r (m) = 1. But (−3 − 2r − 2r ) − (−2R) ≤ 2R − 4 < 2(n − (n − R)), contradiction with condition (I) of Theorem 3.5. If n − r − 1 + r ≤ n − 1, we have moreover u n−r −1+r ,bq −4−2r −2r (m) = 1. Let d = (n−r −1+r −(n− R)) = −1+(R+r −r ) and D = (−4−2r −2r )+2R = 2d −4r −2. If d < 0, condition (I) implies D ≥ −2d + 4, so 0 ≤ D + 2d − 4 = 4d − 4r − 6 < 0, contradiction. If d = 0, condition (I) implies D ∈ 4Z, contradiction as D = −4r − 2. If d > 0, condition (I) implies D ≤ − 4 − 2d, so 0 ≥ D + 4 + 2d = 4d + 2 − 4r = −2 + 4R − 4r and n −r − 1 < n − R < n − R − 1 +r . So the product An−r −1+r ,bq −2−2r −2r · · · An−r −1,bq −2−2r can not appear in m(m )−1 (for example we may use Theorem 4.7 as in the proof of Lemma 5.1), contradiction. Now we suppose that there is b ∈ C∗ such that u n−1,b (m ) ≥ 2. By property (2), we get as in the proof of Lemma 5.4 that m satisfies property (β.1) or (β.2) of Lemma 5.4. For (β.1), we have (2 j1 − 2n + 2 − (−1)) = 2( j1 − n) + 3 < 2(n − j1 ) + 5, contradiction with condition (I) of Theorem 3.5. For (β.2), we have (2 j1 −2n +2−0) = 2( j1 −n)+2 < 2(n − 1 − j1 ) + 6, contradiction with condition (I) of Theorem 3.5. Finally we suppose that there are i ≤ n −2, b ∈ C∗ such that u i,b (m ) ≥ 2. Then m is ({1, . . . , i − 2} ∪ {i} ∪ {i + 2, . . . , n})-dominant. We have (u i−1,d (m ) < 0 ⇒ d = bq 2 ), and (u i+1,d (m ) < 0 ⇒ d = bq 2 ). By applying (3) of Lemma 4.8 and Lemma 5.3 (with bq 2 instead of b and i + 1 instead of j), we get a dominant monomial M ∈ M(L(m)) satisfying one of the conditions (γ .1) (case (1) of Lemma 5.3): u j1 ,bq 2 j1 −2i (M) ≥ 1, u j ,bq 2i−2 j2 +2−r j2 (M) ≥ 1 with 2 j1 < j2 , j1 ≤ i ≤ j2 ≤ n, (γ .2) (case (2) of Lemma 5.3): u j1 ,bq 2 j1 −2i (M) ≥ 1 and u n−1,bq −2n+2i+2 (M) ≥ 1 with j1 ≤ i, (γ .3) (case (3) of Lemma 5.3): u j1 ,bq 2 j1 −2i (M) ≥ 1 and u n,bq 1−2n+2i (M) ≥ 1, u n,bq 3−2n+2i (M) ≥ 1 with j1 ≤ i, (γ .4) (case (4) of Lemma 5.3): u j1 ,bq 2 j1 −2i (M) ≥ 1 and u n−1,bq −2n+2i (M) ≥ 1 with j1 ≤ i. From property (2) we have m = M. For (γ .1), we have 2 j1 −2i −(2i −2 j2 +2−r j2 ) ≤ 2( j1 + j2 ) − 4i ≤ 2( j2 − j1 ), contradiction with condition (I) of Theorem 3.5. For (γ .2), we have 2 j1 − 2i − (−2n + 2i + 2) ≤ 2(n − 1 − j1 ), contradiction with condition (I) of Theorem 3.5. For (γ .3), we have 2 j1 −2i −(3−2n +2i) ≤ 2(n − j1 ), contradiction with condition (I) of Theorem 3.5. For (γ .4), we have 2 j1 −2i −(2i −2n) < 2(n −1− j1 )+4, contradiction with condition (I) of Theorem 3.5. Now we prove property (4) by induction on v(m m −1 ) ≥ 0. Let j be as in property (4). For v(m m −1 ) = 0 we have m = m and the result is clear. We suppose that property (4) is satisfied for m such that v(m m −1 ) < v(m m −1 ). Let R ≥ 0 maximal such that m m −1 ≤ A−1
j,ak q rk λk +2(k− j)
A−1
j,ak q rk λk +2(k− j)−4
· · · A−1
.
j,ak q rk λk +2(k− j)+4−4R
We suppose moreover that m m −1 ≤ A−1
j,ak q rk λk +2(k− j)
A−1
j,ak q rk λk +2(k− j)−4
· · · A−1
j,ak q rk λk +2(k− j)+4−4R
A−1 j,b
with b = ak q rk λk +2(k− j)−4µ and µ = R. By the induction hypothesis on v, m is (I − { j})-dominant, u j,bq 2 (m ) = −1 and (u j,d (m ) < 0 ⇒ d = bq 2 ). By property (3), we
On Minimal Affinizations of Representations of Quantum Groups
247
can apply Lemma 5.3 and we get a dominant monomial M ∈ M(L(m)). From property (2), we have M = m. So we have one of the following situations: Case (1) of Lemma 5.3: m = m A j,b A j+1,bq −2 · · · A j+r,bq −2r , where 0 ≤ r ≤ n − j, and u j+r,bq −2r −r j+r (M) = 1. So R = 0, j + r = k, b = ak q rk λk +2r = ak q rk λk +2(k− j) , contradiction. Case (2) of Lemma 5.3: m = m (A j,b A j+1,bq −2 · · · An−1,bq 2−2n+2 j )An,bq 2−2n+2 j M , where M ∈ Z[A p,bq 2−2n+2 j+2( p−n)−4l ] p
Let R ≥ 0 be maximal such that m m −1 ≤ A−1 n,a
λ nq n
A−1 n,a
λ −2 nq n
· · · A−1 n,a
λ +2−2R nq n
.
We suppose moreover that m m −1 ≤ A−1 n,a
λ nq n
A−1 n,a
λ −2 nq n
· · · A−1 n,a
λ +2−2R nq n
A−1 n,b
with b = an q λn −2µ and µ = R. By the induction hypothesis on v, m is (I − {n})dominant, u n,bq (m ) = −1 and (u n,d (m ) < 0 ⇒ d = bq). So m = An,b m ∈ M(L(m)) is (I − {n − 1})-dominant and (u n−1,d (m ) < 0 ⇒ d = b). If u n−1,b (m ) ≥ 0, m is dominant equal to m, so R = 0 and b = an q λn , contradiction. So u n−1,b (m ) / an−1 q 2λn−1 +4Z < 0, m An−1,bq −2 ∈ M(L(m)) and vn−1,bq −2 (m m −1 ) ≥ 1. So bq −2 ∈ and b ∈ / an q λn +4Z . By Lemma 5.2 there is l ∈ Z such that bq −1−4l = an q λn −1 , so b ∈ an q λn +4Z , contradiction.
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D. Hernandez
5.2.4. Condition (II). We study the general case of condition (II) of Theorem 3.5. Lemma 5.6. Let λ ∈ P + and L(m) be a minimal affinization of V (λ) such that m satisfies condition (II) of Theorem 3.5. Let K = max{i ∈ I |λi = 0}. Then (1) For all m ∈ M(L(m)), if v K (m m −1 ) ≥ 1, then v K ,a
λK K qK
(m m −1 ) ≥ 1.
(2) L(m) is special. (3) L(m) is thin. (4) For all m ∈ M(L(m)) such that vn (m m −1 ) = 0 we have v j,ak q 2(λk + j−k) (m m −1 ) = v j,ak q 2(λk + j−k−2) (m m −1 ) = · · · = v j,ak q 2(λk + j−k−2R) (m m −1 ) = 1, where j = max{i|vi (m m −1 ) = 0}, k = max{i ≤ j|λi = 0} and R = v j (m m −1 ) − 1. Observe that Lemma 5.4, Lemma 5.5 and Lemma 5.6 combined with Corollary 4.13 imply Theorem 3.8 and Theorem 3.10 for type B. In this case we do not need to prove simultaneously the different properties. Proof. Property (4): As vn (m m →(−1) ) = 0, it follows from Lemma 4.5 that m appears in L {1,···n−1} (m). As g J is of type An−1 , the result is exactly property (4) of Lemma 5.1. Property (1) and (2): As property (4) is satisfied, we can use the proof of property (1) and (2) of Lemma 5.1. u (m) Property (3): The monomial M = i∈I,a∈C∗ Y i,a−1 −r ∨ h ∨ satisfies (I), and so it foli,a q lows from Lemma 5.4 and Lemma 5.5 that L(M) is thin. But from Corollary 4.11, σ ∗ L(M) L(m), and so we have property (3) (Lemma 4.10). 5.3. Type G 2
In this section we suppose that g is of type G 2 .
Lemma 5.7. Let m be a dominant monomial satisfying condition (I) of Theorem 3.5. Then L(m) is special. Proof. From Lemma 4.1, M(L(m))⊂M(L(m →(1) )M(L(m →(2) )). From Lemma 2.14, if m is in (M(L(m →(1) ) − {m →(1) })M(L(m →(2) )), then m ≤ m A−1 which is 1,a q 3λ1 1
right-negative, and so m is not dominant. Consider m = m →(1) m 2 , where m 2 ∈ (M(L (m →(2) )) − {m →(2) }). It follows from Theorem 2.14 that m 2 is right-negative. Suppose that m is dominant. In particular m 2 is 2-dominant and (u 1,b (m 2 ) < 0 ⇒ (u 1,b (m 2 ) = − 1 and b ∈ {a1 q 3−3λ1 , a1 q 9−3λ1 , . . . , a1 q 3λ1 −3 })). From Lemma 4.1, m 2 ∈ M(V2 (a2 q 1−λ2 ))M(V2 (a2 q 3−λ2 )) · · · M(V2 (a2 q λ2 −1 )). But for b ∈ C∗ , it follows from [H1, Sect. 8.4.1] (with 1 instead of 2 and 2 instead of 1) that
−1 −1 −1 −1 −1 χq (V2 (b)) =Y2,b + Y2,bq 2 Y1,bq + Y1,bq 7 Y2,bq 4 Y2,bq 6 + Y2,bq 4 Y2,bq 8 + Y1,bq 5 Y2,bq 6 Y2,bq 8 −1 −1 + Y1,bq 11 Y2,bq 10 + Y2,bq 12 . −1 can only appear in χq From condition (I), a1 q −3λ1 +3 = q 7 (a2 q λ2 −1 ). So one Y1,b (V2 (a2 q λ2 −1 )), and so (u 1,b (m 2 ) < 0 ⇒ b = a1 q −3λ1 +3 = q 7 (a2 q λ2 −1 )). As a consequence v1,a1 q −3λ1 (m m −1 ) ≥ 1. From the above explicit description of the χq (V2 (b)), for all
m ∈ M(L(m →(1) )(M(L(m →(2) )) − {m →(2) }),
On Minimal Affinizations of Representations of Quantum Groups
249
if v1,a1 q −3λ1 (m m −1 ) = 0 then l≥0
−3λ1 +3+6l (m 1q −3λ 1 −3+6l 1q
u 1,a
Y1,a
)
= Y1,a
1q
−3λ1 −3
Y1,a1 q 3−3λ1 Y1,a1 q 9−3λ1 · · · Y1,a1 q 3λ1 −3 ,
where ∈ {0, 1}. In particular we can prove as for property (2) of Lemma 5.1 that v1,a1 q −3λ1 (m m −1 ) ≥ 1 implies v1,a1 q 3λ1 (m m −1 ) ≥ 1, contradiction. Lemma 5.8. Let m be a dominant monomial satisfying condition (II) of Theorem 3.5. Then L(m) is special. Proof. It follows from Lemma 4.17 that for m ∈ M(L(m)), if v2 (m m −1 ) = 0 then (m )→(2) is of the form (Y2,a2 q λ2 −1 Y2,a2 q λ2 −3 · · · Y2,a2 q 1−λ2 )Y2,a2 q 1−λ2 −2 · · · Y2,a2 q 1−λ2 −2R , where R ≥ 0 (from condition (II) we have a2 q 1−λ2 −2 = q 5 (a1 q 3λ1 −3 )). So we can use the proof of property (2) of Lemma 5.1. Lemma 5.7 and Lemma 5.8 combined with Corollary 4.13 imply Theorem 3.8 for type G. 5.4. Types C, D and F4 In this subsection we prove Theorem 3.9. From Corollary 4.14, it suffices to consider condition (II). Type C: As λn = 0 and g{1,...,n−1} is of type An−1 , it follows from (1) of Lemma 5.1 that the monomials m ∈ M(L(m)) satisfying vn (m m −1 ) > 0 are right-negative and so not dominant. For the monomials m ∈ M(L(m)) satisfying vn (m m −1 ) = 0, we can use (2) of Lemma 5.1 and Lemma 4.5. Type D: as an = an−1 and λn = λn−1 , all monomials in the set m →(I −{n−1,n}) M(L(m →(n) ))M(L(m →(n−1) )) are right-negative. Moreover we can prove as (1) of Lemma 5.1 that for i = n − 1 or i = n, vi (m m −1 ) > 0 implies vi,ai q λi (m m −1 ) > 0, and so m is right-negative. For the monomials m ∈ M(L(m)) satisfying vn−1 (m m −1 ) = vn (m m −1 ) = 0, we can use (2) of Lemma 5.1 and Lemma 4.5. Type F4 : The proof is analogous to type C by using Lemma 5.6 for g{1,2,3} of type B3 . 6. Applications and Further Possible Developments 6.1. Jacobi-Trudi determinants and Nakai-Nakanishi conjecture In [NN1, Conjecture 2.2] Nakai-Nakanishi conjectured for classical types that the Jacobi-Trudi determinant is the q-character of a certain finite dimensional representation of the corresponding quantum affine algebra. This determinant can be expressed in terms of tableaux (see [BR] for type A, [KOS] for type B, and [NN1, NN2, NN3] for general classical type). The cases considered in [NN1] include all minimal affinizations for type A, and for type B many minimal affinizations (but for example not the fundamental representations Vn (a)). As an application of the present paper, we prove this conjecture for minimal affinizations of type A and B considered in [NN1, Conjecture 2.2] (see the introduction for
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D. Hernandez
previous results). Indeed it can be checked for type A and B that the tableaux expression is special and canceled by screening operators, and so is given by the Frenkel-Mukhin algorithm (see the proofs below; this fact was first announced and observed in some cases in [NN1, Sect. 2.3, Rem. 1]). So from [FM1], Theorem 3.8 proved in the present paper implies that the q-character of a considered minimal affinization is necessarily equal to the corresponding expression. Theorem 6.1. For g of type A, B, the q-character of a minimal affinization considered in [NN1, Conjecture 2.2] is given by the corresponding Jacobi-Trudi determinant. This result is coherent with the thin property proved in this paper. With the same strategy, representations more general than minimal affinizations, and types C, D, will be discussed in a separate publication. Let us recall the tableaux expression of the Jacobi-Trudi determinant and give the proof of Theorem 6.1. We treat the type B (the proof for type A is more simple). We recall that a partition λ = (λ1 , λ2 , · · · ) is a sequence of weakly decreasing non-negative integers with finitely many non-zero terms. The conjugate partition is denoted by λ = (λ 1 , λ 2 , · · · ). For λ, µ two partitions, we say that µ ⊂ λ if for all i ≥ 0, λi ≥ µi . For µ ⊂ λ, the corresponding skew diagram is λ/µ = {(i, j) ∈ N × N|µi + 1 ≤ j ≤ λi } = {(i, j) ∈ N × N|µ j + 1 ≤ i ≤ λ j }. We suppose in the following that d(λ/µ) ≤ n, where d(λ/µ) is the length of the longest column of λ/µ, and that λ/µ is connected (i.e. µi + 1 ≤ λi+1 if λi+1 = 0). Let B = {1, . . . , n, 0, n, . . . , 1}. We give the ordering ≺ on the set B by 1 ≺ 2 ≺ · · · ≺ n ≺ 0 ≺ n ≺ · · · ≺ 2 ≺ 1. As it is a total ordering, we can define the corresponding maps succ and prec. For a ∈ C∗ , let 1 i n 0 n i 1
a
= Y1,a ,
a
−1 = Yi−1,aq 2i Yi,aq 2(i−1)
a
−1 = Yn−1,aq 2n Yn,aq 2n−1 Yn,aq 2n−3 ,
a
−1 = Yn,aq 2n+1 Yn,aq 2n−3 ,
a
−1 −1 = Yn−1,aq 2n−2 Yn,aq 2n+1 Yn,aq 2n−1 ,
a
−1 = Yi−1,aq 4n−2i−2 Yi,aq 4n−2i
a
−1 = Y1,aq 4n−2 .
(2 ≤ i ≤ n − 1),
(2 ≤ i ≤ n − 1),
Observe that we have χq (V1 (a)) =
1
a
+
2
a
+ ··· +
n
a
+
0
a
+
n
a
+ n−1 a + · · · +
1
a.
For T = (Ti, j )(i, j)∈λ/µ a tableaux of shape λ/µ with coefficients in B, let m T,a = Ti, j ∈ Y. 4( j−i) (i, j)∈λ/µ
aq
Let Tab(Bn , λ/µ) be the set of tableaux of shape λ/µ with coefficients in B satisfying the two conditions:
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Ti, j Ti, j+1 and (Ti, j , Ti, j+1 ) = (0, 0), Ti, j ≺ Ti+1, j or (Ti, j , Ti+1, j ) = (0, 0). The tableaux expression of the Jacobi-Trudi determinant [KOS, NN1] is: χλ/µ,a = m T,a ∈ Y. T ∈Tab(Bn ,λ/µ)
u (m) For a monomial m, we denote (m)± = {i∈I,a∈C∗ |±u i,a (m)>0} Yi,ai,a the negative and the positive part of m. We say that (m)− is partly canceled by (m )+ if there is i ∈ I and a ∈ C∗ such that u i,a ((m)− ) = −u i,a ((m)+ ) = 0. Lemma 6.2. Let T ∈ Tab(Bn , λ/µ) and a ∈ C∗ . Let (i, j) = (i , j ) ∈ λ/µ, α = Ti, j − +
and β = Ti , j . If α is partly canceled by β , then (i, j) 4( j −i ) aq 4( j−i) = (i + 1, j ) or ((i, j) = (i + 1, j + 1) and Ti, j = Ti , j = n).
aq
Proof. We study different cases: Case (1): 2 α n and 1 β n − 1. We have α = β + 1 and q 4( j−i)+2α = 4( q j −i )+2(β−1) . So j − i = ( j − i) + 1. If j < j , we have i ≤ i and so Ti, j Ti , j , contradiction. So j ≥ j and i > i . There is ((ir , jr ))1≤r ≤R ∈ (λ/µ) R such that (i 0 , j0 ) = (i , j ) and (i R , j R ) = (i, j) and ((ir +1 , jr +1 ) = (ir + 1, jr ) or (ir +1 , jr +1 ) = (ir , jr + 1)). Let Tr = Tir , jr . As (ir +1 , jr +1 ) = (ir + 1, jr ) implies n Tr +1 Tr , we have TR T1 + (i − i ), and so (i, j) = (i + 1, j ). Case (2): n − 1 α 1 and n β 2. Analog to case (1). Case (3): 2 α n and n β 2. As α
aq 4( j−i)
∈ Z[Yk,aq 2k−2+4r ]k≤n−1,r ∈Z × Z[Yn,aq 2r ]r ∈Z ,
and β
aq 4( j
−i )
∈ Z[Yk,aq 2k+4r ]k≤n−1,r ∈Z × Z[Yn,aq 2r ]r ∈Z ,
we have a contradiction. Case (4): n − 1 α 1 and 1 β n − 1. Analog to case (3). Case (5): α = 0 and β = n. We have q 4( j−i)+2n+1 = q 4( j −i )+2n−3 . So j − i = ( j − i) + 1. As in case (1), we have j ≥ j . So i > i . Consider (ir , jr ), Tr as in case (1). If i ≥ i + 2, there is r1 < r2 such that ir1 +1 = ir1 + 1 and ir2 +1 = ir2 + 1. We have Tr1 = Tr1 +1 = 0 or Tr2 = Tr2 +1 = 0. So there is ( p, q) ∈ λ/µ such that ( p, q + 1), ( p + 1, q + 1) ∈ λ/µ and T p,q+1 = T p+1,q+1 = 0 and T p,q = n. So ( p + 1, q) ∈ λ/µ and T p+1,q = n, contradiction. Case (6): α = 0 and β = 0. We have q 4( j−i)+2n+1 = q 4( j −i )+2n−3 and we can conclude as in case (5). Case (7): α = n and β = 0. We have q 4( j−i)+2n+1 = q 4( j −i )+2n−3 . So j − i = ( j − i) + 1. As in case (1) we have j ≥ j . So i > i . If j > j , as in case (5) we get ( p, q) ∈ λ/µ such that ( p + 1, q), ( p + 1, q + 1) ∈ λ/µ and T p,q = T p+1,q = 0 and T p+1,q+1 = n. So ( p, q + 1) ∈ λ/µ and T p,q+1 = n, contradiction. Case (8): α = n and β = n. We have q 4( j−i)+2n+1 = q 4( j −i )+2n−3 or q 4( j−i)+2n−1 = q 4( j −i )+2n−1 . In the first case j −i = ( j −i)+1. As above we have j ≥ j . So i > i .
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Consider (ir , jr ), Tr as in case (1). If there is r such that ((ir , jr ), (ir +1 , jr +1 ), (ir +2 , jr +2 )) = ((ir , jr ), (ir , jr + 1), (ir + 1, jr + 1)), we have necessarily (Tr , Tr +1 , Tr +2 ) = (n, 0, n). So i = ir and i = ir + 1 = i + 1. We can treat in the same way the situation where there is r such that ((ir , jr ), (ir +1 , jr +1 ), (ir +2 , jr +2 )) = ((ir , jr ), (ir + 1, jr ), (ir + 1, jr + 1)). In the second case j − i = ( j − i). As above we have j > j and i = i + 1. Lemma 6.3. Let T0 = (i − µ j )(i, j)∈λ/µ . Then T0 ∈ Tab(Bn , λ/µ) and m T0 ,a is the unique dominant monomial of χλ/µ,a . Proof. First it is clear that T0 ∈ Tab(Bn , λ/µ) and that m T0 ,a is dominant. Consider T ∈ Tab(Bn , λ/µ) such that T0 = T . So there is (i, j) ∈ λ/µ satisfying the property (i = µ j + 1 and Ti, j = 1) or (i = µ j + 1 and Ti, j = succ(Ti−1, j )).
(4)
From Lemma 6.2 the negative part of the box corresponding to (i, j) is not canceled in m T,a (in case (8) of Lemma 6.2, the negative part of the box can only be partly canceled). Lemma 6.4. For all T ∈ Tab(Bn , λ/µ), a ∈ C∗ , the monomial m T,a is thin. Proof. Let (i, j) = (i , j ) ∈ λ/µ, α = Ti, j and β = Ti , j . We suppose that
+ +
α = β
= 0. We study different cases (by symmetry we 4( j−i) aq aq 4( j −i ) can suppose α β): Case (1): 1 α β n − 1. We have α = β and q 4( j−i)+2(α−1) = q 4( j −i )+2(β−1) . So j −i = ( j −i). If j < j , we have i > i and so Ti, j ≺ Ti , j n −1, contradiction. In the same way we get a contradiction for j > j . Case (2): n α β 2. Analog to case (1). Case (3): 1 α n − 1 and n β 2. Analog to case (3) of Lemma 6.2. Case (4): α = n and β = 0. We have q 4( j−i)+2n−3 = q 4( j −i )+2n−3 . So j −i = j −i. As above, we have j < j . So i < i . We can conclude as in case (5) of Lemma 6.2. Case (5): α = β = 0. We have q 4( j−i)+2n−3 = q 4( j −i )+2n−3 . So j − i = j − i. If j = j , we get ( p, q) ∈ λ/µ such that ( p, q + 1) ∈ λ/µ and T p,q = T p,q+1 = 0, contradiction. Case (6): α = β = n. We have q 4( j−i)+2n−3 = q 4( j −i )+2n−3 or q 4( j−i)+2n−1 = q 4( j −i )+2n−1 . In both cases j − i = j − i and we get a contradiction as in case (1). Finally we can conclude the proof of Theorem 6.1: Lemma 6.5. We have χλ/µ,a ∈ Im(χq ). In the proof we will need the following partial ordering defined on Tab(Bn , λ/µ):for T, T ∈ Tab(Bn , λ/µ) we set: T T ⇔ (∀(i, j) ∈ λ/µ, Ti, j Ti, j ). Also by convention for any α ∈ B, Ti, j = α means ((i, j) ∈ λ/µ ⇒ Ti, j = α). Proof. Let α ∈ I . We want to give a decomposition of χλ/µ as in Proposition 2.15 for J = {α}. From Lemma 6.4, the L α (M) that should appear in this decomposition are thin. It suffices to prove that the set Tab(Bn , λ/µ) is in bijection with a disjoint union of sets M(L α (M)) via T → m T,a .
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First suppose that α ≤ n − 1. Let Mα be the set of tableaux T ∈ Tab(Bn , λ/µ) such that for any (i, j) ∈ λ/µ: Ti, j = α + 1 ⇒ ((i − 1, j) ∈ λ/µ and Ti−1, j = α), Ti, j = α ⇒ ((i − 1, j) ∈ λ/µ and Ti−1, j = α + 1). Then by Lemma 6.2, Mα corresponds to all α-dominant monomials appearing in χλ/µ,a . For T ∈ Mα , let T˜ be the tableaux defined in the following way. For (i, j) ∈ λ/µ: if Ti, j = α and Ti+1, j = α + 1, we set T˜i, j = α + 1, if Ti, j = α + 1 and Ti+1, j = α, we set T˜i, j = α, otherwise we set T˜i, j = Ti, j . Then T˜ ∈ Tab(Bn , λ/µ). For T ∈ Mα , we define: Mα (T ) = {T ∈ Tab(Bn , λ/µ)|T T T˜ }. Then by Lemma 2.9 we have L α (m T,a ) =
m T ,a ,
T ∈Mα (T )
and (Mα (T ))T ∈Mα defines a partition of Tab(Bn , λ/µ). Now we treat the case α = n. Let Mn be the set of tableaux T ∈ Tab(Bn , λ/µ) such that for any (i, j) ∈ λ/µ: Ti, j = 0 ⇒ ((i − 1, j) ∈ λ/µ and Ti−1, j ∈ {0, n}), Ti, j = n ⇒ ((i − 1, j − 1) ∈ λ/µ and Ti−1, j−1 = n). By definition of skew diagram, the last condition implies that (Ti, j = n ⇒ ((i − 1, j), (i, j − 1) ∈ λ/µ and Ti−1, j ∈ {0, n} and Ti, j−1 ∈ {0, n})). This can be rewritten:
n0 nn nn n0 Ti−1, j−1 Ti, j−1 , , , }. ∈{ Ti, j = n ⇒ nn 0n Ti−1, j Ti, j 0n nn Then by Lemma 6.2, Mn corresponds to all n-dominant monomials appearing in χλ/µ,a . For T ∈ Mn , let T˜ be the tableaux defined in the following way. For (i, j) ∈ λ/µ: if Ti, j = n and Ti+1, j+1 = n and Ti+1, j = 0 and Ti+1, j = n, we set T˜i, j = n, if Ti, j = n and Ti+1, j+1 = n and Ti+1, j ∈ {0, n}, we set T˜i, j = 0, if Ti, j = 0 and Ti+1, j = 0 and Ti+1, j = n, we set T˜i, j = n, otherwise we set T˜i, j = Ti, j . Then T˜ ∈ Tab (Bn , λ/µ). For T ∈ Mn , we define: Mα (T ) = {T ∈ Tab(Bn , λ/µ)|T T T˜ }. Then by Lemma 2.9, we have L n (m T,a ) =
m T ,a ,
T ∈Mα (T )
and (Mn (T ))T ∈Mn defines a partition of Tab(Bn , λ/µ).
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6.2. General quantum affinizations The quantum affinization Uq (ˆg) of a quantum KacMoody algebra Uq (g) is defined with the same generators and relations as the Drinfeld realization of quantum affine algebras, but by using the generalized symmetrizable Cartan matrix of g instead of a Cartan matrix of finite type. The quantum affine algebra, quantum affinizations of usual quantum groups, are the simplest examples and have the particular property of being also quantum Kac-Moody algebras. In general these algebras are not a quantum Kac-Moody algebra. In [Mi, Nak1, H2], the category O of integrable representations is studied. For regular quantum affinizations (with a linear Dynkin diagram), one can define analogs of minimal affinizations by using properties (I) and (II) of Theorem 3.5. For example let us consider the type Bn, p (n ≥ 2, p ≥ 2) corresponding to the Cartan matrix (Ci, j )1≤i, j≤n defined as the Cartan matrix of type Bn except that we replace Cn,n−1 = −2 by Cn,n−1 = − p. Then one can prove exactly as for Lemma 5.6 that (an analog of Theorem 4.7 is proved by using [H2, Lemma 5.10]): Theorem 6.6. Let g be of type Bn, p . Then if m satisfies property (I) (resp. (II)), then L(m) is antispecial (resp. special). So the analog of the Frenkel-Mukhin algorithm works for these modules and as an application it should be possible to get additional results for this class of special modules (see also Sect. 6.4 below). 6.3. Multiparameter T -systems The special property of Kirillov-Reshetikhin modules allows to prove a system of induction relations involving q-characters of KirillovReshetikhin modules called T -system (see [Nak5] for the simply-laced cases and [H4] for the general case). Indeed for i ∈ I , k ≥ 1, a ∈ C∗ define the Uq (Lg)-module: ( j) (i) Sr,a =( W −(2k−1)/Ci, j ). {( j,k)|C j,i <0,1≤k≤−Ci, j }
−C j,i +E(ri (r −k)/r j ),aq j
Theorem 6.7 (The T -system). Let a ∈ C∗ , k ≥ 1, i ∈ I . Then we have: (i)
(i)
(i)
(i)
(i)
χq (Wk,a )χq (Wk,aq 2 ) = χq (Wk+1,a )χq (Wk−1,aq 2 ) + χq (Sk,a ). i
i
By analogy, the results of the present paper (special property of minimal affinizations of type A, B, G) should lead to systems of induction relations involving q-characters of minimal affinizations (multiparameter T -systems). Let us look at an example. Let g = sl3 . Then we have the following relation: (1)
(2)
(1)
(2)
χq (L(X 3,q 2 X 2,q 8 ))χq (L(X 3,q 4 X 2,q 10 )) (1)
(2)
(1)
(2)
(2)
(2)
= χq (L(X 4,q 3 X 2,q 10 ))χq (L(X 2,q 3 X 2,q 8 )) + χq (L(X 6,q 6 ))χq (L(X 1,q 9 )). Let us give the idea of the proof for this example: as a q-character is characterized by the multiplicity of his dominant monomials [FM1], it suffices to compare dominant monomials of both sides. By using the process described in Remark 2.16, Theorem 4.7 and arguments of [H4], we get the following results: (1) (2) (1) (2) The dominant monomials of χq (L(X 3,q 2 X 2,q 8 ) ⊗ L(X 3,q 4 X 2,q 10 )) are: 10 122 14 25 27 292 21 1, 10 12 23 25 27 292 21 1, 21 23 25 27 292 21 1, 10 12 124 16 110 27 29 , 10 122 14 110 25 27 29 , 10 12 110 23 25 27 29 , 110 21 23 25 27 29 .
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(2)
The dominant monomials of χq (L(X 4,q 3 X 2,q 10 ) ⊗ L(X 2,q 3 X 2,q 8 )) are: 10 122 14 25 27 292 211 , 10 12 23 25 27 292 211 , 10 122 124 16 110 27 29 , 10 122 14 110 25 27 29 , 10 12 110 23 25 27 29 . (1) (2) (1) (2) The dominant monomials of χq (L(X 3,q 2 X 2,q 8 ) ⊗ L(X 3,q 4 X 2,q 10 )) are: 21 23 25 27 292 211 , 110 2, q23 25 27 29 . We can conclude that the multiplicity of all these monomials is 1. 6.4. Alternative method for the classification of minimal affinizations We explain how to prove certain classification results (included in Theorem 3.5). The proofs here are written in the context of the paper and could be a general uniform strategy for other quantum affinizations. Moreover we get some new refined results on the involved q-characters. Proposition 6.8. Let L(m) be a minimal affinization of V (λ). Then for all i ∈ I , there is ai ∈ C∗ such that m →(i) = X a(i) . i ,λi Proof. For λi ≤ 1 it is clear. Suppose that λi ≥ 2 and that m →(i) in not of this form. Note that λ − αi ∈ P + . It follows from Lemma 4.5 with J = {i} and Proposition 2.9 that dim((L(m))λ−αi ) = dim((L i (m →(i) ))(λi −2)i ≥ 2. (i)
Let a ∈ C∗ and M = m((m)→(i) )−1 X λi ,a . L(M) is an affinization of V (λ). It follows from Lemma 4.5 with J = {i} that dim((L(M))λ−αi ) = 1 so m λ−αi (L(M)) −{i}) = (M)→(I −{i}) , it follows from Lemma 4.5 < m λ−αi (L(m)). Moreover as (m)→(I with J = I − {i} that for µ ∈ λ − j =i Nα j we have dim((L(M))µ ) = dim((L I −{i} (m →(I −{i}) ))µ ) = dim((L(m))µ ) and so m µ (L(M)) = m µ (L(m)). As µ ≤ λ implies µ = λ or µ ≤ λ − αi or µ ∈ λ − j =i Nα j , we have [L(M)] < [L(m)], contradiction. In the following for L(m) a minimal affinization and for i ∈ I such that λi = 0, ai ∈ C∗ denotes the complex number introduced in Proposition 6.8. Let g = sln+1 (n ≥ 2) and λ = λ1 1 +λn n (λ1 , λn ≥ 1). For µ = α1 +α2 +· · ·+αn , (1) (n) we have dim((V (λ))λ−µ ) = n. Let m = X λ1 ,a1 X λn ,an . If L(m) is a minimal affinization of V (λ) then dim((L(m))λ−µ ) = n. For 0 ≤ h ≤ n denote −1 −1 Ai,a Ai,a . mh = m q λn +n−i q λ1 +i−1 i=1,...,h
1
n
i=h+1,...,n
We have different cases: (1) a1 /an ∈ / {q ±(λ1 +λn +n−1) , q λn −λ1 +n−1 , q λn −λ1 +n−3 , . . . , q λn −λ1 −n+1 }. From Remark 2.16, the n +1 monomials m h for 0 ≤ h ≤ n appear in χq (L(m)) and are distinct. So dim((L(m))λ−µ ) ≥ n + 1 and L(m) is not a minimal affinization of V (λ). (2) a1 /an = q λn −λ1 +n+1−2H with 1 = H ≤ n. Then m H = m H −1 . From Remark 2.16, the n − 1 distinct monomials m h for h ∈ / {H − 1, H } appear in χq (L(m)) with multiplicity 1 and m H appears in χq (L(m)) with multiplicity 2. So dim((L(m))λ−µ ) ≥ n + 1 and L(m) is not a minimal affinization of V (λ).
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(3) a1 /an = q λ1 +λn +n−1 . (4) an /a1 = q λ1 +λn +n−1 . From Proposition 4.16, the character is the same in cases (3) and (4). So necessarily these two cases give a minimal affinization with χ (L(m)) = χ (V (λ)). So for λ1 , λn > 0, (1) (n) L(m) is a minimal affinization of V (λ1 1 + λn n ) if and only if m = X λ1 ,a1 X λn ,an with a1 /an = q λ1 +λn +n−1 or an /a1 = q λ1 +λn +n−1 . Now we suppose that g is general and consider J ⊂ I such that g J is of type Ar , 2 ≤ r ≤ n. Denote by i, j ∈ J the two extreme nodes of J . We suppose that we can decompose I = Ii J I j such that Ii ∪ {i} and I j ∪ { j} are connected, and ∀k ∈ Ii , k ∈ J − {i}, Ck,k = 0 and ∀k ∈ I j , k ∈ J − { j}, Ck,k = 0. Observe that Ii or I j may be empty and if J is of type A2 there is always such a decomposition. Proposition 6.9. Let L(m) be a minimal affinization of V (λ) such that λi , λ j ≥ 1 and for k ∈ J − {i, j}, λk = 0. Then one of the two following condition holds: aj ai λ +λ +r −1 λ +λ +r −1 = qi i j or = qi i j . aj ai ±(λi +λ j +r +1) . Proof. We can suppose in the proof that qi = q j = q. Suppose that ai /a j = q Note that λ− k∈J αk ∈ P + . It follows from Lemma 4.5 with J and the above discussion that dim((L(m))λ−k∈J αk ) ≥ r + 1. Let us define
M = m →(Ii ∪{i}) τq λi +λ j +m−1 a a −1 (m →({ j}∪I j ) ). i j
L(M) is an affinization of V (λ). Letus prove that [L(M)] < [L(m)] (which is a contradiction). Let ω ≤ λ. If ω ≤ λ − k∈J αk it follows from Lemma 4.5 with J that dim((L(M))λ−k∈J αk ) < dim((L(m))λ−k∈J αk ). As for J ⊂ J , λ − k∈J αk ∈ / P+ except for J =J or J =∅, we get m λ− k∈J αk (L(M)) < m λ− k∈J αk (L(m)). Otherwise it follows from Lemma 4.5 that dim((L(M))µ ) = dim((L(m))µ ) as (M)→((Ii ∪J )−{ j}) = (m)→((Ii ∪J )−{ j}) and (M)→((I j ∪J )−{i}) = τq λi +λ j +m−1 a a −1 (m →((I j ∪J )−{i}) ). So m µ (L i j
(M)) = m µ (L(m)).
Let g of type Bn (n ≥ 2), λ = λ1 ω1 + λn ωn (λ1 , λn ≥ 1) and µ = α1 + α2 + · · · + αn . (1) (n) Let m = X λ1 ,a1 X λn ,an . For 0 ≤ h ≤ n denote
mh = m
i=1,...,h
We have (L(m))λ−µ =
−1 Ai,a q λ1 +i−1 1
0≤h≤n (L(m))m h .
−1 Ai,a . q 2λn +1+n−i n
i=h+1,...,n
Let us study the different cases:
(1) a1 /an ∈ / {q ±(λ1 +2λn +n) , q 2λn −λ1 +n , q 2λn −λ1 +n−2 , . . . , q 2λn −λ1 −n+2 }. From Remark 2.16 the n + 1 monomials m h for 0 ≤ h ≤ n appear in χq (L(m)) and are distinct. So dim((L(m))λ−µ ) ≥ n + 1. (2) a1 /an = q 2λn −λ1 +n+2−2H with 1 = H ≤ n. Then m H = m H −1 . From Remark 2.16, the n − 1 distinct monomials m h for h ∈ / {H − 1, H } appear in χq (L(m)) with multiplicity 1 and m H appears in χq (L(m)) with multiplicity 2. So dim((L(m))λ−µ ) ≥ n + 1.
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(3) a1 /an = q λ1 +2λn +n . Then dim((L(m))λ−µ ) = n. Indeed, we see as for the proof of point (3) of Lemma 5.1 that for m ∈ M(L(m)), if v1 (m m −1 ) ≥ 1 then v1,a1 q λ1 (m m −1 ) ≥ 1. So m 0 ∈ / M(L(m)) and from Remark 2.16 m 1 , . . . , m H appear in χq (L(m)) with multiplicity 1. (4) an /a1 = q λ1 +2λn +n . As in case (3), dim((L(m))λ−µ ) = n. From Proposition 4.16, the character is the same in cases (3) and (4). Proposition 6.10. For g of type Bn with n ≥ 2 and λ1 , λn > 0, L(m) is a minimal aff(1) (n) inization of V (λ1 1 + λn n ) if and only if m = X λ1 ,a1 X λn ,an with a1 /an = q λ1 +2λn +n or an /a1 = q λ1 +2λn +n . Proof. If m satisfies (1) or (2) and m satisfies (3) or (4), then dim((L(m))λ−µ ) < dim((L(m ))λ−µ ) and for λ ≤ λ if there is j ∈ I such that v j (λ − λ) = 0 then dim((L(m))λ ) =dim((L(m ))λ ) (1)
=dim(Wλ1 ,1 )λ1 1 −k< j vk (λ −λ)αk ) × dim(Wλ(n) ) ). n ,1 λn n − k> j vk (λ −λ)αk
As we have the same character in situations (3) and (4), they correspond necessarily to minimal affinizations. Now we suppose that g is general and consider J ⊂ I such that g J is of type Br , 2 ≤ r ≤ n. Denote by i, j ∈ J the two extreme nodes of J . We suppose that we can decompose I = Ii J I j such that Ii ∪ {i} and I j ∪ { j} are connected, and ∀k ∈ Ii , k ∈ J − {i}, Ck,k = 0 and ∀k ∈ I j , k ∈ J − { j}, Ck,k = 0. Observe that Ii or I j may be empty and if J is of type B2 there is always such a decomposition. Proposition 6.11. Let L(m) be a minimal affinization of V (λ) such that λi , λ j ≥ 1 and for k ∈ J − {i, j}, λk = 0. Then one of the two following condition holds: aj ai λ +2λ +r λ +2λ +r = qi i j or = qi i j . aj ai The proof is analogous to proof of Proposition 6.9. Acknowledgments. The author is very grateful to Evgeny Mukhin for encouraging him to study minimal affinizations in the continuation of the proof of the Kirillov-Reshetikhin conjecture, to Hiraku Nakajima for useful comments and references in an early stage of this research, and to Vyjayanthi Chari, Maxim Nazarov, Alexander Molev, Michela Varagnolo for useful comments and references. A part of this paper was written as the author gave lectures in the East China Normal University in Shanghai, he would like to thank Naihong Hu for the invitation.
References [B] [BL] [BR]
Bourbaki, N.: Groupes et algèbres de Lie, Chapitres IV-VI. Paris:Hermann, 1968 Bernard, D., LeClair, A.: Quantum group symmetries and nonlocal currents in 2d qft. Commun. Math. Phys. 142(1), 99–138 (1991) Bazhanov, V.V., Reshetikhin, N.: Restricted solid-on-solid models connected with simply laced algebras and conformal field theory. J. Phys. A 23(9), 1477–1492 (1990)
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[Cha1] [Cha2] [Che1] [Che2] [CM1] [CM2] [CM3] [CP1] [CP2] [CP3] [CP4] [CP5] [CP6] [CP7] [Do] [Dr1] [Dr2] [DM] [FL] [FM1] [FM2] [FR] [FR] [GK] [H1] [H2] [H3] [H4] [H5] [H6]
D. Hernandez
Chari, V.: Minimal affinizations of representations of quantum groups: the rank 2 case. Publ. Res. Inst. Math. Sci. 31(5), 873–911 (1995) Chari, V.: On the fermionic formula and the kirillov-reshetikhin conjecture. Int. Math. Res. Not. 2001(12), 629–654 (2001) Cherednik, I.: A new interpretation of gelfand-tzetlin bases. Duke Math. J. 54(2), 563–577 (1987) Cherednik, I.: Quantum groups as hidden symmetries of classic representation theory. In: Differential geometric methods in theoretical physics (Chester, 1988), Teaneck, NJ: World Sci. Publishing, 1989, pp. 47–54 Chari, V., Moura, A.: Characters and blocks for finite-dimensional representations of quantum affine algebras. Int. Math. Res. Not. 2005(5), 257–298 (2005) Chari, V., Moura, A.: Characters of fundamental representations of quantum affine algebras. Acta Appl. Math. 90(1–2), 43–63 (2006) Chari, V., Moura, A.: Kirillov–Reshetikhin modules associated to G 2 . math.RT/0604281, 2006 Chari, V., Pressley, A.: Quantum affine algebras. Commun. Math. Phys. 142, 261–283 (1991) Chari, V., Pressley A.: Quantum affine algebras and their representations. In: Representations of groups (Banff, AB, 1994), CMS Conf. Proc, 16, Providence, RI: Amer. Math. Soc., 1995, pp. 59–78 Chari, V., Pressley, A.: Minimal affinizations of representations of quantum groups: the simply laced case, j. Algebra 184(1), 1–30 (1996) Chari, V., Pressley, A.: Minimal affinizations of representations of quantum groups: the nonsimply-laced case. Lett. Math. Phys. 35(2), 99–114 (1995) Chari, V., Pressley, A.: Minimal affinizations of representations of quantum groups: the irregular case. Lett. Math. Phys. 36(3), 247–266 (1996) Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge: Cambridge University Press, 1994 Chari, V., Pressley, A.: Integrable and Weyl modules for quantum affine sl2 . In: Quantum groups and Lie theory (Durham, 1999), London Math. Soc. Lecture Note Ser. 290, Cambridge: Cambridge Univ. Press 2001, pp. 48–62 Dorey, P.: Root systems and purely elastic s-matrices. Nucl. Phys. B 358(3), 654–676 (1991) Drinfeld, V.G.: Quantum groups, In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Providence, RI: Amer. Math. Soc. (1987), pp. 798–820 Drinfeld, V.G.: A new realization of yangians and of quantum affine algebras. Sov. Math. Dokl. 36(2), 212–216 (1988) Delius G.W., MacKay N.J. (2006) Affine quantum groups. Encyclopedia of Mathematical Physics, Oxford:Elsevier, 2006 Fourier, G., Littelmann, P.: Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions. Adv. Math. 211(2):566–593 (2007) Frenkel, E., Mukhin, E.: Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras. Commun. Math. Phys., 216(1), 23–57 (2001) ˆ ∞ . Selecta Math. (N.S.) 8(4), 537–635 Frenkel, E., Mukhin, E.: The hopf algebra r epUq gl (2002) Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras and deformations of w-algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics. Cont. Math. 248, 163–205 (1999) Frenkel, I.B., Reshetikhin, N.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys. 146(1), 1–60 (1992) Grojnowski, I., Kleber, M.: Generalized Kirillov-Reshetikhin Modules for Quantum Affine Algebras. http://www.msri.org/publications/ln/msri/2002/ssymmetry/grojnowski/1/index.html, 2002 Hernandez, D.: Algebraic approach to q, t-characters. Adv. Math. 187(1), 1–52 (2004) Hernandez, D.: Representations of quantum affinizations and fusion product. Transform. Groups 10(2), 163–200 (2005) Hernandez, D.: Monomials of q and q, t-chraracters for non simply-laced quantum affinizations. Math. Z. 250(2), 443–473 (2005) Hernandez, D.: The kirillov-reshetikhin conjecture and solutions of t-systems. J. Reine Angew. Math. 596, 63–87 (2006) Hernandez, D.: Drinfeld coproduct, quantum fusion tensor category and applications. Proc. London Math. Soc. (to appear) math.QA/0504269, 2005 Hernandez, D.: Smallness problem for quantum affine algebras and quiver varieties. math. QA/0607526, 2006
On Minimal Affinizations of Representations of Quantum Groups
[HKOTT] [HKOTY] [J] [Ka] [Kl] [Kn] [KN] [KNH] [KOS] [KR]
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Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Tsuboi, Z.: Paths, crystals and fermionic formulae. In: MathPhys odyssey, 2001, Prog. Math. Phys. 23, Boston, MA, Birkhäuser Boston, 2002, 205–272 Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Remarks on fermionic formula. In: Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math. 248, Providence, RI: Amer. Math. Soc. 1999, pp. 243–291 Jimbo, M.: A q-difference analogue of U (g) and the yang-baxter equation. Lett. Math. Phys. 10(1), 63–69 (1985) Kac V.: (1990) Infinite dimensional Lie algebras. 3rd Edition, Cambridge:Cambridge University Press, 1990 Kleber, M.: Combinatorial structure of finite-dimensional representations of yangians: the simply-laced case. Internat. Math. Res. Notices 1997(4), 187–201 (1997) Knight, H.: Spectra of tensor products of finite-dimensional representations of yangians. J. Alg. 174(1), 187–196 (1995) Kuniba, A., Nakanishi, T.: The bethe equation at q = 0, the möbius inversion formula, and weight multiplicities, II: the X n case. J. Alg. 251(2), 577–618 (2002) Kuniba, A., Nakamura, S., Hirota, R.: Pfaffian and determinant solutions to a discretized toda equation for Br , Cr and Dr . J. Phys. A 29(8), 1759–1766 (1996) (1) Kuniba, A., Ohta, Y., Suzuki, J.: Quantum jacobi-trudi and giambelli formulae for Uq (Br ) from analytic bethe ansatz. J. Phys. A 28(21), 6211–6226 (1995) Kirillov, A.N., Reshetikhin, N.: Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras. J. Sov. Math. 52(3), 3156–3164 (1990); translated from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160, Anal. Teor. Chisel i Teor. Funktsii. 8, 211–221, 301 (1987) Kuniba, A., Suzuki, S.: Analytic bethe ansatz for fundamental representations and yangians. Commun. Math. Phys. 173, 225–264 (1995) Miki, K.: Representations of quantum toroidal algebra Uq (sln + 1, tor)(n ≥ 2). J. Math. Phys. 41(10), 7079–7098 (2000) Molev, A.: On Gelfand-Tsetlin bases for representations of classical Lie algebras. In: Formal power series and algebraic combinatorics (Moscow, 2000), Berlin:Springer 2000, pp. 300–308 Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14(1), 145–238 (2001) Nakajima, H.: T -analogue of the q-characters of finite dimensional representations of quantum affine algebras. In: Physics and combinatorics, 2000 (Nagoya), River Edge, NJ:World Sci. Publishing 2001, pp. 196–219 Nakajima, H.: t–analogs of q–characters of quantum affine algebras of type An , Dn . In: Combinatorial and geometric representation theory (Seoul, 2001), Contemp. Math. 325, Providence, RI:Amer. Math. Soc. 2003, pp. 141–160 Nakajima, H.: Quiver varieties and t-analogs of q-characters of quantum affine algebras. Ann. of Math. 160, 1057–1097 (2004) Nakajima, H.: t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras. Represent. Theory 7, 259–274 (electronic) (2003) Nazarov, M.: Representations of twisted yangians associated with skew young diagrams. Selecta Math. (N.S.) 10(1), 71–129 (2004) Nakai, W., Nakanishi, T.: Paths, tableaux and q-characters of quantum affine algebras: the Cn case. J. Phys. A 39(9), 2083–2115 (2006) Nakai, W., Nakanishi, T.: Paths, tableaux descriptions of Jacubi-Trudi determinant associated with quantum affine algebra of type Dn . J. Algebraic Combin. (to appear) math.QA/0603160, 2006 Nakai, W., Nakanishi, T.: Paths, tableaux descriptions of Jacubi-Trudi determinant associated with quantum affine algebra of type Cn . SIGMA 3, 078 (2007) Nazarov, M., Tarasov, V.: Representations of yangians with gelfand-zetlin bases. J. Reine Angew. Math. 496, 181–212 (1998) Varagnolo, M.: Quiver varieties and yangians. Lett. Math. Phys. 53(4), 273–283 (2000)
Communicated by A. Connes
Commun. Math. Phys. 276: 261–285 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0314-3
Communications in
Mathematical Physics
Super-Derivations and Associated Standard Super-Potentials Huzihiro Araki Research Institute for Mathematical Sciences, Kyoto University, Sakyoku, Kyoto 606-8502, Japan. E-mail: [email protected] Received: 8 December 2006 / Accepted: 18 January 2007 Published online: 23 August 2007 – © Springer-Verlag 2007
Abstract: Abstractly defined super-derivations on Fermionic systems on a lattice are studied. The existence and uniqueness of the associated standard super-potential are shown for every super-derivation with the subalgebra of all local operators as its domain. The relation between the standard super-potential of a super-derivation and the standard potential for the square of the super-potential (which is shown to be a derivation in the case of finite range super-potentials) is obtained (by use of local super-Hamiltonian for the super-derivation and local Hamiltonian for the square). As a consequence, a necessary and sufficient condition for a super-derivation to be nilpotent is obtained in terms of the corresponding standard super potential. Examples of translation invariant nilpotent super-derivations are given in the case of super-potentials of finite ranges on a one-dimensional lattice. A merit of considering the super-potential associated with a super-derivation is that the former can be used as free parameters for the latter. 1. Introduction We consider a system on a lattice with Fermionic grading, which may describe a finite number of Fermions and finite dimensional bosonic objects (such as Pauli spins) on each lattice site. A super-derivation is defined as a derivation with twist by grading [4–7]. For each such super-derivation with the set of all local operators as its domain, we prove the existence of associated super-potentials, which can be made unique by the standardness requirement (relative to a fixed even product state such as the tracial state or the Fermion vacuum state). The proof is obtained by a suitable adaptation of the method used for standard potentials of ∗-derivations recently studied in [3, 1, 2]. A merit of considering the super-potential associated with a super-derivation is that the former can be used as free parameters for the latter. Namely, super-potentials for each finite subset of the lattice can be written out as a linear combination of a finite number of bases (which can be easily listed) and the coefficients are free parameters, without
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any condition for finite range cases and with convergence conditions which restrict only the growth of coefficients as the finite subset become infinitely large. Furthermore, if this set of parameters is different, the corresponding super-derivation is different. These points are explained in more detail in Remarks 4.2, 4.3 and 4.4. A brief outline of the content is as follows. The C ∗ -algebra for the description of the system under consideration along with its local and graded structure, graded commutation relations, a self-adjoint unitary implementer of the grading automorphism, and formulae for intersections and commutants of subalgebras will be introduced in Part A of Sect. 2. After introduction of product states, the conditional expectations (relative to a fixed product state), which provide an essential tool for the present analysis, are defined and their main properties are explained in Part B of Sect. 2. After these preliminaries, super-derivations which is the main subject of this work will be discussed in Sect. 3. They are defined in Definition 3.1 with inner super-derivations (3.2) as a motivation. As a technical tool, the graded commutant (relative to the graded commutator (3.1)) of subalgebras are obtained in Lemma 3.2. Then as the main result of this section, a system of local super-Hamiltonians H s (I ) is shown in Theorem 3.3 to be in one-to-one correspondence with a super-derivation under consideration. In Theorem 4.1, which is the main result of Sect. 4, a standard super-potential satisfying the standardness condition (−2) (equivalently (−2 )) and the convergence condition (−3) is shown to be uniquely associated with a super-derivation under consideration and hence can be used for the description of super-derivations. In Sect. 5, finite range super-derivations are considered and its square is shown to be a derivation, for which the corresponding standard local Hamiltonians are expressed in terms of the local super-Hamiltonians of the super-derivation in Theorem 5.5. As a consequence, a necessary and sufficient condition for a super-derivation to be nilpotent is given in terms of its super-Hamiltonians in Theorems 5.6 and 5.7. Some examples of nilpotent super-derivations are given in terms of the associated super-potentials in Sect. 6. 2. Algebra and Conditional Expectations (A) Algebra. We consider a C ∗ algebra A for the description of the physical system on a lattice L = Zν . For a subset I of L, I c denotes the complement of I , |I | denotes the number of lattice sites in I when I is finite and the notation I ⊂⊂ L is used to indicate that I is a finite subset of L. The C ∗ algebra A has the following two structures. (Local Structure) For each lattice site i, there is a subalgebra A(i) of A, isomorphic to the full matrix algebra Md of all d × d matrices. For each subset I of L, there corresponds a C ∗ subalgebra A(I ) which is generated by all A(i) with i ∈ I . In particular, A = A(L). Due to the graded commutation relations below, A(I ) is isomorphic to the full matrix algebra of d |I | × d |I | matrices if I is finite. We use the following dense ∗ subalgebra of A: A0 = A(I ). (2.1) I ⊂⊂L
Elements of this subalgebra will be called local operators. (Graded Structure) We consider an automorphism of the algebra A satisfying 2 = 1 and (A(i)) = A(i) for all i. (By an automorphism of a C ∗ -algebra, we mean an
Super-Derivations and Associated Standard Super-Potentials
263
algebraic automorphism preserving also the adjoint ∗.) Then (A(I )) = A(I ) for all subsets I of L. Since 2 = 1, any A ∈ A can be split as a sum of even and odd elements, called the even and odd parts of A, uniquely: A = A+ + A− ,
A± = (A ± (A))/2 ∈ A± ,
(2.2)
where A± = {B ∈ A | (B) = ±B}.
(2.3)
A = A+ + A− .
(2.4)
Namely, we have
Correspondingly, we have the following splitting of all subalgebras: A(I ) = A(I )+ + A(I )− , A(I )± = A(I ) ∩ A± .
(2.5)
We assume that each A(i)− is non-trivial (i.e. non-zero). (As 1 ∈ A(i), A(i)+ is automatically non-trivial.) A state ϕ is called even if it is invariant under , namely if ϕ((A)) = ϕ(A) for all A. A state ϕ is even if and only if it vanishes on all odd elements. (Graded Commutation Relations) For any two disjoint subsets I and J of L and for any x± ∈ A(I )± and y± ∈ A(J )± , we assume the following graded commutation relations (σ, σ = + or −): xσ yσ = −yσ xσ if σ = σ = −, xσ yσ = yσ xσ otherwise.
(2.6) (2.7)
(Self-adjoint Unitary Implementer of ) Since each A(i) is isomorphic to a full matrix algebra, there exists a self-adjoint unitary v(i) ∈ A(i) satisfying v(i)∗ Av(i) = (A) for all A ∈ A(i). Such v(i) is unique up to ± and is even. Since it commutes with all elements of A(I c ), v(i) (2.8) v(I ) = i∈I
for any finite I is a self-adjoint unitary implementer of on A(I ). (Formulae for Intersections and Commutants of Subalgebras) As a consequence of the Local Structure and the Graded Commutation Relations, the following formulae for the commutant of the subalgebra A(I ) within A and a related formula are known and useful (see [3], Corollary 4.12, Theorems 4.17, 4.19 and [1], Theorems 2.2, 2.4, 2.5). A(I j ) = A( I j ). (2.9) j
j
If I is finite, then A(I ) = A(I c )+ + v(I )A(I c )− , (A(I )+ ) = A(I c ) + v(I )A(I c ).
(2.10)
If I is not finite, then A(I ) = A(I c )+ ,
(A(I )+ ) = A(I c ).
All the commutants in the above formulae are taken within A.
(2.11)
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(B) Conditional Expectations. (Product states) A state ϕ of A is called a product state if ϕ(x y) = ϕ(x)ϕ(y)
(2.12)
for any x ∈ A(I ) and y ∈ A(J ) for any disjoint subsets I and J . Examples of even product states are the tracial state τ characterized by τ (AB) = τ (B A) for all A, B ∈ A and the vacuum state for Fermions. (Conditional Expectations) Fix an even product state ϕ of A. Then, for any subset I of L, ϕ ϕ there exists a map E I from x ∈ A to E I (x) ∈ A(I ) satisfying and uniquely determined by ϕ
ϕ(a1 xa2 ) = ϕ(a1 E I (x)a2 )
(2.13)
for all a1 and a2 in A(I ). It has the following properties of a conditional expectation: ϕ
(1) E I is linear, positive and unital. (2) For any x ∈ A and b ∈ A(I ), ϕ
ϕ
E I (xb) = E I (x)b,
ϕ
ϕ
E I (bx) = bE I (x).
(2.14)
ϕ
(3) E I is a projection of norm 1 onto A(I ). Furthermore, it has the following properties: (4) For any x ∈ A, ϕ
ϕ
E I ((x)) = (E I (x)),
(2.15)
ϕ
namely, E I commutes with . ϕ (5) E I for different I mutually commute and satisfy the following formula: ϕ
ϕ
ϕ
ϕ
ϕ
E I E J = E J E I = E I ∩J .
(2.16)
ϕ
Here E ∅ is to be interpreted as ϕ. (6) In particular, ϕ
ϕ
E I E I c = ϕ.
(2.17) ϕ
(Continuous Dependence on Subsets) The continuous dependence of E I on the subset I is given by the following formula which converges in the norm topology for any fixed x if K ν → K : ϕ
ϕ
lim E K ν (x) = E K , ν
where K ν → K for a net K ν means K = Kν = Kν , µ
ν≥µ
µ
(2.18)
(2.19)
ν≥µ
in which the second equality is the criterion for convergence of the net K ν . In particular, ϕ
lim E K = 1.
K L
(2.20)
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3. Super-derivation For Q ∈ A− and A ∈ A define the graded commutator [Q, A] = Q A − (A)Q.
(3.1)
Lemma 3.1. For Q ∈ A− , define δ sQ (A) = [Q, A]
(3.2)
for all A ∈ A. Then δ sQ has the following properties: (1) (2) (3) (4) (5) (6)
δ sQ is a linear mapping from A into A. δ sQ (AB) = δ sQ (A)B + (A)δ sQ (B). δ sQ = −δ sQ . δ sQ is linear in Q. s s ∗ ∗ δ sQ (A)∗ = δ−Q ∗ ((A )) = (δ Q ∗ (A )). δ sQ (δ sQ (A)) = [Q 2 , A](= Q 2 A − AQ 2 ).
Proof. Immediate by direct computations.
Definition 3.1. Let Dom δ be a invariant subalgebra of A. A mapping δ from Dom δ to A is called a super-derivation if it satisfies the properties (1), (2) and (3) of Lemma 3.1 . Dom δ is called the domain of δ. The δ sQ in Lemma 3.1 is called an inner super-derivation. As a technical preparation for the proof of the main theorem of this section, we first introduce the notion of super-commutant and prove a lemma about it. Note that in the graded commutator defined by (3.1), Q is assumed to be odd. In the following lemma, Q is an arbitrary element (not restricted to odd elements). So the notation for the graded commutator is not used. Definition 3.2. For any I ⊂ L, the set of all Q ∈ A satisfying Q A − (A)Q = 0
(3.3)
for all A ∈ A(I ) is called the super-commutant of A(I ) and denoted by A(I ) . Lemma 3.2. (1) If I is finite, then
A(I ) = v(I )A(I ) = v(I )A(I c )+ + A(I c )− .
(3.4)
(2) If I is infinite, then
A(I ) = A(I c )− .
(3.5)
(3) If Q ∈ A− satisfies (3.3) for all A ∈ A(I ), then
Q ∈ (A(I ) )− = A(I c )− irrespective of whether I is finite or infinite.
(3.6)
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Proof. (1) Consider a finite I . Eq. (3.3) implies [v(I )Q, A] = v(I )Q A − Av(I )Q = v(I )(Q A − (A)Q) = 0.
(3.7)
Therefore Eq. (2.10) implies v(I )Q ∈ A(I ) = A(I c )+ + v(I )A(I c )− .
(3.8)
Conversely, if this is the case, Eq. (3.3) holds. Hence Lemma 3.2 (1) is proved. (Note that v(I )2 = 1.) (2) Consider an infinite I . Eq. (3.3) with A ∈ A(I )+ and the Eq. (2.11) imply Q ∈ (A(I )+ ) = A(I c )
(3.9)
Let Q = Q + + Q − , Q ± ∈ A(I c )± . Then, Eq. (3.3) with A ∈ A(I )− implies 0 = Q A + AQ = 2Q + A. Since there exist a family of operators Aα ∈ A(I )− such that Remark 3.1 below), we obtain Q+ = (Q + Aα )A∗α = 0.
(3.10)
α
Aα A∗α = 1 (see (3.11)
α
Therefore Q = Q − ∈ A(I c )− . Conversely, if this is the case, Eq. (3.3) holds. Hence Lemma 3.2 (2) is proved. (3) Consider Q ∈ A− , then by proof (1) and (2) above, Q ∈ A(I c )−
(3.12)
is necessary and sufficient for Eq. (3.3) to hold for both finite and infinite I .
Remark 3.1. The family Aα ∈ A in the above proof can be constructed as follows: For a finite I , identify A(I ) with the algebra of all linear operators on C N , N = d |I | . Let {en |n = 1, . . . , N } be an orthonormal basis of C N consisting of eigenvectors of v(I ) and let v(I )en = σn en (σn = + or −).
(3.13)
Let u mn (m, n = 1, . . . , N ) be a self-adjoint system of matrix units satisfying u mn ek = δnk em . They satisfy u kl u mn = δlm u kn ,
N
u kk = 1.
(3.14)
k=1
(They also satisfy u ∗kl = u lk , which we are not going to use.) We now construct Aα for α = n = 1, . . . , N as follows. Fix αe and αo satisfying σαe = + and σαo = −. (Note that A(i)± are non-trivial for each i by our overall assumption.) If σα = +, then set Aα = u ααo . If σα = −, then set Aα = u ααe . Then Aα ∈ A(I )− for all α and α Aα A∗α = 1. We can now state the main theorem of this section.
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Theorem 3.3. Let δ be a super-derivation with Dom δ = A0 . Then there exists a unique H (I ) ∈ A− for each I ⊂⊂ L satisfying the following properties: (1) For every I and every A ∈ A(I ), δ(A) = [H (I ), A] . ϕ (2) E I c (H (I )) = 0. ϕ (3) If I ⊂ J , then H (I ) = H (J ) − E I c (H (J )). Conversely, if a function H (I ) of finite sets I of L with values in A satisfies the above properties (2) and (3), then (1) above defines a super-derivation δ with Dom δ = A0 . Proof. Proof will be carried out in 5 steps. First, we give an explicit candidate h for H (I ) and show that it is odd and satisfies the property (1) for H (I ). Second, we modify h to h so that h is odd and satisfies (1) and (2). Then we set H (I ) = h . Third, we show that odd H (I ) satisfying (1) and (2) is unique. Fourth, we show (3) by using the uniqueness just proved. Finally, for any given A− -valued function H (I ) of I ⊂⊂ L, satisfying properties (2) and (3), the δ defined by the property (1) is shown to be a super-derivation with its domain A0 . (Step 1) Existence of odd H (I ) satisfying (1). For a finite I , let u mn (m, n = 1, . . . , N ) be a self-adjoint system of matrix units for A(I ), constructed in Remark 3.1. Define h = −N −1 σk σn u kn δ(u nk ). (3.15) k,n
Then (h) = −h due to (u kn ) = σk σn u kn and δ = −δ. We show the condition (1) for H (I ) = h. It is enough to show it for A = u αβ : [h, u αβ ] = hu αβ − σα σβ u αβ h = −N −1 σk σn u kn δ(u nk )u αβ + N −1 σk σn σα σβ u αβ u kn δ(u nk ) k,n
= −N
−1
k,n
σk σn u kn (δ(u nk u αβ ) − σn σk u nk δ(u αβ )) + N −1
k,n
= −N −1
σα σn u αn δ(u nβ ) + δ(u αβ ) + N −1
n
σα σn u αn δ(u nβ )
n
σα σn u αn δ(u nβ )
n
= δ(u αβ ), where we have used the property (2) of Lemma 3.1 for the super-derivation δ to obtain the third equality and k u kk = 1 to obtain the fourth equality. (Step 2) Existence of odd H (I ) satisfying (1) and (2). By Step 1, we have for each I ⊂⊂ L, [h, A] = δ(A)
(3.16)
for all A ∈ A(I ). Let ϕ
h = h − E I c (h). Since
ϕ E I c (h)
∈
A(I c )
−,
h
(3.17)
is odd and the graded commutation relation implies ϕ
[E I c (h), A] = 0
(3.18)
for all A ∈ A(I ). Therefore, condition (1) satisfied by h is also satisfied by h . Furtherϕ more, condition (2) is satisfied by h because E I c is a projection. So we take H (I ) = h .
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(Step 3) Uniqueness of H (I ) satisfying (1) and (2). Suppose h 1 and h 2 both satisfy conditions (1) and (2) for H (I ). By property (1), we have [h 1 − h 2 , A] = 0
(3.19)
for all A ∈ A(I ). Due to h 1 − h 2 ∈ A− and Lemma 3.2, this implies h 1 − h 2 ∈ A(I c )− . Hence property (2) implies ϕ
ϕ
ϕ
h 1 − h 2 = E I c (h 1 − h 2 ) = E I c (h 1 ) − E I c (h 2 ) = 0.
(3.20)
Therefore h 1 = h 2 and the uniqueness is proved. (Step 4) Proof of the property (3). Let I ⊂ J and A ∈ A(I ) ⊂ A(J ). By the graded ϕ commutation relation, E I c (H (J )) ∈ A(I c )− implies ϕ
[H (J ) − E I c (H (J )), A] = [H (J ), A] = δ(A). ϕ
(3.21)
ϕ
Furthermore, E I c (H (J ) − E I c (H (J ))) = 0. Hence by the uniqueness proved in Step 3, we obtain property (3). (Step 5) Converse. For a given family of H (I ) we define δ by property (1). Then for each I , it is a super-derivation with domain A(I ) by the proof of Lemma 3.1. If I ⊂ J and A ∈ A(I ), then property (3), of which the second term on the right-hand side has vanishing graded commutators with elements of A(I ), implies [H (I ), A] = [H (J ), A]. For general finite I and J and for A ∈ A(I ) ∩ A(J ) = A(I ∩ J ), we can apply the preceding equality for the pairs I ∩ J ⊂ I and I ∩ J ⊂ J to obtain [H (I ), A] = [H (I ∩ J ), A] = [H (J ), A]. Thus the definitions of δ by different H (I )’s are consistent and altogether define a super-derivation δ on A0 .
Remark 3.2. The existence proof of H (I ) is the super-derivation version of what Sakai ([8]) proved for ∗-derivations. We believe that this is a new result. Remark 3.3. The h given in Step 1 of the above proof can be written as u kl h kl , h=
(3.22)
k,l
where h kl =
σl σm u mk δ(u lm ) − δkl N −1
m
σm σn u mn δ(u nm )
(3.23)
m,n
belongs to v(I )A(I ) and satisfies (h kl ) = −σk σl h kl . This can be shown by the following computation. First, u kl h kl = σl σm u kl u mk δ(u lm ) − N −1 σm σn u kk u mn δ(u nm ) k,l
klm
=
u kk δ(u ll ) − N
k,l
= −N −1
k,n
= h,
−1
kmn
σk σn u kn δ(u nk )
k,n
σk σn u kn δ(u nk )
Super-Derivations and Associated Standard Super-Potentials
where we have used
l
u ll =
k
269
u kk = 1 and δ(1) = 0, the reason for the latter being
δ(1) = δ(12 ) = δ(1)1 + (1)δ(1) = 2δ(1).
(3.24)
Second, we compute the commutator (3.3) for A = u αβ and Q = h kl , irrespective of h kl being even or odd, using again property (2) of Lemma 3.1 for the super-derivation δ. (1) If k = l, h kl u αβ − (u αβ )h kl = σl σm u mk δ(u lm )u αβ − σα σβ u αβ σl σm u mk δ(u lm ) m
=
m
σl σm u mk (δ(u lm u αβ ) − σl σm u lm δ(u αβ )) − σl σα u αk δ(u lβ )
m
= σl σα u αk δ(u lβ ) − σl σα u αk δ(u lβ ) = 0. (2) If k = l, h kk u αβ − (u αβ )h kk = σk σm u mk δ(u km )u αβ − σα σβ u αβ σk σm u mk δ(u km ) m
−N
−1
m
σm σn u mn δ(u nm )u αβ + N
mn
= σk σα u αk δ(u kβ ) − − N −1
−1
σα σβ u αβ σm σn u mn δ(u nm )
mn
u mm δ(u αβ ) − σα σk u αk δ(u kβ )
m
σα σn u αn δ(u nβ ) + δ(u αβ ) + N −1
n
σα σn u αn δ(u nβ )
n
= 0, where for the second equality we have used σk σm u mk δ(u km )u αβ = σk σm u mk (δ(u km u αβ ) − σk σm u km δ(u αβ )) m
m
= σk σα u αk δ(u kβ ) −
u mm δ(u αβ )
m
for the first term on the left-hand side of the equality and the same equation with k replaced by n and summed over n for the third term of the left-hand side of the equality. Hence, by Lemma 3.2(1), h kl is in A(I c )− + v(I )A(I c )+ = v(I )A(I ) .
(3.25)
(h kl ) = −σk σl h kl
(3.26)
Finally we have
due to (u pq ) = σ p σq u pq and δ = −δ. Definition 3.3. A function H (I ) of I ⊂⊂ L satisfying (1), (2) and (3) of Theorem 3.3 is called the family of local super-Hamiltonians for the super-derivation δ.
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H. Araki
4. Standard Super-Potentials We define ϕ
U (I ) = E I (H (I )), which belongs to A(I )− . (A(I ) because of property (2) of H (I ), we have
ϕ EI
(4.1)
and odd because H (I ) is odd.) By
ϕ
ϕ
ϕ(U (I )) = ϕ(E I (H (I ))) = ϕ(H (I )) = ϕ(E I c (H (I ))) = 0.
(4.2)
For the empty set, we define (∅) = U (∅) = 0. ϕ ϕ ϕ ϕ If I ⊂ J , property (3) of H (I ), when multiplied by E I , implies, due to E I = E I E J , ϕ ϕ E I E I c = ϕ and ϕ(H (J )) = 0, ϕ
ϕ
ϕ
U (I ) = E I E J (H (J )) − ϕ(H (J )) = E I (U (J )).
(4.3)
For a general J , we have ϕ
ϕ
ϕ
ϕ
E I (U (J )) = E I E J (U (J )) = E I ∩J U (I ) = U (I ∩ J ). Furthermore, property (3) of H (I ), when multiplied by ϕ
ϕ EJ ,
(4.4)
implies
ϕ
E J H (I ) = U (J ) − E I c (U (J )).
(4.5)
By taking the limit J L, we obtain an expression of H (I ) in terms of U (J ): ϕ
H (I ) = lim (U (J ) − E I c (U (J ))). J L
We now define the (ϕ-)standard super-potential by (I ) = (−1)|I |−|K | U (K ).
(4.6)
(4.7)
K ⊂I
This set of equations (for all I ⊂⊂ L) is equivalent to U (I ) = (K ).
(4.8)
K ⊂I
For the proof, see for example [3]. We can now state the main theorem of this section. Theorem 4.1. (1) The standard super-potential , which is a function of I ⊂⊂ L and is derived from a super-derivation δ through H (I ) of Theorem 3.3, U(I) of Eq. (4.1) and Eq. (4.7), has the following 3 properties: (-1) (I ) ∈ A(I )− , (∅) = 0 . ϕ
(-2) If J ⊂ I and J = I , then E J ((I )) = 0. (-3) For each I ⊂⊂ L, the net
(K )
K ⊂J,K ∩I =∅
converges as J L. Under property (-1), property (-2) is equivalent to
(4.9)
Super-Derivations and Associated Standard Super-Potentials
271
(-2 ) ϕ E J ((I ))
=
(I ), 0,
I ⊂ J, otherwise.
(4.10)
(2) Conversely, any function of I ⊂⊂ L satisfying the 3 properties (-1), (-2) (equivalently (-2 )) and (-3) defines a super-derivation δ with domain A0 by δ(A) = lim [(K ), A] (4.11) J L
K ⊂J
for A ∈ A0 , where the limit on the right-hand side converges in norm. The H (I ) uniquely associated with this δ by Theorem 3.3 is given by H (I ) = lim (K ). (4.12) J L
K ⊂J,K ∩I =∅
(3) The mappings between δ and , given by (1) and (2) above are the inverse of each other and they give a bijective real linear mapping beween the real linear space of all super-derivations with domain A0 and the real linear space of all functions of I ⊂⊂ L satisfying the 3 properties (-1), (-2) and (-3). Remark 4.1. In Eq. (4.11), the summation over K can be (and usually is) limited to those K having non-empty intersection with I when A ∈ A(I ) because terms for other K vanish. Proof. (1) The first property in (-1) follows from Eq. (4.7) and U (K ) ∈ A(K )− . The second property in (-1) is by definition. As for the equivalence of (-2) and (-2 ) under (-1), (-2) is a part of (-2 ) on one hand. On the other hand, if (-2) holds, then with the help of (-1) we have ϕ
ϕ
ϕ
ϕ
E J (I ) = E J E I (I ) = E J ∩I (I ),
(4.13)
which is (I ) if I ⊂ J ∩ I and 0 otherwise due to (-2). Since I ⊂ J ∩ I is the same as I ⊂ J , (-2 ) follows from (-2) and (-1). Substituting Eq. (4.8) into U in Eq. (4.6), using property (-2 ) in which the pair J, I is replaced by the pair I c , K , and noting that K ⊂ I c is equivalent to K ∩ I = ∅, we obtain Eq. (4.12). (2) Let satisfying 3 properties (-1), (-2 ) and (-3) be given. We define (K ), (4.14) H (I ) = lim J L
K ⊂J,K ∩I =∅
where the limit on the right-hand side converges by property (-3). The H (I ) so defined belongs to A− due to (-1). Due to (-2 ), the above defining equation for H (I ) is the same as ϕ H (I ) = lim ((K ) − E I c ((K ))) (4.15) M L
K ⊂M
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H. Araki
because in the sum over K the terms with K ⊂ I c vanish while the other terms ϕ are (K ) both due to (-2 ). Since E I c is a projection, H (I ) obviously satisfies property (2) in Theorem 3.3. To show property (3) of H(I) in the same theorem, we ϕ ϕ ϕ note that if I ⊂ J , then J c ⊂ I c and hence E I c E J c = E J c . This implies ϕ
ϕ
ϕ
((K ) − E J c (K )) − E I c ((K ) − E J c (K )) ϕ
= (K ) − E I c (K ). If we substitute the expression of H (J ) given by the formula (4.15) into the righthand side of property (3) for H (I ) in Theorem 3.3 and use the equation just computed, then we get an expression of H (I ) given by the formula (4.15) . Therefore the property (3) of Theorem 3.3 is shown. By Theorem 3.3, a super-derivation δ is defined by H (I ) through the formula δ(A) = [H (I ), A]
(4.16)
for A ∈ A(I ). Substituting the expression (4.15) into H (I ) of the above equation and noting that the second term in the sum on the right-hand side of (4.15) belongs to A(I c )− and hence its graded commutator with A vanishes, we obtain the expression (4.11) which is guaranteed to converge in norm (due to the convergence of (4.15)). The uniqueness of H (I ) in Theorem 3.3 now proves that our H (I ) corresponds to the super-derivation δ defined by Eq. (4.11). We have completed the proof of (2). (3) Since the bijective real linear correspondence between δ and H is already established by Theorem 3.3, we only have to establish the bijective real linear correspondence between H and . Since H is explicitly given in terms of , we have only to prove that if k , k = 1, 2, both satisfy properties (-1), (-2 ) and (-3) and give the same H , then 1 = 2 . However, from the expression of H (I ) in terms of , one obtains the expression (4.8) for U (I ) by the property (-2 ) of . Since the relation between U and can be solved uniquely for , yielding the expression (4.7) for in terms of U , the uniqueness of for a given H (I ) follows.
Definition 4.1. The function of I ⊂⊂ L with values in A in the above theorem is called the standard super-potential of the super-derivation δ. We now make three remarks (with associated theorems) about the significance of this theorem and then a remark illustrating this theorem in a simple case. The main point of the first three remarks is to show that it is easy to exhibit all possible standard super-potentials explicitly and (infinite number of) parameters for them can be chosen arbitrarily without any constraint binding them (mutually) except for convergence conditions. Since a different choice of the parameters corresponds to a different choice of super-derivation, the former plays a role of free parameters for the latter. This viewpoint is likely to be useful in constructing a super-derivation having a specific property. Remark 4.2. For each fixed I ⊂⊂ L, an arbitrary (I ) satisfying the two properties (-1) and (-2) (the latter being equivalent to (-2 )) can be exhibited as an arbitrary linear combination of the following basis. i− i− i+ For each i ∈ L, A(i) is of a finite (d 2 ) dimension. Let 1, u i+ 1 ,. . ., u N+ , and u 1 ,. . ., u N− , be linear bases of A(i)+ and A(i)− , respectively, where 1 ∈ A(i)+ is taken as one of its
Super-Derivations and Associated Standard Super-Potentials
273
linear basis. The product state ϕ which is assumed to be even, vanishes on u i− k which is odd. On the other hand,
i+ i+ u i+ k = u k − ϕ(u k )1,
(4.17)
k = 1, . . . , N+ together with 1 can be taken to be a linear basis of A(i)+ and satisfies i+ ϕ(u i+ k ) = 0. Thus we may assume that u k , k = 1, . . . , N+ already has this property. We set iσ u ki i , (4.18) u({ki , σi ; i ∈ I }) = i∈I
where each σi is either + or −, the order of the product is according to the (any) fixed order of i ∈ I for each I , and the number of i such that σi = − is restricted to be odd. We can now state the following theorem about a basis of the linear space of all possible standard super-potentials. Theorem 4.2. The collection of elements given in (4.18) is a linear basis of standard super-potentials (I ), namely (1) they satisfy the two properties (-1) and (-2), and (2) exhaust all such possibility. Proof. Because of the graded commutation relations among A(i) with different i, products of their linear bases form a linear basis of A(I ). Equation (4.18) selects those, for which (1) the number of odd elements in the product is odd and (2) the product does not contain 1 as a factor for any i ∈ I . The first condition is obviously equivalent to (-1). Thus it is enough to show that, for a general linear combination of products of bases of A(i), i ∈ I , to satisfy the property (-2), the second property, i.e. vanishing of coefficient of all products containing 1 as a factor for at least one i, is necessary and sufficient. ϕ If Ji = I ∩ {i}c for an i ∈ I , then the application of E Ji on a general linear combinai tion annihilates those terms with the i factor, say u , not equal to 1 because it will produce a factor ϕ(u i ) = 0, while those terms with u i = 1 which belongs to A(Ji ), remains ϕ unchanged. Thus the result of the multiplication of E Ji is now a linear combination of a basis of A(Ji ), and the property (-2) requires it to be 0, namely all its coefficients have to vanish. Hence (I ) must be a linear combination of the product not containing 1 as a factor for any i ∈ I . In the converse direction, consider J ⊂ I, J = I . Then there is some i ∈ I not ϕ ϕ ϕ belonging to J . Then J ⊂ Ji and hence E J = E J E Ji . Since (4.18) is annihilated by ϕ ϕ E Ji , it is annihilated by E J and hence it satisfies (-2).
Remark 4.3. The convergence condition (-3) for the standardness of a potential in Theorem 4.1 is required for all finite I . This is more convenient when using this condition, but the same requirement for only one point sets I = {i} for all i ∈ L is already sufficient, i.e. equivalent to the condition (-3) stated in theorem 4.1, as is formulated and proved in the following theorem. Theorem 4.3. Under conditions (-1) and (-2), the condition (-3) holds if (and only if) it holds only for I = {i} for all i ∈ L.
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H. Araki
Proof. Let I = {i 1 , . . . , i N }, N = |I |. For any integer k ∈ [1, N ], let Ik = {i 1 , i 2 , . . . , i k }. For any finite J ⊂ L containing I , the set of all K ⊂ J with non-empty intersection with I is a disjoint union (over k = 1, 2, . . . , N ) of the set of all K ⊂ J such that i k ∈ K and i 1 , i 2 , . . . , i k−1 ∈ / K . Hence ⎛ ⎞ N ⎝ ⎠. = (4.19) K ⊂J,K ∩I =∅
K i k ,K ⊂J ∩(Ik−1 )c
k=1
Note that i 1 , i 2 , . . . , i k−1 ∈ / K is equivalent to K ⊂ (Ik−1 )c . On the other hand, the condition (-2 ) (which follows from (-1) and (-2)) implies ϕ (K ) = (K ). (4.20) E (Ik−1 )c K i k ,K ⊂J
K i k ,K ⊂J ∩(Ik−1 )c
Hence
(K ) =
K ⊂J,K ∩I =∅
N
⎛ ϕ E (Ik−1 )c
⎝
k=1
⎞ (K )⎠ .
(4.21)
K i k ,K ⊂J
By (-3) for one-point sets I = {i k }, k = 1, . . . , N , the sum in the parentheses ϕ on the right-hand side converges (to H ({i k })) as J L. Due to E (Ik−1 )c = 1, the right-hand side is convergent, which implies (-3) for a general I .
Corollary 4.4. Let I = {i 1 , . . . , i N }, N = |I | and set Ik = {i 1 , i 2 , . . . , i k } (k = 1, . . . , N ). The following identity holds: H (I ) =
N
ϕ
E (Ik−1 )c (H ({i k })),
(4.22)
k=1
where
ϕ E (Ik−1 )c
for k = 1 is taken to be an identity map.
Proof. By taking the limit of (4.21) as J L, we obtain the identity.
Remark 4.4. Suppose that there is a representation of the group of lattice translations i ∈ L −→ i + n ∈ L ,
(4.23)
n ∈ L, by automorphisms τn of A satisfying τn (A(i)) = A(i + n), commuting with , and leaving the product state ϕ invariant. Then the following sense: ϕ
ϕ
τn E I = E I +n τn ,
(4.24) ϕ EI
is covariant in (4.25)
where I + n = {i + n; i ∈ I }. A super-derivation which commutes with τn for all n ∈ L is said to be translation invariant. In this case, the corresponding standard super-potential is shown in the following theorem to be translation covariant (as a function of I under the translation I −→ I + n of its variable I , covariance referring to the change of the value of the
Super-Derivations and Associated Standard Super-Potentials
275
function by the automorphism τn ) and hence the convergence condition for I = {i} for (any) one i only is already sufficient for (-3) (for all I ⊂⊂ L) to hold because the convergence condition for any other j ∈ L follows from the convergence condition for one I = {i} by the translation automorphism τ j−i . Thus the convergence requirement reduces to the convergence of a single summation. Theorem 4.5. Let (I ), H (I ), and U (I ) be the standard super-potential, a system of super-Hamiltnian, and U defined by (4.1), all for a super-derivation δ. The following conditions are equivalent: (1) The super-derivation δ is translation invariant in the sense that δτn = τn δ for all n ∈ L. (2) The standard super-potential is translation covariant in the sense that (I +n) = τn ((I )) for all n ∈ L and for all I ⊂⊂ L. (3) The system of local super-Hamiltonians H are translation covariant in the sense that H (I + n) = τn (H (I )) for all n ∈ L and for all I ⊂⊂ L. (4) U satisfies U (I + n) = τn (U (I )) for all n ∈ L and for all I ⊂⊂ L. Proof. (1)→ (3) Condition (1) implies for any A ∈ A(I ), τn (δ(A)) = δ(τn (A)),
(4.26)
where A = τn (A) exhausts elements of A(I + n) when A runs over A(I ). By the definition of the system of local super-Hamiltonian and τn = τn , we have δ(A ) = τn (δ(A)) = τn ([H (I ), A] ) = [τn (H (I )), A ] . ϕ
(4.27)
ϕ
Due to (I + n)c = I c + n, (4.25) implies E (I +n)c τn = τn E I c and hence ϕ
ϕ
E (I +n)c τn (H (I )) = τn E I c (H (I )) = 0.
(4.28)
By the unique characterization of H (I ) (Theorem 3.3 (1) and (2)), the above two equations imply τn (H (I )) = H (I + n),
(4.29)
which is condition (3). ϕ
ϕ
ϕ
(3) → (4).U (I + n) = E I +n (H (I + n)) = E I +n τn (H (I )) = τn E I (H (I )) = τn (U (I )). (4) → (2). Due to
(4.30) (I ) =
(−1)|I |−|K | U (K ),
(4.31)
K ⊂I
the mapping by τn yields τn ((I )) =
(−1)|I |−|K | U (K + n)
K ⊂I
=
K ⊂I +n
where we set K = K + n.
(−1)|I +n|−|K | U (K ) = (I + n),
(4.32)
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H. Araki
(2)→ (1). By Theorem 4.1 (2), we have for A ∈ A0 , δ(A) = lim [(K ), A] . J L
(4.33)
K ⊂J
The mapping by τn then yields for A = τn (A) ∈ A(I + n), τn (δ(A)) = lim [τn ((K )), τn (A)] J L
= lim
J L
K ⊂J
[(K ), A ] = δ(A )
K ⊂J
= δ(τn (A)) where we denote K + n = K and J + n = J . Hence we have (1).
Remark 4.5. For illustration of standard super-potentials, we compute here the standard super-potential for an inner derivation δ sQ for any Q ∈ A− . Due to the unique characterization of H (I ) for a given δ in terms of properties (1) and (2) of Theorem 3.3 (see Step 2 of its proof), H (I ) for δ sQ is given by ϕ
H (I ) = Q − E I c (Q).
(4.34)
Then ϕ
ϕ
ϕ
U (I ) = E I (H (I )) = E I (Q) − ϕ(Q) = E I (Q), ϕ
(4.35)
ϕ
where the second equality is due to E I E I c = ϕ and the last because Q is odd and ϕ is even, so that ϕ(Q) = 0. By (4.7), we obtain ϕ (I ) = (−1)|I |−|K | E K (Q). (4.36) K ⊂I
One can easily check ϕ ϕ U (I ) = (J ) = (−1)|J |−|K | E K (Q) = E I (Q), J ⊂I
by using Hence
|I |
j=|K | (−1)
(4.37)
J ⊂I K ⊂J
j−|K | |I |−|K | j−|K |
= 0 for K ⊂ I , K = I to obtain the last equality. ϕ
Q = lim E I (Q) = lim I L
I L
(K ).
(4.38)
K ⊂I
This implies δ sQ (A) = lim
I L
[(K ), A] .
(4.39)
K ⊂I
The decomposition of the above type for an element of A has been used in the proof of Lemma 12.5 of [3](for an even element).
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277
5. Super-Derivation of a Finite Range and Nilpotent Super-Derivation Definition 5.1. A standard super-potential is said to have a finite range r if (I ) = 0 whenever diam(I ) > r , where diam(I ) is the maximum distance between two points in I. Lemma 5.1. If the standard super-potential for a super-derivation δ has a finite range, then δ maps A0 into itself, i.e. the range of δ is contained in A0 . Proof. Assume that the standard super-potential has a finite range r . For any finite subset I of L, let [I ]r denote the set of all points of L whose distance from I does not exceed r . Let A ∈ A(I ) and consider the term on the right-hand side of Eq (4.11) corresponding to a finite subset K of L. If diam(K ) > r or I and K does not intersect, then it vanishes. Otherwise, it belongs to A([I ]r ) ⊂ A0 due to I ∪ K ⊂ [I ]r . Hence δ(A) ⊂ A0 .
Remark 5.1. In general, a super-derivation can map A0 into itself even if the corresponding standard super-potential does not have a finite range. (For example, this is the case when (K ) = 0 holds if and only if diam(K ) > dist(0, K ) + 1.) However the following theorem holds. Theorem 5.2. For a translation invariant super-derivation δ with A0 as its domain, it maps the domain into itself if and only if the corresponding standard super-potential is of a finite range. Proof. Since A({0}) is of a finite linear dimension (d 2 ), the union of δ images of its d 2 linear basis are individually in some local algebra and hence δ(A({0})) is contained in their linear span, and hence in some A(J ), J ⊂⊂ L. We add the point 0 to J if not yet contained in J . Let J0 be the minimal choice of such J (take the intersection of all such J ) and set r = diam(J0 ). The explicit construction of h in (Step 1) of the proof of Theorem 3.3 implies h ∈ A(J0 ) in the case of H ({0}). Due to ϕ
ϕ
ϕ
ϕ
E I c (h) = E I c E J0 (h) = E J0 ∩I c (h) ∈ A(J0 ∩ I c ) ⊂ A(J0 ),
(5.1)
we have ϕ
H ({0}) = h = h − E I c (h) ∈ A(J0 ).
(5.2)
We now substitute the expression (4.12) (the case of I = {0}) into H ({0}) in the following quantity which vanishes as a consequence of the above equation: ϕ
0 = H ({0}) − E J0 (H ({0})) (K ). = lim J L
K ⊂J,K 0,K ⊂ J0
By the subsequent lemma, we conclude that (K ) = 0
(5.3)
for all K in the above summand, namely if K contains the point 0 and is not a subset of J0 . By application of τi , we conclude that if K contains a point i, then (K ) = 0 unless K is contained in J0 + i. Since 0 is in J0 by our choice, this means that any point of K must have a distance from an arbitrary point i in K not exceeding r = diam(J0 ), i.e. diam(K ) should not exceed r in order that (K ) = 0. Thus we have shown that the super-potential has a finite range (not exceeding r ).
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H. Araki
Lemma 5.3. Let K be any (finite or infinite) collection of non-empty finite subsets of L and suppose that a standard super-potential satisfies (K ) = 0, (5.4) K ∈K
where the sum is to be norm convergent in any fixed way if K is an infinite set. Then (K ) = 0 for all K ∈ K. Proof. By omitting those K for which (K ) = 0 from the set K, we may assume that (K ) = 0 for all K ∈ K. We will obtain a contradiction if such K is not empty. Among the members of K there is a minimal K (it is minimal if it does not contain a proper subset belonging to K) because of the following reason. Start with any K ∈ K and try to find its proper subset, say K 1 , which belongs to K. If there is no such K 1 , then K is already minimal by definition of minimality. Continue this process obtaining a strictly decreasing sequence of K i belonging to K. Obviously this can not continue more than |K | − 1 times. Let K n be the last one in this sequence. Then it is a minimal member of K and it is not an empty set (because it is a member of K). Thus there exists a minimal member, say K n of K. Because of minimality, there is no proper subset K of K n belonging to K. By the ϕ standardness condition (-2 ), E K n (K ) = 0 if K ∈ K and K = K n . Therefore, ϕ by applying E K n on the equation in the statement of the lemma, we obtain (K n ) = 0 which is a contradiction because it is assumed that (K ) = 0 for all K ∈ K.
Remark 5.2. The above argument works with coefficients in front of each (K ). Therefore, non-zero (K ) with distinct K are linearly independent. Lemma 5.4. Let δ be a super-derivation with A0 as its domain and mapping this domain into itself. Then δ 2 is an even derivation. Proof. As a product of two linear mappings, δ 2 is a linear mapping. The following computation shows that it is a derivation. Let A and B be elements of A0 . Then δ 2 (AB) = δ(δ(A)B + (A)δ(B)) = δ(δ(A))B + (δ(A))δ(B) + δ((A))δ(B) + Aδ(δ(B)) = δ 2 (A)B + Aδ 2 (B), where δ = −δ is used in the last equality. Since δ anticommutes with by the definition of a super-derivation, δ 2 commutes with . Hence it is an even derivation.
An even derivation can be associated with a system of a local Hamiltonian and a standard potential in exactly the same way as an even *-derivation (see [3]) except that they are not necessarily self-adjoint. Let H s (I ) be the local super-Hamiltonian (for I ⊂⊂ L) of the super-derivation δ in Definition 3.3 and let H (I ) be the local Hamiltonian (for I ⊂⊂ L) of the even derivation δ 2 . (We adopt a slightly different convention as compared with the case of *-derivations. We will require δ 2 (A) = [H (I ), A],
(5.5)
while, in the case of *-derivation, i is multiplied on the right-hand side in order to obtain self-adjoint H (I ), which is irrelevant in the present case.) The two are related as follows.
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Theorem 5.5. If a super-derivation δ with A0 as its domain maps this domain into itself, then for each I ⊂⊂ L there exists [I ] ⊂⊂ L containing I such that δ(A(I )) ⊂ A([I ]),
(5.6)
where we define [I ] to be the minimal such [I ]. The following relations hold. For any J such that [I ] ⊂ J ⊂⊂ L, ϕ
H (I ) = H s (J )2 − E I c (H s (J )2 ).
(5.7)
Proof. (Since each A(I ) is of finite dimension, existence of some [I ] containing I and satisfying Eq. (5.6) follows from the assumption on δ. Then take the intersection of all such sets to be [I ] which is then the minimal such set.) We can now make the following computation. For A ∈ A(I ), δ 2 (A) = [H s (J ), δ(A)] = [H s (J ), [H s (J ), A] ] = [H s (J )2 , A],
(5.8)
where the first equality is due to δ(A) ∈ A([I ]) and [I ] ⊂ J , the second due to A ∈ A(I ) and I ⊂ [I ] ⊂ J , and the last by the same computation as the proof of Lemma 3.1(6). Since H s (J ) is odd, H s (J )2 is even. Now the quantity ϕ
H (I ) = H s (J )2 − E I c (H s (J )2 )
(5.9)
satisfies the following 3 properties which characterize the local Hamiltonian of the derivation δ 2 for a finite set I uniquely (see Lemma 5.3 of [3]) and the theorem is proved. (1) ϕ
(H (I )) = ((H s (J ))2 − E I c ((H s (J ))2 ) = H (I ),
(5.10)
due to (H s (J )) = −H s (J ). (2) δ 2 (A) = [H s (J )2 , A] = [H (I ), A]
(5.11)
for any A ∈ A(I ) because the second term on the right-hand side of Eq. (5.7) is an even element of A(I c ) which commutes with A ∈ A(I ). (3) ϕ
E I c (H (I )) = 0 ϕ
(5.12)
ϕ
due to (E I c )2 = E I c .
Theorem 5.6. Let δ be a super-derivation with domain A0 mapping this domain into itself. Let [I ] for I ⊂⊂ L be as in the preceding theorem and H s (I ) be the local super-Hamiltonian for δ. Then δ is nilpotent if and only if H s (J )2 ∈ A(I c ) for one finite J containing [I ] for each finite I , or equivalently for all such J.
(5.13)
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Proof. By the preceding theorem, Eq. (5.13) for one J containing [I ] for each I implies H (I ) = 0 for all I ⊂⊂ L because the right-hand side of (5.7) then vanishes. Hence the nilpotency δ 2 = 0 follows. (Note that H (I ) = 0 implies (5.13) for all finite J containing [I ] due to (5.7) and so, if (5.13) holds for one J containing [I ] for an I , then H (I ) = 0 and (5.13) holds for all J containing [I ] for the same I .) Conversely if δ is nilpotent, then [H (I ), A] = δ 2 (A) = 0 for all A ∈ A(I ) and hence H (I ) belongs to (A(I ) )+ = A(I c )+ . Then (5.7) implies (5.13) for all finite J containing [I ].
Theorem 5.7. Let δ be a super-derivation such that the associated standard superpotential has a finite range r . Let U (J ) = (K ). (5.14) K ⊂J
Then δ is nilpotent if and only if for every finite I , U (J )2 ∈ A(I c )
(5.15)
for one finite J containing all points within distance 2r from I or equivalently for all such finite J . Proof. For finite subsets I, K 1 , K 2 of L, [(K 1 ), [(K 2 ), A] ] = 0
(5.16)
for all A ∈ A(I ) unless diam(K 1 ) and diam(K 2 ) do not exceed r , some point in K 2 is in I and some point in K 1 is in K 2 ∪ I . This implies that every point in K 2 must be within distance r from I and every point in K 1 must be within distance 2r from I . By applying this criterion to Eq. (4.11), we obtain δ 2 (A) = [(K 1 ), [(K 2 ), A] ] , (5.17) K 1 ,K 2
where we can restrict the sum to those K 1 and K 2 satisfying the above stated restrictions. Let J be a finite subset of L containing all points within distance 2r from I. Then all sets K 1 satisfying the above restriction are contained in J . Furthermore, if any other subsets of J is substituted to K 1 , it will give vanishing contribution. Therefore, the sum over K 1 can be extended to all subsets of J. Hence [(K 2 ), A] ] . (5.18) δ 2 (A) = [U (J ), K2
By the same argument, we can extend the sum over K 2 in this equation over all subsets of J and we obtain δ 2 (A) = [U (J ), [U (J ), A] ] = [U (J )2 , A],
(5.19)
where the last equality is by the same calculation as the proof of Lemma 3.1(6). (Note that U (J ) is odd because it is a sum of (K ) which is odd.) From the above expression of δ 2 , we conclude that δ is nilpotent if and only if U (J )2 , which is even, is in (A(I ) )+ = A(I c )+ for at least one J for each I or equivalently for all J for every I .
(5.20)
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If δ 2 (A) = 0 for all A ∈ A(i) for each i ∈ L, then δ 2 (A) = 0 for all A ∈ A0 because A0 is algebraically generated from the union of A(i), i ∈ L. Therefore, we obtain the following result. Corollary 5.8. In the preceding two theorems, the conditions for all I = {i}, i ∈ L are already sufficient for the nilpotency of δ. Suppose that there is a representation of the group of lattice translations i ∈ L −→ i + n ∈ L ,
(5.21)
n ∈ L, by automorphisms τn of A satisfying τn (A(i)) = A(i + n),
(5.22) ϕ
commuting with , and leaving the product state ϕ invariant. Then E I is covariant in the following sense: ϕ
ϕ
τn E I = E I +n τn ,
(5.23)
where I + n = {i + n; i ∈ I }. Consider a super-derivation which is translation invariant in the sense that it commutes with τn for all n ∈ L. Then the associated standard super-potential is covariant in the following sense: τn ((I )) = (I + n).
(5.24)
The associated quantities also satisfy similar properties: τn (U (I )) = U (I + n), τn (H (I )) = H (I + n).
(5.25)
In this situation, if δ 2 (A) = 0 for all A ∈ A(i) for one i ∈ L, then the application of τn on this equation yields the same equation for i + n instead of i and hence we have δ 2 = 0. Thus we obtain the following result. Corollary 5.9. For a translation invariant super-derivation δ, the conditions in the preceding two theorems for only one I with |I | = 1 is already sufficient for its nilpotency. 6. Examples of Nilpotent Super-Derivations Standard super-potentials can be regarded as free parameters for super-derivations and can be used for a systematic construction of their examples. Here, we limit our attention to the system of one kind of Fermion on a one-dimensional lattice L = Z, described by creation and annihilation operators ai∗ and ai at each i ∈ Z, which are odd and satisfy the following CAR (the canonical anticommutation relations): [ai∗ , a ∗j ] = [ai , a j ] = 0, [ai∗ , a j ] = δi j .
(6.1)
The local algebra A(i) at a lattice site i ∈ Z is the linear span of 1, ai , ai∗ and ai∗ ai which give the values 1,0,0,0 under the vacuum state, which we call ψ0 and use for the ϕ product state ϕ of the conditional expectations E I .
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First, we give an infinite-dimensional space of examples of translation invariant nilpotent super-derivations. (Example 1). Consider the following standard potentials for odd |I |: (I ) = c(I )
ai ,
(6.2)
i∈I
where we take the order of the product the same as the order of i in Z and c(I ) is a complex number, vanishing for diam(I) > r as well as for |I | even. It is an odd element of A(I ) and satisfies the standardness condition (-2) and (-2 ). Due to CAR, they mutually anticommute and in particular, the square is 0. Therefore U (J )2 = 0. By Corollary 5.9, the corresponding super-derivation is nilpotent. If c(I ) = c(I + n) for any n ∈ Z, this is a translation invariant nilpotent super-derivation. Because of the choice of c(I ), these examples form an infinite dimensional linear space. (Example 2) If we replace ai by ai∗ in Example 1, we obtain another infinite-dimensional example of translation invariant nilpotent super-potentials. They are conjugate to Example 1 through the Bogoliubov automorphism which interchanges ai with ai∗ for all i ∈ L. We now want to examine the case of a finite range r for small r . (Example 3) The case of r = 0 and hence (I ) = 0 unless |I | = 1. The most general standard super-potential is given by (i) = c1 ai + c2 ai∗ ,
(6.3)
where the coefficients c1 and c2 are independent of i for a translation invariant superderivation. We have (i)2 = c1 c2
(6.4)
which is a constant and so belongs to A({i}c ). Hence all of them give translation invariant nilpotent super-derivations. (Example 4) The case of r = 1 and hence (I ) = 0 unless I = {i} or {i, i + 1}, i ∈ Z. The most general translation covariant standard super-potential is of the following form. For |I | = 1, it is the same as Example 3. For |I | = 2, ∗ ∗ ∗ ai+1 + d2 ai∗ ai+1 ai+1 + d3 ai∗ ai ai+1 + d4 ai∗ ai ai+1 , (i, i + 1) = d1 ai ai+1
(6.5)
where the constants d1 , d2 , d3 , d4 are independent of i. The condition for the nilpotency of the corresponding super-derivation is that U ([−2, 2])2 can be written in the form without terms containing a0 or a0∗ by Corollary 5.9. We have U ([−2, 2]) =
2 i=−2
(i) +
1
(i, i + 1).
(6.6)
i=−2
As is shown in the Appendix, it turns out that this condition is satisfied if and only if all di ’s vanish, namely (i, i + 1) = 0. So we do not obtain a new example in this case.
Super-Derivations and Associated Standard Super-Potentials
Appendix A.
283
Computation of Example 4
The conclusion of Example 4 is obtained by the following (straightforward but tedious) computation. First some preliminary computations. Suppose A and B are some (I )’s which are products of several distinct a’s and a ∗ ’s. If they contain only one factor with index i = 0, then that factor can be brought to either the left or right end of the product to obtain A = a0(∗) A = A a0(∗) and B in the same form with A replaced by B , where the superscript (∗) indicates two possibilities ∗ or non-∗, A and B are even elements of A({0}c ), and hence the second equality. If both contain the factor a0 , then AB = B A = 0 due to (a0 )2 = 0 and the same holds if both contain the factor a0∗ . This include the case A = B. Therefore, in such cases, A2 = B 2 = AB = B A = 0. Now consider the case where A and B are of the above form but A contains the factor a0 and B contains the factor a0∗ . Then AB + B A = A a0 a0∗ B + B a0∗ a0 A = A B + a0∗ a0 [B , A ],
(A.1)
where [B , A ] = B A − A B . If A contains a0∗ and B contains a0 , the same computation with A and B interchanged yields AB + B A = B A + a0∗ a0 [A , B ].
(A.2)
Next consider the case A = a0∗ a0 A and B = a0(∗) B , where A is odd and B is even, both in A({0}c ). By the same type of computation as above, we obtain if B contains a0 , AB = 0,
B A = B a0 A = a0 B A .
(A.3)
AB = −a0∗ A B .
(A.4)
If B contains a0∗ , then we obtain B A = 0,
Finally consider the case A = a0∗ a0 A and B = a0∗ a0 B , where A and B are odd elements of A({0}c ). Then AB + B A = a0∗ a0 (A B + B A ).
(A.5)
This includes the case A = B. Terms in U ([−2, 2]) are all odd and hence for each term there are an odd (and hence non-zero) number of indices i such that it contains either ai alone or ai∗ alone as a factor (∗) with that index. Then one can bring that factor ai to the right or left end (as one pleases). Therefore, if two such terms A and B have a common i for which both have ai (or ai∗ ) as the only factor with the index i, then AB = B A = 0. In particular, the square of all terms are 0. This can be used to select those pairs of terms A and B for which AB + B A can be possibly non-vanishing. We are now ready to compute the square of U ([−2, 2]). We have U ([−2, 2]) = c1 a0 + c2 a0∗ + d1 a0 D1 + d1 a0∗ a0 D1 + d2 a0∗ D2 + d2 a0∗ a0 D2 + d3 a0 D3 + d3 a0∗ a0 D3 + d4 a0∗ D4 + d4 a0∗ a0 D4 + U , (A.6)
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where ∗ , D1 = a1∗ a1 = D2 , D1 = a−1 , D2 = a−1
∗ D3 = a−1 a−1 = D4 , D3 = a1 , D4 = a1∗
(A.7)
and U are the remaining terms which do not contain factors with index 0: ∗ ai + c2 ai∗ + d1 ai ai+1 ai+1 U = c1 i=−2.−1,1,2
+ d2
i=−2,1
i=−2,−1,1.2
∗ ai∗ ai+1 ai+1
+ d3
i=−2,1
ai∗ ai ai+1
i=−2,1
+ d4
∗ ai∗ ai ai+1 .
(A.8)
i=−2,1
Omitting all those terms not containing a0 and a0∗ (which are irrelevant for our purpose), we obtain U ([−2, 2])2 = a0 W1 + a0∗ W2 + a0∗ a0 W3 ,
(A.9)
where W1 = (c1 + d1 D1 + d3 D3 )(d1 D1 + d2 D2 + d3 D3 + d4 D4 ) + [d1 D1 + d3 D3 , U ], (A.10) W2 = −(d1 D1 + d2 D2 + d3 D3 + d4 D4 )(c2 + d2 D2 + d4 D4 ) + [d2 D2 + d4 D4 , U ], (A.11) 2 W3 = (d1 D1 + d2 D2 + d3 D3 + d4 D4 ) + [d2 D2 + d4 D4 , d1 D1 + d3 D3 ] + [d1 D1 + d2 D2 + d3 D3 + d4 D4 , U ]+ , (A.12) where [X, Y ]+ = X Y + Y X denotes the anti-commutator. For the condition U ([−2, 2])2 ∈ A({0}c ) to be satisfied, W1 = W2 = W3 = 0 is necessary and sufficient because 1, a0 , a0∗ , and a0∗ a0 are linearly independent basis of A(0). We obtain our conclusion d1 = d2 = d3 = d4 = 0 already from W3 = 0 alone as follows. By substituting Di and Di , i = 1, 2, 3, 4, we find out that the first term of W3 is d1 d2 + d3 d4 while the second term vanishes (the latter due to D1 = D2 and D3 = D4 ). By substituting U we obtain W3 = (d1 d2 + d3 d4 + d1 c2 + d2 c1 + d3 c2 + d4 c1 ) ∗ ∗ ∗ ∗ − d12 a−2 a−1 − d1 d2 a−2 a−1 + d2 d1 a−2 a−1 + d22 a−2 a−1 ∗ ∗ + (d1 d4 + d2 d3 )a−2 a−2 + (d3 d2 + d4 d1 )a2 a2 + d32 a1 a2 + d3 d4 a1 a2∗ − d4 d3 a1∗ a2 − d42 a1∗ a2∗ .
(A.13)
Since all terms are linearly independent, W3 = 0 implies vanishing of all coefficients. In particular, d12 = d22 = d32 = d42 = 0,
(A.14)
which implies vanishing of d1 , d2 , d3 , and d4 . Acknowledgements. This is a result of work carried out during the author’s stays at the Erwin Schrödinger International Institute for Mathematical Physics, Wien in February and March, 2005 and at Institut für Theoretische Physik, ETH, Zürich in April and May, 2006. The author would like to acknowledge the financial support and stimulating atmosphere of both Institutes.
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References 1. Araki, H.: Conditional expectations relative to a product state and the corresponding standard potentials. Commun. Math. Phys. 246, 113–132 (2004) 2. Araki, H.: Standard potentials for non-even dynamics. Commun. Math. Phys. 262, 161–176 (2006) 3. Araki, H., Moriya, H.: Equilibrium statistical mechanics of Fermion lattice systems. Rev. Math. Phys. 15, 93–198 (2003) 4. Jaffe, A., Lesniewski, A., Osterwalder, K.: Quantum K-theory. Commun. Math. Phys. 118, 1–14 (1988) 5. Jaffe, A., Lesniewski, A., Osterwalder, K.: On super-KMS functionals and entire cyclic cohomology. K-Theory, 2, 675–682 (1989) 6. Jaffe, A., Lesniewski, A., Wisniowski, M.: Deformation of super-KMS functionals. Commun. Math. Phys. 121, 527–540 (1989) 7. Kishimoto, A., Nakamura, H.: Super-derivations. Commun. Math. Phys. 159, 15–27 (1994) 8. Sakai, S.: On one-parameter subgrouops of *-automorphisms on operator algebras and the corresponding unbounded derivations. Amer. J. Math. 98, 427–440 (1976) Communicated by Y. Kawahigashi
Commun. Math. Phys. 276, 287–314 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0336-x
Communications in
Mathematical Physics
Tanaka Theorem for Inelastic Maxwell Models F. Bolley1, , J. A. Carrillo2 1 Institut de Mathématiques, Université Paul Sabatier, F-31062 Toulouse Cedex 9, France 2 ICREA (Institució Catalana de Recerca i Estudis Avançats) and Departament de Matemàtiques,
Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain. E-mail: [email protected] Received: 9 April 2006 / Accepted: 20 March 2007 Published online: 21 September 2007 – © Springer-Verlag 2007
Abstract: We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Even in the elastic classical Boltzmann equation, we give a simpler proof of the Tanaka theorem than the ones in [29, 31]. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance. 1. Introduction This work is devoted to contraction and asymptotic properties of the homogeneous Boltzmann-type equations for inelastic interactions in the Maxwellian approximation introduced in [5] and further analyzed in [16, 6, 7, 10, 1, 11, 2, 8]. We are basically concerned with the Boltzmann equation ∂f = B θ ( f (t)) Q( f, f ) (1.1) ∂t considered in [5] and its variants. Here, f (t, v) is the density for the velocity v ∈ R3 distribution of the molecules at time t, and Q( f, f ) is the inelastic Boltzmann collision operator defined by 1 (ϕ, Q( f, f )) = f (v) f (w) ϕ(v ) − ϕ(v) dσ dv dw (1.2) 4π R3 R3 S 2 for any test function ϕ, where v =
1−e 1+e 1 (v + w) + (v − w) + |v − w|σ 2 4 4
Current address: Université Paris Dauphine, Ceremade, Place du Maréchal de Lattre de Tassigny, F-75775 Paris Cedex 16, France. E-mail: [email protected]
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is the postcollisional velocity, σ ∈ S 2 , v, w ∈ R3 and 0 < e ≤ 1 is the constant restitution coefficient. Equation (1.1) preserves mass and momentum, but makes the kinetic energy (or temperature) 2 1 θ ( f (t)) = v f (t, v) dv f (t, v) dv v − 3 R3 R3 decrease towards 0. In particular, solutions to (1.1) tend to the Dirac mass at the mean velocity of the particles [5]. We refer to [5, 7, 32] for the discussion about the relation of this model to the inelastic hard-sphere Boltzmann equation and different ways of writing √ the operator. Let us just point out that the factor B θ ( f (t)) in front of the operator in (1.1) is chosen for having the same temperature decay law as its hard-sphere counterpart [5] known as Haff’s law. The convergence towards the monokinetic distribution has been made more precise in [7, 10, 2] by means of homogeneous cooling states. They are self-similar solutions of the homogeneous Boltzmann equation (1.1) describing the long-time asymptotics and presenting power-like tail behavior whose relevance was previously discussed in the physics literature [19, 20]. To avoid the collapse of the solution to the Dirac mass, the authors in [16] suggested the introduction of a stochastic thermostat which, at the kinetic level, is modelled by a linear diffusion term in velocity. In this framework, the density f in the velocity space obeys ∂f = B θ ( f (t)) Q( f, f ) + A θ p ( f (t)) v f ∂t
with
0≤ p<
3 · 2
(1.3)
Existence and uniqueness for given mean velocity of a steady state to (1.3) have been shown in [18, 6, 1]. The convergence of solutions towards this steady state in all Sobolev norms has also been investigated and quantified by means of Fourier-based distances between probability measures [1]. Fourier techniques are a good toolbox and have been extremely fruitful for studying Maxwellian models in kinetic theory since Bobylev observed [3, 4] that such equations have closed forms in Fourier variables. Fourier distances are not only suitable technical tools to study the long-time asymptotics of models (1.1) and (1.3), but also they represent the first Liapunov functionals known for inelastic Boltzmann-type equations [1, 2]. In the case of the classical elastic Boltzmann equation for Maxwellian molecules, there is another known Liapunov functional, namely, the Tanaka functional [29], apart from the H -functional for which no counterpart is known in inelastic models. See also [12] for a discussion of Liapunov functionals for the classical Boltzmann equation and another proof of the Tanaka functional in the elastic case. The Tanaka functional is the Euclidean (or quadratic) Wasserstein distance between measures in the modern jargon of optimal mass transport theory. It is defined on the set P2 (R3 ) of Borel probability measures on R3 with finite second moment or kinetic energy as 1/2
1/2 2 W2 ( f, g) = inf |v − w| dπ(v, w) = inf E |V − W |2 , π
R 3 ×R 3
(V,W )
where π runs over the set of joint probability measures on R3 ×R3 with marginals f and g and (V, W ) are all possible couples of random variables with f and g as respective laws. This functional was proven by Tanaka [29] to be non-increasing for the flow of
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the homogeneous Boltzmann equation in the Maxwellian case, see also [12] for a simplified proof. In fact, the Tanaka functional and Fourier-based distances are related to each other [21, 15, 30], and were used to study the trend to equilibrium for Maxwellian gases. On the other hand, related simplified granular models [17] have been shown to be strict contractions for the Wasserstein distance W2 . Other applications of Wasserstein distances to granular models have been shown in [14, 23]. With this situation, a natural question arose as an open problem in [2, Remark 3.3] and [32, Sect. 2.8]: is the Euclidean Wasserstein distance a contraction for the flow of inelastic Maxwell models? The main results of this work answer this question affirmatively. Moreover, we shall not need to introduce Bobylev’s Fourier representation of the inelastic Maxwell models working only in the physical space. We shall show in Sect. 3 the key idea behind the proof of all results concerning contractions in W2 distance for inelastic Maxwell models, namely, the gain part Q + ( f, f ) of the collision operator verifies 3 + e2 W2 (Q + ( f, f ), Q + (g, g)) ≤ W2 ( f, g) 4 for any f, g in P2 (R3 ) with equal mean velocity and any restitution coefficient 0 < e ≤ 1. Based on this property, we shall derive contraction and asymptotic properties both for (1.1) and (1.3) in Subsects. 4.1 and 4.3; for instance we shall prove that the flow for the diffusive equation (1.3) is a strict contraction for W2 . Moreover, Subsect. 4.2 is devoted to the self-similar scaled equation associated to (1.1): in particular we shall show that solutions converge in W2 to a corresponding homogeneous cooling state, without rate but only assuming that initial data have bounded second moment. More precisely, in original variables, let f (t) be a solution to (1.1) with initial datum in P2 (R3 ) with zero mean velocity. If g∞ is the unique stationary solution of the scaled problem with zero mean velocity and unit temperature, and 3
1
f hc (v, t) = θ − 2 ( f (t)) g∞ (v θ − 2 ( f (t))) is the corresponding homogeneous cooling state, then we prove that lim θ ( f (t))−1/2 W2 ( f (t), f hc (t)) = 0.
t→∞
This improves the Ernst-Brito conjecture [19, 20, 7, 10, 2] since it shows that the basin of attraction of the homogeneous cooling state is larger - on f 0 we require bounded moments of order 2 only and not 2 + ε as in previous works - if we do not ask for a rate better than the one dictated by Haff’s cooling law. Moreover, a generalization for non-constant cross sections, including Tanaka’s theorem as a particular case, will be proven in Subsect. 4.4. Finally, we shall also show this generic property for the inelastic Kac model introduced in [27] as a dissipative version of Kac’s caricature of Maxwellian gases [22, 24]. 2. Basic Properties of the Distance W2 and Inelastic Maxwell Models We start by summarizing the main properties of the Euclidean Wasserstein distance W2 that we shall make use of in the rest, referring to [13, 31] for the proofs.
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Proposition 1. The space (P2 (R3 ), W2 ) is a complete metric space. Moreover, the following properties of the distance W2 hold: i) Convergence of measures: Given { f n }n≥1 and f in P2 (R3 ), the following three assertions are equivalent: a) W2 ( f n , f ) tends to 0 as n goes to infinity. b) f n tends to f weakly-* as measures as n goes to infinity and sup |v|2 f n (v) dv → 0 as R → +∞. n≥1 |v|>R
c)
f n tends to f weakly-* as measures and 2 |v| f n (v) dv → |v|2 f (v) dv as n → +∞. R3
R3
ii) Relation to Temperature: If f belongs to P2 (R3 ) and δa is the Dirac mass at a in R3 , then |v − a|2 d f (v). W22 ( f, δa ) = R3
iii) Scaling: Given f in P2 (R3 ) and θ > 0, let us define Sθ [ f ] = θ 3/2 f (θ 1/2 v) for absolutely continuous measures with respect to Lebesgue measure or its corresponding definition by duality for general measures; then for any f and g in P2 (R3 ), we have W2 (Sθ [ f ], Sθ [g]) = θ −1/2 W2 ( f, g). iv) Convexity: Given f 1 , f 2 , g1 and g2 in P2 (R3 ) and α in [0, 1], then W22 (α f 1 + (1 − α) f 2 , αg1 + (1 − α)g2 ) ≤ αW22 ( f 1 , g1 ) + (1 − α)W22 ( f 2 , g2 ). As a simple consequence, given f, g and h in P2 (R3 ), then W2 (h ∗ f, h ∗ g) ≤ W2 ( f, g), where ∗ stands for the convolution in R3 . Here the convolution of the two measures h and f is defined by duality by ϕ(x + y) dh(x) d f (y) (ϕ, h ∗ f ) = R 3 ×R 3
for any test function ϕ on R3 . If f is a Borel probability measure on R3 we shall let v d f (v) = v f (v) dv < f >= R3
R3
denote its mean velocity. We shall use the same notation for densities and measures expecting that the reader will not get confused. It is also interesting to note the following control of averages implied by W2 . As a consequence, the convergence with rate in W2 implies the convergence with rate for all these averages.
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Proposition 2. Given a Lipschitz map ϕ on R3 with Lipschitz constant L, then we have
R
ϕ(v)( f (v) − g(v)) dv ≤ L W2 ( f, g). 3
Proof. Let π be a joint measure on R3 × R3 with marginals f and g. Then
R3
ϕ(v)( f (v) − g(v)) dv =
R 3 ×R 3
(ϕ(v) − ϕ(w)) dπ(v, w).
Using that ϕ is Lipschitz with constant L and estimating by Hölder’s inequality, we get
R3
ϕ(v)( f (v) − g(v)) dv ≤
R 3 ×R 3
|ϕ(v) − ϕ(w)| dπ(v, w)
≤L
R 3 ×R 3
|v − w| dπ(v, w)
≤L
1/2 |v − w| dπ(v, w) 2
R 3 ×R 3
Minimizing over π concludes the argument.
.
Now, let us start by revising the main asymptotic properties of the solutions of (1.1). First of all the mean velocity of the solution is preserved, i.e.,
R3
v f (t, v) dv =
R3
v f 0 (v) dv = U
if f 0 in P2 (R3 ) is the initial datum. By the translational invariance we may assume without loss of generality that U = 0. Then the evolution of the temperature for all solutions of (1.1) is given by Haff’s law of cooling: 1 − e2 dθ ( f (t)) = − B θ ( f (t))3/2 , dt 4
(2.1)
as proven in [5]. Solving explicitly the ODE (2.1) and using the second property in Proposition 1 ensure the following convergence result towards the Dirac mass at 0: Corollary 1. Given a solution f (t, v) to (1.1) with zero mean velocity and initial kinetic energy θ0 corresponding to an initial datum in P2 (R3 ), then W22 ( f (t), δ0 ) =
1−e2 4
12θ0 2 . √ B θ0 t + 2
(2.2)
By Proposition 2, this implies the convergence as t −1 for the averages with Lipschitz functions. More generally it shows the convergence towards total cooling in infinite time. Then, the question is how this cooling is done and what is the typical cooling profile
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for Eq. (1.1). In [19, 20] the existence of a self-similar solution with power-like tails was conjectured (first part of the Ernst-Brito conjecture) which should be asymptotically stable for a large class of initial data (second part of the Ernst-Brito conjecture). The first part of the Ernst-Brito conjecture was answered in [7]: they proved the existence and uniqueness with given initial temperature of self-similar solutions, called homogeneous cooling states (HCS), of the form 3
1
f hc (v, t) = θ − 2 (t) g∞ (θ − 2 (t) v), where g∞ is a solution to ∇ · (g v) = E Q(g, g) with E = 8/(1 − e2 ) and θ (t) is any solution to the evolution (2.1) of the temperature. The results were actually stated in Fourier variables but they can be rewritten in this way. Later, the convergence towards these HCS was obtained in [10, 2]: solutions of the self-similar scaled equation ∂g + ∇ · (g v) = E Q(g, g) ∂τ converge exponentially fast towards the stationary solution g∞ under the assumption that the initial datum has bounded moments of order 2 + ε. Let us remark that in those results the speed of convergence goes to 0 as ε → 0. Also, let us point out that the convergence towards HCS eventually gives the information we are looking for: as times goes on, the solutions to (1.1) cool down towards the Dirac mass at 0 with the corresponding profile f hc (v, t). To conclude this introduction to the Inelastic Maxwell Models, we should mention that more general models allowing non-constant differential cross sections can be considered, as in [8]. Although these models can be written in any dimension, we will stick to the 3-dimensional setting for physical reasons. In any case, the results will generalize equally well to those cases. The operator Q b with variable differential cross section b is defined by
1 v−w (ϕ, Q b ( f, f )) = · σ dσ dv dw, f (v) f (w) [ϕ(v )−ϕ(v)] b 4π R3 R3 S 2 |v − w| (2.3) where again the post-collisional velocity v is given by v =
1−e 1+e 1 (v + w) + (v − w) + |v − w| σ, 2 4 4
and the cross section b satisfies the normalized cut-off assumption 2π π b(k · σ ) dσ = b(cos θ ) sin θ dθ dφ = 1 S2
0
(2.4)
0
for any k in S 2 . 3. Contraction in W2 of the Gain Operators The plan of this section is to obtain contraction properties in the Euclidean Wasserstein distance W2 of the gain operators Q + and Q +b respectively associated to Q and Q b . Since it will be important for later discussions to clarify the cases of equality, and in order to divide difficulties in the proof, we start by deriving this contraction estimate in the case
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of a constant differential cross section Q + ; then we rather quickly generalize the result to general differential cross sections b with the cut-off assumption (2.4). 3.1. Contraction for Q + . Let us write the collision operator Q given in (1.3) as
where Q + ( f, f ) is defined by (ϕ, Q + ( f, f )) =
Q( f, f ) = Q + ( f, f ) − f,
(3.1)
1 f (v) f (w) ϕ(v ) dσ dv dw 4π R3 R3 S 2
(3.2)
for any test function ϕ, where we recall that v =
1−e 1+e 1 (v + w) + (v − w) + |v − w|σ. 2 4 4
In this section we derive a contraction property in W2 distance of the gain operator Q + . For that purpose, let us note that the previous definition of the gain operator can be regarded as follows: given a probability measure f on R3 , the probability measure Q + ( f, f ) is defined by + (ϕ, Q ( f, f )) = f (v) f (w) (ϕ, v,w ) dv dw, R3 R3
where v,w is the uniform probability distribution on the sphere Sv,w with center cv,w =
1−e 1 (v + w) + (v − w) 2 4
and radius rv,w =
1+e |v − w| (Fig. 1). 4
S v,w
v
r v,w σ w
v+w 2
cv,w
Fig. 1. Geometry of inelastic collisions
v
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F. Bolley, J. A. Carrillo
In probabilistic terms, the gain operator is defined as an expectation: Q + ( f, f ) = E V,W , where V and W are independent random variables with law f . Then the convexity of W22 in Proposition 1 implies W22 (Q + ( f, f ), Q + (g, g)) = W22 E V,W , E X,Y
≤ E W22 ( V,W , X,Y ) ,
(3.3) (3.4)
where X and Y are independent random variables with law g. This observation leads us to consider the W2 distance between uniform distributions on spheres. To this aim, we have the following general lemma: Lemma 1. The squared Wasserstein distance W22 between the uniform distributions on the sphere with center O and radius r and the sphere with center O and radius r in R3 is bounded by |O − O|2 + (r − r )2 . Proof. We define a map T : R3 −→ R3 transporting the sphere of center O and radius r > 0 onto the sphere with center O and radius r ≥ r in the following way: – – –
If r = r , then we just let T be the translation map with vector O − O, i.e., T (v) = v + O − O. If O = O , then we just let T be the dilation with factor rr centered at O, i.e., T (v) = rr v. If r = r , then we consider the only point Ω ∈ R3 verifying that 1 1 (O − Ω) = (O − Ω), r r that is, Ω=O+
r
r (O − O). −r
Then we let T be the dilation with factor rr centered at Ω, that is, we let T (v) = Ω + rr (v − Ω). Such a construction of the point Ω and the map T is sketched in Fig. 2 in the case of non-interior spheres.
r’ r Ω
O
Oí v T(v)
Fig. 2. Sketch of the computation of the Euclidean cost of transporting spheres to spheres. Transport lines are just rays from the point Ω
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Let U O,r and U O ,r denote the uniform distributions on the corresponding spheres. Then the transport plan π given by η(v, w) dπ(v, w) = η(v, T (v)) dU O,r (v) R 3 ×R 3
R3
for all test functions η(v, w) has U O,r and U O ,r as marginals by construction of T . Using this transference plan in the definition of the Euclidean Wasserstein distance, we finally conclude
W22 (U O,r , U O ,r )
r − r ≤ |v − T (v)| dU O,r (v) = r R3
2
2
R3
|v − Ω|2 dU O,r (v)
that can be computed explicitly, giving W22 (U O,r , U O ,r ) ≤ |O − O|2 + (r − r )2 and finishing the proof.
Remark. The map T being the gradient of a convex function, Brenier’s theorem (see Ref. [31] for instance) ensures that the transport plan π defined above is optimal in the sense that the squared distance between the two distributions is = |O − O|2 + (r − r )2 . This lemma, using the notation a = v − x and b = w − y, for fixed values v, w, x, y in R3 , implies that W22 ( v,w , x,y ) ≤ |cv,w − cx,y |2 + |rv,w − r x,y |2
3 − e 1 + e 2 1+e 2 a+ b + ≤ |a − b|2 4 4 4 =
5 − 2e + e2 2 (1 + e)2 2 1 − e2 |a| + |b| + a · b; 8 8 4
here a · b denotes the scalar product between a and b in R3 and the bound in
2 1 + e 2 2 |v − w| − |x − y| |rv,w − r x,y | = 4
2 1 + e 2 1+e 2 (v − w) − (x − y) = ≤ |a − b|2 4 4 follows from the Cauchy-Schwarz inequality (v − w) · (x − y) ≤ |v − w| |x − y|.
(3.5)
Therefore, by (3.3), W22 (Q + ( f, f ), Q + (g, g)) ≤
(1 + e)2
5 − 2e + e2 E |V − X |2 + E |W − Y |2 8 8 1 − e2 + E [(V − X ) · (W − Y )] . 4
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F. Bolley, J. A. Carrillo
Let moreover (V, X ) and (W, Y ) be two independent optimal couples in the sense that
W22 ( f, g) = E |V − X |2 = E |W − Y |2 . Then 2 E [(V − X ) · (W − Y )] = E [(V − X )] · E [(W − Y )] = < f > − < g > by independence. Collecting all terms leads to the following key estimate and contraction property: Proposition 3. If f and g belong to P2 (R3 ), then W22 (Q + ( f, f ), Q + (g, g)) ≤
2 3 + e2 2 1 − e2 W2 ( f, g) + < f > − 4 4
for any restitution coefficient 0 < e ≤ 1. As a consequence, given f and g in P2 (R3 ) with equal mean velocity, then 3 + e2 W2 (Q + ( f, f ), Q + (g, g)) ≤ W2 ( f, g). 4 The case of equality is addressed in the following statement: Proposition 4. Let f and g belong to P2 (R3 ) with equal mean velocity and temperature, where g is absolutely continuous with respect to Lebesgue measure with positive density. If 3 + e2 + + W2 (Q ( f, f ), Q (g, g)) = W2 ( f, g) 4 for some restitution coefficient 0 < e ≤ 1, then f = g. Proof. It is necessary that the equality holds at each step of the arguments in Proposition 3. In particular, (3.5) holds as an equality, that is, X −Y V −W = |V − W | |X − Y | almost surely in the above notation. Then, since g is absolutely continuous with respect to Lebesgue measure with positive density, one can proceed as in [29, Lemma 9.1] to show that f = g. We sketch the proof for the sake of the reader. Since g is absolutely continuous with respect to Lebesgue measure, there exists [31] a Borel map u : R3 → R3 such that f is the image measure of g by u, and in probabilistic terms V = u(X ) and W = u(Y ) almost surely. Hence u(x) − u(y) x−y = |u(x) − u(y)| |x − y|
(3.6)
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almost everywhere for Lebesgue measure since X and Y are independent and since their law g has positive density. We leave the reader to check [31, Exercise 7.25] that this implies the existence of constants ω1 and ω2 such that u(x) = ω1 + ω2 x. First of all ω22 = 1 since f and g have the same temperature. Then identity (3.6) forces ω2 = 1, implying ω1 = 0 since < f > = < g >, and finally f = g.
3.2. Contraction for Q +b . In this subsection, we consider the more general case of the operator Q b defined in its weak form in (2.3). We define the gain term Q +b by
1 v−w f (v) f (w) ϕ(v ) b · σ dσ dv dw, 4π R3 R3 S 2 |v − w| (3.7) so that again Q b ( f, f ) = Q +b ( f, f ) − f . We shall prove the following extension of Proposition 3 for non-constant cross sections b. (ϕ, Q +b ( f, f )) =
Theorem 1. If f and g in P2 (R3 ) have equal mean velocity, then W22 (Q +b ( f, f ), Q +b (g, g)) ≤
π 3 + e2 1 − e2 + π b(cos θ ) cos θ sin θ dθ W22 ( f, g). 4 2 0
In the elastic case when e = 1, one recovers Tanaka’s non-strict contraction result [29] for the solutions to the homogeneous elastic Boltzmann equation for Maxwellian molecules, at least under the cut-off assumption, but with a somehow simpler argument than those given in [29, 31]. Proof. By definition (ϕ,
Q +b ( f,
π
f )) = 2π
2π
ϕ(v )
dφ f (v) f (w) dv dw b(cos θ ) sin θ dθ 2π
3 3 0 0 π R R = 2π E (ϕ, UV,W,θ ) b(cos θ ) sin θ dθ,
0
where V and W are independent random variables distributed according to f and, given v, w in R3 , Uv,w,θ is the uniform probability measure on the circle Cv,w,θ with center cv,w,θ
1 = (v + w) + 2
1−e 1+e + cos θ (v − w), 4 4
radius rv,w,θ =
1+e |v − w| sin θ 4
and axis k=
v−w (Fig. 3) · |v − w|
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F. Bolley, J. A. Carrillo
S v,w
θ
w v+w 2
v cv,w, θ
C v,w, θ
Fig. 3. Definition of circles to be transported
Let also g be a Borel probability measure on R3 and X, Y be independent random variables with law g. Then, by the normalization assumption (2.4), the convexity of the squared Wasserstein distance with respect to both arguments ensures that π
2 + + W2 (Q ( f, f ), Q (g, g)) ≤ 2π E W22 (UV,W,θ , U X,Y,θ ) b(cos θ ) sin θ dθ. (3.8) 0
We now let v, w, x, y and θ be fixed in R3 and [0, π ] respectively, and give an upper bound to W22 (Uv,w,θ , Ux,y,θ ). This consists in estimating the transport cost of a circle in R3 onto another one, for which we have the following general bound: Lemma 2. [31] The squared Wasserstein distance between the uniform distributions on the circles with centers c and c , radii r and r and axes k and k is bounded by |c − c |2 + r 2 + r 2 − rr (1 + |k · k |). Hence, using the notations a = v − x, b = w − y, a˜ = v − w and b˜ = x − y in our case we get 2 3 − e 1 + e 1+e 2 W2 (Uv,w,θ , Ux,y,θ ) ≤ + cos θ a + (1 − cos θ ) b 4 4 4
˜ b 1+e 2 2 a ˜ ˜ 2 − |a|| ˜ 1+ · + sin θ |a| ˜ 2 + |b| ˜ b| ˜ 4 |a| ˜ |b| 2 3−e 1+e 1+e 2 2 + cos θ + sin θ |a|2 ≤ 4 4 4
3−e 1+e 2 + cos θ (1 − cos θ ) − sin2 θ a · b +2 4 1+e
2
1+e + (1 − cos θ )2 + sin2 θ |b|2 , (3.9) 4
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299
where we have used the bound ˜ 2 − |a|| ˜ − a˜ · b˜ ≤ |a| ˜ 2 − 2 a˜ · b˜ = |a˜ − b| ˜ 2 = |a − b|2 . |a| ˜ 2 + |b| ˜ b| ˜ 2 + |b| Assume now that (V, X ) and (W, Y ) are two independent couples of random variables, optimal in the sense that
W22 ( f, g) = E |V − X |2 = E |W − Y |2 . Note that E [(V − X ) · (W − Y )] = E [V − X ] · E [W − Y ] = 0, since (V, X ) and (W, Y ) are independent and since f and g have same mean velocity. Then from (3.9):
E W22 (UV,W,θ , U X,Y,θ ) ≤ γ (θ ) W22 ( f, g), where
γ (θ ) = =
3−e 1+e + cos θ 4 4
2 +
1+e 4
2
(1 − cos θ )2 + 2 sin2 θ
3 + e2 1 − e2 + cos θ. 4 4
One concludes after averaging over θ as in (3.8).
4. Contractive Estimates for Inelastic Maxwell Models In this section, we shall derive contractive estimates in the Euclidean Wasserstein distance for solutions to the inelastic Maxwell models both in the non-diffusive and the diffusive cases and with constant and variable differential cross sections. 4.1. The non-diffusive case I: Original Variables. We are first concerned with solutions f (t) to the Boltzmann equation (1.1) with 0 < e < 1. After time scaling defined by B t τ= θ ( f (w)) dw E 0 with E = to
8 , 1−e2
as in [2], we get a function denoted again f (τ ) for simplicity, solution ∂f = E Q( f, f ). ∂τ
(4.1)
Theorem 2. If f 1 and f 2 are two solutions to (4.1) with respective initial data f 10 and f 20 in P2 (R3 ), then 2 W22 ( f 1 (τ ), f 2 (τ )) ≤ e−2τ W22 ( f 10 , f 20 ) + (1 − e−2τ ) < f 10 > − < f 20 > (4.2) for all τ ≥ 0.
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F. Bolley, J. A. Carrillo
Proof. Decomposition (3.1) of the collision operator Q as Q( f, f ) = Q + ( f, f ) − f allows us to represent the solutions to (4.1) by Duhamel’s formula as τ e−E(τ −s) Q + ( f i (s), f i (s)) ds, i = 1, 2. f i (τ ) = e−Eτ f i0 + E 0
Then the convexity of the squared Wasserstein distance in Proposition 1 and Proposition 3 imply W22 ( f 1 (τ ), f 2 (τ ))
≤ e−Eτ
τ
e−E(τ −s) W22 Q + ( f 1 (s), f 1 (s)), Q + ( f 2 (s), f 2 (s)) ds 0
τ 3 + e2 2 W2 ( f 1 (s), f 2 (s)) + X ds; W22 ( f 10 , f 20 ) + E e−E(τ −s) 4 0
≤ e−Eτ W22 ( f 10 , f 20 ) + E
here X=
2 1 − e2 < f 1 (s)> − < f 2 (s)> 4
does not depend on time since the mean velocity is preserved by Eq. (4.1). In other words, the function y(τ ) = e Eτ W22 ( f 1 (τ ), f 2 (τ )) satisfies the inequality
τ
y(τ ) ≤ y(0) + E 0
3 + e2 y(s) + X e Es 4
ds
and then y(τ ) ≤ y(0) eγ Eτ +
X (e Eτ − eγ Eτ ) 1−γ
by Gronwall’s lemma with γ = (3 + e2 )/4. This concludes the argument since (1 − γ )E = 2.
Remark 1. 1. Without further assumptions on the initial data f 10 and f 20 , this result is optimal in the following sense. If f 20 is chosen as the Dirac mass at the mean velocity of f 10 , then inequality (4.2) is actually an equality for all τ ; indeed 2 W2 ( f 1 (τ ), f 2 (τ )) = |v − < f 1 (τ )> |2 f 1 (τ, v) dv = 3 θ ( f 1 (τ )) R3
= 3 e−2τ θ ( f 10 ) = e−2 τ W22 ( f 10 , f 20 ), since
dθ = − 2 θ by Eq. (4.1). dτ
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2. In terms of the original time variable t in (1.1), if f 10 and f 20 are two initial data with the same initial temperature θ0 , then the temperatures of the corresponding solutions f 1 and f 2 to (1.1) follow the law 3 dθ 1 − e2 =− Bθ 2 , dt 4
(4.3)
and hence are both equal to
−2 1 − e2 −1/2 θ (t) = θ0 Bt + . 8 Then estimate (4.2) reads as W22 ( f 1 (t), f 2 (t)) ≤
2 θ (t) 2 0 0 θ (t) < f 10 > − < f 20 > W2 ( f 1 , f 2 ) + 1 − θ0 θ0
for all t ≥ 0, and in the particular case of equal mean velocity we can write it as W22 ( f 1 (t), f 2 (t)) ≤
θ (t) 2 0 0 W2 ( f 1 , f 2 ) θ0
(4.4)
that gives the typical decay towards the delta distribution of all solutions.
4.2. The non-diffusive case II: Self-similar variables. The convergence of the solutions to (1.1) towards the Dirac measure at their mean velocity [5] has been made much more precise in [10, 2] by the introduction of self-similar variables and homogeneous cooling states. There the authors prove that the rescaled solutions g defined by g(τ, v) = θ 3/2 ( f (τ )) f (τ, θ 1/2 ( f (τ )) v)
(4.5)
satisfy the strict contraction property d2+ε (g1 (τ ), g2 (τ )) ≤ e−C(ε)τ d2+ε (g10 , g20 ),
C(ε) > 0
for initial data g10 and g20 in P2 (R3 ) with equal mean velocity and pressure tensor, where ε > 0 and d2+ε is a Fourier-based distance between probability measures. Moreover, for ε = 0 one has C(ε) = 0 giving a non-strict contraction in d2 distance. In fact, by the scaling property in Proposition 1, (4.2) reads as W2 (g1 (τ ), g2 (τ )) ≤ W2 (g10 , g20 )
(4.6)
in the scaled variables. This is consistent with the fact that the distances d2 and W22 are “of the same order” [21, 30, 2] up to moment bounds. In itself this non-strict contraction does not give any information about the convergence towards typical cooling profiles of the system. A measure g(τ, v) defined by (4.5) from a solution f (v, τ ) to (4.1) with initial zero mean velocity has zero mean velocity and unit kinetic energy for all τ , and is a solution to ∂g + ∇ · (g v) = E Q(g, g). (4.7) ∂τ
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Moreover, it is proven in [10, 2] that (4.7) has a unique stationary solution g∞ with zero mean velocity and unit kinetic energy; all measure solutions g(τ, v) to (4.7) with zero mean velocity, unit kinetic energy and bounded moment of order 2 + ε converge to this stationary state g∞ as τ goes to infinity in the d2 sense, that is, in the W2 sense since d2 and W2 metrize the same topology on probability measures [30] up to moment conditions. Moreover, the convergence has exponential rate in the d2 sense, and in the W2 sense if the initial datum has finite fourth order moment as proven in [2]. In turn, this ensures existence and uniqueness of homogeneous cooling states to (1.1) for zero mean velocity and given kinetic energy, by going back to original variables, i.e., 3
1
f hc (v, t) = θ − 2 (t) g∞ (v θ − 2 (t)) with θ (t) any solution to the temperature equation (2.1). Moreover, an algebraic convergence of the solutions f (t) with the same initial kinetic energy towards them in the original variables is obtained in [10, 2]. We conclude this section by proving this convergence result using only the W2 distance, and moreover for initial data which have bounded moments of order 2 only and not 2 + ε as above. This in turn shows that the Euclidean Wasserstein distance W2 between solutions of (4.7) converges to zero as t goes to infinity, improving over (4.6) that does not a priori yield any information on the long-time behavior of the solutions g. As a trade of not assuming more than moments of order two, this argument does not provide any rate of convergence as does the Fourier-based argument in [10, 2]. Theorem 3. Let g10 and g20 be two Borel probability measures on R3 with zero mean velocity and unit kinetic energy, and let g1 (τ ) and g2 (τ ) be the solutions to (4.7) with respective initial data g10 and g20 . Then the map τ → W2 (g1 (τ ), g2 (τ )) is non-increasing and tends to 0 as τ goes to infinity. By taking as one solution, in this theorem, the homogeneous cooling state in scaled variables, i.e., the stationary solution g∞ of (4.7), we improve over the stability part of the Ernst-Brito conjecture shown in [10] and re-addressed in [2]. In terms of the original variables, the scaling properties of W2 given in Proposition 1 and the convergence result lim W2 (g(τ ), g∞ ) = 0
τ →∞
have the following direct consequence, which improves over the decay towards the Dirac mass estimate given in (2.2) and (4.4). Corollary 2. Let f 0 ∈ P2 (R3 ) with zero mean velocity and let f (t) be the solution to (1.1) with initial datum f 0 , then lim θ ( f (t))−1/2 W2 ( f (t), f hc (t)) = 0,
t→∞
where the homogeneous cooling state f hc is given by 3
1
f hc (t) = θ − 2 ( f (t)) g∞ (v θ − 2 ( f (t))).
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Proof of Theorem 3. It is based on the argument in [31] to Tanaka’s theorem. The first statement is a simple consequence of (4.6). Then we turn to the second part of the theorem which by the triangular inequality for the W2 distance is enough to prove when g20 , and hence g2 (τ ), is the unique stationary state g∞ to (4.7) with zero mean velocity and unit kinetic energy. Step 1. Let us first assume that the fourth moment of the initial datum is bounded, i.e., |v|4 g10 (v) dv < ∞. R3
The second step will be devoted to avoid this assumption. Then Proposition 6 in the appendix ensures that sup |v|4 g1 (τ, v) dv < ∞, τ ≥0 R3
so that
sup
τ ≥0 |v|>R
|v|2 g1 (τ, v) dv
tends to 0 as R goes to infinity. Prohorov’s compactness theorem and Proposition 1 imply the existence of a sequence τk → ∞ as k → ∞ and a probability measure µ0 on R3 with zero mean velocity and unit kinetic energy such that W2 (g1 (τk ), µ0 ) → 0 as k → ∞. We want to prove that µ0 = g∞ . Without loss of generality, we can assume that the diverging time sequence satisfies τk + 1 ≤ τk+1 for all k. Now, since g∞ is a stationary solution, it follows from the first part of the theorem that W2 (g1 (τk+1 ), g∞ ) ≤ W2 (g1 (τk + 1), g∞ ) ≤ W2 (g1 (τk ), g∞ ).
(4.8)
On one hand, both W2 (g1 (τk ), g∞ ) and W2 (g1 (τk+1 ), g∞ ) tend to W2 (µ0 , g∞ ) as k goes to infinity by triangular inequality. Then, if µ(τ ) denotes the solution to (4.7) with initial datum µ0 , the first point again ensures that W2 (g1 (τk + 1), µ(1)) ≤ W2 (g1 (τk ), µ0 ) which tends to 0. Hence W2 (g1 (τk +1), g∞ ) tends to W2 (µ(1), g∞ ) by triangular inequality, and finally W2 (µ(1), g∞ ) = W2 (µ0 , g∞ ) by passing to the limit in k in (4.8). By the non-increasing character of W2 along the flow, we deduce that W2 (µ(1), g∞ ) = W2 (µ(τ ), g∞ ) = W2 (µ0 , g∞ ) for all τ ∈ [0, 1]. Consequently µ(τ ) and g∞ are two solutions to (4.7) with zero mean velocity and unit temperature, whose W2 distance is constant on the time interval [0, 1]. This is possible only if equality holds at each step in the proof of Theorem 2 in the original space variables; in particular 3 + e2 W2 (µ(τ ), g∞ ) W2 (Q + (µ(τ ), µ(τ )), Q + (g∞ , g∞ )) = 4
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for all τ , and especially for τ = 0. But µ0 and g∞ have the same mean velocity and temperature, and, according to [7, Theorem 5.3], g∞ is absolutely continuous with respect to Lebesgue measure, with positive density. Hence Proposition 4 ensures that µ0 = g∞ . In particular W2 (g1 (τk ), g∞ ) → 0 as k → ∞, and then W2 (g1 (τ ), g∞ ) → 0 as τ → ∞ since it is a non-increasing function. Step 2. Let us now remove the hypothesis on the boundedness of the initial fourth order moment. Let (g 0n )n be a sequence in P2 (R3 ) with zero mean velocity, unit kinetic energy, finite fourth order moment and converging to g10 in the weak sense of probability measures; in particular it converges to g10 in the W2 distance sense by Proposition 1. Such a g 0n can be obtained by successive truncation of g10 to a ball of radius n in R3 , translation to keep the mean property, and dilation centered at 0 to keep the kinetic energy equal to 1. Then, if g n (τ ) is the solution to (4.7) with initial datum g 0n , the triangular inequality for W2 and (4.6) ensure that W2 (g1 (τ ), g∞ ) ≤ W2 (g1 (τ ), g n (τ )) + W2 (g n (τ ), g∞ ) ≤ W2 (g10 , g 0n ) + W2 (g n (τ ), g∞ ). Given ε > 0, the first term in the right hand side is bounded by ε for some n large enough, and for this now fixed n, the second term is bounded by ε for all τ larger than some constant by the first step. This ensures that W2 (g1 (τ ), g∞ ) tends to 0 as τ goes to infinity.
Let us finally point out that a natural question related to the fact that Eq. (4.7) is a strict contraction with respect to d2+ε is whether a Wassertein distance with larger index, for instance W4 , could be strictly contractive for (4.7). Of course, a similar scheme as above can be performed to verify it, but there is one term we cannot control in the transport of spheres argument and we cannot conclude. It is an open problem to prove or disprove this claim, even for a non-strict contraction in the elastic case.
4.3. The diffusive case. We now turn to the diffusive version (1.3) of (1.1). Again by the change of time τ= with E =
B E
t θ ( f (w)) dw 0
8 we are brought to studying the equation 1 − e2 ∂f = E Q( f, f ) + 2 ( f (τ )) v f, ∂τ
where 2 ( f (τ )) =
EA [θ ( f (τ ))] p−1/2 . B
As in the nonviscous case of (4.1) we shall prove
(4.9)
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305
Theorem 4. If f 1 and f 2 are two solutions to (4.9) for the respective initial data f 10 and f 20 in P2 (R3 ) with same kinetic energy, then 2 W22 ( f 1 (τ ), f 2 (τ )) ≤ e−2τ W22 ( f 10 , f 20 ) + (1 − e−2τ ) < f 10 > − < f 20 > (4.10) for all τ ≥ 0. Proof. We again start by giving a Duhamel’s representation of the solutions. To this aim we write (4.9) as ∂f = E F − E f + 2 ( f (τ )) v f, ∂τ where F = Q + ( f, f ), that is, ∂ fˆ ˆ + E + |k|2 2 ( f (τ )) fˆ = E F. ∂τ Here, we are using the convention µ(k) ˆ =
R3
e−i k·x dµ(x)
for the Fourier transform of the measure µ on R3 . Hence the solutions satisfy τ 2 ˆ k) e−(( f,τ )−( f,s))|k|2 ds, fˆ(τ, k) = e−E τ fˆ0 (k) e−( f,τ ) |k| + E e−E(τ −s) F(s, where ( f, τ ) =
0 τ
2 ( f (s)) ds, and thus
0
τ f (τ, v) = e−E τ ( f 0 ∗ 2( f,τ ) )(v) + E e−E(τ −s) (F(s) ∗ 2(( f,τ )−( f,s)) )(v) ds 0 τ ˜ s, v) ds. := e−E τ f˜(τ, v) + E e−E(τ −s) F(τ, 0
Here α (v) =
1 2 e−|v| /2α (2π α)3/2
is the centered Maxwellian with temperature α/3 > 0. Moreover f 1 and f 2 have the same temperature at all times, so that ( f 1 , τ ) = ( f 2 , τ ). Then the convexity of the squared Wasserstein distance and its non-increasing character by convolution with a given measure, see Proposition 1, imply that τ W22 ( f 1 (τ ), f 2 (τ )) ≤ e−Eτ W22 ( f˜1 (τ ), f˜2 (τ ))+ E e−E(τ −s) W22 ( F˜1 (τ, s), F˜2 (τ, s))ds τ 0 −Eτ 2 0 0 ≤e W2 ( f 1 , f 2 ) + E e−E(τ −s) W22 (F1 (s), F2 (s)) ds. 0
In other words the squared distance W22 ( f 1 (τ ), f 2 (τ )) satisfies the same bound as in the nonviscous case of Theorem 2, and we can conclude analogously.
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Remark 2. 1. As pointed out to us by C. Villani the result can also be obtained by a splitting argument between the collision term and the diffusion term. 2. As proven in [1], the temperature θ ( f (t)) of the solution f in the original time variable t converges towards
θ∞ =
8A B(1 − e2 )
2 3−2 p
as t goes to infinity, and satisfies θ ( f (t)) ≥ min(θ ( f (0)), θ∞ ). In particular B t C1 t τ= θ ( f (s)) ds ≥ E 0 E if C1 = B min(θ ( f (0)), θ∞ )1/2 . Writing (4.10) in the original variable t for initial data with equal mean velocity and temperature, we recover the contraction property W2 ( f 1 (t), f 2 (t)) ≤ W2 ( f 10 , f 20 ) e−(1−γ )C1 t , that coincides with (3.1) in [1] for the Fourier-based d2 distance exactly with the same rate. For p = 1 one can exactly compute τ and also recover (3.2) in [1] but for the distance W2 . 3. The existence of unique diffusive equilibria for each given value of the initial mean velocity can be obtained from this contraction property of the W2 distance analogously to the arguments done in [1] with the Fourier-based distance d2 . 4.4. General cross section. Let f 1 = f 1 (τ, v) and f 2 = f 2 (τ, v) be two solutions to the Boltzmann equation ∂f = Q b ( f, f ) = Q +b ( f, f ) − f ∂τ with respective initial data f 10 and f 20 in P2 (R3 ), where Q +b is defined in (3.7). Then, as in Subsect. 4.1, Duhamel’s representation formula τ f (τ ) = e−τ f 0 + e−(τ −s) Q + ( f (s), f (s)) ds 0
of the solutions and the convexity of W22 ensure the contraction property W2 ( f 1 (τ ), f 2 (τ )) ≤ e−(1−γb )τ/2 W2 ( f 10 , f 20 ) for all τ , where 3 + e2 1 − e2 + π γb = 4 2 is bounded by 1 by (2.4).
π 0
b(cos θ ) cos θ sin θ dθ
(4.11)
Tanaka Theorem for Inelastic Maxwell Models
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5. Inelastic Kac Model In this last section we consider a simple one-dimensional model introduced in [27] which can be seen as a dissipative version of the Kac caricature of a Maxwellian gas [22, 24]. Let us remark that the definition and properties of the Euclidean Wasserstein distance W2 discussed above generalize equally well to any dimension. Tanaka himself [28] showed that the Euclidean Wasserstein distance is a non strict contraction for the elastic classical Kac model. In the inelastic Kac model, the evolution of the density function f is governed by the equation ∂f = Q( f, f ) (5.1) ∂t in which the collision term Q( f, f ) is defined by 2π dθ (ϕ, Q( f, f )) = dv dw f (v) f (w) ϕ(v ) − ϕ(v) 2π R R 0 for any test function ϕ, where v = v cos θ | cos θ | p − w sin θ | sin θ | p is the postcollisional velocity and p > 0 measures the inelasticity. Equation (5.1) preserves mass but makes the momentum and kinetic energy decrease to 0 at an exponential rate, θ ( f (t)) = e−2βt θ ( f 0 ) + (e−2βt − e−2t ) < f 0> with β > 0 given below. In particular, solutions to (5.1) tend to the Dirac mass at 0. The asymptotic behavior of the inelastic Kac model was analyzed in [27] by means of Fourier-based distances in interesting cases where the initial data has infinite kinetic energy, i.e., for Fourier distances ds with index 1 < s < 2. The existence of self-similar solutions with only moments of order 1 is obtained, characterizing long time asymptotics of the scaled solutions for solutions with a certain number of moments bounded. Here, we show that for finite kinetic energy solutions the distance W2 verifies a strict contraction property. This property may be used in particular cases of the Kac equation with self-similar solutions of finite kinetic energy to discuss its stability in the spirit of Theorem 3. As in the inelastic Maxwell model discussed above, we start by deriving a contraction property for the gain operator Q + defined by 2π dθ dv dw. (ϕ, Q + ( f, f )) = f (v) f (w) ϕ(v ) 2π R R 0 Proposition 5. If f and g belong to P2 (R), then 2π dθ W22 ( f, g). | cos θ |2( p+1) + | sin θ |2( p+1) W22 (Q + ( f, f ), Q + (g, g)) ≤ 2π 0 In terms of solutions f (t) and g(t) to the modified Kac equation (5.1) with finite initial energy only, the above proposition yields, as in previous sections, the bound W2 ( f (t), g(t)) ≤ e−βt W2 ( f 0 , g 0 ), where
2β = 1− 0
2π
dθ > 0. | cos θ |2( p+1) + | sin θ |2( p+1) 2π
This bound is optimal without further assumptions on the initial data f 0 and g 0 since equality holds in the case when < f 0 > = 0 and g 0 = δ0 analogously to previous cases.
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Proof. Given a vector (v, w) in R2 , let Cv,w denote the curve {(v (θ ), w (θ )), θ ∈ [0, 2 π ]}, where
v (θ ) = v cos θ | cos θ | p − w sin θ | sin θ | p , w (θ ) = v sin θ | sin θ | p + w cos θ | cos θ | p .
(5.2)
Let also Uv,w be the uniform probability distribution on Cv,w . Given V and W two independent random variables distributed according to f , we note that Q + ( f, f ) is the first marginal on R of E UV,W , but also its second marginal by symmetry. Then, we have the following result, which is analogous to Lemmas 1 and 2 for this model: Lemma 3. Given two vectors (v, w) and (x, y) in R2 , the squared Wasserstein distance between the distributions Uv,w and Ux,y is bounded by (1 − 2β) |v − x|2 + |w − y|2 . Proof. One can transport the curve Cv,w onto Cx,y by the linear map (a, b) → T (a, b) =
r (a cos ω − b sin ω, a sin ω + b cos ω), r
√ where r = v 2 + w 2 , r = x 2 + y 2 and ω is the angle between the vectors (v, w) and (x, y) in case they do not vanish. We leave it to the reader to discuss the case when either (x, y) or (v, w) are zero. Then, analogously to the proof of Lemma 1, one can define a transport plan associated to the transport map T to get 2 W2 (Uv,w , Ux,y ) ≤ |T (a, b) − (a, b)|2 dUv,w (a, b). R2
Furthermore, for all (a, b) in R2 , r 2 r 2 |T (a, b) − (a, b)|2 = (a cos ω − b sin ω) − a + (a sin ω + b cos ω) − b r r r 2 r = − 2 cos ω + 1 a 2 + b2 r r 2 |v − x| + |w − y|2 2 2 = a . + b v 2 + w2 Hence, we deduce |v − x|2 + |w − y|2 (a 2 + b2 ) dUv,w (a, b) 2 v 2 + w2 R |v − x|2 + |w − y|2 2π 2 dθ = (v (θ ) + w (θ )2 ) · v 2 + w2 2π 0
W22 (Uv,w , Ux,y ) ≤
But
v (θ )2 + w (θ )2 = | cos θ |2( p+1) + | sin θ |2( p+1) (v 2 + w 2 )
Tanaka Theorem for Inelastic Maxwell Models
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by (5.2), so that W22 (Uv,w , Ux,y )
≤ 0
2π
2( p+1) 2( p+1) dθ |v − x|2 + |w − y|2 | cos θ | + | sin θ | 2π
which is the bound given by the lemma.
We now continue the proof of Proposition 5. First of all, let (V, X ) and (W, Y ) be two independent couples of random variables, with V and X distributed according to f , W and Y according to g, optimal in the sense that
W22 ( f, g) = E |V − W |2 = E |X − Y |2 . Then, by convexity of the squared Wasserstein distance again, it follows from Lemma 3 that
W22 E UV,W , E U X,Y ≤ E W22 (UV,W , U X,Y )
≤ (1 − 2β) E |V − W |2 + E |X − Y |2 = 2 (1 − 2β) W22 ( f, g).
(5.3)
Next, we remark that the measure UV,W on R2 has first and second marginals equal by symmetry of thecurve CV,W by a π/2 rotation. This implies that the first and second of E UV,W on R2 are equal to Q + ( f, f ), and likewise for the measure marginals E U X,Y with marginals Q + (g, g). We shall conclude the argument of Proposition 5 by using the following general result: Lemma 4. If the Borel probability measures µij on R are the successive one-dimensional marginals of the measure µi on R N , for i = 1, 2 and j = 1, . . . , N , then N
W22 (µ1j , µ2j ) ≤ W22 (µ1 , µ2 ).
j=1 N with marginals µ1 and µ2 . Then its Proof. Let π be any joint measure on RvN × Rw marginal π j on Rv j × Rw j has itself marginals µ1j and µ2j , so 2 1 2 W2 (µ j , µ j ) ≤ |v j − w j |2 dπ j (v j , w j ).
The identity
R ×R
N
j=1 |v j N j=1
− w j |2 = |v − w|2 implies the bound
W22 (µ1j , µ2j ) ≤
R N ×R N
|v − w|2 dπ(v, w).
One concludes the argument by minimizing over π .
In our particular case, Lemma 4 ensures that
2 W22 (Q + ( f, f ), Q + (g, g)) ≤ W22 E UV,W , E U X,Y ,
which concludes the proof of Proposition 5 taking (5.3) into account.
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Appendix: Uniform in Time Propagation of Fourth Order Moments In this appendix we derive a uniform propagation of fourth order moments |v|4 g(τ, v)dv of solutions g to R3
∂g + ∇ · (g v) = E Q(g, g), ∂τ
(5.4)
where the operator Q(g, g) is defined as in (1.2) for 0 < e < 1 and E = This result has been used in the proof of Theorem 3.
8 · 1 − e2
Proposition 6. If g 0 is a Borel probability measure on R3 such that |v|4 g 0 (v) dv < ∞, R3
then the solution g to (5.4) with initial datum g 0 verifies |v|4 g(τ, v) dv < ∞. sup τ ≥0 R3
Proof. Without loss of generality we can assume that g 0 , and hence g(τ ) for all τ ≥ 0, has zero mean velocity. We let m 4 (τ ) = |v|4 g(τ, v) dv R3
denote the fourth order moment of g(τ ). Then, using the weak formulation of the inelastic Boltzmann equation, we have: dm 4 (τ ) 4 = ∇(|v| ) · v g(τ, v) dv + E |v|4 Q(g(τ ), g(τ ))(v) dv. (5.5) dτ R3 R3 While the first term on the right-hand side is simply 4 m 4 (τ ), the second term is computed by Lemma 5. There exist some constants µ1 and µ2 , depending only on e, such that R3
|v|4 Q(g, g)(v) dv = − λ + µ2
R
3
|v|4 g(v) dv + µ1
R 3 ×R 3
2
R3
|v|2 g(v) dv
(v · w)2 g(v) g(w) dv dw
for any probability measure g on R3 with finite moment of order 4 and zero mean velocity, where λ=
1 (1 + 4ε − 7ε2 + 4ε3 − 2ε4 ) 3
and
ε=
1−e · 2
Tanaka Theorem for Inelastic Maxwell Models
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With this lemma in hand, (5.5) reads dm 4 (τ ) = (4 − E λ) m 4 (τ ) + m(τ ), dτ
(5.6)
where m(τ ) is a combination of second order moments, which are bounded in time since the kinetic energy is preserved by Eq. (5.4). Moreover one can check from the expression of E and λ in terms of ε = (1 − e)/2 that 4 − Eλ =
2 [−1 + 2ε + ε2 − 4ε3 + 2ε4 ] 3ε(1 − ε)
which is negative for any 0 < ε < 1/2, that is, for any 0 < e < 1. By Gronwall’s lemma this ensures that m 4 (τ ) is bounded uniformly in time if initially finite, and concludes the argument to Proposition 6.
Let us remark that identity (5.6) is also useful to understand that moments are not created by this equation in contrast to the hard-spheres case [25, 26]. In fact, if initially moments are infinite, they will remain so. Thus, this is another reason why homogeneous cooling states have only a certain number of moments bounded, see [7]. We now turn to the proof of Lemma 5, whose result is given in [5] and [6, Sect. 4] only in the radial isotropic case, i.e., whenever g(v) depends only on |v|. By symmetry we start by writing 1 4 |v| Q(g, g)(v) dv = g(v)g(w) 4π R3 R3 S 2 R3 1 × [|v |4 + |w |4 − |v|4 − |w|4 ] dσ dv dw, 2 where 1−e 1+e 1 (v + w) + (v − w) + |v − w|σ, 2 4 4 1−e 1+e 1 w = (v + w) − (v − w) − |v − w|σ. 2 4 4 v =
Then we introduce the notation u=
v+w v−w 1−e 1+e , U= , ε= , ε = 1 − ε = 2 2 2 2
in which v = u + ε U + ε |U | σ, v = u − ε U − ε |U | σ, v = u + U, w = u − U. Then |v |2 |w |2 |v|2 |w|2
= |u|2 + (ε2 + ε2 )|U |2 + 2 ε ε |U | (U · σ ) + 2 ε (u · U ) + 2 ε |U | (u · σ ), = |u|2 + (ε2 + ε2 )|U |2 + 2 ε ε |U |(U · σ ) − 2 ε (u · U ) − 2 ε |U | (u · σ ), = |u|2 + |U |2 + 2 (u · U ), = |u|2 + |U |2 − 2 (u · U ),
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and eventually 1 4 [|v | + |w |4 − |v|4 − |w|4 ] 2
= (ε2 + ε2 )2 − 1 |U |4 + 2(ε2 + ε2 − 1)|u|2 |U |2 + 4(ε2 − 1)(u · U )2 + 4 ε2 ε2 |U |2 (U · σ )2 + 4 ε2 |U |2 (u · σ )2
+ 4 ε ε |U | |u|2 + (ε2 + ε2 )|U |2 (U · σ ) + 8 ε ε |U | (u · U ) (u · σ ).
Integrating with respect to σ in S 2 and taking the identities dσ dσ dσ |k|2 1 (k · σ ) (k · σ )2 = 1, = 0, = 4π 4π 3 S 2 4π S2 S2 into account, we obtain 1 4 dσ [|v | + |w |4 − |v|4 − |w|4 ] = α |U |4 + β |u|2 |U |2 + γ (u · U )2 , 2 2 4π S where
4 2 α = (ε2 + ε2 )2 − 1 + ε2 ε2 , β = 2 ε2 + ε2 − 1 + ε2 , γ = 4(ε2 − 1). 3 3 Then, by definition of u and U in terms of v and w, the identities
1 m 4 + m 22 + 2 m 22 , |U |4 g(v) g(w) dv dw = 8 R 3 ×R 3
1 m 4 + m 22 − 2 m 22 |u|2 |U |2 g(v) g(w) dv dw = 8 R 3 ×R 3
and
R 3 ×R 3
hold with
(u · U )2 g(v) g(w) dv dw =
m4 = m 22 =
1 [m 4 − m 22 ] 8
3 R
|v|4 g(v) dv, m 2 =
R 3 ×R 3
R3
|v|2 g(v) dv,
(v · w)2 g(v) g(w) dv dw,
since g has zero mean velocity. Collecting all terms, we obtain |v|4 Q(g, g)(v) dv = −λ m 4 + µ1 m 22 + µ2 m 22 , R3
where 1 1 λ = − (α + β + γ ) = (1 + 4 ε − 7 ε2 + 4 ε3 − 2 ε4 ), 8 3 1 1 µ1 = (α + β − γ ) and µ2 = (α − β) 8 4 depend only on ε, that is, only on e. This concludes the proof of Lemma 5.
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Acknowledgements. The authors are grateful to Laurent Desvillettes, Giuseppe Toscani and Cédric Villani for stimulating discussions and fruitful comments. They thank the referees for their relevant comments which helped improve the presentation of the paper. JAC acknowledges the support from DGI-MEC (Spain) project MTM2005-08024 and 2005SGR00611. We acknowledge partial support of the Acc. Integ. program HF20060198.
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27. Pulvirenti, A., Toscani, G.: Asymptotic properties of the inelastic Kac model. J. Stat. Phys. 114, 1453–1480 (2004) 28. Tanaka, H.: An inequality for a functional of probability distributions and its applications to Kac’s one-dimensional model of a Maxwellian gas. Z. Wahrsch. Verw. Gebiete 27, 47–52 (1973) 29. Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46, 1, 67–105 (1978/79) 30. Toscani, G., Villani, C.: Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas. J. Stat. Phys. 94, 619–637 (1999) 31. Villani, C.: Topics in optimal transportation, Graduate Studies in Mathematics, Vol. 58. Providence, RI: Amer. Math. Soc, 2003 32. Villani, C.: Mathematics of granular materials. J. Stat. Phys. 124, 781–822 (2006) Communicated by J.L. Lebowitz
Commun. Math. Phys. 276, 315–339 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0337-9
Communications in
Mathematical Physics
Global Existence and Uniqueness of Solutions to the Maxwell-Schrödinger Equations Makoto Nakamura1, , Takeshi Wada2, 1 Mathematical Institute, Tohoku University, Sendai 980-8578, Japan.
E-mail: [email protected]
2 Department of Mathematics, Faculty of Engineering, Kumamoto University, Kumamoto 860-8555, Japan.
E-mail: [email protected] Received: 16 June 2006 / Accepted: 7 February 2007 Published online: 25 September 2007 – © Springer-Verlag 2007
Dedicated to Professor Hiroki Tanabe on his seventy-fifth birthday Abstract: The time local and global well-posedness for the Maxwell-Schrödinger equations is considered in Sobolev spaces in three spatial dimensions. The Strichartz estimates of Koch and Tzvetkov type are used for obtaining the solutions in the Sobolev spaces of low regularities. One of the main results is that the solutions exist time globally for large data. 1. Introduction The Maxwell-Schrödinger system (MS) in space dimension 3 describes the time evolution of a charged nonrelativistic quantum mechanical particle interacting with the (classical) electro-magnetic field it generates. We can state this system in usual vector notation as follows: i∂t u = (− A + φ)u, −φ − ∂t div A = ρ, A + ∇(∂t φ + div A) = J,
(1.1) (1.2) (1.3)
where (u, φ, A) : R1+3 → C × R × R3 , ∇A = ∇ − iA, A = ∇A 2 , ρ = |u|2 , J = 2 Im u∇ ¯ A u, and ∇, and are the usual gradient, Laplacian and d’Alembertian respectively. Physically, u is the wave function of the particle, (φ, A) is the electromagnetic potential, ρ is the charge density, and J is the current density. The system (MS) formally conserves at least two quantities, namely the total charge Q ≡ u22 and the total energy 1 1 E ≡ ∇A u22 + ∇φ + ∂t A22 + rot A22 . 2 2 Supported by Grant-in-Aid for Young Scientists (B) #16740071 of Japan Ministry of Education, Culture, Sports, Science and Technology. Supported by Grant-in-Aid for Young Scientists (B) #16740075 of Japan Ministry of Education, Culture, Sports, Science and Technology.
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The system (MS) is invariant under the gauge transform (u , φ , A ) = (exp(iλ)u, φ − ∂t λ, A + ∇λ),
(1.4)
and in this paper we mainly study it in the Coulomb gauge div A = 0,
(1.5)
in which we can treat the system most easily. In this gauge, (1.2) and (1.3) become −φ = ρ, A + ∇∂t φ = J.
(1.6)
The first equation of (1.6) is solved as φ = φ(u) = (−)−1 ρ = (4π |x|)−1 ∗ |u|2 , and the term ∇∂t φ in the second equation is dropped by operating the Helmholtz projection P = 1 − ∇ div −1 to both sides of the equation. Therefore in the Coulomb gauge the system (MS) is rewritten as i∂t u = (−A + φ(u))u, A = PJ,
(1.7) (1.8)
which is referred to as (MS-C). To solve (MS-C) we should give the initial condition (u(0), A(0), ∂t A(0)) = (u 0 , A0 , A1 )
(1.9)
in the direct sum of Sobolev spaces X s,σ = {(u 0 , A0 , A1 ) ∈ H s ⊕ H σ ⊕ H σ −1 ; div A0 = div A1 = 0}. The condition (1.5) is conserved under the consistency conditions div A0 = div A1 = 0 since the equation div A = 0 follows from (1.8). Several authors have studied the Cauchy problem and the scattering theory for (MS-C). Nakamitsu-M. Tsutsumi [16] showed the time local well-posedness for (MS-C) in X s,σ with s = σ = 3, 4, 5, . . . . In fact, they treated the case of Lorentz gauge mentioned below, but the Coulomb gauge case can be treated analogously. We remark that their condition can be refined as s = σ > 5/2 by the use of fractional order Sobolev spaces and the commutator estimate by Kato-Ponce [12]. Recently NakamuraWada [17] showed the time local well-posedness for a wider class of (s, σ ) including the case s = σ ≥ 5/3 (precisely see the remark for Theorem 1.1) by using covariant derivative estimates for the Schrödinger part and the Strichartz estimate for the Maxwell part. On the other hand, Guo-Nakamitsu-Strauss [7] constructed a time global (weak) solution in X 1,1 although they did not show the uniqueness. Indeed, in the Coulomb gauge the energy takes the form 1 1 1 E = ∇A u22 + ∇φ22 + ∂t A22 + ∇A22 , 2 2 2 and hence (u, A, ∂t A); X 1,1 does not blow up. Therefore the global existence is proved by parabolic regularization and the compactness method. For the scattering theory, the existence of modified wave operators was proved by Y. Tsutsumi [21], Shimomura [18], and Ginibre-Velo [5, 6].
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317
As we have summarized above, there are several results for the Cauchy problem both at t = 0 or t = ∞. However there are no results concerning the global existence of strong solutions even for small data; the solutions to (MS-C) obtained in [5, 6, 18, 21] exist only for t ≥ 0 and we do not know whether these solutions globally exist or blow-up at finite negative time. The aim of this paper is to answer this problem. Shortly, we prove the global existence of unique strong solutions. To do this, we would need a priori estimates derived from the conservation laws of charge and energy, and hence it is desirable to show the local well-posedness in lower regularity. Therefore we first refine the local theory. To make the statements of the propositions simple, we introduce the notation R∗ = {(s, σ ) ∈ R2 ; σ ≥ max{1; s − 2; (2s − 1)/4}, (s, σ ) = (7/2, 3/2)}, R ∗ = {(s, σ ) ∈ R2 ; σ ≤ min{s + 1; 3s/2; 2s − 3/4}, (s, σ ) = (2, 3)}, and R = R∗ ∩ R ∗ . Theorem 1.1. Let (s, σ ) ∈ R with s ≥ 11/8, σ > 1. Then for any (u 0 , A0 , A1 ) ∈ X s,σ , there exists T > 0 such that (MS-C) with initial condition (1.9) has a unique solution (u, A) satisfying (u, A, ∂t A) ∈ C([0, T ]; X s,σ ). Moreover if s > 11/8 and (s + 1, σ ) ∈ R∗ , then the mapping (u 0 , A0 , A1 ) → (u, A, ∂t A) is continuous as a mapping from X s,σ to C([0, T ]; X s,σ ). σ
s=
11 8
σ = 2s −
3 4
σ=
3s 2
σ =s+1
6 .. .. ... . . ... .. . . 4 . .. ... . . .. ... ... .. .... 3 . . (2, 3) . . . ... .. ... ... . . . . . . . 3 9 . . (2, 4) ... . . . . 2 .... . . . . . 5 3 . ( 2 , 2 ) ... . . . ... ( 72 , 32 ) . . ...... . . ................................................................ . . .. 1 . . . .. ... ... .. .... . . . . . ... .. ... . 0 2 3 4 5 6 ... 1 5
σ =s−1 σ =s−2 σ=
2s+1 4
σ=
2s−1 4
σ=1 s
Fig. 1. A figure of the range of s and σ
Remark. (1) T depends only on s, σ and (u 0 , A0 , A1 ); X s,σ . (2) For any s and σ satisfying the assumption above for the unique existence of the solution, the mapping (u 0 , A0 , A1 ) → (u, A, ∂t A) is continuous in the w*-sense. Namely if a sequence of initial data strongly converges in X s,σ , then the corresponding sequence of solutions also converges star-weakly in L ∞ (0, T ; X s,σ ).
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(3) In [17], we also assume s ≥ 5/3 and 4/3 ≤ σ ≤ (5s − 2)/3 with (s, σ ) = (5/2, 7/2). Generally, in order to construct solutions of dispersive equations in low regularity function spaces, we usually use smoothing effects such as Strichartz estimates. However, the usual Strichartz estimates for Schrödinger equations do not match Eq. (1.7) since we cannot avoid the loss of derivative coming from the term 2iA · ∇u. This is why the preceding results rely on the L 2 -based energy method. In the present work we use a variation of Strichartz estimates first given by Koch-Tzvetkov [15] and refined by KenigKoenig [14] for Benjamin-Ono type equations, and adapted for Schrödinger equations by J. Kato [8]. In the proof of Theorem 1.1, we slightly refine this estimate and combine it with the covariant derivative estimates developed in our previous work [17]. Our local theory does not cover the result for the energy class H 1 , but it is sufficient for our aim. Indeed, we can show the following global result: Theorem 1.2. The solution obtained in Theorem 1.1 exists time globally. By the use of a Koch-Tzvetkov type estimate, we can show 1/2−δ
u; L 2 (0, T ; H6
) ≤ CT 3 ,
where δ > 0 is sufficiently small and the constant C depends only on (u 0 , A0 , A1 ); X 1,1 and δ. Roughly speaking, we can gain 1/2 regularity by this estimate. Indeed, if we control this norm by the Sobolev inequality, we would need u; H 3/2−δ . This estimate, combined with several a priori estimates obtained in the proof of Theorem 1.1, shows that (u, A, ∂t A); X s,σ does not blow up. Next we consider the Lorentz gauge ∂t φ + div A = 0.
(1.10)
(MS) in the Lorentz gauge, which is referred to as (MS-L), is expressed as i∂t u = (−A + φ)u, φ = ρ, A = J. In this case, we need the initial data (u(0), φ(0), ∂t φ(0), A(0), ∂t A(0)) = (u 0 , φ0 , φ1 , A0 , A1 ) ∈ Y s,σ .
(1.11)
Here Y s,σ = {(u 0 , φ0 , φ1 , A0 , A1 ) ∈ H s ⊕ H σ ⊕ H σ −1 ⊕ H σ ⊕ H σ −1 ; div A0 + φ1 = div A1 + φ0 + |u 0 |2 = 0}. The condition (1.10) is conserved under the consistency condition in the definition of Y s,σ since (∂t φ + div A) = ∂t ρ + div J = 0. The first and the second equations respectively follow from the wave equations both for φ and A, and from the conservation of charge derived from the Schrödinger equation. Our result for (MS-L) is the following. Theorem 1.3. Let (s, σ ) ∈ R ∗ with s ≥ 11/8, σ > 1 and σ ≥ s − 1. Then for any (u 0 , φ0 , φ1 , A0 , A1 ) ∈ Y s,σ , there exists T > 0 such that (MS-L) with initial condition (1.11) has a unique solution (u, φ, A) satisfying (u, φ, ∂t φ, A, ∂t A) ∈ C([0, T ]; Y s,σ ). This solution exists time globally. Moreover, if s > 11/8 and (s + 1, σ ) ∈ R∗ , then the mapping (u 0 , φ0 , φ1 , A0 , A1 ) → (u, φ, ∂t φ, A, ∂t A) is continuous as a mapping from Y s,σ to C([0, T ]; Y s,σ ).
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319
We can also consider the temporal gauge φ = 0.
(1.12)
(MS) in the temporal gauge, which is referred to as (MS-T), is expressed as i∂t u = −A u, A + ∇divA = J(u, A). In this case, we need the initial data (u(0), A(0), ∂t A(0)) = (u 0 , A0 , A1 ) ∈ Z s,σ .
(1.13)
Here Z s,σ = {(u 0 , A0 , A1 ) ∈ H s ⊕ H σ ⊕ H σ −1 ; − div A1 = |u 0 |2 }. Our result for (MS-T) is the following. Theorem 1.4. Let (s, σ ) ∈ R ∗ with s ≥ 11/8, σ > 1 and σ ≥ s − 1. Then for any (u 0 , A0 , A1 ) ∈ Z s,σ , there exists T > 0 such that (MS-T) with initial condition (1.13) has a unique solution (u, A) satisfying (u, A, ∂t A) ∈ C([0, T ]; Z s,σ ). This solution exists time globally. Moreover, if s > 11/8 and (s + 1, σ ) ∈ R∗ , then the mapping (u 0 , A0 , A1 ) → (u, A, ∂t A) is continuous as a mapping from Z s,σ to C([0, T ]; Z s,σ ). This paper is organized as follows: In Sect. 2, we first prepare basic estimates used throughout this paper, namely an estimate of Hartree type nonlinearities (Lemma 2.1) and one for covariant derivatives (Lemma 2.2). Next we introduce Strichartz estimates for Klein-Gordon equations (Lemma 2.3) and Koch-Tzvetkov type Strichartz estimates for Schrödinger equations (Lemma 2.4). We also prepare the estimate of the nonlinear term of the Maxwell part, which is based on the Kato-Ponce commutator estimate (Lemmas 2.5 and 2.6). In Sect. 3, we study the linearized Schrödinger equation associated with (1.7). Applying the Koch-Tzvetkov type estimate, we derive a smoothing property of the Schrödinger equation with electro-magnetic potential (Lemma 3.1). Using this estimate together with covariant derivative estimates, we prove the unique solvability of this equation first in H 2 (Lemma 3.2) and next in H s with s ≥ 0 (Lemmas 3.3 and 3.4). In Sect. 4, we discuss the local solvability. We prove the local well-posedness by the contraction mapping principle (Propositions 4.1 and 4.2). The continuous dependence of the solutions on the data is left to Sect. 6, since it is usually the most delicate part of the theory of well-posedness. In Sect. 5, we derive a priori estimates of solutions (Lemma 5.1) and use them in the proof of global existence. Section 6 is devoted to the proof of the continuous dependence of the solutions on the data. In Sect. 7, we prove Theorems 1.3 and 1.4 by using the gauge transform. We conclude this section by giving the notation used in this paper. L p = L p (R3 ) is the usual Lebesgue space and its norm denoted by · p . p = p/( p−1) is the dual exponent of p. This symbol is used only for Lebesgue exponents. H ps = {φ ∈ S (R3 ); (1 − )s/2 φ p < ∞} is the usual Sobolev space. H˙ ps = {φ ∈ S (R3 ); (−)s/2 φ p < ∞} is the homogeneous Sobolev space. The subscript p is omitted if p = 2. For any interval I ⊂ R and Banach space X , L p (I ; X ) denotes the space of X -valued strongly measurable p functions on I whose X -norm belong to L p (I ). This space is often abbreviated to L T X
320
M. Nakamura, T. Wada m, p
for I = (0, T ). Similarly we use the abbreviation C Tm X = C m ([0, T ]; X ) and WT X = W m, p (0, T ; X ), where W m, p (I ; X ) denotes the space of functions in L p (I ; X ) whose derivatives up to the (m − 1)-times are locally absolutely continuous and the derivatives p up to the m-times belong spaces X i , i = 1, . . . , n, we define n to L (I ; X ). For normed n ·; X so that X is also a normed the norm of X = i=1 X i by ·; X = maxi=1 i j,∞ j,∞ = mj=0 WT H σ − j . The space. We define Tm,s = mj=0 WT H s−2 j and Mm,σ T inequality a b means a ≤ Cb, where C is a positive √ constant that is not essential. We write a b if b a as well as a b. a = 1 + a 2 . a ∨ b and a ∧ b denote the maximum and the minimum of a and b respectively. We use the following unusual but convenient symbol: a+ means a ∨ 0 if a = 0, whereas 0+ means a sufficiently small positive number. Namely b ≥ a+ means b ≥ a ∨ 0 if a = 0, and b > 0 if a = 0. It is useful to express sufficient conditions for Sobolev type embeddings Hrs → L p by the inequality (1/r − s/3)+ ≤ 1/ p ≤ 1/r with 1 ≤ r < ∞. 2. Preliminaries In this section we summarize lemmas used in the proof of Theorems 1.1–1.4. The following two lemmas will be repeatedly used in estimates of nonlinear terms. Lemma 2.1. Let s, s1 , s2 , s3 be nonnegative numbers satisfying s ≤ s3 and s1 ∧s2 ≥ s−2 with s1 + s2 > 0. Let s1 + s2 + s3 ∧ (3/2) ≥ s + 1 and the inequality be strict if (i) s j = 3/2 for some 1 ≤ j ≤ 3 or (ii) s = s3 < 3/2. Then the following estimate holds: (−)−1 (u 1 u 2 )u 3 ; H s
3
u j ; H s j .
(2.1)
j=1
Proof. See Lemma 2.1 in [17].
Lemma 2.2. Let (s, σ ) ∈ R∗ with s ≥ 0. Let A ∈ H σ satisfy div A = 0. (i) Let V1 (A, v) = 2iA·∇v+|A|2 v. Then V1 is a continuous mapping from H σ ×H s−1/2 to H s−2 with the estimate V1 (A, v); H s−2 A; H σ 2 v; H s−1/2 .
(2.2)
(ii) The following estimates hold for any v ∈ H s : v; H s + A; H σ α v2 A v; H s−2 + A; H σ α v2 ,
(2.3)
where α = α(s, σ ) is a positive constant independent of v and A. Especially if s = 2, the estimate v; H 2 + A; H˙ 1 4 v2 A v2 + A; H˙ 1 4 v2
(2.4)
holds valid. Proof. For s ≥ 5/2, we can prove (2.2) by the Leibniz rule and the Sobolev inequality. The case s = 0 is the dual of the case s = 5/2, and hence the case 0 < s < 5/2 is proved by interpolation. The continuity of V1 is proved in the same way. Applying 1/2s the interpolation inequality v; H s−1/2 ≤ v; H s 1−1/2s v2 to the estimate of V1 , we can show (2.3). We can show (2.4) if we use the estimate V1 2 ≤ A6 ∇v3 + A26 v6 instead of (2.2).
Global Existence and Uniqueness of Solutions to the MS Equations
321
Next we introduce Strichartz type estimates for Klein-Gordon equations (see for example [2–4, 19]). Lemma 2.3. Let T > 0, σ ∈ R and let (q j , r j ), j = 0, 1, satisfy 0 ≤ 2/q j = 1−2/r j <
σ −1+2/q1
1. Let (A0 , A1 ) ∈ H σ ⊕H σ −1 and F ∈ L T1 Hr1 q
. Then a solution A to the equation
( + 1)A = F with A(0) = A0 , ∂t A(0) = A1 belongs to C T H σ ∩ C T1 H σ −1 and satisfies the estimate σ −k−2/q0
q
max ∂tk A; L T0 Hr0
k=0,1
σ −1+2/q1
(A0 , A1 ); H σ ⊕ H σ −1 + F; L T1 Hr1 q
.
(2.5)
Lemma 2.4. Let T > 0, s ∈ R, α > 0 and 0 ≤ 2/q = 3/2 − 3/r ≤ 1. Let f ∈ L 2T H s−2α . Then a solution u ∈ T1,s to the equation i∂t u = −u + f belongs to
q LT
Hrs−α
and satisfies the estimate
s 1/2 f ; L 2T H s−2α . u; L T Hrs−α u; L ∞ T H +T q
(2.6)
Remark. This kind of estimate was first given by Koch-Tzvetkov [15] for the Benjamin-Ono equation, and it is Kenig-Koenig [14] who formulated the estimate as above. Kato [8] adapted this estimate for Schrödinger equations. However, in [8, 14], s+ to prove (2.6), with the first term in the they need an extra assumption u ∈ L ∞ T H ∞ s+ right-hand side replaced by u; L T H . ∞ Proof. Without loss of generality we assume s = 0. Let u = j=0 u j be the Littlewood-Paley decomposition of the solution u. Namely we take ψ ∈ S (R3 ) such j ˆ that suppψˆ ⊂ {ξ ; 1/2 ≤ |ξ | ≤ 2} and ∞ j=−∞ ψ(ξ/2 ) = 1 for any ξ = 0, and j )u(t, j )u(t, ˆ ˆ put u j = F −1 (ψ(ξ/2 ˆ ξ )) for j ≥ 1, u 0 = F −1 0j=−∞ ψ(ξ/2 ˆ ξ ). Here u(t, ˆ ξ ) is the Fourier transform with respect to the space variable. Similarly let f = ∞ j=0 f j be the Littlewood-Paley decomposition of f . Then u j satisfy the equation i∂t u j = −u j + f j . (2.7) m
j such that 2−2α j T ≤ |I jk | < We divide the interval [0, T ] into disjoint intervals {I jk }k=1 2−2α j+1 T and take t jk ∈ I¯jk at which u j (t)2 attains its minimum in the interval I¯jk . By the standard Strichartz estimate for Schrödinger equations [11, 13, 19, 22],
u j ; L q (I jk ; L r ) u j (t jk )2 + f j ; L 1 (I jk ; L 2 ). Taking the sum with respect to k, we obtain q
u j ; L T L r = (
mj
u j ; L q (I jk ; L r )q )1/q ≤ (
k=1
(
mj
≤(
k=1
u j ; L q (I jk ; L r )2 )1/2
k=1
u j (t jk )22 )1/2 + (
k=1 mj
mj
mj
f j ; L 1 (I jk ; L 2 )2 )1/2
k=1
T
−1 2α j
2
|I jk |u j (t jk )22 )1/2 +(
mj
2−2α j+1 T f j ; L 2 (I jk ; L 2 )2 )1/2
k=1
≤ T −1/2 2α j u j ; L 2T L 2 + (2T )1/2 2−α j f j ; L 2T L 2 .
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M. Nakamura, T. Wada
The first term in the right-hand side is obtained from the fact that t jk are the minimum points and the definition of the integral. We have also used the definition of I jk and the Hölder inequality for the time variable. Therefore u; L T Hr−α { q
∞
(2−α j |u j |)2 }1/2 ; L T L r ≤ ( q
j=0
T −1/2 (
∞
u j ; L 2T L 2 2 )1/2 + T
∞
T
q
j=0 ∞ 1/2
(
j=0 −1/2
2−α j |u j |; L T L r 2 )1/2 2−2α j f j ; L 2T L 2 2 )1/2
j=0
u; L 2T L 2 +
T
1/2
f ; L 2T
H
−2α
,
where we have used the equivalent norms between the Sobolev spaces and the TriebelLizorkin spaces (see p. 29 in [20]). Thus the lemma has been proved. Lemma 2.5. Let σ ≥ 0. Let 1 < p, p1 , p4 < ∞ and 1 < p2 , p3 ≤ ∞ satisfy 1/ p = 1/ p1 + 1/ p2 = 1/ p3 + 1/ p4 . Then the following estimate holds valid: P(u¯ 1 ∇u 2 ); H pσ u 1 ; H pσ1 ∇u 2 p2 + ∇u 1 p3 u 2 ; H pσ4 .
(2.8)
Moreover if 0 ≤ σ ≤ 1, we can omit the second term of the right-hand side. Proof. We can prove (2.8) by the use of the Kato-Ponce commutator estimate [12] and the fact that P∇ = 0; see [17] for details. The last assertion can be checked immediately for σ = 0, 1 and generalized for 0 < σ < 1 by interpolation. Lemma 2.6. Let (s, σ ) ∈ R ∗ with s ≥ 5/4 and σ ≥ 1. Then P(u¯ 1 ∇A u 2 ); L 1T
H
σ −1
T
1/4
T
3/4
σ −1/2 A; L ∞ T H
2 s 2 s−1/2 u j ; L ∞ . T H ∩L T H6 j=1
(2.9) Proof. It suffices to show the following inequalities: P(u¯ 1 ∇u 2 ); L 1T H σ −1 T 1−1/q1 −1/q2 (u 1 ; L T1 Hr1 q
+
s−1/q1
q
s−1/q2
u 2 ; L T2 Hr2
q s−1/q2 q s−1/q1 u 1 ; L T2 Hr2 u 2 ; L T1 Hr1 )
(2.10)
for suitable q j , r j satisfying the conditions 0 ≤ 2/q j = 3/2 − 3/r j ≤ 1 and 2/q1 + 2/q2 ≤ 3/2; σ −1/2 Au¯ 1 u 2 ; L 1T H σ −1 T A; L ∞ T H
2
s u j ; L ∞ T H .
(2.11)
j=1
We first prove (2.10). We use Lemma 2.5 and obtain the estimate P(u¯ 1 ∇u 2 ); H σ −1 u 1 ; H pσ1−1 ∇u 2 p2 + ∇u 1 p2 u 2 ; H pσ1−1 , where the choice of p1 , p2 depends on the value of s. Practically, if s > 2 and σ ≤ s + 1, s−3/2 we choose p1 = 2, p2 = ∞ so that H s → H σ −1 and H6 → L ∞ . Then we obtain (2.10) with 2/q1 = 0, 2/q2 = 1. If 3/2 ≤ s < 2, we choose 1/ p1 = s/3 − 1/6,
Global Existence and Uniqueness of Solutions to the MS Equations s−1/q2
1/ p2 = (2 − s)/3 so that Hr2
s−1/2
≡ H6
323
→ H p12 by the Sobolev inequality.
1 → H pσ1−1 and 2/q1 + Putting r1 = p1 and hence 2/q1 = 2 − s, we obtain Hr1 2/q2 = 3 − s ≤ 3/2 under the condition σ ≤ 3s/2. If 5/4 ≤ s < 3/2, we choose s−1/q2 ≡ H p12 . Putting r1 = p1 1/ p1 = 2(s − 1)/3, 1/ p2 = 1/2 − 2(s − 1)/3 so that Hr2
s−1/q
1 and hence 2/q1 = 7/2 − 2s, we obtain Hr1 → H pσ1−1 and 2/q1 + 2/q2 = 3/2 under the condition σ ≤ 2s − 3/4. Therefore we obtain (2.10) by the Hölder inequality for the time variable. The proof for the case s = 2 has been omitted, but this is covered by the case s ≈ 2. Next we prove (2.11). By the Leibniz rule,
s−1/q
Au¯ 1 u 2 ; H σ −1 A; H3σ −1 u 1 12 u 2 12 + A p3 u 1 ; H pσ4−1 u 2 p5 + A p3 u 1 p5 u 2 ; H pσ4−1 , where 2 ≤ p3 , p4 , p5 < ∞ are numbers satisfying 1/ p3 ≥ {(2 − σ )/3}+ , 1/ p4 ≥ {1/2 − (s − σ + 1)/3}+ , 1/ p5 ≥ (1/2 − s/3)+ and 1/ p3 + 1/ p4 + 1/ p5 = 1/2. Such a choice is possible under the assumption. Thus (2.11) follows from the Sobolev inequality. 3. Linearized Schrödinger Equations In this section we prove some existence theorems and a priori estimates for the linear Schrödinger equation associated with (1.7): i∂t v = (−A + φ)v, 0 < t < T, v(t0 ) = v0 .
(3.1) (3.2)
Here φ = φ(u) ≡ (4π |x|)−1 ∗ |u|2 and 0 ≤ t0 ≤ T . In this section we regard A and u as known functions defined on [0, T ]. Here we clarify the notion of solutions to (3.1). Definition 3.1. A function v is called a weak (resp. strong) H s -solution to (3.1) if v belongs to T1,s (resp. C T H s ∩ C T1 H s−2 ) and satisfies (3.1) for almost every (resp. all) 0 < t < T . A function v is called a weak (resp. strong) H s -solution to (3.1)–(3.2) if v is a weak (resp. strong) H s -solution to (3.1) and satisfies (3.2). For given σ > 1, we fix positive numbers δ, q and r as δ = min{(σ − 1)/2, 1/4}, 1/q = 1/2 − 2δ/3, 1/r = 2δ/3.
(3.3)
This notation will be used throughout the paper. 2 ∞ satisfy σ Lemma 3.1. Let s ≥ 0, σ > 1 and σ ≥ s − 1. Let A ∈ L ∞ T H ∩ LT L ∞ 2 (s−1)∨1 s−1 s and f ∈ L T H . Then a weak H -solution v to div A = 0. Let u ∈ L T H
i∂t v = (−A + φ)v + f, 0 < t < T, s−1/2
belongs to L 2T H6 s−1/2
v; L 2T H6
and satisfies the estimate
σ 2 ∞ ∞ (s−1)∨1 m s T m A; L ∞ v; L ∞ T H ∩ L T L ∨ u; L T H T H
+ T 1/2 f ; L 2T H s−1 , where m is a positive number.
(3.4)
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M. Nakamura, T. Wada
Proof. Applying Lemma 2.4, we obtain s−1/2
v; L 2T H6
s 1/2 v; L ∞ 2iA · ∇v + |A|2 v + φv + f ; L 2T H s−1 . (3.5) T H +T
We shall estimate the second term of the right-hand side. We first estimate 2iA · ∇v. We have ∇vq1 (3.6) A · ∇v; H s−1 A∞ v; H s + A; Hrs−1 1 for 0 < 2/r1 = 1 − 2/q1 ≤ 1, and the second term can be omitted in the case s ≤ 1. Indeed (3.6) without the second term is clearly valid for s = 0 and s = 1, and it is also valid for 0 < s < 1 by interpolation; on the other hand if s > 1, (3.6) is obtained by the Leibniz rule. We define the numbers q1 , r1 as (i) 2/q1 = 1 − 2/r1 = 2 − s if 1 < s < 2; (ii) 2/q1 = 1 − 2/r1 = δ if s = 2; (iii) q1 = ∞, r1 = 2 if s > 2. In any case, by virtue of the Sobolev (Gagliardo-Nirenberg) inequality, there exists an exponent 0 < θ < 1 s−1/2 1−θ with 2/q1 ≤ θ such that ∇vq1 v; H s θ v; H6 . Practically, we can take θ = 2 − s if s < 2, θ = 3δ if s = 2, and θ = 2(s − 2)/(2s − 3) if s > 2. Therefore by the Hölder inequality for the time variable together with the Young inequality, T 1/2 A · ∇v; L 2T H s−1 s 1/2 v; T 1/2 A; L 2T L ∞ v; L ∞ T H +T s−1/2 1−θ
L 2T H6
s θ A; L T Hrs−1 v; L ∞ T H 1 2/θ
s 2 T m {A; L 2T L ∞ + 1−1/θ A; L T1 Hrs−1 1/θ }v; L ∞ T H + v; L T H6 1 q
s−1/2
,
where m is a positive number. We choose > 0 so small that the last term in the right-hand side is absorbed in the left-hand side of (3.5). Next we show the estimate |A|2 v; H s−1 A; H σ A∞ v; H s . If s ≤ 1, this inequality follows from the Sobolev inequality. If s > 1, we use the Leibniz rule to derive A∞ vq2 |A|2 v; H s−1 A3 A∞ v; H6s−1 + A; Hrs−1 2 with 0 < 2/r2 = 1 − 2/q2 ≤ 1. We can choose q2 , r2 such that 1/r2 ≥ (1/2 − (σ − s + 1)/3)+ and 1/q2 ≥ (1/2 − s/3)+ . Therefore we obtain the desired estimate again by the Sobolev inequality. By the Sobolev inequality L 6/(5−2s) → H s−1 for 0 ≤ s ≤ 1, and Lemma 2.1 for s > 1, we have φv; H s−1 u; H (s−1)∨1 2 v; H s . Collecting these estimates, we obtain s−1/2
v; L 2T H6
σ 2 ∞ s−1 ∞ (s−1)∨1 m 1 T m A; L ∞ T H ∩ L T L ∩ L T Hr1 ∨ u; L T H q
s 1/2 × v; L ∞ f ; L 2T H s−1 , T H + T
where m is a positive number, and the third space in the norm of A can be dropped if s ≤ 1. q 2 ∞ σ A; L ∞ To complete the proof, we should show A; L T1 Hrs−1 T H ∩ L T L in the 1 nontrivial case 1 < s ≤ 2. We concentrate on the case 1 < s < 2; we can analogously
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treat the case s = 2. Let A = ∞ j=0 A j be the Littlewood-Paley decomposition. Then by the Hölder inequalities both for the sequence and the space variable, A;
Hrs−1 1
{
∞
|2(s−1) j A j |2 }1/2 r1
j=0
{
∞
|2
σj
A j |2 }1/2 s−1 2 {
j=0
∞
|2− 1 j A j |2 }1/2 2−s ∞ ,
j=0
where 1 = (σ −1)(s−1)/(2−s) > 0. We have { since A j ∞ A∞ . Therefore we obtain
∞
j=0 |2
− 1 j A |2 }1/2 j ∞
≤ C 1 A∞
σ s−1 A; L T1 Hrs−1 A; L ∞ A; L 2T L ∞ 2−s T H 1 q
by the Hölder inequality for the time variable.
1 ∩ satisfy div A = 0. Let u ∈ L ∞ Lemma 3.2. Let σ > 1 and let A ∈ T H . Then for any v0 ∈ H 2 , there exists a unique weak solution v to (3.1)–(3.2), and the solution v satisfies the following estimate:
M1,σ T
L 2T L ∞
2 2 ∞ ˙1 4 v; L ∞ T H ≤ Cv0 ; H A; L T H 2 ∞ ∞ 1 l × exp{C T 1/2 T l A; M1,σ T ∩ L T L ∨ u; L T H }. (3.7)
Here l is a positive number. Moreover, if u ∈ C T H 1 , v is a unique strong solution to (3.1)–(3.2). Proof. From Eq. (3.1), we can immediately show the conservation law of the L 2 -norm v(t)2 = v0 2 . The uniqueness of the solution clearly follows from this identity. The existence of the weak solution follows from the a priori estimate (3.7). Indeed the unique existence of the strong solution has already been known for sufficiently smooth u, A and v0 (see for example [9, 10]). Therefore we approximate these functions by a sequence of smooth ones and consider the corresponding sequence of solutions. If we extract a star-weakly converging subsequence, then the star-weak limit is a weak solution to (3.1)-(3.2). Therefore we formally prove (3.7). Taking Lemma 2.2 into account, we estimate v; HA2 ≡ A v2 + R4 v2 ˙1 instead of v; H 2 , where R ≡ A; L ∞ T H . Taking the time derivative of A v and using Eq. (3.1), we find the equation for A v: i∂t A v = (−A + φ)A v + 2∂t A · ∇A v + [A , φ]v.
(3.8)
Therefore the standard energy method shows that v; L ∞ (t0 , t; HA2 ) ≤ v0 ; HA20 + 2∂t A · ∇A v + [A , φ]v; L 1 (t0 , t; L 2 ), where A0 ≡ A(t0 ). By the Sobolev inequality, we have ∇vr v; H 2 4δ v; 3/2 H6 1−4δ . Applying this inequality together with the Young inequality and Lemma 2.2, we obtain 2∂t A · ∇A v2 ≤ ∂t Aq ∇vr + ∂t A2 A∞ v∞ ≤ v; H6 + C{ (4δ−1)/4δ ∂t A; H σ −1 1/4δ + ∂t A2 A∞ }v; HA2 . 3/2
(3.9)
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M. Nakamura, T. Wada
Here q, r are defined by (3.3) and is a positive number which will be determined later. We can easily handle the term [A , φ]v by Lemma 2.1 or the Hardy-Littlewood-Sobolev inequality; we obtain [A , φ]v2 A; H˙ 1 u; H 1 2 v; H 2 . Therefore v; L ∞ (t0 , t; HA2 ) 3/2
≤ v0 ; HA20 + (t − t0 )1/2 v; L 2 (t0 , t; H6 ) t +C { (4δ−1)/4δ ∂t A; H σ −1 1/4δ + ∂t A2 A∞ t0
+A; H˙ 1 u; H 1 2 }v; HA2 dt .
(3.10)
Taking Lemma 3.1 into account, we choose the positive number so small that the second term in the right-hand side is absorbed in the left-hand side. To this end we 2 ∞ ∞ 1 m σ choose such that T m+1/2 A; L ∞ T H ∩ L T L ∨ u; L T H 1 with m stated in Lemma 3.1. Thus we obtain t ∞ 1 l ∨ u; L H A∞ v; H 2 dt , v; H 2 ≤ CR4 u 0 ; H 2 +CT l A; M1,σ T T t0
where l is some positive number. Applying the Gronwall inequality we obtain (3.7). We proceed to the latter part of the lemma. The weak continuity of A v follows from the construction of v. We shall prove the strong continuity. We take a supreme limit of both sides of (3.10) as t ↓ t0 and obtain lim supt↓t0 A v(t)2 ≤ A0 v0 2 . Therefore s- limt↓t0 A v = A0 v0 in L 2 . This argument shows that A v ∈ C T L 2 ; on the other hand, the fact that v ∈ C T L 2 is similarly proved by the conservation law of the L 2 -norm. 1 Therefore we obtain v ∈ C T H 2 from Lemma 2.2 and the fact A ∈ M1,σ T ⊂ C T H . If 1 2 we also assume u ∈ C T H , we can show ∂t v ∈ C T L taking Eq. (3.1) into account. Definition 3.2. We define the two parameter family of the operator {U (t, τ )}0≤t,τ ≤T as the evolution operator of (3.1). Namely, U (t, t0 )v0 solves (3.1)–(3.2). By the following lemmas, {U (t, τ )} can be extended as a family in H s with s ≥ − 2. We put K s ≡ sup0≤t,τ ≤T U (t, τ ); H s → H s . Remark. In view of Lemma 3.2, we have the estimate 1/2 2 ∞ ∞ 1 l ˙1 4 K 2 ≤ CA; L ∞ T l A; M1,σ T H exp{C T T ∩ L T L ∨ u; L T H }
with some positive numbers C, l. 2 ∞ satisfy div A = 0. Let Lemma 3.3. Let −2 ≤ s ≤ 2, σ > 1 and let A ∈ M1,σ T ∩ LT L ∞ 1 u ∈ L T H . Then we have the following:
(i) {U (t, τ )} can be uniquely extended to a family of operators in H s which solves (3.1), namely U (t, t0 )v0 is a weak H s solution to (3.1)–(3.2), and moreover if s ≥ 0, then U (t, t0 )v0 is a unique weak H s solution; (ii) if u ∈ C T H 1 , U (t, t0 )v0 is a strong H s solution; |s|/2 (iii) 1 ≤ K s ≤ K 2 ;
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327
(iv) if w is a weak L 2 solution to the equation i∂t w = (−A + φ)w + f , where f ∈ L 1T H −2 , then w satisfies the integral form of this equation, namely t w(t) = U (t, t0 )w(t0 ) − i U (t, τ ) f (τ )dτ. t0
Proof. This lemma can be proved in the same way as in Lemmas 3.2 and 3.3 in [17]. 2 ∞ Lemma 3.4. Let (s, σ ) ∈ R∗ with 2 < s < 4 and σ > 1. Let A ∈ M2,σ T ∩ L T L satisfy div A = 0. Let u ∈ T1,s−1 . Then for any v0 ∈ H s , there exists a unique weak solution v to (3.1)–(3.2), and the solution v satisfies the following estimate: 1,σ s s ∞ 1 l v; L ∞ T H ≤ C K 2 v0 ; H A; MT ∨ u; L T H 1,s−1 l 2 ∞ × exp{CK 2 l T l A; M2,σ }. (3.11) T ∩ L T L ∨ u; T
Here l is a positive number. Moreover, if A ∈ C T H σ ∩ C T1 H σ −1 and if u ∈ C T H s−1 , then v is a strong solution to (3.1)–(3.2). Proof. As we have mentioned in the proof of Lemma 3.2, the existence of a weak solution follows from the estimate (3.11), and the uniqueness has already been proved in Lemma 3.2. Instead of (3.11) itself, we first assume 5/2 < s < 4 and prove the estimate 1,σ s s ∞ 1 l v; L ∞ T H ≤ C K 2 v0 ; H A; MT ∨ u; L T H 2 ∞ × exp{CK 2 l T l A; M2,σ T ∩ L T L ∨ u; T
1,s−3/2
l }, (3.12)
which is slightly different from but stronger than (3.11). Then (3.11) follows directly from (3.12) if 5/2 < s < 4 and from interpolating (3.7) and (3.12) if 2 < s ≤ 5/2. Therefore we assume 5/2 < s < 4 and prove (3.12), dividing the proof into several steps. In the proof we estimate ∂t A v; H s−4 instead of v; H s , taking the equivalence of these norms into account. To this end we first prove the equivalence of these norms and prepare some inequalities in Step 1. We also use the smoothing property of the Schrödinger equation; practically, in Step 2 we derive estimates which are consequences of Lemma 3.1. In Step 3 we apply the estimates obtained in the preceding steps to the equation for ∂t A v and derive an integral inequality for v; H s , from which the desired estimate (3.12) follows. In Step 4 we prove the continuity of the solution, namely the latter part of the lemma. Step 1. We can obtain the following estimates for the solution to (3.1): ∂t v; H s−2 + N1 α v2 v; H s + N1 α v2 , ∂t A v; H s−4 + N2 α v2 v; H s + N2 α v2 ,
(3.13) (3.14)
1,σ ∞ 1 ∞ 1 σ where N1 = A; L ∞ T H ∨ u; L T H , N2 = A; MT ∨ u; L T H . To obtain (3.13) and (3.14), we have only to use the following inequalities together with Lemma 2.2 and the standard interpolation inequality:
∇A v; H s−2 A; H σ v; H s−1 ,
(3.15)
φv; H s−2 u; H 1 2 v; H s−1 ,
(3.16)
V2 (A, ∂t A, v); H
s−4
N2 v; H 3
s−1
,
(3.17)
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M. Nakamura, T. Wada
where V2 (A, B, v) = PB · ∇A v. We can prove these inequalities by the Sobolev inequality and Lemma 2.1. We remark that (3.15) and (3.16) holds valid for 0 ≤ s < 4 (for the proof we use the duality argument) and that V2 is a continuous mapping from H σ × H σ −1 × H s−1 to H s−4 . Step 2. We prove s−5/2
A v; L 2T H6
s ≤ CT m N3 m v; L ∞ T H ,
(3.18)
2 ∞ ∞ 1∨(s−5/2) and m is a positive number. We where N3 = A; M1,σ T ∩ L T L ∨ u; L T H remark that the constant C does not depend on T as well as v and N3 . Applying Lemma 3.1 to (3.8), we have s−5/2
A v; L 2T H6
s−2 T m N3 m A v; L ∞ T H
+ T 1/2 2∂t A · ∇A v + [A , φ]v; L 2T H s−3 .
(3.19)
Therefore we obtain (3.18) by (2.3) if we prove the following estimates: ∂t A · ∇A v; H s−3 ∂t A; H σ −1 A; H σ v; H s , |u| v; H
s−3
u; H
∇φ · ∇A v; H
s−3
u; H A; H v; H .
2
1∨(s−5/2)
u; H v; H , 1
σ
1 2
s
s
(3.20) (3.21) (3.22)
These estimates can be proved by the Sobolev inequality, together with the Leibniz rule if s ≥ 3, and moreover the duality argument if s < 3. We need the condition σ ≥ (2s − 1)/4 to bound ∂t A · Av; H s−3 by the right hand side of (3.20). Replacing A v in (3.18) by −i∂t v + φv and using Lemma 2.1, we also obtain s−5/2
∂t v; L 2T H6
s ≤ CT m N3 m v; L ∞ T H .
(3.23)
Step 3. We take the time derivative of (3.8). Then we obtain the following Schrödinger equation for ∂t A v: i∂t2 A v = (−A + φ)∂t A v +2∂t A · ∇A A v + ∂t φ · A v + 2∂t2 A · ∇A v − 2i|∂t A|2 v + 2∂t A · ∇A ∂t v ¯ + 2∇∂t φ · ∇A v − 2i∇φ · ∂t A · v − |u|2 ∂t v + 2∇φ · ∇A ∂t v −2 Re(∂t u · u)v ≡ (−A + φ)∂t A v +
10
f j,
(3.24)
j=1
where we have used the relation ∂t ∇A = ∇A ∂t − i∂t A, ∂t A = A ∂t − 2i∂t A · ∇A , ∂t φ = −2Re∂t u · u, ¯ and [A , φ]v = φ · v + 2∇φ · ∇A v. By the Duhamel principle or Lemma 3.3, we rewrite this equation into integral form as ∂t A v = U (t, t0 )[∂t A v]t=t0 − i Therefore we have W (t) K 2
⎧ ⎨ ⎩
W (t0 ) +
t 10 t0 j=1
t t0
U (t, τ )
10
f j (τ )dτ.
(3.25)
j=1
f j (τ ); H s−4 dτ
⎫ ⎬ ⎭
,
(3.26)
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329
where W (t) = v(t); H s + N2 α v(t)2 . We have used (3.14), Lemma 3.3, and the conservation law of v2 . We estimate f j , j = 1, . . . , 10, and obtain the following estimates: f 1 ; H s−4 ∂t A; H σ −1 A; H σ A v; H6
s−5/2−δ
−1
f2 ; H
s−4
f3 ; H
s−4
f4 ; H
s−4
f5 ; H
s−4
f6 ; H
s−4
f7 ; H
s−4
f8 ; H
s−4
∂t A2 u; H v; H ,
f9 ; H
s−4
u; H ∂t v; H
f 10 ; H
s−4
∂t u; H
∂t2 A;
H
u; H A v; H 1
σ −2
s−2
σ
,
(3.27)
,
(3.28)
A; H v; H ,
∂t A; H
σ −1 2
s
∂t A; H
σ −1
σ
s
(3.29)
v; H ,
(3.30)
A; H ∂t v;
∂t u; H
s−7/2
u; H
s−3/2
∂t u; H
s−7/2
u; H
s−3/2
1 2
s−5/2−δ H6 , s
(3.31)
v; H , σ
(3.32)
A; H v; H , s
s
1 2
s−2
σ
1 2
(3.33) (3.34)
,
(3.35)
A; H u; H ∂t v; H
s−2
.
(3.36)
The inequality (3.27) is obtained by the use of the estimates ∂t A · (∇A w); H s−4 = ∇A · (∂t Aw); H s−4 A; H 1 ∂t Aw; H s−3 for any w ∈ H s−3 , and ∂t AA v; H s−3 ∂t A; H (s−3)∨2δ A v; H6
s−5/2−δ
.
The second inequality is obtained by the Leibniz rule and the Sobolev inequality for s = 3 + 2δ and for s = 4, together with the duality argument for s = 5/2, and it is generalized by interpolation for 5/2 < s < 4. The inequality (3.31) is obtained in the same way. We can obtain the other inequalities principally by the Hölder, the Leibniz, and the Sobolev inequalities, together with the duality argument if Sobolev spaces of negative order appear. We also use Lemma 2.1 for the proof of (3.28), (3.33), (3.34) and (3.36). The condition σ ≥ (2s −1)/4 is needed to bound the terms f 3 and f 4 . We substitute these s−5/2−δ estimates into (3.26), and apply the Gagliardo-Nirenberg inequality A v; H6 s−5/2 s−2 2δ 1−2δ and the corresponding inequality for ∂t v together A v; H A v; H6 with the Young inequality. Then we obtain t
s−5/2 s−5/2 W (t) ≤ K 2 W (t0 ) + A v; H6 + ∂t v; H6 dτ +C
−2δ/(1−2δ)
t0 t
N4 W (τ )dτ l
t0
≤ K 2 W (t0 ) + CεK 2 T m+1/2 N3 m v; L ∞ (t0 , t; H s ) t −2δ/(1−2δ) N4 l W (τ )dτ, + C K2 t0 1,s−3/2 2 ∞ where N4 = A; M2,σ ≥ N3 and and l are positive numbers. T ∩L T L ∨u; T We choose such that CεK 2 T m+1/2 N3 m ≤ 1/2; with this choice we obtain
W (t) ≤ 2K 2 W (t0 ) + CT l N4 ∨ K 2 l
t
t0
W (τ )dτ.
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M. Nakamura, T. Wada
We note that the value of l may differ from that in the previous estimate. Applying the Gronwall lemma, we obtain W (t) ≤ 2K 2 W (t0 ) exp{CT l N4 ∨ K 2 l }, from which we can conclude (3.12). Step 4. We prove the continuity of v under the additional assumptions A ∈ C T H σ ∩ C T1 H σ −1 and u ∈ C T H s−1 . We first remark that v ∈ C T H s−1 since v ∈ C T H 2 ∩ s L∞ U (t, τ ) is H s−4 -strongly T H by virtue of Lemma 3.2 and (3.11). By the assumption, 10 continuous with respect to the parameters t, τ . Moreover, j=1 f j ∈ L 1 H s−4 by Step 3. Applying the Lebesgue convergence theorem to (3.25), we can prove ∂t A v ∈ C T H s−4 . Then we find that A ∂t v = ∂t A v + 2i∂t A · ∇A v ∈ C T H s−4 taking (3.17) into account. Therefore it follows from Lemma 2.2 that ∂t v ∈ C T H s−2 . We go back to Eq. (3.1) and conclude v ∈ C T H s by Lemmas 2.1 and 2.2.
4. Unique Existence of Local Solutions In this section we uniquely solve (MS-C) time locally. To this end we consider the following linearized equation: i∂t v = (−A + φ(u))v, v(0) = u 0 , ( + 1)B = PJ(u, A) + A, B(0) = A0 , ∂t B(0) = A1 ,
(4.1) (4.2)
where φ(u) = (−)−1 |u|2 and J(u, A) = 2 Im u∇ ¯ A u, with the assumptions div A = 0 and (u 0 , A0 , A1 ) ∈ X s,σ . We often consider the equations with (u, A, v, B) replaced by (u , A , v , B ) and (u 0 , A0 , A1 ) by (u 0 , A0 , A1 ). In such a case we often abbreviate φ(u ), J(u , A ) to φ , J . If we define the mapping : (u, A) → (v, B), then the fixed points of solve (MS-C). Proposition 4.1. Let 11/8 ≤ s ≤ 2 and 1 < σ ≤ min{3s/2; 2s − 3/4} with (s, σ ) = (2, 3). Then for any (u 0 , A0 , A1 ) ∈ X s,σ , there exists T > 0 such that (MS-C) with (1.9) s−1/2 ⊕ L ∞ ). has a unique solution satisfying (u, A, ∂t A) ∈ C T X s,σ and (u, A) ∈ L 2T (H6 Moreover, the total energy 1 E = ∇A u22 + {∇φ22 + ∇A22 + ∂t A22 } 2 does not depend on t. Proof. Let 0 < T < 1 and R1 , R2 , R3 > 1. We define the metric space B with the metric induced from the norm ·; B as s 2 B = {(u, A); u ∈ L ∞ T H ∩ L T H6
s−1/2
s 2 u; L ∞ T H ≤ R1 , u; L T H6
2 ∞ , A ∈ M1,σ T ∩ L T L , div A = 0,
s−1/2
2 ∞ ≤ R2 , A; M1,σ T ∩ L T L ≤ R3 },
2 ∞ 1/2 (u, A); B = u; L ∞ ∩ L 4T L 4 . T L ∨ A; L T H
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331
We can easily show that this metric space is complete. We shall show that is a contraction mapping defined on B. On account of Lemmas 2.3, 2.6, 3.1 and 3.3, we have the following estimates for (u, A) ∈ B and (v, B) = (u, A): s s 2s 1/2 (R1 ∨ R3 )l }, v; L ∞ T H ≤ Cu 0 ; H R3 exp{C T s−1/2
v; L 2T H6
s ≤ C(R1 ∨ R3 )m v; L ∞ T H ,
2 ∞ σ σ −1 B; M1,σ + C T 1/4 R3 (R1 ∨ R2 )2 . T ∩ L T L ≤ C(A0 , A1 ); H ⊕ H σ −2/q
To prove the last inequality, we have also used the Sobolev type embedding Hr → L ∞ , where q, r are defined in (3.3). Therefore we can show that is a mapping from B to itself if we choose R1 , R2 , R3 such that R3 ≥ 2C(A0 , A1 ); H σ ⊕ H σ −1 ,
R1 ≥ 2Cu 0 ; H s R32s ,
R2 ≥ C(R1 ∨ R3 )m R1 ,
and if we choose T such that exp{C T 1/2 (R1 ∨ R3 )l } ≤ 2, C T 1/4 (R1 ∨ R2 )2 ≤ 1/2. Next we estimate the difference of (v, B) = (u, A) and (v , B ) = (u , A ). Taking the difference of the equations for v and v , we obtain i∂t (v − v ) = (−A + φ)(v − v ) + 2i(A − A ) · ∇v +(A − A ) · (A + A )v + (φ − φ )v . By the usual L 2 -estimate together with the Sobolev inequality, Lemma 2.1 and the Hölder inequality both for time and space variables, we obtain 2 v − v ; L ∞ T L
≤ 2i(A − A ) · ∇v + (A − A ) · (A + A )v + (φ − φ )v ; L 1T L 2 12 A − A ; L 4T L 4 {T 3/8 ∇v ; L T L 4 + T 3/4 A + A ; L ∞ L 6 v ; L ∞ T L } 8/3
2 ∞ 1 ∞ 1 +T u − u ; L ∞ T L u + u ; L T H v ; L T H
T 3/8 (R1 ∨ R2 ∨ R3 )2 (u − u , A − A ); B. 2 s = (L ∞ )[3/4] . We proHere we have used the interpolation L T H4 T H , L T H6 ceed to the estimate of B − B . We apply Lemma 2.3 to the difference of the equations for B and B taking the relation P∇ = 0 into account. Then 8/3
s−3/8
s−1/2
1/2 B − B ; L ∞ ∩ L 4T L 4 T H
A − A ; L 1T H −1/2 + P(J − J ); L T L 4/3 4/3
1/2 2 T A − A ; L ∞ + T 3/8 ∇(u + u ); L T L 4 u − u ; L ∞ T H T L 8/3
6 ∞ 2 ∞ 12 +T A; L ∞ T L u − u ; L T L u + u ; L T L 4 2 +T 1/2 A − A ; L 4T L 4 u ; L ∞ T L
T 3/8 (R1 ∨ R2 ∨ R3 )2 (u − u , A − A ); B, where we have used the same interpolation relation as above. Therefore we obtain (v − v , B − B ); B ≤ (1/2)(u − u , A − A ); B
(4.3)
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for sufficiently small T > 0. Therefore, is a contraction mapping with the choice of T, R1 , R2 , R3 mentioned above, from which we conclude the unique existence of the solution. Moreover (u, A, ∂t A) ∈ C T X s,σ . Indeed, (A, ∂t A) ∈ C T (H σ ⊕ H σ −1 ) by Lemma 2.3, and u ∈ C T H 1 since solutions to (4.1) belong to T1,s ⊂ C T H 1 by Lemma 3.2; therefore going back to the Schrödinger part, we obtain u ∈ C T H s ∩ C T1 H s−2 by Lemma 3.3. Finally, we prove the conservation of total energy. For H 2 strong solutions, this follows from direct computation. For a solution (u, A) with lower regularity, we consider a sequence of H 2 -solutions {(u j , A j )} j which is an approximation to (u, A). As we obtained (4.3), we can prove that {(u j , A j , ∂t A j )} j converges to 0,1/2 . Since {(u j , A j , ∂ A j )} is bounded in L ∞ X s,σ , this sequence (u, A, ∂t A) in L ∞ t j T X T 1,1 . Therefore the conservation of total energy actually converges to (u, A, ∂t A) in L ∞ X T holds also for (u, A). Proposition 4.2. Let (s, σ ) ∈ R with s > 2 and σ > 1. Then for any (u 0 , A0 , A1 ) ∈ X s,σ , the solution to (MS-C) with (1.9) obtained in Proposition 4.1 actually belongs to C T X s,σ . Proof. Firstly let (s, σ ) ∈ R satisfy 2 < s < 4 and σ < 3. Then the unique solution obtained by Proposition 4.1 belongs to C T X 2,σ . Using Lemma 3.4 at most twice, we can prove that u ∈ C T H s ∩ C T1 H s−2 (remark that ∂t2 A = A + PJ ∈ C T H σ −2 by virtue of Lemma 2.5), and hence we can prove the proposition for such (s, σ ). Next we apply Lemmas 2.3, 2.6 and 3.1 to the solution and obtain the proposition for (s, σ ) ∈ R with 2 < s < 4, σ ≥ 3. The proposition for s ≥ 4 has already been obtained in [17]. Proof Theorem 1.1. We can prove the existence of solutions by combining Propositions 4.1–4.2. The uniqueness without auxiliary conditions (u, A) ∈ L 2T (H s−1/2 ⊕ L ∞ ) is a consequence of Corollary 5.1 in Sect. 5. The continuous dependence of solutions on initial data will be proved by Proposition 6.1 in Sect. 6. 5. Global Existence of Solutions Lemma 5.1. Let 11/8 ≤ s ≤ 2, 1 < σ ≤ 10/9 and (u, A, ∂t A) ∈ C T X s,σ be a solution to (MS-C) obtained in Proposition 4.1. Then the following estimates hold: 1 2 ˙1 (u, A, ∂t A); L ∞ T (H ⊕ H ⊕ L ) ≤ C, 2 A; L ∞ T L q A; L T L r 1/2−δ u; L 2T H6 1,σ 2 ∞ A; MT ∩ L T L
(5.1)
≤ CT ,
(5.2)
≤ CT ,
(5.3)
≤ CT 3 ,
(5.4)
≤ CT .
(5.5)
2
4
Here q, r and δ are given in (3.3). The constants C depend only on σ and (u 0 , A0 , A1 ); X s,σ . Proof. We easily obtain (5.1) by the conservation laws of charge and energy, (5.2) by applying (5.1) after differentiating and integrating A with respect to t. Next we apply Lemma 2.3 to (1.8) and obtain q
q
4/q−1
A; L T L r (A0 , A1 ); H 1 ⊕ L 2 + A; L 1T H 2/q−1 + PJ; L T Hr
.
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The second term in the right-hand side is bounded by CT 2 because of (5.2), and ∞ 1 1 2 the third term is bounded by T 1/q u; L ∞ T H A; L T H by the use of Lemma 2.5 together with the Leibniz rule and the Sobolev inequality. Hence this term is also bounded by CT 2 and (5.3) has been proved. In order to obtain (5.4), we apply Lemma 2.4 to (1.7). Then 1/2−δ
u; L 2T H6
1 1/2 u; L ∞ 2iA · ∇u + |A|2 u + φu; L 2T H −2δ T H +T q r 1 ∞ ˙1 2 ∞ 1 2 T u; L ∞ T H A; L L + A; L T H + u; L T H . T
The right-hand side is estimated by CT 3 by the previous estimates. Therefore (5.4) has been proved. We go back to the Maxwell part and again apply Lemma 2.3 to (1.8). Then q
σ −2/q
A; M1,σ T ∩L T Hr
σ
σ −2/3
σ −1 (A0 , A1 ); H ⊕ H σ −1 + T A; L ∞ + PJ; L T H3/2 T H 6/5
.
By the assumption, σ − 2/3 ≤ 1/2 − δ. Therefore the last term in the right-hand side is bounded by 1/2−δ
T 1/3 u; L 2T H6
1 ∞ ˙1 3+1/3 u; L ∞ . T H A; L T H T σ −2/q
If we use the estimate A; L 2T L ∞ T 1/r A; L T Hr Sobolev inequality, we can show (5.5). q
, which is obtained by the
Corollary 5.1. Let 11/8 ≤ s ≤ 2, 1 < σ ≤ 10/9 and let (u, A) be a solution to (MS-C) s−1/2 satisfying (u, A, ∂t A) ∈ C T X s,σ . Then u ∈ L 2T H6 and A ∈ L 2T L ∞ . Proof. If we check the proof of Lemma 5.1, we find that we can prove A ∈ L 2T L ∞ under the assumption that (u, A) satisfies (MS-C) and that (u, A, ∂t A) ∈ C T X 1,σ . Once we s−1/2 have proved A ∈ L 2T L ∞ , we immediately obtain u ∈ L 2T H6 by Lemma 3.1. Proof Theorem 1.2. We first consider the case s ≤ 2 and σ ≤ 10/9. By Lemma 5.1, 1,σ 2 ∞ 1 u; L ∞ T H and A; MT ∩ L T L are finite as long as the solution exists in 0 < ∞ s t < T . Therefore u; L T H is also finite by virtue of Lemma 3.3. This implies the global existence. For the general case we have only to recover the regularity by using Propositions 4.1 and 4.2. 6. Continuous Dependence on Initial Data In this section we shall complete the proof of Theorem 1.1 by proving the continuous dependence of solutions on data. The argument here is essentially based on Bona-Smith [1]. Lemma 6.1. Let 11/8 ≤ s < 4, σ > 1 and let (s, σ ) ∈ R with (s + 1, σ ) ∈ R∗ . Let (u, A) and (u , A ) be solutions to (MS-C) defined on [0, T ] with the initial data (u 0 , A0 , A1 ) ∈ X s,σ and (u 0 , A0 , A1 ) ∈ X s+1,σ respectively. Let (u, A) satisfy the s−1/2 q σ −2/q estimate u; T1,s ∩ L 2T H6 ∨ A; M1,σ ≤ R and (u , A ) satisfy the T ∩ L T Hr same estimate with (u, A) replaced by (u , A ). Then we have the following estimates:
334
M. Nakamura, T. Wada s 2 u − ; L ∞ T H ∩ L T H6
s−1/2
≤ C(u 0 , A0 , A1 )− ; X A− ; M1,σ T
∩
q LT
q
σ −2/q
∨ A− ; M1,σ T ∩ L T Hr
s,σ
1 + CA− ; L ∞ T H
∩
q LT
σ −2/q−δ Hr u ; T1,s+1 ,
(6.1)
σ −2/q Hr
s 2 ≤ C(A0 , A1 )− ; H σ ⊕ H σ −1 + Cu − ; L ∞ T H ∩ L T H6
s−1/2
.
(6.2)
Moreover, let s > 11/8, 0 < σ − 1 − δ ≤ s − 11/8. Then we also have the following estimates: σ −2/q−δ
−δ s−1 ∨ A− ; M1,σ ∩ L T Hr u − ; L ∞ T H T q
≤ C(u 0 , A0 , A1 )− ; X s−1,σ −δ . (6.3)
Here u − = u − u etc., δ, q, r are defined in (3.3) and the constants C depend on R, T, s and σ . Proof. It suffices to show (6.1)–(6.3) for sufficiently small T = T (R); if not, we divide the interval [0, T ] into small subintervals and repeatedly use the estimates obtained for short intervals. Hence we may assume 0 < T < 1 without loss of generality. We begin with the estimate of the Schrödinger part. Taking the difference of the equations for u and u , we have i∂t u − = (−A + φ)u − + 2iA− · ∇u + A− · A+ u + φ− u ≡ (−A + φ)u − +
3
gj,
(6.4)
j=1
where A+ = A + A . We also need the time derivative of (6.4): i∂t2 u − = (−A + φ)∂t u − +2i∂t A · ∇u − + 2∂t A · Au − + ∂t φu − + 2i∂t A− · ∇u + 2iA− · ∇∂t u +∂t A− · A+ u + A− · ∂t A+ u + A− · A+ ∂t u + ∂t φ− u + φ− ∂t u ≡ (−A + φ)∂t u − +
13
gj.
(6.5)
j=4
In the following, we estimate ∂t u − instead of u − itself in order to obtain (6.1) and (6.3). To this end we introduce here an inequality which shows the equivalence of norms u − ; H s and ∂t u − ; H s−2 . Namely for (s, σ ) ∈ R∗ with s > 1/2 we have u − ; H s + C(R){u − 2 + A− ; H σ } ∂t u − ; H s−2 + C(R){u − 2 + A− ; H σ }. (6.6) We can prove this inequality in the same way as we proved (3.13), namely we use a trivial modification of (2.2) together with Lemma 2.1. We refer to Lemma 6.1 in [17] for details. Next we apply Lemma 3.1 to (6.4) and obtain s−1/2
u − ; L 2T H6
σ −2/q
s ∞ σ ≤ C(R){u − ; L ∞ T H ∨ A− ; L T H ∩ L T Hr q
}.
(6.7)
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Here we have treated 3j=1 g j in the same way as 2iA · ∇v + |A|2 v + φv in the proof of Lemma 3.1. Converting (6.5) into integral form by the use of the propagator U (t, τ ) s−2 -norm and using (6.6), we obtain for (3.1), taking the L ∞ T H 2 ∞ σ u − ; T1,s ≤ C(R){(u 0 , A0 , A1 )− ; X s,σ + u − ; L ∞ T L ∨ A− ; L T H
+
13
g j ; L 1T H s−2 }.
(6.8)
j=4
Here we note that K s ≡ supt,τ ∈[0,T ] U (t, τ ); H s → H s ≤ C(R). We estimate the right-hand side term by term as follows: g4 ; H s−2 ∂t A; H σ −1 u − ; H6
s−1/2
g5 ; H
s−2
g6 ; H
s−2
g7 ; H
s−2
∂t A; H
g8 ; H s−2 g9 ; H
s−2
g10 ; H
s−2
g11 ; H
s−2
g12 ; H
s−2
g13 ; H
s−2
σ −1
A;
,
σ −2/q Hr u − ; s
(6.9) H , s
u; T1,s 2 u − ; H , s−1/2 ∂t A− ; H σ −1 u ; H6 , σ −2/q−δ A− ; Hr ∂t u ; H s−1 , σ −2/q ∂t A− ; H σ −1 A+ ; Hr u ; H s , σ −2/q A− ; Hr ∂t A+ ; H σ −1 u ; H s , σ −2/q−δ σ −2/q A− ; Hr ∩ H 1 A+ ; Hr ∩ H 1 ∂t u ; u − ; T1,s u + ; T1,s u ; H s , u − ; H s u + ; H s u ; T1,s .
(6.10) (6.11) (6.12) (6.13) (6.14) (6.15) H
s−1
, (6.16) (6.17) (6.18)
We remark that we can obtain the estimates above for s ≥ 1 and do not need the assumption s ≥ 11/8. In the proof of (6.9)–(6.18), we mainly use the Leibniz rule if s > 2, the Hölder and the Sobolev inequalities, Lemma 2.1, and the inclusions H σ −1 → L q and σ −2/q−δ Hr → L ∞ together with duality argument if necessary. For example, we can s−1/2 obtain (6.9) for s = 1, 2 by the use of the inclusion H6 → Hrs−1 , and for 1 < s < 2 by interpolation. For s > 2, by the Leibniz rule we obtain g4 ; H s−2 ∂t A; H s−2 u − ∞ + ∂t Aq u − ; Hrs−2 , and hence we obtain (6.9) by using the tools mentioned above. We can analogously estimate g5 , g7 , g9 and g10 . We next estimate g8 . For s > 2, g8 : H s−2 A− ∞ ∂t u ; H s−1 + A− ; H ps−2 ∇∂t u p2 1
(6.19)
by the Leibniz rule, where p1 = max{r ; 3/(s − 2)} and 1/ p2 = 1/2 − 1/ p1 so that σ −2/q−δ Hr → H ps−2 ∩ L ∞ and H s−2 → L p2 . For 1 ≤ s ≤ 2, we can prove (6.19) 1 without the second term in the right-hand side similarly as in the estimate of g4 . Therefore we obtain (6.13). We can analogously estimate g11 . The estimates for g6 , g12 , g13 are 2 easy. We should also estimate u − ; L ∞ T L in (6.8). This can be done as in the proof of Proposition 4.1, namely the inequality 2 ∞ 1/2 u − ; L ∞ ∩ L 4T L 4 ≤ C(R)(u 0 , A0 , A1 )− ; X 0,1/2 T L ∨ A− ; L T H
(6.20)
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is obtained for sufficiently small T . Applying (6.9)–(6.18) and (6.20) to (6.8), using the Hölder inequality for the time variable, and choosing T sufficiently small, we obtain q σ −2/q u − ; T1,s ≤ C(R) (u 0 , A0 , A1 )− ; X s,σ + A− ; M1,σ T ∩ L T Hr q σ −2/q−δ 1 +A− ; L ∞ u ; T1,s+1 . (6.21) T H ∩ L T Hr Next we estimate the Maxwell part. By applying Lemma 2.3 to the equation of the difference A− , σ −2/q
q
A− ; M1,σ T ∩ L T Hr
(A0 , A1 )− ; H σ ⊕ H σ −1 + A− ; L 1T H σ −1 +PJ − ; L 1T H σ −1 .
We have the expression PJ − = 2 Im P u¯ + ∇u − − 2P(A Re(u¯ + u − )) − 2P(A− |u |2 ).
(6.22)
Therefore a slight modification of Lemma 2.6 shows PJ − ; L 1T H σ −1 s 2 T 1/4 {u − ; L ∞ T H ∩ L T H6
s−1/2
s 2 u + ; L ∞ T H ∩ L T H6
s−1/2
σ A; L ∞ T H
s 2 ∞ σ +u ; L ∞ T H A− ; L T H } s 2 ≤ C(R)T 1/4 {u − ; L ∞ T H ∩ L T H6
s−1/2
σ ∨ A− ; L ∞ T H }.
Choosing T sufficiently small, we obtain q
σ −2/q
A− ; M1,σ T ∩ L T Hr
s 2 ≤ C(R){(A0 , A1 )− ; H σ ⊕ H σ −1 + T 1/4 u − ; L ∞ T H ∩ L T H6
s−1/2
},
which is (6.2). Substituting this inequality into (6.21), we can also prove (6.1). We proceed to the proof of (6.3). For the Schrödinger part, we can prove q σ −2/q−δ −δ u − ; T1,s−1 ≤ C(R) (u 0 , A0 , A1 )− ; X s−1,σ −δ +A− ; M1,σ ∩ L H . r T T (6.23) If s > 2, we can prove (6.23) similarly as (6.21). Indeed, for the estimates (6.7)–(6.18) except (6.13) and (6.16), we can replace s with s − 1, and σ with σ − δ respectively since (s − 1, σ − δ) ∈ R∗ . On the other hand, for (6.13) and (6.16) we replace s with s − 1 to obtain g8 ; H s−3 A− ; Hr
σ −2/q−δ
∂t u ; H s−2 ,
g11 ; H s−3 A− ; Hr
σ −2/q−δ
∩ H 1 A+ ; Hr
σ −2/q
∩ H 1 ∂t u ; H s−2 .
In the replacements above, we do not meet the harmful factor ∂t u ; H s−1 . Therefore s−1 . To we obtain (6.23). On the other hand, if s ≤ 2, we directly estimate u − ; L ∞ T H
Global Existence and Uniqueness of Solutions to the MS Equations
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this end, we estimate the L 1T H s−1 -norms of g1 , g2 and g3 in (6.4) similarly as in the proof of Lemma 3.1. Indeed we can show σ −2/q−δ
s 2 u ; L ∞ T H ∩ L T H6
σ −2/q−δ
σ −δ ∩ L T Hr g1 ; L 1T H s−1 T 1/2 A− ; L ∞ T H q
σ −δ g2 ; L 1T H s−1 T 1/2 A− ; L ∞ ∩ L T Hr T H q
g3 ; L 1T
H
s−1
s−1/2
q σ −2/q−δ σ −δ s ×A+ ; L ∞ ∩ L T Hr u ; L ∞ T H T H , s−1 s ∞ s T u − ; L ∞ u + ; L ∞ T H T H u ; L T H .
, (6.24) (6.25) (6.26)
We also have the estimate s−3 ≤ C(R) u − ; L ∞ H s−1 + A− ; L ∞ H 1 ∂t u − ; L ∞ T H using (6.4) and Lemma 2.1. These estimates prove (6.23) for 1 ≤ s ≤ 2. For the Maxwell part, we can show σ −2/q−δ
−δ A− ; M1,σ ∩ L T Hr T q
≤ C(R){(A0 , A1 )− ; H
σ −δ
s−1 ⊕ H σ −1−δ + T 1/r u − ; L ∞ ∩ L 2T H6 T H
s−3/2
}. (6.27)
To show (6.27), we should estimate PJ − written in the form (6.22) term by term. We first estimate P(u¯ + ∇u − ) in the case 0 < θ ≡ σ − 1 − δ ≤ 1. We use the inequality P(u¯ + ∇u − ); H θ ∇u + 4 u − ; H4θ , which is directly proved for θ = 0, 1 and generalized for 0 < θ < 1 by interpolation. Then we have P(u¯ + ∇u − ); H σ −1−δ u + ; H4
s−3/8
s−11/8
u − ; H4
.
We remark that we have used here the assumption s > 11/8 since 0 < σ − 1 − δ ≤ s − 11/8 (and we do not use this assumption elsewhere). This inequality can be proved by the use of Lemma 2.5 even if θ > 1. We next estimate the term A Re(u¯ + u − ) again by the Leibniz rule and the Sobolev inequality: A Re(u¯ + u − ); H σ −1−δ A; H pσ3−1−δ u + ∞ u − p4 + A∞ u + ; H pσ3−1−δ u − p4 +A∞ u + ∞ u − ; H σ −1−δ σ −2/q−δ
A; Hr
s−1/2
u + ; H6
u − ; H s−1 .
Here 1/ p3 = δ/3 and 1/ p4 = 1/2 − δ/3. We can analogously treat the term A− |u |2 . s−3/8 s−1/2 Collecting these estimates, using the fact H4 = (H s , H6 )[3/4] , and the Hölder inequality for the time variable, and choosing T sufficiently small, we obtain (6.27). Substituting (6.27) into (6.23), we obtain (6.3). Proposition 6.1. Let T > 0, s > 11/8, σ > 1 and let (s, σ ) ∈ R with (s + 1, σ ) ∈ R∗ . Then, the mapping defined by (u 0 , A0 , A1 ) → (u, A, ∂t A) is continuous as a mapping from X s,σ to C T X s,σ . Here, (u, A) is the solution to (MS-C) with (1.9) obtained in Propositions 4.1–4.2.
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Proof. We may assume s < 4 since the case s ≥ 4 has already been proved in [17]. Let η be a rapidly decreasing function on R3 satisfying η(x)d x = 1, and let η = −3 η(·/ ). We put u 0 = η ∗ u 0 and A j = η 1/δ ∗ A j , j = 0, 1, and let (u , A ) be a corresponding solution. Then for j = 0, 1, (u 0 , A 0 , A 1 ); X s+ j,σ = O( − j ), (u 0 −u 0 , A0 −A 0 , A1 −A 1 ); X s− j,σ − jδ = o( j ) as ↓ 0. We also have u ; T1,s+1 u 0 ; H s+1 by Lemma 3.4. We use (6.1) and (6.3) for bounding the term u − u ; H s taking (s, σ ) in Lemma 6.1 as (s, σ0 ) with 1 < σ0 ≤ σ , (s, σ0 ) ∈ R, (s + 1, σ0 ) ∈ R∗ , and 0 < σ0 − 1 − δ ≤ s − 11/8. On the other hand, we use (6.2) for bounding the term (A − Aε , ∂t A − ∂t Aε ); H σ ⊕ H σ −1 for (s, σ ). Then we can obtain (u − u , A − A , ∂t A − ∂t A ); X s,σ ≤ C(R){(u 0 − u 0 , A0 − A 0 , A1 − A 1 ); X s,σ +(u 0 − u 0 , A0 − A 0 , A1 − A 1 ); X s−1,σ −δ u 0 ; H s+1 } = o(1) + o( )O( −1 ) = o(1), which proves that (u , A , ∂t A ) converges to (u, A, ∂t A) in C T X s,σ . Next we consider s,σ . We shall prove that the a sequence {(u n0 , An0 , An1 )}∞ n=1 converging to (u 0 , A0 , A1 ) in X n n ∞ n corresponding sequence of the solutions {(u , A , ∂t A )}n=1 converges to (u, A, ∂t A) in C T X s,σ , which is the assertion of the proposition. By the previous step, (u n , An , ∂t An ) converges to (u n , An , ∂t An ) in C T X s,σ uniformly with respect to n as ↓ 0. Moreover for any fixed , (u n , An , ∂t An ) converges to (u , A , ∂t A ) as n → ∞ by virtue of (6.3), since they are sufficiently smooth solutions. Thus we can prove the convergence of {(u n , An , ∂t An )}n by standard argument. 7. The Cases of Lorentz and the Temporal Gauges Proof of Theorems 1.3 and 1.4. We prove the theorems by the gauge transform. For any solution (u L , φ L , AL ) to (MS-L), there exists a solution to (MS-C) which is gauge equivalent to (u L , φ L , AL ). Indeed, let us put λ = −1 div AL , AC = PAL , φ C = (−)−1 ρ(u L ) and u C = e−iλ u L . Then (u L , φ L , AL ) and (u C , φ C , AC ) are connected by the relation (1.4), and (u C , AC ) satisfies (MS-C). Therefore we can prove Theorem 1.3 by Theorem 1.1. The assumption σ ≥ s − 1 is needed to ensure the solution to (MS-C) obtained by the gauge transform having the desired regularity. The case of the temporal gauge can be treated analogously. For details, see [17, Sect. 8]. References 1. Bona, J.L., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278, 555–601 (1975) 2. Brenner, P.: On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations. Math. Z. 186, 383–391 (1984) 3. Ginibre, J., Velo, G.: Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 43, 399–442 (1985) 4. Ginibre, J., Velo, G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995) 5. Ginibre, J., Velo, G.: Long range scattering and modified wave operators for the Maxwell-Schrödinger system. I. The case of vanishing asymptotic magnetic field. Commun. Math. Phys. 236, 395–448 (2003)
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6. Ginibre, J., Velo, G.: Long range scattering for the Maxwell-Schrödinger system with large magnetic field data and small Schrödinger data. Publ. Res. Inst. Math. Sci. 42, 421–459 (2006) 7. Guo, Y., Nakamitsu, K., Strauss, W.: Global finite-energy solutions of the Maxwell-Schrödinger system. Commun. Math. Phys. 170, 181–196 (1995) 8. Kato, J.: Existence and uniqueness of the solution to the modified Schrödinger map. Math. Res. Lett. 12, 171–186 (2005) 9. Kato, T.: Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo Sect. I, 17, 241–258 (1970) 10. Kato, T.: Linear evolution equations of “hyperbolic” type. II. J. Math. Soc. Japan 25, 648–666 (1973) 11. Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 46, 113–129 (1987) 12. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988) 13. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998) 14. Kenig, C.E., Koenig, K.D.: On the local well-posedness of the Benjamin-Ono and modified BenjaminOno equations. Math. Res. Lett. 10, 879–895 (2003) 15. Koch, H., Tzvetkov, N.: On the local well-posedness of the Benjamin-Ono equation in H s (R). Int. Math. Res. Not. 26, 1449–1464 (2003) 16. Nakamitsu, K., Tsutsumi, M.: The Cauchy problem for the coupled Maxwell-Schrödinger equations. J. Math. Phys. 27, 211–216 (1986) 17. Nakamura, M., Wada, T.: Local well-posedness for the Maxwell-Schrödinger equation. Math. Ann. 332, 565–604 (2005) 18. Shimomura, A.: Modified wave operators for Maxwell-Schrödinger equations in three space dimensions. Ann. Henri Poincaré 4, 661–683 (2003) 19. Strichartz, R.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977) 20. Triebel, H.: Theory of function spaces. II. Monographs in Mathematics 84, Basel: Birkhäuser Verlag, 1992 21. Tsutsumi, Y.: Global existence and asymptotic behavior of solutions for the Maxwell-Schrödinger equations in three space dimensions. Commun. Math. Phys., 151, 543–576 (1993) 22. Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987) Communicated by P. Constantin
Commun. Math. Phys. 276, 341–379 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0340-1
Communications in
Mathematical Physics
Local Asymptotic Normality in Quantum Statistics M˘ad˘alin Gu¸ta˘ 1 , Anna Jenˇcová2 1 University of Nottingham, School of Mathematical Sciences, University Park, Nottingham NG7 2RD, U.K.
E-mail: [email protected]
2 Mathematical Institute of the Slovak Academy of Sciences, Stefanikova 49, 814 73 Bratislava, Slovakia
Received: 27 June 2006 / Accepted: 31 March 2007 Published online: 2 October 2007 – © Springer-Verlag 2007
Dedicated to Slava Belavkin on the occasion of his 60th anniversary Abstract: The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕθn +u/√n 0 consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation m relations. √ The convergence holds for all “local parameters” u ∈ Rm such that θ = θ0 + u/ n parametrizes a neighborhood of a fixed point θ0 ∈ ⊂ R . In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For the reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory. 1. Introduction The statistical interpretation of quantum mechanics, also known as the Born rule, is an interface connecting the mathematical framework based on Hilbert space operators and wave functions, with the reality in the form of measurement results. While the Born rule describes the probability distribution of measurement results, quantum statistical inference deals with the inverse problem of estimating quantities related to the preparation of the quantum system, based on the measurement data. The first papers dealing with quantum statistical problems appeared in the seventies [23,56,55,6,24] and tackled issues such as quantum Cramér-Rao bounds for unbiased estimators, optimal estimation for families of states possessing a group symmetry, estimation of Gaussian states, optimal discrimination between non-commuting states. In recent years there has been a renewed interest in the field [21,22,36,5] and the advances in quantum engineering have led to the first practical implementations of theoretical
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methods [1,16,43]. An illustrating example is that of quantum homodyne tomography [53,9,30], a measurement technique developed in quantum optics, which allows the estimation with arbitrary precision [2,7] of the state of a monochromatic beam of light, by repeatedly measuring a sufficiently large number of identically prepared beams [44,42,57]. Asymptotic inference is now a well established topic in quantum statistics, with many papers [32,8,52,12,26,3,20,19,4,11] concentrating on the problem of estimating an unknown state ρ using the results of measurements performed on n quantum systems, identically prepared in the state ρ. For two dimensional systems, or qubits, the optimal state estimation problem has an explicit solution [4] in the special context of Bayesian inference, with invariant priors and figure of merit (risk) based on the fidelity distance between states. However this particular optimization method does not work for more general priors or loss functions and it seems to be limited to the qubit case. In the pointwise approach, Hayashi and Matsumoto [20] showed that the Holevo bound [24] for the variance of locally unbiased estimators can be attained asymptotically, and described a sequence of measurements achieving this purpose. Their results, building on earlier work [18,17], provide the first evidence for the emergence of a Gaussian limit in the problem of optimal state estimation for qubits. This paper together with the closely related works [14,13] extend the results of Hayashi and Matsumoto, and aim at developing quantum statistical analogues of fundamental concepts and tools in asymptotic statistics, such as convergence of statistical experiments and local asymptotic normality. The idea of approximating a sequence of statistical models by a family of Gaussian distributions appeared in [54], and was fully developed by Le Cam [28] who introduced the term “local asymptotic normality”. Among the many applications in mathematical statistics, local asymptotic normality is essential in asymptotic optimality theory and explains the asymptotic normality of certain estimators such as the maximum likelihood estimator. Based on the same principle, the paper [14] shows that a similar phenomenon occurs in quantum statistics: the family of joint states of n identically prepared qubits converges to a family of Gaussian states of a quantum oscillator with unknown displacement. More precisely, there exists a physical transformation (quantum channel) which maps the joint state of the spins into the oscillator state, such that local rotations around a fixed spin direction correspond to displacements of a thermal equilibrium state. In [13], it was further shown that the passage to the limit can be physically implemented by transferring the joint qubits state to an approximate Gaussian state of a Bosonic field through a spontaneous emission coupling. After the transfer, the parameters of the initial qubit state can be estimated by means of standard measurements in the field, which turns out to be optimal with respect to various criteria and a large class of loss functions. In this paper we consider the general set-up of identically prepared finite dimensional quantum systems and prove a different version of the local asymptotic normality principle which we call weak convergence, in analogy with the classical statistics terminology. To motivate the result we build the first elements of a theory of weak convergence of quantum statistical models in close relation with the work of Petz on quantum sufficiency [37,40,35]. Our results add to the accumulating evidence for an underlying theory of quantum statistical experiments and quantum statistical decisions, which parallels the classical framework, but in the same time has new ‘quantum’ features generating a fruitful interaction between Mathematical Statistics, Quantum Information and Operator Algebras.
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Before presenting the structure of the paper, here is a short summary of the key concept and ideas used in the paper. By adopting the terminology introduced by Le Cam [28] we call a quantum statistical experiment a family E := (A, ϕθ : θ ∈ ), of states ϕθ on a von Neumann algebra A indexed by a parameter set . One may think of the quantum system as the carrier of a type of statistical information about the unknown parameter θ encoded by Nature (or an adversary) in the state ϕθ . Quantum decision problems such as state estimation or hypothesis testing can be formulated as a game between Nature who has the choice between different parameters θ and the physicist who tries to extract the maximum amount of information about the chosen θ for a given statistical purpose. Quantum sufficiency deals with the situation when two such experiments E := (A, ϕθ : θ ∈ ),
F := (B, σθ : θ ∈ ),
can be mapped into each other by quantum channels, i.e. there exist unit preserving completely positive maps T : A → B and S : B → A such that ϕθ = σθ ◦ T,
σθ = ϕθ ◦ S, ∀θ.
In this case it is clear that the two experiments are equivalent from a statistical point of view and the solution to any decision problem concerning one experiment can be easily mapped to the other. What if we have two experiments which are not equivalent but are ‘close to each other’ in a statistical sense? In Sect. 3 we enlarge the concept of sufficiency by defining the notion of convergence of experiments whereby a sequence En approaches asymptotically a limit experiment E En → E,
n → ∞.
When convergence holds, statistical problems concerning the experiment En can be cast into problems concerning the potentially simpler experiment E with vanishingly small loss of optimality for large n. An important example is that of local asymptotic normality which means roughly the following: the sequence En of experiments consisting of joint states ϕ n of n identical quantum systems prepared independently in the same state ϕ, converges to a limit experiment E which is described by a family of Gaussian states on an algebra of canonical commutation relations. This paper is intended to be a self-contained introduction to the theory of quantum statistical experiments and local asymptotic normality. In Sect. 2 we give an account of the classical concepts which will later be extended to the quantum domain. Sufficiency and equivalence of statistical experiments are defined in Sect. 2.1. We then show how equivalence classes of experiments can be described using the notion of canonical measure and Hellinger transform (see Sect. 2.2). This enables us to define weak convergence of experiments as the pointwise convergence of the Hellinger transforms for all finite subsets of the parameter space. In parallel with the weak convergence we introduce the stronger topology of the Le Cam distance between two experiments. This distance is based on the existence of a randomization mapping the first experiment as close as possible to the second, and the other way around (see Sect. 2.3). We close the exposition of the classical theory with the exact formulation of local asymptotic normality. Given a “smooth” m-dimensional family of distributions Pθ with θ ∈ ⊂ Rm we
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consider the experiments En consisting of n independent, √ identically distributed variables X 1 , . . . , X n with distribution Pθ , where θ := θ0 + u/ n lies in a local neighborhood of a fixed point θ0 , parametrized by u. Then En converges weakly to a Gaussian shift expe) riment consisting of a single m-dimensional normal variable with distribution N (u, Iθ−1 0 having unknown center u and variance equal to the inverse of the Fisher information of Pθ at θ0 (see Sect. 2.4). Section 3 begins with a brief review of quantum sufficiency followed by the characterization of equivalence classes of experiments through the canonical state (cf. Theorem 3.5). The latter gives the expectation of monomials of Connes cocycles [Dϕθ , Dϕ]t for arbitrary θ ∈ and t ∈ R, and plays a similar role to that of the Hellinger transform of the classical case. Section 3.4 deals with the relation between weak and strong convergence of experiments. We show that for finite parameter sets the weak and strong topologies coincide, under certain assumptions. The quantum Central Limit Theorem which is presented in Sect. 4 is one of the main ingredients of our result. Finally, in Sect. 5 we prove the quantum local asymptotic normality Theorem 5.4 as weak convergence of the i.i.d. experiment ϕθn +u/√n to a quantum Gaussian shift 0 experiment φu , which is the main result of the paper. This theorem holds for smooth families of states on matrix algebras of arbitrary finite dimension, and it is complementary to the result of [14] concerning strong convergence for qubit states. For pedagogical reasons we first prove the result for a unitary family of states in Sect. 5.1, which could be seen as a purely quantum experiment, after which we allow the change in eigenvalues leading to the presence of a classical Gaussian component in the limit experiment. 2. Classical Statistical Experiments In this section we describe the notion of local asymptotic normality and its significance in statistics [28,49,45,50]. Suppose that we observe a sample X 1 , . . . , X n with X i taking values in a measurable space (, ) and assume that X i are independent, identically distributed with distribution Pθ indexed by a parameter θ belonging to an open subset ⊂ Rm . The full sample is a single observation from the product Pθn of n copies of Pθ on the sample space (n , n ). The family of probability distributions Pθn : θ ∈ is called a statistical experiment and the point of local asymptotic normality is to show that for large n such statistical experiments can be approximated by Gaussian experiments √ after a suitable reparametrization. Let us fix a value θ0 , define a local parameter u = n(θ −θ0 ) and rewrite Pθn as Pθn +u/√n seen as a distribution depending on the parameter u. We will 0 show that for large n the experiments m Pθ0 +u/√n : u ∈ Rm and N (u, Iθ−1 , ) : u ∈ R 0 have similar statistical properties for “smooth” models θ → Pθ . The point of this result is that while the original experiment may be difficult to analyze, the limit one is a tractable Gaussian shift experiment which can give us information about the original one, for instance in the form of lower bounds of estimation errors. Let pθ be the density of Pθ with respect to some measure µ. In the second experiment we observe a single sample , where from the normal distribution with unknown mean u and fixed variance Iθ−1 0 Iθ0 i j = Eθ0 ˙θ0 ,i ˙θ0 , j , is the Fisher information matrix at θ0 , with ˙θ,i := ∂ log pθ /∂θi .
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In the following subsections we will introduce the key concepts needed to understand local asymptotic normality: sufficiency, statistical equivalence, canonical measure, convergence of experiments. 2.1. Statistical experiments, sufficiency, randomizations. A typical statistical problem can be formulated as follows: given a sample X from a distribution Pθ over the measure space (, ), find θˆ depending on X , an estimator of the unknown parameter θ ∈ such that the expected value of the distance d(θ, θˆ ) is small. In general the space need not be finite dimensional, for instance in the case of estimating an unknown probability density on R. The estimation problem is an example of a statistical decision problem, a broad framework containing estimation as well as hypothesis testing problems. Clearly it is important to understand how much ‘statistical information’ is contained in the experiment E := (Pθ : θ ∈ ), when is an experiment more informative than another, and when two experiments are close to each other in a statistical sense. Such questions have been the main motivation for the development of the theory of statistical experiments pioneered by Le Cam [28]. In this section we will present some basic ideas of this theory, the converging point being the notion of local asymptotic normality. For more information we refer to the monographs [28,49,45,50]. Let us start by explaining the notion of sufficiency at the hand of an example. Let X 1 , . . . , X n be independent identically distributed random variables with values in {0, 1} and distribution Pθ := (1 − θ, θ ) with θ ∈ (0, 1), and denote En := (Pθn : θ ∈ ) as n X i is an unbiased estimator of θ and moreover before. It is easy to see that X¯ n = n1 i=1 it is a sufficient statistic for En , i.e. the conditional distribution Pθn (·| X¯ n = x) ¯ does not depend on θ ! In other words the dependence on θ of the total sample (X 1 , X 2 , . . . , X n ) is completely captured by the statistic X¯ n which can be used as such for any statistical (n) decision problem concerning En . If we denote by P¯θ the distribution of X¯ n then the (n) experiment E¯n = ( P¯θ : θ ∈ ) is statistically equivalent to En . To convince ourselves that X¯ n does contain the same statistical information as (X 1 , . . . , X n ), we show that we can simulate the latter by using a sample from X¯ n and an additional random variable Y uniformly distributed on [0, 1]. Indeed for every fixed value x¯ of X¯ n there exists a measurable function f x¯ : [0, 1] → {0, 1}n , such that the distribution of f x¯ (Y ) is Pθn (·| X¯ n = x) ¯ or λ( f x¯−1 (x1 , . . . , xn )) = Pθn (x1 , . . . , xn | X¯ n = x), ¯ where λ is the Lebesgue measure on [0, 1]. Then F( X¯ n , Y ) := f X¯ n (Y ), has distribution Pθn . The function F is an example or randomized statistic and it is a particular case of a more general construction called randomization which should be seen as a transformation of an experiment into another which typically contains less information than the original one. We will give a short account of this notion in the case of dominated experiments. An experiment E = (Pθ : θ ∈ ) on (, ) is called dominated if there exists a σ -measure µ such that Pθ µ for all θ . We will often use
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the notation Pθ ∼ µ meaning that for any A ∈ , µ(A) = 0 if and only if Pθ (A) = 0 for all θ . Definition 2.1. A positive linear map M∗ : L 1 (1 , 1 , µ1 ) → L 1 (2 , 2 , µ2 ) is called a stochastic operator or transition if M∗ (g) 1 = g 1 for every g ∈ L 1+ (1 ). Definition 2.2. A positive linear map M : L ∞ (2 , 2 , µ2 ) → L ∞ (1 , 1 , µ1 ) is called a Markov operator if M1 = 1, and if for any f n ↓ 0 in L ∞ (2 ) we have M f n ↓ 0. The pair (M, M∗ ) with M and M∗ as above is called a dual pair if f M(g)dµ1 = M∗ ( f )gdµ2 , for all f ∈ L 1 (1 ) and g ∈ L ∞ (2 ). It is a theorem that for any stochastic operator M∗ there exists a unique dual Markov operator M and conversely, for any Markov operator M there exists a unique dual stochastic operator M∗ . Definition 2.3. Let Ei = (Piθ : θ ∈ ) be two dominated statistical experiments on (i , i ) with Pi ∼ µi , i = 1, 2. Then E2 is a randomization of E1 if any of the following equivalent conditions is satisfied: (i) there exists a stochastic operator M∗ : L 1 (1 , 1 , µ1 ) → L 1 (2 , 2 , µ2 ) such that M∗ (d P1θ /dµ1 ) = d P2θ /dµ2 ,
∀θ ;
(ii) there exists a Markov operator M : L ∞ (2 , 2 , µ2 ) → L ∞ (1 , 1 , µ1 ) such that P2θ = P1θ ◦ M,
∀θ.
A statistic f : 1 → 2 generates a sub−σ −field 0 ⊂ and a randomization which is the restriction of the measures P θ to 0 . At the level of Markov operator this is simply described by the embedding of L ∞ (, 0 , µ) into L ∞ (, , µ). In general by passing to a sub−σ −field some information about the initial distribution is lost. It turns out that the concept of randomization is the proper generalization of sufficiency. Indeed the next theorem shows that 0 is sufficient for a dominated experiment E if this can be recovered by a randomization from the restricted experiment E0 . Theorem 2.4. Let E = (Pθ : θ ∈ ) be a dominated experiment on (, ) and 0 ⊂ a sub-σ -field. Denote by E0 the restriction of Eto 0 . Then 0 is sufficient for E if and only if E is a randomization of E0 . Although the concept of randomization does not have a such a direct statistical meaning as that of randomized statistic, it is a very useful functional analytic generalization of the later and it is important as a mathematical tool due to the compactness of the space of randomizations in a certain weak topology.
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Definition 2.5. Two dominated experiments (Piθ : θ ∈ ), i = 1, 2 are statistically equivalent if each one is a randomization of the other. The idea of statistical equivalence is that for any statistical decision problem the two experiments will have matching statistical procedures with the same risks, and thus contain ‘the same information’. Finally we mention another useful characterization of sufficiency known as the Factorization Theorem [45] which later will be extended to the quantum case. Theorem 2.6. Let E = (Pθ : θ ∈ ) be a dominated experiment on (, ) with Pθ ∼ µ, and let 0 ⊂ be a sub-σ -field. Then 0 is sufficient for E if and only if there exist a measurable function h and for each θ a 0 -measurable function gθ such that d Pθ = gθ h, dµ
µ − almost surely.
2.2. The canonical measure and the Hellinger transform. An important example of a sufficient statistic for (Pθ : θ ∈ ) is the likelihood ratio process. Definition 2.7. Let (Pθ : θ ∈ ) be an experiment over (, ) and suppose that Pθ Pθ0 for some fixed θ0 ∈ and all θ ∈ . The associated likelihood ratio process based at θ0 is
d Pθ θ0 = θ → . d Pθ0 Note that the likelihood ratio process is a rather ‘large’ statistic which takes values in R|| ,
d Pθ θ0 : ω → θ → (ω) , ω ∈ . d Pθ0 The choice of the base point θ0 is not important as long as the distributions Pθ are dominated by Pθ0 . A variation on this can be considered if we restrict to a finite set of parameters. In this case there exists a ‘standard representation’ of statistical experiments such that statistically equivalent experiments have the same representation. Let E = (Pθ : θ ∈ ) on (, ) and define µ := θ∈ Pθ which will play the role of Pθ0 . Then the vector of likelihood ratios V := (d Pτ /dµ)τ ∈ seen as a R|| −valued random variable on (, ) induces the law σE = L(V |µ) called the canonical measure of E. Note that neither µ nor σE is a probability distribution, but they both have mass ||. The experiment in observing V is called the canonical experiment and has law consisting Q θ := L V |P θ . Because the likelihood ratio process is sufficient for E, the canonical experiment is statistically equivalent to E and the distribution Q θ is supported by the simplex
|| S := v = (vθ ) ∈ R+ , vθ = 1 . θ
We can now write Q θ (B) = Eθ 1 B (V ) = Eµ 1 B (V )
d Pθ = Eµ 1 B (V )Vθ = dµ
B
vθ σE (dv),
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which implies that Q θ (dv) = vθ σE (dv), and thus the canonical experiment over the fixed measure space S is uniquely determined by the canonical measure σE . Note that not every measure on the simplex is the canonical measure of some experiment. Theorem 2.8. Two statistical experiments with the same finite parameter space are statistically equivalent if and only if their canonical measures coincide. The canonical measure is at its turn completely characterized by the Hellinger transform which is the function ηE : S → R given by z vθ θ σE (dv). z → ηE (z) = S θ∈
The Hellinger transform is a continuous function on the interior of S taking values in [0, 1]. Note that if = {1, 2} and if z ∈ S is given by z 1 = z 2 = 1/2 then √ d P1 d P2 dµ, ηE (1/2, 1/2) = v1 v2 σE (dv) = dµ dµ S which is the affinity of P1 and P2 appearing in the well known Hellinger distance 2 d P1 d P2 − h(P1 , P2 ) = dµ = 2(1 − ηE (1/2, 1/2)). dµ dµ
2.3. Convergence of statistical experiments. How can we compare two statistical (i) experiments Ei = Pθ : θ ∈ on two different measure spaces (i , i ) for i = 1, 2 ? When can we say that one is more informative than the other, or that the two are very close to each other? More specifically we will be interested in the situation where a sequence of experiments En converges to a fixed one E. A natural route is to compare their canonical measures. (n) Definition 2.9. We say that a sequence of experiments En := Pθ : θ ∈ converges weakly to an experiment E := (Pθ : θ ∈ ) if for every finite I ∈ , the sequence of canonical measures of En converges weakly (in distribution) to the canonical measure of E. Another possibility is to compare the likelihood ratio processes
(n) d Pθ d Pθ (n) , θ0 = θ → = θ → and θ0 (n) d Pθ0 d Pθ0 by demanding convergence in distribution of the marginals of these processes for all finite sets I ⊂ . Theorem 2.10. Let E be such that Pθ Pθ0 for all θ . Then the following are equivalent:
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(i) The sequence En converges weakly to E. (ii) For any finite subset I ⊂ , the sequence of Hellinger transforms η En |I converge to η E |I pointwise on SI . (iii) The sequence of likelihood ratio processes (n) θ0 converges to θ0 marginally in distribution. Example 2.1. Consider a binomial variable with parameters n and success probability (n) θ/n: Pθ (k) = nk (θ/n)k (1 − θ/n)n−k , and the corresponding experiment En with θ ranging over the finite set {θ1 , . . . , θ p }. Then the Hellinger transform is p n p θi vi θi vi ηEn (v1 , . . . , v p ) = + . 1− n n i=1
i=1
As n → ∞ this converges pointwise to η(v1 , . . . , v p ) = exp
p
θivi
−
i=1
p
θi vi ,
i=1
which is the Hellinger transform of an experiment consisting of observing a Poisson variable with mean belonging to the set {θ1 , . . . , θ p }. Example 2.2. The central example of this paper is that of local asymptotic normality. Let En be the experiment consisting in observing a sample X 1 , . . . X n of independent identically distributed random variables with distribution Pθ0 +u/√n , where u ∈ Rm should be seen as the unknown local parameter and we assume sufficient “smoothness” for the map θ → Pθ . The claim is that m En := Pθ0 +u/√n : u ∈ Rm −→ N (u, Iθ−1 , ) : u ∈ R 0 where in the limit experiment we observe a single sample from the normal distribution with unknown mean u and fixed variance Iθ−1 . This claim will be detailed in Sect. 2.4. 0 Although minimalist with respect to the set of required relations, the concept of weak convergence is sufficiently strong to allow the derivation of certain statistical properties of the sequence En from those of the limit experiment E. A stronger convergence concept is that introduced by Le Cam using randomizations. As shown in Sect. 2.1, we can check statistical equivalence of two experiments by finding randomizations which map one experiment into the other. Naturally, when this can be done only approximately we think of the two experiments as being close to each other. Definition 2.11. Let Ei := (Piθ : θ ∈ ) be two statistical experiments dominated by µi for i = 1, 2. The deficiency of E1 with respect to E2 is the quantity δ(E1 , E2 ) := inf sup P1θ ◦ M − P2θ , M
θ
where the infimum is taken over all Markov operators M : L ∞ (2 , 2 , µ2 ) → L ∞ (1 , 1 , µ1 ), and · is the total variation norm. The Le Cam distance between E1 and E2 is defined as (E1 , E2 ) = max {δ(E1 , E2 ), δ(E2 , E1 )} .
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We remind the reader that the total variation norm can be written in terms of the L 1 −norm distance between the probability densities θ θ d P d P 1 1 2 P1θ ◦ M − P2θ = M∗ − . 2 dµ1 dµ2 1
The deficiency measure satisfies the triangle inequality δ(E, F) + δ(F, G) ≥ δ(E, G) but is not symmetric. This is remedied by the Le Cam distance which is a mathematical semi-distance. It can be shown that two experiments are at distance zero from each other if and only if they are statistically equivalent in the sense of Definition 2.5, and thus defines a proper distance on the space of equivalence classes of experiments. The relation between the strong convergence in the Le Cam distance and the weak convergence in the sense of convergence of canonical measures is given by the following theorem. Theorem 2.12. Let be a finite set. Then strong convergence of experiments in the sense of Le Cam is equivalent to weak convergence of the canonical measures. If is not finite then weak convergence implies strong convergence under the additional uniformity assumption: for any > 0 there exists a finite set I ⊂ such that lim sup sup inf Pθ(n) − Pτ(n) < . n→∞
θ τ ∈I
Although the Le Cam distance is very appealing from the mathematical point of view, it is often difficult to calculate and will not play any role in our discussion. However, in a quantum theory of experiments the Le Cam distance should play a central role and some encouraging results in this direction exist already. In [14,13] it is shown that the quantum version of the local asymptotic normality with the Le Cam type convergence holds for identically prepared qubits with the limit experiment being a family of displaced thermal equilibrium states. In [15], the problem of optimal cloning of mixed quantum Gaussian states is solved along lines similar to the solution of the classical problem of finding the deficiency between two Gaussian shift experiments. 2.4. Local asymptotic normality. We return now to the second example of Sect. 2.3. a sufficient √ smoothness property for the family (Pθ : θ ∈ ) is the differentiability of θ → pθ in quadratic mean: there exists a vector of measurable functions ˙θ = ( ˙θ,1 , . . . , ˙θ,k )T such that 2 1 √ √ √ pθ+u − pθ − u T ˙θ pθ dµ = o( u 2 ). 2 √ This condition is satisfied in many models and it is sufficient to have pθ (x) continuously differentiable in θ for almost all x and the Fisher information Iθ continuous in θ . Theorem 2.13. [50] Suppose that is an open set in Rm and that the family (Pθ : θ ∈ ) is differentiable in quadratic mean at θ0 . Then log
n pθ0 +u/√n i=1
pθ0
n 1 T˙ 1 T u Iθ0 u + o Pθ0 (1). (X i ) = √ u θ0 (X i ) − 2n n i=1
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We refer to [50] for the proof of the theorem and outline here only the key points under the stronger assumption that θ (x) = log pθ (x) is twice differentiable with respect to θ for every x ∈ . Assume for simplicity that θ is a one dimensional parameter, then we have the expansion log
n pθ0 +u/√n i=1
pθ0
n n u ˙ 1 u2 ¨ (X i ) = √ θ0 (X i ) + θ0 (X i ) + Remn . 2 n n i=1
i=1
The first term on the right side has mean zero because Pθ θ = 0 and thus it can be written as un,θ0 with n,θ0 converging to a normal distribution of zero mean and variance Iθ0 by the Central Limit Theorem. The second term converges to − 21 u 2 Iθ0 by the Law of Large Numbers. Thus we have the convergence in distribution for X ∼ N (0, Iθ0 ), log
n pθ0 +u/√n i=1
pθ0
1 d N (u Iθ0 , Iθ0 ) (X ). (X i ) → u X − u 2 Iθ0 = log 2 d N (0, Iθ0 )
Theorem 2.14. Let En := (Pθn +u/√n : u ∈ Rm ) be a sequence of experiments satisfying 0
) : u ∈ Rm ). Then local asymptotic normality and E = (N (u, Iθ−1 0 En → E,
n → ∞,
in the sense of weak convergence of experiments. 3. Quantum Statistical Experiments The first steps in developing a quantum analogue of the classical theory of statistical experiments were taken by Petz [37], and the latest results on quantum sufficiency can be found in [40]. We begin this section with the basic notions of quantum sufficiency. Later we will further extend the theory to cover approximate sufficiency through the notion of convergence of quantum statistical experiments. For a review of the complementary theory of quantum statistical inference we refer to [5]. We remind the reader that a quantum mechanical system is modeled by a C ∗ -algebra A, where the observables of the system correspond to self-adjoint elements and the states are represented by normalized positive functionals on A. Let S = (ϕθ : θ ∈ ) be a parametrized family of states on A, then the couple E = (A, S) is called a quantum statistical experiment. We will mostly assume that A is also a von Neumann algebra, in which case the states ϕθ are required to be normal. Von Neumann algebras are the non-commutative analogues of classical algebras of bounded random variables L ∞ (, , µ), and the normal states are the analogue of the probability distributions which are continuous with respect to µ, i.e. their densities span the space L 1 (, , µ). The interest in considering subsets of the whole set of states is that in this way we can encode prior information about the preparation, for instance if we know that the state is pure, or that it has a block diagonal form. Let B be another von Neumann algebra and let α : B → A be a linear map. Then α is a channel if it is completely positive, unit-preserving and normal. Such maps are the quantum versions of Markov operators (see Definition 2.2), and their duals which act on states, are the quantum state transitions. We will further suppose that all the channels are faithful, that is if α(a) = 0 for some positive a then a = 0.
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Let E = (A, ϕθ : θ ∈ ) be an experiment and α : B → A a channel. The induced experiment F = E ◦ α := (B, ϕθ ◦ α : θ ∈ ) is called a randomization of E. If also E is a randomization of F, i.e. there is a channel β : A → B, such that ϕθ ◦ α ◦ β = ϕθ for all θ , then the experiments E and F are statistically equivalent. In this case, we also say that the channel α is sufficient for E. If B ⊂ A is a subalgebra and the inclusion map B → A is sufficient for E, then B is a sufficient subalgebra for E. Note that a sufficient channel is intrinsically related to the quantum experiment, in particular it may not be invertible on the whole set of states of A as we will see in examples. In order to give a characterization of quantum sufficiency, we first need to describe its basic ingredients. We restrict to the case when all the states in S are faithful, and we refer to [40] for the more general situation. We denote the set of all such experiments with parameter space by E(). ϕ
Definition 3.1. Let ϕ be a state on A. There exists a unique group σt of automorphisms of A called the modular group of ϕ such that the following modular condition holds. For each a, b ∈ A, there is a function F ∈ A(J ), such that ϕ
ϕ
F(t) = ϕ(aσt (b)), F(t + i) = ϕ(σt (b)a),
t ∈ R,
where A(J ) denotes the set of functions analytic in the strip J := {z ∈ C, 0 < Im z < 1}, and continuous on the closure J¯. Definition 3.2. Let θ0 , θ be two points in and ϕ := ϕθ0 and ϕθ be the corresponding states. The Connes cocycle derivative u t = [Dϕθ , Dϕ]t is a σ -strongly continuous one parameter family of unitaries in A with the following properties [47]: ϕ
(a) u t satisfies the cocycle condition u s σs (u t ) = u t+s , s, t ∈ R. ϕ ϕ (b) u t σt (a)u ∗t = σt θ (a), a ∈ A, t ∈ R. (c) For all a, b ∈ A, there is a function F ∈ A(J ), such that ϕ
ϕ
F(t + i) = ϕ(au t σt (b)), F(t) = ϕθ (u t σt (b)a),
t ∈ R.
The family of cocycle derivatives ([Dϕθ , Dϕ]t : t ∈ R, θ ∈ ) is the quantum analogue of the likelihood ratio process (see Definition 2.7). Indeed in the commutative case the it modular group is trivial and the above conditions are satisfied by u t = d Pθ /d Pθ0 . In this paper we are particularly interested in the case of type I algebras A which appear more often in physical applications, i.e. matrix algebras M(Cd ), the algebra B(H) for H separable infinite dimensional Hilbert space, and direct sums thereof. Then A admits a trace Tr and each state ϕ is uniquely characterized by its density operator ρ as ϕ(a) = Tr(ρa),
a ∈ A.
Let ρθ be the density operator for ϕθ , then the modular group and the cocycle derivatives are given by ϕ
σt (a) = ρ it aρ −it
and
[Dϕθ , Dϕ]t = ρθit ρ −it .
(1)
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Note that if we put a = b = 1 in (c) and if F is the corresponding √ √ function in A(J ), then F(i1/2) is the transition probability PA (ϕθ , ϕ) := Tr( ρ ρθ ). Moreover, for p ∈ (0, 1), we can define the relative quasi-entropy by S p (ϕθ , ϕ) =
1 1 1− p (1 − F(i p)) = (1 − Tr(ρ p ρθ )). p(1 − p) p(1 − p)
Let A and B be von Neumann algebras and let α : B → A be a channel. Then the multiplicative domain of α is the subalgebra Bα ⊂ B, defined by Bα := {a ∈ B, α(a ∗ a) = α(a)∗ α(a) : α(aa ∗ ) = α(a)α(a)∗ }, and the restriction of α to the multiplicative domain is an isomorphism onto α(Bα ) if α is faithful. Theorem 3.3. [40] Let E = (A, ϕθ : θ ∈ ) be a quantum statistical experiment and let ϕ = ϕθ0 . Let α : B → A be a faithful channel, then the following are equivalent: (i) (ii) (iii) (iv)
α is sufficient for E, S p (ϕθ , ϕ) = S p (ϕθ ◦ α, ϕ ◦ α) for all θ and for some p ∈ (0, 1), [Dϕθ , Dϕ]t = α([D(ϕθ ◦ α), D(ϕ ◦ α)]t ) for all θ and t ∈ R, α(Bα ) is a sufficient subalgebra for E.
Note that in the case that B is a subalgebra in A, the condition (iii) is equivalent to (iii’) [Dϕθ , Dϕ]t ∈ B for all θ ∈ and t ∈ R. This implies that the subalgebra generated by the cocycle derivatives is sufficient for E and it is contained in any other sufficient subalgebra, so that it is minimal sufficient. We will denote this subalgebra by AE . Moreover, the cocycle condition implies that AE ϕ is invariant under the modular group σt . For a channel α : B → A, the conditions of the theorem are equivalent to the fact that the minimal sufficient subalgebra BF for the induced experiment F = E ◦ α is contained in the multiplicative domain of α. Corollary 3.4. Two statistical experiments E := (A, ϕθ : θ ∈ ) and F := (B, σθ : θ ∈ ) are statistically equivalent if and only if there exists an isomorphism α : BF → AE between their minimal sufficient algebras such that ϕθ ◦ α = σθ for all θ . Example 3.1. Let A = Md (C) and let E = (A, ϕθ : θ ∈ θ ) be a quantum experiment. Let A0 ⊂ A be a subalgebra. Then there is a decomposition Cd =
m
HiL ⊗ HiR ,
i=1
m with the projections pi : Cd → HiL ⊗HiR , such that A0 is isomorphic to i=1 B(HiL )⊗ ϕ 1 H R . Let us also suppose that A0 is invariant under σt . Then A0 is sufficient for E if i and only if the density matrices have the form ρθ =
m
L ϕθ ( pi )ρθ,i ⊗ ρiR , θ ∈ ,
(2)
i=1 L ∈ B(H L ), ρ R ∈ B(H R ) are density matrices (cf. [33], see also [40] for where ρθ,i i i i an infinite dimensional version). If A0 is the minimal sufficient subalgebra, then the
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decomposition (2) is the maximal decomposition obtained in [27]. Since any sufficient subalgebra contains the minimal sufficient subalgebra, we may conclude that an arbitrary subalgebra A0 is sufficient if and only if there is an orthogonal sequence of projections { pi } in A0 with i pi = 1, positive elements ρθ,i ∈ A0 and ρi ∈ A with supports pi , commuting for all θ , such that ρθ = ϕθ ( pi )ρθ,i ρi . i
This result is the quantum version of the factorization Theorem 2.6.
3.1. Equivalence classes of experiments. The notion of statistical equivalence of experiments as introduced in the previous Section defines an equivalence relation on E(). In this section, we want to describe the equivalence classes. The aim is to construct quantum analogues of the notions of canonical experiment and canonical measure described in Sect. 2.2. Let E = (A, ϕθ : θ ∈ ) be an experiment in E(). Then the equivalence class of E contains also the restriction E|AE to the minimal sufficient subalgebra AE . We may therefore consider only experiments such that A is generated by the cocycle derivatives. In what follows (AE , HE , ξE ) always denotes the GNS representation of the minimal sufficient subalgebra with respect to the state ϕ = ϕθ0 . Let G = G() be the free group generated by the set of symbols {u t (θ ) : u 0 (θ ) = u t (θ0 ) = e, θ ∈ , t ∈ R}. We denote by L 1 (G) the Banach space of all summable functions f : G → C, with norm f := g∈G | f (g)|. The dual space L 1 (G)∗ can be identified with the space L ∞ (G) of bounded functions over G, equipped with the supremum norm. For each experiment E ∈ E() there is a unique group homomorphism πE : G → U(HE ), u t (θ ) → [Dϕθ , Dϕ]t , ∀θ ∈ , t ∈ R, thus πE is a unitary representation of G on HE . We define a function on G by ωE (g) = ξE , πE (g)ξE = ϕ(πE (g)), g ∈ G. Then ωE is a state, that is a positive definite function on G, satisfying ωE (e) = 1 and will be called the canonical state of the experiment E. Since for any state ω we have |ω(g)| ≤ ω(e) = 1 for all g ∈ G, the set of all states is a subset in the unit ball of L ∞ (G). Clearly, the GNS representation πωE of G with respect to ωE is equivalent with πE . From property (c) of the cocycle derivatives we know that for any θ ∈ and g ∈ G there is a function FE ,g,θ ∈ A(J ) such that FE ,g,θ (t + i) = ϕ(πE (g)[Dϕθ , Dϕ]t ) = ωE (gu t (θ )), and|FE ,g,θ (z)| ≤ 1 for all z ∈ J . We have the following characterization of the equivalence classes of experiments.
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Theorem 3.5. Let E = (A, ϕθ : θ ∈ ) and F = (B, ψθ : θ ∈ ) be experiments in E() with A and B minimal sufficient. Then E is equivalent with F if and only if ωE = ωF . Proof. Let E be equivalent with F, then by Corollary 3.4, there is an isomorphism α : A → B, such that ϕθ = ψθ ◦ α and α([Dϕθ , Dϕ]t ) = [Dψθ , Dψ]t , θ ∈ , t ∈ R. (We remind the reader that AE = A and BF = B.) By uniqueness of πF , it follows that πF = α ◦ πE and ωF = ψ ◦ πF = ψ ◦ α ◦ πE = ωE . To prove the converse, let ωE = ωF =: ω, then πE and πF are equivalent, since they are both equivalent with πω . Hence there is a unitary U : HF → HE , such that πF (g) = U ∗ πE (g)U and the cyclic vectors satisfy U ξF = ξE . In particular [Dψθ , Dψ]t = U ∗ [Dϕθ , Dϕ]t U and it is enough to prove that ψθ = ϕθ ◦ AdU for all θ ∈ . For θ ∈ , g ∈ G, the functions FE ,g,θ and FF ,g,θ are in A(J ) and coincide on R + i, hence they coincide on J . It follows that ψθ (πF (g)) = FF ,g,θ (0) = FE ,g,θ (0) = ϕθ (πE (g)) = ϕθ (U πF (g)U ∗ ), for all g ∈ G. Since the elements {πF (g), g ∈ G} generate B, the proof is finished.
Remark. Let us suppose that E is a binary experiment, that is, consists of two points {θ1 , θ0 }. Let F be the analytic continuation of the function t → ωE (u t (θ1 )). Then the function φE : (0, 1) p → F(i p), can be viewed as a quantum version of the Hellinger transform. If for some binary experiments E and F we have φE = φF , then clearly ωE (u t (θ1 )) = ωF (u t (θ1 )) for all t, but, unlike the classical case, this is not enough to characterize quantum statistical equivalence, since we need the values of the canonical states on all products of u t (θ1 ). This corresponds to the results in [34], where it is proved that, at least in the finite dimensional case, quantum statistical equivalence cannot be determined by the class of quantum f -divergences, unless the experiments are commutative. 3.2. The set of canonical states. As we have seen, E() can be identified with a subset in the unit ball of L ∞ (G()) through the canonical state. In this section we will describe this subset. For each s ∈ R, we define an automorphism on G as the extension of the map αs (u t (θ )) = u s (θ )−1 u t+s (θ ), θ ∈ . Then αs , s ∈ R is a group of automorphisms on G. If ω = ωE is a canonical state, then the cocycle condition implies πE (αs (u t (θ ))) = [Dϕθ , Dϕ]∗s [Dϕθ , Dϕ]t+s = σsϕ (πE (u t (θ ))). It follows that πE (αs (g)) = σsϕ (πE (g)), g ∈ G,
(3)
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so that ω satisfies the modular condition with respect to αs . Moreover, it follows from the properties of the Connes cocycle that for g, h ∈ G and θ ∈ , the functions t → ω(gu t (θ )αt (h)) have an analytic continuation to the strip J˜ ⊂ C which is the reflection of J with respect to the real axis (see Definition 3.1), and they are bounded by 1 in absolute value on J˜. The next theorem shows that this-property completely characterizes the canonical states. Theorem 3.6. Let ω be a state in L ∞ (G). Then ω is the canonical state for some experiment E if and only if for each θ ∈ and g, h ∈ G, there is a function Fg,h,θ ∈ A(J ), |Fg,h,θ (z)| ≤ 1 for z ∈ J , satisfying Fg,h,θ (t + i) = ω(gu t (θ )αt (h)), t ∈ R, Fg,h,θ0 (t) = ω(αt (h)g), g, h ∈ G, t ∈ R, Fe,e,θ (0) = 1, θ ∈ . Proof. Note that the conditions for θ = θ0 imply that ω satisfies the modular condition for αt . If ω is a canonical state, then by Definition 3.2 the function Fg,h,θ (t) = ωθ (u t (θ )αt (h)g),
t ∈R
satisfies the required conditions, where ωθ (g) = ϕθ (πω (g)),
g ∈ G.
(4)
For the converse, let (πω , Hω , ξω ) be the GNS triple for ω and define Mω = πω (G) . We will first show that the state ϕ = ξω , · ξω is faithful on Mω . Suppose that a is a positive element in Mω , such that ϕ(a) = 0. Let C[G] be the algebra of all finite complex-linear combinations of elements of G, then πω extends naturally to C[G] and πω (C[G]) is a strongly dense *-subalgebra in Mω . By the Kaplansky density theorem [25], there is a net {a j } j∈I of positive elements in C[G], such that πω (a j ) converges strongly to a 1/2 . By assumptions, for any b, c ∈ C[G] and j ∈ I, there is a function F j := Fa j b∗ ,c,θ0 ∈ A(J ), such that F j (t + i) = ω(a j b∗ αt (c)),
F j (t) = ω(αt (c)a j b∗ ).
Since ω satisfies the modular condition, it is invariant under αt , so that both F j (t) and F j (i +t) converge uniformly on R. By the maximum modulus principle, F j (z) converges uniformly on J to a function F ∈ A(J ). But since |F j (t + i)|2 ≤ ω(a j b∗ ba j )ω(c∗ c) → 0, F(t + i) = 0, for t ∈ R and hence F(z) = 0 on J¯. It follows that F(0) = πω (c∗ )ξω , a 1/2 πω (b∗ )ξω = 0. As this is true for all b, c ∈ C[G], we get a 1/2 = 0. Let now Ut be the unitary on Hω , given by Ut πω (a)ξω = πω (αt (a))ξω , a ∈ C[G] and let σt = AdUt . Then σt ◦ πω = πω ◦ αt and ϕ satisfies the modular condition for σt on a σ -strongly dense subset in Mω . It follows that σt is the modular group of ϕ [47]. Moreover, for each θ , let Ut (θ ) = πω (u t (θ ))Ut , then Ut (θ )πω (a)ξω = πω (u t (θ )αt (a))ξω . By continuity of the functions Fg,h,θ , the map t → Ut (θ ) is σ -strongly continuous. It follows that πω (u t (θ )) is a σ - strongly continuous family of unitaries, satisfying the cocycle condition. By Theorem 3.8 of [47], there are faithful semifinite normal weights ϕθ , such that πω (u t (θ )) = [Dϕθ , Dϕ]t . By properties of the cocycle derivatives, ϕθ (1) = Fe,e,θ (0) = 1. It follows that E = (Mω , ϕθ : θ ∈ ) is an experiment in E() and ω = ωE .
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3.3. The convex structure of experiments. A convex combination of experiments can be obtained as follows. Let Ei = (Ai , ϕi,θ : θ ∈ ), i = 1, 2 be two experiments in E() and let 0 < λ < 1. Then we define an experiment Eλ ∈ E() by Eλ = (A1 ⊕ A2 , ϕθ = λϕ1,θ ⊕ (1 − λ)ϕ2,θ : θ ∈ ), It is easy to see that [Dϕθ , Dϕ]t = [Dϕ1,θ , Dϕ1 ]t ⊕ [Dϕ2,θ , Dϕ2 ]t , θ ∈ , t ∈ R. and this implies that ωEλ = λωE1 + (1 − λ)ωE2 . We will characterize the extremal points in E(). Theorem 3.7. Let ω, ˜ ω be two canonical states, such that ω˜ ≤ tω for some t > 0. Then there is a positive element T in the center of πω (G) , with T ≤ t, satisfying ωθ (T ) = ω(T ) = 1,
∀θ,
(5)
and such that ω(g) ˜ = ξω , πω (g)T ξω , g ∈ G.
(6)
Conversely, let T ≥ 0 be a central element in π(G) satisfying (5), then (6) defines an experiment in E(). Proof. Let ω˜ ≤ tω, then by standard arguments there is a positive element T ∈ πω (G) , T ≤ t, such that (6) holds. Therefore, ω˜ can be extended to a normal state on πω (G) , which we again denote by ω. ˜ Let a, b be elements in C[G], then by (3), ω(π ˜ ω (a)σsω (πω (b))) = ξω , πω (aαs (b))T ξω = ω(aα ˜ s (b)). Since πω (C[G]) is σ -strongly dense in πω (G) , we obtain from Theorem 3.6 that ω˜ satisfies the modular condition for σtω . This implies that there is a positive central element S in πω (G) , such that ω(a) ˜ = ξω , aSξω for all a ∈ πω (G) . Since ξω is separating for πω (G) , we have T = S. To obtain the condition (5), let F, F˜ ∈ A(J ) be such that F(t + i) = ω(T [Dωθ , Dω]t ), F(t) = ωθ ([Dωθ , Dω]t T ), ˜ + i) = ω(u ˜ F(t ˜ t (θ )), F(t) = ω˜ θ (u t (θ )), where we have used (4) and the properties of the cocycle derivatives. Then F(t + i) = ˜ ˜ + i) for all t and this implies F = F. ˜ In particular, ωθ (T ) = F(0) = F(0) = 1. F(t Conversely, let T ≥ 0 be a central element, satisfying (5), then it is not difficult to check that ω˜ given by (6) satisfies the properties in Theorem 3.6. Corollary 3.8. A canonical state ω is extremal if and only if the center of πω (G) contains no positive element T , satisfying ωθ (T ) = ω(T ) for all θ , other than a multiple of identity.
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Proof. Let the experiment ω be such that πω (G) has the required property and let ω = λω1 + (1 − λ)ω2 . Then ω1 ≤ λ1 ω and by the previous theorem, ω1 is of the form (6) for some positive central element T , satisfying (5). It follows that T = I and we must have ω1 = ω2 = ω. Conversely, suppose that there is a positive element T˜ , other than a multiple of identity, satisfying ωθ (T˜ ) = ω(T˜ ) for all θ . Then by putting T = 1/ω(T˜ )T˜ in Theorem 3.7, we obtain an experiment ω1 ≤ tω, with t = T > 1. Since the vector ξω is separating for πω (G) , we must have ω1 = ω. It follows that ω = 1t ω1 + (1 − 1t )ω2 , where ω2 has the form (6) with the element S = 1/(t − 1)(t − T ). Since S is a positive central element, satisfying (5), ω2 is an experiment. Corollary 3.9. If E ⊂ E() is extremal then the center of πω (G) is of the form Cd with 1 ≤ d ≤ ||. 3.4. Weak and strong convergence of quantum experiments. The strong convergence of quantum experiments is a natural extension of the classical convergence with respect to the Le Cam distance. Definition 3.10. Let E := (A, ϕθ : θ ∈ ) and F := (B, σθ : θ ∈ ) be two quantum statistical experiments. The deficiency δ(E, F) is defined as δ(E, F) = inf sup ϕθ ◦ T − σθ , T
θ
where the infimum is taken over all channels T : B → A. The Le Cam distance between E and F is (E, F) := max (δ(E, F), δ(F, E)). We say that a net Eα := (Aα , ϕθ,α : θ ∈ ), α ∈ I, converges strongly to E if (Eα , E) → 0, i.e. there are channels Tα : Aα → A and Sα : A → Aα , such that supθ∈ ϕθ ◦ Tα − ϕθ,α → 0, supθ∈ ϕθ,α ◦ Sα − ϕθ → 0.
(7) (8)
We say that Eα converges weakly to E if the canonical states converge pointwise ωEα (g) → ωE (g),
∀g ∈ G.
Theorem 3.11. Let A, B be C∗ -algebras and let C P1 (B, A) be the space of unital completely positive maps T : B → A. Then C P1 (B, A) is compact with respect to the topology defined by convergence of the linear functionals T → φ(T (b)) for all b ∈ B and φ ∈ A∗ . Proof. Standard application of Tychonoff’s Theorem.
We will now show that the Le Cam distance is a metric on the space of equivalence classes of quantum statistical experiments. Lemma 3.12. The experiments E := (A, ϕθ : θ ∈ ) and F := (B, σθ : θ ∈ ) are statistically equivalent if and only if (E, F) = 0.
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Proof. The direct implication follows from the definitions. We have to prove that if (E, F) = 0 then there exists a channel T : B → A such that ϕθ ◦ T = σθ for all θ , and similarly in the opposite direction. Let Tα be a sequence (net) of channels such that sup ϕθ ◦ Tα − σθ → 0. θ
By Theorem 3.11 applied to A, B seen as C ∗ -algebras we have that C P1 (B, A) is compact and thus there exists a subnet TI (α) which converges to some unital completely positive map T˜ . The two statements together imply that ϕθ ◦ T˜ = σθ . The only problem is now that T˜ is not necessarily normal. Let us denote by T˜∗ : A∗ → B ∗ the restriction to A∗ of the adjoint map T˜ ∗ , then the map T := (T˜∗ )∗ : B˜ → A is an extension of T˜ to the universal enveloping von Neumann algebra B˜ B ∗∗ of B. Clearly, T is completely positive and unital. Let z 0 be ˜ such that B∗ = B ∗ z 0 , see [46] and let ψ be any state in B∗ . the central projection in B, Define the map S : B → B˜ by S(a) = az 0 + ψ(a)(1 − z 0 ). Then S is completely positive and unital, moreover, ϕ ◦ S ∈ B∗ for all ϕ ∈ B ∗ and ϕ ◦ S = ϕ for ϕ ∈ B∗ . Finally, let T = S ◦ T , then T : B → A is a channel, such that ϕθ ◦ T = σθ . We will now show that our definition of weak convergence coincides with the classical one in the case of commutative statistical experiments with faithful states. Lemma 3.13. Let E = (, , Pθ : θ ∈ ) and En = ((n) , (n) , Pθ(n) : θ ∈ ) be classical statistical experiments with n = 1, 2, . . .. Assuming that all experiments (n) (n) belong to the class E(), that is Pθ ∼ Pθ0 and Pθ ∼ Pθ0 for some θ0 , then the following are equivalent (i) En converges weakly to E in the sense of Definition 2.9 for classical experiments. (ii) En converges weakly to E in the sense of Definition 3.10 for quantum experiments. Proof. Without loss of generality we can consider to be finite. According to Theorem 2.10 En converges weakly to E if and only if the corresponding sequence of likelihood (n) (n) ratio processes θ0 converges in distribution to θ0 . We will show that the latter is equivalent to Definition 3.10. Thus we can represent all experiments as families of ||−1 (n) distributions on R+ with Q θ (dr ) = rθ λ(dr ) and Q (n) θ (dr ) = rθ λ (dr ), where λ, λn are the laws of their respective likelihood ratio processes. The associated von ||−1 ||−1 , λ) and An := L ∞ (R+ , λ(n) ) and the Neumann algebras are A := L ∞ (R+ it cocycle derivatives act by multiplication with the function rθ (for θ = θ0 ): [D Q θ , Dλ]t : f (r ) → f (r ) · rθit . ||−1
Since by assumption, all measures have support in the interior of R+ , we can consider their restriction to this subset without altering the weak convergence property (cf. Theorem 1.3.10 [51]). Assuming (i) and considering that the functions it r → rθ θ , θ=θ0
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are bounded and continuous on the interior of R+ , we obtain rθitθ λ(n) (dr ) → rθitθ λ(dr ), as n → ∞, θ=θ0
θ=θ0
which proves (ii). Conversely, if (ii) holds, we can map r one-to-one into x ∈ R||−1 by xθ = log rθ . Then rθitθ λ(dr ) = ei xθ tθ µ(d x), θ=θ0
θ=θ0
with µ(A) = λ(log−1 (A)). The right-hand side represents the characteristic function of the measure µ and by the Lévy-Cramér Continuity Theorem we get that µn converges weakly to µ. Finally, by the continuity of the x → r transformation we get (i). Proposition 3.14. Let Eα , α ∈ I be a net of experiments in E(), converging weakly to E ∈ E(). Let ωθ , ωθ,α be defined by (4). Then limα ωθ,α (g) = ωθ (g) for all g ∈ G, θ ∈ . Proof. First, note that on the set of states, the pointwise convergence coincides with the weak* convergence in L ∞ (G), and since the unit ball is compact in this topology, it is enough to prove that any convergent subnet ωθ,γ must converge to ωθ . Let g ∈ G and let Fα := FEα ,g,θ , F := FE ,g,θ . Then Fα , F ∈ A(J ) and |Fα (z)| ≤ 1, |F(z)| ≤ 1 for z ∈ J . By assumptions, Fα (t) converges to F(t) for each t. We will use the following family of functions: (t − z)2 1 }, exp{− f β,z (t) = √ β βπ For any β > 0, define φα,β (z) =
β > 0, z ∈ C.
Fα (t) f β,z (t)dt,
φβ (z) =
F(t) f β,z (t)dt.
Then φα,β , φβ are entire analytic and uniformly bounded on compact subsets in C. Moreover, for s ∈ R, |φα,β (s) − φβ (s)| ≤ |Fα (t) − F(t)| f β,s (t)dt → 0 by dominated convergence theorem. It follows that φα,β (z) → φβ (z) for all z ∈ C. Since Fα , F are analytic in J and continuous on J¯, w ∈ J¯. φα,β (z + w) = Fα (t + w) f β,z (t)dt, φβ (z + w) = F(t + w) f β,z (t)dt, In particular, F(t + i) f β,0 (t)dt, Fα (t + i) f β,0 (t)dt = φα,β (i) → φβ (i) =
β > 0.
Suppose now that ωθ,γ is a convergent subnet, then Fγ (t + i) = ωθ,γ (u t (θ )g) by 1. But then converges pointwise to some function ψ(t), bounded again, we have Fγ (t +i) f β,0 (t)dt → ψ(t) f β,0 (t)dt, so that ψ(t) f β,0 (t)dt = F(t +i) f β,0 (t)dt for all β > 0. Letting β → 0, we get limγ ωθ,γ (g) → ωθ (g).
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Remark. Let us choose another point θ ∈ instead of θ0 in the definition of the canonical state. Then by the chain rule for the cocycle derivatives, [Dϕθ , Dϕθ ]t = [Dϕθ , Dϕ]t [Dϕ, Dϕθ ]t , θ ∈ , t ∈ R, so that we obtain the same group G() and the new canonical state is equal to ωθ . The above proposition implies that weak convergence of experiments does not depend on the choice of θ0 . We have shown that our definition of weak convergence corresponds to the classical one, in the commutative case. What is still missing is the relation to the strong convergence, namely that weak and strong convergence are equivalent for finite parameter sets (cf. Theorem 2.12). Note that this would also imply that strong convergence is stronger than the weak one. We will show this equivalence under some conditions. First, we will consider uniformly dominated sets of experiments. Let ψ be any experiment in E() and let B > 0. Let us denote by E(ψ, B) the set of all experiments ω ∈ E(), such that ω ≤ Bψ. By Theorem 3.7, there is a one-to-one correspondence between E(ψ, B) and the set Z(ψ, B) of positive elements in the center of Mψ with T ≤ B and ψθ (T ) = 1, for all θ . Namely, for any ω ∈ E(ψ, B), there is an element T ∈ Z(ψ, B), such that ωθ (g) = ψθ (T g) = ξψθ , T πψ (g)ξψθ ,
g ∈ G, θ ∈ ,
and since ψ is faithful on Mψ , such T is unique. This also implies that ωθ can be extended to a normal state on Mψ . Let us endow E(ψ, B) with the topology of pointwise convergence and Z(ψ, B) with the σ (Mψ , Mψ∗ )- topology. Then Z(ψ, B) is compact. Let Tα be a net in Z(ψ, B), converging to T and let ωα and ω be the corresponding canonical states in E(ψ, B). Then for any g ∈ G, ωα (g) = ξψ , Tα πψ (g)ξψ → ξψ , T πψ (g)ξψ = ω(g), so that the map : Z(ψ, B) T → ψ(T ·) ∈ E(ψ, B) is continuous. It follows that E(ψ, B) is compact. Conversely, let ωα be a net in E(ψ, B), converging to ω and let Tα , T be the corresponding elements in Z(ψ, B). Then for any a, b ∈ C[G], we have πψ (a)ξψ , Tα πψ (b)ξψ = ωα (a ∗ b) → ω(a ∗ b) = πψ (a)ξψ , T πψ (b)ξψ . Since the vectors π(a)ξψ , a ∈ C[G] are dense in Hψ and Tα are uniformly bounded, this implies that Tα converges to T . It follows that the inverse map −1 : E(ψ, B) → Z(ψ, B) is continuous. Moreover, we get that ωθ,α (a) → ωθ (a), for all a ∈ Mψ , θ ∈ . We can summarize as follows: Lemma 3.15. The topology in E(ψ, B) coincides with the topology obtained from the weak topology in Mψ∗ . The set E(ψ, B) is compact, and therefore sequentially compact, by the Eberlein - Smulyan theorem. Now we can state the equivalence theorem, for uniformly dominated sequences of experiments of type I with discrete center.
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Theorem 3.16. Let En := (An , ϕθ,n : θ ∈ ) be a sequence of experiments in E() with a finite set. Assume that the sequence is uniformly dominated, i.e. the canonical states ωn := ωEn ∈ E(ψ, B) for all n, for some fixed experiment ψ and B > 0. Assume further that the minimal sufficient von Neumann algebras of the experiments En are type I with discrete center. Then En converges weakly to E if and only if (En , E) → 0, i.e. there exist sequences of channels αn : An → A, βn : A → An , such that lim ϕθ ◦ αn − ϕθ,n = 0,
n→∞
lim ϕθ,n ◦ βn − ϕθ = 0,
n→∞
∀θ ∈ .
Proof. Let ωn ∈ E(ψ, B), satisfying the assumptions. Then the support pn of ω n in Mψ is a central projection, such that pn Mψ is type I with discrete center. Let ω¯ = n λn ωn , with some λn > 0, n λn = 1, then ω¯ is an experiment in E(ψ, B). Let p be the support of ω, ¯ then Mω¯ pMψ and p = supn pn . It follows that Mω¯ is type I with discrete center and ωn (a) = ωn ( pa) for a ∈ Mψ . Moreover, since ωθ,n have the same support for all θ , ωθ,n ( pa) = ωθ,n (a). Suppose that En → E weakly and let ω := ωE . By the remarks before Lemma 3.15, the normal extensions of ωθ,n converge weakly to ωθ in Mψ∗ . It follows that ωθ ( pa) = ωθ (a) for all a ∈ Mψ , and we can conclude that ωθ,n (a) → ωθ (a), for all a ∈ Mω¯ and θ ∈ . In [10] it is shown that the preduals of the type I von Neumann algebras with atomic center have the Kadec-Klee property: any sequence of normal states ωn converging weakly to a normal state ω is also norm convergent limn ωn − ω = 0. We apply this to the experiments E˜n := (Mω¯ , ωn,θ : θ ∈ ) which by construction are equivalent with the original experiments En and we get limn ωn,θ − ωθ = 0 for all θ . Conversely, suppose that (En , E) → 0. By Lemma 3.15, there is a subsequence En k , converging weakly to some experiment F. By the first part of the proof, (En k , F) → 0. Since also (En k , E) → 0, we have (E, F) = 0 and by Lemma 3.12, this implies that F is equivalent with E, so that ωF = ω. It follows that the whole sequence converges weakly to E. Remark. Our result is complementary to the classical one in two respects. First, the range of covered experiments consists of type I algebras with discrete center, thus the “typical” noncommutative probability spaces. Second, the proof uses the Kadec-Klee property specific to this type of algebras and is not true for general probability spaces. 4. Quantum Central Limit Theorem We have seen that in classical statistics the Central Limit Theorem is an essential ingredient of the proof of local asymptotic normality in its weak version. In the quantum case the situation is similar, so we will proceed in this section to explain the quantum Central Limit Theorem in the simplest situation, that is for a matrix algebra M(Cd ) and a faithful state ϕ on M(Cd ), i.e. a state whose density matrix ρ is strictly positive. However the result holds in the general framework of C ∗ -algebras and we refer to the references [35,38] for more details and proofs. Let L 2 (ρ) = (M(Cd ), ·, ·ρ ) be the complex Hilbert space with inner product X, Y ρ = Tr(ρY ∗ X ),
X, Y ∈ M(Cd ).
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On M(Cd ) we define the symplectic form σ by σ (X, Y ) = Im(X, Y ρ ), and we construct the algebra CC R(M(Cd ), σ ) of canonical commutation relations having as generators the Weyl operators W (X ) for all X ∈ M(Cd ) and satisfying the relations W (X )W (Y ) = W (X + Y ) exp(−iσ (X, Y )). On this algebra we define the quasifree state φ by 1 φ(W (X )) = exp − α(X, X ) , 2 where α is the positive bilinear form α(X, Y ) = Re(X, Y ρ ). By the GNS construction, φ generates a representation of the CCR algebra and for now we denote by W (X ) the Weyl operators in this representation and occasionally express them in terms of the field operators W (X ) = exp(i B(X )). Note that any field operator B(X ) has a Gaussian distribution centered at 0 and with variance X 2ρ = α(X, X ). Consider the tensor product nk=1 M(Cd ) of algebras M(Cd ) which is generated by elements of the form X (k) = 1 ⊗ · · · ⊗ X ⊗ · · · ⊗ 1,
(1)
with X acting on the k th position of the tensor product. We are interested in the asymptotics as n → ∞ of the joint distribution under the state ϕ ⊗n , of ‘fluctuation’ elements of the form n 1 (k) Fn (X ) := √ X . n k=1
Theorem 4.1. Let A1 , . . . , As ∈ M(Cd )sa satisfying ϕ(Al ) = 0, for l = 1, . . . , s. Then we have the following: s s ⊗n lim ϕ Fn (Al ) = φ (B(Al )) , n→∞
lim ϕ ⊗n
n→∞
l=1
s
l=1
exp(i Fn (Al )) = φ
s
l=1
W (Al ) .
l=1
Note that only joint distributions for selfadjoint operators are considered. This is sufficient for the purpose of this paper and for the rest of this section we concentrate on the properties of the subalgebra CC R(M(Cd )sa , σ ) generated by the Weyl operators W (A) with A selfadjoint operator in M(Cd ). This subalgebra will be the key to understanding the limit quantum experiment. In the case of selfadjoint operators the symplectic form becomes σ (A, B) =
i Tr (ρ[A, B]). 2
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The bilinear form α is a positive inner product on M(Cd )sa and from now on we will denote its restriction to this subspace as (A, B)ρ := α(A, B) = Tr (ρ A ◦ B), and the corresponding real Hilbert space by L 2R (ρ) = (M(Cd )sa , (·, ·)ρ ). We write L 2R (ρ) as a direct sum of orthogonal subspaces Hρ ⊕ Hρ⊥ , where Hρ = A ∈ L 2R (ρ) : [A, ρ] = 0 .
In particular if B = B1 ⊕ B2 ∈ L 2R (ρ) then
1 1 φ(W (B)) = exp − (B1 , B1 )ρ exp − (B2 , B2 )ρ . 2 2
(2)
Moreover since σ (A, B) = 0 for A ∈ Hρ and B arbitrary we get the following factorization CC R(M(Cd )sa , σ ) ∼ = CC R(Hρ , σ ) ⊗ CC R(Hρ⊥ , σ ),
(3)
and by (2) the state φ factorizes as φ = φ1 ⊗ φ 2 .
(4)
The left side of the tensor product is a commutative algebra which is isomorphic to L ∞ R|Hρ | carrying a Gaussian state with covariance (A, B)ρ . 5. Local Asymptotic Normality for Quantum States We are now ready to introduce the central result of the paper which extends the concept of local asymptotic normality to the quantum domain and provides also an important example of convergence of quantum statistical experiments. Throughout this section we consider the algebra A = Md (C), a family of strictly positive density matrices ρθ in Md (C) such that the map θ → ρθ has the property that both the eigenvalues and eigenvectors of ρθ are twice continuously differentiable, and denote by ϕθ the corresponding faithful states on A. Consider n quantum systems prepared in the same state ϕθ with √ θ ∈ ⊂ Rm an unknown parameter which will be taken of the form θ = θ0 + u/ n, where u is an unknown parameter belonging to some open, bounded neighborhood of the origin I ⊂ Rm , and θ0 is a fixed and known parameter. We are interested in the asymptotic behavior as n → ∞ of the quantum statistical experiments En = An = A⊗n , ϕu,n = (ϕθ0 +u/√n )⊗n : u ∈ I , whose family of states is indexed by a parameter u ∈ I . Namely, we will show that the sequence En converges weakly to an experiment E, consisting of a family {φ u , u ∈ I } of quasifree states on the CCR algebra M(Cd )sa , σ with σ (A, B) = 2i Tr(ρθ0 [A, B]) (cf. Sect. 4).
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5.1. One parameter unitary family of states. We will first consider a simple model of a one-parameter family of states where the eigenvalues of the density matrices are fixed and only the eigenvectors vary smoothly. This will be helpful in the next section where the general multi-parameter case is considered and it is shown that the quantum local asymptotic normality can be obtained by combining the fixed eigenvalues situation with the classical problem of evaluating the eigenvalues of a density matrix for fixed eigenvectors. For simplicity we consider a local neighborhood around θ0 = 0. Let ρ = ρθ0 be a density matrix on A := M(Cd ) and define ρa = eia H ρe−ia H for a ∈ R, where H is a selfadjoint operator which can be chosen such that ϕ(H ) = 0. Denote by ϕa the corresponding state functionals ϕa (A) := Tr(ρa A). Consider now n quantum systems prepared in the same state ρu/√n , where u is an unknown parameter belonging to some bounded open interval I ⊂ R containing the origin. We are interested in the asymptotic behavior as n → ∞ of the quantum statistical experiments ⊗n√ : u ∈ I , En = (M(Cd ))⊗n , ρu/ n
(1)
whose family of states is indexed by a parameter u ∈ I . As explained in Sect. 2, the likelihood ratio process is a sufficient statistic in the case of classical statistical experiments, and the local asymptotic normality property means that this process converges in distribution to the corresponding likelihood process of the limit experiment. For a quantum experiment however, there is no obvious analogue of the likelihood ratio process. In Sect. 3 we argued that the guiding principle in finding the quantum analog of this process should be to look at operators which are intrinsically related to the quantum experiment in the sense that they generate the minimal sufficient algebra, similarly to the case of the likelihood ratio process. Such operators are the Connes cocycles which in the case of the experiment En are given by it (n) ⊗n√ ⊗n√ ⊗n ⊗n −it ρ , Dϕ ] := ρ . Cu,t = [Dϕu/ t n u/ n We can rewrite this as (n) Cu,t =
√ √ ⊗n it −it ρ ⊗n eiu H/ n ρe−iu H/ n = ⊗n √ √ ⊗n ρ −it eiu H/ n ρ it e−iu H/ n = ⊗n √ √ = eiu H/ n ρ it e−iu H/ n ρ −it √ √ ⊗n eiu H/ n e−iuσt (H )/ n = ⎞ ⎛ n n −iu iu H (k) exp ⎝ √ σt (H )( p) ⎠, exp √ n n k=1
p=1
where σt (H ) := ρ it Hρ −it is the action of the modular group of ϕ on H , and H (k) represents the operator 1 ⊗ · · · ⊗ H ⊗ · · · ⊗ 1 with H acting on the k th term of the tensor product.
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Consider now the expectation values of products of such cocycles with respect to the state ϕ ⊗n : # s $ (n) E (n) (u 1 , t1 , . . . , u s , ts ) := ϕ ⊗n Cul ,tl l=1
# = Tr ρ
⊗n
s
$ exp (iu l Fn (H )) exp −iu l Fn (σtl (H )) .
l=1
We apply now the second part of the central limit Theorem 4.1 to obtain s (n) lim E (u 1 , t1 , . . . , u s , ts ) = φ W (u l H ) W −u l σtl (H ) , n→∞
l=1
where φ is the quasifree state on the algebra CC R(M(Cd )sa , σ ) with symplectic form i Tr(ρ[A, B]). 2 The state φ is defined by φ (W (X )) = exp − 21 α(X, X ) , where α is the real symmetric positive bilinear form α(A, B) = Tr (ρ A ◦ B), where A ◦ B = AB + B A/2. By using the Weyl relations we get s # $ 2 u l lim E (n) (u 1 , t1 , . . . , u s , ts ) = φ ϕ ([H, σt (H )]) . W (u l (H − σt (H ))) exp n→∞ 2 σ (A, B) :=
l=1
In analogy to the classical local asymptotic normality, we would like to interpret the expression on the right side as the expectation of a product of cocycles of the form [Dφ u , Dφ 0 ]t for some family of states {φ u : u ∈ I } with φ 0 = φ, on W := CC R(M(Cd )sa , σ ). Later on we will restrict our attention to the minimal sufficient subalgebra which is generated by the Connes cocycles [40] and still have a statistically equivalent quantum experiment. Let us define the family of translated states on W, φ u (W (A)) = φ (W (u H )W (A)W (−u H )),
A ∈ M(Cd )sa .
The cocycles can be calculated (see e.g. p. 160 of [35]): 2 u u 0 ϕ ([H, σt (H )]) . [Dφ , Dφ ]t = W (u(H − σt (H ))) exp 2
(2)
Thus we obtain the convergence in distribution of the Connes cocycles s s ⊗n lim ϕ ⊗n Dϕu /√n , Dϕ ⊗n Dφ ul , Dφ 0 =φ . n→∞
l
l=1
tl
l=1
tl
Notice that [Dφ u , Dφ 0 ]t do not commute for different times as in general ϕ([H −σt (A), H − σs (H )]) = 0. This implies that the minimal sufficient algebra W0 ⊂ W is noncommutative and is generated by the Weyl operators W (A) with A ∈ K := LinR (H − σt (H ) : t ∈ R). We denote by E the limit experiment in its minimal form E = W0 , φ u : u ∈ I . (3)
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Theorem 5.1. As n → ∞ we have En → E, in the sense of weak convergence of experiments, where En is the sequence defined in (1) and E is the quantum Gaussian shift experiment defined in (3). We will take now a closer look at the limit experiment and in particular at the optimal measurement for estimating the unknown parameter u ∈ I . It is known [24] that asymptotically the optimal procedure for En is to measure the symmetric logarithmic derivative L at the point θ0 = 0 on each of the individual systems separately. As we will see, the optimal procedure for the limit experiment is to measure the corresponding observable B(L) and obtain a classical experiment with Fisher information equal to the quantum Fisher information of E (see also [20]). Let A be an arbitrary element of K . When restricted to the commutative algebra generated by the field B(A), the states φ u give rise to a family of displaced Gaussian distributions on R, PAu := N (−iuϕ([H, A]), ϕ(A2 )). Indeed the expected value of B(A) is φ u (B(A)) = φ(W (u H )B(A)W (−u H )) = φ(B(A) + 2uσ (H, A)1) = −iuϕ([H, A]), and the variance is φ(B(A)2 ) = α(A, A) = ϕ(A2 ). It can be shown that for a Gaussian shift family (N (au, v), u ∈ I ) the Fisher information is given by I = a 2 /v 2 , thus in our case we have I A = ϕ([H, A])2 /ϕ(A2 ).
(4)
Coming back to the original quantum experiment (M(Cd ), ϕa : a ∈ R) we define the symmetric logarithmic derivative at θ0 = 0 by % dρa %% = i[H, ρ]. (5) L◦ρ = da %a=0 Thus iφ([A, H ]) = iTr(ρ[A, H ]) = iTr(A[H, ρ]) = Tr(ρ A ◦ L) = (A, L)ρ , and by inserting into (4) we get I A = | (A, L)ρ |2 / A 2 , which takes its maximum value for A = L. Thus sup I A = IL = Tr(ρL2 ), A
where the last expression is the quantum Fisher information H (ρ) [24]. We will show now that L belongs to the subspace K , so that its corresponding d field belongs to the minimal sufficient algebra W0 . Let ρ = i=1 λi Pi be the spectral decomposition of ρ, then the symmetric logarithmic derivative can be written as ei , Le j = 2i
λi − λ j ei , H e j . λi + λ j
(6)
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By derivating H − σt (H ) with respect to t we obtain that the multiple commutators Cr := [. . . [H, log ρ], . . . , log ρ] belongs to K for any number r of commutators. It is easy to see that ei , Cr e j = ei , H e j (log(λ j /λi ))r , and by writing (6) in the form 1 − elog(λ j /λi ) , 1 + elog(λ j /λi ) we see that L belongs to the linear span of Cr for r ≥ 1 and thus L ∈ K . In conclusion there exists a measurement on the limit experiment such that the Fisher information of the measurement results achieves the upper bound given by the quantum Fisher information. This suggests that the classical statistical experiment ei , Le j = 2iei , H e j
F = (R, PLu : u ∈ I ), ‘contains all the information’ about the asymptotics of the sequence En . We will show that this is not true in the sense that F is not equivalent to E. Indeed if that was the case there would exist a linear positive map S from L 1 (R) to W0∗ , the space of normal functionals on W0 such that u ∈ I. S : PLu → φ u , But S is completely positive and thus E and F can be obtained from each other by quantum randomizations which is impossible as their minimal sufficient subalgebras cannot be isomorphic [40]. In particular this means that there exists a classical statistical decision problem for which the minimax risk of the experiment E is strictly smaller than the minimax risk of the experiment F. An example of such a decision problem [14] is that of distinguishing between two states φ u and φ −u with u = 0 for which the optimal measurement is different from the measurement of L. 5.2. Local asymptotic normality: general case. We pass now the the general case of an m dimensional family of states as described in the beginning of Sect. 5. The main ingredients of the proof are the quantum central limit theorem and the following form of the law of large numbers [35,38]: Let B be the infinite tensor product of copies of A and let ψ be the product state ψ = ϕ ⊗ϕ ⊗· · · Each element a ∈ An can be identified with the element a ⊗ I ⊗ I ⊗. . . in B. For a ∈ A, we denote Sn (a) :=
n 1 (k) a ∈ An n k=0
a (k)
with as in Eq. (1), and similarly for any element b ∈ An , we denote the k-places translated b(k) := 1 ⊗ · · · ⊗ 1 ⊗ b ⊗ 1 ⊗ · · · ∈ B . Let us consider the GNS representation of B with respect to ψ on a Hilbert space H with cyclic vector . We define the contraction V : H → H by V b = b(1) , for b ∈ An . Then we have n 1 k V a = lim Sn (a) = ϕ(a), lim n n n k=0
for all a ∈ A. As a consequence, we get the following lemma.
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Lemma 5.2. Let an , a be selfadjoint elements in A, such that an → a, and let ρ˜n ∈ A∗ be density matrices such that ρ˜n → ρ. Let u n , vn ∈ A be unitaries such that u n → 1 and vn → 1. With the notation wn,t = exp{it (log ρ˜n +
1 an )}ρ˜n−it , n
t ∈ R,
we have ⊗n ⊗n ⊗n ⊗n ⊗n lim ϕ ⊗n (u ⊗n n wn,t vn ) = exp{itϕ(a)} lim ϕ (u n vn ).
n→∞
n→∞
Proof. We will use the Dyson expansion [35] t ∞ −it k = i ds1 . . . exp{it (log D + b)}D 0
k=0
where
σsD (b)
=
D is bD −is .
0
sk−1
dsk σsDk (b) . . . σsD1 (b),
Let us denote bn = an − ϕ(a). We get
⊗n ⊗n ϕ ⊗n (u ⊗n n wn,t vn ) = ⊗n ⊗n −it ⊗n = exp{itϕ(a)}ϕ ⊗n u ⊗n vn n exp{it (log(ρ˜n ) + Sn (bn ))}(ρ˜n ) ⊗n ⊗n ⊗n exp{itϕ(a)}[ϕ u n vn + sk−1 ∞ t ⊗n ⊗n k i ds1 . . . dsk Sn (σsρ˜kn (bn )) . . . Sn (σsρ˜1n (bn ))vn⊗n )]. ϕ (u n k=1
0
0
The term in the last line can be rewritten as xk−1 t ∞ s1 ds1 d x1 . . . d xk i k 0
k=0 0
0
' t × Sn (σxρ˜1n (bn )) . . . Sn (σxρ˜kn (bn ))(u ∗n )⊗n , i Sn (σsρ˜1n (bn ))vn⊗n = ds1 0 & ' × (vn∗ ρ˜n it )⊗n exp{−is1 (log(ρ˜n⊗n ) + Sn (bn ))}(u ∗n )⊗n , i Sn (vn∗ σsρ˜1n (bn )vn ) . &
ρ˜
ϕ
ϕ
The sequence vn∗ σs n (bn )vn converges to σs (a − ϕ(a)) in norm and Sn (σs (a − ϕ(a))) converges to 0, by the weak law of large numbers. Moreover, for all n and s we have ρ˜ Sn (vn∗ σs n (bn )vn ) ≤ bn and bn is bounded. So the last term goes to 0 as n → ∞, by the dominated convergence theorem. Let us now return to the family {ρθ : θ ∈ }, and consider the spectral decomposition ρθ := j λ j,θ P j,θ . By the differentiability of the map θ → ρθ there exist self-adjoint matrices H j,θ ∈ A, such that ∂ P j,θ = i[Hk,θ , P j,θ ] ∂θk
θ ∈ , j = 1, . . . , d, k = 1 . . . , m.
(7)
We fixa point θ0 ∈ and make the notations ρ = ρθ0 , P j = P j,θ0 , Hk = Hk,θ0 , and τθ = j λ j,θ P j . For a smooth function f : R → R, we have % % % % ∂ ∂ % f (ρθ )% = f (τθ )%% + i[Hk , f (ρ)], k = 1, . . . , m. ∂θk ∂θ k θ=θ0 θ=θ0
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The first term commutes with ρ and the second term satisfies Tr a[Hk , f (ρ)] = 0, whenever [a, ρ] = 0. We may suppose that ϕ(Hk ) = 0 for all k. ⊗n ⊗n We will deal with expressions of the form ϕ ⊗n (vn,1 . . . vn,k ), where −it j , θ0 + √1n u j
it j ρ −it j , θ0 + √1n u j
vn, j = ρ
or vn, j = ρ it j ρ
t j ∈ R, u j ∈ I.
(8)
We will first show that the original family of states can be replaced by a simpler one without changing the asymptotics. Lemma 5.3. Let
ρ˜a = exp i
ak Hk τθ0 +a exp −i
k
ak Hk , a ∈ I,
k
⊗n√ : u ∈ I ). Then limn ωEn (g) = limn ωE˜n (g), for all and let E˜n = (An , ρ˜n,u := ρ˜u/ n g ∈ G.
Proof. Let us denote v˜n, j the expression obtained from vn, j by replacing ρθ0 +u j /√n by ρ˜u j /√n , j = 1, . . . , k. We have to show that ⊗n ⊗n ⊗n ⊗n . . . vn,k ) = lim ϕ ⊗n (v˜n,1 . . . v˜n,k ). lim ϕ ⊗n (vn,1 n
n
Let ρn = ρn,0 = ρ˜n,0 . Then it −it it ρn−it = exp{it (log ρ˜u/√n + log ρθ0 +u/√n − log ρ˜u/√n )}⊗n ρ˜n,u ρ˜n,u ρn−it . ρn,u
By considering the Taylor expansion of the functions s → log ρ˜su/√n and s → log ρθ0 +su/√n , we get % % 2 2 % % 1 d d log ρθ0 + √su %% − 2 log ρ˜ √su %% log ρθ0 +u/√n − log ρ˜u/√n = n n 2 ds 2 ds s=sn s=sn with sn , sn ∈ [0, 1] and it can be shown by some computation that the last expression is equal to n1 an , where an converges in norm to 1 a = − ρ −1 [[ρ, H (u)], H (u)], H (u) = u k Hk , 2 k
satisfying ϕ(a) = 0, where we have used the fact that the states ϕθ are faithful and thus ρθ is invertible. The statement can be now proved by a repeated use of Lemma 5.2. We introduce the following notations: ∂ u k lk = uk log τθ |θ=θ0 , l(u) := ∂θk k
h(u) :=
u k ul
k,l
(u) :=
k ∂2
∂θk ∂θl
log τθ |θ=θ0 ,
u k k , k ◦ ρ = i[Hk , ρ],
k
L(u) :=
k
(9)
u k Lk , Lk ◦ ρ =
∂ ρθ |θ=θ0 . ∂θk
(10)
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Note that lk are the logarithmic derivatives in θ0 of the commutative family of states τθ . Similarly k is the symmetric logarithmic derivative of the unitary family obtained by rotating ρ with the unitary. The sum Lk = lk + k is the symmetric logarithmic derivative at θ0 of the original family ρθ . We notice further that ϕ(H (u)) = ϕ(l(u)) = ϕ( (u)) = 0 and −ϕ(h(u)) = ϕ(l(u)2 ) is the Fisher information of the family s → τθ0 +su , at s = 0. We compute now the Connes cocycles for the family ρ˜n,u : ⊗n i i it ρ˜n−it = exp √ H (u) τθit + √1 u exp − √ H (u) (ρ −it )⊗n ρ˜n,u 0 n n n ⊗n ⊗n ⊗n i i it −it it −it τθ + √1 u ρ ρ exp − √ H (u) ρ = exp √ H (u) 0 n n n ⊗n ⊗n ⊗n i i ϕ = exp √ H (u) exp it (log τθ0 +u/√n − log ρ) exp − √ σt (H (u)) . n n Note that τθ0 = ρ and all the elements τθ are mutually commuting. Using again Taylor expansion up to the second order, we get it ρ˜n−it ρ˜n,u
⊗n ⊗n it i ϕ it bn exp √ l(u) + exp − √ σt (H (u)) 2n n n it ϕ = exp (i Fn (H (u))) exp (it (Fn (l(u))) exp Sn (bn ) exp −i Fn (σt (H (u))) , 2
i = exp √ H (u) n
⊗n
where bn =
u k ul
k,l
∂2 log τθ |θ=θn , ∂θk ∂θl
1 θn − θ0 ≤ √ u . n
By continuity of the second derivatives, {bn } converges to h(u) in norm. By the quantum Central Limit Theorem and Lemma 5.2, we can now conclude that the family of cocycles it ρ˜ −it converges to of the modified states ρ˜n,u n it ϕ ϕ(h(u)) W (H (u))W (tl(u))W (−σt (H (u))), := exp 2
Vu,t
where W (A) are the Weyl operators. The convergence holds as usual in the weak sense: for any u 1 , . . . u k ∈ I and t1 , . . . tk ∈ R, ⊗n ⊗n . . . v˜n,k ) = φ(V1 . . . Vk ), lim ϕ ⊗n (v˜n,1 n
where V j is shorthand notation for Vu j ,t j or Vu∗j ,t j , according to (8). In combination with Lemma 5.3 this gives ⊗n ⊗n lim ϕ ⊗n (vn,1 . . . vn,k ) = φ(V1 . . . Vk ). n
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It remains now to identify Vu,t as Connes cocycles of the limit experiment, Vu,t = [Dφ u , Dφ 0 ]t , where φ u are states on the algebra CC R(M(Cd )sa , σ ). Using the fact that [B(l(u)), B(A)] = 0 for any A ∈ M(Cd )sa we can decompose Vu,t into a product 1 Vu,t = W (H (u) − σt (H (u)))) exp ϕ ([H (u), σt (H (u))]) 2 it ×W (tl(u)) exp ϕ (h(u)) , 2
(11)
where the first term is exactly the cocycle appearing in (2) for the unitary family of states and the second term is the ‘classical cocycle’ due to the change in the eigenvalues of the density matrix. We will show that indeed the product of cocycles can be accounted for by a product of transformations such that [Dφ u , Dφ]t = [D (φ ◦ R(u) ◦ L(u)), D (φ ◦ R(u))]t [D (φ ◦ R(u)) , Dφ]t . The inner automorphism R(u) of CC R(M(Cd )sa , σ ) is the ‘translation’ with momentum B(H ), R(u) : W (A) → W (H (u))W (A)W (−H (u)) = W (A) exp{i(A, (u))ρ },
(12)
just like in the unitary case (see Eq. (10)). The transformation L(u) is an outer automorphism of CC R(M(Cd )sa , σ ), i.e. whose generator is a field which does not belong to the algebra as it corresponds to a non-selfadjoint operator, L(u) : W (A) → W (−il(u)/2)W (A)W (il(u)/2) ( ) = W (A) exp i(A, l(u))ρ .
(13)
Using the factorization (3) of CC R(M(Cd )sa , σ ) and the definitions of L(u) and R(u) we get the following picture of the action the product L(u) ◦ R(u): L(u) ◦ R(u) : W (B1 ) ⊗ W (B2 ) → L(u)(W (B1 )) ⊗ R(u)(W (B2 )). Moreover, from (4) we obtain that the state φ u factorizes as well, φ u = φ1 ◦ L(u) ⊗ φ2 ◦ R(u) := φ1u ⊗ φ2u . It is now easy to see that the cocycles for this family of states have the expression (11) and the states φ u are given by [20] # $ ∂ρ θ 1 ui φ (W (A)) = exp − (A, A)ρ exp iTr A 2 ∂θi i 1 = exp − (A, A)ρ + i(A, L(u))ρ . 2 u
(14)
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Theorem 5.4. The sequence
En := M(Cd )⊗n , ϕθ⊗n+u/√n : u ∈ I , 0
of quantum statistical experiments converges weakly as n → ∞ to the limit experiment E := CC R(M(Cd )sa , σ ), φ u : u ∈ I . The latter is a tensor product between a classical Gaussian shift experiment corresponding to the change in the eigenvalues of ρθ , and a non-commutative one corresponding to the rotation of the eigenbasis of ρθ . On the algebraic level we have the isomorphism CC R M(C d )sa , σ ∼ = CC R Hρ ⊗ CC R Hρ⊥ , as described in Sect. 4. With respect to this isomorphism the state φ u given by (14), factorizes as φ u = φ1u ⊗ φ2u = φ1 ◦ L(u) ⊗ φ2 ◦ R(u), with automorphisms R(u), L(u) defined in (12) and (13) respectively. In the reminder of this section we will identify the minimal sufficient algebra W0 ⊂ CC R(M(Cd )sa , σ ) of the experiment E. We know that the Connes cocycles generate the minimal sufficient algebra, and from the expression (11) we get that W0 = CC R(K ), where K is the real linear space, ) ( ϕ K := LinR H (u) − σt (H (u)) + tl(u) : t ∈ R, u ∈ I . By taking derivatives with respect to t and using Eqs. (7) and (9) we get that K is the linear span of the orbits of the logarithmic derivatives log ρk := ∂ log ρθ /∂θk |θ=θ0 under ϕ the modular group σt . Lemma 5.5. The minimal sufficient algebra of the experiment E is given by W0 = CC R(K , σ ) with ( ) ϕ K = LinR {l(u) : u ∈ I } ⊕ LinR H (u) − σt (H (u)) : u ∈ I, t ∈ R . In particular Lk ∈ K and lk ∈ K . Proof. We have Lk ◦ ρ :=
% % ∂λi,θ % ∂ρθ %% % = Pi + iλi [Hk , Pi ], % % ∂θk θ=θ0 ∂θk θ=θ0 i
i
which on the matrix elements becomes % ∂ log λiθ %% ei , Lk e j = δi j % ∂θk %
+ 2i
θ=θ0
1 − elog(λ j /λi ) ei , Hk e j . 1 + elog(λ j /λi )
The logarithmic derivative log ρk is in K and has matrix elements % ∂ log λiθ %% + i log(λ j /λi )ei , Hk e j , ei , log ρk e j = δi j % ∂θk % θ=θ0
(15)
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and by derivating σt log ρk we get that the multiple commutators Cr = [. . . [log ρk , log ρ], . . . , log ρ], are also in K and have the expression ei , Cr e j = i log(λ j /λi )r +1 ei , Hk e j for r ≥ 1. By comparing (15) with the last two equations we conclude that Lk ∈ K for all k = 1, . . . , m, and additionally that l(u) ∈ K for all u ∈ I . Indeed, there exist a finite number of real coefficients {ar , r = 0 . . . } such that d(d−1)
ar log(λ j /λi )r = 0, ∀1 ≤ i = j ≤ d,
r =0
and a0 = 0. With such coefficients we have a0 log ρk +
d(d−1)
ar Cr = a0 lk ∈ K .
r =1
In conclusion K is the linear span of the vectors lk ∈ Hρ and the vectors Hk − σt (Hk ) ∈ Hρ⊥ as desired. Another interesting feature of the minimal sufficient algebra W0 is that apart from the standard symmetric logarithmic derivatives Lk , it contains a broad set of quantum versions of the logarithmic derivative which were investigated in [39] and are defined as follows: LkF = J F ρk , where ρk = ∂ρθ /∂θk |θ=θ0 and J F is an operator on matrices defined as J F = [F(L R −1 )]−1 R −1 . Here L and R are the left and respectively right multiplication by ρ, and F : R+ → R is an operator-monotone function satisfying F(t) = t F(t −1 ) for t > 0 and F(1) = 1. This function is required to satisfy the physical admissibility condition that the associated quantum Fisher information Ikp := Trρk L Fp is monotone under coarse-grainings. Two well-known examples of a quantum score are the symmetric logarithmic derivative Lk , for which F(t) = (1 + t)/2, and the Bogoljubov-Kubo-Mori logarithmic derivative t−1 LkB M K := log ρk for which F(t) = log(t) , and as we have seen they both belong to the subspace K . Lemma 5.6. For any admissible function F the logarithmic derivative LkF belongs to K . Proof. First, we see that ei , LkF e j
= δi j
% ∂ log λiθ %% % ∂θk %
+i θ=θ0
(1 − λi /λ j ) ei , Hk e j . F(λi /λ j )
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Furthermore, for each F we have the integral representation [31] 1−t 1−t = (1 + s)µ(ds), F(t) [0,∞] s + t where µ(s) is a positive finite measure on [0, ∞]. Therefore, % θ% ∂ log λ (1 − λi /λ j ) i % F ei , Hk e j (1 + s)µ(ds), +i ei , Lk e j = δi j % % ∂θk [0,∞] (s + λi /λ j ) θ=θ0
and LkF ∈ K is proved similarly as for Lk .
6. Application to Qubit States In this section we apply the local asymptotic normality results to the simplest situation of a family of qubit states. In Theorem 1.1 of [14] it is shown that in this case local asymptotic normality holds in the strong sense of Definition 3.10. An arbitrary density matrix in M(C2 ) can be written as ρ=
→ → 1+− r − σ , 2
→ → → where − r = (r x , r y , r z ) ∈ R3 is a vector satisfying |− r | ≤ 1, and − σ = (σx , σ y , σz ) are the Pauli matrices. Due to the rotation symmetry, we may choose ρ0 = 1+r2σz correspon→ ding to − r0 = (0, 0, r ) for some fixed r ∈ (0, 1). All the states in a neighborhood of ρ0 can be obtained by a combination of a translation in the radial direction, and a rotation → around an axis in the x-y plane. Thus we can use the local coordinates − u = (r x , r y , a) − → around r0 such that → ρ− u =
→ → → u )− σ 1 + (− r0 + − . 2
Notice that only the coordinate a contributes to the classical part of the experiment calculated and the functions l and h defined in Sect. 5.2 are % % → ∂ log ρ− 1 1 u % la = P+ − P− , = ∂a %u=0 1+r 1−r % % → ∂ 2 log ρ− 1 1 u % =− P+ + P− , h aa = ∂a 2 % (1 + r )2 (1 − r )2 u=0
where P± are the eigenprojectors of ρ0 , and the components corresponding to other derivatives are equal to 0. With the notations defined in Sect. 4, we construct the real Hilbert space L 2R (ρ0 ) with inner product (A, B)ρ0 = Tr (ρ0 A ◦ B),
A, B ∈ M(C2 )sa ,
with respect to which we have the orthogonal decomposition L 2R (ρ0 ) = Hρ0 ⊕ Hρ⊥0 = Lin{1, σz } ⊕ Lin{σx , σ y }.
(1)
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Next, we use the symplectic form σ (A, B) = 2i Tr (ρ0 [A, B]) , to construct the algebra CC R(M(C2 )sa , σ ). We obtain that B(σz ) and B(1) commute with all the other fields and B(σ y ), B(σx ) satisfy the canonical commutation relations [B(σ y ), B(σx )] = 2ir 1. By rescaling we get the usual √ √ quantum oscillator relations [Q, P] = i1 with Q = B(σ y )/ 2r and P = B(σx )/ 2r . Thus CC R(M(C2 )sa , σ ) ∼ = CC R(Lin{1, σz }) ⊗ Alg(Q, P), where the left side of the tensor product is itself a commutative algebra which is naturally isomorphic to L ∞ (R2 ), and the right side is the algebra of a quantum harmonic oscillator with variables Q and P. On this algebra we have a state φ 0 given by φ 0 (W (A)) = exp − 21 Tr(ρ0 A2 ) , which due to (1) splits into a tensor product φ 0 = φ10 ⊗ φ20 . In Sect. 5.2 we have shown that the minimal sufficient algebra of the limit experiment is generated by the fields corresponding to a real linear subspace K ⊂ M(C2 )sa which in this case is K = Rla ⊕ Lin{σx , σ y }. Then the minimal sufficient algebra is of the form CC R(K , σ ) ∼ = L ∞ (R) ⊗ Alg(Q, P) and the family of states defining the limit experiment is r ,r y
φ u = N (Ic a, Ic ) ⊗ φ2x
.
Let us explain the meaning of the right side: Ic = Tr(ρ0 la2 ) = −Tr(ρ0 h aa ) =
1 1 − r2 r ,r y
is the Fisher information corresponding to the parameter a. The state φ2x quantum oscillator can be described through its Wigner function [29] W r x r y (q, p) = exp −r (q − qx )2 + ( p − p y )2 ,
of the
which to a displaced thermal equilibrium state with center (qx , q y ) = √ corresponds √ (r x / 2r , r y / 2r ). 7. Concluding Remarks In this paper we have made a further step in the development of a theory of quantum statistical experiments started by Petz. We believe that the notions which we have introduced are the proper analogues of the classical concepts: weak and strong convergence of experiments, canonical state of an experiment, local asymptotic normality. However the theory is far from complete and the following is a short list of open problems and topics for future work. 1. Extend the theory of statistical experiments to the case of non-faithful states. We expect that the extended space of experiments will be compact under the weak topology.
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2. One of the crucial aspects of the theory is the relation between strong and weak convergence of experiments for finite parameter sets. In Theorem 3.16 we have touched upon this by showing that the two notions are equivalent when the experiments are uniformly dominated and the corresponding algebras are of type I with discrete center. We believe that the same result holds for a much larger class of experiments, where one would have to consider non-trivial channels in order to achieve the convergence in Le Cam sense. One possibility, perhaps too ambitious, would be to construct a quantum version of the Skorohod almost sure representation theorem [41]. Another strategy could be to approximate the quantum experiments by finite dimensional ones, similarly to the treatment of nuclear C∗ -algebras [48]. 3. The work on the previous issue might be simplified by finding alternative characterizations of weak convergence in terms of quantum Radon-Nikodym derivatives. 4. Derive local asymptotic normality under weaker smoothness conditions for the family of states, similar to the differentiability in quadratic mean from the classical set-up [50]. Going beyond the finite parameter, i.i.d. case – which classically is rather standard – remains a challenge for the quantum theory. 5. Develop a quantum statistical decision theory for quantum experiments. This will connect the abstract framework to concrete statistical problems such as estimation and testing. Acknowledgements. We thank Richard Gill, Aad van der Vaart, Denes Petz and Jonas Kahn for fruitful discussions. M˘ad˘alin Gu¸ta˘ acknowledges the financial support received from the Netherlands Organisation for Scientific Research (NWO). Anna Jenˇcová was supported by the Center of Excellence SAS Physics of Information I/2/2005, Science and Technology Assistance Agency under the contract No. APVT-51-032002 and the EU Research Training Network Quantum Probability with Applications to Physics, Information Theory and Biology.
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Commun. Math. Phys. 276, 381–395 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0331-2
Communications in
Mathematical Physics
Scarring on Invariant Manifolds for Perturbed Quantized Hyperbolic Toral Automorphisms Dubi Kelmer Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected] Received: 25 July 2006 / Accepted: 20 April 2007 Published online: 4 September 2007 – © Springer-Verlag 2007
Abstract: We exhibit scarring for the quantization of certain nonlinear ergodic maps on the torus. We consider perturbations of hyperbolic toral automorphisms preserving certain co-isotropic submanifolds. The classical dynamics is ergodic, hence, in the semiclassical limit almost all quantum eigenstates converge to the volume measure of the torus. Nevertheless, we show that for each of the invariant submanifolds, there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold. 1. Introduction A significant problem in quantum chaos is to understand the behavior of eigenstates of classically chaotic systems in the semiclassical limit. In particular, one would like to classify the possible measures on phase space obtained as a quantum limit (e.g., in terms of the corresponding Wigner distributions). The main result in this direction is the Šhnirel’man theorem (also referred to as the Quantum Ergodicity Theorem). This theorem states that for classically ergodic systems almost all sequences of eigenstates converge to the volume measure on the corresponding energy shell [4, 5, 15, 17]. Going beyond the Šhnirel’man theorem, one would like to classify what other possible invariant measures (if any) can be obtained as a quantum limit. For surfaces of constant negative curvature, the Quantum Unique Ergodicity conjecture suggests that the only possible limiting measure is the volume measure [14]. This conjecture has been proved for the case of compact arithmetic surfaces, if one takes into account the arithmetic symmetries of the system [7]. A similar situation also occurs for quantized linear symplectic maps of the two dimensional torus. Here again, the quantized system exhibits arithmetic symmetries. After taking these symmetries into account, the only possible limiting measure is shown to be the volume measure [12]. On the other hand, when considering eigenstates of the propagator without the symmetries, one can construct a thin sequence of eigenstates partially localized on periodic orbits [8].
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D. Kelmer
One can consider also quantized linear symplectic maps of higher dimensional tori. For such a system, if the classical map leaves no invariant rational isotropic subspaces, the only limiting measure (after taking the arithmetic symmetries into account) is again the volume measure on the whole torus [10]. However, if there are rational isotropic invariant subspaces, then there are sequences of quantum states localized on corresponding co-isotropic invariant manifolds. Furthermore, this phenomenon is stable under the arithmetic symmetries of the system [10]. In these notes we show that this localization is also stable under certain nonlinear perturbations, which preserve these co-isotropic manifolds. We briefly review the basic setup: Consider a discrete time dynamical system given by iterating the action of a symplectic linear map A ∈ Sp(2d, Z) on T2d = R2d /Z2d . Assume that A has no eigenvalues of modulus 1, so that the map is Anosov and hence stably ergodic. Further assume that the action of A on Q2d leaves an invariant isotropic subspace V ⊆ Q2d of dimension 1 < d0 ≤ d. Denote by = V ∩ Z2d the integral points of V . For every n ∈ there is a corresponding character of the torus en (x) = exp(2πin · x). For every point ξ ∈ T2d which is fixed by A the manifold X ξ = x ∈ T2d |en (x) = en (ξ ), ∀n ∈ , is a closed co-isotropic submanifold that is invariant under the action of A. Let φ H be a Hamiltonian map on T2d which leaves all the manifolds X ξ invariant, and look at a perturbation = A ◦ φ H (also leaving the X ξ ’s invariant). Furthermore, we can make the perturbation sufficiently small so that the perturbed map remains ergodic. The quantization procedure for maps on the torus is described in Sect. 2. For this procedure, the admissible values of Planck’s constant are inverses of integers h = 1/N , and the space of quantum states is then H N = L 2 [(Z/N Z)d ]. The semiclassical limit is achieved by taking N → ∞. For f ∈ C ∞ (T2d ) a smooth observable, we denote by Op N ( f ) : H N → H N its quantization. Any quantum state ψ ∈ H N can then be interpreted as a distribution on T2d via the Wigner distribution W N (ψ) sending any smooth f to its expectation value Op N ( f )ψ, ψ . The quantization of the map is a family of unitary operators U N () acting on H N , satisfying the Egorov identity in the semiclassical limit. A measure µ on T2d is called a limiting quantum measure, if there is a sequence ψ = ψ (N ) ∈ H N of eigenstates of U N (), such that as N → ∞ the corresponding Wigner distributions W N (ψ) converge (weak∗ ) to µ. We can now state the main theorems, establishing that the volume measures of the manifolds X ξ are all limiting quantum measures. For each fixed point ξ and any integer N divisible by R = det(A − I ), we define H N ,ξ = ψ ∈ H N | Op N (en )ψ = en (ξ )ψ, ∀n ∈ . Theorem 1. The spaces H N ,ξ are of dimension N d−d0 and are invariant under U N (). Furthermore, if ψ j ∈ H N j ,ξ is a sequence of states such that the Wigner distributions converge to some measure µ on T2d , then µ is supported on X ξ . The spaces H N ,ξ are invariant under U N (), and hence have a basis composed of eigenstates. We want to show that there is a sequence of such eigenstates converging to the volume measure of X ξ . In fact, we show that there are many such sequences.
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Theorem 2. For each N (divisible by R) take an orthonormal basis ψi = ψi of H N ,ξ composed of eigenstates of U N (). Then there are subsets S N ⊂ {1, . . . , N d−d0 } N| satisfying lim N →∞ N|Sd−d = 1, such that for any sequence {ψi N } with i N ∈ S N the 0 corresponding Wigner distributions W N (ψi N ), converge to the volume measure dm X ξ concentrated on X ξ . Remark 1.1. In the case that the fixed point ξ = 0, the requirement that N is divisible by R is not necessary, and Theorems 1,2 hold for any sequence of N → ∞. 2. Background: Quantum Maps on the Torus The full details for the quantization of the cat map and its perturbations on T2 can be found in [4, 6, 9, 11]. The generalization of these procedures for higher dimensions are analogous and are accounted for in [2, 3, 10, 13]. We give a short review of these quantization procedures.
2.1. Classical dynamics. We consider a discrete time dynamical system given by the iteration of a symplectic map : T2d → T2d . More precisely, we consider a map = A ◦ φ H that is a composition of a linear symplectic map A ∈ Sp(2d, Z) acting on T2d and a Hamiltonian map φ H (i.e., the evaluation at time one of some Hamiltonian flow). The action of A ∈ Sp(2d, Z) on the torus T2d = R2d /Z2d is induced by the p natural left action of A on the linear space R2d , that is x = ( ) ∈ T2d → Ax q (this is well defined since A preserves Z2d ). The induced action is invertible and area preserving. Furthermore, if A has no eigenvalues that are roots of unity then the induced dynamics are ergodic and mixing. If in addition there are no eigenvalues of modulus 1, then the dynamics is of Anosov type and in particular it is stably ergodic (see [1, 16]). For H ∈ C ∞ (T2d ) a smooth real valued function on T2d , the Hamiltonian flow t φ H : T2d → T2d satisfies the differential equations d f ◦ φ tH = {H, f } ◦ φ tH = {H, f ◦ φ tH }, dt for any smooth f ∈ C ∞ (T2d ), where { f, g} = φ 1H
∂ f ∂g j ( ∂ p j ∂q j
−
∂ f ∂g ∂q j ∂ p j
) is the Poisson
is the evaluation at time t = 1 of this flow. Consequently, brackets. The map φ H = the dynamics induced from the map φ H on T2d is just the evaluation of the corresponding Hamiltonian flow at integral times. As long as the perturbation is sufficiently small, the perturbed map remains an Anosov diffeomorphism. In particular, after the perturbation the dynamics is again ergodic and mixing. 2.2. Quantum mechanics on the torus. For doing quantum mechanics on the torus T2d , the admissible values for Planck’s constant are inverse of integers h = 1/N . The Hilbert
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space of states is finite dimensional of dimension N d . It is convenient to think of it as a space of functions H N = L 2 [(Z/N Z)d ], with inner product given by: ϕ, ψ =
1 Nd
ϕ(Q)ψ(Q).
Q∈(Z/N Z)d
To any classical observable, f ∈ C ∞ (T2d ), we assign a corresponding quantum observable, Op N ( f ), acting on H N by an analog of the Weyl quantization. We define the operators Op N ( f ) first for the Fourier basis en (x) = exp(2πin · x) and then extend by linearity. For n = (n 1 , n 2 ) ∈ Z2d , the corresponding quantum observable act on ψ ∈ H N via: Op N (en )ψ(Q) = e
n · n n · Q 1 2 2 e ψ(Q + n 1 ), 2N N
(2.1)
where we use the notation e(t) = exp(2πit). For any smooth classical observable f ∈ C ∞ (T2d ) with Fourier expansion n∈Z2d fˆ(n)en (x) we thus define Op N ( f ) =
fˆ(n) Op N (en ).
n∈Z2d
The main properties of the elementary observables Op N (en ) are summarized in the following proposition. Proposition 2.1. For any n, m ∈ Z2d , (1) The operator Op N (en ) is a unitary operator. (2) The composition ω(m, n) Op N (en+m ), Op N (em ) Op N (en ) = e 2N
0 I n t is the standard symplectic form. −I 0 (3) The operator Op N (en ) is only dependent on the congruence of n (mod 2N ).
where ω(m, n) = m
These properties are easily derived from the action given in (2.1). Furthermore, they imply that Op N ( f )∗ = Op N ( f¯), and in particular for real valued functions these operators are Hermitian. From the commutation relations of the elementary operators and the fast decay of the Fourier coefficients one can obtain that as N → ∞,
Op ( f ) Op (g) − Op ( f g) = O f,g N N N
1 N
.
(2.2)
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2.3. Quantum dynamics. For φ H : T2d → T2d a Hamiltonian map as in Sect. 2.1, the corresponding quantum propagator is defined i
U N (φ H ) = e Op N (H ) = e2πi N Op N (H ) . This is a unitary operator (because Op N (H ) is Hermitian), and it satisfies the Erogov identity in the semi-classical limit [3]. That is, for any smooth observable f ∈ C ∞ (T2d ),
1
U N (φ H )∗ Op ( f )U N (φ H ) − Op ( f ◦ φ H ) = O f N N N2 (see Appendix 4.2 for more details). For A ∈ Sp(2d, Z) which satisfies certain parity conditions, one can assign unitary operators U N (A), acting on H N and satisfying “Exact Egorov” [2, 10]. That is, for all observables f ∈ C ∞ (T2d ), U N (A)∗ Op N ( f )U N (A) = Op N ( f ◦ A). The quantization of the perturbed map φ = A ◦φ H is the composition of the quantum propagators for A and φ H respectively, U N () = U N (A)U N (φ H ). Consequently, this is again a unitary operator satisfying the Egorov identity in the semiclassical limit
1
U N ()∗ Op ( f )U N () − Op ( f ◦ φ) = O f . N N N2 2.4. Limiting measures. One way to study the eigenstates of the propagator is through their corresponding Wigner distributions. The Wigner distribution W N (ψ) (of a quantum state ψ ∈ H N ) is a distribution on the phase space, T2d , that assigns to a smooth observable its expectation value W N (ψ)( f ) = Op N ( f )ψ, ψ . We call a measure µ on T2d a limiting quantum measure, if there exists a sequence ψ j ∈ H N j of eigenstates of U N j (), such that the corresponding Wigner distributions w∗
W N j (ψ j ) → µ converge to this measure when N j → ∞. 3. Dynamics on Invariant Manifolds 3.1. Invariant manifolds. Let A ∈ Sp(2d, Z) be a symplectic matrix with integer coefficients, and assume that A has no eigenvalues of modulus 1. The natural left action of A on R2d (x → Ax) preserves the lattice Z2d and induces discrete time dynamics on the torus T2d = R2d /Z2d which is invertible, area preserving, and stably ergodic. There is also a natural right action of A on the rational vector space Q2d (n → n A). For any rational subspace V ⊂ Q2d that is preserved by A there is a corresponding closed connected subgroup X 0 ⊆ T2d invariant under the induced dynamics. Let = V ∩ Z2d
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be the lattice obtained by taking the integral points of V , then the corresponding subgroup is X 0 = x ∈ T2d |en (x) = 1, ∀n ∈ . Let dim V = d0 , then X 0 ∼ = T2d−d0 is a subtorus of codimension d0 . Furthermore, for any fixed point of the dynamics ξ ∈ T2d (i.e., Aξ ≡ ξ (mod 1)), the manifold X ξ = ξ + X 0 = x ∈ T2d |en (x) = en (ξ ), ∀n ∈ is also a connected closed sub-manifold that is preserved by the induced dynamics. We say that these manifolds are co-isotropic, when the invariant subspace V is isotropic with respect to the symplectic form.
3.2. Stable ergodicity. It is a well known result of Anosov [1], that hyperbolic automorphisms of the torus are stably ergodic. We now show that the restriction of A to X ξ can be identified with a hyperbolic automorphism of T2d−d0 implying the following Proposition 3.1. The restriction of A to X ξ preserves the volume measure dm X ξ concentrated on X ξ , and is stably ergodic (with respect to dm X ξ ). Proof. Since the map on X ξ is just shifting by ξ of the map on X 0 , it is sufficient to show this for X 0 . Let W = x ∈ R2d |n · x = 0, ∀n ∈ V , then W is invariant under the left action of A. Let = W ∩ Z2d be the lattice of integral points, then is of rank 2d − d0 (because V is rational). The natural injection W → R2d induces an imbedding W/ → T2d with image X 0 . Therefore, X 0 ∼ = W/ and it is sufficient to prove the proposition for the action of A on W/ . Fix an integral basis n 1 , . . . , n 2d−d0 for , and let B ∈ GL(2d − d0 , Z) be the matrix corresponding to the action of A on . By taking coordinates in the integral basis we can identify W/ ∼ = R2d−d0 /Z2d−d0 = T2d−d0 , and under this identification the action 2d−d 0 on T is the automorphism induced by the natural action of B on R2d−d0 . This map is hyperbolic (since eigenvalues of B are also eigenvalues of A) and area preserving (since det(B) = ±1) and hence stably ergodic.
3.3. Perturbation preserving invariant manifolds. We now consider the perturbed map = A ◦ φ H . We require that the manifolds X ξ remain invariant under the perturbed map. For that reason, we choose our Hamiltonian, H , such that the functions en (x) with n ∈ , defining the manifolds X ξ , are constants of motion for the Hamiltonian flow. This is equivalent to requiring that the Poisson brackets {H, en } = 0 for all n ∈ . In ˆ terms of the Fourier this is equivalent to coefficients the requirement that H (m) = 0 ⊥ 2d unless m ∈ = m ∈ Z |ω(n, m) = 0, ∀n ∈ . By Proposition 3.1 (replacing the Hamiltonian H by H if necessary) we can insure that the perturbed map and its restriction to the X ξ ’s remains ergodic.
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4. Scarring on Invariant Manifolds Let A ∈ Sp(2d, Z) be as in the previous section, let V ⊆ Q2d be an invariant isotropic 2d rational subspace of dimension 1 < d0 ≤ d and let = V ∩ Z . For any fixed point ξ ∈ T2d , let X ξ = x ∈ T2d |en (x) = en (ξ ), ∀n ∈ be the corresponding (co-isotropic) invariant manifold as in Sect. 3.1. Let = A ◦ φ H be the perturbed map leaving the manifolds X ξ invariant as in Sect. 3.3, and let U N () be its quantization. 4.1. Proof of Theorem 1. In order to proveTheorem 1, we first show that the spaces H N ,ξ = ψ| Op N (en )ψ = en (ξ )ψ, ∀n ∈ are of dimension N d−d0 and are preserved by U N (φ). Then we show the localization of states from H N ,ξ. Consider the family of operators A N = Op N (en )|n ∈ . This is a commutative family of unitary operators (recall is isotropic). We can thus decompose the Hilbert space H N into joint eigenspaces HN = Hλ , /N
where the sum is over characters of /N ∼ = AN . Lemma 4.1. Any character λ of of order dividing N , is of the form λ(n) = e( ω(n,m) N ) for some m ∈ Z2d . d0 Proof. Notice that if we have an integral basis {n i }i=1 for and corresponding m i ∈ 2d Z , such that ω(n i , m j ) = δi, j , then we are done. Indeed, since we assume the character is of order dividing N , for the basis elements λ(n i ) = e( kNi ) for some ki ∈ Z. Then for m = i ki m i ∈ Z2d , ω(n, m) i, j ω(ai n i , k j m j ) . ai n i = e λ(n) = λ =e N N i
We now proceed by induction on dim V = rank() and construct such an integral basis. If dim V = 1 then choose any primitive vector n ∈ (i.e., gcd(n) = 1). Then there is m ∈ Z2d such that ω(n, m) = gcd(n) = 1. Now for dim(V ) = l > 1 assume that n 1 , . . . , nl−1 are an integral basis with m 1 , . . . m l−1 ∈ Z2d such that ω(n i , m j ) = δi, j . Take an element n˜ l ∈ linearly independent on n 1 , . . . nl−1 , and let nl = n˜ l −
l−1
ω(n˜ l , m i )n i .
i=1
Then n 1 , . . . , nl are still linearly independent and ω(nl , m i ) = 0 for i < l. We can assume that gcd nl = 1 (otherwise divide by any common factors), so there is m˜ l ∈ Z2d such that ω(nl , m˜ l ) = 1. Take m l = m˜ l −
l−1
ω(n i , m˜ l )m i ,
i=1
so indeed ω(m i , n j ) = δi, j for any 1 ≤ i, j ≤ l.
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Corollary 4.2. The dimension of all eigenspaces satisfies dim Hλ = N d−d0 . 2d Proof. Any character of /N is of the form λm (n) = e( ω(m,n) N ) for some m ∈ Z . Hence, the operator Op N (em ) sends the space Hλ0 into the space Hλm with inverse map Op N (e−m ). Hence for all characters dim Hλ = dim Hλ0 and since there are N d0 characters then N d = dim H N = dim Hλ = N d0 dim Hλ0 , λ
concluding the proof.
Let ξ ∈ T2d be a fixed point for A and consider the character of given by λξ (n) = en (ξ ). For R = det(A − I ) we know that Rξ ∈ Z2d . Therefore, if N is divisible by R then the character λξ is a character of /N and the space H N ,ξ = Hλξ . We have thus shown the following Corollary 4.3. dim H N ,ξ = N d−d0 . Proposition 4.4. The spaces H N ,ξ are preserved by U N (). Proof. Recall that the quantum propagator U N () = U N (A)U N (φ H ) is the composition of U N (A) and the quantization of the hamiltonian flow U N (φ H ) = e2πi N Op N (H ) . We will show that both U N (φ H ) and U N (A) preserve H N ,ξ . The quantum observable Op N (H ) = m∈⊥ Hˆ (m) Op N (em ), and hence commutes with Op N (en ) for all n ∈ . Consequently, U N (φ H ) also commutes with Op N (en ) for all n ∈ , and preserves the eigenspace H N ,ξ . Next, for any n ∈ and any ψ ∈ H N ,ξ we have Op N (en )U N (A)ψ = U N (A) Op N (en A )ψ. Because is invariant then n A ∈ as well and Op N (en A )ψ = en A (ξ )ψ. Now, as ξ is a fixed point of A, en A (ξ ) = en (Aξ ) = en (ξ ), so that Op N (en )U N (A)ψ = en (ξ )U N (A)ψ, and the space H N ,ξ is also preserved by U N (A).
Let f ∈ C ∞ (T2d ) and consider the restriction of f to the manifold X ξ . For any x ∈ X ξ and any m ∈ we have em (x) = em (ξ ). We can thus write f (x) = f (n)en (x), fˆ(m)em (x) = m
where f (n) = m∈ fˆ(n + m)em (ξ ) = set of representatives for Z2d /.
n∈
X
f (x)e−n (x)dm X ξ , and ⊆ Z2d is any
Lemma 4.5. There is a choice of representatives ⊆ Z2d for Z2d / such that for any n ∈ and m ∈ we have |ω(n, m)| n + m2 .
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Proof. Let U ⊆ Q2d be the orthogonal complement of V (for the standard inner product). Then there is an integer D ∈ Z (depending on V ) such that the images of the projection maps PV (DZ2d ) and PU (DZ2d ) lie inside Z2d . Denote by [D]2d = {1, . . . , D}2d , and think of it as a set of representatives for Z2d /DZ2d . Then our set of representatives for Z2d / is taken to be = PU (m − r ) + r |m ∈ Z2d , r ∈ [D]2d with r ≡ m (mod D) . This is indeed a set of representatives, since any k ∈ Z2d can be written uniquely as k = PV (k − r ) + PU (k − r ) + r , with r ∈ [D]2d , r ≡ k (mod D) and PV (k − r ) ∈ . Now for any n ∈ and m ∈ let r ≡ n + m (mod D) in [D]2d . Then m = PV (n + m − r ) and n = PU (n + m − r ) + r , hence |ω(n, m)| ≤ n m n + m − r 2 n + m2 . The following proposition concludes the proof of Theorem 1. Proposition 4.6. Let ψ j ∈ H N j ,ξ be a sequence of states such that the Wigner distribuw∗
tions W N j (ψ j ) → µ converge weak∗ to some measure µ on T2d , then µ is supported on X ξ . Proof. Let µ be such a limiting measure. In order to show suppµ ⊆ X ξ , it is sufficient to show that any smooth function f vanishing on X ξ satisfies µ( f ) = lim j→∞ W N j (ψ j ) ( f ) = 0. Fix a smooth function f vanishing on X ξ . Let be a set of representatives for Z2d / as in Lemma 4.5. Then, for any n ∈ , f (n) = f (x)e−n (x)dm X ξ = 0. fˆ(n + m)em (ξ ) = Xξ
m∈
For any fixed N = N j , and ψ = ψ j ∈ H N ,ξ , Op N ( f )ψ, ψ = fˆ(n + m) Op N (en+m )ψ, ψ . n∈ m∈
Replace Op N (en+m )ψ, ψ = e2N (ω(n, m))em (ξ ) Op N (en )ψ, ψ
to get Op N ( f )ψ, ψ =
Op N (en )ψ, ψ
n∈
fˆ(n + m)em (ξ )e2N (ω(n, m)).
m∈
Since we know f (n) = m∈ fˆ(n + m)em (ξ ) = 0 we can subtract it to get Op N (en )ψ, ψ fˆ(n + m)em (ξ )(e2N (ω(n, m)) − 1). Op N ( f )ψ, ψ = n∈,m∈
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D. Kelmer
We can thus bound | Op N ( f )ψ, ψ | ≤
| fˆ(n + m)||(e2N (ω(n, m)) − 1)|,
n∈ m∈
and since |(e2N (ω(n, m)) − 1)|
|ω(n,m)| N
| Op N ( f )ψ, ψ |
n+m2 N
we get
1 1 ˆ , | f (m)| m2 f N N 2d m∈Z
and indeed µ( f ) = lim Op N j ( f )ψ j , ψ j = 0 N j →∞
4.2. Proof of Theorem 2. The proof of Theorem 2 follows the lines of the proof of the Quantum Ergodicity Theorem [4, 17]. The first ingredient is showing that on average the sates in Hξ are evenly distributed in X ξ (Proposition 4.7). Then we use the ergodicity of the restricted map on X ξ to bound the variance (Theorem 3). Theorem 2 is then derived by a standard diagonalization argument. d−d
Proposition 4.7. For any integer N divisible by R, let {ψ j } Nj=1 0 be an orthonormal basis for H N ,ξ ⊂ H N . For any smooth f ∈ C ∞ (T2d ) as N → ∞ the average 1 Op N ( f )ψ j , ψ j −→ f dm X ξ . dim H N ,ξ Xξ j
Proof. We will show that when f is a trigonometric polynomial, for N sufficiently large, there is an equality 1 Op N ( f )ψ j , ψ j = f dm X ξ . dim H N ,ξ Xξ j
The result for arbitrary smooth functions then follows from the rapid decay of the Fourier coefficients. We thus need to show that for any fixed n ∈ Z2d and sufficiently large N , 1 e (ξ ) n ∈ . Op N (en )ψ j , ψ j = en dm X ξ = n 0 otherwise dim H N ,ξ Xξ j
First, for any n ∈ and ψ j ∈ H N ,ξ by definition Op N (en )ψ j = en (ξ )ψ j so the equality is trivial. Next, for n ∈ there are two possible cases, either there is m ∈ such that ω(m, n) = 0 or there is not.
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In the case that such m ∈ exists, since Op N (em )ψ j = em (ξ )ψ j we can write Op N (en )ψ j , ψ j = Op N (en ) Op N (em )ψ j , Op N (em )ψ j
j
j
= Op N (e−m ) Op N (en ) Op N (em )ψ j , ψ j
j
=e
ω(n, m) Op N (en )ψ j , ψ j . N j
1 We can assume N large enough so that e( ω(n,m) N ) = 1 implying that dim H N ,ξ j Op N (en )ψ j , ψ j = 0. If on the other hand ω(m, n) = 0 for all m ∈ , then the operator Op N (en ) commutes with all the operators Op N (em ), m ∈ , and hence preserves the space H N ,ξ . We can thus look at the restriction of Op N (en ) to H N ,ξ and the sum j Op N (en )ψ j , ψ j = Tr(Op N (en )|H N ,ξ ) is the trace of this restricted operator. Now, fix k ∈ Z2d such that ω(n, k) = 0 but ω(m, k) = 0 for any m ∈ . Such a vector exists since we can take ˜ = 0 and then define k = k˜ − ω(k, ˜ m i )m i for some k˜ to be any vector with ω(n, k) integral basis m i of . The operator Op N (ek ) also preserves the space H N ,ξ and so
Tr(Op N (en )|H N ,ξ ) = Tr(Op N (e−k ) Op N (en ) Op N (ek )|H N ,ξ ) ω(n, k) Tr(Op N (en )|H N ,ξ ). =e N Again we can assume that N is sufficiently large so that e( ω(n,k) N ) = 1, implying that Tr(Op N (en )|H N ,ξ ) = 0. For any N (divisible by R) we fix an orthonormal basis of eigenstates ψi ∈ H N ,ξ , i = 1, . . . , N d−d0 . For any smooth function f ∈ C ∞ (T2d ) we define the quantum variance (in this basis) as 1 | Op N ( f )ψi , ψi − f dm X ξ |2 . σ N2 ( f ) = dim H N ,ξ Xξ i
Theorem 3. For f ∈ C ∞ (T2d ), lim N →∞ σ N2 ( f ) = 0. Proof. Without loss of generality we can assume that X f dm X ξ = 0. For T > 0 denote T by f T = T 1+1 t=0 f ◦ φ t the time average of f . For fixed t we have
U N ()−t Op ( f )U N ()t − Op ( f ◦ φ t ) = O f,t 1 . N N N2 Hence for any eigenstate ψ of U N () we have | Op N ( f )ψ, ψ − Op N ( f )ψ, ψ | = O f,T T
1 N2
.
Using Cauchy-Schwarz inequality
2
| Op N ( f T )ψ, ψ |2 ≤ Op N ( f T )ψ ψ2 = Op N ( f T )∗ Op N ( f T )ψ, ψ ,
(4.1)
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D. Kelmer
we get
| Op N ( f T )ψ j , ψ j |2 ≤
j
Op N ( f T )∗ Op N ( f T )ψ j , ψ j .
(4.2)
j
Now, the estimate (2.2) for Op N ( f T )∗ Op N ( f T ) and Op N (| f T |2 ) implies 1 Op N ( f T )∗ Op N ( f T )ψ j , ψ j
dim H N ,ξ j
1 1 T 2 = Op N (| f | )ψ j , ψ j + OT, f . dim H N ,ξ N
(4.3)
j
From the estimates given in (4.1),(4.2) and (4.3) we get σ N2 ( f ) ≤
1 1 , Op N (| f T |2 )ψ j , ψ j + OT, f dim H N ,ξ N j
and in the limit N → ∞,
lim sup σ N2 ( f ) ≤ N →∞
Xξ
2
| f T |2 dm X ξ = f T 2
L (X ξ )
.
(4.4)
This is true for any T > 0. However, since the map = A ◦ φ H induces ergodic dynamics on the manifold X ξ , then in the limit T → ∞ we have L 2 (X ξ )
f T −→ 0, and since the left-hand side of (4.4) has no dependence on T indeed lim N →∞ σ N2 ( f ) = 0. We now give the proof of Theorem 2 from Theorem 3. Proof. For each n ∈ Z2d , and N ∈ N denote by σ N2 (n) = σ N2 (en ). Then, Theorem 3 N →∞
implies that σ N (n) −→ 0. Now for fixed n ∈ use Chebyshev’s inequality to get that 1 i |W N (ψi )(en )| ≥ σ N (n) ≤ σ N (n). d−d 0 N For any integer M > 0 define the set J N (M) = i|∃n ∈ , n ≤ M, |W N (ψi )(en )| ≥ σ N (n) . (M) For fixed M, we have that JNNd−d ≤ 0
n≤M
σ N (n) and in the limit lim N →∞
Consequently, there is a sequence M N → ∞ such that lim N →∞ sets S N = {1, . . . , N d−d0 } − J N (M N ),
are of density one in {1, . . . , N d−d0 }.
J N (M N ) N d−d0
J N (M) N d−d0
= 0.
= 0, and the
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Now let f ∈ C ∞ (T2d ) be a smooth function. Fix > 0 and let M be sufficiently large so that n>M | fˆ(n)| ≤ . Fix N0 sufficiently large so that M N > M for N > N0 . Then, for any ψi we have that f dm X ξ | ≤ | fˆ(n)W N (ψi )(en )| + . |W N (ψi )( f ) − Xξ
n∈,n<M
√ But for i ∈ S N and n ≤ M ≤ M N by definition |W N (ψi )(en )| ≤ σ N (n). We thus get that f dm X ξ | ≤ | fˆ(n)| σ N (n) + . |W N (ψi )( f ) − Xξ
0=n<M
Taking N → ∞ we get lim sup |W N (ψi )( f ) − N →∞
Xξ N →∞
implying that indeed W N (ψi )( f ) −→
Xξ
f dm X ξ | ≤ ,
f dm X ξ .
Appendix A. Proof of Egorov’s Theorem The (semi classical) Egorov Theorem, is a well known result for quantization of Hamiltonian flows on R2d . For Hamiltonian flows on T2d the proof is analogous and is described in [3]. For the sake of completeness, we give a short proof along the same lines. Recall that given a real valued smooth function g ∈ T2d , the associated Hamiltonian flow, φgt : T2d → T2d , satisfies the differential equations: d ( f ◦ φgt ) = {g, f ◦ φgt }, ∀ f ∈ C ∞ (T2d ), dt
(A.1)
and note that the quantum propagator U N (φgt ) = e2πi N Op N (g)t corresponding to this flow satisfies dU N (φgt ) dt
= 2πi N Op N (g)U N (φgt ) = 2πi N U N (φgt ) Op N (g).
(A.2)
The main ingredient in the proof is the connection between the Poisson brackets and quantum commutators.
Lemma A.1. For any f ∈ C ∞ (T2d ) let c f = (−)d+2 f ∞ . Then there is a constant C such that for any f, g ∈ C ∞ (T2d ),
Op ({g, f }) − [2πi N Op (g), Op ( f )] ≤ C cg c f . N N N N2
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Proof. For any n, m ∈ Z2d , Op N ({em , en }) − [2πi N Op N (en ), Op N (em )] 2π ω(m, n) 2 Op N (en+m ). = (2π ) ω(m, n) − 2π 2N sin 2N Hence, 3 3 3
Op ({em , en }) − [2πi N Op (en ), Op (em )] ≤ 4π m n . N N N N2 Decomposing g and f into Fourier series, noting that | fˆ(n)| and |g(n)| ˆ are bounded by cf cg and , one gets (2π n)2d+4 (2π n)2d+4
with C = |
Op ({g, f }) − [2πi N Op (g), Op ( f )] ≤ C cg c f , N N N N2
1 2 n (2π n)2d+1 | .
For f ∈ C ∞ (T2d ) smooth, its composition f ◦ φgs with the Hamiltonian flow is also smooth. We can thus consider
c f ◦φgs = (−)d+2 ( f ◦ φgs ) , ∞
and let C f,g (t) = C · cg · sup (c f ◦φgs ). 0≤s≤t
Theorem 4 (Egorov). For all f, g ∈ C ∞ (T2d ) we have
tC (t)
f,g
.
U N (φgt )∗ Op N ( f )U N (φgt ) − Op N ( f ◦ φgt ) ≤ N2 Proof. Denote by B(s) = U N (φgs ) Op N ( f ◦φgs )U N (φgs )∗ . Since conjugating by a unitary matrix doesn’t change the norm, it is equivalent to bound the norm of
U N (φgt ) Op N ( f ◦ φg )U N (φgt )∗ − Op N ( f ) = B(t) − B(0) . Differentiate B(s), recalling (A.2) and (A.1) to get B (s) = −[2πi N Op N (g), B(s)] + U N (φgs )∗
d(Op N ( f ◦ φgs ))
U N (φgs )∗ ds = U N (φgs )∗ (Op N ({g, f ◦ φgs }) − [2πi N Op N (g), Op N ( f ◦ φgs )])U N (φgs ).
C (t) Consequently, for any 0 ≤ s ≤ t we can bound (using Lemma A.1) B (s) ≤ f,g , N2 implying that
t
tC f,g (t)
B(t) − B(0) = B (s)ds
≤ N2 , 0 as claimed.
Acknowledgements. I thank Jens Marklof for suggesting to extend the localization on invariant manifolds for perturbed maps. This work was supported in part by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.
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References 1. Anosov, D.V.: Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature. Trudy Mat. Inst. Steklov 90, RI: Providence, 1967 2. Bonechi, F., De Bièvre, S.: Controlling strong scarring for quantized ergodic toral automorphisms. Duke Math. J. 117(3), 571–587 (2003) 3. Bouclet, J. M., De Bièvre, S.: Long time propagation and control on scarring for perturbed quantized hyperbolic toral automorphisms. Ann. H. Poincaré 5, 885–913 (2005) 4. Bouzouina, A., De Bièvre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178, 83–105 (1996) 5. De Bièvre, S., Degli Esposti, M.: Egorov theorems and equidistribution of eigenfunctions for sawtooth and Baker maps. Ann. Inst. Poincaré 69, 1–30 (1998) 6. Degli Esposti, M., Graffi, S.: Mathematical aspects of quantum maps. In: M. Degli Esposti, S. Graffi, eds., The mathematical aspects of quantum maps, Volume 618 of Lecture Notes in Physics, Berlin-HeidelbergNew York: Springer, 2003, pp. 49–90 7. Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. (2) 163(1), 165–219 (2006) 8. Faure, F., Nonnenmacher, S., De Bièvre, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239(3), 449–492 (2003) 9. Hanny, J.H., Berry, M.V.: Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating. Phys. D 1, 267–290 (1980) 10. Kelmer, D.: Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus. Ann. of Math., to appear, available at http://arxiv.org/list/math-ph/0510079, 2005 11. Knabe, S.: On the quantisation of Arnold’s cat. J.Phys. A: Math. Gen. 23, 2013–2025 (1990) 12. Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103(1), 47–77 (2000) 13. Rivas, A.M.F., Saraceno, M., Ozoriode Almeida, A.M.: Quantization of multidimensional cat maps. Nonlinearity 13(2), 341–376 (2000) 14. Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math Phys. 161, 195–213 (1994) 15. Šhnirel’man, A.I.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29, 181–182 (1974) 16. Pesin, Y.: Lectures on partial hyperbolicity and stable ergodicity. Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS), 2004 17. Zelditch, S.: Uniform distribution of eigenfunctions om compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987) Communicated by P. Sarnak
Commun. Math. Phys. 276, 397–435 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0341-0
Communications in
Mathematical Physics
On Polynomials Interpolating Between the Stationary State of a O(n) Model and a Q.H.E. Ground State M. Kasatani1 , V. Pasquier2 1 Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan 2 Service de Physique Théorique, C.E.A/ Saclay, 91191 Gif-sur-Yvette, France.
E-mail: [email protected] Received: 7 August 2006 / Accepted: 5 April 2007 Published online: 25 September 2007 – © Springer-Verlag 2007
Abstract: We obtain a family of polynomials defined by vanishing conditions and associated to tangles. We study more specifically the case where they are related to a O(n) loop model. We conjecture that their specializations at z i = 1 are positive in n. At n = 1, they coincide with the Razumov-Stroganov integers counting alternating sign matrices. We derive the CFT modular invariant partition functions labelled by Coxeter-Dynkin diagrams using the representation theory of the affine Hecke algebras. 1. Introduction Much progress has recently been made in the study of the ground state of the X X Z spin chain Hamiltonian when the anisotropy parameter is equal to 1/2 [1]. The Hamiltonian is closely related to a stochastic Hamiltonian of a O(1) fully packed loop model [2, 3]. The components of the O(1) model Hamiltonian stationary state can be normalized to be positive integers, and it is conjectured (the Razumov-Stroganov (R-S) conjecture [4]) that these integers are in bijective correspondence with the states of a square-icemodel with domain wall boundary conditions, or equivalently with certain classes of alternating-sign matrices. In previous works [6, 7], we have introduced the polynomials discussed in this paper. In [6], they were discovered through the study of some representations of the doubleaffine Hecke algebra [8]. In [7], they were obtained by deforming the O(1) transfer matrix eigenstates of Di Francesco and Zinn-Justin [5] in order to generalize the R-S conjecture to the O(n) case. The aim of this paper is to further study these polynomials: • We construct a family of polynomials which transform linearly under the braid group. We single out a basis in correspondence with the flat tangles or patterns. • We define a deformation of the R-S integers by evaluating the basis polynomials at 1. We observe they enjoy positivity properties suggesting that they may coincide with a weighted enumeration sum of objects related to alternating sign matrices.
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The polynomials can be defined by the vanishing conditions which they obey when several variables come close to each other. These vanishing conditions, called the wheel conditions [10], have been classified in [6] in relation with the representation theory of affine Hecke algebras. The number of variables involved in the wheel condition is k + 1 and the wheel condition depends on another parameter r related to the degree with which the polynomial vanishes when the points are put together. In an orthogonal basis, the polynomials are non-symmetric Macdonald polynomials at specialized values of their parameters t k+1 q r −1 = 1.1 Another way to obtain them, giving rise to a different basis, is by deforming the O(1) model transfer matrix ground state. When k = 2, the polynomial representation is dual to a representation of the Temperley and Lieb (T.L.) algebra of a loop model (defined on the disc) parameterized by n = −2 cos (θ ), where n is the fugacity of the loops. The polynomials basis we consider here is dual to the loop model basis. Its specialization at n = 1 coincides with the components of the O(1) model transfer matrix ground state. The basis polynomials are obtained from the action of difference operators [17] on a generating state which in the r = 2 case is simply a product of q-deformed Vandermonde determinants. They can also be defined as the Kazhdan-Lusztig (K-L) basis [15, 16] of this representation.2 At θ = 0, the wheel conditions are precisely the constraint imposed by the interactions on the Q.H.E. wave functions and the generating polynomial of the basis coincides with a wave function of the Quantum Hall Effect (Q.H.E) [13]. In this context, the variables are the coordinates of particles distributed in k layers (or spins) not interacting with each other. We generalize the construction to the case of a cylinder. The T.L. representations acting on link patterns depend on a second parameter n equal to the weight assigned to the loops which wind around the cylinder. We study the simplest case where n = 2 cos (θ/2), and in the case where n = n = 1, we recover the stationary state of a O(1) model considered by Mitra and Nienhuis [20, 21]. The specialization of these polynomials when all their variables are set equal to one turns out to be polynomials in n in the disc case and n in the cylinder case, with integer coefficients. We observe examples and conjecture in general that these integers are positive. When n = n = 1, the disc and the cylinder polynomials become respectively the integers previously conjectured to count alternating sign matrices and half-turn alternating sign matrices in different topological sectors [4, 22]. In the simple case where the Hecke algebra representation is generated from the action of difference operators on a product of q-Vandermonde determinants, we have verified that the evaluation at z i = 1 of the K-L basis polynomials are positive. Although not directly related to these polynomials, we derive the c < 1 unitary modular invariant partition functions of conformal field theories by decomposing certain representations of the affine Temperley and Lieb algebra acting on Dynkin diagrams into irreducible representations. As a byproduct, we define an action of the modular group on the irreducible representations of the affine T.L. (and more generally Hecke) algebra which relates them to a tensor product of Virasoro representations. The paper is organized as follows. 1 The non-symmetrical Macdonald polynomials were first introduced in the context of the CalogeroSutherland models by diagonalizing the affine Hecke generators y¯i (constructed in Appendix C) in the published version of [9]. They have been extensively used by Cherednik [8]. 2 The relation with the Kazhdan-Lusztig basis has been explained to us by A. Lascoux.
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The first part is introductory and serves as a motivation, we describe the representations of the affine T.L. algebra acting on a system of lines projected onto the disk. We study in more detail the case of the punctured disc. The second and third part are the core of the paper and can to a large extent be read independently. In the second part, we obtain representations dual to those of the first part in terms of polynomials obeying the wheel conditions. We give examples where these polynomials are deformed Q.H.E. wave functions. In the third part, we state the positivity conjectures when the variables of the polynomials are set equal to one. In the fourth part, we consider the case n = 1, and we show that the polynomials are the components of a O(1) loop model ground state. In the fifth part, we introduce the representations of the affine T.L. algebra on SOS paths which diagonalize the affine generators. In the sixth part, we consider more specifically the unitary representation when the parameter of the algebra is a root of unity. We decompose certain representations acting on Dynkin diagram paths into irreducible representations. 2. Representation of the Affine T.L. Algebra on Patterns The basic definitions of an affine Hecke algebra (A.H.A.) A N (t), and its T.L. restriction ATN (t), where t is a parameter, are given in Appendix A. In this section, we define tangles and patterns which are the natural objects to represent the action of the generators of ATN . The tangles carry the three dimensional topological information. They can be decomposed into a linear combination of planar tangles called patterns. The patterns provide a basis of a representation of the affine T.L algebra which we describe in this section. 2.0.1. Tangles. A tangle is made by a set of open strings embedded into a three dimensional ball such that their extremities are on the boundary. We also consider the case where the ball is pierced by a flux running through it. A string can eventually connect the center of the ball or the flux to the boundary. The strings cannot cross each other, they cannot cross the flux either. Two tangles are considered equivalent if there is an ambient isotopy of one tangle to the other keeping the boundary of the 3-ball (and the flux) fixed. The extremities of the strings on the boundary are arranged into N marked points 1, 2, . . . , N ordered anticlockwise around the great circle. They are represented by their projection onto the flat disc bounded by the great circle. By noting the under and over crossings, one defines a tangle diagram. To respect isotopy invariance, tangle diagrams are also identified through equivalence relations known as Reidemeister moves. These moves are represented on Fig. 1 (strictly speaking, the first Reidemeister move does not 3 involve the factor −t − 4 ). In the punctured case we project the flux onto the origin and therefore, the projected strings are not allowed to cross the origin. 2.0.2. Linear operators. Linear operators can be represented by an annulus with N ¯ . . . , N¯ ) on its inner boundary and M cyclically cyclically ordered marked points (1, ordered marked points (1, . . . , M) on its outer boundary connected by strings (a string can connect two points of the same boundary). The action of the annulus on the disk with N marked points is null if M = N . It is obtained by gluing its inner boundary to the boundary of the disk so as to identify the marked points with those of the disk:
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1)
−3/4 = −t
3/4 = −t
2)
=
3)
=
Fig. 1. Reidemeister moves corresponding to the equations (2), (3)
¯ . . . , N¯ ), and by joining the strings of the disk with the strings of the (1, . . . , N ) = (1, annulus ending at the same point. For example, the identity is represented by disjoined ¯ open strings connecting i to i. The braid group generators: 1
1
1
gi = t − 4 Ti = t 4 + t − 4 ei , gi−1 = t 4 Ti = t − 4 + t 4 ei , 1
1
1
(1)
are represented by an annulus with N −2 disjoined strings connecting l¯ to l for l = k, k+1, a string connecting k + 1 to k, and a string connecting k¯ to k + 1 passing over it (see Fig. 3). The two first Reidemeister moves, Fig. 1, result from the relations: 3
ei gi = −t − 4 ei , gi gi±1 ei = ei±1 ei ,
(2)
and the third move is the braid relation: gi gi+1 gi = gi+1 gi gi+1 , gi g j = g j gi if |i − j| > 1.
(3)
A map from operators acting on the disc with N marked points to operators acting on the disc with N + 1 marked points consists in adding an additional string to the pattern connecting N + 1 to N + 1 without adding any crossing. Conversely, a partial trace is defined by joining together the extremities N and N¯ without adding any crossings. It maps operators acting on the disc with N marked points to operators acting on the disc with N − 1 marked points. 2.0.3. Patterns. A pattern is a tangle with a flat projection on the disc. It can be represented by disjoint lines connecting pairwise the boundary points of the disk. Also, a line can start vertically down from the inside to reach the boundary without winding (Fig. 2). We can encode a pattern π by a string of letters α or β [16, 20]. We put α for i if the marked point is connected to the inside. In the disc case, we put α for i and β for j
Interpolation of Polynomials Between Stationary Q.H.E. Ground State
3
2
401
2
3
1
1 3
2
1
Fig. 2. The patterns βαα, ααβ, αβα
=
t
1/4
+t−1/4
Fig. 3. Skein relation corresponding to Eq. (1). The figure obtained by rotating each piece by 90 degree is also valid
when i < j are connected by a line. Therefore, when we cut the string into two pieces, the left piece must contain at least as many α’s as β’s. In the the punctured disc case, the lines are oriented so that the domain bounded by it and the great circle surrounds the puncture anticlockwise and we put α for the beginning and β for the end of the line. Given a string of α’s and β’s, by successively erasing factors αβ, one can recover the position of the isolated α’s. It will often be convenient to view a pattern as an infinite periodic string with the identification πi+N = πi . We denote H N , the vector space made by linear combinations of these patterns. Tangles can be projected onto patterns as follows: Using the skein relation of Fig. 3, one can represent a tangle with at least one crossing as a linear combination of two tangles with one less crossing. A loop not surrounding the origin and not crossing any 1 1 other loop is removed by multiplying the weight of a pattern by τ = −(t 2 + t − 2 ). In the punctured disc case, if N is even, a loop surrounding the origin and not crossing any other loop is removed by multiplying it by u + u −1 , where u is a new parameter. If N is odd, we require that if one rotates by an angle 2π around the origin a line which starts from it, the weight is multiplied by u. These transformations preserve the equivalence under Reidemeister move, and using them, a tangle can be projected onto a linear combination of patterns. We define linear operators acting in ATN as for tangles. In particular, the T.L. generators ek are represented by an annulus with N − 2 disjoined lines connecting l¯ to l for l = k, k + 1, a line connecting k¯ to k + 1, and a line connecting k to k + 1. Using the above rules, it is straightforward to verify the T.L. relations with diagrams: ei2 = τ ei ,
402
M. Kasatani, V. Pasquier e e e 5 3 1
=
1
2
3
4
5
6
6 1
5
= 2
4 3
e e e e 5 2 3 1
=
1
3
2
4
5
6
6 1
5
= 2
4 3
e e e e 4 5 3 1
=
1
2
3
5
4
6
6 1
5
= 2
4 3
e e e e e 4 5 2 3 1
=
1
3
2
5
4
6
6 1
5
= 2
4 3
e e e e e e 3 4 5 2 3 1
=
1
4
2
5
3
6
6 1
5
= 2
4 3
Fig. 4. Several representations of the Hilbert space H6 describing six particles on the disc, in terms of: a) T.L. words or paths. b) Young Tableaux where the two columns give the positions of the α’s and β’s of the string. c) Patterns
ei e j = e j ei if |i − j| ≥ 2, ei ei±1 ei = ei .
(4)
2.0.4. Temperley Lieb representations. We give the explicit expression of the T.L. matrices ei in the link pattern basis. We view a pattern as a string of N letters of α and β. If the difference between the number of α’s and β’s is larger than or equal to two, it is not conserved under the action of the T.L. algebra. Here, we restrict to the case where this difference is equal to zero or one (see Fig. 4 for different representations of the patterns). Let us first consider the disc case. Following Lascoux and Schützenberger [16], we identify the link patterns basis with a Kazhdan-Lusztig (K-L) basis (see Sect. 3.5.1). A pattern π carries the label i if it is an eigenstate of ei , in other words if πi πi+1 = αβ. Given a pair of patterns (π, π ), we associate to it a reduced pair (π r , π r ), by successively erasing the factors αβ located at the same place on the two strings. Two patterns are said to be matched if their reduced expression differ only by a change of a factor αβ → βα. It can be encoded into a K-L graph [15], having the patterns as vertices and an edge between two vertices when they are matched (see Fig. 5). We define µ(π, π ) = 1 or 0 indicating if π and π are matched or not.
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αααβββ
ααβαββ
αβααββ
ααββαβ
αβαβαβ
Fig. 5. Kazhdan-Lusztig graph for 6 particles on the disc (H6 )
The expression of the T.L. matrices ei is given by: ei π = τ π, if π has label i, = µ(π, ν)ν, if π has not label i.
(5)
ν has label i
In the punctured disc case, this construction is modified as follows. We view a pattern as an infinite periodic string of letters α, β. The reduction proceeds as in the disc case where we take into account the periodicity of the patterns: πi+N = πi . One still has the expression (5) of the T.L. matrices, where µ(π, π ) = 0 if the patterns are not matched and the value of µ(π, π ) for the matched pattern is given by the following rules. In the N even case, µ(π, π ) = 1, except if the two reduced strings are αβ and βα, where µ(π, π ) = −(u + u −1 ). In the N odd case, µ(π, π ) = 1, except if the two reduced strings are π r = βαα and r π = αβα, where µ(π, π ) = u, π r = βαα and π r = ααβ, where µ(π, π ) = u −1 , and the permuted cases, where µ(π, π ) = µ(π , π )−1 . T which follow from the above In Appendix B, we give the matrices representing A2,3 rules. 2.0.5. Affine generators in the pattern representation . To obtain the affine algebra representation [29], let us define the cyclic operator σ which rotates the ball by −2π/N around its axis. It acts on patterns by shifting the extremities of the strings by one unit clockwise: i → i − 1, so that we have: σ gi = gi−1 σ,
(6)
for i ≥ 2, and one can define a new braid generator g N = σ g1 σ −1 . In the punctured disc case, σ acts by shifting the indices i → i − 1. Let us construct the affine generators yi which form a family of commuting operators, and are therefore useful to characterize the states of a representation. They also have a topological interpretation. We define y1 as: y1 = T1 T2 . . . TN −1 σ.
(7)
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2
3
4
1
Fig. 6. Affine generator y1 N −1
Let us define y1 = t − 4 y1 = σ g2 g3 . . . g N . It acts on tangles by letting the extremity of the string ending at the marked point 1 wind by an angle −2π around the boundary of the disk underneath the strings ending at position j = 1 (see Fig. 6). Similarly, we define yi 2 j−N −1 from the relations defining the affine Hecke algebra (80) of Appendix A, and t 4 y j lets the extremity of the string ending at the marked point j rotate by an angle −2π around the boundary of the disk above the strings ending at j < i and underneath the strings ending at j > i. Let us consider the action of y1 in the pattern representation. If N is even, the line connected to the marked point 1 remains underneath all the other lines and we can decouple it from them. It is therefore sufficient to consider a pattern 3 with only this line. It is straightforward to verify that y1 = −t − 4 if there is no puncture. In the punctured case, the basis has two elements, βα and αβ, where the line surrounds the puncture in two different ways and the matrix has the expression (see Appendix B): 1 0 3 t4 1 y1 = . (8) −t − 4 (u + u −1 )t − 4 It therefore satisfies the quadratic relation: 1
1
(y1 − u −1 t − 4 )(y1 − ut − 4 ) = 0.
(9)
We define a second parameter τ equal to the weight of a loop surrounding the puncture. It is obtained by sandwiching y1 (8) between two e1 : 3
e1 y1 e1 = −τ t − 4 e1 .
(10)
Therefore, in the N even case, we have: τ = u + u −1 .
(11)
In the case where N is odd, we can proceed similarly by considering a basis with three elements involving the patterns of Fig. 2. The matrix representing y1 is given in Appendix B. The relations (9) and (11) are modified and become: 1
(y1 − u −1 t − 2 )(y1 − u) = 0,
1 4
τ = ut + u
−1 − 41
t
.
(12) (13)
In both cases, τ = x + x −1 , where x 2 = y+ /y− is the ratio of the two eigenvalues of y1 .
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2.0.6. Involutions. One can define two natural involutions F and T on tangles. The flip F rotates the ball by half a turn around a horizontal axis passing through its center and relabels the points k → N + 1 − k. It induces the isomorphism taking: gk → g N −k and σ → σ −1 . The reflection T reflects the ball through its equator. It induces the antilinear involution taking: gk → gk−1 , t → t −1 , and σ to itself. 2.0.7. Inclusion and conditional expectation value. There is an imbedding of H N −2 into H N which takes x to e1 x. On patterns, it consists in shifting by two the labels of a pattern of H N −2 , and by adding a line connecting two new points labelled 1 and 2 in such a way that 1, 2, . . . , N are cyclically ordered. Thus, it adds the two letters αβ at the beginning of the string representing the pattern. Conversely, there is a projection E called conditional expectation value [28] from H N to H N −2 defined by: e1 π = τ E(π )e1 .
(14)
E connects the lines ending at 1, 2, so as to produce a pattern starting with αβ, which is identified with a pattern in H N −2 by the preceding imbedding. This projection is hermitian for the scalar product defined next and commutes with the action of ATN −2 . 2.0.8. Scalar product. A scalar product π |π can be defined as follows. Let the reflected pattern of π be the pattern obtained by reflecting the disc with respect to its center. In this way, one obtains a plane with a disc removed around the origin. Place the pattern π to fill the hole so as to identify the marked points k = k of the two patterns on the common boundary. In this way, one obtains loops on the plane and a line connecting the origin to infinity if n is odd. The scalar product π |π is given by τ l1 τ l2 u a , where l1 , (l2 ), is the number of loops which do not (do) encircle the origin, and a is the number of windings the line going to infinity makes around the center. The T.L. generators are hermitian for this scalar product, and the generators gi , yi are unitary if we take the complex conjugation t ∗ = t −1 , u ∗ = u −1 . 3. Polynomial Representations We consider here polynomials in as many variables as there are line extremities attached to the boundary of a pattern. One of our aims is to identify representations of ATN , dual to those of the preceding sections, acting on these polynomials. This is done through the introduction of a vector whose components are polynomials indexed by the patterns, and by requiring that transforms in the same way under the two actions of the generators, on patterns or on polynomials. To obtain these irreducible representations of A N , we restrict the polynomials to obey some conditions called the wheel conditions. 3.1. The vector . The main problem of this section is to obtain a vector : = π ψπ (z i ), π
(15)
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constructed in the following way. The vectors π are basis vectors of a representation on which A N acts from the left. The ψπ are polynomials on which it acts from the right. We denote with a bar the right action of A N on polynomials to distinguish it from the left action. We want to determine the ψπ in such a way that both actions give the same result on the vector : e¯i = ei ,
(16)
σ¯ = σ .
Said differently, we identify the polynomials with the dual basis ψπ (z i ) of the representation. These conditions are equivalent to the conditions (53) introduced later. Using a physical picture, ψπ (z i ) is the amplitude to find the particles labelled by i at positions z i in the Resonance Valance Bond state (R.V.B.) π , where the lines represent the spin singlets. The first condition (16) is the q-deformed conditions for to be a bosonic wave function. For the disc and the punctured disc representations of Sect. 2.0.3, this problem is solved in Sect. 3.5. 3.2. The polynomials. We consider here Laurent polynomials in N variables z i±1 , 1 ≤ i ≤ N , which we identify with the generators z i of the double A.H.A. of Appendix A. We exhibit a representation of the double A.H.A. on these polynomials depending on two parameters t and q. We define permutation operators which permute the labels of the variables z i : z i si = si z i+1 , z i+1 si = si z i , and zl si = si zl if l = i, i + 1.
(17)
Using the construction of Appendix A.1.1, one recovers the Lascoux and Schützenberger [17] expression of Hecke generators e¯i (for 1 ≤ i ≤ N − 1) acting from the right of an expression as follows: 1
1
e¯i = (t 2 z i+1 − t − 2 z i ) (1 − si ) (z i − z i+1 )−1 , 1
1
e¯i − τ = (1 − si )(z i − z i+1 )−1 (t 2 z i − t − 2 z i+1 ).
(18)
Therefore, e¯i projects onto polynomials symmetrical under the exchange of z i and z i+1 , 1 1 and τ − e¯i onto polynomials divisible by (t 2 z i − t − 2 z i+1 ). Notice that the generators e¯i obey the Hecke, not the T.L. relations. It is convenient to define variables z i , i ∈ Z, which are cyclically identified as: z i+N = q −1 z i .
(19)
We can represent the cyclic operator σ¯ as: z i σ¯ = σ¯ z i+1 ,
(20)
and let it act as the identity when it is located left of an expression: 1σ¯ = c0 .
(21)
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Thus, on a homogeneous polynomials of total degree |λ|, σ¯ N = c0N q −|λ| , where c0 needs to be adjusted to satisfy the duality condition (16). In Appendix C we reproduce the results of [9] showing that for the dominant ordering of the monomials, the operators y¯i are represented by triangular matrices and we obtain their spectrum for t, q generic. We summarize the main results here: • A partial ordering on monomials can be defined as follows [12]. Given a monomial z λ = z 1λ1 . . . z λNN , we consider the partition λ+ obtained by reordering the λi in decreasing order. We say that λ ≥ µ if the following conditions are satisfied. The partition λ+ is larger than µ+ for the dominance order defined as: λ+1 + · · · λi+ ≥ µ+1 + · · · µi+ for all i ≥ 1.
(22)
When λ+ = µ+ , µ can be obtained from λ through a sequence of transformations ) → (λ , λ ) with λ > λ . (λi , λi+1 i+1 i i i+1 • The polynomial representations of the A.H.A. depend on the two parameters, t and q, and on a partition λ+ . The affine generators yi are lower triangular for the preceding order. An orthogonal basis is obtained by diagonalizing the yi simultaneously and a basis state Fπ (z i ) is characterized by its highest degree monomial: λ+π
Fπ (z i ) ∝ z 1
1
λ+π
. . . zN N + · · · ,
(23)
where π is a permutation of λ+ : (λπ )i = λ+πi . • Up to an overall normalization factor, the eigenvalues of the operators yk on the polynomials Fπ are obtained by permuting the eigenvalues yˆ a of ya on the highest weight polynomial F1 : Fπ yk = Fπ yˆπk , with :
(24)
yˆa = q −λa t (a−1) . +
3.3. Wheel condition. In this section, we introduce the vanishing conditions obeyed by the polynomials which enable us to construct the vector . These vanishing conditions, called the wheel conditions are studied in [6]. When certain conditions are obeyed by the parameters t, q, the space of polynomial obeying these wheel conditions form a representation of the A.H.A.. We give their definition and we explain why they are preserved under the action of the A.H.A.. We motivate the vanishing conditions from the Q.H.E. point of view by studying some examples in the next section. 3.3.1. Definition of the wheel conditions. Fix two integers, k and r , and two variables t and q related by: t k+1 q r −1 = 1.
(25) k+1
r −1
If m is the largest common divisor of k+1 and r − 1, we take t m q m = ω, with ω a primitive m th root of unity. We say that: A Laurent polynomial P(z i ), in N variables z i±1 satisfies the wheel condition (k, r ), if:
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• For any subset of k + 1 indices {i a }, 1 ≤ a ≤ k + 1. • For any set of k integers baa+1 ∈ N, 1 ≤ a ≤ k, such that: a) baa+1 = 0 ⇒ i a+1 > i a , b)
k
(26) baa+1 ≤ r − 2.
1
• P(z 1 , . . . , z N ) = 0 when we restrict the variables to satisfy the wheel conditions: z ia+1 = tq ba,a+1 z ia .
(27)
A set {i a },{baa+1 } satisfying the conditions (26) defines an admissible wheel, and the vanishing condition specified by this wheel is called a wheel condition. We shall mostly be interested in the simplest cases r = 2 where the rule simplifies drastically. One has q = t −(k+1) and given k + 1 ordered indices, 1 ≤ i 1 ≤ i 2 . . . ≤ i k+1 ≤ N , the polynomial must vanish when (z i1 , z i2 , . . . , z ik+1 ) = (z, t z, t 2 z, . . . , t k z). In Appendix D we show that the wheel conditions are preserved under the action of the A.H.A. 3.3.2. Admissibility conditions. Here, we recall the results of [6] about the polynomial representations of the A.H.A. when the wheel conditions are satisfied. When the condition (25), t k+1 q r −1 = 1, is satisfied, and for t generic, the representation admits an irreducible subrepresentation on polynomials satisfying the wheel condition. The basis states are eigenstates of the affine generators yi (98) and are proportional to the non-symmetrical Macdonald polynomials [9, 25] specialized at t k+1 q r −1 = 1. They are characterized by their highest weight monomial now subject to more restrictive admissibility conditions [6]: • A partition λ+ = (λ+1 , . . . , λ+N ) defines an admissible state if it satisfies: + λa+ − λa+k ≥ r − 1, ∀a ≤ N − k.
(28)
• A highest weight monomial is characterized by the shortest admissible permutation π such that one has: (λπ )i = λ+πi . A permutation π defining λπ is admissible if it satisfies the condition: + λa+ − λa+k = r − 1 ⇒ a = πi , a + k = π j with j > i.
(29)
In other words, the weight λπ = (λ+π1 , . . . , λ+π N ) is a permutation of λ+ such that λa+ + whenever λ+ − λ+ = r − 1. remains to the left of λa+k a a+k In the next section, we describe how the states of this representation can be encoded into tableaux, and in Sect. 6.1, we shall give a path description valid for the case of the two columns tableaux.
Interpolation of Polynomials Between Stationary Q.H.E. Ground State
3
1 2
2
1 3
409
1
2 3
Fig. 7. Tableaux representing the states (k, r ) = (2, 3) obeying the extended wheel condition p = 1 for three particles. The left box is disconnected from the right boxes
3.3.3. Tableaux. The tableaux [38, 39] give a convenient way to represent the states of the A.H.A. representations considered in the last section. An admissible permutation π determining the polynomial Fπ can be encoded by distributing the numbers i, 1 ≤ i ≤ N , into the boxes of a planar diagram. The boxes are labelled from 1 to N and the polynomial Fπ is represented by a tableau putting i in the box πi . We identify the tableau representing the polynomial Fπ with the permutation π . The position of the box occupied by the number i determines the partial degree in the variable z i and the eigenvalue of the affine generator y¯i on Fπ . Denote (xa , xa ), the cartesian coordinates of the box labelled a. Their sum labels the eigenvalue yˆa of ya (24), on the highest-weight polynomial F0 :
t xa +xa = yˆa ,
(30)
The product of the second coordinate with r − 1, (r − 1)xa , is the degree of the highest-weight polynomial F0 in the variable z a . Thus: k + a, r −1 λa+ . xa = r −1
xa = λa+
(31)
As explained in Appendix A.1.1, in order to obtain an irreducible representation, the rule of construction is such that a vertical move of one unit north or a horizontal move of one unit east has the effect to multiply the eigenvalue yˆa by a factor t. The value of xa modulo one, or equivalently the degree modulo r − 1, splits the boxes of a diagram into r − 1 classes, and it is convenient to split the diagram into r − 1 disconnected sub-diagrams. The admissibility condition (29) can be rephrased into the rule: • The numbers are strictly increasing down each column and across each row. These are precisely the rules defining standard tableaux (see Figs. 4 and 7). Starting from any polynomial, one generates the others by acting on it with intertwining operators Yi described in A.1.1. The tableaux are transformed into: Fπ → Fπ Yi = Fπ si , where si exchanges the positions of i and i + 1 in the tableau. Yi acts only if the two boxes are not adjacent in the same row or column. One can start from an arbitrary polynomial Fπ of the basis to generate the other basis elements by acting with the Yi upon it. In practise, the lowest degree polynomial has often a simple factorized expression and is a more convenient generator. In the next section, we exhibit some examples of these lowest degree polynomials.
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3.4. Explicit solutions and Q.H.E. interpretation. We give some explicit solutions of the wheel conditions here. Although we use the Q.H.E. terminology to motivate them from a physical point of view, the polynomials of this section can easily be obtained independently of any connection with the Q.H.E. We consider particles moving in the plane in a strong magnetic field projected to the lowest Landau level [35]. In a specific gauge the orbital wave functions are given by: z z¯ zk ψ N (z) = √ e− 4 , k!
(32)
where z = x + i y is the coordinate of the particle, and the magnetic length scale related to the strength of the magnetic √ field has been set equal to one. These orbitals are concentrated on shells of radius k occupying an area 2π . Each orbital is represented by a monomial z k . The quantum Hall effect [13] ground state is obtained by combining these individual orbitals into a many-body wave function. All the wave functions of system of N parti
z i z¯ i
cles have a common factor e− i 4 which we omit. Thus, a monomial z λ = z 1λ1 . . . z λNN describes a configuration where the particle j occupies the orbital λ j . The wave functions are linear combinations of such monomials. The physical properties are mainly characterized by the inverse filling factor which is the area occupied by a particle measured in units of 2π . It can only be defined in the thermodynamical limit, and is given by the limit when N → ∞ of the maximum degree in each variable is divided by the number of variables. The effect of the interactions is to impose some vanishing conditions when the variables are in contact: ∼ (z i − z j )m with m an integer when z i − z j → 0. A ground state wave function is a polynomial of the minimal degree obeying the constraints. The difference between the polynomials considered here and the Q.H.E. are the wheel conditions which reduce to the Q.H.E. vanishing conditions when t and q tend to 1 (in which case, k + 1 and r − 1 must be prime numbers). Through the examples considered here where the wave function has a product structure, it turns out that k has the interpretation of a spin index and r −1 has the interpretation of an inverse filling factor. The incompressibility condition translates into obtaining the representations of Sect. 3.3.3 having the most compact diagrams. • k = 1, solutions: Let us denote t (z 1 , . . . , z k ) the Vandermonde product: 1 1 t (z 1 , . . . , z k ) = (t 2 z i − t − 2 z j ).
(33)
1≤i< j≤k
A solution obeying the r wheel condition has the simple product structure: ψk=1,r (z i ) =
r −2
tq l (z 1 , . . . , z N ).
(34)
l=0
It obviously satisfies the k = 1, r wheel conditions for wheels with i 1 < i 2 , and if i 2 < i 1 , the wheel condition follows from the fact that (tq b12 )−1 = tq r −1−b12 . This solution is the q-deformation of a Laughlin wave function with an inverse filling factor r − 1 [13].
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Let us pursue the Q.H.E. analogy further. If we insert a magnetic flux in the system at the origin, this has the effect to multiply this wave function by a factor i z i . Thus, the orbital shells are expelled by one unit away from the origin and a region of area 2π is left vacant, which is as if the 1/(r − 1) particle had been removed from the origin. When the flux is inserted, the eigenvalue of the operator y1 gets multiplied by a factor q −1 , which can be interpreted as the phase acquired by the wave function when the particle winds around the flux. • r = 2, solutions with k arbitrary: Let us show in the simplest case r = 2 that k has the interpretation of a layer (spin) index. Consider N particles (variables) split into k layers of xl , 1 ≤ l ≤ k particles each. We denote zli , 1 ≤ i ≤ Nl the coordinates of the particles in the layer l and we order the indices so that li < l j for l < l . We say that two variables are in the same layer if they share the same index l. The particles of the same layer repel each other so that the wave function representing the system vanishes when the variables zl j = t zli for i < j, and the particles belonging to different layers do not interact. A ground state wave function representing this system obeys the (k, 2) wheel condition because given k + 1 variables, two of them necessarily belong to the same layer. A simple wave function obeying the vanishing conditions is thus given by: ψk,r =2 (z i ) =
k
t (zl1 , . . . , zl Nl ).
(35)
l=1
Let us consider the incompressible limit. We look for polynomials of the minimal degree obeying the wheel condition (the degree measures the extension of the wave function). Thus, we split the layers into two sets and set k = k1 + k2 . We fill each of the k1 first layers with N − 1 particles and each of the last k2 layers with N particles. According to the rules of the preceding Sect. 3.3.3, the polynomials belong to the irreducible representation characterized by the skew-diagram, k N /1k1 , with k1 columns of length N −1 and k2 columns of length N . The polynomial (35) coincides with the lowest weight of this representation. Notice that if we do not restrict to the minimal degree case and allow the number of particles to differ by more than one in between different layers, the representation is described by a tableau with more than k columns according to the rules of Sect. 3.3.3. Unless we specify it, we shall not consider these cases here. • k = 2, r = 3 solution: The case k = 2 and r , represents a system of two layers with an inverse filling factor r − 1, and for r = 3, is the q-deformation of the Haldane-Rezayi [14] wave function. We repeat the Q.H.E. construction in the q-deformed case here. There are N = N1 + N2 particles or variables split into 2 layers of N1 , N2 particles each. We label the variables by li, where l = 1, 2 is the layer index and i a particle index within each layer. We order them so that 1i < 2 j. We split the wave function ψk=2,r =3 (z i ) into a product: ψk=2,r =3 (z 1i , z 2 j ) = φ (z 1i , z 2 j ) t (z 11 , . . . , z 1N1 ) t (z 21 , . . . , z 2N2 ).
(36)
Due to the factors t , the wheel condition is satisfied if two indices i a , i a+1 involved in the wheel belong to the same layer and baa+1 = 0. In particular, this covers all the cases where the three variables belong to the same layer.
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By inspection, the wheels left to be considered are those for which: baa+1 = 0 ⇒ i a is in the first layer and i a+1 is in the second layer. Thus, if the two variables 1i and 2 j belonging to different layers participate to a wheel, the ratio z 2 j /z 1i is either t or t 2 q = (tq)−1 . The last two are equal due to the condition t 3 q 2 = 1 (25). Therefore, we must impose φ to vanish in these cases. For φ, we consider an expression of the form: φ(z 1i , z 2 j ) =
N1 N2 i=1 j=1
f (z 1i , z 2 j )
inf{N 1 ,N2 }
f −1 (z 1ik , z 2ik ),
(37)
k=1
where 1i k , 2i k is a maximal pairing between particles of the first layer with those of the second layer and N1 − N2 particles of the first (second) layer are not paired if N1 > N2 (N2 > N1 ). For f we take: f (z, w) = (t z − w)(z − tqw).
(38)
In all the wheel cases considered above, two of the three particles participating to the wheel, say 1i and 2 j, belong to different layers and are unpaired. Since the wave function φ contains the factor f (z 1i , z 2 j ) it vanishes for this wheel. Therefore, the wheel condition is always satisfied. The degree of φ can be reduced by antisymmetrizing it over the variables of the same 2 layer zli and by dividing the result by l=1 i< j (z li − z l j ). If we restrict to the minimal degree cases N1 = N2 = N or N1 = N − 1, N2 = N , using the rules of the preceding Sect. 3.3.3, we see that the polynomial obtained here is the lowest weight state of the representations characterized by the (skew)-diagram, 2 N or 2 N /1, with two columns of length N1 and N2 . When N1 = N2 , the wave function is proportional the Gaudin Determinant [41]: 1 t (z 11 , . . . , z 1r ) t (z 21 , . . . , z 2r ) N1=N2 . ψk=2,r=3 = Det 1 1 1 1 − − i< j (z 1i −z 1 j ) i< j (z 2i −z 2 j ) (t 2 z 1i −t 2 z 2 j )(t 4 z 1i −t 4 z 2 j ) (39) For r > 3, we were not able to obtain a simple trial polynomial obeying the (2, r ) wheel condition with the minimal degree. The polynomials obeying these wheel conditions can nevertheless be obtained by specializing the non-symmetric Macdonald polynomials at t k+1 q r −1 = 1. 3.4.1. Inserting a flux The polynomials with r > 2 allow to describe tangles with a flux inserted. Let us fix an integer p with 1 ≤ p ≤ r −1 and impose the additional wheel condition: • The polynomials vanish at least as p when two arbitrary variables z i and z j approach zero as . This constraint is preserved under the action of the A.H.A. It can be realized on the polynomial representations characterized by a diagram with two disjoined columns vertically shifted by p/(r − 1) with respect to one another. This results from the fact that given a polynomial in this representation and a monomial in its expansion, the sum of any two of its degrees is greater or equal to p. This is
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true for the highest monomial of an eigenstate of the affine generators y¯i , and thus for any monomial in its expansion by the construction of Appendix C.1. N1 ,N2 The p = 1 case can simply be obtained by multiplying the disc polynomial ψk=2 Nl by the product i=1 zli , where l ∈ {1, 2} is the column with the smallest length. 3.5. Link pattern basis. Here, we identify the polynomial basis dual to the pattern basis of Sect. 2.0.3. In the disc case, the basis coincides with the K-L basis and in the punctured disc case, the construction needs to be modified. 3.5.1. Kazhdan-Lusztig basis. We first give the general construction of the K-L basis [15] which we then specialize to the case of the disc patterns [16]. The generating polynomial is a product of q-Vandermonde determinants as in (35): ψ1 (z i ) =
k
t (zl1 , . . . , zl Nl ),
(40)
l=1
and the Nl are sorted decreasingly. It is more convenient here to index the variables from k Nl with li → N1 + · · · Nl−1 + i. 1 to N = l=1 We construct a representation of the Hecke Algebra called Specht representation [19] by letting the generators ei − τ (18) with 1 ≤ i ≤ N − 1 act on this polynomial. It has a K-L basis which we describe here. The basis states are given in terms of standard tableaux associated to a Young diagram as in Sect. 3.3.3. The lowest tableau, denoted 1, has its columns filled with consecutive numbers and corresponds to the polynomial ψ1 (it corresponds to the lowest tableau of Fig. 4). We number the boxes of diagram as in 1. The standard tableau π puts the number i in the box πi . It is generated from the tableau 1 by a succession of elementary permutations π → π si , where si exchanges the position of i and i + 1, and acts only if i + 1 is in a column to the right of i and not in the same line as i. This determines an order on tableaux, the dominance (or Bruhat) order: π ≥ π if it can be written in the preceding way as π = π w with w a word in the si . To the tableau π , we associate an element T¯π in the Hecke algebra obtained by substituting the generator T¯i to the permutation si in the above expression of π . We obtain a polynomial basis of the Specht module, labelled by tableaux, by letting T¯π act on ψ1 : Fπ (z i ) = ψ1 T¯π . 1 2
(41)
We define an antilinear (but preserving the order) involution by T (T¯i ) = T¯i−1 , 1
t → t − 2 and T (ψ1 ) = ψ1 . The K-L basis states ψπ are obtained from the basis states Fπ by a triangular transformation and are defined by the relations: T (ψπ ) = ψπ , 1
1
ψπ − Fπ ∈ ⊕π <π t − 2 Z[t − 2 ]Fπ .
(42)
Let us make a few observations about the K-L basis. The Specht representation coincides with the AHA representation of Sect. 3.3.3 only in the minimal degree case where the length of the columns of the Young diagram differ by at most one. In this case the cyclic permutation σ¯ (20) acts within the representation.
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If we denote by J the subset of {1, · · · , N }, where we omit the numbers {N1 , N1 + N2 , . . . , N }, the polynomial (40) obeys: ψ1 T¯i = t 2 ψ1 , ∀i ∈ J. 1
(43)
One can define a K-L basis for the induced module defined by the relation (43). The Specht representation is a subrepresentation of this induced module. One can define a KL-graph as in Sect. 3.5.1 and expression (5) of the matrices ei holds in the general case. This gives a practical way to construct the K-L basis. The tableau π carries the label i if i is in a column to the left of i + 1. An incomplete K-L graph (the Young graph) is drawn by connecting π to π when π and π differ by the permutation of two consecutive numbers. The trial basis states are obtained recursively by acting with e¯i − τ on ψπ carrying the label i and such that i and i + 1 are not in the same line: ψπ si = ψπ (e¯i − τ ) − µ(π, ν)ψν . (44) ν<π without label i
Using the first relation (16), one constructs the matrices ei − τ in the trial basis and one 1 completes the KL-graph using (44). Due to the factor t 2 in (43), the basis may violate the second condition in (42) (equivalently, ei − τ violates the non-negative integrality condition (5)). This is cured by completing the K-L graph with new links and by correcting the basis states (44) accordingly. The dual basis is also a K-L basis. It is generated by induction from the highest tableau 1 having its lines filled with consecutive numbers. In the dual construction, t − 2 must be 1 substituted to t 2 in the relations (42) and (43) and the conjugate diagram must be substituted to the original diagram. So, the set J is replaced by J = {1, . . . , N }\{N1 , N1 + N2 , . . . , N }, where the Ni are the lengths of the lines. Specializing to the two column diagrams, we identify the tableaux with the link patterns of Sect. 2.0.3. A tableau is the link patterns where the numbers in the left column encode the position of the α’s, and the numbers in the right column the positions of the β’s as in Fig. 4. The lowest tableau is the pattern having all the α’s precede the β’s. Two strings are connected by a link of the Young graph if they differ by a simple transposition of two consecutive α, β, and new links must be added to obtain the KL-graph according to the rules given in Sect. 2.0.4. For example, on Fig. 5, the KL-link connecting αααβββ to αβαβαβ is not a link of the Young graph. By transposing the relations (5) we obtain the action of e¯i − τ on the basis states: ψπ (e¯i − τ ) = −τ ψπ if πi πi+1 = αβ, = µ(ν, π )ψν , if πi πi+1 = αβ,
(45)
νi νi+1 =αβ
which give a convenient way to generate the basis.3 In the case where the number of β is equal to the number of α. The K-L involution T can be identified with the reflection T of Sect. 2.0.6. 3 Another way to proceed is to use the factorized expression of the basis states in terms of a product of Yang-Baxter operators acting on the lowest tableau [18].
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3.5.2. Punctured disc basis. Our aim here is to construct the basis of polynomials dual to the punctured disc patterns of 2.0.3 using the flux representation considered in 3.4.1. The representation cannot be generated using the Hecke algebra only and one must add the generator σ . We must enlarge the involution to include σ and we require that T (σ ) = σ . As a generator of the representation, we take the lowest patterns ω = β . . . α with the β’s preceding the α’s. This pattern is not in the image of any ei , and therefore, the polynomial ψω must be annihilated by the e¯i acting from the right. The minimal degree candidate is given by: ψω (z i ) = t (z 1 , . . . , z N ).
(46)
This polynomial has a highest degree monomial given by z 1N −1 z 2N −2 . . . z 0N . Thus the partition of this representation is given by: λ+ = N − 1, N − 2, . . . , 0.
(47)
ψω obeys the wheel condition k = 2, r = 3, which resumes to require that for triplet i < j < k the polynomial vanishes when z j = t z i or z k = t z i or z k = t z j . It also vanishes when two variables are set equal to zero. It therefore belongs to flux representation p = 1, r = 3 of Sect. 3.4.1. The construction of the basis proceeds as in the disc case where we use the generators ψωσ l = ψω σ¯ l to obtain the other basis states using (45) in each sector determined by l. To characterize the pattern representation, we need to determine the parameter τ (10). The ratio y+ /y− of the two possible eigenvalues of y1 , obtained from (69) and 1 from (24) must be equal. It gives q −1 t −1 = u 2 if N is even, and q −1 t −1 = u 2 t 2 if N is 3 odd, where q = t − 2 by condition (25). Thus, 1
u = t 2 if n is even, u = 1 if n is odd.
(48)
Therefore, in this representation, and if we parameterize τ = −2 cos (θ ), we obtain from (11) and (13): θ . (49) τ = 2 cos 2 We notice that in all cases, σ N = 1. So, in order to satisfy the second duality relation |λ| (16), the normalization factor of σ¯ must be taken equal to: c0 = q n in (21). 3.5.3. Conditional expectation value and involutions . We give two properties of these polynomials parallel to the link patterns properties discussed in Sects. 2.0.6 and 2.0.7: One can realize the involutions dual to those of section (2.0.6) through the transformations: −1 −1 ¯ F()(z 1 , . . . , z N ) = (z N , . . . , z 1 ), T¯ (t )(z 1 , . . . , z N ) = t −1 (z 1−1 , . . . , z −1 N ).
(50)
The projection E dual to the inclusion defined in (14) is defined in Appendix C.2 by specializing the two first variables to take the value 1 and t 2 . It sends polynomials in N
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variables obeying the vanishing condition into polynomials in N − 2 variables obeying the same property. By combining it with the cyclic transformation (20), it gives a way to decompose a polynomial satisfying the wheel condition in the link pattern basis by specializing its variables to be powers of t. 4. Positivity Conjectures The positivity conjectures are motivated by the R-S conjecture [4] which states that at τ = 1, the evaluations of the polynomials considered in the preceding section at z i = 1 count certain classes of alternating sign matrices. We claim that the evaluation of their deformations are positive polynomials in the deformation parameter τ (are in N[τ ]). Similarly, we observe that the evaluation of the cylinder polynomials are in N[τ ], where we set τ = −2 cos (θ ) and τ = 2 cos (θ/2). When τ = τ = 1, these polynomials count certain classes of half-turn symmetric alternating sign matrices [22]. A first observation concerns the K-L basis constructed from the product of Vandermonde determinants (40). We define the evaluation of a polynomial to be: ψ¯ π = N −1 ψπ (1, . . . , 1),
(51)
where the normalization factor N = ψ¯ 1 (1, . . . , 1) in (40). 1 Let us show that the evaluation of K-L basis polynomials ψ¯ π are in Z[t ± 2 ]. The 1 ψπ are polynomials in z i with coefficients in Z[t ± 2 ]. The property is true for ψ1 , and 1 preserved under the action of e¯i − τ (18). Thus, ψ¯ π ∈ Z[t ± 2 ] if ψ(1, . . . , 1) is divisible
by (t − 1)
l
Nl (Nl −1) 2 Nl (Nl −1) l 2
. This results from the fact that when t − 1 and z i − 1 are O()
). Again, the property is satisfied by ψ1 and preserved by the ∀i, ψπ is O( action of e¯i − τ (and σ¯ ). 1 We have verified that they are in fact positive in −t ± 2 . Let us consider more specifically the link pattern polynomials. The disc polynomials without line connected to the center are obtained by considering K-L representations having a Young tableau with two columns of equal length. 1 1 In this case, ψ¯ π is invariant under the transformation t 2 → t − 2 and is therefore in 1 1 Z[τ ]. To show this, let us consider the reflection T which takes z i → z i−1 and t 2 → t − 2 N
and multiplies the result by a factor (z 1 . . . z N )( 2 −1) . If a polynomial ψ invariant under 1 1 a T reflection, then ψ¯ is invariant under the transformation t 2 → t − 2 . T leaves ψ1 3 N invariant and commutes with e¯i −τ (and t − 2 ( 2 −1) σ¯ ). Therefore, the ψπ are left invariant under T and the property ψ¯ π ∈ Z[τ ] follows. We conjecture that the evaluation of the disc polynomials are positive in τ . In Table 1, we evaluate explicitly the polynomials ψ¯ π up to N = 8, and we observe that ψ¯ π ∈ N[τ ]. In Table 2, we evaluate the polynomials ψ¯ π with k = 2, r = 3, and N = 2, 4, (n) obtained from the polynomial ψ2,3 (36). The normalization factor in (51) is taken to be N
1
1
N
1
1
N
N
(−) 2 (t 4 − t − 4 ) N ( 2 −1) (t 4 + t − 4 ) 2 ( 2 −1) . This time, we observe that the evaluation polynomials are positive in the variable τ 2 = 2 − τ . In the case of the disc polynomials with a line connected to the center (when N is odd and the difference between the number of α’s and β’s is equal to one), one must consider the spin basis described in Appendix C.3 rather than the KL basis. In this
Interpolation of Polynomials Between Stationary Q.H.E. Ground State
417
Table 1. Evaluation of the disc polynomials for k = 2, r = 2 and N = 4, 6, 8. We index the polynomials ψ¯ π by the position of the α’s in the notation of 2.0.3. The evaluation is left unchanged under the flip isomorphism defined in 2.0.6 and 3.5.3: ψ¯ π = ψ¯ F (π )
12 13
123 124 134 125 135
1 τ
1 2τ τ2 τ2 τ (τ 2 + 1)
1234 1235 1245 1236 1345 1237 1246 1256 1346 1247 1356 1257 1347 1357
1 3τ 3τ 2 3τ 2 τ3 τ3 τ (5τ 2 + 2) τ4 τ 2 (2τ 2 + 1) τ 2 (2τ 2 + 1) τ 3 (τ 2 + 2) τ 3 (τ 2 + 2) τ (τ 4 + τ 2 + 1) τ 2 (τ 4 + 3τ 2 + 3)
Table 2. Evaluation of the disc polynomials for k = 2, r = 3 and N = 4, 6
12 13
1 + τ 2 2
123 124 134 125 135
τ 6 + 2τ 4 + 3τ 2 + 1 2τ 4 + 8τ 2 + 4 3τ 2 + 4 3τ 2 + 4 τ 4 + 3τ 2 + 10
basis, the reflection T acts by inserting a flux: it multiplies the polynomials by a factor N µi2+1 , where µi = ±1 is the value of the spin. Therefore, in the spin basis, the i=1 z i evaluation at z i = 1 of the components of are in Z[τ ] and we conjecture that they are positive. In the punctured disc case, we have considered the simplest cases where the representation is generated from the Vandermonde determinant (46) (which corresponds to the case k = 2, r = 3, p = 1, in the notations of Sect. 3.5). The normalization factor N
1
1
N ( N −1) 2
in (51) is taken to be N = (−) 2 (t 4 + t − 4 ) 2 1 4
− 14
(N −1)2 4
1 4
− 14
N (N −1) 2 1 2
1
1
(t 4 − t − 4 )
N (N −1) 2
if N is even, and
(t − t ) if N is odd. One can show the same integrability N = (t + t ) 1 conditions as in the disc case with t → t 4 . The explicit evaluations of Table 3 lead us to conjecture that the evaluations are positive in τ . 5. Transfer Matrix 5.1. as an eigenvector of the transfer matrix. In the disc case, at t 3 = q = 1, the vector (15) is an eigenstate of a commuting family transfer matrix T (z 0 ) = T (z 0 |z i ) [5]. The matrix elements of T (z 0 ) are polynomials in the variables z 0 , z i called the spectral parameters. We give a simple proof that in the limiting case where cyclicity is recovered: z i+N = z i , a vector (z i ) obeying the duality relations (16) can be obtained as the stationary state of a transfer matrix. In this case, by specializing the spectral parameters of the transfer matrix, one obtains a Hamiltonian, H = 1N ei , with positive integer matrix elements having (1, . . . , 1)
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Table 3. Evaluation of the punctured-disc polynomials for k = 2, r = 3, p = 1 and N = 2, 3, 4, 5, 6. When N is odd, the natural variable is τ 2
1 2
1 τ
12 13 23
1 1 τ 2
12 13 23 14 24 34
125 123 145 234 135 124 245 235 134 345
1 τ 2 + 2 τ τ τ (τ 2 + 2) τ 4
123 124 125 134 126 234 135 136 235 145 146 245 156 345 236 246 256 346 356 456
1 1 τ 2 τ 2 τ 2 + 3 τ 2 + 3 τ 2 (τ 2 + 3) τ 2 (τ 2 + 3) τ 4 + τ 2 + 2 τ 6
1 τ 2 + 4 (τ 2 + 2)2 (τ 2 + 2)2 τ τ τ 6 + 3τ 4 + 11τ 2 + 10 τ (τ 2 + 4) τ (τ 2 + 4) τ (τ 6 + τ 4 + τ 2 + 2) τ (τ 4 + 3τ 2 + 5) τ (τ 4 + 3τ 2 + 5) τ 4 τ 4 τ (τ 4 + 5τ 2 + 3) τ (τ 6 + 3τ 4 + 11τ 2 + 10) τ 4 (τ 2 + 4) τ 4 (τ 2 + 4) τ 4 (τ 4 + 2τ 2 + 6) τ 9
as a stationary state (notice that the cyclic operator σ must be defined in order to construct e N ). Thus, by the Perron-Frobenius theorem, the components of (1, . . . , 1) can be normalized to be positive integers. These integers are conjectured to count ASM [1]. At this value, the sum of the components of (z i ) is a symmetric polynomial in the z i which can be evaluated explicitly [5]. We give a derivation of this property relying on the fact that the T.L. algebra becomes non-semisimple and can be realized trivially, (ei = 1), at the corresponding values of t. Let us define the following permutation operators acting on the spins at positions i and j: Yi j
zi zj
=
z j Ti j − z i Ti−1 j 1
1
zi t 2 − z j t − 2
.
The vector can also be characterized by the conditions equivalent to (16)[7]: zi Yi (. . . , z i , z i+1 . . .) = (. . . , z i+1 , z i . . .), z i+1 σ (z 1 , . . . , z N −1 , z N ) = (z 2 , . . . , z N , q −1 z 1 ).
(52)
(53)
The first relation (16) can be straightforwardly verified by substituting (52) into (53). Note that the normalization of Yi j has been chosen so that Yii+1 ei = ei , and therefore e¯i (18) projects onto a polynomial symmetric under the exchange of z i , z i+1 . It is convenient to introduce the permutation Pi j which acts by permuting the indices i and j. One then defines the operators X i j = Pi j Yi j ( zzij ) obeying the Yang-Baxter equation (equivalent to the fact that the Yi j are permutation operators): X i j X ji = 1,
Interpolation of Polynomials Between Stationary Q.H.E. Ground State
X i j X kl = X kl X i j , if i, j = k, l, X i j X ik X jk = X jk X ik X i j .
419
(54)
Using the Yang Baxter operator, we can define the commuting operators y˜i by: y˜i = X ii−1 . . . X i1 qˆi . . . X ii+1 ,
(55)
with qˆi acting from the right shifts on the variable z i by a factor q (100). By commuting qˆi to the left, we replace the operators X i j located to its left by X i j+n . When we act from the left with y˜i on , using repeatedly the first equation (53), one sees that the product of X ik substitutes the variables z i to z i+n in . Conversely, the operator qˆi replaces z i with z i+n . So is an eigenvector with eigenvalue 1 of y˜i : y˜i = .
(56)
In what follows, we restrict to the case where qˆi = 1, and we identify the transfer matrix as the generating function of the y˜i . The transfer matrix has the expression:
z0 z0 z0 T (z 0 |z i ) = tr X 01 X 02 . . . X 0n , (57) 0 z1 z2 zN where the partial trace (defined in Sect. 2.0.1) is on the label 0. It follows from (54) that two matrices with different spectral parameters z 0 and w0 and all the other spectral parameters equal commute with each other [41, 42]: [T (z 0 |z i ), T (w0 |z i )] = 0.
(58)
When the shift operator qˆi = 1 which occurs when q = 1, it is straightforward to verify that y˜i is obtained by substituting z i to z 0 in the expression of T (z 0 ) = T (z 0 |z j ) [41]: y˜i = T (z i ).
(59)
In the pattern representation, and in the case τ = τ = u = 1, the T.L. matrices ei (5) transform a pattern into a single pattern with a coefficient equal to one. It follows from this property that the line-vector having all its entries equal to one is a left eigenvector of T (z 0 ) with the eigenvalue 1. As a result, T (z 0 ) has a right eigenvector with the eigenvalue 1 which we determine to be . From (59), (T (z 0 ) − 1) is a rational fraction, with a numerator of degree N in z 0 . It vanishes when z 0 = z i . One also has T (0) = T (∞) = 1, and thus (T (z 0 ) − 1) also vanishes when z 0 = 0, ∞. It is therefore equal to zero and is an eigenvector of T (z 0 ) with the eigenvalue 1. 5.1.1. Sum of the components. At the cyclic point q = 1, it follows from the explicit expression (5) of the matrices ei , that the vector χ with all its entries equal to one is a left eigenvector of the ei ’s with the eigenvalue one. Thus, the scalar product χ ., equal to the sum of the components of , is by the duality relation (16) a symmetric polynomial. We obtain its expression here (see also [27]). Up to the normalization factor, this sum is determined to be the lowest degree symmetrical polynomial obeying the wheel condition.
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For the minimal degree disc representation generated by (35) with k = 2, the polynomial vanishes when three ordered variables are specialized by the r = 2 wheel: z i = z, z j = t z, z k = t 2 z for i < j < k. It has the degree: λ = (N − 1, N − 1, N − 2, N − 2, . . . , 0, 0),
(60)
if the number of variables is even (N1 = N2 = N ), and the first N − 1 is erased if the number of variables is odd (N1 = N , N2 = N − 1). This determines it to be equal to the Schur function sλ [7]. An analogous discussion can be made in the punctured disk case. Z N can be determined by using a recursion argument on N as follows. Let us normalize Z N (z i ) so that its highest degree monomial Z N (z i ) is equal to z 1N −1 z 2N −2 . . . z 0N . Using the projection (107), one has: E (Z N )(z 3 , . . .) = Z N −2 (z 3 , . . .).
(61)
By recursion, this condition determines Z N (z i ) completely to be the product of two Schur functions: Z N (z i ) = Sλ1 ,N Sλ2 ,N ,
(62)
where λ1 = (0, 0, 1, 1, 2, 2 . . .) and λ2 = (0, 1, 1, 2, 2 . . .) and each partition has N rows. Indeed, the product Sλ1 ,N Sλ2 ,N has the same highest degree monomial as Z N , and each Schur function factorizes as follows when one specializes the values of the two first variables: Sλ1 ,n (z 1 = z, z 2 = t z, z 3 , . . .) =
N (t 2 z − z i )Sλ1 ,N −2 (z 3 , . . .), i=3
Sλ2 ,n (z 1 = z, z 2 = t z, z 3 , . . .) = z
N
(63) (t z − z i )Sλ2 ,N −2 (z 3 , . . .). 2
i=3
The product has the same highest monomial as Z N and obeys the same recursion relation (61), it is therefore equal to Z N .
6. Spin and SOS Representations We characterize some representation of the T.L. affine algebra ATN (t) acting on a spin or a path basis. The representation depends on two positive integers r and s, and on a continuous parameter u. When t is generic, and for |r − s| ≤ 1, these representations are isomorphic to the pattern representations of Sect. 2.0.3. In the spin basis, r and s are respectively the number of + and − spins, u is a twist parameter, and the operators yi are realized as triangular matrices. On the other hand, the paths form an orthogonal basis and the affine generators yi are realized as diagonal matrices in this basis.
Interpolation of Polynomials Between Stationary Q.H.E. Ground State
421
6.1. Paths. We describe here the action of the generators of the A.H.A. on the paths of the SOS representation. This is essentially the specialization to the two column tableaux of the construction done in Appendix A.1.1. The spin basis also admits a representation described in Appendix C.3. In the generic case, the two bases are related by a triangular transformation, and the affine generators yi are realized as triangular matrices in the spin basis and as diagonal matrices in the path basis. There is an analogous case of the K-L basis in the spin case known as the canonical basis [24] which we do not discuss here. The path basis states are directed paths π on the square lattice. The path π is a sequence of lattice points πi = (xi+ , xi− ), 0 ≤ i ≤ N . It starts from the origin (0, 0), and moves by steps of one unit towards the north-east or the south-east: − + (xi+1 , xi+1 ) = (xi+ + 1, xi− ) or (xi+ , xi− + 1),
(64)
to reach the final point (x N+ , x N− ) = (N+ , N− ). The path can also be described by (i, h i ) = (xi+ + xi− , xi+ − xi− ), where h i defines the height of the path point i. Thus the path starts from the height 0 to reach the height N+ − N− in N+ up and N− down steps. The affine generator yi acts diagonally on a path by looking at the i th step between i −1 + ) if this step is up (towards the north-east), and to yˆ (x − ) and i. It is equal to yˆ+ (xi−1 − i−1 if it is down (towards the south-east), where yˆ+ (a) = y+ t −a and yˆ− (a) = y− t −a . Let us determine the expression of the T.L. generators so that the relations (86, 80) are satisfied. Since ei commutes with yl for l = i, i + 1, we require that ei acts locally on the piece of path between i − 1 and i + 1. Therefore, the projector ei is equal to zero if h i−1 = h i+1 , and it decomposes into block matrices, ei = ⊕ δh i−1 −h eh , where eh is equal to zero on paths such that h i−1 = h or h i+1 = h. It acts as a two by two matrix on a pair of paths equal everywhere except at the three consecutive points (i − 1, i, i + 1) where their heights take the values: [h, h − 1, h] and [h, h + 1, h]. Then, writing the last relation of (80) in terms of the T.L. generators: 1
1
t − 2 yi − t 2 yi+1 = yi+1 ei − ei yi ,
(65)
and substituting the matrix of eh into this equality, we determine its diagonal elements. Finally, by requiring that ei is proportional to a projector and satisfies (86), we determine eh up to a diagonal similarity transformation to be given by [30]: eh = −
1 Sh−1 Sh+1 , Sh Sh−1 Sh+1
(66)
where Sh is defined as: h
h
Sh = y− t 2 − y+ t − 2 ,
(67)
and obeys the recursion relation: Sh−1 + Sh+1 = −τ Sh .
(68)
With this normalization, the paths form an orthogonal basis and the square of their norm is i Si . In Appendix A.1.1, we define operators Yl which permute the two paths h l = h ± 1 when h l−1 = h l+1 = h.
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M. Kasatani, V. Pasquier
The two possible eigenvalues y± of y1 are defined up to a common factor. It is convenient to fix this normalization by setting: y+ = ut
N+ −1 2
,
y− = u −1 t
N− −1 2
,
(69)
so that we have: y1 y2 . . . y N = σ N = u N+ −N− .
(70)
We can identify this representation with the spin representation of Appendix C.3 with the same value of u by taking N+ to be the number of + spins and N− to be the number of − spins. We also recognize the pattern representations of Sect. 2.0.3 if we identify the values of u, and take: N N , N− = , if N is even, 2 2 N +1 N −1 N+ = , N− = , if N is odd. 2 2 N+ =
(71)
The spin representation coincides with the pattern representation with N+ = N− = if N is even and N+ = N2+1 , N− = N 2−1 if N is odd.
N 2
k
6.1.1. Restricted paths. If parameterize u as u = t 2 and define the spin of the path to − , we can characterize the representation by its basis states given by paths be S z = N+ −N 2 of N steps starting from the initial height h 0 = −k − S z and reaching the final height (N ) h N = −k + S z . We denote ρhh , the representations obtained for h 0 = h, h N = h . The factors y+ , y− in the definition of Sh are absorbed in the redefinition of the height so that one has: h
h
Sh = t 2 − t − 2 ,
(72)
and h
y+ = t − 2 t
N −1 4
h
, y− = t 2 t
N −1 4
.
(73)
If h 0 is integer, for Sh to be defined, the height must be restricted to be strictly positive. We can encode a path into a two line tableau where the numbers i = 1, 2, . . . , n are successively registered in the first or the second line according to whether the i th step is towards the north-east or the south-east. The abscissa of the first line are h 0 , h 0 + 1, . . . and the abscissa of the second line are 1, 2, . . .. For h 0 generic, one obtains in this way the standard tableau with two lines of length N+ and N− described in Appendix A.1.1.
Interpolation of Polynomials Between Stationary Q.H.E. Ground State
423
7. Coxeter-Dynkin Diagram Representations and Action of the Modular Group on the Trace This section lies somewhat outside the scope of the paper. We consider more specifically N of the affine T.L. algebra AT (t), in the root of unity case. We the representations ρhh N construct representations associated to a Coxeter-Dynkin diagram which we decompose N . This decomposition is independent of N . It is consistent with an action of into the ρhh the modular group, which leaves the Coxeter-Dynkin diagram trace invariant but acts on N . It can be viewed as a finite size version of the modular invariant partition function ρhh of conformal the field theories (CFT).4 If the heights defining the paths of Sect. 6.1 are integer, we can restrict the paths to have a strictly positive height. Similarly, when t is a root of unity, S p = 0 in (73) and we can restrict the height to be strictly less than p. The paths obeying these restrictions are called restricted solid on solid (RSOS) paths [42]. (n) In the RSOS case, the basis states of ρhh are the paths of length N starting from the height h and ending at the height h , such that the heights h i obey the constraint 1 ≤ h i ≤ p − 1. We now construct a representation of ATN (t) associated to an arbitrary finite bipartite graph D which we identify with its incidence matrix [30]. The Hilbert HDN is defined by its orthogonal basis given by the closed paths of length N : |a0 , a1 , . . . , a N = a0 , drawn on D, such that the two vertices ai and ai+1 are adjacent on the graph. Let Sa be the components of an eigenvector of D, and τ = −(t + t −1 ) be the corresponding eigenvalue. The T.L. generators el are defined similarly as in (66): el acts locally on the piece of path between l − 1 and l + 1, it is equal to zero if al−1 = al+1 . It decomposes into block matrices el = ⊕ ea , where ea acts on the pieces of path: |al−1 = a, al = b, al+1 = a = |b and is given by: a ebc =−
Sc . Sa
(74)
The cyclic operator σ cyclically shifts the paths by one unit: σ |ai = |ai−1 . These representations are particularly interesting in the case where the incidence matrix of the diagram has a Perron-Frobenius eigenvalue less than two. The diagram is then a Coxeter-Dynkin diagram D ∈ Am , Dm , E 6 , E 7 , E 8 , and we choose Sh to be the Perron-Frobenius eigenvector of D. t is then a primitive root of unity, t = e p is the Coxeter-number of the diagrams given below:
2iπ p
, where
Am , Dm , E 6 , E 7 , E 8 ↔ m + 1, 2(m − 1), 12, 18, 30.
(75)
N : We can decompose the Hilbert space HDN into the irreducible representations ρhh
D N HDN = ⊕γhh ρhh ,
(76)
D ∈ N are the multiplicities. where γhh By adapting the arguments of [31, 32], we compute the multiplicities in Appendix E and we show that they are independent of N when it is large enough. The coefficients of the decomposition coincide with the coefficients of the character decomposition of parafermionic C.F.T. unitary models [33, 34]. More precisely, one can define an action of the 4 A geometrical interpretation of the splitting into two independent numbers h, h can be given from the three dimensional Topological Quantum Field Theory (or annular tensor category) point of view [40].
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M. Kasatani, V. Pasquier
modular group which leaves the trace of the D representations invariant and transforms the trace of ρhh as the tensor product of the two characters χh and χh of the Virasoro algebra entering the decomposition of the partition function. 8. Conclusion We have deformed the stationary state of a O(1) model defined on the disc and the cylinder. This has led us to study polynomial representations of the affine Hecke algebra depending on two complex parameters t and q related by the relation t k+1 q r −1 = 1. These polynomials obey some vanishing conditions and interpolate between the stationary state of a stochastic transfer matrix at t k+1 = 1 and a Q.H.E. wave function at t = 1. In the cases presented here (k = 2), the transfer matrix is that of a O(n) model with n = 1. Another family not considered here interpolates between the stationary state of a O(1) model related to the Birman-Wenzl-Murakami algebra and the Pfaffian state of the Q.H.E. [45]. One can distinguish a basis labelled by link patterns which in the disc case coincides with the Kazhdan-Lusztig basis. We conjecture that the specialization of the basis polynomials at z i = 1 are positive in the loop fugacity. Moreover, when the loop fugacity is equal to one, they count certain classes of alternating sign matrices. It is intriguing that the spectral parameters of a transfer matrix can also be viewed as the coordinates of particles moving in the plane, so that the braiding properties of the polynomials are analogous to the fractional statistics properties of the Q.H.E. wave functions [47, 48]. The vanishing conditions obeyed by the polynomials are the q-deformed vanishing conditions of the Q.H.E. wave functions [49], and we suspect that their positivity properties are in some way related to the incompressible (minimal area for the quantum state which translate into minimal degree condition) properties of the corresponding Q.H.E. wave functions. Finally, in another direction, we have established that the unitary representations of the A.H.A. at roots of unity obey modular properties similar to those of conformal field theories. Acknowledgements. We thank P. Di Francesco, T. Miwa, V. Jones, B. Nienhuis, A.V. Razumov, Y.G. Stroganov, T. Suzuki, K. Walker, P. Zinn-Justin and J-B. Zuber for interesting discussions and comments. We thank A. Lascoux for explanations which helped us to considerably improve the revised version of this paper. This work is partially supported by Grant-in-Aid for JSPS Fellows No. 17-2106 and by the ANR program “GIMP”, ANR-05-BLAN-0029-01. Note added in proof. Since we proposed the positivity conjectures, some progress has been made confirming their validity. P. Di Francesco has conjectured that the sum of the evaluations at z i = 1 of the disc polynomials (Table 1) are related to a q-enumeration of Totally Symmetric Self-Complementary Plane Partitions [51]. In collaboration with A. Lascoux, one of us has established a similar relation for the highest degree coefficients in τ of these evaluations [52].
A. Affine Algebras In this appendix, we give the defining relations of the double A.H.A. The Hecke algebra depending on the parameter t, A N (t) (or A N when there is no possible confusion), is generated by the generators T1 , T2 , . . . , TN −1 ,5 obeying the braid 5 Sometimes, we use the notation T ii+1 for Ti .
Interpolation of Polynomials Between Stationary Q.H.E. Ground State
425
relations: Ti Ti+1 Ti = Ti+1 Ti Ti+1 Ti T j = T j Ti if |i − j| > 1,
(77)
and the quadratic relation: 1
1
(Ti − t 2 )(Ti + t − 2 ) = 0,
(78)
for 1 ≤ i ≤ n − 1. 1 If we define the generators ei = Ti − t 2 , the ei are projectors obeying the Hecke relations: ei2 = τ ei , ei e j = e j ei if |i − j| > 1, ei ei+1 ei − ei = ei+1 ei ei+1 − ei+1 , 1
(79)
1
where τ = −t 2 − t − 2 . The A.H.A. [23], is an extension of the Hecke algebra (78) by the generators yi , 1 ≤ i ≤ n obeying the following relations: yi y j = y j yi , Ti y j = y j Ti if j = i, i + 1, Ti yi+1 =
yi Ti−1
(80)
if i ≤ N − 1.
The double A.H.A. [8, 9], is the extension of the A.H.A. obtained by adjoining to it operators denoted z i . The z i obey the same commutation relations (80) as the affine generators yi with the generators Tk . It depends on an additional parameter q: zi z j = z j zi , Ti z j = z j Ti if j = i, i + 1, Ti z i+1 =
z i Ti−1
(81)
if i ≤ N − 1.
The relations obeyed by the z i and the yi due to Cherednik [8] are given by: y1 z 2 y1−1 z 2−1 = T12 , ⎛ ⎞ ⎛ ⎞ N N yi ⎝ z j⎠ = q ⎝ z j ⎠ yi , ⎛ zi ⎝
j=1 N j=1
⎞
j=1
⎛
y j ⎠ = q −1 ⎝
N
(82)
⎞ y j ⎠ zi .
j=1
A more elementary presentation [9] is in terms of the Ti , z i and of a cyclic generator σ defined as: −1 σ = TN−1 −1 . . . T1 y1 .
(83)
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M. Kasatani, V. Pasquier
We extend the definition of the variables z i , i ∈ Z, by cyclicity: z i+N = q −1 z i .
(84)
Similarly, one can define a braid generator TN by TN = σ T1 σ −1 . Using the braid relations again, one gets σ TN = TN −1 σ , and one can extend the definition of Ti to i ∈ Z: Ti+N = Ti . The defining relations of σ are then: σ Ti = Ti−1 σ,
(85)
σ z i = z i−1 σ.
The double affine Hecke algebra is generated by the generators Ti , z i and σ (85). In Appendix C, we reconstruct the generators yi from this presentation In this paper, we are often concerned with the T.L. quotient ATN (t) of the A.H.A. generated by ei , yi where we constrain the generators ei to obey the restrictions: ei ei±1 ei = ei .
(86)
A.1. Intertwining operators. A.1.1. Tableaux representations of the A.H.A. . Following [36], one can define operators Yl which intertwine the affine generators: y j Yl = Yl y j if j = l, l + 1, yl Yl = Yl yl+1 , yl+1 Yl = Yl yl .
(87)
The relations (87) are satisfied by Yl = yl Tl−1 − yl+1 Tl , 1
1
= (t 2 yl − t − 2 yl+1 ) + (yl − yl+1 )(τ − el ).
(88)
Thus, Yl intertwines the eigenstates of yl and yl+1 . The square of Yl acts diagonally on such states: 1
1
1
1
Yl2 = (t 2 yl − t − 2 yl+1 )(t 2 yl+1 − t − 2 yl ),
(89)
and is null on the states with yl+1 = t ±1 yl . We can use this property to construct a representation of the Hecke algebra on standard tableaux. A standard tableau is a right eigenvector of the yi ’s and the eigenvalue of yi depends on the box of the diagram where the number i is located. Each box is assigned an eigenvalue so that a vertical move of one unit north or a horizontal move of one unit east multiplies the eigenvalue by a factor t. Thus, the two eigenvalues of yl and yl+1 differ by a factor of t when the numbers l and l + 1 belong to adjacent boxes of the same column with yl = t yl+1 , or of the same line with yl+1 = t yl . The tableau is annihilated by Yl in both cases. It is annihilated by el in the column case and by el − τ in the line case.
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427
If l and l + 1 do not belong to adjacent boxes, Yl exchanges the position of l and l + 1. 1
1
We can multiply Yl by a normalization factor equal to (t 2 yl − t − 2 yl+1 )−1 so that the square of the resulting operator, Yl , is equal to one: Yl = 1 +
yl − yl+1 1 2
1
t yl − t − 2 yl+1
(τ − el ).
(90)
If we adopt a normalization so that Yl permutes the two tableaux, we recover the representation [43] of the Hecke algebra on tableaux. Consider two tableaux which are exchanged by Yl . We denote yˆ and yˆ the eigenvalue of yl and yl+1 on the first tableau (and of yl+1 and yl on the second tableau). The expression of Tl in the basis made by these two tableaux is given by: 1 1 1 1 1 (t 2 − t − 2 ) yˆ −t 2 yˆ + t − 2 yˆ . (91) Tl = 1 1 1 1 yˆ − yˆ −t − 2 yˆ + t 2 yˆ (t − 2 − t 2 ) yˆ In the path basis of Sect. 6.1, the paths are annihilated by Yl if h l+1 = h l−1 . Yl acts locally on the piece of path between l − 1 and l + 1 as follows: it is equal to zero if h l−1 = h l+1 , and it decomposes into block matrices Yl = ⊕ Ylh , where Ylh acts on paths with h l−1 = h l+1 = h by exchanging the two paths with the intermediate heights h l = h − 1 and h l = h + 1. All the paths are obtained from one of them under the repeated action of the generators Yl . In the cases where h l are integers, the paths must be restricted to have h l > 0. Furthermore, when t p = 1, the paths must be restricted to have h l < p. A.1.2. Shift operators. Following [36], we give the expression of an operator which changes the degree of a polynomial by keeping it an eigenstate of the operators yi . As in (84) for the coordinates z i , it is convenient to extend the definition of the affine generators yi to i ∈ Z by periodicity: yi+N = q −1 yi .
(92)
Using the affine Hecke relations (81), one can construct a shift operator A¯ which shifts by one unit the affine generators: ¯ A¯ y¯i = y¯i+1 A, ¯ A¯ g¯i = g¯i+1 A.
(93)
It raises the degree of the polynomial by one and its expression is given by A¯ = z 1 σ¯ −1 .
(94)
B. Explicit Matrices in the Cases n=2,3 In the basis αβ, βα, the action of A2T is generated by the matrices: 0 1 τ τ . , σ = e1 = 1 0 0 0
(95)
The T.L. algebra relation (86) is modified into: e1 e2 e1 = τ 2 e1 because the left-hand side creates two lines around the torus in this case.
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In the basis ααβ, αβα, βαα the generators of A3T 0 0 0 0 e1 = 1 τ u , σ = u 0 0 0 0 The matrix y1 is given by:
are given by: 0 1 0 0 . 1 0
(96)
⎛
1 ⎞ u u −1 −t − 2 1 y1 = ⎝ 0 u −1 t − 2 + u −t −1 ⎠ . 1 0 t2 0
(97)
C. Representation of the y¯ j in the Polynomial and the Spin Cases C.1. Polynomial representation. We repeat here the diagonalization of the operators y¯ j done in [9]. The method follows the one initiated by Sutherland [50] in the CalogeroSutherland model context. For a certain ordering of the monomial basis, we obtain an expression of yi in a triangular form by decomposing it into a product of triangular matrices xi j and a diagonal operator qˆi . The expression of y¯i which follows from (83) and (80) is given by: −1 y¯i = T¯i T¯i+1 . . . T¯N −1 σ¯ T¯1−1 . . . T¯i−1 .
(98)
It is convenient to reexpress yi as a product of triangular operators. For this, we decompose the cyclic operator sigma as a product of elementary permutation si which permute the variables z i and z i+1 and an operator qˆ1 which replace the variable z 1 with q −1 z 1 : σ¯ = c0 s N −1 . . . s2 s1 qˆ1 ,
(99)
with qˆi given by: P(z 1 , . . . , z i . . . , z N )qˆi = P(z 1 , . . . , q −1 z i , . . . , z N ).
(100)
We define xi j = T¯i j si j , where si j is the permutation operator which permutes the 1 variables z i and z j . T¯i j is defined in the same way as T¯i = T¯ii+1 = t 2 + e¯i , where we replace the variable z i+1 with the variable z j and the permutation si with permutation si j in the expression (18) of e¯i . The operator xi j takes the form for i < j: 1
1
1
xi j = −t − 2 + (t 2 − t − 2 )(1 − si j )
zj . zi − z j
(101)
After permuting the si through, the expression of y¯i becomes: −1 −1 . . . xi−1i . y¯i = c0 xii+1 xii+2 . . . xi N qˆi x1i
(102)
The operator xi j commutes with z i z j and acts as a triangular matrix on the monomials z im , z mj : 1
1
1
z im xi j = −t − 2 z im + (t 2 − t − 2 )(z im−1 z j + z im−2 z 2j + · · · + z mj ) if m ≥ 0, 1
1
1
) if m > 0. z mj xi j = −t 2 z mj − (t 2 − t − 2 )(z im−1 z j + z im−2 z 2j + · · · + z i z m−1 j
(103)
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Thus, the operators y¯i are also realized as triangular operators in the monomial basis. µ µ We order monomials by saying that z λ = z 1λ1 . . . z 1λ N is larger than z µ = z 1 1 . . . z NN if either µ is obtained from λ by a sequence of squeezing operations {λi , λ j } → {λi − 1, λj + 1} with λi > λj + 1, or µ is a permutation of λ and can be obtained from λ by a ) → (λ , λ ) with λ > λ sequence of permutations (λi , λi+1 i+1 i i i+1 . It follows from the expressions (103) that the action of y j on a monomial produces only monomials which are smaller with respect to this order, and their eigenvalues are given by the diagonal elements in the monomial basis. A common eigenstate of the operators y¯i is characterized by its highest degree monomial z λπ , where π is the shortest permutation such that (λπ )i = λ+πi and λ+ is a partition (λ+1 ≥ λ+2 . . . ≥ λ+N ). The action of y¯ j on a monomial z λπ is given by 1
z λπ y¯ j = c0 (−t − 2 ) N −1 q
−λ+π
j
t π j −1 z λπ + lower terms,
(104)
from which the eigenvalues of y¯ j follow. The global normalization of y¯i is such that: y¯1 y¯2 . . . y¯ N = σ¯ N = c0N qˆ1 . . . qˆ N = c0N q −|λ| .
(105)
C.2. Conditional expectation value in the T.L. cases. When the A.H.A. reduces to a T.L. algebra ATN , in the case of minimal degree polynomials with k = 2, we can define a projection E from the polynomials in N variables to the polynomials in N − 2 variables dual to the inclusion defined in (14). For any polynomial F, E satisfies the conditions: a) E (Fe1 ) = τ E (F), b) E (F x) = E (F)x ∀x ∈ ATN −2 , c) E (Fe1 ) = 0 ⇒ Fe1 = 0,
(106)
and can be realized as: E (F)(z 3 , z 4 , . . .) = φ(z, z 3 , z 4 , . . .)−1 F(z 1 = z, z 2 = t z, z 3 , z 4 , . . .),
(107)
where φ(z, z i ) is a polynomial which removes the z dependence of F in the right-hand side of (107) equal to: φ(z, z i ) = z p
−2 N r
(t 2 q b z − z i ).
(108)
i=3 b=0
C.3. Spin representation. The spin 1/2 representation of A N can be obtained from the representation on polynomials with degree less or equal to 1 in each variable. The monomials z λ are the spin basis elements: λi = 1 if the spin i is plus and 0 if it is minus.
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Let us describe this representation explicitly. The Hilbert space is (C2 ) N with a basis 2 and has the given by sequences of spins | ± ± . . . ±. The matrix ei acts in Ci2 ⊗ Ci+1 following expression in the basis | + +, | + −, | − +, | − − : ⎛
0 0 1 0 1 ⎜ 0 −t − 2 1 ei = ⎝ 0 1 −t 2 0 0 0
⎞ 0 0⎟ ⎠. 0 0
(109)
The permutation operators Pi j permute the spins at positions i and j. It is convenient to introduce the operators ei j having the expression (109) and acting in Ci2 ⊗ C2j . 1
Ti j = t 2 + ei j , and the operators xi j are defined as in (C.1), xi j = Ti j Pi j . In the same basis as above, xi j is realized as triangular matrices as: ⎛
1
0 1 0 0
t2 ⎜0 xi j = ⎝ 0 0
0 1 t − t− 2 1 0 1 2
⎞ 0 0 ⎟. ⎠ 01 t2
(110)
The diagonal matrix i acts on the spin at position i. In the basis |+, |−, it is given by: i =
u 0 . 0 u −1
(111)
The matrix σ is defined by (99) where we substitute i to qˆi and Pii+1 to sii+1 : σ = PN −1N . . . P12 1 .
(112)
It implies the following identification of spins: µi+N = u −µi µi . The Hilbert space is characterized by the total numbers N+ , N− , of plus and minus spins respectively (N+ + N− = N ). The representation also depends on the parameter u of the twist matrix . We can give an alternative definition of the spin representation with the total spin N+ −N− T . Let A(N be a subalgebra of ATN generated by T1 , . . . , TN+ −1 , TN+ +1 , . . . , 2 + ,N− ) TN −1 , and y1 , . . . , y N . We define a one-dimensional representation C1 of A(N+ ,N− ) as follows: 1
Ti 1 = t 2 1, yi 1 = ut yi 1 = u −1 t
N+ −1 2
N− −1 2
(113)
1 if 1 ≤ i ≤ N+ ,
(114)
t i−N+ −1 1 if N+ + 1 ≤ i ≤ N .
(115)
t
i−1
N Then the spin representation is isomorphic to an induced module IndA A(N+ ,N− ) C1, where | + + · · · + − − · · · − corresponds to 1. We can define an ordering on the basis as follows. A state |µ is smaller than |µ, if it can be obtained from |µ through a sequence of permutations of a plus spin at position i and a minus spin at position i + 1.
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D. Stability of the Wheel Condition Under the Action of the A.H.A. We show here that the wheel conditions are preserved under the action of the A.H.A. generators. Let us verify that the ideal of polynomials obeying the condition (27) under the restriction (26a) is preserved under the action of e¯i (18). Consider the polynomial P = P e¯i , where P(z i ) is a polynomial obeying any admissible wheel condition. We show that P obeys the wheel condition specified by any admissible wheel {i a } and {baa+1 }. If there exists a value a, ¯ such that i = i a¯ , i + 1 = i a+1 = 0, then τ P − P ¯ , and ba¯ a+1 ¯ 1 1 − is proportional to t 2 z i − t 2 z i+1 and obeys this wheel condition. By linearity, so does P . If not, the wheel deduced by permutation: i a = si (i a ) and baa+1 = baa+1 ∀a is admissible. Thus, Psi , and by linearity P , also obey this wheel condition. Let us show that the conditions (25) and (26b) imply that the space of polynomials obeying the wheel conditions is preserved under the action of σ¯ (20). This amounts to show that the transformation defined by i a → i a + 1 with {baa+1 } unchanged defines an admissible wheel condition. The identification z N +1 = q −1 z 1 (19), is used when i a + 1 = N + 1. If i a < N ∀a, it is obvious. If i a¯ = N for a value a¯ = k + 1, then i a+1 < N and ba¯ a+1 ≥ 1. The above trans¯ ¯ formation can be recast into the wheel condition: i a = i a + 1 for a = a, ¯ i a¯ = 1, and ba−1 = b + 1, b = b − 1, b = b for a and a + 1 = a. ¯ a−1 ¯ a¯ a¯ a+1 ¯ aa+1 aa+1 ¯ a¯ a¯ a+1 ¯ If i k+1 = N , this transformation can be recast into the wheel condition: i a = i a−1 + 1 for a ≥ 2, i 1 = 1, and baa+1 = ba−1a for a ≥ 2, b12 = r − 2 − k1 baa+1 . To express b12 in this form we have used condition (25), and condition (26b) is necessary to have ≤ r − 2 and condition (26b) is satisfied by the new b12 ≥ 0. We also have: k1 baa+1 wheel.
E. Partition Functions and Traces In this appendix, we give a graphical method to evaluate the trace of operators in ATN (t) following [31]. The trace depends on the representation. We compare the traces between the different representations, and obtain the decomposition of ρD into the representations ρhh , by showing the trace identity: tr D (x) =
hh
D γhh ∀x ∈ ATN (t). tr ρhh (x),
(116)
In this identity, the matrices are finite dimensional. We fix t to be the root of unity, 2iπ
t = e p , where p is the Coxeter number of the diagram D. We also define an action of the modular group which leaves tr D invariant but transD therefore define a modular invariant forms linearly the traces tr ρhh . The coefficients γhh decomposition of the trace. To represent the trace, we use the description of linear operators by a system of lines drawn on the annulus of Sect. 2.0.3. We close the annulus into a torus by identifying the two boundaries. The trace becomes the partition function of a loop model on the torus.
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The contractible loops can be removed by giving them a weight τ . One ends up with a system of non-contractible loops not touching each other and therefore homotopic to the same cycle. In the case of the models defined by a diagram D of Sect. 7, if the number of loops is 2m, the weight is the number of paths of length 2m which can be drawn on the diagram D. In the case of the spin representation of Appendix C.3, the lines carry a spin index and are oriented accordingly. We must then sum over the possible orientations of the loops. We cut the annulus into a rectangle as in Fig. 6. If the total spin across a horizontal cycle is 2S z , and the total spin across a vertical cycle is 2S z , the weight of an oriented z loop configuration is u 2S . We denote by Skz the spin-representation of ATN (t), where the value of the spin is k fixed to be S z and u = t 2 . If |S z | > N2 , we regard Skz as the zero representation. a We define two actions s0 and s1 on a 2-tuple b by: a b = , (117) s1 b a a b− p = . (118) s0 b a+p Setting h = −k − S z and h = −k + S z , we conjecture that: tr ρhh (x) = (−1)l(w) tr w( kz ) (x), ∀x ∈ ATN (t), w∈W
(1)
(119)
S
(1)
where W = W (A1 ) is the affine Weyl group of type A1 . Note that the right-hand side is a finite sum. To establish the identity (116), it is useful to introduce an intermediate representation ρ f defined by a graph D f made of 2 f vertices connected around a circle. We require that f divides p so that τ = −(t + t −1 ) is an eigenvalue of D f . We obtain a representation of ATN (t) if we define the T.L. generators el as in (74) by taking for Sa the eigenvector of D f with the eigenvalue τ . We can also view the trace of this representation as the partition function of a spin model where the total spins 2S z and 2S z across the horizontal and vertical cycles considered above are constrained to be equal to f modulo p. We set p = f f . By representing the constraint on 2S z as a Fourier sum, we obtain the decomposition: af with 0 ≤ a ≤ f − 1, 0 ≤ b ≤ f − 1. ρf = 2 ⊕ (120) bf We can decompose the representations ρD in terms of the ρ f by identifying their traces as in (116). With this interpretation we write ρD = c f ρ f , where f is a divisor of p. From the property of the trace, we determine the coefficients c f by requiring that the number of closed paths of a given length on the graph D is the same as the sum over f of the number of closed paths of the same length on the circular diagram D f multiplied by c f . One obtains [31]: 2ρ An = ρn+1 − ρ1 , 2ρ Dn = ρ2(n−1) − ρn−1 + ρ2 − ρ1 , 2ρ E 6 = ρ12 − ρ6 − ρ4 + ρ3 + ρ2 − ρ1 ,
(121)
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2ρ E 7 = ρ18 − ρ9 − ρ6 + ρ3 + ρ2 − ρ1 , 2ρ E 8 = ρ30 − ρ15 − ρ10 − ρ6 + ρ5 + ρ3 + ρ2 − ρ1 . Finally, by combining (119, 120 and 121), we obtain the following decomposition: ρ An = ⊕nk=1 ρk,k , ρ D2n = ⊕n−1 k=1 ρ(2k−1)+(4n−2k−1),(2k−1)+(4n−2k−1) ⊕ 2ρ2n−1,2n−1 , ρ D2n+1 ρ E6 ρ E7 ρ E8
= = = =
n−1 ⊕2n (122) k=1 ρ2k−1,2k−1 ⊕ ρ2n,2n ⊕k=1 (ρ2k,4n−k ⊕ ρ4n−k,2k ), ρ1+7,1+7 ⊕ ρ4+8,4+8 ⊕ ρ5+11,5+11 , ρ1+17,1+17 ⊕ ρ5+13,5+13 ⊕ ρ7+11,7+11 ⊕ ρ9,9 ⊕ ρ5+13,9 ⊕ ρ9,5+13 , ρ1+11+19+29,1+11+19+29 ⊕ ρ7+13+17+23,7+13+17+23 .,
where ρa+b,c+d = ρa,c ⊕ ρa,d ⊕ ρb,c ⊕ ρb,d . Let us obtain the transformation law of the trace of ρhh under a modular transformation. The spin across the vertical and horizontal cycles is transformed as: z Sz S a b → . (123) c d Sz Sz From the characterization preceding (119) of ρhh , it is straightforward to obtain the following transformation of the traces: 1 2 2 1 1 : tr ρhh → t 4 (h −h ) tr ρhh , 0 1 (124) 1 2 2 1 0 : tr ρhh → t 4 ((r −h) −(r −h ) ) tr ρrr . 1 1 rr
Notice that under a modular transformation, the representation ρhh behaves as a tensor (1) product of affine characters of A1 : ρhh ∼ χl ⊗ χ¯l , where the level k is given by p = 2(k + 2) and the spin l is given by h = 2l + 1. Under a modular transformation of the torus, the partition function of the D-models remains invariant but the partition functions of ρhh transform linearly. Therefore, the multiplicities γhh in (122) are such that the direct sum is left invariant under these transformations. References 1. Razumov, A.V., Stroganov, Y.G.: Spin chains and combinatorics. J. Phys. A 34, 3185–3190 (2001) 2. Pearce, P.A., Rittenberg, V., de Gier, J., Nienhuis, B.: Temperley-Lieb Stochastic Processes. J. Phys. A 35, L661–668 (2002) 3. Batchelor, M.T., de Gier, J., Nienhuis, B.: J. Phys. A 34, L265–270 (2001) 4. Razumov, A.V., Stroganov, Y.G.: Combinatorial nature of ground state vector of O(1) loop models. Theor. Math. Phys. 138, 333–337 (2004) 5. Di Francesco, P., Zinn-Justin, P.: Around the Razumov-Stroganov conjecture: proof of a multiparameter sum rule. Electr. J. Combin. 12, R6 (2005) 6. Kasatani, M.: Subrepresentations in the polynomial representation of the double affine Hecke algebra of type G L n at t k+1 q r −1 = 1. Int. Math. Res. Not. 2005(28), 1717–1742 (2005) 7. Pasquier, V.: Quantum incompressibility and Razumov stroganov type conjectures. Ann. Henri Poincaré 7, 397–421 (2006) 8. Cherednik, I.: Double Affine Hecke Algebras. Cambridge: Cambridge University Press, 2005 9. Bernard, D., Gaudin, M., Haldane, D., Pasquier, V.: Yang-Baxter equation in spin chains with long range interactions. J. Phys. A 26, 5219–5236 (1993)
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10. Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Symmetric polynomials vanishing on the shifted diagonal and Macdonald polynomials. Int. Math. Res. Not. 18, 101 (2003) 11. Macdonald, I.G.: A new class of symmetric functions. Actes 20e Seminaire Lotharingien, p 131–171, Publications I.R.M.A. Strasbourg (1988), 372/S-20 12. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford: Oxford University Press, 1995 13. Prange, E., Girvin, S.: The Quantum Hall effect. Berlin-Heidelberg-New York: Springer-Verlag, 1987 14. Haldane, F.D.M., Rezayi, E.H.: Spin-singlet wave function for the half-integral quantum Hall effect. Phys. Rev. Lett. 60, 956, and E60, 1886 (1988) 15. Kazhdan, D., Lusztig, G.: Representation of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979) 16. Lascoux, A., Schützenberger, M.-P.: Polynômes de Kazhdan & Lusztig pour les grassmaniennes. Asterisque 87–88, 249–266 (1981) 17. Lascoux, A., Schützenberger, M.-P.: Symmetry and Flag manifolds. In: Invariant Theory, Springer L.N 996, Berlin-Heidelberg-New York, 1983, pp. 118–144 18. Kirillov, A. Jr., Lascoux, A.: Factorization of Kazhdan-Lusztig elements for Grassmanians. Adv. Stud. 28, 143–154 (2000) 19. Fulton, W.: Young Tableaux. London Mathematical Society Student Texts 35, Cambridge: Cambridge University Press, 2003 20. Mitra, S., Nienhuis, B.: Osculating Random Walks on Cylinders. In: Discrete Random Walks, C. Banderier, C. Krattenthaler (eds.), Discrete Math. Theor. Computer Sci. Proceed, 2003, pp. 259–264 21. Mitra, S., Nienhuis, B.: Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders. JSTAT, P10006, 2004 22. Razumov, A.V., Stroganov, Y.G.: Enumeration of half-turn symmetric alternating sign matrices of odd order. http://arxiv.org/list/math-phys/0504022, 2005 23. Lusztig, G.: Affine Hecke algebras and their graded version. J. Amer. Math. Soc. 2, 599–635 (1989) 24. Lusztig, G.: Introduction to Quantum Groups. Boston, Birkhauser 1993 25. Cherednik, I.: Nonsymmetric Macdonald Polynomials. Internat. Math. Res. Notices 10, 483–515 (1995) 26. Di Francesco, P., Zinn-Justin, P.: In homogenous model of ossing loops and multidegrees of some algebraic varieties. http://arxiv.org/list/math-ph/0412031, 2001 27. Di Francesco, P. Zinn-Justin, P., Zuber, J.B.: Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain. J. Stat. Mech. p08011 (2006) 28. Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter Graphs and Towers of Algebras. BerlinHeidelberg-New York: Springer-Verlag, 1989 29. Graham, J.J., Lehrer, G.I.: Enseign. Math. The Representation Theory of Affine Temperley-Lieb Algebras 44, 173–218 (1998) 30. Pasquier, V.: Two-dimentional itical systems labelled by Dynkin diagrams. Nucl. Phys. B FS19, 162–172 (1987) 31. Pasquier, V.: Lattice derivation of modular invarient partition functions on the torus. J. Phys. A 20, L1229 (1987) 32. Pasquier, V., Saleur, H.: Common structures between finite systems and conformal field theories through quantum groups. Nucl. Phys. B 330, 525–556 (1990) 33. Cappelli, A., Itzykson, C., Zuber, J.B.: Modular invariant partition functions in two dimensions. Nucl. Phys. B 280([FS18]), 445–465 (1987) 34. Gepner, D., Qiu, Z.: Modular invariant partition functions for parafermionic field theories. Nucl. Phys. B 285([FS19]), 423–453 (1987) 35. Prange, E., Girvin, S.: The Quantum Hall effect. Berlin-Heidelberg-New York: Springer-Verlag, 1987 36. Knop, F., Sahi, S.: A recursion and a combinatorial formula for Jack polynomials. Invent. Math. 128(1), 9–22 (1997) 37. Di Francesco, P., Zinn-Justin, P.: The quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials. J. Phys. A 38, L815–822 (2002) 38. Ram, A.: Affine Hecke algebras and generalized standard Young tableaux. J. Algebra 260(1), 367–415 (2003) 39. Suzuki, T., Vazirani, M.: Tableaux on periodic skew diagrams and irreducible representations of the double affine Hecke algebra of type A. Int. Math. Res. Not. 2005(27), 1621–1656 (2005) 40. Walker, K.: Private communication 41. Gaudin, M.: La Fonction d’Onde de Bethe. Paris: Masson, 1981 42. Baxter, R.J.: Exactly solved Models in Statistical Mechanics. London: Academic, 1982 43. Wenzl, H.: Hecke Algebras of type An and subfactors. Invent. Math. 92, 349–383 (1988) 44. Pasquier, V.: Scattering Matrices and Affine Hecke Algebras. Schladming School 1995, Nucl. Phys. B (Proc. Suppl.) 45A, 62–73 (1996) 45. Pasquier, V.: Incompressible representations of the Birman Wenzl algebra. Ann. Henri Poincaré 7, 603–619 (2006)
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Commun. Math. Phys. 276, 437–472 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0342-z
Communications in
Mathematical Physics
A Multi-Dimensional Lieb-Schultz-Mattis Theorem Bruno Nachtergaele, Robert Sims Department of Mathematics, University of California at Davis, Davis, CA 95616, USA. E-mail: [email protected]; [email protected] Received: 21 August 2006 / Accepted: 26 April 2007 Published online: 9 September 2007 – © B. Nachtergaele and R. Sims 2007
Abstract: For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, with arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem [14] and provides a rigorous proof of the main result in [8].
1. Introduction and Main Result 1.1. Introduction. Ground state properties of Heisenberg-type antiferromagnets on a variety of lattices are of great interest in condensed matter physics and material science. Antiferromagnetic Heisenberg models are directly relevant for the low-temperature behavior of many materials, most notably the cuprates that exhibit high-Tc superconductivity [16]. There are several general types of ground states that are known, or expected, to occur in specific models: a disordered ground state or spin liquid, critical correlations (power law decay), dimerization (spin-Peierls states), columnar phases, incommensurate phases, and Néel order. More exotic phenomena such as chiral symmetry breaking have also been considered [21, 22]. Which behavior occurs in a given model depends on the lattice, in particular the dimension and whether or not the lattice is bipartite, on the type of spin (integer versus half-integer) and, of course, also on the interactions. In this paper we are considering a class of half-integral spin models (or models where the magnitude of at least some of the spins is half-integral). Our aim is to prove a generalization of the Lieb-Schultz-Mattis Theorem [14]. Such a generalization was presented by Hastings in [8] and a substantial part of our proof is based on his work. Our main contribution is to provide what we hope is a more transparent argument which in addition is mathematically rigorous.
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The well known theorem by Lieb and Mattis [13] implies, among other things, that the ground state of the Heisenberg antiferromagnet on a bipartite lattice with isomorphic sublattices, is non-degenerate. For one-dimensional and quasi-one-dimensional systems of even length and with half-integral spin Affleck and Lieb [1], generalizing the original result by Lieb, Schultz, and Mattis [14], proved that the gap in the spectrum above the ground state is bounded above by constant/L. A vanishing gap can be expected to lead to a gapless continuous spectrum above the ground state in the thermodynamic limit. Such an excitation spectrum is generically associated with power-law (as opposed to exponential) decay of correlations. Aizenman and Nachtergaele proved for the spin-1/2 antiferromagnetic chain that if translation invariance is not broken (in particular, when the ground state is unique), the spin-spin correlation function can decay no faster than 1/r 3 [3]. In other words, uniqueness of the ground state implies slow (power-law) decay of correlations. Recently, it was proved rigorously that a non-vanishing spectral gap implies exponential decay of correlations [9, 20]. Therefore, non-exponential decay of correlations implies the absence of a gap. In particular, the result by Aizenman and Nachtergaele implies the absence of a gap in the infinite spin-1/2 antiferromagnetic chain if the translation invariance is not broken, e.g., if the ground state is unique. This result can be generalized to an interesting class of antiferromagnetic chains of half-integer spins [17]. The Lieb-Schultz-Mattis Theorem has also been extended to fermion systems on the lattice [24, 25]. All these results are for one-dimensional systems. The bulk of the applications of the spin-1/2 Heisenberg antiferromagnet is in two-dimensional physics and therefore, the rigorous proof we provide here, based in part on ideas of Hastings [8], should be of considerable interest as it is applicable to higher-dimensional models. The most common argument employed to bound a spectral gap from above uses the variational principle. Often, the variational state is a perturbation of the ground state. The proofs in [14] and [1] are of this kind. However, since the ground state is not known, and no assumptions are made about it except for its uniqueness, these proofs are not a variational calculation in the usual sense. The variational states are defined by acting with suitable local operators A on the (unknown) ground state. For a finite volume Hamiltonian HL generated by a potential Φ of the type we consider (see the paragraph including (1.7) and (1.8) in Sect. 1.2 for the relevant definitions), and with a unique ground state, it is straightforward to show that the gap above the ground state, γ L , is bounded uniformly in L. To see this, note that for any ground state vector Ω and for any site x, there exists a unitary on the state space of x with vanishing expectation in the state Ω, i.e., U Ω ⊥ Ω. Since Ω is the unique ground state by assumption, U Ω is a variational state for the gap. Therefore, we have the bound γ L ≤ Ω, [HL , U ]Ω ≤ 2 inf x
Φ(X ) ≤ 2|||Φ|||1 ,
(1.1)
X x
which is uniform in the system size L. Here, |||Φ|||1 is as defined in (1.13). See Sect. 5.5 for the proof that such a unitary exists. In order to obtain a better bound on the energy of the first excited state one has to exploit the few properties assumed of the ground state, such as its uniqueness and symmetries. Furthermore, one must show that any proposed variational state has a sufficiently large component in the orthogonal complement of the ground state. In Sect. 2.2, we propose a variational state for finite systems of size L and then demonstrate the relevant estimates, as mentioned above, in Sects. 3 and 4. It is interesting to note that the energy estimate we obtain will itself contain the spectral gap of the finite system in
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such a way that assuming a large gap leads to an upper bound less than the assumed gap. From this contradiction one can conclude an upper bound on the finite-volume gaps. Our results apply to a rather general class of models, which we will define precisely in the next section. The application of our general result to spin-1/2 Hamiltonians with translation invariant (or periodic) isotropic finite-range spin-spin interactions on a d-dimensional lattice is easy to state. First, let Λ L = [1, L] × VL with L even and with periodic boundary conditions in the 1-direction, i.e., in the direction that is of even size. It will be important that the number of spins in VL , |VL |, is odd, and satisfies |VL | ≤ cL d−1 , for some d ≥ 1 and a suitable constant c. Assuming that the model defined on Λ L has a unique ground state, we prove that the spectral gap γ L satisfies the bound γL ≤ C
log L , L
(1.2)
where C depends on d and the specifics of the interaction, but not on L. Because of the presence of the factor log L, the bound (1.2) applied to onedimensional models does not fully recover the original Lieb-Schultz-Mattis Theorem in [14] or the bound proved by Affleck and Lieb in [1]. This indicates that in general our bound is not optimal. Our proof uses in an essential way Lieb-Robinson bounds [9, 15, 20], as does Hastings’ argument in [8], and the appearance of the factor log L seems to be an inevitable consequence of this. In fact, it is known that the standard Heisenberg antiferromagnets with spin ≥ 1 on the two-dimensional square lattice or with spin ≥ 1/2 on Zd with d ≥ 3, have Néel ordered ground states [5, 10] and in that case one can show that the gap is bounded by C/L (see, e.g., [11, 12]).
1.2. Setup and main result. The arguments we develop below can be applied to a rather general class of quantum spin Hamiltonians defined on a large variety of lattices. We believe it is useful to present them in a suitably general framework which applies to many interesting models. Attempting to be as general as possible, however, would lead us into a morass of impenetrable notation. Therefore, we have limited the discussion of further generalizations to some brief comments in Sect. 1.5. We assume that the Hamiltonians describe interactions between spins that are situated at the points of some underlying set Λ. For simplicity, one may think of Λ = Zd , but we need only assume that the set Λ has one direction of translational invariance, which we will refer to as the 1-direction. We assume that there is an increasing sequence of d−1 for sets {Λ L }∞ L=1 which exhaust Λ of the form Λ L = [1, L] × VL , where |VL | ≤ cL some d ≥ 1. Here each x ∈ Λ L can be written as x = (n, v), where n ∈ {1, 2, . . . , L} and v ∈ VL , and we will denote by (n, VL ) the set of all x ∈ Λ L of the form x = (n, v) for some v ∈ VL . Estimates on the decay of correlations in the ground state and Lieb-Robinson bounds on the dynamics will play an important role in the proof of the main result. Both are expressed in terms of a distance function on Λ, which we will denote by d. Often, Λ has the structure of a connected graph and d(x, y) is the minimum number of edges in a path from x to y. In any case, we will assume that d is a metric and furthermore that there is a function F : [0, ∞) → (0, ∞) satisfying the following two conditions. Condition F1: F is uniformly integrable over Λ in the sense that F := sup F(d(x, y)) < ∞. (1.3) x∈Λ y∈Λ
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Condition F2: F satisfies F (d(x, z)) F (d(z, y)) < ∞, F (d(x, y)) x,y∈Λ
C(F) := sup
(1.4)
z∈Λ
which means that the “convolution” of F with itself is bounded by a multiple of itself. F1 and F2 are restricitve conditions only when Λ is infinite, however, for finite Λ, the constants F and C(F) will be useful in our estimates. It is also important to note that for any given set Λ and function F that satisfies F1 and F2 above, we can define a one-parameter family of functions, Fλ , λ ≥ 0, by Fλ (x) := e−λx F(x),
(1.5)
and easily verify that F1 and F2 hold for Fλ , with Fλ ≤ F and Cλ (F) ≤ C(F). As a concrete example, take Λ = Zd and d(x, y) = |x − y|. In this case, one may take the function F(x) = (1 + x)−d−ε for any ε > 0. Clearly, (1.3) is satisfied, and a short calculation demonstrates that (1.4) holds with 1 . (1.6) C(F) ≤ 2d+ε+1 d+ε (1 + |n|) d n∈Z
Each x ∈ Λ is assigned a finite-dimensional Hilbert space Hx . For any finite subset X ⊂ Λ, the Hilbert space associated with X is the tensor product H X = x∈X Hx , and the set of corresponding observables supported in X is denoted by A X = B(H X ), the bounded linear operators over H X . These local observables form an algebra, and with the natural embedding of A X 1 in A X 2 for any X 1 ⊂ X 2 , one can define the C ∗ -algebra of all observables, A, as the norm completion of the union of all local observable algebras A X for finite X ⊂ Λ. Since we have assumed that Λ L is of the form [l, r ] × VL with r − l = L − 1, we can define translation automorphisms τn , for n ∈ Z, which map A(m,VL ) into A(n+m,VL ) for all m ∈ Z. An interaction for the system is a map Φ from the finite subsets of Λ to A such that for each finite X ⊂ Λ, Φ(X )∗ = Φ(X ) ∈ A X . For given Λ and F, and any λ ≥ 0, let Bλ (Λ) be the set of interactions that satisfy Φ(X ) Φλ := sup < ∞. (1.7) Fλ (d(x, y)) x,y∈Λ X x,y
All interactions considered in this paper are assumed to belong to Bλ (Λ) for some choice of F and λ > 0. The constant Φλ will show up in many estimates. The finite volume Hamiltonians are defined in terms of the interaction Φ in the usual way by HL = Φ(X ) + boundary terms. (1.8) X ⊂Λ L
We will always assume periodic boundary conditions in the 1-direction and arbitrary boundary conditions in the other directions (i.e., any boundary terms in the other directions are included in the definition of Φ). The condition that Φλ is finite is sufficient to guarantee the existence of the dynamics in the thermodynamic limit as a one-parameter group of automorphisms on A. In particular this means that the limits αtΦ (A) := lim αtΦ,L (A) := lim eit HL Ae−it HL L→∞
L→∞
(1.9)
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exist in norm for all t ∈ R, and all observables A ∈ A X , for any finite X ⊂ Λ. We will often suppress the L or Φ dependence in the notation αtΦ,L . See [4, 18, 23] for more details. Next, we turn to a set of conditions that more specifically describe the class of models to which the Lieb-Schultz-Mattis Theorem may be applied. Condition LSM1: We assume that the interaction is translation invariant in at least one direction, which we will take to be the 1-direction. This means Φ (X + e1 ) = τ1 (Φ(X )) ,
(1.10)
where, for any X ⊂ Λ, X + e1 is translation of all points in X by one unit in the 1-direction. We will consider finite systems with Hamiltonians HL defined with periodic boundary conditions in the 1-direction. For convenience of the presentation we will assume free boundary conditions in the other directions but this is not crucial. Since we have assumed periodicity in the 1-direction, we can implement the translation invariance for finite systems by a unitary T ∈ AΛ L such that Φ(X + e1 ) = T ∗ Φ(X )T , for all X ⊂ Λ L . Here T depends on L, but we suppress this dependence in the notation. Condition LSM2: The interactions are assumed to be of finite range in the 1-direction, i.e., there exists R > 0 (the range), such that if X ⊂ Λ and X xi = (n i , vi ) for i = 1, 2 with |n 1 − n 2 | ≥ R, then Φ(X ) = 0. Condition LSM3: We assume rotation invariance about one axis. More precisely, we assume that there is a hermitian matrix in every A{x} , x ∈ Λ, which we will denote by Sx3 , with eigenvalues that are either all integer or all half-integer (i.e. belonging to Z+1/2). 3 . Define, for θ ∈ R, the unitary U (θ ) ∈ AΛ L by We also require that τm (Sx3 ) = Sx+me 1 3 eiθ Sx . (1.11) U (θ ) = x∈Λ L
The interaction is taken to be rotation invariant in the sense that for each finite X ⊂ Λ, U ∗ (θ )Φ(X )U (θ ) = Φ(X ) for all θ ∈ R.
(1.12)
Condition LSM4: We assume that the Sx3 are uniformly bounded: there exists S such that Sx3 ≤ S, for all x ∈ Λ. The following condition, which we call odd parity, is crucial: of Sx3 are integers, and px = 1/2 define the parity of x, px to be 0 if the eigenvalues if they are half-integers. We assume that v∈VL p(n,v) ∈ Z + 1/2, for all n ∈ Z. The simplest and most important case where this is satisfied is when we have a spin 1/2 at each site, and |VL | is odd. Condition LSM5: The ground state of HL is assumed non-degenerate. This implies it is an eigenvector of the translation T and rotations U (θ ). Without loss of generality we can assume that 1 is the corresponding eigenvalue of T (if the eigenvalue is eiφ , replace T by e−iφ T ). We also assume that the ground state has eigenvalue 1 for the rotations U (θ ). Condition LSM6: We assume that there are orthonormal bases of the Hilbert spaces HΛ L with respect to which Sx3 and Φ(X ) are real, for all x ∈ Λ L , X ⊂ Λ L . This condition is only used in the proof of Lemma 1. Therefore, this condition may be replaced by the property proved in that lemma. We will also use the following quantities: |||Φ|||1 := sup Φ(X ) < ∞, (1.13) x∈Λ X x
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and |||Φ|||2 := sup
x∈Λ X x
|X |
[Sx3 , Φ(X )] < ∞.
(1.14)
x ∈X
It is not hard to show that the conditions F1 and F2 are sufficient to imply that |||Φ|||1 and |||Φ|||2 are finite. We can now state our main result. Theorem 1. Let γ L be the spectral gap, i.e., the difference between the lowest and nextlowest eigenvalue of the Hamiltonian HL of a model satisfying conditions F1, F2, and LSM1-6. Then, there exists a constant C, depending only on properties of Λ (such as the dimension), the constants F and C(F), and the interaction (Φλ , for some λ > 0, |||Φ|||1 , and |||Φ|||2 ), such that γL ≤ C
log L . L
(1.15)
1.3. Structure of the proof. The simplest way to present the proof is as a proof by contradiction. Under the assumption that there exists a sufficiently large constant C > 0, such that γ L exceeds (C log L)/L for large L, we will construct a state orthogonal to the ground state with an energy difference that is boundable by a quantity that is strictly less than the assumed gap for sufficiently large L. Thus, the proof is in essence a variational argument. The variational state is constructed as a perturbation of the ground state, as the solution of the differential equation proposed by Hastings [8] with the ground state as initial condition (see Sect. 2, in particular (2.25), for this equation). The important idea is that this equation will lead to a state which resembles the ground state of the Hamiltonian with twisted rather than periodic boundary conditions (see Sect. 2.1 for the definition of the twists), at least in part of the system, say the left half. In the right half the ground state will be left essentially unperturbed. This state is defined in Sect. 2. After the variational state has been defined, there are two main steps in the proof: estimating its excitation energy and verifying that it is “sufficiently orthogonal” to the ground state. In general, one may also have to consider the normalization of the variational state, but in our case the differential equation defining it will be manifestly norm preserving. Hence, this is not an issue for our proof. The main difficulty is that under the general assumptions we have made, no explicit information about the ground state is available. Its uniqueness, translation, and rotation invariance are the only properties we can use. In combination with the general assumptions on the interactions and the assumption on the magnitude of the spectral gap above the ground state, however, one can obtain an upper bound on the decay of correlations of the ground state in the 1-direction. The recently proved Lieb-Robinson bounds [9, 18, 20] will be essential to show that the effects of the perturbations we define in the left half of the system remain essentially localized there. This allows us to compare the energy of the variational state with the ground state energy of a Hamiltonian, Hθ,−θ introduced in (2.5), which, instead of twisted boundary conditions, has two twists that cancel each other. The twisted Hamiltonian is unitarily equivalent to the original one and therefore has the same ground state energy. We work out this argument in Sect. 3. The result is |ψ1 , HL ψ1 − E 0 | ≤ C L ν e−cγ L L (1 + corrections) ,
(1.16)
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where ψ1 is the normalized variational state we construct, and E 0 is the ground state energy. The dependence of both quantities on L is surpressed in the notation. ν, C and c are positive constants that only depend on properties of the lattice and the interactions. The correction terms appearing above, and also in (1.17) below, can be made explicit by the estimates provided in Sect. 5. They depend on the quantity γ L L in such a way that assuming there exists a constant C > 0 for which γ L L ≥ C for sufficiently large L, they are uniformly bounded in L. Due to the nature of our proof of Theorem 1, see below, we do not write these additional terms out explicitly. For the orthogonality, our strategy is to show that ψ1 is almost an eigenvector of the translation T with eigenvalue −1. Since the ground state ψ0 is an eigenvector of T with eigenvalue 1, by assumption, this shows that ψ1 is nearly orthogonal to ψ0 . In Sect. 4 we obtain a bound on their inner product of the form:
|ψ1 , ψ0 | ≤ C L ν e−c γ L L (1 + corrections) ,
(1.17)
where ν , C and c are positive constants similar to ν, C and c. The proof of Theorem 1 then easily follows. Proof of Theorem 1. Suppose that γ L L ≥ C log L with a sufficiently large constant C. In this case, the correction terms which appear in the bounds (1.16) and (1.17) above are negligible. It is easy to see then that one obtains a contradiction for L large enough. To help the reader see the forest through the trees we have tried to streamline the estimates in Sects. 3 and 4 by collecting some results of a more technical nature in Sect. 5. 1.4. Examples. The conditions LSM1-6 we have imposed on the models are not unreasonable. We will illustrate this by considering various antiferromagnetic Heisenberg models defined on Λ L = [1, L] × VL , where for each L, VL is a finite set. As before, at each x ∈ Λ L , we have a finite-dimensional spin with spin-matrices Sxi , i = 1, 2, 3, and we consider Hamiltonians of the form H= J (x, y)S x · S y , (1.18) x,y∈Λ L ,x= y
where J (x, y) ∈ R are the coupling constants. If VL ⊂ Zd−1 , with d ≥ 1, and such |VL | ≤ cL d−1 , for a suitable constant c, which describes the case for d-dimensional systems defined on subsets of Zd , there exists a function F satisfying Conditions F1 and F2 as we have indicated in the paragraph containing (1.6). It is also easy to see that if VL is a fixed finite set independent of L, in which case the system is (quasi) one-dimensional, any function F that works for the one-dimensional lattice will suffice. All the examples we discuss below will be of this type. Certainly, there are still many Hamiltonians of the form (1.18) that fail to satisfy all six conditions, but this is generally for a good reason. For example, without translation invariance in at least one direction one can easily have a non-vanishing gap above the ground state. Condition LSM2, finite-range, does not need to be satisfied in the strict sense. Sufficiently rapidly decaying interactions could also be treated. For the present
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discussion, let’s assume that the model is translation invariant in the 1-direction, and that the interactions are nearest neighbor in the 1-direction in the sense that for any x = (n 1 , v) and y = (n 2 , u), with |n 1 − n 2 | > 1, we have J (x, y) = 0. The rotation invariance about at least one axis imposed in Condition LSM3 is essential for the type of result we prove. The models (1.18) have full rotation invariance, so they clearly satisfy this condition. Anisotropic models of the XXZ type would still satisfy LSM3. In order to satisfy LSM4, we have to assume a uniform bound on the size of the spin. Clearly, all models with only one kind of spins or a periodic arrangement of spin magnitudes satisfy this condition. Since we already assumed translation invariance in the 1-direction, we can verify the odd parity condition by adding the magnitudes of all spins in the “slice” (1, VL ). If we have only spin 1/2’s, e.g., we simply need that |VL | is odd. For the one-dimensional chains of identical spins of magnitude S, the condition requires that S is half-integral. Haldane’s Conjecture [6, 7] predicts that for integer values of S there exists a non-vanishing gap. There are examples of isotropic integer-spin chains which satisfy all the other conditions and for which the existence of a non-vanishing gap has been rigorously established, such as the AKLT chain [2]. For p-periodic spin chains with a repeating pattern of spin magnitudes S1 , . . . , S p , LSM4 is satisfied if S1 + · · · + S p is half-integral. Similarly, for spin ladders LSM4 is satisfied if the total spin in each rung is half-integral. There is a large class of models for which the uniqueness of the ground state demanded by LSM5 follows from the Lieb-Mattis Theorem [13]. For Hamiltonians of the form (1.18), a simple case where the Lieb-Mattis Theorem applies is the following: if Λ L is the union of two disjoint subsets Λ L ,A and Λ L ,B of equal size, with J (x, y) ≤ 0 whenever x and y do not belong to the same subset, and sufficiently many J (x, y) are non-vanishing such that the graph formed by the edges with non-zero coupling constants is connected. This is satisfied if VL ⊂ Zd−1 is connected and the Hamiltonian is the usual nearest neighbor antiferromagnetic Heisenberg model. All models of the form (1.18) satisfy LSM6. The above discussion demonstrates that there is a large variety of models that satisfy all conditions of our main theorem. In particular, all nearest-neighbor half-integer spin Heisenberg antiferromagnets defined on subsets Λ L = [1, 2L] × VL of d-dimensional hypercubic lattice with |VL | odd and such that |VL | ≤ cL α , for some α ≥ 0 (it is natural but not necessary to assume α = d − 1), have a unique ground state with a gap γ L above it satisfying γ L ≤ C(log L)/L, for some constant C. 1.5. Generalizations. One can envision several generalizations of Theorem 1. An obvious one is to remove the condition that the interaction is strictly finite range in the 1-direction. It is not hard to see that the arguments given in the following sections can be extended to long-range interactions with sufficiently fast decay. One may wonder whether the assumption that L is even is essential. It is used in the proof of near orthogonality of the variational state, which is based on investigating the behavior under translations of the state: the variational state is close to an eigenvector with eigenvalue −1 of the translation operator T , whereas the ground state has eigenvalue 1. Our proof of this fact assumes that the ground state is an eigenvector of the rotations with eigenvalue 1. For L odd, our assumptions preclude the existence of such an eigenvector. However, it seems plausible that for odd L a slight modification of our proof will work to show that the ground state and the variational excited state have opposite eigenvalues for translations.
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The main applications we think of are to SU(2)-invariant Hamiltonians with antiferromagnetic interactions. Affleck and Lieb [1] pointed out that their proof easily extends to a class of models with SU(N) symmetry. There are no obstructions to generalizing our arguments to such models with SU(N) symmetry given by suitable representations. It may also be of interest to consider different topologies of the underlying lattice and/or the twistings. Instead of cylindrical systems with periodic boundary conditions which can be deformed by a twist, one could apply a similar strategy to systems defined on a ball or a sphere. We do not explore such possibilities here. Another question we do not address in this paper is under what circumstances the trial state we construct is actually a good approximation of a low-lying eigenstate with energy close to the first excited state, or even whether it is a state orthogonal to the ground state and with energy bounded by C(log L)/L. We do not believe that statements of this kind hold under the general conditions we impose. It is expected that in some cases the true gap of the system is much smaller than the bound we prove. This is of course not in contradiction with our result, but under such circumstances our method does not provably construct a good variational state. There is no reason to assume that it always would.
2. Construction of the Variational State 2.1. Twisted Hamiltonians. The main motivation behind the construction of the variational excited state is that it should resemble the ground state of the model with twisted (as opposed to periodic) boundary conditions. Therefore, we first describe some elementary properties of a family of perturbations of the Hamiltonian, which we will call twisted Hamiltonians for reasons that will become obvious. Given an interaction Φ which satisfies the general assumptions outlined in Sect. 1.2, we will now define a two parameter family of “twisted Hamiltonians” to analyze. These Hamiltonians will be defined on a finite volume Λ L = [1, L] × VL , where [1, L] is considered with periodic boundary conditions for some even L > 4R, where R > 0 is the range of Φ in the 1-direction. Let Φ L be the periodic extension of Φ restricted to Λ L . Recall that each point x ∈ Λ L can be written as x = (n, v), where n ∈ {1, 2, . . . , L} and v ∈ VL , and we will denote by (n, VL ) = {x ∈ Λ L : x = (n, v) for some v ∈ VL }. For any θ ∈ R and n ∈ {1, 2, . . . , L}, define the “column” rotations Un (θ ) by Un (θ ) =
3
eiθ Sx .
(2.1)
x∈(n,VL )
For m ∈ {1, 2, . . . , L − 1}, we will denote by Vm (θ ) the unitary given by Vm (θ ) =
Un (θ ).
(2.2)
m
The “twisted Hamiltonians” are defined to be perturbations of the initial Hamiltonian with periodic boundary conditions defined by H =
X ⊂Λ L
Φ L (X ).
(2.3)
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The perturbations have the following form: Hθ (m) := Vm (θ )∗ Φ(X )Vm (θ ) − Φ(X ),
(2.4)
X ⊂Λ L
for m ∈ [R, L − R] to avoid interactions across the seam created by identifying L + 1 with 1. Note that here we use the original potential Φ, and not its periodic extension Φ L . Clearly, if X ⊂ m
(2.5)
to be a doubly twisted Hamiltonian. With m fixed, we regard Λ L as the disjoint union of two sets (W )
ΛL = ΛL
(S)
∪ ΛL ,
(2.6)
(W )
where Λ L consists of two windows, one about each column at which a twist has been applied; namely ) (W ) (W ) (W ) Λ(W := Λ (m) ∪ Λ (y) := (n, VL ), (2.7) + L/2) and Λ (m L L L L |n−y|≤ L4 −R (S)
for y ∈ {m, m + L/2}. Moreover, Λ L comprises the remaining strips in Λ L . Given this decomposition of the underlying space, the twisted Hamiltonian can be written as (W )
Hθ,θ = Hθ,θ + H (S) , where H (S) =
(2.8)
Φ L (X ),
(2.9)
X ⊂Λ L : (S) X ∩Λ L =∅
(W ) and Hθ,θ
denotes the remaining terms in Hθ,θ which, due to (2.9), are supported strictly within the windows. There are a variety of useful symmetries the Hamiltonians Hθ,θ , introduced in (2.5), possess. With m ∈ {R, R + 1, . . . , L/2 − R} fixed as above, one may define W (φ) := Un (−φ), (2.10) m
for any real φ. See (2.1) for the definition of the column rotations Un . It is easy to check that for any angles θ, θ , φ ∈ R, one has that W ∗ (φ) Hθ,θ W (φ) = Hθ−φ,θ +φ ,
(2.11)
due to the (term by term) rotation invariance of the interactions. Given this relation, it is clear that along the path θ = −θ the twisted Hamiltonian is unitarily equivalent to the untwisted Hamiltonian, i.e., W (θ )∗ Hθ,−θ W (θ ) = H0,0 = H, which, due to the periodic boundary conditions, is not true for general pairs
(2.12) θ, θ .
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The untwisted Hamiltonian is translation invariant (in the 1-direction), and it is important that the twisted Hamiltonians inherit a “twisted” translation invariance. Define Tθ,θ = T Um (θ ) Um+L/2 (θ ),
(2.13)
where T is the unitary implementing the translation by 1 in the 1-direction. It is then straightforward to verify that ∗ Hθ,θ = Tθ,θ
Hθ,θ Tθ,θ .
(2.14)
Note that under the odd parity condition LSM4 we have T2π,0 = −T,
(2.15)
which will be important in the proof of the almost orthogonality of the trial state in Sect. 4. If we denote by ψ0 the (unique) ground state of H , i.e., H ψ0 = E 0 ψ0 , then by translation invariance, and specifically LSM5, we have that T ψ0 = ψ0 . Moreover, using the unitary equivalence (2.12), we see that the ground state of the twisted Hamiltonian Hθ,−θ satisfies Hθ,−θ ψ0 (θ, −θ ) = E 0 (θ, −θ )ψ0 (θ, −θ ) with E 0 (θ, −θ ) = E 0 and ψ0 (θ, −θ ) = W (θ )ψ0 . Although the twisted ground state ψ0 (θ, −θ ) is not translation invariant, it does satisfy invariance with respect to the twisted translations, i.e., Tθ,−θ ψ0 (θ, −θ ) = ψ0 (θ, −θ ). As a consequence, we have the following simple but important property of E 0 . Lemma 1. Let E 0 (θ, θ ) denote the ground state energy of Hθ,θ .Then, the partial derivatives of E 0 vanish on the line θ = −θ : ∂1 E 0 (θ, −θ ) = ∂2 E 0 (θ, −θ ) = 0.
(2.16)
Proof. First, we note that E 0 is differentiable in its two variables in a neighborhood of (0, 0) by the non-degeneracy condition LSM5. By unitary equivalence E 0 is then differentiable in a neighborhood of the line (θ, −θ ). For ψ, φ ∈ R, let E(ψ, φ) = E 0 (ψ − φ, ψ + φ) denote the ground state energy of Hψ−φ,ψ+φ . Due to the unitary equivalence Eq. (2.11), E depends only on ψ. Hence, ∂φ E(ψ, φ) = 0, for all ψ, φ. Under the additional assumption that the interactions Φ(X ) are real (LSM6), we have that Hθ,θ = H−θ,−θ , and therefore E(ψ, 0) = E(−ψ, 0). Hence, E is an even function of ψ and ∂ψ E(ψ, φ)|ψ=0 = 0. Using these properties and the fact that the partial derivatives of E 0 are linear combinations of the partial derivatives of E, we find that both partial derivatives of E 0 vanish on the line θ = −θ . 2.2. The variational state. Our aim is to construct a state that resembles the ground state of Hθ,−θ in a region surrounding those spins that were twisted by an angle of θ , while it otherwise resembles the ground state of H = H0,0 . From the unitary equivalence (2.12) we have that E 0 (θ, −θ ) is independent of θ , i.e., ∂θ E 0 (θ, −θ ) = 0. Moreover, the partial derivatives of E 0 vanish on the line (θ, −θ ), as was proven in Lemma 1. This property, in general, allows one to derive an equation for the ground state. Consider a differentiable one-parameter family of self-adjoint operators H (x), x ∈ [a, b] ⊂ R, and let E 0 (x) denote the ground state energy of H (x) with a differentiable
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family of ground state eigenvectors ψ0 (x). Suppose ∂x E 0 (x) = 0 for x ∈ [a, b]. Then, it is easy to see that ψ0 (x) ⊥ (∂x H (x))ψ0 (x), from which we obtain: ∂x ψ0 (x) = − [H (x) − E 0 (x)]−1 ∂x H (x) ψ0 (x).
(2.17)
For any vector ψ, this leads to ∞ 1 ψ, ∂x ψ0 (x) = − d ψ, PEx ∂x H (x)ψ0 (x) (2.18) E − E (x) 0 E 0 (x) ∞ ∞ e−(E−E 0 (x))t dt d ψ, PEx ∂x H (x)ψ0 (x) =− E 0 (x) 0 ∞ ψ, αitx (∂x H (x)) ψ0 (x) dt, =− 0
where PEx is the spectral resolution for H (x) and αitx is the imaginary-time evolution corresponding to the Hamiltonian. Motivated by this calculation, we introduce the family of operators B(A, H ), where H is a Hamiltonian for which the dynamics {αt | t ∈ R} exists as a strongly continuous group of ∗-automorphisms and A is any local observable, defined by ∞ B(A, H ) = − αit (A)dt, (2.19) 0
where αit is the imaginary time evolution generated by H . For unbounded Hamiltonians H , it may not be obvious that B(A, H ) can be defined on a dense domain. However, if ψ is a ground state corresponding to the Hamiltonian H , then B(A, H )ψ exists. Moreover, from (2.18), we conclude that ∂x ψ0 (x) = B(x)ψ0 (x),
(2.20)
where B(x) = B (∂x H (x), H (x)). Similarly, in the density matrix formalism, for ρ0 (x) := |ψ0 (x) ψ0 (x)|,
(2.21)
∂x ρ0 (x) = B(x)ρ0 (x) + ρ0 (x)B(x)∗ .
(2.22)
Eq. (2.20) implies that
We will define the proposed excited state ψ as the solution of a differential equation analogous to (2.20). First, we need to introduce some further notation. Let H be a Hamiltonian for which the dynamics {αt } exists; finite volume is sufficient. For any a > 0, t ∈ R \ {0}, and local observable A ∈ A, we may define Aa (it, H ) :=
1 −at 2 e 2πi
∞
−∞
e−as ds. s − it 2
αs (A)
(2.23)
In addition, for T > 0 the quantity
T
Ba,T (A, H ) := − 0
Aa (it, H ) − Aa (it, H )∗ dt,
(2.24)
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449
will play a crucial role. In Lemma 7 of Sect. 5, we will show that when projected onto the ground state of a gapped Hamiltonian H , the quantity Ba,T (A, H ) well approximates B(A, H ) for a judicious choice of parameters, e.g., a = γ L /L and T = L/2; we note that the observable A must also satisfy the constraint that its range is orthogonal to the ground state. With this in mind, consider the solution of the differential equation introduced by Hastings in [8]: ∂θ ψa,T (θ ) = Ba,T (θ ) ψa,T (θ ),
(2.25)
where Ba,T (θ ) = Ba,T (∂θ Hθ,0 , Hθ,−θ ), subject to the boundary condition ψa,T (0) = ψ0 . Note that Ba,T (θ ) is anti-hermitian, and hence any ψa,T (θ ) solving (2.25) will have constant norm. To be explicit, the proposed state ψa,T (θ ) differs from the actual ground state of the doubly twisted Hamiltonian Hθ,−θ , in three essential ways. Compare (2.19) in the case that A = ∂θ Hθ,−θ and H = Hθ,−θ with (2.24) given that A = ∂θ Hθ,0 and H = Hθ,−θ . i) We have introduced a cut-off at T < ∞. ii) We have approximated the imaginary-time evolution of an observable A, αit (A), by Aa (it, H ) − Aa (it, H )∗ . iii) We have replaced the observable ∂θ Hθ,−θ with ∂θ Hθ,0 . The modifications i) and ii) are of a technical nature, i.e., to make the relevant quantities well-defined and amenable to estimation (see Sect. 5). The motivation behind the third replacement is an attempt to approximate the ground state of the singly twisted Hamiltonian Hθ,0 . 3. Energy Estimates As is discussed in the introduction, the goal of this section is to prove an estimate of the form |ψ1 , HL ψ1 − E 0 | ≤ C L ν e−cγ L L ,
(3.1)
see (1.16) and Theorem 3 below, for the proposed variational state. Explicitly, we will take ψ1 = ψa,T (2π ), i.e., the solution of (2.25) evaluated at θ = 2π , with the specific choice of parameters a = γ L /L and T = L/2. Since the operator Ba,T (θ ), defined in (2.25), is anti-hermitian, it is clear that ψ1 remains normalized, and the bound stated above demonstrates that if the gap is sufficiently large, γ L ≥ C log(L)/L, then ψ1 corresponds to a state with small (depending on C) excitation energy. An estimate of the form (3.1), with correction terms, can be proven based on the general results in Sect. 5. In the proof of Theorem 1, which is a proof by contradiction, we will be assuming γ L ≥ C log(L)/L. Therefore, we can assume here, without loss of generality, that there exists a constant c > 0 such that γ L L ≥ c for sufficiently large L. This assumption, which is not necessary, will simplify the presentation in Sects. 3 and 4.
3.1. Local estimates on the states. In this subsection we prove a technical result which estimates, uniformly in θ , the norm difference between the ground state of Hθ,−θ and the proposed state in the left half of the system, more precisely in the window centered around the location, (m, VL ) where the θ -twist occurs. Since the restrictions of the states to the half-systems are described by density matrices, it is natural to use the trace norm
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for this estimate. Recall that for any bounded operator A on a Hilbert space H, the trace norm is defined by √ (3.2) A1 = Tr A∗ A, assuming this quantity is finite. Using the polar decomposition for bounded linear operators, it is easy to see that, alternatively, A1 = sup |Tr AB|.
(3.3)
B∈B(H):
B=1
Recall that the density matrix corresponding to the ground state of the Hθ,−θ Hamiltonian satisfies the equation ∂θ ρ0 (θ, −θ ) = B(θ ) ρ0 (θ, −θ ) + ρ0 (θ, −θ ) B(θ )∗ ,
(3.4)
compare with (2.22), where we have used the notation B(θ ) = B(A(θ ), Hθ,−θ ) for the operator B(A, H ) as defined in (2.19) and the observable A(θ ) = ∂θ Hθ,−θ . Note that by construction ρ0 (θ, −θ ) remains normalized. We will often write A(θ ) = A1 (θ ) − A2 (θ ) where, the observables A1 (θ ) = ∂θ Hθ,0 and A2 (θ ) = ∂θ H0,θ are supported in the window about the twists of angle θ and −θ , respectively. Regarding Hθ,θ as a function of two variables, we may write Ai (θ ) = ∂i Hθ,−θ for convenience. The notation Bi (θ ) = B(Ai (θ ), Hθ,−θ ) will also be useful. The proposed state is the solution of
∂θ ρa,T (θ ) = Ba,T (θ ), ρa,T (θ ) , (3.5) where the operator Ba,T (θ ) = Ba,T (A1 (θ ), Hθ,−θ ) is defined in (2.24) with observable A1 (θ ) = ∂1 Hθ,−θ . The parametrization we choose is a = γ L /L and T = L/2. Since the operator Ba,T (θ ) is anti-hermitian, the solution ρa,T (θ ) is a density matrix. We will denote by Trm c [·] the partial trace over the Hilbert (S) (W ) space corresponding to Λ L ∪ Λ L (m + L/2). Note that the terms in the Hamiltonian that have been twisted by an angle θ are supported in the complementary region ) Λ(W L (m). Given a gap γ L > 0 above the ground state of the H = H0,0 Hamiltonian, we will be able to estimate the trace norm of the difference in the two states restricted
(W ) to Λ L (m). We will show this by estimating ∂θ Tr m c ρa,T (θ ) − ρ0 (θ, −θ ) . Theorem 2. As described in the introduction, we assume F1, F2, and LSM1-6. If there exists a constant c > 0 such that γ L L ≥ c for sufficiently large L and we choose the parameters a = γ L /L and T = L/2, then there exist constants C > 0 and k > 0 so that
sup Tr m c ρa,T (θ ) − ρ0 (θ, −θ ) 1 ≤ C L 2d e−kγ L L , (3.6) θ∈[0,2π ]
for L large enough. Here C and k depend only on the interaction Φ and the underlying set Λ.
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We note that the assumption concerning the existence of a constant c > 0 such that γ L L ≥ c for sufficiently large L is not necessary. We impose it here for simplicity of presentation. Without this additional assumption, one may prove an analogue of (3.6), which contains correction terms, by inserting the bounds proven in Sect. 5 directly into the proof given below. Since we make this assumption, it is convenient to state a lemma which compiles many of the technical results found in Sect. 5 and applies them to the (W ) present set-up. For this, we need two more definitions. Denote by Ba,T (θ ) the operator
(W ) (W ) defined by Ba,T A1 (θ ), Hθ,−θ , where the Hamiltonian Hθ,−θ is the full Hamiltonian Hθ,−θ restricted to the windows about the twists, see (2.8). Lastly, set P0θ to be the projection onto the ground state ψ0 (θ, −θ ) of the twisted Hamiltonian Hθ,−θ .
Lemma 2. Assume F1, F2, and LSM1-6. If there exists a constant c > 0 such that γ L L ≥ c for sufficiently large L and we choose the parameters a = γ L /L and T = L/2, then there exists constants C > 0 and k > 0 for which both (W )
sup Ba,T (θ ) − Ba,T (θ ) ≤ C L 2d e−kγ L L
(3.7)
sup Ba,T (θ ) − B1 (θ ) P0θ ≤ C L d e−kγ L L
(3.8)
θ∈[0,2π ]
and θ∈[0,2π ]
when L is large enough. Proof of Lemma 2. Equation (3.7) follows by combining Lemma 4 and Remark 1. Using Lemma 7 and Remark 3, one obtains (3.8). Proof of Theorem 2. The proof of Theorem 2 follows by deriving a uniform bound on the θ -derivative of the differences in these density matrices. Specifically, the bound is in trace norm, and the uniformity is with respect to θ ∈ [0, 2π ]. (W ) Using (3.4), (3.5), and inserting the local operator Ba,T (θ ) for comparison, one may easily verify that
(W ) Ba,T (θ ), ρa,T (θ ) − ρ0 (θ, −θ ) ∂θ Tr m c ρa,T (θ ) − ρ0 (θ, −θ ) = Tr m c +
3
ri (θ ),
(3.9)
i=1
where the three remainder terms are given by
(W ) Ba,T (θ ) − Ba,T (θ ), ρa,T (θ ) , r1 (θ ) := Tr m c
(W ) Ba,T (θ ), ρ0 (θ, −θ ) − ∂1 ρ0 (θ, −θ ) , r2 (θ ) := Tr m c
(3.10) (3.11)
and r3 (θ ) := Tr m c [∂1 ρ0 (θ, −θ ) − ∂θ ρ0 (θ, −θ )] .
(3.12)
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B. Nachtergaele, R. Sims (W )
As A1 (θ ) is supported near (m, VL ) and Hθ,−θ contains only those interaction terms over (W )
(W )
sets X ⊂ Λ L , it is clear that Ba,T (θ ) is contained in the algebra of local observables (W )
with support in Λ L (m); we will denote this algebra by A(m). Therefore,
(W ) (W ) Tr m c Ba,T (θ ), ρa,T (θ ) − ρ0 (θ, −θ ) = Ba,T (θ ), Tr m c ρa,T (θ ) − ρ0 (θ, −θ ) . (3.13) (W ) Ba,T (θ )
Since is anti-hermitian, we may apply norm preservation, i.e. Theorem 7, to (3.9) and conclude that 3 θ
Tr m c ρa,T (θ ) − ρ0 (θ, −θ ) ≤ ri (θ )1 dθ . (3.14) 1 i=1
0
We need only bound the trace norms of the remainder terms ri (θ ). As ρa,T (θ ) is a density matrix, in particular non-negative with a normalized trace, one has that (W ) (3.15) r1 (θ )1 ≤ 2 Ba,T (θ ) − Ba,T (θ ) ≤ C L 2d e−kγ L L , using Lemma 2 above. To estimate r2 (θ ), we note that as in (3.4), ∂1 ρ0 (θ, −θ ) = B1 (θ ) ρ0 (θ, −θ ) + ρ0 (θ, −θ ) B1 (θ )∗ ,
(3.16)
where ∂1 denotes differentiation with respect to only the first twist angle, namely θ , which is situated near the sites (m, VL ). Here we have also used that ∂1 E 0 (θ, −θ ) = 0, see Lemma 1. A simple norm estimate yields that (W ) r2 (θ )1 ≤ 2 Ba,T (θ ) − B1 (θ ) P0θ (W ) ≤ 2 Ba,T (θ ) − Ba,T (θ ) + 2 Ba,T (θ ) − B1 (θ ) P0θ . (3.17) Appealing again to Lemma 2, we see that r2 (θ ) satisfies the desired bound. Lastly, r3 (θ ) = Tr m c [ ∂2 ρ0 (θ, −θ )]. Since we have shown in Lemma 1 that ∂2 E 0 (θ, −θ ) = 0 as well, the analogue of (3.16) holds for ∂2 ρ0 (θ, −θ ). Thus, r3 (θ )1 = sup Tr O B2 (θ ) ρ0 (θ, −θ ) + ρ0 (θ, −θ ) B2 (θ )∗ O∈A(m):
O=1
≤ 2 sup O∈A(m):
O=1
∞
| ψ0 (θ, −θ ), O αit (A2 (θ )) ψ0 (θ, −θ ) | dt, (3.18)
0
where the observables O are arbitrary elements of A(m), again, the algebra of local (W ) observables with support in Λ L (m). Integrals of this type are bounded using Lemma 6; see also Remark 2. Since the observables we are considering have a separation distance proportional to L, we may estimate r3 (θ )1 ≤ C L 2d e−kγ L L .
(3.19)
Combining the results found on each of the remainders, we arrive at the estimate claimed in (3.6).
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3.2. Bound on the energy. Equipped with Theorem 2 and Lemma 2, we may now bound the excitation energy corresponding to the proposed state. Theorem 3. Assume F1, F2, and LSM1-6. If there exists a constant c > 0 such that γ L L ≥ c for sufficiently large L and we choose the parameters a = γ L /L and T = L/2, then there exists constants C > 0 and k > 0 so that |ψ1 , HL ψ1 − E 0 | ≤ C L 3d−1 e−kγ L L
(3.20)
for large enough L. Here, we take ψ1 = ψa,T (2π ). The proof of this theorem may be understood as follows. Recall that the ground state energy of the doubly twisted Hamiltonian is independent of θ , i.e., E 0 = ψ0 , HL ψ0 = ψ0 (θ, −θ ), Hθ,−θ ψ0 (θ, −θ ).
(3.21)
Moreover, the separation between the twists of angle θ and −θ grows with the volume. Locality should enable one to estimate the energy difference between performing two twists, the ground state, and performing only one twist, the excited state. A rigorous version of this idea is described below. First, we recall some of the notation introduced in Sect. 2.1. We have written the twisted Hamiltonian as the sum of two terms (W )
Hθ,−θ = Hθ,−θ + H (S) .
(3.22)
It is useful to further subdivide the twisted terms as (W )
(W )
Hθ,−θ = Hθ
(W )
(m) + H−θ (m + L/2),
(3.23)
(W ) with support in a window where Hθ(W ) (m) contains all those interaction terms in Hθ,−θ (W )
(W )
about the twist of angle θ , i.e. Λ L (m), and similarly, H−θ (m + L/2) contains all (W )
(W )
those interaction terms in Hθ,−θ with support in Λ L (m + L/2). The untwisted terms in (3.22) are supported in the remaining strips. We refer to Eqs. (2.5)–(2.9) for more details. It was also noted in Sect. 2.1 that W (θ )∗ Hθ,−θ W (θ ) = H,
(3.24)
see (2.12). Now, for any state ψ, one may calculate the expected energy due to a single twist:
(W ) (W ) ψ, Hθ,0 ψ = ψ, Hθ (m)ψ + ψ, H0 (m + L/2) + H (S) ψ = E 0 + R1 (θ ) + R2 (θ ).
(3.25)
Here, we inserted appropriate terms so that we may compare ψ, Hθ,0 ψ to the ground state energy; the remainder terms are given by R1 (θ ) := ψ, Hθ(W ) (m)ψ − ψ0 (θ, −θ ), Hθ(W ) (m)ψ0 (θ, −θ )
(3.26)
and
(W ) (W ) R2 (θ ) := ψ, H0 (m + L/2) + H (S) ψ − ψ0 , H0 (m + L/2) + H (S) ψ0 , (3.27)
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B. Nachtergaele, R. Sims
where ψ0 is the ground state of H = H0,0 . The bound ψ, Hθ,0 ψ − E 0 ≤ |R1 (θ )| + |R2 (θ )|,
(3.28)
readily follows for any state ψ. Proof of Theorem 3. For each fixed θ , the bound (3.28) is valid for our proposed state ψa,T (θ ). We will estimate the resulting remainders uniformly for θ ∈ [0, 2π ] and thereby prove the claimed result. To see this, we first rewrite the remainders in terms of the density matrices of the states restricted to the region containing the first twist. It is clear that (W ) R1 (θ ) = Tr ρa,T (θ ) − ρ0 (θ, −θ ) Hθ (m) (3.29)
(W ) = Trm Trm c ρa,T (θ ) − ρ0 (θ, −θ ) Hθ (m) , where the partial traces are as defined just prior to Theorem 2. Thus
(W ) |R1 (θ )| ≤ Hθ (m) Trm c ρa,T (θ ) − ρ0 (θ, −θ ) 1 ≤CL
3d−1 −kγ L L
e
(3.30)
,
where we have used Theorem 2. In fact, from the assumptions we have made, one verifies that (W ) (W ) Hθ (m) ≤ 2 Φ(X ) ≤ 2|||Φ|||1 |Λ L (m)| ≤ C L d−1 . (3.31) x∈Λ L (m) X x (W )
For the second remainder, (W ) R2 (θ ) = Tr ρa,T (θ ) − ρ0 (0, 0) H0 (m + L/2) + H (S) ,
(3.32)
we observe that the only θ dependence is in the density matrix corresponding to the proposed state. Using the differential equation (3.5), we find that
R2 (θ ) = Tr Ba,T (θ ), ρa,T (θ ) H0(W ) (m + L/2) + H (S)
(W ) = − Tr Ba,T (θ ), H0 (m + L/2) + H (S) ρa,T (θ ) . (3.33) (W )
The first term above is easy to estimate. Recall that the quantity Ba,T (θ ) is supported (W )
(W )
(W )
in Λ L (m), whereas H0 (m + L/2) has support in Λ L (m + L/2). Thus (W ) (W ) (W ) Ba,T (θ ), H0 (m + L/2) = Ba,T (θ ) − Ba,T (θ ), H0 (m + L/2) , (3.34) and moreover, (W ) (W ) Tr Ba,T (θ ) − Ba,T (θ ), H0 (m + L/2) ρa,T (θ ) (W ) (W ) ≤ 2 H0 (m + L/2) Ba,T (θ ) − Ba,T (θ ) ≤ C L 3d−1 e−kγ L . (3.35) (W ) The second term may be similarly estimated. Let H˜ θ,−θ be defined as in (2.8), excepting (W ) L that the windows are slightly smaller: of size 4 − 2R. Then [ B˜ a,T (θ ), H (S) ] = 0, and the argument above applies. We have bounded R2 (θ ).
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455
4. Orthogonality We will now prove that, under the assumptions given in the Introduction, the proposed state is nearly orthogonal to the ground state. As in Sect. 3, we again make the assumption that γ L L ≥ c > 0, for sufficiently large L. The reasoning behind orthogonality is simple. From LSM5, we know that the ground state is an eigenvector of the translation operator with eigenvalue 1, i.e., T ψ0 = ψ0 . On the other hand, the proposed state will very nearly be an eigenvector of T2π,0 , as defined in Sect. 2.1, with eigenvalue 1. Due to the odd parity condition T2π,0 = −T and, hence, we find that the ground state and the proposed state are eigenvectors corresponding to distinct eigenvalues. More concretely, it is easy to check that ψa,T (2π ), ψ0 = T2π,0 ψa,T (2π ), T ψ0 + I − T2π,0 ψa,T (2π ), ψ0 , (4.1) from which the estimate ψa,T (2π ), ψ0 ≤ 1 T2π,0 − I ψa,T (2π ) 2
(4.2)
immediately follows. The remainder of this section will be used to prove a bound on Tθ,0 ψa,T (θ ) − ψa,T (θ ) (4.3) uniformly for θ ∈ [0, 2π ]. This is the content of Theorem 4. 4.1. Observations concerning the twisted ground state. We begin with a warm-up exercise involving the twisted ground state. In Sect. 2.1, we saw that the twisted ground state is invariant with respect to the twisted translations; i.e., Tθ,−θ , ψ0 (θ, −θ ) = ψ0 (θ, −θ ) , and therefore
∂θ Tθ,−θ ψ0 (θ, −θ ) − ψ0 (θ, −θ ) = 0.
(4.4)
One may rewrite this derivative in the form of an operator acting on ψ0 (θ, −θ ), i.e., (4.4) is equivalent to D(θ )ψ0 (θ, −θ ) = 0,
(4.5)
∗ ∗ + Tθ,−θ B(θ ) Tθ,−θ − B(θ ). D(θ ) = ∂θ Tθ,−θ Tθ,−θ
(4.6)
where D(θ ) is given by
Here we have used the differential equation for ψ0 (θ, −θ ), i.e. (2.20), and the notation from the beginning of Sect. 3.1, which will be used throughout this section. It will be easy to see that the operator D(θ ) can be written as the sum of two terms, D1 (θ ) and D2 (θ ), corresponding to the twists at m and m + L/2, respectively. The goal of this subsection is to estimate D1 (θ )ψ0 (θ, −θ ), see Lemma 3 below. Using (2.13), one finds that ∗ 3 3 ∂θ Tθ,−θ Tθ,−θ = i S(m+1,v) −i S(m+L/2+1,v) . (4.7) v∈VL
v∈VL
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B. Nachtergaele, R. Sims
One has that D(θ ) = D1 (θ ) − D2 (θ ), where 3 ∗ D1 (θ ) = i S(m+1,v) + Tθ,−θ B1 (θ ) Tθ,−θ − B1 (θ ),
(4.8)
v∈VL
and D2 (θ ) = i
v∈VL
3 ∗ S(m+L/2+1,v) + Tθ,−θ B2 (θ ) Tθ,−θ − B2 (θ ).
(4.9)
For what follows, we will denote by Aθ = ψ0 (θ, −θ ), Aψ0 (θ, −θ ) the twisted ground state expectation of a local observable A. We have demonstrated in Lemma 1 that 0 = ∂i E 0 (θ, −θ ) = ∂i Hθ,−θ θ = Ai (θ )θ ,
(4.10)
for i = 1, 2. From this, we conclude that ∗ = Bi (θ ) θ = Ai (θ )θ = 0, Tθ,−θ Bi (θ ) Tθ,−θ θ
(4.11)
as well. Moreover, we similarly have that 3 3 Di (θ )θ = 0 as S(x,v) = S(x,v) = 0,
(4.12)
v∈VL
v∈VL
θ
0
for any x ∈ [1, L]. For the last equality above, we used that ψ0 (θ, −θ ) = W (θ )ψ0 , W (θ ) commutes with the third component of the spins, rotation invariance implies that the total spin is zero, and translation invariance in the 1-direction. Since D(θ )ψ0 (θ, −θ ) = 0, we have that D1 (θ )ψ0 (θ, −θ ) = D2 (θ )ψ0 (θ, −θ ) from which it is clear that 0 = D(θ )∗ D(θ )θ = 2D1 (θ )∗ D1 (θ )θ − 2D1 (θ )∗ D2 (θ )θ .
(4.13)
As indicated above, we wish to estimate the first term on the right-hand side above. We do so by estimating the second term. Observe that 3 3 S(m+1,v) S(m+L/2+1,v D1 (θ )∗ D2 (θ )θ = −
) 0
+i
∞
+i
0 0
+
∗ αit (A1 (θ ))∗ αis (A2 (θ )) − αis Tθ,−θ A2 (θ )Tθ,−θ θ ds dt
∞
3 3 αit (A1 (θ ))∗ S(m+L/2+1,v dt
) − S(m+L/2,v ) θ
v ∈VL 0 ∞ ∞
0
0
3 3 S(m,v) αit (A2 (θ )) dt − S(m+1,v) θ
v∈VL 0 ∞ ∞
+
v,v ∈VL
∗ αit (A1 (θ ))∗ αis (A2 (θ )) − αis Tθ,−θ A2 (θ )Tθ,−θ ds dt. θ
(4.14)
That each of these terms is bounded follows from our decay of correlation results found in Sect. 5.3. In fact, we have proven the following lemma.
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457
Lemma 3. Assume F1, F2, and LSM1-6. If there exist a constant c > 0 such that γ L L ≥ c for sufficiently large L, then there exist constants C > 0 and k > 0 so that D1 (θ )ψ0 (θ, −θ )2 ≤ C L 3d−1 e−kγ L L ,
(4.15)
for L large enough. Proof. Clearly, one has that D1 (θ )ψ0 (θ, −θ )2 = D1 (θ )∗ D1 (θ ) θ = D1 (θ )∗ D2 (θ ) θ ,
(4.16)
from (4.13) above. Applying Theorem 6, Lemma 6, and Remark 2, as appropriate, to the terms found in (4.14), one arrives at (4.15). 4.2. Orthogonality of the excited state. We are now ready to provide the orthogonality estimate. Theorem 4. Assume F1, F2, and LSM1-6. If there exist a constant c > 0 such that γ L L ≥ c for sufficiently large L and we choose the parameters a = γ L /L and T = L/2, then there exist constants C > 0 and k > 0 so that ψa,T (2π ), ψ0 ≤ C L 2d e−kγ L L (4.17) when L is large enough. Proof. As is clear from (4.2), Theorem 4 follows from bounding the quantity appearing in (4.3) uniformly for θ ∈ [0, 2π ]. A short calculation, using (2.25), shows that
∂θ Tθ,0 ψa,T (θ ) − ψa,T (θ ) = Ca,T (θ ) Tθ,0 ψa,T (θ ) − ψa,T (θ ) + Da,T (θ ) ψa,T (θ ), (4.18) where ∗ ∗ Ca,T (θ ) = ∂θ Tθ,0 Tθ,0 + Tθ,0 Ba,T (θ ) Tθ,0 ,
(4.19)
∗ ∗ Da,T (θ ) = ∂θ Tθ,0 Tθ,0 + Tθ,0 Ba,T (θ ) Tθ,0 , − Ba,T (θ ),
(4.20)
and
are both anti-Hermitian operators. The first term on the right-hand side of (4.18) is normpreserving, and therefore, we need only bound the norm of the second by Theorem 7. The norm of Da,T (θ )ψa,T (θ ) will now be estimated by rewriting it in terms of quantities for which we have already proven bounds. Each term will be shown to satisfy a bound of the form (4.17). We begin by writing
Da,T (θ ) ψa,T (θ ) 2 = Tr Da,T (θ )∗ Da,T (θ ) ρa,T (θ )
= Tr Da,T (θ )∗ Da,T (θ ) ρ0 (θ, −θ )
+ Tr Da,T (θ )∗ Da,T (θ ) ρa,T (θ ) − ρ0 (θ, −θ ) . (4.21)
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B. Nachtergaele, R. Sims
The first term on the right-hand side above, which is equal to Da,T (θ )ψ0 (θ, −θ )2 , may be estimated by comparing it with the vector D1 (θ )ψ0 (θ, −θ ) introduced in the previous subsection. In fact, Da,T (θ )ψ0 (θ,−θ ) ≤ D1 (θ )ψ0 (θ,−θ )+ Da,T (θ )− D1 (θ ) ψ0 (θ,−θ ). (4.22) We bounded the first term above in Lemma 3. For the second, observe that
∗ (W ) ∗ + Tθ,0 Ba,T (θ ) − Ba,T (θ ) Tθ,0 Da,T (θ ) − D1 (θ ) = Tθ,−θ Ba,T (θ ) − B1 (θ ) Tθ,−θ
(W ) ∗ − Tθ,−θ Ba,T (θ ) − Ba,T (θ ) Tθ,−θ + B1 (θ ) − Ba,T (θ ), (4.23) from which it is clear that (W ) Da,T (θ ) − D1 (θ ) ψ0 (θ, −θ ) ≤ 2 Ba,T (θ ) − Ba,T (θ ) + 2 Ba,T (θ ) − B1 (θ ) P0θ .
(4.24)
That each of these terms satisfies the desired bound follows from Lemma 2. For the final term on the right-hand side of (4.21), we insert and remove (W )
(W )
(W )
∗ ∗ + Tθ,0 Ba,T (θ ) Tθ,0 , − Ba,T (θ ), Da,T (θ ) = ∂θ Tθ,0 Tθ,0
(4.25)
) a local observable supported in Λ(W L (m). Observe that (W ) (W ) ∗ Da,T (θ ) ≤ ∂θ Tθ,0 Tθ,0 + 2Ba,T (θ ) ≤ C L d ,
(4.26)
where we have used Proposition 2. We may write
Tr Da,T (θ )∗ Da,T (θ ) ρa,T (θ ) − ρ0 (θ, −θ ) (W ) (W ) = Tr Da,T (θ )∗ Da,T (θ ) ρa,T (θ ) − ρ0 (θ, −θ ) (W ) (W ) + Tr Da,T (θ )∗ Da,T (θ ) − Da,T (θ )∗ Da,T (θ ) ρa,T (θ ) − ρ0 (θ, −θ ) . (4.27) The first term above may be estimated by
(W ) (W ) Trm Da,T (θ )∗ Da,T (θ ) Trm c ρa,T (θ ) − ρ0 (θ, −θ )
(W ) (W ) ≤ Da,T (θ )∗ Da,T (θ ) Trm c ρa,T (θ ) − ρ0 (θ, −θ ) 1 ≤ C L 4d e−kγ L ,
(4.28)
where for the final inequality above we used Theorem 2 again. For the second term, we rewrite the difference as
∗ (W ) (W ) (W ) Da,T (θ )∗ Da,T (θ )− Da,T (θ )∗ Da,T (θ ) = Da,T (θ )− Da,T (θ ) Da,T (θ )
(W ) (W ) +Da,T (θ )∗ Da,T (θ )− Da,T (θ ) , (4.29)
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459
and apply the norm estimate (W ) (W ) Da,T (θ ) − Da,T (θ ) ≤ 2 Ba,T (θ ) − Ba,T (θ ) .
(4.30)
We find that (W ) (W ) Tr Da,T (θ )∗ Da,T (θ ) − Da,T (θ )∗ Da,T (θ ) ρa,T (θ ) − ρ0 (θ, −θ ) (W ) (W ) ≤ 4 Ba,T (θ ) − Ba,T (θ ) Da,T (θ ) + Da,T (θ ) ,
(4.31)
which satisfies the required bound by Lemma 2 and an estimate analogous to (4.26). This completes the proof of Theorem 4. 5. Auxiliary Results In this section, we collect a number of auxiliary results, technical estimates as well as a few lemmas of a more general nature, which are needed for the proofs in Sects. 3 and 4. We first recall the Lieb-Robinson bounds which are used to demonstrate quasi-locality of the dynamics associated to general quantum spin systems, see Theorem 5. Then, we observe in Proposition 1 that these Lieb-Robinson bounds may be used to compare the dynamics of a Hamiltonian defined on a given system with the dynamics of the same Hamiltonian restricted to a subsystem. Next, we provide in Lemma 4 an explicit bound which applies to the specific type of interactions we consider in this work. In Sect. 5.2, we introduce the operators Ba,T (A, H ) which play a prominent role in our argument. We first discuss a few of their basic properties, and then use Proposition 1 to estimate the difference that arises in defining the operator with the full Hamiltonian as opposed to the Hamiltonian restricted to a subsystem; this is the content of Lemma 5. Lastly, we remark on exactly how this estimate will be used in the main text. We review the Exponential Clustering Theorem in Sect. 5.3, and use it to prove a technical estimate, see Lemma 6. Moreover, in this section we also prove Lemma 7. This result provides an estimate on the quantity Ba,T (A, H ) − B(A, H ) P0 in terms of the parameters a, T , and the spectral gap of H , see (5.64). Here P0 denotes the spectral projection onto the ground state of H , and the bound is valid for local observables A satisfying P0 A P0 = 0. Lastly, we formulate a statement concerning solutions of certain simple differential equations in Sect. 5.4. 5.1. Lieb-Robinson bounds. For what follows, we adopt the same general framework for quantum spin models that was described in Sect. 1.2, including Conditions F1, F2, and the assumption that Φλ < +∞ for some λ > 0 (see (1.7) for the definition of the norm · λ ). We will use the following version of the Lieb-Robinson bound [18], which is a variant of the results proven in [20, 9]. Theorem 5 (Lieb-Robinson Bound). Let λ ≥ 0 and take Φ ∈ Bλ (Λ). For any pair of local observables A ∈ A X and B ∈ AY with X, Y ⊂ Λ, one may estimate [αt (A), B] ≤
2 A B gλ (t) Fλ (d(x, y)) , Cλ (F) x∈X y∈Y
(5.1)
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B. Nachtergaele, R. Sims
for any t ∈ R. Here {αt } is the dynamics generated by Φ, and one may take 2 Φ C (F) |t| λ λ − 1 if d(X, Y ) > 0, e gλ (t) = e2 Φλ Cλ (F) |t| otherwise.
(5.2)
Our proof of the Lieb-Schultz-Mattis theorem relies heavily on comparing the time evolution corresponding to a given Hamiltonian to that of the Hamiltonian restricted to a subsystem. The errors that result from such a comparison can be estimated in terms of a specific commutator to which the Lieb-Robinson bounds readily apply. We begin with some notation. Let λ ≥ 0 and consider Φ ∈ B(Λ). For finite Λ0 ⊂ Λ, the Hamiltonian corresponding to Φ restricted to Λ0 is given by the self-adjoint operator
H0 =
Φ(X ).
(5.3)
X ⊂Λ0 (0)
We will denote by αt the time evolution corresponding to H0 , i.e., for any local ob(0) servable A, αt (A) = eit H0 Ae−it H0 for all t ∈ R. Proposition 1. Let λ ≥ 0 and Φ ∈ Bλ (Λ). Suppose the Hamiltonian corresponding to Φ restricted to a finite volume Λ0 ⊂ Λ is written as the sum of two self-adjoint operators, i.e., H0 = H1 + H2 . Denoting by αt(i) the time evolution corresponding to Hi , for i = 0, 1, 2, then for any local observable A and t ∈ R, one has that (0) αt (A)
−
(1) αt (A)
|t|
≤ 0
[H2 , αs(1) (A)] ds.
(5.4)
Proof. Define the function f : R → A by f (t) := αt(0) (A) − αt(1) (A).
(5.5)
A simple calculation shows that f satisfies the following differential equation: (1) f (t) = i H0 − H1 , αt (A) + i [H0 , f (t)] ,
(5.6)
subject to the boundary condition f (0) = 0. As this is a first order equation, the solution can be found explicitly: f (t) =
(0) αt
t 0
(0) α−s
(1) i H2 , αs (A) ds .
(5.7)
(0)
Using expression (5.7) and the automorphism property of αt , it is clear that f (t) ≤ 0
as claimed.
|t|
[H2 , αs(1) (A)] ds,
(5.8)
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461
To estimate the norm of the commutator appearing in Proposition 1, specifically in the bound (5.4), it is useful to specialize the general Lieb-Robinson bounds described above to the exact context we encounter in the present work. For example, we will be interested in specific finite volume Hamiltonians, those defined in Sect. 2 as Hθ,θ , and particular observables, such as A1 (θ ) = ∂1 Hθ,−θ and A2 (θ ) = ∂2 Hθ,−θ . Let αt be the (W ) time evolution corresponding to the Hθ,θ Hamiltonian, and let αt denote the dynamics (W ) associated with the Hamiltonian Hθ,θ which is defined in (2.8). We use the following estimate several times. Lemma 4. Let Φ ∈ Bλ (Λ), then there exists constants C > 0 and k > 0 for which (W ) (5.9) max sup H (S) , αt (Ai (θ )) ≤ C ek|t| L 2(d−1) e−λL/4 . i=1,2 θ∈[0,2π ]
Here it is important that C and k depend only on the properties of the underlying set Λ and the interaction Φ; they do not depend on the length scale L. Proof. We will estimate the above commutator in the case that the observable is A1 (θ ); an analogous result holds for A2 (θ ). Recall that in (2.9) we wrote H (S) as a sum of interaction terms. Similarly, if one denotes by Pm (θ ; Y ) := Vm (θ )∗ Φ(Y )Vm (θ ) − Φ(Y ), then A1 (θ ) may be written as ∂θ Pm (θ ; Y ) A1 (θ ) = Y ⊂Λ L
= −i
Y ⊂Λ L :
y∈Y+
Pm (θ;Y )=0
Vm (θ )∗ S y3 , Φ(Y ) Vm (θ ),
(5.10)
where Y+ is the set of sites y ∈ Y strictly to the right of m. Inserting both of these expressions into the right-hand side of (5.9) and applying the triangle inequality, it is clear that we must bound many terms of the form
(W ) (5.11) Vm (θ )∗ S y3 , Φ(Y ) Vm (θ ) Φ(X ), αt . Term by term, we apply the Lieb-Robinson bound provided by Theorem 5, and use that the distance between the supports of X and Y is linear in L; concretely for any x ∈ X and y ∈ Y , d(x, y) ≥ d(X, Y ) ≥ L4 − 3R. We find that each term described by (5.11) satisfies an upper bound of the form C(t) Φ(X ) |Y | [S y3 , Φ(Y )] e−λL/4 ,
(5.12)
where C(t) may be taken as C(t) =
2F 2Cλ (F)Φλ |t|+3λR e . Cλ (F)
(5.13)
We need only count the number of terms. The combinatorics of the sums may be naively estimated as follows: H (S) corresponds to a sum of the form X ⊂Λ L : (S) X ∩Λ L =∅
L 4 +R−1
≤
n= L4 −R+1 v∈VL X (m+n,v)
3L 4 +R−1
+
v∈VL X (m+n,v) n= 3L 4 −R+1
,
(5.14)
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B. Nachtergaele, R. Sims
whereas for A1 (θ ) we have that the sum
Y ⊂Λ L :
y∈Y
Pm (θ;Y )=0
m+R
≤
.
(5.15)
n=m−R v∈VL Y (n,v) y∈Y
Putting everything together, we have obtained that L (S) (W ) H , αt (A1 (θ )) ≤ 2 C(t) |||Φ|||1 |||Φ|||2 |VL |2 (2R + 1)(2R − 1) e−λ 4 , (5.16) which proves the claim. Recall, |||Φ|||1 := sup
Φ(X )
(5.17)
[Sx3 , Φ(X )] .
(5.18)
x∈Λ X x
and |||Φ|||2 := sup
x∈Λ X x
|X |
x ∈X
5.2. Approximation of the imaginary time evolution. For our proof of the Lieb-SchultzMattis Theorem, we introduce an operator which, under certain assumptions, approximates the imaginary time evolution corresponding to a given Hamiltonian. In this section, we provide several basic estimates of this operator to which we will often refer in the main text. Let λ ≥ 0, Φ ∈ Bλ (Λ), Λ0 ⊂ Λ be a finite set, and H be the Hamiltonian corresponding to Φ restricted to Λ0 . Denote by αt , for t ∈ R, the time evolution determined by H . For any local observable A, a > 0, M > 0, and t = 0, define 2 M 2 e−at e−as ds, (5.19) Aa,M (it, H ) = αs (A) 2πi −M s − it and set Aa (it, H ) = lim M→∞ Aa,M (it, H ). We use the operator T Ba,T (A, H ) = − Aa (it, H ) − Aa (it, H )∗ dt,
(5.20)
0
to define our variational state in the main text, see (2.25). We begin with some basic properties. Proposition 2 (Shanti’s Bound). Let Φ ∈ Bλ (Λ), A be a local observable, a > 0, and T > 0. The operator Ba,T (A, H ) is anti-hermitian and bounded. In particular, A π . (5.21) Ba,T (A, H ) ≤ 2 a It is important to note that the bound above is independent of the finte volume Λ0 on which the Hamiltonian H is defined.
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463
Proof. That Ba,T (A, H ) is anti-hermitian follows immediately from (5.20). Combining (5.19) and (5.20), one finds that T ∞ i s 2 2 e−a(s +t ) αs (A) 2 2 ds dt, (5.22) Ba,T (A, H ) = π 0 −∞ s +t from which (5.21) easily follows as T 1 A ∞ −as 2 A π Ba,t (A, H ) ≤ . e |s| dt ds ≤ 2 2 π −∞ 2 a 0 s +t
(5.23)
In situations where the local observable A and the Hamiltonian H are fixed, we will often write Aa (it) and Ba,T to simplify notation. The following estimate is a simple consequence of (5.19). Proposition 3. Let Φ ∈ Bλ (Λ) and A be a local observable. One has that T A −a M 2 ≤ T √ A (it) − A (it) dt . e a a,M 2M πa
(5.24)
0
Proof. For any t = 0, e−at Aa (it) − Aa,M (it) = 2πi
2
e−as αs (A) ds, s − it |s|>M 2
and therefore, one has the pointwise estimate Aa (it) − Aa,M (it) ≤ e−at 2 A e−a M 2 2π M Upon integration, (5.24) readily follows.
(5.25)
π . a
(5.26)
We will now prove an analogue of Proposition 1 for the quantities Ba,T (A, H ) introduced in (5.20). The estimate provided below is made explicit in terms of an a priori input, an assumed form of the Lieb-Robinson bound, see (5.27) below. Lemma 5. Let λ ≥ 0 and Φ ∈ Bλ (Λ). Suppose the Hamiltonian corresponding to Φ restricted to a finite volume Λ0 ⊂ Λ is written as the sum of two self-adjoint opera(i) tors, i.e., H0 = H1 + H2 . Denote by αt the time evolution corresponding to Hi , for i = 0, 1, 2. If, for a given local observable A, there exists numbers ci > 0, i = 1, 2, 3, for which (1) (5.27) [ H2 , αt (A) ] ≤ c1 ec2 |t| − c3 , for all t ∈ R, then the following estimate holds: Ba,T (A, H0 ) − Ba,T (A, H1 ) ≤ 2T e−a M 2 M
A c1 M 2 + √ π πa
,
(5.28)
where M has to be chosen as the positive solution of a M 2 + c2 M − c3 = 0.
(5.29)
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B. Nachtergaele, R. Sims
We note that in our applications the numbers ci will depend on the observables A and H2 ; in fact, they will be functions of the length scale L. We articulate this dependence explicitly in Remark 1 below. Proof. One may write
T
Ba,T (A, H0 ) − Ba,T (A, H1 ) = −
Aa (it, H0 ) − Aa (it, H1 ) dt
0 T
Aa (it, H0 )∗ − Aa (it, H1 )∗ dt, (5.30)
+ 0
and therefore Ba,T (A, H0 ) − Ba,T (A, H1 ) ≤ 2
Aa (it, H0 ) − Aa (it, H1 ) dt . (5.31)
T 0
Moreover, the integrand may be expressed as Aa (it, H0 ) − Aa (it, H1 ) = Aa (it, H0 ) − Aa,M (it, H0 ) + Aa,M (it, H0 ) −Aa,M (it, H1 ) + Aa,M (it, H1 ) − Aa (it, H1 ), (5.32) and for j = 0, 1, the bounds
0
T
T A −a M 2 Aa (it, H j ) − Aa,M (it, H j ) dt , ≤ 2M √πa e
(5.33)
follow immediately from Proposition 3. From this we conclude that for any M > 0, Ba,T (A, H0 ) − Ba,T (A, H1 ) ≤ 2
T
0
Aa,M (it, H0 ) − Aa,M (it, H1 ) dt
2T A −a M 2 + . e √ M πa
(5.34)
Clearly, the pointwise estimate Aa,M (it, H0 )− Aa,M (it, H1 ) ≤
e−at 2π
2
M
−M
(0)
(1)
αs (A)−αs (A) −as 2 e ds, (5.35) |s|
follows directly from (5.19). By Proposition 1, we have that αs(0) (A) − αs(1) (A)
|s|
≤ 0
[H2 , αx(1) (A)] d x,
(5.36)
and by assumption (5.27), the integrand satisfies a uniform bound for |s| ≤ M. The implication is that for all |s| ≤ M, αs(0) (A) − αs(1) (A) ≤ c1 ec2 M−c3 . |s|
(5.37)
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465
Putting everything together, we obtain that A −a M 2 Ba,T (A, H0 ) − Ba,T (A, H1 ) ≤ 2T √ e M πa c1 ec2 M−c3 T M −a(t 2 +s 2 ) + e dsdt. π 0 −M
(5.38)
As M here was arbitrary, we chose it as the (positive) solution of the following quadratic equation a M 2 + c2 M − c3 = 0. In this case, A c1 M 2 Ba,T (A, H0 ) − Ba,T (A, H1 ) ≤ 2T e−a M 2 √ (5.39) + M π πa as claimed.
Remark 1. In the main text of the paper, we will use Lemma 5 for the Hamiltonians (W ) H0 = Hθ,−θ and H1 = Hθ,−θ each of which depends on the length scale L; see Sect. 2 for the relevant definitions. It is assumed that H0 has a gap γ L > 0 above the ground state energy. The local observable A will be exactly as in Lemma 4, and therefore, the numbers ci , i = 1, 2, 3, may be taken as follows: c1 = C L 2(d−1) , c2 = k, and c3 = λL/4, where again C and k depend only on the interaction and the underlying set Λ. In this case, we will choose the parametrization a = γ L /L and T = L/2. With this choice, the estimate (5.28) takes the form: 1 (W ) 2d −kγ L L 1 + d√ . (5.40) sup Ba,T (θ ) − Ba,T (θ ) ≤ C L e L γL L θ∈[0,2π ] Here we have used the notation from Sect. 3 and the fact that the gap γ L has a uniform bound from above; see (1.1) in Sect. 1.
5.3. Estimates for gapped systems. We derive two useful results in this subsection. For the first we recall the Exponential Clustering Theorem [9, 20], and use it to prove a technical estimate Lemma 6. The second crucial estimate in this subsection, Lemma 7 below, applies specifically to gapped systems. It provides a bound on the norm of the difference in the operators Ba,T (A, H ) and B(A, H ) when restricted to the space of ground states corresponding to H . The bound applies to local observables A which project off the ground state, i.e. satisfy P0 A P0 = 0, where P0 is the spectral projection of H onto the ground states, and is explicit in the parameters a, T , and the spectral gap of H , see (5.64) below. We will consider Hamiltonians H , of the type introduced in Sect. 5.1, with an additional feature: a gap above the ground state energy. To state the gap condition precisely, we consider a representation of the system on a Hilbert space H. This means that there is a representation π : A → B(H), and a self-adjoint operator H on H such that π(αt (A)) = eit H π(A)e−it H , for all t ∈ R and A ∈ A. For the results which follow, we will assume that H ≥ 0 and that Ω ∈ H is a normalized ground state, i.e., a vector state for which H Ω = 0 and Ω = 1. We say that the system has a spectral gap in this representation if there exists
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B. Nachtergaele, R. Sims
δ > 0 such that σ (H ) ∩ (0, δ) = ∅, where σ (H ) is the spectrum of the operator H . In that case, the spectral gap, γ , is defined to be γ = sup{δ > 0 | σ (H ) ∩ (0, δ) = ∅}.
(5.41)
Let P0 denote the orthogonal projection onto ker H . From now on, we will work in this representation and simply write A instead of π(A). The following result concerning exponential clustering was proven in [20]. Theorem 6 (Exponential Clustering). Fix λ > 0. Let Φ ∈ Bλ (Λ) be an interaction for which the corresponding self-adjoint Hamiltonian has a representation H ≥ 0 with a normalized ground state vector Ω, i.e., H Ω = 0 and Ω = 1. Let PE denote the family of spectral projections corresponding to H . If H has a spectral gap of size γ > 0 above the ground state energy, then there exist µ > 0 such that for any local observables A and B with A ∈ A X , B ∈ AY , d := dist(X, Y ) > 0, and P0 BΩ = P0 B ∗ Ω = 0, the estimate
| Ω, Aαit (B)Ω | ≤ C(A, B) e
−µd
1+
γ 2t2 4µ2 d 2
,
(5.42)
holds for all t : 0 ≤ t (4Φλ Cλ + γ ) ≤ 2 λ d. Here, one may choose ⎞ ⎛ 1 ⎠ 2 F (d(x, y)) + √ C(A, B) = A B ⎝ 1 + π Cλ π µd
(5.43)
x∈X y∈Y
and µ=
λγ . 4Φλ Cλ + γ
(5.44)
The above result easily leads to estimates on integrals of these ground state expectations. We state two such bounds in the next lemma, as they will arise in the proof of our main result. Lemma 6. Under the assumptions of Theorem 6, we have the estimates
∞
| Ω, Aαit (B)Ω | dt ≤
2µd C(A, B) + A B e−µ d
0
e−µd , γ
(5.45)
and
∞ 0
≤
∞
Ω, Aαi(s+t) (B)Ω ds dt
0
e−µd
. (µd)2 C(A, B) + A B 2µd + e−µ d γ2
(5.46)
Proof. Define T by the equation γ T = 2 µ d. We have that
T 0
| Ω, Aαit (B)Ω | dt ≤ C(A, B) T e−µ d ,
(5.47)
A Multi-Dimensional Lieb-Schultz-Mattis Theorem
and also
∞
| Ω, Aαit (B)Ω | dt ≤
T
467
A B −γ T e . γ
Combining these two bounds, we arrive at (5.45). Similarly, one may estimate T /2 T /2 2 Ω, Aαi(s+t) (B)Ω ds dt ≤ C(A, B) T e−µ d , 4 0 0 ∞ T /2 A B T −µ d Ω, Aαi(s+t) (B)Ω ds dt ≤ e , 2γ T /2 0 and finally,
∞ T /2
∞ T /2
Ω, Aαi(s+t) (B)Ω ds dt ≤ A B e−γ T . γ2
(5.48)
(5.49) (5.50)
(5.51)
Remark 2. In our applications, the Hamiltonian H = Hθ,−θ depends on a length scale L and has a gap γ L > 0 above the ground state energy. The support of the observables A and B will have a minimal distance d = L/2 − 2R − 1, and moreover, B = B(θ ) will either be A1 (θ ) or A2 (θ ). In this case, B(θ )θ = 0 by Lemma 1, and therefore, the assumptions of Theorem 6 hold. Here we have used ·θ to denote the ground state expectation corresponding to ψ0 (θ, −θ ). It is easy to see that there exist positive constants C and C
for which C γ L L ≤ µd ≤ C
L, and thus ultimately constants C and k for which the bounds appearing in (5.45) and (5.46) may be estimated by ∞ L d −kγ L L | Aαit (B(θ ))θ | dt ≤ C A |X | sup e , (5.52) γL θ∈[0,2π ] 0 and
∞
∞
sup
θ∈[0,2π ] 0
0
d+1 Aαi(s+t) (B(θ ))θ ds dt ≤ C A |X | L e−kγ L L , (5.53) γ L2
respectively. For the next lemma we will need the following basic estimate involving the decay of certain Fourier transforms. Proposition 4. Let a > 0 and T > 0 be given. Define a function Fa,T : R → C by ∞ −i Es −as 2 T e e 1 2 ds dt. (5.54) e−at Fa,T (E) := 2πi 0 s − it −∞ For all E ∈ R, Fa,T (E) ≥ 0 and the estimate T − E2 e 4a , (5.55) 2 is valid for E ≥ 0. In the parameter range, E ≥ 2aT > 0, one may also show that T T − E2 e 4a . e−Et dt − Fa,T (−E) ≤ (5.56) 2 0 Fa,T (E) ≤
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Proof. One may easily verify that for any t > 0, 1 2πi
∞
−∞
1 e−i Es e−as ds = √ s − it 2 πa 2
∞
e−tw e−
(w+E)2 4a
dw,
(5.57)
0
for all E ∈ R, see e.g. Lemma 1 in [20]. This implies the first claim. Evaluating the Gaussian integral yields ∞ (w+E)2 E2 1 1 e−tw e− 4a dw ≤ e− 4a , (5.58) √ 2 2 πa 0 in the case that E ≥ 0, from which (5.55) is clear. To obtain (5.56), we first recall that the Fourier transform of a Gaussian is a Gaussian, i.e., for all z ∈ C, ∞ 2 x2 1 − z2 e = √ e− 2 e−i x z d x, (5.59) 2π −∞ √ i E z (for E ∈ R), and and therefore, by rescaling √ z → − 2az, multiplying through by e changing variables w = 2ax + E, we have that ∞ (w−E)2 1 2 ei E z e−az = √ eiwz e− 4a dw, (5.60) 2 πa −∞ for all z ∈ C. Now, by direct substitution into (5.57), we have that ∞ T (w−E)2 1 −at 2 Fa,T (−E) = √ e e−tw e− 4a dw. 2 πa 0 0
(5.61)
Applying (5.60), with the special choice of z = it, one sees that 0 T T (w−E)2 1 2 e−Et dt − Fa,T (−E) = √ e−at e−tw e− 4a dw. 2 πa 0 0 −∞
(5.62)
Since w < 0 and t > 0, the integrand above e−tw e−
(w−E)2 4a
E2
= e− 4a e
(E−2at)w 2a
w2
e− 4a
satisfies a trivial bound when E ≥ 2aT . For these values of E, (5.56) holds.
(5.63)
We may now prove the main estimate for gapped systems. Recall the definitions of the operators B = B(A, H ) and Ba,T = Ba,T (A, H ) in (2.19) and (2.24), respectively. Lemma 7. Let H ≥ 0 be a self-adjoint operator and PE denote the family of spectral projections corresponding to H . Suppose H has a gap γ > 0, and let A be a local observable for which P0 A P0 = 0. If 2aT ≤ γ , then one has that ∗ −γ T γ2 Ba,T − B P0 ≤ T e− 4a A P0 + A P0 + e A P0 . (5.64) 2 γ
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Proof. One may rewrite the difference in these operators as T Ba,T − B P0 = (αit (A) − Aa (it)) dt P0 0
∞
+
αit (A) dt P0 +
T
T
Aa (it)∗ dt P0 .
(5.65)
0
Each of these terms may be bounded in norm. For any vectors f and g, one may calculate ∞ ∞ αit (A)dt P0 g = f, e−t H A P0 g dt f, T T ∞ ∞ e−t E d f, PE A P0 g dt, = T
γ
(5.66)
where we have used the spectral theorem to rewrite the time evolution and the fact that P0 A P0 = 0. Clearly then, ∞ ∞ f, ≤ f A P α (A)dt P g g e−γ t dt, (5.67) it 0 0 T
and therefore,
T
∞ T
e−γ T ≤ A P0 . αit (A) dt P0 γ
(5.68)
Likewise, one may similarly calculate T −at 2 ∞ T 2 e e−as ds dt Aa (it)∗ dt P0 g = − αs (A) f, P0 g f, 2πi −∞ s + it 0 0 ∞ = Fa,T (E) d f, PE A∗ P0 g , (5.69) γ
where we have introduced Fa,T (E) = Fa,T (E) as in (5.54) of Proposition 4 above. The estimate T T −γ2 ∗ e 4a A∗ P0 , A (it) dt P (5.70) a 0 ≤ 2 0 readily follows from (5.55) of Proposition 4 and the fact that 0 < γ ≤ E. Lastly, an analogous calculation shows that T f, [αit (A) − Aa (it)] P0 g dt 0 ! ∞ T −Et = e dt − Fa,T (−E) d f, PE A P0 g . γ
(5.71)
0
Thus, for 2aT ≤ γ , we may apply (5.56) of Proposition 4 and establish the bound T T −γ2 (5.72) [αit (A) − Aa (it) ] dt P0 ≤ 2 e 4a A P0 . 0
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Compiling our estimates, we have proven that: if 2aT ≤ γ , then ∗ −γ T γ2 Ba,T − B P0 ≤ T e− 4a A P0 + A P0 + e A P0 , 2 γ as claimed.
(5.73)
Remark 3. Applying Lemma 7 to the operator H = Hθ,−θ , whose spectral projections we denote by PEθ , and the local observable A = A1 (θ ), we find that there exists a constant C > 0 for which, along the parametrization a = γ L /L and T = L/2, " γL L # γ L θ e− 4 d − L4 Ba,T (θ ) − B1 (θ ) P0 ≤ C L e sup 1+ . (5.74) γL L θ∈[0,2π ] 5.4. Norm preserving flows. In this section, we collect some basic facts about the solutions of first order, inhomogeneous differential equations. Definition 1. Let B be a Banach space. For each θ ∈ R, let A(θ ) : B → B be a bounded linear operator, and denote by X (θ ) the solution of the differential equation ∂θ X (θ ) = A(θ ) X (θ )
(5.75)
with boundary condition X (0) = X 0 ∈ B. We say that the family of operators A(θ ) is norm-preserving if the corresponding flow is isometric, i.e., for every X 0 ∈ B, the mapping γθ : B → B which associates X 0 → X (θ ), i.e., γθ (X 0 ) = X (θ ), satisfies γθ (X 0 ) = X 0 for all θ ∈ R.
(5.76)
Two typical examples are the case where B is a Hilbert space and A(θ ) is anti-hermitian and the case where B is a Banach space of linear operators on a Hilbert space with a spectral norm (such as a p−norm with p ∈ [1, +∞]), and where A(θ ) is a symmetric derivation (e.g., i times the commutator with a self-adjoint operator). Theorem 7. Let A(θ ), for θ ∈ R, be a family of norm preserving opeartors in some Banach space B. For any bounded measurable function B : R → B, the solution of ∂θ Y (θ ) = A(θ )Y (θ ) + B(θ ),
(5.77)
with boundary condition Y (0) = Y0 , satisfies the bound θ B(θ ) dθ . Y (θ ) − γθ (Y0 ) ≤
(5.78)
0
Proof. For any θ ∈ R, let X (θ ) be the solution of ∂θ X (θ ) = A(θ ) X (θ )
(5.79)
with boundary condition X (0) = X 0 , and let γθ be the linear mapping which takes X 0 to X (θ ). By variation of constants, the solution of the inhomogeneous equation (5.77) may be expressed as θ −1 Y (θ ) = γθ Y0 + (γs ) (B(s)) ds . (5.80) 0
The estimate (5.78) follows from (5.80) as A(θ ) is norm preserving.
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5.5. Existence of local unitaries with vanishing expectation. Consider a finite system with a Hamiltonian of the form Φ(X ), (5.81) H= X
where Φ is an interaction as defined at the beginning of the paragraph containing Eq. (1.7). In the introduction, (1.1), we stated a simple upper bound for the spectral gap of any such Hamiltonian with a unique ground state. The argument we gave there made use of a one-site unitary U ∈ A{x} with the property that Ω, U Ω = 0. In the following lemma we show that such a unitary always exists. Lemma 8. Let H be a complex Hilbert space of dimension at least 2. Then, for any density matrix ρ on H, there exists a unitary U ∈ B(H) such that TrρU = 0. Proof. First consider the case where dim H is finite and even, or infinite. The odddimensional case has to be treated slightly differently. Let {e0 , e1 , . . .} denote an orthonormal basis of eigenvectors of ρ, with eigenvalues ρi ordered in non-increasing order. If H is not separable, it is sufficient that {e0 , e1 , . . .} contain a basis for the separable subspace ranρ. Then, a suitable unitary U can be defined as follows: $ |e2i+1 e2i | + |e2i e2i+1 |. (5.82) U= i≥0
If dim H is odd (and hence by our assumptions ≥ 3), it is sufficient to change the first summand in (5.82) as follows U = a|e0 e0 | − a|e1 e1 | + b|e1 e0 | + b|e0 e1 | + eiφ |e2 e2 | $ + |e2i−1 e2i | + |e2i e2i−1 |, i≥2
where a, b ∈ C and φ ∈ R, are chosen such that |a|2 + |b|2 = 1 and eiφ ρ2 = aρ0 − aρ1 . This is always possible since ρ22 ≤ (ρ0 + ρ1 )2 . It is straightforward to check that U thus defined has the desired properties.
Acknowledgements This article is based on work supported by the U.S. National Science Foundation under Grant # DMS06-05342. Both authors thank the Erwin Schrödinger Institute for kind hospitality during a fruitful stay.
References 1. Affleck, I., Lieb, E.H.: A proof of part of Haldane’s conjecture on quantum spin chains. Lett. Math. Phys. 12, 57–69 (1986) 2. Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Valence Bond Ground States in Isotropic Quantum Antiferromagnets. Commun. Math. Phys. 115, 477–528 (1988) 3. Aizenman, M., Nachtergaele, B.: Geometric aspects of quantum spin states. Commun. Math. Phys. 164, 17–63 (1994) 4. Bratteli, O., Robinson, D.: Operator algebras and quantum statistical mechanics 2. Second ed., New York: NY: Springer Verlag, 1997
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5. Dyson, F., Lieb, E.H., Simon, B.: Phase transitions in quantum spin systems with isotropic and non-isotropic interactions. J. Stat. Phys. 18, 335–383 (1978) 6. Haldane, F.D.M.: Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model. Phys. Lett. 93, 464–468 (1983) 7. Haldane, F.D.M.: Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983) 8. Hastings, M.B.: Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431–14 (2004) 9. Hastings, M.B., Koma, T.: Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006) 10. Kennedy, T., Lieb, E.H., Shastry, B.S.: Existence of Néel order in some spin 1/2 Heisenberg antiferromagnets. J. Stat. Phys. 53, 1019–1030 (1988) 11. Koma, T., Tasaki, H.: Symmetry breaking and finite-size effects in quantum many-body systems. J. Stat. Phys. 76, 745–803 (1994) 12. Landau, L., Fernando-Perez, J., Wreszinski, W.F.: Energy gap, clustering, and the Goldstone theorem in statistical mechanics. J. Stat. Phys. 26, 755–766 (1981) 13. Lieb, E., Mattis, D.: Ordering energy levels in interacting spin chains. Journ. Math. Phys. 3, 749–751 (1962) 14. Lieb, E., Schultz, T., Mattis, D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. (N.Y.) 16, 407–466 (1961) 15. Lieb, E.H., Robinson, D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972) 16. Manousakis, E.: The spin-1/2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides. Rev. Mod. Phys. 63, 1–62 (1991) 17. Nachtergaele, B.: Quasi-state decompositions for quantum spin systems. Probability Theory and Mathematical Statistics. Proceedings of the 6th Vilnius Conference (Grigelionis, B. et al., ed.), UtrechtTokyo-Vilnius, VSP/Tev, 1994, pp. 565–590 18. Nachtergaele, B., Ogata, Y., Sims, R.: Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124, 1–13 (2006) 19. Nachtergaele, B., Sims, R.: Recent progress in quantum spin systems. Markov Processes Rel. Fields. 13(2), 315–329 (2007) 20. Nachtergaele, B., Sims, R.: Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119–130 (2006) 21. Sachdev, S.: Quantum antiferromagnets in two dimensions. In: Low dimensional quantum field theories for condensed matter physicists. Yu Lu, S., Lundqvist, G., Morandi, eds., Singapore: World Scientific, 1995 22. Sachdev, S., Park, K.: Ground states of quantum antiferromagnets in two dimensions. Annals of Physics (N.Y.) 298, 58–122 (2002) 23. Simon, B.: The statistical mechanics of lattice gases. Princeton Series in Physics, Vol. 1, Princeton, NJ: Princeton University Press, 1993 24. Tasaki, H.: Low-lying excitation in one-dimensional lattice electron system. http://arXiv.org/list/condmat/0407616, 2004 25. Yamanaka, M., Oshikawa, M., Affleck, I.: Nonperturbative approach to Luttinger’s theorem in one dimension. Phys. Rev. Lett. 79, 1110–1113 (1997) Communicated by H.-T. Yau
Commun. Math. Phys. 276, 473–508 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0344-x
Communications in
Mathematical Physics
The Bloch-Okounkov Correlation Functions of Classical Type David G. Taylor, Weiqiang Wang Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA. E-mail: [email protected]; [email protected] Received: 29 August 2006 / Accepted: 21 March 2007 Published online: 26 September 2007 – © Springer-Verlag 2007
Abstract: Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This function has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, ∞ -modules etc., and it can also be interpreted as correlation functions on integrable gl of level one. Such gl∞ -correlation functions at higher levels were then calculated by Cheng and Wang. In this paper, generalizing the type A results, we formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classi∞ of type B, C, D at arbitrary levels. As byproducts, we obtain cal Lie subalgebras of gl new q-dimension formulas for integrable modules of type B, C, D and some fermionic type q-identities. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . 1.1 The earlier works . . . . . . . . . . . . . . 1.2 The goal . . . . . . . . . . . . . . . . . . . 1.3 Our approach . . . . . . . . . . . . . . . . 1.4 Open questions . . . . . . . . . . . . . . . 1.5 Organization and Acknowledgment . . . . . 2. The Preliminaries . . . . . . . . . . . . . . . . 2.1 Classical Lie algebras of infinite dimension 2.2 Classical Lie groups . . . . . . . . . . . . . 2.3 Additional notations . . . . . . . . . . . . . 3. Correlation Functions on d∞ -Modules of Level l 3.1 The Fock space Fl . . . . . . . . . . . . . . 3.2 The (O(2l), d∞ )-Howe duality . . . . . . . 3.3 The main results of [BO, CW1] . . . . . . .
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. . . . . . . . . . . . . . 4.1 The Fock space F 4.2 The (O(2l + 1), d∞ )-Howe duality . . . . . . . 4.3 The n-point d∞ -function of level 21 . . . . . . . 4.4 The n-point d∞ -functions of level l + 21 . . . . 4.5 The q-dimension of a d∞ -module of level l + 21 5. Correlation Functions on c∞ -Modules of Level l . . 5.1 The (Sp(2l), c∞ )-Howe duality . . . . . . . . 5.2 The n-point c∞ -functions of level l . . . . . . . 5.3 The q-dimension of a c∞ -module . . . . . . . . 6. Correlation Functions on b∞ -Modules of Level l . . 6.1 The (Pin(2l), b∞ )-Howe duality . . . . . . . . 6.2 The operator B(t) . . . . . . . . . . . . . . . . 6.3 The n-point b∞ -functions of level l . . . . . . . 6.4 The q-dimension of a b∞ -module . . . . . . . 7. Correlation Functions on b∞ -Modules of Level l + 21 7.1 The (Spin(2l + 1), b∞ )-Howe duality . . . . . 7.2 The n-point b∞ -function of level 21 . . . . . . . 7.3 The n-point b∞ -functions of level l + 21 . . . . 7.4 The q-dimension of a b∞ -module of level l + 21 References . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction 1.1. The earlier works. Bloch and Okounkov [BO] (also see [Ok] for some simplification) introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. Their work was in part motivated by a certain modular invariance property of trace functions of vertex operators and representation theory of the W1+∞ algebra (cf. [Zhu, FKRW, Blo]). Subsequently, this function and its variant have been interpreted as a generating function of the Gromov-Witten invariants of an elliptic curve by Okounkov-Pandharipande [OP], and as a generating function of intersection numbers on Hilbert schemes of points by Li, Qin and the second author [LQW]. We also refer the reader to [Lep, Mil] for formal vertex operator generalizations, [W2] for a neutral fermionic Fock space version, and [CW2] for a q, t-deformation of the Bloch-Okounkov n-point function. From a representation theoretic viewpoint, the Bloch-Okounkov n-point function can be also easily interpreted as correlation functions on integrable modules over the Lie ∞ of level one (cf. [Ok, Mil, CW1]). Along this line, Cheng and the second algebra gl author [CW1] formulated and calculated such n-point correlation functions on integrable ∞ -modules of level l (l ∈ N). gl 1.2. The goal. The goal of this paper is to formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classical
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∞ of type B, C, D at arbitrary level, generalizing the works [BO, Lie subalgebras of gl CW1] in type A. Note that the integrability of these modules implies that the levels have to be positive (half-)integral. By the original works of Date, Jimbo, Kashiwara and Miwa ∞ affords classical Lie subalgebras of type B, C, D, and these ([DJKM1, DJKM2]), gl infinite-dimensional Lie algebras played an important role in connections with solition equations discovered by the Kyoto School in the early 1980’s. ∞ is intimately related to that of the W1+∞ algebra The representation theory of gl (cf. [FKRW] and the references therein). It follows that the Bloch-Okounkov correlation ∞ -modules can be regarded as those for W1+∞ -modules. In the same functions for gl ∞ is closely related vein, the representation theory of the classical Lie subalgebras of gl to that of the classical Lie subalgebras of W1+∞ initiated in [KWY]; the Howe dualities [W1], which are to be used extensively in this work, readily carry over if one replaces ∞ by classical Lie subalgebras of W1+∞ . In this way, the classical Lie subalgebras of gl n-point correlation functions studied in this paper can be in turn regarded as those for modules over classical Lie subalgebras of W1+∞ . 1.3. Our approach. To achieve our goal, the first (main) step here is to relate the correlation functions at higher levels to the correlation functions at the bottom levels (i.e. of level one and/or level 21 ). Our main tool is the free field realization [DJKM1, DJKM2] (also cf. Feingold-Frenkel [FF]) and the Howe duality due to the second author [W1] ∞ and various classical Lie groups (where between the classical Lie subalgebras of gl sometimes disconnected groups and different covering groups are required). We refer to [Ho1, Ho2] for Howe’s original setups, where all Lie algebras and groups involved are finite-dimensional. ∞ appear in these We note that all integrable modules of these Lie subalgebras of gl Howe duality decompositions, and the level of an integrable module matches with the rank of the corresponding Lie group. A detailed knowledge of irreducible modules over various Lie groups (cf. Bröcker-tom Dieck [BtD]) and the determinantal ratio form of the Weyl character formulas for classical Lie algebras (cf. Fulton-Harris [FH]) are also used in this paper in an essential way. A similar approach has actually been applied in [CW1] successfully where the type A ∞ and G L l due to I. Frenkel [Fr] (also cf. [W1]) was used. Howe duality between gl Forced by the new technical features in type B, C, D, we establish in this paper the relations between the n-point functions at higher levels and at the bottom levels in a different way, avoiding the usage of the q-dimension formula for integrable modules in [CW1]. As a byproduct, we obtain neat q-dimension formulas for the corresponding integrable modules over the classical Lie subalgebras of type B, C, D, which are simpler than the ones in [KWY] obtained by a specialization of the Weyl-Kac character formulas. We remark that the idea of using Howe duality to obtain irreducible character formulas has also been applicable in different setups (cf. Cheng-Lam [CL]). Our second step is more straightforward. By using the free field realization we are able to relate the calculation of the n-point function of type B, D of level one to the n-point function of type A of level one which has been computed in [BO]. The type C level one case can be handled by a combination of Howe duality and the connection to the type A level one case. An additional step is needed to take care of the half-integral levels, which occur in type B and D. Using an identification of a pair of complex fermions and two neutral fermions, we obtain formulas, recursive on n, computing the n-point functions of classical type of level 21 in terms of those of level one. Explicit formulas in different forms for
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the 1-point functions of type B and D of level 21 were obtained in [W2] using the method of partition identities. Identifying these different formulas gives rise to two interesting q-identities of fermionic type. It turns out that these identities have been known with a very different proof (cf. e.g. [Kac]). Combining all these steps together, we have calculated all the n-point correlation functions of classical type. The final formulas involve the Weyl groups of the Lie groups appearing in various Howe dualities and the original Bloch-Okounkov function of type A and level one (which in turn is an expression in terms of theta functions). Remarkably, the solutions in type B and type D look almost identical formally though different Lie algebras and different Howe dualities are involved in different types. 1.4. Open questions. The integrable modules whose correlation functions are computed here are occasionally not irreducible (instead it could be a sum of two irreducibles) over the infinite-dimensional Lie algebras of type D, but they can always be regarded as irreducible modules over the corresponding orthogonal groups. This is a familiar phenomenon of spinor vs. half-spinor representations. Nevertheless, it will be interesting to determine completely the (refined) n-point functions for all irreducible integrable modules of type D. In this direction, we have only obtained limited results. By observing an intrinsic connection with the theory of partitions (cf. Andrews [An]), we find an explicit formula for the refined 1-point function of type D of level one. We can also formulate the n-point correlation functions for modules of negative (half-) ∞ and its classical subalgebras, and these modules have appeared in integral levels of gl the Howe duality decompositions (cf. [KR] for type A and [W1] in general). It will be interesting to determine these n-point correlation functions. In light of the developments in [CW1] and in this paper, the main difficulty lies in understanding the cases at level −1, where the connection with the theory of partitions available at level one and at level 21 is now lacking. A more challenging question is to ask for a geometric interpretation of the correlation functions studied in this paper (and also in [CW1]). For example, can they be interpreted as correlation functions in some supersymmetric gauge theory where the classical Lie groups used in this paper appear as gauge groups? 1.5. Organization and Acknowledgment. The paper is organized as follows. In Sect. 2, ∞ and various Lie groups we set up the notations for the classical Lie subalgebras of gl used in later sections. In Sects. 3 and 4 respectively, we formulate and calculate the n-point functions and the q-dimension of integrable d∞ -modules of level l and of level l + 21 respectively. In Sect. 5, we calculate the n-point functions and the q-dimension of integrable c∞ -modules of level l. In Sects. 6 and 7 respectively, we formulate and calculate the n-point functions and the q-dimension of integrable b∞ -modules of level l and of level l + 21 respectively. Various Fock spaces of free fermionic fields and Howe dualities are recalled and used in each of Sects. 3–7. The proofs in Sects. 3 and 4 are given in detail, while the proofs in Sects. 5–7 are often sketchy when they are parallel to the ones in Sects. 3–4. 2. The Preliminaries The purpose of this section is to set up notations for the infinite-dimensional Lie algebras and classical Lie groups which we will use.
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2.1. Classical Lie algebras of infinite dimension. In this subsection we review Lie alge ≡ gl ∞ and various Lie subalgebras of B, C, D type (cf. [DJKM1, DJKM2]). bras gl Denote by gl the Lie algebra of all matrices (ai j )i, j∈Z satisfying 2.1.1. Lie algebra gl. ai j = 0 for |i − j| sufficiently large. Denote by E i j the infinite matrix with 1 at (i, j) and 0 elsewhere and let the weight of E i j be j − i. This defines a Z–principal gradation gl = j∈Z gl j . Denote by gl ≡ gl∞ = gl ⊕ CC the central extension given by the following 2–cocycle with values in C (cf. [DJKM1]): C(A, B) = tr ([J, A]B) ,
(1)
where J = j≤0 E ii . The Z–gradation of Lie algebra gl extends to gl by letting the weight of C to be 0. This leads to a triangular decomposition 0 ⊕ gl − = gl + ⊕ gl gl ± = ⊕ j∈N gl 0 = gl0 ⊕ CC. Let ± j , gl where gl Hia = E ii − E i+1,i+1 + δi,0 C (i ∈ Z). ) the highest weight gl–module ∗0 , where Denote by L(gl; with highest weight ∈ gl ∗ 0 be the fundamental weights, C acts as a scalar which is called the level. Let aj ∈ gl a a with fundamental weights labeled, is i.e. j (Hi ) = δi j . The Dynkin diagram for gl, the following: … ◦ −2
◦ −1
◦ 0
◦ 1
◦… 2
2.1.2. Lie algebra d∞ . Let d ∞ = {(ai j )i, j∈Z ∈ gl | ai j = −a1− j,1−i } be a Lie subalgebra of gl of type D. Denote by d∞ = d ∞ CC the central extension given by the 2-cocycle (1). Then d∞ has a natural triangular decomposition induced with Cartan subalgebra d∞0 = gl 0 ∩ d∞ . Given ∈ d∞ ∗ , we let from gl 0 Hid = E ii + E −i,−i − E i+1,i+1 − E −i+1,−i+1 (i ∈ N), H0d = E 0,0 + E −1,−1 − E 2,2 − E 1,1 + 2C. Denote by id the i th fundamental weight of d∞ , i.e. id (H jd ) = δi j . The Dynkin diagram of d∞ is: ◦ 0@
@ @◦ 2
1 ◦
◦ 3
◦··· 4
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2.1.3. Lie algebra c∞ . Let c∞ = {(ai j )i, j∈Z ∈ gl | ai j = −(−1)i+ j a1− j,1−i } be a Lie subalgebra of gl of type C. Denote by c∞ the central extension of c∞ given a natural triangular decomposition with by the 2-cocycle (1). Then c∞ inherits from gl Cartan subalgebra c∞0 . Given ∈ c∞ ∗0 , we let Hic = E ii + E −i,−i − E i+1,i+1 − E 1−i,1−i (i ∈ N), H0c = E 0,0 − E 1,1 + C. Denote by ic the i th fundamental weight of c∞ , i.e. ic (H jc ) = δi j . The Dynkin diagram of c∞ is: ◦ 0
>
◦ 1
◦ 2
◦ … 3
2.1.4. Lie algebra b∞ . Let b∞ = {(ai j )i, j∈Z ∈ gl | ai j = −a− j,−i } be a Lie subalgebra of gl of type B. Denote by b∞ the central extension of b∞ given by a natural triangular decomposition the 2-cocycle (1). The Lie algebra b∞ inherits from gl with Cartan subalgebra b∞0 . Given ∈ b∞ ∗0 , we let Hib = E ii + E −i−1,−i−1 − E i+1,i+1 − E −i,−i (i ∈ N), H0b = 2(E −1,−1 − E 1,1 ) + 2C. Denote by ib the i th fundamental weight of b∞ , i.e. ib (H jb ) = δi j . The Dynkin diagram of b∞ is: ◦ 0
<
◦ 1
◦ 2
◦ … 3
2.2. Classical Lie groups. We present here a parametrization of irreducible modules of various classical Lie groups. See [BtD] (also [W1]) for more detail. 2.2.1. O(2l). We define O(2l) = {g ∈ G L(2l); t g J g = J } with 0 Il . J= Il 0 The Lie group G L(l) can be identified as a subgroup of O(2l) consisting of matrices of the form diag(g, t g −1 ), where t g denotes the transpose. The Lie algebra so(2l) of S O(2l) consists of matrices of the form α β , (2) γ −t α
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where α, β, γ are l × l matrices and β, γ are skew-symmetric. The Lie algebra gl(l) is identified with the subalgebra of so(2l) consisting of matrices of the form (2) with β = γ = 0. Let b(so(2l)) be the Borel subalgebra of so(2l) which consists of matrices (2) with γ = 0 and α upper triangular matrices, and let h(so(2l)) be the Cartan subalgebra of diagonal matrices diag(t1 , . . . , tl , −t1 , . . . , −tl ), ti ∈ C. Then gl(l) and so(2l) share the same Cartan subalgebra. An irreducible G L(l)-module is parameterized by its highest weight which runs over the set (A) ≡ {(m 1 , m 2 , . . . , m l ) | m 1 ≥ m 2 ≥ . . . ≥ m l , m i ∈ Z}. An irreducible S O(2l)-module is parameterized by its highest weight in {(m 1 , m 2 , . . . , m l ), | m 1 ≥ m 2 ≥ . . . ≥ m l−1 ≥ |m l |, m i ∈ Z}. For notational simplicity, we may identify a highest weight module with its highest weight below. O(2l) is a semi-direct product of S O(2l) by Z2 . Denote by τ ∈ O(2l) \ S O(2l) the 2l × 2l matrix A B (3) B A with A = diag(1, . . . , 1, 0), B = diag(0, . . . , 0, 1). Then τ normalizes the Borel b. If λ is an S O(2l)-module of highest weight (m 1 , m 2 , . . . , m l ), then τ.λ has highest weight λ := (m 1 , m 2 , . . . , −m l ). The induced module of (m 1 , m 2 , . . . , m l ) (m l = 0) to O(2l) is irreducible and its restriction to S O(2l) is a sum of (m 1 , m 2 , . . . , m l ) and (m 1 , m 2 , . . . , −m l ). We denote this O(2l)-module λ by (m 1 , m 2 , . . . , m l ), where m l > 0. If m l = 0, the module λ = (m 1 , m 2 , . . . , m l−1 , 0) extends to two different O(2l)-modules, denoted by λ and λ ⊗ det, where det is the 1-dimensional non-trivial O(2l)-module. We denote (D) = {(m 1 , m 2 , . . . , m l ) | m 1 ≥ m 2 ≥ . . . ≥ m l > 0, m i ∈ Z} ∪ {(m 1 , m 2 , . . . , m l−1 , 0) ⊗ det, (m 1 , m 2 , . . . , m l−1 , 0) | m 1 ≥ m 2 ≥ . . . ≥ m l−1 ≥ 0, m i ∈ Z} . 2.2.2. O(2l + 1). Let O(2l + 1) = {g ∈ G L(2l + 1) | t g J g = J }, where ⎤ ⎡ 0 Il 0 J = ⎣ Il 0 0 ⎦ . 0 0 1 The Lie algebra so(2l + 1) is the Lie subalgebra of gl(2l + 1) consisting of (2l + 1) × (2l + 1) matrices of the form ⎡ ⎤ α β δ ⎣ γ −t α h ⎦ , (4) −t h −t δ 0 where α, β, γ are l × l matrices and β, γ skew-symmetric. The Borel subalgebra b(so(2l + 1)) consists of matrices (4) by putting γ , h, δ to be 0 and α to be upper triangular. The Cartan subalgebra h(so(2l + 1)) consists of diagonal matrices of the form diag(t1 , . . . , tl ; −t1 . . . − tl ; 0), ti ∈ C. An irreducible module of S O(2l + 1) is parameterized by its highest weight (m 1 , . . . , m l ) ∈ Pl , where Pl denotes the set of partitions with at most l non-zero parts.
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It is well known that O(2l + 1) is isomorphic to the direct product S O(2l + 1) × Z2 by sending the minus identity matrix to −1 ∈ Z2 = {±1}. Denote by det the non-trivial one-dimensional representation of O(2l + 1). A representation λ of S O(2l + 1) extends to two different representations λ and λ ⊗ det of O(2l + 1). Then we can parameterize irreducible representations of O(2l + 1) by (m 1 , . . . , m l ) and (m 1 , . . . , m l ) ⊗ det. We shall denote
(B) = Pl ∪ λ ⊗ det | λ ∈ Pl . 2.2.3. Spin(n) and Pin(n). The Pin group Pin(n) is the double covering group of O(n), namely we have 1 −→ Z2 −→ Pin(n) −→ O(n) −→ 1. We then define the spin group Spin(n) to be the inverse image of S O(n) under the projection from Pin(n) to O(n). Case n = 2l. Let 1l = (1, 1, . . . , 1), 1¯ l = (1, 1, . . . , 1, −1) ∈ Zl . An irreducible representation of Spin(2l) which does not factor to S O(2l) is an irreducible representation of so(2l) parameterized by its highest weight λ = 1l /2 + (m 1 , m 2 , . . . , m l )
(5)
λ = 1¯ l /2 + (m 1 , m 2 , . . . , −m l ),
(6)
or
where m 1 ≥ . . . ≥ m l ≥ 0, m i ∈ Z. There are two possibilities. First, an irreducible representation of Pin(2l) factors to that of O(2l), then we can use the parametrization of irreducible representations of O(2l) to parameterize these representations of Pin(2l). Secondly, an irreducible representation of Pin(2l) is induced from an irreducible representation of Spin(2l) with highest weight of (5) or (6). When restricted to Spin(2l), it will decompose into a sum of the two irreducible representations of highest weights (5) and (6). We will use λ = 1l /2 + (m 1 , m 2 , . . . , m l ), m l ≥ 0 to denote this irreducible representation of Pin(2l). Denote by (Pin) = {1l /2 + (m 1 , m 2 , . . . , m l ) | (m 1 , m 2 , . . . , m l ) ∈ Pl }. Case n = 2l + 1. An irreducible representation of Spin(2l + 1) which does not factor to S O(2l + 1) is an irreducible representation of so(2l + 1) parameterized by its highest weight λ ∈ (Pin). 2.2.4. Sp(2l). The Lie group Sp(2l) is the subgroup of G L(2l) which preserves the following skew-symmetric bilinear form 0 Il . −Il 0 Its Lie algebra sp(2l) consists of 2l × 2l matrices of the following form: a b , c −t a
(7)
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where a, b, c are l × l matrices, b, c are symmetric. Let b(sp(2l)) be the Borel subalgebra consisting of matrices of the form (7) with c = 0 and a upper triangular. The Cartan subalgebra h(sp(2l)) consists of diagonal matrices diag(t1 , . . . , tl , −t1 , . . . , −tl ). An irreducible representation of Sp(2l) can be parameterized by its highest weight λ ∈ (C) := Pl . 2.3. Additional notations. The notations introduced in this preliminary section are close to but do not always coincide with [W1]. For example, our b∞ is b˜∞ there. We shall denote by L(x∞ ; ), for x = b, c, d, the irreducible x∞ -module of highest weight . The level can be read off from . Given a classical Lie group G of type X , we shall denote by Vλ (G) the irreducible G-module parameterized by λ ∈ (X ). Let W (X ) be the Weyl group of type X . We will denote the roots of the Lie algebra of G by standard notations εi ± ε j , εi , 2εi etc., and by ( ) the bilinear form such that (εi , ε j ) = δi j . Let ρ denote half the sum of positive roots. 3. Correlation Functions on d∞-Modules of Level l 3.1. The Fock space Fl . Let Z denote =
1 2
+ Z or Z, and set
0, if Z = 21 + Z 1 2 if Z = Z.
Consider a pair of fermionic fields ψ + (z) =
1
ψn+ z −n− 2 + , ψ − (z) =
n∈Z
1
ψn− z −n− 2 + ,
n∈Z
with the following anti-commutation relations: [ψm+ , ψn− ]+ = δm+n,0 ,
[ψm± , ψn± ]+ = 0.
Denote by F the Fock space of the fermionic fields ψ ± (z) generated by a vacuum vector |0 which satisfies 1 1 + Z+ , if Z = + Z; 2 2 − ψn+1 |0 = ψn+ |0 = 0 for n ∈ Z+ , if Z = Z. ψn− |0 = ψn+ |0 = 0 for n ∈
We have the standard charge decomposition (cf. [MJD]) F= F(k) . k∈Z
Each F(k) becomes an irreducible module over a certain Heisenberg Lie algebra. The shift operator S : F(k) → F(k+1) matches the highest weight vectors and commutes with the creation operators in the Heisenberg algebra.
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Now we take l pairs of fermionic fields, ψ ±, p (z) ( p = 1, . . . , l) and denote the corresponding Fock space by Fl . Introduce the following generating series: E(z, w) ≡
E i j z i−1+2 w − j =
i, j∈Z
l
:ψ +, p (z)ψ −, p (w):,
(8)
p=1
where the normal ordering :: means that the operators annihilating |0 are moved to the right with a sign. It is well known that the operators E i j (i, j ∈ Z) generate a with level l. representation in Fl of the Lie algebra gl Let
− p −q
+ p +q + e− = :ψ ψ :, e = :ψr ψ−r :, p = q, (9) r −r pq pq r ∈Z
r ∈Z
and let e pq =
r ∈Z
+p
−q
:ψr ψ−r : + δ pq .
(10)
The operators e+pq , e pq , e− pq ( p, q = 1, · · · , l) generate Lie algebra so(2l) (cf. [FF, W1]). 3.2. The (O(2l), d∞ )-Howe duality. Now let Z = 21 + Z for the remainder of Sect. 3. The representation of the Lie algebra d∞ on Fl is given by (cf. [DJKM2])
(E i, j − E 1− j,1−i )z i−1 w − j i, j∈Z
=
l
:ψ +, p (z)ψ −, p (w): − :ψ +, p (w)ψ −, p (z):.
(11)
p=1
The action of so(2l) can be integrated to the action of the Lie group S O(2l) on Fl . In particular, the operators e pq ( p, q = 1, · · · , l) form a Lie subalgebra gl(l). We identify the Borel subalgebra b(so(2l)) with the one generated by e+pq ( p = q), e pq ( p ≤ q), p, q = 1, . . . , l. Note that τ ∈ O(2l) defined in (3) commutes with the action of d∞ on Fl . The following lemma summarizes Lemmas 3.2, 3.3 in [W1]. Lemma 3.1. The action of the Lie group O(2l) commutes with the action of d∞ on Fl . to
We define a map : (D) −→ d∞ ∗0 by sending λ = (m 1 , · · · , m l ), where m l > 0, (λ) = (l − i)d0 + (l − i)d1 +
i
dm k ,
k=1
sending (m 1 , · · · , m j , 0, . . . , 0), where j < l, to (λ) = (2l − i − j)d0 + ( j − i)d1 +
i
k=1
dm k ,
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483
and sending (m 1 , . . . , m j , 0, . . . , 0) ⊗ det, where j < l, to (λ) = ( j − i)d0 + (2l − i − j)d1 +
i
dm k ,
k=1
if m 1 ≥ . . . m i > m i+1 = . . . = m j = 1 > m j+1 = . . . = m l = 0. Proposition 3.1. [W1, Theorem 3.2]. We have the following decomposition of (O(2l), d∞ )-modules: Fl ∼ Vλ (O(2l)) ⊗ L(d∞ ; (λ)). (12) = λ∈(D)
3.3. The main results of [BO, CW1]. Let t be an indeterminant. We define the following operators in d∞ (cf. [W1]):
1 1 :D(t): = (t k− 2 − t 2 −k )(E k,k − E 1−k,1−k ), k∈N
D(t) = :D(t): +
2 1 2
1
t − t− 2
C.
When acting on Fl , these operators can be written in terms of the operators ψn± by (11) as :D(t): =
l
−, p
+, p
t k (:ψ−k ψk
−, p
+, p
: + :ψ−k ψk :),
p=1 k∈ 1 +Z 2
D(t) =
l
+, p
−, p
t k (ψ−k ψk
−, p
+, p
+ ψ−k ψk ).
p=1 k∈ 1 +Z 2
Recall that Bloch and Okounkov [BO] introduced the following operators in gl: :A(t): =
1
1
t k− 2 E k,k , A(t) = :A(t): +
k∈Z
1 2
1
t − t− 2
C.
We easily verify that D(t) = A(t) − A(t −1 ),
:D(t): = :A(t): − :A(t −1 ):.
(13)
Given λ = (m 1 , . . . , m l ) ∈ (A), we denote by (λ) the gl-highest weight am 1 + (λ)) with highest weight · · · + am l . The energy operator L 0 on the gl-module L(gl; vector v(λ) is characterized by 1 λ2 · v(λ) , 2 [L 0 , E i j ] = (i − j)E i j ,
L 0 · v(λ) =
(14)
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D. G. Taylor, W. Wang
where λ2 := λ21 + λ22 + · · · + λl2 . On Fl , we can realize L 0 as L0 =
l
+, p
−, p
k:ψ−k ψk
:.
p=1 k∈Z+ 1 2
The n-point gl-correlation function of level l associated to λ is defined in [BO] for l = 1 and in [CW1] for general l as L0 Alλ (q; t) ≡ Alλ (q; t1 , . . . , tn ) := tr L(gl ,(λ)) (q A(t1 )A(t2 ) · · · A(tn )).
Here and below we denote t = (t1 , . . . , tn ). Let (a; q)∞ := r∞=0 (1 − aq r ). Define the theta function 1
1
−1
(t) := (t 2 − t − 2 )(q; q)−2 ∞ (qt; q)∞ (qt ; q)∞ , k d
(k) (t) := t
(t), for k ∈ Z+ . dt
(15) (16)
Denote by Fbo (q; t) or Fbo (q; t1 , . . . , tn ) the following expression det
( j−i+1) (tσ (1) ···tσ (n− j) ) ( j−i+1)!
n
1 i, j=1 . · (q; q)∞
(tσ (1) ) (tσ (1) tσ (2) ) · · · (tσ (1) tσ (2) · · · tσ (n) )
(17)
σ ∈Sn
It is understood here that 1/(−k)! = 0 for k > 0, and for n = 1, we have Fbo (q; t) = −1 (q; q)−1 ∞ (t) . The following summarizes the main results of Bloch-Okounkov [BO] for l = 1 and Cheng-Wang [CW1] for general l ≥ 1. Theorem 3.1. Associated to λ = (λ1 , . . . , λl ), where λ1 ≥ . . . ≥ λl and λi ∈ Z, the n-point gl-function of level l is given by Alλ (q; t) = q
λ2 2
(t1 t2 · · · tn )|λ|
(1 − q λi −λ j + j−i ) · Fbo (q; t)l ,
1≤i< j≤l
where |λ| := λ1 + · · · + λl . In the simplest case, i.e. l = n = 1, we have A1λ (q; t)
=q
λ2 2
λ2
q 2 tλ t · Fbo (q; t) = . (q; q)∞ (t) λ
Bloch-Okounkov Correlation Functions of Classical Type
485
3.4. The 1-point d∞ -functions of level l. Definition 3.1. The n-point d∞ -correlation function of level l associated to λ = (λ1 , . . . , λl ) ∈ Pl is Dlλ (q, t ) ≡ Dlλ (q, t1 , . . . , tn ) tr L(d∞ ;(λ)) q L 0 D(t1 ) · · · D(tn ), if λl = 0, := tr L(d∞ ;(λ))⊕L(d∞ ;(λ⊗det)) q L 0 D(t1 ) · · · D(tn ), if λl = 0. (The operator L 0 is defined in the same way as for a gl-module.) Remark 3.1. A justification of this definition when λl = 0 is as follows. The two weights (λ) and (λ ⊗ det) are interchanged by a d∞ -Dynkin diagram automorphism. Thus, the direct sum L(d∞ ; (λ)) ⊕ L(d∞ ; (λ ⊗ det)) can be regarded as an irreducible module of the orthogonal group associated to d∞ . In this subsection, we will restrict to n = 1 for notational simplicity. Theorem 3.2. The 1-point d∞ -function of level l is given by Dlλ (q, t) =
1 (q; q)l∞ (t)l
·
σ ∈W (Dl )
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
l
(t ka + t −ka ),
a=1
where ka = (λ + ρ − σ (ρ), εa ). We first prepare a few lemmas for the proof of this theorem. By a character of a module V of S O(2l) or O(2l), we mean tr V (z 1e11 . . . zlell ). Let |ai j | denote the determinant of a matrix (ai j ). Lemma 3.2. Denote by choλ (z 1 , . . . , zl ) the character of the irreducible O(2l)-module Vλ (O(2l)). If λl = 0, then λi +l−i −(λ +l−i) + zj i z j . choλ (z 1 , . . . , zl ) = l−i −(l−i) z j + z j If λl = 0, then
−(λ +l−i) 2 z λj i +l−i + z j i . choλ (z 1 , . . . , zl ) = l−i −(l−i) z j + z j
Proof. The character of an irreducible S O(2l)-module Vλ (S O(2l)), denoted by chso λ , is well known to be as follows (cf. [FH, pp. 410]): λi +l−i −(λ +l−i) λi +l−i −(λ +l−i) + zj i − zj i z j + z j chso . (18) λ (z 1 , . . . , z l ) = l−i −(l−i) z j + z j Recall that an irreducible O(2l)-module can be a sum of two irreducible S O(2l)-modules or remain to be irreducible as a S O(2l)-module, depending on whether λl is nonzero or not. If λl = 0, the second determinant in the numerator of (18) vanishes and
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hence the character formula for choλ of Vλ (O(2l)) follows from chso λ . If λl = 0, then Vλ (O(2l)) = Vλ (S O(2l)) ⊕ Vλ (S O(2l)). Note that the second determinant terms in the so numerators of chso λ and chλ (cf. (18)) are opposite to each other. Now the formula for o chλ follows. Lemma 3.3. We have 1 k −k k 2 (t + t )q 2 tr F(0) q L 0 D(t), k ∈ Z, 2
(t k + t −k ) k2 tr Fz e11 q L 0 D(t) = tr F(0) q L 0 D(t) · zk q 2 . 2 tr F(k) q L 0 D(t) =
k∈Z
Proof. These two identities are clearly equivalent, and it suffices to prove the first one. We refer to [MJD] or [Ok, Appendix A] for properties of the shift operator S on F. Then, tr F(k) q L 0 D(t) = tr F(0) S−k q L 0 (A(t) − A(t −1 ))Sk k2
= tr F(0) q L 0 + 2 (t k A(t) − t −k A(t −1 )) =q
k2 2
tr F(0) q L 0 (t k A(t) − t −k A(t −1 ))
and tr F(−k) q L 0 D(t) = tr F(0) Sk q L 0 (A(t) − A(t −1 ))S−k k2
= tr F(0) q L 0 + 2 (t −k A(t) − t k A(t −1 )) =q
k2 2
tr F(0) q L 0 (t −k A(t) − t k A(t −1 )).
By (13), t k A(t) − t −k A(t −1 ) + t −k A(t) − t k A(t −1 ) = (t k + t −k )D(t). It follows that tr F(−k) q L 0 D(t) + tr F(k) q L 0 D(t) = (t k + t −k )q
k2 2
tr F(0) q L 0 D(t).
Since F(−k) and F(k) are isomorphic as d∞ -modules and D(t) lies in d∞ , we have tr F(−k) q L 0 D(t) = tr F(k) q L 0 D(t), and the result follows. Lemma 3.4. We have tr F(0) q L 0 D(t) = 2Fbo (q, t) =
2 (q;q)∞ (t) .
Proof. As a special case of Theorem 3.1, we note that A1(0) (q; t) = tr F(0) q L 0 A(t) = Fbo (q; t) =
1 . (q, q)∞ (t)
In particular, tr F(0) q L 0 A(t −1 ) = − tr F(0) q L 0 A(t). Thus, it follows by (13) that tr F(0) q L 0 D(t) = 2Fbo (q; t). Set
cλ =
1, if λl = 0 2, if λl = 0.
We have the following lemma similar to [CW1, Lemma 6].
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487
Lemma 3.5. We have the following identity: λi +l−i i +l−i) l + z −(λ z j
j · Dlλ (q, t). tr Fz ieii q L 0 D(t) = cλ · l−i −(l−i) z j + z j λ i=1 Proof. It follows by computing the trace of z 1e11 · · · zlell q L 0 D(t) of both sides of the (O(2l), d∞ )-duality (12). Note that the factor 2 in choλ in Lemma 3.2 for λl = 0 gives rise to the factor cλ . µ
µ
Let zµ denote z 1 1 · · · zl l for µ = (µ1 , . . . , µl ).
Lemma 3.6. Among all the monomials zµ in the expansion of the determinant z λj i +l−i + −(λ +l−i) zj i , there is exactly one “dominant” monomial with µ1 ≥ . . . ≥ µl ≥ 0, that is, zλ+ρ = li=1 z iλi +l−i . Its coefficient is equal to 2/cλ , or more concretely, it is 1 if λl = 0 and 2 if λl = 0. Proof. The first part is clear by inspection. Note that the coefficient 2 when λl = 0 comes from the last row of the determinant. Proof of Theorem 3.2. It follows by Lemmas 3.3, 3.4, and 3.5 that ⎛ ⎞ l
ka2 1 −(l−i) ⎝ Fbo (q, t)l · (t ka + t −ka )z aka q 2 ⎠ · z l−i + z j 2 j a=1 ka ∈Z
i +l−i) = cλ /2 · z λj i +l−i + z −(λ · Dlλ (q, t). j
(19)
λ
Recall ([FH, (24.38)]) that the Weyl denominator of type Dl is
1 l−i −(l−i) (−1)(σ ) zσ (ρ) . z j + z j = 2
(20)
σ ∈W (Dl )
The theorem follows when we apply Lemma 3.6 to compare the coefficients of the monomial li=1 z iλi +l−i on both sides of (19) with (20) plugged in. 3.5. The n-point d∞ -functions of level l. We now compute the n-point d∞ -function of level l. The 1-point calculation in the previous subsection carries over for general n after suitable modification. Let F(z, q; t1 , . . . , tn ) = tr Fz e11 q L 0 D(t1 ) · · · D(tn ).
(21)
The following lemma is the n-point generalization of Lemma 3.3. Lemma 3.7. We have F(z, q; t1 , . . . , tn ) =
k∈Z
zk q
k2 2
k [ ] · t · Fbo (q; t ),
∈{±1}n
where we denote = ( 1 , 2 , . . . , n ), [ ] = 1 2 · · · n , t = t1 1 · · · tn n , and Fbo (q; t ) = Fbo (q; t1 1 , . . . , tn n ).
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Proof. We calculate using (13) that D(t1 ) · · · D(tn ) =
n A(t j ) − A(t −1 j ) j=1
=
1 2 · · · n A(t1 1 )A(t2 2 ) · · · A(tn n ).
∈{±1}n
It is known (cf. [Ok, CW1]) by the same type of argument as in Lemma 3.3 that tr Fz e11 q L 0 A(t1 ) · · · A(tn ) =
zk q
k2 2
(t1 · · · tn )k Fbo (q, t1 , . . . , tn ).
k∈Z
Now the lemma follows.
Lemma 3.8. We have the following identity: λi +l−i −(λ +l−i) l + zj i z j
· Dlλ (q, t1 , . . . , tn ). F(z i , q; t1 , . . . , tn ) = cλ · l−i −(l−i) z j + z j λ i=1 Proof. The lemma is a straightforward n-point version of Lemma 3.5, which is proved in the same way as before. Theorem 3.3. The n-point d∞ -correlation function of level l associated to λ ∈ Pl is given by Dlλ (q; t1 , . . . , tn ) =
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
σ ∈W (Dl )
l a=1
⎛ ⎝ a
⎞
[ a ](t a )ka Fbo (q; t a )⎠ ,
∈{±1}n
where ka = (λ + ρ − σ (ρ), εa ). Proof. The proof is the same as the proof of Theorem 3.2 by using now Lemmas 3.7 and 3.8. (0)
3.6. A refined 1-point function of level 1. Denote by F± the (±1)-eigenspace of τ ∈ O(2) acting on F(0) where we recall that 0 1 . τ= 1 0 As an (O(2), d∞ )-module, we have (0)
(0)
F = ⊕m∈N (F(m) ⊕ F(−m) ) ⊕ F+ ⊕ F− . (0) (0) As d∞ -modules, all F(m) , F(−m) , F+ , F− are irreducible, and moreover, F(m) ∼ = (−m) . The main result of this subsection is the following. F
Bloch-Okounkov Correlation Functions of Classical Type
489
Theorem 3.4. The tr F(0) q L 0 D(t), which we refer to as a refined 1-point d∞ -correlation ± function, is equal to 1 (q; q)∞ (t) ±(q; q 2 )∞
∞
r =0
1
1
q r +1 t − 2 q r +1 t 2 − 1 − q 2(r +1) t −1 1 − q 2(r +1) t
+
1 1
1
t 2 − t− 2
which is equivalent to 1 1 (−t − 2 ; q)∞ (−qt 2 ; q)∞ 1 d 2 ± (q; q )∞ t ln . 1 1 (q; q)∞ (t) dt (t − 2 ; q)∞ (qt 2 ; q)∞ We denote G(t) = tr F(0) q L 0 D(t) − tr F(0) q L 0 D(t), + − :G(t): = tr F(0) q L 0 :D(t): − tr F(0) q L 0 :D(t):. + − To compute tr F(0) q L 0 D(t) it suffices to compute their difference G(t), since D1(0) (t) = ± tr F(0) q L 0 D(t) + tr F(0) q L 0 D(t) has been calculated. + − Recall that the rank rk(λ) of a partition λ = (λ1 , λ2 , . . .) is the cardinality of the set {i | λi ≥ i}. Lemma 3.9. We have :G(t): = 2
(−1)rk(λ) q |λ|
λ=λt
t λi −i+1/2 − t −(λi −i+1/2) . i
Proof. Note that τ sends ψn+ to ψn− for each n and vice-versa. Thus, F+(0) has a basis given by {ψm−1 · · · ψm−r ψn+1 · · · ψn+r |0 + ψn−1 · · · ψn−r ψm+ 1 · · · ψm+ r |0 | m 1 < · · · < m r < 0, n 1 < · · · < nr < 0, m i = n i for some i} ∪{ψm−1 · · · ψm−r ψm+ 1 · · · ψm+ r |0 | m 1 < · · · < m r < 0, r even} (0)
and F− has a basis given by {ψm−1 · · · ψm−r ψn+1 · · · ψn+r |0 − ψn−1 · · · ψn−r ψm+ 1 · · · ψm+ r |0 | m 1 < · · · < m r < 0, n 1 < · · · < nr < 0, m i = n i for some i} ∪{ψm−1 · · · ψm−r ψm+ 1 · · · ψm+ r |0 | m 1 < · · · < m r < 0, r odd}. (0) F+(0) (resp. F− ) has highest weight vector |0 (resp. ψ − 1 ψ + 1 |0). −2
−2
The action of :D(t): on F(0) can be described explicitly:
:D(t): ψm−1 · · · ψm−r ψn+1 · · · ψn+r |0
t −n i − t n i + t −m i − t m i ψm−1 · · · ψm−r ψn+1 · · · ψn+r |0. = i
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It is well known that F(0) can be identified with an irreducible module of Heisen− berg algebra and its basis is parameterized by partitions. Given an element ψ− p1 · · · − + + (0) ψ− pr ψ−q1 · · · ψ−qr |0 ∈ F , where p1 > . . . > pr > 0, q1 > . . . > qr > 0, the indices ( p1 , . . . , pr | q1 , . . . , qr ) are exactly the Frobenius coordinates of a partition λ (which by our convention here uses half-integers). It is well known that (cf. e.g. [BO, Lemma 5.1]) l r
1 1 t λi −1+ 2 − t −i+ 2 = t pk − t −(−qk ) . i=1
k=1
We then compute that tr F(0) q L 0 :D(t): =X + 2 +
tr F(0) q L 0 :D(t): =X + 2 −
q |λ|
λ=λt rk(λ) even
q |λ|
λ=λt rk(λ) odd
t λi −i+1/2 − t −(λi −i+1/2) , i
t λi −i+1/2 − t −(λi −i+1/2) , i
for the same X whose explicit form is irrelevant here. Now :G(t): is given by the difference of the above two formulas. Proposition 3.2. We have :G(t): = 2(q; q 2 )∞ · = 2(q; q )∞ · 2
1 1 ∞ q 2n−1 t −n+ 2 − t n− 2
n=1 ∞
r =0
1 − q 2n−1 1
1
q r +1 t − 2 q r +1 t 2 − 2(r +1) −1 1−q t 1 − q 2(r +1) t
,
and G(t) = :G(t): + (q; q 2 )∞
2 1 2
1
t − t− 2
.
Proof. There is a canonical bijection between the set of symmetric partitions (i.e. λ such that λt = λ) and the set OSP of odd strict partitions. The bijection is achieved by setting the parts of a new partition µ to be the hook lengths of the diagonal nodes from the original symmetric partition λ (i.e. µi = 2λi − 2i + 1). Here is an example:
→
Under such a bijection which sends a symmetric partition λ to µ ∈ OSP, we have r k(λ) = (µ), |λ| = |µ|, and λi − i + 21 = µi /2. By Lemma 3.9, we have a reformulation
t µk /2 − t −µk /2 . (22) :G(t): = 2 (−1)(µ) q |µ| µ∈OSP
k≥1
Bloch-Okounkov Correlation Functions of Classical Type
491
By Theorem 8 of [W2] (replacing q therein by q 2 ), we have the following identity: ⎛ ⎞ ∞ 2n−1 t n− 21 z
q ⎝z (µ) q |µ| t µk /2 ⎠ = (−qz; q 2 )∞ 1 + . 1 + q 2n−1 z µ∈OSP
n=1
k≥1
Using this identity twice (with the specialization z = −1), we obtain from (22) the first formula for :G(t):. The second formula for :G(t): follows from the identity: 1 1 ∞ ∞
zq 2n−1 t n− 2 (−1)r z r +1 q r +1 t 2 = . 1 + q 2n−1 z 1 − q 2(r +1) t
r =0
n=1
(This identity follows quickly by expanding the left side as a power series, interchanging summations, and then summing up.) Along the same line as the proof of Lemma 3.9 and (22), we can show that
tr F(0) q L 0 − tr F(0) q L 0 = (−1)(µ) q |µ| = (q; q 2 )∞ , −
+
µ∈OSP
where the last equation is the specialization at z = −1 of the identity
z (µ) q |µ| = (−qz; q 2 )∞ . µ∈OSP
Now the formula for G(t) follows from this consideration and (13).
Remark 3.2. The function G(t) is essentially a “super version” of the 1-point d∞ -correlation function of level 21 (cf. [W2, Theorem 9]). Proof of Theorem 3.4. We write tr F(0) q L 0 D(t) = ±
1 1 D(0) (t) ± G(t) . 2
It is known by Theorem 3.2 that D1(0) (t) =
2 . (q; q)∞ (t)
Now the first formula of the theorem follows from Proposition 3.2. To see the equivalence of the second formula, we compute that 1 1 1 ∞ (−t − 2 ; q)∞ qr t − 2 d qr t − 2 1 t ln + = . 1 1 1 dt 2 (t − 2 ; q)∞ 1 + qr t − 2 1 − qr t − 2 r =0
Combining common denominators, one obtains the term
1 1
1
t 2 −t − 2
and the first group of
summands of the first formula. The remaining terms in the ln expression of the second formula yield the second group of summands of the first formula.
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3.7. The q-dimension of a d∞ -module of level l. Given an d∞ -module M, we refer to dimq M := tr M q L 0 as the q-dimension of M. Proposition 3.3. The q-dimension dimq L(d∞ ; (λ)) for λl = 0, or dimq [L(d∞ ; (λ)) ⊕ L(d∞ ; (λ ⊗ det))] for λl = 0, is given by the following (equivalent) formulas:
(ρ)2 1 (σ ) λ+ρ−σ 2 · (−1) q (q; q)l∞ σ ∈W (Dl ) λ2 1 λi −λ j + j−i λi +λ j +2l−i− j 2 1 − q 1 − q . = · q (q; q)l∞ 1≤i< j≤l
Proof. We substitute Lemma 3.3 with the simple identity
k2 tr F(z e11 q L 0 ) = dimq F(0) zk q 2 , k∈Z
and substitute Lemma 3.4 with the identity dimq F(0) = (q; q)−1 ∞ . In this way, the same strategy of establishing the 1-point function in Theorem 3.2 applies and it readily leads to the first q-dimension formula in the proposition. The second formula follows from Lemma 3.10 below and the explicit root system of Dl . Lemma 3.10. Let g be a semisimple Lie algebra with Weyl group W . Set x2 = (x, x), and let λ be a weight. Then,
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
=q
σ ∈W
λ2 2
(1 − q (λ+ρ,α) ).
α∈+
Proof. The Weyl denominator formula reads
(−1)(σ ) q ρ−σ (ρ) = (1 − q α ). σ ∈W
α∈+
The lemma now follows by applying the bilinear pairing with λ + ρ to both sides of this formula of Weyl and noting that 1 1 λ + ρ − σ (ρ)2 = λ2 + (λ + ρ, ρ − σ (ρ)). 2 2 4. Correlation Functions on d∞-Modules of Level l + 1
4.1. The Fock space Fl+ 2 . Recall Z =
1 2
+ Z or Z. Consider the neutral fermion
1 ϕ(z) = ϕn z −n− 2 + 1 2
n∈Z
which satisfies the commutation relation [ϕm , ϕn ]+ = δm,−n .
Bloch-Okounkov Correlation Functions of Classical Type
493
1
We denote by Fl+ 2 the Fock space of one neutral fermion ϕ(z) and l pairs of complex fermions ψ ±, p (z), 1 ≤ p ≤ l, generated by a vacuum vector |0 which satisfies 1 1 + Z+ , if Z = + Z; 2 2 +, p −, p ϕn |0 = ψn |0 = ψn+1 |0 = 0 for n ∈ Z+ , if Z = Z. −, p
+, p
ϕn |0 = ψn |0 = ψn
Let e± p =
|0 = 0 for n ∈
±p
:ψr ϕ−r :, 1 ≤ p ≤ l.
r ∈Z + − It is known (cf. [FF, W1]) that the above operators e+p , e− p together with e pq , e pq , e pq ( p, q = 1, · · · , l) defined in (9, 10) generate the Lie algebra so(2l + 1).
Lemma 4.1. Given a pair of complex fermions ψ ± (z), we let √ √ ϕn := i(ψn+ − ψn− )/ 2. ϕn := (ψn+ + ψn− )/ 2, Then, ϕn and ϕn satisfy the anti-commutation relations: [ϕn , ϕm ]+ = δn,−m , [ϕn , ϕm ]+ = δn,−m , [ϕn , ϕm ]+ = 0, for m, n ∈ Z. Hence, there is an isomorphism of Fock spaces 1 1 F if Z = 21 + Z, ∼ 2 2 F ⊗F = F ⊕ F if Z = Z. Proof. The commutation relations are verified by a direct computation. The multiplicity 2 in the Fock space isomorphism for Z = Z is due to the fact that the central elements ϕ0 , ϕ0 thus defined satisfy ϕ0 |0 = iϕ0 |0. 4.2. The (O(2l + 1), d∞ )-Howe duality. Now let Z = The Lie algebra d∞ acts on Fl by (cf. [DJKM2])
(E i, j − E 1− j,1−i )z i−1 w − j
1 2
+ Z in the remainder of Sect. 4.
i, j∈Z
=
l
:ψ +, p (z)ψ −, p (w): − :ψ +, p (w)ψ −, p (z): + :ϕ(z)ϕ(w):.
(23)
p=1 1
The action of Lie algebra so(2l + 1) on Fl+ 2 can be integrated to an action of the Lie group S O(2l + 1). We identify the Borel subalgebra b(so(2l + 1)) with the one generated by e+pq ( p = q), e pq ( p ≤ q), e+p , where p, q = 1, . . . , l. The element 1
ω := diag(1, . . . , 1, −1) ∈ O(2l + 1) acts on Fl+ 2 by sending ϕn to −ϕn for each n. Lemma 4.2 [W1, Lemmas 4.2]. The action of the Lie group O(2l + 1) commutes with 1 the action of d∞ on Fl+ 2 .
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Define a map from (B) to d∞ ∗0 by sending λ = (m 1 , m 2 , . . . , m l ) to (λ) = (2l + 1 − i − j)d0 + ( j − i)d1 +
i
dm k ,
k=1
and sending λ = (m 1 , m 2 , . . . , m l )
det to
(λ) = ( j − i)d0 + (2l + 1 − i − j)d1 +
i
dm k ,
k=1
where m 1 ≥ . . . ≥ m i > m i+1 = . . . = m j = 1 > m j+1 = . . . = m l = 0. Proposition 4.1 [W1, Theorem 4.1]. We have the (O(2l + 1), d∞ )-module decomposition: 1 Fl+ 2 ∼ Vλ (O(2l + 1)) ⊗ L(d∞ , (λ)). = λ∈(B) 1
By (23), we can write D(t) acting on Fl+ 2 as D(t) =
t
k
k∈ 21 +Z
l
+,i −,i (ψ−k ψk
−,i +,i + ψ−k ψk ) + ϕ−k ϕk
.
i=1
4.3. The n-point d∞ -function of level 21 . Definition 4.1. The n-point d∞ -correlation function of level l + l+ 21
l+ 21
1 2
associated to λ ∈ Pl ,
denoted by Dλ (q, t) or Dλ (q, t1 , . . . , tn ), is tr L(d∞ ;(λ))⊕L(d∞ ;(λ⊗det)) q L 0 D(t1 ) · · · D(tn ). (As a justification of this definition, Remark 3.1 also applies here.) 1
2 When l = 0, D(0) (q, t1 , . . . , tn ) = tr
1
F2
q L 0 D(t1 ) · · · D(tn ), by Proposition 4.1. The aim
of this subsection is to determine this (unique) n-point d∞ -function of level 21 , which will be used in the general level l + 21 case in the following subsection. D(t) = Lemma 4.3. Under the isomorphism F ∼ = F 2 ⊗ F 2 in Lemma 4.1, we have ϕ . D1 (t) + D2 (t), where D1 (t) = k∈Z+ 1 t k ϕ−k ϕk and D2 (t) = k∈Z+ 1 t k ϕ−k k 1
1
2
2
Proof. A simple calculation reveals that − + + ψ−k ψk− + ψ−k ψk = ϕ−k ϕk + ϕ−k ϕk .
Now the lemma follows from the definition of D(t).
Bloch-Okounkov Correlation Functions of Classical Type
495
Given a subset I = (i 1 , . . . , i s ) ⊆ {1, . . . , n}, we denote by I c the complementary set to I , and t I = (ti1 , . . . , tis ). By convention, we let 1
2 (q, t∅ ) = tr D(0)
1
1 F2
q L 0 = (−q 2 ; q)∞
(24)
and recall from (21) that F(z, q; t1 , . . . , tn ) = tr Fz e11 q L 0 D(t1 ) · · · D(tn ). Proposition 4.2. We have
F(1, q; t1 , . . . , tn ) =
1
I ⊆{1,...,n}
1
2 2 D(0) (q, t I )D(0) (q, t I c ).
(25)
Equivalently, we have ⎛
k2 1 1 ⎝ q2 D(0) (q, t) = (−q 2 ; q)−1 ∞ 2 1 2
k∈Z
∈{±1}n
−
k [ ] · t Fbo (q; t )
∅I {1,...,n}
⎞ 1 2
1 2
D(0) (q, t I )D(0) (q, t I c )⎠ .
Proof. By Lemmas 4.1 and 4.3, we have tr Fq L 0 D(t1 ) · · · D(tn ) = tr 1 1 q L 0 (D1 (t1 ) + D2 (t1 )) · · · (D1 (tn ) + D2 (tn )) F 2 ⊗F 2
tr 1 1 q L 0 Di1 (t1 )Di2 (t2 ) · · · Din (tn ). = F 2 ⊗F 2
i∈{1,2}n
This is equivalent to the first formula in the theorem. 1
2 On the right-hand side of (25), there are exactly two terms equal to D(0) (q, t1 , . . . , tn ), which come from I = ∅ and {1, . . . , n}. Now the second formula follows from (24) and Lemma 3.7 which gives a formula for F(1, q; t1 , . . . , tn ).
Proposition 4.2 allows for the determination, which is recursive on n, of all n-point 1
1
2 2 correlation functions D(0) (q, t1 , . . . , tn ). Note that the 1-point function D(0) (q, t) has been computed in [W2] (denoted by S(t) therein) using partition identities. 1
2 Proposition 4.3 [W2, Theorem 9]. The 1-point function D(0) (q, t) is given by
1 2
(−q ; q)∞
1 1
1
t 2 − t− 2
1 1 ∞
(−1)r (q r +1 t) 2 (−1)r (q r +1 t −1 ) 2 + . − 1 − q r +1 t 1 − q r +1 t −1 r =0
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D. G. Taylor, W. Wang
An alternative solution to the 1-point function follows from Proposition 4.2 for n = 1: 1
1
2 (q, t) = (−q 2 ; q)−1 D(0) ∞
q
k2 2
t k Fbo (q, t)
k∈Z
1 2
1
(−q t; q)∞ (−q 2 t −1 ; q)∞
=
1
(−q 2 ; q)∞ (t) 1
1
=
1
1
(t 2 − t − 2 )
·
1
(−q 2 t; q)∞ (−q 2 t −1 ; q)∞ (q; q)2∞ 1
(qt; q)∞ (qt −1 ; q)∞ (−q 2 ; q)∞
,
(26)
where we have used Fbo (q, t −1 ) = −Fbo (q, t) and the Jacobi triple product identity. Comparing this formula with Proposition 4.3 gives us the following. Corollary 4.1. The following q-identity holds: 1
1
(−q 2 t; q)∞ (−q 2 t −1 ; q)∞ (q; q)2∞ 1
(qt; q)∞ (qt −1 ; q)∞ (−q 2 ; q)2∞ 1 1 ∞
1 1 (−1)r (q r +1 t) 2 (−1)r (q r +1 t −1 ) 2 −2 2 − = 1 + (t − t ) · 1 − q r +1 t 1 − q r +1 t −1
.
r =0
4.4. The n-point d∞ -functions of level l + 21 . For λ ∈ Pl , the character of the irreducible O(2l + 1)-module associated to λ and λ ⊗ det is the same, and is given as follows (cf. [FH, p. 408]): λi +l−i+ 12 −(λi +l−i+ 12 ) z − zj j b . (27) chλ (z 1 , . . . , zl ) = 1 l−i+ 12 −(l−i+ 2 ) z − zj j The following lemma is straightforward. µ
µ
1 l Lemma 4.4. Among all the monomials z 1 · · · zl in the expansion of the determinant 1 1 λi +l−i+ 2 −(λi +l−i+ 2 ) z , there is exactly one dominant monomial with µ1 ≥ . . . ≥ − zj j
µl ≥ 0, that is, zλ+ρ =
l
λi +l−i+ 12 . i=1 z i
Its coefficient is equal to 1.
Recall the definition (21) of F(z, q; t1 , . . . , tn ). Lemma 4.5. We have the following q-series identity: tr
1 F2
=
q L 0 D(t1 ) · · · D(tn ) ·
λ∈Pl
l
F(z i , q; t1 , . . . , tn )
i=1 l+ 1
chbλ (z 1 , . . . , zl )Dλ 2 (q; t1 , . . . , tn ).
Bloch-Okounkov Correlation Functions of Classical Type
497
Proof. This follows from the application of tr
e11 · · · zlell q L 0 D(t) to both sides of 1z Fl+ 2 1 1 1 Fl+ 2 ∼ Fl ⊗ F 2 . On the left-hand side,
the Howe duality in Proposition 4.1. Note that = 1 z ieii only acts on the i th tensor factor of Fl and not on F 2 . For the right-hand side, z 1e11 · · · zlell only acts on the first tensor factor and q L 0 D(t) only acts on the second tensor factor. 1
2 Recall that D(0) (q; t) has been computed recursively in the previous subsection.
l+ 1
Theorem 4.1. The n-point d∞ -correlation function of level l + 21 , Dλ 2 (q, t1 , . . . , tn ), is equal to 1
2 D(0) (q; t)
×
(−1)(σ ) q
l
λ+ρ−σ (ρ)2 2
σ ∈W (Bl )
⎛
⎝
⎞ [ a ](t a )ka Fbo (q; t a )⎠ ,
a ∈{±1}n
a=1
where ka = (λ + ρ − σ (ρ), εa ). Proof. The Weyl denominator of type Bl reads that l−i+ 21 −(l−i+ 12 ) = z + z j j
(−1)(σ ) zσ (ρ) .
(28)
σ ∈W (Bl )
It follows by (27), (28), Lemmas 3.7 and 4.5 that
1
2 (−1)(σ ) zσ (ρ) · D(0) (q; t)
σ ∈W (Bl )
×
l a=1
⎛ ⎝
ka ∈Z
z aka q
ka2 2
⎞ k a [ a ] · t a Fbo (q; t a )⎠
a ∈{±1}n
λi +l−i+ 1 1 −(λi +l−i+ 12 ) 2 · Dl+ 2 (q, t). = + zj λ z j λ∈Pl
Now the theorem follows by Lemma 4.4 and by comparing the coefficients of zλ+ρ on both sides of the above identity. 4.5. The q-dimension of a d∞ -module of level l + 21 . In the same manner as in Sect. 3.7, we can derive the following q-dimension formula from the (O(2l + 1), d∞ )-Howe duality in Proposition 4.1. The second formula below is obtained from the first one by using Lemma 3.10 and the explicit root system of type Bl . Recall that L(d∞ ; (λ)) ⊕ L(d∞ ; (λ ⊗ det)) can be regarded as an irreducible module of the orthogonal group corresponding to d∞ (cf. Remark 3.1).
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Proposition 4.4. We have dimq [L(d∞ ; (λ)) ⊕ L(d∞ ; (λ ⊗ det))]
1
=
(−q − 2 ; q)∞ · (q; q)l∞
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
σ ∈W (Bl )
λ2 (−q ; q)∞ 1 − q λi +l−i+1/2 ·q 2 l (q; q)∞ 1≤i≤l × 1 − q λi −λ j + j−i 1 − q λi +λ j +2l−i− j+1 . − 12
=
1≤i< j≤l
5. Correlation Functions on c∞-Modules of Level l 5.1. The (Sp(2l), c∞ )-Howe duality. We again take Z = 21 + Z for the Fock space Fl of fermions ψ ±, p (z), 1 ≤ p ≤ l, in this section. The representation of c∞ on Fl is given by ([DJKM2])
(E i, j − (−1)i+ j E 1− j,1−i )z i−1 w − j i, j∈Z
=
l
:ψ +, p (z)ψ −, p (w): − :ψ +, p (w)ψ −, p (z):.
(29)
p=1
Let e˜− pq =
1
−p
−q
(−1)r − 2 :ψr ψ−r :, e˜+pq =
r ∈ 12 +Z
1
(−1)r − 2 :ψr ψ−r :, +p
+q
r ∈ 12 +Z
and let e˜ pq =
+p
−q
:ψr ψ−r :.
r ∈ 12 +Z
The operators e˜+pq , e˜ pq , e˜− pq ( p, q = 1, . . . , l) generate the Lie algebra sp(2l) and can be integrated to the action of the Lie group Sp(2l) on Fl (cf. [FF, W1]). In particular, the operators e˜ pq ( p, q = 1, . . . , l) form a Lie subalgebra gl(l) in the horizontal of sp(2l). Identify the Borel subalgebra b(sp(2l)) with the one generated by e˜ pq ( p ≤ q), e˜+pq , p, q = 1, . . . , l. It is known by [W1, Lemmas 3.6, Remark 3.7] that the action of the Lie group Sp(2l) commutes with the action of c∞ on Fl . Define the map : Pl −→ c∞ ∗0 by sending λ = (m 1 , . . . , m l ) to (λ) = j (l − j)c0 + k=1 cm k , where j denotes the last non-zero index among m i ’s. Proposition 5.1 [W1, Theorem 3.4]. We have the following decomposition of (Sp(2l), c∞ )-modules: Fl = Vλ (Sp(2l)) ⊗ L(c∞ ; (λ)). (30) λ∈Pl
Bloch-Okounkov Correlation Functions of Classical Type
499
5.2. The n-point c∞ -functions of level l. Introduce the following operators in c∞ :
1 1 (t k− 2 − t 2 −k )(E k,k − E 1−k,1−k ), :C(t): = k∈N
2
C(t) = :C(t): +
1 2
1
t − t− 2
C.
When acting on Fl , these operators can be written in terms of the operators ψn± by (29) as :C(t): =
l
−, p
+, p
t k (:ψ−k ψk
−, p
+, p
: + :ψ−k ψk :),
p=1 k∈ 1 +Z 2
C(t) =
l
−, p
+, p
t k (ψ−k ψk
−, p
+, p
+ ψ−k ψk ).
p=1 k∈ 1 +Z 2
Definition 5.1. The n-point c∞ -correlation function of level l associated to λ ∈ Pl is Clλ (q, t1 , . . . , tn ) = tr L(c∞ ;(λ)) q L 0 C(t1 ) · · · C(tn ). Theorem 5.1. The n-point c∞ -correlation function of level l is given by Clλ (q; t1 , . . . , tn ) =
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
σ ∈W (Cl )
l a=1
⎛ ⎝
⎞ [ a ](t a )ka Fbo (q; t a )⎠ ,
a ∈{±1}n
where ka = (λ + ρ − σ (ρ), εa ). Proof. Note that C(t) = A(t) − A(t −1 ). The proof follows the same strategy which works for Theorems 3.2 and 3.3 for d∞ -correlation functions of level l. We now use instead the combinatorial consequence of the (Sp(2l), c∞ )-Howe duality (30) and the character of irreducible Sp(2l)-modules (cf. [FH, 24.18]) λi +l−i+1 −(λi +l−i+1) − z z j j sp . chλ (z 1 , . . . , zl ) = l−i+1 −(l−i+1) − zj z j Note that the Weyl group W (Cl ) replaces W (Dl ) in the proof and result.
In the case n = 1, the notation can be much simplified. The 1-point c∞ -function of level l is given by Clλ (q, t) = Fbo (q, t)l · where ka = (λ + ρ − σ (ρ), εa ).
σ ∈W (Cl )
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
l a=1
(t ka + t −ka ),
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Let us specialize further to l = 1. The irreducible character of Sp(2) = S L(2) is simply chm (z) = (z m+1 − z −(m+1) )/(z − z −1 ). sp
Then the 1-point c∞ -correlation function of level 1 is given by 2 2 q m /2 t m + t −m − q (m+2) /2 t m+2 + t −(m+2) 1 C(m) (q, t) = . (q; q)∞ (t) In contrast to the d∞ case at level 1 where the charge decomposition of F and the theory of partitions can be used effectively, the description of irreducible c∞ -submodules in F is not explicit and the Howe duality in Proposition 5.1 is essentially used. 5.3. The q-dimension of a c∞ -module. In the same manner as in Sect. 3.7, we can derive the following q-dimension formula from the (Sp(2l), c∞ )-Howe duality in Proposition 5.1. The second formula below is obtained from the first one by using Lemma 3.10 and the explicit root system of type Cl . Proposition 5.2. For λ ∈ Pl , we have dimq L(c∞ ; (λ)) =
1 · (q; q)l∞
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
σ ∈W (Cl )
λ2 1 1 − q 2(λi +l−i+1) ·q 2 l (q; q)∞ 1≤i≤l × 1 − q λi −λ j + j−i 1 − q λi +λ j +2l−i− j+2 .
=
1≤i< j≤l
6. Correlation Functions on b∞-Modules of Level l 6.1. The (Pin(2l), b∞ )-Howe duality. Throughout Sect. 6 we take Z = Z. The action of b∞ on the Fock space Fl is given by ([DJKM2])
i, j∈Z
(E i, j − E − j,−i )z i w − j =
l
+, p :ψ (z)ψ −, p (w): − :ψ +, p (w)ψ −, p (z): . (31) p=1
The Lie algebra so(2l) defined in Sect. 3.2 can be integrated to Spin(2l) and then naturally extended to Pin(2l). Remark 3.5 and Lemma 3.5 of [W1] are summed up by the following lemma. Lemma 6.1. The action of the Lie group Pin(2l) commutes with the action of b∞ on Fl . For λ = 1l /2 + (m 1 , . . . , m l ) in (Pin), define the following map : (Pin) −→ b∞ ∗0 : (λ) = (2l − 2 j)b0 +
j
bm k ,
k=1
where j is such that m 1 ≥ · · · ≥ m j > m j+1 = · · · = m l = 0.
Bloch-Okounkov Correlation Functions of Classical Type
501
Proposition 6.1 [W1, Theorem 3.3]. We have the following decomposition of (Pin(2l), b∞ )-modules: Fl = Vλ (Pin(2l)) ⊗ L(b∞ ; (λ)). λ∈(Pin)
6.2. The operator B(t). Introduce the following operators in b∞ :
:B(t): = (t k − t −k )(E k,k − E −k,−k ), k∈Z+
B(t) = :B(t): +
t +1 C. t −1
When acting on Fl , :B(t): and B(t) can be expressed using (31) as follows: :B(t): =
l
p=1 k∈Z
B(t) =
l
p=1 k∈Z
−, p
+, p
t k (:ψ−k ψk +, p
−, p
−, p
−, p
t k (ψ−k ψk
+, p
: + :ψ−k ψk :), +, p
+ ψ−k ψk ).
We easily verify that 1
1
B(t) = t 2 A(t) − t − 2 A(t −1 ),
1
1
:B(t): = t 2 :A(t): − t − 2 :A(t −1 ):.
(32)
The energy operator L 0 on the b∞ -module L(b∞ , (λ)) with highest weight vector v(λ) is defined by (14) and [L 0 , E i, j − E − j,−i ] = (i − j)(E i, j − E − j,−i ). On Fl , we can realize L 0 as L0 =
l
p=1 k∈Z
+, p
−, p
k:ψ−k ψk
l :+ . 8
(33)
6.3. The n-point b∞ -functions of level l. Definition 6.1. The n-point b∞ -correlation function of level l associated to λ = 1l /2 + (m 1 , . . . , m l ) ∈ (Pin) is Blλ (q, t ) ≡ Blλ (q, t1 , . . . , tn ) = tr L(b∞ ;(λ)) q L 0 B(t1 ) · · · B(tn ). pin
Lemma 6.2. Let λ ∈ (Pin) and denote by chλ (z 1 , . . . , zl ) the character of Vλ (Pin(2l)). Then, −(λ +l−i) 2 z λj i +l−i + z j i pin . chλ (z 1 , . . . , zl ) = l−i −(l−i) z j + z j
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D. G. Taylor, W. Wang
Proof. Recall that λ = 1l /2 + (m 1 , . . . , m l−1 , −m l ). Since Vλ (Pin(2l)) ∼ = pin so , where chso for λ ∈ (Pin) Vλ (Spin(2l))⊕ Vλ (Spin(2l)), we have chλ = chso +ch λ λ λ is also given by (18). Note that the second determinant terms in the numerators of chso λ pin and chso (cf. (18)) are opposite to each other. Now the formula for ch follows. λ λ Let Fb (z, q; t1 , . . . , tn ) := tr Fz e11 q L 0 B(t1 ) · · · B(tn ).
(34)
Lemma 6.3. We have
Fb (z, q; t1 , . . . , tn ) =
zk q
k2 2
k [ ] · t Fbo (q; t ),
∈{±1}n
k∈ 21 +Z
where the notations are as in Lemma 3.7. Proof. The proof is similar to the one for Lemma 3.7, while we have to take into account the difference coming from the integral indices on the fermions ψ ± (z). Denote the charge + ψ − : which acts as 0 on F (0) . By definition, e = C + 1 . We operator C = k∈Z :ψ−k 11 k 2 can check (compare [MJD] and [Ok, Appendix A]) that S−k e11 Sk = e11 + k, S−k A(t)Sk = t k A(t), and 1 1 S−k L 0 S = L 0 + kC + k(k + 1) = L 0 + ke11 + k 2 . 2 2 Then, using (34) and (32), we have Fb (z, q; t1 , . . . , tn ) 1
2 [ ] t tr F(k) z e11 q L 0 A(t1 1 ) · · · A(tn n ) = ∈{±1}n
=
∈{±1}n
=
k∈Z
1 2 [ ] t tr F(0) S−k z e11 q L 0 A(t1 1 ) · · · A(tn n )Sk k∈Z
1 k 1 1 2 [ ] t z k+ 2 q 2 k(k+1) tr F(0) q L 0 t A(t1 1 ) · · · A(tn n ) k∈Z
∈{±1}n
=
z∈Z
1
1
1 2
z k+ 2 q 2 (k+ 2 )
k+ 1 2 [ ] · t Fbo (q; t ).
∈{±1}n
In the last equation we have used the “correction term” in (33).
Lemma 6.4. We have the following identity: l i=1
Fb (z i , q; t1 , . . . , tn ) =
λ∈(Pin)
pin
chλ (z 1 , . . . , zl ) · Blλ (q, t1 , . . . , tn ).
Bloch-Okounkov Correlation Functions of Classical Type
503
Proof. Follows from the (Pin(2l), b∞ )-Howe duality in Proposition 6.1.
Theorem 6.1. The n-point b∞ -correlation function of level l associated to λ ∈ (Pin) is given by Blλ (q; t1 , . . . , tn ) =
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
σ ∈W (Dl )
l
⎛ ⎝
a=1
a
⎞ [ a ](t a )ka Fbo (q; t a )⎠ ,
∈{±1}n
where ka = (λ + ρ − σ (ρ), εa ). pin
Proof. Note that Lemma 3.6 on the dominant monomial of the numerator of chλ remains valid for λ = 1l /2 + (m 1 , . . . , m l ). Now the proof of the theorem is the same as for Theorems 3.2 and 3.3, with the help of Lemmas 6.2, 6.3, and 6.4. Remark 6.1. It is remarkable that the formula for Blλ in Theorem 6.1 coincides with the one for Dlλ given in Theorem 3.3, except that the λ used in these two cases are different. 6.4. The q-dimension of a b∞ -module. In the same manner as in Sect. 3.7, we can derive the following q-dimension formula from the (Pin(2l), b∞ )-Howe duality in Proposition 5.1. Proposition 6.2. For λ ∈ (Pin), we have dimq L(b∞ ; (λ))
(ρ)2 1 (σ ) λ+ρ−σ 2 = · (−1) q (q; q)l∞ σ ∈W (Dl ) λ2 1 λi −λ j + j−i λi +λ j +2l−i− j 2 1 − q 1 − q . = · q (q; q)l∞ 1≤i< j≤l
7. Correlation Functions on b∞-Modules of Level l +
1 2
7.1. The (Spin(2l + 1), b∞ )-Howe duality. Let Z = Z throughout this section. The 1 action of Lie algebra so(2l + 1) on the Fock space Fl+ 2 defined in Sect. 4.1 can be integrated to an action of the Lie group Spin(2l + 1). The action of b∞ on the Fock 1 space Fl+ 2 is given by
(E i, j − E − j,−i )z i w − j i, j∈Z
=
l
:ψ +, p (z)ψ −, p (w): − :ψ +, p (w)ψ −, p (z): + :ϕ(z)ϕ(w):.
p=1 1
Now the operator B(t) acting on Fl+ 2 can be written as B(t) =
l
p=1 k∈Z
+, p −, p −, p +, p + t k ϕ−k ϕk . t k ψ−k ψk + ψ−k ψk
(35)
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D. G. Taylor, W. Wang
Define : (Pin) → b∞ ∗0 by sending λ = 21 1l + (m 1 , m 2 , . . . , m l ) to (λ) = (2l + 1 − 2 j)b0 +
j
bm k
k=1
if m 1 ≥ · · · ≥ m j > m j+1 = · · · = m l = 0. Proposition 7.1 [W1, Theorem 4.2]. We have the (Spin(2l + 1), b∞ )-module decomposition: 1 Fl+ 2 ∼ Vλ (Spin(2l + 1)) ⊗ L(b∞ , (λ)), =2 λ∈(Pin)
where the factor 2 denotes the multiplicity. The energy operator L 0 on the b∞ -module L(b∞ , (λ)) with highest weight vector v (λ) is defined by 1 1 2 L 0 · v (λ) = λ + · v (λ) , 2 16 [L 0 , E i, j − E − j,−i ] = (i − j)(E i, j − E − j,−i ). 1 The convention of shift by 16 will be convenient later on, and it also fits with the standard realization in terms of neutral fermions with integral indices (i.e. Ramond sector) of L 0 of the Virasoro algebra.
7.2. The n-point b∞ -function of level 21 . Definition 7.1. The n-point b∞ -correlation function of level l + λ ∈ (Pin) is
1 2
associated to
l+ 1
Bλ 2 (q, t1 , . . . , tn ) = tr L(b∞ ; (λ)) q L 0 B(t1 ) · · · B(tn ). 1
On Fl+ 2 , we can realize L 0 as L0 =
l
p=1 k∈Z
+, p
−, p
k:ψ−k ψk
:+
k k∈Z
2
:ϕ−k ϕk : +
2l + 1 . 16
(36)
When l = 0, we have by Proposition 7.1 that 1
B 21 (q, t1 , . . . , tn ) = (2)
1 tr 1 q L 0 B(t1 ) · · · B(tn ). 2 F2
(37)
The aim of this subsection is to determine this (unique) n-point b∞ -function of level 21 parallel to Sect. 4.3, which will be used in the general level l + 21 case in the following subsection. The following lemma is straightforward once we recall the setup of Lemma 4.1.
Bloch-Okounkov Correlation Functions of Classical Type
505
∼ F 21 ⊗ F 21 in Lemma 4.1, we have B(t) = Lemma 7.1. Under the isomorphism 2F = ϕ . k B1 (t) + B2 (t), where B1 (t) = k∈Z t ϕ−k ϕk and B2 (t) = k∈Z t k ϕ−k k By convention, we let 1
B 21 (q, t∅ ) = (2)
1 1 tr 1 q L 0 = q 16 (−q; q)∞ . 2 F2
(38)
Proposition 7.2. Recalling Fb from (34), we have
Fb (1, q; t1 , . . . , tn ) = 2
1
1
(2)
(2)
B 21 (q, t I )B 21 (q, t I c ).
I ⊆{1,...,n}
(39)
1
Equivalently, B 21 (q, t) is given by (2)
⎛
1 −1 ⎜ 1 k22 q 16 (−q; q)−1 q ∞ ⎝ 2 2 1 k∈Z+ 2
∈{±1}n
−
k [ ] · t Fbo (q; t )
∅I {1,...,n}
1 2 ( 21 )
B
⎞ 1 2 ( 21 )
(q, t I )B
(q, t I c )⎠ .
Proof. By Lemmas 4.1 and 7.1, we have 2 tr Fq L 0 B(t1 ) · · · B(tn ) = tr 1 1 q L 0 (B1 (t1 ) + B2 (t1 )) · · · (B1 (tn ) + B2 (tn )) F 2 ⊗F 2
tr 1 1 q L 0 Bi1 (t1 )Bi2 (t2 ) · · · Bin (tn ). = i∈{1,2}n
F 2 ⊗F 2
This is equivalent to the first formula in the theorem by (37), since the definitions of L 0 1 1 on F and on F 2 ⊗ F 2 are compatible (see (33) and (36)). On the right-hand side of (39), there are exactly two terms from I = ∅ and {1, . . . , n} 1
which give rise to B 21 (q, t1 , . . . , tn ). Now the second formula follows from (38) and Lemma 6.3.
(2)
Proposition 7.2 allows for the determination, which is recursive on n, of all n-point 1
1
correlation functions B 21 (q, t1 , . . . , tn ) of level 21 . The 1-point function B 21 (q, t) has (2)
1
(2)
been computed in [W2] (denoted by R(t) therein up to a factor q 16 ) using partition identities. 1
Proposition 7.3. [W2, Theorem 4]. The 1-point function B 21 (q, t) is given by 1 16
q (−q; q)∞
(2)
∞
(−1)r q r +1 t t +1 (−1)r q r +1 t −1 . + − 2(t − 1) 1 − q r +1 t 1 − q r +1 t −1 r =0
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An alternative solution to the 1-point function follows from Proposition 7.2 for n = 1: 1
1
B 21 (q, t) = (2)
=
1
4q 16 (−q; q)∞ 1 8
q
k2 2
(t k + t −k )Fbo (q, t)
k∈Z+ 12
1 2
q t (−qt; q)∞ (−t −1 ; q)∞ (q; q)∞ 1
1
2q 16 (−q; q)∞ · t 2 (1 − t −1 ) (t)(q; q)∞ 1
=
q 16 (−qt; q)∞ (−t −1 ; q)∞ (q; q)2∞ , 2(qt; q)∞ (t −1 ; q)∞ (−q; q)∞
where we have used a version of the Jacobi triple product identity
q
k2 2
1
1
t k = q 8 t 2 (q; q)∞ (−qt; q)∞ (−t −1 ; q)∞ .
k∈Z+ 12
Comparing the two formulas of 1-point function, we have the following. Corollary 7.1. The following q-identity holds: (−qt; q)∞ (−t −1 ; q)∞ (q; q)2∞ (qt; q)∞ (t −1 ; q)∞ (−q; q)2∞ ∞
(−1)r q r +1 t (−1)r q r +1 t −1 t +1 . +2 − = t −1 1 − q r +1 t 1 − q r +1 t −1 r =0
The right-hand side of the identity is known (cf. [W2]) to be equal to 2t
2 2 −1 ; q 2 ) 1 (t; q )∞ (q t d ∞ . ln t − 2 dt (qt; q 2 )∞ (qt −1 ; q 2 )∞
7.3. The n-point b∞ -functions of level l+ 12 . The character of the irreducible Spin(2l+1)module associated to λ ∈ (Pin) is also given by chbλ (z 1 , . . . , zl ) in (27). Lemma 7.2. We have the following q-series identity: 1
B 21 (q, t) · (2)
=
l
Fb (z i , q; t1 , . . . , tn )
i=1
λ∈(Pin)
l+ 1
chbλ (z 1 , . . . , zl ) · Bλ 2 (q; t1 , . . . , tn ).
Proof. Follows from the Howe duality in Proposition 7.1 and (37). Note that the cancellation of a factor 2 has occurred.
Bloch-Okounkov Correlation Functions of Classical Type
507 l+ 1
Theorem 7.1. The n-point b∞ -correlation function of level l + 21 , Bλ 2 (q, t1 , . . . , tn ), is equal to 1
B 21 (q; t) (2)
×
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
σ ∈W (Bl )
l a=1
⎛ ⎝
⎞ [ a ](t a )ka Fbo (q; t a )⎠ ,
a ∈{±1}n
where ka = (λ + ρ − σ (ρ), εa ). Sketch of a proof. The proof is completely parallel to the proof for Theorem 4.1, now with the help of (27), (28), Lemmas 4.4, 6.3 and 7.2. 7.4. The q-dimension of a b∞ -module of level l + 21 . In the same manner as in Sect. 3.7, we can derive the following q-dimension formula from the Howe duality in Proposition 7.1. The second formula below is obtained from the first one by using Lemma 3.10 and the explicit root system of type Bl . Proposition 7.4. For λ ∈ (Pin), we have dimq L(b∞ ; (λ)) 1 (−q; q)∞ = q 16 · (q; q)l∞ =q ×
1 16
(−1)(σ ) q
λ+ρ−σ (ρ)2 2
σ ∈W (Bl )
λ2 (−q; q)∞ λi +l−i+1/2 2 1 − q · q (q; q)l∞ 1≤i≤l λi −λ j + j−i 1−q 1 − q λi +λ j +2l−i− j+1 .
1≤i< j≤l
Acknowledgement. This research is partially supported by NSF and NSA grants. We thank Shun-Jen Cheng for helpful discussions and comments.
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Commun. Math. Phys. 276, 509–517 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0343-y
Communications in
Mathematical Physics
On the Periodicity Conjecture for Y-systems Alexandre Yu. Volkov1,2 1 Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium.
E-mail: [email protected]
2 International Solvay Institutes, Campus Plaine (ULB) – CP 231, Bd du Triomphe, B-1050 Brussels, Belgium
Received: 2 September 2006 / Accepted: 14 May 2007 Published online: 26 September 2007 – © Springer-Verlag 2007
Abstract: The conjecture in question (due ultimately to Alexei Zamolodchikov) asserts the periodicity of all the solutions to the so-called Y-systems. Those systems are naturally associated to pairs of indecomposable Cartan matrices of finite type, and the conjectured period is equal to twice the sum of the respective Coxeter numbers. This conjecture has so far been proven only if one of the ranks equals one, in which case the Y-systems are intrinsically related to Fomin-Zelevinsky’s cluster algebras. In this paper, I use elementary projective geometry to prove the case when the two Cartan matrices involved are of type A with both ranks arbitrary. Introduction Following Alexei Zamolodchikov [1], Kuniba and Nakanishi [2], and Ravanini et al [3], one can associate to any pair of indecomposable Cartan matrices of finite type the so-called Y-system of algebraic equations, which reads −a 1 + Yi jk ii Yi
j+1 k
Yi
i =i
j−1 k
= −a , 1 + 1/Yi jk kk
(0.1)
k =k
i, j, k ∈ Z,
1 ≤ i ≤ r,
1 ≤ k ≤ r ,
) are those Cartan matrices, and r and r are the respective ranks. where (aii ) and (akk Then the periodicity conjecture asserts that all solutions to this system are periodic in j, with period equal to twice the sum of the respective Coxeter numbers. Until now this conjecture has only been proven in the case when one of the ranks equals one [4–6]. In this paper, we will take the next logical step and prove the case when the two Cartan matrices are of type A with both ranks arbitrary.
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Recall that Cartan matrices of type A are tridiagonal, with twos on the diagonal and minus ones on the sub- and sup-diagonals. Thus, for 1 < i < r and 1 < k < r we have 1 + Yi+1 j k 1 + Yi−1 j k , Yi j+1 k Yi j−1 k = 1 + 1/Yi j k+1 1 + 1/Yi j k−1 while at the boundaries one or two factors in the right hand side are absent—for instance, 1 + Y2 jk , Y1 j+1 k Y1 j−1 k = 1 + 1/Y1 j k+1 1 + 1/Y1 j k−1 1 + Y2 j 1 . Y1 j+1 1 Y1 j−1 1 = 1 + 1/Y1 j 2 However, they can be added back if we introduce fictitious boundary variables and set them equal to zero or infinity, as appropriate: Y0 jk = Yr +1 j k = 0, Yi j 0 = Yi j r +1 = ∞. Note that Y0 j 0 , Y0 j r +1 , Yr +1 j 0 and Yr +1 j r +1 are thus ill-defined, but they never appear in the right-hand side anyway. Note also that our system consists of two completely decoupled identical subsystems—one involving variables Yi jk with i + j + k even, and the other with i + j + k odd. So we will simply discard the second subsystem and assume that the Yi jk ’s are only defined for i + j + k even. Finally, recall that the Coxeter number for type Ar equals r + 1. Thus, we are going to prove the following: Theorem 1 (Periodicity conjecture, Case AA). All regular solutions (that is, solutions avoiding 0, −1 and ∞ everywhere except the boundaries) to the Y-system 1 + Yi+1 j k 1 + Yi−1 j k , Yi j+1 k Yi j−1 k = (0.2) 1 + 1/Yi j k+1 1 + 1/Yi j k−1 1≤i ≤r
1 ≤ k ≤ r
i + j + k odd ,
where Y0 jk = Yr +1 j k = 0,
Yi j 0 = Yi
j r +1
= ∞,
(0.3)
are 2(r + r + 2)-periodic in j: Yi, j+2(r +r +2),k = Yi jk .
(0.4)
We will prove that by producing a manifestly periodic formula for the general solution. After the necessary preliminaries in Sect. 1, in Sect. 2 we check that this solution does actually satisfy the Y-system (0.2), and in Sect. 3 we complete the proof by showing that it is indeed a general solution. Before we begin, though, a word of explanation is called for as to why such a magic formula should exist at all. Very roughly, this is because the Y-systems are closely related to the so-called Toda field theories. The simplest of those theories is the celebrated Liouville equation, and so our formula for solving the Y-system is basically a generalization of the Liouville formula for solving the Liouville equation. A more detailed explanation will be given elsewhere. This paper is a slightly revised version of [7]. A very different proof of a closely related result appeared in [8]; see the closing remark of Sect. 2.
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1. Cross-Ratio We begin by recalling some textbook facts from projective geometry. Recall that the projective space CPr is the set of all lines through the origin of Cr +1 , or in other words, it is the quotient of Cr +1 \{0} by the equivalence relation wherein two vectors are equivalent iff they are collinear. For a vector x ∈ Cr +1 \{0} we denote the corresponding element (point) of CPr by [x] or simply x, and say that the former is a representative vector of the latter. Points are said to be projectively independent if their representative vectors are linearly independent. The projective span x 1 + · · · + x N of two or more points xn ∈ CPr is the set of all points whose representative vectors lie in the linear span of the vectors x n . Projective spans of r projectively independent points are called hyperplanes. To any invertible linear transformation T of Cr +1 we associate the invertible map t of CPr onto itself defined by t ([x]) = [T x]. Such maps are called projective transformations, and they map projective spans to projective spans: t (x1 + · · · + x N ) = t (x1 ) + · · · + t (x N ). We define r + 2 points in CPr to be in general position, or to form a projective frame, if no r + 1 of them lie in the same hyperplane, or in other words, if no r + 1 of their representative vectors are linearly dependent. Such frames are akin to vector bases in that for any two of them there is a unique projective transformation taking one to the other. For instance, for r = 1 any three distinct points form a projective frame, and thus they can be taken into any other three by a unique projective transformation. This is not the case for four points: four points in CP1 (of which at least three are distinct) have a numeric projective invariant called the cross-ratio, which is defined by X (x1 , x2 , x3 , x4 ) = −
x1 ∧ x2 x3 ∧ x4 . x1 ∧ x4 x2 ∧ x3
Note that division of bivectors makes proper sense here because they all are multiples of one and the same bivector. Explicitly, if xnl (l = 1, 2; n = 1, . . . , 4) are coordinates of the vectors x n relative to some basis e1 , e2 ∈ C2 , and |x m x n | denotes the determinant of the corresponding coordinate matrix, xm1 xn1 |x m x n | = det , xm2 xn2 then x m ∧ x n = |x m x n | e1 ∧ e2 , and hence X (x1 , x2 , x3 , x4 ) = −
(z 1 − z 2 )(z 3 − z 4 ) |x 1 x 2 ||x 3 x 4 | =− , |x 1 x 4 ||x 2 x 3 | (z 1 − z 4 )(z 2 − z 3 )
(1.5)
where z n = xn1 /xn2 ∈ C ∪ {∞}. The last expression makes it particularly easy to determine how the cross-ratio changes under permutations of the four points involved. It turns out that 24 such permutations yield only 6 different values of the cross-ratio: if one of them is ω, then the other five are 1/ω, 1 − ω, 1/(1 − ω), 1 − 1/ω and 1/(1 − 1/ω). In particular, we have X (x1 , x3 , x2 , x4 ) = 1 − X (x1 , x2 , x3 , x4 ), 1 X (x1 , x2 , x4 , x3 ) = , 1 − 1/ X (x1 , x2 , x3 , x4 )
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which is going to explain why the Y-system incorporates, besides the bare Y ’s, also 1 + Y and 1/(1 + 1/Y ). Back to the general case, consider r + 3 points x1 , . . . , xr +3 ∈ CPr such that any three of the first four plus the remaining r − 1 of them are in general position. Let V be the subspace of Cr +1 spanned by the representative vectors of those last r − 1 points x5 , . . . , xr +3 , and π the canonical projection Cr +1 → Cr +1 /V C2 . Then the multi-dimensional cross-ratio is defined by X x5 +···+xr +3 (x1 , x2 , x3 , x4 ) = X (x1 , x2 , x3 , x4 ) ,
(1.6)
where xn = [π(x n )] ∈ CP1 , and, I remind, the sum x5 + · · · + xr +3 denotes the projective span of the points x5 , . . . , xr +3 . In terms of wedge products and determinants, this is written explicitly as X x5 +...+xr +3 (x1 , x2 , x3 , x4 ) x 1 ∧ x 2 ∧ x 5 ∧ . . . ∧ x r +3 x 3 ∧ x 4 ∧ x 5 ∧ . . . ∧ x r +3 x 1 ∧ x 4 ∧ x 5 ∧ . . . ∧ x r +3 x 2 ∧ x 3 ∧ x 5 ∧ . . . ∧ x r +3 |x 1 x 2 x 5 . . . x r +3 ||x 3 x 4 x 5 . . . x r +3 | , =− |x 1 x 4 x 5 . . . x r +3 ||x 2 x 3 x 5 . . . x r +3 | =−
(1.7)
while, clearly, the behaviour under permutation of the points x1 , . . . , x4 remains the same as in one dimension. In particular, we again have X x5 +···+xr +3 (x1 , x3 , x2 , x4 ) = 1 − X x5 +···+xr +3 (x1 , x2 , x3 , x4 ), 1 X x5 +···+xr +3 (x1 , x2 , x4 , x3 ) = . 1 − 1/ X x5 +···+xr +3 (x1 , x2 , x3 , x4 )
(1.8)
We are now in a position to produce the promised explicit formula for the general solution to the Y-system (0.2). This formula involves either r + r + 2 arbitrary points in CPr , xn ∈ CPr ,
xn+r +r +2 = xn ,
or the same number of arbitrary (r + 1)-vectors, x n ∈ Cr +1 ,
x n+r +r +2 = x n ,
(1.9)
and reads Yi jk = − X xa+1 +···+xb−1 + xc+1 +···+xd−1 (xa , xb , xc , xd ) i−1
r −i
|x a . . . x b x c+1 . . . x d−1 ||x a+1 . . . x b−1 x c . . . x d | , = |x a . . . x b−1 x c+1 . . . x d ||x a+1 . . . x b x c . . . x d−1 |
(1.10)
where −i − j − k , 2 i − j +k c =b+k = , 2
a=
i − j −k , 2 −i − j + k d = c+r +1−i = +r + 1. 2 b =a+i =
Note that the order of columns in the determinants here is different from that in Eq. (1.7), but that does not change the total sign.
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Note also that numbers a, b, c, d are all integer because i + j + k is even. Furthermore, the inequalities 0 < i < r + 1 and 0 < k < r + 1 are exactly equivalent to a < b < c < d < a + r + r + 2. This guarantees that formula (1.10) makes sense for all i and k in the range, and none of the determinants involved is identically zero. It is plain to see that these Y ’s are 2(r + r + 2)-periodic in j, and thus Theorem 1 indeed reduces to the following proposition: Proposition 1. a) Formula (1.10) provides a solution to the Y-system (0.2). b) Any solution can be obtained in this way. 2. Proof of Proposition 1, Part (a) As an illustration, we begin with the case r = 1. Clearly, in this case index i can be dropped and the Y-system (0.2) written as 1 Y j+1 k Y j−1 k = 1 + 1/Y j k+1 1 + 1/Y j k−1 j + k even 1 ≤ k ≤ r Y j 0 = Y j r +1 = ∞ . In its turn, formula (1.10) can be rewritten (using Eq. (1.5)) as Y jk =
(z b−1 − z b )(z c − z c+1 ) , (z b−1 − z c+1 )(z b − z c )
j + k odd ,
where b=
−j − k + 1 , 2
c =b+k =
−j + k + 1 , 2
and z n are r + 3 arbitrary complex numbers, z n ∈ C ∪ {∞},
z n+r +3 = z n .
It is then straightforward to check that the latter does indeed solve the former. The general case is a bit more elaborate. Denote the upper left determinant in the right-hand side of Eq. (1.10) by ∆ i+1 j k : ∆ i+1 j k = |x a . . . x b x c+1 . . . x d−1 | . Then, clearly, the remaining three are ∆ i−1 j k , ∆ i Yi jk =
j k+1
and ∆ i
(2.11) j k−1 ,
∆ i+1 j k ∆ i−1 j k , ∆ i j k+1 ∆ i j k−1
and likewise ∆i
j+1 k
= |x a . . . x b−1 x c . . . x d−1 |
∆i
j−1 k
= |x a+1 . . . x b x c+1 . . . x d | .
Hence, by Eqs. (1.8), 1 + Yi jk =
∆i ∆i
∆i j k+1 ∆ i j+1 k
j−1 k j k−1
,
∆ i+1 j k ∆ i−1 j k 1 = , 1 + 1/Yi jk ∆ i j+1 k ∆ i j−1 k
(2.12)
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and therefore, indeed, Yi
j+1 k
∆ i+1 j+1 k ∆ i−1 j+1 k ∆ i+1 j−1 k ∆ i−1 j−1 k ∆ i j+1 k+1 ∆ i j+1 k−1 ∆ i j−1 k+1 ∆ i j−1 k−1 ∆ i+1 j+1 k ∆ i+1 j−1 k ∆ i−1 j+1 k ∆ i−1 j−1 k ∆ i+1 j k+1 ∆ i−1 j k+1 = ∆ i+1 j k+1 ∆ i+1 j k−1 ∆ i−1 j k+1 ∆ i−1 j k−1 ∆ i j+1 k+1 ∆ i j−1 k+1 1 + Yi+1 j k 1 + Yi−1 j k ∆ i+1 j k−1 ∆ i−1 j k−1 , × = ∆ i j+1 k−1 ∆ i j−1 k−1 1 + 1/Yi j k+1 1 + 1/Yi j k−1 Yi
j−1 k
=
at least for 1 < i < r and 1 < k < r . The boundary cases are checked separately using the observation that, by definition (2.11), we always have ∆ 0 jk = ∆ 0 j−1 k−1
∆ r +1 j k = ∆ r +1 j−1 k+1
∆ i j 0 = ∆ i+1 j−1 0 ,
and, if we take into account periodicity property (1.9), also ∆i
j r +1
= ∆ i−1 j−1 r +1 .
Thus, for instance, ∆ 2 j+1 k ∆ 0 j+1 k ∆ 2 j−1 k ∆ 0 j−1 k ∆ 1 j+1 k+1 ∆ 1 j+1 k−1 ∆ 1 j−1 k+1 ∆ 1 j−1 k−1 ∆ 2 j+1 k ∆ 2 j−1 k ∆ 2 j k+1 ∆ 0 j k+1 ∆ 2 j k−1 ∆ 0 j k−1 = ∆ 2 j k+1 ∆ 2 j k−1 ∆ 1 j+1 k+1 ∆ 1 j−1 k+1 ∆ 1 j+1 k−1 ∆ 1 j−1 k−1 1 + Y2 jk , = 1 + 1/Y1 j k+1 1 + 1/Y1 j k−1
Y1 j+1 k Y1 j−1 k =
and ∆ 2 j+1 1 ∆ 0 j+1 1 ∆ 2 j−1 1 ∆ 0 j−1 1 ∆ 1 j+1 2 ∆ 1 j+1 0 ∆ 1 j−1 2 ∆ 1 j−1 0 ∆ 2 j+1 1 ∆ 2 j−1 1 ∆ 2 j 2 ∆ 0 j 2 1 + Y2 j 1 = = , ∆ 2 j 2 ∆ 2 j 0 ∆ 1 j+1 2 ∆ 1 j−1 2 1 + 1/Y1 j 2
Y1 j+1 1 Y1 j−1 1 =
as it should be (see Introduction). The remaining six boundary cases are checked similarly, and this completes this part of the proof. A remark is in order here. In our approach determinants ∆ i jk play a secondary part, but, as it turns out, they satisfy their own system of algebraic equations rather similar to the Y-system. Indeed, note that either of the relations (1.8) is equivalent to the classical bilinear determinant identity |x 1 x 3 x 5 . . . x r +3 ||x 2 x 4 x 5 . . . x r +3 | = |x 1 x 2 x 5 . . . x r +3 ||x 3 x 4 x 5 . . . x r +3 | +|x 1 x 4 x 5 . . . x r +3 ||x 2 x 3 x 5 . . . x r +3 | , also known as the Plücker relation. Hence, the ∆’s satisfy the lattice system ∆i
j+1 k
∆i
j−1 k
= ∆ i+1 j k ∆ i−1 j k + ∆ i
j k+1 ∆ i j k−1 ,
which is variously known as the 3D Hirota equation, Lattice Toda Field Equation, or Octahedron Recurrence. Since the Y-systems and this Hirota equation are related to each other by a simple change of variables (2.12), the periodicity conjecture could be easily reformulated in terms of the latter. It is this alternative version that is considered in [8].
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3. Proof of Proposition 1, Part (b) We begin with numerology. It is easy to figure out that any solution to the Y-system is completely determined by its values at j = −1 and j = 0. There are 2rr of those in the original formulation (0.1), but since we have excluded half of the lattice, we only have half that number left, that is rr . On the other hand, solution (1.10) depends on r + r + 2 points in CPr , but due to projective invariance, r + 2 of them can be chosen arbitrarily. This leaves r points in an r -dimensional space, which also gives rr . So the numbers match up, but now we have to prove that this is not by accident. First, a lemma that describes three different types of initial data for the Y-system. Only the last one will be actually used in the main proof. Lemma 1. Two solutions to the Y-system (0.2) coincide everywhere on the lattice L = {(i, j, k) ∈ Z3 | 1 ≤ i ≤ r, 1 ≤ k ≤ r , i + j + k even} if they coincide on either of the following subsets of this lattice: a) J−1 ∪ J0 , b) Q 0 , c) K 1 ∩ (Q 0 ∪ Q 1 ∪ . . . ∪ Q r −1 ) , where Jn , K n and Q n are intersections of L with the planes j = n, k = n and i + j + k = −2n respectively. The first item is rather obvious. It has already been used in the numerological part, and is of no further use here. To prove item (b), consider first the equation of the Y-system (0.2) with (i, j, k) = (1, −3, 1): Y1 −4 1 =
1 + Y2 −3 1 . Y1 −2 1 (1 + 1/Y1 −3 2 )
All three lattice points involved in the right-hand side belong to Q 0 , while the point (1, −4, 1) in the left-hand side is the only point of the intersection Q 1 ∩ J−4 . Hence, if two solutions coincide on Q 0 , they also coincide on Q 1 ∩ J−4 . But now the next two equations, 1 + Y3 −4 1 1 + Y1 −4 1 Y2 −5 1 = , Y2 −3 1 (1 + 1/Y2 −4 2 ) 1 + Y2 −4 2 , Y1 −5 2 = Y1 −3 2 (1 + 1/Y1 −4 3 ) (1 + 1/Y1 −4 1 ) have the right-hand sides entirely contained in Q 0 ∪ (Q 1 ∩ J−4 ) and the left-hand sides spanning Q 1 ∩ J−5 . Hence, our solutions coincide on Q 1 ∩ J−5 as well. Clearly, this process can be continued step by step to cover the entire Q 1 , and then repeated again and again for all Q n with n > 0. That done, we can start over from Q −1 ∩ J2−r −r and move in the opposite direction to cover all Q n with n < 0 as well. This settles item (b). On to item (c), we begin with the equations 1 + Yi+1 j 1 1 + Yi−1 j 1 1 + 1/Yi j 2 = , −2r + 2 ≤ i + j ≤ −2 . Yi j+1 1 Yi j−1 1
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These guarantee that if two solutions coincide on K 1 ∩ (Q 0 ∪ Q 1 ∪ . . . ∪ Q r −1 ), they also coincide on K 2 ∩ (Q 0 ∪ Q 1 ∪ . . . ∪ Q r −2 ). Next, the equations 1 + Yi+1 j 2 1 + Yi−1 j 2 , 1 + 1/Yi j 3 = −2r + 3 ≤ i + j ≤ −3 Yi j+1 2 Yi j−1 2 1 + 1/Yi j1 give K 3 ∩(Q 0 ∪ Q 1 ∪. . .∪ Q r −3 ), and so it continues all the way to K r ∩ Q 0 . But clearly, the union of these r sets contains Q 0 , and item (c) has thus reduced to item (b). Back to the main proof, fix r + 2 points x0 , . . . , xr +1 ∈ CPr in general position and write out formula (1.10) for (i, j, k) ∈ K 1 ∩ (Q 0 ∪ Q 1 ∪ . . . ∪ Q r −1 ) as the following chain of r equations: |x 0 . . . x i x i+2 . . . x r +1 ||x 1 . . . x i−1 x i+1 . . . x r +2 | = Yi,−i−1,1 , |x 0 . . . x i−1 x i+2 . . . x r +2 ||x 1 . . . x r +1 | |x 1 . . . x i+1 x i+3 . . . x r +2 ||x 2 . . . x i x i+2 . . . x r +3 | = Yi,−i−3,1 , |x 1 . . . x i x i+3 . . . x r +3 ||x 2 . . . x r +2 | .. . |x r −1 . . . x i+r −1 x i+r +1 . . . x r +r ||x r . . . x i+r −2 x i+r . . . x r +r +1 | = Yi,−i−2r +1,1 . |x r −1 . . . x i+r −2 x i+r +1 . . . x r +r +1 ||x r . . . x r +r | Note that every equation here is actually a system of r equations obtained for i = 1, . . . , r , and so the whole chain is a system of rr equations for r unknown points xr +2 , . . . , xr +r +1 ∈ CPr . Clearly, we shall be done if we show that these equations can be solved for any values of the Y ’s in the right-hand sides, except perhaps 0, −1 and ∞. Note that the first of the above systems contains only one unknown point, xr +2 , and so it can be solved by straightforward linear algebra. Indeed, computing the determinants with respect to the basis x 1 , . . . , x r +1 we obtain |x 0 . . . x i x i+2 . . . x r +1 | = (−1)i x0i+1 , |x 1 . . . x i−1 x i+1 . . . x r +2 | = (−1)r +1−i xri +2 , i+1 i |x 0 . . . x i−1 x i+2 . . . x r +2 | = (−1)r +1 (x0i xri+1 +2 − x 0 xr +2 ), |x 1 . . . x r +1 | = 1 ,
where xni denotes the i th coordinate of x n with respect to the said basis. Hence xri +2 xri+1 +2
=
x0i 1 , 1 + 1/Yi,−i−1,1 x0i+1
or a bit more conveniently, xri +2 +1 xrr+2
= λi
x0i x0r +1
,
where λi =
r i =i
1 . 1 + 1/Yi ,−i −1,1
On the Periodicity Conjecture for Y-systems
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Thus, the general solution to the first system is given by x r +2 = c T x 0 , where c is arbitrary, and T is the linear transformation, which has x 1 , . . . , x r +1 as eigenvectors and λi as respective eigenvalues: T x i = λi x i ,
i = 1, . . . , r + 1 .
In projective terms the above means that xr +2 = t (x0 ),
t (xi ) = xi ,
i = 1, . . . , r + 1 ,
where t is the projective counterpart of T . Let me stress that this projective transformation is properly defined because none of the λ’s is zero. So, since it maps the set of points x0 , . . . , xr +1 to x1 , . . . , xr +2 and the former have been chosen in general position, the latter will be in general position as well. But now that point xr +2 is found and shown to be in general position, we can just repeat the above steps to solve the second system for xr +3 , then solve the third one for xr +4 , and so on all the way to the last point xr +r +1 . Thus, solution (1.10) is indeed a general one, and so Proposition 1 and Theorem 1 are now fully proved. As a final remark, note that the change of indices (i, j, k) → (r +1−i, j +r +r +2, r + 1 − k) translates, in terms of the numbers a . . . d, into the pairwise shift/permutation (a, b, c, d) → (c − r − r − 2, d − r − r − 2, a, b). But clearly, the latter has no effect on the general solution (1.10), and so Theorem 1 can be improved somewhat by replacing the periodicity condition (0.4) with the stronger one: Yr +1−i, j+r +r +2,r +1−k = Yi jk . In fact, the periodicity conjecture has always been formulated in this stronger form. The only reason to start with the weaker version was to slightly simplify exposition. Acknowledgement. I wish to thank L. Faddeev, R. Kashaev, I. Krichever, I. Loris, A. Szenes and especially A. Alekseev for helpful discussions. I am also very indebted to the referee for suggesting a number of improvements to the text.
References 1. Zamolodchikov, Al.B.: On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories. Phys. Lett. B 253, 391–394 (1991) 2. Kuniba, A., Nakanishi, T.: Spectra in Conformal Field Theories from the Rogers Dilogarithm. Mod. Phys. Lett. A 7, 3487–3494 (1992) 3. Ravanini, F., Valleriani, A., Tateo, R.: Dynkin TBAs. Int. J. Mod. Phys. A 8, 1707–1727 (1993) 4. Frenkel, E., Szenes, A.: Thermodynamic Bethe ansatz and dilogarithm identities, I. Math. Res. Lett. 2(6), 677–693 (1995) 5. Gliozzi, F., Tateo, R.: Thermodynamic Bethe ansatz and three-fold triangulations. Int. J. Mod. Phys. A 11(22), 4051–4064 (1996) 6. Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. Ann. of Math. 158(3), 977–1018 (2003) 7. Volkov, A.Yu.: On Zamolodchikov’s Periodicity Conjecture. http://arxive.org/list/hep-th/0606094, 2006 8. Henriques, A.: A Periodicity Theorem for the Octahedron Recurrence. http://arxive.org/list/math.CO/ 0604289, 2006 Communicated by L. Takhtajan
Commun. Math. Phys. 276, 519–549 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0346-8
Communications in
Mathematical Physics
A Lie Theoretic Approach to Renormalization Kurusch Ebrahimi-Fard1 , José M. Gracia-Bondía2,3 , Frédéric Patras4 1 I.H.É.S., Le Bois-Marie, 35, Route de Chartres, F-91440 Bures-sur-Yvette, France.
E-mail: [email protected]; URL: http://www.th.physik.uni-bonn.de/th/People/fard/
2 Departamento de Física Teórica I, Universidad Complutense, Madrid 28040, Spain 3 Departamento de Física, Universidad de Costa Rica, San Pedro 2060, Costa Rica 4 Laboratoire J.-A. Dieudonné UMR 6621, CNRS, Parc Valrose, 06108 Nice Cedex 02, France.
E-mail: [email protected]; URL: www.math.unice.fr/~patras Received: 11 September 2006 / Accepted: 13 April 2007 Published online: 19 September 2007 – © Springer-Verlag 2007
Abstract: Motivated by recent work of Connes and Marcolli, based on the Connes– Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann–Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2. Notational Conventions . . . . . . . . . . . . . . . . . 3. The Hopf Algebra of Characteristic Functions . . . . . 4. Logarithmic Derivatives and the Dynkin Operator . . . 5. Algebraic BWH Decomposition of Characters . . . . . 6. On Renormalization Procedures . . . . . . . . . . . . . 7. Locality and the Connes–Kreimer Beta Function . . . . 8. Through the Prism of Other Renormalization Schemes I 9. Through the Prism of Renormalization Schemes II . . . 10. On Connes–Marcolli’s Motivic Galois Theory . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction From its inception, renormalization theory in perturbative quantum field theory (pQFT) had a combinatorial flavour, as well as an analytic one. The former manifests itself in the
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self-similarity of Feynman graphs, the building blocks of pQFT. The intricate combinatorics of extracting and combining subgraphs, required in the renormalization process, is encoded in the Bogoliubov recursion, respectively its solution via Zimmermann’s forest formula [13, 15, 41, 52, 53]. Kreimer’s discovery of a Hopf algebra structure underlying Bogoliubov and Zimmermann’s formulae and illuminating their internal structure [36] was the starting point of a new approach in the field. Then the Connes–Kreimer decomposition à la Birkhoff–Wiener–Hopf (BWH) of Feynman rules [18] captured the process of renormalization in pQFT in the framework of dimensional regularization (DR) with the minimal subtraction (MS) scheme. Further work by Connes, Kreimer and others has since then established various links between the BWH decomposition of characters in renormalizable quantum field theories and relevant mathematical topics, culminating recently in work by Connes and Marcolli on motivic Galois theory [21, 22] and by Bloch, Esnault and Kreimer on the motivic nature of primitive Feynman graphs [6]. In the present work, largely motivated by [21], we return to the origin of the Connes–Kreimer theory and concentrate on algebraic features of renormalization relevant to pQFT, trying to unravel further fundamental properties of renormalization schemes by methods inspired on the classical theory of free Lie algebras (FLAs). It has been known since the mid-nineties that many properties of FLAs, as exposed e.g. in [9, 47], can be lifted to general graded Lie algebras and their enveloping algebras. In other terms Lie theory is relevant to the study of arbitrary graded connected cocommutative or commutative Hopf algebras. In particular, the Solomon algebras of type An [47, Chap. 9] act naturally on arbitrary graded connected commutative Hopf algebras [44, Thm. II.7]. The observation applies to the Hopf algebras of renormalization, yet it has not received the attention it deserves. Here we develop it systematically, considering abstract renormalization Hopf algebras H and commutative target algebras A of quantum amplitudes endowed with a Rota–Baxter operator. We show that some of the deepest combinatorial properties of renormalization schemes are codified in the composition with the Dynkin idempotent of Lie theory, and in its inverse map. We derive in particular from their study the properties of characters under locality assumptions for counterterms in renormalization and, in the process, establish that the data relevant to their computation are contained in the “beta function”. The phenomenon is well known in pQFT; the Lie theoretic approach, however, provides a remarkably efficient way to deduce it from the locality assumptions. Furthermore, the direct sum of Solomon algebras (the descent algebra) is naturally provided with a graded connected cocommutative Hopf algebra structure; the corresponding pro-unipotent group is naturally isomorphic to the universal group U of the Connes–Marcolli Galois theory of renormalization. This isomorphism paves the way to an algebraic and combinatorial approach to the later theory. Some advantages of our method are: (i) It appears to be independent of the DR with MS prescription, applying in principle to any renormalization procedure which can be formulated in terms of a Rota–Baxter structure; (ii) Use of the Dynkin map explains naturally the coefficients of the universal singular frame of [21] —the same as in the Connes–Moscovici index formula [16] for the case of simple dimension spectrum. The article is organized as follows. After settling some notation in the next section, we ponder in Sect. 3 the convolution algebra of linear maps Lin(H, A). It cannot be made into a Hopf algebra in general; but a suitable algebra of characteristic functions can. This is our playground; it encodes, at the Hopf algebra level, the properties of the
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group of A-valued characters of H . Section 4 is the heart of the paper: starting from a short survey on the Dynkin idempotent for cocommutative Hopf algebras, we establish the formal properties of its sibling in the commutative case, then introduce and exhaustively study the inverse Dynkin map. In particular, we show that the latter bijectively sends infinitesimal characters into characters of commutative connected Hopf algebras —applying in particular to the Hopf algebras of Feynman diagrams and rooted trees of renormalization theory and the corresponding Feynman rules. In Sect. 5 we recall the BWH decomposition of characters, choosing once again to obtain it algebraically from Rota–Baxter operator theory and the ‘Baker–Campbell–Hausdorff (BCH) recursion’. After that, our Lie theoretic machine is fully operational. In the rest of the paper, we show the relevance of that machine to pQFT. In Sect. 6 we briefly review some standard lore of renormalization and remind the reader of the dictionary between it and the Connes–Kreimer paradigm. Next we study in Sect. 7 the locality properties for dimensional regularization (DR) schemes by exploiting the properties of the Dynkin pair of maps, together with the BWH decomposition. The main results concerning the Connes–Kreimer beta function and the renormalization group (RG) in DR are rederived from direct computations in the algebra of characteristic functions; hopefully, the role of that beta function is thereby illuminated. Sections 8 and 9 are essays on the same question in other renormalization frameworks; in the second we invoke the BPHZ scheme of renormalization and exhibit the underlying Rota–Baxter algebra structure, exemplifying with the (Ginzburg–Landau–Wilson) ϕ44 scalar model in Euclidean field theory. To finish, in Sect. 10 we go back to the mathematical setting, trying to place our results in the ‘great scheme of things’ of combinatorial Hopf algebra theory. We show there how the Connes–Marcolli “motivic Galois group” of renormalization relates with FLAs as well as the theory of descent algebras. Together with the links between the same group and Connes–Moscovici’s index theorem in noncommutative geometry, these new connections give further evidence for Connes’ and Kreimer’s—already much documented—claim that the divergences of pQFT do reveal the presence of deep mathematical structures. 2. Notational Conventions Let H = ∞ n=0 Hn be a graded connected commutative Hopf algebra (of finite type) over a field k of characteristic zero; this is necessarily free as a commutative alge+ bra [44, Prop. 4.3]. We write ∞for the augmentation from H to H0 = k ⊂ H and H for the augmentation ideal n=1 Hn of H . The identity map of H is denoted I . The product in H is written π or simply by concatenation. The coproduct is written δ; we use n (1) (2) Sweedler’s notation and write h (1) ⊗ h (2) for δ(h), h ∈ Hn ; or i=0 h i ⊗ h n−i when the grading has to be taken into account. The usual restrictions apply, that is, h (1) ⊗ h (2) (1) (2) stands for a sum j∈J h j ⊗ h j and should not be interpreted as the tensor product of two elements of H . The same convention applies in forthcoming notation such as (1) (2) (2) h (1) ⊗g (1) ⊗h (2) ⊗g (2) , that should be understood as j∈J k∈K h (1) j ⊗gk ⊗h j ⊗gk , (1) (2) where δ(g) = k∈K gk ⊗ gk . Graduation phenomena are essential for all our forthcoming computations, since in the examples of physical interest they incorporate information such as the number of loops (or vertices) in Feynman graphs, and the subdivergence structure, relevant for the RG. They are expressed by the action of the grading operator Y : H → H , given
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by: Y (h) =
n∈N
nh n for h =
hn ∈
n∈N
Hn .
n∈N
We write |h n | := n. Notice that Y is usually denoted by D in the FLA literature. In the present article we stick to the notation most often used in the context of the Hopf algebra approach to renormalization [17–19, 21, 25, 30] and reserve D for the Dynkin operator. 3. The Hopf Algebra of Characteristic Functions A character is a map γ of unital algebras from H to the base field k: γ (hh ) = γ (h)γ (h ). It should be clear that the product on the right-hand side is the one in k. We write γn for the restriction of γ to a map from Hn to k. Let A be a commutative k-algebra, with unit 1 A = η A (1), η A : k → A and with product π A , which we sometimes denote by a dot: π A (u ⊗v) =: u ·v. The main examples we have in mind are A = C, A = C[[ε, ε−1 ] and A = H . We extend now the definition of characters and call an (A-valued) character of H any algebra map from H to A. In particular H -valued characters are simply algebra endomorphisms of H . An infinitesimal character is a linear map α from H to k such that: α(hh ) = α(h)(h ) + (h)α(h ). As for characters, we write α(h) = n∈N αn (h n ). The same notational convention will be used in the sequel without further notice: f n stands for the restriction of an arbitrary map f on H to Hn . We remark that by the very definition of characters and infinitesimal characters γ0 (1 H ) = 1, that is γ0 = , whereas α0 (1 H ) = 0. We extend as well in the obvious way the notion of infinitesimal characters to maps from H to a commutative k-algebra A. We have now: α(hh ) = α(h) · e(h ) + e(h) · α(h ), where e := η A ◦ . They can be alternatively defined as k-linear maps from H to A with α0 = 0 that vanish on the square of the augmentation ideal of H . In particular, if α is an infinitesimal character, the linear map α |n , the restriction of which is 0 on all the graded |n infinitesimal character. Thus components of H except for Hn , and αn = αn , is also an α decomposes as a sum of infinitesimal characters α = n>0 α |n . The vector space of infinitesimal characters, written H (A), or just (A), decomposes accordingly as the direct product of its graded components: (A) = n∈N∗ n (A), where n (A) is the linear span of the α |n . Thus we regard (A) as the natural ‘topological’ completion of the graded vector space ⊕n∈N∗ n (A). In more detail, we consider the subspaces ⊕i≥n∈N∗ i (A) and those associated onto homomorphisms, and we take the inverse limit, whose elements are infinite series. This property we agree to abbreviate from now on to “ (A) is a graded vector space”; the sundry objects defined below are graded in that sense —that is, they are actually completions of graded vector spaces, completions of graded Lie algebras, and so on.
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The space Lin(H, A) of k-linear maps from H to A, Lin(H, A) := n∈N Lin(Hn , A), is naturally endowed with an algebra structure by the convolution product: f ∗ g := π A ◦ ( f ⊗ g) ◦ δ :
f ⊗g
δ
πA
H− → H ⊗ H −−→ A ⊗ A −→ A.
The unit for the convolution product is precisely e : H → A. Especially when A = H , the convolution product provides various tools to deal with properties of characters, such as the logarithm of the identity, which is a projector on H with kernel the square of the augmentation ideal. As a corollary, A-valued infinitesimal characters can be characterized as those maps α from H to A such that αn ◦ I ∗k (h n ) = k αn (h n ), for any k, n ∈ N. We refer the reader interested in a systematic study of these phenomena and of their connections to the classical structure theorems for Hopf algebras such as the CartierMilnor-Moore theorem to [12, 43, 44]. The set G H (A) of characters, or just G(A), is a group for the convolution product. The inverse is given explicitly by the formula: γ −1 = e +
n∈N∗
−1 γn
=e+
k∈N∗
(−1)k
∗k γn
.
n∈N∗
The last sum is well-defined as a power series, since only a finite number of terms appear in each degree. We denote as usual by S the convolution inverse of the identity map I in End(H ) := Lin(H, H ). Then γ −1 = γ ◦ S ∈ G(A); the reader unfamiliar with this identity can deduce it easily from the next lemma and other notions introduced soon. Now, Lin(H, A) is not naturally provided with a Hopf algebra structure over the ground field k, except under particular assumptions on the target space A. For example, it is (up to the completion phenomena) a Hopf algebra if A = k. This follows from the usual argument to construct a Hopf algebra structure on the graded dual of a graded connected Hopf algebra of finite type. It is not when A = k[[ε, ε−1 ], that is when the coefficient algebra A is the field of Laurent series —an example relevant to renormalization. However, as will be shown below, a Hopf algebra structure can always be defined on certain remarkable spaces naturally related to Lin(H, A) and, most importantly in view of applications to pQFT, to the group of characters G(A). Lemma 3.1. Assume that, for given φ, ψ ∈ Lin(H, A), there exist elements φ (1) ⊗ φ (2) , respectively ψ (1) ⊗ ψ (2) , in Lin(H, A) ⊗ Lin(H, A) such that, for any h, h ∈ H : φ (1) (h) · φ (2) (h ) = φ(hh ) and ψ (1) (h) · ψ (2) (h ) = ψ(hh ); then φ ∗ ψ(hh ) = φ (1) ∗ ψ (1) (h) · φ (2) ∗ ψ (2) (h ) . Moreover, when ψ ∈ End(H ) and φ ∈ Lin(H, A), with the same hypothesis and ψ (1) , ψ (2) now in End(H ): φ ◦ ψ(hh ) = φ (1) ◦ ψ (1) (h) · φ (2) ◦ ψ (2) (h ) . The last identity in particular holds when A = H , that is, in End(H ).
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Proof. Indeed, we have: φ ∗ ψ(hh ) = φ(h (1) h =φ
(1)
(h
(1)
(1)
(1) (1)
) · ψ(h (2) h
)·φ
(2)
(h
(1)
(1) (2)
(2)
) (2)
)
(2)
)
) · ψ (1) (h (2) ) · ψ (2) (h (2)
(1)
= φ (h ) · ψ (h ) · φ (h ) · ψ = φ (1) ∗ ψ (1) (h) · φ (2) ∗ ψ (2) (h ) ,
(2)
(h
an identity that we also write, for later use, (φ ∗ ψ)(1) ⊗ (φ ∗ ψ)(2) = (φ (1) ⊗ φ (2) ) ∗ (ψ (1) ⊗ ψ (2) ). We also clearly have:
φ ◦ ψ(hh ) = φ(ψ (1) (h) ψ (2) (h )) = φ (1) ◦ ψ (1) (h) · φ (2) ◦ ψ (2) (h ) .
Corollary 3.1. The graded vector space of infinitesimal characters (A) is a graded Lie subalgebra of Lin(H, A) for the Lie bracket induced on the latter by the convolution product. Proof. Indeed, infinitesimal characters are precisely the elements α of Lin(H, A) such that: α (1) ⊗ α (2) = α ⊗ e + e ⊗ α satisfy, for any h, h ∈ H : α (1) (h) · α (2) (h ) = α(hh ). According to the foregoing lemma, for α and β two graded infinitesimal characters, we obtain: [α, β](hh ) := (α ∗ β − β ∗ α)(hh ) = π A [(α ⊗ e + e ⊗ α) ∗ (β ⊗ e + e ⊗ β) −(β ⊗ e + e ⊗ β) ∗ (α ⊗ e + e ⊗ α)](h ⊗ h ) = [α, β](h) · e(h ) + e(h) · [α, β](h ), hence the corollary follows.
Proposition 3.1. The enveloping algebra U ( (A)) of the Lie algebra (A) maps naturally to the convolution subalgebra of Lin(H, A) generated by (A). The existence of that natural algebra map from U ( (A)) to Lin(H, A) follows from the previous lemma and from the universal property of enveloping algebras. Notice that U ( (A)) is also, as the enveloping algebra of a graded Lie algebra, a graded connected cocommutative Hopf algebra, which we call the Hopf algebra Char H (A), or just Char(A), of characteristic functions on H (with values in A). We write ∗ for the product on Char(A) and use for its coproduct. Thus by definition of Char(A) the primitive elements are the infinitesimal A-valued characters of H . Besides providing Hopf algebra tools for the study of Feynman rules, the Hopf algebra of characteristic functions —and the associated pro-unipotent group— will play a crucial role in Sect. 10, when relating the FLA approach to renormalization to the Connes–Marcolli motivic Galois group. Notice that is not defined in general on Lin(H, A), and neither on the image of Char(A) in Lin(H, A), see [34] and [45] for a discussion on the subject in the particular case A = H .
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Proposition 3.2. We have, for any φ ∈ Char(A) and any h, h ∈ H , the reciprocity law: φ(hh ) = φ (1) (h) · φ (2) (h ), where we use the Sweedler notation for (φ), and where the action of φ on H is induced by the natural map from Char(A) to Lin(H, A). Proof. This is true when φ is an infinitesimal character. According to the previous proposition, for φ, φ in Char(A), we have: φ ∗φ (hh ) = φ ∗φ (hh ). Due to the Lemma 3.1, it follows that the identity holds for φ ∗ φ if it holds for φ and φ . Since Char(A) is generated as an associative algebra by infinitesimal characters, the proposition follows. We remark that the reciprocity law can be rewritten: φ ◦ π = π A ◦ (φ). In the cocommutative case, the identity playing a similar role (mutatis mutandis) is [45]: δ ◦ φ = (φ) ◦ δ. As a consequence of Proposition 3.2, the set G (A) of group-like elements in Char(A) maps to characters, that is elements of G(A) —since the identity (φ) = φ ⊗φ in Char(A) translates into the identity φ(hh ) = φ(h)φ(h ) in H . We show now that, as usual, the convolution exponential and logarithm maps are inverse bijections from (A) onto G(A) and from (A) onto G (A). Indeed, for any α ∈ (A), we have in Char(A):
(exp(α)) = exp ( (α)) = exp(α ⊗ e + e ⊗ α) = exp(α ⊗ e) ∗ exp(e ⊗ α) = (exp(α) ⊗ e) ∗ (e ⊗ exp(α)) = exp(α) ⊗ exp(α), which also implies that we have exp(α) ∈ G(A) in Lin(H, A). We have used first that α is a graded infinitesimal character, then that α ⊗ e and e ⊗ α commute. The other way round, if γ is a character:
log(γ )(hh ) = π A log(γ ⊗ γ )(h ⊗ h )
= π A (log(γ ⊗ e) + log(e ⊗ γ ))(h ⊗ h )
= π A (log(γ ) ⊗ e + e ⊗ log(γ ))(h ⊗ h ) = log(γ )(h) · e(h ) + e(h) · log(γ )(h ), whereas if γ ∈ G (A):
(log γ ) = log( γ ) = log(γ ⊗ γ ) = log((γ ⊗ e) ∗ (e ⊗ γ )) = log γ ⊗ e + e ⊗ log γ .
Corollary 3.2. The natural algebra map from Char(A) to Lin(H, A) induces a bijection between the group G (A) of group-like elements in Char(A) and G(A), the group of A-valued characters on H . We identify G(A) with G (A) henceforth. In particular, both the identity map I and the antipode S can be viewed as elements of Char(H ), and we won’t distinguish between I, S and their respective preimages in Char(H ).
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K. Ebrahimi-Fard, J. M. Gracia-Bondía, F. Patras
4. Logarithmic Derivatives and the Dynkin Operator Although the logarithm is the simplest bijection from group-like elements to primitive elements in a graded connected cocommutative Hopf algebra, the most relevant bijection in view of applications to renormalization is a bit subtler. It is a kind of logarithmic derivative, closely related to a Lie idempotent known as the Dynkin idempotent. Presently we survey the properties of the Dynkin operator (the composition of the Dynkin idempotent with the grading map Y ) pertinent to our forthcoming computations, and also obtain new results on the operator, such as the existence of the advertised bijection between G(A) and (A). The results generalize the fine properties of Hopf algebras encapsulated in the notion of descent algebra of a Hopf algebra [44]. They rely as well on the Hopf algebraic treatment of the Dynkin operator given in [45], and on more classical Lie theoretic properties of the operator. We give in particular closed formulas for the inverse map to Dynkin’s operator, i.e., from (A) to G(A). The classical Dynkin operator is defined as follows. Let X = {x1 , . . . , xn , . . .} be an alphabet. The tensor algebra T (X ) := n≥0 Tn (X ) over X is a cocommutative graded Hopf algebra with the set of words xi1 . . . xil as a linear basis in degree l. It is also, canonically, the enveloping algebra of the FLA Lie(X ) over X . The product in T (X ) is induced by concatenation of words, whereas the coproduct is fully characterized by the property that elements of X are primitive in T (X ). The Dynkin operator D : T (X ) → Lie(X ) is given by: D(xi1 . . . xin ) = [. . . [[xi1 , xi2 ], xi3 ], . . . , xin ]; with DT0 (X ) = 0 and DT1 (X ) = id X . According to an idea essentially due to von Waldenfels, this iterated bracketing operator can be represented in a more abstract way as the convolution product of the antipode S of T (X ) with the grading operator Y , acting as the multiplication by n on Tn (X ): D = S ∗ Y ; equivalently I ∗ D = Y. The most famous property of D is the Dynkin–Specht–Wever theorem, stating that an element v in Tn (X ) —a linear combination of words of length n— is in Lie(X ) if and only if D(v) = nv. In effect, such a v is in the primitive part of the tensor algebra, and: nv = Y v = π(I ⊗ D)(1 ⊗ v + v ⊗ 1) = D(v). The converse is also well known. These definitions and properties have been generalized to bialgebras in [45], that we follow mainly here. However, for our purposes we need to give a somewhat detailed account. Indeed, that reference as well as the classical theory of the Dynkin operator do focus on the study of graded connected cocommutative Hopf algebras, whereas we are mainly interested here in commutative Hopf algebras. The interested reader will find further historical and technical information about the Dynkin operator and its relevance to Lie computations, as well as other classical Lie idempotents, in references [31] and [47]. So let H be graded, connected and commutative. Since I ∈ Char(H ), so its graded components In ∈ Char(H ). Notice, for further use, that the subalgebra of Char(H ) generated by the In maps to the descent algebra of H —the convolution subalgebra of End(H ) generated by the In , see [44]. Moreover, the grading operator Y := n∈N n In belongs to Char(H ). Its coproduct is given by:
(Y ) = Y ⊗ I + I ⊗ Y,
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another way of expressing that Y is a derivation of H : Y (hh ) = Y (h)h + hY (h ). Let us adopt the notation Y f for f ◦ Y , according to the custom in distribution theory. Under this guise the operator Y extends naturally to a derivation on Lin(H, A). We find, with f, g ∈ Lin(H, A) and h ∈ H : Y ( f ∗ g)(h) := f ∗ g (Y (h)) = |h|( f ∗ g)(h) = |h| f (h (1) )g(h (2) ) = |h (1) | f (h (1) )g(h (2) ) + |h (2) | f (h (1) )g(h (2) ) = Y f ∗ g (h) + f ∗ Y g (h),
where we used that (Y (h)) = |h| (h) = |h (1) | + |h (2) | h (1) ⊗ h (2) . Proposition 4.1. Convolution of the H -valued character S with any derivation d of H yields an H -valued infinitesimal character. Proof. Indeed, since d is a derivation, we have d(hh ) = d(h)h + hd(h ). Since, furthermore, (S) = S ⊗ S, we get: (S ∗ d)(hh ) = π ◦ [(S ⊗ S) ∗ (d ⊗ I + I ⊗ d)](h ⊗ h ) = π ◦ [(S ∗ d) ⊗ (S ∗ I ) + (S ∗ I ) ⊗ (S ∗ d)](h ⊗ h ) = S ∗ d(h) · e(h ) + e(h) · S ∗ d(h ), hence the proposition follows.
Corollary 4.1. The Dynkin operator D := S ∗ Y is an H -valued infinitesimal character of H . Notice also that D satisfies D ◦ D = D ◦Y —in other terms, D is an idempotent up to a scalar factor depending on the grading, that is, D is a quasi-idempotent. Equivalently, D ◦ Y −1 is an idempotent on H + (also known when H is the cotensor algebra T ∗ (X ) —see below— as the Dynkin idempotent). Indeed, for any h ∈ H , D ◦ D(h) = D ◦ (S ∗ Y )(h) = D S(h (1) )Y (h (2) ) . However, since D is an infinitesimal character, D(hh ) = 0 if h, h ∈ H + and therefore, D ◦ D(h) = D (S(h)Y (1 H ) + S(1 H )Y (h)) = D ◦ Y (h), since Y (1 H ) = 0. Proposition 4.2. Right composition with an infinitesimal character α ∈ (H ) induces a map from G(H ) to (H ). This also holds for G(A) and (A), where A is an arbitrary commutative unital algebra. Proof. Indeed, let γ ∈ G(H ) or G(A), we have: γ ◦ α(hh ) = γ ◦ α(h) e(h ) + e(h) γ ◦ α(h ), by virtue of Lemma 3.1, since γ ◦ e = e for any character γ .
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Corollary 4.2. Right composition with the Dynkin operator D induces a map from G(A) to (A). In general, for γ belonging to G(H ) or G(A) and any f 1 , . . . , f k ∈ End(H ), we have the distributive property: γ ◦ ( f 1 ∗ · · · ∗ f k ) = (γ ◦ f 1 ) ∗ · · · ∗ (γ ◦ f k ). Particularly, γ ◦ D = γ ◦ (S ∗ Y ) = γ −1 ∗ Y γ . Theorem 4.1. Right composition with D is a bijective map from G(H ) to (H ). The inverse map is given by: αk1 ∗ · · · ∗ αkl : α ∈ (H ) −→ ∈ G(H ). (1) ∗ k 1 (k 1 + k 2 ) . . . (k 1 + · · · + kl ) n k1 ,...,kl ∈N k1 +···+kl =n
The theorem also holds if G(H ) and (H ) are replaced by G(A), respectively (A). We show first that is a left inverse to right composition with D. The following lemma has been obtained in [31] in the setting of noncommutative symmetric functions and quasi-determinants. We include the proof, which is elementary. Lemma 4.1. For n ≥ 1 we have: In = k1 ,...,kl ∈N∗ k1 +···+kl =n
Dk 1 ∗ · · · ∗ Dkl . k1 (k1 + k2 ) . . . (k1 + · · · + kl )
Proof. As already remarked, the definition of D implies I ∗ D = I ∗ S ∗ Y = Y . In particular, since D0 = 0: Yn = n In = (I ∗ D)n =
n
In−i ∗ Di .
i=1
Inserting recursively the value of Ii in the right member of the identity, we obtain: Dn + n
n−1
In =
i=1 1≤ j≤n−i
=
n−1 D j ∗ Di Dn + + n j ·n j+i=n i, j=0
=
k1 ,...,kl ∈N∗ k1 +···+kl =n
In−i− j ∗ D j ∗ Di (n − i)n
i=1 1≤ j≤n−i−1
In−i− j ∗ D j ∗ Di (n − i)n
Dk 1 ∗ · · · ∗ Dkl . k1 (k1 + k2 ) . . . (k1 + · · · + kl )
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Now we compute γ = γ ◦ I , where I is expanded according to the previous lemma, yielding: ⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ Dk 1 ∗ · · · ∗ Dkl γ = e+γ ◦ ⎪ k (k + k2 ) . . . (k1 + · · · + kl ) ⎪ ⎪ ⎪ ⎭ ⎩n∈N∗ k1 ,...,kl ∈N∗ 1 1 k1 +···+kl =n
=e+
γ ◦ Dk 1 ∗ · · · ∗ γ ◦ Dkl . k1 (k1 + k2 ) . . . (k1 + · · · + kl )
n∈N∗ k1 ,...,kl ∈N∗ k1 +···+kl =n
As D preserves the grading, it follows that is a left inverse to the right composition with D. Similar calculations help to prove that is character-valued, that is, it is actually a map from (H ) to G(H ). Indeed, let α be any infinitesimal character in (H ). Then we have in Char(H ):
((α)) = e ⊗ e +
= e⊗e+
n>0 k1 ,...,kl ∈N∗ k1 +···+kl =n
(αk1 ∗ · · · ∗ αkl ) k1 (k1 + k2 ) . . . (k1 + · · · + kl )
n>0 k1 ,...,kl ∈N∗ I J ={k1 ,...,kl } k1 +···+kl =n |I |=m,|J |= p
(αi1 ∗ · · · ∗ αim ) ⊗ (α j1 ∗ · · · ∗ α j p ) k1 (k1 + k2 ) . . . (k1 + · · · + kl )
,
where I = {i 1 , . . . , i m }, J = { j1 , . . . , j p } and we have used that the αim are all infinitesimal characters. Particularly, the assertion we intend to prove, that is ((α)) = (α) ⊗ (α), amounts to the equality: ⎛
⎞⊗2
⎜ ⎜e + ⎝
n>0 k1 ,...,kl ∈N∗ k1 +···+kl =n
= e⊗e+
⎟ αk1 ∗ · · · ∗ αkl ⎟ k1 (k1 + k2 ) . . . (k1 + · · · + kl ) ⎠
n>0 k1 ,...,kl ∈N∗ I J ={k1 ,...,kl } k1 +···+kl =n |I |=m,|J |= p
(αi1 ∗ · · · ∗αim ) ⊗ (α j1 ∗ · · · ∗ α j p ) k1 (k1 + k2 ) . . . (k1 + · · · + kl )
.
This follows from the identity:
1 k (k + k2 ) . . . (k1 + · · · + k p+m ) I J =K 1 1 1 1 · , = i (i + i ) . . . (i + · · · + i ) j ( j + j ) . . . ( j1 + · · · + j p ) 2 1 m 1 1 2 ∗ 1 1
(2)
i 1 ,...,i m ∈N j1 ,..., j p ∈N∗
where K runs over all sequences (k1 , . . . , k p+m ) obtained by shuffling the sequences I and J . In turn, the identity follows if the equation ((α)) = (α) ⊗ (α) —that is,
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(α) ∈ G(H )— holds for a particular choice of H and α such that the αi form a family of algebraically independent generators of a subalgebra of the convolution algebra End(H ) —which are therefore also algebraically independent as elements of Char(H ). So, let us consider the particular case H = T ∗ (X ), where H is the graded dual of the enveloping algebra of the FLA over an infinite alphabet X and α = D is the Dynkin operator. Then, we already know that (D) = I , due to the previous lemma, so that (D) is group-like in Char(T ∗ (X )). On the other hand, as is well known, the graded components of the Dynkin operator are algebraically independent in the convolution algebra End(T ∗ (X )), and the identity follows. (For details on the algebraic independence of the graded components of the Dynkin operator, consult [31, 47].) Although we have preferred to give a conceptual proof based on FLAs, let us mention that identity (2) is elementary, and well known in boson Fock space theory. It also follows e.g. from the shuffle identity for the product of two iterated integrals [48]: 1
k p+m −1 x p+m ...
I J =K 0 1
x1k1 −1 d x1 . . . d x p+m
0
xmim −1 . . .
=
x2
0
x2
x1i1 −1 d x1 . . . d xm
0
1 · 0
j −1 y pp . . .
y2
j −1
y11
dy1 . . . dy p ,
0
which generalizes the integration by parts rule. To conclude the proof of Theorem 4.1 we show that is also a right inverse to the composition with D. We contend that, for any h in the augmentation ideal of H and arbitrary α ∈ (H ), the following relation holds: α(h) = (α)−1 ∗ Y (α) (h) or, equivalently, Y (α)(h) = (α) ∗ α (h). Indeed, we have: Y (α)(h) := |h|
k1 ,...,kl ∈N∗ k1 +···+kl =|h|
=
k1 ,...,kl ∈N∗ k1 +···+kl =|h|
αk1 ∗ · · · ∗ αkl (h) k1 (k1 + k2 ) . . . (k1 + · · · + kl )
αk1 ∗ · · · ∗ αkl−1 ∗ αkl (h) k1 (k1 + k2 ) . . . (k1 + · · · + kl−1 )
= (α) ∗ α (h). This together with the fact that (α) ∈ G(H ) for α ∈ (H ) implies: (α) ◦ D = (α)−1 ∗ Y (α) = α. Our task is over. When H is both commutative and cocommutative the convolution product is commutative and γ ◦ D = Y log γ := log(γ ) ◦ Y . In particular in this case D = Y log I . This was put to good use in [7]. Thus clearly D, in the general case, is a noncommutative logarithmic derivative; and the inverse Dynkin operator a extremely powerful tool. We finally remark that Y ∗ S, corresponding, in the free cocommutative case and as an operator from the tensor algebra to the FLA, to the right-to-left iteration of bracketings, is another possible form for the noncommutative logarithmic derivative, leading in particular to γ ◦ (Y ∗ S) = Y γ ∗ γ −1 . More generally, any interpolation of the form S a ∗ Y ∗ S b , with a + b = 1, yields a notion of noncommutative logarithmic derivative.
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5. Algebraic BWH Decomposition of Characters In this section we summarize previous research on Rota–Baxter operators, relevant for our purpose. Let H be still graded, connected and commutative and let again A stand for a commutative unital algebra. Assume the algebra A to split directly, A = A+ ⊕ A− , into the subalgebras A+ , A− with 1 A ∈ A+ . We denote the projectors to A± by R± , respectively. The pair (A, R− ) is a special case of a (weight one) Rota–Baxter algebra [23] since R− , and similarly R+ , satisfies the relation: R− (x) · R− (y) + R− (x · y) = R− (R− (x) · y + x · R− (y)) ,
x, y ∈ A.
(3)
The reader may verify that integration by parts rule is just the weight zero Rota–Baxter identity, that is, the second term on left-hand side of (3) is absent. One easily shows that Lin(H, A) with the idempotent operator R− defined by R− ( f ) = R− ◦ f , for f ∈ Lin(H, A), is a (in general not commutative) unital Rota–Baxter algebra [23]. The subspace L (1) of Lin(H, A) made of linear maps that send the Hopf algebra unit to zero forms a Lie algebra with (A) ⊂ L (1) as a Lie subalgebra. To L (1) does correspond the group G 0 = e + L (1) = exp(L (1) ) of linear maps sending the Hopf algebra unit to the algebra unit. It contains the group of characters G(A) as the subgroup exp( (A)). Due to the characterization of infinitesimal characters as maps that vanish on the square of the augmentation ideal of H , both R+ ( (A)) and R− ( (A)) embed into (A). In particular, both are Lie subalgebras of (A). The Lie algebra decomposition, (A) = R+ ( (A))⊕R− ( (A)) lifts to the group of characters G(A) as follows. Recall the Baker–Campbell–Hausdorff (BCH) formula [47] for the product of two exponentials, that we write: exp(x) exp(y) = exp (x + y + BCH(x, y)) . In [23, 24] the following non-linear map was defined, whose properties where further R explored in [27]. See also [42]. For f ∈ L (1) , define χ R− ( f ) = limn→∞ χn − ( f ), R− where χn ( f ) is given by what we call the BCH recursion: R−
χ0
( f ) = f,
R χn+1− ( f )
R−
χ1
( f ) = f − BCH (R− ( f ), R+ ( f )) , . . . , R R = f − BCH R− χn − ( f ) , R+ χn − ( f ) .
Then the fixed-point map χ R− : L (1) → L (1) satisfies: χ R− ( f ) = f − BCH R− χ R− ( f ) , R+ χ R− ( f ) .
(4)
The superscript R− is justified by the dependency of the limit on the Rota–Baxter operator, and the following result. Lemma 5.1. The map χ R− in (4) solves the simpler recursion: χ R− ( f ) = f + BCH −R− χ R− ( f ) , f , f ∈ L (1) . Following [27], the following factorization theorem holds. Theorem 5.1. For any f ∈ L (1) , we have the unique factorization: exp( f ) = exp R− χ R− ( f ) ∗ exp R+ χ R− ( f ) .
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Uniqueness of the factorization follows from R− being idempotent. In the particular case that f ∈ (A), the BCH recursion (4) takes place inside the Lie algebra (A), and the decomposition of exp( f ) holds therefore inside the group of characters G(A). In particular, it follows ultimately from Theorem 5.1 that G(A) decomposes as a set as the product of two subgroups: G(A) = G − (A) ∗ G + (A), where G − (A) = exp(R− ( (A))), G + (A) = exp(R+ ( (A))).
Corollary 5.1. For any γ = exp(α) ∈ G(A), with α ∈ (A), we have unique α± := R± (χ R− (α)) ∈ R± ( (A)), and unique characters γ± := exp(±α± ) ∈ G ± (A) such that: γ = γ−−1 ∗ γ+ .
(5)
A remark is in order. The factorization in Theorem 5.1 is true for any (filtration preserving) linear map P on Lin(H, A), that is, for χ P ( f ), see [27]. Uniqueness follows from P being idempotent. The Rota–Baxter property (3) implies that both, G ± (A) are subgroups. We may reformulate the last statement about the nature of G ± (A) in the next lemma, saying that the BWH decomposition of Connes and Kreimer, originally found by a more geometrical method [18], is recovered from Theorem 5.1 by using the Rota–Baxter relation [23, 26, 27]: Lemma 5.2. For any γ = exp(α) the unique characters γ± := exp(±α± ) ∈ G ± (A) in the previous corollary solve the equations: γ± = e ± R± (γ− ∗ (γ − e)).
(6)
Proof. There is a sharpened version [24] of Atkinson’s theorem [1], stating that the characters γ , γ± of (5) verify γ− = e−R− (γ− ∗(γ −e)) and γ+ = e−R+ (γ+ ∗(γ −1 −e)). Now: γ+ ∗ (γ −1 − e) = γ− ∗ γ ∗ (γ −1 − e) = γ− ∗ (e − γ ) gives (6).
6. On Renormalization Procedures Prima facie in pQFT, to a Feynman graph F does correspond by the Feynman rules a multiple d-dimensional momentum space integral. Each independent loop in a diagram yields one integration: F → J F ( p) =
|F|
d kl I F ( p, k). d
(7)
l=1
Here |F| is the number of loops, k = (k1 , . . . , k|F| ) are the |F| independent internal N pk = 0, denote the N external (loop) momenta and p = ( p1 , . . . , p N ), with k=1 momenta. In the most relevant kind (d = 4, renormalizable with dimensionless couplings) of field theories, these integrals suffer from ultraviolet (UV) divergencies. Concretely, under a scale transformation, the integrand behaves as |F| l=1
d d (λkl ) I F (λp, λk) ∼ λs(F) ,
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with s(F) the superficial UV degree of divergence of the graph F. Power-counting renormalizable theories are such that all interaction terms in the Lagrangian are of dimension ≤ d; then s(F) is bounded by a number independent of the order of the graph. For instance in the ϕ44 model the superficial UV degree of divergence of a graph with N external legs is: s(F) = 4 − N . The Weinberg–Zimmermann theorem says: “provided all free propagators have nonzero masses, the integral associated to the Feynman graph F is absolutely convergent if its superficial UV degree of divergence and that of each of its one-particle irreducible (1PI) subgraphs is strictly negative”. The BPHZ momentum space subtraction method is rooted in this assertion: the idea is to redefine the integrand I F ( p, k) of a divergent integral by subtracting from it the first terms of its Taylor expansion in the external momenta p, after these subtractions have been performed on the integrands corresponding to the 1PI subgraphs of F that are renormalization parts; in such a way the UV degrees of the integral and its subintegrals are lowered until they become all negative. The com¯ binatorics of those subgraph subtractions leads to Bogoliubov’s recursive R-operation and Zimmermann’s forest formula; we assume the reader acquainted with the former at least [13, 15, 49, 52]. Less straightforward conceptually, but way more practical, is the DR method [51]. This belongs to the class of regularization prescriptions which parameterize the divergencies appearing in J F upon introducing non-physical parameters, thereby rendering them formally finite. In DR one introduces a complex parameter ε ∈ C by changing the integral measure, that is, the space-time dimension, to D ∈ C: dim reg
d d k −−−−→ µε d D k, where ε = (d − D). Henceforth always d = 4. The arbitrary parameter µ = 0 (’t Hooft’s ‘unit-mass’ parameter) is introduced for dimensional reasons. Take the ϕ44 model: if we wrote the interaction term simply in the form gϕ 4 /4!, then the (naive) dimension of g would be [g] = µε . The redefinition gµ ˜ ε ϕ 4 /4! of the original vertex in the Lagrangian includes the mass parameter µ, introduced to make g˜ dimensionless. Now, in any given theory there is a rigid relation between the numbers of loops and vertices, for each given N -point function. For instance in the ϕ44 model, for graphs associated to the 2-point function the number of vertices is just equal to the number of loops. For graphs associated to the 4-point function the number of vertices is equal to the number of loops plus one, and so an extra gµ ˜ ε factor comes in; but, because we are correcting the vertex, this extra factor becomes the coupling constant, and is not expanded in powers of the regularization parameter ε; only the expression multiplying it contributes to the renormalization constant Z g —with a pole term independent of the mass scale µ. The outcome is that in practice one computes as many µε factors as loops: F −→
(ε,µ) J F ( p)
|F|ε
=µ
|F|
d kl I F ( p, k). D
(8)
l=1
See the discussions in [18, 19] and [35, Sects. 7 and 8] as well. The point is important from the algebraic viewpoint, since it makes the grading of Feynman graphs by the number of loops immediately relevant to the renormalization process.
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The next step in DR consists of a specific subtraction rule of those ε-parameterized expressions which allows to take the limit ε ↓ 0. Now, Connes and Kreimer’s BWH decomposition of Feynman rules [18] is extraordinarily well adapted to DR in pQFT. In the Connes–Kreimer paradigm, any renormalizable quantum field theory gives rise to a Hopf algebra H of Feynman graphs, polynomially generated by 1PI Feynman graphs and graded by graph loop number. The coproduct operation of H mirrors the combinatorics of the subgraphs. Looking back to (7), the unrenormalized Feynman integral does define a character because: I F1 ∪F2 ( p1 , p2 , k1 , k2 ) = I F1 ( p1 , k1 )I F2 ( p2 , k2 ) for disjoint graphs F1 , F2 . On the Hopf algebra H the Feynman rules single out a distinguished character γ with values in a suitable target algebra of regularized amplitudes. Precisely, Connes and Kreimer establish the above decomposition G(A) = G − (A) ∗ G + (A), for A the algebra of amplitude-valued Laurent series in the dimension defect ε, using the MS scheme in DR on momentum space. The characters γ± in the decomposition (5) solve Bogoliubov’s renormalization formulae —see Corollary 6.1 below— and may be called the renormalized and counterterm character, respectively. The sought after (ε,µ) (ε,µ) result is given by the ‘positive’ part in (5). To wit, γ+ = γ− ∗ γ (ε,µ) , and the limit (ε↓0,µ) γ+ exists, giving the renormalized Feynman amplitude. In what follows, when dealing with dimensionally regularized characters we drop the superscript ε. We also forget about the other involved variables, that do not enter our considerations, and just write C[[ε, ε−1 ] for A. Thus R− will be the projection onto the subalgebra A− := ε−1 C[ε−1 ]. In summary, A encodes the Feynman rules within the chosen regularization procedure, whereas the splitting of A, hence the projector R− , reflects a renormalization scheme within that choice. Corollary 6.1. The map γ¯ := γ− ∗ (γ − e) = γ+ − γ− in (6) gives Bogoliubov’s ¯ preparation map R. Indeed with the Connes–Kreimer definition of the Hopf algebra of Feynman graphs, Eqs. (6) coincide with Bogoliubov’s recursive formulas for the counterterm and renormalized parts. Recalling the remark after Corollary 5.1 we may characterize the Rota–Baxter structure on A as follows. Theorem 5.1 implies that regularized, i.e. A-valued, Feynman rules on H factorize uniquely upon the choice of an idempotent linear map on A respectively Lin(H, A). The extra demand for a Rota–Baxter structure on A, and hence on Lin(H, A) essentially implies the particular recursive nature of the process of perturbative renormalization as encoded in Bogoliubov’s preparation map R¯ respectively the equations (6).
7. Locality and the Connes–Kreimer Beta Function The results in Sects. 3 to 5 apply to any graded connected commutative bialgebra H and any commutative unital algebra A with a direct splitting. In this section we restrict our consideration to the class of Hopf algebra characters possessing a locality property, with H as before. This will correspond to the example given by the Feynman rules for a renormalizable pQFT in DR, using the framework of the MS and the MS [15, Sect. 5.11.2]
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schemes. There locality comes from the dependency of DR on the arbitrary ‘mass parameter’ µ. It is handy to provisionally fix the value of µ at some convenient reference point µ0 . The difference between both schemes boils down to: eγ E /2 µ0 = µ0 √ , 2 π
(9)
with µ0 , µ0 respectively the MS, MS values and γ E the Euler–Mascheroni constant. Our aim is to recover by our methods the results in [19, 21]; the latter constitute a stronger, algebraic version of a famous theorem by ’t Hooft [51]. Place of pride corresponds to the Connes–Kreimer abstract beta function. This is a conceptually very powerful beast, giving rise to the ordinary beta function through the (tangent morphism to) the morphism (of unipotent groups) from G(C) to the group of transformations of the coupling constants [19]. Any f ∈ Lin(H, A) is now of the form: f (h) =
∞
f :k (h)εk =: f (h; ε)
k=−U
for h ∈ H . Here every f :k ∈ Lin(H, C), the dual of H ; and the last notation is used when we want to emphasize the dependency on the DR parameter ε. If h is a |h|-loop 1PI Feynman graph, a general theorem [49] in DR says that U = |h| at non-exceptional momenta. We define on the group of A-valued characters G(A) a one-parameter action of C∗ t given for h homogeneous in H by: ψ t (h; ε) := t ε|h| ψ(h; ε).
(10)
ε|h|
Physically this amounts to neatly replacing the µ0 factor present in each dimensionally regularized Feynman diagram (8) by (µ0 t)ε|h| ; that is, the mass scale is changed from µ0 to tµ0 —or from µ0 to tµ0 , as the case might be. As ε is a complex variable, there is no harm in taking t complex, too. It is clear that ψ t in (10) is still a character, and that (ψ1 ∗ ψ2 )t = ψ1t ∗ ψ2t . This last property also holds if ψ1 , ψ2 in (10) belong more generally to Lin(H, A). Besides, for future use we store: ∂ ∂ t ψ t = ε|h|ψ t (h; ε) = ε Y ψ t such that t ψ t = ε Y ψ. (11) ∂t ∂t t=1 For any t and any homogeneous h ∈ H we have t ε|h| ∈ R+ (A) = A+ := C[[ε]], so that the one-parameter action on G(A) restricts to a one-parameter action on the group G + (A) : ψ ∈ G + (A) → ψ t ∈ G + (A). We now have for the regularized, but unrenormalized character γ t ∈ G(A) a BWH decomposition: −1
γ t = (γ t )− ∗ (γ t )+ . Notice that we write instead (γ− )t and (γ+ )t for the image of γ− and γ+ under the oneparameter group action. The locality property simply states that for the counterterm the following holds.
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Theorem 7.1. Let γ be a dimensionally regularized Feynman rule character. The t counterterm character in the BWH decomposition γ t = (γ t )−1 − ∗ (γ )+ satisfies: t
∂(γ t )− = 0 or (γ t )− is equal to γ− , i.e. independent of t. ∂t
(12)
We say the A-valued characters γ ∈ G(A) with this property are local characters. The physical reason for this is that the counterterms can be taken mass-independent; this goes back at least to [14]. For this fact, and more details on the physical significance of the locality property in pQFT, we refer the reader to [3, 15, 18, 21]. In the sequel, albeit mustering Dynkin technology, we partly follow the plan in reference [42]. Denote by G loc (A) the subset of local elements of G(A) and G loc − (A) the loc subset of elements in G (A) for which ψ(h) ∈ A− when h has no scalar part. Proposition 7.1. The set G loc (A) decomposes into the product G loc − (A) ∗ G + (A). Proof. Notice first that G + (A) embeds in G loc (A), since ψ t ∈ G + (A) for any ψ ∈ G + (A). Next, if φ is local and ρ ∈ G + (A), then φ ∗ ρ is local. Indeed, we have: −1
φ t ∗ ρ t = (φ t )− ∗ (φ t )+ ∗ ρ t , with polar part the one of φ t , which is constant since φ is local. Let φ still be local. Then −1 −1 , and the proposition follows if we can show φ− ∈ G loc φ ∗ φ+−1 = φ− − (A). Now, if φ is local, we have: −1
−1 t (φ− ) ∗ (φ+ )t = φ t = (φ t )− ∗ (φ t )+ ; −1 t so that the BWH decomposition of (φ− ) is given by: −1
−1 t (φ− ) = (φ t )− ∗ (φ t )+ ∗ ((φ+ )t )−1 ,
with polar part (φ t )− , the one of φ, constant and equal to φ− .
(13)
Now we wish to complete the study of locality by understanding how the decomposition of G loc (A), which is a refinement of the decomposition G(A) = G − (A) ∗ G + (A), is reflected at the Lie algebra level. More precisely, we would like to know if there is a subspace of R− ( (A)) naturally associated to G loc − (A). The answer (quite surprising at first sight) is simple and well known to physicists: beta functions are enough. As shown below, an excellent tool to understand this is the Dynkin operator pair. Let now β ∈ (C) be a scalar-valued infinitesimal character. Notice that β/ε can be regarded as an element of R− ( (A)). Proposition 7.2. With as defined in Eq. (1) of Theorem 4.1, we find: ψβ := (β/ε) ∈ G loc − (A). Proof. From Eq. (1) it is clear that : ⎛ ⎝ ψβ = (β/ε) = n
k1 ,...,kn ∈N∗
⎞ βk1 ∗ · · · ∗ βkn ⎠ 1, k1 (k1 + k2 ) . . . (k1 + · · · + kn ) εn
(14)
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implying ψβ ∈ G − (A). Next we observe ψβt = (β t /ε), which follows simply from (14) and, on use of (11): (ψβt )−1 ∗ t
∂ t −1 ψ = ε(ψβt ) ∗ Y ψβt = εψβt ◦ D = ε(β t /ε) ◦ D = β t . ∂t β
t Now, the BWH decomposition ψβt = (ψβt )−1 − ∗ (ψβ )+ is such that :
(ψβt )−1 ∗ t
∂ t ∂ ∂ −1 −1 ψβ = (ψβt )−1 ∗ t (ψβt )− ∗ (ψβt )+ + (ψβt )−1 ∗ (ψβt )− ∗ t (ψβt )+ ∂t ∂t ∂t ∂ ∂ t −1 t t −1 t t −1 = (ψβ )+ ∗ (ψβ )− ∗ t (ψβ )− ∗ (ψβ )+ + (ψβ )+ ∗ t (ψβt )+ ; ∂t ∂t
hence we find: −1
(ψβt )+ ∗ β t ∗ (ψβt )+ = (ψβt )− ∗ t
∂ ∂ −1 −1 (ψβt )− + t (ψβt )+ ∗ (ψβt )+ . ∂t ∂t
Using ψβ ∈ G − (A) and that β t takes values in A+ , we find by applying the projector R− on both sides of the last equation : ∂ ∂ R− (ψβt )+ ∗ β t ∗ (ψβt )−1 (ψ t )− ∗ (ψβt )−1 = 0 = (ψβt )− ∗ t (ψβt )−1 − = −t − , + ∂t ∂t β implying that (ψβt ) is independent of t; thus (β/ε) is a local character. −
The last proposition is suggestive of the fact that local A-valued characters are essentially determined by ordinary (scalar-valued) infinitesimal characters (beta functions). This is indeed the case. Before proving the main result of this section, we remark that, for any φ ∈ G loc (A), we can write : −1 φ t = φ− ∗ (φ t )+ = φ ∗ h tφ ,
(15)
where h tφ := (φ+ )−1 ∗ (φ t )+ ∈ G + (A). Also, for φ ∈ G loc − (A) we denote φ:−1 by Res φ. Theorem 7.2. The map φ → ε(φ ◦ D), with D the Dynkin operator, sends G loc (A) to (A+ ) and G loc − (A) to (C); explicitly, in the second case: G loc − (A) φ → ε(φ ◦ D) = Y Res φ ∈ (C). Proof. First, recall from Corollary 4.2 that D sends characters to infinitesimal characters and that by Proposition 7.1 any local φ ∈ G loc (A) decomposes as φ a ∗ φ b with b φ a ∈ G loc − (A), φ ∈ G + (A). Therefore, we see that: φ ◦ D = φ −1 ∗ Y φ = (φ b )−1 ∗ (φ a )−1 ∗ Y φ a ∗ φ b + (φ b )−1 ∗ Y φ b ; since (φ b )−1 and Y φ b belong to Lin(H, A+ ), the theorem follows if we can prove that: ε(φ ◦ D) = ε(φ −1 ∗ Y φ) ∈ (C) t t when φ ∈ G loc − (A). Assume the latter is the case and recall the decomposition φ = φ∗h φ t t in (15), with now simply h φ = (φ )+ . So that on the one hand (11) implies :
t
∂ ∂ h tφ = φ −1 ∗ t φ t = φ −1 ∗ ε Y φ = ε(φ ◦ D). ∂t t=1 ∂t t=1
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On the other hand, observe that by using the Bogoliubov formula (6) one finds:
∂ ∂ ∂ t h tφ = t (φ t )+ = t R+ φ− ∗ (φ t − e) ∂t t=1 ∂t t=1 ∂t t=1 ∂ ∂ = t R+ φ −1 ∗ φ t − φ −1 = t R+ φ −1 ∗ t ε|·| φ ∂t t=1 ∂t t=1 = R+ (ε Y φ) = Y Res φ.
(16)
(17)
In the step before the last we took into account that Y (1 H ) = 0 and that φ− ∈ G loc − (A) which implies for h ∈ H + :
where h (1) ⊗h (2) = h⊗1+1⊗h+h are both mapped to zero by t ∂t∂
(1)
(1)
(2)
φ(h
(2)
),
⊗h . Here φ −1 (h) ∈ A− and φ −1 (h R+ .
(1)
)t ε|h
φ −1 ∗ t ε|·| φ(h) = φ −1 (h) + t ε|h| φ(h) + φ −1 (h
)t ε|h
(2)
|
(2)
| φ(h (2) )
t=1
A glance back to Theorem 4.1 and Proposition 7.2 allows us to conclude that for φ ∈ G loc − (A) one has: Y Res φ = φ, (18) ε so indeed the polar part of a local character φ can be retrieved from its beta function, β(φ) := Y Res φ ∈ (C), by the universal formula (1). Equation (18) is equivalent to the ‘scattering type formula’ of [19]; we can refer to [42] for this. Now it is an easy task to recover the other results of [19, 21]. We ought to remark that, for γ a general local character, one defines Res γ = − Res γ− —see in this respect formulae [19, Eq. (11)] or [21, Eq. (2.111)]. Theorem 7.3. For the renormalized character γren (t) := (γ t )+ (ε = 0) it holds: t
∂ γren (t) = (Y Res γ ) ∗ γren (t), ∂t
(19)
the abstract RG equation. Proof. First, in the proof of Theorem 7.2 we saw already that D verifies a cocycle condition [28]: for φ, ψ ∈ G(A): (φ ∗ ψ) ◦ D = ψ −1 ∗ (φ ◦ D) ∗ ψ + ψ ◦ D. −1 This together with Theorem 7.2 implies for φ ∈ G loc − (A) that Res φ = − Res φ . Indeed, this follows by taking the residue Res on both sides of the equation:
0 = (φ −1 ∗ φ) ◦ D = φ −1 ∗ (φ −1 ◦ D) ∗ φ + φ ◦ D = φ −1 ∗
Res φ Res φ −1 ∗φ+ . ε ε
Now, ∈ G loc (A) with BWH decomposition γ t = γ−−1 ∗ (γ t )+ . Recall that (γ t )+ =
let γ ε|·| R+ γ− ∗ t γ maps H + into A+ ⊗ C[[log(t)]] such that: t
∂ ∂(γ t )+ (0) = t γren . ∂t ∂t
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As γ− ∈ G loc − (A), we then find: t
∂ ∂(γ t )+ = γ− ∗ t γ t = γ− ∗ εY γ t = γ− ∗ εY (γ−−1 ∗ (γ t )+ ) ∂t ∂t = γ− ∗ εY (γ−−1 ) ∗ (γ t )+ + εY (γ t )+ = ε(γ−−1 ◦ D) ∗ (γ t )+ + εY (γ t )+ = (Y Res γ−−1 ) ∗ (γ t )+ + εY (γ t )+ = −(Y Res γ− ) ∗ (γ t )+ + εY (γ t )+ = (Y Res γ ) ∗ (γ t )+ + εY (γ t )+ .
Therefore both sides have a limit as ε ↓ 0, yielding the sought after RG equation (19). Equation (19) is solved using the beta function β(γ ) := Y Res γ ∈ (C): γren (t) = exp(ln(t)β(γ )) ∗ γren (1). The last statement and Eq. (13) tell us that: −1 lim γ− ∗ (γ−−1 )t = lim (γ t )+ ∗ ((γ+ )t )−1 = γren (t) ∗ γren (1) = exp(ln(t)β(γ )).
ε→0
ε→0
The scalar-valued characters t (γ ) := exp(ln(t)β(γ )) ∈ G(C) obviously form a one-parameter subgroup in G(A): t1 (γ ) ∗ t2 (γ ) = t1 t2 (γ ), generated by the beta function and controlling the flow of the renormalized Feynman rule character with respect to the mass scale. 8. Through the Prism of Other Renormalization Schemes I We plan now to prospect the usefulness of our approach in other schemes of renormalization. Doubtless DR provides the cleanest formulation of locality in the BWH decomposition for renormalization. However, it is physically clear that in any scheme one has still to parameterize the arbitrariness in separating divergent from finite parts; and that the physical irrelevance of the choices made still gives rise to the RG equation. On the mathematical side, it is worth to recall that the algebraic BWH decomposition of Sect. 5 is not necessarily linked to loops on the Riemann sphere. It is thus legitimate to ponder the question in schemes other than those based on DR. We plan to exemplify with the BPHZ scheme in the next section, but, before dwelling on that, let us briefly indicate other pieces of evidence. A first one concerns old research. In the early seventies, Callan set out to prove that broken scale invariance [10] is all that renormalization was about. He was eventually able to give a treatment of the RG, and proofs of renormalizability of field theories based on the former, by relying entirely in the BPHZ formalism. To derive the beta function, he just set up RG equations by studying the dependency of the N -point functions on the renormalized mass. See in this respect [4, 11]. In a renormalization method without regularization, information about the RG must be stored somehow in the renormalized quantities. Concretely, as hinted at by our last theorem, one finds it in the scaling properties of the renormalized integral. This was noted in the context of Epstein–Glaser renormalization in [33]. In DR this is shown in the RG equation (19).
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A second piece of evidence is furnished by more recent work by Kreimer and collaborators [3,8,38,40]. Indeed, Kreimer has long argued that locality (and renormalizability) is determined by the Hochschild cohomology of renormalization Hopf algebras. This cohomology is trivial in degree greater than one. The coproduct on H can be written recursively in terms of closed Hochschild 1-cochains. Those are grafting maps indexed by primitive 1PI diagrams, that govern the structure of Feynman graphs and give rise through the Feynman rules to integral Dyson–Schwinger equations. Here is not the place for details and we refer the reader to [3, 8, 37, 38, 40], and especially Kreimer’s recent review [39]. The interaction between the Hopf algebra of characteristic functions of H of this paper and the Hochschild 1-cocycles on H is a promising field of interest. In the indicated references the Dynkin operator D (and its close cousins S ∗ Y n ) appears, again defining the residue, in renormalization schemes without regularization. There Green’s functions, = (g, p), are defined in terms of their (combinatorial) Dyson–Schwinger equations using the Hochschild 1-cocycles; g, p denote the coupling constant and external momenta, respectively. Those Green’s functions are expanded as power series in g: =1+ φ(ck )g k , k>0
for Feynman rules φ ∈ G(C) and with order by order divergent Feynman amplitudes φ(ck ) as coefficients. Here the ck ’s are particular linear combinations of graphs of loop order k in H [37]. Renormalization of is achieved by using a renormalized character φren ∈ G(C) defined by subtraction at a specific point p 2 = λ2 —corresponding to Taylor expansion up to zeroth order. Here λ plays the role of the mass-parameter µ. Locality is automatically fulfilled in this approach. The renormalized Green’s functions ren = ren (g, p,λ) can be developed in terms of the parameter L := log( p 2 /λ2 ), hence ren = 1 + k>0 αk (g)L k , with α1 (g) ∈ (C) [40]. Following the above references and adapting partially to our notation, the residue is found to be: ∂ (φren ◦ D) (C) σ1 := = α1 (g). ∂L L=0 In [40] Kreimer and Yeats outline how to derive αk (g), k > 1 recursively from α1 (g). This confirms that, in a deep sense, the beta function is composition with the Dynkin operator. 9. Through the Prism of Renormalization Schemes II Let us now explore the classical BPHZ scheme in the context of the former sections. With I F the integrand of (7) corresponding to the graph F, let us write for the Taylor subtraction employed in BPHZ renormalization: I F ( p, k) → I F ( p, k) − t ps(F) I F ( p, k) := I F ( p, k) −
|α|≤s(F)
pα ∂α I F (0, k). α!
We borrowed here the standard multi-index notation α = {α1 , . . . , αn } ∈ Nn , |α| :=
n i=1
αi , α! =
n i=1
αi ! ;
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each αi takes values between 0 and 3, say. We are assuming that only massive particles are present, otherwise the subtraction at zero momentum gives rise to infrared divergences; the expression of t ps(F) I F in practice simplifies because of Lorentz invariance. Notice that the integral J F ( p) in (7) does originally have a meaning: it is a well defined functional on the linear subspace Ss(F) (R4N ) of Schwartz functions φ on the external momenta, whose first moments p α φ( p) d 4N p up to order |α| ≤ s(F) happen to vanish. The “divergent” loop integrals inside J F ( p) become harmless when coupled exclusively with Schwartz functions of this type. The Taylor ‘jet’ projector map t lp subtracts polynomials, that are functionals of the same type, in such a way that the result (eventually) becomes moreover a tempered distribution. The first question is whether we have a Rota–Baxter algebra in the BPHZ context. Actually, we do have the Rota–Baxter property for t lp . Indeed, the following is obtained by a simple calculation from the definitions. Proposition 9.1. Let I Fi , i = 1, 2 have associated degrees of divergence li , i = 1, 2. Then
t lp11 I F1 t lp22 I F2 = t lp11+l, p22 I F1 t lp22 (I F2 ) + t lp11+l, p22 t lp11 (I F1 ) I F2 − t lp11+l, p22 I F1 I F2 . We leave the verification of this to the care of the reader. In general, if U is a multiplicative semigroup, a family of linear operators Ru , u ∈ U on the algebra A is called a Rota–Baxter family if for any u, v ∈ U and a, b ∈ A, we have Ru (a)Rv (b) = Ruv (a Rv (b)) + Ruv (Ru (a)b) − Ruv (ab),
for all a, b ∈ A.
Thus the l-jets define a Rota–Baxter family. Now, a Rota–Baxter family is almost the same thing as a Rota–Baxter operator. Proposition 9.2. Let A = A[U ] be the semigroup algebra associated to A. Let Ru : A → A, u ∈ U be a Rota–Baxter family. Define au u := Ru (au )u. R : A → A, by R u
u
Then R is a Rota–Baxter operator on A such that R(au) = a u with a in A. Conversely, if R : A → A is a Rota–Baxter operator such that R(au) = a u with a in A, then we obtain a Rota–Baxter family Ru , u ∈ U , by defining Ru (a) = a , where R(a u) = a u. The proof is immediate1 . On the strength of the previous result, we may refer to a Rota–Baxter family as a Rota–Baxter operator. Now, pQFT in practice furnishes an even more radical answer to the question of whether one has here the Rota–Baxter framework. For this is obviously the case when one deals only with logarithmic divergences; and indeed most often only the latter is required. In general, differentiation of an amplitude with respect to an external momentum lowers the overall degree of divergence of a diagram. In DR, the Caswell–Kennedy theorem [13] states that the pole part of any diagram, besides being independent of the scale, is a polynomial in the external momentum. This follows easily from that derivation and the projector R− commute in DR. But even in the BPHZ scheme ∂ p t l = t l−1 ∂ p , and this is enough for the differentiation trick to work. 1 We thank L. Guo for suggesting the notion of Rota–Baxter family.
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Let us then consider the J F ( p) ∈ Ss(F) (R4N ) of (7). Suppose moreover the multi-loop divergent graph F has all its 1PI subgraphs γ made convergent by application ¯ Then the renormalized integral J ren,BPHZ ( p) can of Bogoliubov’s preparation map R. F be defined as
J Fren,BPHZ ( p) =
|F| l=1
=:
|F|
d 4 kl
I F ( p, k) − t ps(F) R¯ I F ( p, k)
d 4 kl R F ( p, k).
(20)
l=1
This recipe is however not unique. We can write as well J Fren,BPHZ ( p) = P s(F) ( p) +
|F|
d 4 kl R F ( p, k),
(21)
l=1
with P s(F) a polynomial of order s(F) in the external momenta. This effects a ‘finite renormalization’, in principle undetermined, that might be put to use to fulfill renormalization prescriptions (again, the form of that polynomial is severely restricted by Lorentz invariance). We now come to the key point. The coefficients of P s(F) in (21) exhibit the ambiguity of renormalization in the context of the BPHZ scheme. On the face of it, the ‘pole terms’ t ps(F) I F ( p, k) do not depend at all on the mentioned coefficients, and thus locality of the BWH decomposition is guaranteed, in a trivial sense. On the other hand, the Galois group approach to renormalization theory [20,21] stems originally from the idea that ambiguities should be, insofar as possible, handled from a group-theoretic point of view, much as classical Galois theory handles the multiple solutions of polynomial equations. Here however the mentioned form of the ambiguity does not apparently lend itself to RG analysis. We contend, however, that the ambiguity is expressed essentially in the same form as before. The Caswell–Kennedy theorem is suggestive of a direct link between the DR and BPHZ formalisms, and next we endeavour to prove the pertinence of the RG to BPHZ renormalization by the most direct possible route: introducing a mass scale in the latter formalism in direct analogy with the former. To express the ambiguity implicit in the P s(F) of (21) in terms of a mass scale, we use the modified BPHZ scheme proposed in [29]. For instance, it is well known that the (‘fish’ graph) giving the first nontrivial contribution to the vertex famous graph correction in the ϕ44 model in the Euclidean yields the amplitude DR Jfish ( p)
= g˜ µ
2 2ε
1 1 d Dk , (2π )4 k 2 + m 2 ( p + k)2 + m 2
where p = p1 + p2 , say, and that, by use of Feynman’s parametrization (see below) and relation (9) one obtains DR Jfish ( p) = g
1 2 µ2 g˜ + dz log + O(ε) . (4π )2 ε p 2 z(1 − z) + m 2 0
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543
Now, the natural ‘zero point’ for the mass scale in this problem is clearly m, and we note DR ( p = 0; µ = m) = 0, R+ Jfish as ε ↓ 0. This, together with the mentioned Caswell–Kennedy theorem, feeds the susren,BPHZ picion that the last expression is just the Jfish ( p) of (20). The suspicion is correct. The computation required for the renormalized fish graph in the BPHZ scheme is 1 1 d 4k 0 . (22) g2 (1 − t ) p (2π )4 k 2 + m 2 ( p + k)2 + m 2 Introduce the Feynman trick, prior to the Taylor subtraction, 1 d 4k 1 2 g dz (1 − t p0 ) ! "2 4 2 2 (2π ) 0 (( p + k) + m )z + (1 − z)(k 2 + m 2 ) 1 d 4k 1 2 =g dz (1 − t p0 ) 2 . 4 2 (2π ) [k + p z(1 − z) + m 2 ]2 0
(23)
The translation k → k − zp, depending on the Feynman parameter, has been made in order to obtain here the same denominator as in DR calculations. With 4 the area of the unit sphere in R4 , the integral (23) now becomes ∞ 4 g 2 1 k3 k3 dz dk − 2 (2π )4 0 [k 2 + p 2 z(1 − z) + m 2 ]2 [k + m 2 ]2 0 1 2 2 m g dz log 2 . = 2 16π 0 p z(1 − z) + m 2 The last step is to convert the p-independent part in the argument of the logarithm into a mass scale: m → µ. With this, we recover on the nose the DR result, in the MS scheme as it turns out. Incidentally, as remarked in [46], this procedure us to give allows the exact 2 1 value of the BPHZ integral (22): the expression 0 dz log 1 + mp 2 z(1 − z) is actually well known in statistical physics, and leads by elementary manipulations involving the golden ratio to ⎛# ⎞ $ 2 2 2 / p2 + 1 4m g 4m ren,BPHZ ⎝ 1+ Jfish ( p) = − log $ − 2⎠ . 16π 2 p2 4m 2 / p 2 − 1 Thus, what we have done above amounts to identify the constant term —recall P s(F) ( p) in (21). the term g 2 /16π 2 times
2 We2 have the right to add to the previous expression ren,BPHZ 2 log µ /m . We note also that one can recover the residue g /8π 2 here from Jfish , as the coefficient of the term logarithmic in the scaling factor, ren,BPHZ ren,BPHZ Jfish (λp) ∼ Jfish ( p) −
g2 log λ, 8π 2
as λ ↑ ∞. The steps of the modified BPHZ procedure are: (i) Introduction of the Feynman parametrization in J FBPHZ (k, p). (ii) Exchange of the integrations. (iii) Translation of
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the integration variables by λp, with λ dependent on the Feynman parameter. (iv) Taylor subtraction. (v) Integration over loops and replacement of the mass m in the p-constant part of the resulting logarithm by a mass scale. There is nothing to forbid the same operations to be performed on any primitive logarithmically divergent graph of any field theory and then we are optimistic that, by use of skeletal expansions and the integral equations, we would be led to a procedure largely parallel to DR, and so to a brute-force proof that the coefficients of the higher powers of the scaling logarithms in BPHZ renormalization are determined by the residues. To verify this with full particulars, however, would take us too far afield. Recapitulating, the Lie-theoretic method shows promise in dealing with renormalization schemes other than DR with MS. The presence of a Rota–Baxter structure is a requisite for the validity of such a framework; it obviously holds for the MS prescription in DR as well. What of other renormalization methods? For massive fields, the BPHZ scheme does verify the required conditions. We have learned, however, that the details are very idiosyncratic: as stated above, locality is moot; and the beta function enters the picture through Theorem 7.3, referring to renormalized quantities, rather than to counterterms. For massless fields, the price of the Rota–Baxter property is relinquishing Lorentz invariance, and this is too heavy to pay. The Taylor subtraction in Epstein–Glaser renormalization has roughly the same properties as the BPHZ scheme, both in regard to massive and massless fields; nevertheless, there one is confronted with the problems of good definition that plague the attempts [2, 32]. Procedures based on analytic renormalization or Hadamard regularization [33] have not been investigated yet from the Rota–Baxter viewpoint. Thus it is too early in the game to draw a list of known schemes that would definitely fit in our approach; we plan to come to this in future work. It is intriguing that the case study of BPHZ renormalization points out the pertinence of the MS prescription in DR. 10. On Connes–Marcolli’s Motivic Galois Theory In the Connes–Kreimer picture of renormalization, group theory and the schemetheoretic equivalent theory of commutative Hopf algebras have become a fundamental tool of pQFT. Connes and Marcolli identified recently [20] a new level at which Hopf algebra structures enter pQFT. Namely, they constructed an affine group scheme U ∗ , universal with respect to physical theories, and pointed out its analogies with number theory, e.g. with the motivic Galois group of the scheme of 4-cyclotomic integers Z[i][ 21 ]. In their work the initial physical problem attacked through the Connes–Kreimer paradigm translates into the classification of equisingular G-valued flat connections on the total space of a principal bundle over an infinitesimal punctured disk (with G the group scheme represented by H ). From the representation theoretic point of view, the classification is provided by representations U ∗ −→ G ∗ , where U ∗ is the semi-direct product with the grading of the pro-unipotent group U , the Lie algebra of which is the free graded Lie algebra with one generator en in each degree n > 0, and similarly for G ∗ . Returning to the geometrical side of the correspondence and featuring the DR setting that leads to the Riemann–Hilbert picture of renormalization, Connes and Marcolli construct a universal singular frame on principal U -bundles over B. A formal expression for it is given by: γ (ε, v) =
n≥0 k j
e(k1 ) · · · e(kn ) v k j ε−n . k1 (k1 + k2 ) · · · (k1 + · · · + kn )
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As already remarked in [20] and our introduction, it is interesting that the coefficients of the frame are essentially those appearing in the index formula of Connes–Moscovici; this would hint at the rooting of noncommutative geometry in quantum field theory, which has been Connes’ contention for a long while. We have already shown that other Hopf algebra structures (or, from the schemetheoretic point of view, pro-unipotent groups) do appear naturally in pQFT, namely the Hopf algebras Char(A) of characteristic functions associated to commutative target algebras, e.g., although not exclusively, of quantum amplitudes. These Hopf algebra structures arise naturally from algebraic-combinatorial constructions on Hopf algebras, and therefore do not immediately relate to the geometrical-arithmetical constructions underlying the definition of the motivic Galois group in [20]. Nevertheless, the formula rendering the universal singular frame in the motivic understanding of renormalization also essentially coincides with our map —the inverse of the Dynkin map. This indicates that the practical consequences for renormalization of the Riemann-Hilbert and/or motivic point of view can be translated to the setting of FLA theory —which, besides being more familiar to many, is independent of the geometry embedded in the DR scheme. As it turns out, the pro-unipotent groups/Hopf algebras Char(H ) and Char(A) are related naturally to the group U . In the remainder of the present section, we would like to make explicit how both viewpoints connect —although the reasons behind this connection certainly ought to be deepened. Let us recall a few general facts from the theory of Solomon algebras —see [44,47] for details. Let σ be a permutation in the symmetric group Sn of order n. The descent set D(σ ) of σ is the set D(σ ) := {i, σ (i) > σ (i + 1)}. Note n ∈ / D(σ ). The descent composition C(σ ) of σ is the composition of n (that is, the sequence (c1 , . . . , ck ) of strictly positive integers of total sum n) such that, when viewed as a word, σ = σ (1) . . . σ (n) can be written u 1 . . . u k , where each word u i is increasing and of length ci , and where k is minimal. For example, D(21534) = {1, 3} and C(21534) = (1, 2, 2). The notions of descent set and descent composition are obviously equivalent, the equivalence being induced by the map: (c1 , . . . , ck ) −→ {c1 , c1 + c2 , . . . , c1 + · · · + ck−1 }. The Solomon algebra n of type An was first introduced as a noncommutative lift to the group algebra of the representation ring of Sn [50]. As a vector space, n is the linear span of the elements D⊆S in Q[Sn ], where S runs over subsets of [n − 1] and D⊆S := σ. σ ∈Sn D(σ )⊆S
Then Solomon’s fundamental theorem states that n is closed under the composition product in Sn . ∗ Now, ∗let X be an infinite alphabet. The dual graded Hopf algebra T (X ) = n∈N Tn (X ) of T (X ) is graded connected commutative, with the shuffle product as the algebra product and the deconcatenation coproduct: xi1 · · · xin −→
n
xi1 · · · xik ⊗ xik+1 · · · xin ,
k=0
where we view xi1 · · · xin as an element of T ∗ (X ) using the usual pairing xi1 · · · xin | x j ···x j x j1 · · · x jk = δxi11···xink . The symmetric group of order n embeds into End(Tn∗ (X )) ⊂
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K. Ebrahimi-Fard, J. M. Gracia-Bondía, F. Patras
End(T ∗ (X )): σ (xi1 · · · xin ) := xiσ −1 (1) · · · xiσ −1 (n) . This map induces an embedding of algebras of n into End(T ∗ (X )), where the product on the latter algebra is the composition of maps. Let us write now D for the descent algebra of T ∗ (X ), that is the convolution subalgebra of End(T ∗ (X )) generated by the projections pn : T ∗ (X ) −→ Tn∗ (X ) on the graded components of T ∗ (X ). The algebra D is naturally graded and provided with a Hopf algebra structure for which the pn form a sequence of divided powers: pi ⊗ p j .
( pn ) = i+ j=n
We write Dn for the component of degree n. Lemma 10.1. The convolution algebra D is also closed under the composition of maps ◦ in End(T ∗ (X )). The result follows from Corollary 9.4 in [47] (where the dual setting is considered, that is, the convolution subalgebra of End(T (X )) generated by the graded projections in T (X )) and also follows directly from [44, Thm II.7]. Proposition 10.1. The embedding of n into (End(T ∗ (X )), ◦) induces an isomorphism of algebras n −→ Dn . The proof follows from Corollary 9.2 in [47] by duality. It basically amounts to observe that, if C(σ ) = (c1 , . . . , ck ), then σ −1 is a (c1 , . . . , ck )-shuffle. For example, if σ = (21534) then C(σ ) = (1, 2, 2) and σ −1 = (21453), so that the word σ (x1 · · · x5 ) = x2 x1 x4 x5 x3 is a shuffle of x1 , x2 x3 and x4 x5 , and appears therefore in the expansion of p1 ∗ p2 ∗ p2 (x1 · · · x5 ). Proposition 10.2. The algebra D is freely generated as an associative algebra by the graded projections pn . Equivalently, it is freely generated by the graded components pn ◦ D of the Dynkin idempotent D = S ∗ Y , regarded as an element of End(T ∗ (X )). The first part of the proof of this proposition is Corollary 9.14 of [47] (stated for , that is, in the dual setting). The second assertion is found e.g. in [31, Sect. 5.2]. Corollary 10.1. Regarded as a pro-unipotent group scheme, the graded dual Hopf algebra D∗ is canonically isomorphic to the ring of coordinates of the Connes–Marcolli group U of renormalization theory. Through this correspondence, by our Lemma 4.1, the coefficients of the universal singular frame are reflected in the coefficients of the expansion of the identity of T ∗ (X ) on the natural linear basis of D viewed as the free associative algebra generated by the graded components of the Dynkin operator. Now, the universal properties of the Galois group U for renormalization, when the group is understood by means of D, follow from the constructions in [44], where it is shown that the descent algebra is an algebra of natural (endo)transformations of the forgetful functor from graded connected commutative Hopf algebras to graded vector spaces —that is, an universal endomorphism algebra
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for graded connected commutative Hopf algebras. In other terms, there is a natural map from D to End(H ), where H is an arbitrary graded connected commutative Hopf algebra. Using the arguments developed in the first sections of this article, one shows easily that this map factorizes through an Hopf algebra map from D to Char(H ); this follows e.g. from the fact that the graded projections generate D as a convolution algebra and form a sequence of divided powers both in D ⊂ End(T ∗ (X )) and in Char(H ). In summary, Corollary 10.2. The descent algebra D acts naturally by right composition on Lin(H, A). Moreover, the group of group-like elements in D acts naturally on the group G(A) of Feynman rules. The second part of the corollary follows from the third identity in Lemma 3.1. Besides providing a natural link between the Galoisian approach to renormalization and the noncommutative representation theory of the symmetric groups [5], the combinatorial approach implies moreover that the Connes–Marcolli universal Galois group U inherits the very rich structure of the descent algebra. The appearance of the descent algebra (or equivalently of the Hopf algebras of noncommutative symmetric functions and quasi-symmetric functions, see [31]), beyond broadening the scope of the mathematical theory of renormalization, should result into new developments in the field, possibly complementary with the arithmetic ones. Acknowledgements. The first named author2 acknowledges greatly the support by the European PostDoctoral Institute and the Institut des Hautes Études Scientifiques (I.H.É.S.). He is also indebted to Laboratoire J. A. Dieudonné at Université de Nice Sophia-Antipolis for warm hospitality. He is very grateful to C. Bergbauer for useful discussions. The second named author acknowledges partial support from CICyT, Spain, through grant FIS2005-02309. He is also grateful for the hospitality of Laboratoire J. A. Dieudonné. The present work received support from the ANR grant AHBE 05-42234. We are pleased to thank the anonymous referee, whose comments prompted us to clarify several aspects of the paper.
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12. Cartier, P.: A primer on Hopf algebras. IHES preprint, August 2006, available at http://www.ihes.fr/ PREPRINTS/2006/M/M-06-40.pdf 13. Caswell, W.E., Kennedy, A.D.: Simple approach to renormalization theory. Phys. Rev. D 25, 392–408 (1982) 14. Collins, J.C.: Structure of the counterterms in dimensional regularization. Nucl. Phys. B 80, 341–348 (1974) 15. Collins, J.C.: Renormalization. Cambridge: Cambridge University Press, 1984 16. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Func. Anal. 5, 174–243 (1995) 17. Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199, 203–242 (1998) 18. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann–Hilbert problem I. The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249–273 (2000) 19. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann–Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216, 215–241 (2001) 20. Connes, A., Marcolli, M.: Renormalization and motivic Galois theory. Internat. Math. Res. Notices 2004(76), 4073–4091 (2004) 21. Connes, A., Marcolli, M.: From Physics to Number Theory via Noncommutative Geometry II: Renormalization, the Riemann–Hilbert correspondence, and motivic Galois theory. In: Frontiers in Number Theory, Physics and Geometry. Berlin Heidelberg-New York: Springer, 2006, p. 269 22. Connes, A., Marcolli, M.: Quantum Fields and Motives. J. Geom. Phys. 56, 55–85 (2006) 23. Ebrahimi-Fard, K., Guo, L., Kreimer, D.: Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT. J. Phys. A 37, 11037–11052 (2004) 24. Ebrahimi-Fard, K., Guo, L., Kreimer, D.: Integrable Renormalization II: the General case. Ann. H. Poincaré 6, 369–395 (2005) 25. Ebrahimi-Fard, K., Kreimer, D.: Hopf algebra approach to Feynman diagram calculations. J. Phys. A 38, R385–R406 (2005) 26. Ebrahimi-Fard, K., Gracia-Bondía, J.M., Guo, L., Várilly, J.C.: Combinatorics of renormalization as matrix calculus. Phys. Lett. B 632, 552–558 (2006) 27. Ebrahimi-Fard, K., Guo, L., Manchon, D.: Birkhoff type decompositions and the Baker–Campbell–Hausdorff recursion. Commun. Math. Phys. 267, 821–845 (2006) 28. Ebrahimi-Fard, K., Manchon, D.: On matrix differential equations in the Hopf algebra of renormalization. Adv. Theor. Math. Phys. 10, 879–913 (2006) 29. Falk, S.: Doktor der Naturwissenschaften Dissertation, Mainz, 2005 30. Figueroa, H., Gracia-Bondía, J.M.: Combinatorial Hopf algebras in quantum field theory I. Rev. of Math. Phys. 17, 881–976 (2005) 31. Gelfand, I.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112, 218–348 (1995) 32. Gracia-Bondía, J.M., Lazzarini, S.: Connes–Kreimer–Epstein–Glaser renormalization. http://arxive.org/ list/hep-th/0006106, 2006 33. Gracia-Bondía, J.M.: Improved Epstein–Glaser renormalization in coordinate space I. Euclidean Framework. Math. Phys. Anal. Geom. 6, 59–88 (2003) 34. Hazewinkel, M.: Hopf algebras of endomorphisms of Hopf algebras. http://arxive.org/list/math.QA/ 0410364, 2004 35. Kleinert, H., Schulte-Frohlinde, V.: Critical Properties of φ 4 -theories. Singapore: World Scientific, 2001 36. Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2, 303–334 (1998) 37. Kreimer, D.: Anatomy of a gauge theory. Annals Phys. 321, 2757–2781 (2006) 38. Kreimer, D.: Étude for linear Dyson–Schwinger Equations. IHES preprint, March 2006, available at http://www.hes.fr/PREPRINTS/2006/p/p-06-23.pdf 39. Kreimer, D.: Dyson–Schwinger Equations: from Hopf algebras to number theory. Fields Institute communications 50, 225–248 (2007) 40. Kreimer D., Yeats K.: An Étude in non-linear Dyson–Schwinger Equations. Nucl. Phys. Proc. Suppl. 160, 116–121, 2006. hep-th/0605096 41. Lowenstein, J.H.: BPHZ renormalization. In: Renormalization theory (Proceedings NATO Advanced Study Institute, Erice, 1975), NATO Advanced Study Institute Series C: Math. and Phys. Sci., Vol. 23, Dordrecht:Reidel, 1976 42. Manchon, D.: Hopf algebras, from basics to applications to renormalization. Comptes-rendus des Rencontres mathématiques de Glanon 2001, http://arxive.org/list/math.QA/0408405, 2009 43. Patras, F.: La décomposition en poids des algèbres de Hopf. Ann. Inst. Fourier 43, 1067–1087 (1993)
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Commun. Math. Phys. 276, 551–569 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0348-6
Communications in
Mathematical Physics
A Simple Proof of Gopakumar-Vafa Conjecture for Local Toric Calabi-Yau Manifolds Pan Peng Mathematics Department, Harvard University, One Oxford Street, Cambridge, MA, 02138, USA. E-mail: [email protected] Received: 25 September 2006 / Accepted: 23 April 2007 Published online: 2 October 2007 – © Springer-Verlag 2007
Abstract: We prove Gopakumar-Vafa conjecture for local toric Calabi-Yau manifolds. It is also proven that the local Gopakumar-Vafa invariants of a given class vanish at large genera. 1. Introduction 1.1. Gopakumar-Vafa conjecture. Let X be a smooth projective variety. For any given β ∈ H2 (X, Z), denote M g,n (X, β)1 to be the moduli space of stable maps from genus g Riemann surface with n marked points to X with image staying in the class β. By Riemann-Roch, dimC [M g,n (X, β)]vir = c1 (T X ) · β + (dimC X − 3)(1 − g) + n. Assume X is a Calabi-Yau threefold. If n = 0, by Calabi-Yau condition and dimC X = 3, dimC [M g (X, β)]vir = 0. This leads to the following definition of the Gromov-Witten invariant: g 1. K β (X ) = [M g (X,β)]vir
In general, M g (X, β) is an orbiford or even a stack. Therefore, Gromov-Witten invariants are in fact rational numbers. From the compactification of M-theory on Calabi-Yau threefold, Gopakumar and Vafa [6, 7] conjectured the following remarkable identity. 1 If there is no marked point involved, namely n = 0, we simply write M (X, β). g
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Conjecture (Gopakumar-Vafa). For the Calabi-Yau threefold X , there exist integral g invariants Nβ such that ∞ β=0 g=0
K β t 2g−2 e−ω·β = g
g ∞ ∞ 2g−2 Nβ 1/2 q − q −1/2 e−kω·β . k
(1.1)
β=0 g=0 k=1
Here ω is the Kähler class, β is taken in H2 (X, Z) and q = e
√
−1t .
g
Nβ are called Gopakumar-Vafa invariants. Gopakumar-Vafa formula (1.1) gives a recursive way of computing these integral invariants as combinations of Gromov-Witten invariants with rational coefficients. Therefore, Gopakumar-Vafa invariants are a priori rational numbers by the definition of Gromov-Witten invariants. So far, there have been several attempts to define Gopakumar-Vafa invariants mathematically. However, none of them has been successful. On the other hand, (1.1) can serve as a definition of Gopakumar-Vafa invariants. Since then, the Gopakumar-Vafa conjecture has become one of the major sources of development in Gromov-Witten theory. 1.2. Main result. In this paper, we will study Gopakumar-Vafa conjecture for local Calabi-Yau geometry. Local Calabi-Yau geometry studies a nonsingular, projective, toric, Fano surface sitting inside a Calabi-Yau threefold. Let S be such a surface. By the Calabi-Yau condition and adjunction formula, it’s not difficult to see that the tubular neighborhood of S in the Calabi-Yau threefold is its canonical bundle K S → S. The Gromov-Witten theory of local Calabi-Yau geometry of S is defined by excess integral [3]. Given β ∈ H2 (X, Z), we have the following diagram: π
ev
M g (X, β) ←− M g,1 (X, β) −→ X, where π forgets the marked point and ev is the evaluation of the stable map at the marked point. The local Gromov-Witten invariant of S can be defined by g K β (S) = ctop (R 1 π∗ ev ∗ K S ), (1.2) [M g (S,β)]vir g
where ctop is the top Chern class. Local Gopakumar-Vafa invariants of S, Nβ (S), will be defined similarly as in (1.1). We will prove the following theorem in this paper: g
Theorem 1. Let S be a smooth, projective, toric, Fano surface and Nβ (S) be its local Gopakumar-Vafa invariants. We have Pβ (x)
∞
g
Nβ (S)x g ∈ Z[x].
(1.3)
g=0
Remark 1.1. The above theorem answered the Gopakumar-Vafa conjecture in the case of local toric Calabi-Yau threefolds. Furthermore, it implies that local GopakumarVafa invariants vanish at large genera. This agrees with the original interpretation of Gopakumar-Vafa invariants in physics.
Simple Proof of Gopakumar-Vafa Conjecture for Local Toric C-Y Manifolds
553
We also discussed some property of this polynomial Pβ for local P2 and Hirzebruch surfaces Fk : The degree of Pβ is the arithmetic genus of a curve representing class β, and the leading coefficient of Pβ is the Euler Characteristics of the holomorphic line bundle corresponding to the divisor β in S except for a possible negative sign determined by β. These examples coincide with some original guesses on Gopakumar-Vafa invariants in mathematics that they might be related to the counting of embedded curves. 1.3. Rough idea of the proof. We start by studying the topological string partition function, namely the exponential of the generating function of the left-hand side of (1.1) based on the two-partition Hodge integral theory developed in [13]. Torus action (C ∗ )2 can extend to the moduli space M g (S, β). The localization technique used on this moduli space leads to a summation of contributions from the fixed points which are certain two-partition Hodge integrals. These two-partition Hodge integrals are related to the Chern-Simons invariants of Hopf link by a convolution formula based on large N duality between Chern-Simons theory and open topological string theory. Thus we will get a closed formula for the topological partition function of local Calabi-Yau threefold K S → S. However, the integrality of Gopakumar-Vafa invariants is still not clear. One important observation comes from the multi-cover contribution. The terms in the topological partition function containing 1/ p factors of degree β have an interesting correspondence with degree β/ p. We then combined p-adic arguments to prove the cancellation of the g 1/2 1/ p factors. At the same time, we show that ∞ − q −1/2 )2g has at most g=0 Nβ (q pole singularity at q = 0 by the computation of Chern-Simons invariants of the Hopf link. This implies that it is a polynomial of (q 1/2 − q −1/2 )2 . Combined with the p-adic argument, we obtain the proof of Theorem 1. Recently, in [12], topological string amplitudes of certain Fano surfaces are regarded as equivariant indices. The Gopakumar-Vafa conjecture is translated into a formula of an infinite product, where the integrality follows. However, the generalization to any local toric Calabi-Yau threefold is not clear so far from their methods. The rest of this paper is organized as follows. We introduce some notations on partition and prove a generalized Möbius formula in Sect. 2. We discuss some results on ChernSimons invariants of Hopf link in Sect. 3. In Sect. 4, we give the formula of topological string partition for local toric Calabi-Yau. Our main result is proved in Sect. 5. Some examples are discussed in Sect. 6. 2. Preliminaries 2.1. Partitions. A partition is a sequence of non-negative integers λ = (λ1 , λ2 , . . . , λr , . . . ) such that
λ1 ≥ λ2 ≥ · · · ≥ λr ≥ · · ·
and containing only finitely many non-zero terms. The degree of λ is defined by |λ| = λ1 + λ2 + · · · . If |λ| = d, we also write it as λ d. The length of λ is defined by l(λ) := Card{ j : λ j = 0}. m i (λ) = Card{ j : λ j = i} is called the multiplicity of i in λ. The transposition
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of λ, denoted by λt = (λt1 , λt2 , . . . ), is defined by λtj = Card{i|λi ≥ j}. Denote by P the set of all partitions. Let c(x) = j − i,
h(x) = λi + λtj − i − j + 1,
n(λ) =
(i − 1)λi . i
For a partition λ, the automorphism group of λ, Aut λ, contains all the permutations of parts of λ which keep λ. Its order is given by |Autλ| =
m i (λ)!.
i≥1
The following notations will be used throughout this paper: l(λ)! , |Autλ|
uλ = kλ =
z λ = |Autλ| ·
λi (λi − 2i + 1),
θλ =
λi , i≥1 (−1)l(λ)−1 (l(λ) − 1)!
i
|Autλ|
,
and we define u (0) = θ(0) = z (0) = 1. Note that kλ = −kλt and kλ is even. 2.2. Infinite series and partition. Given a sequence of variables x = (x1 , x2 , . . . , xn , . . . ) and a partition λ, define xλ = i≥1 xλi , x(0) ≡ 1. Lemma 2.1. Let f (t) =
n.
n≥0 an t
⎛ f⎝
We have the following expansion: ⎞
xi ⎠ =
i≥1
Proof. Note that
i≥1 x i
d
=
l(λ)=d
al(λ) xλ u λ .
λ∈P
xλ u λ .
Apply the above lemma to exp(x) and log(x). Corollary 2.2. We have ⎛ ⎞ bλ x |λ| , exp ⎝ bn x n ⎠ = 1 + |Autλ| n≥1 |λ|≥1 ⎛ ⎞ log ⎝1 + bn x n ⎠ = bλ x λ θλ . n≥1
|λ|≥1
(2.1)
(2.2)
Simple Proof of Gopakumar-Vafa Conjecture for Local Toric C-Y Manifolds
555
2.3. Basis of symmetric functions. Let x = (x1 , x2 , . . . ) be a sequence of variables. Given any partition λ, define Per (λ) = {δ : δ is a permutation of λ}. Define the following symmetric functions: mλ = xδ, pn = m (n) ,
hn =
i
mλ.
λ n
δ∈Per (λ)
Let pλ = follows:
pλi , h λ =
i
h λi and p0 = h 0 = 1. Schur function can be defined as
Sλ = det(h λi −i+ j )1≤i, j≤l(λ) . The following formula is well-known: H (x, t) = hr t r = (1 − xi t)−1 . r ≥0
(2.3)
i
2.4. A generalized Möbius inversion formula. Let µ(n) be the Möbius function. That is: (−1)r , n is a product of r distinct prime numbers; µ(n) = 0, otherwise. Lemma 2.3.
k|n
µ(k) = δ1n .
d piri , p1 , . . . , pd are distinct prime numbers Proof. n = 1, trivial. If n > 1, say n = i=1 and ri > 0, n n (−1)i = (1 − 1)n = 0. µ(k) = i k|n
The proof is completed.
i=0
Let β = (k1 , k2 , . . . , kn ) ∈ Zn≥0 . Denote by n|β if n|ki , ∀1 ≤ i ≤ n. For any r ∈ Q, let rβ (r · k1 , r · k2 , . . . , r · kn ) ∈ Qn . Theorem 2.4. Let f , g be two functions defined on Zn≥0 × Z≥0 . α : N → N satisfies α(n)α(m) = α(nm). If β , nm = g(β, m) (2.4) α(n) f n n|β
holds for any β ∈
Zn≥0 ,
m ∈ Z≥0 , then we have f (β, m) =
n|β
µ(n)α(n)g
β , nm . n
(2.5)
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Proof. α(n)α(m) = α(nm) implies α(1) = 1. Note that f (β, m) is uniquely determined by g(β, m). It is thus sufficient to show (2.5) is a solution of (2.4), β β , nm = , knm α(n) f α(n) µ(k)α(k)g n kn β n|β
n|β
=
k| n
α(n)α(k)µ(k)g
kn|d
= g(β, m) +
= g(β, m) +
α(nk)µ(k)g
p>1 kn= p
β , knm kn
α( p)g
p>1
β , knm kn
β , pm µ(k) p k| p
= g(β, m). In the last step, we’ve used Lemma 2.3.
3. Chern-Simons Invariants of Hopf Link Define: Wλ (q) = q
kλ 4
l(λ) λ
1≤i< j≤d
and
i [λi − λ j + j − i] 1 [ j − i] [v − i + l(λ)]
i=1 v=1
1
3
1
Wλµ (q) = Wλ (q)Sµ (q λ1 − 2 , q λ2 − 2 , . . . , q λn −n+ 2 , . . . ).
Define [m] = q m/2 − q −m/2 ; Wλ (q) can be simplified as kλ 1 Wλ (q) = q 4 [h(x)] x∈λ 1
3
1
= Sλ (q − 2 , q − 2 , . . . , q −n+ 2 , . . . ). Hence where
Wλµ (q) = q − qα
=
(1, q −1 , q −2 , · · · ).
λ+µ 2
Sλ (q α )Sµ (q λ+α ),
(3.1)
It is proven in [17] that
Wλt µt (q) = (−1)|λ|+|µ| Wλµ (q −1 ).
(3.2)
Lemma 3.1. f (q) ∈ Z[q, q −1 ], f (q) = f (q −1 ), then f (q) ∈ Z[[1]2 ]. Proof. f (q) = f (q −1 ), and f ∈ Z[q, q −1 ]. f can be written as N
an (q n + q −n )
n=0
for some N . It’s sufficient to show ∀n, q n + q −n ∈ Z[[1]2 ]. When n = 1, q 1 + q −1 = [1]2 + 2. Now consider n = k. If k odd, q n + q −n = n −n (q + q −1 )(q k−1 + q k−2 + · · · + q −(n−1) ); if k even, q n + q −n = (q 2 + q 2 )2 − 2. So by induction, the proof is completed.
Simple Proof of Gopakumar-Vafa Conjecture for Local Toric C-Y Manifolds
557
By (2.3), ∞
H (q α , t) =
(1 − q −i+1 t)−1
i=1 ∞
=
r
r (r +1) −i 2 i=1 (1 − q )q r tr . 2 [i] i=1
r =0
Therefore, r
α
r (r +1) −i 2 i=1 (1 − q )q r 2 i=1 [i]
h r (q ) =
.
(3.3)
Using the results of H (q α , t), one can calculate H (q λ+α , t) in the following way: H (q λ+α , t) =
∞
(1 − q λi −i+1 t)−1
i=1
⎛ ⎞ l(λ) −i+1 1−q t ⎠ =⎝ · H (q α , t). 1 − q λi −i+1 t i=1
Note that one can expand (1 − q λi −i+1 t)−1 and get the coefficient of t j , which is an element of Z[q, q −1 ]. After collecting the t r terms in the above formula, we have ρ(q) h r (q λ+α ) = r . 2 i=1 [i]
(3.4)
Here ρ(q) ∈ Z[q, q −1 ]. Recall Sλ = det(h λi −i+ j )1≤i, j≤l(λ) . By Lemma 3.1, we know [n]2 is a polynomial of [1]2 with integer coefficients. Combine (3.3) and (3.4); we obtain the following lemma: Lemma 3.2. Sλ (q α ), Sµ (q λ+α ) are of form Z[q, q −1 ].
a(q) , b([1]2 )
where b(x) ∈ Z[x] is monic, a(q) ∈
Let R = (R 1 , R 2 , . . . , R N ) ∈ P N and γ = (γ1 , γ2 , . . . , γ N ) ∈ Z N . The following quantity2 W R,γ = W R N R 1 W R 1 R 2 W R 2 R 3 · · · W R N −1 R N (−1)
N
i=1 γi |R
i|
1
q2
N
i=1 k R i γi
= will be used in the later computation of topological partition function. Denote by R N i | and R t = ((R 1 )t , (R 2 )t , . . . , (R N )t ). Let R N +1 = R 1 ; we rewrite W as |R i=1 R,γ − W R,γ =q
N
i=1 |R
i|
N
S R i (q α )
i=1
×
N +1
S R i (q R
i−1 +α
)(−1)
N
i=1 γi |R
i|
1
q2
N
i=1 k R i γi
.
i=2
Notice that kλ is even for any partition λ, from (3.5) and Lemma 3.2, we have 2 Since γ is determined by the topology of S, we may suppress the dependence of γ .
(3.5)
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Corollary 3.3. W R,γ is of form
a(q) , b([1]2 )
where b(x) ∈ Z[x] is monic, a(q) ∈ Z[q, q −1 ].
−1 Corollary 3.4. W Rt ,γ (q) = W R,γ (q ).
Proof. By Lemma 3.1 and Corollary 3.3, the proof of Corollary 3.4 follows easily from formula (3.2). 4. Local Gopakumar-Vafa Invariants 4.1. Toric Fano surfaces and their toric diagram. Let S be a toric Fano surface associated with T = (C∗ )2 -action. The image µ(S) of the moment map µ : S → R2 of the T -action gives its toric diagram which is a convex polygon. The vertices of the polygon are the images of the fixed points. At each fixed point p of S, T p S = L 1p ⊕ L 2p , where L ip are one-dimensional subspaces on which T acts with weight λ pi . Denote the weights of T pi(v) S by u i(v),i(v)+1 and u i(v),i(v)−1 which satisfies u i(v),i(v)+1 = −u i(v)+1,i(v) . One can extend T -action to M g (S, β). The fixed point components of M g,n (S, β)T are in one-to-one correspondence with a set G g (S, β) of decorated graphs. Each vertex v of the graph ∈ G g (S, β) is assigned with an index i(v) ∈ S T , and a genus g(v). The valence val(v) of v is the number of edges incident at v. If two vertices u and v are joined by an edge e, then i(u) = i(v), and e is assigned with a degree δ(e). Denote by E( ) the set of edges of , V ( ) the set of vertices of . The decorations of satisfy
g(v) + 1 − |V ( )| + |E( )| = g,
v∈V ( )
δe = β.
e∈E( )
If f : C → S represents a fixed point, each vertex v corresponds to a connected component Cv of genus g(v) with val(v) nodal points. The component Cv is mapped by f to the fixed point i(v). When 2g(v) − 2 + val(v) < 0, Cv is simply a point, we have either g(v) = 0, val(v) = 1 or g(v) = 0, val(v) = 2. Each edge e corresponds to a component of C, isomorphic to P1 . Each Ce is mapped to the balloon li(v)i(u) with degree βe . Define M = M g(v),val(v) . v∈V ( )
In this product, M 0,1 and M 0,2 are interpreted as points. There are natural morphisms τ : M → M g (P1 , d). Its image is M /A , where for A we have an exact sequence: 0→
e∈E( )
Zde → A → Aut( ) → 1.
Simple Proof of Gopakumar-Vafa Conjecture for Local Toric C-Y Manifolds
R 1 π∗ ev ∗ K S . By virtual
g
4.2. Topological string partition function. Let Uβ = localization, we have the local Gromov-Witten invariants
g
eT (Uβ ) = [M g (S,β)]vir
∈G g (S,β)
559
τ ∗ (e (U g )) β
T
1 |A |
eT (N )
,
(4.1)
M
where τ ∗ (e (U g ))
T β eT (N )
=
v∈V ( )
M
wv ·
we .
(4.2)
e∈E( )
Here we is some combinatorial formula related to edge e, wv corresponds to a twopartition Hodge integral associated with the vertex v. The two-partition Hodge integral has been related to Chern-Simons invariants of Hopf link [13] by a convolution formula. Define the following generating function of local Gromov-Witten invariants: g 2g−2 t eT (Uβ ) Fβ (t) = g≥0
=
[M g (X,β)]vir
t 2g−2
∈G g (X,β)
g≥0
τ ∗ (e (t g ))
T β
1 |A |
eT (N )
,
M
and its generating series (free energy) F (S) (t) =
Fβ (t)e−ω·β ,
β=0
where ω is the Kähler class. The topological string partition function is defined to be (S) Z top str = exp F (S) (t) . Suppose the toric diagram of S is a polygon of N sides. Combine (4.2) and two-partition Hodge integral theory [13], one rewrites the topological partition function as a generating function of Chern-Simons invariants of Hopf link [2, 9, 5, 16]: (S) Z top str = W R N R 1 W R 1 R 2 W R 2 R 3 . . . W R N −1 R N R 1 ,R 2 ,...,R N
× (−1) =
N
i i=1 γi |R |
1 ,R 2 ,...,R N ) R=(R
q
1 2
N
i=1 k R i γi
W R,γ exp −
exp −
N
ti |R | i
i=1
N
ti |R i | ,
i=1
where γi is the self-intersection numbers of the rational curves associated to the i th edge, ti are linear combinations of Kähler parameters.
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P. Peng
4.3. Local Gopakumar-Vafa invariants. By the Gopakumar-Vafa formula, write F
(S)
g ∞ ∞ Nβ [n]2g−2
=
n
0=β∈H2 (S,Z) n=1 g=0
=
e−ω·β
0=β∈H2 (S,Z)
e−nω·β
∞ 1 g N β [n]2g−2 . n n n|β
g=0
Fix a basis in H2 (S, Z); one can identify H2 (S, Z) = Zm . We only consider β = g g−1 m ∞ m ˜ k , P (S) = . Let (k1 , k2 , . . . , km ) ∈ Z≥0 . Denote by |β| = β i=1 i g=0 Nβ x t ω = (ω1 , ω2 , . . . , ωm ) . Here we represent ω as an m × 1 matrix, β a 1 × m matrix. The relation between T (t1 , t2 , . . . , t N )t and ω is T = A · ω, where A is an N × m = (|R1 |, |R2 |, . . . , |R N |). Then matrix. Denote by | R| ⎛ ⎞ F (S) = log ⎝1 + W R e−| R|·T ⎠ = θλ ηλ . λ
R≥1
Here ηd =
· T , and ηλ = W R exp −| R| ηλi .
|β|=d | R|·A=β
i≥1
Combine the above two formulas, 1 θλ ηλ (q)β , P˜ β ([n]2 ) = n n λ |β|
n|β
where ηλ β is the coefficient of
e−ω·β
in ηλ . Let g Pβ (x) = x P˜β (x) = Nβ x g . g≥0
By Theorem 2.4, we can obtain Pβ ([1]2 ) = [1]2 ·
µ(n) n|β
n
λ |β| n
θλ ηλ (q n ) β .
We fix the following ring in this paper: a(x) : a(x), b(x) ∈ Z[x], b(x) = 0 and b(x) is monic . L[x] = b(x) By Corollary 3.3 and Corollary 3.4, we have: Lemma 4.1. ∀d ∈ N, ηd (q)β ∈ L[x]. Proposition 4.2. λ is a partition, let l = l(λ), then ηλ (q)β = ηλ1 (q)β1 · · · ηλl (q)βl . β1 +β2 +···+βl =β
Then by the above lemma, ηλ (q)β ∈ L[x] Proof. By direct calculation.
(4.3)
n
(4.4)
Simple Proof of Gopakumar-Vafa Conjecture for Local Toric C-Y Manifolds
561
5. The Integrality of Local Gopakumar-Vafa Invariants 5.1. A pattern theorem. Let
R(x) =
c(x) : c(x), d(x) ∈ Z[x] . d(x)
If f (x) ∈ R(x), f (x) = p k · mn · p mn, we write Ord p ( f (x)) = k.
a(x) b(x ,
where a(x), b(x) are primitive polynomials,
Lemma 5.1. a, b ∈ Z, r ∈ N, p is a prime number, f (x) ∈ Z[x]. The following statements hold: a a (A) gcd(a,b) b . r −1
r
(B) pr |(a p − a p ). r r −1 (C) f (x) ∈ Z[x], ( f (x)) p ≡ ( f (x p )) p mod( pr ). r r −1 (D) g(x) ∈ L[x], g(x) p − g(x p ) p = pr · h(x), where h(x) ∈ L[x]. a a a−1 a a Proof. (A) b = b b−1 ⇒ gcd(a,b) b . (B)
If a is p, the claim is obviously true since pr −1 ≥ r . If gcd(a, p) = 1, by the Fermat theorem, a p−1 ≡ 1mod( p). Then r
ap − ap
r −1
r −1
r −1
((kp + 1) p ( p−1) − 1) r −1 r −1 p pr −1 i p =a (kp) i
= ap
i=1
≡ 0 mod ( pr ). (C)
In the last step, we used (A) and the fact i > log(i). By (A) and (B), we have r
( f (x)) p − ( f (x p )) p
(D)
g(x) =
a(x) b(x) ,
r −1
r
≡ ( f (0)) p − ( f (0)) p ≡ 0 mod ( pr ).
r −1
mod( pr )
where b(x) is monic, r
g(x) p − g(x p ) p
r −1
r
=
a(x) p b(x p ) p
r −1
− a(x p ) p
b(x) p b(x p ) p r
r −1
r −1
By (C): r
a(x) p b(x p ) p
r −1
− a(x p ) p
r −1
r
b(x) p ≡ 0 mod ( pr ).
For any partition µ ∈ P, if µ = (µ1 , µ2 , . . . , µl ), define ⎛ ⎞ µ(d) = ⎝µ1 , . . . , µ1 , . . . , µl , . . . , µl ⎠ . d
d
b(x) p
r
.
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P. Peng
Lemma 5.2. For any given λ ∈ P, if θλ = dc is not an integer where gcd(c, d) = 1, then for any k|d, we can find a partition µ such that λ = µ(k) . Proof. Note that θλ =
(−1)l(λ)−1 (l(λ) − 1)! (−1)l(λ)−1 = uλ. | Aut λ| l(λ)
Let
⎛
(5.1)
⎞
λ = ⎝λ1 , . . . , λ 1 , . . . , λ s , . . . , λs ⎠ . m1
ms
One has l(λ) = m 1 + · · · + m s . u λ can be written as follows: l(λ) . uλ = m1, m2, . . . , ms Let n = gcd(m 1 , . . . , m s ) and ⎛
(5.2)
⎞
⎟ ⎜ µ = ⎝ λ 1 , . . . , λ 1 , . . . , λ s , . . . , λs ⎠ . m 1 /n
m s /n
We have λ = µ(n) . By (A) in Lemma 5.1, we have d|n. This implies we can define a n partition µ = µ( k ) and λ = µ(k) for any k|d. With these preparations we have the following theorem: Theorem 5.3. f β (x), gβ(d) (x), ψn (x) satisfy the following conditions: (d)
1. gβ (x) ∈ L[x], ∀d. Let λ = (λ1 , . . . , λl(λ) ) be a partition, define (λl(λ) ) gβλ = gβ(λ11 ) (x) · · · gβl(λ) (x). β1 +β2 +···+βl(λ) =β
2. For any given k, ψk (x) is a monic polynomial of degree k. Moreover, for any k, ψkp (x) − ψk (x) p ≡ 0mod( p), where p is prime. 3. The following equation holds: µ(n) (λ) f β (x) = x · θλ g β (ψn (x)). n n |β| n|β
λ
n
Then, f β (x) ∈ L[x]. Proof. Define by Hβ (x) =
(λ) λ |β| θλ gβ (x). Let’s look at the following lemma at first.
Lemma 5.4. Notations are as above, then for any prime number p, we have 1 Ord p H pβ (x) − Hβ (ψ p (x)) ≥ 0. p
Simple Proof of Gopakumar-Vafa Conjecture for Local Toric C-Y Manifolds
563
Proof. By Lemma 5.2, we have the following observation {λ pd : Ord p (θλ ) < 0} = {µ( p) : µ d}.
(5.3)
Let’s only consider minus as is taken in (5.4), since when plus is taken, it will imply p = 2 and r = 1, which is a rather direct computation. We will match the corresponding terms which involve 1/ p factors in H pβ (x) and 1 p Hβ (ψ p (x)) as in (5.3). Take two general matching terms, pr ·a 1 (λ1 ) (λk ) s1 sk (x)) · · · (g (x)) ζ · (g β1 βk pr · a pr −1 ·a . −η · (gβ(λ11 ) (ψ p (x)))s1 · · · (gβ(λk k ) (ψ p (x)))sk
A (−1) p
r −1 ·a·s
(5.4)
Then Ord p (A) = Ord p (−1)
pr −1 ·a·s
η pr p pr −1 φ(x) − φ(x ) , pr · a
(5.5)
where a, b ∈ Z and gcd( p, a) = 1 and ζ = η=
pr as , pr as1 , pr as2 , . . . , pr ask
pr −1 as , pr −1 as1 , pr −1 as2 , . . . , pr −1 ask (λ )
(λ )
φ(x) = (gβ11 (x))as1 . . . (gβk k (x))ask . In (5.5), we have used Theorem A.2, ψ p (x) ≡ x p mod( p) and pr | p p Lemma 5.1, we have Ord p (A) ≥ 0.
r −1
. By (D) in
Now, we return to the proof of Theorem 5.3. By (5.3), we can show f β (x) ∈ L[x] by proving: ∀ p|β prime, Ord p ( f β (x)) ≥ 0, f β (x) = x ·
µ(n) n|β
=x
n
λ |β| n
(λ)
θλ g β (ψn (x)) n
µ(n) µ( pn) H β (ψn (x)) + x H β (ψ pn (x)). n pn n pn
n|β, p n
n|β, p n
Notice that µ( pn) = −µ(n). Combine Lemma 5.4; we have Ord p ( f β (x)) ≥ 0. By the choice of p, the proof is completed.
564
P. Peng
5.2. Main results. We now give the proof of Theorem 1. Proof. By (4.3), Pβ ([1]2 ) = [1]2
µ(n) n|β
1
n
θλ ηλ (q n ) β .
λ |β| n
n
1
Let x = [1]2 = (q 2 − q − 2 )2 , ψn (x) = [n]2 . To prove ψn satisfies condition (3) in Theorem 5.3, we expand [1]2 p , [1]
2p
p−1 2p 2p k p−k −( p−k) p −p = +q ) + (q + q − 2) + +2 (−1) (q p k =
k=1 p−1 k=1
2p 2p (−1)k (q n−k + q −(n−k) ) + [ p]2 + + 2, k p
equivalently, [ p]2 = −
p−1 2p k=1
k
(−1)k (q n−k + q −(n−k) ) + [1]2 p −
2p − 2. p
By Lemma 3.1, q n + q −n ∈ Z[x], so [ p]2 ∈ Z[x]. By (A) of Lemma 5.1, p| 2kp when 1 ≤ k ≤ p − 1. [1]2 p → 0 as [1]2 → 0 implies as a polynomial of x, [ p]2 has 0 as the constant term. Then ψ p (x) − x p = [ p]2 − [1]2 p ≡ 0mod( p). By Theorem 5.3, Pβ (x) ∈ L[x]. i s /2 − Note that Pβ (x) = a(x) i s (q b(x) , where a(x), b(x) ∈ Z[x]. In fact, b(x) = q −is /2 )2 is a finite product for some i s ’s, which can be obtained from (3.3) and (3.4). g Therefore, q = 0 is a pole of a(x) b(x) . If there are infinitely many g’s such that Nβ is nonzero for the fixed β, then q = 0 is an essential singular point of Pβ (x), which is a contradiction. So Pβ (x) is a polynomial and Pβ (x) ∈ Q[x]. We can write Pβ (x) = h(x) d , where d h(x) ∈ Z[x]. However, we had proved pβ (x) ∈ L[x]. Therefore, there is a(x), b(x) ∈ Z[x], where b(x) is monic, such that a(x) h(x) = . b(x) d We have d|b(x)h(x), which implies d = 1. Thus we proved Pβ (x) ∈ Z[x].
6. Some Examples We can precisely calculate the degree of Pβ (x) using formula (4.3) and the following formula about the Schur function: χλ (ρ) Sλ (x) = pρ (x), zρ |ρ|=|λ|
where χλ is the character of representation corresponding to λ. Let Aβ (S) be the arithmetic genus of the curve representing β ∈ H2 (S, Z). L(β) denotes the holomorphic line bundle associated to the divisor β.
Simple Proof of Gopakumar-Vafa Conjecture for Local Toric C-Y Manifolds
565
Theorem 6.1. (1) We have 2 1 deg Pd(P ) (x) = Ad (P2 ) = (d − 1)(d − 2), 2 kd (Fk ) . deg P(d,m) (x) = A(d,m) (Fk ) = (d − 1) m − 1 − 2 (2) Moreover, A (P2 )
Nd d
(P2 ) = (−1)d χ (L(β)) =
A
(d,m) N(d,m)
(Fk )
(−1)d (d + 1)(d + 2), 2
(Fk ) = (−1)dk χ (L(β)) kd dk . = (−1) (d + 1) m + 1 − 2
Proof. We do a computation for P2 as an example. Let H ∈ H2 (P2 , Z) be the generator; it satisfies H · H = 1. So the topological string amplitudes will be 1 2 3 (P2 ) W R 1 R 2 W R 2 R 3 W R 3 R 1 (−1)|R |+|R |+|R | Z top str = R 1 ,R 2 ,R 3 1 3
×q2
i=1 k R i
exp t H (|R 1 | + |R 2 | + |R 3 |) .
We need to study the following term: 1
W R 1 R 2 W R 2 R 3 W R 3 R 1 (−1)d q 2
3
i=1 k R i
(6.1)
under the condition |R 1 | + |R 2 | + |R 3 | = d. Now Wλµ = q −
|λ|+|µ| 2
Sλ (q α )Sµ (q λ+α ).
By ([14] p. 44, Ex.1) Sλ (q α ) = q −n(λ)
x∈λ
1 . 1 − q −h(x)
We also have Sµ (q λ+α ) =
χµ (ρ) pρ (q λ+α ). zρ
|ρ|=|µ|
Here we can see the degree of q in Sµ (q λ+α ) is not greater than |λ| · |µ| by writing down pρ explicitly and equality holds when λ = (k) for k = |λ|. Hence the degree of q in (6.1) is not greater than 1 k R3 2 3
|R 1 ||R 2 | + |R 2 ||R 3 | + |R 3 ||R 1 | − d +
i=1
566
P. Peng
while 1 n((R i )t ) − n(R i ) ≤ k Ri = n((R i )t ) 2 3
i=1
≤
3
3
i=1
i=1
3 |R i | · (|R i | − 1)
2
i=1
.
The last inequality is obtained from n(λt ) ≤ n((m)), where m = |λ|. This can be proved in the following way. Let l = l(λ). If one subtracts one from λl and adds one to λ1 , one will find n(λt ) is increasing. So we have the inequality. The equality holds when λ = (m). Let |R i | = xi . Then xi = d. In the above inequalities, all equality holds when the corresponding partition is of length 1. So the deg Pd − 1 is 3 xi · (xi − 1) d 2 − 3d − d + (x1 x2 + x2 x3 + x3 x1 ) = , 2 2 i=1
deg Pd = 1 +
(d − 1)(d − 2) d 2 − 3d = . 2 2
The coefficient of the highest degree in Pβ is the number of the non-negative integer solutions of x1 + x2 + x3 = d multiplied by (−1)d , which is (−1)d (d + 1)(d + 2). 2 (Fk ) We do the similar calculation and get the degree of P(d,m) . The coefficient of the highest order term is the number of non-negative integer solutions of x1 + x3 + kx4 = m, x2 + x4 = d multiplied by (−1)dk , which is
(−1)dk
(d + 1)(m + 1 − kd) . 2
The theorem can be easily verified by the adjunction formula and the Riemann-Roch theorem.
A. Appendix Lemma A.1. p is prime, r ≥ 1. Then
r −1 pr a p a − ≡ 0 mod ( p 2r ). r p b pr −1 b
Simple Proof of Gopakumar-Vafa Conjecture for Local Toric C-Y Manifolds
Proof. Consider the ratio pr a pr b pr −1 a pr −1 b
567
pr b
=
(a−b) pr +k k=1 k pr −1 b (a−b) pr −1 +k k=1 k
=
gcd(k, p)=1, 1≤k≤ pr b
(a − b) pr + k k
≡ 1 + pr (a − b)
gcd(k, p)=1, 1≤k≤ pr b
1 mod ( p 2r ). k
Let
A p (n) =
1≤k≤n, gcd(k, p)=1
1 . k
Therefore, the proof of the lemma can be completed by showing A p ( pr b) =
pr c d
for some c, d such that gcd(d, p) = 1. If gcd(k, p) = 1, there exist αk , βk such that αk k + βk pr = 1. Let
B p (n) =
k.
1≤k≤n,gcd(k, p)=1
By the above formula, A p ( pr b) ≡ b A p ( pr ) mod ( pr ) ≡ bB p ( pr ) mod ( pr ), and r
Bp( p ) = r
= =
p
k−p
k=1 pr ( pr
r −1 p
k
k=1
+ 1)
2
p 2r −1 ( p 2
−p
− 1)
pr −1 ( pr −1 + 1) 2
.
Here, we have 2r − 1 ≥ r since r ≥ 1. The proof is then completed.
The following theorem is a simple generalization of the above lemma.
568
P. Peng
n
= a, p is prime, r ≥ 1, then pr a pr a − ≡ 0 mod ( p 2r ). p r a1 , . . . , p r an pr −1 a1 , . . . , pr −1 an
Theorem A.2.
i=1 ai
Proof. We have
pr a pr a − p r a1 , . . . , p r an pr −1 a1 , . . . , pr −1 an k−1 n n r −1 pr (a − k−1 ai ) p (a − i=1 ai ) i=1 − = p r ak pr −1 ak k=1
k=1
≡ 0 mod ( p 2r ). In the last step, we used Lemma A.1. The proof is completed. Lemma A.3. We have
a a . gcd(a, b) b
Proof. Notice that
i.e.
a a a−1 = , b b b−1 a a−1 b a = . gcd(a, b) b gcd(a, b) b − 1
However,
b a , gcd gcd(a, b) gcd(a, b) so
= 1,
a a . gcd(a, b) b
This direct leads to the following corollary: Corollary A.4. a = a1 + a2 + · · · + an . Then a a . gcd(a1 , a2 , . . . , an ) a1 , a2 , . . . , an Acknowledgements. I would like to thank my advisor, Professor Kefeng Liu, for his constant encouragement and a lot of inspiring discussions. I am very grateful to Xiaowei Wang for his helpful comments and friendship. I also want to thank Professor Jian Zhou for pointing out some misleading parts and Professor Amer Iqbal for explaining to me the physical reason for the vanishing property of Gopakumar-Vafa invariants at large genera. It is lucky for me that the referee gave me a lot of helpful suggestions for the final version of this paper. I give special thanks to the referee for hard work and patience in reading my paper.
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References 1. Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005) 2. Aganagic, M., Mariño, M., Vafa, C.: All loop topological string amplitudes from Chern-Simons theory. Commun. Math. Phys. 247, 467–512 (2004) 3. Chiang, T.-M., Klemm, A., Yau, S.-T., Zaslow, E.: Local Mirror Symmetry: Calculations and Interpretations. Adv. Theor. Math. Phys. 3, 495–565 (1999) 4. Diaconescu, D.-E., Florea, B., Grassi, A.: Geometric transitions, DEL PEZZO surfaces and open string instantons. Adv. Theor. Math. Phys. 6, 643–702 (2003) 5. Eguchi, T., Kanno, H.: Topological Strings and Nekrasov’s formulas. JHEP 12, 006 (2003) 6. Gopakumar, R., Vafa, C.: M-theory and topological strings-I. http://arxiv.org/list/hepth/9809187, 1998 7. Gopakumar, R., Vafa, C.: M-theory and topological strings-II. http://arxiv.org/list/hepth/9812127, 1998 8. Gopakumar, R., Vafa, C.: On the gauge theory/goemetry correspondence. Adv. Theor. Math. Phys. 3, 1415 (1999) 9. Iqbal, A.: All Genus Topological String Amplitudes and 5-brane Webs as Feynman Diagrams. http://arxiv. org/list/hepth/0207114, 2002 10. Iqbal, A., Kashani-Poor, A.-K.: Instanton Counting and Chern-Simons Theory. Adv. Theor. Math. Phys. 1, 457–497 (2004) 11. Katz, S., Klemm, A., Vafa, C.: M-theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3, 1445 (1999) 12. Li, J., Liu, K., Zhou, J.: Topological partition function as equivariant indices. Private conversation 13. Liu, C.-C., Liu, K., Zhou, J.: A formula of two-partition Hodge integrals. J. Amer. Math. Soc. 20, 149–184 (2007) 14. MacDonald, I.G.: Symmetric Functions and Hall Polynomials. 2nd edition, London: Clarendon Press, 1995 15. Nekrasov, N.A.: Seiberg-Witten Prepotential from Instanton Counting. In: Proc. of the ICM (Beijing 2002) vol. 3, Beijing: Higher Ed. Press of China, 2003, pp. 477–496 16. Zhou, J.: Localizations on moduli spaces and free field realizations of Feynman rules. http://arxiv.org/ list/math/0310283, 2003 17. Zhou, J.: Curve counting and instanton counting. http://arxiv.org/list/math 0311237, 2003 Communicated by M. R. Douglas
Commun. Math. Phys. 276, 571–610 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0335-y
Communications in
Mathematical Physics
Nucleation of Instability of the Meissner State of 3-Dimensional Superconductors Peter W. Bates1 , Xing-Bin Pan2 1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
E-mail: [email protected]
2 Department of Mathematics, East China Normal University, Shanghai 200062, P. R. China.
E-mail: [email protected] Received: 20 September 2005 / Accepted: 16 March 2007 Published online: 12 October 2007 – © Springer-Verlag 2007
Abstract: This paper concerns a nonlinear partial differential system in a 3-dimensional domain involving the operator curl2 , which is a simplified model used to examine nucleation of instability of the Meissner state of a superconductor as the applied magnetic field reaches the superheating field. We derive a priori C 2+α estimates for a weak solution H, the curl of the magnetic potential, and determine the location of the maximal points of |curl H| which correspond to the nucleation of instability of the Meissner state. We show that, if the penetration length is small, the solution exhibits a boundary layer. If the applied magnetic field is homogeneous, |curl H| is maximal around the points on the boundary where the applied field is tangential to the surface. 1. Introduction In this paper we study the following elliptic system: −λ2 curl 2 A = (1 − |A|2 )A in ,
λ (curl A)T = HeT on ∂,
(1.1)
where is a bounded and smooth domain in R3 , A(x) = (A1 (x), A2 (x), A3 (x)), and He is a given vector field on ∂. Here, the subscript T denotes the tangential component on ∂, respectively, and 0 < λ 1. For reasons that will become clear below, we are interested in the solutions of (1.1) satisfying 1 A L ∞ () < √ . 3
(1.2)
We shall investigate the location of the maximum points of |A(x)| for small λ. Equation (1.1) is an approximation of the Ginzburg-Landau system of superconductivity with large value of Ginzburg-Landau parameter κ, which was derived by Chapman [C1] (also see [Mon]) to describe loss of stability of the Meissner state when the applied magnetic field reaches the superheating field Hsh . In the system He is the applied
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P. W. Bates, X.-B. Pan
magnetic field, A is the magnetic potential, curl A is the induced magnetic field, and λ is the penetration depth, which is small typically. In fact, the instability occurs when the condition (1.2) is violated [C1]. It was first discovered in [C1] (also see [Mon]) that, under the condition (1.2), Eq. (1.1) is equivalent to the following quasilinear system: −λ2 curl F(λ2 |curl H|2 )curl H = H in , HT = HeT on ∂. (1.3) ¯ R3 ) is a solution of (1.1) and satisfies (1.2), More precisely, if A ∈ C 3 (, R3 ) ∩ C 2 (, and letting H = λ curl A, then H solves (1.3), and the following estimate holds: 4 . (1.4) λcurl H L ∞ () < 27 Here the function F is determined by the relation v = F(t 2 )t ⇐⇒ t = (1 − v 2 )v, 4 and F(0) = 1. F is uniquely defined for 0 ≤ t ≤ 27 , i.e., for 0 ≤ v ≤ hand, if in addition is a simply-connected domain, and if H ∈ is a solution of (1.3) and satisfies (1.4), and letting
(1.5)
√1 . On the other 3 ¯ R3 ) C 3 (, R3 )∩C 2 (,
A = −λF(λ2 |curl H|2 )curl H, then A ∈ C 2 (, R3 ) is the unique solution of (1.1) and satisfies (1.2). Moreover, the maximum points of |A(x)| coincide with the maximum points of |curl H(x)|.1 Let us recall that [dG, MS, Fn, Kra, FP1, FP2, C1, BH], for a type II superconductor subjected to an applied magnetic field, the Meissner state is a global minimizer of the Ginzburg-Landau energy and hence globally stable if the applied field is below the first critical field HC1 , it is locally stable if the applied field is between HC1 and Hsh , and it is not stable if the applied field is above Hsh . It is interesting to calculate the superheating field Hsh for a superconductor with general shape ([C1, p.1258]), and to explore how the Meissner state loses its stability as the applied magnetic field increases and reaches Hsh . Consider a cylinder of infinite height with its axis in the x3 -direction, and placed in an applied axial magnetic field He = (0, 0, h). Then the Ginzburg-Landau system is reduced to an elliptic system on the 2-dimensional cross section D of the cylinder. Chapman [C1] derived the following system as a large κ limit of the Ginzburg-Landau system: −λ2 curl 2 A = (1 − |A|2 )A in D, (1.6) ν · A = 0, λ curl A = h on ∂ D, where A = (A1 (x1 , x2 ), A2 (x1 , x2 )), and ν is the outer normal vector of ∂ D. He showed that, if A L ∞ (D) ≤ √1 , (1.6) can be transformed to a quasilinear equation in divergence 3 form for the scalar function H = λ curl A = λ(∂1 A2 − ∂2 A1 ): λ2 div [F(λ2 |∇ H |2 )∇ H ] = H in ,
H = h on ∂.
(1.7)
1 This conclusion can be proved by using the argument in [PK], where the equivalence of two equations (1.6) and (1.7) in the 2-dimensional case was proved.
Instability in Meissner State
573
He also showed that, as the value of h increases, the solution A of (1.6) loses its stability when max |A(x)| reaches √1 , and the instability will occur first at the boundary of the 3 sample. Berestycki, Bonnet and Chapman [BBC] rigorously proved that, if A L ∞ (D) ≤ √1 , then the maximum points of |A(x)| must lie on the boundary ∂ D. Chapman [C2] 3 further showed by formal analysis that, as λ → 0, |A(x)| reaches the maximum value at the points of largest negative curvature of boundary ∂ D. This conclusion was rigorously verified by Pan and Kwek [PK]. Moreover, it was proved in [PK] that the solutions of (1.6) exhibit boundary layer behavior when λ is small, which is recognized as the Meissner effect of superconductivity (see for instance [FP1]). The approximation of Meissner solutions by the solutions of (1.6) was verified by Bonnet-Chapman-Monneau [BCM]. For a bulk superconductor occupying a bounded domain in R3 , Monneau [Mon] derived an approximation of the Ginzburg-Landau system as κ → ∞2 : ⎧ ⎪ − λ2 curl 2 A = (1 − |A|2 )A in , ⎪ ⎨ ¯ curl 2 A = 0 in R3 \ , (1.8) [ν × A] = 0, [ν × curl A] = 0 on ∂, ⎪ ⎪ ⎩ λ curl A − He → 0 as |x| → ∞, where [·] denotes the jump in the enclosed quantity across ∂. System (1.1) can be viewed as an approximation of (1.8) if we restrict ourselves in the domain and take the boundary condition λ (curl A)T = HeT on ∂. Note that if A is a solution of (1.1) in ¯ of the system , and if B is a solution in c = R3 \ λ curl B = He in c ,
BT = AT on ∂,
˜ = A in ˜ = B in c , then A ˜ is a solution of (1.8). ¯ and A and if we define A Monneau [Mon] proved by using the implicit function theorem that, if HeT C 2+α (∂) ¯ R3 ), is small, then (1.3) (in the case where λ = 1) has a unique solution H ∈ C 2+α (, e and |curl H| attains its maximum only on ∂. If HT C 2+α (∂) is large, then (1.3) has no solution. However, the smallness condition of the boundary data depends on λ, and the dependence on λ was not obtained in [Mon]. Thus no criterion has been given for the boundary data that allows the existence of solutions for all λ. Because we wish to determine the location of the points where |A(x)| reaches its maximum, it is necessary to consider the limit as λ → 0. In particular we must establish optimal bounds for boundary data and show solvability for all small λ. This requires the introduction of weak solutions. So, the analysis in [Mon], while very useful, must be significantly extended. In this paper we shall find a bound C of the boundary data such that (1.3) has a solution Hλ for all small λ if HeT C 2+α (∂) < C, and determine the location of the maximum points of |curl Hλ |. We shall also establish an a priori C 2+α estimate of the solutions to (1.1) and (1.3), which is useful in our study of the asymptotic behavior of 2 The Ginzburg-Landau system has two parameters: the penetration depth λ and the coherence length ξ . The ratio κ = λ/ξ is the Ginzburg-Landau parameter (see [GL, CHO]). In this paper we are interested in the behavior of Meissner solutions of the Ginzburg-Landau system with small λ and large κ. For the Meissner state, one may write the order parameter in the form ψ = f eiχ with f > 0. We first fix λ and let κ go to infinity to formally get (1.8) as the limiting equations of the Ginzburg-Landau system (see [C1]); then we investigate the solutions of (1.8) with small λ. We would like to mention that one may rescale the Ginzburg-Landau functional in a way that λ is reduced to 1. This is the case in [Mon]. In that setting, the change in λ in our present paper corresponds to the change of the domain size of the superconductor. See also [DP, Remark 1.4].
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P. W. Bates, X.-B. Pan
solutions. We shall show that, as λ → 0, the points where the maximum of |A(x)| (and of |curl H(x)|) is attained must approach points in ∂(HeT ), where ∂(HeT ) = {x ∈ ∂ : |HeT (x)| = HeT C 0 (∂) }.
(1.9)
In the special case of a homogeneous applied field, namely He = h a constant vector, which is of most importance in applications, HeT = hT = h − (h · ν)ν, and |hT | is maximal at the points where h is tangential to ∂. Thus ∂(hT ) = (∂)h ≡ {x ∈ ∂ : h is tangential to ∂ at x}. In this case the optimal bound of boundary data for solvability is C =
5 18 ,
which
[C1].3
equals the value of the superheating field Hsh found in ¯ R3 ) denote We now state our main results. In this paper H k (, R3 ) and C k+α (, the usual Sobolev space, resp. the usual Hölder space of vector fields, and C k+α (, R3 ) k+α (, R3 ). denotes the space Cloc Theorem 1. Let be a bounded and simply-connected domain in R3 with C 4 boundary, and let HeT satisfy 5 e 2+α 3 e , ν · curl HeT = 0 on ∂. (1.10) HT ∈ C (∂, R ), HT C 0 (∂) < 18 ¯ R3 ) Then for all λ > 0 small, (1.3) has a unique solution Hλ ∈ C 3 (, R3 ) ∩ C 2+α (, which has the following properties: (i) Hλ satisfies (1.4). (ii) For any sequence ρλ such that ρλ ≤ lim
c λ
sup
and ρλ → ∞ as λ → 0, we have
λ→0 dist(x,∂)≥λρλ
|Hλ (x)| = 0.
(1.11)
(iii) Let µ = HeT C 0 (∂) . We have 2 2 1/2 ](1 − 2µ2 ). lim λ2 curl Hλ C 0 (∂) = [1 − (1 − 2µ )
λ→0
(1.12)
(iv) If P λ is a maximum point of |curl Hλ (x)|and if P λ → P0 for a sequence λ = λn → 0, then P0 ∈ ∂(HeT ). (v) In particular, if He = h, a constant vector, and if 5 , (1.13) |h| < 18 then P0 ∈ (∂)h . Using the equivalence between classical solutions of (1.1) and (1.3), we can rewrite Theorem 1 with respect to Eq. (1.1): 3 The optimal bound of h allowing the existence of solutions for all small λ for the 2-dimensional problem 5 , (see [PK]). (1.7) is also equal to 18
Instability in Meissner State
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Theorem 1 . Under the conditions of Theorem 1, for all λ > 0 small, (1.1) has a unique ¯ R3 ) which satisfies (1.2). Moreover, the maximum solution Aλ ∈ C 3 (, R3 ) ∩ C 2+α (, λ points of |A (x)| satisfy the conclusions (iv) and (v) in Theorem 1. Remark 1.1. In order to obtain the optimal bound of boundary data for solvability of (1.3), we need to study the weak limit of solutions as the boundary data approach the bound (see Lemma 7.1), and hence we have to work in the framework of weak solutions. Remark 1.2. It is interesting to compare the phenomena of nucleation of superconductivity and instability of the Meissner state in the 3-dimensional case with that in the 2-dimensional case. In the 2-dimensional case (cylindrical superconductors subject to an axial homogeneous magnetic field): (i) as the applied field decreases from HC3 superconductivity nucleates at the boundary where the curvature is the maximal ([HP]); (ii) as the applied field increases to Hsh the Meissner state loses its stability at the boundary where the curvature is the minimal ([PK]). In the 3-dimensional case (bulk superconductors subjected to a homogeneous magnetic field He = h): (i) as the applied field decreases from HC3 superconductivity nucleates in (∂)h ([LP, P]);4 (ii) as the applied field increases to Hsh the Meissner state loses its stability at points in the set (∂)h .5 Remark 1.3. Chapman [C1, p. 1250] also conjectured that the instability of the Meissner state at Hsh leads to formation of vortices in the sample, and leads to transitions from the Meissner state to the mixed state. It would be interesting to verify the conjecture, and to locate the points of vortex nucleation. Combining our Theorem 1 with Chapman’s conjecture, one may expect that when a type II superconductor changes from the Meissner state to the mixed state, vortices will start to nucleate in (∂)h . In particular, for a superconductor occupying an ellipsoidal domain and placed in a magnetic field directed parallel to its major axis, one may expect that vortices will start to nucleate first at the equator.6 Remark 1.4. From the physical background of our problem, the third condition in (1.10) is natural: For the applied magnetic field He we always assume curl He ≡ 0. Hence ν · curl (HeT ) = ν · curl (He ) = 0 on ∂. In particular, if He = h, a constant vector, then the condition holds. We shall see that this condition forces the solutions of (1.1) to satisfy an additional boundary condition A · ν = 0 on ∂ (see Lemma 2.5). Remark 1.5. Besides the physical motivation, we are interested in systems (1.1) and (1.3) also because of their mathematical structure. For nonlinear differential systems 4 Moreover, the analysis in [P] and [HP] suggests that, superconductivity nucleates at some points in (∂) h where the curvature function P(x) is the minimal among all points in (∂)h . 5 But we do not know yet whether at the location of instability the curvature function P(x) is the maximal among all points in (∂)h . 6 We should mention that Du and Ju [DJ] computed a reduced Ginzburg-Landau equation for a superconducting hollow sphere subjected to a constant applied magnetic field and their computational results show that vortex pairs nucleate on that equator of the sphere, which is everywhere perpendicular to the field.
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with operator curl2 , the existence and regularity of solutions sensitively depends on the nonlinearity. For (1.1) and its 2-dimensional version (1.6), without the smallness condition (1.2), the regularity of the solutions fails. In fact, [PQ, Theorem 4] gives a solution of (1.6) with a singularity in the unit disk; |A(x)| ≡ 1 and curl A is a constant a.e., while div A ∈ L 2 (). The interaction between the nonlinearity of the equations and pointwise degeneracy and the global ellipticity of the operator curl2 is most interesting to us. Due to the pointwise degeneracy of ellipticity of curl2 , the classical Schauder estimates for elliptic systems (see [G, Mor]) do not apply to (1.1) and (1.3). On the other hand, restricted in the space of divergence free vector fields, curl2 is coercive (globally elliptic).7 For linear systems involving curl 2 , one may apply the Hodge decomposition theory and work in the space of divergence free vector fields. One can use the standard difference-quotientmethod to derive H k estimates of the solutions for any positive integer k, and get a C k estimate by using the Sobolev imbedding theorem, see Dautray-Lions [DL]. However, for the quasilinear system (1.3), the difference-quotient-method allows us to obtain only H 2 estimates. The C 2+α estimates require more involved arguments, and we sketch our main ideas here. We begin with considering weak solutions of (1.3) satisfying 4 λcurl H L ∞ () ≤ M < . (1.14) 27 An H 2 estimate and the Sobolev imbedding theorem yield a C δ estimate with 0 < δ < ¯ R3 ) such that 1/2. Since is simply-connected, there exists B ∈ C 1+δ (, λ curl B = H and div B = 0 in , ν · B = 0 on ∂.
(1.15)
From (1.3) and (1.15) we find curl [λF(λ2 |curl H|2 )curl H + B] = 0. So there exists a function ϕ ∈ H 1 () such that λF(λ2 |curl H|2 )curl H + B = ∇ϕ. From the boundary condition and the last equality in (1.10) we have ν · curl H = 0 in the sense of trace in H 1/2 (∂). Thus ϕ is a weak solution of a quasilinear equation div [(1 − |∇ϕ − B|2 )(∇ϕ − B)] = 0 in ,
∂ϕ = 0 on ∂. ∂ν
(1.16)
Condition (1.14) implies that (1.16) is an elliptic equation for ϕ. So we can derive a C 1+δ estimate for ∇ϕ in terms of B, hence in terms of , HeT , λ and M, which in turn produces a C 1+δ estimate for H. Then the regularity theory of (1.15) implies ¯ R3 ) and B ∈ C 2+δ (, BC 2+δ () ¯ ≤ C(, δ)curl BC 1+δ () ¯ .
(1.17)
So we can write (1.16) in a non-divergence form with C 1+δ coefficients and derive the Schauder estimate of ϕ. Finally we establish a C 2+α estimate for H. Remark 1.6. Due to the boundary layer behavior of the solutions of (1.3), the global H 2 estimate blows-up as λ → 0, see (4.1). On the other hand, we have a uniform L ∞ 2 and C 2+α estimates for the rescaled estimate (8.1), which enables us to establish the Hloc loc vector fields in Lemma 8.2. The local estimates are sufficient for us to prove Theorem 1. 7 This is one of the reasons why we prefer studying (1.3) instead of studying (1.1) directly.
Instability in Meissner State
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This paper is organized as follows. In Sect. 2 we provide some properties of the curl operator, and give C 1+α estimates of gradients of vector fields in terms of the divergence, curl and tangential or normal trace of the vector fields. In particular, we shall prove inequality (1.17) for the solutions of (1.15). In Sect. 3 we introduce the definition of weak solutions of (1.1) and (1.3). The H 2 estimate of weak solutions of (1.3) is given in Sect. 4, and the C 2+α estimate is established in Sect. 5. In Sect. 6 we classify the solutions of limiting equations in R3 and in R3+ . In Sect. 7, based on the C 2+α estimate of Sect. 5, we use blow-up arguments to derive a criterion for the boundary data that implies solvability. In Sect. 8 we examine the asymptotic behavior of the solutions as λ → 0, and prove Theorem 1. We would like to mention that (1.15) is a special case of div-curl systems. The solvability and regularity of div-curl systems and the applications in physical and mathematical problems have been studied by many authors, see for instance Kress [Kre], Dautray-Lions [DL], Wahl [W1], Schwarz [S], Bourgain-Brezis [BB] and the references therein. We should also mention that Yin [Y] studied an equation involving curl 2 and obtained C 1+α regularity. Inequalities to control gradients of vector fields by using their divergence and curl and traces have been studied for many years. Control in the framework of Sobolev spaces has been well-established, see Dautray-Lions [DL]. Control of the C α norm of the gradients of vector fields by using C α norms of divergence and curl has been established by Bolik-Wahl [BW], which enables us to derive a C 1+α version of the Bolik-Wahl inequality in Sect. 2. Inequalities of this form play an important role in our study of (1.3). 2. Some Analysis Aspects of the Operator curl In this section we discuss some questions involving the curl operator for vector fields. Our first question is to ask in which context ∇B can be controlled by div B and curl B in , and by either ν · B or BT on ∂. If is a bounded and simply-connected domain in R3 with smooth boundary, the following are known:8 (i) The control is true in H k (, R3 ) for any k ≥ 0: It follows from the classical results of Agmon, Douglis and Nirenberg [ADN] and Dautray and Lions [DL] (Theorem 3 on p. 209, Proposition 6 on p. 237) that, if is a bounded and simply-connected domain with C k+1 boundary, then
ν·B
B H k+1 () ≤ C(, k) div B H k () + curl B H k () +
. (2.1) ν × B k+1/2 H
(∂)
ν·B
Here and in the following
ν × B stands for “either ν · B∗ or ν × B∗ ”. When ∗ k = 0, H 0 () is interpreted as L 2 (). Note that the formula given in [DL, p. 209] contains the L 2 norm of B in the right of the inequality. However, under our assumption on the domain , we have, for any B ∈ H 1 (, R3 ),
ν·B
B L 2 () ≤ C() div B L 2 () + curl B L 2 () +
, (2.2)
ν × B 1/2 H (∂) 8 The results proved by Dautray-Lions [DL], Wahl [W2] and Bolik-Wahl [BW] cover more general cases. These results cited here are limited to the case of simply-connected domains.
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P. W. Bates, X.-B. Pan
(see Appendix A.2 for the proof). Hence the term B L 2 () in the right of the inequality can be dropped.9 ¯ R3 ) for any 0 < α < 1: It follows from the results (ii) The control is true in C α (, of Bolik-Wahl [BW] that, if is a bounded and simply-connected domain with C 2 boundary, then
ν·B
∇BC α () . (2.3) ¯ ≤ C(, α) div BC α () ¯ + curl BC α () ¯ +
ν × B C 1+α (∂) ¯ R3 ) with k ≥ 1. Let ∇∂ For the purposes of this paper, we need control in C k+α (, denote the tangential gradient operator on ∂, and let BC k+α (∂) = BC 0 (∂) +
k−1
j
k ∇∂ BC 0 (∂) + ∇∂ BC α (∂) .
j=1
Proposition 2.1. Let k ≥ 0 be an integer and 0 ≤ α < 1. Assume is a bounded and simply-connected domain in R3 with C k+2 boundary. There exists C(, k, α) > 0 such that
ν · B
BC k+1+α () . ¯ ≤ C(, k, α) div BC k+α () ¯ +curl BC k+α () ¯ +
ν ×B C k+1+α (∂) (2.4) The proof of Proposition 2.1 will be given in Appendix A.3 Our second question is the following: Given H with div H = 0, and let B be a solution of curl B = H and div B = 0 in ,
ν·B=0
on ∂,
(2.5)
what can one say about the regularity of B? For the Sobolev regularity, it is well-known that (see [DL]), if is a bounded and simply-connected domain with smooth boundary, and if H ∈ L 2 (, R3 ), the existence of B satisfying (2.5) is a consequence of the Hodge decomposition; and it follows from (2.1) that, if ∂ is of class C k+2 with k ≥ 0 and if H ∈ H k (, R3 ), then B ∈ H k+1 (, R3 ), and B H k+1 () ≤ C(, k)H H k () .
(2.6)
For the Hölder regularity, the result follows from Proposition 2.1 and a local version of (2.4) (see the proof of Proposition 2.1 in Appendix A.3). Corollary 2.2. Let k be a non-negative integer and 0 ≤ α < 1. Assume that is a bounded, simply-connected domain with C k+2 boundary. ¯ R3 ), then (2.5) has a unique solution B ∈ C k+1+α (, ¯ R3 ), and (i) If H ∈ C k+α (, BC k+1+α () ¯ ≤ C(, k, α)HC k+α () ¯ .
(2.7)
(ii) Furthermore, if H ∈ C k+1+α (, R3 ), then B ∈ C k+2+α (, R3 ). 9 The control is also true in L p (, R3 ) for any 1 < p < ∞ if either ν · B = 0 or B = 0 on ∂, (see T Wahl [W2]).
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Our third question is about extensions of vector fields on ∂ to all of . The existence of such extensions has been proved in [Mon, Lemma 3.1] in the C 2+α case. Here we give a minimality characterization of the extension, and give some estimates. Lemma 2.3. Assume is a bounded and simply-connected domain in R3 with C 4 boundary, and HeT ∈ H 3/2 (∂, R3 ). (i) There exists a vector field H ∈ H 2 (, R3 ) such that div H = 0 in , HT = HeT on ∂,
(2.8)
H H 2 () ≤ C()HeT H 3/2 (∂) .
(2.9)
and
Moreover, H can be chosen such that the L 2 norm of curl H is minimal among all vector fields satisfying (2.8). ¯ R3 ) and (ii) If in addition HeT ∈ C 2+α (∂, R3 ) with 0 ≤ α < 1, then H ∈ C 2+α (, e HC 2+α () ¯ ≤ C(, α)HT C 2+α (∂) .
(2.10)
(iii) Furthermore, if HeT satisfies ν · curl(HeT ) = 0 on ∂,
(2.11)
then H = ∇φ for some function φ. Moreover, φ ∈ H 3 () if HeT ∈ H 3/2 (∂, R3 ), ¯ if He ∈ C 2+α (∂, R3 ). and φ ∈ C 3+α () T The proof of Lemma 2.3 will be given in Appendix A.4. We would like to mention that a minimization problem for the L 2 norm of curl u with prescribed whole trace u = u0 on ∂ has been studied in [PQ]. Our fourth question is to ask in which context the condition BT = BT on ∂,
(2.12)
ν · curl B = ν · curl B on ∂,
(2.13)
allows us to conclude that
¯ it is wellwhere ν is the unit outer normal vector to ∂. For C 1 vector fields on , known that the value of ν · curl B on ∂ depends only on the tangential component BT (see [Mon]), and hence (2.12) implies (2.13). Using a density argument we see that the conclusion remains true for H 2 vector fields. We include it here for reader’s convenience. Lemma 2.4. Assume ∂ is of class C 2 . (i) Let B ∈ C 1 (∂, R3 ) and let BT be its tangential component on ∂. Then the function x → ν · curl(BT )(x) is well-defined on ∂, and its value depends only on ¯ R3 ) such that (2.12) holds the tangential derivatives of BT . For any B ∈ C 1 (, pointwise on ∂, then (2.13) holds pointwise on ∂. (ii) Let B ∈ H 3/2 (∂, R3 ). Then ν · curl BT is well-defined on ∂ as an element in H 1/2 (∂). For any B ∈ H 2 (, R3 ) such that (2.12) holds in the sense of trace in H 1/2 (∂, R3 ), then (2.13) holds in the sense of trace in H 1/2 (∂).
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P. W. Bates, X.-B. Pan
See Appendix A.5 for a proof of (i). As a direct consequence of Lemma 2.4 we have ¯ R3 ) is a Lemma 2.5. Assume HeT ∈ C 2 (∂, R3 ) and satisfies (2.11). If A ∈ C 2 (, solution of (1.1) satisfying A L ∞ () < 1, then A satisfies the additional boundary condition A · ν = 0 on ∂.
(2.14)
¯ R3 ) be a solution of (1.1) satisfying A L ∞ () < 1, and let Proof. Let A ∈ C 2 (, H = λ curl A. From the equation in (1.1) we see that, (2.14) holds if and only if ν·curl H = 0 on ∂. From Lemma 2.4 and using the boundary condition HT = HeT we have ν · curl H = ν · curl (HT ) = ν · curl (HeT ) = 0. Thus (2.14) holds. 3. Weak Solutions Let us define H(, curl) = {A ∈ L 2 (, R3 ) : curl A ∈ L 2 (, R3 )}, H(, curl 0) = {A ∈ L 2 (, R3 ) : curl A = 0}, 1/2 AH(,curl) = A2L 2 () + curl A2L 2 () . Definition 3.1. Let HeT ∈ H 1/2 (∂, R3 ). A is called a weak solution of (1.1) if A ∈ H(, curl) ∩ L ∞ (, R3 ), and for all B ∈ H(, curl) the following equality holds:
2 2 {λ curl A · curl B + (1 − |A| )A · B}d x + λ(HeT × BT ) · νd S = 0. (3.1)
∂
To see that the definition is meaningful, if B ∈ H(, curl), from the trace theorem for H (, curl) (see [DL, p. 204]), the tangential component BT ∈ H −1/2 (∂, R3 ), hence the surface integral in (3.1) is well defined if HeT ∈ H 1/2 (∂, R3 ). Now we give an H 1 estimate. Lemma 3.2. Assume HeT ∈ H 1/2 (∂, R3 ). Let A ∈ H(, curl) be a weak solution of (1.1) which satisfies the equation for a.e. x ∈ and suppose that A L ∞ () ≤ 1. Then H = λ curl A ∈ H 1 (, R3 ), and H2L 2 () ≤ λHeT L 1 (∂) , H2H 1 () ≤ C()λ−1 HeT L 1 (∂) + C()HeT 2H 1/2 (∂) ,
(3.2)
where C() depends only on . Moreover, the equality HT = HeT holds in the sense of trace in H 1/2 (∂, R3 ). Proof. Taking B = A in (3.1), we get
{|H|2 + (1 − |A|2 )|A|2 }d x = λ
∂
(AT × HeT ) · νd S.
(3.3)
Instability in Meissner State
581
From (1.1), λ2 |curl H(x)|2 = (1 − |A(x)|2 )2 |A(x)|2 . Substituting this into (3.3) yields
2 2 2 4 2 {|H| + λ |curl H| + |A| (1 − |A| )}d x = λ (AT × HeT ) · νd S.
∂
¯ we have Since |A(x)| ≤ 1 on , H2L 2 () + λ2 curl H2L 2 () ≤ λHeT L 1 (∂) . Since div H = 0, using (2.1) with k = 0 we get (3.2). Since A satisfies (1.1) for a.e. x ∈ , we can multiply (1.1) by any smooth vector field B and integrate over . Since H ∈ H 1 (, R3 ), we have HT ∈ H 1/2 (∂, R3 ), and we can integrate by parts to get
{λ2 curl A · curl B + (1 − |A|2 )A · B}d x + λ(HT × BT ) · νd S = 0.
∂
From this and (3.1) we see that the surface integrals are equal for any smooth vector field B. Thus HT = HeT in the sense of trace. Under the conditions of Lemma 3.2, it is not always the case that A ∈ H 1 (, R3 ) even in 2-dimensions, (see Remark 1.5). Definition 3.3. Let HeT ∈ H 1/2 (∂, R3 ). H is called a weak solution of (1.3) if the following conditions are satisfied: (i) (ii) (iii) (iv) (v)
H ∈ H 1 (, R3 ); curl H ∈ L ∞ (, R3 ) and (1.4) holds; HT = HeT holds on ∂ in the sense of trace in H 1/2 (∂, R3 ); (curl H)T ∈ H −1/2 (∂, R3 ); for all B ∈ H 1 (, R3 ), the following equality holds:
{λ2 F(λ2 |curl H|2 )curl H · curl B + H · B}d x
+ λ2 F(λ2 |curl H|2 )((curl H)T × BT ) · νd S = 0. ∂
(3.4)
If B ∈ H 1 (, R3 ), then BT ∈ H 1/2 (∂, R3 ), and hence the surface integral in (3.4) is well-defined if H satisfies condition (iv). To see that condition (iv) is reasonable for a weak solution of (1.3), note that we expect the weak solution to satisfy the condition curl [F(λ2 |curl H|2 )curl H] ∈ L 2 (, R3 ). From this and condition (1.4) we have F(λ2 |curl H|2 )curl H ∈ H(, curl). Then it follows from the trace theorem for H(, curl) that F(λ2 |curl H|2 )(curl H)T ∈ H −1/2 (∂, R3 ). Since F(λ2 |curl H|2 ) ≥ 1, H must satisfy (iv). Lemma 3.4. Assume HeT ∈ H 1/2 (∂, R3 ). Let H ∈ H 1 (, R3 ) be a weak solution of (1.3) satisfying (1.14). Then H2L 2 () ≤ λF(M 2 )MHeT L 1 (∂) , and H2H 1 ()
(3.5) ≤ C()λ
−1
F(M
where C() depends only on .
2
)MHeT L 1 (∂)
+ C()HeT 2H 1/2 (∂) ,
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P. W. Bates, X.-B. Pan
Proof. In (3.4) take B = H. Using (1.14) and since 1 ≤ F(λ2 |curl H|2 ) ≤ F(M 2 ) and HT = HeT , we have λ2 curl H2L 2 () + H2L 2 () ≤ λF(M 2 )MHeT L 1 (∂) . Using this and (2.1) we get (3.5).
Lemma 3.5. Given HeT ∈ H 1/2 (∂, R3 ), problem (1.3) has at most one weak solution H satisfying (1.4). Therefore (1.1) has at most one weak solution A such that A satisfies (1.2) and that H = λ curl A is a weak solution of (1.3). The proof is omitted here, as we shall prove in Sect. 6 a uniqueness result in unbounded domains, and the argument there applies to bounded domains as well. 4. H 2 Estimates of Weak Solutions Theorem 4.1. Let be a bounded and simply-connected domain in R3 with C 3 boundary, and HeT ∈ H 3/2 (∂, R3 ). Let H ∈ H 1 (, R3 ) be a weak solution of (1.3) satisfying (1.14). Then H ∈ H 2 (, R3 ), and H2H 2 () ≤ C(, M){(1 + λ−2 )HeT 2H 3/2 (∂) + (λ + λ−3 )HeT L 1 (∂) }, (4.1) where C(, M) depends only on and M. ¯ in the way as stated in Lemma 2.3 Proof. We may assume that HeT has been extended to (i). In particular, div He = 0 in ,
He H 2 () ≤ C()HeT H 3/2 (∂) .
(4.2)
Step 1. Interior estimate. Let e denote one of the coordinate vectors e j , j = 1, 2, 3. For σ > 0 small we define 1 [H(x + σ e) − H(x)], σ Ht,σ (x) = H(x) + t[H(x + σ e) − H(x)] = H(x) + tσ hσ (x), hσ (x) =
(4.3)
u t,σ = |curl Ht,σ |2 . For any subdomain , and for all σ sufficiently small, we have ([G, Proposition 2.1]): hσ L 2 ( ) ≤ C(, )DH L 2 () . From (3.4), for any C 1 vector field B supported in the interior of ,
{λ2 F(λ2 |curl H(x + σ e)|2 )curl H(x + σ e) · curl B + H(x + σ e) · B}d x = 0.
We subtract (3.4) from this to get
λ2 F(λ2 |curl H(x +σ e)|2 )curl H(x + σ e) − F(λ2 |curl H(x)|2 )curl H(x) · curl B + [H(x + σ e) − H(x)] · B d x = 0. (4.4)
Instability in Meissner State
583
Let us write F(λ2 |curl H(x + σ e)|2 )curl H(x + σ e) − F(λ2 |curl H(x)|2 )curl H(x)
1 d = [F(λ2 u t,σ )curl Ht,σ ]dt dt 0
1 F(λ2 u t,σ )curl hσ + 2λ2 F (λ2 u t,σ )(curl Ht,σ · curl hσ )curl Ht,σ dt =σ 0
= σ aσ (x)curl hσ + 2σ Q σ (x)curl hσ , where aσ (x) is a scalar function defined by
1 aσ (x) = F(λ2 u t,σ )dt, 0
and Q σ (x) = (qσ,i j (x)) is a 3 × 3 matrix-valued function with entries
qσ,i j (x) = λ
1
2
F (λ2 u t,σ )(curl Ht,σ )i (curl Ht,σ ) j dt,
0
where (curl Ht,σ )i is the component of curl Ht,σ . Since λ2 u t,σ ≤ M 2 < 4/27, there exists a constant C(M) independent of λ such that ith
aσ (x) ≥ 1, F(λ2 u t,σ ) + F (λ2 u t,σ ) + aσ (x) + |Q σ (x)| ≤ C(M), Q σ (x) is non-negative definite for all x ∈ and λ > 0.
(4.5)
Returning to (4.4) we have, for all B ∈ H 1 (, R3 ) with compact support in ,
λ2 aσ (x)curl hσ · curl B + 2λ2 Q σ (x)curl hσ , curl B + hσ · B d x = 0. (4.6)
Let η be a smooth function with support in , and let B = η2 hσ . From (4.6) we get
λ2 aσ (x)|curl (ηhσ )|2 + 2λ2 Q σ (x)curl (ηhσ ), curl (ηhσ ) + η2 |hσ |2 d x
(4.7) = λ2 aσ (x)|∇η × hσ |2 + 2λ2 Q σ (x)∇η × hσ , ∇η × hσ d x.
Using this and (4.5) we have, for all σ > 0 small, λ2 curl (ηhσ )2L 2 () + ηhσ 2L 2 ()
λ2 aσ (x)|∇η × hσ |2 + 2λ2 Q σ (x)∇η × hσ , ∇η × hσ d x ≤
≤ C(M)λ2 ∇η × hσ 2L 2 () . Note that div (ηhσ ) = ∇η · hσ . Since η = 0 on ∂, we use (2.1) to get ηhσ 2H 1 () ≤ C(){curl (ηhσ )2L 2 () + div (ηhσ )2L 2 () } ≤ C(, M)∇η × hσ 2L 2 () + C()∇η · hσ 2L 2 () .
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P. W. Bates, X.-B. Pan
Hence for any subdomain and σ sufficiently small, hσ H 1 ( ) ≤ C(, , M)hσ L 2 () ≤ C(, , M)DH L 2 () . It follows from this and [G, Proposition 2.1] that ∂ j H ∈ H 1 ( , R3 ), and hσ → ∂ j H in H 1 ( , R3 ) as σ → 0. The above conclusion is true for j = 1, 2, 3. So H ∈ H 2 ( , R3 ), and H H 2 ( ) ≤ C(, , M)H H 1 () .
(4.8)
In particular, D 2 H exists for a.e. x ∈ . Using (3.4) we can show that H satisfies (1.3) for a.e. x ∈ . So curl [λ2 F(λ2 |curl H|2 )curl H] = −H ∈ L 2 (, R3 ). From (1.14), λ2 F(λ2 |curl H|2 )curl H ∈ L ∞ (, R3 ) ⊂ L 2 (, R3 ). Thus λ2 F(λ2 |curl H|2 )curl H ∈ H(, curl ).
(4.9)
By the trace theorem for H(, curl ), λ2 F(λ2 |curl H|2 )curl H ∈ H −1/2 (∂, R3 ). Step 2. Boundary estimate: tangential derivatives. Let b = H − He . Then bT = 0 on ∂. From (4.9), λ2 F(λ2 |curl H|2 )curl b ∈ H −1/2 (∂, R3 ). From (3.4) we have,
λ2 F(λ2 |curl H|2 )curl b · curl B + b · B + λ2 F(λ2 |curl H|2 )curl He · curl B + He · B d x
+ λ2 F(λ2 |curl H|2 )((curl b)T × BT ) · ν ∂ 2 (4.10) + λ F(λ2 |curl H|2 )((curl He )T × BT ) · ν d S = 0. We consider a subset of that is located near ∂. To avoid technical complexity, ¯ is flat. Therefore we may write we assume that = ∂ ∩ + = B2R = {x ∈ B2R : x3 > 0},
2R = {x : |x| < 2R, x3 = 0}.
Let e = e1 or e2 , and set bσ (x) =
1 [b(x + σ e) − b(x)], σ
Hσe (x) =
1 e [H (x + σ e) − He (x)]. σ
+ ∪ Note that bσ T = 0 on 2R . Let B be supported in B2R 2R and satisfy BT = 0 on ∂. From (4.10) we have,
λ2 F(λ2 |curl H(x + σ e)|2 )curl b(x + σ e) · curl B + b(x + σ e) · B + λ2 F(λ2 |curl H(x + σ e)|2 )curl He (x + σ e) · curl B + He (x + σ e) · B d x = 0.
As in Step 1, we subtract (4.10) from this equality and derive
λ2 aσ (x)curl bσ · curl B + 2λ2 Q (1) σ (x)curl bσ , curl B + bσ · B d x
e e =− λ2 aσ (x)curl Hσe · curl B + 2λ2 Q (2) σ (x)curl Hσ , curl B + Hσ · B d x,
(4.11)
Instability in Meissner State
585 (1)
(2)
where aσ is the same as in Step 1, and Q σ and Q σ are defined as follows:
1 (1) F (λ2 u t,σ )(curl bt,σ )i (curl Ht,σ ) j dt, qσ,i j (x) = λ2 0
(2)
qσ,i j (x) = λ2
1
0
e F (λ2 u t,σ )(curl Ht,σ )i (curl Ht,σ ) j dt,
e bt,σ = b(x) + tσ bσ (x), Ht,σ (x) = He (x) + tσ Hσe (x). (1)
(2)
(1)
(1)
Note that Q σ and Q σ are not symmetric. Let (Q σ )t denote the transpose of Q σ . Now we choose B = η2 bσ , where η is a smooth cut-off function supported in B2R . Then (η2 bσ )T = 0 on ∂. Using this in (4.11) we get (comparing with (4.7))
2 2 λ2 aσ (x)|curl (ηbσ )|2 + 2λ2 Q (1) dx σ (x)curl (ηbσ ), curl (ηbσ ) + η |bσ |
= λ2 aσ (x)|∇η × bσ |2 + 2λ2 Q (1) σ (x)∇η × bσ , ∇η × bσ (1) t −2λ2 Q (1) σ (x) − (Q σ (x)) curl (ηbσ ), ∇η × bσ −λ2 aσ (x)(η curl Hσe ) · curl (ηbσ ) + λ2 aσ (x)(η curl Hσe ) · (∇η × bσ ) + η2 bσ · Hσe e (2) e −2λ2 Q (2) d x. σ (x)η curl Hσ , curl (ηbσ ) + Q σ (x)η curl Hσ , ∇η × bσ Using this, (4.5), and applying the Holder inequality, we find
{λ2 |curl (ηbσ )|2 + η2 |bσ |2 }d x ≤ C λ2 bσ 2L 2 (B + ) + curl Hσe 2L 2 (B + ) + Hσe 2L 2 (B + ) , 2R
2R
(4.12)
2R
where C depends on , M and R. Note that bσ L 2 (B +
2R )
≤ CDb L 2 () ≤ C{H H 1 () + He H 1 () },
curl Hσe L 2 (B +
2R )
≤ CDcurl He L 2 () ≤ CHe H 2 () .
Also recall that div bσ = 0 in and (ηbσ )T = 0 on ∂. Now we use (2.1) and (4.12) to get ηbσ 2H 1 () ≤ C curl (ηbσ )2L 2 () + div (ηbσ )2L 2 () ≤ C curl (ηbσ )2L 2 () + ∇η · bσ 2L 2 () ≤ C bσ 2L 2 () + curl Hσe 2L 2 () + Cλ−2 Hσe 2L 2 () ≤ CH2H 1 () + C(1 + λ−2 )He 2H 2 () , where C depends on , M and η only. Letting σ → 0 in (4.13), we see that, for j = 1, 2, ∂ j b ∈ H 1 (B R+ , R3 ), and ∂ j b2H 1 (B + ) ≤ CH2H 1 () + C(1 + λ−2 )He 2H 2 () . R
(4.13)
586
P. W. Bates, X.-B. Pan
Hence ∂ j H ∈ H 1 (B R+ , R3 ), and ∂ j H H 1 (B + ) ≤ CH H 1 () + C(1 + λ−1 )He H 2 () ,
(4.14)
R
where C depends only on , M and R. Step 3. Boundary estimates: normal derivatives. As mentioned in the last part of Step 1, D 2 H exists a.e. in , and H satisfies Eq. (1.3) for a.e. x ∈ . Therefore, λ2 F(λ2 u)H − λ4 F (λ2 u)∇u × curl H = H, for a.e. x ∈ ,
(4.15)
where u(x) = |curl H(x)|2 . Write J(x) = curl H(x) = (J1 , J2 , J3 ). We compute ⎤ ⎡ 2J1 J2 ∂33 H2 − 2J22 ∂33 H1 + f 1 ⎥ ⎢ ∇u × curl H = ⎣ −2J12 ∂33 H2 + 2J1 J2 ∂33 H1 + f 2 ⎦ , f3 where f 1 = −2J2 [J1 ∂23 H3 − J2 ∂13 H3 + J2 (∂13 H2 − ∂23 H1 )] + J3 ∂2 u, f 2 = 2J1 [J1 ∂23 H3 − J2 ∂13 H3 + J2 (∂13 H2 − ∂23 H1 )] − J3 ∂1 u, f 3 = J2 ∂1 u − J1 ∂2 u. Since |J|2 = u ≤ M 2 λ−2 , and |∂ j u| = 2|J · ∂ j J| ≤ 2Mλ−1
|∂i j H|,
(i, j)=(3,3)
we have | f k | ≤ C(M)λ−2
|∂i j H|.
(4.16)
(i, j)=(3,3)
Now we write (4.15) as follows: a∂33 H1 + b∂33 H2 = g1 , b∂33 H1 + c∂33 H2 = g2 , λ2 F(λ2 u)∂33 H3 = g3 , (4.17) where a = λ2 F(λ2 u) + 2λ4 F (λ2 u)J22 , b = −2λ4 F (λ2 u)J1 J2 , c = λ2 F(λ2 u) + 2λ4 F (λ2 u)J12 , g1 = H1 − λ2 F(λ2 u)(∂11 H1 + ∂22 H1 ) + λ4 F (λ2 u) f 1 , g2 = H2 − λ2 F(λ2 u)(∂11 H2 + ∂22 H2 ) + λ4 F (λ2 u) f 2 , g3 = H3 − λ2 F(λ2 u)(∂11 H3 + ∂22 H3 ) + λ4 F (λ2 u) f 3 . We have |a| + |b| + |c| ≤ C(M)λ2 , and ac − b2 = λ4 F(λ2 u)2 + 2λ6 F(λ2 u)F (λ2 u)(J12 + J22 ) ≥ λ4 .
Instability in Meissner State
587
We solve ∂33 Hi from (4.17): ∂33 H1 =
cg1 − bg2 −bg1 + ag2 g3 , ∂33 H2 = , ∂33 H3 = 2 . 2 2 ac − b ac − b λ F(λ2 u)
(4.18)
Now we consider a subdomain located near ∂. As in Step 2 we may assume that + , and = B2R 2R = {x : |x| < 2R, x 3 = 0}. From (4.14) and (4.16),
gk L 2 (B + ) ≤ H L 2 (B + ) + C(M)λ2 R
R
∂i j H L 2 (B + ) R
(i, j)=(3,3)
≤ H L 2 () + C(, M)λ2 {H H 1 () + (1 + λ−1 )He H 2 () }. From this and (4.18) we obtain ∂33 H L 2 (B + ) ≤ C(, M, R) λ−2 H L 2 () + H H 1 () + (1 + λ−1 )He H 2 () . R
(4.19) Step 4. Covering a neighborhood of ∂ by a finite number of subdomains which are diffeomorphic to a half ball, and applying (4.14) and (4.19) on each of such half balls, and then using (4.8), we find H2H 2 () ≤ C(, M) H2H 1 () + (1 + λ−4 )H2L 2 () + (1 + λ−2 )He 2H 2 () . Using this together with (3.5) and (4.2) we get (4.1).
From Theorem 4.1 and the Sobolev imbedding theorem we have: ¯ R3 ) for Corollary 4.2. Under the conditions of Theorem 4.1, the solution H ∈ C α (, 1 any α ∈ (0, 2 ), and 2 −2 e 2 −3 e HC α () ¯ ≤ C(, M, α) (1+λ )HT H 3/2 (∂) + (λ + λ )HT L 1 (∂) , (4.20) where C(, M, α) depends only on , M and α. 5. C 2+α Regularity of the Weak Solutions Theorem 5.1. Assume that is a bounded and simply-connected domain in R3 with C 4 boundary, and HeT ∈ C 2+α (∂),
ν · curl HeT = 0 on ∂,
(5.1)
where 0 < α < 1. Let H ∈ H 1 (, R3 ) be a weak solution of (1.3) satisfying (1.14). ¯ R3 ), and Then H ∈ C 3 (, R3 ) ∩ C 2+α (, e HC 2+α () ¯ ≤ C(, λ, M, α, HT C 2+α (∂) ).
(5.2)
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P. W. Bates, X.-B. Pan
¯ in the way as stated in Lemma 2.3, Proof. We may assume HeT has been extended over (ii). In particular div He = 0 in ,
e He C 2+α () ¯ ≤ C(, α)HT C 2+α (∂) .
(5.3)
In the following, for simplicity of presentation, we only give the proof in the case where λ = 1. Let H be the weak solution of (1.3). Write J = curl H. Step 1. Find a vector field B satisfying (1.15) (with λ = 1). From Corollary 4.2, H ∈ ¯ R3 ) for any 0 < δ < 1/2. For the moment let us fix δ < min{α, 1 }. Since C δ (, 2 div H = 0, from Corollary 2.2 (i) we see that there exists a vector field B ∈ C 1+δ (, R3 ), B = (B1 , B2 , B3 ), such that (1.15) (with λ = 1) holds, and e BC 1+δ () ¯ ≤ C(, δ)HC δ () ¯ ≤ C(, λ, M, δ, H H 2 () ).
(5.4)
Step 2. Find the scalar function ϕ satisfying (1.16). Since H satisfies (1.3) for a.e. x ∈ , we have curl [F(|J|2 )J + B] = 0 for a.e. x ∈ . Thus F(|J|2 )J + B ∈ H(, curl 0). Since is simply-connected, there exists ϕ ∈ H 1 () such that F(|J|2 )J + B = ∇ϕ (see [DL, p. 219, Proposition 2]). From condition (5.1) and using Lemma 2.4 (ii) we have ν · J = ν · curl H = ν · curl HT = ν · curl HeT = ν · curl He = 0 on ∂. Since we also have ν · B = 0 on ∂, it holds that ∂ϕ ∂ν = 0 on ∂. √ From (1.14) and (1.5), there exists b(M) < 1/ 3 such that 1 |∇ϕ − B| = F(|J|2 )|J| ≤ b(M) < √ , 3 |J| = (1 − |∇ϕ − B|2 )|∇ϕ − B|, J=
F(|J|2 ) =
∇ϕ − B = (1 − |∇ϕ − B|2 )(∇ϕ − B). F(|J|2 )
|∇ϕ − B| 1 , (5.5) = |J| 1 − |∇ϕ|2
Since div J = 0 in and ∂ϕ ∂ν = 0 on ∂, ϕ satisfies (1.16). Step 3. C 2+δ regularity for the quasilinear Neumann problem (1.16). Write p = ( p1 , p2 , p3 ),
¯ p ∈ R3 , |p − B(x)| ≤ b}, Qb = {(x, p) : x ∈ ,
A(x, p) = (1 − |p − B(x)|2 )(p − B(x)). Then (1.16) can be written as div A(x, ∇ϕ) = 0 in , We claim that if 0 < b <
√1 , 3
∂ϕ = 0 on ∂. ∂ν
(5.6)
then A(x, p) satisfies the following conditions on Qb :
(i) Ai ∈ C 1+δ (Qb ); (ii)
3 ∂ Ai (x, p)ξi ξ j ≥ λ(b)|ξ |2 for all ξ ∈ R3 ; ∂pj
i, j=1 (5.7) (iii) |A(x, 0)| ≤ m; ∂ Ai 2 ∂ Ai (x, p) + (1 + |p|)|Ai (x, p)| + (1 + |p| ) (x, p) ≤ (b)(1 + |p|2 ), (iv) ∂x j ∂pj
Instability in Meissner State
589
where λ(b) and (b) are positive and depend only on b, and m is positive and depends on B. ¯ R3 ). (ii) is true for λ(b) = 1 − 3b2 > 0. In fact, (i) is true because B ∈ C 1+δ (, ∂ Ai (x, p) = (1 − |p − B(x)|2 )δi j − 2( pi − Bi (x))( p j − B j (x)), ∂pj 3 ∂ Ai (x, p)ξi ξ j = (1 − |p − B(x)|2 )|ξ |2 − 2((p − B(x)) · ξ )2 ∂pj
i, j=1
≥ (1 − 3|p − B(x)|2 )|ξ |2 ≥ (1 − 3b2 )|ξ |2 . Because |A(x, 0)| = |1−|B(x)|2 ||B(x)|, (iii) is true for m = max x∈¯ |1−|B(x)|2 ||B(x)|. To verify (iv), we compute on Qb : |Ai (x, p)| = (1 − |p − B(x)|2 )| pi − Bi (x)| ≤ |p − B(x)|, ∂ Ai 2 ∂ p (x, p) ≤ 1 + |p − B(x)| , j 3 ∂ Ai ∂B ∂B i k = (1 − |p − B(x)|2 ) (x, p) + 2( p − B (x)) ( p − B (x)) i i k k ∂x ∂ x ∂ x j j j k=1
≤ (1 + 3|p − B(x)|2 )∇BC 0 () ¯ . For the number b(M) given in the first line in (5.5), we choose b such that b(M) < ¯ × R3 such that conditions b < √1 . Then we extend A(x, p) to the whole region 3 ¯ and all p ∈ R3 . Then we can apply the classical regu(i)–(iv) are satisfied for all x ∈ ¯ and larity theory for quasilinear elliptic equations to (5.6) to conclude that ϕ ∈ C 2+δ (), ϕC 2+δ () ¯ can be estimated in terms of M, b and BC 1+δ () ¯ , (see [CW, Chapter 5]). Regarding the regularity at the boundary, we would like to mention that, since ν · B = 0 and ∂ϕ ∂ν = 0 on ∂, after straightening a portion of boundary, we can extend B and ϕ by reflection: even extension for ϕ and for the tangential component of B, and odd extension of the normal component of B. Then the regularity at the boundary can be established by applying the interior regularity theory to the extended functions. Since b can be chosen to depend only on M, and using (5.4), we have e ϕC 2+δ () ¯ ≤ C(, λ, M, δ, H H 2 () ).
(5.8)
¯ R3 ), we have J = (1 − |∇ϕ − Step 4. C 2+δ estimates for H. Since B, ∇ϕ ∈ C 1+δ (, 2 1+δ 3 ¯ R ). H satisfies B| )(∇ϕ − B) ∈ C (, curl H = J and div H = 0 in , HT = HeT on ∂.
(5.9)
¯ R3 ), and Applying (2.4) to H we find H ∈ C 2+δ (, e HC 2+δ () ¯ ≤ C(, δ){JC 1+δ () ¯ + HT C 2+δ (∂) } ≤ C,
(5.10)
where C = C(, λ, M, δ, He H 2 () , HeT C 2+δ (∂) ). Step 5. Interior C 3+δ regularity and global C 2+α estimates of H. Using (5.10) we can improve the regularity of B obtained in Step 1: Applying Corollary 2.2 (ii) to Eq. (1.15)
590
P. W. Bates, X.-B. Pan
¯ R3 ). Then the functions Ai (x, p) defined in (with λ = 1) for B we have B ∈ C 2+δ (, 2+δ Step 3 are in C in x. Write Eq. (5.6) in the form 3
ai j (x)
i, j=1
∂ 2ϕ + f (x) = 0 in , ∂ xi ∂ x j
∂ϕ = 0 on ∂, ∂ν
(5.11)
where ai j (x) = (1 − |∇ϕ(x) − B(x)|2 )δi j − 2(
∂ϕ ∂ϕ (x) − Bi (x))( (x) − B j (x)), ∂ xi ∂x j
and f (x) = 2
3 ∂B j ∂ϕ ∂ϕ ( (x) − Bi (x))( (x) − B j (x)) (x). ∂ xi ∂x j ∂ xi
i, j=1
¯ and i,3 j=1 ai j (x)ξi ξ j ≥ λ(b)|ξ |2 . Applying the Schauder Note that ai j , f ∈ C 1+δ (), ¯ and ϕC 3+δ () estimates to (5.11) (see [GT, CW]) we conclude that ϕ ∈ C 3+δ () ¯ is 2+δ ¯ bounded by a constant depending only on , ai j , f and δ. Hence J ∈ C (, R3 ). Using (5.9) and checking the interior estimate in the proof of Proposition 2.1 we see that H ∈ C 3+δ (, R3 ). Now we derive the global C 2+α estimates. If α < 21 , then we can choose δ = α in the above discussions, and we are done. If 21 ≤ α < 1, from the discussion in the last ¯ R3 ) ⊂ C 1+α (, ¯ R3 ). Using (5.9) and (2.4) we find paragraph we have J ∈ C 2+δ (, 2+α 3 ¯ H ∈ C (, R ), and (5.10) holds with δ replaced by α. From Lemma 4.1 and Theorem 5.1 we have the following conclusion: Theorem 5.2. Under the condition of Theorem 5.1, if (1.3) has a weak solution H sat¯ R3 ) satisfying (1.2). Furthermore isfying (1.4), then (1.1) has a solution A ∈ C 2+α (, 5 3+α 3 ¯ if ∂ is of class C then A ∈ C (, R ). Proof. We keep the notations used in the proof of Theorem 5.1, and we shall prove The¯ R3 ). Using orem 5.2 in the case where λ = 1. From Theorem 5.1 we have H ∈ C 2+α (, ¯ R3 ). From the proof of Theorem 5.1 we have ϕ ∈ C 3+α (). ¯ Corollary 2.2, B ∈ C 3+α (, ¯ Applying the Schauder estimates to (5.11) Hence ai j and f in (5.11) are in C 2+α (). ¯ if ∂ is C 5 , and ϕ ∈ C 3+α () ¯ if ∂ is C 4 . Let A = B − ∇ϕ. we see that ϕ ∈ C 4+α () 3+α 3 5 2+α ¯ R ) if ∂ is C and A ∈ C (, ¯ R3 ) if ∂ is C 4 . Using (5.5) we Then A ∈ C (, have curl 2 A = curl 2 B = curl H = J = −(1 − |B − ∇ϕ|2 )(B − ∇ϕ) = −(1 − |A|2 )A. So A is a solution of (1.1). From the first line in (5.5), A satisfies (1.2).
Remark 5.3. In Theorem 5.1, if we further assume that has C k+2 boundary and HeT ∈ ¯ R3 ), and a global C k+α estimate holds. C k+α (∂) with k ≥ 2, then H ∈ C k+α (, Note that (1.16) is of hyperbolic type in the region where √1 < |∇ϕ − B(x)| < 1. 3
Loss of regularity of ϕ when |∇ϕ − B(x)| increases to √1 may be relevant to loss of 3 regularity of the solutions of (1.1) when condition (1.2) is violated.
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6. Classification of Solutions in R3 and in R3+ The uniqueness results established in this section will be essential for exploring the asymptotic behavior of solutions of (1.3) in Sect. 8. For the 2-dimensional case, the uniqueness for C 2 solutions in the entire plane and in the half plane that satisfy √ the condition |A(x)| < 1/ 3 was proved in [PK, Lemma 2.3], and the argument in [PK] could be used in the 3-dimensional case for C 2 solutions in this section. However, our proof below gives the uniqueness result under a weaker condition |A(x)| ≤ 1 − ε0 , and it works also for weak solutions (which can be defined as in Sect. 3). We first consider the problems in the entire space:
and
−curl 2 A = (1 − |A|2 )A in R3 ,
(6.1)
−curl F(|curl H|2 )curl H = H in R3 .
(6.2)
Proposition 6.1. (i) If A is a weak solution of (6.1) satisfying A L ∞ (R3 ) < 1, then A ≡ 0. (ii) If H is a weak solution of (6.2) satisfying 4 curl H L ∞ (R3 ) ≤ , (6.3) 27 then H ≡ 0. Proof. (i) Let A be a solution as stated in the proposition. Choose ε0 > 0 such that A L ∞ (R3 ) < 1 − ε0 . For any smooth vector field B with compact support we have
{curl A · curl B + (1 − |A|2 )A · B}d x = 0. R3
By a density argument, this must also hold for B ∈ H(R3 , curl). Taking B = η2 A, where η is a smooth function with compact support, we get
{|curl (ηA)|2 + (1 − |A|2 )|ηA|2 }d x = |∇η × A|2 d x. (6.4) R3
R3
√
ε
Choose η = e−δr ξ(r ) in (6.4), where r = |x|, δ is a number satisfying 0 < δ < 2 0 , and ξ(r ) is a smooth nonincreasing cut-off function such that ξ(r ) = 1 for r < R, ξ(r ) = 0 for r > R + 1, and |ξ (r )| ≤ 2. Identity (6.4) gives, for some C = C(δ, ε0 ),
e−2δr |A|2 d x ≤2δ 2 e−2δr |A|2 d x + 8 e−2δr d x. ε0 BR
B R+1
B R+1 \B R
Letting R → ∞ we find A ≡ 0. (ii) Let H be a solution as stated in the proposition. For any C 1 vector field B with bounded support,
{F(|curl H|2 )curl H · curl B + H · B}d x = 0. R3
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P. W. Bates, X.-B. Pan
We choose B = η2 H, where η is a smooth function with bounded support, and get
R3
{F(|curl H|2 )|curl (ηH)|2 + |ηH|2 }d x =
R3
|∇η × H|2 d x.
Take η as in the proof of part (i), and find
e−2δr |H|2 d x ≤ 2δ 2
BR
e−2δr |H|2 d x + 8e−2δ R
B R+1
Letting R → ∞ we find H ≡ 0.
R R+1 \B R
|H|2 d x.
Next we discuss the problem in the half-space R3+ = {(x1 , x2 , x3 ) : x3 > 0}: −curl 2 A = (1 − |A|2 )A in R3+ ,
(curl A)T = h on ∂R3+ ,
(6.5)
where h = (h 1 , h 2 , 0) is a constant vector which is tangential to ∂R3+ . We look for a solution of (6.5) satisfying 1 A L ∞ (R3+ ) < √ . 3
(6.6)
If A is such a solution, then H = curl A solves the following equation: −curl [F(|curl H|2 )curl H] = H in R3+ ,
HT = h on ∂R3+ ,
(6.7)
and satisfies curl H L ∞ (R3+ ) <
4 . 27
(6.8)
Recall the function M(ρ) used in [PK]: M(ρ) = [1 − (1 − 2ρ 2 )1/2 ](1 − 2ρ 2 ).
(6.9)
Proposition 6.2. Assume h = (h 1 , h 2 , 0) satisfies (1.13). (i) Equation (6.5) has a smooth solution A which satisfies 1 A L ∞ (R3+ ) = A L ∞ (∂ R3+ ) < √ , 3 curl A L ∞ (R3+ ) = curl A L ∞ (∂ R3+ ) = |h|, curl 2 A L ∞ (R3+ ) = curl 2 A L ∞ (∂ R+ ) =
M(|h|) <
(6.10) 4 , 27
(ii) Equation (6.7) has a unique weak solution satisfying (6.8), and it is a smooth solution.
Instability in Meissner State
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Proof. Step 1. Since curl h = 0, as in the proof of Lemma 2.5 we can show that, if A is a C 2 solution of (6.5) satisfying (6.6) in R3+ , then A must satisfy an additional boundary condition A3 = ν · A = 0 on ∂R3+ . So we look for a solution such that A3 ≡ 0. Write h = ρ(cos θ, sin θ, 0), where the constants ρ ≥ 0 and 0 ≤ θ < 2π . Let A = f (x3 )(− sin θ, cos θ, 0). By computation, H = curl A = − f (x3 )(cos θ, sin θ, 0), curl H = − f (x3 )(− sin θ, cos θ, 0). (6.11) A is a solution of (6.5) if and only if f (t) satisfies f = (1 − f 2 ) f for t > 0,
f (0) = −ρ. (6.12) √ It has been shown in [PK, Lemma 2.4] that, if 0 ≤ ρ < 1/ 2, then (6.12) has a unique solution f (t) satisfying | f (t)| < 1. Moreover, f L ∞ (R+ ) = f (0) = [1 − (1 − 2ρ 2 )1/2 ]1/2 , f L ∞ (R+ ) = | f (0)| = ρ.
Also, f 2L ∞ (R+ ) = | f (0)|2 = M(ρ) if ρ = |h| satisfies condition (1.13). With this condition we have 1 A L ∞ (R3+ ) = f L ∞ (R3+ ) < √ , 3
curl H L ∞ (R3+ ) = f L ∞ (R+ ) <
4 . 27
Thus A satisfies condition (6.10). Let H = curl A. Then H is a smooth solution of (6.7) and satisfies (6.8). Step 2. We show that, if H1 , H2 are weak solutions of (6.7) satisfying (6.8), then for any R > 1,
|H2 − H1 |2 d x ≤ C R 2 , (6.13) B R+
where B R+ = B R ∩ R3+ , and C is a constant independent of R and the solutions. To prove this, let A j be the solution of (6.5) corresponding to H j , namely, H j = curl A j . Then for any smooth vector field B with compact support and for each j, we have an integral equality similar to (3.1). Subtracting one from the other we get
{curl (A2 − A1 ) · curl B + [(1 − |A2 |2 )A2 − (1 − |A1 |2 )A1 ] · B}d x = 0. R3+
Write At (x) = A1 (x) + t (A2 (x) − A1 (x)). We have [(1 − |A2 (x)|2 )A2 (x) − (1 − |A1 (x)|2 )A1 (x)] · B(x)
1 d (1 − |At (x)|2 )At (x) · Bdt = 0 dt
1 = [(1 − |At |2 )(A2 − A1 ) · B − 2(At · (A2 − A1 ))(At · B)]dt. 0
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P. W. Bates, X.-B. Pan
Let η be a smooth cut-off function with bounded support, and let B = η2 (A2 − A1 ). We have
1 {|curl (η(A2 − A1 ))|2 + (1 − |At |2 )|η(A2 − A1 )|2 R3+ 0
− 2[At · (η(A2 − A1 ))]2 }dtd x
= |∇η × (A2 − A1 )|2 d x. R3+
Note that for 0 ≤ t ≤ 1, |At (x)| < √1 . Choose η such that η(x) = 1 for |x| < R, 3 η(x) = 0 for |x| > R + 1, and |∇η| ≤ 2. Then the above identity gives
|curl (A2 − A1 )|2 d x ≤ |∇η × (A2 − A1 )|2 d x + \B + B R+1 R
B R+
≤ 8π [(R + 1)3 − R 3 ] ≤ C R 2 . Thus (6.13) is true. Step 3. We show that (6.7) has a unique weak solution satisfying condition (6.8). H2 be two weak solutions of (6.7) satisfying (6.8). There exists a positive Let H1 and
number M <
4 27
such that curl H j L ∞ (R3+ ) ≤ M.
(6.14)
For any C 1 vector field B with bounded support, and for each j we have an integral equality similar to (3.4). We subtract one from the other, and choose B = η2 (H2 − H1 ), where η is a smooth function with bounded support. This choice of B is allowable by density. Noting that BT = 0 on ∂R3+ , we get
{[F(|curl H2 |2 )curl H2 − F(|curl H1 |2 )curl H1 ] · curl [η2 (H2 − H1 )] R3+ (6.15) 2 2 + η |H2 − H1 | }d x = 0. We compute as in the proof of Theorem 4.1 (see (4.7)) and find
a(x)|curl (η(H2 − H1 )) |2 + 2 Q(x)curl (η(H2 − H1 )) , curl (η(H2 − H1 )) R3+
+ η2 |H2 − H1 |2 d x
= a(x)|∇η × (H2 − H1 )|2 R3+
+ 2 Q(x)∇η × (H2 −H1 ), ∇η × (H2 − H1 )} d x, where Ht (x) = H1 (x) + t (H2 (x) − H1 (x)) , u t (x) = |curl Ht (x)|2 ,
1 a(x) = F(u t (x))dt, Q(x) = (qi j (x)), 0
qi j (x) = 0
1
F (u t (x))(curl Ht (x))i (curl Ht (x)) j dt.
(6.16)
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4 From (6.14), u t (x) ≤ M 2 < 27 . So F(u t (x)) and F (u t (x)), hence a(x) and qi j (x), are 3 uniformly bounded on R+ , a(x) ≥ 1, and Q(x) is non-negative definite for all x. From (6.16) we have
η2 |H2 − H1 |2 d x ≤ C |∇η|2 |H2 − H1 |2 d x.
R3+
R3+
Taking η = e−δr ξ(r ), where ξ(r ) is a cut-off function defined in the proof of Proposition 6.1, and δ 2 = 1/(4C), we have
e−2δr |H2 − H1 |2 d x B R+
≤ 2Cδ 2
+ B R+1
e−2δr |H2 − H1 |2 d x + 8Ce−2δ R
R +R+1 \B R+
|H2 − H1 |2 d x.
Letting R → ∞ and using (6.13) we find H2 − H1 ≡ 0. Thus the only weak solution is actually a smooth solution which has been obtained in Step 1. 7. Existence of Solutions in a Bounded Domain In this section we introduce a parameter µ > 0 into the boundary condition and consider the following boundary value problem: −λ2 curl [F(λ2 |curl H|2 )curl H] = H in ,
HT = µHeT on ∂,
(7.1)
where HeT is given and satisfies (5.1). We shall prove that there exists a constant µ∗ such that (7.1) has a solution for 0 < µ < µ∗ , and give a lower bound estimate of µ∗ when λ is small. This yields the existence part of Theorem 1. In the following we assume that ¯ R3 ) in the way as stated in Lemma 2.3 (ii), HeT has been extended to He ∈ C 2+α (, and satisfies (5.3). From Theorem 5.1, a weak solution of (7.1) satisfying (1.4) must be a classical solution. Lemma 7.1. Let be a bounded and simply-connected domain with C 4 boundary, λ > 0, and let HeT satisfy (5.1). Then there is a constant µ∗ = µ∗ (HeT , λ) > 0 such that the following conclusions hold: (i) For all 0 ≤ µ < µ∗ , Eq. (7.1) has a unique solution Hµ satisfying (1.4), and |curl Hµ (x)| attains its maximum only on ∂. ∗ (ii) µ → curl Hµ C 0 () ¯ is continuous for µ ∈ (0, µ ). (iii) We have 4 . (7.2) lim λcurl Hµ C 0 () ¯ = µ→µ∗ 27 Proof. While (i) and (ii) follow from Monneau [Mon], the proof of (iii) relies on the a priori estimate established in our Theorem 5.1. Step 1. Define µ∗ (HeT , λ) = sup{b > 0 : (7.1) has a solution satisfying (1.4) for each µ ∈ (0, b)}. (7.3)
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P. W. Bates, X.-B. Pan
From [Mon, Theorem 1.2], 0 < µ∗ (HeT , λ) < ∞. From Lemma 3.5, for any 0 < µ < µ∗ (HeT , λ), the solution Hµ of (7.1) is unique. From [Mon, Theorem 1.3], |curl H(x)| attains its maximum value only on ∂. Hence (i) is true for µ∗ = µ∗ (HeT , λ). Step 2. Now we prove (ii). For simplicity we assume λ = 1. We use the implicit function theorem as in [Mon], with a slight modification. Let µ0 ∈ (0, µ∗ ) and let Hµ0 be the solution of (7.1) associated with µ0 . Then we can choose δ0 > 0 such that 4 curl Hµ0 C 0 () ¯ ≤ M ≤ 27 − 2δ0 . For any 0 < δ < δ0 , we define a smooth function Fδ (t) which is equal to F(t) for t ≤
4 27
− δ. Then H = Hµ is a solution of
−curl [Fδ (|curl H|2 )curl H] = H in ,
HT = µHeT on ∂.
(7.4)
Write H = µHe + u and Hµ0 = µ0 He + uµ0 , where He is the extension of HeT . Then (7.4) can be written as an equation in u, −curl [Fδ (|curl (µHe + u)|2 )curl (µHe + u)] = µHe + u in , (7.5) uT = 0 on ∂. Define 2+α ¯ ¯ R3 ) : div u = 0 in , uT = 0 on ∂}, (, div 0) = {u ∈ C 2+α (, Ct0 ¯ div 0) = {Y ∈ C α (, ¯ R3 ) : div Y = 0 in }, C α (, 2+α ¯ ¯ div 0), and (, div 0) → C α (, Fδ : R × Ct0
Fδ (µ, u) = curl [Fδ (|curl (µHe + u)|2 )curl (µHe + u)] + µHe + u. From [Mon], Fδ is continuously differentiable in µ and u in the sense of Frechet, and ∂ 2+α ¯ α ¯ ∂u Fδ (µ0 , uµ0 ) is a homeomorphism from C t0 (, div 0) to C (, div 0). Applying the implicit function theorem to the operator equation Fδ (µ, u) = 0,
(7.6)
we conclude that there exists a > 0 such that for all µ ∈ (µ0 − a, µ0 + a), (7.6) has 2+α (, ¯ div 0). Choosing a small, we have a solution uµ in a neighborhood of uµ0 in Ct0 e + u is a solution of (7.1). From conclusion (i) it uµ − uµ0 C 2+α () ≤ δ. Then µH ¯ µ coincides with Hµ , and curl Hµ − curl Hµ0 C 0 () ¯ ≤ δ. Hence (ii) is true. Step 3. Now we prove (iii). If (7.2) were false, there exists ε > 0 and a sequence µ j → µ∗ such that curl Hµ j C 0 () ¯ ≤ M ≤
4 27
− ε for each j. From Theorem 4.1 we
have a uniform estimate for {Hµ }, and hence from Theorem 5.1 we have a uniform e C 2+α estimate sup j Hµ j C 2+α () ¯ ≤ C, where C = C(, λ, M, α, HT C 2+α (∂) ). Therefore we may pass to a subsequence and assume that, for some 0 < δ < α, ¯ R3 ) as j → ∞, where H∗ is the solution of (7.1) for µ = µ∗ , Hµ j → H∗ in C 2+δ (, and curl H∗ C 0 () ≤ M. Now we apply the implicit function theorem in a neighbor¯ hood of H∗ as in [Mon, Proposition 3.2], with an obvious modification, and find ρ > 0 ¯ R3 ) that satisfies such that, for µ ∈ (µ∗ − ρ, µ∗ + ρ), Eq. (7.1) has a solution in C 2+α (, ∗ (1.4). This contradicts the definition of µ . Hence (iii) is true. H2
Remark 7.2. We conjecture that when µ = µ∗ (HeT , λ), Eq. (7.1) has a unique solution 4 H such that λcurl H L ∞ () = 27 , and the maximum value of |curl H(x)| is obtained
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597
4 only on ∂. Thus, λ|curl H(x)| < 27 for all x ∈ . If this is true, then we can use the comparison in subdomains of to prove the uniqueness of solutions for µ = µ∗ . For a discussion on this point in the 2-dimensional case see [BBC]. Lemma 7.3. Let be a bounded and simply-connected domain with C 4 boundary, and let HeT ∈ C 2+α (∂, R3 ) satisfy the conditions of Theorem 5.1. Then 5 ∗ e lim inf µ (HT , λ) ≥ (HeT C 0 (∂) )−1 . (7.7) λ→0 18 In particular, if He = h is a unit vector, then
∗
lim inf µ (hT , λ) ≥ λ→0
5 . 18
(7.8)
Proof. We use a blow-up argument that was used in [PK] for the two dimensional case,
4 with a slight modification. Let us fix an arbitrary positive number M0 < 27 . Let {λn } be any sequence of positive numbers such that λn → 0 as n → ∞. From Lemma 7.1 (i), (ii), for each n there exists µn , 0 < µn < µ∗ (λn , HeT ), such that (1.3) has a unique solution Hµn which satisfies
λn curl Hµn C 0 () ¯ = M0 .
(7.9)
There exists xn ∈ ∂ such that |curl Hµn (xn )| = curl Hµn C 0 () ¯ . Passing to a subsequence we may assume that xn → x0 and µn → µ˜ as n → ∞. Let us define rescaled fields ˜ µn (y) = Hµn (xn + λn y), y ∈ n = − xn . H λn Using (7.9) we can show that supn Hµn L ∞ () < ∞. Then we can use the arguments in the proofs of Theorems 4.1 and 5.1 to show that, for any fixed R > 0, ˜ µn C 2+α (B (0)∩ ) ≤ C(M0 , R). The proof is omitted here, as it is similar to the H n 2R proof of Lemma 8.1. ˜ µn , such that H ˜ µn → H ˜ in Therefore we can pass to a subsequence, still denoted by H 2+α 3 3 ˜ Cloc (R+ , R ) as n → ∞, and H is a solution of (6.7) on the half-space, with boundary ˜ T = µh ˜ 0 , where h0 = HeT (x0 ) is tangential to ∂R3+ , and curl H˜ L ∞ (R3+ ) = condition H ˜ ˜ 0 = ρ(cos θ, sin θ, 0), |curl H(0)| = M0 . Since M0 > 0, we see that h0 = 0. Write µh ˜ ∞ 3 = where ρ > 0. Then we use Proposition 6.2 and its proof to find M0 = curl H L (R + ) √ 5 M(ρ), where M(ρ) was defined in (6.9). M(ρ) is strictly increasing on [0, 18 ], and 4 , and ρ(M) is strictly hence has an inverse function ρ(M) defined for 0 ≤ M ≤ 27 5 such that ρ˜ = ρ(M0 ). Since the increasing. Thus there exists a unique ρ˜ < 18 solution of the limiting equation (6.7) is unique, the full sequence must converge. So ˜ 0 | = ρ = ρ(M0 ). Therefore limn→∞ µn |HeT (xn )| = µ|h lim inf µ∗ (HeT , λ)HeT C 0 (∂) ≥ lim µn |HeT (xn )| ≥ ρ(M0 ). λ→0
Now we let M0 approach
n→∞
4 27 .
Noting that ρ(
4 27 )
=
5 18 ,
(7.7) is proved.
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P. W. Bates, X.-B. Pan
From Theorems 4.1, 5.1, 5.2 and Lemma 7.3 we get the following Theorem 7.4. Let be a bounded and simply-connected domain with C 4 boundary and HeT satisfy (1.10). Then for all λ > 0 small we have: ¯ R3 ) satisfying (i) Equation (1.3) has a unique solution H ∈ C 3 (, R3 ) ∩ C 2+α (, (1.4), and e HC 2+α () ¯ ≤ C 1 (, λ, HT C 2+α (∂) ).
(7.10)
¯ R3 ) satisfying (ii) Equation (1.1) has a unique solution A ∈ C 3 (, R3 ) ∩ C 2+α (, (1.2), and e AC 2+α () ¯ ≤ C 2 (, λ, HT C 2+α (∂) ).
(7.11)
¯ R3 ) and If ∂ is of class C 5 , then A ∈ C 4 (, R3 ) ∩ C 3+α (, e AC 3+α () ¯ ≤ C 3 (, λ, HT C 2+α (∂) ).
8. Asymptotic Behavior of Solutions with Small λ We use the notation (δ) = {x ∈ : dist(x, ∂) ≥ δ}. Lemma 8.1. Let and HeT satisfy the conditions of Theorem 4.1. For each λ small, let 4 Hλ be a weak solution of (1.3) satisfying (1.14), where M < 27 is independent of λ. Then we have: (i) For any λ0 > 0, there exists C(λ0 ) > 0 such that Hλ L ∞ () ≤ C(λ0 ), for all 0 < λ ≤ λ0 .
(8.1)
(ii) For any sequence {ρλ } satisfying ρλ ≤ λc , ρλ → ∞ as λ → 0, lim
sup
λ→0 x∈(λρλ )
|Hλ (x)| = 0.
(8.2)
¯ R3 ) Proof. Step 1. We first assume (8.1) is true and prove (8.2). Let He ∈ H 2 (, e be the extension of HT as stated in Lemma 2.3 (i). From (1.14) and Corollary 4.2, ¯ R3 ) for any α ∈ (0, 1/2). Let x λ ∈ (λρλ ) and define Hλ ∈ C α (, − xλ . λ
(8.3)
e HλT = HλT on ∂λ ,
(8.4)
Hλ (x) = Hλ (x λ + λx), Hλe (x) = He (x λ + λx), x ∈ λ = Then Hλ is a weak solution of the equation −curl F(|curl Hλ |2 )curl Hλ = Hλ in λ , and satisfies
curl Hλ
L ∞ (
λ)
≤M<
4 . 27
(8.5)
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Since ρλ → ∞ as λ → 0, and since dist(x λ , ∂) ≥ λρλ , for any R > 0, there exists λ(R) > 0 such that B10R ⊂ λ for all 0 < λ < λ(R). Let η R be a cut-off function supported in B2R such that η R = 1 on B R and |∇η R (x)| ≤ C/R. Multiplying (8.4) by 2 H and integrating by parts, we get η4R λ
2 {F(|curl Hλ |2 )|curl (η4R Hλ )|2 + η4R |Hλ |2 }d x B8R
= F(|curl Hλ |2 )|∇η4R × Hλ |2 d x. B8R
From this and using (8.1) and (2.1), Hλ 2H 1 (B ) 2R
≤
η2R Hλ 2H 1 (B ) 4R
≤ C1 (R) B4R
{|curl Hλ |2 + |Hλ |2 }d x ≤ C2 (R).
Then we can apply the interior H 2 estimate (see Step 1 of the proof of Theorem 4.1, with λ = 1) to Hλ on B2R , and find (see (4.8)) Hλ 2H 2 (B
R)
≤ C3 (R, M)Hλ 2H 1 (B
2R )
≤ C4 (R, M).
(8.6)
2 (R3 , R3 ). For any sequence λ → 0, we From (8.6), {Hλ }0<λ≤1 is bounded in Hloc j 2 (R3 , R3 ) can find a subsequence, still denoted by λ j , such that Hλ j → H0 weakly in Hloc 1 (R3 , R3 ). Thus, curl H 1 3 3 and strongly in Hloc λ j → curl H0 weakly in Hloc (R , R ) and strongly in L 2loc (R3 ). So |curl H0 (x)| ≤ M for a.e. x ∈ R3 . Since F(t) is continuous and bounded for t ∈ [0, M], from (8.5) and the Lebesgue dominated convergence thep orem, F(|curl Hλ j |2 )curl Hλ j → F(|curl H0 |2 )curl H0 strongly in L loc (R3 , R3 ) for any 1 < p < ∞. Therefore H0 is a weak solution of (6.2). From Proposition 6.1 (ii) we 2 (R3 , R3 ) and strongly in H 1 (R3 , R3 ) see that H0 ≡ 0. Thus Hλ j → 0 weakly in Hloc loc as j → ∞. Since this is true for any sequences, we must have, for any R > 0 fixed, Hλ H 1 (B R ) → 0 as λ → 0. Now we use the interior H 2 estimate (8.6) and the Sobolev imbedding theorem to find that, as λ → 0,
Hλ C α (B R ) ≤ C5 (R)Hλ H 2 (B R ) ≤ C6 (M, R)Hλ H 1 (B2R ) → 0. In particular, |Hλ (x λ )| = |Hλ (0)| ≤ Hλ L ∞ (B R ) → 0. Thus (8.2) is true. ¯ Step 2. Now we prove (8.1). Suppose (8.1) were not true. We can find λ j → 0, x λ j ∈ , λ λ λ j j j such that m j = |H (x )| = H L ∞ () → ∞. For simplicity we denote λ j by λ. ˆ λ = ελ Hλ , where ελ = 1/m j . For this choice of x λ we define Hλ as in (8.3) and set H ˆ Then Hλ satisfies ˆ λ in λ , ˆλ =H ˆ λT = ελ He on ∂λ , (8.7) H −curl F(|curl Hλ |2 )curl H λT ˆ λ L ∞ (λ ) = 1. As in Step 1 above we can show that, for all λ small, ˆ λ (0)| = H and |H ˆ λ H 1 (B ∩ ) ≤ C7 (R, M, He H 1/2 (∂ ∩B ) ). H λT λ λ R 2R
(8.8)
ˆ λ } is bounded in H 2 : In the following we shall show that {H loc ˆ λ H 2 (B ∩ ) ≤ C8 (R, M, He H 3/2 (∂ ∩B ) ). H λT λ λ R 2R
(8.9)
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P. W. Bates, X.-B. Pan
2 estimate can be established as in Step 1 of the proof of Theorem 4.1. The interior Hloc Write
1 Hλ,t,σ (x) = Hλ (x) + tσ hλ,σ (x), [Hλ (x + σ e) − Hλ (x)], σ u λ = |curl Hλ |2 , u λ,t,σ = |curl Hλ,t,σ |2 , 1 ˆ ˆ λ (x)], ˆ λ,t,σ (x) = H ˆ λ (x) + tσ hˆ λ,σ (x). hˆ λ,σ (x) = [H H λ (x + σ e) − H σ
hλ,σ (x) =
Observe that ˆ λ (x + σ e) − F(|curl Hλ (x)|2 )curl H ˆ λ (x) F(|curl Hλ (x + σ e)|2 )curl H
1 ˆ λ,t,σ dt =σ F(u λ,t,σ )curl hˆ σ + 2F (u λ,t,σ )(curl Hλ,t,σ · curl hλ,σ )curl H 0
=σ
0
1
F(u λ,t,σ )curl hˆ σ + 2F (u λ,t,σ )(curl Hλ,t,σ · curl hˆ λ,σ )curl Hλ,t,σ dt
= σ aσ (x)curl hˆ σ + 2σ Q σ (x)curl hˆ σ , where
aσ (x) =
1
0
F(u λ,t,σ )dt,
qσ,i j (x) =
1 0
Q σ (x) = (qσ,i j (x)),
F (u λ,t,σ )(curl Hλ,t,σ )i (curl Hλ,t,σ ) j dt.
Hence for all B ∈ H 1 (λ , R3 ) with compact support in λ ,
aσ (x)curl hˆ σ · curl B + 2 Q σ (x)curl hˆ σ , curl B + hˆ σ · B d x = 0. λ
2 Then we can proceed as in Step 1 of the proof of Theorem 4.1 to get the interior Hloc estimate. 2 estimate, we assume for simplicity that the = ∂ ∩ ¯ To get the boundary Hloc λ is flat. Using the above observation, we proceed as in Step 2 of the proof of Theorem 4.1 2 estimate for the tangential derivatives. To estimate the normal to get the boundary Hloc derivatives, we compute as in the third step of the proof of Theorem 4.1 and write (8.7) in the form ˆ λ − F (u λ ) ∇|curl H ˆ λ · curl Hλ | × curl Hλ = H ˆ λ, F(u λ )H
ˆ λ in terms of curl Hλ and ∂i j H ˆ λ with (i, j) = (3, 3), from which we can solve for ∂33 H see (4.15) and (4.18). Using (8.5) we get ˆ λ L 2 (B ∩ ) ≤ C(, M, R){H ˆ λ L 2 (B ∩ ) }. ˆ λ L 2 (B ∩ ) + ∂i j H ∂33 H λ λ λ R 2R 2R (i, j)=(3,3)
2 estimate for the tangential derivatives we derive (8.9). From this and the Hloc ˆλ → H ˆ ˆ λ , such that H From (8.9), we can find a subsequence, still denoted by H 2 1 ˆ is a weak solution of the limiting weakly in Hloc and strongly in Hloc as λ → 0, and H
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ˆλ → H ˆ in C α for some α ∈ (0, 1/2). In particular equations of (8.7). Moreover, H loc ˆ |H(0)| = 1. We only need to consider the following two cases. Case 1. limλ→0 λ−1 dist(x λ , ∂) = ∞. Then the limiting equation of (8.7) is (6.2). ˆ ≡ 0, which is a contradiction. From Proposition 6.1 (ii) we have H Case 2. dist(x λ , ∂) ≤ C. Then after a translation in the x3 direction, the limiting ˆ = 0, a equation of (8.7) is (6.7) with h = 0. From Proposition 6.2 (ii) we have H contradiction again. Let ∂(HeT ) and ∂(hT ) be the sets defined in the introduction. Lemma 8.2. Let and HeT satisfy the conditions of Theorem 5.1. For each λ small, let 4 Hλ be a weak solution of (1.3) satisfying (1.14), where M < 27 is independent of λ. (i) Let x λ be any point on ∂ and assume x λ → x 0 as λ → 0. Then in local coordi˜ λ (y) converges in C 2+α (R3+ , R3 ) to the nates near x 0 , the rescaled vector field H loc unique solution of the equation −curl F(|curl H|2 )curl H = H in R3+ , HT = h˜ on ∂R3+ , (8.10) where h˜ = HeT (x 0 ). (ii) Let M(ρ) be the function defined in (6.9). We have lim max λ2 |curl Hλ (x)|2 = M(HeT C 0 (∂) ).
λ→0 x∈ ¯
(8.11)
(iii) Let P λ be a maximum point of |curl Hλ (x)| and assume P λ → P for a sequence λn → 0. Then P ∈ ∂(HeT ). Proof. Step 1. Proof of (i). We consider the rescaled vector field Hλ (x) defined in (8.3). Then Hλ satisfies (8.1) and (8.5). Hence we can apply the arguments in the proofs of Theorems 4.1 and 5.1 (see Step 2 in the proof of Lemma 8.1), and conclude that, for any R > 0, there exists a constant C depending on , HeT C 2+α (∂) , M, R and α, but independent of λ, such that Hλ C 2+α (B + ∩λ ) ≤ C. R
(8.12)
Now for each λ, we adopt the local coordinates y introduced by straightening the boundary in a neighborhood around x λ such that y = 0 corresponds to x λ , and the inner normal vector of ∂ at x λ points in the positive y3 direction (see Appendix A.1). Let ˜ λ (y) = H ˜ λ (λy), which are ˜ λ (y) = Hλ (x). Then we define rescaled vector fields H H well-defined for y ∈ Br+λ , where rλ → ∞ as λ → 0. From (8.5), ˜ λ (y)| ≤ M for all y ∈ Br+ . |curl H λ
(8.13)
From (8.12) we have ˜ λ C 2+α (B + ) ≤ C, H R
HeT C 2+α (∂) ,
where C depends on , M, R and α, but is independent of λ. Thus for any sequence λ j → 0, we can choose a subsequence, still denoted by λ j , such that, for ˜λ →H ˜ 0 in C 2+δ (R3+ , R3 ). any 0 < δ < α, H loc
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P. W. Bates, X.-B. Pan
˜ 0 , we write Eq. (1.3) in the new coordinates to get a quasilinear To find the limit H ˜ λ ∈ C 2+α (R3+ , R3 ), and because of condition (8.13), ˜ λ (y). Because H elliptic system for H loc we can write this system in the form of an elliptic linear system with C 1+α coefficients. ˜ 0 is a Then, as in the proof of Lemma 8.1, we can take the limit λ j → 0 and find that H ˜ classical solution of (8.10). From the choice of y coordinates, h is orthogonal to the y3 direction. Moreover, for a.e. y ∈ R3+ , ˜ 0 (y)| ≤ M, for all y ∈ R3+ . |curl H
(8.14)
˜ 0 ≡ 0. If h˜ = 0, we write From the proof of Proposition 6.2 we see that, if h˜ = 0 then H ˜ h˜ = |h|(cos θ, sin θ, 0) to get ˜ 0 (y) = − f (y3 )(cos θ, sin θ, 0), H ˜ Since the solution to the where f is the solution of (6.12) with f (0) = −ρ = −µ|h|. ˜ λ must converge to H ˜0 limiting equation is unique, we conclude that the full sequence H 2+δ 3 3 in Cloc (R+ , R ) as λ → 0. 2+α (R3 , R3 ). From (8.13) and (8.14) we Now we show that the convergence is in Cloc + see that ˜ 0 (y) + (1 − t)H ˜ λ (y))|2 ) ≤ C, 0 < F (|curl (t H ˜ 0 (y) + (1 − t)H ˜ λ (y))|2 ) ≤ C for all y ∈ R3+ , 0 ≤ t ≤ 1. 0 < F (|curl (t H ˜ 0 we derive an equation for H ˜λ −H ˜ 0, ˜ λ and Eq. (8.10) for H From the equation for H 1+α ˜ ˜ coefficients. Then which can be written as a linear equation for Hλ − H0 with C ˜λ −H ˜ 0 C 2+α (B + ) , and we can apply the Schauder estimates to derive an estimate of H R finally we find that 2+α ˜ 0 in Cloc ˜λ →H (R3+ , R3 ) as λ → 0. H
(8.15)
Thus, conclusion (i) is proved. Step 2. Proof of (ii) and (iii). Let P λ ∈ ∂ be the maximum point of |curl Hλ (x)|. Choose λn → 0 such that P λn → P. Then conclusion (i) is valid. In particular (8.15) holds. Let ˜ qT = h(q). ˜ ˜ h(q) = HeT (q), and let Hq be the solution of (8.10) with boundary data H 2 ˜ From the proof of Proposition 6.2, curl Hq L ∞ (R3 ) = M(|h(q)|). The function M(ρ) + 5 is strictly increasing when 0 < ρ < 18 . Thus ˜ sup curl Hq 2L ∞ (R3 ) = max M(|h(q)|) = max M(HeT C 0 (∂) ),
q∈∂
q∈∂
+
x∈∂
which is achieved on ∂(HeT ). From (8.15) we find 2 2 lim λ2n |curl Hλn (P λn )|2 = lim λ2n curl Hλn C 0 (∂) = lim curl Hλn C 0 (∂)
n→∞
n→∞
˜ λn 2 0 = lim curl H C (∂ n→∞
λn )
=
n→∞
curl H P 2L ∞ (R3 ) +
˜ = M(|h(P)|).
So we have lim λ2 |curl Hλn (P λn )|2 n→∞ n
˜ = max M(|h(P)|) = M(HeT C 0 (∂) ). P∈∂
This verifies (8.11) and proves conclusion (ii). Conclusion (iii) follows from (8.11) and (8.15) immediately.
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Proof of Theorem 1. Under condition (1.10), the existence and uniqueness of the solutions Hλ satisfying (1.4) and their regularity have been proved in Theorem 7.4. So conclusion (i) is true. From the second inequality in (1.10) and Lemma 7.3, we can choose c0 > 1 and small λ∗ > 0 such that µ∗ (HeT , λ) > c0 > 1 for all 0 < λ ≤ λ∗ . Now we claim that
sup λcurl Hλ L ∞ () <
0<λ≤λ∗
4 . 27
Suppose (8.17) were false. Then there exists a sequence λn such that 4 λn λn curl H L ∞ () → . 27
(8.16)
(8.17)
(8.18)
We may assume λn → λ0 . If λ0 > 0, from Lemma 7.1 (iii) we have µ∗ (HeT , λ0 ) = 1, which contradicts (8.16). Hence λ0 = 0, and thus λn → 0 as n → ∞. Fix a positive
From (8.16), (8.18) and the definition of µ∗ (HeT , λ), 4 there exists N (M) > 0 such that, if n > N (M) then M < λn curl Hλn L ∞ () ≤ 27 . Then from Lemma 7.1 (iii), we can choose a positive number µn < 1 such that for ˆ n , and λn curl H ˆ n L ∞ () = M. Now we λ = λn and µ = µn , (7.1) has a solution H argue as in the proof of Lemma 7.3 and show that, after passing to a subsequence, lim inf n→∞ µn HeT C 0 (∂) ≥ ρ(M), where ρ(M) is the inverse function of M(ρ). 4 Since µn < 1, we find HeT C 0 (∂) ≥ ρ(M). Now we let M approach 27 and con 4 5 ) = 18 , contradicting (1.10). So (8.17) is true. clude that HeT C 0 (∂) ≥ ρ( 27 Using (8.17) and Lemma 8.1, we get Conclusion (ii). Using (8.17) and Lemma 8.2 (ii) and (iii), we get Conclusions (iii) and (iv). Hence, Theorem 1 is proved. number M slightly less than
4 27 .
Proof of Theorem 1 . In the proof of Theorem 5.2 (also see Theorem 7.4) we have shown ¯ R3 ) is a solution of (1.3) satisfying (1.4), then (1.1) that, if H ∈ C 3 (, R3 ) ∩ C 2+α (, 3 ¯ R3 ). So Theorem 1 follows from has a solution A = B − ∇ϕ ∈ C (, R3 ) ∩ C 2+α (, Theorem 1. Appendix A. Some Technical Details A.1. Computations in local coordinates near boundary. Let us briefly recall the local coordinates near boundary ∂ determined by a diffeomorphism that straightens a piece of surface, see [P, Sect. 3]. Let us fix a point x0 ∈ ∂, and introduce new variables y1 , y2 such that ∂ can be represented (at least near x0 ) by r = r(y1 , y2 ), and r(0, 0) = x0 . Here and henceforth we let y = (y1 , y2 ) and use the notation r j (y) = ∂ y j r(y), ri j = ∂ yi y j r(y), etc. Let n(y) =
r1 (y) × r2 (y) . |r1 (y) × r2 (y)|
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P. W. Bates, X.-B. Pan
We choose (y1 , y2 ) in such a way that n(y) is the inward normal of ∂, and that the y1 - and y2 -curves on ∂ are the lines of principal curvature; thus, r1 (y) and r2 (y) are orthogonal to each other. Let gi j (y) = ri (y)·r j (y), g(y) = det(gi j (y)) = g11 (y)g22 (y). For scalar functions f , let f , j denote the partial derivative with respect to y j . Let us define a map F by x = F(y, z) = r(y1 , y2 ) + zn(y1 , y2 ). F is a diffeomorphism from a ball B R (0) onto a neighborhood U of the point x0 , and it maps the half ball B R+ (0) onto a subdomain U ∩ , and maps the disc {(y1 , y2 , 0) : y12 + y22 < R 2 } onto a subset of ∂. Let G i j (y, z) = ∂i F · ∂ j F and let G i j (y, z) denote the elements of the inverse of the matrix (G i j (y, z)). Then G j j (y, z) = g j j (y)[1 − κ j (y)z]2 , G j j =
1 , G jj
j = 1, 2,
G 12 = G 13 = G 23 = G 12 = G 13 = G 23 = 0, G 33 = G 33 = 1, G(y, z) ≡ det(G i j (y, z)) = G 11 (y, z)G 22 (y, z). On U we have an orthogonal coordinate framework {E1 , E2 , E3 }, where E j (y, z) =
r j (y) ∂jF = √ , |∂ j F| gjj
j = 1, 2, E3 (y, z) =
∂3 F = n(y). |∂3 F|
¯ for any x0 ∈ ∂, we can write B in a neighborGiven a vector field B defined on , hood of x0 in the new variables (y, z) ∈ B R+ = F −1 (U ∩ ) as follows: ˜ B(y, z) = B(F(y, z)) =
3
G j j (y, z)b j (y, z)∂ j F(y, z) =
j=1
3
B˜ j (y, z)E j (y, z),
j=1
b j (y, z) = B(F(y, z)) · ∂ j F(y, z),
b j (y, z) B˜ j (y, z) = . G j j (y, z)
(A.1)
R˜ j (y, z)E j (y, z),
(A.2)
We compute, at the point x = F(y, z), curl B(x) =
3 j=1
where 1 1 [∂2 ( B˜ 3 G 33 ) − ∂3 ( B˜ 2 G 22 )] = √ [∂2 b3 − ∂3 b2 ], R˜ 1 (y, z) = √ G 22 G 33 G 22 1 1 [∂3 ( B˜ 1 G 11 ) − ∂1 ( B˜ 3 G 33 )] = √ [∂3 b1 − ∂1 b3 ], R˜ 2 (y, z) = √ G 33 G 11 G 11 1 1 [∂1 ( B˜ 2 G 22 ) − ∂2 ( B˜ 1 G 11 )] = √ [∂1 b2 − ∂2 b1 ]. R˜ 3 (y, z) = √ G 11 G 22 G 11 G 22 In the following, to save the notation, we shall write B˜ also by B.
Instability in Meissner State
605
A.2. Proof of (2.2). Suppose (2.2) were false. Then there exists a sequence {B j } with B j L 2 () = 1, such that div B j L 2 () → 0, curl B j L 2 () → 0, and either ν · B j H 1/2 (∂) → 0 or ν ×B j H 1/2 (∂) → 0. From (2.1), {B j } is bounded in H 1 (, R3 ). After passing to a subsequence we may assume that B j → B weakly in H 1 (, R3 ) and strongly in L 2 (, R3 ). Then B L 2 () = 1, div B L 2 () = 0, curl B L 2 () = 0, and either ν · B H 1/2 (∂) = 0 or ν × B H 1/2 (∂) = 0. Since is simply connected and curl B = 0, there exists φ ∈ H 2 () such that B = ∇φ. Then φ = div B = 0. 1/2 (∂). Moreover, either ∂φ ∂ν = 0 or (∇φ)T = 0 on ∂ in the sense of trace in H Then φ must be a constant on , and so B = ∇φ = 0, which is impossible as we have B L 2 () = 1. A.3. Proof of Proposition 2.1. The case where k = 0 has been proved in [BW]. We only prove (2.4) for k = 1. The general case can be proved by induction. ¯ curl B ∈ C 1+α (, ¯ R3 ), and either ν · B or Step 1. Let us assume div B ∈ C 1+α (), ν × B ∈ C 2+α (∂, R3 ). We shall use (2.3) to derive (2.4). Note that, for bounded and simply-connected domains, (2.3) can be written as
ν·B
BC 1+α () . (A.3) ¯ ≤ C(, α) div BC α () ¯ + curl BC α () ¯ +
ν × B 1+α C
(∂)
This is because
BC 0 () ¯
ν·B
≤ C(, α) ∇BC α () . ¯ +
ν × B C α (∂)
(A.4)
Step 2. We first derive an interior estimate of D 2 B. Let u = ∂ j B, j = 1, 2, 3. Let be a subdomain of , and let η be a smooth cut-off function with compact support in such that η = 1 on . Applying (2.3) to ηu we have ∇(ηu)C α () ¯ ≤ C(, α){div (ηu)C α () ¯ + curl (ηu)C α () ¯ } ≤ C(, α){η div uC α () ¯ + ∇η · uC α () ¯ + η curl uC α () ¯ + ∇η × uC α () ¯ }
≤ C(, , α){div uC α () ¯ + curl uC α () ¯ + uC α () ¯ }
≤ C(, , α){div BC 1+α () ¯ + curl BC 1+α () ¯ + ∇BC α () ¯ }.
Now we use (2.3) to control the term ∇BC a () ¯ in the right-hand side, and get
∇BC 1+α (¯ )
ν·B
≤ C(, , α) div BC 1+α () . ¯ +curl BC 1+α () ¯ +
ν × B C 1+α (∂)
(A.5) Step 3. Next we consider a subset D = ∩ B R (x0 ) with x0 ∈ ∂. As in Sect. A.1, we choose R > 0 small but independent of x0 such that on B R (x0 ) there exists a new coordinate system (y1 , y2 , z) which corresponds to an orthonormal framework {E1 , E2 , E3 } such that, restricted to ∂ ∩ B R (x0 ), E1 , E2 are tangential to ∂, and E3 = −ν.
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P. W. Bates, X.-B. Pan
We first consider the “tangential” derivative u = ∂ yi B, i = 1, 2. Let η be a smooth cutoff function supported in B R (x0 ) such that η = 1 on B R/2 (x0 ). Write R = B R (x0 ) ∩ and R = B R (x0 ) ∩ ∂. Applying (2.3) to ηu we have
ην · u
∇(ηu)C α () ¯ ≤ C(, α) div (ηu)C α () ¯ +curl (ηu)C α () ¯ +
ην × u C 1+α (∂) ≤ C(, R, α) div uC α (¯ R ) + curl uC α (¯ R ) + uC α (¯ R )
ν·u
+
ν × u
C 1+α ( R )
+ uC α ( R )
≤ C(, R, α) div BC 1+α (¯ R ) + curl BC 1+α (¯ R ) + BC 1+α (¯ R )
ν·u
+
,
ν × u 1+α C ( )
(A.6)
R
where we have used the fact BC 1+α ( R ) ≤ BC 1+α (¯ R ) . As in Step 2 we can control the term BC 1+α (¯ R ) by using (A.3). Note that ν · u = ∂ yi (ν · B) − (∂ yi ν) · B and ν × u = ∂ yi (ν × B) − (∂ yi ν) × B. So ν · uC 1+α ( R ) ≤∂ yi (ν · B)C 1+α ( R ) + (∂ yi ν) · BC 1+α ( R ) ≤ν · BC 2+α ( R ) + C(, R, α)BC 1+α ( R ) , ν × uC 1+α ( R ) ≤∂ yi (ν × B)C 1+α ( R ) + (∂ yi ν) × BC 1+α ( R ) ≤ν × BC 2+α ( R ) + C(, R, α)BC 1+α ( R ) . Plugging these back into (A.6) and using BC 1+α ( R ) ≤ BC 1+α (¯ R ) , we get ∂ yi BC 1+α (¯ R/2 ) ≤ C(, R, α) div BC 1+α (¯ R ) + curl BC 1+α (¯ R )
ν·B
+BC 1+α (¯ R ) +
.
ν × B 2+α C ( )
(A.7)
R
Step 4. Now we consider the normal derivative. Let w = ∂z B. Then w = −∂ν B on R . For the cut-off function used above, we apply (2.3) to ηw to get ∇(ηw)C α () ¯ ≤ C(, R, α) div BC 1+α ( ¯ R ) + curl BC 1+α ( ¯ R ) + BC 1+α ( ¯ R)
ν × w
+
ν·w
C 1+α ( R )
.
(A.8)
¯ R3 ), We claim that there exists C = C(, R, α) such that, for all B ∈ C 2+α (, ν × ∂ν BC 1+α ( R ) ≤ C{ν × curl BC 1+α ( R ) + ν · BC 2+α ( R ) + BC 1+α ( R ) }, ν · ∂ν BC 1+α ( R ) ≤ C{div BC 1+α ( R ) + ν × BC 2+α ( R ) + BC 1+α ( R ) }.
(A.9)
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607
We will prove (A.9) in Step 5. From (A.8) and (A.9) (either all choose the first option or all choose the second option), and using the obvious facts div BC 1+α ( R ) ≤ div BC 1+α (¯ R ) , ν × curl BC 1+α ( R ) ≤ C(, R, α)curl BC 1+α (¯ R ) , we get ∇∂z BC α (¯ R ) ≤ C(, R, α) div BC 1+α (¯ R ) + curl BC 1+α (¯ R )
ν·B
+BC 1+α (¯ R ) +
.
ν × B 2+α C ( )
(A.10)
R
From (A.5), (A.7), (A.10) (either all choose the first option or all choose the second option) we conclude that ∇BC 1+α () ¯ ≤ C(, α) div BC 1+α () ¯ + curl BC 1+α () ¯ + BC 1+α () ¯
ν·B
+
ν × B
C 2+α (∂)
.
Combing this with (A.3) we finally get (2.4). Step 5. Proof of (A.9). In R we write B as in (A.1). Then on R we have ν × ∂ν B = −∂ν B˜ 1 E2 + ∂ν B˜ 2 E1 +
3
B˜ j ν × ∂ν E j .
j=1
Since ν × curl B =
2 j=1
=−
! " 2 ∂ y j B˜ 3 ∂z B˜ j E j − + κ j (y) B˜ j E j √ gjj
2
j=1
∂ν B˜ j E j +
j=1
$ 2 # ∂ y j (ν · B) ˜ − κ j (y) B j E j , √ gjj j=1
we have ∂ν B˜ 1 E1 + ∂ν B˜ 2 E2 = −ν × curl B +
$ 2 # ∂ y j (ν · B) ˜ − κ (y) B √ j j Ej, gjj j=1
ν × ∂ν BC 1+α ( R )
≤ C1 {∂ν B˜ 1 E2 − ∂ν B˜ 2 E1 C 1+α ( R ) + BC 1+α ( R ) } ≤ C2 {∂ν B˜ 1 E1 + ∂ν B˜ 2 E2 C 1+α ( R ) + BC 1+α ( R ) } ≤ C3 {ν × curl BC 1+α ( R ) + ∂ y1 (ν · B)C 1+α ( R ) + ∂ y2 (ν · B)C 1+α ( R ) + BC 1+α ( R ) } ≤ C4 {ν × curl BC 1+α ( R ) + ν · BC 2+α ( R ) + BC 1+α ( R ) },
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where the C j s depend only on , R and α. This verifies the first inequality in (A.9). Since 2 √ √ 1 div B = √ [ ∂ y j ( G B˜ j ) + ∂z ( G B˜ 3 )], G j=1
we have √ √ 1 ˜ B j ∂ y j G + B˜ 3 ∂z G]. ν · ∂ν B = ∂z B˜ 3 = div B − ∂ y1 B˜ 1 − ∂ y2 B˜ 2 − √ [ G j=1 2
From this and ν × B = B˜ 2 E1 − B˜ 1 E2 , we find ν · ∂ν BC 1+α ( R ) ≤div BC 1+α ( R ) + ∂ y1 B˜ 1 + ∂ y2 B˜ 2 C 1+α ( R ) + C5 BC 1+α ( R ) ≤C6 {div BC 1+α ( R ) + ν × BC 2+α ( R ) + BC 1+α ( R ) }, where the C j s depend only on , R and α. The second inequality in (A.9) follows from this inequality. A.4. Proof of Lemma 2.3. Let us define Ht (, curl, div 0) = {H ∈ L 2 (, R3 ) : curl H ∈ L 2 (, R3 ), div H = 0 in , HT = HeT on ∂}, Ht0 (, curl, div0) = {H ∈ Ht (, curl, div 0) : HT = 0 on ∂}, 1 (, R3 ) = {B ∈ H 1 (, R3 ) : BT = 0 on ∂}. Ht0
Consider the following minimization problem: min
H∈Ht (,curl,div 0)
curl H2L 2 () .
From (2.1) we see that a minimizing sequence is bounded in H 1 (, R3 ). Thus it is easy to see that a minimizer exists. Let H be a minimizer. Then for all B ∈ Ht0 (, curl, div0) it holds that
curl H · curl B d x = 0. (A.11)
1 (, R3 ). Hence It was observed in [Mon, p. 924] that curl Ht0 (, curl, div0) = curl Ht0 1 3 (A.11) holds for all B ∈ Ht0 (, R ), that is, H is a weak solution of the equation
curl 2 H = 0 and div H = 0 in ,
HT = HeT on ∂.
Hence (2.7) follows from (2.4). Now assume that (2.11) holds. Then J = curl H satisfies curl J = 0 and div J = 0 in , and ν · J = 0 on ∂. Hence J = 0. Since is simply-connected, there exists a scalar function φ such that ∇φ = H. The regularity of φ follows from the regularity of H.
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A.5. Proof of Lemma 2.4 (i). We keep the notation used in Sect. A.1. Note that ν = −n. So on ∂ we have ν · curl B(x) = − √
1 [∂1 b2 (y, 0) − ∂2 b1 (y, 0)]. g11 (y)g22 (y)
For x ∈ ∂ near x0 , we represent the tangential component BT by BT (y, 0) = BT (F(y, 0)) =
2 j=1
b j (y, 0) E j (y, 0). G j j (y, 0)
From these two equalities we see immediately that ν · curl B is determined by BT . Thus, conclusion (i) in Lemma 2.4 is true. Acknowledgements. The authors would like to thank the referee for the valuable comments on the first version of this paper. This work was completed when the second author, Pan, was visiting the Department of Mathematics, Michigan State University in the Spring semester of 2005, and he would like to thank the department for hospitality. The main results of this paper were reported by the second author at the Workshop on Ginzburg-Landau Theory and Related Topics, held at Chinese Academy, Beijing in June 28–30 of 2005. This work was partially supported by NSF DMS 0200961 and DMS 0401708 (to Bates), and by the National Natural Science Foundation of China grant no. 10471125, the Science Foundation of the Ministry of Education of China grant no. 20060269012, the National Basic Research Program of China grant no. 2006CB805900, and Shanghai Pujiang Program grant no. 05PJ14039 (to Pan).
References [ADN] [BB] [BBC] [BCM] [BH] [BW] [C1] [C2] [CHO] [CW] [dG] [DGP] [DJ] [DL] [DP] [Fn]
Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, II. Comm. Pure Appl. Math. 12, 623–727 (1959); 17, 35–92 (1964) Bourgain, J., Brezis, H.: On the equation div Y = f and application to control of phases. J. Amer. Math. Soc. 16, 393–426 (2003); announced in C. R. Acad. Sci. Paris, Ser. I 334, 973–976 (2002) Berestycki, H., Bonnet, A., Chapman, S.J.: A semi-elliptic system arising in the theory of type-II superconductivity. Comm. Appl. Nonlinear Anal. 1(3), 1–21 (1994) Bonnet, A., Chapman, S.J., Monneau, R.: Convergence of Meissner minimizers of the GinzburgLandau energy of superconductivity as κ → +∞. SIAM J. Math. Anal. 31, 1374–1395 (2000) Bolley, C., Helffer, B.: Rigorous results on Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field, I, II. Nonlinear Stud. 3, no. 1, pp. 1–29, no. 2, pp. 121–152 (1996) Bolik, J., von Wahl, W.: Estimating ∇u in terms of div u, curl u, either (ν, u) or ν × u and the topology. Math. Methods Appl. Sci. 20, 734–744 (1997) Chapman, S.J.: Superheating fields of type II superconductors. SIAM J. Appl. Math. 55, 1233–1258 (1995) Chapman, S.J.: Nucleation of vortices in type II superconductors in increasing magnetic fields. Appl. Math. Lett. 10(2), 29–31 (1997) Chapman, S.J., Howison, S.D., Ockendon, J.R.: Macroscopic models for superconductivity. SIAM Rev. 34, 529–560 (1992) Chen, Y.Z., Wu, L.C.: Second Order Elliptic Equations and Elliptic Systems. Translations of Math. Monographs, Vol. 174, Providence, R.I.: Amer. Math. Soc., 1998 De Gennes, P.G.: Vortex nucleation in type II superconductors. Solid State Comm. 3, 127–130 (1965) Du, Q., Gunzburger, M., Peterson, J.: Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34, 45–81 (1992) Du, Q., Ju, L.: Numerical simulations of the quantized vortices on a thin superconducting hollow sphere. J. Comput. Phys. 201, 511–530 (2004) Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and technology, Vol. 3, New York: Springer-Verlag, 1990 Du, Y.H., Pan, X.B.: Multiple states and hysteresis for type I superconductors. J. Math. Phys. 46(7), Article no. 073301 (2005) Fink, H.J.: Delayed flux entry into type II superconductors. Phys. Lett. 20(4), 356–357 (1966)
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Fink, H.J., Presson, A.G.: Stability limit of the superheated meissner state due to three-dimensional fluctuations of the order parameter and vector potential. Phys. Review 182(2), 498–503 (1969) Fink, H.J., Presson, A.G.: Superconducting surface sheath of a semi-infinite half-space and its instability due to fluctuations. Phys. Rev. B 1(3), 1091–1096 (1970) Giaguinta, M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Basel: Birkhäuser, 1993 Ginzburg, V., Landau, L.: On the theory of superconductivity. Soviet Phys. JETP 20, 1064–1082 (1950) Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. 2nd edition, Berlin: Springer-Verlag, 1983 Helffer, B., Pan, X.B.: Upper critical field and location of surface nucleation of superconductivity. Ann. L’I.H.P. Analyse Non Linéaire 20, 145–181 (2003) Kramer, L.: Vortex nucleation in type II superconductors. Phys. Lett. 24((11), 571–572 (1967) Kress, R.: Potentialtheoretische randwertprobleme bei tensorfeldern beliebiger dimension und beliebigen ranges. Arch. Rat. Mech. Anal. 47, 59–80 (1972) Lu, K., Pan, X.B.: Surface nucleation of superconductivity in 3-dimension. J. Diff. Eqs. 168, 386–452 (2000) Matricon, J., Saint-James, D.: Superheating fields in superconductors. Phys. Lett. A 24, 241–242 (1967) Monneau, R.: Quasilinear elliptic system arising in a three-dimensional type II superconductor for infinite κ. Nonlinear Anal. 52, 917–930 (2003) Morrey, C.B.: Multiple Integrals in the Calculus of Variations, New York: Springer-Verlag, 1966 Pan, X.B.: Surface superconductivity in 3 dimensions. Trans. Amer. Math. Soc. 356, 3899– 3937 (2004) Pan, X.B., Kwek, K.: On a problem related to vortex nucleation of superconductivity. J. Diff. Eqs. 182, 141–168 (2002) Pan, X.B., Qi, Y.: Asymptotics of minimizers of variational problems involving curl functional. J. Math. Phys. 41, 5033–5063 (2000) Schwarz, G.: Hodge Decomposition– A Method for Solving Boundary Value Problems. Lecture Notes in Math. Vol. 1607, Berlin-Heidelberg: Springer-Verlag, 1995 von Wahl, W.: On necessary and sufficient conditions for the solvability of the equations rot u = γ and div u = with u vanishing on the boundary. In: Lecture Notes in Math. Vol. 1431, J. G. Heywood et al. eds., Berlin: Springer-Verlag, 1990 von Wahl, W.: Estimating ∇u by div u and curl u. Math. Methods Appl. Sci. 15, 123–143 (1992) Yin, H.M.: Regularity of weak solution to a p-curl-system. Diff. Integral Eqs. 19(4), 361–368 (2006)
Communicated by P. Constantin
Commun. Math. Phys. 276, 611–643 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0345-9
Communications in
Mathematical Physics
The Causal Boundary of Spacetimes Revisited José L. Flores Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Teatinos, 29071 Málaga, Spain. E-mail: [email protected] Received: 12 September 2006 / Accepted: 17 April 2007 Published online: 25 September 2007 – © Springer-Verlag 2007
Abstract: We present a new development of the causal boundary of spacetimes, originally introduced by Geroch, Kronheimer and Penrose. Given a strongly causal spacetime (or, more generally, a chronological set), we reconsider the GKP ideas to construct a family of completions with a chronology and topology extending the original ones. Many of these completions present undesirable features, like those which appeared in previous approaches by other authors. However, we show that all these deficiencies are due to the attachment of an “excessively big” boundary. In fact, a notion of “completion with minimal boundary” is then introduced in our family such that, when we restrict to these minimal completions, which always exist, all previous objections disappear. The optimal character of our construction is illustrated by a number of satisfactory properties and examples. 1. Introduction Many properties in Mathematics are usually best understood by attaching an ideal boundary to the target space. This situation is also common in General Relativity, where important physical questions about spacetimes are closely related with properties of their boundaries. In the construction of such boundaries, the causal structure of the spacetime plays a decisive role. The most common method to place an ideal boundary on a spacetime is by embedding it conformally into a larger spacetime and, then, by taking the boundary of the image. The conformal boundary was first introduced for asymptotically simple spacetimes in [22] and, since then, it has provided a number of interesting insights in specific examples. However, this approach presents several important handicaps. The construction imposes strong mathematical restrictions on the spacetime, even though such restrictions are satisfied by many spacetimes of physical interest. On the other hand, there is no systematic and totally general way to carry out the embedding such that the standard character of the conformal boundary is ensured.
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An alternative, but similar, construction based on the new concept of isocausality has been introduced recently in [7]. The more accurate character of the isocausality with respect to the causal structure of the spacetime allows the authors to generalize the conformal approach. As a consequence, this new construction is applicable to larger classes of spacetimes. However, it is unclear if this method will overcome the remaining problems of the conformal method. The existence of a systematic and intrinsic procedure to construct an ideal boundary for general spacetimes was first envisioned by Geroch, Kronheimer and Penrose in [9]. They suggested a construction totally based on the causal structure of the spacetime. In particular, it is invariant under conformal changes. Roughly speaking, they placed a future (past) ideal endpoint for every inextensible future (past) timelike curve, in such a way that it only depends on the curve’s past (future). Then, they characterized these ideal endpoints by means of terminal indecomposable past (future) sets TIPs (TIFs) (see Sect. 2 for definitions). The GKP approach, also called causal completion or c-completion, overcomes the handicaps of the conformal construction. In fact, this method can be applied successfully to any strongly causal spacetime (see, however, [25]), and yields a systematic procedure to construct an unique boundary. However, this method presents an important technical difficulty: in [9] the authors remarked that some TIPs and TIFs must act as the same ideal endpoint. In particular, this makes it necessary to define non-trivial identifications between these sets. There are a number of papers written in order to solve this question, which is closely related to the introduction of a satisfactory topology for the causal completion (see [8] for a detailed review on the subject). The story just begins in [9]. The authors introduced a generalized Alexandrov topology on the initial construction, and then, they suggested the minimum set of identifications necessary to obtain a Hausdorff (T2 ) space, i.e. any two points can be separated by neighborhoods. However, this method fails to produce the “expected” completion in some examples [16, 17, 20, Sect. 5]. On the other hand, although strong separation properties as T2 are desirable, there are no physical reasons to impose it a priori. More annoying, from a topological viewpoint the causal boundary of Minkowski space does not match all the common expectation of a cone [11]. Afterwards, other more accurate attempts have been suggested. The procedure proposed in [23], very close to the GKP approach, also fails in simple examples (see [18]). Another approach proposed in [3], and improved later in [25, 26] via the Szabados relation (Definition 2.6; see also Sect. 8), again presents undesirable properties (see [17, 18, 20, Sects. 2.2, 5]). Recently, Marolf and Ross have introduced in [20] an entirely new use of the Szabados relation, including a new topology for the completion, which overcomes important difficulties in previous attempts (see Sect. 8). In fact, the MR construction satisfies essential requirements to be considered a reasonable completion: (i) the original spacetime becomes densely, chronologically and topologically embedded into the completion and, (ii) any timelike curve in the spacetime has some limit in the completion. However, apart from certain “anomalous” limit behaviors in some examples (see [20]), they also admit an annoying failing: not only is their topology not necessarily Hausdorff (which cannot be regarded as unsatisfactory, as we will see later), but it might not even be T1 , i.e. some point might not be closed (Example 10.5). On the other hand, the MR completion sometimes includes too many ideal points (Example 10.6). Another viewpoint in the study of causal completions was previously inaugurated by Harris in [10]. Prevented by the necessity of non-trivial identifications when considering
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the past and future completions simultaneously, he only focused on the future chronological completion Xˆ (the same study also works for the past). In [10], he showed the universality of this partial completion. In [11], he introduced a topology for Xˆ based ˆ the so called future chronological topology (see Sect. 2). Then, on a limit operator L, a series of satisfactory properties for this topology were shown, including universality when the boundary is spacelike. The specific utility of this approach is checked in some wide classes of spacetimes, as static and multiwarped spacetimes (see [12, 13, 4]; see also [14] for a review). However, the lack of causal information by considering only a partial boundary also implies anomalous limit behaviors in simple cases (Example 10.4). In this article we present a whole revision of the causal boundary of spacetimes by combining in a totally new way the GKP ideas. Our approach is very general, and, indeed, it includes not one, but a family of different completions. By imposing a minimality condition, we will choose between them those completions which are optimal, showing that these minimal completions overcome all the deficiencies which appeared in previous constructions by other authors. We have included in Sect. 2 some basic concepts and preliminary results useful for the next sections. In order to gain in generality, our paper does not treat just spacetimes, but also chronological sets (X, ), Definition 2.1, a mathematical object which abstracts the chronological structure of the spacetime. Our approach essentially begins in Sect. 3. In Definition 3.1 we extend the usual notion of endpoint of a timelike curve in a spacetime, to that of endpoint of an arbitrary chain (totally chronologically related sequence) in a chronological set. Then, we use this definition to introduce a general notion of completion X for a chronological set X , by imposing that any chain in X admits some endpoint in X , Definition 3.2. According to this definition, now a chronological set may admit many different completions, including the GKP pre-completion and the Marolf-Ross construction as particular cases. In order to go further, our completions need also to be made chronological sets. This is done in Sect. 4, where any completion X is endowed with a chronological relation such that the natural inclusion i from (X, ) to (X , ) becomes a dense chronological embedding, Theorem 4.2. With this structure of a chronological set, we can verify the consistency of our notion of completion. This is checked in Sect. 5 by showing that every completion is indeed a complete chronological set, Definition 5.1 and Theorem 5.3. In order to get a deeper analysis of our construction, in Sect. 6 we have endowed any chronological set with a topology, the so-called chronological topology, Definitions 6.1, 6.2, which is inspired by the ideas in [11]: first, we have introduced a (sort of theoreticset) limit operator L on X , and then, defined the closed sets as those subsets C ⊂ X such that L(σ ) ⊂ C for any sequence σ ⊂ C. In particular, every completion now becomes a topological space. Then, a number of very satisfactory properties are shown. With this topology, every chronological set X becomes topologically embedded into X via the natural inclusion i, Theorem 6.3. Therefore, as the manifold topology of a strongly causal spacetime V coincides with its chronological topology, Theorem 6.4, the manifold topology is just the restriction to V of the chronological topology of V . Moreover, the notion of endpoint previously introduced becomes now compatible (even though, non-necessarily equivalent) with the notion of limit of a chain provided by this topology, Theorem 6.5. As a consequence, X is topologically dense in X , and any timelike curve in V has some limit in V , Corollary 6.6. All these properties show that these constructions verify essential requirements to be considered reasonable completions. However, many of these completions are still
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non-optimal, in the sense that they have boundaries formed by “too many” ideal points: clearly, this is the case of the GKP pre-completion in [9], which sometimes attaches two ideal points where we would expect only one. The existence of these spurious ideal points implicitly leads to other undesirable features: non-equivalence between the notions of endpoint and limit of a chain; non-closed boundaries; completions with bad separation properties... In order to overcome these deficiencies, in Sect. 7 we have restricted our attention to those completions X with “the smallest boundary”: that is, those completions which are minimal for a certain order relation based on the “size” of the boundary, Definition 7.1. These minimal completions, which always exist, Theorem 7.2, and, indeed, may be non-unique in certain cases (Example 10.6), are called chronological completions, Definition 7.3. In Theorem 7.4 these completions are characterized in terms of some nice properties (indeed some of them were axiomatically imposed in previous approaches). The rest of Sect. 7 is devoted to show the very satisfactory properties of these completions for strongly causal spacetimes V : the notion of endpoint is now totally equivalent to that of the limit of a chain, Theorem 7.5; the chronological boundaries are closed in the completions, Theorem 7.6; the chronological completions V are always T1 , Theorem 7.7; and, even though non-Hausdorffness is possible, violation of T2 is exclusively restricted to points at the boundary, Theorem 7.9. Section 8 has been devoted to emphasize the optimal character of our approach by comparing it with some previous approaches, see Theorem 8.1 and discussion below. In Sect. 9 we have shown the utility of our approach in Causality Theory by characterizing two levels of the causal ladder in terms of the chronological completion: global hyperbolicity, Theorem 9.1; and causal simplicity, Theorem 9.3. Finally, in Sect. 10 we have described some simple examples illustrating the results and properties stated in previous sections. We have also checked our construction in two important classes of physical spacetimes, standard static spacetimes and plane wave solutions. Before concluding this introduction, we remark that our approach does not include considerations about causal, but non-chronological, relations. We have omitted them because the essential part of the causal structure is exclusively carried out by the chronology of the spacetime. Thus, only chronological relations have been considered here, and so, we have gained in simplicity. Nevertheless, the inclusion of purely causal relations into our framework may be the subject of future investigation. 2. Preliminaries Our construction is intended to be exclusively based on the chronological structure of the spacetime. In order to stress this idea, throughout this paper we will work on the simplest mathematical object carrying this structure; the so-called chronological set (first introduced in [10]). Definition 2.1. A chronological set is a set X with a relation (called a chronological relation) such that is transitive and non-reflexive (x x), there are no isolates (everything is related chronologically to something), and X has a countable set D which is dense: if x y then for some d ∈ D, x d y. When working on a chronological set X , the role of future timelike curves in a spacetime is now played by future chains: sequence ς = {xn } ⊂ X obeying x1 · · · xn xn+1 · · · . As in spacetimes, a subset P ⊂ X is called a past set if it coincides
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Fig. 1. We consider the interior region of a square in Minkowski plane with point p2 and segment L removed: P1 ∪ P2 is a past set which is not indecomposable; P1 , P2 are both IPs; P1 is a PIP (P1 = I − ( p1 )) and P2 is a TIP ( p2 ∈ V ); the common past ↓ F coincides with P ∪ P
with its past, that is, P = I − [P] := {x ∈ X : x x for some x ∈ P}1 . Given a subset S ⊂ X , we define the common past of S as ↓ S := I − [{x ∈ X : x x ∀x ∈ S}]. A past set that cannot be written as the union of two proper subsets both of which are also past sets is called an indecomposable past set IP. (Here, the emptyset ∅ will be assumed a past set which is not indecomposable) An IP which does not coincide with the past of any point in X is called a terminal indecomposable past set TIP. Otherwise, it is called a proper indecomposable past set PIP. In a spacetime the past of a point is always indecomposable, however it is easy to give examples of chronological sets where this does not happen (Example 10.6). See Fig. 1 for an illustration of these definitions. The following adaptation of [11, Prop. 4.1] shows that any past set admits an useful decomposition in terms of IPs: Proposition 2.2. Let X be a chronological set. Every past set ∅ = P ⊂ X can be written as P = ∪α Pα , where {Pα }α is the set of all maximal (under the inclusion relation) IPs included in P. Proof. Consider an arbitrary point x ∈ P = ∅. Let Ax be the set of all IPs included in P which contain x, endowed with the partial order of inclusion. Since P is a past set, we can construct inductively a future chain c starting at x and entirely contained in P. In particular, the past of c is an IP in Ax = ∅. On the other hand, consider {Pi }i∈I ⊂ Ax , I a well-ordered index set with Pi ⊂ Pk for i ≤ k. Then ∪i Pi is clearly an IP into P which also contains x. Whence, ∪i Pi is an upper bound in Ax for {Pi }i∈I . Therefore, Zorn’s Lemma2 ensures the existence of a maximal IP Px into P which contains x. Proposition 2.2 justifies now the following definition: Definition 2.3. Given a past set ∅ = P in a chronological set X , the set dec(P) := {Pα }α of all maximal IPs included in P is called the decomposition of P. By convention, we will assume dec(∅) = ∅. 1 Here we are following standard notation: that is, I − [·] denotes the past of a set of points, while I − (·) is reserved for past of a point. 2 Zorn’s Lemma: Every non-empty partially ordered set in which every totally ordered subset has an upper bound contains at least one maximal element.
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Denote by Xˆ the set of all IPs of X . If X is past-regular (i.e. I − (x) is IP for all x ∈ X ) and past-distinguishing (i.e. I − (x) = I − (x ) implies x = x ), then Xˆ is called the future chronological completion of X . In this case, Xˆ can be endowed with a structure of chronological set and the map x → I − (x) injects X into Xˆ . Therefore, we can ˆ ), ∂(X ˆ ) being the set of all TIPs of X , which is called the future write Xˆ = X ∪ ∂(X chronological boundary of X . More details about the future chronological completion can be found in [10]. It is possible to endow a chronological set with a topology. The heart of the future ˆ chronological topology, firstly introduced in [11], is the following limit operator L: Definition 2.4. Given a sequence σ = {Pn } of past sets in X , an IP P ⊂ X satisfies ˆ ) if P ∈ L(σ (i) P ⊂ LI(Pn ) and (ii) P is maximal I P within LS(Pn ), where LI and LS denote the standard inferior and superior limits of sets: ∞ LI(Pn ) = lim inf n→∞ (Pn ) = ∪∞ n=1 ∩k=n Pk , ∞ LS(Pn ) = lim supn→∞ (Pn ) = ∩n=1 ∪∞ k=n Pk .
The limit operator Lˆ was introduced in [11] in a different way. However, it is not hard to show that both definitions are equivalent (see [14]). Then, the future chronological topology (-topology) of X is introduced by defining ˆ ) ⊂ C for any sequence σ ⊂ C.3 the closed sets as those subsets C ⊂ X such that L(σ ˆ With this definition, L(σ ) is to be thought of as first-order limits of σ . In the particular ˆ case of X = V being a strongly causal spacetime, the L-limit of a sequence coincides with the limit with respect to the manifold topology: Proposition 2.5. Let V be a strongly causal spacetime. For any sequence σ = { pn } ⊂ V , ˆ a point p is L-limit of σ if and only if it is the limit of σ with the topology of the manifold. Proof. See [11, Theorem 2.3].
Of course, the dual notions of the concepts introduced here (past chain, future set, ˇ ↑ S, IF, TIF, PIF, Xˇ , L...), and the corresponding results, can be defined and proved just by interchanging the roles of past and future. We finish this section with a remarkable result coming from [25, Prop. 5.1]. Previously, we recall the following definition: Definition 2.6. If P is maximal IP into ↓ F and F is maximal IF into ↑ P then we say that P, F are S-related, P ∼ S F (see Fig. 1). Proposition 2.7. Let V be a strongly causal spacetime. The unique S-relations involving proper indecomposable sets in V are I − ( p) ∼ S I + ( p) for all p ∈ V . 3 When using the limit operator L, ˆ it is common to implicitly identify the past or future of a point with the point itself.
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3. Completing Chronological Sets A central property to be satisfied by any space X intended to be a completion of X is that any chain in X admits some “limit” in X . So, a natural strategy for completing a chronological set consists of adding to X “ideal endpoints” associated to every “endless” chain in X . In order to develop this idea, we are going to restrict our attention to weakly distinguishing chronological sets; that is, those chronological sets satisfying that any two points with the same past and future must coincide. Observe that this condition is not very restrictive at all, since it is satisfied by any strongly causal spacetime. Denote by X p , X f the sets of all past and future sets of X , resp. Then, the map i : X → Xp × X f x → (I − (x), I + (x))
(3.1)
injects X into X p × X f in a natural way. This injection, joined to the fact that our construction must be exclusively based on the chronological structure of X , makes it natural to conceive X as verifying i[X ] ⊂ X ⊂ X p × X f . So, if we want to completely determine X , we need to establish which elements of X p × X f belong to the completion, or, equivalently, which past and future sets must be paired to form every element of X . According to the central idea suggested at the beginning of this section, this will be done by formalizing the notion of the “endpoint” of a chain. To this aim, we are not going to define a topology on X p × X f . Instead, we will directly deduce a reasonable notion of “endpoint”, which will be justified a posteriori by showing that it is compatible with the topology for X suggested in Sect. 6. Consider a future chain ς = {xn } ⊂ X and assume that it “approaches” some (P, F) ∈ X p × X f , where P and F represent the past and future (computed in X ) of the limit point. If a sequence { pn } converges to some point p in a spacetime, then every point in the past of p is eventually in the past of pn . Therefore, if we translate this property to our situation, we should obtain P ⊂ I − [ς ]. Moreover, since ς is a future chain “approaching” (P, F), it becomes natural to assume ς ⊂ P. In particular, I − [ς ] ⊂ P, and thus, P = I − [ς ]. On the other hand, by transitivity with respect to (P, F), we should also expect F ⊂↑ P. Of course, there is no reason to impose F =↑ P; however, arguing by analogy to what happens in spacetimes, the fact that ς is “approaching” (P, F) also leads to strengthen the inclusion F ⊂↑ P by assuming that any element in ˇ ). Obviously, dual conditions dec(F) is maximal in ↑ P, or, equivalently, dec(F) ⊂ L(ς are deduced in the case of ς being a past chain approaching (P, F). Summarizing, we suggest the following definition: Definition 3.1. A pair (P, F) ∈ X p × X f is an endpoint of a future (resp. past) chain ς ⊂ X if ˇ ) P = I − [ς ], dec(F) ⊂ L(ς
ˆ ), F = I + [ς ]). (resp. dec(P) ⊂ L(ς
(3.2)
We will denote by X end the subset of X p × X f formed by the union of i[X ] with all the endpoints of every chain in X . Now, we are in a position to formulate the notion of completion for a chronological set:
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Definition 3.2. Let X be a weakly distinguished chronological set. A set X satisfying i[X ] ⊂ X ⊂ X end (⊂ X p × X f ) is called a completion of X if any chain in X has some endpoint in X . Then, the boundary of X in X is defined as ∂(X ) := X \ i[X ]. According to this definition, a chronological set will admit different completions (see Example 10.2). We will denote by C X the set of all these completions. A well-known example of completion covered by Definition 3.2 is the GKP pre-completion of spacetimes, first introduced in [9]. In fact, the pre-completion V of a strongly causal spacetime V can be seen as formed by adding to i[V ] the endpoints (I − [ς ], ∅) (resp. (∅, I + [ς ])) for every endless future (resp. past) chain ς .4 An alternative completion is formed by adding to i[V ] the endpoints (P, F) given by P = I − [LI(I − (xn ))] and F = I + [LI(I + (xn ))]
(3.3)
for every endless chain ς = {xn }. In this case, the resulting space V , which, in general, is different from V , may contain pairs (P,F) with some component P or F not necessarily indecomposable (see Example 10.3). Notice also that X end is another example of completion, which, indeed, contains any other completion of the chronological set X . Observe that Definition 3.2 is far from being accurate: for example, it does not avoid the possibility of having completions which remain as completions when some boundary point is removed (see Example 10.2). Of course, we can eliminate this possibility by hand in Definition 3.2; however, this is not sufficient for ensuring that there are no spurious ideal points at the boundary: for example, the GKP pre-completion V , which does not fall under the previous possibility, sometimes attaches two different ideal points where we would intuitively expect only one (see Example 10.2). We will postpone to Sect. 7 the introduction of a non-trivial notion of minimal completion, the so-called chronological completion. Even though many completions included in Definition 3.2 are not optimal, they still satisfy enough properties to justify the name of “completions” for these constructions. The next three sections are devoted to analyze these properties in depth. To this aim, the definition and characterizations below will be very useful: Definition 3.3. Let X be a chronological set. A pair (P, F) ∈ X p × X f is generated by a chain ς = {xn } ⊂ X if equalities (3.3) hold. Of course, every chain ς in X generates an unique pair (P, F) ∈ X p × X f . Proposition 3.4. Let X be a chronological set and consider a chain ς = {xn } ⊂ X and a pair (P, F) ∈ X p × X f . Then, the following statements are equivalent: (i) (P, F) is generated by ς ; (ii) there exists a countable dense set D ⊂ X such that P ∩ D = LI(I − (xn )) ∩ D
and
F ∩ D = LI(I + (xn )) ∩ D;
(3.4)
(iii) (P, F) satisfies ˆ ) = dec(P) L(ς
and
ˇ ) = dec(F). L(ς
4 Actually, V was introduced in [9] by using identifications instead of pairs: that is, V := Vˆ ∪ Vˇ / ∼, where P ∼ F iff P = I − ( p), F = I + ( p), for some p ∈ V .
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Proof. We will follow this scheme: first, we will prove (iii)⇒(ii); then, (i)⇒(iii); and, finally, (ii)⇒(i). ˆ ) = dec(P) and L(ς ˇ ) = dec(F) In order to prove (iii)⇒(ii), first observe that L(ς imply P ⊂ LI(I − (xn )) and F ⊂ LI(I + (xn )). Therefore, we only need to prove that P ∩ D ⊃ LI(I − (xn )) ∩ D
and
F ∩ D ⊃ LI(I + (xn )) ∩ D
(3.5)
for some countable dense set D ⊂ X . To this aim, take any countable dense set D ⊂ X and define D := D \D0 , with D0 = {d ∈ D : I − (d) ⊂ P but d ∈ P}. In order to prove the density of D, consider y y ∈ X . Since D is dense, there exists d ∈ D such that y d y . If d ∈ D0 , necessarily d ∈ D and we finish. Otherwise, consider y d and take d ∈ D such that y d d. Then, necessarily d ∈ D0 since d ∈ I − (d) ⊂ P. Therefore, d ∈ D, and thus, D is dense in X . For the first inclusion in (3.5), assume by contradiction the existence of d ∈ LI(I − (xn )) ∩ D such that d ∈ P ∩ D. From the definition of D, necessarily I − (d) ⊂ P. Therefore, there exists x ∈ I − (d)\P. In particular, x ∈ I − [LI(I − (xn ))] = ∅, and thus, Proposition 2.2 ensures the existence of a maximal IP Px in I − [LI(I − (xn ))] containing x. Taking into account that ς is a chain, Px is also maximal in LS(I − (xn )). Therefore, ˆ ), which contradicts the equality L(ς ˆ ) = dec(P). In conclusion, the first Px ∈ L(ς inclusion in (3.5) holds. We can repeat the same reasoning for the future, but taking the set D instead of D as an initial countable dense set, and removing the elements d ∈ D such that I + (d) ⊂ F but d ∈ F. Then, the resulting set, which we also denote by D, clearly satisfies both inclusions in (3.5). In order to prove (i)⇒(iii), it is clear that P ⊂ LI(I − (xn )). So, assume by contradiction that Pα0 ∈ dec(P) is not maximal in LS(I − (xn )) = LI(I − (xn )). Then, there exists an IP P with Pα0 P ⊂ LI(I − (xn )). In particular, P ⊂ P, and thus, there exist x, x ∈ P \ P such that x x . As x ∈ LI(I − (xn )), necessarily x ∈ I − [LI(I − (xn ))] \ P, which contradicts that P = I − [LI(I − (xn ))]. Therefore, any Pα ∈ dec(P) is maximal ˆ ). in LS(I − (xn )), and thus, dec(P) ⊂ L(ς ˆ ) ˆ To prove L(ς ) ⊂ dec(P), assume by contradiction the existence of an IP P ∈ L(ς such that P ∈ dec(P). Then, necessarily P ⊂ P, since otherwise P would be maximal in P (recall that P = I − [LI(I − (xn ))]), and thus, P ∈ dec(P). Reasoning as in the previous paragraph, there exist x, x ∈ P \ P such that x x . In particular, x ∈ LI(I − (xn )). Therefore, x ∈ I − [LI(I − (xn ))] \ P, which contradicts that ˆ ) = dec(P). Finally, an analogous reasoning P = I − [LI(I − (xn ))]. In conclusion, L(ς ˇ ) = dec(F). proves that L(ς In order to prove (ii)⇒(i), assume by contradiction that P ⊂ I − [LI(I − (xn ))]. Since P is a past set, necessarily P ⊂ LI(I − (xn )). Therefore, there exist x, x ∈ P\LI(I − (xn )) and d ∈ D such that x d x . In particular, d ∈ P but d ∈ LI(I − (xn )). Whence, d contradicts the first equality in (3.4), and thus, P ⊂ I − [LI(I − (xn ))]. For the other inclusion, assume that x ∈ I − [LI(I − (xn ))]. This means x x for certain x ∈ LI(I − (xn )). Therefore, there exists d ∈ D with x d x . In particular,
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d ∈ LI(I − (xn )) ∩ D, which, joined to the first equality in (3.4), implies x d ∈ P. Whence, I − [LI(I − (xn ))] ⊂ P, and the equality follows. Finally, an analogous reasoning also shows F = I + [LI(I + (xn ))]. In the next Sects. 4–6 by X we will understand any completion of X , according to Definition 3.2. 4. The Completions as Chronological Sets Now that we have introduced a family of completions C X for any weakly distinguishing chronological set (X, ), the next step consists of endowing any completion X ∈ C X with a structure of weakly distinguishing chronological set, such that X becomes densely and chronologically embedded into X via the injection i (see (3.1)). To be more precise, let us introduce some definitions: Definition 4.1. A bijection f : X → X between two chronological sets (X, ), (X , ) is a (chronological) isomorphism if f and f −1 preserve the chronological relations. When f is only injective but the image f (X ) ⊂ X endowed with is still isomorph to (X, ) via f , we say that f is a (chronological) embedding of (X, ) into (X , ). Consider the relation (P, F)(P , F ) iff F ∩ P = ∅,
∀(P, F), (P , F ) ∈ X
(first introduced in [25], and used later in [20]). Then, the following results hold: Theorem 4.2. If (X, ) is a weakly distinguishing chronological set then (X , ) is also a chronological set. Moreover, i chronologically embeds (X, ) into (X , ) in such a way that i[X ] is dense in X . Proof. To prove transitivity, assume (P, F)(P , F ) and (P , F )(P , F ). Then, there exist x ∈ F ∩ P and x ∈ F ∩ P . Let ς = {xn } ⊂ X be a chain with endpoint (P , F ) (if (P , F ) = i(x0 ) for some x0 ∈ X , take instead ς = {xn } ≡ {x0 } ⊂ X ). Then, P ⊂ LI(I − (xn )), F ⊂ LI(I + (xn )). In particular, for all n big enough x xn x . But x ∈ F and x ∈ P . Hence, xn ∈ F ∩ P = ∅ for all n big enough, and thus, (P, F)(P , F ). To show that is non-reflexive, assume by contradiction that (P, F)(P, F). Then, there exists x ∈ F ∩ P = ∅. As before, let ς = {xn } ⊂ X be a chain with endpoint (P, F) (again, if (P, F) = i(x0 ) for some x0 ∈ X , take instead ς = {xn } ≡ {x0 } ⊂ X ). Then, P ⊂ LI(I − (xn )) and F ⊂ LI(I + (xn )). In particular, for all n big enough xn x xn . This contradicts that is non-reflexive. To prove that there are no isolates, consider (P, F) ∈ X . Assume for example that x ∈ P = ∅ (if x ∈ F = ∅, the argument is analogous). As P is a past set, there exists x ∈ P such that x x . Then, i(x) ∈ X satisfies i(x)(P, F), since x ∈ I + (x) ∩ P = ∅. The set i[D] = {i(d) : d ∈ D, D countable dense set of (X, )} ⊂ X is a countable dense set of (X , ). In fact, if (P, F)(P , F ) then F ∩ P = ∅. Therefore, there exist x, x ∈ F ∩ P with x x . Let d ∈ D be such that x d x .
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Then, (P, F)i(d)(P , F ), since x ∈ F ∩ I − (d) = ∅ and x ∈ I + (d) ∩ P = ∅. In particular, this also shows that i[X ] is dense in X . Finally, we show that extends without introducing new chronological relations in i[X ]. Assume first that x, x ∈ X satisfy x x . Then, there exists d ∈ D such that x d x . Therefore, d ∈ I + (x) ∩ I − (x ) = ∅ and, thus, i(x)i(x ). Assume now that i(x)i(x ). Then, there exists y ∈ I + (x) ∩ I − (x ) = ∅. Therefore, x y x , and thus, x x . Theorem 4.3. Let (X, ) be a weakly distinguishing chronological set. Then,5 i−1 [I − ((P, F)) ∩ i[X ]] = P, i−1 [I + ((P, F)) ∩ i[X ]] = F
∀ (P, F) ∈ X . (4.1)
In particular, (X , ) is a weakly distinguishing chronological set. Proof. For the first equality in (4.1) consider x ∈ i−1 [I − ((P, F)) ∩ i[X ]]. This means that i(x)(P, F) ∈ X , x ∈ X . Therefore, there exists x ∈ I + (x) ∩ P = ∅, and thus, x x ∈ P. In particular, x ∈ P. Conversely, consider x ∈ P. Since P is a past set, there exists x ∈ X with x ∈ I + (x) ∩ P = ∅. Therefore, i(x)(P, F), and thus, x ∈ i−1 [I − ((P, F)) ∩ i[X ]]. The second equality in (4.1) is proved analogously. In order to prove that (X , ) is weakly distinguishing, assume I − ((P, F)) = − I ((P , F )) and I + ((P, F)) = I + ((P , F )) for some (P, F), (P , F ) ∈ X . From (4.1), P = i−1 [I − ((P, F)) ∩ i[X ]] = i−1 [I − ((P , F )) ∩ i[X ]] = P , F = i−1 [I + ((P, F)) ∩ i[X ]] = i−1 [I + ((P , F )) ∩ i[X ]] = F . Whence, (P, F) = (P , F ).
5. The “Complete Character” of the Completions In the previous section we have showed that given any weakly distinguishing chronological set (X, ), the pair (X , ) is also a weakly distinguishing chronological set. So, we can be tempted to repeat the process on (X , ), and construct a new pair (X , ) with defined by (P, F)(P , F ) ⇐⇒ F ∩ P = ∅,
∀ (P, F), (P , F ) ∈ X .
In this section we are going to justify that completing a completion is unnecessary, in the sense that any completion is already a “complete” chronological set. To this aim, of course we previously need to introduce a reasonable notion of complete chronological set: Definition 5.1. A weakly distinguishing chronological set (Y, ) is (chronologically) complete if i[Y ] itself is a completion of Y , that is, if any chain in Y has some endpoint in i[Y ]. 5 Of course, symbols I − (·), I + (·) in (4.1) refer to the chronological relation instead of .
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Next, we are going to establish a suitable correspondence between the pairs in X p × X f and those in (X ) p × (X ) f , for any completion X of X . Consider the map j : X p × X f → (X ) p × (X ) f (P, F) → ( j (P), j (F)), where j (P) := I − [i[P]]
j (F) := I + [i[F]]
and
(here, I ± [·] are computed in (X , )). From (4.1) and the density of i[X ] into X , it follows ( j (P), j (F)) = (I − ((P, F)), I + ((P, F))) ∀ (P, F) ∈ X . (5.1) Therefore, the map j restricted to X ⊂ X p × X f coincides with the injection i for the chronological set Y = X . Notice also that the inverse map of j is given by: k : (X ) p × (X ) f → X p × X f (P, F) → (k(P), k(F)), where k(P) := i−1 [P ∩ i[X ]]
k(F) := i−1 [F ∩ i[X ]].
and
In fact, first notice that k is well-defined. For example, to check that k(P) is a past set, consider x ∈ k(P). This means i(x) ∈ P. Since P is a past set of X and i[X ] is dense in X , there exists x ∈ X with i(x)i(x ) ∈ P. In particular, x ∈ i−1 [P ∩ i[X ]]. Therefore, x x ∈ k(P), showing that k(P) ⊂ I − [k(P)]. Conversely, assume now that x ∈ I − [k(P)]. Then, x x ∈ k(P), which implies i(x) i(x ) ∈ P. As P is a past set, it follows i(x) ∈ P, and thus, x ∈ k(P). Therefore, I − [k(P)] ⊂ k(P). It remains to show that j and k satisfy the identities k ◦ j = I d X p ×X f ,
j ◦ k = I d(X ) p ×(X ) f .
(5.2)
The first identity is clearly equivalent to the equalities: P = k( j (P)),
F = k( j (F))
∀ (P, F) ∈ X p × X f .
(5.3)
To prove the first equality in (5.3), recall that P is a past set and i a chronological embedding. Therefore, x ∈ P iff i(x) ∈ i[P] ⊂ I − [i[P]] = j (P). But, i(x) ∈ i[X ]. Whence, x ∈ P iff i(x) ∈ j (P)∩i[X ]. In conclusion, x ∈ P iff x ∈ i−1 [ j (P)∩i[X ]] = k( j (P)). The second equality in (5.3) can be proved analogously. On the other hand, the second identity in (5.2) is equivalent to these other equalities: P = j (k(P)),
F = j (k(F))
∀ (P, F) ∈ (X ) p × (X ) f .
(5.4)
To prove the first equality, recall that P is a past set and i[X ] is dense in X . Therefore, (P, F) ∈ P iff there exists x ∈ X with (P, F)i(x) ∈ P. In particular, (P, F) ∈ P iff (P, F)i(x) ∈ i[k(P)]. Therefore, (P, F) ∈ P iff (P, F) ∈ j (k(P)). The second equality in (5.4) can be proved analogously. With these tools, we are now ready to prove Theorem 5.3 below. The hard part of this proof has been extracted in the following lemma: Lemma 5.2. Let X be a completion of a weakly distinguishing chronological set (X, ).
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(i) If (P, F) ∈ X p × X f is an endpoint of a chain δ = {xi } ⊂ X , then the pair ( j (P), j (F)) ∈ (X ) p × (X ) f is an endpoint of the chain i[δ] = {i(xi )}i ⊂ X . (ii) Given a chain ς ⊂ X there exists another chain δ ⊂ X such that i[δ] ⊂ X and ς have the same endpoints. Proof. For (i), we assume without restriction that (P, F) is an endpoint of a future chain δ = {xi } ⊂ X . From Definitions 3.1, 3.3, this implies P = P and dec(F) ⊂ dec(F ),
(5.5)
for (P , F ) being the pair generated by δ. From Proposition 3.4 there exists a countable dense set D ⊂ X such that P ∩ D = LI(I − (xi )) ∩ D, F ∩ D = LI(I + (xi )) ∩ D.
(5.6)
Taking into account that i : X → X is a dense and chronological embedding, from (5.5) and (5.6) we easily obtain
with
j (P) = j (P ) and dec( j (F)) ⊂ dec( j (F )),
(5.7)
j (P ) ∩ i[D] = LI(I − (i(xi ))) ∩ i[D], j (F ) ∩ i[D] = LI(I + (i(xi ))) ∩ i[D].
(5.8)
From (5.8) and Proposition 3.4, we deduce that ( j (P ), j (F )) is generated by i[δ] ⊂ X . This joined to (5.7) proves that ( j (P), j (F)) is an endpoint of i[δ] (recall again Definitions 3.1, 3.3). In order to prove (ii), we can assume without restriction that ς = {(Pn , Fn )}n ⊂ ∂(X ) is a future chain. Let ς o = {(Pno , Fno )} ⊂ X end be a future chain formed by pairs (Pno , Fno ) generated by some chain ς n = {xmn }m ⊂ X admitting some endpoint equal to (Pn , Fn ). In particular, Pn ⊂ Pno and Fn ⊂ Fno for all n.
(5.9)
Denote by (P o , F o ) ∈ (X end ) p × (X end ) f the pair generated by ς o . First we are going to prove the existence of some future chain δ = {xi } ⊂ X generating (k(P o ), k(F o )). From Proposition 3.4 and (5.1) we have P o ∩ i[D] = LI(I − ((Pno , Fno ))) ∩ i[D] = LI( j (Pno )) ∩ i[D], F o ∩ i[D] = LI(I + ((Pno , Fno ))) ∩ i[D] = LI( j (Fno )) ∩ i[D].
(5.10)
Applying i−1 to (5.10), from (5.3) we deduce k(P o ) ∩ D = LI(k( j (Pno ))) ∩ D = LI(Pno ) ∩ D, k(F o ) ∩ D = LI(k( j (Fno ))) ∩ D = LI(Fno ) ∩ D.
(5.11)
Therefore, if ς o ⊂ i[X ] then δ = i−1 [ς o ] is the required sequence. Otherwise, observe that chains ς n = {xmn }m ⊂ X satisfy Pno = I − [LI(I − (xmn ))] and Fno = I + [LI(I + (xmn ))]
(5.12)
(recall Definition 3.3). In order to construct the announced chain δ = {xi }i ⊂ X , we will argue inductively:
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Step 1. Consider d1 ∈ D. If d1 ∈ k(P o ) (resp. d1 ∈ k(F o )), from (5.11) we can define a sequence {n 1k }k ⊂ N by removing from {n}n those elements n with d1 ∈ Pno (resp. d1 ∈ Fno ). Moreover, from (5.12) we can construct a sequence {m 1k,l }l ⊂ N by n1
n1
removing from {m}m those elements m with d1 xmk (resp. xmk d1 ). With these n1
n1
definitions, d1 ∈ I − (xmk1 ) (resp. d1 ∈ I + (xmk1 )) for all k, l. If d1 ∈ k(P o ) ∪ k(F o ) k,l
k,l
define {n 1k }k ≡ {n}n , {m 1k,l }l ≡ {m}m . Step 2: Assume now that {n ik }k , {m ik,l }l ⊂ N have been defined for certain i. Consider di+1 ∈ D. If di+1 ∈ k(P o ) (resp. di+1 ∈ k(F o )), from (5.11) we can define a i i o sequence {n i+1 k }k ⊂ N by removing from {n k }k those elements n k with di+1 ∈ P i (resp. nk
di+1 ∈ F oi ). Moreover, from (5.12) we can construct a sequence {m i+1 k,l }l ⊂ N by removnk
ing from {m ik,l }l those elements m ik,l with di+1 x definitions, di+1 ∈ I − (x
n i+1 k m i+1 k,l
) (resp. di+1 ∈ I + (x
n i+1 k m i+1 k,l
n i+1 k m ik,l
(resp. x
n i+1 k m ik,l
di+1 ). With these
)) for all k, l. If di+1 ∈ k(P o )∪k(F o )
i i+1 i define {n i+1 k }k ≡ {n k }k , {m k,l }l ≡ {m k,l }l . Therefore, we can construct by induction {n ik }k , {m ik,l }l for all i ∈ N. Moreover, it is possible to choose l(i) in such a way that
δ = {xi }i ≡ {x
n ii } m ii,l(i) i
is a future chain contained in k(P o ).
With this definition of δ, the following inclusions hold: LI(Pno ) ∩ D ⊂ LI(I − (xi )) ∩ D ⊂ LI(Pno ) ∩ D, LI(Fno ) ∩ D ⊂ LI(I + (xi )) ∩ D ⊂ LI(Fno ) ∩ D.
(5.13)
In fact, assume d ∈ LI(Pno ) ∩ D. From (5.11) and previous construction, necessarily d x
n ik
m ik,l
for all i big enough and all k, l. In particular, d xi for all i big enough,
and thus, d ∈ LI(I − (xi )) ∩ D. This proves the first inclusion in (5.13). Assume now d ∈ LI(I − (xi )) ∩ D. As δ ⊂ k(P o ), necessarily d ∈ k(P o ). Therefore, from (5.11) we have d ∈ LI(Pno ) ∩ D. For the inclusions in the second line of (5.13), assume first d ∈ LI(Fno ) ∩ D. Reasoning as before, we deduce d x LI(I + (x
n ik
m ik,l
for all i big enough and
LI(I + (x
all k, l, and thus, d ∈ i )) ∩ D. Assume now d ∈ i )) ∩ D. There exists din ∈ k(P o ) ∩ Fno = ∅ for all n. But, din xi for all i ≥ i n . Whence, d din ∈ Fno for all n. This proves the last inclusion in (5.13). In conclusion, from (5.11) and (5.13) we have k(P o ) ∩ D = LI(I − (xi )) ∩ D, k(F o ) ∩ D = LI(I + (xi )) ∩ D.
(5.14)
Therefore, from Proposition 3.4 we conclude that (k(P o ), k(F o )) is generated by δ. Next, it remains to show that i[δ] ⊂ X and ς have the same endpoints. Taking into account that i is a chronological embedding, from (5.4) and (5.14) we deduce P o ∩ i[D] = jk(P o ) ∩ i[D] = LI(I − (i(xi ))) ∩ i[D], F o ∩ i[D] = jk(F o ) ∩ i[D] = LI(I + (i(xi ))) ∩ i[D].
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Whence, Proposition 3.4 ensures that (P o , F o ) is generated by i[δ]. Since (P o , F o ) is also generated by ς o , for every i 0 , n 0 there exists n, i big enough such that o o i(xi0 )(Pn−1 , Fn−1 )
and
(Pno0 +1 , Fno0 +1 )i(xi ).
(5.15)
Moreover, from (5.9) and the fact that ς is a future chain, necessarily o o (Pn−1 , Fn−1 )(Pn , Fn )
and
(Pn 0 , Fn 0 )(Pno0 +1 , Fno0 +1 ).
(5.16)
Therefore, taking into account (5.15) and (5.16) we have proved that for fixed i 0 , n 0 there exists n, i big enough such that i(xi0 )(Pn , Fn )
and
(Pn 0 , Fn 0 )i(xi ).
In conclusion, ς and i[δ] have the same endpoints.
Now, the main result of this section can be proved easily: Theorem 5.3 (Completeness). If (X, ) is a weakly distinguishing chronological set then (X , ) is complete. Proof. Given any chain ς = {(Pn , Fn )} ⊂ X we need to prove that ς has some endpoint in j[X ]. From Lemma 5.2 (ii) there exists some chain δ ⊂ X such that i[δ] ⊂ X and ς have the same endpoints. Let (P, F) be some endpoint of δ in X . From Lemma 5.2 (i), ( j (P), j (F)) is an endpoint of i[δ]. Therefore, ( j (P), j (F)) ∈ j[X ] is also an endpoint of ς , as required. 6. The Chronological Topology In order to provide a deeper description of the relation between a chronological set and its boundary, in this section we are going to introduce a topological structure. More precisely, we are going to endow any chronological set with the chronological topology, a non-trivial generalization of the -topology in [11]. To this aim, first we are going to define a limit operator L for any chronological set Y . We take as a guide property the fact that our topology must turn the endpoints of chains (Definition 3.1) into topological limits. So, the following definition, based on a simple generalization of conditions (3.2), becomes natural: Definition 6.1. Given a sequence σ ⊂ Y , we say that x ∈ L(σ ) if ˆ ) and dec(I + (x)) ⊂ L(σ ˇ ). dec(I − (x)) ⊂ L(σ With this limit operator in hand we can now define the closed sets of Y , which determine the chronological topology (chr -topology): Definition 6.2. The closed sets of Y with the chr -topology are those subsets C ⊂ Y such that L(σ ) ⊂ C for any sequence σ ⊂ C. It is worth noting that the chr -topology has been defined for any chronological set. In particular, it is applicable to both strongly causal spacetimes V and their completions V . So, two natural questions arise: does the manifold topology of a strongly causal spacetime coincide with the chr -topology it inherits when considered as a chronological set?; does the manifold topology of a strongly causal spacetime V coincide with the restriction to V of the chr -topology of V ? The following two theorems will answer positively these questions. For the second one, the hypothesis of strong causality (further than weakly distinguishing) becomes essential.
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Theorem 6.3. Any weakly-distinguishing chronological set (X, ) is topologically embedded into (X , ) via the injection i if both spaces are endowed with the chr -topology. Proof. It suffices to show that a point x ∈ X satisfies x ∈ L(σ ) for some sequence σ = {xn } ⊂ X if and only if i(x) ∈ L(ρ) for ρ = {i(xn )} ⊂ X . Taking into account that i : X → X is a dense and chronological embedding, and equalities (5.3), (5.4), we deduce: j (P) ∈ dec(I − (i(x))), j (P) ⊂ LI(I − (i(xn ))) P ∈ dec(I − (x)), P ⊂ LI(I − (xn )) ⇒ P maximal in LS(I − (xn )) j (P) maximal in LS(I − (i(xn ))), k(P) ∈ dec(I − (x)), k(P) ⊂ LI(I − (xn )) P ∈ dec(I − (i(x))), P ⊂ LI(I − (i(xn ))) ⇒ − P maximal in LS(I (i(xn ))) k(P) maximal in LS(I − (xn )). Analogously, we deduce the corresponding implications for the future. Therefore, the thesis follows from (5.1) and Definitions 2.4, 6.1. Theorem 6.4. The topology of a strongly causal spacetime V as a manifold coincides with the corresponding chr -topology. Proof. From Proposition 2.5 (and its dual), a point p ∈ V is the limit of a sequence σ = { pn } ⊂ V with the topology of the manifold if and only if ˆ ) I − ( p) ∈ L(σ
and
ˇ ). I + ( p) ∈ L(σ
(6.1)
Taking into account that dec(I − ( p)) = {I − ( p)} and dec(I + ( p)) = {I + ( p)}, conditions (6.1) can be written as ˆ ) and dec(I + ( p)) ⊂ L(σ ˇ ). dec(I − ( p)) ⊂ L(σ Therefore, from Definition 6.1, σ converges to p with the manifold topology if and only if p is the L-limit of σ . With this topology, we can also prove that the concept of endpoint is compatible with the notion of limit of a chain: Theorem 6.5. Let ς = {xn } be a chain in a weakly distinguishing chronological set X . Then, the following statements hold: (1) If i(x) ∈ i[X ] is endpoint of ς then x ∈ L(ς ). Moreover, the reciprocal is true if, in addition, X is regular (i.e., past- and future-regular). (2) If (P, F) ∈ X is an endpoint of ς then (P, F) ∈ L(ρ ) for any subsequence ρ ⊂ ρ = {i(xn )}n ⊂ i[X ]. In particular, i[X ] is topologically dense in X . Proof. Statement (1) is a direct consequence of Definitions 3.1, 6.1. For (2), assume without restriction that ς is a future chain. From Definitions 3.1, 3.3, P = P o and dec(F) ⊂ dec(F o ),
(6.2)
where (P o , F o ) is the pair generated by ς . From Proposition 3.4 there exists a countable dense set D ⊂ X such that P o ∩ D = LI(I − (xn )) ∩ D,
F o ∩ D = LI(I + (xn )) ∩ D.
(6.3)
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Taking into account that i is a chronological embedding, from (6.3) we obtain: j (P o ) ∩ i[D] = LI(I − (i(xn ))) ∩ i[D], j (F o ) ∩ i[D] = LI(I + (i(xn ))) ∩ i[D].
(6.4)
Moreover, equalities (6.4) also hold for any subsequence ρ ⊂ ρ = {i(xn )} since ρ is a chain in X . This joined to (6.2) and Proposition 3.4 imply ˆ ), dec( j (F)) ⊂ dec( j (F o )) = L(ρ ˇ ) for any ρ ⊂ ρ, { j (P)} = { j (P o )} = L(ρ and thus, (P, F) ∈ L(ρ ) for any ρ ⊂ ρ.
Finally, as a direct consequence of Theorem 6.5 (1) and Theorem 5.3 we obtain the following result: Corollary 6.6. If ς is a chain in a complete chronological set Y then ς has some limit in Y . In particular, if Y = V is a completion for some strongly causal spacetime V , then any timelike curve γ (≡ i[γ ]) in V (≡ i[V ]) has some limit in V . The results obtained so far in this paper show that our construction satisfies essential requirements to provide reasonable completions for any strongly causal spacetime. However, many of these completions are not optimal, in the sense that they include spurious ideal points. As a consequence: the notions of endpoint and limit of chains, even if compatible, are not totally equivalent (Example 10.2), the boundaries may not be closed in the completions (Example 10.4), the completions may present bad separation properties (Example 10.2)... In the next section we are going to show that all these deficiencies disappear when only completions with minimal boundaries are considered. 7. The Chronological Completions In order to look for completions with minimal boundaries, first we delete from C X those completions which are still completions when some point of its boundary is removed. Denote by C ∗X the resulting set, which is always non-empty: for example, i[X ] joined to those pairs of the form (I − [ς ], ∅) or (∅, I + [ς ]) for any future or past chain ς without endpoints in i[X ], is a completion in C ∗X . Then, we introduce a partial order relation ı j ≤ in C ∗X . Roughly speaking, we will say that X precedes X if there exists a suitable j ı partition of ∂ (X ) by ∂ (X ). More precisely: Definition 7.1. Let X be a weakly distinguishing chronological set and consider two ı j completions X , X ∈ C ∗X with ∂ ı (X ) = {(Pi , Fi ) : i ∈ I } ı
and
∂ j (X ) = {(P j , F j ) : j ∈ J }.
j
Then, we say X ≤ X if there exists some partition ∂ j (X ) = ∪i∈I Si , Si ∩ Si = ∅ if i = i , satisfying the following conditions: (i) if a chain in X has some endpoint in Si then (Pi , Fi ) is also an endpoint of that chain,6 and (ii) for every i ∈ I such that Si = {(P, F)} with (P, F), (Pi , Fi ) endpoints of the same chains, it is dec(Pi ) ⊂ dec(P) and dec(Fi ) ⊂ dec(F). 6 In particular, observe that every S has at most two elements. i
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With this definition the pair (C ∗X , ≤) becomes a partially ordered set (reflexivity and transitivity are direct; antisymmetry needs a simple discussion involving several cases). Furthermore, (C ∗X , ≤) always admits some minimal element: Theorem 7.2. If X is a weakly distinguishing chronological set then (C ∗X , ≤) has some minimal element. Proof. We can assume without restriction that any completion X ∈ C ∗X satisfies that any pair (P, F) ∈ ∂(X ) has dec(P) and dec(F) finite (otherwise, remove from C ∗X those completions which do not verify this property; the minimal elements of the resulting set, which is non-empty, are still minimal elements of C ∗X ). ı ı ı Consider {X α }α∈ ⊂ C ∗X , a well-ordered set with X α ≤ X β for α ≥ β. Fix any α0 ∈ , let S be a set of chains in X such that ∂ ıα0 (X ) = {(Pας0 , Fας0 ) : ς ∈ S}, ς
ς
with (Pα0 , Fα0 ) being some endpoint of ς in X ∂ (X ) = ıα
{(Pας ,
Fας )
ı α0
: ς ∈ S},
. Then, S also satisfies for all α ≥ α0 ,
ς ς ς ς with (Pα , Fα ) being the pre-image in ∂ ıα (X ) of (Pα0 , Fα0 ) via some partition of ∂ ıα0 (X ) ς ς ı α by ∂ (X ) according to Definition 7.1. With these definitions, {Pα }α≥α0 , {Fα }α≥α0 are ∗ ∗ necessarily constants for all α ≥ α , for some α ∈ depending on ς . Therefore, the
set ς
ς
X := i[X ] ∪ {(Pα ∗ , Fα ∗ ) : ς ∈ S} ∈ C ∗X ı
is a lower bound for {X α }α∈ , and thus, Zorn’s Lemma ensures the existence of some minimal completion in C ∗X . We are now ready to introduce the notion of chronological completion: Definition 7.3. A completion X in C X is a chronological completion if it is a minimal element of (C ∗X , ≤). Then, the chronological boundary of X in X is defined as ∂(X ) := X \ i[X ]. Even if it is not very common, there are spacetimes admitting different chronological completions (see Example 10.6). However, if X is complete then C ∗X = {i[X ]}, and thus, i[X ] is the unique chronological completion of X . The next result establishes a series of nice properties (some of them axiomatically imposed in previous approaches) which totally characterize these constructions: Theorem 7.4. Let V be a strongly causal spacetime. Then, a subset ∂(V ) ⊂ V p × V f is the chronological boundary associated to some chronological completion V of V if and only if the following properties hold: (1) Every terminal indecomposable set in V is the component of some pair in ∂(V ). Moreover, if (P, F) ∈ ∂(V ) then P, F are both indecomposable sets if non-empty. (2) If P, F = ∅ satisfy (P, F) ∈ ∂(V ) then P ∼ S F. (3) If (P, ∅) ∈ ∂(V ) (resp. (∅, F) ∈ ∂(V )) then P (resp. F) is not S-related to anything. (4) If (P, F) ∈ ∂(V ) then P, F are both terminal sets if non-empty. (5) If (P, F1 ), (P, F2 ) ∈ ∂(V ), F1 = F2 (resp. (P1 , F), (P2 , F) ∈ ∂(V ), P1 = P2 ) then Fi (resp. Pi ), i = 1, 2, do not appear in any other pair of ∂(V ).
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Proof. First, we will prove the implication to the right. (1) If, for example, some TIP P = ∅ is not the component of any pair in ∂(V ), then any future chain ς ⊂ V with I − [ς ] = P has no endpoint in V , which contradicts that V is a completion. For the second assertion, assume that (P, F) ∈ ∂(V ) is an endpoint of some future chain ς ⊂ V . From Definition 3.1, it is P = I − [ς ], and thus, P is IP. Assume by contradiction that F = ∅ is not IF. Then, if we replace the pair (P, F) in V by (P, ∅), the resulting set is still a completion which contradicts the minimal character of V . (2) By contradiction, assume for example that (P, F) ∈ ∂(V ) is an endpoint of some future chain, but P is not maximal IP into ↓ F. Since V is a completion, there exists some past set P = P such that (P , F) ∈ V . Therefore, if we replace (P, F) in V by (P, ∅), the resulting set is still a completion which contradicts the minimal character of V . (3) By contradiction, assume that (P, ∅) ∈ ∂(V ) but P ∼ S F for some IF F. Then, if we replace (P, ∅) in V by (P, F), the resulting set is still a completion which contradicts the minimal character of V . (4) It directly follows from (2), (3) and Proposition 2.7. (5) Assume by contradiction that (P, F1 ), (P, F2 ), (P , F1 ) ∈ ∂(V ), with F1 = F2 and P = P . Then, if we remove the pair (P, F1 ) from V , the resulting set is still a completion, and thus, contradicts that V ∈ CV∗ . Conversely, consider V := i[V ] ∪ ∂(V ) with ∂(V ) satisfying conditions (1)–(5). From (1) and (2), V is a completion. From (1), (2), (3) and (5), it is V ∈ CV∗ . In order to ı ı prove that V is minimal in (CV∗ , ≤), assume that V ≤ V for some completion V ∈ CV∗ . Then, there exists some partition ∂(V ) = ∪i∈I Si , Si ∩ Si = ∅ if i = i , satisfying condition (i), (ii) in Definition 7.1. From (1), (2) and (3), if (P, F) ∈ Si then P = Pi , ı F = Fi . Whence, Si = {(Pi , Fi )} for all i ∈ I , and thus, V = V . When X = V is a strongly causal spacetime, the chronological completions verify a number of very satisfactory properties. We begin by showing the equivalence between the notions of endpoint and limit of a chain: Theorem 7.5. Let ς = { pn } be a chain in a strongly causal spacetime V . A pair (P, F) is an endpoint of ς in some chronological completion V if and only if (P, F) ∈ L(ρ ) for any subsequence ρ ⊂ ρ = {i( pn )}n ⊂ i[V ]. Proof. From Theorem 6.5 (2), we only need to prove the implication to the left. So, assume (P, F) ∈ L(ρ) for ρ = {i( pn )}n ⊂ i[V ]. If P = ∅ (resp. F = ∅) and ς is a future (resp. past) chain then P = I − [ς ] (resp. F = I + [ς ]), and thus, the implication directly follows from Definitions 3.1, 6.1. So, assume for example that (∅, F) ∈ ∂(V ) is a L-limit of the future chain ρ. From Definition 6.1 and Theorem 7.4 (1), F is maximal IF into ↑ I − [ς ]. Let P be a maximal IP into ↓ F containing I − [ς ]. Then, P ∼ S F, and thus, Theorem 7.4 (3) implies (∅, F) ∈ ∂(V ), a contradiction. Whence, this last possibility cannot happen. Moreover, the chronological boundary is always closed in the corresponding chronological completion: Theorem 7.6. If V is a strongly causal spacetime then ∂(V ) is closed in V . Proof. By contradiction, assume the existence of a sequence σ = {(Pn , Fn )} ⊂ ∂(V ) n} ⊂ V such that i( p) ∈ L(σ ) for some p ∈ V . For every n, consider a chain ς n = { pm m
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with endpoint (Pn , Fn ). Then, ˆ n ) = {I − [ς n ]} = {Pn } or L(ς ˇ n ) = {I + [ς n ]} = {Fn }, either L(ς
(7.1)
is either future or past chain, resp. Let U ⊂ V be a precompact depending on if n } ⊂ V \ U eventually for all m. In neighborhood of p. For every n, necessarily { pm m n fact, otherwise ς converges (up to a subsequence) to certain rn ∈ U with the topology ˆ n ) and I + (rn ) ∈ L(ς ˇ n ) (Proposition 2.5 and its of the manifold, and thus, I − (rn ) ∈ L(ς − dual). Therefore, from (7.1), either Pn = I (rn ) or Fn = I + (rn ), in contradiction with (Pn , Fn ) ∈ ∂(V ) (recall Theorem 7.4 (4)). In conclusion, fixed future and past chains ς = {qk }, ς = {qk } such that I − ( p) = I − [ς ] and I + ( p) = I + [ς ], we can choose nk nk {n k }k , {m k }k satisfying qk pm k qk and pm k ∈ V \ U for all k. Therefore, taking into account that ς , ς converge to p with the topology of the manifold, any sequence nk of future-directed timelike curves joining qk with pm k and then with qk contradicts the strong causality of V . ςn
The chronological completions also satisfy reasonably good separation properties. In fact, from Definition 6.1, Proposition 2.7 and Theorem 7.4 (1), (2), (3) every element (P, F) ∈ V is the unique limit in V of the sequence constantly equal to (P, F), thus: Theorem 7.7. If V is a strongly causal spacetime then V is T1 . Notice however that V is not always T2 (Example 10.3). The lack of Hausdorffness in chronological completions cannot be attributed to a defect of our particular approach. On the contrary, it seems a remarkable property intrinsic to the causal boundary approach itself (see [20] for an interesting discussion on this question). Even if the chronological completions may be non-Hausdorff, there are still some restrictions to the elements of V which can be non-Hausdorff related, Theorem 7.9. In order to prove this result, we need the following proposition: Proposition 7.8. Let V be a chronological completion of a strongly causal spacetime V . If K ⊂ V is compact in V then i[K ] is closed in V . Proof. From Theorems 6.3, 6.4, i[K ] is closed in i[V ]. So, by contradiction, we will assume the existence of (P, F) ∈ ∂(V ), with (P, F) ∈ L(ρ) for a certain sequence ρ = {i( pn )} such that σ = { pn } ⊂ K . Since K is compact, we can also assume that σ = { pn } ⊂ K converges to some p ∈ K . First, observe that P, F = ∅. In fact, by contradiction, assume for example that F = ∅. Let F be a maximal IF in ↑ P containing I + ( p). Then, necessarily P is maximal IP into ↓ F , and thus, P ∼ S F , which contradicts Theorem 7.4 (3). Moreover, from Theorem 7.4 (2) it is also P ∼ S F. Whence, it cannot happen that P ⊂ I − ( pn ) and F ⊂ I + ( pn ) for some n; so, assume for example that P ⊂ I − ( pn ) for infinitely many n.
(7.2)
Let γ be a future-directed timelike curve with P = I − [γ ]. Since (P, F) ∈ L(ρ), necessarily I − [γ ] = P ⊂ LI(I − ( pn )). (7.3) Up to a subsequence of σ , from (7.2) and (7.3) we can choose a future chain ς = {rn }n ⊂ γ with I − [ς ] = I − [γ ], and rn ∈ I˙− ( pn ) for all n ( I˙− ( pn ) denotes the topological boundary of I − ( pn )) such that q pn for all q rn . These conditions necessarily imply P I − ( p) ⊂↓ F, which contradicts that P is maximal IP into ↓ F.
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Theorem 7.9. Let V be a chronological completion of a strongly causal spacetime V . If two elements of V are non-Hausdorff related then they are both in ∂(V ). Proof. Let σ = {(Pn , Fn )} ⊂ V be a sequence such that (P, F), (P , F ) ∈ L(σ ) with (P, F) = (P , F ). Assume by contradiction that (P, F) = (I − ( p), I + ( p)) for some p ∈ V . From Theorems 7.6, 6.3 and 6.4, it is not a restriction to assume (Pn , Fn ) = (I − ( pn ), I + ( pn )) for all n, with { pn }n ⊂ V converging to p with the topology of the manifold. In particular, K = { pn }n ∪ { p} is a compact set in V . Therefore, from Proposition 7.8, i[K] is closed in V , and thus, (P , F ) ∈ i[K ]. Again from Theorems 6.3, 6.4, this contradicts that (P, F) = (P , F ). We finish this section by remarking that all these satisfactory results do not avoid the existence of some “contra-intuitive” limit behaviors for the chr -topology. Consider the situation described in Example 10.4. If we appeal to our intuition, inherited from the natural embedding of this space into Minkowski, we would expect that the sequence σ converges to (P0 , F0 ) ∈ ∂(V ), which is not the case for the chr -topology. Notice however that this intuition is using additional information not exclusively contained in the causal structure of the spacetime. More precisely, if we analyze the chronology of the elements of σ , we observe that all of them have empty future. But the future in V of (P0 , F0 ) is F0 = ∅. Therefore, any topology exclusively based on the chronology must conclude that σ does not converge to (P0 , F0 ), as the chr -topology does (notice that this situation is totally different from that showed by the examples in [17, Sect. III], [18, Sects. II, III], where the chronology of the elements of the sequences have a good limit behavior, but, still, there is no convergence with the topologies involved there). This discussion shows that the causal boundary approach should not be considered an innocent variation of the conformal boundary. On the contrary, it provides a genuine insight on the asymptotic causal structure of the spacetime. In this sense, Examples 10.3 and 10.4 2 tell us that the asymptotic causal structure of L is modified in a very different way if we remove a vertical segment than if we remove a horizontal one (this is reasonable since time evolves just in the vertical direction), in contraposition with the conformal boundary approach, which does not reflect this asymmetry. Therefore, even though this limit behavior does not reproduce the situation in the conformal boundary, we consider this difference very satisfactory. 8. Comparison with Other Approaches A natural question still needs to be investigated in order to emphasize the optimal character of our construction: what is the relation between the chronological completions and the completions suggested by other authors? In order to fix ideas we have chosen what perhaps are the most accurate approaches to the (total) causal boundary, up to date: the Marolf-Ross and the Szabados completions. The Marolf-Ross completion V M R of a strongly causal spacetime V is formed by all the pairs (P, F) composed by an IP P and an IF F, such that: (i) P ∼ S F, or (ii) P = ∅ and P ∼ S F for any IP P , or (iii) F = ∅ and P ∼ S F for any IF F . The chronology adopted here is also (P, F)(P , F ) iff F ∩ P = ∅. So, taking into account Proposition 2.7, the Marolf-Ross construction becomes the biggest completion (according to Definition 3.2) which satisfies properties (1)–(4) in
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Theorem 7.4. In particular, if V admits more than one chronological completion then V M R is strictly greater than any of them, since it includes the union of all of them, as illustrated in the first spacetime of Example 10.6. Remarkably, the strict inclusion V V M R may also hold even when V admits just one chronological completion, as illustrated in the second spacetime of Example 10.6. The authors also adopt a topology for V M R : the topology generated by the subbasis V M R \ L ± (S) for any S ⊂ V M R , where L + (S) = Cl F B [S ∪ L +I F (S)], L − (S) = Cl P B [S ∪ L − I P (S)], with ˆ n ) for (Pn , Fn ) ∈ S}, Cl F B (S) = S ∪ {(P, F) ∈ V M R : F = ∅, P ∈ L(P ˇ n ) for (Pn , Fn ) ∈ S}, Cl P B (S) = S ∪ {(P, F) ∈ V M R : P = ∅, F ∈ L(F and L +I F (S) = {(P, F) ∈ V M R : F = ∅, F ⊂ ∪(P ,F )∈S F }, L− I P (S) = {(P, F) ∈ V M R : P = ∅, P ⊂ ∪(P ,F )∈S P }. With these structures, the following remarkable comparison result can be stated: Theorem 8.1. Given a strongly causal spacetime V , the inclusion defines a continuous and chronological map from every chronological completion V into the MR completion V M R. Proof. Let V be any chronological completion of V . As V ⊂ V M R , the inclusion i : V → V M R is well-defined and is always chronological. In order to prove that i is also continuous, suppose that {(Pn , Fn )}n ⊂ V converges to some (P, F) ∈ V with the chr -topology. We wish to prove that any open set U = V M R \ L ± (S), S ⊂ V M R , of the subbasis which generates the MR topology such that (P, F) ∈ U necessarily contains (Pn , Fn ) for all n big enough. First, notice that (Pn , Fn ) ∈ S ∪ L +I F (S) for all n big enough. In fact, otherwise, F ⊂ LI(Fn ) ⊂ ∪(P ,F )∈S F . So, if F = ∅ then (P, F) ∈ L +I F (S) ⊂ L + (S), a contradiction. If, instead, F = ∅, taking into account that ˆ n ), necessarily (P, F) ∈ Cl F B [S ∪ L + (S)] = L + (S), which is again a contraP ∈ L(P IF diction. Therefore, (Pn , Fn ) ∈ S ∪ L +I F (S) for all n big enough. Furthermore, (Pn , Fn ) ˆ n ), (P n , F n ) ∈ S ∪ L + (S) for all n big enough. In fact, cannot be (Pn , ∅), Pn ∈ L(P k k k IF ˆ n ) for some subsequence {kn }n ⊂ {k}k , with otherwise, it must be F = ∅ and P ∈ L(P kn (Pknn , Fknn ) ∈ S ∪ L +I F (S), and thus, (P, F) ∈ Cl F B [S ∪ L +I F (S)] = L + (S), a contradiction. In conclusion, (Pn , Fn ) ∈ Cl F B [S ∪ L +I F (S)] = L + (S) for all n big enough, as required. A conceptually different approach to the causal boundary of spacetimes consists of using identifications instead of pairs to form the ideal points of the boundary (see [9]; and the subsequent papers [23, 3, 25, 26]). This approach presents important objections (see for example [20, Sect. 2.2] for an interesting discussion); however, sometimes some identifications may be useful to emphasize certain aspects of the original spacetime. Our purpose here is to provide some evidence that chronological completions are the optimal spaces on which to establish eventual identifications. To this aim, we are going to
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give an improved version of the Szabados construction just by establishing some natural identifications on any chronological completion. The Szabados completion V S of a strongly causal spacetime V is formed by taking the quotient of V (as defined in footnote 4) by the minimum equivalence relation R containing ∼ S .7 Whence, each point m ∈ V S is a class [P1 , P2 , . . . ; F1 , F2 , . . .] of R-equivalent IPs and IFs. Szabados writes m m if, for some Fα ∈ π −1 (m) and Pµ ∈ π −1 (m ), Fα ∩ Pµ = ∅. He also endows V S with the quotient topology of T under R, where T is the extended Alexandrov topology defined on V , that is, the coarsest topology such that for each A ∈ Vˇ , B ∈ Vˆ the four sets Aint , B ext , B int , Aext are open sets, where Aint = {P ∗ ∈ V : P ∈ Vˆ and P ∩ A = ∅}, Aext = {P ∗ ∈ V : P ∈ Vˆ and ∀S ⊂ V P = I − [S] ⇒ I + [S] ⊂ A} (the sets B int and B ext have similar definitions with the roles of past and future interchanged). In order to compare any chronological completion V with the Szabados completion V S , the following identifications on V become natural: two pairs in V are R-related iff some of its respective components are R-related. Then, we endow the resulting quotient space V /R with the corresponding quotient structures; that is, the quotient chronology (P1 , F1 ) R (P1 , F1 ) [(P1 , F1 )][(P2 , F2 )] iff (P1 , F1 )(P2 , F2 ) for some (P2 , F2 ) R (P2 , F2 ); and the quotient of the chr -topology. There is an obvious bijection b : V /R → V S which maps every class [(P, F)] ∈ V /R to the class m ∈ V S formed by all the IPs and IFs appearing in some pair in [(P, F)]. With this definition, b is obviously a chronological isomorphism. Furthermore, the only examples where b is not continuous seem to be exclusively caused by “pathologies” of the Szabados topology, and thus, cannot be regarded as an anomaly of our construction. An illustrative example of this situation is showed in [18, Sect. III]. 9. Causal Ladder and the Boundary of Spacetimes The causal boundary approach constitutes an useful tool for examining the causal nature of a spacetime “at infinity”, which usually reflects important global aspects of the causal structure. Therefore, it may be interesting to analyze the causality of a spacetime just by looking at the boundary. An illustrative example of this situation is the following characterization of global hyperbolicity: a spacetime is globally hyperbolic iff there are no elements at the boundary whose past and future are both non-empty. This result, proposed in [24] in a slightly different context, was proved by Budic and Sachs in [3, Th. 6.2]. However, their proof lies on the particular approach of the authors to the causal boundary (developed also in [3]), and thus, it suffers from the same important restriction: it is only valid for causally continuous spacetimes. The main aim of this section consists of extending this characterization to any strongly causal spacetime by using the chronological boundary of spacetimes. More precisely, we prove: 7 According to [26] further identifications between pairs of TIPs or TIFs must be considered; however, they will be omitted in our discussion.
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Theorem 9.1. Let V be a strongly causal spacetime. Then, V is globally hyperbolic if and only if there are no elements (P, F) ∈ ∂(V ) with P, F = ∅. Proof. First, recall that a strongly causal spacetime is globally hyperbolic if and only if the causal diamond J ( p, q) := J + ( p) ∩ J − (q) is compact for any p, q ∈ V . As a direct consequence of [21, p. 409, Lemma 14], this holds if J ( p, q) is included in a compact set K ⊂ V . For the implication to the left, assume that there are no elements (P, F) ∈ ∂(V ) with P, F = ∅. Take K = I ( p, q) ⊃ J ( p, q) with I ( p, q) := I + ( p) ∩ I − (q). Therefore, in order to prove that K is compact we only need to show that any sequence in I ( p, q) admits a subsequence with some limit in V . By contradiction, assume that some σ ⊂ I ( p, q) does not satisfy this assertion. From [4, Theorem 5.11] applied to Vˆ , there exists some subsequence σ ⊂ σ and some TIP P such that ∅ = I − ( p) ⊂ P and ˆ ). Let ∅ = F be a maximal TIF into ↑ P. Let ∅ = P ⊂ P be some maximal P ∈ L(σ TIP into ↓ F. Then, necessarily P ∼ S F. Therefore, from Theorem 7.4 (1), (3), there exists some IF F = ∅, such that (P , F ) ∈ ∂(V ). This contradicts the hypothesis on the boundary. Whence, K is compact. For the implication to the right, assume the existence of (P, F) ∈ ∂(V ) with P, F = ∅. Take points p ∈ P, q ∈ F. Take a chain ς ⊂ V with endpoint (P, F). Then, the elements of ς are eventually contained in I ( p, q). However, from Theorem 6.5 (2) and Theorem 7.9, there cannot exist subsequences of ς converging in V . Whence, J ( p, q) ⊂ V is not compact. A simple illustration of this result is provided by the Minkowski plane L : any element 2 (P, F) ∈ ∂(L ) satisfies either P = ∅ or F = ∅ (Example 10.1); however, when a point is removed, and thus, the spacetime is no longer globally hyperbolic, a pair (P, F) with P, F = ∅ immediately appears (Example 10.2). The causal boundary approach also becomes useful to characterize other levels of the causal ladder as causal simplicity, [3, Cor. 5.2]. For completeness, we are going to prove an extension of this result, again valid for any strongly causal spacetime. To this aim, consider any causal relation ≺ on any chronological completion V such that over i[V ] it satisfies:8 2
i( p) ≺ i(q) iff either I + (q) ⊂ I + ( p) or I − ( p) ⊂ I − (q). Then, the following results hold: Lemma 9.2. For any two points p, q in a strongly causal spacetime V , i( p) ≺ i(q) if and only if, either q ∈ J + ( p) or p ∈ J − (q). Proof. For the implication to the right, assume that i( p) ≺ i(q). If I + (q) ⊂ I + ( p), take any sequence {qn } ⊂ I + (q) such that qn → q. Then, we obtain q ∈ I + (q) ⊂ I + ( p) ⊂ J + ( p). If, instead, I − ( p) ⊂ I − (q), just reason analogously to obtain p ∈ I − ( p) ⊂ I − (q) ⊂ J − (q). For the implication to the left, first assume that q ∈ J + ( p). Take q ∈ I + (q) and a sequence {qn } ⊂ J + ( p) with qn → q. For all n big enough, qn q . Whence, q ∈ I + ( p), and thus, I + (q) ⊂ I + ( p). Therefore, i( p) ≺ i(q). Assume now that p ∈ J − (q). Reasoning analogously we deduce I − ( p) ⊂ I − (q), and thus, i( p) ≺ i(q) holds again. 8 We have introduced here causal relations just to establish Theorem 9.3, but with no further pretension. We postpone to a future paper a more precise definition of causal relations into our framework.
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Theorem 9.3. A strongly causal spacetime V is causally simple if and only if the causality ≺ of V restricted to i[V ] coincides with that of V . Proof. For the implication to the right, Lemma 9.2 implies that relation i( p) ≺ i(q) holds if and only if either q ∈ J + ( p) or p ∈ J − (q), and, from the hypothesis, this holds if and only if q ∈ J + ( p). Therefore, the causality of V restricted to i[V ] coincides with that of V . For the implication to the left, first assume that q ∈ J + ( p). From Lemma 9.2, it is i( p) ≺ i(q), and, from the hypothesis, this implies q ∈ J + ( p). Therefore, J + ( p) is closed. Analogously, if p ∈ J − (q), necessarily p ∈ J − (q), and thus, J − (q) is closed too. Therefore, V is causally simple. Finally, let us remark that the link between the causal ladder and the boundary of spacetimes may also arise in a deeper way. In Example 10.8 we have indicated how the dramatic change in the level of causality of generalized wave type spacetimes (10.1), when the metric coefficient −H leaves the quadratic growth, translates into a low dimensionality of the chronological boundary for these spacetimes. It would be interesting to explore if this intriguing relation between the critical behavior of the causality of a spacetime and the dimensionality of its boundary is generalizable to further classes of spacetimes. 10. Examples In this section we briefly examine our construction in some examples and compare it with previous approaches, putting special emphasis in the differences between them. Example 10.1. Consider Minkowski space V = L (≡ Rn × L ). In order to construct the chronological boundary ∂(V ), in this case it suffices to consider the set of pairs (I − [γ ], ∅) (resp. (∅, I + [γ ])), for every inextensible future-directed (resp. past-directed) lightlike geodesic γ , in addition to the pairs (V, ∅) and (∅, V ). Therefore, ∂(V ) can be identified with a pair of cones on S n−1 with apexes i + , i − (Fig. 2). This is in total n+1
1
Fig. 2. Chronological boundary for L
n+1
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agreement with the image of the (standard) conformal embedding of Minkowski space into the Einstein Static Universe. On the other hand, the limit of every sequence in V with the chr -topology coincides with the set-theoretic limit of the elements of the sequence. Again, this provides just the n+1 into same topology as that inherited from the (standard) conformal embedding of L ESU. Example 10.2. Let V be L with the origin point removed (Fig. 3): 2
V = L \ {(0, 0)}. 2
The GKP pre-completion V of this spacetime attaches at the origin two ideal endpoints given by the pairs (P, ∅), (∅, F). This provides a simple example of the non-equivalence between the notions of limit and endpoint of a chain: the pair (∅, F) ∈ V is a limit of the chain {i( pn )} ⊂ i[V ], however, the unique endpoint of { pn } in V is instead the pair (P, ∅). Consider now the completion resulting from replacing in V the pair (P, ∅) by (P, F). In this case, the sequence constantly equal to (P, F) converges with the chr topology to both, (∅, F), (P, F), showing that this topology is not T1 for this completion. The (unique) chronological completion V of V only attaches at the origin the ideal point (P, F), showing in particular that V is not globally hyperbolic (Theorem 9.1). On the other hand, this example shows that “boundary” need not be metrically infinitely far along geodesics, as illustrated by the curve γ (t) = (0, t), t < 0. Example 10.3. Let V be L with the vertical segment V+ = {(0, t) : t ≥ 0} removed (Fig. 4): 2
V = L \ {(0, t) : t ≥ 0}. 2
Let V be the (unique) chronological completion of V . The pairs (P, Fl ), (P, Fr ) are the unique endpoints in V of the chains {qn }, { pn }, resp. They represent two ideal endpoints attached at the extreme of V+ . These pairs are also endpoints of the future chain {rn }, and thus, limits of {i(rn )} (recall Theorem 6.5 (2)). Therefore, V is non-Hausdorff with the chr -topology. If we extend this analysis to the whole line V+ , we obtain that ∂(V ) contains two copies of V+ , with only the extreme ideal points (P, Fl ), (P, Fr ) being non-Hausdorff related.
Fig. 3. Minkowski plane L with the origin removed 2
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Fig. 4. Minkowski plane L with a vertical segment V+ removed 2
On the other hand, observe that the chain {rn } generates the pair (P, Fl ∪ Fr ) ∈ V p × V f . This pair belongs to the completion V (see Sect. 3), showing in particular that some completions may contain pairs whose components are not necessarily indecomposable. Example 10.4. Let V be L with the horizontal segment H− = {(x, 0) : x ≤ 0} removed (Fig. 5): 2
V = L \ {(x, 0) : x ≤ 0}. 2
Let V be the (unique) chronological completion of V . For x < 0, the unique endpoints in V of the chains {(x, −1/n)} and {(x, 1/n)} are the pairs (Px , ∅) and (∅, Fx ), resp. However, for x = 0 the unique endpoint in V of the chains {(0, −1/n)} and {(0, 1/n)} is the pair (P0 , F0 ). Therefore, in this case the chronological completion V contains two copies of H− with the right extreme points of the copies identified via (P0 , F0 ). On the other hand, we can ask for the limit of the sequence σ = {(Pxn , ∅)}n ⊂ V , with xn = −1/n for
Fig. 5. Minkowski plane L with a horizontal segment H− removed 2
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all n. Surprisingly, σ does not converge to (P0 , F0 ) with the chr -topology, violating the common intuition inherited from the natural embedding of this space into Minkowski. When we consider a completion different from the chronological one, the corresponding boundary may be non-closed. In fact, take for example the completion V end , which contains in particular the endpoint (P, F), with P = P0 ∪ I − ( p) and F = I + ( p), p = (−1, 1). Then, the sequence constantly equal to (P, F), which is obviously contained in the boundary, converges to i( p) ∈ i[V ] with the chr -topology. Finally, consider the future chronological completion Vˆ of this spacetime. Apart from the obvious limit I − ( p), the sequence of PIPs δ = {I − ( pn )} ⊂ Vˆ , pn = (−1 + 1/n, 1) for all n, also converges to P0 with the -topology. This anomalous limit is due to the fact that Vˆ and -topology only retain partial information about the chronology of V . This situation contrasts with our construction, where the full chronology is taken into consideration. In fact, under our approach, Theorems 6.3, 6.4 and 7.9 imply that {i( pn )}n ⊂ V only converges to i( p), as expected. Example 10.5. Consider the spacetime V represented in Fig. 6 (this example comes from [25, Fig. 2]; see also [20, Fig. 7]). Here, infinite null segments {L n }n and the point r have been removed from the Minkowski plane, resulting in a spacetime such that I − [γ ] I − [γ ]. In this case the (unique) chronological boundary ∂(V ) coincides with the MR boundary (see [20]), including the pairs (I − [γ ], F) and (I − [γ ], ∅) as endpoints of γ and γ , resp. However, the MR topology is different from the chr -topology. In fact, as showed in [20], the sequence constantly equal to (I − [γ ], F) also converges to (I − [γ ], ∅) with the MR topology, and thus, it is not T1 . This is not the case for the chr -topology, which is always T1 (Theorem 7.7). Example 10.6. Let X be the disjoint union of Ii = (−∞, +∞)i , i = 1, 2, under the equivalence relation 01 ∼ 02 (Fig. 7). Endow X with the (quotient of the) chronology relation given by either x < x and x, x ∈ Ii for some i; x x iff or x < x and 0 ≥ x ∈ Ii , 0 ≤ x ∈ I j for i = j.
Fig. 6. Minkowski plane L with a sequence of null segments {L n }n , including the limit point r , removed 2
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Fig. 7. Two copies of R with the zeroes identified
Then, X becomes a chronological set satisfying I ± ([0]) = I1± ∪ I2± ,
with
Ii− := (−∞, 0)i . Ii+ := (0, +∞)i
Therefore, the past and future of [0] ∈ X are not indecomposable. Motivated by this example, consider now the (2 + 1)-spacetime V constructed by 3 deleting from L the subsets {x = 0, −1 ≤ t ≤ 0} and {y = 0, 0 ≤ t ≤ 1}. In fact, V has two TIPs P1 , P2 and two TIFs F1 , F2 associated to the removed origin (Fig. 8), which verify P1 ∼ S F1 , F2 and P2 ∼ S F1 , F2 . Therefore, there are two different chronological completions for V : one of them attaches at the origin the ideal endpoints (P1 , F1 ), (P2 , F2 ), while the other one attaches the ideal endpoints (P1 , F2 ), (P2 , F1 ). Both of these chronological completions are different from the MR completion, which attaches at the origin “more” ideal endpoints: (P1 , F1 ), (P1 , F2 ), (P2 , F1 ), 3 (P2 , F2 ). If we additionally delete from L the subset {x > 0, y > 0, t = 0} then the MR completion attaches at the origin the pairs (P1 , F1 ), (P1 , F2 ), (P2 , F2 ). In this case the spacetime admits an unique chronological completion, which attaches at the origin the pairs (P1 , F1 ), (P2 , F2 ). Another example of spacetime whose MR completion also includes spurious ideal points appears in [20, Appendix A]. Example 10.7 (Standard Static Spacetimes). The causal boundary for standard static spacetimes has been studied in [12, 4, 1]. From the conformal invariance of the causal boundary approach, it is not a restriction to assume that these spacetimes can be written as V = M × R,
g = h − dt 2 ,
Fig. 8. Slices showing the cuts made to produce our example, and the corresponding TIPs and TIFs
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where (M, h) is an arbitrary Riemannian 3-manifold. In these spacetimes, the spatial projection c of every inextensible future-directed timelike curve γ (t) = (c(t), t) whose past is different from the whole spacetime is an asymptotically ray-like curve, i.e. an inextensible curve with domain [w, ), ≤ ∞, velocity |c| ˙ ≤ 1 and finite-valued Busemann function bc ; that is, bc : M → R∗ ≡ R ∪ {∞},
bc (·) := lim (t − d(·, c(t))) < ∞. t→
Moreover, the pasts of these curves are totally characterized by the corresponding Busemann functions bc . Therefore, if we denote by B(M) the set of all Busemann functions associated to asymptotically ray-like curves in (M, h), we obtain ˆ ) = B(M) ∪ {∞}. ∂(V In other words, if we define the Busemann boundary of (M, h) as the quotient space ∂B (M) := B(M)/R, which, in particular, includes the Cauchy boundary ∂c (M), then the future chronological boundary of V is a cone with apex i + and base ∂B (M); i.e. ˆ ) = B(M) ∪ {∞} ≡ (∂B (M) × R) ∪ {i + }. ∂(V Analogously, the past chronological boundary of V is a cone with apex i − and base ∂B (M); i.e. ˇ ) = (∂B (M) × R) ∪ {i − }. ∂(V But, what about the (total) chronological boundary? The common future (resp. common past) of any inextensible future-directed (resp. past-directed) timelike curve γ is ˆ ) with p ∈ ∂c (M); moreover, in this case non-empty iff γ approaches some (t, p) ∈ ∂(V the common future (resp. common past) of γ coincides with the future (resp. past) of ˇ ). any past-directed (resp. future-directed) timelike curve approaching also (t, p) ∈ ∂(V So, the (total) chronological boundary is a double cone with base ∂B (M), apexes i + , i − , and future and past copies of lines over the same point p of the Cauchy boundary ∂c (M) identified. Summarizing: ⎧ ˆ ) ⎪ ( p + , t + ) ∈ ∂(V ⎪ ⎨ − − ˇ ) ( p , t ) ∈ ∂(V ˆ ) ∪ ∂(V ˇ ))/ ∼, ∂(V ) = (∂(V with ( p + , t + ) ∼ ( p − , t − ) iff + = p − ∈ ∂ (M) ⎪ p c ⎪ ⎩ + t = t− ∈ R (see Fig. 9). On the other hand, the chr -topology on V coincides with the quotient topology over ∼ of the topology generated by the limits operators Lˆ and Lˇ on Vˆ ∪ Vˇ . Example 10.8 (Locally Symmetric Plane Waves). Consider V = Rn+2 with metric ·, · = ·, · where ·, ·
0
0
+ 2du dv + H (x)du 2 ,
is the canonical metric of Rn and
H (x) = −µ21 x12 − · · · − µ2j x 2j + µ2j+1 x 2j+1 + · · · + µ2n xn2 ,
with j > 0
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Fig. 9. Chronological boundary for Standard Static spacetimes
and µ1 ≥ µ2 ≥ · · · ≥ µ j . The causal boundary of these spacetimes was first analyzed in [19, 20] by using the MR approach. If we denote by L + , L − two copies of the line u ∈ (−∞, ∞), and define L = (L + ∪ L − )/ R,
u R u ⇐⇒ u ∈ L + , u ∈ L − , u = u − π/µ1 ,
the authors found that the MR boundary of V can be represented by the single line ∂(V ) = L ∪ {i + , i − }. Moreover, in this case the MR construction coincides with our construction (see [6]). Therefore, the chronological boundary of V can be also represented by this single line (see Fig. 10). This picture agrees with the result previously obtained in [2] for the maximally symmetric case by using the conformal approach (see [20, Sect. 5] for a brief discussion). This 1-dimensional character of the boundary admits an intriguing interpretation in terms of the global causal behavior of the wave. In fact, in [5] the authors found that the
Fig. 10. Chronological boundary for Locally Symmetric Plane Waves as pictured in [20]
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causality of generalized wave type spacetimes V = M × R2 ,
·, ·
z
= ·, ·
x
+ 2du dv + H (x, u)du 2 ,
(10.1)
where (M, ·, · x ) is a Riemannian n-manifold and H : M × R → R a smooth function, presents a critical behavior with respect to the metric coefficient −H . More precisely, these waves pass from being globally hyperbolic for −H subquadratic (and M complete) to being non-distinguishing for −H superquadratic. This gap in the causal ladder has to do with a sort of degeneracy for the chronology of the wave “at infinity”, which, in this case, translates into a low dimensionality of the boundary. We refer the reader to [19, 15, 6] for a systematic study (of increasing generality) of the causal boundary for spacetimes (10.1), including plane waves and pp-waves. Acknowledgements. I would like to thank Professors Steven Harris and Miguel Sánchez for useful discussions and comments. Part of this work was completed during a research stay at Department of Mathematics, Stony Brook University. Partially supported by MECyD Grant EX-2002-0612, JA regional Grant P06-FQM-01951 and MEC Grants RyC-2004-382 and MTM2007-60731.
References 1. Alaña, V., Flores, J.L.: The causal boundary of product spacetimes. Gen. Relat. Grav., in press, DOI 10.1007/s10714-007-0492-5 2. Berenstein, D., Nastase, H.: On lightcone string field theory from super Yang-Mills and holography. http://arxiv.org/list/hep-th/0205048, 2002 3. Budic, R., Sachs, R.K.: Causal boundaries for general relativistic spacetimes. J. Math. Phys. 15, 1302–1309 (1974) 4. Flores, J.L., Harris, S.G.: Topology of causal boundary for Standard Static spacetimes. Class. Quantum Grav. 24, 1211–1260 (2007) 5. Flores, J.L., Sánchez, M.: Causality and conjugate points in general planes waves. Class. Quantum Grav. 20, 2275–2291 (2003) 6. Flores, J.L., Sánchez, M.: The causal boundary of wave-type spacetimes. Preprint (2007) 7. García-Parrado, A., Senovilla, J.M.: Causal relationship: A new tool for the causal characterization of Lorentzian manifolds. Class. Quantum Grav. 20, 625–664 (2003) 8. García-Parrado, A., Senovilla, J.M.: Causal structures and causal boundaries. Class. Quantum Grav. 22, R1–R84 (2005) 9. Geroch, R.P., Kronheimer, E.H., Penrose, R.: Ideal points in spacetime. Proc. Roy. Soc. Lond. A 237, 545–567 (1972) 10. Harris, S.G.: Universality of the future chronological boundary. J. Math. Phys. 39, 5427–5445 (1998) 11. Harris, S.G.: Topology of the future chronological boundary: universality for spacelike boundaries. Class. Quantum Grav. 17, 551–603 (2000) 12. Harris, S.G.: Causal boundary for Standard Static spacetimes. Nonlinear Analysis 47, 2971–2981 (2001) 13. Harris, S.G.: Discrete group actions on spacetimes: causality conditions and the causal boundary. Class. Quantum Grav. 21, 1209–1236 (2004) 14. Harris, S.G.: Boundaries on spacetimes: an outline. Contemp. Math. 359, 65–85 (2004) 15. Hubeny, V., Rangamany, M.: Causal structures of pp-waves. J. High Energy Phys. 12, 043 (2002) 16. Kuang, Z.-Q., Li, J.-Z., Liang, C.-B.: c-boundary of Taub’s plane-symmetric static vacuum spacetime. Phys. Rev. D 33, 1533–1537 (1986) 17. Kuang, Z.-Q., Liang, C.-B.: On the GKP and BS constructions of the c-boundary. J. Math. Phys. 29, 433–435 (1988) 18. Kuang, Z.-Q., Liang, C.-B.: On the Racz and Szabados constructions of the c-boundary. Phys. Rev. D 46, 4253–4256 (1992) 19. Marolf, D., Ross, S.: Plane Waves: To infinity and beyond!. Class. Quant. Grav. 19, 6289–6302 (2002) 20. Marolf, D., Ross, S.F.: A new recipe for causal completions. Class. Quantum Grav. 20, 4085–4117 (2003) 21. O’Neill, B.: Semi-Riemannian Geometry with applications to Relativity, Series in Pure and Applied Math. 103, N.Y.: Academic Press, 1983 22. Penrose, R.: Conformal treatment of infinity. In: Relativity, Groups and Topology, edited by C.M. de Witt, B. de Witt, New York: Gordon and Breach, 1964; Penrose, R.: Zero rest-mass fields including gravitation: asymptotic behavior. Proc. Roy. Soc. Lond. A 284, 159–203 (1965)
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23. Racz, I.: Causal boundary of space-times. Phys. Rev. D 36, 1673–1675 (1987): Racz, I.: Causal boundary for stably causal space-times. Gen. Relat. Grav. 20, 893–904 (1988) 24. Seifert, H.: The Causal Boundary of Space-Times. Gen. Rel. Grav. 1, 247–259 (1971) 25. Szabados, L.B.: Causal boundary for strongly causal spaces. Class. Quantum Grav. 5, 121–134 (1988) 26. Szabados, L.B.: Causal boundary for strongly causal spacetimes: II. Class. Quantum Grav. 6, 77–91 (1989) Communicated by G.W. Gibbons
Commun. Math. Phys. 276, 645–670 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0304-5
Communications in
Mathematical Physics
Moment Bounds for the Smoluchowski Equation and their Consequences Alan Hammond1, , Fraydoun Rezakhanlou2, 1 Courant Institute, New York University, New York, NY 10012, USA.
E-mail: [email protected]
2 Mathematics Department, University of California, Berkeley, CA 94720, USA.
E-mail: [email protected] Received: 2 October 2006 / Accepted: 6 February 2007 Published online: 25 September 2007 – © Springer-Verlag 2007
Abstract: We prove L ∞ Rd × [0, ∞) bounds on moments X a := m∈N m a f m (x, t) of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than (n + m) d(n) + d(m) , and the diffusion rate d(·) is non-increasing and satisfies m −b1 ≤ d(m) ≤ m −b2 for some b1 and b2 satisfying 0 ≤ b2 < b1 < ∞, then any weak solution satisfies X a ∈ L ∞ Rd × [0, T ] ∩ L 1 Rd × [0, T ] for every a ∈ N and T ∈ (0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. 1. Introduction The Smoluchowski coagulation equation is a coupled system of partial differential equations that describes the evolving densities of a system of diffusing particles that are prone to coagulate in pairs. A sequence of functions f n : Rd × [0, ∞) → [0, ∞), n ∈ N, is a solution of the Smoluchowski coagulation equation if it satisfies ∂ f n x, t = d(n) f n (x, t) + Q n ( f )(x, t), ∂t
(1.1)
with Q n ( f ) = Q +n ( f ) − Q − n ( f ), where Q +n ( f )(x, t)
=
n−1
α(m, n − m) f m (x, t) f n−m (x, t)
m=1 This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C. This work is supported in part by NSF grant DMS0307021.
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and Q− n ( f )(x, t) = 2 f n (x, t)
∞
α(n, m) f m (x, t).
m=1
We will interpret this solution in the weak sense. Namely, we will assume that Q +n and 1 d Q− n belong to L R × [0, T ] for each T ∈ [0, ∞) and n ∈ N, and that d(n) 0 f n (x)
f n (x, t) = St
t
+ 0
d(n)
St−s Q n (x, s)ds,
: n ∈ N denotes the initial data, StD the semigroup associated with the = Du, and where Q n (x, s) means Q n ( f )(x, s). The system (1.1) has two sets of parameter values, the sequence d : N → [0, ∞), where d(n) denotes the diffusion rate of the Brownian particle of mass n, and the collection α : N2 → [0, ∞), where α(m, n) models the average propensity of particles of masses m and n to coagulate. The terms Q +n ( f ) and Q − n ( f ) are gain and loss terms for the presence of particles of mass n that arise from the binary coagulation of particles. The system (1.1) may be augmented by considering the fragmentation of particles into two or more sub-particles. A continuous version of the system, in which particles have real (rather than integer) mass, has also been considered. The data are defined d : [0, ∞) → [0, ∞) and α : [0, ∞)2 → [0, ∞), and the sums are replaced integrals in the definitions of Q +n ( f ) and Q − n ( f ). The spatially homogeneous version of the equations, in which each f n is a function of time alone, are better understood ([1] resolved many of the central questions for the discrete case without fragmentation). Lang and Nguyen [9] considered a system of mass-bearing diffusing particles, whose diffusion rate was chosen to be independent of the mass, that are prone to coagulate in pairs at close range, and demonstrated that, in a kinetic limit, the density of particles evolves macroscopically as a solution of (1.1). A kinetic limit means that, with an initial number N of particles, the order of the interaction range = (N ) of any given particle is chosen so that a typical particle experiences a rate of collision that is bounded away from zero and infinity for all high N . In [7, 8], this kinetic limit derivation was extended to permit more general diffusion rates, and to include a stochastic mechanism of interaction. Suppose that particles of mass n ∈ N have a range of interaction given by r (n), for some increasing function r : N → (0, ∞). That is, = (N ) determines the order of the range as a function of the initial particle number, whereas the function r specifies the relative interaction range of particles of differing masses. The variations to the derivation required by introducing the radial dependence r have been discussed in [12]. It is shown there that, if the dimension d ≥ 3, the macroscopic collision propensity α : N2 → (0, ∞) that appears in the Smoluchowski coagulation equation satisfied by the macroscopic density profile of the particle system satisfies where f n0 equation u t
d−2 α(n, m) ≤ c d(n) + d(m) r (n) + r (m) ,
(1.2)
where c is a constant without dependence on the details of the stochastic interaction. (Note that β rather than α was the notation used in [7, 8]. The change in notation in this paper ensures consistency with the PDE literature.) In fact, the left and right-hand-sides of (1.2) are of the same order provided that the interaction mechanism is strong enough to ensure that a uniformly positive fraction of pairs of particles that come within the interaction range coagulate.
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The growth rate of r : N → (0, ∞) presumably depends on the internal structure of the particles. If they are simply balls, each of the same density, then r (n) = c0 n 1/d . On the other hand, an internal structure that is fractal might give rise to a relation of the form r (n) = n χ , for some χ ≥ 1/d. The physically reasonable range of values of χ would seem to be contained in [1/d, 1]. This is because, whatever fractal structure a particle of mass n ∈ N may have, if contact with another particle is required for the pair to coagulate, then the interaction range r (n) is at most of order n, (with the extremal case being that in which the particle takes the form of a line segment). Indeed, non-trivial fractal structure would suggest that χ is strictly less than one. Monotonically decreasing choices of the diffusion rates d : N → [0, ∞) seem to be physically realistic, since the diffusive motion is presumably stimulated by the bombardment of much smaller elements of an ambient gas. The choice d(n) = n1 in three dimensional space is justified if the particles are modelled as balls. (For a mathematical treatment, see [4], which derives Brownian motion as the long-term behaviour of a ball being struck by elements in a Poisson cloud of point particles of Gaussian velocity.) In this paper, we examine the behaviour of solutions of (1.1). We have directed our attention to parameter values α and d that seem to be justified by the existing kinetic limit derivations of (1.1) (though, in principle, other choices may arise from a derivation of the equations from a quite different model). We mention firstly that, under the assumption α(n, m) = 0 for each n ∈ N, m→∞ m lim
(1.3)
the global existence of a weak solution of (1.1) has been established in a bounded domain in [11] and for Rd in [15], each of these works including fragmentation in the equations. From the physically reasonable assumptions that d(·) is uniformly bounded and r (n) = o(n), we see that (1.3) is satisfied in dimension d = 3 by choices of α satisfying (1.2). (We mention also that, using an assumption analogous to (1.3) as well as either a monotonicity condition on the coagulation rates or a detailed balance condition between the rates of coagulation and fragmentation, [10] proves the global existence of a weak solution in the case of continuous mass.) An important formal property of solutions of (1.1) is mass-conservation. Definition 1.1. Let f solve (1.1) weakly. We say that f conserves mass on the time interval [0, T ] provided that I (t) = I (0) for each t ∈ [0, T ], where I (t) =
m∈N
m
Rd
f m (x, t)d x.
While mass conservation holds formally, the only estimate that is readily available is I (t) ≤ I (0) for t ≥ 0. Indeed, the inequality may be strict, in which case, gelation is said to occur (at the infimum of times at which the inequality is strict). If this happens for a solution of (1.1) which is obtained as a kinetic limit of a particle system, then, after the gelation time, a positive fraction of the mass of particles is contained in particles whose mass is greater than some function that grows to infinity as the initial particle number tends to infinity. In the spatially homogeneous setting, much progress has been made in establishing when gelation occurs. In [6 and 5], continuous versions of the spatially homogeneous
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equations with fragmentation are considered. If the continuous analogue α(x, y) of the coagulation rates α is supposed to satisfy α(x, y) = x a y b + x b y a , x, y ∈ R
(1.4)
with a, b ∈ (0, 1) and a + b > 1, then gelation occurs, unless the inhibiting effect of fragmentation is strong enough. It is natural to postulate from these results that if the microscopic interaction range r (n) we have discussed behaves like r (n) = n χ with 1 χ > d−2 , with a diffusion rate d uniformly bounded below and a bounded domain in place of Rd , then gelation will occur. This is because the formula (1.2), (which, as already noted, may be written as an equality in the case of a reaction mechanism that is not particularly weak), is bounded below by the discrete analogue of the coagulation propensity given in (1.4). As we have commented, however, we do not anticipate such behaviour for α in equations arising from a three-dimensional particle system of the type considered in [12]. Moreover, as discussed in [12], the derivation of (1.1) from a 1 random model performed in [7] is valid only if χ < d−2 : it is thus merely a conjecture 1 that gelation occurs if χ > d−2 . Rigorous sufficient conditions for mass-conservation, or for gelation, have been available in the spatially inhomogeneous setting only under stringent assumptions on parameters. See [3] for the case of constant diffusion rates, and [14] for a criterion that requires uniform boundedness of α and further information about the behaviour of solutions of the system (however, each of these papers includes fragmentation in the equations). In Theorem 1.3, we present a more applicable sufficient condition. The result largely depends on new moment bounds on solutions of (1.1). Theorem 1.1 presents bounds on the L 1 norm of moments of a solution, and Theorem 1.2 provides L ∞ bounds on such moments. Previously, L ∞ estimates on solutions of (1.1), with the effect of fragmentation included, have been obtained (see [13, 14, 2, 3]), under fairly restrictive assumptions on coefficients for coagulation and fragmentation propensity. ∞ norms of f does not generally The dependence on i ∈ N of the bounds obtained on L i ∞ a permit deductions about the L -norm of moments ∞ m=1 m f m (x, t) for any a ≥ 0 (note however that such inferences are made in [3] if the diffusion rate is identically constant, or in [13] if the coagulation rates α(n, m) decay quickly enough). Our final result, Theorem 1.4, provides a criterion for uniqueness of solutions of (1.1). It also relies principally on the moment bounds and is a straightforward adaptation of the uniqueness proof of [1] which applies to the homogeneous case. To our knowledge, the uniqueness question has been largely unsettled: [14] treats the one-dimensional case including fragmentation under quite restrictive hypotheses, while [2] treats the case where d(n) is eventually constant, with a specific choice of α(n, m) whose growth is of the order of nm. We provide a corollary after the theorems that summarises the principal conditions on the coagulation and diffusion coefficients, and on the initial data, under which we may infer uniqueness and mass conservation of the solution. Our results are valid for each dimension d ≥ 1. Each deduction, moment bound, mass conservation, or uniqueness, depends on some regularity in the initial data, and some assumption on the parameters of the system. We now state the various assumptions that we require. Hypothesis 1.1. lim
n+m→∞
α(n, m) = 0. (n + m)(d(n) + d(m))
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More precisely, for every δ > 0, there exists k0 = k0 (δ) > 0 such that if n + m > k0 , then α(n, m) ≤ δ(n + m)(d(n) + d(m)). In addition, the function d is uniformly bounded. Hypothesis 1.2. The function d(n) is a non-increasing function of n and that α(n, m) ≤ C0 (n + m) for a constant C0 . Moreover, there exist positive constants r1 and r2 and nonnegative constants b2 ≤ b1 such that, r1 n −b1 ≤ d(n) ≤ r2 n −b2 . Hypothesis 1.3. The function d is uniformly positive and non-increasing, and there exists a constant C0 such that α(n, m) ≤ C0 (n + m). Our notation for the various moments of f will be X a = X a (x, t) = n a f n (x, t), Xˆ a = Xˆ a (x, t) = n a d(n)d/2 f n (x, t) n
(1.5)
n
and Ya (x, t) =
nm(n a + m a )(d(n) + d(m)) f n (x, t) f m (x, t),
n,m
Yˆa (x, t) =
(n a m + m a n)α(n, m) f n f m .
(1.6)
n,m
⎧ 2−d ⎪ if d ≥ 3, ⎨|x| 1 φ0 (x) = − 2π log |x| 11(|x| ≤ 1) if d = 2, ⎪ ⎩ 1 (1 − |x|) 11(2|x| ≤ 1) if d = 1. 2
We also set
(1.7)
We now state the four theorems. Theorem 1.1 (Moment bound I). Assume Hypothesis 1.1. Then for every a ≥ 2 and positive A and T , there exists a constant C = C(a, A, T ) such that, if X a (x, 0)X 1 (y, 0)φ0 (x − y)d xd y ≤ A, (1.8) n,m
X a (y, 0)φ0 (x − y)dy ≤ A,
ess sup x
then
and
T
T
Ya−1 d xdt ≤ C,
0
X a (x, 0)d x ≤ A, Yˆa−1 d xdt ≤ C,
(1.9)
(1.10)
0
and sup
t∈[0,T ]
X a (x, t)d x ≤ C.
Moreover, the constant C can be chosen to be independent of T when d > 2.
(1.11)
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Theorem 1.2 (Moment bound II). Assume Hypothesis 1.2. Then there exists a function γ (a, b1 , b2 ) with lima→∞ γ (a, b1 , b2 ) = ∞ such that if n e f n0 L ∞ (Rd ) < ∞, X a ∈ L 1 (Rd × [0, T ]), (1.12) n
then
n e f n L ∞ (Rd ×[0,T ]) < ∞,
(1.13)
n
for every e ≤ γ (a, b1 , b2 ). Remark 1.1. In particular, if Hypotheses 1.1–1.2 hold, X a (·, 0) ∈ L 1 (Rd ) and n a f n0 L ∞ (Rd ) < ∞, n
for every a ∈ N, then X a ∈ L ∞ (Rd × [0, T ]) ∩ L 1 (Rd × [0, T ]) for every a ∈ N and T ∈ (0, ∞). We refer to (4.15) for the explicit form of the function γ . Also note that Theorem 1.1 offers sufficient conditions to ensure X a ∈ L 1 (Rd × [0, T ]) (so that this theorem has been invoked in Remark 1.1). Theorem 1.3 (Conservation of Mass). Let f be a weak solution of (1.1). Assume that (1.10) holds for a = 2. Then f conserves mass on the time interval [0, T ]. Assume instead that X 1 (·, 0) ∈ L ∞ (Rd ), X 2 (·, 0) ∈ L 1 (Rd ) and Hypothesis 1.3 holds. Then f conserves mass on the time interval [0, ∞). Theorem 1.4 (Uniqueness). Suppose that, for some c0 > 0, we have that α(n, m) ≤ c0 nm for each n, m ∈ N. Then there is a unique weak solution of (1.1) on the interval [0, T ] among those satisfying X 2 ∈ L ∞ (Rd × [0, T ]). The following corollary states an important consequence of our results. Corollary 1.1. Assume that there exist positive constants a, b, c1 and c2 , with a + b < 1, such that α(n, m) ≤ c1 (n a + m a ) and d(n) ≥ c2 n −b for all n, m ∈ N. If e 0 e 0 n n f n L ∞ (Rd ) < ∞ and n n f n L 1 (Rd ) < ∞ for sufficiently large e, then (1.1) has a unique solution, and this solution is mass conserving. Remark 1.2. Note that Hypothesis 1.1 implies the existence of a weak solution, because (1.3) holds in this case, whereas Hypotheses 1.2 and 1.3 do not. The solution f referred to in Theorems 1.2 and 1.3 is thus not known to exist under the hypotheses of these results, and it should be considered that these results make no assertion if the choice of parameters α and d is such that no solution exists. Remark 1.3. Theorem 1.1 is also true for the continuous version of the system (1.1) with a verbatim proof. In the continuous version, all the summations over n and m are replaced with integrations with respect to dn and dm. On the other hand, we can derive the continuous version of Theorems 1.2 and 1.3 only for a particular solution. The proof of Theorem 1.4 does not readily adapt to the continuous setting. See Remark 4.1 for further comments.
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If the parameters α and d derived from the kinetic limit of a particle system are such that uniqueness among solutions of (1.1) is unknown, then, in principle at least, the particle system may not approximate a single solution of (1.1). It might, for example, approximate several different solutions, each with a positive probability. This would be very peculiar, and so, it is pleasing to be able to rule out the possibility by establishing uniqueness. If we adopt the relation r (n) = n χ for particle interaction range, then, recalling (1.2), we have that, in Rd with d ≥ 3, the macroscopic coagulation propensity arising from the microscopic random model satisfies χ (d−2) α(n, m) ≤ c d(n) + d(m) max n, m . In view of Theorems 1.3 and 1.4, we have a unique solution and mass conservation 1 throughout time provided that χ ∈ 0, d−2 and suitable moments of the initial densities are finite. As such, the discussion following (1.2) rules out the occurrence of gelation in three dimensions in particle systems of the type considered in [7, 8 and 12]. In a similar vein to the comment that follows (1.4), we mention that, in the case where d is uniformly positive, Theorem 1.3 gives a sufficient condition for mass conservation of a solution of (1.1) that is close to being sharp. Indeed, choices of α(n, m) that grow much more quickly than Hypothesis 1.2 permits are bounded below by the expression in (1.4), for some choice of a, b ∈ (0, 1) with a + b > 1. Such a choice of α would thus be expected to show gelation in the spatially homogeneous case. Corollary 8.2 of [6] adapts the argument of the homogeneous case to assert that gelation occurs for any weak solution of the continuous version of (1.1) in a bounded subset of Rd , provided that (1.4) is satisfied for such a and b as above. The diffusive motion of particles in Rd may act to inhibit gelation of a solution of (1.1), though a dense initial condition is likely to ensure it. 2. The Tracer Particle Approach Each of the L ∞ and moment bounds on solutions of (1.1) that we present in this paper will be proved by PDE methods. However, for each of our results, we earlier derived a similar assertion by a quite different approach. A given solution of (1.1) is understood in terms of the random trajectory of a tracer particle, whose behaviour is typical of the many particles that form the density profile of the solution. We have certainly found this random method to be intuitively appealing, and it may find application to other PDE for which kinetic limit derivations have been made. We have thus devoted this section to explaining the tracer particle approach. It will in fact be helpful to recall in more detail the model analysed in [7]. A sequence of microscopic random models indexed by their initial number N ∈ N of particles is given. In the N th model, each of the N particles has an initial location x(0) and integer mass m(0) set independently according to 0 d f (x)d x , (2.1) P m(0) = k = ∞R k 0 n=1 Rd f n (x)d x with x(0) having law Rd
f k0 (·) f k0 (x)d x
,
(2.2)
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conditional on m(0) = k. At any given moment of time t ∈ [0, ∞), particles of mass k ∈ N evolve as Brownian motions with diffusion rate d(k), with d : N → (0, ∞) a given collection of constants. Particles are liable to coagulate in pairs when their displacement is of order = (N ) (we refer the reader to the introduction of [7] for the 1 details of the interaction mechanism). We set ≈ N − d−2 (for d ≥ 3), to ensure that a typical particle experiences a rate of collision that is bounded away from zero and ∞ uniformly in N . At the collision event, the two incoming particles disappear, to be replaced by a third, that assumes the sum of the masses of the ingoing two, and is located in an -vicinity of either of the colliding particles. While the macroscopic behaviour of the system is likely to be independent of the choice of placement of the new particle within this microscopic vicinity of the colliding pair, it was convenient for the derivation performed in [7] to assume that, if the masses of the colliding pair are n and m, then the location of the new particle is taken to be that of one or the other of the pair with n m probabilities n+m and n+m . For what follows, it is convenient to regard a particle that has a collision as surviving it, and becoming the outgoing particle, with a probability proportional to its mass, and disappearing from the model in the other event. Theorem 1 of [7] specifies the macroscopic behaviour of this particle system, when the initial particle number N is taken to be high. To summarise the result without recourse to equations, at typical points (x, t) ∈ Rd × [0, ∞) of space-time, the number of particles of given mass m ∈ N located in a vicinity of x at time t, normalized appropriately, approximates the density f m (x, t), for some solution f m : m ∈ N of (1.1). From the result, we may infer the law of the trajectory of a typical particle, in the limit of high initial particle number, as follows. Consider a particle picked uniformly at random from the N particles that are initially present. We call this the tracer particle in the N th model. In accordance with the preceding description, the initial mass m(0)and location x(0) of the particle are given by (2.1) and (2.2). We know from [7] that f n : Rd × [0, ∞) → [0, ∞), n ∈ N gives the asymptotic particle densities of the microscopic models. Hence, asymptotically in high N , the trajectory of the particle is such that, if at time t ∈ [0, ∞), the particle has location x ∈ Rd and mass m, it evolves as a Brownian motion at rate d(m), and experiences collision with a particle of mass n at rate α(m, n) f n (x, t). The details of the collision event that we specified imply that the tracer particle (x(t), m(t)), on colliding with a mass n particle, survives the collision and assumes mass m(t + ) = m(t) + n with m(t) probability m(t)+n , and disappears from the model otherwise. The limit in high N of the lawof the tracer particle in the N th model is in fact depen dent only on the given solution f n : n ∈ N that the microscopic particle densities approximate, and not on other details of the random models, such as the location of other particles. We this definition: a tracer particle specified in terms of now formalise a given solution f n : n ∈ N of (1.1) and not in terms of the data of any microscopic model. Definition 2.1. Let a solution f n : n ∈ N of (1.1) be given. The tracer particle governed by f is the random process z = (x, m) : [0, ∞) → Rd × N ∪ {c} whose initial law is given by 0 d f (x)d x , P m(0) = m = ∞R m 0 n=1 Rd f n (x)d x with the initial location x(0) having density f m0 (·)/ Rd f m0 (x)d x, conditional on m(0) = m. At time t, the particle’s location x(t) evolves as a Brownian motion at rate d(m),
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653
provided that m(t) = m. Moreover, at time t, and for any n ∈ N, the particle is said to undertake a mass transition m → n + m at rate α(n, m) f n x(t), t . Such a transition succeeds with probability m/(n + m), in which case, m(t) is set equal to n + m, and fails in the other event, in which case, the particle is relegated to the cemetery state c: z(s) is set equal to c for all s ≥ t. Remark 2.1. Certain smoothness (or measurability) assumptions are in fact required to ensure that the process z = (x, m) : [0, ∞) → Rd × N ∪ {c} exists, even locally in time. d Let z : [0, ∞) → R × N ∪ {c} denote the tracer particle governed by a given solution f n : n ∈ N of (1.1). Let gn (x, t) denote the density of its location at time t: P x(t) ∈ A, m(t) = n = gn x, t d x. A
The evolution equation for the system gn : n ∈ N is given by
∂ gn x, t = d(n)gn (x, t) ∂t n−1 ∞ + α(m, n − m) f m (x, t)gn−m (x, t) − 2gn (x, t) α(n, m) f m (x, t). m=1
m=1
(2.3) Note that the choice gn ≡ f n solves this equation. Assuming that there is a unique non-zero solution of (2.3), we find that gn ≡ f n .
(2.4)
Before commenting further on (2.4), we want to emphasise how we have changed our point of view of the tracer particle. At first, we regarded it as a typical particle in a microscopic random model, whose behaviour in the large is described by a solution of (1.1), and then, secondly, as a random trajectory defined purely in terms of such a solution. The former point of view motivates the study of the system (1.1). We will now discuss how the latter is valuable in studying a given solution of (1.1). Indeed, that the tracer particle defined by a given solution of (1.1) may be a useful tool for studying that solution is apparent from (2.4): if we understandthe likely behaviour of the tracer particle, we infer bounds on the density gn : n ∈ N of its location, and, by (2.4), on the solution f n : n ∈ N itself. We have stated the relation (2.4) because doing so permits a more succinct summary of how the tracer particle approach works. However, in making the approach rigorous, we take a different route, which we now summarise. We construct a sequence N f n : Rd × [0, ∞) → [0, ∞), n ∈ N , indexed by N , which will approximate the density of the tracer particle of some solution of (1.1) when N is high. For each N ∈ N, the functions f nN , for n ∈ N, are constructed inductively, on the domains Rd × [0, Ni ), N for each i ∈ N. They are extended to Rd × [0, i+1 N ) by an inductive step in which f n (t) i i+1 for n ∈ N and t ∈ [ N , N ) is defined as the density of the location of a tracer particle governed by the constant data f nN (i/N ) during the time interval [ Ni , t). For each i ∈ N and T ∈ [0, ∞), the function f nN is shown to converge in L 1 Rd ×[0, T ] as N → ∞ to
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a limit f n : Rd × [0, ∞) → [0, ∞).By mimicking the proof of the weak stability result of [11], we infer that the sequence f n : n ∈ N is a solution of (1.1). Tracer particle arguments, of which an example will shortly be given, are then applied directly to the tracer particle densities f nN : n ∈ N , and the resultingbounds, which hold uniformly in N , are inherited in the high N limit by the solution f n : n ∈ N . This method for making the tracer particle approach rigorous does have the drawback of applying to only one solution of (1.1). ∞ How may the tracer particle approach be used to prove an L -bound on the solution f n : n ∈ N of (1.1) whose construction we have just discussed? To simplify the expo sition, suppose that the initial condition takes the form Rd f 10 (x)d x = 1, f m0 (x) = 0 for x ∈ Rd and m > 1. We will now sketch a proof of the following reformulation of 2 Lemma 4.1. Suppose that d : N → (0, ∞) is decreasing, and that d(m) > cm − d (1−α) for some α ∈ [0, 1] and c > 0. Then ∞
m α f m (x, t) ≤ Cu(x, t),
(2.5)
m=1
where u : Rd × [0, ∞) → [0, ∞) solves, ∂u = d(1)u, ∂t u(x, 0) = f 1 (x, 0). The solution f n : n ∈ N of (1.1), being given by the density of the tracer particle (2.4), has the following interpretation: for A ⊆ Rd a Borel set, the quantity ∞ m=1
mα
f m (x, t)d x A
is equal to the expected value of the random variable R = R(A) equal to m α if the tracer particle at time t has mass m and lies inside the set A. In comparing the left- and right-hand-sides of (2.5), we are thus assessing the degree to which the dynamics of the tracer particle may increase the expected values of the random variables R(A) over those obtained by using a simple Brownian particle (for numerous choices of the set A). For example, if the set A is a small ball about x ∈ Rd , and the tracer particle is close to x at some time s satisfying 0 < s t, then a mass transition undertaken by the tracer particle at times shortly after s will serve to increase the expected value of R, because the slower diffusion rate produced by the transition is more likely to leave the particle nearby to x at the later time t. However, if a transition occurs that sharply increases the mass of the particle, then it is likely to fail, in which case, it contributes zero to the expected value of R. This latter effect limits the capacity of the tracer particle to focus towards x. Phrasing the question quantitatively, we ask: what is the expected value of R(A) at time s < t if a mass transition m 1 → m 2 occurs at time s? Assuming that there is no other mass transition, and supposing that the tracer particle is at y ∈ Rd at time s, the expected value of R is 1 (y − x)2 dx exp − m α1 d/2 2(t − s)d(m 1 ) A 2π(t − s)d(m 1 )
Moment Bounds for the Smoluchowski Equation and their Consequences
if no mass transition occurs, whereas, it is m1 α 1 m m 2 2 2π(t − s)d(m 2 ) d/2
exp − A
(y − x)2 dx 2(t − s)d(m 2 )
655
(2.6)
if the transition does occur. (The first factor in (2.6) is the survival probability for the transition). Given that d(m 2 ) ≤ d(m 1 ), the latter exponential term may be bounded pointwise by the former, and we find that E R if mass transition occurs m 1−α d(m 1 )d/2 m 1 d(m 1 )d/2 m α2 1 = . ≤ m 2 d(m 2 )d/2 m α1 d(m 2 )d/2 m 1−α E R if it does not 2 If a whole sequence of mass transition occurs, m i → m i+1 at time ti for i ∈ {1, . . . , n}, with ti ∈ [0, t] an increasing sequence, we similarly find that the ratio of the expected values of R in the case where the sequence of mass transitions occurs and in that where no transition takes place, is bounded above by 1 m 1−α
d(1)d/2 . d(m)d/2
(We have omitted some details of the argument: to make it rigorous, we might use an induction on the total number of mass transitions undertaken prior to time t, and invoke the strong Markov property.) By comparison, the expected value of R if no transition occurs is u(x, t), where ∂u = d(1)u, ∂t with u(x, 0) = f 1 (x, 0). By choosing A = B(x, r ) for each r > 0, we learn that ∞
m α f m (x, t) ≤
m=1
d(1)d/2 u(x, t). inf m∈N m 1−α d(m)d/2
From the hypothesis d(m) > cm −2/d(1−α) , we find that ∞ m=1
m α f m (x, t) ≤
d(1)d/2 u(x, t), c
as we sought. In fact, a more careful tracer particle argument improves this result: the same conclusion (2.5) may be reached under the weaker assumption that d is decreasing and satisfies d(m) > m −(1−α)+ for some α ∈ (0, 1) and ∈ (0, α). This is a reformulation of Corollary 1.1 which is a consequence of Theorem 1.2. We have mentioned that each of our results has an analogue with a derivation that considers the tracer particle governed by a given solution of (1.1). We will not outline the method of proof of the result on mass conservation (see Theorem 1.3), stating only the characterization of gelation in terms of a tracer particle. We alter the definition of the tracer particle governed by f n : n ∈ N , so that the particle survives every mass transition. Given (2.4), mass conservation of the solution f until a given time T ∈ [0, ∞) occurs if and only if the tracer particle experiences only finitely many collisions on the time interval [0, T ].
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3. Moment Bounds under Hypothesis 1.1 Let us first construct an auxiliary function H that will be needed for the proof of Theorem 1.1. Before doing so, let us make an observation. Define ⎧ 2−d ⎪ if d ≥ 3, ⎨c(d)|x| 1 H0 (x) = − 2π log |x| if d = 2, (3.1) ⎪ ⎩− 1 |x| if d = 1, 2 where c(d) = (d − 2)−1 ωd−1 with ωd denoting the (d − 1)–dimensional measure of the unit S d−1 = {x ∈ Rd : |x| = 1}. We then have that H0 = −δ0 , where δ0 denotes the Dirac measure at 0. More precisely, for a test function g, the function u(x) = H0 (x − y)g(y)dy, satisfies u = −g. Note that H0 ≥ 0 when d ≥ 3 and this property is lacking when d ≤ 2. Because of this we can only hope for the existence of a suitable function H such that H ≥ 0 but now H = −δ0 + Err or for an Err or that can be controlled. This is the content of our first lemma. Lemma 3.1. Assume d ≤ 2. There exist functions H and K such that H ≥ 0, K is bounded, K is of compact support, −H (x) = δ0 − K (x),
(3.2)
and the function H − φ0 is bounded. (The function φ0 was defined in (1.7).) Proof. The construction of H for d = 1 is straightforward; we can readily find a nonnegative function H such that H = φ0 in [−1/2, 1/2], H = 0 outside [−1, 1], and H is smooth away from the origin. For the construction of the function H when d = 2, let us start from the function φ0 and make an important observation. Note that if −1 R(x) = J ∗ φ0 (x) = log |x − y|J (y)dy, (3.3) 2π |x−y|≤1 with J continuous and nonnegative, then R ≥ 0 and −R = J − J˜, where
1 J˜(x) = 2π
|z|=1
J (x − z)d S(z) = J ∗ δ˜0 ,
(3.4)
(3.5)
where d S denotes Lebesgue measure on the unit circle and δ˜0 denotes the normalized Lebesgue measure on the unit circle. In (3.3), we may replace J (y)dy with a measure J (dy). Then (3.4) is still valid weakly for an obvious interpretation for (3.5). In particular if we choose J (dy) = δ0 (dy), then −φ0 = δ0 − δ˜0 .
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Our goal is to replace δ˜0 with a bounded function of compact support. For this we set φ1 = δ˜0 ∗ φ0 to obtain φ1 ≥ 0 and −(φ0 + φ1 ) = δ0 − δˆ0 , where δˆ0 = δ˜0 ∗ δ˜0 is now a “function” and is weakly given by 1 h(z + a)d S(z)d S(a) h(z)δˆ0 (dz) = (2π )2 |z|=1 |a|=1 2π 2π 1 h(eiθ1 + eiθ2 )dθ1 dθ2 . = (2π )2 0 0 Note that the Jacobian of the transformation ψ : (θ1 , θ2 ) → (eiθ1 + eiθ2 ) is given by | sin(θ1 −θ2 )|. Since cos(θ1 −θ2 ) = 21 |eiθ1 +eiθ2 |2 −1, we learn that if δˆ0 (dz) = δˆ0 (z)dz, then δˆ0 (z) = 2 =2
1 (2π )2
1
1−
11(|z| ≤ 2) 2−1 2 |z| 2
1
2 1 11(|z| ≤ 2). (2π )2 |z| 4 − |z|2
Here the factor 2 comes from the fact that ψ maps (exactly) two points to one point because ψ(θ1 , θ2 ) = ψ(θ2 , θ1 ). The function δˆ is not bounded. Let us apply the above procedure one more time to define φ2 = φ0 ∗ δˆ0 so that φ2 ≥ 0 and −(φ0 + φ1 + φ2 ) = δ0 − δ¯0 , where δ¯0 (z) =
4 (2π )3
|a|=1
1 11(|z − a| ≤ 2)d S(a). |z − a| 4 − |z − a|2
(3.6)
As we will see, the function δ¯0 is not bounded either. However, the function δ¯0 is less “singular” than δˆ0 . For this we show that in fact δ¯0 has a logarithmic singularity. To do this, first observe that δ¯0 (z) is radially symmetric because both d S and δˆ0 are rotationally invariant. Hence we may assume that z lies on the x1 -axis and z > 0. Observe that the integrand in (3.6) is singular when either |z − a| = 0 or |z − a| = 2. It is not hard to show that there exists a positive constant c1 such that if a = eiθ with θ ∈ (−π, π ], then c1 (|θ | + |1 − |z||) ≤ |z − a| ≤ |θ | + |1 − |z||.
(3.7)
The first inequality will be used to treat the 1/|z − a| singularity in (3.6). We now turn to the singularity which comes from the factor (2 − |z − a|)−1/2 . For this, observe that if a = eiθ , then we have γ (θ ) := 4 − |z − a|2 = 3 − z 2 + 2z cos θ. Let us choose θ0 ∈ [0, π ] so that 3 − z 2 + 2z cos θ0 = 0 or cos θ0 =
z2 − 3 . 2z
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A. Hammond, F. Rezakhanlou
This means that the integrand in (3.6) is singular at θ0 and −θ0 . Note that when z ≥ 0, the singular point θ0 exists only if z belongs to the interval [1, 3]. Also note that if z is neither close to 1 nor 3, then θ0 is neither close to 0 nor π . From the elementary equality γ (θ ) = γ (θ ) − γ (θ0 ) = 2z(cos θ − cos θ0 ) = −4z sin
θ − θ0 θ + θ0 sin , 2 2
we learn that if z is neither close to 1 nor 3, then we can find a positive constant c2 such that 4 − |z − a|2 ≥ c2 |θ − θ0 |, for θ close to θ0 . (Recall that z ∈ [1, 3].) Since this is an integrable singularity with respect to dθ -integration, we deduce that δ¯0 is bounded if z stays away from the circles |z| = 1 and |z| = 3. We now √ assume that |z| is close to 1. In this case, θ0 is close to π and |θ0 − π | is comparable to |1 − |z||. Also, because of |z − a| ≤ 2 and γ (θ ) − γ (θ0 ) > 0 we learn that θ < θ0 . We have 4 − |z − a|2 ≥ c3 [(π − θ ) + (π − θ0 )] [(π − θ ) − (π − θ0 )] ,
(3.8)
for a positive constant c3 . When |z| is close to 1, the integrand in (3.6) is singular at ±θ0 and “almost” singular (see (3.7)) at 0 . From (3.7) and (3.8) we deduce δ¯0 (z) ≤ c4 |log |1 − |z||| , θ whenever |z| is close to 1. (Here we used the fact that 0 0 [(π − θ )2 − (π − θ0 )2 ]−1/2 dθ is of order | log(π − θ0 )|.) √ When |z| is close to 3, θ0 is small (in fact of order O( 3 − |z|)), and the condition |z − a| ≤ 2 forces θ ∈ [−θ0 , θ0 ]. On the other hand, 4 − |z − a|2 ≥ c5 |θ + θ0 ||θ − θ0 |, for a positive constant c5 , implies that δ¯0 (z) is bounded for |z| close to 3. Here we are using θ0 − 1 2 θ02 − θ 2 dθ < ∞. −θ0
Putting all the pieces together, we deduce δ¯0 (z) ≤ c |log |1 − |z||| 11(|z| ≤ 3).
(3.9)
We set φ3 = δ¯0 ∗ φ0 to obtain φ3 ≥ 0 and −(φ0 + φ1 + φ2 + φ3 ) = δ0 − K , where K (z) =
1 2π
|a|=1
(3.10)
δ¯0 (z + a)d S(a).
It is straightforward to use (3.9) to show that K is uniformly bounded. Now (3.2) follows from (3.10) by choosing H = φ0 + φ1 + φ2 + φ3 . It is also straightforward to check that the functions φ1 , φ2 , and φ3 are bounded.
Moment Bounds for the Smoluchowski Equation and their Consequences
659
Let ζ be a nonnegative smooth function of compact support with ζ = 1 and set ζ δ (x) = δ −d ζ (x/δ). We also define f nδ = f n ∗x ζ δ and Q δn = Q n ∗x ζ δ . We certainly have t t δ δ δ f n (x, t) = f n (x, 0) + d(n) f n (x, s)ds + Q δn (x, s)ds. (3.11) 0
0
Also, as it is well-known, φ(n)Q n = α(n, m)(φ(n + m) − φ(n) − φ(m)) f n f m . n
(3.12)
n,m
δ δ The same identity is valid if we replace f n f m with ( f n f m ) ∗x ζ and Q with Q . Using the fact that for φ(n) = n11(n ≤ ), we have n φ(n)Q n ≤ 0 we can readily deduce that sup sup sup n f nδ (x, t)d x < ∞. (3.13) δ
t
n=1
Lemma 3.2. Let H be as in Lemma 2.1. Then there exists a constant c0 such that sup x
n( f nδ ∗x H )(x, t) ≤ sup x
n=1
n( f nδ ∗x H )(x, 0) + c0 t
n=1
for every positive δ. We may choose c0 = 0 when d ≥ 3. Proof. We have
n( f nδ
∗x H )(x, t) =
n=1
n( f nδ
∗x H )(x, 0) −
0 n=1
n=1
+
t 0 n=1
nd(n) f nδ
From the boundedness of K , H ≥ 0 and
t
n( f nδ
∗x H )(x, t) ≤
n=1
n( f nδ
nd(n) f nδ (x, s)ds
∗x K (x, s)ds +
t 0 n=1
n=1 n Q n
∗x H )(x, 0) + c1
≤ 0 we deduce t 0
n=1
n Q δn ∗x H (x, s)ds.
nd(n) f nδ (x, s)d xds.
n=1
We now use (3.13) to bound the last term to complete the proof.
Proof of Theorem 1.1. Set δ
Z (t) =
n
a
f nδ (x, t)
n=1
Z (t) =
n=1
n=1
n f n (x, t) a
n=1
n f nδ
∗x H (x, t) d x,
n f n ∗x H (x, t) d x.
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A. Hammond, F. Rezakhanlou
We have that weakly, d δ Z (t) = − dt
n
a
f nδ (x, t)
n=1
−
n=1
n
a
d(n) f nδ (x, t)
n=1
+
n
a
f nδ (x, t)
+
+
n=1
+
n=1
n=1
n Q δn
n a Q δn (x, t)
n a f nδ (x, t)
∗x H (x, t) d x
n f nδ ∗x H (x, t) d x
nd(n) f nδ ∗x K (x, t) d x
n=1
n a d(n) f nδ (x, t)
dx
n=1
dx
n f nδ (x, t)
n=1
n=1
n=1
nd(n) f nδ (x, t)
n f nδ ∗x K (x, t) d x
n=1
=: 1 + 2 + 3 + 4 + 5 + 6 . We now study the various terms which appear on the right-hand side. We certainly have 1 11(n, m ≤ )[n a md(m) + n a d(n)m + m a nd(n) 1 + 2 = − 2 n,m +m a d(m)n] f nδ f mδ d x 1 =− 11(n, m ≤ )nm(d(n) + d(m))(n a−1 + m a−1 ) f nδ f mδ d x. 2 n,m From n=1 n Q n ≤ 0, we learn that 3 ≤ 0. By boundedness of K , (3.13) and the boundedness of d(·) we deduce that |5 + 6 | ≤ c1 X aδ d x, where X a is defined in (1.5) and X aδ = X a ∗ ζ δ . It remains to bound 4 . Note that n=1
n a Q n (x, t) =
[(n + m)a 11(n + m ≤ ) − n a 11(n ≤ ) n,m
−m a 11(m ≤ )]α(n, m) f n f m ≤ c2 (n a−1 m + m a−1 n)α(n, m)11(n + m ≤ ) f n f m =: c2 Z 4 . n,m
(3.14)
Moment Bounds for the Smoluchowski Equation and their Consequences
661
From this, Lemma 3.2, the boundedness of H − φ0 and (1.9), δ δ 4 ≤ c2 Z 4 n f n ∗x H (x, t) d x ≤ c3 Z 4δ d x, n=1
where Z 4δ = Z 4 ∗ ζ δ . On the other hand, for δ0 > 0, we can find k0 = k0 (δ0 ) such that if k0 < , then Z4 = 11(k0 ≤ n + m ≤ )(n a−1 m + m a−1 n)α(n, m) f n f m n,m
+
11(n + m < k0 )(n a−1 m + m a−1 n)α(n, m) f n f m
(3.15)
n,m
≤ δ0
n,m
+ 2k0a
11(n + m ≤ )(n a−1 m + m a−1 n)(n + m)(d(n) + d(m)) f n f m
11(n + m ≤ )α(n, m) f n f m .
n,m
As a result, d δ 1 Z (t) ≤ − dt 2
11(n, m ≤ )nm(n a−1 + m a−1 )(d(n) + d(m)) f nδ f mδ d x
n,m
+2c3 δ0
11(n, m ≤ )nm(n a−1 + m a−1 )(d(n) + d(m))( f n f m ) ∗ ζ δ d x
n,m
+2c3 k0a
δ
11(n + m ≤ )α(n, m)( f n f m ) ∗ ζ d x + c1
X aδ (x, t)d x.
n,m
Here, we are using the identity, (n + m) n a−2 + m a−2 ≤ 2 n a−1 + m a−1 , which is valid provided that a ≥ 2. We now send δ to 0 to yield d 1 Z (t) ≤ 2c3 δ0 − 11(n, m ≤ )nm(n a−1 + m a−1 )(d(n) + d(m)) f n f m d x dt 2 n,m + 2c3 k0a 11(n + m ≤ )α(n, m) f n f m d x + c1 X a (x, t)d x. (3.16) n,m
Note that the time integral of the second integral is bounded because d dt
n=1
fn d x ≤ −
11(n + m ≤ )α(n, m) f n f m d x,
n,m
which implies, 0
T
n,m
11(n + m ≤ )α(n, m) f n f m d xdt ≤
n=1
n f n0 d x.
(3.17)
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A. Hammond, F. Rezakhanlou
Furthermore, the equality d dt
n f n (x, t)d x = a
n=1
n a Q n (x, t)d x,
(3.18)
n=1
and (3.14) imply that t 0
n a f n (x, s)d xds ≤ t
n=1
n a f n (x, 0)d x + c2
t 0
n=1
≤t
n f n (x, 0)d x + tc2 a
s
Z 4 (x, θ )d xdθ
0
t
Z 4 (x, θ )d xdθ.
0
n=1
From this, (3.17) and (3.16) we deduce that Z (t) − Z (0) is bounded above by
1 2c3 + c1 c2 t δ0 − 2
t
0
n,m≤
nm(n a−1 + m a−1 )
×(d(n) + d(m)) f n f m d xds + c4 (1 + t). We now choose δ0 = δ0 (t) so that 1/2 > (2c3 + c1 c2 t)δ0 . With this choice, the bounds in (1.10) follow. From (1.10), (3.14) and (3.18) we conclude (1.11). We end this section with a variant of Theorem 1.1 that holds under Hypothesis 1.3. Lemma 3.3. Under Hypothesis 1.3, there exists a constant C such that
X 2 (x, t)d x ≤
X 2 (x, 0)d x exp C T
n f n0 L ∞
.
(3.19)
n
Proof. We start from d dt
n f n (x, t)d x = 2
n=1
≤
n=1
2
≤
nmα(n, m)11(n + m ≤ ) f n f m d x
n,m
2C0
nm(n + m)11(n + m ≤ ) f n f m d x
n,m
≤
n 2 Q n (x, t)d x
4C0
n
n11(n ≤ ) f n
m 11(m ≤ ) f m 2
d x.
m
This and Gronwall’s inequality imply (3.19) because we can use the uniform positivity of d(·) and Lemma 3.1 of Sect. 3 to assert that X 1 ∈ L ∞ .
Moment Bounds for the Smoluchowski Equation and their Consequences
663
4. Moment Bounds when d(·) is Non-Increasing This section is devoted to the proof of Theorem 1.2. We start with a lemma. Lemma 4.1. Assume d(·) is non-increasing. Then Xˆ 1 (x, t) =
∞
nd(n)d/2 f n (x, t) ≤ d(1)d/2 u(x, t),
(4.1)
n=1
where u is the unique solution to u t = d(1)u subject to the initial condition u(x, 0) = ∞ n=1 n f n (x, 0). Proof. We first establish
nd(n)
d/2
f n (t) ≤
d(1) d(1)d/2 St
1
n f n0
t
d/2
+ d()
0
1
d() St−s
n Q n (s) ds.
1
(4.2) Here for simplicity, we do not display the dependence on the x-variable. Note that (4.2) implies d(1) nd(n)d/2 f n ≤ d(1)d/2 St n f n0 , (4.3) 1
1
because 1 n Q n ≤ 0. Evidently (4.3) implies (4.1). We establish (4.2) by induction. Equation (4.2) is obvious when = 1 by definition; in fact we have equality. Suppose (4.2) is valid. We would like to deduce (4.2) with replaced with + 1. To do so, first observe that if D1 ≥ D2 and g ≥ 0, then D1 d/2 StD1 g ≥ D2 d/2 StD2 g. From this and (4.2) we learn
nd(n)
d/2
fn ≤
d(1)d/2 Std(1)
1
n f n0
+ d( + 1)
t
d/2 0
1
because d() ≥ d( + 1) and
(4.4)
d(+1) St−s
n Q n (s) ds
1
(4.5)
≤ 0. Applying (4.4) to t d(+1) 0 + St−s Q +1 (s)ds, f +1 (t) = Std(+1) f +1 1 n Qn
0
yields
f +1 (t) ≤
d(1) d( + 1)
d/2
0 Std(1) f +1
t
+ 0 d/2 1)
d(+1) St−s Q +1 (s)ds.
(4.6)
and add the result to (4.5). The We multiply both sides of (4.6) by ( + 1)d( + outcome is +1 +1 t +1 d(+1) d/2 d/2 d(1) 0 d/2 nd(n) f n ≤ d(1) St n f n + d( + 1) St−s n Q n (s) ds. 1
This completes the proof.
1
0
1
664
A. Hammond, F. Rezakhanlou
Remark the continuous version of (1.1), a similar proof leads to an L ∞ bound ∞ 4.1. For d/2 on 0 nd(n) f n dn provided that the function d(·) is piecewise constant, uniformly positive and nonincreasing. If we assume only the second and third conditions on d(·), we can establish the same bound on a solution provided that this solution can be approximated by solutions corresponding to a piecewise constant d(·). Of course, if we already know the uniqueness of solutions to (1.1), then our L ∞ bound applies to all solutions. But we do not: when uniqueness is proved in Sect. 4, we will use Lemma 3.1 and its consequence ∞Theorem 1.2. However, if we only postulate the uniqueness for those solutions with 0 nd(n)d/2 f n dn ∈ L ∞ , then Theorem 1.4 is also valid in the continuous case. Proof of Theorem 1.2. Step 1. Let us simply write L for L (Rd × [0, T ]). We first show that if ≥ 1, then Xˆ a+1 ∈ L 1 ⇒ Xˆ a+1 ∈ L .
(4.7)
Indeed, Xˆ a+1 =
n a nd(n)d/2 f n = Xˆ 1
n
na
n
Xˆ a+1 ≤ Xˆ 1
n a
n
nd(n)d/2 f n , Xˆ 1
nd(n)d/2 f n = Xˆ 1−1 Xˆ a+1 , Xˆ 1
by Hölder’s inequality. This implies (4.7) because by Lemma 3.1, Xˆ 1 ∈ L ∞ . Recall that we are assuming r1 n −b1 ≤ d(n) ≤ r2 n −b2 . From this and (4.7) we deduce that X
a+1−b2 d2
∈ L1 ⇒ X
This means that
a+1−b1 d2
∈ L .
2a+b2 d−2
X a ∈ L 1 ⇒ X b ∈ L 2b+b1 d−2 ,
(4.8)
provided that 2b + b1 d − 2 > 0 and (2a + b2 d − 2)/(2b + b1 d − 2) ≥ 1. Step 2. We now try to bound Q +n with the aid of (4.8). Using α(n, m) ≤ C0 (n + m), we certainly have Q +n = α(n 1 , n 2 ) f n 1 f n 2 n 1 +n 2 =n
≤
11(n 1 ≥ n/2 or n 2 ≥ n/2)α(n 1 , n 2 ) f n 1 f n 2
n 1 ,n 2
where X a (N ) =
≤ 2C0 [X 1 X 0 (n/2) + X 1 (n/2)X (0)] , m≥N
m a f m . Hence
Q +n ≤ c2 n − [X 1 X + X 1+ X 0 ], Q +n L p ≤ c2 n − [X 1 L 1 X L 2 + X 1+ L 3 X 0 L 4 ],
Moment Bounds for the Smoluchowski Equation and their Consequences
665
provided that 1p = 11 + 12 = 13 + 14 . To use (4.8), let us first assume that b1 d > 2 and that 2a + b2 d − 2 ≥ b1 d + 2. Assume X a ∈ L 1 . Then we use (4.8) to assert that X1 ∈ L X ∈ L
2a+b2 d−2 b1 d 2a+b2 d−2 2+b1 d−2
Hence if we set
, X0 ∈ L
2a+b2 d−2 b1 d−2
, X +1 ∈ L
,
2a+b2 d−2 2+b1 d
.
2a + b2 d − 2 , 2b1 d + 2 − 2
p=
(4.9)
and assume that p ≥ 1, b1 d > 2, then we have that X a ∈ L 1 ⇒ Q n L p ≤ c n − .
(4.10)
However, if b1 d ≤ 2, then we have that X 0 ∈ L ∞ because Xˆ 1 ∈ L ∞ . From this, we learn that in this case (4.9) is true but now for 2a + b2 d − 2 . 2 + b1 d
p=
(4.11)
Step 3. Note that if p D (x, t) =
|x|2 (4π Dt)−d/2 exp − 4Dt if t > 0, if t < 0,
0
t D g(x, s)ds = ( p D ∗ g)(x, t), and g is a function with g(x, t) = 0 for t < 0, then 0 St−s where the convolution is in both x and t variables. Also note that dr r T T 1 x r p1 √ , 1 d xdt ( p D (x, t)) d xdt = √ Dt Dt 0 0 T d d (Dt) 2 (1−r ) dt = c(T, r )D 2 (1−r ) = 0
with c(T, r ) < ∞ if and only if r <
2 d
+ 1. We certainly have
f n (x, t) = (Std(n) f n0 )(x) + ( pd(n) ∗ Q n )(x, t). So, f n L ∞ ≤ (Std(n) f n0 )(x) L ∞ + pd(n) L r Q n L p , provided that
1 r
+
1 p
= 1. Since p D ∈ L r with r <
2 d
+ 1, it suffices to have
1 1 + < 1. 1 + 2/d p Choosing p as in (4.9) or (4.11) requires 2b1 d + 2 − 2 b1 d + 2 2 or < . 2a + b2 d − 2 2a + b2 d − 2 d +2
(4.12)
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A. Hammond, F. Rezakhanlou
More precisely,
<
1 d+2 (2a 1 d+2 (2a
+ b2 d − 2) − b1 d + 1 if b1 d > 2, + b2 d − 2) − 21 b1 d if b1 d ≤ 2.
(4.13)
In summary, we need to satisfy (4.13) and r to satisfy d2 (1 − r ) > −1. As a result, if X a ∈ L 1 and satisfies (4.13), then f n L ∞ ≤ An + c3 n − d(n)d(1−r )/2 ≤ An + c4 n − d(n)−1 , where An = (Std(n) ) f n0 (x) L ∞ ≤ f n0 L ∞ (Rd ) . Final Step. We have that n n e f n L ∞ ≤ ∞ if X a ∈ L 1 , n e− d(n)−1 < ∞,
(4.14)
n
and,
n e f n0 L ∞ (Rd ) < ∞.
n
For (4.14) it suffices to have e − + b1 < −1. In other words,
e ≤ γ (a, b1 , b2 ) :=
1 d+2 (2a 1 d+2 (2a
+ b2 d − 2) − b1 (d + 1) if b1 d > 2, + b2 d − 2) − 21 b1 d − b1 − 1 if b1 d ≤ 2.
(4.15)
5. Uniqueness The main result of this section is Theorem 5.1. Theorem 1.4 is an immediate consequence of Theorem 5.1. Theorem 5.1. Assume that α(n, m) ≤ c0 nm and let f and g be two solutions with n2 fn , n 2 gn ≤ A, ∞ ∞ n
where
Lp
abbreviates
n
L
L
L p (Rd
× [0, T ]). Then X (t) := n| f n − gn |(x, t)d x n
satisfies X (t) ≤ e4c0 At X (0),
(5.1)
for t ≤ T . In particular, if f n (·, 0) = gn (·, 0) for all n, then f n (·, t) = gn (·, t) for all n and t ∈ [0, T ].
Moment Bounds for the Smoluchowski Equation and their Consequences
667
We first state a straightforward lemma: Lemma 5.1. Let u be a weak solution of u t = Du + h with u and h ∈ L 1 . Assume that ψ is a continuously differentiable convex function with |ψ (a)| ≤ c1 for a constant c1 and all a ∈ R. Then t (5.2) ψ (u(x, θ ))h(x, θ )d xdθ ψ(u(x, t))d x ≤ ψ(u(x, s))d x + s
whenever 0 < s < t. Lemma 5.1 is established by choosing a smooth mollifier ρ , and showing the inequality (5.2) for u = u ∗ ρ and h = h ∗ ρ . We then pass to the limit → 0. We omit the details. Proof of Theorem 5.1. Choose a continuously differentiable convex function ψδ so that ψδ (r ) = sgn(r ) for r ∈ / (−δ, δ), ψδ ∈ C 2 , |ψδ (r )| ≤ 1, ψδ (0) = 0 and ψδ ≥ 0. We then apply Lemma 5.1 to assert that the expression N
n ψδ ( f n (x, t) − gn (x, t))d x,
(5.3)
n=1
is bounded above by N
n ψδ ( f n (x, s) − gn (x, s))ds
n=1
+
t N s
=
N
n=1
nψδ ( f n (x, s) − gn (x, s))d x
n=1 t
+ s
nψδ ( f n (x, θ ) − gn (x, θ ))(Q n ( f )(x, θ ) − Q n (g)(x, θ ))d xdθ
α(n, m)(n+m − n − m )( f n f m − gn gm )d xdθ,
n,m
where n = n ψδ ( f n − gn )11(n ≤ N ). Observe that if n, m ≤ N , then (n+m − n − m )( f n f m − gn gm ), equals (n+m − n − m )( f n − gn ) f m + (n+m − n − m )( f m − gm )gn ≤ (n + m)| f n − gn | f m − nψδ ( f n − gn )( f n − gn ) f m + m| f n − gn | f m + (n + m)| f m − gm |gn + n| f m − gm |gn − mψδ ( f m − gm )( f m − gm )gn ≤ 2m| f n − gn | f m + 2n11(| f n − gn | < δ)| f n − gn | f m + 2n| f m − gm |gn + 2m11(| f m − gm | < δ)| f m − gm |gn .
(5.4)
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A. Hammond, F. Rezakhanlou
From this, (5.4) and α(n, m) ≤ c0 nm, we learn that the expression (5.3) is bounded above by N
nψδ ( f n (x, s) − gn (x, s))d x
n=1
+ 2c0
t N s
n| f n − gn |
n=1
t N
+ 2c0 δ
s
N
m ( f m + gm ) d xdθ 2
m=1
n 11(| f n − gn | < δ) 2
n=1
N
m( f m + gm ) d xdθ.
m=1
We then send δ → 0 and N → ∞, in this order to obtain ∞
n| f n (x, t) − gn (x, t)|d x
n=1
≤
∞
n| f n (x, s) − gn (x, s)|d x
n=1
+2c0
t ∞ s
n| f n − gn |
n=1
∞
m ( f m + gm ) d xdθ. 2
m=1
The theorem now follows from this and Gronwall’s inequality.
6. Mass Conservation Proof of Theorem 1.3. We first assume that Yˆ1 ∈ L 1 . Evidently, d dt
N
n fn d x = −
{11(n ≤ N < n + m)n
n,m
n=1
+11(m ≤ N < n + m)m}α(n, m) f n f m = −2 11(n ≤ N < n + m)nα(n, m) f n f m n,m
≥ −2
11(n ≥ N /2 or m > N /2)nmα(n, m) f n f m .
n,m
The limit N → ∞ of the time average of the right-hand side is 0 because Yˆ1 ∈ L 1 . From this we can readily deduce that lim
N →∞
N
n f n (x, t) d x −
n=1
This completes the proof when Yˆ1 ∈ L 1 .
N n=1
n f n (x, 0) d x = 0.
Moment Bounds for the Smoluchowski Equation and their Consequences
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We now assume that Hypothesis 1.3 holds. We have, N d n fn d x = − {11(n ≤ N < n+m)n+11(m ≤ N < n + m)m}α(n, m) f n f m dt n,m n=1 ≥ −2C0 11(n ≤ N < n + m)n(n + m) f n f m n,m
≥ −2C0
11(n ≤ N /2, m > N /2)n(n + m) f n f m
n,m
− 2C0
11(n > N /2)n(n + m) f n f m
n,m
= −1 − 2 . We certainly have, 1 L 1
2 L 1
fm ∞ L 1 m>N /2 L +c n fn m fm , 2 n 2 m>N /2 L L ≤ c n fn m fm 2 n>N /2 2 m L L 2 +c n f f , n m ∞ ≤ c n2 fn n
n>N /2
L1
m
L
where L p abbreviates L p (Rd × [0, T ]). Since m m f m ∈ L 2 , m m 2 f m ∈ L 1 , and ∞ n n f n ∈ L , by Lemmas 4.1 and 2.3, we are done. Acknowledgments. We thank Gábor Pete for comments on a draft version.
References 1. Ball, J.M., Carr, J.: The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation. J. Stat. Phys. 61(1–2), 203–234 (1990) 2. Bénilan, P., Wrzosek, D.: On an infinite system of reaction-diffusion equations. Adv. Math. Sci. Appl. 7(1), 351–366 (1997) 3. Collet, J.F., Poupaud, F.: Asymptotic behaviour of solutions to the diffusive fragmentation-coagulation system. Phys. D 114(1–2), 123–146 (1998) 4. Dürr, D., Goldstein, S., Lebowitz, J.L.: A mechanical model of Brownian motion. Commun. Math. Phys. 78(4), 507–530 (1988) 5. Escobedo, M., Laurençot, Ph., Mischler, S., Perthame, B.: Gelation and mass conservation in coagulationfragmentation models. J. Differ. Eqs. 195(1), 143–174 (2003) 6. Escobedo, M., Mischler, S., Perthame, B.: Gelation in coagulation and fragmentation models. Commun. Math. Phys. 231(1), 157–188 (2002) 7. Hammond, A.M., Rezakhanlou, F.: The kinetic limit of a system of coagulating Brownian particles. Arch. Rational Mech. Anal. 185(1), 1–67 (2007)
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8. Hammond, A., Rezakhanlou, F.: Kinetic limit for a system of coagulating planar Brownian particles. J. Stat. Phys. 124(2–4), 997–1040 (2006) 9. Lang, R., Nyugen, X.-X.: Smoluchowski’s theory of coagulation in colloids holds rigorously in the Boltzmann-Grad limit. Z. Wahrsch. Verw. Gebiete 54, 227–280 (1980) 10. Laurençot, Ph., Mischler, S.: The continuous coagulation-fragmentation equations with diffusion. Arch. Ration. Mech. Anal. 162(1), 45–99 (2002) 11. Laurençot, Ph., Mischler, S.: Global existence for the discrete diffusive coagulation-fragmentation equations in L 1. Rev. Mat. Iberoamericana 18(3), 731–745 (2002) 12. Rezakhanlou, F.: The coagulating Brownian particles and Smoluchowski’s equation. Markov Process. Related Fields 12, 425–445 (2006) 13. Wrzosek, D.: Existence of solutions for the discrete coagulation-fragmentation model with diffusion. Topol. Methods Nonlinear Anal. 9(2), 279–296 (1997) 14. Wrzosek, D.: Mass-conserving solutions to the discrete coagulation-fragmentation model with diffusion. Nonlinear Anal. 49(3, Ser. A: Theory Methods), 297–314 (2002) 15. Wrzosek, D.: Weak solutions to the Cauchy problem for the diffusive discrete coagulation-fragmentation system. J. Math. Anal. Appl. 289(2), 405–418 (2004) Communicated by H. Spohn
Commun. Math. Phys. 276, 671–689 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0354-8
Communications in
Mathematical Physics
Opening Mirror Symmetry on the Quintic Johannes Walcher School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, USA. E-mail: [email protected] Received: 2 October 2006 / Accepted: 8 June 2007 Published online: 5 October 2007 – © Springer-Verlag 2007
Abstract: Aided by mirror symmetry, we determine the number of holomorphic disks ending on the real Lagrangian in the quintic threefold. We hypothesize that the tension of the domainwall between the two vacua on the brane, which is the generating function for the open Gromov-Witten invariants, satisfies a certain extension of the Picard-Fuchs differential equation governing periods of the mirror quintic. We verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space. We further check the conjecture by reproducing the first few instanton numbers by a localization computation directly in the A-model, and verifying Ooguri-Vafa integrality. This is the first exact result on open string mirror symmetry for a compact Calabi-Yau manifold.
1. Introduction and Summary It has long been suspected that the enumerative results about holomorphic curves obtained by mirror symmetry [1] could be extended to open Riemann surfaces, provided appropriate boundary conditions are imposed. In the A-model, and at lowest order in the string coupling expansion, the counting of holomorphic disks ending on Lagrangian submanifolds is the central ingredient in the definition of Floer homology and the Fukaya category [2], which appears on one side of the homological mirror symmetry conjecture [3]. From the physics perspective, the chief interest is to determine the superpotential on the worldvolume of D-branes wrapping the Lagrangian, with many applications in studies of N = 1 compactifications of string theory. Until now, the program of extending mirror symmetry to the open string sector has been successfully implemented only in a rather limited set of examples with special, toric, symmetries [4,5]. While certain general structures could be extracted from the results obtained [6–8], and of course much is known in lower-dimensional situations [9,10], it has remained unclear whether and how these ideas could be implemented for
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Table 1. The number (integral invariants) of holomorphic disks in X ending on L, of degree d (only odd d are shown, for reasons explained in the text), and, for comparison, the number of holomorphic spheres in X , according to [1] d
number of disks n d
number of spheres
1 3 5 7 9 11 13 15 17 19
30 1530 1088250 975996780 1073087762700 1329027103924410 1781966623841748930 2528247216911976589500 3742056692258356444651980 5723452081398475208950800270
2875 317206375 229305888887625 295091050570845659250 503840510416985243645106250 1017913203569692432490203659468875 229948856813626664832516010477226554. . . 562465682466848327417948393837157975. . . 146020747145890338745688881159596996. . . 397016669854518762338361058844977288. . .
more general, in particular compact, Calabi-Yau threefolds. This is precisely what we do in this paper. The Calabi-Yau manifold X we will consider is the most popular quintic in CP4 , and our Lagrangian L will be the most canonical real locus inside of it. This Calabi-YauLagrangian pair has been contemplated many times in the literature, starting with [11]. First exact results were obtained in [12], where D-branes wrapping L were identified with certain RS boundary states at the Gepner point [13] (see also [14] for a complementary derivation of this result). In [15,16], the continuation of these boundary states over the moduli space was analyzed using matrix factorizations [17,18] in the mirror B-model Landau-Ginzburg description. In particular, it was explained in [16] that the singularity in the D-brane moduli space at the Gepner point could be interpreted as a degeneration of the Morse-Witten-Floer complex that computes Floer homology. Living in the A-model, the Floer differential differs from the classical Morse differential by corrections from holomorphic disks ending on L [19,2], which suggested that one should be able to turn these results into a computation of the number of holomorphic disks as coefficients in the appropriate large-volume expansion. We will fulfill this promise in the present work, although following a slightly different route. The central technical hypothesis in this work is that the spacetime superpotential on the brane worldvolume, which is the generating function capturing the open string instanton information [20–22], satisfies a differential equation which is a simple extension of the standard Picard-Fuchs differential equation whose solutions are the periods of the holomorphic three-form on the mirror of the quintic. The possible origin of such differential equations is discussed in special circumstances in [7,8] (see also [23,24]). But for a general brane configuration, or when the ambient Calabi-Yau is compact, the existence of this differential equation is, to the very least, surprising. Perhaps the most novel aspect of the equation that we introduce in this paper is that large complex structure is not a singular point of maximal unipotent monodromy. However, this has excellent reasons for being so, as we will explain below. Assuming the existence of the differential equation, the symmetries and monodromies of the superpotential at large volume are sufficient to fix the extended Picard-Fuchs operator. It is then straightforward to extract the open string instanton numbers, and we can check the integrality property conjectured in [21]. We do all this in Sect. 2, and display, for amusement, the results in Table 1.
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It is then also of interest to study the analytic properties of the brane superpotential over the entire Calabi-Yau moduli space, and not just around large volume. Referring to Sect. 3 for details, we would like to point out two salient features here. Firstly, the domainwall tension is invariant under monodromy around the conifold singularity in the moduli space. To appreciate the consistency of this result, one has to remember that the cycle that shrinks to zero volume at the conifold singularity in Kähler moduli space is a holomorphic cycle which can be wrapped by a B-brane, and it would be somewhat non-obvious why an A-brane would feel this singularity. The second interesting feature is that the domainwall tension is not invariant under monodromy around the small-volume Gepner point. This is more surprising because, based on the worldsheet results of [16], one would have naively expected the domainwall tension to vanish at that point where the two vacua on the brane become degenerate. Instead, what happens is that the tension of the domainwall, when analytically continued from large volume, becomes asymptotically equal to a particular closed string period, which measures flux superpotentials. In other words, this domainwall only mediates a transition between different flux sectors, and this is still consistent with the degeneracy of the open string vacua. What it tells us, however, is that it could be much more delicate to understand our results from the worldsheet perspective, which is purportedly insensitive to the flux. It also indicates that it might be appropriate to include some of the flux data into the definition of Floer homology and the Fukaya category. While this derivation of the superpotential and the instanton series can perfectly well stand alone, the confidence in the enumerative results of Table 1 of course increases dramatically if at least some of those numbers can be verified mathematically directly in the A-model. We will do this in Sect. 4. The mathematical definition of open Gromov-Witten invariants in general still appears lacking [2], although several special cases have been treated in the literature. Studies of the local toric situation include [25–28]. Recently, Solomon [29,30] has performed a rigorous study of open Gromov-Witten invariants in the situation in which the Lagrangian providing the boundary conditions arises as the fixed point set of an anti-holomorphic involution. This covers the situation of our interest, so we can be confident that the numbers we are claiming are well-defined. To go ahead with the direct computation of those open Gromov-Witten invariants, one can exploit the fact that, at least in our situation, any holomorphic mapping from the disk into X with boundary on L factors through a holomorphic sphere in X meeting L in a circle. In other words, we can relate the enumeration of holomorphic disks to the enumeration of holomorphic spheres which are invariant under the anti-holomorphic involution. For this problem, we have at our disposal the powerful graph combinatorial method introduced in [31]. This technique computes the Euler characteristic of a particular bundle on the moduli space M(CP4 ) of holomorphic curves in CP4 by using Atiyah-Bott localization with respect to the action of the torus (S 1 )5 ⊂ U (5) inside the symmetry group of CP4 . The anti-holomorphic involution then acts in a natural way on this moduli space and the bundle over it, and one can identify the open Gromov-Witten invariant as the Euler characteristic of the resulting real bundle over the real locus in M(CP4 ) [32]. There are then two key points to appreciate in order to proceed. The first one is that while the anti-holomorphic involution breaks some of the symmetries of the ambient space, it still leaves an (S 1 )2 ⊂ O(5) unbroken. In particular, the fixed points on the real slice with respect to this torus coincide with the real fixed points of the torus in the complex case. The second point is that the Euler class of a real bundle is the squareroot of the Euler class of its complexification, where the sign is determined by the choice
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of orientation. With these two ingredients, it is straightforward to adapt the methods of [31] to develop a graphical calculus which computes the open Gromov-Witten invariants of our interest. We have checked that up to degree 7, these numbers coincide with those obtained using mirror symmetry. The number (30) of holomorphic disks of degree 1 was first computed (without using localization) by Solomon [32,30]. We have also checked the number (1530) of holomorphic disks in degree 3 by taking a real slice of the localization computation of [33] on the space of curves (instead of the space of maps). Besides the many possible applications and extensions of these results that spring to mind, we would like to mention that the numbers we get in this paper can also be viewed as providing lower bounds in real enumerative geometry in the sense of, see, e.g., [34,35].
2. The Problem and its Solution We consider in CP4 the Calabi-Yau hypersurface given as the vanishing locus of a polynomial of degree 5 in the homogeneous coordinates of CP4 : X = {P(z 1 , . . . , z 5 ) = 0} ⊂ CP4 .
(2.1)
The choice of P determines the complex structure of X , and to define a σ -model with target space X , we need to pick a choice of complexified Kähler form B + i J = tω, where we denote by ω the integral generator of H 2 (X, Z) = Z, and t is the Kähler parameter.
2.1. On the real quintic. We want to identify in X a particular Lagrangian submanifold as the fixed-point locus of an anti-holomorphic involution which acts on the ambient CP4 as complex conjugation on the homogeneous coordinates [z 1 : z 2 : · · · : z 5 ] → [¯z 1 : z¯ 2 : · · · : z¯ 5 ].
(2.2)
The complex structure on X will be (anti-)invariant under this involution if the defining polynomial P is real, in the sense that all its coefficients are real (up to a common phase). The fixed point locus, L, where z i = xi is real is then given by the corresponding real equation P(x1 , . . . , x5 ) = 0 inside of RP4 ⊂ CP4 . Straightforwardly, L is a Lagrangian submanifold of X . In fact, L is even special Lagrangian with respect to the holomorphic three-form on X . Now while the topology of X is well-known and independent of the complex structure, the real locus L can have various topologies and singularities, with interesting transitions between them as P is varied. We will not attempt to discuss all the possibilities here, but wish to comment on the consequences. To fix ideas, let us consider the Fermat quintic P = z 15 + z 25 + z 35 + z 45 + z 55 .
(2.3)
Over the reals, z i = xi , we can solve for x5 uniquely in terms of x1 , . . . x4 , not all of which can be zero, lest x5 will be zero too. This identifies L with a copy of RP3 . However, this identification depends on the fact that z 55 = a for real a has only one real root, which will not be useful for a generic P.
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There are at least two things that can happen to the real locus as we vary the complex structure. The first one is familiar from studies of stability conditions on Lagrangian submanifolds, and happens along a real codimension one locus in complex structure moduli space. When crossing such a wall of marginal stability, the special Lagrangian L develops a singularity and reconnects on the other side, changing its topological type (but not its homology class). The second effect is a remnant of the standard conifold singularity in the complex structure moduli space. (It might seem that since the discriminant locus is complex codimension one, it would generically be missed by the half-dimensional real subspace. But this is untrue.) It was shown in [36] using a local model that when crossing such a conifold singularity, the homology class of the real locus always changes by the homology class of the vanishing cycle. The second phenomenon is known to happen on the quintic [37], for example when crossing the standard conifold locus ψ = 1 along the one parameter family P → P − 5ψ z 1 z 2 z 3 z 4 z 5 . Since the Lagrangian is connected at ψ = 0, it implies that we must also be crossing a line of marginal stability somewhere between ψ = 0 and ψ = ∞. In this paper, we are studying L in the A-model, and those aspects should be independent of the complex structure of X , and only depend on the Hamiltonian deformation class of L. Namely, we would expect to only depend on L being Lagrangian, and not the special Lagrangian property. On the other hand, the available definitions of Floer homology for Lagrangians clearly depend on the underlying topology. (For instance, they depend on b1 (L).) It is therefore not a priori clear why there should be a welldefined and invariant notion of Floer homology or of the “number of disks” ending on “the real locus L” which is independent of the complex structure of X . One might worry slightly less about this in regard to the first phenomenon (marginal stability) because at least the homology class is preserved. In this paper, in any case, we will ignore this complication, and just pretend that L ∼ = RP3 . The number of disks we will quote can then be understood as referring to “the generic quintic in a neighborhood of the Fermat point”. For the rest of the paper, we will be concerned with the dependence on the Kähler parameter, t, or its exponentiated version q = e2π it . We begin in the large volume limit q → 0. 2.2. Vacuum structure at large volume. Recall that to wrap an A-brane on L, we also need to specify a U (1) bundle with a flat connection. Since H1 (L; Z) = π1 (L) = Z2 we have two possible choices which are distinguished by a “discrete Wilson line”, W = = ±1. In fact, these two choices correspond to topologically distinct bundles on RP3 , as measured by the first Chern class c1 ∈ H 2 (L; Z). The latter is equal to H1 (L; Z) by Poincaré duality. On the other hand, the K-theory of the quintic does not contain any torsion elements, and the two choices of flat connection can therefore not be distinguished by any topological charge [12]. As a consequence, when wrapping a D6-brane of type IIA string theory on L, the brane worldvolume will support an N = 1 gauge theory with two vacua corresponding to the two possible discrete Wilson lines, which are not distinguished by any conserved charge. We can then ask about the existence of a BPS domainwall that communicates between these two vacua. To represent this domainwall in string theory, it is helpful to understand why the two bundles are topologically equivalent after inclusion in the quintic. Let us consider the situation with “non-trivial” Wilson line = − (we will see in a moment that this isn’t really an invariant notion). The non-trivial first Chern class of the bundle on L
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can be viewed as resulting from dissolving into the D6-brane a D4-brane wrapping the non-trivial one-cycle in H1 (L; Z). But since the quintic does not contain any non-trivial one-cycles, we can also contract it away to nothing. Clearly, then, the BPS domainwall that mediates between the two choices of Wilson line on the D6-brane wrapping on L is a D4-brane wrapping a holomorphic disk D in X with boundary on the non-trivial one-cycle in L and extended along a (2+1)-dimensional subspace of Minkowski space. This D4-brane is a magnetic source on the D6-brane and hence changes the (discrete) magnetic flux on L. The topological classification of D is as a non-trivial relative cohomology class in H2 (X, L; Z) with non-trivial image in H1 (L; Z). It is not difficult to get a first approximation to the tension, T , of this domainwall in the large volume limit (here and throughout the paper, we will refer to the tension as the holomorphic quantity whose absolute value gives the physical tension). Since L is defined as the fixed point locus of an anti-holomorphic involution of X , any holomorphic disk ending on L can be complex conjugated to a second holomorphic disk, and thereby completed to a holomorphic sphere. From the exact sequence H2 (X ; Z) → H2 (X, L; Z) → H1 (L; Z)
(2.4)
we see that in fact also a brane wrapped on twice the generator of H2 (X, L; Z) will not change the vacuum on the brane, and hence be equivalent to a holomorphic sphere. The tension of that sphere being t (the Kähler parameter), we infer 2T ∼ t. To see that this argument was in fact quite incomplete, we need another fact about the relation between the cohomology of X and that of L. Namely, when intersecting a hyperplane in CP4 with the Lagrangian L (the hyperplane has to be represented by a complex linear equation in order to intersect L transversely), we can see that the intersection locus is a non-trivial one-cycle in L. The Poincaré dual statement is that the integral generator of H 2 (X ; Z) restricts on L to the non-trivial element of H 2 (L; Z). Since the gauge invariant gauge field on the brane is B − F, this means that changing the flat B-field on X by one unit is equivalent to exchanging the two flat gauge fields on the brane. A more elementary way to see this is to note that the path-integral contribution of a disk worldsheet wrapped on D has a contribution e2π it/2 = q 1/2 from its area and a contribution = ±1 from its boundary, so changing B → B + 1 is equivalent to changing → −. Taking B → B + 2 does nothing on the brane. In this sense, we can specify the Wilson line on the brane only after fixing the sign of q 1/2 . Now claiming that T ∼ t/2 raises a puzzle because it is not invariant under t → t +2. To resolve this, we have to note that the D4-brane wrapped on D is a magnetic source not only for the gauge field on L, but also for the Ramond-Ramond 3-form field (we actually used this above to derive 2T ∼ t). The change of T under t → t + 2 is then explained by the non-invariance of RR flux under B-field monodromies. So to make the formula for T more precise, and work out the spectrum of domainwalls, we have to include the RR flux quantum numbers in our labeling of the vacua. For the time being, 4-form flux, N4 , and 6-form flux, N6 , (around the unique four and 6-cycle of X ) will suffice, so our vacua are labeled as (N4 , N6 , ). We then require that a domainwall represented by a D4 wrapping an elementary disk D connects −, and that by juxtaposing two such disks we obtain a sphere across which the only change is N4 → N4 + 1, that the B-field monodromy B → B + 1 changes N6 → N6 + N4 , and also → −, but is otherwise a symmetry of the spectrum. We also wish to keep 4- and 6-form flux integrally quantized to avoid concluding with fractional D0-branes.
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It then turns out that, up to parity, there is only one consistent solution to these constraints. The change in 4-form flux across a D4-brane wrapped on D is zero when = − on the left of the domainwall and it is equal to +1 when = + on the left, and, we have to let the B-field monodromy change the 4-form flux, in a way depending on : B → B+1:
(N4 , N6 , −) → (N4 , N6 + N4 , +) . (N4 , N6 , +) → (N4 + 1, N6 + N4 , −)
(2.5)
Let us denote the tension of a domainwall between vacuum (N4 , N6 , ) on the left and vacuum (N4 , N6 , ) on the right by T(N4 ,N6 ,)|(N4 ,N6 , ) . The above constraints are enough to determine all T ’s as a function of t. For example, let us consider the most basic T− ≡ T(0,0,−)|(0,0,+) and T+ ≡ T(0,0,+)|(1,0,−) . Since T− (t + 1) = T+ (t) and T+ + T− = t, we conclude T− =
t 1 − 2 4
T+ =
t 1 + . 2 4
(2.6)
Finally, we can write down the spacetime superpotential, which follows from (2.6) together with T(N4 ,N6 ,)|(N4 ,N6 , ) (t) = W N4 ,N6 , (t) − W N4 ,N6 , (t).
(2.7)
We find W N4 ,N6 ,+ (t) =
t2 + N4 t + N6 4
W N4 ,N6 ,− (t) =
t 1 t2 − + + N4 t + N6 . 4 2 4
(2.8)
Of course, in this section, the discussion has been entirely classical and restricted to the large volume limit t → i∞. We now proceed to study the corrections W quant. from worldsheet instantons. 2.3. Worldsheet instanton corrections. According to general considerations [22,21,20, 4,38], the spacetime superpotential on the worldvolume of a particular supersymmetric brane wrapping a cycle in a Calabi-Yau manifold, X , when expressed in the A-model, and expanded in the appropriate variables, becomes the generating function counting worldsheet instanton corrections from holomorphic disks ending on the Lagrangian, L. Such a statement is in line with the role that holomorphic disks play in the definition of Fukaya’s A∞ category [2], and the relationship between A∞ algebras and D-brane superpotentials [39,40]. More precisely, the spacetime superpotential can be identified with the topological disk partition function and is conjectured to admit an expansion of the general form n d,e W(t, u) = Fdisk (t, u) = n˜ d,e q d y e = q kd y ke . (2.9) k2 d,e
d,e k≥1
Here, the sum is over relative cohomology classes in H2 (X, L), q = e2π it is the (collection of) closed string Kähler parameters of X and y = e2π iu is the (collection of) exponentiated classical open string deformation parameters. The latter come from nonHamiltonian deformations of the Lagrangian. They are b1 (L) in number and are complexified by the Wilson line of the gauge field around the corresponding one-cycles of L. The final transformation in (2.9) is a resummation of multi-cover contributions and the
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central part of the conjecture is that the resulting expansion coefficients n d,e are integers [21] (whereas the n˜ d,e are in general rational numbers). These integers have a spacetime interpretation as counting the “degeneracy of BPS domainwalls” in the class (d, e). The existence and integrality of such an expansion has been checked in many examples involving local toric Calabi-Yau manifolds. Our goal in this paper is to make sense of and evaluate the formula (2.9) for the Calabi-Yau-Lagrangian pair (X, L) = (quintic, real locus). At first sight, the fact that we only have a discrete open string modulus at our disposal is a deficiency because (2.9) makes explicit only rational cohomology. On second thought, however, it’s a blessing. For example, as we have discussed above, domainwalls arising from D4-branes wrapping holomorphic disks are sources for both the Ramond-Ramond field and the gauge field on the brane. But if the disk ends in a rational cycle of L, the gauge flux is non-zero as a differential form. This raises a puzzle because according to the standard worldsheet analysis, gauge fields on Lagrangian A-branes should be flat. From the spacetime perspective, this might well be repaired by a careful analysis of the couplings of the brane to the Ramond-Ramond fields. But it is clearly not obvious to see that from the TFT on the worldsheet. In the cases discussed in the literature (see [4,5] and follow-up work), this problem is avoided because the Lagrangians considered there are non-compact and hence the flux can disperse to infinity. A second advantage of having H1 (L , Z) = Z2 being torsion has to do with certain puzzlements [32] about the multi-cover formula as well as the integral “framing” ambiguity of open string amplitudes discovered in [5]. We do not understand either of those issues sufficiently well enough to usefully discuss here, but the consistency of our results indicates that both problems are absent for H1 (L) = Z2 . Finally, because our Lagrangian is compact, we can also discuss the classical contributions to the superpotential, as we have done in the previous subsection. The structure of these classical terms (which are absent from (2.9)) will help us to normalize the computation by imposing consistency of the monodromies around the various singular loci in the Kähler moduli space (see Sect. 3). So what is the possible structure of worldsheet instanton corrections to our formulas (2.6) for the domainwall tensions? Clearly, the first non-trivial term will arise from worldsheet disks wrapped in the class D generating H2 (X, L; Z) = Z, and will contribute at order q 1/2 . Then there will be higher order terms. Let us call disks contributing at order q d/2 “of degree d”. It is easy to see that the conditions T− (t + 1) = T+ , T+ + T− = t that we have used to derive (2.6) hold also after inclusion of non-perturbative worldsheet corrections. This is because t is essentially defined to be the parameter measuring the tension of the domainwall wrapped on a degree 1 rational curve. The only form of the instanton expansion that is consistent with those constraints is that there are no contributions from even degree disks. This is in fact not unexpected, because disks of even degree have trivial boundary on the Lagrangian, and even though we can contemplate holomorphic disks of even degree ending on L, the triviality of their boundary makes it difficult to keep them there as we vary the complex structure of the quintic. In other words, we do not expect any invariant to exist for even degree. So we expect a result of the form
T± =
1 t ± ± const. n˜ d q d/2 , 2 4 d odd
(2.10)
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where the n˜ d are certain rational numbers such that rewriting them as in (2.9), n d/k n˜ d = , k2
(2.11)
k|d
the n d turn out integer. 2.4. Mirror symmetry and open Picard-Fuchs equation. The easiest way to get an expansion of the form (2.10) is to make use of mirror symmetry. What this means concretely is that we should first identify an object in the D-brane category which appears on the B-model side of the homological mirror symmetry conjecture, and which, via the equivalence of categories and up to auto-equivalences, corresponds to the object of the (derived) Fukaya category that is defined by L. We should then compute the appropriate superpotential/domainwall tension quantity as a function of the mirror parameter ψ and reexpress it in terms of the flat coordinate t. The Calabi-Yau mirror, Y , to the quintic is of course well-known. Itis the resolution of a (Z5 )3 quotient of the one-parameter family of quintics z i5 − 5ψ z i = 0 in CP4 . Equivalently, 5 we can consider a Landau-Ginzburg orbifold model with superpotential W = z i − 5ψ z i and orbifold group (Z5 )4 . The corresponding B-model category which is conjectured [41] to be equivalent to the derived category of Y is the category of (Z5 )4 equivariant matrix factorizations of the superpotential W . (The corresponding equivalence was proven for the quintic itself by Orlov [42].) And in fact, as we have mentioned in the introduction, the matrix factorization which is mirror to the Lagrangian L is known explicitly (see [15,16] for details). Given this identification of the matrix factorization and the equivalence with the derived category, it should be possible in principle to also describe explicitly a coherent sheaf on Y corresponding to L. This would in fact be very interesting, because it would allow making use of some of the well-known machinery of holomorphic vector bundles that applies to problems of this type. In particular, there is an explicit formula for the superpotential, namely, the holomorphic Chern-Simons functional [11] 2 B ¯ W = ShCS (A, A0 ) = ∧ Tr A ∧ ∂ A0 A + A ∧ A ∧ A . (2.12) 3 No such expression is known in the matrix factorization formulation, and although there are formulas for TFT correlators [43,44], they do not appear sufficient to determine the full superpotential. (See, however [45] for recent progress in making the A∞ constraints of [46] useful for this type of question.) Leaving these explicit B-models for future investigations, we will instead obtain sufficient guidance from the non-compact examples of open mirror symmetry introduced in [4], and studied in depth in [5,6,23,24,7,8]. The main simplification that occurs in these examples is that the B-model contains only D5-branes wrapped on curves in the Calabi-Yau. For such a brane configuration, the holomorphic Chern-Simons action (2.12) reduces to a “partial period” integral of the type W(C, C∗ ) = , (2.13) γ
where γ is a three-chain in X with boundary ∂γ = C − C∗ equal to the difference of two possible positions of the D5-branes. (If C and C∗ are holomorphic, (2.13) is literally the
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tension of the domainwall between the two vacua.) In the toric case, one can then further reduce the integral (2.13) to take place on a Riemann surface, so one has essentially a one-dimensional problem. This structure was exploited in [7,8] to show that the differential equations obtained in [6,24] could be viewed as resulting from a certain variation of mixed Hodge structure on a certain relative cohomology. Explicitly, one retains the boundary terms arising in the derivation of the GKZ differential system and converts them into appropriate boundary variations. The upshot is that the open string mirror computations in the local toric case can be cast in a form very similar to the standard, closed string computations, involving Picard-Fuchs differential equations, maximal unipotent monodromy, mirror map, etc.. This is called N = 1 special geometry. We do not know at present whether such considerations make sense for the general B-model situation. The case at hand, however, is sufficiently well constrained by our results so far that assuming the existence of a differential equation with properties as in [7,8], there is essentially a unique candidate. This moreover turns out to produce excellent results. The central idea of N = 1 special geometry is to extend the standard period vector by certain “partial periods” encoding information about the open string sector. We recall that in standard (N = 2) special geometry, we have two periods for every closed string modulus, plus one or two extra ones related to the holomorphic three-form. In N = 1 special geometry, we gain one “partial period” for every classical open string modulus, plus one for every brane vacuum included in the background. Schematically, (tclosed , u open ) = (1, tclosed , ∂t Fclosed , u open , Wbrane , . . .)T ,
(2.14)
where Fclosed is the standard prepotential and the u open are the flat coordinates of the open string sector. The important point is that the period vector (2.14) satisfies a certain extension of the Picard-Fuchs differential equations. This differential system has all of the closed periods as solutions, plus extra ones related to u open and Wbrane . The latter gives the open string instanton expansion according to (2.9). In the case that we have discussed in the previous subsections, we are not adding any classical open string modulus because b1 (RP3 ) = 0, so the only modulus is the Kähler parameter t of X , or equivalently, the mirror variable, z = z(t). Moreover, according to (2.10), we need exactly one non-trivial domainwall tension as a function of t to encode the desired open string expansion. Let us call τ ∼ q 1/2 + · · · the quantum part of the expansion (2.10). Since to leading order z = q = e2π it , we will also have τ (z) ∼ z 1/2 + · · · when expressed as a function of z. Thus, we are simply seeking an ordinary linear differential equation in z, which, in addition to the four known periods of the mirror quintic, has exactly one additional linearly independent solution, τ , with a squareroot behavior at z = 0. The Picard-Fuchs equation governing periods of the mirror quintic being L = θ 4 − 5z(5θ + 1)(5θ + 2)(5θ + 3)(5θ + 4) = 0, (2.15) where θ = z∂z , and z = (5ψ)−5 , virtually the only possible extension that satisfies our constraints is the differential operator (2θ − 1)L = (2θ − 1)θ 4 − 5z(2θ + 1)(5θ + 1)(5θ + 2)(5θ + 3)(5θ + 4).
(2.16)
We will now analyze this differential equation and show that it satisfies all the other desirable properties as well.
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2.5. The instanton sum. We follow conventions of [1]. The differential equation L = 0 has one distinguished solution, called the fundamental period, which has a power series expansion around the large complex structure point z = 0, − w 2 (z) ≡ 0 (z) =
∞ (5m)! m z . (m!)5
(2.17)
m=0
All other solutions contain logarithms as z → 0, large complex structure being a point of maximal unipotent monodromy. The period with a single logarithm, w1 (z), has the information about the mirror map via t = w1 /w 2 , q ≡ e2π it , − 2π iw 1 (z) = 0 (z) log z + 5
∞ (5m)! m z [ (1 + 5m) − (1 + m)] . (m!)5
(2.18)
m=1
Under large complex structure monodromy, z → e2π i z, w 1 → w 1 + w 2 and t → t + 1. There are then two further solutions of (2.15), both of which contain the closed string instanton information, in slightly different forms. Specifically, the solution of (2.15) called F1 in [1] is characterized by the boundary conditions 5 21 (2π i)2 F1 = −5·(2π i)w 1 (z) log z + w 2 (z)(log z)2 − ·(2π i)2 w 1 (z)+O(z). (2.19) 2 2 It transform under large complex structure monodromy as F1 → F1 − 5w 1 − 8w 2 . Finally, the solution called F2 in [1] is characterized by F2 → F2 − F1 − 3w 1 + 5w 2 as t → t + 1. These periods (F1 , F2 , w 1 , w 2 ) can be interpreted as the quantum corrected masses of D4, D6, D2 and D0-brane on the quintic, respectively [12]. They therefore also give the tension of domainwalls mediating between various flux sectors, including the corrections from worldsheet instantons. For example, in the proper Kähler normalization w 2 = 1, one obtains after inverting (2.18) and expanding in q = e2π it , F1 4876875 2 1 5 2 21 2875q + t t + q = − − + · · · . w2 2 2 4π 2 4
(2.20)
The polynomial in t is the classical tension from the geometric volume of the cycles and the power series in q gives the quantum corrections. The rational coefficient N˜ d of q d in this expansion gives the contribution from holomorphic spheres of degree d. They satisfy the property that when reexpressed in terms of Nd via N˜ d =
d Nd/k k|d
k3
,
(2.21)
the Nd are integers. Note that we have here slightly unconventionally expanded the first derivative of the prepotential instead of the prepotential itself or the Yukawa coupling as in [1]. Since periods and brane superpotentials are on equal footing in N = 1 special geometry, this will make the comparison with the open string version (2.25) more natural.
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Turning now to Eq. (2.16), it has, by construction, exactly one additional solution, which we normalize to τ (z) = z 1/2 + · · · . We find τ (z) =
∞ (3/2)5 (5m + 7/2) m+1/2 z . (7/2) (m + 3/2)5
(2.22)
m=0
In the next section, we will determine from monodromy calculations on the Kähler moduli space that τ enters the domainwall tension in the normalization T± (t) =
w1 w2 15 ± ± 2 τ (z). 2 4 π
(2.23)
This then has exactly the expected form (2.10). Consulting (2.6) and its relation with (2.8), we then conclude that the contribution of worldsheet disk instantons to the spacetime superpotential is 30 W quant. = τ (z). (2.24) 4π 2 Dividing by w 2 to go to the canonical normalization of the holomorphic three-form, multiplying by 4π 2 as in (2.20), inverting the mirror map, and doing the expansion, we obtain the open string instanton sum τˆ (q) = 30
τ (z(q)) 4600 3/2 5441256 5/2 = 30q 1/2 + q + q + ··· . 0 (z(q)) 3 5
(2.25)
We can then plug into τˆ (q) the Ooguri-Vafa multi-cover formula (2.9), τˆ (q) =
nd q d/2 (q d , 2, 1/2), q dk/2 = nd 2 k 4
d odd k odd
(2.26)
d odd
where is the Lerch Transcendent. For reasons explained in a previous subsection, we only consider disks of odd degree and their odd multi-covers. The first few n d are indeed integer and displayed in Table 1 in the introduction. It should be stressed that we have strictly speaking not shown that the constant normalization factor in (2.10) is equal to 2π1 2 as claimed. It is, however, the most natural choice and consistent with everything else we know. It would be interesting to derive this value more directly. 3. Analytic Continuation of the Superpotential The purpose of this section is to analytically continue our result for the superpotential/domainwall tension over the entire quantum Kähler moduli space of the quintic, much as was done for the closed string periods in [1]. This will not only help us to fix the normalization factor anticipated in (2.23), but is interesting in its own right as it can shed light on intrinsically stringy aspects of D-brane physics that have hitherto been inaccessible. We will indeed find that the analytic properties of the T± are rather interesting. Recall that the Kähler moduli space of the quintic has three special points: large volume point z → 0 that we have already discussed in depth, the conifold singularity z = 5−5 at which the period F2 vanishes, and the so-called Gepner or Landau-Ginzburg point,
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z → ∞, which is not a singularity of the CFT, but exhibits a Z5 orbifold monodromy. We wish to understand the analytic behavior of W, or equivalently T , around each of these points. We shall work with the ansatz T± (z) =
w 1 (z) w 2 (z) ± ± aτ (z) 2 4
(3.1)
and determine the coefficient a from consistency requirements. The standard tool to do the analytic continuation of solutions of a hypergeometric differential equation of the type (2.16) is the Barnes integral representation. For τ , this representation takes the form π2 1 (−s + 1/2)(5s + 1)(s + 1/2) iπ(s−1/2) s τ (z) = e z , (3.2) 60 2π i C (s + 1)5 where the integration contour is straight up the imaginary axis. For |z| < 5−5 , we close the contour on the positive real axis and recover (2.22). For |z| > 5−5 , we instead close the contour on the negative real axis, and obtain the expansion ∞ π 2 −(−5m − 3/2) −m−1/2 τ (z) = τ1 (z) + τ2 (z) = z 60 (−m + 1/2)5 m=0
∞ −(m/5)e4π im/5 −m/5 −iπ/2 sin π m/5 + z e . 5(m)(1 − m/5)4 cos π m/5 m=1
(3.3) The first term, τ1 (z), is simply the unique solution of (2.16) with a squareroot behavior around z = ∞, and changes sign as we circle around z 1/5 → e−2π i/5 z 1/5 . The second sum in (3.3) is easily verified to be a solution of the ordinary Picard-Fuchs equation, and hence a closed string period. To determine which one, we can compare it with the canonical Z5 symmetric basis of solutions of (2.15) around the Gepner point [1], ( j = 0, . . . , 4), j (z) =
∞ −(m/5)e4π im/5 −m/5 2π i jm/5 z e . 5(n)(1 − m/5)4
(3.4)
m=1
Indeed, the identity sin π m/5 = 2 sin 2π m/5 − 2 sin 4π m/5 cos π m/5 shows that
(3.5)
π2 (3.6) [ 0 + 2 4 + 2 2 ] . 60 According to the results of [1], the small volume period vector = ( 2 , 1 , 0 , 4 )T is related to the large volume basis = (F1 , F2 , w 1 , w 2 )T via = M with ⎞ ⎛3 1 8 − 21 5 5 5 −5 ⎜ 0 −1 1 0 ⎟ ⎟ (3.7) M =⎜ 1 ⎠. ⎝−1 −2 2 5 5 5 5 0 0 −1 0 τ2 (z) =
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This allows us to express τ2 (z) in the integral basis, τ2 (z) =
π2 −4F1 + 8F2 − 22w 1 + 15w 2 . 60
(3.8)
Moreover, by using the known monodromy matrices around the Gepner point, we find that as z −1/5 → e2π i/5 z −1/5 , w 1 → −F2 + w 1 − w 2 ,
w 2 → F2 + w 2 ,
τ → −τ +
π2 F2 . 60
(3.9)
Thus we see that were it not for the quantum corrections of the domainwall tension in 1 2 1 2 (3.1), the Gepner monodromy would take w2 + w4 to w2 − w4 − F42 , and would not induce a symmetry of the domainwall spectrum as it should. Moreover, we see that the lucky number that makes the Gepner monodromy integral is indeed a = π152 . (Strictly speaking, this is only the minimal possibility, a natural choice.) With this value, the Gepner monodromy acts as A:
T+ → T− ,
T− → T+ − w 2 − F2
(3.10)
on the open string periods. Since as discussed in Sect. 2, the large volume monodromy acts by T∞ : T− → T+ , T+ → T− + w 2 , we find by combining the two that the conifold −1 ◦ A−1 acts trivially on both T and T . monodromy about z = 5−5 , T = T∞ + − Let us verify this last assertion explicitly, in order to check that everything is consistent. A straightforward way to compute this monodromy is to compare the divergence of the large volume expansions (2.17) and (2.22) as z approaches the singularity z → z ∗ = 5−5 . We know from [1] that at the conifold, F2 vanishes as F2 ∼ α1 (z −z ∗ )+α2 (z −z ∗ )2 +· · · and 0 behaves as 0 ∼ 2π1 i F2 log(z − z ∗ ) + regular. To determine the coefficient b in τ ∼ 2πb i F2 log(z − z ∗ ) + regular, we compare the second derivatives of 0 and τ as z → z ∗ . Using Stirling’s formula, we find
√ √ 10 5 9 5 1 5 7 · 5 + ··· (3.11) (55 z)m − 0 ∼ 4π 2 4π 2 m m which determines α1 , α2 . Doing the same for τ delivers
√ √ 10 5 7 · 59 5 1 5 m+1/2 5 + ··· . (5 z) − τ ∼ 4π 2 4π 2 m m
(3.12)
This implies b = 1. 2 Thus, we find that the conifold monodromy takes τ → τ + π60 F2 , and since w 2 → w2 − F2 , T± are invariant when we set a = π152 . It is also worth pointing out that for a = π152 , the leading behavior of T± as z → ∞ is the same as that of an integral closed string period. This follows from (3.3) in conjunction with (3.8). As was mentioned in the introduction, this is a further consistency check on our results. It was shown in [15,16] that the two open string vacua associated with the choice of discrete Wilson line (see subsect. 2.2) could be identified with certain matrix factorization in the Landau-Ginzburg B-model. At the Gepner point, z → ∞, the open string spectrum on the brane develops an extra massless state with a cubic superpotential. (This coalescence of open string vacua was first proposed in [12].)
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There should therefore be a domainwall between the two vacua that becomes tensionless as z → ∞. Our result is then that while such a domainwall can indeed exist, it is not the most naive one obtained by wrapping a D4-brane on the primitive disk, but has to be combined with the appropriate integral period from (3.8). To conclude this section, we summarize the results for the action of the monodromies around the Gepner point, conifold point, and large volume point on the extended period vector (we now use T− = −T+ + w 1 ), T = T+ , F1 , F2 , w 1 , w 2 . (3.13) We have: ⎛ −1 ⎜0 ⎜ ⎜0 ⎝0 0
0 1 1 0 0
A 0 3 −4 −1 1
1 5 8 1 0
⎞⎛ 1 0 3 ⎟ ⎜0 ⎟⎜ −5⎟ ⎜0 −1⎠ ⎝0 0 1
0 1 0 0 0
T 0 0 1 0 −1
0 0 0 1 0
⎞ 0 0⎟ ⎟ 0⎟ 0⎠ 1
⎛ −1 ⎜0 ⎜ ⎜0 ⎝0 0
T∞ 0 0 1 1 0 −5 −1 1 −3 0 0 1 0 0 0
⎞ 1 −8⎟ ⎟. 5⎟ 1⎠ 1
(3.14)
These matrices satisfy A · T · T∞ = 1 and A10 = 1, but A5 = 1. Thus we find that the combined open-closed moduli space is a double cover of the quantum Kähler moduli space of the quintic, branched at z = 0 and z = ∞. 4. Localization in the A-Model In this section we shall show how to check the enumerative predictions that we have obtained using mirror symmetry. We have outlined the main strategy in the introduction, so we will attempt to be brief. Details can be filled in from [31] and [19], Chap. 27. Consider the moduli space Md ≡ M0,0 (CP4 , d) of genus zero stable maps to CP4 in degree d. For each point f : → CP4 in M, we can pullback from CP4 the bundle O(5) of quintic polynomials. The global sections of that bundle O(5d) over then fit together to a vector bundle Ed as we vary f over Md . Any particular quintic polynomial P(z 1 , . . . , z 5 ) in the homogeneous coordinates of CP4 gives a section of O(5). The resulting section of Ed vanishes at precisely those genus zero maps into CP4 which happen to be contained in the quintic given by P. This identifies the number of genus zero, degree d maps to the quintic as the Euler class of Ed : ˜ Nd = c5d+1 (Ed ). (4.1) Md
It was shown in [31] that this Euler class can be very efficiently computed using AtiyahBott localization. The entire structure described above carries an (S 1 )5 action inherited from the standard U (5) action on CP4 . On the homogeneous coordinates, this torus acts as (4.2) T5 = (S 1 )5 (ρ1 , . . . , ρ5 ) : [z 1 : · · · : z 5 ]. → [ρ1 z 1 : · · · : ρ5 z 5 ]. (This action can be complexified, of course, but we really only need the real torus.) On CP4 , there are exactly five fixed points, pi , of this torus action, defined by z j = 0, j = i. The fixed point loci on Md can be associated combinatorially with certain decorated tree graphs, . The vertices of these graphs (which can have arbitrary valence, val(v)) correspond to (genus 0) contracted components of the source . They are labeled by one
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of the fixed points pi which tells where the component maps. The edges of the graph correspond to non-contracted rational components of mapping onto the coordinate line joining pi to p j . They are labeled by a positive integer d describing the degree of that map. The constraints on this decoration are that pv = pv for adjacent vertices v, v and that the sum of degrees on the edges be equal to the total degree under consideration. In general, the fixed loci are not isolated points, but consist of certain moduli spaces M arising from the contracted components at the vertices (of valence ≥ 3). One can then compute the (T5 -equivariant) Euler class of the normal bundle of M inside of Md , as well as the Euler class of Ed at the fixed points. The integrals over the M can be done, and what results is a very explicit formula for N˜ d given by a sum over graphs and labellings, divided by the appropriate symmetry factor. We wish to accomplish something similar for the holomorphic maps of disks to the quintic with boundary on the real locus. As we have indicated before, any disk with boundary on the real locus can be completed to a sphere, and the two halves of that sphere contribute in the same relative homology class. Conversely, any sphere of odd degree is cut in two by the real locus in a non-trivial one-cycle.1 Therefore, the number of disks of odd degree d is equal to twice the number of spheres of degree d which are invariant under complex conjugation of source and target. On the real locus MR d ⊂ Md , complex conjugation defines a real structure on the bundle of quintics Ed (and, of course, on the tangent bundle). Since we are interested in maps into a real quintic, we can identify the open Gromov-Witten invariant as [32] n˜ d = 2
MR d
e(EdR ).
(4.3)
In trying to apply localization to this problem, one is naively troubled by the fact that the torus action (4.2) does not commute with the standard complex conjugation (2.2). However, it is easy to realize that there is another real subtorus of U (5) which does. This torus is two-dimensional and is the Cartan torus of O(5) ⊂ U (5). It is the natural four-dimensional analogue of the S 1 action used in [25]. An equivalent way to describe this is to choose the alternative complex conjugation σ :
[z 1 : z 2 : z 3 : z 4 : z 5 ] → [¯z 2 : z¯ 1 : z¯ 4 : z¯ 3 : z¯ 5 ]
(4.4)
which commutes with the subtorus T2 of (4.2) defined by ρ2 = ρ1−1 , ρ4 = ρ3−1 , ρ5 = 1. The nifty thing about this torus is that its fixed points on CP4 are identical to those of (4.2). Moreover, it is not hard to see that the fixed points of T2 acting on MR d are simply those fixed points of T5 acting on Md which are invariant under σ . From this discussion, we see that our task is to take a real section of Kontsevich’s calculation [31] with respect to the complex conjugation σ . A moment’s thought shows why this is feasible: Any σ -invariant decorated graph of odd total degree contains the real locus of at the middle of an edge. In other words, the contracted components of are away from the real locus. The upshot is that the integrals over the fixed loci are identical to those before. To understand the Euler class of the normal bundle and of the bundle of real quintics, we are helped by the following elementary fact: If V is any real vector bundle, then the 1 This is not true for even degrees: There can be real spheres of even degree without real points. In the real problem, they give rise to maps from the crosscap to the quintic. In other words, they will play a role in orientifolds. I am grateful to Jake Solomon for extensive discussions on these issues.
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square of its Euler class is the Euler class of its complexification, e(V ) = e(V ⊗ C).
(4.5)
For bundles of high enough rank, this formula of course only makes sense for the universal bundle, or in equivariant cohomology. The sign of the squareroot in (4.5) is determined by the choice of orientation on V (which does not affect the canonical orientation of V ⊗ C). In our situation, EdR ⊗ C = Ed |MR , and since we already know e(Ed ), d we are done. Our graphical calculus is then very much as in [31]. A moduli space of T2 -invariant disks corresponds to a tree graph with vertices mapping to fixed points pµ(v) (with µ(v) ∈ {1, . . . , 5}) and edges mapping to coordinate lines joining pi to p j . There is one special vertex, call it the first one, on which ends an extra half-edge with odd degree, call it d0 . This restriction is to ensure that the total degree d = d0 + 2 d(e) (4.6) edges
can be odd. Another condition is that the special vertex cannot map to p5 . This arises from the fact that when we reconstruct a σ -invariant sphere by reflecting our graph on the half-edge, the first vertex will be adjacent to its image, and σ ( p5 ) = p5 . In taking a squareroot of the formulas in [31], we have to fix the signs. In principle, this could be done by a careful analysis such as advertised in [29,30]. In practice, the condition that the answer be independent of the torus weights is enough to determine the sign. Explicitly, we have
e(EdR )
M e(NR )
=
5d aλi + (5d − a)λ j d
edges
a=0
(−1)d
(d!)2
(λi − λ j )2d d 2d
d a d −a λi + λ j − λk d d
k=i, j a=0
⎛ ⎞⎛ ⎞val(v)−3 d d val(v)−1 ⎠⎝ ⎠ (λv − λ j ) ·⎝ · λv − λ j λv − λ j (5λv )val(v)−1 vertices flags flags j=v
1
(5d0 −1)/2
·
a=0
aλµ(1) + (5d0 − a)λσ (µ(1)) d0
(d0 −1)/2 a d0 ! d0 − a (−1)(d0 −1)/2 d (λµ(1) − λσ (µ(1)) )d0 λµ(1) + λσ (µ(1)) − λk d0 d0 d0 0 k=µ(1),σ (µ(1)) a=0
.
(4.7)
Here, it is understood that the torus weights satisfy λ2 = −λ1 , λ4 = −λ3 , λ5 = 0. It should be emphasized that setting λ5 to zero introduces zero weight components in the above formula. These always exactly cancel between numerator and denominator, leaving a finite result. In similar situations in other contexts, one would then seek a more generic choice of weights to resolve the ambiguity. This is not possible in the present
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situation, since our choice of weights is already the most general one consistent with the anti-holomorphic involution. The best evidence that our treatment is nevertheless correct is the agreement with the results from the extended Picard-Fuchs equation. But, it is also conceivable that one could exploit the fact that the maximal abelian subgroup of U (5) which commutes with the anti-holomorphic involution is U (1)2 × Z2 . Only the continuous part of this group is seen by the localization formula, resulting in zero weight ambiguities. It turns out that these unfixed directions are odd under the Z2 subgroup, so the fixed point loci under the larger group are in fact isolated. But it would require a more careful analysis to show rigorously that this observation completely fixes the ambiguity. In formula (4.7), it is also understood that in counting the valence of the vertex called 1, the half edge counts full. Our final formula is e(EdR ) 1 n˜ d = 2 . (4.8) R ) | Aut | M e(NM , labellings As in [31], | Aut | is the product of the order of the automorphism group of as a decorated graph times the product of the degrees on the edges (including d0 ). For the first few degrees, one reproduces the results from Eq. (2.25) in Sect. 2. Acknowledgements. My interest in this problem was revived when Jake Solomon told me that the number of degree 1 holomorphic disks was 30. I would like to thank him for several helpful discussions and for sharing parts of his thesis. I am indebted to Katrin Wehrheim for patiently explaining what could (and could not) be learned from FO3 . I would also like to thank Simeon Hellerman, Calin Lazaroiu, Wolfang Lerche, Andy Neitzke, Rahul Pandharipande, and Edward Witten for valuable discussions and Dan Freed and Frank Sottile for helpful correspondence. This work was supported in part by the DOE under grant number DE-FG0290ER40542.
References 1. Candelas, P., De La Ossa, X.C., Green, P.S., Parkes, L.: A Pair Of Calabi-Yau Manifolds As An Exactly Soluble Superconformal Theory. Nucl. Phys. B 359, 21 (1991) 2. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory—anomaly and obstruction. Preprint 2000, book in progress, available at http://www.math.kyoto-u.ac.jp%Efukaya/fukaya.html 3. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of I.C.M., Vol. 1,2 (Zürich, 1994) Basel: Birkhäuser, 1995, pp. 120–139 4. Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. http://arxiv.org/ list/hepth/0012041, 2000 5. Aganagic, M., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1 (2002) 6. Mayr, P.: N = 1 mirror symmetry and open/closed string duality. Adv. Theor. Math. Phys. 5, 213 (2002) 7. Lerche, W., Mayr, P., Warner, N.: Holomorphic N = 1 special geometry of open-closed type II strings. http://arxiv.org/list/hepth/0207259, 2002 8. Lerche, W., Mayr, P., Warner, N.: N = 1 special geometry, mixed Hodge variations and toric geometry. http://arxiv.org/list/hepth/0208039, 2002 9. Polishchuk, A., Zaslow, E.: Categorical mirror symmetry: The Elliptic curve. Adv. Theor. Math. Phys. 2, 443 (1998) 10. Brunner, I., Herbst, M., Lerche, W., Walcher, J.: Matrix factorizations and mirror symmetry: The cubic curve. JHEP 0502, 001 (2005) 11. Witten, E.: Chern-Simons gauge theory as a string theory. Prog. Math. 133, 637 (1995) 12. Brunner, I., Douglas, M.R., Lawrence, A.E., Romelsberger, C.: D-branes on the quintic. JHEP 0008, 015 (2000) 13. Recknagel, A., Schomerus, V.: D-branes in Gepner models. Nucl. Phys. B 531, 185 (1998) 14. Brunner, I., Hori, K., Hosomichi, K., Walcher, J.: Orientifolds of Gepner models. JHEP 0702, 001 (2007) 15. Hori, K., Walcher, J.: F-term equations near Gepner points. HEP 0501, 008 (2005)
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16. Hori, K., Walcher, J.: D-branes from matrix factorizations. Talk at Strings ’04, June 28–July 2 2004, Paris. In: Comptes Rendus Physique 5, 1061 (2004) 17. Kapustin, A., Li, Y.: D-branes in Landau-Ginzburg models and algebraic geometry. JHEP 0312, 005 (2003) 18. Brunner, I., Herbst, M., Lerche, W., Scheuner, B.: Landau-Ginzburg realization of open string TFT. JHEP 0611, 043 (2006) 19. Hori, K., et al.: Mirror Symmetry Clay Mathematics Monographs, Vol. 1, Providence, RI: Amer. Math Soc., (2003) 20. Vafa, C.: Extending mirror conjecture to Calabi-Yau with bundles. http://arxiv.org/list/hepth/9804131, 1998 21. Ooguri, H., Vafa, C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419 (2000) 22. Kachru, S., Katz, S., Lawrence, A.E., McGreevy, J.: Open string instantons and superpotentials. Phys. Rev. D 62, 026001 (2000) 23. Govindarajan, S., Jayaraman, T., Sarkar, T.: Disc instantons in linear sigma models. Nucl. Phys. B 646, 498 (2000) 24. Lerche, W., Mayr, P.: On N = 1 mirror symmetry for open type II strings. http://arxiv.org/list/hepth/ 0111113, 2001 25. Katz, S., Liu, C.C.: Enumerative Geometry of Stable Maps with Lagrangian Boundary Conditions and Multiple Covers of the Disc. Adv. Theor. Math. Phys. 5, 1 (2002) 26. Graber, T., Zaslow, E.: Open string Gromov-Witten invariants: Calculations and a mirror ‘theorem’. http://arxiv.org/list/hepth/0109075, 2001 27. Mayr, P.: Summing up open string instantons and N = 1 string amplitudes. http://arxiv.org/list/hepth/ 0203237, 2002 28. Liu, C.-C.M.: Moduli space of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an S 1 -Equivariant Pair. http://arxiv.org/list/math.sg/0210257, 2002 29. Solomon, J.: Intersection Theory on the Moduli Space of Holomorphic Curves with Lagrangian Boundary Conditions in the Presence of an Anti-symplectic Involution. Introduction to Ph.D. Thesis, MIT, 2006 30. Solomon, J.: Open Gromov-Witten theory. Talk at Columbia University, March 24, 2006 31. Kontsevich, M.: Enumeration of rational curves via torus actions. http://arxiv.org/list/hepth/9405035, 1994 32. Solomon, J.: Private communication 33. Ellingsrud, G., Strømme, S.: Bott’s formula and enumerative geometry. J. AMS 9, 175–193 (1996) 34. Sottile, F.: Enumerative Real Algebraic Geometry. In: Algorithmic and Quantitative Aspects of Real Algbraic Geometry. S. Basu, L. Gonzalez-Vega, eds., DIMACS series 60, Providence, RI: Amer. Math. Soc., 2003. pp. 139–180; also available online at www.math.tamu.edu/~sottile 35. Welschinger, J.-Y.: Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry. http://arxiv.org/list/math.ag/03031452, 2003 36. Hori, K., Hosomichi, K., Page, D.C., Rabadan, R., Walcher, J.: Non-perturbative orientifold transitions at the conifold. JHEP 0510, 026 (2005) 37. Walcher, J.: Unpublished 38. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994) 39. Lazaroiu, C.I.: String field theory and brane superpotentials. JHEP 0110, 018 (2001) 40. Tomasiello, A.: A-infinity structure and superpotentials. JHEP 0109, 030 (2001) 41. Walcher, J.: Stability of Landau-Ginzburg branes. J. Math. Phys. 46, 082305 (2005) 42. Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. http://arxiv.org/list/math.ag/0503632, 2005 43. Kapustin, A., Li, Y.: Topological correlators in Landau-Ginzburg models with boundaries. Adv. Theor. Math. Phys. 7, 727–749 (2004) 44. Herbst, M., Lazaroiu, C.I.: Localization and traces in open-closed topological Landau-Ginzburg models. JHEP 0505, 044 (2005) 45. Herbst, M., Lerche, W., Nemeschansky, D.: Instanton geometry and quantum A(infinity) structure on the elliptic curve. http://arxiv.org/list/hepth/0603085, 2006 46. Herbst, M., Lazaroiu, C.I., Lerche, W.: Superpotentials, A∞ relations and WDVV equations for open topological strings. JHEP 0502, 071 (2005) Communicated by N. A. Nekrasov
Commun. Math. Phys. 276, 691–725 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0351-y
Communications in
Mathematical Physics
Weight Functions and Drinfeld Currents B. Enriquez1 , S. Khoroshkin2 , S. Pakuliak2,3 1 IRMA (CNRS), 7 rue René Descartes, F-67084 Strasbourg, France. E-mail: [email protected] 2 Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia. E-mail: [email protected] 3 Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow reg., Russia.
E-mail: [email protected] Received: 12 October 2006 / Accepted: 2 May 2007 Published online: 6 October 2007 – © Springer-Verlag 2007
Abstract: A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. In representations of the quantum affine algebra it gives off-shell Bethe vectors and is used in the construction of solutions of the qKZ equations. We construct a universal weight function for each untwisted quantum affine algebra, using projections onto the intersection of Borel subalgebras of different types, and study its functional properties.
1. Introduction The first step of the nested Bethe ansatz method ([KR]) consists in the construction of certain rational functions with values in a representation of a quantum affine algebra or 2 ) these its rational or elliptic analogue. In the case of the quantum affine algebra Uq (gl rational functions, known as the off-shell Bethe vectors, have the form B(z 1 ) · · · B(z n )v, where B(u) = T12 (u) is an element of the monodromy matrix (this is a generating series for elements in the algebra) and v is a highest weight vector of a finite dimensional repre2 ). For the quantum affine algebra Uq (gl N ), the off-shell Bethe vectors sentation of Uq (gl were constructed in [KR] by an inductive procedure (the induction is over N ). These Bethe vectors were then used (under the name ‘weight functions’) in the construction of solutions of the q-difference Knizhnik-Zamolodchikov equation ([TV1, S]). Inductive procedures for the construction of Bethe vectors were also used in rational models (where the underlying symmetry algebra is a Yangian) in [BF] and [ABFR], for g = sl N ; in these cases the Bethe vectors were expressed explicitly in the quasi-classical limit or using the Drinfeld twist (see Sect. 3.5). Bethe vectors for rational models with g = o(n) (2) and sp(2k) were studied in [R]; the twisted affine case A2 was treated in [T]. Despite their complicated inductive definition, the weight functions enjoy nice properties, which do not depend on induction steps. These are coalgebraic properties, which
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relate the weight function in a tensor product of representations with weight functions in the tensor components ([TV1]). The goal of this paper is to give a direct construction of weight functions, independent of inductive procedures. For this purpose we introduce the notion of a universal weight function. This is a family of formal Laurent series with values in a quotient of the Borel subalgebra of the quantum affine algebra, satisfying certain coalgebraic properties. The action of the universal weight function on a highest weight vector defines a weight function with values in a representation of the quantum affine algebra, which enjoys coalgebraic properties as in [TV1]. It is well-known that quantum affine algebras, as well as affine Kac-Moody Lie algebras, admit two different realizations ([D2]). In the first realization, the quantum affine algebra is generated by Chevalley generators, satisfying q-analogues of the defining relations for Kac-Moody Lie algebras ([D1]). In the second realization ([D2]), generators are the components of the Drinfeld currents, and the relations are deformations of the loop algebra presentation of the affine Lie algebra. The quantum affine algebra is equipped with two coproducts (‘standard’ and ‘Drinfeld’), each of which expresses simply in the corresponding realization. Both realizations are related to weight functions: on the one hand, the weight function satisfies coalgebraic properties with respect to the standard coproduct structure; on the other hand, the notion of a highest weight vector is understood in the sense of the ‘Drinfeld currents’ presentation. Our construction of a universal weight function is based on the use of deep relations between the two realizations. This connection was done in several steps. The isomorphism between the algebra structures of both sides was proved in [D2] and [DF]. The coalgebra structures were then related in [KT2, KT3, Be, Da, DKP1]; there it was proved that the ‘standard’ and ‘Drinfeld’ coproducts are related by a twist, which occurs as a factor in the decomposition of the R-matrix for the ‘standard’ coproduct. A further description of this twist (in the spirit of the Riemann problem in complex analysis) was suggested in [ER] and developed in [EF, E2, DKP]. Each realization determines a decomposition of the algebra as the product of two opposite Borel subalgebras (so there are 4 Borel subalgebras). In its turn, each Borel subalgebra decomposes as the product of its intersections with the two Borel subalgebras of the other type, and determines two projection operators which map it to these intersections. The twist is then equal to the image of the tensor of the bialgebra pairing between opposite Borel subalgebras by the tensor product of opposite projections. In this paper, we give a new proof of these results. For this, we prove a general result on twists of the double of a finite dimensional Hopf algebra A, arising from the decomposition of A as a product of coideals (Subsect. 2.1); this result has a graded analogue (Subsect. 2.2), which can be applied e.g. to quantum Kac-Moody algebras with their standard coproduct. More importantly, this result has a topological version (Subsect. 2.3). In Sect. 3, we show how this topological version implies that the Drinfeld and standard coproducts are related by the announced twist. The main result of this paper is Theorem 3. It says that the collection of images of products of Drinfeld currents by the projection defines a universal weight function. This allows to compute the weight function explicitly when g = sl2 or sl3 (see [KP1]), or when q = 1 (see identity (4.10)), using techniques of complex analysis and conformal algebras ([DK]). We give a conjecture on the general form of the universal weight function. We describe antisymmetric and functional properties of the weight functions at the formal level, using techniques of ([E1]). We also prove more precise rationality
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results for finite dimensional modules, as a consequence of the conjecture on the form of the universal weight function. The paper is organized as follows. In Sect. 2, we prove results on twists of doubles of Hopf algebras. In Sect. 3, we recall the definition of the untwisted quantum affine algebra Uq ( g ), of its coproducts, the construction of the Cartan-Weyl basis and its relation with the currents realization of Uq ( g ), following [KT2]; we also reprove the twist relation between the two (Drinfeld and standard) coproducts, using Sect. 2. In particular, we introduce Borel subalgebras of different types and the related projection operators. Their definition relies on a generalization of the convexity property of the Cartan-Weyl generators to ‘circular’ Cartan-Weyl generators, see [KT1]; their properties are proved in the Appendix. In Sect. 4, we define and construct universal weight functions, and prove the main theorem. As a corollary, we derive analytical properties of our weight functions. In Sect. 5 we identify them, in the case of Uq ( sl2 ), with the expressions familiar in the algebraic Bethe ansatz theory. 2. Twists of Doubles of Hopf Algebras 2.1. The finite dimensional case. Let A be a finite dimensional Hopf algebra, see [Sw, K] for basic definitions. Assume that A1 , A2 are subalgebras of A such that:1 (a) the map m A : A1 ⊗ A2 → A is a vector space isomorphism, (b) A1 (resp., A2 ) is a left (resp., right) coideal of A, i.e., A (A1 ) ⊂ A ⊗ A1 , A (A2 ) ⊂ A2 ⊗ A. Let Pi : A → Ai be the linear maps such that P1 (a1 a2 ) = a1 ε A (a2 ), P2 (a1 a2 ) = ε A (a1 )a2 for ai ∈ Ai . Then we have m A ◦ (P1 ⊗ P2 ) ◦ A = id A . Let D be the double of A and let R ∈ D ⊗2 be its R-matrix. Set Ri := (Pi ⊗ id)(R). The above identity, together with ( A ⊗ id)(R) = R 1,3 R 2,3 , implies that R = R1 R2 . Let us set2 B := A∗cop , B1 := (Aε1 A2 )⊥ ⊂ B, B2 := (A1 Aε2 )⊥ ⊂ B. Theorem 1 (see [ER, EF, DKP]). i) B1 , B2 are subalgebras of B; the subalgebra B1 (resp., B2 ) is a left (resp., right) coideal of B, i.e., B (B1 ) ⊂ B ⊗ B1 , B (B2 ) ⊂ B2 ⊗ B, and m B : B2 ⊗ B1 → B is a vector space isomorphism. ii) Define Pi : B → Bi by P2 (b2 b1 ) = b2 ε B (b1 ), P1 (b2 b1 ) = ε B (b2 )b1 . Then )(R), for i = 1, 2. In fact, R = (P ⊗ P )(R) ∈ A ⊗ B Ri = (id A ⊗P3−i i i i 3−i . 3−i iii) R2 is a cocycle for D, i.e., R21,2 ( D ⊗ id D )(R2 ) = R22,3 (id D ⊗ D )(R2 ). It follows that D, equipped with the coproduct R2 D (x) := R2 D (x)R2−1 , is a quasitriangular Hopf algebra (which we denote by R2 D) with R-matrix R22,1 R1 . iv) m D (Ai ⊗ Bi ) = m D (Bi ⊗ Ai ) for i = 1, 2, so Di := m D (Ai ⊗ Bi ) ⊂ D are subalgebras of D. v) Ai , Bi have the following coideal properties: R2 D (A1 ) ⊂ A1 ⊗ D1 , R2 D (B1 ) ⊂ D1 ⊗ B1 , R2 D (A2 ) ⊂ A2 ⊗ D2 , R2 D (B2 ) ⊂ D2 ⊗ B2 . vi) Di are Hopf subalgebras of R2 D. The quasitriangular Hopf algebra R2 D is isomorphic to the double of (D1 , ( R2 D )|D1 ), whose dual algebra with opposite coproduct is (D2 , ( R2 D )|D2 ). Proof. 1) For X, Y ⊂ A, set X Y := m A (X ⊗ Y ). We have (id ⊗ε A ) ◦ A = id A , which implies that A (Aε1 ) ⊂ A ⊗ Aε1 + Aε1 ⊗ 1 A . Then A (Aε1 A2 ) ⊂ A (Aε1 ) A (A2 ) ⊂ 1 If X is a Hopf algebra, we denote by m , , S , ε , 1 its operations. X X X X X 2 For X a Hopf algebra, X cop means X with opposite coproduct; if Y ⊂ X , then Y ε := Y ∩ Ker(ε ). X
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(A⊗ Aε1 + Aε1 ⊗1 A )(A2 ⊗ A) ⊂ A⊗ Aε1 A+ Aε1 A2 ⊗ A. Now A = A1 A2 , and Aε1 A1 = Aε1 , so Aε1 A = Aε1 A2 , which implies that Aε1 A2 is a two-sided coideal of A. This implies that B1 is a subalgebra of B. In the same way, B2 is a subalgebra of B. Aε1 A2 is a right ideal of A, which implies that B1 is a left coideal of B. In the same way, B2 is a right coideal of B. We now show that m B : B2 ⊗ B1 → B is a vector space isomorphism. We have vector space isomorphisms Bi A∗3−i , for i = 1, 2, induced by A = A1 ⊕ A1 Aε2 m tB
and A = A2 ⊕ Aε1 A2 . So we will prove that the transposed map A → A1 ⊗ A2 is an isomorphism. Let bi ∈ Ai∗ B3−i . When viewed as elements of B = A∗ , bi satisfy3 b1 , a1 a2 = (1) (1) b1 , a1 ε(a2 ), b2 , a1 a2 = ε(a1 ) b2 , a2 . Then4 b1 b2 , a1 a2 = b1 ⊗ b2 , a1 a2 ⊗ a1(2) a2(2) = b1 , a1(1) ε(a2(1) )ε(a1(2) ) b2 , a2(2) = b1 , a1 b2 , a2 . So the composed map mA
m tB
m tB
A1 ⊗ A2 → A → A1 ⊗ A2 is the identity, which implies that A → A1 ⊗ A2 , and therefore m B : B2 ⊗ B1 → B is an isomorphism. 2) Let Ri ∈ Ai ⊗ B3−i be the canonical element arising from the isomorphism B3−i Ai∗ . Let us show that R = R1 R2 , and Ri = Ri . We have R1 R2 ∈ A ⊗ B. For a ∈ A, let us compute R1 R2 , id ⊗a . We assume that a = a1 a2 , with ai ∈ Ai . Then (1) (1) (2) (2) R1 R2 , id ⊗a = R1 R2 , id ⊗a1 a2 = (R1 )1,2 (R2 )1,3 , id ⊗a1 a2 ⊗ a1 a2 = (1) (1) (2) (2) (R1 )1,2 (R2 )1,3 , id ⊗a1 ε A (a2 ) ⊗ ε A (a1 )a2 = a1 a2 = a. So R1 R2 ∈ A ⊗ B is the canonical element, so it is equal to R. Now Ri = (Pi ⊗ id B )(R) = Ri , since we have (ε A ⊗ id B )(Ri ) = 1 A . We also )(R) = R and (P ⊗ P )(R) = R . compute (id ⊗P3−i i i i 3−i 3) We first prove that for any a ∈ A, b ∈ B, we have a (1) , b(1) b(2) a (2) = a (1) b(1) a (2) , b(2) .
(2.1)
Recall the multiplication formula in the quantum double [D1]: b a = S D (a (1) ), b(1) a (3) , b(3) a (2) b(2) .
(2.2)
Using this equality, we may write the left-hand side of (2.1) as follows: a (1) , b(1) S D (a (2) ), b(2) a (4) , b(4) a (3) b(3) = a (1) ⊗ S D (a (2) ), B (b(1) ) a (4) , b(3) a (3) b(2) = ε(a (1) )ε(b(1) ) a (3) , b(3) a (2) b(2) = a (2) , b(2) a (1) b(1) , which coincides with the right-hand side of equality (2.1). We now prove the cocycle relation R21,2 R212,3 = R22,3 R21,23 (the coproduct is D ). Both sides of this identity belong to A2 ⊗ D ⊗ B1 . Using the pairing, we will identify them with linear maps B1 ⊗ A2 → D. Let us compute the pairing of both sides of this equality with b1 ⊗ id ⊗a2 for arbitrary b1 ∈ B1 and a2 ∈ A2 . For the left-hand side we have 3 We denote by a ⊗ b → a, b = b, a the pairing A ⊗ B → C. 4 Here and below we use Sweedler’s notation (a) = a (1) ⊗ a (2) , ( ⊗ id)((a)) = a (1) ⊗ a (2) ⊗ a (3) ,
etc.
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(1) (2) (1) (2) R21,2 R212,3 , b1 ⊗ id ⊗a2 = R2 (a2 ⊗ a2 ), b1 ⊗ id = ri a2 ⊗ ri a2 , b1 ⊗ id =
ri
(1) (2) ⊗ a2 , b1
(1) (2) ⊗ b1 ri a2
=
i (1) (1) (2) (2) a2 , b1 b1 a2 .
i
Here R2 =
i ri
⊗ ri . On the other hand, for the right-hand side we obtain (1)
(2)
R22,3 R21,23 , b1 ⊗ id ⊗a2 = R2 (b1 ⊗ b1 ), id ⊗a2 = =
(1) ri b1 ri
(2) (1) ⊗ b1 , a2
(2) ⊗ a2
=
(1) (2) ri b1 ⊗ ri b1 , id ⊗a2
i (1) (1) (2) (2) a2 b1 a2 , b1 ,
i
where we used the fact that R2 is the pairing tensor between the subalgebras A2 and B1 . The cocycle identity now follows from (2.1). Here is another proof of iii). Recall that R is invertible, and R −1 = (S D ⊗ id D )(R). It follows that R1 and R2 are invertible. Let us show that R1−1 ∈ A1 ⊗ B2 . We first show that R1−1 ∈ D ⊗ B2 . For this, we let a1 ∈ A1 , a2ε ∈ Aε2 and we compute: R1−1 , id ⊗a1 a2ε = R2 R −1 , id ⊗a1 a2ε = R21,2 (R −1 )1,3 , id ⊗(a1 a2ε )(1) ⊗ (a1 a2ε )(2) = R2 , id ⊗(a1 a2ε )(1) R −1 , id ⊗(a1 a2ε )(2) (1)
(2)
= P2 ((a1 a2ε )(1) )S A ((a1 a2ε )(2) ) = P2 (a1 )(a2ε )(1) S A ((a2ε )(2) )S A (a1 ) = 0, which proves that R1−1 ∈ D ⊗ B2 . In the same way, one proves that R1−1 ∈ A1 ⊗ D, so R1−1 ∈ A1 ⊗ B2 , and then R2−1 ∈ A2 ⊗ B1 . Let us set := R22,3 R21,23 (R21,2 R212,3 )−1 (the coproduct is D ). Then ∈ A2 ⊗ D ⊗ ¯
¯
B1 . Using the quasitriangular identities satisfied by R, we get = (R11,23 R12,3 )−1 R112,3 −1 R11,2 , where the coproduct is now 2,1 D (x) = R D (x)R . The last identity implies that ∈ A1 ⊗ D ⊗ B2 . Since A1 ∩ A2 = C1 A and B1 ∩ B2 = C1 B , we get ∈ 1 D ⊗ D ⊗1 D . The pentagon identity satisfied by then implies that = 1⊗3 D . 4) Recall that for a ∈ A, b ∈ B, we have ab = b(1) , a (1) b(3) , S A (a (3) ) b(2) a (2) , ba = b(1) , S A (a (1) ) b(3) , a (3) a (2) b(2) .
Let us set X Y := m D (X ⊗ Y ), for X, Y ⊂ D. We will show that B1 A1 ⊂ A1 B1 . Let (1) (1) (3) (3) (2) (2) 3 a (i) ∈ a1 ∈ A1 , b1 ∈ B1 . Then b1 a1 = b1 , S A (a1 ) b1 , a1 a1 b1 ; we have ⊗i=1 1 3 b(i) ∈ B ⊗2 ⊗ B , and since b, a = ε (b)ε (a) for a ∈ A , A⊗2 ⊗ A1 and ⊗i=1 1 B A 1 1 (1) (1) (2) (2) (1) (2) b ∈ B1 , we have b1 a1 = b1 , S A (a1 ) a1 b1 ; now b1 ⊗ b1 ∈ B ⊗ B1 and a1(1) ⊗ a1(2) ∈ A ⊗ A1 , which implies that b1 a1 ∈ A1 B1 , as wanted. (One proves in the same way that B1 A1 ⊂ A1 B1 .) It follows that D1 := A1 B1 is a subalgebra of D. In the same way, one shows that D2 := A2 B2 is a subalgebra of D. Note that as above, for a1 ∈ A1 , b1 ∈ B1 , we have (1)
(1)
(3)
(3)
(2) (2)
(1)
(1)
(2) (2)
a1 b1 = b1 , a1 b1 , S A (a1 ) b1 a1 = b1 , a1 b1 a1 , so that B1 A1 = A1 B1 A1 ⊗ B1 . In the same way, B2 A2 = A2 B2 A2 ⊗ B2 .
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5) We have R2 D (A1 ) = R2 D (A1 )R2−1 ⊂ R2 (A ⊗ A1 )R2−1 ⊂ A ⊗ D1 , since −1 R2±1 ∈ A2 ⊗ B1 . On the other hand, R2 D (A1 ) = R1−1 2,1 D (A1 )R1 ⊂ R1 (A1 ⊗ ±1 R 2 A)R1 ⊂ A1 ⊗ D, since R1 ∈ A1 ⊗ B2 . Finally, D (A1 ) ⊂ A1 ⊗ D1 . The other inclusions are proved similarly. 6), 4) and 5) imply that Di are Hopf subalgebras of D. We have now R22,1 R1 ∈ D1 ⊗ D2 . It is a nondegenerate tensor, as it is inverse to the pairing D1 ⊗ D2 B1 A1 ⊗ A2 B2 (A1 ⊗ B1 ) ⊗ (A2 ⊗ B2 ) → C, given by the tensor product of the natural pairings A1 ⊗ B2 → C, A2 ⊗ B1 → C. Let us prove that m D : D1 ⊗ D2 → D is a vector space isomorphism. The map A1 ⊗ B1 ⊗ A2 ⊗ B2 A1 B1 ⊗ A2 B2 = D1 ⊗ D2 → D A ⊗ B is given by (1) (1) (3) (3) (2) (2) a1 ⊗ b1 ⊗ a2 ⊗ b2 → b1 , a2 b1 , a2 a1 a2 ⊗ b1 b2 . One checks that the inverse map is given by A1 ⊗ A2 ⊗ B1 ⊗ B2 A ⊗ B → D1 ⊗ D2 using the same formula, replacing D by R2 D . The statement is now a consequence of the following fact: let (H, R H ) be a quasitriangular Hopf algebra, and Hi , i = 1, 2 be Hopf subalgebras, such that R H ∈ H1 ⊗ H2 is nondegenerate and m H : H1 ⊗ H2 → H is a vector space isomorphism, then ∗cop H2 = H1 and H is the double of H1 (indeed, since R H is nondegenerate, it sets up a vector space isomorphism H1 H2∗ , and since it satisfies the quasitriangularity ∗,cop equations, this is an isomorphism H1 H2 of Hopf algebras; we are then in the situation of the theorem of [D1] on doubles).
2.2. The graded case. In the case when A is a Hopf algebra in the category of N-graded vector spaces with finite dimensional components, the results of the previous section can be generalized as follows. Let (αi j )1≤i, j≤r be a nondegenerate matrix, let A be a N-graded braided Hopf algebra, with finite dimensional components and A [0] C, where the braiding is defined by (q αi j )1≤i, j≤r . Let A := A ⊗ C[Zr ] be the corresponding Hopf algebra. Let B be the graded dual to A and B be the corresponding Hopf algebra. We then have a nondegenerate Hopf pairing A ⊗ B → C. Let D be the quotient of the bicrossproduct of A and B by the diagonal inclusion of C[Zr ]. To explain in what space R lies, we introduce the following notion. If V = ⊕n∈Z V [n] ¯ ¯ := n 1 ,...,n k ∈Z V [n 1 ]⊗· · ·⊗V [n k ], let V ⊗> k ⊂ V ⊗k is a Z-graded vector space, set V ⊗k be the set of all combinations v1 ⊗ · · · ⊗ vk , such that there exists a constant c1 and functions c2 (n 1 ), …, ck (n 1 , . . . , n k−1 ), such that deg(v1 ) ≥ c1 , deg(v2 ) ≥ c2 (deg(v1 )), …, deg(vk ) ≥ ck (deg(v1 ), . . . , deg(vk−1 )). Define R1 ∈ A1 ⊗> B2 as the tensor of the restriction of −, − to A1 ⊗ B2 , R2 ∈ A2 ⊗> B1 as the tensor of the restriction of −, − to A2 ⊗ B1 and R0 as the tensor of the restriction of −, − to C[Z]⊗2 . Then R0 = q r0 , where r0 is inverse to the matrix (αi j ), and the tensor of the restriction of −, − to A2 ⊗ B1 is R2 := R2 R0 (it belongs to a suitable extension of A2 ⊗> B1 ). The R-matrix of D is then R = R1 R2 (also in a suitable extension of A ⊗> B). In Theorem 1, i) is unchanged; ii) is unchanged, with the addition that Bi are now graded subalgebras of B; in iii), the cocycle identity holds in D ⊗> 3 and the next statement is that R2 D defines a topological bialgebra structure, i.e., we have an algebra morphism R2 ⊗> 2 and coassociativity is an identity of maps D → D ⊗> 3 ; iv) is D : D → D unchanged; v) has to be understood in the topological sense; and vi) has to be replaced by the statement that Di are topological subbialgebras of R2 D.
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Example. One may take A = Uq (b+ ), where b+ is the Borel subalgebra of a Kac-Moody Lie algebra, equipped with the principal grading. In some cases, the twisted bialgebra R2 D is an ordinary Hopf algebra, i.e., R2 : D → D ⊗2 . 2.3. The topological case. If V = ⊕n∈Z V [n] is a Z-graded vector space, let V ⊗< k be the image of V ⊗> k by v1 ⊗ · · · ⊗ vk → vk ⊗ · · · ⊗ v1 . We define (V ⊗< k )[n] as the part of V ⊗< k of total degree n and (V ⊗< k ) f s := ⊕n∈Z (V ⊗< k )[n] the ‘finite support’ part of V ⊗< k . We define (V ⊗> k ) f s similarly. Then if V is a Z-graded algebra, we have algebra inclusions V ⊗k ⊂ (V ⊗< k ) f s ⊂ V ⊗< k and V ⊗k ⊂ (V ⊗> k ) f s ⊂ V ⊗> k . We will make the following assumptions. (H1) D is a Z-graded topological bialgebra. Here topological means that the coproduct is an algebra morphism D : D → (D ⊗< 2 ) f s of degree 0 (then the coassociativity is an equality of maps D → (D ⊗< 3 ) f s ). (H2) D ⊃ A, B ⊃ C[Zr ], where A, B are Z-graded topological subbialgebras of D and C[Zr ] ⊂ D0 is equipped with its standard bialgebra structure. We assume that the product map yields an isomorphism of vector spaces A ⊗C[Zr ] B → D, and that we have a nondegenerate bialgebra pairing −, − : A ⊗ B cop → C of degree 0; this means that a1 a2 , b = a1 ⊗ a2 , b(2) ⊗ b(1) , a, b1 b2 = a (1) ⊗ a (2) , b1 ⊗ b2 , (2.3) ai ∈ Ai , bi ∈ Bi , and a, 1 D = ε D (a), 1 D , b = ε D (b); we further require that the identity (2.1) a (1) , b(1) b(2) a (2) = a (1) b(1) a (2) , b(2) holds. (One checks that all the sums involved in these identities are finite.) (H3) A ⊃ A1 , A2 , where Ai are Z-graded subalgebras of A, such that the nontrivial components of A1 (resp., A2 ) are in degrees ≥ 0 (resp., ≤ 0), and the product map A1 ⊗ A2 → A is a linear isomorphism. B ⊃ B1 , B2 , where Bi are Z-graded subalgebras of B, such that the nontrivial components of B1 (resp., B2 ) are in degrees ≥ 0 (resp., ≤ 0) and the product map is an isomorphism B2 ⊗ B1 → B. (H4) Moreover, D (A1 ) ⊂ A ⊗< A1 , D (A2 ) ⊂ A2 ⊗< A, D (B1 ) ⊂ B ⊗< B1 , D (B2 ) ⊂ B2 ⊗< B.
(2.4)
If we define −, − i as the restriction of −, − to Ai ⊗ B3−i → C (i = 1, 2), then this assumption is equivalent to the identity a1 a2 , b2 b1 = a1 , b2 1 a2 , b1 2 ,
(2.5)
where ai ∈ Ai , bi ∈ Bi (i = 1, 2). (H5) The degree zero components are A2 [0] = B1 [0] = C[Zr ], and A1 [0] = B2 [0] = C. We also assume that A2 , B1 contain graded subalgebras A2 , B1 , such that the product induces linear isomorphims A2 ⊗ C[Zr ] A2 , B1 ⊗ C[Zr ] B1 (so A2 [0] = B1 [0] = C). We assume that the homogeneous components of A1 , A2 , B1 , B2 are finite dimensional.
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(H6) Let us denote by −, − 0 the restriction of −, − to A2 [0]⊗B1 [0] C[Zr ]⊗2 → C; we assume that it has the form δi , δ j 0 = q αi j , q ∈ C× and the matrix (αi j ) is nondegenerate (here δi is the i th basis vector of Zr ). Let us denote by −, − 2 the restriction of −, − 2 to A2 ⊗ B1 → C. We assume that a2 a0 , b1 b0 2 = a2 , b1 2 a0 , b0 0 , where a2 ∈ A2 , b1 ∈ B2 , a0 ∈ A2 [0] and b0 ∈ B1 [0]. We assume that the pairings −, − 1 and −, − 2 are non-degenerate, in the sense that each pairing between each pair of finite-dimensional homogeneous components of opposite degrees is nondegenerate. Define R1 ∈ A1 ⊗> B2 as the tensor of the pairing −, − 1 , R2 ∈ A2 ⊗< B1 as the tensor of −, − 2 and R0 as the tensor of −, − 0 . Then R0 = q r0 , where r0 is inverse to the matrix (αi j ), and the tensor of the −, − 2 is R2 := R2 R0 ∈ A2 ⊗< B1 . Actually, R1 , R2 have degree 0, so we have R1 ∈ (A1 ⊗> B2 ) f s , R2 ∈ (A2 ⊗< B1 ) f s . Since R1 has the form 1+ i>0 ai ⊗bi , where deg(ai ) = − deg(bi ) = i, R1 is invertible in A1 ⊗> B2 . In the same way, R2 is invertible. Lemma 2.1. R2 is a cocycle for D , i.e., the identity R21,2 ( D ⊗ id)(R2 ) = R22,3 (id ⊗ D )(R2 ) holds in D ⊗< 3 . In the same way, R1−1 is a cocycle for 2,1 D . Proof. The proof for R2 is the same as the first proof of 3) in Theorem 1. In the case 1,2 2,1 2,3 of R1−1 , we similarly prove that (2,1 D ⊗ id)(R1 )R1 = (id ⊗ D )(R1 )R1 and take inverses. ˜ :D→ We therefore obtain topological bialgebra structures : D → D ⊗< 2 and defined by
D ⊗> 2 ,
˜ = R1−1 D (x)2,1 R1 . (x) := R2 D (x)R2−1 , (x) We now prove: ˜ actually take their values in D ⊗2 , and are equal as maps D → Theorem 2. and D ⊗2 . Let us first briefly summarize the proof. For x ∈ A, (x) ∈ A ⊗< D while ˜ (x) ∈ A ⊗> D. We pair both elements with b ⊗ id, where b ∈ B. Then (x), ˆ of the same convolution algebra ˜ (x) define elements of two completions Hom± (B, D) ⊕(i, j)∈Z2 Hom(Bi , D j ). Using identities in these convolution algebras, and computing degrees carefully to prove that certain maps, a priori valued in D ⊗> 3 or D ⊗< 3 , take in fact their values in D ⊗3 , we prove identity (2.9), which implies that the pairings of (x) ˜ ˜ and (x) with b ⊗ id are the same. This implies (x) = (x) ∈ D ⊗2 ; the proof with x replaced by y ∈ B is similar. Proof. We will consider the convolution algebra ⊕(i, j)∈Z2 Hom(Bi , D j ), where the product is ( f 1 ∗ f 2 )(b) := f 1 (b(2) ) f 2 (b(1) ). This is an associative algebra with identity element 1∗ : b → ε(b)1 D . This algebra is bigraded by Z2 . The convolution product can be ˆ = (i, j)∈Z2 Hom(Bi , D j ). extended as follows. Let Dˆ := i∈Z Di , then Hom(B, D)
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ˆ we define the support of f as supp( f ) := {(i, j)| f i, j = f i, j ∈ Hom(B, D), ˆ are such that the sum map supp( f 1 ) × supp( f 2 ) → 0}. Then if f 1 , f 2 ∈ Hom(B, D) ˆ is defisupp( f 1 ) + supp( f 2 ) has finite fibers, then the convolution f 1 ∗ f 2 ∈ Hom(B, D) ned, and has support contained in supp( f 1 ) + supp( f 2 ). One checks that the convolution ˆ is associative in the restricted sense that if S1 × S2 × S3 → S1 + S2 + S3 in Hom(B, D) has finite fibers, where Si := supp( f i ), then ( f 1 ∗ f 2 ) ∗ f 3 = f 1 ∗ ( f 2 ∗ f 3 ). ˆ (resp., Hom− (B, D)) ˆ as the subset of In particular, we define Hom+ (B, D) ˆ of all the elements f such that supp( f ) is contained is some part of Z2 Hom(B, D) of the form S + N(1, 1) (resp., S + N(−1, −1)),5 where S is a finite subset of Z2 . Then ˆ are both algebras for the convolution. Hom± (B, D) ˆ and (A ⊗> One checks that one has algebra injections (A ⊗< D) f s → Hom+ (B, D) ˆ ˆ D) f s → Hom− (B, D) (denoted x → [x]), extending the map A ⊗ D → Hom(B, D), ˆ a⊗d → (b → b, a d). The intersection of (A⊗< D) f s and (A⊗> D) f s in Hom(B, D) is A ⊗ D. It follows from (2.5) that the images of R1 ∈ (A ⊗> D) f s and R2 ∈ (A ⊗< D) f s ˆ respectively coincide with P2 and P1 , where P2 (b2 b1 ) = b2 ε(b1 ) and in Hom(B, D) P1 (b2 b1 ) = ε(b2 )b1 . ˆ and g := [R −1 ] ∈ Hom+ (B, D). ˆ Then f ∗ Let f := [R1−1 ] ∈ Hom− (B, D) 2 P2 and P2 ∗ f are defined, and f ∗ P2 = P2 ∗ f = 1∗ . This means that for any b ∈ B, f (b(2) )P2 (b(1) ) = P2 (b(2) ) f (b(1) ) = ε(b). For b = b2 b1 , this means that (2) (2) (1) (1) (2) (1) f (b2 b1 )b2 P2 (b1 ) = ε(b) and P2 (b2 ) f (b2 b1 ) = ε(b). The first equality says (2) (2) in particular (setting b1 = 1) that f (b2 )P2 (b2 ) = ε(b2 ); plugging this in the second equality, we get f (b2 )ε(b1 ) = f (b2(2) )ε(b2(1) b1 ) = f (b2(3) )P2 (b2(2) ) f (b2(1) b1 ) = (2) (1) ε(b2 ) f (b2 b1 ) = f (b2 b1 ), so If f =
i, j
f (b2 b1 ) = f (b2 )ε(b1 ). In addition, we have (2)
(1)
(2)
(1)
f (b2 )b2 = P2 (b2 ) f (b2 ) = ε(b2 ). These identities imply f (b(2) )b(1) = P1 (b).
(2.6)
In the same way, one shows that (2)
(1)
(2)
(1)
g(b2 b1 ) = ε(b2 )g(b1 ), g(b1 )P1 (b1 ) = b1 g(b1 ) = ε(b1 ), where all the a priori infinite sums reduce to finite sums. Then we compute (x), b ⊗ id = (P1 ∗ [(x)] ∗ g)(b) = P1 (b(3) )[ D (x)](b(2) )g(b(1) ) = P1 (b2(3) b1(3) ) x (1) , b2(2) b1(2) x (2) g(b2(1) b1(1) ) = P1 (b2(2) )b1(3) x (1) , b2(1) b1(2) x (2) g(b1(1) ) (2)
(3)
(1)
(2)
(1)
= P1 (b2 )b1 x (1) , b2 x (2) , b1 x (3) g(b1 ), 5 N(a, b) = {(0, 0), (a, b), (2a, 2b), . . .}
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(1)
where the fourth equality uses b1 ∈ B1 , b2 ∈ B2 , and the last equality uses the bialgebra pairing rules (2.3), which do not introduce infinite sums. We finally get (2)
(1)
(3)
(2)
(1)
(x), b2 b1 ⊗ id = P1 (b2 ) x (1) , b2 · b1 x (2) , b1 x (3) g(b1 ), and similarly (3)
(2)
(1)
(2)
(1)
˜ (x), b2 b1 ⊗ id = f (b2 ) x (2) , b2 x (1) b2 · x (3) , b1 P2 (b1 ). We now prove that (2)
(1)
(3)
(2)
(1)
P1 (b2 ) x (1) , b2 x (2) = f (b2 ) x (2) , b2 x (1) b2 .
(2.7)
We first show that both sides are finite sums. Let x and b2 be of fixed degree (we denote (1) by |x| the degree of x ∈ D). Then for some constant c, we have |b2 |, |x (1) | ≤ c, (2) (1) |b2 | = |b2 | − |b2 |, |x (2) | = |x| − |x (1) |, so the l.h.s. reduces to the sum of contribu(1) tions with |b2 |, |x (1) | ∈ {−c, . . . , c − 1, c} and is a finite sum. Let us show that the (1) r.h.s. is a finite sum. For some constant c and function c (n), we have |b2 |, |x (1) | ≤ c , (2) (1) (3) (1) (2) |b2 | ≤ c (|b2 |), |x (2) | = |x| − |x (1) | and |b2 | = |b2 | − |b2 | − |b2 |. The nontrivial (2) (3) contributions are for |x (2) | + |b2 | = 0 and |b2 | ≤ 0 (as supp( f ) ⊂ N(−1, −1)). These (1) (2) (2) conditions impose |b2 | + |b2 | ≥ 0, so |b2 | ≥ −c , which leaves only finitely many (2) (1) (2) possibilities for |b2 |; then |b2 | ≥ −|b2 |, which leaves only finitely many possibi(1) (2) (3) lities for (|b2 |, |b2 |, |b2 |). This also implies that only finitely many (|x (1) |, |x (2) |) contribute. We have therefore proven that there are linear maps F, G : B2 ⊗ A → D, (2) (1) (3) (2) (1) F(x ⊗ b2 ) := P1 (b2 ) x (1) , b2 x (2) and G(x ⊗ b2 ) := f (b2 ) x (2) , b2 x (1) b2 . Similarly, one proves that there exists a unique linear map f : B2 ⊗ A → D, such (2)
(1)
that f(b2 ⊗ x) = P1 (b2 ) x, b2 . Then the image of the composed map B2 ⊗ A
id ⊗ D
→
f⊗inc A⊗< 2 →
B2 ⊗ D ⊗< 2 (inc is the canoncial inclusion) is contained in D ⊗2 , so its composition with m D : D ⊗2 → D is well-defined. Then F = m D ◦(f ⊗inc)◦(id ⊗ D ), and (2.7) is expressed as F = G. One checks that there are maps u, v : A ⊗ B → D, such that u(a ⊗ b) := a (1) , b(1) b(2) a (2) and v(a ⊗ b) := a (2) , b(2) a (1) b(1) ; (2.1) can then be expressed by the equality u = v. As above, one checks that the composed map A ⊗ B
id ⊗ f
id ⊗ D
→
w⊗id
A ⊗ B ⊗< 2 → D ⊗<
B → D ⊗< 2 actually takes its values in D ⊗2 for w = u or v. This means that the equality (1)
(2)
(3)
(2)
(1)
(3)
x (1) , b2 b2 x (2) ⊗ f (b2 ) = x (2) , b2 x (1) b2 ⊗ f (b2 ) takes place in D ⊗2 . Applying m D after transposing the factors, we get the identity in D, (3)
(1)
(2)
(3)
(2)
(1)
f (b2 ) x (1) , b2 b2 x (2) = f (b2 ) x (2) , b2 x (1) b2 , which according to (2.6) yields (2.7).
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One proves similarly that the following is an equality between finite sums b1(3) x (1) , b1(2) x (2) g(b1(1) ) = x (1) x (2) , b1(2) P2 (b1(1) ), which one expresses as F = G , where F , G : A⊗ B → D are given by F (x ⊗b2 ) := (3) (2) (1) (2) (1) b1 x (1) , b1 x (2) g(b1 ) and G (x ⊗ b2 ) := x (1) x (2) , b1 P2 (b1 ). As before, there exists a unique linear map g : A ⊗ B1 → D, such that g(x ⊗ b1 ) = (2)
inc ⊗g
D ⊗id
(1)
x, b1 P2 (b1 ). Then the image of the composed map A ⊗ B1 → A⊗< 2 ⊗ B1 → D ⊗< 2 is contained in D ⊗2 , so its composition with m D : D ⊗2 → D is well-defined, and G = m D ◦ (inc ⊗g) ◦ ( D ⊗ id). (2)
We then consider the composed map B ⊗ A ⊗ B (2) D ⊗< 3 (where D = ( D ⊗ id) ◦ D ); this map is (2)
id ⊗ D ⊗id
→
(1)
B ⊗ A⊗< 3 ⊗ B (2)
f⊗id ⊗g
(1)
b2 ⊗ x ⊗ b1 → P1 (b2 ) x (1) , b2 ⊗ x (2) ⊗ x (3) , b1 P2 (b1 ) (1)
→
(2.8) (1)
and actually takes its values in D ⊗3 , since for |x| and |b2 | fixed, we have |b1 |, |b2 |, |x (1) | ≤ c for some c, and the nontrivial contributions are for |b2(1) | = −|x (1) | ≥ −c, which (1) (2) leaves only finitely many possibilities for (|b2 |, |b2 |), and therefore also for |x (1) |. Now for each such |x (1) |, we have |x (2) | ≤ c (|x (1) |) for some function c (n), and |x (3) | = |x| − |x (1) | − |x (2) | ≥ |x| − c − c (|x (1) |). On the other hand, |b1(2) | = (1) (2) |b1 | − |b1 | ≥ |b1 | − c. Since we must have |x (3) | + |b1 | = 0, this leaves only (2) finitely many possibilities for (|b1 |, |x (3) |). Finally, we have finitely many possibilities for (|b1(1) |, |b1(2) |) and for (|x (1) |, |x (2) |), hence for (|x (1) |, |x (2) |, |x (3) |). So the r.h.s. of (2.8) is a finite sum and belongs to D ⊗3 . We then consider the map (2)
(2)
m D ◦ (f ⊗ id ⊗g) ◦ (id ⊗ D ⊗ id)) : B ⊗ A ⊗ B → D (2)
(where m D = (m D ⊗ id) ◦ m D ). On one hand, we have (2) m (2) D ◦ (f ⊗ id ⊗g) ◦ (id ⊗ D ⊗ id)) = m D ◦ (F ⊗ g) ◦ (id ⊗ D ⊗ id) = m D ◦ (G ⊗ g) ◦ (id ⊗ D ⊗ id);
on the other hand, we have (2)
(2)
m D ◦ (f ⊗ id ⊗g) ◦ (id ⊗ D ⊗ id)) = m D ◦ (f ⊗ G ) ◦ (id ⊗ D ⊗ id) = m D ◦ (f ⊗ F ) ◦ (id ⊗ D ⊗ id); so m D ◦ (G ⊗ g) ◦ (id ⊗ D ⊗ id) = m D ◦ (f ⊗ F ) ◦ (id ⊗ D ⊗ id). Explicitly, (2) (1) (2) (1) P1 (b2 ) x (1) , b2 · x (2) · x (3) , b1 P2 (b1 ) (2)
(1)
(3)
(2)
(1)
= P1 (b2 ) x (1) , b2 · b1 x (2) , b1 x (3) g(b1 )
(2.9)
702
and
B. Enriquez, S. Khoroshkin, S. Pakuliak
(2) (1) (2) (1) P1 (b2 ) x (1) , b2 · x (2) · x (3) , b1 P2 (b1 ) (3)
(2)
(1)
(2)
(1)
= f (b2 ) x (2) , b2 x (1) b2 · x (3) , b1 P2 (b1 ), so (2.9) is rewritten as (2)
(1)
(3)
(2)
(1)
P1 (b2 ) x (1) , b2 · b1 x (2) , b1 x (3) g(b1 ) (3)
(2)
(1)
(2)
(1)
= f (b2 ) x (2) , b2 x (1) b2 · x (3) , b1 P2 (b1 ), ˜ ˜ i.e., (x), b2 b1 ⊗ id = (x), b2 b1 ⊗ id . This means that [(x)] = [(x)]. Since ˆ the intersection of (A ⊗< D) f s and (A ⊗> D) f s in Hom(B, D) is A ⊗ D, we get ˜ (x) = (x) ∈ A ⊗ D. ˜ One proves similarly that for y ∈ B, (y) = (y) ∈ D ⊗ B by using the convolution algebra (i, j)∈Z2 Hom(Ai , D j ), where the product is given by ( f 1 ∗ f 2 )(a) = f 1 (a (1) ) f 2 (a (2) ). Proposition 2.2. defines a (nontopological) bialgebra structure on D, quasitriangular with R-matrix R22,1 R1 ∈ D ⊗> 2 (the quasitriangular identities are satisfied in D ⊗> 3 ). Proof. Let us prove that for x ∈ D, R22,1 R1 (x) = 2,1 (x)R22,1 R1
(2.10)
⊗> 2 ) so the l.h.s. is ˜ (equality in D ⊗> 2 ). We have (x) = (x) = R1−1 2,1 D (x)R1 (∈ D 2,1 2,1 ⊗ 2 equal to R2 D (x)R1 (∈ D > ). 2,1 −1 On the other hand, 2,1 (x) = R22,1 2,1 (∈ D ⊗> 2 ) so the r.h.s. is equal D (x)(R2 ) 2,1 2,1 ⊗ 2 to R2 D (x)R1 (∈ D > ). This proves (2.10). We now prove the quasitriangular identity
( ⊗ id)(R) = R 1,3 R 2,3
(2.11)
in D ⊗> 3 . Recall that R21,2 ( D ⊗ id)(R2 ) = R22,3 (id ⊗ D )(R2 ) (equality in D ⊗< 3 ), which gives by applying the transposition x → x 3,2,1 , 2,1 2,1 2,1 2,1 R23,2 (id ⊗2,1 D )(R2 ) = R2 ( D ⊗ id)(R2 )
(2.12)
(equality in D ⊗> 3 ). On the other hand, recall that 1,2 2,1 2,3 (2,1 D ⊗ id)(R1 )R1 = (id ⊗ D )(R1 )R1
(2.13)
(equality in D ⊗> 3 ). Taking the product of (2.12) written in opposite order with (2.13), we get 1,2 3,2 2,1 2,3 R22,1 (2,1 D ⊗ id)(R)R1 = R2 (id ⊗ D )(R)R1 . 1,2 ⊗> 2 ). The Equality (2.11) then follows from R11,2 (x) = 2,1 D (x)R1 (equality in D 1,3 1,2 proof of the identity (id ⊗)(R) = R R is similar.
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703
Proposition 2.3. D1 := A1 B1 , D2 := A2 B2 are subbialgebras of D. We also have (A1 ) ⊂ A1 ⊗ D1 , (B1 ) ⊂ D1 ⊗ B1 , (A2 ) ⊂ A2 ⊗ D2 , (B2 ) ⊂ D2 ⊗ B2 . (2.14) Proof. We first prove that B1 A1 ⊂ A1 B1 . Set R(a, b) := a (1) , b(1) b(2) a (2) − a (2) , b(2) a (1) b(1) and S(a, b) := ba − S D (a (1) ), b(2) a (3) , b(3) a (2) b(2) . We have R(a, b) = a (1) , b(1) S(a (2) , b(2) ) and S(a, b) = a (1) , S D (b(1) ) R(a (2) , b(2) ) so since R(a, b) = 0, we get S(a, b) = 0. Therefore ba = S D (a (1) ), b(2) a (3) , b(3) a (2) b(2) . If now a ∈ A1 , b ∈ B1 , we get a (1) ⊗ a (2) ⊗ a (3) ∈ A⊗< 2 ⊗< A1 and b(1) ⊗ b(2) ⊗ b(3) ∈ B ⊗< 2 ⊗< B1 . For x ∈ A1 , y ∈ B1 , we have x, y = ε(x)ε(y), so ba = S D (a (1) ), b(1) ε(a (3) )ε(b(3) )a (2) b(2) = S D (a (1) ), b(1) a (2) b(2) . Now since D (a1 ) ∈ A ⊗< A1 and D (b1 ) ∈ B ⊗< B1 , we get ba ∈ A1 B1 , as wanted. This implies that D1 := A1 B1 is a subalgebra of D. In the same way, we prove that D2 is a subalgebra of D. Let us prove that (A1 ) ⊂ A1 ⊗ D1 . For x ∈ A1 , (x) = R2 D (x)R2−1 , and since ˜ D1 is an algebra, (x) ∈ A ⊗< D1 . On the other hand, (x) = R1−1 2,1 D (x)R1 ∈ ˜ A1 ⊗> D. Since (x) = (x) ∈ A ⊗ D and (A ⊗ D) ∩ (A ⊗< D1 ) = A ⊗ D1 , we get (x) ∈ A⊗ D1 . In the same way, (x) ∈ A1 ⊗ D. Then (A⊗ D1 )∩(A1 ⊗ D) = A1 ⊗ D1 , which implies (x) ∈ A1 ⊗ D1 , as wanted. The other inclusions (2.14) are proved in the same way. Since Di are generated by Ai , Bi , these inclusions imply that Di are also subbialgebras of (D, ). We will show that the quantum affine algebras, equipped with their currents coproducts, are examples of the situation of Subsect. 2.3. 3. Quantum Affine Algebra Uq ( g) In this paper, q is a complex number, which is neither 0 nor a root of unity. 3.1. Chevalley-type presentation of Uq ( g ). Let g be a simple Lie algebra; let r be its rank and let (bi, j )i, j=1,...,r be its Cartan matrix. Let (ai, j )i, j=0,...,r be the Cartan matrix of the affine Lie algebra g. We denote by = {α1 , . . . , αr } the set of positive simple roots = {α0 , α1 , . . . , αr } the set of positive simple roots of of g and by g. The symmetrized Cartan matrix of g is ((αi , α j ))i, j=0,...,r ; we have (αi , α j ) = di ai, j = d j a j,i (where di = 1, 2 or 3 are coprime). Let δ be the minimal positive imaginary root of g, so n r [n]q ! δ = i=0 n i αi , n i ∈ Z≥0 , n 0 = 1. Let k q = [k]q ![n−k]q ! , [n]q ! = [1]q [2]q · · · [n]q , −n
(α,α)
−q 2 , qi = qα = q di . [n]q = qq−q −1 , qα = q i The quantum (untwisted) affine Lie algebra Uq ( g ) is generated by the Chevalley generators e±αi , kα±1 (i = 0, . . . , r ), the grading elements q ±d , and the central elements i n
±1/2
kδ
, subject to the relations
704
B. Enriquez, S. Khoroshkin, S. Pakuliak ±ai j
[q d , kαi ] = [kαi , kα j ] = 0, q d e±αi q −d = q ±δi,0 e±αi , kαi e±α j kα−1 = qi i ±1/2 2
(kδ
) =
r
1/2 −1/2
i, q d q −d = q −d q d = k k −1 = k −1 k = k kα±n αi αi αi αi δ kδ i
e±α j ,
−1/2 1/2 kδ = 1,
= kδ
i=0
[eαi , e−α j ] = δi j
kαi − kα−1 i
, qi − qi−1 (r ) (−1)r e±α e e(s) = 0, i = j, i ±α j ±αi
(3.15) (k) where e±α = i
r +s=1−ai, j
k e±α i
[k]qi !
.
The standard Hopf structure of Uq ( g ) is given by the formulas: ±1/2
) = kα±1 ⊗ kα±1 , std (kδ std (q ±d ) = q ±d ⊗ q ±d , std (kα±1 i i i
±1/2
) = kδ
±1/2
⊗ kδ
,
std (eαi ) = eαi ⊗ 1 + kαi ⊗ eαi , std (e−αi ) = 1 ⊗ e−αi + e−αi ⊗ kα−1 , i ±1/2
ε(q ±d ) = 1, ε(e±αi ) = 0, ε(kα±1 ) = 1, ε(kδ i
) = 1,
(3.16)
S(eαi ) = −kα−1 eαi , S(e−αi ) = −e−αi kαi , S(kα±1 ) = kα∓1 , i i i ±1/2
S(q ±d ) = q ∓d , S(kδ
∓1/2
) = kδ
,
where std , ε and S are the coproduct, counit and antipode maps respectively. Let Uq (h) be the Cartan subalgebra of Uq ( g ). It is generated by the elements kα±1 i ±d (i = 0, . . . , r ) and q . Denote by Uq (b+ ) the subalgebra of Uq ( g ) generated by the ±1/2
(i = 0, . . . , r ), kδ elements eαi , kα±1 i
and q ±d , and by Uq (b− ) the subalgebra of ±1/2
Uq ( g ) generated by the elements e−αi , kα±1 (i = 0, . . . , r ), kδ and q ±d . i The algebras Uq (b± ) are Hopf subalgebras of Uq ( g ) with respect to the standard coproduct std . They are q-deformations of the enveloping algebras of opposite Borel subalgebras of Lie algebra g. We call them the standard Borel subalgebras. Moreover, Uq (b− ) is the dual, with opposite coproduct, of Uq (b+ ), and Uq ( g) ⊗ Uq (h) is the double of Uq (b+ ) (where Uq (h) is equipped with the standard structure, for which kα±1 i is primitive) The algebras Uq (b± ) contain subalgebras Uq (n± ), which are generated by the elements e±αi , i = 0, . . . , r . The subalgebra Uq (n+ ) is a left coideal of Uq (b+ ) with respect to standard coproduct and the subalgebra Uq (n− ) is a right coideal of Uq (b− ) with respect to the same coproduct, that is std (Uq (n+ )) ⊂ Uq (b+ ) ⊗ Uq (n+ ),
std (Uq (n− )) ⊂ Uq (n− ) ⊗ Uq (b− ).
The algebras Uq (n± ) are q-deformations of the enveloping algebras of the standard pro-nilpotent subalgebras of the affine Lie algebra g. 3.2. Cartan-Weyl basis of Uq ( g). We now construct a Cartan-Weyl (CW) basis of Uq ( g ). Let us denote by + ⊂ the set of positive (resp., all) roots of g and by + ⊂ the set of positive (resp., all) roots of + ⊂ . Recall that α0 g, so + ⊂ , is the affine positive simple root and let δ be the minimal positive imaginary root, so δ = α0 + θ , where θ is the longest root of + .
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705
be the Weyl group + . Let W We consider the following class of normal orderings on of g. It contains the Weyl group W of g and the normal subgroup Q, which is the set of all elements having only finitely many conjugates. There is a unique group morphism Q → h∗ , p → p, ¯ such that the action of p ∈ Q on h∗ is the translation by p. ¯ This map is injective and identifies Q with a subgroup Q¯ of h∗ . Choose p ∈ Q such that ( p, ¯ αi ) > 0 for any i = 1, . . . , r . Choose a reduced decomposition p = sαi0 sαi1 · · · sαim−1 , such that αi0 = α0 is the affine positive root. Extend the sequence i 0 , i 1 , . . . , i m to a periodic sequence . . . , i −1 , i 0 , i 1 , . . . , i n , . . . ,
(3.17)
satisfying the conditions i n = i n+m for any n ∈ Z. We then set γ1 := αi1 , γ2 := sαi1 (αi2 ), . . . , γk := sαi1 . . . sαik−1 (αik ) for k ≥ 1; and γ0 := αi0 , γ−1 := sαi0 (αi−1 ), . . . , γ− := sαi0 . . . sαi1− (αi− ) for ≥ 0.
(3.18)
Then [Be, Da] the order γ1 ≺ γ2 ≺ . . . ≺ γn ≺ . . . ≺ δ ≺ 2δ ≺ . . . ≺ γ−n ≺ . . . ≺ γ−1 ≺ γ0 is normal and satisfies the condition lδ + α ≺ (m + 1)δ ≺ (n + 1)δ − β,
(3.19)
for any positive roots α, β ∈ + , and any l, m, n ≥ 0. From now on, we fix a normal + , given by the procedure above. ordering ≺ on → Z, such that Recall that the principal degree deg is the linear additive map Z + and deg(αi ) = 1 for i = 0, . . . , r . We first construct the CW generators eγ , for γ ∈ , eγ is equal to the corresponding Chevalley generator deg(γ ) ≤ deg(δ) − 1. For γ ∈ of Uq ( g). The CW generator eγ is then constructed by induction on the degree of γ as + and deg(γ ) ≤ deg(δ) − 1, we let [α, β] be minimal for the inclusion, follows: if γ ∈ + are such that γ = α + β (we among the set of all segments [α , β ], where α , β ∈ define the segment [α , β ] as {γ |α γ β }). We then set eγ := [eα , eβ ]q −1 ,
e−γ := [e−β , e−α ]q ,
(3.20)
where6 the q-commutator [eα , eβ ]q means [eα , eβ ]q = eα eβ − q (α,β) eβ eα . The basis elements defined in this way coincide (up to normalization) with those defined by the braid group action, using the same ordering on roots. Since the latter basis is convex, eγ defined above is independent (up to normalization) of the choice of the segment [α, β], and depends only on the ordering of roots. Moreover, it has been shown in [Da], Prop. 11 that eδ−αi is also independent of the choice of a normal ordering (up to normalization). We then put (i)
(i)
eδ = eδ
= [eαi , eδ−αi ]q −1 ,
(i)
e−δ = [eαi −δ , e−αi ]q
6 One can introduce the algebra U˜ ( ˜ q˜ −1 , 1/(q˜ n −1); n ≥ 1] with the same generators q g) over the ring C[q, and relations as Uq ( g); one can show using the CW basis that this is a free module over this ring, and Uq (g) ∗ g) is equipped with the Cartan antiinvolution x → x ∗ , defined by e±α = e∓αi , is its specialization. U˜ q (
kα∗i = kα−1 , q˜ ∗ = q˜ −1 . Then the analogues of e±γ satisfy e−γ = eγ∗ . i
i
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B. Enriquez, S. Khoroshkin, S. Pakuliak
and7 by induction for all k > 0, e±(αi +kδ) = ±
1 1 (i) (i) [e±(αi +(k−1)δ) , e±δ ], e±(δ−αi +kδ) = ± [e , e±(δ−αi +(k−1)δ) ], [2]qi [2]qi ±δ
(i)
(i)
ekδ = [eαi +(k−1)δ , eδ−αi ]q −1 , e−kδ = [e−δ+αi , e−αi −(k−1)δ ]q , (i) e±kδ
=
p1 +2 p2 +···+npn =k
(qi∓1 − qi±1 ) p j −1 ( p j − 1)! (i) p1 (i) pn e±δ · · · e±nδ . p1 ! · · · pn !
(3.21) Then we apply procedure (3.20) again to construct the remaining (real root) CW generators. As before, if γ is real, then eγ depends only on the choice of a normal ordering (up (i) to normalization); the enδ±αi and the enδ are independent of the choice of this ordering (up to normalization). The CW generators of the Borel subalgebras Uq (b± ) satisfy the following properties (see [KT2, Be, Da]): kα eβ kα−1 = q (α,β) eβ , [e±α , e±β ]q −1 =
{γ j },{n j }
[eα , e−α ] = a(α)
kα − kα−1 , q − q −1
±{γ }
nm n1 n2 C{n j } j (q) e±γ e · · · e±γ , m 1 ±γ2
α, β ∈ + , (3.22)
where in (3.22) the sum is over all γ1 , . . . , γm ∈ + , n 1 , . . . , n m > 0 such that α ≺ {γ j } γ1 ≺ γ2 ≺ · · · ≺ γm ≺ β and j n j γ j = α + β and the coefficients C {n j } (q) and
a(α) are rational functions in q in Q[q, q −1 , 1/(q n − 1); n ≥ 1]. The elements kα , are defined according to the prescriptions kα+β = kα kβ , kα = kα±1 , if α = ±αi , α∈ i g) also satisfy e−α = eα∗ . i = 0, . . . , r . The analogues of e±α in U˜ q ( + , satisfy properties The commutators [eα , e−β ], and [e−α , eβ ], where α, β ∈ ±{γ }
analogous to (3.22), but the structure coefficients C{n j } j belong to Uq (h). One can construct slightly different generators in the CW basis such that property (3.22) is still valid with scalar structure constants. For this, the normal ordering ≺ in the system
+ of positive roots is extended to a ‘circular’ normal ordering (which is not a total order) of the system of all roots of g. This order ≺c is defined by: for α, β ∈ + , (α ≺c β) ⇔ (α ≺ β) ⇔ (−α ≺c −β) and (α ≺c −β) ⇔ (β ≺ α) ⇔ (−α ≺c β). The order ≺c can be described as follows. Suppose that γ1 ≺ γ2 ≺ . . . ≺ γ N ≺ . . . is the normal ordering of + . Then we put all the roots of on the circle clockwise in the following order (see Fig. 1): γ1 , γ2 , . . . , γ N , . . . , −γ1 , −γ2 , . . . , −γ N , . . . ,
(3.23)
and say that the root γ ∈ precedes the root γ ∈ , γ ≺c γ , if the segment [γ , γ ] in the circle (3.23) does not contain any of the opposite roots −γ or −γ . (i) (i) ∗ 7 As before, the analogues of e(i) in U˜ ( q g) satisfy e−δ = (eδ ) . ±δ
Weight Functions and Drinfeld Currents
707
of Uq ( Fig. 1. The left figure shows the circle ordering of the system g). It is such that γ1 +m 1 δ ≺c m 2 δ ≺c (m 3 + 1)δ − γ2 ≺c −γ3 − m 4 δ ≺c −m 5 δ ≺c −(m 6 + 1)δ + γ4 ≺c γ5 + m 1 δ, where γi ∈ + and m i > 0. In = + (− + ), related to the subalgebras Uq (n± ). Line the right figure, line 1 shows the decomposition 2 shows the decomposition = E F , related to the currents Borel subalgebras U E and U F discussed in Sect. 3.4
+ put For any positive root γ ∈ eˆγ := eγ ,
eˆ−γ = −kγ e−γ ,
eˇ−γ := e−γ ,
eˇγ = −eγ kγ−1 .
(3.24)
, such that α ≺c β, Proposition 3.1. For any α, β ∈ [eˆα , eˆβ ]q −1 =
{γ j },{n j }
[eˇα , eˇβ ]q −1 =
{γ j },{n j }
{γ }
+ , C {n jj } (q) eˆγn11 eˆγn22 · · · eˆγnmm , if β ∈
(3.25)
{γ } + , C {n jj } (q) eˇγn11 eˇγn22 · · · eˇγnmm , if β ∈ −
(3.26)
where the sums in (3.25) and (3.26) are over all γ1 , γ2 , . . . , γm , n 1 , n 2 , . . . , n m such that α ≺c γ1 ≺c γ2 ≺c · · · ≺c γm ≺c β (meaning α ≺c γ1 , γ1 ≺c γ2 , etc.) and {γ j } {γ j } j n j γ j = α + β; C {n j } (q) and C {n j } (q) are Laurent polynomials of q. Proof. See the Appendix.
such that α ≺c β, define [α, β] := {γ ∈ |α c γ c β}. For α, β ∈ [α,β] g) ⊂ Uq ( g ) as the subalgebra generated by the eˆγ , γ ∈ [α, β], We then define Uq ( + . if β ∈ + , and as the subalgebra generated by the eˇγ , γ ∈ [α, β], if β ∈ − |α c If α, β ∈ are such that α ≺c β, we define the intervals [α, β) := {γ ∈ γ ≺c β} and (α, β] := {γ ∈ |α ≺c γ c β}. We then define the subalgebras [α,β] [α,β) (α,β] g), Uq ( g) and Uq ( g) ⊂ Uq ( g ) as above (using eˆγ or eˇγ depending on Uq ( whether β ∈ + or β ∈ −+ ). [α,β] g) admits a Poincaré-BirkhoffRelations (3.25), (3.26) imply that the algebra Uq ( Witt (PBW) basis, formed by the ordered monomials eˆγn11 eˆγn22 · · · eˆγnmm , where α = γ1 ≺c + , and by the monomials eˇγn11 eˇγn22 · · · eˇγnmm , where α = γ2 ≺c . . . ≺c γm = β, if β ∈ + . Moreover, if α, β, γ ∈ are such that γ1 ≺c γ2 ≺c . . . ≺c γm = β, if β ∈ −
708
B. Enriquez, S. Khoroshkin, S. Pakuliak
+ or β, γ ∈ − + , then the product of α ≺c β, β ≺c γ and α ≺c γ , and either β, γ ∈ Uq ( g ) induces vector space isomorphisms [β,γ ]
Uq[α,β) ( g) ⊗ Uq
[α,γ ]
( g) Uq
(β,γ ] g) ⊗ Uq ( g) Uq[α,β] (
( g),
[α,γ ] Uq ( g),
(β,γ ]
Uq[α,β] ( g) ⊗ Uq
[α,γ ]
( g) Uq
[β,γ ] Uq[α,β) ( g) ⊗ Uq ( g)
( g),
[α,γ ] Uq ( g).
(3.27)
[αi ,δ−θ]
Then Uq (n+ ) coincides with Uq+ ( g) = Uq 1 ( g), where αi1 is the first root + . The algebra Uq (n− ) coincides (necessarily simple) of the normal ordering ≺ of [−αi ,−δ+θ]
g) = Uq 1 ( g). with Uq−+ ( + = [αi1 , δ − θ ] is a disjoint union The segment + = +,e +,F .
Here +,e = {γ + mδ|γ ∈ + , m ≥ 0}, +,F = {mδ|m > 0} {−γ + mδ|γ ∈ + , m > 0}. (3.28) + = [−αi1 , −δ + θ ] is a disjoint union Analogously, − + = −,E −, f , − where −,E = −+,F , −, f = −+,e .
(3.29)
The segments (3.28) and (3.29) can be united in different ways, composing segments E and F : E = +,e −,E = {nδ, γ + mδ| γ ∈ + , n < 0, m ∈ Z}, F = +,F −, f = {nδ, −γ + mδ| γ ∈ + , n > 0, m ∈ Z}.
(3.30)
The subalgebras related to the segments (3.28), (3.29) and (3.30) play a crucial role in our further constructions. g ). In this presentation ([D2]), Uq ( g ) is gene3.3. The ‘currents’ presentation of Uq ( ±1 ±d rated by the central elements C , the grading elements q , and by the elements eα [n], f α [n] (where α ∈ , n ∈ Z) and kα±1 , h α [n] (where n ∈ Z \{0}, α ∈ ). These elements are gathered into generating functions eα [n]z −n , f α (z) = f α [n]z −n , eα (z) = n∈Z
ψα± (z) =
n>0
n∈Z
ψα± [n]z ∓n = kα±1 exp ±(qα − qα−1 )
h α [±n]z ∓n ,
n>0
such that q d q −d = q −d q d = CC −1 = C −1 C = 1 and q d a(z)q −d = a(q −1 z) for any of these generating functions, which we will call currents. Currents are labeled by the simple roots α ∈ of g.
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709
The defining relations of Uq ( g ) in the ‘currents’ presentation are (α, β ∈ ): (z − q (α,β) w)eα (z)eβ (w) = eβ (w)eα (z)(q (α,β) z − w), (z − q −(α,β) w) f α (z) f β (w) = f β (w) f α (z)(q −(α,β) z − w),
−1 q (α,β) C ±1 z − w eβ (w), = ±1 ψα± (z)eβ (w) ψα± (z) C z − q (α,β) w
−1 q −(α,β) C ∓1 z − w f β (w), = ∓1 ψα± (z) f β (w) ψα± (z) C z − q −(α,β) w ψα± (z)ψβ± (w) = ψβ± (w)ψα± (z),
(3.31)
q (α,β) z − C 2 w + q (α,β) C 2 z − w − − (z)ψ (w) = ψ ψ (w)ψα+ (z), β z − q (α,β) C 2 w α C 2 z − q (α,β) w β δα,β 2 + −1 2 − −1 δ(z/C w) ψ (C z) − δ(C z/w)ψ (C w) , [eα (z), f β (w)] = α α qα − qα−1 ni j n ij (−1)r Sym z 1 ,...,z n e±αi (z 1 )· · ·e±αi (zr )e±α j (w)e±αi (zr +1 )· · ·e±αi (z n i j ) = 0. ij r qi r =0
Here αi = α j , n i j = 1 − ai, j , δ(z) = k∈Z z k . We now describe the isomorphism of the two realizations ([D2, CP]). Suppose that the root vector eθ ∈ g, corresponding to the longest root θ ∈ + , is presented as a multiple commutator eθ = λ[eαi1 , [eαi2 , · · · [eαin , eα j ] · · · ]] for some λ ∈ C. The isomorphism is given by the assignment kα±1 → kα±1 , i i
eαi → eαi [0],
kα±1 → C ±2 kθ∓1 = C ±2 0
r
i, kα∓n i
e−αi → f αi [0], ±1/2
kδ
→ C ±1 ,
i = 1, . . . , r, q ±d → q ±d ,
(3.32)
i=1
eα0 → µSi−1 Si−2 · · · Si−n ( f α j [1]),
e−α0 → λSi+1 Si+2 · · · Si+n (eα j [−1]).
Here Si± ∈ End(Uq ( g)) are the following operators of adjoint action (with respect to the coproducts std and (std )2,1 , see (3.16)): Si+ (x) = eαi [0]x − kαi xkα−1 eαi [0], i
Si− (x) = x f αi [0] − f αi [0]kαi xkα−1 , i
(3.33)
and the constant µ is determined by the condition that relation (3.15) for i = j = 0 remains valid in the image. We now describe the inverse isomorphism. Let π : {α1 , . . . , αr } → {0, 1} be the map such that π(αi ) = π(α j ) for (αi , α j ) = 0 and π(α1 ) = 0. Proposition 3.2 (see [KT1, Be, Da]). The inverse of isomorphism (3.32) is such that (−1)nπ(αi ) eαi +nδ , n ≥ 0, eαi [n] → (−1)nπ(αi ) eˆαi +nδ = −(−1)nπ(αi ) k−αi −nδ eαi +nδ , n < 0, (3.34) (−1)nπ(αi ) e−αi +nδ , n ≤ 0, nπ(αi ) eˇ−αi +nδ = f αi [n] → (−1) −(−1)nπ(αi ) e−αi +nδ kαi −nδ , n > 0,
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ψα+i [0] → kαi ,
ψα−i [0] → kα−1 , i
±1/2
C ±1 → kδ − n2
,
(i)
ψα+i [n] → (q − q −1 )(−1)nπ(αi ) kαi kδ enδ , n > 0, − n2
(3.35)
(i)
ψα−i [n] → −(q − q −1 )(−1)nπ(αi ) kα−1 kδ enδ , n < 0. i (i)
(i)
The relation between the imaginary root generators ekδ and ekδ is given by formulas (3.21). Note that the root vectors e±αi +nδ , n ∈ Z, as well as the imaginary root generators + , satisfying the condition (3.19) do not depend on the choice of a normal ordering of ([Da], Prop. 11). So the identification of Proposition 3.2 does not depend on such a normal ordering. The ‘currents’ bialgebra structure (D) on Uq ( g ) is given by: (D) (q ±d ) = q ±d ⊗ q ±d , (D) (C ±1 ) = C ±1 ⊗ C ±1 , (D) ψα± (z) = ψα± (C2±1 z) ⊗ ψα± (C1∓1 z), (D)
eα (z) =
(D)
f α (z) =
eα (z) ⊗ 1 + ψα− (C1 z) ⊗ eα (C12 z) , 1 ⊗ f α (z) + f α (C22 z) ⊗ ψα+ (C2 z),
(3.36) (3.37) (3.38)
where C1 = C ⊗ 1 and C2 = 1 ⊗ C. The counit map is given by: ε(eα (z)) = ε( f α (z)) = 0, ε(ψα± (z)) = ε(q ±d ) = ε(C ±1 ) = 1. The principal degree on Uq ( g ) is defined by |ei [n]| = nν + 1, |ki±1 | = |C| = 0, | f i [n]| = nν − 1, |h α [n]| = nν, where ν = ri=0 n i (recall that δ = ri=0 n i αi ); then (D) : Uq ( g ) → Uq ( g )⊗< 2 is a topological bialgebra, in the sense of Sect. 2.3, (H1). g ), D = (D) . The identification with the situation of this section is D = Uq ( std (D) are related by a twist which can be The coproducts of Subsect. 3.1 and described explicitly (see Proposition 3.8 and [KT1]). 3.4. Intersections of Borel subalgebras. We first describe the Borel subalgebras related to the ‘currents’ realization of Uq ( g ). Let U F be the subalgebra of Uq ( g ) generated by Uq (h) and the elements f α [n] (where α ∈ , n ∈ Z) and h α [m] (where α ∈ , m > 0). In the circular description, this is the subalgebra of Uq ( g ) associated with the segment F (see (3.30)). Analogously, we define U E as the subalgebra of Uq ( g ) generated by Uq (h) and the elements eα [n] (where α ∈ , n ∈ Z) and h α [m] (where α ∈ , m < 0). It corresponds to the subalgebra of Uq ( g ) associated to the segment E . Both U F and U E are Hopf subalgebras of Uq ( g) with respect to the coproduct (D) . We call them the currents Borel subalgebras. We also define U f ⊂ U F as the subalgebra generated by the elements f α [n] (where α ∈ , n ∈ Z) and Ue ⊂ U E as the subalgebra generated by the elements eα [n] (where α ∈ , n ∈ Z). We have Uq ( g ) U E ⊗Uq (h) U F , where the isomorphism is induced by the product map. So D = Uq ( g ), A := U E , B := U F , C[Zr ] := Uq (h) satisfy the first part of (H2) in Sect. 2.3.
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We define U F0 ⊂ U F , U E0 ⊂ U E as the algebras with the same generators, except Uq (h). The relations (3.37) and (3.38) imply that the subalgebra U f is a right coideal of U F with respect to (D) , and the subalgebra Ue is a left coideal of U E with respect to (D) : (D) (U f ) ⊂ U f ⊗ U F ,
(D) (Ue ) ⊂ U E ⊗ Ue .
(3.39)
There is a unique bialgebra pairing −, − : U E ⊗ U F → C, expressed by eα (z), f β (w) =
δα,β δ(z/w) qα−1 − qα
, ψα− (z), ψβ+ (w) =
q (α,β) − z/w , c, d = 1 1 − q (α,β) z/w
in terms of generating series; all other pairings between generators are zero. Proposition 3.3. This pairing satisfies identity (2.1) (see (H2)). Proof. Let us set R(a, b) := a (1) b(1) a (2) , b(2) − b(2) a (2) a (1) , b(1) for a ∈ U E , b ∈ U F . One checks that R(a, b) = 0 if a, b are generators of U E , U F . Moreover, we have the identities R(aa , b) = a (1) R(a , b(1) ) a (2) , b(2) + R(a, b(2) )a (2) a (1) , b(1) , R(a, bb ) = R(a (1) , b)b(1) a (2) , b(2) + b(2) R(a (2) , b ) a (1) , b(1) . Reasoning by induction on the length of a and b (expressed as products of generators), these identities imply that R(a, b) = 0 for any a, b. Therefore −, − satisfies the hypothesis (H2). One also checks that it satisfies hypothesis (H6). We will be interested in intersections of Borel subalgebras of different types. Denote − + by U F+ , U − f , Ue and U E the following intersections of the standard and currents Borel algebras: 0,+ + 0 U− f = U F ∩ Uq (n− ), U F = U F ∩ Uq (b+ ), U F = U F ∩ Uq (n+ ),
Ue+ = U E ∩ Uq (n+ ), U E− = U E ∩ Uq (b− ), U E0,− = U E0 ∩ Uq (n− ). (3.40) The notation is such that the upper sign says in which standard Borel subalgebra Uq (b± ) the given algebra is contained and the lower letter says in which currents Borel subalgebra (U F or U E ) it is contained. This letter is capital if the subalgebra contains imaginary root generators h i [n] and the Cartan subalgebra Uq (h) and is small otherwise. In the + notation of Sect. 3.2, the subalgebras U − f and U F correspond to the segments −, f and +,F ; the subalgebras U E− and Ue+ correspond to the segments −,E and +,e . Proposition 3.4. The product in U E sets up an isomorphism of vector spaces U E
+ Ue+ ⊗ U E− . The product in U F sets up an isomorphism of vector spaces U F U − f ⊗ UF . Proof. This follows from (3.27).
We will set A1 := Ue+ , A2 := U E− , B1 := U F+ , B2 := U − f . Then the hypothesis (H3)
is satisfied. We will set B1 := U F0,+ , A2 := U E0,− , so that (H5) is satisfied. The next proposition describes a family of generators of the intersections of Borel subalgebras.
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Proposition 3.5. (i) The algebra Ue+ (resp., U − f ) is generated by the elements ei [n] (resp., f i [−n]), where i ∈ {1, . . . , r }, n ≥ 0. (ii) The algebra U E− (resp., U F+ ) is generated by Uq (h), the elements ei [n], h i [m], where n, m < 0, i ∈ {1, . . . , r }, and by the root vectors eγ −δ , γ ∈ + (resp., Uq (h), the elements f i [n], h i [m], where n, m > 0, i ∈ {1, . . . , r }, and the root vectors eδ−γ , γ ∈ + ). Proof. The PBW result shows that a basis for Ue+ is given by the ordered monomials in the eˆγ , γ ∈ +,e . These generators are expressed via those listed in the Proposition, by means of successive applications of relations (3.25) and (3.26). The proof is the same in the other cases. Note that the generators listed in the Proposition do not depend on a choice of the normal ordering, satisfying (3.19). This was already stated in Section 3.2 for all of them, except for the root vectors e±(δ−γ ) , γ ∈ + . They are constructed as successive q-commutators of the type [e±γ , e±(δ−γ ) ]q ±1 . By induction of the height of γ , one proves that these q-commutators define the same root vectors. 3.5. Relation between std and (D) . Proposition 3.6. We have (D) (Uq (b− )) ⊂ Uq (b− ) ⊗ Uq (g) and (D) (Uq (b+ )) ⊂ Uq (g) ⊗ Uq (b+ ). Proof. We prove the first statement. It suffices to show that (D) (e−αi ) ∈ Uq (b− ) ⊗ Uq (g), for i = 0, . . . , r . When i ∈ {1, . . . , r }, (D) (e−αi ) = (D) ( f αi [0]) = 1 ⊗ f αi [0] + k≥0 f αi [−k] ⊗ ψα+i [k]C −2n+k ∈ Uq (b− ) ⊗ Uq (g). When i = 0, (D) (e−α0 ) = λ(D) (Si+1 . . . Si+n (eα j [−1])). Let us prove by descending induction on k that (D) (Si+k . . . Si+n (eα j [−1])) ∈ U E− ⊗ Uq (g). This is true for k = n + 1, since (D) (eα j [−1]) = eα j [−1] ⊗ 1 + s≥0 ψα−j [−s]C 2−s ⊗ eα j [s − 1]. If now x = Si+k+1 . . . Si+n (eα j [−1]) is such that (D) (x) ∈ U E− ⊗ Uq (g), we have (D) (Si+ (x)) = (D) (ei [0]x −q (αi ,|x|) xei [0]) = [(D) (eαi [0]), (D) (x)]q = [eαi [0]⊗ 1+ s≥0 ψα−i [−s]C −s ⊗eαi [s], (D) (x)]q ≡ [eαi [0]⊗1, (D) (x)]q modulo U E− ⊗Uq (g) (here |x| is the degree of x). The result now follows from the induction assumption (D) (x) ∈ U E− ⊗ Uq (g) and (3.25). The proof of the second statement is similar. Corollary 3.7. The intersections of Borel subalgebras have the following coideal properties: − (D) (U F+ ) ⊂ U F ⊗< U F+ , (D) (U − f ) ⊂ U f ⊗< U F , (D)
(U E− )
⊂
U E−
(D)
⊗< U E ,
(Ue+ )
⊂
U E ⊗< Ue+ .
Proof. This follows directly from (3.39) and Proposition 3.6.
(3.41) (3.42)
This means that (D, (D) ), Ai , Bi satisfy hypothesis (H4). We can therefore apply Theorem 2 to (D, D ) = (Uq ( g), (D) ), A = U E , A1 = Ue+ , 0,− 0,+ − r A2 = U E , B = U F , B1 = U F , B2 = U f , C[Z ] = Uq (h).
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− + We denote by R1 ∈ Ue+ ⊗ U − f and R2 ∈ U E ⊗ U F the analogues of R1 , R2 . Then g), (D) ), so we get a Hopf algebra structure on Uq ( g), given R2 is a cocycle for (Uq ( tw (D) tw is the analogue of of Sect. 2.3), quasitriangular by (x) = R2 (x)R−1 ( 2 with R-matrix Rtw = R2,1 2 R1 . A proof of the following result (based on the braid group action) was first given in [KT1]. std 2,1 Proposition 3.8. tw = (std )2,1 , therefore we have (D) (x) = R−1 2 ( ) (x)R2 . We also have 2,1 −1 std (D) (x) = R2,1 1 (x)(R1 ) .
(3.43)
Proof. tw defines a Hopf structure on Uq ( g), for which Uq (b+ ) = A1 B1 and Uq (b− ) = A2 B2 are Hopf subalgebras. Since R2 is invariant under Uq (h), we have tw (kα±1 )= i ±1/2
±1/2
)⊗2 and tw (kδ ) = (kδ )⊗2 . (kα±1 i Since (D) and R2 are homogeneous for the principal degree, the same holds for tw . Moreover, R2 = Rh(1+(negative principal degree)⊗(positive principal degree)), where Rh = q Ch and Ch ∈ h⊗2 is the Casimir element of the Cartan algebra. Therefore tw (eαi ) = eαi ⊗ kαi + (degree 0) ⊗ (degree 1), and tw (e−αi ) = 1 ⊗ e−αi + (degree−1) ⊗ (degree 0). −1 (D) )2,1 R , and since R = On the other hand, tw (x) = R2 (D) (x)R−1 1 1 2 = R1 ( 1 + (positive principal degree) ⊗ (negative principal degree), we find tw (eαi ) = 1 ⊗ eαi + (degree 1) ⊗ (degree 0), and tw (e−αi ) = e−αi ⊗ k−αi + (degree 0) ⊗ (degree −1). Combining these results, we get tw = (std )2,1 (D) )2,1 (x)(R2,1 )−1 = (R2,1 )−1 (D) (x)R2,1 , which Then std (x) = R2,1 2 ( 2 1 1 implies (3.43). Proposition 3.9. The intersections of Borel subalgebras have the following coideal properties: − std (U F+ ) ⊂ U F+ ⊗ Uq (b+ ), std (U − f ) ⊂ U f ⊗ Uq (b− ),
(3.44)
std (U E− ) ⊂ Uq (b− ) ⊗ U E− , std (Ue+ ) ⊂ Uq (b+ ) ⊗ Ue+ .
(3.45)
This is a translation of Proposition 2.3. We will denote by P ± the projection operators of the Borel subalgebra U F , corres− + + ponding to the decomposition U F = U − f U F . So for any f + ∈ U F and any f − ∈ U f , P + ( f − f + ) = ε( f − ) f + , The operator
P+
P − ( f − f + ) = f − ε( f + ).
(3.46)
will also be denoted by P (without index).
4. Universal Weight Functions
ε
ε 4.1. The definition. Denote by Ue+ the augmentation ideal of Ue+ , i.e., Ue+ =
ε Ue ∩ Ker(ε). Let J be the left ideal of Uq (b+ ) generated by Ue+ :
ε J = Uq (b+ ) Ue+ = Uq (b+ )eα [n]. (4.1) α∈,n≥0
Recall that the ψα+i [0]±1 , ψα+i [n] and C ±1 (where i ∈ {1, . . . , r }, n > 0) commute in Uq ( g).
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Proposition 4.1. J is a coideal of Uq (b+ ), i.e., std (J ) ⊂ J ⊗ Uq (b+ ) + Uq (b+ ) ⊗ J . The space J is also stable under right multiplication by the ψα+i [0]±1 , ψα+i [n] (where i ∈ {1, .., r }, n > 0), i.e., J ψα+i [0]±1 ⊂ J , J ψα+i [n] ⊂ J . It follows that Uq (b+ )/J is both a coalgebra and a right module over C[ψαi [0]±1 , ψαi [n], C ±1 ; i ∈ {1, . . . , r }, n > 0]. Proof. The second part of (3.45) implies that for n ≥ 0, std (eα [n]) ∈ Uq (b+ ) ⊗ Ue+ . Moreover, (id ⊗ε) ◦ std = id implies that std (eα [n]) − eα [n] ⊗ 1 ∈ Uq (b+ ) ⊗ Ker ε. Therefore std (eα [n]) − eα [n] ⊗ 1 ∈ Uq (b+ ) ⊗ (Ue+ )ε , so (eα [n]) ∈ J ⊗ Uq (b+ ) + Uq (b+ ) ⊗ J . This implies the coideal property of J . The second property follows from the commutation relation of the currents ψα+i (w) and eα (z). Define an ordered -multiset as a triple I¯ = (I, ≺, ι), where (I, ≺) is a finite, totally ordered set, and ι : I → is a map (the ‘coloring map’). Ordered -multisets form a category, where a morphism I¯ → I¯ = (I , ≺ , ι ) is a map m : I → I , compatible with the order relations and such that ι ◦ m = m ◦ ι. If I¯ = (I, ≺, ι) is an ordered -multiset, then a partition I = I1 I2 gives rise to ordered -multisets I¯i = (Ii , ≺i , ιi ), i = 1, 2, where ≺i and ιi are the restrictions of ≺ and ι to Ii . If I¯ = (I, ≺, ι) is an ordered -multiset, and I = {i 1 , . . . , i n }, with i 1 ≺ . . . ≺ i n , then we attach to I¯ an ordered set of variables (ti )i∈I = (ti1 , . . . , tin ). The ‘color’ of tk is ι(i k ) ∈ . For any vector space V denote by V [[t1±1 , . . . , tn±1 ]] the vector space of all formal series Ak1 ,...,kn t1k1 · · · tnkn , Ak1 ,...,kn ∈ V, (k1 ,...,kn )∈Zn
and by V ((t1 )) · · · ((tn )) its subspace corresponding to the maps (k1 , . . . , kn ) → Ak1 ,...,kn , such that there exists an integer a1 and integer valued functions a2 (k1 ), a3 (k1 , k2 ), . . ., an (k1 , . . . , kn−1 ) (depending on the map (k1 , . . . , kn ) → Ak1 ,...,kn ), such that Ak1 ,...,kn = 0 whenever k1 > a1 , or k2 > a2 (k1 ), or k3 > a3 (k1 , k2 ), . . . , or kn > an (k1 , . . . , kn−1 ). A bilinear map V ⊗ W → Z gives rise to a bilinear map V ((t1 )) . . . ((tn )) × W ((t1 )) . . . ((tn )) → Z ((t1 )) . . . ((tn )) using the multiplication of formal series. A universal weight function is an assignment I¯ → W I¯ , where I¯ = (I, ≺, ι) is an )) . . . ((ti−1 )) (I = {i 1 , . . . , i n }, ordered -multiset and W I¯ ((ti )i∈I ) ∈ (Uq (b+ )/J )((ti−1 n 1 with i 1 ≺ . . . ≺ i n ) is such that (a) (Functoriality) If f : I¯ → J¯ is an isomorphism of ordered -multisets, then W I¯ ((ti )i∈I ) = W J¯ ((t f −1 ( j) ) j∈J ); (b) W∅ = 1, where W∅ is the series corresponding to I equal to the empty set; (c) The series W I¯ (ti1 , . . . , tin ) satisfies the relation
W I¯1 ((C22 ti )i∈I1 ) ⊗ W I¯2 ((ti )i∈I2 ) std W I¯ ((ti )i∈I ) = ⎛ × ⎝1 ⊗
I =I1 I2
i∈I1
⎞
+ ψι(i) (Cti )⎠ ×
k,l|k
q −(αι(ik ) ,αι(il ) ) − til /tik 1 − q −(αι(ik ) ,αι(il ) ) til /tik
,
(4.2)
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715
where std is the coproduct of Uq (b+ )/J . Here C2 = 1 ⊗ C, where C is the central element of Uq ( g ), see Sect. 3.3. The summation in (4.2) runs over all possible decompositions of the set I into two disjoint subsets I1 and I2 . g) be the subquotient of Uq ( g ) with trivial central element C = 1 and Remark. Let Uq ( dropped gradation element. Let Uq (b+ ) be the corresponding Borel subalgebra of Uq ( g). The analogue of Proposition 4.1 holds with J replaced by its analogue J . The notion of universal weight function makes sense for the algebra Uq ( g) as well. The conditions (a), (b) remain unchanged, the relation (4.2) of condition (c) is now
std W I¯ ((ti )i∈I ) = W I¯i ((ti )i∈I1 ) ⊗ W I¯2 ((ti )i∈I2 ) ⎛ × ⎝1 ⊗
I =I1 I2
⎞
+ ψι(i) (ti )⎠ ×
i∈I1
q −(αι(ik ) ,αι(il ) ) − til /tik
k,l|k
1 − q −(αι(ik ) ,αι(il ) ) til /tik
(4.3)
,
where std is the coproduct of Uq (b+ )/J . 4.2. Vector-valued weight functions. Let V be a representation of Uq ( g) and v be a vector in V . We call v a singular weight vector of weight {λi (z), i = 1, . . . , r } if eα [n]v = 0, ψα+i (z)v = λi (z)v,
i ∈ {1, . . . , r }, n ∈ Z,
where λi (z) ∈ C[[z −1 ]]× . It is clear that the ideal J defined by (4.1) annihilates any singular weight vector, i.e., J v = 0. For V a representation with the singular weight vector v and I¯ an ordered -multiset, we define a V -valued function ¯
wVI ((ti )i∈I ) = W I¯ ((ti )i∈I )v
(4.4)
which is a V -valued Laurent formal series. For I = ∅ we set wV∅ = v. Let Vi (i = 1, 2) (i) be representations with the singular weight vectors vi , with series {λα (z)|α ∈ }. Then the coproduct property (4.3) of the universal weight function yields the following property of the V1 ⊗ V2 -valued function ¯ ¯ ¯ wVI11 ((ti )i∈I1 ) ⊗ wVI22 ((ti )i∈I2 ) wVI 1 ⊗V2 ((ti )i∈I ) = I =I1 I2
×
i∈I1
(2)
λι(i) (ti )
q −(αι(ik ) ,αι(il ) ) tik − til
k,l|k
− q −(αι(ik ) ,αι(il ) ) til
.
(4.5)
A collection of V -valued functions wV ((ti )i∈I ) for all possible -multisets I¯, and representations V with a singular weight vector v, is called a vector-valued weight function or simply a weight function, if it satisfies the relations (4.5), the initial condition wV (∅) = v, and depends only on the isomorphism class of the ordered -multiset I . It is clear how to modify relation (4.5) to define a vector-valued weight function of the algebra Uq ( g ). Any universal weight function determines a vector-valued weight function by relation (4.4).
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4.3. Main theorems. Let I¯ = (I, ≺, ι) be an ordered -multiset. If I = {i 1 , . . . , i n }, with i 1 ≺ . . . ≺ i n , we set
W I¯ ((ti )i∈I ) := P f ι(i1 ) (ti1 ) · · · f ι(in ) (tin ) . (4.6) Theorem 3. The map I¯ → W I¯ defined by (4.6) is a universal weight function. g ) and Uq ( g). Note that the statement of Theorem 3 is valid for both algebras Uq ( Proof. First one should check that W I¯ ((ti )i∈I ) ∈ Uq (b+ )((ti−1 )) . . . ((ti−1 )). This foln 1 lows from the fact that for any x ∈ U F and αi ∈ , there exists an integer M, such that for any n > M we have P(x f αi [−n])=0. We now prove this fact. Define a degree on U F by deg(h α [n]) = n (n > 0), deg(Uq (h)) = 0, deg( f αi [n]) = n (n ∈ Z). Then U F+ , U − f ⊂ U F are graded subalgebras. Moreover, the nontrivial homogeneous compoε + nents of U F+ all have degree ≥ 0. We have U F = U F+ ⊕ (U − f ) U F . Then we take M to be the largest degree of a nonzero homogeneous component of x. Using the definition of P, one also checks that W I¯ ((ti )i∈I ) ∈ ti−1 Uq (b+ )((ti−1 )) . . . ((ti−1 ))[[ti−1 ]]. n 1 2 1 Let us now show that I¯ → W I¯ satisfies conditions (a)–(c) of Sect. 4.1. Conditions (a) and (b) of Sect. 4.1 are trivially satisfied. Let us show that (c) is satisfied. Theorem 4 and the following relations: (D) f α (z) = 1 ⊗ f α (z) + f α (C22 z) ⊗ ψa+ (C2 z),
−1 q −(α,β) C −1 − w/z f β (w), = ψα+ (z) f β (w) ψα+ (z) 1 − q −(α,β) C −1 w/z g ) ⊗ J . But both sides of (4.2) belong imply that relations (4.2) are satisfied modulo Uq ( to Uq (b+ ) ⊗ Uq (b+ ). Thus they coincide modulo Uq (b+ ) ⊗ J . So, we reduced the proof of Theorem 1 to the following statement. Recall first that std ◦ P defines a map U F → U F+ ⊗ Uq (b+ ), and P ⊗2 ◦ (D) defines a map U F+ → (U F+ )⊗2 . Theorem 4. For any element f ∈ U F ,
std (P( f )) ≡ (P ⊗ P) (D) ( f ) mod U F+ ⊗ J,
(4.7)
where J is the left ideal of Uq (b+ ) defined by (4.1). − ε + + + 8 Proof of Theorem 4. We have U F = U − f U F , therefore U F = U F ⊕ (U f ) U F . So we will prove (4.7) in the two following cases: (a) f ∈ U F+ , and (b) f = x y, where + ε x ∈ (U − f ) and y ∈ U F . Assume first that f ∈ U F+ . According to (3.41), (D) ( f ) ∈ U F ⊗ U F+ , therefore
(P ⊗ P)((D) ( f )) = (P ⊗ id)((D) ( f )). 8 Recall that if A ⊂ U ( ε q g), we set A := A ∩ Ker(ε).
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2,1 −1 std According to (3.43), we have (D) (x) = R2,1 for any x ∈ Uq ( g), 1 (x)(R1 ) 2,1 ±1 − + where (R1 ) ∈ U f ⊗ Ue . It follows that 2,1 −1 std (P ⊗ id)((D) ( f )) = (P ⊗ id)(R2,1 1 ( f )(R1 ) ). −1 ∈ U ⊗ U (b ); it follows Now std ( f ) ∈ U F+ ⊗ Uq (b+ ), therefore std ( f )(R2,1 F q + 1 ) 2,1 −1 std ε + that (P ⊗ id)( ( f )(R1 ) ) is well defined. Now R21 ∈ 1 ⊗ 1 + (U − 1 f ) ⊗ Ue , therefore 2,1 −1 2,1 −1 std std (P ⊗ id)(R2,1 1 ( f )(R1 ) ) = (P ⊗ id)( ( f )(R1 ) ). −1 ∈ 1 ⊗ 1 + U − ⊗ (U + )ε ⊂ 1 ⊗ 1 + U − ⊗ J , we have Since (R2,1 e 1 ) f f −1 std (P ⊗ id)(std ( f )(R2,1 1 ) ) ≡ (P ⊗ id)( ( f )) mod U F ⊗ J.
Recall now that std ( f ) ∈ U F+ ⊗ Uq (b+ ). This implies that (P ⊗ id)(std ( f )) = std ( f ). Finally, since f ∈ U F+ , we have f = P( f ), therefore std ( f ) = std (P( f )). Combining the above equalities and congruence, we get the desired congruence, when f ∈ U F+ . + ε Assume now that f = x y, where x ∈ (U − f ) and y ∈ U F . According to (3.41), (D) ⊗ id) ◦ (D) = id and (D) (x) ∈ U − f ⊗ U F ; on the other hand, the identities (ε g); all this implies that (D) (x) = ε = ε(D) imply that (D) (x) ∈ 1 ⊗ x + Ker(ε) ⊗ Uq ( − ε 1 ⊗ x + i ai ⊗ bi , where ai ∈ (U f ) and bi ∈ U F . Therefore (P ⊗ id)((D) (x y)) = (P ⊗ id) (1 ⊗ x + ai ⊗ bi )(D) (y) . i
(D) (y)
Recall that ∈ η ∈ U F , which implies that (P
ε Now since ai ∈ (U − f ) , we have P(ai η) ⊗ id)((ai ⊗ bi )(D) (y)) = 0. Therefore
U F ⊗ U F+ .
= 0 for any
(P ⊗ id)((D) (x y)) = (P ⊗ id) (1 ⊗ x)(D) (y) .
Applying id ⊗P to this identity, we get
(P ⊗ P)((D) (x y)) = (P ⊗ id) ◦ (id ⊗P) (1 ⊗ x)(D) (y) .
ε Now since x ∈ (U − f ) , we have P(xη) = 0 for any η ∈ U F , which implies that (D) (id ⊗P)((1 ⊗ x) (y)) = 0. Therefore
(P ⊗ P)((D) (x y)) = 0. Since P(x y) = 0, we get std (P(x y)) = (P ⊗ P)((D) (x y)), ε which proves the desired congruence for the elements of the form x y, x ∈ (U − f ) , + y ∈ UF .
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4.4. Functional properties of the universal weight function. Let I¯ = (I, ≺, ι) be an ordered -multiset, let n := |I | and let σ ∈ Sn be a permutation. Let I¯σ = (I, ≺σ , ι) be the -multiset such that if I = {i 1 , . . . , i n }, where i 1 ≺ . . . ≺ i n , then i σ (1) ≺σ . . . ≺σ i σ (n) . We call I¯σ a permutation of I¯. Proposition 4.2. There exists a collection of formal functions (s)
W n 1 ,...,nr ∈ (Uq (b+ )/J )[[u j |s ∈ {1, . . . , r }, j ∈ {1, . . . , n i }]], (s)
symmetric in each group of variables (u j ) j=1,...,n s , such that (σ ) W I¯ ((ti )i∈I ) = i∈I
r
(ti−1 −t −1 j )
s=1 i, j∈Is |i≺ j
ti
(ti−1 −q (ι(i),ι( j)) t −1 j )
−1 W |I1 |,...,|Ir | ((t −1 j1 ) j1 ∈I1 ,. . . , (t jr ) jr ∈Ir ),
i, j∈I |i≺ j
(4.8) where Is := ι−1 (αs ) (it is an ordered set), and σ ∈ Sn is the shuffle permutation of {1, . . . , n} given by {1, . . . , n} I1 . . . Ir → I {1, . . . , n}, where the first and last bijections are ordered, the order relation on I1 . . . Ir is such that I1 ≺ . . . ≺ Ir , and the map I1 . . . Ir → I is such that its restriction to each Is is the natural injection. ,. . ., ti±1 ]] Proof. For any ordered -multiset I¯, denote by FI¯ (ti1 ,. . ., tin ) ∈Uq (b+ )[[ti±1 n 1 the series FI¯ (ti1 , . . . , tin ) = f ι(i1 ) (ti1 ) · · · f ι(in ) (tin ), , . . . , ti−1 ] the product and by A I¯ (ti1 , . . . , tin ) ∈ C[ti−1 n 1 A I¯ (ti1 , . . . , tin ) = (ti−1 − q (ι(ik ),ι(il ) ti−1 ). k l
(4.9)
1≤k
Let I¯ → G I¯ be an assignment taking an ordered -multiset I¯ = (I, ≺, ι) (with , . . . , ti±1 ]]. We say I = {i 1 , . . . , i n } and i 1 ≺ . . . ≺ i n ) to G I¯ (ti1 , . . . , tin ) ∈ V [[ti±1 n 1 that I¯ → G I¯ is antisymmetric, iff for any I¯ and any σ ∈ Sn (where n = |I |), we have G I¯σ ((ti )i∈I ) = (σ )G I¯ ((ti )i∈I ). The defining relations (3.31) imply that the assignment I¯ → F I¯ is antisymmetric, where F I¯ (ti1 , . . . , tin ) = A I¯ (ti1 , . . . , tin )FI¯ (ti1 , . . . , tin ) takes its values in Uq (b+ )[[ti±1 , . . . , ti±1 ]]. n 1 Applying P, we get that the assignment I¯ → W I¯ is antisymmetric, where W I¯ (ti1 , . . . , tin ) := A I¯ (ti1 , . . . , tin )W I¯ (ti1 , . . . , tin ); its takes its values in (Uq (b+ )/J )[[ti±1 , . . . , ti±1 ]]. n 1 According to the proof of Theorem 3, W I¯ (ti1 , . . . , tin ) ∈ ti−1 (Uq (b+ )/J )((ti−1 )) . . . ((ti−1 ))[[ti−1 ]]; n 1 2 1
Weight Functions and Drinfeld Currents
719
therefore W I¯ (ti1 , . . . , tin ) := A I¯ W I¯ (ti1 , . . . , tin ) takes its values in the same space. The antisymmetry of I¯ → W I¯ then implies that it takes its values in the intersection of all (Uq (b+ )/J )((ti−1 )) . . . ((ti−1 ))[[ti−1 ]], where σ ∈ Sn , i.e., in the ti−1 σ (1) σ (n) σ (2) σ (1) (ti1 . . . tin )−1 (Uq (b+ )/J )[[ti−1 , . . . , ti−1 ]]. n 1 If now V is a vector space and I¯ → v I¯ (ti1 , . . . , tin ) ∈ V [[ti−1 , . . . , ti−1 ]] is antisymn 1 (1)
(r )
metric, one shows that there exists a family of formal series v n 1 ,...,nr (u 1 , . . . , u nr ) ∈ (1) (r ) (i) V [[u 1 , . . . , u nr ]], symmetric in each group of variables u α for fixed i, such that v I¯ (ti1 , . . . , tin ) = (σ )
r
s=1 k,l∈Is |k≺l
The result follows.
(tk−1 − tl−1 )v |I1 |,...,|Ir | ((ti−1 )i1 ∈I1 , . . . , (ti−1 )ir ∈Ir ). r 1
Let s, t ∈ {1, . . . , r }, with s = t. Let m := 1 − aαs αt , where (aαβ )α,β∈ is the Cartan matrix of g. Let k1 , . . . , km ∈ {1, . . . , n s } be distinct, and let l ∈ {1, . . . , n t }. Let (s) H(kst1 ,...,km ),l ⊂ ⊕rs=1 Cn s be the subspace of all (u j )s∈{1,...,r }, j∈{1,...,n s } , such that (t)
ul = q −
(m−1)(αs ,αs ) 2
(s)
u k1 = q −
(m−3)(αs ,αs ) 2
(s)
u k2 = . . . = q
(m−1)(αs ,αs ) 2
(s)
u km .
Proposition 4.3. The restriction of W n 1 ,...,nr to H(kst1 ,...,km ),l is identically zero for any s, t, (k1 , . . . , km ), l. Proof. The proof is the same as the proof of the similar statement in [E1], and is based on the quantum Serre relations. When q = 1, W I¯ ((ti )i∈I ) can be computed as follows. Set W I¯Lie ((ti )i∈I ) :=
[[ f ι(i1 ) , f ι(i2 ) ], . . . , f ι(in ) ]+ (ti1 ) , (−1 + tin−1 /tin ) . . . (−1 + ti1 /tin )
where for x, y ∈ g, we set [x, y](t) = [x[0], y(t)] = [x(t), y[0]]; then x(t) = −n , and x + (t) := −n . Then x[n]t n∈Z n>0 x[n]t W I¯ ((ti )i∈I ) =
s≥0,(I1 ,...,Is )|I =I1 ...Is , min(I1 )≺...≺min(Is )
W I¯Lie ((ti )i∈I1 ) . . . W I¯Lie ((ti )i∈Is ), 1
s
(4.10)
where the sum is over all the partitions I = I1 . . . Is , such that min(I1 ) ≺ . . . ≺ min(Is ), and I¯i = (Ii , ≺i , ιi ) is the ordered -multiset induced by I¯. For α ∈ , set where f α+ (z) := n>0 f α [n]z −n . Conjecture 4.4 W I¯ ((ti )i∈I ) is a linear combination of noncommutative polynomials in the f α+ (q k ti ), f α [0] (α ∈ , k ∈ Z), where the coefficients have the form P((ti )i∈I )/ i, j∈I |i≺ j (ti − q −(ι(i),ι( j)) t j ), and P((ti )i∈I ) is a polynomial of degree |I |(|I | − 1)/2.
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B. Enriquez, S. Khoroshkin, S. Pakuliak
+ (t), f We have [[ f ι(i1 ) , f ι(i2 ) ], . . . , f ι(in ) ]+ (t) = [[ f ι(i ι(i 2 ) [0]], . . . , f ι(i n ) [0]], hence 1) the conjecture is true for any g when q = 1. According to [KP1], it is also true when g = sl2 or sl3 for any q = 0. For example, we have in Uq ( sl2 ) (see [KP1]),
P ( f α (t1 ) f α (t2 )) = f α+ (t1 ) f α+ (t2 ) −
2 (q − q −1 )t1 + f (t1 ) . qt1 − q −1 t2 α
n− [z, z −1 ]), Remark. Equation (4.10) is also valid if U (n− [z, z −1 ]) is replaced by U ( where n− is the free Lie algebra with generators f α , α ∈ ; this algebra is presented by the relations (z − w)[ f α (z), f β (w)] = 0 for any α, β ∈ , so the Serre relations do not play a role in the derivation of (4.10) (where q = 1). However, the results of [KP1] use the quantum Serre relations. Let now V be a finite dimensional representation of Uq ( g ) with singular weight ¯ I vector v. Let I¯ be an ordered -multiset and wV ((ti )i∈I ) be the vector-valued weight function
¯ (4.11) wVI ((ti )i∈I ) := P f ι(i1 ) (ti1 ) · · · f ι(in ) (tin ) v. ¯
Proposition 4.5. Assume that Conjecture 4.4 is true. Then wVI ((ti )i∈I ) is the Laurent expansion of a rational function on Cn . There exist rational functions (r ) w n 1 ,...,nr ((u (1) j ) j=1,...,n 1 , . . . , (u j ) j=1,...,nr ),
such that the analogue of identity (4.8) holds. (s) Each function w n 1 ,...,nr is symmetric in each group of variables (u j ) j=1,...,n s . Its (s)
only singularities are poles at u j ∈ Ss , where Ss ∈ C× is a finite subset of C× . It vanishes on the spaces H(kst1 ,...,km ),l . Proof. According to the theory of Drinfeld polynomials, the image of fα+ (t) in End(V ) is ¯ a rational function in t with poles in C× . It follows from Conjecture 4.4 that wVI ((ti )i∈I ) is the Laurent expansion of a rational function, which is regular except for (a) simple poles at ti = q −(ι(i),ι( j)) t j , where i ≺ j, and (b) poles at ti ∈ Si , where Si ⊂ C is a ¯ finite subset. The form of W I¯ proved in Proposition 4.2 also implies that wVI vanishes on the hyperplanes ti = t j , where ι(i) = ι( j) (as a formal function, hence as a rational αβ function), and Proposition 4.3 implies that wVI vanishes on the spaces H(k1 ,...,km ),l (as a formal function, hence as a rational function). Define then wn 1 ,...,nr := W n 1 ,...,nr v. Then the analogue of (4.8) holds. It follows ¯ from the properties of wVI that w n 1 ,...,nr is rational, with the announced poles structure. Since the Laurent expansion of w n 1 ,...,nr is symmetric in each group of variables, so is wn 1 ,...,nr itself. 5. Relation to the Off-Shell Bethe Vectors In this section, we relate the universal weight functions to the off-shell Bethe vectors, in the case of the quantum affine algebra Uq ( sl2 ). The algebra Uq ( sl2 ) is generated by
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721
the modes of the currents e(z), f (z) and ψ ± (z). We will need only the commutation relations between the currents f (z), f (w) and f (z), ψ + (w): (qz − q −1 w) f (z) f (w) = (q −1 z − qw) f (w) f (z), q −2 − w/z f (w)ψ + (z). ψ + (z) f (w) = 1 − q −2 w/z
(5.1) (5.2)
Using formula (5.1) we may calculate the projection P ( f (z 1 ) · · · f (z n )). The algebra Uq ( sl2 ) also has a realization in terms of L-operators ([RS]): ±
L (z) =
1 f ± (z) 0 1
k ± (zq −2 )−1 0 0 k ± (z)
1 0 e± (z) 1
=
A± (z) B ± (z) C ± (z) D ± (z)
which satisfy
R(u/v) · (L (u) ⊗ 1) · (1 ⊗ L (v)) = (1 ⊗ L (v)) · (L (u) ⊗ 1) · R(u/v) with , ∈ {+, −}, and R(z) = (qz − q −1 ) (E 11 ⊗ E 11 + E 22 ⊗ E 22 ) + (z − 1) (E 11 ⊗ E 11 + E 22 ⊗ E 11 ) + (q − q −1 ) (z E 12 ⊗ E 21 + E 21 ⊗ E 12 ) (E i j denotes the matrix unit). According to [DF], the Gauss coordinates of the L-operators are related to the currents as follows: e(z) = e+ (z) − e− (z),
−1 f (z) = f + (z) − f − (z), ψ ± (z) = k ± (zq −2 )k ± (z) .
Let v be a vector such that C + (z)v = 0. The vector-valued function w(z 1 , . . . , z n ) = B + (z 1 ) · · · B + (z n )v
(5.3)
is called an off-shell Bethe vector. Using the equality B + (z) = f + (z)k + (z) and the relation (5.2) we may present the product (5.3) in terms of the product of the halfcurrents f + (z). This gives the relation B + (z 1 ) · · · B + (z n ) =
n n qz i − q −1 z j P ( f (z 1 ) · · · f (z n )) k + (z i ) zi − z j
i< j
(5.4)
i=1
which shows the relation between the off-shell Bethe vectors and the weight function (4.6). For a general quantum affine algebra, the calculation of the weight functions given by the universal weight function (4.6) is a complicated and interesting problem. Such calculations for quantum affine algebras Uq ( sl3 ) and Uq ( sl N +1 ) are given in [KP1] and in [KP2]. The relation between the universal weight function (4.6) and the nested Bethe ansatz ([KR]) will be studied in [KPT].
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B. Enriquez, S. Khoroshkin, S. Pakuliak
Appendix Here we give a proof of the properties (3.25) and (3.26) of circular Cartan-Weyl generators, see Sect. 3.2. The proof uses the braid group approach to the CW generators, which we describe first. Let Ti : Uq ( g ) → Uq ( g ), i = 0, 1, . . . , r , be the Lusztig automorphisms [L2], defined by the formulas Ti (eαi ) = −e−αi kα−1 , T (e ) = (−1) p qis eα( p) eα j eα(s)i i = j, (6.5) i α j i i p+s=−ai, j
Ti (e−αi ) = −kαi eαi ,
Ti (e−α j ) =
(s)
( p)
(−1) p qi−s e−αi e−α j e−αi i = j, (6.6)
p+s=−ai, j ( p)
p
where e±αi = e±αi /[ p]qi !. We attach to the periodic sequence . . . , i −1 , i 0 , i 1 , . . . , i n , . . . given by (3.17) the sequence (wn )m∈Z of elements of the Weyl group, given by w0 = w1 = 1, wk+1 = wk sik for k > 0, and wl−1 = sil wl for l ≤ 0. Let γk be the corresponding positive real roots (3.18). We have a normal ordering γ1 ≺ γ2 ≺ . . . ≺ δ ≺ 2δ ≺ . . . ≺ γ−1 ≺ γ0 of the + . system We define real root vectors e±γk , where k > 0 and e±γl , where l ≤ 0 by the relations e±γk = Twk (e±αik ),
e±γl = Tw−1 (e±αil ), l
(6.7)
that is, e±γn = e±αin for n = 0, 1; e±γk = Ti1 Ti2 · · · Tik−1 (e±αik ) for k > 1, and Ti−1 · · · Ti−1 (e±αil ) for l < 0. The imaginary root vectors are defined by e±γl = Ti−1 0 −1 l+1 the relations (3.21) and (3.21). The imaginary root vectors, related to positive roots, + ⊂ U (n ). It is characterized by the properties [Be] generate an abelian subalgebra UIm q + + ⇔ Tw−1 ( p) ∈ Uq (n+ ) and Twl ( p) ∈ Uq (n+ ) for all k > 0, l ≤ 0. (6.8) p ∈ UIm k
The root vectors (6.7), (3.21) satisfy the property (3.22) (see [Be]) and thus coincide, up to normalization, with the CW generators of Sect. 3.2. Let c be an integer > 0. Let . . . , j−1 , j0 , j1 , . . . be the periodic sequence defined by the rule jn = i n−c for all n ∈ Z, {w˜ n } the related sequence of elements of the Weyl group, given by the rule w˜ 0 = w˜ 1 = 1, w˜ k+1 = w˜ k s jk for k > 0, and w˜ l−1 = s jl w˜ l for l < 0. Let {γ˜n , n ∈ Z} be the corresponding sequence of real positive roots, γ˜k = w˜ k (α jk ), if k ≥ 0 and γ˜l = w˜ l−1 (α jl ), if l ≤ 0. Let {e˜γ } be the CW generators, built by the braid group procedure, related to the sequence { jk }: e˜±γ˜k = Tw˜ k (e±α jk ) if k ≥ 1, and ˜+ e˜±γ˜l = Tw−1 ˜ l (e±α jl ) if k ≤ 0. Let UIm be the subalgebra of Uq (n+ ), generated by the (i)
imaginary root vectors e˜nδ , i = 1, . . . , r , n > 0. We have the correspondence:
n = 1, 2, . . . , c, sαi1−c · · · sαi0 (γn−c ), sαi1−c · · · sαi0 (−γn−c ), n = 1, 2, . . . , c,
= Ti1−c · · · Ti0 eˆγn−c , n ∈ Z,
γ˜n =
e˜γ˜n + + ˜ Im = Ti1−c · · · Ti0 (UIm ). U
(6.9) (6.10)
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723
Indeed, for n = 1, . . . , c we have e˜γ˜n = Ti1−c · · · Ti0 eγn−c by the construction. For −1 −1 n = 1, . . . , c we have e˜γ˜n = Ti1−c · · · Ti0 Twn−c Tin−c Twn−c eαin−c , which is equal to
Ti1−c · · · Ti0 eˆγn−c by (6.5) and (6.6). The relation (6.10) follows from the description + and its analogue for the algebra U˜+ . (6.8) of the algebra UIm Im + , related to the sequence {i n }, ≺c the Let ≺ be the normal ordering of the system , and ≺˜ the normal ordering of + , related to the corresponding circular order in ˜ sequence { jn }. For any α, ˜ β ∈ + we have the correspondence: α˜ ≺˜ β˜ ⇔ α ≺c β,
(6.11)
˜ ˜ and β = sαi0 · · · sαic−1 (β). where α = sαi0 · · · sαic−1 (α), Consider the relation (3.22) for CW generators, related to the sequence { jn }: [e˜α˜ , e˜β˜ ]q −1 =
{˜ν }
C{n jj } (q) e˜νn˜11 e˜νn˜22 · · · e˜νn˜mm , {˜ν }
˜ where C j (q) ∈ C[q, q −1 , 1/(q n − 1); n ≥ 1]. with α˜ ≺˜ ν˜ 1 ≺˜ . . . ≺˜ ν˜ m ≺˜ β, {n j } Due to (6.9), (6.10), (6.8), automorphism properties of the maps Ti , and commutativity of imaginary root vectors, this is equivalent to the relation on circular generators: [eˆα , eˆβ ]q −1 =
{ν } C¯ {n jj } (q) eˆνn11 eˆνn22 · · · eˆνnmm ,
(6.12)
˜ with α ≺c ν1 ≺c . . . ≺c νm ≺c β, where α = sαi0 · · · sαic−1 (α), ˜ and β = sαi0 · · · sαic−1 (β);
{ν } C¯ {n jj } (q) ∈ C[q, q −1 , 1/(q n − 1); n ≥ 1]. This is a particular case of the relation (3.25), when the root α satisfies the condition −δ ≺c α and β is positive. Let d be an integer > 0. Let now { jn } be a periodic sequence, related to the sequence (3.17) by the rule jn = i n+d for all n, {w˜ n } the related sequence of elements of the Weyl group, {γ˜n , n ∈ Z} the corresponding sequence of real positive roots. Let {e˜±γ } be CW ˜ ± the generators, built by braid group procedure, related to the sequence { jk }, and U Im (i) subalgebras of Uq (n± ), generated by imaginary root vectors e˜±nδ , i = 1, . . . , r , n > 0. We have now the following correspondence:
n = 1, 2, . . . d, sαid · · · sαi1 (γn ), sαid · · · sαi1 (−γn ), n = 1, 2, . . . d,
e¯±γn , n ∈ Z, = Ti−1 · · · Ti−1 1 d
γ˜n−d = e˜±γ˜n−d ˜± U Im
=
± Ti−1 · · · Ti−1 (UIm ), 1 d
(6.13) (6.14) (6.15)
where the temporary real root generators e¯−γ˜ are given by the prescription e¯±γn = e±γn for n = 1, 2, . . . , d and e¯±γn = Twn Tin (e±αin ) for n = 1, 2, . . . , d. Again, the normal ordering ≺˜, attached to the sequence { jn }, is in accordance with the circular ordering ˜ Consider ≺c : α˜ ≺˜ β˜ ⇔ α ≺c β, where α = sαi1 · · · sαid (α), ˜ and β = sαi1 · · · sαid (β). the relation (3.22) for CW generators e˜γ˜ , related to negative roots α˜ and β˜ : [e˜α˜ , e˜β˜ ]q −1 =
{˜ν }
C{n jj } (q) e˜νn˜11 e˜νn˜22 · · · e˜νm˜m ,
(6.16)
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˜ ν˜ i , ∈ − ˜ Due to (6.14), (6.15), + , so that −α˜ ≺˜ − ν˜ 1 ≺˜ . . . ≺˜ − ν˜ m ≺˜ − β. with α, ˜ β, it is equivalent to {ν j } [e¯α , e¯β ]q −1 = (6.17) C¯ {n j } (q) e¯νn11 e¯νn22 · · · e¯νnmm , ˜ with α ≺c ν1 ≺c . . . ≺c νm ≺c β, where α = sαi1 · · · sαid (α), ˜ and β = sαi1 · · · sαid (β). Since all the roots in (6.16) are negative, the collection of the roots {α, β, ν1 , . . . , νm } in (6.17) contains negative roots and some positive roots belonging to the set {γ1 , . . . , γd }. Note that e¯ν = eν , if ν is negative, and e¯ν = −kν eν , if ν ∈ {γ1 , . . . , γd }. Now we apply to (6.17) the following automorphism of the algebra Uq ( g ): e−γ → −kγ e−γ ,
eγ → −eγ kγ−1 ,
kγ → kγ ,
+ . for all γ ∈
One can see that this automorphism transforms the relation (6.17) to the particular case of (3.25): {ν j } (6.18) [eˆα , eˆβ ]q −1 = C˜ {n j } (q) eˆνn11 eˆνn22 · · · eˆνnmm , {˜ν }
with α ≺c ν1 ≺c . . . ≺c νm ≺c β and C˜ {n jj } (q) ∈ C[q, q −1 , 1/(q n − 1); n ≥ 1], when the root β satisfies the condition β ≺c δ and α is negative. The relations (6.12) and (6.17) imply (3.25) in full generality. The proof of (3.26) is analogous. Remark. There are analogues of all the relations (3.22), (3.25) and (3.26), in which the order of the products (equivalently, the order of the root vectors) in the monomials in the right-hand sides are reversed. To derive them, it is sufficient to first apply the Cartan antiinvolution ∗ to (3.22), and then to apply the arguments of the Appendix to the result. Acknowledgements. This work was supported by the grant INTAS-OPEN-03-51-3350, the Heisenberg-Landau program, ANR project GIMP N. ANR-05-BLAN-0029-01, the RFBR grant 04-01-00642 and the RFBR grant for scientific schools NSh-8065.2006.2. It was partly done during the visits of S.Kh. and S.P. at MPIM (Bonn) in 2004 and 2005, the visit of S.Kh. at CPM (Marseille) in 2006, the visits of S.P. at LAPTH (Annecyle-Vieux) in 2006 and at IRMA (Strasbourg) in 2003 and 2006. The authors wish to thank these centers for their hospitality and stimulating scientific atmosphere.
References [ABFR] Albert, T.D., Boss, H., Flume, R., Ruhlig, K.: Resolution of the nested hierarchy for rational sl(n) models. J. Phys. A 33, 4963–4980 (2000) [BF] Babudjian, H, Flume, R.: Off-shell bethe ansatz equation for gaudin magnets and solution of knizhnik-zamolodchikov equation. Mod. Phys. Lett. A 9, 2029–2040 (1994) [Be] Beck, J.: Braid group action and quantum affine algebras. Commun. Math. Phys. 165, 555–568 (1994); Convex bases of PBW type for quantum affine algebras. Commun. Math. Phys. 165, 193–199 (1994) [CP] Chari, V., Pressley, A.: Quantum affine algebras and their representations. In: Representations of groups, CMS Conf. Proc. 16, 1994, Providence, RI: Amer. Math. Soc., 1995, pp. 59–78 [Da] Damiani, I.: La r -matrice pour les algèbres quantiques de type affine non tordu. Ann. Scient. Éc. Norm. Sup. 31(4), 493–523 (1998) [DK] Ding, J., Khoroshkin, S.: Weyl group extension of quantized current algebras. Transform. Groups 5(1), 35–59 (2000) 2 ). Theor. [DKP] Ding, J., Khoroshkin, S., Pakuliak, S.: Factorization of the universal r -matrix for Uq (sl and Math. Phys. 124(2), 1007–1036 (2000) [DKP1] Ding, J., Khoroshkin, S., Pakuliak, S.: Integral presentations for the universal r -matrix. Lett. Math. Phys. 53(2), 121–141 (2000)
Weight Functions and Drinfeld Currents
[DF] [D1] [D2] [E1] [E2] [EF] [ER] [K] [KP1] [KP2] [KPT] [KT1] [KT2] [KT3] [KR] [L2] [R] [RS] [S] [Sw] [T] [TV1] [TV2]
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N ). Ding, J., Frenkel, I.B.: Isomorphism of two realizations of quantum affine algebra Uq (gl Commun. Math. Phys. 156(2), 277–300 (1993) Drinfeld, V.: Quantum groups. In: Proc. ICM Berkeley (1986), Vol. 1, Providence, RI: Amer. Math. Soc., 1987, pp. 789–820 Drinfeld, V.: New realization of yangians and quantum affine algebras. Sov. Math. Dokl. 36, 212–216 (1988) Enriquez, B.: On correlation functions of Drinfeld currents and shuffle algebras. Transform. Groups 5(2), 111–120 (2000) Enriquez, B.: Quasi-hopf algebras associated with semisimple lie algebras and complex curves. Selecta Math. (N.S.) 9(1), 1–61 (2003) Enriquez, B., Felder, G.: Elliptic quantum groups E τ,η (sl2 ). Commun. Math. Phys. 195(3), 651–689 (1998) Enriquez, B., Rubtsov, V.: Quasi-hopf algebras associated with sl2 and complex curves. Israel J. Math. 112, 61–108 (1999) Kassel, C.: Quantum Groups. Berlin-Heidelberg-New York: Springer, 1995 3 ). Theor. and Math. Phys. 145(1), Khoroshkin, S., Pakuliak, S.: Weight function for Uq (sl 1373–1399 (2005) Khoroshkin, S., Pakuliak, S.: Method of projections for an universal weight function of the quantum N +1 ). In: Proceedings of the International Workshop Classical and quantum affine algebra Uq (sl integrable systems, January 23–26, 2006, Protvino. Theor. and Math. Phys. 150 (2), 244–258 (2007) Khoroshkin, S., Pakuliak, S., Tarasov, V.: Off-shell Bethe vectors and Drinfeld currents. J. Geom. Phys. 57, 1713–1732 (2007) Khoroshkin, S., Tolstoy, V.: Twisting of quantum (super)algebras. Connection of Drinfeld’s and Cartan-Weyl realizations for quantum affine algebras. MPI Preprint MPI/94-23, http://arxiv.org/lis/hepth/9404036, 1994 Khoroshkin, S., Tolstoy, V.: The Cartan-Weyl basis and the universal R-matrix for quantum Kac-Moody algebras and superalgebras. In: Quantum Symmetries, River Edge, NJ: World Sci. Publ. 1993, pp. 336–351 Khoroshkin, S., Tolstoy, V.N.: On Drinfeld realization of quantum affine algebras. J. Geom. Phys. 11(1–4), 445–452 (1993) Kulish, P., Reshetikhin, N.: Diagonalization of G L(N )-invariant transfer matrices and quantum N -wave system (lee model). J. Phys. A: Math. Gen. 16, L591–L596 (1983) Lusztig, G.: Introduction to quantum groups. Basel: Birkhäuser, 1993 Reshetikhin, N.: Integrable models of one-dimensional quantum magnets with the o(n) and sp(2k) symmetry. Theor. Math. Phys. 63, 347–366 (1985) Reshetikhin, N., Semenov-Tian-Shansky, M.: Central extensions of quantum current groups. Lett. Math. Phys. 19(2), 133–142 (1990) Smirnov, F.: Form factors in completely integrable models of quantum field theory. Adv. Series in Math. Phys., Vol. 14, Singapore: World Scientific, 1992 Sweedler, M.E.: Hopf Algebras. Reading, Ma: Addison-Wesley, 1969 Tarasov, V.O.: An algebraic Bethe ansatz for the Izergin-Korepin r -matrix. Theor. and Math. Phys. 76(2), 793–804 (1988) Tarasov, V., Varchenko, A.: Jackson integrals for the solutions to Knizhnik-Zamolodchikov equation. St. Petersburg Math. J. 6(2), 275–313 (1995) Tarasov, V., Varchenko, A.: Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups. Astérisque 246, 1–135 (1997)
Communicated by L. Takhtajan
Commun. Math. Phys. 276, 727–772 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0352-x
Communications in
Mathematical Physics
A Complete Renormalization Group Trajectory Between Two Fixed Points Abdelmalek Abdesselam Université Paris 13, LAGA, Institut Galilée, CNRS UMR 7539, 99 Avenue J.B. Clément, F93430 Villetaneuse, France. E-mail: [email protected] Received: 19 October 2006 / Accepted: 5 April 2007 Published online: 16 October 2007 – © Springer-Verlag 2007
Abstract: We give a rigorous nonperturbative construction of a massless discrete trajectory for Wilson’s exact renormalization group. The model is a three dimensional Euclidean field theory with a modified free propagator. The trajectory realizes the mean field to critical crossover from the ultraviolet Gaussian fixed point to an analog recently constructed by Brydges, Mitter and Scoppola of the Wilson-Fisher nontrivial fixed point. 1. Introduction In recent years, the mathematical community has shown an increasing interest for the important but difficult topic of quantum field theory [24]. The most comprehensive and insightful, albeit largely conjectural, mathematical framework to address this subject is Wilson’s renormalization group: a grand dynamical system in the space of all imaginable observation scale dependent effective field theories [76, 77]. As emphasized by Wilson himself [77], his approach can be construed as a mathematical theory of scaling symmetry which has yet to be fully unveiled. It generalizes in a very deep way ordinary calculus which matured in the hands of 19th century mathematicians and gave the first rigorous meaning to the notion of ‘continuum’. Formulating precise conjectures about the phase portrait of the RG dynamical system and proving them, an endeavor one could perhaps call the ‘Wilson Program’ [57], is one of the greatest challenges in mathematical analysis and probability theory, and will likely remain so for years to come. When studying a phase portrait, the first features to examine are fixed points, which here mean scale invariant theories. If D is the dimension of space, one expects that for D ≥ 4 there are only two fixed points: the high temperature one and the massless Gaussian one. As one lowers the dimension to the range 3 ≤ D < 4, only one new fixed point should appear: the Wilson-Fisher fixed point [75]. Its existence as well as the construction of its local stable manifold, in the hierarchical approximation, was first rigorously established in [9]; see also [20, 21, 36]. The uniqueness, in the local potential approximation, was shown in [54]. As one continues lowering the dimension to the range
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2 2 < D < 3, past every threshold Dn = 2 + n−1 , n = 3, 4, . . ., a new fixed point appears corresponding to an n-well potential, as was proved in the local potential approximation by Felder [31]. For D = 2, the situation becomes extremely complicated: even a conjectural classification of fixed points corresponding to conformal field theories is not yet complete. Nevertheless, there have been tremendous advances in this area; see e.g. [26] and Gaw¸edzki’s lectures in [24] for an introduction. The next stage in the investigation concerns the various local invariant manifolds around these fixed points and the associated critical exponents. The first such rigorous result, for the Gaussian fixed point, is the work of Bleher and Sinai [8]. For further developments, with emphasis on these dynamical systems aspects, see for instance [20, 36, 37, 49, 62, 74, 44]. Then, in the third stage, one would like to know more global features like how all these local invariant manifolds meet to form separatrices between domains exhibiting qualitatively different behaviours. This question pertains to the active field of the renormalization group theory of crossover phenomena (see e.g. [60, 55] for recent reviews). Our work falls within this third class of problems. The control of a massless RG trajectory between fixed points announced in [1] and for which details are provided here is our contribution to the grand scheme of the Wilson Program. Note that there is extensive physics literature, following the seminal work of Zamolodchikov [78, 79], on such massless RG flows in particular in two dimensions, see e.g. [80, 23, 28, 29] and references therein. However, nonperturbative results substantiated by rigorous mathematical estimates are scarce. To borrow the terminology of the French school of constructive field theory, this is the ‘problème de la soudure’ or the welding problem. One has to control the junction between the ultraviolet and the infrared regimes. For instance, for the two dimensional Gross-Neveu model, the UV regime has been given a rigorous mathematical treatment a long time ago [38, 32]. Likewise, the IR regime with spontaneous mass generation for a UV-cutoff theory is also under control [52]; see also [51] for a similar result on the sigma model. However, the junction, although probably not out of reach of present methods, has proved to be more technically demanding than expected [53]. Note that the model we consider here is simpler in that regard. It does not involve a drastic change of scenery, for instance, from a purely Fermionic theory at the ultraviolet end, to a Bosonic one at the infrared end. Given a small positive bifurcation parameter , we consider a three dimensional φ 4 theory with a modified propagator: the (Φ 4 )3, model of [15], which was also studied in the hierarchical approximation in [36]. Namely, we consider functional integrals of the form (1) dµC˜ (φ) . . . e−V (φ) ,
− def where dµC˜ is the Gaussian measure with covariance C˜ = (−∆) ordered interaction potential def V (φ) = d 3 x g : φ(x)4 :C˜ +µ : φ(x)2 :C˜ .
R3
3+ 4
and Wick
(2)
Over the last two decades, Brydges and his collaborators have devised a general mathematical framework, going beyond the hierarchical and local potential approximations, in order to give a rigorous nonperturbative meaning to the renormalization group dynamical system [17, 11, 12, 15]. The approach, actually involving no approximation whatsoever, is in the spirit of Wilson’s exact renormalization group scheme [56]. Our article which
A Complete Renormalization Group Trajectory
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can be viewed as a direct continuation of [15] takes place in this setting. In very rough terms, the renormalization group map, rather than flow, represents the evolution of the integrand I (φ) of functional integrals such as (1) under convolution and rescaling. The convolution is with respect to the Gaussian measure corresponding to Fourier modes p of the field φ which are restricted to a range of the form L n ≤ | p| ≤ L n+1 , where the integer L ≥ 2 is the scale ratio for one RG step. By rescaling, one can keep the integer n constant, and make the RG transformation autonomous. The latter acts on the integrand I (φ) and produces a new one I (φ). However, the problem with expressing the renormalization group in terms of its action on I (φ) is that I (φ) does not exist in the infinite volume limit. It is essential to express I (φ) in terms of coordinates that (i) are well defined in the infinite volume limit and (ii) carry the exact action of the renormalization group in a tractable form. The key feature that these coordinates have to express is that I (φ) is approximately a product of local functionals of the field and the action of the renormalization group is also approximately local. The first step towards these coordinates is to write I (φ) via the polymer representation: I (φ) = e−V (Λ\X,φ) K (X i , φ), (3) {X i }
i
where Λ is the volume cut-off needed to perform the thermodynamic limit, and {X i } is a collection of disjoint polymers X i in Λ. By polymer we mean a connected finite union of cubes cut by a fixed Z3 lattice inside R3 . The union of the X i has been denoted by X , and the functional V (Λ\X, φ) is given by (2) except that the integration domain is the complement Λ\X instead of R3 . Therefore, the functionals V are determined by the two variables or couplings g and µ. Now the K ’s are local functionals of the field, which means that K (Y, φ) only depends on the restriction of φ to the set Y . The knowledge of the integrand I (φ) amounts to that of the couplings g, µ together with the collection K of all the functionals K (Y, φ) corresponding to all possible polymers Y . One also needs a splitting K = Qe−V + R of these functionals where the Qe−V part is given explicitly in terms of g, µ only. In sum, the integrand is encoded by a triple (g, µ, R). The renormalization group map in [15] is implemented as a mathematically precise transformation (g, µ, R) → (g , µ , R ). The evolution for the : φ 4 : coupling g has the form (4) g = L g − L 2 a(L , )g 2 + ξg (g, µ, R) . The evolution of the mass term or : φ 2 : coupling µ has the form µ = L
3+ 2
µ + ξµ (g, µ, R) .
(5)
Finally the collection R of ‘irrelevant terms’, living in a suitable infinite dimensional space, evolves according to R = L(g,µ) (R) + ξ R (g, µ, R),
(6)
where L(g,µ) is a (g, µ)-dependent contractive linear map in the R direction. The ξ remainder terms are higher order small nonlinearities. An important feature of this formalism is that the polymer representation (3) is not unique. As a result, one has enough freedom when defining the RG map, in order to secure the contractive property of the L(g,µ) . This is the so-called ‘extraction step’ which encapsulates the renormalization subtractions familiar in quantum field theory. The transformation in [15] also carried an extra dynamical variable w with very simple evolution which is independent of the
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Fig. 1. The RG dynamical system
other variables, and converging exponentially fast to a fixed point w∗ . This was introduced in order to make the RG map autonomous. Throughout this article however, we take w = w∗ and incorporate w in the very definition of the RG map. In [15], it was shown that for small > 0 there exists an infrared fixed point (g∗ , µ∗ , R∗ ) which is an analog of the Wilson-Fisher fixed point [75], and which is nontrivial, i.e., distinct from the Gaussian ultraviolet fixed point (g, µ, R) = (0, 0, 0). The local stable manifold of the infrared fixed point was also constructed. Note that if one neglects the ξ remainders, one gets an approximate fixed point (g¯ ∗ , 0, 0), where def
g¯ ∗ =
L − 1 = O() . L 2 a(L , )
(7)
A schematic rendition of the phase portrait of the RG map considered in [15] is provided by Fig. 1. The precise statements of our main results, Theorem 2 and Corollary 1 below, require a substantial amount of machinery to be provided in the next sections. We can nevertheless already give an informal statement. Main Result. In the regime where > 0 is small enough, for any ω0 ∈]0, 21 [, there exists a complete trajectory (gn , µn , Rn )n∈Z for the RG map given by Eqs. (4), (5), and (6), such that lim (gn , µn , Rn ) = (0, 0, 0) the Gaussian ultraviolet fixed point, n→−∞
and lim (gn , µn , Rn ) = (g∗ , µ∗ , R∗ ) the BMS nontrivial infrared fixed point, and n→+∞
determined by the ‘initial condition’ at unit scale g0 = ω0 g¯ ∗ .
(8)
See Fig. 2 for a sketch of such discrete RG orbits Pn = (gn , µn , Rn ), n ∈ Z, which are parametrized by the projection of P0 on the g axis. To the best of our knowledge, the only previous similar result is the construction of the massless connecting heteroclinic orbit going from a UV nontrivial fixed point to the Gaussian IR fixed point for a modified Gross-Neveu model in [39] (see also [22, 30] for related work in the massive case). Our work which essentially amounts to the construction of a nontrivial massless three dimensional Euclidean field theory in the continuum, is probably the first such result in the Bosonic case. This field theory is superrenormalizable in the ultraviolet sector but only barely. Namely, one needs to renormalize divergent Feynman diagrams only
A Complete Renormalization Group Trajectory
731
Fig. 2. The discrete trajectories
up to a finite order in perturbation theory; however this order goes to infinity when the parameter goes to zero. As shown in [15, Sect. 1.1], a proof for the difficult axiom of Osterwalder-Schrader positivity seems feasible on this model, which makes it interesting from the point of view of traditional constructive field theory [43]. Due to the lack of a nonperturbative definition of dimensional regularization, this model is the best available for the mathematically rigorous study of the Wilson-Fisher fixed point [75] which is believed to govern the infrared behavior of the tridimensional Ising model (when = 1). On the technical side, as far as the construction of a global RG trajectory is concerned, one should note that the situation in [39] is facilitated by the availability of a convergent series representation in a whole neighborhood of the Gaussian fixed point which is only possible for a Fermionic theory. In the present situation, the ‘trivial’ fixed point around which the analysis takes place is not so trivial and in fact is highly singular from the point of view of the estimates we use. This is a manifestation of the so-called ‘large field problem’ and the need for the ‘domination procedure’ (see e.g. [64]). The norms needed for the control of R which implement a measurement of the 1 typical size of the field φ ∼ g − 4 through a parameter h appearing in the definition of these norms, create one of the main difficulties we had to overcome: the ‘fibered norm problem’. Namely, the norm for R involves the dynamical variable g. The approach we used is to construct the trajectory s = (gn , µn , Rn )n∈Z via its deviation δs with respect to an approximate trajectory (g¯ n , 0, 0)n∈Z which solves the RG recursion when the ξ terms are thrown out. This is done thanks to a contraction mapping argument in a big Banach space of sequences δs. This approach, in the spirit of Irwin’s proof of the stable manifold theorem [47, 67], was suggested to us by D. C. Brydges. We then realized that one can resolve the vicious circle entailed by the ‘fibered norm problem’ by using the approximate values g¯ n in the definition of the norms. In principle, Wilson’s RG picture reduces deep questions in quantum field theory and statistical mechanics to a chapter in the theory of bifurcations and dynamical systems. In practice, it has proved hard to get away with the application of a ready-made theorem from the corresponding literature, as emphasized in [20, p. 70] from the beginning of the subject and even for the simpler hierarchical models. Most of the works on the rigorous renormalization group use an ad hoc method developed in [9]. An innovation was introduced in [12], by the construction of the stable manifold of the nontrivial fixed point using an iteration in a space of sequences, along the lines of Irwin’s proof. The latter method seems more robust and easier to adapt to our present setting than the more standard Hadamard graph transform method [46, 67]. Formally, the RG map given by (4), (5), and (6), with bifurcation parameter corresponds to a transcritical bifurcation, according to the classification given e.g. in [19, p. 177]. The moving nontrivial fixed point goes through the Gaussian one as one increases the parameter. The negative region is forbidden however, since it would put the nontrivial fixed point in the undefined g < 0
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region. Most pertinent to the construction of a connecting heteroclinic orbit between RG fixed points, in the dynamical systems literature, is the article [50], which is based on Kelley’s center manifold theorem [48, 18, 68]. However, we have so far been unable to apply these methods in the present situation. In the same way [39] is based on the hard analysis estimates of [38], our proof is based on Theorem 1 below which summarizes a slight adaptation of the estimates in [15, Sect. 5] built on the techniques of [17, 11]. With the exception of the proof of this theorem which needs a working knowledge of [15, Sect. 5], our article can be read with only modest prerequisites in functional analysis as covered e.g. in [3, 7, 25], and in the theory of Gaussian probability measures in Hilbert spaces [10, 70]. We provided a completely self-contained definition of the renormalization group map (g, µ, R) → (g , µ , R ) in Sects. 2, 3, and 4. Apart from making the so-called extraction step explicit, this gives us the opportunity to correct some minor sign and numerical factor errors, but also one serious error, namely that in [15] the Banach fixed point theorem was used for a normed space that is not complete. Fortunately, we obtained, through discussions with D. C. Brydges and P. K. Mitter, an amendment which is provided in Sect. 3. It has the advantage that all the estimates in [15, Sect. 5] hold in this new setting without the need for a touch up. For more efficiency, in the sections defining the RG map, we adopted a rather terse style of presentation. We refer the newcomer seeking a proper motivation for this formalism to [56] and the introductory sections of [17, 11, 15]. Note that these definitions are quite involved and by no means the first that would come to one’s mind. Nevertheless, they are about the simplest which give a rigorous nonperturbative meaning to Wilson’s exact renormalization group, and at the same time navigate around the pitfalls of more naïve approaches. These pitfalls have been mapped by the pioneering work of Balaban, Federbush, Feldman, Gallavotti, Gaw¸edzki, Glimm, Jaffe, Kupiainen, Magnen, Rivasseau, Seiler, Sénéor, Spencer, and many others we apologize for not citing [43, 34]. For more ample introduction to the rigorous renormalization group than we provide here, the reader from other areas of mathematics may most profitably read [71, 66, 35, 5] and Gaw¸edzki’s lecture in [24] for a first contact. More technical or specialized material is covered in [6, 43, 64]. 2. The General Setting The ambient space for the field theory we are considering is Euclidean R3 . Given an eledef
3 ment x = (x1 , x2 , x3 ) ∈ R we will use the notation |x|∞ = max(|x1 |, |x2 |, |x3 |) and def |x|2 = x12 + x22 + x32 . Let be a small nonnegative number, then with a slight abuse − 3+
of notation the kernel of the covariance operator C˜ = (−∆) 4 , which is formally d 3 p i p(x−y) 2 − 3+ 4 ˜ ˜ − y) = e ( p ) , (9) C(x, y) = C(x 3 R3 (2π ) is given (see e.g. [40, Sect. II.3.3] for a careful derivation) for noncoinciding points by the Riesz potential κ ˜ − y) = C(x , (10) 3− 2 |x − y|2 with
Γ 3− − 3+ def − 3 4 2 2 (11) ×2 × 3+ . κ = π Γ 4
A Complete Renormalization Group Trajectory
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Let : R3 → R be a pointwise nonnegative C ∞ and rotationally invariant function def
which vanishes when |x|2 ≥ 21 and is equal to one when |x|2 ≤ 41 . Let u˜ = ∗ be the convolution of with itself. It is nonnegative both in direct and momentum spaces, and also rotationally invariant. Since (0) > 0, the integral −3 d 3 z |z|2 2 (z) R3
is strictly positive. We define the function u 0 to be the unique positive multiple of such that 3 −3 d 3 z |z|2 2 u 0 (z) = κ0 = (2π )− 2 . (12) R3
The u 0 function is fixed once and for all in this article. Now define κ
def
λ =
R3
− 3+ 2
d 3 z |z|2
,
(13)
u 0 (z)
and let u (x) = λ u 0 (x). Now we clearly have λ → 1 when → 0 and for x = y in R3 ,
+∞ x−y κ dl − 3− 2 ˜ − y) , l = u = C(x (14) 3− l l 2 0 |x − y|2 i.e., the canonically normalized noncutoff covariance. We now define the scale one UV-cutoff covariance C by
+∞ x−y dl − 3− def 2 l . (15) u C(x − y) = l l 1 Remark 1. In [15] the u is fixed whereas here it is a variable multiple of a fixed function u 0 . Since in the regime where is small the multiplier λ can be assumed to be say between 0.9 and 1.1; this has no effect on the estimates in [15] such as the large field stability bounds: Eq. 2.3, Lemma 5.3 and Lemma 5.4 therein. Let L ≥ 2 be an integer. We will also need the fluctuation covariance
L x−y dl − 3− def 2 Γ (x − y) = l . u l l 1
(16)
Note that in Eq. (11) the letter ‘Gamma’ denoted the usual Euler gamma function; however, from now on the notation will be reserved for the fluctuation covariance (16). The engineering scaling dimension of the field φ which is denoted by [φ] is defined by the ˜ ˜ property C(lx) = l −2[φ] C(x). One can read it off Eq. (14): [φ] = 3− 4 . As in [15] we use the notation def (17) C L (x) = L 2[φ] C(L x) def
for scaling of covariances. We define v (2) (x) = C L (x)2 − C(x)2 and let def a(L , ) = 36 d 3 x v (2) (x) . R3
(18)
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A. Abdesselam
It is a simple exercise in analysis to show that, regardless of the precise shape of the initial cutoff function u 0 , one has log L 18π 2
lim a(L , ) = a(L , 0) =
→0
(19)
as expected for the second order coefficient of the beta function of a marginal (at = 0) coupling. As a result, the approximate fixed point g¯ ∗ =
L − 1 L 2 a(L , )
(20)
satisfies g¯ ∗ ∼ 18π 2
(21)
when → 0. Now consider the lattice Z3 inside R3 . A unit box is any closed cube of the form [m 1 , m 1 + 1] × [m 2 , m 2 + 1] × [m 3 , m 3 + 1] with m = (m 1 , m 2 , m 3 ) ∈ Z3 . The set of all unit boxes is denoted by Box0 . A nonempty connected subset of R3 which is a finite union of unit boxes is called a polymer. The denumerable set of all polymers is denoted def
by Poly0 . We will also need the set Poly−1 = {L −1 X |X ∈ Poly0 } whose elements def
are called L −1 -polymers, as well as Poly+1 = {L .X |X ∈ Poly0 }, whose elements are called L-polymers. Unless otherwise specified, by polymer we will always mean a unit def
polymer, i.e., an element of Poly0 . For a polymer X , we denote |X | = Vol(X ) which is also the number of unit boxes in X . We also define its L-closure X¯ L as the union of all boxes of size L, cut by the (LZ)3 lattice, which contain a unit box in X . This is the same as the smallest L-polymer containing X , which explains the terminology. A polymer X ∈ Poly0 with |X | ≤ 8 is called a small polymer. A polymer X ∈ Poly0 with |X | ≤ 2 is called an ultrasmall polymer. A large polymer simply is one which is not small. We finally define the large set regulator which is a function A : Poly0 → R∗+ , by def
A(X ) = L 5|X | . 3. Functional Spaces 3.1. Sobolev spaces with gluing conditions. To each X ∈ Poly0 , we associate a real separable Hilbert space Fld(X ) where the fields φ : X → R will live. Given any open ◦
◦
unit box ∆, with ∆ ∈ Box0 , we consider the standard Sobolev space W 4,2 (∆) with the norm ⎛ ⎞1 2 def ⎝ ν 2 ⎠ ||∂ φ|| ◦ . (22) ||φ|| 4,2 ◦ = W
(∆)
|ν|≤4
L 2 (∆)
◦
Since obviously ∆ satisfies the so-called strong local Lipschitz condition, by the Sobolev embedding theorem [3, Theorem 4.12] one has a continuous injection ◦
W 4,2 (∆) → C 2 (∆),
A Complete Renormalization Group Trajectory
735
where C 2 (∆) is the real Banach space of functions φ : ∆ → R which are of class C 2 in ◦
the open box ∆ and which are continuous together with their first and second derivatives on all of the closed box ∆. The norm used on C 2 (∆) is the standard one ||φ||C 2 (∆) = sup max |∂ ν φ(x)| . def
(23)
x∈∆ |ν|≤2
Besides there is a constant CSobolev independent of the choice of ∆ in Box0 , such that ||φ||C 2 (∆) ≤ CSobolev ||φ||
◦
W 4,2 (∆)
.
(24) ◦
) to be the finite direct sum of the Hilbert spaces W 4,2 (∆) for ∆ Now define Fld(X ) obtained by imposing the contained in X . We let Fld(X ) be the subspace of Fld(X following gluing conditions. A field φ = (φ∆ )∆⊂X belongs to Fld(X ) if and only if, for any neighbouring boxes ∆1 , ∆2 in X , the C 2 images by the Sobolev embedding of φ∆1 and φ∆2 coincide as well as their first and second derivatives, on the common boundary component ∆1 ∩ ∆2 . Again by the embedding theorem, this is a closed condition, and Fld(X ) is a real Hilbert space with the norm ⎛ def ||φ||Fld(X ) = ⎝
⎞1 2
||∂ ν φ∆ ||2
◦
L 2 (∆)
∆⊂X |ν|≤4
⎠
.
(25)
Note that any polymer X is the closure of its interior. Hence, if one lets as before C 2 (X ) be the space of functions φ : X → R which are of class C 2 in the, possibly disconnected, ◦
open set X and which are continuous together with their first and second derivatives on the closed connected set X ; and if the norm used on C 2 (X ) is again the standard one ||φ||C 2 (X ) = sup max |∂ ν φ(x)| ; def
(26)
x∈X |ν|≤2
then it is not difficult to show that one has an embedding
and an inequality
Fld(X ) → C 2 (X )
(27)
||φ||C 2 (X ) ≤ CSobolev ||φ||Fld(X ) .
(28)
The important thing here is that the constant is independent of X . We will often regard φ as a single function on X . Remark 2. With this definition the lemmata [15, Lemma 5.1, Lemma 5.2] which are used for pointwise estimation of the fields, remain valid. The polygonal line arguments needed in [15, Lemma 5.1] as well as [11, Lemma 15] on which [15, Lemma 5.24] rests, are also preserved. Now we will also need the notation ⎛ def ⎝ ||φ|| X,1,4 =
∆⊂X 1≤|ν|≤4
⎞1 2
||∂
ν
φ∆ ||2 ◦ ⎠ L 2 (∆)
.
(29)
736
A. Abdesselam
This allows, given a parameter κ > 0, to define for any φ ∈ Fld(X ) the large field regulator def G κ (X, φ) = exp κ||φ||2X,1,4 . (30) An important point is that G κ (X, ·) is continuous on Fld(X ). 3.2. Some natural maps. Note that if X 1 ⊂ X 2 are two polymers then there is an obvious linear continuous restriction map Fld(X 2 ) → Fld(X 1 ), φ → φ| X 1 . Indeed one first 1 ). Namely, it projects φ = (φ∆ )∆⊂X 2 2 ) to Fld(X defines this projection from Fld(X onto (φ∆ )∆⊂X 1 . The gluing conditions for the image are automatically satisfied if they hold for the input φ. Now let τ be an isometry of Euclidean R3 which leaves the lattice Z3 globally invariant, and let X be a polymer. One has a natural Hilbert space isometry Fld(X ) → Fld(τ −1 (X )), φ → φ ◦τ . Indeed one first defines this map on elements φ = (φ∆ )∆⊂X ∈ ◦ ), where each component is smooth on ∆, by ordinary composition with τ . Then Fld(X ) → Fld(τ −1 (X )). Finally by density [3, Theorem 3.17], one extends it to a map Fld(X −1 one takes the restriction to Fld(X ) and corestriction to Fld(τ (X )), since the gluing conditions are preserved. We will also need an additional map. Let X ∈ Poly0 . Then L X is also in Poly0 . Given φ ∈ Fld(X ) one can associate to it by a linear continuous map an element ) is such that φ L −1 ∈ Fld(L X ) as follows. First assume that φ = (φ∆ )∆⊂X ∈ Fld(X ◦
def
each φ∆ is smooth on ∆. Then for each ∆ ⊂ X , define (φ∆ ) L −1 (x) = L −[φ] φ∆ (L −1 x) which is smooth in the interior of L∆. Then for any unit box ∆ ⊂ L∆ consider the restriction (φ∆ ) L −1 | ◦ to the interior of ∆ . The collection of all such restrictions for ∆ X ). Then extend the ∆ ⊂ L∆ with ∆ ⊂ X is by definition the image of φ in Fld(L ). Finally the wanted map is obtained by restriction to map, by density, to all of Fld(X Fld(X ) and corestriction to Fld(L X ), since the gluing conditions are easily seen to be preserved.
3.3. Gaussian measures. Now given any polymer X , and using the standard theory of Gaussian probability measures in Hilbert spaces [10, 70], it is not difficult to show that there exists a unique Borel (with respect to the ||.||Fld(X) norm topology) centered Gaussian probability measure dµΓ,X on Fld(X ) such that for any x, y ∈ X , one has (31) dµΓ,X (ζ ) ζ (x)ζ (y) = Γ (x − y), where ζ (x) and ζ (y) are defined using the C 2 (X ) realization of ζ . In other words the covariance of dµΓ,X is the fluctuation covariance Γ . ) → Fld(X ) as follows. If φ = Indeed, one can define a continuous operator S˜ : Fld(X ˜ = ( Sφ) ˜ ∆ ) has smooth components, one defines its image Sφ (φ∆ )∆⊂X ∈ Fld(X ∆⊂X
◦
by letting for any x ∈∆, ˜ ∆ (x) def ( Sφ) =
∆ ⊂X
|α|≤4 ∆
dy (−1)|α| ∂ α Γ (x − y) ∂ α φ∆ (y) .
(32)
A Complete Renormalization Group Trajectory
737
) to a continuous operator with norm It is easy to see that S˜ extends on all of Fld(X bounded by max|α|≤8 ||∂ α Γ || L ∞ (R3 ) . Clearly this operator S˜ has its image contained in the closed subspace Fld(X ). It is also symmetric, and positive. Now define the operator S : Fld(X ) → Fld(X ) by restriction and corestriction. It is easy to show that tr S˜ = tr S = |X | · (−1)|α| ∂ 2α Γ (0) . (33) |α|≤4
As a result S is a continuous symmetric positive trace class operator on Fld(X ), i. e., a covariance operator. By the results in [70, Chap. 1], there exists a unique centered Borel Gaussian probability measure dµΓ,X on Fld(X ) such that for any φ1 , φ2 ∈ Fld(X ), (34) dµΓ,X (ζ ) (φ1 , ζ )(φ2 , ζ ) = (φ1 , Sφ2 ) . ˜ It ) and with S replaced by S. This equality also holds for φ1 , φ2 more generally in Fld(X is not difficult to show that (31) follows from (34). The uniqueness of Gaussian measures satisfying (31) is also easy. Indeed one has the uniqueness of Gaussian measures satis ) fying (34), see [70, Chap. 1]. Besides, consider the continuous linear forms on Fld(X indexed by pairs (∆, x), where ∆ ⊂ X and x ∈ ∆, obtained by evaluating at x the ). Let ψ∆,x ∈ Fld(X ) be the C 2 (∆) image of the component φ∆ of a vector φ ∈ Fld(X corresponding vectors obtained by the Riesz representation theorem. By the injectivity of the Sobolev embedding, it is clear that the subspace generated by the vectors ψ∆,x is ). The uniqueness then follows easily. dense in Fld(X Finally, note that if X 1 ⊂ X 2 are two polymers, then the direct image measure of dµΓ,X 2 , obtained by the restriction map φ → φ| X 1 , coincides with dµΓ,X 1 . 3.4. Polymer activities. Let K denote either the (algebraic) field of real numbers R or that of complex numbers C. The main objects of study in this article are polymer activities or polymer amplitudes. These are functions (or functionals) K (X, ·) from Fld(X ) to K. We will only consider functionals which are n 0 times continuously differentiable in the sense of Frechet between the real Banach spaces Fld(X ) and K [7, Chap. 2], [25, Chap. VIII]. Here n 0 is a nonnegative integer constant which we will actually take to be n 0 = 9 as in [15]. Now consider for any integer n, 0 ≤ n ≤ n 0 , the K-Banach space Ln (Fld(X ), K) of R-multilinear continuous maps W : Fld(X )n → K with the natural norm def
||W || =
|W (φ1 , . . . , φn )| . ||φ || 1 Fld(X ) . . . ||φn ||Fld(X ) φ1 ,...,φn ∈Fld(X )\{0} sup
(35)
Inside it sits the space Ln (Fld(X ), C 2 (X ), K) of W ’s for which the stronger norm def
||W || =
|W (φ1 , . . . , φn )| ||φ || 1 C 2 (X ) . . . ||φn ||C 2 (X ) φ1 ,...,φn ∈Fld(X )\{0} sup
(36)
is finite. We indeed have for any W ∈ Ln (Fld(X ), C 2 (X ), K), n ||W || ≤ CSobolev ||W || .
(37)
It is easy to see that Ln (Fld(X ), C 2 (X ), K) equipped with the sharp norm is a K-Banach space. Let us denote by Cn 0 (Fld(X ), K) the K-vector space of K-valued functionals
738
A. Abdesselam
K (X, ·) defined on all of Fld(X ), which are n 0 times continuously Frechet differentiable in the usual sense [7, 25] with respect to the || · ||Fld(X ) topology. We will also denote the n th Frechet differential at the point φ ∈ Fld(X ) of a polymer activity K (X, ·) by D n K (X, φ). Its evaluation at the sequence of vectors f 1 , . . . , f n of Fld(X ) is ∂n n D (X, φ; f 1 , . . . , f n ) = K (X, φ + s1 f 1 + · · · + sn f n ) , (38) ∂s1 . . . ∂sn s=0 i.e., the corresponding directional or Gateau derivative. We then define the space Cn 0 (Fld(X ), K) of all K (X, ·) ∈ Cn 0 (Fld(X ), K) such that for all φ ∈ Fld(X ) and all integer n, 0 ≤ n ≤ n 0 , the differential D n K (X, φ) belongs to Ln (Fld(X ), C 2 (X ), K), and such that the maps φ → D n K (X, φ) are continuous from (Fld(X ), || · ||Fld(X ) ) to (Ln (Fld(X ), C 2 (X ), K), || · || ). From now on the only norm we will be considering for differentials is the sharp one, therefore we will omit the symbol from the norm notation. Given a parameter h > 0, a functional K (X, ·) ∈ Cn 0 (Fld(X ), K) and a field φ ∈ Fld(X ), we define the local norm def
||K (X, φ)||h =
hn ||D n K (X, φ)|| . n!
(39)
0≤n≤n 0
K (X ) of all K (X ) ∈ C n 0 (Fld(X ), K) for which This allows us to define the space Bh,G κ the norm def (40) ||K (X )||h,G κ = sup G κ (X, φ)−1 ||K (X, φ)||h φ∈Fld(X )
is finite. Now one has the following easy proposition. Proposition 1. For any h, κ > 0, the normed K-vector space K (X ), || · ||h,G κ ) (Bh,G κ
is complete. Now we consider an arbitrary element K = (K (X )) X ∈Poly0 in the product X ∈Poly0
K Bh,G (X ) , κ
and define the norm def
||K ||h,G κ ,A =
sup
∆∈Box0
A(X ) ||K (X )||h,G κ ,
(41)
X ∈Poly0 X ⊃∆
where A is the previously defined large set regulator. Given a parameter h ∗ > 0, we also define the kernel semi-norm, def
|K |h ∗ ,A =
sup
∆∈Box0
X ∈Poly0 X ⊃∆
A(X ) ||K (X, 0)||h ∗ ,
(42)
A Complete Renormalization Group Trajectory
739
where the differentials are taken at the point φ = 0 in each Fld(X ). We now introduce the notion of calibrator: it is a new parameter g¯ > 0. We will use it to set 1
h = c g¯ − 4
(43)
for some fixed constant c > 0 to be adjusted later. We will take def
h∗ = L
3+ 4
.
(44)
The space of all K in the previous product space, such that ||K ||h,G κ ,A and |K |h ∗ ,A are finite, is equipped with the calibrated norm def |||K |||g¯ = max |K |h ∗ ,A , g¯ 2 ||K ||h,G κ ,A , (45) and it is denoted by BBgK ¯ . So as to keep notations under control we only emphasized the dependence on the calibrator g¯ which is the most important one in what follows. One should keep in mind that the calibrated norm depends on g¯ through the g¯ 2 factor in front of || · ||h,G κ ,A , but also through the relation (43) imposed between the h parameter and the calibrator g. ¯ It is easy to see that BBgK ¯ with the norm ||| · |||g¯ , is a K-Banach space. Now let τ be an isometry of Euclidean R3 which leaves the lattice Z3 globally invariant. This transformation can be made to act on an element K of BBgK ¯ by letting for any X ∈ Poly0 , and any φ ∈ Fld(X ), (τ K )(X, φ) = K (τ −1 (X ), φ ◦ τ ), def
(46)
where the map from Fld(X ) to Fld(τ −1 (X )), given by φ → φ ◦ τ is the one defined in Sect. 3.2. We will only consider τ ∈ Transf, where the set Transf is made of all translations by a vector m = (m 1 , m 2 , m 3 ) in Z3 , together with the three orthogonal reflections with respect to the coordinate planes respectively given by the equations x1 = 0, x2 = 0 and x3 = 0. We also define a transformation K → K − of BBgK ¯ by def
letting K − (X, φ) = K (X, −φ). The following lemma is an easy consequence of our previous definitions for norms. Lemma 1. The maps K → τ K , for τ ∈ Transf, as well as the map K → K − , are Banach space isometries of BBgK ¯ . Thanks to this lemma we can finally define the main setting for a single RG map. It K − is the space BBSgK ¯ of all collections of polymer activities K ∈ BBg¯ such that K = K and for any τ ∈ Transf, τ K = K . By the previous lemma it is a closed subspace of BBgK ¯ and therefore a K-Banach space for the norm ||| · |||g¯ . Remark 3. Note that all the calibrated norms, obtained for different values of g, ¯ are ’s are therefore the equivalent. The underlying topological vector spaces of the BBgK ¯ same. The RG map we are interested in is one from a domain in R × R × BBSgR ¯ for some values of the parameters into another analogous triple-product space with a slightly different value of g. ¯ We will need complex versions of these spaces in order to obtain Lipschitz contractive estimates with the least effort. The global trajectory we construct in this article will be obtained by a contraction mapping theorem in a big Banach space of sequences BBSS K to be precisely defined in Sect. 5 below.
740
A. Abdesselam
4. The Algebraic Definition of the RG Map In this section we provide all the formulae which express the RG map studied in [15]. We consider an input (g, µ, R) ∈ C × C × BBS C ; and we will give the algebraic definition for the output (g , µ , R ). Recall that the latter have the form g = L g − L 2 a(L , )g 2 + ξg (g, µ, R) , 3+ 2
µ = L
µ + ξµ (g, µ, R) ,
(g,µ)
R =L
(47) (48)
(R) + ξ R (g, µ, R),
(49)
where a(L , ) has already been defined. We will therefore provide the expressions for the ξ remainders as well as for L(g,µ) (R). 4.1. The local potentials. For any X ∈ Poly0 , any Borel set Z ⊂ R3 , and any φ ∈ Fld(X ), we let def d 3 x : φ(x)4 :C +µ d 3 x : φ(x)2 :C . (50) V (X, Z , φ) = g Z ∩X
Z ∩X
We refer for instance to [43, 66] for a discussion of Wick ordering : • :C . Otherwise the explicit expressions : φ(x)2 :C = φ(x)2 − C(0) (51) and : φ(x)4 :C = φ(x)4 − 6C(0)φ(x)2 + 3C(0)2
(52)
may be used as definitions. Note that in [15] the notation is simplified to V (Z , φ) or even V (Z ) leaving the φ dependence implicit. Here we prefer to keep everything explicit including the first X argument which allows one to keep track of which space Fld(·) the field φ lives in. Also note that the function φ used in the integral formula above is of course the C 2 (X ) realization of φ ∈ Fld(X ) via the embedding (27). Another remark is that although we made the definition sound quite general by allowing Z to be any Borel set, we will only need such Z ’s which are complements of the union of some L −1 -polymers in X . Now define gL = L g , def
def
3+ 2
def
−2[φ]
µL = L C L −1 (x) = L
(53)
µ,
(54)
C(L
−1
x),
(55)
and as in (50) let def V˜ (X, Z , φ) = g
d x : φ(x) :C L −1 +µ 3
Z ∩X
4
Z ∩X
d 3 x : φ(x)2 :C L −1 ,
where Wick ordering is with respect to C L −1 instead of C. Also let def d 3 x : φ(x)4 :C +µ L d 3 x : φ(x)2 :C . V˜ L (X, Z , φ) = g L Z ∩X
Z ∩X
(56)
(57)
A Complete Renormalization Group Trajectory
741
4.2. The w kernels. We now deal with the hidden variable w. Note that by construction the cutoff function u satisfies u (x) = 0 if |x|2 ≥ 1 and a fortiori if |x|∞ ≥ 1. This implies that the fluctuation covariance Γ satisfies Γ (x) = 0 if |x|∞ ≥ L. Now we define w = w∗ = (w (1) , w (2) , w (3) ) to be a triple of real functions w ( p) ∈ W p , where 3p
W p , p = 1, 2, 3, is the weighted L ∞ space L ∞ (R3 , |x|∞2 d 3 x). Namely, f ∈ W p if and only if f : R3 → R is measurable and 3p
def || f || p = ess. sup |x|∞2 | f (x)| (58) x∈R3
is finite. The w’s were constructed in [15, Lemma 5.9] by a Banach fixed point argument. We instead give them explicitly, for x = 0, by def ˜ p w ( p) (x) = C(x) − C(x) p 1 x p dl − 3− 2 l = C(x) + u − C(x) p . l 0 l
(59) (60)
From the last equation it is clear that w( p) (x) = 0 if |x|∞ ≥ 1. Besides, since u ≥ 0, for small one has p ˜ = |w( p) (x)| ≤ C(x)
p
κ
p
|x|2
3− 2
≤
O(1) 3p
.
(61)
|x|∞2
The fixed point property w = w∗ is embodied in the equation ( p)
w ( p) (x) = v ( p) (x) + w L (x)
(62)
for any x = 0, where we used the notation v ( p) (x) = C L (x) p − C(x) p def
and
( p)
w L (x) = L 2 p[φ] w ( p) (L x) . Equation (62) trivially follows from the given definition. def
(63) (64)
4.3. The renormalized expanded quadratic activity Q. For X ∈ Poly0 and φ ∈ Fld(X ) def
we define the activity Q(X, φ) as follows. If X is not ultrasmall we let Q(X, φ) = 0. If X is ultrasmall we introduce an associated integration domain X˜ ⊂ R3 × R3 . If X is def reduced to a single unit box ∆, we let X˜ = ∆ × ∆. If X = ∆1 ∪ ∆2 , where the boxes ∆1 and ∆2 are distinct but neighbouring, we let def X˜ = (∆1 × ∆2 ) ∪ (∆2 × ∆1 ) .
We now write def
Q(X, φ) = g 2
X˜
(65)
d 3 x d 3 y −24w (3) (x − y) : (φ(x) − φ(y))2 :C
−18w (2) (x − y) : (φ(x)2 − φ(y)2 )2 :C + 8w (1) (x − y) : φ(x)3 φ(y)3 :C .
(66)
742
A. Abdesselam
For reference, the Wick ordered expressions are explicitly given by : (φ(x) − φ(y))2 :C = (φ(x) − φ(y))2 − 2C(0) + 2C(x − y) , : (φ(x)2 − φ(y)2 )2 :C = (φ(x)2 − φ(y)2 )2 − 4C(0)φ(x)2 − 4C(0)φ(y)2 +8C(x − y)φ(x)φ(y) + 4C(0)2 − 4C(x − y)2 ,
(67) (68)
and : φ(x)3 φ(y)3 :C = φ(x)3 φ(y)3 − 3C(0)φ(x)φ(y)3 − 3C(0)φ(x)3 φ(y) −9C(x − y)φ(x)2 φ(y)2 + 9C(0)2 φ(x)φ(y) +18C(x − y)2 φ(x)φ(y) + 9C(0)C(x − y)φ(x)2 +9C(0)C(x − y)φ(y)2 − 9C(0)2 C(x − y) −6C(x − y)3 .
(69)
4.4. Integration on fluctuations, reblocking and rescaling. For any unit box ∆ and fields φ, ζ ∈ Fld(∆) we define ˜
P(∆, φ, ζ ) = e−V (∆,∆,φ+ζ ) − e−V (∆,∆,φ) . def
(70)
Now for any X ∈ Poly0 and φ ∈ Fld(X ) we let K (X, φ) = Q(X, φ)e−V (X,X,φ) + R(X, φ) . def
We also define
def
R (X, φ) = as well as (S K ) (X, φ) =
def
dµΓ,L X (ζ )
dµΓ,X (ζ ) R(X, φ + ζ ) ,
⎧ ⎪ ⎨ ⎪ ⎩
(71)
M,N M+N ≥1
1 M!N !
(72)
(∆1 ,...,∆ M ),(X 1 ,...,X N )
M exp −V˜ L X, L X \ ∪i=1 ∆i ∪ ∪ Nj=1 X j , φ L −1 ×
M i=1
⎫ N
⎬ , P ∆i , φ L −1 |∆i , ζ |∆i × K X j , φ L −1 | X j +ζ | X j ⎭
(73)
j=1
where the sum over sequences (∆1 , . . . , ∆ M ) and (X 1 , . . . , X N ) is subjected to the following conditions: 1. The ∆i are distinct boxes in Box0 . 2. The X j are disjoint polymers in Poly0 . 3. None of the ∆i is contained in an X j . 4. The L-closure of the union of all the ∆i and the X j is exactly the set L X . Remark 4. Note that since the X j are closed polymers, the disjointness condition means that they cannot touch each other and have to be at least 1 apart in | · |∞ distance. However, the ∆i are allowed to touch each other or an X j , by sharing no more than a boundary component. Also note that by hypothesis, X and therefore L X is connected. This rules out situations where for instance the (X j ) sequence would be empty, and the (∆i ) sequence would be made of two boxes very far apart.
A Complete Renormalization Group Trajectory
743
4.5. Preparations for the extraction. As a preparation for the crucial so called extraction step we need to introduce for any X ∈ Poly0 the quantities denoted by α˜ 0 (X ), α˜ 2 (X ), α˜ 2,µ (X ) for µ = 1, 2, 3, and α˜ 4 (X ). These are by definition all set to zero if X is large. Now if X is small one lets def
α˜ 0 (X ) = def
α˜ 2 (X ) =
˜
e V (X,X,0) R (X, 0) , |X |
(74)
˜ e V (X,X,0) 2 D (R )(X, 0; 1, 1) + R (X, 0)D 2 V˜ (X, X, 0; 1, 1) , 2|X |
(75)
where the last two arguments of the differentials are given by the constant function equal to 1, seen as an element of Fld(X ). We also let for µ = 1, 2, 3, def
α˜ 2,µ (X ) =
˜ e V (X,X,0) 2 D (R )(X, 0; 1, ∆ X xµ ) |X | +R (X, 0)D 2 V˜ (X, X, 0; 1, ∆ X xµ ) ,
where ∆ X xµ means the function 1 x → xµ − |X |
(76)
d y yµ , 3
X
the deviation from average of the coordinate function xµ on the polymer X , again seen as an element of Fld(X ). Finally one lets def
α˜ 4 (X ) =
˜ e V (X,X,0) 4 D (R )(X, 0; 1, 1, 1, 1) 24|X | +6D 2 (R )(X, 0; 1, 1)D 2 V˜ (X, X, 0; 1, 1) +R (X, 0)D 4 V˜ (X, X, 0; 1, 1, 1, 1) 2 . +3R (X, 0) D 2 V˜ (X, X, 0; 1, 1)
Now given Z ∈ Poly0 , and x ∈ R3 we define def α0 (Z , x) = α˜ 0 (X )L 3 1l L −1 X (x) ,
(77)
(78)
X small, X¯ L =L Z def
α2 (Z , x) =
α˜ 2 (X )L
3+ 2
1l L −1 X (x) ,
(79)
X small, X¯ L =L Z def
α2,µ (Z , x) =
α˜ 2,µ (X )L
1+ 2
1l L −1 X (x) ,
(80)
X small, X¯ L =L Z def
α4 (Z , x) =
α˜ 4 (X )L 1l L −1 X (x),
(81)
X small, X¯ L =L Z
where again µ = 1, 2, 3, and 1l L −1 X denotes the sharp characteristic function of the set L −1 X . Note that these quantities vanish if Z is not small or if x ∈ / Z.
744
A. Abdesselam
Now choose some reference box ∆0 ∈ Box0 . We define
def
α0 = L 3
α˜ 0 (X ),
(82)
X small, X ⊃∆0 def
α2 = L
3+ 2
α˜ 2 (X ),
(83)
X small, X ⊃∆0
α4 = L def
α˜ 4 (X ).
(84)
X small, X ⊃∆0
Note that the latter do not depend on the choice of ∆0 because of the translational invariance imposed on polymer activities in Sect. 3. Also note that in [15, Eq. 4.44] the quantities 1+ def α2,µ = L 2 α˜ 2,µ (X ) (85) X small, X ⊃∆0
for µ = 1, 2, 3, were also defined. However, again by the conditions imposed on polymer activities in Sect. 3, it is easy to see that the latter always vanish. In other words, the RG flow does not create φ∂φ terms in the effective potential. After one has defined def b(L , ) = 48 d 3 x v (3) (x) ; (86) R3
one can at last give some of the outputs of the RG map. Namely, one poses def
ξg (g, µ, R) = −α4 , def ξµ (g, µ, R) = − L 2 b(L , )g 2 + α2 + 6C(0)α4 ,
(87) (88)
as definition of the first two remainder terms. At this point, the new couplings are defined via g = L g − L 2 a(L , )g 2 + ξg (g, µ, R) , def
def
µ = L
3+ 2
µ + ξµ (g, µ, R) .
(89) (90)
What remains is L(g,µ) (R), ξ R (g, µ, R) and their combination R . 4.6. The linear map for R. In order to define the linear part L(g,µ) (R) which was denoted by Rlinear in [15], we need to introduce two polymer activities. For X ∈ Poly0 , and def φ ∈ Fld(X ), we let F˜ R (X, φ) = 0 if X is large; otherwise we let def F˜ R (X, φ) =
d 3 x α˜ 4 (X )φ(x)4 + α˜ 2 (X )φ(x)2 X ⎤ 3 + α˜ 2,µ (X )φ(x)∂µ φ(x) + α˜ 0 (X )⎦ . µ=1
(91)
A Complete Renormalization Group Trajectory
745
Regardless of whether X is small or not, we also let ˜
J (X, φ) = R (X, φ) − F˜ R (X, φ)e−V (X,X,φ) . def
(92)
The previous complicated definitions of the α˜ ... (X ) had no other purpose but to secure the following normalization conditions. For any small polymer X , and for µ = 1, 2, 3, one needs J (X, φ) = 0 , D 2 J (X, 0; 1, 1) = 0 , D 2 J (X, 0; 1, ∆ X xµ ) = 0 ,
(93) (94) (95)
D 4 J (X, 0; 1, 1, 1, 1) = 0 .
(96)
Note that one would have equivalent conditions if one replaced the function ∆ X xµ simply by the coordinate function xµ . These normalization conditions are the analog in the present setting of the BPHZ subtraction prescription (see e.g. [64]). They are the main reason why the map L(g,µ) (·) we are about to define is contractive. Now given X ∈ Poly0 , and φ ∈ Fld(X ), and using constrained sums over polymers Y ∈ Poly0 , we define
L(g,µ) (R)(X, φ) =
def
˜
J (Y, φ L −1 |Y )e−VL (X,X \L
−1 Y,φ)
Y small, Y¯ L =L X
+
˜
R (Y, φ L −1 |Y )e−VL (X,X \L
−1 Y,φ)
.
(97)
Y large, Y¯ L =L X
4.7. The extraction proper. Given X ∈ Poly0 , and x ∈ R3 , we define the function (4) f Q (X, x) as follows. First case. If X is given by a single box ∆ ∈ Box0 , and if x lies in the interior of ∆, we let def (4) f Q (X, x) = d 3 y v (2) (x − y) . (98) ∆
Second case. If X is given by the union of two distinct neighbouring boxes ∆1 , ∆2 ∈ Box0 , and if x lies in the interior of say ∆1 , we let def f Q(4) (X, x) = d 3 y v (2) (x − y) . (99) ∆2
Third case. If none of the first two cases apply, we simply let f Q(4) (X, x) = 0. def
(2)
One can in the same manner define a function f Q (X, x) using v (3) instead of v (2) , as well as a function f Q(0) (X, x) using v (4) which is given by v (4) (z) = C L (z)4 − C(z)4 . Now let X ∈ Poly0 , and Z be a Borel set in R3 , and define def F0,Q (X, Z ) = 12g 2L d 3 x f Q(0) (X, x), (100) def
Z
746
A. Abdesselam
as well as def
d 3 x α0 (X, x) + C(0)α2 (X, x) + 3C(0)2 α4 (X, x)
F0,R (X, Z ) =
(101)
Z
and
def
F0 (X, Z ) = F0,Q (X, Z ) + F0,R (X, Z ) .
(102)
If in addition one has a polymer Y ∈ Poly0 , and a field φ ∈ Fld(Y ), one can also define def (4) d 3 x : φ(x)4 :C f Q (X, x) F1,Q (X, Y, Z , φ) = 36g 2L Z ∩Y 2 +48g L d 3 x : φ(x)2 :C f Q(2) (X, x) (103) Z ∩Y
as well as def
F1,R (X, Y, Z , φ) =
+
Z ∩Y 3
d 3 x α4 (X, x) : φ(x)4 :C α2,µ (X, x) : φ(x)∂µ φ(x) :C
µ=1
+ (α2 (X, x) + 6C(0)α4 (X, x)) : φ(x)2 :C ,
(104)
where : φ(x)∂µ φ(x) :C reduces to φ(x)∂µ φ(x). We finally need def
F1 (X, Y, Z , φ) = F1,Q (X, Y, Z , φ) + F1,R (X, Y, Z , φ) , and
def
F(X, Y, Z , φ) = F0 (X, Z ) + F1 (X, Y, Z , φ) .
(105) (106)
As before the Y argument is for keeping track of which Fld(·) the φ lives in. The Z defines the domain of integration. The new argument X , is here to indicate that the F’s are local counterterms for a polymer activity which originally lived on X . Now given X ∈ Poly0 , and φ ∈ Fld(X ), we let ˜ def K˜ (X, φ) = (S K ) (X, φ) − e−VL (X,X,φ) ×
1 N!
N ≥1 N
!
" exp F Yi , Yi , Yi , φ|Yi − 1 ,
(Y1 ,...,Y N )
(107)
i=1
where the sum is over all sequences of distinct polymers Yi ∈ Poly0 whose union is equal to X . Again given X ∈ Poly0 , a Borel set Z , and a field φ ∈ Fld(X ), we define ⎡ ⎤ def ⎢˜ ⎥ VF (X, Z , φ) = F (Y, ∆, ∆, φ|∆ )⎦ . (108) ⎣VL (∆, ∆, φ|∆ ) − ∆∈Box0 ◦ ∆⊂Z ∩X
Y ∈Poly0 Y ⊃∆
A Complete Renormalization Group Trajectory
747 ◦
Mind the inclusion condition only on the interior ∆ of ∆. Then for X ∈ Poly0 , and φ ∈ Fld(X ), we let
def ˜ E(X, φ) =
M≥1, N ≥0
1 M!N !
(X 1 ,...,X M ), (Z 1 ,...,Z N )
M
M exp −VF X, X \ ∪i=1 Xi , φ × K˜ X i , φ| X i i=1
×
N
M exp −F Z j , Z j , Z j \ ∪i=1 X i , φ| Z j − 1
(109)
j=1
with the following conditions imposed on the X i and Z j : 1. 2. 3. 4. 5. 6.
The X i and Z j are polymers in Poly0 . The X i are disjoint. The Z j are distinct. M X . Every Z j has a nonempty intersection, be it by an edge or a corner, with ∪i=1 i
M Every Z j has a nonempty intersection with X \ ∪i=1 X i . The union of all the X i and Z j is exactly the given polymer X .
Remark 5. We emphasized the condition on the interior of ∆ in (108), and the weak notion of intersection in items (4) and (5) above, since these are the notable modifications to make on the treatment of [11, Sect. 4.2] in order to account for the closed polymers used in [15] and here. The overlap connectedness in [11, Sect. 4.2] is automatically implied by item (6) above and the connectedness of the set X which is assumed a priori. Also note that this notion was defined in [11, Sect. 4.2] based on the idea of having a full box in common, whereas here a nonempty intersection by a boundary component already counts as an overlap. Finally note that if M ≥ 2 then one needs to have N ≥ 1; this is because the X i are forced to be at least 1 apart with respect to the | · |∞ distance, and they need a bridge of Z j ’s joining them. Now given X ∈ Poly0 , and φ ∈ Fld(X ), we let ⎡ def ˜ ⎢ E(X, φ) = E(X, φ) × exp ⎣−
⎤
⎥ F0 (Y, ∆)⎦ .
(110)
∆∈Box0 Y ∈Poly0 ∆⊂X Y ⊃∆
Finally we define Q (X, φ) in exactly the same way as Q(X, φ) in Sect. 4.3 but using the new coupling g obtained in Sect. 4.5 instead of the old one g. Likewise we need a potential V (X, Z , φ) defined in the exact same manner as V (X, Z , φ) in Sect. 4.1 using the new couplings g , µ instead of g, µ. At last one can give the output R of the RG map defined for any X ∈ Poly0 , and φ ∈ Fld(X ) by R (X, φ) = E(X, φ) − Q (X, φ)e−V def
(X,X,φ)
.
(111)
In somewhat of a roundabout manner, the definition of the ξ R remainder is then ξ R (g, µ, R)(X, φ) = R (X, φ) − L(g,µ) (R)(X, φ) . def
(112)
748
A. Abdesselam
The algebraic definition of the RG map is now complete. Note that the Frechet differentiability of the output polymer activities, the justification of the measurability of the integrations over ζ , follow once the proper estimates are established because of the algebraic nature of the operations used in this section. These estimates have been provided in [15, Sect. 5], and their result is summarized in Theorem 1 below. 5. The Dynamical System Construction The RG map for which the defining formulae were given in the previous section is (g, µ, R) → (g , µ , R ), where ⎧ 2 2 ⎨ g = L g − L a(L , )g + ξg (g, µ, R) , 3+ (113) µ = L 2 µ + ξµ (g, µ, R) , ⎩ R = L(g,µ) (R) + ξ R (g, µ, R) . Our aim is to construct a double-sided sequence s = (gn , µn , Rn )n∈Z which solves this recursion and such that lim (gn , µn , Rn ) = (0, 0, 0) the Gaussian ultraviolet n→−∞
fixed point, and lim (gn , µn , Rn ) = (g∗ , µ∗ , R∗ ) the BMS nontrivial infrared fixed n→+∞
point [15]. We proceed as follows. We will simply write a for the coefficient a(L , ) > 0. We also take > 0 small enough so that L ∈]1, 2[. Recall that g¯ ∗ = LL 2−1 > 0 and a consider the function f : [0, g¯ ∗ ] → [0, g¯ ∗ ] (114) x → f (x) = L x − L 2 ax 2 . It is trivial to see that f is a strictly increasing diffeomorphism of [0, g¯ ∗ ]; it is also strictly concave. The only fixed points are 0 and g¯ ∗ , and f (x) > x in the interval ]0, g¯ ∗ [. Given ω0 ∈]0, 1[, there is a unique double-sided sequence (g¯ n )n∈Z in ]0, g¯ ∗ [Z such that g¯ 0 = ω0 g¯ ∗ , and for any n ∈ Z, g¯ n+1 = f (g¯ n ). This sequence is strictly increasing from 0 when n → −∞, to g¯ ∗ when n → +∞. We call g¯ 0 the coupling at unit scale. Once it is chosen it defines the sequence (g¯ n )n∈Z completely. Moreover, if one ignores the remainder terms ξ in (113) then the renormalization group recursion is def
solved by the approximate sequence s¯ = (g¯ n , 0, 0)n∈Z . The true trajectory will be constructed in such a way that g0 = g¯ 0 , and via the construction of the deviation sequence δs = (δgn , µn , Rn )n∈Z with respect to the approximate sequence s¯ . Using the notation δgn = gn − g¯ n , the new recursion which is equivalent to (113) that we have to solve is ! 2 " ⎧ 2 ⎨ δgn+1 = f (g¯ n )δgn + −L a δgn + ξg (g¯ n + δgn , µn , Rn ) , 3+ (115) = L 2 µn + ξµ (g¯ n + δgn , µn , Rn ) , µ ⎩ n+1 Rn+1 = L(g¯n +δgn ,µn ) (Rn ) + ξ R (g¯ n + δgn , µn , Rn ) . The boundary conditions we will need can roughly be stated as: – δg0 = 0. – µn does not blow up when n → +∞. – Rn does not blow up when n → −∞. Also note the behavior of the linear parts of (115) : – When n → +∞, f (g¯ n ) → 2 − L < 1, i.e., one has a deamplification. – When n → −∞, f (g¯ n ) → L > 1, i.e., one has an amplification.
A Complete Renormalization Group Trajectory
749
– One always has an amplification in the ‘relevant’ µ or mass direction. – Once the RG map has been properly defined, one can arrange to always have a deamplification in the ‘irrelevant’ R direction. Based on these observations, it is natural using the standard method of associated ‘discrete integral equations’, used for instance in [47], to rewrite the system (115) as ∀n > 0,
2 δgn = f (g¯ n−1 )δgn−1 + −L 2 a δgn−1 + ξg (g¯ n−1 + δgn−1 , µn−1 , Rn−1 ) , (116)
∀n < 0, δgn = ∀n ∈ Z,
1 1 2 2 δg −L − a δg + ξ ( g ¯ + δg , µ , R ) , n+1 g n n n n n f (g¯ n ) f (g¯ n ) − 3+ 2
− 3+ 2
µn+1 − L ξµ (g¯ n + δgn , µn , Rn ) , µn = L ∀n ∈ Z, Rn = L(g¯n−1 +δgn−1 ,µn−1 ) (Rn−1 ) + ξ R (g¯ n−1 + δgn−1 , µn−1 , Rn−1 ) ,
(117)
(118) (119)
and iterate, i.e., replace the linear term occurrences of the dynamical variables δg, µ, R, in terms of the analogous equations for n − 1 or n + 1, and repeat ad nauseam until one hits a boundary condition. In sum, the true sequence we are seeking will be constructed as a fixed point of a map s → s or rather δs → m(δs), which to a sequence δs = (δgn , µn , Rn )n∈Z associates the new sequence m(δs) = (δgn , µn , Rn )n∈Z which is given as follows. Definition 1. The map on sequences. Leaving the issue of convergence for later, the defining formulae for the map m are : δg0 = 0 , def
(120)
∀n > 0,
⎛
δgn =
def
⎝
0≤ p
⎞
f (g¯ j )⎠ −L 2 a δg 2p + ξg (g¯ p + δg p , µ p , R p ) ,
p< j
(121) ∀n < 0, def δgn =
−
⎛
⎝
n≤ p<0
n≤ j≤ p
⎞ 1 ⎠ 2 2 −L a δg + ξ ( g ¯ + δg , µ , R ) , g p p p p p f (g¯ j ) (122)
∀n ∈ Z, def µn =
−
p≥n
L
− 3+ 2 ( p−n+1)
ξµ (g¯ p + δg p , µ p , R p ) ,
(123)
750
A. Abdesselam
and finally ∀n ∈ Z, Rn =
def
L(g¯n−1 +δgn−1 ,µn−1 ) ◦ L(g¯n−2 +δgn−2 ,µn−2 ) ◦ · · ·
p
··· ◦ L
ξ R (g¯ p + δg p , µ p , R p ) ,
(124)
where the composition ◦ is of course with respect to the R argument. We now come to the definition of the space in which the deviation sequences δs will 3 live. Let us introduce as in [15] the exponent drops δ ∈ [0, 16 ] and η ∈ [0, 16 ] which will be fixed later. We will also define for n ∈ Z, ' def 1 if n ≤ 0 , en = 3 (125) 2 if n ≥ 1 . Now we define the big Banach space of sequences K × K × BBSgK BBSS K ⊂ ¯n
(126)
n∈Z
whose elements are all deviation sequences δs = (δgn , µn , Rn )n∈Z for which the quadruple norm ' ( −( 11 def −en −(2−δ) 4 −η) (127) ||||δs|||| = sup max |δgn |g¯ n , |µn |g¯ n , |||Rn |||g¯n g¯ n n∈Z
is bounded and such that δg0 = 0. Note that the approximate sequence s¯ itself does not belong to BBSS K which somewhat plays the role of a tangent space around it. As an easy consequence of our definitions one has the following proposition. Proposition 2. The space
BBSS K , |||| · ||||
is complete. 6. The BMS Estimates on a Single RG Step The estimates in [15, Sect. 5], slightly modified for the needs of the present construction, can be summarized by Theorem 1 below. Before stating the theorem one can give a brief description of the main ideas behind the estimates of [15, Sect. 5]. Given some a priori hypotheses on the size of the input g, µ, R of the RG map, the goal is to prove estimates on the output g , µ , R . The size of these variables is typically measured in powers of the φ 4 coupling g. However the latter is a dynamical variable of the problem, and in order to avoid a vicious circle one uses instead powers of a predetermined approximation g¯ which we have called the calibrator. The true value of g is allowed to float in a small complex ball centred on g. ¯ In [15, Eq. 5.1] this calibrator is taken equal to the approximate fixed point value which we denoted here by g¯ ∗ and which is of order . Grosso modo the main purpose of [15, Sect. 5] is to show that provided µ is of order g¯ 2 , and R is of order g¯ 3 , then the linear map L(g,µ) is contractive in the R direction, and
A Complete Renormalization Group Trajectory
751
the remainder ξ R remains of order g¯ 3 . In fact, for technical reasons, the exponents are slightly altered and a more precise statement would be: provided µ is of order g¯ 2−δ , and 11 R is of order g¯ 4 −η , then the linear map L(g,µ) is contractive in the R direction, and the 11 remainder ξ R remains of order g¯ 4 . Here δ and η are small nonnegative discrepancies. A nice feature of the estimates [15, Eq. 5.1] is that they allow a bound on the output ξ R which is strictly better than the one on the input R, when η > 0. This is required in the subsequent dynamical system construction, for an effective use of the splitting R = L(g,µ) (R) + ξ R . Two norms are required to measure the R coordinate. The first is the kernel semi-norm | · |h ∗ ,A defined in (42). This norm detects the true power g 3 of the coupling constant inside R. On its own this norm does not carry enough information to control the action of the renormalization group because it only depends on the size of φ derivatives of R at φ = 0. The renormalization group involves convolution by the Gaussian measure µΓ . The role of the second norm · h,G κ ,A is to control R when it is tested on the large fields in the tail of µΓ . A typical polymer amplitude generated by the expansion of [15, Sect. 3.1] (see also our Sect. 4.4) is of the form φ1 · · · φk e−V (φ) , where φ1 , . . . , φk refer to the evaluations of the background field φ at various locations x1 , . . . , xk . The latter eventually are integrated over against a kernel K(x1 , . . . , xk ). Such φi factors usually need to be estimated pointwise. This requires a two-step argument (see [15, Lemma 5.1]). One bounds the difference between φi and the average of φ over some polymer using the large field regulator G κ which only involves L 2 norms of derivatives of φ but not φ itself. Then the average value of φ is controlled, via Hölder’s inequality, thanks to a fraction of the 4 −g φ which is extracted from e−V (φ) by [15, Lemma 5.5]. The cost of the operation e 1 is a large g¯ − 4 factor per φi . Note that by the choice of Q in Sect. 4.3 the action of the renormalization group keeps K = e−V Q fixed up to a trivial rescaling of the coupling constant g, in the second order in perturbation theory approximation. This ensures that the RG map contribution to R is entirely due to third and higher orders of perturbation theory. Now the expansion in [15, Sect. 3.1] typically produces a collection of vertices g[(φ + ζ )4 − φ 4 ] which involve at least one fluctuation field ζ . Therefore, in the worst case scenario, the contri1 1 bution of such a vertex to a || · ||h norm bound is g¯ × (g¯ − 4 )3 = g¯ 4 . The R activities which correspond to remainders beyond second order perturbation theory essentially 3 contain at least three vertices and satisfy a g¯ 4 bound. The last considerations impose 11 3 the g¯ 2 = g¯ 4 × g¯ − 4 multiplicative shift of the || · ||h norm in the definition of the calibrated triple norm (45). This in turn affects the number n 0 of functional derivatives to be accounted for in the norms. This number has to be at least equal to 9 for the needs of [15, Lemma 5.15] which transforms a ||R||h decay into a bound on |R |h∗ , using a Taylor expansion of the polymer activities in the field variable around φ = 0. Once the proper definitions for the polymer activity norms have been made available, the sequence of estimates in [15, Sect. 5] is for the most part reasonably straightforward. It successively provides bounds for activities such as P of (70) and (S K ) of (73) which are intermediates on the way to the final RG product R . Contour integrations are used for conceptual economy when breaking R into pieces to be estimated separately. They ˜ E defined in the extraction step where one would are also used for bounds on the E, otherwise need more cumbersome estimates on derivatives of polymer activities with respect to interpolation parameters.
752
A. Abdesselam
The crucial estimates of [15, Sect. 5] are [15, Corollary 5.25] and [15, Lemma 5.27] which pertain to the linear part of the R → R map, here denoted by L(g,µ) (·). There lies the heart of the renormalization problem in quantum field theory: the action of the renormalization group has expanding (relevant) directions. In the present context these are manifested in (97) which contains a sum over Y small satisfying a constraint. Consider for example the case where Y is a single cube. Then the constraint amounts to summing over all small cubes contained in a fixed cube at the next scale, see the same phenomenon discussed in [64]. The renormalization group inevitably has expanding directions because of the L 3 factor resulting from this summation. In (97) there are two sums and one of them refers to Y large. Typically, for rather intuitive geometrical reasons, the number of cubes in a polymer strictly decreases when it is coarse grained to become the smallest covering by cubes on the next scale. This geometrical effect is exploited in [15, Inequality 2.7] followed by a pin and sum argument [17, Lemma 5.1] to prove that these so-called large polymers are harmless: they are not part of the expanding direction problem. However this purely geometrical effect breaks down in the 7− case of small polymers (see also [2, Lemma 11]). A compensating good factor L − 2 then has to be provided by the scaling behavior of the activity J . The latter corresponds to the R-linear part of what the perturbation expansion produces, when both terms R ˜ and counterterms F˜ R e−V are accounted for. The proper scaling bound on J proceeds by the clever double Taylor expansion argument of [11, Lemma 15] and [15, Corollary 5.25]. Roughly, one expands J in the field variable φ around zero; then one expands the fields or test functions appearing in the low order functional derivative terms, with respect to the space variable x. The normalization conditions [15, Eq. 4.37] eliminate the low order terms in the bigrading given by the degree in φ and the number of spacial 3− derivatives ∂. The surviving terms have enough L − 4 factors provided by the φ’s and −1 L factors given by the ∂’s not only to beat the L 3 volume sum but also to leave an 1− extra L − 2 which secures the contractivity of L(g,µ) (·) for L large, uniformly in . We may now proceed to the statement of the BMS estimates theorem. Mind the order of quantifiers which is important. Theorem 1. ∃κ0 > 0, ∃L 0 ∈ N, 3 ∀κ ∈]0, κ0 ], ∀δ ∈ [0, 16 ], ∀η ∈ [0, 16 ], 1 ∀A g ∈]0, 2 ], ∀Aµ > 0, ∀A R > 0, ∀A g¯ > 0, ∃c0 > 0, ∀c ∈]0, c0 ], ∃Bg > 0, ∃B R L > 0, ∀L ∈ N such that L ≥ L 0 , ∃Bµ > 0, ∃B Rξ > 0, ∃0 > 0, ∀ ∈]0, 0 ], ∀g¯ ∈]0, A g¯ ], if one uses the notations * def ) Dg = g ∈ C| |g − g| ¯ < A g g¯ ,
(128)
def Dµ = µ ∈ C| |µ| < Aµ g¯ 2−δ ,
(129)
def
DR = then
R ∈ BBS C | |||R|||g¯ < A R g¯
11 4 −η
;
(130)
A Complete Renormalization Group Trajectory
753
1. The maps ξg , ξµ , ξ R , are well defined and analytic on the open set Dg × Dµ × D R with values in C, C, and BBS C respectively. 2. The map (g, µ, R) → L(g,µ) (R) is well defined and analytic from Dg × Dµ ×BBS C to BBS C . Besides, for any (g, µ) ∈ Dg × Dµ , the map R → L(g,µ) (R) is linear continuous from BBS C to itself. 3. The maps ξg , ξµ , ξ R send the real cross-section (Dg ∩ R) × (Dµ ∩ R) × (D R ∩ BBS R ) into R, R, and BBS R respectively. 4. The map (g, µ, R) → L(g,µ) (R) sends (Dg ∩ R) × (Dµ ∩ R) × BBS R into BBS R . 5. For any (g, µ, R) ∈ Dg × Dµ × D R one has the estimates |ξg (g, µ, R)| ≤ Bg g¯
11 4 −η
,
|ξµ (g, µ, R)| ≤ Bµ g¯ , 2
|||ξ R (g, µ, R)|||g¯ ≤ B Rξ g¯
11 4
(131) (132)
.
(133)
6. For any (g, µ, R) ∈ Dg × Dµ × BBS C one has the estimate |||L(g,µ) (R)|||g¯ ≤ B R L L
− 1− 2
|||R|||g¯ .
(134)
Remark 6. We suppressed the reference to a calibrator g¯ when mentioning the spaces BBS K . This is because the corresponding statements do not really depend on the choice of one of the equivalent norms ||| · |||g¯ . Also note that the notion of analyticity we used is the standard one in the Banach space context (see for instance [7, Sect. 2.3]). Finally remember that the c quantity is the one involved in the relation (43). For the proof of the theorem we refer to [15, Sect. 5]. The statements about the maps being well defined and analytic will follow from the algebraic nature of the formulae in Sect. 4, once the estimates are established. The statements about the map taking real values are obvious from the formulae in Sect. 4. Now for the estimates, one should say that it is the g¯ ∼ special case of Theorem 1 which is proven in [15]. This is because the analysis takes place in the vicinity of the infrared fixed point where one can assume that the g coupling is almost constant equal to g¯ ∗ = O(). In other words, the small parameter is attributed two roles at once: bifurcation parameter and calibrator. However, by carefully following [15, Sect. 5], one can see that the arguments still apply if one dissociates the two functions. Therefore all one needs is to go over and redo the series of Lemmata from [15, Sect. 5], except that one has to replace the hypothesis in Eqs. (5.1–5.3) of [15] by the new conditions given by the domains Dg , Dµ and D R , namely, |g − g| ¯ < A g g¯ , |µ| < Aµ g¯
2−δ
|||R|||g¯ < A R g¯
(135) ,
11 4 −η
(136) ,
(137)
to which one adds 0 < g¯ ≤ A g¯ ,
(138)
754
A. Abdesselam
knowing that in the end will be taken to be small, after having fixed L. Then instead of using powers of in the bounds, one has to use powers of the calibrator g¯ instead. In Lemmata 5.26 and 5.27 of [15], one has to use bounds in terms of the norms ||R||h,G κ ,A 1 and |R|h ∗ ,A . Note that for [15, Lemma 5.5], one needs (g) 4 h to be small, which can be achieved by taking c small provided g g¯ is bounded from above. This is guaranteed by our assumption (135). Rather than [15, Lemma 5.5], the reader might find it more convenient to use instead specializations of [11, Theorem 1]. The latter needs the ratio g 1 g to be bounded, which again is guaranteed by (135) and the condition A g ≤ 2 . Note 1
that the important [15, Eq. 5.58] on the other hand cannot allow (g) 4 h to be too small either. This is why it seems hard to avoid the fibered norm problem, and we need to keep g rather close to the calibrator g¯ as in (135). Note that a stronger hypothesis was used in [15, Eq. 5.1]. However, as far as [15, Sect. 5] alone is concerned, this hypothesis only serves to show that it reproduces itself, in [15, Corollary 5.18]. We relaxed this conclusion in Theorem 1, and therefore we can drop this hypothesis. 1 Remark that in [15, Sect. 5] the exponents δ, η were taken equal to 64 . The reader who prefers this choice, can simply make the corresponding modifications in our Sect. 3 8. The ranges [0, 16 ] for δ and [0, 16 ] for η which we have given come from the following considerations. First note that the hypothesis δ, η > 0 in [15, Sect. 5] is only used in order to absorb some constant factors in the bounds provided in [15, Theorem 1]. We do not need this, since we allow the B factors above. Then note that each time in [15, Sect. 5] one has a bound with a sum of terms with different powers of , or rather here g, ¯ one has to pick the dominant term in the δ, η → 0 limit. Collecting the inequalities on δ, η which ensure that the term picked is indeed dominant, one can see that δ ≤ 16 and 3 are sufficient for these inequalities to hold. Finally, the modifications introduced η ≤ 16 in our Sect. 3 for the functional analytic setting, do not affect the bounds. One may simply mention that [15, Lemma 5.15] uses the Taylor formula with integral remainder. Of course one first has to apply it in the textbook setting of the space we denoted by Cn 0 (Fld(X ), K); and only then, one can use the sharp norm for the differentials and the || · ||C 2 (X ) norms for the fields when performing the bounds. Armed with the previous remarks, the precise statement of Theorem 1 to aim for, and some patience, the reader with expertise on the techniques from [17, 11, 15] will have no difficulty adapting the arguments of [15, Sect. 5]. 7. Elementary Estimates on the Approximate Sequence This section collects the elementary but crucial estimates on the sequence (g¯ n )n∈Z . 7.1. The discrete step function lemma. We firstly need some basic bounds on the sequence. Lemma 2. The step function behaviour. 1) For any nonnegative integer n,
g¯ ∗ 1 − (1 − ω0 )(1 + ω0 − L ω0 )n ≤ g¯ n ≤ g¯ ∗ 1 − (1 − ω0 )(2 − L )n . 2) For any nonpositive integer n, g¯ ∗ ω0 L
n
≤ g¯ n ≤ g¯ ∗ ω0
+ L +
,
2 L 2 − 4ω0 (L − 1)
(139)
-−n .
(140)
A Complete Renormalization Group Trajectory
755
Remark 7. This simply says that, for n → +∞, g¯ n goes exponentially fast to g¯ ∗ and that, for n → −∞, g¯ n goes exponentially fast to 0, with a transition or ‘step’ in between. These exponential rates are very weak in the → 0 limit. We need as precise estimates on these rates as we can, to be used as input for the following analysis. Indeed, based on these estimates, we will have to determine the winner between close competing effects, as one can see in the next subsections. This is why we included this otherwise trivial lemma. Proof. On the interval [g¯ 0 , g¯ ∗ ] we define the two functions f +h and f +l by f +h (x) = f (g¯ ∗ ) + (x − g¯ ∗ ) f (g¯ ∗ ) , f (g¯ ∗ ) − f (g¯ 0 ) def f +l (x) = f (g¯ 0 ) + (x − g¯ 0 ) × . g¯ ∗ − g¯ 0 def
(141) (142)
Since f is increasing and concave, one has for any x ∈ [g¯ 0 , g¯ ∗ ], g¯ 0 ≤ f +l (x) ≤ f (x) ≤ f +h (x) ≤ g¯ ∗ .
(143)
A trivial iteration then implies ∀n ∈ N, ∀x ∈ [g¯ 0 , g¯ ∗ ], g¯ 0 ≤ ( f +l )n (x) ≤ f n (x) ≤ ( f +h )n (x) ≤ g¯ ∗ .
(144)
Now note that ( f +h )n (x) = g¯ ∗ + (x − g¯ ∗ )[ f (g¯ ∗ )]n = g¯ ∗ + (x − g¯ ∗ )(2 − L )n . Likewise
f +l (x) = g¯ ∗ + (x − g¯ ∗ )
g¯ ∗ − g¯ 1 g¯ ∗ − g¯ 0
(145) (146)
n .
(147)
g¯ 1 = f (g¯ 0 ) = ω0 g¯ ∗ (L − L 2 aω0 g¯ ∗ ) ,
(148)
ω1 = ω0 (L − ω0 (L − 1)) = L ω0 − L ω02 + ω02 ,
(149)
Let g¯ 1 = ω1 g¯ ∗ , for ω1 ∈]0, 1[, then
or so
1 − ω1 g¯ ∗ − g¯ 1 = g¯ ∗ − g¯ 0 1 − ω0 1 − L ω0 + L ω02 − ω02 = 1 − ω0 (1 − ω0 )(1 + ω0 ) − L ω0 (1 − ω0 ) = 1 − ω0 = 1 + ω0 − L ω0 . Thus,
n ( f +l )n (x) = g¯ ∗ + (x − g¯ ∗ ) 1 + ω0 − L ω0 .
(150) (151) (152) (153) (154)
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A. Abdesselam
Now on the interval [0, g¯ 0 ] we also define, using the inverse f −1 , the two functions f −h and f −l by f −1 (g¯ 0 ) , g¯ 0
(155)
f −l (x) = x × ( f −1 ) (0) .
(156)
0 ≤ f −l (x) ≤ f −1 (x) ≤ f −h (x) ≤ g¯ 0 ,
(157)
def
f −h (x) = x × def
One has for any x ∈ [0, g¯ 0 ],
which trivially iterates into ∀n ∈ N, ∀x ∈ [0, g¯ 0 ],
0 ≤ ( f −l )n (x) ≤ ( f −1 )n (x) ≤ ( f −h )n (x) ≤ g¯ 0 .
Now
( f −l )n (x) = L −n x
and ( f −h )n (x) =
g¯ −1 g¯ 0
(158) (159)
n x.
(160)
Let g¯ −1 = ω−1 g¯ ∗ for ω−1 ∈]0, 1[. The latter is the smallest of the two solutions of the quadratic equation L (ω−1 g¯ ∗ ) − L 2 a(ω−1 g¯ ∗ )2 = ω0 g¯ ∗ , (161) i.e.,
2 − L ω−1 + ω0 = 0 ; (L − 1)ω−1
therefore ω−1 = As a result
L −
+
,
L 2 − 4ω0 (L − 1) . 2(L − 1)
-n L 2 − 4ω0 (L − 1) ( f −h ) (x) = x 2ω0 (L − 1) -n + 2 , x. = L + L 2 − 4ω0 (L − 1) n
L −
(162)
(163)
,
(164)
(165)
From the previous considerations, applied to the sequence (g¯ n )n∈Z , the lemma follows. This taken care of, we now proceed to the key lemmata for the construction of a global RG trajectory. Firstly, the forward ‘integral equation’ (122) for δg, or the deviation of the running coupling constant with respect to the reference sequence (g¯ n )n∈Z , requires an explicit bound on ⎧ ⎫ ⎨ 1 ⎬ 1 def Σδg− f (, γ , ν) = sup g¯ νp , (166) γ f (g¯ j ) ⎭ n<0 ⎩ g¯ n n≤ p<0 n≤ j≤ p
A Complete Renormalization Group Trajectory
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where γ , ν are some nonnegative real exponents. Secondly, the backward ‘integral equation’ (121) for δg, requires an analogous bound on ⎫ ⎧ ⎬ ⎨ 1 def Σδg−b (, γ , ν) = sup g¯ νp f (g¯ j ) . (167) γ ⎭ n>0 ⎩ g¯ n 0≤ p
∆n =
1 n− p−1 ν c g¯ p . γ g¯ n p
(171)
Since the sequence (g¯ n )n∈Z contained in ]0, g¯ ∗ [ is increasing, and ν ≥ γ ≥ 0, we trivially have ∆n ≤
1 n− p−1 ν c g¯ n γ g¯ n p
(172)
ν−γ
g¯ n ≤ 1 − cR
(173)
ν−γ
≤
g¯ ∗ . 1 − cR
(174)
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A. Abdesselam
7.3. The forward bound for µ. Again with the assumptions of Sect. 5, we have the following result. Lemma 4. Provided the exponents γ , ν satisfy ν ≥ γ ≥ 0, and ν < def Σµ− f (, γ , ν) ≤ Σ¯ µ− f (, γ , ν) =
3+ 2 ,
we have
ν−γ
g¯ ∗ L
3+ 2
− L ν
.
(175)
Proof. Let n ∈ Z and write
1 − 3+ 2 ( p−n+1) ν L g¯ p γ g¯ n p≥n ⎛ 3+ − ( p−n+1) ν−γ 2 ⎝ = g¯ n L
def
∆n =
p≥n
Now
n< j≤ p
(176) ⎞ν g¯ j ⎠ . g¯ j−1
g¯ j f (g¯ j−1 ) − f (0) = = f (ξ ) > 0 g¯ j−1 g¯ j−1 − 0
(177)
(178)
for some ξ ∈]0, g¯ j−1 [. Since f is concave f (ξ ) ≤ f (0) = L , and therefore ∆n ≤ ≤
ν−γ g¯ n
L
− 3+ 2 ( p−n+1) ν( p−n)
p≥n 3+ ν−γ − 2 g¯ n L
L
1
× 1−L
Since ν − γ ≥ 0,
ν−γ g¯ n
≤
ν−γ g¯ ∗ ,
and we are done.
ν− 3+ 2
.
(179) (180)
7.4. The backward bound for δg. Again with the assumptions of Sect. 5, we have the following result. Lemma 5. For any γ , ν ≥ 0 we have
where def Σ¯ δg−b (, γ , ν) =
Σδg−b (, γ , ν) ≤ Σ¯ δg−b (, γ , ν),
(181)
−γ ν−γ ω0 g¯ ∗ 2(1 − ω0 )(1 + ω0 − L ω0 ) exp . L − 1 ω0 (2 − L )
(182)
Proof. Let n be a strictly positive integer, and denote def 1 ∆n = γ g¯ νp f (g¯ j ) . g¯ n 0≤ p
(183)
Lemma 2 shows that g¯ n → g¯ ∗ when n → +∞. We therefore expect most of the f (g¯ j ) to be very close to f (g¯ ∗ ) = 2 − L . This motivates the rewriting ⎧ ⎫ 1 ⎨ f (g¯ j ) ⎬ ∆n = γ (184) (2 − L )n− p−1 g¯ νp . g¯ n 0≤ p
A Complete Renormalization Group Trajectory
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Since f is decreasing, for any j ≥ 1, f (g¯ j ) L − 2L 2 a g¯ j = >1, 2 − L 2 − L
(185)
and thus p< j
L − 2L 2 a g¯ j f (g¯ j ) ≤ 2 − L 2 − L j≥1 ⎡ ⎤
L − 2L 2 a g¯ j −1 ⎦ . ≤ exp ⎣ 2 − L
(186)
(187)
j≥1
Now
L − 2L 2 a g¯ j 2L 2 a − 1 = × (g¯ ∗ − g¯ j ) 2 − L 2 − L
(188)
g¯ j ≥ g¯ ∗ − g¯ ∗ (1 − ω0 )(1 + ω0 − L ω0 ) j ,
(189)
and Lemma 2 implies
i.e.,
L − 2L 2 g¯ j 2L 2 a −1≤ × g¯ ∗ (1 − ω0 )(1 + ω0 − L ω0 ) j , 2−L 2 − L where 1 + ω0 − L ω0 belongs to ]0, 1[. Hence 2 f (g¯ j ) 2L a g¯ ∗ (1 − ω0 ) (1 + ω0 − L ω0 ) ≤ exp × 2 − L 2 − L 1 − (1 + ω0 − L ω0 )
(190)
(191)
p< j
≤ exp
2(1 − ω0 )(1 + ω0 − L ω0 ) . ω0 (2 − L )
(192)
So we are left with bounding ∆n =
def
1 (2 − L )n− p−1 g¯ νp . γ g¯ n 0≤ p
(193)
To this effect we use the very coarse estimates g¯ n ≥ g¯ 0 = ω0 g¯ ∗ and g¯ p ≤ g¯ ∗ with the result that ∆n ≤ (ω0 g¯ ∗ )−γ (2 − L )n− p−1 g¯ ∗ν (194) 0≤ p
≤ ω0 g¯ ∗
×
1 . 1 − (2 − L )
Inequalities (192) and (195) now imply −γ ν−γ ω g¯ ∗ 2(1 − ω0 )(1 + ω0 − L ω0 ) exp . ∆n ≤ 0 L −1 ω0 (2 − L )
(195)
(196)
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A. Abdesselam
7.5. The forward bound for δg. Once more, with the assumptions of Sect. 5, we have the following result. Lemma 6. For any exponents γ , ν such that 0 ≤ γ ≤ 1, ν > 0 and
def
Υ = we have
L +
,
2L ν L 2 − 4ω0 (L − 1)
∈]0, 1[ ,
Σδg− f (, γ , ν) ≤ Σ¯ δg− f (, γ , ν),
(197)
(198)
where def Σ¯ δg− f (, γ , ν) =
)ν−γ
(ω0 g¯ ∗ 1−Υν
⎤ , ω0 2 − L + L 2 − 4ω0 (L − 1) ⎦ . × exp ⎣ (1 − ω0 ) (L − 2ω0 (L − 1)) ⎡
(199) Proof. Let n be a strictly negative integer, and define def
∆n =
1 1 ν . g¯ p γ g¯ n n≤ p<0 n≤ j≤ p f (g¯ j )
(200)
Lemma 2 shows that g¯ n → 0 when n → −∞. We therefore expect most of the f (g¯ j ) to be very close to f (0) = L . Therefore write ⎛ ⎞ L ⎠ − p−n+1 ν 1 ⎝ ∆n = γ g¯ p . (201) L g¯ n n≤ p<0 n≤ j≤ p f (g¯ j ) Now
1 L = >1. f (g¯ j ) 1 − 2L a g¯ j
(202)
We use n≤ j≤ p
L f (g¯
j)
1 1 − 2L a g¯ j j≤−1 ⎡ ⎤
1 −1 ⎦ ≤ exp ⎣ 1 − 2L a g¯ j j≤−1 ⎤ ⎡ 2L a g¯ j ⎦ . ≤ exp ⎣ 1 − 2L a g¯ j
≤
(203)
(204)
(205)
j≤−1
Now for j ≤ −1, g¯ j ≤ g¯ 0 = ω0 g¯ ∗ ; hence 2L a g¯ j 2L 2 a g¯ j 2L a g¯ j , ≤ = 1 − 2L a g¯ j 1 − 2L a g¯ 0 L − 2ω0 (L − 1)
(206)
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761
and by Lemma 2 2L a g¯ j 2L 2 aω0 g¯ ∗ ≤ 1 − 2L a g¯ j L − 2ω0 (L − 1)
+ L +
,
-− j
2
.
(207)
⎥
⎥ ⎦.
(208)
L 2 − 4ω0 (L − 1)
As a result n≤ j≤ p
L f (g¯ j ) ⎡
⎢ ≤ exp ⎢ ⎣
2ω0 (L − 1) × L − 2ω0 (L − 1)
√
2 L 2 −4ω0 (L −1)
L +
1−
√
L +
⎤
2 L 2 −4ω0 (L −1)
Note that 0<
L
+
,
2 L 2
− 4ω0 (L − 1)
<1,
(209)
because of the global assumptions 1 < L < 2 and 0 < ω0 < 1. A straightforward simplification of the argument of the exponential leads to ⎤ ⎡ , ω0 2 − L + L 2 − 4ω0 (L − 1) L ⎦ . (210) ≤ exp ⎣ f (g¯ j ) (1 − ω0 ) (L − 2ω0 (L − 1)) n≤ j≤ p
Now we are left with bounding ∆n =
def
1 − p−n+1 ν g¯ p . L γ g¯ n n≤ p<0
(211)
We now use Lemma 2 to obtain ∆n ≤ (ω0 g¯ ∗ )−γ L −γ n ×
L
− p−n+1
n≤ p<0
(ω0 g¯ ∗ )
+ ν
L
+
,
-−νp
2 L 2
− 4ω0
i.e., ∆n ≤ (ω0 g¯ ∗ )ν−γ L (γ −1)|n| × L − ×
(L
− 1)
Υν , 1−Υν
,
(212)
(213)
where Υ is the one defined in the statement of the lemma. We now need a bound which is n-independent; this requires the hypothesis γ ≤ 1. Inequalities (210) and (213) now clearly imply ∀n ≤ −1, ∆n ≤ Σ¯ δg− f (, γ , ν) , (214) and the lemma is proved.
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A. Abdesselam
7.6. The → 0 limit. Leaving L , c R , ω0 and the exponents γ , ν fixed, we now analyze the → 0 asymptotics of the previous bounds. Note that in this limit we will have log L a = a(L , ) → 18π 2 . The crux of our construction lies in the following result. Lemma 7. For → 0+ we have 1) Σ¯ R−b (, γ , ν) = ν−γ (K R−b + O()) , where K R−b = provided ν ≥ γ ≥ 0; 2)
ν−γ 1 18π 2 , 1 − cR
(216)
Σ¯ µ− f (, γ , ν) = ν−γ K µ− f + O() ,
where K µ− f = provided ν ≥ γ ≥ 0; 3)
(215)
1 3
L2 −1
(217)
ν−γ 18π 2 ,
(218)
Σ¯ δg−b (, γ , ν) = ν−γ −1 K δg−b + O() ,
where
ν−γ 18π 2
2(1 − ω0 ) exp = γ ω0 ω0 (log L)
(219)
,
(220)
Σ¯ δg− f (, 1, ν) = ν−2 K δg− f + O() ,
(221)
K δg−b provided ν ≥ 0 and γ ≥ 0; 4)
where K δg− f provided ν >
ν−1 ω0ν−1 18π 2 2ω0 , exp = (log L) [ν(1 − ω0 ) − 1] 1 − ω0
(222)
1 1−ω0 .
Proof. Straightforward first year calculus; the only delicate point is in checking condition (197). Simply note the asymptotics
1 Υ = 1 − 1 − ω0 − log L + o() , ν in order to check that Lemma 6 applies, with the above hypothesis on ν.
(223)
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763
8. Fixed Point in the Space of Sequences We start by applying Theorem 1. So we choose some κ0 > 0 and L 0 ∈ N whose existence is guaranteed by the theorem. We set κ = κ0 , and we take 1 , 2 =1, =1, = 19π 2 , 1 = , 6 3 = . 16
Ag =
(224)
Aµ AR A g¯
(225) (226) (227)
δ η
(228) (229)
Now take c to be equal to a c0 provided by the theorem, which also produces some Bg and B R L only depending on the quantities which have been fixed so far. Now choose L ≥ L 0 large enough so that 1 1 (230) B RL L − 4 ≤ . 3 This will guarantee that for any ∈]0, 21 ], B RL L
− 1− 2
≤
1 . 3
(231)
Now the theorem provides us with Bµ , B Rξ , and 0 . We will choose some 1 such that 0 < 1 < min( 21 , 0 ), and such that for all ∈]0, 1 ] one has g¯∗ < A g¯ . This is possible thanks to (21) and (227). We now have the following specialization of Theorem 1. Proposition 3. There exists an 2 ∈]0, 1 ] such that for any ∈]0, 2 ], and for any calibrator g¯ ∈]0, g¯ ∗ [, the conclusions (1)–(6) of Theorem 1 are valid with the inequality in (134) replaced by 1 |||L(g,µ) (R)||| f (g) (232) ¯ ≤ |||R|||g¯ . 2 The proof is an immediate corollary of the following lemma. Lemma 8. Provided
3 1 max L 2 , (2 − L )− 4 ≤ , 2
(233)
which will hold true when → 0, one has for any g¯ ∈]0, g¯ ∗ [, and any R ∈ BBS K , |||R||| f (g) ¯ ≤
3 |||R|||g¯ . 2
(234)
Proof. Let g¯ = f (g). ¯ Since g¯ > g, ¯ and from the definition of the triple norms it is immediate that for any R one has 0 7 − 14 1 g¯ 2 g¯ 4 g¯ |||R|||g¯ ≤ max , , . . . , 1, (235) × |||R|||g¯ . g¯ g¯ g¯
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A. Abdesselam
However, by the mean value theorem, g¯ f (g) ¯ − f (0) = = f (ς ) g¯ g¯ − 0
(236)
for some ς ∈]0, g¯ ∗ [. As a result 2 − L < and the lemma follows.
g¯ < L , g¯
(237)
Now given ω0 ∈]0, 21 [, we construct the sequence (g¯ n )n∈Z as in Sect. 5, as well as the associated spaces (BBSS K , |||| · ||||). Given an element δs ∈ BBSS K , and a positive number β we use the notation BK (δs, β) for the open ball of radius β around δs in BBSS K . We also use B¯ K (δs, β) for the analogous closed ball. We can now state our main theorem. Theorem 2. The Main Theorem. ∃β0 , ∀β ∈]0, β0 ], ∃3 > 0, ∀ ∈]0, 3 ], one has 1. The BBSS C valued map m from Sect. 5 is well defined and analytic on BC (0, β). 2. The image by m of BC (0, β) is contained in B¯ C (0, β6 ). 3. The restriction of m to the closed ball B¯ R (0, β6 ) is a contraction from that ball to itself. 4. There exists a unique fixed point for the map m inside the ball B¯ R (0, β6 ). Proof. Let β > 0 be such that the condition β ≤ 21 = A g is realized. Then by construction, for any n ∈ Z, g¯ n ∈]0, A g¯ [. Therefore, as a consequence of Proposition 3, for any δs = (δgn , µn , Rn )n∈Z ∈ BC (0, β) , all the summands in (121), (122), (123), and (124) are well defined and analytic with respect to δs. The analyticity property required in statement (1) will therefore follow from the uniform absolute convergence of the series. The latter will in turn result from the estimates, required for the statement (2), which we now proceed to establish. Using the notations of Definition 1, we assume that δs is in BC (0, β), and we apply the estimates of Sect. 7, in order to obtain the following results. The backward δg bound . Let n > 0, then ⎞ ⎛ 1 − 23 1 ⎝ |δg |g¯ n ≤ f (g¯ j )⎠ 3 β n β g¯ n2 0≤ p
A Complete Renormalization Group Trajectory
0 × L
765
2
a(L , )β 2 g¯ 3p
+ Bg g¯ p
11 3 4 − 16
1
(238)
3 3 11 1 3 ¯ ¯ . ≤ β L a(L , )Σδg−b , , 3 + Bg Σδg−b , , − 2 β 2 4 16 2
(239) Now by part 3) of Lemma 7 and for any fixed β, the last upper bound goes to zero when → 0. Therefore, by choosing small enough, one will have ∀n > 0 ,
1 − 23 1 |δg |g¯ n ≤ . β n 6
(240)
The forward δg bound . Let n > 0, then in the same vein one will have
11 3 1 1 −1 2 ¯ ¯ |δg |g¯ ≤ β L a(L , )Σδg− f (, 1, 2) + Bg Σδg−b , 1, − . β n n β 4 16
(241)
Now here comes the narrowest passage in the proof. Provided that ω0 ∈]0, 21 [, the limiting case of part 4) in Lemma 7 shows that L a(L , )Σ¯ δg− f (, 1, 2) → 2
ω0 2ω0 exp 1 − 2ω0 1 − ω0
(242)
when → 0. Therefore we need to take 1 − 2ω0 2ω0 β< . exp − 6ω0 1 − ω0
(243)
Then after β is fixed accordingly, the first term in (241) will be strictly less than 16 in the → 0 limit while the second term will go to zero, again by 4) of Lemma 7. We will then have 1 1 (244) ∀n < 0 , |δgn |g¯ n−1 ≤ . β 6 The forward µ bound . Let n ∈ Z, then by the same reasoning one will have 1 −(2− 61 ) 1 |µn |g¯ n ≤ Bµ Σ¯ µ− f β β
1 , 2 − , 2 , 6
(245)
which will go to zero when → 0, as results from case 2) of Lemma 7. We will then have 1 1 −(2− 61 ) (246) ∀n ∈ Z , |µn |g¯ n ≤ . β 6
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A. Abdesselam
The backward R bound . Let n ∈ Z, then proceed in the same manner except that the varying norms require a little care. We have
3 − 11 1 4 − 16 |||Rn |||g¯n × g¯ n β 1 × ≤ |||L(g¯n−1 +δgn−1 ,µn−1 ) ◦ L(g¯n−2 +δgn−2 ,µn−2 ) ◦ · · · 11 3
β g¯ n
4
− 16
(247)
p
· · · ◦ L(g¯ p+1 +δg p+1 ,µ p+1 ) ξ R (g¯ p + δg p , µ p , R p ) |||g¯n 1 n− p−1 3 11 1 × ≤ × × B Rξ × g¯ p4 , 11 3 2 2 4 − 16 p
(248)
where we repeatedly used the inequality (232), as well as (234), and the ξ R estimate in item (5) of Theorem 1. In sum one has
3 − 11 3B Rξ 11 3 11 1 4 − 16 |||Rn |||g¯n × g¯ n × Σ¯ R−b , − , , (249) ≤ β 2β 4 16 4 and this goes to zero when → 0, as shown in part 1) of Lemma 7, with c R = 21 . At this point, statements (1) and (2) of the theorem are proved. The contraction property . Let δs1 = δs2 be two elements of the open ball BC (0, β6 ). Let 2β . (250) r= 3||||δs1 − δs2 |||| Then β (251) ||||δs1 − δs2 |||| ≤ ||||δs1 |||| + ||||δs2 |||| ≤ 3 implies that r ≥ 2. Therefore, if one defines the contour γ as the counterclockwise oriented circle or radius r around the origin in the complex plane; one has by the Cauchy theorem
2 1 1 1 m(δs1 ) − m(δs2 ) = − m (δs2 + z(δs1 − δs2 )) . dz (252) 2πi z−1 z γ
Now for z ∈ γ we have ||||δs2 + z(δs1 − δs2 )|||| ≤ ||||δs2 |||| + r ||||δs1 − δs2 |||| β 2β ≤ + 6 3 <β.
(253) (254) (255)
As a result of the already established statement (2), one has ||||m(δs1 ) − m(δs2 )|||| ≤
1 × max ||||m(δs2 + r eiθ (δs1 − δs2 ))|||| r − 1 0≤θ≤2π (256)
β ≤ 6(r − 1) β ≤ 3r
(257) (258)
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767
because r ≥ 2. Inserting the definition of r shows that ||||m(δs1 ) − m(δs2 )|||| ≤
1 × ||||δs1 − δs2 |||| , 2
(259)
i.e., the contraction property. The real ball stability follows from statements (3) and (4) in Theorem 1/Proposition 3 and Definition 1. Now statement (3) is proved, and (4) follows from the Banach fixed point theorem. This concludes the proof of the main theorem. Corollary 1. The constructed two-sided trajectory (gn , µn , Rn )n∈Z is the unique such sequence inside the ball B¯ R (0, β6 ) of BBSS R which solves the recursion (113). One has lim (gn , µn , Rn ) = (0, 0, 0) ,
n→−∞
(260)
the trivial Gaussian ultraviolet fixed point, and lim (gn , µn , Rn ) = (g∗ , µ∗ , R∗ ) ,
n→+∞
(261)
the BMS nontrivial infrared fixed point [15]. Proof. The proof of the first statement is easy and left to the reader. Note that the statements concerning the limits for Rn are topological and do not depend on a particular choice of a calibrated norm ||| · |||g¯ . The last statement follows from the possibility of making β as small as we want, provided is small enough. Indeed, because of the choice of exponent 23 in (125) at the positive end for n, the convergence of g¯ n to g¯ ∗ when n → +∞ will ensure that for large positive values of n, (gn , µn , Rn ) will fall within the small domain around the approximate IR fixed point where the stable manifold has been constructed, and where the convergence of all one-sided sequences which remain bounded in the future, towards the IR fixed point, has been been established [15, Sect. 6]. 9. Suggestions for Future Work The following is a list of problems which are natural continuations of the present work. 1) The continuous connecting orbit between the two fixed points should be the graph of a function g → (µ(g), R(g)) with g in the range 0 < g < g∗ . In principle, when considering one of the sequences (gn , µn , Rn )n∈Z we constructed as a function of g0 only, this map should correspond to the one giving µ0 and R0 in terms of g0 (which is here provided in the range 0 < g < g¯2∗ ). One could even say that it is also the map giving µn and Rn in terms of gn , for any n, provided one could do the proper inversions. Although we did not yet explore this, it seems likely that by a more refined analysis, one can construct the full invariant curve connecting the two fixed points. This would open the door to the investigation, in a constructive setting, of the old ‘reparametrization’ renormalization group [72, 41]. This has so far remained inaccessible in Bosonic constructive field theory. In contrast, a continuous RG for Fermions has been developed through work initiated in [65] and completed in [27]. 2) If one could answer the first question, then the immediate one that follows is: what would be the regularity of this curve? It seems reasonable to conjecture real analyticity in the range 0 < g < g∗ . An interesting question in this regard raised by K. Gaw¸edzki,
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concerns the C ∞ behavior, or not, of this curve at g = 0+ . A similar question was mentioned in [39], related to a possible explanation of the breakdown of the traditional perturbative argument ruling out nonrenormalizable theories as consistent [63, 73, 61]. To gain insight on this issue, consider the following simplified flow which mimics the behavior of the RG map considered here: . dg 2 dt = αg − βg , dµ 2 dt = γ µ − δg . If one eliminates the time variable then the connecting orbit can be expressed exactly in terms of an incomplete beta function, which admits a convergent hypergeometric series γ representation near g = 0. If one rescales g writing s = βg α and letting ν = α , then the smoothness of the orbit at 0+ is reduced to that of the function s2 π(ν − 1) 1−ν , 2−ν + × 2 F1 s → s ν ;s 3−ν sin[π(ν − 1)] ν − 2 at s = 0, when ν is not an integer. In this case C ∞ behavior is ruled out. In our setting ν is roughly given by
L
3+ 2
−1 , L − 1
which is very large. 3) The RG map considered in [15] and also here is in the so-called ‘formal infinite volume limit’. With more work one can probably perform the true scaling limit of the theory, using an appropriate bare ansatz as in [39] for instance. One should also try to develop a streamlined rigorous RG framework for the handling of correlation functions, including those of more general observables, like composite operators. An important step in this direction was taken in [14]. One should then prove or disprove the existence of anomalous scaling dimensions not only for the field φ(x), but also for composite operators. In the hierarchical model there is no anomalous dimension for the field φ(x) as shown in [37]. A similar result for the full model was recently obtained [57, 58], together with a preliminary perturbative calculation which supports the hypothesis of a nonzero anomalous dimension of order for the composite field φ(x)2 . Justifying this last statement by a rigorous nonperturbative proof, however is a tantalizing open problem. Finally if one can go as far, the investigation by analytical means of Wilson’s operator product expansion would make a nice crowning achievement. 4) Orthogonal to the RG approach by Brydges and collaborators, where one tries to know as little as possible about the irrelevant terms R, there is also the phase space expansion method [42] which has become the trademark of the French school of constructive field theory [32, 33, 64] (see also [5]). In this other approach one, on the contrary, tries to know as much as possible about the explicit structure of these terms [2]. We therefore hope to have the future opportunity of investigating the same model as considered here, with this alternative approach. The lessons learnt with the methods of Brydges and collaborators will be useful in this regard. For instance, in [2] the hypothesis of large L was not used and polymers were allowed which have large gaps in the vertical direction. Albeit esthetically pleasing, these features lead to additional technical difficulties which drive one away from maximal simplicity. The use of strictly short ranged fluctuation
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covariances introduced in [59], exploited in [15] as well as the present article, and systematically developed in [13, 16], should allow major simplifications in the multiscale phase space expansions framework. 5) Important new methods for dealing with φ 4 -type lattice models, based on Witten Laplacian techniques, have been developed recently [45, 69, 4]. It would be desirable to extend their reach to the case of critical theories. Although one should bear in mind that according to RG wisdom, rather than the weakly convex case (no φ 2 in the bare potential), it is the double well case (properly adjusted strictly negative φ 2 coupling) which should entail a power law behaviour of correlations. The result in the present article adds a new confirmation to this picture. Indeed, our trajectory which lies on the critical manifold essentially has µ = O(g 2−δ ). Undoing the Wick ordering, this means that the φ 2 coupling is µ − 6C(0)g < 0. Acknowledgements. I thank D. C. Brydges for giving me this exciting problem to work on, suggesting fruitful directions to explore, and spending many hours explaining the technicalities of the RG formalism he developed together with his collaborators. They laid the groundwork for the result presented here. I thank P. K. Mitter for many useful discussions and in particular for his input concerning the completeness problem solved in Sect. 3. I also thank the anonymous referee for suggesting useful improvements. This research was initiated during an extended visit to the University of British Columbia while I benefited from a relief from teaching duties or Délégation during the academic year 2002/2003 provided by the Centre National de la Recherche Scientifique. Most of the work was done during a longer two year visit to UBC. This was arranged thanks to the help of D. C. Brydges, J. Feldman, and G. Slade. I thank the UBC Mathematics Department for providing ideal working conditions. I also thank my home department, the Laboratoire Analyse, Géometrie et Applications of the Université Paris 13 for giving me leave of absence during that period.
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79. Zamolodchikov, A.B.: Renormalization group and perturbation theory about fixed points in two-dimensional field theory. Sov. J. Nucl. Phys. 46, 1090–1096 (1987) 80. Zamolodchikov, A.B., Zamolodchikov, Al.B.: Massless factorized scattering and sigma models with topological terms. Nucl. Phys. B 379, 602–623 (1992) Communicated by J. Z. Imbrie
Commun. Math. Phys. 276, 773–798 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0353-9
Communications in
Mathematical Physics
Renormalization of Gauge Fields: A Hopf Algebra Approach Walter D. van Suijlekom Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany Received: 24 October 2006 / Accepted: 10 April 2007 Published online: 16 October 2007 – © Springer-Verlag 2007
Abstract: We study the Connes–Kreimer Hopf algebra of renormalization in the case of gauge theories. We show that the Ward identities and the Slavnov–Taylor identities (in the abelian and non-abelian case respectively) are compatible with the Hopf algebra structure, in that they generate a Hopf ideal. Consequently, the quotient Hopf algebra is well-defined and has those identities built in. This provides a purely combinatorial and rigorous proof of compatibility of the Slavnov–Taylor identities with renormalization. 1. Introduction The combinatorial structure underlying renormalization in perturbative quantum field theory was transparent in the original approach of Bogogliubov, Hepp, Parasiuk and Zimmermann (cf. for instance [1, Ch.5] and references therein). It was realized by Kreimer in [5] that this structure is in fact organized by a Hopf algebra of rooted trees. One year later, Connes and Kreimer [2] reformulated this combinatorial structure directly in terms of a (commutative) Hopf algebra of Feynman graphs and understood the BPHZprocedure as a Birkhoff decomposition in the group that is dual to this Hopf algebra. In physics, however, one usually works in the setting of functional calculus since – although defined only formally – functional integrals are particularly well-suited for the perturbative treatment of quantum gauge theories. For example, the Slavnov–Taylor identities that are the reminiscents of the gauge symmetry of the classical field theory, can be gathered elegantly in a single equation known as the Zinn-Justin equation, and involving the effective action. Although this approach is very powerful in showing for example renormalizability of gauge field theories, the applicability of the graph-per-graph approach of the BPHZ-renormalization procedure is not so transparent any more, and the same holds for the combinatorial structure underlying it. Current address: Institute for Mathematics, Astrophysics and Particle Physics, Radboud Universiteit, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands. E-mail: [email protected]
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Recent developments [6, 7] give more insight in the combinatorial aspects of non-abelian gauge theories. In [11], we considered the Hopf algebra of Feynman graphs in quantum electrodynamics and found that certain Ward–Takahashi identities can be imposed as relations on this Hopf algebra. In this article, we continue to explore the combinatorial structure of gauge theories in terms of a Hopf algebra. More precisely, we will show that the Slavnov–Taylor identities between the coupling constants can be implemented as relations on the Hopf algebra, that is, in a way compatible with the counit, coproduct and antipode. This then provides a combinatorial proof of the compatibility of the Slavnov–Taylor identities with renormalization. We start in Sect. 2 with a precise setup of the Hopf algebra of Feynman graphs in a generic theory, including gauge theories, and derive a formula for the coproduct on 1PI Green’s functions. Such Green’s functions are sums of all 1PI graphs with a certain fixed external structure, including symmetry factors. It is the latter that make this derivation slightly involved. Section 3 will be a warming-up for the non-abelian case, by considering quantum electrodynamics, which is an abelian gauge theory. Using the expression for the coproduct on the Green’s functions, we show that certain Ward identities can be imposed as (linear) relations on the Hopf algebra. In other words, they define a Hopf ideal. The case of non-abelian gauge theories will be consided in Sect. 4, where we will show that the Slavnov–Taylor identities define quadratic relations in the Hopf algebra. In fact, as we will see, it is in the very nature of the combinatorial factors that are involved that the Slavnov–Taylor identities appear. We have added two appendices. In the first, we rederive the compatibility of Ward identities with the Hopf algebra structure in QED obtained in Sect. 3 from our previous result on Ward–Takahashi identities in [11]. In the second appendix, we list some useful basic combinatorial identities used throughout the text. 2. General Structure of the Hopf Algebra of Feynman Graphs We start with some definitions on Feynman graphs and their symmetries, thereby making precise several properties needed later. 2.1. Feynman graphs. The Feynman graphs we will consider are built from a certain set of edges and vertices R, and we write R = R V ∪ R E . For example, in φ 3 -theory, the set R V contains the bi- and trivalent vertex and R E the straight line, but more interesting theories such as gauge theories contain different types of edges and vertices (for example involving curly, dotted and straight lines) corresponding to different particles. More precisely, we have the following definition [3]. Definition 1. A Feynman graph is given by a set [0] of vertices each of which is an element in R V and [1] of edges in R E , and maps ∂ j : [1] → [0] ∪ {1, 2, . . . , N },
j = 0, 1,
that are compatible with the type of vertex and edge as parametrized by R V and R E , respectively. Moreover, we exclude the case that ∂0 and ∂1 are both in {1, 2, . . . , N }. −1 The set {1, 2, . . . , N } labels the external lines, so that j card ∂ j (v) = 1 for all v ∈ {1, . . . , N }.
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Sym
=2
775
Sym
=1
Fig. 1. Automorphisms of Feynman graphs respect the type of vertex/edge in R. [1] [1] The set of external lines is ext = ∪i ∂i−1 {1, . . . , N } and its complement int in [1] is the set of internal lines. [1] can thus be labeled as e1 , . . . , e N , where ek := We remark that the elements in ext −1 ∪i ∂i (k) and we understand this labeling as being fixed. With this definition, the notion of a graph automorphism can be defined as follows.
Definition 2. An automorphism of a Feynman graph is given by an isomorphism g [0] from [0] to itself, and an isomorphism g [1] from [1] to itself that is the identity on [1] ext and such that for all e ∈ [1] , ∪ j g [0] (∂ j (e)) = ∪ j ∂ j (g [1] (e)).
(1)
Moreover, we require g [0] and g [1] to respect the type of vertex/edge in the set R. The automorphism group Aut() of consists of all such automorphisms; its order is called the symmetry factor of and is denoted by Sym(). Similarly, there is a notion of an isomorphism of two graphs and as a pair of maps that intertwines the maps ∂i as in Eq. (1). We remark that we correct in this way for the apparent orientation given by the two maps ∂0 and ∂1 and we stress that the fermionic lines are unoriented. We take the complex character of the fermionic fields into account by summing over all possible orientations once we apply the Feynman rules. The above definition of automorphism differs from the usual notion of graph automorphism (cf. for instance [4]) in that the latter might also permute the elements in {1, . . . , N } when understood as external vertices. In the above notation, such an automorphism of would be given by an isomorphism g [0] from [0] ∪ {1, . . . , N } to itself, [1] from [1] to itself such that Eq. (1) holds. and an isomorphism g Note that for = i i the disjoint union of n graphs, the symmetry factor is given by Sym() = n 1 ! · · · n k ! Sym(1 ) · · · Sym(n ), where n i are the numbers of isomorphic (with fixed external lines) connected components of . Equivalently, one has for a 1PI graph , Sym( ) = n(, )Sym()Sym( ), (2) with n(, ) the number of connected components of that are isomorphic to . If is a connected Feynman graph with external lines labeled by {1, . . . , N }, we can construct another graph σ , by permuting the external lines by an element σ ∈ S N , respecting the type of external lines. The graph σ is given by the same sets [0] and [1] but with maps ∂ σj := σ ◦ ∂ j : [1] → [0] ∪ {1, · · · , N }. This permutation affects the labeling of the external lines by {1, . . . N }, which explains the terminology permutation of external lines; we write eσ for the edge in σ corresponding to an edge e ∈ [1] under the permutation σ .
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2
2 σ
−→
1 3
1 3
Fig. 2. The permuation σ = (23) of the external lines of the graph is trivial since reflection in the dotted line induces an automorphism g of such that g [0] |{1,2,3} = σ . Moreover, this is the only trivial permutation so that ||∨ = 3!/2 = 3
Definition 3. A permutation σ of the external lines of is called trivial if there exists an isomorphism between σ and , leaving the labeling of the external lines fixed. The number of non-isomorphic graphs σ obtained by a permutation σ of the external lines of , is denoted by ||∨ and extended to disconnected graphs by | |∨ = ||∨ | |∨ . Lemma 4. A permutation σ of the external lines of is trivial if and only if there exists an automorphism g of the graph not necessarily leaving the external lines fixed, such that g [0] |{1,...,N } = σ . Proof. Firstly, if σ is trivial, there exists an isomorphism f : σ → and the pair ( f [0] ◦ σ, f [1] ◦ σ ) is an automorphism g of (without fixed external vertices), since, ∪ j g [0] (∂ j (e)) = ∪ j f [0] (∂ σj (eσ )) = ∪ j ∂ j ( f [1] (eσ )) = ∪ j ∂(g [1] (e)). On the other hand, such an automorphism g is given by two maps g [0] and g [1] , where g [0] is the product of two permutations of the disjoint sets [0] and {1, . . . , N }, say [1] by permutation, so that also f [0] and σ , respectively. Correspondingly, σ acts on ext [1] [1] g = f ◦ σ . This factorization gives rise to an isomorphism f from σ to , which leaves external lines fixed. For the purpose of renormalization, one is mainly interested in one-particle irreducible Feynman graphs which have residues that are elements in the set R. Definition 5. A Feynman graph is called one-particle irreducible (1PI) if it is not a tree and can not be disconnected by removal of a single edge. Definition 6. The residue res() of a Feynman graph is defined as the vertex/edge the graph corresponds to after collapsing all its internal edges and vertices to a point. For example, we have
We restrict to the class of Feynman graphs for which res() ∈ R and will denote a generic graph with residue r ∈ R by r . If it also has loop number L, we denote it by rL .
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Definition 7. The Hopf algebra H of Feynman graphs is the free commutative algebra generated by all 1PI Feynman graphs, with counit () = 0 unless = ∅, in which case (∅) = 1, coproduct, () = ⊗ 1 + 1 ⊗ + γ ⊗ /γ , γ
and antipode given recursively by, S() = − −
S(γ )/γ .
(3)
γ
2.2. Insertion of graphs. [1] Definition 8. An insertion place for a (connected) graph γ in is the subset of [0] ∪int consisting of vertices/internal n edges of the form r = res(γ ). It can be extended to disconnected graphs γ = i=1 γi by giving n-tuples of insertion places for γ1 , . . . , γn , thereby allowing several insertions of the connected components with residue r in R E on the same internal edge in of the form r . The number of insertion places for γ in is denoted by | γ .
An explicit expression for | γ can be obtained as follows [6]. Let m ,r be the number of vertices/edges r in [0] ∪ [1] , for r ∈ R. Moreover, let n γ ,r be the number of connected components of γ with residue r . Since insertion of a vertex graph (i.e. with residue in R V ) on a v ∈ [0] prevents a subsequent insertion at v of a vertex graph with the same residue, whereas insertion of an edge graph (i.e. with residue in R E ) creates two new edges and hence two insertion places for a subsequent edge graph, we find the following expression: m ,e +n γ ,e −1 | γ = v∈RV n γ ,v ! mn γ,v . e∈R E n γ ,e ! n γ ,e ,v Indeed, the binomial coefficients arise for each vertex v since we are choosing n γ ,v out of m ,v , whereas for an edge e we choose n γ ,e out of m ,e with repetition. We extend this definition to empty graphs by defining | ∅ = ∅ | γ = ∅ | ∅ = 1 for a 1PI graph γ , and ∅ | γ = 0 for a disconnected graph γ . Remark 9. Our expression for | γ differs slightly from the one given in [6] where additional factors of 1/n γ ,r ! are present for r ∈ R. It turns out that the above expression appears naturally in the coproduct on 1PI Green’s functions (see below). A few examples are in place:
Definition 10. An insertion of a connected graph γ at the insertion place x in , is [1] of external lines of γ and the set ∂ −1 (x). If given by a bijection between the set γext [1] −1 x ∈ [0] , ∂ −1 (x) denotes the set of lines attached to the vertex x, and if x ∈ ext , ∂ (x) denotes the set of adjacent edges to any internal point of x. The graph obtained in this way is denoted by ◦(x,φ) γ .
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Two insertions (x, φ) and (x , φ ) are called equivalent if x = x and φ = φ ◦ σ for some trivial permutation σ of the external lines of γ . The set of all insertions of γ in up to equivalence is denoted by X (, γ ); it consists of equivalence classes [x, φ]. This definition of equivalence relation is motivated by the fact that ◦(x,φ) γ ◦(x ,φ ) γ whenever n(x, φ) ∼ (x , φ ). We extend X (, γ ) to disconnected graphs γ as follows. If γ = i=1 γi is the disjoint union of n graphs, the set X (, γ ) of insertions of γ in is defined as the set of n−tuples of pairs ([x1 , φ1 ], . . . , [xn , φn ]), where k γi , γk+1 ) [x1 , φ1 ] ∈ X (, γ1 ) and [xk+1 , φk+1 ] is an element in X ( ◦(x1 ,φ1 )...(xk ,φk ) i=1 which is not part of any of the inserted graphs γ1 , . . . , γk−1 for k = 1, . . . , n − 1. The cardinality of X (, γ ) is the number | γ of insertion places for γ in times the number |γ |∨ of non-trivial permutations of the external lines of γ . We also need the following generalization for the number of insertion places. Definition 11. Let , γ , γ be three (disjoint unions of) 1PI graphs. We define | γ | γ to be the number of places to insert γ into (say, at x using φ) and then subsequently insert γ in ◦(x,φ) γ . In other words, | γ | γ :=
1 |γ |∨
◦(x,φ) γ | γ .
[x,φ]∈X (,γ )
Moreover, we set | ∅ | γ = | γ and ∅ | γ | γ = γ | γ if γ is 1PI and ∅ | γ | γ = 0 if γ is disconnected. The factor 1/|γ |∨ corrects for the overcounting due to the several (non-equivalent) ways to insert γ into at a particular place. Note that automatically | γ | ∅ = | γ and if , γ , γ = ∅, we have | γ | γ = | γ γ + ( | γ )(γ | γ ).
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Suppose γ is a 1PI graph. There is a natural action of Aut() on X (, γ ) given by g · (x, φ) = (gx, g ◦ φ). One readily checks that this action respects the equivalence relation on insertions, and therefore acts on the equivalence classes [x, φ]. Moreover, an element g ∈ Aut() naturally induces an isomorphism ◦(x,φ) γ to ◦g(x,φ) γ . For an element [x, φ] in X (, γ ), we denote by M(x, φ) the number of graphs γ in ◦(φ,x) γ that are images of γ under some element in Aut( ◦(x,φ) γ ). Moreover, N (x, φ) denotes the number of orbits Aut()[x , φ ] such that ◦(x ,φ ) γ ◦(x,φ) γ . Both definitions are independent of the choice of a representative (x, φ) as well as the choice of the element [x , φ ] in the orbit. Indeed, an element g in Aut() will induce a natural isomorphism ◦(x ,φ) γ ◦g(x ,φ ) γ . Lemma 12. Suppose γ is a 1PI graph and let [x, φ] ∈ X (, γ ). The length of the orbit Aut()[x, φ] is given by |Aut()[x, φ]| =
Sym(γ )Sym()M(x, φ) . Sym( ◦(x,φ) γ )
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Proof. We use the orbit-stabilizer theorem, stating in this case that the orbit Aut()[x, φ] is isomorphic to the left cosets of the stabilizer Aut()[x,φ] in Aut(). In particular, we have for its length, |Aut()[x, φ]| = Aut() : Aut()[x,φ] =
|Aut()| . |Aut()[x,φ] |
The order of Aut()[x,φ] can be computed as follows. Let Aut( ◦(x,φ) γ )γ be the subgroup of Aut( ◦(x,φ) γ ) consisting of automorphisms that map γ to itself (but possibly permuting the external lines of γ ). There is a short exact sequence of groups 1 → Aut(γ ) → Aut( ◦(x,φ) γ )γ → Aut()[x,φ] → 1. Indeed, the image g inside Aut() of an element g in Aut( ◦(x,φ) γ )γ is defined by restricting g to − {x} and by the identity map on the vertex x. Then, by Lemma 4, g might permute the edges connected to the vertex x but always in a trivial way, since g induces an automorphism of γ not necessarily leaving its external lines fixed. Therefore, [1] g (x, φ) = (x, φ ◦ σ ) for some trivial permutation σ of γext , so that it is an element in the fixed point subgroup Aut()[x,φ] . Moreover, the kernel of the map that sends such a g to g consists precisely of those elements in Aut( ◦(x,φ) γ )γ that correspond to the identity on ; in other words, these are automorphisms of γ that leave external lines fixed. We conclude that the quotient group Aut( ◦(x,φ) γ )γ /Aut(γ ) is isomorphic to Aut()[x,φ] . Since Aut( ◦(x,φ) γ ) is generated by the elements in Aut( ◦(x,φ) γ )γ and automorphisms that map γ isomorphically to a subgraph γ of , we see that |Aut( ◦(x,φ) γ )γ | =
|Aut( ◦(x,φ) γ )| . M(x, φ)
Combining these results, we conclude that |Aut()[x, φ]| =
|Aut(γ )| |Aut()| Sym(γ )Sym()M(x, φ) = . |Aut( ◦(x,φ) γ )γ | Sym( ◦(x,φ) γ )
As a final preparation to the next section, we will write the coproduct as a sum of maps γ , with γ a disjoint union of 1PI graphs (with fixed external lines). They are given by γ () = /γ , γ ⊂,γ γ
and defined to be zero if contains no subgraphs isomorphic to γ . In particular, ∅ is the identity map, () = ∅ and γ (∅) = 0 if γ = ∅. However, since only subgraphs isomorphic to γ enter in this formula – hence no reference is made to a particular labeling of the external lines of γ – we have to correct by a factor of |γ |∨ if we are to sum over all disjoint unions of 1PI graphs with fixed external lines, =
1 γ ⊗ γ . |γ |∨ γ
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We recall the following combinatorial factor from [2]; for a given , γ , , we denote by n(, γ , ) the number of subgraphs γ γ in such that /γ . With this definition, we can write γ () = n(, γ , ) , (5)
which also yields the following formula for the coproduct, () =
n(, γ , ) γ ⊗ . |γ |∨
γ ,
Remark 13. From this last formula, one easily derives the Lie bracket on Feynman graphs as defined in [2]. Indeed, one can define a pre-Lie product between 1PI graphs 1 , 2 by duality 1 ∗ 2 , := 1 ⊗ 2 , (), with the pairing given by 1 , 2 = 1 if 1 2 and zero otherwise. This pre-Lie product defines a Lie bracket by [1 , 2 ] = 1 ∗ 2 − 2 ∗ 1 with ∗ given explicitly by 1 ∗ 2 =
n(, 1 , 2 )
|1 |∨
.
Lemma 14. If and γ are nonempty (connected) 1PI graphs, then n( ◦(x,φ) γ , γ , ) = M(x, φ)N (x, φ). Proof. We have to count the number of subgraphs γ γ of ◦(x,φ) γ such that there is an isomorphism ( ◦(x,φ) γ )/γ . This isomorphism can be trivial in the sense that there exists an element in Aut(◦(x,φ) γ ) mapping γ to γ . Otherwise, the existence of such an isomorphism implies that there is an isomorphism ◦(x,φ) γ ◦(x ,φ ) γ , with (x , φ ) the image in of res(γ ) in the quotient ( ◦(x,φ) γ )/γ ; such an isomorphism maps γ in ◦(x,φ) γ to a certain subgraph γ of . Moreover, [x, φ] and [x , φ ] are in disjoint Aut()-orbits, since if (x , φ ) = g(x, φ), the isomorphism would be the composition of an element in Aut( ◦(x,φ) γ ) and an element in Aut(). We claim that all subgraphs γ obtained in this way (for disjoint orbits) are all different subgraphs of , and cannot be the image of γ under the action of an element in Aut( ◦(x,φ) γ ). This would then lead to M(x, φ)N (x, φ) many subgraphs γ of ◦(x,φ) γ satisfying ( ◦(x,φ) γ )/γ . Let [x, φ], [x , φ ], [x , φ ] ∈ X (, γ ) be in disjoint orbits and suppose that there are isomorphisms g : ◦(x ,φ ) γ → ◦(x,φ) γ ,
g : ◦(x ,φ ) γ → ◦(x,φ) γ , mapping γ to subgraphs γ and γ in , respectively. If γ and γ coincide (up to an isomorphism h), then the composition (g )−1 ◦ h ◦ g gives an isomorphism from ◦(x ,φ ) γ to ◦(x ,φ ) γ mapping γ to itself. It therefore induces an element in Aut() that sends [x , φ ] to [x , φ ], which cannot be true. We conclude that γ and γ are different subgraphs of .
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On the other hand, if there is an element φ in Aut( ◦(x,φ) γ ) that maps γ to such a subgraph γ ∈ , the composition φ −1 ◦ g would map ◦(x ,φ ) γ to ◦(x,φ) γ isomorphically, sending γ to itself. Again, such a map must be induced by an element in Aut() mapping [x, φ] to [x , φ ], contradicting our assumptions. Lemma 15. Let γ , γ be as above. Then, γ γ =
1 γ γ − ργ γ , n(γ , γ )
where ργ γ is defined by
ργ γ =
[y,ψ]∈X (γ ,γ )
Sym(γ ◦(y,ψ) γ ) γ ◦(y,ψ) γ . Sym(γ γ )
Proof. Consider γ γ () on a 1PI graph ; if γ and γ appear as disjoint subgraphs of , this expression is given by γ γ (), up to a factor of n(γ , γ ) which corrects for the overcounting. Indeed, let γ1 , . . . , γm denote all subgraphs of that are isomorphic to γ . If m ≥ n, then 1 γ n+1 () = /γi1 · · · γin+1 ; (n + 1)! γ n γ () =
{i 1 ,...,i n+1 } ⊂{1,...,m} m i=1
{i 1 ,...,i n } ˆ ⊂{1,...,i,...,m}
1 /γi γi1 · · · γin , n!
leading precisely to the factor n(γ n , γ ) = n + 1. On the other hand, if m < n, then both terms vanish. In the case that contains a subgraph γ such that γ /γ γ , we find a discrepancy between the two terms which is given by the following sum: 1 n( γ , γ , γ ) / γ. ργ γ () = n(γ , γ ) γ ⊂, γ /γ γ
γ that are isomorHere n( γ , γ , γ ) is by definition the number of disjoint subgraphs of phic to γ and such that γ /γ γ , which do indeed all contribute to γ γ (). We replace the above sum by a sum over insertion places of γ in γ , while correcting for the equivalent insertions. The latter correcting factor is given as the number of elements [y , φ ] ∈ X (γ , γ ) such that γ ◦(y ,φ ) γ ◦(y,φ) . Such an isomorphism can be induced by an element g ∈ Aut(γ ), with [y , φ ] = g[y, φ] but leaving γ untouched, leading to a factor of |Aut(γ )[y, φ]|. The number of isomorphisms γ ◦(y ,φ ) γ ◦(y,φ) that are not induced by such an element, is given precisely by the factor N (y, φ). Thus, on inserting the expression for n(γ ◦(y,ψ) γ , γ , γ ) derived in Lemma 14, we infer that, ργ γ () = =
1 n(γ , γ )
[y,ψ]∈X (γ ,γ )
[y,ψ]∈X (γ ,γ )
M(y, ψ)N (y, ψ) γ ◦(y,ψ) γ () N (y, ψ)|Aut(γ )[y, ψ]|
Sym(γ ◦(y,ψ) γ ) γ ◦(y,ψ) γ (), Sym(γ γ )
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where we have applied Lemma 12 in going to the second line. We have also used Eq. (2) to replace n(γ , γ )Sym(γ )Sym(γ ) by Sym(γ γ ).
2.3. The coproduct on 1PI Green’s functions. Our main result of this section is the following. Proposition 16. The coproduct takes the following form on the 1PI Green’s functions: rL
L 1 () = Sym()
K =0 γ K ,rL−K
|γ γ ⊗ , Sym(γ )Sym()
where the sums are over all 1PI graphs with the indicated residue and loop number, and graphs γ at the indicated loop order that are disjoint unions of 1PI graphs. Proof. Since = |γ1|∨ γ γ ⊗ γ , this would follow from the following equality, for γ any disjoint union of 1PI graphs at loop order K < L and γ0 an auxiliary graph, rL
| γ | γ0 | γ0 γ () = . |γ |∨ Sym() Sym(γ )Sym() r
(6)
L−K
Indeed, putting γ0 = ∅ and summing over γ then gives the desired result. We show that Eq. (6) holds by induction on the number of connected components of γ . Lemma 17. If γ is a 1PI graph, then Eq. (6) holds. Proof. If γ = ∅, there is nothing to prove, since γ () = , Sym(∅) = 1 and | ∅ | γ0 ≡ | γ0 . We claim that the following equality holds for γ , = ∅:
=
| γ0 n(, γ , ) |γ |∨ Sym()
[x,φ]∈X ( ,γ )
◦(x,φ) γ | γ0 n( ◦(x,φ) γ , γ , ). |γ |∨ N (x, φ)|Aut( )[x, φ]|Sym( ◦(x,φ) γ )
Indeed, one can replace the sum on the left-hand-side over by a sum over insertion places of γ in (so that ◦(x,φ) γ for some [x, φ] ∈ X (, γ ), and also res( ) = res()), provided one divides by a combinatorial factor counting the number of equivalent insertions. This factor is given as the number of elements [x , φ ] ∈ X (, γ ) such that ◦(x ,φ ) γ ◦(x,φ) γ , in which case Sym( ◦(x,φ) γ ) = Sym( ◦(x ,φ ) γ ) and also ◦(x,φ) γ | γ0 = ◦(x ,φ ) γ | γ0 . ◦(x,φ) γ can be induced by an element in Such an isomorphism ◦(x ,φ ) γ g ∈ Aut( ) with [x , φ ] = g[x, φ] but leaving γ untouched. This leads to division by the length of the orbit Aut( )[x, φ]. Otherwise, an isomorphism from ◦(x,φ) γ to . In that case, it can not be ◦(x ,φ ) γ has to map γ to an isomorphic subgraph γ ⊂ induced by an element in Aut( ), leading precisely to the additional factor of N (x, φ).
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Equation (6) now follows directly by inserting the expressions obtained in Lemma 12 and 14 in the above equation and summing over all 1PI graphs , as in Eq. (5). We also noted on the way that by definition 1 |γ |∨
◦(x,φ) γ | γ0 = | γ | γ0 .
[x,φ]∈X ( ,γ )
The case = ∅ arises whenever K = L and γ , in which case the combinatorial factors | γ0 and ∅ | γ | γ0 coincide. Assume now that Eq. (6) holds for γ a (non-empty) disjoint union of 1PI graphs of loop order K . We will prove that it also holds for the disjoint union γ γ = γ ∪ γ of it with a non-empty 1PI graph γ of loop order K . An application of Lemma 15 yields, | γ0 | γ0 γ γ () = γ γ () n(γ , γ )|γ γ |∨ Sym() ∨ Sym() | γ0 ργ γ (). − |γ γ |∨ Sym()
|γ γ |
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Since γ is a 1PI graph, we can apply Lemma 17 to the first term, which gives for the sum over all graphs rL , 1 | γ0 γ γ () n(γ , γ ) r |γ γ |∨ Sym() L
=
| γ | γ0 1 γ () n(γ , γ ) r |γ |∨ Sym(γ )Sym() L−K
=
| γ γ0 + ( | γ )(γ | γ0 ) 1 γ (), n(γ , γ ) r Sym(γ )|γ |∨ Sym() L−K
using also Eq. (4). The induction hypothesis – that is, validity of Eq. (6) in the case of γ – now yields, 1 | γ0 γ γ () n(γ , γ ) r |γ γ |∨ Sym() L
=
rL−K −K
| γ | γ γ0 + ( | γ | γ )(γ | γ0 ) , Sym(γ γ )Sym()
combining once more the symmetry factors Sym(γ ) and Sym(γ ) with the help of n(γ , γ ). For the second term in Eq. (7), we can use the induction hypothesis on γ ◦(y,ψ) γ to show that rL
| γ0 ργ γ () = |γ γ |∨ Sym()
rL−K −K [y,ψ]∈X (γ ,γ )
| γ ◦(y,ψ) γ | γ0 , |γ |∨ Sym(γ γ )Sym()
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since |γ ◦(y,ψ) γ |∨ = |γ |∨ . We conclude with the following equality: | γ γ | γ0 = | γ | γ γ0 + ( | γ | γ )(γ | γ0 ) 1 | γ ◦(y,ψ) γ | γ0 , − |γ |∨ [y,ψ]∈X (γ ,γ )
which follows easily from Definition 11. Indeed, by definition | γ | γ γ0 + ( | γ | γ )(γ | γ0 ) 1 = |γ γ |∨
( ◦(x,φ) γ ) ◦(x ,φ ) γ | γ0 ,
[x,φ]∈X (,γ ) [x ,φ ]∈X (◦(x,φ) γ ,γ )
which counts the number of places to insert γ γ and then γ0 in . Subtraction of the number of such places with γ sitting inside γ , leads precisely to the number of places to subsequently insert γ γ and γ0 in . 3. Ward Identities in QED This section will be a warming-up of what is to come in the next section concerning nonabelian gauge theories. Quantum electrodynamics is an abelian gauge theory, and as a consequence, the Slavnov–Taylor identities (cf. Def. 25 below) become much more simple. More precisely, they become linear in the graphs and also known as Ward identities [13] (see Def. 19 below). We first make some observations about Feynman graphs in QED. We then proceed to prove compatibility of these Ward identities with the Hopf algebra of renormalization. 3.1. Feynman graphs in QED. In (massless) quantum electrodynamics, there is only the vertex of valence three, describing the interaction of the photon with a pair of electrons. There are two types of edges corresponding to the photon (wiggly edge) and the electron (straight edge). Summarizing, we have in the notation of the previous section: R = R V ∪ R E with = {
};
= {
,
}.
In particular, this means that in the process of renormalization, only three types of graphs are of importance: the vertex graph, the electron self-energy graph and the vacuum polarization. Correspondingly, we define the following 1PI Green’s functions,
Ge = 1 −
e
, Sym()
with and where the sum is over all 1PI Feynman graphs with the indicated residue. When this sum is restricted to 1PI graphs with loop number L, we denote this Green’s function at loop order L by G rL . In particular, G r0 is understood as the 1PI graph with loop number zero, which is the empty graph; hence G r0 = 1.
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Remark 18. The (regularized) Feynman rules can be understood as an algebra map φ from H to the field K of Laurent series in the regularization parameter [2]. Under this map, the above Green’s functions are mapped to the corresponding Feynman amplitudes. In particular, we see that the (unrenormalized) effective action can be written as:
A simplification in QED, is that the number | γ of insertion places only depends on the number of loops, the residue of and γ . Let denote the number of electron lines of r and the number of photon lines. Thus, if has residue r , then is the number of external electron lines of and the number of its external photon lines. Also recall the notation m ,r for the number of vertices/edges in of type r , introduced just below Definition 8. The number of loops L of the graph is then given by,
where the number of vertices instance [9, Sect. 10.1]),
in with residue r can be expressed as (see for
Combining these equalities, we arrive at the following relations:
from which we infer that the number | γ can be expressed in terms of L, r and γ . We denote this number by (L , r ) | γ . Explicitly, we have for example,
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3.2. The Ward identities. In QED, there are relations between the (amplitudes of the) Green’s functions and ; these relations play an important role in the process of renormalization. We will show here that they can be implemented on the Hopf algebra, in a way that is compatible with the coproduct. Definition 19. The Ward elements W L at loop order L > 0 are defined by
where the sum is over 1PI Feynman graphs with loop number L and with the indicated residue. Moreover, we set .
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Before stating our main result on the compatibility of the Ward elements with the coproduct, we introduce the following combinatorial factor:
Note the mixture between the two factors and If the vertex graphs in γ are labeled as γ1 , . . . , n with c(L , γ ) recursively by,
of Eq. (8) above. , one could also define
(9) for a vertex graph γn+1 while setting
if
Theorem 20. For any L ≥ 0, we have
Consequently, the ideal I generated by the Ward elements W L for every L is a Hopf ideal in H , (I ) ⊆ I ⊗ H + H ⊗ I,
(I ) = 0,
S(I ) ⊆ I.
Proof. Essential for the proof will be the relation between the two factors and displayed in Eq. (8). Using Pascal’s rule, Eq. (16), we derive the following relation between the two numbers:
Before inserting this in the expression for derived in Proposition 16, we observe that the second term is just c(L , γ ) with γ the graph γ with one vertex graph γv subtracted, times a factor of . Similarly, the third term is c(L , γ ), where now γ is γ with one electron self-energy graph γe subtracted, times a factor of . Note that in the respective cases and , the above two terms vanish. We then find that,
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The above factors and are precisely canceled when replacing the sum over γ by a sum over γe and γ . Indeed, in doing so, a factor of is needed to correct for overcounting,
by the very definition of n(γ , γe ), and a similar result holds for γv . We thus have respective factors of and in the denominator, and combining n( γ , γe ) with Sym(γe γ ) (and similarly for the analogous expression for γv ) using Eq. (2) yields,
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4. Non-abelian Gauge Theories In this section, we come to the main purpose of this article, and show compatibility of the Slavnov–Taylor identities [10, 12] with the Hopf algebraic structure of renormalization of non-abelian gauge fields. Before that, we carefully describe the setting of non-abelian gauge theories. In particular, we start by describing the graphs that are allowed in such theories and list some combinatorial properties of the number | γ of insertion places defined in Sect. 2.2. These properties will be essential in the proof of compatibility of the Slavnov–Taylor identities with the coproduct.
4.1. Feynman graphs in non-abelian gauge theories. In order to make the following as concrete as possible, we work in the setting of the non-abelian gauge theory quantum chromodynamics (QCD). It describes the interaction between quarks (the fermions) via gluons (the gauge bosons). In contrast with quantum electrodynamics described previously, there are now three vertices of valence three, describing the interaction of the fermion and ghost with the gluon, as well as the cubic gluon self-interaction. In addition, there is the quartic gluon self-interaction. This means that the Feynman graphs are built from the following two sets of vertices and edges:
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where the plain, dotted and curly lines represent the quark, ghost and gluon, respectively. Corresponding to the elements in R = R V ∪ R E , we define 7 (1PI) Green’s functions, Gv = 1 +
v
Ge = 1 −
e
Sym()
(v ∈ R V );
Sym()
(e ∈ R E ),
with the sum over all 1PI Feynman graphs with the indicated residue, i.e. res(r ) = r . Remark 21. As for quantum electrodynamics, the (regularized) Feynman rules for QCD (as listed for instance on p. 34 of [1]) can be understood as an algebra map φ from H to the field K of Laurent series in the regularization parameter [2]. The (unrenormalized) effective action can be written as,
Contrary to the case of QED (cf. Sect. 3.1), there are no relations in QCD expressing the numbers m ,r of vertices/edges of of the type r in terms of the loop number and the residue of a graph . However, we do have relations between them. Denote by Ne (r ) the number of lines attached to rthat are of the type e ∈ R E ; by convention, Ne (e ) = 2δe,e for an edge e . Set N (r ) = e Ne (r ), it is the total number of lines attached to r . Lemma 22. Let be a (QCD) 1PI Feynman graph at loop order L and with residue r . Then, the following relations hold:
Proof. Equation (a) is the usual expression of the Betti number of the graph ; it can be proved by induction on the number of edges of the graph . Equations (b) and (d) follow from the statement that each of these two vertices involve exactly two quark or two ghost lines, respectively. We prove that Eq. (c) holds by induction on the number of edges. First, denote by V3 and V4 the number of all trivalent and quadrivalent vertices of , respectively. Since the type of vertices and edges is irrelevant for Eq. (c), we have to show validity of V3 + 2V4 − N + 2 = 2L ,
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for a graph consisting of V3 trivalent and V4 quadrivalent vertices of a single fixed type. Induction on the number of edges starts with the graph consisting of only one edge, for which N = 2, by definition. Then, if we add an edge to a graph for which Eq. (11) holds, we have the following possibilities to connect the new edge:
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edge - edge edge - edge edge - 3 vertex 3 vertex - 3 vertex edge - ext 3 vertex - ext
δV3 +2 −2 +1 −1
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δV4
δN
+1 +1 +2 +1
δL +1 +1 +1 +1
+1 +1
One readily checks that the corresponding changes in V3 , V4 , N and L leave Eq. (11) invariant. Motivated by these properties, we introduce the following notion of admissibility of a vector m = (m r )r ∈R . Definition 23. A vector m = (m r )r ∈R is called (L , r )-admissible if it satisfies Eqs. (a)–(d) in Lemma 22. Lemma 24. Two (L , r )-admissible vectors differ by a finite number of combinations of the following three steps: I II III
+1
+1 +1
+1 +1
+1 +1 +2
−1 −1 −1
acting by m r → m r + δm r , while retaining admissibility at each step. Proof. Clearly, two admissible vectors differ by integers in each of the entries m e for e ∈ R V . Since Eqs. (a)–(d) in Lemma 22 impose linear constraints on the entries of m, it is enough to consider the cases m e → m e + 1 for each of the three edges individually (thus leaving the other m e invariant). Then, Eqs. (a)–(d) become four independent constraints on the four remaining entries m v , v ∈ R V , the solution of which is displayed in the above table. 4.2. Slavnov–Taylor identities. Contrary to the linear Ward identities that we have encountered in QED, there are now quadratic relations between Green’s functions that reflect the non-abelian nature of the gauge symmetry of the corresponding classical field theory. In addition, there are linear identities between full Green’s functions (cf. for instance [8]). We will not consider the latter here and focus on the (quadratic) Slavnov-Taylor identities between 1PI Green’s functions. For a derivation of these quadratic identities using functional integration methods, we refer the reader to the standard text books on quantum field theory, such as [14, Sect. 17.1] or [15, Sect. 21.4–5]. Pictorially, we have the following three identities:
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where the blob stands for the 1PI Green’s function corresponding to the indicated external structure. In the Hopf algebraic setting of renormalization, this motivates the following definition. Definition 25. The Slavnov–Taylor elements are defined by
Note that these elements involve both a linear and quadratic part, for instance,
The choice for using the same labelling I, II, III for the Slavnov–Taylor identities as for the admissible steps in Lemma 24 is not coincidental, but motivated by the next lemma. First, define for vectors m = (m r )r ∈R and n = (nr )r ∈R the following constant:
mv m e + ne − 1 m . nv ! ne ! c = n nv ne v∈R V
e∈R E
A glance back at Sect. 2.2 makes one realize that whenever m and n arise from two graphs and γ (i.e. m = m , n = nγ ), this constant becomes the number | γ of insertion places of γ in . We also introduce the following standard basis { fr }r ∈R of vectors corresponding to the elements r ∈ R by setting ( fr )r = δrr . Lemma 26. The following equation holds for A = I, II, III:
Proof. First observe that Cases I and II can be treated simultaneously by exchanging external electron lines with external ghost lines. After applying Pascal’s rule (16) eight times, we obtain
Now, the analogue of Eq. (10) in the setting of QCD yields, when applied to these six terms, precisely the above formula in the case A = I (and similarly for A = II). In like manner, one shows that
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from which, with the proper analogue of Eq. (10), one concludes the above formula for A = III. The above lemma will be essential in the proof of our main result, stating compatibility between the Slavnov–Taylor elements and the Hopf algebra structure of renormalization. Theorem 27. The ideal I generated by the Slavnov–Taylor elements is a Hopf ideal, i.e. (I ) ⊆ I ⊗ H + H ⊗ I,
(I ) = 0,
S(I ) ⊆ I.
Proof. By the very definition of the ideal I the second condition is trivially satisfied whereas the third follows from the first, by the recursive formula for the antipode in Eq. (3). It is thus enough to show that (ST A ) ⊆ I ⊗ H + H ⊗ I for A = I, II, III. Clearly, the primitive part of the coproduct (ST A ) is of the desired form, by the very definition of being the primitive part. We will apply Proposition 16 to determine the non-primitive part. Starting with the second term in STI , we find that,
The first two terms combine, since
by an application of Pascal’s rule (16). On the other hand, the second two terms contribute with
Finally, the quadratic terms become ⎛ ⎞
m γ γ m 2 ⎠ m 1 1 + m 2 ⎝ c = , c c n1 n2 n γ Sym(γ ) Sym(γ ) γ
n1 + n 2 = nγ
γ
(13c)
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using Vandermonde’s identity (17). Unfortunately, the three coefficients in Eq. (13) do not coincide, and it is thus not immediately clear how to combine the terms in Eq. (12) in order to obtain on the second leg of the tensor product. However, we claim that they differ by a finite number of combinations of the steps I, II and III, so that they can be combined at the cost of adding terms in I ⊗ H , exactly those that appear in Lemma 26. Indeed, in the case of (13b) one rewrites trivially,
and observes that the vector is -admissible if γ is at loop order K . Also, regarding Eq. (13c), one notes that is admissible. Since m 1 is -admissible by definition, an application of Lemma 24 in combination with Lemma 26 shows that allterms in Eq. (12) can be combined to give plus elements of the form A ST A h A ⊗ h A for some can be written as h, h A and h A in H . In like manner one shows that with that same h and some g A , g A ∈ H . Hence, we can write ST A (g A ⊗ g A − h A ⊗ h A ), (STI ) = h ⊗ STI + A
which is indeed an element in H ⊗ I + I ⊗ H . A completely analogous argument shows that the same conclusion holds for STII and STIII . 5. Conclusions After having derived the necessary combinatorial identities, and obtained a formula for the coproduct on 1PI Green’s functions, we have showed that the Slavnov–Taylor identities in quantum chromodynamics can be implemented on the Hopf algebra of Feynman graphs as relations defining a Hopf ideal I . Let us work in the dimensional regularization scheme with minimal subtraction. Since the map φ : H → K defined by the (regularized) Feynman rules vanishes on the ideal I , it factors through an algebra map from the := H/I to the field K of Laurent series in the regularization quotient Hopf algebra H parameter. Since H is still a commutative connected Hopf algebra, there is a Birkhoff −1 factorization φ = φ− ∗ φ+ [2, Theorem 4] for the convolution product ∗ in the group , K ). The two algebra maps φ± : H → K are given on 1PI graphs by the Hom( H following recursive formulas: ⎛ ⎞ φ− () = −T ⎝φ() + φ− (γ )φ(/γ )⎠ , γ
φ+ () = φ() + φ− () +
φ− (γ )φ(/γ ),
γ
where T is the projection on the pole part in K . It was realized in [2] that φ+ () and φ− () precisely give the renormalized Feynman amplitude and the counterterms, respectively, to K , we conclude corresponding to the graph . Since they are algebra maps from H that they automatically satisfy the Slavnov–Taylor identities.
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Moreover, the compatibility of the Slavnov–Taylor identities with the Hopf algebra structure implies validity of the ‘gauge theory theorem’ in [6, Thm. 5] (see also [7]). γ The latter states there exists a certain sub Hopf algebra and that the map B+ : H → that Hlin , X → n (γ , X, ), where n is a normalized version of n defined just above Eq. (5), is a closed Hochschild cocycle. Acknowledgements. The author would like to thank Matilde Marcolli for several discussions and an anonymous referee for useful comments.
A. An Alternative Proof of Theorem 20 We give an alternative proof of the above theorem stating compatibility of the Ward identities with the coproduct, which is based on our previous result [11] on the compatibility of the so-called Ward–Takahashi identities with the coproduct. Such Ward–Takahashi identities are identities between individual graphs, and an expression of the above Ward elements W L in terms of the Ward–Takahashi elements (to be introduced below) allows us to deduce Theorem 20 from this result. Let us start by introducing a map similar to γ defined in Sect. 2.2. For a disjoint union of 1PI graphs γ and a 1PI electron self-energy graph γe , we define on a 1PI graph , eγ ,γe () :=
/γ γe (e ),
γ γe ⊂ γ γe γ γe
where /γ γe (e ) is the graph /γ γe with an external photon line attached to the edge e corresponding to γe in the quotient. In the case that γ is the empty graph, this map will be denoted by eγe := e∅,γe and when in addition γe = , we set e () = ∅. Two examples that might help the reader to see what is going on are,
The analogue of n(, γ , ) is given by n e (, γ , ); it is defined to be the number of subgraphs γ of the electron self-energy graph that are isomorphic to the electron self-energy graph γ , and such that /γ (e ) is isomorphic to the vertex graph . We also introduce the following combinatorial factor, to be used below. The compatibility of the so-called Ward–Takahashi identities with the coproduct was derived in [11] and will be recalled in Proposition 28 below; for completeness, we also restate its proof. It gives relations between individual graphs, and will be used in the alternative proof of Theorem 20, involving relations between Green’s functions. We first recall some of the notation in that paper. Given any electron self-energy graph , we can label the internal electron edges from 1 to If we fix such a labelling, we denote by (i) the graph with an external photon line attached to the electron line i.
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Proposition 28. Let be a 1PI electron self-energy graphs and define the corre sponding Ward–Takahashi element by W () = i (i) + , with the sum over internal electron lines in . Moreover, set W (∅) = 0. Then,
where the sum is over disjoint unions of 1PI graphs γ (including the empty graph) and 1PI electron self-energy graphs γe . Proof. We start by computing i (i) ; we split the sum over subgraphs into two terms, those for which γ contains the electron line i, and those for which it does not,
where γ E denotes the disjoint union of the electron self-energy graphs in γ . The absence of terms in which i ∈ γ − γ E is due to the fact that there are no vertices with more than one photon line, so that the corresponding graph (γ − γ E )(i) does not appear as a subgraph of (i) in the coproduct. The first term can be written as, γ (i) ⊗ (i)/γ (i) = (γ − γe ) γe (i) ⊗ /γ γe (e), γ ⊆ γe ⊂γ
γ ⊆ i∈γ E
i∈γe
by decomposing the sum over all i that are part of the subgraph γ E into the connected components γe of γ E , thereby also noting that if i ∈ γe , the quotient (i)/γe (i) is nothing other than a /γe (e). On the other hand, the second term can be split into two parts, one for which i is an external electron edge for one of the electron self-energy graphs in γ , and one for which it is not, ⎞ ⎛ ⎝ γ ⊗ (i)/γ = γ ⊗ (i)/γ + γ ⊗ /γ (i)⎠ γ ⊆ i ∈γ /
γ ⊆
=
i∈∂γ E
γ ⊆ γe ⊂γ
i ∈γ / ∪∂γ E
(γ − γe )γe ⊗ /(γ − γe )γe (e) + γ ⊗
/γ ( j).
j∈/γ
Here we used the fact that although an electron self-energy graph has two external electron edges, they might be in common with another electron self-energy graph that is also part of γ . This means that for a block of electron self-energy graphs that sit inside by concatenation, there is a term (γ − γe )γe ⊗ /(γ − γe )γe (e) for each such graph, plus one for the closing “fence pole”. Indeed, the number of external electron edges of such a block is precisely the number of its constituents plus one. The extra term combines with the second term, since it precisely gives the term /γ ( j) with j the electron edge corresponding to γe so as to complete the sum over all internal electron edges in the last term. Proposition 29. There are the following relations between W () and the Ward elements WL ,
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Proof. Since every vertex graph can be written as (i) for an electron self-energy graph , we only have to check that the symmetry factors turn out right. We first observe that Sym((i)) coincides with the order of the isotropy group Aut()i of the electron edge i in Aut(). Moreover, two graphs (i) and (i ) are isomorphic if and only if i and i are in the same orbit under the action of Aut(). An application of the orbit-stabilizer theorem shows that such an orbit has length |Aut()i| = Sym()/|Aut()i |, so that,
from which the result follows at once.
Before continuing our derivation of Theorem 20 from Proposition 28, we prove a result analogous to Lemma 15 for the map eγ ,γe . Instead of the insertion of graphs of Definition 8 that appeared in Lemma 15, it involves insertion of an electron self-energy graph into a vertex graph, defined as follows. Definition 30. An insertion of a 1PI electron self-energy graph γ into a 1PI vertex graph [1] to the two electron lines connected to the ver is given by an isomorphism φ from γext tex to which the external photon line is attached, after removal of this photon line. The resulting electron self-energy graph is denoted by #φ γ . For example,
Lemma 31. Let γ and γe be as above and let graphs in γ as before. Then,
denote the number of 1PI vertex
eγ ,γe = γ eγe − ρ γ ,γe , where ρ γ ,γe is defined in terms of the vertex graph
in γ by,
Proof. Similar to the proof of Lemma 15, we introduce a map ρ γ ,γe ; it corrects for the overcounting in the cases that there is a subgraph γ of containing γe γe , such that γ /γe (e ) γ . Indeed, quotienting by such a γ does not appear in eγ ,γe (), although e it does in γ γe (). We then write,
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since each such graph γ has to be isomorphic to the graph obtained by inserting γe (using #) into one of the vertex graph components of γ , with an additional factor of n e (γl #φ γe , γe , γl ) due to the multiplicity arising on the left-hand-side. On the other hand, division by n(γ − γl , γl ) corrects for insertions of γe into isomorphic γl ; indeed, if γl γl then clearly γl #φ γe γl #φ γe . Finally, the factor |γl #φ γe |∨ /|γl |∨ arises from the difference of summing over subgraphs isomorphic to γl and subgraphs isomorphic to γl #φ γe . Theorem 20. For any L ≥ 0, we have
Consequently, the ideal I generated by the Ward elements W L for every L is a Hopf ideal in H , (I ) ⊆ I ⊗ H + H ⊗ I,
(I ) = 0,
S(I ) ⊆ I.
Proof. Firstly, with W L being related to W () by the above Eq. (29) (and understood to be zero for L = 0), Proposition 28 implies that
The second term can be easily written in the desired form since with Proposition 16,
which is indeed an element in H ⊗ I . For the first term, we derive the result from the following equation:
for two graphs γ and γe at respective loop order K and K . In fact, validity of this equation implies that
which, on its turn, is an element in I ⊗ H . We prove that Eq. (14) holds by induction on the number of 1PI vertex graphs in γ . If this number is zero, Lemma 31 takes a simple form and we conclude that in this case,
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(15) since for each 1PI vertex graph , there are precisely |γe |∨ |Aut()i | = |γe |∨ Sym((i)) many electron self-energy graphs that result in after quotienting by a subgraph isomorphic to γe and connecting an external photon line to the corresponding internal electron line. Thus, an application of Eq. (6) shows validity of Eq. (14), since by definition Let us then suppose that Eq. (14) holds for and number the vertex graphs in γ by γ1 , . . . , γn . If γn+1 is another 1PI vertex graph, then Lemma 31 yields
after insertion of the definition of ρ and an application of Definition 3. The first term can be reduced by applying Eq. (15) to eγe and Eq. (6) to γ γn+1 . For the second term, we can apply the induction hypothesis, since γ γn+1 −γl has n 1PI vertex graph components. With the following two equalities for the combinatorial factors: n(γ γn+1 − γl , γl )Sym(γl )Sym(γ γn+1 − γl ) = Sym(γ γn+1 ); Sym(γl #φ γe ) = Sym(γe )Sym(γl ) n e (γl #φ γe , γe , γl ), which can be obtained from Eq. (2) and the observation that n e (γ #φ γe , γe , γ ) counts precisely the automorphisms of γl #φ γe that do not come from an automorphism of γl or γe , we derive
Note that the sum over [φ] has been cancelled against the factor |γe |∨ in the denominator. A glance back at the recursive definition of c in Eq. (9) then completes the proof. B. Some Combinatorial Identities For completeness, we list some combinatorial identities used throughout the text. Firstly, there is the well-known Pascal rule:
k+1 k k = + , (16a) l l l −1 which can be conveniently rewritten as
k +l −2 k +l −1 k +l −2 = − , l l l −1 for combinations with repetition.
(16b)
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Another identity used in the text is due to Vandermonde, stating that
q k1 k2 k1 + k2 = , l q −l q
(17a)
l=0
which can be proved by expanding both sides of (1 + t)k1 +k2 = (1 + t)k1 (1 + t)k2 . The analogous result for combinations with repetition is,
q k1 + l − 1 k2 + q − l − 1 l=0
l
q −l
k1 + k2 + q − 1 = . q
(17b)
This can be proved by equating both sides of (1−t)−k1 −k2 = (1−t)−k1 (1−t)−k2 , where n+r −1 r −n t is the generating function for combinations with repetition. (1 − t) = r r References 1. Collins, J.: Renormalization. Cambridge: Cambridge University Press, 1984 2. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann- Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249–273 (2000) 3. Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. In press, 2007 4. Diestel, R.: Graph theory. Volume 173 of Graduate Texts in Mathematics. New York: Springer-Verlag, 1997 5. Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2, 303–334 (1998) 6. Kreimer, D.: Anatomy of a gauge theory. Ann. Phys. 321, 2757–2781 (2006) 7. Kreimer, D.: Dyson–Schwinger equations: From Hopf algebras to number theory. In: I. Binder, D. Kreimer, eds., Fields Inst. Commun. 50, Providence, RI: Amer. Math. Soc., 2007, pp. 225–248 8. Pascual, P., Tarrach, R.: QCD: Renormalization for the Practitioner. Volume 194 of Lecture Notes in Physics, Boston: Springer-Verlag, 1984 9. Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Boston, MA: Addison-Wesley, 1995 10. Slavnov, A.A.: Ward identities in gauge theories. Theor. Math. Phys. 10, 99–107 (1972) 11. van Suijlekom, W.D.: The Hopf algebra of Feynman graphs in QED. Lett. Math. Phys. 77, 265–281 (2006) 12. Taylor, J.C.: Ward identities and charge renormalization of the Yang-Mills field. Nucl. Phys. B33, 436–444 (1971) 13. Ward, J.C.: An identity in quantum electrodynamics. Phys. Rev. 78, 182 (1950) 14. Weinberg, S.: The quantum theory of fields. Vol. II. Cambridge: Cambridge University Press, 1996 15. Zinn-Justin, J.: Quantum field theory and critical phenomena. Volume 77 of International Series of Monographs on Physics. New York: The Clarendon Press Oxford University Press, 1989 Communicated by A. Connes
Commun. Math. Phys. 276, 799–825 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0349-5
Communications in
Mathematical Physics
Gauss Decomposition of the Yangian Y (glm|n ) Lucy Gow School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia. E-mail: [email protected] Received: 24 October 2006 / Accepted: 30 April 2007 Published online: 2 October 2007 – © Springer-Verlag 2007
Abstract: We describe a Gauss decomposition for the Yangian Y (glm|n ) of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y (A(m − 1, n − 1)) (for m = n) defined and studied in papers by Stukopin, and suggests natural definitions for the Yangians Y (sln|n ) and Y (A(n, n)). We also show that the coefficients of the quantum Berezinian generate the centre of the Yangian Y (glm|n ). This was conjectured by Nazarov in 1991. 1. Introduction The Yangian Y (glm|n ) is the Z2 -graded associative algebra over C with generators {ti(rj ) | 1 ≤ i, j ≤ m + n; r ≥ 1} and defining relations (r )
(s)
[ti j , tkl ] = (−1)i
j+i k+ j k
min(r,s)−1
( p) (r +s−1− p)
(tk j til
(r +s−1− p) ( p) til ),
− tk j
(1)
p=0
where i is the parity of the index i. We take i = 0 for i ≤ m; and i = 1 for i ≥ m + 1, and write square brackets for the super-commutator. We define the formal power series (1)
(2)
ti j (u) = δi j + ti j u −1 + ti j u −2 + · · · and a matrix T (u) =
m+n i, j=1
ti j (u) ⊗ E i j (−1) j(i+1) ,
(2)
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where E i j is the standard elementary matrix. (Here we identify an operator Ai j ⊗ m+n j(i+1) −1 m|n E i j (−1) in Y (glm|n )[[u ]] ⊗ End C with the matrix Ai j i, j=1 . The extra sign ensures that the product of two matrices can still be calculated in the usual way). Then, as for the Yangian Y (gln ) (see for example [2, 15]), the defining relations may be expressed by the matrix product R(u − v)T1 (u)T2 (v) = T2 (v)T1 (u)R(u − v), where 1 P12 (u − v) j and P12 is the permutation matrix: P12 = i,m+n j=1 E i j ⊗ E ji (−1) . We also have the following equivalent form of the defining relations: R(u − v) = 1 −
[ti j (u), tkl (v)] =
(−1)i j+i k+ j k (tk j (u)til (v) − tk j (v)til (u)). (u − v)
(3)
The Yangian Y (glm|n ) is a Hopf algebra with comultiplication ∆ : ti j (u) →
m+n
tik (u) ⊗ tk j (u),
(4)
k=1
antipode S : T (u) → T (u)−1 and counit : T (u) → 1. Throughout this article we observe the following notation for entries of the inverse of the matrix T (u): n . T (u)−1 =: ti j (u) i, j=1
A straightforward calculation yields the following relation in Y (glm|n ): m+n m+n (−1)i j+i k+ j k [ti j (u), tkl (v)] = · δk j tis (u)tsl (v) − δil tks (v)ts j (u) . (5) (u − v) s=1
s=1
We may define two different filtrations on the Yangian Y (glm|n ). These are defined by setting the degree of a generator as follows: deg1 (ti(rj ) ) = r ;
deg2 (ti(rj ) ) = r − 1.
(6)
Let gr1 Y (glm|n ) and gr2 Y (glm|n ), respectively, denote the corresponding graded algebras. There is an injective homomorphism ι : U (glm|n ) → Y (glm|n ) given by i ι : E i j → ti(1) j (−1) .
The injectivity of ι follows from the fact that its composition with a surjective homomorphism π : Y (glm|n ) → U (glm|n ) is the identity map on the universal enveloping algebra U (glm|n ). This map π is given as follows: π : ti j (u) → δi j + E i j (−1)i u −1 .
(7)
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Thus we regard the universal enveloping algebra U (glm|n ) as a subalgebra of Y (glm|n ). The Yangian Y (glm|n ) was introduced in [16]. It has applications in mathematical physics because it describes symmetry in integrable models of Calogero-Sutherland systems [1, 12], superstrings in Ad S5 × S 5 [11], and in the hierarchy of a form of the non-linear super-Schrödinger equation with m bosons and n fermions [4]. The centre of the Yangian Y (glm|n ) is conveniently described using a formal power series called the quantum Berezinian (see Sect. 7). Vladimir Stukopin [19, 20] has introduced Yangians for classical simple Lie superalgebras. In this article we provide a new presentation for the Yangian Y (glm|n ) that allows us to relate it to the Yangian Y (A(m − 1, n − 1)) (where m = n) studied by Stukopin. This leads us to introduce a natural definition of the Yangian Y (sln|n ) as a subalgebra of the Yangian Y (gln|n ), as well as a definition of Y (A(n − 1, n − 1)) (see Sect. 8). The Yangian that features in D = 4 superconformal Yang-Mills theory [6] is that associated with the supergroup P SU (4, 4), which has a Lie superalgebra of type A(3, 3), so the results presented here may be relevant. Our paper follows similar treatments of the Yangian Y (gl N ) given in papers by Brundan and Kleshchev, and Crampé, and of the super-Yangian Y (gl1|1 ) in the work of Jin-fang Cai, Guo-xing Ju, Ke Wu and Shi-kun Wang (see [2, 3, 5]). 2. The Poincaré-Birkhoff-Witt Theorem for Super Yangians In this section we prove the Poincaré-Birkhoff-Witt theorem for the Yangian Y (glm|n ). The proof is based very closely on that of the corresponding theorem for Y (gl N ) given in [2]. For each positive integer l ≥ 1, we define a homomorphism κl := (π ⊗ · · · ⊗ π ) ◦ ∆(l) : Y (glm|n ) → U (glm|n )⊗l , where ∆(l) : Y (glm|n ) → Y (glm|n )⊗l is the coproduct iterated (l − 1) times and π is the map given in (7). Then (r ) i + i 1 + i 2 +···+i r −1 r] κl (ti j ) = E ii[s11 ] E i[s1 i22] · · · E i[sr −1 , j (−1) 1≤s1 <···<sr ≤l 1≤i 1 ,...,ir −1 ≤m+n
(r )
⊗(s−1) ⊗ E ⊗ 1⊗(l−s) . For any r > l ≥ 1, we have κ (t ) = 0. where E i[s] ij l ij j =1 (r )
Theorem 1. Suppose we have fixed some ordering on the generators ti j (1 ≤ i, j ≤ m + n; r ≥ 1) for the Yangian Y (glm|n ). Then the ordered products of these, containing no second or higher order powers of the odd generators, form a basis for Y (glm|n ). Proof. 1 By relation (1), the graded algebra gr1 Y (glm|n ) is supercommutative, and thus the set of all ordered monomials in the generators ti(rj ) (with no second or higher order powers of the odd generators) span the Yangian Y (glm|n ). It remains to show that they are linearly independent. We show that, for every l ≥ 1, the corresponding monomials (r ) in {κl (ti j ) | 1 ≤ r ≤ l} are linearly independent in κl (Y (glm|n )). Consider the filtration F0 U (glm|n )⊗l ⊆ F1 U (glm|n )⊗l ⊆ F2 U (glm|n )⊗l ⊆ · · · 1 This theorem was stated in [21] but the proof there is incomplete.
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on U (glm|n )⊗l defined by setting each generator E i[rj ] to be of degree 1. Then the associated graded algebra gr U (glm|n )⊗l is the polynomial algebra on supersymmetric generators xi[rj ] := gr 1 E i[rj ] , where xi[rj ] is even if i + j = 0 and odd if i + j = 1. The map κl preserves the filtration on (r )
the Yangian given by setting deg1 (ti j ) = r , and thus defines a homomorphism between the corresponding graded algebras. It is enough to show that the same monomials in the (r ) (r ) elements yi j := grr κl (ti j ) in the graded algebra are linearly independent. But for this, (r )
it is enough to show that the superderivatives dyi j are linearly independent at a point. We have: (r ) i +i 1 +···+i r −1 r] yi j = xii[s11 ] xi[s1 i22] · · · xi[sr −1 . j (−1) 1≤s1 <···<sr
(r ) → dy has We will show that the matrix dφ corresponding to the map d xi[s] ij j non-zero determinant at a point. It suffices to show that the determinant of this matrix is (s) nonzero even when the variables are specialized to xkl = δkl cs (−1)k for some distinct cs (s ≥ 1). When the variables are specialized as described, we find: dyi(rj ) =
l
cs1 cs2 · · · csr −1 (−1)i d xi[s] j .
s=1 1≤s1 <···<sr −1 ≤l si =s
Let J be the (m + n) × (m × n) matrix J = δi j (−1)i . Then dφ = J ⊗ X l , where ⎛
⎞ 1 1 ... 1 + c3 + · · · + cl ) (c1 + c3 + · · · + cl ) . . . (c1 + c2 + · · · + cl−1 ) ⎟ ⎜ (c2 ⎜ ( ⎟ c c ) ( c c ) . . . ( i j i j ⎜ ⎟. i, j = 1 i, j = 2 i, j=l ci c j ) Xl = ⎜ ⎟ .. .. ⎝ ⎠ . . c1 c3 c4 · · · cl ... c2 c3 · · · cl−1 c2 c3 · · · cl We show by induction that detX l = Π1≤i< j≤l (ci − c j ) = 0, and hence detdφ = 0. Indeed, row-reducing X l gives the following matrix: ⎛
1 ... 1 (cl − c1 ) ... (cl −cl−1 ) ⎜ ⎜ (c − c ) i, j=1 ci c j . . . (cl − cl−1 ) i, j=l−1 ci c j 1 ⎜ l ⎜ i, j
⎞ 1 0⎟ 0⎟ ⎟ ⎟, .. ⎟ ⎠ . 0
which clearly has determinant (c1 − cl )(c2 − cl ) · · · (cl−1 − cl )detX l−1 . Now, suppose we have some non-trivial linear combination P of the ordered mono(r ) mials in ti j (with no second or higher order powers of the odd generators) and take l to
Gauss Decomposition of the Yangian Y (glm|n )
803 (r )
be any number greater than all the r that occur in P. Since the monomials in κl (ti j ) are linearly independent in κl (Y (glm|n )), we must have κl (P) = 0. Therefore, P = 0 in the Yangian. Let glm|n [t] denote the algebra glm|n ⊗ C[t] with basis {E i j t r }1≤i, j≤m+n;r ≥0 . Corollary 1. The graded algebra gr2 Y (glm|n ) is isomorphic to the universal enveloping algebra U (glm|n [x]), via the map gr 2 Y (glm|n ) → U (glm|n [x]) (r )
grr2−1 ti j → E i j x r −1 (−1)i (1 ≤ i, j ≤ m + n, r ≥ 1). 3. Gauss Decomposition of T (u) Here we describe a decomposition of the matrix T (u) in terms of the quasideterminants of Gelfand and Retakh [8]. Definition 1. Let X be a square matrix over a ring with identity such that its inverse matrix X −1 exists, and such that its ( j, i)th entry is an invertible element of the ring. Then the (i, j)th quasideterminant, |X |i j , of X is defined by the formula x11 |X |i j = xi1 xn1
· · · x1n ··· −1 · · · xin := (X −1 ) ji . ··· · · · xnn
· · · x1 j ··· · · · xi j ··· · · · xn j
By Theorem 4.96 in [8], the matrix T (u) defined in (2) has the following Gauss decomposition in terms of quasideterminants: T (u) = F(u)D(u)E(u) for unique matrices ⎛ ⎜ ⎜ D(u) = ⎜ ⎜ ⎝
d1 (u) .. . 0
··· d2 (u)
⎛ ⎞ 1 e12 (u) · · · e1,m+n (u) ⎜ ⎟ .. ⎜ . e2,m+n (u)⎟ ⎜ ⎟, E(u) =⎜ ⎟ .. .. ⎝ ⎠ . . 0 1
···
..
.
⎞
0 .. .
⎟ ⎟ ⎟, ⎟ ⎠
dm+n (u) ⎛
⎞ ··· 0 ⎜ .. ⎟ .. ⎜ f 21 (u) . .⎟ ⎜ ⎟, F(u) =⎜ ⎟ .. . . ⎝ ⎠ . . f m+n,1 (u) f m+n,2 (u) · · · 1 1
804
L. Gow
where
t11 (u) . di (u) = .. ti1 (u)
· · · t1,i−1 (u) t1i (u) .. .. . . , · · · ti,i−1 (u) tii (u) t11 (u) · · · t1,i−1 (u) t1 j (u) .. .. .. .. . . . . −1 ei j (u) = di (u) , ti−1,i (u) · · · ti−1,i−1 (u) ti−1, j (u) ti1 (u) · · · ti,i−1 (u) ti j (u) t11 (u) · · · t1,i−1 (u) t1i (u) .. .. .. .. . . . . f ji (u) = di (u)−1 . ti−1,1 (u) · · · ti−1,i−1 (u) ti−1,i (u) t ji (u) · · · t j,i−1 (u) t ji (u)
We use the following notation for the coefficients: (r ) (r ) di u −r ; (di (u))−1 = di u −r ; di (u) = r ≥0
ei j (v) =
ei(rj ) v −r ;
r ≥1
f ji (v) =
(8)
r ≥0
f ji(r ) v −r .
(9)
r ≥1
It is easy to recover each generating series ti j (u) by multiplying together and taking commutators of the series di (u), e j (u) := e j, j+1 (u), and f j (v) := f j+1, j (u) for 1 ≤ i ≤ m + n, 1 ≤ j ≤ m + n − 1. Indeed, for each pair i, j such that 1 < i + 1 < j ≤ m + n − 1, we have: (r ) (r ) (1) (r ) (1) (r ) ei j = (−1) j−1 ei, j−1 , e j−1 ; f ji = (−1) j−1 f j−1 , f i, j−1 . (10) Thus the Yangian Y (glm|n ) is generated by the coefficients of the series di (u), e j (u), f j (u) | 1 ≤ i ≤ m + n; 1 ≤ j ≤ m + n − 1 . 4. Maps between Yangians For Yangians Y (glm|n ) with small m and n, such as Y (gl1|1 ) and Y (gl2|1 ), it is feasible to use this matrix relationship T (u) = F(u)D(u)E(u) to translate the defining relations (3) into relations between the generating series di (u), e j (u) and f j (u). However, in order to transfer these results to the general case of Y (glm|n ) we must define various homomorphisms between Yangians. Lemma 1. The map ρm|n : Y (glm|n ) → Y (gln|m ) defined by ρm|n (ti j (u)) = tm+n+1−i,m+n+1− j (−u) is an associative algebra isomorphism. Proof. We check that the map ρm|n preserves the defining relation (3).
Gauss Decomposition of the Yangian Y (glm|n )
805
Note where we have swapped m and n in the above. We use the same symbols for the generators of both Y (glm|n ) and Y (gln|m ). It should be clear from the context which algebra ti j (u) belongs to. Proposition 1. Let ζm|n : Y (glm|n ) → Y (gln|m ) be the associative algebra isomorphism given by ζm|n = ρm|n ◦ ωm|n , where ωm|n is the Y (glm|n ) automorphism given by ωm|n : T (u) → T (−u)−1 . That is, ζm|n : ti j (u) → tm+n+1−i,m+n+1− j (u).
Then: ζm|n
⎧ ⎨ di (u) → (dm+n−i+1 (u))−1 , : ek (u) → − f m+n−k (u), ⎩ f k (u) → −em+n−k (u),
for 1 ≤ i ≤ m + n and 1 ≤ k ≤ m + n − 1. Proof. We multiply out the matrix products T (u) = F(u)D(u)E(u) and
T (u)−1 = E(u)−1 D(u)−1 F(u)−1 .
These show that for all 1 ≤ i < j ≤ m + n, tii (u) = di (u) + f ik (u)dk (u)eki (u), k
ti j (u) = di (u)ei j (u) +
f ik (u)dk (u)ek j (u),
k
t ji (u) = f ji (u)di (u) +
f jk (u)dk (u)eki (u),
k
and tii (u) = di (u)−1 +
eik (u)dk (u)−1 f ki (u),
k>i
ti j (u)
=
ei j (u)d j (u)−1
=
d j (u)−1 f ji (u) +
+
eik (u)dk (u)−1 f k j (u),
k> j
t ji (u)
ejk (u)dk (u)−1 f ki (u),
k> j
where ei j (u) =
(−1)s ei0 i1 (u)ei1 i2 (u) · · · eis−1 is (u)
i=i 0
and f ji (u) =
i=i 0
(−1)s f is is−1 (u) · · · f i2 i1 (u) f i1 i0 (u).
(11)
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L. Gow
Then we have immediately that ζm|n (d1 (u)) = dm+n,m+n (u)−1 , ζm|n (e1 j (u)) = f m+n,m+n+1− j (u), and ζm|n ( f j1 (u)) = em+n+1− j,m+n (u). By induction on i, we derive: ζm|n (di (u)) = (dm+n+1−i (u))−1 , ζm|n (ei j (u)) = f m+n+1−i,m+n+1− j (u), ζm|n ( f ji (u)) = em+n+1− j,m+n+1−i (u).
The result stated in the proposition is the special case of this where j = i + 1. When it is reasonable we will write simply ζ for the map ζm|n . The map ζm|n restricts to the isomorphism U (glm|n ) → U (gln|m ) defined by E i j → E m+n+1−i,m+n+1− j . It can be calculated explicitly (using induction and basic properties of quasideterminants) for any 1 ≤ i, j ≤ m + n to give the following result: (r ) ζ (tm+n+1−i,m+n+1− j) =
r1 +···+r p =r r1 ,...,r p >0
(−1) p
m+n k1 ,...,k p−1 =1
(r
)
(r1 ) (r2 ) p−1 tik t . . . tk p−1 j . 1 k1 k2
Also, ζ is not a Hopf algebra map between the two Yangians, but instead has the following property. Proposition 2. Let τ : Y (gln|m ) ⊗ Y (gln|m ) → Y (gln|m ) ⊗ Y (gln|m ) be the map given by τ (y1 ⊗ y2 ) = y2 ⊗ y1 (−1) y 1 y 2 for all homogeneous elements y1 , y2 ∈ Y (gln|m ). Then: (ζ ⊗ ζ ) ◦ ∆ = τ ◦ ∆ ◦ ζ. Proof. Recall that ∆ : T (u) → T[1] (u)T[2] (u), where following [15] we write T[1] (u) =
m+n
ti j (u) ⊗ 1 ⊗ E i j (−1) j(i+1) ,
i, j=1
T[2] (u) =
m+n
1 ⊗ ti j (u) ⊗ E i j (−1) j(i+1) .
i, j=1
Then since ∆ is an algebra homomorphism and we must have that ∆ : T (u)−1 → T[2] (u)−1 T[1] (u)−1 , which gives explicitly: ∆(ti j (u)) =
m+n
tk j (u) ⊗ tik (u)(−1)(i+k)( j+k) .
k=1
It is easy to see that this coincides with ((ζ ⊗ ζ ) ◦ τ ◦ ∆ ◦ ζ ) (ti j (u)).
Gauss Decomposition of the Yangian Y (glm|n )
807 (r )
Finally, let ϕm|n : Y (glm|n ) → Y (glm+k|n ) be the inclusion which sends each ti j ∈ (r )
Y (glm|n ) to the generator tk+i,k+ j ∈ Y (glm+k|n ); and let ψk : Y (glm|n ) → Y (glm+k|n ) be the injective homomorphism defined by ψk = ωm+k|n ◦ ϕm|n ◦ ωm|n .
(12)
Then, for any 1 ≤ i, j ≤ m + n (see Lemma 4.2 of [2]) we have: t11 (u) · · · t1k (u) t1,k+ j (u) .. .. .. .. . . . . ψk (ti j (u)) = tk1 (u) · · · tkk (u) tk,k+ j (u) tk+i,1 (u) · · · tk+i,k (u) tk+i,k+ j (u)
.
As an immediate consequence we have the following lemma. Lemma 2. For k, l ≥ 1, we have ψk (dl (u)) = dk+l (u), ψk (el (u)) = ek+l (u), ψk ( fl (u)) = f k+l (u). (r ) Notice that the map ψk sends ti j(r ) ∈ Y (glm|n ) to the element tk+i,k+ j in Y (glm+k|n ). Thus (r )
the subalgebra ψk (Y (glm|n )) is generated by the elements {tk+s,k+t }ns,t=1 . Then, by (5), all elements of this subalgebra commute with those of the subalgebra generated by the elements {ti(rj ) }i,k j=1 . This implies in particular that for any i, j ≥ 1, the quasideterminants di (u) and d j (v) commute. 5. Gauss Decomposition of Y (gl2|1 ) We begin by describing a presentation of the Yangian Y (gl2|1 ) using the Gauss decomposition. We will then use this to give the more general result in the next section. We use the matrix relationship T (u) = F(u)D(u)E(u) to convert the defining relations (3) into relations between the generating series di (u), e j (v) and f j (v). Note that in the Yangian Y (gl1|1 ), and in the Yangian Y (gl2 ), we have the following: d1 (u) e1 (u) d1 (u) , (13) T (u) = f 1 (u)d1 (u) f 1 (u)d1 (u)e1 (u) + d2 (u) d1 (v)−1 + e1 (v)d2 (v)−1 f 1 (v) −e1 (v) d2 (v)−1 T (v)−1 = , (14) −d2 (v)−1 f 1 (v) d2 (v)−1 whereas in the Yangian Y (gl2|1 ), ⎛ ⎞ d1 (u) d1 (u)e1 (u) d1 (u)e13 (u) ⎠, f 1 (u)d1 (u)e1 (u)+d2 (u) ∗ T (u) = ⎝ f 1 (u)d1 (u) f 31 (u)d1 (u) f 31 (u)d3 (u)e1 (u)+ f 2 (u)d3 (u) ∗ ⎛ ⎞ ∗ ∗ e13 (v)d3 (v)−1 T (v)−1 = ⎝ ∗ d2 (v)−1 + e2 (v)d3 (v)−1 f 2 (v) −e2 (v)d3 (v)−1 ⎠. −1 d3 (v) f 31 (v) −d3 (v)−1 f 2 (v) d3 (v)−1 These expressions for the entries of T (u) allow us to derive the following relations.
808
L. Gow
Lemma 3. We have the following identities in Y (gl2|1 ): (u − v)[di (u), e j (v)] =
(δi, j − δi, j+1 ) di (u)(e j (v) − e j (u)), if j = 1; (δi, j + δi, j+1 ) di (u)(e j (v) − e j (u)), if j = 2;
(15)
− (δi j − δi, j+1 )( f j (v) − f j (u))di (u), if j = 1; − (δi j + δi, j+1 )( f j (v) − f j (u))di (u), if j = 2; δ jk d j (u)−1 d j+1 (u) − d j (v)−1 d j+1 (v) , if j = 1; (u − v)[e j (u), f k (v)] = − δ jk d j (u)−1 d j+1 (u) − d j (v)−1 d j+1 (v) , if j = 2; (e j (v) − e j (u))2 , if j = 1; (16) (u − v)[e j (u), e j (v)] = 0, if j = 2; − ( f j (v) − f j (u))2 , if j = 1; (u − v)[ f j (u), f j (v)] = 0, if j = 2; (u − v)[di (u), f j (v)] =
(u − v)[e1 (u), e2 (v)] = e1 (u)e2 (v) − e1 (v)e2 (v) − e13 (u) + e13 (v); (u − v)[ f 1 (u), f 2 (v)] = − f 2 (v) f 1 (u) + f 2 (v) f 1 (v) + f 31 (u) − f 31 (v); [[ei (u), e j (v)], e j (w)] + [[ei (u), e j (w)], e j (v)] = 0, if |i − j| = 1; [[ f i (u), f j (v)], f j (w)] + [[ f i (u), f j (w)], f j (v)] = 0, if |i − j| = 1; where unless otherwise indicated the indices i, j, k range over i = 1, 2, 3 and j, k = 1, 2. Proof. We give a proof of just the first equation (15), since the rest are proven similarly. First, note that by the remarks at the end of the previous section, d3 (u) = ψ2 (d1 (u)) commutes with e1 (v) = t11 (v)−1 t12 (v). Similarly, e2 (v) = ψ1 (e1 (v)) commutes with d1 (u). Now consider the quasideterminants d1 (u), d2 (u) and e1 (v) in the algebra Y (gl2 )[[u −1 , v −1 ]]. Here, we have the matrices T (u), T (v)−1 as in (13) and (14). By (5), (u − v)[t11 (u), t12 (v)] = t11 (u)t12 (v) + t12 (u)t22 (v),
but this is the same as (u − v)[d1 (u), −e1 (v)d2 (v)−1 ] = −d1 (u)e1 (v)d2 (v)−1 + d1 (u)e1 (u)d2 (v)−1 . Cancelling d2 (v) on the right gives the desired equation when i = j = 1, but in Y (gl2 )[[u −1 , v −1 ]]. We deduce the relation in Y (gl2|1 )[[u −1 , v −1 ]] by following the natural inclusion Y (gl2 ) → Y (gl2|1 ) which sends generators in Y (gl2 ) to those of the same name in Y (gl2|1 ). (u), t (v)] in the For the result when i = 2, j = 1, we consider the commutator [t22 12 −1 −1 algebra Y (gl2 )[[u , v ]] and make the same deduction. For the case j = 2, we find the relations between d1 (u), d2 (u) and e1 (v) in the algebra Y (gl1|1 )[[u −1 , v −1 ]], and map these into the algebra Y (gl2|1 )[[u −1 , v −1 ]], by following ψ1 : Y (gl1|1 ) → Y (gl2|1 ). Theorem 2. The algebra Y (gl2|1 ) is generated by the even elements d1(r ) , d2(r ) , d3(r ) , (r ) (r ) (r ) (r ) (r ) (r ) (r ) d1 , d2 , d3 , e1 , f 1 , and odd elements e2 , f 2 , with r ≥ 1, subject only to the
Gauss Decomposition of the Yangian Y (glm|n )
809
following relations: (0)
di r
(t)
(r −t)
di di
= 1, = δr 0 ,
t=0 (r )
(s)
[di , dl ] = 0, [di(r ) , e(s) j ]= [di(r ) , f j(s) ] = [e(rj ) , f k(s) ] =
(δi j − δi, j+1 )
r −1
rt=1 −1
(17) (t) (r +s−1−t)
di e j
, if j = 1;
(δi j + δi, j+1 ) t=1 di(t) e(rj +s−1−t) , if −1 (r +s−1−t) (t) −(δi, j −δi, j+1 ) rt=1 fj di , r −1 (r +s−1−t) (t) −(δi, j +δi, j+1 ) t=1 f j di , r +s−1 (t) (r +s−1−t) − δ jk t=0 d j d j+1 , if j = r +s−1 (t) (r +s−1−t) δ jk t=0 d j d j+1 , if j =
j = 2; if j = 1; if j = 2; 1; 2;
[e1(r ) , e1(s+1) ] − [e1(r +1) , e1(s) ] = e1(r ) e1(s) + e1(s) e1(r ) , (r +1) (s) (r ) (s+1) [ f1 , f1 ] − [ f1 , f1 ] (r )
(s)
=
(r ) (s) f1 f1
(r )
(s)
+
(18) (19) (20)
(21)
(s) (r ) f1 f1 ,
(22)
[e2 , e2 ] = 0, [ f 2 , f 2 ] = 0, (r +1)
[e1
(r +1)
[ f1
(s)
(r )
(s+1)
] = e1 e2 ,
(s+1)
] = − f2 f1 ,
, e2 ] − [e1 , e2 (s)
(r )
, f2 ] − [ f1 , f2
(r ) (s)
(s) (r )
[[e1(r ) , e2(s) ], e2(t) ] + [[e1(r ) , e2(t) ], e2(s) ] = 0, (r ) (s) (t) (r ) (t) (s) [[ f 1 , f 2 ], f 2 ] + [[ f 1 , f 2 ], f 2 ] (r ) (s) (t) (r ) (t) (s) [[e2 , e1 ], e1 ] + [[e2 , e1 ], e1 ] (r ) (s) (t) (r ) (t) (s) [[ f 2 , f 1 ], f 1 ] + [[ f 2 , f 1 ], f 1 ]
(23)
= 0, = 0,
(24)
= 0,
for all i, l = 1, 2, 3, j, k = 1, 2 and all r, s, t ≥ 1. Remark 1. Relations (21) and (22) are equivalent to the following relations: (r )
(s)
s−1
(s)
t=1 r −1
[ei , ei ] = (r )
[ fi , fi ] =
t=1
(t) (r +s−1−t)
ei ei
−
(r +s−1−t) (t) fi
fi
r −1
(t) (r +s−1−t)
ei ei
t=1 s−1
−
,
(r +s−1−t) (t) fi .
fi
t=1
Proof. We follow the method given in the proof of Theorem 5.2 in [2]. First, we show that the corresponding coefficients of quasideterminants in the Yangian satisfy the relations given in the theorem. The first three relations are obvious from the fact that the di (u)’s commute and the definition of the series di (u) := (di (u))−1 . The rest follow from the relations in Lemma (3). We show the proof of only (18) and (22) since the rest are derived similarly.
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L. Gow
Observe that for any formal series g(u) =
r ≥0
g (r ) u −r we have the identity
g(u) − g(v) =− g (r +s−1) u −r v −s . u−v r,s≥1
Then, by (15),
⎛
[di (u), e j (v)] = (δi j − (−1)δi,2 δi, j+1 ) ⎝
⎞⎛ (t)
di u −t ⎠ ⎝
t≥1
⎞
( p+s−1) − p −s ⎠
ej
u
v
.
p,s≥1
Taking coefficients of u −r v −s gives (18). Now consider (16). In the case where j = 1, this expands out as follows: ⎞2 ⎛ (r ) (s) (u − v)[e1 (u), e1 (v)] = ⎝ e1 u −r − e1 v −s ⎠ r ≥1
=−
s≥1 (r ) (s) e1 e1 u −r v −s
r,s≥1
+
−
(s) (r )
e1 e1 u −r v −s
r,s≥1
(r ) (s) e1 e1 u −r −s
+
r,s≥1
(r ) (s)
e1 e1 v −r −s .
r,s≥1
Taking coefficients of u −r v −s on both sides gives the relation (22). be the algebra defined by the relations in Theorem 2. We have shown Now let Y → Y (gl2|1 ) taking each generathat there is an associative algebra homomorphism Y tor in Y to the quasideterminant coefficient of the same name in the Yangian. By (10) these elements generate the Yangian, so this homomorphism is surjective. We will now is spanned as a vector space by certain monomials, and that show that the algebra Y the images of these monomials form a basis for the Yangian Y (gl2|1 ). It follows that the homomorphism is an isomorphism. (r ) (r ) defined by Let e13 and f 31 be the elements of Y (r )
(r )
(1)
e13 = [e1 , e2 ],
(r )
(r )
(1)
f 31 = [ f 1 , f 2 ]
(c.f. (10)).
is spanned by the set of ordered monomials in We want to show that the algebra Y (r )
(r )
(r )
(r )
(r )
(r )
(r )
(r )
(r )
{ f 31 , f 2 , f 1 , d1 , d2 , d3 , e1 , e2 , e13 | r ≥ 1}, taken in some order so that the f ’s come before all the d’s, which come before all the e’s. It is clear from the relations (17), (18), (19) and (20) that the monomials in the above elements, where f ’s come before d’s and d’s come before e’s, with the d’s taken in some . fixed order, do indeed span Y + of Y generated by elements So our problem is to show that the subalgebra Y (r ) (r ) (r ) (r ) {ei }i=1,2 is spanned by the monomials in {e1 , e2 , e13 ; r ≥ 1} taken in some fixed − generated by elements { f (r ) }i=1,2 is spanned order, and similarly that the subalgebra Y i (r ) (r ) (r ) + . by the monomials in { f 31 , f 2 , f 1 ; r ≥ 1} taken in some fixed order. Consider Y Define a filtration + ⊆ L 1 Y + ⊆ · · · L 0Y
Gauss Decomposition of the Yangian Y (glm|n )
811
(r )
+ by setting the degree of e equal to (r − 1). Let gr L Y + be the associated graded on Y i (r ) + for each i = 1, 2, 13. Then we have the algebra, and let ei (r ) := grrL−1 ei ∈ gr L Y following: (r )
(s)
(r )
(s)
[e1 , e1 ] = 0, [e2 , e2 ] = 0, ) (s) (r ) (s) [e(r 13 , e1 ] = 0, [e13 , e2 ] = 0, (r )
(s)
(r )
(s)
(r +s−1)
[e13 , e13 ] = 0, [e1 , e2 ] = e13
.
Indeed, the first two identities are clear by the relations in the remark above. For the next two, first note that (r +1)
[e1
(s)
(r )
(s+1)
(1)
(r )
, e2 ] = [e1 , e2
].
(25)
Then (r )
(s)
(r )
(1)
(s)
(s)
[e13 , e12 ] = [[e12 , e23 ], e12 ] = [[e12 , e23 ], e12 ] (r )
(1)
(s)
(r )
(s)
(1)
= −[[e23 , e12 ], e12 ] = −[[e23 , e12 ], e12 ] =
(r +s−1) (1) (1) −[[e23 , e12 ], e12 ]
=0
(by (24))
(by (24) again).
Similarly, ) (s) (r ) (1) (s) [e(r 13 , e23 ] = [[e12 , e23 ], e23 ] (r )
(s)
(1)
= −[[e12 , e23 ], e23 ] (by (23)) (r +s−1)
= [[e12
(1)
(1)
, e23 ], e23 ] = 0.
The fifth relation is an easy consequence of these and the super-Jacobi identity: ) (s) (r ) (1) (s) (r ) (s) (1) (r ) (1) (s) [e(r 13 , e13 ] = [[e12 , e23 ], e13 ] = [[e12 , e13 , e23 ]] + [e12 , [e23 , e13 ]] = 0.
The final relation is just another extended application of (25). Given these calculations, + is spanned by the set of all ordered monomials it is clear that the graded algebra gr L Y (r ) + is itself spanned by the corin {ei j }1≤i< j≤3;r ≥1 taken in some fixed order. Hence Y (r )
responding monomials in {ei j }1≤i< j≤3;r ≥1 . The result for the subalgebra Y − is shown similarly. Now we want to show that the monomials in (r )
(r )
(r )
{di }1≤i≤3; r ≥1 ∪ {ei j , f ji }1≤i< j≤3; r ≥1 taken in some fixed order so that f ’s come before d’s and d’s come before e’s form a basis for the Yangian Y (gl2|1 ). By Corollary 1, we may identify the associated graded algebra gr2 Y (glm|n ) with U (glm|n [t]). By the definition of the quasideterminants, under (r +1) (r +1) (r +1) , grr2 ei j , and grr2 f ji are identified, respectively, with this identification, grr2 di E ii (−1)i t r , E i j (−1)i t r , and E ji (−1) j t r . Then the result follows from the PoincaréBirkhoff-Witt theorem for Lie superalgebras ([17]).
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L. Gow
6. Gauss Decomposition of Y (glm|n ) Lemma 4. The following relations hold in the algebra Y (glm|n )[[u −1 , v −1 ]]: [di (u), d j (v)] = 0 for all 1 ≤ i, j ≤ m + n, ⎧ ⎨ (δi j − δi, j+1 )di (u)(e j (v) − e j (u)), (u − v)[di (u), e j (v)] = (δi j + δi. j+1 )di (u)(e j (v) − e j (u)), ⎩ −(δ − δ ij i, j+1 )di (u)(e j (v) − e j (u)),
if j ≤ m − 1, if j = m, if j ≥ m + 1,
⎧ ⎨ −(δi j − δi, j+1 )( f j (v) − f j (u))di (u), (u − v)[di (u), f j (v)] = −(δi j + δi, j+1 )( f j (v) − f j (u))di (u), ⎩ (δ − δ ij i, j+1 )( f j (v) − f j (u))di (u),
(27) if j ≤ m − 1, if j = m, if j ≥ m + 1,
−1
(26)
−1
(u − v)[ei (u), f j (v)] = (−1) δi j di (u) di+1 (u) − di (v) di+1 (v) , j+1 e (v) − e (u) 2 , if j = m, j j (u − v)[e j (u), e j (v)] = (−1) 0, if j = m, j+1 f (v) − f (u) 2 , if j = m, j j (u − v)[ f j (u), f j (v)] = −(−1) 0, if j = m, (u − v)[e j (u), e j+1 (v)] = (−1) j+1 e j (u)e j+1 (v) − e j (v)e j+1 (v) − e j, j+2 (u) + e j, j+2 (v) , (u − v)[ f j (u), f j+1 (v)] = −(−1) j+1 f j+1 (v) f j (u) − f j+1 (v) f j (v) − f j+2, j (u) + f j+2, j (v) , [ei (u), e j (v)] = 0 for |i − j| > 1; [ f i (u), f j (v)] = 0 for |i − j| > 1; j+1
(28) (29) (30) (31)
(32)
(33) (34) (35)
Proof. The relations for i, j between 1 and m are an easy consequence of those already found for the Yangians Y (glm ) in [2] and Y (gl2|1 ) in Sect. 3, and the fact that the natural inclusions Y (glm ) → Y (glm|n ) and Y (gl2|1 ) → Y (glm|n ) are homomorphisms. The remaining relations follow by applying the map ζn|m to the corresponding relations in Y (gln|m ). Lemma 5. In addition, we have the following relations in Y (glm|n ) when m > 1 and n > 1. For any r, s ≥ 1, (r )
(s)
(r )
(s)
(1) (1) ], [em , em+1 ]] = 0; and [[ f m−1 , f m(1) ], [ f m(1) , f m+1 ]] = 0. [[em−1 , em
(36)
Proof. We prove the result in Y (gl2|2 ), and then map this result into the Yangian Y (glm|n ) via the map ψm−2 . First we show the following relation: [e13 (u), e2 (z)e3 (z) − e2,4 (z)] = 0. Indeed, we have: [e13 (u), e2 (z)e3 (z) − e24 (z)] = [e13 (u), e24 (w)] = [t11 (u)−1 t13 (u), −t24 (w)t44 (w)−1 ] = 0.
(37)
Gauss Decomposition of the Yangian Y (glm|n )
813
Now we find the super-commutator (u − v)(w − z)[[e1 (u), e2 (v)], [e2 (w), e3 (z)]]. By (32), this is equal to the following: [e1 (u)e2 (v) − e1 (v)e2 (v) − e13 (u) + e13 (v), −e2 (w)e3 (z) + e24 (w) + e2 (z)e3 (z) − e24 (z)]. Taking the coefficient of u −r z −s and using (37) we find the first relation in (36). The other part follows from this with the use of the map ζ . Now we can state our main result. The proof takes the same line of reasoning as the proof of Theorem 2 but is somewhat longer and more complicated. Again it is very closely based on the proof of Theorem 5.2 in [2]. Theorem 3. The Yangian Y (glm|n ) is isomorphic as an associative superalgebra to the (r ) (r ) (r ) (r ) algebra with even generators di , di , f j , e j , (for 1 ≤ i ≤ m + n, 1 ≤ j ≤ (r )
(r )
m + n − 1, j = m, r ≥ 1) and odd generators em , f m (where again r ≥ 1) and the following defining relations: (0)
di r
(t)
(r −t)
di di
= 1; = δr,0 ;
t=0 (r )
(s)
[di , dl ] = 0; ⎧ r −1 (t) (r +s−1−t) , for j ≤ m − 1, ⎪ ⎨ (δi, j − δi, j+1 ) t=0 di e j (r ) (s) r −1 (t) (r +s−1−t) [di , e j ] = (δi, j + δi, j+1 ) t=0 di e j , for j = m, ⎪ r −1 (t) (r +s−1−t) ⎩ −(δi, j − δi, j+1 ) t=0 di e j , for j ≥ m + 1, ⎧ r −1 (r +s−1−t) (t) di , for j ≤ m − 1; ⎪ ⎨ −(δi, j − δi, j+1 ) t=0 f j (r ) (r ) −1 (r +s−1−t) (t) [di , f j ] = −(δi, j + δi, j+1 ) rt=0 fj di , for j = m; ⎪ −1 (r +s−1−t) (t) ⎩ (δi, j − δi, j+1 ) rt=0 fj di , for j ≤ m + 1; r +s−1 (t) (r +s−1−t) −δ j,k t=0 d j d j+1 , for j ≤ m − 1; (r ) (s) [e j , f k ] = r +s−1 (t) (r +s−1−t) +δ j,k t=0 d j d j+1 , for j ≥ m; (r ) (s) [em , em ] = 0,
[ f m(r ) , f m(s) ] = 0; s−1 r −1 (t) (r +s−1−t) (r ) (s) (t) (r +s−1−t) j [e j , e j ] = (−1) ej ej − ej ej , [ f j(r ) ,
f j(s) ]
= (−1)
j
t=1 r −1 t=1
f j(t) f j(r +s−1−t)
t=1 s−1
−
t=1
(38)
(39)
(40)
(41) for j = m;
(42)
f j(t) f j(r +s−1−t)
,
for j = m; (43)
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L. Gow
(r )
(s+1)
(r +1)
[e j , e j+1 ] − [e j (r +1)
[ fj
(s)
(s)
(r ) (s)
, e j+1 ] = −(−1) j e j e j+1 ;
(r )
(s+1)
(s)
(r )
, f j+1 ] − [ f j , f j+1 ] = −(−1) j f j+1 f j ;
(r ) (s) (r ) (s) [e j , ek ] = 0; and [ f j , f k ] (r ) (s) (t) (r ) (t) (s) [[e j , ek ], ek ] + [[e j , ek ], ek ] (r ) (s) (t) (r ) (t) (s) [[ f j , f k ], f k ] + [[ f j , f k ], f k ] (r ) (1) (1) (s) , em ], [em , em+1 ] ] [[em−1 (r ) (s) (1) (1) [[ f m−1 , f m ], [ f m , f m+1 ] ]
(44) (45)
= 0, if | j − k| > 1;
(46)
= 0, if j = k;
(47)
= 0, if j = k;
(48)
= 0,
(49)
= 0,
(50)
for all r, s, t ≥ 1 and all admissible i, j, k. m|n be the associative algebra given by the relations in the theorem. By Proof. Let Y m|n to the Yangian Y (glm|n ) that sends every Lemma 4 and Lemma 5 the map from Y element of Ym|n to the element of the same name in the Yangian is a homomorphism. We have already stated in Sect. 3 that Y (glm|n ) is generated by the elements:
(r ) (r ) (r ) di , e j , f j 1 ≤ i ≤ m + n, 1 ≤ j ≤ m + n − 1, r ≥ 1 .
Thus this homomorphism is surjective. We need to show that it is injective. Our method m|n is spanned as a vector space by the monois as follows: we show that the algebra Y (r ) (r ) (r ) mials in the elements f ji , di , ei j with 1 ≤ i < j ≤ m + n, r ≥ 1, taken in some fixed order so that the f ’s come before d’s and d’s come before e’s. (These elements are (r ) (r ) (r ) (r ) defined inductively by f i+1,i = f i ; ei,i+1 = ei and (r )
(1)
(r )
(r )
(r )
(1)
f j,i = [ f j, j−1 , f j−1,i ] (−1) j−1 ; ei, j = [ei, j−1 , e j−1, j ] (−1) j−1 , for j > i + 1). Since the image of these monomials in the Yangian form a basis for Y (glm|n ), it follows that the map is an isomorphism. − and Y 0 be the subalgebras of Y m|n generated by all elements of the + , Y Let Y m|n m|n m|n (r )
(r )
(r )
form ei , f i and di , respectively. By the defining relations (38), (39) and (40), we m|n is spanned by the monomials where all f ’s come before all d’s and know that Y all d’s come before all e’s. Also, since the d’s commute, we may assume that they are + is spanned by the written in some fixed order. If we can show that the subalgebra Y m|n (r )
monomials in ei j written in some fixed order, then by applying the map ζ we can show − is similarly spanned by the monomials in f (r ) written in some that the subalgebra Y m|n ji fixed order. This will then complete the proof. + by setting deg(e(r ) ) = r − 1, and denote by Define an ascending filtration on Y m|n
i
+ the corresponding graded algebra. Let e(r ) be the image of e(r ) in the (r − 1)th gr L Y m|n ij ij + . We claim that these images satisfy: component of the graded algebra gr L Y m|n (r )
(s)
(r +s−1)
[ei j , ekl ] = (−1) j δk j eil
− (−1)i
j+ j k+i k
(r +s−1)
δil ek j
.
(51)
Gauss Decomposition of the Yangian Y (glm|n )
815
+ is spanned by the monoFrom this relation it follows that the graded algebra gr L Y m|n + is itself spanned by the monomials mials in e(r ) taken in some fixed order. Hence Y m|n
ij
(r )
in ei j taken in some fixed order. So now it remains only to prove the claim (51). We begin by noting the following relations: (r ) [ei,i+1 , e(s) k,k+1 ] = 0,
(r ) ei j
=
(r +1) (s) [ei,i+1 , ek,k+1 ] (r ) (s) (t) [ei,i+1 , [ei,i+1 , ek,k+1 ]] (r ) (1) [ei, j−1 , e j−1, j ] (−1) j−1
= = =
if |i − k| = 1,
(52)
(r ) (s+1) [ei,i+1 , ek,k+1 ], if |i − k| = 1, (s) (r ) (t) −[ei,i+1 , [ei,i+1 , ek,k+1 ]], if |i − k| = (1) (r ) [ei,i+1 , ei+1, j ] (−1)i+1 , for j > i + 1.
(53) 1, (54) (55)
Here, (52) is a consequence of (46); (53) is a consequence of (44); and (54) is a consequence of (47). The first part of the last relation (55) follows from the definition of (r ) the elements ei j . The second part of (55) follows from the first part using (53) and induction on the difference j − i. Now we break up the problem of showing (51) in cases. We assume without loss of (r ) (s) generality that i ≤ k. If j < k, then [ei j , ekl ] = 0 by (52) and (55). Consider the case where j = k. By (53) and (55) we have ) (s) j (r +s−1) [e(rj−1, j , e j, j+1 ] = (−1) e j−1, j+1 . (1) (1) We bracket both sides of this with e(1) j+1, j+2 , e j+2, j+3 , . . ., el−1,l in turn to obtain: (r )
(s)
(r +s−1)
[e j−1, j , e jl ] = (−1) j e j−1,l , (1)
(1)
then bracket both sides of this new equation with e j−2, j−1 , . . . , ei,i+1 to get the relation: j (r +s−1) [ei,(r j) , e(s) . j,l ] = (−1) ei,l
Before we consider the case j > k in detail, we prove the following special cases: (r )
(s)
[ei,i+2 , ei+1,i+2 ] = 0,
for 1 ≤ i ≤ m + n − 2,
(56)
(r ) (s) [ei,i+1 , ei,i+2 ] = 0, (r ) (s) [ei,i+2 , ei+1,i+3 ] = 0 [ei(rj ) , e(s) k,k+1 ] = 0 for
for 1 ≤ i ≤ m + n − 2,
(57)
for 1 ≤ i ≤ m + n − 3,
(58)
1 ≤ i < k < j ≤ m + n.
(59)
Indeed, for (56), we have: (r ) (s) , ei+1,i+2 ] (−1)i+1 [ei,i+2 (r ) (1) (s) = [[ei,i+1 , ei+1,i+2 ], ei+1,i+2 ] by (55) (r ) (s) (1) = −[[ei,i+1 , ei+1,i+2 ], ei+1,i+2 ] by (54) (r +s−1) (1) (1) = −[[ei,i+1 , ei+1,i+2 ], ei+1,i+2 ] by (53),
which is 0 by (54). The relation (57) is shown in a very similar way.
816
L. Gow
When i + 1 = m, the relation (58) follows directly from (49). On the other hand, when i + 1 = m, the left-hand side of (58) equals (r )
(1)
(1)
(s)
(−1)(i+1 + i+2) [[ei,i+1 , ei+1,i+2 ], [ei+1,i+2 , ei+2,i+3 ]] (1) (r ) (1) (s) = (−1)(i+1 + i+2) [ei+1,i+2 , [ei,i+1 , ei+1,i+2 ], ei+2,i+3 ]] (1)
(r )
(1)
(s)
= (−1)(i+1 + i+2) [ei+1,i+2 , [ei,i+1 , [ei+1,i+2 , ei+2,i+3 ]]] (1) (r ) (1) (s) = (−1)(i+1 + i+2) [[ei+1,i+2 , ei,i+1 ], [ei+1,i+2 , ei+2,i+3 ]], (r )
(1)
(1)
(s)
= − (−1)(i+1 + i+2) [[ei,i+1 , ei+1,i+2 ], [ei+1,i+2 , ei+2,i+3 ]]. Hence the commutator is zero. Here we have used (47) and the super-Jacobi identity, and the fact that since i + 1 = m, no two of the elements we are concerned with are odd. Finally, we use (55) relation to reduce the problem of showing (59) to that of showing (r ) [ei,k+1 , e(s) k,k+2 ] = 0, and (r ) [ei,k+1 , e(s) k,k+1 ] = 0,
for all i ≤ k. The first of these relations follows from (57) and (58) by induction on the difference k − i, using (55). The second follows from (56), again by induction on k − i, using the relation (55). Now we properly begin the case j > k. We break this into the following subcases: Case 1:
(s)
i < k, j = l. Expanding ek j by (55) and then using the super-Jacobi identity and (59), we have: (r )
(s)
(1)
(r )
(s)
[ei j , ek j ] = ±[ek,k+1 , [ei, j , ek+1, j ]]. Continuing on in this fashion, we find: (1) (r ) (s) = ± e , . . . [e , e ] . . . , ei(rj ) , e(s) kj k,k+1 ij j−1, j (r )
(s)
so our problem reduces to showing that [ei j , e j−1, j ] = 0. We now expand
out the ei(rj ) in this using (55) and apply the super-Jacobi identity to reduce (r )
Case 2:
(s)
this problem to that of showing that [e j−2, j , e j−1, j ] = 0. Then we have the result in this case by (56). (s) i < k, j > l. We expand out ekl using (55) and then apply the super-Jacobi identity and (59) to find: (1) (r ) (s) [ei(rj ) , e(s) kl ] = ±[ek,k+1 , [ei j , ek+1,l ]].
Repeating this process as many times as is necessary we eventually get (r )
(s)
(1)
(r )
(s)
[ei j , ekl ] = ±[ek,k+1 , . . . , [ei j , el−1,l ] . . .], which is 0 by (59).
Gauss Decomposition of the Yangian Y (glm|n )
Case 3:
817
i < k, j < l. We prove this case by induction on the difference l − j. When l − j = 1, we have by expanding out e(s) k, j+1 and using the super-Jacobi identity that (r )
(s)
(r )
(s)
(1)
(r )
(r )
(s)
(1)
(s)
(1)
[ei j , ek, j+1 ] = [[ei j , ek j ], e j, j+1 ] (−1) j +[ek j , [ei j , e j, j+1 ]] (−1)i (s)
j+ j k+i k
(r )
= [[ei j , ek j ], e j, j+1 ] (−1) j + [ei, j+1 , ek j ](−1)( j+ j+1)( j+k) . The first term is 0 by Case 1 and the second term is 0 by Case 2. When l − j > 1, (r )
(s)
(r )
(s)
(1)
[ei j , ekl ] = [[ei j , ek,l−1 ], el−1,l ](−1)l−1 , Case 4: Case 5:
which is 0 by the induction hypothesis. i = k, j < l. We use (55) (and (52) and Case 2) to reduce this case to (57). i = k, j = l. If j = i + 1, then this is (52). Otherwise, we can expand out one term with (55) to find: (r )
(s)
(r )
(s)
(1)
(r )
(1)
(s)
[ei j , ei j ] = ±[[ei, j−1 , ei j ], e j−1, j ] + ±[ei, j−1 , [e j−1, j , ei j ]]. Case 6:
The first term is 0 by Case 4 and the second term is 0 by Case 1. i = k, j > l. This follows immediately from Case 4.
This completes the proof of the claim (51), which completes the proof of the theorem. 7. The Centre of Y (glm|n ) The quantum Berezinian was defined by Nazarov [16] as the following power series with coefficients in the Yangian Y (glm|n ): bm|n (u) :=
sgn(ρ) tρ(1)1 (u)tρ(2)2 (u − 1) · · · tρ(m)m (u − m + 1)
ρ∈Sm
×
σ ∈Sn
sgn(σ ) tm+1,m+σ (1) (u − m + 1) · · · tm+n,m+σ (n) (u − m + n). (60)
Recall from [9] that we may also write the quantum Berezinian in the following form. bm|n (u) = d1 (u) d2 (u − 1) · · · dm (u − m + 1) −1
× dm+1 (u − m + 1)
· · · dm+n (u − m + n)
(61) −1
.
We shall prove that the coefficients of this formal power series generate the centre of the Yangian. This was conjectured by Nazarov who proved that the quantum Berezinian was central [16]. A new proof of the centrality of the quantum Berezinian was also given in [9]. m+n Lemma 6. Let glm|n [x] be the polynomial current algebra and I = i=1 E ii . The centre of U (glm|n [x]) is generated by I, I x, I x 2 , . . ..
818
L. Gow
Proof. We reduce the problem to that of the well-known even case considered for example in Lemma 7.1 of [2]. First note that the supersymmetrization map gives an isomorphism between the glm|n [x]-modules U (glm|n [x]) and S(glm|n [x]), where S(glm|n [x]) denotes the supersymmetric algebra of glm|n [x]. The natural action of glm|n [x] on S(glm|n [x]) is obtained by extending the adjoint action. The Lie algebra glm|n has the root space decomposition: glm|n = h ⊕
k
gαi
i=1
where h is the Cartan subalgebra, {α1 , . . . , αk } is the set of roots relative to h, and gαi is the one-dimensional root space corresponding to the root αi . Let eαi be a root vector corresponding to root αi . Suppose P ∈ S(glm|n [x]) is an arbitrary glm|n -invariant element and M is the maximal integer such that eαi x M occurs in P for some root αi . Then we may write: s1 sk P= As eα1 x M . . . eαk x M , (62) s
where we sum over tuples of positive integers s = (s1 , . . . , sk ), and for each such s, the As is a monomial in elements hx r for h ∈ h, r ≥ 0, and eαi x r for r < M. For any h ∈ h, we have by assumption that: 0 = [hx, P] s1 sk = [hx, As ] eα1 x M . . . eαk x M s
+
k
si αi (h)
s1 si −1 sk eαi x M+1 . As eα1 x M . . . eαi x M . . . eαk x M
s
i=1
Then taking the coefficient of eαi x M+1 we find that for all h ∈ h, and for all roots αi that: s1 si −1 sk si αi (h) As eα1 x M . . . eαi x M . . . eαk x M = 0. s
Since αi (h) is not zero for all h ∈ h, and the monomials corresponding to different s are linearly independent, we must have that si = 0. Thus P is a sum of monomials in hx r , where h ∈ h and r ≥ 0. The Cartan subalgebra h contains only even elements, and so the action of glm|n [x] on invariant elements P is the same as the action of glm+n [x]. Then we may use Lemma 7.1 of [2] to obtain our desired result. Theorem 4. The coefficients of the quantum Berezinian generate the centre of Y (glm|n ). Proof. Our proof is based on that of Theorem 2.13. in [15]. Write bm|n (u) = 1 + br u −r . r ≥1
Recall from Corollary 1 that the graded algebra gr2 Y (glm|n ) is isomorphic to U (glm|n [x]). We show that for any r = 1, 2, . . . , the coefficient br has degree r − 1 with respect
Gauss Decomposition of the Yangian Y (glm|n )
819
to deg2 (.) and that its image in the (r − 1)th component of gr2 Y (glm|n ) coincides with I x r −1 . Indeed, if we expand out the expression (61) for the quantum Berezinian, using the fact from [8] that d j (u) = t j j (u) − t jk (u)(|T (u){1,2,..., j−1},{1,2,..., j−1} |lk )−1 tl j (u), k,l< j
we find br =
(l ) (l )
(l
)
(l
)
(lm ) m+n m+1 t111 t222 · · · tmm · (− tm+1,m+1 ) · · · (− tm+n,m+n )
l1 +l2 +···+lm+n =r
+ terms of lower degree. Then it is clear that the terms with li = r for some i = 1, . . . , m + n have degree r − 1, and all the other terms have lower degree. Then (r )
(r )
(r )
(r ) br = t11 + · · · + tmm − tm+1,m+1 − · · · − tm+n,m+n + terms of lower degree.
The result follows when we evaluate the image of the graded part of this under the isomorphism in Corollary (1). 8. The Yangian Y (slm|n ) Recall that the special linear Lie superalgebra slm|n is the subalgebra of glm|n consisting of matrices with zero supertrace. It may be defined explicitly by the following presentation [10, 18]. We take generators {h i , x +j , x − j | 1 ≤ i ≤ m + n − 1}. ± The generators h i , x ± j are declared even for all i and all j = m; the generators x m are declared odd. The defining relations are:
[h i , h j ] = 0; [xi+ , x − j ] = δi, j h i ; ± [h i , x ± j ] = ±ai j x j ;
[xm± , xm± ] = 0;
[xi± , x ± j ] = 0, if |i − j| > 1; [xi± , [xi± , x ± j ]] = 0, if |i − j| = 1; ± ± [[xm−1 , xm± ], [xm+1 , xm± ]] = 0,
m+n−1 for all i, j between 1 and m +n −1. Here A = ai j i, j=1 is the symmetric Cartan matrix of the Lie superalgebra slm|n , with entries aii = 2 for all i < m; amm = 0; aii = −2 for all i > m; ai+1,i = ai,i+1 = −1 for all i < m; ai+1,i = ai,i+1 = 1 for all i ≥ m; and all other entries are 0. We define the Yangian Y (slm|n ) associated to the special linear Lie superalgebra as the following subalgebra of Y (glm|n ): Y (slm|n ) := {y ∈ Y (glm|n ) | µ f (y) = y for all f },
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where we take µ f as defined as in [15]. In other words, for a formal power series f = 1 + f 1 u −1 + f 2 u −2 + · · ·
∈ C[[u −1 ]],
the map µ f is the automorphism of Y (glm|n ) given by µ f : T (u) → f (u)T (u). This is justified by analogy with the definition of the Yangian Y (sl N ) as a subalgebra of the Yangian Y (gl N ) in [15]. Also, in the case where m = n our definition agrees with that arrived at by Stukopin [19] through a quantization of the Lie bi-superalgebra U (slm|n [x]) (see Proposition 5). Proposition 3. Let Z m|n denote the centre of the Yangian Y (glm|n ). Then for m = n, we have Y (glm|n ) ∼ = Z m|n ⊗ Y (slm|n ). Proof. We assume that m > n. (The result for n < m follows from this by the application of the map ζ .) The proof of this result is very similar to that of Proposition 2.16 in [15]. We use the fact, stated there, that for any commutative associative algebra A and any formal series, a(u) = 1 + a1 u −1 + a2 u −2 + · · · ∈ A[[u −1 ]], and any positive integer K there exists a unique series a(u) ˜ = 1 + a˜ 1 u −1 + a˜ 2 u −2 + · · · ∈ A[[u −1 ]] such that a(u) = a(u) ˜ a(u ˜ − 1) · · · a(u ˜ − K + 1).
(63)
We take a(u) = bm|n (u) and K = m − n in the commutative subalgebra Y 0 ⊂ Y (glm|n ) (r ) generated by the elements di for i = 1, . . . , m + n and r ≥ 1. Write ˜ b(u ˜ − 1) · · · b(u ˜ − m + n + 1). bm|n (u) = b(u) By the definition of the map µ f we have that µ f (bm|n (u)) = f (u) f (u − 1) · · · f (u − m + n + 1)bm|n (u). ˜ ˜ = f (u)b(u) for all It follows from the uniqueness of the expansion (63) that µ f (b(u)) ˜ generate the centre Z m|n since f . Also, the coefficients b˜k , (k ≥ 1) of the series b(u) we may recover the coefficients of the series bm|n (u) from them. The remaining parts of the proof are exactly the same as in [15]. Lemma 7. For any m, n ≥ 0, the coefficients of the series d1 (u)−1 di+1 (u), ei (u), f i (u), generate the subalgebra Y (slm|n ).
for 1 ≤ i ≤ m + n − 1,
(64)
Gauss Decomposition of the Yangian Y (glm|n )
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Proof. It is clear that the coefficients of the series d1 (u) together with those of the series listed above generate the Yangian Y (glm|n ). Also, or any f , the map µ f leaves the coefficients of the series in (7) fixed and maps µ f (d1 (u)) = f (u)d1 (u). By the PoincaréBirkhoff-Witt theorem, any element P of Y (glm|n ) is a polynomial in d1(1) , d1(2) , d1(3) , . . . and the other generators that are fixed by µ f for all f . We can assume further that in (r ) (r ) each monomial in P the generators are ordered so that the f i ’s come before the di ’s, (r ) which come before the ei ’s. Suppose that P ∈ Y (slm|n ) and that R is the maximum r (r ) (r ) such that d1 occurs in P, and K is the maximum power of d1 occurring in P for any −R r . Fix f = 1 + λu , where λ is an arbitrary nonzero complex number. Then we can write: (1) a1 (2) a2 (R) a R d1 Fa Da d1 . . . d1 Ea , P= a1 ,a2 ,...,a R
where Fa , Da and E a are monomials in the generators fixed by µ f , and we sum over all R-tuples a = (a1 , a2 , . . . , a R ) of positive integers not exceeding K . Then a R (1) a1 (2) a2 (R) d1 Fa Da d1 . . . d1 + λ E a = P. µ f (P) = a1 ,a2 ,...,a R
By the linear independence of the different monomials and the fact that λ is an arbitrary (R) complex number, we see that in fact d1 cannot occur in P ∈ Y (slm|n ). Recall from [13] that the family A(m, n) of classical Lie superalgebras is defined by: A(m − 1, n − 1) = slm|n for m = n; m, n ≥ 1; A(n − 1, n − 1) = sln|n /I , for n > 1,
(65) (66)
where I is the one-dimensional ideal consisting of scalar matrices λI , (λ ∈ C). We define the Yangian of the classical Lie superalgebra A(n − 1, n − 1) as the following quotient: ! Y (A(n − 1, n − 1)) := Y (sln|n )/ bn|n (u) = 1 = Y (sln|n )/B, (67) where B is the ideal in Y (sln|n ) generated by the coefficients b1 , b2 , . . . of the quantum Berezinian. This definition is justified to a certain extent by Proposition 4 below. Lemma 8. For n > 1, the centre of U (A(n, n)[x]) is trivial. Proof. We follow the argument of Lemma 6 using the properties of the root-space decomposition given in [13]. Proposition 4. For n > 1, the centre of the Yangian Y (A(n, n)) is trivial. Proof. We show that gr2 Y (sln|n ) ∼ = U (sln|n [x]), and that gr(Y (A(n − 1, n − 1))) ∼ = U (A(n − 1, n − 1)[x]). Then the result follows from Lemma 8. Here the filtration on Y (A(n − 1, n − 1)), C = A−1 ⊂ A0 ⊂ A1 ⊂ . . . ⊂ Ai ⊂ . . . ,
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is defined by setting Ai = Yi + B, where Yi is the set of elements a ∈ Y (sln|n ) with deg2 (a) ≤ i, and gr(Y (A(n − 1, n − 1))) is the corresponding graded algebra. The restriction of the map in Corollary 1 to gr2 Y (sln|n ) is injective onto its image in U (sln|n ). By Lemma 7, this is the image of the coefficients of the series d1 (u)−1 di+1 (u), ei (u) and f i (u), for i = 1, . . . , 2n − 1. Now, for any r ≥ 1, the coefficients of u −r these series are, respectively: (r )
(r )
ti+1,i+1 − t11 + elements of lower degree, (r )
ti,i+1 + elements of lower degree, (r )
ti+1,i + elements of lower degree. The image of these elements in U (gln|n [x]) is: (−1)i+1 E i+1,i+1 x r −1 − E 11 x r −1 , (−1)i E i,i+1 x r −1 , (−1)i+1 E i+1,i x r −1 . These elements generate precisely the subalgebra U (sln|n [x]). Thus we find that gr Y (sln|n ) ∼ = U (sln|n [x]). 2
The natural projection map p : Y (sln|n ) → Y (A(n − 1, n − 1)) satisfies p(Yi ) ⊂ Ai , and thus gives a natural surjective mapping ∼ U (sln|n [x]) → grY (A(n − 1, n − 1)), gr Y (sln|n ) = 2
with kernel the ideal I = I, I x, I x 2 , . . . ⊂ U (sln|n [x]). Then grY (A(n − 1, n − 1)) ∼ = U (sln|n [x])/I ∼ = U (A(n − 1, n − 1)[x]). Corollary 2. For n > 1, the centre of the subalgebra Y (sln|n ) is generated by the coefficients of the quantum Berezinian bn|n (u). 9. Presentation of Y (slm|n ) Set h i (u) = di (u + 21 (−1)i (m − i))−1 di+1 (u + 21 (−1)i (m − i)), xi+ (u) = f i (u + 21 (−1)i (m − i)),
(68)
xi− (u) = (−1)i ei (u + 21 (−1)i (m − i)), for 1 ≤ i ≤ m + n − 1, and use the following notation for the coefficients: h i (u) := 1 + h i,s u −s−1 , xi+ (u)
:=
s≥0 + −s−1 xi,s u ,
s≥0
xi− (u)
:=
− −s−1 xi,s u .
s≥0
Then we have the following presentation for the subalgebra Y (slm|n ).
(69)
Gauss Decomposition of the Yangian Y (glm|n )
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Proposition 5. The subalgebra Y (slm|n ) is isomorphic to the associative superalgebra ± over C defined by the generators xi,s and h i,s for 1 ≤ i ≤ m + n − 1 and s ∈ Z+ , and by the relations [h i,r , h j,s ] = 0, + , x− [xi,r j,s ] = δi j h i,r +s , ± [h i,0 , x ± j,s ] = ±ai j x j,s , ±ai j ± ± [h i,r +1 , x ± (h i,r x ± j,s ]−[h i,r , x j,s+1 ] = j,s + x j,s h i,r ), for i, j not both m, 2 ± [h m,r +1 , xm,s ] = 0, ±ai j ± ± ± ± ± ± ± [xi,r (xi,r x j,s + x ± +1 , x j,s ] − [x i,r , x j,s+1 ] = j,s x i,r ), for i, j not both m, 2 ± ± [xm,r , xm,s ] = 0, ± , x± [xi,r j,s ] = 0, if |i − j| > 1, ± ± ± ± ± [xi,r , [xi,s , x± j,t ]] = −[x i,s , [x i,r , x j,t ]], if |i − j| = 1, ± ± ± ± [[xm−1,r , xm,0 ], [xm,0 , xm+1,s ]] = 0,
where r , s and t are arbitrary positive integers and ai j are the elements of the Cartan ± are odd and all other generators are even. matrix above. The generators xm,s Proof. For the duration of this proof we refer to the algebra given by the presentation "(slm|n ). By Lemma 4 we have a homomorphism ϕ : Y "(slm|n ) → in Proposition 5 as Y ± Y (slm|n ) given by sending the elements h i,s , xi,s to those defined in Y (slm|n ) by (68) and (69). By Lemma 7 this homomorphism is surjective. We need to show ϕ is injective. We do this by constructing a set of monomials that span Y˜ (slm|n ), and whose image under ϕ is a basis for the Yangian Y (slm|n ). Following [14, 19] we construct this basis as follows. Let α be a positive root of slm|n and α = αi1 + · · · + αi p a decomposition of α into a sum of roots such that xα± = [xi±1 , [xi±2 , . . . , [xi±p−1 , xi±p ] . . .]] is a nonzero root vector in slm|n . Suppose s > 0 and we have a decomposition s = ± s1 + · · · + s p of s into p non-negative integers. Then define the root vector xα,s in 1 +···+s p the Yangian by ± = [xi±1 ,s1 , [xi±2 ,s2 , . . . , [xi±p−1 ,s p−1 , xi±p ,s p ] . . .]]. xα,s 1 +···+s p
(70)
With respect to the second filtration defined in (6), the degree of an element h i,s or ± ± xi,s is equal to its second index s, and deg2 (xα,s ) = s. If s = s1 + · · · + s p is 1 +···+s p another decomposition of s into non-negative integers, then (since the defining relations in Proposition 5 are satisfied by the elements of the Yangian) we have ± ± deg2 (xα,s +···+s − x α,s1 +···+s p ) ≤ s − 1. 1
p
(71)
Now for each s > 0 fix the decomposition s = 0 + · · · + 0 + s to be used always ± = x± and write xα,s α,0+···+0+s . Also any positive root α is just α = i − j for some
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± 1 ≤ i ≤ j − 1 ≤ m + n − 1. We then write: xi,±j;s = xα,0+···+0+s . Now choose any total ordering ≺ on the set
{xi,−j;q , h i,r , xi,+ j;s | 1 ≤ i ≤ j − 1 ≤ m + n − 1, q, r, s > 0} and define Ω(≺) to be the set of ordered monomials in these elements, where the odd elements (xi,±j;r with i ≤ m but j > m) occur with power at most 1.
Define the length l(M) of a monomial in xi,−j;q , h i,r , xi,+ j;s as the number of factors of M and note that by the relations in Proposition 5, if we rearrange the factors of M, then we obtain additional terms of either smaller degree, or the same degree but smaller length. Then by induction on the degree d of a polynomial, and for fixed degree d, induction on the maximal length of its terms, we see that Y (slm|n ) is spanned by the elements of Ω(≺). (This argument is given in [14] for the Yangian Y (sl N ).) Now suppose that some linear combination Σ of the monomials in Ω(≺) is equal to 0, and that the highest degree of a monomial term in Σ is r . The degree r part of Σ must be equal to zero. This will be the sum of products of the highest degree parts of elements xi,−j;r , h i,r , xi,+ j;r , which by the isomorphism gr2 Y (slm|n ) ∼ = U (slm|n [x]) get mapped to the elements εi,−j E i j x r −1 ,
(−1)i E ii − (−1)i+1 E i+1,i+1 ; εi,+ j E ji x r ,
respectively, where εi,±j is some power of −1. Together these elements form the basis for slm|n [x], and so by the Poincaré-Birkhoff-Witt theorem for Lie superalgebras [17], the set of ordered monomials in them, containing powers of at most one of the odd elements, are linearly independent. This implies that the highest degree part of Σ must in fact be trivial. Thus Ω(≺) is a basis for Y (slm|n ). "(slm|n ) by the same formulas as in (70), except now " Now, we define a set Ω(≺) in Y "(slm|n ). We define a filtration on we take the symbols to represent the elements of Y ± " equal to its second index s. Y (slm|n ) by setting the degree of an element h i,s or xi,s All the arguments required to show that Ω(≺) span the Yangian depended only on the "(slm|n ). Then Ω(≺) " " relations in Proposition 5, and thus hold true for Ω(≺) in Y is a "(slm|n ), and whose image under φ, Ω(≺), is a basis for set of monomials that span Y Y (slm|n ). This is the presentation given by Stukopin [19, 20], except that the last relation has been corrected. Stukopin derives this presentation of the Yangian Y (slm|n ) according to the definition of Yangian given in [7], as the quantization of the Lie bi-superalgebra slm|n [x]. He names it after the series of classical Lie superalgebras A(m − 1, n − 1) and defines it only for the case m = n, since in the case where m = n the Lie superalgebra slm|n [x] does not have a canonical Lie bi-superalgebra structure. Stukopin defines the root vectors given in the proof of Proposition 5 and gives a Poincaré-Birkhoff-Witt theorem for the Yangian Y (slm|n ) using the same general argument as Levendorskii [14]. The linear independence part of this PBW theorem may now also be obtained as a corollary of Proposition 5. Acknowledgements. Thanks to my PhD supervisor Alex Molev for his patient assistance. He made many suggestions for improving this paper. Thanks also to Vladimir Stukopin for explaining details of his work to me through email.
Gauss Decomposition of the Yangian Y (glm|n )
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References 1. Ahn, C., Koo, W.M.: gl(n|m) color Calogero-Sutherland models and super Yangian algebra. Phys. Lett. B 365(1-4), 105–112 (1966) 2. Brundan, J., Kleshchev, A.: Parabolic presentations of the Yangian Y (gln ). Commun. Math. Phys. 254(1), 191–220 (2005) 3. Cai, J.-f., Ju, G.-x., Wu, K., Wang, S.-k.: Super Yangian double DY (gl(1|1)) and its Gauss decomposition. J. Phys. A 30(11), L347–L350 (1997) 4. Caudrelier, V., Ragoucy, E.: Lax pair and super-Yangian symmetry of the nonlinear super-Schrödinger equation. J. Math. Phys. 44(12), 5706–5732 (2003) 5. Crampé, N.: Hopf structure of the Yangian Y (sln ) in the Drinfeld realization. J. Math. Phys. 45(1), 434–447 (2004) 6. Dolan, L., Nappi, C.R., Witten, E.: Yangian symmetry in D = 4 superconformal Yang-Mills theory. In: Quantum theory and symmetries, World Sci. Publ., Hackensack, NJ, 2004, pp. 300–315 7. Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Providence, RI: Amer. Math. Soc., 1987, pp. 798–820 8. Gelfand, I., Gelfand, S., Retakh, V., Wilson, R.L.: Quasideterminants. Adv. Math. 193(1), 56–141 (2005) 9. Gow, L.: On the Yangian Y (glm|n ) and its quantum Berezinian. Czech. J. Phys. 55(11), 1415–1420 (2005) 10. Grozman, P., Leites, D.: Defining relations for Lie superalgebras with Cartan matrix. Czech. J. Phys., 51(1), 1–21 (2001) 11. Hatsuda, M., Yoshida, K.: Classical integrability and super Yangian of superstring on AdS5 × S 5 . Adv. Theor. Math. Phys. 9(5), 703–728 (2005) 12. Ju, G.-x., Wang, S.-k., Wu, K.: The algebraic structure of the gl(n|m) color Calogero-Sutherland models. J. Math. Phys. 39(5), 2813–2820 (1998) 13. Kac, V.G.: Lie superalgebras. Advances in Math. 26(1), 8–96 (1977) 14. Levendorski˘ı, S.Z.: On PBW bases for Yangians. Lett. Math. Phys. 27(1), 37–42 (1993) 15. Molev, A., Nazarov, M., Olshanskii, G.: Yangians and classical Lie algebras. Russ. Math. Surv. 51, 205–282 (1996) 16. Nazarov, M.L.: Quantum Berezinian and the classical Capelli identity. Lett. Math. Phys. 21(2), 123–131 (1991) 17. Scheunert, M.: The theory of Lie superalgebras, Volume 716 of Lecture Notes in Mathematics. Berlin: Springer, 1979 18. Scheunert, M.: The presentation and q deformation of special linear Lie superalgebras. J. Math. Phys. 34(8), 3780–3808 (1993) 19. Stukopin, V.A.: Yangians of Lie superalgebras of type A(m, n). Funct. Anal. Appl. 28(3), 217–219 (1994) 20. Stukopin, V.: Yangians of classical lie superalgebras: Basic constructions, quantum double and universal R-matrix. Proceedings of the Institute of Mathematics of NAS of Ukraine 50(3), 1195–1201 (2004) 21. Zhang, R.B.: The gl(M|N ) super Yangian and its finite-dimensional representations. Lett. Math. Phys. 37(4), 419–434 (1996) Communicated by L. Takhtajan
Commun. Math. Phys. 276, 827–861 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0308-1
Communications in
Mathematical Physics
Dobrushin Interfaces via Reflection Positivity Senya Shlosman1,2 , Yvon Vignaud1 1 Centre de Physique Theorique, UMR 6207 CNRS, Luminy Case 907, 13288 Marseille Cedex 9, France.
E-mail: [email protected]; [email protected]
2 Institute for the Information Transmission Problems, Moscow, Russia.
E-mail: [email protected] Received: 26 October 2006 / Accepted: 16 February 2007 Published online: 19 September 2007 – © Springer-Verlag 2007
Abstract: We study the interfaces separating different phases of 3D systems by means of the Reflection Positivity method. We treat discrete non-linear sigma-models, which exhibit power-law decay of correlations at low temperatures, and we prove the rigidity property of the interface. Our method is applicable to the Ising and Potts models, where it simplifies the derivation of some known results. The method also works for large-entropy systems of continuous spins. 1. Introduction The first example of a pure state describing the coexistence of phases separated by an interface was discovered by R. Dobrushin in 1972, [D72]. There he was studying the low temperature 3D Ising model. He was considering the Ising spins in a cubic box VN with (±)-boundary condition σ ± : all spins of σ ± are (+) in the upper half-space and (−) in the lower half-space. Such a boundary condition forces an interface into VN , separating the (+)-phase from the (−)-phase. Dobrushin has shown that in the thermodynamic limit N → ∞ the distribution of goes to a proper limit (in contrast with the 2D case). This limit describes the behavior of the surface separating the (+)- and the (−)-phases. His method of analysis was what is now called the cluster expansion, based on Pirogov-Sinai Contour Functional theory. Later on, this approach was applied to other discrete models in [HKZ, CK, GG]. The question of coexistence of phases for systems with continuous symmetry was addressed in [FP]. It was found there that the analogous states for the X Y -model do not exist, and that the surface tension between two magnetized phases vanishes. Other systems were not studied in the literature. There are probably two reasons for that: 1. most systems with continuous symmetry do not display the above Ising-type rigid interface separating different phases,
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2. the Pirogov-Sinai theory (PS) “does not work” for continuous symmetry systems, while the (only) alternative method – the Reflection Positivity (RP) – works just for periodic boundary conditions, and therefore one can not handle boundary conditions of the type σ ± needed in order to create the interface. In order to illustrate the first point, let us consider the low-temperature 3D classical X Y model, defined by the Hamiltonian H (ϕ) = − (1) cos ϕx − ϕ y , x,y∈Z3 , |x−y|=1
where the spins ϕ· take values on the circle S1 = R1 mod (2π ). As was established in the seminal paper [FSS], this model has a continuum of low-temperature magnetized phases, ·α , α ∈ S1 . One can try to create a state of coexistence of two phases by using the (±)-boundary condition ϕ ± , which assigns the value 0 to spins in the upper half-space and the value π in the lower half-space. However, as the comparison with the Gaussian case shows, one expects the thermodynamic limit of that state to be the mixture state, 21 ·π/2 + ·3π/2 , with no interface emerging. The X Y Z model is defined by the same Hamiltonian (1), but the variables ϕ-s take values on the sphere S2 ∈ R3 , and the difference ϕx − ϕ y is just the angle between ϕx and ϕ y . Let (ψ, θ ) be the “Euler angles” coordinates on S2 , ψ ∈ S1 , θ ∈ − π2 , π2 . Again, at low temperatures there are extremal Gibbs states ·(ψ,θ) , (ψ, θ ) ∈ S2 . The (±)-boundary condition ϕ ± is now the configuration assigning the value θ = π2 (north pole) to the upper half-space, and θ = − π2 (south pole) to the lower half-space. We expect that the corresponding finite-volume state ·ϕN± converges weakly, as N → ∞, to the mixture ·(ψ,θ=0) dψ. Still, we believe that Dobrushin states for some systems with continuous symmetry do exist. One likely example is the so-called non-linear sigma-model, considered recently in [ES1, ES2]. Its Hamiltonian is given by p 1 + cos ϕx − ϕ y H (ϕ) = − , (2) 2 d x,y∈Z , |x−y|=1
with ϕx ∈ S1 . For p large enough – i.e. when the potential well is narrow enough – this model exhibits the following behavior: at high temperatures it has unique Gibbs state (the chaotic state). At low temperatures in 2D it presumably has the Kosterlitz-Thouless phase with power-law correlation decay, which can be obtained by the methods of the paper [FS]. At low temperatures in 3D it should have infinitely many ordered Gibbs states, indexed by magnetization, as the results of [FSS] suggest. Moreover – and that is the main result of [ES1] – there exists a critical temperature Tc = Tc ( p, d), at which we have the coexistence of the chaotic state and the ordered state(s). (Of course, all these states are translation-invariant.) The results of [ES1] are valid in any dimension d ≥ 2. We believe that in dimension d = 3 at the critical temperature Tc the system possesses also non-translation-invariant states, describing the coexistence of ordered states and chaotic state, with the rigid interface separating them. The present paper was started as an attempt to prove the above conjecture. Unfortunately, we are currently unable to complete this program. (Our partial results in this direction are briefly described at the end of this introduction.) However, we are able
Dobrushin Interfaces via Reflection Positivity
829
to study the interfaces in some discrete approximations of the non-linear sigma-model and other models of this type. By discrete
approximation we mean here the following. Let H (ϕ) = − x,y∈Z3 , U ϕx − ϕ y be the Hamiltonian for the continuous spin |x−y|=1
model, ϕx ∈ S1 , with free measure dϕ. Then its discrete approximation is given by the Hamiltonian
(3) H (σ ) = − U σ x − σ y , x,y∈Z3 , |x−y|=1
with σx ∈ Zq ⊂ S1 , where the group Zq is equipped with counting measure. The integer q is the parameter of the approximation. (One can call the resulting model as the clock-model, corresponding to the interaction U.) If the function U has unique nondegenerate minimum on S1 , then the resulting Zq model at low temperatures is Potts-like, and thus has properties quite different from the continuous symmetry system. The situation becomes much more interesting if the minimum of U is degenerate and, moreover, the minimal value is attained along a (small) segment, while the discretization parameter q is large. Then the properties of such a system are quite similar to the one with continuous symmetry. Unlike the Potts model, the ground states of our Hamiltonian (3) are infinitely degenerate. We believe that in the 3D case at low temperatures (as well as at zero temperature) such a model exhibits spontaneous magnetization, while the truncated correlations decay as a power law. Hopefully one can establish this conjectured behavior by a suitable version of the infrared bounds. In the 2D case we believe that “Mermin-Wagner” theorem holds, so that the magnetization is zero, even at zero temperature. We expect the correlation decay to be a power law. Our 2D conjecture at zero temperature is close in spirit to the results of R. Kenyon [K] on 2D tilings, while for positive low temperatures its behavior looks to us to be similar to that of the intermediate phase of the classical clock-model, established in [FS]. Another model with similar features was considered by M. Aizenman, [A]. The methods of the cited papers [ES1, ES2] can be easily adapted to prove that in dimension d ≥ 2 the structure of the phase diagram for the Hamiltonian (3), with the function U having deep and ε-narrow well (possibly with a flat bottom) and q large enough, has the same features as for the “very” non-linear sigma-model: at high temperatures it has unique Gibbs state, while at low temperatures it has (one or more) Gibbs states “with
local order”, which means that the probability of seeing the discrepancy:
σi − σ j ≥ 3ε at two n.n. sites is small. Moreover, there exists a temperature Tc (q) at which the high-T chaotic state coexists with the low-T locally-ordered state(s). The main result of the present paper is the rigidity property of the chaos/order interface once the dimension d is at least 3. Namely, we show that if the two phases are put into coexistence at the transition temperature Tc (q) by applying suitable boundary conditions in a given volume, then the interface between them is rigid, and its height function exhibits the long-range order. Since the proof of this result is quite involved, we will establish it in the present paper only for the simplest model of the above type, defined below, (6). We will now comment on the method we use to study our problem. Presently there are two techniques to study phase transitions: the Pirogov-Sinai theory and the method of Reflection Positivity. It seems unlikely that our model can be treated by PS-theory, since we have here infinite degeneracy of the ground states and we expect power-law decay of correlations. On the other hand, the applications of the RP method rely on the study of the states with periodic boundary conditions. In the phase coexistence regime such
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a state is not ergodic, and its ergodic decomposition allows one to study various pure states – but only some states, since the non-translation-invariant states do not contribute to the state with periodic boundary conditions. Notwithstanding the above discussion, our method of proof will be that of Reflection Positivity. But in order to study the chaos/order interface, we will use RP not with periodic boundary conditions, but with mixed ones; namely, we impose periodic boundary conditions only in two (horizontal) dimensions, and we leave the third (vertical) dimension “free” to impose fixed spins boundary conditions in the third dimension. In other words, we consider the cylindric boundary conditions topology, T2N × [0, L], and we impose ordered boundary conditions on the top of the cylinder and chaotic boundary conditions on its bottom. Of course, the resulting state will be RP only with respect to reflections in vertical planes, but that will be sufficient for our purposes. This restricted Reflection Positivity is the main technical innovation of this paper. Our main result will be that the so constructed state at T = Tc necessarily possesses an interface, separating the ordered and the disordered phases, which interface is rigid in the sense of [D72]: it has a well-defined (random) global height, while the deviations from it happen at any given location with a small probability. One usual advantage of the RP method and the chess-board estimates is that their technical implementations are usually quite simple, as compared with the Pirogov-Sinai theory. In this respect we have to note that the restricted RP is already more involved technically and requires a detailed study of various spatially extended defects, not present in the usual applications of RP. Our technique enables one to study also the continuous symmetry case, and to obtain similar results in a 3D slab Z2 × [0, L]: with order-disorder boundary conditions and for a suitable narrow-well interaction U one has the chaos/order rigid interface at the critical temperature Tc . However, the technical limitations of our approach are such that the width ε of the potential well depends on the width L of the slab, with ε (L) → 0 as L → ∞. Therefore, in contrast to the discrete symmetry case, we can not take the full thermodynamic limit N → ∞, L → ∞. The details will be published separately, see [V1]. Other models. Finally we remark that our technique, applied to the 3D Ising or Potts models, allows one to obtain simpler proofs of the rigidity of their interfaces. Indeed, since in these models the ground states are non-degenerate, our machinery simplifies a lot, and the resulting proofs are relatively short. We can also treat various 3D real valued random fields. For example, we can study the double-well case, defined by the Hamiltonian 2 H (ψ) = ψs2 − 1 + (4) (ψs − ψt )2 . s
s,t n.n.
We can show that at low temperatures this system possesses rigid interface separating the plus-phase, where ψ ≈ +1, from the minus-phase, where ψ ≈ −1. Another case of interest is the model with extra local minimum of the energy, considered in [DS], where H (ψ) = (ψs ) + (5) (ψs − ψt )2 . s
s,t n.n.
Here the potential has a (unique) global minimum, which is narrow, and an additional local one, which has to be relatively wide. Then, as it is shown in [DS], such a model undergoes a phase transition at some temperature Tcr ( ), at which temperature one has a coexistence of the low-energy phase, corresponding to the global minimum, with the
Dobrushin Interfaces via Reflection Positivity
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high entropy phase, corresponding to the local minimum. In dimension 3 we can show that at this temperature this model exhibits a rigid interface separating the above two phases. We want to stress that the above stated results for the models (4) and (5) are technically simpler than the corresponding statements for the system (3) and its discrete versions. Indeed, while in the models (4) and (5) one has exponential decay of correlation due to the positive mass of the potential wells, in (3) and its discrete version we expect power law decay. This is why in the present paper we concentrate on the last model. The corresponding results for (4) and (5) will be published separately, [SV]. The organization of the paper is the following: The next section contains the definition of the model we study and the formulation of the main result. Section 3 contains the main steps of the proof. We introduce there the gas of defects of the interface, and we use Reflection Positivity and the chess-board estimates to reduce the study of the local defects to the study of defect sheets. Some defects do not contribute to the weight of the interface, so to control these we have to glue them in pairs by means of the gluing transformation. Sections 4 and 5 contain the needed combinatorial-energy properties of various defect sheets. The last Sect. 6 contains the final steps of the proof of our main result. 2. The Main Result In what follows we will consider the 3D lattice model with spins σi taking values in the additive group Zq = Z/qZ. We will equip Zq with the counting measure. Let
σ = σi : σi ∈ Zq , i ∈ Z3 be a configuration of our model. The Hamiltonian of our system is given by H (σ ) = − I|σi −σ j |≤1 , (6) i∼ j
where the summation goes over nearest neighbors. Clearly, the interaction and the Hamiltonian are Zq -invariant. (In terms of Sect. 1, the Hamiltonian (6) corresponds to the model (3) with interaction having a well of width ε = q3 .) Let us define the notion of order: Definition 1 (Ordered bonds). A bond i ∼ j of our lattice N ,L is called ordered in
σ iff σi − σ j ≤ 1. Otherwise it is called disordered. Using a technique similar to [ES1, ES2], one can show that for q large enough the above model undergoes a first-order phase transition in temperature. Namely, the following theorem holds: Theorem 2. There exists a temperature Tc = Tc (q), at which the Hamiltonian (6) has at least two Gibbs states: the ordered state ·oTc and the disordered state ·dTc . They are characterized by the properties: d I|σi −σ j |≤1 ≤ p (q) , (7) Tc o I|σi −σ j |≤1 ≥ 1 − p (q) , (8) T c
where i, j is any bond of Z3 , while p (q) goes to zero as q → ∞. (Incidentally, the critical temperature Tc (q) goes to zero as q → ∞.)
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Note. We believe that in 3D the state ·oTc is not pure, and is a mixture of q states with different values of magnetization. The purpose of our work is the study of the interface between the ordered and disordered phases of the Hamiltonian (6) at the critical temperature Tc (q), put into coexistence by suitable boundary conditions. The construction of the corresponding non-translationinvariant states will be discussed in another publication, [V]. To study the interfaces we will consider special boxes and we will impose special boundary conditions, which will force the interface into the box. Namely, we will take the boxes N ,L ⊂ Z3 :
N ,L = {(x, y, z); 0 ≤ x, y ≤ N ; 0 ≤ z ≤ L + 1} , and we will impose the periodic boundary conditions in x and y directions. In other words, we think about the box N ,L as a product of the torus T N and a segment. In what follows we suppose that N is even. The boundary of N ,L has two components, and we denote them by P o = N ,L ∩ {z = L + 1}
and
P d = N ,L ∩ {z = 0} .
We will impose boundary conditions on P o and P d , which (hopefully) would bring the order-disorder interface into N ,L . So we fix a value s ∈ Zq , and we impose on P o the ordered boundary condition σord = σi, j, L+1 ≡ s . We also fix four values: s00 = 0, s01 = [q/4], s10 = [3q/4] and s11
= [q/2] in Zq , and we impose the strongly disordered boundary condition σdisord = σa+2i, b+2 j, 0 = sab , a, b = 0, 1 on P d . The resulting boundary condition will be called the order-disorder b.c. In what follows we will be interested in the Gibbs states in N ,L , corresponding to the Hamiltonian (6), with these order-disorder b.c. at inverse temperature β. They will β,q β,q be denoted by µ N ,L , while by Z N ,L we denote the corresponding partition function. To formulate our results we need some more definitions. Let a configuration σ in
N ,L be fixed. Definition 3 (Pure and frustrated cubes). We will call an elementary cube of our lattice
N ,L frustrated in σ, if it has both ordered and disordered bonds among its (twelve) bonds. Otherwise it will be called pure. Any pure cube is either ordered or chaotic, in the obvious sense. The set of all frustrated cubes of σ will be denoted by F(σ ). Definition 4 (Contours, 3D-interfaces). A connected component of F(σ ) is called a 3D interface, iff it separates P o and P d . Otherwise it is called a contour. Remark 5. Here two cubes are called connected, if they share at least one bond. The union of all the 3D interfaces of σ will be denoted by I (σ ). The complement
N ,L \I (σ ) has several connected components; each one of them is occupied by a phase – ordered or chaotic. The type of the phase in any of these components is defined by the type of the elementary cube on its inner boundary; inside the components the phases might have of course frustrated contours. We need the following topological fact: Proposition 6 (Existence of a 3D-interface). With the order-disorder b.c., defined above, each configuration has at least one 3D-interface.
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833
This obvious claim in fact requires a proof, as was pointed out by G. Grimmett, [G]. One is given in [GG], though it also can be deduced from known results of homotopy theory, see, e.g. [D]. Now we will define the boundary surface, which rigidity we will prove below: Definition 7 (2D-interface). Let σ be a configuration in N ,L , with order-disorder b.c. imposed. The complement N ,L \I (σ ) has several (at least two – containing P d and P o ) connected components. Let us consider all its disordered components. (There is at least one such component – the one containing the boundary P d ⊂ N ,L .) We denote their union by D(σ ); this is the disordered phase region. Denote by ∂D all the plaquettes which belong both to elementary cubes in D and to elementary cubes in I (σ ). It can have several connected components. Let B(σ ) be the union of these components, each of which separates P d and P o . It will be called the 2D-interface, or just the interface. A collection of plaquettes B will be called admissible if B = B(σ ) for some configuration σ . Let us denote by : B(σ ) → P the orthogonal projection onto the plane P = ¯ of the surface B(σ ) will be called regular, if the preimage of its pro{z = 0}. A point M jection −1 M¯ consists of exactly one point, which is M¯ itself. The plaquette p of B, containing M¯ will be then also called regular, as well as the point M = M¯ ∈ P and its plaquette. A ceiling is a maximal connected component of regular plaquettes. We split the complement of ceilings of B into connected components, which will be called walls. Note that all plaquettes of a ceiling C necessarily belong to the same horizontal plane {z = h(C)}, so the height of a ceiling h(C) is well defined. The height h (M) of the regular point M ∈ P is defined in the obvious way. If the point M ∈ P is not regular, we put h (M) = ∞ by definition. The regular points M of the plane P also can be splitted into connected components. Let R (σ ) ⊂ P be the one with the largest area. (If there are several such, we choose one of them.) The set R (σ ) will be called the rigidity set of σ. The preimage C¯ (σ ) = −1 (R (σ )) is (contained in one of) the largest ceiling of B (σ ). Our main result states that, typically, the rigidity set is very big: Theorem 8. • Rigidity. Let q > q0 , with q0 being large enough. Let our box N ,L have even width
N , while the height L does not exceed exp N 2/3 . Then for every β, |R (σ )| β,q µ N ,L > 1 − a (q) → 1 N2 as N → ∞, for some a (q) > 0, with a (q) → 0 as q → ∞. In particular, the surface B has typically only one connected component. • Long-range order. The function h (M) is the long-range order parameter: if M , M
are two arbitrary points in P, then the probability of the event β,q µ N ,L h M = h M
and are finite → 1 as q → ∞, uniformly in N and M , M
and for every β. Of course, for most values of the temperature this result is not very surprising. Indeed, if T > Tcr , say, then the box N ,L will be filled with disordered phase, while the surface B(σ ) is pressed to stay in the vicinity of the P o -component of the boundary. Our
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result is of real interest precisely at criticality, since at T = Tcr the surface B(σ ) stays away from the boundaries of the box N ,L due to the entropic repulsion. We expect that at criticality the location h C¯ (σ ) of the interface B(σ ) is distributed approximately uniformly in the segment [C ln N , L − C ln N ]. The details will be given in [V]. We would like to comment that the power of the RP method lies in the property that one can make statements about the behavior at the critical point by establishing some features for general temperatures. Indeed, it would be very difficult for us to work precisely at the critical temperature, since we do not even know its exact value. The main step towards the proof of rigidity is the control of the fluctuations of the interface with respect to the optimal flat shape. We thus make the following definition: Definition 9. Let B be the interface, and D ⊂ B be any collection of plaquettes. We define the weight of D to be w(D) = |D|−| (D)|, where |·| is the number of plaquettes in the collection. We have the following estimate: Theorem 10 (Peierls estimate). Suppose that N is even. Then, for all β, L and all collections of plaquettes D, β,q
µ N ,L (B : D ⊂ B) ≤ a w(D) , where a = a(q) goes to 0 when q → ∞. 3. Proof of Theorem 10 3.1. Settings for reflection positivity, construction of the blobs. In order to set the framework for reflection positivity, we consider the system as a spin-system on the 2dimensional torus T N , where at each site of T N we have a random variable taking values L in Zq (we recall that Zq = Z/qZ). β,q It is straightforward to see that µ N ,L is reflection positive with respect to the group generated by the reflections in the lines passing through the sites of the torus, see any of the RP papers [FL, FILS], or the review [S]. Let p ⊂ T N be any plaquette. Its full preimage c = −1 ( p) ⊂ N ,L will be called a column. The set of all columns will be denoted by C N . Any horizontal plaquette N ,L belongs to a well defined column, but for (some) vertical plaquettes we will make a σ -dependent choice. We assume the following convention: let P be a vertical plaquette, separating a frustrated cube of configuration σ from a pure disordered one; then we say that P belongs to the column containing the frustrated cube but not to the column containing the pure one. Now for any column c ∈ C N , we define Bc = Bc (σ ) to be the set of plaquettes of B (σ ) contained in c. Definition 11. We define the blobs of σ in c to be the connected components of Bc . We will denote by B(Bc ) = (b1 , . . . , br ) the set of blobs in the column c for the collection B (σ ), enumerated upwards. 3.2. Application of the chessboard estimate. In the first three subsections of this section we will reduce the Peierls estimate – the estimate of a local event, see (9)– to an estimate
Dobrushin Interfaces via Reflection Positivity
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N of a global event πˆ τ c , see (14). The remaining two subsections describe the splitting N of πˆ τ c into defects and their pairing. Let σ be some configuration. We distinguish several kinds of blobs in B(Bc ) = (b1 , . . . , br ), as we move upwards. The blob bi has: • type h− (h+), if bi begins (ends), as one ascends, with a horizontal plaquette, the rest being vertical; if bi consists of just a single plaquette, then it is of type h− (h+) if the cube below (above) it is pure disordered; • type h − +, if bi begins and ends with a horizontal plaquette, the rest being vertical (in that case the first cubes above it and below it have to be pure disordered); • type v: bi is a pack of vertical facets. Note that because of the convention we took for vertical plaquettes, there are no other cases. Moreover, from bottom to top we have the following rules: • there exists at least one signed blob, and the blob-signs are alternating; • the first and the last signs are −, • the first signed blob after a v-blob is of the type h + . Remark 12. If Bc is made of exactly one horizontal plaquette, there is only one blob in c, and it is of type h−. This blob is called trivial. 3.2.1. Defining Defects. Let us now consider the set F (σ ) of all frustrated cubes, attached to B (σ ). We will denote by Fc (σ ) the intersection F (σ ) ∩ c. Let C N (σ ) be the set of all columns c, such that Bc contains at least two plaquettes. For c ∈ C N (σ ) let Fi ⊂ Fc (σ ), i = 1, . . . , r be connected components of Fc (σ ). These segments of frustrated cubes will be called defects of σ. Now, every blob b j is contained in some defect Fi , but since some Fi -s can contain several blobs, we have r ≤ r. The set of all defects of σ is denoted by π (σ ), while πc (σ ) ⊂ π (σ ), c ∈ C N (σ ) will be those belonging to the column c. Our immediate goal will be the proof of the following Proposition 13. Let D ⊂ N ,L be any collection of cubes. Then β,q
µ N ,L (σ : D ⊂ π (σ )) ≤ a |D|−| (D)| ,
(9)
where a = a(q) goes to 0 when q → ∞. The Peierls estimate evidently follows from this. The rough idea of proving Proposition 13 is the following. We will try to show that the cost of having a defect with k frustrated cubes is of the order of ck , c < 1. This is indeed true, and we will show that for all defects with k ≥ 2 the price behaves as ck−1 . However, for some defects with k = 1 there is no price to pay at all, due to our choice of boundary conditions, which force the interface - and hence the defects - into the system. We will show then that if there are several such problematic defects – i.e. defects with k = 1 – then one can pair them, and extract the cost contribution of the order of c for every pair. This will be enough for our purposes. Note. The reader who would like to understand first the easy part of the proof – the one dealing with non-problematic defects – can go after Definition 14 below straight to Sect. 3.3.1. To implement the above strategy we need to impose some more structure on the defects. First of all, we define their signs. Namely, each defect F contains several blobs.
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Let us add all the signs of all the blobs in F. The resulting sign will be called the sign of F, sgn(F). It takes values +, − or 0. Since in the string of blobs in c the signs are alternating, the sign of F is well defined. We will also need some information about the vicinity of the defects. So we will spatially extend the defects, fixing to a certain extent the configuration at their ends. Then, of course, we will have to perform the summation over all extensions. In the process of extension some defects might coagulate into a single bigger defect, in which case we always will treat the result as a single defect. 3.2.2. Extending Defects. Here we will describe the process of extending the defects. The extension will depend on σ, of course. On the first step we extend each defect F j ⊂ c to a longer segment of cubes φ1 F j , F j ⊂ φ1 F j ⊂ c, which is a minimal segment containing F j , which contains, apart from F j , only frustrated cubes, except two end-cubes, which are pure cubes. (The added cubes need not touch the interface.) In the case that the defect F is attached to the boundary of N ,L , the extended defect has at most one pure end-cube. Evidently, theoperation φ1 is well-defined. It can happen that some resulting segments φ1 F j and φ1 F j have an elementary pure cube in common. In that case we merge them into a single defect: we will consider the connected components of the family {φ1 (Fi )}, and by a slight abuse of terminology we still call the resulting segments defects (or extended defects). The sign of the merger is defined to be the sum of the constituents. Now any two defects have no cubes in common (though they can share a facet). From now on we will deal exclusively with extended defects, so in what follows we will omit the symbol φ1 and will write just F for the extended defects. We also fix the nature of every bond in the defect, i.e. whether the bond is ordered or disordered. Definition 14 (Problematic defects). Among the defects we single out those with the property that every bond not belonging to the two end-cubes is disordered. (Note that at least one of these end-cubes has then to be ordered.) If this defect is signed, it was built from a blob consisting of just one horizontal plaquette; if it is not signed, it was built from the coagulation of two consecutive signed blobs, both consisting of just one horizontal plaquette. In both cases these defects will be called problematic. If both end-cubes of a problematic defect are ordered, the defect consists of 5 cubes, 3 of which are pure; if only one end-cube is ordered, the defect consists of 3 cubes, 2 of which are pure. Other defects, which will be called exceptional problematic defects, appear among defects attached to the bottom (disordered) boundary. Such a defect is called e-problematic, if it has the following three properties: 1. It consists from one or two frustrated cubes, followed by one ordered cube at the top of the defect, 2. The bottom cube has at least 3 vertical disordered bonds, 3. The corresponding blob consists of exactly one plaquette, which is the horizontal plaquette at the bottom of the box N ,L . In particular, any e-problematic defect has sign (−). All other defects will be called non-problematic. See Figs. 1,2 for (two-dimensional!) sketches of non-problematic and problematic defects.
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837
o
d
d
Fig. 1. A piece of interface, which generates a (non-problematic) defect; frustrated cubes are shaded. The third picture shows one possible outcome of the extension of the defect
o d
d o
o d
Fig. 2. A piece of interface, which generates three problematic defects
Thus we have assigned to every configuration σˆ with B σˆ = B (σ ) = B and to every column c ∈ C N (σ ) the extension πˆ c of the initial set πc (σ ), this extension including the order-disorder specification of every bond of πˆ c . The set of all possible extensions πˆ of π will be denoted by E (π ). Evidently, we have the partition
{σ : B (σ ) = B, π (σ ) = π } = ∪πˆ ∈E (π ) σ : B (σ ) = B, πˆ (σ ) = πˆ , so β,q
µ N ,L (σ : π (σ ) = π ) =
πˆ ∈E (π )
β,q µ N ,L σ : πˆ (σ ) = πˆ .
We will also use the notation σ ∈ πˆ , in the obvious sense. A straightforward combinatorial counting of the possible extensions of a given defect shows that to prove (9) it is enough to show that β,q µ N ,L σ : πˆ (σ ) = πˆ ≤ a πˆ −| (πˆ )| (10)
(with some smaller a), where πˆ is the number of frustrated cubes in πˆ , and πˆ is the number of plaquettes in the projection πˆ .
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3.2.3. Fixing the boundary conditions for defects. The last phase of fixing the environment of the defect consists in fixing the type of the configuration on ordered plaquettes P at the boundaries of the defect. If the plaquette P = (x, y, z, w) is fully ordered, with σ (x) − σ (y) = a, σ (y) − σ (z) = b, σ (z) − σ (w) = c, and σ (w) − σ (x) = d, we say that σ is of (a, b, c, d)-type on P; we notice that since a, b, c, d ∈ {−1, 0, 1}, there are at most 34 = 81 possible ordered types for σ on P. We denote by T the set of all possible types. Each defect F is delimited by two horizontal plaquettes: the top one, F t , and the bottom one, F b ; we define ∂ F = F t ∪ F b . Each of these plaquettes can be either fully ordered or fully disordered; we denote by ∂ o F ⊂ ∂ F the ordered plaquettes of ∂ F (the subset ∂ o F depends on πˆ ). For every collection πˆ of extended defects, πˆ ∈ E (π ), we define ∂ πˆ = ∪ F∈πˆ ∂ F and ∂ o πˆ = ∪ F∈πˆ ∂ o F. We refine the partition E (π ) by specifying the types the configuration o σ has on every plaquette from the set ∂ o πˆ : if τ ∈ T ∂ πˆ , we define
πˆ τ = σ ∈ πˆ : ∀P ⊂ ∂ o πˆ , σ is of type τ (P) on P , so
πˆ =
τ ∈T ∂
πˆ τ .
o πˆ
We notice that for any column c ∈ C N (σ ), containing a non-trivial blob,we have
for the corresponding defect, that the number of plaquettes ∂ o πˆ c ≤ 3 πˆ c − 1 (with equality iff πˆ c consists of two problematic defects – the first one with order-disorder b.c., the second with order-order b.c. and with one frustrated cube each). In particular, β,q β,q β,q µ N ,L πˆ τ ≤ (81)3(πˆ −| (πˆ )|) sup µ N ,L πˆ τ . µ N ,L σ : πˆ (σ ) = πˆ = τ ∈T ∂
τ ∈T ∂
o πˆ
o πˆ
(11) N (The estimate (11) is helpful in the discrete case, since the reflected event πˆ τ c (see below) has a relatively simple structure. This is not so in the continuous symmetry case.) o In the following πˆ ∈ E (π ) and τ ∈ T ∂ πˆ will be fixed, and we will estimate from
β,q above the µ N ,L -probability of the event σ ∈ πˆ τ . We have
σ ∈ πˆ τ = ∩c∈C N (σ ) σ ∈ πˆ τ c , (12) where the event πˆ τ c consists of configurations σ which in the column c have their o pattern of extended defects equal
to πˆ c , while their restriction to the plaquettes ∂ πˆ ∩ c
have types defined by τc ≡ τ o . ∂ πˆ ∩c The application of the chess-board estimate (see [FILS], relation (4.4)) reduces the β,q problem of getting the upper bound for the probability µ N ,L σ ∈ πˆ τ to that for all N N β,q probabilities µ N ,L σ ∈ πˆ τ c , c ∈ C N (σ ), where the event πˆ τ c is the result of N applying multiple reflections to πˆ τ c . (The reflected event πˆ τ c is described in details in the following subsection.) Namely, the chess-board estimate claims that N N12 β,q β,q µ N ,L σ ∈ πˆ τ c µ N ,L σ ∈ πˆ τ ≤ . c
(13)
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We will prove that uniformly in τ , 2 β,q N µ N ,L πˆ τ c ≤ a N (πˆ c −1) ,
(14)
provided that “the interface B is not regular in the column c ”; that means that for any σ ∈ πˆ τ c the collection of blobs B(Bc ) = (b1 , . . . , br ) of the interface B (σ ) in the column c for the collection B (σ ) is not just one trivial blob. (We do not care for the situation with the trivial blob, since it does not contribute to (9) anyway.) We will call such patterns non-trivial. Then (14) , (13) and (11) imply the relation (10). N 3.2.4. Description of the reflected event πˆ τ c . The column c is now fixed. The event πˆ τ c consists of collection of (extended) defects F1 , F2 , . . . , Fs in the column c, o each of these equipped with a boundary condition τi ∈ T ∂ Fi . Let the slab i = {(x, y, z); 0 ≤ x, y ≤ N , ai ≤ z ≤ bi } be the smallest one containing the defect Fi . N The event σ ∈ πˆ τ c happens if the following two conditions hold: • in every column c the pattern of order/disorder bonds of configuration σ agrees with θc,c (F1 , F2 , . . . , Fs ), where θc,c is any composition of the reflections in the lines passing through the sites of the torus, which takes c to c , • on every ordered plane z = ai (resp. z = bi ), i = 1, . . . , s, the configuration σ is of N N N “reflected” type τib (resp. τit ), where in column c the type τib is defined to be θc,c τib (resp. θc,c τit ). We denote by FiN the repeated reflection of the defect Fi , i.e. FiN = ∩c θc,c (Fi ) . It is a pattern of order/disorder bonds in i . We put L i = bi − ai − 1, and we define m i to be the number of frustrated cubes in Fi . Since every point (x, y, z) with ai < z < bi belongs to at least one frustrated cube of FiN , we have L i ≤ 2m i ,
(15)
s is a collection of which will be of importance later. The complement N ,L \ ∪i=1 i
slabs i = {(x, y, z); 0 ≤ x, y ≤ N , bi ≤ z ≤ ai+1 }, i = 0, . . . , s, with the conventions that b0 = 0 and as+1 = L + 1. N We now fix the values η of the configuration σ on ∂ πˆ τ c , i.e. on each plane z = ai N N or z = bi . The set of η-s which are compatible with πˆ τ c is denoted by B πˆ τ c . N We choose some η ∈ B πˆ τ c , and define
N N
πˆ τ c (η) = πˆ τ c ∩ σ
= η . N
∂πc
We obviously have β,q
µ N ,L
N πˆ τ c =
η∈B
( )
N πˆ τ c
β,q
µ N ,L
β,q
Uniformly in τ, η, we will get an estimate on µ N ,L
N πˆ τ c (η) .
N πˆ τ c (η) .
(16)
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S. Shlosman, Y. Vignaud
We will denote by ηib (resp. ηit ) the restriction of η to the plane z = ai (resp. β,q N z = bi ). Clearly, the partition function Z N ,L πˆ τ c (η) , computed over the set N σ ∈ πˆ τ c (η) , factors: β,q
Z N ,L
s s ηb ,ηt ηt ,ηb N πˆ τ c (η) = Z ii i FiN Z i i+1 , i=1
i=0
(17)
i
where the superscripts in the partition functions denote the corresponding boundary conb = σ ), while the presence ditions for slabs (with the convention that η0t = σdisord , ηs+1 ord N of arguments Fi describes the corresponding periodic order-disorder pattern of bonds. (We note for clarity that it can happen that bi = ai+1 for some i, in which case the slab b ηt ,ηi+1
i degenerates to a plane, and the partition function Z i i we put it to be 1 by definition.) Our goal is now to prove that s
is taken over the empty set;
s 2 s η η Z ii FiN ≤ a N [( i=1 m i )−1] Z ii ,
i=1
(18)
i=1
where m i is the number of frustrated cubes in Fi , and we use the shorthand notation η ηb ,ηt ηb ,ηt η Z ii FiN ≡ Z ii i FiN , Z ii ≡ Z ii i . Since, obviously, s
η∈B
( )
N πˆ τ c
i=1
η
Z ii
s i=0 β,q
b ηt ,ηi+1
Z i
i
Z N ,L
≤ 1,
the relations (18) and (16) imply (14). We can easily deal with each non-problematic defect Fi , and we will show that they satisfy the estimate: 2 η η Z ii FiN ≤ a m i N · Z ii . (19) However, no reasonable estimate can be obtained for a single problematic defect. To produce the cost factor needed, we will have to treat the problematic defects in pairs, 2 and we will produce a factor a 2N for every such pair. Let us explain the heuristics behind the above claim. Consider for example a nonproblematic defect, which, in ascending order, has the following pattern of cubes (see Fig. 3): (. . . , d, d, f, f, f, f, o, o, . . .) , which means that we consider a defect sheet of width 4, sandwiched between the disordered and ordered phases. We will show in Sect. 3.3.1 that the replacement of it by one of the two following narrower defect sheets: (. . . , d, d, d, d, d, f, o, o, . . .) or (. . . , d, d, f, o, o, o, o, o, . . .)
Dobrushin Interfaces via Reflection Positivity
841
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
d
d
d
o
o
o
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
Low T
High T
Fig. 3. From left to right: a defect sheet of width four and its replacements at low and high temperatures. Ordered (disordered) bonds are represented by straight (wriggled) segments
leads to the increase of the probability. Which of the last two patterns gives the needed increase depends on the temperature; in the high temperature region the first scenario (the advance of disorder) wins over the frustration strip, while at low temperatures the second one takes over the frustration. Moreover – and that is of crucial importance – the two temperature regions are intersecting, and at the common temperature each of the two scenarios gets a higher probability than the thick frustration sheet. Note also, that the frustration sheet can not disappear completely: in every column there should be at least one frustrated cube between the ordered and the disordered phase, which is the reason for the problematic defects to be treated separately. In more details, our strategy will be the following: we consider all signed defects in the column c, which from now on will have their special notation: G 1 , G 2 , . . . , G 2k−1 . Note that we always have an odd number of them; moreover, their signs alternate, with sgn(G 1 ) = (−). Some of G i -s can be problematic. The remaining neutral defects will be denoted by H1 , H2 , . . . , Hl ; some of them can also be problematic. We pair signed defects as follows: (G 1 , G 2k−1 ) , (G 2 , G 2k−2 ) , . . . while neutral defects are paired in the following way: (H1 , H2 ) , (H3 , H4 ) , . . . . If l is odd, we finally pair the remaining neutral defect Hl with G k ; if l is even, the defect G k is left unpaired. Notice that the two paired signed defects have the same sign. Note also that for a non-trivial pattern it can not happen that we have just one defect of problematic or e-problematic type. The above pairing will be essential for us only when both defects in the pair are problematic – i.e. when we have a problematic pair. In that case we will treat them together via gluing construction, explained below. The pairing of the remaining defects is inessential, since each pair contains at least one non-problematic defect, so we can distribute the cost of the latter over the pair. In particular, if both are non-problematic, we will just add the two separate contributions. N 3.2.5. Gluing process. In this section we will construct for every layered event πˆ τ c N another layered event, φ2 πˆ τ c , of a similar type. The new event will have less
842
S. Shlosman, Y. Vignaud
frustrated layers, and, what is most important, it will have no problematic pairs of defects. More precisely, we prove the following: N Lemma 15. For any event πˆ τ c with l problematic pairs one can construct the event N φ2 πˆ τ c , such that: 1. β,q
µ N ,L
2 β,q N N πˆ τ c ≤ a 2l N µ N ,L φ2 πˆ τ c ,
N πˆ τ c can be paired in such a way that no pair is problematic, N 3. the number of frustrated layers in φ2 πˆ τ c is πˆ c − 2l. 2. all defects of φ2
Proof. We proceed by induction on the number l of problematic pairs, successively 2 removing every such pair and producing instead a factor a 2N . We consider first the case when the two defects paired are problematic (or e-problematic) signed defects G i and G 2k−i , with 1 ≤ i ≤ k − 1. We assume that the sign of G i (and therefore of G 2k−i ) is minus; the plus case is even simpler, since both defects are then non-exceptional problematic defects. We remind the reader that G 2k−i consists of a sequence of 3 cubes: in ascending order we first meet one pure disordered cube, then one frustrated, followed by one pure ordered cube. All the bonds not in the ordered cube are disordered. G i may be of problematic or e-problematic type, when i = 1. In the first case it consists of li = 3 cubes. In the second case it will be convenient for us to include in the count of the cubes also the “virtual” disordered cube in the layer {−1 ≤ z ≤ 0}, so we put l1 to be 3, when the e-defect has one frustrated and one ordered cube, and we put l1 = 4 when the e-problematic defect has two frustrated cubes plus one ordered on the top. Note that in any case the first frustrated cube of the defect has at least 3 vertical disordered bonds. Each of G j -s comes with the boundary condition – a configuration η j ∈ ∂G j . The first step of the gluing process is to make a global rotation, 1 , of the spin system in the slab Si = {ai + 2 ≤ z ≤ a2k−i }, so as to make the configuration ηit – the configuration on the plane {z = bi }, the top boundary condition of the lower defect G i – to be t closer to η2k−i , the top boundary condition of the defect G 2k−i . If the defect G 1 happens to be an e-problematic defect, then the slab S1 = {1 ≤ z ≤ a2k−1 } by definition. t The configurations ηit and η2k−i are two periodic ordered configurations, defined by their restriction to any given plaquette, so we write symbolically that ηit = (s1 , s2 , s3 , s4 ) t and η2k−i = (s1 , s2 , s3 , s4 ), where all s1 , . . . , s4 are just points of the discrete circle Zq .
t Since ηit and η2k−i are ordered, we can choose s ∈ {s1 , s2 , s3 , s4 } and s ∈ s1 , s2 , s3 , s4 such that for all j = 1, 2, 3, 4,
s − s j ≤ 1, and
s − s
≤ 1. j We will call the values s, s the dominant values of the boundary conditions. Now for N every σ ∈ πˆ τ c , we define 1 (σ ) by if z∈ ai + 2, a2k−i , σ (x, y, z) + s − s [ 1 (σ )] (x, y, z) = σ (x, y, z) otherwise.
Dobrushin Interfaces via Reflection Positivity
843
n+1
n+1
n
n
n−1
n−1
n−2
n−2
n−3
3 4 5
5 4 3
n−3
2
2
1
1
0
0
Fig. 4. Second step of the gluing transformation. From left to right: two problematic defects (with minus signs), generated by a piece of interface; the corresponding reflected event (made of two defect sheets); the result of the gluing operation
The transformation 1 is bijective. t are relaThe result thus achieved is that the configurations 1 ηit and 1 η2k−i tively close to each other. The second (and the last) step of the gluing process is to apply to the system in the slab Si the reflection 2 in its middle horizontal plane, thus bringing the upper part of 1 (G i ) in contact with 1 (G 2k−i ): σ (x, y, ai + a2k−i + 2 − z) if z∈ ai + 2, a2k−i , [ 2 (σ )] (x, y, z) = σ (x, y, z) otherwise. See Fig. 4 for a sketch of this second step. (Again, for G 1 being e-problematic, the reflec 1+a2k−1 + 1 .) tion is done in the slab S1 = {1 ≤ z ≤ a2k−1}, with respect to the plane z = 2 N The composition = 2 1 is bijective. Note that for every configuration σ ∈ πˆ τ c 2
all the bonds connecting the slab Si with its outside are disordered, except at most N4 vertical bonds when G 1 is an e-problematic defect. Thus can increase the energy of 2 the resulting configuration by N4 units, which is the possible number of ordered bonds 2
turning into disordered ones after the rotation: H (σ ) − H ( (σ )) ≥ − N4 . Therefore we get βN2 N β,q N β,q µ N ,L πˆ τ c (η) ≤ e 4 µ N ,L πˆ τ c (η) . (20) N Let us describe the event πˆ τ c (η) . Consider the images F˜ jN = F jN . If F j is between G i and G 2k−i , it is clear that F˜ j has exactly the same as F j , properties t b b up to shift and reversal of pattern. Moreover, we will have η˜ j = η j , η˜ j = ηtj
844
S. Shlosman, Y. Vignaud
as boundary conditions around F˜ j . If F j is before G i or after G 2k−i , we have F˜ j = F j and η˜ j = η j . The pattern τ˜ = (τ ) is defined in the following natural way: it coincides with τ outside the slab Si , and with a reflection of τ inside this slab. N N We will now focus on what happened to G iN and G 2k−i ; we denote by G˜ 2k−i the N restriction of πˆ τ c to the slab ˜ 2k−i = {a2k−i − li + 2 ≤ z ≤ b2k−i = a2k−i + 3} ,
which is at most 5-cubes wide, since li ≤ 4. N If G˜ 2k−i occurs, we have two slabs – {a2k−i − li + 2 ≤ z ≤ a2k−i − li + 3} and {a2k−i + 2 ≤ z ≤ a2k−i + 3} – filled with ordered bonds, and one slab – {a2k−i + 1 ≤ z ≤ a2k−i + 2} – filled with disordered bonds; actually, the pattern of the bonds is fixed, except for the N 2 vertical t of the slab {a2k−i ≤ z ≤ a2k−i + 1}. bonds around this defect are very close Since the boundary conditions η˜ 2k−i = ηit , η2k−i to each other, we will be able to derive the following estimate: e leading to e
βN2 4
βN2 4
η˜
2k−i
Z ˜2k−i
2 η˜ N G˜ 2k−i (η) ≤ a 2N · Z ˜2k−i ,
2k−i
2 N β,q β,q µ N ,L πˆ τ c (η) ≤ a 2N · µ N ,L (π˜ τ˜ )cN (η) ˜ ,
(21)
(22)
where (π˜ τ˜ )cN (η) ˜ is the event that for all j such that F j ∈ / {G i , G 2k−i }, F˜ jN occurs, that the type on plaquettes of ∂ o F˜ j is given by τ˜ = (τ ) and that at the boundaries of F˜ N j
the configuration agrees with η˜ Nj . The remaining case of a pair of problematic defects F j , Fk with one of them – say, the upper one, Fk – having both end-cubes ordered, is even simpler. Namely, it is enough to perform a global rotation in a suitable slab, which will make the two (ordered) boundary conditions of the defect Fk close enough, as was the case in the first step above. After that, the defect Fk can be treated in precisely the same way as the defect G˜ iN of the was treated. To define the rotation needed we take any slab
preceding paragraph d j ≤ z ≤ d j + 1 inside the defect F j , which has at least 3 disordered vertical bonds. Such a slab clearly exists by definition. Then we do the rotation of all the spins in the slab d j + 1 ≤ z ≤ ak + 2 by the angle s t −s b , where s t and s b are the dominant values of the boundary conditions ηkt and ηkb of the defect Fk , leaving all other spins unchanged. 2
Since does not increase the energy by more than N4 units, we have reduced our case to the one already considered. Applying the above arguments to each problematic pair, we get rid of all of them, 2 getting a factor of a 2N for each pair. We denote by φ2 the composition of the several transformations described above, which were needed through the gluing process. N Then φ2 F j will be the family of remaining reflected defects, not yet treated, with N φ2 (ηc ) being their boundary conditions. We denote by φ2 πˆ τ c the event that all these defects occur and that the configuration takes the prescribed values φ2 (ηc ) on corresponding planes. Summarizing, the lemma follows from (20, 22), the proof of (21) being deferred to the next section.
Dobrushin Interfaces via Reflection Positivity
845
3.3. Estimating defects. The estimates proceed differently for problematic and nonproblematic defects. We begin with the case of non-problematic ones. 3.3.1. Non-problematic defects: Proof of (19). Thanks to the previous analysis, the proof of our main theorem is reduced to estimating a non-problematic defect. The analysis will be divided into three cases, according to the nature of boundary conditions around the defect: disordered, mixed, or ordered. In the reflected defect FiN , we denote by K i the number of chaotic sites, which are 2
sites with 6 adjacent disordered bonds; notice that K i = ki N4 , with ki being an integer (or zero), due to the periodic structure of FiN . We denote by Di the number of inner disordered bonds of FiN , (those of the configurations ηi are not included). Let us consider the connected components of the graph made by ordered bonds of FiN . Some of these components are vertical segments, not attached to the boundary; let Q i be their 2 number. Again, Q i = qi N4 with integer qi . The number of other connected components of this graph is at most 2L i N . Indeed, every such component contains at least one full horizontal line (and there are 2L i N such lines). Note that the number of sites to which at least one ordered bond is attached is L i N 2 − K i ≤ 2m i N 2 , while the number of connected components in this ordered bonds graph is at most Q i + 2L i N . We have therefore the following simple universal upper bound: 2 2 η Z ii FiN ≤ 32m i N q K i +Q i +2L i N eβ (3L i +1)N −Di . Indeed, let us pick a point in every connected component of the ordered bond graph. Then the factor q Q i +2L i N estimates the number of possible spin configurations κ on 2 these sites, while 32m i N is the estimate on the number of configurations on the ordered bond graph, given κ. (If the spin value at one end of the ordered bond is fixed, then at the other end the spin can have 3 different values, see (6).) The factor q K i is the number of configurations on chaotic sites. Finally, (3L i + 1) N 2 − Di is the energy estimate. We will use different lower bounds, depending on the boundary conditions and the temperature. They use some (heavy) combinatorics of the defects. We postpone the proof of the relevant combinatorial statements till the end of the paper. Order–disorder. In this subsection we consider non-problematic defects with ordered boundary condition at one end of the defect and disordered boundary condition at the other. We have the bound η
2
2
Z ii ≥ (q − 18) L i N + e3β L i N ; here the first term estimates the partition function taken over fully disordered configurations, while the second one – the partition function taken over fully ordered configurations. (In fact, it is enough to take just one ordered configuration.) If eβ ≤ q 1/3 , we have (omitting unimportant terms, not depending on q): η K i +Q i +2L i N eβ (3L i +1)N 2 −Di Z ii FiN mi N 2 q ≤9 η 2 Z ii (q − 18) L i N Di −N 2 /3−K i −Q i −2L i N mi N 2 1 ≤9 . q
846
S. Shlosman, Y. Vignaud
If eβ ≥ q 1/3 ,
η Z ii FiN η
Z ii
mi N 2 q
≤9
mi N 2
≤9
K i +Q i +2L i N eβ (3L i +1)N 2 −Di
1 q
e3β L i N
2
2
Di −N /3−K i −Q i −2L i N
.
By (26) below we can take α > 0 such that Di − N 2 /3 − K i − Q i ≥ 2α m i N 2 . Since L i ≤ 2m i , for all N large enough and all order–disorder defects Fi , Di − N 2 /3 − K i − Q i − 2L i N ≥ α m i N 2 . Therefore, for all β and all such defects, η Z ii FiN 2
2 ≤ 9m i N q −α m i N , η Z ii
and the desired estimate is valid with a(q) = 9q −α . Order–Order. As in the order–disorder case, we have η
2
2
Z ii ≥ (q − 18) L i N + e3β L i N , so we will be done by the previous analysis, if the estimate Di − N 2 /3 − K i − Q i ≥ 2α m i N 2 still holds for the order-order case. This is indeed so, see again (26). Therefore for all β > 0, η Z ii FiN 2
2 ≤ 9m i N q −α m i N . ηi Z i Bulk Disorder–Disorder. We have η
Z ii ≥ (q − 18) L i N + eβ(3L i −1)N . 2
2
If eβ ≤ q L i /(3L i −1) , we have (omitting unimportant terms, not depending on q): η K i +Q i +2L i N eβ (3L i +1)N 2 −Di Z ii FiN mi N 2 q ≤9 η 2 Z ii (q − 18) L i N L i Di −2N 2 − 3L i −1 (K i +Q i ) −2L i N 3L i −1 Li 1 2 ≤ 9m i N . q
Dobrushin Interfaces via Reflection Positivity
If eβ ≥ q L i /(3L i −1) , η Z ii FiN η
Z ii
mi N 2 q
≤9
≤ 9m i
N2
847
K i +Q i +2L i N eβ (3L i +1)N 2 −Di
eβ(3L i −1)N
Li 1 3L i −1 q
2
Di −2N 2 −
3L i −1 L i (K i +Q i )
−2L i N
.
By (27) we can take α > 0 such that Di − 2N 2 −
3L i − 1 (K i + Q i ) ≥ 6α m i N 2 . Li
Since L i ≤ 2m i , for all N large enough and all disorder–disorder defects Fi , Li 3L i − 1 2 Di − 2N − (K i + Q i ) − 2L i N ≥ α m i N 2 . 3L i − 1 Li Therefore, for all β and all such defects, η Z ii FiN 2
2 ≤ 9m i N q −α m i N , ηi Z i
and the desired estimate is valid with a(q) = 9q −α . Boundary Disorder–Disorder. We have η
2
Z ii ≥ (q − 18) L i N + e
β 3L i − 34 N 2
.
(We have 43 in the energy estimate 3L i − 43 N 2 due to the fact that at least one quarter of the boundary bonds will be ordered.) If eβ ≤ q
L i / 3L i − 34
η
N
Z ii Fi η
Z ii
, mi N 2 q
≤9
K i +Q i +2L i N eβ (3L i +1)N 2 −Di
(q − 18) L i N
Li 1 mi N 2 1 6 Li − 4 ≤9 q If
eβ
≥q
2
2Di − 27 N 2 −
6 L i − 14 Li
(K i +Q i ) −2L i N
.
L i / 3L i − 34 η
, N
Z ii Fi η
Z ii
mi N 2 q
≤9
K i +Q i +2L i N eβ (3L i +1)N 2 −Di β 3L i − 34 N 2 e
Li 1 mi N 2 1 6 Li − 4 ≤9 q
2Di − 27 N 2 −
6 L i − 14 Li
(K i +Q i ) −2L i N
.
848
S. Shlosman, Y. Vignaud
Below in (36) we will show that for some α > 0, 6 L i − 41 7N 2 − 2Di − (K i + Q i ) ≥ 12α m i N 2 . 2 Li Since L i ≤ 2m i , for all N large enough and all disorder-disorder boundary defects Fi , Li 7 2 6 L i − 41 (K i + Q i ) − 2L i N ≥ α m i N 2 . 2Di − N − 2 Li 6 L i − 41 Therefore for all β and all such defects η Z ii FiN 2
2 ≤ 9m i N q −α m i N , ηi Z i
and the desired estimate is valid with a(q) = 9q −α .
3.3.2. Glued pair of problematic defects: Proof of (21, 22). We will analyze the defect G˜ iN , generated by the gluing process, and will prove the estimates (21, 22). The defect G˜ iN is at most 5-cubes wide, both end-layers are ordered, and all vertical bonds attached to the top cube are disordered; we notice that some vertical bonds in the third layer from the top may be ordered, possibly in a non-periodic way. We fix the pattern V of these extra vertical ordered bonds, G˜ iN (V ) denoting the restriction of G˜ iN to configurations η˜ agreeing with the pattern V . We will now estimate the partition function Z ˜i G˜ iN (V )
i ˜ i = a˜ i ≤ z ≤ b˜i , and write L˜ i = b˜i − a˜ i − 1 ≤ 4. in its slab
˜ i , such that the event G˜ N (V ) occurs, is The number of configurations in the slab
˜
˜
2
i
bounded from above by 3 L i N q K +Q+2 L i N , where K , Q depend on V ; every such con(i) (i) (i) figuration σ has energy H (σ ) = D − 3 L˜ i + 1 N 2 , where D also depends on V . Combining this we get: η˜ Z ˜i
i
2 N L˜ i N 2 K +Q+2 L˜ i N β (3 L˜ i +1)N −D ˜ G i (V ) ≤ 3 q e . η˜
Now we need a lower bound on Z ˜i . We will use one consisting of two contributions: the
i first is obtained by summing over high temperature configurations, while the second – by summing over low temperature ones. For high temperatures, we just integrate over configurations with zero energy, the set ˜ 2 of such configurations containing at least (q − 18) L i N configurations. For low temperatures, we simply take one single configuration with minimal energy 2 ˜ under given boundary conditions. Let us check that this equals to −(3 L i +1)N . minimum Indeed, since the (periodic) configurations (η˜ i )t on z = b˜i and (η˜ i )b on {z = a˜ i } have by dominant value, s, the constant configuration σs ≡ s in construction the common ˜ i , taken with boundary conditions η˜ i , has all a˜ i + 1 ≤ z ≤ b˜i − 1 – the interior of
˜ i ordered. bonds in
Dobrushin Interfaces via Reflection Positivity
849
Gathering all this we have: ˜
η˜
i
˜
Z ˜i ≥ (q − 18) L i N + e(3 L i +1)N β . ˜
˜
2
˜
η˜
i
2
If eβ ≤ q L i /(3 L i +1) , we use Z ˜i ≥ (q − 18) L i N to get
e
1 2 4βN
·
η˜
i
Z ˜i
G˜ iN (V )
≤
η˜
i
Z ˜i ˜
˜
2
D− 1 N 2 − 3 L i +1 (K +Q+2 L˜ i N ) L˜ i N 2 4 L˜ L˜ i − ˜i . (23) q 3 L i +1 ˜
3q q − 18
˜
η˜
i
If eβ ≥ q L i /(3 L i +1) , we use Z ˜i ≥ e(3 L i +1)β N to get
e
1 2 4βN
·
η˜
i
Z ˜i
G˜ iN (V ) η˜
i
Z ˜i
2
˜
≤ 3Li N
2
q
−
L˜ i 3 L˜ i +1
D− 1 N 2 − 3 L˜ i +1 (K +Q+2 L˜ i N ) L˜ i
4
.
(24)
We will use the following Lemma 16. For any pattern V of ordered bonds in the third layer from the top, and all N large enough, D−
3 L˜ i + 1 (K + Q + 2 L˜ i N ) ≥ L˜ i
1 1 + 4 5
N 2.
(25)
Proof. We recall that the ordered cubes at end-points of the defect are always disconnected in the ordered graph corresponding to V , because of the vertical disordered bonds in the second layer from the top; using Lemma 19 below we get D − 3(K + Q) ≥ N 2 . ˜ Moreover, it is clear that K + Q ≤ ( L˜ i − 2)N 2 and thus D − 3 L˜i +1 (K + Q) ≥ N 2 −
L˜ i −2
L˜ i 1 2 4N
Li
+
N2
=
1 2 5N
2 L˜ i
N 2.
Our lemma now follows from L˜ i ≤ 4 since
+ 2 L˜ i N , provided N is large enough.
2 L˜ i
N2 ≥
2
1 2 2N
≥
Since there are 2 N possible patterns V , Lemma 16 together with (23, 24) give for all β:
e
1 2 4βN
with a(q) =
·
η˜
i
Z ˜i
G˜ iN
η˜ Z ˜i
i
≤2
N2
3q q − 18
L˜ i q
− 15 ·
! 4 3q 2 q−18 q −3/50 , since L˜ i ≤ 4 and also
L˜ i 3 L˜ i +1
L˜ i 3 L˜ i +1
N 2 2
≤ a 2N ,
≥
3 10 .
850
S. Shlosman, Y. Vignaud
4. Combinatorial Estimates for Bulk Defects We prove here the needed combinatorial estimates on non-problematic defects, restricting the proof to defects in the bulk of the system (i.e. when the defect is not stuck to the bottom boundary), and divide this proof into three parts according to the nature of the boundary conditions around the defect. We introduce the number d, which equals the number of disordered cubes at the ends of our defect, i.e. ⎧ ⎨ 0 for the order-order bc, d = 1 for the order-disorder bc, ⎩ 2 for the disorder-disorder bc. The case of boundary defects is more involved and is deferred to the next section. For a d = 0 or d = 1 non-problematic bulk defect F with m ≥ 1 frustrated cubes, and its reflection F N we will prove the relation 2D − 6K − 6Q ≥ 2N 2 + αm N 2
(26)
for some universal α > 0, where D, K and Q are the characteristics of F N , introduced above. For d = 2 non-problematic bulk defect F with m ≥ 1 frustrated cubes we will prove (3L − 1) (K + Q) ≥ 4N 2 + αm N 2 . 2D − 2 (27) L We introduce the set K of chaotic sites and the set D of disordered bonds in F, |K| = K , |D| = D, and we rewrite 6K as a double sum Ie∈D , 6K = x∈K e:x∈e
to get
2D = 6K + d N 2 + ∂ 1 O + 2 ∂ 2 O , where O is the graph of ordered bonds in F N , and ∂ n O denotes the set of disordered bonds with n vertices belonging to O, n = 1, 2; the term d N 2 comes from the d N 2 vertical disordered bonds in the boundary chaotic cubes (this is precisely where we use the fact that the defect is in the bulk). We rewrite it as
2
∂ X j + (28) 2D = 6K + d N 2 +
∂ X j , j
j
where X j -s are the connected components of the ordered-bond graph of F N , ∂ X j is the set of disordered bonds touching X j , and ∂ 2 X j the set of disordered bonds with both vertices in X j . When the number m of frustrated cubes in the defect is small, we will use for the derivation of (26) the above relation (28) directly. For large m-s we will utilize its corollary, which we will derive now. Lemma 17. The relation (28) implies that 1 2D − 6K − 6Q ≥ d N 2 + m N 2 + 2Q. 2
(29)
Dobrushin Interfaces via Reflection Positivity
851
Proof. If X j is a vertical segment, not touching the boundary, we have ∂ X j = n j + 6, where n j is the number of frustrated cubes sharing a bond with X j ; also, ∂ 2 X j = 0. Let us denote the set of these j-s by J. For other components we use the estimate:
∂ X j +
∂ 2 X j
≥ 1 n j . 2
(30)
To see it hold, we first note that
∂ X j ∩ c +
∂ 2 X j ∩ c
,
∂ X j +
∂ 2 X j
≥ 1 4 c
(31)
where the summation goes over all cubes c, contributing to n j ; we have the factor 14 due to the fact that
belongs to at most 4 cubes. We claim now that for every
every bond cube c we have ∂ X j ∩ c + ∂ 2 X j ∩ c ≥ 2. Indeed, either c has at least two bonds from ∂ X j , or just one such bond, f. In the latter case, all other (eleven) bonds of c belong to X j , and therefore f belongs not only to ∂ X j , but also to ∂ 2 X j . That proves (30). Gathering all this leads to: 2D − 6K ≥ d N 2 + 6Q +
nj +
j∈J
1 nj 2
1 ≥ d N 2 + 8Q + n j, 2
j ∈J /
j
since for every j ∈ J we have 21 n j ≥ 2. Finally, every frustrated cube in F N contributes to at least one n j , so we arrive to 1 2D − 6K − 6Q ≥ d N 2 + m N 2 + 2Q. 2 4.1. Order–disorder (d = 1): Proof of (26). Here we consider a non-problematic defect with an ordered (disordered) cube at the top (bottom). 1. If m ≥ 3, (29) gives 2D − 6(K + Q) − 2N 2 ≥
m − 1 N 2 ≥ N 2, 2 6
m
which is what we need. 2. Assume m ≤ 2. Let us consider the ordered connected component X 0 , containing the upper ordered cube; its boundary ∂ X 0 has at least N 2 bonds, with equality if and only if X 0 is the result of multiple reflections of the upper ordered cube. Thus 2 (28) shows that 2D − 6(K
+ Q) ≥ 2N , with equality if and only if Q = 0, 2 |∂ X 0 | = N , and the set X j consists of only one component – X 0 . The equality therefore can occur only if the defect is problematic. Hence in the case considered 2 2D − 6(K + Q) ≥ 2N 2 + N2 .
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4.2. Disorder–disorder (d = 2): Proof of (27). Here we consider a non-problematic reflected defect F N surrounded by two chaotic layers. We want to obtain the bound (27). In fact, for most defects the stronger statement holds: 2D − 6(K + Q) ≥ 4N 2 + αm N 2 .
(32)
Indeed, the relation (29) reads 1 2D − 6K − 6Q ≥ 2N 2 + m N 2 + 2Q, 2
(33)
so the estimate (32) holds once m ≥ 5. So we assume in the following that m ≤ 4; if K = Q = 0, the simple fact that D ≥ 6N 2 is enough to get (32), so we assume it is not the case. But then, it is enough to show that 2D − 6(K + Q) ≥ 4N 2 ; indeed, (27) will 2 follow from L ≤ 2m = 8 and K + Q ≥ N4 . Next we note the following simple Lemma 18. For any bulk defect with disorder–disorder b.c. (d = 2), the existence of an ordered horizontal bond e implies |∂ X i | ≥ 2N 2 . i ∈J /
Proof. Indeed, in its column c the bond e has two horizontal adjacent bonds e , e
. If both of them are disordered, their reflections produce N 2 horizontal bonds belongN2 ing to ∪i ∈J / ∂ X i , while the reflections of the bond e contain 2 sites, each of which has a disordered bond from ∪∂ X i above it and another one below it. If e is ordered 2 2 and e
is disordered, we get similarly N2 horizontal bonds and 3N2 vertical bonds in the boundaries. If both e and e
are ordered, we get 2N 2 vertical bonds in the boundaries. The previous lemma, combined with (28), reduces the analysis to the case where there is no ordered horizontal bonds. In this last case we have 2D−6K = 2N 2 +6Q+ j∈J n j , where n j is the number of frustrated cubes sharing a bond with X j . From this, we get 2D −6(K + Q) ≥ 2N 2 +m N 2 ≥ 4N 2 , if m ≥ 2. If m = 1, then L = 2, K = 2N 2 −2Q, 2 D = 7N 2 − Q with Q ≥ N4 , so that 2D − 2 3L−1 L (K + Q) = 2D − 5(K + Q) = 4N 2 + 3Q ≥ 4N 2 +
3N 2 4 .
4.3. Order–order (d = 0): Proof of (26). We consider the reflection F N of a nonproblematic defect F with m frustrated cubes, surrounded by two ordered cubes. Since every defect by definition contains a disordered plaquette, every order-order defect has m ≥ 2. We want to establish the relation (26) : 2D − 6(K + Q) ≥ 2N 2 + αm N 2 . For m ≥ 5 it follows immediately from (29), so we assume that m ≤ 4. In this case the relation (26) follows from the following two lemmas: Lemma 19. For all defects with order–order b.c. and such that the ordered cubes at the ends of the defect are disconnected in the ordered graph, 2D − 6(K + Q) ≥ 2N 2 , with equality if and only if it is problematic.
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853
Proof. We denote by X 0 and X 1 the two connected components corresponding to the extreme ordered cubes. Then |∂ X 0 | + |∂ X 1 | ≥ 2N 2 and (28) shows the inequality, and we see that the case of equality is precisely the problematic defect. Lemma 20. For all defects with order–order b.c. such that ordered cubes at the ends of the defect belong to the same component, 2D − 6(K + Q) ≥ 3N 2 . Proof. We denote by X 0 the component containing both ordered cubes. Our assumption means that our defect contains a vertical disordered plaquette P. Indeed, all the blobs defining our defect have only vertical plaquettes, since the defect does not contain disordered cubes. Looking at the two vertical lines passing through P, we see that each of them is either completely ordered outside P, its unique disordered bond then belonging to ∂ 2 X 0 , or else
it has two bonds in ∂ X 0 ; therefore, the contribution of vertical bonds to |∂ X 0 | + ∂ 2 X 0
2
is at least 2 N2 = N 2 .
We shall now prove that the horizontal contribution to |∂ X 0 | + ∂ 2 X 0 is at least 2N 2 . Let us look at the horizontal plaquette P , which contains the bottom horizontal bond of
P. Of course,
this bond is disordered. If some ordered bonds of P belong to X 0 , then
∂ X 0 ∩ P + ∂ 2 X 0 ∩ P ≥ 2, as a simple counting shows. Otherwise, since there is an ordered path through the defect, there is a vertical bond in X 0 touching P at a vertex x. By assumption, the two bonds of P containing x are disordered (since otherwise they would belong to X 0 ), so they both are in ∂ X 0 . The same holds for the horizontal plaquette P , which shares the top horizontal bond with P, which proves our claim. 5. Combinatorial Estimates for Boundary Defects Now we deal with the case when the defect is stuck to the bottom of the box. 5.1. Order–disorder: Proof of (26). Let us denote by D b the number of vertical disordered bonds attached to the bottom boundary of F N and replacing in (29) the term d N 2 by D b , we have the analog of (28),
2
∂ X j + (34) 2D = 6K + D b +
∂ X j , j
j
and the analog of (29) : 1 2D − 6K − 6Q ≥ D b + m N 2 + 2Q. 2
(35)
(We remark for clarity that here Q is the number of ordered vertical segments, not touching any boundary of the defect F N .) We remind the reader that we aim to prove the relation (26) for non-e-problematic defects. Note that the only m = 1 boundary defect with order–disorder b.c. is e-problematic. So in what follows we assume that m ≥ 2. If m ≥ 5, the relation (26) follows directly from (35). For smaller m we will use the following three lemmas.
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Lemma 21. For any m ≤ 4 boundary defect with order-disorder b.c. the strong disorder b.c. implies that ∂ X 0 contains at least 43 N 2 vertical bonds, provided q is large enough. (Here X 0 is the ordered component of the top ordered cube.) Proof. Since m ≤ 4, the defect is at most 8-cubes wide, and therefore contains at most 36 sites. If ∂ X 0 had less than 3 vertical bonds in the column, then two sites u, v of the bottom plaquette would belong to X 0 . Denoting by M the size of the largest possible path in a graph with 36 sites, we would get |σu − σv | ≤ M, therefore contradicting the strong disorder b.c. for q large enough. Lemma 22. Consider any boundary defect with any b.c. on the top. If D b ≤ strong disorder b.c. imply
|∂ X i | + ∂ 2 X i ≥ 3N 2 .
N2 4 ,
the
i ∈J /
Proof. We start with the case D b = 0. Since the b.c. are strongly disordered, all horizontal bonds of the first layer have to be disordered as well, and each of them belongs to ∂ 2 X i for some i ∈ / J , so their contribution to the sum above is 4N 2 . b In the case D = N 2 /4, the strong disordered b.c. implies that three or four horizontal bonds in the first layer are disordered. If we have 4 such disordered bonds, they all belong to some ∂ X i , and two of them actually belong to some ∂ 2 X i ; if we have only three such bonds, they all belong to some ∂ 2 X i . In any case, they contribute 3N 2 to the sum above. Lemma 23. For any boundary defect with order–disorder b.c. with a horizontal disordered bond at the level z = 1, the contribution of horizontal bonds to
|∂ X i | + ∂ 2 X i , i ∈J /
is at least N 2 . Proof. If all four bonds of the horizontal plaquette P at z = 1 are disordered, there has to be a vertical ordered bond touching the boundary (because the first cube is frustrated). It touches two horizontal bonds of P; all their reflections contribute N 2 to the sum. If the plaquette P has two or three disordered bonds, at least two of them belong to the boundary of some X j . Finally, if P has only one disordered bond, then it belongs to ∂ 2 X j for some j, and so contributes twice to the sum above. 2
2
If D b ≤ N4 , we can apply Lemma 22 and (34) to get (26). If D b = N2 , the strong disorder b.c. prevent the horizontal plaquette at z = 1 from being completely ordered, so we can apply Lemma 23 together with Lemma 21, getting (26). 2 If D b ≥ 3N4 and m ≥ 3, we apply (35) to get the relation (26). 2
In the remaining case D b ≥ 3N4 and m = 2 we know that the blob corresponding to the defect had at least 2 plaquettes, because it would be e-problematic otherwise. If this extra (disordered!) plaquette is horizontal, then the second cube is pure disordered; otherwise it is vertical. In any case the horizontal plaquette at z = 1 cannot be completely ordered. Thus we can apply Lemma 23 and (34) to get (26).
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5.2. Disorder–disorder: Proof of (36). Now we prove the relation 6 L − 41 7N 2 2D − (K + Q) ≥ + αm N 2 . L 2
(36)
In fact, for most defects we will prove the stronger statement (32) : 2D − 6(K + Q) ≥ 7N 2 2 2 + αm N . We start with the identity
|∂ X i | + (37) 2D − 6K = D b + N 2 +
∂ 2 X i , i
i
where D b is the number of vertical disordered bonds attached to the bottom boundary of F N . (The term N 2 equals the number of vertical disordered bonds attached to the top boundary.) From this we deduce, as above, that 2D − 6K − 6Q ≥ D b + N 2 +
mN2 + 2Q. 2
(38)
The desired estimate is directly derived from this for m ≥ 6. We now deal with the case m ≤ 5. The case m = 1 is completely explicit. We have L = 1, D = 3N 2 + D b , Q = 0, K = D b and D b ≤ 43 N 2 (because the first cube is frustrated), so that 2D − 6 L−1/4 L K = 6N 2 + 2D b − 29 D b = 6N 2 − 25 D b ≥ 4N 2 + 18 N 2 , and thus we assume m ≥ 2. Also, if K = Q = 0, the simple fact that D ≥ 3N 2 is enough to get (32), so we 2 assume K + Q ≥ N4 . Lemma 24. For any boundary defect with m ≥ 2 and all horizontal bonds disordered,
∂ X j ≥ 3N 2 + 6Q. Db + Proof. Our assumption implies that all ordered components are vertical segments ; since the first cube must be frustrated, D b ≤ 43 N 2 . For j ∈ / J , X j starts from the boundary
X j + 1 ≥ 10Q + 5 |J c |. Since |J c | = N 2 − D b , we 4 and ∂ X j ≥ 10Q + j ∈J /
have D b + ∂ X j ≥ 10Q + 5N 2 − 4D b ≥ 10Q + 2N 2 . The lemma is then proved if Q ≥ 41 N 2 , so we assume Q = 0. We now pick one ordered bond in the second frustrated cube, which is vertical
by assumption. Since Q = 0, the corresponding ordered component X j satisfies X j ≥ 2. 2
2
After reflections, there are N4 such segments, and 3N4 − D b other segments. Then 2
D b + ∂ X j ≥ D b + 9 N 2 + 5 3N − D b = 6N 2 − 4D b and the lemma follows 4
from D b ≤ 43 N 2 .
4
Lemma 25. For any boundary defect with disorder-disorder b.c., and for any ordered horizontal bond e,
3
|∂ X e | + ∂ 2 X e ≥ N 2 + N 2 , 4 where X e denotes the ordered component containing e.
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Proof. We denote by P the horizontal plaquette containing e. If P is completely ordered, the strong disorder b.c. force ∂ X e ≥ N 2 + 43 N 2 (compare with Lemma 21). If P is not completely ordered, either two horizontal bonds of P belong to ∂ X e or one of them is in ∂ 2 X e ; moreover, due to the strong disorder b.c. at least 3 vertical bonds of the column belong to the boundary of X e . If no horizontal bond is ordered, we can combine (37) with Lemma 24 to get 2D − 6(K + Q) ≥ 4N 2 . Otherwise, we can find a bond e to which we apply Lemma 25, to get 2D−6(K + Q) ≥ 2N 2 + D b + 43 N 2 . If D b ≥ 21 N 2 , this shows that 2D − 6(K + Q) ≥ 3N 2 + 41 N 2 . Since 2D − 6(K + Q) is an integer multiple of 21 N 2 , we actually have 2D − 6(K + Q) ≥ 27 N 2 . 2
Equation (36) now follows from L ≤ 2m = 10 and K + Q ≥ N4 . If D b ≤ 41 N 2 we apply Lemma 22 and (37) to get 2D − 6(K + Q) ≥ 4N 2 .
6. Proof of the Main Theorem 8 In this section, we derive our main results from the Peierls estimate. We start with the question of the interface uniqueness. Lemma 26. For any b > 1 there exists a q0 < ∞ such that the following holds: For any 2 β,q q ≥ q0 and any sequence L N ≤ b N the probability µ N ,L N (discn) of the event that the interface B is disconnected, vanishes as N → ∞. Proof. Let 0 ∈ T be the origin. Denote by l (B) the quantity min {z : (0, z) ∈ B} ; it is the height of the interface B at the origin. Let m (B) be the number of frustrated cubes having at least one plaquette in common with B. Let B be disconnected. Then it has at least three connected components, which are interfaces themselves. Let B1 , B2 , B3 be the first three of them. Clearly, Pr (B is disconnected) = Pr (B1 , B2 , B3 : l (B1 ) = l1 , l (B2 ) = l2 , l (B3 ) = l3 ) . l1
Applying Proposition 13 we have Pr (B1 , B2 , B3 : l (B1 ) = l1 , l (B2 ) = l2 , l (B3 ) = l3 ) ≤ a m(B1 )+m(B2 )+m(B3 )−N . 2
Note that for any B the number m (B) ≥ N 2 , and the number of interfaces B with m (B) = m and with l (B) fixed is at most C m for some C. Therefore Pr (B has at least three components) 2 3 2 m 2 N 3 ≤ LN Ca 3 = const · L Ca 3 , m≥3N 2
2 −1 2 which goes to zero as N → ∞ once L N < b N with b < Ca 3 .
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In what follows we will treat only connected interfaces. We will now show that typically the interface does not have a wall which winds around the torus. The reason is that such walls contain so many plaquettes that they appear very seldom, as estimates from previous sections will show. We say that a wall γ is winding if the projection (γ ) contains a non-trivial loop of the torus. In that case γ contains at least N plaquettes, so β,q µ N ,L N ( there is a winding wall ) ≤ 2N 2 L N C l a l/2 , l≥N
−1 which goes to zero as N → ∞, once L N < b N with b < Ca 1/2 . Let M ∈ T N be a point in the 2D torus, and γ be a wall of some interface in the 3D box N ,L . Denote by γ˜ the projection (γ ). We will say that γ surrounds M, iff M ∈ γ˜ ∪ Int (γ˜ ). Evidently, the rigidity property of the interface that we want to prove, would follow from Proposition 27. β,q
µ N ,L (M is surrounded by some wall) ≤ c (q) ,
(39)
with c (q) → 0 as q → ∞. Remark. The long-range claim of our main Theorem 8 also follows from Proposition 27. Indeed, if the heights h M = h M
or one of them is infinite, then at least one of the points M , M
is surrounded by a wall. Proof. From the Peierls estimate we know that the probability of the presence of an interface wall γ satisfies β,q
µ N ,L (γ ) ≤ a w(γ ) . However, we need evidently the estimate on the probability of the larger event γ ∗ = ∪τ γ τ , where γ τ is the wall obtained from γ by a vertical shift along the vector (0, 0, τ ). Since there are about L values of τ for which γ τ ⊂ N ,L , the estimate we have thus far is β,q µ N ,L γ ∗ ≤ La w(γ ) , and since L is diverging with N , the above estimate seems to be not enough for our purposes. β,q Yet, we know more about our measure µ N ,L . Namely, we know also that if is a collection (γ1 , . . . , γk ) of walls belonging to the same interface, then β,q
µ N ,L () ≤ a w() , with w () = w (γi ). Therefore if E xt = (γ1, . . . , γk ) is a collection of exterior walls in some interface, and E xt∗ = γ1∗ , . . . , γk∗ is the event ∪τ1 ,...,τk γ1τ1 , . . . , γkτk of observing the collection γ1τ1 , . . . , γkτk to be exterior walls of an interface, then necessarily τ1 = . . . = τk , and so β,q µ N ,L E xt∗ ≤ La w() .
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This estimate is helpful to eliminate long walls, but it is useless for dealing with collections of walls of finite total length l, when L is large. Note however that such β,q a collection can surround only a finite total area ≤ l 2 . Since our measure µ N ,L is translation invariant, the probability to see at any given location can be estimated by l2 . In what follows we will make these heuristic arguments rigorous. N2 We start with the following simplified model, which contains all the essential features of our problem. Let T R be a R × R discrete torus, and let ξt = 0, 1 be a random field indexed by t ∈ T R . Let µ be the distribution of the field ξ. Lemma 28. Suppose that • µ is translation-invariant, • there exists a value K such that for all k ≥ K , µ ξ t1 = ξ t2 = . . . = ξ tk = 1 ≤ α k ,
(40)
for small enough α. Then there exists a function R (α), such that for any t ∈ T R ,
provided R ≥
%
µ (ξt = 1) ≤ 3α, K 3α
+ R (α).
Proof. Let us show first that for any k, ⎛ ⎞ ⎛ ⎞ k µ ⎝ξ0 = 1, ξt = k ⎠ = 2 µ ⎝ ξt = k ⎠ . R t∈T R
(41)
t∈T R
To see this let X ⊂ T R be any subset with |X | = k, and let X˜ be the event that the support Supp (ξ ) coincides with some shift X + t of X, t ∈ T R . Note that for any X ,
k
µ ξ0 = 1 X˜ = 2 . R This is immediate if the set X is not periodic, i.e. if all the shifts X +t are different subsets. In case X is periodic we have to consider the sublattice L X ⊂ T R of its periods
and its fun
∩P X | , damental parallelogram P X ⊂ T R . By the same reasoning µ ξ0 = 1 X˜ = |X|P X| ∩P X | while evidently |X|P = Rk2 . Finally, µ ξ0 = 1, t∈T R ξt = k = X˜ µ ξ0 = 1, X˜ = X| k ˜ = k2 µ µ X ξ = k . t ˜ 2 t∈ T X R R R We now prove our lemma. From (41) we know that for all M ≥ 1, ⎛ ⎞ ⎛ ⎞ M k M µ ⎝ξ0 = 1, 1 ≤ ξt ≤ M ⎠ = µ⎝ ξt = k ⎠ ≤ 2 . (42) 2 R R
t∈T R
k=1
t∈T R
In the region t∈T R ξt ≥ M we would like to use the “Peierls estimate” (40). Our choice will be M = b R 2 with some b ≥ 2α. Then we have ⎛ ⎞ 2 R ⎝ ⎠ αM. µ ξ0 = 1, ξt ≥ M ≤ M t∈T R
(43)
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Once R satisfies b R 2 > K , we can use both (42) and (43) to conclude that 2 R 2 αb R . µ (ξ0 = 1) ≤ b + 2 bR Finally, introducing c = 1 − b, we have by Stirling, that 2 R2 R 2R b R2 b R2 α ∼√ b R 2 c R 2 α b R2 c R2 2π bc R 2 b R 2 2 b R α 1 =√ . 2π bc R 2 bb cc
A straightforward check shows that for b = 3α the ratio as R → ∞, which concludes the proof.
αb bb cc
< 1, so
R2 b R2 →0 b R2 α
Returning to the proof of Proposition 27, we take two large numbers, R and Q, to be chosen later, and we consider the box N ,L with N = Q R. Let O be the origin, O ∈ T N . The probability that O is surrounded by a wall with the weight w ≥ Q satisfies β,q
µ N ,L (O is Q-surrounded) ≤ La Q
(44)
(modulo an unimportant constant). So once Q ln L , this probability is small. We are left with the event that O is surrounded by a wall with the weight w < Q. To estimate its probability we will use the above lemma. Let us consider the torus sublattice T R ⊂ T N . For every point t ∈ T R we define the random variable ξt by 1 if t is inside some γ˜ with w (γ ) < Q, ξt = 0 otherwise. Evidently, the field {ξt , t ∈ T R } is translation-invariant. Let us estimate the probability of the event ξ T = {ξt = 1 for all t ∈ T ⊂ T R } . As was explained above, our Peierls estimate gives β,q µ N ,L ξ T ≤ L a w(γ1 )+...+w(γ|T | ) , γ1 ,...,γ|T |
where the summation goes over all collections γ1 , . . . , γ|T | of exterior walls with base at the given level – say, L/2 – such that every γi surrounds precisely one point of T. Since always w (γ ) ≥ 4, β,q µ N ,L ξ T ≤ La 4|T | . Therefore the condition (40) holds with α = a 2 and K = % ln L ln a1 R≥ a
1 2
ln L ln a1 . Hence for (45)
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we have µ N ,L (ξ O = 1) ≤ 3a 2 . On the other hand, our bound La Q from (44) satisfies La Q < a 2 once 1 (46) (Q − 2) ln > ln L . a If we take Q = R 2 , then both inequalities (45) and (46) will be satisfied, provided ln L < N 2/3 . So under this condition we can conclude that (39) is satisfied with c (q) = 4a 2 . 7. Conclusions In this work we have developed a version of the Reflection Positivity method suitable for the investigation of the rigidity property of the interfaces between coexisting phases of certain 3D systems. It is applicable to various known models, such as the Ising, Potts or FK models. However, the main advantage of the method is that it works also for models with non-trivial structure of the ground states, which can not be treated by the PS theory, one example being the clock version of the “very non-linear σ -model”. We hope to be able to extend our methods to systems with continuous symmetry. Acknowledgement. In the course of this work we have benefitted by discussions with many colleagues, including M. Biskup, L. Chayes, A. van Enter, K. Khanin, S. Miracle-Sole, Ch. Pfister, S. Pirogov, to whom we express our gratitude. The generous support of GREFI MEFI is gratefully acknowledged. S.S. also acknowledges the support of the Grant 05-01-00449 of RFFR.
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Aizenman, M.: On the slow decay of o(2) correlations in the absence of topological excitations: remark on the Patrascioiu-Seiler model. J. Stat. Phys. 77, 351–359 (1994) Cerny, J., Kotecky, R.: Interfaces for random cluster models. J. Stat. Phys. 111, 73–106 (2003) Dobrushin, R.L.: Gibbs state, describing the coexistence of phases in the three-dimensional Ising model. Th. Prob. and Its Appl. 17, 582–600 (1972) Dobrushin, R.L., Shlosman, S.: Phases corresponding to the local minima of the energy. Selecta Math. Soviet. 1(4), 317–338 (1981) Dold, A.: Lectures on Algebraic Topology. Berlin-Heidelberg-New York: Springer, 1995 Fröhlich, J., Israel, R., Lieb, E., Simon, B.: Phase transitions and reflection positivity I. Commun. Math. Phys. 62, 1–34 (1978) Fröhlich, J., Lieb, E.H.: Phase transitions in anisotropic lattice spin systems. Commun. Math. Phys. 60(3), 233–267 (1978) Fröhlich, J., Pfister, C.-E.: Spin waves, vortices, and the structure of equilibrium states in the classical xy model. Commun. Math. Phys. 89, 303–327 (1983) Frohlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50, 79 (1976) Frohlich, J., Spencer, T.: The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas. Commun. Math. Phys. 81, 527–602 (1981) Grimmett, G.: Private communication Gielis, G., Grimmett, G.: Rigidity of the interface in percolation and Random-Cluster models. J. Stat. Phys. 109(1–2), 1–37 (2002) Holicky, P., Kotecky, R., Zahradnık, M.: Rigid interfaces for lattice models at low temperatures. J. Stat. Phys. 50, 755–812 (1988) Kenyon, R.: Dominos and the Gaussian free field. Ann. Prob. 29(3), 1128–1137 (2001) van Enter, A.C.D., Shlosman, S.: First-order transitions for n-vector models in two and more dimensions: rigorous proof. Phys. Rev. Lett. 89, 285702 (2002) van Enter, A.C.D., Shlosman, S.: Provable first-order transitions for nonlinear vector and gauge models with continuous symmetries. Commun. Math. Phys. 255(1), 21–32 (2005) Shlosman, S.: The method of reflection positivity in the mathematical theory of first-order phase transitions. Russ. Math. Surv. 41(3), 83–134 (1986)
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Shlosman, S., Vignaud, Y.: Rigidity of the interface between low-energy and high-entropy phases. In preparation Vignaud, Y.: Entropic repulsion and entropic attraction. In preparation Vignaud, Y.: Rigidity of the interface for a continuous symmetry model in a slab. In preparation
Communicated by M. Aizenman
Commun. Math. Phys. 276, 863–899 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0322-3
Communications in
Mathematical Physics
Free Meixner States Michael Anshelevich Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA. E-mail: [email protected] Received: 8 February 2007 / Accepted: 6 March 2007 Published online: 18 August 2007 – © Springer-Verlag 2007
Abstract: Free Meixner states are a class of functionals on non-commutative polynomials introduced in [Ans06]. They are characterized by a resolvent-type form for the generating function of their orthogonal polynomials, by a recursion relation for those polynomials, or by a second-order non-commutative differential equation satisfied by their free cumulant functional. In this paper, we construct an operator model for free Meixner states. By combinatorial methods, we also derive an operator model for their free cumulant functionals. This, in turn, allows us to construct a number of examples. Some of these examples are shown to be trivial, in the sense of being free products of functionals which depend on only a single variable, or rotations of such free products. On the other hand, the multinomial distribution is a free Meixner state and is not a product. Neither is a large class of tracial free Meixner states which are analogous to the simple quadratic exponential families in statistics.
1. Introduction The subject of this paper are states and orthogonal polynomials in non-commuting variables. The definition is straightforward. The usual orthogonal polynomials are obtained by starting with a measure µ on Rd , thinking of R[x1 , x2 , . . . , xd ] as a vector space with the (pre-)inner product P, Q =
Rd
P(x)Q(x) dµ(x),
and applying the Gram-Schmidt procedure to the monomials xu(1) xu(2) . . . xu(n) . In the non-commutative case, one starts directly with a positive linear functional (state) ϕ This work was supported in part by NSF grant DMS-0613195.
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on the algebra of non-commutative polynomials Rx1 , x2 , . . . , xd , and orthogonalizes the monomials in non-commuting variables with respect to the inner product P, Q = ϕ P ∗ (x)Q(x) . Among the general “non-commutative measures” and polynomials orthogonal with respect to them, there is a specific class of what is appropriate to call free Meixner states. The classical Meixner class [Mei34] consists of familiar distributions—normal, Poisson, gamma, negative binomial, Meixner, and binomial—which, somewhat less familiarly, share a number of common properties: their orthogonal polynomials have exponentialform generating functions [Mei34], they satisfy a quadratic regression property [LL60], they generate quadratic natural exponential families [Mor82], they are quadratic harnesses [Wes93], they are induced by representations of su(1, 1) [KVDJ98], and they have explicit linearization coefficient formulas [KZ01]. The multivariate Meixner distributions have also been investigated, frequently in the guise of quadratic exponential families [Cas96, Pom96], though a complete classification is still lacking. Even the ´ infinite-dimensional case was considered [Sni00, Lyt03]. In [Ans03], I introduced the free Meixner polynomials, which are a family of orthogonal polynomials in one variable. The term “free” refers to their relation to free probability, see [VDN92, NS06] for an introduction. As a matter of fact, these polynomials have been found independently both before and after my work, for example in [Sze22, CT84, Fre98, SY01, KKN06]. They share a number of the Meixner properties listed above, as long as these are properly translated into the “free” context, see my original paper and also [BB06]. Some of the corresponding distributions also appear in random matrix theory, as the limiting distributions in the Gaussian, Wishart, and Jacobi ensembles. In [Ans06] I started the investigation of multivariate free Meixner distributions, which are states on the algebra of non-commutative polynomials. I continue their study in Sect. 3. The main new tool is to represent these states as joint distributions of certain operators on a Fock space, following the more general construction in [Ans07]. I use this machinery, in combination with combinatorial methods, to find explicit formulas for the free cumulants of these states. This provides an explanation for the one-variable results in Sect. 3.1 of [Ans03] and Proposition 2.2 of [BB06], and is the first main result of the paper. The operator representation of the state also allows me to handle states that are not necessarily faithful, thus answering a question of the referee of [Ans06], where only faithful free Meixner states were considered. Having an explicit representation for the cumulants, and being able to handle non-faithful states, allows me to describe a number of examples, which is done in Sect. 4. Among the usual multivariate Meixner distributions, two are familiar, namely the multivariate normal and the multinomial distributions. It is well known that the free analog of the multivariate normal distribution is the distribution of a free semicircular system, see Sect. 4.2. The second question treated in this paper is: what is the “free” multinomial distribution? I show that the basic multinomial distribution itself also belongs to the free Meixner class. In particular, this allows me to calculate the distribution of a free sum of d-tuples of orthogonal projections. Among states on non-commutative algebras, traces form an important class. The final result in this paper provides a way to construct a large family of non-trivial, tracial free Meixner states. These turn out to be analogs of simple quadratic exponential families.
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2. Preliminaries Variables in this paper will typically come in d-tuples, which will be denoted using the bold font: x = (x1 , x2 , . . . , xd ), and the same for z, S, etc. 2.1. Polynomials. Let Rx = Rx1 , x2 , . . . , xd be all the polynomials with real coefficients in d non-commuting variables. Multi-indices are elements u ∈ {1, . . . , d}k for k ≥ 0; for |u| = 0 denote u by ∅. Monomials in non-commuting variables (x1 , . . . , xd ) are indexed by such multi-indices: x u = xu(1) . . . xu(k) . For V ⊂ N, we will call the elements of {1, . . . , d}V multi-indices indexed by V . Note that our use of the term “multi-index” is different from the usual one, which is better suited for indexing monomials in commuting variables. For two multi-indices u, v, denote by (u, v) their concatenation. For u with |u| = k, denote (u)op = (u(k), . . . , u(2), u(1)). Define an involution on Rx via the R-linear extension of (x u )∗ = x(u)op . A monic polynomial family in x is a family {Pu (x)} indexed by all multi-indices ∞
u ∈ {1, . . . , d}k
k=1
(with P∅ = 1 being understood) such that Pu (x) = x u + lower-order terms. Note that Pu∗ = P(u)op in general. Definition 1. A state on Rx is a functional ϕ : Rx1 , x2 , . . . , xd → R that is linear, compatible with the ∗-operation, that is for any P, ϕ [P] = ϕ P ∗ , unital, that is ϕ [1] = 1, and positive, that is for any P, ϕ P ∗ P ≥ 0. A state is faithful if in the preceding equation, the equality holds only for P = 0. Unless noted otherwise, the states in this paper are not assumed to be faithful. The numbers ϕ [x u ] are called the moments of ϕ.
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A state ϕ induces the pre-inner product P, Qϕ = ϕ P ∗ Q = Q, Pϕ and the seminorm P ϕ =
ϕ [P ∗ P].
Throughout the paper, we will typically drop ϕ from the notation, and denote the inner product and norm it induces simply by ·, ·, · . We may think of ϕ is a “joint distribution” of “random variables” (x1 , x2 , . . . , xd ). In the remainder of the paper, as we did in [Ans06], we will assume that under the state ϕ, the variables have zero mean and identity covariance, ϕ xi x j = δi j . ϕ [xi ] = 0, The last assumption is made primarily so that Eq. (14) has a clean form. In Sect. 3.1 we briefly describe how to modify the results if that assumption is dropped. 2.2. Monic orthogonal polynomials states. Definition 2. A state has a monic orthogonal polynomial system, or MOPS, if for any multi-index u, there is a monic polynomial Pu with leading term x u , such that these polynomials are orthogonal with respect to ϕ, that is, Pu , Pv = 0 for u = v. Note that the same abbreviation is used in [DES04] to denote a class of multivariate orthogonal polynomial systems (in commuting variables) which is different from ours. States that have MOPS were characterized in [Ans07]. We briefly summarize the results of that paper which we will use in the next section. 2.2.1. Fock space construction I. Let H = Cd , with the canonical orthonormal basis e1 , e2 , . . . , ed . Define the (algebraic) full Fock space of H to be Falg (H) =
∞
H⊗k .
k=0
Falg (H) can be identified with the vector space of non-commutative polynomials in e1 , e2 , . . . , ed . Following convention, we will denote the generating vector in H⊗0 = C by Ω instead of 1. For i = 1, 2, . . . , d, define ai+ and ai− to be the usual (left) free creation and annihilation operators, ai+ eu(1) ⊗ eu(2) ⊗ . . . ⊗ eu(k) = ei ⊗ eu(1) ⊗ eu(2) ⊗ . . . ⊗ eu(k) ,
ai− (e j ) = ei , e j Ω = δi j Ω, ai− eu(1) ⊗ eu(2) ⊗ . . . ⊗ eu(k) = ei , eu(1) eu(2) ⊗ . . . ⊗ eu(k) .
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For each k ≥ 2 let C (k) be an operator C (k) : H⊗k → H⊗k . We think of each C (k) as a d k × d k matrix. Assume that for each k, C (k) is diagonal and C (k) ≥ 0. It is convenient to also take C (1) = I ; this corresponds to the identity covariance. Similarly, for each i = 1, 2, . . . , d and each k ≥ 1, let Ti (k) be an operator Ti
(k)
: H⊗k → H⊗k .
(k)
Assume that Ti and C ( j) satisfy a commutation relation (see [Ans07]). We will de(k) note by Ti and C the operators acting as Ti and C (k) on each component. Finally, let a˜ i− = ai− C and Xi = ai+ + Ti + a˜ i− . With the appropriate choice of the inner product ·, ·C on the completion FC (H) of the quotient of Falg (H), all the operators ai+ , Ti , a˜ i− factor through to FC (H), and each Xi is a symmetric operator on it. Theorem 1. (Part of Theorem 2 of [Ans07]). Let ϕ be a state on Rx. The following are equivalent: a. The state ϕ has a monic orthogonal polynomial system. b. There is a family of polynomials {Pu } such that ϕ [Pu ] = 0 for all u = ∅ and they satisfy recursion relations xi = Pi + Bi,∅,∅ , xi Pu = P(i,u) +
d
Bi,w,u Pw + δi,u Cu ,
w=1
xi Pu = P(i,u) +
Bi,w,u Pw + δi,u(1) C u P(u(2),u(3),...,u(k)) ,
|w |=|u|
with C u ≥ 0 and, denoting s j = (s( j), . . . , s(k)), Bi,s,u
k
C s j = Bi,u,s
j=1
k
Cu j .
j=1
c. For some choice of the matrices C (k) and Ti space representation ϕC ,{Ti } as
(k)
as in Sect. 2.2.1, the state ϕ has a Fock
ϕ [P(x1 , x2 , . . . , xd )] = Ω, P(X1 , X2 , . . . , Xd )Ω. We will also need the following relation between the operators in part (c) and coefficients in part (b) of the theorem: Bi,w,u ew(1) ⊗ . . . ⊗ ew(k) (1) Ti (eu(1) ⊗ . . . ⊗ eu(k) ) = |w |=k
and C(eu(1) ⊗ . . . ⊗ eu(k) ) = C u eu(1) ⊗ . . . ⊗ eu(k) .
(2)
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M. Anshelevich
2.3. Fock space construction II. The following construction is a particular case of the construction in Sect. 2.2.1, but this time we provide full details. As before, let H = Cd , with the canonical basis e1 , e2 , . . . , ed , denote its (algebraic) full Fock space by Falg (H), and the generator of the zeroth component by Ω. Let C be an operator on H ⊗ H, which we identify with its d 2 × d 2 matrix in the standard basis. Assume that C is diagonal and (I ⊗ I ) + C ≥ 0,
(3)
where I will always denote the identity operator on H. On Falg (H), define a new inner product η, ζ C = η, K C ζ , where ·, · is the usual tensor inner product, and K C is the non-negative kernel K C = I ⊗(k−2) ⊗ (I ⊗2 + C) . . . I ⊗ (I ⊗2 + C) ⊗ I ⊗(k−3) (I ⊗2 + C) ⊗ I ⊗(k−2) on each H⊗k . Denote the completion of Falg (H) with respect to this inner product FC (H). If the inner product is degenerate, first factor out the subspace of vectors of length zero, and then complete. For i = 1, 2, . . . , d, let ai+ and ai− be the usual (left) free creation and annihilation operators as defined in Sect. 2.2.1. Let T1 , . . . , Td be operators on H which we identify with their d × d matrices. Assume that each Ti is symmetric and (Ti ⊗ I )C = C(Ti ⊗ I ). With a slight abuse of notation, we will denote Ti = Ti ⊗ I ⊗(k−1) on H⊗k and a˜ i = ai− (C ⊗ I ⊗(k−2) ) on H⊗k . Note that ai− Ω = Ti Ω = a˜ i Ω = 0 and a˜ i = 0 on H.
(4)
It follows from the general construction in Sect. 2.2.1 (or [Ans07]) that all the operators X i = ai+ + ai− + Ti + a˜ i factor through to FC (H). Definition 3. The Fock state ϕ = ϕC,{Ti } on Rx determined by such C and Ti is the state ϕ [P(x1 , x2 , . . . , xd )] = Ω, P(X 1 , X 2 , . . . , X d )Ω = Ω, P(X 1 , X 2 , . . . , X d )ΩC . Lemma 1. Each X i is symmetric with respect to the C-inner product and bounded with respect to the C-norm, hence self-adjoint.
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Proof. The symmetry of X i is proved exactly as in Proposition 1 of [Ans07], or can be deduced from it. Each of ai+ , ai− , Ti , a˜ i is bounded with respect to the norm · coming from the usual tensor inner product ·, · on Falg (H), therefore so is X i . In particular, it has an adjoint with respect to this inner product, which we denote by X i ,
η, X i ζ C = η, K C X i ζ = X i K C η, ζ = X i η, ζ C = X i η, K C ζ = K C X i η, ζ , so X i K C = K C X i .
(5)
2 (K C X i2 )(K C X i2 ) = K C X i2 (X i )2 K C ≤ X i 2 X i K C2 ≤ X i 4 K C2 .
(6)
Also,
But using (5), the operator K C X i2 = X i K C X i ≥ 0 is positive, so we can take the square root of Eq. (6) to get K C X i2 ≤ X i 2 K C . Finally, 2 X i η, X i ηC = η, X i2 η = η, K C X i2 η ≤ X i 2 η, K C η = X i 2 η C , C
so X i C ≤ X i . Therefore X i is bounded and symmetric, and so can be taken to be self-adjoint.
2.4. Non-crossing partitions. A partition π of a set V ⊂ N is a collection of disjoint subsets of V (classes of π ), π = (B1 , B2 , . . . , Bk ), whose union equals V . Most of the time we will be interested in partitions of {1, 2, . . . , n}. Partitions form a partially ordered set (in fact a lattice) under the operation of refinement, so that the largest partition is 1ˆ = {1, 2, . . . , n} and the smallest partition is 0ˆ = {1} , {2} , . . . , {n} . We will use π
i ∼ j to denote that i, j lie in the same class of π . Let NC(V ) denote the collection of non-crossing partitions of V , which are partitions π such that π
π
π
i ∼ i , j ∼ j , i ∼ j, i < j < i ⇒ i < j < i . Equivalently, a partition is non-crossing if and only if one of its classes is an interval and the restriction of the partition to the complement of this class is non-crossing. Noncrossing partitions are a sub-lattice of the lattice of all partitions. For each n, let NC(n) denote the lattice of non-crossing partitions of the set {1, 2, . . . , n}. We will also denote
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by NC 0 (V ) all non-crossing partitions with no singletons (one-element classes), and by NC (V ) all the partitions π such that π
min V ∼ max V. Equivalently, partitions in NC (V ) have a single outer class—the one that contains both min V and max V —in the terminology of [BLS96]. (A class B ∈ π is outer if there do π not exist i, i ∈ B, j ∈ B with i ∼ i and i < j < i .) See [NS06] or [Sta97] for more details on the relevant combinatorics. 2.5. Free cumulants. The free cumulant functional R corresponding to a state ϕ is the linear functional on Rx defined recursively by R [1] = 0 and for |u| = n, R [x u ] = ϕ [x u ] − R xu(i) , (7) π ∈NC(n), B∈π π =1ˆ
i∈B
which expresses R [x u ] in terms of the joint moments and sums of products of lowerorder free cumulants. From these, we can form the free cumulant generating function of ϕ via R(z 1 , z 2 , . . . , z d ) =
∞
R [x u ] z u ,
(8)
n=1 |u|=n
where z = (z 1 , . . . , z d ) are non-commuting indeterminates. One can also define R using an implicit functional relation involving the moment generating function of ϕ, see Corollary 16.16 of [NS06]. 2.6. Words and partitions. In this section, we collect a number of facts that will be useful in the proof of the next two theorems. Note that in many places, operators are considered as acting on Falg (H), with a degenerate inner product, rather than on FC (H). Lemma 2. Let u be a multi-index indexed by a set V ⊂ N, and W = i∈V W (i) be a − + ,T ˜ u(i) . If word with W (i) equal to au(i) u(i) , au(i) , or a Ω,
W (i)Ω = 0,
i∈V
then − + W (min V ) = au(min V ) , W (max V ) = au(max V ) , ∀i ∈ V, − or W ( j) = a ˜ j ∈ V | j ≥ i, W ( j) = au( u( j) j) + ≤ j ∈ V | j ≥ i, W ( j) = au( j) , − + j ∈ V |W ( j) = a = or W ( j) = a ˜ j ∈ V |W ( j) = au( u( j) u( j) , (9) j)
Free Meixner States
and
871
− or W ( j) = a ˜ j ∈ V | j ≥ i, W ( j) = au( u( j) j) + = j ∈ V | j ≥ i, W ( j) = au( j) − ⇒ W (i) = au(i) .
(10)
Proof. This follows from the fact that if η ∈ H⊗k , then ai+ (η) ∈ H⊗(k+1) , Ti (η) ∈ H⊗k , and ai− (η), a˜ i (η) ∈ H⊗(k−1) , and Eq. (4). In combinatorics, Eq. (9) is related to the notion of a Motzkin path. More generally, our operator representations are closely related to a common way of representing moments as sums over lattice paths [Fla80, Vie85], but in the multivariate case we find the operator formulation more useful. Notation 1. Let
Wn (u) = W = W (1)W (2) . . . W (n)
satisfying conditions (9) and (10) for V = {1, . . . , n} ,
and for a general subset V ⊂ N, define WV (u) similarly. Lemma 3. For a multi-index u, partition π ∈ NC 0 (n), π = (V1 , V2 , . . . , Vk ) and partitions σ j ∈ NC 0 (V j ), j = 1, 2, . . . , k, define a word W = βu (π ; σ1 , . . . , σk ) by ⎧ + , i ∈ B ∈ σ , i = max B, au(i) ⎪ j ⎪ ⎪ ⎨a − , i ∈ V , i = min V , j j u(i) W (i) = (11) ⎪ , i ∈ B ∈ σ , i = min B, i = min V j , a ˜ j u(i) ⎪ ⎪ ⎩ Tu(i) , otherwise. Then W ∈ Wn (u), and for each V ∈ π , W restricted to V is in WV (u : V ), where (u : V ) is the sub-multi-index of u indexed by the elements of V . Moreover, for each u, βu is a bijection onto Wn (u). Proof. Let W = βu (π ; σ1 , . . . , σk ). Condition (9) for the whole set {1, 2, . . . , n} (respectively, for V j ) follows from the definition of β and the fact that π, σ1 , . . . , σk (respectively, σ j ) are non-crossing. Condition (10) follows from the definition that the minima of the outer classes of the partition π (respectively, σ j ) are all a − . Conversely, let W ∈ Wn (u). Let Λ ⊂ {1, 2, . . . , n}, Λ = j|W ( j) = Tu( j) . It follows from Proposition 2.13 and Exercise 8.23 of [NS06] that, as long as W restricted to Λ satisfies condition (9), there is a unique non-crossing pair partition π ∈ NC(Λ) such that for any B ∈ π , − or a˜ u(i) , i = min B ⇔ W (i) = au(i) + i = max B ⇔ W (i ) = au(i
).
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M. Anshelevich
− Note that W (1) = au(1) , so (1, i ) ∈ π for some i > 1. Moreover,
W (i + 1) . . . W (n)Ω ∈ H⊗0 , −
so by condition (10), W (i + 1) = au(i
+1) . Thus (i + 1, j ) ∈ π for some j , etc. π
ending with (s, n) ∈ π . It follows that for any j ∈ {1, . . . , n}, there exist i ∼ i such π
− . For each j, choose the largest i such that i ∼ i , that i ≤ j ≤ i and W (i) = au(i) π
π
− i ≤ j ≤ i , and W (i) = au(i) , and require that i ∼ j ∼ i . Similarly, for each class π
Vs ∈ π and each j ∈ Vs , choose the largest i ∈ Vs such that i ∼ i and i ≤ j ≤ i , and σs σs require that i ∼ j ∼ i . Pictorially, we draw the integers 1, 2, . . . , n on a line, draw the
pair classes or π as arcs connecting each i with the corresponding i above the line, and then connect each of the other elements to the arc immediately above it. Lemma 4. For any W ∈ Wn (u), let (π ; σ1 , . . . , σk ) = βu−1 (W ) with π = (V1 , V2 , . . . , Vk ). Then Ω, W (1)W (2) . . . W (n)Ω =
k
Ω,
i=1
W ( j)Ω .
j∈Vi
Proof. Since π is a non-crossing partition, it has a class V that is an interval, V = [i, i ] = j|i ≤ j ≤ i . Since π restricted to {1, . . . , n} \V is still a non-crossing partition, it suffices to show that i−1 n i Ω, W (1)W (2) . . . W (n)Ω = Ω, W ( j) W ( j)Ω Ω, W ( j)Ω . j=1
j=i +1
Denote η = W (i + 1) . . . W (n)Ω ∈ H⊗m . We now show that for any i < j ≤ i , W ( j) . . . W (n)Ω = ζ j ⊗ η for ζ j = W ( j) . . . W (i )Ω. The proof is by induction. +
W (i )η = au(i
) η = eu(i ) ⊗ η = (W (i )Ω) ⊗ η.
j=i
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873
+ ,ζ =e If W ( j) = au( u( j) ⊗ ζ j+1 . If W ( j) = Tu( j) , then ζ j = Tu( j) ζ j+1 . W ( j) cannot j) j − equal au( j) . Finally, it follows from condition (10) applied to V = [i, i ] that for all j, i < j ≤ i ,
W ( j + 1) . . . W (n)Ω ∈ H⊗s with s > m. Thus W ( j) may equal a˜ u( j) only if s ≥ m + 2, otherwise W ( j)W ( j + 1) . . . W (n)Ω ∈ H⊗m . − But if s ≥ m + 2, ζ j = au( j) Cζ j+1 . It follows that also
W (i) . . . W (n)Ω = (W (i) . . . W (i )Ω) ⊗ η = Ω, W (i) . . . W (i )Ω η. Thus
Ω, W (1)W (2) . . . W (n)Ω = Ω, W (1) . . . W (i − 1) Ω, W (i) . . . W (i )Ω η
= Ω, W (1) . . . W (i − 1)η Ω, W (i) . . . W (i )Ω i−1 n i W ( j) W ( j)Ω Ω, W ( j)Ω . = Ω, j=1
j=i +1
j=i
Notation 2. For V ⊂ N, denote − + WV (u) = {W ∈ WV (u)|W (min V ) = au(min V ) , W (max V ) = au(max V ) ,
− and none of the other W (i) are equal to au(i) }.
The partition π corresponding to any such W has only one class, π = (V ) ∈ NC 0 (V ), and βu−1 (WV (u)) = ((V ), σ )|σ ∈ NC 0 (V ) ∼ = NC 0 (V ). Denote Θ(σ ; V, u) = Ω, βu ((V ), σ )Ω . Lemma 5. If W ∈ Wn (u) and βu−1 (W ) = (π, σ1 , . . . , σk ), π = (V1 , V2 , . . . , Vk ), then Ω, W (1) . . . W (n)Ω =
k j=1
Ω,
i∈V j
W (i)Ω =
k
Θ(σ j ; V j , (u : V j )),
j=1
where (u : V j ) is the sub-multi-index of u indexed by the elements of V j . Proof. This follows from Lemma 4 using Notation 2.
874
M. Anshelevich
− Lemma 6. Suppose that W ∈ Wn (u) such that W (1) = au(1) = a− j and W (2) = −1 a˜ u(2) = a˜ i . Then βu (W ) = (({1, . . . , n}), σ ). It follows from condition (11) that for + 2 ∈ B ∈ σ , we have 2 = min B. Let k = max B. Then W (k) = au(k) and
Ω, W (1)W (2) . . . W (k) . . . W (n)Ω + + = Ω, a − a ˜ W (3) . . . a W (k + 1) . . . W (n − 1)a Ω i u(k) u(n) j − + + = Ci j Ω, a j W (k + 1) . . . W (n − 1)au(n) Ω Ω, ai− W (3) . . . au(k) Ω , where Ci j is the ((i, j), (i, j)) entry of the matrix C. Moreover, the map − = a− ˜ u(2) = a˜ i } {W ∈ Wn (u)|W (1) = au(1) j , W (2) = a
→ ∼ =
n−1
(u : {1, k + 1, . . . , n}) × W{2,...,k} (u : {2, . . . , k}) W{1,k+1,...,n}
k=3 n−1
NC 0 ({1, k + 1, . . . , n}) × NC 0 ({2, . . . , k})
k=3
is a bijection. Proof. By the same method as in Lemma 4, we deduce that + + W (3) . . . au(k) W (k + 1) . . . W (n − 1)au(n) Ω + + Ω . = W (3) . . . au(k) Ω ⊗ W (k + 1) . . . W (n − 1)au(n)
Thus
+ + Ω, a − a ˜ W (3) . . . a W (k + 1) . . . W (n − 1)a Ω u(k) u(n) j i + + = ei ⊗ e j , C W (3) . . . au(k) W (k + 1) . . . W (n − 1)au(n) Ω + + = C(ei ⊗ e j ), W (3) . . . au(k) W (k + 1) . . . W (n − 1)au(n) Ω − + + = Ci j Ω, a − j W (k + 1) . . . W (n − 1)au(n) Ω Ω, ai W (3) . . . au(k) Ω .
Lemma 7. Suppose that
Ci j = ei ⊗ e j , C(ei ⊗ e j ) = c
for all i, j. Let σ ∈ N C0 (n), σ = b1,1 , . . . , b1, j (1) , . . . , bk,1 , . . . , bk, j (k) , where each class is ordered and b1,1 = 1. Then Θ(σ ; {1, . . . , n} , u) = ck−1
k
eu(bi,1 ) , Tu(bi,2 ) . . . Tu(bi, j (i)−1 ) eu(bi, j (i) ) .
i=1 − Proof. This follows from the definition of β, noting that W (b1,1 ) = W (1) = au(1) , − + W (bi,1 ) = a˜ u(bi,1 ) = cau(bi,1 ) for i = 1, W (bi, j (i) ) = au(bi, j (i) ) , and the rest of the terms are Tu(bi,l ) .
Free Meixner States
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3. Main Theorems Theorem 2. For each i, let Si = ai+ + Ti + a˜ i = X i − ai− be an operator on Falg (H). Then the free cumulants of the Fock state ϕC,{Ti } from Definition 3 are given by the formula R [xi ] = 0, R xi P(x)x j = ei , P(S)e j = ei , P(S)e j C . + +T ˜ u(i) , and using Notation 2, for |u| = n, Proof. Since Su(i) = au(i) u(i) + a
− + eu(1) , Su(2) . . . Su(n−1) eu(n) = Ω, au(1) Su(2) . . . Su(n−1) au(n) Ω Ω, W Ω =
W ∈Wn (u)
=
Ω, βu−1 ({1, . . . , n}, σ )Ω
σ ∈NC 0 (n)
=
Θ(σ ; {1, . . . , n}, u).
σ ∈NC 0 (n) − + +T ˜ u(i) , using Lemmas 2, 3, 5 and the preceding Similarly, since X u(i) = au(i) u(i) +au(i) + a equation,
Ω, X u(1) X u(2) . . . X u(n) Ω Ω, W (1)W (2) . . . W (n)Ω = W ∈Wn (u)
=
n k=1
=
k
i=1 π ∈NC 0 (n) π =(V1 ,V2 ,...,Vk )
n k=1
k
Θ(σi ; Vi , (u : Vi ))
π ∈NC 0 (n) σ j ∈NC 0 (V j ) i=1 π =(V1 ,V2 ,...,Vk ) j=1,...,k
n k=1
=
k
⎛ ⎝
⎞ Θ(σi ; Vi , (u : Vi ))⎠
σi ∈NC 0 (Vi )
e(u:Vi )(min Vi ) ,
i=1 π ∈NC 0 (n) π =(V1 ,V2 ,...,Vk )
S(u:Vi )( j) e(u:Vi )(max Vi ) .
j∈Vi j =min Vi j =max Vi
Thus ϕ [x u ] =
π ∈NC 0 (n) V ∈π
e(u:V )(min V ) ,
j∈V j =min V, j =max V
S(u:V )( j) e(u:V )(max V ) .
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M. Anshelevich
Since R [xi ] = ϕ [xi ] = 0, the conclusion of the theorem now follows from the defining relation for the free cumulants, namely ϕ [x u ] = R xu(i) . (12) π ∈NC(n) V ∈π
i∈V
Notation 3. Let z = (z 1 , . . . , z d ) be non-commuting indeterminates, which commute with x. For a non-commutative power series G in z and i = 1, . . . , d, define the left non-commutative partial derivative Di G by a linear extension of Di (1) = 0, Di z u = δiu(1) z u(2) . . . z u(n) . Denote by DG = (D1 G, . . . , Dd G) the left non-commutative gradient. For a non-commutative power series G, denote by G −1 its inverse with respect to multiplication. For a d-tuple of non-commutative power series G = (G 1 , . . . , G d ), denote by G−1 its inverse with respect to composition (which is also a d-tuple). Theorem 3. Let ϕ be a state on Rx with a monic orthogonal polynomial system (MOPS), zero means and identity covariance. The following are equivalent. a. There exists a non-commutative power series F(z) = 1 + (terms of degree ≥ 2) and a d-tuple of non-commutative power series U, Ui (z) = z i + higher-order terms, such that the polynomials defined via their generating function −1 Pu (x)z u = F(z) 1 − x · U(z) |u|≥0
are a MOPS for ϕ. b. The polynomials with the generating function −1 Pu (x)z u = 1 − x · (DR)−1 (z) + R (DR)−1 (z)
(13)
|u|≥0
are a MOPS for ϕ, where R is the free cumulant generating function (8) of ϕ. c. The free cumulant generating function of ϕ satisfies, for each i, j, a (non-commutative) second-order partial differential equation Di D j R(z) = δi j +
d
Bikj Dk R(z) + Ci j Di R(z)D j R(z),
(14)
k=1 j
where Ci j ≥ −1, Bikj = Bik , and for each j, k, either Bikj = 0 for all i, or C ju = Cku for all u.
Free Meixner States
877
d. There exists a family of polynomials {Pu } such that ϕ [Pu ] = 0 for all u = ∅ and they satisfy recursion relations xi = Pi , xi P j = P(i, j) +
d
Bikj Pk + δi j ,
k=1
xi P( j,u) = P(i, j,u) +
d
Bikj P(k,u) + δi j (1 + Ci,u(1) )Pu ,
k=1
where Ci j , Bikj satisfy the same conditions as in part (c). e. There exist symmetric matrices Ti and a diagonal non-negative matrix C with (Ti ⊗ I )C = C(Ti ⊗ I ) such that ϕ has a representation ϕC,{Ti } as a Fock state of Definition 3. We call such states free Meixner states. Proof. The equivalence (a) ⇔ (b) follows from Lemma 4 of [Ans06] and Theorem 3.21 of [Ans04], neither of which relied on the assumption that ϕ is faithful. The equivalence (d) ⇔ (e) follows from the equivalence between the more general Fock space construction and the more general recursion relations in Theorem 1. (e) ⇒ (c). By Theorem 2,
R(z) =
d $
e j , el z j zl +
% e j , Su el z j z u zl .
|u|≥1
j,l=1
Therefore
D j R(z) =
d $
e j , el zl +
%
e j , Su el z u zl
|u|≥1
l=1
and % d $
Di D j R(z) = e j , ei + e j , Si el zl + e j , Si Su el z u zl |u|≥1
l=1
= e j , ei +
d $
e j , Ti el zl +
l=1
% e j , (Ti + a˜ i )Su el z u zl .
|u|≥1
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M. Anshelevich
This can be decomposed as
% d $
e j , Ti el zl + = e j , ei + e j , Ti Su el z u zl |u|≥1
l=1
+
d
e j , a˜ i Su el z u zl
l=1 |u|≥1
=
d $ d l=1
d
e j , Ti ek ek , el zl +
%
e j , Ti ek ek , Su el z u zl
|u|≥1 k=1
k=1
d
+ e j , ei + e j , a˜ i Su el z u zl , l=1 |u|≥1
where in the last step we have used the fact that {ek } form an orthonormal basis. Using Lemma 6, for n ≥ 4 and u a multi-index on {3, . . . , n − 1},
e j , a˜ i Su el = Ω, a − ˜ i Su el j a = Ci j
n−1
k=3 W ∈W 1 {1,k+1,...,n} j,( u:{k+1,...,n−1}),l
W2 ∈W{2,...,k} i,(u:{3,...,k})
= Ci j
n−1
Ω, W1 Ω Ω, W2 Ω
e j , W1 el ei , W2 es ,
k=3 W1 ∈W{1,k+1,...,n} ( j,w ,l) W2 ∈W {2,...,k} (i,v )
where v = (u : {3, . . . , k − 1}), s = u(k), and w = (u : {k + 1, . . . , n}). Thus
Di D j R(z) = e j , ei ⎛ ⎞ d d d
⎝ e j , Ti ek ek , el zl + e j , Ti ek ek , Su el z u zl ⎠ + l=1
+ Ci j
|u|≥1 k=1
k=1 d
l=1 u=(v ,s,w ) W1 ∈W{1,k+1,...,n} ( j,w ,l)
W2 ∈W{2,...,k} (i,v )
e j , W1 el ei , W2 es z v z s z w zl ,
Free Meixner States
879
which is finally simplified to d
e j , Ti ek Dk R(z) = e j , ei + k=1
+
d
l=1
v ,s,w |v |,|w |≥0
Ci j ei , Sv es e j , Sw el z v z s z w zl
d
= e j , ei + e j , Ti ek Dk R(z) + Ci j Di R(z)D j R(z). k=1
The conditions on the coefficients in part (c) are equivalent to the conditions on the matrices in part (e). (c) ⇒ (e). Since the states are assumed to have zero means, the corresponding free cumulant generating functions have no linear terms. In that case, a free cumulant generating function R, and so the corresponding state ϕ, are completely determined by Eq. (14). k Moreover, for any choice of Ci j , Bi j subject to the conditions of part (c), if
Ti (e j ) =
d
Bikj ek
k=1
and C(ei ⊗ e j ) = Ci j ei ⊗ e j , then those equations are satisfied by RϕC,{T } . So the states whose free cumulant generai ting functions satisfy the equations in part (c) are exactly the states in part (e). (k) (b) ⇒ (c). ϕ has a MOPS, so by Theorem 1, ϕ = ϕC ,{Ti } for some C (k) , Ti . Thus, ϕ is the joint distribution of the operators (X1 , . . . , Xd ) on the Hilbert space FC (H), with Xi = ai+ + Ti + ai− C. Note that since ϕ has means zero and identity covariance, Ti notation from Sect. 2.2.1, and denoting
(0)
(D R)u (z) = Du(1) R(z) . . . Du(|u|) R(z) and eu = eu(1) ⊗ . . . ⊗ eu(|u|) ,
= 0 and C (1) = I . Using
880
M. Anshelevich
we see that (D R)u (z)eu 1 − X · z + R(z) Ω + u
=Ω−
d
z i ei + R(z)Ω +
u
i=1
−
(D R)u (z)eu + R(z) (D R)u (z)eu
d
u
z i (D R)u (z)e(i,u)
i=1 u
−
d
z i Di R(z)Ω +
d
z i (D R)u (z)Ti (eu ) −
d
u
i
u
i=1
−
z i (Di R)(z)(D R)u (z)eu
z i (D R)u (z)ai− (C − I )eu .
u
i
Since for any function G with zero constant term, in particular for R, d
z i Di G(z) = G(z),
(15)
i=1
the preceding expression equals
=Ω−
d
z i ei +
u
i=1
−
d
d (D R)u (z)eu − z i (D R)u (z)e(i,u) i=1 u
z i (D R)u (z)Ti (eu ) −
u
i
d i
z i (D R)u (z)ai− (C − I )eu .
u
Using the expansions (1) and (2) from Theorem 1, we now continue the equation as
=Ω−
d
z i ei +
u
i=1
−
d
i=1 u
z i Bi,k, j Dk R(z)e j + Bi,(k,u),( j,w) Dk R(z)(D R)u (z)e( j,w)
d
u,w
i, j,k=1
−
d (D R)u (z)eu − z i (D R)u (z)e(i,u)
z i (C(i, j) − 1)Di R(z)D j R(z)e j
i, j=1
+
(C(i, j,u) − 1)Di R(z)D j R(z)(D R)u (z)e( j,u) u
Free Meixner States
881
or equivalently =Ω+
d &
D j R(z)
j=1
−
d d ' z i δi j + Bi,k, j Dk R(z) + (C(i, j) − 1)Di R(z)D j R(z) e j i=1
+
d &
k=1
D j R(z)(D R)u (z)
j=1 u
−
d d z i δi j (D R)u (z) + Bi,(k,u),( j,w) Dk R(z)(D R)w (z) k=1 w
i=1
+ (C(i, j,u) − 1)Di R(z)D j R(z)(D R)u (z)
' e( j,u) .
Using Eq. (15) again, this equals d
=Ω+
& z i Di D j R(z)
i, j=1 d ' − δi j + Bi,k, j Dk R(z) + (C(i, j) − 1)Di R(z)D j R(z) e j k=1
+
d
& z i Di D j R(z)(D R)u (z)
(16)
i, j=1 u d − δi j (D R)u (z) + Bi,(k,u),( j,w) Dk R(z)(D R)w (z) k=1 w
+ (C(i, j,u) − 1)Di R(z)D j R(z)(D R)u (z)
'
e( j,u) .
If the polynomials {Pu } with the generating function (13) from part (b) are orthogonal, then −1 Pu (x)(D R)u (z) = 1 − x · z + R(z) , |u|≥0
and
so that
Pu (X)Ω = eu ,
(17)
1 − X · z + R(z) Ω + (D R)u (z)eu = Ω.
(18)
u
Equating to zero the coefficient of e j starting with z i in Eq. (16), we get exactly Eq. (14) from part (c), with Bikj = Bi,k, j and Ci j = C(i, j) − 1. The conditions on the coefficients follow from the general conditions in Theorem 1.
882
M. Anshelevich
(e) ⇒ (b). If ϕ = ϕC,{Ti } , it follows that in Eq. (16), Bi,(k,u),( j,w) = Bikj δu,w and Ci, j,u = 1 + Ci j . Then that expression equals =Ω+
d i, j=1
&
d ' z i Di D j R(z) − δi j + Bikj Dk R(z) + Ci j Di R(z)D j R(z) k=1
' & × ej + (D R)u (z)e( j,u) = Ω, u
since part (e) ⇒ (c). So Eq. (18) holds. As a result, for polynomials with the generating function (13), 1+ Pu (X)z u Ω = Ω + z u eu . |u|≥0
|u|≥0
Thus Eq. (17) holds. Since the vectors {eu } are orthogonal, so are the polynomials {Pu }. 3.1. Nontrivial covariance and other extensions. In this section we consider a number of constructions and examples that involve free Meixner states with non-trivial covariances. We still assume that they have zero means (for simplicity); if desired, the means p1 , . . . , pd can easily be incorporated into the operator model by considering the operators (X 1 + p1 , . . . , X d + pd ) instead, and the corresponding combinatorics will involve all non-crossing partitions NC(n) rather than the non-crossing partitions without singletons NC 0 (n). On the other hand, Theorem 2 of [Ans07] requires that for any state with MOPS, the covariance matrices have to be diagonal. But now we allow & ' ψ xi2 = ti2 . (19) Note that degenerate variances ψ xi2 = 0 are still not permitted. 3.1.1. Dilations. Let ϕ be a Meixner state, and fix positive numbers (t1 , t2 , . . . , td ). Let ψ be the state defined by the R-linear extension of ψ [P(x1 , . . . , xd )] = ϕ [P(t1 x1 , . . . , td xd )] . Note that Eq. (19) holds. It is easy to see that if {Pu } is a MOPS for ϕ, then Q u (x1 , . . . , xd ) = tu Pu (x1 /t1 , . . . , xd /td ) is a MOPS for ψ. We now briefly state how the results of Theorem 3 get modified for ψ. The generating function for the MOPS still has the same “resolvent” form, and any state with MOPS and such a generating function arises as a dilation of a free Meixner state, Rψ [P(x1 , . . . , xd )] = Rϕ [P(t1 x1 , . . . , td xd )] ,
Free Meixner States
883
which shows how to modify the differential equation satisfied by the free cumulant generating function. Similarly, the MOPS satisfy the recursion relations xi Q ( j,u) = Q (i, j,u) +
d ti t j k=1
Bikj Q (k,u) + δi j ti2 (1 + Ci,u(1) )Q u ,
tk
where Bikj , Ci j were the corresponding coefficients for ϕ. Finally, suppose that ϕ = ϕC,{Ti } , represented as the joint distribution of (X 1 , X 2 , . . . , X d ). In the Hilbert space H = Cd with an orthonormal basis {ei }, let f i = ti ei . On the Fock space FC (H), let ae+i := ai+ , aei := ai− , Tei := Ti , and extend these definitions C-linearly to a +f , a −f , T f for any f ∈ H. Let X fi = a +fi + T fi + a −fi + a −fi C = ti X i .
(20)
Then ψ is the joint distribution of (X f1 , X f2 , . . . , X fd ), ψ [P(x)] = Ω, P(Xf )Ω. 3.1.2. Free convolution semigroups. Let ϕ be a free Meixner state. For t > 0, define a linear functional ϕ t via its free cumulant functional using relation (12): Rϕ t [P(x)] = t Rϕ [P(x)] and ϕ
t
[x u ] =
t
π ∈NC(n)
|π |
Rϕ
V ∈π
xu(i) .
i∈V
Note that ϕ t xi2 = t. The notation reflects the fact that ϕ s ϕ t = ϕ (s+t) , where is the operation of (additive) free convolution; we will not use this property in the paper. Using the methods of Theorem 2, it is easy to see that ϕ t is a state (and so positive) if and only if t + min Ci j ≥ 0, i, j
in other words if t (I ⊗ I ) + C ≥ 0. In particular, by assumption (3), ϕ t is always a state for t ≥ 1; this is typical behavior for free convolution, as indicated by Corollary 14.13 in [NS06]. ϕ t is a state for all t > 0 if and only if C ≥ 0; in this case we say that ϕ is freely infinitely divisible. Again, ϕ t has a MOPS and the generating function for the MOPS still has the same “resolvent” form, Di D j Rϕ t = δi j t +
d k=1
Bikj Dk Rϕ t + (Ci j /t)Di Rϕ t D j Rϕ t ,
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M. Anshelevich
and the MOPS for ϕ t satisfy xi P( j,u) = P(i, j,u) +
d
Bikj P(k,u) + δi j (t + Ci,u(1) )Pu ,
k=1
where Bikj , Ci j were the corresponding coefficients for ϕ. Finally, suppose that ϕ = ϕC,{Ti } . On the algebraic Fock space Falg (H), define an inner product using the kernel (t) K C = I ⊗(k−2) ⊗ (t I ⊗2 + C) . . . . . . I ⊗ (t I ⊗2 + C) ⊗ I ⊗(k−3) (t I ⊗2 + C) ⊗ I ⊗(k−2) t = t k K C/t on H⊗k , and denote the completion of Falg (H) with respect to this inner product FC(t) (H). Let (t)
X i = ai+ + Ti + tai− + a˜ i = ai+ + Ti + tai− (I + C/t). (21) (t) (t) Then ϕ t is the joint distribution of X 1 , . . . , X d . As constructed above, X i(t) are represented on different Hilbert spaces for different ´ t. We can combine this construction with an idea from Sect. 7.2 of [Sni00] to represent a whole family of functionals ϕ t |0 < t < 1 on a single space. We will assume that C ≥ 0. A subset Λ ⊂ {1, 2, . . . , n − 1} can be identified with an interval partition π(Λ) ∈ Int(n): if Λ = {i(1), i(2), . . . , i(k)}, then π(Λ) = {1, . . . , i(1)}, {i(1) + 1, . . . , i(2)} , . . . , {i(k) + 1, . . . , n} . Consider the vector space H = H ⊗ L ∞ ([0, 1], d x) as a subspace of the Hilbert space H ⊗ L 2 ([0, 1], d x), with the inner product 1 η ⊗ f, ζ ⊗ g = η, ζ f (x)g(x) d x. 0
On its algebraic Fock space Falg (H ), define the inner product (η1 ⊗ f 1 ) ⊗ . . . ⊗ (ηl ⊗ fl ), (ζ1 ⊗ g1 ) ⊗ . . . ⊗ (ζn ⊗ gn )C c η1 ⊗ . . . ⊗ ηn , C Λ (ζ1 ⊗ . . . ⊗ ζn ) = δln Λ⊂{1,...,n−1} π(Λ)=(V1 ,V2 ,...,Vk )
×
k j=1
⎛ ⎝
R
⎡ ⎣
⎤
f i (x)gi (x)⎦ d x ⎠,
i∈V j
where Λc is the complement {1, . . . , n − 1} \Λ, and c I ⊗(i−1) ⊗ C ⊗ I ⊗(n−i−1) . CΛ = i∈Λc
⎞
Free Meixner States
885
Complete Falg (H ) with respect to this inner product to get the Hilbert space FC (H ). On this space, define operators (t) ai+ (η1 ⊗ f 1 ) ⊗ . . . ⊗ (ηn ⊗ f n ) = (ei ⊗ 1[0,t) ) ⊗ (η1 ⊗ f 1 ) ⊗ . . . ⊗ (ηn ⊗ f n ), ⊗ fn ) f 1 (x) d x (η2 ⊗ f 2 ) ⊗ . . . ⊗ (ηn ⊗ f n ), = ei , η1
(t) ai− (η1 ⊗ f 1 ) ⊗ . . . ⊗ (ηn t 0 (t) Ti (η1 ⊗ f 1 ) ⊗ . . . ⊗ (ηn
⊗ fn )
= (Ti η1 ⊗ f 1 1[0,t) ) ⊗ (η2 ⊗ f 2 ) ⊗ . . . ⊗ (ηn ⊗ f n ), (t) a˜ i (η1 ⊗ f 1 ) ⊗ . . . ⊗ (ηn ⊗ f n ) = (ai− C(η1 ⊗ η2 )) ⊗ ( f 1 1[0,t) f 2 ) ⊗ (η3 ⊗ f 3 ) ⊗ . . . ⊗ (ηn
⊗ f n ),
where 1[0,t) is the indicator function of the interval [0, t), and let (t)
X i = ai+
(t)
(t)
+ Ti
+ ai−
(t)
(t)
+ a˜ i .
´ By combining Lemma 1 with (a slight modification of) Theorem 6 from [Sni00], it (t) follows that each X i is self-adjoint on FC (H ). Note that if all f i = gi = 1[0,t) , then (η1 ⊗ f 1 ) ⊗ . . . ⊗ (ηn ⊗ f n ), (ζ1 ⊗ g1 ) ⊗ . . . ⊗ (ζn ⊗ gn )C c η1 ⊗ . . . ⊗ ηn , C Λ (ζ1 ⊗ . . . ⊗ ζn ) t k = =
Λ⊂{1,...,n−1} π(Λ)=(V1 ,V2 ,...,Vk ) t n η1 ⊗ . . . ⊗ ηn , ζ1
⊗ . . . ⊗ ζn C/t .
(t)
Moreover, each X i restricted to
(t) FC (H ⊗ Span 1[0,t) ) ∼ = FC (H) (t) (t) is given by Eq. (21), and so ϕ t is the joint distribution of X 1 , . . . , X d . 3.1.3. Rotations. Let O = (Oi j ) be an orthogonal d × d matrix. Let , d d OT x = Oi1 xi , . . . , Oid xi i=1
and
i=1
' & ϕ O [P(x)] = ϕ P(O T x) .
(22)
We call ϕ O a rotation of ϕ. ϕ O is the joint distribution of (X f1 , . . . , X fd ) from (20), where we take f j = O(e j ) =
d i=1
Oi j ei .
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M. Anshelevich
ϕ O need not have a MOPS, since the matrix C need not be diagonal in the basis { f 1 , . . . , f d }. In fact, it follows from Lemma 9 of [Ans06] that ϕ O has a MOPS for all O if and only if Ci j = c for all i, j, and that in this case ϕ O is also a free Meixner state. It is easy to see that more generally, if Λ ⊂ {1, . . . , d} and Ci j = c for all i, j ∈ Λ, then ϕ O is a free Meixner state whenever O(ek ) = ek for all k ∈ Λ. 3.1.4. Linear transformations. Finally, one can consider a general invertible change of variables , d d AT x = Ai1 xi , . . . , Aid xi i=1
i=1
and the corresponding state ϕ A defined as in Eq. (22). ϕ A is the joint distribution of .d operators from (20), where we take f j = A(e j ) = i=1 Ai j ei . As an alternative to our definition, one can call free Meixner states all states obtained by a linear transformation of a free Meixner state with MOPS (compare with [Pom96]). 4. Examples 4.1. Free products. In preparation for the examples in this section, for the reader’s convenience we explain a key notion from free probability. Again, see [VDN92, NS06] for more details and other approaches. Let ϕ1 , . . . , ϕd be one-dimensional states on R[x1 ], . . . , R[xd ], respectively. There is a canonical way to define their free product state ϕ on Rx1 , . . . , xd . Combinatorially, a natural way to define ϕ is via its MOPS. Let Pn(i) be the MOPS for ϕi . For a multi-index u, decompose i(1) i(2)
i(k)
x u = xv(1) xv(2) . . . xv(k) , where the consecutive indices v( j) = v( j + 1), although non-consecutive indices may coincide. Then the MOPS {Pu } for ϕ are defined by Pu (x) =
k
(v( j))
Pi( j) (xv( j) ).
j=1
For example, (1)
(2)
(1)
(2)
P1,1,2,1,2 (x) = P2 (x1 )P1 (x2 )P1 (x1 )P1 (x2 ). Note that if one considers polynomials in commuting variables and assumes that all v( j) above are different, one gets the usual (Cartesian) product of measures. Also, ϕ is a free product state if and only if the elements x1 , x2 , . . . , xd are freely independent with respect to ϕ, in the sense of Voiculescu. This can be taken as the definition of free independence; note that for random variables independent in the usual probabilistic sense, their joint distribution is a product measure. Finally, the crucial property of free cumulant generating functions is their relation to free products: a state ϕ is a free product
Free Meixner States
887
state of ϕ1 , . . . , ϕd if and only if the free cumulant generating function of ϕ decomposes as Rϕ (z) =
d
Rϕi (z i ).
i=1
This is often stated as the “mixed free cumulants are zero” condition. It is related to the familiar property that the Fourier transform of the joint distribution of independent random variables is the product of their individual Fourier transforms. It is easy to see that free product free Meixner states are exactly the free products of one-dimensional free Meixner states, see Remark 6 of [Ans06]. Recall that these one-dimensional distributions, as described in that remark, Theorem 4 of [Ans03] and Sect. 2.2 of [BB06], are √ known. With variance t, they are: the semicircular (free Gaus1 sian) distributions 2π t 4t − x 2 d x, the Marchenko-Pastur (free Poisson) distributions √ 4t−(x−b)2 1 2π t 1+(b/t)x d x + possibly one atom, and more generally 4(t + c) − (x − b)2 1 d x + zero, one, or two atoms, 2π t 1 + (b/t)x + (c/t 2 )x 2 depending on the particular values of b, c, t. 4.2. Semicircular systems. Let C = 0 and all Ti = 0. Then Si = ai+ and R xi x u x j = ei , Su e j = 0 for |u| ≥ 1. Thus in distribution, all Si ∼ 0. Only second-order free cumulants of (X 1 , X 2 , . . . , X d ) are non-zero, and ϕ is the distribution of a freely independent semicircular system, the free analog of the standard d-dimensional Gaussian distribution. 4.3. Free Poisson states. Let C = 0 and Ti arbitrary. Then Si = ai+ + Ti and R xi x u x j = ei , Su e j = ei , Tu e j . Thus in distribution, (S1 , S2 , . . . , Sd ) ∼ (T1 , T2 , . . . , Td ). It is appropriate to say that in this case, the joint distribution ϕ of (X 1 , X 2 , . . . , X d ) is d-dimensional free Poisson. In [Ans06] we showed that if ϕ is tracial, then ϕ is a rotation of a free product of onedimensional free Poisson distributions. Whether or not ϕ is tracial, the vector Ω is cyclic and separating for the von Neumann algebra W ∗ (X 1 , X 2 , . . . , X d ).
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4.4. Free product states. For two vectors f, g, denote by E f,g the corresponding rank one operator, E f,g (h) = f g, h. For an orthonormal basis { f i }, E fi , f j are the corresponding matrix units. For the standard basis {ei }, we will denote these simply by E i j . In particular, E ii is the orthogonal projection onto ei . Let C(ei ⊗ e j ) = ci δi j (ei ⊗ e j ), and let Ti = bi E ii . Then Si acts entirely on the subspace Fci (Span (ei )), on which it equals Si = ai+ + bi + ci ai− . ai+ +ci ai− has the centered semicircular distribution with variance ci (note that on FC (H), this operator is not symmetric, so its star-distribution is different from the semicircular one). Therefore Si has the semicircular distribution with mean bi and variance ci . Also, it follows that R xi x u x j = ei , Su e j = 0 unless i = u(1) = . . . = u(n) = j. In other words, all the mixed free cumulants of (X 1 , X 2 , . . . , X d ) are zero. This says precisely that their joint distribution ϕ is a free product of the distributions of each of X 1 , X 2 , . . . , X d . Each of these, in turn, is a one-dimensional free Meixner distribution, whose free cumulants are, up to a shift of index, the moments of the semicircular distribution with mean bi and variance ci . 4.5. Exponentiated semicircular systems. Let C(ei ⊗ e j ) = ci (ei ⊗ e j ) and Ti = bi I . Then Si = ai+ + ci ai− + bi I, again the distribution of Si is the semicircular distribution with mean bi and variance ci , but now the operators Si themselves are freely independent with respect to the state ϕ, so that their joint distribution is a free product. The joint distribution of (X 1 , X 2 , . . . , X d ) is not a free product (typically, not even tracial); it was described in the last section of [Ans06]. Note that the preceding two examples make sense for −1 ≤ ci < 0, except that one loses the interpretation of Si as having a semicircular distribution with variance ci , and the resulting states are not freely infinitely divisible. Remark 1. The constructions in the previous two examples coincide in the one-dimensional case. That case, and in particular the corresponding free cumulants, were also considered in [Ans03] and described completely in [BB06]. Moreover, many one-dimensional free Meixner distributions arise as limits in the central and Poisson limit theorems for the t-transformed free convolution, in the sense of [BW01]; that paper also contains a Fock space construction which coincides with the one-dimensional version of the one in Sect. 2.3.
Free Meixner States
889
4.6. Free multinomial states. It is well known that the Bernoulli distribution (1 − p)δ0 + pδ1 is a Meixner distribution. It was noted in [Ans03] that it is also a (one-dimensional) free Meixner distribution. Moreover, the binomial distributions, which are convolution powers
(1 − p)δ0 + pδ1
∗n
=
n $ % n (1 − p)n−k p k δk k k=0
of the Bernoulli distribution, are all Meixner, and the free binomial distributions, which n are free convolution powers (1 − p)δ0 + pδ1 of the Bernoulli distribution are free t Meixner. In fact, it was noted in [BB06] that (1 − p)δ0 + pδ1 are free Meixner for all real t ≥ 1. It is also well-known that the multinomial distributions are Meixner [Pom96]. In particular, the basic multinomial distribution p 1 δe 1 + p 2 δe 2 + · · · p d δe d
(23)
on Rd has this property. We now show that it, and so the free semigroup it generates, also induce free Meixner states. In this example, it is natural to consider the state with non-trivial means and a non-diagonal covariance matrix; an actual free Meixner state can be obtained from it by an affine transformation as in Sect. 3.1. In the Fock space construction of Sect. 2.3, take dim H = d − 1 rather than d, and C(ei ⊗ e j ) = −ei ⊗ e j for all i, j, so that Ci j = −1. In this case the induced inner product on the Fock space Falg (H) is degenerate, and the vector space factors through to simply FC (H) = C ⊕ H. In this example only, choose (linearly dependent) vectors { f i |i = 1, 2, . . . d} in H that are not orthonormal, but instead satisfy f i , f i = pi (1 − pi ),
f i , f j = − pi p j , where pi > 0,
i = 1, 2, . . . , d,
p1 + p2 + · · · + pd = 1.
Since these numbers are the covariances of the centered version of the basic multinomial distribution (23), the corresponding matrix is positive semi-definite (it has rank d − 1) and so the { f i } can be chosen in this fashion. Let Ti ( f i ) = (1 − 2 pi ) f i , Ti ( f j ) = − pi f j − p j f i , and define X i = ai+ + Ti + ai− + ai− C
890
M. Anshelevich
as usual, except that ai+ = 0 on H. In other words, for Yi = X i + pi , Yi Ω = f i + pi Ω, Yi f i = (1 − pi )[ f i + pi Ω], Yi f j = − p j [ f i + pi Ω].
(24)
Proposition 4. Yi is an orthogonal projection of C ⊕ H onto Span ( f i + pi Ω). These projections are orthogonal among themselves and their sum is the identity operator. Their joint distribution with respect to the state ϕC,{Ti } from Definition 3 is the basic multinomial distribution (23). In particular, ϕ [Yi ] = pi . The free cumulant generating function of ϕ satisfies the differential equations Di D j R = (δi j pi − pi p j ) + (δi j − p j )Di R − pi D j R − Di R D j R = δi j (Di R + pi ) − (Di R + pi )(D j R + p j ). Proof. The image of Yi is Span ( f i + pi Ω) and
f i + pi Ω, f j + p j Ω = − pi p j + pi p j = 0, so these subspaces are orthogonal. Moreover, Yi ( f i + pi Ω) = (1 − pi )[ f i + pi Ω] + pi [ f i + pi Ω] = f i + pi Ω and Yi ( f j + p j Ω) = − p j [ f i + pi Ω] + pi [ f i + pi Ω] = 0. Therefore each Yi is an orthogonal projection onto its image (in particular, it is selfadjoint), and these projections are orthogonal to each other. C ⊕ H has dimension d, .d therefore the sum i=1 Yi is identity. It also follows that / pi , i = u(1) = u(2) = . . . , ϕ [Yu ] = 0, otherwise. Therefore, the joint distribution of (Y1 , Y2 , . . . , Yd ) with respect to ϕ is the basic multinomial distribution. The last part follows from the operator representation. Definition 4. Free multinomial states are the states ϕ t |t ≥ 1 , where ϕ is the basic multinomial distribution (23). Note that ϕ n is the joint distribution of the sum of n d-tuples of orthogonal projections. Remark 2. Since the Bernoulli distribution is both classical and free Meixner, one may conjecture that it is in some sense also q-Meixner (for 0 ≤ q ≤ 1, with q = 1 corresponding to the classical case and q = 0 corresponding to the free case). The meaning of this term is not well-defined, but see for example Sect. 4.3 of [Ans04] or [BY06]. Indeed, the recursion relation for its orthogonal polynomials is of the q-Meixner form x P0 = P1 + p, x P1 = P2 + (1 − p)P1 + p(1 − p)P0 , x Pn = Pn+1 + (1 − p)[n]q Pn + [n]q p(1 − p)(1 − [n − 1]q )Pn−1 ,
Free Meixner States
891
independently of q, as long as the degree of the polynomial n ≤ 1, which suffices since L 2 (1 − p)δ0 + pδ1 is 2-dimensional. One may also hope that its q-cumulant generating function would then satisfy the equation Dq2 R (q) = Dq R (q) − (Dq R (q) )2 , where Dq is the q-derivative Dq ( f )(z) =
f (z) − f (qz) , (1 − q)z
and R (q) =
∞ n=1
1 αn z n . [n]q !
The corresponding recursion for its q-cumulants is αn+2
n 0 1 n = αn+1 − α α i q i+1 n−i+1 i=0
(compare with Remark 5.4 of [BB06]), with the initial condition α1 = p. Using Maple, it is easy to calculate the first 5 cumulants. Unfortunately, the fifth q-cumulant of the Bernoulli distribution calculated in this fashion differs from its fifth q-cumulant in the sense of Sect. 6 of [Ans01]. Proposition 5 (Free Poisson limit theorem). Let ϕ be a version of the free multinomial state corresponding to the measure (1 −
d i=1
pi )δ0 +
d
pi δei ,
(25)
i=1
.d with each pi > 0 and i=1 pi ≤ 1. As n → ∞, the states ϕ n with parameters pi = converge pointwise to a re-scaled free Poisson state.
ai n
Proof. The distribution (25) has the same operator representation as (23), except that .d the vectors { f 1 , . . . , f d } are, if i=1 pi < 1, linearly independent, and H has dimension t d. The MOPS for ϕ satisfy recursion relations xi xi Pi xi P j xi P(i,u) xi P( j,u)
= = = = =
Pi + t pi , P(i,i) + (t pi + (1 − 2 pi ))Pi + t pi (1 − pi ), P(i, j) − p j Pi + (t − 1) pi P j − t pi p j , P(i,i,u) + (t pi + (1 − 2 pi ))P(i,u) + (t − 1) pi (1 − pi )Pu , P(i, j,u) − p j P(i,u) + (t − 1) pi P( j,u) − (t − 1) pi p j Pu .
892
Setting t = n, pi =
M. Anshelevich ai n
and taking the limit as n → ∞ produces the recursions xi = Pi + ai , xi P(i,u) = P(i,i,u) + (1 + ai )P(i,u) + ai Pu , xi P( j,u) = P(i, j,u) + ai P( j,u) .
The mean of xi is ai ; subtracting them, we get xi = Pi , xi P( j,u) = P(i, j,u) + δi j P(i,u) + δi j ai Pu . Finally, rescaling by
√1 ai
, we get the recursion
1 xi P( j,u) = P(i, j,u) + δi j √ P(i,u) + δi j Pu . ai In other words, these polynomials are orthogonal with respect to a state ϕC,{Ti } with C = 0 and Ti = √1a E ii , which is a free Poisson free product state (see Sects. 4.3 i and 4.4). Convergence of recursion relations implies convergence of moments and so pointwise convergence of functionals on Rx. Note that one cannot hope to get all free Poisson states in this fashion. Indeed, (see the next section) since the free multinomial states are tracial, so is any limit of such states, and tracial free Poisson states are all rotations of free product states. 4.7. Tracial examples. If ϕ is a state on a non-commutative algebra A, one says that ϕ is tracial, or a trace, if for any x, y ∈ A, ϕ [x y] = ϕ [yx] . Tracial states play a crucial role, for example, in the theory of von Neumann algebras. Lemma 8. Let ϕ = ϕC,{Ti } be a free Meixner state, represented as the joint distribution of operators (X 1 , . . . , X d ). Suppose that ϕ is tracial. Then for all i, j, Ti e j = T j ei
(26)
Ti T j − T j Ti = C ji E i j − Ci j E ji .
(27)
and
Proof. Since ϕ is tracial, for all i, j, k, ϕ X i X j X k = ei , T j ek = e j , Tk ei = ei , Tk e j , so for all j, k, T j ek = Tk e j .
Free Meixner States
893
Similarly, for all i, j, k, l,
ϕ X i X j X k X l = ei , e j ek , el + ei , el e j , ek (1 + Ckl ) + ei , T j Tk el
= e j , ek el , ei + e j , ei ek , el (1 + Cli ) + e j , Tk Tl ei , so
ei , el e j , ek Ckl + ei , T j Tk el = e j , ei ek , el Cli + ei , Tl Tk e j .
Using Eq. (26) and the orthonormality of {ei },
ei , el e j , ek C jl + ei , T j Tl ek = e j , ei ek , el Cl j + ei , Tl T j ek , so
e j , ek C jl el + T j Tl ek = ek , el Cl j e j + Tl T j ek
and T j Tl − Tl T j = Cl j E jl − C jl El j . Example 1. A general theorem of Voiculescu (Proposition 2.5.3 in [VDN92]) implies that the free product states from Example 4.4 are tracial. It is also easy to see that any rotation of a tracial state is tracial. Lemma 9. If ϕ is tracial, then ϕ t is tracial for all t for which it is defined. Proof. It is easy to see directly from the defining equation (7) that a state ϕ is tracial if any only if its free cumulant generating functional Rϕ is tracial. The combinatorial reason is that if a partition π is non-crossing when points {1, 2, . . . , n} are placed on a line, it is also non-crossing when they are placed on a circle. The lemma follows from this fact and the defining relation for ϕ t , Rϕ t [x u ] = t Rϕ [x u ] . Proposition 6. All free multinomial states are tracial. Proof. Since the operators {Yi } defined in Eq. (24) commute, the basic multinomial distribution is tracial; in fact, it factors through to a state on commutative polynomials R[x1 , . . . , xd ] corresponding to the basic multinomial measure (23). It follows from Lemma 9 that all the other free multinomial states are tracial as well. We conclude the paper with three further results on when free Meixner states are traces. The first two show that under one set of general assumptions, the only tracial free Meixner states are the trivial ones, namely the rotations of free product states. It generalizes Proposition 11 of [Ans06]. The last one provides a way to construct a large class of tracial examples which do not come from free products. It generalizes the multinomial example above. Proposition 7. Let ϕ = ϕC,{Ti } be a tracial free Meixner state with C diagonal as a d × d matrix, Ci j = δi j ci . Then ϕ is a rotation of a free product state.
894
M. Anshelevich
Proof. If Ci j = δi j ci , then Eq. (27) states that all Ti , T j commute. Combining this with Eq. (26), we see that moreover, for some orthonormal basis { f 1 , f 2 , . . . , f d }, Tj =
d
αi f i , e j E fi , fi .
(28)
i=1
From C(T j ⊗ I )(ek ⊗ el ) = (T j ⊗ I )C(ek ⊗ el ) it follows that d
αi f i , e j f i , ek f i , el cl (el ⊗ el )
i=1 d
= δkl
αi f i , e j f i , ek f i , em cl (em ⊗ el ).
i,m=1
Thus for all k = l, d i=1
αi f i , e j f i , ek f i , el cl
,
= ek ,
d
αi f i , e j E fi , fi
-
el cl = ek , T j el cl = 0.
i=1
It follows that whenever cl = 0, T j el ∈ Span (el ), and one can take fl = el . So if Λ = {l|cl = 0}, then Span (el |l ∈ Λ) = Span ( fl |l ∈ Λ) is an invariant subspace for all T j . We can choose an orthogonal transformation O so that O(ei ) = f i , in other words / ei for i ∈ Λ, that is ci = 0, O(ei ) = f i for i ∈ Λ, that is ci = 0. Following the comments at the end of Sect. 3.1.3, the state ϕ O , which is the joint distribution of (X f1 , . . . , X fd ), is still a tracial free Meixner state. From Eq. (28), T fk =
d
αi f i , e j e j , f k E fi , fi = αk E fk , fk .
i, j=1
Finally, C(η ⊗ ζ ) = 0 whenever one of η, ζ ∈ Span (el |l ∈ Λ) = Span ( fl |l ∈ Λ), so / C(ei ⊗ e j ) = ci δi j if i, j ∈ Λ, C( f i ⊗ f j ) = 0 if one of i, j ∈ Λ. Thus C, T fk in the basis { f k } have the form in Example 4.4, and so ϕ O is a free product state.
Free Meixner States
895
Corollary 8. Let ϕ = ϕC,{Ti } be a tracial free Meixner state and Ti = 0 for all i. Then ϕ is a free product state. Proof. If Ti = 0 for all i, then it follows from Eq. (27) that Ci j = δi j ci , and the preceding proposition applies. In this case, a rotation is unnecessary. Meixner states correspond to quadratic natural exponential families. The following states are free versions of simple quadratic natural exponential families in the terminology of [Cas96], where all such (classical) families were classified. Proposition 9. Let C be a constant matrix, Ci j = c for all i, j. Then the necessary conditions (26) and (27) for ϕ to be a tracial free Meixner state, namely that {Ti } are symmetric matrices, Ti e j = T j ei and (Ti T j − T j Ti ) = c E i j − E ji , are also sufficient. Proof. If c = 0, the result follows from Proposition 7. So we will assume that c = 0. For each n, let π A(n) = π ∈ NC 0 (n)|1 ∼ 2 . For any partition σ ∈ NC 0 (n)\A(n), define the partition l(σ ) ∈ A(n) as follows: if 1 ∈ B ∈ σ and 2 ∈ C ∈ σ , let l(σ ) be the partition with the same classes as σ except that B ∪ C is a class of l(σ ). Conversely, any such σ can be obtained by starting with π ∈ A(n) with the (unique) outer class B, choosing i ∈ B, 2 < i < n (if it exists), and taking σ to have the same classes as π except that B ∩ ({1} ∪ {i + 1, . . . , n}) and B ∩ {2, . . . , i} are classes of σ . For π ∈ N C0 (n), define the partition ρ(π ) ∈ NC 0 2, 3, . . . , n, 1¯ (where 1¯ is identified with n + 1) by π
ρ(π )
i ∼ j ⇔ i ∼ j for i, j = 1, ρ(π ) π 1 ∼ j ⇔ 1¯ ∼ j.
Clearly
and
π ρ(NC 0 (n)) = π ∈ NC 0 ( 2, . . . , n, 1¯ )|n ∼ 1¯ , π π ρ(A(n)) = π ∈ NC 0 ( 2, . . . , n, 1¯ )|2 ∼ n ∼ 1¯ .
In particular, ρ(A(n)) ⊂ NC 0 ( 2, . . . , n, 1¯ ). For any partition σ ∈ NC 0 ( 2, . . . , n, 1¯ )\ρ(A(n)),
896
M. Anshelevich
define the partition r (σ ) ∈ ρ(A(n)) as follows: if 1¯ ∈ B ∈ σ and n ∈ C ∈ σ , let r (σ ) be the partition with the same classes as σ except that B ∪ C is a class of r (σ ). Conversely, any such σ can be obtained by starting with π ∈ ρ(A(n)) with the (unique) outer class B, choosing i ∈ B, 2 < i < n (if it exists), and taking σ to have the same classes as π except B ∩ {2, . . . , i} ∪ 1¯ and B ∩ {i + 1, . . . , n} are classes of σ . We will use the usual commutator notation [T, T ] = T T − T T . Using Eq. (26),
eu(1) , Tu(2) Tu(3) Tu(4) . . . Tu(n−2) Tu(n−1) eu(n)
= eu(2) , Tu(1) Tu(3) Tu(4) . . . Tu(n−2) Tu(n−1) eu(n)
= eu(2) , Tu(1) , Tu(3) Tu(4) . . . Tu(n−1) eu(n) + · · ·
+ eu(2) , Tu(3) Tu(4) . . . Tu(n−2) Tu(1) , Tu(n−1) eu(n)
+ eu(2) , Tu(3) Tu(4) . . . Tu(n−1) Tu(n) eu(1) . Now using Eq. (27), [Ti , T j ] = c(E i j − E ji ),
eu(1) , Tu(2) Tu(3) Tu(4) . . . Tu(n−2) Tu(n−1) eu(n)
= c eu(2) , eu(1) eu(3) , Tu(4) . . . Tu(n−1) eu(n)
+ c eu(2) , Tu(3) eu(1) eu(4) , Tu(5) . . . Tu(n−1) eu(n)
+ c eu(2) , Tu(3) . . . Tu(n−2) eu(1) eu(n−1) , eu(n) + · · ·
+ eu(2) , Tu(3) . . . Tu(n) eu(1)
− c eu(2) , eu(3) eu(1) , Tu(4) . . . Tu(n−1) eu(n)
− c eu(2) , Tu(3) eu(4) eu(1) , Tu(5) . . . Tu(n−1) eu(n) − · · ·
− c eu(2) , Tu(3) . . . Tu(n−2) eu(n−1) eu(1) , eu(n) . The left-hand-side of the preceding equation is equal to Θ(1ˆ n ; {1, . . . , n} , u), where 1ˆ n ∈ NC 0 (n) is the partition with a single class. Similarly, using Lemma 7, the preceding equation itself states that Θ(1ˆ n ; {1, . . . , n} , u) = Θ(ρ(1ˆ n ); 2, . . . , n, 1¯ , ρ(u)) + −
n−2
Θ( 2, . . . , i, 1¯ , {i + 1, . . . , n} ; 2, . . . , n, 1¯ , ρ(u))
i=2 n−1
Θ( {2, . . . , i}, {1, i + 1, . . . , n} ; {1, . . . , n}, u),
i=3
where for a multi-index u = ((u(1), . . . , u(n)), we denote by ρ(u) = (u(2), . . . , u(n), u(1))
Free Meixner States
897
the multi-index on 2, . . . , n, 1¯ . Using the descriptions of l(σ ), r (σ ) at the beginning of the proof, this in turn equals to = Θ(ρ(1ˆ n ); 2, . . . , n, 1¯ , ρ(u)) + Θ(τ ; 2, . . . , n, 1¯ , ρ(u)) τ :r (τ )=ρ(1ˆ n )
−
(29)
Θ(σ ); {1, . . . , n} , u).
σ :l(σ )=1ˆ n
Using Lemma 7 again and Eq. (29) applied to the class of π containing 1, we conclude that for π ∈ A(n), c−(|π |−1) Θ(π ; {1, . . . , n}, u) = c−(|π |−1) Θ(ρ(π ); 2, . . . , n, 1¯ , ρ(u)) +c c−(|τ |−1) Θ(τ ; 2, . . . , n, 1¯ , ρ(u)) τ :r (τ )=ρ(π )
−c
c−(|σ |−1) Θ(σ ; {1, . . . , n}, u),
σ :l(σ )=π
or, since |ρ(π )| = |π | and |τ | = |σ | = |π | + 1, Θ(π ; {1, . . . , n}, u) = Θ(ρ(π ); 2, . . . , n, 1¯ , ρ(u)) + Θ(τ ; 2, . . . , n, 1¯ , ρ(u)) τ :r (τ )=ρ(π )
−
Θ(σ ); {1, . . . , n}, u).
σ :l(σ )=π
Therefore
R [x u ] = eu(1) , Su(2) . . . Su(n−1) eu(n) = Θ(π ; {1, . . . , n}, u) π ∈NC 0 (n)
Θ(π ; {1, . . . , n}, u) +
=
π ∈A(n)
Θ(σ ; {1, . . . , n}, u)
σ :l(σ )=π
Θ(ρ(π ); 2, . . . , n, 1¯ , ρ(u)) + = π ∈A(n)
=
Θ(τ ; 2, . . . , n, 1¯ , ρ(u))
τ :r (τ )=ρ(π )
Θ(τ ; 2, . . . , n, 1¯ , ρ(u))
τ ∈NC 0 ({2,...,n,1¯ })
= eu(2) , Su(3) . . . Su(n) eu(1) . Using Theorem 2, we conclude that the free cumulant functional Rϕ , and so ϕ itself, is tracial.
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M. Anshelevich
Example 2. For d = 2, one can take % $ c+1 0 , T1 = 0 1 For d = 3, one can take ⎛ ⎞ 0c0 T1 = ⎝c 0 0⎠, 000 For d = 4, one can take ⎛ ⎞ ⎛ c010 0 ⎜0 0 0 0 ⎟ ⎜0 T1 = ⎝ , T = 1 0 0 0⎠ 2 ⎝0 0001 0
T2 =
⎛
$ % 01 . 10
⎞ c 0 0 T2 = ⎝0 c + 1 0⎠, 0 0 1
0 0 c 0
0 c 0 0
⎞ ⎛ 0 1 0⎟ ⎜0 , T = 0⎠ 3 ⎝0 0 0
⎛
⎞ 000 T3 = ⎝0 0 1⎠. 010
⎞ ⎛ 0 0 0 0 c 0 0⎟ ⎜0 , T = 0 c + 1 0 ⎠ 4 ⎝0 0 0 1 1
0 0 0 0
0 0 0 1
⎞ 1 0⎟ . 1⎠ 0
Remark 3. If Ci j = c for all i, j, the corresponding Fock space is an interacting Fock space in the sense of [AB98]. If Ci j = c and in addition all Ti = 0, the von Neumann algebras W ∗ (X 1 , . . . , X d ) were described in [Ric06]. References [AB98]
Accardi, L., Bo˙zejko, M.: Interacting Fock spaces and Gaussianization of probability measures. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1(4), 663–670 (1998) [Ans01] Anshelevich, M.: Partition-dependent stochastic measures and q-deformed cumulants. Doc. Math. 6, 343–384 (2001) (electronic) [Ans03] Anshelevich, M.: Free martingale polynomials. J. Funct. Anal. 201(1), 228–261 (2003) [Ans04] Anshelevich, M.: Appell polynomials and their relatives. Int. Math. Res. Not. 65, 3469–3531 (2004) [Ans06] Anshelevich, M.: Orthogonal polynomials with a resolvent-type generating function. http://arxiv:org/list/math.CO/0410482, 2004, to appear in Transactions of the AMS, 2007 [Ans07] Anshelevich, M.: Monic non-commutative orthogonal polynomials. http://arXiv.org/list/math. CO/0702157, 2007, to appear in Proceedings of the AMS, 2007 [BB06] Bo˙zejko, M., Włodzimierz, B.: On a class of free Lévy laws related to a regression problem. J. Funct. Anal. 236(1), 59–77 (2006) [BLS96] Bo˙zejko, M., Leinert, M., Speicher, R.: Convolution and limit theorems for conditionally free random variables. Pacific J. Math. 175(2), 357–388 (1996) [BW01] Bo˙zejko, M., Wysocza´nski, J.: Remarks on t-transformations of measures and convolutions. Ann. Inst. H. Poincaré Probab. Statist. 37(6), 737–761 (2001) [BY06] Bo˙zejko, M., Yoshida, H.: Generalized q-deformed Gaussian random variables. In: Quantum probability (Bêdlewo, 2004), Banach Center Publ., Vol. 73, Warsaw: Polish Acad. Sci., 2006, pp. 127–140 [Cas96] Casalis, M.: The 2d + 4 simple quadratic natural exponential families on Rd . Ann. Statist. 24(4), 1828–1854 (1996) [CT84] Cohen, J.M., Trenholme, A.R.: Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis. J. Funct. Anal. 59(2), 175–184 (1984) [DES04] Dumitriu, I., Edelman, A., Shuman, G.: MOPS: Multivariate orthogonal polynomials (symbolically). http://arxiv.org/list/math-ph/0409066, 2004, to appear in J. Symbolic Comput. (2007) [Fla80] Flajolet, P.: Combinatorial aspects of continued fractions. Discrete Math. 32(2), 125–161 (1980) [Fre98] Freeman, J.M.: A strategy for determining polynomial orthogonality. In: Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), Progr. Math., Vol. 161, Boston, MA: Birkhäuser Boston, 1998, pp. 239–244 [KKN06] Kubo, I., Kuo, H.-H., Namli, S.: Interpolation of Chebyshev polynomials and interacting Fock spaces. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(3), 361–371 (2006) [KVDJ98] Koelink, H.T., Van Der Jeugt, J.: Convolutions for orthogonal polynomials from Lie and quantum algebra representations. SIAM J. Math. Anal. 29(3) 794–822 (1998) (electronic)
Free Meixner States
[KZ01] [LL60] [Lyt03] [Mei34] [Mor82] [NS06] [Pom96] [Ric06] ´ [Sni00] [Sta97] [SY01] [Sze22] [VDN92] [Vie85] [Wes93]
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Kim, D., Zeng, J.: A combinatorial formula for the linearization coefficients of general Sheffer polynomials. European J. Combin. 22(3), 313–332 (2001) Laha, R.G., Lukacs, E.: On a problem connected with quadratic regression. Biometrika 47, 335–343 (1960) Lytvynov, E.: Polynomials of Meixner’s type in infinite dimensions—Jacobi fields and orthogonality measures. J. Funct. Anal. 200(1), 118–149 (2003) Meixner, J.: Orthogonale polynomsysteme mit einer besonderen gestalt der erzeugenden funktion. J. London Math. Soc. 9, 6–13 (1934) Morris, C.N.: Natural exponential families with quadratic variance functions. Ann. Statist. 10(1), 65–80 (1982) Nica, A., Speicher, R.: Lectures on the combinatorics of free probability. London Mathematical Society Lecture Note Series, Vol. 335, Cambridge: Cambridge University Press, 2006 Pommeret, D.: Orthogonal polynomials and natural exponential families. Test 5(1), 77–111 (1996) Ricard, É.: The von Neumann algebras generated by t-Gaussians. Ann. Inst. Fourier (Grenoble) 56(2), 475–498 (2006) ´ Sniady, P.: Quadratic bosonic and free white noises. Commun. Math. Phys. 211(3), 615–628 (2000) Stanley, R.P.: Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, Vol. 49, Cambridge: Cambridge University Press, 1997, Corrected reprint of the 1986 original Saitoh, N., Yoshida, H.: The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. Math. Statist. 21(1), 159–170 (2001) Szegö, G.: Über die entwicklung einer willkürlichen funktion nach den polynomen eines orthogonalsystems. Math. Z. 12(1), 61–94 (1922) Voiculescu, D.V., Dykema, K.J., Nica, A.: Free random variables. CRM Monograph Series, Vol. 1, Providence, RI: Amer. Math. Soc. 1992 Viennot, G.: A combinatorial theory for general orthogonal polynomials with extensions and applications. In: Orthogonal polynomials and applications (Bar-le-Duc, 1984), Lecture Notes in Math., Vol. 1171, Berlin: Springer, 1985, pp. 139–157 Wesołowski, J.: Stochastic processes with linear conditional expectation and quadratic conditional variance. Probab. Math. Statist. 14(1), 33–44 (1993)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 276, 901–905 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0355-7
Communications in
Mathematical Physics
Explicit A Priori Bounds on Transfer Operator Eigenvalues Oscar F. Bandtlow, Oliver Jenkinson School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK. E-mail: [email protected]; [email protected] Received: 19 March 2007 / Accepted: 24 April 2007 Published online: 16 October 2007 – © Springer-Verlag 2007
Abstract: We provide explicit bounds on the eigenvalues of transfer operators defined in terms of holomorphic data. Linear operators of the form L f = i∈I wi · f ◦ Ti , so-called transfer operators (see e.g. [Bal, Rue1, Rue2]), arise in a number of problems in dynamical systems. If the Ti are inverse branches of an expanding map T , and the weight functions wi are positive, the spectrum of L has well-known interpretations in terms of the exponential mixing rate of an invariant Gibbs measure (see [Bal]). Applications also arise when the wi are real-valued (e.g. [CCR, JMS, Pol]) or complex-valued (e.g. [Dol, PS]). In this article we suppose that Ti and wi are analytic functions of d variables, for each i in some countable1 index set I. Under suitable hypotheses on Ti and wi the transfer operator L defines a compact operator on Hardy space H 2 (B), and we can give completely explicit bounds on its eigenvalue sequence2 {λn (L)}∞ n=1 : Theorem 1. Suppose there is acomplex Euclidean ball B ⊂ Cd such that each wi : B → C is holomorphic with i∈I supz∈B |wi (z)| < ∞, and each Ti : B → B is holomorphic with ∪i∈I Ti (B) contained in the ball concentric with B whose radius is r < 1 times that of B. Then L : H 2 (B) → H 2 (B) is compact and √ d W d 1/d 1/d n (d−1)/(2d) r d+1 (d!) n for all n ≥ 1 , r d (1 − r 2 )d/2 where W := supz∈B i∈I |wi (z)|. |λn (L)| <
(1)
1 Subsequent results are new even when I is finite, but it is convenient to also allow countably infinite I. 2 Precisely, {λ (L)}∞ denotes the sequence of all eigenvalues of L counting algebraic multiplicities and n n=1
ordered by decreasing modulus, with the usual convention (see e.g. [Pie, 3.2.20]) that distinct eigenvalues with the same modulus can be written in any order.
902
O. F. Bandtlow, O. Jenkinson
If d = 1 then W r (n−1)/2 for all n ≥ 1 . |λn (L)| ≤ √ 2 1−r
(2)
Remark 2. 1/d
(i) An estimate of the form |λn (L)| ≤ Cθ n for some (undefined) constants C > 0, θ ∈ (0, 1) is asserted, either implicitly or explicitly, in the work of several authors (e.g. [FR, Fri, GLZ]); the novelty here is that careful derivation of this bound renders explicit the constants C, θ . 1/d (ii) Using different techniques, the bound |λn (L)| ≤ Cθ n can also be established d in the case where B is an arbitrary open subset of C (see [BJ]), though here our expressions for C, θ are more complicated. 2 1 1 Example 3. If L f (z) = ∞ (the Perron-Frobenius operator for the f n+z n=1 n+z Gauss map x → 1/x (mod 1), cf. [May]), B ⊂ C may be chosen as the open−2disc of radius 3/2 centred at the point 1. In this case W = supz∈B ∞ = n=1 |n + z| ∞ −2 = π 2 /2 and r = 2/3, so (2) yields (n − 1/2) n=1 3π 2 |λn (L)| ≤ √ (2/3)(n−1)/2 for all n ≥ 1 . 2 5 Notation 4. For an open ball D ⊂ Cd , let H ∞ (D) denote the Banach space consisting of all bounded holomorphic C-valued functions on D, with norm f H ∞ (D) := supz∈D | f (z)|. Hardy space H 2 (D) (see [Kra, Ch. 8.3]) is the L 2 (∂ D, σ )-closure of the set of those f ∈ H ∞ (D) which extend continuously to the boundary ∂ D, where σ denotes (2d −1)dimensional Lebesgue measure on ∂ D, normalised so that σ (∂ D) = 1. In particular, H 2 (D) is a Hilbert subspace of L 2 (∂ D, σ ) with each element f ∈ H 2 (D) having a natural holomorphic extension to D (see [Kra, Ch. 1.5]). In the sequel, no generality is lost by taking B in the statement of Theorem 1 to be the unit ball B1 , and the smaller concentric ball to be Br , the ball of radius r centred at 0. If L : X 1 → X 2 is a continuous operator between Banach spaces then for k ≥ 1, its k th approximation number ak (L) is defined as ak (L) = inf{ L − K | K : X 1 → X 2 linear and continuous with rank(K ) < k} . The proof of Theorem 1 hinges on the following two lemmas. Lemma 5. If J : H 2 (B1 ) → H ∞ (Br ) denotes the canonical embedding, then J and L are compact and for all n ≥ 1, |λn (L)| ≤ W
n
ak (J )1/n .
(3)
k=1
Proof. If f ∈ H 2 (B1 ) and z ∈ Br then | f (z)| ≤ (2/(1 − r ))d/2 by [Rud, Thm. 7.2.5], so { f | f H 2 (B1 ) ≤ 1} is a normal family in H ∞ (Br ), hence relatively compact in H ∞ (Br ) by Montel’s Theorem (see [Nar, Ch. 1, Prop. 6]), thus J is compact.
Transfer Operator Eigenvalues
903
Next observe that if f ∈ H ∞ (B1 ) then f ∈ H 2 (B1 ) by [Rud, Thm. 5.6.8] and the canonical embedding Jˆ : H ∞ (B1 ) → H 2 (B1 ) is continuous of norm 1, because σ (∂ B1 ) = 1. We claim that Lˆ f := i∈I wi · f ◦ Ti defines a continuous opera∞ ∞ ˆ tor L : H (Br ) → H (B1 ). In order to see this, fix f ∈ H ∞ (Br ) and note that wi · f ◦Ti ∈ H ∞ (B1 ) with wi · f ◦ Ti H ∞ (B1 ) ≤ wi H ∞ (B1 ) f H ∞ (Br ) for every i ∈ I. But since Lˆ f H ∞ (B1 ) ≤ i∈I wi H ∞ (B1 ) f H ∞ (Br ) and i∈I wi H ∞ (B1 ) < ∞ by hypothesis, we conclude that Lˆ f ∈ H ∞ (B1 ) and that Lˆ is continuous. Now | f (Ti (z))| ≤ f H ∞ (Br ) for every z ∈ B1 , i ∈ I, so it follows that Lˆ f H ∞ (B1 ) = supz∈B1 |(Lˆ f )(z)| ≤ supz∈B1 i∈I |wi (z)| | f (Ti (z))| ≤ W f H ∞ (Br ) , and hence that ˆ ≤ W . Now clearly L = JˆLJ ˆ , so L is compact, and L ˆ k (J ) ≤ W ak (J ) for all k ≥ 1 , ak (L) ≤ JˆL a
(4)
since in general ak (L 1 L 2 ) ≤ L 1 ak (L 2 ) whenever L 1 and L 2 are bounded operators between Banach spaces (see [Pie, 2.2]). Moreover, since L is a compact operator n |λ on Hilbert space, Weyl’s inequality (see [Pie, 3.5.1], [Wey]) asserts that k=1 k (L)| ≤ n a (L) for all n ≥ 1. Together with (4) this yields (3), because |λn (L)| ≤ k k=1 n 1/n . |λ (L)|
k=1 k Lemma 6. If h d (k) := k+d d then for all n ≥ 1, an (J )2 ≤
∞
h d−1 (l)r 2l where k ≥ 0 is such that h d (k − 1) < n ≤ h d (k) .
(5)
l=k
Proof. H 2 (B1 ) has reproducing kernel K (z, ζ ) = (1 − (z, ζ )Cd )−d (see [Kra, Thm. 1.5.5]3 ), where (·, ·)Cd denotes the Euclidean inner product, and K (z, ζ ) = ∞ ∞ 2 n=1 pn (z) pn (ζ ) whenever { pn }n=1 is an orthonormal basis for H (B1 ), the series converging pointwise for every (z, ζ ) ∈ B1 × B1 (see [Hal, p. 19]). Define Jn : H 2 (B1 ) → H ∞ (Br ) by Jn f = n−1 k=1 ( f, pk ) pk . If z ∈ Br , then ∞ 2 2 2 |J f (z) − Jn f (z)| = | f (z) − Jn f (z)| = ( f, pk ) pk (z) k=n
≤
∞
|( f, pk )|2
k=n
∞
| pk (z)|2 ≤ f 2H 2 (B ) (K (z, z) −
n−1
1
k=n
so
k=1
an (J ) ≤ sup 2
z∈Br
| pk (z)|2 ) ,
K (z, z) −
n−1
| pk (z)|
2
.
(6)
k=1
∞ h d−1 (l)r 2l = If n = 1 then k = 0, in which case (5) follows from (6) since l=0 d 2 −d (1 − r ) . Now define the orthonormal basis { pn | n ∈ N0 } by (cf. [Rud, Prop. 1.4.8, 1.4.9]) pn (z) = K n z n (n ∈ Nd0 ) , 3 Note that the extra factor (d −1)!/(2π d ) appearing in [Kra, Thm. 1.5.5] is due to a different normalisation of the measure σ on ∂ B1 .
904
O. F. Bandtlow, O. Jenkinson
(|n|+d−1)! nd n1 n where K n = (d−1)! n! , n = (n 1 , . . . , n d ), z = z 1 · · · z d , n! = n 1 ! · · · n d !, |n| = n1 + · · · + nd . If n ≥ 2 then there are k+d−1 multinomials of degree less than or equal to k − 1, so d ⎛ ⎞ ∞ ∞ (l + d − 1)! 2l pn (z) 2⎠ = sup pn (z) 2 ≤ an (J )2 ≤ sup ⎝K (z, z)− r (d − 1)! l! z∈Br z∈Br l=k l=k |n |=l |n |≤k−1 , because |n |=l for all n > k+d−1 d theorem.
1 n!
|z n |2 ≤
1 2l l! r
for z ∈ Br by the multinomial
Proof of Theorem 1. By Lemma 5 it suffices to bound the geometric means ( nk=1 ak )1/n , where ak := ak (J ). From Lemma 6 it follows that ˜
r 2βn (1 − r 2 )d
an2 ≤ α˜ n
for all n ≥ 1 ,
(7)
where α˜ n := h d−1 (k) β˜n := k
for h d (k − 1) < n ≤ h d (k) ,
because ∞
h d−1 (l)r 2l = h d−1 (k)r 2k
l=k
∞ h d−1 (l + k) l=0
≤ h d−1 (k)r 2k
∞
h d−1 (k)
h d−1 (l)r 2l = h d−1 (k)
l=0
r 2l
r 2k . (1 − r 2 )d
Combining (7) with Lemma 5 gives, for all n ≥ 1, |λn (L)| ≤ W αn
r βn , (1 − r 2 )d/2
(8)
where αn :=
n
1/(2n) α˜ l ,
l=1
n 1 β˜l . βn := n l=1
To obtain (1) and (2) from (8) we require an upper bound on αn and a lower bound on βn . We start with the bounds for αn . Observe that α˜ 1 = h d−1 (0) = 1 , and α˜ l ≤ d(l − 1)1−1/d for l ≥ 2 .
(9)
To see this note that h d−1 (k) (d!)1−1/d = h d (k − 1)1−1/d (d − 1)!
d−1
d l=1 (k + l) d−1 d−1 l=0 (k + l)
1/d =
d−1 l 1/d (d!)1−1/d 1+ (d − 1)! k l=1
Transfer Operator Eigenvalues
is decreasing in k, so if h d (k − 1) < n ≤ h d (k) then h d−1 (1) h d (0)1−1/d
905 α˜ l (l−1)1−1/d
≤
h d−1 (k) h d (k−1)1−1/d
≤
=d. The estimate (9) now yields the upper bound n √ nn(d−1)/(2dn) √ (d−1)/(2d) 1/(2n) √ α˜ i ≤ d((n − 1)!)(d−1)/(2dn)≤ d 2 ≤ dn ,(10) αn = e i=1 n where, for n > 1, we have used the estimate (n − 1)! ≤ 2 ne (i.e. log(n − 1)! ≤ n x=2 log x d x ≤ n log n − n + log 2). We now turn to the bounds for βn . If h d (k − 1) < l ≤ h d (k), so that β˜l = k, then l ≤ h d (k) ≤ (d!)−1 (k + d)d , which implies β˜l = k ≥ (d!)1/d l 1/d − d . Therefore n n 1 1/d d 1/d 1 β˜l ≥ −d + (d!)1/d n , l > −d + (d!)1/d (11) n n d +1 l=1 l=1 n 1/d n d 1+1/d l > x=0 x 1/d d x = d+1 n . where we have used l=1 Assertion (1) now follows from (8), (10), and (11). Finally, if d = 1 then βn = 1 n 1 n ˜ β = (l − 1) = (n − 1)/2, and (10) becomes α n ≤ 1, so substituting l=1 l l=1 n n into (8) yields (2).
βn =
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Communicated by G. Gallavotti
