Commun. Math. Phys. 274, 1–30 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0272-9
Communications in
Mathematical Physics
Boson Stars as Solitary Waves Jürg Fröhlich1 , B. Lars G. Jonsson1,2 , Enno Lenzmann3,4 1 Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland.
E-mail:
[email protected];
[email protected]
2 Division of Electromagnetic Engineering, School of Electrical Engineering, Royal Insitute of Technology,
SE-100 44 Stockholm, Sweden
3 Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland 4 Department of Mathematics, MIT, Cambridge, MA 02139, USA.
E-mail:
[email protected] Received: 14 December 2005 / Accepted: 11 December 2006 Published online: 19 June 2007 – © Springer-Verlag 2007
Abstract: We study the nonlinear equation −∆ + m 2 − m ψ − (|x|−1 ∗ |ψ|2 )ψ on R3 , i∂t ψ = which is known to describe the dynamics of pseudo-relativistic boson stars in the meanfield limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, ψ(t, x) = eitµ ϕv (x − vt), for some µ ∈ R and with speed |v| < 1, where c = 1 corresponds to the speed of light in our units. Due to the lack of Lorentz covariance, such travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with v = 0). To overcome this difficulty, we introduce and study an appropriate variational problem that yields the functions ϕv ∈ H1/2 (R3 ) as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentration-compactness-type arguments. In addition to their existence, we prove orbital stability of travelling solitary waves ψ(t, x) = eitµ ϕv (x − vt) and pointwise exponential decay of ϕv (x) in x. 1. Introduction In this paper and its companion [4], we study solitary wave solutions — and solutions close to such — of the pseudo-relativistic Hartree equation 1 ∗ |ψ|2 ψ on R3 . −∆ + m 2 − m ψ − (1.1) i∂t ψ = |x| Here ψ(t, x) is a complex-valued wave field, and the symbol ∗ stands for √ convolution √ on R3 . The operator −∆ + m 2 − m, which is defined via its symbol k 2 + m 2 − m in Fourier space, is the kinetic energy operator of a relativistic particle of mass, m ≥ 0, and the convolution kernel, |x|−1 , represents the Newtonian gravitational potential in appropriate physical units.
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
As recently shown by Elgart and Schlein in [3], Eq. (1.1) arises as an effective dynamical description for an N -body quantum system of relativistic bosons with two-body interaction given by Newtonian gravity. Such a system is a model system for a pseudorelativistic boson star. That is, we √ consider a regime, where effects of special relativity (accounted for by the operator −∆ + m 2 − m) become important, but general relativistic effects can be neglected. The idea of a mathematical model of pseudo-relativistic boson stars dates back to the works of Lieb and Thirring [10] and of Lieb and Yau [11], where the corresponding N -body Hamiltonian and its relation to the Hartree energy functional H(ψ) = 2E(ψ) are discussed, with E(ψ) defined in (1.3), below. Let us briefly recap the state of affairs concerning Eq. (1.1) itself. With help of the conserved quantities of charge, N (ψ), and energy, E(ψ), given by N (ψ) = |ψ|2 dx, (1.2)
R3
1 1 1 2 ∗ |ψ|2 |ψ|2 dx, ψ −∆ + m − m ψ dx − (1.3) E(ψ) = 2 R3 4 R3 |x| results derived so far can be summarized as follows (see also Fig. 1 below). – Well-Posedness: For any initial datum ψ0 ∈ H1/2 (R3 ), there exists a unique solution ψ ∈ C0 [0, T ); H1/2 (R3 ) ∩ C1 [0, T ); H−1/2 (R3 ) , (1.4)
for some T > 0, where Hs (R3 ) denotes the inhomogeneous Sobolev space of order s. Moreover, we have global-in-time existences (i. e., T = ∞) whenever the initial datum satisfies the condition N (ψ0 ) < Nc , (1.5) N = Nc
E
I
II
0
N
ground states at rest
III
E = − 12 mN
Fig. 1. Qualitative diagram for the boson star Eq. (1.1) with positive mass parameter m > 0. Here N = N (ψ0 ) and E = E(ψ0 ) denote charge and energy for the initial condition ψ0 ∈ H1/2 (R3 ). In region I, all solutions are global in time and the (unboosted) ground states are minimizers of E(ψ) subject to fixed N (ψ0 ) = N with 0 < N < Nc . If N exceeds Nc , the energy E can attain values below − 21 m N . As shown in [5] for 3 spherically symmetric ψ0 ∈ C∞ c (R ) that belong to region III, we have in fact blow-up of ψ(t) within a finite time. Finally, the qualitative behavior of solutions with initial conditions in region II seems to be of indefinite nature
Boson Stars as Solitary Waves
3
where Nc > 4/π is some universal constant; see [7] for a detailed study of the Cauchy problem for (1.1) with initial data in Hs (R3 ), s ≥ 1/2. – Solitary Waves: Due to the focusing nature of the nonlinearity in (1.1), there exist solitary wave solutions, which we refer to as solitary waves, given by ψ(t, x) = eitµ ϕ(x),
(1.6)
where ϕ ∈ H1/2 (R3 ) is defined as a minimizer of E(ψ) subject to N (ψ) = N fixed. Any such minimizer, ϕ(x), is called a ground state and it has to satisfy the corresponding Euler-Lagrange equation 1 ∗ |ϕ|2 )ϕ = −µϕ, (1.7) −∆ + m 2 − m ϕ − |x| for some µ ∈ R. An existence proof of ground states, for 0 < N (ϕ) < Nc and m > 0, can be found in [11]. The method used there is based on rearrangement inequalities that allow one to restrict one’s attention to radial functions, which simplifies the variational calculus. But in order to extend this existence result to so-called boosted ground states, i. e., x in (1.6) is replaced by x −vt and Eq. (1.7) acquires the additional term, i(v · ∇)ϕ, we have to employ concentration-compactness-type methods; see Theorem 2.1 and its proof, below. 3 – Blow-Up: Any spherically symmetric initial datum, ψ0 ∈ C∞ c (R ), with 1 E(ψ0 ) < − mN (ψ0 ) 2
(1.8)
leads to blow-up of ψ(t) in a finite time, i. e., we have that limtT ψ(t) H1/2 = ∞ holds, for some T < ∞. We remark that (1.8) implies that the smallness condition (1.5) cannot hold. See [5] for a proof of this blow-up result.1 In physical terms, finite-time blow-up of ψ(t) is indicative of “gravitational collapse” of a boson star modelled by (1.1); the constant Nc appearing in (1.5) may then be regarded as a “Chandrasekhar limit mass.” We now come to the main issues of the present paper which focuses on existence and properties of travelling solitary waves for (1.1). More precisely, we consider solutions of the form ψ(t, x) = eitµ ϕv (x − vt) (1.9) with some µ ∈ R and travelling velocity, v ∈ R3 , such that |v| < 1 holds (i. e., below the speed of light in our units). We point out that, since Eq. (1.1) is not Lorentz covariant, solutions such as (1.9) cannot be directly obtained from solitary waves at rest (i. e., we set v = 0) and then applying a Lorentz boost. To circumvent this difficulty, we plug the ansatz (1.9) into (1.1). This yields 1 ∗ |ϕv |2 ϕv = −µϕv , −∆ + m 2 − m ϕv + i(v · ∇)ϕv − (1.10) |x| which is an Euler-Lagrange equation for the following minimization problem i Ev (ψ) := E(ψ) + ψ(v · ∇)ψ dx = min! subject to N (ψ) = N . 2 R3
(1.11)
1 In [5] the energy functional, E(ψ), is shifted by + 1 mN (ψ). Thus, condition (1.8) reads E(ψ ) < 0 in 0 2
[5].
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
We refer to such minimizers, ϕv ∈ H1/2 (R3 ), as boosted ground states throughout this paper. Indeed, we will prove existence of boosted ground states when |v| < 1 and 0 < N < Nc (v) holds, as well as non-existence when N ≥ Nc (v); see Theorem 2.1, below. Our existence proof rests on concentration-compactness arguments which for our √ problem need some technical modifications, due to the pseudo-differential operator −∆ + m 2 . Apart from existence of boosted ground states, we are also concerned with properties such as “orbital stability” and exponential decay of ϕv (x) in x; see Theorems 3.1 and 4.1, below. We remark that both properties rely crucially on the positivity of the mass parameter, i. e., we have m > 0 in (1.1). By contrast, it is shown, for instance, in [5] that (resting) solitary waves become unstable when m = 0, due to nearby initial data leading to blow-up solutions. In a companion paper [4], we will explore the effective dynamics of (slowly) travelling solitary waves in an external potential; see also Sect. 5 for a short summary of these result. The plan of this paper is as follows: – In Sect. 2, we set-up the variational calculus for problem (1.11) and we prove existence of boosted ground states, ϕv ∈ H1/2 (R3 ), for 0 < N (ϕv ) < Nc (v) and |v| < 1, as well as their nonexistence if N (ϕv ) ≥ Nc (v); see Theorem 2.1, below. – Section. 3 addresses “orbital stability” of travelling solitary waves ψ(t, x) = eitµ ϕv (x − vt); see Theorem 3.1, below. – In Sect. 4, we derive pointwise exponential decay and regularity of boosted ground states; see Theorem 4.1, below. – In Sect. 5, we sketch the main result of [4] describing the effective dynamics of travelling solitary waves in an external potential. – In App. A–C, we collect and prove several technical statements which we refer to throughout this text. Notation. Lebesgue spaces of complex-valued functions on R3 will be denoted by L p (R3 ), with norm · p and 1 ≤ p ≤ ∞. We define the Fourier transform for f ∈ S(R3 ) (i. e., Schwartz functions) according to 1 (F f )(k) = f (k) = f (x)e−ik·x dx, (1.12) (2π )3/2 R3 where F extends to S (R3 ) (i. e., the space of tempered distributions) by duality. For s ∈ R, we introduce the operator (1 − ∆)s via its multiplier (1 + |k|2 )s in Fourier −1 [(1 + |k|2 )s F f ]. Likewise, we define the operator space, i. e., we set (1 − ∆)s f = F √ 2 −∆ + m through its multiplier |k|2 + m 2 in Fourier space. We employ Sobolev spaces, Hs (R3 ), of fractional order s ∈ R defined by Hs (R3 ) := f ∈ S (R3 ) : (1 − ∆)s/2 f ∈ L2 (R3 ) (1.13) and equipped with the norm f Hs := (1 − ∆)s/2 f 2 . Since we exclusively deal with R3 , we often write L p and Hs instead of L p (R3 ) and s H (R3 ) in what follows. A further abbreviation we use is given by f dx := f (x) dx. (1.14) R3
R3
Boson Stars as Solitary Waves
5
We equip L2 (R3 ) with a complex inner product, ·, · , defined as f, g := f¯g dx. R3
(1.15)
Operator inequalities (in the sense of quadratic forms) are denoted by A ≤ B, which means that ψ, Aψ ≤ ψ, Bψ holds for all ψ ∈ D(|A|1/2 ) ⊆ D(|B|1/2 ), where A and B are self-adjoint operators on L2 (R3 ) with domains D(A) and D(B), respectively. Finally, we remark that we employ the notation v · ∇ = 3k=1 vk ∂xk , where v ∈ R3 is some fixed vector. 2. Existence of Boosted Ground States We consider the following minimization problem: E v (N ) := inf Ev (ψ) : ψ ∈ H1/2 (R3 ), N (ψ) = N ,
(2.1)
where N (ψ) is defined in (1.2), and N > 0, v ∈ R3 , with |v| < 1, denote given parameters. Furthermore, we set Ev (ψ) :=
i 1 ψ, −∆ + m 2 − m ψ + ψ, (v · ∇)ψ 2 2 1 1 − ∗ |ψ|2 |ψ|2 dx. 3 4 R |x|
(2.2)
Any minimizer, ϕv ∈ H1/2 (R3 ), for (2.1) has to satisfy the corresponding Euler-Lagrange equation given by 1 ∗ |ϕv |2 ϕv = −µϕv , −∆ + m 2 − m ϕv + i(v · ∇)ϕv − |x|
(2.3)
with some Lagrange multiplier, −µ ∈ R, where this sign convention turns out to be convenient for our analysis. In what follows, we refer to such minimizers, ϕv , for (2.1) as boosted ground states, since they give rise to moving solitary waves ψ(t, x) = eitµ ϕv (x − vt),
(2.4)
for (1.1) with travelling speed v ∈ R3 with |v| < 1. Concerning existence of boosted ground states, we have the following theorem which generalizes a result derived in [11] for minimizers of (2.1) with v = 0. Theorem 2.1. Suppose that m > 0, v ∈ R3 , and |v| < 1. Then there exists a positive constant Nc (v) depending only on v such that the following holds. i) For 0 < N < Nc (v), problem (2.1) has a minimizer, ϕv ∈ H1/2 (R3 ), and it satisfies (2.3), for some µ ∈ R. Moreover, every minimizing sequence, (ψn ), for (2.1) with 0 < N < Nc (v) is relatively compact in H1/2 (R3 ) up to translations, i. e., there exists a sequence, (yk ), in R3 and a subsequence, (ψn k ), such that ψn k (· + yk ) → ϕv strongly in H1/2 (R3 ) as k → ∞, where ϕv is some minimizer for (2.1). ii) For N ≥ Nc (v), no minimizer exists for problem (2.1), even though E v (N ) is finite for N = Nc (v).
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Remarks. 1) It has been proved in [11] that (2.1) with v = 0 has a spherically symmetric minimizer, which can be chosen to be real-valued and nonnegative. But the proof given in [11] crucially relies on symmetric rearrangement arguments that allow to restrict to radial functions in this special case. For v = 0, such methods cannot be used and a general discussion of (2.1) needs a fundamental change of methods. Fortunately, it turns out that the concentration-compactness method introduced by P.-L. Lions in [12] is tailor-made for studying (2.1). To prove Theorem 2.1, we shall therefore proceed along √ the lines of [12]. But — due to the presence of the pseudodifferential operator −∆ + m 2 in (2.2) — some technical modifications have to be taken into account and they are worked out in detail in App. A. 2) A corresponding existence result for boosted ground states can also be derived when −1/|x| in (2.2) is replaced by some other attractive two-body potential, e. g., a Yukawa type potential Φ(x) = −e−µ|x| /|x| with µ > 0. But then minimal L2 -norm of minimizers may also arise, i. e., the condition N > N∗ (v; Φ) enters for some N∗ (v; Φ) > 0. 3) We do not know whether we have uniqueness of minimizers (up to phase and translation) for problem (2.1). Even the simpler case, where one assumes v = 0, has not been settled yet.
2.1. Setting up the Variational Calculus. Before we turn to the proof of Theorem 2.1, we collect and prove some preliminary results. First one easily verifies that Ev (ψ) is real-valued (by using, for instance, Plancherel’s theorem for the first two terms in (2.2)). Moreover, the inequality √ 1
∗ |ψ|2 |ψ|2 dx ≤ Sv ψ, ( −∆ + iv · ∇)ψ ψ, ψ , (2.5) R3 |x| which is proven in App. B, ensures that Ev (ψ) is well-defined on H1/2 (R3 ). As stated in Lemma B.1, inequality (2.5), with |v| < 1, has an optimizer, Q v ≡ 0, which satisfies √
−∆ Q v + i(v · ∇)Q v −
1 ∗ |Q v |2 Q v = −Q v |x|
(2.6)
and yields the best constant, Sv , in terms of Sv =
2 . Q v , Q v
(2.7)
Correspondingly, we introduce the constant, Nc (v), by Nc (v) :=
2 . Sv
(2.8)
By Lemma B.1, we also have the bounds Nc ≥ Nc (v) ≥ (1 − |v|)Nc ,
(2.9)
where Nc (v = 0) = Nc > 4/π is, of course, the same constant that appeared in Sect. 1. We now state our first auxiliary result for (2.1).
Boson Stars as Solitary Waves
7
Lemma 2.1. Suppose that m ≥ 0, v ∈ R3 , and |v| < 1. Then the following inequality holds:
N √ ψ, −∆ + iv · ∇ ψ − m N 2Ev (ψ) ≥ 1 − (2.10) Nc (v) for all ψ ∈ H1/2 (R3 ) with N (ψ) = N . Here Nc (v) is the constant introduced in (2.8) above. Moreover, we have that E v (N ) ≥ − 21 m N for 0 < N ≤ Nc (v) and E v (N ) = −∞ for N > Nc (v). Finally, any minimizing sequence for problem (2.1) is bounded in H1/2 (R3 ) whenever 0 < N < Nc (v). Proof (of Lemma 2.1). Let the assumption on m√and v stated above be satisfied. Estimate √ (2.10) is derived by noting that −∆ + m 2 ≥ −∆ and using inequality (2.5) together with the definition of Nc (v) in (2.8). Furthermore, that E v (N ) ≥ − 21 m N for N ≤ Nc (v) is a consequence of (2.10) itself. To see that E v (N ) = −∞ when N > Nc (v), we recall from Lemma B.1 that there exists an optimizer, Q v ∈ H1/2 (R3 ), with N (Q v ) = Nc (v), for inequality√(2.5). Using that Q v turns (2.5) into an equality and noticing that √ −∆ + m 2 − m ≤ −∆ holds, a short calculation yields E v (N ) ≤ Ev (λQ v )
m=0
=−
1 λ2 (λ2 − 1) ∗ |Q v |2 |Q v |2 dx. 4 R3 |x|
(2.11)
For N > Nc (v), we can choose λ > 1 which implies that the right-hand side is strictly negative and, in addition, by L 2 -norm preserving rescalings, Q v (x) → a 3/2 Q v (ax) with a > 0, we find that E v (N ) ≤ Ev (λa 3/2 Q v (a·)) = aEv (λQ v ) → −∞, (2.12) m=0
m=0
with λ > 1 fixed and as a → ∞. Thus, we deduce that E v (N ) = −∞ when N > Nc (v). To see the H1/2 (R3 )-boundedness of any minimizing √ √ sequence, (ψn ), with 0 < N < Nc (v), we note that −∆ + iv · ∇ ≥ (1 − |v|) −∆ holds. Hence we see that √ supn ψn , −∆ψn ≤ C < ∞, thanks to (2.10). This completes the proof of Lemma 2.1. As a next step, we derive an upper bound for E v (N ), which is given by the nonrelativistic ground state energy, E vnr (N ), defined below. Here the positivity of the mass parameter, m > 0, is essential for deriving the following estimate. Lemma 2.2. Suppose that m > 0, v ∈ R3 , and |v| < 1. Then we have that E v (N ) ≤ −
1 1 − 1 − v 2 m N + E vnr (N ), 2
(2.13)
where E vnr (N ) is given by E vnr (N ) := inf Evnr (ψ) : ψ ∈ H1 (R3 ), N (ψ) = N , Evnr (ψ)
√ 1 1 − v2 1 ∗ |ψ|2 |ψ|2 dx. := |∇ψ|2 dx − 4m 4 R3 |x| R3
(2.14) (2.15)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Proof (of Lemma 2.2). To prove (2.13), we pick a spherically symmetric function, φ ∈ H1 (R3 ) with N (φ) = N , and we introduce the one-parameter family φλ (x) := eiλv·x φ(x) = eiλ|v|z φ(x), with λ > 0.
(2.16)
Here and in what follows, we assume (without loss of generality) that v is parallel to the z-axis, i. e., v = |v|ez . One checks that
i λv 2 φλ , (v · ∇)φλ = − N, 2 2
(2.17)
using the fact that φ, ∇φ = 0 holds, by spherical symmetry of φ(x). Hence, we find that
i|v| 1 φλ , ∂ z φλ Ev (φλ ) = φλ , −∆ + m 2 − m φλ + 2 2 1 1 2 2 − ∗ |φλ | |φλ | dx 4 R3 |x|
1 1
1 2 2 φλ , −∆ + m − m φλ − v λN − ∗ |φ|2 |φ|2 dx = 2 4 R3 |x| =: A + B. (2.18) To estimate A in (2.18), we recall the operator inequality
−∆ + m 2 ≤
1 (−∆ + m 2 + λ2 ), 2λ
(2.19)
which follows from the elementary inequality 2ab ≤ a 2 + b2 . Thus, we are led to
1 1 1 φλ , (−∆ + m 2 + λ2 )φλ − m N − v 2 λN 4λ 2 2 1 1 2 2 1 2 2 = λ v N + φ, −∆φ + (m + λ )N − m N − v 2 λN . 4λ 2 2
A≤
(2.20)
By minimizing the upper bound (2.20) with respect √ to λ > 0, which is a matter of elementary calculations, we obtain with λ∗ = m/ 1 − v 2 the estimate 1 Ev (φλ∗ ) ≤ − 1 − 1 − v 2 m N 2 √ 1 1 1 − v2 + φ, −∆φ − ∗ |φ|2 |φ|2 dx 4m 4 R3 |x| 1 = − 1 − 1 − v 2 m N + Evnr (φ). 2
(2.21)
Next, we remark that Evnr (ψ) is the energy√ functional for the non-relativistic boson star problem with mass parameter m v = m/ 1 − v 2 . Indeed, it is known from [8] that Evnr (ψ) subject to N (ψ) = N has a spherically symmetric minimizer, φ∗ ∈ H1 (R3 ), with E vnr (N ) = Evnr (φ∗ ) < 0, (2.22) which completes the proof of Lemma 2.2.
Boson Stars as Solitary Waves
9
By making use of Lemma 2.2, we show that the function E v (N ) satisfies a strict subadditivity condition. This is essential to the discussion of (2.1) when using concentrationcompactness-type methods. Lemma 2.3. Suppose that m > 0, v ∈ R3 , and |v| < 1. Then E v (N ) satisfies the strict subadditivity condition E v (N ) < E v (α) + E v (N − α) (2.23) whenever 0 < N < Nc (v) and 0 < α < N . Here Nc (v) is the constant of Lemma 2.1. Moreover, the function E v (N ) is strictly decreasing and strictly concave in N , where 0 < N < Nc (v). Remarks. 1) Condition m > 0 is necessary for (2.23) to hold. To see this, note that if m = 0 then Ev (ψλ ) = λEv (ψ) holds, where ψλ = λ3/2 ψ(λx) and λ > 0. This leads to the conclusion that E v (N ) is either 0 or −∞ when m = 0. 2) The fact that E v (N ) is strictly concave will be needed in our companion paper [4] when making use of the symplectic structure associated with the Hamiltonian PDE (1.1). More precisely, the strict concavity of E v (N ) will enable us to prove the nondegeneracy of the symplectic form restricted to the manifold of solitary waves. nr (N ) < 0 holds, Proof (of Lemma 2.3). By Lemma 2.2 and the fact that E vnr (N ) ≤ E v=0 by (2.22), we deduce that
E v (N ) < −
1 1 − 1 − v2 m N . 2
(2.24)
Next, we notice the following scaling behavior: E v (N ) = N ev (N ),
(2.25)
where ev (N ) :=
1
i −∆ + m 2 − m ψ + ψ, (v · ∇)ψ 2 2 N 1 ∗ |ψ|2 |ψ|2 dx . − (2.26) 4 R3 |x| inf
ψ∈H1/2 , ψ 22 =1
ψ,
This shows that ev (N ) is strictly decreasing, provided that we know that we may restrict the infimum to elements such that 1 (2.27) ∗ |ψ|2 |ψ|2 dx ≥ c > 0 R3 |x| holds for some c. Suppose now that (2.27) were not true. Then there exists a minimizing sequence, (ψn ), such that 1 ∗ |ψn |2 |ψn |2 dx → 0, as n → ∞. (2.28) 3 R |x| But on account of the fact that (cf. App. C)
ψ, −∆ + m 2 − m ψ + i ψ, (v · ∇)ψ ≥ − 1 − 1 − v 2 m N ,
(2.29)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
we conclude that E v (N ) = N ev (N ) ≥ −
1 1 − 1 − v2 m N , 2
(2.30)
which contradicts (2.24). Thus ev (N ) is strictly decreasing. Returning to (2.25) and noting that ev (N ) < 0 holds, by (2.24), we deduce that E v (ϑ N ) < ϑ E v (N ), for ϑ > 1 and 0 < N < Nc (v).
(2.31)
By an argument presented in [12], this inequality leads to the strict subadditivity condition (2.23). Finally, we show that E v (N ) is strictly decreasing and strictly concave on the interval (0, Nc (v)). To see that E v (N ) = N ev (N ) is strictly decreasing, we notice that ev (N ) is strictly decreasing and negative. Furthermore, we remark that ev (N ) = inf{linear functions in N } has to be a concave function. Therefore it follows that E v (N ) = N ev (N ) is strictly concave, since the left- and right-derivatives, D ± E v (N ), exist and are found to be strictly decreasing, by using that ev (N ) is concave and strictly decreasing. 2.2. Proof of Theorem 2.1. We now come to the proof of Theorem 2.1 and we suppose that m > 0, v ∈ R3 , and |v| < 1 holds. Proof of Part i) Let us assume that 0 < N < Nc (v),
(2.32)
where Nc (v) is the constant defined in (2.8). Furthermore, let (ψn ) be a minimizing sequence for (2.1), i. e., lim Ev (ψn ) = E v (N ), with ψn ∈ H1/2 (R3 ) and N (ψn ) = N for all n ≥ 0. (2.33)
n→∞
By Lemma 2.1, we have that E v (N ) > −∞ and that (ψn ) is a bounded sequence in H1/2 (R3 ). We now apply the following concentration-compactness lemma. Lemma 2.4. Let (ψn ) be a bounded sequence in H1/2 (R3 ) such that N (ψn ) = R3 |ψn |2 dx = N for all n ≥ 0. Then there exists a subsequence, (ψn k ), satisfying one of the three following properties: i) Compactness: There exists a sequence, (yk ), in R3 such that, for every > 0, there exists 0 < R < ∞ with |ψn k |2 dx ≥ N − . (2.34) |x−yk |
ii) Vanishing: lim sup
k→∞ y∈R3 |x−y|
|ψn k |2 dx = 0, for all R > 0.
Boson Stars as Solitary Waves
11
iii) Dichotomy: There exists α ∈ (0, N ) such that, for every > 0, there exist two bounded sequences, (ψk1 ) and (ψk2 ), in H1/2 (R3 ) and k0 ≥ 0 such that, for all k ≥ k0 , the following properties hold: ψn − (ψ 1 + ψ 2 ) ≤ δ p (), for 2 ≤ p < 3, (2.35) k k k p with δ p () → 0 as → 0, and |ψk1 |2 dx − α ≤ and R3
R3
|ψk2 |2 dx − (N − α) ≤ ,
dist (supp ψk1 , supp ψk2 ) → ∞, as k → ∞. Moreover, we have that
lim inf ψn k , T ψn k − ψk1 , T ψk1 − ψk2 , T ψk2 ≥ −C(), k→∞
where C() → 0 as → 0 and T := v ∈ R3 , |v| < 1.
(2.36) (2.37)
(2.38)
√ −∆ + m 2 − m + i(v · ∇) with m ≥ 0 and
Remark. We refer to App. A for the proof of Lemma 2.4. Part i) and ii) are standard, but part iii) requires some technical arguments, due to the presence of the pseudo-differential operator T . Invoking Lemma 2.4, we conclude that a suitable subsequence, (ψn k ), satisfies either i), ii), or iii). We rule out ii) and iii) as follows. Suppose that (ψn k ) exhibits property ii). Then we conclude that 1 ∗ |ψn k |2 |ψn k |2 dx = 0, (2.39) lim 3 k→∞ R |x| by Lemma A.1. But as shown in the proof of Lemma 2.3, this implies E v (N ) ≥ −
1 1 − 1 − v2 m N , 2
(2.40)
which contradicts (2.24). Hence ii) cannot occur. Let us suppose that iii) is true for (ψn k ). Then there exists α ∈ (0, N ) such that, for every > 0, there are two bounded sequences, (ψk1 ) and (ψk2 ), with α − ≤ N (ψk1 ) ≤ α + , (N − α) − ≤ N (ψk2 ) ≤ (N − α) + ,
(2.41)
for k sufficiently large. Moreover, inequality (2.38) and Lemma A.2 allow us to deduce that E v (N ) = lim Ev (ψn k ) ≥ lim inf Ev (ψk1 ) + lim inf Ev (ψk2 ) − r (), k→∞
k→∞
k→∞
(2.42)
where r () → 0 as → 0. Since (ψk1 ) and (ψk2 ) satisfy (2.41), we infer E v (N ) ≥ E v (α + ) + E v (N − α + ) − r (),
(2.43)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
using that E v (N ) is decreasing in N . Passing to the limit → 0 and by continuity of E v (N ) in N (recall that E v (N ) is a concave function on an open set), we deduce that E v (N ) ≥ E v (α) + E v (N − α)
(2.44)
holds for some 0 < α < N . This contradicts the strict subadditivity condition (2.23) stated in Lemma 2.3. Therefore iii) is ruled out. By the discussion so far, we conclude that there exists a subsequence, (ψn k ), such that i) of Lemma 2.4 is true for some sequence (yk ) in R3 . Let us now define the sequence k := ψn k (· + yk ). ψ
(2.45)
k ) is a bounded sequence in H1/2 (R3 ), we can pass to a subsequence, still Since (ψ k ), such that (ψ k ) converges weakly in H1/2 (R3 ) to some ϕv ∈ H1/2 (R3 ) denoted by (ψ k → ϕv strongly in L p (R3 ) as k → ∞, for as k → ∞. Moreover, we have that ψ loc 2 ≤ p < 3, thanks to a Rellich-type theorem for H1/2 (R3 ) (see, e. g., [9, Theorem 8.6] for this). But on account of the fact k |2 dx ≥ N − , |ψ (2.46) |x|
for every > 0 and suitable R = R() < ∞, we conclude that k → ϕv strongly in L p (R3 ) as k → ∞, for 2 ≤ p < 3. ψ
(2.47)
Next, by the Hardy–Littlewood–Sobolev and Hölder’s inequality, we deduce that 1 1 k |2 |ψ k |2 dx − ∗ |ψ ∗ |ϕv |2 |ϕv |2 dx R3 |x| R3 |x| k 312/5 + ϕv 312/5 ψ k − ϕv 12/5 . ≤ C ψ (2.48) k converges strongly to ϕv in L12/5 (R3 ), as k → ∞, and From (2.47), we have that ψ therefore 1 1 k |2 |ψ k |2 dx = ∗ |ψ ∗ |ϕv |2 |ϕv |2 dx. (2.49) lim k→∞ R3 |x| R3 |x| Moreover, we have that k ) ≥ Ev (ϕv ) ≥ E v (N ), E v (N ) = lim Ev (ψ
(2.50)
T (ψ) := ψ, −∆ + m 2 ψ + i ψ, (v · ∇)ψ ,
(2.51)
k→∞
since the functional
is weakly lower semicontinuous on H1/2 (R3 ), see Lemma A.4 in App. A. Thus, we have proved that ϕv ∈ H1/2 (R3 ) is a minimizer for (2.1), i. e., we have E v (N ) = Ev (ϕv ) and N (ϕv ) = N . To prove the relative compactness of minimizing sequences in H1/2 (R3 ) (up to transk ) = lations), we notice that there has to be equality in (2.50), which leads to limk→∞ T (ψ T (ϕv ). By Lemma A.4, this fact implies a posteriori that k → ϕv strongly in H1/2 (R3 ) as k → ∞, ψ which completes the proof of part i) of Theorem 2.1.
(2.52)
Boson Stars as Solitary Waves
13
Proof of Part ii) To complete the proof of Theorem 2.1, we address its part ii). Clearly, no minimizer exists if N > Nc (v), since in this case we have that E v (N ) = −∞, by Lemma 2.1. Next, we show that E v (N ) = − 21 m N holds if N = Nc (v), which can be seen as follows. We take an optimizer, Q v ∈ H1/2 (R3 ), for inequality (2.5); see Lemma B.1 and recall that N (Q v ) = Nc (v). Then √
1 1 (λ) E v (N ) ≤ Ev (Q (λ) Q v , −∆ + m 2 − −∆ Q (λ) (2.53) − mN, v )= v 2 2 (λ)
(λ)
for N = Nc (v), where Q v (x) := λ3/2 Q v (λx) with λ > 0, so that N (Q v ) = N (Q v ) = Nc (v). Using Plancherel’s theorem and by dominated convergence, we deduce that √ (λ) (λ)
2 v (k)|2 λ2 k 2 + m 2 − λ|k| dk Q v , −∆ + m − −∆ Q v = |Q R3
→ 0 as λ → ∞.
(2.54)
Thus, we conclude that E v (N ) ≤ − 21 m N for N = Nc (v). In combination with the estimate E v (N ) ≥ − 21 m N for N ≤ Nc (v) taken from Lemma 2.1, this shows that 1 E v (N ) = − m N for N = Nc (v). 2
(2.55)
Finally, we prove that there does not exist a minimizer for (2.1) with N = Nc (v). We argue by contradiction as follows. Suppose that ϕv ∈ H1/2 (R3 ) is a minimizer for (2.1) √ with √ N = Nc (v). For ϕv ∈ H1/2 (R3 ), ϕv ≡ 0, and m > 0, we use ϕv , −∆ + m 2 ϕv > ϕv , −∆ ϕv to obtain 1 1 1 − m N = Ev (ϕv )m>0 > Ev (ϕv )m=0 − m N ≥ − m N , 2 2 2
(2.56)
which is a contradiction. Here we use Lemma 2.1 to estimate Ev (ϕv )|m=0 ≥ 0 for N (ϕv ) = Nc (v). Hence no minimizer exists for (2.1) if N ≥ Nc (v). This completes the proof of Theorem 2.1. 3. Orbital Stability The purpose of this section is to address “orbital stability” of travelling solitary waves ψ(t, x) = eitµ ϕv (x − vt),
(3.1)
where ϕv ∈ H1/2 (R3 ) is a boosted ground state. By the relative compactness of minimizing sequences (see Theorem 2.1) and by using a general idea presented in [2], we are able to prove the following abstract stability result. Theorem 3.1. Suppose that m > 0, v ∈ R3 , |v| < 1, and 0 < N < Nc (v). Let Sv,N denote the corresponding set of boosted ground states, i. e., Sv,N := ϕv ∈ H1/2 (R3 ) : Ev (ϕv ) = E v (N ), N (ϕv ) = N , which is non-empty by Theorem 2.1.
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Then the solitary waves given in (3.1), with ϕv ∈ Sv,N , are stable in the following sense. For every > 0, there exists δ > 0 such that inf
ϕv ∈Sv,N
ψ0 − ϕv H1/2 ≤ δ implies that sup inf
t≥0 ϕv ∈Sv,N
ψ(t) − ϕv H1/2 ≤ .
Here ψ(t) denotes the solution of (1.1) with initial condition ψ0 ∈ H1/2 (R3 ). Remark. It is an interesting and open question whether uniqueness (modulo phase and translation) of boosted ground states holds, i. e., we have that Sv,N is of the form {eiγ ϕv (· − y) : γ ∈ R, y ∈ R3 }, for some fixed ϕv ∈ Sv,N . Proof (of Theorem 3.1). Let m and v satisfy the given assumptions. Since we have N < Nc (v) ≤ Nc , we can choose δ > 0 sufficiently small such that inf φ∈Sv,N ψ0 − φ H1/2 ≤ δ guarantees that N (ψ0 ) < Nc . By the global well-posedness result for (1.1) derived in [7], we have that the corresponding solution, ψ(t), exists for all times t ≥ 0. Thus, taking supt≥0 is well-defined. Let us now assume that orbital stability (in the sense defined above) does not hold. Then there exists a sequence on initial data, (ψn (0)), in H1/2 (R3 ) with inf ψn (0) − ϕ H1/2 → 0, as n → ∞,
ϕ∈Sv,N
(3.2)
and some > 0 such that inf ψn (tn ) − ϕ H1/2 > , for all n ≥ 0,
ϕ∈Sv,N
(3.3)
for a suitable sequence of times (tn ). Note that (3.2) implies that N (ψn (0)) → N as n → ∞. Since N < Nc by assumption, we can assume — without loss of generality — that N (ψn (0)) < Nc holds for all n ≥ 0, which guarantees (see above) that the corresponding solution, ψn (t), exists globally in time. Next, we consider the sequence, (βn ), in H1/2 (R3 ) that is given by βn := ψn (tn ).
(3.4)
By conservation of N (ψ(t)) and of Ev (ψ(t)), whose proof can be done along the lines of [7] for the conservation of E(ψ(t)), we have that N (βn ) = N (ψn (0)) and Ev (βn ) = Ev (ψn (0)), which, by (3.2), implies lim Ev (βn ) = E v (N ) and
n→∞
lim N (βn ) = N .
n→∞
Defining the rescaled sequence √ n := an βn , where an := N /N (βn ), β and using the fact (βn that n H1/2 ≤ C|1 − an | → 0, as n → ∞. βn − β 1/2 By continuity of Ev : H (R3 ) → R, we deduce that n ) = E v (N ) and N (β n ) = N , for all n ≥ 0. lim Ev (β
(3.5)
(3.6)
) has to be bounded in H1/2 (R3 ), by virtue of Lemma 2.1, we infer
n→∞
(3.7) (3.8)
n ) is a minimizing sequence for (2.1) which, by Theorem 2.1 part i), has to Therefore (β n k ), that strongly converges in H1/2 (R3 ) (up to translations) to contain a subsequence, (β some minimizer ϕ ∈ Sv,N . In particular, inequality (3.3) cannot hold when βn = ψn (tn ) n . But in view of (3.7), this conclusion is easily extended to the sequence is replaced by β (βn ) itself. Thus, we are led to a contradiction and the proof of Theorem 3.1 is complete.
Boson Stars as Solitary Waves
15
4. Properties of Boosted Ground States Concerning fundamental properties of boosted ground states given by Theorem 2.1, we have the following result. Theorem 4.1. Let m > 0, v ∈ R3 , |v| < 1, and 0 < N < Nc (v). Then every boosted ground state, ϕv ∈ H1/2 (R3 ), of problem (2.1) satisfies the following properties. i) ϕv ∈ Hs (R3 ) for all s ≥ 1/2. √ ii) The corresponding Lagrange multiplier satisfies µ > (1 − 1 − v 2 )m. Moreover, we have pointwise exponential decay, i. e., |ϕv (x)| ≤ Ce−δ|x|
(4.1)
holds for all x ∈ R3 , where δ > 0 and C > 0 are suitable constants. iii) For v = 0, the function ϕv (x) can be chosen to be radial, real-valued, and strictly positive. Remarks. 1) By part i) and Sobolev embeddings, any boosted ground state is smooth: ϕv ∈ C∞ (R3 ). Moreover, we have that ϕv ∈ L1 ∩ L∞ , due to part ii). 2) Part iii) follows from the discussion presented in [11], except for the strict positivity which we will show below. 3) For a more precise exponential decay estimate for ϕv (x), see Lemma C.1 in App. C. Proof (of Theorem 4.1). Part i): We rewrite the Euler-Lagrange equation (3.1) for ϕv as (H0 + λ)ϕv = F(ϕv ) + (λ − µ)ϕv , for any λ ∈ R, where H0 := −∆ + m 2 − m + i(v · ∇),
F(ϕv ) :=
1 ∗ |ϕv |2 ϕv . |x|
(4.2)
(4.3)
By [7, Lemma 3], we have that F : Hs (R3 ) → Hs (R3 ) for all s ≥ 1/2 (F is indeed locally Lipschitz). Thus, the right-hand side in (4.2) belongs to H1/2 (R3 ). Since H0 is bounded from below, we can choose λ > 0 sufficiently large such that (H0 + λ)−1 exists. This leads to ϕv = (H0 + λ)−1 F(ϕv ) + (λ − µ)ϕv . (4.4) Noting that (H0 +λ)−1 : Hs (R3 ) → Hs+1 (R3 ), we see that ϕv ∈ H3/2 (R3 ). By repeating the argument, we conclude that ϕv ∈ Hs (R3 ) for all s ≥ 1/2. This proves part i). Part ii): The exponential decay follows from Lemma C.1, provided that the Lagrange multiplier, −µ, satisfies −µ < − 1 − 1 − v 2 m, (4.5) which means that −µ lies strictly below the essential spectrum of H0 ; see App. C. To prove (4.5), we multiply the Euler-Lagrange equation by ϕ v and integrate to obtain 1 1 ∗ |ϕv |2 |ϕv |2 dx = −µN . (4.6) 2E v (N ) − 2 R3 |x| Using the upper bound (2.24) for E v (N ), we conclude that −µN < − 1 − 1 − v 2 m N ,
(4.7)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
which proves (4.5). Part iii): For the sake of brevity, we write ϕ(x) := ϕv=0 (x). By [11] problem (2.1), with v = 0, has a minimizer that equals its symmetric-decreasing rearrangement, i. e., ϕ(x) = ϕ ∗ (x). In particular, ϕ(x) is a spherically symmetric, real-valued, nonincreasing function with ϕ(x) ≥ 0. It remains to show that ϕ(x) > 0 holds. To see this, we put λ = µ in (4.4), which is possible by the proof of ii), and we obtain −1 ϕ= −∆ + m 2 − m + µ F(ϕ). (4.8) √ By using functional calculus for the self-adjoint operator −∆ + m 2 on L2 (R3 ) with domain H1 (R3 ), we find that ∞ √ −1 2 −∆ + m 2 − m + µ = e−tµ e−t ( −∆+m −m) dt. (4.9) 0
Referring to the explicit formula (C.10) for v = 0, we see that the integral kernel, √ 2 −m) −t ( −∆+m e (x, y), is strictly positive. In view of (4.8), (4.9), and the fact that F(ϕ) ≥ 0, we conclude that ϕ(x) > 0 holds for almost every x ∈ R3 . But since ϕ(x) is a nonincreasing and continuous function, we deduce that ϕ(x) > 0 has to be true for all x ∈ R3 . This completes the proof of Theorem 4.1. 5. Outlook Our analysis presented so far serves as a basis for the upcoming work in [4] which explores the effective motion of travelling solitary waves in an external potential. More precisely, we consider 1 i∂t ψ = ∗ |ψ|2 ψ on R3 . −∆ + m 2 − m ψ + V ψ − (5.1) |x| Here the external potential V : R3 → R is assumed to be a smooth, bounded function with bounded derivatives. Note that its spatial variation introduces the length scale ext = ∇V −1 ∞.
(5.2)
In addition, another length scale, sol , enters through the exponential decay of ϕv (x), i. e., we have that sol = δ −1 , (5.3) where δ > 0 is the constant taken from Theorem 4.1. On intuitive grounds, one expects that if we have that sol ext (5.4) holds, then solutions, ψ(t, x), of (5.1) that are initially close to some ϕv (x) should approximately behave like point-particles, at least on a large (but possibly finite) interval of time. We now briefly sketch how this heuristic picture of point-particle behavior of solitary waves is addressed by rigorous analysis in [4]. There we introduce a nondegeneracy assumption on the linearized operator L1 0 L := (5.5) 0 L2
Boson Stars as Solitary Waves
17
acting on L2 (R3 ; R2 ) with domain H1 (R3 ; R2 ), where L 1 ξ :=
1 1 ∗ ϕ2 ξ − 2 ∗ (ϕξ ) ϕ, −∆ + m 2 − m + µ0 ξ − |x| |x|
L 2 ξ :=
1 ∗ ϕ 2 ξ. −∆ + m 2 − m + µ0 ξ − |x|
(5.6) (5.7)
Here ϕ(x) = ϕv=0 (x) is some ground state at rest, whose corresponding Lagrange multiplier we denote by µ0 ∈ R. Note that we can assume that ϕv=0 (x) is spherically symmetric and real-valued, see Theorem 4.1. The nondegeneracy condition introduced in [4] now reads 0 ∂x1 ϕ ∂x2 ϕ ∂x3 ϕ ker(L) = span , , , . (5.8) ϕ 0 0 0 Under this kernel assumption, we then construct in [4] (by an implicit-function-type argument) a map (v, µ) → ϕv,µ , so that ϕv,µ ∈ H1/2 solves Eq. (1.7) and (v, µ) belongs to the small neighborhood around (0, µ0 ). The main result proven in [4] can now be sketched as follows. We consider suitable external potentials of the form V (x) := W (x). (5.9) Furthermore, let ϕv0 ,µ0 with |v0 | 1 be given and choose 1 so that (5.4) holds. Then for any initial datum, ψ0 (x), such that |||ψ0 − eiϑ0 ϕv0 ,µ0 (· − a0 )||| ≤ , for some ϑ0 ∈ R and a0 ∈ R3 ,
(5.10)
where ||| · ||| is some weighted Sobolev norm, the corresponding solution, ψ(t, x), of (5.1) can be uniquely written as ψ(t, x) = eiϑ [ϕv,µ (x − a) + ξ(t, x − a)], for 0 ≤ t < C −1 .
(5.11)
Here |||ξ ||| = O() holds and the time-dependent functions {ϑ, a, v, µ} satisfy equations of the following form: N˙ = O( 2 ), ϑ˙ = µ − V (a) + O( 2 ), (5.12) 2 a˙ = v + O( ), γ (µ, v)v˙ = −∇V (a) + O( 2 ), where N = N (ϕv,µ ). The term γ (µ, v) can be viewed as an “effective mass” which takes relativistic effects into account. Finally, we remark that the proof of (5.11) and (5.12) makes extensive use of the Hamiltonian formulation of (5.1) and its associated symplectic structure restricted to the manifold of solitary waves. Moreover, assumption (5.8) enables us to derive additional properties of ϕv (x), for |v| 1, such as cylindrical symmetry with respect to the v-axis, which is of crucial importance in the analysis presented in [4]. A. Variational and Pseudo-Differential Calculus In this section of the appendix, we collect and prove results needed for our variational and pseudo-differential calculus.
18
J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
A.1. Proof of Lemma 2.4. Let (ψn ) be a bounded sequence in H1/2 (R3 ) with ψn 22 = N for all n. Along the lines of [12], we define the sequence, (Q n ), of Lévy concentration functions by Q n (R) := sup |ψn |2 dx, for R ≥ 0. (A.1) y∈R3 |x−y|
As stated in [12], there exists a subsequence, (Q n k ), such that Q n k (R) → Q(R) as k → ∞ for all R ≥ 0,
(A.2)
where Q(R) is a nonnegative, nondecreasing function. Clearly, we have that α := lim Q(R) ∈ [0, N ]. R→∞
(A.3)
If α = 0, then situation ii) of Lemma 2.4 arises as a direct consequence of definition (A.1). If α = N , then i) follows, see [12] for details. Assume that α ∈ (0, N ) holds, and let > 0 be given. Suppose that ξ, φ ∈ C∞ (R3 ) with 0 ≤ φ, ξ ≤ 1 such that ξ(x) ≡ 1 for 0 ≤ |x| ≤ 1,
ξ(x) ≡ 0 for |x| ≥ 2,
(A.4)
φ(x) ≡ 0 for 0 ≤ |x| ≤ 1,
φ(x) ≡ 1 for |x| ≥ 2.
(A.5)
Furthermore, we put ξ R (x) := ξ(x/R) and φ R (x) := φ(x/R), for R > 0, and we introduce (A.6) ψk1 := ξ R1 (· − yk )ψn k and ψk2 := φ Rk (· − yk )ψn k . As shown in [12, Proof of Lemma III.1], there exists
and a sequence, (Rk ), with
R1 () → ∞, as → 0,
(A.7)
Rk → ∞, as k → ∞,
(A.8)
such that (ψk1 ) and (ψk2 ) satisfy (2.36) and (2.37) in Lemma 2.4. Moreover, we have that
R3
|ψn k − (ψk1 + ψk2 )|2 dx ≤ 4,
(A.9)
for k sufficiently large. By [9, Theorem 7.16], we see that ψk1 and ψk2 defined in (A.6) are bounded in 1/2 H (R3 ). More precisely, using the technique of the proof given there and the explicit formula √ | f (x) − f (y)|2 dx dy, for f ∈ H1/2 (R3 ), (A.10) f, −∆ f = (const.) 4 3 3 |x − y| R ×R we deduce that
g f H1/2 ≤ C g ∞ + ∇g ∞ f H1/2 .
(A.11)
1 1 ψk1 H1/2 ≤ C 1 + and ψk2 H1/2 ≤ C 1 + , R1 Rk
(A.12)
Thus, we find that
Boson Stars as Solitary Waves
19
for some constant C = C(M), where M = supk≥0 ψn k H1/2 < ∞. Thus, (ψk1 ) and (ψk2 ) are bounded sequences in H1/2 (R3 ). This fact together with Hölder’s and Sobolev’s inequalities leads to ψn − (ψ 1 + ψ 2 ) ≤ δ p (), for 2 ≤ p < 3, k k k p
(A.13)
where δ p () → 0 as → 0. This proves (2.35) in Lemma 2.4. It remains to show property (2.38) in Lemma 2.4. Since
lim inf ψn k , (−m)ψn k − ψk1 , (−m)ψk1 − ψ 2 , (−m)ψk2
(A.14)
≥ −m N + m(α − ) + m(N − α − ) ≥ −2m → 0, as → 0,
(A.15)
k→∞
we observe that it suffices to prove the claim
lim inf ψn k , Aψn k − ψk1 , Aψk1 − ψk2 , Aψk2 ≥ −C(), k→∞
(A.16)
for some constant C() → 0 as → 0, where A :=
−∆ + m 2 + i(v · ∇) + λ,
(A.17)
with m ≥ 0, v ∈ R3 , |v| < 1, and λ > 0 is some constant so that √ A ≥ (1 − |v|) −∆ + λ ≥ λ > 0.
(A.18)
In view of (A.14), adding any fixed λ can be done without loss of generality. Next, we recall definition (A.6) and rewrite the left-hand side in (A.16) as follows
lim inf ψn k , (A − ξk Aξk − φk Aφk )ψn k ,
(A.19)
ξk (x) := ξ R1 (x − yk ) and φk (x) := φ Rk (x − yk ).
(A.20)
k→∞
where Using commutators [X, Y ] := X Y − Y X , we find that A − ξk Aξk − φk Aφk = A(1 − ξk2 − φk2 ) − [ξk , A]ξk − [φk , A]φk √ √ √ √ = A(1 − ξk2 − φk2 ) A − A[ A, (ξk2 + φk2 )] −[ξk , A]ξk − [φk , A]φk . (A.21) Note that
√
A > 0 holds, due to A > 0. By applying Lemma A.3, we obtain [ξk , A]
L2 →L2
[φk , A]
L2 →L2
≤ C ∇ξk ∞ ≤
C , R1
(A.22)
≤ C ∇φk ∞ ≤
C . Rk
(A.23)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
To estimate the remaining commutator in (A.21), we use (A.58) in the proof of Lemma A.3 to find that ∞ √ √ 1 2 [ A, (ξ 2 + φ 2 )] 2 2 ≤ C 1 + 1 2 2 ds s (A.24) k k L →L R1 Rk (s + A) L →L 0 ∞ √ 1 s 1 ≤C + ds (A.25) R1 Rk (s + λ)2 0 1 1 . (A.26) + ≤C R1 Rk Returning to (A.19) and using that ψn k H1/2 ≤ C, we conclude, for k large, that √ √ ψn k , (A − ξk Aξk − φk Aφk )ψn k ≥ Aψn k , (1 − ξk2 − φk2 ) Aψn k 1 1 (A.27) −C + R1 Rk 1 1 , (A.28) ≥ −C + R1 Rk since (1 − ξk2 − φk2 )(x) ≥ 0 when k is sufficiently large. Finally, we note that Rk → ∞ as k → ∞ as well as R1 () → ∞ as → 0 holds, which leads to
(A.29) lim inf ψn k , (A − ξk Aξk − φk Aφk )ψn k ≥ −C() → 0, as → 0. k→∞
The proof of Lemma 2.4 is now complete.
A.2. Technical Details for the Proof of Theorem 2.1. Lemma A.1. Let (ψn ) satisfy the assumptions of Lemma 2.4. Furthermore, suppose that there exists a subsequence, (ψn k ), that satisfies part ii) of Lemma 2.4. Then 1 ∗ |ψn k |2 |ψn k |2 dx = 0. lim k→∞ R3 |x| Remark. A similar statement can be found in [12] in the context of other variational problems. For the sake of completeness, we present its proof for the situation at hand. Proof (of Lemma A.1). Let (ψn k ) be a bounded sequence in H1/2 (R3 ) such that |ψn k |2 dx = N , for all k ≥ 0, (A.30) R3
and assume that (ψn k ) satisfies part ii) in Lemma 2.4, i. e., |ψn k |2 dx = 0, for all R > 0. lim sup
(A.31)
For simplicity, let ψk := ψn k . We introduce f δ (x) := |x|−1 χ (x){|x|−1 ≥δ} , with δ > 0,
(A.32)
k→∞ y∈R3 |x−y|
Boson Stars as Solitary Waves
21
where χ A denotes the characteristic function of the set A ⊂ R3 . This definition leads to
1 |ψk (x)|2 |ψk (y)|2 f δ (x − y) dx dy, ∗ |ψk |2 |ψk |2 dx ≤ δC + 3 3 3 |x| R R ×R
(A.33)
where C is some constant. For R > 0 and δ > 0, let gδR (x) := min{ f δ (x), R}, f δR (x) := max{ f δ (x) − R, 0}χ (x){|x|≤R} + f δ (x)χ (x){|x|>R} .
(A.34) (A.35)
Notice that f δ ≤ gδR χ{|x|≤R} + f δR holds. In view of (A.33), this leads to 1 ∗ |ψk |2 |ψk |2 dx ≤ δC + |ψk (x)|2 |ψk (y)|2 R3 |x| R 3 ×R 3 × gδR (x − y)χ (x − y){|x−y|≤R} dx dy
+ ψk 48/3 f δR 2 =: δC + I + I I,
(A.36)
using Young’s inequality and that f δR ∈ L2 (R3 ). By our assumption on (ψk ), we find that 2 |ψk (x)| dx |ψk (y)|2 dy → 0, as k → ∞. I ≤R R3
|x−y|≤R
Furthermore, we have that I I ≤ C f δR 2 ,
(A.37)
by Sobolev’s inequalities and the fact that (ψk ) is bounded in H1/2 (R3 ). Thus, we obtain 0≤
R3
1 ∗ |ψk |2 |ψk |2 dx ≤ δC + C f δR 2 + r (k), for all δ, R > 0, |x|
(A.38)
where r (k) → 0 as k → ∞. Since f δR 2 → 0 as R → ∞, for each fixed δ > 0, the assertion of Lemma A.1 follows by letting R → ∞ and then sending δ to 0. Lemma A.2. Suppose that > 0. Let (ψn ) satisfy the assumptions of Lemma 2.4 and let (ψn k ) be a subsequence that satisfies part iii) with sequences (ψk1 ) and (ψk2 ). Then, for k sufficiently large,
1 1 2 2 − ∗ |ψn k | |ψn k | dx ≥ − ∗ |ψk1 |2 |ψk1 |2 dx 3 |x| R3 |x| R 1 − ∗ |ψk2 |2 |ψk2 |2 dx − r1 (k) − r2 (), R3 |x| where r1 (k) → 0 as k → ∞ and r2 () → 0 as → 0.
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Proof (of Lemma A.2). Let > 0 and suppose that (ψn k ), (ψk1 ), and (ψk2 ) satisfy the assumptions stated above. Introducing βk := ψn k − (ψk1 + ψk2 )
(A.39)
and expanding the squares, we find that 1 1 2 2 ∗ |ψn k | |ψn k | dx = ∗ |ψk1 |2 |ψk1 |2 dx 3 3 |x| |x| R R 4 1 22 2 2 ∗ |ψk | |ψk | dx + In , (A.40) + R3 |x| n=0
where I0 =
I1 = I2 =
I3 = I4 =
1 22 1 1 2 ∗ |ψk | |ψk | dx + 4 ∗ (Re ψ¯ k1 ψk2 ) (Re ψ¯ k1 ψk2 ) dx 2 3 3 |x| |x| R R 1 1 2 1 2 +4 ∗ |ψk | (Re ψ¯ k ψk ) dx R3 |x| 1 ∗ |ψk2 |2 (Re ψ¯ k1 ψk2 ) dx, (A.41) +4 3 |x| R 1 ∗ |ψk1 + ψk2 |2 )(Re β¯k (ψk1 + ψk2 )) dx, 4 (A.42) 3 R |x| 1 ∗ (Re β¯k (ψk1 + ψk2 )) (Re β¯k (ψk1 + ψk2 )) dx 4 R3 |x| 1 ∗ |ψk1 + ψk2 |2 |βk |2 dx, (A.43) +2 3 |x| R 1 ∗ |βk |2 (Re β¯k (ψk1 + ψk2 )) dx, 4 (A.44) 3 |x| R 1 ∗ |βk |2 |βk |2 dx. (A.45) R3 |x|
To estimate I0 , we notice that if k is sufficiently large then ψk1 and ψk2 have disjoint supports receding from each other, i. e., dk := dist (supp ψk1 , supp ψk2 ) → ∞, as k → ∞;
(A.46)
see the proof of Lemma 2.4 in Sect. A.1. Thus, the last three terms of the right-hand side in (A.41) equal 0 if k is large, since ψ¯ k1 ψk2 = 0 a. e. if k is sufficiently large. Also by (A.46), we infer 1 |ψk2 (y)|2 dx dy |ψk1 (x)|2 3 3 |x − y| R ×R χ{|x−y|≥dk } (x − y) 2 = |ψk1 (x)|2 |ψk (y)|2 dx dy 3 3 |x − y| R ×R 1 2 2 2 −1 C ≤ ψk 2 ψk 2 |x| χ{|x|≥dk } (x)∞ ≤ → 0, as k → ∞, (A.47) dk
Boson Stars as Solitary Waves
23
using Young’s inequality. Thus we have shown that |I0 | ≤ r1 (k) → 0, as k → ∞.
(A.48)
The remaining terms I1 –I4 are controlled by the Hardy-Littlewood-Sobolev inequality and Hölder’s inequality as follows: |I1 | ≤ C( ψk1 312/5 + ψk2 312/5 ) βk 12/5 ,
(A.49)
|I2 | ≤ C( ψk1 212/5 + ψk2 212/5 ) βk 212/5 ,
(A.50)
|I3 | ≤ |I4 | ≤
C( ψk1 12/5 C β 412/5 .
+ ψk2 12/5 ) βk 312/5 ,
(A.51) (A.52)
We notice that ψk1 12/5 and ψk2 12/5 are uniformly bounded, by Sobolev’s inequality and the H1/2 -boundedness of these sequences. Furthermore, we have that βk 12/5 ≤ r2 () → 0, as → 0,
(A.53)
by part iii) of Lemma 2.4. Hence we conclude that |I1 + · · · + I4 | ≤ r2 () → 0, as → 0, which proves Lemma A.2.
(A.54)
A.3. Commutator Estimate. An almost identical result is needed in [5], but we provide its proof again. √ Lemma A.3. Let m ≥ 0, v ∈ R3 , and define Av := −∆ + m 2 + i(v · ∇). Furthermore, suppose that f (x) is a locally integrable and that its distributional gradient, ∇ f , is an L∞ (R3 ) vector-valued function. Then we have that [Av , f ] L2 →L2 ≤ Cv ∇ f ∞ , for some constant Cv that only depends on v. Remark. This result can be deduced by means of Calderón–Zygmund theory for singular integral operators and its consequences for pseudo-differential operators (see, e. g., [13, Sect. VII.3]). We give an elementary proof which makes good use of the spectral theorem, enabling us to write the commutator in a convenient way. Proof (of Lemma A.3). Since [i(v · ∇), f ] = iv · ∇ f holds, we have that [i(v · ∇), f ] L2 →L2 ≤ |v| ∇ f ∞ .
(A.55)
Thus, it suffices to prove our assertion for A := Av=0 , i. e., A := p 2 + m 2 , where p = −i∇.
(A.56)
Since A is a self-adjoint operator on L2 (R3 ) (with domain H1 (R3 )), functional calculus (for measurable functions) yields the formula 1 ∞ 1 ds A−1 = . (A.57) √ 2 π 0 s A +s
24
J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Due to this fact and A = A2 A−1 , we obtain the formula √ 1 ∞ s ds [A, f ] = [A2 , f ] 2 . 2 π 0 A +s A +s
(A.58)
Clearly, we have that [A2 , f ] = [ p 2 , f ] = p · [ p, f ] + [ p, f ] · p, which leads to √ 1 ∞ s ds [A, f ] = p · [ p, f ] + [ p, f ] · p 2 . (A.59) π 0 p2 + m 2 + s p + m2 + s Moreover, since [ p, f ] = −i∇ f holds, we have that 1 2 , [ p, f ] L 2 →L 2 ≤ ∇ f ∞ . (A.60) p2 + m 2 + s s 3 Hence we find, for arbitrary test functions ξ, η ∈ C∞ c (R ), that √ ∞
ds s [ p, f ] · p η ξ, p2 + m 2 + s p2 + m 2 + s 0 √ ∞
s ds ≤ [ p, f ]ξ, p η ( p 2 + m 2 + s)2 0 √ ∞
p s ds 1 + ξ, , [ p, f ] · η p2 + m 2 + s p2 + m 2 + s 0 √ ∞ p s ds ≤ [ p, f ]ξ 2 η 2 2 2 2 ( p + m + s) 0 ∞ p ds . +2 ξ 2 ∇ f ∞ η √ 2 s( p + m 2 + s) 2 0
Evaluation of the s-integrals yields (A.61) ≤ C ∇ f ∞ ξ 2 √
p p 2 +m 2
η2 ≤ C ∇ f ∞ ξ 2 η 2 .
(A.61)
(A.62)
The same estimate holds if [ p, f ] · p is replaced by p · [ p, f ] in (A.61). Thus, we have found that ξ, [A, f ]η ≤ C ∇ f ∞ ξ 2 η 2 , for ξ, η ∈ C∞ (R3 ), (A.63) c 3 2 3 with some constant C independent of m. Since C∞ c (R ) is dense in L (R ), the assertion 2 for the L -boundedness of [A, f ] now follows. This completes the proof of Lemma A.3.
A.4. Lower Semicontinuity. Lemma A.4. Suppose that m > 0, v ∈ R3 , with |v| < 1. Then the functional
T (ψ) := ψ, −∆ + m 2 ψ + ψ, i(v · ∇)ψ is weakly lower semicontinuous on H1/2 (R3 ), i. e., if ψk ψ weakly in H1/2 (R3 ) as k → ∞, then lim inf T (ψk ) ≥ T (ψ). k→∞
Moreover, if limk→∞ T (ψk ) = T (ψ) holds, then ψ k → ψ strongly in H1/2 (R3 ) as k → ∞.
Boson Stars as Solitary Waves
25
Proof (of Lemma A.4). Assume that m > 0, v ∈ R3 , with |v| < 1 holds. By Fourier transform and Plancherel’s theorem, we have that
(k)|2 |ψ k 2 + m 2 − (v · k) dk. (A.64) T (ψ) = R3
We notice that
c1 (|k| + m) ≤
k 2 + m 2 − (v · k) ≤ c2 (|k| + m),
(A.65)
for √ some suitable constants c1 , c2 > 0, where the lower bound follows from the inequality k 2 + m 2 ≥ (1 − δ)|k| + δm, with 0 < δ < 1, and the fact that |v| < 1 holds. Thus, ψ T := T (ψ) (A.66) defines a norm that is equivalent to · H1/2 . Consequently, the notion of weak and strong convergence for these norms coincide. Finally, by (A.64), we identify ψ T with the taken with respect to the integration measure L 2 -norm of ψ
k 2 + m 2 − (v · k) dk. (A.67) dµ = The assertion of Lemma A.4 now follows from corresponding properties of the L2 (R3 , µ)-norm; see, e. g., [9, Theorem 2.11] for L p (, µ)-norms, where is a measure space with positive measure, µ, and 1 < p < ∞. B. Best Constant and Optimizers for Inequality (2.5) Lemma B.1. For any v ∈ R3 with |v| < 1, there exists an optimal constant, Sv , such that
1 √ ∗ |ψ|2 |ψ|2 dx ≤ Sv ψ, −∆ + iv · ∇ ψ ψ, ψ (B.1) R3 |x| holds for all ψ ∈ H1/2 (R3 ). Moreover, we have that Sv =
2 , Q v , Q v
(B.2)
where Q v ∈ H1/2 (R3 ), Q v ≡ 0, is an optimizer for (B.1) and it satisfies √ 1 ∗ |Q v |2 Q v = −Q v . −∆ Q v + i(v · ∇)Q v − |x|
(B.3)
In addition, the following estimates hold: Sv=0 <
π and Sv=0 ≤ Sv ≤ (1 − |v|)−1 Sv=0 . 2
(B.4)
Proof (of Lemma B.1). Let v ∈ R3 with |v| < 1 be fixed and consider the unconstrained minimization problem √ 1 ψ, ( −∆ + iv · ∇)ψ ψ, ψ . (B.5) := inf −1 ∗ |ψ|2 )|ψ|2 dx Sv ψ∈H1/2 (R3 ),ψ≡0 R3 (|x| For v = 0, a variational problem equivalent to (B.5) is studied in [11, App. B] by using strict rearrangement inequalities that allow restriction to radial functions. For v = 0, we
26
J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
have to depart this line of argumentation and we employ (similarly to the discussion of (2.1) in Sect. 2) concentration-compactness-type methods. By√scaling properties of (B.5), it suffices to prove the existence of a minimizer with ψ, ( −∆+iv·∇)ψ and ψ, ψ fixed. Thus, we introduce the constrained minimization problem, which is equivalent to (B.5), as follows: 1 ∗ |ψ|2 |ψ|2 dx : ψ, ψ = α, Iv (α, β) := inf − 3 |x| √R (B.6) ψ, ( −∆ + iv · ∇)ψ = β , where α > 0 and β > 0. In particular, it is sufficient to show that Iv (α = 1, β = 1) is finite and attained so that Sv = −Iv (1, 1). (B.7) In fact, we will show that all minimizing sequences for I (1, 1) are relatively compact in H1/2 (R3 ) up to translations. In turn, this relative compactness implies that all minimizing sequences for problem (B.5) are relatively compact in H1/2 (R3 ) up to translations and rescalings: For any minimizing sequence, (ψn ), for (B.5), there exist sequences, {(yk ), (ak ), (bk )}, with yk ∈ R3 , 0 = ak ∈ C, 0 = bk ∈ R, such that (B.8) ak ψn k bk (· + yk ) → Q v strongly in H1/2 (R3 ) as k → ∞, along a suitable subsequence, (ψn k ), and Q v minimizes (B.5). First we show that I (α, β) is indeed finite. The Hardy–Littlewood–Sobolev inequality implies √ 1 ∗ |ψ|2 |ψ|2 dx ≤ C |ψ|2 26/5 = C ψ 412/5 ≤ C ψ, −∆ψ ψ, ψ , (B.9) R3 |x| √ where we use Sobolev’s inequality ψ 23 ≤ C ψ, −∆ ψ in R3 and Hölder’s inequa√ √ lity. Since ψ, −∆ψ ≤ (1 − |v|)−1 ψ, ( −∆ + iv · ∇)ψ , we deduce that I (α, β) ≥ −Cαβ > −∞,
(B.10)
for some constant C. On the other hand, we have that
I (α, β) < 0,
(B.11)
since R3 (|x|−1 ∗ |ψ|2 )|ψ|2 dx = 0 when ψ ≡ 0. Next, we show that Iv (1, 1) is attained. Let (ψn ) be a minimizing sequence for Iv (1, 1). In order to invoke Lemma 2.4, we notice that R3 |ψn |2 dx = 1 and that (ψn ) is √ √ bounded in H1/2 (R3 ), since ψ, ( −∆ + iv · ∇)ψ is equivalent to ψ, −∆ψ when |v| < 1, by (A.65) with m = 0. Let us suppose now that case ii) of Lemma 2.4 occurs. Referring to Lemma A.1, we conclude that I (1, 1) = 0 holds, which contradicts (B.11). Next, let us assume that dichotomy occurs for a subsequence of (ψn ), i. e., property iii) of Lemma 2.4 holds. Using Lemma A.2 and the lim inf-estimate stated in iii) of Lemma 2.4 and by taking the limit → 0, we conclude that Iv (1, 1) ≥ Iv (α, β) + Iv (1 − α, 1 − β),
(B.12)
for some α ∈ (0, 1) and β ∈ [0, 1]. On the other hand, we have the scaling behaviour Iv (α, β) = αβ Iv (1, 1) < 0,
(B.13)
Boson Stars as Solitary Waves
27
which follows from (B.6) and rescaling ψ(x) → aψ(bx) with a, b > 0. Combining (B.12) with (B.13) we get a contradiction. Therefore dichotomy for minimizing sequences is ruled out. In summary, we see that any minimizing sequence, (ψn ), for Iv (1, 1) contains a subsequence, (ψn k ), with a sequence of translations, (yk ), satisfying property i) of Lemma v strongly 2.4. Similarly to the proof of Theorem 2.1, we conclude that ψn k (· + yk ) → Q 1/2 3 1/2 3 in H (R ) as k → ∞, where Q v ∈ H (R ) is a minimizer for Iv (1, 1). To show that the best constant, Sv , is given by (B.2) with Q v minimizing (B.5) and v . Since satisfying (B.3), let us denote the minimizer constructed above for I (1, 1) by Q v also minimizes the unconstrained problem (B.5), it has to satisfy the corresponding Q Euler-Lagrange equation which reads as follows: √
v + iv · ∇ Q v − −∆ Q
2 1 v |2 Q v = 0, v + Q ∗ |Q Sv |x|
(B.14)
√ v , ( −∆ + iv · ∇) Q v = 1 and Q v , Q v = 1 holds. By putting where we use that Q √ −1/2 Q v = 2Sv Q v , we see that Q v minimizes (B.5) and satisfies (B.2). Moreover, we have that Q v , Q v = 2/Sv holds. Finally, we turn to the estimates for Sv stated in Lemma B.1. That Sv=0 < π/2 holds follows from the√appendices in [11, 7].√To see that Sv ≤ (1 − |v|)−1 Sv=0 is true, we use the estimate −∆ ≤ (1 − |v|)−1 ( −∆ + iv · ∇). Moreover, it is known from the discussion in [7] that if v = 0 the minimizer, Q v=0 , for (2.2) can be chosen to be radial (by symmetric rearrangement). This implies that Q v=0 , ∇ Q v=0 = 0, which leads to Sv=0 ≤ Sv . C. Exponential Decay In this section, we prove pointwise exponential decay for solutions, ϕ ∈ H1/2 (R3 ), of the nonlinear equation
1 ∗ |ϕ|2 ϕ = −µϕ. −∆ + m 2 − m ϕ + i(v · ∇)ϕ − |x|
(C.1)
Clearly, ϕ(x) is an eigenfunction for the Schrödinger type operator H = H0 + V, where H0 :=
1 ∗ |ϕ|2 ). −∆ + m 2 − m + i(v · ∇) and V := − |x|
(C.2)
(C.3)
By using the bootstrap argument for regularity (presented in the proof of Theorem 4.1)), we have that ϕ ∈ Hs (R3 ) for all s ≥ 1/2, which shows in particular that ϕ is smooth. Investigating the spectrum of H0 we find that σ (H0 ) = σess (H0 ) = [Σv , ∞), where the bottom of the spectrum is given by Σv = ( 1 − v 2 − 1)m.
(C.4)
(C.5)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
To see this, we remark that the function f (k) = (k 2 + m 2 )1/2 − m − v · k
(C.6)
√ √ obeys f (k) ≥ ( 1 − v 2 )m − m with equality for k = (mv/ 1 − v 2 ). We have the following result. Lemma C.1. Suppose that m > 0, v ∈ R3 , and |v| < 1. Furthermore, let ϕ ∈ H1/2 (R3 ) Σv +µ √ be a solution of (C.1) with −µ < Σv . Then, for every 0 < δ < min m, , there 1−v 2 exists 0 < C(δ) < ∞ such that |ϕ(x)| ≤ Ce−δ|x| holds for all x ∈ R3 . Proof (of Lemma C.1). We rewrite (C.1) as follows : ϕ = −(H0 + µ)−1 V ϕ,
(C.7)
where H0 and V are defined in (C.3). Note that (H0 + µ)−1 exists, since we have that µ ∈ σ (H0 ) holds, by the assumption that −µ < Σv . We consider the Green’s function, G µ (x − y), given by 1 (x − y), (C.8) G µ (x − y) = F −1 √ k2 + m2 − m − v · k + µ √ where F : S → S denotes the Fourier transform. Since 1/ k 2 + m 2 . . . does not belong to L1 (R3 ), we cannot use Payley–Wiener type theorems directly to deduce pointwise exponential decay for G µ (z) in |z|. To overcome this difficulty, we first notice that ∞ ∞ √ 2 2 (H0 + µ)−1 = e−tµ e−t H0 dt = e−t (µ−m) e−t ( p +m −v· p) dt, (C.9) 0
0
by self-adjointness of H0 and functional calculus. Here and in what follows, we put p = −i∇ √ for convenience. By using the explicit formula for the Fourier transform of exp{−t k 2 + m 2 } (see e. g., [9]) in R3 and by analytic continuation, we obtain from (C.9) the formula ∞ t m t 2 + (z + itv)2 dt. (C.10) e−t (µ−m) 2 K G µ (z) = Am 2 2 t + (z + itv) 0 Here K 2 (z) stands for the modified Bessel function of the third kind, and Am > 0 denotes some constant. Notice that w = t 2 + (z + itv)2 = (1 − v 2 )t 2 + z 2 + 2itv · z
(C.11)
is a complex number with |arg w| < π/2. Next we analyze G µ (z) for |z| ≤ 1/m and for |z| > 1/m separately. From [1] we recall the estimate C |K 2 (mw)| ≤ , for |arg w| < π/2, (C.12) |w|2
Boson Stars as Solitary Waves
29
which implies that G µ (z) with |z| ≤ 1/m satisfies the bound ∞ t |G µ (z)| ≤ C e−t (µ−m) 2 2 (1 − v )t + |z|2 + 2itv · z 0 K 2 m (1 − v 2 )t 2 + |z|2 + 2itv · z dt ∞ t ≤C e−t (µ−m) dt. 2 )t 2 + |z|2 ]2 [(1 − v 0
(C.13)
Since µ − m ≥ 0, the t-integral is finite for z = 0 and we obtain C , for |z| ≤ 1/m, |z|2 √ √ where we use that |a + ib| ≥ |a| and | a + ib| ≥ |a| holds for a, b ∈ R. To estimate G µ (z) for |z| > 1/m, we use the bound |G µ (z)| ≤
(C.14)
e−mw e−m|Re w| ≤ C |K 2 (mw)| ≤ C , for |arg w| < π/2 and |w| > 1, (C.15) |w|2 |w|2 √ taken from [1]. By means of the inequality a 2 + b2 ≥ (1 − )|a| + |b|, for any 0 < ≤ 1, we proceed to find that ∞ √ t 2 e−t (µ−m+(1−) 1−v m) dt, (C.16) |G µ (z)| ≤ Ce−m|z| 2 [(1 − v )t 2 + |z|2 ]2 0 for |z| > 1/m. The assumption on µ allows us to choose ∈ (0, 1] such that the exponent in the t-integral is nonpositive. The best is given by Σv + µ ∈ (0, 1], (C.17) = min 1, √ m 1 − v2 and hence |G µ (z)| ≤ Ce−m|z| ≤C
0
∞
t [(1 − v 2 )t 2
+ |z|2 ]2
e−m|z| , for |z| > 1/m. |z|2
dt (C.18)
Combining now (C.14) and (C.18), we see that |G µ (z)| ≤ C
e−m|z| , for z ∈ R3 , |z|2
(C.19)
where is given by (C.17) and C is some constant. This shows that G µ (z) exhibits exponential decay; in particular, we have that G µ ∈ L p (R3 ) if 1 ≤ p < 3/2. Returning to (C.7), we notice that ϕ(x) = − G µ (x − y)V (y)ϕ(y) dy. (C.20) R3
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Moreover, the function V (x) = −(|x|−1 ∗ |ϕ|2 )(x) obeys V ∈ C0 (R3 ) and
lim V (x) = 0,
|x|→∞
(C.21)
since f ∗ g is a continuous function vanishing at infinity, provided that f ∈ L p and g ∈ L p with 1/ p + 1/ p = 1 and p > 1; see, e. g., [9]. Here we note that, e. g., |x|−1 ∈ L2 (R3 ) + L4 (R3 ) and in particular |ϕ|2 ∈ L4/3 (R3 ) ∩ L2 (R3 ) since ϕ ∈ Hs (R3 ) for all s ≥ 1/2 (cf. beginning of App. C). Using (C.20), (C.19) and (C.21), the claimed pointwise exponential decay of ϕ(x) follows from a direct adaptation of an argument by Slaggie and Wichmann for exponential decay of eigenfunctions for Schrödinger operators; see, e. g., [6] for a convenient exposition of this method. This completes the proof of Lemma C.1. Acknowledgement. The authors are grateful to I. M. Sigal and M. Struwe for useful discussions, as well as the referee for some helpful comments. Lastly, E. L. also thanks D. Christodoulou.
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Commun. Math. Phys. 274, 31–64 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0271-x
Communications in
Mathematical Physics
Unoriented WZW Models and Holonomy of Bundle Gerbes Urs Schreiber, Christoph Schweigert, Konrad Waldorf Fachbereich Mathematik, Schwerpunkt Algebra und Zahlentheorie, Universität Hamburg, Bundesstraße 55, D–20146 Hamburg, Germany. E-mail:
[email protected];
[email protected];
[email protected] Received: 24 January 2006 / Accepted: 26 July 2006 Published online: 13 June 2007 – © Springer-Verlag 2007
Abstract: The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models. manche meinen lechts und rinks kann man nicht velwechsern werch ein illtum Ernst Jandl [Jan95] Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Bundle Gerbes with Jandl Structures . . . . . . . . . . . . . 2.1 Bundle gerbes and stable isomorphisms . . . . . . . . . 2.2 Jandl structures . . . . . . . . . . . . . . . . . . . . . . 2.3 Classification of Jandl structures . . . . . . . . . . . . . 2.4 Local data . . . . . . . . . . . . . . . . . . . . . . . . . 3. Holonomy of Gerbes with Jandl Structure . . . . . . . . . . . 3.1 Double coverings, fundamental domains and orientations 3.2 Unoriented surface holonomy . . . . . . . . . . . . . . .
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K.W. is supported with scholarships by the German Israeli Foundation (GIF) and by the Rudolf und Erika Koch–Stiftung.
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3.3 Holonomy in local data . . . . . . . . . 3.4 Examples . . . . . . . . . . . . . . . . 4. Gerbes and Jandl Structures in WZW models 4.1 Oriented and orientable WZW models . 4.2 Unoriented WZW models . . . . . . . . 4.3 Crosscaps and the trivial line bundle . . 4.4 Examples of target spaces . . . . . . . .
U. Schreiber, C. Schweigert, K. Waldorf
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1. Introduction Wess-Zumino-Witten (WZW) models are one of the most important classes of (twodimensional) rational conformal field theories. They describe physical systems with (non-abelian) current symmetries, provide gauge sectors in heterotic string compactifications and are the starting point for other constructions of conformal field theories, e.g. the coset construction. Moreover, they have played a crucial role as a bridge between Lie theory and conformal field theory. It is well-known that for the Lagrangian description of such a model, a WessZumino term is needed to get a conformally invariant theory [Wit84]. Later, the relation of this term to Deligne hypercohomology has been realized [Gaw88] and its nature as a surface holonomy has been identified [Gaw88, Alv85]. More recently, the appropriate differential-geometric object for the holonomy has been identified as a hermitian U (1) bundle gerbe with connection and curving [CJM02]. Already the case of non-simply connected Lie groups with non–cyclic fundamental group, such as G := Spin(4n)/Z2 × Z2 shows that gerbes and their holonomy are really indispensable, even when one restricts one’s attention to oriented surfaces without boundary. The original definition of the Wess-Zumino term as the integral of a three form H over a suitable three-manifold cannot be applied to such groups; moreover, it could not explain the well-established fact that to such a group two different rational conformal field theories that differ by “discrete torsion” can be associated. Bundle gerbes will be central for the problem we address in this paper. A long series of algebraic results indicate that the WZW model can be consistently considered on unorientable surfaces. Early results include a detailed study of the abelian case [BPS92] and of SU (2) [PSS95b, PSS95a]. Sewing constraints for unoriented surfaces have been derived in [FPS94]. Already the abelian case [BPS92] shows that not every rational conformal field theory that is well-defined on oriented surfaces can be considered on unoriented surfaces. A necessary condition is that the bulk partition function is symmetric under exchange of left and right movers. This restricts, for example, the values of the Kalb-Ramond field in toroidal compactifications [BPS92]. Moreover, if the theory can be extended to unoriented surfaces, there can be different extensions that yield inequivalent correlation functions. This has been studied in detail for WZW theories based on SU (2) in [PSS95b, PSS95a]; later on, this has been systematically described with simple current techniques [HS00, HSS99]. Unifying general formulae have been proposed in [FHS+ 00]; the structure has been studied at the level of NIMreps in [SS03]. Aspects of these results have been proven in [FRS04] combining topological field theory in three-dimensions with algebra and representation theory in modular tensor categories. As a crucial ingredient, a generalization of the notion of an algebra with involution, i.e. an algebra together with an algebra-isomorphism to the opposed algebra, has been
Unoriented WZW Models and Holonomy of Bundle Gerbes
33
identified in [FRS04]; the isomorphism is not an involution any longer, but squares to the twist on the algebra. An algebra with such an isomorphism has been called a Jandl algebra. A similar structure, in a geometric setting, will be the subject of the present article. The success of the algebraic theory leads, in the Lagrangian description, to the quest for corresponding geometric structures on the target space. From previous work [BCW01, HSS02, Bru02] it is clear that a map k : M → M on the target space with the additional property that k ∗ H = −H will be one ingredient. Examples like the Lie group S O(3), for which two different unoriented WZW models with the same map k are known, already show that this structure does not suffice. We are thus looking for an additional structure on a hermitian bundle gerbe which allows to define a Wess-Zumino term, i.e. which allows to define holonomy for unoriented surfaces. For a general bundle gerbe, such a structure need not exist; if it exists, it will not be unique. In the present article, we make a proposal for such a structure. It exists whenever there are sufficiently well-behaved stable isomorphisms between the pullback gerbe k ∗ G and the dual gerbe G ∗ . If one thinks about a gerbe as a sheaf of groupoids, the formal similarity to the Jandl structures in [FRS04] becomes apparent, if one realizes that the dual gerbe plays the role of the opposed algebra. For this reason, we term the relevant structure a Jandl structure on the gerbe. We show that the Jandl structures on a gerbe on the target space M, if they exist at all, form a torsor over the group of flat equivariant hermitian line bundles on M. As explained in Sect. 4.3, this group always contains an element L k−1 of order two. We show that two Jandl structures that are related by the action of L k−1 provide amplitudes that just differ by a sign that depends only on the topology of the worldsheet. Such Jandl structures are considered to be essentially equivalent. We finally show that a Jandl structure allows to extend the definition of the usual gerbe holonomy from oriented surfaces to unoriented surfaces. We derive formulae for these holonomies in local data that generalize the formulae of [GR02, Alv85] for oriented surfaces. To give a concrete impression of a Jandl structure, we write out the local data of a Jandl structure for a given gerbe G on the target space M. To this end, we first recall the local data of a hermitian bundle gerbe in a good open cover {Vi }i∈I of M: we have a 2-form Bi for each open set Vi , a 1-form Ai j on each intersection Vi ∩ V j and a U (1)valued function gi jk on each triple intersection Vi ∩ V j ∩ Vk . They are required to satisfy the following constraints:
A jk
−1 g jkl · gikl · gi jl · gi−1 jk = 1, − Aik + Ai j + dlog gi jk = 0,
−d Ai j + B j − Bi = 0. To write down the local data of a Jandl structure for a given involution k : M → M in a succinct manner, we make the simplifying assumption that we have a cover {Vi }i∈I that is invariant under k, k(Vi ) = Vi , and that is still good enough to provide local data. The local data of a Jandl structure then consist of a U (1)-valued function ji : Vi → U (1) for each open subset, a U (1)-valued function ti j : Vi ∩V j → U (1) on two-fold intersections and a 1-form Wi ∈ 1 (Vi ). They relate the pullbacks of the gerbe data under k to the local data of the dual gerbe as follows:
34
U. Schreiber, C. Schweigert, K. Waldorf
k ∗ Bi = −Bi + dWi , k ∗ Ai j = −Ai j − dlog(ti j ) + W j − Wi , −1 k ∗ gi jk = gi−1 jk · t jk · tik · ti j .
The local data of a Jandl structure are required to be equivariant under k in the sense that k ∗ Wi = Wi − dlog( ji ), k ∗ ti j = ti j · j −1 j · ji , k ∗ ji = ji−1 . It should be appreciated that the functions ti j are not transition functions of some line bundle; as we will explain in Sect. 2.4, they are rather the local data describing an isomorphism of line bundles appearing in the Jandl structure. The notion of a Jandl structure naturally explains algebraic results for specific classes of rational conformal field theories. It is well-known that both the Lie group SU (2) and its quotient S O(3) admit two Jandl structures that are essentially different (i.e. that do not just differ by a sign depending on the topology of the surface). In the case of SU (2), this is explained by the fact that two different involutions are relevant: g → g −1 and g → zg −1 , where z is the non-trivial element in the center of SU (2). Indeed, since SU (2) is simply-connected, we have a single flat line bundle and hence for each involution only two Jandl structures which are essentially the same. The two involutions of SU (2) descend to the same involution of the quotient S O(3). The latter manifold, however, has fundamental group Z2 and thus twice as many equivariant flat line bundles as SU (2). The different Jandl structures of S O(3) are therefore not explained by different involutions on the target space but rather by the fact that one involution admits two essentially different Jandl structures. Needless to say, there remain many open questions. A discussion of surfaces with boundaries is beyond the scope of this article. The results of [FRS04] suggest, however, that a Jandl structure leads to an involution on gerbe modules. Most importantly, it remains to be shown that, in the Wess-Zumino-Witten path integral for a surface , the holonomy we introduced yields amplitudes that take their values in the space of conformal blocks associated to the complex double of , which ensures that the relevant chiral Ward identities are obeyed. To this end, it will be important to have a suitable reformulation of Jandl structures at our disposal. Indeed, the holonomy we propose in this article also arises as the surface holonomy of a 2-vector bundle with a certain 2-group; this issue will be the subject of a separate publication. 2. Bundle Gerbes with Jandl Structures 2.1. Bundle gerbes and stable isomorphisms. In preparation of the following sections, in this section we define an equivalence relation on the set of stable isomorphisms between two fixed bundle gerbes. To this end, we first set up the notation concerning bundle gerbes and stable isomorphisms. We mainly adopt the formalism used by Murray and collaborators, see [CJM02] for example, as well as by Gaw¸edzki and Reis [GR02]. Definition 1. A hermitian U (1) bundle gerbe G with connection and curving over a smooth manifold M consists of the following data: a surjective submersion π : Y → M, a hermitian line bundle p : L → Y [2] with connection, an associative isomorphism ∗ ∗ ∗ L ⊗ π23 L −→ π13 L µ : π12
(1)
Unoriented WZW Models and Holonomy of Bundle Gerbes
35
of hermitian line bundles with connection over Y [3] , and a 2-form C ∈ 2 (Y ) which satisfies π2∗ C − π1∗ C = curv(L). (2) Here Y [ p] denotes the p-fold fiber product of π : Y → M, which is a smooth manifold since π is a surjective submersion. For example π12 : Y [3] → Y [2] is the projection on the first two factors. Remark 1. From now on we will use the following conventions: the term line bundle refers to a hermitian line bundle with connection, and an isomorphism of line bundles refers to an isomorphism of hermitian line bundles with connection. Accordingly, we refer to Definition 1 by the term gerbe. The 2-form C is called curving, and the isomorphism µ is called multiplication. One can show that there is a unique 3-form H ∈ 3 (M) with π ∗ H = dC; this 3-form is called the curvature of the gerbe and is denoted by H = curv(G). To each gerbe G, we associate the dual gerbe G ∗ . It has the same surjective submersion π : Y → M, but the dual line bundle L ∗ → Y [2] with multiplication ∗ ∗ ∗ ∗ ∗ ∗ L ⊗ π23 L −→ π13 L , (µ∗ )−1 : π12
(3)
and the negative curving −C. Accordingly, the curvature of the dual gerbe satisfies curv(G ∗ ) = −curv(G).
(4)
Even more, the classes of G and the one of G ∗ in Deligne hypercohomology are inverses. For a smooth map f : N → M and a pullback diagram Yf
f˜
πf
N
/Y π
f
/M
,
(5)
∗ π f : Y f → N is a surjective submersion, and together with the line bundle f˜ L over ∗ ˜∗ ˜∗ Y [2] f , the multiplication f µ and the curving f C, we have defined a gerbe f G. If f : M → M is a diffeomorphism, Y f is canonically isomorphic to Y , such that f˜ = idY and π f = f −1 ◦ π . The curvature of the pullback gerbe is
curv( f ∗ G) = f ∗ curv(G).
(6)
Remark 2. As we did in the last paragraph, whenever there is a map f˜ : Y f → Y , we will use the same letter for the induced map on higher fiber products. Definition 2. A trivialization T = (T, τ ) of a gerbe G is a line bundle T → Y , together with an isomorphism τ : L ⊗ π2∗ T −→ π1∗ T (7) of line bundles over Y [2] , which is compatible with the isomorphism µ of the gerbe.
36
U. Schreiber, C. Schweigert, K. Waldorf
We call a gerbe G trivial, if it admits a trivialization. A choice of a trivialization T gives the 2-form C − curv(T ) ∈ 2 (Y ), which descends to a unique 2-form ρ ∈ 2 (M) with π ∗ ρ = C − curv(T ). This 2-form satisfies dρ = H , so the curvature H of a trivial gerbe is an exact form. If there are two trivializations T1 = (T1 , τ1 ) and T2 = (T2 , τ2 ) of the same gerbe G, one obtains an isomorphism α := τ1−1 ⊗ τ2∗ : π1∗ (T1 ⊗ T2∗ ) −→ π2∗ (T1 ⊗ T2∗ ),
(8)
Y [2] .
of line bundles over From the compatibility condition between the multiplication µ and both τ1 and τ2 the cocycle condition ∗ ∗ ∗ π23 α ◦ π12 α = π13 α
(9)
follows. Such an isomorphism determines a unique descent line bundle N → M with connection together with an isomorphism ν : π ∗ N → T1 ⊗T2∗ [Bry93]. The two 2-forms ρ1 and ρ2 coming from the two trivializations are related by ρ2 = ρ1 + curv(N ). Definition 3. Let G and
G
(10)
be two gerbes. A stable isomorphism A : G −→ G
consists of a line bundle A → Z over the fiber product Z :=
(11) Y
× M Y with curvature
curv(A) = p ∗ C − p ∗ C,
(12)
α : p ∗ L ⊗ p ∗ L ∗ ⊗ π2∗ A −→ π1∗ A
(13)
and an isomorphism of line bundles over Z [2] , which is compatible with the multiplications µ and µ of both gerbes. Here p and p denote the projections from Z to Y and to Y respectively. Since the pullbacks of the curvings C and C to Z differ by a closed 2-form, the curvatures of stably isomorphic gerbes, defined by the differential of C, are equal. Definition 4. Let G and G be two gerbes, and A1 and A2 two stable isomorphisms from G to G . A morphism β : A1 =⇒ A2 (14) is an isomorphism β : A1 → A2 of line bundles over Z , which is compatible with α1 and α2 in the sense that the diagram p ∗ L ⊗ p ∗ L ∗ ⊗ π2∗ A1
α1
/ π ∗ A1 1
1⊗1⊗π2∗ β
p ∗ L ⊗ p ∗ L ∗ ⊗ π2∗ A2
α2
π1∗ β
(15)
/ π ∗ A2 1
of isomorphisms of line bundles over Z [2] commutes. The definition of such a morphism of stable isomorphisms already appeared in [Ste00]. We call two stable isomorphisms equivalent, if there is a morphism between them. This defines an equivalence relation on the set of stable isomorphisms between two fixed gerbes G and G .
Unoriented WZW Models and Holonomy of Bundle Gerbes
37
2.2. Jandl structures. Recall that for a group K acting on a manifold M by diffeomorphisms k : M → M, a K -equivariant structure on a line bundle L → M is a family ϕ k k∈K of isomorphisms ϕ k : k ∗ L −→ L
(16)
of line bundles, which respect the group structure of K in the sense that ϕ 1 : L → L is the identity, and the multiplication law ϕ k1 k2 = ϕ k2 ◦ k2∗ ϕ k1
(17)
is satisfied. Remember that according to our convention in Remark 1 all line bundles have connections, and all isomorphisms of line bundles preserve them. In this article, we only consider the group K = Z2 for the sake of simplicity. Let G be a gerbe over M and let K = Z2 act on M. Denote the action of the non-trivial element k by k : M → M. Assume that there is a stable isomorphism A = (A, α) : k ∗ G → G ∗ . Recall that in this particular situation, A is a line bundle over the space Z = Yk × M Y , where Yk := Y and πk := k −1 ◦ π as in our discussion of the pullback of G by a diffeomorphism k. We still denote the projections from Z to Y and to Yk by p and p respectively. Define the surjective submersion π Z := π ◦ p : Z → M. As k 2 = id M , the permutation map (18) k˜ : Z −→ Z : (yk , y) −→ (y, yk ) gives the following commuting diagram: Z
k˜
πZ
M
/Z πZ
k
(19)
/M
Furthermore, since also k˜ 2 = id Z , we even have a lift of the action of K into Z . Definition 5. A Jandl structure on G is a collection J = (k, A, ϕ) consisting of • a smooth action of K = Z2 on M, where we denote the non-trivial element and the diffeomorphism associated to that non-trivial element by k : M → M. • a stable isomorphism of gerbes A = (A, α) : k ∗ G → G ∗ . • a K -equivariant structure ϕ := ϕ k on the line bundle A, which is compatible with the stable isomorphism A in the sense that the diagram p ∗ L ⊗ p ∗ L ⊗ π2∗ A
α
1
1⊗1⊗π2∗ ϕ
p ∗ L ⊗ p ∗ L ⊗ k ∗ π2∗ A
/ π∗ A π1∗ ϕ
k∗α
of isomorphisms of line bundles over Z [2] commutes.
/ k∗π ∗ A 1
(20)
38
U. Schreiber, C. Schweigert, K. Waldorf
We can immediately deduce a necessary condition for the existence of a Jandl structure for a given gerbe G, namely the condition that the gerbes k ∗ G and G ∗ are stably isomorphic. Since the curvatures of stably isomorphic gerbes are equal, this in turn demands k ∗ H = −H (21) for the curvature H = curv(G) of G. In particular, there will be gerbes on manifolds with involution which do not admit a Jandl structure. Definition 6. Two Jandl structures J and J on the same gerbe G are equivalent, if the following conditions are satisfied: • the actions are the same, i.e. k and k are the same diffeomorphisms, • there is a morphism β : A ⇒ A of stable isomorphisms in the sense of Definition 4 such that • β : A → A is even an isomorphism of K -equivariant line bundles on Z . Next, we show that Jandl structures behave well under the pullback of gerbes along a smooth map f : N → M. Let J = (k, A, ϕ) be a Jandl structure on G. Assume that there is an action of K = Z2 on N by a diffeomorphism g, such that the diagram f
N
/M
g
(22)
k
N
/M
f
commutes. Consider the pullback of G by f as discussed before, and define Z f := (Y f )g × N Y f
(23)
and the permutation map g˜ : Z f → Z f . Then f˜
Zf
}} }} } } ~} }
Zf
/Z k˜
g˜
f˜
/Z
πZ
πZ f πZ f
{{ {{ { { { }{ N
N
g
f
πZ
(24)
f /M } }} }}k } ~}} /M
is a cube with commuting faces. It follows that f ∗ A := ( f˜∗ A, f˜∗ α) is a stable isomorphism from g ∗ f ∗ G to f ∗ G ∗ . Furthermore, f˜∗ ϕ is a K -equivariant structure on f˜∗ A, where K acts by g. ˜ In summary, f ∗ J := (g, f˜∗ A, f˜∗ ϕ) defines a pullback Jandl structure on
f ∗ G.
(25)
Unoriented WZW Models and Holonomy of Bundle Gerbes
39
2.3. Classification of Jandl structures. If a gerbe G admits a Jandl structure, it is natural to ask, how many inequivalent choices exist. So we are interested in the set Jdl(G, k) of equivalence classes of Jandl structures J = (k, −, −) with a fixed action of K = Z2 via k. This will be crucial in the discussion of the unoriented WZW model in Sect. 4. To approach this task, we first investigate the set Hom(G, G ) of equivalence classes of stable isomorphisms between G and G . We start by recalling the following Lemma 1 ([CJM02]). (i) If N → M is a flat line bundle and A = (A, α) is a stable isomorphism, then N .A := (A ⊗ π Z∗ N , α ⊗ 1) is also a stable isomorphism. (ii) If A1 = (A1 , α1 ) and A2 = (A2 , α2 ) are two stable isomorphisms, then there is a unique flat line bundle N → M such that A1 and N .A2 are equivalent as stable isomorphisms. Proof. For the first part we note that because N is flat, A and A ⊗ π Z∗ N have the same curvature, so that (12) is satisfied. For the second part, we use the isomorphism α1−1 ⊗ α2∗ : π1∗ (A1 ⊗ A∗2 ) −→ π2∗ (A1 ⊗ A∗2 )
(26)
which satisfies the cocycle condition because of the compatibility of α1 and α2 with µ and µ . This determines a unique line bundle N → M with connection together with an isomorphism ν : π Z∗ N → A1 ⊗ A∗2 . Because (12) requires the curvatures of both A1 and A2 to be the same, N is flat. Now ν determines an isomorphism A1 → A2 ⊗ π Z∗ N , which is a morphism A1 ⇒ N .A2 .
We denote the group of isomorphism classes of flat line bundles over M by Pic0 (M). It is a subgroup of the Picard group Pic(M) of isomorphism classes of hermitian line bundles with connection over M. Lemma 2. The set Hom(G, G ) of equivalence classes of stable isomorphisms is a torsor over the flat Picard group Pic0 (M). Proof. We will (a) define the action and show that it is (b) transitive and (c) free. (a) We act [N ].[A] := [N .A], where the right-hand side was defined in Lemma 1 (i). This definition is independent of the choice of representatives N and A: an isomorphism N → N gives an isomorphism N .A → N .A, which in fact is a morphism of stable isomorphisms N .A ⇒ N .A. On the other hand, a morphism A ⇒ A of stable isomorphisms induces a morphism N .A ⇒ N .A . Because N .A is defined using the group structure on the group of isomorphism classes of line bundles with connection, it respects the group structure on Pic0 (M), and hence defines an action. (b) The transitivity follows directly from Lemma 1 (ii). (c) Let [A] be an element in Hom(G, G ), let N be a flat line bundle and let us assume that N .A and A are equivalent, in particular A ⊗ π Z∗ N is isomorphic to A. Since N is unique by Lemma 1 (ii), it is the trivial line bundle. Hence the action is free.
This lemma allows us to make use of the flat Picard group Pic0 (M). Remember that line bundles are, according to our convention in Remark 1, line bundles with connection.
40
U. Schreiber, C. Schweigert, K. Waldorf
It is well understood [Bry93], that the Picard group Pic(M) of isomorphism classes of line bundles fits into the exact sequence 0
/ Pic(M)
/ H1 (M, U (1))
curv
/ 2 (M) .
(27)
In particular this means Pic0 (M) ∼ = H1 (M, U (1)). This cohomology group can be computed using the universal coefficient theorem 0
/ Ext(H0 (M), U (1))
/ H1 (M, U (1))
/ Hom(H1 (M), U (1))
/ 0 . (28)
If M is connected, the Ext-group is trivial and we obtain Pic0 (M) ∼ = Hom(π1 (M), U (1)).
(29)
An equivariant version of Lemma 2 applies to Jandl structures. We denote the group of isomorphism classes of flat K -equivariant line bundles by Pic0K (M) and call it the flat K -equivariant Picard group. In this equivalence relation isomorphisms are isomorphisms of equivariant line bundles with connection. Theorem 1. The set Jdl(G, k) of equivalence classes of Jandl structures on G with involution k is a torsor over the flat K -equivariant Picard group Pic0K (M). Proof. (a) We first describe the action of a flat line bundle N over M with equivariant structure ν on a Jandl structure J = (k, A, ϕ). According to diagram (19), π Z∗ ν : π Z∗ N → k˜ ∗ π Z∗ N is a K -equivariant structure on π Z∗ N . Now, by taking the tensor product of A and π Z∗ N as K -equivariant line bundles, we obtain an equivariant structure ϕ ⊗ π Z∗ ν on the line bundle of N .A. So we define N .J := (k, N .A, ϕ ⊗ π Z∗ ν). Since Z [2] π2
π1∗ π Z∗ ν
Z
π1
/Z πZ
πZ
π2∗ π Z∗ ν.
(30)
/M
(31)
commutes, we have = This shows that condition (20) for Jandl structures is satisfied for N .J . The arguments in the proof of Lemma 2 (a) apply here too and show that this defines an action on equivalence classes. (b) Let two equivalence classes of Jandl structures be represented by J1 and J2 . We already know from Lemma 1 (ii) that there is a flat line bundle N → M together with an isomorphism β : A1 → A2 ⊗ π Z∗ N , which is a morphism of stable isomorphism β : N .A1 ⇒ A2 . We have to show that there is an equivariant structure on N such that β is an isomorphism of equivariant line bundles. Remember that we defined N by a descent isomorphism α1−1 ⊗ α2∗ in (26). Because the equivariant structures on A1 and A2 are compatible with α1 and α2 respectively due to the property (20) of Jandl structures, the descent isomorphism is an isomorphism of equivariant line bundles. Thus N is an equivariant line bundle, and β is an isomorphism of equivariant line bundles.
Unoriented WZW Models and Holonomy of Bundle Gerbes
41
(c) Let J = (k, A, ϕ) represent a Jandl structure on G, and let N be a flat line bundle over M with equivariant structure ν, such that N .J and J are equivalent. It follows from Lemma 2 that N is the trivial line bundle. Furthermore, π Z∗ ν is the trivial equivariant structure on π Z∗ N , so that ν is the trivial equivariant structure on N.
For an action of a discrete group K on M, an equivariant version of the sequence (27) is derived in [Gom03], namely 0
/ H1 (M, U (1)) K
/ Pic K (M)
curv
/ 2 (M) K .
(32)
Here, H1K (M, U (1)) is the equivariant cohomology of M, i.e. the cohomology of the associated Borel space. In particular, we get for flat equivariant line bundles Pic0K (M) ∼ = H1K (M, U (1)).
(33)
2.4. Local data. Let G be a gerbe over M and V = {Vi }i∈I be a good open cover of M. Let MV be the disjoint union of all the Vi ’s. The p-fold fiber product of MV over M is just the disjoint union of all p-fold intersections of the Vi ’s. Recall from [CJM02] how to extract local data from G: A choice of local sections si : Vi → Y gives a fiber preserving map s : MV → Y by (x, i) → si (x). Pull back the line bundle L → Y [2] with its connection ∇ along s to a line bundle on the double intersections, and choose local sections σi j : Vi ∩ V j → s ∗ L. Pull back the isomorphism µ of the gerbe, too. Then define local data, namely smooth functions gi jk : Vi ∩ V j ∩ Vk → U (1), real-valued 1-forms Ai j ∈ 1 (Vi ∩ V j ) and 2-forms Bi ∈ 2 (Vi ) by the following relations: ∗ ∗ ∗ σi j ⊗ π23 σ jk = gi jk · π13 σik , (34) s ∗ µ π12 1 s ∗ ∇(σi j ) = Ai j ⊗ σi j , (35) i Bi = si∗ C. (36) ˇ These local data give elements g, A, B in the Cech-Deligne double complex for the cover V, and the cochain (g, A, B) satisfies the Deligne cocycle condition D (g, A, B) = (1, 0, 0) ,
(37)
or equivalently in components
A jk
−1 · gi jl · gi−1 g jkl · gikl jk = 1, − Aik + Ai j + dlog gi jk = 0,
(39)
−d Ai j + B j − Bi = 0.
(40)
dBi = H |Vi ,
(41)
(38)
Furthermore, it satisfies where the 3-form H is the curvature of the gerbe. The dual gerbe and the pullback gerbe f ∗ G along some map f : N → M can be conveniently expressed in local data as follows: by choosing the same si and the dual
42
U. Schreiber, C. Schweigert, K. Waldorf
sections σi∗j , one gets (g −1 , −A, −B) = −(g, A, B) as local data of G ∗ . Furthermore, if we induce a cover { f −1 Vi }i∈I of N , and choose the pullback sections f ∗ si and f˜∗ σi j , then we obtain ( f ∗ g, f ∗ A, f ∗ B) = f ∗ (g, A, B) as local data of f ∗ G. We next need to derive local data of trivializations and stable isomorphisms. So, let T = (T, τ ) be a trivialization of G. Since T is a line bundle over Y , we can pull it back with s : MV → Y to a line bundle over the open subsets, and choose local sections σi : Vi → s ∗ T . We also pull back the isomorphism τ to an isomorphism s ∗ τ : s ∗ L ⊗ π2∗ s ∗ T −→ π1∗ s ∗ T . Then we obtain smooth functions h i j : Vi ∩ V j → U (1) by s ∗ τ σi j ⊗ π2∗ σ j = h i j · π1∗ σi .
(42)
(43)
Let be the connection of T . It defines connection 1-forms Mi ∈ 1 (Vi ) by s ∗ (σi ) =
1 Mi ⊗ σi . i
(44)
ˇ The local data h and M are again elements in the Cech-Deligne double complex. Now the compatibility of τ and µ in Definition 2 is equivalent to −1 gi jk = h i j · h ik · h jk ,
and the condition, that the isomorphism τ respect connections, is equivalent to Ai j = −dlog h i j + M j − Mi .
(45)
(46)
Furthermore, the local 2-form ρ = Bi + dMi coincides with the 2-form ρ obtained from Definition 2. The last three properties of h and M are equivalent to the Deligne coboundary equation (47) (g, A, B) = (1, 0, ρ) + D (h, M) . Now consider a stable isomorphism A : G → G of gerbes over M. With respect to the good open cover {Vi }i∈I we may have chosen local sections si , σi j and si , σij to get local data (g, A, B) and (g , A , B ) of G and G respectively. We construct a map s˜ : MV −→ Y × M Y : (x, i) −→ (si (x), si (x)),
(48)
and pull the line bundle A → Y × M Y of the stable isomorphism together with its connection back to MV. We also pull back the isomorphism α and get an isomorphism s˜ ∗ α : s ∗ L ⊗ s ∗ L ∗ ⊗ π2∗ s˜ ∗ A −→ π1∗ s˜ ∗ A.
(49)
Then we choose local sections σi : Vi → s˜ ∗ A. We obtain local data in the form of smooth functions ti j : Vi ∩ V j → U (1) and connection 1-forms Wi ∈ 1 (Vi ) by the following relations: s˜ ∗ α σi j ⊗ σi∗j ⊗ π2∗ σ j∗ = ti j · π1∗ σi , (50) s˜ ∗ (σi ) =
1 Wi ⊗ σi . i
(51)
Unoriented WZW Models and Holonomy of Bundle Gerbes
43
Note that the functions ti j are not transition functions of some bundle but are defined by the isomorphism α. ˇ These local data t and W are elements in the Cech-Deligne double complex. The compatibility of α with the isomorphisms µ and µ of both gerbes as isomorphisms of hermitian line bundles with connection according to Definition 3 is equivalent to −1 gi jk · gi−1 jk = t jk · tik · ti j ,
Ai j −
Ai j
= −dlog(ti j ) + W j − Wi ,
(52) (53)
while the condition (12) on the curvature of A is equivalent to Bi − Bi = dWi .
(54)
The three last equations are in turn equivalent to the Deligne coboundary equation (55) (g, A, B) − g , A , B = D (t, W ) . This formalism of local data reproduces results on bundle gerbes and their stable isomorphisms, for example Lemma 1 (ii). Consider again two gerbes G and G , and now two stable isomorphisms A1 and A2 both from G to G . We may have extracted local data (t1 , W1 ) of A1 and (t2 , W2 ) of A2 such that Eq. (55) holds for both. It follows D(t · t −1 , W − W ) = (1, 0, 0),
(56)
which is the Deligne cocycle condition for a flat hermitian line bundle over M. This is the bundle N constructed in Lemma 1 (ii). We are now in a position to derive the local data of a Jandl structure J = (k, A, ϕ) on a gerbe G. Recall that k : M → M is the action of the non-trivial element of K = Z2 acting on M, in particular k 2 = id M . We simplify the situation by considering an open cover V = {Vi }i∈I of M, which is invariant under k, i.e. k(Vi ) = Vi , and which is still good enough to enable us to extract local data. The generalization to other covers is straightforward, but makes the notation somewhat more cumbersome. Recall further that A is a stable isomorphism from k ∗ G → G ∗ . Let (t, W ) be local data of A, obtained by pulling back the line bundle A → Z by s˜ : MV → Z from Eq. (48) and choosing local sections σi : Vi → s˜ ∗ A. As we derived for the local data of the dual gerbe and the pullback gerbe, Eq. (55) here appears as k ∗ (g, A, B) = −(g, A, B) + D(t, W ),
(57)
k ∗ Bi = −Bi + dWi , k ∗ Ai j = −Ai j − dlog(ti j ) + W j − Wi ,
(58) (59)
−1 k ∗ gi jk = gi−1 jk · t jk · tik · ti j .
(60)
or equivalently:
Now recall that a part of a Jandl structure is a K -equivariant structure ϕ : k ∗ A → A on A. By pullback with s˜ , we obtain s˜ ∗ ϕ : k ∗ s˜ ∗ A −→ s˜ ∗ A.
(61)
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U. Schreiber, C. Schweigert, K. Waldorf
Now, because σi is a section of s˜ ∗ A, k ∗ σi = σi ◦k is a section of k ∗ s˜ ∗ A on the same patch Vi , since the latter is invariant under k. This allows us to extract a local U (1)-valued function ji : Vi → U (1), defined by s˜ ∗ ϕ(σi ) = ji · σi ◦ k.
(62)
The compatibility of ϕ with α in the sense of diagram (20) is equivalent to k ∗ (t, W ) = (t, W ) − D ( j) ,
(63)
k ∗ Wi = Wi − dlog( ji ),
(64)
or in turn equivalently ∗
k ti j = ti j ·
j −1 j
· ji .
(65)
By definition of an equivariant structure, the K = Z2 group law (17) is satisfied. In terms of local data, this is equivalent to k ∗ ji = ji−1 .
(66)
In summary, the Jandl structure J = (k, A, ϕ) gives rise to local data (t, W ) and j which satisfy the following three conditions: k ∗ (g, A, B) = −(g, A, B) + D(t, W ), k ∗ (t, W ) = (t, W ) − D ( j) , k ∗ ji = ji−1 .
(67) (68) (69)
Again, using local data, we can reproduce results on Jandl structures like Theorem 1. In detail, let J be a Jandl structure on G with local data (t, W ) and j. Let N be a flat K -equivariant hermitian line bundle over M with transition functions n i j : Vi ∩ V j → U (1) and local connection 1-forms Ni ∈ 1 (Vi ) with D(n, N ) = (1, 0, 0).
(70)
The equivariant structure on N determines smooth functions νi : Vi → U (1) with k ∗ (n, N ) = (n, N ) − D(ν)
(71)
(t , W ) := (t, W ) + (n, N ), j := j · ν
(72) (73)
and k ∗ ν = ν −1 . Then,
are local data of the Jandl structure N .J . Indeed, Eq. (67) is satisfied because of the Deligne cocycle condition (70). Compute k ∗ (t , W ) = k ∗ (t, W ) + k ∗ (n, N ) = (t, W ) − D ( j) + (n, N ) − D(ν) = (t , W ) − D( j ),
(74)
this is Eq. (68), and the last equation (69) for j is just a consequence from the conditions on j and ν.
Unoriented WZW Models and Holonomy of Bundle Gerbes
45
Let now J and J be two Jandl structures on G with local data (t, W ), j and respectively; (75) (n, N ) := (t, W ) − (t , W )
(t , W ), j
are the local data of the flat descent line bundle N , and using Eq. (67), we get its cocycle condition D(n, N ) = (1, 0, 0). (76) Now compute k ∗ (n, N ) = k ∗ (t, W ) − k ∗ (t , W ) = (t, W ) − D( j) − (t , W ) + D( j ) = (n, N ) − D(ν),
(77)
where we defined ν := j · j −1 . Hence, N and k ∗ N are isomorphic as hermitian line bundles with connection via an isomorphism represented by ν. By definition, we have k ∗ ν = ν −1 , this means, that ν is a K -equivariant structure. 3. Holonomy of Gerbes with Jandl Structure 3.1. Double coverings, fundamental domains and orientations. Let us first recall the setup that allows to define holonomy around closed oriented surfaces. This is a gerbe G over M and a closed oriented surface together with a smooth map φ : → M. Following [CJM02], we pull back G along φ to a gerbe over . For dimensional reasons, φ ∗ G is trivial. As explained in Sect. 2.1, a trivialization T determines a 2-form ρ ∈ 2 (), while another trivialization T determines a 2-form ρ = ρ + curv(N ). Since curv(N ) defines an integral class in cohomology, we have ρ = ρ mod 2π Z. (78)
So the integral is independent of the choice of a trivialization up to 2π Z, and admits therefore the following Definition 7. The holonomy of G around the closed oriented surface φ : → M is defined as holG (φ, ) := exp i ρ ∈ U (1). (79)
We state three important properties of this definition: • The dual gerbe has inverse holonomy, holG (φ, ) = holG ∗ (φ, )−1 .
(80)
• If A : G → G is a stable isomorphism, we have holG (φ, ) = holG (φ, ).
(81)
¯ we denote the same manifold with the opposite orientation; then we obtain • By ¯ −1 . holG (φ, ) = holG (φ, )
(82)
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U. Schreiber, C. Schweigert, K. Waldorf
Obviously, the orientation on is essential for this definition. In this section we will define the holonomy around unoriented or even unorientable surfaces. The most important property of this definition will be, that it reduces to Definition 7 if is orientable and an orientation is chosen. One of the main tools will be an orientation covering. Let be a smooth manifold (without orientation). ˆ → with an Definition 8. An orientation covering of is a double covering pr : ˆ such that the canonical involution σ : ˆ → ˆ is orientationoriented manifold , reversing. Recall three basic properties of orientation coverings (some of them can be found for example in [BG88]): • It is unique up to orientation-preserving diffeomorphisms of covering spaces. ˆ → ˆ preserves fibers and permutes the the sheets. • The canonical involution σ : ˆ is connected if and only if is not • Under the assumption that is connected, orientable. ˆ we will from now on refer to this unique orientation Due to the first point, by ˆ M)σ,k we denote the space cover. Let k : M → M be an involution on M. By C ∞ (, ˆ → M for which the diagram of smooth maps φˆ : ˆ
φˆ
σ
ˆ
/M k
φˆ
/M
(83)
commutes in the category of smooth manifolds (neglecting orientations). Let be orientable. Lemma 3. An orientation on defines a bijection ˆ M)σ,k −→ C ∞ (, M). C ∞ (,
(84)
ˆ consists of two disjoint copies of with opposite orienProof. Since is orientable, ˆ in the covering pr : ˆ → . tations. An orientation on is a global section or : → ˆ → M be a map. Define its image as φ := φˆ ◦ or. On the other hand, Now let φˆ : ˆ separately given a map φ : → M, we define the preimage φˆ on the two sheets of ˆ ˆ as φ|or() := φ and φ|σ or() := k ◦ φ respectively.
If is not orientable or no orientation of is chosen, we will make use of the following generalization of an orientation. ˆ is a submanifold F ⊂ ˆ possibly with Definition 9. A fundamental domain for in (piecewise smooth) boundary, satisfying the following two conditions as sets: (i) F ∩ σ (F) = ∂ F, ˆ (ii) F ∪ σ (F) = .
Unoriented WZW Models and Holonomy of Bundle Gerbes
47
Fig. 1. The construction of a fundamental domain by local orientations for a dual triangulation
This is a generalization of an orientation on in the sense that any orientation on ˆ which in turn defines a fundamental domain, namely gives a global section or : → ˆ F := or(), one of the two copies of in . We show the existence of such a fundamental domain for an arbitrary closed surface by an explicit construction, which we will also use in Sect. 3.3. Let U = {Ui }i∈I be ˆ One can think of such an open cover of , which admits local sections ori : Ui → . sections as local orientations. Choose a dual triangulation T of , subordinate to the cover U, together with a subordinating map i : T → I . So, for each face f ∈ T there is an index i( f ) with f ⊂ Ui( f ) , as well as for each edge e ∈ T and for each vertex v ∈ T . Because we have a dual triangulation, each vertex is trivalent. Consider a common edge e = f 1 ∩ f 2 of two faces f 1 and f 2 . We call the edge e orientation-preserving, if ori( f1 ) (e) = ori( f2 ) (e), (85) otherwise we call it orientation-reversing. So the set of edges splits in a set E of orientation-preserving, and a set E¯ of orientation-reversing edges. If v is a vertex, the number of orientation-reversing edges ending in v must be even, and since we started with a dual triangulation, it is either zero or two. Hence, the edges in E¯ form non-intersecting closed lines in , Define the subset
F := ori( f ) ( f ) (86) f ∈T
ˆ and endow it with the subspace topology. The boundary of F is exactly the union of of the preimages of orientation-reversing edges under the covering map,
∂F = pr −1 (e), (87) e∈ E¯
and hence a disjoint union of piecewise smooth circles. This shows that F is a subˆ with piecewise smooth boundary. It satisfies the two properties of a manifold of fundamental domain, and hence shows the existence of such a fundamental domain. ˆ The following observation will Let now F be any fundamental domain for in . be essential.
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U. Schreiber, C. Schweigert, K. Waldorf
Fig. 2. The orientation on ∂ F
Lemma 4. The quotient ∂ F := ∂ F/σ is a 1-dimensional oriented closed submanifold of . Proof. We act with σ on property (i) of the fundamental domain F: σ (∂ F) = σ (F ∩ σ (F)) = F ∩ σ (F) = ∂ F.
(88)
ˆ without fixed This shows that σ restricts to an involution on ∂ F. Since σ acts on points, the quotient ∂ F/σ is a submanifold of , and as ∂ F is closed, so is the quotient. ˆ induces an orientation on F. Because σ is orientation-reversing, The orientation of the orientation of σ (F) is opposite to the one induced on σ (F) as a submanifold of ˆ Hence, ∂ F and ∂(σ (F)) are equal as sets as well as oriented submanifolds. Thus σ . preserves the orientation on ∂ F.
3.2. Unoriented surface holonomy. The setup for the definition of holonomy around closed unoriented surfaces is • a gerbe G over a smooth manifold M with Jandl structure J = (k, A, ϕ) • a closed surface ˆ M)σ,k . • a map φˆ ∈ C ∞ (, ˆ along φ, ˆ The idea of the definition is the following: Pull back the gerbe G to 2 ˆ choose a trivialization and determine the 2-form ρˆ ∈ () as in Definition 7. Choose ˆ The integral a fundamental domain F for in . exp i ρˆ (89) F
is independent neither of the choice of the trivialization – which enters in ρˆ – nor of the choice of the fundamental domain F. The Jandl structure, however, allows to correct (89) by a boundary term in such a way that the holonomy becomes well-defined. We will now give a detailed definition of this boundary term, and then show that it gives rise to a well-defined holonomy. Recall that a gerbe G consists of the following data: a surjective submersion π : Y → M, a line bundle L → Y [2] , an isomorphism µ, and a 2-form C ∈ 2 (Y ). Recall that
Unoriented WZW Models and Holonomy of Bundle Gerbes
49
the pullback gerbe φˆ ∗ G consists of a pullback Yφ
φ˜
/Y
πφ
ˆ
π
φˆ
/M
,
(90)
the pullback line bundle φ˜ ∗ L, isomorphism φ˜ ∗ µ and 2-form φ˜ ∗ C. Accordingly, a trivialization T of φˆ ∗ G is a line bundle T → Yφ together with an isomorphism τ : φ˜ ∗ L ⊗ πφ ∗2 T −→ πφ ∗1 T
(91)
ˆ with of line bundles over Yφ[2] . It determines a 2-form ρˆ ∈ 2 () πφ∗ ρˆ = φ˜ ∗ C − curv(T ).
(92)
Due to the commutativity of diagram (83), φˆ ∗ J = (σ, φ˜ ∗ A, φ˜ ∗ ϕ) is a Jandl structure on φˆ ∗ G. Recall that part of the data is a line bundle φ˜ ∗ A → Z φ over the space Z φ := (Yφ )σ ׈ Yφ , and an isomorphism φ˜ ∗ α : p ∗ φ˜ ∗ L ⊗ p ∗ φ˜ ∗ L ∗ ⊗ πφ ∗2 φ˜ ∗ A −→ πφ ∗1 φ˜ ∗ A
(93)
of line bundles over Z φ[2] , where p and p are the projections in Zφ p
p
πφ
Yφ σ
/ Yφ
σ ◦πφ
/ ˆ
.
(94)
Further, the action of K by σ lifts to Z φ via the permutation map σ˜ , and φˆ ∗ J contains an K -equivariant structure φ˜ ∗ ϕ on φ˜ ∗ A. Combining the trivialization with the Jandl structure, we define a line bundle R := φ˜ ∗ A ⊗ p ∗ T ∗ ⊗ p ∗ T ∗
(95)
over Z φ . In addition, we define an isomorphism r := φ˜ ∗ α −1 ⊗ p ∗ τ ∗ ⊗ p ∗ τ ∗ : πφ ∗1 R −→ πφ ∗2 R
(96)
of line bundles over Z φ[2] . The compatibility of τ and α with the isomorphism µ of G guarantees the cocycle condition πφ ∗23r ◦ πφ ∗12 r = πφ ∗13r
(97)
50
U. Schreiber, C. Schweigert, K. Waldorf
ˆ together with an over Z φ[3] , hence R determines a unique descent line bundle Rˆ → , ∗ ˆ isomorphism π Z R → R. We shall compute the curvature of these bundles, namely φ
(95) curv (R) = φ˜ ∗ curv (A) − p ∗ curv (T ) − p ∗ curv (T ) (12)
= p ∗ (φ˜ ∗ C − curv(T )) + p ∗ (φ˜ ∗ C − curv(T ))
(92)
=
p ∗ πφ∗ ρˆ
+
p ∗ πφ∗ ρˆ
(99) (100)
(94)
= π Z∗ φ (σ ∗ ρˆ + ρ). ˆ
Hence the curvature of Rˆ is
(98)
(101)
ˆ = σ ∗ ρˆ + ρ. curv( R) ˆ
(102) ˆ The next step is to define the σ -equivariant structure on R. Note that the canonical permutation of tensor products is an equivariant structure on p ∗ T ∗ ⊗ p ∗ T ∗ , since the permutation map σ˜ exchanges p and p . Together with the equivariant structure φ˜ ∗ ϕ on φ˜ ∗ A, the tensor product (95) is the tensor product of two equivariant line bundles. By definition of a Jandl structure ϕ is compatible with α, which means that the descent isomorphism r is an isomorphism of equivariant line bundles. Hence, also the descent ˆ is endowed with an equivariant structure. bundle Rˆ over It is a standard fact [Gom03, Bry00], that if K is discrete and acts freely, ˆ defines a unique line bundle Q on the quotient a K -equivariant line bundle Rˆ → ˆ /K = . ˆ Now choose a fundamental domain F of in . Definition 10. The holonomy of the gerbe G with Jandl structure J around the unoriented closed surface is defined as ˆ ) := exp i holG ,J (φ, ρˆ · hol Q (∂ F)−1 . (103) F
In this definition, the compensating term hol Q (∂ F) is the holonomy of the line bundle Q around the one-dimensional closed oriented submanifold ∂ F. Theorem 2. The holonomy defined in Definition 10 depends neither on the choice of the fundamental domain F nor on the choice of the trivialization T . Proof. Let F be another fundamental domain. We define the set B := Int(F) ∩ σ (Int(F )),
(104)
where Int denotes the interior. As the intersection of two open sets, B is open and hence ˆ It contains those parts of F, which are not contained in F (cf. a submanifold of . Fig. 3). Because we excluded the boundaries of F and F , we have B ∩ σ (B) = ∅,
(105)
ˆ with image B. such that there is a unique section or B : pr(B) → From Fig. 3, we have ˆ ρˆ = ρˆ − ρˆ + ρˆ = ρˆ − curv( R), F
F
B
σ (B)
F
B
(106)
Unoriented WZW Models and Holonomy of Bundle Gerbes
51
Fig. 3. The difference between two fundamental domains
since σ is orientation-reversing. By Stokes’ theorem, the exponential of the integral of the curvature of Rˆ over B is nothing but the holonomy of that line bundle around ∂ B. Thus, ˆ = hol ˆ (∂ B)−1 = hol Q (pr(∂ B))−1 . exp −i curv( R) R B
This is the term which is compensated by the boundary term, which is hol Q (∂ F )−1 = hol Q (∂ F)−1 · hol Q (pr(∂ B)). In summary
exp i
F
ρˆ · hol Q
(∂ F )−1
= exp i ρˆ · hol Q (∂ F)−1 ,
(107)
(108)
F
i.e. the holonomy is independent of the choice of the fundamental domain. Now let T = (τ , T ) be another trivialization of φˆ ∗ G. As discussed in Sect. 2.1, ˆ together with an isomorphism ν : π ∗ N ⊗ T → T , such there is a line bundle N → φ that the 2-forms ρˆ and ρˆ are related by ρˆ = ρˆ + curv(N ).
(109)
For the line bundle Rˆ defined in (95) this means R = R ⊗ π Z∗ σ ∗ N ⊗ π Z∗ N ,
(110)
and its descent line bundle Rˆ is Rˆ = Rˆ ⊗ σ ∗ N ⊗ N .
(111)
This is an equation of σ -equivariant line bundles, where Rˆ and Rˆ obtain equivariant structures from the Jandl structure as described before, and K := σ ∗ N ⊗ N carries the canonical σ -equivariant structure by permuting the order in the tensor product. Hence, Eq. (111) pushes into the quotient, namely Q = Q ⊗ K¯ .
(112)
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U. Schreiber, C. Schweigert, K. Waldorf
The holonomy of the descent bundle K¯ satisfies hol K¯ (∂ F) = hol N (∂ F) = holσ ∗ N (∂ F).
(113)
This finally means ρˆ · hol Q (∂ F)−1 exp i F (109) = exp i ρˆ + curv(N ) · hol Q⊗ K¯ (∂ F)−1 F (113) = exp i ρˆ · hol N (∂ F) · hol N (∂ F)−1 · hol Q (∂ F)−1 F ρˆ · hol Q (∂ F)−1 , = exp i
(114) (115) (116)
F
thus the holonomy is independent of the choice of the trivialization.
The following lemma asserts that the definition of holonomy is compatible with the definition of equivalence of Jandl structures. Lemma 5. The holonomy of a gerbe G with Jandl structure J only depends on the equivalence class of J . Proof. Let J = (k, A, ϕ) and J = (k, A , ϕ ) be two equivalent Jandl structures on G. It is shown in Theorem 1 that there is a unique flat equivariant line bundle N on M, such that N .A ∼ = A as equivariant line bundles. Because the action of Pic0K (M) is free, and A and A are isomorphic, N is the trivial equivariant line bundle. Remember the definition of the bundle R → Z in Eq. (95). For the two Jandl structures we get ˆ Since N is the R = R ⊗ π Z∗ N , and hence the descent bundles Rˆ = Rˆ ⊗ N over . trivial equivariant line bundle, Rˆ and Rˆ are isomorphic as equivariant line bundles, and thus define isomorphic line bundles Q and Q over . Isomorphic line bundles have the same holonomies, so Definition 10 is independent of the equivalence class of J .
An important condition for any notion of unoriented surface holonomy is its compatibility with ordinary surface holonomy for oriented surfaces: Theorem 3. If is orientable, for any choice of an orientation, the holonomy defined in Definition 10 reduces to the ordinary holonomy defined in Definition 7, ˆ ) = holG (φ, ), holG ,J (φ,
(117)
where φ and φˆ are related by the bijection of Lemma 3. In particular, if G admits a Jandl structure, the holonomy of G does not depend on the orientation. ˆ be a choice of an orientation on . Then F := or() is a Proof. Let or : → fundamental domain with empty boundary ∂ F = ∅. Choose a trivialization T of φˆ ∗ G to 2 ˆ obtain the 2-form ρˆ ∈ (). Then the left-hand side is equal to exp i or() ρ, ˆ because ∗ ∗ ˆ ˆ of Theorem 2. Because φ and φ correspond to each other, or φ G is the same gerbe as φ ∗ G, and or ∗ T is a trivialization with 2-form ρ = or ∗ ρ. ˆ Thus, the right-hand side is equal to exp i ρ and therefore equals the ordinary holonomy.
Unoriented WZW Models and Holonomy of Bundle Gerbes
53
3.3. Holonomy in local data. Let {Vi }i∈I be an open cover of M. To avoid notation, we assume that it is invariant under k and still good enough to admit all the local sections necessary to extract local data (g, A, B) of the gerbe G and (t, W, j) of the Jandl strucˆ →M ture J , as we explained in Sect. 2.4. We pull back the cover {Vi }i∈I along φˆ : −1 and obtain a cover {Uˆ i }i∈I with Uˆ i := φˆ (Vi ), together with pullback local data. Next, choose local data (h, M) of the trivialization T of the pullback gerbe and a 2-form ˆ so that ρˆ ∈ 2 (), φˆ ∗ g, φˆ ∗ A, φˆ ∗ B = 1, 0, ρˆ + D (h, M) (118) ˆ holds. Following the definition of the bundle R → Z in Eq. (95), the bundle Rˆ → has local data (119) (r, R) := φˆ ∗ (t, W ) − σ ∗ (h, M) − (h, M); the condition that Rˆ descends is equivalent to the Deligne cocycle condition D(r, R) = (1, 0),
(120)
which follows from Eqs. (118) and (67). ˆ M)σ,k , the pullback cover is invariant under σ . Because φˆ is an element of C ∞ (, Hence it projects to a cover of with open sets Ui := pr(Uˆ i ). Choose local sections ˆ and a dual triangulation T of , subordinate to the cover {Ui }i∈I , together ori : Ui → with a subordinating map i : T → I . As we did in Sect. 3.1 we choose the fundamental domain
F := ori( f ) ( f ), (121) f ∈T
where the f ’s are the faces of the triangulation. We now introduce three abbreviations. Let ωi2 ∈ 2 (Uˆ i ), ωi1j ∈ 1 (Uˆ i ∩ Uˆ j ) and ωi jk : Uˆ i ∩ Uˆ j ∩ Uˆ k → U (1) be some local data. First we denote the integral over a face f by ⎞ ⎛ 2 1 ⎠ I f (ω, ω1 , ω2 ) := exp ⎝i ωi( ωi( f) + i f )i(e) ·
v∈∂e
ori( f ) ( f )
e∈∂ f
ε( f,e,v)
ωi( f )i(e)i(v) (ori( f ) (v)),
ori( f ) (e)
(122)
where ε( f, e, v) ∈ {1, −1} indicates whether v is the end or the starting point of the edge e with respect to the orientation ori( f ) . Second, we denote the integral of some local data ωi1 ∈ 1 (Uˆ i ) and ωi j : Uˆ i ∩ Uˆ j → U (1) along an edge e of a face f by ε( f,e,v) 1 ωi(e) ωi(e)i(v) (ori( f ) (v)). (123) Ie, f (ω, ω1 ) := exp i · ori( f ) (e)
v∈∂e
Recall that the set of edges in T splits into the set E of orientation-preserving edges and the set E¯ of orientation-reversing edges. For an orientation-preserving edge e ∈ f 1 ∩ f 2 we have Ie, f1 (ω, ω1 ) = Ie, f2 (ω, ω1 )−1 , (124)
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while for an orientation-reversing edge Ie, f1 (ω, ω1 ) = Ie, f2 (σ ∗ ω, σ ∗ ω1 )
(125)
holds. In the latter case, since e is orientation-reversing, we have either ori(e) (e) = ori( f1 ) (e) or ori(e) (e) = ori( f2 ) (e), so that we can write just Ie (ω, ω1 ), where for f the choice of the face with the coinciding orientation is understood. Third, if v is a vertex of an edge e, we define for some smooth function ωi : Uˆ i → U (1) ε( f,e,v) Iv,e, f (ω) := ωi(v) (ori( f ) (v)). (126) ¯ we call Now if v is the common vertex of two orientation-reversing edges e1 , e2 ∈ E, v orientation-preserving, if ori(e1 ) (v) = ori(e2 ) (v) and orientation-reversing otherwise. Let us denote the set of orientation-reversing vertices by V¯ . If v is such a vertex, we just write Iv (ω) instead of Iv,e, f (ω), where for e the choice of the edge as well as for f the face with the coinciding orientation is understood. Now the first factor in the holonomy formula (103) is ⎞ ⎛ (127) exp i ρˆ = exp ⎝i φˆ ∗ Bi( f ) + dMi( f ) ⎠ . F
f ∈T
ori( f ) ( f )
Following [CJM02], by using Stoke’s theorem, Eq. (118) and our abbreviations, we end up with exp i ρˆ = I f (φˆ ∗ g, φˆ ∗ A, φˆ ∗ B) · Ie, f (h, M)−1 . (128) F
f ∈T
f ∈T e∈∂ f
Here the second factor collects the boundary contributions that appear in the application of Stokes’ theorem. Let us assume for the moment that is oriented, and all sections ori coincide with the global orientation restricted to Ui . In this situation, we have only orientation preserving edges, and each of them appears twice in the second factor. Since the contributions are inverse by (124), the second factor vanishes. We obtain the local holonomy formula expressed only by the local data of the gerbe, as it appeared originally in [Alv85]. If is not oriented, the second factor still consists of two contributions for each ¯ which are orientation-reversing edge e ∈ E, Ie, f1 (h, M) · Ie, f2 (h, M) = Ie (h · σ ∗ h, M + σ ∗ M). Hence, in the general case, the second factor of (128) is Ie, f (h, M)−1 = Ie (h · σ ∗ h, M + σ ∗ M)−1 . f ∈T e∈∂ f
(129)
(130)
e∈ E¯
For the second factor of the holonomy formula (103) we have to compute the holonomy of the descent line bundle Q around ∂ F. Note that
ori(e) (e) (131) Eˆ¯ := e∈ E¯
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Fig. 4. Assignment of local data. The middle layer shows and the subordinated indices; the top and lower ˆ layer show parts of the two sheets of
is a fundamental domain of ∂ F in ∂ F with boundary consisting of the preimages of the orientation-reversing vertices v ∈ V¯ . Now the holonomy of Q around ∂ F is equal to ˆ¯ where at the boundary points the equivariant structure of the holonomy of Rˆ around E, ˆ R is used; this is Ie (r, R) · Iv (φˆ ∗ j). (132) hol Q (∂ F) = e∈ E¯
v∈V¯
Since e is orientation-reversing, Ie (r, R) = Ie (φˆ ∗ t · σ ∗ h −1 · h −1 , φˆ ∗ W − σ ∗ M − M) = Ie (φˆ ∗ t, φˆ ∗ W ) · Ie (h · σ ∗ h, M + σ ∗ M)−1 .
(133) (134)
The second factor of (134) cancels (130) so that all the local data coming from the trivialization drops out. It remains ˆ = holG ,J (, φ) I f (φˆ ∗ g, φˆ ∗ A, φˆ ∗ B) · Ie (φˆ ∗ t, φˆ ∗ W )−1 · Iv (φˆ ∗ j), (135) f ∈T
e∈ E¯
v∈V¯
depending only on the local data of the gerbe and of the Jandl structure. We visualize this formula in Fig. 4.
3.4. Examples. In the next two subsections we will apply the general formula (135) to some examples of surfaces , and we will simplify the situation considerably by starting with the pullback gerbe φˆ ∗ G which allows us to choose a triangulation adapted to .
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Fig. 5. Klein bottle
Fig. 6. Klein bottle with a dual triangulation
Fig. 7. A fundamental domain for the Klein bottle in its double covering
Fig. 8. The real projective plane
3.4.1. Klein bottle Think of the Klein bottle as a rectangle with the identifications of the boundary indicated by arrows as in Fig. 5. The identification by the vertical arrows is orientation-preserving, while the one by the horizontal arrows is orientation-reversing. A dual triangulation is shown in Fig. 6. Note that this is a triangulation with only one face. We choose a local section from that face into the double cover, and define the fundamental domain F as its image, as indicated in Fig. 7. Here we dropped the arrows, but the identifications are still to be understood, so that both points labelled by v are identified. This means that we can choose the local orientations of the edges such that the orientation-reversing edges form a closed line, as indicated by the thick line. So there is no orientation-reversing vertex, and the local datum j of the Jandl structure is not relevant for the holonomy around the Klein bottle. 3.4.2. The real projective plane We proceed in the same way as for the Klein bottle, so think of the real projective plane RP 2 as a two-gon with the identification on the boundary indicated by arrows in Fig. 8. The identification is orientation-reversing. An example of a dual triangulation is for example shown in Fig. 9. Now we choose local sections from these two faces into the double cover, for example as shown in Fig. 10. ˆ and v is an orientation-reversing Note that here the thick line is not a closed line in , vertex. According to the local holonomy formula (135) here the local datum j of the Jandl structure enters in the holonomy.
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Fig. 9. A dual triangulation of the real projective plane with two faces
Fig. 10. A fundamental domain of the real projective plane in its double covering
4. Gerbes and Jandl Structures in WZW models 4.1. Oriented and orientable WZW models. In the following we are concerned with Lie groups M, and we will use the following notation. The left multiplication with a group element h is denoted by lh : M → M, and the map which assigns to h the inverse group element h −1 is denoted by Inv : M → M. The left invariant Maurer-Cartan form is denoted by θ , and the right invariant form by θ¯ . We call a gerbe G over M left invariant, if it is stably isomorphic to the gerbe lh∗ G for each h ∈ M, and similar for right and bi-invariance. A WZW model is a theory of maps φ : → M from a worldsheet into a target space M, which is a Lie group together with additional structure, called the background fields. It assigns to each map φ an amplitude, i.e. a number in U (1), as the weight of this map in a path integral. To be more precise: Definition 11. An oriented WZW model consists of a compact connected Lie group M, which is equipped with an Ad-invariant metric g = −, − on its Lie algebra and a bi-invariant gerbe G. It assigns an amplitude Aortd g,G (φ, ) := exp (iSkin (φ)) · holG (, φ)
(136)
to a map φ : → M from a closed oriented conformal worldsheet to M, where the kinetic term is ∗ 1 φ θ ∧ φ ∗ θ . (137) Skin (φ) := 2 Note that the conformal structure and the orientation on determine the Hodge star. In [Wit84] Witten discussed this theory for M = SU (2), which is an example for a compact, simple, connected and simply-connected Lie group. In this particular situation, the holonomy can be written as the exponential of the Wess-Zumino term, (138) holG (, φ) = exp i φ˜ ∗ H , B
so that we can express the amplitudes as Aortd g,G (φ, ) = exp(iSWZW (φ))
(139)
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with the action functional
φ˜ ∗ H .
SWZW (φ) := Skin (φ) +
(140)
B
Here B is a 3-dimensional manifold with boundary , φ˜ is an extension of φ on B, and H is the curvature of the gerbe G. Witten observed two symmetries of the WZW model on the type of Lie groups he considered. The first is translation symmetry: the action functional SWZW (φ) is invariant under the translation φ → lh ◦ φ. The associated conserved Noether current is given by J (φ) := −(1 + )φ ∗ θ ,
(141)
which is a 1-form on with values in the Lie algebra of M. To obtain this conserved, non-abelian current, Witten derived a specific relative normalization of the kinetic and the Wess-Zumino term, which was also adapted here. The second symmetry Witten observed is the invariance of the action functional SWZW (φ) under what he called parity transformation: reverse the orientation on and replace φ by φ¯ := Inv ◦ φ. Accordingly, the conserved current J (φ) for and the one ¯ the manifold with reversed orientation, namely for , ¯ = (1 − )φ ∗ θ¯ , J¯(φ)
(142)
are often called equivalent. Note that here the right invariant Maurer-Cartan form appears. In that sense, the parity transformation exchanges left and right movers. We now want to generalize this equivalence to any compact connected Lie group M. It is a simple consequence of the properties of the holonomy of G, that the parity symmetry ortd ¯ Aortd (143) g,G (φ, ) = A g,G (Inv ◦ φ, ) holds, if the gerbes Inv∗ G and G ∗ are stably isomorphic. Note that this is a condition on the gerbe G. It should not come as a surprise that in Witten’s discussion there is no such condition: Lemma 6. If G is a bi-invariant gerbe over a compact, simple, connected and simply connected Lie group, then Inv∗ G and G ∗ are stably isomorphic. Proof. Because stably isomorphic gerbes have the same curvatures, the curvature H of the bi-invariant gerbe G is a bi-invariant 3-form. It is a theorem by Cartan that on compact, simple, connected, simply connected Lie groups M the space of bi-invariant 3-forms is the span of the canonical 3-form ν, which satisfies Inv∗ ν = −ν. Hence Inv∗ G and G ∗ have the same curvature. Because the set of stable isomorphism classes of gerbes of the same curvature form a torsor over H2 (M, U (1)) [GR02], which here is the trivial group, the gerbes Inv∗ G and G ∗ are stably isomorphic.
We now give an even more general definition of parity transformations of a target space M with metric g and gerbe G. Definition 12. A parity transformation map is an isometry k : M → M of the metric g of order two, such that k ∗ G and G ∗ are stably isomorphic. We denote the set of parity transformation maps by P(M, g, G).
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Consider an oriented WZW model with target space M, Ad-invariant metric g and bi-invariant gerbe G. If k ∈ P(M, g, G) is a parity transformation map, we obtain the parity symmetry ortd ¯ Aortd (144) g,G (φ, ) = A g,G (k ◦ φ, ). We already discussed that k = Inv is a parity transformation map in the sense of Definition 12, if the gerbes Inv∗ G and G ∗ are stably isomorphic. However, for oriented WZW models on compact connected Lie groups there are more such parity transformation maps. Because the gerbe G is supposed to be bi-invariant, we try an ansatz k := lh ◦ Inv for some group element h ∈ M. The condition k 2 = id M restricts h to be an element of the center Z (M). So, the set P(M, g, G) of parity transformation maps for a compact connected Lie group M and a bi-invariant gerbe G, such that G ∗ is stably isomorphic to Inv∗ G, contains at least {l z ◦ Inv | z ∈ Z (M)} ⊂ P(M, g, G).
(145)
In particular, P(M, g, G) is not empty in the situation we are interested in. As a preparation for the unoriented case, we now relate parity symmetry to the ˆ Start with an oriented WZW model on together with a parity orientation cover : transformation map k. Let φ : → M be a map. By Lemma 3, there is a unique map ˆ M)k,σ . Once we have the orientation cover ˆ and the map φ, ˆ we may forget φˆ ∈ C ∞ (, their origin, in particular the orientation on . Then we may give the following Definition 13. An orientable WZW model consists of a compact connected Lie group M, which is equipped with an Ad-invariant metric g on its Lie algebra, a bi-invariant gerbe G and a parity transformation map k ∈ P(M, g, G). To a closed orientable conˆ M)k,σ , the following amplitude Aorble (φ, ˆ ) formal surface and a map φˆ ∈ C ∞ (, g,G is assigned. Choose any orientation on , and obtain a map φ : → M by Lemma 3. Define ortd ˆ (146) Aorble g,G (φ, ) := A g,G (φ, ). The amplitude is well-defined: if we had chosen the other orientation, we would get the same amplitudes, due to the fact that k is a parity transformation map and satisfies Eq. (144).
4.2. Unoriented WZW models. In the last section we gave the definition of an orientable ˆ M)k,σ makes use WZW model. The derivation of the amplitude of a map φˆ ∈ C ∞ (, of the existence of an orientation on both in the kinetic term and in the holonomy term. In this section, we want to overcome this obstruction. ˆ for a Let us first discuss the kinetic term. We want to define the kinetic term Skin (φ) ∞ k,σ ˆ M) in such a way that if is orientable, it reduces to the kinetic map φˆ ∈ C (, term Skin (φ) of the corresponding map φ. Note that ˆ := L(φ)
1 ∗ φˆ θ ∧ φˆ ∗ θ 2
(147)
ˆ which satisfies is a 2-form on , ˆ = −L(φ). ˆ σ ∗ L(φ)
(148)
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ˆ defines a 2-density Lden (φ) ˆ [BT82, BG88] on . The This property tells us that L(φ) integral of a 2-density over a surface is defined without respect to the orientability of this surface, so we define ˆ := Skin (φ)
ˆ Lden (φ).
(149)
To make the integral (149) more explicit, choose a triangulation T of , and for each ˆ where U f is some open neighborhood of f face f ∈ T a local section or f : U f → , in . By definition of the integral of a density, ˆ = ˆ Skin (φ) L(φ). (150) f ∈T
or f ( f )
One immediately checks that this definition is independent of the choice of the local sections: if one chooses for one face f the other orientation, namely σ (or f ), the corresponding term in the sum (150), ∗ ˆ ˆ ˆ L(φ) = − σ L(φ) = L(φ), (151) σ (or f ( f ))
or f ( f )
or f ( f )
gives the same contribution. It is also independent of the choice of the triangulation. Furthermore, if is orientable, we can choose a triangulation with a single face f = ˆ = Skin (φ), which was precisely our requirement on Skin (φ). ˆ and get Skin (φ) We have already discussed in Sect. 3 how to define surface holonomies for an arbitrary ˆ M)k,σ : we have to choose a Jandl structure J closed surface with a map φˆ ∈ C ∞ (, ˆ ) is defined in Definition 10 in such a way that if is orientable, on G. Then holG ,J (φ, it coincides by Theorem 3 with holG (φ, ). Remember that a necessary condition on the existence of a Jandl structure J = (k, −, −) was that the gerbes k ∗ G and G ∗ are stably isomorphic. We already have encountered this condition for the orientable WZW model, so that it does not come as an additional restriction. This leads us to the following Definition 14. An unoriented WZW model consists of a compact connected Lie group M, which is equipped with an Ad-invariant metric g on its Lie-algebra and a bi-invariant gerbe G with Jandl structure J , whose action of Z2 on M is a parity transformation map k ∈ P(M, g, G). To a closed conformal surface and a map ˆ M)k,σ the amplitude φˆ ∈ C ∞ (, ˆ ) := exp iSkin (φ) ˆ · holG ,J (φ, ˆ ) Aunor ( φ, (152) g,G ,J is assigned. According to the definition of both factors, if is orientable, we have orble ˆ ˆ Aunor g,G ,J (φ, ) = A g,G (φ, ).
(153)
If is even oriented, by Eq. (146) we have ortd ˆ Aunor g,G ,J (φ, ) = A g,G (φ, ).
(154)
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4.3. Crosscaps and the trivial line bundle. In the following two sections we use the classification of Jandl structures to classify unoriented WZW models with a fixed gerbe G and a fixed parity transformation map k ∈ P(M, g, G). By Theorem 1, the set of equivalence classes of Jandl structures of G with the action of K = Z2 on M defined by k is a torsor over the flat K -equivariant Picard group Pic0K (M). In this section we discuss a special element of this group. On any manifold, there is the trivial line bundle L 1 := M × C with the trivial hermitian metric and the trivial connection, which is flat. It represents the unit element of the flat Picard group Pic0 (M). Recall the following facts concerning equivariant line bundles [Gom03]. There are two obstructions for a given line bundle to admit equivariant structures: the first depends on the bundle and the group action, namely that k∗ L ⊗ L ∗ ∼ = L 1,
(155)
which is still to be understood as an equation of hermitian line bundles with connection. 2 (K , U (1)). Now, The second obstruction is a class in the group cohomology group HGrp if both obstructions are absent, the possible equivariant structures are parameterized by 1 (K , U (1)) which is just the group of one-dimensional the group cohomology group HGrp characters of K . In our case K = Z2 we have 1 (K , U (1)) = Z2 , HGrp
(156)
2 (K , U (1)) HGrp
(157)
= 0,
so that the second obstruction vanishes, and every line bundle L, which satisfies the remaining obstruction (155) admits exactly two K -equivariant structures. In particular L 1 itself satisfies (155). We exhibit its two equivariant structures explicitly. Remember from Sect. 2.2, that we have to choose an isomorphism ϕ : k∗ L 1 → L1
(158)
of line bundles, such that ϕ ◦ k ∗ ϕ = id L 1 . So both choices are either ϕ1 = id M×C or ϕ−1 : (x, z) → (x, −z). We denote L 1 together with the equivariant structure ϕ1 by L 1K . It represents the unit element of Pic0K (M). We denote L 1 together with the equivariant K . Note that L K ⊗ L K = L K as equivariant line bundles. Hence structure ϕ−1 by L −1 −1 −1 1 it represents a non-trivial element of order two in Pic0K (M). The whole construction is completely independent of M, so Pic0K (M) always contains at least these two elements. As a consequence, if a gerbe G admits a Jandl structure K .J is another, inequivalent Jandl structure on G. We will now investigate J , then L −1 the difference between the corresponding unoriented WZW models. We work with local data, so let {Vi }i∈I be a good open cover of M. Choose all the sections that have been introduced in Sect. 2.4, and extract local data (t, W ), j of the Jandl structure J . We also explained how to extract a local datum νi : Vi → U (1) from an equivariant structure on a line bundle over M. The local datum of L 1K is the constant K is the constant global function global function ν1 = 1, and the local datum of L −1 ν1 = −1. According to the definition of the action of Pic0K (M) on Jdl(G, k), the local data of K L −1 .J are (t, W ) and − j. Now observe the occurrences of the local datum j in the local holonomy formula (135): it appears for each orientation-reversing vertex v ∈ V¯ .
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Following our example in Sect. 3.4.2, this happens in the presence of a crosscap. We conclude that the amplitudes of both unoriented WZW models with Jandl structures J K .J differ by a sign for each crosscap in . and L −1 4.4. Examples of target spaces. We would like to discuss three examples of target spaces, namely the Lie groups SU (2), S O(3), where the Ad-invariant metric on their Lie algebras is given by their Killing forms, and the two-dimensional torus T 2 = S 1 × S 1 with the euclidean scalar product. The gerbes are supposed to be bi-invariant. 4.4.1. The Lie group SU (2) Following our general discussion, the actions of Z2 on SU (2) we have to consider are given by k : g → g −1 and k : g → −g −1 , where −1 ∈ Z (SU (2)) is the non-trivial element in the center. The same maps were considered in [HSS02, Bru02, BCW01]. Fix a bi-invariant gerbe G over SU (2). Up to stable isomorphism, this is G = G0⊗n , where G0 is the basic gerbe over SU (2) [Mei02]. By Lemma 6, both k’s are parity transformation maps. The set Jdl(G, k) is a torsor over Pic0K (SU (2)) by Theorem 1. In order to compute the group of equivariant flat line bundles, we first observe Pic0 (M) = Hom(π1 (M), U (1)) = 0,
(159)
since SU (2) is simply connected. So up to isomorphism there is only one flat line bundle, the trivial one. Hence there are exactly two inequivalent Jandl structures for each map k and each bi-invariant gerbe G; this is in agreement with the results of [PSS95a, PSS95b]. 4.4.2. The Lie group S O(3) The center of S O(3) is trivial, so that we have only one action to consider, namely by k : g → g −1 . Let G be a bi-invariant gerbe over S O(3), such that k ∗ G and G ∗ are stably isomorphic. Such gerbes for example are constructed up to stable isomorphism in [GR03]. We have to investigate the group Pic0K (S O(3)) of flat equivariant line bundles. Again we first consider the group Pic0 (S O(3)) of flat line bundles and classify equivariant structures on them. By π1 (S O(3)) = Z2 we have Hom(π1 (S O(3)), U (1)) = Hom(Z2 , U (1)) = Z2 ,
(160)
so there are - up to isomorphism - two flat line bundles. We will give them explicitly: As S O(3) is the quotient of SU (2) by q : g → −g, the two flat line bundles over S O(3) K . correspond to the two equivariant flat line bundles over SU (2), namely L 1K and L −1 K ˜ Clearly, L 1 descends to the trivial flat line bundle L 1 → S O(3), which admits equivariant structures, more precisely, according to the discussion in Sect. 4.3, there are two K descends to a non-trivial flat line bundle L ˜ −1 → S O(3), and we have to of them. L −1 ask whether it admits equivariant structures, which is equivalent to the condition that (161) d L˜ −1 := k ∗ L˜ −1 ⊗ L˜ ∗−1 ∼ = L˜ 1 . Now d L˜ −1 is a flat line bundle, and hence either isomorphic to L˜ −1 or to L˜ 1 . Because Pic0 (S O(3)) is a group of order two, we have L˜ −1 ⊗ L˜ −1 = L˜ 1 . The assumption d L˜ −1 ∼ = L˜ −1 would therefore mean k ∗ L˜ −1 ∼ = L˜ 1 , which is a contradiction since L˜ 1 is ∗ ˜ the trivial bundle and k L −1 is not. Hence (161) is true, and L˜ ∗−1 admits two equivariant structures. All together, there are four equivariant flat line bundles over S O(3) and hence four Jandl structures on G; again, this is in agreement with [PSS95a, PSS95b].
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4.4.3. The two-dimensional Torus T 2 For dimensional reasons, all gerbes over T 2 are trivial and have curvature H = 0. This allows us to discuss an example with a parity transformation map k, which is not of the form k = l z ◦ Inv but simply the identity map k = id. This allows us to make contact with [BPS92]. Now let G be a bi-invariant gerbe over T 2 . The set Jdl(G, id) is a torsor over Pic0K (T 2 ) by Theorem 1, which is isomorphic to H1K (T 2 , U (1)) by Eq. (33). The Borel space associated to the trivial K -action is TK2 = EZ2 × T 2 . With EZ2 = RP ∞ we have H1K (T 2 , U (1)) = H 1 (TK2 , U (1)) = H1 (RP ∞ , U (1)) ⊕ H1 (T 2 , U (1)) = Z2 ⊕ U (1) ⊕ U (1) = Z2 ⊕ T 2 . We now assume that the gerbe G admits a Jandl structure J = (id, A, ϕ). In particular, A = (A, α) is a stable isomorphism from G to G ∗ . Recall that a gerbe G consists of the following data: a surjective submersion π : Y → M, a line bundle L → Y [2] , an isomorphism µ, and a 2-form C ∈ 2 (Y ). Recall further that here A is a line bundle over Z = Y [2] , and both projections p and p from Z to Y coincide with π2 , π1 : Y [2] → Y . The condition on the curvature of A in Definition 3 now reads curv(A) = π1∗ C + π2∗ C.
(162)
Furthermore, since for all gerbes the curving C satisfies −π2∗ C + π1∗ C = curv(L), we have 2π2∗ C = curv(A) − curv(L), (163) which is an equation of 2-forms on Y [2] . On the right-hand side we have a closed 2-form which defines an integral class in cohomology. Since π2 is a surjective submersion, also 2C defines a class in H2 (Y, Z). Because the gerbe G is trivial, we can choose a trivialization T and obtain the 2-form B ∈ 2 (M) as in Definition 7, which satisfies π ∗ B = C + curv(T ) and dB = H = 0. Usually one chooses T such that B is constant, then it is nothing but the Kalb-Ramond “B-Field”. Because π is also a surjective submersion it follows that 2B defines a class in H2 (M, Z). Thus we have derived the quantization condition that the B-Field has half integer valued periods. This condition was originally found in [BPS92] by an analysis of the bulk spectrum of right and left movers. References [Alv85]
Alvarez, O.: Topological quantization and cohomology. Commun. Math. Phys. 100, 279– 309 (1985) [BCW01] Bachas, C., Couchoud, N., Windey, P.: Orientifolds of the 3-sphere. JHEP 12, 003 (2001) [BG88] Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves, and Surfaces. Volume 115 of Graduate Texts in Mathematics, Berlin-Heidelberg-New York: Springer, 1988 [BPS92] Bianchi, M., Pradisi, G., Sagnotti, A.: Toroidal compactification and symmetry breaking in openstring theories. Nucl. Phys. B 376, 365–386 (1992) [Bru02] Brunner, I.: On orientifolds of WZW models and their relation to geometry. JHEP 01, 007 (2002) [Bry93] Brylinski, J.-L.: Loop spaces, Characteristic Classes and Geometric Quantization. Volume 107 of Progress in Mathematics, Basel: Birkhäuser, 1993 [Bry00] Brylinski, J.-L.: Gerbes on complex reductive Lie Groups. http://arxiv.org/list/math/0002158, 2000 [BT82] Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, Volume 82 of Graduate Texts in Mathematics, Berlin-Heidelberg-New York: Springer, 1982
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[CJM02] Carey, A.L., Johnson, S., Murray, M.K.: Holonomy on D-Branes. J. Geom. Phys. 52(2), 186– 216 (2002) [FHS+ 00] Fuchs, J., Huiszoon, L.R., Schellekens, A.N., Schweigert, C., Walcher, J.: Boundaries, crosscaps and simple currents. Phys. Lett. B 495(3–4), 427–434 (2000) [FPS94] Fioravanti, D., Pradisi, G., Sagnotti, A.: Sewing constraints and non-orientable open strings. Phys. Lett. B 321, 349–354 (1994) [FRS04] Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators ii: unoriented world sheets. Nucl. Phys. B 678(3), 511–637 (2004) [Gaw88] Gaw¸edzki, K.: Topological Actions in two-dimensional Quantum Field Theories. In: Nonperturbative Quantum Field Theory, London: Plenum Press, 1988 [Gom03] Gomi, K.: Equivariant smooth deligne cohomology. Osaka J. Math. 42(2), 309–337 (2003) [GR02] Gaw¸edzki, K., Reis, N.: Wzw branes and gerbes. Rev. Math. Phys. 14(12), 1281–1334 (2002) [GR03] Gaw¸edzki, K., Reis, N.: Basic gerbe over non-simply connected compact groups. J. Geom. Phys. 50(1–4), 28–55 (2003) [HS00] Huiszoon, L.R., Schellekens, A.N.: Crosscaps, boundaries and t-duality. Nucl. Phys. B 584(3), 705–718 (2000) [HSS99] Huiszoon, L.R., Schellekens, A.N., Sousa, N.: Klein bottles and simple currents. Phys. Lett. B 470(1), 95–102 (1999) [HSS02] Huiszoon, L.R., Schalm, K., Schellekens, A.N.: Geometry of wzw orientifolds. Nucl. Phys. B 624(1–2), 219–252 (2002) [Jan95] Jandl, E.: Lechts und rinks. Munich: Luchterhand Literaturverlag, 1995 [Mei02] Meinrenken, E.: The basic gerbe over a compact simple lie group. Enseign. Math., II. Sér. 49 (3–4), 307–333 (2002) [PSS95a] Pradisi, G., Sagnotti, A., Stanev, Y.S.: The open descendants of nondiagonal su(2) wzw models. Phys. Lett. B 356, 230–238 (1995) [PSS95b] Pradisi, G., Sagnotti, A., Stanev, Y.S.: Planar duality in su(2) wzw models. Phys. Lett. B 354, 279– 286 (1995) [SS03] Sousa, N., Schellekens, A.N.: Orientation matters for nimreps. Nucl. Phys. B 653(3), 339– 368 (2003) [Ste00] Stevenson, D.: The Geometry of Bundle Gerbes, PhD thesis, University of Adelaide, http:// arxiv.org/list/math.DG/0004117, 2000 [Wit84] Witten, E.: Nonabelian bosonization in two dimensions. Commun. Math. Phys. 92, 455–472 (1984) Communicated by M.R. Douglas
Commun. Math. Phys. 274, 65–80 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0275-6
Communications in
Mathematical Physics
Mass Under the Ricci Flow Xianzhe Dai1,2, , Li Ma3, 1 Department of Mathematics, University of California, Santa Barbara, CA 93106, USA.
E-mail:
[email protected]
2 Chern Institute of Mathematics, Tianjin, China 3 Department of Mathematical Science, Tsinghua University, Peking 100084, People’s Republic of China.
E-mail:
[email protected] Received: 25 January 2006 / Accepted: 25 January 2007 Published online: 6 June 2007 – © Springer-Verlag 2007
Abstract: In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let (M, g) be an ALE manifold of dimension n = 3. If m(g) = 0, then the Ricci flow starting at g can not have Euclidean space as its (uniform) limit. 1. Introduction Ricci flow is an important geometric evolution equation in Riemannian Geometry. It was introduced by R. Hamilton in 1982 (see [8]) and used extensively by him to prove some outstanding results on 3-manifolds and 4-manifolds. Recently it has been used spectacularly by G. Perelman [14] to study the geometrization conjecture on 3-manifold. The flow has also been very useful in the study of pinching results and the metric smoothing process. As a natural geometric tool, Ricci flow should be used to study properties of physically meaningful objects such as mass, entropy, etc. In this paper we would like to understand the behavior of mass under the Ricci flow. In general relativity, isolated gravitational systems are modeled by spacetimes that asymptotically approach Minkowski spacetime at infinity. The spatial slices of such spacetimes are then the so-called asymptotically flat or asymptotically Euclidean (AE in short) manifolds. That is, Riemannian manifolds (M n , g) such that M = M0 ∪ M∞ (for simplicity we deal only with the case of one end; the case of multiple ends can be n dealt with similarly) with M0 compact and M∞ R − B R (0) for some R > 0 so that in the induced Euclidean coordinates the metric satisfies the asymptotic conditions gi j = δi j + O(r −τ ), ∂k gi j = O(r −τ −1 ), ∂k ∂l gi j = O(r −τ −2 ). Partially supported by NSF and NSFC.
(1.1)
The research is partially supported by the National Natural Science Foundation of China 10631020 and
SRFDP 20060003002.
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Here τ > 0 is the asymptotic order and r is the Euclidean distance to a base point. The total mass (the ADM mass) of the gravitational system can then be defined via a flux integral [1, 11], 1 m(g) = lim (∂i gi j − ∂ j gii ) ∗ d x j . (1.2) R→∞ 4ωn S R Here ωn denotes the volume of the (n − 1)-sphere and S R the Euclidean sphere with radius R centered at the base point. By [3], when the scalar curvature is integrable and τ > n−2 2 , the mass m(g) is well defined and independent of the coordinates at infinity, and therefore is a metric invariant. The famous Positive Mass Theorem, proved by Schoen-Yau [19] (later Witten gave an elegant spinor proof [20]), says that the mass m(g) ≥ 0 if the scalar curvature is nonnegative (and the manifold is spin). Moreover, m(g) = 0 if and only if M is the Euclidean space. There is also the notion of asymptotically locally Euclidean, or ALE, manifolds. For our purpose we will use the following characterization of the ALE property of a complete non-compact Riemannian manifold (M, g). Namely we use the curvature decay condition |Rm|(x) = O(d(x)−(2+τ ) )
(1.3)
for some τ > 0 as d(x) → ∞, and the volume growth condition V ol(Br , g) ≥ V r n
(1.4)
for some constant V > 0. Here Rm is the Riemannian curvature tensor of the metric g, and d(x) is the distance function from the base point. According to [3, Theorem (1.1)], (1.3), (1.4) imply that (M, g) is asymptotically lon cally Euclidean. Namely, M = M0 ∪M∞ with M0 compact and M∞ R − B R (0) / , n where ⊂ O(n) is a finite group acting freely on R − B R (0), so that the asymptotic conditions (1.1) hold. For an ALE manifold (M, g), the mass m(g) can be defined by (1.2) again, except that S R should be taken as the distance sphere or, equivalently, the quotient of the Euclidean sphere by . An ALE manifold (M, g) is actually AE if the asymptotic volume ratio µ = 1. Here µ = lim V (Br , g)/ n r n , r →∞
(1.5)
where n is the volume of the unit ball in the standard n-dimensional Euclidean space Rn . As we mentioned, we would like to investigate the behavior of the mass m(g) under the Ricci flow. Recall that the Ricci flow is a family of evolving metrics g(t) such that ∂ g(t) = −2Rc(g(t)), ∂t
(1.6)
on M with g(0) = g, where Rc(g(t)) is the Ricci tensor of the metric g(t). To make sure that the mass is well defined under this evolving flow g(t), we first need to show that the ALE property is preserved along the flow. Our main result is the following Theorem 1. Let g(t), 0 ≤ t ≤ T < +∞, be a Ricci flow on M with (M, g(0)) being an ALE (AE resp.) manifold of dimension n. Assume that g(t) has uniformly bounded sectional curvature. Then
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A) The ALE (AE resp.) property is preserved along the flow. B) The integrability condition R ∈ L 1 is also preserved along the flow provided (4.25) and (4.26) hold, which is the case if either R = O(r −q ), q > n and τ > n−4 2 or }. τ > max{n − 4, n−2 2 C) Under the conditions above, the mass m(t) = m(g(t)) is well defined, and m (t) = Ri d S i , (1.7) Sr →∞
where d S i = 4ω1 n ∗ d xi and Ri denotes the covariant derivative of R. Furthermore, n−2 if R = O(r −q ) for q > n and τ > n−2 2 , or τ > max{n − 3, 2 }, the mass is invariant under the Ricci flow. In particular, the mass is invariant in dimension 3 (τ > 21 ). D) Assume that the initial metric g(0) = g satisfies the additional decay condition ∂k ∂l ∂ p gi j = O(r −τ −3 ),
(1.8)
then the mass is invariant under the Ricci flow if n ≤ 6 (and τ > n−2 2 ) or if τ > n−2 max{n − 4, 2 }. That the ALE property is preserved will be obtained by using the maximum principles of Ecker-Huisken [5] and W.X.Shi [15], Cf. also P.Li and S.T.Yau [12]. To compute the changing rate of the mass of the evolving metric g(t), the key part is to get a decay estimate of the spatial derivative of the scalar curvature function R(x, t) of the metric g(t) at infinity, which is furnished by Shi’s local gradient estimate [15]. Theorem 2. Let g(t), 0 ≤ t < +∞, be a Ricci flow on M with uniformly bounded sectional curvature. Assume further that each g(t) is ALE and g(t) converges uniformly to an ALE metric g∞ as t goes to infinity. Then lim m(g(t)) = m(g∞ ).
t→∞
The notion of uniform convergence is introduced by using the space Mτ of [11], see Definition 13. A direct consequence of Theorems 1 and 2 is the following Corollary 3. Let (M, g) be an ALE manifold of dimension n and of asymptotic order τ > max{n − 3, n−2 2 }. If m(g) = 0, then the Ricci flow starting at g can not converge uniformly to a Euclidean space. Note that the Ricci flow preserves nonnegative scalar curvature [8]. This raises the prospect of proving the positive mass theorem of Schoen and Yau via Ricci flow, which we intend to investigate elsewhere. On the other hand, one can also look for applications of our results by combining with the positive mass theorem. For example, one sees that there are no complete non-compact Riemannian manifolds satisfying the hypothesis of the Main Theorem in [15] in dimension 3 by using the long time convergence result of [15], Theorems 1 and 2, and the positive mass theorem, see also [4, 7]. Let us explain why Theorem 1 comes so natural. We begin by recalling some basic facts about Ricci flow on complete non-compact Riemannian manifold (M, g) with bounded sectional curvature K 0 . Let g(t) be a family of the metrics evolving under the Ricci flow on M with initial data g, 0 ≤ t ≤ T < +∞. We shall write by ∇g(t) and Ri jkl (t) the Riemannian connection and Riemannian curvature tensor of g(t) respectively. Hamilton proved in [8] that
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the asymptotic volume ratio µ(t) = µ(g(t)) of (1.5) is a constant under the Ricci flow with bounded curvature and nonnegative Ricci curvature, where |Rm| → 0 at infinity on the complete non-compact Riemannian manifold. This result tells us that if µ = 1 at t = 0, then the Ricci flow can not have Euclidean space as its limit. It is well known that Ricci flow smoothes out the metric. W.X.Shi [16] showed that there exists a positive constant T > 0 such that for any integer α ≥ 0 and any 0 < t ≤ T , there exist constants c(n, K 0 ), c(n, K 0 , T ) and c(n, , K 0 , α, t) such that e−c(n,K 0 )t g ≤ g(t) ≤ ec(n,K 0 )t g, |∇g − ∇g(t) | ≤ c(n, K 0 )t, |Ri jkl | ≤ c(n, K 0 , T ), and α Ri jkl (t)| ≤ c(n, K 0 , α, t). |∇g(t)
All these facts will be implicitly used in this paper. It is clear that the volume growth condition (1.4) is preserved along the Ricci flow. In Sect. 18 in [8], Hamilton further proved that if the curvature Rm → 0 as s → +∞ for the initial metric, where s is the distance function to a fixed point of the metric g, the same is true for each g(t). So it is very natural for one to expect that if the curvature of the initial metric has decay at infinity, then the same is true for the evolving metric g(t). With this understanding, one would like to know the change of the mass, a natural invariant of the metric, under the flow. Finally, we refer the reader to [6] for related discussions under the (worldsheet) RG flow. Also, in a recent preprint [13], a different approach is used to study these questions that we address here. Throughout this paper we will denote by C, c various constants depending only on dimension. 2. Preliminaries In this section we briefly introduce some facts on Ricci flow. We shall use notations from [9]. Let M be a manifold of dimension n, g(t) a family of metrics evolving by Ricci flow (1.6). Then the curvature tensor evolves by the equation ∂ Rm = Rm + Rm ∗ Rm, ∂t
(2.9)
where Rm ∗ Rm denotes a quadratic expression of the curvature tensor. It follows then ∂ |Rm|2 = |Rm|2 − 2|∇ Rm|2 + Rm ∗ Rm ∗ Rm, ∂t which yields ∂ |Rm|2 ≤ |Rm|2 + C|Rm|3 . ∂t
(2.10)
The evolution equation for the scalar curvature is much simpler, and one has ∂ R = R + 2|Rc|2 . ∂t
(2.11)
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Now let X be a point in M. Let Y = {Ya }, 1 ≤ a ≤ n be a frame at X . In local coordinates X = {x i }, we have Ya = yai ∂/∂ x i . Let gab = g(Ya , Yb ), and let ∇ba = ybi ∂/∂ yai be the vector fields tangent to the fibers of the frame bundle. Write by Da the vector field on the frame bundle F(M) which is the lift of the vector Ya at Y ∈ F(M). Then we have j
Da = yai [∂ x i − ikj yb ∂/∂ ybk ], where ikj ’s are the Christoffel symbols of the connection. Under the Ricci flow, we can define the evolving orthonormal frame on M such that ∂t Fai = g i j R jk Fak , where (g i j ) is the inverse matrix of (gi j ). Then we set Dt = ∂t + Rab g bc ∇cb . Note that Dt gab = 0. This says that Dt is the unique tangent vector field to the orthonormal bundle. Choose a metric on F(M) such that Da , ∇cb are an orthonormal basis. Then we can see that Dt − ∂t is a space-like vector orthonormal to the orthonormal frame bundle. A useful fact for us is that for a smooth function u on M × (0, T ), we have (Dt − )Da u = Da (∂t − )u.
(2.12)
We now recall Hamilton’s argument [8]. Assume that a K -bounded smooth function u satisfies the heat equation u t = u, in M
(2.13)
with |Du|2 ≤ δ at t = 0. Then we have by (2.12), Dt Da u = Da u, and thus ∂t |Du|2 = |Du|2 − 2|D 2 u|2 . By the maximum principle of Shi [15] we have |Du|2 (x, t) ≤ δ.
(2.14)
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Let F = t|D 2 u|2 + |Du|2 . Then by a direct computation, we have ∂t F ≤ F − (1 − cK t)|D 2 u|2 . Hence, by the maximum principle of Shi [15] again we get for t ≤ 1/cK , F(x, t) ≤ δ 2 , which implies that √ |D 2 u| ≤ δ/ t. Note that | u|2 ≤ n|D 2 u|2 . Using the heat equation we obtain that for t ≤ 1/cK , √ √ |u t | ≤ nδ/ t. Therefore, we have
Take δ ≤
√
√ √ |u(x, t) − u(x, 0)| ≤ 2 nδ t. K 2 and assume that lim u(x, 0) = 0,
x→∞
in M
uniformly. Then we can conclude using an iteration argument that for any t ∈ [0, T ], lim u(x, t) = 0,
x→∞
in M
uniformly. In fact, Hamilton [8] showed that for any δ > 0 and for any bounded smooth function u 0 ∈ C 1 (M) with lim x→∞ u 0 (x) = 0, one can find a bounded smooth solution u(x, t) to the heat equation such that u 0 (x) ≤ u(x, 0) and |Du|2 (x, t) ≤ δ on M ×[0, T ]. 3. ALE is Preserved In this section we study the ALE property under the Ricci flow and show that it is preserved. The question can be reduced to the study of the non-negative solutions to the heat equation u t = u, in M
(3.15)
with initial data u(0) = u 0 , where = g(t) is the Laplacian operator of the family of metrics g(t). We assume that u 0 has a decay O(d(x)−σ ) for some σ > 0. Theorem 4. Let g(t) be the solution of the Ricci flow (1.6) over [0, T ]. Assume that g(t) has uniform curvature bound |Rm(g(t))| ≤ K . Then non-negative solutions to (3.15) have the same decay rate as the initial data u 0 . The main tools here are the maximum principles, especially the maximum principle of [5, Theorem 4.3]. For the reader’s convenience, we quote the result here (the superscript ‘t’ is put in here to emphasize the t-dependence from the metric g(t)).
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Theorem 5 (Ecker-Huisken). Suppose that the complete non-compact manifold M n with Riemannian metric g(t) satisfies the uniform volume growth condition volt (Brt ( p)) ≤ exp k(1 + r 2 )
(3.16)
for some point p ∈ M and a uniform constant k > 0 for all t ∈ [0, T ]. Let w be a function on M × [0, T ] which is smooth on M × (0, T ] and continuous on M × [0, T ]. Assume that w and g(t) satisfy i) the differential inequality ∂ w − t w ≤ a · ∇w + bw, ∂t
(3.17)
where the vector field a and the function b are uniformly bounded sup M×[0,T ] | a| ≤ α1 , sup M×[0,T ] | b| ≤ α2
(3.18)
for some constants α1 , α2 < ∞; ii) the initial data w( p, 0) ≤ 0
(3.19)
for all p ∈ M; iii) the growth condition 0
T
exp −α3 d t ( p, y)2 |∇w|2 (y)dµt dt < ∞
(3.20)
M
for some constant α3 > 0; iv) bounded variation condition in metrics
∂
sup M×[0,T ] g(t)
≤ α4 ∂t
(3.21)
for some constant α4 < ∞. Then, we have w≤0
(3.22)
on M × [0, T ]. In our situation, with the metric g(t) coming from the Ricci flow (1.6), the condition (3.21) is clearly satisfied by the uniform curvature bound. The uniform volume growth condition (3.16) also follows immediately from the volume comparison theorem via the curvature bound. The differential inequality will be coming from a modification of the solution of the heat equation (3.15). To see that the coefficients are uniformly bounded per (3.18) requires the following lemma.
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Lemma 6. Let g(t) be the solution of the Ricci flow (1.6) over [0, T ] with g(0) = g being an ALE. Assume that g(t) has uniform curvature bound |Rm(g(t))| ≤ K . Then, for sufficiently large R, there is a smooth positive function f on M such that f (x) = C0 >> 1, for x ∈ B R ; c d t (x) ≤ f (x) ≤ Cd t (x) for x ∈ M − B R . Moreover f ≥ C0 , |∇ t f | ≤ C1 , | t f | ≤ C2 . Proof. Since (M, g) is ALE, we have coordinates at infinity, which we denote by x. Let |x| be the Euclidean distance function. Choose a smooth increasing function φ(s) on R such that φ(s) = C0 = R − 1, if s ≤ R − 1; φ(s) = s, if s ≥ R, and |φ | ≤ 1,
|φ | ≤ 2.
We define our function f to be f (x) = φ(|x| ). Then clearly f ≥ C0 . Since the metrics g(t) are all equivalent and g(0) = g is ALE, we can use the Euclidean norm in estimating |∇ t f | and | t f |. Then |∇ t f | = |φ ||∇ t |x| | ≤ C2 . Similarly the estimate | t f | ≤ C2 follows from the coordinate expression of the Laplacian 1 ∂ t = √ det g(t) ∂ xi and the known estimate for g(t).
det g(t)g i j (t)
∂ ∂x j
We now suppress the superscript ‘t’ with the understanding that all covariant derivatives and the Laplacian are taken with respect to g(t). In our application of Theorem 5 to the proof of Theorem 4, we will let w = f σ u, where σ > 0. Then the growth condition (3.20) follows from the gradient bound (2.14), which implies that |∇w| ≤ C(T ) f σ +1 . We now turn to the proof of Theorem 4.
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Proof. For simplicity, we assume that (M, g0 ) is an ALE with one end. Let u 0 (x) = O(d(x)−σ ) as d(x) → ∞, where the distance function is with respect to a fixed point o in M. Choose a global smooth positive function f (x) on M as in Lemma 6 and let h(x) = f (x)σ . Set w(x, t) = h(x)u(x, t). Then, by a direct computation we have that wt = hu t , wi = h i u + hu i , and w = hu + 2∇h∇u + h u. Hence, (∂t − )w = Bw − 2∇ log h∇w, where B(x, t) = 2|∇h|h 2−h h . Note that the coefficients B and ∇ log h are uniformly bounded by Lemma 6. In particular, |B| ≤ b. Since 2
w(x, 0) = d(x)σ u 0 (x) ≤ D < +∞, and (∂t − )(w − Detb ) ≤ B(w − Detb ) − 2∇ log h∇(w − Detb ), we have by the maximum principle of Ecker-Huisken, Theorem 5, (see also the proofs of Theorem 18.2 [8] and Theorem 4.3 in [5], see also [15]) that there exists a uniform constant C1 > 0 such that max MT w ≤ C1 , where MT = M × [0, T ]. This implies the desired decay for u(x, t).
We are now in a position to prove the first part of Theorem 1. In fact, since |Rm(g0 )| ≤ C0 d(x)−σ for d(x) >> 1, where σ = 2 + τ , we can choose a bounded smooth function u 0 , which dominates the function |Rm(g0 )|2 such that it is C0 d(x)−2σ for d(x) >> 1 and has bounded gradient. Let u be the solution of the heat equation as above. Then under Ricci flow, we have from (2.10) and the uniform curvature bound, ∂t |Rm|2 ≤ |Rm|2 + C K |Rm|2 while ∂t (eC K t u) = (eC K t u) + C K eC K t u. Therefore, we have by the maximum principle of Shi [15] and Theorem 4 that |Rm|2 ≤ eC K t u ≤ eC K t d(x)−2σ , on M. Thus, under the Ricci flow, the ALE property is preserved.
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Remark 7. That the ALE property is preserved does not follow from [10, Remark 0.11], as was claimed there. The same is true for AE, as we have the following analog of a theorem of Hamilton [8]. Corollary 8. Let g(t), 0 ≤ t ≤ T , be a Ricci flow on M with uniformly bounded sectional curvature. Assume further that g(0) is ALE. Then the asymptotic volume ratio µ(t) = µ(g(t)) is constant along the Ricci flow. Proof. If (M, g) is ALE, it follows in [3] that M = M0 ∪ M∞ nfrom the characterization with M0 compact and M∞ R − B R (0) / , where ⊂ O(n) is a finite group n acting freely on R − B R (0), so that the asymptotic conditions (1.1) hold, where the asymptotic coordinates come from the projection of the Euclidean coordinates under Rn − B R (0) → Rn − B R (0) / . Therefore, (M, g) has the asymptotic volume ratio µ=
1 . ||
Since ALE is preserved along Ricci flow by Theorem 1, we deduce that µ(t) is a constant by the continuity of µ(t), as shown in the following lemma. Lemma 9. Let g(t), 0 ≤ t ≤ T , be a Ricci flow on M with uniformly bounded scalar curvature. Assume that the asymptotic volume ratio µ(t) = µ(g(t)) are well defined. Then µ(t) is a continuous function of t. Proof. The volume of a ball changes according to the formula d R(g(t))dvg(t) . V (Br , g(t)) = − dt Br
(3.23)
Hence |
d V (Br , g(t))| ≤ K V (Br , g(t)), dt
where K denotes the uniform bound on the scalar curvature. Therefore, e−K (t−t0 ) V (Br , g(t0 )) ≤ V (Br , g(t)) ≤ e K (t−t0 ) V (Br , g(t0 )), from which the continuity follows.
4. The Changing Rate of Mass For the mass to be well defined, one needs the integrability condition R ∈ L 1 in addition to the requirement that the asymptotic order τ > n−2 2 [2, 17]. We have seen that the ALE property is preserved along the Ricci flow. We now examine the integrability condition. One thing that in particular guarantees the integrability is the decay condition R = O(r −q ), We have
q > n.
(4.24)
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Theorem 10. Let g(t) be the solution of the Ricci flow (1.6) over [0, T ] with uniformly bounded curvature. Assume that g(0) is ALE with asymptotic order τ > 0 and its scalar curvature satisfy the decay condition R(0) = O(r −q ), q > 0. Then the scalar curvature of g(t) satisfies the decay condition
R(t) = O(r −q ), q = min{q, 2(τ + 2)}. Proof. This is similar to our proof of the ALE property. The scalar curvature satisfies the evolution equation ∂ R = R + 2|Rc|2 . ∂t By the assumption and our result on the ALE property, we have |Rc|2 = O(r −2(τ +2) ). Let f be the function in Lemma 6 and w = f q R. Then (∂t − )w ≤ Bw + 2q ∇ log f ∇w + C,
where f q |Rc|2 ≤ C. Hence, by the argument in the proof of ALE property, we have max w ≤ C1 . MT
In particular, when the order of decay of the initial scalar curvature q > n and the asymptotic order τ > n−4 2 , then the order of decay of the evolving scalar curvature also satisfies q > n. In general, without assuming the ALE conditions, we show that under the natural condition T |Rc|2 < +∞, (4.25) 0
the property R ∈
L1
M
is preserved under the Ricci flow if the decay condition R(t) = O(r −σ )
(4.26)
holds uniformly for some σ ≥ n − 2 and all t ∈ [0, T ]. We remark that both conditions (4.25) and (4.26) are always true for 0 < T < +∞ if the initial metric is ALE, provided the order σ = 2+τ ≥ n −2 (i.e. τ ≥ n −4) and 2σ > n in the curvature decay condition (1.3), as it follows from our result on ALE property and Theorem 10. Theorem 11. Let g(t) be the solution of the Ricci flow (1.6) over [0, T ] with uniformly bounded curvature. Assume that the conditions (4.25) and (4.26) hold. Then the property R ∈ L 1 is preserved under the Ricci flow. Proof. Recall that on M, R = Rt − 2|Rc|2 . Let p = 1 + with small > 0. Let φ be a non-negative cut-off function such that 0 ≤ φ ≤ 1 on M, φ = 1 on Br (o), φ = 0 outside B2r (o), and |∇φ|2 ≤ 4φ/r 2 .
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Then
dt φ 2 |R| p−1 (Rt − 2|Rc|2 ) sgnR M 0 t = dt φ 2 |R| p−1 ( R) sgnR M 0 t = −2 dt φ|R| p−1 < ∇φ, ∇ R > sgnR M 0 t − ( p − 1) dt φ 2 |R| p−2 |∇ R|2 M 0 t 2 2( p − 1) t ≤ dt |∇φ|2 |R| p − dt φ 2 |∇(|R| p/2 )|2 p−1 0 p2 M M 0 t 2 2( p − 1) t p ≤ dt φ|R| − dt φ 2 |∇(|R| p/2 )|2 ( p − 1)r 2 0 p2 M M 0 t 2 2C T r n−2− pσ →0 ≤ dt φ|R| p ≤ 2 ( p − 1)r 0 p−1 M t
as r → ∞. Here we have used the decay condition R = O(r −σ ) for some σ > n − 2. By direct computation, we have
t
φ 2 |R| p−1 (Rt − 2|Rc|2 ) sgnR M 0 t t = −2 dt φ 2 |R| p−1 |Rc|2 sgnR + dt φ 2 |R| p−1 Rt sgnR M M 0 0 t 1 t d 2 p−1 2 = −2 dt φ |R| |Rc| sgnR + dt φ 2 |R| p p dt M M 0 0 1 t 2 p + dt φ |R| R p 0 M t 1 1 dt φ 2 |R| p−1 |Rc|2 sgnR + φ 2 |R| p (t) − φ 2 |R| p (0) = −2 p M p M 0 M 1 t dt φ 2 |R| p R. + p 0 M dt
Hence, we have
1 dt φ 2 |R| p−1 |Rc|2 sgnR + φ 2 |R| p (t) p M M 0 1 t 1 φ 2 |R| p (0) + dt φ 2 |R| p R − p M p 0 M 2C T r n−2− pσ ≤ . p−1
−2
t
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77
Sending r → +∞, we get t 1 1 dt |R| p−1 |Rc|2 sgnR + |R| p (t) − |R| p (0) −2 p M p M M 0 1 t dt |R| p R ≤ 0. + p 0 M Sending p → 1, we have that t t dt |Rc|2 sgnR + |R|(t) − |R|(0) + dt |R|R ≤ 0. −2 M
0
M
M
M
0
That is,
|R(t)| − M
|R(0)| ≤ M
T
2
dt
R +2 M
0
which implies that R ∈ L 1 for each t > 0.
T
|Rc|2 ,
dt 0
M
Using a similar argument, we can show that the property |Rm| ∈ L p ( p ≥ 1) is preserved under the Ricci flow with bounded curvature. We now look at the change of mass under the Ricci flow. Let S be a hypersurface in M. Without loss of generality, we can assume that M is oriented. We can take the local frame Fa such that F1 , ..., Fn−1 are tangent to S and Fn = ν is orthogonal to S at X . Let ωa be a local frame dual to Fa . Then the area form on S is d S = ω1 ∧ ... ∧ ωn−1 . Let f ba = ∂/∂t ωa (Fb ). Then we have f ba = −ωa (∂/∂t Fb ), which is a decay term of the same order as R jk = O(r −σ ), and ∂/∂t ωa = f ba ωb . Hence, ∂/∂t d S = O(r −σ ). It is also clear that ∂/∂t g(Fn , Fa ) = O(r −σ ). Hence ∂/∂t d S i = O(r −σ ).
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Therefore,
∂ ∂ gi j, j − g j j,i d S i m (t) = ∂t Sr →∞ ∂t ∂ gi j, j − g j j,i + d Si ∂t Sr →∞
and the second term in the above equation is zero. So, under the flow, we have m (t) = −2 (Ri j, j − R j j,i )d S i . Sr →∞
By using the contracted first Bianchi identity 2Ri j, j = Ri , we have that
m (t) = −2 =
Sr →∞
1 Ri − Ri d S i 2
Ri d S i .
Sr →∞
Note that, by using the local gradient estimate of Shi (see Theorem 13.1 in [8]), we have that Ri = O(|x|−σ ) for σ = 2 + τ . Hence, we have m (t) = 0 when n = 3. The same is true if τ > n − 3 for any dimension n ≥ 3. Similarly, if the initial metric satisfies the additional decay condition 1.8, then one has a better estimate Ri = O(|x|−σ ) for σ = 3 + τ . Therefore, m (t) = 0 provided n−2 τ + 3 > n − 1. This will be the case if n ≤ 6 and τ > n−2 2 , or τ > min{n − 4, 2 }. On the other hand, if R = O(r −q ), q > n, then Theorem 10 applies and we once again have m (t) = 0. Combining the results in Sect. 3, and 4, we have proved Theorem 1. Let us now make an observation. Using the divergence theorem we have 1 m (t) = Rdvg(t) . 4ωn Br →∞ Comparing this with the formula R = Rt − 2|Rc|2 , and d dt we obtain that m (t) =
1 4ωn
Rdvg(t) = Br
Rt dvg(t) − Br
R 2 dvg(t) , Br
(Rt − 2|Rc|2 )dvg(t) Br →∞
1 r →∞ 4ωn
= lim
d dt
Br
This yields the following result.
|Rc|2 dvg(t) . (4.27)
R 2 dvg(t) − 2
Rdvg(t) + Br
Br
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Proposition 12. Under the Ricci flow for ALE metrics with (4.25) we have d 2|Rc|2 − R 2 dvg(t) , Rdvg(t) = dt M M provided that n = 3 or τ > n − 3. 5. Uniform Convergence We now turn our attention to Theorem 2. First, we introduce the notion of uniform conk,q vergence in our context. To this end, we now discuss the weighted Sobolev spaces Wτ and a certain related space Mτ on an ALE space M, see [2, 11]. q For q ≥ 1 and τ ∈ R, the weighted Lebesque space L τ (M) consists of locally integrable functions u on M for which the norm 1/q −τ q −n uq,τ = |r u| r dvol M k,q
is finite. For nonnegative integer k, the weighted Sobolev space Wτ (M) is the set of u q for which |∇ i u| ∈ L τ (M) for 0 ≤ i ≤ k, with the norm uq,k,τ =
k
∇ i uq,τ .
i=0 ∞ metrics g on M such that, in For τ > n−2 2 , we define Mτ to be the set of all C some asymptotic coordinates,
gi j − δi j ∈ Wτ1,q (M),
R(g) ∈ L 1 (M).
(5.28)
We equip Mτ with the norm gMτ = gi j − δi j W 1,q + R(g) L 1 . τ
(5.29)
Now, consider the Ricci flow (M, g(t)) which we assume to exist for all time 0 ≤ t < ∞. Furthermore, suppose that each (M, g(t)) is ALE of asymptotic order τ > n−2 2 and that the scalar curvature R(t) of g(t) is integrable on M (so that the ADM mass is well defined), i.e. g(t) ∈ Mτ . Definition 13. We say that g(t) converges uniformly to g∞ ∈ Mτ as t → ∞ if g(t) converges to g∞ in Mτ . i.e., lim g(t) − g∞ Mτ = 0.
t→∞
We now prove Theorem 2. Proof. This follows from the argument of [11, Lemma 9.4]. The key here is the following identity, first observed in [17, 2]. In terms of the asymptotic coordinate, R(g) = ∂ j (∂i gi j − ∂ j gii ) + O(r −2τ −2 ), where the O(r −2τ −2 ) is controlled by the Wτ -norm of g. 1,q
(5.30)
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Let η be a cut off function which is identically 1 for large r and 0 for r ≤ 1 and inside. Then, by the Divergence Theorem 1 η(∂i gi j − ∂ j gii ) ∗ d x j m(g) = lim R→∞ 4ωn S R −η∇ ∗ β + β, ∇η dvol, = M
where β = (∂i gi j − ∂ j gii )∂ j is the mass density vector. Theorem 2 now follows from the formula above and (5.30). Acknowledgements. Part of the work is done while both authors were visiting the Nankai Institute of Mathematics, Tianjin, China. We would like to thank Nankai Institute and its Director Weiping Zhang for the hospitality. The first author thanks Gary Horowitz and Rick Ye for interesting discussions and Gary for bringing the reference [6] to his attention.
References 1. Arnowitt, S., Deser, S., Misner, C.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122, 997–1006 (1961) 2. Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39, 661–693 (1986) 3. Bando, S., Kasue, A., Nakajima, H.: On a construction of coordiantes at infinity on manifold with fast curvature decay and maximal volume growth. Invent. Math. 97, 313–349 (1989) 4. Chen, B., Zhu, X.: Complete Riemannian manifolds with pointwise pinched curvature. Invent. Math. 140(2), 423–452 (2000) 5. Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991) 6. Gutperle, M., Headrick, M., Minwalla, S., Schomerus, V.: Spacetime Energy Decreases under Worldsheet RG Flow. JHEP 0301, 073 (2003) 7. Greene, R., Petersen, P., Zhu, S.: Riemannian manifolds of faster-than-quadratic curvature decay. Internat. Math. Res. Notices 9, 363–377 (1994) 8. Hamilton, R.: The formation of Singularities in the Ricci flow. Surveys in Diff. Geom. 2, 7–136 (1995) 9. Hamilton, R.: The Harnack estimate for the Ricci flow. J. Diff. Geom. 37, 225–243 (1993) 10. Kapovitch, V.: Curvature bounds via Ricci smoothing. Illinois J. Math. 49(1), 259–263 (2005) 11. Lee, J.M., Parker, T.: The Yamabe problem. Bull. AMS 17, 37–91 (1987) 12. Li, P., Yau, S.T.: On the parabolic kernel of the Schrodinger operators. Acta. Math. 158, 153–201 (1986) 13. Oliynyk, T., Woolgar, E.: Asymptotically Flat Ricci Flows. http://arxiv.org/list/math.DG/0607438, 2006 14. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. http:// arxiv.org/list/math.DG/math.DG/0211159, 2002 15. Shi, W.X.: Ricci deformation of the metric on complete noncompact Riemannian manifolds. J. Diff. Geom. 30, 303–394 (1989) 16. Shi, W.X.: Deforming the metric on complete Riemannian manifolds. J. Diff. Geom. 30, 223–301 (1989) 17. Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Topics in Calculus of Variations, LNM 1365, Berlin-Heidelberg-New York: Springer-Verlag, 1989 18. Schoen, R., Yau, S.T.: Lectures on Differential Geometry. Cambridge, MA: International Press (1994) 19. Schoen, R., Yau, S.T.: On the proof the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979) 20. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981) Communicated by G.W. Gibbons
Commun. Math. Phys. 274, 81–122 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0274-7
Communications in
Mathematical Physics
Translation-Invariance of Two-Dimensional Gibbsian Point Processes Thomas Richthammer Mathematisches Institut der Universität München, Theresienstraße 39, D-80333 München, Germany. E-mail:
[email protected] Received: 7 March 2006 / Accepted: 11 January 2007 Published online: 6 June 2007 – © Springer-Verlag 2007
Abstract The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian particle systems with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. We start with the special case of pure hard core repulsion in order to show how to treat hard cores in general. 1. Introduction Gibbsian processes were introduced by R. L. Dobrushin (see [D1] and [D2]), O. E. Lanford and D. Ruelle (see [LR]) as a model for equilibrium states in statistical physics. (For general results on Gibbs measures on a d-dimensional lattice we refer to the books of H.-O. Georgii [G], B. Simon [Sim] and Y. G. Sinai [Sin], which cover a wide range of phenomena.) The first results concerned existence and uniqueness of Gibbs measures and the structure of the set of Gibbs measures related to a given potential. The question of uniqueness is of special importance, as the non-uniqueness of Gibbs measures can be interpreted as a certain type of phase transition occurring within the particle system. A phase transition occurs whenever a symmetry of the potential is broken, so it is natural to ask, under which conditions symmetries are broken or conserved. The answer to this question depends on the type of the symmetry (discrete or continuous), the number of spatial dimensions and smoothness and decay conditions on the potential (see [G], Chapters 6.2, 8, 9 and 20). It turns out that the case of continuous symmetries in two dimensions is especially interesting. The first progress in this case was achieved by M. D. Mermin and H. Wagner, who showed for special two-dimensional lattice models that continuous internal symmetries are conserved ([MW] and [M]). In [DS] R. L. Dobrushin and S. B. Shlosman established conservation of symmetries for more general potentials which satisfy smoothness and decay conditions, and C.-E. Pfister improved this in [P]. Later also continuum systems were considered: S. Shlosman obtained results
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for continuous internal symmetries ([Sh]), while J. Fröhlich and C.-E. Pfister treated the case of translation of point particles ([FP1] and [FP2]). All these results rely on the smoothness of the interaction, but in [ISV] D. Ioffe, S. Shlosman and Y. Velenik were able to relax this condition. Considering a lattice model they showed that continuous internal symmetries are conserved, whenever the interaction can be decomposed into a smooth part and a part which is small with respect to L 1 -norm, using a perturbation expansion and percolation theory. We generalised this to a point particle setting ([Ri1]). Here we will investigate the conservation of translational symmetry for non-smooth, singular or hard-core potentials in a point particle setting. While we treat non-smoothness by generalising ideas used in [Ri1], we will give an approach to singular potentials which is different from the one given in [FP1] and [FP2]. The advantage of our approach is that the integrability condition (2.13) of [FP2] is simplified and relaxed and the case of hard-core potentials can easily be included. Thus we are able to show the conservation of translational symmetry for the pure hard core model, for example. In Sect. 2 we will first confine ourselves to this special case of pure hard core repulsion. The corresponding result (Theorem 1) is of interest on its own and its proof shows how to deal with hard cores in the general case. For this general case we then define a suitable class of potentials (Definition 1), give some sufficient conditions for potentials to belong to that class (Lemmas 1 and 2) and state the general result obtained (Theorem 2). The precise setting is then given in Sect. 3. The proofs of the lemmas from Sects. 2 and 3 are relegated to Sect. 4. In Sects. 5 and 7 we will give the proofs of Theorems 1 and 2 respectively. The proofs of the corresponding lemmas are relegated to Sects. 6 and 8 respectively. In the proof of the general case arguments of the special case have to be modified and refined by new concepts and ideas at several instances. So for sake of clarity we will repeat arguments from the proof of Theorem 1 in the proof of Theorem 2 whenever necessary.
2. Result We consider particles in the plane R2 without internal degrees of freedom. The chemical potential − log z of the system is given via an activity parameter z > 0. The interaction between particles is modelled by a translation-invariant pair potential U , i.e. a measurable function U : R2 → R := R ∪ {∞}, which is assumed to be symmetric in that U (x) = U (−x) for all x ∈ R2 . The potential of two particles x1 , x2 ∈ R2 is then given by U (x1 − x2 ). We first consider the particular case of pure hard core repulsion, where the size and the shape of the hard core are given by a norm |.|h on R2 . The corresponding pure hard-core potential Uhc is defined by Uhc (x) :=
∞ 0
for for
|x|h ≤ 1 |x|h > 1.
Theorem 1. Let z > 0 be an activity parameter, |.|h be a norm on R2 and Uhc be the corresponding pure hard-core potential. Then every Gibbs measure corresponding to Uhc and z is translation-invariant.
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83
The proof of Theorem 1, which is given in Sect. 5, will show how to deal with hard cores in the general case presented below. In order to describe a class of potentials for which translational symmetry is conserved we will define important properties of sets, functions and potentials. A set A ⊂ R2 is called symmetric if A = −A. We call U a standard potential if U is a measurable, symmetric pair potential and its hard core K U := {U = +∞} is bounded. Usually the hard core will be empty, {0} or a disc, but in our setup we are able to treat fairly general hard cores. For a given function ψ : R2 → R+ we say that a standard potential U has ψ-dominated derivatives on the set A if ∂i2 U (x + tei ) ≤ ψ(x)
for all x ∈ A, t ∈ [−1, 1] s.t. x + tei ∈ A
for i = 1, 2. Here e1 = (1, 0), e2 = (0, 1) and ∂i is the partial derivative in direction ei . The above definition is meant to imply that these derivatives exist. In the context of ψ-domination we will use the notion of a decay function, which is defined to satisfy ψ < ∞
and
ψ(x)|x|2 d x < ∞.
This definition of course does not depend on the choice of norm |.|, but for sake of definiteness let |.| be the maximum norm on R2 . If U is a potential, z is an activity parameter and X0 is a set of boundary conditions, we say that the triple (U, z, X0 ) is admissible if all conditional Gibbs distributions corresponding to U and z with boundary condition taken from X0 are well defined, see Definition 2 in Sect. 3.3. Important examples are the cases of superstable potentials with tempered boundary configurations and nonnegative potentials with arbitrary boundary conditions, see Sect. 3.4. For admissible (U, z, X0 ) the set of Gibbs measures GX0 (U, z) corresponding to U and z with full weight on configurations in X0 is a well defined object. Finally we need bounded correlations: For admissible (U, z, X0 ) we call ξ ∈ R a Ruelle bound if the correlation function of every Gibbs measure µ ∈ GX0 (U, z) is bounded by powers of ξ in the sense of (3.3) in Sect. 3.3. Definition 1. Let (U, z, X0 ) be an admissible triple with Ruelle bound ξ , where U : R2 → R is a translation-invariant standard potential. We say that U is smoothly approximable if there is a decomposition of U into a smooth part U¯ and a small part u in the following sense: We have a symmetric, compact set K ⊃ K U , a decay function ψ and measurable symmetric functions U¯ , u : K c → R such that U = U¯ − u and u ≥ 0 on K c , U¯ has ψ-dominated derivatives on K c , 2 2 U u(x)|x| ˜ d x < ∞ and λ (K \ K ) + Kc
where u˜ := 1 − e−u ≤ u ∧ 1.
Kc
u(x) ˜ dx <
1 , zξ
(2.1)
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The class of smoothly approximable standard potentials is a rich class of potentials. A smoothly approximable standard potential U may have a singularity or a hard core at the origin, and the type of convergence into the singularity or the hard core is fairly arbitrary, as we have not imposed any condition on U in the set K \ K U . For small activity z the last condition of (2.1) holds for large sets K , which relaxes the conditions on U . The small part u of U is not assumed to satisfy any regularity conditions, so that U doesn’t have to be smooth or continuous. We note that Definition 1 does not depend on the choice of the norm |.|. If we know a potential to be smooth outside of its hard core the above conditions simplify: Lemma 1. Let (U, z, X0 ) be an admissible triple with Ruelle bound ξ , where U : R2 → R is a translation-invariant standard potential. Suppose we have a symmetric compact set K ⊃ K U and a decay function ψ such that U has ψ-dominated derivatives on K c and λ2 (K \ K U ) < 1/(zξ ). Then U is smoothly approximable. This is an immediate consequence of Definition 1. In the non-smooth case, the following lemma gives important examples of smoothly approximable potentials: Lemma 2. Let (U, z, X0 ) be an admissible triple with Ruelle bound ξ , where U : R2 → R is a translation-invariant standard potential such that K U is compact and U is continuous in (K U )c . Suppose we have a decay function ψ and a compact set K˜ ⊂ R2 such that one of the following properties holds: (a) U has ψ-dominated derivatives in K˜ c . (b) There is a standard potential U˜ ≥ 0 such that |U | ≤ U˜ in K˜ c , U˜ has ψ-dominated c ˜ ˜ derivatives in K and K˜ c U (x)|x|2 d x < ∞. Then U is smoothly approximable. For example, (a) holds trivially when U has finite range, and (b) includes the case that there are > 0 and k ≥ 0 such that |U (x)| ≤ k/|x|4+ for large |x|. Our main result is now the following: Theorem 2. Let (U, z, X0 ) be admissible with Ruelle bound, where U : R2 → R is a translation-invariant standard potential. If U is smoothly approximable then every Gibbs measure µ ∈ GX0 (U, z) is translation-invariant. For a generalisation of the above result to the case of particles with inner degrees of freedom, i.e. Gibbsian systems of marked particles, we refer to [Ri2]. 3. Setting 3.1. State space. We will use the notations N := {0, 1, . . .}, R+ := [0, ∞[, R := R ∪ {+∞}, r1 ∨ r2 := max{r1 , r2 } and r1 ∧ r2 := min{r1 , r2 }
for r1 , r2 ∈ R.
On R2 we consider the maximum norm |.| and the Euclidean norm |.|2 . For > 0 the -enlargement of a set A ⊂ R2 is defined by A := {x + x : x ∈ A, |x |2 < }.
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85
The state space of a particle is the plane R2 . The Borel-σ -algebra B2 on R2 is induced by any norm on R2 . Let B2b be the set of all bounded Borel sets and λ2 be the Lebesgue measure on (R2 , B2 ). Integration with respect to this measure will be abbreviated by d x := dλ2 (x). Often we consider the centred squares r := [−r, r [2 ⊂ R2
(r ∈ R+ ).
We also want to consider bonds between particles. For a set X we denote the set of all bonds in X by E(X ) := {A ⊂ X : # A = 2}. A bond will be denoted by x x := {x, x }, where x, x ∈ X such that x = x . For a bond set B ⊂ E(X ) (X, B) is an (undirected) graph, and we set X,B
x ←→ x
:⇔
∃ m ∈ N, x0 , . . . , xm ∈ X : x = x0 , x = xm , xi−1 xi ∈ B for all 1 ≤ i ≤ m.
This connectedness relation is an equivalence relation on X whose equivalence classes are called the B-clusters of X . Let X,B C X,B (x) := {x ∈ X : x ←→ x } and C X,B () := C X,B (x ) x ∈X ∩
denote the B-clusters of a point x and a set respectively. Primarily we are interested in the case X = R2 . On the corresponding bond set E(R2 ) we consider the σ -algebra F E(R2 ) := {{x1 x2 ∈ E(R2 ) : (x1 , x2 ) ∈ M} : M ∈ (B2 )2 }. Every symmetric function u on R2 can be considered a function on E(R2 ) via u(x x ) := u(x − x ). 3.2. Configuration space. A set of particles X ⊂ R2 is called finite locally finite
if # X < ∞, and if #(X ∩ ) < ∞ for all ∈ B2b ,
where # denotes the cardinality of a set. The configuration space X of particles is defined as the set of all locally finite subsets of R2 , and its elements are called configurations of particles. For X, X¯ ∈ X let X X¯ := X ∪ X¯ . For X ∈ X and ∈ B2 let X := X ∩ (restriction of X to ), X := {X ∈ X : X ⊂ } (set of all configurations in ) and N (X ) := # X (number of particles of X in ). The counting variables (N )∈B2 generate a σ -algebra on X, which will be denoted by FX. For ∈ B2 let FX , be the σ -algebra on X obtained by restricting FX to X , −1 FX, be the σ -algebra on X obtained from FX and let FX, := e , by the restriction
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mapping e : X → X , X → X . The tail σ -algebra or σ -algebra of the events far from the origin is defined by FX,cn . FX,∞ := n≥1
Let ν be the distribution of the Poisson point process on (X, FX), i.e. 1 −λ2 () d x1 . . . d xk f ({xi : 1 ≤ i ≤ k}), ν(d X ) f (X ) = e k! k k≥0
for any FX, -measurable function f : X → R+ , where ∈ B2b . For ∈ B2b and X¯ ∈ X, let ν (.| X¯ ) be the distribution of the Poisson point process in with boundary condition X¯ , i.e. ¯ ν(d X ) f (X X¯ c ) ν (d X | X ) f (X ) = for any FX-measurable function f : X → R+ . It is easy to see that ν is a stochastic kernel from (X, FX,c ) to (X, FX). The configuration space of bonds E is defined to be the set of all locally finite bond sets, i.e. E := {B ⊂ E(R2 ) : #{x x ∈ B : x x ⊂ } < ∞ for all ∈ B2b }. On E the σ -algebra FE is defined to be generated by the counting variables N E : E → N, B → #(E ∩ B) (E ∈ F E(R2 ) ). For a countable set E ∈ E one can also consider the Bernoulli-σ -algebra B E on E E := P(E) ⊂ E, which is defined to be generated by the family of sets ({B ⊂ E : e ∈ B})e∈E . Given a family ( pe )e∈E of reals in [0, 1] the Bernoulli measure on (E E , B E ) is defined as the unique probability measure for which the events ({B ⊂ E : e ∈ B})e∈E are independent with probabilities ( pe )e∈E . It is easy to check that the inclusion (E E , B E ) → (E, FE) is measurable. Thus any probability measure on (E E , B E ) can trivially be extended to (E, FE). 3.3. Gibbs measures. Let U : R2 → R be a potential and z > 0 an activity parameter. For finite configurations X, X ∈ X we consider the energy terms U (x1 − x2 ) and W U (X, X ) := U (x1 − x2 ). H U (X ) := x1 x2 ∈E(X )
x1 ∈X x2 ∈X
) The last definition can be extended to infinite configurations X whenever W U (X, X 2 2 converges as ↑ R through the net Bb . The Hamiltonian of a configuration X ∈ X in ∈ B2b is given by HU (X ) := H U (X ) + W U (X , X c ) = U (x1 − x2 ), x1 x2 ∈E (X )
where E (X ) := {x1 x2 ∈ E(X ) : x1 x2 ∩ = ∅}.
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
The integral U,z ¯ ( X ) := Z
87
U ν (d X | X¯ ) e−H (X ) z # X
is called the partition function in ∈ B2b for the boundary condition X¯ c ∈ X. In order to ensure that the above objects are well defined and the partition function is finite and positive we need the following definition: Definition 2. A triple (U, z, X0 ) consisting of a potential U : R2 → R, an activity parameter z > 0 and a set of boundary conditions X0 ∈ FX,∞ is called admissible if for all X¯ ∈ X0 and ∈ B2b the following holds: W U ( X¯ , X¯ c ) has a well defined value U,z ¯ in R and Z ( X ) is finite. If (U, z, X0 ) is admissible, ∈ B2b and X¯ ∈ X0 then W U (X , X¯ c ) ∈ R is well defined for every X ∈ X, because X0 ∈ FX,∞ implies X X¯ c ∈ X0 . As a consequence U,z ¯ ( X ) is well defined. Furthermore by definition it is finite and the partition function Z by considering the empty configuration one can show that it is positive. The conditional Gibbs distribution γU,z (.| X¯ ) in ∈ B2b with boundary condition X¯ ∈ X0 is thus well defined by U 1 U,z ¯ γ (A| X ) := U,z ν (d X | X¯ ) e−H (X ) z # X 1 A (X ) for A ∈ FX. ¯ Z (X ) γU,z is a probability kernel from (X0 , FX0 ,c ) to (X, FX). Let GX0 (U, z) := {µ ∈P1 (X, FX) : µ(X0 ) = 1 µ(A|FX,c ) =
γU,z (A|.)
and µ-a.s. ∀ A ∈ FX, ∈ B2b }
be the set of all Gibbs measures corresponding to U and z with whole weight on boundary conditions in X0 . It is easy to see that for any probability measure µ ∈ P1 (X, FX) such that µ(X0 ) = 1 we have the equivalence µ ∈ GX0 (U, z)
⇔
(µ ⊗ γU,z = µ ∀ ∈ B2b ).
So for every µ ∈ GX0 (U, z), f : X → R+ measurable and ∈ B2b we have µ(d X ) f (X ) = µ(d X¯ ) γU,z (d X | X¯ ) f (X ).
(3.1)
If we consider a fixed potential and a fixed activity we will omit the dependence on U U,z . As a consequence of (3.1) the hard core K U of a and z in the notations γU,z and Z potential U implies that particles are not allowed to get too close to each other, i.e. for admissible (U, z, X0 ) and µ ∈ GX0 (U, z) we have µ({X ∈ X : ∃x, x ∈ X : x = x , x − x ∈ K U }) = 0.
(3.2)
For admissible (U, z, X0 ) and a Gibbs measure µ ∈ GX0 (U, z) we define the correlation function ρ U,µ by U U ¯ ρ U,µ (X ) = e−H (X ) µ(d X¯ ) e−W (X, X )
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for any finite configuration X ∈ X. If there is a ξ = ξ(U, z, X0 ) ≥ 0 such that ρ U,µ (X ) ≤ ξ # X
for all finite X ∈ X and all µ ∈ GX0 (U, z),
(3.3)
then we call ξ a Ruelle bound for (U, z, X0 ). Actually we need this bound on the correlation function in the following way: Lemma 3. Let (U, z, X0 ) be admissible with Ruelle bound ξ . For every Gibbs measure µ ∈ GX0 (U, z) and every measurable f : (R2 )m → R+ , m ∈ N we have = µ(d X ) f (x1 , . . . , xm ) ≤ (zξ )m d x1 . . . d xm f (x1 , . . . , xm ). (3.4) x1 ,...,xm ∈X
We use = as a shorthand notation for a multiple sum such that the summation indices are assumed to be pairwise distinct. 3.4. Superstability and admissibility. Now we will discuss some conditions on potentials which imply that (U, z, X0 ) is admissible and has a Ruelle bound whenever the set of boundary conditions X0 is suitably chosen. Apart from purely repulsive potentials such as the pure hard-core potential considered in Theorem 1 we also want to consider superstable potentials in the sense of Ruelle, see [R]. Therefore let r := r + [−1/2, 1/2[2 ⊂ R2
(r ∈ Z2 )
be the unit square centred at r and let Z2 (X ) := {r ∈ Z2 : N r (X ) > 0} be the minimal set of lattice points such that the corresponding squares cover the configuration. A potential U : R2 → R is called superstable if there are real constants a > 0 and b ≥ 0 such that for all finite configurations X ∈ X, H U (X ) ≥ [a N r (X )2 − bN r (X )]. r ∈Z2 (X )
U is called lower regular if there is a decreasing function : N → R+ with ∞ such that W U (X, X ) ≥ −
r ∈Z2 (X ) s∈Z2 (X )
r ∈Z2
(|r |) <
1 1
(|r − s|) [ N r (X )2 + N s (X )2 ] 2 2
for all finite configurations X, X ∈ X. So superstability and lower regularity give lower bounds on energies in terms of particle densities. In order to be able to control these densities, a configuration X ∈ X is defined to be tempered if s¯ (X ) := sup sn (X ) < ∞, where sn (X ) := n∈N
1 (2n + 1)2
N 2r (X ).
r ∈Z2 ∩n+1/2
By Xt we denote the set of all tempered configurations. We note that Xt ∈ FX,∞ .
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Lemma 4. Let z > 0 and U : R2 → R be a translation-invariant pair potential. (a) If U is purely repulsive, i.e. U ≥ 0, then (U, z, X) is admissible with Ruelle bound ξ := 1. (b) If U is superstable and lower regular then (U, z, Xt ) is admissible and admits a Ruelle bound. The first assertion is a straightforward consequence of the fact that all energy terms are nonnegative. For the second assertion see [R]. 3.5. Conservation of translational symmetry. Every τ ∈ R2 gives a translation on the configuration space X via gτ (X ) := X + τ := {x + τ : x ∈ X }. We say that a measure µ on (X, FX) is τ-invariant if µ ◦ gτ−1 = µ, and µ is translationinvariant if it is τ-invariant for every τ ∈ R2 . The following lemma gives a sufficient condition for the conservation of τ-symmetry. Lemma 5. Let (U, z, X0 ) be admissible, where U : R2 → R is a translation-invariant potential. If for all cylinder events D ∈ FX,m (m ∈ N) and all Gibbs measures µ ∈ GX0 (U, z) we have µ(D + τ) + µ(D − τ) ≥ µ(D),
(3.5)
then every Gibbs measure µ ∈ GX0 (U, z) is τ-invariant. We further note that R2 is generated by the set {τi ei : 0 ≤ τi < 1/2, i ∈ {1, 2}}, so we only have to consider translations of this special form in order to establish translationinvariance of a set of Gibbs measures. 3.6. Concerning measurability. We will consider various types of random objects, all of which have to be shown to be measurable with respect to the considered σ -algebras. However we will not prove measurability of every such object in detail. Instead we will now give a list of operations that preserve measurability. Lemma 6. Let X, X ∈ X, B, B ∈ E, x ∈ R2 and p ∈ be variables, where (, F) is a measurable space. Let f : × R2 → R and g : × E(R2 ) → R be measurable. Then the following functions of the given variables are measurable with respect to the considered σ -algebras: f ( p, x ), X ∩ X , X ∪ X , X \ X , X + x, (3.6) x ∈X
g( p, b ), B ∩ B , B ∪ B , B \ B , B + x,
(3.7)
b ∈B
inf f ( p, x ), {x ∈ X : f ( p, x ) = 0}, C X,B (x), E(X ),
x ∈X
the number of different clusters of (X, B).
(3.8) (3.9)
Using this lemma and well known theorems, such as the measurability part of Fubini’s theorem, we can check the measurability of all objects considered.
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4. Proof of the Lemmas from Sections 2 and 3 4.1. Smoothly approximable potentials: Lemma 2. Let (U, z, X0 ), ξ , ψ, K˜ and U˜ (in case (b)) be as in the formulation of Lemma 2. By compactness of K U we can choose an > 0 such that the -enlargement K := (K U ) of the hard core K U has the property c := 1/(zξ ) − λ2 (K \ K U ) > 0. In case (a) let U1 := U and in case (b) let U1 := U˜ . Let R ≥ 1 such that
K ∪ K˜ ⊂ R
and furthermore
cR
2U˜ (x)|x|2 d x <
c 2
in case (b).
In both cases U1 serves as an approximation of U on cR . Let C := R+1 \ K , δ > 0 and f δ : R → R+ be a symmetric smooth probability density with support in the |.|2 -disc B2 (δ), e.g. f δ (x) :=
1 2 2 −1 1 B2 (δ) (x)e−(1−|x|2 /δ ) , cδ
where cδ :=
e−(1−|x|2 /δ 2
B2 (δ)
2 )−1
d x.
Then
d x f δ (x )U (x − x )
U2 (x) := U ∗ f δ (x) :=
is a smooth approximation of U on C. By continuity of U and compactness of C a sufficiently small δ guarantees |U2 (x) − U (x)| < c :=
c 4λ2 (C)
for x ∈ C.
Let g : R2 → [0, 1] be a smooth symmetric function such that g = 0 on R and g = 1 on cR+1 . Now we can define U¯ , u : K c → R by U¯ := (1 − g)(U2 + c ) + gU1
and
u := U¯ − U.
It is easy to verify that the constructed objects have all the properties described in Definition 1 in both cases (a) and (b). 4.2. Property of the Ruelle bound: Lemma 3. For every n ∈ N, every measurable g : Xn → R+ and every X¯ ∈ X0 we have
νn (d X | X¯ )
f (x1 , . . . , xm ) g(X )
x1 ,...,xm ∈X n
=
=
n m
d x1 . . . d xm f (x1 , . . . , xm )
νn (d X | X¯ ) g({x1 , . . . , xm }X ).
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Combining this with (3.1), the definition of the conditional Gibbs distribution and the definition of the correlation function we get = µ(d X ) f (x1 , . . . , xm ) =
x1 ,...,xm ∈X n
µ(d X¯ )
=
n
m
1 U,z ¯ Z (X ) n
νn (d X | X¯ )
=
f (x1 , . . . , xm ) e−Hn (X ) z # X n U
x1 ,...,xm ∈X n
d x1 . . . d xm f (x1 , . . . , xm ) z m ρ U,µ ({x1 , . . . , xm }).
Now we use (3.3) to estimate the correlation function by the Ruelle bound ξ . Letting n → ∞ the assertion follows from the monotone limit theorem. 4.3. Sufficient condition: Lemma 5. The lemma can be shown exactly as Proposition (9.1) in [G] and we will only outline the proof: We first note that (X, FX) is a standard Borel space, which follows from [DV], Theorem A2.6.III. Hence the point particle version of Theorem (7.26) in [G] implies that every Gibbs measure can be decomposed into extremal Gibbs measures. Thus without loss of generality we may assume µ to be extremal. Suppose now that µ is not τ-invariant, i.e. µ ◦ gτ−1 = µ, which also implies µ ◦ gτ = µ. As the extremality of µ implies the extremality of µ ◦ gτ−1 and µ ◦ gτ , the point particle version of Theorem (7.7) guarantees the existence of sets A− , A+ ∈ FX,∞ such that µ ◦ gτ−1 (A− ) = 0, µ ◦ gτ (A+ ) = 0 and µ(A− ) = µ(A+ ) = 1. Hence for A := A− ∩ A+ we have µ ◦ gτ (A) + µ ◦ gτ−1 (A) = 0 < 1 = µ(A). On the other hand by assumption (3.5) we know that µ ◦ gτ + µ ◦ gτ−1 ≥ µ on the algebra of all cylinder events. By the monotone class theorem this inequality even holds on all of FX, which contradicts the above inequality. 4.4. Measurability: Lemma 6. Details concerning measurability of functions of point processes can be found in [DV, K or MKM], for example. The first part of (3.6) is the measurability part of Campbell’s theorem. For the rest of (3.6) it suffices to observe that for ∈ B2b we have 1{x=x ∈} , N (X \ X ) = N (X ) − N (X ∩ X ), N (X ∩ X ) = x∈X x ∈X
N (X + x) =
1 (x + x) and N (X ∪ X ) = N (X ) + N (X \ X ).
x ∈X
For (3.7) we can argue similarly. For c ∈ R, ∈ B2b , x ∈ R2 and L ∈ F E(R2 ) , inf f ( p, x ) < c ⇔ 1{ f ( p,x )
x ∈X
N ({x ∈ X : f ( p, x ) = 0}) =
x ∈X
1{ f ( p,x )=0,x ∈} ,
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T. Richthammer
N (C X,B (x)) =
1{x ∈C X,B (x),x ∈} ,
x ∈X
x ∈ C X,B (x) ⇔
1{x=x0 ,x =xm }
m≥0 x0 ,...,xm ∈X
N L (E(X )) =
1 2
m
1{xi xi+1 ∈B} ≥ 1
and
i=1
1 L (x1 x2 ).
x1 ∈X x2 ∈X \{x1 }
Using these relations, the measurability of the terms in (3.8) follows easily. For (3.9) it suffices to observe that there are at most k different clusters of (X, B) iff
1{X \(C X,B (x1 )∪...∪C X,B (xk ))=∅} ≥ 1.
x1 ,...,xk ∈X
5. Proof of Theorem 1: Main Steps 5.1. Basic constants. Let z > 0. Let |.|h be a norm on R2 and U := Uhc the corresponding pure hard-core potential. As U is purely repulsive we know that (U, z, X) is admissible with Ruelle bound ξ := 1 by Lemma 4, part (a). Let K := K U and > 0. If we choose sufficiently small we have cξ := λ2 (K \ K U ) <
1 , zξ
(5.1)
where K is the -enlargement of K . Let f K : R2 → R be a function such that f K is smooth,
f K = 0 on K
and
f K = 1 on (K )c .
Furthermore we need the following finite constants: c K := sup{|x| : x ∈ K }
and
c f := sup{| f K (x)| : x ∈ R2 }.
(5.2)
On R2 let ≤ be the lexicographic order and let the partial order ≤e1 be defined by (r1 , r2 ) ≤e1 (r1 , r2 )
:⇔
r1 ≤ r1 , r2 = r2 .
In order to show the conservation of translational symmetry we fix a Gibbs measure µ ∈ GX0 (U, z) and a cylinder event D ∈ FX,n −1 , where n ∈ N, see Subsect. 3.5. As mentioned there it suffices to consider translations τ e, where τ ∈ [0, 1/2] and e = e1 or e2 . Hence we fix τ ∈ [0, 1/2], and by symmetry we may assume that e = e1 . We also fix an arbitrarily small real δ > 0 in order to control probabilities close to 0. As all the above objects are fixed for the whole proof we will ignore dependence on them in our notations.
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5.2. Generalised translation. Let n > n and X ∈ X. We consider the bond set K n := {x1 x2 ∈ E(X ) : x1 x2 ∩ n = ∅, x1 − x2 ∈ K }. Every time we use this notation it will be clear from the context which configuration X it refers to. Note that K n is finite as X is locally finite and K is bounded. For a bounded set ∈ B2b let rn,X () = sup{|y | : y ∈ C X,K n ()} denote the range of the corresponding K n -cluster. In the following lemma we consider the case = n , where n ∈ N is the number fixed in Sect.5.1. Lemma 7. We have sup µ(d X ) rn,X (n ) < ∞. n>n
By the Chebyshev inequality we therefore can choose an integer R > n , such that for every n > n we have δ for G n := {X ∈ X : rn,X (n ) < R} ∈ FX. µ(G n ) ≥ 1 − 2 For n > R we define the functions q : R+ → R,
Q : R+ → R, r : R × R+ → R
1 , 1 ∨ (s log(s)) k q(s ) r (s, k) := ds , (s∨0)∧k Q(k) q(s) :=
τn : R → R
and
Q(k) :=
by
k
q(s)ds, 0
τn (s) := τ r (s − R, n − R).
For a sketch of the graph of τn see Fig. 1. Some important properties of τn are the following: τn (s) = τ for s ≤ R, τn (s) = 0 for s ≥ n and τn is decreasing.
(5.3)
For X ∈ X and x ∈ X we define an,X (x) to be the point of C X,K n (x) with maximal |.|-distance to the origin. (If there is more than one such point we choose the maximal one with respect to the lexicographic order for the sake of definiteness.) Then (5.3) implies |an,X (x)| ≥ |x|
and
τn (|an,X (x)|) = min{τn (|x |) : x ∈ C X,K n (x)}.
The transformation Tn0 : R2 → R2 , Tn0 (x) := x + τn (|x|)e1 can also be viewed as a transformation on X, such that every point x of a configuration X is translated the distance τn (|x|) in direction e1 . We would like to use this generalised translation Tn0 as a tool for our proof just as in [FP1] and [FP2].
Fig. 1. Graph of τn
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T. Richthammer
5.3. Good configurations. In order to deal with the hard core we will replace the above translation Tn0 by a transformation Tn : X → X which is required to have the following properties: (1) For X ∈ X the transformed configuration X˜ = Tn (X ) is constructed by translating every x ∈ X a certain distance tn,X (x) in direction e1 . We note that we do not require the particles to be translated independently. (2) Particles in the inner region n −1 are translated by τ e1 , and particles in the outer region n c are not translated at all. (3) Tn is bijective, the density of the transformed process with respect to the untransformed process under the measure ν can be calculated explicitly and we have a suitable estimate on this density. (4) The Hamiltonian HUn (X ) is invariant under Tn , i.e. particles within hard core distance remain within hard core distance and particles at larger distance remain at larger distance. Property (2) implies that the translation of the chosen cylinder event D is the same as the transformation of D by Tn . Properties (3) and (4) imply that the density of the transformed process with respect to the untransformed process under the measure µ can be estimated. Therefore a transformation with these properties seems to be a good tool for proving (3.5). However, in general it is difficult to construct a transformation with all the given properties. For example properties (2) and (4) cannot both be satisfied if X is a configuration of densely packed hard-core particles. If n > R and X ∈ G n then such a situation can not occur, and by Lemma 7 this is the case with high probability. Similar problems arise for the other properties, so we will content ourselves with a transformation satisfying the above properties only for configurations X from a set of good configurations 3
G n := X ∈ G n : i (n, X ) < 1 ∈ FX. (5.4) i=1
The functions i (n, X ) will be defined whenever we want good configurations to have a certain property. In Lemma 13 we then will prove that the set of good configurations G n has probability close to 1 when n is big enough. Up to that point we consider a fixed n ≥ R + 1. 5.4. Modifying the generalised translation. With a view to properties (1) and the second part of (2) we define the transformation Tn : X → X by k k + τn,X e1 : 1 ≤ k ≤ m(X )} = {x + tn,X (x)e1 : x ∈ X } Tn (X ) := X cn ∪ {Pn,X k k for every X ∈ X, where m(X ) := # X n , {Pn,X : 1 ≤ k ≤ m(X )} = X n , τn,X is k the translation distance of Pn,X and the translation distance function tn,X : X → R is k ) := τ k defined by tn,X (x) := 0 for x ∈ X cn and tn,X (Pn,X n,X for 1 ≤ k ≤ m(X ). We k k . In order to are left to identify the points Pn,X of X and their translation distances τn,X k k if it simplify notation we will omit the dependence on X in m(X ), tn,X , Pn,X and τn,X
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Fig. 2. Every point Pnk is translated by τnk e1
Fig. 3. Construction of tnk
is clear which configuration is considered. In our construction we would like to ensure that the points Pnk are ordered in a way such that 0 =: τn0 ≤ τn1 ≤ . . . ≤ τnm .
(5.5)
This relation will be an important tool for showing the bijectivity of the transformation as required in property (3) of the last subsection. As required in (4) we also would like to have x1 , x2 ∈ X, x1 − x2 ∈ K ⇒ tn,X (x1 ) = tn,X (x2 ), x1 , x2 ∈ X, x1 − x2 ∈ / K ⇒ (x1 + tn,X (x1 )e1 ) − (x2 + tn,X (x2 )e1 ) ∈ / K.
(5.6) (5.7)
With these properties in mind we will now give a recursive definition of Pnk and τnk for k a fixed configuration X ∈ X using a translation distance function tnk := tn,X : R2 → R in each step. In the kth construction step (1 ≤ k ≤ m) let tnk := tn0 ∧ m Pni ,τni = tnk−1 ∧ m Pnk−1 ,τnk−1 , 0≤i
where
tn0
:= τn (|.|)
and
m Pn0 ,τn0 :=
m x,0 .
x∈X c
n
The auxiliary functions m x ,t will be defined later. Let Pnk be the point of X n \ {Pn1 , . . . , Pnk−1 } at which the minimum of tnk is attained. If there is more than one such point then take the smallest point with respect to the lexicographic order for the sake of definiteness. Let τnk := tnk (Pnk ) be the corresponding k := T k := id + t k e . minimal value of tnk and Tn,X n n 1 k 0 tn is defined to be tn modified by local distortions m x ,t . On the one hand we have thus ensured that tnk − tn0 is small, i.e. τnk ≈ τn (|Pnk |), which will give us hold on the density
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T. Richthammer
Fig. 4. One-dimensional sketch of the graph of m x ,t
in property (3). On the other hand the auxiliary functions of the form m x ,t slow down the translation locally near every point x with known translation distance t, see Fig. 3. This will ensure properties (5.6) and (5.7). ¯ be given by For x ∈ R2 and t ∈ R let the auxiliary function m x ,t : R2 → R t if h x ,t c f > 21 m x ,t (x) := t + h x ,t f K (x − x ) + ∞ 1{ f K (x−x )=1} else, where
h x ,t := |τn (|x | − c K ) − t|.
Note that the first case in the definition of m x ,t has been introduced in order to bound the slope of m x ,t . In Sect. 6.2 we will show important properties of this auxiliary function, but for the moment we will content ourselves with the intuition given by Fig. 4. Using Lemma 6 one can show that all the above objects are measurable with respect to the considered σ -algebras. In the rest of this section we will convince ourselves that the above construction has indeed all the required properties. Lemma 8. The construction satisfies (5.5), (5.6) and (5.7). Lemma 9. For good configurations X ∈ G n we have (Tn X − τ e1 )n −1 = X n −1
and (Tn X )n c = X n c .
(5.8)
Lemma 10. The transformation Tn : X → X is bijective. Actually in the proof of Lemma 10 we construct the inverse of Tn . This is needed in the proof of Lemma 11, where we will show for every X¯ ∈ X that νn (.| X¯ ) is absolutely −1 continuous with respect to νn (.| X¯ ) ◦ T−1 n with density ϕn ◦ Tn , where ϕn (X ) :=
m(X )
1 + ∂1 t k (P k ) . n,X n,X
(5.9)
k=1
The proof will also show that definition (5.9) makes sense νn ( . | X¯ )-a.s., in that the considered derivatives exist. Lemma 11. For every X¯ ∈ X and every FX-measurable function f ≥ 0, dνn (.| X¯ ) f. dνn (.| X¯ ) ( f ◦ Tn · ϕn ) =
(5.10)
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¯ n and ϕ¯n be defined Considering (3.5) we also need the backwards translation. So let T analogously to the above objects, where now e1 is replaced by −e1 . The previous lemmas ¯ n is not the apply analogously to this deformed backwards translation. We note that T inverse of Tn . 5.5. Final steps of the proof. From (3.1) and Lemma 11 we deduce µ(Tn (D ∩ G n )) U 1 ¯ = µ(d X ) νn (d X | X¯ ) 1Tn (D∩G n ) (X ) z # X n e−Hn (X ) ¯ Z n ( X ) 1 ¯ = µ(d X ) νn (d X | X¯ ) Z n ( X¯ ) 1Tn (D∩G n ) ◦ Tn (X ) z #(Tn X )n e−Hn (Tn X ) ϕn (X ). U
By Lemma 10 Tn is bijective, by (5.8) #(Tn X )n = # X n and by (5.6) and (5.7) we have HUn (Tn X ) = HUn (X ). Hence the above integrand simplifies to 1 D∩G n (X ) z # X n e−Hn (X ) ϕn (X ), U
¯ n . So and we have an analogous expression for the backwards transformation T ¯ n (D ∩ G n )) + µ(Tn (D ∩ G n )) − µ(D ∩ G n ) µ(T U 1 ¯ = µ(d X ) νn (d X | X¯ ) 1 D∩G n (X ) z # X n e−Hn (X ) ¯ Z n ( X )
× ϕ¯n (X ) + ϕn (X ) − 1]. We note that for X ∈ G n we have 1
1
ϕ¯n (X ) + ϕn (X ) ≥ 2 (ϕ¯ n (X )ϕn (X )) 2 ≥ 2 e− 2 ≥ 1, where we have used the arithmetic-geometric-mean inequality in the first step and the following estimate in the second step: Lemma 12. For X ∈ G n we have log ϕ¯n (X ) + log ϕn (X ) ≥ −1.
(5.11)
¯ n (D ∩ G n )) + µ(Tn (D ∩ G n )) ≥ µ(D ∩ G n ). µ(T
(5.12)
Hence we have shown that
In (5.12) we would like to replace D ∩ G n by D, and for this we need G n to have high probability: Lemma 13. If n ≥ R + 1 is chosen big enough, then µ(G cn ) ≤ δ.
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For the proof of Theorem 1 we choose such an n ≥ R + 1. Because of D ∈ FX,n −1 and (5.8) we have ∀ X ∈ D ∩ Gn :
(Tn X − τ e1 )n −1 ∈ D,
i.e. Tn X ∈ D + τ e1 ,
and an analogous result for the backwards transformation. Hence Tn (D ∩ G n ) ⊂ D + τ e1
¯ n (D ∩ G n ) ⊂ D − τ e1 . T
and
Using these inclusions and Lemma 13 we deduce from (5.12), µ(D − τ e1 ) + µ(D + τ e1 ) ≥ µ(D) − δ. δ > 0 was chosen to be an arbitrary positive real, so we get the estimate (3.5) by taking the limit δ → 0. Now the claim of the theorem follows from Lemma 5. 6. Proof of the Lemmas from Section 5 6.1. Cluster bounds: Lemma 7. For n > n and X ∈ X we want to estimate rn,X (n ). For any path x0 , . . . , xm in the graph (X, K n ) such that x0 ∈ n we have |xm | ≤ |x0 | +
m
|xi − xi−1 | ≤ n + mc K .
i=1
By considering all possibilities for such paths we obtain rn,X (n ) ≤ n +
=
1{x0 ∈n } mc K
m≥1 x0 ,...,xm ∈X
m
1{xi xi−1 ∈K n } .
i=1
Using the hard core property (3.2) and Lemma 3 we get Rn := µ(d X )rn,X (n ) − n ≤
µ(d X )
1{x0 ∈n } mc K
x0 ,...,xm ∈X
m≥1
≤
=
1 K \K U (xi − xi−1 )
i=1
(zξ )m+1
m
d x0 . . . d xm 1{x0 ∈n } mc K
m≥1
m
1 K \K U (xi − xi−1 ).
i=1
By (5.1) we can estimate the integrals over d xi in the above expression beginning with i = m. This gives m times a factor cξ and the integration over d x0 gives an additional factor λ2 (n ) = (2n )2 . Thus Rn ≤ (2n )2 zξ c K
m(cξ zξ )m < ∞,
m≥1
where the last sum is finite because cξ zξ < 1.
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6.2. Properties of the auxiliary function . Let f : I → R be a function on an interval. f is called 1/2-Lipschitz-continuous if 1 |r − r | for all r, r ∈ I. 2 f is called piecewise continuously differentiable if it is continuous and if | f (r ) − f (r )| ≤
∃ countable and closed M ⊂ I :
f is continuously differentiable on I \ M.
As M is closed, the connected components of I \ M are countably many intervals. For a strictly monotone piecewise continuously differentiable transformation f on R we can apply the Lebesgue transformation theorem: The derivative f is well defined λ1 -a.s. and for every B1 -measurable function g ≥ 0 we have g(x )d x . (6.1) g( f (x))| f (x)|d x = The above properties are inherited as follows: Lemma 14. Let f 1 , f 2 : I → R be functions on an interval I . (a) If f 1 and f 2 are 1/2-Lipschitz-continuous, then so is f 1 ∧ f 2 . (b) If f 1 and f 2 are piecewise continuously differentiable, then so is f 1 ∧ f 2 . For the proof of these easy facts we refer to [Ri2]. A function f : R2 → R is called 1/2e1 -Lipschitz-continuous or piecewise continuously e1 -differentiable if for all r2 ∈ R the function f (., r2 ) is 1/2-Lipschitz-continuous or piecewise continuously differentiable respectively. Lemma 15. For x ∈ R2 and t ∈ R the function τn (|.|) ∧ m x ,t is 1/2-e1 -Lipschitz-continuous and piecewise continuously e1 -differentiable. For details of the proof we again refer to [Ri2]. Basically Lemma 15 follows from Lemma 14. The only difficulty is to show the continuity of τn (|.|) ∧ m x ,t , which might be a problem because of the jump to infinity of m x ,t in case of h x ,t c f ≤ 1/2. But if x ∈ ∂{m x ,t < ∞} = ∂{ f K (. − x ) < 1} then x − x is contained in the closure of K . Hence |x − x | ≤ c K , which implies |x | − c K ≤ |x|. As τn is decreasing we obtain τn (|x|) ≤ τn (|x | − c K ) ≤ t + h x ,t ≤ m x ,t (x) by definition of h x ,t , which implies the claimed continuity. 6.3. Properties of the construction: Lemma 8. We will first investigate monotonicity and regularity properties of tnk and Tnk : Lemma 16. For X ∈ X and k ≥ 0, tnk is 1/2-e1 -Lipschitz-continuous and piecewise cont. e1 -differentiable, Tnk
is ≤e1 -increasing and bijective.
(6.2) (6.3)
Proof. tnk is the minimum of finitely many functions of the form τn (|.|) ∧ m x ,t , where x ∈ R2 and t ∈ R. Hence (6.2) is an immediate consequence of Lemmas 15 and 14. Statement (6.2) implies that Tnk is e1 -continuous and ≤e1 -increasing, and hence bijective. This shows (6.3).
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For (5.5) it suffices to observe that for every 2 ≤ k ≤ m we have τnk = tnk (Pnk ) = tnk−1 (Pnk ) ∧ m Pnk−1 ,τnk−1 (Pnk ) ≥ τnk−1 . This follows from the definition of τnk and tnk , from tnk−1 (Pnk ) ≥ τnk−1 by the definition of Pnk−1 and from m x ,t ≥ t. For (5.6) and (5.7) let x1 , x2 ∈ X . Without loss of generality j we may suppose that x1 = Pn and x2 = Pni , where 0 ≤ i ≤ j. Here Pn0 is interpreted j to be any point of X cn . We first observe that Pn ∈ i := {x ∈ R2 : tni (x) ≥ τni } and j ∀ x ∈ (Pni + K ) ∩ i : tn (x) = tni (x) ∧ m Pnk ,τnk (x) = τni . (6.4) i≤k≤ j
This holds as tni (x) ≥ τni by definition of i , m Pnk ,τnk ≥ τni by (5.5) and m Pni ,τni (x) = τni by j
j
j
j
j
x ∈ Pni +K . If Pn −Pni ∈ K , then Pn ∈ (Pni +K )∩i , so (6.4) implies τn = tn (Pn ) = τni , j j / K . We have Pn ∈ i \ (Pni + K ) and which shows (5.6). For (5.7) suppose Pn − Pni ∈ j j j τn = tn (Pn ) by definition, so it suffices to show j
Tn (i \ (Pni + K )) = i \ (Pni + K ) + τni e1 .
(6.5)
In order to show this we fix r ∈ R. Continuity of tni (., r ) implies tni = τni on ∂i (., r ). j j Just as in the proof of (6.4) it follows that tn = τni on ∂i (., r ). But Tn (., r ) is increasing, continuous and bijective by (6.3), so j
Tn (i ) = i + τni e1 , and combining this with (6.4) we are done. 6.4. Properties of the deformed translation: Lemma 9. The following lemma shows how to estimate the translation distances τnk . Lemma 17. For X ∈ X and k ≥ 0 we have
τnk ≤ tn0 (Pnk ) for all k ≥ k, τnk
≥
tn0 (an,X (Pnk ))
if X ∈ G n .
(6.6) (6.7)
Proof. Assertion (6.6) follows from the definition of Pnk and from tnk ≤ tn0 . For the proof of (6.7) let X ∈ G n . We first would like to show that X,K n
∀x, x ∈ X : |x| ≤ |x |, x ←→ x ⇒ |τn (|x| − c K ) − τn (|x |)|c f ≤ 1/2.
(6.8)
Defining 1 (n, X ) :=
x,x ∈X
1{|x|≤|x |} 1
X,K n
{x ←→x }
2 4 τn (|x| − c K ) − τn (|x |) c2f ,
(6.9)
we have 1 (n, X ) < 1 by definition of the set G n of good configurations in (5.4) and by X ∈ G n . Hence every summand of 1 is less than 1, which implies (6.8). We now can prove (6.7) by induction on k. For k = 0 we have equality if the right-hand side is
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defined to be 0. For the inductive step k − 1 → k let i ≤ k − 1. By (6.6), the inductive hypothesis and (6.8) we have 0 ≤ τn (|Pni | − c K ) − τni c f ≤ τn (|Pni | − c K ) − τn (|an,X (Pni )|) c f ≤ 1/2, / K . Thus so h Pni ,τni c f ≤ 1/2. Therefore m Pni ,τni (Pnk ) = ∞ whenever Pnk − Pni ∈ τnk = tnk (Pnk ) = tn0 (Pnk ) ∧ m Pni ,τni (Pnk ) ≥ tn0 (an,X (Pnk )), i
where the last step follows from m Pni ,τni (Pnk ) ≥ τni ≥ tn0 (an,X (Pni )), which holds by induction hypothesis, and an,X (Pni ) = an,X (Pnk ) for Pnk − Pni ∈ K . In the proof of (6.7) we have also shown that good configurations X ∈ G n have the following property: In the construction of Tn (X ) we have h Pnk ,τnk c f ≤ 1/2 for every k, i.e. in the definition of m Pnk ,τnk we always have the second case. Now we will prove Lemma 9. It suffices to show for all X ∈ G n and x ∈ X that x ∈ n ⇒ tn,X (x) = τ, x ∈ cn ⇒ tn,X (x) = 0,
x ∈ cn ⇒ x + tn,X (x)e1 − τ e1 ∈ / n −1 , and x ∈ n ⇒ x + tn,X (x)e1 ∈ n .
(6.10)
So let X ∈ G n and x ∈ X . We first note that 0 ≤ τn (|an,X (x)|) ≤ tn,X (x) ≤ τn (|x|) ≤ τ,
(6.11)
which is an immediate consequence of (6.7) and (6.6). We observe x ∈ n ⇒ an,X (x) ∈ R ⇒ τn (|an,X (x)|) = τ ⇒ tn,X (x) = τ, where we have used the definition of R, X ∈ G n , (5.3) and (6.11). This gives the first assertion of (6.10). The second assertion is an immediate consequence of 0 ≤ τ − tn,X (x) ≤ 1, which follows from (6.11) and τ ≤ 1. The third assertion follows from (6.11) and (5.3), and for the fourth assertion let x ∈ n . As x ≤e1 x + tn,X (x)e1 ≤e1 Tn0 (x) by (6.11), it suffices to show that also Tn0 (x) ∈ n . This however follows from Tn0 = id on n c and the bijectivity of Tn0 from (6.3). 6.5. Bijectivity of the transformation: Lemma 10. We will construct the inverse trans˜ n recursively just as in the construction of Tn , i.e. from a given configuration formation T ˜ X we will choose points P˜nk and translate them by τ˜nk in direction −e1 . To get an idea how to define the inverse transformation we start with X ∈ X and set X˜ := Tn (X ). In the construction of X˜ we defined points Pnk and translation distances τnk . We denote the corresponding image points by P˜nk := Pnk + τnk e1 , see Fig. 5. For the construction of the inverse transformation we have to find a method to identify the points P˜nk among the points of X˜ without knowing X . Suppose now that we have already found P˜n1 , . . . , P˜nk−1 . Then inductively we are able to construct the translation j j distances τni for all 1 ≤ i < k, because tni is defined in terms of Pn and τn , where j < i, Tni = id + tni e1 , Pni = (Tni )−1 ( P˜ni ) and τni = tni (Pni ). So in particular we know the transformation functions tnk and Tnk . Thus the following lemma gives a characterisation of P˜nk just as needed:
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˜ n of Tn Fig. 5. Construction of the inverse T
Lemma 18. Let 1 ≤ k ≤ m. For every x˜ ∈ X˜ n \ { P˜n1 , . . . , P˜nk−1 } we have tnk ◦ (Tnk )−1 ( P˜nk ) ≤ tnk ◦ (Tnk )−1 (x). ˜ For all x˜ for which equality occurs we have (Tnk )−1 ( P˜nk ) ≤ (Tnk )−1 (x). ˜ Proof. We first observe that for all k by definition of Tnk we have (Tnk )−1 + tnk ◦ (Tnk )−1 e1 = id.
(6.12)
Since tnk+1 ≤ tnk , we also have Tnk+1 ≤e1 Tnk , and therefore (Tnk )−1 ≤e1 (Tnk+1 )−1 by the e1 -monotonicity of (Tnk+1 )−1 from (6.3). Together with (6.12) this implies tnk+1 ◦ (Tnk+1 )−1 ≤ tnk ◦ (Tnk )−1 .
(6.13)
Now let 1 ≤ k ≤ m and x˜ ∈ X˜ n \ { P˜n1 , . . . , P˜nk−1 }, i.e. x˜ = P˜nl for some l ≥ k. By definition we have tnl (Pnl ) = τnl , Tnl (Pnl ) = P˜nl and P˜nk = Tnk (Pnk ). Using (5.5) and (6.13) we deduce tnk (Tnk )−1 ( P˜nk ) = τnk ≤ τnl = tnl (Pnl ) = tnl (Tnl )−1 (x) ˜ ≤ tnk (Tnk )−1 (x). ˜ If for the given x˜ we have equality, all inequalities in the previous line have to be equalities, so τnk = τnl and tnl (Tnl )−1 (x) ˜ = tnk (Tnk )−1 (x). ˜ Combining this with (6.12) we get l l −1 k −1 ˜ = (Tn ) (x), ˜ so Tnk (Pnl ) = x, ˜ and thus tnk (Pnl ) = τnl = τnk . By Pn = (Tn ) (x) k k −1 k k ˜ definition of Pn we conclude (Tn ) ( Pn ) = Pn ≤ Pnl and we are done. Lemma 18 tells us exactly how to construct the inverse of Tn recursively. So let X˜ ∈ X. Let m˜ = m( ˜ X˜ ) := # X˜ n , t˜0 ˜ = τn (|.|) and τ˜ 0 ˜ := 0. In the kth construction n, X n, X step (1 ≤ k ≤ m) ˜ let k t˜n, := t˜k−1 where m P˜ 0 −τ˜ 0 ,τ˜ 0 := m x,0 ˜ . ˜ ∧ m P˜ k−1 −τ˜ k−1 ,τ˜ k−1 , X˜ n, X
n, X˜
n, X˜
n, X˜
n, X˜
n, X˜
n, X˜
x∈ ˜ X˜ c
n
Let T˜ k ˜ n, X
= id + t˜k ˜ e1 and let P˜ k ˜ be the point of X˜ n \ { P˜ 1 ˜ , . . . , P˜ k−1 } at which the n, X n, X n, X˜ n, X minimum of t˜k ˜ ◦ (T˜ k ˜ )−1 is attained. If there is more than one such point then take n, X n, X the point y such that (T˜ k ˜ )−1 (y) is minimal with respect to the lexicographic order ≤. n, X Let τ˜ k ˜ := t˜k ˜ ◦ (T˜ k ˜ )−1 ( P˜ k ˜ ) be the corresponding minimal value. In the above n, X n, X n, X n, X
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notations we will omit dependencies on X˜ if it is clear which configuration is considered. We need to show that the above construction is well defined, i.e. that T˜ k ˜ is invertible n, X in every step. Furthermore we need some more properties of the construction: Lemma 19. Let X˜ ∈ X and k ≥ 0. Then t˜nk is 1/2-e1 -Lipschitz-continuous, T˜nk is bijective and ≤e1 -increasing, (6.14) (T˜nk )−1 + t˜nk ◦ (T˜nk )−1 e1 = id, (6.15) 2 k k −1 k ∀ c ∈ R, x ∈ R : t˜n ◦ (T˜n ) (x) ≥ c ⇔ t˜n (x − ce1 ) ≥ c, (6.16) t˜nk ≤ t˜nk−1
and τ˜nk−1 ≤ τ˜nk .
(6.17)
Proof. The definitions of t˜nk and T˜nk are similar to those of tnk and Tnk , so we can show (6.14) and (6.15) just as the corresponding properties in (6.2), (6.3) and (6.12). For (6.16) we note that for c ∈ R and x ∈ R2 the equivalence t˜nk ◦ (T˜nk )−1 (x) ≥ c
⇔ ⇔
(T˜nk )−1 (x) ≤e1 x − ce1 x ≤e1 T˜nk (x − ce1 ) = x − ce1 + t˜nk (x − ce1 )e1
follows from (6.15) and (6.14). The first part of (6.17) is obvious and for the second part we observe that t˜nk−1 ◦(T˜nk−1 )−1 ( P˜nk ) ≥ τ˜nk−1 ⇒ t˜nk−1 ( P˜nk − τ˜nk−1 e1 ) ≥ τ˜nk−1 ⇒ t˜nk ( P˜nk − τ˜nk−1 e1 ) ≥ τ˜nk−1 ⇒ τ˜nk = t˜nk ◦ (T˜nk )−1 ( P˜nk ) ≥ τ˜nk−1 , where the first statement holds by definition of P˜nk−1 , the first and the third implication hold by (6.16) and the second holds by definition of t˜nk . Let t˜n, X˜ ( P˜ k ˜ ) := τ˜ k ˜ and t˜n, X˜ (x) = 0 for x ∈ X˜ cn . This defines a translation distance n, X n, X ˜ n : X → X be defined by function t˜ ˜ : X˜ → R. Let T n, X
˜ n ( X˜ ) := X˜ c ∪ { P˜ k − τ˜ k e1 : 1 ≤ k ≤ m} = {x − t˜ ˜ (x)e1 : x ∈ X˜ }. T n, X n n, X˜ n, X˜ By Lemma 6 we again see that all above objects are measurable with respect to the considered σ -algebras. The only difficulty is to show that the functions (T˜ k ˜ )−1 (x) are n, X measurable, which follows from the e1 -monotonicity of T˜ k . n, X˜
˜ n really is the inverse of Tn we need an analogue of Lemma 18. In order to show that T k k ˜ n ( X˜ ) and Let X˜ ∈ X. Let t˜n , T˜n , P˜nk and τ˜nk (0 ≤ k ≤ m) ˜ as above and denote X := T Pnk := P˜nk − τ˜nk e1 , see Fig. 5. Lemma 20. Let 1 ≤ k ≤ m. ˜ For every x ∈ X n \ {Pn1 , . . . , Pnk−1 } we have t˜nk (Pnk ) ≤ t˜nk (x). For all x for which equality occurs we have Pnk ≤ x.
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Proof. Let 1 ≤ k ≤ m˜ and x ∈ X n \ {Pn1 , . . . , Pnk−1 }, i.e. x = Pnl for some l ≥ k. By definition of τ˜nk and τ˜nl and (6.15) we have (T˜nk )−1 ( P˜nk ) = Pnk and (T˜nl )−1 ( P˜nl ) = x. Using (6.17) we obtain t˜nk (Pnk ) = τ˜nk ≤ τ˜nl = t˜nl (T˜nl )−1 ( P˜nl ) = t˜nl (x) ≤ t˜nk (x). If for the given x we have equality, all inequalities in the previous line have to be equalities, so τ˜nk = τ˜nl and t˜nk (x) = τ˜nl , i.e. T˜nk (x) = x + τ˜nl e1 = P˜nl . This gives τ˜nk = τ˜nl = t˜nk (x) = t˜nk ◦ (T˜nk )−1 ( P˜nl ). So Pnk = (T˜nk )−1 ( P˜nk ) ≤ (T˜nk )−1 ( P˜nl ) = x by definition of P˜nk and we are done. ˜ n ◦ Tn = id Lemma 21. On X we have T
˜ n = id. and Tn ◦ T
Proof. For the first part let X ∈ X and X˜ := Tn (X ). We have m( ˜ X˜ ) = m(X ) by construction and we have X cn = X˜ cn by (5.8). Now it suffices to prove k k k k k k k k = tn,X , T˜n,k X˜ = Tn,X , τ˜n, = τn,X and P˜n, = Pn,X + τn,X t˜n, X˜ X˜ X˜
for every k ≥ 0 by induction on k. Here P˜ 0
n, X˜
(6.18)
0 +τ 0 is interpreted as X c = X ˜ c . = Pn,X n n,X n
The case k = 0 is trivial. For the inductive step k − 1 → k we observe that t˜nk = tnk by induction hypothesis, and T˜nk = Tnk is an immediate consequence. Combining this with Lemma 18 and the definition of P˜nk we get P˜nk = Pnk + τnk and τ˜nk = τnk . ˜ n ( X˜ ). As above it suffices to show (6.18) For the second part let X˜ ∈ X and X := T by induction on k. Here X˜ cn = X n follows from an analogue of (5.8) and the inductive step follows from Lemma 20. 6.6. Density of the transformed process: Lemma 11. By definition the left-hand side of (5.10) equals 2 1 e−4n I (k), where I (k) = d x( f ◦ Tn · ϕn )( X¯ x ), k! n k k≥0
using the shorthand notation X¯ x := {xi : i ∈ J } ∪ X¯ cn for x ∈ nJ . To compute I (k) i among the we need to calculate Tn ( X¯ x ), and for this we must identify the points Pn, X¯ x particles x j . So let be the set of all permutations η : {1, . . . , k} → {1, . . . , k}. For η ∈ let
j Ak,η := x ∈ nk : ∀ 1 ≤ j ≤ k : xη( j) = Pn, X¯ and x
j A˜ k,η := x ∈ nk : ∀ 1 ≤ j ≤ k : xη( j) = P˜n, X¯ , x
where P˜n, X¯ are the points from the construction of the inverse transformation in Subx sect. 6.5. Now we can write I (k) = I (k, η), where I (k, η) = d x 1 Ak,η (x)( f ◦ Tn · ϕn )( X¯ x ). j
η∈
n k
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If x ∈ Ak,η we can derive a simple expression for Tn ( X¯ x ): For x ∈ nk we define a i (x )) formal transformation Tη (x) := (Tη,x i 1≤i≤k , where η( j)
η( j)
η( j)
j
Tη,x := id + tη,x e1 , tη,x := tn, X¯
x η, j−1 := (xη(i) )1≤i≤ j−1 .
and
x η, j−1
η( j)
Clearly, Tη,x doesn’t depend on all components of x, but only on those xη(l) such that l ≤ j − 1. By definition we now have j η( j) Tn ( X¯ x ) = X¯ Tη (x) and Tn, X¯ = Tη,x for all j ≤ k.
∀ x ∈ Ak,η :
(6.19)
x
Furthermore we observe that for all x ∈ (R2 )k we have Tη (x) ∈ A˜ k,η .
⇔
x ∈ Ak,η
(6.20)
Here “⇒” holds by (6.19) and (6.18) from the proof of Lemma 21. For “⇐” let x ∈ (R2 )k ˜ n ( X¯ T (x) ), where T˜ n is the inverse of Tn as defined such that Tη (x) ∈ A˜ k,η and let X := T η in the last subsection. By induction on j we can show η( j)
j
∀1 ≤ j ≤ k :
Tn,X = Tη,x
and
j
xη( j) = Pn,X .
In the inductive step j − 1 → j the first assertion follows from the induction hypothesis j and the second follows from the bijectivity of Tn,X and j η( j) j Tn,X (xη( j) ) = Tη,x (xη( j) ) = P˜n, X¯
j
Tη (x)
j
j
j
= Pn,X + τn,X = Tn,X (Pn,X ),
j η( j) which follows from Tn,X = Tη,x , the definition of A˜ k,η and (6.18) from the proof of Lemma 21. This completes the proof of the above assertion and we conclude X¯ x = X , j j which implies xη( j) = Pn,X = Pn, X¯ . Thus (6.20) holds. x
Now let g :
(R2 )k
→ R, g(x) := 1 A˜ k,η (x) f ( X¯ x ). Then (6.19) and (6.20) imply
I (k, η) =
k
η( j) d xη( j) 1 + ∂1 tη,x (xη( j) ) g(Tη (x)),
j=1
where we have also inserted the definition of ϕn (5.9). Now we transform the integrals. i x , where i := η( j). The transformation only conFor j = k to 1 we substitute xi := Tη,x i i (., r¯ ). cerns the first component of xi = (ri , r¯i ). For fixed r¯i ri is transformed by id +tη,x i i From (6.2) we know that tη,x (., r¯i ) is 1/2-Lipschitz-continuous and piecewise continui (., r¯ ) is strictly increasing and piecewise continuously ously differentiable, so id + tη,x i differentiable. Therefore the Lebesgue transformation theorem (6.1) gives i (xi ) . d xi = d xi 1 + ∂1 tη,x Thus I (k, η) =
k
j=1
d xη( j)
g(x ) =
n k
d x 1 A˜ k,η (x) f ( X¯ x ),
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and we are done as the same arguments show that the right-hand side of (5.10) equals e−4n
2
1 d x 1 A˜ k,η (x) f ( X¯ x ). k! n k k≥0
η∈
An analogous argument shows that the density function is well defined: ∀ X¯ ∈ X :
νn (“ϕn is well defined | X¯ ) k 2 1 = e−4n d x 1 Ak,η (x) 1{∂ t η( j) (x ) exists} . 1 η,x η( j) k! n k k≥0
η∈
j=0
η( j)
As tη,x is piecewise continuously e1 -differentiable, we have for arbitrary r ∈ R, k, η η( j) and x as above that ∂1 tη,x (., r ) exists λ1 -a.s.. So we may replace all indicator functions in the above product by 1 using Fubini’s theorem. Hence the above probability equals 1. 6.7. Estimation of the densities: Lemma 12. Let X ∈ G n . By the 1/2-e1 -Lipschitzcontinuity from (6.2) we have k k (Pn,X )| ≤ 1/2. |∂1 tn,X
Using − log(1 − a) ≤ 2a for 0 ≤ a ≤ 1/2 we obtain f n (X ) := − log ϕ¯n (X ) − log ϕn (X ) k k k k = − log 1 − (∂1 tn,X (Pn,X ))2 ≤ 2(∂1 tn,X (Pn,X ))2 . 1≤k≤m
1≤k≤m
If ∂1 tnk (Pnk ) exists it equals either ∂1 tn0 (Pnk ) or ∂1 m x,tn (x) (Pnk ) for some x ∈ X such that x = Pnk and Pnk ∈ x + K . By using (6.7) we see that |∂1 m x,tn (x) (Pnk )| ≤ (τn (|x| − c K ) − tn (x))c f ≤ τn (|x| − c K ) − τn (|an,X (x)|) c f . Furthermore |∂1 tn0 (Pnk )| ≤ τ q(|Pnk | − R)/Q(n − R) by definition of tn0 = τn (|.|), so we can estimate f n (X ) by the sum of the two following terms: 2 (n, X ) := 2τ 2 3 (n, X ) :=
1{x∈n }
x∈X = 2 2c f x,x ∈X x ∈X
q(|x| − R)2 , Q(n − R)2
1 K (x − x)1
X,K n
{x ←→x }
×(τn (|x| − c K ) − τn (|x |))2 . Using these terms in the definition (5.4) of G n we are done.
1{|x|≤|x |}
(6.21)
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6.8. Set of good configurations: Lemma 13. The functions i (n, X ) from the definition of the set of good configurations G n in (5.4) have been specified in (6.9) and (6.21). Using the shorthand τn (x, x ) := 1{|x|≤|x |} |τn (|x| − c K ) − τn (|x |)|2 q
we have
1 = 4c2f 3 =
1
τn (x, x ), 2 = 2τ 2 q
X,K n
{x ←→x } x,x ∈X = 2c2f 1 K (x x,x ∈X x ∈X
1{x∈n }
x∈X
− x)1
q(|x| − R)2 Q(n − R)2
and
τn (x, x ). q
X,K n
{x ←→x }
We will show that the expectation of every i can be made arbitrarily small when n is chosen big enough. But first we will give some relations used later. Let n ≥ R + 1. For s > s such that s > R and s < n we have s ∧n q(t − R) q(s − R) 0 ≤ r (s − R, n − R)−r (s − R, n− R) = dt ≤ (s − s) Q(n− R) R∨s Q(n − R) by the monotonicity of q. Defining n¯ := n + c K and R¯ := R + c K we thus have τn (x, x ) ≤ 1{x∈n¯ } τ 2 (|x | − |x| + c K )2 q
¯ 2 q(|x| − R) ¯ 2 Q(n¯ − R)
for x, x ∈ R2 ,
(6.22)
using the substitution s := |x | and s := |x| − c K . (If s ≤ R or s ≥ n then τn (x, x ) = 0.) The following relations will give us control over the relevant terms of the right-hand side of (6.22). We first observe that ¯ 2 ≤ 16 R¯ 2 + 32Q(n¯ − R) ¯ for n¯ ≥ 2 R. ¯ d x q(|x| − R) (6.23) q
n¯
Indeed, writing s := |x| we obtain n¯
¯ 2 d x q(|x| − R)
≤ 16 R¯ 2 + 32
≤
2 R¯
ds 8s + 0
n− ¯ R¯
q(s)ds
≤
n− ¯ R¯ R¯
2 ¯ ds 8(s + R)q(s)
¯ 16 R¯ 2 + 32Q(n¯ − R).
0
In the first step we used q ≤ 1, and in the second step R¯ ≤ s and sq(s) ≤ 2. We observe lim Q(n) = ∞, which is a consequence of log log n ≤ Q(n) for n > 1. Therefore by n→∞ (6.23), ¯ 2 q(|x| − R) lim c(n) = 0 for c(n) := dx . (6.24) ¯ 2 n→∞ Q(n¯ − R) n¯ Finally, for x0 , . . . , xm ∈ R2 such that xi − xi−1 ∈ K we have |xi − xi−1 | ≤ c K , so (|xm | − |x0 | + c K )2 ≤ (m + 1)2 c2K .
(6.25)
108
T. Richthammer
Now we will use the ideas of the proof of Lemma 7. For X ∈ X we can estimate the summands of 1 (n, X ) by considering all paths x0 , . . . , xm in the graph (X, K n ) connecting x = x0 and x = xm . By (6.22) and (6.25) we can estimate 1 (n, X ) by a constant c times
=
(m + 1)2
1{x0 ∈n¯ }
x0 ,...,xm ∈X
m≥0
m ¯ 2 q(|x0 | − R) 1{xi xi−1 ∈K n } . ¯ 2 Q(n¯ − R) i=1
Using Lemma 3 we can thus proceed as in the proof of Lemma 7: µ(d X ) 1 (n, X ) ≤ zξ c (m + 1)2 (cξ zξ )m c(n). m≥0
Likewise,
µ(d X )2 (n, X ) ≤ 2zξ τ 2 c(n).
Finally, we can estimate 3 (n, X ) by a constant c times
(m + 1)2
=
1{x0 ∈n¯ }
x0 ,...,xm ∈X
m≥0
×
m
1{xi xi−1 ∈K n }
i=1
¯ 2 q(|x0 | − R) ¯ 2 Q(n¯ − R)
1 K (x − x0 ) +
x ∈X,x =xi ∀ i
m
1 K (x j − x0 ) .
j=1
The second sum in the brackets can be estimated by m. As above µ(d X ) 3 (n, X ) ≤ zξ c (m + 1)2 (cξ zξ )m c(n)(zξ cξ + m). m≥0
In the bounds on the expectations of 1 and 3 the sums over m are finite by (5.1). Collecting all estimates and using (6.24) we thus find that µ(d X )
3 i=1
i (n, X ) ≤
δ 2
for sufficiently large n, and µ(G cn ) ≤ δ follows from the high probability of G n , the Chebyshev inequality and the definition of G n in (5.4). 7. Proof of Theorem 2: Main Steps 7.1. Basic constants. Let (U, z, X0 ) be admissible with Ruelle bound ξ , where U : R2 → R is a translation-invariant, smoothly approximable standard potential. We choose K , ψ, U¯ and u according to Definition 1. W.l.o.g. we may assume 0 ∈ K , U¯ = U and u = 0 on K . We then let > 0 so small that 1 . (7.1) u(x)d ˜ x< cξ := λ2 (K \ K U ) + zξ Kc
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
109
In addition to the function f K and the constants c K and c f introduced in Sect. 5.1 we also define 2 u(x)|x| ˜ d x and cψ := ψ ∨ d x ψ(x)(|x|2 ∨ 1). (7.2) cu := Kc
These constants are finite by our assumptions. Finally, we fix a Gibbs measure µ ∈ GX0 (U, z), a cylinder event D ∈ FX,n −1 , where n ∈ N, a translation distance τ ∈ [0, 1/2], the translation direction e1 and a real δ > 0. 7.2. Decomposition of µ and the bond process. For n ∈ N and X ∈ X we consider the bond set E n (X ) := E n (X ) = {x1 x2 ∈ E(X ) : x1 x2 ∩ n = ∅}. On (E E n (X ) , B E n (X ) ) we introduce the Bernoulli measure πn (.|X ) with bond probabilities (u(b)) ˜ b∈E n (X )
where
u(b) ˜ := 1 − e−u(b) ,
using the shorthand notation u(x1 x2 ) := u(x1 − x2 ) for x1 , x2 ∈ R2 . We note that 0 ≤ u(b) ˜ < 1 for all b ∈ E n (X ) as 0 ≤ u < ∞. As remarked earlier πn (.|X ) can be extended to a probability measure on (E, FE). For all D ∈ FE, πn (D|.) is FXmeasurable, so πn is a probability kernel from (X, FX) to (E, FE). Lemma 22. Let n ∈ N. We have
µ ⊗ νn (G n ) = 1 and µ(G n ) = 1 for G n := {X ∈ X0 :
u(b) ˜ < ∞}.
b∈E n (X )
For X ∈ G n the Borel-Cantelli lemma implies that every bond set is finite πn (.|X )-a.s., so πn ({B}|X ) = 1, B⊂E n (X )
where the summation symbol We have πn ({B}|X ) = u(b) ˜ b∈B
indicates that the sum extends over finite subsets only.
(1 − u(b)) ˜ = e−Hn (X ) u
b∈E n (X )\B
(eu(b) − 1),
b∈B
so for every X ∈ G n the Hamiltonian Hu n (X ) is finite, and thus the decomposition of the potential gives a corresponding decomposition of the Hamiltonian ¯
HUn (X ) = HUn (X ) − Hu n (X ). Using (3.1) we conclude that for every FX ⊗ FE-measurable function f ≥ 0, 1 µ(d X¯ ) f (X, B) dµ ⊗ πn f = νn (d X | X¯ ) Z n ( X¯ ) B⊂E n (X ) U¯ (eu(b) − 1). ×z # X n e−Hn (X ) b∈B
(7.3)
110
T. Richthammer
Here by Lemma 22 on both sides we have X ∈ G n with probability one, thus the equality follows from the above decomposition. If f does not depend on B at all, the integral on the left-hand side of (7.3) is just the µ-expectation of f , as πn is a probability kernel, and from the right-hand side we learn that the perturbation u of the smooth potential U¯ can be encoded in a bond process B such that the perturbation affects only those pairs of particles with x1 x2 ∈ B. On (E E n (X ) , B E n (X ) ) we denote the counting measure concentrated on finite bond sets by πn (.|X ). Again πn can be considered as a probability kernel from (X, FX) to (E, FE). For all FE-measurable functions f ≥ 0 we have f (B). πn (d B|X ) f (B) = B⊂E n (X )
7.3. Generalised translation. First of all, we need to augment each bond set B by additional bonds between all particles that are close to each other. That is, for n > n , X ∈ X and B ⊂ E n (X ) we introduce the K -enlargement of B by B+ := B ∪ {x1 x2 ∈ E n (X ) : x1 − x2 ∈ K }. We then consider the range of the B+ -cluster of ∈ B2b , rn,X,B+ () = sup{|x | : x ∈ C X,B+ ()}. Lemma 23. We have sup µ ⊗ πn (d X, d B) rn,X,B+ (n ) < ∞. n>n
By the Chebyshev inequality we therefore can choose an integer R > n , such that for every n > n the event G n := {(X, B) ∈ X × E : rn,X,B+ (n ) < R, B ⊂ E n (X ) finite} ∈ FX ⊗ FE has probability
µ ⊗ πn (G n ) ≥ 1 − δ/2.
For n > R we define the functions q, Q, r and τn exactly as in Sect. 5.2. For X ∈ X, B ∈ E n (X ) and x ∈ X we define an,X,B+ (x) to be a point of C X,B+ (y) such that |an,X,B+ (x)| ≥ |x|, τn (|an,X,B+ (x)|) = min{τn (|x |) : x ∈ C X,B+ (y)} and an,X,B+ (x) is a measurable function of x, X and B. 7.4. Good configurations. In order to deal with the hard core and the perturbation encoded in the bond process, we will introduce a transformation Tn : X × E → X × E which is required to have the following properties: (1) Whenever B is a set of bonds between particles in X , the transformed configura˜ = Tn (X, B) is constructed by translating every particle x ∈ X by a tion ( X˜ , B) certain distance tn,X,B (x) in direction e1 , and by translating bonds along with the corresponding particles.
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
111
(2) Particles in the inner region n −1 are translated by τ e1 , and particles in the outer region n c are not translated at all. (3) Particles connected by a bond in B are translated the same distance. (4) Tn is bijective, and the density of the transformed process with respect to the untransformed process under the measure ν ⊗ πn can be calculated explicitly. ¯ ¯ (5) We have suitable estimates on this density and on HUn ( X˜ ) − HUn (X ). For the last assumption we need particles within hard core distance to remain within hard core distance and particles at larger distance to remain at larger distance. Property (2) implies that the translation of the chosen cylinder event D is the same as the transformation of D by Tn . Properties (3)-(5) are chosen with a view to the right-hand side of (7.3): If Tn has these properties then the density of the transformed process with respect to the untransformed process under the measure µ ⊗ πn can be estimated. We will content ourselves with a transformation satisfying the above properties only for (X, B) from a set of good configurations 5
G n := (X, B) ∈ G n : i (n, X, B) < 1/2 ∈ FX ⊗ FE.
(7.4)
i=1
The functions i (n, X, B) will be defined whenever we want good configurations to have a certain property. In Lemma 28 we then will prove that the set of good configurations G n has probability close to 1 when n is big enough. Up to that point we consider a fixed n ≥ R + 1. 7.5. Modifying the generalised translation. The construction of the deformed translation Tn will go along the same lines as the corresponding construction in Sect. 5.4. However, here we also have to consider bonds between particles, and by property (3) from the last section we know that we have to translate not just particles, but whole B-clusters. For a rigorous recursive definition of Tn (X, B) we first consider the case that B is a 0 0 finite subset of E n (X ). Let tn,X,B := τn (|.|), Cn,X,B be the B-cluster of the outer region 0 0 c n , m = m(X, B) the number of different B-clusters of X \ Cn,X,B and τn,X,B := 0. In the kth construction step (1 ≤ k ≤ m) let k−1 k 0 := tn,X,B ∧ m x,τ k−1 = tn,X,B ∧ m x,τ i , tn,X,B k−1 x∈Cn,X,B
n,X,B
0≤i
n,X,B
¯ is defined as in Sect. 5.4. Let the pivotal where the auxiliary function m x ,t : R2 → R k−1 k k 0 point Pn,X,B be the point of X \ (Cn,X,B ∪ . . . ∪ Cn,X,B ) at which the minimum of tn,X,B is attained. If there is more than one such point then take the smallest point with respect k k k to the lexicographic order for the sake of definiteness. Let τn,X,B := tn,X,B (Pn,X,B ) be k k k the corresponding minimal value of tn,X,B , Cn,X,B the B-cluster of the point Pn,X,B k k m+1 , but then the and Tn,X,B := id + tn,X,B e1 . For k = m + 1 we can still define tn,X,B m 0 ∪ . . . ∪ Cn,X,B ) = ∅. In the above notations we will recursions stops as X \ (Cn,X,B omit dependence on X and B if it is clear which configuration is considered. Now for k k x ∈ Cn,X,B let tn,X,B (x) := τn,X,B be the deformed translation distance function and let
112
T. Richthammer
Tn,B (X ) :=
m(X,B)
k k (Cn,X,B + τn,X,B e1 ) = {x + tn,X,B (x)e1 : x ∈ X }
and
k=0
Tn,X (B) := {(x + tn,X,B (x)e1 )(x + tn,X,B (x )e1 ) : x x ∈ B}. If B is not a finite subset of E n (X ) we define Tn,B = id and Tn,X = id. The deformed transformation can now be defined to be Tn : X × E → X × E, Tn (X, B) := (Tn,B (X ), Tn,X (B)). Using Lemma 6 one can show that all the above objects are measurable with respect to the considered σ -algebras. In the rest of this section we will convince ourselves that the above construction has indeed the required properties. Lemma 24. For good configurations (X, B) ∈ G n we have (Tn,B X − τ e1 )n −1 = X n −1
and (Tn,B X )n c = X n c .
(7.5)
Lemma 25. The transformation Tn : X × E → X × E is bijective. In the proof of Lemma 25 we again construct the inverse of Tn , which is needed in the proof of the following lemma. There we will also show that ϕn (X, B) :=
m(X,B)
1 + ∂1 t k
k n,X,B (Pn,X,B )
(7.6)
k=1
is well defined νn ⊗ πn ( . | X¯ )-a.s., in that the considered derivatives exist. Lemma 26. For every X¯ ∈ X and every FX ⊗ FE-measurable function f ≥ 0, dνn ⊗ πn (.| X¯ ) f. dνn ⊗ πn (.| X¯ ) ( f ◦ Tn · ϕn ) =
(7.7)
¯ n, T ¯ n,B , T ¯ n,X and ϕ¯n be defined analWe also need the backwards translation. So let T ogously to the above objects, where now e1 is replaced by −e1 . The previous lemmas ¯ n is not the apply analogously to this deformed backwards translation. We note that T inverse of Tn . 7.6. Final steps of the proof. From (7.3) and Lemma 26 we deduce 1 µ(d X¯ ) µ ⊗ πn (Tn (D ∩ G n )) = νn ⊗ πn (d X, d B| X¯ ) Z n ( X¯ ) U¯
1Tn (D∩G n ) ◦ Tn (X, B) z #(Tn,B X )n ϕn (X, B)e−Hn (Tn,B X ) × (eu(b) − 1). b∈Tn,X B
Here we have identified D and D × E. By the bijectivity of Tn from Lemma 25, by (7.5) and by construction of Tn,X the above integrand simplifies to U¯ (eu(b) − 1). 1 D∩G n (X, B) z # X n elog ϕn (X,B)−Hn (Tn,B X ) b∈B
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
113
¯ n can be treated analogously, hence The backwards transformation T ¯ n (D ∩ G n )) + µ ⊗ πn (Tn (D ∩ G n )) − µ ⊗ πn (D ∩ G n ) µ ⊗ πn (T 1 = µ(d X¯ ) (eu(b) − 1) νn ⊗ πn (d X, d B| X¯ ) 1 D∩G n (X, B) z # X n Z n ( X¯ ) b∈B
U¯ (T¯ U¯ (T U¯ (X ) log ϕ¯n (X,B)−H X ) log ϕ (X,B)−H X ) −H n n,B n,B n n . +e − e n × e We note that for (X, B) ∈ G n we have U¯
U¯
¯
elog ϕ¯n (X,B)−Hn (Tn,B X ) + elog ϕn (X,B)−Hn (Tn,B X ) U¯
1
U¯
¯
≥ 2 e 2 (log ϕ¯n (X,B)+log ϕn (X,B)−Hn (Tn,B X )−Hn (Tn,B X )) U¯
1
≥ 2 e− 2 −Hn (X )
U¯
e−Hn (X ) ,
≥
where we have used the convexity of the exponential function in the first step and the following estimates in the second step: Lemma 27. For (X, B) ∈ G n we have ¯
¯
¯
¯ n,B X ) + H U (Tn,B X ) ≤ 2H U (X ) + 1/2 and HUn (T n n
(7.8)
log ϕ¯n (X, B) + log ϕn (X, B) ≥ −1/2.
(7.9)
Hence we have shown that ¯ n (D ∩ G n )) + µ ⊗ πn (Tn (D ∩ G n )) ≥ µ ⊗ πn (D ∩ G n ). µ ⊗ πn (T
(7.10)
In (7.10) we would like to replace D ∩ G n by D, and for this we need G n to have high probability: Lemma 28. If n ≥ R + 1 is chosen big enough, then µ ⊗ πn (G cn ) ≤ δ. For the proof of Theorem 2 we choose such an n ≥ R + 1. The rest of the argument is then the same as that in Sect. 5.5. 8. Proof of the Lemmas from Section 7 8.1. Convergence of energy sums: Lemma 22. Let n ∈ N. For every X ∈ X we have = Hu˜ n (X ) = u(b) ˜ ≤ 1{x1 ∈n } u(x ˜ 1 − x2 ), and so x1 ,x2 ∈X
b∈E n (X )
νn (d X | X¯ )Hu˜ n (X ) ≤
n
d x1
n
d x2 u(x ˜ 1 − x2 ) +
u(x ˜ 1 − x2 )
x2 ∈ X¯ c n
for all X¯ ∈ X. By Lemma 3 we get u˜ d x1 µ ⊗ νn (d X )Hn (X ) ≤ n
≤
n
n
d x2 u(x ˜ 1 − x2 ) + zξ
d x1 (1 + zξ )cξ
≤
nc
d x2 u(x ˜ 1 − x2 )
4n 2 (1 + zξ )cξ
<
∞,
114
T. Richthammer
where we have estimated the integrals over x2 by cξ using (7.1). Thus we have proved the first assertion. However, µ is absolutely continuous with respect to µ ⊗ νn , which follows from (3.1) and the definition of the conditional Gibbs distribution. So the first assertion implies the second one.
8.2. Cluster bounds: Lemma 23. Let us refine the argument of Sect. 6.1 as follows. For n > n , X ∈ X and B ⊂ E n (X ) we consider a path x0 , . . . , xm in the graph (X, B+ ) such that x0 ∈ n , and we consider an integer k such that 1 ≤ k ≤ m and the bond xk−1 xk has maximal |.|-length among all bonds on the path. We have |xm | ≤ |x0 | +
m
|xi − xi−1 | ≤ n + m|xk − xk−1 |.
i=1
By considering all paths and bonds of maximal length we obtain rn,X,B+ (n ) ≤ n +
m
=
1{x0 ∈n } m|xk − xk−1 |
m≥1 k=1 x0 ,...,xm ∈X
m
1{xi xi−1 ∈B+ } .
i=1
Under the Bernoulli measure πn (d B|X ), the events {xi xi−1 ∈ B+ } are independent, and for g := 1 K \K U + u˜ we have (8.1) πn (d B|X )1{xi xi−1 ∈B+ } ≤ 1 K U (xi − xi−1 ) + g(xi − xi−1 ). Using the hard core property (3.2) and Lemma 3 we thus find Rn := µ(d X ) πn (d B|X )rn,X,B+ (n ) − n ≤ ≤
m m≥1 k=1 m
µ(d X )
=
1{x0 ∈n } m|xk − xk−1 |
x0 ,...,xm ∈X
(zξ )
g(xi − xi−1 )
i=1
m+1
m
d x0 . . . d xm 1{x0 ∈n } m|xk − xk−1 |
m≥1 k=1
m
g(xi − xi−1 ).
i=1
Setting cg := (1 + c2K )cξ + cu we conclude from (7.1) and (7.2) that g(x)|x| d x ≤ g(x)(1 + |x|2 ) d x ≤ cg and g(x) d x ≤ cξ ,
(8.2)
hence we can estimate the integrals over d xi in the above expression beginning with i = m. This gives m − 1 times a factor cξ and once a factor cg . Finally the integration over d x0 gives an additional factor λ2 (n ) = (2n )2 . Thus m 2 (cξ zξ )m−1 . Rn ≤ (2n zξ )2 cg m≥1
The last sum is finite because cξ zξ < 1.
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
115
8.3. Properties of the deformed translation: Lemma 24. We will show properties of the construction which are analogous to properties of the corresponding objects from the proof of the special case in Sects. 6.3 and 6.4. Additionally we need a way to calculate the translation distance of an arbitrary particle x ∈ Cnk without knowing Pnk . This can be done using the first relation of the following lemma. Lemma 29. For X ∈ X, finite B ⊂ E(X ), k ≥ 0, x, x ∈ X and s ∈ [−1, 1], τnk = tnk+1 (x) if x ∈ Cnk ,
(8.3)
τnk ≤ τnk+1 , tnk is 1/2-e1 -Lipschitz-continuous and piecewise cont. e1 -differentiable, Tnk is ≤e1 -increasing and bijective, x − x ∈ K ⇒ tn,X,B (x) = tn,X,B (x ), x − x ∈ / K ⇒ x − x + s(tn,X,B (x) − tn,X,B (x ))e1 ) ∈ / K,
(8.4)
τnk ≤ tn0 (x) for all x ∈ Cnk such that k ≥ k, τnk
≥
tn0 (an,X,B+ (Pnk ))
(8.5) (8.6) (8.7) (8.8) (8.9)
if (X, B) ∈ G n .
(8.10)
Proof. For (8.3) let x ∈ Cnk . By definition of Pnk we have tnk (x) ≥ τnk , so m x ,τnk (x) = τnk , tnk+1 (x) = tnk (x) ∧ x ∈Cnk
where we have also used m x ,τnk (x) ≥ τnk and m x,τnk (x) = τnk . The other assertions can be shown as in Sects. 6.3 and 6.4. Here for the proof of (8.7) and (8.8) we have to use (8.3), j+1 j+1 and the key observations are the following: For i ≤ j, xi ∈ Cni and Tn,s := id +s ·tn e1 , we have j+1 ∀ x ∈ K (xi ) ∩ i : tn (x) = tni (x) ∧ m x ,τnk (x) = τni , i≤k≤ j x ∈Cnk
j+1
Tn,s (i ) = i + sτni e1
j+1
Tn,s (i \ K (xi )) = i \ K (xi ) + sτni e1 .
and
To obtain (8.10) here we specify the function 2 1 (n, X, B) := 1{|x|≤|x |} 1 X,B+ 4 τn (|x| − c K ) − τn (|x |) c2f {x ←→x }
x,x ∈X
used in the definition of G n .
(8.11)
Lemma 24 follows from (8.9) and (8.10), just as in the proof of Lemma 9. 8.4. Bijectivity of the transformation: Lemma 25. The construction of the inverse transformation is analogous to the one in Sect. 6.5. Let X˜ ∈ X and B˜ ⊂ E n ( X˜ ) be finite. ˜ ˜ the number of different B-clusters ˜ of cn , m˜ = m( ˜ X˜ , B) of Let C˜ 0 ˜ ˜ be the B-cluster n, X , B 0 0 0 th X˜ \ C˜ , t˜ = τn (|.|) and τ˜ := 0. In the k construction step (k ≥ 1) let n, X˜ , B˜
n, X˜ , B˜
n, X˜ , B˜
k t˜n, := t˜k−1 ˜ ˜ ∧ X˜ , B˜ n, X , B
x∈C˜ k−1 −τ˜ k−1 n, X˜ , B˜ n, X˜ , B˜
m x,τ˜ k−1 . n, X˜ , B˜
116
T. Richthammer
Let T˜ k
e and P˜ k ˜ ˜ be the point of X˜ \ (C˜ 0 ˜ ˜ ∪ . . . ∪ C˜ k−1 ) at n, X , B n, X˜ , B˜ n, X , B k −1 which the ◦ (T˜ ˜ ˜ ) is attained. If there is more than one such n, X , B point then take the point x such that (T˜ k )−1 (x) is minimal with respect to the lexn, X˜ , B˜
= id + t˜k
n, X˜ , B˜ 1 minimum of t˜k ˜ ˜ n, X , B
n, X˜ , B˜
icographic order ≤. Let := t˜k ˜ ˜ ◦ (T˜ k ˜ ˜ )−1 ( P˜ k ˜ ˜ ) be the corresponding n, X , B n, X , B n, X , B ˜ minimal value and C˜ k ˜ ˜ be the B-cluster of the pivotal point P˜ k ˜ ˜ . The recursion n, X , B n, X , B stops for k = m˜ + 1. In the above notations we will omit dependence on X˜ and B˜ if it is clear which configuration is considered. We need to show that the above construction is well defined, i.e. that T˜ k ˜ ˜ is invertible in every step. Furthermore we need some n, X , B more properties of the construction. All this is done in the following lemma: τ˜ k ˜ ˜ n, X , B
Lemma 30. Let X˜ ∈ X, B˜ ⊂ E n ( X˜ ) finite and k ≥ 0. Then t˜nk is 1/2-e1 -Lipschitz-continuous, T˜nk is bijective and ≤e1 -increasing, (T˜nk )−1 + t˜nk ◦ (T˜nk )−1 e1 = id,
(8.12)
∀ c ∈ R, x ∈ R : t˜nk ◦ (T˜nk )−1 (x) t˜nk ≤ t˜nk−1 and τ˜nk−1 ≤ τ˜nk , ∀ x ∈ C˜ nk : t˜nk+1 ◦ (T˜nk+1 )−1 (x) =
(8.14)
2
≥ c ⇔
t˜nk (x
(8.13)
− ce1 ) ≥ c,
(8.15) τ˜nk .
(8.16)
Proof. Assertions (8.12)–(8.15) can be shown exactly as the corresponding assertions from Lemma 19. For (8.16) let x ∈ C˜ nk . We have t˜nk ◦(T˜nk )−1 (x) ≥ τ˜nk ⇒
t˜nk (x − τ˜nk e1 ) ≥ τ˜nk t˜nk+1 (x − τ˜nk e1 ) = τ˜nk ⇒ t˜nk+1 ◦ (T˜nk+1 )−1 (x) = τ˜nk , ⇒
where the first statement holds by definition, and the implications follow from (8.14), x − τ˜nk e1 ∈ C˜ nk − τ˜nk e1 and (8.13) respectively. For x ∈ C˜ k ˜ ˜ let t˜n, X˜ , B˜ (x) := τ˜ k ˜ ˜ be the distance the particle x is translated. We n, X , B n, X , B define ˜ ˜ ( X˜ ) := T n, B
m k=0
k k (C˜ n, − τ˜n, e ) = {x − t˜n, X˜ , B˜ (x)e1 : x ∈ X˜ } X˜ , B˜ X˜ , B˜ 1
and
˜ ˜ ( B) ˜ := {(x − t˜ ˜ ˜ (x)e1 )(x − t˜ ˜ ˜ (x )e1 ) : x x ∈ B}. ˜ T n, X n, X , B n, X , B ˜ ˜ = id and T ˜ ˜ = id. Tn is Now if B˜ is a not a finite subset of E n ( X˜ ) we define T n, B n, X then defined by ˜ n (X, B) := (T ˜ ˜ (X ), T ˜ ˜ (B)). ˜ n : X × E → X × E, T T n, B n, X By Lemma 6 we see again that all above objects are measurable with respect to the ˜ n is indeed considered σ -algebras. The following two lemmas are the key to show that T the inverse of Tn . The proofs differ from the proofs of Lemmas 18 and 20 only, in that we have to use (8.3) and (8.16), whenever we want to calculate the translation distance of a point explicitly. In the first lemma we consider X ∈ X, finite B ⊂ E n (X ), tnk , Tnk , Cnk , Pnk and τnk (0 ≤ k ≤ m) as in the construction of Tn (X, B), and we define ˜ := Tn (X, B), P˜nk := Pnk + τnk e1 and C˜ nk := Cnk + τnk e1 , see Fig. 6. ( X˜ , B)
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
117
˜ n of Tn . Fig. 6. Construction of the inverse T
Lemma 31. Let 1 ≤ k ≤ m. For every x˜ ∈ X˜ \ (C˜ n0 ∪ . . . ∪ C˜ nk−1 ) we have ˜ tnk ◦ (Tnk )−1 ( P˜nk ) ≤ tnk ◦ (Tnk )−1 (x). ˜ For all x˜ for which equality occurs we have (Tnk )−1 ( P˜nk ) ≤ (Tnk )−1 (x). For the second lemma we consider X˜ ∈ X, finite B˜ ⊂ E n ( X˜ ), t˜nk , T˜nk , C˜ nk , P˜nk and τ˜nk ˜ n ( X˜ , B), ˜ n ( X˜ , B), ˜ and we define (X, B) := T ˜ (0 ≤ k ≤ m) ˜ as in the construction of T Pnk := P˜nk − τ˜nk e1 and Cnk := C˜ nk − τ˜nk e1 , see Fig. 6. Lemma 32. Let 1 ≤ k ≤ m. For every x ∈ X \ (Cn0 ∪ . . . ∪ Cnk−1 ) we have t˜nk (Pnk ) ≤ t˜nk (x). For all x for which equality occurs we have Pnk ≤ x. Now the following lemma follows exactly as in the proof of Lemma 21. ˜ n ◦ Tn = id Lemma 33. On X × E we have T
˜ n = id. and Tn ◦ T
8.5. Density of the transformed process: Lemma 26. By definition the left-hand side of (7.7) equals 2 1 e−4n I (k), where I (k) = dx ( f ◦ Tn · ϕn )( X¯ x , B), k! n k k≥0
B⊂E n ( X¯ x )
using the shorthand notation X¯ x = {x1 , . . . , xk }∪ X¯ cn . We would like to fix the bond set B before we choose the positions xi of the particles. Thus we introduce bonds between indices of particles instead of bonds between particles. Let Nk := {1, . . . , k}, X¯ k := Nk ∪ X¯ cn and E n ( X¯ k ) := {x1 x2 ∈ E( X¯ k ) : x1 x2 ∩ Nk = ∅}. For B ⊂ E n ( X¯ k ) and x ∈ nI (I ⊂ Nk ) we define Bx to be the bond set constructed from B by replacing the point i ∈ I by xi in every bond of B and by deleting every bond B that contains a point i ∈ Nk \ I . Analogously let X¯ x := {xi : i ∈ I } ∪ X¯ cn be the
118
T. Richthammer
configuration corresponding to the sequence and let ( X¯ , B)x := ( X¯ x , Bx ). Using this notation we obtain I (k) = I (k, B), where I (k, B) := d x ( f ◦ Tn · ϕn )( X¯ , B)x . n k
B⊂E n ( X¯ k )
To compute I (k, B) we need to calculate Tn ( X¯ , B)x , and for this we must identify the i points Pn, among the particles x j . So let m B be the number of different B-clusters of X¯ x ,Bx k c X¯ \ C X¯ k ,B (n ), C X¯ k ,B (η(0)) := C X¯ k ,B (cn ) ∩ Nk and (B) be the set of all mappings η : {1, . . . , m B } → ( X¯ k \ C X¯ k ,B (cn )) such that every η(i) is in a different B-cluster. For η ∈ (B) let
j and Ak,B,η := x ∈ nk : ∀ 1 ≤ j ≤ m B : xη( j) = Pn, X¯ ,B x x
j A˜ k,B,η := x ∈ nk : ∀ 1 ≤ j ≤ m B : xη( j) = P˜n, X¯ ,B , x
x
j P˜n, X¯ ,B x x
where are the pivotal points from the construction of the inverse transformation in Subsect. 8.4. Now we can write I (k, B) = d x 1 Ak,B,η (x)( f ◦ Tn · ϕn )( X¯ , B)x , k η∈(B) n
and we denote the summands in the last term by I (k, B, η). If x ∈ Ak,B,η we can derive a simple expression for Tn ( X¯ , B)x . For x ∈ nk and η ∈ we define a formal i transformation TB,η (x) := (TB,η,x (xi ))1≤i≤k , where η( j)
η( j)
j
t B,η,x := tn,( X¯ ,B)
x η, j−1
η( j)
η( j)
i , TB,η,x := id + t B,η,x e1 and TB,η,x := id + t B,η,x (xη( j) )e1
for 0 ≤ j ≤ m B and i ∈ C X¯ k ,B (η( j)), i = η( j). Here x η, j is defined to be the sub η, j sequence of x corresponding to the index set C X¯ k ,B := i≤ j C X¯ k ,B (η(i)). Clearly, for
i i ∈ C X¯ k ,B (η( j)), TB,η,x doesn’t depend on all components of x, but only on those xl η, j−1
such that l ∈ C X¯ k ,B and additionally on xη( j) if i = η( j). By definition we now have x ∈ Ak,B,η ⇒
Tn ( X¯ , B)x = ( X¯ , B)TB,η (x) j Tn, X¯ ,B x x
=
η( j) TB,η,x
and
for all j ≤ m B .
(8.17)
Furthermore we observe that for all x ∈ (R2 )k we have x ∈ Ak,B,η
⇔
TB,η (x) ∈ A˜ k,B,η .
(8.18)
This can be shown exactly as (6.20) in Sect. 6.6. Let g : (R2 )k → R, g(x) := 1 A˜ k,B,η (x) f ( X¯ x , Bx ). Then (8.17) and (8.18) imply I (k, B, η) =
mB
j=0
i∈C X¯ k ,B (η( j))
η( j) d xi 1 + ∂1 t B,η,x (xη( j) ) g(TB,η (x)),
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
119
where we have also inserted the definition of ϕn (7.6). Now we transform the integrals. i For j = m B to 1 and i ∈ C X¯ k ,B (η( j)) we substitute xi := TB,η,x xi . For i = η( j) i TB,η,x is a translation by a constant vector, so d xi = d xi . For i = η( j) the Lebesgue transformation theorem (6.1) gives η( j) d xη( j) = 1 + ∂1 t B,η,x (x η( j) ) d x η( j) as in Sect. 6.6. Thus I (k, B, η) =
mB
d xi g(x ) =
n k
j=0 i∈C X¯ k ,B (η( j))
d x 1 A˜ k,B,η (x) f ( X¯ x , Bx )
and we are done as the same arguments show that the right-hand side of (7.7) equals 1 −4n 2 e d x 1 A˜ k,B,η (x) f ( X¯ x , Bx ). k k! n k k≥0
B⊂E n ( X¯ ) η∈(B)
Combining the above ideas with the reasoning in Sect. 6.6 also shows that the density function is well defined. 8.6. Key estimates: Lemma 27. For all x ∈ R2 and ϑ ∈ [−1, 1] such that x + se1 ∈ / K for all s ∈ [−ϑ, ϑ], we have U¯ (x + ϑe1 ) + U¯ (x − ϑe1 ) − 2U¯ (x) ≤
sup s∈[−ϑ,ϑ]
∂12 U¯ (x + se1 )ϑ 2 ≤ ψ(x)ϑ 2 ,
by Taylor expansion of U¯ at x using the e1 -smoothness of U¯ and by the ψ-domination of the derivatives. Let (X, B) ∈ G n . W.l.o.g. we may assume that the right-hand side of (7.8) is finite. Introducing ηx,x := x − x , ϑx,x := tn,X,B (x ) − tn,X,B (x) for x, x ∈ E n (X ) and E n,K (X ) := {x x ∈ E n (X ) : x − x ∈ / K } for X ∈ X we have ¯
¯
¯
¯ n,B X ) + H U (Tn,B X ) − 2H U (X ) HUn (T n n = [U¯ (ηx,x + ϑx,x e1 ) + U¯ (ηx,x − ϑx,x e1 ) − 2U¯ (ηx,x )] x x ∈E n,K (X )
≤
x x ∈E
ψ(x − x ) (tn,X,B (x) − tn,X,B (x ))2
=:
f n (X, B).
n,K (X )
In the first step we have used that for x − x ∈ K we have ϑx,x = 0. In the second step we are allowed to apply the above Taylor estimate, as for x − x ∈ / K we have x − x + se1 ∈ / K for all s ∈ [−ϑx,x , ϑx,x ] by (8.8). The arithmetic-quadratic mean inequality gives 2 1 (tn,X,B (x) − τn (|x|)) + (τn (|x|) − τn (|x |)) + (τn (|x |) − tn,X,B (x )) 3 ≤ (tn,X,B (x) − τn (|x|))2 + (τn (|x|) − τn (|x |))2 + (τn (|x |) − tn,X,B (x ))2 ,
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T. Richthammer
and thus
=
f n (X, B) ≤ 6
ψ(x − x ) (τn (|x|) − tn,X,B (x))2
x,x ∈X =
+3
1{|x|≤|x |} ψ(x − x ) (τn (|x|) − τn (|x |))2 .
x,x ∈X
In the first sum on the right-hand side we estimate 2 (τn (|x|) − tn,X,B (x))2 ≤ τn (|x|) − τn (|an,X,B+ (x)|) ≤ 1{|x|≤|x |} 1 X,B+ (τn (|x|) − τn (|x |))2 {x ←→x }
x ∈X
using (8.10). By distinguishing the cases x = x, x and x = x we thus can estimate f n (X, B) by the sum of the two following expressions: = ψ(x − x )1{|x|≤|x |} |τn (|x| − c K ) − τn (|x |)|2 , 2 (n, X ) := 9 x,x ∈X =
3 (n, X, B) := 6
x,x ,x ∈X
1
X,B+
{x ←→x }
ψ(x − x )1{|x|≤|x |}
×|τn (|x| − c K ) − τn (|x |)|2 .
(8.19)
Inserting these sums into the definition of G n in (7.4), we obtain assertion (7.8). Assertion (7.9) can be proved as in Sect. 6.7 using 4 (n, X ) := 2τ 2 5 (n, X, B) :=
1{x∈n }
x∈X = 2 2c f x,x ∈X x ∈X
q(|x| − R)2 , Q(n − R)2
1 K (x − x )1
X,B+
{x ←→x }
1{|x|≤|x |}
×(τn (|x| − c K ) − τn (|x |))2
(8.20)
in the definition (7.4) of G n . 8.7. Set of good configurations: Lemma 28. The functions i (n, X, B) from the definition of the set of good configurations G n in (7.4) have been specified in (8.11), (8.19) and (8.20). Using the shorthand τn (x, x ) := 1{|x|≤|x |} |τn (|x| − c K ) − τn (|x |)|2 , q
we have 1 = 4c2f
x,x ∈X
3 = 6 5 =
=
X,B+ {x ←→x }
1
2 = 9
=
ψ(x − x )τn (x, x ), q
x,x ∈X
ψ(x − x )τn (x, x ), 4 = 2τ 2 q
X,B+ {x ←→x }
x,x ,x ∈X = 2c2f x,x ∈X x ∈X
τn (x, x ), q
1
x∈X
1 K (x − x )1
X,B+
{x ←→x }
q τn (x, x ).
1{x∈n }
q(|x| − R)2 , Q(n − R)2
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
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To estimate these sums we set n¯ := n + c K and R¯ := R + c K and use the assertions (6.22) and (6.24) of Sect. 6.8. As a refinement of (6.25), we note that for x0 , . . . , xm ∈ R2 , m m |xi − xi−1 | + c K ≤ (m + 1)(1 ∨ c K ) 1 ∨ |xi − xi−1 | , |xm | − |x0 | + c K ≤ m i=1
so
i=1
(|xm | − |x0 | + c K )2 ≤ (m + 1)2 (1 ∨ c2K )
m
(1 ∨ |xi − xi−1 |2 ).
(8.21)
i=1
For the estimation of the expectations of i we combine the ideas from Sect. 6.8 and from the proof of Lemma 23. Using (6.22), (6.25), (7.1), (7.2),(8.1) and (8.2) we obtain µ(d X ) πn (d B|X ) i (n, X, B) ≤ ci c(n), where ci are finite constants. By (6.24) we find that µ ⊗ πn (d(X, B))
5 i=1
i (n, X, B) ≤
δ 4
for sufficiently large n. Now µ ⊗ πn (G cn ) ≤ δ follows from the high probability of G n , the Chebyshev inequality and the definition of G n in (7.4). Acknowledgements. I would like to thank H.-O. Georgii for suggesting the problem and many helpful discussions and F. Merkl for helpful comments.
References [D1]
Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theor. Prob. Appl. 13, 197–224 (1968) [D2] Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theor. Prob. Appl. 15, 458–486 (1970) [DS] Dobrushin, R.L., Shlosman, S.B.: Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics. Commun. Math. Phys. 42, 31–40 (1975) [DV] Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. New York, Springer (1988) [FP1] Fröhlich, J., Pfister, C.-E.: On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems. Commun. Math. Phys. 81, 277–298 (1981) [FP2] Fröhlich, J., Pfister, C.-E.: Absence of crystalline ordering in two dimensions. Comm. Math. Phys. 104, 697–700 (1986) [G] Georgii, H.-O.: Gibbs measures and phase transitions, de Gruyter Studies in Mathematics 9. Berlin, Walter de Gruyter & Co. (1988) [ISV] Ioffe, D., Shlosman, S., Velenik, Y.: 2D models of statistical physics with continuous symmetry: the case of singular interactions. Commun. Math. Phys. 226, 433–454 (2002) [K] Kallenberg O.: Random Measures. Berlin & London: Akademie-Verlag and Academic Press, 1986 [LR] Lanford, O.E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 194–215 (1969) [M] Mermin, N.D.: Absence of ordering in certain classical systems. J. Math. Phys. 8, 1061–1064 (1967) [MKM] Matthes, K., Kerstan, J., Mecke, J.: Infinitely divisible point processes. Chichester, John Wiley (1978) [MW] Mermin, N.D., Wagner, H.: Absence of ferromagnetism or antiferromagnetism in one- or twodimensional isotropic Heisenberg models. Phys. Rev. Letters 17, 1133–1136 (1966)
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[P]
Pfister, C.-E.: On the symmetry of the gibbs states in two dimensional lattice systems. Commun. Math. Phys. 79, 181–188 (1981) Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970) Richthammer, T.: Two-dimensional Gibbsian point processes with continuous spin symmetries. Stoch. Proc. Appl. 115, 827–848 (2005) Richthammer, T.: Translation-invariance of two-dimensional Gibbsian systems of particles with internal degrees of freedom. http://arXiv.org/list/math.PR/0603140, (2006) Shlosman, S.: Continuous models with continuous symmetries in two dimensions. In: J. Fritz, J. L. Lebowitz, D. Szasz (eds.), Random fields Vol. 2, Amsterdam: North Holland, pp. 949–966, (1979) Simon, B.: The statistical mechanics of lattice gases. Princeton, NJ, Princeton University Press (1993) Sinai, Y.G.: Theory of phase transitions: Rigorous results. Oxford, Pergamon Press (1982)
[R] [Ri1] [Ri2] [Sh] [Sim] [Sin]
Communicated by H. Spohn
Commun. Math. Phys. 274, 123–140 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0279-2
Communications in
Mathematical Physics
Cluster Expansion for Abstract Polymer Models. New Bounds from an Old Approach Roberto Fernández1 , Aldo Procacci1,2 1 Labo. de Maths Raphael SALEM, UMR 6085 CNRS-Univ. de Rouen, Avenue de l’Université, BP.12,
76801 Saint Etienne du Rouvray, France. E-mail:
[email protected]
2 Dep. Matemática-ICEx, Universidade Federal de Minas Gerais, CP 702, Belo Horizonte MG,
30.161-970, Brazil. E-mail:
[email protected] Received: 12 May 2006 / Accepted: 8 January 2007 Published online: 13 June 2007 – © Springer-Verlag 2007
Abstract: We revisit the classical approach to cluster expansions, based on tree graphs, and establish a new convergence condition that improves those by Kotecký-Preiss and Dobrushin, as we show in some examples. The two ingredients of our approach are: (i) a careful consideration of the Penrose identity for truncated functions, and (ii) the use of iterated transformations to bound tree-graph expansions. 1. Introduction Cluster expansions, originally developed to express thermodynamic potentials as power series in activities, are at the heart of important perturbative arguments in statistical mechanics and other branches of mathematical physics. The classical approach to obtain convergence conditions was based on combinatorial considerations [10, 20], which were greatly simplified through the use of tree-graph bounds [4, 2]. A completely new inductive approach originated in the work of Kotecký and Preiss [8], later refined by Dobrushin [5, 6] and many others [12, 1, 11, 19, 21, 18]. This later approach is mathematically very appealing and, in its original version [8], it even disposes of any reference to power series, becoming, in Dobrushin’s words, a “no-cluster-expansion” approach. The combinatorial approach, however, kept its adepts who reformulated it in a very clear and compact way [13] and showed how it can lead to bounds at least as good as those given by Kotecký and Preiss [15]. In this paper, we revisit the classical combinatorial approach and point out that it can be used, in a rather simple and natural way, to produce improved bounds on the convergence region and the sum of the expansion. Our approach has two ingredients. First, we exploit an identity, due to Oliver Penrose [14], relating the coefficients of the expansion to a family of trees determined by compatibility constraints. (As a matter of fact, we learnt this identity from the nice exposition in [13, Sect. 3].) Successive approximations are obtained by considering larger families of trees that neglect some of the constraints. If only the very basic constraint is kept (links in the tree must relate
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incompatible objects), the Kotecky-Preiss condition emerges. To the next order of precision (branches must end in different objects) Dobrushin’s condition is found. By refining this last constraint (branches’ ends must be mutually compatible rather than just different) we obtain a new convergence condition which leads to improvements in several well-studied cases. In particular, for polymers on a graph—for which compatibility means non-intersection—our criterion yields the original polymer condition due to Gruber and Kunz [7, formula (42)]. This somehow forgotten condition—which is better than the ones usually applied—was obtained in the very paper that introduced the polymer formalism, through the use of Kirkwood-Salzburg equations. Our second ingredient is a strategy to sum tree-graph expansions that is complementary to the classical one. The latter is based on an inductive “defoliation” of tree diagrams, which are summed “from the leaves in” with the help of the convergence condition. Here, we show instead that tree expansions are generated by successive applications of a transformation defined by the convergence condition. Besides leading to an improved convergence criterion, this point of view presents, in our opinion, several advantageous features. On the conceptual side, it shows a direct link between the convergence of tree expansions and inequalities involving the functions found in Kotecký-Preiss and Dobrushin (and our) conditions: The inequalities ensure that the iterative procedure leads to a finite expansion. From a more practical point of view, it is easy to see that finite iterations of the transformations yield progressively sharper bounds on the tree expansions. Thus, our approach produces, for each convergence condition, an associated sequence of upper bounds for the pinned free energy. In particular the majorizing tree expansions are shown to be fix points of the corresponding transformations. All this information is absent in previous treatments. Finally, regarding future work, our approach leaves ample room for extensions and improvements. To emphasize this fact, we state a general result (Proposition 7) showing how bounds on truncated functions translate into convergence criteria and associated results. To establish our new criterion we used the Penrose identity in the most natural and immediate way. Improvements should come from the incorporation of additional tree conditions contained in the Penrose identity or, for specific models, through a more accurate description of the compatibility constraints. Also, as emphasized in [18] and reviewed in Sect. 4.1, there is a generalized Penrose identity which allows the use of trees other than Penrose’s to characterize truncated functions. These alternative choices may turn out to be of interest in particular settings. Penrose identity, in its original or generalized form—and thus our approach—is valid only for hard-core interactions (incompatibilities). The extension of our treatment to polymer systems subjected to softer interactions is another direction for further research. 2. Set Up and Previous Results We adopt the following abstract polymer setting. The starting point is an unoriented graph G = (P, E)—the interaction graph—on a countable vertex set. The vertices γ ∈ P are called polymers for historical reasons [7]. The name is misleading; Dobrushin [6] proposes to call them animals, but the traditional name holds on. The edge set corresponds to an incompatibility relation: Two polymers γ , γ are incompatible if {γ , γ } ∈ E, in which case we write γ γ . Otherwise they are compatible and we write γ ∼ γ . (Unfortunately, this notation—well established within the mathematicalphysics community—is the opposite to that adopted in graph theory.) The set of edges is arbitrary, except for the assumption that it contains all pairs of the form {γ , γ }, that is,
Cluster Expansion for Abstract Polymer Models. New Bounds
125
every polymer is assumed to be incompatible with itself. In particular vertices can be of infinite degree (each polymer can be incompatible with infinitely many other polymers). This happens, for instance, for graphs associated to gases of low-temperature contours or “defects”. The physical information of each polymer model is given by the incompatibility relation and a family of activities z = {z γ }γ ∈P ∈ CP . For each finite family ⊂ P, these ingredients define complex weights on the set of subsets of : W {γ1 , γ2 , . . . , γn } =
1 z γ1 z γ2 · · · z γn 11{γ j ∼γk } (z)
(2.1)
j
for n ≥ 1 and W (∅) = 1/ , where (z) = 1 +
1 n! n≥1
z γ1 z γ2 . . . z γn
(γ1 ,...,γn )∈n
11{γ j ∼γk } .
(2.2)
j
In physical terms, the measure (2.1) corresponds to the grand-canonical ensemble of a polymer gas with activities z and hard-core interaction defined by the incompatibility relation. The abstract formalism makes it equivalent to a lattice gas on the graph G with self- and nearest-neighbor hard-core repulsion. The normalization constant (2.2) is the grand-canonical partition function in the “volume” . Cluster expansions allow the control of the measures (2.1) uniformly in and absolutely in the activities. [Thus, the control extends to the unphysical region of non-positive (complex) activities, where the expressions on the right-hand side of (2.1) do not define probability measures.] The basic cluster expansion is the formal power series (“F”) of the logarithm of the partition function, which takes the form (Mayer expansion, see e.g. [17]) F
log (z) =
∞ 1 n! n=1
with
φ T (γ1 , . . . , γn ) =
⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(γ1 ,...,γn
φ T (γ1 , . . . , γn ) z γ1 . . . z γn
(2.3)
)⊂n
n=1
(−1)|E(G)| n ≥ 2 , G{γ1 ,...,γn } connected
G⊂G{γ ,...,γn } 1 G conn. spann.
, (2.4)
n ≥ 2 , G{γ1 ,...,γn } not connected
where G{γ 1 ,...,γn } is the graph of vertices {1, . . . , n} and edges {i. j} : γi γ j , 0 ≤ i, j ≤ n and G ranges over all its connected spanning subgraphs; here E(G) is the edge set of G. The functions φ T are the truncated functions of order n (also called Ursell functions). The families {γ1 , . . . , γn } such that G{γ1 ,...,γn } is connected are the clusters. A telescoping argument shows that the properties of the measures (2.1) are determined by the one-polymer ratios (“pinned” expansions) log
0
∞
F 1 (z) = \{γ0 } n!
n=1
(γ1 ,...,γn )⊂n ∃i: γi =γ0
φ T (γ1 , . . . , γn ) z γ1 . . . z γn
(2.5)
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R. Fernández, A. Procacci
for each γ0 ∈ . A more efficient alternative is to consider instead the formal series ∞ ∂
1 F log (z) = 1 + ∂z γ0 n! n=1
(γ1 ,...,γn
φ T (γ0 , γ1 , . . . , γn ) z γ1 . . . z γn . (2.6)
)⊂n
This leads to the (infinite volume) formal power series γ0 (ρ) := 1 +
∞ 1 n! n=1
T φ (γ0 , γ1 , . . . , γn ) ργ1 . . . ργn ,
(γ1 ,...,γn
(2.7)
)∈P n
for ρ ∈ [0, ∞)P —in which φ T is replaced by φ T —and its finite-volume versions γ0 obtained by restricting the sum to polymers in . The finiteness of the positive-term series (2.7) for a certain ρ implies the absolute convergence of (2.3), (2.5) and (2.6), uniformly in for |z| ≤ ρ. This leads to the control of the measures (2.1) and their → P limit [6]. [Throughout this paper, operations and relations involving boldface symbols should be understood componentwisely, for instance ρ ≤ µ indicates ργ ≤ µγ , |z| = {z γ }γ ∈P , etc.] γ ∈ P ; −z = {−z γ }γ ∈P ; ρ = {ργ γ }γ ∈P ; The truncated functions satisfy the alternating-sign property (2.8) φ T (γ0 , γ1 , . . . , γn ) = (−1)n φ T (γ0 , γ1 , . . . , γn ) . (This is a well known result, that appears, for instance, in [17, Theorem 4.5.3] where it is attributed to Groeneveld [9]. Other proofs can be found in [11, 18] and in Proposition 5 below.) Thus, (2.6) and the -restriction of (2.7) are related in the form γ0 (ρ) = F
∂ log (−ρ) ∂z γ0
(ρ ∈ [0, ∞)P ).
(2.9)
In the sequel we focus on the convergence of the series (2.7) for positive activities. Its convergence allows the removal of the label “F” in all precedent identities, and it implies the inequalities ∂
∂ log log (− |z|) = (z) ≤ γ0 (|z|) ≤ γ0 (|z|), (2.10) ∂z ∂z γ0 γ0 and | log
\{γ0 }
(z) ≤ − log
\{γ0 }
(− |z|) ≤ z γ0 γ0 (|z|) ≤ z γ0 γ0 (|z|). (2.11)
A rather detailed study of different properties of these objects can be found in [18]. In the present general setting, two benchmark convergence conditions were published in 1986 [8] and 1996 [5]. For comparison purposes it is useful to write them in the following form. Suppose that for some ρ ∈ [0, ∞)P there exists µ ∈ [0, ∞)P such that
µγ ≤ µγ0 (Kotecký-Preiss) (2.12) ργ0 exp γ γ0
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or ργ0
1 + µγ ≤ µγ0
(Dobrushin)
(2.13)
γ γ0
for each γ0 ∈ P. [Please note that the sum and product over γ here include γ0 , which is always incompatible with itself.] Then the power series (2.7) converges for such ρ and, moreover, ργ0 γ0 (ρ) ≤ µγ0
(2.14)
for each γ0 ∈ P. The reader may be more familiar with the following forms of these conditions. The change of variables µγ = ργ eaγ shows that condition (2.12) is equivalent to the existence of a ∈ [0, ∞)P such that ργ eaγ ≤ aγ0 (Kotecký-Preiss) (2.15) γ :γ γ0
for each γ0 ∈ P, and (2.14) becomes ≤ e a . The substitution µγ = eαγ − 1, on the other hand, makes (2.13) equivalent to the existence of a ∈ [0, ∞)P such that ργ0 ≤ eαγ0 − 1 exp − (Dobrushin) (2.16) αγ γ :γ γ0
for each γ0 ∈ P. The inequality
1 + µγ ≤ exp µγ
γ γ0
(2.17)
γ γ0
shows that the Dobrushin condition is an improvement over Kotecký-Preiss’. Nevertheless, the latter is particularly suited for some applications (see, for instance, [19]) and, furthermore, can be extended to polymers with soft self- and two-body interactions. By contrast, the Dobrushin condition can be extended to systems with soft two-body interaction [19] but requires hard-core self-interaction. Looking to inequality (2.17) we see that the difference between both criteria lies in factors µγ at powers higher than two, which are absent in the left-hand-side. A quick illustration of the consequences of this fact is provided by polymers subjected only to self-exclusion (each polymer is compatible with everybody else, except itself). In this case = γ ∈ (1 + z γ ) and γ0 (ρ) =
n≥0
F
ργn0 =
1 . 1 − ργ0
(2.18)
The Kotecký-Preiss condition requires the existence of µ > 0 such that ργ0 eµγ0 ≤ µγ0 for each γo ∈ P, and this yields a radius of convergence for γ0 equal to supµγ µγ0 e−µγ0 = e−1 . The Dobrushin condition, on the other hand, provides the 0 sharp estimate supµγ µγ0 /(1 + µγ0 ) = 1. 0
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3. Results 3.1. New convergence criteria. Our new criterion involves the grand-canonical partition functions Nγ∗ , associated to the polymer families Nγ∗0 = {γ ∈ P : γ γ0 }, γ0 ∈ P 0 (G-neighborhood of γ0 , including γ0 ). These functions, defined in (2.2), can also be written in the form 1 Nγ∗ (µ) = 1 + µγ1 µγ2 . . . µγn (3.1) 0 n! (γ ,...,γ )∈P n n≥1
n 1 γ0 γi , γi ∼γ j , 1≤i, j≤n
because compatible polymers are different. Here is the practitioner’s version of our criterion (a more detailed statement is given in Theorem 4 below). Theorem 1. Let ρ ∈ [0, ∞)P . If there exists a µ ∈ [0, ∞)P such that ργ0 Nγ∗ (µ) ≤ µγ0 , ∀γ0 ∈ P, 0
(3.2)
then the series γ0 (ρ) [defined in (2.7)] converges for such ρ and satisfies ργ0 γ0 (ρ) ≤ µγ0 . The inequality Nγ∗ (µ) ≤ 0
1 + µγ
(3.3)
γ γ0
shows that condition (3.2) is an improvement over Dobrushin’s condition—which in turn is an improvement over Kotecký-Preiss’ condition. The improvement comes from the fact that only monomials involving mutually compatible polymers are allowed in the left-hand side. Such improvement comes, therefore, from two sources: (I1) (I2)
In Nγ∗ there are no monomials involving triangle diagrams in G, namely pairs 0 of neighbors of γ0 that are themselves neighbors. In Nγ∗ , the only monomial containing µγ0 is µγ0 itself, because γ0 is incompa0 tible with all other polymers in Nγ∗0 .
Improvement (I2) is present whichever the graph G, and makes inequality (3.3) strict except for the non-interacting example discussed circa (2.18). The terms corresponding to (I1) and (I2) can be neatly separated by writing Nγ∗ (ρ) = ργ0 + Nγ0 (ρ), 0
(3.4)
where Nγ0 = Nγ∗0 \ {γ0 } (∅ := 1). Using a bound similar to (3.3) but for Nγ0 we obtain another criterion—halfway between ours and Dobrushin’s—which may be useful in some settings. Corollary 2. Let ρ ∈ [0, ∞)P . If there exists a µ ∈ [0, ∞)P such that 1 + µγ ≤ µγ0 , (improved Dobrushin) ργ0 µγ0 +
(3.5)
γ γ0 γ =γ0
for all γ0 ∈ P, then the series γ0 (ρ) converges for such ρ and satisfies ργ0 γ0 (ρ) ≤ µγ0 .
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Our condition (3.2) coincides with (3.5) for triangle-free graphs G (ex. trees, Zd ), and it is maximally better for complete (“triangle-full”) graphs. This and other examples will be analyzed below. Summing up, available convergence conditions are of the form ργ0 ϕγ0 (µ) ≤ µγ0 with
ϕγ0 (µ) =
⎧ ⎪ exp µγ ⎪ γ γ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1+µ γ γ0
(Kotecký-Preiss)
(Dobrushin) γ . γ γ0 1 + µγ (improved Dobrushin)
⎪ ⎪ µγ0 + ⎪ ⎪ γ =γ0 ⎪ ⎪ ⎪ ⎩ Nγ∗ (µ) 0
(3.6)
(3.7)
(ours)
Each condition is strictly weaker than the preceding one except for the facts that the improved Dobrushin condition coincides with Dobrushin’s if the polymers are non-interacting (only self-excluding) and with our condition if G does not include any triangle diagram. The corresponding criteria yield information on two issues: (i) regions of convergence, and (ii) upper bounds on each γ0 . Regarding the first issue, it is known that the region of absolute convergence of cluster expansions has the properties of being a “down-region”—convergence for ρ entails ≤ ρ—and log-convex. The latter means that if the series converges convergence for ρ then it converges for ρ λ ρ 1−λ for 0 ≤ λ ≤ 1 [18]. It is reassuring to verify for ρ and ρ that these properties also hold for the regions of validity of conditions (3.6)/(3.7). Indeed, the “down” character is obvious, and the log-convexity property is a consequence of the following proposition: Proposition 3. Suppose 0 ≤ λ ≤ 1 and let us denote RCD = (ρ, µ) ∈ [0, ∞)∞ × [0, ∞)∞ condition CD is satisfied ,
(3.8)
where “CD” stand for each of the conditions in (3.6)/ (3.7). Then, ) ∈ RCD =⇒ 1−λ , µλ µ 1−λ ∈ RCD . (ρ, µ) , ( ρ, µ ρλ ρ
(3.9)
Proof. Given the form (3.6) of the conditions, we see that it is enough to prove that 1−λ ) ϕγ0 (µ)λ ϕγ0 ( µ)1−λ ≥ ϕγ0 (µλ µ
(3.10)
for each of the functions ϕγ0 in (3.7). For the last three functions this is a consequence of the Hölder inequality in the form n n n λ 1−λ ai bi ≥ aiλ bi1−λ i=1
i=1
(3.11)
i=1
(ai , bi ≥ 0, i = 1, . . . , n). For the Kotecký-Preiss function, (3.10) is a consequence of the inequality λa + (1 − λ)b ≥ a λ b1−λ , valid for a, b ≥ 0 (this is an elementary inequality, see [16, p. 112]).
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Our results on the second issue (upper bound on ) are contained in the following strengthening of Theorem 1. Its formulation relies on the map iterates used in Sect. 4.2 to sum tree-graph expansions. For each fixed ρ ∈ [0, ∞)P let us consider the map Tρ : [0, ∞)P −→ [0, ∞]P defined by Tρ (µ) := ρ ϕ(µ),
(3.12)
where ϕ is any of the functions defined in (3.7). Denote Tρn = Tρ (Tρn−1 ) the successive compositions of Tρ with itself. Theorem 4. Let ρ ∈ [0, ∞)P be fixed and let Tρ be a map of the form (3.12)/ (3.7). Assume there exists µ ∈ [0, ∞)P satisfying (3.6), that is, Tρ (µ) ≤ µ.
(3.13)
Then: (i) There exists ρ ∗ ∈ [0, ∞)P such that Tρn (ρ) ρ ∗ and Tρ (ρ ∗ ) = ρ ∗ . (ii) For each n ∈ N, ρ ≤ ρ ∗ ≤ Tρn+1 (µ) ≤ Tρn (µ) ≤ µ.
(3.14)
The deepest statement in this theorem is the first inequality in (3.14). The rest of the theorem follows from the fact that for all choices (3.7)) of ϕ the map Tρ is monotonicitypreserving and satisfies ρ ≤ Tρ (ρ) ≤ Tρ (µ) ≤ µ. 3.2. Comparison with previous criteria. To test our criterion we compare the estimates of the regions of convergence provided by the criteria (3.6)–(3.7) for two families of benchmark examples. Polymer graphs with bounded maximum degree. These are examples where G has maximum degree < ∞. We shall suppose that all polymers have equal activity ργ ≡ ρ for all γ ∈ G, and therefore we search for equally constant functions µγ ≡ µ. The preceding criteria take the form ρ ≤ µ/ϕ(µ) for appropriate functions ϕ, and the maximization of the right-hand side with respect to µ yields the best lower bounds of the radius of convergence of (2.7) [and hence of (2.3)]. In Table 1 we summarize both convergence criteria and best estimates on the convergence radii obtained with Kotecký-Preiss, Dobrushin and improved Dobrushin conditions. The only feature of the graph G relevant for these criteria is the maximal degree of the vertices. Therefore they provide the sharpest results for graphs which lack any other feature and whose vertices have all degree . These are the regular trees with branching rate − 1. This fact—trees supply a worst-case condition that can be used whenever we ignore, or decide to ignore, any topological information on the graph—has been emphasized in [19] (see, also, Remark 6). For regular trees, the weak Dobrushin condition coincides with ours, and there is a further, optimal condition, due to Scott and Sokal [18], which we have included in the last line of the table. This condition is derived through a sequence of volume-dependent Dobrushin conditions. It would be interesting to see whether a similar strategy could be developed within our approach. In Table 2 we show the improved results obtained from the application of our criteria to some popular examples. The values of R in the first two lines are to be compared
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Table 1. Convergence criteria and lower bounds (R ) on the radius of convergence when G is a graph with maximal degree . A star indicates that the value is exact for the ( − 1)-regular tree Condition
Criterion
Kotecký-Preiss
ρ ≤ µ e−(+1)µ
Dobrushin
ρ≤
µ (1 + µ)+1
Improved Dobrushin µ ρ≤ µ + (1 + µ)
=(3.2) for (−1)-reg. tree
Scott-Sokal
[18, Theorem 5.6]
R
R6
1 ( + 1) e ⎧ ⎨ ≥1 +1 ⎩ ( + 1) 1 (∗) =0 ⎧ ⎪ ⎨
1+ ( − 1)−1 ⎪ ⎩ ( + 1)−1 (∗) ( − 1)(−1) (∗)
0.0525 0.0566
−1 ≥2
0.0628
= 0, 1 0.067
Table 2. Convergence criteria and lower bounds (R) on the radius of convergence obtained with condition (3.2) for some graphs G of finite degree. A star indicates an exact value Model
Criterion
R
ρ≤
0.0769
µ 1 + 7µ + 9µ2 µ ρ≤ 1 + 7µ + 8µ2 + 2µ3 µ ρ≤ 1 + ( + 1)µ
Domino in Z2 Triangular lattice (+1)-complete graph
4R 3 + 8R 2 = 1, R ≈ 0, 078 ( + 1)−1 (∗)
with the values for R6 in Table 1, and that of the complete graph with the values of R . The source of these improvements is, of course, the sensitivity of our new criterium to triangle diagrams. In particular, our criterion gives the exact value of the radius of convergence for the complete graph, for which = [1 − ( + 1)µ]−1 . Polymers on a graph. This is the general example of cluster expansions for graphs with vertices of infinite degree. Applications include contour ensembles of low-temperature phases, geometrical objects of high-temperature expansions, random sets of the FortuinKasteleyn representation of the Potts model, …The general setup for these models is a polymer family formed by the finite parts of a given set V with incompatibility defined by overlapping. (Usually, V is formed by the vertices of a graph with respect to which polymers form connected sets.) For these systems it is useful and traditional to pass to exponential weight functions a(γ ) defined by µγ = ργ ea(γ ) . Condition (3.2) becomes
1+
n
n≥1
{γ1 ,...,γn }⊂P γ0 ∩γi =∅ , γi ∩γ j =∅ , 1≤i, j≤n
i=1
ργi ea(γi ) ≤ ea(γ0 ) .
(3.15)
From the constraint in the sum we only keep the fact that each of the polymers γ1 , . . . , γn must intersect different points in γ0 (otherwise they would overlap). This implies: (i) n ≤ |γ0 |, and (ii) there are n different points in γ0 touched by γ1 ∪ · · · ∪ γn . These points
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Table 3. Convergence conditions for general polymer models. Our condition (3.17) with a(γ ) = a |γ | coincides with that by Gruber and Kunz Kotecký-Preiss sup ργ ea|γ | ≤ a x
Dobrushin
sup x
γ ∈P :γ x
can be chosen in 1+
|γ0 | n
n
γ ∈P :γ x
Gruber-Kunz
sup x
ργ ea|γ | ≤ ea − 1
γ ∈P :γ x
ways. Hence, the left-hand side of (3.15) is less than or equal to
|γ0 | |γ0 | n=1
1 + ργ ea|γ | ≤ ea
sup x∈γ0
ργ ea(γ )
n
=
1 + sup
γ ∈P γ x
x∈γ0
|γ0 | ,
(3.16)
γ ∈P γ x
which leads us to the following sufficient condition for (3.15): sup ργ ea(γ ) ≤ ea(γ0 )/|γ0 | − 1. x∈γ0
ργ ea(γ )
(3.17)
γ ∈P γ x
This condition entails the finiteness of : γ0 (ρ) ≤ ea(γ0 ) .
(3.18)
In practice, the function a(γ ) is chosen to be of the form a(γ ) = a |γ |, with a a positive constant. This choice, which in many cases is the expected optimal asymptotic behavior of a(γ ) for large polymers, simplifies the procedure reducing it to the determination of the single constant a. Our emphasis in a general dependence is not just mathematical finesse. As dominant contributions come from the smallest polymers, a dependence of a(γ ) dealing more accurately with them would improve precision. Also, the criteria are usually presented in the slightly weaker form obtained by replacing the supremum over x ∈ γ0 by a supremum over x ∈ V. In this form, a condition like (3.17) is, in fact, present in the seminal paper by Gruber and Kunz [7] [formula (42) with normalization φ(x) = 1 and parametrization ξ0 = ea ]. Table 3 lists the different conditions with the preceding usual choices. 4. Proofs The argument has two distinct parts. First, we use the Penrose tree identity for the truncated functions to turn (2.7) into a sum over trees—a tree-graph expansion. In the second part, we control this expansion through a natural iterative procedure defined by the functions (3.7). 4.1. Partitionability and the Penrose identity. Formula (2.4) involves a huge number of cancellations. Penrose [14] realized that they can be optimally handled through what is now known as the property of partitionability of the family of connected spanning subgraphs. While his original argument involved a particular partition scheme, it works equally well for any other choice, as emphasized in [18]. For the sake of completeness, and due to its potential use for extensions and alternative versions of our criterion, we start by reproducing this simple but deep argument. Our exposition is based on [18, Sect. 2.2].
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Let us consider a finite graph G = (U, E) and denote CG the set of all connected spanning subgraphs of G and TG the family of trees belonging to CG . Further, we consider CG partial ordered by bond inclusion: G≤G
⇐⇒
E(G) ⊂ E(G).
(4.1)
let us denote [G, G] the set of G ∈ CG such that G ≤ G ≤ G. Let us call a If G ≤ G, partition scheme for the family CG any map R : TG → CG : τ → R(τ ) such that (i) E R(τ ) ⊃ E(τ ), and (ii) CG is the disjoint union of the sets [τ, R(τ )], τ ∈ TG . A number of such partition schemes are by now available (see references in [18, Sect. 2.2]). The one proposed by Penrose is constructed in the following way: Let us fix an enumeration v0 , v1 , . . . , vn for the vertices of G, and for each τ ∈ TG (thought of as a tree rooted in v0 ). For any vertex vi of τ , let d(i) be the tree distance of the vertex vi to v0 and let vi be de precedessor of vi , i.e. d(i ) = d(i) − 1 and {vi , vi } ∈ E(τ ). Penrose’s scheme associates to τ the graph RPen (τ ) formed by adding (only once) to τ all edges {vi , v j } ∈ E \ E(τ ) such that either: (p1) (p2)
d(i) = d( j) (edges between vertices of the same generation), or d( j) = d(i) − 1 and i < j (edges between vertices one generation away).
For a partition scheme R, let us denote T R := τ ∈ TG R(τ ) = τ
(4.2)
(set of R-trees). In particular, T RPen is the set of Penrose trees. The following is the generalized version of Penrose identity. Proposition 5.
(−1)|E(G)| = (−1)|V|−1 T R
(4.3)
G∈CG
for any partition scheme R. Proof. For any numbers xe , e ∈ E, we have xe = xe G∈CG e∈E(G)
τ ∈TG e∈E(τ )
=
τ ∈TG e∈E(τ )
xe
F ⊂E(R(τ ))\E(τ ) e∈F
xe
(4.4)
(1 + xe ).
e∈E(R(τ ))\E(τ )
The first equality is due to property (ii) of partition schemes. If xe = −1, the last factor killsthe contributions of any tree τ with E(R(τ )) \ E(τ ) = ∅. Furthermore, for any tree E(τ ) = |V| − 1. We see that the hard-core condition is crucial for the identity. For polymer models with soft repulsion, only |1 + xe | ≤ 1 is guaranteed, and this leads to the inequality |xe | ≤ |TG |. (4.5) xe ≤ G∈CG e∈E(G)
τ ∈TG e∈E(τ )
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This much weaker inequality is the one used in traditional treatments of the tree expansion [4, 2]. The previous proposition applied to the Penrose scheme implies T 11{τ ∈TPen (γ0 ,γ1 ,...,γn )} , (4.6) φ (γ0 , γ1 , . . . , γn ) = τ ∈Tn0
where Tn0 is the set of (labeled) trees with vertices {0, 1, . . . , n} rooted in 0, and TPen (γ0 , γ1 , . . . , γn ) denotes the Penrose trees on the graph G{γ0 ,γ1 ,...,γn } (with the canonical enumeration of vertices). Similar formulas are valid replacing “Pen” by any partition scheme R. Remark 6. As the number of Penrose trees grows with the disappearance of triangle diagrams, the value of (resp. the region of convergence of the cluster expansion) for a given graph G is bounded above by (resp. contains) that of a tree where each vertex has a degree larger than or equal to that at G. Furthermore, the latter is bounded above (resp. contains) that of a homogeneous tree with branching rates equal to the maximal rate. 4.2. Trees and convergence. Replacing (4.6) in (2.7) we obtain a sum in terms of trees. Traditionally, such expansions have been inductively summed a la Cammarota [4], namely “from the leaves in”. Conditions of the type (3.6) guarantee the reproducibility of the inductive hypothesis. Here we present a complementary approach, based on generating the expansion through repeated application of a nonlinear map Tρ . Conditions (3.6) prevent the successive partial sums to diverge. The end product of this section is the following proposition. Each τ ∈ Tn0 is uniquely defined by the branching factor si of each vertex i and the labels i 1 , . . . , i si of its descendants. Proposition 7. Let G = (P, E) be a polymer system and assume there exist functions cn : P n+1 → [0, ∞), for n ∈ N, invariant under permutations of the last n arguments such that n T csi (γi , γi1 , . . . , γisi ). (4.7) φ (γ0 , γ1 , . . . , γn ) ≤ τ ∈Tn0 i=0
Consider the function ϕ : [0, ∞)P → [0, ∞]P defined by ϕγ0 (µ) = 1 +
1 n! n≥1
cn (γ0 , γ1 , . . . , γn ) µγ1 . . . µγn
(4.8)
(γ1 ,...,γn )∈P n
for each γ0 ∈ P. Assume that, for a given ρ ∈ [0, ∞)P there exists µ ∈ [0, ∞)P such that ργ0 ϕγ0 (µ) ≤ µγ0
(4.9)
for each γ0 ∈ P. Then, (a) The cluster expansion (2.3) for the system G converges absolutely and uniformly in and in the activities z with |z| ≤ ρ.
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(b) Furthermore, if Tρ = ρ ϕ is the map defined as in (3.12) but with ϕ given by (4.8), then (i) There exist ρ ∗ , Tρ∞ (µ) ∈ [0, ∞)P such that Tρn (ρ) ρ ∗ , Tρn (µ) Tρ∞ (µ). n→∞
(4.10)
n→∞
(ii) Tρ (ρ ∗ ) = ρ ∗ . (iii) For each n ∈ N, ρ ≤ ρ ∗ ≤ Tρ∞ (µ) ≤ Tρn+1 (µ) ≤ Tρn (µ) ≤ µ.
(4.11)
The proof requires only elementary manipulations which, however, require some previous considerations to introduce the necessary notation. It is useful to visualize the maps (3.12) in diagrammatic form •1 •2 Tρ (µ) = ◦ + ◦ ◦ .. + · · · . γ0 γ0 γ0@ . γ0 @•n The sum is over all single-generation rooted trees. In each tree, open circles represents a factor ρ, bullets a factor µ and vertices other than the root must be summed over all possible polymers γ . At each vertex with n descendants, a “vertex function” cn /n! acts, having as arguments the ordered n + 1-tuple formed by the polymer at the vertex, the polymer at the top offspring, the polymer at the next offspring from the top,…, in that order. With this representation, the iteration Tρ 2 (µ) corresponds to replacing each of the bullets by each one of the diagrams of the expansion for Tρ . This leads to rooted trees of up to two generations, with open circles at first-generation vertices and bullets at second-generation ones. In particular, all single-generation trees have only open circles. Notice that the two drawings of Fig. 1 appear in two different terms of the expansion, and hence should be counted as different diagrams. More generally, the kth iteration of Tρ involves all possible rooted tree diagrams, counting as different those obtained by permutations of non-identical branches. We shall call these diagrams planar rooted trees. In each term of the expansion, vertices of generation k are occupied by bullets and all the others by open circles. Formally, the definition of planar rooted trees is determined by a labeling choice which we fix as follows. There is a special vertex, labeled 0 (the root), placed, say, at the leftmost position of the drawing. From there s0 branches emerge ending at the
•1 •1 + ◦ + ··· + γ0HH H•2
(a)
(b)
Fig. 1. Planar rooted trees defined by (a) s0 = s(0,1) = 2 and s(0,2) = s(0,1,1) = s(0,1,2) = 0; (b) s0 = s(0,2) = 2 and s(0,1) = s(0,2,1) = s(0,2,2) = 0
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first-generation vertices. The value s0 = 0 describes the trivial tree with the root as its only vertex. Otherwise these vertices are drawn along a vertical line at the right of the root and labeled (0, 1), . . . (0, s0 ) with the second subscript increasing from the top to the bottom of the line. The construction continues rightwards: Each of the vertices (0, i), gives rise to a family of second-generation vertices (0, i, 1), . . . (0, i, s(0,i) ) and so on. The vertex v is of generation if its label has the form v = (0, i 1 , . . . , i ) with 1 ≤ i j ≤ s(0,i1 ,...,i j−1 ) , 1 ≤ j ≤ (i 0 ≡ 0). The sequence of such branching factors s(0,i1 ,...,i ) ∈ N ∪ {0} define the planar rooted tree. Let us denote T0,k the set of trees with maximal generation number k; T0,0 being the trivial tree. Figure 1 shows two different trees of T0,2 . We enumerate the vertices following the generation number and the “top to bottom” order in case of equal generation. [This amounts to declaring (0, i 1 , . . . , i ) < (0, i 1 , . . . , i ) if < and using lexicographic order if = .] A straightforward inductive argument shows that
Tρk (µ)
γ0
k−1
= ργ0
=0
Rγ() (ρ) + Rγ(k) (ρ, µ) 0 0
(4.12)
with Rγ() (ρ) = 0
|Vt | 1 cs (γv , γ(vi ,1) , . . . , γ(vi ,svi ) ) svi ! vi i
t∈T0, (γv1 ,...,γv|V | )∈P |Vt | i=0 t
×ργ(vi ,1) . . . ργ(vi ,sv
(4.13)
i)
(k)
and Rγ0 (ρ, µ) has a similar expression but with the activities of the vertex of the kth generation weighted by µ. In this expression Vt denotes the set of non-root vertices of t and we agree that c0 (γv ) ≡ 1 and ∅ ≡ 1. We are interested in the k → ∞ limit of (4.12). Let us denote T0 = ∪ T0, . These considerations make almost immediate the proof of the following lemma which, together with a simple combinatorial argument, proves Proposition 7. Proposition 8. For some fixed ρ ∈ [0, ∞)P , let Tρ be a map of the form (3.12)/ (4.8) and assume there exists µ ∈ [0, ∞)P such that Tρ (µ) ≤ µ. Then Tρn (ρ) ρ ∗ ∈ [0, ∞)P as n → ∞, with ργ∗0
:= ργ0
|Vt | 1 cs (γv , γ(vi ,1) , . . . , γ(vi ,svi ) ) svi ! vi i
t∈T0 (γv1 ,...,γv|V | )∈P |Vt | i=0 t
×ργ(vi ,1) . . . ργ(vi ,sv
i)
(4.14)
for each γ0 ∈ P. Furthermore, (i) Tρ (ρ ∗ ) = ρ ∗ . (ii) There exists Tρ∞ (µ) ∈ [0, ∞)P such that Tρn (µ) Tρ∞ (µ) as n → ∞. (iii) For all , n ∈ N, µ ≥ Tρn (µ) ≥ Tρn+1 (µ) ≥ Tρ∞ (µ) ≥ ρ ∗ ≥ Tρ+1 (ρ) ≥ Tρ (ρ) ≥ ρ. (4.15)
Cluster Expansion for Abstract Polymer Models. New Bounds
137
Proof. The map Tρ is obviously monotinicity preserving in the coordinatewise partial order of [0, ∞]P and µ ≥ Tρ (µ) ≥ Tρ (ρ) ≥ ρ.
(4.16)
[The first inequality is by hypothesis, the second one by monotonicity and the third one is immediate from the definition of Tρ .] Therefore, by induction, µ ≥ Tρn (µ) ≥ Tρn+1 (µ) ≥ Tρn++1 (µ) ≥ Tρn++1 (ρ) ≥ Tρ+1 (ρ) ≥ Tρ (ρ) ≥ ρ (4.17) for all , n ∈ N. This shows that, for each γ ∈ G, the series Tρ (ρ) is increasing and bounded above while Tρn (µ) n is decreasing and bounded below. Thus, the limits ρ ∗ := sup Tρ (ρ) and Tρ∞ (µ) := inf Tρ (µ) exist and are finite and, by letting alternatingly → ∞ and n → ∞ in (4.11), we obtain the inequalities (4.15). The fact that Tρ∞ (ρ) = ρ ∗ is immediate from expression (4.12). Finally, ρ ∗ = lim Tρ Tρn (ρ) = Tρ lim Tρn (ρ) = Tρ (ρ ∗ ), (4.18) n→∞
n→∞
where the middle identity is by monotone convergence. = ργ∗0 + limk→∞ Rγ(k) We notice that Tρ∞ (µ) 0 (µ). The last limit is in fact an γ0 infimum because R(k) (ρ, µ) ≤ R(k−1) ρ, Tρ (µ) ≤ R(k−1) (ρ, µ). Proof of Proposition 7. The sum in (4.14) can be written in the form ργ∗0 = ργ0 Wγ0 (t).
(4.19)
t∈T0
The symmetry of the vertex functions cn (γ0 , γ1 , . . . , gn ) implies that the weights W (t) are invariant under permutations of the branches of the planar tree t. That is, they depend only on the underlying labeled tree τ obtained by neglecting the order of the vertices. Formally, if Tn0 is the set of rooted trees on {0, 1, . . . , n} (=labelled trees of n + 1 vertices), there is a map Tn0 τ → tτ ∈ T0n , where tτ is the planar tree obtained by drawing branches starting on the root according to the order given by the labels of the first offspring, and continuing in this way for branches within branches. This map is many-to-one, in fact, the cardinality of the preimage of a tree t (=number of ways of labelling the |Vt | non-root vertices of a planar rooted tree with |Vt | distinct labels consistently with the rule “from high to low”) is |Vt |! βt = |V | t i=0 svi !
(4.20)
(see e.g. Theorem 145B in [3]). Thus, we can replace the sum in (4.19) by a sum over trees τ on the set T 0 = ∪n Tn0 of rooted trees: ργ∗0 = ργ0
Wγ0 (tτ ) . βtτ 0
τ ∈T
(4.21)
138
R. Fernández, A. Procacci
If we expand W and permute the sum over trees with the sum over polymer sequences (allowed operation for a series of positive terms), we obtain ργ∗0
= ργ0
1 n! n≥0
n
(γ1 ,...,γn )∈P n τ ∈Tn0 i=0
csi (γi , γi1 , . . . , γisi ) ργ1 · · · ργn .
(4.22)
Comparing this expression with (2.7), we see immediately that hypothesis (4.7) implies that ρ ≤ ρ ∗ . The remaining statements are a consequence of Proposition 8. 4.3. Proof of Theorem 4. We just have to show that the different convergence conditions can be written in the form (4.8) for vertex functions cs satisfying (4.7). Theorem 4 then follows from Proposition 7. We use Penrose identity (4.6) to obtain a bound of the form (4.7). For this, we keep only the vertex constraints of a Penrose tree τ : The descendants of a given vertex may not be linked by an edge in the initial graph G. Otherwise [by condition (p1) in Section 4.1], the graph RPen (τ ) would include such an edge and would, therefore, differ from τ . That is, we consider the larger family of trees such that If {i, i 1 } and {i, i 2 } are edges of τ , then γi1 ∼ γi2 .
(4.23)
In this way we obtain bounds of the form (4.7) with cn (γ0 , γ1 , . . . , γn ) =
n
11{γ0 γi }
i=1
n
11{γi ∼γ j } ,
(4.24)
j=1
and Proposition 7 applies with ϕγ0 (µ) = 1 +
1 n! n≥1
(γ1 ,...,γn )∈P n γ0 γi , γi ∼γ j , 1≤i, j≤n
µγ1 . . . µγn , = Pγ0 (µ).
(4.25)
This proves the criterion of Theorem 1. If we replace in (4.23) the condition γi γ j by the weaker requirement γi = γ j we obtain cnDob (γ0 , γ1 , . . . , γn ) =
n
11{γ0 γi }
i=1
n
11{γi =γ j } ,
(4.26)
j=1
and (µ) = 1 + ϕγDob 0
1 n! n≥1
(γ1 ,...,γn )∈P n γ0 γi , γi =γ j , 1≤i, j≤n
µγ1 . . . µγn =
γ γ0
(1 + µγ ), (4.27)
which corresponds to the Dobrushin condition. The improved Dobrushin condition is obtained by strengthening (4.26) through the further requirement that γi = γ0 for i = 1, . . . , n and n ≥ 2.
Cluster Expansion for Abstract Polymer Models. New Bounds
139
Finally, if requirement (4.23) is ignored altogether, cnKP (γ0 , γ1 , . . . , γn )
=
n
11{γ0 γi } ,
(4.28)
i=1
and ϕγKP (µ) = 1 + 0
1 n! n≥1
µγ1 . . . µγn = exp
(γ1 ,...,γn )∈P n γ0 γi , 1≤i≤n
yields the criterion of Kotecký and Preiss.
γ γ0
µγ
(4.29)
Acknowledgements. We are indebted to the two referees for detailed and constructive criticism that prompted us to a substantial rewriting of the introductory sections and a clearer layout of our proof. We are thankful to Benedetto Scoppola, Roman Kotecký and Alan Sokal for long and fruitful discussions which led us to a number of notions that have enriched our presentation (in particular, partitionability of complexes of edges, log-convexity of convergence regions and the fixed-point character of ρ ∗ ). The authors thank Warwick University and the Università di Roma “Tor Vergata” (AP) for hospitality during these discussions. It is a pleasure to thank Aernout van Enter for useful comments and for a critical reading of the manuscript, and Charles-Edouard Pfister and Daniel Ueltschi for encouragement. The work of AP was supported by a visitor grant of CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brasil). He also thanks the Mathematics Laboratory Raphael Salem of the University of Rouen for the invitation that started the project and for hospitality during its realization. The work of RF was partially supported by the project GIP-ANR NT05-3-43374 (Agence Nationale de la Recherche, France).
References 1. Bovier, A., Zahradnk, M.: A simple inductive approach to the problem of convergence of cluster expansions of polymer models. J. Statist. Phys. 100, 765–78 (2000) 2. Brydges, D.C.: A short course in cluster expansions. In: Critical Phenomena, Random Systems, Gauge Theories. Osterwalder, K., Stora, R. (eds.), Amsterdam: Elsevier, 1984, pp. 129–83 3. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. New York: Wiley-Interscience Publication, 1987 4. Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems. Commun. Math. Phys. 85, 517–28 (1982) 5. Dobrushin, R.L.: Estimates of semiinvariants for the Ising model at low temperatures In: Topics in Statistics and Theoretical Physics. Amer. Math. Soc. Transl. (2) 177, 59–81 (1996) 6. Dobrushin, R.L.: Perturbation methods of the theory of Gibbsian fields. In: Ecole d’Eté de Probabilités de Saint-Flour XXIV – 1994, Lecture Notes in Mathematics 1648, Berlin–Heidelberg–New York: SpringerVerlag, 1996, pp. 1–66 7. Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys. 22, 133–61 (1971) 8. Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491– 498 (1986) 9. Groeneveld, J.: Two theorems on classical many-particle systems. Phys. Letters 3, 50–1 (1962) 10. Malyshev, V.A.: Cluster expansions in lattice models of statistical physics and quantum theory of fields. Russ. Math. Surv. 35, 1–62 (1980) 11. Miracle-Solé, S.: On the convergence of cluster expansions. Physica A 279, 244–9 (2000) 12. Nardi, F.R., Olivieri, E., Zahradnk, M.: On the Ising model with strongly anisotropic external field. J. Stat. Phys. 97, 87–144 (1999) 13. Pfister, Ch.-E.: Large deviation and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64, 953–1054 (1991) 14. Penrose, O.: Convergence of fugacity expansions for classical systems. In: Statistical mechanics: foundations and applications, A. Bak (ed.), New York: Benjamin, 1967 15. Procacci, A., Scoppola, B.: Polymer gas approach to N -body lattice systems. J. Statist. Phys. 96, 49–68 (1999) 16. Royden, H.L.: Real Analysis (2nd. ed.). New York: Macmillan Publishing Co., 1968
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17. Ruelle, D.: Statistical mechanics: Rigorous results. New York-Amsterdam: W. A. Benjamin, Inc., 1969 18. Scott, A., Sokal, A.D.: The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Stat. Phys. 118, 1151–261 (2005) 19. Sokal, A.: Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Combin. Probab. Comput. 10, 41–77 (2001) 20. Seiler, E.: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Physics 159, Berlin–Heidelberg–New York, Springer-Verlag, 1982 21. Ueltschi, D.: Cluster expansions and correlation functions. Mosc. Math. J. 4, 511–22, 536 (2004) Communicated by H. Spohn
Commun. Math. Phys. 274, 141–186 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0277-4
Communications in
Mathematical Physics
Periodicity and the Determinant Bundle Richard Melrose1 , Frédéric Rochon2 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
E-mail:
[email protected]
2 Department of Mathematics, State University of New York, Stony Brook, New York 11794, USA.
E-mail:
[email protected] Received: 2 August 2006 / Accepted: 28 November 2006 Published online: 16 June 2007 – © Springer-Verlag 2007
Abstract: The infinite matrix ‘Schwartz’ group G −∞ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G −∞ . We show that while the higher (even, Schwartz) loop groups of G −∞ , again classifying for odd K-theory, do not carry multiplicative determinants generating the first Chern class, ‘dressed’ extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so doing we relate two distinct extensions of the eta invariant to self-adjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding τ invariant is a determinant in this sense. Contents 1. Determinant Line Bundle . . . . . . . 1.1 Bundles of groups . . . . . . . . . 1.2 Classifying principal bundles . . . 1.3 Associated bundles . . . . . . . . 1.4 Det(P) . . . . . . . . . . . . . . . 1.5 Quillen’s definition . . . . . . . . 1.6 Metric on Det(P) . . . . . . . . . 1.7 Primitivity . . . . . . . . . . . . . 2. Classes of Pseudodifferential Operators 2.1 m (X ; E, F) . . . . . . . . . . . m 2.2 sus( p) (X ; E, F) . . . . . . . . .
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The first author acknowledges the support of the National Science Foundation under grant DMS0408993, the second author acknowledges support of the Fonds québécois sur la nature et les technologies and NSERC while part of this work was conducted.
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m,m 2.3 psus( p) (X ; E, F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 m 2.4 iso(2n,) (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
m,m 2.5 iso(2n,) (X ; E, F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
m,m 2.6 psus(2n) (X ; E, F)[[]] . . . . . . . . 3. Adiabatic Determinant . . . . . . . . . . . 3.1 Isotropic determinant . . . . . . . . . 3.2 Asymptotics of det . . . . . . . . . . 3.3 Star product . . . . . . . . . . . . . . 3.4 Adiabatic determinant . . . . . . . . . 4. Periodicity of the Numerical Index . . . . 4.1 Product-suspended index . . . . . . . 4.2 Periodicity . . . . . . . . . . . . . . . 5. Periodicity of the Determinant Line Bundle 5.1 Adiabatic determinant bundle . . . . . 5.2 Isotropic determinant bundle . . . . . 5.3 Adiabatic limit of Det( Dˆ n ) . . . . . 6. Eta Invariant . . . . . . . . . . . . . . . . 6.1 Product-suspended eta . . . . . . . . 6.2 η(A + iτ ) = η(A) . . . . . . . . . . . 7. Universal η and τ Invariants . . . . . . . . 8. Geometric η and τ Invariants . . . . . . . 9. Adiabatic η . . . . . . . . . . . . . . . . . Appendix A. Symbols and Products . . . . . . Appendix B. Product Suspended Operators . . . Appendix C. Mixed Isotropic Operators . . . . References . . . . . . . . . . . . . . . . . . . .
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Introduction The Fredholm determinant is a character for the group of invertible operators of the form Id +T with T of trace class on a Hilbert space. Transferred to invertible operators of the form Id +A with A smoothing on the compact fibres of a fibration it induces the determinant bundle of families of elliptic pseudodifferential operators. For suspended families of smoothing operators, depending in a Schwartz fashion on an even number of Euclidean parameters, we introduce an adiabatic determinant with similar topological properties and use it to prove periodicity properties for the determinant bundle. The corresponding suspended eta invariants are also discussed and in a subsequent paper will be used to describe cobordism of the determinant bundle in a pseudodifferential setting, extending the result of Dai and Freed [6] that the eta invariant in the interior defines a trivialization of the determinant bundle on the boundary. The basic notion of determinant is that on finite rank matrices. If M(N , C) is the algebra of N × N complex matrices then the determinant is the entire (polynomial) multiplicative map det : M(N , C) −→ C, det(AB) = det(A) det(B), which is determined by the condition on its derivative at the identity d det(Id +s A)s=0 = Tr(A), A ∈ M(N , C). ds
Periodicity
143
It has the fundamental property that det(A) = 0 is equivalent to the invertibility of A, so GL(N , C) = {A ∈ M(N , C); det(A) = 0} = det −1 (C∗ ). As is well-known, such a map into C∗ determines, through the winding number, an integral 1-cohomology class: α(c) = wn(det : c −→ C∗ ), α ∈ H1 (GL(N , C); Z).
(1)
Conversely for any path-connected space H 1 (X ; Z) ≡ {α : π1 (X ) −→ Z; α(c1 ◦ c2 ) = α(c1 ) + α(c2 )}, so each integral 1-cohomology class may be represented by a continuous function f : X −→ C∗ such that α(c) is the winding number of f restricted to a curve representing c. Even if X is a group and the class is invariant, it may not be possible to choose this function to be multiplicative. Each integral 1-cohomology class on X may also be represented as the obstruction to the triviality of a principal Z bundle over X. Such a bundle, with total space P, always admits a ‘connection’ in the sense of a map h : P −→ C such that h(np) = h( p) + n for the action of n ∈ Z. Given appropriate smoothness, the function on X associated to the connection, f = exp(2πi h), fixes the obstruction 1-class as a deRham form α=
1 −1 f d f = dh. 2πi
In particular the triviality of the Z-bundle is equivalent to the existence of a continuous (normalized) logarithm for f, that is a function l : X −→ C such that h − φ ∗l is locally constant, where φ : P −→ X is the bundle projection. Returning to the basic case of the matrix algebra and GL(N , C), these spaces can be naturally included in the ‘infinite matrix algebra’ which we denote abstractly −∞ . For the moment we identify −∞ = {a : N2 −→ C; sup (i + j)k |ai j | < ∞ ∀ k ∈ N}. i, j∈N
The algebra structure is just the extension of standard matrix multiplication (ab)i j =
∞
ail bl j .
l=1
Now, although M(N , C) −→ −∞ is included as the subalgebra with ai j = 0 for i, j > N , for the determinant this is not natural, in part because −∞ is non-unital. Namely, we consider instead the isomorphic space Id + −∞ which may be identified with −∞ with the product a ◦ b = a + b + ab. Then the inclusion M(N , C) a −→ (Id −π N ) + π N aπ N
144
R. Melrose, F. Rochon
is multiplicative and the determinant is consistent for all N with the Fredholm determinant which is the entire multiplicative function det Fr : Id + −∞ −→ C satisfying the normalization ∞
d det Fr (Id +sa)s=0 = Tr(a) = aii . ds i=1
Again for a ∈ −∞ the condition det Fr (Id +a) = 0 is equivalent to the existence of an inverse Id +b, b ∈ −∞ and this defines the topological group G −∞ = {Id +a; a ∈ −∞ , det Fr (Id +a) = 0} in which the GL(N , C) are included as subgroups. Since these determinants are consistent we generally drop the distinction between the finite and Fredholm determinants. Now, G −∞ is a classifying group for odd K-theory, K 1 (X ) = 0 { f : X −→ G −∞ }, where the maps can be taken to be either continuous or smooth. As such, {0} l even −∞ l (G ) = Z l odd. The odd Chern forms (see for example [15]), β2k−1 =
(k − 1)! 1 Tr[((Id +a)−1 da)2k−1 ], k ∈ N, (2πi)k (2k − 1)!
give an explicit isomorphism h 2k−1 : 2k−1 (G
−∞
) [ f ] −→
S2k−1
f ∗ β2k−1 ∈ Z,
(2)
(3)
where [ f ] ∈ 2k−1 (G −∞ ) is represented by a smooth map f : S2k−1 −→ G −∞ . The cohomology classes [β2k−1 ] ∈ H 2k−1 (G −∞ ); C) generate H ∗ (G −∞ ; C) as an exterior algebra over C, H ∗ (G −∞ ; C) = C (β1 , β3 , . . . , β2k−1 , . . .). However, as noted by Bott and Seeley in [5], even though they give integers when integrated over the corresponding spherical homology class, the classes [β2k−1 ] are not all integral. When k = 1, the isomorphism (3) shows that as in the case of the matrix groups, a loop along which the winding number of the determinant is 1 generates 1 (G −∞ ) and H1 (G −∞ ) is generated by the deRham class α=
1 Tr((Id +a)−1 da). 2πi
(4)
Bott periodicity corresponds to the fact that the (reduced) loop groups of G −∞ are also classifying spaces for odd or even K-theory. Consistent with the ‘smooth’ structure
Periodicity
145
championed here, we consider loop groups of ‘Schwartz’ type. In fact we can first identify −∞ above as the expansion of an operator with respect to the eigenvectors of the harmonic oscillator on Rn to identify −∞ ←→ −∞ (Rn ) = S(R2n ), where the product on S(R2n ) is the operator product (ab)(x, y) = a(x, z)b(z, y)dz. Rn
With this identification the loop groups become n p −∞ G −∞ (Rn ); f = Id +a, a ∈ S(R p+2n )}. sus( p) (R ) = { f : R −→ G
Thus G −∞ sus( p) is a classifying group for K-theory of the parity opposite to that of p. In − p−1 and Bott periodicity as fact we may regard G −∞ sus( p) as classifying for the groups K giving the identification between these for all even and all odd orders. The analogues of the forms (2) are given by ( p) β2k−1− p ( f ) = f ∗ β2k−1 , p ≤ 2k − 1, k ∈ N. Rp
For p = 1, this gives the even forms k! 1 (1) −1 2k −1 da β2k = dτ, k ∈ N0 , Tr (a da) a (2πi)k+1 (2k)! R dτ where τ is the suspension parameter (cf.[15]). It is equally possible to use the eigenbasis of a Laplacian on the sections of a vector bundle over a compact Riemannian manifold without boundary (or of any self-adjoint elliptic pseudodifferential operator of positive order) to identify −∞ with −∞ (X ; E), the space of smoothing operators. Then the loop groups are realized as G −∞ sus( p) (X ; E) = { f : R p −→ G −∞ (X ; E); f = Id +a, a ∈ S(R p × X × X ; Hom(E))}. Here, the space of Schwartz sections is defined for any vector bundle which is the pullback to R p × Z of a vector bundle over a compact manifold Z . Now, the basic issue considered here is the existence of a determinant on the spaces G −∞ sus(2k) . One can simply look for a smooth multiplicative function which generates the 1-dimensional homology through the winding number formula (1). In what is really the opposite side of the ‘Miracle of the loop group’ of Pressley and Segal [17] there is in fact no such function as soon as k > 0. As we show below, there is a multiplicative function closely related to the determinant but which has a global logarithm (if k > 0). However, as we also show below, there is a determinant function, the ‘adiabatic determinant’ in this sense, provided the group G −∞ sus(2k) is ‘dressed’ by replacing it by an extension with respect to a star product, of which G −∞ sus(2k) is the principal term. This extension is homotopically trivial, i.e. still gives a classifying space for K-theory.
146
R. Melrose, F. Rochon k More precisely, consider the space sus(2n) (X ; E)[[ε]] of formal power series ∞
k aµ εµ , aµ ∈ sus(2n) (X ; E)
µ=0
(see (2.9) for the definition) equipped with the star-product ⎛ ⎞ ∞ ∞ (A ∗ B)(u) = ⎝ aµ ε µ ⎠ ∗ bν εν µ=0
=
∞ ∞ µ=0 ν=0
ν=0
⎞ ∞ p iε p ⎠ ⎝ ω(Dv , Dw ) aµ (v)bν (w) 2 p p! ⎛
εµ+ν
p=0
(5) v=w=u
∗ (X ; E)[[ε]], where ω is the standard symplectic form on R2n . This for A, B ∈ sus(2n) gives a corresponding group −∞ G −∞ sus(2n) (X ; E)[[ε]] = {Id +Q; Q ∈ sus(2n) (X ; E)[[ε]], −∞ 0 (X ; E)[[ε]], (Id +Q) ∗ (Id +P) = Id ∈ sus(2n) (X ; E)[[ε]]} ∃ P ∈ sus(2n)
with group law given by the star-product (5). Then G −∞ sus(2n) (X ; E) is a retraction of −∞ G sus(2n) (X ; E)[[ε]]. Our first main result is the following. Theorem 1. There is a multiplicative ‘adiabatic’ determinant function ∗ deta : G −∞ sus(2n) (X ; E)[[ε]] −→ C ,
deta (A ∗ B) = deta (A) deta (B) ∀ A, B ∈ G −∞ sus(2n) (X ; E)[[ε]], which generates H1 (G −∞ sus(2n) (X ; E)[[ε]]). This is proven in §3 by considering a corresponding determinant for mixed isotropic operators and taking the adiabatic limit. Given a (locally trivial) fibration of compact manifolds Z
M
(6)
φ
B and a family of elliptic 2n-suspended operators
k (M/B; E, F) D ∈ sus(2n)
with vanishing numerical index, one can construct an associated determinant line bundle Deta (D) → B as described in §3, the definition being in terms of (a slightly extended notion of) principal bundles; a related construction can be found in [17]. More generally,
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this construction can be extended to a fully elliptic family of product-suspended operators (see the Appendix and §2 for the definition)
k,k D ∈ psus(2n) (M/B; E, F)
with vanishing numerical index. Our second result is to relate this determinant line bundle with Quillen’s definition via Bott periodicity. Let D0 ∈ 1 (M/B; E, F) be a family of elliptic operators with vanishing numerical index. Define, by recurrence for n ∈ N, the fully elliptic product-suspended families by
∗ itn − τn Dn−1 1,1 ∈ psus(2n) Dn (t1 , . . . , tn , τ1 , . . . , τn ) = (M/B; 2n−1 (E ⊕ F)), Dn−1 itn + τn where 2n−1 (E ⊕ F) is the direct sum of 2n−1 copies of E ⊕ F. In §5 we prove Theorem 2 (Periodicity of the determinant line bundle). For each n ∈ N, there is an isomorphism Deta (Dn ) ∼ = Det(D0 ) as line bundles over B. In §6, we investigate the counterpart of the eta invariant for the determinant of Theorem 1. After extending the definition given in [13] to product-suspended operators, we relate this invariant (denoted here ηsus ) to the extension of the original spectral definition of Atiyah, Patodi and Singer given by Wodzicki [21]. Namely consider ηz (A) =
sgn(a j )|a j |−z ,
(7)
j
where the a j are the eigenvalues of A in order of increasing |a j | repeated with multiplicity. Theorem 3. If A ∈ 1 (X ; E) is an invertible self-adjoint elliptic pseudodifferential 1,1 operator and A(τ ) = A + iτ ∈ psus (X ; E) is the corresponding product-suspended family then ηsus (A(τ )) = regz=0 ηz (A) = η(A)
(8)
is the regularized value at z = 0 of the analytic extension of (7) from its domain of convergence. The eta invariant for product-suspended operators is, as in the suspended case discussed in [13], a log-multiplicative functional
k,k l,l ηsus (AB) = ηsus (A) + ηsus (B), A ∈ psus (X ; E), B ∈ psus (X ; E).
Finally, in §7, we show (see Theorem 4) that in the appropriate context, this eta invariant can be interpreted as the logarithm of the determinant of Theorem 1. To discuss these results, substantial use is made of various classes of pseudodifferential operators, in particular product-type suspended operators and mixed isotropic operators. An overview of the various classes used in this paper is given in §2 and some of their properties are discussed in the Appendix.
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1. Determinant Line Bundle Quillen in [18] introduced the determinant line bundle for a family of ∂ operators. Shortly after, Bismut and Freed in [4] and [3] generalized the definition to Dirac operators. We will show here that this is induced by the Fredholm determinant, as a representation of the group G −∞ . To do so we need to slightly generalize the standard notion of a principal bundle. 1.1. Bundles of groups. Definition 1.1. Let G be a topological group (possibly infinite dimensional). Then a fibration G → B over a compact manifold B with typical fibre G is called a bundle of groups with model G if its structure group is contained in Aut(G), the group of automorphisms of G. The main example of interest here is the bundle of smoothing groups, with fibre G −∞ (Z b ) on the fibres of a fibration (6). In this case the group is smooth and the bundle inherits a smooth structure. Definition 1.2. Let φ : G −→ B be a bundle of groups with model G, then a (right) principal G-bundle is a smooth fibration π : P −→ B with typical fibre G together with a continuous (or smooth) fibrewise group action h : Pb × Gb ( p, g) −→ p · g −1 ∈ Pb which is continuous (or smooth) in all variables, locally trivial and free and transitive on the fibres. An isomorphism of principal G-bundles is an isomorphism of the total spaces which intertwines the group actions. The fibre actions combine to give a continuous map from the fibre product P × B G = {( p, g) ∈ P × G; π( p) = φ(g)} −→ P. Definition 1.2 is a generalization of the usual notion of a principal bundle for a group G in the sense that a principal G-bundle π : P −→ B is naturally a principal G-bundle for the trivial bundle of groups G = G × B → B. Any bundle of groups G → B is itself a principal G-bundle and should be thought of as the trivial principal G-bundle. Thus a principal G-bundle P → B is trivial, as a principal G-bundle, if it is isomorphic as a principal G-bundle to G. 1.2. Classifying principal bundles. Lemma 1.3. If G has a topological classifying sequence of groups G −→ E G −→ B G
(1.1)
(so E G is weakly contractible) which is a Serre fibration, G is a bundle of groups modelled on G with structure group H ⊂ Aut(E G, G), the group of automorphisms of E G restricting to automorphisms of G, then, principal G-bundles over compact bases are classified up to G-isomorphism by homotopy classes of global sections of a bundle G(B G) of groups with typical fibre B G.
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Proof. The assumption that the structure group of G is a subgroup of Aut(E G, G) allows the bundle of groups G to be extended to a bundle of groups with model E G. Namely taking an open cover of X by sets over which G is trivial, the fibres may be extended to E G, the transition maps then extend to the larger fibres and the cocycle condition continues to hold. Denote the resulting bundle of groups, G(E G) ⊃ G, with typical fibre E G. The quotient bundle G(B G) = G(E G)/G is a bundle of groups with typical fibre B G and structure group Aut(E G, G) acting on B G. Similarly, any (right) principal G-bundle, P, has an extension to a principal G(E G)bundle, P(E G), P(E G)x = Px × G(E G)x /Gx , ( p, e) ≡ ( pg −1 , eg −1 ). Since the group E G is, by hypothesis, weakly contractible, and the base is compact, the extended bundle P(E G) has a continuous global section. As in the case of a traditional principal bundle, the quotient of this section by the fibrewise action of G gives a section of G(B G). Since all sections of a bundle with contractible fibre are homotopic, the section of G(B G) is well-defined up to homotopy. Bundles isomorphic as principal G bundles give homotopic sections and the construction can be reversed as in the standard case. Namely, given a continuous section u : B −→ G(B G) we may choose a ‘good’ open cover, {Ui } of B, so that each of the open sets is contractible and G is trivial over them. By assumption, the sequence (1.1) is a Serre fibration, and the fibre is weakly contractible, so it follows that u lifts to a global section u˜ : B −→ G(E G). The subbundle, given by the fibres G ⊂ E G in local trivializations, is well-defined and patches to a principal G bundle from which the given section can be recovered.
1.3. Associated bundles. As in the usual case there is a notion of a vector bundle associated to a principal G-bundle. Suppose given a fixed (real or complex) vector space V and a smooth bundle map r : G × V → B × V which is a family of representations, rb : Gb × V → V of the Gb . Then, from P and r, one can form the associated vector bundle P ×r V with fibre (P ×r V )b = Pb × V / ∼b , where ∼b is the equivalence relation ( pg, rb (g −1 , v)) ∼b ( p, v).
1.4. Det(P). Consider again the fibration of closed manifolds (6) and let D ∈ m (M/B; E), D : C ∞ (M; E + ) → C ∞ (M; E − )
(1.2)
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be a family of elliptic operators parameterized by the base B. Then G −∞ (M/B; E + ) B with fibres
G −∞ (Z b ; E b+ ) = Id +Q; Q ∈ −∞ (Z b ; E + (b)), Id +Q b is invertible
is a bundle of groups, with model G −∞ . To the family D we associate the bundle G −∞
P(D)
(1.3)
B of invertible perturbations of D by smoothing operators where the fibre at b is Pb (D) = Db + Q b ; Q b ∈ −∞ (Z b ; E + , E − ), Db + Q b is invertible . The assumption that the numerical index vanishes implies that Pb (D) is non-empty. In fact, for each b ∈ B, the group G −∞ (Z b ; E + (b)) acts freely and transitively on the right on Pb (D) to give P(D) the structure of a principal G −∞ (M/B; E + )-bundle. On the other hand, the Fredholm determinant gives a smooth map det : G −∞ −→ C∗ ∼ = GL(1, C) which restricts to a representation in each fibre. Thus the construction above gives a line bundle associated to the principal bundle (1.3); for the moment we denote it Det(P). 1.5. Quillen’s definition. Proposition 1.4. For an elliptic family of pseudodifferential operators of order m > 0 with vanishing numerical index, the determinant line bundle of Quillen, Det(D), is naturally isomorphic to the line bundle, Det(P), associated to the bundle (1.3) and the determinant as a representation of the structure group. Proof. First we recall Quillen’s definition (following Bismut and Freed [3]). Since it extends readily we consider a pseudodifferential version rather than the original context of Dirac operators. So, for a fibration as in (6), let D be the smooth family of elliptic pseudodifferential operators of (1.2). We also set E ± (b) = E Z , Eb± = C ∞ (Z b , E ± (b)) b and consider the infinite dimensional bundles E ± over B. By assumption, Db has vanishing numerical index. Choosing inner products on E ± and a positive smooth density on the fibres of M allows the adjoint D ∗ of D to be defined. Then, for each b ∈ B, Db∗ Db : Eb+ −→ Eb+ and Db Db∗ : Eb− −→ Eb− have a discrete spectrum with nonnegative eigenvalues. They have the same positive eigenvalues with Db an isomorphism of the corresponding eigenspaces. Given λ > 0, the sets Uλ = b ∈ B; λ is not an eigenvalue of Db∗ Db
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− + are open and H[0,λ) ⊂ E + and H[0,λ) ⊂ E − , respectively spanned by the eigenfunctions ∗ ∗ of Db Db and of Db Db with eigenvalues less than λ, are bundles over Uλ of the same − + ⊕ H[0,λ) is a superbundle to which we dimension, k = k(λ). Now, H[0,λ) = H[0,λ) associate the local determinant bundle − + Det(H[0,λ) ) = (∧k H[0,λ) )−1 ⊗ (∧k H[0,λ) ). − + A linear map P : H[0,λ) → H[0,λ) induces a section − + det(P) = ∧m P : ∧m H[0,λ) −→ ∧m H[0,λ)
(1.4)
of Det(H[0,λ) ). + For 0 < λ < µ, H[0,µ) = H[0,λ) ⊕ H(λ,µ) over Uλ ∩ Uµ , where H(λ,µ) = H(λ,µ) ⊕ − − + H(λ,µ) and H(λ,µ) and H(λ,µ) are respectively the local vector bundles spanned by the eigenfunctions of Db∗ Db and Db Db∗ with associated eigenvalues between λ and µ. Thus, + if D(λ,µ) denotes the restriction of D to H(λ,µ) , then (1.4) leads to transition maps φλ,µ : Det(H[0,λ) ) s −→ s ⊗ det(D(λ,µ) ) ∈ Det(H[0,µ) ) over Uλ ∩ Uµ . The cocycle conditions hold over triple intersections and the resulting bundle, which is independent of choices made (up to natural isomorphism), is Quillen’s determinant bundle, Det(D). Let Q b ∈ −∞ (Z b ; E + , E − ), for b ∈ U ⊂ B open, be a smooth family of perturbations such that Db + Q b is invertible; it therefore gives a section of P over U. The associated bundle Det(P) is then also trivial over U with U b −→ (Db + Q b , 1) being a non-vanishing section. For λ > 0, let P[0,λ) be the projection onto H[0,λ) , and − − + + denote by P[0,λ) and P[0,λ) the projections onto H[0,λ) and H[0,λ) respectively. Then, on − + U ∩ Uλ for λ large enough, P[0,λ) (Db + Q b )P[0,λ) is invertible, and one can associate to the section Db + Q b of P the isomorphism FU ,λ : Det(P) [(Db + Q b , c)] −→
− + det(P[0,λ) (Db + Q b )P[0,λ) ) det(A(Q b , λ))c ∈ Det(D), (1.5)
− + ) is defined by (1.4), where det(P[0,λ) (Db + Q b )P[0,λ) − + Q b P[0,λ) )−1 (Db + Q b ) ∈ G −∞ A(Q b , λ) = (Db + P[0,λ) b ,
(1.6)
and det(A(Q b , λ)) ∈ C∗ is the determinant defined on G −∞ b . The map FU ,λ induces a global isomorphism of the two notions of determinant bundle since it is independent of choices. Indeed, it is compatible with the equivalence relation − + ∼b in the sense that for each g ∈ G −∞ (Z b ; E + ) such that both P[0,λ) (Db + Q b )P[0,λ) − + and P[0,λ) (Db + Q b )g P[0,λ) are invertible, FU ,λ ((D + Q b )g, det(g −1 )c) = FU ,λ ((D + Q b ), c).
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It is also compatible with increase of λ to µ in that φλ,µ ◦ FU ,λ = FU ,µ on U ∩ Uλ ∩ Uµ . This is readily checked − + (Db + Q b )P[0,λ) ) det(A(Q b , λ))c] φλ,µ ◦ FU ,λ (Db + Q b , c) = φλ,µ [det(P[0,λ)
− + + = det(A(Q b , λ)) det(P[0,λ) (Db + Q b )P[0,λ) )⊗ det(D(λ,µ) )c − − + + = det(A(Q b , λ)) det(P[0,µ) (Db + P[0,λ) Q b P[0,λ) )P[0,µ) )c
− + (Db + Q b )P[0,µ) )× = det(A(Q b , λ)) det(P[0,µ)
− − + + det((Db + P[0,µ) Q b P[0,µ) )−1 (Db + P[0,λ) Q b P[0,λ) ))c
− + (Db + Q b )P[0,µ) )c = det(A(Q b , µ)) det(P[0,µ)
= FU ,µ (Db + Q b , c).
(1.7)
1.6. Metric on Det(P). The Quillen metric has a rather direct expression in terms of the definition of the determinant bundle as Det(P). Namely, if (Db + Q b ) is a section of P over the open set U ⊂ B, then
1 |(Db + Q b , 1)| Q = exp − ζb (0) , (1.8) 2 where ζb is the ζ -function associated to the self-adjoint positive elliptic operator (Db + Q b )∗ (Db + Q b ) as constructed by Seeley [19]. When Ab = Id +Rb ∈ G −∞ with b + + Rb : H[0,λ) → H[0,λ) for some λ > 0, Proposition 9.36 of [2], adapted to this context, shows that |(Db + Q b )Ab | Q = | det(Ab )| |Db + Q b | Q ,
(1.9)
but then by continuity the same formula follows in general. Moreover, in the form (1.8), Quillen’s metric generalizes immediately to the case of an arbitrary family of elliptic pseudodifferential operators with vanishing numerical index.
1.7. Primitivity. Lemma 1.5. The determinant bundle is ‘primitive’ in the sense that there is a natural isomorphism Det(P Q) Det(P) ⊗ Det(Q)
(1.10)
for any elliptic families Q ∈ m (M/B; E, F), P ∈ m (M/B; F, G) of vanishing numerical index. Proof. Let P and Q denote the principal bundles of invertible smoothing perturbations of P and Q. Let Pb + Rb and Q b + Sb be local smooth sections over some open set U. Certainly L b = (Pb + Rb )(Q b + Sb ) is a local section of the principal bundle for P Q and (L b , 1) as a local section of Det(P Q) may be identified with the product of the sections
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(Pb + Rb , 1) and (Q b , Sb , 1) as a section of Det(P) ⊗ Det(Q). Changing the section of P to (Pb + Rb )gb modifies the section L b to L b gb , gb = (Q b + Sb )−1 gb (Q b + Sb ). Since det(gb ) = det(gb ), the identification is independent of choices of sections and hence is global and natural. Later, it will be convenient to restrict attention to first order elliptic operators. This is not a strong restriction since for k ∈ Z, let D ∈ k (M/B; E, F) be a smooth family of elliptic pseudodifferential operators with vanishing numerical index. Let M/B ∈ 2 (M/B; F) be an associated family of Laplacians, so that M/B + Id is a family of invertible operators. k−1
Corollary 1.6. The family D = ( M/B + Id)− 2 D ∈ 1 (M/B; E, F) has determinant bundle isomorphic to the determinant bundle of D. 2. Classes of Pseudodifferential Operators Since several different types, and in particular combinations of types, of pseudodifferential operators are used here it seems appropriate to quickly review the essentials. 2.1. m (X ; E, F). On a compact manifold without boundary the ‘traditional’ algebra (so consisting of ‘classical’ operators) may be defined in two steps using a quantization map. The smoothing operators acting between two bundles E and F may be identified as the space −∞ (X ; E, F) = C ∞ (X 2 ; Hom(E, F) ⊗ R ).
(2.1)
Here Hom(E, F)x,x = E x ⊗ Fx is the ‘big’ homomorphism bundle and = π R∗ is the lift of the density bundle from the right factor under the projection π R : X 2 −→ X. The space m (X ; E, F) may be identified with the conormal sections, with respect to the diagonal, of the same bundle m (X ; E, F) = Iclm (X 2 , Diag; Hom(E, F) ⊗ R ).
(2.2)
More explicitly Weyl quantization, given by the inverse fibre Fourier transform from T ∗ X to T X, qg : ρ −m C ∞ (T ∗ X ; π ∗ hom(E, F)) a −→ −n (2π ) χ exp (iv(x, y) · ξ )a(m(x, y), ξ )dξ dg y ∈ m (X ; E, F) T∗X
(2.3)
is surjective modulo −∞ (X ; E, F). Here a Riemann metric, g, is chosen on X and used to determine a small geodesically convex neighbourhood U of the diagonal in X 2 which is identified as a neighbourhood U of the zero section in T X by mapping (x, y) ∈ U to m(x, y), the mid-point of the geodesic joining them in X and to v(x, y) ∈ Tm(x,y) X, the tangent vector to the geodesic at that mid-point in terms of the length parameterization of the geodesic from y to x. The cut-off χ ∈ Cc∞ (U ) is taken to be identically equal
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to 1 in a smaller neighbourhood of the diagonal. Connections on E and F are chosen and used to identify Hom(E, F) over U with the lift of hom(E, F) to U , dξ is the fibre density from g on T ∗ X and dg y is the Riemannian density on the right (in the y variable). The symbol a is a classical symbol of order k on T ∗ X realized as ρ −k a , where a ∈ C ∞ (T ∗ X ) with T ∗ X the compact manifold with boundary arising from the radial compactification of the fibres of T ∗ X and ρg = |ξ |−1 g outside a compact set in T ∗ X is a boundary defining function for that compactification. Then qg (a) ∈ −∞ (X ; E, F) if and only if a ∈ C˙∞ (T ∗ X ) is a smooth function vanishing to all orders on the boundary of T ∗ X , i.e. is a symbol of order −∞. This leads to the short exact ‘full symbol sequence’ σg
−∞ (X ; E, F) −→ ∞ (X ; E, F) −→ C ∞ (S ∗ X ; hom(E, F))[[ρ, ρ −1 ]] (2.4) with values in the Laurent series in ρ (i.e. formal power series in ρ with finite factors of ρ −1 ). The leading part of this is the principal symbol σm
m−1 (X ; E, F) −→ m (X ; E, F) −→ C ∞ (S ∗ X ; hom(E, F) ⊗ Rm ),
(2.5)
where Rm is the trivial bundle with sections which are homogeneous of degree m over T ∗ X \0. Pseudodifferential operators act from C ∞ (X ; E) to C ∞ (X ; F) and composition gives a filtered product,
m (X ; F, G) ◦ m (X ; E, F) ⊂ m+m (X ; E, G),
(2.6)
which induces a star product on the image spaces in (2.4), a g b = ab +
∞
B j (a, b),
(2.7)
j=1
where the B j are smooth bilinear differential operators with polynomial coefficients on T ∗ X lowering total order, in terms of power series, by j. The leading part gives the multiplicativity of the principal symbol. m m 2.2. sus( p) (X ; E, F). There is a natural Fréchet topology on (X ; E, F), corres∞ ponding to the C topology on the symbol and the kernel away from the diagonal. Thus, smoothness of maps into this space is well-defined. The p-fold suspended operators are a subspace p m m ∞ sus( (2.8) R ; (X ; E, F) p) (X ; E, F) ⊂ C
in which the parameter-dependence is symbolic (and classical). In terms of the identification (2.2) this reduces to m sus( p) (X ; E, F) = p M 2 p FR−1p Icl, (X × R , Diag ×{0}; Hom(E, F) ⊗ ) , M = m + . (2.9) R S 4
Here we consider conormal distributions on the non-compact space X 2 × R p but with respect to the compact submanifold Diag ×{0}; the suffix S denotes that they are to be
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Schwartz at infinity and then the inverse Fourier transform is taken in the Euclidean variables R p giving the ‘symbolic’ parameters. The shift of m to M is purely notational. These kernels can also be expressed directly as in (2.3) with a replaced by (2.10) a ∈ ρ −m T ∗ X × R p ; π ∗ hom(E, F) . Composition, mapping and symbolic properties are completely analogous to the ‘unsuspended’ case. Note that we use the abbreviated notation for suffixes sus(1) = sus . If D is a first order elliptic differential operator acting on a bundle on X then D +iτ ∈ 1 (X ; E) is elliptic in this sense and invertible, with inverse in −1 (X ; E), if D is sus sus self-adjoint and invertible. However this is not the case for general (elliptic self-adjoint) D ∈ 1 (X ; E); we therefore introduce larger spaces which will capture these operators and their inverses.
m,m 2.3. psus( p) (X ; E, F). By definition in (2.9), before the inverse Fourier transform is taken, the singularities of the ‘kernel’ are constrained to Diag ×{0} ⊂ X 2 × R p . For product-type (really partially-product-type corresponding to the fibration of X × R p with base R p ) the singularities are allowed to fill out the larger submanifold
X 2 × {0} ⊃ Diag ×{0}.
(2.11)
Of course they are not permitted to have arbitrary singularities but rather to be conormal with respect to these two, nested, submanifolds
m,m psus( p) (X ; E, F) = M M 2 p 2 (X × R , X × {0}, Diag ×{0}; Hom(E, F) ⊗ ) , FR−1p Icl, R S
M =m+
p p n , M = m + − . 4 4 2
(2.12)
The space of classical product-type pseudodifferential operators is discussed succinctly in an appendix below. Away from Diag ×{0} the elements of the space on the right are just classical conormal distributions at {0} × R p , so if χ ∈ C ∞ (X 2 ) vanishes near the diagonal (or even just to infinite order on it)
m,m −m ∞ C (R p × X 2 ; Hom(E, F) ⊗ R ) K ∈ psus( p) (X ; E, F) =⇒ χ K ∈ ρ
(2.13)
is just a classical symbol in the parameters depending smoothly on the variables in X 2 . Conversely, if χ ∈ C ∞ (X 2 ) has support sufficiently near the diagonal then the kernel is given by a formula as in (2.3), χ exp (iv(x, y) · ξ )a(m(x, y), ξ, τ )dξ dg y , χ K = (2π )−n −m
a ∈ (ρ )
T∗X −m ∞
(ρ )
C (S; π ∗ hom(E, F)), S = [T ∗ X × R p ; 0T ∗ X × ∂R p ]. (2.14)
Here the space on which the ‘symbols’ are smooth functions (apart from the weight factors) is the same compactification as in (2.10) but then blown up (in the sense of [11]) at the part of the boundary (i.e. infinity) corresponding to finite points in the cotangent bundle. Then ρ is a defining function for the ‘old’ part of the boundary and ρ for the
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new part, produced by the blow-up. Conversely (2.14) and (2.13) together (for a partition of unity) define the space of kernels. From the general properties of blow-up, if ρ ∈ C ∞ (T ∗ X × R p ) is a defining function for the boundary then ρ = ρ ρ after blow-up. From this it follows easily that m,m m sus( p) (X ; E, F) ⊂ psus( p) (X ; E, F).
(2.15)
Again these ‘product suspended’ operators act from S(X × R p ; E) to S(X × R p ; F) and have a doubly-filtered composition m 1 ,m
m 2 ,m
m 1 +m 2 ,m 1 +m 2
1 2 psus( p) (X ; F, G) ◦ psus( p) (X ; E, F) ⊂ psus( p)
(X ; E, G).
(2.16)
The symbol map remains, but now only corresponds to the part of the amplitude in (2.14) at ρ = 0,
σm
m−1,m m,m m,m psus( p) (X ; E, F) −→ psus( p) (X ; E, F) −→ Spsus( p) (X ; E, F)
(2.17)
with
m,m (X ; E, F) = C ∞ ([S(T ∗ × R p ); 0 × S p−1 ]; hom(E, F) ⊗ Rm,m ) Spsus(d)
the space of smooth sections of a bundle over the sphere bundle corresponding to T ∗ X × R p , blown up at the image of the zero section and with Rm,m a trivial bundle capturing the weight factors. The other part of the amplitude corresponds to a more global ‘symbol map’ called here the ‘base family’,
βm
m,m −1 m,m ∞ p−1 psus( ; m (X ; E, F) ⊗ Rm ), p) (X ; E, F) −→ psus( p) (X ; E, F) −→ C (S
(2.18) taking values in pseudodifferential operators on X depending smoothly on the parameters ‘at infinity’, i.e. in S p−1 with the appropriate homogeneity bundle (over S p−1 ). These two symbol maps are separately surjective and jointly surjective onto pairs satisfying the natural compatibility condition σm (βm (A)) = σm (A)∂
(2.19)
that the symbol family, restricted to the boundary of the space on which it is defined, is the symbol family of the base family. An operator in this product-suspended class is ‘fully elliptic’ if both its symbol and its base family are invertible. If it is also invertible then its inverse is in the corresponding space with opposite orders. An elliptic suspended operator is automatically fully elliptic when considered as a product-suspended operator using (2.15).
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m 2.4. iso(2n,) (Rn ). The suspension variables for these product-suspended operators are purely parameters. However, for the adiabatic limit constructions here, on which the paper relies heavily, we use products which are non-local in the parameters. In the trivial case of X = {pt} we are dealing just with symbols above and the corresponding non-commutative product is the ‘isotropic’ algebra of operators on symbols on R2n , as operators on Rn , for any n. This is variously known as the Weyl algebra or the Moyal product (although both often are taken to mean slightly different things). The isotropic pseudodifferential operators of order k act on the Schwartz space S(Rn ) and, using Euclidean Weyl quantization, may be identified with the spaces ρ −k C ∞ (R2n ). Thus, in terms of their distributional kernels on R2n , this space of operators is given by essentially the same formula as (2.3),
qW : ρ −k C ∞ (R2n ) b −→ qW (b)(t, t ) = (2π )−n
Rn
ei(t−t )·τ b
t + t −k , τ dτ ∈ iso (Rn ). 2
(2.20)
This map is discussed extensively in [10]. In this case qW , with inverse σW , is an isomorphism onto the algebra and restricts to an isomorphism of the ‘residual’ algebra −∞ iso (Rn ) = qW (S(R2n )). The corresponding star product is the Moyal product. The full product on symbols on R2n may be written explicitly as a ◦ω b(ζ ) = π −2n eiξ ·ξ +iη·η +2iω(ξ ,η ) a(ζ + ξ )b(ζ + η)|ωξ |n |ωη |n , (2.21) R8n
where the integrals are not strictly convergent but are well defined as oscillatory integrals. Here ω is the standard symplectic form on R2n . By simply using linear changes of variables, it may be seen that this product and the more general ones in which ω is replaced by an arbitrary non-degenerate antisymmetric bilinear form on R2n are all isomorphic. In fact the product depends smoothly on ω as an antisymmetric bilinear form, even as it becomes degenerate. When ω ≡ 0 the product reduces to the pointwise, commutative, product of symbols. In fact it is not necessary to assume that the underlying Euclidean space is even dimensional for this to be true; of course in the odd-dimensional case the form cannot be non-degenerate and correspondingly there is always at least one ‘commutative’ variable. The adiabatic limit here corresponds to replacing the standard symplectic form ω by ω and allowing ↓ 0. As already noted, this gives a family of products on the classical symbol spaces which is smooth in and is the commutative product at = 0. We denote m the resulting smooth family of algebras by iso(2n,) (Rn ).
m,m 2.5. iso(2n,) (X ; E, F). Now, we may replace the parameterized product on the productsuspended algebra by ‘quantizing it’ as in (2.21), in addition to the composition in X itself. For the ‘adiabatic’ choice of ω this induces a one parameter family of quantized products m 1 ,m
m 2 ,m
m 1 +m 2 ,m 1 +m 2
1 2 [0, 1] × psus(2n) (X ; F, G) × psus(2n) (X ; E, F) −→ psus(2n)
(X ; E, G). (2.22)
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The suspended operators still form a subalgebra. The Taylor series as ↓ 0 is given by ∞ (i)k ω(D , D )A(v)B(w) . (2.23) (A ◦ B)(u) ∼ v w k 2 k! v=w=u k=0
A more complete discussion of product suspended operators and the mixed isotropic product may be found in the appendix.
m,m 2.6. psus(2n) (X ; E, F)[[]]. This is the space of formal power series in with coeffi
m,m (X ; E, F). The product (2.23) projects to induce a product cients in psus(2n) m 1 ,m
m 2 ,m
m 1 +m 2 ,m 1 +m 2
1 2 (X ; F, G)[[]] × psus(2n) (X ; E, F)[[]] −→ psus(2n) psus(2n)
(X ; E, G)[[]] (2.24)
which is consistent with the action on formal power series
m,m psus(2n) (X ; E, F)[[]] A : C ∞ (X ; E)[[]] −→ C ∞ (X ; F)[[]].
3. Adiabatic Determinant Let E −→ X be a complex vector bundle over a compact manifold X. Consider the infinite dimensional group −∞ G −∞ sus(2n) (X ; E) = {Id +Q; Q ∈ sus(2n) (X ; E), Id +Q is invertible}
of invertible 2n-suspended smoothing perturbations of the identity. A naive notion of determinant would be given by using the 1-form d log d(A) = Tr sus(2n) (A−1 d A), where Tr sus(2n) (B) =
1 Tr(B(t, τ ))dtdτ (2π )2n R2n
is the regularized trace for suspended operators as defined in [13]. The putative determinant is then given by
1
dγ d(A) = exp Tr sus(2n) γ −1 ds , (3.1) ds 0 where γ : [0, 1] → G −∞ sus(2n) (X ; E) is any smooth path such that γ (0) = Id and γ (1) = A. Although d(A) is multiplicative, it is topologically trivial, in the sense that for any smooth loop γ : S1 → G −∞ sus(2n) (X ; E), one has
1 −1 dγ −1 dγ ds = ds dt dτ = 0. (3.2) Tr sus(2n) γ Tr γ ds (2π )2n R2n S1 ds S1 So this is not a topological analogue of the usual determinant.
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3.1. Isotropic determinant. To obtain a determinant which generates the 1-dimensional cohomology, we instead use the isotropic quantization of § B. At the cost of slightly deforming the composition law on G −∞ sus(2n) (X ; E), this determinant will be multiplicative as well. Notice first that because of the canonical identification −∞,−∞ −∞ psus(2n) (X ; E) = sus(2n) (X ; E)
there is no distinction between G −∞ (X ; E) and the group −∞,−∞ G −∞,−∞ psus(2n) (X ; E) = {Id +Q; Q ∈ psus(2n) (X ; E), Id +Q is invertible},
so in this context, we can interchangeably think in terms of suspended or productsuspended operators. For ∈ [0, 1], we use the ◦ -product of Theorem 5 to define the group 0,0 −∞ G −∞ iso(2n,) (X ; E) = {Id +Q; Q ∈ sus(2n) (X ; E), ∃ P ∈ psus(2n) (X ; E),
P ◦ (Id +Q) = (Id +Q) ◦ P = Id}. (3.3) For = 0, we have the canonical group isomorphism −∞ G −∞ iso(2n,0) (X ; E) = G sus(2n) (X ; E). −∞ so that On the other hand, for > 0, the group G −∞ iso(2n,) (X ; E) is isomorphic to G it is possible to transfer the Fredholm determinant to it.
Proposition 3.1. For > 0, there is a natural multiplicative determinant ∗ det (A) : G −∞ iso(2n,) (X ; E) → C
defined for A ∈ G −∞ iso(2n,) (X ; E) by
det (A) = exp
1 0
dγ ds , Tr γ −1 ◦ ds
where γ : [0, 1] → G −∞ iso(2n,) (X ; E) is any smooth path with γ (0) = Id and γ (1) = A so d log det (A) = Tr (A−1 ◦ d A). Proof. To show that det is well-defined and multiplicative, it suffices to show that it reduces to the Fredholm determinant under a suitable identification of G −∞ iso(2n,) (X ; E) −∞ −∞ with G . From Appendix C it follows that G iso(2n,) (X ; E) acts on S(X × Rn ; E). Fix a Riemannian metric g on X and a Hermitian metric h on E. Let ∈ 2 (X ; E) be the corresponding Laplace operator. Then consider the mixed isotropic operator n ∂2 2 2 = + (X ; E), − 2 + ti ∈ iso(2n,) ∂t i i=1
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∂2 2 is the harmonic oscillator on Rn . As an operator acting on − + t i=1 i ∂t 2
n
i
S(X × Rn ; E), has a positive discrete spectrum. Let {λk }k∈N be the eigenvalues, in non-decreasing order, with corresponding eigensections f k = λk f k ,
f k ∈ S(X × Rn ; E)
such that { f k }k∈N is an orthonormal basis of L2 (X × Rn ; E). This gives an algebra isomorphism −∞ (X ; E) A −→ f i , A f j L2 ∈ −∞ , F : iso(2n,) −∞ . Under these and a corresponding group isomorphism F : G −∞ iso(2n,) (X ; E) → G isomorphisms, one has
Tr (A) = Tr(F (A)) and consequently det (Id +A) = det Fr (F (Id +A)).
(3.4)
3.2. Asymptotics of det . Now, for any δ > 0 we can consider the group of sections, 2n ∞ −∞ (R2n × X )); G −∞ iso ([0, δ] × R × X ; E) = {A ∈ C ([0, δ]; Id + 2n A() ∈ G −∞ iso, (R × X ; E) ∀ ∈ [0, δ]}. (3.5)
Proposition 3.2. The determinant with respect to the ◦ product defines 2n ∞ d et : G −∞ iso ([0, δ] × R × X ; E) −→ C ((0, δ])
which takes the form n−1 k−n 2n ak (A) F (A) ∀ A ∈ G −∞ d et(A)() = exp iso ([0, δ] × R × X ; E)
(3.6)
(3.7)
k=0
where 2n ∞ F : G −∞ iso ([0, δ] × R × X ; E) A −→ F (A) ∈ C ([0, δ]),
(3.8)
2n ∞ functions and the a only depend and ak : G −∞ k iso ([0, δ] × R × X ; E) −→ C are C on the Taylor series of A.
Proof. Since the group is open (for each ∈ [0, 1] and also for the whole group) the tangent space at any point is simply C ∞ ([0, δ]; −∞ (R2n × X ; E)). With the usual identifications for a Lie group the form A−1 ◦ d A therefore takes values in C ∞ ([0, δ]; −∞ (R2n × X ; E)). On the other hand, the trace functional is not smooth down to = 0. In fact it is rescaled by a factor of −n . Thus, d log det (A) = Tr (A−1 ◦ d A) ∼
∞ k=0
is −n times a smooth function.
αk k−n
(3.9)
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For any smooth map 2n f : S1 → G −∞ iso ([0, δ] × R × X ; E),
the integral S1 f ∗ d log det (A) also has an asymptotic expansion S1
f ∗ d log det (A) ∼
∞
ck k−n , ck =
k=0
S1
αk ∈ C.
(3.10)
On the other hand, by (3.4), this is a winding number so cannot depend on . Hence ck = αk = 0 for k = n. (3.11) S1
So, for k = n, αk is exact and then (3.7) follows directly by integration along any path 2n γ : [0, 1] → G −∞ iso ([0, δ] × R × X ; E) with γ (0) = Id and γ (1) = A. The range space is path-connected, so
1
ak (A) =
γ ∗ αk , k < n
0
is independent of the path and well-defined.
3.3. Star product. The restriction map at = 0 −∞ 2n R : G −∞ iso ([0, δ] × R × X ; E) −→ G sus(2n) (X ; E)
(3.12)
2n × X ; E) be the is surjective. From this it follows that if we let G˙ −∞ iso ([0, δ] × R subgroup of those elements which are equal to the identity to infinite order at = 0 then the quotient 2n 2n ˙ −∞ G −∞ iso ([0, δ] × R × X ; E)/G iso ([0, δ] × R × X ; E) = −∞ (R2n × X ; E)[[]] (3.13) G −∞ sus(2n) (X ; E) +
is the obvious formal power series group, namely with invertible leading term and arbitrary smoothing lower order terms. The composition law is the one induced by the ◦ -product. Since the higher order terms in amount to an affine extension of the leading part, this formal power series group is also a classifying group for odd K-theory.
k,k Definition 3.3. We denote by psus(2n) (X ; E)[[ε]], k, k ∈ R ∪ {−∞}, the space of formal series ∞
aµ ε µ
µ=0
k,k with coefficients aµ ∈ psus(2n) (X ; E), where ε is a formal parameter.
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k,k l,l For A ∈ psus(2n) (X ; F, G)[[ε]] and B ∈ psus(2n) (X ; E, F)[[ε]] the ∗-product
k+l,k +l (X ; E, G)[[ε]] is A ∗ B ∈ psus(2n) ⎞ ⎛ ∞ ∞ µ ν aµ ε ⎠ ∗ bν ε A ∗ B(u) = ⎝ µ=0
=
∞ ∞ µ=0 ν=0
⎛
ν=0
⎞ ∞ pε p i ⎠, εµ+ν ⎝ ω(Dv , Dw ) p A(v)B(w) 2 p p! v=w=u p=0
where u, v, w ∈ R2n . Since this is based on the asymptotic expansion (C.3) of Appendix B, its associativity follows immediately from the associativity of the ◦ -product. 3.4. Adiabatic determinant. This product is consistent with that of the quotient group in (3.13), so Lemma 3.4. The quotient group G −∞ sus(2n) (X ; E)[[]] of (3.13) is canonically isomorphic to −∞ −∞ G −∞ sus(2n) (X ; E)[[ε]] = {(Id +Q); Q ∈ sus(2n) (X ; E)[[ε]], ∃ P ∈ sus(2n) (X ; E)[[ε]] 0 (X ; E)[[ε]]}. such that (Id +Q) ∗ (Id +P) = Id ∈ sus(2n)
We can now prove Theorem 1 stated in the Introduction. Theorem 1. The functional 2n ∗ F0 (A) : G −∞ iso ([0, δ] × R × X ; E) → C
induces a multiplicative determinant deta on the formal power series group G −∞ sus(2n) (X ; E)[[ε]] in the sense discussed above, i.e. it is a smooth multiplicative function which generates H1 . Proof. From (3.9),
F0 (A) = exp
γ
αn ,
(3.14)
where γ : [0, 1] → G −∞ sus(2n) (X ; E)[[ε]] is any smooth path with γ (0) = Id and γ (1) = A. In the expansion (3.9), the only non-trivial cohomological contribution comes from αn . Since det corresponds to the Fredholm determinant under the identification of −∞ the integral of α along a generator of the fundamental G −∞ n iso(2n,) (X ; E) with G group is ±2πi. Thus, the determinant induced by F0 (A) has the desired topological behavior.
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For the multiplicativity, from (3.9), Tr((AB)−1 d(AB)) = Tr(B −1 A−1 d AB + B −1 A−1 Ad B) = Tr(A−1 d A) + Tr(B −1 d B),
(3.15)
where the ∗-product is used to compose elements and define the inverses. From the ε-expansion of (3.15), αn (A ∗ B) = αn (A) + αn (B). As a consequence, the determinant defined in (3.1) is multiplicative.
(3.16)
This determinant can be used to define the determinant line bundle of a fully elliptic k,k (M/B; E, F)[[ε]] of fibrewise product 2n-suspended pseudodiffamily D ∈ psus(2n) ferential operators on a fibration (6). Full ellipticity here corresponds to ellipticity of the leading term D0 and its invertibility for large values of the parameters. Assume k,k (Z b ; E b , Fb )[[ε]] can be perturbed in addition that for each b ∈ B, Db ∈ psus(2n) −∞ by Q b ∈ sus(2n) (Z b ; E b , Fb )[[ε]] to be invertible, where invertibility is equivalent to invertibility of the leading term. Then over the manifold B, consider the bundle of invertible smoothing perturbations with fibres −∞ Pb (D) = {Db + Q b ; Q b ∈ sus(2n) (Z b ; E b , Fb )[[ε]], −k,−k ∃ P ∈ psus(2n) (Z b ; Fb , E b )[[ε]], P ∗ (Db + Q b ) = (Db + Q b ) ∗ P = Id}. (3.17)
Let G −∞ sus(2n)
/ G −∞ (M/B; E) sus(2n)
(3.18)
φ
B be the bundle of groups with fibre at b ∈ B,
−∞ G −∞ sus(2n) (Z b ; E b )[[ε]] = {Id +Q; Q ∈ sus(2n) (Z b ; E b )[[ε]], 0,0 ∃ P ∈ psus(2n) (Z b ; E b )[[ε]]P ∗ (Id +Q) = (Id +Q) ∗ P = Id}. (3.19)
Then P(D) is a principal G −∞ sus(2n) (M/B; E)[[ε]]-bundle in the sense of Definition 1.2. Definition 3.5. The adiabatic determinant line bundle associated to the family D of product 2n-suspended elliptic pseudodifferential operators is Deta (D) = P(D) ×deta C induced by the adiabatic determinant as representation of the bundle of groups (3.18). 4. Periodicity of the Numerical Index In the next section, we establish a relation between the determinant line bundles of a family of standard elliptic pseudodifferential operators and the determinant line bundle just defined for families of 2n-suspended operators. Here we consider the corresponding question for the numerical index.
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4.1. Product-suspended index. A product-suspended operator
m,m P ∈ psus(k) (Z ; E, F)
is fully elliptic if both its symbol in the usual sense and its base family are invertible. Here the base family, elliptic because of the invertibility of the symbol, is parameterized by Sk−1 . As a family of operators over Rk , P has a families index. Since by assumption the family is invertible at, and hence near, infinity the family defines an index class in compactly-supported K-theory Z k even 0 k (4.1) ind(P) ∈ K (R ) = {0} k odd. Thus by choosing a generator (i.e. Bott element) in K 0 (R2n ) a product 2n-suspended family has a numerical index which we will denote indsus(n) (since it only arises for even numbers of parameters). The families index of Atiyah and Singer does not apply directly to this setting although it does apply if the operator is in the ‘suspended’ subspace (and so in particular m = m.) Using the properties of the suspended eta invariant we will show in §9 that the suspended index can be expressed in terms of the ‘adiabatic’ η invariant discussed below. Namely, suppose a linear decomposition R2n = R × R2n−1 is chosen in which the variables are written τ and ξ. Then, for some R ∈ R, P(τ, ξ ) is invertible for |τ | ≥ R for all ξ ∈ R2n−1 . Furthermore, by standard index arguments we may find a family of smoothing operators, A, of compact support in (τ, ξ ) such that P = P + A is invertible for all τ ≤ R. Then 1 indsus(n) (P) = − ηa(n-1) (P τ =R ) − ηa(n-1) (P τ =R ) . (4.2) 2 4.2. Periodicity. Here we show that there is a ‘Bott map’ from ordinary pseudodifferential operators into product-type suspended operators which maps the usual index to the suspended index (although most of the proof is postponed until later). Thus if D ∈ 1 (Z ; E, F) is an elliptic operator then
it − τ D∗ 1,1 2 ˆ , Dˆ ∈ psus(2) (Z ; E ⊕ F) (4.3) R (t, τ ) −→ D(t, τ ) = D it + τ is an associated twice-suspended fully elliptic operator. In [14], such a family is realized explicitly as the indicial family of a product-suspended cusp operator. The ellipticity of Dˆ follows from the fact that
∗ 0 D D + t2 + τ 2 2,2 ∈ psus(2) (Z ; E ⊕ F) (4.4) Dˆ ∗ Dˆ = 0 D D∗ + t 2 + τ 2 is an elliptic family which is invertible for t 2 + τ 2 > 0. Definition 4.1. Given an elliptic operator D ∈ 1 (Z ; E, F), we define by recurrence 1,1 on n ∈ N0 , elliptic product-suspended operators Dn ∈ psus(2n) (Z ; 2n−1 (E ⊕ F) by
∗ Dn−1 itn − τn Dn (t1 , . . . , tn , τ1 , . . . , τn ) = Dn−1 itn + τn with D0 = D.
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165
Lemma 4.2. If D is elliptic then Dn is a totally elliptic product 2n-suspended operator for all n and indsus(n) (Dn ) = ind(D). Proof. Both the ordinary index and the n-suspended index (on fully elliptic 2n-suspended operators) are homotopy invariant. Since the map D −→ Dn maps invertible operators to invertible operators it follows that ind(D) = 0 implies indsus(n) (Dn ) = 0. Indeed, ind(D) = 0 means there exists a smoothing operator Q ∈ −∞ (Z ; E, F) such that D + Q is invertible. Then (D + s Q)n is a homotopy of fully elliptic 2n-suspended operators which is invertible for s = 1 so indsus(n) (Dn ) = 0. The actual equality of the index is proved below in §9, using (4.2). 5. Periodicity of the Determinant Line Bundle 5.1. Adiabatic determinant bundle. Returning to the setting of a fibration with compact fibres, φ : M → B, as in (6), let D ∈ 1 (M/B; E, F) be a family of elliptic pseudodifferential operators with vanishing numerical index. From Lemma 4.2 (the part that is already proved), the suspended index of the fully elliptic family Dn ∈ 1,1 psus(2n) (M/B; E, F), given by Definition 4.1, also vanishes. Thus the fibres −∞ Ppsus(2n) (Dn )b = {Dn,b + Q b ; Q b ∈ sus(2n) (Z b ; 2n−1 (E b ⊕ Fb ))[[ε]], −1 ∃ ( Dˆ n,b + Q b )−1 ∈ psus(2n) (Z b ; 2n−1 (E b ⊕ Fb ))[[ε]]} (5.1) n−1 (E ⊕ F))[[ε]]are non-empty and combine to give a principal-G −∞ sus(2n) (M/B; 2 bundle as in (3.17). Since we have defined an adiabatic determinant on these groups we have an associated determinant bundle
Det sus(2n) (D) = Deta (Dn ) = Ppsus 2n (D) ×deta C.
(5.2)
5.2. Isotropic determinant bundle. One can make a different, but similar, construction using the isotropic quantization of Dn . 1,1 Definition 5.1. For > 0, let Dˆ n ∈ iso(2n,) (M/B; 2n−1 (E ⊕ F)) be the isotropic quantization of Dn as in Appendix C, so giving an operator on S(Rn ×X ; 2n−1 (E b ⊕Fb )).
As discussed earlier for families of standard elliptic operators, there are two equivalent definitions of the determinant line bundle for Dˆ n . Namely, Quillen’s spectral definition or as an associated bundle to the principal bundle of invertible perturbations. In the latter n−1 (E ⊕ F))[[ε]]-bundle has fibre case, the principal G −∞ iso(2n,) (M/B; 2 −∞ (Z b ; 2n−1 (E b ⊕ Fb )), Piso(2n,) ( Dˆ en )b = { Dˆ 2n,b + Q b ; Q b ∈ iso(2n,) −1,−1 ∃ ( Dˆ 2n,b + Q b )−1 ∈ iso(2n,) (Z b ; 2n−1 (E b ⊕ Fb ))}. (5.3)
Note that this fibre is non-empty as soon as the original family D has vanishing numerical index. Indeed, we know that Dn then has vanishing suspended index and hence has an invertible perturbation by a smoothing operator (in the suspended sense). The isotropic product is smooth down to = 0, where it reduces to the suspended product (pointwise in the parameters). Thus such a perturbation is invertible with respect to the isotropic product for small > 0. Since these products are all isomorphic for > 0, it follows that perturbations as required in (5.3) do exist.
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Proposition 5.2. Let D ∈ 1 (M/B; E, F) be an elliptic family with vanishing numerical index, then for each n ∈ N0 and > 0, the determinant line bundle Det( Dˆ n ) is naturally isomorphic to the determinant line bundle Det(D). Proof. The proof is by induction on n ∈ N0 starting with the trivial case n = 0. We proceed to show that Det( Dˆ n+1 ) ∼ = Det( Dˆ n ). In Quillen’s definition of the determinant line bundle, only the eigenfunctions of the low eigenvalues are involved and the strategy is to identify the eigensections of the low eigenvalues of Dˆ n with those of Dˆ n+1 . The isotropic quantization of the polynomial τn2 + tn2 , is the harmonic oscillator, Hn , so D ˆ ∗ Dˆ n,b + H − 0 ˆ∗ ˆ n+1 n,b Dn+1,b Dn+1,b = D ˆ n,b Dˆ ∗ + H + . (5.4) 0 n+1 n,b are positive, with the smallest being simple. The eigensections The eigenvalues of Hn+1 ∗ ∗ with small eigenvalues are of the form of Dˆ n+1,b Dˆ n+1,b and Dˆ n+1,b Dˆ n+1,b
0 ϕn+1 ⊗ f + ( f b ) = , − ( f b ) = , (5.5) 0 ϕn+1 ⊗ Dˆ n,b f ∗ D ˆ n,b with eigenvalue less than 2. Note also that where f is an eigenfunction of Dˆ n,b ˆ on such an eigenfunction, Dn+1,b acts as ∗ D ˆ∗ iCn+1 n,b + ( f ) = − ( f ) (5.6) b b D ˆ n,b iCn+1 ∗ ϕ since Cn+1 n+1 = 0. For 0 < λ < 2, consider the open set
Uλ = {b ∈ B; λ is not an eigenvalue of Db∗ Db }.
(5.7)
+,k ∗ D ˆ k,b Let H[0,λ) denote the vector bundle over Uλ spanned by the eigenfunctions of Dˆ k,b
−,k with eigenvalues less than λ. Let H[0,λ) denote the vector bundle over Uλ spanned by ∗ the eigenfunctions of Dˆ k,b Dˆ k,b with eigenvalues less than λ. Then there are natural identifications ±,n ±,n+1 ± FU±,n ,λ : H[0,λ) f b −→ ( f b ) ∈ H[0,λ) .
(5.8)
Thus, directly from Quillen’s definition of the determinant bundle Dˆ n+1,b and Dˆ n,b have isomorphic determinant line bundles. 5.3. Adiabatic limit of Det( Dˆ n ). m (M/B; E, F) is a family of fully elliptic operators Proposition 5.3. If P ∈ psus(2n) with vanishing numerical index then the bundle over B × [0, 1] with fibre −∞ Pb, = Q ∈ sus(2n) (Z b , E b ); −m −1 (Z b ; Fb , E b ) for the isotropic product (5.9) ∃ (P + Q) ∈ psus(2n)
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167
is a principal G-bundle for the bundle of groups with fibre −∞ −∞ (Z b , E b ) = Id +A, A ∈ sus(2n) (Z b , E b ); Gsus(2n), −∞ (Z b , E b ) for the isotropic product ∃ (Id +A)−1 = Id +B, B ∈ sus(2n)
(5.10)
and the associated determinant bundle defined over > 0 extends smoothly down to = 0 and at = 0 is induced by the adiabatic determinant. Proof. This is just the smoothness of the ‘rescaled’ determinant (i.e. with the singular terms removed) down to = 0. We will now complete the proof of Theorem 2 in the Introduction which we slightly restate as Theorem 2 (Periodicity of the determinant line bundle). Let D ∈ 1 (M/B; E, F) be an elliptic family with vanishing numerical index, then for n ∈ N and > 0, Deta (Dn ) ∼ = Det( Dˆ n ) ∼ = Det(D). Proof. The existence of the second isomorphism follows from Proposition 5.2. The first follows from Proposition 5.3. 6. Eta Invariant In [13] a form of the eta invariant was discussed for elliptic and invertible once-suspended families of pseudodifferential operators. Applied to the spectral family (on the imaginary axis) of a self-adjoint invertible Dirac operator this new definition was shown to reduce to the original definition, of Atiyah, Patodi and Singer in [1] of the eta invariant of a single operator. Here, the definition in [13] is shown to extend to (fully) elliptic, invertible, product-suspended families. In §9 it is further extended to such productsuspended families in any odd number of variables. The extension to single-parameter product-suspended operators allows us to apply the definition to A + iτ, τ ∈ R, for A ∈ 1 (X ; E) an invertible elliptic selfadjoint pseudodifferential operator and check that this reduces to the spectral definition, now as given by Wodzicki ([21]). Again the extended (and below also the ‘adiabatic’) eta invariant gives a log-multiplicative function for invertible families η(AB) = η(A) + η(B)
(6.1)
and this allows us to show quite directly that the associated τ invariant is a determinant in the sense discussed above.
m,m (X ; E) is a product-suspended family it 6.1. Product-suspended eta. If B ∈ psus satisfies
∂N m−N ,m −N B(τ ) ∈ psus (X ; E) ∀ N ∈ N0 . ∂τ N
(6.2)
This implies that for N large, say N > dim X + m, the differentiated family takes values in operators of trace class on L 2 .
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Proposition 6.1. For any m, m ∈ Z, if N ∈ N is chosen sufficiently large then,
N ∂ m,m (X ; E) =⇒ Tr E B(τ ) ∈ C ∞ (R p ) B ∈ psus ∂τ N
(6.3)
has a complete asymptotic expansion (possibly with logarithms) as τ → ±∞ and the coefficient of T 0 in the expansion as T → ∞ FB,N (T ) =
T −T
Tr(B) = LIM FB,N (T ), T →∞
N tN t1 ∂ ... Tr E B(s) ds dt N . . . dt1 ∂s N 0 0
(6.4)
is independent of the choice of N and defines a trace functional Z,Z Z,Z Tr : psus (X ; E) −→ C, Tr([A, B]) = 0 ∀ A, B ∈ psus (X ; E)
which reduces to
(6.5)
−∞,−∞ Tr E (B(τ )) dτ ∀ A ∈ psus (X ; E). (6.6) R As already noted, ∂sN B(s) is a continuous family of trace class operators as soon
Tr(B) =
Proof. as N > dim X + m. Then (6.3) is a continuous function and further differentiation again gives a continuous family of trace class operators so the trace is smooth. To see that this function has a complete asymptotic expansion we appeal to the discussion of the structure of the kernels of such product-suspended families in Appendix B. −n−1,0 It suffices to consider the trace of a general element B ∈ psus (X ; E). Since the ∞ 2 kernels form a module over C (X ) we can localize in the base variable (not directly in the suspended variable since that has global properties). Localizing near a point away from the diagonal gives a classical symbol in the suspending variable with values in the smoothing operators. Since the trace is the integral over the diagonal this makes no contribution to (6.3). Thus it suffices to suppose that B is supported in the product of a coordinate neighbourhood with itself over which the bundle E is trivial. Locally (see (2.3)) the kernel is given by Weyl quantization of a product-type symbol so the trace becomes the integral of the sum of the diagonal terms and hence we need only consider 1 a(x, ξ, τ )d xdξ , (6.7) (2π )n where a is compactly supported in the base variables x. Now by definition, a is a smooth function, with compact support, on the product Rn × [R × Rn ; ∂(R × {0})]. Thus we can further localize the support of a on this blown up space. There are three essentially different regions, corresponding to the part of the boundary which arises from the radial compactification, the part arising from the blow up and the corner. The first of these regions corresponds to a true suspended family, as considered in [13]. In this region the variable |ξ | dominates, and |τ | ≤ C|ξ | on the support so the integral takes the form 1 1 φ(r τ )r n+1 f (x, ω, r, r τ )r −n−1 d xdωdr (2π )n 0 1 = sφ(R) f (x, ω, Rs, R)d xdωd R, s = 1/τ. (6.8) 0
Periodicity
169
Here, φ has compact support and f (with the factor of r n+1 representing the order −n−1) is smooth. The result is smooth in s = 1/τ, which corresponds to a complete asymptotic expansion with only non-negative terms. The second region corresponds to boundedness of the variable ξ with the function being a classical symbol (by assumption of order at most 0) in τ so integration simply gives a symbol 1 a(x, ξ, τ )d xdξ. (6.9) (2π )n The third region is the most problematic. Here the two boundary faces of the compactification are defined by r = 1/|ξ | and |ξ |/τ and with polar variables ω = ξ/|ξ |. Thus the integral takes the form r n+1 f (x, ω, r, s/r )r −n−1 d xdωdr ∈ C ∞ ([0, 1)s ) + (log s)C ∞ ([0, 1)s ), (6.10) where f is smooth and with compact support near 0 in the last two variables. This is a simple example of the general theorem on pushforward under b-fibrations in [12], or the ‘singular asymptotics lemma’ of Brüning and Seeley (see also [9]) and is in fact a type of integral long studied as an orbit integral. In any case the indicated regularity follows and this proves the existence of a complete asymptotic expansion, possibly with single logarithmic terms. It follows that the integral in (6.4) also has a complete asymptotic expansion as T → ∞; where in principle there can be factors of (log T )2 after such integration. Thus the coefficient of T 0 does exist, and defines Tr(B). Now if N is increased by one in the definition, the additional integral gives the same formula (6.4) except that a constant of integration may be added by the first integral. After N additional integrals, this adds a polynomial, so the result is changed by the integral over [−T, T ] of a polynomial. This is an odd polynomial, so has no constant term in its expansion at infinity. Thus the definition of Tr(B) is in fact independent of the choice of N . The trace identity follows directly from (6.4), since if B = [B1 , B2 ], then any derivative is a sum of commutators between operators with order summing to less than −n and the trace of such a term vanishes. Thus applied to a commutator (6.4) itself vanishes. Using this trace functional on product-suspended operators we extend the domain of the eta invariant. Proposition 6.2. The eta invariant for any fully elliptic, invertible element A ∈ m,m psus (X ; E) defined using the regularized trace 1 ˙ A˙ = ∂ A Tr(A−1 A), πi ∂τ is a log-multiplicative functional, in the sense of (6.1). η(A) =
(6.11)
Proof. Certainly (6.11) defines a continuous functional on elliptic and invertible productsuspended families. The log-multiplicativity, (6.1), follows directly since if B is another invertible product-suspended family then ∂(AB) ˙ + B −1 B˙ = B −1 A−1 AB ∂τ ˙ = η(A). and the trace identity shows that Tr(B −1 A−1 AB) (AB)−1
(6.12)
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6.2. η(A + iτ ) = η(A). To relate this functional on product-suspended invertible operators to the more familiar eta invariant for self-adjoint elliptic pseudodifferential operators we rewrite the definition in a form closer to traditional zeta regularization, starting with the regularized trace. Consider the meromorphic family t+−z of tempered distributions with support in [0, ∞). This family has poles only at the positive integers, with residues being derivatives of the delta function at the origin. For Re z sufficiently positive and non-integral, t+−z can be paired with the function FB,N (t) in (6.4), since this is smooth and of finite growth at infinity. This pairing gives a meromorphic function in Re z > C, with poles only at the natural numbers since the poles of t+−z are associated with the behaviour at 0, where FB,N is smooth. In fact this pairing g(z) = T+−z−1 , FB,N (T )
(6.13)
extends to be meromorphic in the whole complex plane. Indeed, dividing the pairing into two using a cut-off ψ ∈ Cc∞ ([0, ∞)) which is identically equal to 1 near 0, g(z) = T+−z−1 , ψ(T )FB,N (T ) + T+−z−1 , (1 − ψ(T )))FB,N (T ),
(6.14)
the first term is meromorphic with poles only at z ∈ N and the poles of the second term arise from the terms in the asymptotic expansion of FB,N (T ). Notice that there is no pole at z = 0 for the first term since the residue of T+−z−1 at z = 0 is a multiple of the delta function and FB,N (0) = 0. The pole at z = 0 for the second term arises exactly from the coefficient of T 0 in the asymptotic expansion so we see that Tr(B) = resz=0 g(z).
(6.15)
Any terms ak (log T )k for k ∈ N, in the expansion do not contribute to the residue since they integrate to regular functions at z = 0 plus multiples of z −k .
m,m Proposition 6.3. For B ∈ psus (X ; E) and any N > m − dim X − 1, the regularized trace is the residue at z = 0 of the meromorphic continuation from Re z >> 0, z ∈ / Z, of (t + i0) N −z (t − i0) N −z (−1) N +1 N , Tr E (∂t B(t) . (6.16) + (N − z) . . . (1 − z)(−z) 1 + e−πi z 1 + eπi z
Proof. Consider the identity t+−z−1 =
d N +1 −z+N 1 t . (N − z) . . . (1 − z)(−z) dt N +1 +
(6.17)
After inserting this into (6.14), integration by parts is justified (since (6.17) holds in the sense of distributions on the whole real line, supported in [0, ∞)), and shows that
N
N 1 ∂ ∂ g(z) = t+−z+N , (−1) N +1 Tr E B (t)−Tr B (−t), E N (N −z) . . . (1−z)(−z) ∂t ∂t N (6.18) where the pairings are defined, and holomorphic, for Re z large and z non-integral. Using the identity t+z =
(t + i0)z (t − i0)z + , 1 − e2πi z 1 − e−2πi z
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171
(6.18) becomes g(z) = D(t, z) =
(−1) N +1 D(t, z), Tr E (N − z) . . . (1 − z)(−z)
∂N B (t), ∂t N
(t + i0) N −z (t − i0) N −z + 1 − e2πi(N −z) 1 − e−2πi(N −z) (−t + i0) N −z (−t − i0) N −z + (−1) N . + (−1) N 2πi(N −z) 1−e 1 − e−2πi(N −z)
Now, (−t − i0)−z = eπi z (t + i0)−z so
eπi z 1 (t + i0) N −z + D(t, z) = 1 − e−2πi z 1 − e2πi z
1 e−πi z (t − i0) N −z , + + 1 − e2πi z 1 − e−2πi z which reduces (6.20) to (6.16).
(6.19)
(6.20)
This allows us to prove a result of which Theorem 3 in the introduction is an immediate corollary. Theorem 3. If A ∈ 1 (X ; E) is a self-adjoint elliptic and invertible pseudodifferential operator then η(A + iτ ), defined through (6.11) reduces to the (regularized) value at z = 0 of the analytic continuation from Re z >> 0 of sgn(a j )|a j |−z , (6.21) j
where the a j are the eigenvalues of A, in order of increasing |a j | repeated with multiplicities. Proof. With A(τ ) = A + iτ the eta invariant defined by (6.11) reduces to 1 1 η(A + iτ ) = Tr (A + iτ )−1 = resz=0 h(z), π π
(6.22)
where h(z) is the function (6.16) with B(t) = (A + it)−1 . Computing the N th derivative ∂N −1 (A + iτ ) = i(−1) N +1 N !(τ − i A)−N −1 . (6.23) ∂τ N The trace is therefore given, for any N > n, by
N ∂ −1 tr E = i(−1) N +1 N ! (A + iτ ) (τ − ia j )−N −1 . (6.24) N ∂τ j
This converges uniformly with its derivatives so can be inserted in the pairing (6.16) and the order exchanged. Thus a N (z)i(−1) N +1 N ! h(z) = lim τ N −z (τ − ia j )−N −1 dτ ↓0 R+i 1 + e−πi z j N +1 N! a N (z)i(−1) + lim τ N −z (τ − ia j )−N −1 dτ, (6.25) ↓0 R−i 1 + eπi z j
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where a N (z) =
(−1) N +1 . (N − z) . . . (1 − z)(−z)
Each of these contour integrals is actually independent of > 0 for smaller than the minimal |a j |. By residue computation, in the first sum by moving the contour to infinity in the upper half plane and in the second by moving the contour into the lower half plane ±2πi (N −z)···(1−z) e∓πi z/2 |a j |−z ±a j > 0 N! . (6.26) τ N −z (τ − ia j )−N −1 dτ = 0 ±a j < 0 R±i Inserting this into (6.25) shows that η(A + iτ ) is the residue at z = 0 of 1 sgn(a j )|a j |−z . z cos(π z/2)
(6.27)
j
By definition, the usual eta invariant, η(A), is the value at z = 0 of the continuation of the series in (6.27). This series is the analytic continuation of the trace of an entire family of classical elliptic operators of complex order −z (namely A−z ( + − − ), where ± are the projections onto the span of positive and negative eigenvalues) which can have only a simple pole at z = 0. In fact, here, it is known that there is no singularity, i.e. the residue vanishes. Even without invoking this we conclude the desired equality, since the explicit meromorphic factor in (6.27) is odd in z, so a pole in the continuation of the series would not affect the residue. 7. Universal η and τ Invariants That the differential of the eta invariant of a family of self-adjoint Dirac operators is a multiple of the first (odd) Chern class of the index, in odd cohomology of the base, of the family is well-known. In the case of the suspended eta invariant discussed in [13] and above, we show that the η invariant is, in appropriate circumstances, the logarithm of a determinant, which is to say a multiplicative function giving the first odd Chern class. Initially we show this in the context of classifying spaces for K-theory, then in the geometric context of (2n + 1)-fold suspended odd elliptic families. Consider again the algebra of once-suspended isotropic pseudodifferential operators of order 0 on Rn , with values in smoothing operators on a compact manifold X. This can be identified with the smooth functions on R2n+1 × X 2 and the subspace (7.1) I+ = A ∈ C ∞ (R2n+1 × X 2 ); A ∼ = 0 in {t ≤ 0} ∩ S2n × X 2 , is a subalgebra. Here, t is the suspending parameter and equality is in the sense of Taylor series at infinity on the compactified Euclidean space. Thus the subalgebra is just the sum of the smoothing ideal (identified with the functions vanishing to infinite order everywhere at the boundary) and the subalgebra of functions vanishing in t < 0. In fact I+ is also an ideal. We consider the corresponding group G+ = {B = Id +A, A ∈ I+ , B −1 = Id +B , B ∈ I+ }.
(7.2)
Periodicity
173
Now we may use the suspending variable t to identify the upper half-sphere {t > 0} ∩ S2n of the boundary of R2n+1 with R2n , {t > 0} ∩ S2n [(t, x, ξ )] −→ (X, ) = (x/t, ξ/t) ∈ R2n .
(7.3)
The inverse image under pull-back of S(R2n ) is then naturally identified with {a ∈ C ∞ (S2n ); a = 0 in t < 0}, where S2n is the boundary of the radial compactification of R2n+1 . This allows the space of formal power series S(R2n )[[t]] to be identified with the formal power series at the boundary of the subspace of C ∞ (R2n+1 ) consisting of the functions vanishing in t < 0. The same identifications carry over to the case of functions valued in the smoothing operators and so gives a short exact sequence of algebras −∞ (Rn × X ) sus
/ I+
/ S(R2n × X 2 )[[t]] .
(7.4)
Lemma 7.1. In (7.4), the product induced on the quotient is the standard product (valued in smoothing operators on X ) on R2n (i.e. the ‘Moyal product’). n ˆ Bˆ ∈ 0 Proof. Let A, B ∈ I+ be the symbols of two operators A, psus(1) (R × X ). Then the asymptotic expansion at infinity of the symbol of Aˆ Bˆ is given by the standard product ∞ 1 k ˆ ∼ σ ( Aˆ B) (D D − D D ) A(t, x, ξ )B(t, y, η) . (7.5) x η y ξ k!(2i)k k=0
x=y,η=ξ
Under the map (7.3), the asymptotic expansion (7.5) becomes an asymptotic expansion at {t > 0} ∩ S2n ⊂ R2n+1 , ∞ 1 1 k ˆ ∼ σ ( Aˆ B) (D D − D D ) A(t, t X, t)B(t, tY, t) (7.6) . X Y k!(2i)k t 2k X =Y,=
k=0
Thus, if A(t, t X, t) ∼
∞ ∞ 1 1 a (X, ), B(t, t X, t) ∼ bk (X, ) k tk tk k
(7.7)
k
are the asymptotic expansions of A and B at {t > 0} ∩ S2n ⊂ R2n+1 , then 1 1 k ˆ ∼ (D D − D D ) a (X, )b (Y, ) σ ( Aˆ B) X Y l m X =Y,= k!(2i)k t 2k+l+m k,l,m≥0
(7.8) ˆ at {t > 0} ∩ S2n ⊂ R2n+1 . But the right-hand is the asymptotic expansion of σ ( Aˆ B) side is precisely the standard product on S(R2n × X 2 )[[ε]] with ε = t12 . Corresponding to this exact sequence of algebras is the exact sequence of groups consisting of the invertible perturbations of the identity n −∞ G −∞ (Rn × X )[[t]]. sus (R × X ) −→ G+ −→ G
(7.9)
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Theorem 4. In this ‘delooping’ sequence, the first group is classifying for even K-theory, the central group is (weakly) contractible and the quotient is (therefore) a classifying group for odd K-theory; the eta invariant, defined as in (6.11), η : G+ −→ C
(7.10)
restricts to twice the index on the normal subgroup and eiπ η = deta is the adiabatic determinant on G −∞ (Rn × X )[[t]]. Proof. As a first step in the proof we consider the behaviour of the regularized trace. Lemma 7.2. The regularized trace Tr on the central algebra in (7.4) restricts to the integrated trace on the smoothing subalgebra and
∂b Tr b2n d X d (7.11) = ∂t R2n for any b ∈ I+ , where bk is the term of order k in the formal power series of the image in (7.4). Proof. When the parameter t is fixed, an element b ∈ I+ is actually a smoothing operator, since the asymptotic behavior on the surface where t is constant is determined by the equatorial sphere t = 0 at infinity. Thus the definition, from (6.4), of Tr(b) for any element b ∈ I+ may be modified by dropping all N integrals, i.e. we may take N = 0. Indeed, taking N > 0 and then integrating results in the case N = 0, plus a polynomial which, as noted earlier, does not affect the result. Carrying out the last integral by the fundamental theorem of calculus,
˙ = LIM Tr(b) b(T, x, ξ )d xdξ − b(−T, x, ξ )d xdξ , (7.12) T →∞
R2n
R2n
where LIM stands for the constant term in the asymptotic expansion. The second term in (7.12) corresponds to t < 0 where b is rapidly decreasing so does not contribute to the asymptotic expansion. Now, making the scaling change of variable in (7.3), transforms (7.12) to ˙ = LIM T −2n ˜ Tr(b) b(T, X, )d X d, (7.13) T →∞
R2n
where b˜ is the transformed function. Thus (7.13) picks out the term of homogeneity 2n ˜ This gives exactly (7.11). (in T ) in the formal expansion of b. 1 Now, by definition, the eta invariant is πi Tr(a −1 a). ˙ It follows directly that restricted to the smoothing subgroup this lies in 2Z. Thus D = eiπ η does indeed descend to the quotient group in (7.9). This group is connected, so to check that it reduces to the ‘adiabatic’ determinant defined earlier we only need check the variation formula, both being 1 on the identity. Along a curve a(s),
1 d d a˙ da d −1 da η(a(s)) = − a −1 a −1 a˙ = Tr (a ) . (7.14) Tr a −1 ds πi ds ds dt ds
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175
Thus the identity (7.11) shows that
d a˜ d η(a(s)) = Tr (a(s) ˜ )2n , ds ds
(7.15)
where a˜ is the image of a in the third group in (7.9). The identity term in a does not affect the argument since it is annihilated by d/ds. Since the right hand side of (7.15) is the variation formula for the logarithm of the adiabatic determinant this proves the theorem. 8. Geometric η and τ Invariants Returning to the ‘geometric setting’ of a fibration (6) with compact fibres, consider a m,m totally elliptic family A ∈ psus (M/B; E, F). Although we allow for operators between different bundles here, (6.11) is still meaningful as a definition of the eta invariant if A is invertible. Consider the principal bundle, of the type discussed above, G −∞ sus (M/B; E)
A
(8.1)
ν
B
with fibre −∞ −m,−m Ab = A + Q; Q ∈ sus (Z b ; E b , Fb ),(A + Q)−1 ∈ psus (Z b ; Fb , E b ) . (8.2) Proposition 8.1. The eta invariant, defined by (6.11), is a smooth function on A such that for the fibre action of the structure group at each point η(A(Id +L)) = η(A) + 2 ind(Id +L)
(8.3)
τ = eiπ η : B −→ C∗
(8.4)
so projects to
which represents the first odd Chern class of the index bundle of the family A. In particular this result applies to an elliptic, self-adjoint, family of pseudodifferential operators of order 1 by considering the spectral family. Proof. That η : A −→ C is well-defined follows from the discussion above as does the multiplicativity (8.3). Thus, τ is well-defined as a function on B and it only remains to check the topological interpretation. Note that the fibre of A is non-empty at each point of the base. In fact it is always possible to find a global smoothing perturbation to make the family invertible, although only when the families index vanishes is this possible with a smoothing perturbation of compact support in the parameter space. Thus, in complete generality, it is possible to choose a smooth map Q + : R −→ −∞ (M/B; E, F) such that Q + (t) = 0 for t << 0, Q + (t) = Q + (T ) for t ≥ T >> 0,
(A(t) + Q(t))−1 ∈ −m,−m (M/B; E, F) ∀ t ∈ R.
(8.5)
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This follows directly from the fact that the index bundle, over R× B, is trivial for t << 0 and so is trivial over R × B, but defines a generally non-trivial index class in K 1 (B). In fact the index class of the family A is represented by the map (A(T ) + Q + (T ))−1 A(T ) ∈ G −∞ (M/B; E).
(8.6)
Namely, if this family is deformable to the identity in this bundle of groups then there is a perturbation of compact support in t making the original family invertible. The existence of Q + may be directly related to a larger principal bundle with bundle −∞ of structure groups Gsus,+ (M/B; E) with fibre
Id +Q ; Id +Q b ∈ C ∞ (R; G −∞ (Z b ; E b ), ρ(t)Q(t) ∈ S(R; −∞ (Z b ; E b ), ∃ Q 0 ∈ −∞ (Z b ; E b ), (1 − ρ(t))(Q(t) − Q 0 ) ∈ S(R; −∞ (Z b ; E b )) . (8.7)
Here ρ(t) ∈ C ∞ (R) is equal to 1 in t < −1 and vanishes in t > 1. Thus the short exact sequence of groups π∞
−∞ −∞ G −∞ (M/B; E) sus (M/B; E) −→ G +,sus (M/B; E) −→ G
(8.8)
is the ‘delooping sequence’ for G −∞ (M/B; E). In particular the central group is weakly contractible and we may consider the enlarged principal bundle G −∞ +,sus (M/B; E)
A+
(8.9)
B −∞ defined by replacing G −∞ sus above by G +,sus . The existence of Q + shows that this bundle is trivial, i.e. has a global section
q : B −→ A+ which induces a ‘classifying bundle map’ −∞ −∞ (M/B; E), q(A ˜ b + Q b ) = (Ab + Q +,b )−1 (Ab + Q b ) ∈ G+,sus (Z b ; E b ). q˜ : A −→ G+,sus
Now, the definition and basic properties of the eta invariant given by (6.11) are quite insensitive to the enlargement of A to A+ and so still define a smooth function η+ : −∞ A+ −→ C. The same is true for the group G+,sus (M/B; E), defining the corresponding −∞ function η˜ : G+,sus (M/B; E) −→ C and the discussion of multiplicativity shows that ˜ η = η+ ◦ q ◦ ν + η˜ ◦ q.
(8.10)
From the fundamental theorem of calculus, ∗ d log det, iπ d η˜ = π∞
(8.11)
so we conclude from (8.10) that τ = eiπ η = eiπ η
+ ◦q◦π
(π∞ q) ˜ ∗ det
(8.12)
defines the same cohomology class as the determinant on the classifying group, i.e. the first odd Chern class of the index bundle.
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177
9. Adiabatic η We may further extend the discussion above by replacing the once-product-suspended spaces by (2n + 1)-times product-suspended spaces using the isotropic quantization in 2n of the variables, as in Theorem 5 applied to a decomposition R2n+1 = R × R2n with the standard symplectic form used on R2n . Let A[[]] be the principal bundle of invertible perturbations for the family A with respect to the star product from (C.3).
m,m Proposition 9.1. If A ∈ psus(2n+1) (M/B; E, F) is a fully elliptic family and (6.11) is used, with the product interpreted as the parameter-dependent product of Theorem 5 for the symplectic form on R2n then the resulting eta invariant on the bundle of smoothing perturbations has an asymptotic expansion as ↓ 0 which projects to
η : A[[]] −→ −n C[[]]
(9.1)
which has constant term the adiabatic eta invariant ηa(n) : A[[]] −→ C
(9.2)
which generates the first odd Chern class of the index bundle. Proof. This is essentially a notational extension of the results above.
In particular (4.2) is a consequence of this result and Bott periodicity. Namely, given an 2n product-suspended family we may always choose a smoothing family, analogous to Q + in (8.5) which is Schwartz in the second 2n − 1 variables and in the first is Schwartz at −∞ and of the form Q 0 + Q with Q Schwartz at +∞ and Q 0 constant in the first variable (and Schwartz in the remainder). By Bott periodicity, the even index of the family is the odd K-class on R2n−1 × B given by the product (A(t) + Q 0 )A(t)−1 for t large. Then (4.2) follows by an elementary computation and the proof of Lemma 4.2 follows directly. Appendix A: Symbols and Products By choice of a quantization map, spaces of pseudodifferential operators on a compact manifold can be identified, modulo smoothing operators, with the appropriate spaces of symbols on the cotangent bundle as in (2.3). It is important to discuss, and carefully distinguish between, several classes of such symbols and operators. To prepare for this we describe here classes of product-type symbols for a pair of vector spaces; subsequently this is extended to the case of vector bundles. For a real vector space V, the space of classical symbols of order 0 on V is just C ∞ (V ), the space of smooth functions on the radial compactification. In terms of any 1 Euclidean metric on V, ρ(v) = (1 + |v|2 )− 2 is a defining function for the boundary of V and the space of symbols of any complex order z on V is S z (V ) = ρ −z C ∞ (V ).
(A.1)
If W is a second real vector space then we may consider the radial compactification V × W and corresponding symbol spaces S z (V × W ). The natural projection πW : V × W −→ W does not extend to a map from V × W to W and correspondingly classical symbols on W do not generally lift to be classical symbols on V × W. Rather
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V → V × W may be considered as an embedded submanifold, simply the closure (of the preimage in V × W ) of V × {0}. On the other hand there is certainly a smooth projection from V × W to W ; the smooth functions, S 0 (V ; S 0 (W )) = S 0 (W ; S 0 (V )) = C ∞ (V × W ))
(A.2)
on this space are symbols on V with values in the symbols on W (or vice-versa). The main space we wish to consider here has some properties between these two compactifications of V × W. Namely, in terms of radial (real) blow-up (in the sense of [11]), we set V
V × W = [V × W ; ∂(V × {0})].
(A.3)
This manifold with corners has two boundary faces (unless one or both of the factors is one-dimensional in which case either or both of the boundary hypersurfaces may have two components). We use a superscript V to refer to the new boundary hypersurface produced by the blow-up in (A.3). Lemma A.1. The projection πW : V × W −→ W extends to a smooth map π W : V V × W −→ W which is a fibration (with fibres which are manifolds with boundary) and in terms of Euclidean metrics on V and W the functions
ρV (v, w) =
1 + |w|2 1 + |v|2 + |w|2
21
1
and ρr (v, w) = (1 + |w|2 )− 2
extend from V × W to be smooth functions on V V × W and are defining functions for the two boundary faces. Proof. To check the first statement of the lemma, notice that the projection V × W → V has a smooth extension pW : V × W \ ∂(V × {0}) → W which is a fibration with typical fibre given by V. Blowing up the submanifold ∂(V ×{0}) in V × W exactly allows us to extend pW to a fibration π W : V V × W −→ W with typical fibre given by V . Indeed, in V × W near the submanifold ∂(V × {0}), we can consider the generating functions (i.e. everywhere containing a coordinate system) vˆ =
v 1 w , σV = , w = = σV w. 1 1 |v| 2 (1 + |v| ) 2 (1 + |v|2 ) 2
The blow up amounts to introducing polar coordinates r V = (σV2 + w 2 ) 2 , (ϕ, θˆ ) = ( 1
σV w , ) rV rV
Periodicity
179
so that the blow-down map is given locally by V
V × W = [V × W ; ∂(V × {0})] (v, ˆ r V , ϕ, θˆ ) −→ (v, ˆ σV = r V ϕ, w = r V θˆ ).
In these polar coordinates, and for r V > 0, the fibration pW is given by ˆ r V , ϕ, θˆ ) = pW (v,
θˆ
ϕ
, 1 1 (ϕ 2 + |θˆ |2 ) 2 (ϕ 2 + |θˆ |2 ) 2
∈ W,
(A.4)
where we have used the identification of W with the upper half-sphere which is the closure of the image W w −→ (
1 (1 + |w|2 )
1 2
,
w 1
(1 + |w|2 ) 2
) ∈ {(a, b) ∈ R × W ; a ≥ 0, a 2 + |b|2 = 1}.
Thus, pW extends to r V = 0 to give the desired fibration. 1 It follows from this that a defining function for the boundary of W such as (1+|w|2 )− 2 lifts from W to be smooth and to define the ‘old’ boundary hypersurface, the one not 1 produced by the blow up. Now (1 + |w|2 + |v|2 ) 2 is a smooth boundary defining function on V × W . It therefore lifts under the blow up in (A.3) to be the product of defining functions for both boundary hypersurfaces and so
ρV (v, w) =
1 + |w|2 1 + |v|2 + |w|2
21
is a boundary defining function for the new boundary produced by the blow-up.
Now we define general spaces of ‘partial-product’ symbols by
S z,z (V V × W ) = ρrz ρVz C ∞ (V V × W ).
(A.5)
Directly from this definition,
S z,z (V V × W ) · S ζ,ζ (V V × W ) = S z+ζ,z +ζ (V V × W ).
(A.6)
Two of the ‘remainder’ classes have simpler characterizations. Namely S −∞,z (V V × W ) = C˙∞ (W ; S z (V )), S −∞,−∞ (V V × W ) = C˙∞ (V × W ) = S(V × W ).
(A.7)
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Appendix B: Product Suspended Operators We can now introduce a generalization of the ‘suspended’ algebra considered in [13] and in [15] (an algebra similar to the suspended algebra was already introduced by Shubin in [20]). The d-fold suspended pseudodifferential algebra on a compact manifold X may be viewed as a space of smooth maps from Rd into k (X ; E, F) in which the parameters (which we think of as the base variables for a fibration) appear as ‘symbolic variables’. The inverse Fourier transform identifies the suspended space k d ˇk sus(d) (X ; E, F) ⊂ (R × X ; E, F)
directly, as is done in [13], with the elements which are translation-invariant in Rd and have convolution kernels vanishing rapidly at infinity, with all derivatives, in these variables; this space may also be defined directly as in (2.3). The subspace of smoothing operators is −∞ sus(d) (X ; E, F) = S(Rd × X 2 ; Hom(E, F) ⊗ R )
in terms of the Schwartz space. Then the finite-order operators may be specified, up to smoothing terms, by Weyl quantization as qg : ρ −k C ∞ (Rd × T ∗ X ; π ∗ hom(E, F)) a −→ k (2π )−n χ eiv(x,y)·ξ a(m(x, y), ζ, ξ )dξ dg ∈ sus(d) (X ; E, F), T∗X
(B.1)
where the symbol space is compactified in the joint fibre Rd × Tx∗ X. The resulting full symbol sequence is as in (2.4) except that the formal power series have coefficients on the sphere bundle of R p × T ∗ X ; the parameters do not affect the operators B j , acting on T ∗ X, appearing in the product. If A ∈ 1 (X ; E) is a first order pseudodifferential operator and τ is the suspension 1 1 variable for sus(1) (X ; E), then A + iτ is not in general an element of sus(1) (X ; E). 1 In fact, A + iτ ∈ sus(1) (X ; E) if and only if A is a differential operator. Similarly, for A ∈ 1 (X ; E, F), the operator
it + τ A∗ (B.2) A it − τ 1 is in sus(2) (X ; E ⊕ F) if and only if A is a differential operator. This restriction to differential operators is unfortunate since the operator A +iτ arises in the alternative definition of the eta invariant as described in Sect. 6, while in Sect. 5 the operator (B.2) is used to implement Bott periodicity for determinant line bundles. For these reasons, and others, we pass to the wider context of product-suspended operators. We first need to enlarge the space of symbols as in Appendix A. Identifying X with the zero section of T ∗ X , consider the blown-up space X
Rd × T ∗ X = [Rd × T ∗ X ; ∂Rd × X ],
where Rd × T ∗ X is the radial compactification of Rd × T ∗ X fibre by fibre and Rd × X ⊂ Rd × T ∗ X
(B.3)
Periodicity
181
is the closure of Rd × X in Rd × T ∗ X . In terms of a Riemannian metric g and the Euclidean metric on Rd , Lemma A.1 generalizes directly to Lemma B.1. The projection Rd × T ∗ X → T ∗ X extends to a smooth map π T ∗ X : X Rd × T ∗ X −→ T ∗ X which is a fibration with typical fibre Rd , and the smooth functions 1
ρsus (v, w) =
(1 + |w|2 ) 2
1
(1 + |v|2 + |w|2 )
1 2
, ρr (v, w) = (1 + |w|2 )− 2 , v ∈ Rd , w ∈ T ∗ X,
define the two boundary faces. Proof. This results from the invariance of the construction in Appendix A under those linear transformations of V × W which leave V invariant, so Lemma A.1 extends to the case of a vector bundle. For z, z ∈ C, the space of (partially) product-type symbols with values in a vector bundle over X is then
−z ∞ X d S z,z ( X Rd × T ∗ X ; U ) = ρr−z ρsus C ( R × T ∗ X ; U ).
(B.4) 1
On Rd × X × X , consider the boundary defining function ρτ (τ ) = (1 + |τ |2 )− 2 . Let E and F be smooth complex vector bundles on X. For z ∈ C set
−∞,z (X ; E, F) = ρτ−z C ∞ (Rd × X × X ; Hom(E, F) ⊗ R X ), psus(d)
(B.5)
where R X = π3∗ X, π3 being the projection on the third factor, and X being the bundle of densities on X. This is the space of smoothing operators (defined as usual through their kernels) on X depending symbolically on d parameters; which we identify as the product-suspended operators of order −∞ on X. Definition B.2. The general spaces of product d-suspended pseudodifferential operators of order k, k ∈ Z acting from S(Rd × X ; E) to S(Rd × X ; F) is
k,k −∞,k psus(d) (X ; E, F) = qg (S k,k ( X Rd × T ∗ X ; hom(E, F))) + psus(d) (X ; E, F),
where qg is the Weyl quantization (B.1) applied to these more general symbol spaces. We limit attention to integral orders here only because it is all that is needed. Pseudodifferential operators are included in the product-suspended operators k,0 (X ; E, F), k (X ; E, F) ⊂ psus(d)
being independent of the parameters. For integers l ≤ k, l ≤ k , there are inclusions
l,l k,k (X ; E, F) ⊂ psus(d) (X ; E, F). psus(d)
Furthermore, as we will see below in Theorem 5, product d-suspended operators compose in the expected way
k,k l,l k+l,k +l psus(d) (X ; E, F) ◦ psus(d) (X ; G, E) ⊂ psus(d) (X ; G, F).
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Suspended operators are particular instances of product suspended operators, k,k k sus(d) (X ; E, F) ⊂ psus(d) (X ; E, F), k ∈ Z,
and −∞,−∞ −∞ (X ; E, F) = psus(d) (X ; E, F). sus(d)
Product d-suspended pseudodifferential operators are intimately related with the algebra of product-type operators introduced in [16]. More precisely, consider the projection φ : Rd × X → Rd
(B.6)
as a fibration. If E and F are smooth complex vector bundles on X, then as discussed in [16], to such a fibration one can associate the space of product-type pseudodifferential operators of order (k, k )
k,k d φ− p (R × X ; E, F)
acting from Cc∞ (Rd × X ; E) to C ∞ (Rd × X ; F). Given τ ∈ Rd , let Tτ : Rd × X → Rd × X denote the translation in the first factor Tτ (t, x) = (t − τ, x). We can consider the product-type pseudodifferential operators which are translation-invariant in the Euclidean variable, that is, satisfying Tτ∗ (A f ) = ATτ∗ f, ∀ τ ∈ Rd , f ∈ Cc∞ (Rd × X ; E).
(B.7)
In terms of the Schwartz kernel K A of A, this means that K A acts by convolution in the first factor A f (x, t) = K A (t − s, x, x ) f (x , s)ds, Rd
X
is a density in the x variable. Now one can ask in addition that this convolution
where K A kernel decay to all orders at infinity
K A ∈ Cc−∞ (Rd × X 2 ; Hom(E, F) ⊗ R X ) + S(Rd × X 2 ; Hom(E, F) ⊗ R X ). (B.8) This leads to the following characterization of product d-suspended operators. Lemma B.3. Fourier transformation in the suspension variables ˆ ( A(τ ) f )(x) = e−itτ K A (t, x, x ) f (x )dt, τ ∈ Rd X
Rd
is an isomorphism of the space of translation-invariant product-type pseudodifferential operators satisfying (B.8) onto the d-parameter product-suspended pseudodifferential operators; it preserves products.
Periodicity
183
Proof. Modulo small changes of notation, this is the same as for suspended operators. One advantage of the alternative definition through Lemma B.3 is that the Fredholm theory for product d-suspended operators follows almost immediately from the corresponding Fredholm theory for product-type operators. Indeed, the principal symbol map and the base family map for product-type operators gives via the inclusion (using the ˇ psus(d) (X ; E, F) ⊂ k,k (Rd × X ; E, F) a corresponding inverse Fourier transform) φ− p symbol map and base family map for product d-suspended operators. For the convenience of the reader, we will define these directly without referring to product-type operators. Of the two boundary faces of X R p × T ∗ X , the ‘old’ boundary, or really its blow-up, Bσ = [S(Rd × T ∗ X ); S(Rd ) × X ] with X being the zero section of T ∗ X, carries the replacement for the usual principal symbol. In terms of a quantization map as above, this is given by the restriction of the full symbol a ∈ S m,m (Rd T ∗ X ; E, F) of an operator A = qg (a) to this boundary face,
m,m m,m σm,m : psus(d) (X ; E, F) −→ Spsus(d) (X ; E, F)
(B.9)
with
m,m Spsus(d) (X ; E, F) = C ∞ (Bσ ; hom(E, F) ⊗ N −m ⊗ Nff−m ),
where N is the normal bundle to Bσ and and Nff is the normal bundle of the ‘new’ boundary, which is canonically identified with the normal bundle to the boundary of Bσ . Both are trivial bundles. This corresponds to the multiplicative short exact sequence
σm,m
m−1,m m,m m,m 0 −→ psus(d) (X ; E, F) −→ psus(d) (X ; E, F) −→ Spsus(d) (X ; E, F)→0.
(B.10)
k,k (X ; E, F) is elliptic if its principal A product d-suspended operator A ∈ psus(d) symbol σm,m (A) is invertible. Ellipticity alone does imply that the family is Fredholm for each value of the parameter but, as for product-type operators, it does not suffice to allow the construction of a parametrix modulo Schwartz-smoothing errors. There is a second symbol map which takes into account the behavior of the operator for large values of the suspension parameters. Let Bsus ⊂ X Rd × T ∗ X denote the ‘new’ boundary, which is the ‘front face’ produced by the blow up. The fibration of Lemma B.1 gives a canonical identification of Bsus with S(Rd ) × T ∗ X . Thus, the restriction map (using a boundary defining function ρsus for Bsus ) becomes
R : S k,k ( X Rd × T ∗ X ; hom(E, F)) a −→ ρm a ∈ C ∞ (S(Rd ); S k,k (T ∗ X ; hom(E, F))). sus
Bsus
k,k
(B.11)
−∞,k
Given A = qg (a1 ) + A2 ∈ psus(d) (X ; E, F) with A2 ∈ psus(d) (X ; E, F) and
a1 ∈ S k,k ( X Rd × T ∗ X ; hom(E, F)) the base family is defined by k A2 ) B ∈ C ∞ (S(Rd ); m (X ; E, F)). L(A) = qg (R(a1 )) + ρsus sus
(B.12)
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Proposition B.4. The base family (B.12) is independent of choices and corresponds to the multiplicative short exact sequence
k,k −1 k,k 0 −→ psus(d) (X ; E, F) −→ psus(d) (X ; E, F) −→ C ∞ (S(Rd ); k (X ; E, F)) −→ 0, L
(B.13) so
m,m k,k L(A ◦ B) = L(A) ◦ L(B), A ∈ psus(d) (X ; E, F), B ∈ psus(d) (X ; G, E).
Proof. The fact that there is a short exact sequence is essentially by definition of L . The fact that L is a homomorphism follows by very simple ‘oscillatory testing’. Namely, if k,k u ∈ C ∞ (X ; E) and A ∈ psus( p) (X ; E, F) then Au ∈ ρτ−k C ∞ (R p × X ; F) and L(A)u = ρτk Au ∂ R p ∈ C ∞ (S p−1 × X ; F).
(B.14)
k,k Definition B.5. The joint symbol J (A) of an operator A ∈ psus(d) (X ; E, F) is the combination of its principal symbol and its base family J (A) = (σ (A), L(A)) where σ (L(A)) = σ (A) B . σ
An operator A is said to be fully elliptic if its joint symbol is invertible. The important feature that motivates the introduction of product-suspended operators (as opposed to suspended operators) is the following lemma. 1,1 Lemma B.6. If A ∈ 1 (X ; E) then the one-parameter family τ −→ A + iτ ∈ psus(1) (X ; E) and if B ∈ 1 (X ; E, F), then the two-parameter family
B∗ 1,1 ˆ τ ) = it + τ ∈ psus(2) (X ; E ⊕ F). (t, τ ) −→ B(t, B it − τ
Moreover if A is self-adjoint and elliptic (respectively B is elliptic) then A + iτ (respecˆ is fully elliptic. tively B) In fact, it suffices that all the eigenvalues of the symbol of A have a nonvanishing real part for A + iτ to be fully elliptic. Proof. Fix a quantization qg . In the first case a ∈ ρ −1 C ∞ (T ∗ X ; π ∗ hom(E)) exists such that (A − qg (a)) ∈ −∞ (X ; E). Then −∞,1 (X ; E), a + iτ ∈ S 1,1 ( X R × T ∗ X ; E) and A + iτ − qg (a + iτ ) ∈ psus(1) 1,1 which shows that A + iτ ∈ psus(1) (X ; E). The symbol of A + iτ is invertible if σ (A)
has no eigenvalues in iR and its base family is ±i Id at the two components of ∂Rτ . Thus A + iτ is fully elliptic. In the second case, choose b ∈ ρ −1 C ∞ (T ∗ X ; π ∗ hom(E, F)) such that B − qg (b) ∈ −∞ (X ; E).
Periodicity
Then
185
it + τ bˆ = b
b∗ it − τ
∈ S 1,1 ( X R2 × T ∗ X ; E, F)
ˆ ∈ −∞,1 (X ; E, F), which shows that Bˆ ∈ 1,1 (X ; E, F). To see and Bˆ − qg (b) psus(2) psus(1) that Bˆ is fully elliptic when B is elliptic, consider the invertible operator
∗ 0 B B + t2 + τ 2 + 1 2,2 ∗ ˆ ˆ ∈ psus(2) Q = B B+1= (X ; E ⊕ F). 0 B B∗ + t 2 + τ 2 + 1 Then −2,−2 (Q −1 Bˆ ∗ ) Bˆ − Id E⊕F = −Q −1 ∈ psus(2) (X ; E ⊕ F),
ˆ −1 = J (Q −1 Bˆ ∗ ) exists, which shows that Bˆ is fully elliptic. so that J ( B)
Appendix C: Mixed Isotropic Operators Next we proceed to the ‘parameter quantization’ of these spaces of product suspended operators. That is, we introduce a new product depending on the choice of an antisymmetric form on R p . These products are used above in the identification of the determinant bundle, as constructed in the product 2n-suspended case, with the determinant bundle as introduced by Quillen. To do so we use an adiabatic limit, with a parameter which passes from the quantized to the unquantized case discussed above; for the isotropic algebra itself such degenerations are treated in [7] and as shown there implements Bott periodicity. So, to introduce these spaces we simply combine (2.3) and its Euclidean analogue (2.20). Note that the quantization map will be global in the Euclidean variables but can only be local near the diagonal in the manifold. In defining these spaces we use the formula for the action of an operator by Weyl quantization in (2.21). Proposition C.1. Let X be a compact manifold E and F complex bundles over X , then for any p ∈ N combining (2.21) with the operator product gives a smooth family of associative products m 1 ,m
m 2 ,m
m 1 +m 2 ,m 1 +m 2
1 2 2 (Rn ) × psus(2n) (X ; F, G) × psus(2n) (X ; E, F) −→ psus(2n)
(X ; E, G). (C.1)
This follows by combining essentially standard treatments of the composition of pseudodifferential operators with those of the ‘isotropic’ operators on Rn . We are especially interested in the ‘adiabatic limit’ where the general ω is replaced by ω for a fixed antisymmetric form. The cases which occur above are where p is even and ω is non-degenerate, or where p is odd and ω has maximal rank. In this case we state the corresponding corollary of the result above (see also [10] and [7]). Theorem 5. For any fixed antisymmetric form on R p , the composition (C.1) induces a smooth 1-parameter family of quantized products
k,k l,l k+l,k +l [0, 1] × psus( p) (X ; F, G) × psus( p) (X ; E, F) −→ psus( p) (X ; E, G)
(C.2)
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and as ↓ 0 there is a Taylor series expansion A ◦ B(u) ∼
∞ (−i)k k=0
2k k!
ω(Dv , Dw )k A(v)B(w)v=w=u ,
(C.3)
in particular, when = 0 the product reduces to the usual parameterized product of suspended operators.
References 1. Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Camb. Phils. Soc. 78(3), 405–432 (1975) 2. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Springer-Verlag, Berlin (1992) 3. Bismut, J.-M., Freed, D.: The analysis of elliptic families, II. Commun. Math. Phys. 107, 103–163 (1986) 4. Bismut, J.-M., Freed, D.: The analysis of elliptic families: Metrics and connections on determinant bundles. Commun. Math. Phys. 106, 159–176 (1986) 5. Bott, R., Seeley, R.: Some remarks on the paper of Callias. Commun. Math. Phys. 62, 235–245 (1978) 6. Dai, X., Freed, D.S.: and determinant lines. J. Math. Phys. 35(10), 5155–5194 (1994) 7. Epstein, C.L., Melrose, R.B.: The Heisenberg algebra, index theory and homology. This became [8] without Mendoza as coauthor 8. Epstein, C.L., Melrose, R.B., Mendoza, G.: The Heisenberg algebra, index theory and homology. In preparation 9. Grieser, D., Gruber, M.J.: Singular asymptotics lemma and push-forward theorem. In: Approaches to singular analysis (Berlin, 1999), Oper. Theory Adv. Appl., Vol. 125, Basel: Birkhäuser, 2001, pp. 117– 130 10. Hörmander, L.: The Weyl calculus of pseudo-differential operators. Comm. Pure Appl. Math. 32, 359–443 (1979) 11. Melrose, R.B.: Analysis on manifolds with corners. In preparation 12. Melrose, R.B.: Calculus of conormal distributions on manifolds with corners. Internat. Math. Res. Notices 1992(3), 51–61 (1992) 13. Melrose, R.B.: The eta invariant and families of pseudodifferential operators. Math. Res. Lett. 2(5), 541–561 (1995) 14. Melrose, R.B., Rochon, F.: Boundaries, eta invariant and the determinant bundle. Preprint, http://arxiv.org/ list/math.DG/0607480, 2006 15. Melrose, R.B., Rochon, F.: Families index for pseudodifferential operators on manifolds with boundary. IMRN (22), 1115–1141 (2004) 16. Melrose, R.B., Rochon, F.: Index in K-theory for families of fibred cusp operators. K-Theory 37, 25–104 (2006) 17. Pressley, A., Segal, G.: Loop groups. Oxford Science publications, Oxford Univ. Press, Oxford (1986) 18. Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl. 19, 31–34 (1985) 19. Seeley, R.T.: Complex powers of an elliptic operator. Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Providence, R.I.: Amer. Math. Soc., 1967, pp. 288–307 20. Shubin, M.A.: Pseudodifferential operators and spectral theory. Berlin-Heidelberg-New York: SpringerVerlag, 1987, Moscow, Nauka: 1978 21. Wodzicki, M.: Spectral asymmetry and zeta functions. Invent. Math. 66, 115–135 (1982) Communicated by L. Takhtajan
Commun. Math. Phys. 274, 187–216 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0261-z
Communications in
Mathematical Physics
Fast Soliton Scattering by Delta Impurities Justin Holmer, Jeremy Marzuola, Maciej Zworski Mathematics Department, Evans Hall, University of California, Berkeley, CA 94720, USA. E-mail:
[email protected] Received: 9 August 2006 / Accepted: 22 November 2006 Published online: 31 May 2007 – © Springer-Verlag 2007
Abstract: We study the Gross-Pitaevskii equation with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L 2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons. 1. Introduction We study the Gross-Pitaevskii equation (NLS) with a repulsive delta function potential (q > 0) i∂t u + 21 ∂x2 u − qδ0 (x)u + u|u|2 = 0 (1.1) u(x, 0) = u 0 (x). As initial data we take a fast soliton approaching the impurity from the left: u 0 (x) = eivx sech(x − x0 ), v 1, x0 0.
(1.2)
Because of the homogeneity of the problem this covers the case of the general soliton profile Asech(Ax). The quantum transmission rate at velocity v is given by the square of the absolute value of the transmission coefficient, see (2.2) below, Tq (v) = |tq (v)|2 =
v2
v2 . + q2
(1.3)
For the soliton scattering the natural definition of the transmission rate is given by Tqs (v) = lim
t→∞
u(t)x>0 2L 2 u(t)2L 2
=
1 lim u(t)x>0 2L 2 , 2 t→∞
(1.4)
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provided that the limit exists. We expect that it does and that for fixed q/v, there is a σ > 0 such that Tqs (v) = Tq (v) + O(v −σ ), as v → +∞. (1.5) Based on the comparison with the linear case (see (2.21) below) and the numerical evidence [9] we expect (1.5) with σ = 2. Towards this heuristic claim we have Theorem 1. Let δ satisfy 23 < δ < 1. If u(x, t) is the solution of (1.1) with initial condition (1.2) and x0 ≤ −v 1−δ , then for fixed q/v, 3 1 v2 |u(x, t)|2 d x = 2 + O(v 1− 2 δ ), as v → +∞, (1.6) 2 2 x>0 v +q uniformly for |x0 | + v −δ ≤ t ≤ (1 − δ) log v. v We see that by taking δ very close to 1, we obtain an asymptotic rate just shy of v −1/2 . More precisely, we show that there exists v0 = v0 (q/v, δ), diverging to +∞ as δ ↑ 1 and q/v → +∞, such that for fixed q/v, if v ≥ v0 , then 1 3 v 2 2 ≤ cv 1− 2 δ . |u(x, t)| d x − 2 2 + q2 v x>0 The constant c appearing here is independent of all parameters (q, v, and δ). We have conducted a numerical verification of Theorem 1 – see Fig. 2. It shows that the approximation given by (1.6) is very good even for velocities as low as ∼ 3, at least for def
0.6 ≤ α = q/v ≤ 1.4. A more elaborate numerical analysis will appear in our forthcoming paper [9]. Our second result shows that the scattered solution is given, on the same time scale, by a sum of a reflected and a transmitted soliton, and of a time decaying (radiating) term – see the fourth frame of Fig. 1. This is further supported by a forthcoming numerical study [9]. In previous works in the physics literature (see for instance [2]) the resulting waves were only described as “soliton-like”. Theorem 2. Under the hypothesis of Theorem 1 and for |x0 | + 1 ≤ t ≤ (1 − δ) log v, v we have, as v → +∞,
3 u(x, t) = u T (x, t) + u R (x, t) + O L ∞ (t − |x0 |/v)−1/2 + O L 2x (v 1− 2 δ ), x u T (x, t) = eiϕT ei xv+i(AT −v 2
2 )t/2
u R (x, t) = eiϕ R e−i xv+i(A R −v 2
A T sech(A T (x − x0 − tv)),
2 )t/2
A R sech(A R (x + x0 + tv)),
(1.7)
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Fig. 1. Numerical simulation of the case q = v = 3, x0 = −10, at times t = 0.0, 2.7, 3.3, 4.0. Each frame is a plot of amplitude |u| versus x
Fig. 2. A plot of the numerically obtained transmission Tqs (v) versus velocity v for five values of α = q/v = 0.6, 0.8, 1.0, 1.2, 1.4. The dashed lines are the corresponding theoretical v → +∞ asymptotic values given by 1/(1 + α 2 ).
where A T = (2|tq (v)| − 1)+ , A R = (2|rq (v)| − 1)+ , and ϕT = arg tq (v) + ϕ0 (|tq (v)|) + (1 − A2T )|x0 |/2v, ϕ R = arg rq (v) + ϕ0 (|rq (v)|) + (1 − A2R )|x0 |/2v,
(1.8)
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reflection rate
0.6 0.5 0.4 transmission rate
0.3 0.2
soliton reflection soliton transmission
0.1 0
0
0.5
1
1.5
2
2.5
3
def
Fig. 3. Comparison of linear and nonlinear scattering coefficients as functions of α = q/v.
∞
ϕ0 (ω) = 0
ζ sin2 π ω log 1 + dζ. 2 2 cosh π ζ ζ + (2ω − 1)2
Here tq (v) and rq (v) are the transmission and reflection coefficients of the delta-potential (see (2.2)). When 2|tq (v)| = 1 or 2|rq (v)| = 1 the first error term in (1.7) is modified to 1 ((log(t − |x0 |/v))/(t − |x0 |/v)) 2 ). OL ∞ x Here and later we use the standard notation k a a ≥ 0, a+k = 0 a < 0.
(1.9)
This asymptotic description holds for v greater than some threshold depending on q/v and δ, as in Theorem 1. The implicit constant in the O L 2x error term is entirely independent of all parameters (q, v, and δ), although the implicit constant in the O L ∞ x error term depends upon q/v, or more precisely, the proximity of |tq (v)| and |rq (v)| to 1 2. A comparison of the transmission and reflection coefficients (1.3) of the δ potential, and of the soliton transmission and reflections coefficients appearing in (1.7), is shown in Fig. 3. Scattering of solitons by delta impurities is a natural model explored extensively in the physics literature – see for instance [2, 8], and references given there. The heuristic insight that at high velocities “linear scattering” by the external potential should dominate the partition of mass is certainly present there. In the mathematical literature the dynamics of solitons in the presence of external potentials has been studied in high velocity or semiclassical limits following the work of Floer and Weinstein [6], and Bronski and Jerrard [1] – see [7] for recent results and a review of the subject. Roughly speaking, the soliton evolves according to the classical motion of a particle in the external potential. That is similar to the phenomena in other settings, such as the motion of the Landau-Ginzburg vortices.
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The possible novelty in (1.6) and (1.7) lies in seeing quantum effects of the external potential strongly affecting soliton dynamics. As shown in Fig. 2, Theorem 1 gives a very good approximation to the transmission rate already at low velocities. Figure 1 shows time snapshots of the evolution of the soliton, and the last frame suggests the soliton resolution (1.7). We should stress that the asymptotic solitons are resolved at a much larger time – see [9]. The proof of the two theorems, given below in §3–4, proceeds by approximating the solution during the “interaction phase” (the interval of time during which the solution significantly interacts with the delta potential at the origin) by the corresponding linear flow. This approximation is achieved, uniformly in q, by means of Strichartz estimates established in §2. The use of the Strichartz estimates as an approximation device, as opposed to say, energy estimates, is critical since the estimates obtained depend only upon the L 2 norm of the solution, which is conserved and independent of v. Thus, v functions as an asymptotic parameter; larger v means a shorter interaction phase and a better approximation of the solution by the linear flow. Theorem 2 combines this analysis with the inverse scattering method. The delta potential splits the incoming soliton into two waves which become single solitons. 2. Scattering by a Delta Function Here we present some basic facts about scattering by a delta-function potential on the real line. Let q ≥ 0 and put Hq = −
1 d2 + q δ0 (x). 2 dx2
We define special solutions, e± (x, λ), to (Hq − λ2 /2)e± = 0, using notation given in (1.9): 0 0 e± (x, λ) = tq (λ)e±iλx x± + (e±iλx + rq (λ)e∓iλx )x∓ , (2.1) where tq and rq are the transmission and reflection coefficients: tq (λ) =
iλ q , rq (λ) = . iλ − q iλ − q
(2.2)
They satisfy two equations, one standard (unitarity) and one due to the special structure of the potential: (2.3) |tq (λ)|2 + |rq (λ)|2 = 1, tq (λ) = 1 + rq (λ). We use the representation of the propagator in terms of the generalized eigenfunctions– see for instance the notes [16] covering scattering by compactly supported potentials. The resolvent Rq (λ) = (Hq − λ2 /2)−1 , def
has kernel given by Rq (λ)(x, y) =
1 e+ (x, λ)e− (y, λ)(x − y)0+ + e+ (y, λ)e− (x, λ)(x − y)0− . iλtq (λ)
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This gives an explicit formula for the spectral projection, and hence the Schwartz kernel of the propagator: ∞ 1 2 e−itλ /2 e+ (x, λ)e+ (y, λ) + e− (x, λ)e− (y, λ) dλ. (2.4) exp(−it Hq ) = 2π 0 The propagator for Hq is described in the following Lemma 2.1. Suppose that ϕ ∈ L 1 and that supp ϕ ⊂ (−∞, 0]. Then exp(−it Hq )ϕ(x) = 0 , exp(−it H0 )(ϕ ∗ τq )(x)x+0 + (exp(−it H0 )ϕ(x) + exp(−it H0 )(ϕ ∗ ρq )(−x))x− (2.5)
where 0 ρq (x) = −q exp(q x)x− , τq (x) = δ0 (x) + ρq (x).
(2.6)
Proof. All we need to do is to combine (2.1) and (2.4). Using the support property of ϕ we compute, ϕ(y)e+ (y, λ)dy = rq (−λ)ϕ(−λ) ˆ + ϕ(λ), ˆ ϕ(y)e− (y, λ)dy = tq (−λ)ϕ(−λ), ˆ so that
1 exp(−it Hq )ϕ x>0 = 2π
∞ 0
e−itλ
2 /2
tq (λ)eiλx (rq (−λ)ϕ(−λ) ˆ + ϕ(λ)) ˆ ˆ dλ + (rq (λ)eiλx + e−iλx )tq (−λ)ϕ(−λ)
1 2 iλx = e−itλ /2 tq (λ)ϕ(λ)e ˆ dλ 2π R = exp(−it H0 )(τq ∗ ϕ)(x), τq (λ) = tq (λ),
where we used the fact that rq (−λ)tq (λ) + rq (λ)tq (−λ) = 0. Similarly, using rq (−λ)rq (λ) + tq (−λ)tq (λ) = 1, we have ∞
1 2 iλx −iλx ˆ dλ e−itλ /2 ϕ(λ)e + rq (λ)ϕ(λ)e ˆ exp(−it Hq )ϕ x<0 = 2π 0 = exp(−it H0 )ϕ(x)+exp(−it H0 )(ϕ ∗ ρq )(−x), ρq (λ) =rq (λ). A simple computation gives (2.6) concluding the proof.
We have two simple applications of Lemma 2.1: the Strichartz estimates (Proposition 2.2) and the asymptotics of the linear flow exp(−it Hq ) as v → +∞ (Proposition 2.3). We start with the Strichartz estimate, which will be used several times in the various approximation arguments of §3. Since it is particularly simple in our setting, we give a complete proof (see [11] for references and the general version).
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Proposition 2.2. Suppose q ≥ 0 and i∂t u(x, t) + 21 ∂x2 u(x, t) − qδ0 (x)u(x, t) = f (x, t), u(x, 0) = ϕ(x).
(2.7)
Let the indices p, r , p, ˜ r˜ satisfy 2 ≤ p, r ≤ ∞, 1 ≤ p, ˜ r˜ ≤ 2,
1 2 1 + = , p r 2
2 1 5 + = p˜ r˜ 2
(2.8)
.
(2.9)
and fix a time T > 0. Then u L p
r [0,T ] L x
≤ cϕ L 2 + c f L p˜
r˜ [0,T ] L x
The constant c is independent of q and T . Moreover, in (2.7), we can take f (x, t) = g(t)δ0 (x) and, on the right-hand side of (2.9), replace f L p˜ L r˜ with g 4 . [0,T ] x
3 L [0,T ]
def
Proof. We put Uq (t) = exp(−it Hq ), so that Uq (t) is a unitary group on L 2 (R). For ϕ ∈ L 1 (R) we have, using Lemma 2.1, 0 Uq (t)(ϕx± ) L ∞ Uq (t)ϕ L ∞ ≤ ±
≤
±
0 0 U0 (t) L 1 →L ∞ ((ϕx± ) ∗ τq L 1 + (ϕx± ) ∗ ρq L 1 )
(2.10)
1 ≤√ (1 + 2ρq L 1 )ϕ L 1 π |t| 3 ϕ L 1 . ≤√ π |t|
By the Riesz-Thorin interpolation theorem (see for instance [10, Theorem 7.1.12]) we have 1 1 − 21 1− r2 + = 1. , 1 ≤ r ≤ 2, (2.11) U (t) L r →L r ≤ C|t| r r The estimate (2.9) with f ≡ 0 reads U (t)g L tp L r ≤ Cg L 2 (R) , x
which by duality is equivalent to
U (−s)F(s)ds
R
L 2 (R )
≤ CF
p
L t L rx
.
(2.12)
The two equivalent estimates together give ((2.12) is applied with p , r replaced by p, ˜ r˜ – it is easily checked that (2.8) still holds)
U (t − s)F(s)ds ≤ F L p˜ L r˜ .
R
p
L t L rx
t
x
Putting F(s) = 1l[0,t] (s) f (s, x) we obtain (2.9) for u 0 = 0. Hence it suffices to prove (2.12).
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Put def
T F(x) =
R
U (−s)F(s, x)ds,
and note that T ∗ g(s, x) := U (s)g(x). The estimate (2.12) is equivalent to T ∗ T G, F L 2 ≤ CG t,x
p
L t L rx
F
which is the same as ≤ CG G(t), U (t − s)F(s) dtds R R
p
L t L rx
p
,
L t L rx
F
p
L t L rx
.
(2.13)
To obtain (2.13) from (2.11) we apply the Hardy-Littlewood-Sobolev inequality which says that if K a (t) = |t|−1/a and 1 < a < ∞ then K a ∗ F L α (R) ≤ CF L β (R) ,
1 1 1 = − 1 + , 1 < β < α, α β a
see for instance [10, Theorem 4.5.3]. We apply it with 1 2 1 1− , α = p, β = p , = a 2 r which is the admissibility condition (2.8).
We now turn to the large velocity asymptotics of the linear flow exp(−it Hq ). Proposition 2.3. Let θ ∈ C ∞ (R) be bounded, together will all of its derivatives. Let ϕ ∈ S(R), v > 0, and suppose supp[θ (•)ϕ(• − x0 )] ⊂ (−∞, 0]. Then for 2|x0 |/v ≤ t ≤ 1, e−it Hq [ei xv ϕ(x − x0 )] = t (v)e−it H0 [ei xv ϕ(x − x0 )] −it H0
+ r (v)e + e(x, t),
[e
−i xv
(2.14)
ϕ(−x − x0 )]
where, for any k ≥ 0, e(·, t) L 2 ≤
1 ∂x [θ (x)ϕ(x − x0 )] L 2 v ck + xk ϕ(x) H k (tv)k + 4(1 − θ (x))ϕ(x − x0 ) L 2x .
In §3, Proposition 2.3 will be applied with θ (x) a smooth cutoff to x < 0, and ϕ(x) = sechx with x0 = −v 1−δ 0. Before proving Proposition 2.3, we need the following
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Lemma 2.4. Let ψ ∈ S(R) with supp ψ ⊂ (−∞, 0]. Then 0 e−it Hq [ei xv ψ(x)](x) = e−it H0 [ei xv ψ(x)](x) x− −it H0
(2.15)
+ t (v)e [e ψ(x)](x) x+0 −it H0 −i xv + r (v)e + e(x, t),
i xv
[e
0 ψ(−x)](x) x−
where e(x, t) L 2x ≤
1 ∂x ψ L 2 v
uniformly in t. Proof of Lemma 2.4. By (2.5) with ϕ(x) = ei xv ψ(x), e(x, t) = [e−it H0 (ϕ ∗ (τ − t (v)δ0 ))(x)] x+0 0 + [e−it H0 (ϕ ∗ (ρ − r (v)δ0 ))(−x)] x− ,
and thus it suffices to show e−it H0 (ϕ ∗ (τ − t (v)δ0 ))(x) L 2x ≤
1 ∂x ψ L 2x v
(2.16)
and e−it H0 (ϕ ∗ (ρ − r (v)δ0 ))(x) L 2x ≤
1 ∂x ψ L 2x . v
The proofs of these two estimates are similar, so we only carry out the proof of (2.16). By unitarity of e−it H0 and Plancherel’s identity, ˆ − v)(t (λ) − t (v)) 2 . eit H0 [ϕ ∗ (τ − t (v)δ0 )](x) L 2x = ψ(λ L λ
(2.17)
Since t (λ) − t (v) =
−iq(λ − v) (iλ − q)(iv − q)
we have |t (λ) − t (v)| ≤ (λ − v)/v. Using this to estimate the right-hand side of (2.17) and applying Plancherel’s identity again yields (2.16). Proof of Proposition 2.3. Apply (2.15) to ψ(x) = θ (x)ϕ(x − x0 ) to obtain 0 e−it Hq [ei xv ϕ(x − x0 )](x) = e−it H0 [ei xv ϕ(x − x0 )](x) x−
(2.18)
+ t (v)e−it H0 [ei xv ϕ(x − x0 )](x) x+0 0 + r (v)e−it H0 [e−i xv ϕ(−x − x0 )](x) x− + e1 (x, t) + e2 (x, t),
where e1 (x, t) is as in Lemma 2.4 and (putting f (x) = ei xv (1 − θ (x))ϕ(x − x0 )) 0 e2 (x, t) = + e−it Hq f (x) − e−it H0 f (x) x−
0 − t (v)e−it H0 f (x) x+0 − r (v)e−it H0 [ f (−x)](x) x− .
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By Lemma 2.4, e1 (x, t) L 2x ≤
1 ∂x [θ (x)ϕ(x − x0 )] L 2x v
uniformly for all t, and by unitarity of the linear flows, e2 (x, t) L 2x ≤ 4(1 − θ (x))ϕ(x − x0 ) L 2x also uniformly in all t. Now restrict to the time interval 2|x0 |/v ≤ t ≤ 1. By (2.18), it remains to show that ck xk ϕ(x) Hxk , x<0 (tv)k ck e−it H0 [e−i xv ϕ(−x − x0 )](x) L 2 ≤ xk ϕ H k . x>0 (tv)k e−it H0 [ei xv ϕ(x − x0 )](x) L 2
≤
(2.19)
The second of these is in fact equivalent to the first, since for any function g(x), e−it H0 [g(−x)](x) = e−it H0 [g(x)](−x). Now we establish (2.19). Since [ei•v ϕ(• − x0 )]ˆ(λ) = e−i x0 (λ−v) ϕ(λ ˆ − v),
e
−it H0
[e
i xv
1 ϕ(x − x0 )](x) = 2π
ˆ − v) dλ ei xλ e−i x0 (λ−v) e−itλ /2 ϕ(λ 1 2 2 = e−itv /2 ei xv ˆ dλ. eiλ(x−x0 −tv) e−itλ /2 ϕ(λ) 2π 2
By k applications of integration by parts in λ, e
iλ(x−x0 −tv) −itλ2 /2
e
ϕ(λ) ˆ dλ =
i x − x0 − tv
k
eiλ(x−x0 −tv) ∂λk [e−itλ
2 /2
ϕ(λ)] ˆ dλ.
Since 2|x0 |/v ≤ t, we have −x0 − tv < 0 and thus |x − x0 − tv| ≥ | − x0 − tv| ≥ tv/2 for x < 0. Hence 0 |e−it H0 [ei xv ϕ(x − x0 )](x)|2 d x −∞
ck iλ(x−x0 −tv) k −itλ2 /2
e ≤ ∂λ [e ϕ(λ)] ˆ dλ (2.20)
2 k (tv) Lx
ck
k −itλ2 /2 = [e ϕ(λ)] ˆ
2
∂ λ k Lλ (tv) from which the result follows by applying the Leibniz product rule and the Plancherel identity once again (and using that t ≤ 1).
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Remark. Suppose that u(x, t) = e−it Hq [ei xv ψ(x)], ψ ∈ S(R), supp ψ ⊂ (−∞, 0), ψ L 2 = 1. Then for t 1 and as v → +∞, ∞ |u(x, t)|2 d x = 0
v2 1 . +O v2 + q 2 v2
(2.21)
In fact using (2.5) and an estimate similar to (2.20) we see that for t ≥ 1, ∞ |u(x, t)|2 d x = e−it H0 ((ei•v ψ) ∗ τq )x+0 2 = (ei•v ψ) ∗ τq 22 + O(v −∞ ) 0
1 ˆ − v)/(iλ − q)2 + O(v −∞ ) iλψ(λ 2π λ2 1 ˆ − v)|2 dλ + O(v −∞ ). |ψ(λ = 2π |λ−v|≤√v λ2 + q 2 =
An expansion in powers of (λ − v)/v gives (2.21). 3. Soliton Scattering In this section, we prove Theorem 1. We recall the notation for operators from Sect. 2 and introduce short-hand notation for the nonlinear flows: • H0 = − 21 ∂x2 . The flow e−it H0 is termed the “free linear flow”. • Hq = − 21 ∂x2 + qδ0 (x). The flow e−it Hq is termed the “perturbed linear flow”. • NLSq (t)ϕ, termed the “perturbed nonlinear flow” is the evolution of initial data ϕ(x) according to the equation i∂t u + 21 ∂x2 u − qδ0 (x)u + |u|2 u = 0. • NLS0 (t)ϕ, termed the “free nonlinear flow” is the evolution of initial data ϕ(x) according to the equation i∂t h + 21 ∂x2 h + |h|2 h = 0. From Sect.1 we recall the form of the initial condition: u 0 (x) = ei xv sech(x − x0 ), v 1, x0 ≤ −v 1−δ , 23 < δ < 1, and we put u(x, t) = NLSq (t)u 0 (x). We begin by outlining the scheme, and will then supply the details. The O notation always means the L 2x difference, uniformly on the time interval specified, and up to a multiplicative factor that is independent of q, v, and δ (any such dependence will be exhibited explicitly). Phase 1 (Pre-interaction). Consider 0 ≤ t ≤ t1 , where t1 = |x0 |/v − v −δ so that x0 + vt1 = −v 1−δ . The soliton has not yet encountered the delta obstacle and propagates according to the free nonlinear flow u(x, t) = e−itv
2 /2
eit/2 ei xv sech(x − x0 − vt) + O(qe−v
1−δ
), 0 ≤ t ≤ t1 .
(3.1)
The analysis here is valid provided v is greater than some absolute threshold (independent 1−δ of q, v, or δ). But if we further require that v be sufficiently large so that v −3/2 ev ≥ α 1−δ ≤ v −1/2 ≤ v −δ/2 . This is the error that arises in the (recall α = q/v), then qe−v main argument of Phase 2 below. Phase 2 (Interaction). Let t2 = t1 + 2v −δ and consider t1 ≤ t ≤ t2 . The incident soliton, beginning at position −v 1−δ , encounters the delta obstacle and splits into a transmitted
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J. Holmer, J. Marzuola, M. Zworski
component and a reflected component, which by time t = t2 , are concentrated at positions v 1−δ and −v 1−δ , respectively. More precisely, at the conclusion of this phase (at t = t2 ), u(x, t2 ) = t (v)e−it2 v + r (v)e + O(v
2 /2
−it2
− 12 δ
eit2 /2 ei xv sech(x − x0 − vt2 )
v 2 /2
e
it2 /2 −i xv
e
(3.2)
sech(x + x0 + vt2 )
).
This is the most interesting phase of the argument, which proceeds by using the following three observations: • The perturbed nonlinear flow is approximated by the perturbed linear flow for t1 ≤ t ≤ t2 . • The perturbed linear flow is split as the sum of a transmitted component and a reflected component, each expressed in terms of the free linear flow of soliton-like waveforms. • The free linear flow is approximated by the free nonlinear flow on t1 ≤ t ≤ t2 . Thus, the soliton-like form of the transmitted and reflected components obtained above is preserved. The brevity of the time interval [t1 , t2 ] is critical to the argument, and validates the approximation of linear flows by nonlinear flows. Phase 3 (Post-interaction). Let t3 = t2 + (1 − δ) log v, and consider [t2 , t3 ]. The transmitted and reflected waves essentially do not encounter the delta potential and propagate according to the free nonlinear flow, u(x, t) = e−itv +e
2 /2
eit2 /2 ei xv NLS0 (t − t2 )[t (v)sech(x)](x − x0 − tv)
−itv 2 /2
(3.3)
eit2 /2 e−i xv NLS0 (t − t2 )[r (v)sech(x)](x + x0 + tv)
3
+ O(v 1− 2 δ ),
t2 ≤ t ≤ t3 .
This is proved by a perturbative argument that enables us to evolve forward a time (1−δ) log v at the expense of enlarging the error by a multiplicative factor of e(1−δ) log v = 3 v 1−δ . The error thus goes from v −δ/2 at t = t2 to v 1− 2 δ at t = t3 . Now we turn to the details. 3.1. Phase 1. Let u 1 (x, t) = NLS0 (t)u 0 (x) and u(x, t) = NLSq (t)u 0 (x). Let w = u − u 1 . Recall that t1 = |x0 |/v − v −δ so that x0 + vt1 = −v 1−δ . Note that u 1 (x, t) = e−itv
2 /2
eit/2 ei xv sech(x − x0 − tv).
We will need the following perturbation lemma. Lemma 3.1. If ta < tb , tb − ta ≤ c1 , and w(·, ta ) L 2 + qu 1 (0, t) L ∞ ≤ 1, then [t ,t ] a b
w L ∞
2 [ta ,tb ] L x
≤ c2 (w(·, ta ) L 2x + qu 1 (0, t) L ∞ ), [t ,t ] a b
where the constants c1 and c2 depend only on constants appearing in the Strichartz estimates and are, in particular, independent of q and v.
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Proof. w solves i∂t w + ∂x2 w − qδ0 (x)w = −|w + u 1 |2 (w + u 1 ) + |u 1 |2 u 1 + qδ0 (x)u 1 = − |w|2 w − (2u 1 |w|2 + u¯ 1 w 2 ) − (2|u 1 |2 w + u 21 w) ¯ +qδ0 (x)u 1 . cubic
quadratic
linear
From this equation, w is estimated using Proposition 2.2. For the cubic nonlinear term we take p˜ = r˜ = 6/5 and estimate by Hölder as |w|2 w L 6/5
6/5 [ta ,tb ] L x
≤ (tb − ta )1/2 w2L 6
6 [ta ,tb ] L x
w L ∞
2 [ta ,tb ] L x
.
Since complex conjugates become irrelevant in the estimates, both quadratic terms are treated identically. In Proposition 2.2, we take p˜ = r˜ = 6/5 and estimate by Hölder as u 1 w 2 L 6/5
6/5 [ta ,tb ] L x
≤ (tb − ta )1/2 w2L 6 L 6 u 1 L ∞ L 2x [ta ,tb ] [ta ,tb ] x √ 1/2 2 ≤ 2(tb − ta ) w L 6 L 6 . [ta ,tb ] x
For the linear terms (both of the form u 21 w), we take p˜ = r˜ = 6/5 in Proposition 2.2 and estimate as u 21 w L 6/5
6/5 [ta ,tb ] L x
≤ (tb − ta )1/2 w L 6
6 [ta ,tb ] L x
≤ 2(tb − ta )2/3 w L 6
u 1 L 6
6 [ta ,tb ] L x
6 [ta ,tb ] L x
u 1 L ∞
2 [ta ,tb ] L x
.
The delta term is estimated by the concluding sentence of Proposition 2.2 as qu(0, t) L 4/3
[ta ,tb ]
≤ q(tb − ta )3/4 u(0, t) L ∞ . [t ,t ] a b
Since tb −ta ≤ 1, collecting the above estimates we have (taking w X = w L ∞ L 2x + [ta ,tb ] w L 6 L 6 ), [ta ,tb ] x
. w X ≤ cw(·, ta ) L 2x + c(tb − ta )1/2 (w X + w2X + w3X ) + cqu(0, t) L ∞ [t ,t ] a b
Provided (tb − ta )1/2 ≤ 1/(2c) above, the linear term on the right can be absorbed by the left as . w X ≤ 2cw(·, ta ) L 2x + 2c(tb − ta )1/2 (w2X + w3X ) + 2cqu(0, t) L ∞ [t ,t ] a b
Continuity of w X (tb ) as a function of tb shows that provided 2c(tb − ta )1/2 (4cw(·, ta ) L 2 + 4cqu 1 (0, t) L ∞ ) ≤ 1/2, the above estimate implies [t ,t ] a b
w X ≤ 4cw(·, ta ) L 2x + 4cqu(0, t) L ∞ , [t ,t ] a b
concluding the proof.
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Now we proceed to apply Lemma 3.1. The constants c1 and c2 will, for convenience of exposition, be taken to be c1 = 1 and c2 = 2. Let k ≥ 0 be the integer such that k ≤ t1 < k + 1. (Note that k = 0 if the soliton starts within a distance v of the origin, i.e. −v − v 1−δ ≤ x0 ≤ −v 1−δ , and the inductive analysis below is skipped.) Apply Lemma 3.1 with ta = 0, tb = 1 to obtain (since w(·, 0) = 0) w L ∞
2 [0,1] L x
≤ 2qu 1 (0, t) L ∞ ≤ 2qsech(x0 + v). [0,1]
Apply Lemma 3.1 again with ta = 1, tb = 2 to obtain w L ∞
2 [1,2] L x
≤ 2(w(·, 1) L 2x + qu 1 (0, t) L ∞ ) [1,2] ≤ 22 qsech(x0 + v) + 21 qsech(x0 + 2v).
We continue inductively up to step k, and then collect all k estimates to obtain the following bound on the time interval [0, k]: w L ∞
2 [0,k] L x
≤ 2q
k
2k− j sech(x0 + jv).
j=1
The estimate sechα ≤ 2e−|α| reduces matters to bounding 2k qe x0 +v
k−1
2− j e jv
j=0
and, after summing the geometric series, we obtain w L ∞
2 [0,k] L x
≤ c2k e x0 +v
(2−1 ev )k − 1 ≤ cqe x0 +kv , 2−1 ev − 1
where the last inequality requires 2−1 ev ≥ 2. Finally, applying Lemma 3.1 on [k, t1 ], w L ∞
2 [0,t1 ] L x
≤ c(qe x0 +kv + qsech(x0 + t1 v)) ≤ cqe−v
1−δ
.
As a consequence, (3.1) follows. 3.2. Phase 2. We shall need a lemma stating that the free nonlinear flow is approximated by the free linear flow, and that the perturbed nonlinear flow is approximated by the perturbed linear flow. Both estimates are consequences of the corresponding Strichartz estimates (Proposition 2.2). Crucially, the hypotheses and estimates of this lemma depend only on the L 2 norm of the initial data ϕ. Below, (3.5) is applied with ϕ(x) = u(x, t1 ), and u(x, t1 ) L 2x = u 0 L 2 is independent of v; thus v does not enter adversely into the analysis. Lemma 3.2. If ϕ ∈ L 2 and 0 < tb such that tb < c1 ϕ−4 , then L2 NLS0 (t)ϕ − e−it H0 ϕ L ∞
≤ c2 tb ϕ3L 2 ,
NLSq (t)ϕ − e−it Hq ϕ L ∞
≤ c2 tb ϕ3L 2 ,
2 [0,tb ] L x 2 [0,tb ] L x
1/2
(3.4)
1/2
(3.5)
where c1 and c2 depend only on constants appearing in the Strichartz estimates. In particular, they are independent of q.
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Proof. Estimate (3.4) is in fact a special case of (3.5) obtained by taking q = 0. Let h(t) = NLSq (t)ϕ so that i∂t h + 21 ∂x2 h − qδ0 (x)h + |h|2 h = 0 with h(x, 0) = ϕ(x). Let us define 2 6 6 X = L∞ [0,tb ] L x ∩ L [0,tb ] L x ,
with the natural norm, • X . We apply Proposition 2.2 with, in the notation of that proposition, u(t) = h(t) − e−it Hq ϕ, f = −|h|2 h, p = r = 6, p˜ = r˜ = 6/5, and then again with p = ∞, r = 2, p˜ = r˜ = 6/5, to obtain h(t) − e−it Hq ϕ X ≤ c|h|2 h L 6/5
6/5 [0,tb ] L x
.
The generalized Hölder inequality, 1 1 1 1 = + + , p q1 q2 q3
h 1 h 2 h 3 p ≤ h 1 q1 h 2 q2 h 3 q3 ,
applied with h j = h, p = 6/5 and q1 = q2 = 6, q3 = 2, gives h(t) − e−it Hq ϕ X ≤ Ch2L 6
6 [0,tb ] L x
h L 2
1/2
≤ Ctb h2L 6
6 [0,tb ] L x
2 [0,tb ] L x
h L ∞
2 [0,tb ] L x
(3.6)
1/2
≤ Ctb h3X . Another application of the homogeneous Strichartz estimate shows that e−it Hq ϕ X ≤ Cϕ L 2 , and consequently, 1/2
h X ≤ cϕ L 2 + ctb h3X . 1/2
By continuity of h X (tb ) in tb , if ctb (2cϕ L 2 )2 ≤ 1/2, h X ≤ 2cϕ L 2 . Substituting into (3.6) yields the result.
Now we proceed to apply Lemma 3.2. Set t2 = t1 + 2v −δ , and apply (3.5) on [t1 , t2 ] to obtain u(·, t) = NLSq (t − t1 )[u(·, t1 )] = e−i(t−t1 )Hq [u(·, t1 )] + O(v −δ/2 ). By combining this with (3.1), u(·, t) = e−it1 v
2 /2
eit1 /2 e−i(t−t1 )Hq [ei xv sech(x − x0 − t1 v)] + O(v −δ/2 ).
(3.7)
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By Proposition 2.3 with θ (x) = 1 for x ≤ −1 and θ (x) = 0 for x ≥ 0, ϕ(x) = sech(x), and x0 replaced by x0 + t1 v, e−i(t2 −t1 )Hq [ei xv sech(x − x0 − vt1 )](x) = t (v)e−i(t2 −t1 )H0 [ei xv sech(x − x0 − vt1 )](x) + r (v)e−i(t2 −t1 )H0 [e−i xv sech(x + x0 + vt1 )](x)
(3.8)
+ O(v −1 ). By combining (3.7), (3.8) and (3.4), u(·, t) = t (v)e−it1 v
2 /2
+ r (v)e−it1 v
eit1 /2 NLS0 (t2 − t1 )[ei xv sech(x − x0 − vt1 )](x)
2 /2
eit1 /2 NLS0 (t2 − t1 )[e−i xv sech(x + x0 + vt1 )](x)
+ O(v −δ/2 ). By noting that NLS0 (t2 − t1 )[ei xv sech(x − x0 − t1 v)] = e−i(t2 −t1 )v
2 /2
ei(t2 −t1 )/2 ei xv sech(x − x0 − t2 v)
and NLS0 (t2 − t1 )[e−i xv sech(x + x0 + t1 v)] = e−i(t2 −t1 )v
2 /2
ei(t2 −t1 )/2 e−i xv sech(x + x0 + t2 v),
we obtain (3.2). 3.3. Phase 3. Let t3 = t2 + (1 − δ) log v. Label u tr (x, t) = e−itv
2 /2
eit2 /2 ei xv NLS0 (t − t2 )[t (v)sech(x)](x − x0 − tv)
for the transmitted (right-traveling) component and u ref (x, t) = e−itv
2 /2
eit2 /2 e−i xv NLS0 (t − t2 )[r (v)sech(x)](x + x0 + tv)
for the reflected (left-traveling) component. By Appendix A, for each k ∈ N, there is a constant c(k) > 0 and an exponent σ (k) > 0 such that u tr (x, t) L 2
x<0
+ u ref (x, t) L 2
x>0
+ |u tr (0, t)| + |u ref (0, t)| ≤
c(k)(log v)σ (k) v k(1−δ)
(3.9)
uniformly on the time interval [t2 , t3 ]. We shall need the following perturbation lemma, again a consequence of the Strichartz estimates.
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Lemma 3.3. Let w = u − u tr − u ref . If ta < tb , tb − ta ≤ c1 , and w(·, ta ) L 2x + then
c(k)q(log v)σ (k) ≤ 1, v k(1−δ)
w L ∞
2 [ta ,tb ] L x
≤ c2 w(·, ta ) L 2x
c(k)q(log v)σ (k) + v k(1−δ)
.
The constants c1 , c2 depend only on constants appearing in the Strichartz estimates and are in particular independent of q and v. Proof. We write the equation satisfied by w: i∂t w + 21 ∂x2 w − qδ0 (x)w = − |w + u tr + u ref |2 (w + u tr + u ref ) + |u tr |2 u tr + |u ref |2 u ref + qδ0 (x)u tr − qδ0 (x)u ref = − |w|2 w − (2(u tr + u ref )|w|2 + (u¯ tr + u¯ ref )w 2 ) ¯ − (2|u tr + u ref |2 w + (u tr + u ref )2 w) − (u 2tr u¯ ref + 2u ref |u tr |2 + u 2ref u¯ tr + 2u tr |u ref |2 ) + qδ0 (x)u tr − qδ0 (x)u ref , tr−delta
tr−ref interaction
ref−delta
and we estimated w using Proposition 2.2. The cubic, quadratic, and linear in w terms on the first line are estimated exactly as was done in the proof of Lemma 3.1. For the “tr − ref interaction terms” (taking u ref |u tr |2 as a representative example), we apply Proposition 2.2 with p˜ = 4/3, r˜ = 1 and estimate as u ref |u tr |2 L 4/3
1 [ta ,tb ] L x
u tr L 2x =
≤ c(tb − ta )3/4 u tr L ∞
2 [ta ,tb ] L x
u ref u tr L ∞
2 [ta ,tb ] L x
,
(3.10)
√ 2|t (v)| by mass conservation for the free nonlinear flow, and
u tr u ref L ∞
2 [ta ,tb ] L x
≤ u tr u ref L ∞
2 [ta ,tb ] L x<0
≤ u ref L ∞ [t ,t
L∞ a b] x
+ u tr u ref L ∞
2 [ta ,tb ] L x>0
u tr L ∞
2 [ta ,tb ] L x<0
+ u tr L ∞ [t ,t
L∞ a b] x
u ref L ∞
2 [ta ,tb ] L x>0
.
Now ∞ = NLS0 (t)[r (v)sech](x) L ∞ L ∞ u ref L ∞ t Lx t x
1/2
1/2
≤ NLS0 (t)[r (v)sech](x) L ∞ L 2 ∂x NLS0 (t)[r (v)sech](x) L ∞ L 2 t
x
t
x
≤c ∞ ≤ c. by mass and energy conservation of the free nonlinear flow. Similarly, u tr L ∞ t Lx By this and (3.9), the above yields
u tr u ref L ∞
[ta ,tb
2 ]Lx
≤
c(k)(log v)σ (k) . v k(1−δ)
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J. Holmer, J. Marzuola, M. Zworski
Thus, by (3.10),
c(k)(log v)σ (k) (3.11) v k(1−δ) and similarly for all other “tr−ref interaction” terms. Now we address the “tr−delta” and “ref−delta” terms (working with qδ0 (x)u tr as the representative of both). By Proposition 2.2, we estimate as u ref |u tr |2 L 4/3
1 [ta ,tb ] L x
qu tr (0, t) L 4/3
[ta ,tb ]
≤
≤ c(tb − ta )3/4 qu tr (0, t) L ∞ . [t ,t ] a b
By (3.9),
(log v)σ (k) . (3.12) [ta ,tb ] v k(1−δ) Collecting (3.11), (3.12), and the estimates for cubic, quadratic, and linear terms in w (as exposed in Lemma 3.1), we have, with w X = w L ∞ L 2x + w L 6 L 6 , qu tr (0, t) L 4/3
≤ c(k)q(tb − ta )3/4
[ta ,tb ]
[ta ,tb ] x
w X ≤ cw(·, ta ) L 2 + c(tb − ta )1/2 (w X + w2X + w3X ) +
c(k)q(log v)σ (k) . v k(1−δ)
If c(tb − ta )1/2 ≤ 21 , then the first-order w-term on the right side can be absorbed by the left, giving w X ≤ 2cw(·, ta ) L 2 + 2c(tb − ta )1/2 (w2X + w3X ) + By continuity of w X (tb ) in tb , if 2c(tb − ta )1/2 4cw(·, ta ) L 2
2c(k)q(log v)σ (k) . v k(1−δ)
4c(k)q(log v)σ (k) + v k(1−δ)
≤
1 2
we have w X ≤ 4cw(·, ta ) L 2 + completing the proof.
4c(k)q(log v)σ (k) , v k(1−δ)
Assume that α = q/v has been fixed. Choose k = k(δ) large so that k(1 − δ) ≥ 3. Then the coefficient appearing in Lemma 3.3 is bounded by q (log v)σ (k) c(k)q(log v)σ (k) ≤ c(k) . v v2 v k(1−δ) Now take v sufficiently large in terms of q/v and k (thus in terms of δ) so that the above is bounded by v −1 . Now we implement Lemma 3.3. For convenience of exposition, we take c1 = 1, c2 = 2. Let be the integer such that < (1 − δ) log v < + 1. We then apply Lemma 3.3 successively on the intervals [t2 , t2 + 1], . . . , [t2 + − 1, t2 + ] as follows. Applying Lemma 3.3 on [t2 , t2 + 1], we obtain w(·, t) L ∞
2 [t2 ,t2 +1] L x
≤ 2(w(·, t2 ) L 2x + v −1 ).
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Applying Lemma 3.3 on [t2 + 1, t2 + 2] and combining with the above estimate, w(·, t) L ∞
2 [t2 +1,t2 +2] L x
≤ 22 w(·, t2 ) L 2x + (22 + 2)v −1 .
Continuing up to the th step and then collecting all of the above estimates, w(·, t) L ∞
2 [t2 ,t3 ] L x
≤ 2 w(·, t2 ) L 2x + (2 + · · · + 2)v −1 .
Since w(·, t2 ) L 2x ≤ v −δ/2 and 2 ≤ v 1−δ , 3
w(·, t) L ∞
2 [t2 ,t3 ] L x
≤ cv 1− 2 δ ,
(3.13)
thus proving (3.3). Now we complete the proof of the main theorem and obtain (1.6). By (3.13) and (3.9), u(·, t) − u tr (·, t) L 2
x>0
≤ w(·, t) L 2
x>0
+ u ref (·, t) L 2
x>0
3
≤ cv 1− 2 δ .
(3.14) 3
Since u tr (·, t) L 2x = t (v), (3.9) implies u tr (·, t) L 2 = t (v) + O(v 1− 2 δ ), which x>0 combined with (3.14) gives (1.6) and proves Theorem 1. 4. Resolution of Outgoing Waves In this section, we prove Theorem 2. We note that the proof of Theorem 1 presented in §3 in fact provided a more complete long-time description of the solution: u(x, t) = e−itv
2 /2
eit2 /2 ei xv NLS0 (t − t2 )[t (v)sech](x − x0 − tv)
(4.1)
−itv 2 /2 it2 /2 −i xv
NLS0 (t − t2 )[r (v)sech](x + x0 + tv) 3 |x0 | + v −δ ≤ t ≤ c(1 − δ) log v, + O L 2x (v 1− 2 δ ), v where t (v), r (v) are defined in (2.2) and NLS0 (t)ϕ denotes the solution to the NLS equation i∂t h + 21 ∂x2 h + |h|2 h = 0 (without potential) and initial data h(x, 0) = ϕ(x). It thus suffices to obtain the resolution of NLS0 (t − t2 )[t (v)sech] and NLS0 (t − t2 )[r (v)sech] into solitons plus radiation decaying in L ∞ x . By the phase invariance of the free nonlinear flow t (v) NLS0 (t − t2 )[|t (v)|sech] NLS0 (t − t2 )[t (v)sech] = |t (v)| +e
e
e
and similarly for NLS0 (t − t2 )[r (v)sech]. Since 0 ≤ |t (v)|, |r (v)| ≤ 1, we apply asymptotics (B.1) proved of Appendix B using the inverse scattering method. When |t (v)| or |r (v)| is equal to 1/2 we use the result of [12] recalled in (B.2). The result obtained by these substitutions differs from that stated in Theorem 2 by a factor of 1 − A2T −δ exp i ·v (4.2) 2 for u T (x, t), owing to the fact that t2 = |x0 |/v + v −δ . But (4.2) differs from 1 by ∼ v −δ , and thus omitting it only introduces a discrepancy of v −δ in both L 2x and L ∞ x . There is a similar inconsequential disparity in the u R (x, t) part.
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Appendix A. Spatial Localization of the Free Nonlinear Propagation Let ϕ ∈ S and
i∂t h + 21 ∂x2 h + |h|2 h = 0, h(x, 0) = ϕ(x).
(A.1)
Notational conventions. We denote ∂x by ∂ hereafter. The x and t dependence of h(x, t) will be routinely dropped. The constants c(k), σ (k), and the polynomials gγ (t) that appear below may change (enlarge) from one line to the next without comment. The constants c(k) depend on the fixed function ϕ ∈ S. The solution h satisfies conservation of mass and conservation of energy, which means that the integrals E0 = |h|2 d x, E 2 = − (|∂h|2 − |h|4 )d x, R
R
are independent of time t. Since h2L ∞ ≤ h L 2 ∂h L 2 , we have h4L 4 ≤ h3L 2 ∂h L 2 and it follows from the E 2 and E 0 conservation that ∂h L 2 ≤ c, where c depends on ϕ L 2 and ∂ϕ L 2 . In fact, there are an infinite number of conserved integrals, E k , with integrands defined inductively as follows: 1 fk + f j1 f j2 , (A.2) f 0 = |h|2 , f k+1 = h∂ h j1 + j2 =k−1
see [17, §8] for a proof of this fact (rescaling time and putting κ = 2 produces an agreement with our slightly different convention). The inductive definition of f k and the Sobolev embedding theorem can now be used to show that, for ≥ 2, E 2 = (−1) |∂ h(x)|2 d x + O((1 + h Hx−1 )2+2 ), (A.3) R
and hence for ≥ 0, we have
∂ h L 2 ≤ c(),
(A.4)
where c() depends upon Sobolev norms of the initial data ϕ of at most order . We now elaborate on how to obtain (A.3). An inductive argument using (A.2) shows that for k ≥ 0, f k is of the form f k = h∂ k h¯ + h p(2 j + 1, k − 2 j), (A.5) j≥1, 2 j≤k
where p(n, m) indicates a linear combination of terms of degree n and cumulative order m, or more precisely terms of the form ˜ α1 + · · · + αn = m ∂ α1 h˜ ∂ α2 h˜ · · · ∂ αn h,
(A.6)
¯ To prove (A.3) for ≥ 2, one uses (A.5) for k = 2 and it only and h˜ is either h or h. remains to verify that for any n ≥ 4 and m ≤ 2 − 2, (A.7) p(n, m) d x ≤ hnH −1 . We now show this. Note that in (A.6), we may assume without loss of generality that α1 ≤ α2 ≤ · · · ≤ αn .
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Case 1. αn ≤ − 1. It follows that α j ≤ − 2 for all j ≤ n − 2 and αn−1 ≤ − 1. We estimate as: ⎛ ⎞ n−2 ∂ α1 h˜ ∂ α2 h˜ · · · ∂ αn h˜ d x ≤ ⎝ ∂ α j h L ∞ ⎠ ∂ αn−1 h L 2 ∂ αn h L 2 ≤ chnH −1 j=1
by Sobolev embedding. Case 2. αn ≥ . In this case, we begin by integrating by parts to obtain ˜ ∂ −1 h˜ d x. (−1)αn −+1 ∂ αn −+1 (∂ α1 h˜ · · · ∂ αn−1 h)
(A.8)
The Leibniz rule expansion is ˜ = ∂ αn −+1 (∂ α1 h˜ · · · ∂ αn−1 h)
˜ cµ ∂ µ1 +α1 h˜ · · · ∂ µn−1 +αn−1 h,
(A.9)
where the sum is over (n − 1)-tuples µ such that µ1 + · · · + µn−1 = αn − + 1 and cµ is some constant depending on µ. By adding the α and µ constraints, we obtain that (µ1 + α1 ) + · · · + (µn−1 + αn−1 ) ≤ − 1 and thus there is at most one index j∗ (1 ≤ j∗ ≤ n − 1) such that µ j∗ + α j∗ = − 1 and for all remaining j (1 ≤ j ≤ n − 1, j = j∗ ) we have µ j + α j ≤ − 2. (If no such j∗ exists, take j∗ to be any fixed index 1 ≤ j∗ ≤ n − 1.) By substituting (A.9) into (A.8), we estimate as ∂ α1 h˜ ∂ α2 h˜ · · · ∂ αn h˜ d x ⎛ ⎞ n−1 ≤⎝ ∂ µ j +α j h L ∞ ⎠ ∂ µ j∗ +α j∗ h L 2 ∂ −1 h L 2 ≤ chnH −1 j=1, j= j∗
again by Sobolev embedding. This concludes the proof of (A.7), thus (A.3), and thus (A.4). Using that the commutator [(x + it∂), i∂t + 21 ∂ 2 ] = 0 and some integration by parts manipulations, we have the pseudoconformal conservation law: t 2 2 4 4 |(x + it∂)h(x, t)| d x −t |h(x, t)| d x + s |h(x, s)| d xds = |xϕ(x)|2 d x. x
x
0
x
x
From this, (A.4) for = 0, 1, and the Gagliardo-Nirenberg estimate hL44 ≤ hL32 ∂h L 2 , we have xh L 2 ≤ ct, where c depends on xϕ L 2 , ϕ L 2 , and ∂ϕ L 2 . We want to show that more generally, for each k ∈ Z, k ≥ 0, we have x α ∂ β h L 2 ≤ c(k)tσ (k)
for α + β = k, α, β ≥ 0, α, β ∈ Z.
(A.10)
Here c(k) is a constant depending on k and weighted Sobolev norms of the initial data ϕ (up to order 2k), and σ (k) is a positive exponent depending upon k. We are not concerned
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with obtaining the optimal value of σ (k); the mere fact that the bound in (A.10) is power-like in t, as opposed to exponential in t, suffices for our purposes. In our proof, both c(k) and σ (k) will be increasing with k, and will go to +∞ as k → +∞. Let 0 = ∂ and 1 = (x + it∂). Note that both operators have the commutator property [ j , (i∂t + 21 ∂ 2 )] = 0, j = 0, 1. (A.11) We first claim that for each k ≥ 0, there exists a constant c(k) > 0 and an exponent σ (k) > 0 such that j1 j2 · · · jk h L 2 ≤ c(k)tσ (k) for all j1 , . . . jk ∈ {0, 1}.
(A.12)
When we wish to consider a composition of the form j1 j2 · · · jk and do not care to report whether each operator in the composition is 0 or 1 , we will instead write the composition as k . We prove (A.12) by induction on k . When k = 0, (A.12) is just the mass conservation law. Suppose that (A.12) holds for 0, . . . , k − 1; we aim to prove it holds for k. The main ingredient (in addition to the inductive hypothesis) is (A.4). Fix j1 , . . . jk ∈ {0, 1}, and apply the operator j1 · · · jk to the equation, pair with −i j1 · · · jk h, integrate in x, take twice the real part, and appeal to (A.11) to obtain (A.13) ∂t j1 · · · jk h2L 2 = 2 Re i j1 · · · jk |h|2 h j1 · · · jk h d x. Note that ¯ = ∂ F(h, h) ¯ h + F(h, h) ¯ 0 h 0 F(h, h)h and ¯ = it∂ F(h, h) ¯ h + F(h, h) ¯ 1 h. 1 F(h, h)h Both of these product rules take the form ¯ h + F(h, h) ¯ h, |h|2 h = g(t)∂ F(h, h) where g(t) is a polynomial in t of degree ≤ 1. Thus we see that j1 · · · jk |h|2 h = |h|2 j1 · · · jk h + gγ (t)∂ γ1 |h|2 γ2 h, γ1 +γ2 =k γ2 ≤k−1
where gγ (t) is a polynomial in t. Substituting into (A.13), we obtain two terms: the first is zero since it is the real part of a purely imaginary number; the second is estimated by the Hölder inequality to obtain: | ∂t j1 · · · jk h2L 2 | ≤ c(k)tσ (k) ×
sup ∂ j |h|2 L ∞ j≤k−1
sup h L 2 j1 · · · jk h L 2 . j
j≤k−1
By Sobolev embedding estimates, (A.4), and the induction hypothesis, we have | ∂t j1 · · · jk h2L 2 | ≤ c(k)tσ (k) j1 · · · jk h L 2 from which (A.12) follows.
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Now to deduce (A.10) from (A.12), we just note that since x = 1 − it0 , there are polynomials g j (t) such that the following relation holds: xα∂β = g j (t) j1 · · · jα+β . j∈{0,1}α+β
Let us now consider the application of (A.10) to obtain (3.9) in the Phase 3 analysis. We have x0 + tv ≥ v 1−δ for t ≥ t2 . If x < 0, then v k(1−δ) ≤ (x0 + tv)k ≤ |x − x0 − tv|k . Thus v k(1−δ) u tr (x, t) L 2
x<0
≤ (x − x0 − tv)k NLS0 (t − t2 )[t (v)sech](x − x0 − tv) L 2x = x k NLS0 (t − t2 )[t (v)sech](x) L 2x ≤ c(k)t − t2 σ (k)
by (A.10), which gives the first estimate in (3.9). The second is obtained similarly. To obtain the third, we note that for t ≥ t2 , |u tr (0, t)|2 = |NLS0 (t − t2 )[t (v)sech](−x0 − tv)|2 0 =− ∂x |NLS0 (t − t2 )[t (v)sech](x − x0 − tv)|2 d x −∞
and this can be estimated by NLS0 (t −t2 )[t (v)sech](x −x0 − tv) L 2 ∂x NLS0 (t −t2 )[t (v)sech](x −x0 −tv) L 2 . x<0
x<0
Using (A.10) as before establishes j
v k(1−δ) ∂x NLS0 (t − t2 )[t (v)sech](x − x0 − tv) L 2
x<0
≤ c(k + j)t − t2 σ (k+ j) .
Replacing (c(k)c(k + 1))1/2 by c(k) and 21 (σ (k) + σ (k + 1)) by σ (k), we obtain the bound (3.9). Finally, we note that the fourth bound in (3.9) is similar to the third. Appendix B. Free Nonlinear Evolution of α sech This appendix is devoted to showing that for 0 < α < 1, 1 eiϕ(α) NLS0 ((2α − 1)sech((2α − 1)•)) + O L ∞ (t − 2 ) 1/2 < α < 1, NLS0 (α sech) = 1 O L ∞ (t − 2 ) 0 < α < 1/2, (B.1) where ϕ(α) is given in (B.23). A more precise understanding of error terms is possible thanks to advances in the study of asymptotics for integrable nonlinear waves [4, 12]. Since we do not know an exact reference for (B.1) we present a proof of this simpler asymptotic result. It is based on the now classical work on the inverse scattering method initiated for NLS by Zakharov-Shabat [17] – see [4, 5] for discussion and references. For the reader’s
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convenience, especially in view of different conventions used in different sources for our argument, we review all the needed aspects of the method. In the case of α = 1/2 we can use the result of [12] to conclude that 1
NLS0 (α sech) = O L ∞ ((log t/t) 2 ).
(B.2)
A slightly inaccurate statement similar to (B.1) was given in [15] and the calculation of the scattering matrix in that paper was our starting point in obtaining (B.1).
B.1. Inverse scattering method. We present a quick review of this celebrated method. Thus, let us consider two operators acting on S (R; C2 ): L = −i J ∂x + i J Q, A = J ∂x2 − where
1 1 J Q x − J Q∂x − Q 2 J, 2 2
0 u(t, x) , u(t, •) ∈ S(R), Q = Q(t, x) = 0 −u(t, x)
−1 J= 0
0 . 1
Then i [L , A] = − Q x x + i Q 3 2 which is checked by using J Q J = −Q, J 2 = I, Q 2 =
−|u|2 0 . 0 −|u|2
It is now the case that 1 ∂t L = i[L , A] ⇐⇒ i∂t u + ∂x2 u + |u|2 u = 0, 2
(B.3)
and, since we are solving NLS, we assume that these equivalent equations hold. We now consider scattering theory for the time dependent operator L. For that we introduce special solutions to Lψ = λψ with prescribed asymptotic behaviour: −i xλ 0 e ¯ ψ(x, λ) , ψ(x, λ) i xλ , x −→ +∞, 0 e −i xλ (B.4) 0 e , ϕ(x, ¯ λ) i xλ , x −→ −∞, ϕ(x, λ) 0 e def
see instance [5, Sect.I.5]. Here for vector valued functions, ϕ¯ = [ϕ¯2 , −ϕ¯1 ]t , if ϕ = [ϕ1 , ϕ2 ]t . Each pair of solutions forms a basis for the solution set and, for λ ∈ R, ¯ ϕ(x, λ) = a(λ)ψ(x, λ) + b(λ)ψ(x, λ), ¯ ψ(x, ¯ ϕ(x, ¯ λ) = a(λ)ψ(x, ¯ λ) − b(λ) λ), |a(λ)| + |b(λ)| = 1. 2
2
(B.5)
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Another consequence comes from (B.3). If L(t)ψ(t) = λψ(t) then we see that (L − λ)(i∂t ψ − Aψ) = 0, and hence ¯ i∂t ψ(t) − Aψ(t) = c1 (t)ψ(t) + c2 ψ(t). Now we note that for u(t, •) ∈ S(R), A J ∂x2 , as |x| → ∞, and the asymptotic behaviour (B.4) gives c1 (t) ≡ λ2 , c2 (t) ≡ 0. More generally we conclude that ¯ i∂t ψ = (A + λ2 )ψ, i∂t ψ¯ = (A − λ2 )ψ, ¯ i∂t ϕ = (A + λ2 )ϕ, i∂t ϕ¯ = (A − λ2 )ϕ.
(B.6)
The solutions ψ and ϕ have analytic extensions in λ to the upper half plane and ψ¯ and ϕ¯ to the lower half plane. Same is true for a(λ) and a(λ) respectively. Except in very special cases (such as our potential α sechx) b(λ) does not have an analytic extension off the real axis. The reflection coefficient is defined as r (λ) =
b(λ) . a(λ)
We assume that a(λ) has at most one zero and that it can only lie in Im λ > 0, Re λ = 0. That zero, λ0 , corresponds to an L 2 eigenfuction of L, and at λ = λ0 , the two solutions are proportional: ϕ(x, λ0 ) = γ0 ψ(x, λ0 ). (B.7) The scattering data is given by the triple u(t, x) −→ {r (λ, t), λ0 , γ0 (t)}.
(B.8)
The evolution of the scattering data is easily obtained from (B.6): 2
2
r (λ, t) = e2itλ r (λ, 0), γ0 (t) = e2itλ γ0 (0).
(B.9)
In fact, we can use (B.5) (B.6) to see (A − λ2 )ϕ = i∂t ϕ = i∂t a ψ¯ + ai∂t ψ¯ + i∂t bψ + bi∂t ψ = i∂t a ψ¯ + (i∂t b + 2λ2 b)ψ + (A − λ2 )ϕ. Independence of ψ and ψ¯ shows that, remarkably, ∂t a(λ, t) = 0, ∂t b(λ, t) = 2iλ2 b(λ, t), which gives the first part of (B.9). From (B.7) we see that, γ0 (A − λ2 )ψ(λ0 ) = i∂t ϕ(λ0 ) = i∂t γ0 ψ(λ0 ) + γ0 (A + λ2 )ψ(λ0 ), so that, ∂t γ0 (t) = 2iλ2 γ0 (t). That gives (B.9). The justification of this formal calculation depends on u(t, •) ∈ S(R) and we refer to, for instance, [5, Sect. I.7] for a full proof.
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B.2. The Riemann Hilbert problem. It is now universally acknowledged that the best way to obtain long time asymptotics for the inverse of (B.8) and (B.9) is by solving a Riemann-Hilbert problem [4], [5, Chap. II]. To recall this method let us consider the following matrix valued function of λ ∈ C \ R, depending parametrically on x ∈ R: ⎧ ⎨ [a(λ)−1 ϕ(x, λ)eiλx , ψ(x, λ)e−iλx ], Im λ > 0 def (λ, x) = (B.10) ⎩ ¯ [ψ(x, λ)eiλx , a( ¯ λ¯ )−1 ϕ(x, ¯ λ)e−iλx ], Im λ < 0. The properties of ψ and ϕ (see for instance [5, Sect. I.5]) imply that (λ) = I + O(|λ|−1 ), |λ| → ∞,
(B.11)
where the decay rate may depend on x, uniformly in compact sets. From (B.5) we see that 1 0 iλx −iλx ¯ , (λ + i0, x) = [ψ(x, λ)e , ψ(x, λ)e ] 2 r (λ)e2i(λx+λ t) 1 −2i(λx+λ2 t) 1 −¯ r (λ)e iλx −iλx ¯ (λ − i0, x) = [ψ(x, λ)e , ψ(x, λ)e , λ ∈ R. ] 0 1 Hence, the boundary values of (λ) satisfy (λ + i0) = (λ − i0)Vx,t (λ), λ ∈ R, ! 2 1 + |r (λ)|2 r¯ (λ)e−2i(λx+λ t) def Vx,t = . 2 r (λ)e2i(λx+λ t) 1
(B.12)
If a(λ) has no zeros in Im λ ≥ 0 then the Riemann-Hilbert problem is to construct (λ) satisfying (B.11) and (B.12). Liouville’s theorem readily shows that it is unique. If a(λ) has a zero in Im λ > 0, and in our presentation we allow at most one, λ0 , we have to consider a Riemann-Hilbert problem in which (λ) is allowed to have singularities at λ0 and λ¯ 0 . The structure of that singularity can be seen in (B.10): 0 0 , Resλ=λ0 = (λ0 ) 2iλ0 x γ0 0 e (B.13) −2i λ¯0 x γ¯ γ0 def 0 , γ = Resλ=λ¯ 0 = (λ¯ 0 ) 0 e . 0 0 0 a (λ0 ) Since a(λ) can be reconstructed from r (λ) and λ0 (see for instance [5, Chap. I, (6.23)]; in our case it will be explicit) the Riemann-Hilbert problem in the case of one singularity is to find which in addition to (B.11) and (B.12) satisfies (B.13). A standard way to read off u(t, x) from (λ, x) follows from high frequency asymptotics of ψ(x, λ) (see for instance [17, (18)]): 1 " u(t, x) 1 1 −iλx + ψ(x, λ)e +O , (B.14) ∞ 2 0 2iλ x |u(t, y)| dy |λ|2 so that u(x, t) = lim 2iλ12 (λ, x). λ→∞
(B.15)
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We conclude this brief review by describing a reduction of the problem with prescribed singularities (B.13) to a problem with analytic in C \ R. To do that we follow [5, Sect. II.2] by considering a reformulation of the Riemann-Hilbert problem: 1 0 def def def ¯ ± Im λ > 0, , a+ (λ) = a(λ), a− (λ) = a( G ± (λ)∓1 = (λ) ¯ λ), 0 a± (λ)∓1 ¯ def 1 −b(λ) . G + (λ + i0)G − (λ − i0) = G(λ), G(λ) = −b(λ) 1 ¯ but their The operators G ± (λ) are now analytic in Im λ± > 0 (in fact, G ∗+ (λ) = G − (λ)) ranks drop precisely at λ = λ0 and λ¯ 0 respectively. The condition (B.13) becomes ¯ 1 −γ¯0 e−2i λ0 x . (B.16) ¯ , ker G ( λ ) = span Im G + (λ0 ) = spanC − 0 C γ0 e2iλ0 x 1 We now look for B(λ), analytic in Im λ > 0, with B(λ)−1 analytic in Im λ < 0, B(λ) = I + O(1/|λ|), and such that #− (λ) = B(λ)G + (λ), ± Im λ > 0, #+ (λ) def = G + (λ)B(λ)−1 , G G are nonsingular matrices. We note this requires B(λ)−1 to have a pole at λ0 and B(λ), at λ¯ 0 . That is natural since we are adding to the ranks of G ± (λ). The condition (B.16) means that ¯ 1 −γ¯0 e−2i λ0 x . #−1 # ¯ ¯ λ , ker B( (λ ) ) = span ( λ ) G Im B(λ0 ) = spanC G 0 0 C − 0 + γ0 e2iλ0 x 1 (B.17) This determines B(λ) uniquely as a Blaschke-Potapov factor: λ¯ 0 − λ0 P, P ∗ = P, P 2 = P, λ − λ¯ 0 2 −2i λ¯ 0 x + G #(12) #(11) ¯ ¯ G 1 |β| β − (λ0 , x)γ0 e − (λ0 , x) P= , , β(x) = ¯ 2 (21) (22) ¯ β 1 1 + |β| #− (λ¯ 0 , x) #− (λ¯ 0 , x)γ0 e−2i λ0 x + G G B(λ) = I +
(B.18)
see [5, Chap. II, (2.17)–(2.27)]. Hence to solve the Riemann-Hilbert problem (B.12),(B.13) we first solve the problem # (λ + i0) = # (λ − i0)Vx,t (λ), # = I + O(1/|λ|), # analytic in C \ R. (B.19) We then define
#− (λ) = # (λ) 1 G 0
0 #+ (λ) = G #− (λ¯ )∗ , Im λ < 0, , G a( ¯ λ¯ )∓1
from which, using (B.18), we construct B(λ). Then # (λ), (λ) = B(λ)−1
(B.20)
and we can finally use (B.15) to obtain u(t, x). In particular the long time behaviour of # (λ). u(t, x) is determined by the longtime behaviour of
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# solving (B.19) B.3. Manakov ansatz. The basic structure of the long time behaviour of can be obtained from the Manakov ansatz for the solution of (B.19) – see [4, Sect. 2] and references given there. To describe it we define r± (ζ )δ(ζ + i0)±1 δ(ζ − i0)±1 ∓2(itζ 2 +xζ ) def 1 m ± (λ, x, t) = e dζ, 2πi R ζ −λ where r+ (ζ ) = r (ζ ), r− (ζ ) = r¯ (ζ ), and $ −x/2t 1 log(1 + |r (ζ )|2 ) def dζ , δ(λ, x, t) = exp 2πi −∞ ζ −λ solves a scalar Riemann-Hilbert problem, δ(ζ − i0)(1 + |r (ζ )|2 ), ζ < −x/2t δ(ζ + i0) = δ(ζ − i0), ζ > −x/2t, see [3, Prop. 2.12] for a detailed list of properties of δ(z) (stated in the defocusing case log(1 − |r (ζ )|2 )). The Manakov ansatz is then given by 1 1 m + (λ, x, t) δ(λ, x, t) def (λ, x, t) = . (B.21) m − (λ, x, t) 1 1 δ(λ, x, t)−1 we use the following elementary lemma: To see the properties of Lemma B.1. Suppose that f ∈ C ∞ ((0, R)), f ∈ L 1 ([0, ∞), x k f (l) ∈ L ∞ ([1, ∞)), for all k and l. Then, as λ → ±∞, ⎧ 1 2 ∞ 2 ⎨ f (x)eiλx + O(|λ|− 2 |x|−1 ) x > 0 and λ > 0 f (y)eiλy 1 = 2πi 0 y − x − i0 ⎩ 1 O(|λ|− 2 |x|−1 ) otherwise. Using this lemma one checks that (λ − i0, x, t)−1 (λ + i0, x, t) = Vx,t (λ) + O √
1 , t|λ + x/2t|
from which it follows as in [4, Sect. 2] that √ # (λ) = (I + O(1/ t)) (λ).
√ # in (B.15) this shows that u(t, If we defined u(t, ˜ x) by putting ˜ x) = O(1/ t), and in fact a more precise statement can be obtained by using the second component in (B.14). We can now use (B.18) to obtain an approximation to B(λ) as t → ∞: λ¯ 0 − λ0 B(λ) = B(λ) + O(t −1/2 |λ|−1 ), B(λ) = I − P, λ − λ¯ 0 is as in (B.18) with β replaced by where P 0 $ 1 log(1 + |r (ζ )|2 ) dζ . βˆ = γ0 exp(−2λ¯ 0 x) exp πi −∞ ζ − λ0
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Hence to obtain a long time approximation for a solution we apply the procedure of [5, Sect. II.5] to B0 (λ) since that corresponds to using (B.15) with # (λ) = B(λ)−1 (I + O(t −1/2 )) (λ) = B0 (λ)−1 (I + O(t −1/2 |λ|−1 )). (λ) = B(λ)−1 This gives Lemma B.2. Suppose that u(0, •) ∈ S(R) and that the scattering data (B.8) for u(0, •) is given by r (λ), γ0 and λ0 , Im λ > 0. Then u(x, t) = eiϕ0 NLS0 e2i Re λ0 • 2 Im λ0 sech(2 Im λ0 (• − x0 )) + O L ∞ (t −1/2 ), (B.22) where 1 x0 = 2 Im λ0 Im λ0 − π
log |γ0 | − log |a (λ0 )| − log(2 Im λ0 )
0 −∞
log(1 + |r (ζ )|2 ) dζ , (ζ − Re λ0 )2 + λ20
and ϕ0 = arg γ0 − arg a (λ0 ) +
1 π
0
−∞
log(1 + |r (ζ )|2 ) (ζ − Re λ0 )dζ. (ζ − Re λ0 )2 + λ20
We state this important result as a lemma to stress the fact that better error estimates seem available if more advanced methods [3, 4] are used. B.4. Scattering matrix. We now apply Lemma B.2 to obtain (B.1). For that we need to find the scattering data (B.8) for u(0, x) = αsechx. That is done by a well known computation [15],[14, Sect.3.4] which reappears in many scattering theories, from the free S-matrix in automorphic scattering, to Eckhardt barriers in quantum chemistry. We quote the results: a(λ) =
( 21 − iλ) ( 21
+α
− iλ)( 21
− α − iλ)
, b(λ) = i
b(λ) sin π α , r (λ) = . cosh π λ a(λ)
We note that in this special case b and r are meromorphic in C (with infinitely many “nonphysical” poles). Also, λ0 = i(α − 21 ) if 1/2 < α < 1, and γ0 = b(λ0 ) = i. We need to compute x0 and ϕ0 . In general when u(0, x) is even then x0 = 0 by symmetry considerations. Here we see it by using [5, Chap. II, (2.6)] which shows that Im λ0 ∞ log(1 + |r (ζ )|2 ) log |a (λ0 )| = log(2 Im λ0 ) + dζ 2π −∞ ζ 2 + Im λ20 Im λ0 0 log(1 + |r (ζ )|2 ) dζ. = π ζ 2 + Im λ20 −∞
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Thus the formula in Lemma B.2 results in x0 = 0. To compute ϕ0 we need to find the following integral: ∞ ζ sin2 π α log 1 + dζ, 1/2 < α < 1. (B.23) 2 2 cosh π ζ ζ + (2α − 1)2 0 Acknowledgement. We would like to thank Mike Christ, Percy Deift, and Michael Weinstein for helpful discussions during the preparation of this paper. The work of the first author was supported in part by an NSF postdoctoral fellowship, and that of the second and third author by NSF grants DMS-0354539 and DMS-0200732.
References 1. Bronski, J.C., Jerrard, R.L.: Soliton dynamics in a potential. Math. Res. Lett. 7, 329–342 (2000) 2. Cao, X.D., Malomed, B.A.: Soliton-defect collisions in the nonlinear Schrödinger equation. Phys. Lett. A 206, 177–182 (1995) 3. Deift, P.A., Zhou, X.: Long-time asymptotics for solutions of the NLS equation with initial data in weighted Sobolev spaces. Comm. Pure Appl. Math. 56, 1029–1077 (2003) 4. Deift, P.A., Its, A.R., Zhou, X.: Long-time asymptotics for integrable nonlinear wave equations. In: Important developments in soliton theory, Springer Ser. Nonlinear Dynam., Berlin Springer, 1993 pp. 181–204 5. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons, Part One. Berlin Heidelberg New York: Springer Verlag, 1987 6. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986) 7. Fröhlich, J., Gustafson, S., Jonsson, B.L.G., Sigal, I.M.: Solitary wave dynamics in an external potential. Commun. Math. Phys. 250, 613–642 (2005) 8. Goodman, R.H., Holmes, P.J., Weinstein, M.I.: Strong NLS soliton-defect interactions. Physica D 192, 215–248 (2004) 9. Holmer, J., Marzuola, J., Zworski, M.: Numerical study of soliton scattering by delta impurities. In preparation 10. Hörmander, L.: The Analysis of Linear Partial Differential Operators, Vol.I,II. Berlin Heidelberg New York: Springer Verlag, 1983 11. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998) 12. Kamvissis, S.: Long time behavior for the focusing nonlinear Schroedinger equation with real spectral singularities. Commun. Math. Phys. 180, 325–341 (1996) 13. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968) 14. Maimistov, A.I., Basharov, M.: Nonlinear Optical Waves. Fundamental Theories of Physics 104, Dordrecht-Boston-London: Kluwer Academic Publishers 1999 15. Miles, J.W.: An envelope soliton problem. SIAM J. Appl. Math. 41(2), 227–230 (1981) 16. Tang, S.H., Zworski, M.: Potential scattering on the real line. Lecture notes, http://www.math.berkeley. edu/∼zworski/tz1.pdf 17. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media. Soviet Physics JETP 34(1), 62–69 (1972) Communicated by P. Constantin
Commun. Math. Phys. 274, 217–231 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0260-0
Communications in
Mathematical Physics
On the Absence of Ferromagnetism in Typical 2D Ferromagnets Marek Biskup1 , Lincoln Chayes1 , Steven A. Kivelson2 1 Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA. E-mail:
[email protected] 2 Department of Physics, Stanford University, Stanford, CA 94305-4045, USA
Received: 18 August 2006 / Accepted: 21 November 2006 Published online: 16 May 2007 – © M. Biskup, L. Chayes and S. A. Kivelson 2007
Abstract: We consider the Ising systems in d dimensions with nearest-neighbor ferromagnetic interactions and long-range repulsive (antiferromagnetic) interactions that decay with power s of the distance. The physical context of such models is discussed; primarily this is d = 2 and s = 3 where, at long distances, genuine magnetic interactions between genuine magnetic dipoles are of this form. We prove that when the power of decay lies above d and does not exceed d + 1, then for all temperatures the spontaneous magnetization is zero. In contrast, we also show that for powers exceeding d + 1 (with d ≥ 2) magnetic order can occur. 1. Introduction While most of our knowledge of statistical mechanics is derived from studies of model problems with short-range forces, in nature interactions more often fall off only in proportion to an inverse power of the distance, U (r ) ∼ 1/r s . This includes systems interacting via Coulomb forces (s = 1), dipolar interactions (s = 3) as well as interactions caused by collective effects such as strain induced interactions in solids or the effective entropic interactions (analogous to Casimir forces) in lipid films. When the interactions are sufficiently long-range, i.e., when s ≤ d, where d is the spatial dimension, the very definition of the thermodynamic limit is different than for short-ranged models. However, even when s > d there can be qualitatively new, or at least unexpected, phenomena, cf, e.g., [2–4, 29]. In the present paper we study a class of systems with long-range forces; namely, the Ising models on Zd , d ≥ 1, which are defined by the (formal) Hamiltonians 1 J σi σ j + K i, j σi σ j . (1.1) H =− 2 i, j
i, j
© 2007 by M. Biskup, L. Chayes and S.A. Kivelson. Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.
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Here σi ∈ {+1, −1}, i and j index sites in Zd and i, j denotes a nearest neighbor pair. The above notation expresses the relevant signs of all the couplings: J > 0 is the shortrange ferromagnetic interaction while K i, j ≥ 0 represents the antiferromagnetic long range interaction which we assume decays with power s of the distance between i and j. We investigate the question of presence, and absence, of spontaneous magnetization in such models. The motivation for this work was provided by a paper of Spivak and one of us [27] where it was conjectured that, in the presence (or absence) of an external field, discontinuous transitions permitting coexisting states of different magnetization are forbidden for antiferromagnetic power law interactions with range d < s ≤ d + 1. A heuristic proof by contradiction was presented based on the explicit construction of a “microemulsion” phase which has a lower free energy than the state of macroscopic two-phase coexistence. Simply put, the anticipated surface tension between the two pure phases would be negative—and divergent. The proof is heuristic in the sense that it makes the physically plausible assumption that correlations in the putative coexisting phases have reasonable decay and that there is a well defined interface. As it turns out, versions of the above conjecture are actually more than 20 years old. For example, on the physics side, modulated phases in 2D dipolar ferromagnets were analyzed in [1, 11, 18]. On the mathematics side, in [8], models with extreme anisotropic repulsive interactions which have very slow decay, but only among a sparse set of spins, were considered and absence of spontaneous magnetism was proved. The isotropic case, U (r ) ∼ 1/r s , was also mentioned in [8] and the significance of the interval d < s < d + 1 for the absence of magnetization was highlighted (with no mention of s = d + 1). Related problems were described in [17] for systems with longer range, e.g., Coulomb, interactions and in [8, 9] for the current setup with O(n)-spins. Furthermore, general theorems demonstrating instability of phase coexistence under the addition of generic long-range interactions have been proved in [7, 16, 26]. In the present paper we provide a full proof of the absence of ferromagnetism in the model (1.1) with d < s ≤ d + 1, thereby vindicating completely the arguments of [27]—at least for h = 0. The mathematical result presented in this note has the following consequence for 2D physics: Two-dimensional magnetic systems often have strong “crystal field” effects which orient the electron spins (largely or entirely) in the z direction, perpendicular to the plane in which they reside. This gives the problem of magnetic ordering an Ising character. Interactions between nearby spins—quantum mechanical and somewhat complicated—are, often enough, of the ferromagnetic type and considerably stronger than the direct magnetic dipolar interactions (which are a relativistic effect). Thus, it seems reasonable to study Ising ferromagnets in 2D contexts and conclude that there is a definitive possibility for ferromagnetism. However, while possibly weak, there is always the long-range 1/r 3 repulsive interaction. The conclusion of this note is that, no matter how small its relative strength may be, this interaction will preclude the possibility of ferromagnetism among the z-components. We remark that the absence of magnetization certainly does not disallow other types of ordering. Indeed, a large body of physics literature [1, 5, 6, 10, 11, 14, 18, 19, 21–23, 27], points in the direction of modulated (striped and/or bubble) states in this and related systems. (For an extremely insightful review of the phases produced by models of this sort and many experimentally clear realizations of the corresponding physics, see [24].) From the perspective of mathematics, recent rigorous estimates on ground-state energies [13], which are asymptotic in d ≥ 2 and exact in d = 1, also indicate striped order
Absence of Ferromagnetism in 2D Ferromagnets
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in the ground state. In fact, for certain special cases of the 1D ground-state problem, this has been established completely. The organization of the rest of this paper is as follows: In the next section we define all necessary background and state the main results. In Sect. 3 we derive some estimates on the strength of the long-range interaction between a box and its complement. These are assembled into the proof of the main result in Sect. 4. Section 5 contains some open problems and further discussion. 2. Statement of Main Results As mentioned, for the problem of central interest we have K i, j ∼ |i − j|−3 in d = 2, where |i − j| is the Euclidean distance, but we may as well treat all powers for which the interaction is absolutely summable. To be definitive we will simply take, for s > d, K i, j =
1 , |i − j|s
(2.1)
with the proviso K i,i = 0. We remark that more generality than (2.1) is manifestly possible as is also the case with the ferromagnetic portion of the interaction in (1.1). However, these generalities would tend to obscure the mechanics of the proofs and so we omit them. In order to define the corresponding Gibbs measures, let ⊂ Zd be a finite set and, d given a configuration σ ∈ {+1, −1}Z , let H = H (σ , σc ) denote the Hamiltonian in which is obtained from (1.1) by pitching out all terms with both i and j outside . Since s > d, the corresponding object is bounded uniformly in σ . Then the DLR ford malism tells us that a probability measure on {+1, −1}Z —equipped with the product σ -algebra—is a Gibbs measure if the regular conditional distribution of σ = (σi )i∈ given a configuration σc = (σi )i∈c in the complement c = Zd \ is of the form Z (σc )−1 e−β H (σ ,σc ) ,
(2.2)
where Z (σc ) =
e−β H (σ ,σc )
(2.3)
σ
is the partition function. We will use the notation − to denote expectations with respect to Gibbs measures (which may often stay implicit). We wish to establish that all Gibbs measures corresponding to the above Hamiltonian have zero average magnetization once s ∈ (d, d + 1]. We will employ some thermodynamic arguments based, ultimately, on the notion of the free energy. To define this quantity, let Z ,h (σc ) denote the partition function in with the Hamiltonian H (σ , σc ) − h
σi ,
(2.4)
i∈
i.e., for the model in homogeneous external field h. Let L = [−L , L]d ∩ Zd .
(2.5)
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Then there exists L = o(| L |)—with little-o uniform in h—such that for all σ, d σ˜ ∈ {−1, 1}Z , Z (σ c ) log L ,h L ≤ L . (2.6) Z L ,h (σ˜ cL ) In particular, the limit f (β, h) = −
1 1 lim log Z L ,h (σcL ) β L→∞ | L |
(2.7)
exists and is independent of the boundary condition. Furthermore, the function h → f (β, h) is concave for all h. The independence of the free energy on the boundary condition is standard and follows from the uniform bound on energy per site; cf. [25, Theorem II.3.1]. In Sect. 3 we will show that, perhaps not surprisingly, L is order L max{2d−s,d−1} with a logarithmic correction at s = d + 1. The concavity of the free energy now permits us to define the spontaneous magnetization m = m (β) via the right-derivative of h → f (β, h) at h = 0: ∂ f m = − + . (2.8) ∂h h=0 It is clear that, by the plus-minus symmetry built into the model, the corresponding left derivative equals −m . The statement of our main result is as follows: Theorem 2.1. Consider the interaction described by the Hamiltonian in (1.1–2.1). Then for all s ∈ (d, d + 1] and all β ∈ (0, ∞), the spontaneous magnetization, m , is zero. The regime d < s < d + 1 of exponents for the vanishing of the spontaneous magnetization was surmised already in [8]; the present work covers this and, in addition, the somewhat subtle borderline case s = d + 1. The above is about as strong a statement as possible concerning the absence of magnetic order from a thermodynamic perspective; the implications for statistical mechanics are similar in their finality. Indeed, the following standard conclusions are implied for the properties of equilibrium states: Corollary 2.2. Let s ∈ (d, d + 1] and let µ be any infinite-volume Gibbs measure for the Hamiltonian in (1.1–2.1) at inverse temperature β ∈ (0, ∞). Let L be as in (2.5). For each > 0 there exists δ > 0 such that for all L sufficiently large and µ-almost every boundary condition σcL , σi > | L |σcL ≤ e−δ| L | . (2.9) µ i∈ L
In particular, µ-almost all configurations σ have zero block-average magnetization, 1 σi = 0. L→∞ | L | lim
(2.10)
i∈ L
Finally, in any translation-invariant (infinite volume) Gibbs state, the expectation of the spin at the origin is zero.
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The last statement should not be interpreted as a claim that the state is disordered. In fact, as already mentioned, one expects the occurrence of “striped states” at sufficiently low temperatures; see our discussion in Sect. 1 and also Sect. 5. Note that no restrictions are put on the nearest-neighbor coupling J ; the theorem works for all J ∈ R. To complement our “no-go” Theorem 2.1, we note that for exponents s > d + 1, spontaneous magnetization will occur under the “usual” conditions: Theorem 2.3. Let d ≥ 2, pick s > d + 1 and consider the interaction as described in (1.1–2.1). Then there exist J0 = J0 (s, d) ∈ (0, ∞) and C0 = C0 (d) ∈ (0, ∞) such that for all β(J − J0 ) ≥ C0 , m > 0.
(2.11)
In particular, under such conditions, there exist two distinct, translation-invariant extremal Gibbs states −+ and −− such that σ0 + = −σ0 − > 0.
(2.12)
Strictly speaking, this result could be proved by directly plugging in a theorem from [12, Sect.3], which is based on an enhanced Peierls estimate. Instead, we provide an independent way to estimate the contour-flip energy which is technically no more demanding and permits the use of sharp contour-counting arguments [20] to derive good estimates on J0 and the critical value of β at which the transition occurs. As a result, the corresponding constants can be bounded as follows: J0 ≤ C
πd log d and C0 ≤ C , d +1−s d
(2.13)
where πd is the “surface” measure of the unit sphere in Rd , and C is a constant of order unity. 3. Estimates on Interaction Strength In this section we will perform some elementary but in places tedious calculations that are needed for the proof of our main results. We begin by an estimate on the energy cost of turning large magnetized blocks to opposite magnetization: Proposition 3.1. Let L be as above and, for the couplings K i, j described in (2.1), consider the discrete sum TL = K i, j . (3.1) i∈ L j∈cL
Then, as L tends to infinity: (i) For d < s < d + 1, TL ∼ L 2d−s Q, where Q ∈ (0, ∞) is the integral Q= with S1 = {x ∈ Rd : |x|1 ≤ 1}.
x∈S1 y∈S1c
dx dy . |x − y|s
(3.2)
(3.3)
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(ii) For s = d + 1, there exists a constant A = A(d) ∈ (0, ∞) such that TL ∼ AL d−1 log L .
(3.4)
In both (i) and (ii) the symbol ∼ is interpreted to mean that the ratio of the two sides tends to unity in the stated limit. To prove this claim, we will instead consider the quantity TL ,a which is defined in the same fashion as TL except that the “inside sum” now ranges over L−a instead of L , providing us with a cutoff scale a. Of course we must allow a → ∞ and, for s ∈ (d, d + 1), not much more is actually required but, to save work, we shall insist that a/L d+1−s → 0. (Indeed, we remark that while most of the up and coming is not strictly necessary for these cases, it will allow for a unified treatment later.) For the marginal case of s = d + 1 we need to implement the stronger requirement that a/ log L → 0. Our claim is that the augmented quantities have the asymptotics that was stated for their unadorned counterparts. This is sufficient since, keeping in mind the above requirements, TL ,a ≤ TL ≤ TL ,a + 2dEs L d−1 a, where Es =
(3.5)
K 0, j < ∞
(3.6)
j
denotes the maximum antiferromagnetic energy associated with a single spin flip. For the purposes of explicit calculations, it will be convenient to replace K i, j with the quantities K˜ i, j obtained by “smearing” the interaction about the unit cells surrounding the sites i and j: dx dy K˜ i, j = . (3.7) |x−i|∞ ≤1/2 |x − y|s 1 |y− j|∞ ≤
/2
It is noted that since all distances exceed (the large quantity) a, the approximation is not severe: K i, j K i, j ≤ K˜ i, j ≤ , (3.8) −1 s (1 + θa ) (1 − θa −1 )s where θ is a number of order unity. Thus, to prove the asymptotics for the TL ,a , we may insert the K˜ i, j and then perform blatant continuum integration. As a technical step, for the proof we will need to calculate the total (long-range) interaction between the line segment (−L , −a) on the x-axis and the half-space in Rd containing all points with positive x-coordinate: Lemma 3.2. Consider the integral L dx I1 (L , a) = a
∞
dy 0
Rd−1
dz
[(x +
1 . + |z|2 ]s/2
y)2
(3.9)
In the limit when a/L → 0 (with L ≥ 1) when s < d + 1 and |log a|/ log L → 0 when s = d + 1, C1 L d+1−s , if d < s < d + 1, I1 (L , a) ∼ (3.10) C1 log L , if s = d + 1, where C1 = C1 (d, s) ∈ (0, ∞).
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Proof. Scaling z by x + y yields I1 (L , a) = C˜ 1
L
a
where C˜ 1 =
∞
dx Rd−1
dy (x + y)d−1−s ,
(3.11)
0
dz . [1 + |z|2 ]s/2
From here the result follows by direct integration.
(3.12)
Now we are ready to prove the s < d + 1 part of Proposition 3.1: Proof of Proposition 3.1(i). For r < 1 let Qr denote the integral (3.3) with x restricted to a cube Sr instead of S1 . Let T˜L ,a denote the quantity TL ,a with K i, j replaced by K˜ i, j . A simple scaling yields T˜L ,a = L 2d−s Q1−a/L .
(3.13)
Hence, all we need to show is that Qr remains finite as r ↑ 1. This in turn boils down to the absolute convergence of the integral defining Q. To show that Q < ∞ we note that the quantity (L − a)d−1 I1 (L − a, a) in Lemma 3.2 may be interpreted as the integral of |x − y|−s over x ∈ L−a and over y ranging through the half-space marked by the hyperplane passing through a given side of the cube L . This implies T˜L ,a ≤ 2d L d−1 I1 (L − a, a)
(3.14)
Q ≤ 2d I1 (1, 0).
(3.15)
and, more importantly,
By Lemma 3.2 and the Monotone Convergence Theorem, I1 (1, 0) < ∞ when s < d +1.
The proof of the critical case, s = d + 1, is more subtle. The following lemma encapsulates the calculations that are needed on top of those in Lemma 3.2: Lemma 3.3. Let s ∈ (d, d + 1] and consider the integral L L ∞ ∞ 1 I2 (L , a) = dx dy d x˜ d y˜ dz . 2 + (y + y˜ )2 + |z|2 ]s/2 d−2 [(x + x) ˜ R a a 0 0 (3.16) There exists C2 = C2 (d, s) < ∞ such that for L a 1, I2 (L , a) ≤ C2 L d+2−s .
(3.17)
Similarly to the quantity I1 (L , a) in Lemma 3.2, the integral I2 (L , a) may be interpreted as the total interaction between the square (−L , −a) × (−L , −a) in the (x, y)-plane and the quarter-space in Rd containing all points with positive x and y coordinates.
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Proof of Lemma 3.3. Applying the bound (x + x) ˜ 2 + (y + y˜ )2 + |z|2 ≥ x 2 + x˜ 2 + y 2 + y˜ 2 + |z|2
(3.18)
and scaling z by the root of x 2 + x˜ 2 + y 2 + y˜ 2 we get
L
I2 (L , a) ≤ O(1)
L
dx
∞
dy
a
a
0
d x˜
∞
d−2−s d y˜ x 2 + x˜ 2 + y 2 + y˜ 2 2 . (3.19)
0
Writing r 2 = x 2 + y 2 and ρ 2 = x˜ 2 + y˜ 2
(3.20)
we pass to the polar coordinates in both pairs of variables—with ρ ∈ (0, ∞) and, as an upper bound, r ∈ (a/2, 2L)—yielding the result I2 (L , a) ≤ O(1)
2L
dr r a/2
∞
dρ ρ[r 2 + ρ 2 ]
d−2−s 2
= O(1)
0
2L
dr r d+1−s . (3.21)
a/2
Here we scaled ρ by r and integrated ρ out to get the last integral. Since s ≤ d + 1, the integral over r is order L d+2−s .
Proof of Proposition 3.1(ii). In this case we cannot simply set a = 0 and apply scaling. Notwithstanding, we still have the bound T˜L ,a ≤ 2d L d−1 I1 (L − a, a).
(3.22)
By Lemma 3.2, we have T˜L ,a ≤ 2dC1 L d−1 log L 1 + o(1) ,
L → ∞.
(3.23)
We claim that this bound is asymptotically sharp. Indeed, (3.23) overcounts by including (the integral over y in) the intersection of two halfspaces—marked by two neighboring sides of L —multiple times. In light of the aforementioned interpretation of I2 (L , a), the contribution from each such intersection is bounded by L d−2 I2 (L , a). By Lemma 3.3, this is at most order L d−1 . Hence we have (3.4) with A = 2dC1 .
Theorem 2.3 will require us to show that, for s > d + 1, the total strength of the long-range interaction through the boundary of a finite set is of order boundary: Proposition 3.4. Let s > d + 1. Then there is a constant C3 = C3 (d, s) < ∞ such that if ⊂ Zd is finite and connected, then
K i, j ≤ C3 |∂|,
(3.24)
i∈ j∈c
where |∂| denotes the number of bonds with one endpoint in and the other in c .
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Proof. Let V ⊂ Rd denote the union of unit cubes centered at the sites of . Let
(3.25) W = y ∈ V c : dist(y, V ) ≥ 1 . In light of (3.8), it suffices to show that, for some C < ∞, 1 dy dx ≤ C(∂ V ), |x − y|s W V
(3.26)
where denotes the surface measure on ∂ V . (Indeed, we have (∂ V ) = |∂|.) To this end we note that the function x → (d − s)|x|−s is the divergence of the vector field x → x/|x|s . The Gauss-Green formula thus tells us that for all y ∈ (V c )◦ , τ (x) · (x − y) 1 1 dx = (dx), (3.27) s |x − y| d − s |x − y|s V ∂ where τ (x) is the unit outer normal to the surface at point x (which is well defined -a.e. because ∂ V is piecewise smooth). But |τ (x) · (x − y)| ≤ |x − y| and so 1 1 1 dx ≤ (dx). (3.28) s |x − y| s − d ∂ |x − y|s−1 V But s > d + 1 ensures that y → |x − y|−(s−1) is integrable over {y ∈ Rd : |y − x| ≥ 1} and so integrating over y, applying Fubini’s theorem, extending the y-integral from y ∈ W to {y : |y − x| ≥ 1}, and setting |z|1−s 1{|z|≥1} dz, (3.29) C = (s − d)−1 Rd
we get (3.26).
4. Proofs of Main Results Here we will prove the results from Sect. 2; we begin with Theorem 2.1. In our efforts to rule out that m > 0, it is useful to have a definite state that exhibits the magnetization. Our choice will be the limit of states at positive external field that are constructed on the torus. Definition 4.1. Let h > 0 and let −T;h denote an infinite volume state for the interaction described in (1.1–2.1) at inverse temperature β and external field h that is constructed as a limit of finite volume states with toroidal boundary conditions. We define −T to be any h ↓ 0 weak limit of the states −T;h . When the occasion arises, we will denote the measure associated with this state by wT . Lemma 4.2. The measure wT is a Gibbs measure for the interaction described in (1.1–2.1) at inverse temperature β. Moreover, wT is translation invariant, it satisfies σ0 T = m and if m L denotes the block magnetizations, 1 mL = σi , (4.1) | L | i∈ L
then for any µ with 0 < µ < m , lim wT (m L > µ) = 1.
L→∞
(4.2)
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Proof. These are standard results from the general theory of Gibbs states. Indeed, translation invariance follows by construction while the fact that wT is Gibbs is a result of the absolute summability of interactions; cf. [25, Corollary III.2.3]. To compute the expectation σ0 T we recall that concavity of the free energy ensures that for any h < h < h and any translation-invariant Gibbs state −h at external field h, −
∂f ∂f (β, h ) ≤ σ0 h ≤ − −(β, h ). ∂h + ∂h
(4.3)
The definition of m —and the construction of −T —then implies σ0 T = m . Finally, we claim that m L → m in wT -probability, implying (4.2). Indeed, if the random variable m L were not asymptotically concentrated, then c L : = wT (m L > m + )
(4.4)
would be uniformly positive (at least along a subsequence) for some > 0. But then the DLR conditions and (2.6–2.7) would imply that, for any h > 0,
h| |m Z L ,h h| L |(m +) L L ≤ e = e−| L |[ f (β,h)− f (β,0)+o(1)] . (4.5) cL e T = Z L ,0 T Hence we would conclude f (β, h) − f (β, 0) ≤ −(m + )h, in contradiction with (2.8).
(4.6)
We now define the random analogue of the quantity TL denoted by T L . In each configuration this quantity measures the antiferromagnetic interaction between the inside and outside of a box of scale L: TL = K i, j σi σ j . (4.7) i∈ L j∈cL
The central estimate—from which Theorem 2.1 will be readily proved—is as follows: Proposition 4.3. Consider the interaction described by (1.1–2.1) with s ∈ (d, d +1] and β ∈ (0, ∞) and let m denote the spontaneous magnetization corresponding to these parameters. For each λ ∈ (0, 1) there is L 0 < ∞ such that for L ≥ L 0 , T L T ≥ λm 2 TL .
(4.8)
To facilitate the proof we will state and prove a small lemma concerning the averaging behavior of the K i, j ’s: Lemma 4.4. Let and a be such that a and let V1 and V2 be two translates of d such that dist(V1 , V2 ) ≥ a. Then for any σ ∈ {+1, −1}Z and any i 0 ∈ V1 and j0 ∈ V2 , K σ σ − K σi σ j ≤ C K i0 , j0 | |2 . (4.9) i, j i j i 0 , j0 a i∈V1 j∈V2
i∈V1
j∈V2
Here C is a constant independent of a, , σ , i 0 or j0 .
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Proof. This is a simple consequence of the bound K i, j − K i , j ≤ C K i , j 0 0 a 0 0
(4.10)
which follows by (discrete) differentiation of the formula (2.1) and using the fact that the distance between V1 and V2 is at least a, while the difference between the minimum and maximum separation of V1 and V2 is a number of order and a.
Proof of Proposition 4.3. For a = a(L) tending to infinity in the fashion described in the proof of Lemma 3.1, it is sufficient to establish the inequality in (4.8) with TL replaced by TL ,a and T L replaced by its random analogue, T L ,a , defined by the corresponding modification of (4.7). We will need to introduce one more length scale, namely = (L) which will also tend to infinity but in such a way that /a → 0. We will assume that L, a and are such that both L−a and cL may be tiled by disjoint copies of . (Technically this only proves the result for a subsequence but the extension is trivial.) Let V1 and V2 denote translates of with V1 ⊂ L−a and V2 ⊂ cL and let us pick i 0 ∈ V1 and j0 ∈ V2 . Let q = wT (m > µ).
(4.11)
The following is now easily derived using Lemma 4.4: On the event that the average magnetization in both V1 and V2 exceeds µ (which has probability at least 2q − 1) the contribution of i ∈ V1 and j ∈ V2 to the random variable T L ,a is at least 1 + O(/a ) K i0 , j0 | |2 µ2 .
(4.12)
On the other hand, on the complementary event (which has probability 1 − q ) the contribution can be as small as − 1 + O(/a ) K i0 , j0 | |2 .
(4.13)
This means that the blocks V1 and V2 contribute to T L ,a T at least
1 + O(/a ) K i0 , j0 | |2 µ2 (2q − 1) − (1 − q ) .
(4.14)
Finally, Lemma 4.4 also gives K i0 , j0 | |2 = 1 + O(/a ) K i, j .
(4.15)
i∈V1 j∈V2
Noting that the error O(/a ) holds uniformly in the position of V1 and V2 , we may now sum over all (disjoint) translates of V1 and V2 in L−a and cL , respectively, to get T L ,a T ≥ 1 + O(/a ) µ2 (2q − 1) − (1 − q ) TL ,a .
(4.16)
Since we assumed /a → 0 and q → 1 as L → ∞, the right-hand side exceeds λµ2 TL once L 1.
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Proof of Theorem 2.1. By the inherent spin-reversal symmetry, an enhancement of the standard Peierls contour (de)erasement procedure yields, for any κ > 0, wT (T L ≥ κ TL ) ≤ e−2β[κ TL −2d J L
d−1 ]
.
(4.17)
Indeed, considering the probability conditioned on the configuration outside L , we may split the energy into two parts: the energy inside E in (σ ) and the energy E bdry (σ ) across the boundary of L . The important difference between these objects is that E in is invariant under the (joint) reversal of all spins in L , while E bdry changes sign. Using the fact that the conditional measure has the Gibbs-Boltzmann form, and restricting the partition function in the denominator to configurations obeying T L ≤ −κ TL , we get −β[E in (σ )+E bdry (σ )] σ : T L ≥κ TL e wT (T L ≥ κ TL |σc ) ≤ . (4.18) −β[E in (σ )+E bdry (σ )] σ : T L ≤−κ TL e Now let us reverse all spins in L in the lower sum; this yields −β[E in (σ )+E bdry (σ )] σ : T L ≥κ TL e wT (T L ≥ κ TL |σc ) ≤ . −β[E in (σ )−E bdry (σ )] σ : T L ≥κ TL e
(4.19)
But E bdry (σ ) ≥ κ TL − 2d J L d−1
(4.20)
for every σ in these sums and so (4.17) holds pointwise for wT (T L ≥ κ TL |σc ). Integrating over the boundary condition, we get (4.17). To finish the proof, we now note T L T ≤ TL wT (T L ≥ κ TL ) + κ TL wT (T L < κ TL ).
(4.21)
We learned in Proposition 4.3 that for any λ < 1 the left-hand side is bounded below by λm 2 TL for all L large enough. Thus we have, ∀κ ∈ (0, 1) and ∀λ ∈ (0, 1) λm 2 − κ ≤ wT (T L ≥ κ TL ) (1 − κ)
(4.22)
once L 1. But Proposition 3.1 tells us TL L d−1 and so, in light of (4.17), the L → ∞ limit forces λm 2 ≤ κ. Taking κ ↓ 0 yields m = 0 as claimed.
Proof of Corollary 2.2. Let µ be an arbitrary Gibbs state. A variant of the inequality in (4.5) tells us that, for any h > 0, µ(m L > |σcL ) ≤ e−h| L |
Z L ,h (σcL ) Z L ,0 (σcL )
.
(4.23)
Since m = 0, the ratio of the partition functions behaves like Z L ,h (σcL ) Z L ,0 (σcL )
= exp{| L | [o(h) + o(1)]}
(4.24)
and so, choosing 0 < h 1, the right-hand side decays exponentially in | L |. An analogous derivation (involving h < 0) shows a bound on µ(m L < −). The second part of the claim now follows by the Borel-Cantelli lemma.
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We will also finish the proof of the existence of magnetic order for s > d + 1: Proof of Theorem 2.3. The proof is a simple modification of the standard Peierls argument. Consider the box L and let µ+L denote the Gibbs measure in L with plus boundary condition in cL . We claim that µ+L (σ0 = −1) 1 once J and β are sufficiently large (in d ≥ 2). Indeed, given a connected set ⊂ L whose component is connected and which contains the origin, let A denote the event that σ0 = −1 and that ∂ is the outer boundary of the connected component of −1’s containing the origin. (In other words, c is the unique infinite connected component in the complement of the connected component of −1’s containing 0.) Given σ ∈ A , let σ be the result of flipping all spins in (including the +1’s). We have H L (σ ) − H L (σ ) ≥ 2J |∂| − 2 K i, j . (4.25) i∈ j∈c
By Proposition 3.4 the second term in the exponent is bounded by C3 |∂|. Letting J0 = C3 and applying the argument in (4.18–4.19), we thus get µ+L (A ) ≤ e−2β(J −J0 )|∂| .
(4.26)
But µ+L (σ0 = −1) can be written as the sum of µ+L (A ) over all connected ⊂ L (with connected complement) containing the origin. The standard Peierls argument shows that this sum is dominated by the = {0} term once e+2β(J −J0 ) exceeds the connectivity constant for the so-called Peierls contours. It follows that µ+L (σ0 = −1) 1 for J > J0 and β sufficiently large, uniformly in L. Taking the weak limit L → ∞ produces a magnetized infinite volume Gibbs measure µ+ and, by symmetry, a counterpart negatively-magnetized state µ− .
5. Open Problems We finish by some comments and a few open problems. First, the present paper shows the absence of magnetization at h = 0. A natural question is now as follows: Problem 1. Let s ∈ (d, d + 1]. Characterize the values h = 0 at which the free energy is continuously differentiable in homogeneous external field h. An answer to this question depends strongly on the precise structure of lowtemperature states. In particular, if there is a rigid stripe order (see Problem 3) it is possible that, for some particular values of h, there will be phase coexistence between different arrangements of stripes. Whether that has an effect on the continuity of the magnetization is not clear. To move to our next problem, let us recall the main reason why the exponent s = d +1 is critical for the disappearance of magnetic order: For s ≤ d + 1, the gain to be obtained from the antiferromagnetic interaction “through” the boundary of a volume of scale L is order L 2d−s which—including the log L correction when s = d + 1—overpowers the short-range surface cost of order L d−1 . However the short-range calculation only applies under the conditions where one envisions a surface tension, e.g., discrete spins. If we replace the Ising spins by, say, plane rotors, the cost due to local interactions for turning over a block now scales as L d−2 . Various exponents will readjust accordingly. Thus we pose:
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Problem 2. For the Ising spins replaced by O(n)-spins, and the spin-spin interactions given by the dot product, find the range of exponents s for which the spontaneous magnetization vanishes. The problem is interesting due to competing effects in the vicinity of the (purported) interfaces. It has been stipulated in [8] that, in these cases, magnetism will not occur for d < s < d + 2. See [9] for some relevant calculations. As for our next problem we note that, as already mentioned, absence of magnetism is far from ruling out other types of order, with striped states being a prime candidate. Thus we ask: Problem 3. Prove the existence of striped states at low temperatures for interactions of the type discussed in this note. Some mathematical progress [13] and a great deal of physical progress [1, 5, 6, 10, 11, 14, 18, 19, 21–23, 27] in this direction has been made for the ground state problem. But, at present, the positive-temperature case is far from resolved. Finally, we recall that much of our proof was based on thermodynamic arguments which, to begin with, require the existence of thermodynamics. Notwithstanding, analogous results should hold even for interactions that decay so slowly that the standard techniques ensuring the existence of the free energy fail. An instance of some genuine interest arises from Ref. [17]: Consider the model with the Hamiltonian as in (1.1) but with the long-range interaction term modified into K i, j (σi − ρ)(σ j − ρ). (5.1) i, j
The quantity ρ plays the role of “background charge” density; the spin configurations are restricted to have average ρ (otherwise their energy diverges). Problem 4. Suppose K i, j ∼ |i − j|−1 in d = 2, 3 (and, in general, K i, j ∼ |i − j|−s with sd < s ≤ d and d ≥ 2). Prove that the free energy is differentiable in ρ, at ρ = 0. On the basis of [28] one can infer that the lower bound, sd , on the region of exponents in the previous open problem satisfies sd ≤ d − 1. However, it is noted that, for s = d − 2, there is a (complicated) counterexample to differentiability [15] so, presumably, sd ≥ d − 2. Acknowledgement. The authors wish to thank Aernout van Enter for many useful comments on the content and literature. This research was partially supported by the grants NSF DMS-0505356 (M.B.), NSF DMS-0306167 (L.C.) and DOE DE-FG03-00ER45798 (S.K.).
References 1. Abanov, Ar., Kalatsky, V., Pokrovsky, V.L., Saslow, W.M.: Phase diagram of ultrathin ferromagnetic films with perpendicular anisotropy. Phys. Rev. B 51(2), 1023–1038 (1995) 2. Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the magnetization in onedimensional 1/|x − y|2 Ising and Potts models. J. Stat. Phys. 50(1-2), 1–40 (1988) 3. Aizenman, M., Newman, C.: Discontinuity of the percolation density in one-dimensional 1/|x − y|2 percolation models. Commun. Math. Phys. 107(4), 611–647 (1986) 4. Anderson, P.W., Yuval, G., Hamann, D.R.: Exact results in the Kondo problem. II. Scaling theory, qualitatively correct solution, and some new results on one-dimensional classical statistical models. Phys. Rev. B 1(1), 4464–4473 (1970)
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5. Bak, P., Bruinsma, R.: One-dimensional Ising model and the complete devil’s staircase. Phys. Rev. Lett. 49(4), 249–251 (1982) 6. Chayes, L., Emery, V.J., Kivelson, S.A., Nussinov, Z., Tarjus, G.: Avoided critical behavior in a uniformly frustrated system. Physica A 225(1), 129–153 (1996) 7. Daniëls, H.A.M., van Enter, A.C.D.: Differentiability properties of the pressure in lattice systems. Commun. Math. Phys. 71(1), 65–76 (1980) 8. van Enter, A.C.D.: A note on the stability of phase diagrams in lattice systems. Commun. Math. Phys. 79(1), 25–32 (1981) 9. van Enter, A.C.D.: Instability of phase diagrams for a class of “irrelevant” perturbations. Phys. Rev. B 26(3), 1336–1339 (1982) 10. Fogler, M.M., Koulakov, A.A., Shklovskii, B.I.: Ground state of a two-dimensional electron liquid in a weak magnetic field. Phys. Rev. B 54(3), 1853–1871 (1996) 11. Garel, T., Doniach, S.: Phase-transitions with spontaneous modulation: the dipolar ferromagnet. Phys. Rev. B 26(1), 325–329 (1982) 12. Ginibre, J., Grossmann, A., Ruelle, D.: Condensation of lattice gases. Commun. Math. Phys. 3(3), 187– 193 (1966) 13. Giuliani, A., Lebowitz, J.L., Lieb, E.H.: Ising models with long-range dipolar and short range ferromagnetic interactions. Phys. Rev. B 74(6), 064420 (2006) 14. Grousson, M., Tarjus, G., Viot, P.: Evidence for “fragile” glass-forming behavior in the relaxation of Coulomb frustrated three-dimensional systems. Phys. Rev. E 65(6), 065103 (2002) 15. Huse, D.: private communication 16. Israel, R.B.: Generic triviality of phase diagrams in spaces of long-range interactions. Commun. Math. Phys. 106(3), 459–466 (1986) 17. Jamei, R., Kivelson, S., Spivak, B.: Universal aspects of Coulomb-frustrated phase separation. Phys. Rev. Lett. 94(5), 056805 (2005) 18. Kashuba, A.B., Pokrovsky, V.L.: Stripe domain structures in a thin ferromagnetic film. Phys. Rev. B 48(14), 10335–10344 (1993) 19. Koulakov, A.A., Fogler, M.M., Shklovskii, B.I.: Charge density wave in two-dimensional electron liquid in weak magnetic field. Phys. Rev. Lett. 76(3), 499–502 (1996) 20. Lebowitz, J.L., Mazel, A.E.: Improved Peierls argument for high-dimensional Ising models. J. Statist. Phys. 90(3-4), 1051–1059 (1998) 21. Löw, U., Emery, V.J., Fabricius, K., Kivelson, S.A.: Study of an Ising model with competing long- and short-range interactions. Phys. Rev. Lett. 72(12), 1918–1921 (1994) 22. Ng, K.-O., Vanderbilt, D.: Stability of periodic domain structures in a two-dimensional dipolar model. Phys. Rev. B 52(3), 2177–2183 (1995) 23. Ortix, C., Lorenzana, J., Di Castro, C.: Frustrated phase separation in two-dimensional charged systems. Phys. Rev. B 73(24), 245117 (2006) 24. Seul, M., Andelman, D.: Domain shapes and patterns: The phenomenology of modulated phases. Science 267(5197), 476–483 (1995) 25. Simon, B.: The Statistical Mechanics of Lattice Gases. Vol. I, Princeton Series in Physics, Princeton, NJ: Princeton University Press (1993) 26. Sokal, A.D.: More surprises in the general theory of lattice systems. Commun. Math. Phys. 86(3), 327– 336 (1982) 27. Spivak, B., Kivelson, S.: Phases intermediate between a two-dimensional electron liquid and Wigner crystal. Phys. Rev. B 70(15), 155114 (2004) 28. Spivak, B., Kivelson, S.: Transport in two dimensional electronic micro-emulsions. Ann. Phys. 321(9), 2071–2115 (2006) 29. Thouless, D.J.: Critical region for the Ising model with a long-range interaction. Phys. Rev. 181(2), 954– 968 (1969) Communicated by M. Aizenman
Commun. Math. Phys. 274, 233–241 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0257-8
Communications in
Mathematical Physics
Discrete Path Integral Approach to the Selberg Trace Formula for Regular Graphs P. Mnëv St Petersburg Department of the Steklov Institute of Mathematics, St Petersburg 191023, Russia. E-mail:
[email protected] Received: 6 September 2006 / Accepted: 3 January 2007 Published online: 9 May 2007 – © Springer-Verlag 2007
Abstract: We give a new proof of the Selberg trace formula for regular graphs. Our approach is inspired by path integral formulation of quantum mechanics, and calculations are mostly combinatorial. 1. Introduction The famous Selberg trace formula first appeared in [1]. On a compact hyperbolic surface it relates the eigenvalue spectrum of the Laplace operator to the length spectrum of closed geodesics. A version of this formula for finite regular graphs was obtained by Ahumada [2] (cf. also Ihara [3]). Trace formulae are known to have many implications. For instance, they can be considered as nonabelian generalizations of the Poisson summation formula. In case of finite graphs, since one can find the eigenvalue spectrum of the Laplacian for a given graph explicitly, the trace formula lets one find the numbers of closed geodesics of any length (see (34)). In physics trace formulae indicate the cases when semi-classical evaluation of the path integral for state sum of a quantum free particle in some background is exact. The idea of deriving the original Selberg trace formula for hyperbolic surfaces from a path integral belongs to Gutzwiller [6]. The Selberg trace formula is also known to bear much resemblance to the Riemann-Weil formula in number theory. The original proof of the trace formula for regular graphs (30) was in the framework of “discrete harmonic analysis”. We propose another way to derive it, inspired by the path integral approach in quantum mechanics [5]. We consider the trace Z (t) = tr et as a state sum of the quantum free particle living on the graph. We rewrite it as a sum over closed paths, which is a discrete version of the usual path integral over loops for the quantum mechanical state sum. Then we divide the set of closed paths into classes of homotopically equivalent paths. There is a class of contractible paths, and one homotopy Supported by grants: RFBR 05-01-00922, RAS Presidium program “Mathematical Problems of Nonlinear Dynamics”.
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class for each closed geodesic on the graph. We explicitly calculate the contribution of each homotopy class to the state sum, thus rewriting it as a contribution of contractible paths plus the sum over “nontrivial” geodesics (of nonzero length) of contributions of their individual homotopy classes. This is analogous to the stationary phase calculation of the path integral. Geodesics serve as stationary points of the action in the space of loops. The homotopy class of a geodesic serves as a neighbourhood of the stationary point. Thus we arrive at the known trace formula for a regular graph, with a specific, physically relevant, choice of test function for the eigenvalue spectrum of the Laplacian. We wish to thank P. Zograf who inspired this work. 2. Notations and Definitions Let be a finite regular connected non-oriented graph with vertices of valence q + 1 ≥ 2 with no multiple edges and no edges connecting a vertex with itself. Denote by V () and E() the set of vertices of and the set of edges respectively. Let || = #V () be the number of vertices. Further denote the space of complex-valued functions on vertices by Fun() = CV () . A basis function (vector) |v > associated with vertex v equals 1 on v and 0 on the other vertices. We further adopt the quantum-mechanical notations and denote the transposed basis vector by < v| = |v >T . We call the set of vertices connected to v by edges its link and denote it Lk(v). The averaging operator T : Fun() → Fun() acts as follows: for f ∈ Fun(),
(T f )(v) =
f (v ).
(1)
f (v ) − val(v) f (v),
(2)
v ∈
Lk(v)
The Laplace operator on is defined by ( f )(v) =
v ∈
Lk(v)
where val(v) is the valence of v. Since we consider a regular graph , differs from T by a multiple of identity: = −(q + 1)1 + T , where 1 is the identity map Fun(T ) → Fun(T ). The matrix of the averaging operator is just the adjacency matrix of the graph: < v |T |v >= 1 if v and v are connected by an edge and 0 otherwise. The diagonal elements of T are zero. The physically interesting quantity is the trace of the heat kernel (the state sum) et , Z (t) = tr exp(t) = e−(q+1)t Z T (t),
(3)
Z T (t) = tr exp(t T ).
(4)
where ||
It turns out that Z T (t) is more convenient for our calculation than Z (t). If {λ j } j=1 is the set of eigenvalues of T then Z T (t) =
|| j=1
eλ j t .
(5)
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If we introduce the spectral density function for T , ρ(s) =
||
δ(s − λ j ),
(6)
j=1
we may express Z T (t) as a Laplace transform of ρ(s): ∞ ρ(s)est ds. Z T (t) = −∞
(7)
Eigenvalues λ j are known to satisfy −q − 1 ≤ λ j ≤ q + 1. Moreover, q + 1 is always an eigenvalue, while −q − 1 may be an eigenvalue and may not be. If it is, then the distribution of eigenvalues is necessarily even ρ(−s) = ρ(s). 3. Sum over Paths Let us define a closed path of length l as a sequence of vertices (v1 , . . . , vl ) such that for every j = 1, . . . , l the v j is connected to v j+1 by an edge (we identify vl+1 with v1 ). We will usually omit the word “closed” in the following, since all paths, walks, trajectories etc. will be supposed to be closed. Denote the set of paths by P and the length of a path p ∈ P by | p|. We also denote the number of closed paths of length l by pl . It is convenient to identify paths of length 0 with vertices of . Lemma 1. Z T (t) =
∞ t | p| tl = pl . | p|! l! p∈P
(8)
l=0
This expression may be viewed as a discrete version of path integral over loops for | p| the state sum, with |t p|! being the analogue of the measure e−S on loops. We give two different explanations of (8). The first one is more lengthy, but done in the spirit of usual derivation of path integral representation in quantum mechanics. The second is absolutely straightforward and evident. 3.1. First proof of Lemma 1. Let us evaluate Z T (t) in the following manner: t < v|et T |v >= lim < v|(1 + T ) N |v > Z T (t) = N →∞ N v∈V () v∈V () t t < v1 |(1 + T )|v N >< v N |(1 + T )|v N −1 > = lim N →∞ N N v1 ,...,v N ∈V ()
× · · · < v2 |(1 +
t T )|v1 >; N
(9)
we are summing here over all sequences of N vertices v1 , . . . , v N . Notice that matrix elements < vi+1 |(1 + Nt T )|vi > equal 1 if vi+1 = vi ; Nt if vi+1 and vi are connected by an edge; and 0 otherwise. Let us call a walk of length N a sequence of vertices (v1 , . . . , v N ) such that each pair of successive vertices v j , v j+1 are either connected by
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an edge or coincide. Let W N be the set of walks of length N . For a walk w ∈ W N denote the number of values of j for which v j and v j+1 are connected by an edge by |w|. The only nonzero terms in the last line of (9) are those with the sequence w = (v1 , . . . , v N ) being a walk. For these terms the summand is (t/N )|w| . Thus we have (t/N )|w| . (10) Z T (t) = lim N →∞
w∈W N
This is also a sort of discrete path integral representation for Z T . To transform it to the form (8), we need a projection π N : W N → P which leaves only those vertices in a walk for which v j = v j+1 , and forgets the others. For a walk w ∈ W N the result of projection π N (w) is a path of length |w|. Each path of length l has C lN walks as preimages under π N (C lN is a binomial coefficient). So | p| Z T (t) = lim C N (t/N )| p| . (11) N →∞
p∈P
Using | p|
lim
N →∞
we obtain (8).
CN 1 , = | p| N | p|!
(12)
3.2. Second proof of Lemma 1. One can arrive at (8) in a more straightforward way: we may just expand the exponent in the definition of Z T (t) in a Taylor series in the variable t: ∞ l t tr T l , (13) Z T (t) = tr et T = l! l=0
then
tr T l =
< v1 |V |vl >< vl |V |vl−1 > · · · < v2 |V |v1 >;
(14)
v1 ,...,vl ∈V ()
the terms in this sum with (v1 , . . . , vl ) ∈ P equal 1, all the others vanish; hence tr T l = pl , and we obtain (8).
(15)
4. Sum over Geodesics We use the term “closed trajectory” for equivalence class of closed paths under cyclic permutations of vertices along the path. So a trajectory is a path with information on the starting point forgotten. An elementary homotopy is a transformation of trajectories of the following kind: (v1 , . . . , v j , . . . , vn ) → (v1 , . . . , v j , v , v j , . . . , vn ), where v ∈ Lk(v j ). Two trajectories are called homotopic if they can be connected by a chain of elementary homotopies (with arrows either forward or backward). The shortest
Discrete Path Integral Approach to Selberg Trace Formula for Regular Graphs
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representative in a homotopy class is called a geodesic trajectory (or just geodesic). An alternative definition of geodesic trajectory is as a trajectory satisfying vi = vi+2 for all i. Denote the set of all geodesics on by G. Two paths are called homotopic if their trajectories are homotopic. If γ = (v1 , . . . , vn ) is a trajectory of length n then its r th power is defined as a trajectory of length r n obtained as γ walked around r times: γ r = (v1 , . . . , vn , . . . , v1 , . . . , vn ). A trajectory γ is called primitive if it is not a (nonunit) power of any trajectory. A geodesic trajectory with one of its vertices chosen as a starting point is a geodesic path. A path homotopic to a path of length 0 is called contractible. We call geodesics of length 0 trivial, and geodesics of length >0 nontrivial. We proceed now to the calculation of the contribution of contractible paths to (8) (one may also call it the contribution of trivial geodesics). 4.1. Contribution of contractible paths. Let us denote ¯ the covering tree for and call some point C ∈ ¯ the center. The function dist on vertices of the covering tree dist : ¯ → N0 returns the minimal number of edges one must pass to reach a given vertex from the center. Any contractible closed path on can be lifted to a closed path on ¯ (and all closed paths there are contractible, since ¯ is a tree) and we adjust the lift so that it starts and ends in C. Since each edge passed in one direction by a closed path on ¯ must be passed in the opposite direction, the length of the path must be even. ¯ the set of closed paths on ¯ of length 2k starting and ending in C; Denote by P2k () ¯ ⊂ P2k () ¯ consisting of closed paths not returning to C we also need a subset P˜2k () (except for the starting point and the end point). Recall a concept of Dyck path of length 2k (see e.g. [4]): it is a sequence of integers (α1 , α2 , . . . , α2k+1 ) with α1 = α2k+1 = 0, αi ≥ 0 and αi+1 = αi ± 1. We denote the set of Dyck paths of length 2k as D2k ; # D2k = Catk (the k th Catalan number). There is a ¯ → D2k−2 . It acts as follows: projection πkD : P˜2k () (v1 = C, v2 , . . . , v2k , v2k+1 = C) → (dist(v2 ) − 1, . . . , dist(v2k ) − 1).
(16)
The number of preimages for any Dyck path under πkD equals (q + 1)q k−1 since there are q + 1 choices to make the step from v1 = C to v2 ; q choices for each step, increasing dist; steps, decreasing dist are done uniquely (since for any vertex v = C of one edge from it leads inward, while the q others lead outward). So we have obtained that ¯ = (q + 1)q k−1 Catk−1 . # P˜2k ()
(17)
¯ is a simple modification of The generating function for the numbers of paths # P˜2k () the usual generating function for Catalan numbers: ∞ 1 − 4qs 2 1 − 2k −1 ¯ s = (1 + q ) F P˜ (s) = . (18) # P˜2k () 2 k=1
For numbers of paths that may pass through the center we have ∞ (q + 1) 1 − 4qs 2 − q + 1 1 2k ¯ F P (s) = = , # P2k () s = 1 − F P˜ (s) 2 (1 − (q + 1)2 s 2 )
(19)
k=0
and hence we obtain the contribution to Z T (t) from contractible paths: A+i∞ ∞ ¯ 2k # P2k () 1 1 1 t = || · [Z T (t)]con = || · ds est F P ( ), (2k)! 2πi A−i∞ s s k=0
(20)
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where the real part of A is greater than real parts of all singular points of the integrand, as usual for the inverse Laplace transform. The factor of || in front is due to the fact that a contractible path can start from any vertex of (we recall that || denotes the number of vertices in ). Further evaluating the integral we wrap the contour of integration around √ √ the cut s ∈ [−2 q, 2 q]. Thus we proved Lemma 2. The contribution of contractible paths to (8) is √ 4q − s 2 q +1 2 q st [Z T (t)]con = || · ds e . √ 2π −2 q (q + 1)2 − s 2
(21)
In other words, we derived a contribution to the spectral density of T on from contractible paths: 4q − s 2 q +1 (22) [ρ(s)]con = || · 2π (q + 1)2 − s 2 √ √ in the interval s ∈ [−2 q, 2 q] and 0 outside it. 4.2. Contribution of nontrivial geodesics. Suppose we have a (nontrivial) primitive geodesic γ of length l on . To calculate the contribution of its homotopy class to Z T (t) we need to find the number of paths of length k homotopic to γ : pγ ,k . Since a path is a trajectory with some point on it chosen as a start/end point, pγ ,k = k tγ ,k , where tγ ,k is the number of trajectories of length k homotopic to γ . Note that this is not true for non-primitive γ . We may find the numbers tγ ,k using the following combinatorial construction. If γ = (v1 , . . . , vl ) (with the periodic condition v0 = vl ), any trajectory that can be contracted to γ can be represented as a closed contractible path from v1 to v1 never going along the edge v1 − v0 ; then a step v1 − v2 ; then a closed contractible path from v2 to v2 , never returning along the edge v2 − v1 ; then step v2 − v3 and so on. All in all it is l closed contractible paths with one direction prohibited and l unit steps. Thus we find the generating function for tγ ,k : Fγ(tra j.) (s) =
∞
tγ ,k s k = s |γ | (Fˆ P (s))|γ | ,
(23)
k=|γ |
where the superscript (tra j.) indicates that we are counting trajectories, Fˆ P (s) is the generating function for the numbers of contractible closed paths with one direction prohibited: 1 1 − 1 − 4qs 2 ˆ F P (s) = , (24) = q 2qs 2 1 − q+1 F P˜ (s) and hence
Fγ(tra j.) (s) =
1−
1 − 4qs 2 2qs
l .
For the numbers of paths homotopic to γ we have ∞
∂ |γ | Fγ( paths) (s) = pγ ,k s k = s Fγ(tra j.) (s) = ∂s 1 − 4qs 2 k=|γ |
(25)
1−
|γ | 1 − 4qs 2 . 2qs (26)
Discrete Path Integral Approach to Selberg Trace Formula for Regular Graphs
239
Now we would like to calculate the numbers of paths homotopic to non-primitive geodesic γ = (γ )r , that is a primitive geodesic γ passed r ≥ 2 times. It turns out that if we carry out the scheme above in this case, every path becomes calculated r times. For any geodesic γ denote by (γ ) the length of the underlying primitive geodesic. If γ is primitive itself, we set (γ ) = |γ |. Thus for any geodesic we have Fγ( paths) (s)
(γ )
= 1 − 4qs 2
1−
1 − 4qs 2 2qs
|γ | .
(27)
Now we have all the information to write down the contribution of a nontrivial geodesic γ to Z T (t): 1 [Z T (t)]γ = 2πi
A+i∞
1 ( paths) 1 F ( )= s γ s
ds est
A−i∞
1 = 2πi
A+i∞
ds e A−i∞
st
(γ )
s 2 − 4q
s−
s 2 − 4q 2q
|γ | .
(28)
The last integral reduces to the modified Bessel function of the first kind I|γ | . So we deduced. Lemma 3. For every nontrivial geodesic γ ∈ G the contribution of its homotopic class in P to (8) equals √ [Z T (t)]γ = (γ )q −|γ |/2 I|γ | (2 qt). (29) Collecting together (21) and (29) we obtain the full trace formula: Theorem (Trace formula for regular graphs). Let be a finite connected regular graph of valence q + 1 ≥ 2 with || vertices, without multiple edges and loops. Let T be the averaging operator on and Z T (t) = tr et T with t a complex variable. Let also G be the set of nontrivial closed geodesics on . For each γ ∈ G, |γ | is the length of γ and (γ ) is the length of the underlying primitive geodesic: for γ = (γ )r with primitive γ we set (γ ) = |γ |. Then q +1 Z T (t) = || · 2π
√ 2 q √ −2 q
ds e
st
4q − s 2 √ + (γ )q −|γ |/2 I|γ | (2 qt). (30) 2 2 (q + 1) − s γ ∈G
Another useful form of the same result is √ ∞ 4q − s 2 q +1 2 q √ st Z T (t) = || · ds e + gpl q −l/2 Il (2 qt), 2π −2√q (q + 1)2 − s 2
(31)
l=3
where gpl is the number of geodesic paths of length l. To pass from (30) to (31) one must notice that the number of geodesic paths corresponding to a given geodesic trajectory γ is (γ ). We may interpret (30) in terms of spectral density ρ(s).
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Corollary 1. ρ(s)=||·
s ) T|γ | ( √ 4q − s 2 q +1 2 )+ −|γ |/2 2 q · θ(4q − s 2 ), (32) · θ (4q − s (γ )q 2π (q + 1)2 − s 2 π 4q − s 2 γ ∈G
where Tl (x) = cos(l arccos(x)) =
1 (x + x 2 − 1)l + (x − x 2 − 1)l 2
(33)
is the Chebyshev polynomial of the first kind of degree l. The sum on the right of (32) is to be understood in the generalized function sense (it does not exist in the ordinary sense since numbers gpl grow too fast). Notation θ is used for the unit step function. A remarkable fact is that although each √geodesic √ gives a smooth contribution to ρ(s) with support on the interval s ∈ [−2 q, 2 q] (with singularities on the endpoints), the sum of all contributions is a generalized function with support on eigenvalues of T , scattered across a wider interval s ∈ [−q − 1, q + 1]. We may invert (32) in a sense to reproduce the numbers of geodesic paths gpl from the spectrum of averaging operator: Corollary 2. For l ≥ 1, gpl = 2q l/2
|| j=1
Tl
λj √ 2 q
+
1 + (−1)l (q − 1) || 2
(34)
where λ j are eigenvalues of the averaging operator T . In particular since we know the highest eigenvalue λmax = q + 1, we immediately get (for q ≥ 2) from (34) the asymptotic law for numbers of geodesic paths: gpl ∼ q l as l → ∞ if −q − 1 is not an eigenvalue. If −q − 1 belongs to the spectrum of T (i.e. if the graph is bipartite), the asymptotic is gpl ∼ (1 + (−1)l ) q l . For comparison the numbers of all paths behave like pl ∼ (q + 1)l . 4.3. Case q = 1. This is a simple example where we can check (30) explicitly. The graph is necessarily a polygon with L ≥ 3 angles. Each nontrivial geodesic γ is characterized by the winding number r ≥ 1 and its direction of movement around the polygon: either clockwise or counterclockwise; (γ ) = L for all γ . The spectrum of T can be found easily: λ j = 2 cos 2πL j and the trace formula (30) gives Z T (t) =
L j=1
e2t cos
2π j L
= L I0 (2t) + 2L
∞ r =1
Ir L (2t) = L
∞
Ir L (2t).
(35)
r =−∞
This identity can checked by Poisson resummation. In the limit L → ∞, t = L 2 τ (keeping τ fixed) we recover a special case of the modular transformation for Jacobi theta function: ∞ ∞ r2 1 2 2 e−4π j τ = √ e− 4τ . (36) 4π τ r =−∞ j=−∞
Discrete Path Integral Approach to Selberg Trace Formula for Regular Graphs
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4.4. Remark. The original trace formula for a regular graph in [2] (rewritten in our notations and for the special case of trivial character of the fundamental group of ) states that for any function g : Z → C such that g(n) = −g(n) for all n ∈ Z and ∞
|g(n)| q n/2 < ∞,
(37)
n=1
the following holds: ||
g(z ˆ j ) = || ·
j=1
q 2πi
|z|=1
g(z) ˆ
1 − z 2 dz + (γ )q −|γ |/2 g(|γ |), q − z2 z
(38)
γ ∈G
where g(z) ˆ = ∞ g(n)z −n and numbers z j are defined by eigenvalues λ j by the √ n=−∞ −1 equation λ = q (z + z ). √ Formula (30) follows from (38) if we choose g(n) = In (2 qt) with corresponding √ −1 g(z) ˆ = et q(z+z ) . Integrals representing the contribution of contractible paths convert √ into one another with the change of variables s = q (z + z −1 ). 4.5. Remark. The trace formula (30) is actually valid for graphs with multiple edges and loops. The condition of absence of multiple edges and loops was chosen to simplify the combinatorial constructions. References 1. Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–82 (1956) 2. Ahumada, G.: Fonctions periodiques et formule des traces de Selberg sur les arbres. C. R. Acad. Sci. Paris 305, 709–712 (1987) 3. Ihara, Y.: On discrete subgroup of the two by two projective linear group over p-adic field. J. Math. Soc. Japan 18(3), 219–235 (1966) 4. Stanley, R.P.: Enumerative Combinatorics. Vol. 1. Monterey, CA: Wadsworth and Brooks, 1986 5. Feynman, R.P., Hibbs, A.R.: Quantum Physics and Path Integrals. New York: McGraw-Hill, 1965 6. Gutzwiller, M.C.: Physics and Selberg’s trace formula. In: The Selberg trace formula and related topics (Brunswick, Maine, 1984), Contemp. Math. 53, Providence, RI: Amer. Math. Soc., 1986, pp. 215–251 Communicated by L. Takhtajan
Commun. Math. Phys. 274, 243–252 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0264-9
Communications in
Mathematical Physics
Destruction of Absolutely Continuous Spectrum by Perturbation Potentials of Bounded Variation Yoram Last Institute of Mathematics, The Hebrew University, Givat Ram, 91904 Jerusalem, Israel. E-mail:
[email protected] Received: 13 October 2006 / Accepted: 9 November 2006 Published online: 16 June 2007 – © Springer-Verlag 2007
Abstract: We show that absolutely continuous spectrum of one-dimensional Schrödinger operators may be destroyed by adding to them decaying perturbation potentials of bounded variation.
1. Introduction In this paper we study discrete one-dimensional Schrödinger operators on 2 (N) of the form HV = H0 + V,
(1.1)
where H0 = + V0 is a discrete Schrödinger operator and V is a decaying perturbation potential. More explicitly, is the discrete Dirichlet Laplacian on 2 (N), defined by (ψ)(n) = ψ(n + 1) + ψ(n − 1)
(1.2)
for n > 1 and (ψ)(1) = ψ(2). V0 and V are discrete potentials, that is, (V0 ψ)(n) = V0 (n)ψ(n),
(V ψ)(n) = V (n)ψ(n),
n = 1, 2, . . . ,
(1.3)
where {V0 (n)} and {V (n)} are sequences of real numbers. We say that a potential V is decaying if V (n) → 0 as n → ∞. For operators H = HV of the form (1.1), we define ac (H ), the essential support of the absolutely continuous spectrum of H , as the equivalence class, up to sets of zero Lebesgue measure, of the set E ∈ R lim Im δ1 , (H − E − i)−1 δ1 exists and is finite and non-zero , ↓0
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where · , · denotes the scalar product in 2 (N) and δ j (n) is 1 if j = n and 0 otherwise. The absolutely continuous spectrum of H , σac (H ), coincides with the essential closure of ac (H ), namely, σac (H ) = E ∈ R |ac (H ) ∩ (E − , E + )| > 0 ∀ > 0 , where | · | denotes Lebesgue measure. In what follows we may write equalities of the form ac (H ) = S, where S may be a concrete subset of R. Since ac (·) is an equivalence class of sets rather than a concrete subset of R, such equalities should be understood as equalities up to sets of zero Lebesgue measure, or more precisely, as saying S ∈ ac (H ). We say that a potential V is of bounded variation if ∞
|V (n + 1) − V (n)| < ∞.
n=1
Recall Weidmann’s classical result [23]: Theorem 1.1 (Weidmann’s theorem). If V0 = 0 and V is a decaying potential of bounded variation, then ac (HV ) = ac () = (−2, 2). Remark. Weidmann [23] actually proves an analog of this for continuous Schrödinger operators on L 2 ([0, ∞), d x). For a proof of the discrete case, see Dombrowski-Nevai [7] or Simon [19]. Weidmann’s theorem bears some similarity to the following well-known consequence of the Birman-Kato theory (see [18, Chapter XI.3]): Theorem 1.2. For any V0 , if ∞ n=1 |V (n)| < ∞, then ac (HV ) = ac (H0 ). A notable difference, though, is that Theorem 1.2 allows an arbitrary V0 , while Weidmann’s theorem applies only to V0 = 0. The purpose of this paper is to answer the following natural question: Does Weidmann’s theorem extend to the general case of V0 = 0? Namely, is it true that for a general V0 and a decaying V of bounded variation, ac (HV ) = ac (H0 )? The answer turns out to be negative, as indicated by our first result: Theorem 1.3. If H0 has nowhere dense spectrum, then there exists a decaying potential V of bounded variation for which ac (HV ) = ∅. The point here is that there are known examples of V0 ’s for which the spectrum of H0 is both nowhere dense and absolutely continuous. Such examples, with limit periodic V0 ’s, have been constructed by Avron-Simon [1]. Another example is the weakly coupled irrational almost Mathieu operator, namely, the case V0 (n) = λ cos(2π αn + θ ), where λ, α, θ ∈ R, |λ| < 2 and α ∈ R \ Q (see, e.g., [16] and references therein). Thus, Theorem 1.3 provides a fairly broad class of H0 ’s whose absolutely continuous spectrum can always be destroyed by adding to them a decaying perturbation potential of bounded variation. However, the requirement for H0 to have nowhere dense spectrum along with the fact that ac () is an interval raises again a natural question: Does Weidmann’s theorem extend at least to the case of V0 ’s for which ac (H0 ) is made of intervals? The answer turns out to be negative, again, as indicated by our second result: Theorem 1.4. There exist a decaying potential V0 and a decaying potential of bounded variation V , such that ac (H0 ) = ac () = (−2, 2), but ac (HV ) = ∅.
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We note that in spite of Theorems 1.3 and 1.4, Weidmann’s theorem does have natural extensions and generalizations. In particular, we note the following result of GolinskiiNevai [8]: Theorem 1.5. If V0 = 0, V is decaying, and for some q ∈ N, ∞
|V (n + q) − V (n)| < ∞,
(1.4)
n=1
then ac (HV ) = ac () = (−2, 2). Remarks. 1. The result is actually more general. In particular, it covers cases where V isn’t decaying, so that (1.4) implies its convergence to a periodic potential of period q. (This is equivalent, of course, to the case where V is decaying and obeys (1.4) and V0 is some periodic potential of period q.) The result says, in this case, that ac (HV ) coincides with the spectrum of the limiting periodic operator, and thus, in particular, it extends Theorem 1.1 to the case of any periodic V0 . 2. Before [8], some related results were obtained by Stolz [21, 22]. We believe that the following stronger statement should be true: Conjecture 1.6. If V0 = 0, V is decaying, and for some q ∈ N, ∞
|V (n + q) − V (n)|2 < ∞,
(1.5)
n=1
then ac (HV ) = ac () = (−2, 2). We note that this conjecture for the special case q = 1 has also been made by Simon [20, 12]. Recently Kupin [13] came quite close to establishing it by showing that Chap. ∞ 2 if |V ∞ n=1 (np+ 1) − V (n)| < ∞ and in addition, there exists some p > 0 for which n=1 |V (n)| < ∞, then ac (HV ) = ac () = (−2, 2). Kupin’s result is among the strongest of several recent results (see [12, 14, 24]) in which an p requirement for V itself and an 2 requirement for some type of variation of V combine to ensure that ac (HV ) = ac () = (−2, 2). In most other results of this genre, p is a concrete number, such as 3 or 4. We consider the proving (or disproving) of the full Conjecture 1.6 to be an interesting (and potentially hard) open problem. It may have some connection to the results of Christ-Kiselev [4] for continuous Schrödinger operators, although it isn’t fully clear what should be considered a discrete analog of their results and how connected to Conjecture 1.6 it may be. We also recall the following conjecture of Kiselev-Last-Simon [11]: 2 Conjecture 1.7. For any V0 , if ∞ n=1 |V (n)| < ∞, then ac (HV ) = ac (H0 ). For the special case V0 = 0, it has been established by Deift-Killip [5], and for periodic V0 , by Killip [10]. A related result concerning general V0 has been recently obtained by Breuer-Last [2], who show stability of absolutely continuous spectrum associated with bounded generalized eigenfunctions under random decaying 2 perturbation potentials (also see a related result by Kaluzhny-Last [9]). We believe that Theorems 1.3 and 1.4 elucidate a fundamental difference between spectral stability under perturbation potentials decaying sufficiently fast (namely, statements like Theorem 1.2 and Conjecture 1.7) and “Weidmann type” statements like
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Theorem 1.5 and Conjecture 1.6, which appear not to extend much beyond cases where the unperturbed operator is periodic. The rest of this paper is organized as follows. In Sect. 2 we filter some results from [17] that we use later. In Sect. 3 we prove Theorem 1.3 and in Sect. 4 we prove Theorem 1.4. 2. Preliminaries Given a discrete Schrödinger operator of the form H˜ = + V˜ , we denote its associated 2 × 2 transfer matrices as follows: E − V˜ (n) −1 Tn (E) = , 1 0 m,n (E) = Tn (E)Tn−1 (E) · · · Tm (E) , n (E) = 1,n (E) = Tn (E)Tn−1 (E) · · · T1 (E) . The following is an immediate consequence of [17, Theorem 3.10]: Proposition 2.1. For Lebesgue a.e. E ∈ ac ( H˜ ), lim sup N →∞
N 1 n (E)2 < ∞ . N (log N )2 n=1
The following is an easy consequence of [17, Theorem 4.2] and its proof: Proposition 2.2. Let H1 = + V1 be another discrete Schrödinger operator on 2 (N) or 2 (Z). Suppose that for some m, k ∈ N, k ≥ 4, we have V˜ (n) = V1 (n) for every n ∈ {m, m +1, . . . , k}, and that for some E ∈ R and δ > 0, σ (H1 )∩(E −δ, E +δ) = ∅, where σ ( · ) denotes the spectrum of the operator. Then for every ∈ {4, 5, . . . , k}, m,m+ (E) ≥
−3 1 2 δ (1 + δ 2 ) 2 . 2
3. Proof of Theorem 1.3 Proof of Theorem 1.3. If V0 is unbounded, then so is V0 + V for any decaying V , and thus, by well known results (see, e.g., [17, Theorem 4.1]), we would get ac (HV ) = ∅ for any decaying V . We may thus assume, without loss of generality, that V0 is bounded, so that the spectrum of H0 , σ (H0 ), is a compact set. Its complement, R \ σ (H0 ), is an open set and thus a union of countably many disjoint open intervals: R \ σ (H0 ) = ∞ ν=1 Iν , Iν = (aν , bν ). We call the Iν ’s gaps in σ (H0 ), and denote the collection of all these gaps by G, that is, G = {Iν }∞ ν=1 . Since σ (H0 ) is compact, G contains exactly two elements of infinite length, which are (−∞, min σ (H0 )) and (max σ (H0 ), ∞). The other elements of G are open intervals of finite length and contained in (min σ (H0 ), max σ (H0 )). For every δ > 0, we let Gδ denote the collection of elements in G of length larger than δ, that is, Gδ = {Iν ∈ G | |Iν | > δ}, where |Iν | = bν − aν . We note that for each δ > 0, Gδ is a finite set, and denote m(δ) = #Gδ . We reorder the elements of Gδ , so that
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they are ordered by their order of occurrence on the real line. That is, we assume that Gδ = {I1 , . . . , Im(δ) }, where aν+1 > bν for every 1 ≤ ν < m(δ). We define a positive function η(δ) to be the maximal distance between neighboring elements of Gδ , that is, η(δ) = max{aν+1 − bν | Iν , Iν+1 ∈ Gδ } . Clearly, η(δ) is monotonely decreasing. Moreover, since σ (H0 ) is nowhere dense, one easily sees that η(δ) → 0 as δ → 0. We can thus pick a monotonely decreasing sequence ∞ {δ j }∞ j=1 (η(δ j ) + δ j ) < ∞. We also define, for each j, j=1 ⊂ (0, 1), such that m j = [2η(δ j )/δ j ] + 3 , where [ · ] denotes the integer part. We will now show that we can construct recursively a sequence {L j }∞ j=1 ⊂ N, numbers {n j,1 , . . . , n j,m j }, so that n j,k+1 > n j,k , n j,1 = L j , n j,m j = L j+1 , and a bounded monotonely increasing step potential V , so that H0 + V has no absolutely continuous spectrum. The resulting V will not be decaying, but converge to some constant limn→∞ V (n). Since we can simply subtract this constant from V to obtain a decaying potential, Theorem 1.3 would follow. To accomplish the recursive construction, let L 1 = 1, V (1) = 0, and assume that L j and V (n) for 1 < n ≤ L j are determined. Define V (n), for L j < n ≤ L j+1 , by V (n) = V (L j ) + kδ j /2 for n j,k < n ≤ n j,k+1 , where n j,2 , . . . , n j,m j will be determined later. Note that, by the construction, V (L j+1 ) − V (L j )= m j δ j /2 ≤ η(δ j ) + 3δ j /2, and thus V is positive and supn V (n) = limn→∞ V (n) ≤ ∞ j=1 (η(δ j ) + 3δ j /2). Consider now some fixed k ∈ {1, . . . , m j }. Suppose that n j,2 , . . . , n j,k are already determined and that E ∈ R obeys (E − δ j /4, E + δ j /4) ∩ σ (H0 + V (L j ) + kδ j /2) = ∅ .
(3.1)
Since V (n) coincides with the constant V (L j ) + kδ j /2 for n j,k < n ≤ n j,k+1 , it follows from Proposition 2.2 that for 4 < ≤ n j,k+1 − n j,k , n j,k +1,n j,k + (E) ≥
−4 1 (δ j /4)2 (1 + (δ j /4)2 ) 2 , 2
(3.2)
where we use the notations of Sect. 2 for V˜ = V0 +V . If n j,k (E) > ( jn j,k )1/2 log n j,k , then we clearly have N 1 n (E)2 > j N (log N )2
(3.3)
n=1
for N = n j,k . Otherwise, since n j,k +1,n j,k + (E) = n j,k (E)−1 n j,k + (E) ≤ n j,k (E)−1 n j,k + (E) and det( n j,k (E)) = 1 implies n j,k (E)−1 = n j,k (E), (3.2) implies n j,k + (E) ≥
n j,k +1,n j,k + (E) n j,k (E)
≥
−4 (δ j /4)2 (1 + (δ j /4)2 ) 2 . 2( jn j,k )1/2 log n j,k
(3.4)
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This implies that, by choosing n j,k+1 to be sufficiently large, we can ensure that for any E obeying (3.1), (3.3) will hold either for N = n j,k or for N = n j,k+1 . Now given any fixed E ∈ R, the definitions of η(δ) and m j ensure that there will be some k ∈ {1, . . . , m j } for which (3.1) holds. We thus see that we can choose n j,2 , . . . , n j,m j so that for every E ∈ R, there will be some L j ≤ N ≤ L j+1 for which (3.3) holds. Thus, by choosing n j,2 , . . . , n j,m j appropriately for each j, we ensure that lim sup N →∞
N 1 n (E)2 = ∞ , N (log N )2
(3.5)
n=1
for every E ∈ R. By Proposition 2.1, this implies ac (HV ) = ∅, which completes the proof. While we formulated and proved Theorem 1.3 for discrete Schrödinger operators, the theorem also applies to continuous Schrödinger operators on L 2 ([0, ∞)), as well as to cases where H0 is a more general tridiagonal operator on 2 (N), which may have unbounded absolutely continuous spectrum. The proof for these cases is very similar to the above. The main difference is that we cannot assume σ (H0 ) to be bounded. This technical difference can be easily accommodated by considering in the j th stage of the construction only E ∈ (− j, j). More explicitly, instead of considering Gδ and η(δ) as above, we consider G j,δ = {Iν ∈ G | |Iν | > δ , Iν ∩ (− j, j) = ∅} and η j (δ), which is defined to be the maximal distance between neighboring elements ∞ of G j,δ . Since, for a fixed j, limδ→0 η j (δ) = 0, we can find {δ j }∞ ⊂ (0, 1) so that j=1 (η j (δ j )+δ j ) < ∞. j=1 By proceeding as above with η j (δ j ) instead of η(δ j ), we then obtain that for every E ∈ (− j, j) (instead of every E ∈ R), (3.3) holds for some L j ≤ N ≤ L j+1 . It then follows that (3.5) holds for every E ∈ R, and thus ac (HV ) = ∅. 4. Proof of Theorem 1.4 The potential V0 will be constructed in conjunction with three sequences, {q j }∞ j=1 , ∞ ∞ { j } j=1 , {L j } j=1 ⊂ N, connected by L 1 = 0, L j+1 = L j + j q j , and a sequence of coupling constants {λ j }∞ j=1 ⊂ (0, 1). We let V0 (n) = λ j cos(2π(n − L j )/q j )
(4.1)
for L j < n ≤ L j+1 . For the sequence {q j }∞ j=1 , we could take, in principle, any strictly increasing sequence of positive integers, but it is more convenient to take q j ’s that are −1 2 odd and obey ∞ j=1 q j < ∞. We thus fix, once and for all, q j = (2 j + 1) , and so ∞ { j }∞ j=1 uniquely determines {L j } j=1 . We have the following: Theorem 4.1. For V0 as above, there exists a sequence {λ j }∞ j=1 ⊂ (0, 1), obeying λ j → 0 as j → ∞, such that for any choice of the sequence { j }∞ j=1 , ac ( + V0 ) = (−2, 2). Remark. The proof below actually establishes a more general theorem in the following sense: {q j }∞ j=1 may be any arbitrary sequence of positive integers and cos(2π(n − L j )/q j ), for L j < n ≤ L j+1 , can be replaced by any real-valued periodic sequence of period q j and norm one.
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Proof. Given λ ∈ (0, 1) and q ∈ N, we denote by Vλ,q the periodic potential given by Vλ,q (n) = λ cos(2π n/q), n ∈ Z, and let Hλ,q = + Vλ,q on 2 (Z). We also denote, for = 0, . . . , q, E q, = 2 cos(π /q) and S(λ, q) = [−2, 2] \
q
[E q, − λ, E q, + λ] .
=0
The periodic spectrum σ (Hλ,q ) is well known to consist of q bands (closed intervals) separated by q − 1 gaps. The edges of these bands are well known to coincide with eigenvalues of certain q × q matrices (see, e.g., [15]), from which it is easy to see (and also well known) that all of them are contained in the set q
[E q, − λ, E q, + λ] .
=0
Thus, denoting by σ˜ λ,q the union of the interiors of the bands that make up σ (Hλ,q ), we have S(λ, q) ⊂ σ˜ λ,q ⊂ σ (Hλ,q ). Moreover, if λ ∈ (0, λ0 ), then S(λ0 , q) ⊂ σ˜ λ,q . For any ∈ N, let (q, λ, E) be the transfer matrix for Hλ,q from 1 to , namely, (q, λ, E) =
E − Vλ,q () −1 E − Vλ,q (1) −1 ··· . 1 0 1 0
σ (Hλ,q ) is well known to coincide with the set {E ∈ R | Tr q (q, λ, E) ≤ 2} and σ˜ λ,q = {E ∈ R | Tr q (q, λ, E) < 2}. Thus, for E ∈ σ˜ λ,q , q (q, λ, E) has two complex eigenvalues of the form e±iqθ . Let (ϕq (λ, E), 1)T be an appropriately normalized eigenvector of q (q, λ, E), corresponding to the eigenvalue eiqθ , then (ϕq∗ (λ, E), 1)T , where · ∗ denotes complex conjugation, is an eigenvector corresponding to the eigenvalue e−iqθ . It is thus easy to see that for E ∈ σ˜ λ,q , we have the equality q (q, λ, E) = Uq (λ, E) R(qθ ) Uq (λ, E)−1 , where R(qθ ) =
iqθ 0 e , 0 e−iqθ
Uq (λ, E) =
ϕq (λ, E) ϕq∗ (λ, E) 1 1
.
−2 −1 Consider now some j ∈ N, E ∈ S( j −2 q −1 j , q j ), and λ ∈ (0, j q j ). As λ → 0
˜ λ,q j and thus ϕq j (λ, E) is well defined and converges (contininside (0, j −2 q −1 j ), E ∈ σ iθ(E) , where θ (E) ∈ (0, π ) is determined by E = 2 cos θ (E). uously) to ϕq j (0, E) = e Similarly, max1≤≤q j (q j , λ, E) converges to the corresponding value for λ = 0, which is bounded from above by 1/ sin θ (E). By the Egoroff theorem, there is a mea−2 −1 −2 ≥ 4 − 3 j −2 , surable subset S j ⊂ S( j −2 q −1 j , q j ), with |S j | > |S( j q j , q j )| − j
on which these convergences are uniform. Thus, we can pick λ j ∈ (0, j −2 q −1 j ), so iθ(E) −2 | < j and max1≤≤q j (q j , λ j , E) < that, for every E ∈ S j , |ϕq j (λ j , E) − e 1 + (sin θ (E))−1 .
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−2 Now let j0 , j1 ∈ N, 2 < j0 < j1 , and let E ∈ ∞ j= j0 S j be such that sin θ (E) > j0 . The transfer matrix for + V0 from L j0 + 1 to L j1 is given by L j0 +1,L j1 (E) =
j 1 −1
( q j (q j , λ j , E)) j .
j= j0
Since q j (q j , λ j , E) = Uq j (λ j , E) R(q j θ j ) Uq j (λ j , E)−1 ,
(4.2)
( q j (q j , λ j , E)) j = Uq j (λ j , E) (R(q j θ j )) j Uq j (λ j , E)−1 , and thus, since R(q j θ j ) = 1, we see that L j0 +1,L j1 (E) is bounded from above by Uq j0 (λ j0 , E)Uq j1 −1 (λ j1 −1 , E)−1
j 1 −2
Uq j (λ j , E)−1 Uq j+1 (λ j+1 , E) .
j= j0
Denote ϕ j = ϕq j (λ j , E), then we see by a simple calculation that ϕ j+1 − ϕ ∗j ϕ ∗j+1 − ϕ ∗j 1 Uq j (λ j , E)−1 Uq j+1 (λ j+1 , E) = , ∗ ϕ j − ϕ ∗j ϕ j+1 − ϕ j ϕ j − ϕ j+1 and thus, since |ϕ j − eiθ(E) | < j −2 , we obtain Uq j (λ j , E)−1 Uq j+1 (λ j+1 , E) ≤ 1 +
2 j −2 . sin θ (E) − j −2
Similarly, it’s easy to see that Uq j0 (λ j0 , E) < 3 and Uq j1 −1 (λ j1 −1 , E)−1 <
3 , 2(sin θ (E) − ( j1 − 1)−2 )
and so we can conclude that L j0 +1,L j1 (E) ≤
j 1 −2
9 2(sin θ (E) − j0−2 )
j= j0
2 j −2 1+ sin θ (E) − j −2
.
Since the product in the last expression clearly converges as j1 → ∞, we conclude that lim sup j1 →∞ L j0 +1,L j1 (E) < ∞. Consider now any L j1 < n < L j1 +1 . By using (4.2), one easily sees that L j1 +1,n (E) ≤ Uq j1 (λ j1 , E)Uq j1 (λ j1 , E)−1 max (q j1 , λ j1 , E) 1≤≤q j1
and thus, since max1≤≤q j1 (q j1 , λ j1 , E) < 1 + (sin θ (E))−1 , L j1 +1,n (E) ≤
9(1 + (sin θ (E))−1 ) 2(sin θ (E) − j1−2 )
.
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Since L j0 +1,n (E) ≤ L j0 +1,L j1 (E) L j1 +1,n (E) , we conclude that lim supn→∞ L j0 +1,n (E) < ∞. Now given any E ∈ lim inf j→∞ S j ⊂ (−2, 2), there will be some j0 for which −2 E∈ ∞ j= j0 S j and sin θ (E) > j0 . Thus, we conclude that for any E ∈ lim inf j→∞ S j , lim supn→∞ 1,n (E) < ∞. By well known results (see, e.g., [17, 19]), this implies lim inf j→∞ S j ⊂ ac ( + V0 ). Since |(−2, 2) \ S j | ≤ 3 j −2 , it follows from the BorelCantelli lemma that |(−2, 2) \ lim inf j→∞ S j | = 0 and thus ac ( + V0 ) = (−2, 2) as required. Proof of Theorem 1.4. We let V0 be as above and fix {λ j }∞ j=1 ⊂ (0, 1) to be some sequence of the type whose existence is ensured by Theorem 4.1, namely, λ j → 0 as j → ∞ and ac ( + V0 ) = (−2, 2) for any choice of { j }∞ j=1 . Theorem 1.4 would now follow by showing that we can simultaneously pick the sequence { j }∞ j=1 and construct a bounded monotonely increasing potential V , such that ac ( + V0 + V ) = ∅. This construction is very similar to (and will rely upon) the one done in the proof of Theorem 1.3. Consider j ∈ N and suppose that 1 , . . . , j−1 and thus L j are determined and that V (n) is determined for 1 ≤ n ≤ L j . By a result of Choi-Elliott-Yui [3], σ (Hλ j ,q j ) is known to have q j − 1 open gaps and the length of each of these gaps is larger than δ j = (λ j /16)q j . By a general result of Deift-Simon [6, Corollary 1.5], the distance between every two neighboring gaps is bounded from above by η j = 4π/q j . Denote m j = [2η j /δ j ] + 3 . We will construct the potential V (n), for L j < n ≤ L j+1 , in conjunction with numbers {n j,1 , . . . , n j,m j }, so that n j,k+1 > n j,k , (n j,k+1 − n j,k )/q j is an integer, n j,1 = L j and n j,m j = L j+1 . Note that j will be determined by this construction as j = (n j,m j − n j,1 )/q j . Similarly to the proof of Theorem 1.3, we define V (n) = V (L j ) + kδ j /2 for n j,k < n ≤ n j,k+1 , and determine n j,2 , . . . , n j,m j recursively. Note that V (L j+1 ) − V (L j ) = m j δ j /2 ≤ η j + 3δ j /2, and thus, since ∞ j=1 (η j + δ j ) < ∞, V will be bounded. Assuming that n j,2 , . . . , n j,k are determined, we can repeat the considerations of (3.1)–(3.4), except that we replace (3.1) by (E − δ j /4, E + δ j /4) ∩ σ (Hλ j ,q j + V (L j ) + kδ j /2) = ∅ ,
(4.3)
and use the fact that V0 (n) coincides with Vλ j ,q j (n) for L j < n ≤ L j+1 , so that n j,k +1,n j,k + (E) doesn’t change if we change the potential from V0 to Vλ j ,q j . We thus show that we can choose n j,k+1 to be sufficiently large, so that for any E obeying (4.3), (3.3) will hold either for N = n j,k or for N = n j,k+1 . Another difference from the proof of Theorem 1.3 is that we now also need to ensure that (n j,k+1 − n j,k )/q j is an integer, but this is clearly not a problem. Now given any fixed E ∈ R, the definitions of δ j , η j , and m j ensure that there will be some k ∈ {1, . . . , m j } for which (4.3) holds. We thus see that we can choose n j,2 , . . . , n j,m j so that for every E ∈ R, there will be some L j ≤ N ≤ L j+1 for which (3.3) holds. Thus, by choosing n j,2 , . . . , n j,m j appropriately for each j, we ensure that (3.5) holds for every E ∈ R. By Proposition 2.1, this implies ac (HV ) = ∅.
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Acknowledgement. It is a pleasure to thank Barry Simon for useful discussions. This research was supported in part by The Israel Science Foundation (Grant No. 188/02) and by Grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.
References 1. Avron, J., Simon, B.: Almost periodic Schrödinger operators. I. Limit periodic potentials. Commun. Math. Phys. 82, 101–120 (1982) 2. Breuer, J., Last, Y.: Stability of spectral types for Jacobi matrices under decaying random perturbations. J. Funct. Anal. 245, 249–283 (2007) 3. Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math. 99, 225– 246 (1990) 4. Christ, M., Kiselev, A.: WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials. Commun. Math. Phys. 218, 245–262 (2001) 5. Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Commun. Math. Phys. 203, 341–347 (1999) 6. Deift, P., Simon, B.: Almost periodic Schrödinger operators, III. The absolutely continuous spectrum in one dimension. Commun. Math. Phys. 90, 389–411 (1983) 7. Dombrowski, J., Nevai, P.: Orthogonal polynomials. measures and recurrence relations. SIAM. J. Math. Anal. 17, 752–759 (1986) 8. Golinskii, L., Nevai, P.: Szeg˝o difference equations. transfer matrices and orthogonal polynomials on the unit circle. Commun. Math. Phys. 223, 223–259 (2001) 9. Kaluzhny, U., Last, Y.: Purely absolutely continuous spectrum for some random Jacobi matrices. In: Proceedings of “Probability and Mathematical Physics” a conference in honor of Stanislav Molchanov’s 65th birthday, to appear 10. Killip, R.: Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not. 2002, 2029–2061 (2002) 11. Kiselev, A., Last, Y., Simon, B.: Stability of singular spectral types under decaying perturbations. J. Funct. Anal. 198, 1–27 (2003) 12. Kupin, S.: On a spectral property of Jacobi matrices. Proc. Amer. Math. Soc. 132, 1377–1383 (2004) 13. Kupin, S.: Spectral properties of Jacobi matrices and sum rules of special form. J. Funct. Anal. 227, 1–29 (2005) 14. Laptev, A., Naboko, S., Safronov, O.: On new relations between spectral properties of Jacobi matrices and their coefficients. Commun. Math. Phys. 241, 91–110 (2003) 15. Last, Y.: On the measure of gaps and spectra for discrete 1D Schrödinger operators. Commun. Math. Phys. 149, 347–360 (1992) 16. Last, Y.: Spectral theory of Sturm-Liouville operators on infinite intervals: a review of recent developments. In: Amrein, W.O., Hinz, A.M., Pearson, D.B. (eds.), Sturm-Liouville Theory: Past and Present, Basel: Birkhäuser, (2005), pp 99–120 17. Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999) 18. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. Scattering Theory, New York: Academic Press (1979) 19. Simon, B.: Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators. Proc. Amer. Math. Soc. 124, 3361–3369 (1996) 20. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. AMS Colloquium Series, Providence, RI: American Mathematical Society, 2005 21. Stolz, G.: Spectral theory for slowly oscillating potentials. I. Jacobi Matrices. Manuscripta Math 84, 245– 260 (1994) 22. Stolz, G.: Spectral theory for slowly oscillating potentials. II. Schrödinger operators. Math. Nachr. 183, 275–294 (1997) 23. Weidmann, J.: Zur Spektraltheorie von Sturm-Liouville-Operatoren. Math. Z. 98, 268–302 (1967) 24. Zlatoš, A.: Sum rules for Jacobi matrices and divergent Lieb-Thirring sums. J. Funct. Anal. 225, 371– 382 (2005) Communicated by B. Simon
Commun. Math. Phys. 274, 253–276 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0268-5
Communications in
Mathematical Physics
Some Remarks on Group Bundles and C ∗ Dynamical Systems Ezio Vasselli Dipartimento di Matematica, University of Rome “La Sapienza”, P.le Aldo Moro, 2-00185 Roma, Italy. E-mail:
[email protected] Received: 20 October 2006 / Accepted: 30 November 2006 Published online: 22 May 2007 – © Springer-Verlag 2007
Abstract: We introduce the notion of fibred action of a group bundle on a C(X ) -algebra. By using such a notion, a characterization in terms of induced C*-bundles is given for C*-dynamical systems such that the relative commutant of the fixed-point C*-algebra is minimal (i.e., it is generated by the centre of the given C*-algebra and the centre of the fixed-point C*-algebra). A class of examples in the setting of the Cuntz algebra is given, and connections with superselection structures with nontrivial centre are discussed. 1. Introduction A result by S. Doplicher and J.E. Roberts ([6, Thm.1]) characterizes any compact C*-dynamical system (B, G) with fixed-point algebra A := B G and conditions A ∩ B = B ∩ B , A ∩ A = C1, in terms of an induced bundle of C*-algebras over a homogeneous space. The fibre of such a bundle is a C*-dynamical system (F, H ) with H ⊆ G , satisfying the conditions F H A , A ∩ F = C1. It is of interest to generalize the above theorem to the case in which the centre of A is non-trivial. From a mathematical point of view, our motivation is a duality theory for (noncompact) groups acting on Hilbert bimodules ([21]). From the point of view of mathematical physics, the original motivation was the following: given our C*-dynamical system (B, G) , define C := B ∩ B , Z := A ∩ A , and assume that A ∩ B is generated as a C*-algebra by Z and C ; we look for a C*-dynamical system (F, H ) , H ⊆ G , such that F H = A , A ∩ F = Z. (1.1) The above relations play an important role in the context of certain C*-dynamical systems arising from superselection structures with nontrivial centre, called “Hilbert C*-systems” (see [2]). From a physical viewpoint, (1.1) appeared in low-dimensional The author was partially supported by the European Network “Quantum Spaces - Noncommutative Geometry” HPRN-CT-2002-00280.
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conformal quantum field theory as a principle (see [16, §1, p.142]): the C*-algebra F plays the role of a field algebra, A is a universal algebra from which the “real” observable algebra can be recovered, and H is the “gauge group”, which in the above reference turns out to be actually a quantum group. In particular, the above interpretation of (1.1) has been recently recognized ([4, (4.34)]) in the Streater-Wilde model ([19]). Now, at the algebraic level F appears as the image of a C*-epimorphism η : B → F , and H is the stabilizer of ker η w.r.t. the G -action. In the case in which the centre of A is nontrivial, it is easy to construct examples such that H reduces to the trivial group (see Sect.6.2). This unpleasant fact led us to introduce the following construction. At first, it is well-known that if an Abelian C*-algebra C(X ) is a unital subalgebra of the centre of a C*-algebra B , then we may regard B as a bundle of C*-algebras with base space X . Given a bundle G → X of compact groups, we introduce the notion of fibred action of G on B , in such a way that each fibre of G acts on the corresponding fibre of B (Def.3.2). Then, we adopt this point of view for the case of our C*-dynamical system (B, G) , by choosing C(X ) := C ∩ A . In this way, we find that F has the structure of a bundle with base space X , on which a bundle H acts in the above sense, with H ⊆ X × G . The fibred action of H turns out to be non-trivial also in cases in which the stabilizer H is trivial (Rem.3.1). Roughly speaking, we obtain that F is a “field algebra” carrying an action by a bundle H of “local gauge groups” (Thm.6.1, Thm.6.2). In analogy with the original motivation of Doplicher and Roberts, we apply our construction to symmetric endomorphisms arising from superselection structures with non-trivial centre (Thm.7.2, Thm.7.4); in this setting, C(X ) is interpreted as the C*subalgebra of A ∩ A which is left invariant by the action of such endomorphisms. Since in low-dimensional quantum field theory unitary braidings arise, we hope to develop in a future work an analogous construction, involving Hopf C(X ) -algebras ([1]) instead of group bundles. Groups of sections (of trivial bundles) acting on C*-algebras have been also considered in [3]. In this case, the main motivation arises from local quantum gauge field theory, and X is interpreted as a (compactified) space-time. The present work is organized as follows. In Sect.2, we give some basic properties about bundles p : Y → X , regarded as the topological counterparts of the commutative C0 (X ) -algebras C0 (Y ) . Then, we consider group bundles, and the associated groups of sections (for brevity called section groups, see Def.2.3); in the case in which the group bundle is locally trivial, we prove the existence of C0 (X ) -valued invariant functionals, playing the role of the Haar measure (Def.4.1, Prop.4.3). In Sect.3, we introduce the notion of fibred action of a group bundle G → X on a C0 (X ) -algebra (Def.3.2); we establish some basic properties, and make use of invariant means arising from the invariant C0 (X ) -valued functionals (Sect.4). Usual group actions on C*-algebras are recovered as fibred actions by trivial group bundles (Cor.3.4). In Sect.5, we discuss the case of fibred actions on Abelian C*-algebras. In Sect.6, we provide a generalization of [6, Thm.1] to the non-trivial centre case: every fibred C*-dynamical system (B, G) is isomorphic to an induced bundle with base space the spectrum of C (Prop.6.4); moreover, if A ∩ B is generated by C and Z , then F satisfies the analogue of (1.1) with a group bundle H playing the role of H (Thm.6.1). Existence of a section s : X → is our main assumption for the above results; this characterizes as a homogeneous bundle in the sense of Sect.2.1. We provide a class of examples in the context of the Cuntz algebra, from which it is evident that existence and unicity of (F, H) are not ensured (Sect.6.2, Sect.8).
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In Sect.7, we apply our results to Hilbert C*-systems and Doplicher-Roberts endomorphisms associated with superselection sectors. Let A be a C*-algebra with centre Z , and ρ an endomorphism satisfying the special conjugate property in the sense of [10, §2], [9, §4]. Then, we prove that fibred Hilbert C*-systems (F, G) satisfying F G = A , A ∩ F = Z are in one-to-one correspondence with the set of sections of a suitable homogeneous bundle (Thm.7.2). In the particular case of endomorphisms studied by Baumgärtel and Lledó, we are also able to prove existence and unicity of (F, G) (Thm.7.4), in accord with [2, Thm.4.13]. Keywords. Standard notions about topological and group bundles can be found in [13]. Let X be a locally compact Hausdorff space; a C0 (X ) -algebra is a C*-algebra A equipped with a nondegenerate morphism C0 (X ) → Z M(A) , where Z M(A) is the centre of the multiplier algebra M(A) ([15, §2]); in the sequel, we will identify the elements of C0 (X ) with their images in M(A) . C0 (X ) -algebras correspond to upper-semicontinuous bundles of C*-algebras ([17, Thm.2.3]), thus generalize the classical notion of continuous field (bundle) of C*-algebras ([5, §10]). If x ∈ X , the fibre Ax of A over x is defined as the quotient of A by the ideal ker xA := { f a, f ∈ C0 (X ), f (x) = 0, a ∈ A}. Thus, for every x ∈ X there is an evaluation epimorphism πx : A → Ax , in such a way that a = supx πx (a) , a ∈ A . If U ⊆ X is an open set, then CU (X ) := { f ∈ C0 (X ) : f | X −U = 0} is an ideal of C(X ) ; we define the restriction AU := closed span { f a, f ∈ CU (X ), a ∈ A}, which is a closed ideal of A . Note that AU is a CU (X ) -algebra in the natural way. If W ⊆ X is closed, then the restriction of A over W is defined by the epimorphism πW : A → AW := A/A X −W . A C0 (X ) -morphism from a C0 (X ) -algebra A into a C0 (X ) -algebra B is a C*-algebra morphism η : A → B such that η( f a) = f η(a) , f ∈ C0 (X ) , a ∈ A . Tensor products in the setting of C0 (X ) -algebras are defined in [1, §3.2], and are denoted by ⊗ X . 2. Bundles, Section Groups A bundle is a surjective, continuous map of locally compact Hausdorff spaces p : Y → X . We denote by Yx := p −1 (x) the fibre of Y over x ∈ X . If U ⊆ X is an open set, then YU := p −1 (U ) ⊆ Y defines itself a bundle p|YU : YU → U , called the restriction of Y on U . If p : Y → X is a bundle, a bundle morphism is a continuous map F : Y → Y such that p ◦ F = p ; if F is also a homeomorphism, then we say that Y , Y are isomorphic. The fibred product of bundles p : Y → X , p : Y → X is defined as Y × X Y := (y, y ) ∈ Y × Y : p(y) = p (y ) ; note that Y × X Y has a natural structure of a bundle over X . Let X , Y0 be locally compact Hausdorff spaces; the natural projection p : X ×Y0 → X defines in a natural way a bundle. A bundle isomorphic to X × Y0 is called trivial. More in general, a bundle is locally trivial if for every x ∈ X there is a neighborhood U x with a bundle isomorphism αU : p −1 (U ) → U × Y0 , where Y0 is a fixed locally compact Hausdorff space. In our terminology, a bundle p : Y → X is the topological counterpart of the commutative C0 (X ) -algebra C0 (Y ) ; the associated structure morphism is i p : C0 (X ) → M(C0 (Y )) , i p ( f ) := f ◦ p , and the fibres are C0 (Y )x = C0 (Yx ) , x ∈ X ; if z ∈ C0 (Y ) , the image of z w.r.t. the epimorphism πx : C0 (Y ) → C0 (Yx ) is given by the restriction z x := z|Yx .
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Definition 2.1. Let p : Y → X be a bundle. A positive C0 (X ) -functional is a linear, positive C0 (X ) -module map ϕ : C0 (Y ) → C0 (X ) , i.e. ϕ( f z) = f ϕ(z) , ϕ(z ) ≥ 0 for every z ∈ C0 (Y ) , f ∈ C0 (X ) , 0 ≤ z ∈ C0 (Y ) . Lemma 2.2. Let p : Y → X be a bundle, ϕ : C0 (Y ) → C0 (X ) a positive C0 (X ) -functional. Then, for every x ∈ X there is a unique positive functional ϕx : C0 (Yx ) → C such that [ϕ(z)] (x) = ϕx (z x ) , z ∈ C0 (Y ) . Proof. We prove that the map ϕx : C0 (Yx ) → C , ϕx (z x ) := (ϕ(z)) (x) , x ∈ X , is well defined. Let z ∈ C0 (Y ) such that z x = z x , i.e. z −z ∈ ker πx , and C x (X ) ⊂ C0 (X ) be the ideal of functions vanishing on x ; since ker πx is a nondegenerate Banach C x (X ) bimodule, by [1, Prop.1.8] there is f ∈ C x (X ) such that z − z = f w , w ∈ ker πx . Thus ϕx (z x )−ϕx (z x ) = ( f ϕ(w)) (x) = f (x) (ϕ(w)) (x) = 0 , and ϕx is well-defined. Let id X denote the identity map over X . A (not necessarily continuous) map s : X → Y such that p ◦ s = id X is called a selection of Y . In particular, a section is a continuous selection s : X → Y , p ◦ s = id X . From the C*-algebraic point of view, sections correspond to C0 (X ) -epimorphisms φs : C0 (Y ) → C0 (X ), z → φs (z) := z ◦ s. In general, existence of a section is not ensured. On the other side, a frequently verified property is existence of local sections, i.e. maps sU : U → Y , p ◦ sU = idU , where U ⊂ X is an open set. We say that Y has local sections if for every x ∈ X there is a neighborhood U x with a section sU : U → Y . Locally trivial bundles have local sections. We denote by S X (Y ) the set of sections of Y . Now, S X (Y ) is endowed with the “uniform convergence” weak topology such that for every z ∈ C0 (Y ) the map {s → z ◦ s} is norm continuous. Of course, if X reduces to a single point we get the usual Gel’fand topology over Y ≡ S{x} (Y ) . A closed subset S ⊆ S X (Y ) is said to be total if for every y ∈ Y there is a section s ∈ S with y = s ◦ p(y) . In this case, we say that Y is full. A group bundle is a bundle p : G → X such that every fibre G x := p −1 (x) , x ∈ X , is a locally compact group w.r.t. the topology induced by G . For every x ∈ X , we denote by ex the identity of G x , and consider the identity selection (2.1) e := {X x → ex ∈ G} . Moreover for every g, g ∈ S X (G) , the following selections are defined: gg : gg (x) := g(x)g (x) g −1 : g −1 (x) := g(x)−1 , x ∈ X.
(2.2)
Definition 2.3. Let p : G → X be a group bundle. A section group for G is a total subset G ⊆ S X (G) , which is also a topological group w.r.t. the structure (2.1,2.2). Example 2.1. Let G 0 be a locally compact group and G := X × G 0 ; then, S X (G) can be identified with the group C(X, G 0 ) of continuous maps from X into G 0 . It is clear that S X (G) is a section group, so that G is full. Let G ⊂ S X (G) be the subgroup of constant maps from X into G 0 ; then, G is a section group, isomorphic to G 0 .
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2.1. Homogeneous bundles. Let p : G → X be a group bundle, and H ⊆ G a group subbundle. Then, every fibre Hx := H ∩ p −1 (x) , x ∈ X , is a subgroup of G x := p −1 (x) . We denote by H\G the quotient space defined by theequivalence relation induced by H on G . For every y ∈ G , we denote by y H := hy : h ∈ H p(y) the associated element of H\G . Now, H\G is a locally compact Hausdorff space, endowed with the surjective map p H : G → H\G, p H (y) := y H .
(2.3)
Moreover, H\G has a natural bundle structure q : H\G → X , q(y H ) := p(y) , with fibre the space p −1 H (x) := Hx \G x of left Hx -cosets in G x . Thus, p H is a bundle morphism, i.e. q ◦ p H = p . If G is full then the same is true for H\G , in fact the set G H := {g H := p H ◦ g, g ∈ S X (G)} ⊂ S X (H\G) is total for H\G . To be more concise, we define := H\G , so that we have a bundle q : → X . The map (2.3) defines itself a bundle p : G → , that we are going to describe. Let us consider the restriction p,x : G x → x , x ∈ X . Then, we may regard x as the coset space Hx \G x , so that p,x can be interpreted as the natural projection from G x onto Hx \G x , induced by the quotient topology. We have the following commutative diagram: ⊆
p
/G // H@ @@ @@ p q p @@ X
(2.4)
from which it follows that p : G → has fibres H p(ω) := G p(ω) ∩ H , ω ∈ . Note that if X reduces to a single point, then (2.4) is simply a quotient map of the type K → K → K \L , where K ⊆ L is an inclusion of compact groups. Lemma 2.4. Let p : G → X be a compact group bundle, H ⊆ G a compact group subbundle. If p : G → , := H\G , has local sections, then there is a finite open cover {Wk } of with bundle morphisms
such that
−1 δk : Gk := p (Wk ) → Hk := H ∩ Gk
(2.5)
δk (hy) = hδk (y), h ∈ Hk , y ∈ Gk .
(2.6)
Proof. For every ω ∈ , there exists a neighborhood W ⊆ , ω ∈ W , with a section sW : W → G , p ◦ sW = idW . Since is compact, we obtain a finite open subcover {Wk }, with sections sk : Wk → G . The morphisms δk are defined by δk (y) := y · (sk ◦ p (y))−1 . Lemma 2.5. Let p : G → X be a locally trivial bundle with fibre a compact Lie group L , H ⊆ G a compact group subbundle. Then, the bundle p : G → , := H\G has local sections. Proof. Let q : → X denote the natural projection, and ω ∈ ; then, there is a neighborohood W of ω such that U := q(W ) trivializes G , so that we may identify GU := p −1 (U ) with the trivial bundle U × L . The previous remarks imply that HU := GU ∩ H is “subtrivial”, in the following sense: there is a family {Ui ⊆ U }i∈I such that Ui ∩ U j = ∅, i Ui = U , and HU is the disjoint union i Ui × Hi , where each Hi is
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a closed subgroup of L . This also implies q −1 (U ) = i Wi , Wi = Ui × (Hi \L) , and ω ∈ i Wi . We now proceed with the proof of the existence of local sections. By construction, there is a set of indexes J ⊆ I such that ω ∈ ∩i∈J W i ∩i∈J U i × (Hi \L) , where W i (resp. U i ) denotes the closure of Wi (resp. Ui ). Let now i, j ∈ J such that Ui ∩ U j contains some element x . We consider a sequence {(xn , yn )} ⊂ U j × H j converging to (x, y) ∈ i Ui × Hi . Since H j is closed, we find y ∈ H j ; moreover, since x ∈ Ui ∩ U j , we find y ∈ Hi . Since we may pick an arbitrary converging sequence {yn } ⊆ H j , we find that H j ⊆ Hi . By applying the same procedure to every pair of elements of J , we conclude that the set {Hi }i∈J is totally ordered w.r.t. the inclusion, thus there is an index i 0 such that Hi ⊆ Hi0 for every i ∈ J . In order for more concise notations, we define H0 := Hi0 , U0 := Ui0 , W0 := U0 × H0 \L . In this way, we have inclusions U × (H0 \L) ⊆ i Ui × (Hi \L) ; moreover, we may pick V0 ⊆ H0 \L with a section s : V0 → L ([12, Thm.3.4.3]). In this way, we obtain a section idU × s ∈ SU ×V0 (U × L) . Since we may identify U × V0 with a neighborhood of ω , say W0 , we conclude that there exists a section from W0 into U × L . 3. Actions on C(X) -Algebras In the present section, we introduce the notion of fibred action of group bundle on a unital C(X ) -algebra. For this purpose, we briefly recall some results on the categorical equivalence between C0 (X ) -algebras and a class of topological objects called C*-bundles. In order to simplify the exposition, we consider the case in which X is compact and the given C(X ) -algebra unital. 3.1. C*-bundles. Let X be a compact Hausdorff space. A C*-bundle is a Hausdorff space endowed with a surjective, open, continuous map Q : → X such that every fibre x := Q −1 (x) , x ∈ X , is homeomorphic to a C*-algebra. We assume that is full, i.e. for every σ ∈ there is a section a ∈ S X ( ) such that a ◦ Q(σ ) = σ . Let Q : → X be a C*-bundle. A morphism from into is a continuous map φ : → such that: (1) Q ◦ φ = Q ; this implies that φ( x ) ⊆ x , x ∈ X ; (2) φx := φ| x : x → x is a C*-algebra morphism for every x ∈ X . The map φ is said to be an isomorphism (resp. monomorphism, epimorphism) if every φx , x ∈ X , is a C*-algebra isomorphism (resp. monomorphism, epimorphism). Now, every a ∈ S X ( ) defines a family {a(x) ∈ x } ; thus, the set of sections of is endowed with a natural structure of C(X ) -algebra w.r.t. the pointwise-defined *-algebra operations, sup-norm, and pointwise multiplication by elements of C(X ) . On the other side, let A be a C(X) -algebra with evaluation epimorphisms πx : A → Ax , x ∈ X . := Then, the set A x∈X Ax , endowed with the natural surjective map Q : A → X , becomes a C*-bundle if equipped with the basis : Q(σ ) ∈ U and σ − π Q(σ ) (a) < ε , (3.1) TU,a,ε := σ ∈ A where U ⊆ X is open, a ∈ A , ε > 0 . Some examples follow: the C*-bundle associated = X × A0 , with Q(x, a0 ) := x , x ∈ X , a0 ∈ A0 .; with A := C(X ) ⊗ A0 is clearly A if A := C(Y ) is Abelian, then there is a surjective map q : Y → X , and the C*-bundle → X has fibres Ax = C(q −1 (x)) , x ∈ X . A } defines a functor: if η : A → A is a C(X ) -algebra morphism, The map { A → A is defined in such a way that → A then the associated morphism η:A η ◦ πx (a) = πx ◦ η(a) , a ∈ A , x ∈ X , where πx : A → Ax denotes the evaluation epimorphism.
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For the proof of the following theorem, see [11, Thm.5.13]; in order to understand the terminology used in the above reference, we recommend the reader to [11, §1 , §5]. } induces Theorem 3.1. Let X be a compact Hausdorff space. The functor { A → A an equivalence from the category of C(X ) -algebras into the category of C*-bundles with base space X . For every C(X ) -morphism η : A → A , it turns out that ker η is a C(X ) -algebra. We denote by ker η → X the associated C*-bundle. It is clear that there is an inclusion such that (ker ker η⊆A η)x ⊆ ker ηx , x ∈ X . We conclude the present section with a remark on C(X ) -algebras and C*-bundles. Let A be a unital C(X ) -algebra with fibre epimorphisms πx : A → Ax , x ∈ X , and → X p : Y → X a compact bundle, so that C(Y ) is a unital C(X ) -algebra. Let P : A be the C*-bundle associated with A . We consider the set S X (Y, A) of continuous maps such that P ◦ F = p , F := sup y F(y) < ∞ . S X (Y, A) is endowed F :Y →A with a natural structure of C*-algebra w.r.t. the pointwise-defined operations, and it is easily verified that there is a natural isomorphism [S(z ⊗ a)] (y) := z(y) π p(y) (a). S : C(Y ) ⊗ X A → S X (Y, A),
(3.2)
3.2. Group bundles acting on C(X ) -algebras. Let G be a locally compact group; as remarked in [17, §4], every action α : G → aut X A induces a family of actions {α x : G → autAx }x , such that πx ◦ αg = αgx ◦ πx , x ∈ X, a ∈ A, g ∈ G.
(3.3)
The above elementary remark suggests to introduce the following notion. Definition 3.2. Let X be a compact space, p : G → X a group bundle, A a C(X ) algebra. We say that G acts fiberwise on A if there is a family of strongly continuous actions α := {α x : G x → autAx , x ∈ X } , such that the map → A, (y, σ ) → α yx (σ ), x := p(y) = Q(σ ), α : G ×X A
(3.4)
is continuous. In such a case, we say that α is a fibred action, and that the triple (A, G, α) is a fibred C(X ) -system. If G is full, then we say that α is full. The fixed-point algebra w.r.t. α is the C(X ) -subalgebra Aα := {a ∈ A : α yx ◦πx (a) = πx (a) ∀x ∈ X, ∀y ∈ G x } . Proposition 3.3. Let (A, G, α) be a full fibred C(X ) -system. Then, for every section group G ⊆ S X (G) there is a strongly continuous action α G : G → aut X A
(3.5)
x αg(x) ◦ πx = πx ◦ αgG , g ∈ G, x ∈ X.
(3.6)
such that On the converse, if a strongly continuous action (3.5) endowed with a family {α x : G x → autAx }x satisfying (3.6) is given, then there is a unique fibred C0 (X ) -system (A, G, α) such that α G is associated with α in the above sense.
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Proof. Suppose that a fibred action α is given. If g ∈ S X (G) and a ∈ A , then by sg,a (x) := α x ◦ πx (a) , belongs to S X (A) . continuity of α the map sg,a : X → A, g(x) Elementary remarks show that the map αgG (a) := sg,a defines the desired automorphic action. Let now α G be an action satisfying (3.5,3.6). Since G is full, for every → A, y ∈ G there is g ∈ G with g( p(y)) = y . We define the map α : G × X A x , α(g(x), σ ) := αg(x) (σ ) , σ ∈ Ax . Note that (3.6) ensures that α(y, σ ) , y ∈ G , σ ∈ A is well-defined, in fact it does not depend on the choice of g ∈ G satisfying g(x) = y . , We now verify that α is continuous. For this purpose, let us consider (y, σ ) ∈ G × X A with x := p(y) = Q(σ ) ; we pick a ∈ A and g ∈ G such that g(x) = y and πx (a) = σ , so that πx ◦ αgG (a) = α yx (σ ) . Moreover, we define b := αgG (a) , and consider a neighborhood of the type TU,b,ε for α yx (σ ) , with U ⊂ X , U x (see (3.1)). Since α G is strongly continuous, there is a neighborhood V y with the following property: for every g ∈ G such that g(U ) ⊆ V , and for every cutoff λ ∈ C W (X ) , U ⊂ W , λ|U ≡ 1, it turns out that λb − λαgG (a) < ε/2 , so that for every x ∈ U we find πx (b) − α yx ◦ πx (a) < ε/2 , y := g (x ) ∈ V . Thus, for every (y , σ ) ∈ V × X TU,a,ε/2 , it turns out, with x := p(y ) = Q(σ ) ∈ U , πx (b) − α yx (σ ) ≤ πx (b) − α yx ◦ πx (a) + πx (a) − σ ≤ ε/2 + ε/2.
Thus, α yx (σ ) ∈ TU,b,ε , and we conclude that α is continuous.
Corollary 3.4. Let A be a unital C*-algebra, G a locally compact group, and α : G → autA a strongly continuous action. We define C(X ) := (A ∩ A )α (so that A is → A a C(X ) -algebra) and G := X × G . There is a unique fibred action α : G ×X A such that α is the action associated with α in the sense of the previous proposition. Proof. G appears as a section group of constant sections for G := X × G , so that (3.3) can be regarded as a special case of (3.6). The proof now follows from Prop.3.3. We introduce a natural notion of equivariance. If (A, G, α) , (B, G, β) are fibred C(X ) -systems and η : A → B is a C(X ) -morphism, we say that η is G -equivari. In this case, we use the notation ant if η ◦ α(y, σ ) = β(y, η(σ )) , y ∈ G , σ ∈ A η : (A, α) → (B, β) . The proof of the following lemma is trivial, therefore we omit it. Lemma 3.5. Let p : G → X be a group bundle, (A, G, α) a fibred C(X ) -system, and η : A → F a C(X ) -algebra epimorphism with associated family {ηx : Ax → Fx }x∈X . If H ⊆ G is a group subbundle such that α(y, σ ) ∈ ker η, y ∈ H, σ ∈ ker η,
(3.7)
→ F such that then there is a unique fibred action β : H × X F η ◦ α(y, σ ) = . β(y, η(σ )) , y ∈ H , σ ∈ A Corollary 3.6. Let A be a C(X ) -algebra, G a compact group, and α : G → aut X A a strongly continuous action. If η : A → F is a C(X ) -epimorphism, then every element of the stabilizer H := {h ∈ G : αh (ker η) = ker η} defines a section of the group bundle H ⊆ X × G of elements satisfying (3.7) w.r.t. the induced fibred action on A .
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Remark 3.1. According to the previous corollary, there is a fibred action (F, H, β) ; anyway also a strongly continuous action β H : H → aut X F is defined. Note that in general H is not a section group for H ; for example, if we consider X = [0, 1] , H = {(x, h) ∈ X × G : h = 1 for x ≥ 1/2} , then we obtain H = {1}. Thus, we may have the “extreme” situation in which H reduces to the trivial group, whilst H is non-trivial (this happens in the case in which S X (H) reduces to the identity selection). Roughly speaking, fibred actions are better behaved than usual strongly continuous actions w.r.t. C(X ) -epimorphisms. Example 3.1. Let p : G → X be a compact group bundle. We denote by C(G) the G → X the associated C(X ) -algebra of continuous functions of G , and by Q : A C*-bundle with fibres AG ,x = C(G x ) , x ∈ X . It is clear that there are actions λx : G x → autC(G x ) , λhx ξ(y) := ξ(h −1 y) , ρ x : G x → autC(G x ) , ρhx ξ(y) := ξ(yh) , ξ ∈ C(G x ) , h, y ∈ G x . Thus, there are fibred actions G → A G , λ(y, ξ ) := λ yp(y) (ξ ) λ:G×A . (3.8) G , ρ(y, ξ ) := ρ yp(y) (ξ ) G → A ρ :G×A We call λ , ρ respectively the left and right translation actions over G . Let p : G → X be a group bundle, and (A, G, α) a fibred C(X ) -system. We denote by X the spectrum of (A ∩ A)α , and identify C(X ) with (A ∩ A)α . The obvious inclusion C(X ) → C(X ) induces a surjective map q : X → X . Now, it is clear → X , and there is a that A is a C(X ) -algebra with associated C*-bundle Q : B C(X ) -isomorphism τ : A → B := S X (B) . At the level of C*-bundles, it is easy to → B . Let q∗ G := X × X G ; recognize that τ induces an epimorphism τ : X ×X A then, the following map is well-defined: → B, β((x , y), β : q∗ G × X B τ (x , σ )) := τ (x , α(y, σ )).
(3.9)
By construction (B, q∗ G, β) is a fibred C(X ) -system, and (B ∩ B)β = C(X ) . We conclude that fibred C(X ) -systems (A, G, α) can be always “rescaled” in such a way that C(X ) coincides with (A ∩ A)α . 4. Invariant Means We remind the reader of the notation used in Lemma 2.2 and Ex.3.1. Definition 4.1. Let p : G → X be a group bundle. A positive C0 (X ) -functional ϕ : C0 (G) → C0 (X ) is said to be left (resp. right) invariant if ϕx ◦ λ(y, z x ) = ϕx (z x ) (resp. ϕx ◦ ρ(y, z x ) = ϕx (z x ) ),
(4.1)
y ∈ G , x := p(y) , z ∈ C0 (G) . If ϕ is left and right invariant, then ϕ is said to be invariant. In the next lemma, we establish a unicity result for invariant functionals. We denote by Cb (X ) (= M(C0 (X )) ) the C*-algebra of bounded continuous functions on X , and by Cb+ (X ) the space of positive functions. Lemma 4.2. Let p : G → X be a group bundle, ϕ a left (resp. right) invariant positive C0 (X ) -functional. Let µx : C0 (G x ) → C denote a left (resp. right) Haar measure on G x , x ∈ X . Then, ϕ is unique up to a multiplicative factor τ ∈ Cb (X )+ .
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Proof. Let {ϕx ∈ C0 (G x )∗ }x be the family of states defined as in Lemma 2.2. The equality (4.1) implies that ϕx is a left (resp. right) G x -invariant state on C0 (G x ) ; the left (resp. right) G x -invariance of ϕx implies that ϕx = χx µx for some χx ∈ R+ . This also implies that if ϕ : C0 (G) → C0 (X ) is a left (resp. right) invariant positive C0 (X ) -functional then we find χx · (ϕ(z)) (x) = χx · (ϕ (z))(x) , z ∈ C0 (G) . The previous equality implies that ϕ(z) , ϕ (z) ∈ C0 (X ) have the same support for every z ∈ Z . We define the map τ : X → C , τ (x) := χx−1 χx , x ∈ X , and prove that it is continuous; for this purpose, we consider an approximate unit {λi }i ⊂ C0 (G) , and note that τ is a limit in the strict topology τ = limi ϕ (λi )ϕ(λi )−1 ; since the net in the r.h.s. of the previous equality is contained in C0 (X ) , we conclude that τ ∈ Cb (X ) . Let G 0 be a locally compact group, K a topological group acting continuously by proper homeomorphisms on G 0 . Then, K acts by automorphisms on C0 (G 0 ) ; in the sequel, we will identify elements of K with the corresponding automorphisms of C0 (G 0 ) . Proposition 4.3. Let X be a locally compact, paracompact Hausdorff space, p : G → X a locally trivial group bundle with fibre G 0 and structure group K . Let ϕ0 : C0 (G 0 ) → C denote a left (resp. right) Haar measure such that ϕ0 ◦ α = ϕ0 , α ∈ K . Then, there is a left (resp. right) invariant positive C0 (X ) -functional on C0 (G) . Proof. Let {Ui }i∈I be a locally finite, trivializing open cover for G . Then C0 (G) is a locally trivial continuous bundle with local charts αi : C0 ( p −1 (Ui )) → C0 (Ui ) ⊗ C0 (G 0 ) , and K -cocycle αi j : Ui ∩ U j → K ⊆ autC0 (G 0 ) . Let idi be the identity on C0 (Ui ) ; we define ϕi : C0 ( p −1 (Ui )) → C0 (Ui ) , ϕi := (idi ⊗ ϕ0 ) ◦ αi . It is clear that every ϕi is a left (resp. right) invariant C0 (Ui ) -functional on C0 ( p −1 (Ui )) . Now, since ϕ0 is K -invariant we find that if Ui ∩U j = ∅, then (idi j ⊗ ϕ0 ) ◦ (idi j ⊗ α) = idi j ⊗ ϕ0 , α ∈ K . This implies that ϕi (F) = ϕ j (F) , F ∈ C0 ( p −1 (Ui )) ∩ C0 ( p −1 (U j )) . Thus, if {λi } is a partition of unity subordinate to {Ui }, then the following left (resp. right) invariant positive C0 (X ) -functional is defined: ϕ : C0 (G) → C0 (X ) , ϕ(F) := i ϕi (λi F) , F ∈ C0 (G) . It is clear that every trivial group bundle admits an invariant C0 (X ) -functional. Moreover, by applying the previous result we may also easily prove that the same is true for principal bundles, and for bundles of unitary, vector-bundle morphisms ([14, I.4.8]). Let A be a C(X ) -algebra with fibre epimorphisms πx : A → Ax , x ∈ X , and p : G → X a compact group bundle. To be concise, we define AG := C(G) ⊗ X A . AG has a natural structure of C(X ) -algebra, with fibres AGx = C(G x ) ⊗ Ax , x ∈ X . G → X the associated C*-bundle. The translation actions defined We denote by P : A in Ex.3.1 extend in a natural way on AG : ,x −1 G → A G , λA λA : G × A y F(y ) := F(y y ), (4.2) A ,x A G G →A , ρ y F(y ) = F(y y), ρ :G×A AG (see (3.2)), y, y ∈ G , x := p(y) = p(y ) . We call (4.2) where F ∈ S X (G, A) left (resp. right) translation G -action over C(G) ⊗ X A . The following lemma is just the translation in terms of fibred systems of a well-known basic property of C*-algebra actions, thus we omit the proof. Lemma 4.4. Let p : G → X be a compact group bundle, and (A, G, α) a fibred C(X ) system. Then, there is an equivariant C(X ) -monomorphism T : (A, α) → (AG , ρ A ) such that T a(y) := α(y, πx (a)) , a ∈ A , y ∈ G , x := p(y) .
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Let p : G → X be a compact group bundle with a right invariant positive C(X ) -functional ϕ : C(G) → C(X ) , and A a C(X ) -algebra. Then, the following positive functional is defined ϕA : AG → A : ϕA (z ⊗ a) := ϕ(z) a,
in such a way that the equality πx ◦ϕA (F) = G x F(y) dµx (y) holds for every F ∈ AG, x := p(y) ∈ X ; µx denotes a Haar measure of G x , according to Lemma 4.2. Let → A be a fibred action. Then, a G -invariant mean m G : A → Aα , α : G ×X A m G (a) := ϕA ◦ T a , is defined, where T a ∈ AG is defined as in Lemma 4.4. For every x ∈ X , the following equality holds: πx ◦ m G (a) = α(y, πx (a)) dµx (y). (4.3) Gx
We define m G ,x : Ax → (Ax )
αx
as the l.h.s. of (4.3), so that m G ,x ◦ πx = πx ◦ m G .
5. Abelian Fibred C(X) -Systems Structural properties of centers of C*-algebras carrying a group action will be important in the sequel. For this reason, we establish some properties of fibred actions in the Abelian case. We fix some notations for the rest of the present section. Let X be a compact Hausdorff space, C := C() a unital, Abelian C(X ) -algebra, so that there is a surjective map q : → X ; we define x := q −1 (x) , x ∈ X . In such a way, the C(X ) -algebra structure of C is described by the monomorphism i q : C(X ) → C , i q ( f ) := f ◦ q . We denote the evaluation epimorphisms by ηx : C → Cx C(x ) , x ∈ X . Let p : G → X be a compact group bundle. We consider a fibred action α : G× X C → C, such that C α = i q (C(X )) C(X ) . By applying the Gel’fand transform, we obtain an action by bundle automorphisms α ∗ : G × X → , α ∗ (y, ω) := ω ◦ α yx , x := q(ω).
(5.1)
Let (C( ), G, β) be a fibred C(X ) -system. A (bundle) morphism F : → is G -equivariant if F ◦ α ∗ (y, ω) = β ∗ (y, F(ω)) , y ∈ G , ω ∈ ; in this case, we write F : (, α ∗ ) → ( , β ∗ ) . In the sequel of the present section, we assume that there exists a left invariant positive C(X ) -functional ϕ : C(G) → C(X ) . Lemma 5.1. C is a continuous bundle of Abelian C*-algebras over X ; for every x ∈ X , there is a closed group Hx ⊆ G x , unique up to isomorphism, such that the fibre Cx is isomorphic to C(Hx \G x ) . Proof. Let ω ∈ , x := p(ω) ∈ X . For every x ∈ X there is an action α x : G x → autCx , with Cx C(x ) . We now prove that G x acts transitively on x . Let f ∈ C(x ) be G x -invariant; by the Tietze theorem, there is f ∈ C() with f |x = f ; by applying the G -invariant mean m G : C → C α , we obtain a function f 0 := m G ( f ) ∈ C α . By (4.3), we obtain f 0 |x = m G ( f )|x = m G ,x ( f |x ) = m G ,x ( f ) = f . Since f 0 ∈ i q (C(X )) = C α , we obtain that f 0 is a constant function on x for every x ∈ X . x This proves that f is a constant map; thus C(x )α = C1, i.e. G x acts transitively
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on x . The isomorphism C(x ) C(Hx \G x ) is obtained by defining Hx as the stabilizer of some ωx ∈ x . It remains to verify that C is a continuous bundle over X . For this purpose, we consider the right Hilbert C(X ) -module M := L 2 (C, m G ) , defined by completion of C w.r.t. the C(X ) -valued scalar product c, c := m G (c∗ c ) ; by construction, there is a C(X ) -module map C c → vc ∈ M. For every x ∈ X , we consider the state ωx := x ◦ m G : C → C , and the associated GNS representation πx : C → L(Mx ) . By definition, Mx coincides with the fibre of M over x in the sense of [1, Def.14]. Now, C acts by multiplication on M, so that there is a C(X ) morphism π : C → L(M) , π(c)vc := vcc , c, c ∈ C . Since m G is positive, we find that π is injective. If c, c ∈ C and ηx (c − c ) = 0 , then there is f ∈ C(X ) such that f (x) = 0 and c − c = i q ( f )c for some c ∈ C (see [1, Prop.1.8]). This implies ωx (c−c ) = f (x)m G (c ) = 0 , i.e. πx (c−c ) = 0 . We conclude that ηx (c) = ηx (c ) ⇒ πx (c) = πx (c ) ; thus, πx = πx ◦ ηx for some representation πx : Cx → L(Mx ) . The previous considerations imply that π defines a field {πx }x∈X of faithful representations in the sense of [1, Def.2.11], and [1, Prop.3.11] implies that C is a continuous bundle of C*-algebras over X . Lemma 5.2. Suppose that there is a section s ∈ S X () . Then, there is a compact group subbundle H ⊆ G , with a bundle isomorphism H\G (see Sect.2.1). Proof. We define H := {h ∈ G : α ∗ ( h, s( p(h)) ) = s( p(h))}; moreover, we note that for every y ∈ G , h ∈ H , it turns out α ∗ ( hy, s( p(hy)) ) = α ∗ (y, s( p(y))) ; thus, the map τ : H\G → , τ (y H ) := α ∗ (y, s( p(y))) (5.2) is well-defined, and it is trivial to check that it is a bundle isomorphism.
We conclude that if (C(), G, α) is a fibred C(X ) -system endowed with an invariant C(X ) -functional ϕ : C(G) → C(X ) and a section s ∈ S X () , then there exists a compact group subbundle Hs ⊆ G with an isomorphism Hs \G . Thus, the considerations of Sect.2.1 apply, in particular Lemma 2.4 and Lemma 2.5. Note that the isomorphism class of Hs depends on s : by choosing a different section s ∈ S X () , we may get a group bundle Hs not isomorphic to Hs (Sect. 8). 6. Induced C*-Bundles In the present section, we denote by p : G → X a compact group bundle. Moreover, we consider a fibred C(X ) -system (B, G, α) . We fix our notation with A := B α , Z := A ∩ A , C := B ∩ B . By (3.9), we may eventually “rescale” X , and assume that C(X ) = C α . We denote by C ∨ Z the C*-subalgebra of B generated by C and Z . We denote by πx : B → Bx , x ∈ X , the epimorphisms associated with B as an upper semicontinuous bundle over X . Note that A , Z , C inherit from B the structure of C(X ) -algebra. We denote by the spectrum of C , and by q : → X the natural projection induced by the C(X ) -algebra structure of C ; in particular, we define x := q −1 (x) , x ∈ X . Since C is stable w.r.t. α , by applying the Gel’fand transform we obtain p(y) a fibred action α ∗ : G × → , α ∗ (y, ω) := ω ◦ α y . We now expose the main results of the present section. Together with Prop.6.4, they generalize [6, Thm.1] to the nontrivial centre case. Theorem 6.1. Suppose that G admits a left invariant positive C(X ) -functional, and consider the fibred C(X ) -system (B, G, α) . Then, for every section s ∈ S X () there
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is a compact group subbundle Hs ⊆ G and a fibred C(X ) -system (Fs , Hs , β) , with a C(X ) -epimorphism η : B → Fs . Moreover, the following properties are satisfied: 1. C is a continuous bundle of C*-algebras over X , with fibres C(Hx \G x ) , x ∈ X ; there is a C(X ) -algebra isomorphism C C(Hs \G) ; 2. η is injective on A , and A η(A) = Fs β ; 3. assume that A ∩ B = C ∨ Z ; if G → has local sections, or Hs is endowed with a right-invariant positive C(X ) -functional, then Z η(Z) = η(A) ∩ Fs . Some remarks follow. Existence of a (left) right invariant positive C(X ) -functional is verified under quite general conditions, in particular when G → X is trivial (see Prop.4.3); recall that compact group actions on C*-algebras in the usual sense can be regarded as fibred actions by trivial group bundles (Cor.3.4). Existence of the section s ∈ S X () is the necessary assumption to define the group bundle Hs , on which our construction is based; fibred C(X ) -systems such that → X does not admit sections can be constructed also in the case in which G → X is trivial, as we shall see in Sect.8. About the assumptions to prove Point 3 of Thm.6.1, we note that existence of local sections for G → is ensured under quite general conditions (Lemma 2.5); anyway, there are examples for which G → does not admit local sections also in the case in which X reduces to a single point: in explicit terms, there are compact groups G with compact subgroups H ⊆ G such that the projection G → H \G does not admit local sections (see [18, I.7.5]); but in this last case, we can use the Haar measure of H and apply Lemma 6.7, as done (implicitly) in [6]. Finally, unicity of Hs is not ensured: by choosing a different section s : X → we may get a group bundle Hs not isomorphic to Hs ; explicit examples are given in Sect.6.2, by choosing as in Sect.8. Let G be a compact Lie group. Then, the trivial bundle G := X × G admits an invariant C(X ) -functional; moreover, for every group subbundle H ⊆ G we find that p : G → has local sections (Lemma 2.5). Thus, we obtain the following result. Theorem 6.2. Let G be a compact Lie group, B a unital C*-algebra, and α : G → autB a strongly continuous action. Define A := B α , Z := A ∩ A , C() := B ∩ B , C(X ) := C() ∩ A (so that, a bundle → X is defined), and suppose that A ∩ B = C() ∨ Z . For every section s ∈ S X () , the following properties are satisfied: 1. there is a compact group bundle Hs → X , Hs ⊆ X × G , with a fibred C(X ) -system (Fs , Hs , β) ; 2. there is a C(X ) -epimorphism η : B → Fs , which is injective on A , and such that β η(A) = Fs , η(A) ∩ Fs = η(Z) . 6.1. Proof of Theorem.6.1. Let s ∈ S X () . We denote by φs : C → C(X ) , φs (c) := c◦s , the C(X ) -epimorphism associated with s . Moreover, we consider the group bundle Hs ⊆ G defined by Hs := h ∈ G : α ∗ (h, s ◦ p(h)) = s ◦ p(h) , (6.1) and denote by Hx := Hs ∩ G x , x ∈ X , the fibres of Hs . In the sequel, we will denote by y, y , . . . (resp. h, h , . . . ) generic elements of G (resp. Hs ). Note that if s , h ∈ H , x := p(h) ; in c ∈ ker φs , then by definition of Hs we find αhx (cx ) ∈ ker φ x ∗ fact, αh ◦ cx (s(x)) = cx ◦ α (h, s(x)) = cx (s(x)) = 0 . Let us now suppose that there is a left invariant positive C(X ) -functional ϕ : C(G) → C(X ) . By Lemma 5.1, Lemma 5.2, there exists an isomorphism C C(Hs \G) with
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x Hx \G x , x ∈ X . In the sequel, we will identify with the homogeneous bundle Hs \G .This implies that we have a group bundle p : G → , defined according to (2.3). The above considerations supply a proof of Point 1 of Thm.6.1. We now consider the closed C(X ) -ideal ker φs B ⊆ B generated by {cb, c ∈ ker φs , b ∈ B} , and denote by (6.2) η : B → Fs := B (ker φs B) the associated C(X ) -epimorphism. We denote by τx : Fs → Fs,x , x ∈ X , the evaluation epimorphisms associated with Fs . Note that if we regard B as a C() -algebra, then η is the restriction of B over the closed subset s(X ) ⊂ . By construction, ker φs B is a non-degenerate ker φs -bimodule, thus every b ∈ ker φs B admits a factorization b = cb , c ∈ ker φs , b ∈ ker φs B . By definition of Hs , we find αhx (cx πx (b)) = αhx (cx ) αhx ◦ πx (b) ∈ ker η , x := p(h) . This implies that Hs satisfies (3.7), thus we have a fibred action s → F s : η ◦ α(h, σ ) = β(h, η(σ )), σ ∈ B. β : Hs × X F Lemma 6.3. η is injective on A . Proof. In order to prove that η is injective on A , it suffices to verify that each ηx : Bx → Fs,x , ηx := η|Bx , x ∈ X , is injective on πx (A) ⊆ Bx . For this purpose, note that B is a C() -algebra, thus it is easy to verify that each Bx , x ∈ X , is a C(x ) -algebra; for every ω ∈ , we consider Bx,ω := Bx /[Cω (x )Bx ] , and denote by σω ∈ Bx,ω the image of σ ∈ Bx w.r.t. the quotient epimorphism Bx → Bx,ω . It is clear that σ = supω∈x σω . Moreover, since x is a homogeneous space w.r.t. the action α ∗ , we find σ = sup σω(y) , (6.3) y∈G x
where ω(y) := α ∗ (y, s(x)) , x := p(y) . Note that ω(ex ) = s(x) , where ex is the identity of G x . Now, by definition of η we have ηx (σ ) = σs(x) ; for every y ∈ G x , σ 0 ∈ πx (A) , by using the fact that σ 0 is α x -invariant, we find σ 0 +Cω(y)(x )Bx = 0 σ 0 + α yx−1 (Cs(x) (x )Bx ) = α yx−1 (σ 0 + Cs(x) (x ) . Thus, we obtain σω(y) = inf σ,c 0 σ + cσ = inf σ,c σ 0 + α yx−1 (c σ ) = inf σ,c α yx−1 (σ 0 + c σ ) = inf σ,c 0 0 = η (σ 0 ) , where c ∈ B , c ∈ C σ + c σ = σs(x) x x ω(y) (x ) , c ∈ C s(x) (x ) . 0 = 0 for every y ∈ G x , The above computation shows that if ηx (σ 0 ) = 0 , then σω(y) so that the lemma is proved.
Let us now consider the C(X ) -algebra tensor product FsG := C(G) ⊗ X Fs , and the G = C(G ) ⊗ F , x ∈ X . By sG → X with fibres Fs,x associated C*-bundle Q : F x s,x G (3.2), we identify Fs with S X (G, Fs ) , and define the following C(X ) -subalgebra of FsG :
s ) := F ∈ FsG : F(hy) = β(h, F(y)), h ∈ Hs , y ∈ G (6.4) Sβ (G, F s ) will be (note that we assume p(y) = p(h) ). The C*-bundle associated with Sβ (G, F G G denoted by Fs,β → X . Let α : G × X Fs → Fs denote the right translation G -action s ) then α ) is a fibred C(X ) -system. If F ∈ Sβ (G, F α yx ◦ F(hy ) = (4.2), so that (FsG , G,
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s,β is stable w.r.t. the action F(hy y) = βhx ◦ F(y y) = βhx ◦ α yx ◦ F(y ) . Thus, F α, and a fibred C(X ) -system s ), G, ( Sβ (G, F α) (6.5) is defined. We call (6.5) the induced C(X ) -system associated with (B, G, α) . Some elementary remarks, analogous to the one in [6, p.71], show that the fixed-point algebras α α β s ) s ) Sβ (G, F and Fs coincide, in the sense that elements of Sβ (G, F are exactly those β of the type F(y) := τ p(y) (F) , F ∈ Fs . Proposition 6.4. There is an equivariant C(X ) -isomorphism s ), T : (B, α) → (Sβ (G, F α ), T b(y) := η ◦ α(y, πx (b)), where b ∈ B , y ∈ G , x := p(y) . , we find that T b Proof. By continuity of α , η , and since b defines a section of B belongs to S X (G, Fs ) . In order to economize in notations, we define σ := πx (b) , so that T b(y) = ηx ◦α yx (σ ) . We compute T b(hy) = ηx ◦αhx ◦α yx (σ ) = βhx ◦ηx ◦ α yx (σ ) = s ) . Let now T → F s,β be the morphism induced :B βhx ◦ T b(y) ,thus T b ∈ Sβ (G, F σ )] (y ) = σ as a map from G x into Fs,x , and [ α (y, T by T . Then, we may regard T x x x T σ (y y) = ηx ◦ α y (α y (σ )) = [T ◦ α y (σ )] (y ) . So that α (y, T σ ) = T ◦ α(y, σ ) , and T is equivariant. We now verify that T is injective. For this purpose, we remind the reader of the C(x ) -algebra structure of Bx , and make use of the notation introduced in the proof of Lemma 6.3. Let y ∈ G , ω(y) := α ∗ (y, s(x)) ; then, for every σ ∈ Bx , we find α yx (σ + Cω(y −1 ) (x )Bx ) = α yx (σ ) + Cs(x) (x )Bx . By recalling the definition of the fibre epimorphism Bx → Bx,ω , we conclude that α yx defines an isomorphism from Bx,ω(y −1 ) onto Bx,s(x) . This implies σω(y −1 ) = (α yx (σ ))s(x) . On the other σ (y) . Thus, we conclude side, (α yx (σ ))s(x) = ηx ◦ α yx (σ ) = T T σ (y) = σω(y −1 ) , y ∈ G x . Now, every ω ∈ x is of the type ω = ω(y) for some y ∈ G x . Thus (by σ = 0 if and only if σ = 0 , and T is injective (i.e., T is injective). We now (6.3)) T s ) . Since every ηx is surjective, for every prove that T is surjective. Let F ∈ Sβ (G, F y ∈ G there is bx ∈ B such that T bx (y) = η ◦ α(y, πx (bx )) = F(y) , x := p(y) . Let ε > 0 ; for every y ∈ G , we consider the open set (6.6) U y := y ∈ G : F(y ) − T bx (y ) < ε . Now, for every h ∈ Hs we have βhx (F(y)) = F(hy) , βhx (T bx (y )) = T bx (hy ) , and this implies U y = Uhy . By construction, U y y is an open cover for G ; we pick a finite subcover U yk k , with the corresponding bk := b p(yk ) ∈ B satisfying (6.6). Let us now consider the projection p : G → ; since U yk = Uhyk , h ∈ Hs , there k } ⊂ C such that {λk ◦ p } ⊂ C(G) is subordinate is a partition of unit {λ to U := . We define b ε k ∈ B . If y ∈ G , then F(y) − T bε (y) ≤ k λk b yk F(y) λ ( p (y)) − T b (y) ≤ k k k k λk (y)ε = ε . This proves that the image of T s ) . is dense in Sβ (G, F Let ex ∈ G x denote the identity. By definition of η and T , we find τx ◦ η(b) = T b(ex ), x ∈ X.
(6.7)
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Corollary 6.5 (Point 2 of Thm.6.1). The map T induces a C(X ) -algebra isomorphism A Fs β . Moreover, Fs β = η(A) . s ) is of Proof. Since T is covariant, we find that every α -invariant element of Sβ (G, F the type T a for some a ∈ A . But T a is α -invariant if and only if T a(y y) = T a(y ) for every y ∈ G ; thus, if h ∈ Hs then T a(h) = T a(hex ) = βhx (T a(ex )) = T a(ex h) = T a(ex ) . In other terms, by (6.7), βhx ◦ τx (η(a)) = τx (η(a)), x ∈ X, β
β
so that we conclude η(A) ⊆ Fs . On the other side, it is clear that every F ∈ Fs defines s ) , say F , by defining F(y) an α -invariant element of Sβ (G, F := τ p(y) (F) , y ∈ G ; in fact, F being fiberwise constant, we have F(hy) = F(y) = τx (F) , x := p(y) = = T a for some a ∈ A , i.e. p(hy) , so that βhx ◦ F(y) = F(y) = F(hy) . Thus, F F = η(a) . Since η(A) is isomorphic to A (Lemma 6.3), the corollary is proved. Now, C(X ) may be regarded as a C*-subalgebra of η(Z) (recall that Fs is a C(X ) algebra), and η(C) = φ(C) = C(X ) ⊆ η(Z) . Thus, we find η(A ∩ B) = η(C ∨ Z) = η(Z).
(6.8)
Lemma 6.6 (Point 3 of Thm.6.1). Suppose that p : G → has local sections. If A ∩ B = C ∨ Z , then η(A) ∩ Fs = η(Z) Z . Proof. It is clear that η(Z) ⊆ η(A) ∩ Fs . On the converse, let F ∈ η(A) ∩ Fs . In order to prove that F ∈ η(Z) , it suffices to verify that F = η(b) for some b ∈ A ∩ B . In fact, in such a case (6.8) implies that F ∈ η(Z) . Let us now consider a finite open of Lemma 2.4. We consider a partition of cover {Wk } of satisfying the properties
−1 unity {λk } ⊂ C(G) subordinate to p (Wk ) ⊂ G , so that λk (hy) = λk (y) , h ∈ Hs , y ∈ G . We consider the bundle morphisms (2.5), and define the map T b(y) :=
λk (y) · β( δk (y), τ p(y) (F) ), y ∈ G.
k
s , and by continuity of β , δk , we find T b ∈ Since F defines a continuous section of F s ) . Moreover, by applying (2.6), we find T b(hy) = βh (T b(y)) ; thus, T b ∈ S X (G, F s ) , i.e. b ∈ B . Since F ∈ η(A) ∩ Fs , we find T b ∈ T (A) ∩ Sβ (G, F s ) = Sβ (G, F T (A ∩ B) . By applying (6.7), we find τx ◦ η(b) = T b(ex ) = τx (F) for every x ∈ X , and this implies F = η(b) . Lemma 6.7 (Point 3 of Thm.6.1). Suppose that there is a right-invariant positive C(X ) functional δ : C(Hs ) → C(X ) . If A ∩ B = C ∨ Z , then η(A) ∩ Fs = η(Z) Z . Proof. As for the previous lemma, it suffices to prove that for every F ∈ η(A) ∩ Fs there is b ∈ A ∩ B with η(b) = F . To be more concise, we define Fs := η(A) ∩ Fs ; ∈ C(Hs ) ⊗ X Fs , it is clear that Fs is stable w.r.t. β . Let us consider the map F F(h) := βhx ◦ τx (F) , x := p(h) . Since Hs ⊆ G is closed, by the Tietze extension theorem for continuous bundles ([5, Chp.10]) we find that there is b ∈ C(G) ⊗ X Fs . In particular, b(ex ) = τx (F) , x ∈ X . Let us now consider the such that b|Hs = F
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Hs -invariant mean m : Fs → (Fs )β induced by δ ; then, for every x ∈ X , F ∈ Fs , we find τx ◦ m(F ) = Hx βhx ◦ τx (F )dµx (h) (see (4.3)). We define the map s , y ∈ G ; (βhx )−1 ◦ b(hy) dµx (h) ∈ F b(y) := Hx
b ∈ C(G) ⊗ X Fs ⊆ C(G) ⊗ X Fs ; moreover, since since b ∈ C(G) ⊗ X Fs , we find that x )−1 ◦b(hh y)dµ (h) = β x ◦ dµx is right-invariant, we find b(h y) = βhx ◦ (βhh x h b(y) . This implies b ∈ Sβ (G, Fs ) , thus b = T b for some b ∈ B . Since b takes values in x −1 F
s , we find b ∈ A ∩ B ; finally, T b (ex ) = b(ex ) = (βh ) ◦ F(h)dµx (h) = τx (F)dµx (y) = τx (F) , so that (by (6.7)) we find η(b ) = F , and the lemma is proved. 6.2. A class of examples. Let d ∈ N , d ≥ 2 , G ⊆ SU(d) be a compact Lie group. We consider an Abelian C(X ) -algebra C() carrying a strongly continuous action ρ : G → aut X C() , such that C(X ) = C()ρ . Starting from this data, we construct a C*-dynamical system (B, G, α) satisfying the hypothesis of Thm.6.2. We now start our construction. By Lemma 5.1, every fibre x := q −1 (x) is homeomorphic to a homogeneous space Hx /G , x ∈ X . In order to simplify our notation, we write ωg := ω ◦ ρg , ω ∈ , g ∈ G . Let us now consider the order d Cuntz algebra d Od , generated by a set {ψi }i=1 of d orthogonal isometries satisfying ψi ψi∗ = 1. (6.9) ψi∗ ψ j = δi j 1, i
It is well-known that G acts by automorphisms on Od : if g ∈ G , then there exists a unique g ∈ autOd such that g (ψi ) = gψi := gi j ψ j , (6.10) ij
where gi j ∈ C are the matrix coefficients of g w.r.t. the canonical basis of Cd . We define B := C() ⊗ Od . B is a trivial continuous bundle of Cuntz algebras over ; moreover, B is also a C(X ) -algebra, with fibres Bx C(x ) ⊗ Od , x ∈ X . In the sequel, we will regard elements of B as continuous maps from into Od . We introduce the action g (b(ωg −1 )), α : G → aut X B, [αg (b)](ω) := → B , where b ∈ B , g ∈ G , ω ∈ . We can now regard α as a fibred action α : G × X B with G := X × G . We define A := B α ; by construction A = {A ∈ B : g (A(ω)) = A(ωg), g ∈ SU(d)} . Lemma 6.8. A ∩ B = C() , and A ∩ A = C(X ) . Proof. It is clear that C() ⊆ A ∩ B . Let OG denote the fixed-point algebra of Od ∩ O = C1. This also w.r.t. the action (6.10). By [7, Cor.3.3], we conclude that OG d ∩ B = C() (we identify O with the set of constant maps from implies that OG G into OG ). Let us now consider the constant map A(ω) := a ∈ OG , A ∈ B . Then
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[αg (A)](ω) = g (a) = a = A(ω) , g ∈ G , ω ∈ . This proves that OG ⊆ A , so ∩ B = C() . We conclude that A ∩ B = C() . In particular, if that A ∩ B ⊆ OG b ∈ C() is α -invariant (i.e., b ∈ A ∩ A ), then by definition b is ρ -invariant. We conclude that b ∈ C()ρ = C(X ) . We now assume that there is a section s ∈ S X () , and consider the associated C(X ) morphism φs : C() → C(X ) . By applying Thm.6.2, we obtain a fibred C(X ) -system (F, Hs , β) , where Hs ⊆ X × G is defined according to (6.1). We now verify that for every h ∈ Hs , F ∈ C(X ) ⊗ Od , x := p(h) , the following relations hold: ⎧ F C(X ) ⊗ Od ⎪ ⎪ ⎨ x βh ◦ F(x) = h (F(x)) . (6.11) β A = F ⎪ ⎪ ⎩ A ∩ A = A ∩ F = C(X ) About (6.11.1): we have ker φs := {c ∈ C() : c(s(x)) = 0, x ∈ X } , thus a continuous map b : → Od belongs to ker φs B if and only if b ◦ s(x) = 0 , x ∈ X . The quotient of B w.r.t. such an ideal is the set of continuous maps from s(X ) into Od , and η is the restriction on s(X ) . Since s(X ) is homeomorphic to X , we find (6.11.1). In order to simplify our notation, in the following lines we identify F with C(X ) ⊗ Od . About (6.11.2): if b ∈ B , then (η(b)) (x) = b ◦ s(x) , x ∈ X . By definition of Hs , we find s(x)h = h for every h ∈ Hx , x ∈ X . Moreover, the fibred action β satisfies βhx ◦ ηx = ηx ◦ αhx , h ∈ H , x := p(h) . If F ∈ F and F = η(b) for some b ∈ B , then h(b(s(x)h −1 )) = h(b ◦ s(x)) = h(F(x)) . βhx ◦ F(x) = βhx ◦ b ◦ s(x) = αhx (b(s(x))) = The equality (6.11.3) follows from Thm.6.2. About (6.11.4), note that (6.11.2) implies ∩ O = C(X ) ⊗ OG ⊆ F β . Thus, (F β ) ∩ F ⊆ (C(X ) ⊗ OG ) ∩ F ; since OG d β C1, we conclude that (C(X ) ⊗ OG ) ∩ F = C(X ) , so that (F ) ∩ F = C(X ) . This also implies (F β ) ∩ F β = C(X ) . Note that Hs may be not full; in particular, the stabilizer H of ker η in G may be trivial (Rem.3.1). Moreover, a different s ∈ S X () may define a group bundle Hs ⊆ G not isomorphic to Hs (see Sect.8) Of course, the case in which does not admit sections is possible (see Sect.8, where actually an action T → aut X C() is considered; anyway, we may take G := U (T) , where U : T → SU(d) is a faithful representation). In this case, the group bundle Hs cannot be defined. 7. Superselection Structures It is well-known that the field algebra describing a superselection structure of quantum localized observables is constructed in mathematical terms as a crossed product by a semigroup of C*-endomorphisms having permutation symmetry ([9, §3],[10, §2]). In the present section, we give a generalization of the above-cited crossed product to the nontrivial centre case, at least for the case of a single endomorphism (Thm.7.2). In particular, we also cover the case of Hilbert C*-systems considered in [2] (Thm.7.4). Let A be a unital C*-algebra, ρ ∈ endA a unital endomorphism. We define Z := A ∩ A , and C(X ) := { f ∈ Z : ρ( f ) = f }. Let us assume that there is a C*-algebra B with identity 1, carrying an inclusion A ⊂ B of unital C*-algebras, such that: d , d ∈ N , of isometries 1. B is generated as a C*-algebra by A and a set {ψi }i=1 satisfying the Cuntz relations (6.9). Thus, an endomorphism σB ∈ endB , σB (b) := ψi bψi∗ , b ∈ B (7.1) i
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is defined. By universality of the Cuntz algebra, there is a unital monomorphism j : Od → B . 2. The following relations hold: ρ(a) = σB (a), a ∈ A.
(7.2)
Note that the previous relations imply that if f ∈ C(X ) , then f commutes with ψ1 , . . . , ψd , thus f ∈ B ∩ B . In other words, B is a C(X ) -algebra (also note that σB is a C(X ) -endomorphism). 3. There is a strongly continuous action α : SU(d) → aut X B , such that B α = A , αg ◦ j (t) = j ◦ g (t) , g ∈ SU(d) , t ∈ Od , where g ∈ autOd is defined as in (6.10). We denote by C the centre of B , and by the spectrum of C . Let OSU(d) ⊂ Od be the fixed-point algebra w.r.t. the SU(d) -action (6.10); then, the above considerations imply that j (OSU(d) ) ⊆ A . For every r, s ∈ N , we consider the intertwiner spaces r s (ρ , ρ ) := {t ∈ A : ρ s (a)t = tρ r (a), a ∈ A} . (σBr , σBs ) := w ∈ B : σBs (b)w = wσBr (b), b ∈ B In particular, for r = 0 , we define σB0 := ιB (the identity on B ) and ρ 0 := ιA (the identityon A ). Let I := {i 1 , . . . , ir } be a multiindex of length |I | = r ∈ N . We define ψ I := r ψir ∈ (ιB , σBr ) ; note that ψ I ψ I∗ = 1, ψ I∗ ψ I = δ I I 1. (7.3) I
Let J be a multiindex with length s ∈ N . By [7, §2], the following “flip” operators belong to the j (OSU(d) ) ⊆ A : ε(r, s) := ψ J ψ I ψ J∗ ψ I∗ ∈ (σBr +s , σBr +s ); IJ
note that in particular ε(r, s) ∈ (ρ r +s , ρ r +s ) . To be coherent, we also define ε(0, s) = ε(r, 0) = 1. By using (7.3), we compute ψi ψ I ψi∗ = ψi ψ J (ψ J∗ ψ I )ψi∗ = ε(1, r )ψ I . (7.4) σB (ψ I ) = i
iJ
With the same argument used for the ε(r, s) ’s, we conclude that if P(r ) is the permutation group of r objects and ε( p) := ψi1 · · · ψir ψi∗p(r ) · · · ψi∗p(1) , then ε( p) ∈ (ρ r , ρ r ) . Let us now define (ρ r , ρ s )ε := t ∈ (ρ r , ρ s ) : ρ(t)ε(r, 1) = ε(s, 1)t . By definition of ε(r, s) (r = s = 0 ), it turns out (ιA , ιA )ε = C(X ) ; moreover, a simple computation shows that ε(r, s) ∈ (ρ r +s , ρ r +s )ε , r, s ∈ N . Note that every (ρ r , ρ s )ε is a C(X ) -bimodule w.r.t. left and right multiplication by elements of C(X ) . We say that ρ has permutation quasi-symmetry if every t ∈ (ρ r , ρ s ) admits a decomposis r s r s tion t = i ρ (z i )ti , z i ∈ Z , ti ∈ (ρ , ρ )ε (note that since ti ∈ (ρ , ρ ) , then ρ s (z i )ti = ti ρ r (z i ) ). In particular, if (ρ r , ρ s ) = (ρ r , ρ s )ε , r, s ∈ N , then we say that ρ has permutation symmetry, according to [9, §4].
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Proposition 7.1. If ρ has permutation quasi-symmetry, then A ∩ B = C ∨ Z . Proof. We proceed in a order of ideas similar to [8, Lemma 5.1]. As a first step, note that A ∩B is SU(d) -stable; thus, by Fourier analysis, A ∩B is generated as a C*-algebra by sets of “irreducible tensors” of the type T1 , . . . , Tn ∈ A ∩ B , satisfying the relations αg (Ti ) =
N
T j u ji (g),
(7.5)
j
where g ∈ SU(d) and u ji (g) ∈ C are matrix elements in some irreducible, unitary representation of SU(d) (in particular, note that u ji (g) = u i∗j (g) are the matrix elements of the inverse of (u i j (g))i j ). Now, every irreducible representation of SU(d) is a subrepresentation of some tensor power of the defining representation. It follows from (6.10) that SU(d) acts on Hd := span {ψi }i ⊂ (ιB , σB ) as the defining representation. Moreover, for every r ∈ N we can identify Hrd := span ψ I ∈ (ιB , σBr ), |I | = r with the r -fold tensor power of Hd , and SU(d) acts on Hrd as the r -fold tensor power of the defining representation. Thus, there is n ∈ N and an orthonormal set {ϕh } ⊂ Hnd (i.e., ϕi∗ ϕ j = δi j 1), such that ϕ j u ji (g). (7.6) αg (ϕi ) = j
∗ Let W := i Ti ϕi∗ ; by using (7.5) and (7.6), we find αg (W )= i hk Th ϕk u hi (g) ∗ u ki (g) = hk δhk Th ϕk = W . We conclude that W ∈ A . Moreover, Ti = W ϕi , where W ∈ (ρ n , ιA ) , ϕ ⊂ (ιB , σBn ) . Thus, by permutation quasi-symmetry, W = pd n k ρ (z k )Wk , where z k ∈ Z and Wk ∈ (ρ , ιA )ε . So that we obtain Ti = k z k Wk ϕi . Let us now define T i,k := Wk ϕi ; we prove that σB (T i,k ) = T i,k (so that T i,k commutes with ψ1 , . . . , ψd and T i,k ∈ C ). By permutation quasi-symmetry and (7.4), we obtain σB (T i,k ) = ρ(W k )σB (ϕi ) = W k ε(1, n) · ε(n, 1)ϕi = T i,k , and this implies T i,k ∈ C . Since Ti = n z k T i,k , we conclude that A ∩ B = C ∨ Z . We now look for C*-epimorphisms η : B → F which are injective on A , and such that η(A) ∩ F = η(Z) . Pairs (F, η) of the above type are called Hilbert extensions of A . Every Hilbert extension corresponds to the crossed product of A by ρ in the sense of [9, §4], thus can be interpreted as a “field algebra” associated with (A, ρ) . We discuss existence and unicity of the Hilbert extension in two important cases. Doplicher-Roberts endomorphisms (with non-trivial centre). Let A be a unital C*-algebra with centre Z , and ρ ∈ endA an endomorphism satisfying the special conjugate property in the sense of [9, §4] (i.e., ρ has permutation symmetry, and there is an isomed ) satisfying the special conjugate equations R ∗ ρ(R) = (−1)d−1 d −1 1, try R ∈ (ιA , ρ ε R R ∗ = d!−1 p∈Pd sign( p)ε( p) ). Then, we can construct a C*-algebra B satisfying the above properties, as the crossed product of A by the dual SU(d) -action induced by ρ ([9, Thm.4.2]); we maintain the notation C ≡ C() for the centre of B , and C(X ) for the C*-algebra of ρ -invariant elements of Z . Since in particular ρ has permutation symmetry the previous proposition applies, so that A ∩ B = C ∨ Z . Theorem 7.2. Let ρ ∈ endA be an endomorphism satisfying the special conjugate property. Sections s ∈ S X () are in one-to-one correspondence with Hilbert extensions
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(Fs , ηs ) of A . For every s ∈ S X () , there exists a group bundle Gs ⊆ X × SU(d) such that (Fs , Gs , β) is a fibred C(X ) -system satisfying ηs (A) = Fs β , ηs (A) ∩ Fs = ηs (Z) . Proof. If there is s ∈ S X () , then existence of a Hilbert extension (Fs , ηs ) follows from Thm.6.2, and a fibred action (Fs , Gs , β) is defined, satisfying the required properties. On the other side, if (F, η) is a Hilbert extension, then we note that η is injective on j (Od ) ⊂ B , in fact Od is simple and η(t) = 0 for every t ∈ j (Od ) ∩ A . Let us now consider the restriction φ := η|C . Then, φ(C) ⊆ η(Z) , thus for every c ∈ C there is f ∈ Z with φ(c) = η( f ) . Since η is injective on Z , we find that f is unique. Moreover, by using the fact that φ(c) ∈ F ∩ F , we find η ◦ ρ( f ) = η ◦ σB ( f ) = η(ψ )η( f )η(ψi )∗ = η( f ) . We conclude that η( f − ρ( f )) = 0 , i.e. f = ρ( f ) ; i i by definition of C(X ) , this implies f ∈ C(X ) . Thus, we have a C*-epimorphism φ : C → η(C(X )) , which is injective on C(X ) ⊆ C . By applying the Gel’fand transform, we obtain the desired section s ∈ S X () . Note that Gs may be a non-trivial bundle, and may depend on the choice of s (see Sect.8). The question of unicity of Gs may be approached by regarding Fs as a crossed s in the sense of [21]. Anyway there are some further complications, due product A G to the fact that we may choose a Hilbert C(X ) -bimodule instead of Hd := span {ψi } to construct an analogue of the C*-algebra B (see [20, §1]). For this reason, we postpone a complete discussion to a forthcoming paper. Baumgärtel-Lledó endomorphisms. The following class of C*-endomorphisms has been studied in [2, §4]. Let A be a unital C*-algebra; we say that ρ is a canonical endomorphism of A if ρ has permutation quasi-symmetry, with the additional property that every (ρ r , ρ s )ε is a free C(X ) -bimodule generated by a finite-dimensional vector space, say (ρ r , ρ s )C . The family of vector spaces (ρ r , ρ s )C , r, s ∈ N , is required to be ρ -stable, closed for multiplication (i.e., (ρ k , ρ s )C (ρ r , ρ k )C ⊆ (ρ r , ρ s )C ) and involution, and such that (ιA , ιA )C = C , ε(r, s) ∈ (ρ r +s , ρ r +s )C . Let us now suppose that there is R ∈ (ι, ρ d )ε satisfying the special conjugate equations; in such a case, we say that ρ is a special canonical endomorphism. The ρ -stable C*-subalgebra of A generated by {R, ε(r, s)}r,s , is isomorphic to OSU(d) ([7, Thm.4.1]): such an isomorphism defines a dual action µ : OSU(d) → A . Again, B can be constructed as the crossed product of A by µ in the sense of [9, §3], and Prop.7.1 applies. Lemma 7.3. There is a closed group G ⊆ SU(d) with a bundle isomorphism X × G\SU(d) . The group G is unique up to conjugation in SU(d) . Proof. We retain the same notation used in the proof of Prop.7.1. As a preliminary step, we consider C0 of B generated by the set {W ∗ ϕ : W ∈ (ιA , ρ n )C , the C*-subalgebra n n n ϕ ∈ Hd . Since Hd ⊂ (ιB , σB ) , we find that W ∗ ϕa = W ∗ ρ n (a)ϕ = aW ∗ ϕ , a ∈ A . Moreover, by using (7.4), we find σB (W ∗ ϕ) = W ∗ ε(1, n)ε(n, 1)ϕ = W ∗ ϕ : this means that W ∗ ϕ commutes with ψi for every i = 1, . . . , d (recall (7.1)). We conclude that C0 is contained in C . Morever, W1∗ ϕ1 W2∗ ϕ2 = W1∗ ρ n 1 (W2 )ϕ2 ϕ1 = (W1 W2 )∗ ϕ1 ϕ2 , for every W1∗ ϕ1 , W2∗ ϕ2 ∈ C0 : thus, the set of linear combinations of terms of the type W ∗ ϕ (and the adjoints ϕ ∗ W ) is dense in C0 . Let now c := i Wi∗ ϕi such that αg (c) = c , g ∈ SU(d) ; by averaging w.r.t. the Haar mean, say m : B → A , we find c = m(c) = i Wi∗ m(ϕi ) . Now, m(ϕi ) is an α -invariant element of some Hnd i . Since αg ◦ j = j ◦ g , g ∈ SU(d) , we conclude that m(ϕi ) belongs to j (OSU(d) ) ⊆ A . Now, j (OSU(d) ) is generated by elements belonging to the family (ρ r , ρ s )C , r, s ∈ N ,
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∗ thus m(ϕi ) ∈ (ιA , ρ n i )C . This implies that c = m(c) = i Wi m(ϕi ) belongs to C1, in fact Wi∗ m(ϕi ) ∈ (ρ n i , ιA )C · (ιA , ρ n i )C ⊆ (ιA , ιA )C = C1. This proves that C0α C1; thus, by well-known results, if we pick an element ω0 of the spetrum 0 of C0 , and define G ⊆ SU(d) as the stabilizer of ω0 , then we get a homeomorphism 0 G\SU(d) . Note that G is unique up to conjugation in SU(d) . We now introduce the map : C(X ) ⊗ C0 → C , ( f ⊗ c0 ) := f c0 . Let W ∗ ϕ ∈ C0 ; then, f W ∗ ϕ2 = ( f ∗ f )(ϕ ∗ W W ∗ ϕ) . Now, note that cϕ,W := (ϕ ∗ W W ∗ ϕ) belongs to (ιA , ιA )C ; thus, cϕ,W is a (positive) multiple of the identity, and it is now clear that cϕ,W = W ∗ ϕ2 . 2 Moreover, f W ∗ ϕ2 = f ∗ f cϕ,W = f 2 cϕ,W = f 2 W ∗ ϕ2 . This proves that is isometric. We now prove that is surjective. With the same argument as the previous proposition, C is the closed vector space spanned by multiplets of the type n {ci }i=1 , such that αg (ci ) = k ck u ki (g) , g ∈ SU(d) ; the coefficients u ki (g) ∈ C are matrix elements of some irreducible representation of SU(d) , say u . Thus, there is a set {ϕi } ⊂ (ι, σBn ) , ϕi∗ ϕ j = δi j 1 such that W ∗ := i ci ϕi∗ belongs to A (i.e., αg (W ∗ ) = W ∗ ); note that ci = W ∗ ϕi . Moreover, it is clear that W ∗ ρ n (a) = aW ∗ , i.e. W ∈ (ιA , ρ n ) . Since ρ is a canonical endomorphism, we find W = h f h Wh , Wh ∈ (ι, ρ n )C . Thus, ci = kh f h∗ (Wh∗ ϕi ) , and is surjective. It is now clear that X × 0 X × G\SU(d) . Theorem 7.4. Let ρ ∈ endA be a special canonical endomorphism. Then, there exists a Hilbert extension (F, η) , and a closed group G ⊆ SU(d) unique up to conjugation in SU(d) , with an action β : G → autF such that A η(A) = F β , η(A) ∩ F = η(Z) . Proof. It suffices to apply Thm.6.2 to the C*-dynamical system (B, SU(d)) , by choosing s ∈ S X (X × (G\SU(d))) of the type s(x) := (x, ω0 ) , ω0 ∈ G\SU(d) . 8. Appendix Bundles of homogeneous spaces lacking of sections. Let us denote by T := {z ∈ C : |z| = 1} the torus. For every n ∈ N , we consider the compact subgroup Rn := {z ∈ T : z n = 1} of roots of unity. Elementary computations show that the coset space pn ⊂ Rn \T can be identified with T, and that there is an exact sequence 1 → Rn −→ T −→ T → 1, where 1 is the trivial group, and pn (z) := z n , z ∈ N . Let S 2 denote the 2 -sphere. Principal T -bundles over S 2 are classified by the cohomology group H 1 (S 2 , T) Z: for every h ∈ Z, we denote by h → S 2 the associated principal T-bundle, having fibre h,x T . The bundle h → S 2 is trivial if and only if h = 0 . Let k = 1, . . . , n ; it follows from [13, Prop.7.1.7] that there is an action ρ k : T → aut X C(h ), which fiberwise behaves as the multiplication ρ k,x : T → autC(h,x ) , ρzk,x c(ω) := c(ωz k ) , c ∈ C(h,x ) , ω ∈ T h,x . Since multiplication by z k , z ∈ T, defines an ergodic action on T, we conclude that the fixed-point algebra of C(h,x ) w.r.t. ρ k,x reduces to the complex numbers. Thus, C()ρ C(S 2 ) . By general properties of principal bundles ([13, Thm.6.2.3]), the unique principal T -bundle over S 2 which admits a section is the trivial bundle 0 S 2 × T . Non-isomorphic group subbundles, with isomorphic associated homogeneous bundles. Let n ∈ N , and O(n) ⊂ Mn (R) denote the orthonomormal group with unit 1n .
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For every m < n , we have that O(m) may be regarded as a closed subgroup of O(n) via the embedding O(m) ⊕ 1n−m ⊂ O(n) . The quotient O(m)\O(n) is the homogeneous space known as the Stiefel manifold ([13, 7.1]). Let X be a compact Hausdorff space. For every n ∈ N , we denote by Tn := X × Rn the trivial rank n vector bundle, and by Gn := X × O(n) the trivial group bundle with fibre O(n) . For every rank d real vector bundle E → X , we consider the associated bundle of orthonormal endomorphisms OE → X with fibre O(d) ([14, I.4.8]). It is well-known that if E → X is a real rank d vector bundle, then OE is isomorphic to OE if and only if E = E ⊗ L for some real line bundle L → X . Thus, if X is a space such that the cohomology group H 1 (X, Z2 ) is trivial, then every real line bundle L → X is trivial, and E E if and only if OE OE . Let now E → X be a non-trivial rank d vector bundle such that E ⊕ Tn Td+n for some n ∈ N (i.e., E has trivial class in K -theory). Since E is non-trivial, OE is non-trivial; on the other side, OTd+n = Gd+n is trivial. Moreover, the natural monomorphism j : E → Td+n induces a monomorphism ad j : OE → Gd+n . So that the trivial bundle Gd+n has two non-isomorphic group subbundles with fibre O(d) , i.e. the trivial bundle Gd and H := ad j (OE) . This happens despite the fact that we have isomorphisms Gd \Gd+n H\Gd+n := X × (O(d)\O(d + n)) ;
(8.1)
in fact, is a bundle having the (trivial) cocycle associated with Gd+n as a set of transition maps (see [13, Thm.6.4.1] and following remarks). Now, C() is endowed with an automorphic action ρ : O(d + n) → aut X C() such that C()ρ = C(X ) , and is clearly full (in fact, it is a trivial bundle over X ). This implies that → X is a homogeneous bundle. Anyway, (8.1) implies that can be recovered as a quotient of Gd+n w.r.t. non-isomorphic subbundles, namely Gd and H . For example, all the above considerations apply in the case in which X is the sphere S k , k = 1, 3, 7, and E is the the tangent bundle T S k ([14, I.5.5]). Acknowledgement. The author would like to thank S. Doplicher and C. Pinzari for stimulating discussions, and Gerardo Morsella for precious help.
References 1. Blanchard, E.: Déformations de C*-algébres de Hopf. Bull. Soc. Math. France 124, 141–215 (1996) 2. Baumgärtel, H., Lledó, F.: Duality of compact groups and Hilbert C*-systems for C*-algebras with a nontrivial center. Int. J. Math. 15, 759–812 (2004) 3. Carey, A., Grundling, H.: Amenability of the Gauge Group. Lett. Math. Phys. 68, 113–120 (2004) 4. Ciolli, F.: Massless scalar free Field in 1+1 dimensions I: Weyl algebras Products and Superselection Sectors, Preprint http://arxiv.org/list/ math-ph/0511064, 2005 5. Dixmier, J.: C*-algebras. Amsterdam - New York - Oxford North-Holland Publishing Company, 1977 6. Doplicher, S., Roberts, J.E.: A Remark on Compact Automorphism Groups of C*-Algebras. J. Funct. Anal. 66, 67–72 (1986) 7. Doplicher, S., Roberts, J.E.: Duals of Compact Lie Groups Realized in the Cuntz Algebras and Their Actions on C*-Algebras. J. Funct. Anal. 74, 96–120 (1987) 8. Doplicher, S., Roberts, J.E.: Compact Group Actions on C*-Algebras. J Operator Theory 91, 227– 284 (1988) 9. Doplicher, S., Roberts, J.E.: Endomorphisms of C*-algebras, Cross Products and Duality for Compact Groups. Ann. Math. 130, 75–119 (1989) 10. Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51–107 (1990) 11. Gierz, G.: Bundles of topological vector spaces and their duality. Lecture Notes in Mathematics 955, Berlin-Heidelberg-NewYork: Springer-Verlag, 1982
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12. Hirzebruch, F.: Topological Methods in Algebraic Geometry. Berlin-Heidelberg-Newyork: Springer-Verlag, 1966 13. Husemoller, D.: Fiber Bundles. Newyork: Mc Graw-Hill Series in Mathematics, 1966 14. Karoubi, M.: K-Theory. Berlin - Heidelberg - New York: Springer Verlag, 1978 15. Kasparov, G.G.: Equivariant K K -Theory and the Novikov Conjecture. Invent. Math. 91, 147–201 (1988) 16. Mack, G., Schomerus, V.: Conformal field algebras with quantum symmetry from the theory of superselection sectors. Commun. Math. Phys. 134, 139–196 (1990) 17. Nilsen, M.: C*-Bundles and C0 (X ) -algebras. Indiana Univ. Math. J. 45, 463–477 (1996) 18. Steenrod, N.: Topology of Fibre Bundles. Princeton Mathematical Series, Princeton, NJ: Princeton University Press, 1965 19. Streater, R.F., Wilde, I.F.: Fermion states of a boson field. Nucl. Phys. B 24, 561–575 (1970) 20. Vasselli, E.: Crossed Products by Endomorphisms, Vector Bundles and Group Duality. Int. J.Math. 16(2), 137–171 (2005) 21. Vasselli, E.: Crossed Products by endomorphisms, Vector Bundles and Group Duality, II. Int. J. Math. 17(1), 65–96 (2006) Communicated by Y. Kawahigashi
Commun. Math. Phys. 274, 277–295 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0273-8
Communications in
Mathematical Physics
Quasi-Local Mass and the Existence of Horizons Yuguang Shi1, , Luen-Fai Tam2, 1 Key Laboratory of Pure and Applied Mathematics, School of Mathematics Science,
Peking University, Beijing 100871, People’s Republic of China. E-mail:
[email protected]
2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China.
E-mail:
[email protected] Received: 30 November 2005 / Accepted: 26 February 2007 Published online: 23 June 2007 – © Springer-Verlag 2007
Abstract: In this paper, we obtain lower bounds for the Brown-York quasilocal mass and the Bartnik quasilocal mass for compact three manifolds with smooth boundaries. As a consequence, we derive sufficient conditions for the existence of horizons for a certain class of compact manifolds with boundary and some asymptotically flat complete manifolds. The method is based on analyzing Hawking mass and inverse mean curvature flow.
1. Introduction In this work, we will discuss the relations between different kinds of quasi-local mass and use them to derive sufficient conditions for the existence of horizons in the time symmetric case. In the time symmetric case, a horizon is defined to be a compact minimal surface which is the boundary of some open sets (see §3 for more precise definition). In 1972, Thorne made the following conjecture, which later became known as the hoop conjecture (see [10]): Black holes with horizons form when and only when a mass M gets compacted into a region whose circumference in every direction is C ≤ 4π M. The conjecture is loosely formulated. Several concepts such as mass, circumference, etc. are not clearly defined. Hence, this conjecture allows many different precise interpretations. In 1983, Schoen and Yau in [22], and later Yau in [26] defined a kind of radius of a bounded region, and derived an upper bound for this radius for a region in spacetime without apparent horizons in terms of the lower bound of mass density. By this and the study of the obstruction to the existence of regular solution to the Jang equation, they obtained some important results for the existence of black holes in the spirit of the hoop conjecture. Research Partially Supported by 973 Program (2006CB805905) and Fok YingTong Education Foundation. Research partially supported by Earmarked Grant of Hong Kong #CUHK403005.
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In the time symmetric case, the method does not work because the Jang equation always has a solution. For this case, we will use the inverse mean curvature flow of Huisken and Ilmanen [14]. We will define a quantity m() on a compact three manifold with boundary which involves Hawking mass of some subsets of and some other geometric quantities of , see the definition in (2.19). Then we compare this with the Brown-York mass m BY (∂) of ∂ of a simply connected compact manifold with nonnegative scalar curvature and with connected smooth boundary which has positive Gauss curvature and positive mean curvature with respect to the outward normal. It is well-known that ∂ can be isometrically embedded in R3 . The first result is as follows: If contains no horizons, then m BY (∂) ≥ m(). From this one can prove that if m() ≥ 2R, where R is the radius of the smallest circumscribed ball of ∂ in R3 , then must contain a horizon. In particular, if m() ≥ 2 diam(∂), then contains a horizon. Examples satisfying these conditions are given. We will prove that if contains a round sphere ∂ E in the sense of [8] such that its Hawking mass is larger than m BY (∂), then also contains a horizon. However, it is unclear if one can find a round sphere satisfying the condition. There is another important definition of quasi-local mass m B () (see the definition in §4) introduced by Bartnik for a compact manifold (, g) with smooth boundary which has an admissible extension. Since m B () is defined to be the infimum of ADM mass of all admissible extensions of , it is an interesting question to find a lower bound for Bartnik’s quasi-local mass. It turns out that m B () is also bounded below by m(). On the other hand, it was observed by Walter Simon (see [3]) that if (, g) has an admissible extension and if (, g) is isometrically embedded in an asymptotically flat and complete manifold M with nonnegative scalar curvature so that the ADM mass m AD M (M) of M is less than m B (), then M must contain a horizon. Our lower bound for m B () implies that if m AD M (M) < m(), then M must contain a horizon. We should emphasis that the quantity m() defined in (2.19) is nontrivial, in the sense that m() ≥ 0. It is zero only if it is locally flat and is actually a domain in R3 if it is simply connected. The basic outline of the paper is as follows. In Sect. 2, we first construct examples to motivate the definition of m(). Then we prove the positivity of m() (see Theorem 2.5). In Sect. 3, we compare m(), m BY () to give sufficient conditions for the existence of horizons for compact manifold with boundary as mentioned above. We will give another lower bound for the Brown-York mass. In particular, we prove that the Hawking mass of the boundary is dominated by the Brown-York mass of the domain. In Sect. 4, we prove that m() is bounded above by the Bartnik mass m B () for which has an admissible extension. We will also discuss some properties of Bartnik quasi-local mass including the conjecture of Bartnik that m B () is realized by a static admissible extension, see [3] for more details of the conjecture. The main analytical tool in this paper is the inverse mean curvature flow which has been studied by Huisken and Ilmanen [14]. We obtain our results by studying the obstruction for the monotonicity of Hawking mass under this flow. 2. Hawking Mass of Subsets of a Domain In this section, we will introduce a quantity involving Hawking mass of some subsets of a compact three manifold (, g) with smooth boundary which will be used to give a condition for the existence of stable minimal spheres on a compact manifold with bound-
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ary. All Riemannian manifolds in this work are assumed to be oriented and connected with dimension three. To motivate the definition, let us construct some examples. Since they are rather technical, readers already familiar with the examples may skip ahead to the definition of m() and the discussion of Theorem 2.5. Proposition 2.1. There exist asymptotically flat metrics g1 , g2 with nonnegative scalar curvature on R3 such that g1 = g2 outside some compact set, (R3 , g1 ) contains a stable minimal sphere but (R3 , g2 ) does not contain any compact minimal surfaces. Remark 2.2. Examples in the proposition can also be constructed using the methods in [9] and the results in [5]. Recall that an asymptotically flat (AF) three manifold (M, g) with one end is a complete manifold with nonnegative scalar curvature which is in L 1 (M) such that for some compact set K of M, M \ K is diffeomorphic to R3 \ B R (0) for some R > 0 and in the standard coordinates in R3 , the metric g satisfies:
with
gi j = δi j + bi j
(2.1)
|bi j | + r |∂bi j | + r 2 |∂∂bi j | = O(r −1 ),
(2.2)
where r and ∂ denote the Euclidean distance and standard derivative operator on R3 . Recall also that the ADM mass of M is defined as 1 m AD M (M) = lim (gii, j − gi j,i )ν j dσr , (2.3) r →∞ 16π S r where Sr is the Euclidean sphere, ν is the outward unit normal of Sr in R3 and the derivatives are taken with respect to the Euclidean metric. Proof of Proposition 2.1. We first construct metric g2 which contains no compact minimal surface such that g2 is Schwarzschild near infinity. Let m > 0 be any positive constant, ρ0 > m, and let ρ1 > ρ0 . Fix a smooth nonincreasing function h on (0, ∞) m such that h(r ) = 0 for any r ∈ (0, ρ0 ), h(r ) = − m2 for any r ∈ (ρ1 , ∞). Let c0 = 1+ 2ρ . 1 3 Define a function u on R as: r h(τ ) dτ (2.4) u(x) = c0 + 2 ρ1 τ for |x| = r . If |x| = r ≤ ρ0 , then
u(x) = c0 −
ρ0
because h ≤ 0. If |x| = r ∈ (ρ0 , ρ1 ), then m − u(x) = 1 + 2ρ1 If |x| = r ≥ ρ1 , then u(x) = 1 +
u =
m 2r .
ρ1
r
h(τ ) dτ > 0 τ2 ρ1
h(τ ) dτ > 0. τ2
Hence u is smooth and positive. Moreover,
∂ 2 u 2 ∂u h + ≤ 0. = ∂r 2 r ∂r r2
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Let g2 = u 4 g0 , where g0 is the Euclidean metric. Then (R3 , g2 ) is an AF manifold which is Schwarzchild near infinity. To prove that (R3 , g2 ) contains no compact minimal surfaces, it is sufficient to show that each Euclidean sphere with center at the origin has positive mean curvature in g2 . Let H be the mean curvature of the Euclidean sphere Sr = {|x| = r } with respect to g2 . Then 1 2 4 ∂u ∂u 2 H= 2 + = 3 u + 2r . (2.5) u r u ∂r ru ∂r If r ≤ ρ0 , then u = then
∂u ∂r
= 0. Since that u > 0 is positive, we have H > 0. If r ≥ ρ0 ,
r m h(τ ) 2h(r ) 2r u + u = +1+ + dτ 2 r 2ρ1 ρ1 τ m 2h(r ) 1 1 +1+ = + h(ξ ) − r 2ρ1 ρ r 1 m 2h(r ) 1 1 +1+ ≥ + h(r ) − r 2ρ1 ρ r 1 m 1 1 = 1+ + h(r ) + 2ρ1 ρ r 1 m m 1 1 = 1+ − + 2ρ1 2 ρ1 r m ≥ 1− 2ρ0 >0
(2.6)
for some ξ between r and ρ1 , here we have used the fact that h is nonincreasing, h ≥ − m2 and r ≥ ρ0 > m. Hence we also have H > 0. In [19], Miao has constructed a scalar flat AF metric g1 on R3 which contains a stable minimal sphere and is Schwarzschild at infinity. By rescaling, we see that the proposition is true. However, because of later application, we will construct g1 directly using similar method as in the construction of g2 . Basically, we glue the standard sphere to the Schwarzschild manifold. Under the stereographic projection, the metric on the sphere minus a point is: dsS22 =
1 (1 + 41 ρ 2 )2
(dρ 2 + ρ 2 dσ 2 ),
(2.7)
where dρ 2 +ρ 2 dσ 2 is the standard Euclidean metric. On the other hand the Schwarzschild metric is given by m 2 dsSch = (1 + )4 (dr 2 + r 2 dσ 2 ), (2.8) 2r defined on R3 minus the origin, where m > 0 is a constant. We need to rescale the metric so that the compact minimal surface {r = 1/2m} is near ρ = ∞ in the metric (2.7). Namely, let r = ρ, then 2 dsSch = 2 (1 +
m 4 ) (dρ 2 + ρ 2 dσ 2 ). 2 ρ
(2.9)
Quasi-Local Mass and the Existence of Horizons
For ρ0 > 0, define
k(ρ) =
281
− ρ4 (1 + 41 ρ 2 )− 2 , m ρ −2 , − 2√ 3
ρ ≤ ρ0 ; ρ ≥ 2ρ0 .
(2.10)
We want to find > 0 and ρ0 so that k can be defined to be nonincreasing. Let = m 2 /64, ρ0 = 1/(4m). Then k(ρ0 ) = −
32m 2 3
(1 + 64m 2 ) 2
,
k(2ρ0 ) = −64m 2 . Hence there exists m 0 > 0 such that k(ρ0 ) > k(2ρ0 ) for all 0 < m < m 0 . For such m we can define k satisfying (2.10), and is smooth and nonincreasing. Next define ρ u m (x) = b0 + k(τ )dτ (2.11) 0
√ if |x| = ρ, where b0 is chosen such that u m (x) = (1 + m/(2 ρ)) for ρ ≥ 2ρ0 . More precisely, 2ρ0 √ m + − k(τ )dτ b0 = √ 4 ρ0 0 2ρ0 65 8 =m +1− −√ k(τ )dτ. (2.12) 8 1 + 64m 2 ρ0 Note that
0≥
2ρ0 ρ0
k(τ )dτ ≥ −16m.
(2.13)
Hence
⎧
1 2ρ 65 ⎪ √ 8 m − ρ0 0 k(τ )dτ + (1 + 41 ρ 2 )− 2 , − ⎪ ⎪ 8 2 1+64m ⎪ ⎨
ρ u m (x) = m 65 − √ 8 + 1 − ρ0 k(τ )dτ, 8 2 ⎪ 1+64m ⎪ ⎪ ⎪ ⎩ √ (1 + m ), 2 ρ
ρ ≤ ρ0 ; ρ0 ≤ 2ρ0 ;
(2.14)
ρ ≥ 2ρ0 ,
for |x| = ρ. Hence choosing a smaller m 0 > 0, we have u m (x) > 0 for all x if 0 < m < m 0 . As before, since k < 0 and k ≤ 0, we have u m ≤ 0. The metric dsm2 = u 4m (dρ 2 + ρ 2 dσ 2 ) is an AF metric with nonnegative scalar curvature such that on ρ ≥ 2ρ0 , the metric is 2 (1 +
m 4 m ) (dρ 2 + ρ 2 dσ 2 ) = (1 + )4 (dr + r 2 dσ 2 ) 2 ρ 2r
which is Schwarzschild. Moreover, there is a minimal sphere at ρ=
32 m = > 2ρ0 . 2 m
Hence R3 with the metric dsm2 has a horizon and is Schwarzschild at infinity.
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From the proposition, in order to find a sufficient condition for the existence of compact minimal surfaces, we need to know information in the interior of the domain. This motivates us to introduce the following quantity using Hawking mass of some compact surfaces inside a domain. Let E be an open set in a Riemannian manifold with compact C 1 boundary, then one can define the mean curvature H of ∂ E in the weak sense, see [14]. Recall that the Hawking mass m H (∂ E) of ∂ E is defined as: m H (∂ E) =
1 |∂ E| 2 1− H , 16π 16π ∂ E
(2.15)
where |∂ E| is the area of ∂ E. In our convention, the standard unit sphere in R3 has mean curvature 2 with respect to the unit outward normal. Let us recall the idea of minimizing hull introduced in [14]. Let (, g) be a Riemannian manifold. Definition 2.3. Let E be a set in with locally finite perimeter. E is said to be a minimizing hull in if |∂ ∗ E ∩ K | ≤ |∂ ∗ F ∩ K | for any set F with locally finite perimeter such that F ⊃ E and F \ E ⊂⊂ and for any compact set K with F \ E ⊂ K ⊂ . Here ∂ ∗ E and ∂ ∗ F are the reduced boundaries of E and F respectively. E is said to be a strictly minimizing hull if equality implies E ∩ = F ∩ a.e. Suppose E is an open set of such that there is a strictly minimizing hull in containing E, then define E to be the intersection of all strictly minimizing hulls containing E. E is called the strictly minimizing hull of E. Let E ⊂⊂ be an open set. Suppose is compact with smooth boundary which has positive mean curvature with respect to the outward normal, then E exists and E ⊂⊂ . Let 1 ⊂⊂ 2 ⊂ such that 1 and 2 have smooth boundaries. We need the following lemma from [17, Lemma 1]. Lemma 2.4 [Meeks-Yau]. With the above notations and let d be the distance between 1 and ∂2 . Let ι be the infinmum of the injectivity radius of points in {x ∈ 2 | d(x, ∂2 ) > d 4 }. Let K > 0 be the upper bound of the curvature of 2 . Suppose N is a minimal surface and suppose x ∈ N is a point satisfying d(x, ∂2 ) = d2 and d(x, ∂ N ) ≥ d2 , then |N ∩ Bx (r )| ≥ 2π K −2
r
τ −1 (sin K τ )2 dτ,
(2.16)
0
where r = min{ d4 , ι}. The statement of original result in [17] is more simple, namely, it does not involve 1 . We formulate it in this way in order to define the following quantity α1 ;2 . For such 1 , 2 , let 2 α 1 ;2
= min
2π K −2
r 0
τ −1 (sin K τ )2 dτ ,1 . |∂1 |
(2.17)
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2 Some results in §3 may not be true if we simply take α to be 1 ;2
2π K −2
r 0
τ −1 (sin K τ )2 dτ |∂1 |
which may be larger than 1. Let F2 be the family of precompact connected minimizing hulls with C 2 boundary in 2 . Define sup m H (E). (2.18) m(1 ; 2 ) = E∈F2 ,E⊂1
Define m() = sup α1 ;2 m(1 ; 2 ),
(2.19)
where the supremum is taken over all 1 ⊂⊂ 2 ⊂ with smooth boundaries. Here and below i is always assumed to be nonempty. In general, the Hawking mass of a compact surface may be negative. However, one can prove that m() ≥ 0. Theorem 2.5. Let (, g) be a compact manifold with smooth boundary. Then F2 = ∅ for any 2 ⊂⊂ and m() ≥ 0. If can be embedded in R3 , then m() = 0. On the other hand, suppose has nonnegative scalar curvature and m() = 0, then is locally flat. In particular, if is simply connected, then is a domain in R3 . Proof. Since has smooth boundary, by taking a collar of ∂, we can embed in a compact manifold 3 with smooth boundary. Taking a double of 3 , we may assume that is isometrically embedded in a compact manifold 4 without boundary. Take a small geodesic disk in 4 \ and glue it to the exterior of some compact set of R3 , we may assume that is embedded in a complete noncompact manifold M with only one end and such that near infinity of M is isometric to the exterior of a compact set in R3 . Suppose 2 ⊂ . Take a point x0 ∈ 2 and let 3r > 0 be such that Bx0 (3r ) ⊂ 2 , 3r is less than the injectivity radius of x0 and ∂ Bx0 (ρ) has positive mean curvature for all 0 < ρ < 3r . We claim that there exists 0 < 0 < r such that Bx0 (ρ) is a strictly minimizing hull in 2 for all 0 < ρ < 0 . Let 0 < ρ < r and let F be the strictly minimizing hull of Bx0 (ρ) in M. F exists because M is Euclidean near infinity. Moreover, there is R > 3r independent of ρ such that F ⊂ B p (R), where p ∈ M is a fixed point. Suppose F ∩ (M\Bx0 (2r )) = ∅. Then there is a point x ∈ ∂ F and x ∈ Bx0 (R)\Bx0 (2r ). Moreover, Bx (r ) ∩ Bx0 (r ) = ∅. Since ∂ F \ Bx0 (r ) is minimal surface, see [14]. By [17, Lemma 1], we have |∂ F| ≥ c > 0 for some constant c > 0 depending only on the upper bound of the curvature and lower bound of the injectivity radius of Bx0 (2R) and is independent of ρ. Now choose r > 0 > 0 such that |∂ Bx0 (ρ)| < c for all 0 < ρ < 0 . Then for 0 < ρ < 0 , the strictly minimizing hull of Bx0 (ρ) must be a subset of Bx0 (3r ). However, since ∂ Bx0 (s) has positive mean curvature for all 0 < s < 3r , we conclude that F = ∂ Bx0 (ρ). This proves the claim. In particular, Bx0 (ρ) ∈ F2 for all 0 < ρ < 0 and so F2 = ∅. Let Hρ be the mean curvature of ∂ Bx0 (ρ), then it is easy to see that Hρ = ρ2 + O(1) as ρ → 0 and |∂ Bx0 (ρ)| = 1. ρ→0 4πρ 2 lim
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Hence we have lim ρ→0 m H (∂ Bx0 (ρ)) = 0. From this it is easy to see that m(1 ; 2 ) ≥ 0 for all 1 ⊂⊂ 2 ⊂ , and so m() ≥ 0. It is well known that m H () ≤ 0 for any compact surface in R3 , see [25]. Hence if is a domain in R3 , then m() = 0. To prove the last two assertions, we first observe that by the definition of m(), m( ) ≤ m() if ⊂ . Now suppose m() = 0. Let x0 ∈ and ρ > 0 such that Bx0 (ρ) ⊂ . Then 0 ≤ m(Bx0 (ρ)) ≤ m() = 0 and so m(Bx0 (ρ)) = 0. We want to prove is flat near x0 using the idea in [14]. Suppose it is not flat near x0 , let M be as above. Let ρ0 > 0 be such that (i) ρ0 < r , where 3r is the injectivity radius of x0 ; (ii) ∂ Bx0 (ρ) has positive mean curvature for all 0 < ρ < 3r ; (iii) for all 0 < ρ < ρ0 , m(Bx0 (ρ)) = 0 and Bx0 (ρ) is a strictly minimizing hull in M. Moreover by the proof above we may assume that for any open set E ⊂ Bx0 (ρ0 ) with C 2 boundary, then the strictly minimizing hull E of E in M satisfies E ⊂ Bx0 (ρ0 ). ρ Let ρ0 > ρ > 0 and let E t , 0 ≤ t < ∞ be the weak solution of inverse mean curvature flow with initial condition Bx0 (ρ) given by a locally Lipschitz proper function u ρ . Such a solution exists by [14]. Moreover, for x ∈ Bx0 (3r ), |∇u ρ (x)| ≤ C1 d −1 (x)
(2.20)
for some constant C1 independent of ρ and d(x) is the distance between x and x0 . ρi By [14, p.421-422], we can find ci → ∞, ρi → 0 such that Nt+c converges for a.e. i 1 t ∈ (−∞, ∞) in C to Nt which is a solution of inverse mean curvature flow in M\{x0 } ρi ρi given by a locally Lipschitz proper function u. Here Nt+c = ∂ E t+c = ∂{u ρi < t + ci } i i t and Nt = ∂ E t = ∂{u < t}. Moreover, Nt is nearly equal to ∂ Bx0 (e 2 ) as t → −∞. ρi Since Nt+ci is connected for all i, Nt is also connected. ρi Let b > −∞ be such that E t+c ⊂⊂ Bx0 (ρ0 ) for all t ≤ b and for all i. Then as i in [14, p.427], one can show that m H (Nt ) > 0 for all t < b. Let us outline the idea. ρi ρi Let t < b be such that Nt+c converges to Nt = ∂ E t in C 1 . Observe that m H (Nt+c )≥ i i m H (∂ Bx0 (ρi )) ≥ −δ0 , here δ0 is a constant independent of i by the monotonicity forρi mula [14, Theorem 5.8] because has nonnegative scalar curvature. Since |Nt+c | also i converges to |Nt |, we have H 2 ≤ C2 (2.21) ρ i Nt+c i
ρ
ρ
i converges to Nt in C 1 , the topology of Nt0i+ci for some C2 independent of i. Since Nt+c i is the same as that Nt0 provided i is large enough. By (2.21), [14, Lemma 5.5] and its proof, we have |A|2 ≤ C3 (2.22) ρ i Nt+c i
for some constant C3 for all i large enough, where A is the second fundamental form of ρi Nt+c . By lower semicontinuity i |A|2 < ∞. (2.23) Nt
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Hence the monotonicity formula [14, Theorem 5.8] for Nt holds. Note that by upper semicontinuity m H (Nt ) ≥ 0 for such t. Hence if for such a t, m H (Nt ) = 0, then one can prove that m H (Ns ) = 0 for a.e. s < t. Since there is no compact minimal surfaces in Bx0 (3r ), one can argument as in [14] that is flat near x0 . This is a contradiction. ρ Now choose t0 < b such that Nt0i+ci converge in C 1 to Nt0 . Note that even though m H (Nt0 ) > 0 and E t0 is a minimizing hull by [14], Nt0 may not have C 2 boundary. In order to show that m(Bx0 (ρ)) > 0 for ρ small, we need some approximation. As before, we have |A|2 < ∞. N t0
By (5.19) in [14], we can find an open set F ⊂ Bx0 (ρ0 ) with smooth boundary such that m H (∂ F) > 0, ∂ F is connected. Let F be the strictly minimizing hull of F. Then F ⊂ Bx0 (ρ0 ) and
m H (∂ F ) = ≥
1 |∂ F | 1− H2 16π 16π ∂ F |∂ F | m H (∂ F) |∂ F|
>0
(2.24)
because the mean curvature of ∂ F is zero on ∂ F \∂ F and the mean curvature of ∂ F is equal to the mean curvature of ∂ F a.e. on ∂ F ∩ ∂ F. Since Bx0 (3r ) is foliated by geodesic sphere with positive mean curvature, ∂ F cannot be a minimal surface. Since F is connected, F is connected. Since ∂ F is C 1,1 , by Lemma 5.6 in [14] we can find G ⊂ Bx0 (ρ0 ) with smooth boundary such that G is a strictly minimizing hull in M and m H (G) > 0. G can be chosen to be connected because F is connected. Therefore G ∈ F Bx0 (ρ0 ) , we have m(Bx0 (ρ0 )) > 0. This is a contradiction because m(Bx0 (ρ0 )) = 0. Hence is locally flat. If is simply connected, then is a domain in R3 , see [24, p.42-44]. 3. Sufficient Conditions on the Existence of Compact Minimal Surfaces Let (, g) be a compact orientable three manifold with smooth boundary which consists of finitely many components and with nonnegative scalar curvature. In this section, we always assume that the boundary of has positive Gauss curvature and positive mean curvature with respect to the outward normal. Hence each boundary component is diffeomorphic to a sphere. We want to obtain a sufficient condition on the existence of outermost horizon which is defined as in [4] as follows: Let 1 , . . . , k be the components of ∂. For a fixed i, by adding points to each component except i , we obtain a topological space i . Let Si be the collection of surfaces which are smooth boundaries of precompact open sets in i containing those points. The boundary ∂ E of a set E with ∂ E ∈ Si is called outer minimizing if |∂ E| ≤ |∂ F| for all F ⊃ E. A horizon (relative to i ) is a surface ∂ E ∈ Si with zero mean curvature. A horizon is said to be outermost if it is not enclosed by another horizon, see [4, p.185]. Suppose ∂ E is a horizon. Let E
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be its strictly minimizing hull in i , which exists because the mean curvature of i is positive. Then ∂ E is also a horizon. Let (, g) and i as before. By Weyl embedding theorem, see [21], each i can be isometrically embedded in R3 and the embedding is unique up to an isometry of R3 . Let H be the mean curvature of i with respect to the outward normal and let H0 be the mean curvature of i when embedded in R3 . The Brown-York mass [6, 7] of i is defined to be 1 m BY (i ) = (H0 − H )dσ, (3.1) 8π i where dσ is the volume element on i induced by g. In this section we will give some lower bounds of the Brown-York mass in terms of the Hawking mass of subsets of . We will also give a sufficient condition of existence of horizons in using the Brown-York mass and m() defined in the previous section. The first lower bound for the Brown-York mass is the following: Theorem 3.1. Let (, g) be a compact three manifold with connected smooth boundary and with nonnegative scalar curvature. Assume that is simply connected and suppose ∂ has positive Gauss curvature and positive mean curvature with respect to the outward normal. Then the following are true: (a) m BY (∂) ≥ m H (∂ E)
(3.2)
for any connected minimizing hull E in where E ⊂⊂ with C 1,1 boundary. Moreover, equality holds for some minimizing hull E with the above properties if and if is a domain in R3 and E is a standard ball in . (b) m BY (∂) ≥ m H (∂) and equality holds if and only if is a standard ball in R3 . Proof. To prove (a), let E be a minimizing hull in with C 1,1 boundary such that E ⊂⊂ . We want to prove that m BY (∂) ≥ m H (∂ E). Since m BY (∂) ≥ 0 by the result of the authors [23], it is sufficient to prove the case when m H (∂ E) > 0. Isometrically embed = ∂ in R3 . As in [23], one can glue to the exterior of ∂ in R3 to form a manifold (M, h) such that the metric h satisfies: h| = g; h M\ is smooth up the boundary; the scalar curvature of h in M \ is zero; h is Lipshitz near ∂; the mean curvatures of ∂ with respect to the outward normal are the same for metrics inside and outside ; (vi) h is asymptotically flat; (vii) if r is the surface consisting of points in the exterior to in R3 and with Euclidean distance r from , then r has positive mean curvature with respect to h, which is bounded from below by C/(1 + r ) for some C > 0. (i) (ii) (iii) (iv) (v)
Given > 0, by (i)-(vi) and [18], there is a smooth AF metric h on M with nonnegative scalar curvature such that (1 − )h (v, v) ≤ h(v, v) ≤ (1 + )h (v, v)
(3.3)
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for all tangent vector v on M, and such that lim m AD M (h ) = m AD M (h),
→0
(3.4)
where m AD M (h ) and m AD M (h) are the ADM masses of (M, h ) and (M, h) respectively. Let θ > 0 be given. We can find a connected open set F ⊃ E with smooth boundary such that F ⊂⊂ , |∂ E| − θ ≤ |∂ F| ≤ |∂ E| + θ ; m H (∂ E) ≤ m H (∂ F) + θ ; and m H (∂ F) > 0.
(3.5)
For any > 0, let F be the strictly minimizing hull of F in (M, h ). F exists, precompact in M and has C 1,1 boundary. Since F is connected, F is connected. Moreover M is simply connected because is simply connected and ∂ is homeomorphic to the sphere by the Seifertvan Kampen Theorem. Hence ∂ F is connected, see [13, p. 107]. Choose δ > 0 small enough, such that the boundary of t for all 0 < t ≤ δ has positive mean curvature in g, where t = {x ∈ | dg (x, ∂) > t}. Here dg is the distance with respect to g. There is 0 < t0 < δ such that F ⊂⊂ t0 . Suppose F \t0 = ∅. Since M \δ is foliated by compact surfaces of positive mean curvature with respect to h, if we denote the unit outward normal vector in h of the surfaces by ν, we have 0< =
F \t0
div h νd Vh
∂ F ∩(M\t0 )
h(ν, µ) − |F ∩ ∂t0 |g ,
(3.6)
where µ is the unit outward normal of ∂ F in h. Here div h is the divergence with respect to h. Hence we have
|∂ F ∩ (M \t0 )|h > |F ∩ ∂t0 |g .
(3.7)
Now F ∩ t0 ⊃ E is a set of finite perimeter and E is a minimizing hull in (, g), we have |∂(F ∩ t0 )|g ≥ |∂ E|g .
(3.8)
From (3.7) and (3.8) we conclude that |∂ F |h = |∂ F ∩ (M \t0 )|h + |∂ F ∩ t0 |h
≥ |F ∩ ∂t0 |g + |∂ F ∩ t0 |g = |∂(F ∩ t0 )|g ≥ |∂ E|g .
(3.9)
¯ t0 , then (3.9) is still true by the definition of E. If F ⊂ Moreover the mean curvature of ∂ F in h is zero on ∂ F \∂ F and the mean curvature of ∂ F and ∂ F are equal a.e. in ∂ F ∩ ∂ F. Using the inverse mean curvature flow with
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initial data F , we obtain by the Penrose inequality in [14] 1 |∂ F |h 1− m AD M (h ) ≥ H 2 dσh 16π 16π ∂ F |∂ F|g 1 |∂ F |h 1− · H 2 dσh ≥ |∂ F|g 16π 16π ∂ F |∂ F |h = m H (∂ F) |∂ F|g |∂ F |h (m H (∂ E) − θ ) ≥ |∂ F|g |∂ F|h |∂ F |h m H (∂ E) − θ ≥ (1 − ) |∂ F|g |∂ F|g 1 |∂ E|g ≥ (1 − ) m H (∂ E) − θ (1 + ) 2 |∂ F|g 1 |∂ E|g ≥ (1 − ) m H (∂ E) − θ (1 + ) 2 . |∂ E|g + θ
(3.10)
Here, we have used the assumption m H (∂ E) ≥ 0 in the fifth inequality above, and dσh and dσg are the area elements on surfaces induced by h and g respectively and we have used (3.5), and (3.9). Let → 0, and then let θ → 0, we have m AD M (h) ≥ m H (∂ E). By [23], we have m BY (∂) ≥ m AD M (h) ≥ m H (∂ E). (3.11) If m BY (∂) = m H (∂ E) for some E with the properties in the theorem, then m BY (∂) = m AD M (h) by (3.11). Hence is a domain in R3 by [23] and m H (∂ E) = m BY (∂) = 0. By a result of Willmore [25, p.109], is a standard ball in R3 . Since m H (∂ E) = 0, E is also a standard ball in R3 . Conversely, if is a domain in R3 , since each sphere has zero Hawking mass, and a ball which is small enough is a minimizing hull, we have m BY () = m H (∂ E) = 0 for some small ball. To prove (b), we observe that t is a minimizing hull in if t > 0 is small. Here t = {x ∈ | d(x, ∂) > t}. Hence m BY (∂) ≥ limt→0 m H (∂t ) = m H (∂). In fact, from the proof above, m BY (∂) > limt→0 m H (∂t ) unless is a domain in R3 . Hence by [25, p.109] again, m BY (∂) = m H (∂) implies that ∂ is a standard sphere so is a standard ball in R3 . In the above theorem, we do not assume that (, g) contains no horizons. In order to obtain a sufficient condition that (, g) contains a horizon, we need another estimate of the Brown-York mass. Let m() be as defined in (2.19). We have the following: Theorem 3.2. Let (, g) be a compact manifold with connected smooth boundary and with nonnegative scalar curvature. Assume that is simply connected and suppose ∂
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has positive Gauss curvature and positive mean curvature with respect to the outward normal. Suppose (, g) has no horizons. Then m() ≤ m BY (∂). Equality holds if and only if is a domain in R3 . Remark 3.3. If ∂ has more than one component, then it is easy to see that will have an outermost minimizing horizon with respect to any boundary component because ∂ has positive mean curvature. If ∂ is connected and is not simply connected, then contains a minimal sphere or real projective space by [16] and [11, Lemma 2]. Proof of Theorem 3.2. Assume that has no horizons. Suppose is not diffeomorphic to an open 3-ball in R3 , then must contain a minimal sphere S or projective plane [16] and [11, Lemma 2]. Since is simply connected, S must be a minimal sphere and is the boundary of some precompact open set E, see [13, p.107]. So S is a horizon. This is a contradiction. Hence is diffeomorphic to an open 3-ball in R3 . Let 1 ⊂⊂ 2 ⊂ with smooth boundaries. Let E be a connected precompact minimizing hull in 2 with C 2 boundary such that E ⊂ 1 . We want to prove that m BY (∂) ≥ α1 ;2 m H (∂ E). Again, we may assume that m H (∂ E) ≥ 0. Let E be the strictly minimizing hull of E in . E exists because ∂ has positive mean curvature. Moreover, E has C 1,1 boundary because E has C 2 boundary. Note that E and ∂ E are connected because is diffeomorphic to the unit ball in R3 . By Theorem 3.1, m BY (∂) ≥ m H (∂ E ) 1 |∂ E | = 1− H2 16π 16π ∂ E
(3.12)
because ∂ E \∂ E is minimal and the mean curvatures of ∂ E and ∂ E are equal a.e. on their common part. Suppose E ⊂⊂ 2 , then |∂ E | ≥ |∂ E| and we have m BY (∂) ≥ m H (∂ E). Suppose E is not a precompact set in 2 , then ∂ E ∩ (M \2 ) = ∅. On the other hand, if ∂ E ∩ ∂ E = ∅ then ∂ E would be a horizon contradicting the assumption that contains no horizons. Hence ∂ E ∩ 1 ⊃ ∂ E ∩ ∂ E = ∅ and there is a point x ∈ ∂ E and d(x, ∂2 ) = and ∂2 . By Lemma 2.4, we have
d 2,
where d is the distance between 1
2 2 |∂ E | ≥ α |∂1 | ≥ α |∂ E| 1 ;2 1 ;2
(3.13)
because E is a minimizing hull in 2 . Hence by (3.12), (3.13) we have m BY (∂) ≥ α1 ;2 m H (∂ E).
(3.14)
This completes the proof of the first part. If equality holds, then from the proof above, for any > 0 we can find a strictly minimizing hull E in with C 1,1 boundary such that m BY (∂) ≤ m H (∂ E ) + . From the proof of Theorem 3.1, one can conclude that must be a domain in R3 . Let be as in the theorem. Isometrically embed ∂ in R3 . Let R be the radius of the smallest circumscribed ball of ∂ in R3 . We have the following:
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Corollary 3.4. Let (, g) be a compact manifold with nonnegative scalar curvature and with connected boundary which has positive mean curvature and positive Gauss curvature. Suppose is simply connected and suppose m() ≥ m BY (∂), then there is a horizon in unless is a domain in R3 . Hence if m() ≥ 2R, then contains a horizon. In particular, if m() ≥ 2diam(∂) then contains a horizon. Here diam(∂) is the diameter of ∂ with respect to the metric induced by g. Proof. The first statement is just a restatement of Theorem 3.2. Since m BY () ≤
1 8π
∂
H0 ≤ 2R < 2diam(∂)
(3.15)
by the Minkowski integral formula [15, p.136], the fact that ∂ has positive Gauss curvature and the Gauss-Bonnet formula. There are examples so that m() > m BY (). Consider the metrics (R3 , dsm2 = + ρ 2 dσ 2 )) defined in the proof of Proposition 2.1, where dρ 2 + ρ 2 dσ 2 is the Euclidean metric. Since as m → 0, dsm2 converges uniformly on the Euclidean ball B4 = {x ∈ R3 | |x| < 4} to the standard metric on the unit sphere, we see that the mean 4 2 curvature of ∂ Bτ is equal to u −2 m (log u m ρ ) |ρ=τ , and by a direct computation we know that it is positive for sufficiently small m > 0 and 0 < τ < 2. Hence, the boundary of Bτ is mean convex for all 0 < τ < 2 with respect to dsm2 for m sufficiently small. B1 is a minimizing hull in B2 . Moreover, there is δ > 0 such that the Hawking mass m H (∂ B1 ) of ∂ B1 with respect to dsm2 is at least δ, provided m > 0 is small enough. Hence for all m > 0 small enough, α1 ;2 m(B1 ; B2 ) ≥ δ. Now consider the domain = Bρ , where ρ = 8/m > 2ρ0 . It is easy to see that ∂ has positive mean curvature and positive Gauss curvature with diameter d ≤ Cm, here C is a universal constant. Hence u 4m (dρ 2
m() ≥ α1 ;2 m(B1 ; B2 ) ≥ δ > 2d ≥ m BY (), provided m is small enough. There is another sufficient condition for the existence of horizons in terms of the Hawking mass of other kind of surfaces. Suppose E ⊂⊂ is bounded and connected with volume v0 . As in [8], ∂ E is called a round sphere if ∂ E is smooth and connected and has least area among all subsets of with locally finite perimeter with volume v0 . The volume of a set of locally finite perimeter is simply the measure of the set with respect to the volume element of . If the scalar curvature of is nonnegative, and ∂ E is a round sphere, then the Hawking mass m H (∂ E) of ∂ E is nonnegative by [8]. We have the following: Theorem 3.5. Let (, g) be a compact Riemannian three manifold with smooth and connected boundary and with nonnegative scalar curvature. Assume that is simply connected and suppose that both the Gauss curvature and the mean curvature of the boundary (with respect to outward normal) are positive. If there is a round sphere ∂ E in such that m H (∂ E) > m BY (∂), then there is a horizon in .
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First, we prove the following lemma: Lemma 3.6. Let (, g) be a compact Riemannian three manifold with smooth boundary and with nonnegative scalar curvature such that ∂ has positive mean curvature. Suppose E ⊂⊂ such that ∂ E is a round sphere. Then either contains a horizon or E is a minimizing hull in . Proof. Suppose E is not minimizing hull in . Since ∂ has positive mean curvature, there exists a strictly minimizing hull E containing E with E ⊂⊂ . Since E is not a minimizing hull, |∂ E | < |∂ E|. Let v0 be the volume of E, then vol(E ) > v0 because E is a round sphere. Choose 0 > 0 small enough so that ∂ is smooth and has positive mean curvature for all 0 < < 2 0 , where = {x ∈ | d(x, ∂) > }. Moreover, 0 is chosen so that E ⊂⊂ 0 . Let be the family of precompact subsets F with locally finite perimeter of 0 with the properties that vol(F) > v0 , and |∂ F| < 21 (|∂ E |+|∂ E|). is not empty because E ∈ . Let v1 = inf vol(F). F∈
(3.16)
Note that v1 ≥ v0 . We claim that v1 > v0 . In fact, let Fi ∈ be such that limi→∞ vol(Fi ) = v1 . By compactness theorem for sets of locally finite perimeter, we may assume that the characteristic functions of Fi converges in L 1 () to the characteristic function of a subset F of 0 with locally finite perimeter such that vol(F) = v1 . By lower semicontinuity, |∂ F| ≤ 21 (|∂ E | + |∂ E|) < |∂ E|. (For simplicity, we use |∂ F| to denote the area of the reduced boundary of F). Hence v1 = vol(F) > v0 by the definition of round spheres. Let a = inf |∂ F|. (3.17) F∈
Choose a sequence Fi ∈ such that limi→∞ |∂ Fi | = a. Since v1 > v0 and |∂ Fi | < 1 2 (|∂ E | + |∂ E|), by the approximation result of sets of locally finite perimeter, see [12], we may choose Fi such that ∂ Fi is smooth. Choose 2 0 > 1 > 0 such that vol( 0 \ 1 ) ≤ 21 (v1 − v0 ). Consider the set F˜i = Fi ∩ 1 , then |∂ F˜i | ≤ |∂ Fi | < 1 2 (|∂ E | + |∂ E|) since ∂ has positive mean curvature for all 0 < < 2 0 . Moreover, 1 vol( F˜i ) ≥ vol(Fi ) − vol( 0 \ 1 ) ≥ (v1 + v0 ) > v0 (3.18) 2 by (3.16). Hence F˜i ∈ and limi→∞ |∂ F˜i | = a. Passing to a subsequence if necessary, we may assume that the characteristic functions of F˜i converge in L 1 () to the charac˜ ≥ teristic function of a subset F˜ of 1 with locally finite perimeter. Moreover, vol( F) 1 1 ˜ 2 (v1 + v0 ) > v0 by (3.18) and |∂ F| = a. Since E ∈ , a ≤ |∂ E | < 2 (|∂ E | + |∂ E|). Hence F˜ ∈ . Let Bx (r ) ⊂⊂ 0 with vol(Bx (r )) < 21 (v1 − v0 ). Suppose F is any set of locally finite perimeter such that the symmetric difference F F˜ ⊂⊂ Bx (r ), then F ⊂⊂ 0 , ˜ − vol(Bx (r )) ≥ vol(F) ≥ vol( F)
1 (v1 + v0 ) > v0 2
˜ then F ∈ because |∂ F| ˜ < 1 (|∂ E | + |∂ E|). This is imposby (3.16). If |∂ F| < |∂ F| 2 ˜ Hence one can see that F˜ is minimizing in any geodesic sible by the construction of F. ball inside 0 with radius small enough and F˜ is a compact minimal surface. That is to say contains a horizon.
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Proof of Theorem 3.5. Let ∂ E be a round sphere. E is not a minimizing hull in by Theorem 3.1 and the assumption that m H (∂ E) > m BY (∂). By Lemma 3.6, we see that contains a horizon. This completes the proof of the theorem. 4. On the Bartnik Quasi-Local Mass In this section, we will discuss the relation between m() and the quasi-local mass introduced by Bartnik [1]. We will also discuss some properties of the Bartnik quasi-local mass. We use the definition of the Bartnik quasi-local mass m B () in [2] of a compact manifold (, g) with smooth boundary and with nonnegative scalar curvature. Let PMo be the set of complete noncompact AF manifolds with nonnegative scalar curvature which admit no minimal two spheres or projective planes. For M ∈ PMo , ⊂⊂ M means that is isometrically embedded in M. Then the Bartnik quasi-local mass m B () of is defined as: m B () =
inf
M∈PMo
{m AD M (M)| ⊂⊂ M}.
(4.1)
If ⊂⊂ M ∈ PMo , then M is said to be an admissible extension of . We should remark that for M ∈ PMo , M is topologically R3 , by [16], see also [11]. We have the following lower bound of m B (). Theorem 4.1. Suppose has an admissible extension. Then m B () ≥ m().
(4.2)
Proof. Let (M, g) ∈ PMo with ⊂⊂ M. We want to prove that the ADM mass m g of M is no less than m(). Since M is AF, the coordinates sphere ∂ B(R) in a coordinates system in the definition of AF manifold has positive Gauss curvature and positive mean curvature for R large enough. Choose such an R so that ⊂ B(R). As in the proof of (3.11), if E ⊂⊂ B(R) is a minimizing hull in B(R) with C 1,1 boundary, then m g ≥ m H (∂ E).
(4.3)
Since B(R) is an open 3-ball in R3 , using (4.3) instead of Theorem 3.1 we may proceed as in the proof of (3.12)-(3.14) to conclude that m g ≥ m(B(R)). Since m() ≤ m(B(R)) by the definition of m(·), the result follows.
(4.4)
It was proved in [14] that if m B () = 0, then is locally flat. Hence it is a domain in R3 if ∂ is topologically a sphere. If the boundary of has several components, then one can show that if m B () = 0, then is still a domain in R3 provided that each component of ∂ is topologically a sphere. This follows from the lemma: Lemma 4.2. Let (, g) be a compact three manifold with boundary with nonnegative scalar curvature such that has an admissible extension. Suppose m B () = 0. Let 1 , . . . , k be the boundary components of . Then can be extended to a manifold with compact closure with nonnegative scalar curvature such that has only one boundary component which is equal to i for some i and m B () = 0.
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Proof. Let Ml be a minimizing sequence of admissible extensions of for m B (). Then each Ml is diffeomorphic to R3 . Each j separates R3 into two components. Hence by taking a subsequence if necessary, we may assume that is in the interior of 1 , say, be the interior of 1 in M1 . Then ⊂ , the scalar curvature of is for all l. Let = 1 . We want to prove that m B ( ) = 0. nonnegative, and ∂ l be the manifold obtained in the following way: M l = Ml on Ml \interior of Let M l . in the interior of 1 , so that in a neighborhood of 1 , Ml = M 1 and replace by We want to prove that if l is large, Ml will not contain any stable minimal spheres. Choose a > 0 be small enough, such that {x ∈ | d(x, 1 ) = } is also topologically a sphere and so that if U = {x ∈ | d(x, 1 ) < }, then U ∩ j = ∅ for all l . If N ∩ ( \U ) = ∅, then N ⊂ Ml . This is j = 1. Let N be an outermost horizon of M impossible by the fact that Ml is admissible and the fact that each component of N is a \U ) = ∅ and there is a constant c > 0 stable minimal sphere, see [14, 4]. Hence N ∩ ( independent of i, such that |N | ≥ c > 0 by [17, Lemma 1]. On the other hand, l ) = lim m AD M (Ml ) = 0. lim m AD M ( M
l→∞
l→∞
Hence by the Penrose inequality in [14, 4], no such N exists if l is large enough. l is homeomorphic to R3 . Hence M l contains no minimal projective plane. Note that M l contains a minimal sphere S, then a strictly minimizing hull of the interior of S If M l is an admissible is an outermost horizon. Hence no such S exists. We conclude that M and so m B ( ) = 0. extension of From this, we obtain the following: Proposition 4.3. Let (, g) be as in the lemma. Suppose m B () = 0 and each i is topologically a sphere, then is a domain in R3 . is diffeomorphic to the unit as in Lemma 4.2. Since each i is a sphere, Proof. Let ) = 0. By Theorem 2.5, and hence ball in R3 . By Lemma 4.2 and Theorem 4.1, m( 3 is a domain in R . In case the boundary of ∂ is not a sphere, it is still unknown if m B () = 0 will imply is a domain in R3 . In the following, using Theorem 4.1 we are going to construct a locally flat compact manifold which either has no admissible extension or its Bartnik mass is positive. Example 4.4. Consider the cylinder in R3 : C = {(x, y, z) ∈ R3 |x 2 + y 2 ≤ r 2 , 0 ≤ z ≤ l}. Let T be obtained by identifying (x, y, 0) and (x, y, l) in C. If l is small enough depending only on r , then the Hawking mass of T is positive. Moreover, if T has an admissible extension, then the Bartnik mass is also positive. To prove that m H (T ) > 0, it is easy to see that the mean curvature H of ∂C is r1 , and 2πl . H2 = r ∂C Suppose l < 8r , hence
∂C
We have m H (T ) > 0.
H 2 < 16π.
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Now suppose that T has an admissible extension; we may conclude that m B (T ) > 0 because of the following consequence of Theorem 4.1: Proposition 4.5. Suppose (, g) has an admissible extension so that ∂ is connected. Suppose ∂ has positive mean curvature and suppose m H (∂) > 0. Then m B () > 0. Proof. Choose 0 > 0 be small enough such that ∂ is smooth with positive mean curvature for all 0 < < 0 , where = {x ∈ | d(x, ∂) > }. Moreover, by choosing 0 small enough, we may also assume that m H (∂ ) > 0 for all 0 < < 0 . Then each is a minimizing hull in because \ 0 is foliated by positive mean curvature surfaces. Hence m( ; ) > 0 provided > 0 is small enough. In particular, m() > 0. By Theorem 4.1, the proposition follows. Even though it is unclear if T in Example 4.4 has an admissible extension, one can prove that T has no smooth static extension. Namely, there is no AF manifold (M, g) with one end satisfying the following: (a) g is smooth. (b) T is isometrically embedded in M. (c) There is a smooth function u defined on M\interior of T such that u → 1 at infinity, Ric = u −1 ∇ 2 u and u = 0. In fact, if (M, g) is such an extension, then by [20], u = 1 − mr + O(r −2 ), where r is the Euclidean distance in a coordinate system near infinity in the definition of the AF manifold and m is the ADM mass of M. Moreover, let = ∂ T , then on (see [20]):
u + H
∂u = 0, ∂ν
(4.5)
where is the Laplacian on , ν is the unit outward normal and H > 0 is the mean curvature of . Here we have used the fact that is flat, H is constant and T is locally flat. Hence we have 0=
u M\T ∂u = 4π m − ∂ν
u = 4π m + H −1 = 4π m.
Hence M must be R3 . This is impossible because the boundary of T is a flat torus which cannot be isometrically embedded in R3 . Acknowledgements. The authors would like to thank Robert Bartnik and Piotr T. Chrusciel for their interest in the work. They would also like to thank the referees for their useful comments.
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References 1. Bartnik, R.: New definition of quasilocal mass. Phys. Rev. Lett. 62, 2346–2348 (1989) 2. Bartnik, R.: Energy in general relativity. Tsing Hua lectures on geometry and analysis (Hsinchu, 1990– 1991), Cambridge, MA: Internat. Press 1997, pp. 5–27 3. Bartnik, R.: Mass and 3-metrics of non-negative scalar curvature. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Beijing: Higher Ed. Press 2002, pp. 231–240 4. Bray, H.L.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59, 177–267 (2001) 5. Beig, R.Ó., Murchadha, N.: Trapped surfaces due to concentration of gravitational radiation. Phys. Rev. Lett. 66, 2421–2424 (1991) 6. Brown, J.D., York, J.W.: Quasilocal energy in general relativity in Mathematical aspects of classical field theory (Seattle, WA, 1991), Contemp. Math. 132, Providence, RI: Amer. Math. Soc., 1992, pp. 129–142 7. Brown, J.D., York, J.W.: Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D (3), 47(4), 1407–1419 (1993) 8. Christodoulou, D., Yau, S.T.: Some remarks on quasi-local mass. In: Contemporary Mathematics 71, Mathematics and General Relativity, J. Isenberg, ed. Providence RI: Amer. Math. Soc., 1988, pp. 9–14 9. Corvino, J., Schoen, R.M.: On the asymptotics for the vacuum Einstein constraint equations. J. Differ. Geom. 73, 185–217 (2006) 10. Flanagan, E.: Hoop conjecture for black-hole horizon formation. Phys. Rev. D 44, 2409 (1991) 11. Galloway, G.J.: On the topology of black holes. Commun. Math. Phys. 151, 53–66 (1993) 12. Giusti, E.: Minimal surfaces and functions of bounded variation. Notes on pure mathematics 10, Canberra: Department of Pure Mathematics, 1977 13. Hirsch, M.W.: Differential Topology. New York: Springer-Verlag, 1976 14. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001) 15. Klingenberg, W.: A course in differential geometry. Graduate Texts in Mathematics, Vol. 51, New YorkHeidelberg: Springer-Verlag, 1978 16. Meeks, W.H., Simon, L., Yau, S.-T.: Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature. Ann. of Math. 116, 621–659 (1982) 17. Meeks, W.H., Yau, S.-T.: Topology of three-dimensional manifolds and the embedding problems in minimal surface theory. Ann. of Math. 112, 441–484 (1980) 18. Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6(6), 1163–1182 (2002) 19. Miao, P.: Asymptotically flat and scalar flat metrics on R3 admitting a horizon. Proc. Amer. Math. Soc. 132, 217–222 (electronic) (2004) 20. Miao, P.: A remark on boundary effects in static vacuum initial data sets. Classical Quantum Gravity 22(11), L53–L59 (2005) 21. Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6, 337–394 (1953) 22. Schoen, R., Yau, S.-T.: The existence of a black hole due to condensation of matter. Commun. Math. Phys. 90, 575–579 (1993) 23. Shi, Y.-G., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62, 79–125 (2002) 24. Spivak, M.: A Comprehensive introduction to differential geometry. v. 3, Berkeley, CA: Publish or Perish, 1970–75 25. Willmore, T.J.: Total curvature in Riemannian geometry. Chichester: E. Horwood, 1982 26. Yau, S.-T.: Geometry of three manifolds and existence of black hole due to boundary effect. Adv. Theor. Math. Phys. 5(2001) Communicated by G.W. Gibbons
Commun. Math. Phys. 274, 297–341 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0278-3
Communications in
Mathematical Physics
Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket K. Bering Institute for Theoretical Physics & Astrophysics, Masaryk University, Kotláˇrská 2, CZ-611 37 Brno, Czech Republic. E-mail:
[email protected] Received: 4 April 2006 / Accepted: 8 February 2007 Published online: 23 June 2007 – © Springer-Verlag 2007
Abstract: We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the “big bracket” of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2. Homotopy Lie Algebras . . . . . . . . . . . . . . . . 2.1 Sign conventions . . . . . . . . . . . . . . . . . 2.2 Connection to Lie algebras . . . . . . . . . . . . 2.3 Rescaling . . . . . . . . . . . . . . . . . . . . . 2.4 Polarization . . . . . . . . . . . . . . . . . . . . 3. The Koszul Bracket Hierarchy . . . . . . . . . . . . . 3.1 Basic settings . . . . . . . . . . . . . . . . . . . 3.2 Review of the commutative case . . . . . . . . . 3.3 The intermediate case: Im() ⊆ Z (A) . . . . . 3.4 Nilpotency relations . . . . . . . . . . . . . . . . 3.5 Off-shell with respect to the nilpotency condition 3.6 The general non-commutative case . . . . . . . .
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4. The Derived Bracket Hierarchy . . . . . . 4.1 Definitions . . . . . . . . . . . . . . . 4.2 Nilpotency versus square relations . . 4.3 Solution . . . . . . . . . . . . . . . . 4.4 Ward-like and Jacobi-like identities . . 5. The Courant Bracket . . . . . . . . . . . . 5.1 Review of the operator representation 5.2 Symplectic structure . . . . . . . . . . 5.3 Anti-symplectic structure . . . . . . . 5.4 Derived brackets . . . . . . . . . . . . 5.5 Discussion . . . . . . . . . . . . . . . 5.6 Higher brackets . . . . . . . . . . . . 5.7 B-transforms . . . . . . . . . . . . . 6. Supplementary Formalism . . . . . . . . . 6.1 Pre-Lie products . . . . . . . . . . . . 6.2 A co-product . . . . . . . . . . . . . 6.3 A lifting . . . . . . . . . . . . . . . . 6.4 Normalization . . . . . . . . . . . . . 6.5 The Ward solution revisited . . . . . . 7. Conclusions . . . . . . . . . . . . . . . .
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1. Introduction It is well-known [34] that in general the symmetrized, multiple nested Lie brackets [[. . . , [[Q, a1 ], a2 ], . . .], an ],
(1.1)
where Q is a fixed nilpotent Lie algebra element [Q, Q] = 0, do not obey the original homotopy Lie algebra definition of Lada and Stasheff [22]. Several papers have been devoted to tackle this in special situations. For instance, Voronov considers the projection of the above nested, so-called derived brackets (1.1) into an Abelian subalgebra [34, 17, 35, 2]. In this paper we stay in the non-Abelian setting and observe that although a multiple nested bracket (1.1) does not obey the generalized Jacobi identities of Lada and Stasheff [22], it is – after all – very close. It turns out that one may organize the nested Lie brackets (1.1) in such a way that all the terms in the generalized Jacobi identities of Lada and Stasheff appear, but as a caveat, with different numerical prefactors related to the Bernoulli numbers. The paper is organized as follows. In Sect. 2 we widen the definition of a homotopy Lie algebra by basing it on a generalized bracket product to allow for more general prefactors. In Sect. 3 we consider the Koszul bracket hierarchy [21, 1, 4, 11], and solve a long-standing problem of providing an ordering prescription for a construction of higher Koszul brackets for a non-commutative algebra A, in such a way that the higher brackets form a homotopy Lie algebra. It turns out that in the non-commutative Koszul construction, the emerging homotopy Lie algebra is of the original type considered by Lada and Stasheff [22], cf. Theorem 3.5. On the other hand, the new types of (generalized) homotopy Lie algebras with non-trivial prefactors will be essential for the derived bracket hierarchies (1.1) studied in Sect. 4. Section 5 is devoted to the Courant bracket [13], which is a two-bracket defined on a direct sum of the tangent and the cotangent bundle
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T M ⊕ T ∗ M over a manifold M,
1 [X ⊕ ξ, Y ⊕ η] H = [X, Y ] ⊕ L X η − LY ξ + d i Y ξ − i X η + i X i Y H , 2
(1.2)
where H is a closed “twisting” three-form. This bracket has many interesting applications, for instance Hitchin’s generalized complex geometry [15, 16]. In hindsight, the importance of the Courant bracket can be traced to the fact that it belongs to a derived homotopy Lie algebra [28–30] related to the exterior de Rham complex. Section 6 contains further theoretical aspects of homotopy Lie algebras. The pre-Lie property of a bracket product is investigated in Subsect. 6.1, and the co-algebraic structures are studied in Subsect. 6.2-6.3. Finally, Sect. 7 has our conclusions. 2. Homotopy Lie Algebras Let Sym• A := T • A/I denote a graded 1 symmetric tensor algebra over a graded vector • space A, where I ⊆ T A ≡ n≥0 T n A is the two-sided ideal generated by the set (2.1) b⊗a − (−1)(a +)(b +) a⊗b a, b ∈ A ⊆ T 2 A ≡ A⊗A. Here ∈ {0, 1} modulo 2 is a fixed “suspension parity”. We let the symbols “⊗” and “” denote the un-symmetrized and the symmetrized tensor product in the tensor algebras T • A and Sym• A, respectively. In practice we shall focus on the symmetric tensor product “”, and the only important thing is, that two arbitrary vectors a, b ∈ A, with Grassmann parities a and b , commute in Sym2 A up to the following sign convention: ba = (−1)(a +)(b +) ab ∈ Sym2 A.
(2.2)
A •-bracket : Sym• A → A is a collection of multi-linear n-brackets n : Symn A → A, where n ∈ {0, 1, 2, . . .} runs over the non-negative integers. In addition a •-bracket carries an intrinsic Grassmann parity ∈ {0, 1}. Detailed explanations of sign conventions are relegated to Subsect. 2.1. We now introduce a •-bracket product denoted with a “◦”. Definition 2.1. Let there be given a set of complex numbers ckn , where n ≥ k ≥ 0. The “◦” product ◦ : Sym• A → A of two •-brackets , : Sym• A → A is then defined as ( ◦ )n (a1 , . . . , an ) := n
ckn (−1)π,a n−k+1 k (aπ(1) , . . . , aπ(k) ), aπ(k+1) , . . . , aπ(n) (2.3) k!(n−k)! k=0
π ∈Sn
for n ∈ {0, 1, 2, . . .}. Definition 2.2. The “◦” product is non-degenerate if the complex coefficients ckn , n ≥ k ≥ 0, satisfy ∀n ∈ {0, 1, 2, . . .} ∃k ∈ {0, 1, . . . , n} : ckn = 0. (2.4) 1 Adjectives from supermathematics such as “graded”, “super”, etc., are from now on implicitly implied. We will also follow commonly accepted superconventions, such as, Grassmann parities are only defined modulo 2, and “nilpotent” means “nilpotent of order 2”.
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A priori we shall not assume any other properties of this product, like for instance associativity or a pre-Lie property. See Subsect. 6.1 for further discussions of potential product properties. The aim of this paper is to determine values of the ckn coefficients that lead to useful products, guided by important examples. We first generalize an important definition of Lada and Stasheff [22]. Definition 2.3. A vector space A with a Grassmann-odd •-bracket : Sym• A → A is a (generalized) homotopy Lie algebra if the •-bracket is nilpotent with respect to a non-degenerate “◦” product, ◦ = 0,
= 1.
(2.5)
The infinite hierarchy of nilpotency relations behind (2.5) are also known as “generalized Jacobi identities” [23] or “main identities” [36, 11]. The first few relations will be displayed in detail in Subsect. 2.2. In the original homotopy Lie algebra definition of Lada and Stasheff [22] the product coefficients are fixed to be ckn = 1,
(2.6)
cf. Subsect. 2.2. We shall not assume (2.6) because important examples are incompatible with this restriction, cf. Sect. 4. Instead we adapt the non-degeneracy condition (2.4). The A∞ definition [31] can similarly be generalized. We note that we shall in general lose an auxiliary description of a homotopy Lie algebra in terms of a nilpotent co-derivation, cf. Subsect. 6.3. Our goal is to determine universal values of the ckn coefficients that generate important classes of homotopy Lie algebras. By the word “universal” we mean that a particular set of ckn coefficients works within an entire class of •-brackets. For instance, the Bernoulli numbers will play an important rôle for the so-called derived brackets, cf. Sect. 4. The above algebraic homotopy Lie algebra construction has a geometric generaliza tion to vector bundles E = p∈M E p over a manifold M, where each fiber space E p is a homotopy Lie algebra. However, for most of this paper, it is enough to work at the level of a single fiber. 2.1. Sign conventions. The sign factor (−1)π,a in the product definition (2.3) arises from introducing a sign (2.2) each time two neighboring elements of the symmetric tensor a1 . . . an are exchanged to form a permuted tensor aπ(1) . . . aπ(n) , i.e. working in Symn A, we have aπ(1) . . . aπ(n) = (−1)π,a a1 . . . an ∈ Symn A.
(2.7)
In detail, the sign conventions are ◦ = + , (n (a1 , . . . , an )) =
n
(2.8)
ai + (n − 1) + ,
(2.9)
i=1
n (a1 , . . . , ai , ai+1 , . . . , an ) = (−1)(i +)(i+1 +) n (a1 , . . . , ai+1 , ai , . . . , an ), (2.10) n (a1 , . . . , ai λ, ai+1 , . . . , an ) = (−1)λ n (a1 , . . . , ai , λai+1 , . . . , an ), (λa1 , a2 , . . . , an ) = (−1) n
λ
λ (a1 , a2 , . . . , an ), n
(2.11) (2.12)
Non-Commutative B-V Algebras, Homotopy Lie Algebras and the Courant Bracket
n (a1 , . . . , an λ) = n (a1 , . . . , an )λ,
301
(2.13)
aλ = (−1)λ a λa,
(2.14)
where λ is a supernumber of Grassmann parity λ . It is useful to memorize these sign conventions by saying that a symbol “” carries Grassmann parity , while a “comma” and tensor-symbols “⊗” and “” carry Grassmann parity . Note however that the zero-bracket 0 has Grassmann parity +. The bracket product “◦” is Grassmanneven, (◦) = 0, cf. Eq. (2.8). While the sign implementation may vary with authors and applications, we stress that the Grassmann-odd nature of a •-bracket is an unavoidable, characteristic feature of a homotopy Lie algebra, cf. Eq. (2.5). We remark that one could in principle bring different kinds of Grassmann parities (i) into play, where an upper index i ∈ I labels the different species. In that case Eq. (2.2) should be replaced by ba = ab
(i) (i) (i) (i) (−1)(a + )(b + ) .
(2.15)
i∈I
As an example the exterior form degree could be assigned to a different type of parity. This could provide more flexible conventions for certain systems. Nevertheless, we shall only consider one type of parity in this paper for the sake of simplicity.
2.2. Connection to Lie algebras. The importance of the homotopy Lie algebra construction is underscored by the fact that the two-bracket 2 (a, b) of a Grassmann-odd •-bracket gives rise to a Lie-like bracket [·, ·] of opposite parity := 1 − ,
[a, b] := (−1)(a + ) 2 (a, b), a, b ∈ A, = 1.
(2.16)
(This particular choice of sign is natural for a derived bracket, cf. Sect. 4. Note that in the context of the Koszul bracket hierarchy and Batalin-Vilkovisky algebras the opposite sign convention is usually adapted, i.e. [a, b] := (−1)(a +) 2 (a, b), cf. Sect. 3.) The bracket (2.16) satisfies bi-linearity and skewsymmetry ([a, b]) = a + b + , [λa, bµ] = λ[a, b]µ,
(2.17) (2.18)
[aλ, b] = (−1) λ [a, λb], [b, a] = −(−1)
(a + )(b + )
(2.19) [a, b],
(2.20)
where λ, µ are supernumbers. The failure (if any) of the Jacobi identity
(−1)(a + )(c + ) [[a, b], c] = (−1)b + +(a + )(c + ) Jac(a, b, c)
(2.21)
a,b,c cycl.
is measured by the Jacobiator Jac : Sym3 A → A, defined as Jac(a1 , a2 , a3 ) :=
1 (−1)π,a 2 2 (aπ(1) , aπ(2) ), aπ(3) . 2 π ∈S3
(2.22)
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The first few nilpotency relations (2.5) are c00 1 (0 ) = 0,
c01 2 (0 , a) + c11 1 1 (a) = 0,
c02 3 (0 , a, b) + c12 2 1 (a), b + (−1)a + 2 a, 1 (b)
+c22 1 2 (a, b) = 0,
c03 4 (0 , a, b, c) + c23 Jac(a, b, c) + c33 1 3 (a, b, c)
+c13 3 1 (a), b, c + (−1)a + 3 a, 1 (b), c
+(−1)a +b 3 a, b, 1 (c) = 0,
(2.23) (2.24)
(2.25)
(2.26)
and so forth. If all the ckn coefficients are equal to 1 this becomes the homotopy Lie algebra of Lada and Stasheff [22]. (We shall ignore the fact that Lada and Stasheff [22] do not include a zero-bracket 0 in the definition and they use another sign convention.)If c01 = 0 the one-bracket 1 becomes nilpotent, cf. Eq. (2.24), so in this case (ignoring the fact that we have not defined an integer grading) the one-bracket 1 essentially gives rise to a complex (A, 1 ). Note that the Jacobi identity (2.26) is modified by the presence of higher brackets. In this paper we work under the hypothesis that the characteristic features of a homotopy Lie algebra is formed by the Grassmann-odd and nilpotent nature of the •-bracket , i.e. the mere existence of the ckn coefficients, rather than what particular values those ckn coefficients might have. 2.3. Rescaling. A couple of general remarks about the prefactors in the “◦” product (2.3) is in order. First of all, the denominator k!(n−k)! has been included to conform with standard practices. (Traditionally homotopy Lie algebras are explained via unshuffles [22], a notion we shall not use in this paper. The combinatorial factor k!(n−k)! disappears when recast in the language of un-shuffles.) It is convenient to introduce an equivalent scaled set of coefficients bkn that are always assumed to be equal to the ckn coefficients multiplied with the binomial coefficients, n n ckn , 0 ≤ k ≤ n. (2.27) bk ≡ k We shall often switch back and forth between the “b” and the “c” picture using Eq. (2.27). Secondly, we remark that the nilpotency relations (2.31), and hence the coefficients ckn , may always be trivially scaled ckn −→ λn ckn ,
(2.28)
where λn , n ∈ {0, 1, 2, . . .}, are non-zero complex numbers. Also, if one allows for a re-normalization of the bracket definition n → n /λn , and one scales the coefficients ckn → λk λn−k+1 ckn accordingly, the nilpotency relations are not changed. In practice, one works with a fixed convention for the normalization of the brackets, so the latter type of scaling is usually not an issue, while the former type (2.28) is a trivial ambiguity
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inherent in the definition (2.5). When we in the following make uniqueness claims about the ckn coefficients in various situations, it should always be understood modulo the trivial scaling (2.28). 2.4. Polarization. The product definition (2.3) may equivalently be written in a diagonal form ( ◦ )n (a n ) =
n
bkn n−k+1 k (a k )a (n−k) , (a) = .
(2.29)
k=0
That Eq. (2.29) follows from the product definition (2.3) is trivial. The other way follows by collecting the string a1 . . . an of arguments into a linear combination a =
n
λk ak , (a) = ,
(2.30)
k=0
where the supernumbers λk , k ∈ {0, 1, . . . , n}, have Grassmann parity (λk ) = (ak ) + . The product definition (2.3) then follows by inserting the linear combination (2.30) into Eq. (2.29), considering terms proportional to the λ1 λ2 . . . λn monomial, and using the appropriate sign permutation rules. The general lesson to be learned is that the diagonal carries all information. This polarization trick is not new, as one can imagine; see for instance Ref. [3] and Ref. [35]. The crucial point is that the nilpotency relations (2.5) may be considered on the diagonal only, ( ◦ )n (a n ) = 0, (a) = , n ∈ {0, 1, 2, . . .}.
(2.31)
Equation (2.31) will be our starting point for subsequent investigations. We mention in passing that a construction involving a pair of •-brackets a , a ∈ {1, 2}, sometimes referred to as an “Sp(2)-formulation” [11], can always be deduced from pola2 λa a , (λa ) = 0. rization of = a=1 3. The Koszul Bracket Hierarchy The heart of the following construction goes back to Koszul [21, 1, 4] and was later proven to be a homotopy Lie algebra in Ref. [11]. 3.1. Basic settings. Consider a graded algebra (A, ·) of suspension parity ∈ {0, 1}, satisfying bi-linearity and associativity, (a · b) = a + b + , (λa) · (bµ) = λ(a · b)µ, (aλ) · b = (−1)
λ
a · (λb),
(a · b) · c = a · (b · c),
(3.1) (3.2) (3.3) (3.4)
where λ, µ are supernumbers and a, b, c ∈ A are algebra elements. Let there also be given a fixed algebra element e of Grassmann parity (e) = and a Grassmann-odd, linear operator : A → A, also known as a Grassmann-odd endomorphism
304
K. Bering
∈ End(A). Note that the algebra product “·” carries Grassmann parity, cf. Eq. (3.1). This implies for instance that a power a ·n := a · . . . · a of an element a ∈ A has Grassmann parity (a ·n ) = n(a) + (n−1), n ∈ {1, 2, 3, . . .}. Let L a , Ra : A → A denote the left and the right multiplication map L a (b) := a · b and Ra (b) := b · a with an algebra element a ∈ A, respectively. The Grassmann parity of the multiplication maps L a , Ra ∈ End(A) is in both cases (L a ) = (a) + = (Ra ). 3.2. Review of the commutative case. In this Subsect. 3.2 we assume that the algebra A is commutative. Definition 3.1. If the algebra A is commutative, the Koszul n-bracket n is defined [11] as multiple, nested commutators acting on the algebra element e, n (a1 , . . . , an ) := [[. . . [, L a1 ], . . .], L an ] e, 0 := (e).
(3.5)
n commutators
Here [S, T ] := ST − (−1) S T T S denotes the commutator of two endomorphisms S, T ∈ End(A) under composition. One easily verifies that this definition is symmetric in the arguments (a1 , . . . , an ) by using the Jacobi identity for the commutator-bracket in End(A). Proposition 3.2. In the commutative case the Koszul •-bracket satisfies a recursion relation with only three terms [1] n n n+1 (a1 , . . . , an , an+1 ) = (a1 , . . . , an · an+1 ) − (a1 , . . . , an ) · an+1 (n +)(n+1 +) n −(−1) (a1 , . . . , an+1 ) · an (3.6)
for n ∈ {1, 2, 3, . . .}. Note that the one-bracket 1 can not be expressed recursively in terms of the zerobracket 0 alone. Proof of Proposition 3.2.
Observe that
n[,L a L b ] (a1 , . . . , an ) − n[[,L a ],L b ] (a1 , . . . , an ) = (−1)a + nL a [,L b ] (a1 , . . . , an ) +(−1)(a +)(b +) (a ↔ b) (3.7) for n ∈ {0, 1, 2, . . .}. The wanted Eq. (3.6) emerges after relabelling of Eq. (3.7) and use of the definition (3.5). We mention for later that Eq. (3.7) also makes sense in a non-commutative setting. The main example of the Koszul construction is with a bosonic suspension parity = 0, cf. Eq. (2.2), with e being an algebra unit, and where ∈ End(A) is a nilpotent, Grassmannodd, linear operator, 2 = 0, () = 1. This is called a generalized Batalin-Vilkovisky algebra by Akman [1, 6, 10]. If furthermore the higher brackets vanish, i.e. n = 0, n ≥ 3, then the operator is by definition an operator of at most second order, and (A, ) becomes a Batalin-Vilkovisky algebra [14, 26]. We give an explicit example in Eq. (5.46). The zero-bracket 0 = (e) typically vanishes in practice. For a BatalinVilkovisky algebra with a non-vanishing zero-bracket 0 , see Ref. [12].
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3.3. The intermediate case: Im() ⊆ Z (A). We would like to address the following two questions: 1. Which choices of the bracket product coefficients ckn turn the Koszul construction (3.5) into a (generalized) homotopy Lie Algebra? 2. Does there exist a non-commutative version of the Koszul construction? As we shall see in Subsect. 3.6 the answer to the second question is yes. For practical purposes, it is of interest to seek out intermediate cases that are no longer purely commutative, but where the non-commutative obstacles are manageable. In this and the following Subsects. 3.3–3.5 we make the simplifying assumption that the image of the operator lies in the center of the algebra, i.e. Im() ⊆ Z (A), where the center Z (A) is, as usual, Z (A) := a ∈ A ∀b ∈ A : b · a = (−1)(a +)(b +) a · b .
(3.8)
(3.9)
The fully non-commutative case is postponed until Subsect. 3.6. The case (3.8) clearly includes the commutative case, and it turns out that the treatment of the first question from this intermediate perspective is completely parallel to the purely commutative case. Definition 3.3. If the assumption (3.8) is fulfilled, the Koszul n-bracket is defined as symmetrized, nested commutators acting on the algebra element e, n (a1 , . . . , an ) :=
1 (−1)π,a [[. . . [, L aπ(1) ], . . .], L aπ(n) ] e, 0 := (e). n! π ∈Sn
n commutators
(3.10) In general, all the information about the higher brackets is carried by the diagonal, n (a n )
n n (−L a )k L an−k (e), (a) = . = [[. . . [, L a ], . . .], L a ] e = k n commutators
k=0
(3.11) Proposition 3.4. If the assumption (3.8) is fulfilled, the Koszul brackets satisfy the recursion relation ⎤ n ⎡ n
1 n (a1 , . . . , an ) = (−1)(i +)(k +) ⎣ (−1)( j +)( +) ⎦ n(n−1) = j+1
1≤i< j≤n k=i+1 (i +)( j +) ×n−1 a , . . . , a , . . . , a , . . . , a , a · a ai · a j 1 i j n j i + (−1) n
2 (−1)(i +)(k +) n−1 ai , . . . , an ) · ai − (a1 , . . . , n 1≤i≤n
k=i+1
(3.12) for n ∈ {2, 3, 4, . . .}.
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K. Bering
For instance the two-bracket 2 can be defined via the one-bracket 1 as 2 (a, b) =
1 1 (a · b) − 1 (a) · b + (−1)(a +)(b +) (a ↔ b). 2
(3.13)
The recursion relations are more complicated than in the commutative case. Whereas the commutative recursion relations (3.6) involve only three terms, the number of terms now grows quadratically with the number n of arguments. Loosely speaking, one may say that the recursion relations dissolve as one moves towards full-fledged non-commutativity, cf. Subsect. 3.6. This is fine since recursion relations are anyway not an essential ingredient of a homotopy Lie algebra, although at a practical level they can be quite useful. Proof of Proposition 3.4. Note that Eq. (3.7) still holds in this case: n[,L 2 ] (a n ) − n[[,L a ],L a ] (a n ) = n2L a [,L a ] (a n ), (a) = , a
(3.14)
for n ∈ {0, 1, 2, . . .}. This leads to (n+1) n+1 ) = n ((a · a)a (n−1) ) − 2a · n (a n ) (a = n (a (n−1) (a · a)) − 2n (a n ) · a, (a) = , (3.15)
for n ∈ {1, 2, 3, . . .}. The recursion relation (3.12) now follows from polarization of Eq. (3.15), cf. Subsect. 2.4.
3.4. Nilpotency relations. We now return to the first question in Subsect. 3.3. More precisely, we ask which coefficients ckn could guarantee the nilpotency relations (2.31), if one is only allowed to additionally assume that the operator is nilpotent in the sense that Re = 0 ? (3.16) Note that the criterion (3.16) reduces to the usual nilpotency condition 2 = 0 if e is a right unit for the algebra A. The generic answer to the above question is given by the following Theorem 3.5. Theorem 3.5. Let there be given a set of ckn product coefficients, n ≥ k ≥ 0. The nilpotency relations (2.31) are satisfied for all Koszul brackets that have a nilpotent operator (in the sense of Eq. (3.16)), if and only if ∀n ∈ {0, 1, . . .} : ckn = cn
(3.17)
is independent of k ∈ {0, . . . , n}. Bearing in mind the trivial rescaling (2.28), this solution (3.17) is essentially ckn = 1 in perfect alignment with the requirement (2.6) in the original definition of Lada and Stasheff [22]. So there is no call for a new bracket product “◦” to study the Koszul bracket hierarchy. This no-go statement obviously remains valid when considering the general non-commutative case, cf. Subsect. 3.6, since the general bracket (3.30) should in particular reproduce all the severely limiting situations where the condition (3.8) holds.
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Proof of Theorem 3.5 when assuming Eq. (3.8). We start with the “only if” part. To see Eq. (3.17), first note that for two mutually commuting elements a, b ∈ A with (a) = , n n n n+1 (ba ) = [,L b ] (a ) n n (−L a )i L an−i (b · e) − (−1)b + b · n (a n ). (3.18) = i i=0
Putting b = k (a k ) with Grassmann parity b = 1 − , the element b commutes with a because of Eq. (3.8), and the n th square bracket becomes ( ◦ ) (a n
n
)=
n
bkn
n−k
n−k (−L a )i L an−k−i k (a k ) · e i
k=0 i=0 n (n−k) bkn k (a k ) · n−k ). + (a k=0
(3.19)
The two sums on the right-hand side of Eq. (3.19) are of different algebraic natures, because the two ’s are nested in the first sum, while in the second sum they are not. In general, to ensure the nilpotency relations (2.31), one should therefore impose that the two sums vanish separately. (To make this argument sound one uses that nilpotency relations (2.31) hold for all possible choices of A, and e satisfying Eq. (3.16).) The vanishing of the second sum just imposes a symmetry n bkn = bn−k
(3.20)
among the bkn coefficients, because the family of brackets k (a, . . . , a), k ∈ {0, 1, . . . , n}, mutually commute in a graded sense, which in plain English means: anti-commute. We shall see shortly that the symmetry (3.20) is superseded by stronger requirements coming from the first sum. After some elementary manipulations the first sum reads n
(−L a )
i
i=0
n−i
n−i− j L a
j=0
n i, j, n−i − j
n−i− j n−i − j j n [ (−1)k ]Re L a (e). ck+ j k k=0
(3.21) Terms where non-zero powers of L a are sandwiched between the two ’s are bad, as the nilpotency condition (3.16) does not apply to them. Accordingly, the expression inside the square brackets in Eq. (3.21) must vanish for such terms. In detail, there should exist complex numbers c(nj) , 0 ≤ j ≤ n, such that ∀i, j, n :
0 ≤ i, j ≤ n
⇒
n−i− j
n ck+ j
k=0
n−i − j k
(−1)k = c(nj) δn,i+ j . (3.22)
(The complex numbers ckn and c(nj) should not be confused.) Putting j = 0 and m = n − i ∈ {0, . . . , n}, the Eq. (3.22) reduces to ∀m, n :
0≤m≤n
⇒
m k=0
ckn
m k
n (−1)k = c(0) δm,0 .
(3.23)
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K. Bering
This in turn implies that ckn can only depend on n, ∀k ∈ {0, . . . , n} :
n , ckn = c(0)
(3.24)
which establishes the claim (3.24). For completeness let us mention that if one inserts the solution (3.24) back into Eq. (3.22) one gets similarly, ∀ j ∈ {0, . . . , n} :
n . c(nj) = c(0)
(3.25)
The “if” part of the proof follows easily by going through above reasoning in reversed order, but it is also a consequence of Theorem 3.6 below. 3.5. Off-shell with respect to the nilpotency condition. One may summarize the discussions of the last Subsect. 3.4 in the following Theorem 3.6. Theorem 3.6. A Koszul •-bracket satisfies a square identity [11] ◦ = Re ,
(3.26)
where “◦” here refers to the ordinary bracket product (2.3) with
ckn
= 1.
We stress that this identity holds without assuming the nilpotency condition (3.16). It is instructive to see a direct proof of this square identity (3.26) that uses a generating function and polarization to minimize the combinatorics. In the case of the Koszul •-bracket the generating function is just the ordinary exponential function “exp”. This is implemented as a formal series of “exponentiated brackets”, ∞
1 n n (a ) = e−[L a ,·] e = e−L a e L a (e), (a) = . (3.27) n! n=0
Conversely, one may always extract back the n th bracket n by identifying terms in Eq. (3.27) that has homogeneous scaling degree n under scaling a → λa of the argument a. Proof of Theorem 3.6 when assuming Eq. (3.8). First note that for two mutually commuting elements a, b ∈ A with (a) = , ∞ ∞ 1 n+1 1 n n (ba ) = (a n ) n! n! [,L b ] n=0
n=0
= e−L a e L a (b · e) − (−1)b + b · e−L a e L a (e). (3.28)
Putting b = e−L a e L a (e) with Grassmann parity b = 1 − , the element b commutes with a because of Eq. (3.8), the element b is nilpotent b · b = 0, and the exponentiated left-hand side of Eq. (3.26) becomes n ∞ ∞
1 1 ( ◦ )n (a n ) = n−k+1 k (a k )a (n−k) n! k!(n−k)! n=0 n=0 k=0
= e−L a e L a e−L a e L a (e) · e + e−L a e L a (e) · e−L a e L a (e) ∞ 1 n = e−L a Re e L a (e) = (a n ), n! Re n=0
which is just the exponentiated right-hand side of Eq. (3.26).
(3.29)
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309
3.6. The general non-commutative case. We now consider the general case without the assumption (3.8). Definition 3.7. In the general non-commutative case the Koszul n-brackets are defined as Bi,k n (a1 , . . . , an ) := (−1)π,a +π(1) +...+π(i) +i aπ(1) · . . . · aπ(i) i! j!k! i, j, k ≥ 0 i + j +k = n
π ∈Sn
· aπ(i+1) · . . . · aπ(i+ j) · e · aπ(i+ j+1) · . . . · aπ(n) ,
(3.30)
where the Bk, coefficients are given through the generating function ∞
x k y x−y = B(y, x) = x k! ! e − ey i, j=0
x y 1 x 2 2x y y 2 1 x 2 y x y 2 = 1− + + + + − + 2 2 2 6 3 6 3! 2 2 4 3 2 2 3 4 x 2x y 4x y 2x y y 1 − + + + − + .... + 4! 30 15 5 15 30
B(x, y) =
Bk,
(3.31)
The Bk, coefficients are related to the Bernoulli numbers Bk via B(x, y) = e−x B(y −x) = e−y B(x − y),
(3.32)
cf. Eq. (4.14), or in detail, Bk,
k k Bi+ = B,k , k, ∈ {0, 1, 2, . . .}. = (−1) i k
(3.33)
i=0
The first few brackets read 0 = (e),
(3.34)
1 1 1 (a) = (a · e) − (e) · a − (−1)a + a · (e), (3.35) 2 2 1 1 1 1 2 (a, b) = (a · b · e) − (a · e) · b − (−1)a + a · (b · e) + (e) · a · b 2 2 2 12 1 1 + (−1)a + a · (e) · b + (−1)a +b a · b · (e) 3 12 +(−1)(a +)(b +) (a ↔ b). (3.36) In general, all the information about the higher brackets is carried by the diagonal, n Bi,k a ·i · (a · j · e) · a ·k , n (a n ) = (a) = . (3.37) i, j, k i, j, k ≥ 0 i + j +k = n
The definition (3.37) is consistent with the previous definition (3.11) for the intermediate case (3.8). This is because B(x, x) = e−x , or equivalently, n n Bk,n−k = (−1)n . (3.38) k k=0
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K. Bering
The formal series of exponentiated brackets may be compactly written ∞ 1 n n (a ) = B(L a , Ra )e L a (e) = B(Ra −L a )e−L a e L a (e), n!
(a) = .
n=0
(3.39) The latter expression shows that all the “·” products in the bracket definition (3.30) can be organized as commutators from either A or End(A), except for the dot “·” in front of the fixed element e. In the general non-commutative case the n+1 bracket can not be expressed recursively in terms of the n bracket alone, although there are exceptions. Most notably, the three-bracket 3 can be expressed purely in terms of the two-bracket 2 , 1 1 3 (a1 , a2 , a3 ) = (−1)π,a 2 aπ(1) · aπ(2) , aπ(3) − 2 aπ(1) , aπ(2) · aπ(3) 6 2 π ∈S3 1 − (−1)π(1) + aπ(1) · 2 aπ(2) , aπ(3) . (3.40) 2 (Of course, one may always replace appearances of in definition (3.30) with zero and one-brackets, i.e. (e) = 0 , and (a · e) = 1 (a) + 21 0 · a + 21 (−1)a + a · 0 , and in this way express the n-bracket n in terms of lower brackets, in this case 0 and 1 .) Our main assertion is that the square identity (3.26) in Theorem 3.6 holds for the fully non-commutative bracket definition (3.30), i.e. without assuming Eq. (3.8). Theorem 3.5 is also valid in the general situation. Proof of Theorem 3.6 in the general case. First note that for two elements a, b ∈ A with (a) = , ∞ 1 n+1 (ba n ) = B(L a , Ra )(E(L a , Ra )b · e) n! n=0
+
∞
(R)
Bi, j,k a ·i · e L a (e) · a · j · b · a ·k
i, j,k=0
+(−1)b +
∞
(L)
Bi, j,k a ·i · b · a · j · e L a (e) · a ·k , (3.41)
i, j,k=0
where E(x, y) :=
∞ x k y ex − e y = , x−y (k ++1)!
(3.42)
k,=0
Bi, j+k+1 , i!( j +k +1)! ∞ B(x, y) − B(x, z) i j k B (R) (x, y, z) := Bi,(R) , j,k x y z = y−z (R)
Bi, j,k :=
(3.43) (3.44)
i, j,k=0
(L)
Bi, j,k :=
Bi+ j+1,k , (i + j +1)!k!
(3.45)
Non-Commutative B-V Algebras, Homotopy Lie Algebras and the Courant Bracket ∞
B (L) (x, y, z) :=
(L)
Bi, j,k x i y j z k =
i, j,k=0
B(x, z) − B(y, z) . x−y
311
(3.46)
Putting b = B(L a , Ra )e L a (e) with Grassmann parity b = 1 − , the exponentiated left-hand side of Eq. (3.26) becomes n ∞ ∞
1 1 k (a k )a (n−k) ( ◦ )n (a n ) = n−k+1 n! k!(n −k)! n=0 n=0 k=0
= B(L a , Ra ) E(L a , Ra )B(L a , Ra )e L a (e) · e ∞
+
i, j,k=0 ∞
−
(R) Bi, j,k a ·i · e L a (e) · a · j · B(L a , Ra )e L a (e) · a ·k (L) Bi, j,k a ·i · B(L a , Ra )e L a (e) · a · j · e L a (e) · a ·k .
i, j,k=0
(3.47)
This should be compared with the exponentiated right-hand side of Eq. (3.26), ∞ 1 n (a n ) = B(L a , Ra )Re e L a (e). n! Re
(3.48)
n=0
The two sides (3.47) and (3.48) are equal provided that the following two conditions are met: E(x, y)B(x, y) = 1, B
(R)
(x, y, z)B(y, z) = B(x, y)B
(3.49) (L)
(x, y, z).
(3.50)
The first Eq. (3.49) has a unique solution for B(x, y) given by 1/E(x, y), leaving no alternative to Eq. (3.31). It is remarkable that this unique solution (3.31) satisfies the non-trivial second criterion (3.50) as well, as one may easily check by inspection, thereby ensuring the existence of the non-commutative Koszul construction.
4. The Derived Bracket Hierarchy In this section we consider an important class of •-brackets that naturally requires a nontrivial bracket product (2.3) in order to satisfy the nilpotency relations (2.5), namely the so-called derived •-brackets. As we shall soon see in Eq. (4.6) below, the derived brackets are composed of nested Lie brackets in a simple manner. This should be contrasted with the non-trivial definition (3.30) of the non-commutative Koszul hierarchy that – among other things – involved the Bernoulli numbers. Nevertheless, in a strange twist, while the ckn coefficients in the nilpotency relations (2.5) are all simply 1 for the non-commutative Koszul hierarchy, the ckn coefficients will be considerably more complicated for the derived hierarchy and involve – of all things – the Bernoulli numbers!
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K. Bering
4.1. Definitions. We abandon the associative “·” structure considered in Subsect. 3.1, and consider instead a Lie algebra (A, [ , ]) of parity ∈ {0, 1}, satisfying bi-linearity, skewsymmetry and the Jacobi identity, ([a, b]) [λa, bµ] [aλ, b] [b, a]
= a + b + , = λ[a, b]µ, = (−1)λ [a, λb], = −(−1)(a +)(b +) [a, b], (−1)(a +)(c +) [[a, b], c], 0=
(4.1) (4.2) (4.3) (4.4) (4.5)
a,b,c cycl.
where λ, µ are supernumbers. Let there be given a fixed Lie algebra element Q ∈ A. Definition 4.1. The derived n-bracket nQ , n ∈ {0, 1, 2, . . .}, is defined as [7] nQ (a1 , . . . , an ) :=
1 (−1)π,a [[. . . [Q, aπ(1) ], . . .], aπ(n) ], 0Q := Q. (4.6) n! π ∈Sn
n Lie brackets
Note that the zero-bracket 0Q = Q is just the fixed Lie algebra element Q itself. Since we are ultimately interested in •-brackets Q that carry an odd intrinsic Grassmann parity ( Q ) = 1, cf. Eq. (2.5), we shall demand from now on that the Grassmann parity Q of the fixed Lie algebra element Q is the opposite of the suspension parity , Q = 1 − .
(4.7)
All the information is again carried by the diagonal, (n−1) ), a] = [[. . . [Q, a], . . .], a] = (−ada)n Q, (a) = , nQ (a n ) = [n−1 Q (a n Lie brackets
(4.8) where we have defined the adjoint action ad : A → End(A) by (ada)(b) := [a, b]. Proposition 4.2. A derived •-bracket Q satisfies the recursion relation n n 1 n (i +)(k +) n−1 a1 , . . . , aˆ i , . . . , an , ai (−1) Q (a1 , . . . , an ) = Q n i=1 k=i+1 (4.9) for n ∈ {1, 2, 3, . . .}. Proof of Proposition 4.2. The recursion relation (4.9) follows from polarization of Eq. (4.8), cf. Subsect. 2.4. As we saw in Subsect. 2.2 there is a Lie-like bracket of opposite parity Q = 1 − given by 1 1 [[a, Q], b] + [a, [Q, b]] 2 2 = −(−1)(a + Q )(b + Q ) [b, a] Q .
[a, b] Q : = (−1)a + Q 2Q (a, b) =
(4.10)
Non-Commutative B-V Algebras, Homotopy Lie Algebras and the Courant Bracket
313
Thus the derived •-bracket Q gives rise to an interesting duality [ , ] → [ , ] Q between Lie-like brackets of even and odd parity [8]. The suspension parity was introduced in the first place in Eq. (2.2) to bring the even and odd brackets on equal footing, and we see that the formalism embraces this symmetry. The bracket (4.10) is known as a (skewsymmetric, inner) derived bracket [18–20]. The outer, derived •-bracket hierarchies are modeled after the properties of the inner hierarchies, and will be discussed elsewhere.
4.2. Nilpotency versus square relations. We would like to analyze which coefficients ckn could guarantee the nilpotency relations (2.31), if we are only allowed to additionally assume that Q is nilpotent in the Lie bracket sense, [Q, Q] = 0.
(4.11)
The n th square bracket ( Q ◦ Q )n (a n ) is nothing but a linear combination of terms built out of n+1 nested Lie brackets [ , ], whose n+2 arguments consist of two Q’s and n a’s. The only such term that the nilpotency condition (4.11) annihilates is the term n[Q,Q] (a n ) = [[. . . [[Q, Q], a], . . .], a] = (−ada)n [Q, Q] n+1 Lie brackets
n n n−k (a (n−k) ), kQ (a k ) , (a) = , = Q k
(4.12)
k=0
for n ∈ {0, 1, 2, . . .}. Therefore, instead of imposing the nilpotency condition (4.11), it is equivalent to let the n th square bracket ( Q ◦ Q )n (a n ) be proportional to the term (4.12), i.e. ( Q ◦ Q )n (a n ) = αn n[Q,Q] (a n ), (4.13) where αn is a proportionality factor that depends on n ∈ {0, 1, 2, . . .}. This off-shell strategy with respect to the nilpotency condition (4.11) has also been promoted in Ref. [9] in a similar context. Since one may trivially scale the nilpotency relations (2.31) with a non-zero complex number, cf. Subsect. 2.3, it is enough to study the square relation (4.13) with a proportionality factor equal to either αn = 1 or αn = 0. The case αn = 1 is a set of coupled, non-homogeneous linear (also known as affine) equations in the ckn product coefficients. The analogous homogeneous problem corresponds to letting the proportionality factor be αn = 0, while continuing not to require nilpotency (4.11) of Q.
4.3. Solution. In this subsection we present the complete solution to the square relation (4.13). To this end, let Bk be the Bernoulli numbers, k ∈ {0, 1, 2, . . .}, generated by ∞ 1 x4 Bk k x 1 x2 x = x = 1− + − + O(x 6 ), B(x) = x e −1 k! 2 6 2! 30 4!
(4.14)
k=0
and let us for later convenience define the negative Bernoulli numbers as zero, 0 = B−1 = B−2 = B−3 = . . . .
(4.15)
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K. Bering
Theorem 4.3. Let there be given a set of ckn coefficients with n ≥ k ≥ 0. The square relations Q ◦ Q = [Q,Q] (4.16) are satisfied for all derived •-brackets Q , if and only if n(H )
ckn = Bk + δk,1 + ck n(H )
where the homogeneous part ck
, n ≥ k ≥ 0,
(4.17)
solves the corresponding homogeneous equation Q ◦ Q = 0.
(4.18)
For a given n ∈ {0, 1, 2, . . .}, the solution space for the homogeneous problem (4.18) is n(H ) n−1 [ n+1 2 ] dimensional. A basis ck(m) of solutions, labelled by an integer m ∈ {0, 1, . . . , [ 2 ]}, is k k n(H ) Bk−m − Bk−n+m = − (m ↔ n−m). ck(m) = (4.19) m n−m In practice, it is easier to let the label m ∈ {0, 1, . . . , n} run all the way to n, and work with an over-complete set of solutions. Also introduce a generating polynomial n(H )
ck
(t) :=
n
n(H )
ck(m) t m .
(4.20)
m=0
One may reformulate the solution Eqs. (4.17) and (4.19) with the help of a generating function ∞ ∞ bkk+ k x k y c(x, y) := ckk+ (4.21) ≡ x y . k! ! (k +)! k,=0
The particular solution
ckn
k,=0
= Bk + δk,1 is then
y y c(x, y) = B(x)e x+y = B(−x)e = (B(x)+x)e 2 1 x2
x 1 x y 3x y 2 +y + + x y + y2 + + + y3 = 1+ 2 2 6 3! 2 2 4 x 1 − + x 2 y 2 + 2x y 3 + y 4 + . . . , (4.22) + 4! 30
and the homogeneous solution is c(H ) (x, y, t) = B(x)e xt+y − B(xt)e x+yt ,
(4.23)
where B is the Bernoulli generating function, cf. Eq. (4.14). Equations (4.22) and (4.23) are our main result of Sect. 4. We emphasize that the square relation (4.16) and its homogeneous counterpart (4.18) are satisfied with these ckn product solutions without assuming the nilpotency condition (4.11). Also note that none of these solutions are consistent with the ordinary product (2.6). The solution shows that attempts to fit the derived bracket hierarchy (4.6) into the original homotopy Lie algebra definition (2.6) are bound to be unnatural and will only work in special situations.
Non-Commutative B-V Algebras, Homotopy Lie Algebras and the Courant Bracket
315
Proof of Theorem 4.3. To derive the solution Eqs. (4.17) and (4.19), first note that for two elements a, b ∈ A with (a) = , 1 [[. . . [[Q (a ), b], a], . . .], a] n+1 n
n n+1 Q (ba ) =
=0
n−+1 Lie brackets
n 1 = (−ada)n− [Q (a ), b] n+1
(4.24)
=0
for n ∈ {0, 1, 2, . . .}. Letting b = kQ (a k ) this becomes n+1 Q
n
1 n k k n (n−i) k+i (k+i) n−i Q (a )a = (a ), (a ) . (4.25) Q Q i +1 i i=0
After some elementary manipulations the n th bracket product reads ( Q ◦ Q )n (a n ) =
n
k k (n−k) bkn n−k+1 (a )a Q Q
k=0
n k n bk−i n−k +i (n−k) k k n−k = (a ), (a ) Q Q i i +1 k=0 i=0 n k n ck−i n k n−k (a (n−k) ), kQ (a k ) . (4.26) = Q k i +1 i k=0
i=0
Combining Eqs. (4.12), (4.16) and (4.26) with Q = Q, one derives ∀k, n :
0≤k≤n
⇒
k n ck−i k − 1 = − (k ↔ n−k). i +1 i
(4.27)
i=0
The right-hand side of Eq. (4.27) comes from the symmetry kQ (a k ), Q (a ) = (k ↔ ), k, ∈ {0, 1, 2, . . .}.
(4.28)
This symmetry is the origin of the non-trivial homogeneous solutions. Let us first assume that the left-hand side of (4.27) vanishes. The unique solution for this case reads ckn = Bk + δk,1 ,
(4.29)
which establishes Eq. (4.17). We now focus on the homogeneous part of Eq. (4.27). First note that for a fixed m ∈ {0, . . . , n}, the equation k n ck−i k = δk,m , i +1 i
(4.30)
i=0
with a Kronecker delta function source on the right-hand side, has the unique solution k Bk−m . ckn = (4.31) m
316
K. Bering
Hence a (k ↔ n−k) skewsymmetric version of the Kronecker delta function source, i.e. k n ck−i k (4.32) = δk,m − δk,n−m i +1 i i=0
will have the unique solution Eq. (4.19).
4.4. Ward-like and Jacobi-like identities. We now discuss particular useful solutions, i.e. non-trivial identities with as few terms as possible. The n th square bracket ( Q ◦ Q )n (a, . . . , a) typically has n + 1 terms on the diagonal. Here we shall use the freedom in the homogeneous part to kill most of these terms. Consider first the Ward solution n(H ) = δk,1 + δk,n , ckn = Bk + δk,1 − ck(0)
n ≥ k ≥ 0,
(4.33)
or equivalently using Eq. (4.21), c(x, y) = xe y + e x .
(4.34)
It corresponds to a hierarchy of Ward identities
n nQ 1Q (a)a (n−1) + 1Q nQ (a n ) = n[Q,Q] (a n ),
(a) = , (4.35)
where n ∈ {0, 1, 2, . . .}. (The name “Ward identity” refers to a similar identity encountered in String Theory.) After polarization the Ward identities are n
(−1)1 +...+k−1 +(k−1) nQ a1 , . . . , ak−1 , 1Q (ak ), ak+1 , . . . , an k=1
+1Q nQ (a1 , . . . , an )
= n[Q,Q] (a1 , . . . , an ).
(4.36)
The first few Ward identities read, (4.37) 1Q (0Q ) = 0[Q,Q] ,
21Q 1Q (a) = 1[Q,Q] (a), (4.38)
2Q 1Q (a), b + (−1)a + 2Q a, 1Q (b) + 1Q 2Q (a, b) = 2[Q,Q] (a, b). (4.39) Consider next the solution ckn = Bk + δk,1 −
n 2 n(H ) m n(H ) ck(1) + c n n k(m) m=0
4 2 = δk,2 + δk,n−1 − δk,n , n n or equivalently using Eq. (4.21),
0 = n ≥ k ≥ 0,
c(x, y) − 1 = 2x 2 E (y) + (2y − x)E(x) 2 y x =2 1 + (y − 1)e y + 2 − 1 e x − 1 . y x
(4.40)
(4.41)
Non-Commutative B-V Algebras, Homotopy Lie Algebras and the Courant Bracket
317
Here we have defined E(x) :=
∞ ex − 1 xk 1 = = . x (k +1)! B(x)
(4.42)
k=0
The solution (4.40) corresponds to generalized Jacobi identities
(n+1) 2n nQ 2Q (aa)a (n−1) + 22Q nQ (a n )a − 1Q n+1 (a ) Q (n+1) ), = n+1 [Q,Q] (a
(4.43)
where n ∈ {0, 1, 2, . . .}. The first few read,
22Q 0Q , a − 1Q 1Q (a) = 1[Q,Q] (a), (4.44)
2Q 1Q (a), b + (−1)a + 2Q a, 1Q (b) + 1Q 2Q (a, b) = 2[Q,Q] (a, b), (4.45)
2 Jac(a, b, c) − 1Q 3Q (a, b, c) = 3[Q,Q] (a, b, c), (4.46) where Jac(a, b, c) is the Jacobiator, cf. Eq. (2.22). It is worth mentioning that the zerobracket 0Q , which normally complicates a homotopy Lie algebra, cf. Eqs. (2.24), (2.25) and (2.26), here decouples, cf. the corresponding Eqs. (4.38), (4.39) and (4.46). Therefore, in the nilpotent case [Q, Q] = 0, the Grassmann-odd one-bracket 1Q is nilpotent (4.38), and it obeys a Leibniz rule (4.36) with respect to the n-bracket nQ . And perhaps most importantly, the two-bracket satisfies a generalized Jacobi identity (4.46) that only differs from the original Jacobi identity by a 1Q -exact term. By transcribing the work of Courant [13] to this situation, one may define the notion of a Dirac subalgebra. Definition 4.4. Assume that Q is nilpotent [Q, Q] = 0. A Dirac subalgebra is a subspace L ⊆ A that is: 1. Maximal Abelian with respect to the original Lie bracket [ , ], 2. Closed under the two-bracket 2Q , i.e. 2Q (L, L) ⊆ L, or equivalently, [L, L] Q ⊆ L, cf. Eq. (4.10). It follows immediately from the bracket definition (4.6) that all the higher brackets nQ , n ∈ {3, 4, 5, . . .} vanish on a Dirac subalgebra L. In particular, the Jacobi identity for the two-bracket 2Q is satisfied in a Dirac subalgebra L, cf. Eq. (4.46). We also point out a connection to Courant algebroids, where the generalized Jacobi identity (4.46) translates into the first (out of five) defining properties for the Courant algebroid, cf. Ref. [24].
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K. Bering
5. The Courant Bracket The Courant bracket [13] has received much interest in recent years primarily due to Hitchin generalized complex geometry [16]. A broad introduction to the subject can be found in the Ph.D theses of Roytenberg [29] and Gualtieri [15]. In this section we give three different constructions of the (skewsymmetric) Courant bracket as a derived bracket, and we elaborate on its connection to homotopy Lie algebras [28, 29]. One wellknown construction [18–20, 29] relies on an operator representation, see Subsect. 5.1, and two partially new constructions rely on an even and an odd Poisson bracket [25, 29, 30], respectively, see Subsects. 5.2–5.4. The Courant bracket is defined on vectors and exterior forms as [X, Y ] H = [X, Y ] + (−1)Y i X i Y H = − (−1) X Y [Y, X ] H , [X, η] H [ξ, η] H
X, Y ∈ (T M), (5.1) 1 (−1) X (−1) X di X )η = (L X − di X )η = (L X + i X d)η = (i X d + 2 2 2 X η = −(−1) [η, X ] H , (5.2) = 0, ξ, η ∈ ( • (T ∗ M)), (5.3)
with a closed twisting form H ∈ ( (T ∗ M)). (We shall ignore the fact that the twisting form H is zero in the original Courant bracket [13].) The Courant bracket (5.1)-(5.3) does not satisfy the Leibniz rule nor the Jacobi identity. It is therefore natural to ask: what is the significance of these formulas, in particular Eq. (5.2)? In hindsight the answer is that there exists a homotopy Lie algebra structure behind the Courant bracket, that makes the Jacobi identity valid modulo Q-exact terms, cf. Eq. (4.46). And underneath the homotopy Lie algebra structure, there is a Grassmann-odd nilpotent Hamiltonian vector field that generalizes the de Rham exterior derivative. As we shall see in Subsect. 5.6 below, the higher Courant brackets are naturally defined on multi-vectors and exterior forms via the derived bracket hierarchy (4.6). When restricted to only vector fields, functions and one-forms, the L ∞ hierarchy truncates and becomes an L 3 algebra [23]. Roytenberg and Weinstein [28, 29] define a related set of higher brackets through a homological resolution [5]. Finally, let us mention for completeness that people often consider a nonskewsymmetric version of the Courant bracket (which is also called a Dorfman bracket and is related to Loday/Leibniz algebras), partly to avoid handling the Jacobi identity directly, and partly to simplify the axioms for a Courant algebroid, cf. Ref. [29]. Nevertheless, at the end of the day, it is often the skewsymmetrized bracket that is relevant for applications. More importantly, our underlying derived bracket definition (4.6) is manifestly skewsymmetric, and hence we shall here only treat skewsymmetric brackets. We shall take advantage of polarization to shortcut the lengthy calculations that are normally associated with the skewsymmetric bracket, cf. Subsect. 2.4.
5.1. Review of the operator representation. Perhaps the simplest realization of the Courant bracket as a (skewsymmetric, inner) derived bracket is the following operator construction due to Kosmann-Schwarzbach [18–20, 29]. Consider a d-dimensional bosonic manifold M, and let x i , i ∈ {1, 2, . . . , d}, denote local bosonic coordinates in some coordinate patch U ⊆ M. To avoid cumbersome notation, we shall often take the liberty to write local objects, which formally only live on U , as if they are living on the
Non-Commutative B-V Algebras, Homotopy Lie Algebras and the Courant Bracket
whole manifold M. Furthermore, we shall regard exterior forms η = η(x i , ci ) ∈ ( • (T ∗ M)) ∼ E ≡ T M, = C ∞ (E),
319
(5.4)
as functions in the variable (x i , ci ) on the parity-inverted tangent bundle E ≡ T M by identifying the basis of one-forms d x i ≡ ci with the fermionic variables ci . Notice that an 1 n-form η = n! ηi1 ...in ci1 . . . cin has Grassmann parity η = n modulo 2, if the coordinate functions ηi1 ...in = ηi1 ...in (x) are bosonic. Now let the Lie algebra A of Sect. 4 be the Lie algebra A = End(C ∞ (E)) of operators acting on the above functions (5.4), and let the Lie bracket for A be given by the commutator bracket [·, ·]. We shall often identify an exterior form η with the left multiplication operator L η ∈ A, i.e. the operator that multiplies from the left with η. For the fixed Lie algebra element of Sect. 4, one now chooses a twisted version D := d + Hodd ∈ A (5.5) of the de Rham exterior derivative d on the manifold M, d = ci
∂ , ∂ xi
(d) = 1.
(5.6)
Here Hodd ∈ ( odd (T ∗ M)) is an odd closed exterior form, i.e. linear combinations of forms of odd form-degree, that are closed. The oddness condition ensures that D carries definite Grassmann-parity. The D operator is nilpotent, [D, D] = 2d Hodd = 0,
(5.7)
because the exterior form Hodd is closed. Parity-inverted vector fields X˜ (where the tilde “∼” denotes parity-inversion, cf. Eq. (5.27) below) are now represented as first-order differential operators, iX = Xi
∂l , ∂ci
(i X ) = 1− X .
(5.8)
One may now construct the derived •-bracket hierarchy of D. The zero-bracket 0D = D is the generator D itself. The one-bracket 1D = [D, ·] is a nilpotent, Grassmann-odd operator, cf. Eq. (4.38). It is just the de Rham exterior derivative on exterior forms, 1D (η) = [D, η] = dη,
(5.9)
independent of Hodd . On vectors, the one-bracket 1D becomes a sum of a Lie derivative and a contracted term, (−1) X 1D (i X ) = [i X , D] = L X + i X Hodd .
(5.10)
Here and below, we repeatedly make use of the Cartan relations 2d 2 = [d, d] = 0,
[i X , i Y ] = 0,
L X = [i X , d],
i [X,Y ] = [L X , i Y ]. (5.11)
The two-bracket 2D gives rise to an odd version [·, ·] D of the Courant bracket [18–20], cf. Eq. (4.10).
320
K. Bering Table 1. The Poisson bracket {·, ·} xj
pj
cj
xi
0
−δ ij
0
pi
δi
ci bi
j
− R˜ i j
bj 0 j
−ik ck
0
ijk ck
0
− kji bk
0 j
δi
ikj bk
δ ij 0
Definition 5.1. The odd Courant bracket [·, ·] D is defined as the derived bracket, 1 1 [[a, D], b] + [a, [D, b]] 2 2 = −(−1)(a +1)(b +1) [b, a] D , a, b ∈ A = End(C ∞ (E)). (5.12)
[a, b] D := (−1)a +1 2D (a, b) =
In this formulation, the Courant bracket (5.1)-(5.3) reads [i X , i Y ] D = i [X,Y ] + (−1)Y i X i Y Hodd , X, Y ∈ (T M), 1 [i X , η] D = (L X + i X d)η, 2 ξ, η ∈ ( • (T ∗ M)). [ξ, η] D = 0,
(5.13) (5.14) (5.15)
5.2. Symplectic structure. It is also possible to build the Courant bracket (5.1)-(5.3) as a derived bracket with the help of a Poisson bracket {·, ·}. Consider a d-dimensional bosonic base manifold M. Let x i , i ∈ {1, 2, . . . , d}, denote local bosonic coordinates in some coordinate patch, and let pi be the local bosonic basis vectors pi ≡ ∂i ∈ (T M) for symmetric multi-vectors, i ∈ {1, 2, . . . , d}. We now define a Poisson bracket on the cotangent bundle T ∗ M as j
{ pi , x j } = δi = −{x j , pi }, {x i , x j } = 0,
(5.16)
where we are going to fix the fundamental Poisson bracket { pi , p j } a little bit later. Within the (x i , p j ) sector, it is consistent to put { pi , p j } to zero, but an extension below to other sectors of the Whitney sum E ≡ T ∗ M ⊕ E ⊕ E ∗ ⊇ T ∗ M will complicate matters, cf. Eq. (5.20) and Table 1. (The sign of the Poisson bracket (5.16), which is opposite of the standard physics conventions, has been chosen to minimize appearances of minus signs.) We next introduce the notation bi ≡ ∂˜i ∈ (E) for the local fermionic basis of skewsymmetric multi-vectors, where as before the tilde “∼” represents a parity-inversion, cf. Eq. (5.27). The skewsymmetric multi-vectors π = π(x i , bi ) ∈ ( • (T M)) ∼ E∗ ∼ (5.17) = C ∞ (E ∗ ), = T ∗ M, can be identified with functions on the parity-inverted cotangent bundle T ∗ M ∼ = E ∗. i i ∗ Similarly, as explained in Subsect. 5.1, we write c ≡ d x ∈ (E ) for the local fermionic one-forms that constitute a basis for the exterior forms. Recall also that exterior forms (5.4) can be regarded as functions on the parity-inverted tangent bundle E ≡ T M. Note
Non-Commutative B-V Algebras, Homotopy Lie Algebras and the Courant Bracket
321
Table 2. The Poisson bracket {·, ·} in Darboux coordinates xj
Pj
cj
bj
xi
0
−δ ij
0
0
Pi ci
δi 0
0 0
0 0
0 δ ij
bi
0
0
δi
j
j
0
j that the natural symmetric pairing ∂˜i , d x j + = δi = d x j , ∂˜i + between exterior forms and multi-vectors can equivalently be viewed as a canonical Poisson bracket of fermions, j
{bi , c j } = δi = {c j , bi }, {ci , c j } = 0 = {bi , b j }.
(5.18)
The idea is now to regard the two Poisson brackets (5.16) and (5.18) as part of the same symplectic structure on a 4d dimensional manifold, which we take to be the total space E ≡ T ∗ M ⊕ E ⊕ E ∗ of the 3d dimensional vector bundle E → M with local coordinates (x i ; pi , ci , bi ), and let the combined symplectic structure play the rôle of the Lie algebra structure [·, ·] of Sect. 4. Obviously this idea implies that one should fix the Poisson bracket in the cross-sectors between the even and the odd coordinates. In the end, it turns out that the cross-sector assignments do not matter, as they do not enter the Courant bracket in pertinent sectors. However to be specific, we shall model the cross-sectors over an arbitrary connection ∇ : (T M) × (T M) → (T M). In detail, the Poisson brackets between bosonic and fermionic variables are [25] {x i , b j } = 0 = {b j , x i }, {x i , c j } = 0 = {c j , x i }, { pi , b j } = ikj bk = −{b j , pi },
(5.19)
j
−{ pi , c j } = ik ck = {c j , pi }, cf. Table 1. To ensure the Jacobi identity for the {·, ·} bracket, one finally defines the { pi , p j } sector to be (5.20) −{ pi , p j } = R k i j c bk =: R˜ i j , where R k i j =
∂ kj ∂ xi
k + im mj − (i ↔ j)
(5.21)
is the Riemann curvature tensor. For instance, the second Bianchi identity 0 =
(
i, j,k cycl.
∂ R m ni j m + k R ni j − kn R m i j ) ∂xk
(5.22)
guarantees that the Jacobi identity holds in the ( pi , p j , pk ) sector. A simpler picture emerges if one introduces the momentum variables [32, 33] Pi := pi + ikj c j bk .
(5.23)
322
K. Bering Table 3. The symplectic two-form ω in Eq. (5.24) consists of the inverse matrix of Table 1
dx i
dx j
dpj
dc j
db j
−∂[i kj] c bk
j δi
−ikj bk
ik ck
0
0
0
0
0
δi
0
δ ij
d pi
−δ ij
dci
− kji bk ijk ck
dbi
j
j
0
Remarkably, the quadruple (x i , Pi , ci , bi ) are local Darboux coordinates for the Poisson bracket {·, ·}, cf. Table 2. The corresponding symplectic two-form is just the canonical two-form ω = dx i ∧ d Pi + dci ∧ dbi , dω = 0, (5.24) cf. Table 3. Here d denotes the de Rham exterior derivative on the Whitney sum E, → → → → ∂ ∂ ∂ i ∂ d = dx + d pi + dc i + dbi , i ∂x ∂ pi ∂c ∂bi i
(d) = 0,
(5.25)
which should not be confused with the de Rham exterior derivative d on the base manifold M, cf. Eq. (5.6). One may choose the symplectic potential ϑ to be −ϑ = Pi dx i − bi dci = dx i Pi + dci bi ,
dϑ = ω.
(5.26)
Let us address the issue of coordinate transformations in the base manifold M. Because of the presence of the ikj symbol in definition (5.23), the momentum variables Pi do not have the simple co-vector transformation law that for instance the variables pi and bi enjoy. Nevertheless, the local expression (5.26) for the one-form ϑ is invariant. Hence ϑ is a globally defined symplectic potential for the (2d|2d) symplectic manifold E with an exact symplectic two-form ω = dϑ. Moreover, the Whitney sum E ∼ = T ∗ E may ∗ be identified with the cotangent bundle T E, where E ≡ T M is the parity-inverted tangent bundle with local coordinates (x i , ci ). More precisely, the momentum variables (Pi , bi ) can be identified with the fiber coordinates of T ∗ E, and (Pi , bi ) are co-vectors in that sense [32, 33]. On the other hand, if one instead had started with the cotangent bundle T ∗ E rather than the Whitney sum E, one would have gotten the symplectic structure (5.24) for free, without the use of a connection ∇. Depending on the application, it is useful to do just that. The catch is that the Pi variables follow a more complicated set of transformation rules than the pi variables. There is a natural tilde isomorphism ∼: (T M) → ( T M), which maps vectors to parity-inverted vectors, (T M) X = X i pi X
∼
−→
X i bi =: X˜ ∈ ( T M), X˜ = 1 − X .
(5.27)
In particular, vectors X and parity-inverted vectors X˜ are bosons and fermions, respectively, if the coordinate functions X i = X i (x) are bosonic, as is normally the case. More generally, we define the tilde operation ∼: C ∞ (E) → C ∞ (E) to be the right derivation ← ← ∂r ∂r a˜ := (a )bi +(a ) pi , ∂ pi ∂bi
a = a(x i , pi , ci , bi ) ∈ C ∞ (E),
(∼) = 1. (5.28)
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323
Table 4. The anti-bracket (·, ·) xj
bj
cj
xi
0
−δ ij
0
bi
δi
−Ri j
−ik ck
ci pi
j
pj 0 j
0
ijk ck
0
− kji pk
ikj pk −δ ij
0 j δi
0
The Poisson bracket on vectors and exterior forms mimics the Lie bracket, the covariant derivative and the interior product (=contraction), {X, Y } = =: {X, Y˜ } = { X˜ , Y˜ } = {X, η} = { X˜ , η} = {ξ, η} =
∂Y j X Y p − (−1) (X ↔ Y ) − X i Y j R˜ i j j ∂ xi [X, Y ] − R(X, Y )∼ , ∂Y j X i ( i b j + ikj Y j bk ) =: (∇ X Y )k bk =: (∇ X Y )∼ , ∂x 0, X, Y ∈ (T M), → → j k ∂ i ∂ )η =: ∇ X η, X ( i − ik c ∂x ∂c j ∂l η X i i =: i X η, ∂c 0, ξ, η ∈ ( • (T ∗ M)). Xi
(5.29) (5.30) (5.31) (5.32) (5.33) (5.34)
The sign conventions are, (i X ) = 1− X = X˜ ,
i λX η = λ i X η,
(λX )∼ = λ X˜ ,
where λ is a supernumber.
X λ = (−1)λ X λX, (5.35)
5.3. Anti-symplectic structure. There is a dual formulation in terms of an odd Poisson bracket, also known as an anti-bracket and traditionally denoted as (·, ·) in the physics literature. The symbol {·, ·} will be reserved for the even Poisson bracket introduced in the last Subsect. 5.2. The anti-bracket (·, ·) on E ∗ ⊆ E is given as [25] j
δi = −(x j , bi ), (bi , x j ) = i j (x , x ) = 0, −(bi , b j ) = R k i j c pk =: Ri j .
(5.36)
It is well-known that in the flat case R k i j = 0, this anti-bracket (5.36) is just the SchoutenNijenhuis bracket ← → ← → ∂r ∂ ∂r ∂ (π, ρ) S N := π( − i )ρ, (5.37) i ∂bi ∂ x ∂ x ∂bi
324
K. Bering Table 5. The anti-bracket (·, ·) in Darboux coordinates xj
Bj
cj
pj
xi
0
−δ ij
0
0
Bi ci
δi 0
0 0
0 0
0 −δ ij
pi
0
0
δi
j
j
0
Table 6. The anti-symplectic two-form ω˜ in Eq. (5.42) consists of the inverse matrix of Table 4
dx i
dx j
db j
dc j
dpj
−∂[i kj] c pk
j δi
ikj pk
ik ck
0
0
0
0
0
δi
0
−δ ij
dbi
−δ ij
dci
− kji pk − ijk ck
d pi
j
j
0
when restricting to skewsymmetric multi-vectors π, ρ ∈ ( • (T M)) ∼ = C ∞ (E ∗ ). j j Similar to Eq. (5.18), the natural skewsymmetric pairing ∂i , d x − = δi = −d x j , ∂i − can be modelled over an anti-bracket, j
( pi , c j ) = δi = −(c j , pi ), (ci , c j ) = 0 = ( pi , p j ).
(5.38)
The anti-bracket (·, ·) is extended to the Whitney sum E by fixing the cross-sectors as (x i , p j ) = 0 = ( p j , x i ), (x i , c j ) = 0 = (c j , x i ), (bi , p j ) = ikj pk = −( p j , bi ),
(5.39)
j
−(bi , c j ) = ik ck = (c j , bi ), cf. Table 4. When one defines the anti-field variables Bi := bi + ikj c j pk ,
(5.40)
the quadruple (x i , Bi , ci , pi ) becomes local Darboux coordinates for the anti-bracket (·, ·), cf. Table 5. The corresponding anti-symplectic two-form reads ω˜ = dx i ∧ dBi + dci ∧ d pi ,
dω˜ = 0,
(5.41)
cf. Table 6. One may choose the following globally defined anti-symplectic potential: −ϑ˜ = Bi dx i + pi dci = dx i Bi + dci pi ,
dϑ˜ = ω. ˜
(5.42)
Note that the anti-symplectic potential ϑ˜ is – as the name suggests – equal to the symplectic potential (5.26) acted upon with the tilde operator “∼”, cf. Eq. (5.28). The (2d|2d) anti-symplectic manifold E ∼ = T ∗ E may be identified with the parity-inverted cotangent bundle T ∗ E. More precisely, the anti-field variables (Bi , pi ) can be identified with the fiber coordinates of T ∗ E.
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Table 7. A list of manifolds in diagram (5.43) Manifold
Local coordinates
Base manifold M
(x i )
Structure
Cotangent bundle T ∗ M
(x i , pi )
Parity-inv. tang. bdl. E ≡ T M
(x i , ci )
Parity-inv. cot. bdl. T ∗ M ∼ = E∗
(x i , bi )
Parity-inverted tangent bundle T (E ∗ )
(x i , bi ; ci , −Pi )
Cotangent bundle T ∗ (E ∗ )
(x i , bi ; Pi , ci )
Symplectic potential −Pi dx i + ci dbi
Cotangent bundle T∗E
(x i , ci ; Pi , bi )
Symplectic potential ϑ = −Pi dx i + bi dci
Whitney sum E ≡ T ∗ M ⊕ E ⊕ E ∗ equipped with a connection ∇ : (T M) × (T M) → (T M)
Pi = pi + ikj c j bk (x i ; pi , ci , bi ) Bi = bi + ikj c j pk
Parity-inverted cotangent bundle T ∗ E
(x i , ci ; Bi , pi )
Anti-symplectic potential ϑ˜ = −Bi dx i − pi dci
Parity-inverted cotangent bundle T ∗ (T ∗ M)
(x i , pi ; −Bi , ci )
Anti-symplectic potential −Bi dx i + ci d pi
Parity-inverted tangent bundle T (T ∗ M)
(x i , pi ; ci , Bi )
Sympl. pot. − pi dx i
Anti-sympl. pot. −bi dx i
We here display a commutative diagram of various bundle isomorphisms, so-called Legendre transformations and canonical projection maps possible [29, 30, 32, 33] Legendre ∼ T ∗ (E ∗ ) T∗E =
(if ∇) (if ∇) ∼ T ∗ E = E ∼ =
Anti Legendre ∼ T ∗ (T ∗ M) =
"
#
↓
%
↓
#
↓
%
T (E ∗ )
→
E ⊕ E∗
→
M
←
T ∗M ⊕ E
←
" T (T ∗ M) (5.43)
cf. Table 7. We shall actually only use the bundle isomorphism T ∗ E ∼ = (E; ∇) ∼ = T ∗ E, corresponding to the upper middle part of the diagram (5.43). These bundle identifications will from now on often be used without explicitly mentioning it, as it will be clear from the context whether they have been applied or not. The rest of the diagram (5.43) is shown for the sake of completeness. The Grassmann-odd tilde transformation (5.28) (which exchanges the fiber coordinates pi ↔ bi and Pi ↔ Bi ), is responsible for the apparent reflection symmetry along a vertical symmetry axis in the diagram (5.43). We mention in passing that the anti-bracket may be encoded in a commutative Koszul hierarchy (−1)a (a, b) = 2 (a, b) = [[, L a ], L b ]1,
a, b ∈ C ∞ ( T ∗ E), (5.44)
326
K. Bering
of an odd, nilpotent, second-order operator → → → → ∂ ∂ ∂ ∂ + i =− i ∂ x ∂ Bi ∂c ∂ pi → → → → → → → ∂ ∂ ∂ 1 ∂ ∂ ∂ j j k ∂ k + ik c + Ri j ) + i , = (− i − i j − i j pk j ∂x ∂ pj ∂c 2 ∂b j ∂bi ∂c ∂ pi
(5.45)
(5.46)
cf. Subsect. 3.2. Both formulas (5.45) and (5.46) are invariant under coordinate transformations in the base manifold M without the use of a volume form [12]. This is due to a balance of bosonic and fermionic degrees of freedom. The latter formula (5.46), which uses the bundle isomorphism T ∗ E ∼ = E, cf. diagram (5.43), is presumably new. Note that the higher Koszul brackets and the Koszul zero-bracket vanish, i.e. n = 0 for n ≥ 3 and for n = 0, so this is an example of a commutative Batalin-Vilkovisky algebra. The anti-bracket on vectors and exterior forms mimics the covariant derivative, the Lie bracket and the interior product, as was the case for the even Poisson bracket {·, ·}. However in the anti-bracket case the rôles of X and X˜ are exchanged, (X, Y ) = 0,
X, Y ∈ (T M),
(5.47)
∂Y ( X˜ , Y ) = X i ( i p j + ikj Y j pk ) =: (∇ X Y )k pk =: ∇ X Y, ∂x ∂Y j i X Y ˜ ˜ b j − (−1) (X ↔ Y ) − X i Y j Ri j (X , Y ) = X ∂ xi = [X, Y ]∼ − R(X, Y ), ∂l η (X, η) = X i i =: i X η, ∂c → → ∂ ∂ j ( X˜ , η) = X i ( i − ik ck j )η =: ∇ X η, ∂x ∂c (ξ, η) = 0, ξ, η ∈ ( • (T ∗ M)). j
(5.48)
(5.49) (5.50) (5.51) (5.52)
5.4. Derived brackets. We now use the even and odd symplectic structure {·, ·} and (·, ·) from Subsect. 5.2-5.3 to define a homotopy Lie algebra structure on C ∞ (T ∗ E) and C ∞ ( T ∗ E), respectively. Using the bundle isomorphisms of diagram (5.43), we then obtain two homotopy Lie algebra structures on the same algebra C ∞ (E). The even and odd symplectic structure {·, ·} and (·, ·) will here both play the rôle of the Lie algebra structure [·, ·] of Sect. 4. It comes in handy that we have left the parity ∈ {0, 1} of the [·, ·] bracket open, so that we readily can model both even and odd brackets. Similarly, we will have two generators for the derived •-brackets, a Grassmann-odd generator Q and a Grassmann-even generator S, which (despite the notation) both will play the rôle of the fixed Lie algebra element Q of Sect. 4. The new Q and S are defined as [25] Q := ci Pi + Hodd = ci pi + 21 ci c j Tikj bk + Hodd ,
Q = 1,
S := ci Bi + Heven = ci bi + 21 ci c j Tikj pk + Heven ,
S = 0,
(5.53)
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327
where the exterior forms Hodd ∈ ( odd (T ∗ M)) (resp. Heven ∈ ( even (T ∗ M))) are closed Grassmann-odd (resp. Grassmann-even) forms. The local expressions for Q ∈ C ∞ (T ∗ E) and S ∈ C ∞ ( T ∗ E) are invariant under coordinate transformations in the base manifold M, and hence define global scalars. Here Tikj in Eq. (5.53) is the torsion tensor, Tikj = ikj − kji . (5.54) The generators Q and S are nilpotent in the Poisson and anti-bracket sense, ∂ Hodd = 2d Hodd = 0, (5.55) ∂ xi ∂ Heven (S, S) = 2ci = 2d Heven = 0, (5.56) ∂ xi respectively, because Hodd and Heven are closed. (The nilpotency (5.55) and (5.56) are also due to the first Bianchi identity, {Q, Q} = 2ci
R
ki j
i, j,k cycl.
=
(
i, j,k cycl.
∂ Tij ∂xk
+ km Timj ),
(5.57)
if one starts from the latter expressions in Eq. (5.53) that depend on pi and bi explicitly.) We stress that the nilpotent generators Q and S, and similarly, the even and odd Poisson structures {·, ·} and (·, ·), are on completely equal footing, regardless of what the notation might suggest. (The notation is inspired by the physics literature on constrained dynamics; for instance Eq. (5.56) resembles the Classical Master Equation of Batalin and Vilkovisky.) The generators Q and S turn C ∞ (T ∗ E) and C ∞ ( T ∗ E) into homotopy Lie algebras with derived •-brackets Q and S , respectively. The zero-bracket 0Q = Q is the generator Q itself. The one-bracket 1Q = {Q, ·} is a nilpotent, Grassmann-odd, Hamiltonian vector field on T ∗ E, cf. Eq. (4.38), Similarly for the S bracket hierarchy. For exterior forms η ∈ C ∞ (E) (which can be viewed as functions on any of the three bundles E, T ∗ E and T ∗ E mentioned in Table 7), the one-brackets are just the de Rham exterior derivative d, ⎫ 1Q (η) = {Q, η} ⎬ ∂η (5.58) = ci i = dη, ⎭ ∂ x 1 S (η) = (S, η) independent of Hodd , Heven and ∇. On vectors X ∈ (T M), the one-bracket 1Q reads 1Q (X ) = {Q, X } = d X + ci ( t )ikj X j Pk − ci c j ∂i ( t )kj bk X − (−1) X ∇ X Hodd = ∇ t X − (R t )k bk X − (−1) X ∇ X Hodd , 1S ( X˜ )
= (S, X˜ ) = d X + c ∼
i
( t )ikj X j Bk
X i j
− (−1) c c
(5.59)
∂i ( t )kj
pk X
X
−(−1) ∇ X Heven = ∇ t X ∼ − (−1) X (R t )k pk X − (−1) X ∇ X Heven , 1Q ( X˜ )
= {Q, X˜ } = (−1) X Pi + d X + (−1) i X Hodd ∈ C (T E) = (−1) X X + ∇ t X ∼ + (−1) X i X Hodd , 1 S (X ) = (S, X ) = Bi X i + d X + (−1) X i X Heven ∈ C ∞ ( T ∗ E) = (−1) X X˜ + ∇ t X + (−1) X i X Heven , X
i
∼
X
∞
(5.60)
∗
(5.61) (5.62)
328
K. Bering
where ∇ = ci ∇i is the connection one-form, R k = 21 ci c j R k i j is the curvature twoform, and ∇ t denotes the transposed connection, which is defined as ( t )ikj := kji . The geometric importance of these one-brackets is underscored by the fact that their adjoint actions (in the Poisson or anti-bracket sense) reproduce the Lie derivative on exterior forms ⎫ {1Q ( X˜ ), η} = {Q, { X˜ , η}} + (−1) X { X˜ , {Q, η}} ⎬ = [d, i X ]η = (−1) X L X η, ⎭ (1S (X ), η) = (S, (X, η)) + (−1) X (X, (S, η)) (5.63) independent of Hodd , Heven and ∇. The two-brackets 2Q and 2S give rise to an odd and an even Courant bracket, (·, ·) Q and {·, ·} S , respectively, cf. Eq. (4.10). Definition 5.2. The odd and even Courant brackets (·, ·) Q and {·, ·} S are defined as the derived brackets, 1 1 {{a, Q}, b} + {a, {Q, b}} 2 2 = −(−1)(a +1)(b +1) (b, a) Q , a, b ∈ C ∞ (T ∗ E), 1 1 {a, b} S := (−1)a 2S (a, b) = ((a, S), b) + (a, (S, b)) 2 2 = −(−1)a b {b, a} S , a, b ∈ C ∞ ( T ∗ E),
(a, b) Q := (−1)a +1 2Q (a, b) =
(5.64)
(5.65)
respectively. Notice the complete democracy among the even and odd brackets in Definition 5.2. By restricting to vector fields and exterior forms one finds the celebrated formulas (5.1)(5.3) for the Courant bracket,
{X, η} S
{X, Y } S = [X, Y ] + (−1)Y i X i Y Heven , X, Y ∈ (T M), (5.66) ∼ Y (5.67) ( X˜ , Y˜ ) Q = [X, Y ] + (−1) i X i Y Hodd , 1 = ( X˜ , η) Q = (L X + i X d)η, (5.68) 2 {ξ, η} S = 0, (5.69) • ∗ (ξ, η) Q = 0, ξ, η ∈ ( (T M)), (5.70)
which do not use the bundle isomorphisms of diagram (5.43), and hence are independent of ∇; plus one finds the following formulas: 1 1 (−1)Y ∇ X [∇ t Y ] − ∇it X (∇ t Y )i − ∇ X ∇Y Hodd 2 2 2 2 ( X +1)(Y +1) +O(c b) − (−1) (X ↔ Y ), (5.71) Y
1 (−1) ∇it X˜ (∇ t Y )i − ∇ X ∇Y Heven { X˜ , Y˜ } S = ∇ X [∇ t Y˜ ] + 2 2 +O(c2 p) − (−1)( X +1)(Y +1) (X ↔ Y ), (5.72) X 1 (−1) = { X˜ , η} S = ([∇ X , d] + ∇ X d)η = ∇ X dη − d∇ X η, (5.73) 2 2 (X, Y ) Q =
(X, η) Q
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329
which do use the bundle isomorphisms of diagram (5.43), and whose right-hand sides are second-order differential operators and not particularly illuminating. The lesson to be learned is that one should stick to vector fields without parity-inversion X ∈ (T M) ⊆ C ∞ ( T ∗ E) for the S brackets on the algebra C ∞ ( T ∗ E), and to vector fields with parity-inversion X˜ ∈ (E) ⊆ C ∞ (T ∗ E) for the Q brackets on the algebra C ∞ (T ∗ E). 5.5. Discussion. Obviously, we have only scratched the surface. One can for instance also calculate what the Courant brackets should be on skewsymmetric multi-vectors. For instance, the one-bracket 1Q on a multi-vector π ∈ ( • (T M)) ∼ = C ∞ (E ∗ ) ⊆ C ∞ (T ∗ E) reads 1Q (π )
← → ∂r ∂ = dπ + (Pi + Hodd i )( π ) ∈ C ∞ (T ∗ E) ∂c ∂bi ← → ∂r ∂ t π ), π ∈ ( • (T M)). = ∇ π + ( pi + Hodd i )( ∂c ∂bi
(5.74)
The Courant two-bracket (·, ·) Q becomes a twisted version of the Schouten-Nijenhuis bracket (5.37), ← → ← → ∂r ∂r ∂ ∂ (π, ρ) Q = (π, ρ) S N + (π )( i Hodd j )( ρ) ∈ C ∞ (T ∗ E), ∂bi ∂c ∂c ∂b j π, ρ ∈ ( • (T M)),
(5.75)
which does not depend on the Pi variables. However, the Courant bracket (π, η) Q between a higher-order skewsymmetric multi-vector, π , and a higher-order form, η, does not close on such objects, but will in general also depend on the Pi variables. A complete treatment of the Courant bracket would therefore include an investigation of form-valued multi-vectors, where the word “multi-vector” here should be understood in a generalized sense that depends on both the bosonic and fermionic generators, pi ≡ ∂i ∈ (T M) and bi ≡ ∂˜i ∈ ( T M), or the analogues obtained via the bundle identifications of diagram (5.43). It is then natural, in turn, to allow the twisting Hodd (resp. Heven ) to be an odd (resp. even) form-valued multi-vector as well. The even or odd Poisson construction may be generalized further to an arbitrary Poisson or anti-Poisson manifold with a nilpotent function Q or S satisfying {Q, Q} = 0 or (S, S) = 0, respectively. Finally, let us mention that the odd bracket (·, ·) Q may be viewed as a classical counterpart of the odd operator bracket [·, ·] D in the spirit of deformation quantization. In detail, one defines a hat quantization map “∧” that takes functions a (also known as symbols) to normal-ordered differential operators a, ˆ C ∞ (T ∗ E) a = a(x i , ci , Pi , bi )
∧
−→ → → ← → ← → r ∂ ∂ ∂ ∂ ∂ r ∂ aˆ = : a(x i , ci , i , i ) : = a exp( + ) i ∂ x ∂c ∂ Pi ∂ x ∂bi ∂ci
∈ End(C ∞ (E)).
Pi ,bi =0
(5.76)
330
K. Bering
The colons in Eq. (5.76) indicate normal-ordering, which means that the derivatives are ordered to the rightmost position. Examples of the “∧” quantization map are ηˆ = η,
Xˆ˜ = i X ,
Qˆ = D,
→ → ∂ k j ∂ pˆ i = − c = ∇i , i j ∂ xi ∂ci
(5.77)
cf. Eqs. (5.4), (5.8), (5.53) and (5.5). The algebra isomorphism aˆ ◦ bˆ = a# b between corresponding associative algebras is given by the product, ← → ← → ∂r ∂ ∂r ∂ a b = (a exp( + ) b) ∈ C ∞ (T ∗ E), ∂ Pi ∂ x i ∂bi ∂ci
a, b ∈ C ∞ (T ∗ E). (5.78)
The outer parenthesis on the right-hand side of Eq. (5.78) indicates that the expression should be interpreted as a function, i.e. a zeroth-order differential operator. The “∧” quantization map (5.76) is clearly related to the symplectic structure (5.24). In fact, the commutator [a, b] ≡ a b − (−1)a b b a is a quantum deformation of the Poisson bracket {a, b}, where the Planck constant here is set equal to the value −2πi, that is, i h¯ = 1. In the same way we get the new result, that the Courant operator bracket ˆ D= [a, ˆ b]
1 ([[a, Q] , b] + [a, [Q, b] ] )∧ , 2
a, b ∈ C ∞ (T ∗ E),
(5.79)
is a quantum deformation of the classical Courant bracket (a, b) Q = 21 {{a, Q}, b} + 21 {a, {Q, b}}. Recall that the bracket (·, ·) Q and the operator counterpart [·, ·] D are both odd brackets. They both realize the Courant bracket (5.1)-(5.3) with the help of parity-inverted vectors, either directly via X˜ ∈ C ∞ (T ∗ E), or via the contraction i X ∈ End(C ∞ (E)). They are also both capable of reproducing the same twisting with an odd closed form Hodd . On the other hand, the possibility of twisting the Courant bracket with an even closed form Heven seems to have gone unnoticed so far in the literature. The even bracket {·, ·} S realizes just this possibility. 5.6. Higher brackets. It is a major point that the derived bracket construction (4.6) of the Courant bracket automatically provides us with an infinite tower of higher Courant brackets and a host of nilpotency relations (4.13) corresponding to all the allowed values of the ckn product coefficients found in Sect. 4, like for instance the Ward identity (4.36) and the generalized Jacobi identity (4.43). In this subsection we calculate the higher Courant brackets in pertinent sectors. The connection ∇ and the bundle isomorphisms of diagram (5.43) are not used in the rest of the paper. Proposition 5.3. The higher Courant brackets among vectors read nS (X (1) , . . . , X (n) ) = i X (1) . . . i X (n) Heven , $ nQ ( X˜ (1) , . . . , X˜ (n) ) = i X (1) . . . i X (n) Hodd , nD (i X (1) , . . . , i X (n) )
X (1) , . . . , X (n) ∈ (T M), (X (1) ) = 0, . . . , (X (n) ) = 0, (5.80)
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331
for n ∈ {3, 4, 5, . . .}. The Courant three-bracket between two vectors X, Y ∈ (T M) with Grassmann-parity X = 0 = Y and one exterior form η ∈ ( • (T ∗ M)), is 3S (X, Y, η) = 3Q ( X˜ , Y˜ , η) = 3D (i X , i Y , η) 1 1 = + (i L − i Y L X ) + i X i Y d η i 3 [X,Y ] 2 X Y 1 1 = (i X LY − i Y L X ) + di X i Y η 2 3 1 1 = i X i Y d − di X i Y + i [X,Y ] η. 2 3
(5.81)
The higher Courant brackets of vectors and one exterior form are given recursively as n+1 ˜ n+1 ˜ n+1 S (X (1) , . . . , X (n) , η) = Q ( X (1) , . . . , X (n) , η) = D (i X (1) , . . . , i X (n) , η) 1 = (−1)i−1 i X (i) nQ ( X˜ (1) , . . . , X˜ (i) , . . . , X˜ (n) , η) n+1 1≤i≤n
(5.82) for n ∈ {3, 4, 5, . . .} and (X (1) ) = 0, . . ., (X (n) ) = 0 bosonic, or directly as ˜ ˜ n+1 Q ( X (1) , . . . , X (n) , η) =
6 (n+1)n(n−1)
(−1)(n−1−i)+(n− j)
×i X (1) . . . i X (i) . . . i X ( j) . . . i X (n) 3Q X˜ (i) , X˜ ( j) , η 1≤i< j≤n
(5.83) for n ∈ {2, 3, 4, . . .}. Brackets between vectors and exterior forms with more than one exterior form as an argument vanish. We note that the (n+1)-brackets in Eqs. (5.82) and (5.83) vanish if the form degree of η is ≤ n−2. Proof of Proposition 5.3. The calculations are most efficiently done along the diagonal X n with the vector X taken to be a fermion, X = 1. Hence the parity-inverted vector X˜ and i X are bosons. One finds 2S (X X ) = −[X, X ] + i X2 Heven , 2Q ( X˜ X˜ ) = −[X, X ]∼ + i X2 Hodd , 2D (i X i X ) nS (X n ) % nQ ( X˜ n ) nD (i X n )
= =
−i [X,X ] + i X2 Hodd , (−i X )n Heven ,
= (−i X )n Hodd ,
(5.84) (5.85) (5.86) (5.87)
X = 1,
(5.88)
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for n ∈ {3, 4, 5, . . .}. These facts imply that the expansion Eq. (4.24) truncates after only three terms, n−1 1 (−ad X˜ )n−1−k kQ ( X˜ k ), η nQ ( X˜ (n−1) η) = n k=0
2 1 (−i X )n−1−k kQ ( X˜ k ), η = n k=0 3 = (−i X )n−3 3Q ( X˜ X˜ η), n
(5.89)
with X = 1 and n ∈ {3, 4, 5, . . .}. Similarly for the S and D hierarchies. In the n = 3 case this reads 1 2 i X d + i X L X − i [X,X ] η 3S (X X η) = 3Q ( X˜ X˜ η) = 3D (i X i X η) = 3 1 2 1 2 1 2 = i X L X + di X η = X = 1. i X d − di X − i [X,X ] η, 3 2 3 (5.90) n λ(i) X (i) with (λ(i) ) = 1 and (X (i) ) = 0, cf. SubNow apply polarization X = i=1 section 2.4. The vanishing of higher brackets with two or more exterior forms follows from Eqs. (5.34) and (5.52). 5.7. B-transforms. In this subsection the B-transforms [15, 16] are generalized to the higher brackets. The B-transforms in this context are canonical or anti-canonical transformations generated by even or odd exterior forms Beven or Bodd , respectively. It follows immediately from the derived bracket definition (4.6) that
e−[Beven ,·] nD (a1 , . . . , an ) = nD e−[Beven ,·] a1 , . . . , e−[Beven ,·] an ,
e−{Beven ,·} nQ (a1 , . . . , an ) = nQ e−{Beven ,·} a1 , . . . , e−{Beven ,·} an ,
e−(Bodd ,·) nS (a1 , . . . , an ) = nS e−(Bodd ,·) a1 , . . . , e−(Bodd ,·) an , (5.91) where we have defined D := e−[Beven ,·] D,
Q := e−{Beven ,·} Q, S := e−(Bodd ,·) S.
(5.92)
The B-transforms are a symmetry of the derived brackets, i.e. D = D,
Q = Q, S = S,
(5.93)
if the B-forms are closed, [D, Beven ] = d Beven = 0,
{Q, Beven } = d Beven = 0,
(S, Bodd ) = d Bodd = 0. (5.94) In this way it becomes obvious, that closed B-transforms are algebra automorphisms for the full Courant •-bracket hierarchy [15, 16].
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6. Supplementary Formalism Section 6 is an open-ended investigation, where we make contact to some notions that could be useful for future studies. More specifically, we go back to the original setup of Sect. 2 and touch on some of the theoretical aspects of a homotopy Lie algebra A, such as properties of the bracket product “◦” and co-algebraic structures on Sym• A. 6.1. Pre-Lie products. What properties should one demand of the “◦” product? Associativity is too strong: This is not even fulfilled for the ordinary product with coefficients ckn = 1. The next idea is to let the product “◦” be pre-Lie. To measure the non-associativity one usually defines the associator ass(, , ) := ( ◦ ) ◦ − ◦ ( ◦ ).
(6.1)
Definition 6.1. The bracket product “◦” is pre-Lie if for all •-brackets , , : Sym• A → A the associator is symmetric in the last two entries, ass(, , ) = (−1) ass(, , ).
(6.2)
For a pre-Lie product “◦” the commutator [, ] := ◦ − (−1) ◦
(6.3)
becomes a Lie bracket that satisfies the Jacobi identity, hence the name “pre-Lie”. One may simplify the pre-Lie condition (6.2) by polarization into ass(, , ) = 0,
= 1,
(6.4)
cf. Subsect. 2.4. We now give a necessary and sufficient condition in terms of the ckn coefficients for the “◦” product to be pre-Lie. Proposition 6.2. A “◦” product is pre-Lie if and only if the ckn product coefficients satisfy the following two conditions: ∀k, , m ≥ 0 :
⎧ +m+1 = ck++m ck+ , ⎨ ckk++m c+1 k k+ ⎩
ckk++m c+m+1 = (k ↔ ).
(6.5)
Proof of Proposition 6.2. To see Eq. (6.5), first note that for two vectors a, b ∈ A with (a) = , ( ◦ )n+1 (ba n ) =
n k +1 n+1 n−k+1 k+1 bk+1 (ba k )a (n−k) n+1 k=0 n n−k +1 n+1 n−k+2 k k bk (a )ba (n−k) + n+1 k=0
(6.6)
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K. Bering
for n ∈ {0, 1, 2, . . .}. Therefore n ( ◦ ) ◦ (a n )
+m+1 +1 m+1 +1 k (a k )a a m = bkn b+1 +m +1 k, , m ≥ 0 k ++m = n
+ b+m+1
m +1 m+2 k k m (a ) (a )a . +m +1
On the other hand, n ◦ ( ◦ ) (a n ) =
(6.7)
n bk+ bkk+ m+1 +1 k (a k )a a m .
k, , m ≥ 0 k ++m = n
(6.8) Next insert the two expressions (6.7) and (6.8) into the pre-Lie condition (6.4) with = odd. By comparing coefficients one derives ⎧ k++m +m+1 +1 k++m bk+ , b+1 +m+1 = bk+ ⎨ bk k (6.9) ∀k, , m ≥ 0 : ⎩ k++m +m+1 m+1 b = (k ↔ ). bk +m+1 The two conditions (6.5) follow by translating (6.9) into the “c” picture with the help of Eq. (2.27). We would like to find the possible ckn coefficients that solve the two necessary and sufficient pre-Lie conditions (6.5). The full problem turns out to be quite involved. For simplicity, we shall work within the following generic case: c00 = 0 ∧ c01 = 0.
(6.10)
Theorem 6.3. The “◦” product is pre-Lie and satisfies the condition (6.10), if and only if there exist non-zero complex numbers λn , n ∈ {0, 1, 2, . . .}, such that ckn =
λk λn−k+1 . λn
(6.11)
In other words, a generic pre-Lie product is essentially just an ordinary product ckn = 1 with rescaled brackets n → λn n . Note that the solution (6.11) is non-degenerate, cf. Eq. (2.4), and implies that c1n = cnn does not depend on n ∈ {0, 1, 2, . . .}. It turns out that in the special case c00 = 0 ∨ c01 = 0, there exists infinitely many disconnected solutions to be classified elsewhere. Proof of Theorem 6.3: It is simple to check that the solution (6.11) satisfies the two pre-Lie conditions (6.5) and the technical condition (6.10). Now let us prove the other direction. Putting k = 0 in the first of the two conditions (6.5), one gets after relabelling ∀k, n :
0≤k≤n
⇒
n+1 c0n ck+1 = c0k ckn .
(6.12)
∀n ∈ {0, 1, 2, . . .} : c0n c0n+1 c2n+2 = c0n c00 c01 .
(6.13)
One may apply Eq. (6.12) twice to produce
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335
Equation (6.13) and assumption (6.10) lead to ∀n ∈ {0, 1, 2, . . .} : c0n = 0
(6.14)
by an inductive argument. Therefore one may define non-zero numbers λn :=
n−1
c0k = c0n−1 λn−1 , λ0 := 1.
(6.15)
k=0
Repeated use of (6.12) leads to the solution (6.11). Interestingly in the generic case (6.10), one does not need the second of the two pre-Lie conditions (6.5) to derive the solution (6.11). 6.2. A co-product. Similar to the “◦” bracket product construction (2.3) one may define a co-product ' on Sym• A. Definition 6.4. Let there be given a set of complex numbers γkn with n ≥ k ≥ 0. The coproduct ' : Sym• A → Sym• A⊗Sym• A is then defined as '(a1 . . . an ) := n γn k
k=0
k!(n−k)!
(−1)π,a (aπ(1) . . . aπ(k) )⊗(aπ(k+1) . . . aπ(n) )
(6.16)
π ∈Sn
for n ∈ {0, 1, 2, . . .}. We denote the coefficients γkn , n ≥ k ≥ 0, with a Greek γ to stress that they in general differ from the “◦” bracket product coefficients ckn . Recall that “⊗” and “” denote the un-symmetrized and symmetrized tensor product in the tensor algebras T • A and Sym• A, respectively, cf. Eq. (2.1). The Grassmann parity of the co-product ', and the bracket product “◦” are assumed to be bosonic, (') = (◦) = 0,
(6.17)
while the parity of the symmetrized and the un-symmetrized tensor products “” and “⊗” follows the suspension parity, () = (⊗) = .
(6.18)
The co-product definition (6.16) is by polarization equivalent to '(a n ) =
n
βkn a k ⊗a (n−k) , (a) = ,
(6.19)
k=0
where we assume that an analogue of Eq. (2.27) holds for the Greek co-product coefficients βkn and γkn , n γkn , 0 ≤ k ≤ n. βkn ≡ (6.20) k The standard co-product ' on Sym• A corresponds to coefficients γkn = 1, cf. Ref. [27].
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K. Bering
Definition 6.5. A co-product ' is co-associative if (1⊗')' = ('⊗1)'.
(6.21)
Co-associativity is equivalent to k++m k+ ∀k, , m ≥ 0 : βkk++m β+m = βk+ βk ,
which generically looks like βkk+ = n . if βkn = βn−k
λk λ λk+ .
(6.22)
Similarly, a co-product ' is co-commutative
Definition 6.6. A linear operator δ : Sym• A → Sym• A is a co-derivation [27] if 'δ = (δ⊗1 + 1⊗δ) '.
(6.23)
˜ for •-brackets. 6.3. A lifting. We now describe a lifting map “∼”: → ˜ −→ Sym• A
Sym• A
(6.24)
# A
One co-product plays a special rôle in this lifting. This is the co-product that corresponds to the “◦” bracket product itself, i.e. when the co-product coefficients γkn are equal to the “◦” product coefficients ckn . Let this particular co-product be denoted by a triangle ◦
' with a “◦” on top. Definition 6.7. Let there be given a bracket product “◦” and a •-bracket ˜ : Sym• A → Sym• A is defined as : Sym• A → A. The lifted bracket ◦
˜ := Sym• (⊗1) ',
(6.25)
or written out, ˜ 1 . . . an ) := (a n cn k
k=0
k!(n−k)!
(−1)π,a k (aπ(1) . . . aπ(k) )aπ(k+1) . . . aπ(n)
π ∈Sn
(6.26) for n ∈ {0, 1, 2, . . .}. The definition (6.26) is by polarization equivalent to ˜ a ) = (e
ck+ k k (a k )a , k!!
k,≥0
(a) = ,
(6.27)
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337
where we have defined a formal exponentiated algebra element as ea :=
1 a n , n!
(a) = .
(6.28)
n≥0
In case of a standard bracket product “◦” with coefficients ckn = 1, the lifting (6.26) ˜ becomes the standard lifting [27] of a •-bracket to a co-derivation δ = . Proposition 6.8. Let there be given a bracket product “◦” and a co-product '. All lifted ˜ are co-derivations, if and only if brackets ⎧ +m+1 = γ k++m ck+ , ⎨ ckk++m γ+1 k k+ ∀k, , m ≥ 0 : (6.29) ⎩ k++m +m+1 ck γ = γk++m ckk+m . The two co-derivation conditions (6.29) become identical if the co-product ' is cocommutative. The Proposition 6.8 suggests that one should adjust the co-product ' according to which “◦” product one is studying. For instance, if the bracket product coefficients ckn = λk λn−k+1 /λn are of the generic pre-Lie form (6.11), it is possible to satisfy the two co-derivation conditions (6.29) by choosing co-product coefficients of the form γkn = λk λn−k /λn . This choice of co-product is at the same time both co-associative and co-commutative. Proof of Proposition 6.8. First note that for two vectors a, b ∈ A with (a) = , n n k +1 n+1 n−k +1 n+1 k βk+1 (ba k )⊗a (n−k) + β a ⊗(ba (n−k) ) n+1 n+1 k k=0 k=0 (6.30) for n ∈ {0, 1, 2, . . .}. Therefore
n n +m+1 +1 ˜ '(a ) = k (a k )a ⊗a m bk β+1 +m +1
'(ba n ) =
k, , m ≥ 0 k ++m = n
+ β+m+1
m +1 k k a ⊗ (a )a m . +m +1
(6.31)
On the other hand, n ˜ (⊗1)(a ) =
n βk+ bkk+ k (a k )a ⊗a m ,
(6.32)
βn bkk+m a ⊗ k (a k )a m .
(6.33)
k, , m ≥ 0 k ++m = n
and n ˜ (1⊗)(a ) =
k, , m ≥ 0 k ++m = n
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Next insert the three expressions (6.31), (6.32) and (6.33) into the co-derivative definition ˜ By comparing coefficients one derives (6.23) with δ = . ⎧ k++m +m+1 +1 k++m bk+ , β+1 +m+1 = βk+ ⎨ bk k ∀k, , m ≥ 0 : (6.34) ⎩ k++m +m+1 m+1 k++m bk+m . β bk k +m+1 = β The two conditions (6.29) follow by translating (6.34) into the “c” picture with the help of Eq. (2.27). Proposition 6.9. A bracket product “◦” is pre-Lie, if and only if the lifting map “∼” is an algebra homomorphism, i.e. for all brackets , : Sym• A → A, ˜ ˜ . ( ◦ )∼ =
(6.35)
Proof of Proposition 6.9. First note that for two vectors a, b ∈ A with (a) = , a ˜ )= (be
ck++1 ck++1 k+1 k k+1 (ba k )a + k (a k )ba . k!! k!!
k,≥0
k,≥0
(6.36) ˜ and ˜ is Therefore the composition of two lifted brackets
c+m+1 ck++m k +1 +1 k (a k )a a m k!!m!
˜ ˜ (ea ) =
k,,m≥0
+
c+m+1 ck++m k (a )k (a k )a m . k!!m!
(6.37)
ck++m ck+ k k+ +1 k (a k )a a m . k!!m!
(6.38)
k,,m≥0
On the other hand, ( ◦ )∼ (ea ) =
k,,m≥0
By comparing coefficients in Eqs. (6.37) and (6.38) one sees that the condition (6.35) is equivalent to the two pre-Lie conditions (6.5). 6.4. Normalization. One may get back the •-bracket : Sym• A → A from its lifted ˜ : Sym• A → Sym• A with the help of the projection maps πn : Sym• A → bracket n Sym A. In particular, ˜ 1 . . . an ) = cnn (a1 . . . an ). π1 ◦ (a
(6.39)
Definition 6.10. A “◦” bracket product is normalized if ∀n ∈ {0, 1, 2, . . .} : cnn = 1.
(6.40)
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339
Obviously, a normalized “◦” product is non-degenerate, cf. Eq. (2.4). Moreover, in the normalized case one may sharpen Eq. (6.39) into ˜ = π1 ◦ ,
(6.41)
so that the following diagram is commutative. ˜ −→
Sym• A
#
Sym• A (6.42)
%
π1 A In the normalized case the product of brackets (2.3) may be related to composition of the lifted brackets as ˜ ˜ . ˜ = π1 ◦ = (6.43) This carries the advantage that composition, unlike the bracket product, is born associative. It is natural to ask what ckn coefficients would satisfy a nilpotency condition for a lifted bracket ˜ ˜ = 0, (6.44) = 1 ? Contrary to the nilpotency relations (2.5) for the “◦” product, which are first-order equations in the ckn coefficients, the nilpotency conditions (6.44) are quadratic in the ckn coefficients. We end this discussion with a corollary. Corollary 6.11. For a normalized pre-Lie product “◦”, the bracket is nilpotent with ˜ is nilpotent with respect to the bracket product “◦”, if and only if the lifted bracket respect to composition, i.e. ◦ = 0
⇔
˜ ˜ = 0.
(6.45)
6.5. The Ward solution revisited. As an example let us consider the Ward solution (4.33) of the derived bracket hierarchy, but this time normalized according to Eq. (6.40), ) 1 for k = 1 ∨ k = n, n ck = (6.46) 0 otherwise, or equivalently using Eq. (4.21), c(x, y) = x(e y − 1) + e x .
(6.47)
This solution is identical to the original solution (4.33) and (4.34), except for the fact that we have divided the first Ward identity (4.38) with 2, which is always permissible, cf. Subsect. 2.3. Proposition 6.12. The Ward solution (6.46) is pre-Lie, normalized and satisfies the nilpotency relations for the derived bracket hierarchy, i.e. [Q, Q] = 0 ⇒ Q ◦ Q = 0. We conclude that there is a non-empty overlap between the derived solutions found in Sect. 4, the pre-Lie property (6.2) and the normalization condition (6.40). Proof of Proposition 6.12. The Ward solution (6.46) is obviously normalized, cf. Eq. (6.40). We saw in Sect. 4 that it satisfies the nilpotency relations for the derived bracket hierarchy. The pre-Lie property (6.5) may either be checked directly, or perhaps more
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K. Bering
enlightening, one may consider product coefficients ckn = λk λn−k+1 /λn of the generic pre-Lie form (6.11) with ) 1 for k ∈ {0, 1}, λk = (6.48) k for k ∈ {2, 3, 4, . . .}, where is a non-zero complex number. One may easily see that the solution (6.11) becomes the Ward solution (6.46) in the limit → 0. Hence the Ward solution (6.46) is also pre-Lie by continuity. 7. Conclusions The paper contains the following main new results: • We found a non-commutative generalization (3.30) of the higher Koszul brackets, such that they form a homotopy Lie algebra. • We found the most general nilpotency relations for the derived bracket hierarchy, cf. Theorem 4.3. • We defined and calculated the higher Courant brackets, cf. Proposition 5.3. A common platform for all of these topics is provided by a (generalized) homotopy Lie algebra that allows for arbitrary non-degenerate prefactors ckn in the nilpotency relations. These prefactors can equivalently be viewed as a non-standard bracket product “◦”, cf. Eq. (2.3). The generalization is desirable because the original definition (2.6) of Lada and Stasheff [22] excludes important systems, for instance the derived bracket hierarchy, which in all other respects has the hallmarks of an L ∞ algebra, cf. Sect. 4. In Sect. 6 we have analyzed the (generalized) homotopy Lie algebras further, and we have displayed their co-algebra structures. The question remains whether the (generalized) homotopy Lie algebra definition (2.4) considered in this paper is the final say. For instance, should one demand the pre-Lie property (6.2) of a homotopy Lie algebra definition? As we saw in Subsect. 6.5 the pre-Lie property carries a small, but non-empty, overlap with the solutions (4.22) and (4.23) to the derived bracket hierarchy. A complete answer will require further studies and definitely more examples. Acknowledgements. Special thanks go to J.D. Stasheff and the referees for remarks on the first version of the text. The work of K.B. is supported by the Ministry of Education of the Czech Republic under the project MSM 0021622409.
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Communicated by Y. Kawahigashi
Commun. Math. Phys. 274, 343–355 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0281-8
Communications in
Mathematical Physics
Supersymmetry and Combinatorics E. Onofri1,2 , G. Veneziano3,4 , J. Wosiek5 1 2 3 4 5
Dipartimento di Fisica, Università di Parma, Parma, Italy I.N.F.N., Gruppo Collegato di Parma, 43100 Parma, Italy Theory Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail:
[email protected] Collège de France, 11 place M. Berthelot, 75005 Paris, France M. Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Cracow, Poland
Received: 10 April 2006 / Accepted: 20 February 2007 Published online: 23 June 2007 – © Springer-Verlag 2007
Abstract: We show how a recently proposed supersymmetric quantum mechanics model leads to non-trivial results/conjectures on the combinatorics of binary necklaces and linear-feedback shift-registers. Pauli’s exclusion principle plays a crucial role: by projecting out certain states/necklaces, it allows to represent the supersymmetry algebra in the resulting subspace. Some of our results can be rephrased in terms of generalizations of the well-known Witten index. 1. Introduction In a recent series of papers [1–3] two of us have introduced a supersymmetric quantum mechanical matrix model and studied some of its intriguing properties. The model is defined as the N → ∞ limit of a quantum mechanical system whose degrees of freedom are bosonic and fermionic N ×N creation and destruction operator matrices. The model’s supersymmetry charges and Hamiltonian are explicitly given by: 2
Q = Tr[ f a † (1 + ga † )], Q † = Tr[ f † (1 + ga)a], H = {Q † , Q} = H B + H F , 2
Q 2 = Q † = 0, 2
H B = Tr[a † a + g(a † a + a † a 2 ) + g 2 a † a 2 ], H F = Tr[ f † f + g( f † f (a † + a) + f † (a † + a) f ) +g 2 ( f † a f a † + f † aa † f + f † f a † a + f † a † f a)],
(1) (2) (3) (4)
where bosonic and fermionic destruction and creation operators satisfy [ai j , a † kl ] = δil δ jk ; { f i j f kl† } = δil δ jk ; i, j, k, l = 1, . . . N ,
(5)
all other (anti)commutators being zero. While taking the large-N limit, one keeps, as usual [4], the ’t Hooft coupling, λ ≡ g 2 N , fixed. Note that the Hamiltonian (4) conserves
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(commutes with) the fermionic number F = Tr[ f † f ]. Hence the system can be studied separately for each eigenvalue of F. By contrast, H does not commute with the bosonic number operator B = Tr[a † a] except in the trivial g → 0 limit. The model exhibits a number of interesting properties: (i) It is exactly soluble in the F = 0, 1 sectors, i.e. the complete energy spectrum and the eigenstates are available in analytic form, in particular it exhibits a discontinuous phase transition at λ = λc = 1. At this point the otherwise discrete spectrum loses its energy gap and becomes continuous. (ii) An exact weak-strong duality holds in the F = 0, 1 sectors relating spectra at λ and 1/λ. (iii) It exhibits unbroken supersymmetry, i.e. its E = 0 eigenstates consist of degenerate boson-fermion doublets. (iv) In the weak coupling phase, λ < 1, there is only one (unpaired) zero-energy state (also referred to as a SUSY vacuum). It lies in the F = 0 sector and is nothing else but the empty Fock state |0 while for λ > 1 there are two zero-energy states in each bosonic (even F) sector of the model. For F = 0 the Fock vacuum continues to be a zero energy eigenstate, but it is joined by another, non-trivial, analytically known ground state. For each higher even F, the two non-trivial “vacua” appear suddenly at λ > 1. Some understanding of these unexpected states was obtained by considering the λ → ∞ limit of the model [5]. In that same limit, the model can be connected to two interesting one-dimensional statistical mechanics quantum systems [5]. In the appropriate large-N limit defined above, the Hilbert space of the model can be restricted to the one corresponding to the action of single-traces of products of creation operators acting on the Fock vacuum. As such the vectors of the large-N Hilbert space can be put in one-to-one correspondence with binary necklaces, with the two beads representing bosonic and fermionic matrices. However, Fermi statistics provides a welldefined “Pauli razor”, which projects out a subset of all binary necklaces. As we shall see, only after this projection is performed, the resulting space does allow for a faithful representation of supersymmetry1 . The purpose of this paper is to illustrate how supersymmetry in our physical model gives non-trivial results on the combinatorics of binary necklaces and how, vice versa, known combinatorics results on the latter allow to determine the way supersymmetry is realized. In particular, combinatorics will allow us to understand where the null eigenstates lie and to compute the value of the Witten index [7] –and generalizations thereof– in different regions of λ. The rest of the paper is organized as follows: in Sect. 2 we explain how singletrace states are connected to necklaces and describe how they can be enumerated taking into account Fermi statistics; the concept of Pauli allowed or forbidden necklaces is introduced together with some examples. We also introduce there the connection with “linear feedback shift registers” which helps in finding the correct answer. In Sect. 3 we provide a generalization of Polya’s formula, by giving the number of forbidden necklaces with a prescribed number of bosonic and fermionic beads. In Sect. 4 we show how supersymmetry suggests combinatorial identities, which can also be proven by classical arguments, but are otherwise difficult to envisage. The Appendix provides a technical proof of a corollary of the main theorem. 1 After this work was completed we learned from M. Bianchi that some of the results presented here had already been derived (or guessed) by other methods in Refs. [6]. We wish to thank M. Bianchi for the information and for instructive discussions about that work.
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2. Fock States, Necklaces and Linear Feedback Registers The Hilbert space of our model (2) is spanned by states created by single trace operators, e.g. Tr[a † a † f † . . . a † a † f † f † ]|0. These are also eigenstates of the above-mentioned fermion and boson number operators F and B. Such states too can be labeled by binary numbers, e.g. (001 . . . 0011),
(6)
with 0 (1) corresponding to bosonic (fermionic) creation operators. Because of the cyclic property of a trace, all n binary sequences, related by cyclic shifts, describe the same state with n quanta, and consequently should be identified. Therefore our states correspond to what mathematicians define as necklaces - periodic chains made of different beads. In our model only two kinds of beads occur, hence only binary necklaces will be encountered here. From now on, if not specified otherwise, the term necklace will mean a binary necklace. Since we are dealing with fermions, some of the above necklaces will not be allowed by the Pauli exclusion principle which will turn out to be crucial for supersymmetry as already mentioned in the introduction. We therefore define the allowed and forbidden necklaces as those which are allowed and forbidden by the Pauli principle. Hence the set of all necklaces is the union of allowed and forbidden ones.
2.1. Counting states/necklaces. We begin by recalling the classical results on counting all binary necklaces. The total number N (n) of necklaces with n beads is given by MacMahon’s formula 1 N (n) = ϕ(d) 2n/d , (7) n d|n
where d|n means that d divides n and ϕ(d) is Euler’s “totient” function, counting the numbers in 1, 2, . . . , d −1 relatively prime to d. The slightly “more differential” number N (B, F) of necklaces with B and F separate beads (B beads of type “0” and F of type “1”) is given by Polya’s formula [8, 9]: B/d + F/d 1 ϕ(d) , (8) N (B, F) = B+F F/d d|B,F
where d|B, F means that d is a divisor of both B and F. Of course the numbers given in (8) sum up to those in (7).
2.2. Allowed vs. forbidden necklaces. Let us now look in more detail at how the antisymmetry excludes some of the planar states/necklaces. For instance the sequence 0101 corresponding to the operator Tr[a † f † a † f † ]
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vanishes identically, since by anticommutation of f one has Tr[a † f † a † f † ] = −Tr[ f † a † f † a † ] = −Tr[a † f † a † f † ] = 0. On the other hand the sequence aa f f survives, since Tr[a † a † f † f † ] = −Tr[ f † a † a † f † ] = +Tr[ f † f † a † a † ] = +Tr[a † a † f † f † ]. Our problem is to find in a systematic way which necklaces and how many of them survive the Pauli principle. The distinction into allowed and forbidden necklaces crucially depends on whether a necklace has even or odd number of fermionic quanta. Therefore, from now on, we reserve the term fermionic necklace to one with an odd number of fermions (1’s in its binary representation), while a bosonic necklace will denote a necklace with even number of fermionic quanta F. It follows that fermionic necklaces are always allowed, since a cyclic shift consists of an even number of fermionic anticommutations, while some of the bosonic necklaces may be forbidden. To see this consider two longer necklaces (011011),
and
(01010101),
(9)
both of them are bosonic, however only the first one is allowed. Both states have an additional symmetry (Z 2 and Z 4 , respectively), but the number of fermionic transpositions needed to shift them into themselves is different. A little thought allows now to identify the necessary and sufficient condition for a necklace to be forbidden: A necklace with Z k symmetry, k even, and Fk odd, is forbidden and vice versa. Because of the cyclic invariance of a trace and of Fermi statistics, we find that these states are equal to their opposite and hence vanish. The above condition splits the space of all necklaces in the way sketched in Fig. 1, where the necklaces are divided into four groups according to whether they contain an even or odd number of fermionic and bosonic beads. The exclusion principle is only effective in the even-even group where some necklaces are Pauli-forbidden. Supersymmetry manifests itself in terms of the existence of doublets of energy eigenstates (with non-vanishing eigenvalue) consisting of a boson (a bosonic necklace) and a fermion (a fermionic necklace). Precisely the removal of the forbidden necklaces from the even-even sector should give back the balance between even-even and odd-odd sectors required by supersymmetry.
2.3. Allowed necklaces and linear feed-back registers. It turns out that fermionic necklaces are closely related to linear feedback shift registers (LFSR) - yet another class of objects well known in combinatorics [10]. In order to avoid lengthy definitions we illustrate hereafter the concept of a LFSR, of length n, and its relation to odd necklaces of length n (giving n = 4 as an example): i) Definition of a LFSR: • Take an arbitrary binary number with n − 1 (here 3) digits: (000, 001, 010, 011, . . . , 111).
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Fig. 1. A map of the space of all necklaces for even and odd B and F showing where Pauli-forbidden necklaces lie. The double arrows indicate the way supersymmetry connects allowed necklaces of opposite statistics at weak coupling. The connection with linear feedback shift registers (LFSR) is explained in Subsect. 2.3
• Start adding digits to its right by the following (linear feedback) rule: add a 0 if the sum of the three digits is odd and a 1 if the sum is even. This gives: (0001, 0010, 0100, 0111, . . . , 1110). By construction, the sum of the 4 digits is always odd. • Repeat the procedure by applying the rule to the new last three figures. The result is (00010, 00100, 01000, 01110, . . . , 11101). Clearly the 5th figure coincides with the first. If we keep going, we get a series that is periodic with period 4. ii) Claim of equivalence The claim is that the distinct LFSR thus obtained are in one-to-one correspondence with fermionic necklaces of length n. Proof. We have already argued that elementary cells of length n have an odd sum. Also, if two cells of length n are related by a cyclic transformation, they lead to the same infinite periodic structure. Thus every inequivalent, odd necklace of length n gives a distinct infinite sequence of period n and vice versa. This proof is illustrated by the following three infinite sequences: 010001000100010001 . . . , 100010001000100010 . . . , 110111011101110111 . . . ,
(10)
generated by our rule out of three different initial “data”. The first two infinite sequences are considered to be equivalent: they correspond to the same 4-digit periodic structure (up to a cyclic permutation) repeating itself indefinitely, and corresponds to the odd necklace of Fig. 2 (left side), while the third sequence gives the only other inequivalent odd necklace of length four, also shown in Fig. 2 (right side).
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Fig. 2. The two inequivalent necklaces corresponding to the infinite sequences (10)
The number of linear feed-back registers of length n is catalogued as A000016(n) in Sloane’s library (we shall often refer to this remarkable tool [11], which was very helpful at a certain stage of our work). In conclusion Nfermionic (n) = NLFSR (n) = A000016(n) =
1 ϕ(d) 2n/d . 2n
(11)
d|n d odd
2.4. Separate, global counting of allowed and forbidden necklaces. Supersymmetry requires that the numbers of allowed bosonic and fermionic necklaces are the same for given n. However this general condition has different consequences for even and odd n. If n is odd all necklaces are allowed (cf. Fig. 1) and consequently the number of bosonic necklaces coincides with the number of fermionic ones. In this case the restriction “d odd” in the last equation is superfluous and we get the total count of necklaces given by MacMahon’s formula (7). On the contrary, when n is even we still have the value of Nfermionic given by Eq. (11), but now supersymmetry requires this to match the number of allowed bosonic necklaces in the even-even sector. This yields the general (i.e. valid for all n) rule Nallowed (n) =
1 ϕ(d) 2n/d , n
(12)
d|n d odd
implying a total number of forbidden necklaces Nforbidden (n) =
1 ϕ(d) 2n/d , n
(13)
d|n d even
for any n. For small even n (2 ≤ n ≤ 32), the formula gives Nforbidden (n) = 1, 2, 2, 4, 4, 8, 10, 20, 30, 56, 94, 180, 316, 596, 1096. This result follows by supersymmetry; it will be also derived by traditional combinatorial arguments later on (see Appendix). It is perhaps amusing that the obvious algebraic fact that odd n has only odd divisors while even n admits both (i.e. even and odd) divisors, directly corresponds to the existence of the necklaces allowed and forbidden by the Pauli principle!
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3. Generalization of Polya’s Formula for Forbidden Necklaces We would like to find the counterpart of Polya’s formula Eq. (8) which holds for the bosonic/fermionic necklaces with fixed numbers of separate beads . Equivalence between classical necklaces and susy necklaces when F is odd tells us that, in this case, we simply have: 1 B/d + F/d Nallowed (B, F) = ϕ(d) B+F F/d d|B,F 1 B/d + F/d = ϕ(d) , F odd. (14) B+F F/d d|B,F d odd
By an obvious symmetry, the same formula holds if B is odd and F is even. The only tricky case, again, is the one where both B and F are even: here we want to distinguish allowed from forbidden necklaces and count them separately for given values of B and F. It turns out to be easier to find first the general formula for the number of forbidden necklaces, which, when combined with Polya’s Eq. (8), will produce as a corollary also the number of allowed necklaces. Our claim is as follows: Theorem 1. Let r be the unique positive integer (if it exists) for which f = F/2r is odd and b = B/2r is an integer. The number of Pauli Forbidden Necklaces is given by Nforbidden (B, F) = N (B/2r , F/2r ) =
1 b/d + f /d ϕ(d) . b+ f f /d
(15)
d|b, f
If such an r does not exist then Nforbidden (B, F) = 0. Proof. The Pauli principle is active in deleting necklaces which are Z p -symmetric with p even and F/ p odd; then it is clear that, by writing p = 2r q with q odd, F/2r = f must be odd and B/2r = b must be an integer. If we now consider any sequence of length b + f (a cell repeated 2r times along the whole necklace), we see that such a cell is itself an arbitrary necklace with b bosons, f fermions and symmetry Z q with q any odd number. Since f is odd, such a Z q symmetry covers all possible cases, and therefore the number of inequivalent cells is indeed given by Polya’s formula; notice that a different cyclic permutation of the elementary cell gives the same necklace, because a cyclic permutation of the cell is equivalent to a cyclic permutation of the whole necklace. As an example, consider the string Tr[ f † a † a † f † a † a † f † a † a † f † a † a † ] corresponding to the necklace of Fig. 3. The elementary cell in this case is ( f aa) and the number of forbidden necklaces coincides with N (2, 1) = 1, which corresponds to the necklaces depicted in Fig. 3. In fact, rotating the necklace until it matches the initial configuration, one has an odd number of fermionic commutations yielding the minus sign which kills the necklace. Finally, by taking the difference between Eq. (8) and (15), we conclude that: Nallowed (B, F) = N (B, F) − N (B/2r , F/2r ),
(16)
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Fig. 3. Counterclockwise rotating the necklace until it matches the initial configuration (from left to right): ( f a a f a a f a a f a a), (a f a a f a a f a a f a), (a a f a a f a a f a a f)
where r is as defined in Theorem 1. The values of Nallowed (B, F) for F + B ≤ 26 are reported2 in the form of a (B, F) array in Table 1. At this point, if our supersymmetrybased argument is correct, summing over the number of allowed necklaces given by Eq. (16) for fixed n should reproduce exactly the same number as twice the sum over the fermionic supersymmetric partners, i.e. just Eq. (12). We have verified numerically that this is the case up to n = 3000 and later found a direct mathematical proof reported in the Appendix. The existence of such a proof confirms the solidity of the supersymmetrybased arguments, as well as their considerable heuristic value. 4. Witten-like Indices The formula for Nallowed verifies a number of checks coming from the properties of the supersymmetric model at weak and strong coupling. In the first, weak-coupling regime, which is fully under control, supersymmetry tells us that the allowed necklaces with a given n = B + F should organize themselves in supersymmetry doublets, each of which consists of a necklace with some B and F and one with B = B ±1 and F = F ∓1. Since the number of such pairs is always non-negative, we obtain the following inequalities for graded partial sums (a kind of generalization of Witten’s index [7]): W (n; m) ≡ (−1) F−m Nallowed (B, F) ≥ 0, W (n; n) = 0, (17) B+F=n 0≤F≤m
where the last equality corresponds to that between even and odd allowed necklaces with a given n. The above consequences of supersymmetry have been explicitly checked up to n ∼ 5000), while, so far, we have not been able to construct a direct proof of them by more standard techniques. The strong (’t Hooft) coupling limit of the model of [1] can be shown [5] to imply instead that allowed necklaces must also organize in supersymmetry doublets whose partners have the same value of B + 2F (and again differ by one, positive or negative, unit of F). A look at Table 1 shows that, along diagonals at fixed B + 2F, the balance between even and odd allowed necklaces is not always satisfied. This implies that, along those diagonals, there must be, at large coupling, (unpaired) E = 0 states. The large-coupling limit unfortunately is not fully under control yet. Therefore, in this case, the connection between eigenstates and allowed necklaces can be used in 2 This table and the following one were produced by a sieve method for B + F ≤ 26, independently of Theorem 1, and were used to check the general formulae.
0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12
1 1 2 4 5 7 10 12 15 19 22 26 31 35 40 46 51 57 64 70 77 85 92 100
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3
B↓ 0 1 2
F→
0 1 3 5 9 14 22 30 42 55 73 91 115 140 172 204 244 285 335 385 445 506 578
4
1 1 3 7 14 26 42 66 99 143 201 273 364 476 612 776 969 1197 1463 1771 2126 2530
5
0 1 3 10 20 42 76 132 212 335 497 728 1028 1428 1932 2586 3384 4389 5601 7084 8844
6
1 1 4 12 30 66 132 246 429 715 1144 1768 2652 3876 5538 7752 10659 14421 19228 25300
7 0 1 5 15 43 99 217 429 809 1430 2438 3978 6310 9690 14550 21318 30666 43263 60115
8 1 1 5 19 55 143 335 715 1430 2704 4862 8398 14000 22610 35530 54484 81719 120175
9 0 1 5 22 70 201 497 1144 2424 4862 9226 16796 29372 49742 81686 130752 204248
10 1 1 6 26 91 273 728 1768 3978 8398 16796 32066 58786 104006 178296 297160
11 0 1 7 31 115 364 1038 2652 6308 14000 29414 58786 112716 208012 371516
12 1 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400024
13 0 1 7 40 168 612 1932 5538 14520 35530 81686 178296 371384
14
Table 1. Nallowed (B, F) as generated with the sieve method
1 1 8 46 204 776 2586 7752 21318 54484 130752 297160
15 0 1 9 51 245 969 3399 10659 30667 81719 204347
16 1 1 9 57 285 1197 4389 14421 43263 120175
17
0 1 9 64 330 1463 5601 19228 60060
18
1 1 10 70 385 1771 7084 25300
19
0 1 11 77 445 2126 8866
20
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either direction to infer properties of one in terms of known properties of the other. For instance, some evidence has been accumulated on where zero-energy states lie in the B, F plane. On the basis of this evidence we can conjecture new checks on our formulae for Nallowed by the following property of a second Witten-like index: δ F,B+1 +δ F,B−1 (1+(−1) F ) ≥ 0, W˜ (n; m) ≡ (−1) F−[m/2] Nallowed (B, F)− 2 B+2F=n F≤m/2
(18) (with m ≤ n) and, in particular, W˜ (n; n) = 0 ⇒ (−1) F Nallowed (n − 2F, F) = δn≡1(mod 6) + δn≡−1(mod 6) .
(19)
F
Our formulae passed the test of these (in)equalities for n ≤ 5000. Actually, the validity of Eq. (19) for all values of n follows from an explicit expression for the generating function of Nallowed (B, F) recently obtained by D. Zagier: allowed (x, y; n) ≡
n F=0
Nallowed (n − F, F) x n−F y F =
n/d 1 ϕ(d) x d − (−y)d , n d|n
after setting y = −x 2 and summing over n.3 When B + 2F is small, the zero-energy eigenstates causing the imbalance can be uniquely identified in Table 1, while, for the moment, their identification can only be guessed at (and verified later) for B + 2F large. This is how we arrived at the conjecture [5] that, at strong coupling, there is one and only one zero-energy eigenstate for each even value of F and B = F ± 1, a conjecture leading precisely to Eqs. (18) and (19). Finally, it is amusing to notice that the total number of strong-coupling eigenstates at these special locations (forming a kind of magic staircase in Table 1) is given by the sequence 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, . . . which is easily recognized as being that of Catalan’s numbers: 2n 1 NCatalan = . n+1 n Catalan’s numbers are ubiquitous, 66 appearances of them being listed in Stanley’s treatise [9]. It is easy to convince oneself that to every necklace with |B − F| = 1 one can associate an infinite sequence of ups and downs describing a mountain profile, the number of which is precisely given by Catalan numbers [12]. These entries belong to the subset with Nforbidden = 0, since either B or F is odd. Other diagonals can be identified with known sequences; for instance, Nallowed (F ± 2, F) is identical to the number of plane trees with odd/even number of leaves (A071684, A071688 [11]).
3 We are very grateful to Professor Zagier for informing us of this result, and for giving us permission to report it here.
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
B↓
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2
F→
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
4
1 1 2 4 5 7 10 12 15 19 22 26 31 35 40 46 51 57 64 70 77
6
1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
8
1 1 3 7 14 26 42 66 99 143 201 273 364 476 612 776 969 1197 1463 1771 2126
10
1 0 1 0 2 0 4 0 5 0 7 0 10 0 12 0 15 0 19 0 22
12
1 1 4 12 30 66 132 246 429 715 1144 1768 2652 3876 5538 7752 10659 14421 19228 25300 32890
14 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
16 1 1 5 19 55 143 335 715 1430 2704 4862 8398 14000 22610 35530 54484 81719 120175 173593 246675 345345
18 1 0 1 0 3 0 7 0 14 0 26 0 42 0 66 0 99 0 143 0 201
20 1 1 6 26 91 273 728 1768 3978 8398 16796 32066 58786 104006 178296 297160 482885 766935 1193010 1820910 2731365
22 1 0 0 0 1 0 0 0 2 0 0 0 4 0 0 0 5 0 0 0 7
24 1 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400024 742900 1337220 2340135 3991995 6653325 10855425 17368680
26 1 0 1 0 4 0 12 0 30 0 66 0 132 0 246 0 429 0 715 0 1144
28 1 1 8 46 204 776 2586 7752 21318 54484 130752 297160 643856 1337220 2674440 5170604 9694845 17678835 31429068 54587280 92798380
30 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
32
1 1 9 57 285 1197 4389 14421 43263 120175 312455 766935 1789515 3991995 8554275 17678835 35357670 68635478 129644790 238819350 429874830
34
Table 2. Nforbidden , the number of Pauli-forbidden necklaces calculated from Eq. (15)(entries with odd F and/or odd B vanish identically)
1 0 1 0 5 0 19 0 55 0 143 0 335 0 715 0 1430 0 2704 0 4862
36
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To summarize our main results: • We have been able to divide all binary necklaces in two disjoint classes, which we termed (Pauli)-allowed and (Pauli)-forbidden . • At the most “inclusive” level, the number of binary necklaces with a total number n of beads, as given by MacMahon’s formula (7), is split into allowed and forbidden necklaces by restricting the divisor d in (7) to odd and even values, respectively. • At a more “differential” level, the number of necklaces with B bosonic and F fermionic beads is rewritten in terms of allowed necklaces with different values of B and F via Eq. (16), which can also be rewritten as: N (B, F) = Nallowed (B, F) + N (B/2r , F/2r ), N (B/2r , F/2r ) = Nallowed (B/2r , F/2r ),
(20)
where r is as defined in Theorem 1. We have verified numerically (and then proved directly, see the Appendix) that the appropriate sum performed on (20) reproduces the above-mentioned relation at fixed n = B + F. • Supersymmetry implies several non-trivial constraints on Nallowed (B, F) and thus, through (20), also on N (B, F). Examples have been given in Sect. 4, but we stress that, by suitably extending the supersymmetric model under consideration, it is quite conceivable that many more constraints will emerge, not only for binary necklaces, but also for their generalization to more than two kinds of beads. This new game (that we may dub “super-combinatorics”) should reserve further surprises both for physicists and for mathematicians. Acknowledgements. GV would like to acknowledge interesting discussions with Professors Jean-Christophe Yoccoz and Don Zagier. JW thanks A. Kotanski for instructive discussions. EO warmly thanks G. Cicuta for interesting discussions. This work is partially supported by the grant of the Polish Ministry of Education and Science P03B 024 27 (2004–2007).
Appendix: Proof of Consistency between Eqs. (13) and (15) Theorem 2. n
Nforbidden (n − F, F) =
F=0
1 φ(d)2n/d . n d|n d even
Proof. Let n = 2r q, with q odd. We have r 1 1 r −m 2r q/d φ(d)2 = φ(2m d)22 q/d 2r q r 2r q m=1 d|q
d|2 q d even
1 m−1 1 1 −m −m 2 φ(d)22 n/d = n φ(d)22 n/d = NLFSR (2−m n) r m 2 q 2 2 r
=
m=1
r
d|q
m=1
r
d|q
m=1
(we use φ(2m ) = 2m−1 and in the second step we put d = d1 d, d1 |2r , d|q).
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On the other hand the only non-vanishing contributions to nF=0 Nforbidden (n− F, F) come from F = 2m q , where m = 1, 2, . . . , r and q = 1, 3, 5, . . . , q, so that we have: n
Nallowed (n − F, F) =
F=0
n
Nforbidden (2r q − F, F)
F=0
=
q r
Nforbidden (2r q − 2m q , 2m q )
m=1 q =1 q odd
=
q r m=1 q =1 q odd
NBNL (2r −m q − q , q ) =
r
NLFSR (2−m n).
m=1
References 1. Veneziano, G., Wosiek, J.: Planar Quantum Mechanics: an Intriguing Supersymmetric example. JHEP 0601, 156 (2006). [hep-th/0512301] 2. Veneziano, G., Wosiek, J.: Large-N, Supersymmetry … and QCD, in Sense of Beauty in Physics - A volume in honour of Adriano Di Giacomo. Edited by M. D’Elia, K. Konishi, E. Meggiolaro, P. Rossi, Pisa Ed. PLUS, Pisa University Press, 2006. [hep-th/0603045] 3. Veneziano, G., Wosiek, J.: A supersymmetric matrix model: II. Exploring Higher Fermion-number sectors. JHEP 0610, 033 (2006). [hep-th/0607198] 4. ’t Hooft, G.: A Planar Diagram Theory For Strong Interactions. Nucl. Phys. B72, 461 (1974); see also Veneziano, G.: Some Aspects Of A Unified Approach To Gauge, Dual And Gribov Theories. Nucl. Phys. B117, 519 (1976) 5. Veneziano, G., Wosiek, J.: A supersymmetric matrix model: III. Hidden SUSY in Statistical Systems. JHEP 0611, 030 (2006). [hep-th/0609210] 6. Beisert, N., Bianchi, M., Morales, J.F., Samtleben, H.: On the spectrum of AdS/CFT beyond supergravity. JHEP 0402, 001 (2004) and references therein 7. Witten, E.: Nucl. Phys. B185, 513 (1981); ibid. B202, 253 (1983) 8. van Lint, J.H., Wilson, R.M.: A Course in combinatorics. Cambridge: Cambridge U.P., 1992 9. Stanley, R.P.: Enumerative combinatorics, Vol. 2, Cambridge: Cambridge U.P., 1999 10. Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications. New York: Cambridge U.P., 1968 11. Sloane, N.J.A.: On-line encyclopedia of integer sequences. http://www.research.att.com/∼njas 12. Conway, J.H., Guy, R.K.: The book of numbers. N.Y.: Springer-Verlag, 1996 Communicated by N.A. Nekrasov
Commun. Math. Phys. 274, 357–379 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0280-9
Communications in
Mathematical Physics
The Manifold of Compatible Almost Complex Structures and Geometric Quantization T. Foth1, , A. Uribe2, 1 Department of Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7.
E-mail:
[email protected]
2 Mathematics Department, University of Michigan, Ann Arbor, Michigan 48109, USA.
E-mail:
[email protected] Received: 19 May 2006 / Accepted: 26 January 2007 Published online: 23 June 2007 – © Springer-Verlag 2007
Abstract: Let (M, ω) be an integral symplectic manifold. We study a family of hermitian vector bundles on the space J of almost complex structures on M compatible with ω, whose fibers consist of nearly holomorphic sections of powers of a prequantum line bundle. We obtain asymptotics of the curvature of a natural connection in these bundles. These results, together with Toeplitz operator theory, provide another proof of Donaldson’s result that the action of Ham(M) on J is hamiltonian with moment map the scalar curvature. We also give an example involving Teichmüller space and discuss a relationship between parallel transport and the Schrödinger equation. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . 2. Proof of Theorem 1.2 . . . . . . . . . . . . . . . . 2.1 Preliminaries on the linear case . . . . . . . . . 2.2 Hermite FIOs . . . . . . . . . . . . . . . . . . 2.3 The curvature as an operator on Z . . . . . . . 2.4 The end of the proof . . . . . . . . . . . . . . . 3. Moment Maps . . . . . . . . . . . . . . . . . . . . 3.1 The L 2 connection . . . . . . . . . . . . . . . 3.2 The action of Aut(L) . . . . . . . . . . . . . . 4. Final Remarks . . . . . . . . . . . . . . . . . . . . 4.1 Example: Teichmüller Space . . . . . . . . . . 4.2 Parallel transport and the Schrödinger equation Appendix A: The Euclidean Case . . . . . . . . . . . . . Appendix B: Notation Index . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . T.F. supported in part by NSF grant DMS-0204154 and by NSERC.
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1. Introduction Let L → M be a hermitian line bundle with connection, and assume that its curvature, ω, is everywhere non-degenerate so that (M, ω) is a symplectic manifold. Let J denote the infinite-dimensional space of all almost complex structures on M that are compatible with the symplectic form ω: J = { J : T M → T M ; J 2 = −I, ω(J., J.) = ω(., .), ω(·, J.) > 0 }. J is an infinite-dimensional Kähler manifold naturally associated with the symplectic manifold (M, ω). The Kähler structure of J can be described as follows. The tangent space to J at one of its points, J , is easily seen to be T J J = {A ∈ End (T M) ; J ◦ A + A ◦ J = 0 and ω(A(·), J (·)) + ω(J (·), A(·)) = 0}. (1.1) The Kähler form, , on J , is given by 1 (A, B) = − 2
Tr(A x Jx Bx ) M
ωnx , n!
where n = dim(M)/2, while the complex structure is given by composition with J . (For each x ∈ M the integrand above is the symplectic structure on the linear compatible complex structures on Tx M.) In addition, the group Ham(M) of hamiltonian symplectomorphisms of M acts on J , with moment map a constant times the scalar curvature, [8]. In this paper we raise the issue of pre-quantizing J , that is, finding a hermitian line bundle with connection L → J whose curvature is the Kähler form, , of J . It is natural to expect that the line bundle L will be a determinant line bundle, with a Quillen metric and connection. There is indeed a natural candidate for L of this sort, namely, the determinant line bundle associated with the family of Spin-c Dirac operators indexed by J ∈ J . The purpose of this paper, however, is to show that the theory of BerezinToeplitz operators (in the general symplectic context of almost-Kähler quantization, [4]) provides us with a pre-quantization of J , at least asymptotically. More precisely, over any open set U ⊂ J bounded in the C 3 topology, we will exhibit a natural family of hermitian line bundles with connection, L N → U, whose curvature is asymptotic as N → ∞ to N n (here N is a positive integer). The construction is as follows. Given J ∈ J , the associated Riemannian metric on M, g J = ω(·, J ), and the structures on L define a metric Laplacian on sections of L ⊗N , NJ = (∇ (N ) )∗ ◦ ∇ (N ) , where ∇ (N ) is the induced connection on L ⊗N . Let NJ := NJ − N n. As shown in [12, 17], there exist positive constants C1 and C2 such that Spec NJ ⊂ (−C1 , C1 ) (N C2 , ∞), so that if N is large enough there is a well-defined band of eigenvalues near the origin. Moreover, the number d N of such low-lying eigenvalues (counted with multiplicities) equals the Riemann-Roch number of (M, N ω): dN = e N ω τ ∼ Vol (M)N n , M
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where τ is the Todd class of M. (The proof in [17] shows that the constants C1 and C2 depend only on the curvature of g J , and so can be chosen uniformly on an open subset, U, as described above.) In the integrable case, these low-lying eigenvalues are all zero and the corresponding eigenspace consists precisely of the holomorphic sections of L ⊗N . The almost-Kähler quantization of (M, ω, J ) is defined as the sequence, {E JN }, where for all N , E JN ⊂ L 2 (M, L ⊗N ) is the subspace spanned by the eigenvectors associated with the first d N eigenvalues of NJ . Note that there may be an ambiguity in the definition of the E JN for small values of N , but, since our results are asymptotic as N → ∞, we will ignore this issue. Formally, we will speak of the vector bundles EN → J whose fibers are the E JN . The pre-quantum line bundle of (J , N ) is defined as the associated determinant line bundle: Definition 1.1. L N := ∧d N E N . For each J ∈ J , the fiber E JN is a subspace of the J -independent Hilbert space 2 L (M, L ⊗N ). For this reason, the bundles E N (and hence L N ) have natural connections that we will call the L 2 connections. We will prove: Theorem 1.2. If N is the curvature of the L 2 connection on L N , then as N → ∞, N = N n + O(N n−1 ). This theorem follows from the study of the large N asymptotics of the curvature of the L 2 connection on the bundles E N → J . We show (Theorem 2.1) that curvature, evaluated on a pair of tangent vectors, is a Berezin-Toeplitz operator and compute its symbol. This gives a precise measure of the non-centrality of the L 2 connection on E N → J , a fact first established by A. Viña, [19]. However, L. Charles has shown that if one includes the metaplectic correction then the curvature is asymptotically zero, [6]. We will denote by Aut(L) the group of automorphisms of the Hermitian line bundle L → M that preserve the connection. In §4 we will consider the natural action of Aut(L) on L N → J . Recall that the Lie algebra of Aut(L) group is C ∞ (M). Let NJ : L 2 (M, L ⊗N ) → E JN
(1.2)
be the orthogonal projector. We will prove: Proposition 1.3. For each N , the natural action of Aut(L) on the bundle L N → J is Hamiltonian with respect to N , with moment map J → (C ∞ (M))∗ , J → H → Tr( NJ THN NJ ) , (N )
(1.3)
where THN = 1i ∇ H + N H is the Kostant-Souriau operator on C ∞ (M, L ⊗N ) associated to the Hamiltonian H .
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As we will see,
Tr( NJ THN NJ ) = N
(N )
M
H ρJ
ωn + O(N n−1 ), n!
(1.4)
(N )
where ρ J ∈ C ∞ (X ) is the restriction to the diagonal of the Schwartz kernel of NJ (see Lemma 3.4 and remarks following it). Therefore, the moment map (1.3) can be identified by duality with a map of the form J → C ∞ (M), (N ) J → Nρ J + O J (N n−1 ).
(1.5) (N )
On the other hand, as obtained by X. Ma and G. Marinescu [18], ρ J (N )
ρJ
= Nn +
is of the form
1 n−1 N κ J + O(N n−2 ), 4
(1.6)
where κ J is the “symplectic scalar curvature” 1 , κ J := r J +
1 LC 2 |∇ J | , 2
defined in terms of the Riemannian scalar curvature, r J , and the Levi-Civita connection, ∇ LC , of the metric g J . (Ma and Marinescu’s results generalize to the almost Kähler case results of Zhiqin Lu in the Kähler case, [16].) In other words, we obtain a moment map for (J , N ) of the form J → C ∞ (M), J → 41 N n κ J + O J (N n−1 ).
(1.7)
These results imply the following corollary, first proved by Donaldson, [8, 9]: Corollary 1.4 ([8]). A moment map for the action of Ham(M) on (J , ), J →
1 κ J ∈ C ∞ (M). 4
(1.8)
We end this introduction by putting our results in the context of the literature. The L 2 connection on the bundles E N → J has been extensively studied in the case M = R2n with its standard symplectic form, and J is the manifold of linear compatible complex structures on M. We are aware of an early paper by I. Daubechies in 1980, [7], the recent work of Kirwin and Wu, [15]. In this case the curvature of the L 2 connection is projectively flat, and if one introduces the metaplectic correction it becomes flat. We recall that a similar situation holds in the context of Chern-Simons theory, in which case M is a moduli space of flat connections on a surface , and J consists of the complex structures on M that come from complex structures on , [1] . The L 2 connection on the bundles E N → Jint over the space of integrable structures has been studied by A. Viña, [19], who proved that this connection is not flat. (In Theorem (1.2) below we compute the semi-classical limit of the curvature). However, recently L. Charles has shown that if one includes the metaplectic correction, then the L 2 connection is asymptotically flat, 1 Our metric g differs from that of [18] by a factor of 2π , which accounts for the different numerical J factor in the second term in (1.6).
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[6]. In [11], V. Ginzburg and R. Montgomery show that there is not a natural connection on this bundle that is projectively flat. The set of all Szegö-type projectors on a contact manifold was considered by Epstein and Melrose, [10], in the context of an index problem for Fourier integral operators, who also considered how J changes with J . Zelditch’s article, [21], contains results on the change of J as J is transformed under a time-dependent Hamiltonian. The present context suggests natural questions regarding Berry’s phase. Although we won’t go into details, results analogous to those obtained by Viña in the integrable case, [19] §8, should hold in the present case as well. 2. Proof of Theorem 1.2 Theorem 1.2 is a consequence of the following theorem on the large N asymptotics of the curvature of the L 2 connection on the bundles E N → J . Let J ∈ J and A, B ∈ T J J . Then the curvature of the natural connection on E N evaluated on (A, B), ϒ J(N ) (A, B), is an operator (N )
ϒ J (A, B) : E J,N → E J,N . The semi-classical limit of the curvature is described by the following: Theorem 2.1. Assume that for each J ∈ J the sequence of orthogonal projectors, J,N : L 2 (M, L ⊗N ) → E J,N , is associated to a Fourier integral operator of Hermite type in the unit circle bundle, Z ⊂ L ∗ , symbolically identical to the Szegö projector and depending smoothly on J . Then, given variations of complex structures A, B as above, (N ) the sequence of operators, ϒ J (A, B), is a Berezin-Toeplitz operator with symbol χ A,B (x) = −
1 Tr(A x Jx Bx ). 2
Remarks: (1) The technical assumption on the ( N ,J ) holds if M is almost Kähler, J is the space of compatible complex structures on M, and the quantization scheme is by the span of the eigensections corresponding to low–lying eigenvalues of the rescaled Laplacian, , as described in the introduction ([4]). The hypotheses also hold in the case when M is Kähler and J is the space of integrable complex structures. (2) The function χ A,B is general, in the following sense: for any J and χ ∈ C ∞ (M), one can find A and B such that χ = χ A,B . (3) A corollary of Theorem 2.1 is that the curvature of E N → J is highly noncentral. Perhaps the simplest way to see this is to observe that, by the Szegö limit theorem for Toeplitz operators and Theorem 2.1, the normalized spectral measure (N ) of ϒ J (A, B) converges weakly to the push-forward of Liouville’s measure on M by the function χ A,B . Centrality, in the semi-classical limit, would amount to the function χ A,B being constant. On the other hand the curvature preserves microsupport. The curvature of L N → J is the trace of the curvature of E N → J . Since the semiclassical limit of the traces of Berezin-Toeplitz operators is given by the integral of the symbol, Theorem 1.2 follows from Theorem 2.1. The rest of this section is devoted to the proof of Theorem 2.1.
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2.1. Preliminaries on the linear case. Let (V, ω) be a symplectic vector space, and let L denote the set of linear compatible complex structures on V . It will be useful to have several different descriptions of L: (1) L is the symmetric space for the group G = Sp(V ) of linear symplectic transformations of V . (2) L is a Grassmannian: We can identify L with the set of positive complex lagrangian subspaces of the complexification V ⊗ C, by mapping a complex structure to the subspace of vectors of type (0, 1) with respect to that structure. We now investigate the consequences of these two points of view. Let us pick a compatible complex structure on V , J0 , which will serve as a base point: If we let U ⊂ Sp(V ) be the isotropy subgroup of J0 (the unitary group of (V, ω, J0 )) then we can identify: L∼ = Sp(V )/U. At the Lie algebra level the choice of J0 induces a Cartan decomposition (see Appendix A for a matrix description): sp = u + p. Elements of sp are the linear transformations A : V → V such that ω(A(·), ·) + ω(·, A(·)) = 0. The elements of p (resp. u) are those A that, in addition, satisfy: A J0 = −A
resp. A J0 = A.
For any A ∈ sp the vector field V v → A(v) is associated to the Hamiltonian 1 ω(A(v), v). (2.1) 2 Therefore, elements of sp can also be thought of as (arbitrary) quadratic forms on V (the Hamiltonians). From this point of view p is the space of quadratic forms that are the real part of J0 -complex quadratic forms. In a similar spirit, u consists of J0 -Hermitian quadratic forms. The vector space p has a symplectic structure: q A (v) :=
1 Tr(A J0 B), 2 where we now think again of A, B ∈ p as linear transformations of V . This structure extends to the (Sp(V )-invariant) symplectic form of L. Let B be a quantization of (V, ω): to be specific, let us pick (A, B) = −
B = { f : V → C;
f J0 -holomorphic } ∩ L 2 (V, exp(−ω(v, J0 (v)) ωn ),
the Bargmann space. This Hilbert space carries a (Heisenberg) representation of the Heisenberg group of V . A lagrangian subspace, ⊂ V ⊗ C can be thought of as a maximal abelian subalgebra of the Lie algebra of the complex Heisenberg group of V , heis(V )C . If is positive, the subspace, R ⊂ B, which is the joint kernel of all the operators induced by vectors of a positive is one-dimensional. The spaces R are the fibers of a line bundle, R → L. Since all the fibers are subspaces of the fixed Hilbert space, B, we can endow this bundle with the L 2 connection. Below we will need:
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Lemma 2.2. The curvature of R → L is the symplectic form of L. Proof. We sketch a proof, invoking the existence of the metaplectic representation of V . Thus let Mp(V ) be the metaplectic group of V (the double cover of Sp(V ) on which the determinant has a smooth square root), and let ρ denote the representation of Mp(V ) on B. By its definition, ρ induces an action on the bundle R → L, where ρ acts on L via Sp(V ). More specifically, if for each ∈ L P : B → B denotes the orthogonal projection onto R , then ∀g ∈ Mp(V )
ρ(g) ◦ P ◦ ρ(g)−1 = Pg· .
(2.2)
Let A, B ∈ p. By the formula above, the variation of P by the infinitesimal action of A is δ A P = [dρ(A) , P ]. By Lemma 3.1 and the formula above, the curvature at J0 evaluated at A, B ∈ p is Tr P [[dρ(B) , P ] , [dρ(A) P ]],
(2.3)
where is the space of J0 -(0, 1) vectors. (Taking the trace is here used to identify the curvature as an operator on a 1-dimensional vector space with a scalar.) A short calculation shows that (2.3) is equal to dρ([A, B])e , e , where e ∈ B is a unit vector in R . This inner product is indeed the symplectic form of L evaluated at (A, B) (Corollary A.7 in Appendix A, see also [15, 1]). 2.2. Hermite FIOs. We need now to be more explicit about the quantization scheme we are assuming. Let Z ⊂ L ∗ be the unit circle bundle and α the connection form on Z . Then α is a contact form on Z and dα n ∧ α is a volume form on Z . If we let L 2 (Z ) = ⊕k∈Z L 2 (Z )k be the decomposition of L 2 (Z ) into S 1 isotypes, then one has a tautological isomorphism: L 2 (Z )k ∼ = L 2 (M, L ⊗k ). For each J ∈ J , let H J :=
∞
E J,N
N =1
(Hilbert space direct sum), and let J : L 2 (Z ) → H J be the orthogonal projector. Let us also introduce the manifolds
= { ( p, r α p ) ; p ∈ Z , r > 0 }, which is a symplectic submanifold of T ∗ Z , and
× = { (( p, σ ) , ( p, −σ )) ; ( p, σ ) ∈ } which is an isotropic submanifold of T ∗ Z × T ∗ Z . The assumptions on our quantization scheme are:
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(1) For each J ∈ J , the orthogonal projector J : L 2 (Z ) → H J is in the class
I 1/2 (Z × Z , × ) of Boutet de Monvel and Guillemin, [5], with a symbol to be described below. (2) The map J → J is differentiable. These assumptions hold, for example, if J is the space of integrable structures and H J,N is the space of holomorphic sections of L ⊗N . We now need to recall the notion of symbol of J . Since is a symplectic submanifold of T ∗ Z , the bundle S → , where Sσ = (Tσ )o /Tσ
(where o denotes the symplectic orthogonal) is a bundle of symplectic vector spaces over . S is called the symplectic normal bundle, and there is a natural isomorphism (induced by the projection, π : Z → M, lifted to T T ∗ Z ): Sσ = Tπ(σ ) M − . We will denote by Wσ a Hilbert space which carries the metaplectic representation of the metaplectic group of Sσ . Then, the symbol of a Fourier integral operator in a class
I · (Z × Z , × ) is an object of the following type: For each σ ∈ , the symbol at σ is an operator Wσ → Wσ , (if we take into consideration the metaplectic structure then we have to work with the half forms; however the half forms will play no role in our calculations). In particular, the symbol of J is a rank-one projector onto a ground state, e J,σ ∈ Wσ : |e J,σ e J,σ |. 2.3. The curvature as an operator on Z . Let J ∈ J and A, B ∈ T J J . As mentioned, for each N , the curvature of our connection on E N → J , evaluated at (A, B), is an (N ) operator ϒ J (A, B) : E J,N → E J,N . Let us define: ϒ J (A, B) :=
∞
ϒ J(N ) (A, B) : H J → H J ,
N =1
and extend ϒ J (A, B) to be an operator on L 2 (Z ) by defining it to be zero on H⊥ J . (Recall that H J = ⊕ N E J,N .) We will prove: Proposition 2.3. The operator ϒ J (A, B) is a Fourier integral operator of Hermite type,
in the class I 1/2 (Z × Z , × ), and with symbol equal to the function χ A,B , pulled back to , times the symbol of J . Proof. We begin by proving that ϒ J (A, B) is an FIO in the stated class. Let J (s) be a d J (s)|s=0 = A. Introduce the notation: smooth curve on J such that J (s)|s=0 = J and ds δ A J :=
d J (t) |t=0 , dt
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and analogously for δ B J . Then, by Lemma 3.1, ϒ J (A, B) = J [δ A J , δ B J ] J . By the smooth dependence of J on J , δ A J and δ B J are both in the class I 1/2 (Z ×
Z , × ). It follows directly from Theorem 9.6 of [5] that ϒ J (A, B) is in this class as well. It remains to compute the symbol of ϒ J (A, B). Let σ ∈ . Denote by x ∈ M the projection of σ , and let V = Tx M. This is a symplectic vector space, with symplectic form ωx and a compatible linear complex structure, Jx . The values A x and Bx of A and B at x are infinitesimal variations of the linear structure Jx . Let B be a Hilbert space quantizing V ; for example we can take B to be the Bargmann space B(V,Jx ,ωx ) = { f : V → C ; f Jx -holomorphic } ∩ L 2 (V, exp(−ωx (v, Jx (v)) ωnx ), and then the symbol of J at σ is projection, P : B → B, onto the line in B spanned by |e J , σ a (normalized) constant function. Let us denote by δ A P, resp. δ B P the derivative of P with respect to A x , resp. Bx . By the general symbol calculus of Hermite FIOs, [5], the calculation of the symbol of ϒ J (A, B) at σ amounts to the calculation of the operator P ◦ [δ B P , δ A P] ◦ P.
(2.4)
Thus the symbol calculus reduces the problem to the linear case that we discussed in §2.1. 2.4. The end of the proof. With Proposition 2.3 at hand it is a simple matter to finish the proof of the main theorem. We keep the same notation and assumptions as in the statement of Theorem 2.1, except that we will suppress the subindex J , for simplicity. Let denote the horizontal lift to Z of the Hamilton vector field of χ , and consider the first-order operator on Z , Q = −i + χ . Then Q is an Hermite FIO in the same class and with the same symbol as ϒ(A, B), and therefore the difference R := ϒ(A, B) − Q
is in the class I 0 (Z × Z , × ). Furthermore, it is clear that R commutes both with the circle action on Z and with the projector, . The latter property implies that the symbol of R must be a function, 1 , times the symbol of . Now the difference R1 = R − Q 1 , where Q 1 is a Toeplitz operator of order (−1) and symbol 1 , is an Hermite FIO of degree (−2) and having the same properties as R. We can proceed in this fashion to all orders, which implies that ϒ(A, B) itself must be a Toeplitz operator.
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3. Moment Maps 3.1. The L 2 connection. We gather here some needed facts about the L 2 connection. We begin with a general expression for the curvature of the natural connection of a bundle, E → J , such that ∀J ∈ J the fiber, E J , is a closed subspace of a fixed Hilbert space, H. It will be useful to introduce the following notation: For each J ∈ J , we let J : H → E J be the orthogonal projection, and if v ∈ T J J we let δv denote the operator on H, d γ (t) (ψ), dt where γ (t) is a curve on J whose velocity at time t = 0 is v. ∀ψ ∈ H,
δv (ψ) =
Lemma 3.1. If v, w ∈ T J J , the curvature of the natural connection on E at (J, v, w) is the operator ϒ J (v, w) = J [δw , δv ] : E J → E J .
(3.1)
Proof. This statement is contained, in particular, in [11]. We sketch a proof. Let V, W be two commuting vector fields on J , and let ψ be a section of E N . We wish to compute ϒ(V, W )(ψ) = ∇V (∇W ψ) − ∇W (∇V ψ). If we denote by ∂V ψ(J ) ∈ H N ,J the derivative of ψ with respect to V (J ) inside the ambient Hilbert space L 2 (M, L ⊗N ), then ϒ(V, W )(ψ) = ∂V (∂W ψ) − ∂W (∂V ψ). Leibniz rule gives the desired result. We put on L → J the induced connection. Explicitly, let γ : t → e1 (t) ∧ · · · ∧ ed (t) , J (t) be a section of L over a curve t → J (t) in J . Then the covariant derivative of this section is, by definition, ∇ J˙ γ =
d
e1 ∧ · · · ∧ J (e˙ j ) ∧ · · · ∧ ed .
j=1
Let us now describe the horizontal distribution on L. Notice that we can think of the total space, E, as a subset of C ∞ (M, L) × J : E = { (e, J ) ∈ C ∞ (M, L) × J ; e ∈ H J }. We can therefore identify the tangent space, T(e,J ) E, with a subspace of C ∞ (M, L) × TJ J : ˙ }. T(e,J ) E = { (e˙ , J˙) ∈ C ∞ (M, L) × T J J ; e˙ = ˙ (e) + J (e) J
(Here J˙ is the derivative of J → J with respect to J˙.) Let us assume that, at time t = 0, the sections e1 (0), . . . , ed (0) form an orthonormal basis of H J (0) . Then, decomposing J (e˙ j ) in this basis and substituting, we obtain ∇ J˙ γ =
d e˙ j , e j e1 ∧ · · · ∧ ed , j=1
both sides evaluated at t = 0.
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Lemma 3.2. Let e1 (0), . . . , ed (0) form an orthonormal basis of H J (0) . Then the horizontal subspace of Te1 ∧···∧ed L consists precisely of vectors ⎛ ⎞ d ⎝ e1 ∧ · · · ∧ e˙ j ∧ · · · ∧ ed , J˙⎠ j=1
such that e˙ = J˙ (e) + J (e) ˙ and d d 1 d e j 2 = e˙ j , e j = 0. 2 dt j=1
j=1
Notice that we are identifying the total space of L with a subset of ∧d C ∞ (M, L) ×J : L = { (e1 ∧ · · · ∧ ed , J ) ; e j ∈ H J }. 3.2. The action of Aut(L). Recall that Aut(L) denotes the automorphism group of the Hermitian line bundle with connection, L → M. Aut(L) is a central extension of Ham(M) whose Lie algebra is C ∞ (R). Aut(L) acts naturally on E N → M, and hence on L N → M, by pull back. In this section we prove Proposition 1.3, that is, we compute the moment map of this action with respect to the curvature, N , of the metric connection. Given a function H ∈ C ∞ (M), we will denote by H and φt : M → M the associated Hamilton vector fields and flow, and by φ˜ t : L N → L N an automorphism of the bundle L N that covers φt . φ˜ t induces an operator on sections of L N (by pulling back sections), and the infinitesimal generator of this action on sections is by the operator K H :=
1 (N ) + H. ∇ i N H
(3.2)
Let us denote by t : J → J the flow on J induced by φt . Pulling back sections by φ˜ −t induces a bundle map ˜ t : EN → EN ˜ t is the vector field that covers t . The infinitesimal generator of (e, J ) = (K H (e) , ξ JH ). ˜ t induce bundle maps The bundle maps ˜ t : LN → LN , Ft := ∧top by ˜ t (e1 ) ∧ · · · ∧ ˜ t (ed ), t (J )) Ft (e1 ∧ · · · ∧ ed , J ) = (
(3.3)
(where d = dimH J ). Differentiating (3.3) with respect to t and setting t = 0, we see that the infinitesimal generator of Ft is
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(e1 ∧···∧ed ,J ) =
d
e1 ∧ · · · ∧ K H (e j ) ∧ · · · ∧ ed , ξ JH .
(3.4)
j=1
We will now identify the vertical component of . Let us split K H (e j ) in its H J component and the rest: K H (e j ) = J (K H (e j )) + (I − J )(K H (e j )). Note that THJ := J ◦ K H : H J → H J is precisely the Toeplitz operator associated with H in the almost complex structure J . This leads to a decomposition d
e1 ∧ · · · ∧ K H (e j ) ∧ · · · ∧ ed = Tr(THJ ) e1 ∧ · · · ∧ ed + r,
(3.5)
j=1
where r=
d
e1 ∧ · · · ∧ (I − J )K H (e j ) ∧ · · · ∧ ed .
j=1
We summarize: Lemma 3.3. The decomposition, (3.5), is precisely the decomposition of (e1 ∧···∧ed ,J ) into vertical and horizontal components. In particular, the vertical component is equal to −i Tr(THJ ) times the infinitesimal generator of the circle action on the unit circle bundle of L. ˜ t preserves the connection and the Hermitian structure, Since the one-parameter group its projection, t , is automatically Hamiltonian, and the Hamiltonian is the vertical ˜ t . We have just seen that this function is component of the infinitesimal generator of the trace, J → Tr(THJ ). This proves Proposition 1.3. Proof of Corollary 1.4. Let us verify that the moment map above is in agreement with Donaldson’s moment map for the action of Ham(M) on (J , ). Here Ham(M) is the group of symplectic transformations of M that are the time-one maps of time-dependent Hamiltonians on M; its Lie algebra is C ∞ (M)/R. Standard symbolic arguments show that, as N → ∞, H dµ + O(1/N ) , Tr(THJ,N ) ∼ N · dim(H NJ )
(3.6)
M
where µ is a normalized Liouville measure. Notice that the leading term of this expansion is independent of J , as it should be, since H ∈ C ∞ (M) and H − M H dµ induce the same dynamics on M and therefore on J . (This is also in agreement with the scaling: N ∼ dim(H NJ ) and the extra factor of N in (3.6).) We now compute the second term in the expansion of Tr(THJ,N ).
Manifold of Compatible Almost Complex Structures and Geometric Quantization
Lemma 3.4. For any H ∈ C ∞ (M) and any J ∈ J , 1 THN = NJ N H − H NJ + O(1/N ) . 2
369
(3.7)
Proof. This formula holds in the integrable case, [2] , p.284, without a remainder term. However, the subprincipal symbol calculus in the non-integrable case is exactly the same as in the integrable case, and therefore (3.7) holds in general. Since M H dµ = 0, the second term in the expansion of Tr(THJ,N ) is the same as the second term in the expansion of N N Tr J H J = N H (x) J (x, x) dµ, (3.8) M
where J (x, x) is the restriction to the diagonal of the Schwartz kernel of J . Now, as N → ∞, NJ (x, x) has an asymptotic expansion NJ (x, x) ∼ N n +
N n−1 κ J (x) + l.o.t., 4
where κ J is the Hermitian scalar curvature [18]. Notice that the leading order term is a constant function, so that, asymptotically, the dynamics on J are unchanged if we use the moment map J →
Nn κ J (x) 4
(notice the factor of N in (3.8)). On the other hand we have seen that, if N is the curvature of the L 2 connection on J , then N ∼ N n , where is the standard symplectic form on J . Therefore N n (ξ H , ·) ∼ N (ξ H , ·) ∼ and we recover Donaldson’s result that J → Ham on (J , ).
1 4 κJ
Nn κ J (·), 4
is a moment map for the action of
4. Final Remarks 4.1. Example: Teichmüller Space. In this section we compute explicitly the symbol of the operator in Theorem 2.1 in the case when M is a compact surface of genus g > 1 and the vector bundle E N is restricted to the Teichmüller space Tg embedded into J as a particular slice. Let be a compact smooth surface of genus g > 1. Choose and fix a hyperbolic 2 metric σ (z, z¯ )dzd z¯ on . Here σ (z, z¯ ) = − (z−¯ . Denote by Q(C) the space of holoz )2 morphic quadratic differentials on the hyperbolic Riemann surface C:=( , σ (z, z¯ )dzd z¯ ). Recall that the Teichmüller space is Tg = Met−1 ( )/Di f f 0 ( ), and Met−1 ( ) ∼ = Met ( )/Con f ( ) parametrizes all complex structures on . Consider an explicit
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realization of the Teichmüller space Tg as the smooth family of Riemannian metrics on
¯ z¯ 2 ], ds 2 = f [dz 2 + ρ Edzd z¯ + d
(4.1)
where w = w(z, z¯ ) is uniquely determined by the condition that the map ( , σ (z, z¯ ) dzd z¯ ) → ( , ρ(w, w)dwd ¯ w) ¯ which is the identity as the map → is harmonic [20, 14], ρ(w, w) ¯ := − f :=
∂w ∂ w¯ ∂w ∂ w¯ + , ∂z ∂ z¯ ∂ z¯ ∂z ∂w ∂ w¯ , , (z) = ρ(w, w) ¯ ∂ w¯ ∂z ∂z − ∂w ) ∂ z¯ ∂z
2 , (w − w) ¯ 2 σ (z, z¯ )
∂ w¯ ρ(w, w)( ¯ ∂w ∂z ∂ z¯
E :=
(z)dz 2 ∈ Q(C), and this is an explicit bijection between Tg and Q(C). Note that E > 0. All metrics (4.1) have the same area form σ (z, z¯ )d Re(z)∧d I m(z), this is achieved via introducing the conformal factor f which is not present in the parametrization of Tg given in [20] (p.456) or [14] (p.16), (4.1) provides an embedding ι : Tg → Met ( ), note that ι(Tg ) ∩ Met−1 ( ) consists of one point (corresponding to C). Now pick a point on the slice ι(Tg ) represented by 0 dz 2 ∈ Q(C). The corresponding ¯ 0 d z¯ 2 ], denote metric is f 0 [0 dz 2 + ρ0 E 0 dzd z¯ +
¯ 0 + ρ0 E 0 ¯ 0) 0 + i(0 − g0 = f 0 ¯ 0) ¯ 0 + ρ0 E 0 . −0 − i(0 − ¯ t d z¯ 2 ] Choose two one-parameter families of metrics on C: f t [t dz 2 + ρt E t dzd z¯ + 2 2 ¯ and f s [s dz + ρs E s dzd z¯ + s d z¯ ] with t |t=0 = 0 , s |s=0 = 0 . We have:
¯ t + ρt E t ¯ t) i(t − t + g J (t) = f t ¯ t) ¯ t + ρt E t , i(t − −t − g J (s)
¯ s + ρs E s s + = fs ¯ s) i(s −
g J (t) |t=0 = g J (s) |s=0 = g0 . Denote u 1 = Tg0 ι(Tg ).
¯ s) i(s − ¯ s + ρs E s , −s − dg J (t) dt |t=0 ,
u2 =
dg J (s) ds |s=0 .
So u 1 , u 2 ∈
Proposition 4.1. For variations u 1 , u 2 at g0 the symbol of the Toeplitz operator in Theorem 2.1 is ¯ s) d( f s d( f t t ) 8 |t=0 |s=0 ). I m( (4.2) χ =− σρ0 f 0 E 0 dt ds Remark 4.2. If 0 ≡ 0 (i.e. g0 is the hyperbolic metric σ (z, z¯ )dzd z¯ of C) then f 0 ≡ 1, E 0 ≡ 1, ρ0 ≡ σ , and the pull-back to T[C] Tg of the natural symplectic form on Tg0 ι(Tg ) obtained via integrating (4.2) over with respect to the area form of g0 is, up to a constant factor, the (1, 1)-form associated to the Weil-Petersson metric t on Tg . Indeed, the expression (4.2) becomes − σ82 I m(ϕ1 ϕ¯2 ), where ϕ1 = d dt |t=0 , 2 2 s ϕ2 = d ds |s=0 , ϕ1 (z)dz , ϕ2 dz are holomorphic quadratic differentials on C representing two variations of the complex structure on C (we use that T[C] (Tg ) Q(C), and also T0 Q(C) Q(C) because Q(C) is a finite-dimensional complex vector space).
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Proof of the Proposition. First consider R2 with the symplectic form ω = cd x∧dy, where c is a non-zero constant. Let us denote the matrices of the symplectic form ω, a complex structure J , and the Riemannian
metric g J defined by g J (u, v) := ω(u, J v) by 0 −1 the same letters. Denote J0 = . We have: ω = c J0−1 , g J = ω J , in particular 1 0
1 0 g J0 = c . Consider a 1-parameter family J (t) such that J (0) = J0 . Denote 0 1 X := d Jdt(t) |t=0 ∈ p, where sp(1, R) sl(2, R) = k + p. The Riemannian metric obtained from the Killing form is, up to a positive factor, B(X 1 , X 2 ) = tr (X1 X 2 ). 1 1 [13] The invariant complex structure on p is X → Z X Z −1 , where Z := √1 2 −1 1 −1 p. 323. Therefore the symplectic form on p is given by (X 1 , X 2 ) = −B(X 1 , Z X 2 Z ) = −tr (X 1 Z X 2 Z −1 ). A tangent vector to the space of linear complex structures on R2 dg viewed as the set of metrics with the area form ω is dtJ (t) |t=0 = d(ωdtJ (t)) |t=0 = ωX = c J0−1 X . Now pick a point p ∈ and apply the above considerations to T p (R2 , g0 |T p ). Everything is assumed to be restricted to T p . We have: c = σ , g0 = ωh J0 h −1 = σ J0−1 h J0 h −1 , where
¯ 0 − ρ0 E 0 ) ¯ 0) 1 i f 0 (0 − −σ + f 0 (0 + h=√ ¯ 0) ¯ 0 + ρ0 E 0 ) , −σ − f 0 (0 + i f 0 (0 − 2σ (σ + f 0 ρ0 E 0 ) h ∈ Sp(1, R), dg J (t) dg J (t) 1 |t=0 h = h −1 J0 |t=0 h, dt σ dt dg J (s) dg J (s) 1 |s=0 h = h −1 J0 |s=0 h, X 2 = h −1 ω−1 ds σ ds X 1 = h −1 ω−1
and χ =−
1 dg J (t) dg J (s) |t=0 h Z h −1 J0 |s=0 h Z −1 ). tr (h −1 J0 σ2 dt ds
A straightforward computation results in (4.2).
4.2. Parallel transport and the Schrödinger equation. We now turn to the action of the group Symp0 , of Hamiltonian diffeomorphisms of M, on the bundles E N → J by the “pull-back” operation. We will then compare this action with parallel transport along trajectories of Symp0 on J . This has also been discussed in [21]. Our calculations will be independent of N , so we will suppress it from the notations. Let Ht : M → R be a time-dependent Hamiltonian, and let φt : M → M be its associated Hamilton flow. Thus φ0 is the identity and if we let ξt be the vector field on M given by: d(φt )x (ξt )x =
d φt (x), dt
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then ιξt ω = −d Ht . The one-parameter family of symplectic transformations of M, φt , can be lifted to the bundle L → M. Specifically, if Z ⊂ L ∗ denotes the unit circle bundle, let us define a time-dependent vector field on Z , ηt , by: ηt = ξ˜t + Ht ∂θ , where ξ˜ is the horizontal lift of ξ with respect to the given connection and ∂θ is the generator of the S 1 action on Z . We let f t : Z → Z be the one-parameter family of diffeomorphisms generated by ηt . It is clear that for each t, f t commutes with the circle action, and therefore it defines an automorphism of the bundle L which we continue to denote by f t . We will denote by Vt : L 2 (M, L) → L 2 (M, L) the (unitary) operator of pull-back with respect to f t . As is known, one has: V˙t ◦ Vt−1 = ∇ξt + i Ht .
(4.3)
Let us now pick a base point, J0 ∈ J , and let Et denote the fiber of E → J over Jt = ψt∗ J0 = d(ψt−1 ) ◦ J0 ◦ d(ψt ), and by t the orthogonal projector onto Et . In the holomorphic (Kähler) case it is clear that Vt maps E0 to Et and, Vt being unitary, we therefore have: t = Vt−1 ◦ 0 ◦ Vt .
(4.4)
We will assume that this identity holds as well for the chosen almost-holomorphic quantization scheme (naturality of the scheme). Proposition 4.3. Denote by Pt : L 2 (M, L) → L 2 (M, L) the operator defined by Pt |E0 : E0 → Et is parallel transport along {Jt } and Pt |E ⊥ = 0. Let A : L 2 (M, L) → L 2 (M, L) denote the Schrödinger operator 0
At = 0 ◦ −i∇ξt + Ht ◦ 0 , and let St : L 2 (M, L) → L 2 (M, L) denote the solution to the problem: S0 = 0 . Then Pt = Vt−1 ◦ St .
(4.5) d dt St
= i At St ,
(4.6)
Thus the fundamental solution of the Schrödinger equation arises as the operator that intertwines parallel transport and the operation of pull-back by the classical Hamilton flow. (Notice, by the way, that if J (t) ≡ J (0) then Vt = St , i.e. the fundamental solution is the pull-back operator.) Acknowledgements. We wish to thank Laurent Charles for suggesting improvements to an earlier version of this paper, and for sharing with us a preprint of [6]. We also thank George Marinescu for explaining to us aspects of his paper, [18].
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Appendix A: The Euclidean Case This appendix contains the computation of curvature of the connection in the Euclidean case, needed in the proof of Lemma 2.2. The result which is used there is Corollary A.7 stated at the end of this appendix. n Consider V = R2n with the standard symplectic form ω = x ∧dy j = j=1 d j
0n −1n n i n n . We have: j=1 dz j ∧d z¯ j , x, y ∈ R , z = x + i y ∈ C . Denote σ = 1 2 n 0n ω(u, v) = u t σ t v. Notations: G = Sp(n, R) = {X ∈ G L(2n,R)|X t σ X = σ }, g = A B sp(n, R) = {X ∈ Mat (2n, R)|X t σ + σ X = 0} = { |B = B t , C = C t }, C −At a maximal compact subgroup of G K = Sp(n, R) ∩ U (2n), Cartan automorphism : g → g, X → −X t , the Cartan decomposition is g = k + p,
(A.1)
where k = sp(n, R) ∩ u(2n) = {X ∈ Mat (2n, R)|X t σ + σ X = 0, X + X t = 0}
A B ={ |A = −At , B = B t } = {X ∈ g|(X ) = X }, −B A and
A p = {X ∈ g|(X ) = −X } = { B
B |A = At , B = B t }. −A
Also we have: G = P K , where P = exp p, and gC = kC + pC . The decomposition of an arbitrary element X ∈ g according to (A.1) is given by X=
1 1 (X − X t ) + (X + X t ). 2 2
The algebra g is isomorphic to the algebra of quadratic Hamiltonians (with the Poisson bracket). For X ∈ g the corresponding quadratic form is H (v) = 21 ω(v, X v) = 1 t t 1 2 v σ X v. The Poisson bracket is given by {H1 , H2 }(v) = 2 ω(v, [X 2 , X 1 ]v) = ω(X 1 v, X 2 v). For our purposes we would like to decompose H (v) according to (A.1). We obtain: 1 1 H (v) = v t ( σ t (X − X t ) + σ t (X + X t ))v 4 4 1 t = Re(i z (A + i B )¯z + i z t (A − i B )z), (A.2) 4
A B ∈ g, B = B t , C = C t , A = A − At , B = B − C t , where X = C −At A = A + At , B = B + C t . We observe: A + i B ∈ u(n), and A − i B is a symmetric complex matrix. The Hamiltonian vector field ξ H is determined by ξ H ω = d H , in local coordinates n n ∂H ∂ ∂H ∂ ∂H ∂ ∂H ∂ = 2i . ξH = − − ∂yj ∂x j ∂x j ∂yj ∂z j ∂ z¯ j ∂ z¯ j ∂z j j=1
j=1
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We shall denote by Lξ H the derivation on C ∞ (V ) induced by ξ H , the Lie bracket is [Lξ H1 , Lξ H2 ] = Lξ H1 Lξ H2 − Lξ H2 Lξ H1 . The space J of all linear complex structures on V compatible with the standard symplectic form ω is J = {J ∈ G L(2n, R)|J 2 = −1, J t σ J = σ, v t σ t J v > 0 f or v = 0} and is naturally identified with the bounded symmetric domain of type III: J ∼ = Sp(n, R)/ U (n), G acts on J by conjugation. Any complex structure J ∈ J is obtained from the standard complex structure σ as J = gσ g −1 for g ∈ Sp(n, R), in fact g can always be chosen in P. A function ψ : V → C is called J -holomorphic if dψ ◦ J = idψ. For a positive integer N define the Bargmann space 2 (N ) − N2 |z|2 n Hσ = { f (z)e | f : C → C is entire, | f (z)|2 e−N |z| dµ(z) < ∞} = L 2holo (Cn ; e−N |z| dµ(z)), 2
where dµ(z) is the standard Lebesgue measure. It is in natural bijection with the space of σ -holomorphic sections of the line bundle L ⊗2N , where L → Cn is the trivial line |z|2
bundle with Hermitian structure defined by (s, s) = e− 2 , where s is the unit section. 1 L is a quantizing line bundle on Cn in the sense that c1 (L) = 2π [ω]. Denote by P the (N ) (N ) ∗ unit circle bundle in L . Denote by π = πσ the orthogonal projector L 2 (P) → Hσ . We recall that 2 − N2 |z|2 − N2 |z|2 π( f (z, z¯ )e ) = ce f (w, w)e ¯ N z w¯ e−N |w| dµ(w), where c = c(n, N ) is a constant [3]. The following lemma is proved using integration by parts; the proof is a short straightforward computation. N
(N )
Lemma A.1. For f (z)e− 2 |z| ∈ Hσ 2
the following identities hold:
1 ∂ f − N |z|2 e 2 , N ∂z s N 1 ∂ 2 f − N |z|2 2 π(¯z s z¯r f (z)e− 2 |z| ) = 2 e 2 . N ∂z s ∂zr N
π(¯z s f (z)e− 2 |z| ) = 2
We observe: if g ∈ G, J = gσ g −1 , and ψ is σ -holomorphic, then d(ψ ◦ g −1 ) ◦ J = dψ ◦ σ ◦ g −1 = idψ ◦ g −1 = id(ψ ◦ g −1 ), so ψ ◦ g −1 is J -holomorphic. The map (N ) (N ) ψ → ψ ◦ g −1 establishes a unitary isomorphism between Hσ and the space H J of ⊗N J -holomorphic sections of L . We shall drop N from the notation and write simply πJ . Denote by J0 the fixed point of K in J . Proposition A.2. Let J1 (t) = g1 (t)σ g1 (t)−1 and J2 (t) = g2 (t)σ g2 (t)−1 be two paths in J such that J1 (0) = J2 (0) = J0 , g1 (t) ∈ P, g2 (t) ∈ P, g1 (0) = g2 (0) = I . Denote by τg1 (t) and τg2 (t) the corresponding endomorphisms V → V ; also denote τ˙1 = τ˙g1 (0) , τ˙2 = τ˙g2 (0) . The curvature of the connection at J0 is equal to π [τ˙2 , τ˙1 ]π − [π τ˙2 π, π τ˙1 π ].
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Proof. For π J1 (t) = τg1 (t) ◦ π ◦ τg−1 we obtain: 1 (t) π˙ J1 (t) = τ˙g1 (t) ◦ π ◦ τg−1 − τg1 (t) ◦ π ◦ τg−1 ◦ τ˙g1 (t) ◦ τg−1 , 1 (t) 1 (t) 1 (t) d where τ˙g(t) (ϕ)(v) = dt ϕ(g(t)v), and the hamiltonian operator τ˙1 acts by τ˙1 (ϕ)(v) = d | ϕ(g(t)v). So π ˙ |t=0 = [τ˙1 , π ], and similarly π˙ J2 (t) |t=0 = [τ˙2 , π ]. Finally t=0 J (t) 1 dt
Cur v(∇) = π [[τ˙1 , π ], [τ˙2 , π ]]π = π(−τ˙1 τ˙2 + τ˙1 π τ˙2 + τ˙2 τ˙1 − τ˙2 π τ˙1 )π = π [τ˙2 , τ˙1 ]π − [π τ˙2 π, π τ˙1 π ]. Denote Cur v H1 ,H2 = π [Lξ H2 , Lξ H1 ]π − [π Lξ H2 π, π Lξ H1 π ]. Decomposition (A.2) shows that computation of curvature for τ˙1 ∈ p, τ˙2 ∈ p is reduced to the following proposition. Proposition A.3. (i) If H1 = z m zl , H2 = zr z s , then Cur v H1 ,H2 = 0, (ii) If H1 = z¯ m z¯l , H2 = z¯r z¯ s , then Cur v H1 ,H2 = 0, (iii) If H1 = z m zl , H2 = z¯r z¯ s , then Cur v H1 ,H2 = 4(δmr δls + δms δlr ). Remark A.4. These quadratic Hamiltonians are associated to elements of pC . Proof. (i) We have: ξ H1 = 2i(zl
∂ ∂ ∂ ∂ + zm ), ξ H2 = 2i(zr + zs ), [Lξ H2 , Lξ H1 ] = 0, ∂ z¯ m ∂ z¯l ∂ z¯ s ∂ z¯r k
k
π Lξ H1 π Lξ H2 π( f (z)e− 2 |z| ) = −2ikπ Lξ H1 π( f (z)zr z s e− 2 |z| ) 2
k
2
= −4k 2 f (z)zr z s zl z m e− 2 |z| . 2
Similarly k
k
π Lξ H2 π Lξ H1 π( f (z)e− 2 |z| ) = −4k 2 f (z)zr z s zl z m e− 2 |z| , 2
2
therefore Cur v H1 ,H2 = 0. (ii) ∂ ∂ ∂ ∂ + z¯ m ), ξ H2 = −2i(¯zr + z¯ s ), [Lξ H2 , Lξ H1 ] = 0, ∂z m ∂zl ∂z s ∂zr k k ∂f ∂f 2 2 + z¯r − k z¯ s z¯r f (z))e− 2 |z| ) π Lξ H1 π Lξ H2 π( f (z)e− 2 |z| ) = −2iπ Lξ H1 π((¯z s ∂zr ∂z s 2i ∂ 2 f − k |z|2 = − π Lξ H1 ( e 2 ) k ∂z s ∂zr k 4 ∂ ∂2 f ∂ ∂2 f ∂2 f 2 = − π((¯zl + z¯ m − k z¯l z¯ m )e− 2 |z| ) k ∂z m ∂z s ∂zr ∂zl ∂z s ∂zr ∂z s ∂zr ∂ 2 f − k |z|2 4 ∂2 =− 2 e 2 . k ∂zl ∂z m ∂z s ∂zr ξ H1 = −2i(¯zl
k
Computing π Lξ2 π Lξ1 π( f (z)e− 2 |z| ) we obtain the same expression, hence Cur v H1 ,H2 = 0. 2
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(iii) ∂ ∂ ∂ ∂ + zm ), ξ H2 = −2i(¯zr + z¯ s ), ∂ z¯ m ∂ z¯l ∂z s ∂zr ∂ ∂ ∂ ∂ + z¯r ) + δrl (−z m + z¯ s )) [Lξ H2 , Lξ H1 ] = 4(δsl (−z m ∂zr ∂ z¯ m ∂z s ∂ z¯ m ∂ ∂ ∂ ∂ + z¯r ) + δr m (−zl + z¯ s )). + δsm (−zl ∂zr ∂ z¯l ∂z s ∂ z¯l
ξ H1 = 2i(zl
We compute: k
k ∂f ∂f 2 + z¯r − k f (z)¯z s z¯r )e− 2 |z| ) ∂zr ∂z s k 2i ∂ 2 f − k |z|2 ∂2 f 2 = − π Lξ H1( e 2 ) = −4π( zl z m e− 2 |z| ), k ∂z s ∂zr ∂z s ∂zr
π Lξ H1 π Lξ H2 π( f (z)e− 2 |z| ) = −2iπ Lξ H1 π((¯z s 2
hence k
π Lξ H1 π Lξ H2 π( f (z)e− 2 |z| ) = −4 2
k ∂2 f 2 zl z m e− 2 |z| . ∂z s ∂zr
(A.3)
Also k
k
π Lξ H2 π Lξ H1 π( f (z)e− 2 |z| ) = −2ikπ Lξ H2 π( f (z)zl z m e− 2 |z| ) 2
= −4kπ((¯z s
2
∂( f (z)zl z m ) ∂( f (z)zl z m ) + z¯r ∂zr ∂z s k
−k f (z)zl z m z¯ s z¯r )e− 2 |z| ), 2
so k
2
k
2
π Lξ H2 π Lξ H1 π( f (z)e− 2 |z| ) = −4
∂ 2 ( f (z)zl z m ) − k |z|2 e 2 . ∂zr ∂z s
(A.4)
Also we have: π [Lξ H2 , Lξ H1 ]π( f (z)e− 2 |z| ) = −4(zl (δsm
k ∂f ∂f ∂f ∂f 2 + δr m ) + z m (δls + δrl ))e− 2 |z| . ∂zr ∂z s ∂zr ∂z s
(A.5)
Combining together (A.3), (A.4), and (A.5), we obtain: k
k
Cur v H1 ,H2 ( f (z)e− 2 |z| ) = 4(δmr δls + δms δlr ) f (z)e− 2 |z| . 2
2
The subspace of the algebra of quadratic hamiltonians corresponding to p is generated + := z z + z¯ z¯ and H − := i(z z − z¯ z¯ ), 1 ≤ l ≤ n, 1 ≤ m ≤ n. (over R) by Hm,l m l m l m l m l m,l + (resp. Denote by p+ (resp. p− ) the subspace of p corresponding to the linear span of Hm,l − Hm,l ) in the algebra of quadratic hamiltonians. We have the decomposition p = p+ ⊕p− .
Manifold of Compatible Almost Complex Structures and Geometric Quantization
Remark A.5. dim g = 2n 2 + n, dim k = n 2 , dim p = n 2 + n, dim p+ = dim p− = and dim a = n, where p = a + n.
377 n(n+1) 2 ,
We see that the curvature restricted to p+ (and to p− ) is zero, indeed, Proposition A.3 implies: Corollary A.6. Cur v H +
− m,l ;Hr,s
= −8i(δmr δls + δms δlr ),
+ ;H + = Cur v − Cur v Hm,l H r,s
− m,l ;Hr,s
= 0.
Proof. Cur v H +
− m,l ;Hr,s
= i(Cur vz m zl ,zr z s + Cur vz¯ m z¯l ,zr z s − Cur vz m zl ,¯zr z¯ s − Cur vz¯ m z¯l ,¯zr z¯ s ) = i(−Cur vzr z s ,¯z m z¯l − Cur vz m zl ,¯zr z¯ s ) = −8i(δmr δls + δms δlr ).
Similarly + ;H + = Cur vz z ,z z + Cur vz¯ z¯ ,z z + Cur vz z ,¯z z¯ + Cur vz¯ z¯ ,¯z z¯ Cur v Hm,l m l r s m l r s m l r s m l r s r,s
= −Cur vzr z s ,¯z m z¯l + Cur vz m zl ,¯zr z¯ s = −4(δmr δls + δms δlr ) + 4(δmr δls + δms δlr ) = 0, and Cur v H −
− m,l ;Hr,s
= −(Cur vz m zl ,zr z s − Cur vz¯ m z¯l ,zr z s − Cur vz m zl ,¯zr z¯ s + Cur vz¯ m z¯l ,¯zr z¯ s )
= −(Cur vzr z s ,¯z m z¯l −Cur vz m zl ,¯zr z¯ s )= − (4(δmr δls +δms δlr ) − 4(δmr δls +δms δlr )) = 0. Corollary A.7. If J ∈ J , u, v ∈ T J J , and q1 , q2 are the quadratic forms representing u, v, and X 1 , X 2 are the elements of p corresponding to q1 , q2 respectively, then the (k) curvature of the connection (which, once u, v are fixed, is an element of End(H J )) is (k) equal to − 21 i T r (X 1 σ X 2 )I d, where I d denotes the identity operator on the fiber H J . Proof. First we see that due to Corollary A.6 this is true if J = J0 (the proof is a straightforward verification). The curvature and the symplectic form on Sp(n, R)/U (n) are both G-invariant, therefore the equality holds at every other point of J . Appendix B: Notation Index L → M: A prequantum Hermitian line bundle with connection. ω:
Symplectic form on M, equal to the curvature of L → M.
J:
The space of compatible almost complex structures on (M, ω).
:
The symplectic form on J .
gJ :
For J ∈ J , denotes the Riemannian metric g J = ω(·, J ).
NJ :
The metric Laplacian on sections of L N associated to g J .
NJ :
= NJ − N n is the rescaled Laplacian, where n is half the dimension of M.
E JN :
The subspace of C ∞ (M, L N ) spanned by the eigensections of NJ of bounded eigenvalues.
378
dN :
T. Foth, A. Uribe
The complex dimension of E JN .
E N → J : The bundle whose fibers are the E JN . LN :
= ∧d N E N → L is the top exterior power of E N → J .
N :
The curvature of the L 2 connection on L N .
NJ :
The orthogonal projector L 2 (M, L N ) → E JN .
H :
The Hamilton vector field on M associated to H ∈ C ∞ (M) through the symplectic form ω.
∇ (N ) :
The natural connection on L N ; its curvature form is N ω.
K HN :
=
THN :
=
κJ :
=
(N ) 1 ∞ N i N ∇ H + H acting on C (M, L ). NJ ◦ K HN ◦ NJ Toeplitz operator acting on C ∞ (M, L N ). r J + 21 |∇ LC J |2 it the Hermitian scalar curvature, where r J
and ∇ LC
are the scalar curvature and Levi-Civita connection of g J , respectively. χ:
For each A, B ∈ T J J , χ A,B is the symbol of the curvature of the L 2 connection on {E N → J } evaluated at J, A, B.
References 1. Axelrod, S., de la Pietra, S., Witten, E.: Geometric quantization of Chern-Simons gauge theory. J. Diff. Geom. 33, 781–902 (1991) 2. Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz Quantization of Kähler Manifolds and gl(N ), N → ∞ limits. Commun. Math. Phys. 165, 281–296 (1994) 3. Borthwick, D.: Introduction to Kähler quantization. In First Summer School in Analysis and Mathematical Physics (Cuernavaca Morelos, 1998), Contemp. Math. 260, Providence, RI: Amer. Math. Soc., 2000, pp. 91–132 4. Borthwick, D., Uribe, A.: The semiclassical structure of low-energy states in the presence of a magnetic field. to appear in Trans. AMS 359, 1875–1888 (2007) 5. Boutet de Monvel, L., Guillemin, V.: The spectral theory of Toeplitz operators. Annals of Mathematics Studies No. 99, Princeton, NJ: Princeton University Press (1981) 6. Charles, L.: Semi-classical properties of geometric quantization with metaplectic correction. http://arxiv.org/list/math.SG/0602168, 2006 7. Daubechies, I.: Coherent states and projective representation of the linear canonical transformations. J. Math. Phys. 21(6), 1377–1389 (1980) 8. Donaldson, S.: Remarks on gauge theory, complex geometry and 4-manifold topology. In: Fields Medallists’ lectures, World Sci. Ser. 20th Century Math. 5, River Edge, NJ: World Sci. Publishing, 1997, pp. 384–403 9. Donaldson, S.: Scalar curvature and projective embeddings. I. J. Diffe. Geom. 59(3), 479–522 (2001) 10. Epstein, C., Melrose, R.: Contact degree and the index of Fourier integral operators. Math. Res. Lett. 5(3), 363–381 (1998) 11. Ginzburg, V., Montgomery, R.: Geometric quantization and no-go theorems. In: Poisson geometry (Warsaw, 1998), Banach Center Publ. 51, Warsaw: Polish Acad. Sci., 2000, pp. 69–77 12. Guillemin, V., Uribe, A.: The Laplace operator on the nth tensor power of a line bundle: eigenvalues which are uniformly bounded in n. Asymptotic Analysis 1, 105–113 (1988) 13. Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press (1962) 14. Jost, J.: Harmonic maps and curvature computations in Teichmüller theory. Ann. Acad. Sci. Fen., Ser. A. I. Math. 16, 13–46 (1991) 15. Kirwin, W., Wu, S.: Geometric Quantization, Parallel Transport and the Fourier Transform. http://arxiv.org/list/math.SG/0409555, 2004 16. Zhiqin, Lu.: On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math. 122(2), 235–273 (2000)
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17. Ma, X., Marinescu, G.: The Spinc Dirac operator on high tensor powers of a line bundle. Math Z. 240(3), 651–664 (2002) 18. Ma, X., Marinescu, G.: Generalized Bergman kernels on symplectic manifolds. C. R. Math. Acad. Sci. Paris 339(7), 493–498. (2004) Announced in: Available at http://arixv.org/list/math.DG/0411559, 2004 19. Viña, A.: Identification of Kähler quantizations and the Berry phase. J. Geom. Phys. 36(3–4), 223–250 (2000) 20. Wolf, M.: The Teichmüller theory of harmonic maps. J. Diff. Geom. 29, 449–479 (1989) 21. Zelditch, S.: Index and dynamics of quantized contact transformations. Ann. Inst. Fourier 47(1), 305–363 (1997) Communicated by P. Sarnak
Commun. Math. Phys. 274, 381–397 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0282-7
Communications in
Mathematical Physics
Resistivity of an Infinite Three Dimensional Stationary Random Electric Conductor Jérôme Depauw Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Université Rabelais, Parc de Grandmont, F-37000 Tours, France. E-mail:
[email protected] Received: 22 May 2006 / Accepted: 8 February 2007 Published online: 21 June 2007 – © Springer-Verlag 2007
Abstract: The aim of this paper is to prove that the resistivity of an infinite stationary random medium can be computed from one realization. This is accomplished through the application of the pointwise degree 2 ergodic theorem for divergence-free stationary random fields (see [4]). 1. Introduction We consider a probability space (Ω, T , µ), and a three dimensional random tensor of resistivity ρ˜ defined on it, i.e. a random field ρ˜ : R3 × Ω → S3+ (x, ω) → ρ(x, ˜ ω) of symmetric positive definite 3 × 3-matrices. We suppose that its stochastic law is invariant under shifts in R3 . Actually, we assume that there exists a measure preserving action T of R3 , T : R3 × Ω → Ω (x, ω) → Tx ω and a positive definite symmetric random matrix ω → ρ(ω) defined on Ω, such that ρ(x, ˜ ω) = ρ(Tx ω). For details on this representation, which is not really restrictive, we refer the reader to Chapter 7 of the book of Jikov et al. [7]. We restrict ourselves to the elliptic case, i.e. we suppose that there are two constants 0 < c < C such that the three eigenvalues of ρ(ω) are between c and C for any ω. We assume moreover that the dynamical system (Ω, T , µ, (Tx )x∈R3 ) on which ρ is defined is ergodic. The datum is the expectation I = (I1 , I2 , I3 ) of the flux of the current in the medium. We seek a stationary flux which can be represented by a field x → f (Tx ω) satisfying divT f = 0, in the weak sense (law of conservation of the current), where the operator divT is defined as follows:
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Definition 1. Let f be an integrable random vector and g be an integrable random variable. We say that g is the divergence of f if, ω-almost surely, for any C ∞ -function φ with compact support in R3 we have 3 ∂φ (x) · f i (Tx ω) dλ(x) = ∂ xi i=1 − φ(x) · g(Tx ω) dλ(x), where λ is the Lebesgue measure on R3 . We denote this property by divT ( f ) = g. When divT ( f ) = 0, we say that the random vector f is divergence-free in the weak sense. The dissipated power density x → P(Tx ω) is given by the Ohm formula P = f , ρ f (multidimensional and continuous version of the law “the power is equal to the resistance multiplied by the square of the intensity of current”). It is assumed that electricity passes through the medium in such a way that the expectation of the power density is minimal. It can easily be checked, by a standard Lax-Milgram method of orthogonal projection, that the system satisfied by f can be written as ⎧ ⎪ f dµ = I ⎨ Ω (S0 ) , ⎪ ⎩ curlT ρ f = 0 divT f = 0 where the operator curlT is defined as follows. Definition 2. Let g, h be integrable random vectors. We say that h is the curl of g if, ω-almost surely, we have for any C ∞ -function φ with compact support, ∂φ ∂φ (x)g j (Tx ω) − (x)gk (Tx ω) dλ(x) = φ(x)h i (Tx ω) dλ(x), ∂ xk ∂x j for any cyclic permutation (i, j, k) of (1, 2, 3). We will denote h = curlT g. The field x → ρ(Tx ω) f (Tx ω) represents the gradient of the electric potential in the medium. The existence of a unique solution in L2 (Ω) was proved by Golden and Papanicolaou (see [6]). In addition Jikov et al. showed in [7] the existence of a positive definite symmetric matrix ρ0 such that P dµ = I, ρ0 I. Ω
Let ω be a realization. We want to calculate the effective resistivity of the medium. We consider, in the medium compressed with a factor R, a cubic cell, with edges parallel to the axes, length c and center x. From the macroscopic point of view, we can consider that it is crossed by a current J R (c, x, I , ω) = f (TRy ω), ei dσ (y) , Si
i=1,...,3
each integral being computed on the section Si of the cell passing by its center x and with normal vector the i th vector ei of the canonical basis of R3 . The power dissipated in this cell is given by PR (c, x, I, ω) = P(TRy ω) dλ(y),
Resistivity of an Infinite Three Dimensional Stationary Random Electric Conductor
383
the domain of integration being the cell itself. According to the Ohm law, the resistance of the cell for a direction of current parallel to J R (c, x, I, ω) is the quantity equal to the power divided by the square of the intensity of current PR (c, x, I , ω) . J R (c, x, I , ω)2 Its resistivity is its resistance multiplied by its cross sectional area c2 and divided by its length c, c
PR (c, x, I, ω) . J R (c, x, I , ω)2
Assume for the moment that, in order to impose the current in the cell, it is possible for any J ∈ R3 to choose the datum I such that J R (c, x, I , ω) = J (the proof of this fact is given below). Denote by I R (c, x, J, ω) the solution I, and P˜R (c, x, J , ω) the corresponding power dissipated in the cell. The limit of the resistivity tensor of the cell is the matrix ρ0 if, for any direction of current J, the limit of the resistivity of the cell 0 J is J,ρ , hence if we have the convergence J 2 c
J, ρ0 J P˜R (c, x, J, ω) −→ J2 J2
for R tending to infinity. Moreover, the resistivity of the medium itself converges if the above convergence holds for any cell. As it is explained below, the existence of a full-probability set of realizations on which the above convergence holds for any cell (contained in the ball of center 0 and radius 1, see Corollary 2) is a consequence of the pointwise degree 2 ergodic theorem for divergence-free random vectors (see [4]). In order to apply this theorem to the random vector f , it should be checked that this last lies in the Lorentz space L2,1 . This is what we prove in this paper. Theorem 1. The solution of the system (S0 ) lies in L p (Ω) for some p > 2. We state now the degree 2 ergodic theorem for f . We denote by (A, B, C) the surface delimited by the triangle with vertices A, B, C. Theorem 2 (Degree 2 Ergodic Theorem). Let f be a divergence-free random field for an ergodic stationary R3 -action. We suppose that f ∈ L2,1 . We denote by B the ball of center 0 and radius 1. Then, for almost all ω, the integral of the field x → f (Tx ω) is defined through any triangle, and we have the uniform convergence
(A,B,C)
f (TRx ω), dσ (x) −→
R→+∞
(A,B,C)
Ω
f dµ , dσ (x)
on A, B, C ∈ B, where dσ denotes the normal infinitesimal field of the triangle . We point out that a similar version can be stated for the usual Wiener ergodic theorem on balls (see [10]), or rather on tetrahedra, which appears to be the degree 3 ergodic theorem in [4]. This is
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Theorem 3 (Degree 3 Ergodic Theorem). Assume that P is an integrable function defined on Ω. For A, B, C, D ∈ B, we denote the domain delimited by the tetrahedron of vertices A, B, C, D by τ (A, B, C, D). Then for almost all ω, the integral of the function x → P(Tx ω) is defined on any tetrahedron, and we have the uniform convergence P(TRx ω) dλ(x) −→ P dµ dλ(x) τ (A,B,C,D)
R→+∞
τ (A,B,C,D)
Ω
on A, B, C, D ∈ B. In our framework, these results mean that the flux of the current through triangles and the power dissipated in tetrahedra almost surely converge to their expectations, convergences being uniform on B. In these theorems, the datum I is fixed. Since the system of equations (S) is linear, the principle of superposition of the solutions works, and I can vary. Convergences are now also uniform on I, with the proviso of imposing that the norm I remains bounded. According to our notations for the power and current associated to a cell, the two above ergodic theorems can thus be rewritten as Corollary 1. Let M be a positive constant. For almost all ω, we have the following convergences J R (c, x, I , ω) −→ c2 I; R→+∞
PR (c, x, I , ω) −→ c3 I, ρ0 I, R→+∞
(1)
uniformly with respect to I satisfying I ≤ M, and x, c such that the associated cell is contained in the ball B. In order to impose the current J R (c, x, I, ω) passing through the cell, it is possible to choose a suitable datum I. More precisely, we have Lemma 1. Let c0 be a positive constant. For almost all ω, there exists R0 > 0 such that for any c > c0 and x such that the associated cell belongs to the ball B, for any J ∈ R3 , R > R0 , the equation J R (c, x, I , ω) = J has a unique solution I. Proof. Since the map I → J R (c, x, I, ω) is linear from R3 on itself, it is enough to check that the determinant det J R (c, x, ei , ω) i=1,...3
does not vanish. According to Corollary 1, it is close to c6 . Now assumption c > c0 guarantees that it is bounded below by c06 /2, for R large enough.
We denote by I R (c, x, J , ω) the solution I given by the above lemma, and the corresponding power in the cell by P˜R (c, x, J , ω). We have
Resistivity of an Infinite Three Dimensional Stationary Random Electric Conductor
385
Corollary 2. Assume c0 > 0. For almost all ω, we have c
J, ρ0 J P˜R (c, x, J, ω) −→ , 2 R→+∞ J2 J
uniformly with respect to J , and with respect to c > c0 and x such that the associated cell is contained in the ball B. Proof. We first remark that multiplying J by a scalar does not change the ratios which appear in the convergence stated by our corollary. Hence we may assume that J = 1. According to Corollary 1, the datum I is close to J/c2 . Thus, the assumption c > c0 guarantees that I remains bounded. Now it follows from Corollary 1 that the power ˜ J) ∼ J, ρ0 J, and ˜ J) = P(I) is close to c3 I, ρ0 I. This leads obviously to c P( P( completes the proof.
As it was already said, this convergence means that the limit of the resistivity of any cell J, ρ0 J for a current of direction J is , therefore the limit of the tensor of resistivity is J2 ρ0 . Moreover the full-probability set of convergence depends neither on the cell nor on the direction of the current . Theorem 1 is obtained in this paper by applying methods from harmonic analysis. The point is to find L p estimates ( p ≥ 2) for the solution g of the following auxiliary system: ⎧ ⎪ ⎨ g dµ = U Ω (Sh ) 0 ⎪ ⎩ curlT g = divT g = divT h. The Golden-Papanicolaou method to obtain L2 estimates was based on an orthogonal decomposition, which does not work on L p , for p > 2. In this paper we pass through an explicit expression of the solution g, using a “Riesz’s fashion” operator. This operator is an analog of the classical Riesz operator Ri , i = 1, . . . , 3 on R3 , modified to be defined on dynamical systems (Ω, T , T ). Similar methods were already used in our preceding work [3], which is related to discrete networks. Nevertheless, the proofs are different. 3 Indeed, a fundamental property of Riesz operators is the equality i=1 Ri2 f = − f . This property was obtained in paper [3] by using the fact that the square of our “Riesz’s fashion operator” on discrete dynamical systems is a kernel operator. This is not true anymore on continuous dynamical systems. Hence in this paper the analogous equality is obtained by using a property of duality. See Sect. 5 below for details on this remark. 2. Statement of the Problem with the Potential The solution of our system is more simply expressed on the gradient g of the electrical potential g = ρ f . We denote by A the tensor of conductivity A = ρ −1 . The datum is now the expectation U = (U1 , U2 , U3 ) of g. The system satisfied by g is thus written as ⎧ ⎪ ⎨ g dµ = U Ω (S) ⎪ ⎩ curlT g = 0 divT Ag = 0.
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The existence of a unique solution in L2 (Ω), as the equality U = ρ0 I, is proved for instance in [7] (Sect. 7.2, Eq. 7.10). According to the ellipticity assumption, Theorem 1 can be rewritten as Theorem 1. There exists p > 2 such that the unique L2 (Ω)-solution of the above system (S) lies in L p (Ω). The sequel of the paper is devoted to the proof of Theorem 1. 3. System with Constant Coefficients We first study the system with constant coefficients, and with the right-hand side divT h, where h ∈ L p , namely ⎧ ⎪ ⎨ g dµ = 0 Ω (Sh ) ⎪ ⎩ curlT g = 0 divT g = divT h. Since h is not supposed to admit a divergence, the last equation is to be understood as divT (g − h) = 0. We remark that for p = 2, the spaces 2 Vpot = {g ∈ L2 ; g dµ = 0 and curlT g = 0} Ω
L2sol = {g ∈ L2 ; divT (g) = 0} constitute an orthogonal decomposition of L2 (Ω) (see Lemma 7.3 in Jikov et al. [7]). This gives Proposition 1 (Jikov et al.). Assume that h ∈ L2 (Ω). The system (Sh ) has a unique 2 . We have the inequality solution g ∈ L2 , which is the orthogonal projection of h on Vpot g2 ≤ h2 . We want to solve the system (Sh ) for p > 2, and establish an inequality of the form g p ≤ κ p h p (in the vectorial case h = (h 1 , h 2 , h 3 ), the norm L p (Ω) is defined by p
h p =
3 i=1
Ω
|h i | p dµ
).
The resolution of the similar question on R3 passes through the Riesz operator (see Giaquinta [5]). Hence we will define an operator of “Riesz’s fashion” on the dynamical system (Ω, µ, T ). 3.1. Operator of “Riesz’s fashion”. Formally, the analog of the Riesz operator is defined for any 1 ≤ i ≤ 3 by yi Ri ( f )(ω) = f (T−y ω) 2 dλ(y), 4 π y y where λ denotes the Lebesgue measure on R3 , and y = (y1 , y2 , y3 ). But the function under the integral does not belong to L1 (dλ) neither at 0 nor at infinity. The following
Resistivity of an Infinite Three Dimensional Stationary Random Electric Conductor
387
section is devoted to giving sense to this integral. We note that in the discrete case of an action of Z3 , there is no problem at 0 (see [1, 3]), and for the traditional Riesz operator on yi R3 , there is no problem at infinity, because the norm Lq (R3 ) of the kernel 1y>ε π 2 y 4 p is finite for all 1 < q < ∞, and y → f (y) is assumed belonging to L (dλ) at infinity (see [8]). 3.1.1. Definition, continuity We start by defining the operator R j on a space of functions having suitable regularity, then we will extend it by continuity. We denote by C T the space of functions h ∈ L∞ (Ω) such that for almost all ω, the function h ω defined on R3 by h ω : x → h(Tx ω) is a C ∞ -function having partial derivatives of all orders bounded on x and ω. Finally let us denote by E the space defined by E=
f =
3
∂i h i ; h i ∈ C T , i = 1, . . . 3 ,
i=1 ω where ∂i h(ω) = ∂h ∂ xi (0). Next, consider the operator Ri,ε defined on E by the following lemma:
Lemma 2. Assume that f ∈ E and ε > 0. The integral yi R Ri,ε ( f )(ω) = f (T−y ω) 2 dλ(y) π y4 ε<y
3 R<y
∂h j,ω ∂x j
and deriving
h j,ω (−y)
yi hω (−y), dσ R (y) − π 2 y4
yi ), π 2 y4
we obtain the following
yi ∂ dλ(y)+ 2 ∂ x j π y4
y=R
yi hω (−y), dσ R (y), π 2 y4
where dσ R is the infinitesimal normal field of the sphere of radius R. Passing to polar |yi | yi 1 ∂ 4 coordinates we have y 4 ≤ r 3 , and | ∂ x j y4 | ≤ r 4 . Integrating (again in polar coordinates), we thus obtain the upper bound 1 −2 −1 −1 × 3. h 4π 4r dr + R + R ∞ π2 R
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Our next claim is Lemma 3. For all 1 < p < ∞, there exists a constant c p such that for all f ∈ E and all ε > 0 we have Ri,ε ( f ) p ≤ c p f p . Proof. According to the “transfer principle”, the proof consists in using the analogous result on R3 , due to Calderon and Zygmund (see [2], another good reference is Stein’s book [8], Theorem 3 in Chapter II). Theorem 4 (Calderon and Zygmund). For all 1 < p < ∞, there exists a constant c p such that, for all ε > 0, the operator Ri,ε defined on L p (R3 ) by yi Ri,ε (φ)(x) = φ(x − y) dλ(y) 2 4 y>ε π y satisfies Ri,ε φ p ≤ c p φ p . Moreover the limit lim Ri,ε (φ)(x) = Ri (φ)(x) exists in L p (R3 ) norm, and satisfies Ri φ p ≤ c p φ p .
ε→0
Application of the transfer principle can be done by using the usual pointwise ergodic theorem on balls (see Wiener [10]). We present this method now. Thus assume ε > 0 and f ∈ E. For any ω, for all R > 0, the function φω,R defined by φω,R (x) = f (Tx ω)·1x
ε x−y
and, for ρ > 0, Rρε ( f )(Tx ω) =
ε<y<ρ
K (y) f (Tx−y ω) dλ(y).
If ρ < R and x < R − ρ, the first domain of integration contains the second one. Hence we have ρ Rε (φω,R )(x) = Rε ( f )(Tx ω) + K (y) f (Tx−y ω) dλ(y). y>ρ x−y
According to the proof of Lemma 2, and considering h ∈ C T such that f = divh, the last integral above is bounded by h∞ 72π −1 ρ −1 . Hence by considering the norm L p (D, dλ) of the domain D = {x, x < R − ρ}, we have, according to the triangular inequality, 1/ p |Rρε ( f )(Tx ω)| p dλ(x) ≤ x
Resistivity of an Infinite Three Dimensional Stationary Random Electric Conductor
389
Since φω,R is zero except on the domain x < R, this integral is equal to 1/ p | f (Tx ω)| p dλ(x) . cp x
Plugging this in to Inequality (2), dividing each term by (4π R 3 /3)1/ p and using the usual ergodic theorem, we obtain for R tending to infinity, Rρε ( f ) p ≤ c p f p + h∞ 72π −1 ρ −1 . Finally, for ρ tending to infinity, we obtain the desired bound Rε ( f ) p ≤ c p f p .
This lemma allows us to extend the operator Ri,ε to the closure of E in L p (Ω). However we have Lemma 4. The closure of E for the L p (Ω) norm is the space of L p (Ω)-functions of p vanishing expectation. This space is denoted by L0 (Ω) in the sequel. Proof. Let f be a function in L p (Ω) with vanishing expectation. We want to approach it, for the L p (Ω) norm, by a function belonging to E. Since L∞ (Ω) is dense in L p (Ω), we can suppose that f is bounded. Consider the function h R defined by 3 h j,R (ω) = h j,x (ω) dλ(x), 4π R 3 x
1
h j,x (ω) = −x j
f (Tsx ω) ds.
0
It can be easily checked that, in the weak sense, we have divT h x = f − f ◦ Tx . This leads to 3 divT h R (ω) = f (ω) − f (Tx ω) dλ(x). 4π R 3 x 0, denote by ψη the function defined by ψη (x) = η13 ψ( ηx ). Consider finally the function (ψη T h R ) defined by (ψη T h R )(ω) = ψη (y)h R (T−y ω) dλ(y). The following formula can be easily checked a.s. ω: ∂i (ψη T h R ) = (
∂ψη T h R ) ∂ xi
(3)
(by dominated convergence, since h R is bounded and ψ has compact support). It is now clear that a.s. ω, the function x → (ψη T h R )(Tx ω) admits partial derivatives of order 1,
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which are bounded on ω. By induction, we show in a similar way that this function is C ∞ and has all its partial derivatives bounded on ω. Thus the function fη,R = divT (ψη T h R ) lies in E. It remains to be checked that f η,R (ω) converges to f R (ω) for the L p (Ω) norm when η tends to 0. However we also have, according to formula (3) and Definition 1 , divT (ψη T h R ) = ψη T divT (h R ) which tends to divT (h R ) in L p norm for η tending to 0, according to the usual Wiener local ergodic theorem on balls. Since f R is as close as we want to f in L p (Ω), the lemma is proved.
p
We always denote by Ri,ε the operator extended to L0 (Ω). p
Proposition 2. For any p > 1 and for any function f ∈ L0 (Ω), the limit Ri ( f )(ω) = lim Ri,ε ( f ) →0
exists in
p L0 (Ω)
norm. We have moreover Ri ( f ) p < c p f p .
Proof. Since the norm of the operator Ri,ε does not depend on ε, it is enough to check convergence for functions f ∈ C T of null expectation. However we have Ri,ε ( f )(ω) − Ri,ε ( f )(ω) = f (Ty ω)K (−y) dλ(y). ε<y<ε
Since the integral of the kernel K (y) on any sphere is zero, it is still equal to f (Ty ω) − f (ω) yK (−y) dλ(y). y ε<y<ε The above ratio being bounded (because f ∈ C T ), and function y → yK (y) being integrable on a neighborhood of 0, the family of functions Ri,ε ( f ) constitutes a Cauchy p sequence in L0 (Ω). This proves the desired convergence, and proposition.
3.1.2. Properties of duality Weak operators curlT and divT which appear in the system (Sh ) were defined, by analogy with the theory of distributions, by formulas of duality. To solve the system (Sh ) using Ri , it is thus necessary to study the properties of duality of these operators. We start by extending the convolution φ T f . As it was seen in the proof of Lemma 4, it is defined by the formula φ(−x) f (Tx ω) dλ(x). φ T f (ω) = This integral converges for almost all ω if φ is integrable on R3 , and f is bounded. More1 over, if f ∈ C T and if φ and all its derivatives are bounded by x 4 up to a constant, we have, by dominated convergence, the two formulas ∂ (φ T f ) = φ T ∂ f ∂φ = T f. ∂ x We extend now this convolution for f ∈ L p (Ω).
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Lemma 5. Assume 1 ≤ p ≤ ∞. Assume φ ∈ L1 (R3 ) and f ∈ L p (Ω). The formula φ(−x) f (Tx ω) dλ(x) φ T f (ω) = defines an L p (Ω)-function, and we have φ T f p ≤ φ1 f p . Proof. The inequality is an easy consequence of the Jensen inequality and of the sta tionarity of T . Since L p (Ω) is complete, this proves that φ T f is well defined.
We state now the first duality property of Ri . Lemma 6. Let ψ be a C ∞ -function We suppose that ψ and all its derivatives are bounded 1 by x 4 up to a constant. Then ∞ −4 up to a 1. Ri ( ∂∂ψ x j ) and all its derivatives are C - functions bounded by x → x constant. 2. We have, for any h ∈ C T , the formula of duality a.s. ω
ψ T Ri (∂ j h)(ω) = Ri (
∂ψ ) T h(ω). ∂x j
(5)
p
Remark 1. We can not extend this lemma for h ∈ L0 (Ω), since the operator ∂ j is not continuous. Proof. In the sequel we denote by a “prime” the derivatives with respect to x j . To begin with, we study Ri (∂ j h)(Tx ω). This quantity is the limit, for R tending to infinity and ε tending to 0, of R Ri,ε (∂ j h)(Tx ω) = ∂ j h(Tx−y ω)K (y) dλ(y). ε<y
Splitting this integral according to y < 1 and y > 1, we have, for R tending to infinity, Ri,ε ∂ j h(Tx ω) = ∂ j h(Tx−y ω)K (y) dλ(y)+ ε<y<1 h(Tx−y ω)K (y) dλ(y)+ (6) 1<y − h(Tx−y ω)K (y) dσ j (y). y=1
where dσ = (dσk )k=1,...,3 is the normal infinitesimal field of the sphere, and the two last terms are coming from an integration by parts. Hence Ri,ε ∂ j h is bounded, since all these integrals are convergent, and h ∈ C T . Similarly, computing derivatives by dominated convergence, we have Ri,ε ∂ j h ∈ C T . Now, since ψ ∈ L1 (R3 ), the convolution ψ T Ri,ε ∂ j h is bounded, and is equal to ψ T Ri,ε ∂ j h(ω) = ∂ j h(Tx−y ω)K (y)ψ(−x) dλ(y)dλ(x) x ε<y<1 + h(Tx−y ω)K (y)ψ(−x) dλ(y)dλ(x) x 1<y + h(Tx−y ω)K (y)ψ(−x) dσ j (y)dλ(x). x
y=1
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By the change of variables u = x − y, v = y in these three integrals, and then integrating by parts with respect to u in the first one, we obtain ψ T Ri,ε ∂ j h(ω) = Ri,ε ψ T h(ω), where
Ri,ε ψ (−u) =
ε<v<1
+
K (v)ψ (−u − v) dλ(v) K (v)ψ(−u − v) dλ(v)
+
(7)
1<v
v=1
K (v)ψ(−u − v) dσ j (v).
Since the domains of the first and third integral are contained in u − v ≤ 1, these integrals have the same order as ψ (−u) and ψ(−u) for u tending to infinity, hence are bounded by Mε max(u−4 , 1). The analogous upper bound on the second integral follows from the classical formula v−4 u − v−4 dλ(v) ≤ Mu−4 v>1 u−v>1
for u large enough. Now, since the integral of K on any sphere vanishes, the first term of the above decomposition of Ri,ε ψ (−u) can be rewritten as ψ (−u − v) − ψ (−u) dλ(v). vK (v) v ε<v<1 This is bounded by M max(u−4 , 1) independently of ε, since v → vK (v) is integrable on v < 1. Hence, for ε tending to 0, we have |Ri ψ (u)| < M max(u−4 , 1). By similar arguments, this kind of upper bound holds for any derivatives of Ri ψ , and Paragraph 1 is proved. Moreover, it follows from the fact that M does not depend on ε, that the convergence Ri,ε ψ → Ri ψ for ε tending to 0 is dominated, hence holds in L1 (R3 ). By an analogous computation, the convergence Ri,ε ∂ h → Ri ∂ h for ε tending to 0 holds in L∞ (Ω). Now, according to Lemma 5 with p = +∞, we can take ε tending to infinity in (7), which gives (5). This achieves the proof of Lemma 6.
In order to obtain a duality formula for Ri2 , we have to generalise Lemma 6 for functions p f ∈ L0 . Lemma 7. Let φ be a C ∞ -function with compact support. Then ∞ −4 up 1. Ri ( ∂∂φ x j ) and all its partial-derivatives are C -functions bounded by x → x to a constant. p 2. We have, for all f ∈ L0 (Ω), the duality formula a.s. ω,
∂φ ∂φ T Ri ( f )(ω) = Ri ( ) T f (ω). ∂x j ∂x j
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Proof. Paragraph 1 follows from Lemma 6. Next, applying ∂ to (5) we have ∂ (φ T Ri (∂ j h)) = ∂ (Ri (
∂φ ) T h). ∂x j
Applying (4), we obtain φ T ∂ (Ri (∂ j h)) = Ri (
∂φ ) T ∂ h. ∂x j
(9)
Since ∂ (Ri (∂ j h)) = Ri (∂ j ∂ h) = ∂ j (Ri (∂ h)), the left hand side of (9) is ∂ j (φ T Ri (∂ h)). From (4), it is still ∂∂φ x j T (Ri (∂ h). It follows ∂φ ∂φ T Ri (∂ h)(ω) = Ri ( ) T ∂ h(ω). ∂x j ∂x j p
By density of E in L0 (Ω), continuity of Ri , and Lemma 5, this gives (8).
These lemmas will be applied in particular in the following way: if ψ = Ri ( ∂∂φ xk ), where φ is a C ∞ -function with compact support, and h ∈ C T , it follows from Lemma 6, Ri (
∂φ ∂ ∂φ ) T R j (∂ h) = R j ( Ri ( )) T h, ∂ xk ∂ x ∂ xk
and from Lemma 7 with f = R j ∂ h, ∂φ ∂φ T Ri (R j ∂ h) = Ri ( ) T R j ∂ h. ∂ xk ∂ xk
(10)
Finally we have Corollary 3. For any C ∞ -function φ with compact support and for any h ∈ C T , we have Rj(
∂ ∂φ ∂φ Ri ( )) T h = T Ri (R j ∂ h). ∂ x ∂ xk ∂ xk
3.2. Resolution of (Sh ). These results have as an immediate consequence the expression of the solution of the system (Sh ) using the operators R j . Proposition 3. Assume that p ≥ 2 and that h ∈ L0 . The system (Sh ) admits as a unique solution in L p (Ω) the field g defined by p
gi = −
3
Ki j (h j ),
j=1
with Ki j = Ri R j . Moreover there exists a constant κ p not depending on h such that g p ≤ κ p h p .
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Proof. The case p = 2 was done at the beginning of this section. It was also seen that κ2 = 1. By inclusion of L p ⊂ L2 , this gives uniqueness for p > 2. To check that the suggested solution is appropriate, we can suppose that h ∈ E, i.e. h = j j ∂ h , j with h , j ∈ C T . The first equation Ω g dµ = 0 is trivial. We check first that curlT g = 0. Let φ be a C ∞ -function with compact support; let us compute
∂φ ∂φ (x)gk (Tx ω) − (x)gi (Tx ω) dλ(x). ∂ xi ∂ xk
According to the expression of g given by Proposition 3, and Equality (10), it is equal to
∂φ ∂φ (x) − Ri (x) R j h j (Tx ω) dλ(x). ∂ xi ∂ xk 3
Rk
j=1
The first bracket vanishes (this is a classical property of the Riesz operator Ri on R3 , which follows from an obvious computation on Fourier transforms). This shows that curlT g = 0. We compute divT g. This random variable is defined by
3 ∂φ (−x)gi (Tx ω) dλ(x). φ(−x) · divT g(Tx ω) dλ(x) = ∂ xi i=1
According to the expression of g given by Proposition 3, and Corollary 3, it is equal to −
3 3 3 =1 j=1 i=1
Rj
∂ ∂φ Ri (−x) h , j (Tx ω) dλ(x), ∂ x ∂ xi
which is still equal, according to a property of the Riesz operator Ri on R3 easily checkable on Fourier transforms, to 3 3 ∂ 2φ (−x) h , j (Tx ω) dλ(x). ∂ x j ∂ x =1 j=1
Integrating by parts, we obtain 3 3 ∂φ (−x) ∂ h , j (Tx ω) dλ(x). ∂x j =1
j=1
According to the equality h j = proves Proposition 3.
3
=1 ∂ h , j ,
this shows that divT g = divT h, and
Resistivity of an Infinite Three Dimensional Stationary Random Electric Conductor
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4. Initial System Let us return to the system (S). We consider the sequence of fields g n defined by induction by g 0 ≡ 0, and for n + 1, the field g n+1 is the solution of the system (Sh n ) of right-hand side hn = g n − A C (g n + U). The system (S hn ) being linear, the field g n+1 − g n is the solution corresponding to the second member hn − hn−1 . According to what precedes, we have then
g n+1 − g n p ≤ κ p hn − hn−1 p . It follows from the expression of hn and from the assumption of ellipticity on A that it is still bounded by κ p (1 −
c )g n − g n−1 p , C
and therefore by induction still by (κ p (1 − Cc ))n g 1 p . However according to the Riesz interpolation theorem (see [9]), the norm κ p is convex, therefore continuous, with respect to p. However κ2 = 1, therefore, for p close enough to 2, we have κ p (1 − Cc ) < 1. The series ∞ n=1 g n+1 − g n p is thus convergent, and therefore the sequence of the fields g n is a Cauchy sequence, hence converges to a field denoted g ∞ . Now, for n + 1, the third equation of the system (Sh n ) is written divT (g n+1 − hn ) = 0, i.e. A divT (g n+1 − g n ) − (g n + U) = 0. C The operator divT is not continuous, however its kernel is closed. We thus obtain, for n tending to infinity, divT (A(g ∞ + U)) = 0. In the same way, the second equation of the system (Sh n ) gives, for n tending to infinity, curlT (g ∞ ) = 0. Thus g ∞ + U is an L p -solution of (S). Hence by uniqueness, the L2 -solution lies in L p (Ω) for p close enough to 2. And the proof of Theorem 1 is complete.
5. Integral Expression of K i j Operators Since the operators Ri are defined by an integral expression, a natural question is: does such an expression exist for Ki j = Ri R j . The answer is stated in the following proposition. p
Proposition 4. For any f ∈ L0 (Ω), we have ⎧ 3yi y j ⎪ f (T−y ω) dλ(y) if i = j ⎨ 4π y5 Ki j f (ω) = 2yi2 − yk2 − y2 ⎪ ⎩ −1 f (ω) + f (T−y ω) dλ(y) if i = j, 3 4π y5 where {k, } = {1, 2, 3} \ {i}. Remark 2. The kernel under both the integrals is the derivative ∂ x∂i ∂ x j 4π1x of the Green function. The term − f (ω)/3 may be understood as follows: at x = 0, the derivative
∂2 1 is the Dirac distribution φ → −φ(0)/3. ∂ xi2 4π x 2
x=0
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Proof. Consider the case i = j, which is more interesting. To begin with, we check the analogous formula for the classical Riesz operator. For any C ∞ -function φ, we denote by K˜ ii (φ) the function defined by 2yi2 − yk2 − y2 ˜ K ii (φ)(x) = φ(x − y) dλ(y). 4π y5 According to Theorem 5 Chapter III of [8], its Fourier transform is η →
−2ηi2 + ηk2 + η2 ˆ φ(η). 3η2
−η ˆ Since the Fourier transform of Ri2 (φ) is the function η → ηi2 φ(η), this leads to −1 ˆ 2 ˜ (R (φ) − K ii (φ))ˆ(η) = φ(η) and proves the desired equality 2
i
3
1 Ri2 φ = − φ + K˜ ii φ. 3
(11)
p Next, we consider the operator K˜ ii defined for any f ∈ L0 (Ω) by 2yi2 − yk2 − y2 K˜ ii ( f )(ω) = f (T−y ω) dλ(y). 4π y5
We have the duality formula ∂φ ∂φ K˜ ii ( ) T f = T K˜ ii ( f ) ∂x j ∂x j
(12)
(the proof of Lemma 7 works without modification). Thus it follows from this equality, Corollary 3, and equality (4), that for any C ∞ -function φ with compact support, any j ∈ {1, 2, 3}, and any h ∈ C T , ∂φ 1 T Ri2 (∂ h) − K˜ ii (∂ h) + ∂ h = ∂x j 3 ∂ ∂φ ∂ ˜ ∂φ 1 ∂ 2 φ Ri T h. Ri − + K ii ∂ x ∂ x j ∂ x ∂ x j 3 ∂ x ∂ x j It is zero, according to the equality (11). This implies by density of E that for any p f ∈ L0 (R3 ), ∂φ 1 T Ri2 ( f ) − K˜ ii ( f ) + f = 0. ∂x j 3 According to the assumption Ω f dµ = 0, the proof of Proposition 4 is completed by the following lemma. Lemma 8. Assume f ∈ L p (Ω). Assume that for any C ∞ -function φ with compact support, and any j ∈ {1, 2, 3}, we have
Then f is constant: f =
Ω
∂φ T f = 0. ∂x j f dµ.
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Proof. It follows from the ergodicity assumption that it is sufficient to prove that f is T -invariant. Assume x ∈ R3 . According to the Wiener local ergodic theorem, it is sufficient to prove that φ T ( f − f ◦ Tx ) = 0, for any C ∞ -function φ with compact support. We have (φ(y) − φ(x + y) f (T−y ω) dλ(y). φ(y) f (T−y ω) − f (Tx−y ω) dλ(y) = By computation of the derivative of s → φ(sx + y), this is equal to
1
1
s=0 j=0
xj
∂φ (sx + y) ds f (T−y ω) dλ(y), ∂x j
which is zero, according to the assumption (13) and the Fubini Theorem. This completes the proof of Lemma 8 and therefore of Proposition 4.
References 1. Boivin, D.: Weak convergence for reversible random walks in a random environment. Ann. Probab., 21(3), 1427–1440 (1993) 2. Calderon, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math., 88, 85–139 (1952) 3. Depauw, J.: Flux moyen d’un courant électrique dans un réseau aléatoire stationnaire de résistances. Ann. Inst. H. Poincaré Probab. Statist. 35(3), 355–370 (1999) 4. Depauw, J.: Degree two ergodic theorem for divergence-free random fields. Israel J. Math., 157, 283–308 (2007) 5. Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, volume 105 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ: (1983) 6. Golden, K., Papanicolaou, G.: Bounds for effective parameters of heterogeneous media by analytic continuation. Commun. Math. Phys., 90(4), 473–491 (1983) 7. Jikov, V.V., Kozlov, S.M., Ole˘ınik, O.A.: Homogenization of differential operators and integral functionals. Berlin:Springer-Verlag, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif yan] 8. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton, N.J.: Princeton University Press, 1970 9. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton, N.J.: Princeton University Press, 1971. 10. Wiener, N.: The ergodic theorem. Duke Math. J. 5(1), 1–18 (1939) Communicated by M. Aizenman
Commun. Math. Phys. 274, 399–408 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0283-6
Communications in
Mathematical Physics
On the Buchdahl Inequality for Spherically Symmetric Static Shells Håkan Andréasson Mathematical Sciences, Chalmers and Göteborg University, S-41296 Göteborg, Sweden. E-mail: [email protected] Received: 1 June 2006 / Accepted: 13 February 2007 Published online: 21 June 2007 – © Springer-Verlag 2007
Abstract: A classical result by Buchdahl [6] shows that for static solutions of the spherically symmetric Einstein equations, the ADM mass M and the area radius R of the boundary of the body, obey the inequality 2M/R ≤ 8/9. The proof of this inequality rests on the hypotheses that the energy density is non-increasing outwards and that the pressure is isotropic. In this work neither of Buchdahl’s hypotheses are assumed. We consider non-isotropic spherically symmetric shells, supported in [R0 , R1 ], R0 > 0, of matter models for which the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ ρ, ≥ 1. We show a Buchdahl type inequality for shells which are thin; given an < 1/4 there is a κ > 0 such that 2M/R1 ≤ 1 − κ when R1 /R0 ≤ 1 + . It is also shown that for a sequence of solutions such that R1 /R0 → 1, the limit supremum of 2M/R1 of the sequence is bounded by ((2 + 1)2 − 1)/(2 + 1)2 . In particular if = 1, which is the case for Vlasov matter, the bound is 8/9. The latter result is motivated by numerical simulations [3] which indicate that for non-isotropic shells of Vlasov matter 2M/R1 ≤ 8/9, and moreover, that the value 8/9 is approached for shells with R1 /R0 → 1. In [1] a sequence of shells of Vlasov matter is constructed with the properties that R1 /R0 → 1, and that 2M/R1 equals 8/9 in the limit. We emphasize that in the present paper no field equations for the matter are used, whereas in [1] the Vlasov equation is important. 1. Introduction The metric of a static spherically symmetric spacetime takes the following form in Schwarzschild coordinates: ds 2 = −e2µ(r ) dt 2 + e2λ(r ) dr 2 + r 2 (dθ 2 + sin2 θ dϕ 2 ), where r ≥ 0, θ ∈ [0, π ], ϕ ∈ [0, 2π ]. Asymptotic flatness is expressed by the boundary conditions lim λ(r ) = lim µ(r ) = 0,
r →∞
r →∞
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and a regular centre requires λ(0) = 0. The Einstein equations read e−2λ (2r λr − 1) + 1 = 8πr 2 ρ, e−2λ (2r µr + 1) − 1 = 8πr 2 p, 1 µrr + (µr − λr )(µr + ) = 4πqe2λ . r
(1) (2) (3)
Here ρ is the energy density, p the radial pressure and q is the tangential pressure. If the pressure is isotropic, i.e. 2 p = q, a solution will satisfy the well-known TolmanOppenheimer-Volkov equation for equilibrium pr = −µr ( p + ρ).
(4)
In the case of non-isotropic pressure this equation generalizes to 1 pr = −µr ( p + ρ) − (2 p − q). r
(5)
In 1959 Buchdahl [6] showed that static fluid spheres satisfy the inequality 2M/R ≤ where M is the ADM mass,
∞
M=
8 , 9
4π η2 ρ(η)dη,
0
and R is the outer boundary (in Schwarzschild coordinates) of the fluid sphere. The fluid spheres considered by Buchdahl have isotropic pressure and are in addition assumed to have an energy density which is non-increasing outwards. Note that the isotropy assumption implies that also the pressure is monotonic. This follows from (4) since µr ≥ 0, which is a consequence of (1) and (2). It is sometimes argued that the assumption of non-increasing energy density is natural in the sense that the fluid sphere is unstable otherwise [10]. However, at least for Vlasov matter this is certainly not the case. The existence of stable, spherically symmetric static shells of Vlasov matter (i.e. the matter is supported in [R0 , R1 ], R0 > 0) has been demonstrated numerically e.g. in [2]. Moreover, these shells are in general non-isotropic. Static solutions of Vlasov matter can also be constructed which do not have the shell structure [9], i.e. R0 = 0. Also for these solutions, the hypotheses of non-increasing energy density and isotropic pressure are in general violated. The method of proof by Buchdahl rests on the monotonicity of both ρ and p, and a natural question is then if there is a Buchdahl type inequality also for static solutions where neither ρ nor p is monotonic? Quite surprisingly, numerical results in [3] support that 2M/R1 < 8/9 for any static solution of the Einstein Vlasov system and that there are solutions with 2M/R1 arbitrary close to 8/9. (R1 will always denote the outer boundary of the static solution.) An analytic investigation for Vlasov matter is given in [1]. In this work we will however consider static shells of any matter model (which has static solutions) for which ρ and p are non-negative and p + q ≤ ρ, ≥ 1.
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401
In particular, by taking = 3, any matter model with p non-negative, which satisfies the dominant energy condition is admitted. Note that the condition that ρ ≥ 0 can be replaced by requiring that the weak energy condition is satisfied. In the case of Vlasov matter ρ, p and q are all non-negative and (6), with strict inequality, is satisfied with = 1. Remark. The condition that p ≥ 0 can be replaced by − p ≤ cρ, 0 < c < 1, and > 0 is sufficient in order to derive a Buchdahl type inequality, but the formulation of the results and their proofs would be slightly more cumbersome so we have chosen to use the conditions above. Bondi [5] has investigated (not rigorously) if isotropic solutions, without the assumption of non-increasing energy density, obey a Buchdahl type inequality. He considers models for which ρ ≥ 0, ρ ≥ p, or ρ ≥ 3 p, and gets 0.97, 0.86 and 0.70 respectively as upper bounds of 2M/R1 . The isotropic condition is crucial though, since, as a consequence of [1] (and numerically in [3]), the second and third bounds are violated by non-isotropic steady states of Vlasov matter. Note that ρ ≥ p always holds for Vlasov matter. Remark. We point out that steady states of Vlasov matter which are isotropic and which have non-increasing energy density do exist but the analysis of such states is identical to Buchdahl’s original analysis. Non-isotropic solutions have previously been studied in [4] where it is shown that 2M/R1 < 1, and in [7] static shells where the density is concentrated at a single radius and the source in the Einstein equations is distributional, were found to satisfy 2M/R1 ≤ 24/25. Here we will consider shells which are thin in the sense that R1 /R0 ≤ 1 + , and we show that for any < 1/4 there is a κ > 0 such that 2M/R1 ≤ 1 − κ. We also show that for a sequence of static shells such that R1 /R0 → 1, the limit supremum of 2M/R1 is bounded by 8/9. The outline of the paper is as follows. In the next section our main results are presented in detail and the proofs are given in Sect. 3. 2. Main results Theorem 1. Let a > 0, K ≥ 5/4 + 3/4, and let κ=
1 (2 + 1 +
2(+1)2 2+1 (K
+ a))2
.
Consider a solution to the Einstein equations such that p + q ≤ ρ, with support in [R0 , R1 ], and such that R1 /R0 ≤ (1 + ), where < min {
1 − (2 + 3)κ − ( + 1)(1 − κ)e−4K /9 √ , a κ}. 2( + 1)
Then 2M/R1 ≤ 1 − κ, and sup r
2m ≤ 1 − κ/2, if ≤ κ/2. r
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The condition on K is to ensure that the first term in the expression for is positive and can be sharpened. The numbers κ and are related in the sense that a larger can be chosen if κ is taken smaller and vice versa. In view of the next theorem, which gives a much improved value of κ when is made arbitrary small, it seems more interesting to make large even if κ then becomes smaller, since it gives us an estimate on the possible thickness of the shells that our method allows in order to admit a Buchdahl inequality at all. We see immediately from the formula above that our method cannot handle ≥ 1/4, but by fixing K large and then taking a sufficiently large we can have arbitrary close to 1/4. A choice which favours a large value of rather than a large κ, in the case when = 1, is K = 6 and a = 9, which implies that when < 1/5 it holds that 2M/R1 ≤ 1 − 1/432 . j j j j Next we consider a sequence of shells supported in [R0 , R1 ], such that R1 /R0 → 1. We find that, as the support gets thinner, the value (2 + 1)2 − 1 (2 + 1)2 of 2M/R1 can not be exceeded. In particular, if = 1, then the same bound 8/9 as in Buchdahl’s original work is obtained, and if = 2, then the bound is 24/25 which agrees with the value found in [7]. The latter agreement is not surprising since the case considered in [7] is an infinitely thin shell with radial pressure zero which satisfies the dominant energy condition. In our terminology this means = 2. The former result is more surprising since the steady state that realizes 8/9 in Buchdahl’s original work is the well-known interior solution with constant energy density which is completely different from our situation where the energy density gets more and more peaked at the radius R1 . Theorem 2. Assume that {(ρ j , p j , q j , µ j )}∞ j=1 is a sequence of solutions to the static j
j
Einstein equations such that p j + q j ≤ ρ j , with support in [R0 , R1 ], and such that j
lim
j→∞
R1 j
R0
= 1.
(7)
Then lim sup j→∞
2M j j
R1
≤
(2 + 1)2 − 1 . (2 + 1)2
(8)
The hypothesis (7) holds for a class of static shell solutions of the Einstein-Vlasov system [1]. Numerical simulations [3] (where = 1) also indicate that for a sequence of shell solutions for which 2M/R1 increase and take as large values as possible (< 8/9) the hypothesis (7) is satisfied. In [1] a sequence of shells of Vlasov matter with the property (7) is indeed constructed which attains 8/9 in the limit and thus improves statement (8) for Vlasov matter. 3. Proofs of Theorem 1 and 2 Before starting to prove the theorems let us collect a couple of facts concerning the system (1)-(3). A consequence of Eq. (1) is that e−2λ = 1 −
2m(r ) , r
On the Buchdahl Inequality for Spherically Symmetric Static Shells
403
and from (2) it then follows that µr = (
m + 4πr p)e2λ . r2
Adding (1) and (2) and using the boundary conditions at r = ∞ gives ∞ 4π η(ρ + p))e2λ dη. µ(r ) + λ(r ) = −
(9)
r
In particular if R1 is the outer radius of support of the matter then eµ(r )+λ(r ) = 1, when r ≥ R1 . Hence, eµ(r ) = e−λ(r ) =
1 − 2m(r )/r, r ≥ R1 .
The facts above will be used in the proofs without further comment. Next we note that the generalized Tolman-Oppenheimer-Volkov equation (5) implies that a solution satisfies m 1 r µ+λ ( 2 + 4πr p)e = 2 4π η2 eµ+λ (ρ + p + q)dη. (10) r r 0 Indeed, let ψ = (m + 4πr 3 p)eµ+λ . Using (5) we get ψr = 4πr 2 (ρ + p + q)eµ+λ , and the claim follows since ψ(0) = 0. We have chosen to treat Theorem 2 before Theorem 1. Proof of Theorem 2. First we will show that a uniform bound on eλ = √
1 , 1 − 2m/r
will imply (8). Hence, assume that there is a κ, 1 > κ > 0, such that sup η
2m j (η) ≤ 1 − κ, for all j. η
(11)
Given > 0 we choose j sufficiently large so that j
R1 j
R0
− 1 ≤ .
Below we drop the index j. Since p(R1 ) = 0, m(R1 ) = M and e(µ+λ)(R1 ) = 1, we get by taking r = R1 in (10) and using that f is supported in [R0 , R1 ], M=
R1 R0
4π η2 eµ+λ (ρ + p + q) dη ≤ ( + 1)
R1 R0
4π η2 eµ+λ ρ dη.
(12)
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H. Andréasson
Here we also used that p + q ≤ ρ. Now µ is increasing so the right hand side is less than or equal to R1 µ(R1 ) R1 4π ηeλ ρdη ( + 1)e R0
= ( + 1)eµ(R1 ) R1
R1
(4π ηρ(η) −
R0
+( + 1)e
µ(R1 )
R1
R1 R0
m(η) λ )e dη η2
m(η) λ e dη =: S1 + S2 . η2
Since λr = (4πrρ(r ) −
m(r ) 2λ )e , r2
we get S1 = −( + 1)eµ(R1 ) R1
R1 R0
Using that eµ(R1 ) =
√
d −λ e dη = ( + 1)eµ(R1 ) R1 (1 − 1 − 2M/R1 ). dη
1 − 2M/R1 , we thus get
√ 2( + 1)M 1 − 2M/R1 S1 = ( + 1) 1 − 2M/R1 R1 (1 − 1 − 2M/R1 ) = . √ 1 + 1 − 2M/R1 The idea is now to show that S2 is approaching zero as j tends to infinity, since note that if S2 = 0 we have √ 2( + 1)M 1 − 2M/R1 , (13) M≤ √ 1 + 1 − 2M/R1 or
1 ≤ (2 + 1) 1 − 2M/R1 ,
which implies that 2M (2 + 1)2 − 1 ≤ . R1 (2 + 1)2 To estimate S2 we use (11) and that eλ = (1 − 2m/r )−1/2 to obtain R1 m(η) S2 =( + 1)eµ(R1 ) R1 dη √ 2 1 − 2m(η)/η η R0 R1 dη µ(R1 ) −1/2 R1 Mκ ≤ ( + 1)e 2 R0 η R1 ≤ ( + 1)eµ(R1 ) Mκ −1/2 ( − 1) R0 ≤ ( + 1)M 1 − 2M/R1 κ −1/2 .
(14)
On the Buchdahl Inequality for Spherically Symmetric Static Shells
405
To obtain a similar form of the inequality (13) also when S2 = 0 we simply use that 2 ≥ 1, 1 − 2M/R1
(15)
√ 2( + 1)M(1 + κ −1/2 ) 1 − 2M/R1 . √ 1 + 1 − 2M/R1
(16)
1+
√
and with (14) we thus get M≤ This yields 2M (2 + 1 + 2( + 1)κ −1/2 )2 − 1 ≤ . R1 (2 + 1 + 2( + 1)κ −1/2 )2 j
Hence, a uniform bound on 1/(1 − 2m/r ) implies that the limit supremum of 2M j /R1 , as j → ∞, is bounded by (2 + 1)2 − 1 . (2 + 1)2 A uniform bound is given by Theorem 1, which we prove below, and which completes the proof of Theorem 2. Proof of Theorem 1. Let us first show that the second statement in the theorem is a consequence of the first if ≤ κ/2. Since m(r ) = 0 when r < R0 and m(r ) = M when r > R1 it is sufficient to take r ∈ [R0 , R1 ]. If ≤ κ/2, we then have 2m/r ≤ 2M/r ≤ 2M/R0 ≤ R1 (1 − κ)/R0 ≤ (1 + )(1 − κ) < 1 − κ + ≤ 1 − κ/2. Thus we only have to prove that 2M/R1 ≤ 1 − κ. The proof is by contradiction. Hence, assume that 2M/R1 > 1 − κ, and let η∗ := min{η ∈ [R0 , R1 ] :
2m(η) = 1 − κ}. η
Consider (12) again. The term S1 above can be estimated as before (note that the result in [4] shows that 2m/r < 1 for a given solution so there are no regularity problems) but for S2 we split the integration over [R0 , η∗ ] and [η∗ , R1 ]. The integral over the former interval is estimated as in (14) and we get η∗ m(η) µ(R1 ) ( + 1)e R1 1 − 2M/R1 √ . dη ≤ . . . ≤ ( + 1)M √ 2 1 − 2m(η)/η κ η R0 If it now holds that
R1
η∗
η2
√
m(η) KM , dη ≤ R1 1 − 2m(η)/η
then the estimate for S2 reads S2 ≤ ( + 1)M 1 − 2M/R1 √ + ( + 1)K M 1 − 2M/R1 . κ
(17)
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This leads to an inequality similar to (16). More precisely we obtain √ 2(+1)2 2M (2 + 1 + 2+1 (K + / κ))2 − 1 ≤ √ 2 2 R1 (2 + 1 + 2(+1) 2+1 (K + / κ)) 1 ≤1 − = 1 − κ, 2(+1)2 (2 + 1 + 2+1 (K + a))2
(18)
where we used that ≤ aκ 1/2 , together with a slightly different form of (15) using 2M/R1 ≥ 1 − κ > 8/9, i.e. 3(1 +
√
4 ≥ 1. 1 − 2M/R1 )
(19)
Now we have assumed that (18) is not true, so in fact the following inequality must hold: R1 m(η) KM . (20) dη > √ 2 1 − 2m(η)/η ∗ R1 η η Let us now consider (12) again. We will show that (20), together with the hypotheses in the theorem, imply that the right-hand of this inequality is less than M, which gives a contradiction. First we need a couple of facts. Since R1 R1 d −λ m λ e dη = e−λ(r ) − e−λ(R1 ) , (4π ηρ − 2 )e dη = − η dη r r we get
R1
λ
4π ηρe dη = e
−λ(r )
−e
−λ(R1 )
R1
+
r
r
Since λ ≥ 0 and ρ = p = 0 when r > R1 , we have R1 2λ 4π η(ρ + p)e dη ≤ − (µ + λ)(r ) = − r
m λ e dη. η2
R1
(21)
4π ηρeλ dη.
(22)
r
Now we split the integral in the inequality for M in two parts R1 η∗ 2 µ+λ ( + 1) 4π η e ρ dη =( + 1) 4π η2 eµ+λ ρ dη R0
R0
+( + 1)
R1 η∗
4π η2 eµ+λ ρ dη
=: T1 + T2 . Using the two facts above we get η∗ 4π η2 eµ+λ ρ dη ≤ ( + 1) T1 =( + 1) =( + 1)
R0 η∗ R0
η∗
(23)
4π η2 ρe−
R η
1
4π σρeλ dσ
dη
R0
4π η2 ρee
−λ(R1 ) −e−λ(η) − R1 η
m(σ )eλ /σ 2 dσ
dη.
(24)
On the Buchdahl Inequality for Spherically Symmetric Static Shells
407
We have e−λ(R1 ) ≤ e−λ(η) , since η ≤ η∗ , and 2M/R1 > 1 − κ. We thus get due to (20), again since η ≤ η∗ ,
η∗
T1 ≤ ( + 1)
4π η2 ρe−K M/R1 dη ≤ ( + 1)Me−K M/R1 .
(25)
R0
Note that M/R1 > 4/9 so that T1 is small if K is large. Let σ ∗ ∈ [η∗ , R1 ] be such that e2λ(σ
∗)
≤ e2λ(σ ) , ∀σ ∈ [η∗ , R1 ].
Using again that (µ + λ)(η) = −
R1 η
4π σ (ρ + p)e2λ dσ,
we get T2 ≤ ( + 1)R1 ≤ ( + 1)R1 ≤ ( + 1)R1
R1 η∗ R1
η∗ R1 η∗
4π ηeµ+λ ρ dη 4π ηρe−e 4π ηρe−e
2λ(σ ∗ )
R
2λ(σ ∗ )
R
η
η
1
4π σ (ρ+ p) dσ
1
4π σρ dσ
dη
dη
2λ(σ ∗ ) R1 4π σρ dσ d η dη e−e η∗ dη ∗ R ∗ −e2λ(σ ) η∗1 4π σρ dσ =( + 1)R1 e−2λ(σ ) 1 − e
=( + 1)R1 e−2λ(σ
∗)
≤ ( + 1)R1 e−2λ(σ
∗)
R1
= ( + 1)R1 (1 − 2m(σ ∗ )/σ ∗ ).
Now since m(r ) is increasing we have for any σ ∈ [η∗ , R1 ], 2m(η∗ ) η∗ 2m(η∗ ) 2m(σ ) 2m(η∗ ) ≥ ≥ = σ σ R1 R1 η ∗ ∗ R0 2m(η ) R0 1−κ . ≥ = (1 − κ) ≥ ∗ R1 η R1 1+ Using this estimate twice we obtain 1−κ ) 1+ R1 κ + (1 + ) (κ + ) = ( + 1) M ≤ 2( + 1)M M 1+ (1 − κ) (1 + ) 2( + 1)M(κ + ) = . 1−κ
T2 ≤ ( + 1)R1 (1 − 2m(σ ∗ )/σ ∗ ) ≤ ( + 1)R1 (1 −
(26)
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In conclusion the following inequality holds, using again that M/R1 > 4/9, together with the condition on , 2( + 1)M(κ + ) 1−κ 2( + 1)M(κ + ) −4K /9 + ≤( + 1)Me < M. 1−κ
M ≤( + 1)Me−K M/R1 +
Hence we have obtained a contradiction and the proof is complete. Acknowledgements. The author wants to thank Gerhard Rein for reading and commenting the manuscript, and Alan Rendall for carefully answering my questions.
References 1. Andréasson, H.: On static shells and the Buchdahl inequality for the spherically symmetric EinsteinVlasov system. Commun. Math. Phys. 2. Andréasson, H., Rein, G.: A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein-Vlasov system. Class. Quantum Grav. 23, 3659–3677 (2006) 3. Andréasson, H., Rein, G.: On the steady states of the spherically symmetric Einstein-Vlasov system. Class. Quantum Grav. 24, 1809–1832 (2007) 4. Baumgarte, T.W., Rendall, A.D.: Regularity of spherically symmetric static solutions of the Einstein equations. Class. Quantum Grav. 10, 327–332 (1993) 5. Bondi, H.: Massive spheres in general relativity. Proc. R. Soc. A 282, 303–317 (1964) 6. Buchdahl, H.A.: General relativistic fluid spheres. Phys. Rev. 116, 1027–1034 (1959) 7. Fraundiener, J., Hoenselaers, C., Konrad, W.: A shell around a black hole. Class. Quantum Gravity 7, 585–587 (1990) 8. Rein, G.: Static shells for the Vlasov-Poisson and Vlasov-Einstein systems. Indiana Univ. Math. J. 48, 335–346 (1999) 9. Rein, G., Rendall, A.D.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics, Math. Proc. Camb. Phil. Soc. 128, 363–380 (2000) 10. Weinberg, S.: Gravitation and cosmology. Newyork: John Wiley and Sons, Inc., 1972 Communicated by M. Aizenman
Commun. Math. Phys. 274, 409–425 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0285-4
Communications in
Mathematical Physics
On Static Shells and the Buchdahl Inequality for the Spherically Symmetric Einstein-Vlasov System Håkan Andréasson Mathematical Sciences, Chalmers and Göteborg University, S-41296 Göteborg, Sweden. E-mail: [email protected] Received: 1 June 2006 / Accepted: 13 February 2007 Published online: 20 June 2007 – © Springer-Verlag 2007
Abstract: In a previous work [1] matter models such that the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ ρ, ≥ 1, were considered in the context of Buchdahl’s inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, [R0 , R1 ], R0 > 0, satisfies R1 /R0 < 1/4. Moreover, given a sequence of solutions such that R1 /R0 → 1, then the limit supremum of 2M/R1 was shown to be bounded by ((2 + 1)2 − 1)/(2 + 1)2 . In this paper we show that the hypothesis that R1 /R0 → 1, can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of 2M/R1 is bounded, but that the limit is ((2 + 1)2 − 1)/(2 + 1)2 = 8/9, since = 1 for Vlasov matter. Thus, static shells of Vlasov matter can have 2M/R1 arbitrary close to 8/9, which is interesting in view of [3], where numerical evidence is presented that 8/9 is an upper bound of 2M/R1 of any static solution of the spherically symmetric Einstein-Vlasov system. 1. Introduction Under the assumptions of isotropic pressure and non-increasing energy density outwards, Buchdahl [6] has proved that a spherically symmetric fluid ball satisfies 8 2M ≤ , R1 9 where M and R1 is the total ADM mass and the outer boundary of the fluid ball respectively. In [1] the Buchdahl inequality was investigated for spherically symmetric static shells with support in [R0 , R1 ], R0 > 0, for which neither of Buchdahl’s hypotheses hold. We refer to the introduction in [1] for a review of previous results on Buchdahl type
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inequalities. The matter models considered in [1] were assumed to have non-negative energy density ρ and pressure p, and to satisfy the following inequality: p + q ≤ ρ, ≥ 1,
(1)
where q is the tangential pressure. It was shown that given < 1/4, there is a κ > 0 such that any static solution of the spherically symmetric Einstein equations satisfies 2M ≤ 1 − κ. R1 j
j
Furthermore, given a sequence of static solutions, indexed by j, with support in [R0 , R1 ] where j
j
R1 /R0 → 1 as j → ∞, it was proved that lim sup j→∞
2M j j R1
≤
(2 + 1)2 − 1 , (2 + 1)2
(2)
where M j is the corresponding ADM mass of the solution with index j. The latter result is motivated by numerical simulations [3] of the spherically symmetric Einstein-Vlasov system. For Vlasov matter = 1 and the inequality (1) is strict, and the bound in (2) becomes 8/9 as in Buchdahl’s original work. We will see that for Vlasov matter, j j a sequence can be constructed such that R1 /R0 → 1, and such that the value 8/9 of 2M/R1 is attained in the limit. It should be emphasized that the static solution which attains the value 8/9 in Buchdahl’s case is an isotropic solution with constant energy density, whereas the limit state of the sequence that we construct is an infinitely thin shell which has p j /q j → 0 as j → ∞, which means that it is highly non-isotropic. Before describing in more detail the numerical results in [3], which provide the main motivation for this work, let us first introduce the spherically symmetric Einstein-Vlasov system. The metric of a static spherically symmetric spacetime takes the following form in Schwarzschild coordinates: ds 2 = −e2µ(r ) dt 2 + e2λ(r ) dr 2 + r 2 (dθ 2 + sin2 θ dϕ 2 ), where r ≥ 0, θ ∈ [0, π ], ϕ ∈ [0, 2π ]. Asymptotic flatness is expressed by the boundary conditions lim λ(r ) = lim µ(r ) = 0,
r →∞
r →∞
and a regular centre requires λ(0) = 0. Vlasov matter is described within the framework of kinetic theory. The fundamental object is the distribution function f which is defined on phase-space, and models a collection of particles. The particles are assumed to interact only via the gravitational field created by the particles themselves and not via direct collisions between them. For an introduction to kinetic theory in general relativity and the Einstein-Vlasov system in
Static Shells and Buchdahl Inequality for Spherically Symmetric E-V System
411
particular we refer to [2] and [15]. The static Einstein-Vlasov system is given by the Einstein equations e−2λ (2r λr − 1) + 1 = 8πr 2 ρ, e−2λ (2r µr + 1) − 1 = 8πr 2 p, 1 = 4πqe2λ , µrr + (µr − λr ) µr + r together with the (static) Vlasov equation w L ∂r f − µr ε − 3 ∂w f = 0, ε r ε where ε = ε(r, w, L) =
(3) (4) (5)
(6)
1 + w 2 + L/r 2 .
The matter quantities are defined by π ∞ ∞ ε f (r, w, L) d Ldw, ρ(r ) = 2 r −∞ 0 π ∞ ∞ w2 f (r, w, L) d Ldw, p(r ) = 2 r −∞ 0 ε π ∞ ∞L q(r ) = 4 f (r, w, L) d Ldw. r −∞ 0 ε The variables w and L can be thought of as the momentum in the radial direction and the square of the angular momentum respectively. Let E = eµ ε, the ansatz f (r, w, L) = (E, L),
(7)
then satisfies (6) and constitutes an efficient way to construct static solutions with finite ADM mass and finite extension, cf. [13, 14]. It should be pointed out that spherically symmetric static solutions which do not have this form globally exist, cf. [16], which contrasts with the Newtonian case where all spherically symmetric static solutions have the form (7), cf. [4]. As a matter of fact, the solutions we construct in Theorem 1 below are good candidates for solutions which are not globally given by (7). Here the following form of will be used: (E, L) = (E 0 − E)k+ (L − L 0 )l+ ,
(8)
where l ≥ 1/2, k ≥ 0, L 0 > 0, E 0 > 0, and x+ := max{x, 0}. In the Newtonian case with l = L 0 = 0, this ansatz leads to steady states with a polytropic equation of state. Note that when L 0 > 0 there will be no matter in the region L0 , (9) r< (E 0 e−µ(0) )2 − 1
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H. Andréasson
since there necessarily E > E 0 and f vanishes. The existence of solutions supported in [R0 , R1 ], R0 > 0, with finite ADM mass has been given in [13], and we shall call such configurations static shells of Vlasov matter. It will be assumed that is always as above, which in particular means that L 0 > 0, so that only shells are considered. The case L 0 = 0 is left to a future study. Let the matter content within the sphere of area radius r be defined by r 4π η2 ρdη. m(r ) = 0
Note that the ADM mass M = limr →∞ m(r ). We also note that Eq. (3) implies that e−2λ = 1 −
2m(r ) , r
so that µ alone can be regarded as the unknown metric function of the spherically symmetric Einstein-Vlasov system. Numerical evidence is presented in [3] that the following hold true: i)
For any solution of the static Einstein-Vlasov system
:= sup r
ii)
iii)
8 2m(r ) < . r 9
(10)
The inequality is sharp in the sense that there is a sequence of steady states such that = 2M/R = 8/9 in the limit. These statements hold for both shells (L 0 > 0) and non-shells (L 0 = 0). More information on the latter case is given in [3]. In the former case the sequence which realizes = 8/9 is obtained numerically in [3] by constructing solutions where the inner boundary of the shells tend to zero. The outer boundary of these shells also tend to zero, and in [3] numerical support is obtained for the following claim: j j j There is a sequence of static shells supported in [R0 , R1 ], such that R0 → 0, and j j R1 /R0 → 1, as j → ∞.
Our main results are described in the next section and concern issues (ii) and (iii) which will be proved. Of course, issue (i) is a very interesting open problem, and we believe that the results in this paper are important for proving also issue (i), in view of (ii). Let us end this section with a brief discussion on the possible role of the Buchdahl inequality for the time dependent problem with Vlasov matter. The cosmic censorship conjecture is fundamental in classical general relativity and to a large extent an open problem. In the case of gravitational collapse the only rigorous result is by Christodoulou who has obtained a proof in the case of the spherically symmetric Einstein-Scalar Field system [8]. One key result for this proof is contained in [7], cf. also [9], and states roughly that if there is a sufficient amount of matter within a bounded region then necessarily a trapped surface will form in the evolution. If a trapped surface forms, then Dafermos [10] has shown under some restrictions on the matter model, that cosmic censorship holds. In particular, Dafermos and Rendall [11] have proved that spherically symmetric Vlasov matter satisfies these restrictions. Thus cosmic censorship holds for the spherically symmetric Einstein-Vlasov system if there is a trapped surface in spacetime. Now, assume that a Buchdahl inequality holds in general for this system, then in view of the
Static Shells and Buchdahl Inequality for Spherically Symmetric E-V System
413
result by Christodoulou mentioned above, it is natural to believe that if 2m/r exceeds the value given by such an inequality, then a trapped surface must form. The outline of the paper is as follows. In the next section our main results are presented in detail. Some preliminary results on static shells of Vlasov matter are contained in Sect. 3, and in Sect. 4 the proofs of the theorems are given. 2. Main Results In view of (9) it is clear that the region where f necessarily vanishes can be made arbitrary small if the values of E 0 and µ(0) are such that E 0 e−µ(0) is large. That this is always possible can be seen as follows. Set E 0 = 1, and construct a solution by specifying an arbitrary non-positive value µ(0), in particular e−µ(0) can be made as large as we wish. The metric function µ is then obtained by integrating from the centre using Eq. (4), r m µ(r ) = µ(0) + ( 2 + 4π ηp)e2λ dη. 0 η This implies that the boundary condition at ∞ of µ will be violated in general. How˜ ) := µ(r ) − µ(∞), then µ, ˜ and the distribution ever, by letting E˜ 0 = eµ(∞) , and µ(r ˜ function f associated with µ˜ and E 0 , will solve the Einstein-Vlasov system and satisfy the boundary condition at infinity, and in view of (13)–(15), the matter terms will be identical to the original solution since ˜ ) = eµ(r ) . E˜ 0 eµ(r
Hence we will always take E 0 = 1 and obtain arbitrary small values of R0 by taking −µ(0) sufficiently large. Let us define L0 R0 := . −2µ(0) e −1 It will be clear from the proofs that in fact f (r, ·, ·) > 0, when r is sufficiently close to but larger than R0 . Hence, given any number R0 > 0 we can construct a solution having inner radius of support equal to R0 . The following result proves issue (iii). The constants q, C1 , C2 and C3 which appear in the formulation of the theorem are specified in Eq. (22)–(24). Theorem 1. Consider a shell solution with a sufficiently small inner radius of support R0 . The distribution function f then vanishes within the interval
(q+3)/(q+1) 2/(q+1) −1 (q+2)/(q+1) R0 + B2 R0 , R0 + B0 R0 , 1 − B1 R0 where B0 , B1 and B2 are positive constants which depend on C1 , C2 and C3 . The solution can thus be joined with a Schwarzschild solution at the point where f vanishes and a static shell is obtained with support within [R0 , R1 ], where (q+2)/(q+1)
R1 = so that R1 /R0 → 1 as R0 → 0.
R0 + B2 R0
2/(q+1)
1 − B1 R0
,
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H. Andréasson
This result is interesting in its own right since it gives a detailed description of the support of a class of shell solutions to the Einstein-Vlasov system. Moreover, the solutions constructed in Theorem 1 can be used to obtain a sequence of shells of Vlasov matter with the property that 2M/R = 8/9 in the limit. j
j
Theorem 2. Let ( f j , µ j ) be a sequence of shell solutions with support in [R0 , R1 ], and j j j such that R1 /R0 → 1 and R0 → 0, as j → ∞, and let M j be the corresponding ADM mass of ( f j , µ j ). Then lim
2M j
j→∞
j R1
=
8 . 9
(11)
3. Static Shells of Vlasov Matter When the distribution function f has the form f (r, w, L) = (E, L),
(12)
the matter quantities ρ, p and q become functionals of µ, and we have ρ=
2π r2
2π p= 2 r 2π q= 4 r
E 0 e−µ L 1+ 20
r
E 0 e−µ L 1+ 20
r e−µ
E0 L 1+ 20 r
r 2 (s 2 −1)
(eµ s, L)
d Lds,
(13)
(eµ s, L) s 2 − 1 − L/2 d Lds,
(14)
L0
r 2 (s 2 −1)
s2 s 2 − 1 − L/2
L0
r 2 (s 2 −1) L0
(eµ s, L)
L s2
− 1 − L/2
d Lds.
(15)
Here we have kept the parameter E 0 but recall that E 0 = 1 in what follows. If (8) is chosen for these integrals can be computed explicitly in the cases when k = 0, 1, 2, ... and l = 1/2, 3/2, ... as the following lemma shows. Let γ = −µ −
1 L0 log (1 + 2 ). 2 r
Lemma 1. Let k = 0, 1, 2, . . . and let l = 1/2, 3/2, 5/2, . . . , then there are positive j constants πk,l , j = 1, 2, 3 such that when γ ≥ 0, L 0 l+2 γ ) (e − 1)l+k+3/2 Pl+5/2−k (eγ ), r2 L0 2 2l r (1 + 2 )l+2 (eγ − 1)l+k+5/2 Pl+3/2−k (eγ ), p = πk,l r L 0 3 2l r (1 + 2 )l+1 (eγ − 1)l+k+3/2 Pl+1/2−k (eγ ). z = πk,l r
1 2l ρ = πk,l r (1 +
(16) (17) (18)
If γ < 0 then all matter components vanish. Here Pn (eγ ) is a polynomial of degree n and Pn > 0, and z := ρ − p − q.
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415
Remark. The restriction l ≥ 1/2 is made so that the matter terms get the form above which is convenient. We believe however that the cases 0 ≤ l < 1/2 can be treated by similar arguments as presented below. Sketch of proof of Lemma √ 1. To evaluate the L−integration in the expressions for ρ and z we substitute x := L − L 0 and use that for n > 0, √ x n−1 ax 2 + b (n − 1)b xn x n−2 = d x. − √ √ na na ax 2 + b ax 2 + b The same substitution is made for p together with (n − 1)b x n−1 (ax 2 + b)3/2 n 2 − x ax + b = x n−2 ax 2 + bd x. (n + 2)a (n + 2)a The s−integration is straightforward and the claimed expressions follow by integration by parts and using that E 0 e−γ . eµ = 1 + L 0 /r 2 Next we show that if R0 is small then also γ is small. Lemma 2. Given k and l there is a cγ > 0 and a δγ > 0 such that eγ (r ) − 1 ≤ cγ r 2/(l+k+5/2) , for all r ∈ [R0 , 5R0 ], when R0 ≤ δγ . Proof of Lemma 2. Since γ = 0 at r = R0 we can consider an interval I := [R0 , y], y > R0 , such that γ ≤ 1 on this interval. Hence on I, P2−k (eγ ) > C P for some C P > 0. We have m L0 L0 γ (r ) = −µ (r ) + = − + 4πr p e2λ + r (r 2 + L 0 ) r2 r (r 2 + L 0 )
m C(eγ − 1)l+k+5/2 2λ L0 . (19) ≤ − 2 + 4πr e + 4 2 r r r (r + L 0 ) Here the constant C depends on C P and L 0 . Since γ (R0 ) = 0 and e2λ ≥ 1, this implies that l+k+5/2 γ r2 , ≤ e −1 4πC since if γ l+k+5/2 r2 , e −1 ≥ 4πC
(20)
on some subinterval [r1 , r2 ], with equality at r = r1 , it holds that γ (r ) < 0 on [r1 , r2 ], but in view of (20) γ must increase at r = r1 since the right-hand side is increasing in r, which contradicts γ (r1 ) < 0. It follows that eγ (r ) ≤ Cr 2/(l+k+5/2) + 1, for R0 ≤ r ≤ 5R0 , if R0 is sufficiently small, since γ is then less than or equal to one on [R0 , 5R0 ], which was an assumption of the argument.
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Remark. The assumption γ ≤ 1 is not needed if l + 3/2 − k ≥ 0. The choice [R0 , 5R0 ] was arbitrary and can be replaced by [R0 , N R0 ], N > 0, by taking R0 accordingly. Since we will show that the interval of support [R0 , R1 ] is such that R1 /R0 → 1, as R0 → 0, the interval [R0 , 5R0 ] in the lemma is no restriction and therefore we can always assume that eγ (r ) ≤ Cr 2/(l+k+5/2) + 1. Since R0 is small this implies that γ is small so that in particular γ ≤ eγ − 1 ≤ 2γ on [R0 , 5R0 ]. Thus, from Lemma 1 it follows that there are positive constants CU and C L such that CL
γ l+k+3/2 γ l+k+3/2 ≤ ρ ≤ CU , 4 r r4
(21)
and analogously for p and z. For non-integer values on k ≥ 0 and l ≥ 1/2, one can use the strategy described in the sketch of the proof of Lemma 1 and obtain sufficient information to conclude that the claim in Lemma 2, i.e. that γ is small whenever r is small, holds also for non-integer values of k and l. It is then straightforward to make upper and lower estimates on the matter terms with respect to the corresponding matter terms for integer values on k and l. Therefore, since upper and lower estimates as in (21) are sufficient for all the arguments below we will for simplicity, and without loss of generality, assume that for some positive constants C1 , C2 , and C3 , γq , r4 γ q+1 p = C2 4 , r γq z = C3 2 , r
ρ = C1
(22) (23) (24)
where q = l + k + 3/2 ≥ 2. 4. Proofs of Theorems 1 and 2 Proof of Theorem 1. We will always take R0 ≤ 1 sufficiently small so that the statement in Lemma 2 holds and so that L0 ≥ 5/6, for r ∈ [R0 , 5R0 ]. r2 + L0
(25)
It will also be tacitly assumed that all intervals we consider below are subsets of [R0 , 5R0 ]. It will be clear from the arguments that this can always be achieved by taking R0 sufficiently small. The positive constant C can change value from line to line. The proof of Theorem 1 will follow from a few lemmas and a proposition. Lemma 3. Let Cm = max {1, C1 , 4πC2 } and let (q+3)/(q+1)
δ≤
R0 , Cm 81/q+1
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then γ (r ) ≥
1 , 2r
for r ∈ [R0 , R0 + δ]. Proof of Lemma 3. We have γ (r ) = −µ (r ) +
L0 . r (r 2 + L 0 )
Let σ ∈ [0, δ], since γ (R0 ) = 0, it follows that γ (R0 + σ ) = γ (R0 + σ ) − γ (R0 ) ≤ σ γ (ξ ) ≤
δ , R0
(26)
where ξ ∈ [R0 , R0 + σ ]. Hence, by (22) we get C1 δ q q , for r ∈ [R0 , R0 + δ]. r 4 R0
ρ≤
Using that ρ = 0 when r < R0 , we obtain for σ ∈ [0, δ], R0 +σ C1 δ q 2 C1 δ q+1 C1 δ q σ ≤ . η dη = m(R0 + σ ) ≤ q q+1 q+1 η 4 R0 R0 R0 (R0 + σ ) R0 (R0 + σ ) Hence, m(R0 + σ ) C1 δ q+1 C1 δ q+1 ≤ q+1 ≤ . q+3 R0 + σ R0 (R0 + σ )2 R0 From (23) we also have p(r ) ≤
C2 δ q+1 q+1
r 4 R0
, for r ∈ [R0 , R0 + δ].
q+3
Note that by taking δ q+1 ≤ R0 /8Cm , we have m(R0 + σ )/(R0 + σ ) ≤ 1/8 so that e2λ(Ro +σ ) ≤ 4/3, and q+3
4πr 2 p ≤
R0
q+1 8r 2 R0
≤
1 . 8
Thus for r ∈ [R0 , R0 + δ], r µ (r ) =
m(r ) 2λ(r ) e + 4πr 2 p(r )e2λ(r ) ≤ 1/6 + 1/6 = 1/3. r
In view of (25) we thus have γ (r ) ≥ −
L0 1 + ≥ 1/2r, 2 3r r (r + L 0 )
for r ∈ [R0 , R0 + δ], and the lemma follows.
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The lemma implies that for σ ∈ [0, δ], γ (R0 + σ ) ≥ γ (R0 ) + σ inf γ (R0 + s) ≥ s∈[0,σ ]
σ , 2(R0 + σ )
(27)
where we again used that γ (R0 ) = 0. Let 2q+1 1+ q(q+1)
σ∗ := C0 R0
,
where C0 satisfies the conditions C0 ≤ 1/4 and C0 ≤ 1/(Cm 81/(q+1) ). Now, since 1+
q +3 1 q +3 2q + 1 = + ≥ , q(q + 1) q + 1 q(q + 1) q +1
we have in view of the second assumption on C0 that σ∗ < δ. Define γ∗ by γ∗ =
σ∗ . 2(R0 + σ∗ )
It is clear that γ (R0 + σ∗ ) ≥ γ∗ in view of (27). We will show that γ must reach the γ∗ −level again at r2 (i.e. a second time) close to R0 + σ∗ . We point out that the choice of the exponent in the definition of σ∗ is crucial, and our arguments provide almost no room to choose it differently. We have q
Lemma 4. Let κ = 32 22q /(C1 C0 ) and let = max {C0 , κ}, and consider a solution 1/(q+1)
≤ 1. Then there is a point r2 such that γ (r ) ↓ γ∗ as r ↑ r2 , with R0 such that R0 (q+2)/(q+1) . and r2 ≤ R0 + σ∗ + κ R0 Proof of Lemma 4. Since γ (R0 +σ∗ ) ≥ γ∗ , and since Lemma 3 gives that γ (R0 +σ∗ ) > 0, the point r2 must be strictly greater than R0 + σ∗ . Let [R0 + σ∗ , R0 + σ∗ + ], for some > 0, be such that γ ≥ γ∗ on this interval. We will show that (q+2)/(q+1)
≤ κ R0
.
We have from (22), m(r ) ≥ C1
R0 +σ∗ + R0 +σ∗
q
σ∗ r 2 dr 4 q r 2 (R0 + σ∗ )q
q
σ∗ . = C1 q 2 (R0 + σ∗ )q+1 (R0 + σ∗ + ) Hence, q
σ∗ m(R0 + σ∗ + ) ≥ C1 q . q+1 R0 + σ∗ + 2 (R0 + σ∗ ) (R0 + σ∗ + )2 The assumption in the lemma guarantees that σ∗ ≤ R0 since 2q + 1 1 ≥ . q(q + 1) q +1
(28)
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Substituting for σ∗ then gives 2q+1 q 1+ q(q+1)
m(R0 + σ∗ + ) q R ≥ C1 C0 0 q+1 R0 + σ∗ + (R0 + σ∗ + )2 22q+1 R0 = (q+2)/(q+1)
Let = κ R0 It follows that
q/(q+1) R0 q C1 C0 2q+1 . 2 (R0 + σ∗ + )2
, and note that the assumptions of the lemma imply that ≤ R0 . q
R2 C1 C0 κ m(R0 + σ∗ + ) q ≥ C1 C0 κ 2q+1 0 = 2 2q+1 . R0 + σ∗ + 2 (3R0 )2 3 2 The definition of κ implies that m(R0 + σ∗ + ) 1 ≥ . R0 + σ∗ 2 This is impossible since it is proved in [5] that all static solutions have 2m/r < 1, and (q+2)/(q+1) . This completes the proof therefore r2 must be strictly less than R0 + σ∗ + κ R0 of the lemma. We will next show that if R0 is sufficiently small then 2m/r will attain values arbitrarily close to 8/9. Proposition 1. Let r2 be as in Lemma 4 for a sufficiently small R0 . Then the corresponding solution satisfies m(r2 )/r2 ≥ 2/5, and if R0 → 0, then 2m(r2 )/r2 → 8/9. Proof of Proposition 1. We now consider the fundamental equation (10) in [1] which reads m 1 r µ+λ + 4πr p e = 4π η2 eµ+λ (ρ + p + q)dη. (29) r2 r2 0 This equation is a consequence of the generalized (for non-isotropic pressure) Oppenheimer-Tolman-Volkov equation. For r = r2 we then have r2 (µ+λ)(r2 ) m(r2 )e = 4π η2 eµ+λ (ρ + p + q) dη − 4πr23 pe(µ+λ)(r2 ) R0 r2 = 4π η2 eµ+λ (2ρ − z) dη − 4πr23 pe(µ+λ)(r2 ) . (30) R0
Here we used that p + q = ρ − z. From (24) and (22) we get r2 C 3 r2 4π η2 eµ+λ z dη ≤ 4π η4 γρeµ+λ dη C 1 R0 R0 C3r23 r2 4π ηρeµ+λ dη ≤ C 1 R0 r2 4π ηρeλ dη. ≤ Cr23 eµ(r2 ) R0
(31)
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Here we used that eµ is increasing. Next we observe that the result of Theorem 1 in [1] can be applied to the interval [R0 , r2 ] when √ R0 is small enough. Indeed, replace M by m(r2 ), use the fact that eµ(r2 ) ≤ e−λ(r2 ) = 1 − 2m(r2 )/r2 , and use the fact that p(r2 ) ≥ 0 so that the second term on the left-hand side of Eq. (29) can be dropped. Therefore, since r2 /R0 → 1, Theorem 1 in [1] implies that for a sufficiently small R0 there exists a positive number k, less than one, such that 2m(r ) ≤ 1 − k, r r ∈[R0 ,r2 ] sup
√ so that λ ≤ − log k =: Cλ on the interval [R0 , r2 ]. (Note also that Cλ can be made arbitrary close to log 3 by Theorem 2 in [1] by taking R0 sufficiently small.) Now we write
r2 R0
r2 d −λ meλ − e dη + 4π ηρe dη = dη 2 dr R0 R0 η r 2 − R0 2m(r2 ) ≤ 1− 1− + eCλ m(r2 ) r2 r 2 R0 (q+2)/(q+1) 2m(r2 ) Cm(r2 )R0 ≤ 1− 1− + r2 R02
λ
r2
2m(r2 ) Cm(r2 ) + q/(q+1) √ r2 (1 + 1 − 2m(r2 )/r2 ) R 0 2m(r2 ) Cm(r2 ) Cm(r2 ) ≤ + q/(q+1) ≤ . R0 R0 R
(32)
=
(33)
0
(q+2)/(q+1)
(q+2)/(q+1)
Here we used that r2 ≤ R0 + σ∗ + κ R0 , and that σ∗ ≤ C R0 , for some C > 0 when R0 is small. From Lemma 4 we have that γ approaches γ∗ from above and therefore γ (r2 ) ≤ 0, which implies that µ (r2 ) ≥ 5/(6r2 ), in view of (19) and (25). Now, since µ =
m r2
+ 4πr p e2λ ,
and since (23) gives (2q+1)/q
r2 p(r2 ) = C2 r2
q+1 C R0 γ∗ ≤ 4 r2 r23
≤
C (q−1)/q R0
,
we necessarily have m(r2 )/r2 ≥ 1/4 if R0 is small enough. Indeed, if m(r2 )/r2 ≤ 1/4, then e2λ ≤ 2, and we get µ (r2 ) =
m r2
1 C + 4πr p e2λ ≤ + , 2r2 R (q−1)/q 0
Static Shells and Buchdahl Inequality for Spherically Symmetric E-V System
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and this is smaller than 5/(6r2 ) when r2 , or equivalently R0 , is small. Hence, r2 ≤ 4m(r2 ), and from (30), (31) and (33) we obtain (using R0 ≤ r2 ≤ 3R0 ) r2 4π η2 eµ+λ ρ dη − Ceµ(r2 )r22 m(r2 ) − Cr22 m(r2 ) pe(µ+λ)(r2 ) m(r2 )e(µ+λ)(r2 ) ≥ 2 R0 r2 ≥ 2eµ(R0 ) R0 4π ηeλ ρ dη − Ce
R0 2 R0 m(r2 ) − C R02 m(r2 ) pe(µ+λ)(r2 ) .
µ(r2 )
(34)
Here we again used that eµ is increasing. For the integral term of the right-hand side we use the computation in (33), but now we estimate from below (since the sign is the opposite) and thus we do not need to estimate the integral r2 meλ dη, 2 R0 η which we drop and we get r2 4π ηeλ ρ dη ≥
r2 (1 +
R0
Thus we obtain m(r2 )e(µ+λ)(r2 ) ≥ eµ(R0 )
R0 r2
2m(r2 ) . √ 1 − 2m(r2 )/r2 )
1+
√
(35)
4m(r2 ) 1 − 2m(r2 )/r2
− Ceµ(r2 ) R02 m(r2 ) − C R02 m(r2 ) pe(µ+λ)(r2 ) .
(36)
Now we use again that (2q+1)/q
p(r2 ) = C2
q+1 C R0 γ∗ ≤ 4 r2 r24
≤
C (2q−1)/q R0
,
together with the fact that λ ≥ 1, and obtain R0 4e−λ(r2 ) 1/q 1 ≥ eµ(R0 )−µ(r2 ) − C R02 − C R0 . √ r2 1 + 1 − 2m(r2 )/r2
(37)
Let us now consider µ(R0 ) − µ(r2 ). Since r m µ(r ) = µ(0) + + 4π ηp e2λ dη, η2 0 we have
µ(R0 ) − µ(r2 ) = −
r2 R0
m + 4π ηp e2λ dη. η2
We will show that the integral goes to zero as R0 → 0. From Lemma 2 we have that γ ≤ Cr 2/(q+1) (where we use the fact that r is small so that at least eγ − 1 ≤ 2γ ) and it follows by (23) that p≤
C . r2
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Since λ ≤ Cλ on [R0 , r2 ], and m/r ≤ 1/2 always, we get m P + 4πr p e2λ ≤ , r2 r where P is a constant depending on Cλ and C2 . Hence, µ(R0 ) − µ(r2 ) ≥ −P log (r2 /R0 ). This estimate implies that (37) can be written P+1 √ R0 4 1 − 2m(r2 )/r2 1/q 1≥ − C R02 − C R0 . √ r2 1 + 1 − 2m(r2 )/r2
(38)
(39)
Since R0 /r2 ↑ 1, as R0 → 0, we can write this inequality as √ 4 1 − 2m(r2 )/r2 1 ≥ (1 − (R0 )) − C (R0 ), √ 1 + 1 − 2m(r2 )/r2 where (R0 ) ↓ 0 as R0 → 0. This yields 1 + 1 − 2m(r2 )/r2 ≥ (1 − (R0 )) 4 1 − 2m(r2 )/r2 − C (R0 ), so that
1 ≥ 3 1 − 2m(r2 )/r2 − C (R0 ).
Squaring both sides and solving for 2m(r2 )/r2 gives 2m(r2 ) 8 ≥ − C (R0 ). r2 9
(40)
It is now clear that m(r2 )/r2 ≥ 2/5, when R0 is sufficiently small, and that 2m(r2 )/r2 → 8/9 as R0 → 0, which completes the proof of the proposition. We can now finish the proof of Theorem 1. First of all note that f cannot vanish for (q+3)/(q+1)
r ≤ R0 +
R0 , Cm 81/(q+1) (q+3)/(q+1)
by Lemma 3. Thus the claim that f will not vanish before r = R0 + B0 R0 follows with B0 := 1/Cm 81/(q+1) . The main issue is of course to prove that f vanishes before (q+2)/(q+1)
R0 + B2 R0
2/(q+1)
1 − B1 R0
,
where B1 and B2 are positive constants. Inspired by an idea of T. Makino introduced in [12], we show that γ necessarily must vanish close to the point r2 if R0 is sufficiently small. Let x :=
m(r ) . r γ (r )
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Using that m (r ) = 4πr 2 ρ, it follows that r x =
x2 x L0 4πr 2 ρ −x+ − . γ 1 − 2γ x γ (r 2 + L 0 )
In our case r > R0 and γ > 0 and we will show that γ (r ) = 0 for some r < (1 + (R0 ))r2 , where has the property as in the proof of Proposition 1. Since γ > 0 and ρ ≥ 0 the first term can be dropped and we have r x ≥ −x +
x L0 x2 2x 2 x L0 x2 − = −x + − . 2 1 − 2γ x γ (r + L 0 ) 3(1 − 2γ x) 3(1 − 2γ x) γ (r 2 + L 0 ) (41)
Take R0 sufficiently small so that m(r2 )/r2 ≥ 2/5 by Proposition 1. Let r ∈ [r2 , 16r2 /15], then since m is increasing in r we get m(r ) m(r2 ) r2 m(r2 ) 3 15 2 ≥ = · = . ≥ r r r r2 16 5 8 Now by the definition of x it follows that m 2λ 3 m 3 x m = e = ≥ , when ≥ . 1 − 2γ x γr γ r (1 − 2m/r ) 2γ r 8 Thus on [r2 , 16r2 /15], x L0 2x 2 − ≥ 0, 3(1 − 2γ x) γ (r 2 + L 0 ) so that on this interval r x ≥
4 x2 − x ≥ x 2 − x, 3(1 − 2γ x) 3
(42)
where we used that 1 m 3 1 = ≥ 4when ≥ . 1 − 2γ x 1 − 2m/r r 8 Lemma 2 gives an upper bound of γ which implies that x(r2 ) =
2 C m(r2 ) ≥ · 2/(q+1) . r2 γ (r2 ) 5 r 2
Thus x(r2 ) → ∞ as R0 → 0, and we take R0 sufficiently small so that x(r2 ) 16 ≤ . x(r2 ) − 3/4 15 In particular x(r2 ) ≥ 3/4. Solving (42) yields r (4x(r2 )/3 − 1) −1 , on r ∈ [r2 , 16r2 /15), x(r ) ≥ 1 − 4r2 x(r2 )/3
(43)
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and we get that x(r ) → ∞ as r → R1 , where R1 ≤ r 2
x(r2 ) 16r2 ≤ . x(r2 ) − 3/4 15
(44)
Now in view of (43), 1 x(r2 ) ≤ → 1, as R0 → 0, 2/(q+1) x(r2 ) − 3/4 1 − B1 R0 (q+2)/(q+1)
, if R0 is sufficiently small, it for some positive constant B1 . Since σ∗ ≤ κ R0 (q+2)/(q+1) , and the is clear that there is a positive constant B2 such that r2 ≤ R0 + B2 R0 proof of Theorem 1 is complete. Proof of Theorem 2. From Theorem 2 in [1] we get with = 1 that 2M 8 ≤ . R1 9
lim sup R0 →0
The arguments in Propostion 1 leading to (40) can be applied when r = R1 instead of r = r2 . Thus we also have lim inf R0 →0
and the claim of the theorem follows.
2M 8 ≥ , R1 9
Acknowledgement. I want to thank Gerhard Rein for discussions and for commenting the manuscript, and Alan Rendall for drawing my attention to the Buchdahl inequality in connection to a result by Christodoulou [7] on the formation of trapped surfaces.
References 1. Andréasson, H.: On the Buchdahl inequality for spherically symmetric static shells. Commun. Math. Phys. 2. Andréasson, H.: The Einstein-Vlasov System/Kinetic Theory. Living Rev. Relativ. 8 (2005) 3. Andréasson, H., Rein, G.: On the steady states of the spherically symmetric Einstein-Vlasov system. Class. Quantum Grav. 24, 1809–1832 (2007) 4. Batt, J., Faltenbacher, W., Horst, E.: Stationary spherically symmetric models in stellar dynamics. Arch. Rat. Mech. Anal. 93, 159–183 (1986) 5. Baumgarte, T.W., Rendall, A.D.: Regularity of spherically symmetric static solutions of the Einstein equations. Class. Quantum Grav. 10, 327–332 (1993) 6. Buchdahl, H.A.: General relativistic fluid spheres. Phys. Rev. 116, 1027–1034 (1959) 7. Christodoulou, D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math. 44, 339–373 (1991) 8. Christodoulou, D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math. 149, 183–217 (1999) 9. Christodoulou, D.: On the global initial value problem and the issue of singularities. Class. Quantum Grav. 16, A23–A35 (1999) 10. Dafermos, M.: Spherically symmetric spacetimes with a trapped surface. Class. Quantum Grav. 22, 2221–2232 (2005) 11. Dafermos, M., Rendall, A.D.: An extension principle for the Einstein-Vlasov system in spherical symmetry. Ann. Henri Poincaré 6, 1137–1155 (2005) 12. Makino, T.: On spherically symmetric stellar models in general relativity. J. Math. Kyoto Univ. 38, 55–69 (1998)
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13. Rein, G.: Static shells for the Vlasov-Poisson and Vlasov-Einstein systems. Indiana Univ. Math. J. 48, 335–346 (1999) 14. Rein, G., Rendall, A.D.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128, 363–380 (2000) 15. Rendall, A.D.: An introduction to the Einstein-Vlasov system. Banach Center Publ. 41, 35–68 (1997) 16. Schaeffer, J.: A class of counterexamples to Jeans’ theorem for the Vlasov-Einstein system. Commun. Math. Phys. 204, 313–327 (1999) Communicated by G.W. Gibbons
Commun. Math. Phys. 274, 427–455 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0276-5
Communications in
Mathematical Physics
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model Estelle L. Basor1, , Torsten Ehrhardt2 1 Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407, USA.
E-mail: [email protected]
2 Department of Mathematics, University of California, Santa Cruz, CA 95064, USA.
E-mail: [email protected] Received: 31 July 2006 / Accepted: 19 December 2006 Published online: 3 July 2007 – © Springer-Verlag 2007
Abstract: We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomermonomer correlation function. The model depends on a parameter interpolating between the square lattice (t = 0) and the triangular lattice (t = 1), and we obtain the asymptotics for 0 < t ≤ 1. For 0 < t < 1 we apply the Szegö Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form. 1. Introduction A dimer is a bond connecting two nearest neighbors on a lattice. There is a long history in statistical mechanics of the study of the configurations of the bonds on planar lattices and mathematical results involving determinants. The most classical connection, due to Fisher and Stephenson [9], is the reformulation of the two-dimensional Ising model as a dimer model on a decorated planar lattice. Since the dimer model can be reduced, by means of the theorem of Kasteleyn [12, 14], to the computation of a certain Pfaffian, the problem is then equivalent to computing a determinant. Recently, there has been increased interest in the dimer statistics due to the investigation of quantum dimer models. These models are important in the study high-temperature superconductivity. It has been proposed that the superconducting state evolves out of a short-range resonating valence bond (RVB) state. The underlying Hilbert space of these models consists of all configurations of pairings of the spins on the lattice into singlet bonds. In the work of Rokhsar and Kivelson on the square lattice, it was shown that this quantum dimer model exhibits a critical point in which the equal-amplitude superpositions of all dimer configurations of each topological sector are the unique zero-energy ground states. However, this “RK” point turned out to be an isolated point between two solid phases, rather than an actual RVB phase [13, 16, 17, 19]. Supported in part by NSF Grants DMS-0200167 and DMS-0500892.
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E. L. Basor, T. Ehrhardt
For the triangular lattice, the situation is quite different. In [15, 18] it was shown that there is an RVB phase characterized by liquid correlations. It turns out that ground state correlations at the “RK” point are the same as those computed from the correlations of the classical dimer model. Hence it is important to have information about such correlations. This was the motivation for the results obtained in this paper. In the hard-core, close-packed dimer model, each site is paired with exactly one of its nearest neighbors. If the dimers have equal fugacity, each configuration is equally likely and the partition function is simply the number of configurations. For planar lattices this can be computed using classical Pfaffian techniques that go back to the work of Kasteleyn and Fisher [9, 12]. On the planar lattice a directed graph is constructed so that an arrow connects all nearest neighbors with the property that the number of clockwise arrows around any closed circuit is odd. Then a modified adjacency matrix Mi, j is constructed so that Mi, j = 1 if an arrow points from site i to site j, Mi, j = −1 if the arrow points from site j to i and 0 otherwise. For the triangular lattice with arrows pointing to the right/upwards, this approach is modified so that the same matrix is considered, except that in the lattice, alternating factors of ±i are placed on alternating rows. In either case the result is that the number of configurations is (det M)1/2 . To interpolate, the triangular bonds are weighted by t so that t = 0 corresponds to the square lattice and t = 1 corresponds to the triangular lattice case. The monomer-monomer correlation is the ratio of square roots of determinants described above, but on different lattices. The denominator is for lattice points equally spaced. The numerator is the same, but for the lattice with two sites removed. For this paper, the sites are n spaces apart in adjacent rows. If this ratio, denoted by P (mm) (n), tends to zero as n tends to infinity, it is said that the model is confining. If it tends to a non-zero limit, it is said to be deconfining and is a desired feature for the 2d quantum models. In this paper we show that the limit is non-zero for t > 0. We also note that our computation yields a result for correlations with variable bonds, a result that we believe is only rarely explicitly obtainable. The starting point for our work is the computation from [8], by Fendley, Moessner, and Sondhi which describes the monomer-monomer correlation function P (mm) (n), as the determinant of a block matrix ⎛ ⎞ R Q 1 ⎠, P (mm) (n) = det Mn , Mn = ⎝ (1) 2 Q R where R is an n × n matrix with entries depending on the difference of their indices, and Q is an n × n matrix with entries depending on the sum of the indices. The authors computed this correlation function numerically for t = 1 as the size of the matrix increases and found that it converged to a constant value of around 0.1494 . . .. For all the details of the dimer model computation, the reader is referred to [8]. We will not only compute the determinant asymptotically for all values of t between zero and one but in addition we will be able to determine the asymptotics for all complex parameters t with Re (t) > 0. This covers all physically interesting cases except t = 0. Our method is to convert the determinant of the block matrix Mn that arises in the dimer problem into a determinant of a block Toeplitz matrix and then to find a way to explicitly compute the asymptotics of the block Toeplitz determinant. To be more specific the dimer matrix (1) has n × n matrix blocks R and Q whose entries are given by R jk = 2(−1)[(k− j)/2] Rk− j+1 + θ ( j − k)t j−k−1 ,
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model
429
Q jk = 2i(−1)[( j+k)/2] Q n+1− j−k , (1 ≤ j, k ≤ n) and where the expressions Rk , Q k , and θ (k) are defined by the following. For even k, π π cos y cos(kx + y) 1 d yd x, Rk = 2 2 8π −π −π cos x + cos2 y + t 2 cos2 (x + y) and Q k = 0. For odd k, 1 Qk = 8π 2 and Rk =
t 8π 2
π
−π
π −π
π
−π
cos2
cos x cos kx d yd x, x + cos2 y + t 2 cos2 (x + y)
cos2
cos(x + y) cos(kx + y) d yd x. x + cos2 y + t 2 cos2 (x + y)
π
−π
The expression θ (k) equals 1 for k > 0 and 0 otherwise. In the Appendix we show that the determinant of the dimer matrix Mn is the same as the determinant of an n × n block Toeplitz matrix. A finite block Toeplitz matrix is one of the form Tn (φ) = (φ j−k ), 0 ≤ j, k ≤ n − 1, where φk =
1 2π
2π
φ(ei x )e−ikx d x
0
are the (matrix) Fourier coefficients of an N × N matrix valued function φ defined on the unit circle T = { z ∈ C : |z| = 1 }. In the dimer case we have N = 2, and for 0 < t < 1 we obtain ⎛ ⎞ c d ⎠, det Mn = det Tn (φ) with φ = ⎝ (2) d˜ c˜ where (t cos x + sin2 x) , √ (t − e−i x ) t 2 + sin2 x + sin4 x sin x d(ei x ) = √ . 2 t + sin2 x + sin4 x c(ei x ) =
(3) (4)
Here and in what follows a(e ˜ i x ) = a(e−i x ). In the case 0 < t < 1, we are able to find the asymptotic expansion of the determinant of Tn (φ) by using the Strong Szegö Limit Theorem. The main difficulty is that the constant term in the asymptotics for block Toeplitz matrices is generally only given in a rather abstract form. Thus we are required to find a more explicit expression for the constant, which fortunately, in this case, can be done. For this purpose we will use an expression for the constant which holds in very special cases and which is due to Widom [20]. We will also use another determinantal identity which is perhaps new and by the
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help of which more general (yet special) cases can be reduced to the situation covered by Widom. We will also give a new proof of Widom’s result using the Borodin-OkounkovCase-Geronimo identity [2, 4, 10]. All these results are described in Sect. 2 and are of independent interest. The application to the dimer case still requires a lot of elementary, yet tedious computations which will be done in Sects. 4 and 5. The final result is that the limit of the monomer-monomer correlation function P (mm) (∞) := lim P (mm) (n)
(5)
n→∞
exists and is given by the formula P
(mm)
1 (∞) = 2
t 2t (2 + t 2 ) + (1 + 2t 2 )
. √ 2 + t2
(6)
At this point we have proved this in the case 0 < t < 1. In the case t = 1, we will also obtain this asymptotic formula, which agrees with the numerical calculation [8]. In this case we cannot use the Szegö Limit Theorem because the function c has a singularity. This singularity is due to the second term in the definition of the entries of R in terms of Rk . The entries of Mn are well-defined and analytic for t ∈ C with Re (t) > 0. For these values we will actually be able to prove the same asymptotic formula as above. This will be done in Sect. 6. The idea is to transform the matrix Tn (φ) into a matrix which is a finite block Toeplitz matrix plus a certain perturbation (which is of fixed finite rank). A generalization of the block version of the Segö Limit Theorem then gives the expected asymptotics. The statement and the proof of this generalized version is given in Sect. 3, and is also of independent interest. 2. Determinant Identities We begin with some preliminary facts about Toeplitz operators and Toeplitz matrices. Let φ ∈ L ∞ (T) N ×N be an essentially bounded N × N matrix valued function defined on the unit circle with Fourier coefficients φk ∈ C N ×N . The Toeplitz and Hankel operators are defined on 2 (Z+ ), Z+ = {0, 1, . . .}, by means of the semi-infinite infinite matrices T (φ) = (φ j−k ), H (φ) = (φ j+k+1 ),
0 ≤ j, k < ∞, 0 ≤ j, k < ∞.
For φ, ψ ∈ L ∞ (T) N ×N the identities ˜ T (φψ) = T (φ)T (ψ) + H (φ)H (ψ), ˜ H (φψ) = T (φ)H (ψ) + H (φ)T (ψ)
(7) (8)
are well-known. It follows from these identities that if ψ− and ψ+ have the property that all their Fourier coefficients vanish for k > 0 and k < 0, respectively, then T (ψ− φψ+ ) = T (ψ− )T (φ)T (ψ+ ), H (ψ− φ ψ˜ + ) = T (ψ− )H (φ)T (ψ+ ).
(9) (10)
In the following sections we compute an explicit expression for one of the constants that appears in the version of the Strong Szegö Limit Theorem for matrix-valued symbols,
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model
431
for certain symbols having a special form. In order to state the classic Strong Szegö Limit Theorem for matrix-valued symbols, let B stand for the set of all functions φ ∈ L 1 (T) such that the Fourier coefficients satisfy φB :=
∞
|φk | +
∞
k=−∞
|k| · |φk |2
1/2
< ∞.
(11)
k=−∞
With the norm (11) and pointswise defined algebraic operations on T, the set B becomes a Banach algebra of continuous functions on the unit circle (see, e.g., [4, 21]). Theorem 2.1 (Szegö-Widom [21]). Let φ ∈ B N ×N and assume that the function det φ does not vanish on T and has winding number zero. Then det Tn (φ) ∼ G(φ)n E(φ), where G(φ) = exp
as n → ∞,
1 2π
log det φ(ei x )d x 2π 0
and E(φ) = det T (φ)T (φ −1 ). In connection with this theorem, let us explain why the definitions of the constants G(φ) and E(φ) make sense. First of all, one can show that each nonvanishing function in B with winding number zero possesses a logarithm which belongs to B. Hence G(φ) is well defined as the geometric mean of det φ. The constant E(φ) has to be understood as an operator determinant. In fact, we have T (φ)T (φ −1 ) = I − H (φ)H (φ˜ −1 ), where the product of the Hankel operators is a trace class operator. Observe that we have defined B in such a way that Hankel operators with symbols in B are Hilbert-Schmidt. For general information about trace class operators and operator determinants see, e.g., [11]. In the scalar case (N = 1), there exists an explicit expression for E(φ) given by E(φ) = exp
∞
k[log φ]k [log φ]−k .
k=1
In general (N ≥ 2) a more explicit expression for E(φ) does not exist apart from very special cases. For more information about related results we refer the reader to [4]. One of the few results concerning an explicit expression for E(φ) in the matrix case goes back to Widom who showed that if φk vanish for k < n or for k > n (for some fixed n), then E(φ) = det Tn (φ −1 )G(φ)n (see [20], Theorem 5.1). Thus the constant E(φ) is reduced to the computation of a finite determinant, which for small values of n is very computable. If one can somehow reduce the computation of our E(φ) to this case, then an explicit formula for E(φ) is possible. The next theorem and its corollary will facilitate such a reduction and since we cannot expect to do it except for very special cases, it requires special assumptions on the symbol. Proposition 2.2. Let R ∈ B N ×N . Then the operator determinant det(T (e R )e−T (R) ) is well defined. Moreover, f (λ) = det(T (eλR )e−T (λR) ) is an entire function. Proof. This is proved in [7, Sect. 7].
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E. L. Basor, T. Ehrhardt
In what follows let I N denote the N × N identity matrix. Theorem 2.3. Let Q ∈ B N ×N , trace Q = 0, and a ∈ B. Then
N trace H (a)H (a) ˜ · det(T (e Q )e−T (Q) ). det(T (ea I N +Q )e−T (a I N +Q) ) = exp 2 Proof. This proof is modeled on the ones given in [6, 7] and a more detailed account of why the various determinants and derivatives are defined can be found there. We give only a sketch here. Let R = a I N + Q and define f (λ) = det(T (eλR )e−λT (R) ). The function f (λ) is analytic on C and it is nonzero if and only if the Toeplitz operator T (eλR ) is invertible. In particular, f (λ) is nonzero except on a discrete subset of C. Using the fact that (log det F(λ)) = trace F (λ)F −1 (λ) we have that
f (λ)/ f (λ) = trace T (ReλR )T −1 (eλR ) − T (eλR )T (R)T −1 (eλR ) . This implies
f (λ)/ f (λ) = trace T −1 (eλR )T (ReλR ) − T (R) . Differentiating again we have that
( f (λ)/ f (λ)) = trace T −1 (eλR )T (R 2 eλR )−T −1 (eλR )T (ReλR )T −1 (eλR )T (ReλR ) . At the points where f (λ) = 0, the symbol ψ λ = eλR has the property that it factors into a product λ λ ψ+ ψ λ = ψ−
such that the factors ψ− and ψ+ as well as their inverses belong to B N ×N and have Fourier coefficients that vanish for k > 0 and k < 0 respectively. Thus, by (9), λ T (eλR ) = T (ψ− )T (ψ+λ ).
This yields λ −1 ) ). T −1 (eλR ) = T ((ψ+λ )−1 )T ((ψ−
After simplifying we obtain
λ −1 2 λ λ −1 λ ( f (λ)/ f (λ)) = trace T ((ψ− ) R ψ− ) − T 2 ((ψ− ) Rψ− ) = trace(H ()H ()), ˜ where λ −1 λ λ −1 λ = (ψ− ) Rψ− = a I N + (ψ− ) Qψ− .
Now define g(λ) = det(T (eλQ )e−λT (Q) )
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model
433
and compute the above expression once again with R replaced by Q (i.e., put a = 0). Then the difference ˜ ( f (λ)/ f (λ)) − (g (λ)/g(λ)) = N · trace H (a)H (a). The reason for this is that if we expand trace H ()H () ˜ λ λ ) has trace zero since we have four terms. A term of the form H (a I N )H ((ψ− )−1 Qψ− λ −1 λ each matrix Fourier coefficient of (ψ− ) Qψ− has this property and multiplication by a matrix coefficient of a I N corresponds to a scalar multiplication. The term λ −1 λ λ λ) trace H ((ψ− ) Qψ− )H ((ψ− )−1 Qψ− λ cancels with the corresponding term for g since the corresponding functions ψ− λa I +λQ λQ (obtained from e N and e , resp., by factorization) only differ by a scalar function. Finally the “N ” comes from the fact that a I N is a N × N matrix. Thus
log f (λ) − log g(λ) has a constant second derivative and satisfies (log f − log g)(0) = 0 and ( f / f )(0) − (g /g)(0) = 0. From this the theorem follows easily.
Corollary 2.4. Let Q ∈ B N ×N , trace Q = 0, and a ∈ B. Then
E(ea I N +Q ) = exp N · trace H (a)H (a) ˜ · E(e Q ). Proof. We can write E(ea I N +Q ) = det T (ea I N +Q )T (e−a I N −Q ) = det T (ea I N +Q )e−T (a I N +Q) · det e T (a I N +Q) T (e−a I N −Q ) and apply the previous theorem twice.
The following theorem and proposition yields an alternative proof to Widom’s result in the case where either the positive or negative Fourier coefficients vanish for all but a finite number of indices. Proceeding as in the proofs of the Borodin-OkounkovCase-Geronimo (BOCG) identity [2, 10] given in [1] and [3] one obtains the following proposition. Widom’s result is stated in our Theorem 2.7. ˜ are invertible Proposition 2.5. Suppose that ψ ∈ B N ×N such that both T (ψ) and T (ψ) on (2 (Z+ )) N . Then det Tn (ψ −1 ) =
E(ψ) −n −1 ˜ ˜ z −n )T −1 (ψ) . · det I − H (z ψ)T ( ψ)H ( ψ G(ψ)n
(12)
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E. L. Basor, T. Ehrhardt
˜ implies the invertibility of ψ and the existence Proof. The invertibility of T (ψ) and T (ψ) of a left and a right canonical factorization (in B N ×N ) ψ −1 = u − u + = v+ v− . Proceeding as in the proof of the BOCG-identity one obtains −1 )Pn , det Tn (ψ −1 ) = G(ψ −1 )n · det Pn T (v+−1 )T (u − )T (u + )T (v− 2 where Pn is the projection {xk }∞ k=0 → {x 0 , . . . , x n−1 , 0, 0, . . .} acting on (Z+ ). Using the identity
det Pn A Pn = det(Q n + Pn A Pn ) = det(Q n + A Pn ) = det A · det(A−1 Q n + Pn ) = det A · det(Q n A−1 Q n )
(13)
in which Q n = I − Pn and A is an invertible operator of the form identity plus trace class, it follows that −1 )Pn det Pn T (v+−1 )T (u − )T (u + )T (v−
−1 −1 = det T (v+−1 )T (u − )T (u + )T (v− ) · det Q n T (v− )T (u −1 + )T (u − )T (v+ )Q n .
The first term on the right is easily seen to be det T (ψ)T (ψ −1 ) while the second term is equal to −1 −1 n −n −1 −n det T (z −n v− u −1 + )T (u − v+ z ) = det(I − H (z v− u + )H (u˜ − v˜ + z )), −1 −1 −1 n ˜ where we use z −n v− u −1 + u − v+ z = I N . Now we substitute u + = ψu − and u˜ − = u˜ + ψ and pull out the remaining factors from the Hankel operators (see (10)), which give the inverses of the Toeplitz operators.
Proposition 2.6. Let ψ ∈ B N ×N be invertible. Then det Tn (ψ −1 ) = 0 if and only if the operator
) z −n ) T (ψ H (ψ T (ψ) H (ψ z −n ) has a non-trivial kernel (or, equivalently, a non-trivial cokernel). Proof. The proof relies on the general fact that the kernel (resp. cokernel) of an operator P A P + Q is trivial if and only if the kernel (resp. cokernel) of the operator Q A−1 Q + P is trivial, where A is assumed to be an invertible operator, P is a projection operator and Q = I − P. Indeed, this follows from the fact that both operators are equivalent, i.e., there exist invertible S1 and S2 such that P A P + Q = S1 (Q A−1 Q + P)S2 . (One can also say that P A P and Q A−1 Q are equivalent after extension. See (13) for the underlying line of argumentation.) We apply this statement in the setting A = M(ψ −1 ) being the Laurent operator 2 N
acting on (2 (Z)) N , and P being the projection {xk }k∈Z → {xk }n−1 k=0 on ( (Z)) . Theorem 2.7 (Widom [20]). Let ψ ∈ B N ×N be such that the function det ψ does not vanish on T and has winding number zero. Assume that ψk = 0 for all k > n or that ψ−k = 0 for all k > n. Then E(ψ) = G(ψ)n det Tn (ψ −1 ). Note the above result holds also for n = 0 when stipulating det T0
(14) (ψ −1 )
= 1.
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model
435
Proof. The winding number condition implies that both T (ψ) and T (ψ −1 ) are Fredholm ) are equivalent operators with index zero. Since one can show that T (ψ −1 ) and T (ψ after extension, T (ψ ) is also a Fredholm operator with index zero. )) are invertible. Hence if E(ψ) = 0, then both T (ψ) and T (ψ −1 ) (and also T (ψ Now Proposition 2.5 settles the assertion. If E(ψ) = 0 then (since T (ψ)T (ψ −1 ) = I −compact), the product T (ψ)T (ψ −1 ) has a non-trivial kernel and cokernel. This implies that T (ψ) has non-trivial kernel and )) has non-trivial kernel and cokernel. cokernel, or, that T (ψ −1 ) (hence T (ψ Consider the case that ψk = 0 for all k > n. (The other case can be treated analogously.) Then the operator considered in Proposition 2.6 takes the form
) H (ψ t −n ) T (ψ . 0 T (ψ) ) has a non-trivial kernel, the operator above has a non-trivial In the case where T (ψ kernel. However, if T (ψ) has a non-trivial kernel (hence a non-trivial cokernel), the above operator has a non-trivial cokernel, too.
3. A Generalized Szegö-Widom Limit Theorem When dealing with the case of parameters t ∈ / (0, 1) in Sect. 6, we are going to employ a generalization of the matrix version of the Szegö-Widom limit theorem. The reader who is not interested in the details of this case can skip this section. In the analysis of this case we are led to compute the asymptotics of the determinants of block Toeplitz matrices which are perturbed in some way by trace class operators. Although this generalization is very much straightforward, we give the details of the proof because our matrices depend analytically on the parameter t and we want to show that the limiting constant also depends analytically on t. This will enable us to identify the constant. The kind of sequences for which we are going to establish the limit theorem are the N × N block versions of the sequences described as follows. Recalling first the definition of the Banach algebra B of smooth functions given in the previous section, let F stand for the set of all sequences (An )∞ n=1 of n × n matrices which are of the from An = Tn (a) + Pn K Pn + Wn L Wn + Cn , where a ∈ B, the operators K and L are trace class operators on 2 , and Cn are n × n matrices tending to zero in the trace norm. The set of such sequences (Cn )∞ n=1 will be denoted by N . The operators Pn and Wn acting on 2 are defined by Pn : {xk }∞ k=0 → {x 0 , x 1 . . . , x n−1 , 0, 0, . . .}, Wn : {xk }∞ k=0 → {x n−1 , x n−2 , . . . , x 0 , 0, 0, . . .}, and Pn K Pn and Wn L Wn are identified with n × n matrices in the natural way. Now we are precisely in the setting considered in [7]. (There is a change of notation, namely, that our B corresponds to S of [7].) The set F is a Banach algebra with algebraic operations defined elementwise and a norm (An )F = aB + K 1 + L1 + sup Cn 1 , n≥1
where · 1 refers to the trace norm. The subset N is a closed two-sided ideal of F.
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E. L. Basor, T. Ehrhardt
Let GB N ×N stand for the group of all invertible elements in the Banach algebra and denote by G1 B N ×N the connected component of GB N ×N containing the identity element. We remark that for a ∈ G1 B N ×N the constant
B N ×N ,
G(a) = exp
1 2π
log det a(ei x ) d x 2π 0
is well-defined [7, Sect. 6]. N ×N which depend In what follows we are going to consider elements (An,t )∞ n=1 ∈ F N ×N analytically on a parameter t ∈ . By this we mean that the map t → (An,t )∞ n=1 ∈ F N ×N is an analytic F -valued function. If ∞ N ×N (An,t )∞ n=1 = (Tn (at ) + Pn K t Pn + Wn L t Wn + C n,t )n=1 ∈ F
(15)
this is equivalent to requiring that the maps t → at , t → K t , t → L t , and t → (Cn,t ) are analytic (because of the definition of the norm in F N ×N ). N ×N , Theorem 3.1. Let be an open subset of C. For each t ∈ let (An,t )∞ n=1 ∈ F ∞ N ×N is analytic. Moreover, suppose and assume that the map t ∈ → (An,t )n=1 ∈ F that at ∈ G1 B N ×N , where at is given by (15). Then for each t ∈ the limit
det An,t n→∞ G(at )n
E t = lim
(16)
exists, the convergence is locally uniform on , and E t depends analytically on t. Proof. By the Vitali-Porter Theorem on induced convergence [5, Chap. 9], it suffices to show that the sequence G(at )−n · det An,t converges pointwise and is locally uniformly bounded on . Because of the assumption at ∈ G1 B N ×N , we have a pointwise representation of the form a t = e b1 · · · e b R ,
b1 , . . . , b R ∈ B N ×N
with R possibly depending on t. A moment’s thought reveals that for each point t0 ∈ one can find a representation of the above kind on a sufficiently small neighborhood (t0 ) of t0 such that b1 , . . . , b R depend analytically on t. In what follows we will consider any fixed, but arbitrary point t0 ∈ and a corresponding (t0 ). For t ∈ (t0 ) consider
Bn,t = e−Tn (b1 ) · · · e−Tn (b R ) Tn (at ) + Pn K t Pn + Wn L t Wn + Cn,t . The determinant of Bn,t equals G(at )−n · det An,t since the trace of Tn (b1 ) + . . . + Tn (b R ) equals n times the zeroth Fourier coefficient of trace(b1 +. . .+b R ) = log det at . Moreover, since (Bn,t ) is built from elements in F N ×N depending analytically on t, it also belongs to F N ×N and depends analytically on t. Proceeding as in [7, Prop. 9.2], one can show that the sequence Bn,t is of the form n,t t Pn + Wn Bn,t = Pn + Pn K L t Wn + C
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model
437
t , n,t )∞ belonging to N N ×N , and with analytic with K L t being trace class, (C n=1 dependence. Applying Lemma 9.3 of [7], it follows that t ) · det(I + lim det Bn,t = det(I + K Lt )
n→∞
pointwise. Since | det(I + A)| ≤ exp A1 for trace class operators A, it follows easily that det Bn,t is locally uniformly bounded.
One might think of giving another simpler proof of the previous theorem by considering the limit of the quotient det An,t = det(Pn + Tn−1 (at )Pn K Pn + Wn Tn−1 (a˜ t )L Wn ). det Tn (at ) In order to make this work one needs the assumption that Tn (at ) is stable (i.e. uniformly invertible as n → ∞). It is known that this equivalent to the operators T (at ) and T (a˜ t ) both being invertible. Unfortunately, the assumption on at made in the theorem only guarantees that these operators are Fredholm with index zero. 4. The Basic Computations for 0 < t < 1 The first step in the calculation of the asymptotics of the correlation function in the dimer model is to show how the block matrix considered by the authors in [8] has a determinant that can be computed using the theorems of the previous section. As mentioned in the introduction, in [8] the authors show the correlation is one-half the square root of the determinant of the matrix Mn given in (1). In Appendix A we show how to first convert the above coefficients as Fourier coefficients of certain functions in the case 0 < t < 1. From this we are able to identify the symbol of the block Toeplitz matrix with N = 2, and obtain that ⎛ ⎞ c d ⎠, det Mn = det Tn (φ) with φ = ⎝ (17) ˜ d c˜ where t cos x + sin2 x , √ (t − e−i x ) t 2 + sin2 x + sin4 x sin x d(ei x ) = √ . 2 t + sin2 x + sin4 x c(ei x ) =
(18) (19)
Recall that a(e ˜ iθ ) = a(e−iθ ). Since we now have a block Toeplitz matrix, our next step is to apply the Szegö Limit Theorem. For 0 < t < 1 the symbol φ is a smooth function. If we compute the determinant of the symbol we find that after simplifying, det φ = cc˜ − d d˜ =
1 , t 2 − 2t cos x + 1
(20)
and thus det φ does not vanish and has winding number zero. In addition the geometric mean G(φ) is one since π π 1 1 log det φ(ei x ) d x = − log((1 − tei x )(1 − te−i x ))d x = 0. 2π −π 2π −π
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Hence the Szegö-Widom Limit Theorem implies that the asymptotics are given by det Tn (φ) ∼ E(φ),
n → ∞,
(21)
and the limit of the monomer-monomer correlation function (5) exists and equals P (mm) (∞) =
1 E(φ). 2
(22)
Note this agrees with the result from [8] that the determinants should approach a constant. Now we wish to apply Corollary 2.4 and Theorem 2.7 to identify the constant E(φ). We write ⎞ ⎛ p q ⎠, φ=σ⎝ q˜ p˜ where p(ei x ) = (t cos x + sin2 x)(t − ei x ),
q(ei x ) = sin x(1 − 2t cos x + t 2 ),
(23)
and σ (ei x ) = (t 2 + sin2 x + sin4 x)−1/2 (1 − 2t cos x + t 2 )−1 .
(24)
The following lemma is needed in order to verify the assumptions of Corollary 2.4. Lemma 4.1. For 0 < t < 1, the matrix function φ can be written as φ = ea I2 +bQ with a, b ∈ B, Q ∈ B 2×2 , trace Q = 0. In particular, ⎛ p− p˜ ⎞ q 2 1 ⎠. a = − log(1 − 2t cos x + t 2 ) + πi, Q=⎝ 2 p− ˜ p q˜ 2 Proof. Clearly, a and Q satisfy the stated conditions. Let us mention that b is given by
1 p + p˜ ( p − p) ˜ 2 b= −a + πi + log σ + log − − , = + q q, ˜ 2 4 and let us first show that b ∈ B (if the logarithm is chosen appropriately). Notice that p + p˜ = (t cos x + sin2 x)(t − cos x), 2 = i sin x (1 − 2t cos x + t 2 )2 + (t cos x + sin2 x)2 . Thus for α(x) = − p+2 p˜ − , we have Im α(x) < 0 for 0 < x < π , Im α(x) > 0 for −π < x < 0, and both α(0) = t (1 − t) and α(π ) = t (1 + t) are real positive. Hence α(x) is a continuous function with winding number zero, which possesses a continuous logarithm which we normalize in such a way that the logarithm is real positive for x = 0 and x = π . With this normalization the numerator in the expression for b becomes zero at x = 0 and x = π , which cancels with the zero in the denominator (as one can easily show). Hence b ∈ B.
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From the definition of b we further conclude that
p + p˜ + . eb = e−a σ 2 We claim that e−b = e−a σ
p + p˜ − . 2
This can be seen most easily by considering the product and noting that ( p + p) ˜ 2 − 2 = (t 2 + sin2 x + sin4 x)(1 − 2t cos x + t 2 ) = e2a σ −2 . 4 It follows that cosh(b) = e−a σ
p + p˜ , 2
It can be verified straightforwardly that
p + p˜ φ=σ I2 + Q , 2
sinh(b) = e−a σ .
Q 2 = −(det Q)I2 = 2 I2 .
Hence ebQ = cosh(b)I2 +
sinh(b) Q.
Combining all this yields ebQ = e−a σ and thus φ = ea I2 +bQ .
p + p˜ I2 + Q 2
In the following computation the factorization of the function t 2 + sin2 x + sin4 x, which appears in the function σ , will play an important role. We make a substituion z = ei x and can rewrite t 2 + sin2 x + sin4 x =
1 4 (z − 8z 2 + (14 + 16t 2 ) − 8z −2 + z −4 ). 16
Making the obvious substitution z = y 2 we can now factor y 2 − 8y + (14 + 16t 2 ) − 8y −1 + y −2 = ξ1−1 ξ2−1 (1 − ξ1 y)(1 − ξ2 y)(1 − ξ1 y −1 )(1 − ξ2 y −1 ), where ξ1 and ξ2 are defined by ξ1 := 2 + µ − 2 1 − t 2 + µ, with
µ :=
ξ2 := 2 − µ − 2 1 − 4t 2 .
1 − t2 − µ
(25)
(26) (27)
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Notice that |ξ1 | < 1 and |ξ2 | < 1. Their inverses are given by ξ1−1 = 2 + µ + 2 1 − t 2 + µ, ξ2−1 = 2 − µ + 2 1 − t 2 − µ.
(28)
If 1/2 < t < 1, then ξ1 and ξ2 are complex conjugate of each other, whereas if 0 < t < 1/2 they are distinct real numbers. Also notice that ξ1 + ξ1−1 = 4 + 2µ,
ξ2 + ξ2−1 = 4 − 2µ.
(29)
Indeed, in order to verify formula (25) it suffices to remark that the right hand side equals (ξ1 + ξ1−1 − y − y −1 )(ξ2 + ξ2−1 − y − y −1 ) = 16 − 4µ2 − 8(y + y −1 ) + (y + y −1 )2 , which is the same as the left hand side. Proposition 4.2. For 0 < t < 1, we have E(φ) = (1 − ξ12 )(1 − ξ22 )(1 − ξ1 ξ2 )2 (1 − t 2 ξ1 )(1 − t 2 ξ2 )E(σ −1 φ), 1 G(σ −1 φ) = . 16ξ1 ξ2
(30) (31)
Proof. First of all notice that, by Lemma 4.1, φ = ea1 I2 +bQ and σ −1 φ = ea2 I2 +bQ with 1 a1 = − log(1 − 2t cos x + t 2 ) + πi, 2
a2 = a1 − log σ.
We apply Corollary 2.4 twice (with bQ instead of Q) and obtain
E(φ) = exp 2 trace(H (a1 )H (a˜ 1 ) − H (a2 )H (a˜ 2 )) E(σ −1 φ). Introducing α1 = log(1 − 2t cos x + t 2 ) and α2 = log(t 2 + sin2 x + sin4 x), we have a1 = −α1 /2 + πi, a2 = a1 + α1 + α2 /2 = (α1 + α2 )/2 + πi (see (24)), and thus
exp 2 trace(H (a1 )H (a˜ 1 ) − H (a2 )H (a˜ 2 )) 1
= exp − trace(H (α1 )H (α˜ 2 ) + H (α2 )H (α˜ 1 ) + H (α2 )H (α˜ 2 )) . 2 With the substitution z = ei x we can write α1 (z) = log((1−t z)(1−t z −1 )), α2 (z) = log((z 4 −8z 2 +(14+16t 2 )−8z −2 + z −4 )/16). Since we can decompose z 4 −8z 2 +(14+16t 2 )−8z −2 +z −4 = ξ1−1 ξ2−1 (1−ξ1 z 2 )(1−ξ2 z 2 )(1−ξ1 z −2 )(1−ξ2 z −2 ), we have the Fourier coefficients (k = 0) |k|
[α1 ]k = −
|k|
ξ + ξ2 t |k| , [α2 ]2k = − 1 , [α2 ]2k+1 = 0. |k| |k|
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We obtain ∞ ξ k + ξ2k traceH (α1 )H (α˜ 2 ) = traceH (α2 )H (α˜ 1 ) = t 2k 1 k k=1
= − log((1−t ξ1 )(1−t ξ2 )) 2
2
and traceH (α2 )H (α˜ 2 ) =
∞ 2(ξ1k + ξ2k )2 = −2 log((1 − ξ12 )(1 − ξ22 )(1 − ξ1 ξ2 )2 ). k k=1
This implies E(φ) = (1 − ξ12 )(1 − ξ22 )(1 − ξ1 ξ2 )2 (1 − t 2 ξ1 )(1 − t 2 ξ2 )E(σ −1 φ). In order to compute G(σ −1 φ) notice that det(σ −1 φ) =
det φ = (t 2 + sin2 x + sin4 x)(1 − 2t cos x + t 2 ) = exp(α2 + α1 ). σ2
The arithmetic mean of α1 is zero while the arithmetic mean of α2 is log(1/(16ξ1 ξ2 )) as can be seen from the factorization.
Clearly, the previous proposition reduces the computation of E(φ) to the computation of E(ψ) where ⎛ ⎞ p q ⎠. ψ := σ −1 φ = ⎝ (32) q˜ p˜ Since the Fourier coefficients ψk vanish if |k| > 3, the constant E(ψ) can be computed from Theorem 2.7 with n = 3: E(ψ) = G(ψ)3 det T3 (ψ −1 ).
(33)
We know in addition G(ψ) = G(σ −1 φ) so all that remains is the computation of the determinant of a block 3 × 3 Toeplitz matrix whose symbol is the inverse of ψ −1 . The next section is devoted to this task because even the determinant of a 3 × 3 block matrix (which is in fact 6 × 6 in size) can be difficult to compute. 5. The Computation of det T3 (ψ −1 ) The symbol ψ −1 is given by ⎛ ψ −1 = η ⎝
p˜
q˜
q
p
⎞
⎛
⎠ =: ⎝
a˜
b˜
b
a
⎞ ⎠,
(34)
where η is the even function η = ( p p˜ − q q) ˜ −1 = (det(σ −1 φ))−1 = (1 − 2t cos x + t 2 )−1 (t 2 + sin2 x + sin4 x)−1 .
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E. L. Basor, T. Ehrhardt
Using the fact that b is an odd function, it follows that the block Toeplitz matrix T3 (ψ −1 ) has the following structure: ⎞ ⎛ a0 0 a1 b1 a2 b2 ⎟ ⎜ ⎜ 0 a0 −b1 a−1 −b2 a−2 ⎟ ⎟ ⎜ ⎟ ⎜ a −b a 0 a b ⎟ ⎜ −1 1 0 1 1 ⎟. T3 (ψ −1 ) = ⎜ ⎜ b a1 0 a0 −b1 a−1 ⎟ ⎟ ⎜ 1 ⎟ ⎜ ⎜a 0 ⎟ ⎠ ⎝ −2 −b2 a−1 −b1 a0 b2 a2 b1 a1 0 a0 What we will next attempt is to show is that the special form of this matrix allows us to reduce its determinant to that of a 2 × 2 scalar matrix. The first thing to notice is that if c1 = det A and c2 = det B then the above matrix has the form ⎞ ⎛ a0 I 2 A B ⎟ ⎜ ⎜ c1 A−1 a0 I 2 A ⎟ ⎠. ⎝ c2 B −1 c1 A−1 a0 I2 Now if we factor out the a0 terms and multiply the top block row from the left by c1 a0−1 A−1 and subtract this from the second block row we are left with ⎛ ⎞ I2 a0−1 A a0−1 B ⎜ ⎟ ⎜ 0 (1 − c1 a0−2 )I2 a0−1 A − c1 a0−2 A−1 B ⎟ ⎝ ⎠. c1 a0−1 A−1 I2 c2 a0−1 B −1 Except for the factor a06 this has the same determinant as the one of interest. Multiplying the top block row on the left by c2 a0−1 B −1 and then subtracting yields in the same manner a determinant equal to that of the matrix ⎛ ⎞ I2 a0−1 A a0−1 B ⎜ ⎟ ⎜0 a0−1 A − c1 a0−2 A−1 B ⎟ (1 − c1 a0−2 )I2 ⎝ ⎠. (1 − c2 a0−2 )I2 0 c1 a0−1 A−1 − c2 a0−2 B −1 A The determinant of the above is given by det((1 − c1 a0−2 )(1 − c2 a0−2 )I2 − (c1 a0−1 A−1 − c2 a0−2 B −1 A)(a0−1 A − c1 a0−2 A−1 B)) or det((1 − 2c1 a0−2 − c2 a0−2 )I2 + c2 a0−3 B −1 A2 + c12 a0−3 A−2 B). Now any 2 × 2 matrix of the form (det M)M −1 + M is a constant times the identity. This is easy to see if one thinks of the inverse of the matrix. In fact if the matrix is given by ⎛ ⎞ m 1 m 12 ⎠, M =⎝ m 21 m 2
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443
then the constant is m 1 + m 2 . The matrix c2 B −1 A2 + c12 A−2 B is of this form and it follows that the matrix given in the last determinant expression is a constant times the identity. Recalling the definition of the terms A and B we see that this constant is 2 a0−3 (a03 − 2c1 a0 − c2 a0 + a−2 (a12 − b12 ) + 2b1 b2 (a1 + a−1 ) + a2 (a−1 − b12 )).
Since we originally had a factor of a06 the a0−3 term cancels (since this comes from a 2 × 2 matrix). Thus, at this point, we have shown that det T3 (ψ −1 ) = 2
(35)
with 2 := a03 − 2c1 a0 − c2 a0 + a−2 (a12 − b12 ) + 2b1 b2 (a1 + a−1 ) + a2 (a−1 − b12 ).
(36)
Now we will investigate the Fourier coefficients. Going back to (23) and (34), the functions a and b are given by b(ei x ) =
sin x , t 2 + sin2 x + sin4 x
a(ei x ) =
(t cos x + sin2 x) . (37) (t − e−i x )(t 2 + sin2 x + sin4 x)
With the substitution z = ei x the function b can be written as 8i(z −1 − z) + (14 + 16t 2 ) − 8z −2 + z −4 8iξ1 ξ2 (z −1 − z) . = 2 (1 − ξ1 z )(1 − ξ2 z 2 )(1 − ξ1 z −2 )(1 − ξ2 z −2 )
b(z) =
z4
− 8z 2
The Fourier coefficients of this can be easily computed and a little algebra shows that b2 = 0 and b1 =
−8iξ1 ξ2 . (1 + ξ1 )(1 + ξ2 )(1 − ξ1 ξ2 )
With b2 = 0 we also can simplify the constant (36) to 2 = a03 − 2(a1 a−1 + b12 )a0 − a2 a−2 a0 + a−2 (a12 − b12 ) + a2 (a−1 − b12 ).
(38)
The function a can be rewritten as a(z) =
8t (z + z −1 ) − 4(z + z −1 )2 . (t − z −1 )(z 4 − 8z 2 + (14 + 16t 2 ) − 8z −2 + z −4 )
Using the factorization employed for finding the coefficients for b it can be shown in a messy, yet elementary, computation that the following holds: Let α :=
4ξ1 ξ2 (1 − ξ1 ξ2 )(ξ1 − ξ2 )
(39)
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E. L. Basor, T. Ehrhardt
and 1 x 2 − 4x − 1 = k(x) := (1 − t 2 x)(1 − x 2 ) 1 − t2x l(x) :=
2 2 − −1 , 1+x 1−x
(1 − 2t 2 )x 2 + (−2 − 2t 2 )x + 1 2 = k(x) + . 2 2 (1 − t x)(1 − x ) 1−x
(40) (41)
Then the Fourier coefficients of a are given by a0 = tα(ξ1 k(ξ1 ) − ξ2 k(ξ2 )), a1 = α(l(ξ1 ) − l(ξ2 )), a−1 = α(ξ1l(ξ1 ) − ξ2 l(ξ2 )), a2 = tα(k(ξ1 ) − k(ξ2 )), a−2 = tα(ξ12 k(ξ1 ) − ξ22 k(ξ2 )). With this notation our constant (38) becomes = α 3 (ξ1 − ξ2 )2 t 3 ξ1 k(ξ1 )2 k(ξ2 ) − t 3 ξ2 k(ξ2 )2 k(ξ1 ) + tk(ξ1 )l(ξ2 )2 − tk(ξ2 )l(ξ1 )2
+ 4(1+ξ1 )−2 (1+ξ2 )−2 (tk(ξ1 )(ξ1 +1)2 − tk(ξ2 )(ξ2 +1)2 ) . Simplifying this expression is the last remaining task. For simplicity we will assume that 1/2 < t < 1. In this case the roots ξ and ξ2 are conjugate and µ is complex. The case 0 < t < 1/2 is similar, but we omit the details. The answer however is valid for all 0 < t < 1 because of analyticity of the expressions with respect to t. We begin by proving that the expression t 3 ξ1 k(ξ1 )2 k(ξ2 ) − t 3 ξ2 k(ξ2 )2 k(ξ1 ) + tk(ξ1 )l(ξ2 )2 − tk(ξ2 )l(ξ1 )2 +
(42)
4tk(ξ1 ) 4tk(ξ2 ) − 2 (1 + ξ2 ) (1 + ξ1 )2
is equal to 8µ √ , (1 + ξ1 )(1 + ξ2 ) ω
(43)
ω := (1 − t 2 ξ1 )(1 − t 2 ξ2 ) > 0
(44)
where First recall the identities ξ1 + ξ1−1 = 4 + 2µ,
ξ2 + ξ2−1 = 4 − 2µ,
µ2 = 1 − 4t 2
(45)
from which it is easy to conclude that (ξ1 − 1)(ξ1−1 − 1)(ξ2 − 1)(ξ2−1 − 1) = 16t 2 .
(46)
In order to evaluate the expression in (42) we first simplify t 3 ξ1 k(ξ1 )2 k(ξ2 ) − t 3 ξ2 k(ξ2 )2 k(ξ1 ) + tk(ξ1 )l(ξ2 )2 − tk(ξ2 )l(ξ1 )2 .
(47)
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445
Using (40) and (41) the expression t 3 ξ1 k(ξ1 )2 − tl(ξ1 )2 becomes
t 3 ξ1 k(ξ1 )2 − t k(ξ1 ) +
2 1 − ξ1
2 = t 3 ξ1 k(ξ1 )2 − tk(ξ1 )2 −
4tk(ξ1 ) 4t − . 1 − ξ1 (1 − ξ1 )2
A little more algebra shows that this is the same as
2 2 4t − +1 − . tk(ξ1 ) − 1 + ξ1 1 − ξ1 (1 − ξ1 )2 Since we have only used the identity for l(x) in terms of k(x) the above is also true if we replace ξ1 with ξ2 . Using this last above equation (for both ξ1 and ξ2 terms) we have that (47) is given by
4tk(ξ2 ) 2 2 2 2 4tk(ξ1 ) − − + + + , tk(ξ1 )k(ξ2 ) − 2 1 + ξ1 1 − ξ1 1 + ξ2 1 − ξ2 (1 − ξ1 ) (1 − ξ2 )2 which is of course
4tk(ξ1 )k(ξ2 )
ξ22 − ξ12 (1 − ξ12 )(1 − ξ22 )
−
4tk(ξ1 ) 4tk(ξ2 ) + . (1 − ξ1 )2 (1 − ξ2 )2
(48)
We now proceed to simplify the above. From (45) it is easy to show that (1 − µξ1 )2 = 4ξ1 (1 − t 2 ξ1 ),
(1 + µξ2 )2 = 4ξ2 (1 − t 2 ξ2 ),
which using (46) implies ω=
(1 − µξ1 )2 (1 + µξ2 )2 (1 − µξ1 )2 (1 + µξ2 )2 (ξ1 − 1)2 (ξ2 − 1)2 = . 16ξ1 ξ2 (16ξ1 ξ2 t)2
(49)
Moreover, using 2(1 − µξ1 ) = 1 − ξ12 + 4ξ1 and 2(1 + µξ2 ) = 1 − ξ22 + 4ξ2 (which also follows from (45)) we conclude k(ξ1 ) =
−8ξ1 , (1 − µξ1 )(1 − ξ12 )
k(ξ2 ) =
−8ξ2 . (1 + µξ2 )(1 − ξ22 )
With this we have that k(ξ1 )k(ξ2 ) =
64ξ1 ξ2 . (1 − µξ1 )(1 + µξ2 )(1 − ξ12 )(1 − ξ22 )
Using (49) it then follows that 4 k(ξ1 )k(ξ2 ) = √ . t ω(1 + ξ1 )(1 + ξ2 ) Hence the term (47) (which is also (48)) is √
16(ξ22 − ξ12 ) ω(1 + ξ1 )(1 + ξ2 )(1 − ξ12 )(1 − ξ22 )
−
4tk(ξ1 ) 4tk(ξ2 ) + , (1 − ξ1 )2 (1 − ξ2 )2
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and the expression for the constant (42) is the above plus the additional terms of 4tk(ξ1 ) 4tk(ξ2 ) − . 2 (1 + ξ2 ) (1 + ξ1 )2 In other words we still need to evaluate √
16(ξ22 − ξ12 ) ω(1 + ξ1 )(1 + ξ2 )(1 − ξ12 )(1 − ξ22 )
4tk(ξ2 ) 4tk(ξ1 ) 4tk(ξ1 ) 4tk(ξ2 ) − + + − . 2 2 2 (1 − ξ1 ) (1 − ξ2 ) (1 + ξ2 ) (1 + ξ1 )2
(50)
If we combine the terms 4tk(ξ1 ) 4tk(ξ1 ) + (1 − ξ2 )2 (1 + ξ2 )2 we have 8tk(ξ1 )(1 + ξ22 ) (1 − ξ2 )2 (1 + ξ2 )2 which becomes after using the identity for k(ξ1 ) −64tξ1 (1 + ξ22 ) (1 − µξ1 )(1 − ξ12 )(1 − ξ2 )2 (1 + ξ2 )2
.
Using (49) and (45) this becomes −4(ξ2−1 + ξ2 )(1 + µξ2 ) −8(2 − µ)(1 + µξ2 ) =√ √ ω(1 + ξ1 )(1 − ξ2 )(1 + ξ2 )2 ω(1 + ξ1 )(1 + ξ2 )(1 − ξ22 ) or −16 + 4µ(2 − 4ξ2 + 4µξ2 ) √ ω(1 + ξ1 )(1 + ξ2 )(1 − ξ22 ) and finally −16 4µ . +√ √ ω(1 + ξ1 )(1 + ξ2 ) ω(1 + ξ1 )(1 + ξ2 )(1 − ξ22 ) We still have another term the same as the above but with the ξ1 and ξ2 switched, µ replaced by −µ and an additional sign change. However when we add this new term to the above and then substitute in (50) we have that our constant (42) is √ +√
16(ξ22 − ξ12 ) ω(1 + ξ1 )(1 + ξ2 )(1 − ξ12 )(1 − ξ22 )
−16 16 +√ 2 ω(1 + ξ1 )(1 + ξ2 )(1 − ξ2 ) ω(1 + ξ1 )(1 + ξ2 )(1 − ξ12 ) +√
8µ ω(1 + ξ1 )(1 + ξ2 )
(51)
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447
which nicely reduces to 8µ √ ω(1 + ξ1 )(1 + ξ2 ) as promised. Thus the constant (38) is 8µα 3 (ξ1 − ξ2 )2 4ξ1 ξ2 . = √ with α = (1 − ξ1 ξ2 )(ξ1 − ξ2 ) ω(1 + ξ1 )(1 + ξ2 )
(52)
Now we can put the final pieces together. Theorem 5.1. For 0 < t < 1, we have E(φ) =
t 2t (2 + t 2 ) + (1 + 2t 2 )
. √ 2 + t2
(53)
Proof. We have from Proposition 3.2 and formulas (32), (33), (35), and (52) that E(φ) = (1 − ξ12 )(1 − ξ22 )(1 − ξ1 ξ2 )2 (1 − t 2 ξ1 )(1 − t 2 ξ2 )(16ξ1 ξ2 )−3
2 8µα 3 (ξ1 − ξ2 )2 × √ . ω(1 + ξ1 )(1 + ξ2 ) This can be reduced to 45 t 2 µ2 ξ14 ξ24 (1 − ξ12 )(1 − ξ22 )(1 − ξ1 ξ2 )4 (ξ1 − ξ2 )2
.
This uses only elementary algebra and formula (46). To evaluate the remaining terms we note (45) implies that (1 − ξ1 ξ2 )(ξ1 − ξ2 ) = −ξ1 + ξ2 − ξ1−1 + ξ2−1 = −4µ ξ1 ξ2 and (1 + ξ1 )(1 + ξ1−1 )(1 + ξ2 )(1 + ξ2−1 ) = (6 − 2µ)(2 + 2µ) = 16(2 + t 2 ). From the last equation together with (47) we can conclude that (1 − ξ12 )(1 − ξ22 ) = 16 t 2 (2 + t 2 )ξ1 ξ2 . Thus from the line above and (54) it follows that E(φ) = √
4tξ1 ξ2 2 + t 2 (1 − ξ1 ξ2 )2
.
To complete the computation notice again (45), hence (ξ1 + ξ1−1 )(ξ2 + ξ2−1 ) = (4 + 2µ)(4 − 2µ) = 12 + 16t 2 . Also we know
(ξ1−1 − ξ1 )(ξ2−1 − ξ2 ) = 16 t 2 (2 + t 2 ),
(54)
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so we can check that ξ1 ξ2 + 1/ξ1 ξ2 = 6 + 8t 2 + 8 t 2 (2 + t 2 ). This yields that ξ1 ξ2 1 . = 2 2 (1 − ξ1 ξ2 ) 4 + 8t + 8 t 2 (2 + t 2 ) From this the theorem follows.
6. The Asymptotics for More General Parameters In the previous sections we have shown that det Mn ∼ E(φ),
n → ∞,
(55)
if 0 < t < 1 and we have identified the constant E(φ). The goal of this section is to show that (55) holds for all values of parameters t belonging to the set := t ∈ C : Re (t) > 0 under the assumption that the entries of Mn are continued by analyticity in t onto . Indeed, if we recall the definition of the entries of Mn in terms of Rk and Q k (as stated in the introduction) one only needs to observe that the denominator of the fraction under the integrals is nonzero as long as t 2 is not real non-positive number, i.e., if Re (t) = 0. Since we have related Mn to Tn (φ) by simple row and column operations (see (2) or (17)) we are going to elaborate on Tn (φ), the entries of which also depend analytically on t. Recall that φ is given by ⎛ ⎞ ct dt ⎠ φ=⎝ d˜t c˜t with t cos x + sin2 x , √ (e−i x − t) t 2 + sin2 x + sin4 x sin x dt (ei x ) = √ . 2 t + sin2 x + sin4 x ct (ei x ) =
Moreover, we introduce the function et+ (ei x ) =
(e−i x
t cos x + sin2 x 1 − −i x . √ 2 4 2 e −t − t) t + sin x + sin x
Proposition 6.1. For each t ∈ the functions dt and et+ belong to B, and the dependence on t is analytic.
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model
449
Proof. The statement concerning the function t is easy to see. Indeed, the function sin2 x + sin4 x belongs to B and has its spectrum equal to [0, 2]. By functional calculus we can √ define√the reciprocal of the square-root of t 2 + sin2 x + sin4 x whenever t ∈ C \ [−i 2, i t] and also at t = ∞. This domain (extended with the point at infinity) is simply connected and we can choose an analytic branch. Clearly, we choose the branch in such a way that the square-root is positive for real, positive t. In regard to the function et the assertion is obvious except for |t| = 1. We rewrite this function as √ 2 x) − t 2 + sin2 x + sin4 x (t cos x + sin et (ei x ) = . √ (e−i x − t) t 2 + sin2 x + sin4 x We are going to show that term e−i x − t cancels with a term in the numerator for |t| = 1, Re (t) > 0, and that the resulting function belongs to B and depends analytically on t. To see this rewrite the numerator (with the substitution z = ei x ) as a quarter times n(t, z) = 2t (z + z −1 ) − (z − z −1 )2 −
z 4 − 8z 2 + 14 + 16t 2 − 8z −2 + z 4 .
Allow, for a moment, z to take complex values in a neighborhood of the unit circle. More specifically, it is easy to see that for each ε > 0 there exists a δ > 0 such that the function under the square-root is nonzero (and of course analytic in both z and t) whenever Re (t) > ε, |t| < 1 + ε, 1 − δ < |z| < 1 + δ. Denote the set of all (t, z) satisfying these conditions by Uε . Hence n(t, z) is analytic on Uε . Consider the subset Mε = {(t, z) ∈ Uε : z = 1/t}. Then n(t, t −1 ) = (t 2 − t −2 + 4) −
(t 2 − t −2 + 4)2 .
For t = 1, keeping in mind the proper choice of the branch of the square-root, this equals zero. Hence by analytic continuation and since Mε is connected and the function under the square-root does not vanish on Mε , it follows that n(t, z) = 0 for all (t, z) ∈ Mε . Hence n(t, z) = (z −1 − t)m(t, z) with some function m(t, z) which is analytic on Uε . From this the assertion follows.
Since the function dt depends analytically on t ∈ , the entries of the matrix Tn (dt ) are also analytic in t. In regard to the matrix Tn (ct ) we decompose for 0 < t < 1, + Tn (ct ) = Tn (et+ ) + K n,t ,
(56)
+ is the n × n Toeplitz matrix with entries where K n,t
+ K n,t
jk
=
0 if j ≤ k t j−k−1 if j > k.
This decomposition holds since K n,t is the Toeplitz matrix with the generating function equal to (e−i x − t)−1 for |t| < 1. The right hand side of (56) is well-defined and analytic for t ∈ , and hence we can define + Bn,t := Tn (et+ ) + K n,t ,
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E. L. Basor, T. Ehrhardt
which is the analytic continuation of Tn (ct ) onto . From this it is easy to see that the analytic continuation of Tn (φ) onto t ∈ is the matrix ⎛ ⎞ Bn,t Tn (dt ) ⎠, Bˆ n,t = ⎝ T Tn (dt )T Bn,t where A T stands for the transpose of A. Theorem 6.2. For each t ∈ , we have lim det Bˆ n,t = E t with E t =
n→∞
t 2t (2 + t 2 ) + (1 + 2t 2 )
√
2 + t2
.
(57)
Moreover, the convergence is locally uniform on . Proof. We make the decomposition φ = φ+ + φ0 with ⎛ + ⎛ −i x ⎞ et dt − t)−1 (e ix ⎝ ⎝ ⎠ φ+ = , φ0 (e ) = d˜t e˜t+ 0 In view of the definition of Bn,t we obtain Bˆ n,t = Tn (φ+ ) + Kˆ n+ , Now let
⎛ + (ei x ) = ⎝
⎛
Kˆ n+ := ⎝
(ei x
− t)−1
⎠.
⎞
+ K n,t
0
0
+ )T (K n,t
⎠.
⎞
1 − tei x
0
0
1 − te−i x
A straightforward computation shows that
⎞
0
⎛
Tn (+ ) Kˆ n+ = Tn (+ φ0 ), (+ φ0 )(ei x ) = ⎝
⎠.
ei x
0
0
e−i x
⎞ ⎠.
Moreover, + )Pn − Wn H ( + )H (φ+ )Wn . Tn (+ )Tn (φ+ ) = Tn (+ φ+ ) − Pn H (+ )H (φ Adding the above two equations, taking the determinant and noting that det Tn (+ ) = 1, it follows that
:= + φ+ + + φ0 = + φ. (58) ) + Pn K Pn + Wn L Wn , φ det Bˆ n,t = det Tn (φ belongs to B 2×2 and depends analytically on t ∈ . Moreover, det φ = 1, The function φ ∈ GB 2×2 for each t ∈ . For 0 < t < 1, Lemma 4.1 implies that φ ∈ G1 B 2×2 , hence φ and since is connected this holds for each t ∈ . The operators K and L are trace class operators (in fact, rank one operators), which also depend analytically on t ∈ . Hence all the assumptions of Theorem 3.1 are fulfilled, and applying it yields the desired assertion.
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model
451
Appendix A: The Basic Identity As mentioned in the introduction, the block matrix considered by the authors in [8] is of the form ⎛ ⎞ R Q ⎠, Mn = ⎝ (59) Q R where R andQ are n × n matrices with entries given by R jk = 2(−1)[(k− j)/2] Rk− j+1 + θ ( j − k)t j−k−1 , Q jk = 2i(−1)[( j+k)/2] Q n+1− j−k (1 ≤ j, k ≤ n), and the expressions Rk , Q k , and θ (k) are defined by the following: For even k, π π cos y cos(kx + y) 1 d yd x, Rk = 2 2 8π −π −π cos x + cos2 y + t 2 cos2 (x + y) and Q k = 0. For odd k,
1 Qk = 8π 2
π
−π
π
−π
cos x cos kx d yd x, cos2 x + cos2 y + t 2 cos2 (x + y)
and Rk =
t 8π 2
π −π
π
−π
cos2
cos(x + y) cos(kx + y) d yd x. x + cos2 y + t 2 cos2 (x + y)
The expression θ (k) equals 1 for k > 0 and 0 otherwise. We begin with the following lemma which describes the above coefficients as Fourier coefficients of certain functions. This is our starting place, since once this is done, we use some simple matrix algebra to express the determinant of the above matrix as the determinant of a block Toeplitz matrix. Throughout what follows we will assume that 0 < t < 1 because the second part of the matrix R becomes unbounded in the limit n → ∞ in the case t = 1. Lemma 6.3. Consider the functions t S(e ) = 4π
ix
1 T (e ) = − 4π ix
π −π
cos(x + y − π/2)ei(x+y−π/2) dy, cos2 (x − π/2) + cos2 y + t 2 cos2 (x + y − π/2)
π −π
cos yei(x+y) dy. cos2 (x − π/2) + cos2 y + t 2 cos2 (x + y − π/2) k
These functions have Fourier coefficients, Sk and Tk , satisfying Sk +Tk = (−1)[− 2 ] R−k+1 , where Rk is as defined above.
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E. L. Basor, T. Ehrhardt
Proof. Replacing x by x + π in the integral defining S shows that S(ei(x+π ) ) = S(ei x ) and hence that Sk = 0 for odd k. For k even we have π π cos(x + y − π/2)ei(x+y−π/2) e−ikx t d yd x. Sk = 8π 2 −π −π cos2 (x − π/2) + cos2 y + t 2 cos2 (x + y − π/2) If we replace x with x + π/2 then this integral becomes
k
t (−1) 2 8π 2 k
t (−1) 2 = 8π 2
π
π
−π
−π
π
−π
π
−π
cos(x + y)ei(y−(k−1)x) d yd x cos2 x + cos2 y + t 2 cos2 (x + y) cos(x + y) cos(y − (k − 1)x) d yd x. cos2 x + cos2 y + t 2 cos2 (x + y)
The last equality follows since the inner integral as a function of x is even and this k is exactly (−1)− 2 R−k+1 for even k. If we replace x by x + π in the definition of T , that shows that the even coefficients vanish. A very similar computation to the above k shows that for odd k, Tk = (−1)[− 2 ] R−k+1 and thus combining the two cases we have k
Sk + Tk = (−1)[− 2 ] R−k+1 for all k. Lemma 6.4. For 0 < t < 1, the matrix R is an n × n Toeplitz matrix with symbol 2(S + T ) + U , where S and T are defined in Lemma 1 and U (ei x ) = e−i1x −t . Proof. This follows directly from Lemma 1 and the fact that the Fourier coefficients of 1 are t k−1 θ (k).
e−i x −t Lemma 6.5. Define n
V (ei x ) = (−1)[ 2 ]+1
1 4π
π
−π
cos(x − π/2) dy. cos2 (x − π/2) + cos2 y + t 2 cos2 (x + y − π/2)
Then Vk is zero for k even and for k odd we have Vk = i(−1)[
n+1−k 2 ]
Qk .
Proof. To see that Vk is zero for k even, replace x by x + π as before. For the remaining equation we have π π n 1 cos(x − π/2)e−ikx d yd x. Vk = (−1)[ 2 ]+1 2 8π −π −π cos2 (x − π/2) + cos2 y + t 2 cos2 (x + y − π/2) Making the substitution x + π/2 for x and simplifying yields the result.
Our next step is to take the functions given by these integal representations and simplify them. To this end, notice that the denominator in all three integrals, cos2 (x − π/2) + cos2 y + t 2 cos2 (x + y − π/2) can be written as sin2 x + cos2 y + t 2 sin2 (x + y)
Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model
453
which is in turn 1 ((2 + t 2 − cos 2x) + (1 − t 2 cos 2x) cos 2y + t 2 sin 2x sin 2y). 2 If we combine the integrals for S and T and use the obvious trigonometric identities we see that the numerator is of the form 1 ((t − ei x ) − (te2i x + ei x ) cos 2y − i(te2i x + ei x ) sin 2y). 2 Thus combining S and T we see that we have an integral of the form 1 4π
2π 0
D + E cos 2y + F sin 2y dy, A + B cos 2y + C sin 2y
where the coefficients depend on x and t and are given by the above two equations for the numerator and denominator. It is fairly straightforward to evaluate such an integral. First we replace 2y by y which results in no change because of periodicity. Second we define cos φ = √
B B2 + C 2
,
sin φ = √
C B2 + C 2
and change variables using y = z + φ. At this point we have an integral of the form 1 4π
2π
0
D + G cos z + H sin z dz. A + K cos z
The contribution from the sin-term is zero and the rest can be easily evaluated since it reduces to one of the form
2π 0
dz 2π , =√ 1 + a cos z 1 − a2
providing that |a| < 1. Using this equation we can compute the integral for S + T . In the same way we are also able to evaluate the integral for the function V . This computation is even more straightforward and we leave the details to the reader. Notice that the denominator in the defining integral is the same as for S and T and the numerator only depends on x. Lemma 6.6. We have t cos x + sin2 x 1 + , √ −i x 2(t − e−i x ) t 2 + sin2 x + sin4 x 2(t − e ) (−1)[n/2]+1 sin x . V = √ 2 t 2 + sin2 x + sin4 x
S+T =−
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E. L. Basor, T. Ehrhardt
Hence the generating function for the Toeplitz matrix R is given by 2(S + T ) + U =
(e−i x
t cos x + sin2 x . √ − t) t 2 + sin2 x + sin4 x
Notice finally that the ( j, k)-entry (1 ≤ j, k ≤ n) of the Q matrix is given by twice the n + 1 − j − k Fourier coefficient of V . Now that we have identified the entries in each of the blocks of the matrix (59) as Fourier coefficients of certain functions, it is fairly easy to see how the determinant of (59) can be computed as the determinant of a block Toeplitz matrix. As a preliminary step, we define an operator Wn on the finite-dimensional complex vector space of dimension n by Wn (a1 , . . . , an ) = (an , . . . , a1 ), and denote the identity operator on the same space by In . Multiplication of a matrix by Wn on the right results in changing the ( j, k)-entry into the ( j, n + 1 − k)-entry and on the left changes the entry to the (n + 1 − j, k)-entry. Also note that det Wn2 = 1. Hence the block matrix product ⎛ ⎞⎛ ⎞⎛ ⎞ In 0 0 R Q In ⎝ ⎠⎝ ⎠⎝ ⎠ 0 Wn 0 Wn Q R has the same determinant as the one given in formula (59) and thus we can replace it with the product. The advantage to this is that since ⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ In 0 0 R Q In R QWn ⎝ ⎠⎝ ⎠⎝ ⎠=⎝ ⎠ 0 Wn 0 Wn Q R Wn Q Wn RWn the ( j, k)-entry of QWn is now the k − j Fourier coeffcient of the function V and the ( j, k)-entry of Wn Q is j − k, and thus the upper-right and lower-left matrices are now Toeplitz in structure. The matrix Wn P Wn still remains Toeplitz, but with symbol (2(S + T ) + U )(ei x ) replaced by (2(S + T ) + U )(e−i x ). The end result of this is that we have shown that (for 0 < t < 1) ⎛ ⎞ c d ⎠, det Mn = det Tn (φ) with φ = ⎝ (60) d˜ c˜ where t cos x + sin2 x , √ (e−i x − t) t 2 + sin2 x + sin4 x sin x d(ei x ) = √ , 2 t + sin2 x + sin4 x c(ei x ) =
˜ i x ) = d(e−i x ). Notice that dropping the sign when passing from c(e ˜ i x ) = c(e−i x ), d(e V to d does not influence the value of the determinant. Acknowledgement. The authors would like to thank Paul Fendley for introducing the topics of this paper to us and also Jim Delany who helped us with mathematica computations in our early investigations.
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References 1. Basor, E.L., Widom, H.: On a Toeplitz determinant identity of Borodin and Okounkov. Int. Eqs. Operator Th. 37(4), 397–401 (2000) 2. Borodin, A., Okounkov, A.: A Fredholm determinant formula for Toeplitz determinants. Int. Eqs. Operator Th. 37, 386–396 (2000) 3. Böttcher, A.: One more proof of the Borodin-Okounkov formula for Toeplitz determinants. Int. Eqs. Operator Th. 41(1), 123–125 (2001) 4. Böttcher, A., Silbermann, B.: Analysis of Toeplitz operators: 2nd ed. Berlin, Springer (2006) 5. Burckel, R.B.: An introduction to classical complex analysis. Basel, Birkhäuser (1979) 6. Ehrhardt, T.: A generalization of Pincus’ formula and Toeplitz operator determinants. Arch. Math. 80, 302– 309 (2003) 7. Ehrhardt, T.: A new algebraic proof of the Szegö-Widom limit theorem. Acta Math. Hungar. 99(3), 233– 261 (2003) 8. Fendley, P., Moessner, R., Sondhi, S.L.: Classical dimers on the triangular lattice. Phys. Rev. B 66, 214513 (2002) 9. Fisher, M.E., Stephenson, J.: Statistical Mechanics of Dimers on a Plane Lattice. II. Dimer Correlations and Monomers. Phys. Rev. 132, 1411 (1963) 10. Geronimo, J.S., Case, K.M.: Scattering theory and polynomials orthogonal on the unit circle. J. Math. Phys. 20, 299–310 (1979) 11. Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Transl. Math. Monographs, Vol. 18, Providence. In: Amer. Math. Soc., (1969) 12. Kasteleyn, P.W.: Dimer statistics and phase transitions. J. Math. Phys. 4, 287 (1963) 13. Kivelson, S.A., Rokhsar, D.S., Sethna, J.P.: Topology of the resonating valence-bond state: Solitons and high-Tc superconductivity. Phys. Rev. B35, 8865 (1987) 14. McCoy, B.M., Wu, T.T.: The two-dimensional Ising model. Cambridge, MA: Harvard Univ. Press (1973) 15. Moessner, R., Sondhi, S.L.: Resonating valence bond phase and the triangular lattice quantum dimer model. Phys. Rev. Lett. 86, 1881 (2001) 16. Moessner, R., Sondhi, S.L.: Ising and dimer models in two and three dimensions. Phys. Rev. B 68, 054405 (2003) 17. Moessner, R., Sondhi, S.L.: Three-dimensional resonating-valence-bond liquids and their excitations. Phys. Rev. B 68, 184512 (2003) 18. Moessner, R., Sondhi, S.L., Fradkin, E.: Short-ranged resonating valence bond physics, quantum dimer models, and Ising gauge theories. Phys. Rev. B 65, 024504 (2002) 19. Rokhsar, D.S., Kivelson, S.A.: Superconductivity and the quantum hard-core dimer gas. Phys. Rev. Lett. 61, 2376 (1988) 20. Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants. Adv. in Math. 13(3), 284–322 (1974) 21. Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants. II. Adv. in Math. 21(1), 1–29 (1976) Communicated by H. Spohn
Commun. Math. Phys. 274, 457–486 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0200-z
Communications in
Mathematical Physics
An Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes, and Bohr’s Frequency Condition Volker Bach1 , Jürg Fröhlich2,3 , Alessandro Pizzo2 1 FB Mathematik, Universität Mainz, D-55099 Mainz, Germany.
E-mail: [email protected]
2 Inst. f. Theoretische Physik, ETH Hönggerberg, CH-8093 Zürich, Switzerland.
E-mail: [email protected]
3 IHES, Bures-sur-Yvette, France.
E-mail: [email protected] Received: 3 August 2006 / Accepted: 14 September 2006 Published online: 18 April 2007 – © Springer-Verlag 2007
Abstract: In this paper, we rigorously justify Bohr’s frequency condition in atomic spectroscopy. Moreover, we construct an algorithm enabling us to calculate the transition amplitudes for Rayleigh scattering of light at an atom, up to a remainder term of arbitrarily high order in the finestructure constant. Our algorithm is constructive and circumvents the infrared divergences that invalidate standard perturbation theory. I. Description of the Problem and Summary of Main Results In this paper, we present a mathematical justification of Bohr’s frequency condition in atomic spectroscopy. Since the physical value, α ∼ = 1/137, of the finestructure constant is very small, it suffices to expand scattering amplitudes for Rayleigh scattering of light at an atom to leading order in α in order to reach a precise understanding of Bohr’s frequency condition. We accomplish more than that: We provide a constructive algorithm enabling us to calculate the scattering amplitudes up to (finite) remainder terms of arbitrarily high order in the finestructure constant. For two reasons, this is a nontrivial result. The rate of convergence of the interpolating electromagnetic field to the asymptotic field crucially enters the control of the Duhamel expansion of propagators appearing in reduction formulae for the scattering matrix elements. It is therefore not obvious, a priori, that one can construct an algorithm to determine these matrix elements to arbitrarily high order in α. Infrared divergences invalidate a straightforward Taylor expansion of the groundstate and the groundstate energy. Since the scattering amplitudes for Rayleigh scattering depend on the atomic groundstate and the groundstate energy, it is thus far from obvious how to calculate these amplitudes to arbitrarily high order in α. The convergence to the asymptotic field is sufficiently rapid (faster than any inverse power of time t) to allow for a complete control of the expansion of the propagator in powers of α, provided some technical subtleties, connected to the vector nature of the interaction in QED, are properly taken into account. The expansion of the groundstate is a more delicate issue. In fact, we require an iterative construction (see [1, 2]), based
458
V. Bach, J. Fröhlich, A. Pizzo
on a multiscale analysis, to remove an infrared cut-off in photon momentum space in our construction of the atomic groundstate. As a result of our analysis, we have a mathematical tool to calculate contributions to the scattering amplitudes up to finite remainder terms of arbitrarily high order in α. Because of the infrared features of the theory, naive perturbation theory is infrared divergent at some finite order in α. But if the finestructure constant were not as small as it is in nature experimental data could only be reproduced accurately by the theory if radiative corrections of very high order in α were taken into account. We therefore construct an algorithm to calculate such corrections. In the following, an atom is described as a quantum-mechanical bound state consisting of a static, positively charged, pointlike nucleus surrounded by electrons. The electrons are described as nonrelativistic, pointlike quantum-mechanical particles with electric charge −e and spin 21 , as originally proposed by Pauli. They are bound to the nucleus by the electrostatic Coulomb force, and they interact with the transverse soft modes of the quantized electromagnetic field. We eliminate ultraviolet divergences by imposing an ultraviolet cutoff on the interaction term. To keep our exposition as simple as possible, we consider a hydrogen atom consisting of a single, static proton of charge e accompanied by only one electron. The spin of the electron then turns out to be an inessential complication. We neglect the coupling of the magnetic moment of the electron to the quantized magnetic field. It is, however, not difficult to include the Zeeman term in our analysis. Throughout our paper we follow the notation and conventions of [1]. Next, we recall the mathematical definition of our model system. The Hilbert space of pure state vectors is given by H := Hel ⊗ F,
(I.1)
where Hel = L 2 (R3 ) is the Hilbert space appropriate to describe states of a single electron (neglecting its spin), and F is the Fock space used to describe the states of the transverse modes of the quantized electromagnetic field, i.e., the photons. More explicitly, F :=
∞
F (N ) ,
F (0) = C ,
(I.2)
N =0
where is the vacuum vector (i.e., the state of the electromagnetic field without any excited field modes), and F (N ) := S N
N
h,
N ≥ 1,
(I.3)
j=1
where the Hilbert space, h, of state vectors of a single photon is given by h := L 2 [R3 × Z2 ].
(I.4)
In (I.4), R3 is the photon momentum space, and Z2 accounts for the two independent transverse polarizations, or helicities, of a photon. In Eq. (I.3), S N denotes the orthogonal projection onto the subspace of Nj=1 h of totally symmetric N -photon wave functions, in accordance with the fact that photons satisfy Bose-Einstein statistics. Thus, F (N ) is
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
459
the subspace of F of state vectors of configurations of exactly N photons. It is convenient to represent the Hilbert space H as the space of square-integrable wave functions on electron position space, R3 , with values in the photon Fock space F, i.e., H ∼ (I.5) = L 2 R3 ; F . The dynamics of the system is generated by the Hamiltonian H :=
x + α 3/2 A (α x) − i∇
2
1 − V ( x ) + Hˇ , V ( x ) := . | x|
(I.6)
x denotes the gradient with respect to the electron position variable x ∈ R3 , Here, ∇ ∼ x ) denotes the vector potential of the α = 1/137 is the finestructure constant, A ( transverse modes of the quantized electromagnetic field in the Coulomb gauge, x · A ( x ) = 0, ∇
(I.7)
and with an ultraviolet cutoff imposed on the high-frequency modes, V is the Coulomb potential of electrostatic attraction of the electron to the nucleus. A general class of potentials, V , for which our analysis can be carried out, is characterized in the following hypothesis. We define the atomic Hamiltonian, Hel , by Hel := −x − V ( x ),
(I.8)
where x is the Laplacian. Hypothesis 1. The form domain of V includes the form domain, H1 (R3 ), of the Laplacian −, and, for any ε > 0, there exists a constant bε < ∞, such that ±V ≤ ε (−) + bε · 1
(I.9)
on H1 (R3 ). Moreover, lim|x |→∞ V ( x ) = 0, and eel := inf σ (Hel ) < 0 is an isolated eigenvalue of multiplicity one, with corresponding normalized eigenvector ϕel ∈ Hel .
(I.10)
In Eq. (I.6), Hˇ is the Hamiltonian of the quantized, free electromagnetic field. This operator is given by λ) |k| a(k, λ), Hˇ := (I.11) d 3 k a ∗ (k, λ=±
λ) and a(k, λ) are the usual photon creation- and annihilation operators where a ∗ (k, obeying the canonical commutation relations λ) , a ∗ (k , λ )] = [a(k, λ) , a(k , λ )] = 0, [a ∗ (k, ∗ λ) , a (k , λ )] = δλλ δ(k − k ), [a(k, λ) = 0, a(k, k ∈ R3 and λ, λ ∈ Z2 ≡ {±}. for all k,
(I.12) (I.13) (I.14)
460
V. Bach, J. Fröhlich, A. Pizzo
The regularized vector potential in the Coulomb gauge is given by x ) := A ( d 3k 1 x ∗ x λ)∗ ei k· ε(k, λ)e−i k·
a (k, λ) + ε(k, a(k, λ) , (I.15) (k) 3/2 (2π ) λ=± 2 |k| ≤ κ} (or a nonnegative, is the characteristic function of the ball {k ∈ R3 | |k| where (k) +), ε(k, −) are photon polarization vectors, i.e., smooth approximation thereof), and ε(k, two unit vectors in C ⊗ R3 satisfying µ) = δλµ , k · ε(k, λ) = 0, λ)∗ · ε(k, ε(k,
(I.16)
λ) = 0 expresses the Coulomb gauge condition. for λ, µ = ±. The equation k · ε(k, The function (k) ensures that modes of the electromagnetic field corresponding to ≥ κ do not interact with the electron; i.e., is an ultraviolet wave vectors k with |k| cutoff that will be kept fixed throughout our analysis. Next, we recall some well known properties of the Hamiltonian H and of its spectrum used in the following sections. For sufficiently small values of α, the Hamiltonian H is selfadjoint on its domain, D(H ) = D(H0 ), where D(H0 ) is the domain of the selfadjoint operator H0 := −x − V ( x ) + Hˇ .
(I.17)
The operator H is bounded from below, and the infimum of the spectrum is a non-degenerate eigenvalue, the groundstate energy, E gs , corresponding to a unique eigenvector, φgs . There is an ionization threshold , > E gs , above which the spectrum is absolutely continuous and the electron is not bound to the nucleus, anymore. For an analysis of resonances corresponding to the eigenstates of the Hamiltonian H0 with energy in the interval (E gs , ), we refer the reader to [4] and references given there. Next, we summarize the organization and the main results of this paper. In Sect. II, we construct asymptotic electromagnetic field operators applied to vectors in the spectral subspace, H −δ := χ (H < − δ)H,
(I.18)
of the Hamiltonian corresponding to energies below − δ, where is the ionization threshold and δ > 0 is arbitrarily small. We exploit the fact that, in such states, the electron is (exponentially) well localized near the nucleus, which yields an estimate of the rate of convergence of the interpolating field operators to the asymptotic field operators. In Sect. III, we rigorously establish general reduction formulae (see [5]) for the S-matrix elements of Rayleigh scattering in our model, i.e., for the matrix elements Sαm ,m ({ fi }, { h j }) :=
m
i=1
Aout [ fi ]φgs ,
m
Ain [h j ]φgs ,
(I.19)
j=1
where the asymptotic states, on the right side of (I.19), are constructed in Sect. II and m and { h j }m in are assumed to belong to H −δ . The vector-valued functions { fi }i=1 j=1 (I.19) are positive energy solutions of the free wave equation whose Fourier transforms are smooth and vanish at the origin of momentum space. The reduction process amounts
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461
to expressing the scalar product (I.19) in terms of integrals of expectation values of timeordered products of interpolating fields. A precise formulation of our result is given in Proposition III.2 (see Sect. III). In Sect. IV, starting from the general expressions in Proposition III.2, we develop a modified reduction procedure useful to calculate the S−matrix elements (I.19), up to a remainder of arbitrarily high order in α. Our algorithm for calculating the matrix elements (I.19) uses, as an ingredient, the infrared-finite algorithm developed in [1, 2], for the construction and re-expansion of the groundstate φgs and the groundstate energy E gs . In Sect. IV.2, we provide a rather detailed outline of the re-expansion procedure for φgs and E gs that enables us to circumvent infrared divergences appearing in standard perturbation theory. Our analysis culminates in the following results for the connected parts of the S−matrix elements Sαm ,m ({ fi }, { h j }). Main Result. For α ≤ α, ¯ with α¯ ≡ α¯ N small enough and N -dependent, the S-matrix elements Sαm ,m ({ fi }, {h j }), with ( f i , h j ) = 0 ∀i, j, have expansions of the form Sαm ,m ({ fi }, {h j })conn =
2N =3(m+m )
Slm ,m ({ fi }, {h j }; α)α 2 + o(α N )
(I.20)
with
lim α δ |Slm ,m ({ fi }, {h j }; α)| = 0, for arbitrary δ > 0,
α→0
(I.21)
for N = 3, 4, 5, . . . and N ≥ 23 (m + m ). The coefficients Slm ,m ({ fi }, {h j }; α) are computable in terms of finitely many convergent integrals, for arbitrary l < ∞ (with l ≥ 3(m + m ) ≥ 6). The point of Expansion (I.20) is that Eq. (I.21) accounts for the possible appearance of powers of ln[1/α] (“infrared logarithms”). We expect that infrared logarithms are not an artefact of our algorithm, but are an expression of infrared divergences in naive perturbation theory for the groundstate φgs and the groundstate energy E gs . In the last section of our paper, Sect. V, we calculate the scattering amplitude corresponding to a process where an incoming photon excites the atom from the groundstate to an excited (resonance) state, whereupon the atom relaxes to the groundstate by emitting one outgoing photon. As has been known since the birth of quantum electrodynamics, the lowest order contribution to the transition amplitude is significantly different from zero only for photon energies close to the difference of the energy, En , of an excited bound state and the groundstate energy, E0 , of H0 . In particular, when the wave functions of the incoming and of the outgoing photon coincide, the imaginary part of the matrix element of the operator T := i(S − 1) (where S is the scattering matrix and 1 is the identity operator) is not zero, in leading order, only if the photon wave function does not vanish for energies corresponding to {En − E0 ; n ∈ N}, {En } being the energy levels of the Coulomb system. Only the imaginary part of this matrix element matters in the computation of the total cross section for the given incoming photon state. We provide a recipe to calculate higher order corrections of the scattering amplitudes. Our results represent a rigorous justification of Bohr’s frequency condition. To our knowledge, a mathematically controlled expansion, accurate to an arbitrary order in the finestructure constant α, with a finite remainder term, of the scattering amplitudes for Rayleigh scattering of light at an atom in nonrelativistic QED has not been provided in the literature before. The novelty of our results is that they turn infrared divergences in naive perturbation theory into powers of ln[ α1 ].
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II. Asymptotic Fields Positive energy solutions, f t , of the free wave equation 2 y f t (y ) − ∂ f t (y ) = 0, y · ∇ ∇ ∂t 2
of fast decay in |y |, for fixed t, are given by y −i|k|t+i k· fˆ(k)e f t (y ) :=
(2π )
d 3k √ 3/2
(II.1)
(II.2)
2|k|
∈ C ∞ (R3 \{0}). We construct a vector of test functions with fˆ(k) 0 ft (y ) :=
−i|k|t+i k·y λ)∗ fˆλ (k)e ε(k,
λ=±
(2π )
d 3k √ 3/2
2|k|
(II.3)
satisfying the wave equation (II.1), with
λ)∗ fˆλ (k) =: ˆf(k) ∈ C0∞ (R3 \{0}; C3 ). ε(k,
λ=±
An asymptotic vector potential is constructed as an LSZ (t → ±∞) limit of interpolating field operators
ft , t] := i A[ ft , t] := −i A[−
y , t) · ∂ f t (y ) − ∂ A(y , t) · ft (y ) d 3 y, A( ∂t ∂t
(II.4)
y , t) · ∂ f t (y ) − ∂ A(y , t) · ft (y ) d 3 y, A( ∂t ∂t
with ft as in (II.3) and y , t) := ei H t A( y )e−i H t , A(
A := A≡1 .
(II.5)
If the photon were a massive particle, the smeared field operator (II.4) would converge strongly, as t → ∞, on a dense linear subspace of the Hilbert space of the system. For massless photons, convergence of (II.4) has only been proven on a subspace of vectors in the Hilbert space whose maximal energy is so small that the propagation speed of the electron is strictly below the speed of light, e.g., on the space H , see [7, 11]. The existence of strong limits of the smeared field operators in Eq. (II.4), as t → ±∞, implies the existence of asymptotic creation- and annihilation operators ∗ λ) , aout/in (k, µ)}, {aout/in (k,
(II.6)
defined on a dense subspace of H (see (I.18)) and obeying the canonical commutation relations. In fact, the limits of the operators in (II.4), as t → ±∞, correspond to ∗ ∗ λ) fˆλ (k)d 3 k =: aout/in ft , t] = (k, ( f ) (II.7) Aout/in [ f] := lim A[ aout/in t→±∞
λ=±
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
463
and, similarly,
λ) fˆλ (k)d 3 k =: aout/in ( f ). ft , t] = aout/in (k, Aout/in [− ¯f] := lim A[− t→±∞
λ=±
(II.8)
Using estimates proven in [7] and in Lemma II.1, below, one can prove that, for fˆλ (k), 3 −1 3 λ ∈ L 2 R ; (1 + |k| )d k the following relations hold in the sense of quadratic hˆ (k) forms, on H : i) ∗ [aout/in ( f ), aout/in (h)] = ( f, h) =
hˆ λ (k)d 3 k, fˆλ (k)
(II.9)
λ=±
∗ ∗ [aout/in ( f ), aout/in (h)] = [aout/in ( f ), aout/in (h)] = 0;
ii) ∗ ∗ eit H aout/in (h)e−it H = aout/in (eitω h) ,
eit H aout/in (h)e−it H = aout/in (e−itω h)
(II.10)
where the Fourier transform of each component of eitω h is obtained from the ones := |k|. of h by multiplying by eitω(k) , ω(k) Since, in this paper, we are interested in Rayleigh scattering, we restrict our attention to the construction of asymptotic states describing an atom below the ionization threshold, . (They have spectral support strictly below the ionization threshold.) For such states, the position of the electron remains close to the one of the proton for all times. More precisely, such states exhibit exponential decay in the distance between the electron and the nucleus. This implies that the convergence of (II.4), as time t → ±∞, is faster than any inverse power of time t. This implies that when applied to vectors in H −δ the operators defined in (II.4) converge faster than any inverse power of time t, as t → ∞. This has important consequences for the expansion of Rayleigh scattering amplitudes in the finestructure constant. In the following, we make use of the field equation 2 y , t) − ∂ A(y , t) = − Jtr (y , t), y · ∇ y A( ∇ ∂t 2
(II.11)
where
α 3/2 y −α x(t)) 3 tr λ)(|k|)e −i k·( λ)∗ )ε (k, J (y , t) := − d k + h.c. ( v (t) · ε(k, (2π )3 λ=±
(II.12) x + α A (α x))e−i H t and x(t) = ei H t xe−i H t . with v(t) = ei H t ve−i H t = ei H t (−i ∇ Equation (II.11) is meaningful as an equation between densely defined operator-valued distributions. By Cook’s argument, the existence of the limits 3 2
ft , t]ψ, lim A[
t→±∞
(II.13)
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V. Bach, J. Fröhlich, A. Pizzo
for ψ ∈ H , follows from the existence of the integral ±∞ ∂ A[ f t , t]ψdt ∂t 0 ±∞ y , t) t (y ) ∂ A( ∂ ∂ f y , t) · − · ft (y ) d 3 yψdt = i A( ∂t ∂t ∂t 0 ±∞ 2 A( y , t) ∂ y , t) − y · ∇ y A( = i ∇ · ft (y ) d 3 yψdt, ∂t 2 0
(II.14) (II.15)
where, in passing from (II.14) to (II.15), we use the wave equation (II.1), and, taking into account the rapid spatial decay of ft (y ), we then integrate by parts twice in y. Introducing the notation tr J [ f t , t] := Jtr (y , t) · ft (y )dy, (II.16) and using the field equation (II.11), we may rewrite (II.14) as follows: ±∞ ±∞ ∂ A[ f t , t]ψdt = −i Jtr [ ft , t]ψdt. ∂t 0 0
(II.17)
The following lemma guarantees the convergence of the integral over time t on the right side of Eq. (II.17). Lemma II.1. Let ψ be a vector belonging to the subspace H −δ defined in (I.18). For λ)∗ fˆλ (k) = ˆf(k) ∈ C ∞ (R3 \{0}; C3 ) the following estimate holds: (k, λ=± ε 0 Jtr [ ft , t]ψ ≤
Cm , 1 + |t|m
(II.18)
for any m ∈ N, where Cm is a finite constant (depending on m). Proof. We write the identity operator in electron position space as 1
x − β|t|), x := ( x · x + 1) 2 1 = χ ( x − β|t|) + χ c (
(II.19)
for 0 < β < 1, where the function χ (y)(= 1−χ c (y)) is a non-negative C ∞ (R)-function equal to 1, for y < −1, and equal to 0, for y > 1. The operator Jtr [ ft , t] is given by Jtr [ ft , t] =
3
gtl (α x(t))vl (t),
(II.20)
l=1
where the vector function gt is as in Eq. (II.3). We use the partition of unity (II.19) to obtain the inequality 3 3 gtl (α x(t))vl (t)ψ ≤ gtl (α x(t))vl (t)χ ( x (t) − β|t|)ψ l=1
l=1 3
+
l=1
(II.21)
gtl (α x(t))vl (t)χ c ( x (t) − β|t|)ψ. (II.22)
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
465
We treat each term in the two sums in Eqs. (II.21) and in (II.22) separately. To bound the terms in (II.21), the following norm inequality suffices: gtl (α x(t))vl (t)χ ( x (t) − β|t|)ψ ∂χ ( x (t) − β|t|) ≤ gtl (α x(t)) ψ ∂ x l (t) x (t) − β|t|) · vl (t)ψ. + gtl (α x(t))χ (
(II.23) (II.24)
Similarly gtl (α x(t))vl (t)χ c ( x (t) − β|t|)ψ c x (t) − β|t|) x (t)−m ) ∂(χ ( · x (t)m ψ ≤ gtl (α x(t)) l ∂ x (t) + gtl (α x(t))χ c ( x (t) − β|t|) x (t)−m · vl (t) x (t)m ψ.
(II.25) (II.26) (II.27)
In order to prove (II.18), it is enough to notice that i) the norms vl (t)ψ and vl (t) x (t)m ψ are bounded, uniformly in time, because ψ ∈ H −δ , see, e.g., [9]; ii) the bound on the right side of Eq. (II.18) holds for sup|y |<β|t| |gtl (y )|, see, e.g., [12]. III. Reduction Formulae for S-Matrix Elements In this section we rigorously derive reduction formulae for the S-matrix elements corresponding to Rayleigh scattering. We study scattering processes, where the incoming state and the outgoing state describe a finite number of incoming and outgoing photons, respectively, and a hydrogen atom in its groundstate. Thus, we consider amplitudes of the form
m
Aout [ fi ]φgs ,
m
i=1
Ain [h j ]φgs ,
(III.1)
j=1
where { fi }, {h j } are vector test functions as in (II.3). In order to be able to apply the results of the previous section, we must impose the following condition on the supports of the test functions { fi }, {h j }:
χ (H ≥ − δ) χ (H ≥ − δ)
m i=1 m
Aout [ fi ]φgs = 0, (III.2) Ain [h j ]φgs = 0
j=1
for an arbitrary δ > 0, where χ (H ≥ − δ) is the spectral projector on values larger than or equal to − δ.
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V. Bach, J. Fröhlich, A. Pizzo
Let | k ∈ supp ˆfi (k)}, ωi := sup{|k| | k ∈ supphˆ (k)}. ω := sup{|k|
(III.3)
j
(III.4)
ωi ≤ − δ
(III.5)
ω j ≤ − δ.
(III.6)
j
Then (III.2) and (II.10) imply that
E gs +
m i=1
and E gs +
m j=1
Assumption (III.2) implies that the states
ψmout
:=
m
Aout [ fi ]φgs
(III.7)
Ain [h j ]φgs
(III.8)
i=1
and ψmin :=
m j=1
exhibit exponential decay in the distance between the electron and the proton. Hence in by Lemma II.1, one further asymptotic creation operator can be applied to ψmout , ψm , respectively. In the presence of an arbitrarily small infrared cutoff in the interaction term of the Hamiltonian, asymptotic completeness of Rayleigh scattering has been proven in [8]; the first result about asymptotic completeness has been proven in [6], for massive scalar in bosons. This implies that states ψmout and ψm , as in Eqs. (III.7), (III.8), respectively, satisfying (III.2), with δ = 0, span the space H . It is expected, but not proven, that this result remains true when the infrared cutoff is removed. The reduction formulae enable us to express the matrix elements in Eq. (III.1) in terms of (integrals of) time-ordered products of interpolating fields. These formulae will serve as a starting point to derive an infrared-convergent algorithm to explicitly calculate the scattering amplitudes, up to a remainder term of arbitrarily high order in α. We first explain the reduction procedure for a matrix element describing a process with only one incoming photon in a wave function h and one outgoing photon in a wave function f, i.e., for φgs , Aout [− ¯f] Ain [h]
(III.9)
where ·φgs denotes the expectation value with respect to the groundstate φgs . Exploiting Eq. (II.17), we find that φgs Aout [− ¯f] Ain [h]
(III.10)
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
φgs + Aout [− ¯f]( Ain [h] − Aout [h]) φgs = Aout [− ¯f] Aout [h] +∞ Aout [− ¯f] Jtr [hs , s]φgs ds = ( f, h) + i −∞ +∞ −i Jtr [hs , s] Ain [− ¯f]φgs ds,
467
(III.11) (III.12)
−∞
where the third term in (III.12) can be added for free, because it actually vanishes. This is seen by noticing that the groundstate is a vacuum for the asymptotic annihilation operators; see, e.g., [8]. Now, using again Eq. (II.17), we get φgs = Aout [− ¯f] Ain [h] (III.13) +∞ +∞ = ( f, h) − ft (z )T ( Jtr (z , t) Jtr (y , s)φgs hs (y )d 3 yd 3 zdsdt −∞ −∞ +∞ ¯fs , s], Jtr [hs , s]]φgs ds, +i [ A[− (III.14) −∞
where T is the time ordered product. A scalar product between vector quantities depending on the same position variables is understood, here and in the following. The equal time commutator in Eq. (III.14) corresponds to z , s) tr ∂ A( (III.15) − fs (z ) , J (y , s) hs (y )d 3 yd 3 z ∂s that is, in general, a non-vanishing function of the electron position. In order to get a more symmetrical expression that can be easily generalized to scattering amplitudes with an arbitrary finite number of photons, starting from the expression in Eq. (III.12) we follow the standard reduction formulae procedure. Due to the results achieved in the previous sections, we can control the mathematical quantities that will be derived. First we choose a smooth real function, ξ , of the time variable s with the following properties: ξ(s) = 1,
for
ξ(s) = 0,
for
R R ≤ s ≤ α − − , 2 2 R R − and s ≥ α − + , s ≤ −α − 2 2
−α − +
(III.16) (III.17)
where 0 < 1 and R > 0 is an α-independent number. Remark. This is a crucial step for the expansion in α we are going to carry out. Because of the fast convergence proved in Lemma II.1, we can choose a “short” time scale, α − , as a time cutoff, provided the cutoff-function ξ is smooth, uniformly in α. It follows from Lemma II.1, Inequality (II.18), that +∞ i Aout [− ¯f] Jtr [hs , s]φgs ds (III.18) −∞
can be approximated by i
+∞
−∞
Aout [− ¯f] Jtr [hs , s]φgs ξ(s)ds,
(III.19)
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V. Bach, J. Fröhlich, A. Pizzo
up to an error term of order o(α N ), for any N ∈ N. After some integrations by parts, we end up with +∞ y , s)φgs (hs (y )ξ(s))d 3 yds. (III.20) i Aout [− ¯f] A( −∞
Similarly −i is approximated by −i
+∞
−∞
+∞ −∞
Jtr [hs , s] Ain [− ¯f]φgs ds
y , s) Ain [− ¯f]φgs (hs (y )ξ(s))d 3 yds, A(
(III.21)
(III.22)
up to an error term of order o(α N ). We propose to express the sum of the two quantities (III.18) and (III.21) as the integral of a time-ordered product of the fields. For general S-matrix elements, we will basically apply the two operations i) and ii) explained below. i) The sum of the two terms (III.18) and (III.21) is given by +∞ ← → 2 y , s)φgs −limt→+∞ i z , t) ∂ ft (z )) A( ( A θt,s ( −∞
× (hs (y )ξ(s))d 3 zd 3 yds +∞ ← → y , s)( A + limt→−∞ i 2 z , t) ∂ ft (z ))φgs A( θs,t ( −∞
(III.23)
(III.24)
× (hs (y )ξ(s))d 3 zd 3 yds, where z , t) z , t) := A( θ (t − s), A θt,s (
(III.25)
z , t) ∂ ft (z ) ∂ A ← → θt,s ( − ( z , t) ∂ ( z ) := A ( z , t) f ft (z ), A t t θt,s θt,s ∂t ∂t
(III.26)
and θt,s = θ (t − s) is a non-negative, α-independent, C ∞ approximation of the Heaviside step function. We apply the fundamental theorem of calculus to (III.23), (III.24). From the derivative with respect to t of the expression ← → z , t) ∂ ft (z ), A θt,s (
(III.27)
with |t| large enough, we get a current, Jtr [ ft , t] applied to the groundstate; (an integration by parts in the z −coordinates is involved here). Thanks to Estimate (II.18) in Lemma II.1, we can choose the function ξ(t) in order to cutoff the t-integration. This introduces an error of order o(α N ), for an arbitrarily large N . z , t) and ii) Since the groundstate vector φgs is in the domain of the fields A θt,s ( z , t) when smeared out in space, we can rewrite the difference of the two ∂t A θt,s ( limits (III.23) and (III.24) as
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
−i 2
+∞ +∞
−∞
−∞
469
dtdsd 3 zd 3 yξ(t)×
∂ ← → y , s)φgs (hs (y )ξ(s)) A z , t) ∂t ft (z ) A( θt,s ( ∂t ← → y , s) A + A( z , t) ∂t ft (z )φgs (hs (y )ξ(s)) θs,t (
×
(III.28)
up to an error term of arbitrarily high order in α. After integrations by part, we finally conclude that if ( f, h) = 0 the expression in Eq. (III.9) is given by +∞ +∞ 2 y , s))φgs (hs (y )ξ(s))d 3 yd 3 zdsdt, i z , t) A( ( ft (z )ξ(t))T θ ( A( −∞
−∞
(III.29)
up to an error of order o(α N ), for any N ∈ N, where T θ denotes the smooth time-ordered product obtained when replacing the Heaviside function by the smooth function θ. In order to generalize this result to an arbitrary, finite number of asymptotic photons, we have to control the norm of vectors of the form +∞ +∞ yn , sn ))φgs × ·· T θ ( A( ·· y1 , s1 ) · · · A( −∞
×
−∞
n
(hi,si (yi )ξ(si ))
i=1
n
d 3 yi dsi .
(III.30)
i=1
Some operator domain problems might, in principle, arise, because the vector potential (smeared in space) is an unbounded operator. However, because of the time integrations, and because the groundstate belongs to D(H m ), for any m ∈ N, the vector in Eq. (III.30) turns out to be well defined, and its norm is bounded uniformly in α. Lemma III.1. Let the function ξ be smooth and of compact support and such that m yl , sl ) =: hl,sl (yl )|l = sups∈R | d dsξ(s) m | is α−independent, for any m ∈ N. Let {h l ( 1, . . . , n} be smooth solutions of the free wave equation with properties as in Eq. (II.3). Then the following operator is bounded in the operator norm, uniformly in α: +∞ +∞ ··· ··· T θ ( X (y1 , s1 ) · · · X (yn , sn )) × −∞
×
−∞
n
(hl,sl (yl )ξ(sl ))
l=1
n
d 3 yl dsl
l=1
1 , (H + i)n
(III.31)
˙ y , s) and y , s) or A( where X (y , s) is either A( T θ ( X (y1 , s1 ) · · · X (yn , sn )) := X (y p(1) , s p(1) ) · · · X (y p(n) , s p(n) )× = p∈Pn
× θ (s p(1) − s p(2) ) · · · θ (s p(n−1) − s p(n) ), Pn being the group of permutations of n elements.
(III.32)
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V. Bach, J. Fröhlich, A. Pizzo
Proof. The proof is by induction in n. Given the permutation ( p(1), . . . , p( j)), we assume that, for 1 ≤ j ≤ n − 1, the following statements are true: H1) The operator +∞ +∞ ·· ·· X (y p(1) , s p(1) ) · · · X (y p( j) , s p( j) ) × −∞ j−1
×
−∞
θ (s p(l) − s p(l+1) )
l=1
j
(h p(l),s p(l) (y p(l) )ξ(s p(l) ))
l=1
j
d 3 y p(l) ds p(l)
l=1
1 (H + i) j (III.33)
is bounded uniformly in α; k H2) For u ∈ R, and for functions ζ , g belonging to the families { d ξ(sk l ) |l = 1, .., n ; k = k { ∂ hl (ykl ,sl ) |l ∂sl
1, 2, 3, · · ·} and
+∞
−∞
dsl
= 1, .., n ; k = 1, 2, 3, · · ·}, respectively, the operator
(H + i) j−1 X (y , s) θ (u − s) gs (y )ζ (s)d 3 yds
1 (H + i) j
(III.34)
is bounded, uniformly in u and in α. We first prove that H2) holds when j is replaced by j +1. For this purpose we consider the scalar product +∞ 1 ψ, φ , (III.35) (H + i) j X (y , s) θ (u − s) gs (y )ζ (s)d 3 yds (H + i) j+1 −∞ where ψ ∈ D(H m ), for any m ∈ N, and φ is an arbitrary vector. The expression in Eq. (III.35) can be written as (recall θu,s = θ (u − s))
−i
+∞
(ψ, (H + i) j−1 X (y , s) θu,s gs (y )ζ (s)
−∞ +∞ −∞
(ψ, (H + i) j−1
1 φ)d 3 yds (H + i) j
∂ X (y , s) 1 θu,s gs (y )ζ (s) φ)d 3 yds. ∂s (H + i) j+1
(III.36) (III.37)
Integrating by parts in the time variable s, the expression in Eq. (III.37) is seen to be given by +∞ 1 ∂ i θu,s gs (y )ζ (s) φ)d 3 yds. (III.38) (ψ, (H + i) j−1 X (y , s) ∂s (H + i) j+1 −∞ By the induction hypothesis H2), we conclude that the absolute value of the scalar product (III.37) is bounded by Cψφ,
(III.39)
where C is a positive constant independent of ψ and φ. Due to Riesz’ Lemma, the operator in (III.35) is bounded. Because of our assumptions on θu−s , gs (y ), ζ (s), the constant C can be chosen to be independent of α and u.
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
471
Turning to H1), we first introduce a shorthand notation: Expression (III.33) is abbreviated by X ( p(1), . . . , p( j)), and the expression in Eq. (III.34) by Au ( g , ζ ). Because of property H2), for j + 1, and assuming H1) holds for j, the following inequality X ( p(1), . . . , p( j + 1)) ∂ h p( j+1) (y p( j+1) , s p( j+1) ) dξ(s p( j+1) ) ≤ 2X ( p(1), . . . , p( j))sups p( j) As p( j) , ∂s p( j+1) ds p( j+1) d 2 ξ(s p( j+1) ) (III.40) + X ( p(1), . . . , p( j))sups p( j) As p( j) h p( j+1) , ds 2p( j+1) implies that H1) holds for j + 1, too. Since H1) and H2) are obviously true for j = 1, they hold for any j ≤ n. Lemma III.1 follows from H1) and H2). In our derivation of reduction formulae we neglect forward scattering, i.e., we assume that ( f i , h j ) = 0, for arbitrary i, j. The general case can easily be derived from the result below, at the price of more complicated expressions. Proposition III.2. Under the assumptions in Eq. (III.2) and of Lemma III.1 on {h j }, { fp } and ξ , and if ( f p , h j ) = 0, for all p and j, the S-matrix element
m
Aout [ fp ]φgs ,
p=1
m
Ain [h j ]φgs
(III.41)
j=1
is given by i m+m
+∞
−∞
··
+∞ −∞
··
×
m
z p, tp) A(
p=1
m
T θ
( fp,t p (z p )ξ(t p )
y j , s j )φgs A(
j=1
m
p=1
m
(h j,s j (y j )ξ(s j ))
m
d 3 z p dt p
p=1
j=1
m
(III.42) d 3 y j ds j ,
j=1
up to an error term of arbitrarily high order in α. Proof. The proof is by induction. If there are only one incoming and one outgoing photon, we have proven this result at the beginning of the section. Notice that, using the same arguments, we can eliminate one incoming and one outgoing photon from the asymptotic states in the scalar product (III.41). The inductive assumption is that l + l = n − 1(< m + m ) photons can be eliminated yielding the expression
−l m
out
A
[ fp ]φgs , T
×
m−l
Ain [h j ]φgs
j=1 m j=m−l+1
m
m
z p, tp) A(
p=m −l +1
p=1
×
m
θ
yj, sj) × A(
j=m−l+1
( fp,t p (z i )ξ(t p ))×
p=m −l +1
(h j,s j (y j )ξ(s j ))
m p=m −l +1
d 3 z p dt p
(III.43) m
j=m−l+1
d 3 y j ds j ,
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V. Bach, J. Fröhlich, A. Pizzo
up to an error term of arbitrarily high order in α. To prove (III.43) for l + l + 1 = n, another photon, for example an outgoing photon, must be eliminated from the outgoing state in Expression (III.41), with l + l = n − 1. This can be accomplished by repeating operations i) and ii) at the beginning of this section, with the following modifications: a) The counterpart of the expectation value in (III.23) is given by: out → z m −l , tm −l )← ψm −l −1 , T θ ( A( z m , tm ) · · A( ∂ f m −l ,tm −l (z m −l ) ym−l+1 , sm−l+1 )ψ in , · · A( (III.44) m−l where ψmout −l −1
:=
m −l −1
Aout [ fp ]φgs ,
(III.45)
p=1 in ψm−l :=
m−l
Ain [h j ]φgs ;
(III.46)
j=1
b) As before, using Lemma III.1, we may cutoff the integration over tm −l by introducing the smooth function of compact support (see (III.16),(III.17)), ξ(tm −l ), up to an error of, at most, order o(α N ). We then integrate by parts, which is legitimate because of Lemma III.1, applied to the product of l + l = n − 1 fields. IV. Asymptotic Expansion of the Scattering Amplitudes The expression in Eq. (III.42) derived in the previous section must be evaluated in terms of explicit convergent integrals, up to an error term that, as we will prove, can be chosen to be of arbitrarily high order in the finestructure constant α. The reduction formulae derived in the previous section are not a particularly convenient starting point to develop an algorithm for calculating S-matrix elements. However, with a slight modification that depends on the desired order, o(α N ), N = 1, 2, 3, .., of accuracy of the algorithm, we can, in essence, repeat the reduction procedure in Proposition III.2. This procedure gives rise to time-ordered products. Our modified version of the reduction procedure does not rely on Lemma III.1, and, more importantly, it will yield an expression that can be expanded up to error terms of o(α N ), by making use of the Duhamel expansion of the propagators and the infrared-finite algorithm developed in [2] for the calculation of the groundstate and the groundstate energy. IV.1. A modified reduction procedure. Let o(α N ) be the desired order of the error term in 3 the calculation of the scattering amplitude (III.41). Since the leading order is α 2 (m+m ) , we may assume that N ≥ 23 (m + m ). The N −dependent, modified reduction procedure differs from the usual one in the following way: Before eliminating a photon from one of the asymptotic states, e.g. from the incoming state ψmin =
m j=1
Ain [h j ]φgs ,
(IV.1)
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
473
we apply the operator (H + i)n+1 , n = [N − 23 (m + m ) + 1] (where [·] is the integer part) to obtain n+1 m m m m n+1 (l ,l ,··,l ) (E gs + i)n+1− p ··· Ain [h j 1 2 p ]φgs p l1 =1 l2 =1
p=1
+ (E gs + i)n+1
m
l p =1 j=1
Ain [h j ]φgs ,
(IV.2)
j=1
where (l ,l ,··,l ) h j 1 2 p (y ) :=
( λ)∗ |k| ε(k,
p
q=1 δlq , j )
y +i k· hˆ λj (k)e
λ=±
d 3k
(2π )3/2 2|k|
, (IV.3)
where δlq , j is Kronecker’s symbol. The original vector can be written as ψmin = = (H + i) ×
m
··
l1 =1
−n−1 m
n+1 n+1 (E gs + i)n+1− p × p
(IV.4)
p=1 m
(l ,l ,··,l ) Ain [h j 1 2 p ]φgs
l p =1 j=1
+ (H + i)−n−1 (E gs + i)n+1
m
Ain [h j ]φgs .
(IV.5)
j=1
We now eliminate the m th photon from the state in Eq. (IV.2), using the procedure already employed in Proposition III.2. The same manipulations must be repeated for each photon. The final expression consists of finitely many terms similar to the expression in Eq. (III.42) except that 1) the test functions are the ones obtained in Eqs. (IV.2),(IV.3); and 2) the time-ordered product is given by T Nθ
m
zl , tl ) A(
l=1
=Tθ
m
yi , si ) : = A(
(IV.6)
j=1
m l=1
zl , tl ) A(
1 1 yj, sj) . A( n+1 n+1 (H + i) (H + i) m
(IV.7)
j=1
The operation T Nθ in (IV.6) can be expanded in α. In fact, a factor of (H1+i) can be put next zl , 0), and a factor of 1 n remains next to each propagator to each field operator A( (H +i)
e−i H (tl −tl+1 ) . Then we use the Duhamel expansion
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V. Bach, J. Fröhlich, A. Pizzo
1 ei H t = (H + i)n t 1 i 1 i H0 t e + ei H0 τ1 = H I e−i H0 τ1 dτ1 ei H0 t n n−1 (H + i) (H + i) (H + i) 0 ······················ t n−2 τl n−1 1 i n−1 H I e−i H0 τn− j × + ei H0 τn− j (H + i) 0 (H + i) 0 l=1
(IV.8)
j=1
×dτn−1 · · · dτ1 e τl n−1 1 H I e−i H0 τn × ei H τn (H + i) 0 i H0 t
t
+ in 0
×
(IV.9)
l=1
n−1
ei H0 τn− j
j=1
1 H I e−i H0 τn− j dτn · · · dτ1 ei H0 t , (H + i)
where H0 has been defined in Eq. (I.17), and H I := H − H0 . We then apply the Neumann series expansion of the resolvent n−1 j 1 1 1 − HI = (H + i) (H0 + i) (H0 + i)
(IV.10)
j=0
+
n 1 1 − HI , (H + i) (H0 + i)
(IV.11)
and we exploit the α-independence on the norm bounds of the operators 1
3
α− 2
(H0 + i)
1 2
HI
1 (H0 + i)
zl , 0) fl (zl , 0)d 3 zl A(
1
3
1 2
, α− 2 1
1 2
HI
1
(IV.12)
1
(H + i) (H0 + i) 2 zl , 0) fl (zl , 0)d 3 zl , A( 1
(H0 + i) 2
1 1
(H + i) 2
.
(IV.13)
Using the fact that the time-integrations extend over intervals of length proportional to α − , we conclude that the remainder term (IV.9) in the Duhamel expansion is bounded by 3 const ·α n( 2 −) . Therefore the operator (IV.6) can be approximated, up to an error term of o(α N ), by finitely many expressions only involving the propagator and the resolvent of the Hamiltonian H0 , besides the groundstate φgs and the groundstate energy E gs . These latter quantities can be calculated using the algorithm developed in [2] and outlined in the next section. We can therefore state the main result of the paper:
Theorem IV.1. For α ≤ α, ¯ with α¯ ≡ α¯ N small enough, the S-matrix elements Sαm ,m ({ f i }, {h j }), where ( f i , h j ) = 0, have expansions of the form
Sαm ,m ({ fi }, {h j })conn =
2N =3(m+m )
Slm ,m ({ fi }, {h j }; α)α 2 + o(α N )
(IV.14)
with
lim α δ |Slm ,m ({ fi }, {h j }; α)| = 0, for arbitrary δ > 0,
α→0
(IV.15)
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
475
for N = 3, 4, 5, . . . and N ≥ 23 (m + m ). The coefficients Slm ,m ({ fi }, {h j }; α) are computable in terms of finitely many convergent integrals, for arbitrary l < ∞ (with l ≥ 3(m + m ) ≥ 6). IV.2. Expansion of the groundstate and the groundstate energy. The final and technically most subtle step in the calculation of the S-matrix elements concerns the calculation of the groundstate and the groundstate energy. Because of infrared divergences, which invalidate a straightforward Taylor expansion, an iterative construction must be employed to remove an infrared cutoff in photon momentum space and to devise a convergent algorithm. Such a construction has been developed in [2] on the basis of results in [10] and [1]. In the following, we describe the results of [2] without providing proofs; but see [1, 2] and [10]. The main ideas underlying the construction of the groundstate will be discussed, and the strategy of the re-expansion will be outlined. IV.2.1. Notation. In the following part of this section, we simplify our notation by setting λ), ω(k) ≡ |k| := |k|, λ) d 3 k, k := (k, and f (k) dk := f (k, λ=±
for any integrable functions f (·, λ), λ = ±. Given an operator-valued function F: R3 × Z2 → B(Hel ), we write F(k) ⊗ a ∗ (k)dk, (IV.16) a ∗ (F) := a(F) := F(k)∗ ⊗ a(k)dk. (IV.17) This allows us to write the velocity operator v (rescaled by 2) as + a(G), x + a ∗ (G) v := −i ∇
(IV.18)
: R3 × Z2 → B(Hel )3 are the multiplication operators defined by where G G(k) :=
(k) −iα k· α 3/2 x ε(k). e √ (2π )3/2 2 |k|
(IV.19)
In terms of the velocity operator, the Hamiltonian assumes the simple form H = v 2 − V ( x ) + Hˇ .
(IV.20)
We define a decreasing sequence, (σn )∞ n=0 , of energy scales by setting σn := κ α n .
(IV.21)
Because of the assumptions on α (small enough) discussed in [1, 2], for all n ∈ N0 := N ∪ {0}, we have that σn+1 ≤ κα < 1. To cut the interaction Hamiltonian into slices corresponding to ever lower energy scales, we make use of the operators m n (k) := 1 σn ≤ |k| G(k), (IV.22) and G G n (k) := 1 σn ≤ |k| < σm G(k),
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V. Bach, J. Fröhlich, A. Pizzo
= ∞ n for all k ∈ R3 × Z2 and m, n ∈ N0 , with m < n. Note that G n=0 G n+1 and that n is the coupling function of the interaction Hamiltonian cutoff in the infrared region, G |k| ≤ σn . We factorize Fock space F = F(h) into tensor products by introducing suitable subspaces corresponding to the one-photon Hilbert space corresponding to different energy scales: m 2 hn := L 2 Kn and hm (IV.23) n := L Kn , where,
τ ) ∈ R3 × Z2 (k, τ ) ∈ R3 × Z2 for 0 ≤ m < n ≤ ∞, Knm := (k, for 0 ≤ n ≤ ∞, Kn :=
σn ≤ ω(k) , (IV.24) σn ≤ ω(k) < σm . (IV.25)
Note that Kn0 ⊂ Kn is a proper subset. For integers 1 ≤ m < n < ≤ ∞, we have the disjoint decomposition K = Km ∪ Knm ∪ Kn , and hence the direct sum n h ∼ = hm ⊕ hm n ⊕ h ,
(IV.26)
which gives rise to the isomorphism F ∼ = Fm ⊗ Fnm ⊗ Fn ,
(IV.27)
with Fn := F(hn ) and Fnm := F(hm n ). In particular, for any n ∈ N, n n+1 F = F∞ ∼ ⊗ F∞ . = Fn ⊗ Fn+1
(IV.28)
Hn := Hel ⊗ Fn and Hnm := Hel ⊗ Fnm .
(IV.29)
We set
For energy-scale indices m, n ∈ N0 , with m < n, we define the velocity operator vn , the field-energy operators Hˇ n , Hˇ nm , and the Hamiltonian Hn by n ) + a(G n ), + a ∗ (G vn := −i ∇ x Hˇ n := 1(σn ≤ |k|) ω(k) a ∗ (k)a(k) dk, m ˇ Hn := 1(σn ≤ |k| < σm ) ω(k) a ∗ (k)a(k) dk, Hn := vn2 − V ( x ) + Hˇ n ,
(IV.30) (IV.31) (IV.32) (IV.33)
as operators on Hn and Hnm , respectively. We introduce the groundstate energy at scale n and groundstate energy differences E n := inf σ (Hn ) and E nm := E m − E n .
(IV.34)
To compare Hamiltonians, Hn and Hn+1 , at successive energy scales, it is convenient to n+ on Hn and Hn+1 , respectively, by define positive operators Hn+ and H Hn+ := Hn − E n , n n+ := Hn+ ⊗ 1nn+1 + 1n ⊗ Hˇ n+1 , H
(IV.35) (IV.36)
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
477
where we denote the identity operator on Hn and on Fnm by 1n and 1m n , respectively. We identify vn with vn ⊗ 1nn+1 acting on Hn+1 . Note that, for n ∈ N0 , nn+1 ) + a(G nn+1 ). vn+1 = vn + a ∗ (G
(IV.37)
Similarly, given ψn ∈ Hn , we define a vector ψ˜ n := ψn ⊗ nn+1 ∈ Hn+1 ,
(IV.38)
m where n and m n denote the vacuum vectors in Fn and Fn , respectively. With these notations, we have that + n n n+ + Wn+1 Hn+1 = H + E n+1 ,
(IV.39)
where n Wn+1 := ( vn+1 )2 − ( vn )2
nn+1 ) · vn + 2 vn · a(G nn+1 ) + a ∗ (G nn+1 ) + a(G nn+1 ) 2 = 2 a ∗ (G nn+1 ) · vn + 2 vn · a(G nn+1 ) + a ∗ (G nn+1 ) · a ∗ (G nn+1 ) = 2 a ∗ (G nn+1 ) + 2 a ∗ (G nn+1 ) · a(G nn+1 ) + G nn+1 2 , nn+1 ) · a(G + a(G
(IV.40)
n 2 n := |G (k)|2 dk. In Eq. (IV.40), we make use of the Coulomb gauge with G n+1 n+1 · A(x) condition ∇ = 0, which implies that nn+1 ) · vn = vn · a ∗ (G nn+1 ) a ∗ (G
and
nn+1 ) · vn = vn · a(G nn+1 ). (IV.41) a(G
IV.2.2. Preliminary results and outline of strategy. Here we describe some results derived in [1], concerning the construction of the groundstate of H , that will be used again and again in the re-expansion procedure. A key ingredient used in our construction of the groundstate of H is the simple identity 2 v = i [H , x],
(IV.42)
which implies that the interaction term in H , which is marginal in the infrared region (in the sense of power counting), is actually infrared-irrelevant on the subspace of all those states where the electron is bound to the nucleus. Proposition IV.2. Assume Hypothesis 1. Then there exist constants 0 < C ≤ C, C ≥ 4 1 such that, for all α < 2C , E n is an eigenvalue of multiplicity one. Moreover 3 inf σ (Hn+ ) \ {0} =: gapn ≥ 1 − Cα σn , 4 n+ ) \ {0} =: g apn = σn+1 , inf σ ( H
(IV.43) (IV.44)
and sup
z∈n+1
1 1/2 1 1/2 n − W ≤ C α, n+1 + + Hn − z Hn − z
(IV.45)
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V. Bach, J. Fröhlich, A. Pizzo
n := |φ˜ n φ˜ n | where n+1 := { 41 σn+1 eiϑ ∈ C|ϑ ∈ [0, 2π )}. The spectral projections P ˜ and Pn+1 := |φn+1 φn+1 |, where φn and φn+1 are groundstates of Hn and Hn+1 , respectively, correspond to dz n+1 n = −1 , (IV.46) P n+ − z n+1 2πi n+1 H and Pn+1 =
−1 2πi
n+1
dz n+1 , Hn+1 − E n+1 − z n+1
(IV.47)
respectively, with n := { 41 σn eiϑ ∈ C|ϑ ∈ [0, 2π )}. Their difference has a norm convergent series expansion ∞ (ν) n = −1 Pn+1 − P (−1)ν Yn+1 (z n+1 ) dz n+1 , (IV.48) 2πi n+1 ν=1
where (ν) (z) := Yn+1
1 ν 1 n , W n+1 n+ − z n+ − z H H
(IV.49)
and n ≤ C α (n+2)/2 . Pn+1 − P
(IV.50)
The eigenvalue E ∞ ≡ E gs is non-degenerate. If we are interested in deriving an explicit expression for φgs up to a remainder term of order o(α N ), we may as well consider the vector φ2N −1 , because φ2N −1 − φgs ≤ o(α N ).
(IV.51)
The bound (IV.51) follows from Proposition IV.2. Up to normalization, the vector φ2N −1 is given by the product P2N −1 P2N −2 · · · P1 of the projections {P2N −1 , . . . ., P1 } applied to the groundstate, φ0 , of the bare Hamiltonian H0+ , i.e. φ2N −1 ∝ P2N −1 · · · ·P1 φ0 .
(IV.52)
For a more precise version of Formula (IV.52), see [2]. From Proposition IV.2, more precisely from estimate (IV.45), we infer that the expansion (IV.48) can be truncated at a finite order in such a way that the remainder term is o(α N ). Thus, the vector T T P2N −1 · · · ·P1 φ0 ,
(IV.53)
where (Pm )T := −
N j 1 1 1 (−Wmm−1 ) + dz m + , (IV.54) 2πi Hm−1 − z m Hm−1 − z m j=0 m
is an approximate expression for φgs , up to a remainder term o(α N ) and up to a normalization factor. The truncation in the definition of PmT only depends on N .
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
479
From now on, we focus our attention on the analysis of the finitely many resolvents appearing on the right side of (IV.54), for m = 2N − 1. Our derivation of explicit finite expressions for the vector (IV.53) relies on three operations to be iterated a finite number of times: i) The photon creation- and annihilation operators are Wick-ordered, shell by shell; ii) the identity operator, 1n , in the space Hn is decomposed into the sum Pn + Pn⊥ ; iii) two slightly different, truncated Neumann expansions, which we call A and B, are used to re-expand the contributions associated with Pn⊥ coming from operation ii). Re-expansion procedure. As a first step, we expand the resolvents + H 2N −2
1 − z 2N −1
(IV.55)
T appearing in the truncated projection P2N −1 in Expression (IV.53) until only resolvents + of the Hamiltonian H2N −3 are left. We then put the expressions obtained in a form that enables us to iterate the operation, until only resolvents of the bare Hamiltonian H0+ are left. This can be accomplished if we take into account that the operator in Eq. (IV.55) is applied to a vector containing only a finite, N -dependent, but α-independent, number 2N −2 T of photons with momenta in the shell K2N −1 , thanks to the truncation in P2N −1 . After 2N −2 Wick-ordering of the photon operators in the shell K2N −1 , the original resolvent (IV.55) is replaced by a finite, N -dependent number of resolvents of the form
1 (IV.56) + H2N − z 2N −1 + j |k j | −2 applied to a vector in H2N −2 , where the sum, j |k j |, of energies |k j |, {k j : σ2N −1 ≤ |k j | ≤ σ2N −2 } is finite and depends only on N . The key idea underlying the re-expansion of φgs is to split the operator (IV.56) into two pieces, 1 + H2N −2
− z 2N −1 +
j
|k j |
P2N −2 and
1 + H2N −2
− z 2N −1 +
j
|k j |
⊥ P2N −2 .
(IV.57) The first one is proportional to the projection P2N −2 , the factor of proportionality being an explicit number. Up to a remainder term of order o(α N ), P2N −2 can be expanded by using Eq. (IV.48) and then truncated as in Eq. (IV.54). The second term in (IV.57) is analyzed by using the Neumann expansion below, which we call of type A: ⊥ P2N −2
1 ⊥ P2N −2 + − z + |k | H 2N −1 j j 2N −3
⊥ + P2N −2
×
∞ j=1
1 × P⊥ + H2N −3 − z 2N −1 + j |k j | 2N −2 2N −3
2N −3 − W2N −2 − E 2N −2
1 + H 2N −3 − z 2N −1 +
j j
|k j |
⊥ P2N −2
.
(IV.58)
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V. Bach, J. Fröhlich, A. Pizzo
2N −3 This expansion converges, because the absolute value of the energy shift E 2N −2 is very small, namely of order O(α 2N ); and the expression
1 ⊥ P2N −2 + − z + |k | H 2N −1 j j 2N −3
(IV.59)
−1 ⊥ is bounded in norm by σ2N −2 , because of the orthogonal projection P2N −2 . The expansion in Eq. (IV.58) yields additional powers of α. It can therefore be truncated at some ⊥ N −dependent order. Finally, P2N −2 = 12N −2 − P2N −2 has to be expanded and truncated similarly to P2N −2 . As a result of the previous operations (decomposition (IV.57), and Neumann expansion A), the resolvent (IV.56) is represented by a polynomial in the following operators:
• the resolvents 1 , + H2N −3 − z 2N −1 + j |k j |
1 ; + H2N −3 − z 2N −2
(IV.60)
2N −3 • the slice interaction W2N −2 ; 2N −3 • and the energy shift E 2N −2 .
Returning to the expression (IV.53), we then Wick-order the photon operators corre2N −3 sponding to photon momenta in the shell K2N −2 . This yields finitely many terms, and the resolvents in Eq. (IV.60) are replaced by resolvents of the form + H2N −3
1 , − z 2N −1 + j |k j | + i |qi |
1 + H2N −3
− z 2N −2 +
i
|qi |
,
(IV.61)
2N −3 where the photon momenta qi all belong to the shell K2N −2 , and the number of terms in the sum i |qi | is bounded by an N −dependent number, thanks to the truncation T in P2N −2 . As before, we decompose the resolvents into two pieces, using the projec⊥ tions P2N −3 and P2N −3 . This yields terms proportional to the projection P2N −3 , with an explicit factor of proportionality, and terms of the form
1 + H2N −3
− z 2N −2 +
i
|qi |
⊥ P2N −3 ,
+ H2N −3
1 P⊥ . − z 2N −1 + j |k j | + i |qi | 2N −3 (IV.62)
The first term in Eq. (IV.62) is analyzed by using the Neumann expansion of type A, see (IV.58), with 2N − 2 replaced by 2N − 3. The second term in Eq. (IV.62) is treated by applying the following expansion, which we call of type B: 1 P⊥ + H2N −4 − z 2N −2 + i |qi | 2N −3 1 ⊥ + P2N × P⊥ −3 + H2N −4 − z 2N −2 + i |qi | 2N −3
⊥ P2N −3
×
N j=1
2N −4 2N −4 − W2N −3 − E 2N −3 )
1 P⊥ + H2N −4 − z 2N −2 + i |qi | 2N −3
(IV.63)
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
+ z 2N −1 − z 2N −2 − |k j | j
481
j 1 ⊥ P2N −3 + H2N −4 − z 2N −2 + i |qi |
1 P⊥ + + × H2N −3 − z 2N −1 + i |qi | + j |k j | 2N −3 1 2N −4 2N −4 × − W2N P⊥ −3 − E 2N −3 ) + H2N −4 − z 2N −2 + i |qi | 2N −3 N +1 1 ⊥ + z 2N −1 − z 2N −2 − |k j | + . P2N −3 H2N −4 − z 2N −2 + i |qi | j
(IV.64)
In this truncated expansion, the remainder term (IV.64) is proven to be of order o(α N ) with respect to the original expression in (IV.62). To see this, we use the bounds on the shift of the integration variable, z 2N −1 − z 2N −2 , and on the sum j |k j |: Both quantities are of order σ2N −2 , the first one by definition ofz 2N −2 and z 2N −1 , and the second one because of the fact that the number of terms in j |k j | is finite and α-independent. The new features of our expansion of type B, as compared to an expansion of type A, are as follows: The replacement of the integration variable z 2N −1 by the variable z 2N −2 is possible because of Estimate (IV.44) on the spectral gap: One then expands in the energies |k j |, which are bounded by const · σ2N −2 . −1 The norm of the remaining resolvents is bounded by const · σ2N −3 . Finally we trun⊥ cate P2N −3 . Hence, starting with a resolvent of the form given in Eq. (IV.56), and then performing the operations just described, we end up with a polynomial in the following operators: ∗ Resolvents, 1 1 ; , + + H2N −4 − z 2N −2 + i |qi | H2N −4 − z 2N −3
(IV.65)
2N −4 ∗ slice interactions W2N −3 ; 2N −4 ∗ energy shifts E 2N −3 , difference of integration variables, z 2N −1 − z 2N −2 , and energy sums, j |k j |.
We proceed by Wick-ordering the photon creation- and annihilation operators in the 2N −4 shell K2N −3 in every term obtained so far, starting from the vector in Eq. (IV.53), using (IV.54), for m = 2N − 1, and then expanding the resolvents as described above. Two types of resolvents result: 1 1 + |, H+ ; H2N − z + |q | + |q − z 2N −2 2N −3 + i i i i |qi | −4 2N −4 i
(IV.66)
2N −4 where the new sum, i |qi |, corresponds to photon momenta, qi , in the shell K2N −3 , and the number of terms is bounded by an N -dependent, but α-independent integer. ⊥ After inserting the partition of unity, 12N −4 = P2N −4 + P2N −4 , we arrive at the expressions in (IV.62), but with 2N −3 replaced by 2N −4. Thus , our re-expansion procedure, based on truncated Neumann expansions of type A and B and on Wick-ordering, can be iterated.
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V. Bach, J. Fröhlich, A. Pizzo
By applying the re-expansion procedure, scale by scale, to all the resolvents appearing in the truncated projections of Expression (IV.53), we eventually end up with an expansion of (IV.53) involving only “bare” resolvents, i.e., resolvents of the form H0+
− z2 +
1 1 , + , i |qi | + i |qi | H0 − z 1 + i |qi |
(IV.67)
with momenta {qi }, {qi } belonging to the shells K21 and K10 , respectively. The arguments described above represent the essential ingredients in the re-expansion procedure developed in [2], where the mathematical details are presented. The main difficulty in the inductive proof of convergence of the re-expansion procedure is related to determining the energy shifts E mn more explicitly. In re-expanding φ2N −1 , all the energy shifts up to scale 2N − 2 appear in our formulas. The energy shifts can be expressed in terms of the groundstate vectors on scales up to 2N − 3. Explicit expressions for the energy shifts E mn are obtained from the re-expansions of {φi |i = 1, . . . , 2N − 2}, up to remainder terms of o(α N ). The result of the re-expansion of the groundstate and the groundstate energy presented in [2] is: For α ≤ α, ¯ with α¯ ≡ α¯ N small enough, the groundstate energy E gs ≡ E gs (α) and the 1
groundstate φgs ≡ φgs (α 2 ) have expansions of the form E gs (α) = E 0 +
N
ε (α) α + o(α N ),
(IV.68)
ϕ (α) α /2 + o(α N ),
(IV.69)
=3 1
φgs (α 2 ) = φ0 +
2N =3
with lim α δ |ε (α)| = 0 ∀δ > 0,
(IV.70)
lim α δ ϕ (α) = 0 ∀δ > 0,
(IV.71)
α→0
and α→0
for arbitrary N = 3, 4, 5, . . . . The coefficients ε (α) and ϕ (α) are computable in terms of finitely many convergent integrals, for any 3 ≤ < ∞. Equations (IV.70), (IV.71) account for the possible appearance of powers of ln[1/α] (“infrared logarithms”). We expect that infrared logarithms are not an artefact of our algorithm, but are an expression of infrared divergences in naive perturbation theory: 1 The quantities E gs (α) and φgs (α 2 ) are not analytic, nor even smooth, at α = 0; rather, 1
derivatives in α 2 of sufficiently high order of these quantities diverge, as α → 0. By combining the expansion of the modified reduction formulae developed in Sect. IV.1 with Expressions (IV.68) and (IV.69), Expansion (I.20) for the S-matrix elements of Rayleigh scattering is established. V. Bohr’s Frequency Condition In this last section, we explicitly compute the scattering amplitude in Eq. (III.9) to leading order in α.
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
483
In Sect. III, Eq. (III.13), we have derived that φgs − i( f, h) =: ( f, T h) i Aout [− ¯f] Ain [h] corresponds to +∞ +∞ −i f t (z )T ( Jtr (z , t) Jtr (y , s)φgs hs (y )d 3 yd 3 zdsdt −∞ −∞ +∞ ¯fs , s], Jtr [hs , s]]φgs ds, − [ A[− −∞
(V.1)
(V.2) (V.3)
where we have used the standard definition for T , the T-matrix operator. The leading order term is of order α 3 and arises from the expression in Eq. (V.2). In fact, due to one power of α multiplying the electron position operator, x, in the transverse current, the term in Eq. (V.3) is of order α 4 . In computing the leading term, we first rewrite the integrand in Expression (V.2) as ei E gs (t−s) ft (z ) Jtr (z , 0)e−i H (t−s) Jtr (y , 0)φgs θ (t − s)hs (y ) +e
i E gs (s−t)
tr
ft (z ) J (y , 0)e
J (z , 0)φgs θ (s − t)hs (y ).
−i H (s−t) tr
(V.4) (V.5)
Then we approximate Jtr (y , 0) by −
α 3/2 y 3 λ)(|k|)e −i k· λ)∗ )ε (k, d k + h.c., ( p · ε(k, (2π )3
(V.6)
λ=±
x , and we approximate φgs by ϕel ⊗ , ψ0 = ϕel being the ground where p := −i ∇ state of the atomic system alone with corresponding Hamiltonian Hel := −x − V ( x ).
(V.7)
Finally, we replace the propagator e−i H (t−s) by e−i H0 (t−s) , and we rewrite the identity operator between the two currents as a sum of projections onto eigenstates (and generalized eigenstates) of the Hamiltonian Hel tensor the vacuum state . To compute the contribution of order α 3 to the transition amplitudes corresponding to an intermediate eigenstate ψn of the Hamiltonian Hel , we first analyze the contribution of the term in Eq. (V.4) to order α 3 , by using the identity p = 2i [Hel , x]: −i
1 (2π )3
×e
+∞ +∞ −∞
−∞
(n E)2
−i(En −E0 −|k|)t −i(−En +E0 +| q |)s
e
λ)ψn )(ψn , x · ε( (ψ0 , x · ε(k, q , λ )∗ ψ0 ) × (V.8) λ,λ
hˆ λ ( q ) (|k|)(| fˆλ (k) q |) 3 3 d kd qdsdt, θ (t − s)
2 | q| |k|
where n E := En − E0 and En is the energy of the eigenstate ψn ; in passing from Eq. (V.4) to Eq. (V.8), we have also approximated E gs by E0 . Introducing the variable u := t − s > 0, the integral in Eq. (V.8) can be written as
484
V. Bach, J. Fröhlich, A. Pizzo
i
1 (2π )3 ×e
+∞ +∞
−∞
(n E)2
0
λ)ψn )(ψn , x · ε( (ψ0 , x · ε(k, q , λ )∗ ψ0 ) ×
λ,λ
q |)t i(−En +E0 +| i(|k|−| q |)u
e
hˆ λ ( q ) (|k|)(| fˆλ (k) q |) 3 3
d kd qdudt. 2 | q| |k|
(V.9)
We insert regularizing factors e−|t| , > 0, and e−µ|u| , µ > 0, and then pass to the limits → 0 and µ → 0 of
i
1 (2π )3 ×e
+∞ +∞
−∞
(n E)2
0
λ)ψn )(ψn , x · ε( (ψ0 , x · ε(k, q , λ )∗ ψ0 ) × λ,λ
q |)t −|t| i(−En +E0 +| i(|k|−| q |)u −µu
e
e
e
hˆ λ ( q ) (|k|)(| fˆλ (k) q |) 3 3
d kd qdudt. 2 | q| |k| (V.10)
An explicit calculation gives
lim lim i
→0 µ→0
1 (2π )3
(En )2
λ)ψn )(ψn , x · ε( (ψ0 , x · ε(k, q , λ )∗ ψ0 ) × λ,λ
(V.11) × = i
q) hˆ λ (
i (|k|)(| q |) 3 3
d kd q = E0 − En + | q |+iµ 2 | q| |k| λ)ψn )(ψn , x · ε( (ψ0 , x · ε(k, q , λ )∗ ψ0 ) × (V.12) (n E)2 2
− | (|k| q |)2 1 (2π )3
fˆλ (k)
+ 2
·
λ,λ
hˆ λ ( q ) (|k|)(| fˆλ (k) q |) 3 3 − | d qd k × 2π 2 δ(|k| q |)δ(E0 − En + | q |)
2 | q| |k| 1 3 −P d q d 3 k(n E)2 × E0 − En + | q| λ)ψn )(ψn , x · ε( × (ψ0 , x · ε(k, q , λ )∗ ψ0 ) × λ,λ
hˆ λ ( q ) (|k|)(| fˆλ (k) q |) − | , × (2π )−2 δ(|k| q |)
2 | q| |k|
(V.13)
Infrared-Finite Algorithm for Rayleigh Scattering Amplitudes
485
where P stands for the principal part of E0 −E1n +|q | . By similar calculations we see that the contribution in Eq. (V.5) gives: 1 2 λ)ψn )(ψn , x · ε( ( E) (ψ0 , x · ε(k, q , λ )∗ ψ0 ) × (V.14) i n (2π )3 λ,λ
hˆ λ ( q ) (|k|)(| fˆλ (k) q |) 3 3 − | d qd k × 2π 2 δ(|k| q |)δ(−E0 + En + | q |)
2 | q| |k| 1 3 − d q (V.15) d 3 k(n E)2 × −E0 + En + | q| λ)ψn )(ψn , x · ε( × (ψ0 , x · ε(k, q , λ )∗ ψ0 ) × λ,λ
hˆ λ ( q ) (|k|)(| fˆλ (k) q |) − | . × (2π )−2 δ(|k| q |)
2 | q| |k| We observe that the real part of a (connected) scattering amplitude of the type calculated above, with f = h, is different from zero, in leading order, only if the photon wave function does not vanish for photon energies corresponding to a difference, En − E0 , of the energy En of an excited boundstate and the groundstate energy E0 . In fact, it is given by Expression (V.12). Assuming the validity of the optical theorem, the total cross section for an incoming photon with wave function f is proportional to the imaginary part of ( f, T f ). Therefore, in leading order, the total cross section for an incoming photon with wave function f vanishes if f ( q ) = 0 when | q | = En − E0 , for arbitrary n. This is Bohr’s frequency condition! Since we have assumed that the spectral support, with respect to H , of the initial (and the final) state is strictly below the ionization threshold
, transitions corresponding to intermediate states in the continuous spectrum do not contribute to the total cross section to leading order in α. References 1. Bach, V., Fröhlich, J., Pizzo, A.: Infrared-Finite Algorithms in QED I. The Groundstate of an Atom Interacting with the Quantized Radiation Field. Commun. Math. Phys. 264(1), 145–165 (2006) 2. Bach, V., Fröhlich, J., Pizzo, A.: Infrared-Finite Algorithms in QED II. The Expansion in the Coupling Constant of the Groundstate of an Atom Interacting with the Quantized Radiation Field. http://www.ma.utekas.edu/ mp-arc 06–140 (2006) 3. Bach, V., Fröhlich, J., Sigal, I.M.: Renormalization group analysis of spectral problems in quantum field theory. Adv. in Math. 137, 205–298 (1998) 4. Bach, V., Fröhlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999) 5. Bjorken, J.D., Drell, S.: Relativistic Quantum Fields. New York: McGraw-Hill, 1965 6. Derezinski, J., Gerard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11(4), 383–450 (1999) 7. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic electromagnetic fields in models of quantummechanical matter interacting with the quantized radiation field. Adv. Math. 164(2), 349–398 (2001) 8. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. H. Poincaré, 3, 107–170 (2002) 9. Griesemer, M.: Exponential decay and ionization threshold in non-relativistic quantum electrodynamics. J. Funct. Anal. 210(3), 321–340 (2004) 10. Pizzo, A.: One-particle (improper) states in Nelson’s massless model. Ann. H. Poincaré, 4(3), 439– 486 (2003)
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11. Pizzo, A.: Scattering of an Infraparticle: the One-Particle Sector in Nelson’s massless model. Ann. H. Poincaré 6, 553–606 (2005) 12. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. III, Scattering Theory. New York Academic Press 1979 Communicated by G. Gallavotti
Commun. Math. Phys. 274, 487–551 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0269-4
Communications in
Mathematical Physics
Classification of Minimal Actions of a Compact Kac Algebra with Amenable Dual Toshihiko Masuda1 , Reiji Tomatsu2 1 Graduate School of Mathematics, Kyushu University, 6-10-1 Hakozaki, Fukuoka 812-8581, Japan.
E-mail: [email protected]
2 Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro,
Tokyo 153-8914, Japan. E-mail: [email protected] Received: 16 October 2006 / Accepted: 18 November 2006 Published online: 27 June 2007 – © Springer-Verlag 2007
Abstract: We show the uniqueness of minimal actions of a compact Kac algebra with amenable dual on the AFD factor of type II1 . This particularly implies the uniqueness of minimal actions of a compact group. Our main tools are a Rohlin type theorem, the 2-cohomology vanishing theorem, and the Evans-Kishimoto type intertwining argument. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Quick review of theory of Kac algebras . . . . . . . . . . . . . 2.3 Amenability of a Kac algebra . . . . . . . . . . . . . . . . . . . 2.4 Actions and cocycle actions . . . . . . . . . . . . . . . . . . . . 2.5 Crossed products and dual functionals . . . . . . . . . . . . . . 3. Ultraproduct von Neumann Algebras . . . . . . . . . . . . . . . . . 3.1 Liftable and semiliftable homomorphisms . . . . . . . . . . . . 3.2 Strongly free cocycle actions . . . . . . . . . . . . . . . . . . . 4. Cohomology Vanishing I . . . . . . . . . . . . . . . . . . . . . . . 4.1 2-cohomology vanishing in ultraproduct von Neumann algebras 4.2 Cocycle actions on central sequence algebras . . . . . . . . . . 4.3 Intertwining cocycles . . . . . . . . . . . . . . . . . . . . . . . 5. Rohlin Type Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Local quantization principle . . . . . . . . . . . . . . . . . . . 5.2 Tower bases and diagonal elements . . . . . . . . . . . . . . . . 5.3 Rohlin type theorem . . . . . . . . . . . . . . . . . . . . . . . . 6. Cohomology Vanishing II . . . . . . . . . . . . . . . . . . . . . . . 6.1 2-cohomology vanishing in McDuff factors of type II1 . . . . . 6.2 Shapiro unitary . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction This paper presents the uniqueness of minimal actions of a compact Kac algebra with amenable dual. More precisely, any two minimal actions of a compact Kac algebra with amenable dual on the approximately finite dimensional (AFD) factor of type II1 are conjugate. Note that every compact group action is a particular example of an action of such a compact Kac algebra. Here we say that an action is minimal if it has the full spectrum and the relative commutant of its fixed point algebra is trivial [ILP]. After the completion of classification of discrete amenable group actions, it is natural to focus our attention to actions of continuous groups. Although it is very difficult to analyze actions of continuous groups in general, compact group actions have been extensively studied among them since the dual of a compact group is discrete. Indeed as emphasized in [W1], it gives us important insight about compact group actions to consider actions of compact group duals, namely, coactions of groups [NT], or Roberts actions [Ro]. Actions of compact abelian groups have been completely classified in [JT] and [KT] by combining results in [Oc1, ST and KST] with the Takesaki duality [T]. For compact non-abelian groups, classification of all actions is still very far from completion, but a few kinds of actions, such as ergodic actions and minimal actions, have been studied. For example, ergodic actions have been studied in [OPT] for the abelian case, and A. Wassermann has dealt with general ergodic actions in [W2, W3 and W4]. In particular, he has finished the classification of ergodic actions of SU (2). It is a central theme to classify all minimal actions in the study of compact group actions. The notion of minimality corresponds to outerness of actions in the discrete case. So far several attempts have been made for this classification problem. In an unpublished work [Oc2], Ocneanu has announced the uniqueness of minimal actions of a compact group by developing the method used in [Oc1]. A different approach has been proposed by S. Popa and A. Wassermann in [PW], which is based on the classification of subfactors by Popa [P1]. They have applied the main theorem of [P1] to Wassermann’s subfactors [W1], and concluded the uniqueness of minimal actions of a compact Lie group. Unfortunately, the details of both adorable theories have not been available to the authors. We explain our approach to this problem. Most of this paper is devoted to classification of centrally free actions of amenable discrete Kac algebras. We can utilize the framework of amenable discrete Kac algebras for treating the duals of compact groups. The uniqueness of minimal actions follows through the duality between compact Kac algebras and discrete Kac algebras. Throughout this paper, the most fundamental tools are the ultraproducts and central sequence technique. Here one notes that actions of Kac algebras never preserve central sequences. To overcome this difficulty, we mainly treat approximately inner actions. Then we can use the central sequence technique as demonstrated in [M2]. The first half of our arguments is similar to those in [C1, Oc1], that is, we formulate a Rohlin type theorem for actions of discrete amenable Kac algebras, and show approximate 1-cohomology vanishing and 2-cohomology vanishing by the Shapiro type
Classification of Minimal Actions of a Compact Kac Algebra with Amenable Dual
489
argument. Combining these results with the stability of minimal actions (cf. [W2]), we show that every minimal action is dual. The traditional method for classification theory of actions on von Neumann algebras is the model action splitting argument, that is, we construct the model action as an infinite tensor product type action at first, and pull out pieces of the model action from a given action by means of the Rohlin type theorem and cohomology vanishing. It is not so difficult to construct the infinite tensor product type model action for cyclic groups or finite groups as in [C1, C3 and J], but it is necessary to apply the paving theorem of D. Ornstein and B. Weiss [OW1, OW2] to construct the model action for a general discrete amenable group. Because of use of the paving theorem, Ocneanu’s Rohlin type theorem [Oc1] takes a complicated form, and it seems difficult to generalize it to a discrete amenable Kac algebra case. Of course, it is easy to construct the model action for coactions of finite groups, and the classification given in [M2] is based on the model action splitting argument. So we do not take the model action splitting argument in the final stage of classification. Instead of that, we use the intertwining argument, which was initiated by D. E. Evans and A. Kishimoto in [EK], and has been further developed in [N] and [I2] for group actions on C ∗ -algebras. It also works for von Neumann algebras as is shown in [M1], enables us to avoid using a paving theorem, and makes our arguments simpler than those in [Oc1].
2. Preliminaries 2.1. Notations. We treat only separable von Neumann algebras and Kac algebras except for ultraproduct von Neumann algebras. We denote by R0 the AFD factor of type II1 . Let M be a von Neumann algebra. For a subset S ⊂ M, we denote by W ∗ (S) the von Neumann subalgebra of M generated by S. We denote the sets of unitaries, projections, positive elements, the unit ball and the center of M by U (M), Proj(M), M+ , M1 and Z (M), respectively. For a weight φ on M and x ∈ M, we set |x|φ = φ(|x|), xφ = φ(x ∗ x)1/2 and xφ = 2−1/2 (φ(x ∗ x) + φ(x x ∗ ))1/2 . For each a ∈ M and φ ∈ M∗ , aφ, φa ∈ M∗ are defined by aφ(y) = φ(ya) and φa(y) = φ(ay) for y ∈ M. Set [φ, x] = φx − xφ. For x ∈ M+ , s(x) denotes its support projection. For von Neumann algebras M and N , we denote by Mor(M, N ) the set of unital faithful normal ∗-homomorphisms from M to N . For von Neumann algebras M and N , M ⊗ N is the tensor product von Neumann algebra. We denote by z the real part of a complex number z. We write A B when A is a finite subset of B. Let X be a linear space and consider the n-fold tensor product X ⊗n . Then the symmetric group Sn acts on X ⊗n canonically. The action is written by σ . Note that we also use the symbol σ for an irreducible representation of a compact Kac algebra.
2.2. Quick review of theory of Kac algebras. Our basic references on theory of Kac algebras are [BS] and [ES]. A compact Kac algebra is a triple G = (A, δ, h), where A is a von Neumann algebra, δ ∈ Mor(A, A ⊗ A) is a coproduct and h is a faithful invariant tracial state, that is, they satisfy (δ ⊗ id) ◦ δ = (id ⊗δ) ◦ δ, (θ ⊗ h)(δ(a)) = θ (1)h(a) = (h ⊗ θ )(δ(a)) for all θ ∈ A∗ , a ∈ A.
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Let σ ∈ Aut(A ⊗ A) be the flip automorphism. Then the map σ ◦ δ is denoted by δ opp . Let {πh , 1ˆ h , Hh } be the GNS representation of A with respect to the state h, which means Hh is a Hilbert space, πh ∈ Mor(A, B(Hh )) and 1ˆ h is the GNS cyclic vector. We always regard A as a subalgebra of B(Hh ) via the map πh . The canonical tracial weight on B(Hh ) is denoted by Tr. We define a unitary V ∈ B(Hh ⊗ Hh ) by V ∗ (x 1ˆ h ⊗ y 1ˆ h ) = δ(y)(x 1ˆ h ⊗ 1ˆ h ) for x, y ∈ A. Then V satisfies the following pentagonal equality, and is called a multiplicative unitary, V12 V13 V23 = V23 V12 . By definition, V ∈ A ⊗ B(Hh ). There exists the antipode I on A which is an antiautomorphism on A satisfying (I ⊗ id)(V ) = V ∗ . Define Aˆ by the σ -weak closure of the ˆ Then linear space {(θ ⊗ id)(V ) | θ ∈ A∗ } and a map (x) = V (x ⊗ 1)V ∗ for x ∈ A. ˆ ˆ ˆ ˆ A is actually a von Neumann algebra and ∈ Mor( A, A ⊗ A). Let ϕ be the Plancherel ˆ Then the weight on Aˆ induced by a compact Kac algebra G. In fact, ϕ = Tr holds on A. = ( A, ˆ , ϕ) is a discrete Kac algebra. The tracial weight ϕ is invariant for , triple G that is, (θ ⊗ ϕ)((x)) = θ (1)ϕ(x) = (ϕ ⊗ θ )((x)) for all θ ∈ Aˆ ∗ , x ∈ Aˆ + . ˆ we make use For a more convenient description of von Neumann algebras A and A, of unitary representations of G. Let K be a Hilbert space and v ∈ U (A ⊗ B(K )). The pair π = (v, K ) is called a unitary representation of G if (δ ⊗ id)(v) = v13 v23 . The unitary representation 1 = (1, C) is called the trivial representation. For two unitary representations π = (v, K ) and σ = (v , K ), an element S ∈ B(K , K ) is called an intertwiner from π to σ if (1 ⊗ S)v = v (1 ⊗ S). The set of intertwiners from π to σ is a linear space and denoted by (π, σ ). If (π, σ ) contains an isometry, then we write π ≺ σ . If (π, σ ) contains a unitary, π and σ are said to be equivalent and we write π ∼ σ . Of course π ≺ σ and σ ≺ π implies π ∼ σ . We define the tensor product representation , K ⊗ K ). by π · σ = (v12 v13 For a unitary representation π = (v, K ), we define the conjugate unitary representation π c = (v c , K ) as follows. Let K be the conjugate Hilbert space of K with the conjugation map j : K → K . Define the transpose map t : B(K ) → B(K ) by t (x) = j x ∗ j −1 for all x ∈ B(K ). Then set v c = (I ⊗ t)(v). The relation δ ◦ I = (I ⊗ I ) ◦ δ opp implies that π c is a unitary representation. For π = (v, K ), the set (π, π ) is a C ∗ -subalgebra of B(K ). We say that π is irreducible if (π, π ) = C. The irreducibility of π = (v, K ) implies finite dimensionality of K . We write dπ for dim K . Let σ be another unitary representation. Then the intertwiner space (π, σ ) is a Hilbert space with the inner product (S, T )1π = T ∗ S for all S, T ∈ (π, σ ), where 1π denotes the unit of B(K ). When we fix an orthonormal basis of (π, σ ), we denote it by ONB(π, σ ). We denote by Irr(G) the set of equivalence classes of all irreducible unitary representations. We denote by [π ] the equivalence class of a unitary representation π . The set Irr(G) has the conjugation operation defined by [π ] = [π c ] for all [π ] ∈ Irr(G). We fix a representative π = (vπ , Hπ ) for each [π ] ∈ Irr(G) as follows. If [π ] = [π ], we take representatives π and π c for [π ] and [π ], respectively. In this case, we often write π = (vπ , Hπ ) for π c = (vπc , Hπ ) with the conjugation jπ : Hπ → Hπ . If [π ] is self-conjugate, that is, [π ] = [π ], then we take a representative π for [π ].
Classification of Minimal Actions of a Compact Kac Algebra with Amenable Dual
491
Take a self-conjugate [π ] ∈ Irr(G). Then there exists a unitary νπ ∈ B(Hπ , Hπ ) such that vπ = (1 ⊗ νπ )vπc (1 ⊗ νπ∗ ). Taking the conjugation of both sides, we have vπc = (1 ⊗ νπ )vπ (1 ⊗ νπ ∗ ), where νπ = jπ νπ jπ ∈ B(Hπ , Hπ ). Hence νπ νπ is in (π, π ) and we have νπ νπ = π 1π or equivalently νπ νπ = π 1π for some π ∈ C with | π | = 1. Taking the conjugation of νπ νπ = π 1π , we have νπ νπ = π 1π , and hence
π = π , that is, π = ±1. We assign π = 1 for a nonself-conjugate [π ] ∈ Irr(G). Now for each [π ] ∈ Irr(G), fix a finite index set Iπ with |Iπ | = dπ and an orthonormal basis {επi }i∈Iπ of Hπ . Then we introduce another orthonormal basis of Hπ {επ i }i∈Iπ as follows. If [π ] = [π ], set επ i = jπ επi ∈ Hπ . If [π ] = [π ], then set επ i = νπ jπ επi ∈ Hπ . For each [π ] ∈ Irr(G), we define an isometric intertwiner Tπ ,π ∈ (1, π · π ) by Tπ,π =
i∈Iπ
1 √ επ i ⊗ επi . dπ
We claim the following equality:
Tπ ,π = π
i∈Iπ
1 √ επ i ⊗ επi . dπ
For a nonself-conjugate [π ], it is trivial. We verify the equality for a self-conjugate [π ]. By invariance of the summation in changing bases, we have Tπ ,π =
i∈Iπ
1 √ επ i ⊗ επi dπ
= (νπ ⊗ 1)
i∈Iπ
= (νπ ⊗ 1)
i∈Iπ
= (νπ νπ ⊗ 1) = π
i∈Iπ
1 √ jπ επi ⊗ επi dπ 1 √ jπ νπ jπ επi ⊗ νπ jπ επi dπ
i∈Iπ
1 √ επi ⊗ επi dπ
1 √ επi ⊗ επi . dπ
On Tπ,π we have the following equalities: ∗ (1π ⊗ Tπ,π )(Tπ,π ⊗ 1π ) =
π ∗ 1π = (Tπ,π ⊗ 1π )(1π ⊗ Tπ,π ). dπ
From now on, we simply write π for [π ]. Hence π means an element of Irr(G) and an irreducible unitary representation. We take the systems of matrix units {eπi, j }i, j∈Iπ and {eπ i, j }i, j∈Iπ for B(Hπ ) and B(Hπ ) coming from the bases {επi }i∈Iπ and {επ i }i∈Iπ , respectively. We decompose vπ as vπ = vπi, j ⊗ eπi, j . i, j∈Iπ
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Then the elements {vπi, j | i, j ∈ Iπ , π ∈ Irr(G)} are linearly independent and the linear span of them is σ -weakly dense in A. In fact the following orthogonal relations of them hold. For all π, ρ ∈ Irr(G), i, j ∈ Iπ and k, ∈ Iρ , h(vπ∗i, j vρk, ) = dπ−1 δπ,ρ δi,k δ j, . For each π ∈ Irr(G) and i, j ∈ Iπ , set επi, j = dπ hvπ∗i, j ∈ A∗ and put f πi, j = (επi, j ⊗ id)(V ). Then f πi, j f ρk, = δπ,ρ δ j,k f πi, and f π∗i, j = f π j,i hold. Since { f πi, j }i, j∈Iπ ,π ∈Irr(G) ˆ we have identification by putting f πi, j = eπi, j , generates A,
Aˆ =
B(Hπ ).
π ∈Irr(G)
By this identification, we have
ϕ=
dπ Tr π
π ∈Irr(G)
and V =
vπi, j ⊗ eπi, j ,
π ∈Irr(G) i, j∈Iπ
where Tr π is the non-normalized trace on B(Hπ ). We also use the tracial state τπ on ˆ we often use the notation xπ = x(1 ⊗ 1π ). The support of x ∈ Aˆ B(Hπ ). For x ∈ A, is the subset of Irr(G) which consists of π ∈ Irr(G) such that xπ = 0. The support of x ∈ Aˆ is denoted by supp(x). We write Aˆ 0 for the set of finitely supported elements of ˆ We denote by Projf( A) ˆ the set of projections in Aˆ 0 . A. The representation Aˆ ⊂ B(Hh ) is standard as is seen below. Let (πϕ , ϕ , Hϕ ) be the GNS representation of Aˆ with respect to ϕ. We define a unitary Hϕ → Hh which maps ϕ (eπi, j ) to dπ vπi, j 1ˆ h for all π ∈ Irr(G) and i, j ∈ Iπ . Then this unitary intertwines πϕ and the identity representation on Hh . Thus we always identify Hh with Hϕ . Then we obtain V (ϕ (x) ⊗ ϕ (y)) = ϕ⊗ϕ ((x)(1 ⊗ y)) for all x, y ∈ Aˆ 0 . We also use a unitary W defined by W ∗ (ϕ (x) ⊗ ϕ (y)) = ϕ⊗ϕ ((y)(x ⊗ 1)) for x, y ∈ Aˆ 0 . The unitary W also satisfies the pentagonal equality. We call V and W the right regular respectively. Then V ∈ A⊗ Aˆ and representation and the left regular representation of G, ˆ W ∈ A ⊗ A . By definition of V and W , we have V (x ⊗ 1)V ∗ = (x) = W ∗ (1 ⊗ x)W ˆ for all x ∈ A. We give a description of in terms of intertwiners. We write π ρ (x) for (x)(1π ⊗ 1ρ ) for each π, ρ ∈ Irr(G). Let π, ρ, σ ∈ Irr(G) and then π ρ (x)S
ˆ S ∈ (σ, π · ρ), = Sxσ for all x ∈ A,
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in particular, π π (x)Tπ ,π
ˆ = x1 Tπ,π for all x ∈ A.
By complete decomposability of π · ρ, 1π ⊗ 1ρ =
SS ∗ ,
σ ≺π ·ρ S∈ONB(σ,π ·ρ)
where the index σ runs in Irr(G). This equality implies that Sxσ S ∗ . π ρ (x) = σ ≺π ·ρ S∈ONB(σ,π ·ρ)
Putting x = e1 , we have π π (e1 )
∗ = Tπ ,π Tπ,π 1 = eπ ⊗ eπi, j dπ i, j i, j∈Iπ
=
1 eπ ⊗ eπi, j . dπ i, j
i, j∈Iπ
Let π, ρ and σ be elements in Irr(G). Then there exists a conjugate unitary map · from (σ , ρ · π) to (σ, π · ρ) as is defined below. For S ∈ (σ , ρ · π ), ∗ )(1σ ⊗ 1ρ ⊗Tπ∗,π ⊗ 1ρ )(1σ ⊗ S ⊗ 1π ⊗ 1ρ )(Tσ,σ ⊗ 1π ⊗ 1ρ ). S ∗ = dσ dπ dρ (1σ ⊗ Tρ,ρ The modular conjugations Jh and Jϕ are defined by Jh x 1ˆ h = x ∗ 1ˆ h ,
ˆ Jϕ ϕ (y) = ϕ (y ∗ ) for x ∈ A, y ∈ A.
ˆ define Iˆ(x) = Jh x ∗ Jh . Then Iˆ is an Set a unitary U = Jh Jϕ = Jϕ Jh . For x ∈ A, The equality (I ⊗ Iˆ)(V ) = V ˆ and called the antipode of G. antiautomorphism of A, ˆ yields Iˆ(1π ) = 1π . We often write x for Iˆ(x) for x ∈ Z ( A). Since we want to illustrate Kac algebras as function algebras on noncommutative spaces, we prepare the new notations L ∞ (G) = A,
= A, ˆ L ∞ (G)
L 2 (G) = Hh = Hϕ = L 2 (G).
ˆ is the flip automorphism Define the coproduct opp = σ ◦ , where σ ∈ Aut( Aˆ ⊗ A) opp = ( A, ˆ opp , ϕ) is called the opposite discrete Kac algebra of G. and then the triple G ×G opp = (L ∞ (G ×G opp ), opp , ϕ opp ) as We define a discrete Kac algebra G G× G G× G ×G opp ) is L ∞ (G) ⊗ L ∞ (G). The coproduct follows. The von Neumann algebra L ∞ (G opp (y) . The invariant tracial weight G opp is given by G opp (x ⊗ y) = (x)13 ×G ×G 24 ϕG opp is equal to ϕ ⊗ ϕ. ×G The map ( ⊗ id) ◦ = (id ⊗) ◦ is often denoted by (2) . For subsets F and K ⊂ Irr(G), the subset F · K ⊂ Irr(G) is defined as {π ∈ Irr(G) | π ≺ ρ · σ for some ρ ∈ F, σ ∈ K}.
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2.3. Amenability of a Kac algebra. The amenability of Kac algebras has been studied by many authors. We refer readers to [Ru] and the references therein. A discrete Kac = (L ∞ (G), , ϕ) is called amenable if there exists a left invariant state algebra G ∞ ∗ m ∈ L (G) , here the left invariance means m((θ ⊗ id)((x))) = θ (1)m(x) for all ∗ and x ∈ L ∞ (G). When G comes from a compact group, then G is amenaθ ∈ L ∞ (G) ble by [Ru, Theorem 4.5], which is directly proved by using the Kakutani-Markov fixed The amenability is also characterized point theorem due to the cocommutativity of G. by existence of projections with approximate invariance. We follow [Oc2] for a notion of approximate invariance of a projection. and ε > 0. A projection S ∈ Projf(L ∞ (G)) is Definition 2.1. Let F ∈ Projf(L ∞ (G)) said to be (F, ε)-invariant if we have |(F ⊗ 1)(S) − F ⊗ S|ϕ⊗ϕ < ε|F|ϕ |S|ϕ . satisfies the Følner condition if for any We say that a Kac algebra G and ε > 0, there exists an (F, ε)-invariant K ∈ Projf(Z (L ∞ (G))). F∈ Projf(Z (L ∞ (G))) is amenable if and only if it satisfies the Thanks to [Ru, Theorem 4.5], a Kac algebra G Følner condition. Moreover for any F, ε > 0 as above, we can take an (F, ε)-invariant such that K ≥ e1 . We sketch the proof of this fact as follows. K ∈ Projf(Z (L ∞ (G))) ∈ Projf(Z (L ∞ (G))). 1π = 0. Take an (F, ε)-invariant K Take π ∈ Irr(G) such that K and it has sufficient invariance )) ≤ 1. Actually x is in Z (L ∞ (G)) Set x = (id ⊗τπ )(( K for F. Then applying the Day-Namioka-Connes trick [C2, Theorem1.2.2], we obtain a central projection K which is sufficiently invariant for F. Moreover since xe1 = e1 , e1 still remains after the trick, that is, K ≥ e1 . 2.4. Actions and cocycle actions. Let M be a von Neumann algebra and , ϕ) a discrete Kac algebra. Let α ∈ Mor(M, M ⊗ L ∞ (G)) and u = (L ∞ (G), G ∞ ∞ on a unitary in M ⊗ L (G) ⊗ L (G). The pair (α, u) is called a cocycle action of G M if we have the following three conditions: (1) (α ⊗ id) ◦ α = Ad u ◦ (id ⊗) ◦ α, (2) (u ⊗ 1)(id ⊗ ⊗ id)(u) = (α ⊗ id ⊗ id)(u)(id ⊗ id ⊗)(u), (3) u 1,π = 1 ⊗ e1 ⊗ 1π , u π,1 = 1 ⊗ 1π ⊗ e1 for all π ∈ Irr(G). By definition, α1 = id. The unitary u is called a 2-cocycle. If u = 1, we say that α is an is defined by action. A perturbation u of u by v ∈ U (M ⊗ L ∞ (G)) u = (v ⊗ 1)(α ⊗ id)(v)u(id ⊗)(v ∗ ). Then the pair (Ad v ◦ α, u ) is a cocycle action perturbed by v. If u = 1, we say that is called a 1-cocycle or u is a 2-coboundary. For an action α, v ∈ U (M ⊗ L ∞ (G)) α-cocycle if we have (v ⊗ 1)(α ⊗ id)(v) = (id ⊗)(v). Note that if v perturbs an action (α, 1) to the action (Ad v ◦ α, 1), v is an α-cocycle. A perturbation of an α-cocycle v by w ∈ U (M) is v = (w ⊗ 1)vα(w∗ ). If v = 1, v is a 1-coboundary. We say that a cocycle is small if it is close to 1. To simplify notations, we often omit the symbol ⊗ id after symbols of cocycle actions such as α, β, γ and so on. For example, we write α(u) for (α ⊗ id ⊗ id)(u). Decompose u as u= u πi, j ,ρk, ⊗ eπi, j ⊗ eρk, . π,ρ∈Irr(G) i, j∈Iπ k, ∈Iρ
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Then each element u πi, j ,ρk, is called the entry of u. We simply say that an element x ∈ M commutes with u if it does with all entries of u. Definition 2.2. Let M be a von Neumann algebra and K a finite dimensional Hilbert space. Let α ∈ Mor(M, M ⊗ B(K )). A faithful normal unital completely positive map : M ⊗ B(K ) → M is called a left inverse of α if it satisfies ◦ α = id. We denote by Mor 0 (M, M ⊗ B(K )) the subset consisting of an element in Mor(M, M ⊗ B(K )) with a left inverse. Note that if is a left inverse of α, α ◦ is a conditional expectation from M ⊗ B(K ) to α(M). Although a left inverse is not uniquely determined in general, we always treat the following left inverses for cocycle actions. on M and π ∈ Irr(G), we define the Definition 2.3. For a cocycle action (α, u) of G map απ : M ⊗ B(Hπ ) → M by απ (x) = (1 ⊗ Tπ∗,π )u ∗π ,π (απ ⊗ id)(x)u π ,π (1 ⊗ Tπ,π ) for all x ∈ M ⊗ B(Hπ ). We simply write π for απ if no confusion arises. The next lemma shows that π is actually a left inverse of απ . on M and π ∈ Irr(G). Lemma 2.4. Let (α, u) be a cocycle action of G (1) The map π is faithful, normal, unital, and completely positive. (2) π (απ (a)bαπ (c)) = aπ (b)c for all a, c ∈ M and b ∈ M ⊗ B(Hπ ). (3) For any x ∈ M, the following equality holds: ∗ (π ⊗ id)(u π,π (x ⊗ Tπ,π Tπ,π )u ∗π,π ) = dπ−2 απ (x).
Proof. (1) It is clear that π is a normal unital completely positive map. Assume π (x ∗ x) = 0, and then we have (απ ⊗ id)(x)u π ,π (1 ⊗ Tπ ,π ) = 0. Applying απ to the first leg, we have (απ ⊗ id) ◦ απ ⊗ id (x)(απ ⊗ id ⊗ id)(u π ,π )(1 ⊗ 1π ⊗ Tπ ,π ) = 0. The equality implies ∗ ⊗ 1π )(u ∗π,π ⊗ 1π ) (απ ⊗ id) ◦ απ ⊗ id (x) 0 = (1 ⊗ Tπ,π · (απ ⊗ id ⊗ id)(u π ,π )(1 ⊗ 1π ⊗ Tπ ,π ) ∗ = (1 ⊗ Tπ,π ⊗ 1π )(id ⊗π π ⊗ id) (α ⊗ id)(x)
· (u ∗π,π ⊗ 1π )(απ ⊗ id ⊗ id)(u π,π )(1 ⊗ 1π ⊗ Tπ ,π )
∗ ⊗ 1π )(u ∗π,π ⊗ 1π )(απ ⊗ id ⊗ id)(u π ,π )(1 ⊗ 1π ⊗ Tπ ,π ) = x(1 ⊗ Tπ,π
∗ ⊗ 1π )(id ⊗π π ⊗ id)(u)(id ⊗ id ⊗π π )(u ∗ )(1 ⊗ 1π ⊗ Tπ ,π ) = x(1 ⊗ Tπ,π
∗ ⊗ 1π )(1 ⊗ 1π ⊗ Tπ,π )u π,1 = xu 1,π (1 ⊗ Tπ,π
= dπ−1 π x. Hence π is faithful.
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(2) It follows from π (απ (a)b) = (1 ⊗ Tπ∗,π )u ∗π ,π (απ ⊗ id)(απ (a)b)u π ,π (1 ⊗ Tπ ,π )
= (1 ⊗ Tπ∗,π )(id ⊗π π )(α(a))u ∗π ,π (απ ⊗ id)(b)u π ,π (1 ⊗ Tπ,π ) ∗ )u ∗π ,π (απ ⊗ id)(b)u π ,π (1 ⊗ Tπ,π ) = a(1 ⊗ Tπ,π
= aπ (b). (3) For x = 1, we have ∗ (π ⊗ id)(u π,π (1 ⊗ Tπ,π Tπ,π ))u ∗π,π )
= (π ⊗ id)(u π,π (1 ⊗ π π (e1 ))u ∗π,π )
= (1 ⊗ Tπ∗,π ⊗ 1π )(u ∗π ,π ⊗ 1π )(απ ⊗ id ⊗ id)(u π,π )
· (1 ⊗ 1π ⊗ π π (e1 ))(απ ⊗ id ⊗ id)(u ∗π,π )(u π ,π ⊗ 1π )(1 ⊗ Tπ,π ⊗ 1π )
= (1 ⊗ Tπ∗,π ⊗ 1π )(id ⊗π π ⊗ id)(u)(id ⊗ id ⊗π π )(u ∗ )
· (1 ⊗ 1π ⊗ π π (e1 ))(id ⊗ id ⊗π π )(u)(id ⊗π π ⊗ id)(u ∗ )(1 ⊗ Tπ ,π ⊗ 1π ) = (1 ⊗ Tπ∗,π ⊗ 1π )(1 ⊗ 1π ⊗ π π (e1 ))(1 ⊗ Tπ ,π ⊗ 1π )
= dπ−2 . Then the desired equality follows from ∗ (π ⊗ id)(u π,π (x⊗Tπ,π Tπ,π ))u ∗π,π )
= (π ⊗ id)(u π,π (1 ⊗ π π (e1 ))(id ⊗π π )(α(x))u ∗π,π )
= (π ⊗ id)(u π,π (1 ⊗ π π (e1 ))u ∗π,π (απ ⊗ id)(απ (x))) = (π ⊗ id)(u π,π (1 ⊗ π π (e1 ))u ∗π,π )απ (x)
= dπ−2 απ (x). On a composition of left inverses, the next lemma holds. Lemma 2.5. Let {π }π ∈Irr(G) be the left inverses of a cocycle action (α, u) as before. Then for all x ∈ M ⊗ B(Hπ ) ⊗ B(Hρ ), one has ρ (π ⊗ id)(u π,ρ xu ∗π,ρ ) =
σ ≺π ·ρ σ ∈Irr(G)
T ∈ONB(σ,π ·ρ)
dσ σ ((1 ⊗ T ∗ )x(1 ⊗ T )). dπ dρ
Remark 2.6. This summation does not depend on the choice of an orthonormal basis of (σ, π · ρ). Proof. We use the leg notations indexed by irreducible representations to represent positions in tensor products. Set Tπ ·ρ = Tπ,π Tρ,ρ = Tρ,ρ Tπ,π ∈ (1, ρ · π · π · ρ). We verify the desired equality as follows: ρ (π ⊗ id)(uxu ∗ ) = (1 ⊗ Tπ∗·ρ )u ∗ρ,ρ αρ (u ∗π,π )αρ (απ (uxu ∗ ))αρ (u π ,π )u ρ,ρ (1 ⊗ Tπ ·ρ )
= (1 ⊗ Tπ∗·ρ )u ∗ρ,ρ αρ (u ∗π,π )u ρ,π ((id ⊗ρ π ) ◦ α(uxu ∗ ))u ∗ρ,π αρ (u π ,π )u ρ,ρ (1 ⊗ Tπ ·ρ )
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= (1 ⊗ Tπ∗·ρ )u ∗ρ,ρ (id ⊗ idρ ⊗π π )(u)(id ⊗ρ π ⊗ idπ )(u ∗ )((id ⊗ρ π ) ◦ α(uxu ∗ )) · (id ⊗ρ π ⊗ idπ )(u)(id ⊗ idρ ⊗π π )(u ∗ )u ρ,ρ (1 ⊗ Tπ ·ρ ) = (1 ⊗ Tπ∗·ρ )u ∗ρ,ρ (id ⊗ρ π ⊗ idπ )(u ∗ )((id ⊗ρ π ) ◦ α(uxu ∗ ))
· (id ⊗ρ π ⊗ idπ )(u)u ρ,ρ (1 ⊗ Tπ ·ρ ) = (1 ⊗ Tπ∗·ρ )u ∗ρ,ρ (id ⊗ρ π ⊗ idπ ⊗ idρ ) (u ∗ ⊗ 1ρ )α(uxu ∗ )(u ⊗ 1ρ ) u ρ,ρ (1 ⊗ Tπ ·ρ ) = (1 ⊗ Tπ∗·ρ )u ∗ρ,ρ (id ⊗ρ π ⊗ idπ ⊗ idρ ) (id ⊗ ⊗ id)(u)(id ⊗ id ⊗)(u ∗ )α(x) · (id ⊗ id ⊗)(u)(id ⊗ ⊗ id)(u ∗ ) u ρ,ρ (1 ⊗ Tπ ·ρ ) = (1 ⊗ Tπ∗·ρ )u ∗ρ,ρ (id ⊗(2) ⊗ idρ )(u)(id ⊗ρ π ⊗ π ρ )(u ∗ )(id ⊗ρ π )(α(x)) · (id ⊗ρ π ⊗ π ρ )(u)(id ⊗(2) ⊗ idρ )(u ∗ )u ρ,ρ (1 ⊗ Tπ ·ρ ) = (1 ⊗ Tπ∗·ρ )u ∗ρ,ρ u ρ,ρ (id ⊗ρ π ⊗ π ρ )(u ∗ )(id ⊗ρ π )(α(x)) · (id ⊗ρ π ⊗ π ρ )(u)u ∗ρ,ρ u ρ,ρ (1 ⊗ Tπ ·ρ )
= (1 ⊗ Tπ∗·ρ )(id ⊗ρ π ⊗ π ρ )(u ∗ )(id ⊗ρ π )(α(x))(id ⊗ρ π ⊗ π ρ )(u)(1 ⊗ Tπ ·ρ ) = (1 ⊗ Tπ∗·ρ )(id ⊗ρ π ⊗ idπ ⊗ idρ ) (id ⊗ id ⊗π ρ )(u ∗ )α(x)(id ⊗ id ⊗π ρ )(u) · (1 ⊗ Tπ ·ρ )
=
σ ≺π ·ρ S∈ONB(σ ,ρ·π )
=
(1 ⊗ Tπ∗·ρ )(1 ⊗ S ⊗ 1π ⊗ 1ρ )(id ⊗ idσ ⊗π ρ )(u ∗ )ασ (x)
σ ≺π ·ρ S∈ONB(σ ,ρ·π )
=
σ ≺π ·ρ S∈ONB(σ ,ρ·π )
=
σ ≺π ·ρ S∈ONB(σ ,ρ·π )
=
σ ≺π ·ρ T ∈ONB(σ,π ·ρ)
· (id ⊗ idσ ⊗π ρ )(u)(1 ⊗ S ∗ ⊗ 1π ⊗ 1ρ )(1 ⊗ Tπ ·ρ ) dσ (1 ⊗ Tσ∗,σ )(1 ⊗ 1σ ⊗ S ∗ )(id ⊗ idσ ⊗π ρ )(u ∗ )ασ (x) dπ dρ S)(1 ⊗ Tσ ,σ ) · (id ⊗ idσ ⊗π ρ )(u)(1 ⊗ 1σ ⊗ dσ (1 ⊗ Tσ∗,σ )u ∗σ ,σ (1 ⊗ 1σ ⊗ S ∗ )ασ (x)(1 ⊗ 1σ ⊗ S) dπ dρ · u σ ,σ (1 ⊗ Tσ ,σ ) dσ σ ((1 ⊗ S ∗ )x(1 ⊗ S)) dπ dρ dσ σ ((1 ⊗ T ∗ )x(1 ⊗ T )). dπ dρ
Next we recall the notion of freeness for a cocycle action. on a von Neumann algebra M. Definition 2.7. Let (α, u) be a cocycle action of G Then it is said to be free if for any π ∈ Irr(G) \ {1}, there exists no nonzero element a ∈ M ⊗ B(Hπ ) with a(x ⊗ 1π ) = απ (x)a for all x ∈ M. Note that freeness is stable under perturbation, that is, a perturbed cocycle action of a free cocycle action is also free. The following lemma is essentially proved in [I1, Lemma 5.1].
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on a von Neumann algebra M. Lemma 2.8. Let (α, u) be a free cocycle action of G Then απ (M) ∩ (M ⊗ B(Hπ )) = απ (Z (M)) holds for any π ∈ Irr(G). Proof. Take an element a in απ (M) ∩ (M ⊗ B(Hπ )). Then for any x ∈ M, the equality aαπ (x) = απ (x)a holds. Applying απ to the first leg, we have (απ ⊗ idπ )(a)(απ ⊗ idπ )(απ (x)) = (απ ⊗ idπ )(απ (x))(απ ⊗ idπ )(a). By definition of a cocycle action, u ∗π ,π (απ ⊗ idπ )(a)u π ,π (id ⊗π π )(α(x))
= (id ⊗π π )(α(x))u ∗π ,π (απ ⊗ idπ )(a)u π ,π .
Then (1 ⊗ Tπ∗,π )u ∗π ,π (απ ⊗ idπ )(a)u π ,π (id ⊗π π )(α(x))
= (1 ⊗ Tπ∗,π )(id ⊗π π )(α(x))u ∗π ,π (απ ⊗ idπ )(a)u π ,π
= x(1 ⊗ Tπ∗,π )u ∗π ,π (απ ⊗ idπ )(a)u π ,π . Multiplying S ∈ (σ, π · π ) from the right, we have (1 ⊗ Tπ∗,π )u ∗π ,π (απ ⊗ idπ )(a)u π ,π (1 ⊗ S)ασ (x)
= x(1 ⊗ Tπ∗,π )u ∗π ,π (απ ⊗ idπ )(a)u π ,π (1 ⊗ S).
∗ )u ∗ (α ⊗ id )(a)u Since (1 ⊗ Tπ,π π π ,π (1 ⊗ S) ∈ M ⊗ B(Hσ , H1 ), this is equal to 0 π ,π π for σ = 1 by freeness of α. Using the equality 1π ⊗ 1π = SS ∗ , σ ≺π ·π S∈ONB(σ,π ·π )
we obtain ∗ ∗ (1 ⊗ Tπ,π )u ∗π ,π (απ ⊗ idπ )(a)u π ,π =(1⊗ Tπ,π )u ∗π ,π (απ ⊗ idπ )(a)u π ,π (1⊗ Tπ,π Tπ∗,π )
=π (a)(1 ⊗ Tπ∗,π ).
This equality implies that π (ab) = π (a)π (b) for all a, b ∈ απ (M) ∩ (M ⊗ B(Hπ )). Note that π (a) ∈ Z (M). Hence on απ (M) ∩ (M ⊗ B(Hπ )), the map π is a faithful ∗-homomorphism to Z (M). Since π maps απ (Z (M)) onto Z (M), we have απ (M) ∩ (M ⊗ B(Hπ )) = απ (Z (M)) by the faithfulness of π . As an application of the previous lemma, we can show that a free cocycle action of a discrete Kac algebra preserves a center, although this result is unnecessary for on M. By Lemma 2.8, we have our study. Let (α, u) be a free cocycle action of G απ (M) ∩(M ⊗ B(Hπ )) = απ (Z (M)). Clearly Z (M)⊗C1π ⊂ απ (M) ∩(M ⊗ B(Hπ )) holds. Hence Z (M) ⊗ C1π ⊂ απ (Z (M)). In fact, they are equal as is shown below. For any z ∈ Z (M), there exists θπ (z) ∈ Z (M) such that απ (θπ (z)) = z ⊗ 1π . Applying π to both sides, we have θπ (z) = π (z ⊗ 1π ). Thus θπ (θπ (z)) = π (θπ (z) ⊗ 1π ) = (1 ⊗ Tπ∗,π )u ∗π ,π (απ (θπ (z)) ⊗ 1π )u π ,π (1 ⊗ Tπ,π ) = (1 ⊗ Tπ∗,π )u ∗π ,π (z ⊗ 1π ⊗ 1π )u π ,π (1 ⊗ Tπ,π )
= z.
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Hence θπ ∈ Aut(Z (M)) and απ (z) = θπ (z) ⊗ 1π for all z ∈ Z (M) and π ∈ Irr(G). Moreover by definition of a cocycle action, for z ∈ Z (M) and π, σ ∈ Irr(G) we have θπ (θσ (z)) ⊗ 1π ⊗ 1σ = απ (θσ (z)) ⊗ 1σ = (απ ⊗ idσ )(ασ (z)) = u π,σ (id ⊗π σ )(α(z))u ∗π,σ = u π,σ (id ⊗π σ )(αρ (z))u ∗π,σ ρ≺π ·σ
=
ρ≺π ·σ
u π,σ θρ (z) ⊗ π σ (1ρ )u ∗π,σ .
Applying Ad u π,σ to the above equality, we obtain θρ (z) ⊗ π σ (1ρ ). θπ (θσ (z)) ⊗ 1π ⊗ 1σ = ρ≺π ·σ
Hence if ρ, ρ ≺ π · σ , then θρ = θρ . Summarizing these arguments, we have the following lemma. on M. Then there exists a map Lemma 2.9. Let (α, u) be a free cocycle action of G θ : Irr(G) → Aut(Z (M)) such that (1) απ (z) = θπ (z) ⊗ 1π for all z ∈ Z (M) and π ∈ Irr(G). (2) θπ ◦ θπ = id for all π ∈ Irr(G). (3) If ρ ≺ π · σ , then θπ ◦ θσ = θρ . In particular, α preserves the center Z (M). We close this subsection with a criterion on the invariance of a trace. Proposition 2.10. Let (α, u) be a free cocycle action on a finite von Neumann algebra M with a normalized trace τ . (1) The equality (τ ⊗ τπ ) ◦ απ ◦ π = τ ⊗ τπ holds for all π ∈ Irr(G). (2) If u = 1 and (τ ⊗ τπ ) ◦ απ = τ for all π ∈ Irr(G), then the action α preserves τ , i.e. (τ ⊗ id)(α(x)) = τ (x)1 holds for all x ∈ M. Proof. (1) Since απ (M) ∩ (M ⊗ B(Hπ )) = απ (Z (M)), απ ◦ π is the unique conditional expectation from M ⊗ B(Hπ ) to απ (M). Hence the map preserves the tracial state τ ⊗ τπ , that is, (τ ⊗ τπ ) ◦ απ ◦ π = τ ⊗ τπ . (2) By (1), the equality τ ◦ π = τ ⊗ τπ holds. Since ∗ π (x ⊗ eπi, j ) = (1 ⊗ Tπ,π )(απ (x) ⊗ eπi, j )(1 ⊗ Tπ,π )
∗ = (1 ⊗ Tπ,π )(1 ⊗ eπ i,i ⊗ 1π )(απ (x) ⊗ 1π )(1 ⊗ eπ j, j ⊗ 1π )(1 ⊗ Tπ ,π )
= dπ−1 απ i, j (x), we have dπ−1 τ (απ i, j (x)) = (τ ⊗ τπ )(x ⊗ eπi, j ) = dπ−1 δi, j τ (x).
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2.5. Crossed products and dual functionals. We collect well-known results on crossed product von Neumann algebras. Let M be a von Neumann algebra and α an action of G 2 on M. The crossed product is the von Neumann subalgebra M α G in M ⊗ B(L (G)) defined by = α(M) ∨ (C ⊗ L ∞ (G)). M α G ⊗ L ∞ (G). Then in fact the crossed prodSet an element λπ = 1 ⊗ Vπ ∈ (M α G) is the weak closure of the linear space spanned by {α(x)λπi, j | x ∈ M, uct M α G by π ∈ Irr(G), i, j ∈ Iπ }. Now define a unitary V = (1 ⊗ U )V (1 ⊗ U ), V ⊗ L 2 (G). Then V is a multiplicative unitary in where is the flip unitary on L 2 (G) ⊗ L ∞ (G) and satisfies L ∞ (G) ∗ 13 V12 V 13 = V12 V32 . V
It actually preserves M α G, )(· ⊗ 1) on M ⊗ B(L 2 (G)). Consider the action Ad(1 ⊗ V It is called the dual action of α and and the restriction gives an action of G on M α G. denoted by α. ˆ By definition, we have α(α(x)) ˆ = α(x) ⊗ 1, α(λ ˆ πi, j ) = λπi,k ⊗ vπk, j k∈Iπ
for all π ∈ Irr(G), i, j ∈ Iπ . Recall a normal functional επi, j ∈ L ∞ (G)∗ defined in §2.2. Set linear maps Pπi j and by Pπi, j = (id ⊗ id ⊗επi, j ) ◦ αˆ and Pπ = Pπ on M α G i∈Iπ Pπi,i , respectively. In particular, P1 = (id ⊗h) ◦ αˆ is a faithful normal conditional expectation from M α G α ˆ . We write E αˆ for P1 . The following equalities onto the fixed point algebra (M α G) are directly verified by easy calculation. Let π, ρ ∈ Irr(G), i, j ∈ Iπ and k, ∈ Iρ . Then we have (i) Pπi, j (α(a)λρk, α(b)) = δπ,ρ δ j, α(a)λπk,i α(b), (ii) Pπ (α(a)λρk, α(b)) = δπ,ρ α(a)λπk, α(b), (iii) Pπ2 = Pπ . onto α(M). The In particular, the map E αˆ is a conditional expectation from M α G above equalities yield ∗ (i) E αˆ α(b)λρk, α(a)λπi, j = δπ,ρ δ j, α(π (b∗ a ⊗ eπk,i )), ∗ (ii) E αˆ α(a)λπi, j α(b)λρk, = δπ,ρ δi,k δ j, dπ−1 α(ab∗ ). The first equality implies E αˆ (Pπ (x)∗ y) = E αˆ (x ∗ Pπ (y)) for all x, y ∈ M α G. −1 ˆ For a normal functional θ ∈ M∗ , put θ = θ ◦ α ◦ E αˆ . It is called a dual functional of θ . When θ is a state, θˆ is called a dual state. From the equalities on E αˆ , we have ∗ (i) θˆ α(b)λρk, α(a)λπi, j = δπ,ρ δ j, θ (π (b∗ a ⊗ eπk,i )), ∗ = δπ,ρ δi,k δ j, dπ−1 θ (ab∗ ). (ii) θˆ α(a)λπi, j α(b)λρk, ) In constructing a tower base in §5, we need these equalities for a dual state of a trace. The next proposition characterizes when a dual state of a trace is also a trace.
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Proposition 2.11. Let M be a finite von Neumann algebra with a faithful normal tracial state τ . Then the dual state τˆ is tracial if and only if all the left inverses {π }π ∈Irr(G) preserve the trace τ . If α is a free action on a finite factor, then the left inverse π preserves the trace for all π ∈ Irr(G) by Proposition 2.10. Hence a dual state of the unique tracial state is also tracial. for Finally in this subsection, we study the relative commutant α(A) ∩ (M α G) a free cocycle action (α, u) and a von Neumann subalgebra A ⊂ M. In order to avoid ambiguous arguments on formal expansion of elements in a crossed product, we intro duce the map F, which picks up coefficients of arbitrary elements of M α G. Lemma 2.12. Set L ∞ (G)π = span{vπi, j | i, j ∈ Iπ } for each π ∈ Irr(G). (1) For each π ∈ Irr(G), the elements {λπi, j }i, j∈Iπ are a basis for the M-left module α(M)(C ⊗ L ∞ (G)π ). (2) For each π ∈ Irr(G), the linear space α(M)(C ⊗ L ∞ (G)π ) is σ -weakly closed. In = α(M)(C ⊗ L ∞ (G)π ). particular, Pπ (M α G) consider the element F(x) = (Pπ (x))π in the linear (3) For any element x ∈ M α G, → space α(M)(C ⊗ L ∞ (G)π ). Then the map F : M α G π ∈Irr(G) ∞ π ∈Irr(G) α(M)(C ⊗ L (G)π ) is an injective linear map. Proof. (1), (2) We know the equality E αˆ α(a)λπi, j λ∗σk, = dπ−1 δπ,σ δi,k δ j, α(a). For define an element Q πi, j (x) ∈ M by α(Q πi, j (x)) = dπ E αˆ (xλ∗π ). Hence x ∈ M α G, i, j the map Q πi, j catches the coefficient of λπi, j , so that they give a basis over M. Indeed, the σ -weak continuity of this map shows the σ -weak closedness of α(M)(C ⊗ L ∞ (G)π ). (3) Assume Pπ (x) = 0 for all π ∈ Irr(G). Since E αˆ (Pπ (x)∗ y) = E αˆ (x ∗ Pπ (y)) for E αˆ (x ∗ Pπ (y)) = 0 for all π ∈ Irr(G) and y ∈ M α G. Since the all y ∈ M α G, π ∈Irr(G) is dense in M α G and E αˆ is faithful, linear space generated by {Pπ (M α G)} we have x = 0. Let A be a von Neumann subalgebra of M. Its global invariance for α is not assumed Since G trivially acts on α(A), it preserves R. here. Set R = α(A) ∩ (M α G). Hence the von Neumann algebra R is the weak closure of the linear space generated by {Pπ (R)}π ∈Irr(G) . Assume that for some π ∈ Irr(G), Pπ (R) is not zero. Take a α(ai,∗ j )λπi, j in Pπ (R). For x ∈ A, aα(x) = α(x)a holds. Using nonzero a = i, j∈Iπ
λπi, j (α(x)) =
k∈Iπ
Then a =
α(απi,k (x))λπk, j , we have
ai,∗ j απi,k (x) = xa ∗j,k .
i∈Iπ
ai, j ⊗ eπi, j satisfies a(x ⊗ 1π ) = απ (x)a for all x ∈ A. Summarizing
i, j∈Iπ
these arguments, we have the following lemma. Lemma 2.13. Let A be a von Neumann subalgebra of M. Then the relative commutant is not equal to α(A ∩ M) if and only if there exists π ∈ Irr(G) \ {1} α(A) ∩ (M α G) and a nonzero a ∈ M ⊗ B(Hπ ) such that a(x ⊗ 1π ) = απ (x)a for all x ∈ A.
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Applying it to the case that α is free and A = M, we can derive the following well-known results. on a von Neumann algebra M. Then it is free if Theorem 2.14. Let α be an action of G is equal to α(Z (M)). and only if the relative commutant α(M) ∩ (M α G) on a factor M of type II1 with the tracial Corollary 2.15. Let α be a free action of G is trivial. In fact, M α G is state τ . Then the relative commutant α(M) ∩ (M α G) a factor of type II1 whose tracial state is given by τˆ . 3. Ultraproduct von Neumann Algebras After Connes’s classification of cyclic group actions on the AFD factor of type II1 , we have enjoyed benefits of ultraproduct techniques in studying actions of various groups. In those circumstances, it is a key that an automorphism on a von Neumann algebra induces an automorphism on its central sequence algebra. Although general Kac algebra actions do not have such a property, the ultraproduct technique plays an essential and important role for approximately inner actions. We begin with the notion of a convergence of homomorphisms. 3.1. Liftable and semiliftable homomorphisms. Fix a free ultrafilter ω on N. For a separable von Neumann algebra M, we recall the definition of the ultraproduct von Neumann algebras M ω and Mω . Let Tω be the set of bounded sequences converging to 0 strongly* in the ultralimit. We denote by N(Tω ) the C ∗ -subalgebra of ∞ (N, M) normalizing Tω . An element (xn )n in ∞ (N, M) is called ω-centralizing if lim [φ, xn ] = 0 for all φ ∈ M∗ .
n→ω
The C ∗ -algebra of ω-centralizing sequences is denoted by Cω which is a C ∗ -subalgebra of N(Tω ). Set the quotient C ∗ -algebras M ω = N(Tω )/Tω and Mω = Cω /Tω . The quotient map is denoted by q. Then they also have the preduals and hence are von Neumann algebras. We say that (xn )n ∈ ∞ (N, M) is a representing sequence of x ∈ M ω if x = q((xn )n ). For u ∈ U (M ω ), we always take a representing sequence (u n )n of u such that u n ∈ U (M) for all n. Define a map τ ω : M ω → M by τ ω (x) = lim xn for n→ω
a representing sequence (xn )n of x ∈ M ω . The convergence is taken in the σ -weak topology of M. Then it is a faithful normal conditional expectation from M ω onto M. Note that τ ω is tracial on Mω and τ ω (Mω ) = Z (M). The restriction of τ ω on Mω is denoted by τω . Now we define the notion of convergence of homomorphisms and their left inverses. Definition 3.1. Let M be a von Neumann algebra and K a finite dimensional Hilbert space. Let αn , β ∈ Mor 0 (M, M ⊗ B(K )), n ∈ N, with left inverses n and , respectively. We say that the sequence of the pairs {(αn , n )}n∈N converges to (β, ) if lim φ ◦ n − φ ◦ = 0 for all φ ∈ M∗ .
n→∞
For a finite dimensional Hilbert space K , we always identify (M ⊗ B(K ))ω with ω M ⊗ B(K ) in a natural way. Let αn , β ∈ Mor 0 (M, M ⊗ B(K )), n ∈ N, with left inverses n , β , n ∈ N, respectively. Assume that (αn , n ) converges to (β, β ). Define the maps α : ∞ (N, M) → ∞ (N, M) ⊗ B(K ) and : ∞ (N, M) ⊗ B(K ) → ∞ (N, M) by α((xn )n ) = (αn (xn ))n , ((xn )n ) = (n (xn ))n .
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Lemma 3.2. In the above situation, the following conditions hold: (1) α(Tω ) ⊂ Tω ⊗ B(K ), (Tω ⊗ B(K )) ⊂ Tω . (2) α(N(Tω )) ⊂ N(Tω ) ⊗ B(K ), (N(Tω ) ⊗ B(K )) ⊂ N(Tω ). Proof. (1) Let (xn )n ∈ Tω with supn xn ≤ 1 and φ ∈ M∗ a faithful normal state. Then 2
2αn (xn )φ◦β = φ ◦ β (αn (xn∗ xn + xn xn∗ )) = (φ ◦ β − φ ◦ n )(αn (xn∗ xn + xn xn∗ )) + (φ ◦ n )(αn (xn∗ xn + xn xn∗ )) ≤ φ ◦ β − φ ◦ n · 2xn 2 + φ(xn∗ xn + xn xn∗ ) 2
≤ 2φ ◦ β − φ ◦ n + 2xn φ →0
as n → ω. It shows that α preserves Tω . We next show that preserves Tω . Let (xn )n ∈ Tω ⊗ B(K ) with supn xn ≤ 1 and φ be a faithful normal state on M. Then 2
2n (xn )φ = φ(n (xn∗ )n (xn ) + n (xn )n (xn∗ )) ≤ φ(n (xn∗ xn + xn xn∗ ))
= (φ ◦ n − φ ◦ β )(xn∗ xn + xn xn∗ ) + φ ◦ β (xn∗ xn + xn xn∗ ) 2
≤ 2φ ◦ n − φ ◦ β + 2xn φ◦β →0 as n → ω. Hence (Tω ⊗ B(K )) ⊂ Tω . (2) Let (xn )n ∈ N(Tω ) with supn xn ≤ 1 and (yn )n ∈ Tω ⊗ B(K ) with supn yn ≤ 1. Then we have yn αn (xn )2φ◦β = φ ◦ β (αn (xn∗ )yn∗ yn αn (xn )) = (φ ◦ β − φ ◦ n )(αn (xn∗ )yn∗ yn αn (xn )) + φ ◦ n (αn (xn∗ )yn∗ yn αn (xn )) ≤ φ ◦ β − φ ◦ n + φ(xn∗ n (yn∗ yn )xn ). By (1), (n (yn∗ yn ))n ∈ Tω . Hence the right-hand side converges to 0 as n → ω. Similarly we can show that yn∗ α(xn∗ ) converges to 0 strongly as n → ω. Hence α preserves N(Tω ). Next we show that (N(Tω ) ⊗ B(K )) ⊂ N(Tω ). Let (xn )n ∈ N(Tω ) ⊗ B(K ) with supn xn ≤ 1 and (yn )n ∈ Tω with supn yn ≤ 1. Then yn n (xn )2φ = φ(n (xn∗ )yn∗ yn n (xn ))
= φ(n (xn∗ αn (yn∗ ))n (αn (yn )xn )) ≤ φ ◦ n (xn∗ αn (yn∗ yn )xn )
= (φ ◦ n − φ ◦ β )(xn∗ αn (yn∗ yn )xn ) + φ ◦ β (xn∗ αn (yn∗ yn )xn ) ≤ φ ◦ n − φ ◦ β + αn (yn )xn 2φ◦β . Since (αn (yn ))n ∈ Tω ⊗ B(K ), the right-hand side converges to 0 as n → ω. Similarly we can show that (yn∗ n (xn∗ ))n strongly converges to 0 as n → ω. Hence (N(Tω ) ⊗ B(K )) ⊂ N(Tω ).
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By the previous lemma, we can induce the maps α : M ω → M ω ⊗ B(K ) and : M ω ⊗ B(K ) → M ω defined by α(x) = (q ⊗ id)((αn (xn ))n ), (x ⊗ y) = q((n (xn ⊗ y))n ) for all x = q((xn )n ) ∈ M ω and y ∈ B(K ). Note that preserves Mω , that is, (Mω ⊗ C) ⊂ Mω . Indeed, for any (xn )n ∈ Cω , φ ∈ M∗ and y ∈ M we have [φ, n (xn ⊗1)](y) = [φ ◦ n , xn ⊗ 1](αn (y)) and hence [φ, n (xn ⊗ 1)] ≤ [φ ◦ n , xn ⊗ 1] ≤ [φ ◦ n − φ ◦ β , xn ⊗ 1] + [φ ◦ β , xn ⊗ 1] ≤ 2φ ◦ n − φ ◦ β xn + [φ ◦ β , xn ⊗ 1] →0 as n → ω. We verify α ∈ Mor(M ω , M ω ⊗ B(K )) and is a left inverse of α. The nontrivial points are faithfulness and normality of them. Since ◦ α = id, α is faithful. For , we claim that the following equality holds: τ ω ◦ = β ◦ (τ ω ⊗ id). Once we prove this equality, the faithfulness and the normality of immediately follow. Moreover with the equality ◦ α = id, it also derives the normality of α. Now we prove the claim as follows. Let φ ∈ M∗ and x ∈ M ω ⊗ B(K ) with a representing sequence (xn )n . Then we have φ(τ ω ◦ (x)) = lim φ(n (xn )) n→ω
= lim (φ ◦ n − φ ◦ β )(xn ) + φ ◦ β (xn ) n→ω β
= φ( ((τ ω ⊗ id)(x))), where we have used lim φ◦n −φ◦β = 0 and the normality of β . We summarize n→∞ these arguments in the following lemma. Lemma 3.3. Let M be a von Neumann algebra. Consider αn , β ∈ Mor 0 (M, M ⊗B(K )), n ∈ N, with left inverses n , β , respectively. Assume that (αn , n ) converges to (β, β ) and define the maps α and as before. Then α ∈ Mor 0 (M ω , M ω ⊗ B(K )) and is a left inverse of α. Moreover we have τ ω ◦ = β ◦ (τ ω ⊗ id). Definition 3.4. Let α ∈ Mor 0 (M ω , M ω ⊗ B(K )) with a left inverse . (1) We say that the pair (α, ) is semiliftable if there exists (αn , n ) and (β, β ) which induce (α, ) as in the previous lemma. (2) We say that the pair (α, ) is liftable if the pair is semiliftable and we can take αn = β and n = β for all n ∈ N. In this case we write β ω for α. on M ω , we say that it is semiliftable or liftable if (3) For a cocycle action (β, w) of G β the pairs (βπ , π ), π ∈ Irr(G), are semiliftable or liftable, respectively. Definition 3.5. Let (α, ) be a pair of α ∈ Mor 0 (M, M ⊗ B(K )) and a left inverse of α.
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(1) We say that the pair is approximately inner if there exists a sequence of unitaries {u n }n ⊂ M ⊗ B(K ) such that the pair (Ad u n (· ⊗ 1), (id ⊗τ K ) ◦ Ad u ∗n ) converges to (α, ). on M, we say that it is an approximately inner (2) For a cocycle action (α, u) of G cocycle action if for each π ∈ Irr(G), the pair (απ , π ) is approximately inner. In the above situation, the pairs {(Ad u n , (id ⊗τ K ) ◦ Ad u ∗n )}n induce the semiliftable pair (Ad U, (id ⊗τ K ) ◦ Ad U ). By the previous lemma, we have (τ ω ⊗ τ K ) ◦ Ad U ∗ = ◦ (τ ω ⊗ id). If we apply this equality to α ω , we have the following equality by using (τ ω ⊗ id) ◦ α ω = α ◦ τ ω, (τ ω ⊗ τ K ) ◦ Ad U ∗ ◦ α ω = τ ω . Next we prepare a useful lemma which characterizes elements in Cω ⊗ B(K ). Lemma 3.6. Let M be a von Neumann algebra, K a finite dimensional Hilbert space and τ K a normalized tracial state on B(K ). Then an element (an )n ∈ ∞ (N, M)⊗ B(K ) is in Cω ⊗ B(K ) if and only if lim [ψ ⊗ τ K , an ] = 0, ψ ∈ M∗ . n→ω
Proof. The following equality is easily verified. [ψ ⊗ τ K , an ] =
[ψ, (an )i, j ] ⊗ (τ K )ei, j ,
i, j
where {ei, j }i, j is a matrix unit of B(K ). Hence if (an )n ∈ Cω ⊗ B(K ), then lim [ψ ⊗ n→∞
τ K , an ] = 0. Suppose lim [ψ ⊗ τ K , an ] = 0. Since [ψ, (an )i, j ](y) = [ψ ⊗ n→∞
τ K , an ](y ⊗ e j,i ), we have [ψ, (an )i, j ] ≤ [ψ ⊗ τ K , an ]. Hence (an )n ∈ Cω ⊗ B(K ). If (α, ) is approximately inner, there exists a unitary U ∈ M ω ⊗ B(K ) as before. Then consider the map γ = Ad U ∗ ◦ α ω . By definition of the approximate innerness, (γ , ◦ Ad U ) is semiliftable. Indeed, the pairs (Ad u ∗n ◦ α, ◦ Ad u n ) converge to (· ⊗ 1, id ⊗τ K ). Since α = Ad U (· ⊗ 1) on M, γ fixes M and hence preserves M ∩ M ω . In fact, γ preserves Mω as is shown in the following lemma. Lemma 3.7. Let α ∈ Mor 0 (M, M ⊗ B(K )) be a ∗-homomorphism with a left inverse . Assume that the pair (α, ) is approximately inner. Take a sequence of unitaries {u n }n ⊂ M ⊗ B(K ) with lim (φ ⊗ τ K ) ◦ Ad u ∗n − φ ◦ = 0 for all φ ∈ M∗ .
n→∞
Set U := (u n )n ∈ ∞ (N, M ⊗ B(K )) and then the following statements hold: (1) U ∈ N(Tω ) ⊗ B(K ). (2) The ∗-homomorphisms Ad U ∗ ◦ α and Ad U ∗ (· ⊗ 1) from ∞ (N, M) to ∞ (N, M) ⊗ B(K ) preserve Tω , N(Tω ) and Cω .
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Proof. (1) Let (xn )n ∈ Tω ⊗ B(K ). Fix a faithful normal state φ on M. We show lim u n xn φ◦ = lim xn u n φ⊗τ K = 0. It is trivial that lim u n xn φ⊗τ K = 0 = n→ω
n→ω
n→ω
lim u ∗n xn∗ φ⊗τ K . On xn u n φ⊗τ K , we have
n→ω
xn u n 2φ⊗τ K = (φ ⊗ τ K )(u ∗n xn∗ xn u n )
= (φ ⊗ τ K ) ◦ Ad u ∗n (xn∗ xn ) = ((φ ⊗ τ K ) ◦ Ad u ∗n − φ ◦ )(xn∗ xn ) + φ ◦ (xn∗ xn ) ≤ (φ ⊗ τ K ) ◦ Ad u ∗n − φ ◦ xn 2 + xn 2φ◦
→0 as n → ω. Next we show that {xn∗ u ∗n } converges to 0 σ -strongly as follows: xn∗ u ∗n 2φ◦ = φ((u n xn xn∗ u ∗n ))
= (φ ◦ − (φ ⊗ τ K ) ◦ Ad u ∗n )(u n xn xn∗ u ∗n ) + (φ ⊗ τ K )(xn xn∗ ) ≤ φ ◦ − (φ ⊗ τ K ) ◦ Ad u ∗n xn 2 + xn∗ 2φ⊗τ K
→0 as n → ω. Hence U = (u n )n ∈ N(Tω ) ⊗ B(K ). (2) Set γn1 := Ad u ∗n ◦α and n (x) := (u n xu ∗n ). Note that n is a left inverse of γn1 . By assumption, we have lim ψ ◦ n − ψ ⊗ τ K = 0. Let (an )n ∈ Cω be a centralizing n→∞
sequence. By the previous lemma, it suffices to show lim [γn1 (an ), ψ ⊗ τ K ] = 0. We may assume an ≤ 1. Then for x ∈ M ⊗ B(K ),
n→ω
|[γn1 (an ), ψ ⊗ τ K ](x)| = |[γn1 (an ), ψ ◦ n ](x)| + |[γn1 (an ), ψ ⊗ τ K − ψ ◦ n ](x)| ≤ [an , ψ]n (x) + 2ψ ⊗ τ K − ψ ◦ n x ≤ [an , ψ]x + 2ψ ⊗ τ K − ψ ◦ n x. Hence we have [γn1 (an ), ψ ⊗ τ K ] ≤ [an , ψ] + 2ψ ⊗ τ K − ψ ◦ n , and it follows that lim [γn1 (an ), ψ ⊗ τ K ] = 0 because (an )n ∈ Cω . n→ω
Next set γn2 = Ad u ∗n (·⊗1) ∈ Mor(M, M ⊗ B(K )). Let (an )n ∈ Cω with supn an ≤ 1. Then [φ ⊗ τ K , γn2 (an )] ≤ [φ ⊗ τ K − φ ◦ n , γn2 (an )] + [φ ◦ n , γn2 (an )] ≤ 2φ ⊗ τ K − φ ◦ n + [φ ◦ , an ⊗ 1] ◦ Ad u n = 2φ ⊗ τ K − φ ◦ n + [φ ◦ , an ⊗ 1] →0 as n → ω.
In the end of this subsection, we state a simple criterion of approximate innerness of homomorphisms of the AFD factor R0 of type II1 . For the sake of this, we prove the following lemma. Lemma 3.8. Let α, αn ∈ Mor 0 (M, M ⊗ B(K )), n ∈ N with left inverses and n , n ∈ N, respectively. Fix a faithful normal state φ ∈ M∗ . The following conditions are equivalent:
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(1) lim ψ ◦ n − ψ ◦ = 0 for all ψ ∈ M∗ . n→∞
(2) lim φ ◦ n − φ ◦ = 0 and lim αn (a) = α(a) strongly for all a ∈ M. n→∞
n→∞
√ Proof. First note that ψa ≤ ψa, aψ ≤ ψa, φa ≤ φa ∗ φ and a2φ ≤ aφa for ψ ∈ M∗ and a positive φ ∈ M∗ . (1) ⇒ (2). We will show lim αn (a) − α(a)φ◦ = 0 as follows: n→∞
αn (a) − α(a)2φ◦ ≤ (αn (a) − α(a)) (φ ◦ ) αn (a) − α(a) ≤ 2aαn (a) · (φ ◦ − φ ◦ n ) + 2aαn (a) · (φ ◦ n ) − α(a) · (φ ◦ ) ≤ 2a2 φ ◦ − φ ◦ n + 2a(aφ) ◦ n − (aφ) ◦ →0 as n → ∞. (2) ⇒ (1). At first we verify lim (φa) ◦ − (φa) ◦ n = 0. This is shown as n→∞ follows: (φa) ◦ − (φa) ◦ n = (φ ◦ ) · α(a) − (φ ◦ n ) · αn (a) ≤ (φ ◦ ) · α(a) − (φ ◦ ) · αn (a) + (φ ◦ ) · αn (a) − (φ ◦ n ) · αn (a) ≤ α(a ∗ ) − αn (a ∗ )φ◦ + φ ◦ − φ ◦ n a →0 as n → ∞. Since {φa}a∈M is dense in M∗ , we are done.
Lemma 3.9. Let R0 be the AFD factor of type II1 with the tracial state τ . Let α ∈ Mor(R0 , R0 ⊗ B(K )) with a left inverse , where K is a finite dimensional Hilbert space. Assume that preserves the trace, i.e. τ ◦ = τ ⊗ τ K . Then the pair (α, ) is approximately inner. Proof. Let M1 ⊂ M2 ⊂ . . . be an ascending sequence of finite dimensional subfactors of R0 whose union is dense in R0 . Let {ei,n j }i, j be a matrix unit for Mn . By the uniqueness n ) and en ⊗ 1 of the trace on R0 , (τ ⊗ τ K ) ◦ α = τ holds. Hence the projections α(ei,i i,i are equivalent in R0 ⊗ B(K ) for all i. Take an element i 0 and a partial isometry u n in R0 ⊗ B(K ) such that u n u ∗n = α(ein0 ,i0 ) and u ∗n u n = ein0 ,i0 ⊗ 1. Set un =
n α(ei,i )u n (ein0 ,i ⊗ 1). 0
i
Then it is a unitary in R0 ⊗ B(K ) and α(x) = Ad u n (x ⊗ 1) holds for all x ∈ Mn . Set n = (id ⊗τ K ) ◦ Ad u ∗n . We show that the pair (Ad u n (· ⊗ 1), n ) converges to (α, ). It is easy to see that Ad u n (x ⊗ 1) strongly converges to α(x) for all x ∈ R0 . The condition lim τ ◦ n − τ ◦ = 0 is trivial because the maps n and preserve n→∞
the trace τ . Hence by the previous lemma, (α, ) is approximately inner.
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3.2. Strongly free cocycle actions. In this paper, we frequently make use of the next two results. Since they are proved in a similar way to proofs of [Oc1, Lemma 5.3, Lemma 5.5] with a little modification, we omit proofs. Lemma 3.10 (Fast Reindexation Trick). Let M be a von Neumann algebra. Let N and S be countably generated von Neumann subalgebras of M ω , and B a countable family of liftable homomorphisms whose elements are of the form β ω with β ∈ Mor 0 (M, M ⊗ B(K β )), where K β is a finite dimensional Hilbert space. Take a countably generated ⊂ M ω satisfying β ω (N ) ⊂ N ⊗ B(K β ) for all β ω ∈ B von Neumann subalgebra N ω . Then there exists a map ∈ Mor( N , M ) such that and N ⊂ N ∩ M, (1) is identity on N ∩ Mω ) ⊂ S ∩ Mω , (2) ( N , a ∈ S, (3) τ ω (a(x)) = τ ω (a)τ ω (x) for all x ∈ N (4) β ω ((x)) = ( ⊗ id)(β ω (x)) for all x ∈ N , β ω ∈ B. Lemma 3.11 (Index Selection Trick). Let M be a von Neumann algebra. Let C be a separable C ∗ -subalgebra of ∞ (N, M ω ), and B a countable family of semiliftable homomorphisms, which is of the form β ∈ Mor(M ω , M ω ⊗ B(K β )) with a finite dimensional Hilbert space K β and also acts term by term on ∞ (N, M ω ). Take a C ∗ -algebra C containing C and preserved by each β ∈ B. Then there exists a ∗-homomorphism : C→ M ω such that for any x = (xn )n ∈ C, (1) τ ω (( x )) = lim τ ω (xn ) weakly, n→ω
(2) ( x ) = x if xn = x for all n, (3) ( x ) ∈ Mω if xn ∈ Mω for all n, (4) ( ⊗ id)( y) = β(( x )) for β ∈ B and y = (β(xn ))n ∈ ∞ (N, M ω ⊗ B(K β )). We define the notion of strong outerness of homomorphisms, which plays a central role in making Rohlin projections. Definition 3.12. Let M be a von Neumann algebra and γ ∈ Mor(M ω , M ω ⊗ B(K )). Then we say that γ is strongly outer on Mω (or simply strongly outer) if for any countably generated von Neumann subalgebra S ⊂ M ω , there exists no nonzero element a ∈ M ω ⊗ B(K ) with γ (y)a = a(y ⊗ 1) for all y ∈ S ∩ Mω . For β ∈ Mor 0 (M, M ⊗ B(K )), β is said to be strongly outer if β ω is strongly outer on Mω . Actually the strong outerness does not depend on ω ∈ βN \ N. Definition 3.13. Let M be a von Neumann algebra and (α, u) a cocycle action of G on M. We say that (α, u) is strongly free if απ is strongly outer for any nontrivial π ∈ Irr(G). It is easy to see that strong freeness implies freeness, but the converse does not hold in general. Note that if M is the AFD factor of type II1 , then the strong freeness and freeness of a cocycle action are equivalent, which is obtained by a similar argument to [C3, Lemma 3.4] (see Corollary 8.6 in Appendix). 4. Cohomology Vanishing I = (L ∞ (G), , ϕ) an amenable discrete Kac Let M be a von Neumann algebra and G on M. On M ⊗ B(L 2 (G)), we set the algebra. Consider a cocycle action (α, u) of G
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map γ = σ23 ◦ (α ⊗ id) and the unitary u 134 . Then (γ , u 134 ) is a cocycle action on It is well-known that (γ , u 134 ) is stabilized to an action. However if M ⊗ B(L 2 (G)). Irr(G) is infinite, the stabilization is not appropriate for our work on finite von Neumann algebras. The amenability gives us a prescription for the problem. Indeed, by the ame we can take a sufficiently large finitely supported projection in L ∞ (G). nability of G, by the projection and stabilize a 2-cocycle approximately. We cut B(L 2 (G))
4.1. 2-cohomology vanishing in ultraproduct von Neumann algebras. By making use for an (F, δ)-invariant projection K we can conclude of the relation ϕ = Tr on L ∞ (G), the approximate commutativity of F ⊗ K and the multiplicative unitary W . Lemma 4.1. Let F, K ∈ Projf(Z (L ∞ (G))). If K is (F, δ)-invariant, then (1) (2) (3) (4)
W (F ⊗ K ) − (F ⊗ K )W ϕ⊗Tr < δ 1/2 Fϕ K ϕ , (F ⊗ K )W (F ⊗ K ) − (F ⊗ K )W ϕ⊗Tr < δ 1/2 Fϕ K ϕ , (F ⊗ K )W (F ⊗ K ) − W (F ⊗ K )ϕ⊗Tr < δ 1/2 Fϕ K ϕ , (F ⊗ K )W ∗ (F ⊗ K ) · (F ⊗ K )W (F ⊗ K ) − F ⊗ K ϕ⊗Tr < δ 1/2 Fϕ K ϕ .
Proof. (1) Use Powers-Størmer inequality ([PS]) to the left-hand side of |W (F ⊗ K )W ∗ − (F ⊗ K )|ϕ⊗Tr < δ|F|ϕ |K |ϕ . This inequality immediately implies conditions (2) and (3). (4) This is shown as follows: (F ⊗ K )W ∗ (F ⊗ K ) · (F ⊗ K )W (F ⊗ K ) − F ⊗ K ϕ⊗Tr = (F ⊗ K )(K )(F ⊗ K ) − F ⊗ K ϕ⊗ϕ ≤ F ⊗ K (F ⊗ 1)(K ) − F ⊗ K ϕ⊗ϕ < δ 1/2 Fϕ K ϕ . The next lemma shows that we can perturb a given 2-cocycle to a smaller 2-cocycle. The outline is as follows. Let F, K be projections as in the previous lemma. By [J, Lemma ⊗ K B(L 2 (G))K 3.2.1], we can take a unitary v ∈ L ∞ (G)F with v − (F ⊗ K )W (F ⊗ K )ϕ⊗Tr < 4δ 1/2 Fϕ K ϕ . If F ≥ e1 , v is taken as v(e1 ⊗ K ) = e1 ⊗ K . The unitary v plays a role of finite dimensional cut of the left regular representation W . The error coming from the cut is controlled by the trace norm, that is, strong operator topology. By making use of the unitary v, we can approximately stabilize a 2-cocycle for a cocycle action on a von Neumann algebra of type II1 . Lemma 4.2. Let M be a von Neumann algebra of type II1 with a faithful tracial state τ . on M. Then for any ε > 0 and F ∈ Projf(Z (L ∞ (G))), Let (α, u) be a cocycle action of G satisfying w1 = 1 ⊗ e1 and there exists a unitary w ∈ M ⊗ L ∞ (G)
(w ⊗ 1)α(w)u(id ⊗)(w∗ ) − 1 (1 ⊗ F ⊗ F)
< ε. τ ⊗ϕ⊗ϕ
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Proof. Step A. We construct a unitary close to a cut of W which acts on sufficiently large finite dimensional subspace of L 2 (G). Let F be the support of F and F the central projection whose support is F · F. We holds. Let δ > 0 with 21δ 1/2 Fϕ F ϕ < ε and K a may assume 1 ∈ F, and F ≤ F 2 finitely supported central ( F, δ)-invariant projection. Set H = K L (G). Let τH be the normalized trace on B(H). From now in this proof, we use τH on B(H) for measure F ⊗ B(H) ments of norms. Then by [J, Lemma 3.2.1], we can take a unitary v in L ∞ (G) satisfying
v − ( F ⊗ K )W ( F ⊗ K )
ϕ⊗τH
. < 4δ 1/2 F ϕ
By (2) and (3) in Lemma 4.1, we also have the inequalities ⊗ K )W ϕ⊗τH < 5δ 1/2 F ϕ, v − ( F 1/2 ϕ. ⊗ K )ϕ⊗τH < 5δ F v − W ( F Step B. We regard B(H) ⊂ M and perturb (α, u) to ( α, u ) which fixes B(H) ⊂ M. Since M is of type II1 , we can take a unital embedding B(H) into M. Let {ei, j }i, j be a system of matrix units generating B(H). For all π ∈ Irr(G), {απ (ei, j )}i, j is also a system of matrix units in M ⊗ B(Hπ ). Hence projections απ (ei,i ) and e j, j ⊗ 1π are equivalent. Then there exists a unitary wπ in M ⊗ B(Hπ ) such that απ = Ad w∗π (· ⊗ 1π ) on B(H). Set w = (w π )π ∈Irr(G) and we have α = Ad w ∗ (· ⊗ 1) on B(H). Then perturb (α, u) by the unitary w and we obtain a cocycle action ( α, u ) on M which fixes B(H). in order to approximately stabilize the 2-cocycle Step C. We utilize B(H) like B(L 2 (G)) u. Let M = N ⊗ B(H) be the tensor product decomposition. Since α fixes B(H), ⊗ L ∞ (G). Then we have a cocycle action (γ , u) of G on N such u ∈ N ⊗ C ⊗ L ∞ (G) ⊗ B(H). ⊥ ⊗ K ∈ L ∞ (G) that α = σ23 ◦ (γ ⊗ id) and u = σ234 (u ⊗ 1H). Set v = v + F defined by w = σ23 ((1 ⊗ v)u ∗ ). Then Consider a unitary w in M ⊗ L ∞ (G) (w F ⊗ F) α F (w F ) u (id ⊗ F F )(w ∗ ) = (1 ⊗ 1 ⊗ F ⊗ F)σ23 ((1 ⊗ v)u ∗ ⊗ 1) · σ23 ((γ ⊗ id ⊗ id)(σ23 ((1 ⊗ v)u ∗ ))) · σ234 (u ⊗ 1) · σ234 ((id ⊗ ⊗ id)(u(1 ⊗ v ∗ ))) = (1 ⊗ 1 ⊗ F ⊗ F)σ23 ((1 ⊗ v)u ∗ ⊗ 1) · σ23 σ34 ((1 ⊗ 1 ⊗ v)γ (u ∗ )) · σ234 (u ⊗ 1) · σ234 ((id ⊗ ⊗ id)(u(1 ⊗ v ∗ ))) = σ234 (1 ⊗ F ⊗ F ⊗ 1)((1 ⊗ v)u ∗ ⊗ 1)1243 · (1 ⊗ 1 ⊗ v) · γ (u ∗ )(u ⊗ 1) · (id ⊗ ⊗ id)(u(1 ⊗ v ∗ )) = σ234 (1 ⊗ F ⊗ F ⊗ 1)((1 ⊗ v)u ∗ ⊗ 1)1243 · (1 ⊗ 1 ⊗ v) · (id ⊗ id ⊗)(u)(1 ⊗ ( ⊗ id)(v ∗ )) ∗ ∗ = σ234 (1 ⊗ F ⊗ F ⊗ 1)v24 v34 · v34 u 124 v34 · (id ⊗ id ⊗)(u)(1 ⊗ ( ⊗ id)(v ∗ )) . (4.1)
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We estimate the size of the difference of the right-hand side and 1 ⊗ K ⊗ F ⊗ F as follows. Since (F ⊗ F ⊗ K )(W13 W23 − ( ⊗ id)(v))(F ⊗ F ⊗ K )ϕ⊗ϕ⊗τH = (F ⊗ F ⊗ K )(( ⊗ id)(W ) − ( ⊗ id)(v))(F ⊗ F ⊗ K )ϕ⊗ϕ⊗τH = ( F F ⊗ id)(v − (1 ⊗ K )W (1 ⊗ K ))ϕ⊗ϕ⊗τH
⊗ K )W ( F ⊗ K ))
= ( F F ⊗ id)(v − ( F ϕ⊗ϕ⊗τH
⊗ K )W ( F ⊗ K ))
≤ ( F ⊗ id)(v − ( F ϕ⊗ϕ⊗τH
= Fϕ v − ( F ⊗ K )W ( F ⊗ K ) ϕ⊗τ H
ϕ, < 4δ 1/2 Fϕ F we have
(F ⊗ F ⊗ K )(v 13 v 23 − ( ⊗ id)(v))ϕ⊗ϕ⊗τH ≤ (F ⊗ F ⊗ K )(v13 − W13 )v23 ϕ⊗ϕ⊗τH + (F ⊗ K ⊗ K )W13 (v23 − W23 (1 ⊗ F ⊗ K ))ϕ⊗ϕ⊗τH + (F ⊗ F ⊗ K )(W13 W23 − ( ⊗ id)(v))(F ⊗ F ⊗ K )ϕ⊗ϕ⊗τH < Fϕ v − (F ⊗ K )W (F ⊗ K )ϕ⊗τH + Fϕ v − W (F ⊗ K )ϕ⊗τH ϕ + 4δ 1/2 Fϕ F ϕ < 4δ 1/2 F2ϕ + 5δ 1/2 F2ϕ + 4δ 1/2 Fϕ F ϕ. ≤ 13δ 1/2 Fϕ F
(4.2)
Set u K = u(1 ⊗ 1 ⊗ K ) and then ∗ ∗ (1 ⊗ F ⊗ F ⊗ 1)(v34 u 124 v34 − (id ⊗ id ⊗)(u K∗ ))τ ⊗ϕ⊗ϕ⊗ϕ ∗ ∗ ≤ (1 ⊗ F ⊗ F ⊗ K )(v34 − ((F ⊗ K )W ∗ (F ⊗ K ))34 u 124 v34 τ ⊗ϕ⊗ϕ⊗τH
∗ ∗
+ (1 ⊗ F ⊗ F ⊗ K )W34 u 124
· ((F ⊗ K )v(F ⊗ K ))34 − ((F ⊗ K )W (F ⊗ K ))34 τ ⊗ϕ⊗ϕ⊗τ H
∗ + (1 ⊗ F ⊗ F ⊗ K )(W34 (u K∗ )124 W34 − (id ⊗ id ⊗)(u K∗ ))τ ⊗ϕ⊗ϕ⊗τH ≤ Fϕ (F ⊗ K )v(F ⊗ K ) − (F ⊗ K )W (F ⊗ K )ϕ⊗τH + Fϕ (F ⊗ K )v(F ⊗ K ) − (F ⊗ K )W (F ⊗ K )ϕ⊗τH
< 4δ 1/2 F2ϕ + 4δ 1/2 F2ϕ ϕ. = 8δ 1/2 Fϕ F
(4.3)
By using (4.1), (4.2) and (4.3), we have (w F ⊗ F) α F (w F ) u (id ⊗ F F )(w ∗ ) − 1 ⊗ K ⊗ F ⊗ Fτ ⊗ϕ⊗ϕ
∗ ∗ = σ234 (1 ⊗ F ⊗ F ⊗ 1)v24 v34 · v34 u 124 v34 · (id ⊗ id ⊗)(u)(1 ⊗ ( ⊗ id)(v ∗ ))
− 1 ⊗ K ⊗ F ⊗ F
∗ ∗ = v24 v34 · v34 u 124 v34
τ ⊗τH ⊗ϕ⊗ϕ
· (id ⊗ id ⊗)(u)(1 ⊗ ( ⊗ id)(v ∗ ))(1 ⊗ F ⊗ F ⊗ K ) − 1 ⊗ F ⊗ F ⊗ K τ ⊗ϕ⊗ϕ⊗τH
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ϕ < 13δ 1/2 Fϕ F ∗ ∗ + (1 ⊗ ( ⊗ id)(v)) · v34 u 124 v34 · (id ⊗ id ⊗)(u)(1 ⊗ ( ⊗ id)(v ∗ )) · (1 ⊗ F ⊗ F ⊗ K ) − 1 ⊗ F ⊗ F ⊗ K τ ⊗ϕ⊗ϕ⊗τH 1/2 ϕ + 8δ 1/2 Fϕ F ϕ < 13δ Fϕ F ϕ = 21δ 1/2 Fϕ F < ε. We have shown that any 2-cocycle can be approximately stabilized in Lemma 4.2. When we consider the stabilization problem in an ultraproduct von Neumann algebra, approximate stabilization yields exact stabilization by the Index Selection Trick. Lemma 4.3. Let M be a von Neumann algebra such that Mω is of type II1 and (γ , w) a ∞ (G). on M ω preserving Mω and w ∈ Mω ⊗L ∞ (G)⊗L Assume that cocycle action of G ω ∞ and β ∈ Mor(M ω , M ω ⊗ γ is of the form γ = Ad U ◦ β, where U ∈ U (M ⊗ L (G)) with a semiliftable βπ for all π . Then the 2-cocycle w is a coboundary in Mω . L ∞ (G)) ∞ Proof. Take an increasing sequence of projections {Fn }∞ n=1 in Projf(Z (L (G))) with ∞ Fn → 1 strongly, and decreasing positive numbers {εn }n=1 with εn → 0. Let φ be a faithful normal state on M and set ψ = φ ◦ τ ω , which is a trace on Mω . By using the satisfying previous lemma, for each n ∈ N we can find a unitary vn in Mω ⊗ L ∞ (G)
(vn ⊗ 1)γ (vn )w(id ⊗)(v ∗ ) − 1 (1 ⊗ Fn ⊗ Fn )
< εn . n ψ⊗ϕ⊗ϕ
and w = (U )n in ∞ (N, M ω ⊗ L ∞ (G)) Then set a unitary v = (vn )n and U = ∞ ω ∞ ∞ Let C be a C ∗ -subalgebra generated by (w)n in (N, M ⊗ L (G) ⊗ L (G)). vπi, j , πi, j , w πi, j ,ρk, for all π, ρ ∈ Irr(G), i, j ∈ Iπ and k, ∈ Iρ . Let B = {β}. Then applyU ing the Index Selection Trick, we have a ∗-homomorphism : C→ M ω satisfying the and conditions in Lemma 3.11 for C and B. Set v = ( ⊗ id)( v ) ∈ Mω ⊗ L ∞ (G) x = ( v ⊗ 1)γ ( v ) w (id ⊗)( v ∗ ) − 1 ⊗ 1 ⊗ 1. By definition of , ( ⊗ id ⊗ id)( x ) = (v ⊗ 1)γ (v)w(id ⊗)(v ∗ ) − 1 ⊗ 1 ⊗ 1. The right-hand side is equal to 0. Indeed for any π, ρ ∈ Irr(G),
(vπ ⊗ 1ρ )γπ (vρ )wπ,ρ (id ⊗π ρ )(v ∗ ) − 1 ⊗ 1π ⊗ 1ρ 2 ψ⊗ϕ⊗ϕ = (ψ ⊗ ϕ ⊗ ϕ)(|( ⊗ id ⊗ id)( xπ,ρ )|2 ) xπ,ρ |2 ) = (φ ◦ τ ω ◦ ⊗ ϕ ⊗ ϕ)(| xn )π,ρ |2 ) = lim (φ ◦ τ ω ⊗ ϕ ⊗ ϕ)(|( n→ω
≤ lim εn2 n→ω
= 0.
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The previous 2-cohomology vanishing result yields two results about approximately inner (cocycle) actions, which play crucial roles in our study. We separately discuss them in the following subsections. We prepare the equivalence relation ∼ for sequences of normal functionals. Let (φn )n and (ψn )n be sequences of normal functionals on a von Neumann algebra. We write (φn )n ∼ (ψn )n or simply φn ∼ ψn when lim φn−ψn = 0. n→ω
4.2. Cocycle actions on central sequence algebras. Let M be a von Neumann algebra on such that Mω is of type II1 and (α, u) an approximately inner cocycle action of G α M. Let π be the left inverse of (α, u) and π ∈ Irr(G). Then there exists a unitary satisfying the conditions in Lemma 3.7. Then α = Ad U (· ⊗ 1) U ∈ M ω ⊗ L ∞ (G) on M. Set γ = Ad U ∗ ◦ α ω and a unitary w = (U ∗ ⊗ 1)α ω (U ∗ )u(id ⊗)(U ). Then (γ , w) is a cocycle action on M ω fixing M. Note that each γπ is semiliftable. Indeed, let (u n )n be a representing sequence of U . By the proof of Lemma 3.7, we know that (Ad u ∗n ◦ απ , απ ◦ Ad u n ) converges to (· ⊗ 1π , id ⊗τπ ). This result and the equality ω γ γ π = απ ◦ Ad U yield the semiliftability of (γπ , π ). The map γπ preserves Mω by Lemma 3.7. Since γ fixes M, w ∈ M ∩ M ω ⊗ L ∞ (G)⊗ but in fact w is a 2-cocycle whose entries are evaluated in Mω . L ∞ (G), ⊗ L ∞ (G). Lemma 4.4. The unitary w is in Mω ⊗ L ∞ (G) Proof. Let π, ρ ∈ Irr(G) and φ ∈ M∗ . Let (U n )n be a representing sequence of U and set w n = (U n∗ ⊗ 1)α ω (U n∗ )u(id ⊗)(U n ). Then (w n )n is a representing sequence of n w. We show that lim [φ ⊗ τπ ⊗ τρ , wπ,ρ ] = 0. This is verified as follows: n→ω
n∗ n (φ ⊗ τπ ⊗ τρ )wπ,ρ wπ,ρ
= (id ⊗)(U n∗ )u ∗ α ω (U n )(U n ⊗ 1)(φ ⊗ τπ ⊗ τρ )(U n∗ ⊗ 1)α ω (U n∗ )u(id ⊗)(U n ) ω
∼ (id ⊗)(U n∗ )u ∗ α ω (U n )(φ ◦ απ ⊗ τρ )α ω (U n∗ )u(id ⊗)(U n ) ω = (id ⊗)(U n∗ )u ∗ (U (φ ⊗ τρ )U ∗ ) ◦ (απ ⊗ idρ ) u(id ⊗)(U n ) ω ∼ (id ⊗)(U n∗ )u ∗ φ ◦ αρ ◦ (απ ⊗ idρ ) u(id ⊗)(U n ) dσ ω (id ⊗)(U n∗ ) (1 ⊗ T )φ ◦ ασ (1 ⊗ T ∗ ) (id ⊗)(U n ) = dπ dρ σ ≺π ·ρ =
T ∈ONB(σ,π ·ρ)
σ ≺π ·ρ T ∈ONB(σ,π ·ρ)
∼
σ ≺π ·ρ T ∈ONB(σ,π ·ρ)
dσ ω (1 ⊗ T )φ ◦ (Uσn∗ ασ Uσn )(1 ⊗ T ∗ ) dπ dρ dσ (1 ⊗ T )(φ ⊗ τσ )(1 ⊗ T ∗ ) dπ dρ
= φ ⊗ τπ ⊗ τρ , where we have used the composition rule of the left inverses in Lemma 2.5 and the property of U in Lemma 3.7. Hence the restriction of (γ , w) on Mω is a cocycle action. Since γ is semiliftable, we such that can apply Lemma 4.3. Hence there exists a unitary v ∈ Mω ⊗ L ∞ (G) (v ⊗ 1)γ (v)w(id ⊗)(v ∗ ) = 1.
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Since w = (U ∗ ⊗ 1)α ω (U ∗ )u(id ⊗)(U ), we have (vU ∗ ⊗ 1)α ω (vU ∗ )u(id ⊗)(U v ∗ ) = 1. Setting V = U v ∗ , we have the following lemma. on a von Neumann Lemma 4.5. Let (α, u) be an approximately inner cocycle action of G algebra M such that Mω is of type II1 . Then there exists a unitary V ∈ M ω ⊗ L ∞ (G) such that (1) For a representing sequence (vn )n of V , we have lim (φ ⊗ τπ ) ◦ Ad vn∗ − φ ◦ απ = 0 for all φ ∈ M∗ , π ∈ Irr(G),
n→ω
where απ is the left inverse of (α, u). on M ω fixing M and preserving Mω . (2) γ = Ad V ∗ ◦ α ω is an action of G ∗ (3) u = (V ⊗ 1)γ (V )(id ⊗)(V ). γ γ (4) τ ω ◦ π = (τ ω ⊗ τπ ) for all π ∈ Irr(G), where π is the left inverse of γ . (5) (τ ω ⊗ τπ ) ◦ γπ (x) = τ ω (x) for all x ∈ M ω and π ∈ Irr(G). Proof. Let V = U v ∗ as before. Let (U n )n and (v n )n be representing sequences of U and v, respectively. Set (v n )n = (U n v n )∗n which represents V . (1) It is verified as (φ ⊗ τπ ) ◦ Ad v n∗ = (φ ⊗ τπ ) ◦ Ad v n U n∗ ∼ (φ ⊗ τπ ) ◦ Ad U n∗ ∼ φ ◦ απ . Conditions (2) and (3) have been already shown. ω γ (4) The left inverse of γ is given by π = απ ◦ Ad V . Then for φ ∈ M∗ and ω x ∈ M ⊗ B(Hπ ), φ(τ ω (γπ (x))) = φ(τ ω (απ (V x V ∗ ))) = lim φ(απ (vn xn vn ∗ )) n→ω
= lim (φ ⊗ τπ )(xn ) n→ω
= φ((τ ω ⊗ τπ )(x)). (5) It is a direct consequence of (4). Let (α, u) be an approximately inner cocycle action on M. By the previous lemma, such that there exists a unitary V ∈ M ω ⊗ L ∞ (G) (V ∗ ⊗ 1)α ω (V ∗ )u(id ⊗)(V ) = 1. Therefore we can perturb the cocycle action to an action on M ω . By taking a representing sequence of V , we can make u close to 1 with an arbitrarily small error in M. A problem is that we have no estimates of perturbation unitaries. For the sake of solving that, we will use the Rohlin type theorem presented in Theorem 5.9.
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4.3. Intertwining cocycles. In this subsection, we study two approximately inner actions. By the 2-cohomology vanishing result, we can take a 1-cocycle intertwining them in an ultraproduct von Neumann algebra. Lemma 4.6. Let M be a von Neumann algebra such that Mω is of type II1 . Let α and on M. Then there exists an α ω -cocycle W in β be approximately inner actions of G with β = Ad W ◦ α on M ⊂ M ω . M ω ⊗ L ∞ (G) We prove this result after proving Lemma 4.7. We denote the left inverses of α and β β by {απ }π ∈Irr(G) and {π }π ∈Irr(G) , respectively. Take unitaries U and V in M ω ⊗ L ∞ (G) such that they satisfy the conditions in Lemma 3.7 for α and β, respectively. Then we have α = Ad U (· ⊗ 1), β = Ad V (· ⊗ 1) on M. Define a map γ ∈ Mor(M ω , M ω ⊗ ×G opp )) by L ∞ (G ∗ ω α (V ∗ )(α ω (x) ⊗ 1)α ω (V )U12 for x ∈ M ω . γ (x) = U12
G opp on M. Since γ is the perturbation Set α (x) = α(x)⊗1 and then α is an action of G× ∗ α ω (V ∗ ), a 2-cocycle w ∈ M ω ⊗ L ∞ (G ×G opp )⊗2 is of the action α by the unitary U12 given by ∗ ω ∗ ∗ ω ω α (V ∗ ) α ω U12 α (V ) (id ⊗G w = U12 opp )(α (V )U12 ). ×G ×G opp on M ω . The map γ is a composition of Then (γ , w) is a cocycle action of G ∗ ω ∗ the maps Ad U12 ◦ α and Ad V (· ⊗ 1). Since they preserve Mω , so does γ . We prove ×G opp )⊗2 as follows. w ∈ M ω ⊗ L ∞ (G ×G opp )⊗2 . Lemma 4.7. The unitary w is in Mω ⊗ L ∞ (G Proof. Let (u n )n and (v n )n be representing sequences of U and V , respectively. Set n∗ n∗ n n α u n∗ wn = u n∗ opp )(α(v )u 12 ). ×G 12 α(v ) 12 α(v ) (id ⊗G Then (w n )n is a representing sequence of w. Let φ ∈ M∗ and π, ρ, σ, ζ ∈ Irr(G). We show lim [φ ⊗ τπ ⊗ τρ ⊗ τσ ⊗ τζ , w n ] = 0. In order to do, we estimate n→ω
ψn = w n∗ (φ ⊗ τπ ⊗ τρ ⊗ τσ ⊗ τζ )w n . β
α n n∗ Use u nπ (θ ⊗ τπ )u n∗ π ∼ θ ◦ π , vπ (θ ⊗ τπ )vπ ∼ θ ◦ π for all θ ∈ M∗ and then n∗ n∗ α α(v n )u n12 α(v n ) ψn ∼ (id ⊗G opp )(u 12 α(v )) ×G
· (φ ◦ απ ⊗ τρ ⊗ τσ ⊗ τζ ) n∗ n n · α(v n∗ ) α u n∗ opp )(α(v )u 12 ) ×G 12 α(v ) (id ⊗G n∗ n∗ n n n = (id ⊗G opp )(u 12 α(v ))απ α(v )u 12 1245 απ (v ) ×G
· (φ ◦ απ ⊗ τρ ⊗ τσ ⊗ τζ ) n∗ n n · απ (v n∗ )απ u n∗ opp )(α(v )u 12 ) ×G 12 α(v ) 1245 (id ⊗G n∗ n∗ n n = (id ⊗G opp )(u 12 α(v ))απ α(v )u 12 1245 ×G · (v n (φ ⊗ τρ )v n∗ ⊗ τσ ⊗ τζ ) ◦ (απ ⊗ idρ ⊗ idσ ⊗ idζ ) n∗ n n · απ u n∗ opp )(α(v )u 12 ) ×G 12 α(v ) 1245 (id ⊗G
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n∗ n∗ n n ∼ (id ⊗G opp )(u 12 α(v ))απ α(v )u 12 1245 ×G
· (φ ◦ βρ ⊗ τσ ⊗ τζ ) ◦ (απ ⊗ idρ ⊗ idσ ⊗ idζ ) n∗ n n · απ u n∗ opp )(α(v )u 12 ) ×G 12 α(v ) 1245 (id ⊗G n∗ n∗ n = (id ⊗G opp )(u 12 α(v ))απ α(v ) 1245 ×G α · u n13 (φ ◦ βρ ⊗ τσ ⊗ τζ )u n∗ 13 ◦ (π ⊗ idρ ⊗ idσ ⊗ idζ ) n n · απ α(v n∗ ) 1245 (id ⊗G opp )(α(v )u 12 ) ×G n∗ n∗ n ∼ (id ⊗G opp )(u 12 α(v ))απ ασ (vζ ) 1245 ×G · (φ ⊗ τζ ) ◦ (βρ ⊗ idζ ) ◦ ασ ◦ (απ ⊗ idρ ⊗ idσ ⊗ idζ ) n n · απ ασ (vζn∗ ) 1245 (id ⊗G opp )(α(v )u 12 ) ×G
n∗ n∗ = (id ⊗G opp )(u 12 α(v )) ×G · vζ · (φ ⊗ τζ ) ◦ (βρ ⊗ idζ ) · vζ∗ ◦ ασ ◦ (απ ⊗ idρ ⊗ idσ ⊗ idζ ) n n · (id ⊗G opp )(α(v )u 12 ) ×G n∗ n∗ ∼ (id ⊗G opp )(u 12 α(v )) ×G β
· φ ◦ βρ ◦ ζ ◦ ασ ◦ (απ ⊗ idρ ⊗ idσ ⊗ idζ ) n n · (id ⊗G ×G opp )(α(v )u 12 ).
By Lemma 2.5, we have ασ ◦ απ =
ξ ≺π ·σ S∈ONB(ξ,π ·σ )
dξ (1 ⊗ S)αξ (1 ⊗ S ∗ ) dπ dσ
on M ⊗ B(Hπ ) ⊗ B(Hσ ) and β
βρ ◦ ζ =
η≺ζ ·ρ T ∈ONB(η,ζ ·ρ)
dη (1 ⊗ ρ,ζ T )βη (1 ⊗ (ρ,ζ T )∗ ) dζ dρ
on M ⊗ B(Hρ ) ⊗ B(Hζ ), where ρ,ζ is the flip unitary of Hρ and Hζ . Note that ρ sits right from ζ in that lemma, but it does left here. The flip arises for this reason. Also note that ρ,ζ T may not be an intertwiner between η and ρ · ζ . Using these, we have β
βρ ◦ ζ ◦ ασ ◦ (απ ⊗ idρ ⊗ idσ ⊗ idζ ) dξ dη (1 ⊗ ρ,ζ T )βη (1 ⊗ (ρ,ζ T )∗ ) ◦ (1 ⊗ S)αξ (1 ⊗ S ∗ ) = dπ dσ dζ dρ ξ,η S,T
dξ dη = ρ,ζ (1 ⊗ S · T ) βη ◦ (αξ ⊗ idη ) (1 ⊗ (S · T )∗ )ρ,ζ , dπ dσ dζ dρ ξ,η S,T
where the indices S, T runs ONB(ξ, π · σ ) and ONB(η, ζ · ρ), respectively and S · T ∈ ×G opp ) and then we (ξ · η, π · ζ · σ · ρ) is naturally defined via S, T . Let x ∈ L ∞ (G have G opp (x)ρ,ζ (S · T ) = ρ,ζ (S · T )x ξ,η . ×G
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Hence we have dξ dη n∗ ψ∼ ρ,ζ (1 ⊗ S · T ) · u n∗ ξ αξ (vη ) dπ dσ dζ dρ ξ,η S,T · φ ◦ βη ◦ (αξ ⊗ idη ) · αξ (vηn )u nξ · (1 ⊗ (S · T )∗ )ρ,ζ dξ dη = ρ,ζ (1 ⊗ S · T ) · u n∗ ξ dπ dσ dζ dρ ξ,η S,T · (vηn∗ (φ ◦ βη ) · vηn ) ◦ (αξ ⊗ idη ) ·u nξ · (1 ⊗ (S · T )∗ )ρ,ζ dξ dη n α ∼ ρ,ζ (1 ⊗ S · T ) · u n∗ ξ · (φ ⊗ τη ) ◦ (ξ ⊗ idη ) · u ξ dπ dσ dζ dρ ξ,η S,T
· (1 ⊗ (S · T )∗ )ρ,ζ
∼
dξ dη ρ,ζ (1 ⊗ S · T ) · (φ ⊗ τξ ⊗ τη ) · (1 ⊗ (S · T )∗ )ρ,ζ dπ dσ dζ dρ ξ,η S,T
= φ ⊗ τπ ⊗ τρ ⊗ τσ ⊗ τζ . G opp is amenable, we can apply Proof of Lemma 4.6. Since the discrete Kac algebra G× × Lemma 4.3 to the cocycle action (γ , w). Then there exists a unitary c ∈ Mω ⊗ L ∞ (G opp ) such that G ∗ (4.4) c123 γ (c)w(id ⊗G opp )(c ) = 1. ×G By definition we have Set the unitaries c = c·⊗1 , cr = c1⊗· in Mω ⊗ L ∞ (G). γ·⊗1 = Ad U ∗ ◦ α ω , γ1⊗· = Ad V ∗ (· ⊗ 1). Hence applying 1 ⊗ 1 ⊗ e1 ⊗ 1 ⊗ e1 , 1 ⊗ e1 ⊗ 1 ⊗ e1 ⊗ 1 to (4.4), we have
∗ ω c12 U12 α (c )U12 w·⊗1⊗·⊗1 (id ⊗)(c ∗ ) = 1, r ∗ r c12 V12 c13 V12 w1⊗·⊗1⊗· (id ⊗opp )(cr ∗ ) = 1.
The equalities ∗ ω ∗ ∗ α (U ∗ )(id ⊗)(U ), w1⊗·⊗1⊗· = V12 V13 (id ⊗opp )(V ) w·⊗1⊗·⊗1 = U12
yield
∗ ω ∗ r ∗ r ∗ U12 α (c U )(id ⊗)(U c ∗ ) = 1, c12 V12 c13 V13 (id ⊗opp )(V cr ∗ ) = 1. c12
Set v = c U ∗ Hence c U ∗ is an α ω -cocycle and V cr ∗ is a unitary representation of G. r ∗ v ω and v = V c . Set the perturbed action α = Ad v ◦ α . We claim that the unitary representation v is fixed by α v , and then it follows that W = vv is an α ω -cocycle. If we prove this claim, the unitary W is a desired one. Indeed, for x ∈ M we have Ad W ◦ α(x) = Ad v ◦ α v (x) = vvα ω (x)v ∗ v ∗ = vvU (x ⊗ 1)U ∗ v ∗ v ∗ = vc (x ⊗ 1)c ∗ v ∗ = v(x ⊗ 1)v ∗ = V (x ⊗ 1)V ∗ = β(x).
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We prove the claim as follows. Applying 1 ⊗ 1 ⊗ e1 ⊗ e1 ⊗ 1 to (4.4), we have
∗ ω r ∗ c12 U12 α (c )U12 w·⊗·⊗1⊗1⊗· c123 = 1.
The equality w·⊗·⊗1⊗1⊗· = 1 yields
∗ ω r ∗ c12 U12 α (c )U12 c123 = 1.
(4.5)
Again applying 1 ⊗ e1 ⊗ 1 ⊗ 1 ⊗ e1 to (4.4), we have r ∗ ∗ V12 c13 V12 w·⊗1⊗·⊗·⊗1 c132 = 1. c12
The equality ∗ ∗ ω w·⊗1⊗·⊗·⊗1 = V12 U13 α (V )132 U13
and (4.5) implies r ∗ ∗ ∗ ω ∗ 1 = c12 V12 c13 V12 V12 U13 α (V )132 U13 c132 r ∗ ∗ ω ∗ = c12 V12 c13 U13 α (V )132 U13 c132 r ∗ ∗ ω ∗ ω r∗
∗ = c12 V12 c13 U13 α (V )132 U13 · U13 α (c )132 U13 c13 r ∗ ∗ ω
∗ = c12 V12 c13 U13 α (V cr ∗ )132 U13 c13 = v ∗12 α v (v)132 .
Therefore α v (v) = v 13 and we have proved Lemma 4.6.
5. Rohlin Type Theorem The Rohlin theorem in [Oc1, Theorem 6.1] has been a main ingredient to show vanishing results on 1 and 2-cohomology for strongly free cocycle actions of discrete amenable groups. Even for amenable discrete Kac algebras we can prove the Rohlin type theorem which is, however, not a generalization of the Rohlin theorem in [Oc1]. As one difference, which comes from difficulty of reducing a cocycle action to an action, we give a Rohlin tower which has a good estimate only for cocycle actions whose 2-cocycles are very small. Another difference is that we treat not paving families but one sufficiently large projection. Since our classification result is deduced from the Evans-Kishimoto type intertwining argument, we do not need a model action splitting method. Hence it is unnecessary to utilize a paving family. Also even in proving vanishing results on 2-cohomology, we do not need such a family. We use the Rohlin type theorem only to find a unitary perturbing a 1-cocycle to a smaller 1-cocycle by using the Shapiro lemma. Although it may seem to be an incomplete form, it is in fact a sufficiently powerful tool for our strategy.
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5.1. Local quantization principle. We begin with the local quantization principle proved by Popa [P1, Lemma A.1.1], [P2, Theorem A.1.2]. Theorem 5.1 (Popa). Let A ⊂ B be an inclusion of finite von Neumann algebras. Let n τ be a faithful normal trace of B. Assume that elements {xi }i=1 ⊂ B are orthogonal to A ∨ (A ∩ B) with respect to τ . Then for any ε > 0, there exists a finite index set J and a partition of unity {qr }r ∈J ⊂ A satisfying
qr xi qr < ε for all 1 ≤ i ≤ n.
τ
r ∈J
be a strongly free Let M be a von Neumann algebra. Let γ : Mω → Mω ⊗ L ∞ (G) semiliftable action with the left inverses {π }π ∈Irr(G) . We assume that τ ω ◦π = τ ω ⊗τπ on Mω ⊗ B(Hπ ) for all π ∈ Irr(G). Take a faithful state φ ∈ M∗ and set ψ = φ ◦ τ ω . Then the dual state ψˆ is tracial by Proposition Consider the inclusion Mω ⊂ Mω γ G. 2.11. Let S be a countably generated von Neumann subalgebra of M ω , A = S ∩ Mω Since γ is strongly free, we have the inclusion A ∨ (A ∩ B) ⊂ Mω and B = Mω γ G. → Mω be the conditional expectation defined by by Lemma 2.13. Let E γˆ : Mω γ G ˆ Let E A∨(A ∩B) be the trace averaging the dual action γˆ , and then it preserves the trace ψ. preserving conditional expectation from B onto A ∨ (A ∩ B). Then it factors through Mω with E γˆ . Hence by definition of E γˆ , we have E A∨(A ∩B) (λπi, j ) = 0 for all π = 1 and i, j ∈ Iπ . Let F be a finite subset of Irr(G) \ {1}. We apply the local quantization principle to the above A, B and {λπi, j }π ∈F ,i, j∈Iπ . Then for any ε > 0, we get a finite partition of unity {qr }r ∈J ⊂ S ∩ Mω satisfying
γ (qr )λπi, j γ (qr ) < ε
ψˆ
r ∈J
for all π ∈ F and i, j ∈ Iπ . Since λπi, j γ (qr ) = k∈Iπ γ (γπi,k (qr ))λπk, j and ∗ ψˆ γ (qr γπi, (qr ))λπ , j γ (qr γπi,k (qr ))λπk, j = dπ−1 ψ π |qr γπi,k (qr )|2 ⊗ eπ ,k = dπ−1 (ψ ⊗ τπ ) |qr γπi,k (qr )|2 ⊗ eπ ,k = δ ,k dπ−1 ψ |qr γπi,k (qr )|2 , we have
2
γ (qr )λπi, j γ (qr )
ˆ
i∈Iπ
ψ
r ∈J
2
=
γ (qr )λπi, j γ (qr ) ˆ
ψ
i∈Iπ r ∈J
2
= γ (qr γπi,k (qr ))λπk, j
ˆ i∈Iπ r ∈J
=
r ∈J i,k, ∈Iπ
=
ψ
k∈Iπ
r ∈J i,k, ∈Iπ
∗ ψˆ γ (qr γπi, (qr ))λπ , j γ (qr γπi,k (qr ))λπk, j
δ ,k dπ−1 ψ |qr γπi,k (qr )|2
(qr ⊗ 1π )γπ (qr ) 2 . = ψ⊗τ r ∈J
π
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Thus for π ∈ F,
(qr ⊗ 1π )γπ (qr ) 2 < dπ ε2 . ψ⊗τ π
r ∈J
Summing up the above inequality with π ∈ F, we obtain
(qr ⊗ 1π )γπ (qr ) 2 < dπ ε2 . ψ⊗τ π
r ∈J π ∈F
π ∈F
We use a Chebyshev inequality as follows. Define an index subset
2
(qr ⊗ 1π )γπ (qr ) 2 J0 = r ∈ J | < ( d )εq π r ψ . ψ⊗τ π
π ∈F
π ∈F
For r ∈ J0 , we have 2 (qr ⊗ 1π )γπ (qr ) (qr ⊗ 1π )γπ (qr )2 ≤|F| ψ⊗τ ψ⊗τ π
π ∈F
π ∈F
≤|F|
π ∈F
π
2
γπ (qr )2ψ⊗τπ (qr ⊗ 1π )γπ (qr ) ψ⊗τ
=|F|qr 2ψ <|F|qr 2ψ =|F|
π ∈F
π
(qr ⊗ 1π )γπ (qr ) 2 ψ⊗τ
π ∈F
π ∈F
π
dπ εqr 2ψ
dπ ε|qr |2ψ .
Hence the following inequality holds: 1/2 (qr ⊗ 1π )γπ (qr ) ≤ |F| dπ ε1/2 |qr |ψ . ψ⊗τ π
π ∈F
On the size of
r ∈J \J0
r ∈J \J0 qr ,
π ∈F
we have
−1
(qr ⊗ 1π )γπ (qr ) 2 |qr |ψ < dπ ε−1 ψ⊗τ π ∈F
≤
dπ
π ∈F
−1
ε−1
(qr ⊗ 1π )γπ (qr ) 2 ψ⊗τ r ∈J π ∈F
−1 < dπ ε−1 dπ ε2 π ∈F
=ε. We summarize these arguments.
π
r ∈J \J0 π ∈F
π ∈F
π
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Lemma 5.2. Let M be a von Neumann algebra. Let γ be a strongly free semiliftable on Mω whose left inverse π satisfies τ ω ◦π = τ ω ⊗τπ for all π ∈ Irr(G). action of G Let φ be a faithful normal state on M and set ψ = φ ◦ τ ω . Then for any countably generated von Neumann algebra S ⊂ M ω , any finite subset F ⊂ Irr(G), 1 ∈ / F and any 0 < ε < 1, there exists n ∈ N and a partition of unity {qr }rn=0 ⊂ S ∩ Mω with the following properties: (1) |q0 |ψ < ε. (qr ⊗ 1π )γπ (qr ) (2) < ε|qr |ψ for all 1 ≤ r ≤ n and π ∈ F. ψ⊗τ π
π ∈F
We can strengthen this result as follows. Lemma 5.3. Let M be a von Neumann algebra. Let γ be a strongly free semiliftable on Mω whose left inverse π satisfies τ ω ◦π = τ ω ⊗τπ for all π ∈ Irr(G). action of G Let φ be a faithful normal state on M and set ψ = φ ◦ τ ω . Then for any countably generated von Neumann algebra S ⊂ M ω , any finite subset F ⊂ Irr(G), 1 ∈ / F and any 0 < δ < 1, there exists n ∈ N and a partition of unity {er }rn=0 ⊂ S ∩ Mω with the following properties: (1) |e0 |ψ ≤ δ. (2) (er ⊗ 1π )γπ (er ) = 0 for all 1 ≤ r ≤ n and π ∈ F. Proof. This is proved by a similar argument to one in [Oc1] as follows. We may assume S ⊂ M ω is γ -invariant by considering a von Neumann algebra generated by S and γρk, (S) for all ρ ∈ Irr(G) and k, ∈ Iρ . Step A. Let µ > 0 and f ∈ Proj(S ∩ Mω ), f = 0. We show that there exists f ∈ Proj(S ∩ Mω ), 0 = f ≤ f , such that ( f ⊗ 1π )γπ ( f )
ψ⊗τπ
π ∈F
< 2µ| f |ψ ,
for all π ∈ F. Let S be a von Neumann subalgebra in M ω which is generated by S and γρi, j ( f ) for all ρ ∈ Irr(G) and i, j ∈ Iρ . By the previous lemma, there exists a partition of unity f 0 , f 1 , . . . , f m in S ∩ Mω such that (1) | f 0 |ψ ≤ 2−1 | f |ψ , ( f i ⊗ 1π )γπ ( f i ) (2) < µ| f |ψ | f i |ψ , 1 ≤ i ≤ m. ψ⊗τ π
π ∈F
Let f¯i = f f i ∈ Proj(S ∩ Mω ). Since f i ⊗ 1ρ commutes γρ ( f ) for all ρ ∈ Irr(G), we have (f¯i ⊗ 1ρ )γ ρ ( f¯i ) = ( f ⊗ 1 ρ )γρ ( f )( f i ⊗ 1ρ )γρ ( f i ). Hence we have ( f¯i ⊗ 1ρ )γρ ( f¯i )ψ⊗τ ≤ ( f i ⊗ 1ρ )γρ ( f i )ψ⊗τ . Suppose that for each i = 1, . . . , m, π
π
( f¯i ⊗ 1π )γπ ( f¯i ) ≥ 2µ| f¯i |ψ . ψ⊗τ
π ∈F
π
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Then m m ( f i ⊗ 1π )γπ ( f i ) ( f¯i ⊗ 1ρ )γρ ( f¯i ) ≥ ψ⊗τ ψ⊗τ π
i=1 π ∈F
π
i=1 π ∈F m
| f¯i |ψ
≥2µ
i=1
=2µ|(1 − f 0 ) f |ψ ≥2µ(| f |ψ − | f 0 |ψ ) ≥µ| f |ψ . On the other hand, we have m ( f i ⊗ 1π )γπ ( f i ) <µ| f |ψ | f i |ψ ψ⊗τ π
i=1 π ∈F
i=1
≤µ| f |ψ . This derives a contradiction and hence for some i ∈ {1, . . . , m}, the equality ( f¯i ⊗ 1π )γπ ( f¯i ) < 2µ| f¯i |ψ ψ⊗τ π
π ∈F
holds and we take f = f¯i . Step B. We show that for any f ∈ Proj(S ∩ Mω ) and any µ > 0, there exists e ∈ Proj(S ∩ Mω ) with (1) e ≤ f , (2) (e ⊗ 1π )γπ (e)ψ⊗τ ≤ µ|e|ψ for all π ∈ F, π (3) |e|ψ ≥ (1 + π ∈F ∪F 2dπ2 )−1 | f |ψ . Set Tπ (x) = π (x ⊗ 1π ) for x ∈ M ω . Note Tπ (S ∩ Mω ) ⊂ S ∩ Mω for all π ∈ Irr(G) as is seen below. Let x ∈ S ∩ Mω . Since Tπ preserves Mω , Tπ (x) ∈ Mω . Take an element y ∈ S. The γ -invariance of S implies yTπ (x) = π (γπ (y)(x ⊗ 1π )) = π ((x ⊗ 1π )γπ (y)) = Tπ (x)y. This shows Tπ (x) ∈ S ∩ Mω . Now the family of projections e ∈ S ∩ Mω satisfying (1) and (2) is not empty and inductively ordered, so let e be maximal with these properties. We show that e also satisfies e∨
π ∈F
s(Tπ (e)) ∨
s(Tπ¯ (e)) ∨ (1 − f ) = 1.
(5.1)
π ∈F
Otherwise let e be a nonzero projection in S ∩ Mω orthogonal to the left member of (5.1). We claim that (e ⊗ 1π )γπ (e ) = 0 = (e ⊗ 1π )γπ (e) for π ∈ F ∪ F. Since π γπω (e )(e ⊗ 1π )γπω (e ) = e π (e ⊗ 1π )e = e Tπ (e)e = 0,
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we have (e ⊗ 1π )γπ (e ) = 0 for π ∈ F ∪ F by faithfulness of π . Since π γπ (e)(e ⊗ 1π )γπ (e) = eπ e ⊗ 1π e = e(1 ⊗ Tπ∗,π )(γπ (e ) ⊗ 1π )(1 ⊗ Tπ ,π )e ∗ = (1 ⊗ Tπ,π ) (e ⊗ 1π )γπ (e )(e ⊗ 1π ) ⊗ 1π (1 ⊗ Tπ,π ) = 0, ∪ F. By Step we also have (e ⊗ 1π )γπ (e) = 0 for π ∈ F A, under e we can find a nonzero projection e ∈ S ∩ Mω with (e ⊗ 1π )γπ (e ) ψ⊗τ ≤ µ|e |ψ . Then e π
satisfies e ⊥ e, e ≤ f and (e ⊗ 1π )γπ (e ) = 0 = (e ⊗ 1π )γπ (e) for all π ∈ F ∪ F. Then we have (e¯ ⊗ 1π )γπ (e) ¯ ψ⊗τ = (e ⊗ 1π )γπ (e) + (e ⊗ 1π )γπ (e )ψ⊗τ π π = (e ⊗ 1π )γπ (e)ψ⊗τ + (e ⊗ 1π )γπ (e )ψ⊗τ ≤µ|e|ψ + µ|e |ψ =µ|e| ¯ ψ.
π
π
This is a contradiction to the maximality of e, and hence (5.1) holds. Now we estimate the size of e. Here we make use of tensor products to treat translations of projecω = tions. We index F ∪ F as {π1 , · · · , π N }. Consider the tensor product algebra M N N ω M ⊗ B(⊗k=1 Hπk ). Set a product state ψ = ψ ⊗ ⊗k=1 τπk , which is tracial on Mω ⊗ N H ). We regard M ω and M ω ⊗ B(H ) as subalgebras of M ω via the natural B(⊗k=1 πk πk ∼ embedding. For each πk , take a finite group Uk = Zdπk Zdπk consisting of unitaries in B(Hπk ) which acts on Hπk irreducibly. Our claim is the following one: e∨
N
(1 ⊗ v)γπk (e)(1 ⊗ v ∗ ) ∨ (1 − f ) = 1.
(5.2)
k=1 v∈Uk
ω which is orthogonal to the above If not so, we have a nonzero projection p ∈ M left projection. Hence p ≤ f − e and satisfies p · (1 ⊗ v)γπk (e)(1 ⊗ v ∗ ) = 0 for all k = 1, . . . , N and v ∈ Uk . Irreducibility of the action of Uk implies the equality τπk = dπ−2 v∈Uk Ad v on B(Hπk ). Hence pTπ k (e) = p · (id ⊗τπk )(γπk (e)) = 0 for k all k. It shows that p must be orthogonal to e, 1 − f and ∨π ∈F ∪F s(Tπ (e)), but this to both contradicts (5.1). Therefore the above claim holds. Applying the product trace ψ sides of (5.2), we obtain 1 ≤ |e|ψ + |1 − f |ψ +
N
(ψ ⊗ τπk ) (1 ⊗ v)γπk (e)(1 ⊗ v ∗ )
k=1 v∈Uk
≤ |e|ψ + 1 − | f |ψ +
N
(ψ ⊗ τπk ) γπk (e)
k=1 v∈Uk
= 1 − | f |ψ + (1 +
N k=1
2dπ2k )|e|ψ .
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Hence |e|ψ ≥ (1 +
N
2dπ2k )−1 | f |ψ .
k=1
Step C. Let q ∈ N be such that (1 − (1 + π ∈F ∪F 2dπ2 )−1 )q < δ. We show that for any q µ > 0 there exists a partition of unity {ek }k=0 ⊂ S ∩ Mω such that (1) (2)
|e 0 |ψ ≤ δ. (ek ⊗ 1π )γπ (ek ) < µ|ek |ψ , for all k = 1, . . . , q and π ∈ F. ψ⊗τ π
Set f 1 = 1. According to Step B, we construct projections ek and f k+1 successively for k = 1, . . . , q such that ek ≤ f k , f k+1 = f k − ek , (ek ⊗ 1π )γπ (ek ) ≤ µ|ek |ψ for all π ∈ F, ψ⊗τπ 2 −1 (4) |ek |ψ ≥ (1 + π ∈F ∪F 2dπ ) | f k |ψ . Then we have | f q+1 |ψ ≤ (1 − (1 + π ∈F ∪F 2dπ2 )−1 )q < δ and letting e0 = f q+1 , Step C is proved. Step D. We finish the proof by using the Index Selection Trick. Note that the partition number q depends not on µ but on δ and F. Letting µ = 1/n, q take a partition of unity {ekn }k=0 ⊂ S ∩ Mω for each n ∈ N such that (1) (2) (3)
(1) (2)
n |e 0 n|ψ ≤ δ, n (e ⊗ 1π )γπ (en ) k k ψ⊗τ < (1/n)|ek |ψ , for all k = 1, . . . , q and π ∈ F. π
Set the elements ei = (ein )n in ∞ (N, S ∩ Mω ) for 0 ≤ i ≤ q. Then apply the Index q Selection Trick for C = C ∗ ({ei }i=0 ) and B = {γπ }π ∈Irr(G) . Let be the index selection map with respect to them. Set ei = (ei ) which is in S ∩ Mω . Then we have |e0 |ψ ≤ δ and (ei ⊗ 1π )γπ (ei ) = (ψ ⊗ τπ ) (ei ⊗ 1π )γπ (ei ) ψ⊗τπ = (φ ◦ τ ω ◦ ⊗ τπ ) (ei ⊗ 1π )γπ (ei ) = lim (φ ◦ τ ω ⊗ τπ ) (en ⊗ 1π )γπ (en ) n→ω
i
i
≤ lim 1/n n→ω
=0 for π ∈ F.
5.2. Tower bases and diagonal elements. In the previous subsection, we have obtained a projection in Mω which behaves like a tower base with respect to an action γ as in the case of group actions. Here one shall note that it gives projections in not Mω but von Neumann algebras Mω ⊗ B(Hπ ), π ∈ Irr(G). In the following lemma, we clarify the special properties of such a projection in a general situation. on M. Lemma 5.4. Let M be a von Neumann algebra and (α, u) a cocycle action of G If a projection e in M satisfies [e ⊗ 1 ⊗ 1, u] = 0 and Let K be a finite subset of G. (e ⊗ 1ρ )αρ (e) = 0 for all ρ ∈ K · K \ 1, then
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(1) (απ (e) ⊗ 1σ )(ασ (e))1,3 = 0 for all π = σ ∈ K. (2) (e ⊗ 1π ⊗ 1π )(απ ⊗ id)(απ (e)) = u π ,π (e ⊗ π π (e1 ))u ∗π ,π for all π ∈ K. (3) The element dπ2 π (e ⊗ 1π ) is a projection which is equal to qπ = inf{q ∈ Proj(M) | (q ⊗ 1π )απ (e) = απ (e), [q ⊗ 1 ⊗ 1, u] = 0}. ∗ )u ∗ (α (e) ⊗ 1 ) = α (e) ⊗ 1 . (4) For all π ∈ K, dπ2 (απ (e) ⊗ 1π )u π,π (1 ⊗ Tπ,π Tπ,π π π π π,π π
Proof. (1) It suffices to show (π ⊗ id)(απ (e)12 ασ (e)13 απ (e)12 ) = 0. Since π = σ , 1 ⊀ π · σ and then (e ⊗ 1π ⊗ 1σ )απ (ασ (e)) = (e ⊗ 1π ⊗ 1σ )u π ,σ (id ⊗π σ )(α(e))u ∗π ,σ
= u π ,σ (e ⊗ 1π ⊗ 1σ )(id ⊗π σ )(α(e))u ∗π ,σ = u π ,σ (e ⊗ 1π ⊗ 1σ )(id ⊗π σ )(αρ (e))u ∗π ,σ ρ≺π ·σ
= 0. Using this equality, we have (π ⊗ id)(απ (e)12 ασ (e)13 απ (e)12 ) = (e ⊗ 1σ )(π ⊗ id)(ασ (e)13 )(e ⊗ 1σ ) = (e ⊗ 1σ )(1 ⊗ Tπ∗,π ⊗ 1σ )(u ∗π,π ⊗ 1σ )απ (ασ (e))124 (u π,π ⊗ 1σ ) = (1 ⊗
∗ Tπ,π
· (1 ⊗ Tπ,π ⊗ 1σ )(e ⊗ 1σ ) ⊗ 1σ )((e ⊗ 1π ⊗ 1σ )απ (ασ (e)))124 (u π ,π ⊗ 1σ )
⊗ 1σ )(u ∗π ,π
· (1 ⊗ Tπ,π ⊗ 1σ )(e ⊗ 1σ ) = 0. Since π is faithful, we have απ (e)12 ασ (e)13 = 0. (2) It is verified as follows: (e ⊗ 1π ⊗ 1π )απ (απ (e)) = (e ⊗ 1π ⊗ 1π )u π ,π (id ⊗π π )(α(e))u ∗π ,π = u π ,π (id ⊗π π )((e ⊗ 1ρ )αρ (e))u ∗π ,π ρ≺π ·π
= u π ,π (id ⊗π π )((e ⊗ e1 )α1 (e))u ∗π ,π = u π ,π (e ⊗ π π (e1 ))u ∗π ,π .
(3) Applying the map π ⊗ id to the equality in (2), by Lemma 2.4 we have (dπ2 π (e ⊗ 1π ) ⊗ 1π )απ (e) = απ (e). Set pπ = dπ2 π (e ⊗ 1π ). Since e commutes with u, so does with α(u). Hence pπ commutes with u. Then pπ2 = pπ dπ2 π (e ⊗ 1π )
∗ )u ∗π,π (απ (e) ⊗ 1π )u π,π (1 ⊗ Tπ,π ) = pπ dπ2 (1 ⊗ Tπ,π ∗ )u ∗π,π (( pπ ⊗ 1π )απ (e) ⊗ 1π )u π,π (1 ⊗ Tπ,π ) = dπ2 (1 ⊗ Tπ,π ∗ )u ∗π,π (απ (e) ⊗ 1π )u π,π (1 ⊗ Tπ,π ) = dπ2 (1 ⊗ Tπ,π
= pπ .
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Hence pπ is a projection and pπ ≥ qπ . The inequality pπ ≤ qπ is verified as follows: qπ pπ = qπ dπ2 π (e ⊗ 1π )
∗ )u ∗π,π (απ (e) ⊗ 1π )u π,π (1 ⊗ Tπ,π ) = dπ2 qπ (1 ⊗ Tπ,π ∗ )u ∗π,π ((qπ ⊗ 1)απ (e) ⊗ 1π )u π,π (1 ⊗ Tπ,π ) = dπ2 (1 ⊗ Tπ,π ∗ )u ∗π,π (απ (e) ⊗ 1π )u π,π (1 ⊗ Tπ,π ) = dπ2 (1 ⊗ Tπ,π
= pπ . ∗ )u ∗ (α (e) ⊗ 1 )u (4) Since dπ2 π (e ⊗ 1π ) = dπ2 (1 ⊗ Tπ,π π π,π (1 ⊗ Tπ,π ) is a proπ,π π jection, the operator v := dπ (απ (e) ⊗ 1π )u π,π (1 ⊗ Tπ,π ) is a partial isometry. Hence f := vv ∗ is a projection. Clearly we have f ≤ απ (e) ⊗ 1π . In fact they are equal as is shown below,
(π ⊗ id)( f ) = dπ2 (π ⊗ id)((απ (e) ⊗ 1π )u π,π (1 ⊗ π π (e1 ))u ∗π,π (απ (e) ⊗ 1π )) = dπ2 (e ⊗ 1π )(π ⊗ id)(u π,π (1 ⊗ π π (e1 ))u ∗π,π )(e ⊗ 1π ) = e ⊗ 1π = (π ⊗ id)(απ (e) ⊗ 1π ), where we have used Lemma 2.4. The faithfulness of π yields f = απ (e) ⊗ 1π .
We call a projection e a tower base of the tower along with K if e satisfies the conditions from (1) to (4) in this lemma. on a von Neumann algebra M. The Definition 5.5. Let (α, u) be a cocycle action of G defined by diagonal of u is the element a in M ⊗ L ∞ (G) (a ⊗ 1)(1 ⊗ (e1 )) = u(1 ⊗ (e1 )). has the following explicit form. For all π ∈ Irr(G), The diagonal a ∈ M ⊗ L ∞ (G) aπ = dπ π (1 ⊗ 1π ⊗ Tπ∗,π )(u π,π ⊗ 1π )(1 ⊗ Tπ,π ⊗ 1π ), where π ∈ {±1} is defined in §2.2. Lemma 5.6. Let aπ , π ∈ Irr(G), be as above. Then one has (1) (απ ⊗ id)(aπ∗ )(aπ ⊗ 1π )(1 ⊗ Tπ,π ) = 1 ⊗ Tπ ,π , (2) (id ⊗τπ )(aπ∗ aπ ) = 1, (3) π (aπ aπ∗ ) = 1, (4) dπ2 (π ⊗ idπ )(x ⊗ (e1 )) = aπ∗ απ (x)aπ for all x ∈ M, (5) aπ∗ aπ = dπ2 (π ⊗ idπ )(1 ⊗ π π (e1 )). Proof. (1) Since u is a 2-cocycle, we have (1 ⊗ 1 ⊗ (e1 ))(α(a ∗ ) ⊗ 1)(a ⊗ 1 ⊗ 1)(1 ⊗ (e1 ) ⊗ 1) = (1 ⊗ 1 ⊗ (e1 ))α(u ∗ )(u ⊗ 1)(1 ⊗ (e1 ) ⊗ 1) = (1 ⊗ 1 ⊗ (e1 ))α(u ∗ )(u ⊗ 1)(id ⊗ ⊗ id)(u)(1 ⊗ (e1 ) ⊗ 1) = (1 ⊗ 1 ⊗ (e1 ))(id ⊗ id ⊗)(u)(1 ⊗ (e1 ) ⊗ 1) = (1 ⊗ 1 ⊗ (e1 ))(1 ⊗ (e1 ) ⊗ 1).
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Applying id ⊗ id ⊗ id ⊗ϕ to both sides, we obtain the desired equality by using (id ⊗ϕ)((e1 )) = 1. (2) It is verified as ∗ )(aπ∗ aπ ⊗ 1π )(1 ⊗ Tπ,π ) (id ⊗τπ )(aπ∗ aπ ) = (1 ⊗ Tπ,π ∗ )(u ∗π,π u π,π )(1 ⊗ Tπ,π ) = (1 ⊗ Tπ,π
∗ )(1 ⊗ Tπ,π ) = (1 ⊗ Tπ,π
= 1. (3) It is verified as π (aπ aπ∗ ) = dπ2 π (id ⊗ id ⊗τπ )((a ⊗ 1)(1 ⊗ (e1 ))(a ∗ ⊗ 1)) = dπ2 (π ⊗ τπ )(u π,π (1 ⊗ (e1 ))u ∗π,π ),
= 1, where we have used Lemma 2.4. (4) It is verified as dπ2 (π ⊗ id)(x ⊗ π π (e1 ))
∗ ⊗ 1π )(u ∗π,π ⊗ 1π )(απ (x) ⊗ π π (e1 ))(u π,π ⊗ 1π )(1 ⊗ Tπ,π ⊗ 1π ) = dπ2 (1 ⊗ Tπ,π ∗ ⊗ 1π )(aπ∗ ⊗ 1π ⊗ 1π )(απ (x)⊗ π π (e1 ))(aπ ⊗ 1π ⊗ 1π )(1⊗Tπ,π ⊗ 1π ) = dπ2 (1⊗ Tπ,π
= aπ∗ απ (x)aπ .
(5) It is obtained by putting x = 1 in (4).
With diagonals, we obtain the following result for a tower base: Lemma 5.7. If a projection e ∈ M satisfies the condition of Lemma 5.4, then dπ2 (απ (e)aπ ⊗ 1π )(1 ⊗ π π (e1 ))(aπ∗ απ (e) ⊗ 1π ) = απ (e) ⊗ 1π . απ (e)aπ aπ∗ απ (e) = απ (e), in particular απ (e)aπ is a partial isometry. 2 ∗ dπ (π ⊗ idπ )(e ⊗π ∗π (e1 )) = aπ απ (e)a π is a projection. dπ απ (e)aπ i,k dπ aπ απ (e) , j = δk, dπ απ (e) i, j for all π ∈ Irr(G) and i, j, k,
∈ Iπ . (5) Decompose a ∗ α(e)a K as (1) (2) (3) (4)
a ∗ α(e)a K =
π ∈K i, j∈Iπ
dπ−1 f π i, j ⊗ eπi, j .
Then one has f π∗i, j = f π j,i ,
f π i, j f ρ k, = δπ,ρ δ j,k f π i,
for all π, ρ ∈ Irr(G), i, j ∈ Iπ and k, ∈ Iρ .
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Proof. (1) It is a direct consequence of Lemma 5.4 (4). (2) Apply id ⊗ id ⊗ϕ to both sides of (1). (3) It is derived by using (2) and Lemma 5.6 (4). (4) Recall a system of matrix units {eπ i, j }i, j∈Iπ defined in §2.2. Set functionals ωπi, j = Tr π eπ j,i , ωπ k, = Tr π eπ ,k . Apply id ⊗ωπi, j ⊗ ωπ k, to both sides of (1). Then the left-hand side is equal to (id ⊗ωπi, j ⊗ ωπ k, )(dπ2 (απ (e)aπ ⊗ 1π )(1 ⊗ π,π (e1 ))(aπ∗ απ (e) ⊗ 1π )) = dπ2 (id ⊗ωπi, j )(απ (e)aπ dπ−1 (1 ⊗ eπk, )aπ∗ απ (e)) = dπ απ (e)aπ i,k aπ∗ απ (e) , j and the right-hand side is equal to δk, απ (e) i, j . (5) The self-adjointness yields f π∗i, j = f π j,i . By (4), f π i, j = dπ aπ∗ απ (e)aπ i, j satisfies the desired equality. In particular, (5) shows the similarity of projections a ∗ α(e)a K and K (e1 ). We pre sume that the projection a ∗ α(e)a K makes a copy of finite dimensional algebra L ∞ (G)K in M. This is a reason for giving indices not f πi, j but f π i, j . We close this subsection with the next useful lemma. It shows that the support projection of each tower element is invariant by perturbation if e commutes with u and α(v). on a von Neumann algebra M. Let Lemma 5.8. Let (α, u) be a cocycle action of G ∞ v ∈ U (M ⊗ L (G)) and ( α, u ) be the perturbed cocycle action of (α, u) by v. Let a and a be the diagonals of u and u , respectively. Let K be a finite subset of Irr(G). Assume that e ∈ Proj(M) satisfies [e ⊗ 1 ⊗ 1, u] = 0 = [e ⊗ 1 ⊗ 1, α(v)] and (e ⊗ 1ρ )αρ (e) = 0 for all ρ ∈ K · K \ {1}. Then one has (id ⊗ϕ)(a ∗ α(e)aπ ) = (id ⊗ϕ)( a ∗ α (e) aπ ). Proof. By Lemma 5.4 and Lemma 5.7, the elements qπ = (id ⊗ϕ)(a ∗ α(e)aπ ) and qπ = (id ⊗ϕ)( a ∗ α (e) aπ ) are the projections which are given by qπ = inf{q ∈ Proj(M) | (q ⊗ 1π )απ (e) = απ (e), [q ⊗ 1 ⊗ 1, u] = 0}, qπ = inf{q ∈ Proj(M) | (q ⊗ 1π ) απ (e) = απ (e), [q ⊗ 1 ⊗ 1, u ] = 0}. Since qπ = dπ2 π (e ⊗ 1π ), we have (qπ ⊗ 1)v = (dπ2 π (e ⊗ 1π ) ⊗ 1)v = dπ2 (π ⊗ id)((e ⊗ 1π ⊗ 1)απ (v)) = dπ2 (π ⊗ id)(απ (v)(e ⊗ 1π ⊗ 1)) = v(dπ2 π (e ⊗ 1π ) ⊗ 1) = v(qπ ⊗ 1). Hence qπ commutes with v. Similarly we can show that qπ commutes with α(v), and so does with u . Then we have (qπ ⊗ 1π ) απ (e) = απ (e). It yields qπ ≥ qπ . We prove qπ ≤ qπ . Since e commutes with α (v) = (v ⊗ 1)α(v)(v ∗ ⊗ 1), qπ commutes with v and α(v), and hence so does with u, and then we have ( qπ ⊗ 1π )απ (e) = απ (e). This equality yields qπ ≤ qπ .
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5.3. Rohlin type theorem. We present a Rohlin type theorem. In order to simplify our arguments, we treat only McDuff factors. Theorem 5.9. Let M be a McDuff factor and (α, u) an approximately inner and strongly such that on M. Take a unitary v ∈ M ω ⊗ L ∞ (G) free cocycle action of G (i) α = Ad v on M ⊂ M ω , (ii) (v ∗ ⊗ 1)(α ω ⊗ id)(v ∗ )u(id ⊗)(v) = 1, (iii) γ = Ad v ∗ ◦ α ω is an action preserving Mω , (iv) (τ ω ⊗ τπ ) ◦ γπ = τ ω for all π ∈ Irr(G). Let φ be a faithful normal state on M and set ψ = φ ◦ τ ω . Let 0 < δ, κ < 1 and F ∈ Projf(Z (L ∞ (G))). Take K ∈ Projf(Z (L ∞ (G))) which is (F, δ)-invariant and satisfies K ≥ e1 . Set F = supp(F) and K = supp(K ). Assume that the 2-cocycle u is small in the following sense: u π,ρ − 1 ⊗ 1π ⊗ 1ρ ψ⊗ϕ⊗ϕ < κ for all π ∈ F ∪ K and ρ ∈ K. Then for any countable set S ⊂ M ω , there exists a satisfying the following conditions: projection E in Mω ⊗ L ∞ (G) (1) E = E(1 ⊗ K ). (2) (approximate equivariance) γ F (E) − (id ⊗ F K )(E) < 5δ 1/2 |F|ϕ . ψ⊗ϕ⊗ϕ (3) Decompose E as E=
ρ∈K i, j∈Iρ
dρ−1 f ρ i, j ⊗ eρi, j .
Then { f ρ i, j }i, j∈Iρ is a system of matrix units. Moreover, they are orthogonal in the following sense. For all ρ = π ∈ K, i, j ∈ Iρ and k, ∈ Iπ , f ρ i, j f π k, = 0. (4) (joint property of U ) Let a be the diagonal of u. Set an operator U = a ∗ v E and decompose as dρ−1 f ραi, j ⊗ eρi, j , a ∗ v Ev ∗ a K = ρ∈K i, j∈Iρ
U =
ρ∈K i, j∈Iρ
dρ−1 µρ i, j ⊗ eρi, j .
Then we have µ∗ρ i, j µπ k, = δρ,π δi,k f ρ j, , µρ i, j µ∗π k, = δρ,π δ j, f ραi,k , for all ρ, π ∈ K, i, j ∈ Iρ and k, ∈ Iπ . In particular U ∗ U = E. (5) Ev ∗ aa ∗ v E = E. (6) For each ρ ∈ K, the projection (id ⊗ϕρ )(E) is in S ∩ Mω and satisfies (id ⊗ϕρ )(E) = (id ⊗ϕρ )(a ∗ v Ev ∗ a).
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(7) (partition of unity) (id ⊗ϕ)(E) = 1 = (id ⊗ϕ)(a ∗ v Ev ∗ a). (8) (Shapiro lemma) Set µ = (id ⊗ϕ)(U ) and then µ is a unitary. If the weights ψ ⊗ ϕ and ψ ⊗ ϕ ⊗ ϕ are invariant for Ad v and Ad(id ⊗)(v), respectively, then the following inequality holds: |v F γ Fω (µ) − µ ⊗ F|ψ⊗ϕ < (9δ 1/4 + 3κ 1/2 )|F|ϕ . In this situation, we call a projection E and a unitary µ a Rohlin projection and a Shapiro unitary, respectively. Lemma 5.10. Let π ∈ K. If u π,π − 1 ⊗ 1π ⊗ 1π ψ⊗ϕ⊗ϕ < κ, then we have aπ − 1 ⊗ 1π ψ⊗ϕ < κ. ∗ ·T Proof. Set a functional θπ = Tπ,π π,π on B(Hπ ⊗ Hπ ). Then we have
(ψ ⊗ ϕπ )(|aπ − 1 ⊗ 1π |2 ) = dπ2 (ψ ⊗ θπ )(|aπ − 1 ⊗ 1π |2 ⊗ 1π ) = dπ2 (ψ ⊗ θπ )(|u π,π − 1 ⊗ 1π ⊗ 1π |2 ) ≤ (ψ ⊗ ϕ ⊗ ϕ)(|u π,π − 1 ⊗ 1π ⊗ 1π |2 ) < κ 2, where we have used dπ2 θπ ≤ ϕπ ⊗ ϕπ .
satisfying the Define a set J which consists of a projection E in Mω ⊗ L ∞ (G) conditions (1), (3), (4), (5), (6) in Theorem 5.9, and in addition, the following ones: (a) (τ ω ⊗ id)(E) = (τ ω ⊗ id)(v Ev ∗ ) ∈ CK , (b) (id ⊗ϕρ )(E) = (id ⊗ϕρ )(a ∗ v Ev ∗ a) for all ρ ∈ K. Define functions a and b from J to R+ by a E = |F|−1 ϕ γ F (E) − (id ⊗ F K )(E) ψ⊗ϕ⊗ϕ , b E = |E|ψ⊗ϕ . Lemma 5.11. Let E be an element of J. Assume b E < 1 − δ 1/2 . Then there exists E ∈ J satisfying the following inequalities: (1) a E − a E ≤ 3δ 1/2 (b E − b E ), (2) 0 < (δ 1/2 /2)|E − E|ψ⊗ϕ ≤ b E − b E . Proof. We may assume that the entries of E, v and u are in S and S is α ω -invariant. Take a projection e from S ∩ Mω such that (e ⊗ 1ρ )γρ (e) = 0 for all ρ ∈ K · K \ {1} by Lemma 5.3. Since e commutes with v, (e ⊗ 1ρ )αρω (e) = 0 also holds. By Lemma 5.8, we have (id ⊗ϕ)(γρ (e)) = (id ⊗ϕ)(aρ∗ αρω (e)aρ ) ∈ S ∩ Mω . Let N be a von Neumann subalgebra in M ω which is generated by M and the entries of {αρω (e)}ρ∈Irr(G) . Applying the Fast Reindexation Trick for N and S, we have a map as in Lemma 3.10. Set f = (e) and then f ∈ S ∩ Mω . , M ω ⊗ L ∞ (G)) ∈ Mor( N
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Since ( f ⊗ 1ρ )αρω ( f ) = ( ⊗ id)((e ⊗ 1ρ )αρω (e)) = 0 for ρ ∈ K · K \ {1}, the equality ( f ⊗ 1ρ )γρ ( f ) = 0 holds. Then by Lemma 5.8, we have (id ⊗ϕ)(γρ ( f )) = (id ⊗ϕ)(aρ∗ αρω ( f )aρ ) = (id ⊗ϕ)(aρ∗ αρω (e)aρ ) . This shows the following splitting property of τ ω for x ∈ S ⊗ L ∞ (G), (τ ω ⊗ id) x((id ⊗ϕ)(γρ ( f )) ⊗ 1) = (τ ω ⊗ id) x (id ⊗ϕ)(aρ∗ αρω (e)aρ ) ⊗ 1 = (τ ω ⊗ id)(x) · (τ ω ⊗ id) (id ⊗ϕ)(aρ∗ αρω (e)aρ ) ⊗ 1 = (τ ω ⊗ id)(x) · τ ω ((id ⊗ϕ)(γρ ( f ))) ⊗ 1 . Set a projection f = (id ⊗ϕ)(γ K ( f )) in S ∩Mω . The equality (τ ω ⊗τρ )◦γρ = τ ω yields τ ω ((id ⊗ϕ)(γρ ( f ))) = τω ( f )dρ2 for ρ ∈ K, in particular | f |ψ = | f |τω = τω ( f )|K |ϕ . by Then set a projection E ∈ Mω ⊗ L ∞ (G) E = E( f ⊥ ⊗ 1) + γ K ( f ). We verify that E is a desired projection. At first, we will show E ∈ J. The condition (1) in Theorem 5.9 is trivial. The conditions (3), (4) and (5) hold by Lemma 5.7. The condition (6) follows from Lemma 5.8 on γ K ( f ). We verify the remained conditions (a) and (b). We claim (τ ω ⊗ id)(γ K ( f )) = τω ( f )K . Indeed, by applying Proposition 2.10 to the action γ on Mω , we have (τω ⊗ id) ◦ γρ = τω . Then the condition (a) is verified as (τ ω ⊗ id)(E ) = (τ ω ⊗ id)(E( f ⊥ ⊗ 1) + γ K ( f )) = (τω ⊗ id)(E)(τω ( f ⊥ ) ⊗ 1) + τω ( f )K and (τ ω ⊗ id)(v E v ∗ ) = (τ ω ⊗ id)(v Ev ∗ ( f ⊥ ⊗ 1) + α ωK ( f )) = (τ ω ⊗ id)(v Ev ∗ )(τω ( f ⊥ ) ⊗ 1) + τω ( f )K = (τω ⊗ id)(E)(τω ( f ⊥ ) ⊗ 1) + τω ( f )K . Next we verify the condition (b). Since (id ⊗ϕ)(γρ ( f )) = (id ⊗ϕ)(a ∗ α ω (e)aρ ), we have (id ⊗ϕ)(E ρ ) = (id ⊗ϕ)(E ρ ) f ⊥ + (id ⊗ϕ)(γρ ( f )) = (id ⊗ϕ)(a ∗ v Ev ∗ aρ ) f ⊥ + (id ⊗ϕ)(a ∗ α ω (e)aρ ) = (id ⊗ϕ)(a ∗ v E v ∗ aρ ). Hence E ∈ J. Now we estimate a E and b E . First we have |E − E|ψ⊗ϕ = | − E( f ⊗ 1) + γ K ( f )|ψ⊗ϕ ≤ |E( f ⊗ 1)|ψ⊗ϕ + |γ K ( f )|ψ⊗ϕ = |(id ⊗ϕ)(E) f |ψ + | f |ψ ≤ 2| f |ψ .
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Since b E = |E( f ⊥ ⊗ 1) + γ K ( f )|ψ⊗ϕ = |E( f ⊥ ⊗ 1)|ψ⊗ϕ + |γ K ( f )|ψ⊗ϕ = ψ( f ⊥ )|E|ψ⊗ϕ + | f |ψ = ψ( f ⊥ )b E + | f |ψ , we have the condition (2) in Lemma 5.11 as follows. b E − b E = ψ( f )(1 − b E ) > δ 1/2 | f |ψ ≥
δ 1/2 |E − E|ψ⊗ϕ . 2
Secondly we verify the condition (1) in Lemma 5.11. By direct calculation, we have (γ F ⊗ id)(E ) − (id ⊗ F K )(E ) = (γ F ⊗ id)(E( f ⊥ ⊗ 1)) − (id ⊗ F K )(E( f ⊥ ⊗ 1)) + (γ F ⊗ id)(γ K ( f )) − (id ⊗ F K )(γ K ( f )) = (γ F ⊗ id)(E)(γ F ( f ⊥ ) ⊗ 1 − f ⊥ ⊗ 1 ⊗ 1) ⊥
+ (γ F ⊗ id)(E)( f ⊗ 1 ⊗ 1) − (id ⊗ F K )(E( f + (id ⊗ F K )(γ K ⊥ ( f )).
(5.3) ⊥
⊗ 1))
(5.4) (5.5)
We estimate the trace norms of the above three terms. On (5.3), we know [(γ F ⊗ id)(E), γ F ( f ⊥ ) ⊗ 1 − f ⊥ ⊗ F ⊗ 1] = 0, and then |(5.3)|ψ⊗ϕ⊗ϕ = (γ F ⊗ id)(E)(γ F ( f ⊥ ) ⊗ 1 − f ⊥ ⊗ F ⊗ 1)ψ⊗ϕ⊗ϕ = (ψ ⊗ ϕ ⊗ ϕ) (γ F ⊗ id)(E) γ F ( f ⊥ ) − f ⊥ ⊗ F ⊗ 1 = (ψ ⊗ ϕ) γ F ((id ⊗ϕ)(E))γ F ( f ⊥ ) − f ⊥ ⊗ F ≤ (ψ ⊗ ϕ) γ F ( f ⊥ ) − f ⊥ ⊗ F = (ψ ⊗ ϕ) γ F ( f ) − f ⊗ F = (id ⊗ id ⊗ϕ) (γ F ⊗ id)(γ K ( f )) − (id ⊗ F )(γ K ( f )) ψ⊗ϕ = (id ⊗ id ⊗ϕ) − (id ⊗ F K ⊥ )(γ K ( f )) + (id ⊗ F K )(γ K ⊥ ( f )) ψ⊗ϕ ≤ (id ⊗ F K ⊥ )(γ K ( f ))ψ⊗ϕ⊗ϕ + (id ⊗ F K )(γ K ⊥ ( f ))ψ⊗ϕ⊗ϕ holds. On the last terms, we have (id ⊗ F ⊥ )(γ K ( f )) K
ψ⊗ϕ⊗ϕ
= (ψ ⊗ ϕ ⊗ ϕ) (id ⊗ F K ⊥ )(γ K ( f )) = (ϕ ⊗ ϕ) (τ ω ⊗ F K ⊥ )(γ K ( f )) = τω ( f )(ϕ ⊗ ϕ)((F ⊗ K ⊥ )(K )) < | f |ψ δ|F|ϕ |K |ϕ = δ|F|ϕ | f |ψ ,
Classification of Minimal Actions of a Compact Kac Algebra with Amenable Dual
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where we have used the (F, δ)-invariance of K . Similarly we get (id ⊗ F K )(γ ⊥ ( f )) < δ|F|ϕ | f |ψ . K
ψ⊗ϕ⊗ϕ
Hence we obtain |(5.3)|ψ⊗ϕ⊗ϕ < 2δ|F|ϕ | f |ψ . On (5.4), we have
|(5.4)|ψ⊗ϕ⊗ϕ = (γ F ⊗ id)(E) − (id ⊗ F K )(E) ( f ⊥ ⊗ 1 ⊗ 1)ψ⊗ϕ⊗ϕ ≤ (γ F ⊗ id)(E) − (id ⊗ F K )(E)ψ⊗ϕ⊗ϕ = |F|ϕ a E .
On (5.5), we have
|(5.5)|ψ⊗ϕ⊗ϕ = (id ⊗ F K )(γ K ⊥ ( f ))ψ⊗ϕ⊗ϕ = ψ( f ) F K (K ⊥ ) ϕ⊗ϕ
< δ|F|ϕ | f |ψ . Summarizing these calculations, we have |F|ϕ a E ≤ |(5.3)|ψ⊗ϕ⊗ϕ + |(5.4)|ψ⊗ϕ⊗ϕ + |(5.5)|ψ⊗ϕ⊗ϕ < 2δ|F|ϕ | f |ψ + |F|ϕ a E + δ|F|ϕ | f |ψ = |F|ϕ a E + 3δ|F|ϕ | f |ψ ≤ |F|ϕ a E + 3δ 1/2 |F|ϕ (b E − b E ). Hence the condition (1) in Lemma 5.11 holds.
Proof of Theorem 5.9. Consider a subset S ⊂ J whose element E satisfies a E ≤ 3δ 1/2 b E . We order S by E ≺ E if E = E or the inequalities (1) and (2) in Lemma 5.11 hold. Since S contains 0, S is nonempty. The order of S is inductive as is shown below. By Lemma 5.11 (2), the map b is an order preserving isomorphism on a totally ordered subset L ⊂ S onto a subset in [0, 1]. Hence L is cofinal. Then again with (2), the cofinal subsequence of L strongly converges to a projection. We can easily observe that S is strongly closed. Hence the supremum of L exists in S. By Zorn’s lemma, there exists a maximal element E in S. Assume b E¯ < 1 − δ 1/2 and then by Lemma 5.11 we can take an element E ∈ J which satisfies the conditions in the lemma for E. It is easy to see that E ∈ S, E¯ ≺ E and E¯ = E , but this is a contradiction. Hence we have b E¯ ≥ 1 − δ 1/2 . Set a projection p = 1 − (id ⊗ϕ)(E) = 1 − (id ⊗ϕ)(a ∗ v Ev ∗ a) in S ∩ Mω and then we have τω ( p) ≤ δ 1/2 . Then set a projection E = E + p ⊗ e1 . Since K ≥ e1 , E = E(1 ⊗ K ). We verify all the conditions in Theorem 5.9. The conditions (3), (4), (5), (6), (7) are immediately verified. On the condition (2), we estimate as follows: γ F (E) − (id ⊗ F K )(E) ψ⊗ϕ⊗ϕ ≤ γ F (E) − (id ⊗ F K )(E) ψ⊗ϕ⊗ϕ + γ F ( p) ⊗ e1 − p ⊗ F K (e1 )ψ⊗ϕ⊗ϕ ≤ 3δ 1/2 |F|ϕ b E + 2| p|ψ |F|ϕ ≤ 5δ 1/2 |F|ϕ .
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Finally we show the condition (8). By the conditions (4) and (7), the element µ = (id ⊗ϕ)(U ) is a unitary. We claim the following inequalities: (a ∗ − 1 ⊗ K )v Eψ⊗ϕ < κ, γ F (a ∗ − 1 ⊗ K )v E ψ⊗ϕ⊗ϕ < κFϕ ,
(5.6)
(u F,K − 1 ⊗ F ⊗ K )(id ⊗)(v E)ψ⊗ϕ⊗ϕ < (4δ 1/2 + κ 2 )1/2 Fϕ .
(5.8)
(5.7)
The inequality (5.7) is an immediate consequence from (5.6) since ψ ⊗ τπ ◦ γ = ψ for all π ∈ Irr(G). The inequality (5.6) is proved as follows: (a ∗ − 1 ⊗ K )v E2ψ⊗ϕ = (ψ ⊗ ϕ)(Ev ∗ |a ∗ − 1 ⊗ K |2 v E) = (ψ ⊗ ϕ)(Ev ∗ aa ∗ v E + E) − 2(ψ ⊗ ϕ)(Ev ∗ av E) = 2|E|ψ⊗ϕ − 2(ψ ⊗ ϕ)(v ∗ av E) = 2|E|ψ⊗ϕ − 2(ψ ⊗ ϕ)(av Ev ∗ ) = 2|E|ψ⊗ϕ − 2(φ ◦ τ ω ⊗ ϕ)(av Ev ∗ ) = 2|E|ψ⊗ϕ − 2(φ ⊗ ϕ) a(τ ω ⊗ id)(v Ev ∗ + p ⊗ e1 ) = 2|E|ψ⊗ϕ − 2(φ ⊗ ϕ) a(b E |K |−1 ϕ (1 ⊗ K ) + τω ( p)(1 ⊗ e1 ) = 2|E|ψ⊗ϕ − 2b E |K |−1 ϕ (φ ⊗ ϕ)(a K ) − 2τω ( p) = b E |K |−1 ϕ (2|K |ϕ − 2(φ ⊗ ϕ)(a K )) 2 ≤ |K |−1 ϕ a K − 1 ⊗ K φ⊗ϕ
< κ 2, where we have used Lemma 5.10. The inequality (5.8) is obtained as follows: (u F,K − 1 ⊗ F ⊗ K )(id ⊗)(v E)2ψ⊗ϕ⊗ϕ = (ψ ⊗ ϕ ⊗ ϕ)((id ⊗)(Ev ∗ )|u F,K − 1 ⊗ F ⊗ K |2 (id ⊗)(v E)) = (ψ ⊗ ϕ ⊗ ϕ)(|u F,K − 1 ⊗ F ⊗ K |2 (id ⊗)(v Ev ∗ )) = (φ ◦ τ ω ⊗ ϕ ⊗ ϕ)(|u F,K − 1 ⊗ F ⊗ K |2 (id ⊗)(v Ev ∗ )) = (φ ⊗ ϕ ⊗ ϕ)(|u F,K − 1 ⊗ F ⊗ K |2 (τ ω ⊗ )(v Ev ∗ )) = (φ ⊗ ϕ ⊗ ϕ)(|u F,K − 1 ⊗ F ⊗ K |2 · (b E |K |−1 ϕ (1 ⊗ (K )) + τω ( p)(1 ⊗ (e1 )))) 2 ≤ b E |K |−1 ϕ u F,K − 1 ⊗ F ⊗ K ψ⊗ϕ⊗ϕ
+ τω ( p)(u F,K − 1 ⊗ F ⊗ K )(1 ⊗ (e1 ))2ψ⊗ϕ⊗ϕ 2 2 ≤ b E |K |−1 ϕ κ |F||K| + 4τω ( p)1 ⊗ F K (e1 )ψ⊗ϕ⊗ϕ
≤ κ 2 b E |K |−1 ϕ |F|ϕ |K |ϕ + 4τω ( p)|F|ϕ < κ 2 |F|ϕ + 4δ 1/2 |F|ϕ . Let v F γ F (µ) − µ ⊗ F = w|v F γ F (µ) − µ ⊗ F|
Classification of Minimal Actions of a Compact Kac Algebra with Amenable Dual
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Then we have be the polar decomposition with the partial isometry w ∈ M ω ⊗ L ∞ (G). |v F γ F (µ) − µ ⊗ F|ψ⊗ϕ = (ψ ⊗ ϕ)(w ∗ (v F γ F (µ) − µ ⊗ F)) = (ψ ⊗ ϕ) w ∗ (id ⊗ id ⊗ϕ)((v F ⊗ K )γ F (a ∗ v E) − (id ⊗ F )(a ∗ v E)) = (ψ ⊗ ϕ ⊗ ϕ) (w ∗ ⊗ K )(v F ⊗ K )γ F ((a ∗ − 1 ⊗ K )v E) + (w ∗ ⊗ K ) (v F ⊗ K )γ F (v)(γ F (E) − (id ⊗ F K )(E)) + (w ∗ ⊗ K ) (v F ⊗ K )γ F (v) − (id ⊗ F K )(v) (id ⊗ F K )(E) + (w ∗ ⊗ 1) (id ⊗ F K )(v E) − (id ⊗ F )(v E) + (w ∗ ⊗ 1)(id ⊗ F )((1 ⊗ K − a ∗ )v E) .
(5.9) (5.10) (5.11) (5.12) (5.13)
On (5.9), using (5.7), we have |(5.9)| = (ψ ⊗ ϕ ⊗ ϕ) (w ∗ ⊗ K )(v F ⊗ K )γ F ((a ∗ − 1 ⊗ K )v E) = (ψ ⊗ ϕ ⊗ ϕ) γ F (E)(w ∗ ⊗ K )(v F ⊗ K )γ F ((a ∗ − 1 ⊗ K )v E)
∗
≤ (v F ⊗ K )(w ⊗ K )γ F (E) ψ⊗ϕ⊗ϕ γ F ((a ∗ − 1 ⊗ K )v E) ψ⊗ϕ⊗ϕ
≤ Fϕ γ F ((a ∗ − 1 ⊗ K )v E) ψ⊗ϕ⊗ϕ < κ|F|ϕ . On (5.10), the term γ F (E)−(id ⊗ F K )(E) is in the centralizer of the weight ψ ⊗ϕ ⊗ϕ. Using approximate equivalence of E, we have |(5.10)| = (ψ ⊗ ϕ ⊗ ϕ) (w ∗ ⊗ K ) (v F ⊗ K )γ F (v)(γ F (E) − (id ⊗ F K )(E)) ≤ (w∗ ⊗ K )(v F ⊗ K )γ F (v)γ F (E) − (id ⊗ F K )(E) ψ⊗ϕ⊗ϕ
< 5δ
1/2
|F|ϕ .
On (5.11), using u = (v ⊗ 1)γ (v)(id ⊗)(v ∗ ) and (5.8), we have |(5.11)| = (ψ ⊗ ϕ ⊗ ϕ) (w ∗ ⊗ K ) (v F ⊗ K )γ F (v) − (id ⊗ F K )(v) (id ⊗ F K )(E) = (ψ ⊗ ϕ ⊗ ϕ) (id ⊗ F K )(E)(w ∗ ⊗ K )(u F,K − 1 ⊗ F ⊗ K )(id ⊗ F K )(v E) ≤ (w ⊗ K )(id ⊗ F K )(E)ψ⊗ϕ⊗ϕ (u F,K − 1 ⊗ F ⊗ K )(id ⊗ F K )(v E)ψ⊗ϕ⊗ϕ < Fϕ (4δ 1/2 + κ)1/2 Fϕ = (4δ 1/2 + κ)1/2 |F|ϕ .
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On (5.12), we have |(5.12)| = (ψ ⊗ ϕ ⊗ ϕ) (w ∗ ⊗ 1) (id ⊗ F K )(v E) − (id ⊗ F )(v E) = (ψ ⊗ ϕ ⊗ ϕ) − (w ∗ ⊗ 1)(id ⊗ F K ⊥ )(v E) = (ψ ⊗ ϕ ⊗ ϕ) − (id ⊗ F K ⊥ )(E)(w ∗ ⊗ 1)(id ⊗ F K ⊥ )(v E) ≤ (w ⊗ 1)(id ⊗ F K ⊥ )(E)ψ⊗ϕ⊗ϕ (id ⊗ F K ⊥ )(v E)ψ⊗ϕ⊗ϕ ≤ Fϕ (id ⊗ F K ⊥ )(E)ψ⊗ϕ⊗ϕ 1/2 = Fϕ (ψ ⊗ ϕ ⊗ ϕ) (id ⊗ F K ⊥ )(E + p ⊗ e1 ) 1/2 = Fϕ (φ ◦ τ ω ⊗ ϕ ⊗ ϕ) (id ⊗ F K ⊥ )(E + p ⊗ e1 ) 1/2 = Fϕ (φ ⊗ ϕ ⊗ ϕ) (id ⊗ F K ⊥ )(b E |K |−1 ϕ 1 ⊗ K + τω ( p) ⊗ e1 ) 1/2 ⊥ ≤ Fϕ b E |K |−1 ϕ (ϕ ⊗ ϕ)((F ⊗ K )(K )) + τω ( p)|F|ϕ 1/2 < Fϕ b E |K |−1 ϕ δ|F|ϕ |K |ϕ + τω ( p)|F|ϕ ≤ |F|ϕ (δ + δ 1/2 )1/2 < 21/2 δ 1/4 |F|ϕ . Finally on (5.13), using (5.6), we have |(5.13)| = (ψ ⊗ ϕ ⊗ ϕ) (w ∗ ⊗ 1)(id ⊗ F )((1 ⊗ K − a ∗K )v E) = (ψ ⊗ ϕ ⊗ ϕ) (id ⊗ F )(E)(w ∗ ⊗ 1)(id ⊗ F )((1 ⊗ K − a ∗K )v E)
≤ (w ⊗ 1)(id ⊗ F )(E)ψ⊗ϕ⊗ϕ (id ⊗ F )((1 ⊗ K − a ∗K )v E)ψ⊗ϕ⊗ϕ ≤ Fϕ Fϕ (1 ⊗ K − a ∗K )v Eψ⊗ϕ < κ|F|ϕ .
Therefore we obtain |v F γ F (µ) − µ ⊗ F|ψ⊗ϕ ≤ |(5.9)| + |(5.10)| + |(5.11)| + |(5.12)| + |(5.13)| < κ + 5δ 1/2 + (4δ 1/2 + κ)1/2 + 21/2 δ 1/4 + κ |F|ϕ ≤ (9δ 1/4 + 3κ 1/2 )|F|ϕ . 6. Cohomology Vanishing II 6.1. 2-cohomology vanishing in McDuff factors of type II1 . In Lemma 4.3, we have proved the 2-cohomology vanishing result in an ultraproduct von Neumann algebra. This result ensures the existence of a unitary v which is able to perturb a 2-cocycle u to a much smaller 2-cocycle u , but the problem is how small v is. In the construction of v in the proof of Lemma 4.2, we see that v is not small even when u is small. It is, however, an approximate 1-cocycle and then the Rohlin type theorem enables us to perturb v to a small new unitary v by a 1-coboundary constructed from a Shapiro unitary. Since the perturbation of v does not change 2-cocycle u essentially, we can perturb u to make it much smaller by a small unitary v . Successive perturbations yield a vanishing result of
Classification of Minimal Actions of a Compact Kac Algebra with Amenable Dual
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2-cocycles in the original von Neumann algebra. We mention that this strategy has been seen in the context of [Oc2]. From now on, we assume that M is a McDuff factor of type II1 with the tracial state τ . Then the technical assumption in Theorem 5.9 (8) automatically stands up for φ = τ . The trace τ ◦ τ ω on M ω is also denoted by τ . We choose a suitable net of Følner sets as follows. If Irr(G) is finite, we set F0 = K 0 = 1 and δ0 = 0. When Irr(G) is infinite, and for each n ≥ 0 we will take finitely supported central projections Fn , K n in L ∞ (G) δn > 0 inductively such that ∞ (1) {Fn }∞ n=0 and {K n }n=0 are increasing and strongly converge to 1, ∞ (2) {δn }n=0 is decreasing and lim δn = 0, n→∞
(3) K n is (Fn , δn )-invariant and K n ≥ e1 , 1/4 1/2 (4) (9δn + 3δn+1 )|Fn |ϕ < (1/2)δn−1 (n ≥ 1). ∞ Fix a sequence of ascending finitely supported central projections {Sn }∞ n=0 in L (G) ∞ with S0 = e1 and ∪n=0 supp(Sn ) = Irr(G). Let F0 = e1 = K 0 and δ0 = 1. First take F1 1/4 and δ1 > 0 such that F1 ≥ F0 ∨ K 0 ∨ K 0 ∨ S1 and 9δ1 |F1 |ϕ < (1/2)δ0 . Then take an (F1 , δ1 )-invariant finitely supported central projection K 1 with K 1 ≥ e1 . Second take 1/4 1/2 F2 and δ2 > 0 such that F2 ≥ F1 ∨ K 1 ∨ K 1 ∨ S2 and (9δ1 + 3δ2 )|F1 |ϕ < (1/2)δ0 1/4 and 9δ2 |F2 |ϕ < (1/2)δ1 . Suppose we have chosen finitely supported central projec and δn > 0 with 9δn1/4 |Fn |ϕ < (1/2)δn−1 . Then take Fn+1 and tions Fn , K n in L ∞ (G) 1/4 1/2 δn+1 such that Fn+1 ≥ Fn ∨ K n ∨ K n ∨ Sn+1 , (9δn + 3δn+1 )|Fn |ϕ < (1/2)δn−1 and 1/4 9δn+1 |Fn+1 |ϕ < (1/2)δn . Then take K n+1 which is (Fn+1 , δn+1 )-invariant and K n+1 ≥ e1 . Set Fn = supp(Fn ) and Kn = supp(K n ). Since Fn ≥ Sn , ∪∞ n=0 Fn = Irr(G). on M. By Lemma 4.5, we Let (α, u) be an approximately inner cocycle action of G such that α = Ad v on M ⊂ M ω , can take a unitary v in M ω ⊗ L ∞ (G)
(v ∗ ⊗ 1)α ω (v ∗ )u(id ⊗)(v) = 1, lim (φ ⊗ τπ ) ◦ Ad vπ∗ ◦ απ − φ = 0 for all φ ∈ M∗ , π ∈ Irr(G).
n→∞
Then γ = Ad v ∗ ◦ α ω is an action on M ω fixing M and preserving Mω . Assume that u π,ρ − 1 ⊗ 1π ⊗ 1ρ τ ⊗ϕ⊗ϕ < δn+1 for all π, ρ ∈ Fn+1 . Then v is an approximate coboundary, and by Theorem 5.9 there exists a unitary µ ∈ M ω such that 1/4
|v Fn γ Fn (µ) − µ ⊗ Fn |τ ⊗ϕ < (9δn
1/2
+ 3δn+1 )|Fn |ϕ < δn−1 .
Then we have |(µ ⊗ Fn )v ∗Fn α ωFn (µ∗ ) − 1 ⊗ Fn |τ ⊗ϕ = |(µ ⊗ Fn )γ Fn (µ∗ )v ∗Fn − 1 ⊗ Fn |τ ⊗ϕ = |µ ⊗ Fn − v Fn γ Fn (µ)|τ ⊗ϕ < δn−1 . Set a perturbed unitary v = (µ ⊗ 1)v ∗ α ω (µ∗ ) and then we have ( v ⊗ 1)α ω ( v )u(id ⊗)( v ∗ ) = 1, | v Fn − 1 ⊗ Fn |τ ⊗ϕ < δn−1 .
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T. Masuda, R. Tomatsu
Let ( vm )m be a representing sequence of v . Then there exists m ∈ N such that ∗ (( vm )π ⊗ 1ρ )απ (( vm )ρ )u π,ρ (id ⊗π ρ )( vm ) − 1 ⊗ 1π ⊗ 1ρ τ ⊗ϕ⊗ϕ < δn+2 , | vm (1 ⊗ Fn ) − 1 ⊗ Fn |τ ⊗ϕ < δn−1
for all π, ρ ∈ Fn+2 . Hence we obtain the following lemma. Lemma 6.1. Let M be a McDuff factor of type II1 and (α, u) an approximately inner on M. If the inequality strongly free cocycle action of G u π,ρ − 1 ⊗ 1π ⊗ 1ρ τ ⊗ϕ⊗ϕ < δn+1 such that holds for all π, ρ ∈ Fn+1 , then there exists a unitary w ∈ M ⊗ L ∞ (G) (1) for all π, ρ ∈ Fn+2 ,
(wπ ⊗ 1ρ )απ (wρ )u π,ρ (id ⊗π ρ )(w ∗ ) − 1 ⊗ 1π ⊗ 1ρ
< δn+2 , τ ⊗ϕ⊗ϕ (2) |w Fn − 1 ⊗ Fn |τ ⊗ϕ < δn−1 . Then similar to the proof of [Oc1, Theorem 7.6], we obtain a 2-cohomology vanishing result in M. Theorem 6.2 (2-cohomology vanishing theorem). Let M be a McDuff factor of type II1 with the tracial state τ . Let (α, u) be an approximately inner strongly free cocycle action on M. Then u is a coboundary. Moreover, assume for fixed n ≥ 2, the inequality of G u π,ρ − 1 ⊗ 1π ⊗ 1ρ τ ⊗ϕ⊗ϕ < δn+1 such that holds for all π, ρ ∈ Fn+1 . Then there exists a unitary w ∈ M ⊗ L ∞ (G) (1) (w ⊗ 1)α(w)u(id ⊗)(w∗ ) = 1, (2) |w Fn − 1 ⊗ Fn |τ ⊗ϕ < δn−2 .
Proof. It suffices to prove the theorem only in the case that u π,ρ −1⊗1π ⊗1ρ τ ⊗ϕ⊗ϕ < δn+1 holds for all π, ρ ∈ Fn+1 . By the previous lemma, there exists a unitary w ∈ such that M ⊗ L ∞ (G) (i) for all π, ρ ∈ Fn+2 ,
(wπ ⊗ 1ρ )απ (wρ )u π,ρ (id ⊗π ρ )(w ∗ ) − 1 ⊗ 1π ⊗ 1ρ
< δn+2 , τ ⊗ϕ⊗ϕ (ii) |w Fn − 1 ⊗ Fn |τ ⊗ϕ < δn−1 . Set u n = u, w n = w, α n+1 = Ad w n ◦α and u n+1 = (wπ ⊗1ρ )απ (wρ )u(id ⊗π ρ )(w ∗ ). Then (α n+1 , u n+1 ) is a strongly free cocycle action on M with u n+1 π,ρ − 1 ⊗ 1π ⊗ 1ρ τ ⊗ϕ⊗ϕ < δn+2 for all π, ρ ∈ Fn+2 . With a repetition of the above argument, we get a family of cocycle actions {(α m , u m )}m≥n and unitaries {w m }m≥n satisfying the following conditions: (1.m) (2.m) (3.m) (4.m)
α m+1 = Ad w m ◦ α m , u m+1 = (w m ⊗ 1)α m (w m )u m (id ⊗)(w m∗ ), u m π,ρ − 1 ⊗ 1π ⊗ 1ρ τ ⊗ϕ⊗ϕ < δm+1 for all π, ρ ∈ Fm+1 , |w m Fm − 1 ⊗ Fm |τ ⊗ϕ < δm−1 .
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Then set w m = w m w m−1 . . . wn and we have α m+1 = Ad w m ◦ α and u m+1 = (w m ⊗ 1)α(w m )u(id ⊗)(wm∗ ). By the condition (4.m), it is easy to see that the sequence of unitaries {w m }m≥n strongly converges to a unitary w. Then by (3.m), (w ⊗ 1)α(w)u(id ⊗)(w∗ ) = 1 holds. Moreover we have n |w mFn − 1 ⊗ Fn |τ ⊗ϕ ≤ |w m Fn − 1 ⊗ Fn |τ ⊗ϕ + · · · + |w Fn − 1 ⊗ Fn |τ ⊗ϕ
< δm−1 + · · · + δn−1 ≤ δn−1 (1 + 1/2 + 1/22 + . . . ) < δn−2 . Hence we are done.
By virtue of 2-cohomology vanishing, we can show the following results. Corollary 6.3. Let M be a McDuff factor of type II1 with the tracial state τ and α an on M. Let v be a unitary in M ⊗ L ∞ (G). approximately inner strongly free action of G Assume for fixed n ≥ 2, the inequality (vπ ⊗ 1ρ )απ (vρ )(id ⊗π ρ )(v ∗ ) − 1 ⊗ 1π ⊗ 1ρ τ ⊗ϕ⊗ϕ < δn+1 such that holds for all π, ρ ∈ Fn+1 . Then there exists a unitary w ∈ M ⊗ L ∞ (G) (1) (wv ⊗ 1)α(wv)(id ⊗)(((wv)∗ ) = 1, (2) |w Fn − 1 ⊗ Fn |τ ⊗ϕ < δn−2 . Proof. Let α = Ad v ◦ α and u = (v ⊗ 1)α(v)(id ⊗)(v ∗ ). Apply the previous theorem to the cocycle action ( α, u ) and it is done. With Lemma 4.6 and the previous corollary, we can show the following result. Corollary 6.4. Let M be a McDuff factor of type II1 with the tracial state τ , and α, β approximately inner strongly free actions. Then for any ε > 0, finite sets F Irr(G) and T M, there exists an α-cocycle v satisfying
βπ (x) − Ad vπ (απ (x))
<ε τ ⊗ϕ for all π ∈ F and x ∈ T . with β = Ad W ◦ α Proof. By Lemma 4.6, we can find an α ω -cocycle W ∈ M ω⊗ L ∞ (G) ∞ ν on M. Let (w )ν=0 be a representing sequence of W . Since (W ⊗1)α ω (W )(id ⊗)(W ∗ ) = 1, for all π, ρ ∈ Irr(G) we have
lim (wπν ⊗ 1ρ )απ (wρν )(id ⊗π ρ )(w ν∗ ) − 1 ⊗ 1π ⊗ 1ρ τ ⊗ϕ⊗ϕ = 0. ν→ω
√ 1/2 sup (1 + 2 2x)δn−2 < ε. Also take a large ν Take a
large n so that F ⊂ Fn and x∈T
so that βπ (x) − Ad wπν (απ (x)) τ ⊗ϕ < δn−2 and
ν
(w ⊗ 1ρ )απ (w ν )(id ⊗π ρ )(w ν∗ ) − 1 ⊗ 1π ⊗ 1ρ
< δn+1 π ρ τ ⊗ϕ⊗ϕ
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for all π, ρ ∈ Fn+1 and x ∈ T . Apply the previous corollary and then we have a unitary such that |v Fn − 1 ⊗ Fn |τ ⊗ϕ < δn−2 and vw ν is an α-cocycle. Set v ∈ M ⊗ L ∞ (G) ν v = vw and then
βπ (x) − Ad vπ (απ (x))
τ ⊗ϕ
ν
≤ βπ (x) − Ad wπ (απ (x)) τ ⊗ϕ + Ad wπν (απ (x)) − Ad vπ (απ (x)) τ ⊗ϕ
≤ βπ (x) − Ad wπν (απ (x)) τ ⊗ϕ + 2wπν − vπ τ ⊗ϕ x
= βπ (x) − Ad wπν (απ (x)) τ ⊗ϕ + 21 ⊗ 1π − v π τ ⊗ϕ x √
1/2 ≤ βπ (x) − Ad wπν (απ (x)) τ ⊗ϕ + 2 2|1 ⊗ 1π − v π |τ ⊗ϕ x √ 1/2 < δn−2 + 2 2δn−2 x <ε for all π ∈ Fn and x ∈ T .
6.2. Shapiro unitary. We represent a Rohlin type theorem for two actions in order to study a commutation property of a Shapiro unitary, which is stated in Theorem 6.5 (9). We will explain a motivation for the study by considering a simple case. Let α be an action on a von Neumann algebra M and v an α-cocycle. Put β = Ad v ◦ α. Let K ∈ Proj(Z (L ∞ (G))) as before. Assume that α ω has a Rohlin projection E ∈ ω ∞ M ⊗ L (G)K in the sense of Theorem 5.9. In addition, we assume that the projection v Ev ∗ is a Rohlin projection for β. Now take a finite subset T ⊂ M. Suppose that [E ρ ⊗ 1ρ , αρ (αρ (x))] = 0 and [vρ ⊗ 1ρ , αρ (αρ (x))] = 0 for all x ∈ T and ρ ∈ K. Then the Shapiro unitary µ = (id ⊗ϕ)(v E) commutes with all x ∈ T . We present an approximate version of the above argument. Theorem 6.5. Let M be a McDuff factor of type II1 with the tracial state τ and α an on M. Let 0 < δ < 1, ε > 0 and approximately inner strongly free action of G with K ≥ e1 . F ∈ Projf(Z (L ∞ (G))). Take an (F, δ)-invariant K ∈ Projf(Z (L ∞ (G))) and set Set F = supp(F) and K = supp(K ). Let v be a unitary α-cocycle in M ⊗ L ∞ (G) a perturbed strongly free action β = Ad v ◦ α. Then for any countable set S ⊂ M ω and satisfying a finite subset T ⊂ M1 , there exist projections E α and E β in M ω ⊗ L ∞ (G) the following conditions: (1) E α = E α (1 ⊗ K ), E β = E β (1 ⊗ K ). (2) The following splitting properties of τ ω : (τ ω ⊗ id)(x E α ) = (τ ω ⊗ id)(x)(τ ω ⊗ id)(E α ), (τ ω ⊗ id)(x E β ) = (τ ω ⊗ id)(x)(τ ω ⊗ id)(E β ) : hold for all x ∈ S ⊗ L ∞ (G)K (3) (approximate equivariance) ω α α (E ) − (id ⊗ F K )(E α ) < 5δ 1/2 |F|ϕ , F τ ⊗ϕ ω β β (E ) − (id ⊗ F K )(E β ) < 5δ 1/2 |F|ϕ . F τ ⊗ϕ
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(4) Decompose E α and E β as Eα =
ρ∈K i, j∈Iρ
β
E =
ρ∈K i, j∈Iρ
dρ−1 f ραi, j ⊗ eρi, j , β
dρ−1 f ρ i, j ⊗ eρi, j .
β
Then { f ραi, j } and { f ρ i, j } satisfy f ραi, j f παk, = δρ,π δ j,k f ραi, ,
β
β
β
f ρ i, j f π k, = δρ,π δ j,k f ρ i,
for all ρ, π ∈ K, i, j ∈ Iρ and k, ∈ Iπ . (5) (joint property of U ) Set U = v E α and decompose U as U= dρ−1 µρ i, j ⊗ eρi, j . ρ∈K i, j∈Iρ
Then we have µ∗ρ i, j µπ k, = δρ,π δi,k f ραj, , β
µρ i, j µ∗π k, = δρ,π δ j, f ρ i,k for all ρ, π ∈ K, i, j ∈ Iρ and k, ∈ Iπ . In particular, U ∗ U = E α and UU ∗ = E β holds. β (6) For each ρ ∈ K, the projections (id ⊗ϕ)(E ρα ) and (id ⊗ϕ)(E ρ ) are equal. In addition, they are in S ∩ Mω . (7) (partition of unity) (id ⊗ϕ)(E α ) = 1 = (id ⊗ϕ)(E β ). (8) (Shapiro lemma) Set µ = (id ⊗ϕ)(U ) and then µ is a unitary satisfying |v F α ω (µ) − µ ⊗ F|τ ⊗ϕ < 9δ 1/4 |F|ϕ . (9) Further assume βπ (απ (x)) − απ (απ (x))τ ⊗ϕ < ε for all x ∈ T and π ∈ K, then the unitary µ satisfies |[µ, x]|τ < ε for all x ∈ T . such that α = Ad V on Proof. As in Lemma 4.5, we take a unitary V ∈ M ω ⊗ L ∞ (G) ∗ ω M and V is an α -cocycle. Set the strongly free action γ = Ad V ∗ ◦ α ω . We use the same notations in Theorem 5.9. We may assume that S contains the entries of απ (x) for all π ∈ Irr(G) and x ∈ T . Recall the set J defined in §5.3. We denote by I the subset of J whose elements satisfy the following conditions. For E ∈ I, (i) [E, γ (x)] = 0 for all x ∈ S.
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T. Masuda, R. Tomatsu α
β
(ii) The projections E := V E V ∗ and E := vV E V ∗ v ∗ satisfy the conditions (1), (2), (4), (5) and (6) in Theorem 6.5. α β (iii) The equality (id ⊗ϕ)(E ρ ) = (id ⊗ϕ)(E ρ ) = (id ⊗ϕ)(E ρ ) holds for all ρ ∈ K. α
β
ω (iv) (τ ω ⊗ id)(E ) = b E |K |−1 ϕ K = (τ ⊗ id)(E ).
Recall a subset S ⊂ J defined in the proof of Theorem 5.9. Then we can easily see that I ∩ S is an inductive ordered set as similar to the proof of Theorem 5.9. Let E be a maximal element of I. Since the proof of Lemma 5.11 is applicable for I with the additional assumption (i), the projection p = 1 − (id ⊗ϕ)(E) satisfies τω ( p) ≤ δ 1/2 . Let E α = V E V ∗ and E β = vV E V ∗ v ∗ , where E = E + p ⊗ e1 . Then the conditions (1), (2), (4), (5), (6), (7) of Theorem 6.5 are satisfied. Since V ∗ is an α ω -cocycle, the condition (3) follows from the tracial property of τ . On (8), a similar proof to that of Theorem 5.9 (8) is applicable to (α, E α ) and an α-cocycle v. Finally we verify (9). For ρ ∈ K and x ∈ T , α
[vρ ⊗ 1ρ , αρ (αρ (x))](E ρ ⊗ 1ρ )2τ ⊗ϕ⊗ϕ α α = (τ ⊗ ϕ ⊗ ϕ) (E ρ ⊗ 1ρ )|[vρ ⊗ 1ρ , αρ (αρ (x))]|2 (E ρ ⊗ 1ρ ) α = (τ ⊗ ϕ ⊗ ϕ) |[vρ ⊗ 1ρ , αρ (αρ (x))]|2 ((τ ω ⊗ id)(E ρ ) ⊗ 1ρ ) = (τ ⊗ ϕ ⊗ ϕ) |[vρ ⊗ 1ρ , αρ (αρ (x))]|2 (b E |K |−1 ϕ 1ρ ⊗ 1ρ )
2
= b |K |−1 [vρ ⊗ 1ρ , αρ (αρ (x))]
<ε
E 2
ϕ
τ ⊗ϕ⊗ϕ
|K |−1 ϕ .
Using it, for ρ ∈ K and x ∈ T , |[Uρ ⊗ 1ρ , αρ (αρ (x))]|τ ⊗ϕ⊗ϕ = |[v E α ⊗ 1ρ , αρ (αρ (x))]|τ ⊗ϕ⊗ϕ α
= |[v E ⊗ 1ρ , αρ (αρ (x))]|τ ⊗ϕ⊗ϕ α
≤ |[vρ ⊗ 1ρ , αρ (αρ (x))](E ρ ⊗ 1ρ )|τ ⊗ϕ⊗ϕ α
+ |(vρ ⊗ 1ρ )[E ρ ⊗ 1ρ , αρ (αρ (x))]|τ ⊗ϕ⊗ϕ α
= |[vρ ⊗ 1ρ , αρ (αρ (x))](E ρ ⊗ 1ρ )|τ ⊗ϕ⊗ϕ α
≤ [vρ ⊗ 1ρ , αρ (αρ (x))](E ρ ⊗ 1ρ )τ ⊗ϕ⊗ϕ α
· E ρ ⊗ 1ρ τ ⊗ϕ⊗ϕ α
< εK −1 ϕ E ρ τ ⊗ϕ dρ −1 2 = εK −1 ϕ b E K ϕ dρ 1/2
≤ εdρ2 |K |−1 ϕ . ∗ ·T 2 Set a state θρ = Tρ,ρ ρ,ρ on B(Hρ ⊗ Hρ ). Then we have ϕρ (a) = dρ θρ (a ⊗ 1ρ ) for 2 all a ∈ B(Hρ ) and dρ θρ ≤ ϕρ ⊗ ϕρ as positive functionals. We claim that
dρ2 |(id ⊗θρ )(x)|τ ≤ |x|τ ⊗ϕρ ⊗ϕρ for all x ∈ M ω ⊗ B(Hρ ⊗ Hρ ).
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Let (id ⊗θρ )(x) = w|(id ⊗θρ )(x)| and x = w |x| be the polar decompositions. Then the claim is verified as follows: dρ2 |(id ⊗θρ )(x)|τ = dρ2 τ (w ∗ (id ⊗θρ )(x)) = dρ2 (τ ⊗ θρ )((w ∗ ⊗ 1ρ ⊗ 1ρ )x) = dρ2 (τ ⊗ θρ )((w ∗ ⊗ 1ρ ⊗ 1ρ )w |x|1/2 |x|1/2 ) 1/2 ∗ ≤ dρ2 (τ ⊗ θρ ) (w ∗ ⊗ 1ρ ⊗ 1ρ )w |x|w (w ⊗ 1ρ ⊗ 1ρ ) (τ ⊗ θρ )(|x|)1/2 ∗ 1/2 ∗ ≤ (τ ⊗ ϕρ ⊗ ϕρ ) (w ⊗ 1ρ ⊗ 1ρ )w |x|w (w ⊗ 1ρ ⊗ 1ρ ) (τ ⊗ ϕρ ⊗ ϕρ )(|x|)1/2 1/2 ∗ = (τ ⊗ ϕρ ⊗ ϕρ ) |x|1/2 w (ww ∗ ⊗ 1ρ ⊗ 1ρ )w |x|1/2 (τ ⊗ ϕρ ⊗ ϕρ )(|x|)1/2 ≤ (τ ⊗ ϕρ ⊗ ϕρ )(|x|)1/2 (τ ⊗ ϕρ ⊗ ϕρ )(|x|)1/2 = |x|τ ⊗ϕρ ⊗ϕρ . Then we obtain |[µ, x]|τ = |[(id ⊗ϕ)(U ), x]|τ dρ2 (id ⊗θρ )(Uρ ⊗ 1ρ ), x = ρ∈K
τ
= dρ2 (id ⊗θρ )([Uρ ⊗ 1ρ , αρ (αρ (x))]) ρ∈K
≤
τ
d 2 (id ⊗θρ )([Uρ ⊗ 1ρ , αρ (αρ (x))]) ρ τ
ρ∈K
≤
[Uρ ⊗ 1ρ , αρ (αρ (x))] τ ⊗ϕ⊗ϕ
ρ∈K
≤
ρ∈K
εdρ2 |K |−1 ϕ
= ε. Although a 1-cohomology does not vanish in M in general, it approximately vanishes. The following theorem shows the approximate vanishing with a commutation property of a Shapiro unitary. Since it is easily proved by considering a representing sequence of µ in the previous theorem, we omit the proof. Theorem 6.6 (Approximate vanishing of 1-cohomology). Let M be a McDuff factor on M. Let F ∈ of type II1 with the tracial state τ and α a strongly free action of G ∞ with Projf(Z (L (G))) and δ, ε > 0. Take an (F, δ)-invariant K ∈ Projf(Z (L ∞ (G))) K ≥ e1 . Let T be a finite subset in the unit ball of M. If an α-cocycle v satisfies Ad vρ ◦ αρ (αρ (x)) − αρ (αρ (x))τ ⊗ϕ⊗ϕ < ε for all x ∈ T and ρ ∈ K , then there exists a unitary w in M satisfying (i) |v F − (w ⊗ 1)α F (w ∗ )|τ ⊗ϕ < 9δ 1/4 |F|ϕ , (ii) |[w, x]|τ < ε, x ∈ T .
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T. Masuda, R. Tomatsu
7. Main Theorem 7.1. Intertwining argument. For a proof of the cocycle conjugacy of two actions, we make use of a so-called intertwining argument initiated by Evans-Kishimoto in [EK]. The results Corollary 6.4 and Theorem 6.6 are necessary for the argument. We briefly explain the outline. Let γ 0 := α and γ −1 := β be approximately inner strongly free actions on M. First by Corollary 6.4, we perturb the action γ −1 to γ 1 by a γ −1 -cocycle v 1 so that γ 1 is close to γ 0 . Second by Corollary 6.4, we perturb the action γ 0 to γ 2 by a γ 0 -cocycle v 2 so that γ 2 is close to γ 1 . We construct families of actions and 1-cocycles inductively and achieve the equality at the limit. However in that process, we have to use Theorem 6.6 in order to treat successive multiplications of unitaries. Theorem 7.1. Let M be a McDuff factor of type II1 . Let α and β be approximately inner on M. Then they are cocycle conjugate, that is, there exist an strongly free actions of G automorphism θ in Int(M) and an α-cocycle v with Ad v ◦ α = (θ −1 ⊗ id) ◦ β ◦ θ. ∞ be a strongly dense countable subset of the unit ball of M. Proof. Let S = {ai }i=1 n Put Sn = {ai }i=1 and εn = 2−n . Take the sequences of the finitely supported central ∞ ∞ 0 projections {Fn }∞ n=1 , {K n }n=1 and the positive numbers {δn }n=1 as in §6.1. Set γ = α, γ −1 = β, u −1 = u 0 = 1, θ0 = θ−1 = id ∈ Int(M) and T0 = {1} M. For each n ≥ 1, we construct inductively the following members: on M, (i) an action γ n of G (ii) wn ∈ U (M), (iii) θn ∈ Int(M), (iv) an Ad(wn ⊗ 1) ◦ γ n−2 ◦ Ad wn∗ -cocycle u n , (v) a (θn ⊗ id) ◦ γ n ◦ θn−1 -cocycle u n , where n is equal to 0 or −1 according to that n is even or odd respectively, (vi) a finite subset Tn M.
The induction conditions are (1,n) γρn (x) − γρn−1 (x)τ ⊗ϕ < εn for x ∈ Sn and ρ ∈ Fn (n ≥ 1), (2,n) γρn (γρk (x)) − γρn−1 (γρk (x))τ ⊗ϕ⊗ϕ < εn for all x ∈ Tn−1 , ρ ∈ Kn ∪ Kn+1 and 1 ≤ k ≤ n − 1 (n ≥ 2), 1/2 (3,n) |u nFn − 1 ⊗ Fn |τ ⊗ϕ < 9δn |Fn |ϕ (n ≥ 3), (4,n) |[wn , x]|τ < εn−2 , for x ∈ Tn−2 (n ≥ 3), (5,n) u n = u n (wn ⊗ 1)u n−2 (wn∗ ⊗ 1) (n ≥ 1), (6,n) θn = Ad wn ◦ θn−2 (n ≥ 1), (7,n) γ n = Ad u n ◦ Ad(wn ⊗ 1) ◦ γ n−2 ◦ Ad wn∗ (n ≥ 1), (8,n) Tn = Tn−1 ∪ Sn ∪ θn (Sn ) ∪ {u nπi, j | π ∈ Fn , i, j ∈ Iπ } (n ≥ 1). −1 1st step. Since α and β are approximately inner
and strongly free, there exists a γ cocycle v 1 with Ad vρ1 (γρ−1 (x)) − γρ0 (x) τ ⊗ϕ < ε1 for all x ∈ S1 and ρ ∈ F1 by Applying 6.6 to v 1 , we can take a unitary w1 ∈ M with 1 Corollary 6.4. −1 Theorem 1/4 ∗ v − (w1 ⊗ 1)γ (w ) < 9δ |F1 |ϕ . Then set u 1 = u 1 = v 1 γ −1 (w1 )(w ∗ ⊗ 1) F1
F1
1
τ ⊗ϕ
1
1
and it is an Ad(w1 ⊗ 1) ◦ γ −1 ◦ Ad w1∗ -cocycle. Set an action γ 1 = Ad v 1 ◦ γ −1 = Ad u 1 ◦ Ad(w1 ⊗ 1) ◦ γ −1 ◦ Ad w1∗ , an automorphism θ1 = Ad w1 and T1 as in (8, 1).
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2nd step. Next take a γ 0 -cocycle v 2 with
Ad v 2 (γ 0 (x)) − γ 1 (x)
< ε2 for all x ∈ S2 , ρ ∈ F2 , ρ ρ ρ τ ⊗ϕ
Ad v 2 ◦ γ 0 (γ k (x)) − γ 1 (γ k (x))
< ε2 for all x ∈ T1 , ρ ∈ K2 ∪ K3 , k = 0, 1 ρ ρ ρ ρ ρ τ ⊗ϕ by Corollary 6.4. Then also by Theorem 6.6, we can take a unitary w2 with v 2F2 − (w2 ⊗ 1/4 < 9δ |F2 |ϕ . Set u 2 = v 2 γ 0 (w2 )(w ∗ ⊗ 1) and γ 2 = Ad v 2 ◦ γ 0 . Set 1)γ 0 (w ∗ ) F2
2
τ ⊗ϕ
2
2
θ2 = Ad w2 , u 2 = u 2 and T2 as in (8, 2). (n + 1)st step. Suppose that we have done up to n th step. By Corollary 6.4, we can take a γ n−1 -cocycle v n+1 with
Ad v n+1 (γ n−1 (x)) − γ n (x)
< εn+1 for all x ∈ Sn+1 , ρ ∈ Fn+1 ρ ρ ρ τ ⊗ϕ and for 0 ≤ k ≤ n,
Ad v n+1 ◦ γ n−1 (γ k (x)) − γ n (γ k (x))
< εn+1 for all x ∈ Tn , ρ ∈ Kn+1 ∪Kn+2 . ρ ρ ρ ρ ρ τ ⊗ϕ⊗ϕ Since we have the condition (2, n) (with k = n − 1), i.e.,
n n−1
γ (γ (x)) − γρn−1 (γρn−1 (x)) τ ⊗ϕ⊗ϕ < εn for all x ∈ Tn−1 , ρ ∈ Kn ∪ Kn+1 , ρ ρ we obtain for all x ∈ Tn−1 and ρ ∈ Kn+1 ,
Ad v n+1 ◦ γ n−1 (γ n−1 (x)) − γ n−1 (γ n−1 (x))
< εn + εn+1 . ρ ρ ρ ρ ρ τ ⊗ϕ⊗ϕ Then by Theorem 6.6, there exists a unitary wn+1 in M with n+1 1/4 n−1 ∗ v Fn+1 − (wn+1 ⊗ 1)γ Fn+1 (wn+1 ) τ ⊗ϕ < 9δn+1 |Fn+1 |ϕ and |[wn+1 , x]|τ ⊗ϕ < εn + εn+1 < εn−1 for all x ∈ Tn−1 . ∗ ⊗ 1) and γ n+1 = Ad v n+1 ◦ γ n−1 . Set u n+1 , θ Set u n+1 = v n+1 γ n−1 (wn+1 )(wn+1 n+1 and Tn+1 as in (5, n + 1), (6, n + 1) and (8, n + 1), respectively. Then all the conditions have been verified. Thus we have constructed the members in (i),…,(vi) inductively. We show the existence of lim θ2m and lim θ2m+1 . Let x ∈ Sn . For m ≥ n + 2, we m→∞ m→∞ have ∗ θm (x) − θm−2 (x)τ = wm θm−2 (x)wm − θm−2 (x)τ = [wm , θm−2 (x)]τ √ ≤ 2|[wm , θm−2 (x)]|1/2 τ √ 1/2 < 2εm−2 . √ 1/2 −1 (x)τ < 2εm−2 . Hence the strong limits lim θ2m (x) Similarly we have θm−1 (x)−θm−2 m→∞
and lim θ2m+1 (x) exist for all x ∈ S. It clearly derives the existence of the limits for m→∞
all x ∈ M. Let θ 0 = lim θ2n and θ 1 = lim θ2n+1 . Then they are approximately inner n→∞ n→∞ automorphisms.
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T. Masuda, R. Tomatsu
Next we show the existence of lim u 2m and lim u 2m+1 . For m ≥ n we have m→∞
m→∞
m+2 m+2 m m ∗ m u Fn − u Fn τ ⊗ϕ = u Fn (wm+2 ⊗ 1)u Fn (wm+2 ⊗ 1) − u Fn τ ⊗ϕ m ≤ u m+2 Fn − 1 ⊗ Fn τ ⊗ϕ + [wm+2 ⊗ Fn , u Fn ] τ ⊗ϕ ≤ u m+2 |[wm+2 , u m πi, j ] ⊗ eπi, j |τ ⊗ϕ Fm+2 − 1 ⊗ Fm+2 τ ⊗ϕ + π ∈Fn i, j∈Iπ
1/4 < 9δm+2 |Fm+2 |ϕ
+
π ∈Fn i, j∈Iπ 1/4 < 9δm+2 |Fm+2 |ϕ
+
dπ |[wm+2 , u m πi, j ]|τ dπ εm+2
π ∈Fn i, j∈Iπ 2 ≤ δm+1 + εm+2 |Fn |ϕ . 2n−1 ∞ ∞ }n=1 are Cauchy sequences for all π ∈ Irr(G), and the Hence {u 2n π }n=1 and {u π 0 strong limits uˆ = lim u 2n and uˆ 1 = lim u 2n+1 exist. It is easy to see that γ 2n = n→∞
n→∞
−1 −1 and γ 2n+1 = Ad u 2n+1 ◦ (θ2n+1 ⊗ id) ◦ β ◦ θ2n+1 . Since Ad u 2n ◦ (θ2n ⊗ id) ◦ α ◦ θ2n 2m+1 2m for fixed n, lim γπ (x) − γπ (x)τ ⊗ϕ = 0 for all x ∈ Sn and π ∈ Irr(G), the next m→∞ equality holds on ∪n≥1 Sn , and so does on M, −1
−1
Ad uˆ 0 ◦ (θ 0 ⊗ id) ◦ α ◦ θ 0 = Ad uˆ 1 ◦ (θ 1 ⊗ id) ◦ β ◦ θ 1 . −1
−1
Moreover, uˆ 0 and uˆ 1 are 1-cocycles for (θ 0 ⊗ id) ◦ α ◦ θ 0 and (θ 1 ⊗ id) ◦ β ◦ θ 1 , respectively. Therefore α and β are cocycle conjugate. Since strong freeness and freeness are equivalent notions for the AFD factor of type II1 (see Appendix), we obtain the following result. Corollary 7.2. Any two free actions of an amenable discrete Kac algebra on the AFD factor of type II1 are cocycle conjugate. 7.2. Classification of minimal actions. We show the uniqueness of minimal actions of a compact Kac algebra G = (L ∞ (G), δ, h) with amenable dual on the AFD factor of type II1 . Lemma 7.3. Let M be a finite von Neumann algebra, α an action of a compact Kac algebra G on M. If M α G is a factor, then any α-cocycle is a coboundary. Proof. The proof is similar to that of [W2, Theorem 12]. Let w be an α-cocycle. Set N := M2 (C) ⊗ M, α := id ⊗α and w := e11 ⊗ w + e22 ⊗ 1 ⊗ 1. Then w is an α -cocycle, ∼ and β := Ad w ◦ α is an action of G on N . Since M2 (C)⊗(Mα G) ∼ = N α G = N β G, N β is a factor by [S, Corollary 5] or [Y, Corollary 3.9], and the restriction of any trace on M2 (C) ⊗ M is the unique trace on N β . Since e11 ⊗ 1, e22 ⊗ 1 ∈ N β and their values of trace are 1/2, they are equivalent. Let v ∈ N β be such that v ∗ v = e11 ⊗ 1 and vv ∗ = e22 ⊗ 1. Then v is of the form v = e12 ⊗ u for some u ∈ U (M), and v ∈ N β implies that w = (u ⊗ 1)α(u ∗ ).
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Recall that an action α of a compact Kac algebra G on M is said to be minimal if (M α ) ∩ M = C and the linear span of {(φ ⊗ id)(α(M)) | φ ∈ M∗ } is weakly dense in L ∞ (G) [ILP]. Since this definition is equivalent to (M α ) ∩ M = C and the factoriality of M α G by [S, Corollary 7] and [Y, Corollary 3.10], any 1-cocycle for a minimal action is a coboundary. Readers are referred to [HY] or [V] for constructions of minimal actions. Corollary 7.4. Let G = (L ∞ (G), δ, h) be a compact Kac algebra with amenable dual. Let M be the AFD factor of type II1 and α a minimal action of G on M. Then α is dual. Proof. On M⊗B(Hπ ), consider a minimal action α π defined by α π (x) = (α⊗id)(x)132 . defined in §2.2 and §2.5, respectively. Since Recall the multiplicative unitaries V and V 23 V21 V ∗ = V21 V31 , 1 ⊗ (Vπ )21 ∈ M ⊗ B(Hπ ) ⊗ L ∞ (G) is an α π -cocycle. By V 23 Lemma 7.3, there exists vπ ∈ U (M ⊗ B(Hπ )) with (vπ∗ ⊗ 1)α π (vπ ) = 1 ⊗ (Vπ )21 , that by v(1 ⊗ 1π ) = vπ for all is, α(vπ ) = (vπ )13 (1 ⊗ Vπ ). Define v ∈ U (M ⊗ L ∞ (G)) π ∈ Irr(G). Set β(x) := Ad v(x ⊗ 1) for x ∈ M α . We claim that β is a cocycle action on M α with a 2-cocycle u := v12 v13 (id M ⊗)(v ∗ ). Since for x ∈ M α , we have of G (α ⊗ id)(β(x)) = α(v)(α(x) ⊗ 1)α(v ∗ ) ∗ = v13 (1 ⊗ V )(x ⊗ 1 ⊗ 1)(1 ⊗ V ∗ )v13 = β(x)13 . ⊗ L ∞ (G) as follows: Hence β preserves M α . We verify u ∈ M α ⊗ L ∞ (G) α(u) = α(v)123 α(v)124 (id ⊗ id ⊗)(α(v ∗ ))1234 ∗ ∗ = v13 V23 v14 V24 (id ⊗ id ⊗)(V23 v13 )1234 ∗ = v13 V23 v14 V24 (V23 V24 ) (id ⊗)(v ∗ )134 = v13 v14 (id ⊗)(v ∗ )134 = u 134 . Hence (β, u) is a cocycle action on M α . Next we show the freeness of the cocycle action (β, u). Assume that for an element π ∈ Irr(G) \ {1}, there exists a nonzero a ∈ M α ⊗ B(Hπ ) such that βπ (x)a = a(x ⊗ 1π ) for all x ∈ M α . Then since vπ∗ a ∈ ((M α ) ∩ M) ⊗ B(Hπ ) = C ⊗ B(Hπ ), there exists b ∈ B(Hπ ) such that a = vπ (1 ⊗ b). Applying α ⊗ id to both sides, we have a13 = (vπ )13 (1 ⊗ Vπ )(1 ⊗ 1 ⊗ b) and hence 1 ⊗ 1 ⊗ b = (Vπ )23 (1 ⊗ 1 ⊗ b) ∈ C ⊗ L ∞ (G)π ⊗ B(Hπ ), but this is a contradiction. Therefore (β, u) is a free cocycle action on the AFD factor M α of type II1 . By Theorem with (w ⊗ 1)(β ⊗ id)(w)u(id ⊗)(w∗ ) = 6.2, there exists w ∈ U (M α ⊗ L ∞ (G)) (w12 v12 )(w13 v13 )(id ⊗)(v ∗ w ∗ ) = 1. Then wu is a unitary representation of L ∞ (G). Since α(v) = v13 V23 and α(w) = w13 , it follows α(wv) = (wv)13 V23 . Hence α is a dual action for a free action Ad w ◦ β on M α . In the end, we prove the following main theorem. Theorem 7.5. Let M be the AFD factor of type II1 , and G = (L ∞ (G), δ, h) a compact Kac algebra with amenable dual. Let α and β be minimal actions of G on M. Then they are conjugate. Proof. By the previous corollary, a minimal action of G on the AFD factor of type II1 on is dual, and α and β are of the form γˆ0 and γˆ1 where γ0 and γ1 are free actions of G M α and M β respectively. Since M α and M β are injective factors of type II1 , they are isomorphic by Connes’s result [C2]. By Corollary 7.2, γ0 and γ1 are cocycle conjugate. Hence their dual actions α and β are conjugate.
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8. Appendix Let M be a von Neumann algebra, and K a finite dimensional Hilbert space. For β ∈ Mor 0 (M, M ⊗ B(K )), we prepare several properties. Definition 8.1. We say that β is (1) properly outer if there exists no nonzero a ∈ M ⊗ B(K ) such that β(x)a = a(x ⊗ 1) for all x ∈ M, (2) centrally trivial if β ω (x) = x ⊗ 1 for all x ∈ Mω , (3) centrally nontrivial if β is not centrally trivial, (4) properly centrally nontrivial if there exists no nonzero element a ∈ M ⊗ B(K ) such that β ω (x)a = (x ⊗ 1)a for all x ∈ Mω . Lemma 8.2. A map β ∈ Mor 0 (M, M ⊗ B(K )) is properly centrally nontrivial if and only if it is strongly outer. Proof. The “if” part is trivial. We show the “only if” part. Let β be a properly centrally nontrivial homomorphism. Assume that β is not strongly outer. Then there exists a nonzero a ∈ M ω ⊗ B(K ) and a countably generated von Neumann algebra S ⊂ M ω such that β ω (x)a = a(x ⊗ 1) holds for all x ∈ S ∩ Mω . We claim that the proper central nontriviality implies 1= s (τ ω ⊗ id)(|β ω (z) − z ⊗ 1|2 ) . z∈Mω
Indeed, if b ∈ M ⊗ B(K ) satisfies (τ ω ⊗ id)(|β ω (z) − z ⊗ 1|2 )b = 0 for all z ∈ Mω , then (β ω (z) − z ⊗ 1)b = 0, but this is a contradiction. Hence there exists z ∈ Mω such that it satisfies (τ ω ⊗ id)(|a ∗ |2 )(τ ω ⊗ id)(|β ω (z) − z ⊗ 1|2 ) = 0. Let {ei, j }i,n j=1 be a system of matrix units for B(K ). Decompose a as a = i,n j=1 ai, j ⊗ ei, j . Set S = W ∗ (S, {ai, j }1≤i, j≤n ) and B = {β ω }. Then by Lemma 3.10, there N = W ∗ (z), , M ω ) such that exists ∈ Mor( N (1) (z) ∈ S ∩ Mω , (2) τ ω (b(z)) = τ ω (b)τ ω (z) for all b ∈ S, (3) β ω ((z)) = ( ⊗ id)(β ω (z)). Set y = (z) ∈ S ∩ Mω and then β ω (y)a = a(y ⊗ 1) = (y ⊗ 1)a. However the following equality: (τ ω ⊗ id)(|a ∗ |2 |β ω (y) − y ⊗ 1|2 ) = (τ ω ⊗ id) |a ∗ |2 ( ⊗ id)(|β ω (z) − z ⊗ 1|2 ) = (τ ω ⊗ id)(|a ∗ |2 )(τ ω ⊗ id)(|β ω (z) − z ⊗ 1|2 ) = 0 implies (β ω (y) − (y ⊗ 1))a = 0. This is a contradiction.
Hence β is strongly outer ⇔ properly centrally nontrivial ⇒ centrally nontrivial.
(8.1)
A map β ∈ Mor(M, M ⊗ B(K )) is said to be irreducible if β(M) ∩(M ⊗ B(K )) = C.
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Lemma 8.3. Let β ∈ Mor 0 (M, M ⊗ B(K )) be irreducible. Then β is centrally nontrivial if and only if β is properly centrally nontrivial. Proof. The “if” part is trivial. We show the “only if” part. Let β be a centrally nontrivial homomorphism. Assume that there exists a ∈ M ⊗ B(K ) satisfying β ω (x)a = (x ⊗ 1)a for all x ∈ Mω . Let a = v|a| be the polar decomposition. Then it is easy to see that β ω (x)vv ∗ = (x ⊗ 1)vv ∗ for all x ∈ Mω . Put p = vv ∗ ∈ M ⊗ B(K ). If u ∈ U (M), then β ω (x)β(u) pβ(u ∗ ) = (x ⊗ 1)β(u) pβ(u ∗ ). Therefore the projection z = ∨u∈U (M) β(u) pβ(u ∗ ) satisfies β ω (x)z = (x ⊗ 1)z for all x ∈ Mω . Since z ∈ β(M) ∩ M ⊗ B(K ) = C, we have z = 1, and β is properly centrally nontrivial. Hence under the assumption on irreducibility, all the properties of (8.1) are equivalent. In addition, if M is the AFD factor of type II1 , they are equivalent to proper outerness. Lemma 8.4. Let R0 be the AFD factor of type II1 . Let β ∈ Mor 0 (R0 , R0 ⊗ B(K )) be irreducible. Then the following properties on β are equivalent: (1) (2) (3) (4)
central nontriviality, proper central nontriviality, strong outerness, proper outerness.
Proof. We know the equivalence of (1), (2) and (3). It is trivial that (3) implies (4). We show that (4) implies (1). Let τ be the trace on R0 . We assume the following lemma for a moment, and we prove this implication. Let K1 ⊂ K2 ⊂ . . . be an increasing net of finite dimensional subfactors in R0 with (∪n≥1 Kn ) = R0 . For each n, there exists a unitary u n ∈ Kn ∩ R0 with β(u n ) − u n ⊗ 1τ ⊗τ K > 1/2. Then the sequence (u n )n is central which defines u in (R0 )ω . It satisfies β ω (u) − u ⊗ 1τ ⊗τ K ≥ 1/2. Hence β ω is not trivial on (R0 )ω . We adapt [C3, Lemma 3.4] to the case of a homomorphism as follows. Lemma 8.5. Let M be a factor of type II1 and K a finite dimensional Hilbert space. Let β ∈ Mor(M, M ⊗ B(K )) be irreducible. If there exists a finite dimensional subfactor K ⊂ M with sup{β(u) − u ⊗ 1τ ⊗τ K | u ∈ U (K ∩ M)} < 1, then β is not properly outer. Proof. Consider a weakly closed convex set in M ⊗ B(K ), C = cow {(u ⊗ 1)β(u ∗ ) | u ∈ U (K ∩ M)}. In C take a unique point y0 attaining the minimal distance from 0 with respect to · τ . By assumption, y0 − 1τ < 1, in particular, y0 = 0. Unicity yields (u ⊗ 1)y0 = y0 β(u) for any u ∈ U (K ∩ M). Hence we have (x ⊗ 1)y0 = y0 β(x) for any x ∈ K ∩ M. Let {ei, j }i,n j=1 be a system of matrix units for K. Since β(e1,1 ) and e1,1 ⊗ 1 are equivalent in M ⊗ B(K ), there exists a partial n isometry v ∈ M ⊗ B(K ) with β(e1,1 ) = vv ∗ and e1,1 = v ∗ v. Set a unitary u = i=1 β(ei,1 )ve1,i in M ⊗ B(K ). Then we have u(ei, j ⊗ 1) = β(ei, j )u. Hence u(x ⊗ 1) = β(x)u for all x ∈ K. Let γ = Ad(u ∗ ) ◦ β and y1 = y0 u. Then γ is trivial on K and y1 γ (x) = (x ⊗ 1)y1 holds for x ∈ K ∩ M. Let E : M ⊗ B(K ) → (K ∩ M) ⊗ B(K ) be a faithful conditional expectation. Then there exists an element a in K with z = E((a ⊗ 1)y1 ) = 0. Since γ (K ∩ M) ⊂ (K ∩ M) ⊗ B(K ), z satisfies zγ (x) = (x ⊗ 1)z for x ∈ K ∩ M. In fact, this equality is valid for any x ∈ M because γ is trivial on K. This shows γ is not outer. Hence β = Ad(u) ◦ γ is not outer.
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on a von Neumann algebra M. We call (α, u) Let (α, u) be a cocycle action of G centrally free if απ is properly centrally nontrivial for each π ∈ Irr(G) \ {1}. on a von Neumann algebra M. Then Corollary 8.6. Let (α, u) be a cocycle action of G the following properties of (α, u) are equivalent: (1) central freeness, (2) strong freeness. In addition, if M is the AFD factor of type II1 , they are also equivalent to (3) freeness. Proof. We know that (1) and (2) are equivalent and (2) implies (3). We show that (3) implies (1). By Lemma 2.8, each map απ , π ∈ Irr(G), is irreducible. Then by Lemma 8.4, απ is properly centrally nontrivial. Acknowledgements. The authors are grateful to Masaki Izumi and Yasuyuki Kawahigashi for permanent encouragement and fruitful discussions. The second named author was supported in part by Research Fellowship for Young Scientists of the Japan Society for the Promotion of Science.
References [BS] [C1] [C2] [C3] [ES] [EK] [HY] [I1] [I2] [ILP] [J] [JT] [KST] [KT] [M1] [M2] [NT]
Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés de C ∗ -algèbres. Ann. Sci. École Norm. Sup. (4) 26(4), 425–488 (1993) Connes, A.: Outer conjugacy classes of automorphisms of factors. Ann. Sci. École Norm. Sup. (4) 8(3), 383–419 (1975) Connes, A.: Classification of injective factors, cases II1 , II∞ , IIIλ , λ = 1. Ann. of Math. (2) 104(1), 73–115 (1976) Connes, A.: Periodic automorphisms of the hyperfinite factor of type II1 . Acta Sci. Math. (Szeged) 39(1–2), 39–66 (1977) Enock, M., Schwartz, J.-M.: Kac algebras and duality of locally compact groups. Berlin: SpringerVerlag (1992) Evans, D.E., Kishimoto, A.: Trace scaling automorphisms of certain stable AF algebras. Hokkaido Math. J. 26(1), 211–224 (1997) Hayashi, T., Yamagami, S.: Amenable tensor categories and their realizations as AFD bimodules. J. Funct. Anal. 172(1), 19–75 (2000) Izumi, M.: Canonical extension of endomorphisms of type III factors. Amer. J. Math. 125(1), 1–56 (2003) Izumi, M.: Finite group actions on C ∗ -algebras with the Rohlin property I. Duke Math. J. 122(2), 233–280 (2004) Izumi, M., Longo, R., Popa, S.: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal. 155(1), 25–63 (1998) Jones, V.F.R.: Actions of finite groups on the hyperfinite type II1 factor. Mem. Amer. Math. Soc. 28(237) (1980) Jones, V.F.R., Takesaki, M.: Actions of compact abelian groups on semifinite injective factors. Acta Math. 153(3–4), 213–258 (1984) Kawahigashi, Y., Sutherland, C.E., Takesaki, M.: The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions. Acta Math. 169(1– 2), 105–130 (1992) Kawahigashi, Y., Takesaki, M.: Compact abelian group actions on injective factors. J. Funct. Anal. 105(1), 112–128 (1992) Masuda, T.: Evans-Kishimoto type argument for actions of discrete amenable groups on McDuff factors. To appear in Math. Scand., ArXiv: math.OA/0505311 (2005) Masuda, T.: Classification of actions of duals of finite groups. http//arxiv.org/list/math.OA/ 0601601 (2006) Nakagami, Y., Takesaki, M.: Duality for crossed products of von Neumann algebras. Lecture Notes in Mathematics, 731, Berlin: Springer, 1979
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[N] [Oc1] [Oc2] [OPT] [OW1] [OW2] [P1] [P2] [PW] [PS] [Ro] [Ru] [S] [ST] [T] [V] [W1] [W2] [W3] [W4] [Y]
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Nakamura, H.: Aperiodic automorphisms of nuclear purely infinite simple C ∗ -algebras. Ergodic Theory Dynam. Systems 20(6), 1749–1765 (2000) Ocneanu, A.: Actions of discrete amenable groups on von Neumann algebras. Lecture Notes in Mathematics 1138, Berlin: Springer-Verlag, (1985) Ocneanu, A.: Prime actions of compact groups on von Neumann algebras. Unpublished Olsen, D., Pedersen, G., Takesaki, M.: Ergodic actions of compact abelian groups. J. Operator Theory 3(2), 237–269 (1980) Ornstein, D., Weiss, B.: Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.) 2(1), 161–164 (1980) Ornstein, D., Weiss, B.: Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math. 48, 1–141 (1987) Popa, S.: Classification of amenable subfactors of type II. Acta Math. 172(2), 163–255 (1994) Popa S.: Classification of subfactors and their endomorphisms. CBMS Regional Conference Series in Mathematics 86, Published for the Conference Board of the Mathematical Sciences, Washington, DC; Providence, RI: Amer. Math. Soc., 1995 Popa, S., Wassermann, A.: Actions of compact Lie groups on von Neumann algebras (English, French summary). Acad, C.R. Sci. Paris Sér. I Math. 315(4), 421–426 (1992) Powers, R.T., Størmer, E.: Free states of the canonical anticommutation relations. Commun. Math. Phys. 16, 1–33 (1970) Roberts, J.E.: Cross products of von Neumann algebras by group dual. Symp. Math. XX: 335 (363), 335–363 (1976) Ruan, Z.-J.: Amenability of Hopf von Neumann algebras and Kac algebras. J. Funct. Anal. 139(2), 466–499 (1996) Sekine, Y.: An analogue of Paschke’s theorem for actions of compact Kac algebras. Kyushu J. Math. 52(2), 353–359 (1998) Sutherland, C.E., Takesaki, M.: Actions of discrete amenable groups on injective factors of type IIIλ , λ = 1. Pacific J. Math. 137(2), 405–444 (1989) Takesaki, M.: Duality for crossed products and the structure of von Neumann algebras of type III. Acta. Math. 131, 249–310 (1973) Vaes, S.: Strictly outer actions of groups and quantum groups. J. Reine Angew. Math. 578, 147– 184 (2005) Wassermann, A.: Coactions and Yang-Baxter equations for ergodic actions and subfactors. Operator algebras and applications. Vol. 2, London Math. Soc. Lecture Note Ser. 136, Cambridge: Cambridge Univ. Press, 1988, pp. 203–236 Wassermann, A.: Ergodic actions of compact groups on operator algebras. I. General theory. Ann. of Math. ( 2( 130(2), 273–319 (1989) Wassermann, A.: Ergodic actions of compact groups on operator algebras. II. Classification of full multiplicity ergodic actions. Canad. J. Math. 40(6), 1482–1527 (1988) Wassermann, A.: Ergodic actions of compact groups on operator algebras. III. Classification for SU(2). Invent. Math. 93(2), 309–354 (1988) Yamanouchi, T.: The Connes spectrum for actions of compact Kac algebras and factoriality of their crossed products. Hokkaido Math. J. 28(2), 409–434 (1999)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 274, 553 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0266-7
Communications in
Mathematical Physics
Erratum
The Map Between Conformal Hypercomplex/ Hyper-Kähler and Quaternionic(-Kähler) Geometry Eric Bergshoeff1 , Stefan Vandoren2 , Antoine Van Proeyen3 1 Center for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,
The Netherlands
2 Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3508 TD Utrecht, The Netherlands 3 Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D,
B-3001 Leuven, Belgium. E-mail: [email protected] Received: 29 September 2006 / Accepted: 8 February 2007 Published online: 26 June 2007 – © Springer-Verlag 2007 Commun. Math. Phys. 262, 411–457 (2006)
The sentence before (3.5) “The integrability conditions for (1.1) and (3.2) then read” should be replaced by “We demand, here and everywhere below, that the vectors k and k are ‘symmetry generators’ in the sense of (5.1), i.e., which is mathematically the statement that they define affine transformations. This leads to”
W kX R = 0, XY Z
W k X R = 0. XY Z
(3.5)
When the connection is metric, then these equations are integrability conditions for (1.1) and (3.2) using the symmetries of the Riemann tensor. Communicated by M. Aizenman
The online version of the original article can be found at http://dx.doi.org/10.1007/s00220-005-1475-6
Commun. Math. Phys. 274, 555–626 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0284-5
Communications in
Mathematical Physics
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators Jean Bricmont1 , Antti Kupiainen2, 1 UCL, FYMA, chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium 2 Department of Mathematics, Helsinki University, P.O. Box 4, 00014 Helsinki, Finland.
E-mail: [email protected] Received: 22 May 2006 / Accepted: 30 January 2007 Published online: 17 July 2007 – © Springer-Verlag 2007
Abstract: We consider a Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice Z2N × Z2 , and subjected to stochastic forcing mimicking heat baths of temperatures T1 and T2 on the hyperplanes at 0 and N . We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two-point stationary correlation functions. We prove that these equations have a unique solution which, for N large, is approximately a local equilibrium state satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile. 1. Introduction Fourier’s law states that a local temperature gradient is associated with a flux of heat J which is proportional to the gradient: J (x) = −k(x)∇T (x),
(1.1)
˜ (x)). where the heat conductivity k(x) is a function of the temperature at x : k(x) = k(T Fourier’s law is experimentally observed in a variety of materials from gases to solids at low and at high temperatures. It also belongs to basic textbook material. However, a first principle derivation of the law is missing and, many would say, is not even on the horizon. The quantities T and J in (1.1) are macroscopic variables, statistical averages of the variables describing the microscopic dynamics of matter. A first principle derivation of (1.1) entails a definition of T and J in terms of the microscopic variables and a proof of the law in some appropriate limit. An example of an idealized physical situation would be a crystal occupying the region [0, N ] × R2 in R3 . The crystal is heated at the two boundaries by uniform temperatures, Partially supported by the Academy of Finland.
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T1 on {0} × R2 and T2 on {N } × R2 . Then, for large N , the relation (1.1) should hold, with corrections o(1/N ) and ∇T = O(1/N ) = J . The proper description of the crystal would be quantum mechanical, but, being a macroscopic law, (1.1) is expected to hold for a classical system as well and the quantum corrections are expected to be small except at low temperatures. An example of a classical toy model of such a system is given by coupled oscillators organized on a d-dimensional lattice Zd . Consider a subset = [0, N ] × Zd−1 of Zd and let x ∈ index the dynamical variables, coordinates qx and momenta px . The dynamics in the phase space R2 = {(qx , px ) | x ∈ } is defined in terms of the Hamiltonian H (q, p) =
1 2 p + V (q), 2 x x
i.e. the Hamiltonian flow is given by the system q˙ x = px , ∂V . p˙ x = − ∂qx
(1.2) (1.3)
We can think of this model as describing atoms with unit mass with equilibrium positions at x ∈ and qx being the deviation of the position of the atom indexed by x from its equilibrium (qx should of course be in Rd but we simplify and take qx ∈ R), while px is the momentum (velocity) of the atom. The potential V should describe the forces between the atoms and possible restoring or pinning forces pulling qx to the equilibrium qx = 0. Let, for unit lattice vectors eα , ∇α q(x) = q(x + eα ) − q(x) denote the discrete derivative. Then, the interactions are described by a potential U (∇q) whereas the pinning is described by W (q), and V = U + W . In the simplest case, U and W are local: W (q) = w(qx ), x∈
U (∇q) =
u ∇α q(x) ,
x,x+eα ∈
and, to model small oscillations, w and u are given by a low order polynomial: w(q) = aq 2 + bq 3 + cq 4 , and u similarily. In addition to the Hamiltonian dynamics of the system, we want to model the heating of the system at the boundary. The simplest way to model this is to add stochastic forces to (1.3) for x ∈ ∂, see Sect. 2 for details. Then the deterministic flow (1.2, 1.3) is replaced by a Markov process (q(t), p(t)) and the Fourier law will be a statement about the stationary state of this Markov process.
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The first question one would like to answer is the existence and uniqueness of the stationary state. When T1 = T2 = T such a state exists and is a Gibbs state, given formally by 1 −β H (q, p) dq dp (1.4) e Z (see Sect. 2), with β = 1/T . This is customarily referred to as the equilibrium state in contrast to the non-equilibrium situation T1 = T2 . In the latter case, there is no simple formula like (1.4) and, indeed, in our setup, even the existence of a stationary state is an open problem. In the d = 1 case of a finite chain of N + 1 oscillators the existence is proved provided the interaction potential U dominates the pinning one W (see [9–12, 19, 13, 26–28]). In this case, uniqueness is also proved, i.e. the Markov process converges to this state as t → ∞. Supposing that we have a stationary state µ, let us formulate the statement (1.1). The Hamiltonian flow (1.2, 1.3) preserves the total energy H . In particular, if we write H as a sum of local terms, each one pertaining to a single oscillator: H= Hx , x∈
then, under the flow (1.2, 1.3), H˙ x = −
∇α jα (x) ≡ −∇ · j(x),
α
where the microscopic heat current j(x) will depend on p y and q y for y near x (see (7.2) for a concrete expression). Let also t (x) = 21 px2 be the kinetic energy of the oscillator indexed by x. Then, the macroscopic temperature and heat current in Eq. (1.1) are defined by T (x) = Eµ t (x), J (x) = Eµ j (x), where Eµ denotes expectation in the stationary state. We do not attempt here to give a comprehensive review of the status of (1.1), but refer the reader to the reviews [6, 22 and 31]. There is also substantial amount of work on Fourier’s law for fully stochastic models (i.e. where there is noise in the bulk too), going back to [17, 20], see eg. [7, 3–5, 14, 15]. The only rigorous results in our model are for the harmonic case where U and W are quadratic [29, 30]. In that case, Fourier’s law does not hold: the current j is O(1) as N → ∞, whereas ∇T = 0 except near the boundary. If the model has pinning, i.e. W = 0 and if U or W are not harmonic, the law seems to hold in simulations in all dimensions [1]. In the unpinned W = 0 anharmonic case, conductivity seems anomalous in low dimensions: k in (1.1) depends on N as N α in d = 1 and logarithmically in d = 2. It is a major challenge to explain the α which, numerically, seems to be in the interval [1/3, 2/5] (see [23–25] for theories on α). One way to try to get hold of the stationary state µ is via its correlation functions. Denote u x = (qx , px )T and choose U, W even for simplicity. Then, the Hamiltonian vector field (1.2, 1.3) is a sum of a linear and a cubic term in u. Therefore, the correlation functions G n (x1 , x2 , . . . , xn ) = E µ u x1 ⊗ . . . ⊗ u xn
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J. Bricmont, A. Kupiainen
satisfy a linear set of equations ∂t G n = An G n + Vn G n+2 + Cn G n−2 ,
(1.5)
where An , Vn , Cn are linear operators coming respectively from the linear and cubic terms of (1.2, 1.3), and the term coming from the noise via Ito’s formula. See Sect. 3 for a detailed derivation of these equations. Such equations for the correlation functions are known as the BBGKY hierarchy in particle systems or the Hopf equations in turbulence. Although linear, the system (1.5) is intractable due to the appearance of G n+2 in the equation for G n . In this paper we will consider the situation where the equilibrium T1 = T2 Gibbs measure is close to a Gaussian measure. This holds if the anharmonicity in u and w (the coefficients b and c) is weak and the harmonic part in w (i.e. the pinning) is large (i.e. the Gibbs measure is far from critical), as we will assume. In such cases, we expect that the non-equilibrium measure is also close to a Gaussian. In such a situation, one can attempt a closure of the Hopf equations, i.e. to express the higher order correlation functions G n+2 in terms of G m with m ≤ n, thereby obtaining a finite set of equations for G m , m ≤ n. We will introduce such a closure and solve the closed equations. The simplest closure would be to write, for n = 2, the equation G4 =
G 2 ⊗ G 2 + G c4 ,
and set G c4 = 0, thereby obtaining a closed quadratic equation for G 2 . It turns out that this is too simple: the solution will be qualitatively similar to the one of the harmonic case. Our closure is done to the G 4 -equation by setting the connected 6-point function to zero. This is an uncontrolled approximation that we do not know how to justify rigorously. An analogous approximation was studied in both classical and quantum systems in [31] and has been used in [21] in a model similar to ours, but in a translation invariant setting and in one-dimension; our model, in one-dimension, was further studied, theoretically and numerically, in [2]. Our motivation for studying the closure equations in detail is on the one hand in the interesting picture of the local equilibrium state that emerges and on the other hand in building approaches that go beyond this approximation. Traditionally one arrives to such a closure in an appropriate limit, the “kinetic limit”, which in our case means taking the −1
anharmonicity proportional to N 2 and re-scaling distances by N , see [31]. One then arrives at a Boltzmann equation (for phonons) and after a further limit [31] to the Fourier law. This last limit is analogous to the derivation of the Navier-Stokes equations from the Boltzmann equation for particles, which, together with the derivation of Euler’s equation from the one of Boltzmann, has been the subject of intense investigations, see e.g. [16, 18, 34]. However, most of the structure of the stationary state correlations disappear in these limits. In our case we arrive to approximate expressions of stationary correlation functions without any limits (admittedly with no control of the corrections!) and then can study how the Fourier law emerges from these expressions. In this paper, we work with N fixed and, apart from the closure approximation, we assume only that the nonlinearity is weak, the harmonic pinning strong, and the boundary noise small. Interesting phenomena emerge too. In particular, one expects the presence of very long range spatial correlations in the stationary state even though the equilibrium state
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559
has exponential decay of correlations. Although we do not demonstrate the presence of these long range correlations, because we obtain only upper bounds on the decay rates, not lower ones, we show how to handle the technical problems caused by these slowly decaying correlations, and that could be useful in other contexts. Finally, we believe some of the methods developed in this paper could be of use in trying to prove the existence of the kinetic limit and Fourier’s law therein. The outline of the paper (to which the reader can return later) is as follows: in the next section, we define our model and we derive the Hopf equation (or BBGKY hierarchy) in Sect. 3. In Sect. 4, we explain the particular closure that we will study, leading to our final equations (see (4.18)–(4.20) below), and we state a not too technical version of our main result. Section 5 is devoted to several changes of variables: we first apply a Fourier transformation, and then we introduce variables that could be called slow and fast varying, namely the one of which the non-translation invariant part of of the correlation functions depends (slow) and the one related to the translation invariant part (fast). Next, we outline our arguments and discuss heuristically our main result (Sect. 6). To prove the latter, we first derive (in Sect. 7) identities satisfied by our equations, which take the form of current conservation equations, consisting of an energy conservation law and a number conservation law (the presence of the latter being, to some extent, a consequence of our approximations, i.e. of our closure). We also write down the stationary states in the translation invariant case. These do not reduce, for our closed equations, to the usual Gibbs states, but depend on two parameters, the temperature, as one would expect, but also a “chemical potential”, corresponding to the number conservation law. These conservation laws are related to the presence of zero modes in the linearization of our equations, which are discussed also in Sect. 6. In fact, the current conservation equations coincide with the projection of the full equations on the zero modes. In Sect. 8, we define precisely the spaces in which our equations are solved and we state our main result in a more technical form. The solution that we shall construct is the sum of a modified stationary state, with coefficients (temperature and chemical potential) slowly varying in space, and a perturbation. The main technical problem that we face is that the nonlinear terms in our equations involve collision kernels that are delta functions (or principal values) (see (5.12)– (5.14) below). Since we want to solve our equations by using a fixed point theorem, we need to show that the nonlinear terms belong to the space that we introduced, and, because of the presence of the delta functions, this is rather technical. Section 9 is devoted to solving those problems, but most of the proofs of that section are given in Appendices B and C. Another problem is that the linear operator in our equations is not invertible, because of the zero modes. In Sect. 10, we show that our linear operator can be inverted in the complement of the zero modes. This uses the fact that this operator is a sum of a multiplication operator and a convolution. To show invertibility, it is useful to know that the convolution operator is compact and this in turn follows from Hölder regularity properties that are proven in Sect. 9. Finally, in Sect. 11, we prove our main result, which consist in using the result of Sect. 10 to solve the equations in the complement of the zero modes with the solution to the current conservation equations, i.e. of the projection of the full equations onto the zero modes. The first equations lead to Fourier’s law, namely an expression of the conserved currents in terms of the parameters of the modified stationary state (temperature and chemical potential ). And the second equations determine the spatial dependence of those parameters.
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2. Lattice Dynamics with Boundary Noise Let us define the model we consider in more detail. Instead of working in the strip [0, N ] × Zd−1 it is convenient to double it to the cylinder V = Z2N × Zd−1 , where Z2N are the integers modulo 2N . The noise is put on the “boundary” {0}×Zd−1 ∪{N }×Zd−1 . We consider the phase space (q, p) ∈ R2V , i.e. q = (qx )x∈V , qx = qx+y for y = (2N , 0), and similarly for px . The dynamics is given by the stochastic differential equations dqx = px dt, ∂H − γx px dt + dξx , dpx = − ∂qx
(2.1)
where H (q, p) =
λ 4 1 2 1 px + (q, ω2 q) + qx , 2 2 4 x∈V
(2.2)
x∈V
γx = γ (δx1 0 + δx1 N ),
(2.3)
and the random variables ξx (t) are Brownian motions with covariance Eξx ξ y = 4γ δx y (T1 δx1 0 + T2 δx1 N )t.
(2.4)
The Hamiltonian (2.2) describes a system of coupled anharmonic oscillators with coupling matrix ω2 : (q, ω2 q) =
qx q y ω2 (x − y).
x,y∈V
Our analysis requires that the Fourier transform ω2 (k) of ω2 is smooth and ω2 (k) = m 2 + ρ(k) with ρ(k) = O(k 2 ) as k → 0, and m 2 > 2ρ∞ . Moreover we will need some regularity properties that will be checked explicitly for ω2 = (− + m 2 )2 , i.e. ω(k) = 2
d
(1 − cos kα ) + m 2 ,
(2.5)
α=1
see the proof of Proposition 9.3. As explained in the Introduction, Eqs. (2.1) describe a Hamiltonian dynamics subjected to stochastic heat baths on the “boundaries” of V , with temperature T1 on the hyper-plane x1 = 0, and temperature T2 on the hyper-plane x1 = N . This defines a Markov process (q(t), p(t)) in the phase space R2V and we are interested in the stationary states for this process.
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561
If the temperatures are equal, T1 = T2 = T , an explicit stationary state is given by the Gibbs state at temperature T of the Hamiltonian H . This probability measure is given as a weak limit ⎤ ⎡ 1 λ νT = lim exp ⎣− qx4 ⎦ µT (dp, dq), (2.6) M→∞ Z M T x∈VM
where VM is defined by |xi | < M, i = 2, . . . , d, and µT is the Gaussian measure with covariance E px p y = T δx y , E px q y = 0,
(2.7)
Eqx q y = T ω−2 (x − y).
(2.8)
For small λ, the Gibbs measure νT is nearly Gaussian, with (2.7) still true and small O(λ) corrections in (2.8). It is very well understood via cluster expansions. Physically, the fact that the Markov process reaches the stationary distribution νT means that the heat introduced at the boundary spreads inside the system, which reaches equilibrium at temperature T . When T1 = T2 , things are very different. Even the existence of a stationary state is not known rigorously (not even in finite volume, M < ∞). However, physically, one expects a unique stationary state ν to exist. In this paper we assume this and inquire about the properties of ν. In particular one would like to understand how the heat from the boundary now spreads inside the system: what is its stationary temperature distribution and what sort of flux of heat exists in it. 3. Hopf Equations Let us introduce a more compact notation for the stochastic differential equation (2.1). Denote (q, p)T = u, (u) = −λ(0, q 3 )T , (u)x = (0, γx px )T and η = (0, ξ )T . Then (2.1) becomes du(t) = (A − )u + (u) dt + dη(t), (3.1) where
A=
0 1 . −ω2 0
Define the correlation functions G n (x1 , . . . , xn , t) = E u x1 (t) ⊗ . . . ⊗ u xn (t) ∈ R2nV . By Ito’s formula, we get G˙ n = (An − n )G n + n G n+2 + Cn G n−2 , where An = A ⊗ 1 ⊗ . . . ⊗ 1 + . . . 1 ⊗ 1 ⊗ . . . ⊗ A
(3.2)
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J. Bricmont, A. Kupiainen
and n is defined similarly. Moreover, n G n+2 =
n
Eu x1 ⊗ . . . ⊗ (u)xi ⊗ . . . ⊗ u xn ,
i=1
Cn G n−2 =
Cxi x j G n−2 (x1 , . . . , xˆi , . . . , xˆ j . . . xn ),
i< j
where the arguments xˆi , xˆ j are missing. This defines linear operators from R2(n+2)V → R2nV and R2(n−2)V → R2nV respectively. C equals one-half the time derivative of the covariance of η, i.e. 0 0 C= , 0C with C x y = 2γ δx y (T1 δx1 0 + T2 δx1 N ).
(3.3)
Suppose ν is a stationary state of the process u(t). Then Eq. (3.2) leads to a linear set of equations for the stationary correlation functions
n G n (x1 , . . . , xn ) = ⊗i=1 u xi ν(du), (3.4) (An − n )G n + n G n+2 + Cn G n−2 = 0.
(3.5)
Equations (3.5) have the drawback that they do not “close”: to solve for G n , we need to know G n+2 . For λ small, the equilibrium T1 = T2 measure is close to Gaussian. When T1 = T2 we expect this to remain true; however, the measure will not satisfy E px q y = 0. We look for a Gaussian approximation to Eq. (3.5) for small λ by means of a closure, i.e. expressing the G n in terms of G 2 . Since G n = 0 only for n even, the first equation in (3.5) reads: (A2 − 2 )G 2 + 2 G 4 + C = 0.
(3.6)
The simplest closure would be to replace G 4 in (3.6) by the Gaussian expression G 2 (xi , x j ) ⊗ G 2 (xk , xl ), (3.7) p
where the sum runs over the pairings of {1, 2, 3, 4}. Equations (3.6) and (3.7) lead to a nonlinear equation for G 2 . It turns out that the solution to this equation is qualitatively similar to the λ = 0 case, i.e. G 2 does not exhibit a temperature profile nor a finite conductivity. The only effect of the nonlinearity is a renormalization of ω. We will therefore not discuss this closure any further. The next equation is (A4 − 4 )G 4 + 4 G 6 + C4 G 2 = 0.
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
Write G4 =
563
G 2 ⊗ G 2 + G c4
p
and G6 =
G2 ⊗ G2 ⊗ G2 +
G 2 ⊗ G c4 + G c6 ,
p
p
where G c4 and G c6 are the connected correlation functions describing deviation from Gaussianity and the sums run over the usual partitions of indices. After some algebra, we may write the first two Hopf equations in the following form: (A2 − 2 + 2 )G 2 + 2 G c4 + C = 0,
(3.8)
(A4 − 4 + 4 )G c4 + b(G 2 ) + 4 G c6 = 0,
(3.9)
where the operators 2 and 4 are 2 (G 2 )G 2 = 2 4 (G 2 )G c4 =
G2 ⊗ G2,
(3.10)
p
4 G 2 ⊗ G c4 ,
p
and 4 means the following: G 2 ⊗ G c4 belongs to (R2V )⊗2 ⊗ (R2V )⊗4 ; 4 is a sum of terms i jk 4 = 4 , i< j
where 4 acts with 4 in the spaces i, j, k and as identity in the rest. 4 then has at least one of the indices i, j, k equal to either 1 or 2. Finally, b(G 2 ) = 4 G 2 ⊗ G 2 ⊗ G 2 , i jk
p
where 4 is similar to 4 , but with 4 acting on all the three factors G 2 . Explicitly, denote i = (α, x) ∈ {1, 2} × R V , so that u = (u i ), u (1,x) = qx , u (2,x) = px . Then, i jk
b(G 2 )i1 i2 i3 i4 = 6
j2 j3 j4
i1 j2 j3 j4
4
(G 2 )ia ja + (i 1 → i b , b = 2, 3, 4)
a=2
with j1 j2 j3 j4 = −λδα1 2
4 a=2
δαa 1 δx1 xa .
(3.11)
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J. Bricmont, A. Kupiainen
Equation (3.9) may be solved for G c4 :
G c4 = −(A4 − 4 − 4 )−1 b(G 2 ) + 4 G c6 ,
(3.12)
provided that A4 −4 −4 is invertible. Substitution of (3.12) in (3.8) yields a nonlinear equation for G 2 with dependence on G c6 : with
(A2 − 2 + 2 )G 2 + N (G 2 , G c6 ) + C = 0
(3.13)
N (G 2 , G c6 ) = −2 (A4 − 4 + 4 )−1 b(G 2 ) + 4 G c6 .
(3.14)
If we set G c6 = 0 in (3.13) we get a closed equation for G 2 . However, before defining the closure equation to be studied, let us discuss in more detail the term b(G 2 ). 4. Closure The leading term (in powers of λ and γ ) in (3.14) is N = −2 A−1 4 b(G),
(4.1)
where we write, as we shall do from now on, G for G 2 , since we shall only deal with G 2 . Let us write this more concretely. To define the inverse of A4 we define the stationary state correlation functions G n as limits → 0 of the stationary state G n where, in Eq. (2.1), we add a term px dt, and in (2.4) a term 4tδx y , i.e. we put a noise and a friction of size everywhere. Then, the matrix A becomes 0 1 A= (4.2) −ω2 − and, letting R(t) = et A , we have, −A−1 4 =
∞
et A4 dt =
0
(4.3) ∞
R(t)⊗4 dt.
0
Thus, b(G) − A−1 4
i1 i2 i3 i4
=6
j
∞
4
dt R(t)
i 1 j2 j3 j4
0
R(t)G
a=2
i a ja
+(3 permutations),
with j = ( j2 , j3 , j4 ) and, then, with i = (i 2 , i 3 , i 4 ), −2 A−1 4 b(G)
ii
+3
j i1 i2 i3
∞ 0
=6 j
∞
4 R(t)G dt R(t) i i ij
0
i
dti i R(t)
i1 j
3 a=2
R(t)G
i a ja
i a ja
a=2
R(t)G
i j4
+ (i ↔ i ).
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565
Recalling the expression (3.11), some calculation gives
∞ Nαβ R(t)G (x, z)2 (x, y) = 6λ2 δα2 dt 0
11
z
· 3R12 (t, x − z) R(t)G (y, z) + R(t)G (x, z)Rβ2 (t, z − y) + tr, (4.4) β1
11
where tr means that both α, β and x, y are interchanged and where the translation invariance of R was used. Since ∂t R = A R we have, from (4.2), ∂t R1α = R2α
(4.5)
Rβ2 = −∂t (RG 0 )β1 ,
(4.6)
and, using ∂t R = R A, we get:
where
G0 =
ω−2 0 . 0 1
(4.7)
Inserting (4.6) in (4.4), and integrating by parts, (4.4) becomes: Nαβ (x, y) = Nαβ (x, y) + 6λ2 δα2 δβ1 Q(x, z)3 ω−2 (z − y) + tr,
(4.8)
z
where we denote G 11 = Q and
∞
2 R(t)G (x, z) Nαβ (x, y) = 18λ2 δα2 lim dt →0 0
11
z
· R12 (t, x −z) R(t)G (y, z)+ R(t)G (x, z) R(t)G 0 (z−y) +tr. β1
21
β1
(4.9) The closure equation that we will study is the replacement of the exact equation (3.13) by (A2 − 2 )G + N (G) + C = 0,
(4.10)
i.e. we drop the terms 2 , 4 , 4 , G c6 , as well as the second term in (4.8). Before discussing the motivation for this approximation, let us write (4.10) explicitly. Let Q H G= (4.11) HT P so, e.g. H (x, y) = Eqx p y and H T (x, y) = H (y, x). Then, (4.10) can be written as: H + H T = 0,
(4.12)
P − ω2 Q − H + N12 = 0,
(4.13)
−ω2 H − H T ω2 − P − P + N22 + C = 0,
(4.14)
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J. Bricmont, A. Kupiainen
where x y = γ δx y (δx1 0 + δx1 N ),
(4.15)
and C was defined in (3.3). Let J=
1 (H − H T ), 2
so by (4.12), H = J , H T = −J , and we write: Q J . G= −J P
(4.16)
(4.17)
Then (4.13) and (4.14) can be written as: T 2P = ω2 Q + Qω2 + J − J − N12 − N12 ,
(4.18)
T ω2 Q − Qω2 + J + J + N12 − N12 = 0,
(4.19)
ω2 J − J ω2 + P + P − N22 − C = 0.
(4.20)
Equations (4.18)–(4.20) is a system of nonlinear equations for the correlation functions Q, J, P. An important property of N in Eq. (4.9) is that, for all T , N (T G 0 ) = 0 for
G0 =
ω−2 0 . 0 1
(4.21)
(4.22)
Indeed, R(t)G 0 = R(t)21 ω−2 = −R(t)12 (see formula (5.2) below). Thus, at 21 γ = 0, our set of Eqs. (4.18)–(4.20) has a 1-parameter family of solutions Q = T ω−2 , P = T, J = 0 and, for the equilibrium case, with γ = 0, T1 = T2 , only one of these persists, namely the one with T = T1 . Note that the true equilibrium Gibbs state has P = T , J = 0, and −1 = T ω(k)2 + σ (k, λ, T ) Q(k) with σ = O(λ). The terms 2 , 4 and the second term in (4.8) would contribute to changing the σ = 0 of our closure solution to a σ which would agree with the true σ to O(λ2 ). Dropping these terms, as well as 4 , is done for convenience and should not change our analysis qualitatively. The uncontrolled approximation consists in dropping G c6 . Presumably, for small λ, this would not make a qualitative difference. It would be √ interesting to try to prove this in the kinetic limit λ = O(1/ N ). In the rest of this paper, we will discuss only Eqs. (4.18)–(4.20).
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567
We will now outline what we are going to prove about this model; this will be stated in more detail in Theorem 2 of Sect. 8 below, but we want to state here our result in a less technical form. Since the matrix G is translation invariant in all directions except the first one, let us define G(x, y) = g(x1 + y1 , x − y), and, with some abuse of notation, let G(x, k), with x ∈ Z2N be the Fourier transform of g(x1 + y1 , x − y) with respect to its second argument (with x in G(x, k) equal to x1 + y1 ). We say that x is away from the boundaries 0, N , if |x| ≥ O(N 1− ),
|x − N | ≥ O(N 1− ),
(4.23)
for some > 0. We assume that τ = |T1 − T2 |, λ are small enough and that N is large; we also take γ small in an N -dependent way (see (8.17)), and we take m 2 > O(1). Then we have the following result, which is a direct consequence of Theorem 2 below: Theorem 1. Let x be away from the boundaries. Then the function Q(x, k) takes the form: −1 1 Q(x, k) = T (x) ω(k)2 − A(x)ω(k) + o( ), (4.24) N where A(x) = O(τ λ2 ) and T (x) = T1 +
|x| N (τ
+ O(τ 2 + τ λ2 )).
Remarks. 1. For A = 0, the first term in (4.24) would simply be the temperature dependent equilibrium solution, T (x)ω−2 (x − y). The occurrence of A(x), which can be thought of as a sort of chemical potential (for phonons) is due to the existence of a conserved quantity other than the energy in this model (the phonon number), and will be discussed in Sect. 6 below. 2. Corresponding to the two conserved quantities in this model, we have two currents ( j, j ) (defined precisely in Sect. 7) that are related to the discrete gradients ∇T, ∇ A, for x away from the boundaries, by: −2 1 T ∇T (x) j (x) = −κ(A(x)) (4.25) + o( ), −1 j (x) T ∇ A(x) N where κ(A(x)) is a 2 × 2 matrix whose matrix elements are analytic functions of A(x) and are O(λ−2 ). Moreover, for x = 0, N , j (x + 1) − j (x) = 0,
(4.26)
and, for x away from the boundaries, j (x + 1) − j (x) = f (T (x), A(x), ∇T (x), ∇ A(x)) + o(
1 ), N2
(4.27)
where f is analytic in its arguments and is quadratic in ∇T (x), ∇ A(x), i.e. of order N12 . Equations (4.25) and (4.27) can be viewed as a (discrete) nonlinear elliptic system for the leading terms of T and A. However we don’t solve them directly but rather derive a coupled system of equations for T , A and the rest of G which we solve by fixed point methods in a suitable space. For the derivation of (4.25) and (4.27) from that solution, see the end of Sect. 11. 3. The temperature profile T (x) is non-linear. Due to the complicated functions κ and f in Eqs. (4.25) and (4.27) there is no closed formula. However, in the kinetic limit λ =
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J. Bricmont, A. Kupiainen
√ O(1/ N ) (actually λ = O(N − ) suffices) these formulae simplify. A will disappear and the inverse temperature has a linear profile: T (x)−1 = T2−1 +
|x| −1 (T − T2−1 ) N 1
which results from the Fourier law j (x) = −κ T (x)−2 ∇T (x) + o(
1 ) N
with κ a constant, the conductivity given by the kinetic theory (κ = κ11 (0) in (11.48)) [31]. 5. Changes of Coordinates For large N we expect the solution to Eqs. (4.18)–(4.20) to be translation invariant in the directions perpendicular to the 1-direction with a slowly varying dependence of the first coordinate x1 . It is therefore convenient to represent G in coordinates that are suited to such behavior. We first write Eqs. (4.18)–(4.20) in terms of the Fourier transform of G. Recall that x ∈ Zd with x1 ∈ Z2N . Introduce momentum variables: q = (q, q), and write
G(x, y) =
with the shorthand notation
where q ∈
π N Z2N
ei(q x+q
y)
q )dqdq G(q,
1
dq , dq = 2N (2π )d−1 I q
and I = [0, 2π ]d−1 . Then R becomes a Fourier multiplier ˜ ∂t + 1 sin ω(q)t e−t/2 , R(t, q) = ∂t (∂t + ) ∂t ω(q) ˜
where ω(q) ˜ = (ω2 (q) −
2 1/2 . 4)
Then, letting → 0 in the matrix, we get 1 cos ω(q)t ω(q) sin ω(q)t e−t/2 q) = R(t, −ω(q) sin ω(q)t cos ω(q)t 1 (isω(q)−/2)t 1 −isω(q)−1 , = e isω(q) 1 2 s=±1
so,
(5.1)
1 (isω(q)−/2)t 1 , R(t)G (q, q ) = e Ws (q, q ) isω(q) ·1 2 ∧
s=±1
(5.2)
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569
where q ) + isω(q)−1 J(q, q ). Ws (q, q ) = Q(q,
(5.3)
αβ (q, q ) = 9 (2π )3d λ2 (Nαβ (q, q ) + Nβα (q , q)) N 8
(5.4)
Thus, (4.9) becomes
with
N (q, q ) = i
dµ
s
4
−1 si ω(qi ) + i
i=1
2
Wsi (qi , qi ) ·
i=1
0 0 1 is4 ω(q )
−is3 ω(q3 )−1 δ(q3 + q3 )Ws4 (q4 , q4 ) + is3 ω(q3 )Ws3 (q3 , q3 )ω(q4 )−2 δ(q4 + q4 ) , (5.5) where
dµ = δ q −
3
4 4 qi δ qi δ(q − q4 ) dqi dqi ,
1
1
(5.6)
1
and we replaced 2 by . We got in (5.4) a factor (2π )d for each lattice sum in the αβ (q, q ), and a factor 1 for each of the four sums over si . Note that, definition of N 2 because of the sum over s in (5.5), only terms that are even in s contribute, which means −1 4 gives rise to a delta function if the integrand is that the factor i=1 si ω(qi ) + i even in s, and a principal value if it is odd. We will look for solutions to (4.18)–(4.20) which are translation invariant in the directions orthogonal to the 1-direction. Thus, we look for solutions of the form q ) = (2π )d−1 δ(q + q )g(q, q , q) G(q, and
G(x, y) =
ei(qx+q y)+iq(x−y) g(q, q , q)dqdq dq,
where we write x = (x, x), and similarly for y. It will be convenient to change coordinates in the 1-direction. Let g be a 2π periodic π π function on N Z2N × N Z2N . Define g(p, ˜ k) = g(p + k, p − k) on the set {p, k ∈
π π Z4N | p + k ∈ Z2N }. 2N N
Then, 1 2 1 2 iqx+iq y e g(q, ˆ q)= eip(x+y)+ik(x−y) g(p, ˜ k). (5.7) √ 2N 2 2N q,q
p,k
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J. Bricmont, A. Kupiainen
Indeed, each pair (q, q ) gets counted exactly twice in the (p, k)-sum (because the pair (p, k) gives the same contribution as √ the pair (p + 2N , k + 2N ), with addition modulo sides tend 4N), which accounts for the factor 2. Note that, in the N → ∞ limit, √ both π to [−π,π ]2 ·(2π )−2 since the lattice spacing in the RHS of (5.7) is 2 · 2N . Let p = (p, 0). We have then
p, k)dpdk, (5.8) G(x, y) = ei p(x+y)+ik(x−y) G( p, k) = g(p + k, p − k, k) and the integrals over the first components are where G( is 2π periodic in Riemann sums, with the same convention as in (5.7). The function G all the variables and, moreover, p + π˜ , k − π˜ ) = G( p, k), G(
(5.9)
where π˜ = (π, 0). ˜ p, k), We will, from now on, work in the p, k variables, drop, in general, the tilde in G( and we shall not distinguish between p and p, unless we need to stress that p is a number. Since the components of p (unlike those of k) other than the first one are always 0, this abuse of notation is harmless. Equations (5.5) then becomes 2
0 0 N ( p, k) = Wsi ( pi , ki ) · dν( si ω( pi + ki ) + i)−1 1 is4 ω( p4 + k4 ) s i=1
s3 ω( p3 + k3 ) ω( p3 + k3 )−2 δ(2 p3 )Ws4 ( p4 , k4 ) − ω( p4 + k4 )−2 δ(2 p4 )Ws3 ( p3 , k3 ) , (5.10) where dν = δ(2 p −
( pi + ki ))δ( ( pi − ki ))δ( p − k − p4 − k4 )dpdk,
(5.11)
4 , p = ( p )4 , and where we used the identity (see (5.6)) : and k = (ki )i=1 i i=1
q−
3
qi = p + k − (
i=1
4 4 ( pi + ki ) − q4 ) = p + k − ( ( pi + ki ) − q ) i=1
= p+k−
i=1 4
4
i=1
i=1
( pi + ki ) + p − k = 2 p −
( pi + ki ).
We can then write 9 9 N12 ( p, −k) ≡ (2π )3d λ2 n 1 ( p, k), N22 ( p, k) ≡ (2π )3d λ2 (n 2 ( p, k) + n 2 ( p, −k)) 8 8 (5.12) with 2 n1 1 n(W )( p, k) = = Wsi ( pi , ki − pi )s3 ω(k3 ) is4 ω(k4 ) n2 s i=1
· ω(k3 )−2 δ(2 p3 )Ws4 ( p4 , k4 − p4 )−(3 ↔ 4) νs pk (d p dk),
(5.13)
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
where νs pk (d p dk) =
si ω(ki ) + i
i
× δ(2 p −
571
−1 δ 2( p − pi ) i
ki )δ( p − k − k4 )d pdk.
(5.14)
i
In (5.13) we have shifted the ki -integrals by − pi compared to (5.10). Let us introduce the convenient notation 1
1
ω( p, k) = ( 2 (ω( p + k)2 + ω( p − k)2 )) 2
(5.15)
δω2 ( p, k) = ω( p + k)2 − ω( p − k)2 .
(5.16)
and
Then Eqs. (4.18)–(4.20) read as follows in the p, k variables: ω( p, k)2 Q + 2 ((J − J )∼ − N12 ( p, k) − N12 ( p, −k)) = P,
(5.17)
δω2 ( p, k)Q + (J + J )∼ + N12 ( p, −k) − N12 ( p, k) = 0, p, k) = 0. δω2 ( p, k)J + (P + P)∼ − N22 ( p, k) − C(
(5.18) (5.19)
1
We look for a solution that satisfies in x-space G(x, y) = G(−x, −y), Q(x, y) = Q(y, x) and J (x, y) = −J (y, x), i.e. Q( p, k) = Q( p, −k) = Q(− p, k), (5.20) J ( p, k) = −J ( p, −k) = −J (− p, k). (5.21) This is consistent since Nαβ ( p, k, W ) = Nαβ − p, −k, W (−·, −·) . We will see below that the lim→0 N is well defined for bounded W ( p, k). Before that, let us outline our arguments. 6. Heuristics Before getting into the details we give an outline of the argument. Since the nonlinear term N depends only on Q and J , P can be solved from (5.17) in terms of them. Then (5.18) and (5.19) are two equations for the two unknown functions Q and J which we write as J 0 2 0 1 + N (Q, J ) + N (Q, J ) = , (6.1) δω Q 10 C where
N (Q, J ) =
N12 ( p, −k) − N12 ( p, k) −N22 ( p, k)
and
N (Q, J ) =
J + J P + P
(6.2)
,
(6.3)
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J. Bricmont, A. Kupiainen
where P is expressed in terms of J and Q in (5.17). N collects the friction terms that have both linear and nonlinear contributions but will be treated as a perturbation. For large N the solution to this equation will be locally in the x1 coordinate a perturbation of the equilibrium state (4.22) given in the new variables by Q T ( p, k) = T ω( p + k)−2 δ(2p),
(6.4)
δ(p) = 2N δp,0 .
(6.5)
where
Hence the importance to study the linearization of N around that state. It turns out (see Sect. 9.2) that it is given by an operator which is a multiplier in the variable p: J ( p, ·) J DN (Q T , 0) (k). (6.6) ( p, k) = L p Q( p, ·) Q L p is a matrix of operators Lp =
L11 ( p) L12 ( p) . L21 ( p) L22 ( p)
(6.7)
The translation invariant equilibrium (6.4) has support at p = 0 (and due to the periodicity (5.20) at p = π ). The non-equilibrium solution will also have most of its mass in the neighborhood of these points. Thus it is important to understand L0 . It will turn out that Li j (0) = Li j (π ) = 0, i = j,
(6.8)
whereas L11 (0) is invertible. Invertibility of L0 would then follow from invertibility of L22 (0). This, however, is not the case: L22 (0) has two zero modes. One of them is easy to understand. Equation (6.4) is a one-parameter family of solutions to the γ = 0 equations. Hence the derivative with respect to the parameter T is a zero eigenvalue eigenvector of the linearization L22 (0), i.e. L22 (0)ω−2 = 0.
(6.9)
There is, however, a second zero mode for L22 (0): L22 (0)ω−3 = 0.
(6.10)
While the first zero mode has to persist for the full Hopf equations due to the one parameter family of Gibbs states that solve them for γ = 0, the second one is an artifact of the closure approximation. The nonlinear terms in the closure equations can be interpreted as describing phonon scattering and in our case only processes where two phonons scatter. These processes conserve phonon energy, leading to the first zero mode, and also phonon number, leading to the second one. The connected six-point correlation function which was neglected in the closure approximation would produce terms that violate phonon conservation and remove the second zero mode. However, for weak anharmonicity its eigenvalue would be close to zero and should be treated with some perturbation of the present analysis.
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
573
The second zero mode leads one to expect that our equations have in the γ = 0 limit a two parameter family of stationary solutions which indeed is the case. These are given by (see Sect. 7.3) Q T,A ( p, k) = T (ω( p + k)2 − Aω( p + k))−1 δ(2 p),
(6.11)
with J = 0 and and P given by (4.18). For (6.11) to be a well defined covariance we need positivity of the denominator which holds if A < m 2 . The zero-mode (6.10) is proportional to the derivative of Q T,A with respect to A, at A = 0. These considerations lead to the following ansatz for the solution Q(x, y) = Q 0 (x − y) + r (x, y),
(6.12)
Q 0 (x − y) = Q T (x),A(x) (x − y) + Q 1 (x, y).
(6.13)
where
The first term here is of local equilibrium form with slowly varying temperature and “chemical potential” profiles T (x) and A(x) (A, or more precisely T · A, can be considered as being related to a chemical potential, because it arises from the conservation of a number current, see Sect. 7.2), and the second is a small perturbation (see (9.4) below and the remark following it). T (x) and A(x) are determined from the current conservation laws which are projections of Eqs. (6.1) as follows. The operator L22 (0) has two left zero modes. Projecting Eq. (6.1) onto those modes yields two nonlinear elliptic equations for the functions T (x) and A(x) coupled to the rest of the variables i.e. J and r . For the latter taken in a subspace complementary to the zero modes the linear operator δω2 + L p is invertible and J and r can be determined by fixed point arguments in a suitable Banach space as functionals of T (x) and A(x). The projected equations then allow us to determine the latter. Our main result, stated more precisely in Sect. 8, says that Q(x, y) is as above, while the currents corresponding to the two conservation laws discussed above and defined in Sect. 7, are linearly related, to leading order in |T1 − T2 |, to the gradients of T (x) and A(x). T (x) is, to leading order in |T1 − T2 |, linear in x, and therefore the currents are, 2| to leading order, O( |T1 −T ). N 7. Current Conservation In the Introduction we recalled that the Hamiltonian structure of the dynamics leads to a local conservation law. We will show that our closure too has such a conservation law.
7.1. Heat-current. We start with a simple identity Lemma 7.1. N22 ( p, k)dk = 0 for all (bounded) W . Proof. From (5.11) we see that dkdν p,k is symmetric under the simultaneous interchanges s3 ↔ s4 , ( p3 , k3 ) ↔ ( p4 , k4 ). Hence, the integral over k of the second term in (5.12) vanishes, because the integrand in (5.13) is antisymmetric and the rest of the integrand is symmetric under those simultaneous interchanges.
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J. Bricmont, A. Kupiainen
Note that, by (5.8),
N22 (x, x) =
e2i px N22 ( p, k)dpdk
(7.1)
i.e., by Lemma7.1, N22 (x, x) = 0, ∀x. This follows also by inspection from (4.9) = −R(t)12 (by (5.2)) and R(t)12 (z − y) = R(t)12 (x − z) if x = y. since R(t)G 0 21
Consider now Eq. (4.20) restricted to the diagonal x = y. Let J = ω J + J ω. Then, 2 ω J − J ω2 = ω J − J ω. Defining, for eµ the unit vector in the µ-direction, jµ (x) = J (x − eµ , x),
(7.2)
we have, for ω given in (2.5): (ω2 J − J ω2 )(x, x) = (− J + J )(x, x) = −J (x − eµ , x) − J (x + eµ , x) + J (x, x − eµ ) µ
=2
+ J (x, x + eµ )
(J (x, x + eµ ) − J (x − eµ , x)) = 2∇ · j (x),
µ
where (∇µ f )(x) = f (x + eµ ) − f (x). Note that, for other functions ω, there will also be a current, but its form will depend on ω. In view of Lemma 7.1, (4.20), for x = y, reads ∇ · j (x) + γ P(x, x)(δx0 + δxN ) = γ (T1 δx0 + T2 δxN ), or, since P(x, x) depends only on x, ∇ · j (x) = γ ((T1 − P(0, 0))δx0 + (T2 − P(N , N ))δxN ),
(7.3)
which is a current conservation equation; the heat current j has sources on the boundary. Since j depends only on x, we define j (x) = j1 (x) and, so, ∇ · j (x) = j (x + 1) − j (x).
(7.4)
It is a useful exercise to rewrite this in the ( p, k)-variables. We have, see (2.5), ω( p + k) − ω( p − k) = 2 cos(p − k) − cos(p + k) = 4 sin p sin k, and so δω2 ( p, k) = 4 sin p sin k(ω(p + k) + ω(p − k)). Thus
(7.5)
where
j (p) = −i
δω2 (p, k)J (p, k)dk = 2(e2ip − 1) j (2p),
(7.6)
dke−ip/2 sin k(ω(p/2 + k) + ω(p/2 − k))J (p/2, k)
(7.7)
π Z2N . In (7.6) we used the fact that the k integral is π -periodic in p, due to for p ∈ N (5.9), to write it as a function of 2p. It may be checked directly that this is the Fourier transform of j1 (x) given by (7.2).
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575
7.2. Number current . As stated in Sect. 6, the closure equations possess another, approximate, conservation law which we will derive now. Let ρ( p, k) be given by ρ( p, k) = ω( p, k)−1 ,
(7.8)
with ω( p, k) given by (5.15), and project N22 now onto ρ instead of the function 1 as in Lemma 7.1. Let
9 p p p p 3d 2 ρ 2 , k n 2 2 , k dk. (7.9) θ (p) = ρ( 2 , k)N22 2 , k dk = (2π ) λ 4 Then θ is 2π periodic. Unlike what happened in Lemma 7.1 θ is not zero, but it will turn out to be very regular, see Proposition 9.7 and Appendix C below, due to the fact that (7.8) at p = 0 is a left zero eigenvector of the linearization of N22 . We will now integrate Eq. (5.19) multiplied by a linear combination of ρ and 1. For this, write it in the ( p, k) representation. The covariance is p, k) = 2γ (T1 + T2 e−2i N p ) C( and the friction term
∧
( P + P) (q1 , q2 ) = γ
(7.10)
q2 ) + (1 + ei(q−q2 )N ) P(q 1 , q) . dq (1 + ei(q−q1 )N ) P(q,
So, after shifting q in the first integral by q22 and in the second by q21 , we get:
∼ q,q +k− p + P q , − q + p + k (1 + ei(q−2 p)N ). ( P + P) ( p, k) = γ dq P 2 2 2 2 (7.11) Let
η( p, k) = ρ( p, k) −
ρ( p, k)dk,
(7.12)
with ρ given by (7.8). Integrating Eq. (5.19) multiplied by η, we get j (x + 1) − j (x) − θ (x) + γ (x) = 0, where j (p) = −i
(7.13)
dke−ip/2 η(p/2, k) sin k ω(p/2 + k) + ω(p/2 − k) J (p/2, k)(7.14)
and
γ (p) = γ
p , k (1 + ei(q−p)N ), dkdpψ(p, k, q) P 2
where ψ( p, k, q) = η( p, k +
p 2
q − 2 ) + η p, k −
p 2
+
q 2
(7.15)
(7.16)
˜ p, k) is a smooth function. Note that the covariance dropped out from (7.13) because C( is independent of k, and we subtract from ρ its average in the definition of η. Also, the latter does not contribute to θ due to Lemma 7.1. Equation (7.13) is again a conservation law: γ is a boundary term and, as we will see, θ vanishes in the limit of translation invariance. We call the current j the particle number current.
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J. Bricmont, A. Kupiainen
7.3. Generalized Gibbs states. We finish this section by checking that the states (6.11), with J = 0, are indeed solutions to the γ = 0 equations. Indeed, now, Ws ( p, k − p) = Q T,A (k)δ(2 p), with Q T,A = T (ω(k)2 − Aω(k))−1 . So, from (5.13), we get: 1 n( p, k) = δ(2 p) Q T,A (ki )s3 ω(k3 ) is4 ω(k4 ) s 1
× ω(k3 )−2 Q T,A (k4 ) − ω(k4 )−2 Q T,A (k3 ) −1 · si ω(ki ) + i δ 2p − ki δ( p − k − k4 )dk.
2
(7.17)
n 1 is obviously even in k, thus (5.18) holds. As for (5.19), write the [-] in (7.17) as
−AT −1 Q T,A (k3 )Q T,A (k4 ) ω(k3 )−1 − ω(k4 )−1 , and use (x + i)−1 = P
1 x
− 2πiδ(x) to get:
n 2 ( p, k) = −2π δ(2 p)AT −1 n˜ 2 ( p, k), 4 Q T,A (ki )s3 s4 [ω(k4 ) − ω(k3 )] n˜ 2 ( p, k) = s
·δ
(7.18)
1
4
si ω(ki ) δ(2 p − ki )δ( p − k − k4 )dk.
(7.19)
1
(By s → −s symmetry only the delta function contributes.) The integral in (7.17) is supported on
si ω(ki ) = 0.
(7.20)
We will choose m 2 in (2.4) large enough so that (7.20) forces
si = 0.
(7.21)
3 si and, by (7.21) also by − 13 s4 . Again, by By symmetry, we may replace s3 by 13 i=1 3 symmetry, s3 ω(k3 ) may be replaced by 13 i=1 si ω(ki ) and by (7.20) by − 13 s4 ω(k4 ). Doing the first replacement for s3 ω(k4 ) in the first term in the [·] in (7.19), and the second for s3 ω(k3 ) in the second term, we see that (7.19) vanishes. Note that n˜ 2 vanishes for all p and not only for 2 p = 0 mod 2π . We shall need this later (see the derivation of (9.27) below). Hence (6.11) solves (5.17)–(5.19) if γ = 0. Note however that these are not equilibrium T1 = T2 solutions, unless A = 0. This is because for A = 0, the noise and friction terms in (4.20) will not balance each other any more.
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577
8. The Space of Local Equilibrium Solutions We will now describe the space where (4.18)–(4.20) are solved. To motivate our choice, consider the current conservation equation (7.3). Summing over x, we get, since j (x) is periodic: T1 − P(0, 0) = − T2 − P(N , N ) ≡ j0 , (8.1) and so j (x + 1) − j (x) = j0 (δx0 − δx N ) which is solved by j (x) =
⎧ ⎨ j (0) + j0
x ∈ [1, N ]
⎩ j (0)
x ∈ [−N + 1, 0]
(8.2)
.
Since j (−x) = − j (x) we get j (0) = − 21 j0 . The Fourier transform of this is ⎧ π 1 − eipN ⎪ ⎨ j0 p ∈ Z2N , p = 0 ip − 1 e N j (p) = ⎪ ⎩ 0 p=0
(8.3)
(8.4)
(note that eipN takes only the values ±1). The current j0 will turn out to be O(1/N ). We now describe the space to which functions such as J ( p, k) belong. This space has to encode the 1/p singularity at origin in Eq. (8.4) as well as the factor eipN coming from the fact that in x space there are two special points in the first coordinate, the origin and N . Let H be the space of continuous functions f ( p, k) on = {( p, k) | p, k ∈
π π Z4N , p + k ∈ Z2N , k ∈ [−π, π ]d−1 } 2N N
(8.5)
that are 2π -periodic in all the variables, and invariant under (p, k) → (p + π, k + π ). We denote by f ∞ the sup norm in H. Let π + = {( p, k) ∈ | p ∈ Z2N } N and − = \ + . Then H = H+ ⊕ H− with f ( p, k) = f + ( p, k)σ+ (2p) + f − ( p, k)σ− (2p)
(8.6)
σ± (p) = 1 ± eipN
(8.7)
with
and f ± =
1 2
f |± .
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J. Bricmont, A. Kupiainen
Define the discrete C α -norm for f ∈ H± as f α = sup | f ( p, k)| + |λ|−α | f ( p + λe1 , k) − f ( p, k)| p,k,λ,µ
+|µ|−α | f ( p, k + µe1 ) − f ( p, k)| , where λ, µ ∈
π N Z2N
(8.8)
\ 0 (note that p + λ ∈ ± if p ∈ ± ). Set then, for f ∈ H, f α ≡ f + α + f − α .
Let d(p)−1 ≡
⎧ ⎨ (eip − 1)−1 ⎩
p = 0 (8.9)
0
p = 0,
and let, with some abuse of notation, d(p) = eip − 1.
(8.10)
We will then consider a space S of functions J (p, k) of the form J = N −1 δ(2p)J0 + (N (d(2p))−1 J1 + N α/2−1 J2 + (N d(2p))−3/2 J3 ,
(8.11)
where Ji (p, k) are in H. Define J S = max{Ji α , J3 ∞ , i = 0, 1, 2}.
(8.12)
More properly, S is the space H⊕4 of 4-tuples (J0 , . . . , J3 ) ≡ J which is a Banach space with this norm. We identify J and J when no confusion arises. Remarks. 1. Due to the friction and the noise terms (see (7.10) and (7.11)) we end up with the factors e2ipN . This leads to a factor eipN in functions such as j (p) and j (p). A Hölder continuous function f , of exponent α, decays in x-space at least as |x|−α (see (9.11) below, which gives a more general result) whereas eipN f produces decay away from x = N , i.e. |x − N |−α . Thus f α ≤ O(1) gives rise to functions of x that are localized near the boundaries, x = 0 or N . 2. The Fourier transform of (8.11) is as follows. The first term is constant in the 1-direction and O(1/N ). The second one is also of O(1/N ) because it involves a (discrete) Hilbert transform of a Hölder continuous function. However, e.g. for the current it can produce opposite constant values for positive and negative x. The fourth term is also of this order whereas the third is O(N −1+α/2 (| x |−α + | N − x |−α )), i.e. O(N −1−α/2 ) for x far away from the boundaries (see (4.23)). The J2 and J3 terms will be sub-leading corrections to the current and the profile. Note that no smoothness in the transverse variable k is assumed so that correlations will have a very slow decay in x space. This is due to the poor regularity properties of the “collision kernel” that enters the nonlinear terms, see Proposition 9.3. 3. The J correlation function is odd in p and hence J0 = 0. However, with Q, we will encounter functions with a nonzero first term in (8.11).
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579
π Finally, we will extend these definitions to the functions like j (p) defined on N Z2N , the only difference being the occurrence of σ± (2p) in (8.6) and d(2p) in (8.11), instead of d(p) here. We denote the space now by S. Hence j ∈ S is of the form
j (p) = N −1 δ(p) j0 + (N d(p))−1 j1 (p) + N α/2−1 j2 (p) + (N d(p))−3/2 j3 (p) (8.13) π with j0 a constant. Of course, here, since j is defined on N Z2N , the space is finite dimensional and all norms are equivalent, but (8.13) is a convenient way to record the dependence on N of various terms. We will look for solutions to (4.18)–(4.20) with J ∈ S and Q as follows. Q will have a leading “local equilibrium” term which we now describe. Recall that our equations have the 2-parameter family (6.11) of solutions when the friction vanishes. The leading term in Q will be of this form, where the constants T and A will be replaced π by p-dependent functions. More precisely let T (p), A(p) be functions on N Z2N , of the form
T (p) = T0 δ(p) + d(−p)−1 t (p), A(p) = A0 δ(p) + d(−p)
−1
a(p),
(8.14) (8.15)
where δ(p) is defined in (6.5) and t, a ∈ S. Let Q T (x),A(x) be obtained from the generalized Gibbs state in (6.11) by spatially varying the parameters T and A:
−1 . (8.16) Q T (x),A(x) (x − y) = T (x) dkeik(x−y) ω(k)2 − A(x)ω(k) We make now the following assumptions. We take γ = N −1+α/4
(8.17)
and m 2 large enough so that (7.20) implies (7.21)). We assume that 0 < λ and |T1−T2 | are small enough and that N > N (λ). Then, we have:
=τ
Theorem 2. Under the above assumptions, Eqs. (4.18)–(4.20) have a unique solution with J ∈ S and Q = Q T (x),A(x) + R with R ∈ S, J S = O(τ λ−2 ), R S = O(τ λ−2 ), and R(x, k) = o( N1 ), for x away 2 2 2 from the boundaries. Moreover, T (x) = T1 + |x| N (τ + O(τ + τ λ ), A0 = O(τ λ ), a ∈ S, and a S = O(τ λ2 ). Remark. We choose N large so that we can use bounds like N −α ≤ λ, for any given α > 0 appearing in the proofs. In the proofs, we shall, purely for convenience, assume that λ is small compared to τ . The choice of γ in (8.17) is somewhat artificial but is convenient. It makes the noise and the dissipation small enough so that they can be treated as a negligible perturbation in the bulk of the system, while their effect on the boundaries is strong enough to create a non-equilibrium profile (see (11.21), (11.32), and the derivation of (11.83) below to see how the choice of γ enters our estimates). In the proofs, C or c will denote constants that can change from place to place. The value of α is determined by the degree of Hölder continuity that one obtains in Proposition 9.3 below. We do not try to optimize that value (any α > 0 suffices); in fact, it will be convenient sometimes to assume that α is not too close to one, so that, e.g. the power on N in (11.21) is negative, and we shall implicitly assume that.
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9. Nonlinear Terms In this section we study the nonlinear terms of our equations given in (5.12) and (5.13). Our goal is to show that their linearization defines a bounded operator on S and that the remaining nonlinearities define suitable Lipschitz functions on S. Since λ2 enters as a multiplication factor in the nonlinear terms, see (5.12), it will be convenient to discuss the λ independent nonlinearities n(W ), and introduce explicitly λ2 in Sect. 10 (because only part of the linear operator there is multiplied by λ2 ) and in Sect. 11, where λ2 is used to make some Lipschitz constants less than 1. It is convenient to introduce a space for the functions T , A , analogous to S but with stronger singularity at p = 0. We let E denote the pairs (T0 , t) with T0 ∈ R and t ∈ S with t0 = 0 in (8.13). They parametrize functions T ( p) = T0 δ( p) + d(− p)−1 t ( p),
(9.1)
(from now on, we shall identify p and p). We use the norm T E = |T0 | + t S ,
(9.2)
which has the convenient property, used often below, that dT S ≤ T E . In this section, we shall work with (T, A) ∈ B , where B = {(T, A) ∈ E × E| T E ≤ C, A E ≤ },
(9.3)
where C is arbitrary and is chosen small enough so that various series below converge. Define Q 0 ( p, k) =
∞
(T ∗ A∗n )(2p)ω( p, k)−2−n ,
(9.4)
n=0
where ∗ is the convolution and ω( p, k) is defined in (5.15). Remark. Let Q T,A be the generalized Gibbs state in (6.11), at p = 0,
−1 . Q T,A (x − y) = T dkeik(x−y) ω(k)2 − Aω(k)
(9.5)
Comparing (9.4) with T (ω2 − Aω)−1 =
∞
T An ω−2−n
(9.6)
n=0
it is not hard to show that the Fourier transform of (9.4) is Q 0 (x, y) = Q T (x),A(x) (x − y) + Q 1 (x, y),
(9.7)
where the first term is a local (generalized) equilibrium state, and where (the Fourier transform of) Q 1 is in S, with a norm of order τ , and, away from the boundaries (see (4.23)), Q 1 (x, y) = o(
1 ). N
(9.8)
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Indeed, Q T (x),A(x) (x − y) is obtained by replacing ω( p, k) in (9.4) by ω(0, k). Since ω( p, k) is smooth, periodic and even in p, the correction can be written as d( p) f , where f belongs to S, with a norm of order τ . Now, observe that, by (8.11), (8.12), functions of the form g = |d( p)|α f , with f S ≤ O(1), satisfy: g(x) = o(
1 ), N
(9.9)
for x away from the boundaries: indeed, the first term in (8.11) vanishes, the third term is o(N −1 ) away from the boundaries, as noticed in Remark 2 after (8.12), the fourth term is O(N −1−α ) everywhere, using the easy bound
|d( p)|α |d( p)|−k dp ≤ C max(1, N k−1−α ), k − α = 1, (9.10) which holds because |d( p)| ≥ c| p|. For the second term, we observe that
|(eiλx − 1) ei px g( p)dp| = | ei px (g( p − λ) − g( p))dp| ≤ C|d(λ)|α ,
(9.11)
|g( p−λ)−g( p)| is integrable. Then, with λ = 1/x, this implies that | ei px g( p)dp| ≤ |d(λ)|α C|x|−α . Now put g( p) = |d( p)|α−1 J , with J ∈ H, let α = α/2, and use (9.10), with if
α replaced by α/2. So, we obtain (9.8), and, more generally, (9.9).
We expand (5.13) around Q 0 defined by (9.4). Let W = Q 0 + w, ˜ n(W ) = n(Q 0 ) + Dn(Q 0 )w + n(w).
(9.12)
We discuss the three terms on the right-hand side in turn. 9.1. n(Q 0 ). Consider first n(Q 0 ). Let us start with a lemma, whose proof is given in Appendix B, and which controls convolutions between elements of E, S and among themselves: Lemma 9.1. (a) Let T ∈ E and j ∈ S. Then T ∗ j ∈ S and T ∗ j S ≤ CT E j S .
(9.13)
(b) Let T, A ∈ E. Then T ∗ A ∈ E and T ∗ A E ≤ CT E A E .
(9.14)
(c) Let j, k ∈ S. Then j ≡ j ∗ k ∈ S with | j0 | ≤ C N −1 j S k S , j1 α j2 α | j3 ( p)|
≤ CN
−1
≤ CN
−1+α/2
≤ CN
−1
j S k S , j S k S ,
log(N | p|) j S k S ,
(9.15) (9.16) (9.17) (9.18)
and j S ≤ C N −1+α/2 j S k S .
(9.19)
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Using this lemma, we get: Proposition 9.2. The function n(Q 0 ) satisfies the following bounds, for (T, A) ∈ B : n 1 (Q 0 ) =
∞
T ∗3 ∗ A∗n (2 p)gn ( p, k) + m
(9.20)
n=1
with gn smooth functions bounded together with their derivatives (to any given order) by C n and m ∈ S. Moreover mS ≤ C(t S + A E ), n 1 (Q 0 )( p, k) − n 1 (Q 0 )(− p, k)S ≤ C(t S + A E ), n 2 (Q 0 )S ≤ C(t S + A E ).
(9.21) (9.22) (9.23)
Furthermore the functions on the LHS of (9.21), (9.22) and (9.23) are uniformly Lipschitz in T and A, for (T, A) ∈ B . Proof. From (5.13) and (9.4), we get n(Q 0 )( p, k) =
2 n
s
T ∗A∗n i (2 pi )ω( pi , ki − pi )−2−n i s3 ω(k3 )
i=1
1 is4 ω(k4 )
· ω(k3 )−2 δ(2 p3 )T ∗A∗n 4 (2 p4 )ω( p4 , k4 − p4 )−2−n 4 − (3 ↔ 4) νs pk (d p dk)
(9.24) 4 and ω( p, k) given in (5.15). We have, see (5.15), with νs pk given by (5.14), n = (n i )i=1
ω( p, k − p) = ω(k) + (e2i p − 1)O(1),
(9.25)
where the O( p) term, is written in an unusual form, which records the fact that it vanishes also at p = π , and which will be convenient later. Inserting this into (9.24), the leading term is 1 T ∗A (2 pi )ω(ki ) s3 ω(k3 ) is4 ω(k4 ) n s i=1
· ω(k3 )−2 δ(2 p3 )T ∗A∗n 4 (2 p4 )ω(k4 )−2−n 4 − [3 ↔ 4] νs pk (dp dk).
2
∗n i
−2−n i
We may do the pi -integrals to get =
∞ n=0
T
∗3
∗n
∗ A (2 p)
2 s n i =n
i=1
ω(ki )
−2−n i
· ω(k3 )−2 ω(k4 )−2−n 3 − (3 ↔ 4) ν˜ s pk (dk) ∞ gn ( p, k) , T ∗3 ∗ A∗n (2 p) ≡ f n ( p, k) n=1
1 s3 ω(k3 ) is4 ω(k4 )
(9.26)
where ν˜ s pk is like νs pk above, but without δ(2( p− pi )), and where, in the last equality, we used the fact that the n = 0 term has n 3 = 0 and therefore the [·] factor vanishes.
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Using (9.6), we see that f n ( p, k) are the Taylor coefficients of the expansion in A of n˜ 2 ( p, k), given by (7.18, 7.19). Since n˜ 2 vanishes identically, we get f n ( p, k) = 0.
(9.27)
For gn we need to study n 1 in (7.17) in more detail. Proceeding as with n 2 , we see that gn are the Taylor coefficients of the expansion in powers of the constant A of 4 −1 4 Q 0 (ki )P si ω(ki ) n˜ 1 ( p, k) = s
1
1
ki δ( p − k − k4 )dk. · s3 1 − ω(k3 )ω(k4 )−1 δ 2 p − Here, again because of the s → −s symmetry, only the principal value contributes. Consider first those terms with si = 0. We may replace s3 by − 13 s4 and s3 ω(k3 ) by 1 1 1 si ω(ki ) = − s4 ω(k4 ) + si ω(ki ). 3 3 3 3
4
1
1
Thus these terms give
4 1 Q T,A (ki )δ 2 p − ki δ( p − k − k4 )dk 3 si =0
1
which is smooth in p. For the terms with si = 0, P(·) has no singularity, (see (7.20), 7.21)), and they are smooth. Thus, the functions gn are smooth and bounded together with their derivatives (to any given order) by C n . The contribution to (9.24) of the (e2i p − 1)O(1) term in (9.25) is of form 2 m= T ∗A∗n i (2 pi )(e2i p3 − 1)T ∗A∗n 3 (2 p3 )δ(2 p4 ) n
s
i=1
pi νs pk (dp dk), ·G s ( p, k, p, k)δ 2 p − 2
(9.28)
where G s is smooth. By an easy extension of Lemma 9.1 to convolutions of functions in E and S, each summand is in S and (9.21) follows easily. The series over n converge for A E ≤ small enough. To get (9.22) we use the smoothness and periodicity to get gn ( p, k) − gn (− p, k) = (e2i p − 1)O(1), and hence each summand is now in S. The bound (9.23) is obtained in the same way as the bound on m (by (9.27), the only contribution to n 2 (Q 0 ) is similar to m). Remark. It will be useful later to analyze n 2 (Q 0 ) in x space in more detail, specially away from the boundaries (see (4.23)). As we noticed, n 2 (Q 0 ) is given by terms of the form (9.28). Let F( p, p) denote the integral over k of G s ( p, k, p, k) νs pk (dpdk); F( p, p) is C α in p and smooth in p. Moreover, the contribution to n 2 (Q 0 ) coming from (9.28) with F( p, p) evaluated at (0, 0) that are linear in p3 (or, p1 , p2 ) are odd on p, hence must vanish by symmetry: using (5.20), (5.21), we see that all the terms in (5.19) are even in p, and n 2 (Q 0 ) here equals N22 ( p, k) in (5.19).
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So, we are left with terms of the form O(|d( p)|α ) f , with f ∈ S or convolutions of three factors of the form T ∗ A∗n i (2 pi ) multiplied by pi p j for i = j or by pi2 for i = 1, 2, 3 (or higher order terms). For the O(|d( p)|α ) f terms, we use (9.9). Since T ∗ A∗n belongs to E, pi (T ∗ A∗n i )(2 pi ) belongs to S and, using (9.19) terms with pi p j , i = j, has a S-norm which is O(N −1+α/2 ), which means that it is o( N1 ) for all x (see Remark 2 after the definition of the S norm (8.12)). Finally, for the terms of the form pi2 T ∗ A∗n i (2 pi ), they are of the form pi f ( pi ), with f ∈ S, and we can use (9.9) again. Combining all this, we get that n 2 (Q 0 )(x, k) is o( N1 ) for x away from the boundaries. 9.2. Linearization. Let us turn to the second term in (9.12), i.e. the linearization of n at Q 0 . We wish to separate a leading term when T2 − T1 is small. Therefore, we write, using (8.14) and (9.4), Q 0 = T+ δ(2 p)ω−2 ( p, k) + Q˜ 0
(9.29)
with ω given by (5.15), and T+ = 21 (T1 + T2 ) being the average temperature. Insert this into (5.13). The [ - ] term vanishes for the first term in (9.29), so Dn(Q 0 )w = Lw + L w, with (Lw)( p, k) =
2T+2
−2 s3 ω(k3 ) ω(k1 )ω(k2 )
s
(9.30)
1 is4 ω(k4 )
ω(k3 )−2 ws4 ( p, k4 − p) − ω(k4 )−2 ws3 ( p, k3 − p) −1 si ω(ki ) + i δ 2p − ki δ( p − k − k4 )dk (9.31)
(where we used ω(0, k) = ω(k), and w( p + π, ki − p − π ) = w( p, ki − p), i = 3, 4). The operator L is defined as the rest; it will be discussed in the next subsection, see Proposition 9.5, but here, we will analyze further L. Since k4 − p = −k and k3 − p = k − k1 − k2 (coming from k1 + k2 + k3 = 2 p − k4 = 2 p − ( p − k) = p + k), we see that L is a multiplication operator in the p-variable (Lw)( p, k) = L˜ p w( p, ·) (k), (9.32) and L˜ p in turn is a sum of an operator which acts as a multiplication operator in the subspace of even or of odd functions, and an integral operator: L˜ p = M˜ p + K˜ p ,
(9.33)
where, after shifting k3 by p, ( K˜ p w)(k) = −2T02 ω( p − k)−2
s
ws3 ( p, k3 )
s3 ω(k3 + p) ω(k1 )2 ω(k2 )2
1 dµs pk , is4 ω( p − k) (9.34)
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with dµs pk =
2
−1 si ω(ki ) + s3 ω(k3 + p) + s4 ω( p − k) + i
δ k−
3 1
i=1
ki
3
dki
i=1
(9.35) and (remember that k4 − p = −k so that the operator acts as a multiplication operator only on functions of definite parity): (9.36) M˜ p (k) = −ω( p − k)2 K˜ p ω−2 ( p + ·) (k), when it acts on even functions, and M˜ p (k) = ω( p − k)2 K˜ p ω−2 ( p + ·) (k),
(9.37)
when it acts on odd functions. Here, we integrated over k4 = p − k, and we used: 2p −
4 i=1
ki = 2 p −
3
ki − p + k,
i=1
3 which, after shifting k3 by p equals k − i=1 ki . Separating the real and imaginary parts of ws (see (5.3)), one can also view the operator L p as a 2 × 2 matrix of operators the (J, Q) variables, where here Q denotes an arbitrary even function, and J an arbitrary odd one. We write it as: L˜ ( p)J + L˜ 12 ( p)Q J = ˜ 11 L˜ p ; (9.38) Q L21 ( p)J + L˜ 22 ( p)Q this defines the operators L˜ i j ( p), i, j = 1, 2. From (9.33), we see that each L˜ i j ( p) is a sum of a multiplication operator M˜ i j ( p) and an integral operator K˜ i j ( p). The operators here are acting on functions defined on , see (8.5). However, it is convenient to consider them as acting on functions defined on [−π, π ]d , which is always possible, by extending, say in a piecewise linear way, a function on to functions on [−π, π ]d . With that identification, we may consider these operators to be acting, for all N , on the same space. Moreover, a discrete Hölder continuous function (see (8.8)) becomes, with such an extension, an ordinary Hölder continuous function. The main property of these operators is: Proposition 9.3. M˜ p (k) is C α in p, k, for some α > 0, and the operators K˜ p are compact operators mapping C α p (k) , where p (k) = {k|( p, k) ∈ }, into itself. Moreover, in the norm of bounded operators on C α , they are uniformly bounded in N α ˜ and C α in p, uniformly in N . M˜ p (k) converges as N → ∞ to a C function, while K p α d converges to a compact operator mapping C [−π, π ] into itself. Obviously, this proposition implies that the operators Li j ( p) define also bounded operators from S into itself.
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Proof. Looking at (9.34–9.35, 9.38), we see that K˜ p is a two by two matrix of integral operators whose kernel is a sum of terms of the form
A p (k, k ) =
2
si ω(ki ) + s3 ω(k + p) + s4 ω( p − k)
i=1
(9.39) · δ(k − k1 − k2 − k )ρs (k1 , k2 , k , k, p)dk1 dk2 , 1 where (x) = δ(x) or P x , ρs is C ∞ in all its arguments and we write k for k3 . Let us consider first (x) = δ(x), integrate over k2 , and choose s1 = +1, s2 = −1 (all other terms can be treated similarly). We obtain the integral
δ ω(k1 ) − ω(k − k1 − k ) + s3 ω(k + p) + s4 ω( p − k) (9.40) × ρs (k1 , k − k1 − k , k , k, p)dk1 . Now, by (5.1), dk1 is actually a discrete sum over the first component of k1 and an integral over the last two components. Fix a value of k11 , replace k − k by k , write k12 = − π2 + q2 , k13 = − π2 + q3 and use lower indices, k1 , k2 , k3 for the components of k . We get, using the explicit formula (2.5) for ω,
(9.40) = δ sin q2 − sin(q2 − k2 ) + sin q3 − sin(q3 − k3 ) + f (k2 , k3 , λ) ×ρs (q2 , q3 , λ)dq2 dq3
(9.41)
where we write λ for the set of variables (k11 , k1 , k, p), the integral is an ordinary, not discrete, one and f (k2 , k3 , λ) = sin q1 − sin(q1 − k1 ) + s3 ω(k + p) + s4 ω( p − k).
(9.42)
We shall now study the singularities of (9.41) in k2 , k3 and the smoothness of (9.41) in λ away from the singularities. For notational simplicity, we set ρs = 1; since ρs is smooth, this does not affect our arguments. Let us denote by the argument of the delta function and shift the integration variables: let qi = ri−1 + yi−1 , where yi−1 = 21 ki . Then 2 2 = (sin(ri + yi ) − sin(ri − yi )) + f (2y, λ) = 2 cos ri sin yi + f (2y, λ). i=1
i=1
(9.43) We change variables to cos rα = sα (1 − xα ) with sα = 1 in the region qα ∈ [− π2 , π2 ], and sα = −1 in the region |qα | > π2 . In both cases xα ∈ [0, 1]. Our integral becomes a sum of integrals
− 21 Is (k, k ) = δ(2 xα sα sin yα − g(y, λ))) xα h(xα )d xα (9.44) [0,1]2
− 12
with h(x) = (2 − x)
α
α
and
g(y, λ)) = f (2y, λ) + 2(s1 sin y1 + s2 sin y2 ).
(9.45)
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For sin yi = 0 and g = 0, I is bounded by |I (y, λ)| ≤ C| sin y1 sin y2 |
− 21
(1 + | log |g(y, λ)||).
(9.46)
From (9.42) and (9.45) we have g(y, λ) = 2s3 (cos(2y1 ) + cos(2y2 ) + 2(sin y1 + sin y2 ) + g (λ) with g smooth. Thus the bound (9.46) is integrable in y, uniformly in λ. Moreover, the Hölder derivative of orderα in λ also remains integrable for α small enough. Hence for all bounded functions f, A p (k, k ) f (k )dk is C α in λ = (k11 , k1 , k, p). This in turn implies that A p (k, k ) f (k )dk dk11 (where now the integral includes a Riemann sum over k1 , k11 ) is C α in k, p, since each term in the Riemann sum is C α in k, p. This means that each matrix element of K p maps bounded functions into Hölder continuous ones. Moreover, all the bounds are uniform in N , since the bound (9.46) is independent of N , and taking the Riemann sums preserves that property. To obtain compactness, let α be the Hölder exponent obtained above and take α < α . Then A p is bounded from C α into itself and maps C α into C α . Since the latter is compactly embedded A p is a compact operator from C α into itself. in the former, Obviously A p (k, k )dk is also C α in p, k, so that each matrix element of M p (k) is C α in p, k. Next, we get:
|A p f (k) − A p f (k) − (A p f (k ) − A p f (k ))| ≤ C f ∞ min(|k − k |α , | p − p |α )
≤ C f ∞ |k −k |α /2 | p− p |α /2, (9.47) so that, choosing α = α /2, we get: A p − A p α ≤ C| p − p |α ,
(9.48)
where · α is the operator norm on bounded operators of C α into itself. Finally, it is easy to see that the Riemann sum of a C α function converges to the corresponding integral, with, for the sum in (5.1), an error O(N −α ). This then implies the claims on convergence as N → ∞ made in the proposition.
In Appendix A, we will extend this result. 9.3. Nonlinearities. Let us now turn to the n i ’s defined in (5.13). They are linear combinations of functionals of the following form:
u( f 1 , f 2 , f 3 )( p, k) =
G
v( f 1 , f 2 , f 3 )( p, k) =
G
3 i=1 2
f i ( pi , ki − pi )δ(2 p −
3
2 pi )dpµ(dk),
(9.49)
1
f i ( pi , ki − pi ) f 3 ( p3 , p − k − p3 )
i=1
δ(2 p −
3 1
2 pi )dpµ(dk),
(9.50)
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where G is some smooth function of all the variables (defined in the continuum (N = ∞) and restricted to the discrete) and µ p,k (dk) =
3
3 −1 si ω(ki )+ s4 ω( p− k)+ i δ( ki − p− k)dk1 dk2 dk3 .
1
(9.51)
1
Indeed, (9.49) corresponds to the second term in the bracket in (5.13), after integrating over k4 , p4 , while (9.50) corresponds to the first term, after integrating over k4 (i.e. replacing k4 by p − k) and p3 , and relabeling p4 as p3 . Since Q = Q 0 + r , r ∈ S and Q 0 is given by the series (9.4) we only need to consider f i ( p, k) = F(2 p) for F ∈ E or f i ∈ S. We have then Proposition 9.4. Let m be of the form (9.49) or (9.50). Then (a) Let f i ( p, k) = Fi (2 p) with Fi ∈ E. Then m( f 1 , f 2 , f 3 ) E ≤ C
3
Fi E .
(9.52)
i=1
(b) Let f i ∈ S, f j ( p, k) = F j (2 p) with F j ∈ E for j = i. Then m( f 1 , f 2 , f 3 )S ≤ C f i S F j E .
(9.53)
j=i
(c) Let f i , f j ∈ S, f k ∈ E. Then 1
m( f 1 , f 2 , f 3 )S ≤ C N − 2 +α f k E
fl S .
(9.54)
l=k
For the proof, see Appendix B. The operators L in (9.30) are not multiplication operators in p. Inserting the expansion (9.4) in (5.13), we see that L w is a sum of terms of the form discussed in Proposition 9.4.(b) with Fi of form T ∗A∗n with n > 0 or d(−2 p)−1 t (2 p). Proposition 9.4.(b) then gives Proposition 9.5. The operator L : S → S is bounded, for (T, A) ∈ B , in operator norm by L ≤ C(t S + A E ),
(9.55)
and it is uniformly Lipschitz in T and A for (T, A) ∈ B . Proposition 9.4.(c) gives immediately for the term n(w) ˜ in (9.12): 1
− +α 2 Proposition 9.6. For w S ≤ O(τ λ−2 ), and (T, A) ∈ B , n(w) ˜ S ≤ C N 2 w S , 1
and it is Lipschitz in w, T and A with constant C N − 2 +α , where C may depend on λ−2 .
We still need to discuss the function θ given by (7.9). Its main property, proven in Appendix C, is: Proposition 9.7. d −1 θ is in S, with, for (T, A) ∈ B , w S ≤ O(τ λ−2 ) and t S ≤ τ , d −1 θ S ≤ Cλ2 (t S + A E )(1 + J S ),
(9.56)
and it is uniformly Lipschitz in T and A for (T, A) ∈ B , with constants Cτ , and in J and r with constants Cτ λ2 . Moreover, |θ (0)| ≤ C N −1 .
(9.57)
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10. Solution of the Linear Problem In order to solve Eqs. (5.18), (5.19) we need to study the invertibility of the linear operator 01 2 δω ( p, k) + Lp, (10.1) 10 where δω2 ( p, k) = ω2 ( p + k) − ω2 ( p − k) and L p denotes the linearization around the first term in (9.29) of the nonlinear terms in (5.18, 5.19). L p is given explicitly by adding or subtracting to (9.31) a term with k → −k, see (5.18, 5.12), and multiplying it by 98 (2π )3d λ2 , see (5.12). The fact that L in (9.30) is a small perturbation of L follows from Proposition 9.5, for τ small enough, since, as we shall see in the next section, we shall solveour equations in a space where the RHS of (9.55) is of order τ . So, to invert 0 1 δω2 ( p, k) + D N (Q 0 ), it is enough to concentrate our attention to (10.1), where 10 L p is written as a two by two matrix as in (9.38). Because of the zero modes, in order for (10.1) to be invertible, it needs to be restricted to the orthogonal complement of the zero modes (which occur at p = 0, π ). Let H p be π N Z2N
× [−π, π ]d−1 , ω( p, k)2 dk and let P ⊥ be the projection to the orthogonal complement of ω( p, k)−2 , ω( p, k)−3 in H p , and P = 1 − P ⊥ . Note that the scalar product of f (k) with ω( p, k)−2 equals f (k)dk, which implies the Hilbert space L 2
P ⊥ 1 = 0.
(10.2)
We shall study the operator D p , defined as: D p = δω2 σ1 + L p , where we use the shorthand notations σ1 = =
(10.3)
01 , 10
1 0 0 P⊥
,
for p ∈ E 0 , where E 0 = [− p0 , p0 ] ∪ [π − p0 , π + p0 ],
(10.4)
with p0 = Bλ2 , and B a number that will be chosen large below (see (10.17)); (10.5) D p = δω2 σ1 + L p , for p ∈ / E0 . We shall make the following assumptions that we shall verify later for the operator Lp : 1.
Li j (0) = Li j (π ) = 0, i = j.
(10.6)
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2. ∃ c > 0 such that L11 (0) < −cλ2 , (10.7) P L22 (0)P ⊥ > cλ2 , (10.8) π where the inequalities hold for operators in L 2 N Z2N × [−π, π ]d−1 , ω2 (k)dk restricted to functions that are odd in k (for L11 ) or even in k (for L22 ). The same inequalities hold for L11 (π ), L22 (π ). 3. ∃ C < ∞, independent of N , such that, ∀ p, p , ∀i, j = 1, 2, ⊥
Li j ( p) − Li j ( p ) ≤ C| p − p |α ,
α π
where · is the operator norm of bounded operators mapping C N Z2N into itself. 4. ∃ c > 0 such that, ∀ p, ∀k, ∀ f ∈ R2 , δω2 σ1 + M( p, k) f ≥ c(λ2 + | sin p| | sin k| )| f |,
(10.9) × [−π, π ]d−1
(10.10)
where | f | = | f 1 | + | f 2 | π 5. The kernels K i j (k, ·) ∈ L 1+η ( N Z2N × [−π, π ]d−1 ), for some η > 0, and a norm 2 O(λ ). Proposition 10.1. Under Assumptions (1–5) above, for any B in (10.4), ∃λ0 such that, π for λ ≤ λ0 and for all p ∈ 2N Z2N , D p is invertible, and ∃C < ∞ such that ∀λ ≤ λ0 , D−1 p ≤ where the norm is the one π ⊕ Cα N Z2N × [−π, π ]d−1 .
of
C , λ2
operators
(10.11) in
Cα
π
N Z2N
× [−π, π ]d−1
Proof. We consider | p| ≤ π2 . For | p| > π2 the proof below can be repeated with 0 replaced by π . Let us first consider | p| ≤ Bλ2 , where the constant B, independent of λ, will be specified later. Write p = aλ2 , with |a| ≤ B. We have, for | p| ≤ Bλ2 , δω2 (k, p) = aλ2 ϕ(k) + O(λ4 ),
(10.12)
where we expand δω2 (k, p) (which vanishes at p = 0) in p, the first term in (10.12) is the one linear in p, with ϕ an odd function of k alone, and the second is O( p 2 ) for | p| ≤ Bλ2 . Moreover, due to (10.9), Li j ( p) = Li j (0) + O(λ2+α ),
(10.13)
where O(λ2+α ) is a bound on the operator norm (Li j ( p) has a factor λ2 ). So, we have δω2 σ1 + L p = λ2 aϕ(k)σ1 + L˜ 0 + O(λ2+α ) (10.14) with L˜ 0 = L0 /λ2 , which is λ-independent. Thus if we show that ∃ C(B) such that ∀a, |a| ≤ B, −1 ≤ C(B) < ∞, (10.15) aϕ(k)σ1 + L˜ 0
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we obtain, from (10.14) and a resolvent expansion the bound (10.11) for | p| ≤ Bλ2 , provided that λ is small enough, given B. To prove (10.15), we observe that the spectrum of the multiplication part of aϕ(k) σ1 + L˜ 0 , i.e. aϕ(k)σ1 + M(0,k) lies outside a ball of fixed radius around zero. This follows λ2 from (10.10), using the approximations (10.12) and (10.13) for δω2 and M( p, k). Hence, since adding to this a compact operator does not change the essential spectrum (note that the projection operator P ⊥ adds a rank 2 operator), (10.15) will hold, provided that ˜ we show that aσ1 + L(0) has no zero eigenvalue. But solving for f 2 the first of the two equations f 1 (k) L˜ 11 (0) aϕ(k) =0 f 2 (k) aϕ(k) L˜ 22 (0) (we use the fact that, by (10.6), L˜i j (0) = 0, i = j), substituting inthe second equation,
and taking a scalar product in L 2
π N Z2N
× [−π, π ]d−1 , ω2 (k)dk , with f 1 , we get:
⊥ P ⊥ f 1 , L˜ 11 (0)P ⊥ f 1 − a 2 P ⊥ f 1 , ϕ L˜ −1 =0 (0)ϕ P f 1 22
which is impossible (for all a’s) because of (10.7) and (10.8). This finishes the proof of (10.15) and therefore of (10.11) for | p| ≤ Bλ2 . Consider now | p| > Bλ2 . Since adding a compact operator does not change the essential spectrum, which is bounded away from zero by (10.10), it is enough to show that the equation (δω2 σ1 + L p ) f = µf
(10.16)
f 1 (k) , does not have non-zero solutions for |µ| ≤ cλ2 , and c > 0. Let f 2 (k) E = {k sin k| ≤ b} for some constant b to be specified later. Then, from (10.10), (10.16) and the fact that L p = M p + K p , we get, for k ∈ E:
with f =
c Bbλ2 | f (k)| ≤ |(δω2 σ1 + M p ) f (k)| ≤ K p f ∞ + |µ| f ∞ ≤ (Cλ2 + |µ|) f ∞ , using the fact (see the proof of Proposition 9.3) that K p maps bounded functions into C α ones which, in particular, are bounded, and that K p ≤ Cλ2 . Given β > 0, c > 0, c > 0 C < ∞ and b > 0, we can choose B = B(β, c, c , C, b), independent of λ, so that this implies, if |µ| ≤ cλ2 , | f (k)| ≤ β f ∞
(10.17)
for k ∈ E. Consider now k ∈ E. Then, we get from (10.10), (10.16): cλ2 | f (k)| ≤ |(δω2 σ1 + M p ) f (k)|
≤ K (k, k ) f (k ) + K (k, k ) f (k )dk + |µ| f ∞ . k ∈E
k ∈ E
(10.18)
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We bound, using (10.17),
K (k, k ) f (k )dk ≤ Cλ2 β f ∞ ,
k ∈ E
(10.19)
and since k ∈ E means, for b small, that k must be close to zero or close to π , we get, using Assumption 5 and Hölder’s inequality,
k ∈E
K (k, k ) f (k )dk ≤ K (k, ·)1+η (cb)η/1+η f ∞ ≤ Cλ2 bη/1+η f ∞ . (10.20)
|µ| Inserting (10.19) and (10.20) in (10.18), we get | f (k)| ≤ C β + cλ 2 f ∞ , for k ∈ E, by choosing b small enough. This, combined with (10.17) implies f = 0, for β 2 small enough, if, say, |µ| < cλ2 . Thus, there is no non-zero solution of (10.16) in that ball and (10.11) holds. Let us now check that the operator L p has the properties (1–5) above. To do that, we must first write explicitly the operators Li j (0), Li j (π ). Let us start with p = 0. Using (9.31, 9.38), we get, since only terms that are even in s give a non-zero contribution (see (5.5) and comments afterwards): L12 (0)Q(k) = 2T02
−2
ω(k1 )ω(k2 ) s3 ω(k3 ) ω(k)−2 Q(k3 ) − ω(k3 )−2 Q(−k) s
·P
−1 δ ki δ(k + k4 )dk − (k → −k), si ω(ki )
(10.21)
where the (k → −k) term comes from (5.18); (10.21) vanishes because Q and ω are even in k, see (5.20), −2 −2 s3 J (k3 ) −2 s4 J (−k) −ω(k3 ) s3 ω(k3 ) ω(k) ω(k1 )ω(k2 ) ω(k3 ) ω(k) s ·δ si ω(ki ) δ ki δ(k + k4 )dk − (k → −k), (10.22)
L11 J (k) = 2T02
which equals twice the first term, since J is odd in k (see (5.21)) and ω even. L21 (0)J vanishes by symmetry, like (10.21) (there is a + (k → −k) term in (5.12) and J is odd). This proves property 1, for p = 0. To prove point 2, write: (L22 Q)(k) = 4T02
−2 s3 ω(k3 )s4 ω(k4 ) ω(k1 )ω(k2 ) s
× ω(k4 )−2 Q(k3 ) − ω(k3 )−2 Q(k) ·δ si ω(ki ) δ ki δ(k + k4 )dk.
(10.23)
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π Taking the scalar product of (10.23) with Q(k) in L 2 N Z2N ×[−π, π ]d−1 , ω2 (k)dk , 3 1 and replacing in the second term in [—], s3 ω(k3 ) by si ω(ki ) , using symmetry, 3 i=1 1 and then by − s4 ω(k4 , using the delta function, we get: 3 2 1 6 3 3 ¯ s3 ω (k3 )s4 ω (k4 )Q(k3 ) Q(k)+ ω (k4 )Q(k) ω(ki ) 3 s i=1 ·δ si ω(ki ) δ ki δ(k + k4 )dk. (10.24)
(Q, L22 Q) = 4T02
4
−2
Using k = −k4 , the symmetry between the ki ’s and the evenness of ω, Q, we can write (10.24) as:
4 2 T02 si ω(ki )3 Q(ki ) δ ω(ki )−2 si ω(ki ) δ ki δ(k + k4 )dk. 3 s i=1
(10.25) A similar computation, starting with (10.22), leads to
4 2 T02 ω(ki )2 J (ki ) ω(ki )−2 3 s i=1 ×δ si ω(ki ) δ ki δ(k + k4 )dk.
(J, L11 J ) = −
(10.26)
To conclude the proof of (10.7), (10.8), we need the following Lemma 10.2. Let f be a Hölder continuous function from Td into R, with d ≥ 3, satisfying f (k1 ) + f (k2 ) = f (k3 ) + f (k4 ) on the set 4 | ki ∈ Td , i = 1, . . . , 4, k1 + k2 = k3 + k4 , ω(k1 ) + ω(k2 ) = ω(k3 ) + ω(k4 )}, {(ki )i=1
then, f (k) = aω(k) + b.
(10.27)
Now, consider first (10.25) and (10.26) in the N → ∞ limit, i.e. with the sum in (5.1) replaced by an integral. To apply the lemma, use the fact that, by (5.21) and relabeling indices, we may assume that s1 = s2 = 1, s3 = s4 = −1 and change k3 → −k3 , k4 → −k4 . Then, the lemma applied to f = ω(k)3 Q(k) or f = ω(k)2 J (k) implies that (10.25) and (10.26) cannot equal zero unless J = 0, since J (k) is odd, or unless Q(k) = aω(k)−2 +bω(k)−3 which, for Q = P ⊥ Q, i.e. for Q orthogonal to ω(k)−2 and to ω(k)−3 implies a = b = 0. Since each Lii (0) is the sum of a multiplication operator which is bounded away from zero (see (10.29), (10.30) below for p = 0) and a compact operator, either there is a zero
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eigenvalue or inequalities (10.7)–(10.8) hold. Since we showed that there cannot be a zero eigenvalue, (10.7)–(10.8) hold for p = 0, and N = ∞. To conclude the proof for N large, but finite, we simply use the convergence results stated in Proposition 9.3, which imply the convergence of (10.25) and (10.26) to their N = ∞ limit. The statements for p = π follow from (9.31) and the fact that w(π, k−π ) = w( p, k). The bound (10.9) follows from Proposition 9.3, and point 5 follows easily from the bounds (9.46) on the kernels K i j ( p)(k, ·), given by (9.39). Finally (10.10) holds because, for given p, k, δω2 σ1 + M( p, k) is a 2 × 2 matrix, and the lower bound (10.10) holds if the eigenvalues of that matrix satisfy |µi ( p, k)| ≥ c(λ2 + | sin p|| sin k|), i = 1, 2. To prove this lower bound, it is enough to prove it for the square root of the absolute value of the determinant of the matrix δω2 σ1 + M( p, k), which equals M11 (k, p)M22 (k, p) − δω2 + M12 (k, p) δω2 + M21 (k, p) .
(10.28)
We can check, from the explicit formulas (10.22), (10.23), in which the multiplication operator corresponds to the last term in the [—], that, for all p, M11 (k, p) < −cλ2 , M22 (k, p) > cλ2
(10.29) (10.30)
for c > 0. The signs and the factor λ2 are obvious, and to get a non-zero contribution, we need only to check that si ω(ki ) vanishes for some k1 , k2 , k3 , k4 , with the constraints 3 k4 = p − k, ki = p + k. Choosing s1 = +1, s2 = −2, s3 = +1, s4 = −1 (which can i=1
always be obtained by relabeling indices), and inserting the constraints, this means ω(k1 ) − ω(k2 ) + ω( p + k − k1 − k2 ) − ω( p − k) which vanishes for k1 = k2 = k. Now, since Mi j (k, p) for i = j vanishes at p = 0 or π and at k = 0 or π , we get, using Proposition 9.3, α/2 |Mi j (k, p)| ≤ Cλ2 | sin k| | sin p|
(10.31)
for i = j. Inserting (10.29), (10.30), (10.31) into (10.28), using |δω2 | = 4| sin k| | sin p|, we get that Mi j , i = j is small compared to |δω2 | if | sin k| | sin p| ≥ c λ2 , for c small
(in which case, both terms in (10.28) are negative), and that δω2 + Mi j (k, p) is small
compared to λ2 , i.e. compared to Mii , otherwise. We are left with the
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Proof of Lemma 10.2. The proof follows closely the one in [32], which itself is inspired by [8]. Let us first assume that f is twice continuously differentiable. The hypothesis of the lemma implies that f (k) + f (k ) is constant ∀k, k ∈ Td , with k + k constant and ω(k) + ω(k ) constant. Therefore, there exists g : R × Td → R, of class C 2 , such that f (k) + f (k ) = g ω(k) + ω(k ), k + k . Writing k = (k α )dα=1 , k α ∈ T, ω = ω(k ), ω = ω(k ), we get ∂α f (k) = ∂ω g(ω + ω , k + k )∂α ω + ∂α g(ω + ω , k + k ), ∂α f (k ) = ∂ω g(ω + ω , k + k ), ∂α ω + ∂α g(ω + ω , k + k ), ∂ where ∂α = ∂k∂ α , ∂ω = ∂ω . Subtracting these two equations, we get ∂α f (k) − ∂α f (k ) = ∂ω g(ω + ω , k + k ) ∂α ω(k) − ∂α ω(k ) . (10.32) Multiplying first (10.32) by ∂β ω(k) − ∂β ω(k ) , then rewriting the resulting equation by exchanging α and β we get, for all α, β: ∂α f (k)− ∂α f (k ) ∂β ω(k)− ∂β ω(k ) = ∂β f (k)− ∂β f (k ) ∂α ω(k)− ∂α ω(k ) .
If we differentiate this identity with respect to kγ , we get: ∂α ∂γ f (k) ∂β ω(k) − ∂β ω(k ) + ∂α f (k) − ∂α f (k ) ∂β ∂γ ω(k) = ∂β ∂γ f (k) ∂α ω(k) − ∂α ω(k ) + ∂β f (k) − ∂β f (k ) ∂α ∂γ ω(k). Differentiating now this with respect to kδ , we get: ∂α ∂γ f (k)∂β ∂δ ω(k ) + ∂α ∂δ f (k )∂β ∂γ ω(k) = ∂β ∂γ f (k)∂α ∂δ ω(k ) + ∂β ∂δ f (k )∂α ∂γ ω(k).
(10.33)
Now, for ω(k) as in (2.5), we have ∂α ∂β ω(k) = δαβ cos kα . Using this and choosing in (10.33) α = γ = β = δ, we get ∂α ∂γ f (k) = 0 for α = γ . This holds first on a dense set kγ = ± π2 (cos kγ = 0) and then, by continuity, everywhere on Td . Then, choosing α = γ = β = δ, we get ∂α2 f (k)∂β2 ω(k ) = ∂β2 f (k )∂α2 ω(k), which implies that ∂α2 f (k) = a∂α2 ω(k) for a constant a. Integrating, we get f (k) = aω(k) + b + ck, for a, b, c ∈ R, and we get c = 0 from the fact that f is a continuous function on Td . This finishes the proof of f of class C 2 . For f merely Hölder continuous, we interpret all the (linear) identities above, and all the derivatives in the sense of distributions, and we obtain the same conclusion. Remark. The lemma holds by assuming only that f is a distribution, but we do not need this. It is crucial here that the dimension d ≥ 3 in order to be able to choose α = γ = β. For counterexamples in d = 1, see [21].
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11. Proof of Theorem 2 In this section we solve Eq. (6.1). Due to the presence of zero modes in the operator L22 (0) we need to consider separately the projection of (6.1) onto the zero modes, and Eq. (6.1) in the complement of these zero modes, at least for p ∈ E 0 , where E 0 was defined in (10.4). The solution to the complementary equations leads to the Fourier Law, i.e. an expression of the currents ( j, j ) in terms of the temperature T and chemical potential A. The solution to the projected equation determines finally T and A. We look for a solution of the form W = Q 0 + w, where Q 0 = Q 0 (T, A), for some functions T, A, is given by (9.4), and w, also written as a pair (J, r ) (so that, in Theorem 2, R = Q 1 + r ) , is as follows: let ws = w χ ( p ∈ E 0 ), and let P be the projection in L 2 ω( p, k)2 dk to the span of ω( p, k)−2 , ω( p, k)−3 and P ⊥ = 1 − P. We demand P ⊥rs ( p, ·) = rs ( p, ·), p ∈ E 0 . Given a function f that is Hölder continuous in p on E 0 , let f˜ denote a linear extenπ sion of f to N Z2N . We have f˜α ≤ C f α . We proceed similarly with elements of S or of E. Now write w = w˜ s + w , where w˜ s is the extension of ws defined above. The function w satisfies w ( p) = 0 for p ∈ E0 . Since from (7.10, 10.2) we have PC = C, Eq. (6.1) can be written, for p ∈ E 0 , as a pair of equations: 0 δω2 J + N (Q, J ) + N (Q, J ) = 0, (11.1) Q δω2 0 where
=
1 0 0 P⊥
and P(δω2 J + N2 + N2 ) = PC. For p ∈ / E 0 , we will solve directly (6.1): 0 δω2 0 J . + N (Q, J ) + N (Q, J ) = C Q δω2 0
(11.2)
(11.3)
˜ ∈ B1 , We look for solutions where (T, A), defined for p ∈ E 0 , is such that (T˜ , A) where B1 ⊂ E × E is the ball (11.4) T − T+ δ( p) ≤ B1 τ, E
A E ≤ B1 τ λ2 ,
(11.5)
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with 1
T+ = 2 (T1 + T2 ),
(11.6)
and B1 can be chosen to be O(1). For τ small enough, we have B1 ⊂ B , defined in (9.3), so that the estimates of Sect. 9 can be used. As for w, we look for ws with w˜ s in the ball B2 ⊂ S ⊕ S, given by B2 = {( J˜, r˜ ) | ( J˜, r˜ ) − ( J˜0 , r˜0 )S ≤ B2 τ },
(11.7)
where (J0 , r0 ) is given in (11.37) and bounded in (11.38). Finally, we shall choose w ∈ B3 , where B3 = {(J, r ) | J S + r S ≤ B3 λ−2 τ, J p = r p = 0 for p ∈ E 0 }.
(11.8)
Theorem 2 is a consequence of the proposition below (remembering that R = Q 1 + r and using the fact that the S norm of Q 1 is of order τ , see (9.7)), and the remark following it. Proposition 11.1. Let λ, τ , N be as in Theorem 2. There exist constants B1 ,B2 , B3 in the definitions of B1 , B2 , B3 , so that: ˜ ∈ B1 , and given w ∈ B3 , there (a) Given (T, A) defined for p ∈ E 0 , such that (T˜ , A) ˜ + w˜ s + w solves (11.1) exists a unique ws , such that w˜ s ∈ B2 and such that Q 0 (T˜ , A) for p ∈ E 0 . Moreover, w˜ s is Lipschitz in (T, A) with Lipschitz constant O(λ−2 ) and in w with constant O(N −α/4 ). ˜ ∈ (b) Given w ∈ B3 , there exists a unique (T, A) defined for p ∈ E 0 , such that (T˜ , A) ˜ + w˜ s + w solves (11.2), for p ∈ E 0 , where ws (T, A, w ) is B1 , and such that Q 0 (T˜ , A) the solution obtained in (a). The pair (T, A) is Lipschitz in w with Lipschitz constant O(τ ). ˜ w˜ s (T, A, w )+w solves (11.3), (c) There exists a unique w ∈ B3 such that Q 0 (T˜ , A)+ for p ∈ / E 0 , where (T, A) = (T, A)(w ) is the solution obtained in (b). Remark. Moreover, the precise bounds stated in Theorem 2 will be given in the course of the proof: see (11.84), (11.54), (11.89), for the statements about T and A, and, for J and r , see (11.7), (11.38) and the argument given after (11.89). 11.1. Fourier’s Law. Let us prove first part (a) of Proposition 11.1. For simplicity of ˜ and drop the tilde on T, A. Recall that Q = notation, we shall write Q 0 for Q 0 (T˜ , A), Q 0 + r . The leading inhomogeneous term in (11.1) is −δω2 Q 0 . Using (9.4), we write: δω2 Q 0 = t (2 p)ρ1 ( p, k) + d(−2 p)(T ∗ A)(2 p)ρ2 ( p, k) + ρ3 ( p, k),
(11.9)
where t ( p) = d(− p)T ( p), ρ1 ( p, k) = d(−2 p)−1 δω2 ( p, k)ω( p, k)−2 , ρ2 ( p, k) = d(−2 p)−1 δω2 ( p, k)ω( p, k)−3 are smooth functions and ρ3 ( p, k) = δω2 ( p, k)
∞ n=2
T ∗ A∗n (2 p)ω( p, k)−2−n
(11.10) (11.11)
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is in S with ρ3 S ≤ CA2E ,
(11.12)
and Lipschitz in A, with constant of order τ λ2 , since ρ3 is at least quadratic in A (see (11.12)), and we work in the ball defined by (11.5). The nonlinear term N (Q, J ) in (11.1, 11.3) was studied in Sect. 9. It is given by (see Eq. (5.12, 5.18, 5.19)) 9 n 1 ( p, k) − n 1 ( p, −k) , (11.13) N ( p, k) = (2π )3d λ2 n 2 ( p, k) + n 2 ( p, −k) 8 where n is the sum (9.12), and where we reintroduced λ2 . From Proposition 9.2, we infer (using the symmetry n 1 ( p, −k) = n 1 (− p, k)), that N (Q 0 , 0)S ≤ Cλ2 (tS + A E ), and is Lipschitz in t, A, with constant of order λ2 . From (9.30), (9.32), the definition of L p in Sect. 10 and Proposition 9.5, we get DN (Q 0 , 0) − L p ≤ Cλ2 (tS + A E ),
(11.14)
and is Lipschitz in t, A, with constant of order λ2 , where on the LHS the norm is the operator norm in S ⊕ S. Finally, Proposition 9.6 gives 1
N − N (Q 0 , 0) − DN (Q 0 , 0)(J, r )T S ≤ C N − 2 +α (J S + r S )2 , (11.15) 1
with a Lipschitz constant in J , r , of order N − 2 +α . So, combining those estimates, we get that, for T , A, J , r , as in Theorem 2: N ( p, k) − L p (J, r )T S ≤ Cτ λ2
(11.16)
and N is Lipschitz as a function of t, A with constant Cλ2 , and as a function of J , r , 1 with a constant C N − 2 +α . Consider next the function N , defined in (6.3): N = ( J + J , P + P)T .
(11.17)
Let us start with J + J , given by (7.11) with P replaced by J . Recalling the definition (8.11), using
|d( p)|−1 dp ≤ C log N , (11.18) that follows from |d( p)|−1 ≤ C| p|−1 , for p = 0 and | p| ≥ cN −1 , we get
|J (q, q + k − p)|dq ≤ C N −1+α/2 J S ,
(11.19)
uniformly in k, p, since the singularities in (8.11) affect only the first variable and log N ≤ N α/2 . Now, use the fact that, for functions on the π/2N lattice, f α ≤ C N α f ∞ ,
(11.20)
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599
and identify J + J as an element of S of the form J + J = (0, 0, , 0), to get:
J + J S ≤ Cγ N α N 1−α/2 |J (q, q + k − p)|dq∞ ≤ Cγ N α J S ≤ C N −1+5α/4 J S ,
(11.21)
where in the last inequality, we use the definition (8.17) of γ . J + J is of course Lipschitz, with constant C N −1+5α/4 . For the term P + P, we need to study P, given in (5.17): 9 1 P = ω( p, k)2 Q + 2 (J − J ) − (2π )3d λ2 n 1 ( p, k) + n 1 ( p, −k) . (11.22) 8 We have ω( p, k)2 Q = T (2 p) + (T ∗ A)(2 p)ω( p, k)−1 + p1 + ω( p, k)2 r, where p1 collects the n ≥ 2 terms coming from the expansion (9.4). From Proposition 9.2, the last term in (11.22) at J = r = 0 is given by (9.20). Propositions 9.3, 9.5 and 9.6 control the corrections. The (J − J ) term is bounded as in (11.21). To summarize, let us call PS the sum of 21 (J − J ) + ω( p, k)2 r , of the corrections to the J = r = 0 term in − 98 (2π )3d λ2 n 1 ( p, k) + n 1 ( p, −k) and of the term corresponding to m in (9.20). Let PE be the rest, i.e. T (2 p) + (T ∗ A)(2 p)ω( p, k)−1 + p1 , and what corresponds to the sum in (9.20). Collecting the bounds established for these various terms, we get: Proposition 11.2. P can be written as P = PE + PS with PE ( p, k) = T (2 p) + (T ∗ A)(2 p)ω( p, k)−1 + P E ( p, k),
(11.23)
where P E ( p, k) =
∞
Fn (2 p)h n ( p, k),
(11.24)
n=1
where Fn ∈ E, with Fn E ≤ C n AnE , for n ≥ 2, F1 E ≤ Cλ2 A E , and where the functions h n are smooth with h n ∞ ≤ C n ,
(11.25)
PS S ≤ C(r S + λ2 (J S + tS + A E )).
(11.26)
PS ∈ S and
Moreover, PS , PE are Lipschitz functions of their arguments, with constants O(1).
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We may now bound P ⊥ (P + P). The PS term is in S, like J and can be bounded, as in (11.21), by the RHS of (11.26) times N −1+5α/4 , and a similar Lipschitz constant. The T (2 p) in (11.23) drops out from P ⊥ , since it is constant in k and we use (10.2). The other terms in (11.23) are bounded using
(11.27) dq|F(q)| ≤ CF E , which follows easily from |F( p)| ≤ δ( p) +
1 1 1 + 3/2 5/2 F E , + 1−α/2 2 N |d| N |d| N |d|
(11.28)
using (11.18) and
|d( p)|−k dp ≤ C N k−1
k > 1,
(11.29)
which is proven like (11.18). Thus P ⊥ ( PE + PE )( p, k) is smooth, since, looking at (7.11) we see that the dependence on p, k in P ⊥ ( PE + PE )(q, q + k − p) is only through the second argument of PE , in which PE is smooth, by Proposition 11.2. Moreover, using (11.27) and (11.23), (11.24),
(11.30) | P ⊥ ( PE + PE )(q, q + k − p)| ≤ Cγ A E . Since P ⊥ ( PE + PE )( p, k) is smooth, let us identify it with an element of S of the form (0, 0, , 0). Then, by (8.17), P ⊥ ( PE + PE ) ∈ S and P ⊥ ( PE + PE )S ≤ C N −1+α/4 N 1−α/2 A E = C N −α/4 A E .
(11.31)
Combining the above bounds, we get: P ⊥ ( P + P)S ≤ C N −α/4 A E + C N −1+5α/4 (r S + J S + tS ) (11.32) and P ⊥ ( PE + PE ) is Lipschitz in A, t, J , r , with constants O(N −α/4 ). This, combined with (11.21), implies that P ⊥ N (x, y) is o( N1 ), for x, y away from the boundaries (see (4.23)). Combining (11.21) and (11.32), we obtain that N , with N given by (11.17), satisfies, for T, A, J, r as in Proposition 11.1, N ≤ C N −α/4 ,
(11.33)
where the norm is in S ⊕ S. We may summarize this discussion by rewriting Eq. (11.1); consider T, A given, for p ∈ E 0 and w given for p ∈ / E 0 . Let us denote s( p) = d(− p)(T ∗ A)( p) ≡ d(− p)S( p),
(11.34)
and D˜ = D p + 98 (2π )3d λ2 L = (δω2 σ1 + DN (Q 0 )) (D˜ is not a multiplication operator in p, because of the L term, which is bounded in Proposition 9.5). Then,
(11.35) (Js , rs )T = −D˜ −1 (ρ1 t (2 p) + ρ2 s(2 p) + ρ3 , 0)T + R
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601
for p ∈ E 0 , using the fact that rs = P ⊥rs , and therefore, (Js , rs )T = (Js , rs )T . Here, ˜ s , rs )T and the R ∈ S includes all the terms in (11.1), apart from the linear term D(J 2 term δω Q 0 , see (11.9). Combining (11.16), (11.33), RS is bounded, for T , A, J , r as in Theorem 2, by: RS ≤ C B1 τ λ2 ,
(11.36)
and is Lipschitz in J, r with constant O(N −α/4 ), and in t, A with constant O(λ2 ). D−1 p is the operator defined by (10.3), and bounded in Proposition 10.1, while 98 (2π )3d λ2 L is controlled by Proposition 9.5, which implies (11.14), i.e. its norm is of order τ λ2 , so that D˜ −1 satisfies a bound like (10.11). Let, for p ∈ E 0 , (J0 , r0 )T = −D˜ −1 (ρ1 t (2 p) + ρ2 s(2 p) + ρ3 , 0)T .
(11.37)
Then, by Proposition 10.1 and Lemma 9.1.b, ( J˜0 , r˜0 )S ≤ Cλ−2 (tS + sS ) ≤ Cλ−2 (tS + A E ).
(11.38)
Now, given w˜ s ∈ B2 and w ∈ B3 we have, from Proposition 10.1 and (11.36), D˜ −1 RS ≤ C B1 τ,
(11.39)
i.e. D˜ −1 R ∈ B2 for B2 > C B1 . It is Lipschitz in J, r, w with constant O(N −α/4 ), and in T, A with constant O(1). Thus, combining the contraction mapping principle, and the fact that the leading term (J0 , r0 ) in (11.35) is Lipschitz in (T, A) with Lipschitz constant O(λ−2 ), and remembering that, by assumption, inverse powers of N are small compared to λ2 , we get: Proposition 11.3. Given T, A, w as above, Eq. (11.35) has a unique solution (Js , rs ) with ( J˜s , r˜s ) ∈ B2 , which is Lipschitz in (T, A) with Lipschitz constant O(λ−2 ) and in w with Lipschitz constant O(N −α/4 ). This proves part (a) of Proposition 11.1. Now assume that we have proven part (b) of that proposition. This will be done in the next subsection. We want to prove here part (c), since it is done in the same spirit as part (a). Thus, consider Eq. (11.3), for p ∈ / E 0 . Following the argument that led from (11.1) to (11.35), we may rewrite it as:
(11.40) (J, r )T = −D˜ −1 (ρ1 t (2 p) + ρ2 s(2 p) + ρ3 , 0)T + R , where D˜ = D p + 98 (2π )3d λ2 L , with D p defined in (10.5), for p ∈ / E 0 , which is invertible by Proposition 10.1. We can write (11.40) as:
(11.41) (J , r )T = −( J˜s , r˜s )T − D˜ −1 (ρ1 t + ρ2 s + ρ3 , 0)T + R , where ( J˜s , r˜s ) is the Lipschitz function of w , given by Proposition 11.3. The S-norm of the RHS is bounded by C B1 τ λ−2 and so, taking B3 = C B1 , (11.41) is in B3 , provided we show it vanishes for p ∈ E 0 . By assumption, T, A here is such that (11.2) holds for p ∈ E 0 (part (b) of the proposition, to be proven below), and, by part (a), we know that (11.1) holds for p ∈ E 0 .
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Putting (11.1) and (11.2) together, we see that the full equation, (11.3), is satisfied by Q T,A + w˜ s + w for p ∈ E 0 . But (11.40) is merely a rewriting of (11.3), whenever D˜ is invertible. Since we can assume, by choosing, if necessary, B in Proposition 10.1 larger than here, that D± p0 and Dπ ± p0 are invertible, and since, by assumption, Q T,A + w˜ s + w = Q T,A + w˜ s for p ∈ E 0 , we know that Q T,A + w˜ s solves (11.40) for p = ± p0 or p = π ± p0 . But that means that the RHS of (11.41) vanishes for those values of p. We can define both sides to be zero for other values of p ∈ E 0 , if we want, without affecting the fact that J , r are in S, and thus we obtain part (c) of Proposition 11.1, by applying the contraction mapping principle to the fixed point equation (11.41) to (J , r ) ∈ B3 (since R is Lipschitz in w , see the comment after Eq. (11.39)). Remark. The function J (T, A) is a general form of the Fourier Law, expressing the Eqx p y correlation function as a function of the local temperature and chemical potential. In particular, by (11.40), and remembering that acts as the identity on the first component, J ( p, k) = κ1 ( p, k)t (2 p) + κ2 ( p, k)s(2 p) + O(
τ2 ) + O(τ ) λ2
(11.42)
with T κ j = −D−1 p (ρ j , 0) ,
(11.43)
and where the term which is O( τλ2 ) (in S norm) comes from expanding the L part of D˜ −1 and using Proposition 9.5, while the O(τ ) comes from D−1 p acting on ρ3 , see (11.12, 11.5), and on R, see (11.39). Since, as we shall see now, the first term in (11.42) is of order λτ2 , these last two terms are a small correction to the first ones. 2
Inserting (7.7) and (7.14) we get
j ( p), j ( p)
T
T τ2 = κ( p) t ( p), s( p) + O( 2 ) + O(τ ), λ
(11.44)
where κ( p) is C α in p. Since D p is of order λ2 , κ( p) is of order λ−2 . In the next subsection, we shall need the fact that κ( p) is invertible for p ∈ E 0 . To show that, let us first compute κ(0). From (10.3) and (6.8), we get L11 (0) 0 . (11.45) D0 = 0 L22 (0) Inserting (7.5) to (11.10) and (11.11) we have ρ j (0, k) = 4i sin kω(k)− j for j = 1, 2, and so, defining ψ j (k) = 4 sin kω(k)− j ,
j = 1, 2,
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
we have
603
κ j (0, k) = −i L11 (0)−1 ψ j (k).
Inserting this into (7.7) we have, for j = 1, 2,
κ1 j (0) = −2 dk sin k ω(k)(L11 (0)−1 ψ j )(k). From (7.14) we get
κ2 j (0) = −2
dk sin k η(0, k)ω(k)(L11 (0)−1 ψ j )(k),
where η(0, k) equals ω(k)−1 −
(11.46)
(11.47)
dkω(k)−1 . Now, note that 1 κ1 j (0) = − ψ1 , L−1 , (0)ψ j 11 2
(11.48)
where (·, ·) is the scalar product in H0 , and that 1 1 ψ1 , L−1 (11.49) κ2 j (0) = − ψ2 , L−1 11 (0)ψ j + β0 11 (0)ψ j , 2 2 where β0 = dkω(k)−1 . Computing the determinant of the 2 × 2 matrix κ(0), we see that it equals the one with β0 = 0, and the latter does not vanish because L11 (0) is a strictly negative operator (see (10.7)). Using the Hölder continuity of κ( p) in p, we get that the invertibility of κ(0) implies the one of κ( p), for p small, i.e. for p ∈ E 0 , for λ small enough. Finally, observe that, since L11 (0) is a strictly negative operator, κ( p) is a positive matrix, for p ∈ E 0 . To understand the connection with the Fourier law (1.1), note that t ( p) = d(− p)T ( p), and, in x-space, d(− p) is −∇x . 11.2. Solving the conservation laws. We are left with the proof of part (b) of Proposition 11.1. This reduces to solving the two conservation laws, Eqs. (7.3) and (7.13), which are equivalent to (11.2). Indeed, (7.3) is (4.20) for x = y, which is the same as integrating (5.19) with dk, or taking the scalar product of (5.19) with ω( p, k)−2 in H p . For (7.13), it amounts to taking the scalar product of (5.19) with a linear combination of ω( p, k)−2 and ω( p, k)−3 . Let us introduce a more compact notation. We set J = ( j, j ) and write (7.3) and (7.13) as d( p)J ( p) + F( p) − ( p) = 2γ (T1 + ei N p T2 , 0)T
(11.50)
for p ∈ E 0 with E 0 given by (10.4), where we can assume that κ( p) is invertible. Equation (11.50) has the friction term, see (7.15),
F( p) = γ dkdq P(q/2, k)(1 + ei(q+ p)N )(2, ψ( p, k, q))T , (11.51) where we use ei pN = e−i pN , and the projection of N22 : ( p) = (0, θ ( p))T ,
(11.52)
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J. Bricmont, A. Kupiainen
where θ is given by (7.9). J is given by the Fourier law, i.e. the solution of (11.40), which we may write as (see (11.44)): J ( p) = κ( p)[(t ( p), s( p))T + z( p)],
(11.53)
where s is given in Eq. (11.34) and z includes the corrections coming from the L , R and the ρ3 terms in (11.35). J , F and are functions of T, A ∈ E, and w ∈ B3 (including, indirectly, as functions of ws , which is a function of T , A, w , by part a of Proposition 11.1). So, for wl fixed (11.50) is a nonlinear, nonlocal elliptic equation for T and A. We look for a solution to (11.50) in the ball B1 ⊂ E × E, defined in (11.4, 11.5). For such a T , the map A → T ∗ A ≡ S is invertible (because T in (11.4) is close to a delta function, which is the identity for the convolution) and S − T+ A E ≤ Cτ λ2 .
(11.54)
Thus we can use T, S as the unknowns, in the ball B1 . The following proposition collects the properties of the functions z and θ , studied in Propositions 11.3, 9.5 and 9.7 (since |θ (0)| ≤ C N −1 , we have that d −1 (θ − θ (0)) ∈ S) : Proposition 11.4. z and d −1 ( − (0)) are Lipschitz functions B1 × B3 → S ⊕ S with z = (z 1 , z 2 )T , z 1 S ≤ Cτ 2 , z 2 − |d( p)|α z 20 S ≤ Cτ λ2 , z 20 S ≤ Cτ 2 , −1
(11.55)
d ( − (0)) S ≤ Cτ λ , |(0)| ≤ C N −1 , 2
(11.56) (11.57)
and Lipschitz constants bounded by Cτ . Proof. Since the operator bound on L is of order τ , by Proposition 9.5, and since L is applied to (t, s), which is of order τ , and since R and ρ3 are of order τ λ2 (see (11.36), (11.12) and (11.5)), we easily get a bound on z of order τ 2 (for λ2 small compared to τ ). For z 2 , we use the fact that the norm of s, like the one of A, is of order τ λ2 , so we get a bound Cτ λ2 , except for an off-diagonal contribution from L , which is of order τ |d( p)|α , since this off-diagonal contribution vanishes at p = 0, see (6.8). Both z and depend on t, A, directly, but also indirectly through their dependence on rs , Js , that are Lipschitz in t, A, but with constants Cλ−2 . Since ρ3 is quadratic in A (see (11.12)), it satisfies the bounds above. The R term in z is Lipschitz in rs , Js with a constant O(N −α/4 ), which by assumption, controls the factor Cλ−2 , so that we can get an overall Lipschitz constant Cτ for z. For , we get, from Proposition 9.7, a Lipschitz constant Cτ for T and A and a Lipschitz constant Cτ λ2 for J and r , so that the overall constant is O(τ ). Recall next that f ∈ S is decomposed as (see (8.6)) f ( p, k) = f + ( p, k)σ+ (2 p) + f − ( p, k)σ− (2 p), with σ± ( p) = 1 ± ei N p , and f ± = 1 + ei(q+ p)N
1 2
f . Insert 1 σ+ ( p)σ+ (q) + σ− ( p)σ− (q) = 2
(11.58)
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
into Eq. (11.51) and use σ+ (q)σ− (q) = 0 for q ∈
π N Z2N
605
to get:
F( p) = F+ ( p)σ+ ( p) + F− ( p)σ− ( p), with
F± ( p) = γ
±
dq
dk P± (q/2, k)(2, ψ( p, k, q))T ,
(11.59)
where ± dq = 21 dqσ± (q), i.e. the q-sum in ± runs over the odd (for −) or even, π non-zero (for +), multiples of N . We used here the fact that σ± (q)2 = 0, 4 to cancel the factors of 21 in (11.58) and in P± = 21 P. Thus, (11.50) becomes two equations, one (+) valid on the even sub-lattice, and the other one (−) on the odd sub-lattice: dJ± + F± − ± = 2γ T± (1, 0)T
(11.60)
with T+ the average temperature (11.6) and 1
T− = 2 (T1 − T2 ).
(11.61)
It is useful to separate from (11.59) the part which is constant in p: F± ( p) = F± (0) + f ± ( p),
(11.62)
± ( p) = ± (0) + ± ( p).
(11.63)
and similarly for ± ( p): Since ψ in (7.16) is smooth, we may write: f ± ( p) = d( p)g± ( p),
(11.64)
and we may estimate g± as in the derivation of (11.31), and obtain (there is no P ⊥ here, so we have a term linear in T ): Proposition 11.5. The functions g± ∈ S are Lipschitz in (T, S) ∈ B1 with g± S and the Lipschitz constants O(N −α/4 ). Using (11.53), we may write (11.60) as d( p)κ( p) (t± ( p), s± ( p))T + W± ( p) = 2γ T± (1, 0)T − F± (0) + ± (0), (11.65) where W± ( p) = z ± ( p) + κ( p)−1 (g± ( p) − d( p)−1 ± ( p)), is defined for p ∈ E 0 , where κ( p) is invertible (see the remark at the end of Subsect. 11.1). So, combining Propositions 11.4 and 11.5, W± is a pair (W1± , W2± ), with Wi± satisfying bounds like (11.55), and the functions Wi± are Lipschitz in (T, S), with constants Cτ . Let us next analyze F± (0). Using Proposition 11.2, S = T ∗ A and (7.8), we write F± (0) − ± (0) = φ± + ψ±
φ± = γ dq dk T (q) + ρ(q/2, k)S(q) (2, ψ(0, k, q))T (11.66) ±
dq dk P E (q/2, k) + PS (q/2, k) (2, ψ(0, k, q))T −± (0). ψ± = γ ±
(11.67)
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For ψ± , we proceed as in the proof of (11.31). The contribution coming from PS is small, see (11.21), while the one coming from P E is to leading order, see (11.24) and (11.5), O(γ τ λ4 ). The contribution of ± (0) is O(N −1 ), by Proposition 9.7. So, ψ± are bounded by |ψ± | ≤ Cγ τ λ4
(11.68)
and are Lipschitz in (T, S) with constant Cγ λ2 . It is instructive to solve first the simplified equation (11.65) with W± dropped and F± (0) − ± (0) replaced by φ± . Then, for p = 0, (11.69) κ( p) t± ( p), s± ( p) = d( p)−1 ξ± with ξ± = 2γ T± (1, 0)T − φ±
(11.70)
and, for p = 0 (where obviously only the equation with index + holds), 2γ T+ (1, 0)T = φ+ .
(11.71)
Equations (11.70) and (11.71) imply ξ+ = 0
(11.72)
so, by (11.69), (t+ , s+ ) = 0 and, from (11.69) for −, we can write t− ( p), s− ( p) = d( p)−1 τ ( p), ζ ( p) with
T κ( p) τ ( p), ζ ( p) = ξ−
(11.73)
(11.74)
constant in p, i.e.:
τ ( p), ζ ( p)
T
T = κ( p)−1 κ(0) τ (0), ζ (0) ,
(11.75)
where (τ (0), ζ (0) is defined by extending (11.74) (which was derived on the odd sublattice) to p = 0. Hence, since s( p) = d(− p)S( p), t ( p) = d(− p)T ( p), T T U ( p) ≡ T ( p), S( p) = U0 δ( p) + |d( p)|−2 τ ( p), ζ ( p) σ− ( p). (11.76) The unknowns are U0 = (T0 , S0 )T , τ (0) and ζ (0) and they will be determined from (11.70) and (11.71), where the functions φ± , given by (11.66), are functions of U0 , τ (0), ζ (0) via (11.76):
T φ+ = γ dk T0 + ρ(0, k)S0 2, ψ(0, k, 0) , (11.77)
T φ− = 2γ dq dk|d(q)|−2 τ (q)+ρ q/2, k ζ (q) 2, ψ(0, k, q) , (11.78) −
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607
where the prefactor of 2 comes from the fact that, in (11.76), σ− ( p) = 0, 2. Equations (7.8), (7.12) and (7.16) imply ψ(0, k, 0) = 2(ρ(0, k) − dkρ(0, k)), with ρ(0, k) = ω(k)−1 . So,
ψ(0, k, 0)dk = 0,
ρ(0, k)dk ≡ β1 > 0,
2 1 −2 dk − ω(k)−1 dk ≡ β2 > 0. ρ(0, k)ψ(0, k, 0)dk = ω(k) 2 Hence (11.77) gives φ+ = 2γ (T0 + β1 S0 , β2 S0 )T and, from (11.71), we get: 1
S0 = 0 , T0 = T+ = 2 (T1 + T2 ).
(11.79)
To analyze (11.78), we need the straightforward π Lemma 11.6. Let a be a function on N Z2N . Then
|d(q)|−2 a(q)dq = N I± a(0) + O(N −α )aα , 2 ±
where I+ =
1 12
(11.80)
and I− = 3I+ = 41 .
The constants I+ , I− follow from the fact that (see (5.1)) 2 ± |d(q)|−2 dq equals, to leading order in N , N times the sum over even or odd non-zero integers n (positive and negative) of π 21n 2 . The even sum equals a quarter of the sum over all integers, and the latter equals 13 . Inserting (11.75) into (11.78), and applying Lemma 11.6 to the result, we obtain 1 β1 τ (0) φ− = 2γ N I− + O(N −α ) . (11.81) 0 β2 ζ (0) T = 2γ T− (1, 0)T − φ− (see (11.74), (11.70)), Recalling that ξ− = κ(0) τ (0), ζ (0) we can write: τ (0) 1 β1 1 τ (0) −α = 2γ T− . (11.82) + κ(0) + O(N ) 2γ N I− ζ (0) 0 ζ (0) 0 β2 Since γ = N −1+α/4 , we get: τ (0) = (N I− )−1 (T− + O(N −α/4 )), ζ (0) = O(N −1−α/4 ).
Combining this with (11.75), (11.76), (11.79), and κ( p)−1 κ(0) = 1 + O(|d( p)|α ) (from Hölder continuity of κ), we get that the simplified problem is solved by T− (T 0 , S 0 ) ≡ (T+ , 0)δ( p) + |d( p)|−2 σ− ( p) (1, 0) + O(|d( p)|α ) + O N −α/4 . N I− (11.83)
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J. Bricmont, A. Kupiainen
In x-space, this is a linear profile: use, for p = 0 the explicit formula (8.4) with j0 replaced by NT−I− = 2(T1N−T2 ) , and observe that, in (8.3), j (x) = j20 for x ∈ [1, N ]; remember also that t ( p) = d(− p)T ( p), and, in x-space, d(− p) is −∇x . So, we get: T 0 (x) = T1 +
|x| (T2 − T1 ) + O(N −α/4 ). N
(11.84)
Moreover, ∇x T 0 (x) = Nτ + o( N1 ), for x away from the boundaries. The first term in (11.84) comes from integrating (11.83) over p, using (11.80) for the second term, and T1 = T+ + T− . Let us now return to the full Eq. (11.65) and incorporate the corrections W± in (11.65) and ψ± given by (11.67). Let T u ± ( p) = t± ( p), s± ( p) and U± ( p) = U0 δ( p) + d(− p)−1 u ± ( p). Write
u ± ( p) = κ( p)−1 d( p)−1 ξ± + v± ( p).
(11.85)
In the absence of W± , ψ± , we have v± = 0 and ξ± are as above. Consider the equation v± ( p) = −W± ( p),
(11.86)
as a fixed point problem for v± ( p). Since the functions W± are Lipschitz in U with constants of order τ , (11.86) has a unique solution v± ( p) = (v1± ( p), v2± ( p)), with vi± ∈ S satisfying bounds like (11.55), that were satisfied by Wi± (see the discussion after (11.65)), and vi± is Lipschitz in U0 , ξ± , with constant O(τ ). We still need to determine U0 and ξ± (in the simplified model, determining ξ− was equivalent to determining (τ (0), ζ (0)), by (11.74)). With the v± obtained here, (11.65) becomes, as in (11.70) and (11.71) and (11.72): ξ+ = 0,
2γ T+ (1, 0)T = φ+ + ψ+ ,
ξ− = 2γ T− (1, 0)T − φ− − ψ− .
(11.87)
Proceeding as in the simple case, φ+ = 2γ (T0 + β1 S0 , β2 S0 )T + (O(γ τ 2 ), O(γ τ λ2 ))T , 1 β1 + O(N −α ) κ −1 (0)ξ− + (O(γ τ 2 ), O(γ τ λ2 ))T, (11.88) φ− = 2γ N I− 0 β2 where the term (O(γ τ 2 ), O(γ τ λ2 ))T collects the contributions from v± that are bounded by (11.55) applied to vi± , and is Lipschitz in T0 , S0 , ξ− . Here, we use Lemma 11.6 to bound the integral of the contribution to φ± coming from the |d( p)|α z 20 term in (11.55) by O(N −α ). Since ψ± is bounded by (11.68), and is Lipschitz in T0 , S0 , ξ− , we can solve Eqs. (11.88) for T0 , S0 , ξ− , by the contraction mapping theorem, and obtain T − T 0 E ≤ Cτ 2 ,
(11.89)
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
S − S 0 E ≤ Cτ λ2 ,
609
(11.90)
which, combined with (11.83), and the fact that W± is Lipschitz in (T, S) with constant Cτ , yields the claim of Proposition 11.1 (b). Finally, let us prove the estimates on R(x, k) for x away from the boundaries (see (4.23)). Since Q = Q 0 +r , we have, see (9.7), R = Q 1 +r . For Q 1 , we use the bound (9.8). To estimate r (x, k) for x away from the boundaries, we need to analyze Eq. (11.35). Observe that D˜ has a leading term at p = 0 which is diagonal, plus a correction O(|d( p)|α ), so that the first term in the RHS of (11.35) gives a contribution to r of the form |d( p)|α f , with f ∈ S, i.e. which is o( N1 ) away from the boundaries (see (9.9)), and the same holds for the contribution to r coming from the first component of R, since RS is bounded, see (11.36). So, we need to consider only the second component of R. Going back to (11.1), we see that R contains the N term, which is o( N1 ) by (11.33), the nonlinear terms which are also o( N1 ) by (11.15) and finally the n 2 (Q 0 ) term which is o( N1 ) away from the boundaries by the remark made at the end of Subsect. 11.1. Proof of (4.25), (4.27). To justify the claims made in (4.25), (4.27), we need to identify the leading contribution in N1 for x away from the boundaries. The currents were defined, by integrating over k J ( 2p , k), times some smooth function, see (7.7), (7.14). J ( p, k) was obtained by applying ((δω2 σ1 + DN (Q 0 )))−1 to (ρ1 t (2 p) + ρ2 s(2 p) + ρ3 , 0)T + R, see (11.35), (11.40). We know that the result belongs to S and so we know its behavior in x from the remark made after Eq. (8.12). To study the behavior for x away from the boundaries, we need to analyze this in more detail for p small. J is a sum of terms like (9.24), Let us consider DN (Q 0 ) for p small: DN (Q 0 ) r ∗n i but with one of the T ∗ A factors replaced by J or r . As we discussed in the remark at the end of Subsect. 9.1, the leading behavior is obtained by replacing each ω( pi , ki − pi ) in (9.24) by ω(ki ), see (9.25). The result, integrated over k with νs pk (d pdk) is C α in p; hence, we may replace it by its value at p = 0, with an error which is of the form |d( p)|α f , with f ∈ S, and then use (9.9). So we are left with a series of integrals (up to permutations of the arguments), of the form:
T ∗ A∗n 1 (2 p1 )T ∗ A∗n 2 (2 p2 ) f (2 p3 , k3 − p3 )δ(2 p4 )G(k, k3 )δ(2( p− pi ))d pdk3 (11.91) with f = J or r and G a function that is C α in k and smooth in k3 . Since f is C α in its arguments, we may replace it by f (2 p3 , k3 ), up to an error of the form | p3 |α g, with g ∈ S (and, for the error, we use again (9.9)). If we perform the integrals over k3 and p and then consider its Fourier transform, we see that (11.91) is an operator proportional to T (x)2 A(x)n 1 +n 2 . Now consider the terms in (ρ1 t (2 p) + ρ2 s(2 p) + ρ3 , 0)T + R to which ((δω2 σ1 + DN (Q 0 )))−1 is applied. Since d(− p) corresponds, in x space, to −∇, ρ1 t (2 p) + ρ2 s(2 p) + ρ3 is, for x away from the boundaries, of the form −∇T (x)h 1 (A(x))a1 (k) − T (x)∇ A(x)h 2 (A(x))a2 (k)
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for smooth functions a1 (k), a2 (k) and with h 1 , h 2 analytic. The R term is o( N1 ) for x away from the boundaries: for the N part of R, we use (11.33), for the nonlinear part we use (11.15), and for the N (Q 0 , 0) term, we use the remark at the end of Subsec. 11.1. Combining all the previous observations, we get that, for x away from the boundaries,
j (x) j (x)
= −κ(A(x))T
−2
(x)
∇T (x) T (x)∇ A(x)
+ o(
1 ), N
where κ(A(x)) is a 2×2 matrix, whose elements are analytic in A(x); this proves (4.25). Finally (4.26) follows trivially from (7.3). For (4.27), we use (7.13) and show first that θ (x) is of the form of the RHS of (4.27), by proceeding as above: If we consider (7.9), we can write N22 ( p, k) as a sum of a N (Q 0 , 0) term, a DN (Q 0 ) term and a nonlinear term. We may then combine the analysis at the end of Subsect. 11.1 for N (Q 0 , 0), the one of DN (Q 0 ) made here and (11.15) to show that θ (x) is of the required form. Finally, going back to (7.8)–(7.16), we see that γ (x) contributes to the o( N1 ) term, since it is a convolution, in x space, between a function localized on the boundary (the Fourier transform of (7.11)) and a rapidly decaying function (the Fourier transform of ψ, which is analytic in its arguments). A. Hölder Regularity In this appendix, we prove some refinements of the Hölder continuity of the kernels proven in Sect. 10. We start with a corollary of Proposition 9.3 that will be needed in Appendix B. For this, let G ∈ C 0 (T3d ) and consider the function
g( p, k) =
G(k)µ p,k (dk)
(A.1)
on T1+d , where µ p,k (dk) is defined by (9.51): µ p,k (dk) =
3
si ω(ki ) + s4 ω( p − k) + i
3 −1 δ( ki − p − k)dk1 dk2 dk3 . (A.2)
1
1
Denote g = I (G).
(A.3)
Then we have Corollary A.1. (a) If G is smooth, I (G) is in C α (T1+d ). (b) Let G(k) = H (k1 , k2 )h(k3 ) with H smooth and h ∈ C 0 (Td ). Then I (G) is in C α (T1+d ) and I (G)α ≤ C(H )h∞ . (c) I is a bounded map from C 0 (T3d ) to C 0 (T1+d ). Furthermore, if G depends smoothly on some parameters so does g.
(A.4)
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611
Proof. By comparing (A.2) and (9.39), we see that, if we replace k3 in (A.2) by k3 + p, we may identify k3 here and k in (9.39). But then, the function G in (a) only changes ρs in (9.39), and the statement (a) can be proven just as the one on M( p, k) in Proposition 9.3. In (b), the function H again affects only ρs , and h is the function on which the operator K p in Proposition 9.3 acts. Hence, (b) follows from the claims made on K p . Finally (c) follows because we can bound the integral I (G) by the sum norm of G and the resulting integral is continuous on T1+d , for the same reason that M( p, k) is continuous. We need one more regularity result for the analysis of the θ in (7.9) that will be made in Appendix C. For this, define the function
4 ki χ (k)dk. (A.5) I ( p) = δ ω(k1 ) + ω(k2 ) − ω(k3 ) − ω(k4 ) δ p − 1
Then Lemma A.2. Let χ in (A.5) be smooth and let I ( p) be the discrete derivative on Z2N . Then, for some α > 0, I α ≤ C
π 2N
(A.6)
uniformly in N . For the proof we need a further lemma: 3 Lemma A.3. Let χ (x) be smooth on T and F(x) = sin(x1 + x2 + x3 ) − sin x1 − sin x2 − sin x3 . Then g( p) ≡ δ(F(x) − p)χ (x)d x is smooth if p = 0 and
|g( p)| ≤ C(log p)2 , log p |. |g ( p)| ≤ C | p Proof of Lemma A.2. In (A.5) write ki = π2 + ki , i = 1, 2 and k3 = − π2 + k3 which imply k4 = p − k1 − k2 − k3 − π2 and the argument of the δ-function equals 3
sin(k1α + k2α + k3α ) − sin k1α − sin k2α − sin k3α + P,
α=2
where P = sin(k1 + k2 + k3 − p) − sin k1 − sin k2 − sin k3 . Thus,
I (p) =
with
dk J P(k, p), k ,
d xd y δ F(x) + F(y) + λ χ (k, x, y)
= dt d x g(t, k, x)δ F(x) + t + λ ,
J (λ, k) =
(A.7)
(A.8)
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where
g(t, k, x) =
δ F(y) − t χ (k, x, y)d y,
which by Lemma A.3. is smooth in all variables, for t = 0, and is bounded by |g| ≤ C(log t)2 . Similarily the x-integral yields
J (λ, k) = dtds h(t, s, k)δ(t + s + λ), with h smooth if t, s = 0 and
log |t| (log |s|)2 , |∂t h| ≤ C t
and similarly for ∂s h. Then one gets |∂λ J | ≤ C(log |λ|)4 .
(A.9)
Consider first the N → ∞ limit of (A.8). Then by (A.9) |I ( p)| ≤ C dk| log P|4 . Since P is an analytic function, | log P|4 is integrable. It is easy to do the argument for the Riemann sum. In the same vein, one can extract a little Hölder continuity for I ( p). We leave the details for the reader. Proof of Lemma A.3. a) We have ∂α F = cos(x1 + x2 + x3 ) − cos xα , so ∇ F = 0 ⇐⇒ cos x1 = cos x2 = cos x3 = cos(x1 + x2 + x3 ) and thus x1 = x2 = −x3 and permutations thereof. At these points F = 0. Hence F(x) = 0 ⇒ ∇ F(x) = 0. Thus, given x ∈ F −1 ( p) there is a neighbourhood U of x and a smooth diffeomorphism: φ p : B (0) → U such that (F ◦ φ p )(y) = p + y1 and φ p is smooth in p. Given ψ ∈ C0∞ (U ),
gψ ( p) ≡ δ(F − p)ψd x = dy2 . . . dyd (ψ ◦ φ −1 p )(0, y2 , . . . , yd ) det Dφ p is smooth. By a partition of unity, we conclude that g is smooth for p = 0. b) The Hessian of F is ∂α ∂β F = δαβ sin xα − sin(x1 + x2 + x3 ). At the critical points x1 = x2 = −x3 it equals ⎛
⎞ 011 H = − sin x1 ⎝ 1 0 1 ⎠ . 112
H has eigenvalues 0, sin x1 , −3 sin x1 . By a partition of unity argument it suffices to study two cases: 1◦ χ has support in a small ball around a critical point with x1 = 0, x1 = π . 2◦ supp χ is a small ball around the origin, around (π, π, −π ) or permutations thereof.
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Case 1◦ By scaling and rotation, there exists a local coordinate z = (u, v, w) such that F = uv + f (u, v, w)
(A.10)
with f analytic, O(z 3 ) and f (0, 0, w) ≡ 0 ≡ Du,v f (0, 0, w). Indeed, set x1 = x + u + w, x2 = x + v + w and x3 = x − w, then F = sin x(cos(u + v + w) − cos(u + w) − cos(v + w) + cos w) + cos x(sin(u + v + w) − sin(u + v) − sin(v + w) + sin w), which, upon scaling, is of the form (A.10). Writing (u, v) ≡ y, f (u, v, w) = (y, A(z)y) with A = O(z). Since uv is non-degenerate, there exist an analytic diffeomorphism g close to identity such that F ◦ g = uv. Hence
g( p) = δ(uv − p)χ˜ (u, v, w)dudvdw, ˜ then, with χ˜ ∈ C0∞ (B (0)). Let φ = χdw,
∞ g( p) = − du log u∂u (φ(u, p/u) + φ(−u, − p/u)) 0
∞
∞ du∂u (ψ( pu, 1/u) − du log u∂u (ψ( pu, 1/u)), = − log | p| 0
0
with ψ(u, v) = φ(u, v) + φ(−u, −v). Thus g( p) = a( p) log | p| + b( p) with a and b smooth. The claim follows. Case 2◦ We may suppose x = 0, the other points being similar. We have: F = s1 c2 c3 + c1 s2 c3 + c1 c2 s3 − s1 s2 s3 − s1 − s2 − s3 , where si = sin xi , ci = cos xi . The diffeomorphism near the origin yi = sin xi leads to F = (y1 + y2 )(y2 + y3 )(y1 + y3 ) + O(y 5 ), and letting z 1 = y1 + y2 , z 2 = y2 + y3 , z 3 = y1 + y3 , we get: F = z 1 z 2 z 3 + f (z), with f analytic in B (0), f = O(z 5 ), and symmetric under permutations of coordinates. We want to bound
h( p) = δ(F(z) − p)ψ(z)dz, (A.11) for ψ ∈ C0∞ (B (0)) and small enough. By symmetry we may insert into (A.11) 6χ (|z 1 | ≤ |z 2 | ≤ |z 3 |). Expand f (z) = f 0 (z 2 , z 3 ) + f 1 (z 2 , z 3 )z 1 + f 2 (z)z 12 .
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Since f 1 = O(z 2 , z 3 )4 , by a diffeomorphism φ close to identity we have (z 2 z 3 + f 1 )◦φ = z 2 z 3 , i.e. may assume f 1 = 0 and bound χ from above by χ (|z 1 | ≤ 2|z 2 | ≤ 3|z 3 |): ˜ · χ |z 1 | ≤ 2|z 2 | ≤ 3|z 3 | dz, h( p) ≤ δ z 1 z 2 z 3 + f 0 (z 2 , z 3 ) + f 2 (z)z 12 − p ψ(z) (A.12) with ψ˜ ∈ C0∞ (B (0)). On the support of χ and ψ˜ | f 2 | ≤ C|z 3 |3 ≤ C 2 |z 3 | and thus |z 2 z 3 + f 2 z 1 | ≥ (1−O( 2 )|z 2 z 3 |, since |z 1 | ≤ 2|z 2 |. So, given z 2 , z 3 and p, the argument of the δ function in (A.12) either is nonzero for all z 1 ∈ B (0) or vanishes at z 1 (z 2 , z 3 , p) satisfying p − f0 1 p − f 0 . ≤ |z 1 | ≤ 2 2 z2 z3 z2 z3 Moreover z 2 z 3 + ∂1 ( f 2 z 12 ) ≥ 1 − O( 2 ) |z 2 z 3 |. Thus
p − f 0 −1 ˜ 1 , z 2 , z 3 )dz 2 dz 3 . < C ψ(z h( p) ≤ C |z 2 z 3 | χ |z 2 | ≤ 2|z 3 | χ z2 z3 (A.13) Since f 0 (z 2 , z 3 ) = f 0 (0, z 3 ) + z 2 f 3 (z 2 , z 3 ), where | f 3 | ≤ C z 34 ≤ C 3 |z 3 |, the second factor χ in (A.13) is bounded by p − f 0 (0, z 3 ) < C . χ z2 z3 Now, f 0 (0, z 3 ) is analytic, so
f 0 (0, z 3 ) = az 3n 1 + O(z 3 ) ,
for some a = 0, n < ∞ (actually n = 5). Insert this into (A.13), write χ = χ p (z 2 ) + χ − χ p (z 2 ) , with n−1 χ p (z 2 ) = χ |z 2 | > C| p| n . Then h = h 1 + h 2 with
dz dz 3 1 2 χ (z 2 ) χ |z 3 | > |z 2 | χ p z ∈ B (0) h 1 ( p) ≤ |z 2 | |z 3 | 2 ≤ C(log p)2 .
For |z 2 | ≤ C| p|
n−1 n
3) < C , , we note that, since, in the support of χ p− zf02(0,z z3 | p − az 3n | ≤ C|z 2 z 3 |,
we may write z 3 = ( p/a)1/n + x, with p n−1 n |x| ≤ C|z 2 | | p|1/n , a
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615
where a is absorbed into C in the RHS, or |x| ≤ C| p|−
n−2 n
|z 2 |.
So, doing the z 3 integral, we get:
n−1 n−2 dz 2 χ |z 2 | < C | p| n | p|−1/n |z 2 | | p|− n h 2 ( p) ≤ C |z 2 | ≤ C. We conclude that |g( p)| ≤ C(log p)2 , i.e. the claim holds for g. The claim for g is similar: ∂ p brings an extra |z 2 z 3 |−1 and
C log p dz 2 dz 3 . χ |z z | > ( p) ≤ 2 3 (z 2 z 3 )2 p Remark. I ( p) is not smooth for p = 0. By some algebra one can show ∇k P = 0, P = 0 ⇐⇒ k1 = k2 = k3 , p = 2k1 , and permutations. Thus zeros of P and ∇k P occur for nonzero p too. B. Nonlinear Estimates Here we prove the estimates on the nonlinear terms that were made in Sect. 9. They are based mostly on an analysis of convolutions of functions in E or S, whose results were stated in Lemma 9.1. We start with the proof of this lemma. Proof of Lemma 9.1. Recall first that j = j+ σ+ + j+ σ+ and T similarly. Let T ∗ j = j . Then j+ = T+ ∗ j+ + T− ∗ j− , j− = T+ ∗ j− + T− ∗ j+ . Consider eg. T+ ∗ j+ and drop the +. Recall that j is defined through the 4-tuple j in (8.13). We need to define the 4-tuple j . We set j0 = 2 (T ∗ j)(0), 1
where the factor
1 2
(B.1)
follows from our conventions (8.13) and (6.5). Let, for p = 0,
j3
j1 = T ∗ j1 , j2 = T ∗ j2 ,
(N d)3/2
=T∗
j3 (N d)3/2
(B.2) (B.3) +T ∗
j1 Nd
−
1 (T Nd
∗ j1 ).
Let us estimate these in turn, using two simple observations:
f ( p − p )g( p )dp α ≤ f α g L 1 ,
(B.4)
(B.5)
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where − L 1 is the L 1 norm, and
f ( p − p )d( p )−1 dp α ≤ f α
(B.6)
which holds because the Hilbert transform of a Hölder continuous function is Hölder continuous (for α < 1, see [33], Theorem 106), and this is true also when p −1 is replaced by d( p)−1 , and when we have discrete sums as here. Then, using T L 1 ≤ CT E , (see (11.27)), and (B.5), we get: j1 α ≤ CT E j1 α , j2 α ≤ CT E j2 α .
(B.7) (B.8)
d( p) = d( p ) + d( p − p ) + d( p )d( p − p ),
(B.9)
For j3 , we use the identity
to write: T∗
j1 Nd
−
1 (T Nd
∗ j1 ) ( p) =
j N d(T ∗ N1d ) − (T ∗ j1 ) ( p)
1 = Nd j1 ( p − p )(d( p − p )−1 d( p )T ( p )dp
(B.10) + j1 ( p − p )d( p )T ( p )dp . 1 Nd
Hence, using (11.28), (B.10) is bounded by
1 CT E j S (N | p|)−1 ( N | p | + N −1+α/2 +
1 )(1 + | p (N | p |)3/2
− p |)−1 )dp .(B.11)
The integral is bounded by C log |N p|( N | p| + N −1+α/2 ). 1
Combining with (B.11) we arrive at the bound |(B.10)| ≤ CT E j S
1 . |N d( p)|3/2
(B.12)
Similar calculations give the bound |T ∗
1 | |N d|3/2
≤ T E
1 . |N d|3/2
Combining with (B.12) we get j3 ( p)∞ ≤ CT E j S .
(B.13)
|T ( p) j ( p)|dp ≤ CT E j S
(B.14)
Finally (B.1) is bounded by
1 2
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
using (11.29), e.g.
617
(N 2 | p|3 )−1 dp ≤ C.
Estimates (B.7) , (B.8), (B.13) and (B.14) give the claim (9.13). Let us next consider part (c). We set j0 = 2 ( j ∗ k)(0) 1
and, for p = 0, j1 = N d j3
j1 Nd
∗
k1 Nd
+ H1 +
(B.15)
1 N
( j0 k1 + k0 j1 ),
j2 = j ∗ k2 + k ∗ j2 − N −1+α/2 j2 ∗ k2 ,
(N d)3/2
=
j3 (N d)3/2
∗
k3 (N d)3/2
+ H2 +
1 N
(B.16) (B.17)
( j0 k3 + k0 j3 ),
(B.18)
where we write Nj1d ∗ (N d)k33/2 + Nk1d ∗ (N d)j33/2 = H1 + H2 , and the Hi ’s are defined after (B.23) below. To proceed, we need the following bounds: using (B.5), (B.6), we get that,
f ( p − p )g( p )d( p )−1 dp , (B.19) h 1 ( p) ≡ satisfies h 1 α ≤ C f α gα .
(B.20)
This holds because we can write in (B.19) g( p ) = g( p ) − g(0) + g(0); since g is in C α , |g( p ) − g(0)||d( p )−1 | ≤ gα | p |−1+α , i.e. (g( p ) − g(0))d( p )−1 is integrable; so (B.20) follows from (B.6) applied to the g(0) term above and (B.5) to the g( p ) − g(0) term. Using (B.9), we then get that
f ( p − p )d( p − p )−1 g( p )d( p )−1 dp (B.21) h 2 ( p) ≡ d( p) also satisfies h 2 α ≤ C f α gα .
(B.22)
Now, consider (B.16). The bound (B.22) applied to the first term implies that its C α norm is O(N −1 ) j1 α k1 α . Now, use (B.9) to write j
N d( N1d ∗
k3 ) (N d)3/2
= j1 ∗
k3 (N d)3/2
+
j1 Nd
∗
k3 (N d)1/2
+
j1 N
∗
k3 . (N d)1/2
(B.23)
Let the sum of the first and third term in (B.23), plus the corresponding terms in N d( Nk1d ∗ j3 ) define N d H1 and let H2 = (N d)−1 ( Nj1d ∗ (N d)k31/2 ) plus the corresponding terms (N d)3/2 in
k1 Nd
∗
j3 (N d)3/2
. Since (N d)k33/2 L 1 ≤ C N −1 k3 ∞ , we get from (B.5) that j1 ∗
k3 (N d)3/2 α
≤ O(N −1 ) j1 α k3 ∞ ,
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which controls the contribution of the first term in (B.23) to N d H1 . This bound holds also for the last term in (B.23), of course (which is even O(N −3/2 ) j1 α k3 ∞ ). The terms N1 ( j0 k1 + k0 j1 ) are trivially bounded, so we get: j1 α ≤ C N −1 j S k S .
(B.24)
Using (B.5) and the fact that (Nj1d) L 1 ≤ N −1 log N j1 ∞ for all terms in (B.17), we get j2 α ≤ C N −1+α/2 .
(B.25)
Consider now (B.18). Going back to the discrete sum (see (5.1)), it is easy to see that the first term is bounded by C N −1 |N p|−3/2 j3 ∞ k3 ∞ , i.e. that the contribution of this term to | j3 ( p)| is less than C N −1 j S k S . Consider now (N d)−1 ( Nj1d ∗ k3 ), contributing to H2 . Going back to the discrete sum, it easy to bound it by (N d)1/2
C N −1 |N p|−3/2 log |N p|. Finally, the bound on
1 N
( j0 k3 + k0 j3 ) is trivial and we get:
| j3 ( p)| ≤ C N −1 log(N | p|) j S k S .
(B.26)
Finally, using repeatedly (11.29), we get | j0 | ≤ C N −1 j S k S .
(B.27)
Combining (B.24), (B.25), (B.26), (B.27), proves part c) of the lemma, since the bound on j S follows from the previous ones. Let us finally turn to (9.14). Note that, using (B.9), d(T ∗ A) = dT ∗ A + T ∗ d A + dT ∗ d A Since dT ∈ S if T ∈ E the claim, for p = 0, follows from (a) and (c). For p = 0, we simply apply (11.29) to all the terms in (T ∗ A)(0). Using this lemma, it is rather easy to give the proof of the main estimate of Sect. 9. Proof of Proposition 9.4. (a) Apply (a) of Corollary A.1 to (9.49) and (9.50) to get
m( p, k) =
g( p, k, p)
3 i=1
Fi (2 pi )δ(2 p −
3
2 pi )dp,
(B.28)
1
where g is smooth in p and C α in p, k. m may now be defined as an element of E as in Lemma 9.1.b, the smooth function not affecting the bounds. (b) If m = u or if m = v and i = 3 (in which case i = 1, 2, and, by symmetry, we can choose i = 1) we have m( p, k) =
3 i=2
Fi (2 pi )(
G f 1 ( p1 , k1 − p1 )µ p,k (dk))δ(2 p −
3
2 pi )dp
1
with G smooth, and f 1 in S. Using the representation (8.11) for f 1 we write G f 1 ( p1 , k1 − p1 ) as a sum of terms, with singularities in p1 ; by Corollary A.1.b, we get that integrating each of those terms with µ p,k (dk) gives rise to a function that is C α in p, k. Thus,
(B.29) G f 1 ( p1 , k1 − p1 )µ p,k (dk) = f ( p1 , k; p, p2 , p3 ),
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where f is in S in the variables p1 , k depending smoothly on p2 , p3 and C α in p. The convolutions with the F j ’s can then be estimated as in Lemma 9.1. If m = v and i = 3 we need to study
2
Fi (2 pi ) f ( p3 , p − k − p3 )(
Gµ p,k (dk))δ(2 p −
3
2 pi )dp,
1
i=1
where f ∈ S. Shifting p3 by p, this becomes
g( p, k, p)
2
Fi (2 pi )δ(
3
2 pi ) f 3 ( p + p3 , −k − p3 )dp
1
i=1
here g, defined in (A.1) is, by Corollary A.1.a, C α in p, k and smooth in p. Performing the pi integrals for i = 1, 2, we get
1 1 F( p, k, p3 ) f ( p + 2 p3 , −k − 2 p3 )d p3 , (B.30) where F is C α in the first two arguments and belongs to E as a function of the third. We may proceed now as in Lemma 9.1.(a) to define and estimate the quadruple g in S corresponding to (B.30). E.g. the component of index 1, see (B.2), is given by
1 1 g1 ( p, k) = F( p, k, p3 ) f 1 ( p + 2 p3 , −k − 2 p3 )dp3 , (B.31) where f 1 is C α in both arguments. Again, since F is integrable in the third argument and C α in the others the integral is C α in p, k. The other components of g can be bounded as in Lemma 9.1.a. (c) We have to specify again the quadruple m ∈ S corresponding to m. We define m i = 0 and m 3 ( p, k) to be the integral (9.49) or (9.50) corresponding to m. We use Corollary A.1.c to do the ki integrals and estimate the pi integrals by brute force. Since only the singularity at pi = 0 (or, by periodicity at π ) matters, we need the easy bounds (remember that the variable p is discrete!) |N d( p)|−1 ∗ |N d( p)|−1 ≤ C N
− 21
log N |N d( p)|−3/2 ,
N −1+α/2 ∗ |N d( p)|−1 ≤ C log N N −2+α/2 ≤ C N N −1+α/2 ∗ N −1+α/2 ≤ C N
− 12 +α
− 21 +α/2
log N |N d( p)|−3/2 ,
|N d( p)|−3/2 ,
together with |N d( p)|−3/2 ≤ |N d( p)|−1 (since N | p| ≥ π ), which allows to use the first inequality here in order to bound the convolutions with the last term in (8.11), to conclude − 1 +α fl S m( p, k)∞ ≤ C N 2 f k E l=k
which is the claim (9.54) since we defined m = (0, 0, 0, m).
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J. Bricmont, A. Kupiainen
C. Estimates on θ Finally, we prove the estimates on the function θ defined in (7.9). Proof of Proposition 9.7. By the (3 ↔ 4) symmetry in (5.13), we get (leaving out the factor 94 (2π )3d λ2 ), θ ( p) = i
3 s
p ρ 2,
Wsi ( pi , ki − pi )s3 s4 ω(k3 )ω(k4 )−1
1 p 2
p − k3 − ρ 2 ,
p 2
− k4
νs pk (dp dk)
(C.1)
and νs pk =
si ω(ki ) + i
−1
δ(2 p4 )δ( p − 2
i
pi )δ( p −
i
ki )d pdk.
i
Then the [ ] in (C.1) equals 1 1 − + (ei p − 1) r ( p, k) 2 ω(k3 ) ω(k4 )
(C.2)
with r smooth. The integral in (C.1) has singularities when si ω(ki ) = 0.
(C.3)
Recall that (C.3) forces
si = 0,
(C.4)
see (7.20). Consider the s such that (C.4) holds in (C.1), and, replace [ ] in (C.1) by the first term of (C.2). We define θ1 ( p) ≡ 2i
3
si =0
1
Wsi
s3 s4 s3 ω(k3 )s4 − ω(k4 ) ω(k4 )2
· νs pk (dp dk).
(C.5)
3 By symmetry, we may replace s3 by 13 i=1 si and, by (C.4) also by − 13 s4 . Again, 3 si ω(ki ). So, the parenthesis equals by symmetry, s3 ω(k3 ) may be replaced by 13 i=1 −1 −s4 4 s ω(k ), and the sum cancels the factor s ω(k ) + i in ν . Hence, i i i i 2 i=1 i 3ω(k4 ) (C.5) equals
3 s4 2i Wsi δ(2 p )δ( p − 2 p )δ( p − ki )dp dk. θ1 ( p) = − 4 i 3 ω(k4 )2 si =0
1
i
i
(C.6) We decompose θ as θ ( p) = θ1 ( p) + θ2 ( p) + pθ3 ( p),
(C.7)
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
621
where θ2 has the terms of (C.1) with si = 0 and the first term of (C.2) inserted, while θ3 corresponds to the insertion of the second term of (C.2). Consider first θ1 given by (C.6). Remember that Ws ( p, k − p) = Q( p, k − p) + isω(k)−1 J ( p, k − p). The terms with an odd number of Q factors vanish by the s → −s symmetry. Consider then the term linear in J . We insert Q = Q 0 + r and start with the term with no r . After shifting the ki variables by 2 pi (and using 2 p4 = 0), we obtain a sum of terms of the form
T ∗ A∗n 1 (2 p1 )T ∗ A∗n 2 (2 p2 )J ( p3 , k3 + p3 )
ω(k4 )
−2
ω(k3 + 2 p3 )
−1
δ(2 p4 )δ p − 2
2 −2−n i ω(ki + 2 pi ) + ω(ki ) · i=1
pi δ ki dk dp.
(C.8)
Now, use the fact that ω(ki + 2 pi ) − ω(ki ) = (e2i p − 1)O(1), which implies −2−n i −2−n i ω(ki + 2 pi ) + ω(ki ) = 2ω(ki ) + n i (e2i pi − 1)O(1).
(C.9)
For the second term on the RHS of (C.9), let us choose i = 1, which we can do by symmetry, and insert it in (C.8), to obtain, after integrating over k1 , k2 , k4 :
I1 ( p) = n 1 (e2i p1 − 1)T ∗ A∗n 1 (2 p1 )T ∗ A∗n 2 (2 p2 )J ( p3 , k3 + p3 ) · f ( p, k3 )δ(2 p4 )δ p − 2 pi dpdk3 , (C.10) with f smooth. For the first term on the RHS of (C.9), we obtain:
I˜1 ( p) = T ∗ A∗n 1 (2 p1 )T ∗ A∗n 2 (2 p2 )J ( p3 , k3 + p3 ) · f˜( p, k3 )δ(2 p4 )δ p − 2 pi dpdk3 ,
(C.11)
with f˜ smooth and even in k3 . Doing the k3 -integral, we get, for (C.10),
J ( p3 , k3 + p3 ) f ( p, k3 )dk3 = g(2 p3 , 2 p),
(C.12)
where we used the π -periodicity of the result. g is in S as a function of p3 , depending smoothly on p, with g S ≤ CJ S . For (C.11), we get, since J is odd in k3 and f˜ even, that the integral vanishes if J ( p3 , k3 + p3 ) is replaced by J ( p3 , k3 ), and thus, since f˜ is smooth, the integral can be written as:
˜ J ( p3 , k3 ) f˜( p, k3 − p3 )dk3 J ( p3 , k3 + p3 ) f ( p, k3 )dk3 = = (e2i p3 − 1)g(2 ˜ p3 , 2 p),
(C.13)
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J. Bricmont, A. Kupiainen
using again the π -periodicity of the result. g˜ is in S as a function of p3 , depending smoothly on p, with g ˜ S ≤ CJ S . We write, using the constraints δ(2 p4 ), δ p − 2 pi , e2i p3 − 1 = (ei p − 1 + 1)(e−2i p1 − 1 + 1)(e−2i p2 − 1 + 1) − 1.
(C.14)
Expanding the product we see that the integral (C.11) equals the sum of terms of the form I1 and of the form:
˜ p3 , 2 p)δ(2 p4 )δ p−2 pi dp dk3 . I2 = d( p) T ∗ A∗n 1 (2 p1 )T ∗ A∗n 2 (2 p2 )g(2 (C.15) The integral in (C.15) is a convolution of two functions in E with one in S, hence, by Lemma 9.1, it is in S. The prefactor d( p) cancels the d −1 , so that this contribution satisfies d −1 I2 S ≤ (Cδn 1 +n 2 ,0 t S + (CA E )n 1 +n 2 )J S , and also I2 (0) = 0. Going back to I1 , see (C.10), (C.12), we obtain:
I1 ( p) = n 1 pi dp f ( p1 )F( p2 )g( p3 , p)δ(2 p4 )δ p −
(C.16)
(C.17)
with f ∈ S and F ∈ E. So, we have a convolution of two elements of S and one of E, i.e. the convolution of two elements of S. Going back to the definition (8.11), we see that we can write I1 = I1 + I1 + I1
(C.18)
corresponding to the ji , i = 1, 2, 3 terms in the convolution of two elements of S that are bounded in part c) of Lemma 9.1. From that lemma, we get: |I1 ( p)| ≤
1 (Cδn 1 +n 2 ,0 t S + (CA E )n 1 +n 2 )J S . N 2 |d( p)|
(C.19)
If we identify d( p)−1 I1 with an element of S of the form (0, 0, 0, ), we get from (C.19) and (N |d( p)|)−1/2 ≤ C, d −1 I1 S ≤ (Cδn 1 +n 2 ,0 t S + (CA E )n 1 +n 2 )J S .
(C.20)
Next, we get, also from Lemma 9.1.c, together with the definition (8.11): I1 α ≤ N −2+α (Cδn 1 +n 2 ,0 t S + (CA E )n 1 +n 2 )J S .
(C.21)
Now, identify d( p)−1 I1 with an element of S of the form (0, , 0, 0), we get, writing d( p)−1 I1 = (N d( p))−1 N I1 , that d −1 I1 S ≤ N −1+α (Cδn 1 +n 2 ,0 t S + (CA E )n 1 +n 2 )J S .
(C.22)
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For I1 , we use (9.18), and identify d( p)−1 I1 with an element of S of the form (0, 0, 0, ). Since (N |d( p)|)−1 log(N | p|) ≤ C, we get d −1 I1 S ≤ (Cδn 1 +n 2 ,0 t S + (CA E )n 1 +n 2 )J S .
(C.23)
Combining these estimates, we get: d −1 I1 S ≤ (Cδn 1 +n 2 ,0 t S + (CA E )n 1 +n 2 )J S .
(C.24)
We also get, by (9.15): |I1 (0)| ≤ C N −1 .
(C.25)
The terms with one J and one r are sums of terms of the form:
−2−n 1 T ∗ A∗n 1 (2 p1 )r ( p2 , k2 − p2 )J ( p3 , k3 − p3 ) ω(2 p1 − k1 ) + ω(k1 ) ki dp dk. ω(k3 )−1 ω(k4 )−2 δ(2 p4 )δ p − 2 pi δ p − Doing the k1 and k4 integrals, and shifting k2 , k3 , this equals
I3 = T ∗ A∗n 1 (2 p1 )r ( p2 , k2 )J ( p3 , k3 ) f ( p, k2 , k3 , p)δ(2 p4 ) ×δ p − pi dp dk2 dk3
(C.26)
with f smooth and even in p. Note that, if we replace here f ( p, k2 , k3 , p) by f ( p, k2 , k3 , 0), the integral vanishes, because it is odd in p (T , A, r , are even and J is odd), while θ is even in p (every term in (5.19) is even in p). So, we may write f ( p, k2 , k3 , p) = pg( p, k2 , k3 , p), with g smooth. The factor p cancels the d −1 factor, and the remaining integral, being a convolution of T ∗ A∗n 1 ∈ E with r, J ∈ S, can be bounded using (9.54): 1
d −1 I3 S ≤ (CA E )n 1 N − 2 +α r S J S .
(C.27)
I3 (0) = 0.
(C.28)
We also have:
In a similar way, we may analyze θ2 , i.e. (C.1) with si = 0 and the first term in (C.2) inserted. We note that the terms that are odd in Q vanish since, for those terms, because of the s → −s symmetry, the measure is proportional to δ si ω(ki ) , which vanishes for si = 0. Starting again with the term linear in J and with r = 0, it is given by
T ∗ A∗n 1 (2 p1 )T ∗ A∗n 2 (2 p2 )J ( p3 , k3 )h( p3 +k3 , k3 )δ(2 p4 )δ p−2 pi dp dk3 (C.29)
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J. Bricmont, A. Kupiainen
with h( p, k) = h(− p, −k) smooth. By oddness of J in k, we may replace h by h(k3 + p3 , k3 ) − h(k3 − p3 , k3 ) i.e., near p3 = 0, h is O( p3 ). Similarly, using J ( p + π, k + π ) = J ( p, k) and the 2π -periodicity of h,
J ( p3 , k3 )h( p3 + k3 )dk3
1 = J ( p3 − π, k3 ) [h(k3 + p3 − π, k3 + π ) −h(k3 − ( p3 − π ), k3 + π )] , 2 i.e. (C.29) may be written as
˜ p3 , k3 )δ(2 p4 ) T ∗ A∗n 1 (2 p1 )T ∗ A∗n 2 (2 p2 )(e2i p3 − 1)J ( p3 , k3 )h( ×δ p − 2 pi dp dk3 . (C.30) Writing e2i p3 − 1 as in (C.14), and expanding the product, we see that the integral (C.30) equals the sum of terms of the form I1 and of the form I2 , i.e.
˜ p3 , k3 )δ(2 p4 ) I˜2 = d( p) T ∗ A∗n 1 (2 p1 )T ∗ A∗n 2 (2 p2 )J ( p3 , k3 )h( ×δ p − 2 pi dp dk3 (C.31) with h˜ smooth. The integral in (C.31) is in S and the prefactor d( p) cancels the d −1 , so that I˜2 has the same bound as in (C.16). Finally, the term with one J and one r is again of the form (C.26). The remaining terms in θ1 and θ2 are of type J 3 and Jr 2 . These are bounded by brute force by log N J (r 2 + J 2 ), N2
(C.32)
and considered as elements of S of the form (0, 0, 0, ). Since |d|−1 ≤ C N we obtain by combining Eqs. (C.24), (C.27), (C.31) and (C.32), 1
d −1 (θ1 + θ2 ) S ≤ CJ S (t S + A E + N − 2 +α (r S + J 2S )).
(C.33)
We still need to estimate pθ3 ( p) in (C.7), i.e. the contribution to (C.1) of the second term in (C.2), which we can write as: ω−1 (k3 − p) − ω−1 (k3 ) − (3 ↔ 4) = p f (k3 , k4 ) + O( p 2 ),
(C.34)
with f odd. Consider first the terms where Wsi = Q 0 , ∀i. We get a sum of terms of the form
3
Fn i (2 pi )ω−2−n i ( pi , ki − pi )s3 s4 ω(k3 )ω(k4 )−1 .
i=1
ω(k3 − p)−1 − ω(k3 )−1 + ω(k4 − p)−1 − ω(k4 )−1 ν (d p dk).
(C.35)
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625
Write: ω−2−n i ( pi , ki − pi ) = ω(ki )−2−n i + (e2i pi − 1)n i O(1). Therefore (C.35) gives rise to two contributions: the one coming from ω(ki )−2−n i ; after inserting (C.34) in the [-] in (C.35), and writing p = −id( p) + O( p 2 ), this contribution can be written as: 1 d( p)(Fn 1 ∗ Fn 2 ∗ Fn 3 )( p) I ( p) − I (− p) + O( p 2 )(Fn 1 ∗ Fn 2 ∗ Fn 3 )( p), 2 with Fn = T ∗ A∗n ,
I ( p) =
δ ω(k1 ) + ω(k2 ) − ω(k3 ) − ω(k4 ) δ
p−
4
(C.36)
ki φ(k)dk, (C.37)
1
where φ is smooth, and I is odd (since f (k3 , k4 ) above, and hence φ, is odd). By Lemma 9.1, (Fn 1 ∗ Fn 2 ∗ Fn 3 )( p) is in E, so d( p)(Fn 1 ∗ Fn 2 ∗ Fn 3 )( p) is in S. By Lemma A.2., the first term in the RHS of (C.36) multiplied by d −1 is in S and so is the last one, since O( p 2 )(Fn 1 ∗ Fn 2 ∗ Fn 3 )( p) is N −1 times a C α function. The second contribution, coming from (e2i pi − 1)n i O(1), is of the form: d( p)n i
3
Fn j (2 p j )(e2i p1 − 1)ψ( p, k, p)ν (d p dk),
(C.38)
j=1
where ψ is smooth and i = 1, 2, 3. The integral in (C.38) is in S with norm bounded by (CA E ) n i . The other terms in θ3 are simpler to bound: as before, we get a factor p that controls the factor d −1 ( p) and the remaining integrals belong to S by Proposi1 tion 9.4, the nonlinear terms in J , r having a factor N − 2 +α in their upper bound. We get: 1
d −1 θ3 S ≤ C(t S + A E )(1 + J S + N − 2 +α (r S J S + J 3S )). (C.39) Of course, pθ3 ( p) vanishes at p = 0. Equations (C.33) and (C.39) together with (C.15), (C.25), (C.28), yield the bound (9.56), remembering that T E , A E are O(1), while r S , J S are O(τ λ−2 ), which is smaller than N α . The statements on Lipschitz bounds follow easily from the above bounds and this last observation. Acknowledgements. We thank Joel Lebowitz, Raphaël Lefevere, Jani Lukkarinen, Alain Schenkel and Herbert Spohn for useful discussions. A.K. thanks the Academy of Finland for funding.
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5. Basile, G., Bernardin, C., Olla, S.: Thermal conductivity for a momentum conserving model. http://arxiv.org/abs/cond-mat/0601544, 2006 6. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier Law: A challenge to Theorists. In: Mathematical Physics 2000, London: Imp. Coll. Press, 2000, pp. 128–150 7. Bonetto, F., Lebowitz, J.L., Lukkarinen, J.: Fourier’s Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs. J. Stat. Phys. 116, 783–813 (2004) 8. Cercignani, C., Kremer, G.M.: On relativistic collisional invariants. J. Stat. Phys. 96, 439–445 (1999) 9. Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201, 657–697 (1999) 10. Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Entropy production in non-linear, thermally driven Hamiltonian systems. J. Stat. Phys. 95, 305–331 (1999) 11. Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212, 105–164 (2000) 12. Eckmann, J.-P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235, 233– 253 (2003) 13. Eckmann, J.-P.: Non-equilibrium steady states. In: Proceedings of the International Congress of Mathematicians, Beijing, Vol. III, Beijing: Higher Education Press, 2002, pp. 409–418 14. Eckmann, J.-P., Young, L.-S.: Temperature profiles in Hamiltonian heat conduction. Europhys. Lett. 68, 790–796 (2004) 15. Eckmann, J.-P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1-D models. To appear in Commun. Math Phys 16. Esposito, R., Pulvirenti, M.: From particles to fluids. In: Friedlander S., Serre D. (eds) Handbook of Mathematical Fluid Dynamics, Vol. III, Amsterdam: Elsevier Science, 2004 17. Galves, A., Kipnis, C., Marchioro, C., Presutti, E.: Nonequilibrium measures which exhibit a temperature gradient; study of a model. Commun. Math. Phys. 81, 127–147 (1981) 18. Golse, F., Sant-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161 (2004) 19. Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with high degree potential. Arch. Rat. Mech. Anal. 171, 151, 218 (2004) 20. Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27, 65– 74 (1982) 21. Lefevere, R., Schenkel, A.: Normal heat conductivity in a strongly pinned chain of anharmonic oscillators. J. Stat. Mech., L02001 (2006), available on: http://www.iop.org/EJ/toc/1742-5468/2006/02, 2006 22. Lepri, S., Livi, R., Politi, A.: Thermal conductivity in classical low-dimensional lattices. Phys. Reports 377, 1–80 (2003) 23. Lepri, S., Livi, R., Politi, A.: On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett. 43, 271 (1998) 24. Narayan, O., Ramaswamy, S.: Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys. Rev. Lett. 89, 200601 (2002) 25. Pereverzev, A.: Fermi-Pasta-Ulam β lattice: Peierls equation and anomalous heat conductivity. Phys. Rev. E. 68, 056124 (2003) 26. Rey-Bellet, L., Thomas, L.E.: Asymptotic behavior of thermal non-equilibrium steady states for a driven chain of anharmonic oscillators. Commun. Math. Phys. 215, 1–24 (2000) 27. Rey-Bellet, L., Thomas, L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225, 305–329 (2002) 28. Rey-Bellet, L.: Nonequilibrium statistical mechanics of open classical systems. In: XIVTH International Congress on Mathematical Physics, edited by Jean-Claude Zambrini, Singapore: World Scientific, 2006 29. Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of a harmonic crystal in a stationary non-equilibrium state. J. Math. Phys. 8, 1073–1085 (1967) 30. Spohn, H., Lebowitz, J.L.: Stationary non-equilibrium states of infinite harmonic systems. Commun. Math. Phys. 54, 97–120 (1977) 31. Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124, 1041–1104 (2006) 32. Spohn, H.: Collisional invariants for the phonon Boltzmann equation. J. Stat. Phys. 124, 1131– 1135 (2006) 33. Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Oxford: Clarendon Press, 1948 34. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D., (eds.) Handbook of Mathematical Fluid Dynamics, Vol. I. Amsterdam: Elsevier Science, 2002 Communicated by H. Spohn
Commun. Math. Phys. 274, 627–658 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0295-2
Communications in
Mathematical Physics
Coisotropic Deformations of Associative Algebras and Dispersionless Integrable Hierarchies B. G. Konopelchenko1 , F. Magri2 1 Dipartimento di Fisica, Universita di Lecce and Sezione INFN, 73100 Lecce, Italy.
E-mail: [email protected]
2 Dipartimento di Matematica ed Applicazioni, Universita di Milano Bicocca, 20126 Milano, Italy
Received: 4 July 2006 / Accepted: 9 January 2007 Published online: 6 July 2007 – © Springer-Verlag 2007
Abstract: The paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies, like the dispersionless KP and modified KP hierarchies and the universal Whitham hierarchy of genus zero. It stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras. It discusses the link between the structure constants and Hirota’s tau function, and shows that the dispersionless Hirota bilinear equations are, within this approach, a way of writing the associativity conditions for the structure constants in terms of the tau function. It also suggests a simple interpretation of the algebro-geometric construction of the universal Whitham equations of genus zero due to Krichever.
1. Introduction The purpose of this paper is to introduce the concept of “coisotropic deformations” of associative algebras and to show its relevance for the theory of integrable hierarchies of dispersionless PDE’s. The concept of coisotropic deformation originates from a melting of ideas borrowed from commutative algebra and Hamiltonian mechanics. From commutative algebra comes the idea of structure constants Ckl j defining the table of multiplication p j pk =
n
C ljk pl
(1)
l=1
of a commutative associative algebra with unity. They obey the commutativity conditions C ljk = Ckl j
(2)
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B. G. Konopelchenko, F. Magri
and the associativity conditions n
p
C ljk Clm =
l=1
n
p
l Cmk Cl j .
(3)
l=1
Furthermore, if the algebra is infinite-dimensional, they are assumed to vanish for l sufficiently large, so that the sums are always over a finite number of terms. From deformation theory comes the idea of regarding the structure constants Ckl j as functions of a certain number of deformation parameters x j . From Hamiltonian mechanics come the ideas of introducing a deformation parameter x j for each generator p j of the algebra, and of considering the pairs (x j , p j ) as pairs of conjugate canonical variables. In this frame, the characteristic trait of the theory of coisotropic deformations is to associate with the structure constants C ljk (x1 , x2 , . . . , xn ) the set of quadratic Hamiltonians f jk = − p j pk +
n
C ljk (x1 , x2 , . . . , xn ) pl ,
(4)
l=1
the polynomial ideal J = < f jk >
(5)
generated by these Hamiltonians, and the submanifold = {(x j , p j ) ∈ R 2n | f jk = 0},
(6)
where these Hamiltonians vanish. It lives in R 2n endowed with the canonical Poisson bracket i=n ∂ f ∂g ∂g ∂ f . − { f, g} = ∂ xi ∂ pi ∂ xi ∂ pi i=1
Definition 1. If the ideal J is closed with respect to the Poisson bracket {J, J } ⊂ J,
(7)
so that is a coisotropic submanifold of R 2n , the functions C ljk of the deformation parameters x j are said to define a coisotropic deformation of the associative algebra. Owing to the identity { f jk , flr } =
n
[C, C]mjklr pm +
m=1 n
−
m=1
n n ∂Clrm ∂Clrm f jm + f km ∂ xk ∂x j
m=1
∂C mjk ∂ xr
flm −
m=1
n ∂C m jk m=1
∂ xl
fr m ,
(8)
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defining the Schouten-type bracket n s ∂C sjk ∂C s m m ∂Clr m − Csr Csmj lr + Csk [C, C] jklr = ∂ xk ∂x j ∂ xl s=1
∂C sjk m
− Csl
∂ xr
m ∂C lr s + Cs − C , ∂ xs lr ∂ xs jk ∂C mjk
(9)
one readily sees that the structure constants define a coisotropic deformation if and only if they satisfy the system of partial differential equations [C, C]mjklr = 0
(10)
for any value of the indices ( j, k, l, m, r ) ranging from zero to the dimension of the algebra. This system and the associativity conditions (3) are central in our approach. For this reason they will be afterwards referred to as the central system of the theory of coisotropic deformations. The thesis of the present paper is that the central system conceals a lot of interesting examples of dispersionless integrable hierarchies of soliton theory, and that it provides a new interesting route to understand their integrability. This thesis is argued in six sections according to the following plan: • Section 2 explains the origin of the new viewpoint from the theory of generalized dispersionless KP equations. In particular it explains how to write these equations in the form of a central system. • Section 3 is a short study of the structure constants associated with the dispersionless KP hierarchy, in search of the property of these constants which make the dKP equations integrable. • Section 4 presents the first noticeable outcome of the new approach. It shows that, for certain classes of algebras, the associative conditions (3) and the coisotropy conditions (10) nicely interact to produce the existence of a tau function seen as a potential for the structure constants C ljk . It also shows that the well-known dispersionless Hirota bilinear equations are nothing else than the associativity conditions (3) written in terms of this potential. This result should make easier the comprehension of the link between the Hirota equations and the associativity equations of Witten, Dijkgraaf, Verlinde, Verlinde. • Section 5 is a brief study of the symmetry properties of the central system, which has an infinite-dimensional Abelian symmetry group. The study of the orbits of this group in the space of structure constants allows to enlarge the arsenal of interesting systems of structure constants at our disposal, and to prove the invariance of the tau function under the action of the symmetry group. • Section 6 presents the second noticeable outcome of the new approach. The technique of quotient algebras is used to glue together several copies of the algebras associated with the dKP and dmKP hierarchies in such a way to obtain new interesting examples of central systems. The class of integrable equations covered by the new central systems is sufficiently large to encompass the universal Whitham’s hierarchy of genus zero studied by Krichever. • Section 7 is a terse study of coisotropic deformations from the viewpoint of the geometry of the submanifolds previously introduced. The paper ends with the indication of a few further possible developments, and with two appendices containing the details of the computations presented throughout the paper.
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2. Introduction to the Idea of Coisotropic Deformations The examples of dispersionless KP and mKP equations are well suited to illustrate the connection between coisotropic deformations and integrable hierarchies. In this section we recast these equations as a central system for the structure constants of a simple commutative associative algebra with unity. In the early eighties, when interest in the dispersionless limit of soliton equations increased (see e.g. [1–38]), it was well-known that the standard dKP and dmKP equations arise as compatibility conditions of the linear problem (see e.g. [39, 40]) ∂ψ ∂ 2ψ ∂ψ = + u 1 (x1 , x2 , x3 ) + u 0 (x1 , x2 , x3 )ψ, ∂ x2 ∂ x1 ∂ x12 ∂ψ ∂ 3ψ ∂ 2ψ ∂ψ = + v (x , x , x ) + v1 (x1 , x2 , x3 ) + v0 (x1 , x2 , x3 )ψ. 2 1 2 3 3 2 ∂ x3 ∂ x1 ∂ x1 ∂ x1 It was therefore natural to assume that their dispersionless limits are the compatibility conditions of the pair of time dependent Hamilton-Jacobi equations ∂S ∂S 2 ∂S = + u 1 (x1 , x2 , x3 ) + u 0 (x1 , x2 , x3 ), ∂ x2 ∂ x1 ∂ x1 ∂S 2 ∂S ∂S 3 ∂S = + v2 (x1 , x2 , x3 ) + v1 (x1 , x2 , x3 ) + v0 (x1 , x2 , x3 ). ∂ x3 ∂ x1 ∂ x1 ∂ x1 The compatibility conditions were obtained, as in the dispersive case, by imposing the equality of the second-order mixed derivatives of the function S(x1 , x2 , x3 ), and the result was the following system of four partial differential equations: ∂ (2v2 − 3u 1 ) ∂ x1 ∂ ∂v2 ∂u 1 ∂v2 (2v1 − 3u 0 ) − + u1 − 2v2 ∂ x1 ∂ x2 ∂ x1 ∂ x1 ∂v0 ∂u 0 ∂v1 ∂u 1 ∂u 1 ∂v1 2 − − 2v2 + u1 − v1 + ∂ x1 ∂ x2 ∂ x1 ∂ x1 ∂ x1 ∂ x3 ∂v0 ∂v0 ∂u 0 ∂u 0 − + u1 − v1 + ∂ x2 ∂ x1 ∂ x1 ∂ x3
= 0, = 0, (11) = 0, = 0.
From it the dKP and dmKP equations [10] are obtained by setting u1 = 0
v2 = 0
v1 = 3/2u 0 ,
u0 = 0
v0 = 0
v2 = 3/2u 1
and
respectively. This selection of particular classes of equations by means of suitable additional constraints was called “gauge fixing”. Looking critically at this procedure, one may remark that the method of compatibility conditions leaves a wide freedom in the choice of the form of the auxiliary problem, allowing to substitute the above pair of Hamilton-Jacobi equations by any equivalent system of partial differential equations, without which this change modifies the
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631
compatibility conditions and hence the equations one is interested in. This inherent freedom is poorly reflected by the above approach. For this reason we prefer to follow a different route, by replacing the above pair of Hamilton-Jacobi equations with the polynomial ideal J = < h2, h3 > generated by the pair of Hamiltonian functions h 2 = − p2 + p12 + u 1 (x1 , x2 , x3 ) p1 + u 0 (x1 , x2 , x3 ), h 3 = − p3 + p13 + v2 (x1 , x2 , x3 ) p12 + v1 (x1 , x2 , x3 ) p1 + v0 (x1 , x2 , x3 ). The use of this ideal allows to account simultaneously for all the possible forms of the auxiliary problems. The compatibility conditions are easily recovered as the conditions by saying that the ideal J is closed with respect to the classical Poisson bracket in R 6 . This classical viewpoint throws us immediately into the theory of coisotropic deformations. To arrive to the central system, it is sufficient to follow the construction of the dKP and dmKP hierarchies. The first higher equation in the dKP and dmKP hierarchies is defined by the compatibility conditions expressing the closure with respect to the classical Poisson bracket in R 8 of the polynomial ideal generated by the three Hamiltonians h 2 = − p2 + p12 + u 1 (X ) p1 + u 0 (X ), h 3 = − p3 + p13 + v2 (X ) p12 + v1 (X ) p1 + v0 (X ), h 4 = − p4 + p14 + w3 (X ) p13 + w2 (X ) p12 + w1 (X ) p1 + w0 (X ), where X = (x1 , x2 , x3 , x4 ) is the new set of coordinates. Continuing the process, at each step one introduces a new pair of coordinates (x j , p j ) and a new Hamiltonian h j = − p j + P j ( p1 )
f or
j ≥ 2,
where P j is a monic polynomial of degree j . The sequence of polynomials P j , defining the hierarchy, does not contain the polynomials P0 = 1 and P1 = p1 , and therefore does not form a basis of the ring of polynomials. To remedy this defect without changing the system of compatibility conditions, it is expedient to introduce four new coordinates (x0 , p0 , x, p) and two new Hamiltonians h 0 = − p0 + 1,
h 1 = − p1 + p,
and to regard the polynomials P j as polynomials in p rather than in p1 . The ideal K generated by the Hamiltonians h j = − p j + P j ( p)
f or
j ≥0
(12)
has the same closure conditions of the previous ideal J . Indeed the conditions {h 0 , K } ⊂ K
{h 1 , K } ⊂ K
simply entail that x0 is a cyclic coordinate, and that x appears always in the form x + x1 . By this property the remaining closure conditions {h j , K } ⊂ K
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B. G. Konopelchenko, F. Magri
give exactly the compatibility conditions of J . The advantage of the completion of the basis is that one may now consider the structure constants C ljk (X ) defined by P j ( p)Pk ( p) =
l= j+k
C ljk (X )Pl ( p).
(13)
l=0
By construction they depend polynomially on the coefficients of the Hamiltonians h j , and therefore they also satisfy a system of partial differential equations. The main task of this section is to identify these equations. If one tries to attack the problem directly, by using the explicit form of the dKP and dmKP equations, one may get easily lost. The best strategy is to use the properties of the ideal K , and to notice that the double sequence of polynomials f jk = − p j pk +
C ljk (X ) pl
l 0
belong to K since f jk = h j h k −
C ljk (X )h l + p j h k + pk h j ,
(14)
l 0
as one can easily check by using the definition of the Hamiltonians h j . For the condition {K , K } ⊂ K , each Poisson bracket { flm , f pq } belongs to the ideal, and therefore the structure constants C ljk satisfy the equations [C, C]mjklr = 0 as explained in the Introduction. Since they obviously satisfy also the associativity conditions, it is proved that the structure constants of the dKP and dmKP equations satisfy the central system. To understand how the dKP and dmKP equations are sitting inside this system, it is worth at this point to give a closer look at the huge set of partial differential equations [C, C]mjklr = 0. Not to be lost, it is convenient to fix at first the values of the indices ( j, k, l, r ). In this way one may exploit the condition C ljk = 0
for
l > j +k
(15)
to reduce drastically the number of equations. For instance, for j = k = l = 1 and r = 2, the infinite sequence of equations s ∂ s ∂ m ∂ m ∂ C12 C m − C11 C m − Cs2 C s − Cs1 Cs ∂ xs 11 ∂ xs 12 ∂ x1 11 ∂ x2 11 s≥0 m ∂ s m ∂ s = 0, + C1s C12 + C1s C12 ∂ x1 ∂ x1 contracts to four equations s=3 s ∂ m s ∂ m m ∂ s m ∂ C12 C11 − C11 C12 − Cs2 C11 − Cs1 Cs ∂ xs ∂ xs ∂ x1 ∂ x2 11 s=0 m ∂ s m ∂ s = 0 m = 0, 1, 2, 3 + C1s C12 + C1s C12 ∂ x1 ∂ x1
(16)
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633
owing to the triangularity condition (15). In the same way the choice j = k = l = 1 and r = 3 gives rise to five equations, and so on. The next question is to give these equations a sense. The way is to have recourse to the table of multiplication P1 P1 = P2 − u 1 P1 − u 0 P0 , P1 P2 = P3 + (u 1 − v2 )P2 + (u 0 − u 21 + u 1 v2 − v1 )P1 + (v2 u 0 − v0 − u 1 v0 )P0 , P1 P3 = P4 + (v2 − w3 )P3 + (v1 − w2 − v22 + w3 v2 )P2 + (v22 u 1 − u 1 v2 w3 − v1 v2 + v1 w3 − u 1 v1 + u 1 w2 + v0 − w1 )P1 − (w0 + v0 v2 + v0 w3 + u 0 v1 − u 0 w2 )P0 , P2 P2 = P4 + (2u 1 − w3 )P3 + (2u 0 − w2 + u 21 − 2u 1 v2 + w3 v2 )P2 + (2u 21 v2 − 2u 1 v1 − u 1 v2 w3 + v1 w3 − u 31 + u 1 w2 − w1 )P1 + (w0 + u 0 w2 − u 20 − u 0 u 21 + v0 w3 − 2u 1 v0 − u 0 v2 w3 2u 0 u 1 v2 )P0 of the polynomials P j . From it one may read the structure constants C ljk as functions of the coefficients of the Hamiltonians h j . For instance 2 = u 1 − v2 , C12 1 C12 = u 0 − v1 − u 21 + u 1 v2 , 0 C12 = u 0 v2 − v0 − u 1 v0 .
The insertion of these expressions into the first fragment (16) of the central system allows to realize that these equations coincide with the set of four compatibility conditions (11) used to defined the dKP and dmKP equations. Similarly, one may check that the fragment corresponding to j = k = l = 1 and r = 3 coincides with the compatibility conditions defining the next members of the dKP and dmKP hierarchies. In general one may prove that for j = k = l = 1 and r arbitrary one obtains the full dKP and dmKP hierarchies. Equivalently the same equations can be obtained by imposing the vanishing of the Poisson brackets { f 11 , f 1r } on the submanifold f 1k = 0, for k = 1, . . . , r . At the end of this introductory discussion, it is interesting to emphasize again that the discovery of the second interpretation of the dispersionless KP hierarchy, as coisotropic deformation of the structure constants of a specific associative algebra, is the outcome of the replacement of the scheme of zero-curvature representations with the scheme of coisotropic ideals, imposed by the desire of making the theory covariant with respect to all allowed changes of the auxiliary problems. The latter scheme encompasses the former one, and allows to understand that there is a unique mechanism behind the different representations of the dispersionless KP hierarchy. Each representation is the expression of the coisotropy of the ideal K in a different system of generators. By choosing the system of generators h j one obtains the standard zero-curvature representation. By choosing instead the system of generators f jk one arrives to see them as equations controlling the evolution in time of the structure constants of a certain associative algebra. This freedom in the choice of the representation is a powerful tool for the study of the integrable hierarchies. Soon we shall see that it allows to account very easily for the passage from the zero-curvature representation to Hirota’s representation of the dKP hierarchy.
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B. G. Konopelchenko, F. Magri
3. The Structure Constants of dKP Theory The plan of this section is to study the form of the structure constants associated with the dispersionless KP hierarchy. According to the viewpoint of the previous section, this hierarchy is defined as the system of coisotropy conditions of the submanifold formed by the zeroes of the Hamiltonian functions h j = −pj +
l= j−1
u jl p1l ,
l=0
and its structure constants are defined by the table of multiplication (h j + p j )(h k + pk ) =
n
C ljk (h l + pl ).
l=1
The point to be noticed is that the coisotropy conditions split in two classes. Certain among them have the form of algebraic constraints on the coefficients of the Hamiltonian functions, and the question is to see the effect of these constraints on the form of the structure constants C ljk . To have a reasonable control of the question it is sufficient to consider the first few Hamiltonians: h 2 = − p2 + p12 + u 0 (X ), h 3 = − p3 + p13 + v1 (X ) p1 + v0 (X ), h 4 = − p4 + p14 + w2 (X ) p12 + w1 (X ) p1 + w0 (X ), h 5 = − p5 + p15 + z 3 (X ) p13 + z 2 (X ) p12 + z 1 (X ) p1 + z 0 (X ), and the algebraic constraints v1 w2 w1 z3 z2 z1
= 3/2u 0 , = 2u 0 , = 4/3v0 , = 5/2u 0 , = 5/3v0 , = 5/4w0 + 5/8u 20 ,
originating from the coisotropy conditions of the ideal J generated by them. These constraints may be encoded into the definition of a special class of polynomials, henceforth called Faa’ di Bruno polynomials. The first six polynomials are P0 ( p) = P1 ( p) = P2 ( p) = P3 ( p) = P4 ( p) =
1, p, p2 + u 0 , p 3 + 3/2u 0 p + v0 , p 4 + 2u 0 p 2 + 4/3v0 p + w0 ,
P5 ( p) = p 5 + 5/2u 0 p 3 + 5/3v0 p 2 + (5/4w0 + 5/8u 20 ) + z 0 .
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635
Our interest is in their table of multiplication. The simplest part is P1 P1 = P2 − [u 0 ]P0 , P1 P2 = P3 − [1/2u 0 ]P1 − [v0 ]P0 , P1 P3 = P4 − [1/2u 0 ]P2 − [1/3v0 ]P1 − [w0 − 1/2u 20 ]P0 , P1 P4 = P5 − [1/2u 0 ]P3 − [1/3v0 ]P2 − [1/4w0 − 1/8u 20 ]P1 − [z 0 − 5/6u 0 v0 ]P0 . The coefficients of these equations are the footprints of the structure constants we are looking for. One may notice that each coefficient becomes stationary exactly one step after it appears for the first time. To account for this phenomenon it is useful to rewrite the table in the form P1 P1 = P2 − [1/2u 0 ]P0 − [1/2u 0 ]P0 , P1 P2 = P3 − [1/2u 0 ]P1 − [1/3v0 ]P0 − [2/3v0 ]P0 , P1 P3 = P4 − [1/2u 0 ]P2 − [1/3v0 ]P1 − [1/4w0 − 1/8u 20 ]P0 − [3/4w0 − 3/8u 20 ]P0 , P1 P4 = P5 − [1/2u 0 ]P3 − [1/3v0 ]P2 − [1/4w0 − 1/8u 20 ]P1 − [1/5z 0 − 1/6u 0 v0 ]P0 − [4/5z 0 − 2/3u 0 v0 ]P0 , showing that P1 P j = P j+1 +
j
Hl1 P j−l +
l=1
1
j
Hl P1−l
l=1
for two suitable sequences of constants (Hl1 , H1l ). Things go similarly for the second table of multiplication: P2 P1 = P3 − ([2/3v0 ]P0 ) − ([1/2u 0 ]P1 + [1/3v0 ]P0 ), P2 P2 = P4 −([2/3v0 ]P1 +[1/2w0 − 1/2u 20 ]P0 ) − ([2/3v0 ]P1 + [1/2w0 − 1/2u 20 ]P0 ), P2 P3 = P5 −([2/3v0 ]P2 +[1/2w0 − 1/2u 20 ]P1 +[∗ ∗ ∗∗]P0 ) − ([∗ ∗ ∗]P1 + [∗ ∗ ∗]P0 ), showing that P2 P j = P j+2 +
j
Hl2 P j−l +
l=1
2
j
Hl P2−l
l=1
for two new sequences of constants (Hl2 , H2l ). In general one may expect (and prove subsequently) that Pk P j = Pk+ j +
j l=1
Hlk P j−l
+
k
j
Hl Pk−l .
(17)
l=1
By this formula we have reached our aim, discovering that the structure constants of the Faa’ di Bruno polynomials have the form j
l k Ckl j = δk+ j + H j−l + Hk−l ,
(18)
where δkl is the Kronecker symbol. This formula is the starting point of the algebraic analysis of Hirota’s bilinear formulation of the dispersionless KP hierarchy.
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4. Tau Function and Hirota’s Equations The important question which remains unanswered at the bottom of the algebraic approach is to understand the link of coisotropic deformations to integrability. So far we have realized that certain integrable partial differential equations may be written in the form of a central system, but we do not yet know if all that makes sense. In this section we point out a second occurrence that partly answers the previous question. We assume that the structure constants have the form suggested by the examination of the dKP hierarchy, without specifying the coefficients H jk , and we investigate the implications of the process of coisotropic deformation for this class of algebras. Our purpose is to show that the associativity conditions (3) and the coisotropy conditions (10) cooperate to produce the existence of a tau function, which is the first sign of the integrability of the given set of partial differential equations. The first step is to implement the associativity conditions on the coefficients H jk . By direct substitution of expressions (18) into Eqs. (3), and by the use of the identity k−1
k−1
i m Hk−l Hl−n =
l=n−1
m i Hk−l Hl−n ,
(19)
l=n−1
one is led to the system of equations m i+k i i−k Hi+k−n + Hm−n − Hm+k−n − Hm+n k i + θ (n − m)Hi+m−n + θ (n − m)Hk+m−n k m − θ (n − i)Hi+m−n − θ (n − i)Hi+k−n i−1
+
i−1
k m Hi−l Hl−n +
k l Hi−l Hm−n
l=n−1
l=1
n−1
m−1
+
i l Hk−l Hm−n −
l=1
−
m−1
(20) k i Hm−l Hl−n
l=n−1 k l Hm−l Hi−n
−
l=1
k−1
m l Hk−l Hi−n = 0,
l=1
where θ (n) = 1 or θ (n) = 0 according to whether n 0 or n < 0. However these equations are not all independent. The analysis of Appendix A allows to reduce the number of equations, and shows that the structure constants C ljk obey the associative conditions if and only if the bracket i k − Hi+m + [H, H ]ikm := Hmi+k − Hm+k
i−1 l=1
k Hi−l Hml +
k−1 l=1
i Hk−l Hml −
m−1
k Hm−l Hli
l=1
vanishes identically for any choice of the indices (i, k, m) ∈ N . An interesting consequence of this result can be drawn by contraction. Indeed one may check that the equations [H, H ]ikm = 0 (21) m+k= p
Algebraic Foundations of a Theory of Dispersionless Integrable Hierarchies
637
and the use of the identities (19) lead to the contracted identities p H pi = H pi +
p−1
i+ p−k
(Hk
k − Hi+ p−k ) −
k=1
p−1 i−1
Hl
p−k
Hki−l ,
l=1 k=1
entailing the useful symmetry relations p
p H pi = i Hi .
(22)
As we shall see in a moment, this remarkable outcome of the associativity conditions is at the basis of Hirota’s formulation. Next one has to implement the coisotropy conditions. In terms of the coefficients Hki the ensuing equations are, at first sight, rather complicated, and they are presented in Appendix A (as a particular case of Eq. (58)). However a closer scrutiny shows that they are simplified drastically on account of the associativity conditions just obtained. Indeed in Appendix A it is proved that the coisotropy conditions may be reduced to the equations ∂[H, H ]l,n,i− j ∂[H, H ]l,n,k− j ∂[H, H ]i,k,l− j ∂[H, H ]i,k,n− j + − − = 0, ∂ xk ∂ xi ∂ xn ∂ xl which are automatically fulfilled owing to the associativity conditions, and to the linear equations ∂ H pi ∂ xl
=
∂ H pl ∂ xi
.
So, to summarize, the associativity and coisotropy conditions in the case of the structure constants of the form (18) are together equivalent to the set of quadratic algebraic equations [H, H ]ikl = 0,
(23)
entailing the symmetry conditions p
p H pi = i Hi ,
(24)
and to the set of linear differential equations ∂ H pi ∂ xl
=
∂ H pl ∂ xi
(25)
having the form of a system of conservation laws. Equations. (23) and (25) give the specific form of the central system of the dispersionless KP hierarchy. It encodes all the information about the hierarchy. In particular it entails that for any solution of the central system one has st { f ik , fln } = K ikln f st , (26) s,t≥1
where
∂ ∂ ∂ ∂ st n l k i (Hl−s K ikln = δit + δkt + Hn−s ) − (δnt + δlt )(Hi−s + Hk−s ). ∂ xk ∂ xi ∂ xl ∂ xn
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B. G. Konopelchenko, F. Magri
From this formula one sees that the Hamiltonians f jk of the dispersionless KP hierarchy form a Poisson algebra. The above central system can be seen also as the dispersionless limit of the central system of the full dispersive KP hierarchy [41]. There are presently two strategies to decode the information contained in the central system. According to the first strategy, one first tackles the associativity conditions (23), noticing that they allow to compute the coefficients (Hk2 , Hk3 , . . .) as polynomial functions of Hk1 . For instance, the symmetry conditions H12 = 2H21
H13 = 3H31
give (H12 , H13 ), and then the condition (23) with i = k = 1, l = 2, i.e. H13 − H31 − H22 + H11 H11 = 0 gives H22 , and so forth. Renaming the free coefficients as suggested by the table of multiplication of the previous section, that is by setting H11 = −1/2u 0 ,
H21 = −1/3v0 ,
H31 = −1/4w0 + 1/8u 20 ,
one gets H12 = −2/3v0 ,
H22 = −1/2w0 + 1/2u 20 ,
H13 = −3/4w0 + 3/8u 20 .
At this point one may plug these expressions into the simplest linear coisotropy conditions ∂ H11 ∂ H12 − = 0, ∂ x2 ∂ x1
∂ H21 ∂ H22 − = 0, ∂ x2 ∂ x1
∂ H11 ∂ H13 − = 0, ∂ x3 ∂ x1
arriving to the equations ∂v0 ∂u 0 = 3/4 , ∂ x1 ∂ x2 ∂v0 ∂u 0 ∂w0 = 3/2 − 3u 0 , ∂ x2 ∂ x1 ∂ x1 ∂u 0 ∂u 0 ∂w0 = 3/2 − 3/2u 0 . ∂ x3 ∂ x1 ∂ x1 0 The elimination of ∂w ∂ x1 leads finally to the dispersionless KP equation and to the higher equations, if one insists enough in the computations. By this strategy one comes back to the hierarchy in its standard formulation. A simple inversion in the order in which the equations are considered leads instead to Hirota’s formulation. It is enough to remark that Eqs. (25) entail the existence of a sequence of potentials Sm such that
Hmi =
∂ Sm . ∂ xi
Then the symmetry conditions (24), oblige the potentials Sm to obey the constraints i
∂ Sl ∂ Si =l , ∂ xl ∂ xi
Algebraic Foundations of a Theory of Dispersionless Integrable Hierarchies
639
which in turn entail the existence of a superpotential F(x1 , x2 , . . .) such that Si = −1/i
∂F , ∂ xi
Hmi = −1/m
∂2 F . ∂ xi ∂ xm
This result provides a second parametrization of the structure constants, after that described before. The insertion of the new parametrization into the full set of associativity conditions finally leads to the system of equations 1 1 1 Fi+k,m + Fi,k+m + Fk,i+m m m+k i +m i−1 k−1 m−1 (27) 1 1 1 Fk,i−l Fl,m + Fi,k−l Fl,m − Fk,m−l Fi,l = 0, + m(i − l) m(k − l) i(m − l) −
l=1
l=1
l=1
where Fi,k stands for the second-order derivative of F with respect to xi and xk . They are equivalent to the celebrated Hirota bilinear equations for the tau function of the dispersionless KP hierarchy (see e.g. [12, 13, 17, 29]). For instance, for (i = k = 1, m = 2) or (i = m = 1, k = 2) or (m = k = 1, i = 2) one obtains directly the first Hirota equation −1/2F2,2 + 2/3F1,3 − (F1,1 )2 = 0. For (i = 1, k = 2, m = 2) or (i = 2, k = 1, m = 2) or (i = 1, k = 1, m = 3) it gives instead the second Hirota equation 1/2F1,4 − 1/3F2,3 − F1,1 F1,2 = 0. The choices (i = 1, k = 1, m = 4) and (i = 1, k = 2, m = 3) lead to the equations −1/4F2,4 + 2/5F1,5 − 2/3F1,1 F1,3 − 1/4(F1,2 )2 = 0 and −1/3F3,3 + 1/5F1,5 + 1/4F2,4 + 1/3F1,1 F1,3 − 1/2F1,1 F2,2 − 1/2(F1,2 )2 = 0 respectively, which do not separately coincide with Hirota’s bilinear equations, but which are together equivalent to a pair of standard Hirota equations of the same weight. The process of identification may be continued indefinitely. The observation that the Hirota equations of the dispersionless KP hierarchy are the associativity equations for the structure constants (18) immediately recalls the associativity equations of Witten, Dijkgraaf, Verlinde, Verlinde [42, 43], first derived in the frame of two-dimensional topological field theories and subsequently interpreted by Dubrovin [44] in the frame of his theory of Frobenius manifolds. In spite of the ideological resemblance there are, however, essential differences between the present approach and that of Frobenius manifolds developed by Dubrovin. A comparative analysis of these two approaches will be given elsewhere. One may also note that the central system represents itself an interesting linearization of the dKP hierarchy. In terms of the coefficients of the Hamiltonians h j this system is just the dKP hierarchy of nonlinear PDE’s. In terms of variables Hki the dKP hierarchy is represented by linear exactness conditions on the family of quadrics (21).
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5. The Symmetry Action So far we have considered a relatively narrow class of algebras and of dispersionless integrable equations. To enlarge this class we shall follow two different routes, in this and the next section, having recourse to the action of a symmetry group and to a process of gluing. The transformation of the structure constants j d ab C (X ) = C ljk (X )Aa (X )Akb (X )Ald (X ), (28) a,b,l>0
induced by a change of basis p˜ a =
Ala (X ) pl
(29)
l>0
depending on deformation parameters X = (x1 , x2 , . . .), obviously preserves the symmetry conditions (2) and the associativity conditions (3), but violates in general the coisotropy conditions (9). The class of transformations preserving the last conditions can be worked out by turning to the correspondence between deformation parameters and generators, on one side, and coordinates and momenta, on the other side, which has been used to establish the Hamiltonian interpretation of the process of coisotropic deformation. Shifting from commutative algebra to Hamiltonian mechanics, one then substitutes the above change of basis in the associative algebra with the transformation of coordinates x˜a = δal xl , l>0
p˜ a =
Ala (X ) pl
(30)
l>0
on the symplectic manifold related to the algebra. This change of coordinates acts on the Hamiltonians f jk according to the transformation law j Aa (X )Akb (X ) f jk (X ), (31) f˜ab (X ) = j,k>0
which shows that the functions f˜ab still belong to the ideal J generated by the functions f jk . This ideal is thus left invariant by the transformation (30), but it loses in general the property of being closed with respect to the classical Poisson bracket since the bracket is modified by the coordinate transformation. So to preserve the coisotropy conditions we need to restrict the change of basis in such a way that the transformation (30) is canonical. This is a severe restriction and the unique solution is p˜ a = pa +
∂φ p0 , ∂ xl
(32)
where φ is an arbitrary function of the deformation parameters. The conclusion is that there exists an infinite-dimensional Abelian symmetry group of the equations defining the coisotropic deformations of associative algebras depending on an arbitrary function φ. This group is the subject of this section.
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The action of the group on the space of structure constants is defined by ljk = C ljk + δkl ∂φ + δlj ∂φ , C ∂x j ∂ xk ∂φ 0jk = C 0jk + ∂φ ∂φ − C C mjk ∂ x j ∂ xk ∂ xm
(33) (34)
m>0
for j, k, l > 0. Therefore the equations of the orbit passing through the point corresponding to the structure constants of the dispersionless KP hierarchy are k ljk = δlj+k + H j + H j−l jk , C + δ0l D k−l
(35)
with the understanding that the coefficients are given by ki = Hki , H 0i = ∂φ , H ∂ xi k0 = δk0 , H m=i−1 m=k−1 ∂φ ∂φ k i ik = − ∂φ − ∂φ ∂φ − D Hi−m − Hk−m , ∂ xi+k ∂ xi ∂ xk ∂ xm ∂ xm m=1
m=1
0k = −δk0 D l closely resemble the for i, k > 0. One may notice that the new structure constants C jk original ones, and one may infer from this fact that they also derive from a tau function. The first problem we are interested in is to inquire how the symmetry group acts at the level of tau functions. 1 , x2 , . . .) for the new structure conThe proof of the existence of a tau function F(x l stants C jk strictly follows the pattern of the previous section. The strategy is always to ik of the conditions of coisotropy i and D work out the implications on the coeffifients H k and associativity. In Appendix A it is shown that the conditions of associativity give rise to two sets of constraints, affecting separately the two families of coefficients. One set according to fixes the form of the coefficients D ik ik = − H 0i+k + H 0i H 0k − D
l=k
i l k−l H H0 −
l=1
i−1 l=1
k l i−l H Hm +
k l i−l H H0 ,
l=1
while the second set requires that the coefficients i k mi+k − H m+k i+m H −H +
l=i
i H k
k−1 l=1
verify the already known conditions
i l k−l H Hm −
m−1
k m−l li = 0. H H
l=1
A closer scrutiny of these conditions shows that they do not contain terms of the form i , since these terms cancel in pairs, proving that the coefficients H i , for i, k > 0, H 0 k satisfy again the symmetry conditions . pi = i H pH i p
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Similarly the coisotropy conditions also split in two sets that, owing to the associativity conditions, take the form l i ∂H ∂H 0 0 = , ∂ xi ∂ xl pl pi ∂H ∂H = ∂ xi ∂ xl for i, l, p > 0. One infers from the last conditions the existence of a sequence of potentials φ = S0 and Sk , such that 0i = ∂φ , H ∂ xi
ki = ∂ Sk . H ∂ xi
By the same argument discussed in Sect. 4, the potentials Sk lead to the existence of a 1 , x2 , . . .) such that single function F(x 2 ki = − 1 ∂ F . H k ∂ xi ∂ xk
l The existence of the potential φ shows instead that the new structure constants C jk belong to the orbit passing through the point representing the KP hierarchy. Due to the i along the orbit (i, h > 0), it turns out that the tau invariance of the coefficients H k function is invariant along the orbit. Thus this function is a property of the orbit rather than of the points of the orbit. The second problem of our concern is to investigate the properties of a second remarkable point belonging to the orbit, whose definition is suggested by the equations of the orbit. They entail, in particular, that 0jk = H j + H jk + D jk . C k 0 vary along the orbit, and therefore This formula shows that the structure constants C jk one can ask if there exists a point on the orbit where these structure constants vanish. The answer is affirmative owing to the first half of the associativity conditions. Indeed 0 = 0 is equivalent to search a function φ such that to demand that C jk m=i−1 m=k−1 ∂φ ∂φ ∂φ ∂φ ∂φ k i + + Hi−m + Hk−m = Hki + Hik , ∂ xi+k ∂ xi ∂ xk ∂ xm ∂ xm m=1
m=1
if one takes into account the invariance of the coefficients Hki along the orbit. The system just written is an overdetermined system of nonlinear partial differential equations on the single unknown function φ whose compatibility conditions are exactly the first half of the associativity conditions. Accordingly there exists a function φ which allows to reach the desired point from the point corresponding to dispersionless KP. It is readily seen that the new point corresponds to the structure constants of the dispersionless mKP hierarchy. It is sufficient to interpret the last equations as the hierarchy of dispersionless Miura transformations relating the dispersionless KP and mKP hierarchies.
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This interpretation is motivated by the remark that, at the lowest level i = k = 1, the equation of the symmetry action has the form ∂φ 2 ∂φ +( ) = 2H11 , ∂ x2 ∂ x1 and therefore becomes u 0 = 1/2
∂ −1 ∂u 1 ( ) − 1/4u 21 , ∂ x1 ∂ x2
upon the insertion of the standard parametrization u 0 = −2H11 ,
u 1 = −2H01 = −2
∂φ . ∂ x1
The above equation is the dispersionless limit of the well-known Miura transformation between the KP and mKP equations (see e.g. [23, 24]). The lesson is that the dispersionless Miura transformation is just a particular instance of the symmetry action generated by the special changes of basis (29) in the associative algebra. At the level of structure constants the Miura transformations have therefore a very simple meaning. It remains to identify the meaning of the other points of the orbits. A possible way is to l obeying the equations carefully study the parametrization of the structure constants C jk of the orbit. We omit this study and we limit ourselves to give the final answer. As the reader may expect at this point, one may prove that the structure constants of the orbit passing through the dKP hierarchy correspond to the dispersionless generalized dKP hierarchy briefly discussed in Sect. 2. This remark closes the study of the orbit defined by the symmetry action. 6. The Process of Gluing An elementary way of gluing together two algebras of polynomials, one in the variable p and the other in the variable q, is to add the relation pq = ap + bq + c which allows to write the product of a polynomial in p by a polynomial in q as a sum of two polynomials of the same variables. The new associative algebra is known as the quotient of the algebra of polynomials in two variables p and q with respect to the ideal generated by the polynomial (− pq + ap + bq + c). The leading idea of this section is to apply this procedure to two copies of the algebra of Faa di Bruno polynomials encountered in Sect. 3 and slightly generalized in Sect. 5. In practice this means that we complete the tables of multiplication p j pk =
p j+k +
k l=0
q j qk =
q j+k +
j
Hl pk−l +
k l=0
j
Hlk p j−l + D jk ,
l=0
j qk−l + H l
j l=0
(36) jk lk q j−l + D H
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by adding the relation p1 q1 = ap1 + bq1 + c,
(37)
and consistently computing the products p j qk . The aim is to show that the structure constants of the enlarged table of multiplication admit interesting coisotropic deformations reproducing the universal Whitham equations of genus zero, obtained by Krichever by his technique of meromorphic functions on the Riemann sphere with punctures [14]. The process of gluing is thus the algebraic implementation of the pasting of meromorphic functions on the Riemann sphere (or of meromorphic differentials on Riemann surfaces) characteristic of the algebro-geometric approach of Krichever. What for him is the introduction of a new pole, for us is simply the gluing of an additional copy of the same algebra. To give an idea of the potentialities of the new procedure, let us consider the fragment of the above table of multiplication consisting of the three simplest equations p1 p1 = p2 − (vp1 + u), q1 q1 = q2 − (v˜ p1 + u), ˜ p1 q1 = ap1 + bq1 + c,
(38)
where new names have been used for the structure constants to simplify the notation. According to the scheme of coisotropic deformations, we introduce two sets of space coordinates x j and y j and their conjugate momenta p j and q j , and we transform the partial table of multiplication (38) into the definition of three Hamiltonians f = − p2 + p12 + vp1 + u, f˜ = −q2 + q12 + v˜ p1 + u, ˜ g = − p1 q1 + ap1 + bq1 + c
(39)
on the symplectic manifold R 8 endowed with the classical Poisson bracket for canonical coordinates (x1 , x2 , y1 , y2 , p1 , p2 , q1 , q2 ). Finally we demand that the ideal J generated by these Hamiltonians be closed with respect to the Poisson bracket, and we obtain three sets of equations. From the study of { f, g} we obtain the equations v y1 + 2ax1 = 0, ax2 − 2(ab + c)x1 − (1/4v 2 + u) y1 = 0, cx2 − (cv)x1
bx2 − (bv + b2 + u)x1 = 0, − 2cbx1 + au x1 + bu y1 = 0,
while the study of { f˜, g} leads to 2b y1 + v˜ x1 = 0, ˜ y1 = 0, a y2 + (a v˜ − a 2 − u) b y2 − 2(ab + c) y1 − (u˜ + 1/4v˜ 2 )x1 = 0, c y2 − (cv) ˜ y1 − 2ca y1 + a u˜ x1 + bu˜ y1 = 0,
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and the study of { f, f˜} gives
2cv y1
2av y1 − 2a v˜ x1 + vv ˜ y1 − v y2 − 2u˜ x1 = 0, 2bv y1 − 2bv˜ x1 − v v˜ x1 − v˜ x2 + 2u y1 = 0, − 2cv˜ x1 + vu ˜ y1 − u y2 − v u˜ x1 + u˜ x2 − u y2 = 0.
If these equations are satisfied then { f, g} = (v + 2b)x1 g, { f˜, g} = (v˜ + 2a) y1 g, { f, f˜} = 2 v˜ x1 − v y1 g. These equations admit several interesting reductions. For instance, by setting v = 0, u˜ = 0, a = 0 the first system becomes u y1 + 2cx1 = 0,
bx2 − (b2 + u)x1 = 0,
cx2 − 2(bc)x1 = 0,
which is the simplest example of a generalized Benney system introduced in [11, 14, 16]. At the same time the second system becomes the modified Benney system [11] v˜ x1 + 2b y1 = 0,
b y2 − 1/4(v˜ 2 )x1 − 2c y1 = 0,
c y2 − (cv) ˜ y1 = 0,
and the third system becomes the Miura type transformation v˜ x2 + 2(u + b2 ) y1 = 0,
u y2 + 2(cv) ˜ x1 = 0
between them [11]. If instead one sets u = 0, u˜ = 0, c = 0 from the first two systems of coisotropy conditions one obtains ax2 − 2(ab)x1 − 1/4(v 2 ) y1 = 0, bx2 − (b2 + bv)x1 = 0, v y1 + 2ax1 = 0, ˜ y1 = 0, a y2 − (a 2 + a v) b y2 − 2(ab) y1 − 1/4(v˜ 2 )x1 = 0, v˜ x1 + 2b y1 = 0 which is, in fact, the dispersionless limit of the equations for the components of the wave function of the Dawey-Stewartson equation discussed in [45]. It is remarkable that the new associative algebra, defined by the process of gluing, is isomorphic to the algebra of meromorphic functions on the Riemann sphere with two punctures. Indeed resolving the coupling relation with respect to q1 , one gets q1 =
ab + c + a. p1 − b
Hence any polynomial in q1 becomes a rational function in p1 . So the algebra (36), (37) may be identified with the algebra of rational functions with poles in b and in the point at infinity. This algebra is just the algebra of meromorphic functions on the Riemann sphere with two punctures used by Krichever [14]. The general case of n puntures is equivalent to the gluing of n copies of algebras of the initial type.
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To give an example of equations coming from the gluing of n copies of the algebra of Faa di Bruno polynomials, let us consider the fragment of the table of multiplication consisting simply of the 1/2n(n − 1) relations pα pβ = Hαβ pα + Hβα pβ
α = β
which serve to glue together the n copies of the algebra. In this example the indices α, β = 1, 2, . . . , n identify the copies of the algebra, and the symbol pα stands for pα,1 . If n is greater than two, the coefficients Hβα must verify an appropriate set of associativity conditions. In Appendix B it is shown that the part of the associativity conditions pertaining to the coefficients Hβα is γ
Hαβ Hαγ − Hγβ Hαγ − Hβ Hαβ = 0,
(40)
where the indices α, β, γ take distinct values. In the same appendix it is also shown that the coisotropy conditions for the Hamiltonians gαβ = − pα pβ + Hαβ pα + Hβα pβ , α = β, contains a subset of conditions pertaining only to the coupling coefficients Hβα , and that this subset simplifies into the by now familiar form ∂ Hγα ∂ xβ
β
−
∂ Hγ = 0, ∂ xα
α = β = γ = α,
owing to the previous associativity conditions. The equations just written describe the coisotropic deformations of the coupling coefficients Hβα alone. As in the case of the dispersionless KP hierarchy, one may treat these equations in two alternative ways. If one solves first the associativity conditions by setting, for instance, Hβα =
uα vα − vβ
α = β,
where u α and vα are arbitrary functions of the coordinates xα , one obtains from the coisotropy conditions the final system uβ ∂ uα ∂ ( )− ( ) = 0. ∂ xβ vα − vγ ∂ xα vβ − vγ If instead one chooses the second strategy, and solves first the coisotropy conditions by setting Hβα =
∂ Fβ , ∂ xα
α = β,
one obtains from the associativity conditions the dispersionless Darboux system ∂ Fα ∂ Fα ∂ Fα ∂ Fγ ∂ Fα ∂ Fβ − − =0 ∂ xβ ∂ xγ ∂ xγ ∂ xβ ∂ xβ ∂ xγ
Algebraic Foundations of a Theory of Dispersionless Integrable Hierarchies
647
introduced in [32]. It represents the dispersionless limit of the well-known Darboux system ∂(ln γ ) ∂ α ∂(ln β ) ∂ α ∂ 2 α − − =0 ∂ yβ ∂ yγ ∂ yγ ∂ yβ ∂ yβ ∂ yγ describing conjugate nets of curves in R n ( see e.g. [46]). Indeed, putting in this system ∂ ∂ Fα (x) =ε ,
α = exp ∂ yα ∂ xα ε and taking the limit ε → 0, one obtains the above dispersionless system. The second example is the n-component (2+1)-dimensional Benney system introduced by Zakharov [16]. It can be recovered by gluing one copy of the algebra associated with the dispersionless KP hierarchy to (n + 1) copies of the algebras associated with the dispersionless mKP hierarchy. The Benney systems can be then obtained as coisotropy conditions of the ideal J generated by the first Hamiltonian of the dispersionless KP hierarchy f = − p2 + p12 − u, by the n functions kα = p1 qα − aα qα − να ,
α = 1, . . . , n,
which serve to glue the algebra of the dispersionless KP hierarchy to n copies of the algebra of the dispersionless mKP hierarchy, and by the function g = qn+1 −
n
qα ,
α=1
which serves to glue the (n + 1)th copy of this algebra to the previous ones. To apply the technique of coisotropic deformations we have to simultaneously introduce new deformation parameters. Let us denote by (x1 , x2 , yα , τ ) the coordinates conjugate to the momenta ( p1 , p2 , qα , qn+1 ) respectively, and let us work out explicitly the coisotropy conditions of the ideal J . One may easily find that the condition { f, kα } = 0 on gives u yα + 2ναx1 = 0, u x1 + (aα2 )x1
ναx2 + aα u yα
− aαx2 = 0, − 2να aαx1 = 0.
(41) α = 1, . . . , n,
(42) (43)
Similarly the condition { f, g} = 0 on implies uτ −
n
u yα = 0
α=1
while the conditions {kα , g} = 0 on give aατ −
n ∂aα =0 ∂ yβ
β=1
(44)
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B. G. Konopelchenko, F. Magri
and νατ −
n ∂να = 0. ∂ yβ
β=1
From Eqs. (41) and (44) one gets u τ + 2(
n
να )x1 = 0.
α=1
Then from Eqs. (41) and (43) one obtains ναx22 − 2(να aα )x1 = 0. The last two equations together with (42) are just the (2+1)-dimensional n-component Benney system [16]. If one wants to glue n copies of the full algebras j
pk p j = pk+ j +
Hlk p j−l
+
l=1
k
j
Hl pk−l + D jk ,
l=1
by means of the gluing relations pα,i pβ,k =
n γ =1 l≥1
γl
0 Cαβik pγ l + Cαβik p0 ,
where p0 is the unit element, α, β = 1, . . . , n and i, k, l = 1, 2, . . ., one is obliged to consider the full system of structure constants γl
β,k
α,i l + δγ α Hα,i−l + δγβ Hβ,k−l . Cαβik = δγ α δγβ δi+k
Coisotropic deformations for this algebra can be constructed according to our general scheme, but they will be studied elsewhere. 7. Coisotropic and Lagrangian submanifolds Our purpose in this section is to stress an important geometrical difference between the usual approach to integrable dispersionless equations, based on the study of compatibility conditions of a system of Hamilton-Jacobi equations, and the present approach of coisotropic deformations. A characteristic trait of the usual approach is to assume p j = ∂∂xSj since the beginning, and therefore one is unwittingly confined to a Lagrangian submanifold inside the symplectic manifold M 2n . As a consequence one loses the canonical symplectic 2-form ω=
n
dpi ∧ d xi
i=1
which vanishes on the Lagrangian submanifold. In the present approach, instead, the main role is given the coisotropic submanifold (for the definition of coisotropic
Algebraic Foundations of a Theory of Dispersionless Integrable Hierarchies
649
submanifolds see e.g. [47–49]). It is interesting to see some examples of them. Any solution of the integrable hierarchies discussed in the paper provides us with a coisotropic submanifold. A simple example corresponds to the following solution of the dKP equation ([6]): x22 x1 , + 4/9 (1 + 2x3 ) (1 + 2x3 )2 x23 x1 x2 − 4/9 . v0 = 2/3 (1 + 2x3 )2 (1 + 2x3 )3
u 0 = −2/3
The associated coisotropic submanifold of dimension 4 in the six dimensional symplectic space R 6 , with coordinates (x1 , x2 , x3 , p1 , p2 , p3 ) is defined by the equations x22 x1 − 4/9 = 0, (1 + 2x3 ) (1 + 2x3 )2 x22 x1 x1 x2 + 2/9 = − p1 p2 + p3 − p1 (−1/3 ) − 2/3 (1 + 2x3 ) (1 + 2x3 )2 (1 + 2x3 )2 x23 + 4/9 = 0. (1 + 2x3 )3
f 11 = − p12 + p2 + 2/3 f 12
If instead one takes the zero-set of the first three Hamiltonians ( f 11 , f 12 , f 13 ) of the dKP hierarchy, and one considers the common solution u 0 (x1 , x2 , x3 , x4 ) of the dKP and first higher dKP equations, one gets a five dimensional coisotropic submanifold in R 8 . Continuing in this process, one obtains an infinite tower on coisotropic submanifolds associated with dKP hierarchy. In general they have dimension (n + 1) in a symplectic space of dimension 2n. In this sense they are minimal since they are the closest to Lagrangian submanifolds which have dimension n. Furthermore, the restriction of the symplectic 2-form ω to does not vanish, but is a presymplectic 2-form. As is well-known, its kernel is spanned by the (n-1) Hamiltonian vector fields associated with the functions f jk defining the coisotropic submanifold. These vector fields define the so-called characteristic foliation of , whose space of leaves is called the reduced phase space r ed (see e.g. [47–49]). In our case the restriction of ω to has rank two and hence ω = dL ∧ dM.
(45)
The canonical variables L and M play an important role in the theory of dispersionless hierarchies. Various Lagrangian submanifolds are obtained by setting a constraint on these variables. An obvious example is provided by the constraint L = z = const.
(46)
As usual the Lagrangianity implies the existence of a generating function S(z, x) such that (see e.g. [50–52]) p j (x) =
∂ S(z, x) for ∂x j
z = const.
(47)
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One also readily concludes that the restriction of M to the Lagragian submanifold is given by M|z = M(z) =
∂ S(z, x) . ∂z
(48)
With this identification the formula (45) coincides with that obtained in [12, 14]. Thus one may say that the foliation of the coisotropic submanifold by Lagrangian submanifolds parametrized by z corresponds to the “algebraic orbits” of Whitham’s equations of genus zero considered in [14]. A more general class of Lagrangian submanifolds are provided by the constraint g(L, M) = 0,
(49)
where g is an arbitrary function. Finally we would like to note that the nonlinear ∂¯ equation, which is the basic equation for the quasiclassical ∂¯ method [30–32], also has a simple geometric meaning within the present approach, at least if one considers the complexified version of the scheme, where all the variables are complex. In this case the rank of ω| is equal to four (complex two), and ˜ ω| = dL ∧ dM + d L¯ ∧ d M,
(50)
where the bar denotes, as usual, the complex conjugation. In this case the class of Lagrangian submanifolds is defined by the constraint ¯ M, M) ˜ = 0, W (L, L,
(51)
where W is an arbitrary complex function. Using the parametrization L = z, L¯ = z¯ and = ∂ S , one writes Eq. (51) as the formulae M = ∂∂zS , M ∂ z¯ W (z, z¯ ,
∂S ∂S , )=0 ∂z ∂ z¯
(52)
which is exactly the nonlinear ∂¯ equation used in the papers [30–32, 34, 35]. So many of the known methods to solve dispersionless integrable equations deal, actually, with different classes of Lagrangian submanifolds contained inside the coisotropic submanifold . 8. Final Remarks The aim of this paper was to point out a new phenomenon (the appearance of structure constants inside the theory of dispersionless integrable hierarchies), without pretention of completeness or systematicity. The paper is, indeed, a first exploration of a territory which remains largely unknown. A few other directions of explorations are known. Some are routine work, consisting in encompassing a different system of structure constants and therefore different classes of integrable hierarchies, such as , for instance, the dispersionless Harry Dym hierarchy. Some have the theoretical aim of probing more deeply the basis of Hirota’s bilinear formulation, by understanding for what kind of associative algebras the associativity conditions allow to reduce the coisotropy conditions to the form of a system of conservation laws. In the same vein, the other interesting question
Algebraic Foundations of a Theory of Dispersionless Integrable Hierarchies
651
is to understand systematically if the concept of coisotropy has nothing to do with a possible Hamiltonian (or bihamiltonian) structure of the integrable hierarchy defined by the central system on the structure constants. A last exciting direction, finally, points towards more a general type of algebras. We know that to encompass, for instance, the dispersionless Veselov-Novikov hierarchy in the present scheme, one has to abandon the associative commutative algebra with unit and to consider Jordan’s triple systems. So of the scheme presented in this paper one should take mainly the spirit rather than the form, and to consider it as a possible point of departure for new interesting investigations in a field which has not yet exhausted its source of surprises, despite intensive investigations during the last twenty years or more. 9. Appendix A Here we will derive equations (23), (24) and (25) for the generalised dKP hierarchy. Thus we consider an algebra with the structure constants of the form j
k C ljk = δlj+k + H j−l + Hk−l + δ0l D jk ,
(53)
j
where H0l = 0, Hk = 0 k ≤ −1. For such structure constants the associativity condition takes the form m i n Rikmn + H−n Dik − H−n Dmk + δm Dik − δin Dmk + δ0n Di+k,m − Di,m+k + Dik D0m − Dmk D0i k (54) i k m n i k m k + δ0 Hk−l Dlm + Hi−l Dlm − Hk−l Dli − Hm−l Dli = 0, l=0
l=0
l=0
l=0
where Rikmn is given by l.h.s of Eq. (20) with H0i = 0 and substitution l=1 → 0 . Equation (54) in the case when all indices i,k,m,n are distinct and different from zero is of the form Rikmn = 0.
(55)
These equations are easily seen to be equivalent to the system [H, H ]ikm = 0,
(56)
m+k= p
where i k − Hi+m + [H, H ]ikm := Hmi+k − Hm+k
i
k Hi−l Hml +
l=0
k
i Hk−l Hml −
l=0
m
k Hm−l Hli .
l=0
Note that these equations do not contain H0i . Equations (54) with n = m = 0 and all other indices distinct are equivalent to Dik + H0i+k − H0i H0k +
k l=1
i Hk−l H0l +
i l=1
i Hi−l H0l = 0.
(57)
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At n = i = 0 one gets equations which are equivalent to these ones. Finally at n = 0 Eq. (54) is reduced to Di+k,m − Di,m+k +
k
i Hk−l Dlm +
l=0
i
k Hi−l Dlm −
l=0
k
m Hk−l Dli −
l=0
m
k Hm−l Dli = 0.
l=0
It is a straightforward but cumbersome check that these equations are verified by virtue of the previous ones. So the associativity conditions for the structure constants (35) are given by Eqs. (56) and (57). The coisotropy conditions { f ik , fln } = 0 on for the Hamiltonians defined by the above structure constants are equivalent to the system Tiklnm + δ0m Q ikln − δnm
0 ∂Cik ∂C 0 ∂C 0 ∂C 0 − δlm ik + δkm ln + δim ln = 0, ∂ xl ∂ xn ∂ xi ∂ xk
where 0 = Hki + Hik + Dik , Cik
and Tiklnm
k ∂ Hi
∂ H k−m i−m s l n δl+n = + Hn−s + Hl−s + ∂ xs ∂ xs s=1 n ∂ Hl
∂ Hl−m n−m s i k δi+k + Hk−s + Hi−s + − ∂ xs ∂ xs s=1 i k ∂ Hk−s ∂ Hi−s m s n δs+n + Hn−m + Hs−m + − ∂ xl ∂ xl s=1 k ∂ Hi
∂ Hi−s k−s m s l δl+s + Hl−m + Hs−m + − ∂ xn ∂ xn s=1 k ∂ Hl
∂ H l−s n−s m k s δk+s + Hs−m + Hk−m + + ∂ xi ∂ xi s=1 n ∂ Hl
∂ Hl−s n−s m i s δi+s + Hs−m + Hi−m + , + ∂ xk ∂ xk s=1
and Q ikln =
∂ Dik ∂ Dln ∂ Dik ∂ Dln l n i k − + [(Hn−s + Hl−s ) − (Hk−s + Hi−s ) ∂ xl+n ∂ xi+k ∂ xs ∂ xs s=0
− Dsn (
i ∂ Hk−s
k ∂ Hi−s
) − Dsl (
i ∂ Hk−s
k ∂ Hi−s
) ∂ xl ∂ xl ∂ xn ∂ xn n n ∂ Hl−s ∂ Hl−s ∂ Hl ∂ Hl + Dsk ( n−s + ) + Dsi ( n−s + )]. ∂ xi ∂ xi ∂ xk ∂ xk +
+
(58)
Algebraic Foundations of a Theory of Dispersionless Integrable Hierarchies
653
For arbitrary indices i, k, l, n and for m > i, k, l, n; m > n + i; m > l + i; m > l + k; m ≤ n + k Eqs. (58) are reduced to −
i ∂ Hk+n−m
∂ xl
+
l ∂ Hk+n−m
∂ xi
= 0.
Since k + n − m ≥ 0 one has therefore ∂ H pi ∂ xl
=
∂ H pl
(59)
∂ xi
for p ≥ 0 and i, l ≥ 1. Using this condition, one can show that Tiklnm can be represented as Tiklnm =
∂ [H, H ]l,n,k−m ∂ [H, H ]l,n,i−m ∂ [H, H ]i,k,l−m ∂ [H, H ]i,k,n−m + − − . ∂ xi ∂ xk ∂ xn ∂ xl
So Eqs. (58) take the form ∂
∂
0 0 + [H, H ]l,n,k−m + δkm Cln [H, H ]l,n,i−m + δim Cln ∂ xi ∂ xk (60) ∂
∂
0 0 − + δ0m Q ikln = 0. − [H, H ]i,k,l−m + δlm Cik [H, H ]i,k,n−m + δnm Cik ∂ xn ∂ xl For arbitrary indices i, k, l, n and for m < i, k, l, n ; m = 0 these equations are reduced to ∂ [H, H ]i,k,l−m ∂ [H, H ]i,k,n−m ∂ [H, H ]l,n,k−m ∂ [H, H ]l,n,i−m + − − = 0. ∂ xi ∂ xk ∂ xn ∂ xl These equations are satisfied by virtue of the associativity conditions. For arbitrary indices i, k, l, n and for m = k ; m < i, l, n ; m = 0 Eq. (60) is ∂
0 = 0. [H, H ]l,n,0 + Cln ∂ xi It is satisfied if 0 = 0. [H, H ]l,n,0 + Cln
Since [H, H ]l,n,0 = H0l+n − Hnl − Hln − H0l H0n +
l s=0
n Hl−s H0s +
n
l Hn−s H0s
s=0
the above equations just follow from the associativity conditions. In the same manner one can show that in the cases m = 0 and m = i or m = l or m = n Eqs. (60) are satisfied owing to the associativity conditions. At m = 0 and arbitrary i, k, l, n = 0 Eqs. (60) take the form ∂ [H, H ]i,k,l ∂ [H, H ]i,k,n ∂ [H, H ]l,n,k ∂ [H, H ]l,n,i + − − + Q ikln = 0, ∂ xi ∂ xk ∂ xn ∂ xl
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and, consequently, they reduce to Q ikln = 0. It is a direct but quite cumbersome check that these equations also are satisfied by virtue of Eqs. (56),(57), and (59). Thus, the coisotropic deformations of the generalized dKP algebra defined by the structure constants (35) are given by the associativity conditions (56), (57), and the exactness conditions (59). The dKP case corresponds to the particular choice of H0i = 0 and D jk = 0. 10. Appendix B For the algebra given by 0 pα pβ = Hαβ pα + Hβα pβ + Cαβ α = β
(61)
the associativity conditions give rise to the system 0 Cαβ − Hγα Hγβ + Hγα Hαβ + Hγβ Hβα = 0
α = β = γ = α
(62)
and γ
0 0 0 0 Hαβ Cαγ + Hβα Cβγ − Hβ Cαγ − Hγβ Cαγ = 0.
(63)
Equations (62) imply 1/2n(n − 1)(n − 2) equations: β
β
Hγα Hγβ − Hγα Hαβ − Hγβ Hβα = Hγα˜ Hγ˜ − Hγα˜ Hαβ − Hγ˜ Hβα ,
(64)
where α, β, γ , γ˜ are all distinct. Then it is a simple check that (63) is satisfied due to (62) and (64).Thus the associativity conditions for the algebra (61) are given by Eqs. (62) 0 vanish they are reduced to Eqs. (40). and (64). In the particular case when all Cαβ In order to find the coisotropy conditions, let us first compute the Poisson brackets { f αβ , f γ δ } for pairs of Hamiltonians corresponding to indices α, β, γ , δ all distinct. Since xα , pα , for α = 1, 2, . . . , n are pairs of conjugate canonical variables, we have γ γ β ∂ Hβα ∂ Hδ ∂ Hδ ∂ Hα { f αβ , f γ δ } = − − f αδ + f βδ ∂ xβ ∂ xγ ∂ xα ∂ xγ β ∂ Hβα ∂ Hγδ ∂ Hγδ ∂ Hα + − − f αγ + f βγ ∂ xβ ∂ xδ ∂ xα ∂ xδ γ β β β β ∂ Hγδ γ ∂ Hα δ ∂ Hα γ ∂ Hα δ ∂ Hα γ δ ∂ Hδ + −Hα − Hα + Hγ − Hδ + Hα + Hα pα ∂ xγ ∂ xδ ∂ xγ ∂ xδ ∂ xβ ∂ xβ (65) α γ γ γ γ β β ∂ Hβ β ∂ Hδ α ∂ Hα α ∂ Hδ β ∂ Hδ α ∂ Hδ + −Hδ − Hδ + Hδ + Hδ − Hα − Hβ pδ ∂ xγ ∂ xγ ∂ xα ∂ xβ ∂ xα ∂ xβ α γ δ ∂ Hβα ∂ Hβα ∂ Hβα γ ∂ Hβ γ ∂ Hγ δ δ δ δ ∂ Hδ − −Hβ − Hβ + Hγ + Hγ + Hβ + Hβ pβ ∂ xγ ∂ xδ ∂ xγ ∂ xδ ∂ xα ∂ xα β ∂ Hβα ∂ Hγδ ∂ Hγδ ∂ Hγδ ∂ Hγδ α ∂ Hα β β α β α + −Hγ − Hγ + Hγ + Hγ − Hα − Hβ pγ , ∂ xδ ∂ xδ ∂ xα ∂ xβ ∂ xα ∂ xβ
Algebraic Foundations of a Theory of Dispersionless Integrable Hierarchies
655
where there is not summation over repeated indices. The Poisson brackets of Hamiltonians having equal indices, like { f αβ , f βδ }, may be obtained from this formula by means of β the following substitutions: Hβ = 0 and f ββ → p2β + f 11β + vβ pβ + u β , where f 11β = pβ2 − p2β − vβ pβ − u β is the lowest Hamiltonian for the β th dKP hierarchy. The coisotropy condition requires that the r.h.s of (65) vanishes for the values of (xα , pα ) for which f αβ = 0 for all α = β. In the case when all indices in (65) are distinct it is satisfied if the coefficients in front of pα , pβ , pγ , pδ all vanish. It is not difficult to check that all these equations are satisfied by virtue of the system Hαδ
β
β
β
∂ Hα ∂ Hα ∂ Hα + Hαγ − Hγδ ∂ xγ ∂ xδ ∂ xγ
β ∂ Hγδ ∂ Hγδ ∂ Hα − Hαγ − Hαδ = 0, ∂ xδ ∂ xβ ∂ xβ α = β = γ = δ = α. γ
− Hδ
(66)
The coisotropy conditions corresponding to a pair of Hamiltonians with coinciding indices, say { f αβ , f βδ } require that the coefficients in front of p2β , of pα and pδ , and of pβ vanish. The first requirement leads to ∂ Hβδ ∂ xα
−
∂ Hβα ∂ xδ
=0
α = β = δ = α.
(67)
The second requirement gives Eq. (66) with β = γ , and finally the third requirement gives the equation relating Hβα with the functions vβ , u β for the β th KP hierarchy. So the coisotropy conditions contain the subset of Eqs. (66), (67) containing only the functions Hβα for α = β. Using (67), one shows that Eqs. (66) can be recast in the form ∂ γ δ γ Hα Hα − Hαδ Hδ − Hαγ Hγδ = 0. ∂ xβ
(68)
Thus this subset of coisotropy conditions is equivalent to the associativity conditions (40) and to the exactness conditions (67). Note that in terms of the functions Fα introduced in Sect. 4 one has { f αβ , f γ δ } =
∂ 2 (Fδ − Fβ ) ∂ 2 (Fδ − Fα ) f αδ + f βδ ∂ xβ ∂ xγ ∂ xα ∂ xβ ∂ 2 (Fγ − Fα ) ∂ 2 (Fγ − Fβ ) + f αγ + f βγ . ∂ xβ ∂ xδ ∂ xα ∂ xδ
(69)
This representation of the dDarboux system is the quasiclassical limit of the operator representation of the original Darboux system. The coisotropic submanifold defined by the equations 0 f αβ = pα pβ − Hαβ pα − Hβα pβ − Cαβ = 0,
α = β,
(70)
and the quadrics defined by the associativity conditions have quite remarkable properties. First, the quadrics (70) are transformed into quadrics of the same type under a Cremona transformation pα → ξα = p1α . Indeed one gets 0 αβ ξα − H βα ξβ − C αβ = 0, ξα ξβ − H
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where β
αβ = − Hα H 0 Cαβ
1 0 αβ = 0 . C Cαβ
0 = 0 Cremona’s transformation linearises the quadric (70), In the particular case Cαβ since in this case
Hαβ ξβ + Hβα ξα − 1 = 0
for
α, β = 1, 2, . . . , n α = β.
(71)
So for the dDarboux system any section of the coisotropic submanifold with xα = const is , in fact, the intersection of the planes (71). Secondly, the equations defining the associativity quadrics can be rewritten as γ
β
Hγ
β
+
Hα
Hβ
γ
Hα
− 1 = 0,
α = β = γ = α.
In the case n = 3 , in terms of the variables z 1 , z 2 , z 3 defined by z1 =
H21 H31
,
z2 =
H32 H21
,
z3 =
H13 H23
,
the above equations become z1 +
1 = 1, z2
z2 +
1 = 1, z3
z3 +
1 = 1, z1
with the obvious constraint z 1 z 2 z 3 = −1. These equations define a curve in R 3 which is the intersection of three cylinders which are generated by the above hyperbolas. Acknowledgements. The authors are grateful to A. Moro for the help in preparation of the paper. The work has been partly supported by the grants COFIN 2004 “Sintesi”, and COFIN 2004 “Nonlinear Waves and Integrable systems”.
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9. Dubrovin, B.A., Novikov, S.P.: Hydrodynamics of weakly deformed soliton lattices: differential geometry and Hamiltonian theory. Russ. Math. Surv. 44, 35–124 (1989) 10. Kuperschmidt, B.A.: The quasiclassical limit of the modified KP hierarchy. J. Phys. A: Math. Phys. 23, 871–886 (1990) 11. Kuperschmidt, B.A.: Factorization of quasiclassical integrable systems. Physica D 44, 565–574 (1990) 12. Takasaki, K., Takebe, T.: Sdiff(2) KP hierarchy. Int. J. Mod. Phys. A 7, 889–922 (1992) 13. Takasaki, K., Takebe, T.: Integrable hierarchies and dispersionless limit. Rev. Math. Phys. 7, 743–818 (1995) 14. Krichever, I.M.: The tau-function of the universal Whitham hierarchy, Matrix models and Topological field theory. Commun. Pure Appl. Math. 47, 437–475 (1994) 15. Ercolani, N.M., Gabitov, I., Levermore, D., Serre, D. (eds.): Singular limits of dispersive waves. Nato Advance Sci. Inst. Ser. B Physics, Vol. 320, New York: Plenum, 1994 16. Zakharov, V.E.: Dispersionless limit of integrable systems in 2+1-dimensions. In: Singular limits of dispersive waves. (Ercolani, N.M. et al., eds.) New York: Plenum, 1994 17. Carroll, R., Kodama, Y.: Solutions of the dispersionless Hirota equations. J. Phys. A: Math. Gen. 28, 6373– 6387 (1995) 18. Strachan, I.A.B.: The Moyal bracket and dispersionless limit of the KP hierarchy. J. Phys. A.: Math. Gen. 28, 1967–1975 (1995) 19. Dubrovin, B.A.: Hamiltonian formalism of Whitham type hierarchies and topological Landau-Ginzburg models. Commun. Math. Phys. 145, 195–203 (1992) 20. Jin, S., Levermore, C.D., McLaughlin, D.W.: The semiclassical limit of defocusing NLS hierarchy. Commun. Pure Appl. Math 52, 613–654 (1999) 21. Gibbons, J., Tsarev, S.P.: Reductions of the Benney equations. Phys. Lett. A 211, 19–23 (1996) 22. Gibbons, J., Tsarev, S.P.: Conformal maps and reductions of the Benney equations. Phys. Lett. A 258, 263– 271 (1999) 23. Li, L.C.: Classical r -matrices and compatible Poisson structures for Lax equations on Poisson algebras. Commun. Math. Phys. 203, 573–592 (1999) 24. Chang, J.-H., Tu, M.-H.: On the Miura map between the dispersionless KP and dispersionless modified KP hierarchies. J. Math. Phys. 41, 5391–5406 (2000) 25. Mineev-Weinstein, M., Wiegmann, P.B., Zabrodin, A.: Integrable structures of interface dynamics. Phys. Rev. Lett. 84, 5106–5108 (2000) 26. Wiegmann, P.B., Zabrodin, A.: Conformal maps and integrable hierarchies. Commun. Math. Phys. 213, 523–538 (2000) 27. Takhtajan, L.A.: Free bosons and tau function for compact Riemann surfaces and smooth Jordan curves: Current correlation functions. Lett. Math. Phys. 56, 181–228 (2001) 28. Dunaiski, M., Mason, L.J., Tod, P.: Einstein-Weyl geometry, the dKP equation and twistor theory. J. Geom. Phys. 37, 63–93 (2001) 29. Boyaskj, A., Marshakov, A., Ruchayskiy, O., Wiegmann, P.B., Zabrodin, A.: Associativity equations in dispersionless integrable hierarchies. Phys. Lett. 515, 483–492 (2001) ¯ 30. Konopelchenko, B.G., Martinez, A.L.: ∂-equations,integrable deformations of quasiconformal mappings and Whitham hierarchy. Phys. Lett. A 286, 161–166 (2001) 31. Konopelchenko, B.G., Martinez, A.L.: Dispersionless scalar hierarchies,Whitham hierarchy and tha quasi¯ classical ∂-method. J. Math. Phys. 43, 3807–3823 (2003) 32. Konopelchenko, B.G., Martinez, A.L.: Nonlinear dynamics in the plane and integrable hierarchies of infinitesimal deformations. Stud. Appl. Math. 109, 313–336 (2002) 33. Manas, M., Martinez, A.L., Medina, E.: Reductions and hodograph solutions for dispersionless KP hierarchy. J. Phys. A: Math. Gen. 35, 401–417 (2002) 34. Bogdanov, L.V., Konopelchenko, B.G.: Nonlinear Beltrami equation and tau function for dispersionless hierarchies. Phys. Lett. A 322, 330–337 (2004) 35. Konopelchenko, B.G., Moro, A.: Integrable equations in nonlinear geometrical optics. Stud. Appl. Math. 113, 325–352 (2004) 36. Ferapontov, E.V., Khusnutdinova, K.R.: On the integrability of 2+1-dimensional quasilinear systems. Commun. Math. Phys. 248, 187–206 (2004) 37. Ferapontov, E.V., Khusnutdinova, K.R., Tsarev, S.P.: On a class of three-dimenional integrable Lagrangians. Commun. Math. Phys. 261, 225–243 (2006) 38. Krichever, I., Mineev-Weinstein, M., Wiegmann, P.B., Zabrodin, A.: Laplacian growth and Whitham equations in soliton theory. Physica D 198, 1–28 (2004) 39. Zakharov, V.E., Manakov, S.V., Novikov, S.P., Pitaevski, L.P.: The theory of solitons: the inverse problem method. New York: Plenum, 1984 40. Ablowitz, M., Clarkson, P.: Nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press, 1991
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41. Falqui, G., Magri, F., Pedroni, M.: Bihamiltonian geometry, Darboux coverings, and linearization of the KP hierarchy. Commun. Math. Phys. 197, 303–324 (1998) 42. Witten, E.: On the structure of topological phase of two-dimensional gravity. Nucl. Phys. B 340, 281– 332 (1990) 43. Dijkgraaf, R., Verlinde, H., Verlinde, E.: Topological strings in d <1. Nucl. Phys. B 352, 59–86 (1991) 44. Dubrovin, B.A.: Geometry of 2D topological field theories. Lect, Notes in Math., Vol. 1620. Berlin: Springer, 1996, p. 1620 45. Konopelchenko, B.G.: Soliton eigenfunction equations: the IST integrability and some properties. Rev. Math. Phys. 52, 399–440 (1990) 46. Darboux, G.: Lecons sur les systemes orthogonaux et les coordonne’es curvilignes. Paris: Hermann, 1910 47. Weinstein, A.: Coisotropic calculus and Poisson groupoids. J. Math. Soc. Japan 40, 705–727 (1988) 48. Karasev, M.V., Maslov, V.P.: Nonlinear Poisson brackets,geometry and quantization. Trans. Math. Monogr. Vol 119, Providence, RI: Amer. Math. Soc., 1993 49. Berndt, R.: An introduction to symplectic geometry. Providence, RI: Amer. Math. Soc., 2001 50. Arnold, V.I.: Mathematical methods of Classical Mechanics. New York: Springer, 1978 51. Arnold, V.I., Novikov, S.P.: Dynamical Systems IV: Symplectic geometry and its applications. Berlin: Springer, 1990 52. Weinstein, A.: Lectures on Symplectic manifolds. Providence, RI: Amer. Math. Soc., 1979 Communicated by L. Takhtajan
Commun. Math. Phys. 274, 659–689 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0291-6
Communications in
Mathematical Physics
A Formal Model of Berezin-Toeplitz Quantization Alexander V. Karabegov Department of Mathematics, Abilene Christian University, ACU Box 28012, Abilene, TX 79699-8012, USA. E-mail: [email protected] Received: 14 July 2006 / Accepted: 21 February 2007 Published online: 13 July 2007 – © Springer-Verlag 2007
Abstract: We give a new construction of symbols of the differential operators on the sections of a quantum line bundle L over a Kähler manifold M using the natural contravariant connection on L. These symbols are the functions on the tangent bundle T M polynomial on fibres. For high tensor powers of L, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on T M corresponding to some pseudo-Kähler structure on T M. Surprisingly, this star product is intimately related to the formal symplectic groupoid with separation of variables over M. We extend the star product on T M to generalized functions supported on the zero section of T M. The resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the Bergman projection operator. Using this idempotent, we define an algebra of Toeplitz elements and show that it is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on M.
1. Introduction Deformation quantization of a Poisson manifold (M, {·, ·}) is an associative algebra structure on the space C ∞ (M)[ν −1 , ν]] of formal functions on M with the product (named the star product) φψ =
∞
ν r Cr (φ, ψ),
r =0
where Cr are bidifferential operators on M such that C0 (φ, ψ) = φψ and C1 (φ, ψ) − C1 (ψ, φ) = i{φ, ψ}.
(1)
660
A. V. Karabegov
Here we use the following definition of formal vectors. Given a vector space V , we denote by V [ν −1 , ν]] the space of formal Laurent series of the form v=
ν r vr ,
r ≥n
where vr ∈ V and n is possibly negative, and call its elements formal vectors. We assume that the unit constant 1 is the unity of the star algebra (C ∞ (M)[ν −1 , ν]], ). Two star products 1 and 2 are called equivalent if there exists a formal differential operator B = 1 + ν B1 + ν 2 B2 + · · · such that φ 1 ψ = B −1 (Bφ 2 Bψ) for any formal functions φ, ψ ∈ C ∞ (M)[ν −1 , ν]]. A star product can be localized to any open subset U ⊂ M. Deformation quantization of Poisson manifolds was introduced in the seminal work [1]. The existence and classification of deformation quantizations on the symplectic manifolds were established in a number of papers (see [10, 12, 26, 11, 13, 24, 4, 29]). Kontsevich showed the existence and gave the classification of deformation quantizations on an arbitrary Poisson manifold in [22]. The concept of a star product is related to the notion of operator symbols and their composition. A star product φ ψ can be thought of as the asymptotic expansion of a family of symbol products φ ∗h ψ depending on a small parameter h as h → 0, where the asymptotic parameter h is replaced with the formal parameter ν. For instance, this way one obtains Moyal deformation quantization from the composition of Weyl symbols. In some cases the space of h-dependent operator symbols does not carry a natural symbol product, but the symbol-operator correspondence has a well-defined classical limit as h → 0 which allows to define a star product (see [14, 28]). The examples of star products related to symbols are mostly obtained from covariant and contravariant symbols on Kähler manifolds, introduced by Berezin (see [2, 3, 7, 8, 23, 18, 5, 28, 21]). These star products on Kähler manifolds enjoy a special property that the bidifferential operators Cr in (1) differentiate their first argument in holomorphic directions and the second argument in antiholomorphic ones or vice versa. The deformation quantizations with this property are called deformation quantizations with separation of variables or of the Wick type (see [16, 6, 27]). It was shown in [16] that all deformation quantizations with separation of variables on a Kähler manifold can be explicitly constructed and bijectively parameterized by the formal deformations of the Kähler form (see also [25]). In this paper we consider a formal model of the construction of Berezin-Toeplitz quantization (see [5, 28]). Berezin-Toeplitz quantization on a compact Kähler manifold M uses the following data. Let L be a quantum line bundle on M (see the details in the main body of the paper). Denote by (N ) the orthogonal projector onto the space of holomorphic sections of L ⊗N , the N th tensor power of L (the Bergman projector). To a given (N ) function φ ∈ C ∞ (M) there corresponds the Toeplitz operator Tφ = (N ) ◦ φ ◦ (N ) on the sections of L ⊗N (here ◦ denotes composition of operators). The function φ is (N ) then called a contravariant symbol of the operator Tφ . The symbol-operator mapping (N )
φ → Tφ is not injective and therefore there is no natural product of contravariant symbols. However, one can extract a star product from the asymptotics of the Toeplitz
Berezin-Toeplitz Quantization
661
operators as N → ∞. It was proved in [28] that there exists a unique deformation quantization (1) on M, the Berezin-Toeplitz deformation quantization, such that for each k, ||Tφ(N ) Tψ(N ) −
k−1 1 (N ) 1 . T || = O N r Cr (φ,ψ) Nk r =0
In [21] the Berezin-Toeplitz deformation quantization was completely identified. It was shown that the deformation quantization with the opposite star product is a deformation quantization with separation of variables whose characterizing formal deformation of the Kähler form was explicitly calculated. In this paper we define symbols of the differential operators on the sections of L ⊗N and study the corresponding symbol product ∗1/N . These symbols are the fibrewise polynomial functions on the tangent bundle T M. The asymptotic expansion of the symbol product ∗1/N as N → ∞ leads to the star product ∗ of a deformation quantization with separation of variables on the tangent bundle T M endowed with some pseudo-Kähler structure. It is not clear how to extend the symbols of differential operators introduced in this paper to wider classes of operators that would include, in particular, the Bergman projector (N ) . However, one can expect from semiclassical considerations that, as N → ∞, such a symbol of (N ) might have a well-defined limit which would be a generalized function supported at the zero section Z of the tangent bundle T M. We show that the star product ∗ can be naturally extended to a class of generalized functions supported at Z and that this class contains an idempotent element. Using this idempotent, one can define an algebra of Toeplitz elements. We show that this algebra is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on M. 2. Deformation Quantizations with Separation of Variables Let M be a complex manifold endowed with a Poisson bivector field η of type (1,1) with respect to the complex structure. We call such manifolds Kähler-Poisson. If η is nondegenerate, M is a pseudo-Kähler manifold. On a coordinate chart U ⊂ M with ¯ local holomorphic coordinates {z k , z¯ l }, we write η = iglk ∂z∂ k ∧ ∂∂z¯l . The condition that ¯
η is Poisson is expressed in terms of the tensor glk as follows: ¯
glk
¯
¯ lm g nm nk ¯ g = g ∂z k ∂z k
and
¯
glk
¯ nk ¯ g nm ¯ g lm = g . ∂ z¯ l ∂ z¯ l
The corresponding Poisson bracket on M is given locally by the formula ∂φ ∂ψ ∂ψ ∂φ ¯ . {φ, ψ} M = iglk − ∂z k ∂ z¯ l ∂z k ∂ z¯ l
(2)
(3)
Deformation quantization (1) on a Kähler-Poisson manifold M is called a deformation quantization with separation of variables if the bidifferential operators Cr differentiate their first argument in antiholomorphic directions and its second argument in holomorphic ones. Denote by L φ the operator of star multiplication by a function φ from the left and by Rψ the operator of star multiplication by a function ψ from the right, so that
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A. V. Karabegov
L φ ψ = φ ψ = Rψ φ. Recall that the associativity of the star product is equivalent to the condition that [L φ , Rψ ] = 0 for any φ, ψ ∈ C ∞ (M)[ν −1 , ν]]. With the assumption that the unit constant 1 is the unity of the star product, the condition that is a star product with separation of variables can be reformulated as follows. For any local holomorphic function a and antiholomorphic function b the operators L a and Rb are the operators of pointwise multiplication by the functions a and b, respectively, L a = a, Rb = b. It is easy to check that ¯
C1 (φ, ψ) = glk
∂φ ∂ψ . ∂ z¯ l ∂z k
(4)
The Laplace-Beltrami operator given locally by the formula ¯
= glk
∂2 ∂z k ∂ z¯ l
(5)
is coordinate invariant and thus globally defined on M. For a given star product with separation of variables on M there exists a unique formal differential operator B on M such that B (ab) = b a
(6)
for any local holomorphic function a and antiholomorphic function b. The operator B is called the formal Berezin transform of the star product (see [17]). We see from Eqs. (4) and (5) that B = 1 + ν + · · · . In particular, B is invertible. One can recover the star product of a deformation quantization with separation of variables from its formal Berezin transform. Using B as an equivalence operator, one can define a star product on (M, η) as follows: φ ψ = B−1 (B φ B ψ). We call the star product opposite to the dual star product to the star product and denote it by ˜ so that φ ˜ ψ = ψ φ. It was shown in [20] that ˜ defines a deformation quantization with separation of variables on the complex manifold M endowed with the opposite Poisson bivector field −η, while the star product defines a deformation quantization with separation of variables on the manifold M¯ with the opposite complex structure and the same Poisson bivector field η. The formal Berezin transform of the dual star product ˜ is B−1 and the star product dual to ˜ is . It follows from (6) that B a = a
and
B b = b.
(7)
In particular, B 1 = 1. It was proved in [20] that for any local holomorphic function a and antiholomorphic function b, B a B−1 = Ra
and
B bB−1 = L b .
Berezin-Toeplitz Quantization
663
We call a formal differential operator A = A0 + ν A1 + · · · natural if the operator Ar is of order not greater than r for any r ≥ 0. A star product (1) is called natural if for each r the bidifferential operator Cr is of order not greater than r with respect to each of its arguments (see [15]). For a natural star product the operators L f and R f are natural for any f ∈ C ∞ (M). A star product with separation of variables on a Kähler-Poisson manifold is natural (see [20, 6, and 25]). Let (M, ω−1 ) be a pseudo-Kähler manifold. We say that a formal closed (1, 1)-form ω = (1/ν)ω−1 + ω0 + νω1 + · · · is a formal deformation of the form ω−1 . Deformation quantizations with separation of variables on M are bijectively parametrized by the formal deformations of the pseudo-Kähler form ω−1 as follows. Assume that ω is a formal deformation of the form ω−1 . On a contractible coordinate chart U ⊂ M the formal form ¯ There exists ω has a potential = (1/ν) −1 + 0 + ν 1 + · · · such that ω = −i∂ ∂ . a unique star product on (M, ω−1 ) such that on each contractible coordinate chart U for any holomorphic function a and antiholomorphic function b the following formulas hold: ∂ ∂ ∂
−
e , L a = a, L ∂ = k + k = e ∂z ∂z ∂z k ∂z k ∂ ∂ ∂
−
Rb = b, R ∂ = l + l = e e . ∂ z¯ ∂ z¯ ∂ z¯ l ∂ z¯l Namely, the centralizer of the operators Rb and R∂ /∂ z¯l in the algebra of formal differential operators on U can be identified with the algebra of left multiplication operators with respect to some star product on U . This star product does not depend on the choice of the potential and determines a global deformation quantization with separation of variables on M parameterized by the formal form ω. The star product ˜ dual to gives a deformation quantization with separation of variables on the pseudo-Kähler manifold (M, −ω−1 ). It is parameterized by a formal form ω˜ = −(1/ν)ω−1 + ω˜ 0 + ν ω˜ 1 + · · ·. A formal density ρ = r ≥n ν r ρr (where n is possibly negative) is called a trace density of a star product on M if for any functions φ, ψ ∈ C ∞ (M) such that at least one of them has compact support, the following identity holds: φψρ = ψ φ ρ. M
M
Recall that a star product on a symplectic manifold has local derivations of the form δ = d/dν + A, where A is a formal differential operator (it does not contain derivatives with respect to the formal variable ν). They are called ν-derivations. On a symplectic manifold each star product has formal trace densities which differ by formal constant factors. There exists a canonical trace density µ uniquely determined by the following two requirements (see [17]): (a) The leading term of the formal series µ is given by the formula 1 ν m m!
m ω−1 ,
(8)
where m is the complex dimension of M. (b) Given any open subset U ⊂ M with a ν-derivation δ of the star product on it and any function f ∈ C0∞ (U ), the following identity holds: d f µ= δ( f ) µ. dν U U
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A. V. Karabegov
We call a formal function φ ∈ C ∞ (M)[ν −1 , ν]] invertible if it can be represented as φ= ν r φr r ≥n
for some integer n and with φn nonvanishing. Locally the function φ can be represented in the form φ = eθ , where θ = n log ν + θ0 + νθ1 + · · · . Let be the star product of the deformation quantization with separation of variables on a pseudo-Kähler manifold (M, ω−1 ) parameterized by a formal form ω. In [17] we gave an explicit construction of the canonical trace density of the star product . Denote by B the formal Berezin transform of the star product . Let U be a contractible coordinate chart and = (1/ν) −1 + · · · a formal potential of ω on U . There exists a potential of the dual form ω˜ expressed as = −m log ν − (1/ν) −1 + 0 + ν1 + · · · and such that ∂ ∂
∂ ∂
d d
= − = − =− . B , B and B k k l l ∂z ∂z ∂ z¯ ∂ z¯ dν dν
(9)
The potential is determined by Eq. (9) uniquely up to a constant summand. The first two equations in (9) determine only up to a formal constant. The canonical trace density of the star product can be expressed on U as follows: µ = Ce( +) dzd z¯ ,
(10)
where the constant factor C is uniquely determined by requirement (8). The star products , , and ˜ have the same trace densities and the same canonical trace density. 3. Symbols of Differential Operators on a Quantum Line Bundle In this section we will relate to a pseudo-Kähler manifold M endowed with a pseudo-Kähler form ω−1 a family of associative products on the fibrewise polynomial functions on the tangent bundle T M. Let U be a contractible coordinate chart on M with holomorphic coordinates {z k , z¯ l }. Consider a holomorphic Hermitian line bundle L on U with the fibre metric | · | whose metric-preserving covariant connection has the curvature ω−1 . Such a line bundle is called ‘quantum’. Let s be a nonvanishing holomorphic sec¯ −1 . tion of L. Then −1 = − log |s|2 is a potential of the form ω−1 so that ω−1 = −i∂ ∂
¯ ∂ 2 −1 k l In coordinates, ω−1 = −igkl¯dz ∧ d z¯ , where gkl¯ = ∂z k ∂ z¯l . The inverse matrix (glk ) ¯
of (gkl¯) determines a (global) Poisson bivector field η = iglk ∂z∂ k ∧ Eq. (2). For a given positive integer N we will set h=
1 . N
∂ ∂ z¯ l
and satisfies
Berezin-Toeplitz Quantization
665
Consider the N th tensor power L ⊗N of the line bundle L and denote h =
1
−1 . h
In the trivialization of L ⊗N determined by the section s N the metric-preserving covariant connection ∇• is as follows: ∇k =
∂ ∂h ∂ − , ∇l¯ = l . k k ∂z ∂z ∂ z¯
(11)
Introduce a contravariant connection ∇ • on L ⊗N by lifting the index in (11) via the ¯ tensor hglk : ∂ ∂h ¯ ∂ ¯ ∇ p = −hgl p l , ∇ q¯ = hg qk . − ∂ z¯ ∂z k ∂z k Let {ηk , η¯ l } be the fibre coordinates on the tangent bundle T U corresponding to the base coordinates {z k , z¯ l }. For a function f = f (z, z¯ ) on U denote the pointwise multiplication operator by f by the same symbol. Introduce the following operators on the sections of L ⊗N : p = ∇ p , η fˆ = f, η ¯ q = ∇ q¯ . We make a crucial observation that, according to Eq. (2), the operators z 1 , . . . , z m , ¯ ∇ 1 , . . . , ∇ m pairwise commute and the operators z¯ 1 , . . . , z¯ m , ∇ 1 , . . . , ∇ m¯ pairwise commute as well. We can treat these commuting families as ‘coordinate’ and ‘momentum’ operators on the sections of L ⊗N and define symbols of differential operators on L ⊗N via the normal ordering. Denote by Pr (T U ) the space of fibrewise polynomial functions on the tangent bundle T U whose fibrewise degree is not greater than r and set P(T U ) = ∪r Pr (T U ). To a fibrewise polynomial function P= u i (η) f i vi (η) ¯ i
from P(T U ), where u i and vi are monomials in the fibre variables {ηk } and {η¯ l }, respectively, we relate a differential operator Pˆ on L ⊗N represented in the normal form Pˆ = u i (η) ˆ f i vi ( η). ¯ i
The symbol-operator mappings P → Pˆ is a bijection of the space of fibrewise polynomial functions on T U onto the space of differential operators on the sections of the line bundle L ⊗N . The symbol product will be denoted ∗h so that for symbols P, Q ∈ P(T U ), ˆ (P ∗h Q) = Pˆ Q. Denote by L P and R P the operators of left and right multiplication by a symbol P with respect to the product ∗h , respectively. Notice that for functions φ, ψ on U and monomials u = u(η), v = v(η), ¯ φ ∗h ψ = φψ,
L u = u(η),
and
Rv = v(η). ¯
(12)
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A. V. Karabegov
Due to the commutation relations ∂f p , fˆ] = −h gl¯ p [η ∂ z¯ l
and
¯ ∂f [η ¯ q , fˆ] = h g qk , ∂z k
(13)
where f = f (z, z¯ ) is a function on U , we get that ¯
f ∗h η p = η p f + h g l p
∂f ∂ z¯ l
and
¯ η¯ q ∗h f = f η¯ q + h g qk
∂f . ∂z k
(14)
It follows from (12) and (14) that the symbols which do not depend on the antiholomorphic fibre variables {η¯ l } form a subalgebra of the algebra (P(T U ), ∗h ) which will be denoted A A. Denote by L A P , R P the operators of left and right multiplication by a symbol P in the algebra A, respectively. Similarly, B will denote the subalgebra of symbols which do not depend on the variables {ηk } and L BP , R B P the left and right multiplication operators by a symbol P in the algebra B. For a function f on U , RA f = f
and
L Bf = f.
and
q qk ¯ LB η¯ q = η¯ + h g
(15)
Formulas (12) and (14) imply that ¯
RηAp = η p + h gl p
∂ ∂ z¯ l
∂ . ∂z k
(16)
Let D(T U ) denote the space of differential operators on T U . It has a decreasing filtration by the subspaces Dr (T U ) ⊂ D(T U ) of differential operators annihilating Pr (T U ). ˆ U ) the completion of the space D(T U ) with respect to this filtration. Denote by D(T ˆ U ) act on the space P(T U ) as differential operators of infinite The elements of D(T ˆ U ) by the formulas order. Introduce operators Jh , K h ∈ D(T ∂2 ∂2 ¯ ¯ and K h = exp hglk k l . Jh = exp hglk k l ∂η ∂ z¯ ∂z ∂ η¯ Here the exponentials are given by convergent series in the topology determined by the filtration. We need the following lemma. Lemma 1. For any functions φ, ψ on U the operators Jh φ Jh−1 and K h φ K h−1 commute with the operator of point-wise multiplication by the function ψ,
Jh φ Jh−1 , ψ = K h φ K h−1 , ψ = 0. ˆ U ) the ring of operators of the form A = A(z, z¯ , ∂ ), i.e., Proof. Denote by S ⊂ D(T ∂η generated by the multiplication operators by the functions f = f (z, z¯ ) and the derivations ∂η∂ k . The inclusion
∂2 hg ,S ⊂ S ∂ηk ∂ z¯ l ¯ lk
implies that Jh S Jh−1 ⊂ S. The statement that the operators Jh φ Jh−1 and ψ commute follows from the fact that the ring S is commutative and contains the multiplication operators by the functions f = f (z, z¯ ). The rest of the lemma can be checked similarly.
Berezin-Toeplitz Quantization
667
Notice that for a function f = f (z, z¯ ) and monomials u = u(η), v = v(η), ¯ Jh f = Jh−1 f = K h f = K h−1 f = f, Jh u = Jh−1 u = u, K h v = K h−1 v = v.
(17)
Using Eqs. (2) and (16) one can check that RηAp = Jh η p Jh−1
and
q −1 LB η¯ q = K h η¯ K h .
Taking into account Eqs. (17) and (18) we get
f ∗h u(η) = RuA f = u RηA f = Jh u(η)Jh−1 f = Jh (u f ) = Jh f Jh−1 u.
(18)
(19)
This calculation allows to determine the operator L f . Lemma 2. The operators of left and right multiplication by a function f = f (z, z¯ ) in the algebra (P(T U ), ∗h ) can be expressed as follows: L f = Jh f Jh−1
and
R f = K h f K h−1 .
Proof. We will prove the first formula using Eqs. (12), (15), and (19). For a function φ = φ(z, z¯ ) and monomials u = u(η), v = v(η) ¯ we get, taking into account Lemma 1 ¯ and that the operator Jh commutes with the operator of pointwise multiplication by v(η), that
f ∗h (uφv) = (( f ∗h u) ∗h φ) ∗h v = Jh f Jh−1 u φv = Jh f Jh−1 (uφv). The second formula in the lemma can be checked similarly. Introduce a potential h on U by the following formula: h = h +
∂h k ∂h l η + η¯ + log g, ∂z k ∂ z¯ l
(20)
where g = det(gkl¯). We need to calculate a number of commutation relations in the algebra of differential operators on L ⊗N . Lemma 3. The following formulas and commutation relations hold: h ∂h ∂ = , p ∂η ∂z p h ∂ ∂h = , q ∂ η¯ ∂ z¯ q
2 h ∂ ∂ ¯ ∂ h ∂ lk = − hg ; ∂z p ∂z p ∂z k ∂z p ∂ z¯ l 2 h ∂ ∂ ¯ ∂ h ∂ = − q + hglk l q k q ∂ z¯ ∂ z¯ ∂ z¯ ∂ z¯ ∂z
2 ∂h ¯ ∂ h ∂h ; + q − hglk l q ∂ z¯ ∂ z¯ ∂ z¯ ∂z k ∂h l ∂h k ∂h l ∂h k k l , η = δp, , η¯ = −δq , ,η = , η¯ = 0. ∂η p ∂ η¯ q ∂z p ∂ z¯ q
(21)
(22)
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A. V. Karabegov
Proof. We have h ∂ ∂h ∂ ∂ 2 h ¯ ∂ lk = + log g − hg ◦ ∂z p ∂z p ∂z p ∂ z¯ l ∂z k ∂z p ∂ 1 ∂h ¯ + g pl¯hglk − k h ∂z ∂z k =
2 ∂h ∂ ¯ ∂ h ∂ lk + log g − hg ∂z p ∂z p ∂z k ∂z p ∂ z¯ l 2 ∂ ∂h ∂ ¯ ∂ ¯ ∂ h ∂ lk − glk k g pl + p − = − hg . ∂z ∂z ∂z p ∂z p ∂z k ∂z p ∂ z¯ l
∂ h A similar calculation provides the formula for the operator ∂ z¯ q from Eq. (22). Further,
∂h ∂h k ¯ ∂ ¯ lk = = g pl¯ glk = δ kp . , η , −hg ∂η p ∂z p ∂ z¯ l The following calculation is based on the identity ¯
∂glk ¯ ∂gm n¯ nk = −glm g¯ ∂z p ∂z p and Eq. (2): ∂h ∂ ∂ 2 h ∂ 1 k ¯ nk ¯ ∂ lm = , − − hg , g η ∂z p h ∂z p ∂z m ∂z p ∂ z¯ l ∂ z¯ n =
¯ ¯ ∂ ∂ 2 h ∂g nk ∂ ∂g nk ¯ lm − hg ∂z p ∂ z¯ n ∂z m ∂z p ∂ z¯ l ∂ z¯ n ¯ + h g nk
¯
∂glm ∂ 2 h ∂ ¯ ∂gm n¯ ∂ ¯ lm + g nk g = 0. ∂ z¯ n ∂z m ∂z p ∂ z¯ l ∂z p ∂ z¯ l
The rest of the lemma can be proved by similar calculations. Using Lemmas 2 and 3 we will calculate the left and right multiplication operators with respect to the product ∗h for a number of symbols. Lemma 4. For a holomorphic function a and an antiholomorphic function b on U the following formulas hold: L a = a, L η p = η p , Rb = b, Rη¯ q = η¯ q , ∂ ∂h ∂ ∂h L ∂h = L ∂h = + = + , p p p p p ∂η ∂η ∂η ∂η p ∂η ∂z ∂ ∂h ∂ ∂h R ∂h = R ∂h = q + q = q + , ∂ η¯ ∂ z¯ ∂ η¯ ∂ η¯ q ∂ η¯ q ∂ z¯ q ∂ ∂h ∂ ∂h L ∂h = + , R ∂h = q + q . p p ∂z ∂z ∂ z¯ ∂ z¯ ∂z p ∂ z¯ q
Berezin-Toeplitz Quantization
669
Proof. The four formulas in the first line are obvious. The formulas for L ∂h /∂η p and L ∂h /∂ η¯ q follow from Lemma 2. To prove the formula for L ∂h /∂z p we will calculate first its particular case with the use of Eqs. (14) and (15): 2 1 ∂h ∂ k ∂ h f + g pl¯ ∗h η¯ l ∗h f L ∂h f = + p log g + η p k p p ∂z ∂z ∂z ∂z h ∂z 1 ∂h ∂f ∂h ¯ ∂f f + g pl¯h glk k = f + p. (23) = ∂z p h ∂z ∂z p ∂z It follows from Eq. (23) that h ∂h ∂ ∂f ˆ f + p. f = ∂z p ∂z p ∂z
(24)
Consider a symbol P = u(η) f v(η), ¯ where u(η) and v(η) ¯ are monomials in the variables {ηk } and {η¯ l }, respectively. Taking into account Eq. (24) and Lemma 3, we obtain h h ∂ ∂ ˆv( ˆ = ∂h u(η) ˆ f η) ¯ = u( η) ˆ P fˆv( η) ¯ ∂z p ∂z p ∂z p ∂h ∂h ∂f ∂P = u(η) ˆ f v(η) ¯ + u(η) ˆ v(η) ¯ = P + p, p p p ∂z ∂z ∂z ∂z whence the formula for L ∂h /∂z p follows. The formula for L ∂h /∂ z¯ q can be checked by similar calculations. Our next task is to find an explicit expression for the operator L η¯ q which is possible due to Lemmas 2 and 4. It follows from Lemma 4 that ∂h ∂ 2 h ∂h ∂ 1 k = + log g + η ∗ + g pl¯ ∗h η¯ l . h p p p k p ∂z ∂z ∂z ∂z ∂z h Therefore, L ∂h = L ∂h + ∂z p
∂z p
∂ ∂z p
log g
+ ηk L
∂ 2 h ∂z k ∂z p
+
1 Lg L l . h pl¯ η¯
(25)
Assuming summation over repeated indices and taking into account Eq. (12), we may write: q L gq¯ p L g pl¯ = L gq¯ p ∗h g pl¯ = L gq¯ p g pl¯ = L δq = δl . (26) l
Applying Eq. (26) to Eq. (25) we obtain an expression for the operator L η¯ q : L η¯ q = h L gq¯ p
L ∂h − L ∂h + ∂z p
∂z p
∂ ∂z p
log g
− ηk L
∂ 2 h ∂z k ∂z p
.
(27)
It can be shown that the product ∗h defined locally on the fibrewise polynomial functions on T M is coordinate invariant and determines a global product on P(T M) even when there is no global Hermitian line bundle on M with the global differential operators on it which would have these functions as their symbols. Denote by D(T M)h the space of series of the form A= h r Ar , r ≥0
670
A. V. Karabegov
where Ar ∈ D(T M) and limr →∞ Ar = 0 in the topology determined by the filtration {Dr (T M)}. These series form an algebra. Lemma 2, Eqs. (12) and (27) imply that for any symbol P ∈ P(T M) the operator L P is in the space D(T M)h. In particular, for P, Q ∈ P(T M) the product P ∗h Q = L P Q is a polynomial in h, i.e., an element of P(T M)[h]. Thus the space P(T M)[h] is an algebra with respect to the product ∗h and the mapping P → L P is a homomorphism from P(T M)[h] to D(T M)h. Given a local potential −1 of the pseudo-Kähler form ω−1 on M, denote 1
−1 . ν on T M as follows: =
Introduce a local potential −1
−1 = −1 +
∂ −1 k ∂ −1 l η + η¯ ∂z k ∂ z¯ l
and set
∂ ∂ 1 −1 + log g = + k ηk + l η¯ l + log g. ν ∂z ∂ z¯ A simple check shows that the form =
(28)
−1 = −i∂T M ∂¯ T M −1 is a globally defined pseudo-Kähler form on T M. The restriction of the form −1 to the zero section Z of the tangent bundle T M coincides with the form ω−1 (under the obvious identification of the zero section Z with the manifold M). Thus the pseudo-Kähler manifold M is realized as a submanifold of the pseudo-Kähler manifold (T M, −1 ) with the induced pseudo-Kähler structure. It is well known that the form ωcan = −i∂ ∂¯ log g
(29)
is globally defined on M. Denote by 0 its lift to the tangent bundle T M and set 1 −1 + 0 . ν There exists a deformation quantization with separation of variables on (T M, −1 ) with the characterizing form . The corresponding star product will be denoted ∗. In the rest of the paper we will extend the star product ∗ on T M to singular symbols supported on the zero section Z which can be interpreted as a reduction to a deformation quantization with separation of variables on M. Consider the ‘formalization’ mapping F that replaces h with ν (it will be used somewhat loosely). The formalizer F determines a homomorphism of the algebra D(T M)h to the algebra of formal differential operators on T M. In particular, it maps all left and right multiplication operators with respect to the product ∗h from Lemma 4 to the corresponding left and right multiplication operators with respect to the star product ∗. Thus the image of the algebra of left multiplication operators of the algebra (P(T M)[h], ∗h ) with respect to F commutes with the operators Rb , Rη¯ q , R∂/∂ z¯ q , and R∂/∂ η¯ q and therefore belongs to the algebra of left multiplication operators with respect to the star product ∗. This implies that F determines a homomorphism of the algebra (P(T M)[h], ∗h ) to the star algebra (C ∞ (T M)[ν −1 , ν]], ∗). The pseudo-Kähler metric corresponding to the form −1 has the signature (2m, 2m), where m is the complex dimension of M. Since this metric is indefinite, there are no known analytic constructions of symbols on T M generalizing the algebraic construction given in this section to wider classes of symbols even if M is Kähler. =
Berezin-Toeplitz Quantization
671
4. Deformation Quantization with Separation of Variables on the Tangent Bundle T M A number of formulas from Sect. 3 have their formal analogues (with h replaced by ν) which can be proved directly. We list those of them which will be used in the sequel but give no proofs. The formal analogue of the first equation from (12) is that for φ, ψ ∈ C ∞ (M), φ ∗ ψ = φψ. (30) The formal analogues of the other two equations from (12) follow from the definition of deformation quantization with separation of variables. Denote by [·, ·]∗ the commutator with respect to the star product ∗. The formal analogue of Eq. (13) is as follows:
¯ ∂φ ηk , φ = −νglk l ∗ ∂ z¯
and
¯ ∂φ η¯ l , φ = νglk k . ∗ ∂z
(31)
Formulas (31) can be written in an equivalent form: ¯
f ∗ η p = η p f + ν gl p
∂f ∂ z¯ l
and
¯ η¯ q ∗ f = f η¯ q + ν g qk
∂f . ∂z k
(32)
Given a function f on M (which is identified with its lift to T M), one can express the operators L f and R f in terms of the locally defined formal differential operators ¯ J = exp νglk as follows:
∂2 ∂ηk ∂ z¯ l
L f = J f J −1
and
and
¯ K = exp νglk
∂2 ∂z k ∂ η¯ l
R f = K f K −1 .
(33)
(34)
Denote by B∗ the formal Berezin transform of the star product ∗. Using Eq. (30) we can prove the following lemma: Lemma 5. Given a local function f = f (z, z¯ ) and monomials u = u(η), v = v(η), ¯ the formulas B∗ (u f ) = f ∗ u, B∗ ( f v) = v ∗ f,
and
B∗ ( f ) = f
hold. Proof. It is sufficient to prove the lemma for a function f of the form f = ab, where a = a(z) is a holomorphic and b = b(¯z ) an antiholomorphic function. It follows from Eqs. (6) and (30) that B∗ (u f ) = B∗ (uab) = b ∗ (ua) = b ∗ (a ∗ u) = (b ∗ a) ∗ u = (ab) ∗ u = f ∗ u. The second formula can be proved similarly. They both imply the third one. In the rest of this section we will calculate the canonical formal trace density of the star product ∗ on T M. Given a potential (28) of the formal form , we will show that the potential ˜ = −2m log ν − −
∂ k ∂ l η − l η¯ + log g ∂z k ∂ z¯
672
A. V. Karabegov
satisfies Eq. (9) rewritten in the notations adapted to the pseudo-Kähler manifold T M as follows: ˜ ˜ ˜ ∂ ∂ ∂ ∂ ∂ ∂ B∗ = − p , B∗ = − p , B∗ =− q, ∂z p ∂z ∂η p ∂η ∂ z¯ q ∂ z¯ ˜ ˜ ∂ d ∂ d B∗ = − q , and B∗ =− . (35) q ∂ η¯ ∂ η¯ dν dν The calculations below are based on Lemma 5. The following calculation proves the first formula: 2 ˜ ∂ 1 ∂ ∂ k ∂ l B∗ − g − η − η ¯ + log g = B ¯ ∗ ∂z p ∂z p ∂z k ∂z p ν pl ∂z p ∂ 2 1 ∂ ∂ − k p ∗ ηk − η¯ l ∗ g pl¯ + p log g p ∂z ∂z ∂z ν ∂z ∂ 2 ∂ 1 ∂ ∂ ¯ ∂g ¯ ¯ ∂g pl¯ = − p − ηk k p − glk kpl − g pl¯η¯ l −glk k + p log g = − p . ∂z ∂z ∂z ∂z ν ∂z ∂z ∂z =−
The following one proves the second formula: B∗
˜ ∂ ∂η p
∂ ∂ ∂ = B∗ − p = − p = − p . ∂z ∂z ∂η
The next two formulas can be proved similarly. The proof of the last formula in (35) is as follows: ˜ d 1 1 ∂ k 1 ∂ l m B∗ + η + η¯ − 2 = B∗ dν ν ν ∂z k ν ∂ z¯ l ν =
1 ∂ 1 1 1 ∂ m + ∗ ηk + η¯ l ∗ l − 2 = k ν ν ∂z ν ∂ z¯ ν ν d 1 ∂ k 1 lk 1 ∂ l 1 lk m ¯ ¯ . + η + g gkl¯ + η¯ + g gkl¯ − 2 = − ν ∂z k ν ν ∂ z¯ l ν ν dν
According to Eq. (10), the canonical trace density µ∗ of the star product ∗ is given by the formula
˜ λm + dzd z¯ dηd η¯ = 2m g 2 dzd z¯ dηd η, ¯ (36) µ∗ = λm e ν where λm is a constant and dηd η¯ = dη1 . . . dηm d η¯ 1 . . . d η¯ m . Eventually, we obtain from Eq. (8) that 1 2m , µ∗ = 2m (37) ν (2m)! −1 which means that the star-product ∗ is ‘closed’ (see [9]). The constant λm is thus determined by Eqs. (36) and (37) and can be explicitly calculated.
Berezin-Toeplitz Quantization
673
5. A Fibrewise Fourier Transformation In this section we will use the following terminology and facts. A generalized function is a functional on the smooth compactly supported densities. If E → M is a fibre bundle then a fibrewise generalized function (i.e. a smooth family of generalized functions on the fibres of E) is a generalized function on the total space E. The action of a differential operator on the functions on E extends to the space of fibrewise generalized functions on E and to the space of all generalized functions on E. Denote by E r the space of fibrewise generalized functions on the tangent bundle T M supported on the zero section of T M of order not greater than r and set E = ∪r E r . Consider the subspace Eν ⊂ E[ν −1 , ν]] consisting of the elements such that for each A ∈ Eν there is an integer s for which A can be represented as A=
ν r Ar
r ≥s
with Ar ∈ E r −s for all r ≥ s. Natural formal differential operators act on the space Eν . We will prove a more general statement. Lemma 6. A formal differential operator B given by the ν-adically convergent series B=
ν r Br ,
r ≥k
where k is a (possibly negative) integer and Br are natural formal differential operators, leaves invariant the space Eν . Proof. Given A ∈ Eν , there is an integer l such that A can be represented as A=
ν s As
s≥l
with As ∈ E s−l . Represent each natural formal differential operator Br as Br =
ν t Br,t ,
t≥0
where Br,t is a differential operator of order not greater than t. Now B(A) =
ν r +s+t Br,t (As ).
r ≥k t≥0 s≥l
Set n = r + s + t. The order of the generalized function Br,t (As ) is not greater than t + s − l. The lemma follows from the fact that t + s − l = n − (r + l) ≤ n − (k + l) for any n. We identify the elements of the space C ∞ (T ∗ M, Z ) of functions on the formal neighborhood (T ∗ M, Z ) of the zero section Z of the cotangent bundle T ∗ M with the formal series A = A0 + A1 + A2 + · · ·, where Ar is a fibrewise homogeneous polynomial
674
A. V. Karabegov
of degree r in the fibre variables on T ∗ M (see the Appendix for the details on formal neighborhoods). Denote by P r the space of sums P=
r 1 Ps , νs s=0
where Ps is a fibrewise homogeneous polynomial of degree s on the cotangent bundle T ∗ M. The space C ∞ (T ∗ M, Z )[ν −1 , ν]] can be alternatively described as the set of formal series of the form F= ν r Pr , r ≥s
where s is some integer and Pr ∈ P r −s for all r ≥ s. To see it, represent Pr as Pr =
r −s 1 Pr,t , νt t=0
where Pr,t is a fibrewise homogeneous polynomial of degree t in the fibre variables on T ∗ M. Then F can be rewritten as an element of C ∞ (T ∗ M, Z )[ν −1 , ν]] as follows: ⎛ ⎞ r −s 1 F= νr P νr ⎝ Pr +t,t ⎠ . = t r,t ν r ≥s r ≥s t=0
t≥0
The pseudo-Kähler metric gkl¯ on M defines a global fibrewise density gdηd η¯ on T M, where g = det(gkl¯). Using this density and the natural pairing of T M and T ∗ M, define a fibrewise Fourier transformation of the elements of the space E by the formula i ˜ z¯ , ξ, ξ¯ ) = e ν (ηk ξk +η¯ l ξ¯l ) A(z, z¯ , η, η)gdηd A(z, ¯ η. ¯ (38) It is an isomorphism of E r onto P r and therefore it extends to an isomorphism of the space Eν onto C ∞ (T ∗ M, Z )[ν −1 , ν]]. It transfers the differential operators on T M to operators on T ∗ M as follows. For f = f (z, z¯ ) denote the pointwise multiplication operator by f by the same symbol. Then ∂ ∂ ∂ i , η¯ l → −iν , → − ξk , k ∂ξk ∂η ν ∂ ξ¯l ∂ ∂ ∂ ∂ ∂ ∂ → k − k log g, → l − l log g. ∂z k ∂z ∂z ∂ z¯ l ∂ z¯ ∂ z¯
f → f, ηk → −iν ∂ i → − ξ¯l , l ∂ η¯ ν
(39)
Notice that the pointwise multiplication operator by a function u(η) transfers to a formal differential operator ∞
∞
r =0
∂ , u r −iν ∂ξ
where r =0 u r is the Taylor series of the function u(η) at the origin with u r a homogeneous polynomial of degree r .
Berezin-Toeplitz Quantization
675
Since the star product ∗ on T M is natural, the left and right multiplication operators L A and R A are natural for any A ∈ C ∞ (T M). It follows from Lemma 6 that for A ∈ C ∞ (T M)[ν −1 , ν]] the action of the operators L A and R A can be extended to the space Eν . This action can be transferred to the space C ∞ (T ∗ M, Z )[ν −1 , ν]] via the Fourier transformation (38). Denote the corresponding operators on the space C ∞ (T ∗ M, Z )[ν −1 , ν]] by L˜ A and R˜ A , respectively, so that for B ∈ Eν , ˜ A B˜ L AB = L
and
˜ R A B = R˜ A B.
Thus both Eν and C ∞ (T ∗ M, Z )[ν −1 , ν]] are bimodules over the star algebra (C ∞ (T M) [ν −1 , ν]], ∗). It turns out quite surprisingly that the operators L˜ A and R˜ A are naturally expressed in terms of the so called ‘formal symplectic groupoid with separation of variables’ over the pseudo-Kähler M (see [20]). Consider the standard Poisson structure on T ∗ M given on A, B ∈ C ∞ (T ∗ M) as follows: {A, B}T ∗ M =
∂A ∂B ∂B ∂A ∂A ∂B ∂B ∂A − + − . ∂ξk ∂z k ∂ξk ∂z k ∂ ξ¯l ∂ z¯ l ∂ ξ¯l ∂ z¯ l
There exist a Poisson and an anti-Poisson (global) morphisms S, T :C ∞ (M)→ C ∞ (T ∗ M, Z ), respectively, given locally by the formulas Sφ = e
¯
−iξk glk
∂ ∂ z¯l
φ
Tψ = e
and
¯
−i ξ¯l glk
∂ ∂z k
ψ.
These are the source and target mappings of the formal symplectic groupoid with separation of variables over the pseudo-Kähler manifold M. The images of the source and target mappings Poisson commute, {Sφ, T ψ}T ∗ M = 0. The mapping S ⊗ T : φ ⊗ ψ → (Sφ)(T ψ) extends to a Poisson isomorphism ¯ Mdiag ) → C ∞ (T ∗ M, Z ), S ⊗ T : C ∞ (M × M,
(40)
where the factor M¯ is endowed with the Poisson structure opposite to (3) and the product M × M¯ by the product Poisson structure (see [19]). For a local holomorphic function a and antiholomorphic function b on M Sa = a
and
T b = b.
Given a function f ∈ C ∞ (M) (which is identified with its lift to T ∗ M), its pullback via the mapping S ⊗ T is the function δ f , where δ is the formal analytic extension mapping (see the Appendix). To check it, it is sufficient to consider a function of the form ai bi , f = i
where ai and bi are local holomorphic and antiholomorphic functions, respectively. Then (S ⊗ T )(δ f ) = (S ⊗ T ) δ ai bi i
= (S ⊗ T )
i
ai ⊗ bi
=
i
S(ai )T (bi ) =
i
ai bi = f.
676
are
A. V. Karabegov
Using the fact that the Fourier transforms of the operators J and K given by Eq. (33) ¯ ∂ ¯ ∂ lk lk ˜ ˜ ¯ J = exp −iξk g and K = exp −i ξl g , (41) ∂ z¯ l ∂z k
we can prove the following lemma. Lemma 7. Given a function f on M, the Fourier transforms of the operators L f and R f are pointwise multiplication operators given by the formulas L˜ f = S( f ) and
R˜ f = T ( f ).
Proof. Since the fibrewise Fourier transform of the pointwise multiplication operator by f is also the multiplication operator by f , we get from Eqs. (34) and (41) that −iξ g L˜ f = J˜ f J˜−1 = e k
¯ ∂ lk ∂ z¯l
f = S f.
The formula for R˜ f can be proved similarly. Given a function f on M, denote by l f and r f the pullbacks of the operators L˜ f and R˜ f via the isomorphism (40), respectively. According to Lemma 7, lf = f ⊗1
and
r f = 1 ⊗ f.
(42)
For a Hamiltonian A ∈ C ∞ (T ∗ M) we will denote the corresponding Hamiltonian vector field on T ∗ M by H A so that for B ∈ C ∞ (T ∗ M), H A B = {A, B}T ∗ M . Similarly, we will denote by h φ the Hamiltonian vector field on M corresponding to a Hamiltonian function φ so that h φ ψ = {φ, ψ} M for ψ ∈ C ∞ (M). Our next task will be to calculate several left and right multiplication operators, their Fourier transforms and pullbacks via the mapping S ⊗ T . These calculations will be used in the rest of the paper. Since L η p = η p and Rη¯ q = η¯ q we see from Eq. (39) that L˜ η p = −iν ∂ξ∂ p = iν Hz p = iν HS(z p ) = iνe− HS(z p ) e and
(43) R˜ η¯ q = −iν ∂∂ξ¯ = iν Hz¯ q = iν HT (¯z q ) = iνe− HT (¯z q ) e . q
We see from Eq. (43) and the fact that S ⊗ T is a Poisson morphism that lη p = iν (h z p ⊗ 1) = iνe−δ (h z p ⊗ 1) eδ and
(44) rη¯ q = −iν (1 ⊗ h z¯ q ) =
−iνe−δ
(1 ⊗ h z¯
q ) e δ .
Formula (28) implies that L
∂ ∂z p
=
∂ ∂ ∂ ∂ 2 k 1 ∂ ∂ + = + η + g pl¯η¯ l + p log g + p . ∂z p ∂z p ∂z p ∂z p ∂z k ν ∂z ∂z
(45)
Berezin-Toeplitz Quantization
677
Notice that H
S
∂ ∂z p
=−
∂ i ∂ ∂ 2 ∂ 1 − p k − g pl¯ . p ν ∂z ∂z ∂z ∂ξk ν ∂ ξ¯l
(46)
It follows from Eqs. (39), (45), and (46) that L˜
∂ ∂z p
= iν H
S
∂ ∂z p
+
∂ = iν e− H ∂ e . S ∂z p ∂z p
(47)
A similar calculation shows that R˜ ∂q = iν e− H
T
∂ z¯
∂ ∂ z¯ q
e .
Pulling these operators back via the mapping S ⊗ T we obtain that
l ∂p = iν e−δ h ∂p ⊗ 1 eδ ∂z
∂z
and r ∂q = ∂ z¯
−iν e−δ
1 ⊗ h ∂q
∂ z¯
(48) eδ ,
where δ is the formal analytic extension mapping. Given a function φ on M, denote by A(φ) the following function on T M: A(φ) =
∂φ k ∂φ l η + l η¯ . ∂z k ∂ z¯
Notice that A(z k ) = ηk and A(¯z l ) = η¯ l . There is a formal analogue (with h replaced by ν) of long and indirect formula (27) for the operator L η¯ q . It turns out that the Fourier transform of this operator has a nice expression. We will find a general formula for the operator L˜ A(φ) by working first with its pullback l A(φ) via the mapping S ⊗ T . Using Eqs. (3), (30), and (31) we get that ∂φ ∂φ [A(φ), ψ]∗ = ηk ∗ k + l ∗ η¯ l , ψ ∂z ∂ z¯ ∗
∂φ ∂φ l
k = η , ψ ∗ k + l ∗ η¯ , ψ = iν{φ, ψ} M = iνh φ ψ. (49) ∗ ∂z ∗ ∂ z¯ Using Eq. (42), we obtain from Eq. (49) that l A(φ) , ψ ⊗ 1 = iνh φ ψ ⊗ 1.
(50)
Since the left multiplication operators commute with the right multiplication operators, we see from Eqs. (42) and (50) that the operator B(φ) = l A(φ) − iνe−δ h φ ⊗ 1 eδ commutes with the functions ψ ⊗ 1 and 1 ⊗ ψ for any ψ ∈ C ∞ (M) and thus with the ¯ Mdiag ). pointwise multiplication operators by the elements of the space C ∞ (M × M, Thus B(φ) is a pointwise multiplication operator itself. It follows from Eqs. (44) and (48) that the multiplication operator B(φ) commutes with the operators
rη¯ q = −iνe−δ (1 ⊗ h z¯ q ) eδ and r ∂q = −iνe−δ 1 ⊗ h ∂q eδ , ∂ z¯
∂ z¯
678
A. V. Karabegov
and therefore with the operators 1 ⊗ h z¯ q
and
1 ⊗ h ∂q . ∂ z¯
This implies that the function B(φ) is of the form C(φ) ⊗ 1. To identify the mapping φ → C(φ) we will push forward the formula l A(φ) = iνe−δ h φ ⊗ 1 eδ + C(φ) ⊗ 1 via the mapping S ⊗ T obtaining that
L˜ A(φ) = iνe− HS(φ) e + S(C(φ)).
(51)
The following formula can be obtained by straightforward calculations with the use of Eqs. (30) and (31). For φ, ψ ∈ C ∞ (M), ∂φ ∂ψ ∂ψ ∂φ ¯ lk . (52) + A(φψ) = φ ∗ A(ψ) + ψ ∗ A(φ) − νg ∂z k ∂ z¯ l ∂z k ∂ z¯ l Calculating L˜ A(φψ) in two different ways using Lemma 7, Eqs. (51) and (52) we get that iνe− HS(φψ) e + S(C(φψ)) = iν S(φ)e− HS(ψ) e ∂φ ∂ψ ∂ψ ∂φ ¯ − lk . (53) HS(φ) e − S νg + iν S(ψ)e + ∂z k ∂ z¯ l ∂z k ∂ z¯ l Simplifying Eq. (53) we arrive at the statement that the Hochschild differential of the operator C is ∂φ ∂ψ ∂ψ ∂φ ¯ dHoch C(φ, ψ) = φC(ψ) − C(φψ) + C(φ)ψ = νglk + ∂z k ∂ z¯ l ∂z k ∂ z¯ l and thus coincides with the Hochschild differential of the operator −ν, where is the Laplace-Beltrami operator (5) on M. Therefore C = −ν + D, where D is a derivation. To determine D, we use Eqs. (43) and (47). We see from Eqs. (43) and (51) that Dz k = 0, i.e. that D differentiates in antiholomorphic directions only. It follows from Eq. (31) that ∂ 2 ∂ ηk , k p = − p log g, ∂z ∂z ∗ ∂z whence
∂ ∂ ∂ 2 1 = + ∗ ηk + g pl¯ ∗ η¯ l . p p k p ∂z ∂z ∂z ∂z ν We see from Eqs. (43), (47), and (54) that 2 ∂ ∂ −
+ iν S e− HS(z p ) e iν e H ∂ e = S S ∂z p ∂z p ∂z k ∂z p
1 + S g pl¯ iνe− HS (z¯l ) e + S D(¯z l ) . ν
(54)
(55)
Berezin-Toeplitz Quantization
679
Simplifying Eq. (55) we get that ¯
D(¯z l ) = −νglk
∂ , ∂z k
which means that ¯
D = −νglk
∂ ∂ . ∂z k ∂ z¯ l
Eventually we obtain a formula for the operator L˜ A(φ) : ¯ ∂ ∂φ ˜L A(φ) = iνe− HS(φ) e − ν S φ + glk . ∂z k ∂ z¯ l Similarly, one can obtain a formula for the operator R˜ A(φ) : ¯ ∂ ∂φ R˜ A(φ) = iνe− HT (φ) e − νT φ + glk l k . ∂ z¯ ∂z
(56)
(57)
6. A Product on the Singular Symbols We want to show that there exists a natural construction of an associative product • on the space C ∞ (T ∗ M, Z )[ν −1 , ν]] compatible with the bimodule structure over the star algebra (C ∞ (T M)[ν −1 , ν]], ∗) so that one can define an associative product on the direct sum C ∞ (T M)[ν −1 , ν]] ⊕ C ∞ (T ∗ M, Z )[ν −1 , ν]]. The compatibility conditions are as follows. For F ∈ C ∞ (T M)[ν −1 , ν]] and A, B ∈ C ∞ (T ∗ M, Z )[ν −1 , ν]], ( L˜ F A) • B = L˜ F (A • B), ( R˜ F A) • B = A • ( L˜ F B), A • ( R˜ F B) = R˜ F (A • B).
and
(58)
Assume now that there exists such an associative product • on the space C ∞ (T ∗ M, Z ) [ν −1 , ν]] satisfying the compatibility conditions (58). We want to study the properties that this product must have. We identify the functions on T ∗ M which do not depend on the fibre variables ξk , ξ¯l with the functions on M. Thus the space C ∞ (M)[ν −1 , ν]] can be treated as a subspace of C ∞ (T ∗ M, Z )[ν −1 , ν]]. Since L˜ ηk = −iν ∂ξ∂ k and R˜ η¯ l = −iν ∂∂ξ¯ , then it follows from Eqn. (58) that l
for φ ∈ C ∞ (M)[ν −1 , ν]] and A ∈ C ∞ (T ∗ M, Z )[ν −1 , ν]] the product φ • A does not depend on the holomorphic fibre variables ξk and A • φ does not depend on the antiholomorphic fibre variables ξ¯l . Therefore C ∞ (M)[ν −1 , ν]] is closed with respect to the product •. We will make an assumption that the algebra (C ∞ (M)[ν −1 , ν]], •) has a unity ε which is an invertible formal function, ε= ν r εr , r ≥n
where n is an integer and εn is nonvanishing. In the whole algebra (C ∞ (T ∗ M, Z ) [ν −1 , ν]], •) the element ε will be an idempotent. Taking into account Eq. (58) we see
680
A. V. Karabegov
that for f ∈ C ∞ (M)[ν −1 , ν]] the element ε • (S( f )ε) = (T ( f )ε) • ε also belongs to the space C ∞ (M)[ν −1 , ν]]. Define a linear operator Q on the space C ∞ (M)[ν −1 , ν]] by the formula Q( f )ε = ε • (S( f )ε) = (T ( f )ε) • ε. (59) Notice that Q(1) = 1. We will assume that Q is a formal differential operator. Define an associative product ◦ on C ∞ (M)[ν −1 , ν]] by the following formula. For φ, ψ ∈ C ∞ (M)[ν −1 , ν]] set (φ ◦ ψ)ε = (φε) • (ψε). The unit constant 1 is the unity in the algebra (C ∞ (M)[ν −1 , ν]], ◦). We will make a further assumption that the operation ◦ is a star product on M with respect to some Poisson structure on M. The assumptions we have made allow us to identify this star product. Since for a local, holomorphic function a and an antiholomorphic function b L˜ a = S(a) = a
and
R˜ b = T (b) = b,
we see from Eq. (58) that (a ◦ ψ)ε = (aε) • (ψε) = a(ε • (ψε)) = aψε, whence a ◦ ψ = aψ and, similarly, φ ◦ b = φb. This means that the star product ◦ defines a deformation quantization with separation of variables with respect to some Kähler-Poisson structure on M given by a bivector field η of type (1, 1) with respect to the complex structure on M. Denote by B◦ the formal Berezin transform corresponding to the star product ◦ so that B◦ (ab) = b ◦ a, where a, b are as above. Introduce an equivalent star product ◦ on M by the formula φ ◦ ψ = B◦−1 (B◦ (φ) ◦ B◦ (ψ)). (60) This is the star product of a deformation quantization with separation of variables on ¯ η) with the opposite complex structure, so that locally the Kähler-Poisson manifold ( M, b ◦ f = b f and f ◦ a = a f . Denote by ◦˜ the opposite star product, ◦˜ = (◦ )opp , so that φ ◦˜ ψ = B◦−1 (B◦ (ψ) ◦ B◦ (φ)).
(61)
This is the star product of a deformation quantization with separation of variables on (M, −η). The star products ◦ and ◦˜ are dual. It follows from the definition (59) of the operator Q and compatibility conditions (58) that Q(ab)ε = ε • (S(ab)ε) = (T (b)ε) • (S(a)ε) = (bε) • (aε) = (b ◦ a)ε = B◦ (ab)ε. Therefore Q = B◦ and we obtain the following formula: ε • (S( f )ε) = (T ( f )ε) • ε = B◦ ( f )ε,
(62)
where f ∈ C ∞ (M)[ν −1 , ν]]. Introduce a formal form 1 ω˜ = − ω−1 + ωcan , ν
(63)
where ωcan is given by Eq. (29). Denote by ˜ the star product of the deformation quantization with separation of variables on the pseudo-Kähler manifold (M, −ω−1 )
Berezin-Toeplitz Quantization
681
whose characterizing form is ω˜ and by the opposite star product, = (˜)opp . Let denote the star product dual to ˜ . It is a deformation quantization with separation of variables on the pseudo-Kähler manifold (M, ω−1 ) whose characterizing form will be denoted ω. Our goal is to show that the star products ◦ and must coincide. Fix a contractible coordinate chart U ⊂ M with holomorphic coordinates {z k , z¯ l }. ¯ −1 and set Denote by −1 a potential of the form ω−1 on U so that ω−1 = −i∂ ∂
= (1/ν) −1 . The formal invertible function ε can be represented on U as ε = eθ for some formal function θ = n log ν + θ0 + νθ1 + · · ·. Set
= + θ = (1/ν) −1 + n log ν + θ0 + νθ1 + · · ·
(64)
so that ε = e − . We see from Eq. (56) that ¯ ∂ L˜ η¯ l = iνe− HS(¯zl ) e − S νglk k . ∂z
(65)
We will need the following two technical lemmas. Lemma 8. Given a function f = f (z, z¯ ) on the chart U , the following formulas hold: ¯ ∂f HS(¯zl ) S( f ) = {S(¯z l ), S( f )}T ∗ M = S −iglk k ; ∂z ∂ f ¯ HS(¯zl ) f = {S(¯z l ), f }T ∗ M = S(−iglk ) k . ∂z Proof. {S(¯z ), S( f )} l
T∗M
¯ ∂f lk . = S({¯z , f } M ) = S −ig ∂z k l
Using the first formula, we obtain ∂f ∂ S(¯z l ) ∂ f = {S(¯z l ), z k }T ∗ M k ∂ξk ∂z k ∂z ∂f ¯ ∂f = {S(¯z l ), S(z k )}T ∗ M k = S(−iglk ) k , ∂z ∂z
{S(¯z l ), f }T ∗ M =
which concludes the proof. Lemma 9. Given local functions φ, ψ on M, the following formula holds:
ε • (S(φ)ψε) = (B◦ (φ) ◦ ψ) ε = B◦ B◦−1 (φ)˜◦ψ ε. Proof. It is sufficient to consider the function ψ of the form ai bi , ψ= i
682
A. V. Karabegov
where ai and bi are local holomorphic and antiholomorphic functions, respectively. Then, using Eqs. (60) and (61), we get that ε • (S(φ)ψε) = ε • (S(φai )T (bi )ε) =
i
(ε • S(φai )ε) T (bi ) =
i
=
i
B◦ (φ ◦ ai ) ◦ bi ε =
B◦ (φai )bi ε
i
B◦ (φ) ◦ (ai ◦ bi ) ε i
= (B◦ (φ) ◦ ψ) ε = B◦ φ ◦ B◦−1 (ψ) ε = B◦ B◦−1 (ψ)˜◦φ ε, which proves the lemma. Since R˜ η¯ l = −iν∂/∂ ξ¯l and the function ε does not depend on the fibre variables ξ, ξ¯ , we obtain from formulas (62), (65), and Lemma 8 that for an arbitrary formal function f ∈ C ∞ (U )[ν −1 , ν]], 1 1 ˜ ˜ g ¯ f ε = ε • L η¯ l S g ¯f ε 0 = ( Rη¯ l ε) • S ν pl ν pl 1 ¯ ∂ lk −
−ε•S g g ¯f e g ¯f = ε • iνe HS(¯zl ) S ν pl ∂z k pl
l ∗ ε + ε • i S g f {S(¯ z ),
} ε = ε • i S(¯z l ), S g pl¯ f ¯ T M pl T∗M ∂ ¯ ∂ f ε = ε • S glk k (g pl¯ f ) ε (66) −ε • S ∂z p ∂z ∂ ∂ ¯ ε − B f ε + ε • S g pl¯ f S glk ◦ ∂z k ∂z p ∂f ∂ ∂ ∂
+ f log g ε + ε • S( f ) ε − B f ε. = B◦ ◦ ∂z p ∂z p ∂z p ∂z p We conclude from Eq. (66) that ∂f ∂ ∂
B◦ + f (− + log g) ε = −ε • S( f ) ε . ∂z p ∂z p ∂z p Using Lemma 9 we obtain that ∂
−1 ∂
ε • S( f ) p ε = B◦ B◦ ◦˜ f ε. ∂z ∂z p Formulas (67) and (68) imply that for any formal function f on U , ∂f ∂ −1 ∂
−B◦ ◦˜ f = p + f p (− + log g). ∂z p ∂z ∂z A calculation similar to (66) that starts with the observation that 1 1 gk q¯ f ε • ( L˜ ηk ε) = R˜ ηk T gk q¯ f ε • ε 0=T ν ν
(67)
(68)
(69)
Berezin-Toeplitz Quantization
shows that
683
∂
∂ ∂f f ◦˜ −B◦−1 = q + f q (− + log g). q ∂ z¯ ∂ z¯ ∂ z¯
(70)
Since − +log g is a potential of the form ω, ˜ it immediately follows from Eq. (69) or (70) and the description of the star products with separation of variables on a pseudo-Kähler manifold that the star product ◦˜ must coincide with ˜ and thus the star products and ◦ must coincide as well. Denote by B◦˜ , B , and B˜ the formal Berezin transforms of the star products ◦˜ , , and ˜ , respectively. Thus we must have that B = B◦ = B◦˜−1 = B˜−1 . Setting f = 1 in Eqs. (69) and (70) and replacing B◦−1 with B˜ we get that ∂
∂ = − p (− + log g) B˜ ∂z p ∂z and (71) ∂
∂ = − q (− + log g). B˜ ∂ z¯ q ∂ z¯ Formulas (71) imply that must be a potential of the form ω and the density e − gdzd z¯ must be a local trace density of the star product . There exists a constant κm such that 1 m ω = κm gdzd z¯ . m! −1
(72)
m must be a global trace density of Since ε = e − , we conclude from (72) that εω−1 the star product , which determines ε up to a formal constant factor. We see that the assumptions made in this section determine what the product ◦ and the formal function ε might be. Now we will give an explicit definition of the product •. Denote by µ the canonical formal trace density of the star product and fix an arbitrary nonzero formal constant C(ν). There exists a unique invertible formal function ε= ν r εr (73) r ≥n
on M for some integer n and with εn nonvanishing such that m µ = C(ν)εω−1 .
(74)
We want to define an operation • on C ∞ (T ∗ M, Z )[ν −1 , ν]] such that ε • (S( f )ε) = (T ( f )ε) • ε = B ( f )ε,
(75)
as suggested by Eq. (62). Formulas (58) and (75) allow to define the product • on the elements of C ∞ (T ∗ M, Z )[ν −1 , ν]] of the form S(φ)T (ψ)ε with φ, ψ ∈ C ∞ (M)[ν −1 , ν]] as follows: S(φ1 )T (ψ1 )ε • S(φ2 )T (ψ2 )ε = S(φ1 )B (ψ1 φ2 )T (ψ2 )ε. (76) Using the method explained in the Appendix one can extend the product • to the whole space C ∞ (T ∗ M, Z )[ν −1 , ν]]. One can also show applying the technique used in this paper that the product • satisfies compatibility conditions (58). Now we will prove the associativity of this product.
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Lemma 10. The product • is associative. Proof. Because of the compatibility conditions (58) it is sufficient to prove that for any functions ψ1 , φ2 , ψ2 , φ3 ∈ C ∞ (M), (T (ψ1 )ε • S(φ2 )T (ψ2 )ε) • S(φ3 )ε = T (ψ1 ) • (S(φ2 )T (ψ2 )ε • S(φ3 )ε) or, equivalently, that B (ψ1 φ2 )T (ψ2 )ε • S(φ3 )ε = T (ψ1 )ε • S(φ2 )B (ψ2 φ3 )ε.
(77)
Using Eq. (58), simplify Eq. (77) as follows: B (ψ1 φ2 )T (ψ2 φ3 )ε • ε = ε • S(ψ1 φ2 )B (ψ2 φ3 )ε.
(78)
Setting φ = ψ1 φ2 and ψ = ψ2 φ3 rewrite Eq. (78) as B (φ)T (ψ)ε • ε = ε • S(φ)B (ψ)ε.
(79)
It is sufficient to prove Eq. (79) for φ = a1 b1 and ψ = a2 b2 , where a1 , a2 are local holomorphic and b1 , b2 local antiholomorphic functions on M. The left-hand side of Eq. (79) takes the form a1 b1 T (ψ)ε • ε = S(a1 )T (b1 ψ)ε • ε = S(a1 )B (b1 ψ)ε = B (a1 ) B (b1 ψ)ε = B (a1 b1 ψ)ε = B (φ ψ)ε. A similar calculation shows that the right-hand side of Eq. (79) also equals B (φ ψ)ε, which proves the lemma. For any formal function f ∈ C ∞ (M)[ν −1 , ν]] define an element Q f ∈ C ∞ (M) −1 [ν , ν]] by the formula Q f = f ε. Theorem 1. The mapping f → Q f is an isomorphism of the algebra (C ∞ (M)[ν −1 , ν]], ) onto the algebra (C ∞ (M)[ν −1 , ν]], •). Proof. For functions φ, ψ ∈ C ∞ (M)[ν −1 , ν]] we have to prove that Qφ • Qψ = Qφψ .
(80)
It is sufficient to prove Eq. (80) locally for functions φ, ψ of the form φ = a1 b1 and ψ = a2 b2 , where a1 , a2 are local holomorphic and b1 , b2 local antiholomorphic functions. It follows from Eqs. (58) and (75) that Qφ • Qψ = (φε) • (ψε) = (a1 b1 ε) • (a2 b2 ε) = (S(a1 )T (b1 )ε) • (S(a2 )T (b2 )ε) = S(a1 ) (ε • S(b1 a2 )ε) T (b2 ) = a1 (B (b1 a2 )ε) b2 = (a1 b1 a2 b2 ) ε = ((a1 b1 ) (a2 b2 )) ε = (φ ψ)ε = Qφψ , which concludes the proof.
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Setting f = 1 in Eq. (75) we see that the element ε is an idempotent in the algebra (C ∞ (T ∗ M, Z )[ν −1 , ν]], •), ε • ε = ε. For any function f ∈ C ∞ (M)[ν −1 , ν]] we call the element of the space C ∞ (T ∗ M, Z ) [ν −1 , ν]] given by Eq. (75) the Toeplitz element corresponding to the function f and denote it T f . Thus
T f = ε • L˜ f ε = R˜ f ε • ε, which is analogous to the definition of a Toeplitz operator. The Toeplitz elements in C ∞ (T ∗ M, Z )[ν −1 , ν]] are exactly the elements which do not depend on the fibre variables ξ, ξ¯ and thus can be identified with the elements of C ∞ (M)[ν −1 , ν]]. Remark. Here we would like to give more heuristic arguments to corroborate the analogy between the Toeplitz operators on the sections of a quantum line bundle over M and the Toeplitz elements in the algebra (C ∞ (T ∗ M, Z )[ν −1 , ν]], •). Assume that (M, ω−1 ) is a compact Kähler manifold and L is a global quantum line bundle. The Hilbert structure on the sections of the N th tensor power L ⊗N of L is given by the norm || · ||h such that the norm of a section s is 2 m ||s||h = |s|2h ω−1 , where | · |h is the Hermitian fibre metric (it is implied that h = 1/N ). The symbol mapping P → Pˆ constructed in Sect. 3 is involutive. Namely, the complex conjugate symbol P¯ corresponds to the Hermitian conjugate operator Pˆ ∗ . Let β = f k (z, z¯ )dz k be a global differential form of type (1, 0) on M. Then f k ηk is a global function on T M. Consider the global symbol ¯
P = Jh ( f k ηk ) = f k ηk + hglk
∂ fk = f k ∗h η k ∂ z¯ l
on T M. The corresponding global differential operator Pˆ = f k ∇ k = ∇ β annihilates the holomorphic sections of L ⊗N . The range of the conjugate operator Pˆ ∗ with the symbol P¯ = η¯ l ∗h f¯l , given by the formula ¯ Pˆ ∗ = ∇ l ◦ f¯l ,
is orthogonal to the space of holomorphic sections of L ⊗N (here ◦ denotes composition (N ) of operators). Thus, for any Toeplitz operator Tφ , Pˆ Tφ(N ) = Tφ(N ) Pˆ ∗ = 0. This statement has an obvious analogue for the Toeplitz elements, L˜ fk ∗ηk Tφ = R˜ η¯ l ∗ f¯l Tφ = 0, which is equivalent to the fact that the Toeplitz elements do not depend on the fibre variables ξ, ξ¯ . Theorem 1 has the following
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Corollary 1. The mapping f → T f induces an isomorphism of the algebra (C ∞ (M) [ν −1 , ν]], ) onto the algebra (C ∞ (M)[ν −1 , ν]], •) of Toeplitz elements. Proof. For a function f ∈ C ∞ (M)[ν −1 , ν]] we have from Eq. (75) that T f = B ( f )ε = Q B ( f ) . The statement of the corollary follows from the fact that the formal Berezin transform B is an equivalence operator for the deformation quantizations corresponding to the star products and . As it was shown in [21], the Berezin-Toeplitz star product on a Kähler manifold (M, ω−1 ) coincides with the star product whose opposite star product ˜ determines the deformation quantization with separation of variables on (M, −ω−1 ) with the characterizing form ω˜ given by Eq. (63). Thus the construction presented in this paper can be thought of as a formal model of Berezin-Toeplitz quantization. This construction remains valid for any invertible formal constant C(ν) in the definition of the idempotent ε given by Eq. (74). In the rest of the section we will show that there is a natural normalization of ε which determines it uniquely. Given a formal function f ∈ C ∞ (M)[ν −1 , ν]], one can define an element E f ∈ Eν by the following local formula: Ef = f
ε δ(η)δ(η), ¯ g
where δ(η)δ(η) ¯ is the delta-function at the origin η = η¯ = 0 so that δ(η)δ(η)dηd ¯ η¯ = 1. The fibrewise Fourier transform (38) of the element E f is E˜ f = f ε = Q f . Therefore, according to Theorem 1, the mapping f → E f is a homomorphism of the algebra (C ∞ (M)[ν −1 , ν]], ) to the space Eν endowed with the pullback of the product • via the Fourier transformation (38). If f has a compact support, then, using Eqs. (36), (37), (72), and (74), we can pair the generalized function E f with the canonical trace density µ∗ of the star product ∗ as follows: λm λm m E f , µ∗ = 2m (81) f εg dzd z¯ = 2m f εω−1 ν ν κm m! λm = 2m f µ . ν κm m!C(ν) We see from Eq. (81) that if the formal constant C(ν) is set to be C(ν) =
λm , ν 2m κm m!
then
E f , µ∗ =
f µ ,
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which means that the canonical trace density µ∗ on T M induces the canonical trace density µ on M via the mapping f → E f . Taking into account Eqs. (8) and (73) and equating the leading terms on both sides of Eq. (74) we see that 1 ν m m!
m = ω−1
λm m , ν n εn ω−1 2m ν κm m!
whence it follows that n = m and εm =
κm . λm
7. Appendix Let N be a submanifold of a manifold M and I be the ideal of smooth functions on M vanishing on N . We call C ∞ (M, N ) = C ∞ (M)/(∩r I r ) the space of functions on the formal neighborhood (M, N ) of the submanifold N in M. Assume that M is a complex manifold and M¯ is the same manifold with the opposite complex structure. Take a local chart U with holomorphic coordinates {z k , z¯ l } on M and its copy U¯ with coordinates {w k , w¯ l }. We will cover the diagonal Mdiag of M × M¯ by the Cartesian squares U × U¯ so that on the diagonal z k = w k and z¯ l = w¯ l . There is a mapping ¯ Mdiag ) δ : C ∞ (M) → C ∞ (M × M, that maps a function f (z, z¯ ) on M to its formal analytic extension δ f = f (z, w) ¯ on ¯ Mdiag ) which is a unique solution of the equation (M × M, (∂¯ z + ∂w )(δ f ) = 0 with the initial condition δ f | Mdiag = f. Given functions φ, ψ ∈ C ∞ (M), we will denote by φ ⊗ ψ both a function on M × M¯ ¯ Mdiag ) which will be called factorizable. and the corresponding element in C ∞ (M × M, Let A be a differential operator on M. In this appendix we will explain how to extend the bilinear operation (φ1 ⊗ ψ1 , φ2 ⊗ ψ2 ) → (φ1 ⊗ ψ2 ) · δ(A(ψ1 φ2 ))
(82)
from the (linear combinations of) factorizable elements to the whole space C ∞ (M × ¯ Mdiag ). A local model of C ∞ (M × M, ¯ Mdiag ) on a chart U × U¯ can be given in the M, coordinates z k , z¯ l , τ k = w k − z k , τ¯ l = w¯ l − z¯ l as C ∞ (U )[[τ, τ¯ ]],
(83)
where τ k , τ¯ l are treated as formal variables. Using this model one can introduce the operation F(z, z¯ , w, w) ¯ → F|w=z = F(z, z¯ , z, w) ¯ (84) ∞ ¯ on C (M × M, Mdiag ) by setting τ = 0 in the formal series representing F in (83). Operation (84) is the extention of the operation φ ⊗ ψ → (φ ⊗ 1) · δ(ψ)
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¯ Mdiag ). Similarly, one can from the factorizable elements to the whole C ∞ (M × M, introduce the operation F → F|z¯ =w¯ ¯ Mdiag ) which extends the operation on C ∞ (M × M, φ ⊗ ψ → δ(φ) · (1 ⊗ ψ) from the factorizable elements. Denote by B the bidifferential operator on M such that B(φ, ψ) = A(φ · ψ). In local coordinates
K
∂ ∂ z¯
ψ(z, z¯ ) .
B(φ, ψ) = B K L¯ P Q¯ (z, z¯ ) ·
∂ ∂z
P
∂ ∂z Q
∂ L φ(z, z¯ ) ∂ z¯
Here we assume that, say, K = (k1 , . . . , km ) and K ∂ ∂ k1 ∂ km = . . . , ∂z ∂z 1 ∂z m where m is the complex dimension of M. Now, operation (82) can be extended to the ¯ Mdiag ) as follows: space C ∞ (M × M, ∂ K ∂ L ¯ F1 (z, z¯ , w, w) ¯ |w=z (F1 , F2 ) → B K L¯ P Q¯ (z, w) ∂w ∂ w¯ ∂ P ∂ Q · F2 (z, z¯ , w, w) ¯ |z¯ =w¯ . ∂z ∂ z¯ References 1. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Physics 111(1), 61–110 (1978) 2. Berezin, F.A.: Quantization. Math. USSR-Izv. 8, 1109–1165 (1974) 3. Berezin, F.A.: Quantization in complex symmetric spaces. Math. USSR-Izv. 9, 341–379 (1975) 4. Bertelson, M., Cahen, M., Gutt, S.: Equivalence of star products. Geometry and physics. Class. Quantum Grav. 14(1A), A93–A107 (1997) 5. Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of Kähler manifolds and gl(n), n → ∞ limits. Commun. Math. Phys. 165, 281–296 (1995) 6. Bordemann, M., Waldmann, S.: A Fedosov star product of the Wick type for Kähler manifolds. Lett. Math. Phys. 41(3), 243–253 (1997) 7. Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds I: Geometric interpretation of Berezin’s quantization. JGP 7, 45–62 (1990) 8. Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds II. Trans. Amer. Math. Soc. 337, 73–98 (1993) 9. Connes, A., Flato, M., Sternheimer, D.: Closed star-products and cyclic cohomology. Lett. Math. Phys. 24, 1–12 (1992)
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10. De Wilde, M., Lecomte, P.B.A.: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7(6), 487–496 (1983) 11. Deligne, P.: Déformations de l’algébre des fonctions d’une variété symplectique: comparison entre Fedosov et De Wilde, Lecomte. Selecta Math. (N.S.) 1(4), 667–697 (1995) 12. Fedosov, B.: A simple geometrical construction of deformation quantization. J. Diff. Geom. 40(2), 213–238 (1994) 13. Fedosov, B.: Deformation quantization and index theory. Mathematical Topics, 9. Berlin: Akademie Verlag (1996) 14. Guillemin, V.: Star products on pre-quantizable symplectic manifolds. Lett. Math. Phys. 35, 85–89 (1995) 15. Gutt, S., Rawnsley, J.: Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66, 123–139 (2003) 16. Karabegov, A.: Deformation quantizations with separation of variables on a Kähler manifold. Commun. Math. Phys. 180, 745–755 (1996) 17. Karabegov, A: On the canonical normalization of a trace density of deformation quantization. Lett. Math. Phys. 45, 217–228 (1998) 18. Karabegov, A.: Pseudo-Kähler quantization on flag manifolds. Commun. Math. Phys. 200, 355–379 (1999) 19. Karabegov, A.: On the dequantization of Fedosov’s deformation quantization. Lett. Math. Phys. 65, 133–146 (2003) 20. Karabegov, A.: Formal symplectic groupoid of a deformation quantization. Commun. Math. Phys. 258, 223–256 (2005) 21. Karabegov, A., Schlichenmaier, M.: Identification of Berezin-Toeplitz deformation quantization. J. Reine Angew. Math. 540, 49–76 (2001) 22. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66, 157–216 (2003) 23. Moreno, C.: ∗-products on some Kähler manifolds. Lett. Math. Phys. 11, 361–372 (1986) 24. Nest, R., Tsygan, B.: Algebraic index theorem. Commun. Math. Phys. 172(2), 223–262 (1995) 25. Neumaier, N.: Universality of Fedosov’s construction for star products of Wick type on Pseudo-Kähler manifolds. Rep. Math. Phys. 52, 43–80 (2003) 26. Omori, H., Maeda, Y., Yoshioka, A.: Weyl manifolds and deformation quantization. Adv. Math. 85, 224–255 (1991) 27. Reshetikhin, N., Takhtajan, L.: Deformation quantization of Kähler manifolds. L. D. Faddeev’s Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 201, Providence, RI: Amer. Math. Soc., 2000, pp. 257–276 28. Schlichenmaier, M.: Berezin-Toeplitz quantization of compact Kähler manifolds. In: Quantization, Coherent States and Poisson Structures, Proc. XIVth Workshop on Geometric Methods in Physics (Bialowieza, Poland, 9–15 July 1995), A. Strasburger, S. T. Ali, J.-P. Antoine, J.-P. Gazeau, A. Odzijewicz (eds.) Warsaw:Polish Scientific Publisher PWN, 1998, pp. 101–115 29. Xu, P.: Fedosov ∗-products and quantum momentum maps. Commun. Math. Phys. 197(1), 167–197 (1998) Communicated by L. Takhtajan
Commun. Math. Phys. 274, 691–715 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0292-5
Communications in
Mathematical Physics
A Mathematical Theory for Vibrational Levels Associated with Hydrogen Bonds I: The Symmetric Case George A. Hagedorn1, , Alain Joye2,3 1 Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics,
Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0123, USA. E-mail: [email protected] 2 Institut Fourier, Unité Mixte de Recherche CNRS-UJF 5582, Université de Grenoble I, BP 74, F–38402 Saint Martin d’Hères Cedex, France 3 Laboratoire de Physique et Modélisation des Milieux Condensés, UMR CNRS-UJF 5493, Université de Grenoble I, BP 166, 38042 Grenoble, France Received: 25 July 2006 / Accepted: 7 March 2007 Published online: 17 July 2007 – © Springer-Verlag 2007
Abstract: We propose an alternative to the usual time–independent Born–Oppenheimer approximation that is specifically designed to describe molecules with symmetrical Hydrogen bonds. In our approach, the masses of the Hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. Consequently, anharmonic effects play a role in the leading order calculations of vibrational levels. Although we develop a general theory, our analysis is motivated by an examination of symmetric bihalide ions, such as F H F − or Cl H Cl − . We describe our approach for the F H F − ion in detail. 1. Introduction In standard Born–Oppenheimer approximations, the masses of the electrons are held fixed, and the masses of the nuclei are all assumed to be proportional to −4 . Approximate solutions to the molecular Schrödinger equation are then sought as expansions in powers of . For the time–independent problem, the electron energy level surface is also assumed to behave asymptotically like a quadratic function of the nuclear variables near a local minimum. In this paper and in a future one [4], we propose an alternative approximation for molecules that contain Hydrogen atoms as well as some heavier atoms, such as Carbon, Nitrogen, or Oxygen. Our motivation is to develop an approach that is specifically tailored to describe the phenomenon of Hydrogen bonding. In this paper, we examine the specific case of systems with symmetric Hydrogen bonds, such as F H F − . In [4], we plan to study non–symmetric cases, where the structure of the typical electron energy surface is very different. The mathematical analysis of that situation is consequently completely different. Partially Supported by National Science Foundation Grants DMS–0303586 and DMS–0600944.
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The model we present here differs from the usual Born–Oppenhimer model in two ways: 1. We scale the masses of the Hydrogen nuclei as −3 instead of −4 . This is physically appropriate. If the mass of an electron is 1, and we define −4 to be the mass of a C 12 nucleus, then = 0.0821, and the mass of a H 1 nucleus is 1.015 −3 . 2. We do not assume that the electron energy level is well approximated by an –independent quadratic function near a local minimum. Instead, we allow it to depend on and to take a particular form that we specify below. The particular form we have chosen is motivated by a detailed examination of the lowest electronic potential energy surfaces for F H F − and Cl H Cl − . Although symmetric bihalide ions are quite special, our approach is flexible enough to describe more general phenomena. For example, the lowest electron energy surface for F H F − has a single minimum with the Hydrogen nucleus mid–way between the two Fluorines. Our model can handle situations with single or double wells in the coordinates for a Hydrogen nucleus that participates in Hydrogen bonding. We hope that the ideas in this paper and [4] might provide some insight into some properties of Hydrogen bonded systems. Our model leads to a different expansion from the usual Born–Oppenheimer approximation. For Hydrogen nuclei not involved in Hydrogen bonding, the vibrational energies are of order 3/2 , while the vibrational energies for the other nuclei and the Hydrogen nuclei involved in the symmetric Hydrogen bonding are of order 2 . Furthermore, anharmonic effects must be taken into account for a Hydrogen nucleus involved in Hydrogen bonding at their leading order, 2 . In the standard Born–Oppenheimer model, all vibrational energies appear in a harmonic approximation at order 2 . Anharmonic corrections enter at order 4 . We present our ideas only in the simplest possible situation. In that situation, there are only 3 nuclei, and they are constrained to move along a fixed line. We plan to study more general possibilities, such as bending of the molecule, in the future. The paper is organized as follows: In Sect. 2, we present the formal expansion. In Sect. 3 we state our rigorous results as Theorems 3.7 and 3.8. The proofs of some technical results are presented in Sect. 4. 2. Description of the Model We study a triatomic system with two identical heavy nuclei A and B, and one light (Hydrogen) nucleus C. We begin by describing the Hamiltonian for this system in Jacobi coordinates. We let x A and x B be the positions of the heavy nuclei, and let xC be the position of the light nucleus C. We let their masses be m A = m B and m C . We m A x A + m B x B + m C xC let R = denote the center of mass of all three nuclei, and let m A + m B + mC xA + xB denote the center of mass of the heavy nuclei. We let W = x B − x A be x AB = 2 the vector from nucleus A to nucleus B and let Z = xC − x AB be the vector from the center of mass of A and B to C. We assume the electronic Hamiltonian h e only depends on the vectors between the nuclei, and we set m AB = m A +m B and M = m A +m B +m C . In the original variables, the Hamiltonian has the form −
1 1 1 x A − x B − xC + h e (x B − x A , xC − x A , xC − x B ). 2mA 2 mB 2 mC
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In these Jacobi coordinates, it has the form −
1 m AB M R − W − Z + h e (W, Z + W/2, Z − W/2). 2M 2 mA mB 2 m AB m C
Since we are interested in bound states, we discard the kinetic energy of the center of mass. We take the electron mass to be 1, and the masses of the heavy nuclei to be m A = m B = −4 µ, for some fixed µ. The mass of the light nucleus is m C = −3 ν, for some fixed ν. The electronic Hamiltonian h e then becomes h e (W, Z +W/2, Z −W/2) ≡ h(W, Z ), so that the Hamiltonian of interest is −
4 3 W − µ 2ν
1+
ν 2µ
Z + h(W, Z ).
This computation is exact and valid in any dimension. ν To simplify the exposition, we drop the term in the factor that multiplies Z . 2µ It gives rise to uninteresting, regular perturbation corrections. Also, for simplicity, we assume µ = 2 and ν = 1. This can always be accomplished by trivial rescalings of W and Z . To describe our ideas in the simplest situation, we restrict W and Z to one dimension. Thus, we are not allowing rotations or bending of the molecule. Furthermore, we introduce dependence of the electronic Hamiltonian to model the pecularities of symmetric Hydrogen bonds that we describe below. These considerations lead us to study the Hamiltonian H1 () = −
4 ∂ 2 3 ∂ 2 − + h(, W, Z ). 2 2 ∂W 2 ∂ Z2
(2.1)
The electron Hamiltonian h(, W, Z ) is an operator in the electronic Hilbert space that depends parametrically on (, W, Z ) and includes the nuclear repulsion terms. For convenience, we assume that h(, W, Z ) is a real symmetric operator. We now describe the specific dependence of h(, W, Z ) that we assume. Although the electron Hamiltonian does not depend on nuclear masses, the parameter is dimensionless, and thus may play more than one role. The dependence of h on we allow is motivated by the smallness of a particular Taylor series coefficient we observed in numerical computations for the ground state electron energy level for the real system F H F − . We allow only the ground state eigenvalue to depend on . Otherwise, our electron Hamiltonian is –independent. With the physical value of inserted in our Hamiltonian, we obtain the true physical Hamiltonian. From numerical computations of E(W, Z ) for F H F − , we observed that the Z 2 coefficient in the Taylor expansion about the minimum (W0 , 0) of the ground state potential energy surface had a small numerial value, on the order of the value of = 0 , where 0 was defined by setting 0−4 equal to the nuclear mass of the C 12 isotope of Carbon. The value of 0 is roughly 0.0821. We define a2 so that the true Z 2 Taylor series term is a2 0 Z 2 . We then obtain h(, W, Z ) by adding ( − 0 ) a2 Z 2 to the ground state eigenvalue E(W, Z ). We make no other alterations to the electron Hamiltonian. When = 0 , our h(, W, Z ) equals the true physical electron Hamiltonian h(0 , W, Z ).
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Thus, we assume the ground state electron level has the specific form E 1 (, W, Z ) = E 0 + a1 (W − W0 )2 + a2 − a3 (W − W0 ) Z 2 + a4 Z 4 + · · · ,
(2.2)
with a j = O(1). As we shall see, the leading order behavior of the energy and the wave functions for the molecule are determined from the terms written explicitly in (2.2). The terms not explicitly displayed are of orders (W − W0 )α Z 2β , where α and β are non-negative integers that satisfy α + β ≥ 3. They play no role to leading order, but contribute to higher order corrections. We assume a1 , a3 , and a4 are positive, but that a2 can be positive, zero, or negative. When a2 is negative, E 1 (, W, Z ) has a closely spaced double well near (W0 , 0) instead of a single local minimum. To ensure that the leading part of E 1 (, W, Z ), 1 (, W, Z ) = E 0 + a1 (W − W0 )2 + a2 − a3 (W − W0 ) Z 2 + a4 Z 4 , E is bounded below, we assume that either a32 < 4 a1 a4 ,
(2.3)
or a32 = 4 a1 a4
and
a2 ≥ 0.
(2.4)
1 (, W, Z ) ≥ − C for some C, since These conditions are equivalent to the property E we can write 2 2 a a 3 1 (, W, Z ) = a1 (W − W0 ) − Z2 + a4 − 3 Z 4 + a2 Z 2 . E 2a1 4a1 By rescaling with w = (W − W0 )/ and z = Z / 1/2 , we see that the Hamiltonian −
3 ∂ 2 4 ∂ 2 1 (, W, Z ) − + E 2 2 ∂W 2 ∂ Z2
is unitarily equivalent to 2 times the –independent Normal Form Hamiltonian HNF = −
1 ∂2 1 ∂2 − + E NF (w, z), 2 ∂w 2 2 ∂z 2
where
E NF (w, z) = a1 w + 2
(2.5)
a2 − a3 w
z 2 + a4 z 4 .
(2.6)
Remark. Although we do not use it, further scaling shows that HNF is essentially a three–parameter model, since the change of variables w = α s, z = α t, yields 1 ∂2 1 ∂2 2 2 2 4 , − + α s + α t − α s t + t HNF α −2 − 1 2 3 2 ∂s 2 2 ∂t 2
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with −1/6
α = a4
, α1 =
a1 2/3 a4
, α2 =
a2 2/3 a4
, and α3 =
a3 5/6
a4
.
Under conditions (2.3) or (2.4), HNF is essentially self-adjoint on C0∞ (R2 ) and has purely discrete spectrum. This last property is easy to verify under condition (2.3), or condition (2.4) with a2 > 0, because E NF (w, z) tends to infinity as (w, z) → ∞. When (2.4) is satisfied with a2 = 0, the result is more subtle because E NF (w, z) attains its minimum value of zero along a parabola in (w, z). In that case we prove that the spectrum is discrete in Proposition 3.1. Explicit Computations for F H F − . The expression (2.2) is clearly special. Our computations for F H F − that motivate this expression have roughly the following values, where distances are measured in Angstroms and energies are measured in Hartrees: W0 E0 a1 a2 a3 a4
= = = = = =
2.287, −200.215, 0.26, 1.22 ( if = 0.0821 ), 1.29, 1.62.
These results came from fitting the output from Gaussian 2003 using second order Moller–Plesset theory with the aug–cc–pvtz basis set. We observed that the process of fitting the data was numerically quite unstable, and that condition (2.3) was barely satisfied by these a j . The experimentally observed values [11] for the excitation energies to the first symmetric stretching vibrational mode and the first asymmetric vibrational mode of F H F − are 583.05 cm−1 and 1331.15 cm−1 , respectively. With the values of a j above, the leading order calculation from our model predicts 600 cm−1 and 1399 cm−1 . By leading order, we mean E 0 + 2 E2 in the expansion we present below. These values depend sensitively on precisely how we fit the potential energy surface, which itself depends sensitively on the electron structure calculations. By comparison, Gaussian 2003 with the aug-cc-pvdz basis set predicts harmonic frequencies of 608 cm−1 and 1117 cm−1 . We could not obtain frequencies for the aug-cc-pvtz basis set from Gaussian because of our computer limitations. For some very recent numerical results for vibrational frequencies of F H F − that appeared as we were finishing this paper, see [2]. We now mimic the technique of [3] to obtain an expansion for the solution to the eigenvalue problem for (2.1). We could have used the technique of [5], but that would have led to more complicated formulas. For convenience, we replace the variable W by W − W0 , so that henceforth, W0 = 0. The technique of [3] uses the method of multiple scales. Instead of searching directly for an eigenvector (, W, Z ) for (2.1), we first search for an eigenvector ψ(, W, Z , w, z) for an operator that acts in more variables. When we have determined ψ, we obtain by setting (, W, Z ) = ψ(, W, Z , W/, Z / 1/2 ).
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Displacement of H from F − F Center of Mass
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
2.1
2.2
2.3
2.4
2.5
F − F Distance Fig. 1. Contour plot of the ground state electronic potential energy surface in the Jacobi coordinates (W, Z). It is obviously not well approximated by a quadratic. Our technique exploits the flatness of the surface in the Z direction near the minimum
This is motivated physically by the following observation: The dependence of the electrons on the nuclear coordinates occurs on the length scale of (W, Z ), while the semiclassical quantum fluctuations of the nuclei occur on the length scale of (w, z). To leading order in , these effects behave independently. The equation for ψ is formally H2 () ψ(, W, Z , w, z)
=
E() ψ(, W, Z , w, z),
(2.7)
where 2 ∂ 2 ∂2 4 ∂ 2 3 ∂ 2 − − 3 − 2 2 2 ∂W ∂ W ∂w 2 ∂w 2 ∂ Z2 2 2 2 ∂ ∂ − − 5/2 ∂ Z ∂z 2 ∂z 2 + [ h(, W, Z ) − E(, W, Z ) ] + E(, w, 1/2 z) ∞ + m/2 Tm/2 (W, Z ) − Tm/2 ( w, 1/2 z) .
H2 () = −
m=6
(2.8)
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The functions Tm/2 in this expression will be chosen later. Different choices yield equally valid expansions for (, W, Z ), although they alter the expressions for ψ(, W, Z , w, z) by converting (W, Z ) dependence into (w, z) dependence. In (2.8), we expand both E(, w, 1/2 z) and Tm/2 (w, 1/2 z) in Taylor series in powers of 1/2 . We then make the Ansatz that (2.7) has formal solutions of the form ψ(, W, Z , w, z) = ψ0 (W, Z , w, z) + 1/2 ψ1/2 (W, Z , w, z) + 1 ψ1 (W, Z , w, z) + · · · ,
(2.9)
with E() = E0 + 1/2 E1/2 + 1 E1 + · · · .
(2.10)
We substitute these expressions into (2.7) and solve the resulting equation order by order in powers of 1/2 . Note. The description in this section is purely formal. In particular, it does not take into account the cutoffs that are necessary for rigorous results. The mathematical details are dealt with in the next section. Order 0. The order 0 terms require [ h(, W, Z ) − E(, W, Z ) ] ψ0 + E 0 ψ0 = E0 ψ0 . We solve this by choosing E0 = E 0 , and ψ0 (W, Z , w, z) = f 0 (W, Z , w, z) (W, Z ), where (W, Z , · ) is a normalized ground state eigenvector of h(, W, Z ). Under our assumptions, we can choose (W, Z , · ) to be real, smooth in (W, Z ), and independent of . This choice satisfies (W, Z , · ), ∇W,Z (W, Z , · ) Hel = 0,
(2.11)
where the inner product is in the electronic Hilbert space. We assume that f0 (W, Z , w, z) is not identically zero. Order 1/2. The order 1/2 terms require [ h(, W, Z ) − E(, W, Z ) ] ψ1/2 + E 0 ψ1/2 = E0 ψ1/2 + E1/2 ψ0 . The components of this equation in the (W, Z ) direction in the electronic Hilbert space require E1/2 = 0. The components of the equation orthogonal to (W, Z ) in the electronic Hilbert space require [ h(, W, Z ) − E(, W, Z ) ] ψ1/2 = 0,
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so ψ1/2 (W, Z , w, z) = f 1/2 (W, Z , w, z) (W, Z ). Orders 1 and 3/2. By similar calculations, the order 1 and 3/2 terms yield E1 = E3/2 = 0, ψ1 (W, Z , w, z) = f 1 (W, Z , w, z) (W, Z ), ψ3/2 (W, Z , w, z) = f 3/2 (W, Z , w, z) (W, Z ).
and
Order 2. The order 2 terms that are multiples of (W, Z ) in the electronic Hilbert space require −
1 ∂ 2 f0 1 ∂ 2 f0 (W, Z , w, z) − (W, Z , w, z) + E NF (w, z) f 0 (W, Z , w, z) 2 ∂w 2 2 ∂z 2 = E2 f 0 (W, Z , w, z), (2.12)
where E NF (w, z) is given by (2.6). Because of the form of E NF (w, z), (2.12) does not separate into two ODE’s. We do not know E2 or f 0 exactly, although accurate numerical approximations can be found easily. These eigenvalues and eigenfunctions describe the coupled anharmonic vibrational motion of all three nuclei in the molecule. As we commented earlier, hypotheses (2.3) or (2.4) guarantee that the eigenvalues E2 are discrete and bounded below, with normalized bound states f 0 (W, Z , w, z) in (w, z) for any (W, Z ). Later in the expansion, we choose the operator T3 so that f 0 has no (W, Z ) dependence. With this in mind, Eq. (2.12) determines E2 and a normalized function f 0 (w, z) (up to a phase) for any given vibrational level. The terms of order 2 that are orthogonal to (W, Z ) require [ h(, W, Z ) − E(, W, Z ) ] ψ2 = 0. Thus, ψ2 = f 2 (W, Z , w, z) (W, Z ). We split the scalar functions f α (W, Z , w, z) with α > 0 into two contributions: f α (W, Z , w, z) = f α (W, Z , w, z) + f α⊥ (W, Z , w, z),
where for each fixed W and Z , f α (W, Z , ·, · ) is a multiple of f 0 (·, · ), and f α⊥ (W, Z , ·, · ) perpendicular to f 0 (·, · ) in L 2 (R2 , dw dz). Furthermore, we choose the operators T3+m/2 later in the expansion so that f α (W, Z , ·, · ) has no (W, Z ) dependence. We will not precisely normalize our approximate eigenfunctions, so we henceforth assume f α (W, Z , w, z) = 0 for all α > 0. Order m/2 with m > 4. We equate the terms of order m/2 and then separately examine the projections of the resulting equation into the (W, Z ) direction in the electron Hilbert space and into the direction perpendicular to (W, Z ). From the terms in the (W, Z ) direction, we obtain the value of Em/2 and an expres⊥ (W, Z , w, z). When m = 6 we choose T3 sion for f (m−4)/2 (W, Z , w, z) = f (m−4)/2 so that f 0 can be chosen independent of (W, Z ). When m > 6, we choose Tm/2 , so that f (m−6)/2 can be taken to be zero.
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The terms orthogonal to (W, Z ) in the electronic Hilbert space give rise to an equation for [ h(, W, Z ) − E(, W, Z ) ] ψm/2 . This equation has a solution of the form ⊥ ψm/2 (W, Z , w, z) = f m/2 (W, Z , w, z) + f m/2 (W, Z , w, z) (W, Z ) ⊥ (W, Z , w, z), + ψm/2 ⊥ is obtained by applying the reduced resolvent operator [ h(, W, Z ) − where ψm/2 E(, W, Z ) ]r−1 to the right hand side of the equation. In the next section, we prove that this procedure yields a quasimode whose approximate eigenvalue and eigenvector each have asymptotic expansions to all orders in 1/2 .
3. Mathematical Considerations In this section we present a mathematically rigorous version of the expansion of Sect. 2. This involves inserting cutoffs and proving that many technical conditions are satisfied at each order of the expansion. Proposition 3.1. Assume (2.3) or (2.4). 1 ∂2 1 ∂2 Then, the spectrum of HN F = − − + E N F (w, z) is purely discrete. 2 ∂w 2 2 ∂z 2 This proposition is an easy consequence of the following general criterion [12] that guarantees that the spectrum of − + V is discrete for certain polynomials V . This criterion finds its roots in earlier work on hypoelliptic operators. (See e.g., [9].) Proposition 3.2. ([12], Thm 1.3). Let V (x) ≥ 0 be a non-negative polynomial in x ∈ Rn . Define m ∗V (x) = 1 + |D α V (x)|, α∈Nn
where D α = ∂xα11 ∂xα22 · · · ∂xαnn , and the sum is finite. Let H = − + V be selfadjoint on a domain in L 2 (Rn ). Then the resolvent of H is compact if and only if lim|x|→∞ m ∗V (x) = ∞. Proof of Prop. 3.1. One easily checks that lim|x|→∞ m ∗E N F (x) = ∞, so Prop. 3.1 is an immediate consequence of Prop. 3.2. In the usual Born–Oppenheimer approximation, the semiclassical expansion for the nuclei is based on Harmonic oscillator eigenfunctions. They have many well-known properties. Our expansion relies on the analogous properties for eigenfunctions of H N F . The following proposition establishes some of the properties we need in an even more general setting. Proposition 3.3. Let V be a non-negative polynomial, such that H = − + V has purely discrete spectrum. Let ϕ(x) be an eigenvector of H , i.e., an L 2 (Rn ) solution of H ϕ = Eϕ, where E > 0. Then, ϕ ∈ C ∞ (Rn ) and ∇ϕ ∈ L 2 (Rn ). Moreover, for any a > 0, ϕ ∈ D(eax ), ∇ϕ ∈ D(eax ), and ϕ ∈ D(eax ),
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n
where x = 1 + x 2j , and D(eax ) denotes the domain of multiplication by j=1
eax . Proof. Since V ∈ C ∞ , elliptic regularity arguments (see e.g., [8], Thm. 7.4.1) show that all eigenfunctions are C ∞ . We first show that the ∇ϕ is L 2 . Since V ≥ 0, the quadratic form defined by √ √ h(ϕ, ψ) = ∇ϕ, ∇ψ + V ϕ, V ψ on Q(h) = Q(−) ∩ Q(V ), is closed and positive. Here Q(A) means the quadratic form domain of the operator A. Since D(H ) ⊂ Q(h), any eigenvector of H belongs to Q(−) = { ϕ ∈ L 2 (Rn ) : ∇ϕ < ∞ }. Thus, ∇ϕ ∈ L 2 . Next, we prove ϕ ∈ D(eax ), for any a > 0 by a Combes–Thomas argument, as presented in Theorem XII.39 of [15]. We describe the details for completeness. Let α ∈ R, and let v denote x j for any j ∈ {1, · · · , n}. We consider the unitary group W (α) = eiαv for α ∈ R, and compute H (α) = W (α) (− + V ) W (α)−1 = H + i α ∂v + α 2 . The operator i∂v is H -bounded, with arbitrary small relative bound, since V ≥ 0. Thus {H (α)} extends a self-adjoint, entire analytic family of type A, defined on D(H ). We note that since H (0) = H has purely discrete spectrum, its resolvent, R0 (λ) is compact, for any λ ∈ ρ(H ) ≡ C \ σ (H ). Hence, Rα (λ) = (H (α) − λ)−1 is compact for any α ∈ R, and hence, for all α ∈ C, if λ ∈ ρ(H (α)). It is jointly analytic in α and λ. The eigenvalues of H (α) are thus analytic in α, except at crossing points, where they may have algebraic singularities. Since for α real, W (α) is unitary, the eigenvalues are actually independent of α, and σ (H (α)) = σ (H ), for any α. Let P be the finite rank spectral projector corresponding to an eigenvalue E of H N F . Then, for α ∈ R, P(α) = W (α)P W (α)−1 is the spectral projector corresponding to the eigenvalue E of H (α). By Riesz’s formula and the properties of the resolvent, P(α) extends to an entire analytic function that satisfies W (α0 )P(α)W (α0 )−1 = P(α0 + α) for any α0 ∈ R. By O’Connor’s Lemma (Sect. XIII.11 of [15]), this yields information about the eigenvectors. If ϕ = Pϕ, the vector ϕ α = W (α)ϕ, defined for α ∈ R has an analytic extension to the whole complex plane, and is an analytic vector for the operator v. Therefore, ϕ ∈ D(ea|v| ), for any a > 0. By taking all possible x j ’s for v, and noting that D(eax ) = D(ea( j |x j |) ), we see that ϕ ∈ D(eax ). From this, it follows that ϕ ∈ D(eax ) for any a > 0 as well, since for any δ > 0,
e2ax |ϕ(x)|2 d x n R
= e2ax | (V (x) − E) ϕ(x) |2 d x Rn
≤ (V − E)2 e−δ· ∞ e(a+δ/2)· ϕ(·) 2 < ∞.
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Finally, Lemma 3.4 below shows that ∇ϕ ∈ D(eax ). To apply this lemma in our situation, we let p(x) = eax and note that for any a > 0, (∇eax )/eax = a∇x = ax/x is uniformly bounded.
Lemma 3.4 requires some notation. Letting p(x) be a positive weight function, we introduce the space
2 2 2 2 | f (x)| + | f (x)| p(x) d x < ∞ . f : f F 2 = Fw = w
Rn
We write f 2w = Rn | f (x)|2 p(x) d x, for any f ∈ L 2 (Rn , p(x)d x), and f 2 = 2 Rn | f (x)| d x when the weight is one. Lemma 3.4. Let p ∈ C 1 be positive, and assume that there exists a constant C < ∞, such that |(∇ p(x))/ p(x)| ≤ 2 C for all x ∈ Rn . Then, for any f ∈ Fw2 , ∇ f w ≤ C f w + f w f w + C 2 f 2w . (3.1) We present the proof of this technical lemma in Sect. 4. We now state and prove the following corollary to Proposition 3.3: Corollary 3.5. Assume the hypotheses of Proposition 3.3. Let R(λ) be the resolvent of H = − + V for λ ∈ / σ (H ), and let PE be the finite dimensional spectral projector −1 of H on E. Let r (E) = (H − E)|(I−PE )L 2 be the reduced resolvent at E. Then, eax R(λ) e−ax and eax r (E) e−ax are bounded on L 2 (Rn ). Proof. We use the notation of the proof of Proposition 3.3. We know that Rα (λ) is compact and analytic in α ∈ C, if λ ∈ σ (H ). Hence, for any ψ1 , ψ2 ∈ C0∞ , the map from R × ρ(H ) to C given by (a, λ) → ψ1 , eav R0 (λ) e−av ψ2 is uniformly bounded by C ψ1 ψ2 on any given compact set of R×ρ(H ) for some C. From this we infer that for any a > 0, eax R(λ) e−ax is bounded in L 2 (Rn ), uniformly for λ in compact sets of ρ(H ). Since the reduced resolvent r (E) can be represented as
1 1 r (E) = dλ, (3.2) R0 (λ) 2πi CE λ−E where CE is a loop in the resolvent set encircling only E, the boundedness of eax r (E) e−ax follows. To show that the terms of our formal expansion all belong to L 2 , we use the following generalization of Proposition 3.3. We present its proof in Sect. 4. Proposition 3.6. Assume the hypotheses of Proposition 3.3 and let ϕ be an L 2 solution of (− + V − E) ϕ = 0. Then, for any a > 0, and any multi-index α ∈ Nn , D α ϕ ∈ D(eax ), where D α = ∂xα11 ∂xα22 · · · ∂xαnn .
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Remark. Exponential decay of eigenfunctions is a well known and well studied property for Schrödinger operators. (See e.g., the review [6].) However, we were unable to find any references dealing with the exponential decay of all successive derivatives of eigenfunctions in our framework. We now prove that our formal expansion leads to rigorous quasimodes for the Hamiltonian H1 () given by (2.1). Theorem 3.7 summarizes this result for the leading order, while Theorem 3.8 handles the arbitrary order results. Theorem 3.7. Let h(, W, Z ) be defined as in Sect. 2 with W shifted so that W0 = 0. We assume h(, W, Z ) on Hel is C 2 in the strong resolvent sense for (W, Z ) near the origin. We assume its non-degenerate ground state is given by 2 E 1 (, W, Z ) = E 0 + a1 W + a2 − a3 W Z 2 + a4 Z 4 + S(, W, Z ) ˜ ≡ E 0 + E(, W, Z ) + S(, W, Z ),
(3.3)
under hypothesis (2.3) or (2.4), and we denote the corresponding normalized eigenstate by (W, Z ). Suppose the remainder term S is uniformly bounded below by some r > −∞ and that |S| satisfies a bound of the form | S(, W, Z ) | ≤ C | W α Z 2β | (3.4) α+β≥3
for (W, Z ) in a neighborhood of the origin. Here C is independent of , the sum is finite, and α and β are non-negative integers. Let f 0 (w, z) be a normalized non-degenerate eigenvector of H N F , i.e., (−∂w2 /2 − ∂z2 /2 + E N F (w, z)) f 0 = E2 f 0 , with
E N F (w, z) = a1 w 2 +
a2 − a3 w
z 2 + a4 z 4 .
Then, for small enough , there exists an eigenvalue E() of H1 () which satisfies E() = E 0 + 2 E2 + O( ξ ), for some ξ > 2 as → 0. Remarks. 1. At this level of approximation, it is not necessary to require the eigenvector
to satisfy condition (2.11) or to require h(, W, Z ) be real symmetric. 2. We have stated our results for the electronic ground state, but the analogous results would be true for any non-degenerate state that had the same type of dependence on . Proof. In the course of the proof, we denote all generic non-negative constants by the same symbol c. Our candidate for the construction of a quasimode is √ Q (, W, Z ) = F(W/ δ1 ) F(Z / δ2 ) f 0 (W/, Z / ) (W, Z ), (3.5) where F : R → [0, 1] is a smooth, even cutoff function supported on [−2, 2] which is equal to 1 on [−1, 1]. One should expect the introduction of these cutoffs not to affect
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the expansion at any finite order because the eigenvectors of H1 () are localized near the minimum of E 1 (, W, Z ). Thus, the properties of the electronic Hamiltonian for large values (W, Z ) should not matter. The choice of a different cutoff for each variable is required because these variables have different scalings in . We determine the precise values of the positive exponents δ1 and δ2 in the course of the proof. We also use the notation F(, W, Z )
=
F(W/ δ1 ) F(Z / δ2 ).
(3.6)
We first estimate the norm of Q :
√ |F(, W, Z ) f 0 (W/, Z / )|2 (W, Z )2H Q 2 = el 2
R √ 2 = | f 0 (W/, Z / )| dW d Z R2
√ − (1 − F 2 (, W, Z )) | f 0 (W/, Z / )|2 dW d Z . R2
The first term of the last expression equals 3/2 , by scaling, since f 0 is normalized. If δ1 < 1 and δ2 < 1/2, the negative of the second term is bounded above by
√ | f 0 (W/, Z / )|2 dW d Z |W |≥ δ1 |Z |≥ δ2
=
≤ =
3/2 e−2a(1/ O( ∞ ),
3/2 |w|≥ 1−δ1 |z|≥ 1/2−δ2
e−2a(|w|+|z|) e2a(|w|+|z|) | f 0 (w, z)|2 dw dz
(1−δ1 ) +1/ (1−δ2 ) )
ea(|·|+|·|) f 0 2
since f 0 ∈ D(ea(W,Z ) ). Hence, Q = 3/4 (1 + O( ∞ )), where the O( ∞ ) correction is non-positive. (3.7) Next we compute (H1 () − (E 0 + 2 E2 )) Q (, W, Z ) = S(, W, Z ) f 0 (w, z)|W,Z F(, W, Z ) (W, Z ) 4 2 3 2 F(, W, Z ) (W, Z ) f 0 (w, z)|W,Z , − ∂ + ∂ 2 W 2 Z − 3 ∂w f 0 (w, z)|W,Z ∂W (F(, W, Z ) (W, Z )) − 5/2 ∂z f 0 (w, z)|W,Z ∂ Z (F(, W, Z ) (W, Z )),
(3.8)
√ where we have introduced the shorthand f 0 (w, z)|W,Z = f 0 (W/, Z / ) and used the identity √ ˜ E(, W, Z ) − 2 E N F (W/, Z / ) ≡ 0.
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Also 1 F (W/ δ1 ) F(Z / δ2 ), δ1 1 ∂ Z F(, W, Z ) = δ F(W/ δ1 ) F (Z / δ2 ), 2
∂W F(, W, Z ) =
µ
and, by assumption, ∂W ∂ Zν (W, Z )Hel is continuous and of order 0 in a neigborhood of the origin, for µ + ν ≤ 2. Therefore, sup ∂W (F(, W, Z ) (W, Z )) Hel R2
sup ∂ Z (F(, W, Z ) (W, Z )) Hel R2
2 sup ∂W (F(, W, Z ) (W, Z )) Hel
R2
sup ∂ Z2 (F(, W, Z ) (W, Z )) Hel R2
c , δ1 c ≤ , δ2 c ≤ , 2δ 1 c ≤ , 2δ2 ≤
(3.9)
where all vectors are supported in { (W, Z ) : |W | ≤ 2/ δ1 , |Z | ≤ 2/ δ2 }. Each of these vectors appears in (3.8), multiplied by one of the scalar functions f 0 (w, z)|W,Z , (∂w f 0 (w, z)) |W,Z , or (∂z f 0 (w, z)) |W,Z . In turn, each of these functions belongs to L 2 (R2 ) by Proposition 3.3, and each one has norm of order 3/4 because of scaling, e.g., R2
√ | (∂w f 0 )(W/, Z / ) |2 dW d Z
1/2 =
3/4 ∂w f 0 L 2 (R2 ) .
Therefore, the norms of the last three vectors in (3.8) are of order 3/4 times the corresponding power of stemming from (3.9). We now estimate the norm of the term that arises from the error term S. From our hypothesis on the behavior of S, we have
√ 2 S F f 0 = |W |≤2/ δ1 | f 0 (W/, Z / ) S(W, Z )|2 dW d Z |Z |≤2/ δ2
≤c
|W |≤2/ δ1 |Z |≤2/ δ2
α+β≥3
≤c
α+β≥3
=c
2 √ | f 0 (W/, Z / )|2 W α Z 2β dW d Z
2(αδ1 +2βδ2 )
R2
√ | f 0 (W/, Z / )|2 dW d Z
2(αδ1 +2βδ2 ) 3/2 ,
α+β≥3
where the sums are finite. Collecting these estimates and inserting the allowed values of α and β, we obtain (H1 () − (E 0 + 2 E2 )) Q ≤ c 3/4 3δ1 + 2(δ1 +δ2 ) + δ1 +4δ2 + 6δ2 + 4−2δ1 + 3−2δ2 + 3−δ1 + 5/2−δ2 .
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We further note that δ1 < 1 and δ2 < 1/2 imply 4−2δ1 3−δ1 and 3−2δ2 5/2−δ2 . This, together with (3.7), shows that for small enough , (H1 () − (E 0 + 2 E2 )) Q Q 3δ1 ≤ c + 2(δ1 +δ2 ) + δ1 +4δ2 + 6δ2 + 3−δ1 + 5/2−δ2 . We still must show that all terms in the parenthesis above can be made asymptotically smaller than 2 . This can be done if there exist choices of δ1 and δ2 such that all exponents in the parenthesis above are strictly larger than 2. The inequalities to be satisfied are 0 < δ1 < 1, δ1 > 2/3, δ1 + δ2 > 1, 0 < δ2 < 1/2, δ2 > 1/3, δ1 + 4δ2 > 2. Satisfying these is equivalent to satisfying 2/3 < δ1 < 1 1/3 < δ2 < 1/2 which defines the set of allowed values. The best value, ξ = max min { 3δ1 , 2(δ1 + δ2 ), δ1 + 4δ2 , 6δ2 , 3 − δ1 , 5/2 − δ2 } > 0, 0<δ1 <1 0<δ2 <1/2
is obtained by straightforward optimization and is given by ξ = 15/7, obtained for 5/7 < δ1 < 6/7 and δ2 = 5/14. With such a choice, there exists an eigenvalue E() of H1 () that satisfies E() = E 0 + 2 E2 () + O( ξ ), with ξ = 2 + 1/7.
We now turn to the construction of a complete asymptotic expansion for the energy level E() of H1 (), as → 0. Theorem 3.8. Assume the hypotheses of Theorem 3.7 with the additional condition that h(, W, Z ) on Hel is C ∞ in the strong resolvent sense in the variables (, W, Z ). Then the energy level E() of H1 () admits a complete asymptotic expansion in powers of 1/2 . The same conclusion is true for the corresponding quasimode eigenvector. Proof. Our candidate for the quasimode is again the formal expansion (2.9) truncated at order N /2 and multiplied by the cutoff function (3.6), i.e., Q (, W, Z ) = F(, W, Z )
N
√ j/2 ψ j/2 (W, Z , W/, Z / ).
j=0
We shall determine ψ j/2 and T j/2 in (2.8) explicitly, but first we introduce some notation for certain Taylor series. Expanding in powers of 1/2 , we write T j/2 (W, Z ) − T j/2 (w, 1/2 z)
= T j/2 (W, Z ) − T j/2 (0, 0) − 1/2 ∂ Z T j/2 (0, 0)z + ∂W T j/2 (0, 0)w + ∂ Z2 T j/2 (0, 0)z 2 /2 + · · · ≡ T j/2 (W, Z ) − T j/2 (0, 0) +
∞ k=1
(k/2)
τ j/2 (w, z) k/2 .
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Next, our hypotheses imply that the function S(, W, Z ) in (3.3) E 1 (, W, Z )
=
˜ E 0 + E(, W, Z ) + S(, W, Z )
is C ∞ in (, W, Z ). Using (3.4), we write E 1 (, w, 1/2 z)
=
E 0 + 2 E N F (w, z) +
∞
m/2 Sm/2 (w, z).
m≥6
Note. Because we have assumed E 1 (, W, Z ) is even in Z , Sm/2 (w, z) = 0 when m is odd, but the notation is somewhat simpler if we include these terms. We use this notation and substitute the formal series (2.9) and (2.10) into the eigenvalue equation (2.7), with H2 given by (2.8). For orders n/2 with n ≤ 4, we find exactly what we obtained in Sect. 2. When n ≥ 5, we have to solve [h(, W, Z ) − E 1 (, W, Z )] ψn/2 + E N F (w, z)ψ(n−4)/2 + S6/2 (w, z)ψ(n−6)/2 + S7/2 (w, z)ψ(n−7)/2 + · · · + Sn/2 (w, z)ψ0 (1)
(2)
2
2
( n−6 2 )
2 2 + (T n2 (W, Z ) − T n2 (0, 0))ψ0 − τ n−1 (w, z)ψ0 −τ n−2 (w, z)ψ0 − · · ·−τ 6
( n−7 2 )
(1/2)
+ (T n−1 (W, Z ) − T n−1 (0, 0))ψ1/2 − τ(n−2)/2 (w, z)ψ1/2 − · · · − τ 6 2
(w, z)ψ0
2
2
(w, z)ψ1/2
2
.. . (1)
+ (T 7 (W, Z ) − T 7 (0, 0))ψ n−7 − τ 6 2 (w, z)ψ(n−7)/2 2
2
2
2
+ (T 6 (W, Z ) − T 6 (0, 0))ψ(n−6)/2 2
2
1 1 1 2 2 − w,z ψ(n−4)/2 − ∂ Z2 ,z ψ(n−5)/2 −∂W,w ψ(n−6)/2 − ∂ Z2 ,Z ψ(n−6)/2 − ∂W,W ψ(n−8)/2 2 2 2 = E2 ψ(n−4)/2 + E5/2 ψ(n−5)/2 + · · · + En/2 ψ0 , (3.10) with the understanding that the quantities S, T and τ that appear with indices lower than those allowed in their definitions are equal to zero. We solve (3.10) by induction on n. We assume that
E j/2 , ψ ⊥ j/2 (W, Z , w, z), T j/2 (W, Z ) f j/2 (W, Z , w, z)
for j ≤ n − 1, for j ≤ n − 5
and
have already been determined, with f j/2 (W, Z , w, z) = 0, for j ≥ 1. We project (3.10) into the (W, Z ) direction and the orthogonal direction in the electronic Hilbert space to obtain two equations that must each be solved. First, we take the scalar product of (3.10) with (W, Z ) in the electronic Hilbert space to obtain
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E N F (w, z) f (n−4)/2 + S6/2 (w, z) f (n−6)/2 + S7/2 (w, z) f (n−7)/2 + · · · + Sn/2 (w, z) f 0 (1)
(2)
( n−6 2 )
2
2
2 ( n−7 2 ) 6 2
2 2 + (T n2 (W, Z ) − T n2 (0, 0)) f 0 −τ n−1 (w, z) f 0 −τ n−2 (w, z) f 0 −· · · − τ 6
(1/2)
+ (T n−1 (W, Z ) − T n−1 (0, 0)) f 1/2 − τ(n−2)/2 (w, z) f 1/2 − · · · − τ 2
2
(w, z) f 0
(w, z) f 1/2
.. . (1)
+ (T 7 (W, Z ) − T 7 (0, 0)) f n−7 − τ 6 2 (w, z) f (n−7)/2 2
2
2
2
+ (T 6 (W, Z ) − T 6 (0, 0)) f (n−6)/2 2
2
1 2 − w,z f (n−4)/2 − (W, Z ), ∂ Z2 ,z ψ(n−5)/2 Hel − (W, Z ), ∂W,w ψ(n−6)/2 Hel 2 1 1 2 − (W, Z ), ∂ Z2 ,Z ψ(n−6)/2 Hel − (W, Z ), ∂W,W ψ(n−8)/2 Hel 2 2 = E2 f (n−4)/2 + E5/2 f (n−5)/2 + · · · + En/2 f 0 . (3.11) We further project (3.11) into the f 0 direction and the orthogonal direction in L 2 (R2 ) to obtain two equations that must each be solved. We take the scalar product of (3.11) with f 0 in L 2 (R2 ). Using f j/2 = 0 for j ≥ 1 and (− 21 w,z + E N F (w, z)) f 0 = E2 f 0 , we obtain En/2 = T n2 (W, Z ) − T n2 (0, 0) +
n
f 0 , S j/2 f ( j−6)/2 L 2 (R2 ) −
j=6
n−7 n−6−k k=0
(n−( j+k))/2)
f 0 , τ j/2
f k/2 L 2 (R2 )
j=1
2 − f 0 , (W, Z ), ∂ Z2 ,z ψ(n−5)/2 Hel + (W, Z ), ∂W,w ψ(n−6)/2 Hel 1 1 2 + (W, Z ), ∂ Z2 ,Z ψ(n−6)/2 Hel + (W, Z ), ∂W,W ψ(n−8)/2 Hel . 2 2 L 2 (R 2 ) (3.12) We can solve this equation for En/2 if the right-hand side is independent of (W, Z ). This will be true, if we choose T n2 (W, Z ) = −
n
f 0 , S j/2 f ( j−6)/2 L 2 (R2 )
j=6
+
n−7 n−6−k k=0
(n−( j+k))/2)
f 0 , τ j/2
f k/2 L 2 (R2 )
j=1
2 + f 0 , (W, Z ), ∂ Z2 ,z ψ(n−5)/2 Hel + (W, Z ), ∂W,w ψ(n−6)/2 Hel 1 + (W, Z ), ∂ Z2 ,Z ψ(n−6)/2 Hel 2 1 2 + (W, Z ), ∂W,W ψ(n−8)/2 Hel . 2 L 2 (R 2 )
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We then are forced to take En/2 = − T n2 (0, 0). The first non-zero T j/2 (W, Z ) is T6/2 (W, Z ) =
1 (W, Z ), ∂ Z2 (W, Z ) Hel − f 0 , S3 f 0 L 2 (R2 ) . (3.13) 2
So, 1 (0, 0), (∂ Z2 )(0, 0) Hel . 2 We next equate the components on the two sides of (3.11) that are orthogonal to f 0 in L 2 (R2 ). The resulting equation can be solved by applying the reduced resolvent r N F (E2 ), which is the inverse of the restriction of (− 21 w,z + E N F − E2 ) to the subspace orthogonal to f 0 . We thus obtain ⎡ n−5 n ⊥ E(n− j)/2 f j/2 − (S j/2 (w, z) f ( j−6)/2 )⊥ f (n−4)/2 = r N F (E2 ) ⎣ E3 = f 0 , S3 f 0 L 2 (R2 ) −
j=1
+
n−7 n−6−k k=0
+
j=6
(n−( j+k))/2)
(τ j/2
(w, z) f k/2 )⊥
j=1
n−6
⊥ (T(n− j)/2 (0, 0) − T(n− j)/2 (W, Z )) f j/2
j=1 2 ⊥ + (W, Z ), ∂ Z2 ,z (n−5)/2 ⊥ Hel + (W, Z ), ∂W,w (n−6)/2 Hel
⎤
2 ⊥ ⎦ + (W, Z ), ∂ Z2 ,Z (n−6)/2 ⊥ Hel + (W, Z ), ∂W,W (n−8)/2 Hel . ⊥ orthogonal to f 0 , as claimed in Sect. 2. The first This solution has f (n−4)/2 = f (n−4)/2 non-trivial f j/2 , for j ≥ 1 is
f 1⊥ (W, Z , w, z) = − r N F (E2 ) (S3 (w, z) f 0 (w, z))⊥ .
(3.14)
Next, we equate the components of (3.10) that are orthogonal to (W, Z ) in Hel . ⊥ by applying the reduced resolvent r (W, Z ) of We solve the resulting equation for ψn/2 h(, W, Z ) at E 1 (, W, Z ). This yields ⎡ n−5 n ⊥ = r (W, Z ) ⎣ E(n− j)/2 ψ ⊥ S j/2 (w, z) ψ(⊥j−6)/2 ψn/2 j/2 − j=1
+
n−7 n−6−k k=0
+
n−6 j=1
(n−( j+k))/2)
τ j/2
j=6 ⊥ (w, z)ψk/2
j=1
(T(n− j)/2 (0, 0) − T(n− j)/2 (W, Z ))ψ ⊥ j/2
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2 + (∂ Z2 ,z ψ(n−5)/2 )⊥ + ((∂W,w + ∂ Z2 ,Z ) ψ(n−6)/2 )⊥ 2 + (∂W,W ψ(n−8)/2 )⊥
⎤ 1 ⊥ ⎦. − ( w,z + E N F (w, z) − E2 ) ψ(n−4)/2 2
(3.15)
The first non-zero component ψ ⊥ j/2 with j ≥ 0, is ⊥ ψ5/2 (W, Z , w, z)
=
(∂z f 0 )(w, z) r (W, Z ) (∂ Z )(W, Z ).
(3.16)
Finally, Proposition 3.10 below shows that each ψ j/2 in this expansion belongs to √ D(ea(|W |/+|Z |/ ) ). As a result, whenever a derivative acts on the cutoff, it yields a contribution whose L 2 norm is exponentially small. This way, we can neglect such terms. For example √ (∂W F(, W, Z )) ψ j/2 (W, Z , W/, Z / ) √ = −δ1 F (W/ δ1 ) F(Z / δ2 ) ψ j/2 (W, Z , W/, Z / ). The square of the L 2 norm of this term is bounded by a constant times
√ √ 1−δ1 |ψ j/2 (W, Z , W/, Z / )|2 e2a(|W |/+|Z |/ ) e−2a(1/ ) −2δ1 |W |/ δ1 ≥1,|Z |/ δ2 ≤2 ∞
dW d Z = O( ). Thus, we have constructed the non-zero quasimode (3.5) that satisfies the eigenvalue equation up to an arbitrary high power of 1/2 . The proof of Proposition 3.10 relies on the following lemma. Lemma 3.9. Let V be a polynomial that is bounded below, such that the spectrum of H = − 21 + V purely discrete. Let ϕ ∈ C ∞ (Rn ) satisfy D α ϕ ∈ D(eax ), for all α ∈ Nn and any a > 0. If R(λ) denotes the resolvent of H , then D α R(λ)ϕ ∈ D(eax ) for all α ∈ Nn and all λ in ρ(H ). The same is true for D α r (E)ϕ, where r (E) is the reduced resolvent at E. Proof. We first note that elliptic regularity implies that the resolvent R(λ) maps C ∞ functions to C ∞ functions. Next, applied to smooth functions in L 2 , we have the identity [ ∂x j , R(λ) ] = R(λ) (∂x j V ) R(λ). We claim that the operators on the two sides of this equation have bounded extensions to all of L 2 . To see this, note that D β V is relatively bounded with respect to V for any β ∈ Nn , because V is a polynomial. Furthermore, since H ≥ V , we see that D β V is relatively bounded with respect to H , which implies the claim. Hence, for ϕ as in the lemma, we have ∂x j R(λ) ψ = R(λ) ∂x j ϕ + R(λ) (∂x j V ) R(λ) ϕ.
(3.17)
The first term on the right-hand side of this equation belongs to D(eax ) since R(λ) maps exponentially decaying functions to exponentially decaying functions (see Corollary 3.5). The same is true for the second term, with a possible arbitrarily small loss on the exponential decay rate, due to the polynomial growth of ∂x j V . This provides the starting
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point for an induction on the order of the derivative that appears in the conclusion of the lemma. We now assume that for some α ∈ Nn , D α R(z)ϕ is a linear combination of smooth functions of the form R(z)(D γ1 V )R(z) · · · R(z)(D γ j−1 V )R(z)D γ j ϕ, all of which belong to D(eax ), for any a > 0. We assume every γ that occurs here has |γ | ≤ |α|. Then ∂x j D α R(z)ϕ is a linear combination of elements of the form ∂x j (R(z)(D γ1 V )R(z) · · · R(z)(D γ j−1 V )R(z)D γ j ϕ). Applying (3.17) successively, we see that the structure is preserved. Since all D γk V are polynomial, Corollary 3.5 implies the result. The statement for the reduced resolvent follows from the representation (3.2). Proposition 3.10. Assume the hypotheses of Theorem 3.8. Let ψ j/2 (W, Z , w, z) be determined by the construction above, where (W, Z ) belongs to a closed neighborhood of the origin and (w, z) ∈ R2 . Then ψ j/2 is C ∞ , and the function G(w, z) = sup(W,Z )∈ |ψ j/2 (W, Z , w, z)| belongs to D(ea(w,z) ). Proof. The hypothesis on the Hamiltonian and the properties of the normal form H N F proven above imply that (W, Z ) and r (W, Z ) are smooth, and that r N F (E2 ) maps smooth functions to smooth functions. We also know that the non-degenerate eigenstate f 0 is smooth and belongs to D(ea(w,z) ). The smoothness of ψ j/2 (W, Z , w, z) follows trivially in × R2 . Concerning the exponential decay, we observe that the (w, z) dependence of ψ j/2 stems from the successive actions of derivatives, reduced resolvents, and multiplications by polynomials in (w, z), acting on the eigenstate f 0 . Lemma 3.9 applied in conjunction with Proposition 3.6 shows that the exponential decay properties are preserved under such operations. 4. Technicalities In this section, we present the proofs of Lemma 3.4 and Proposition 3.6. Proof of Lemma 3.4. We first note that the hypothesis on p implies p(x) > 0 for any x ∈ Rn , and that e−2C|x−y| ≤ p(x)/ p(y) ≤ e2C|x−y| .
(4.1)
Let B R ∈ Rn be a ball of radius R > 0. We first show that f ∈ L 2 (B R+1 ) and f ∈ L 2 (B R+1 ) imply f ∈ H 2 (B R ), where H 2 (B R ) = { f ∈ L 2 (B R ), ∇ f ∈ L 2 (B R ), and f ∈ L 2 (B R ) }. We denote the usual H 2 (B R ) norm by · H 2 (B R ) . We now show the existence of a constant K (R) > 0, which depends only on R, such that
2 | f |2 + | f |2 . (4.2) |∇ f | ≤ K (R) BR
B R+1
Note. This estimate does not hold in general if the balls over which one integrates have the same radius.
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We set g = f on B R+1 and g(x) = 0 if |x| > R + 1. We can then decompose f = f 1 + f 2 with f 1 and f 2 solutions to f 1 |∂ B R+3 = 0, f 1 = g, f 2 = 0, |x| ≤ R + 1. Thus, f 1 ∈ H 2 (B R+3 ), and there exists a constant c1 (R), which depends only on R, such that f 1 H 2 (B R+3 ) ≤ c1 (R) f 1 L 2 (B R+3 ) = c1 (R) f L 2 (B R+1 ) ,
(4.3)
so that ∇ f 1 L 2 (B R+1 ) ≤ f 1 H 2 (B R+3 ) ≤ c1 (R) f L 2 (B R+1 ) . By the mean value property for harmonic functions, f 2 also satisfies estimate (4.2), for some constant K 2 (R) with f 2 = 0 (see e.g., Chapter 8 of [1]). Combining these arguments, we see that for c2 (R) = c1 (R) + K 2 (R),
|∇( f 1 + f 2 )|2 ≤ c2 (R) (| f 1 |2 + | f 2 |2 ) BR B R+1
≤ c2 (R) (| f |2 + 2(| f |2 + | f 1 |2 )). B R+1
But B R+1 | f 1 |2 ≤ f 1 2H 2 (B ) , so (4.3) implies that (4.2) holds for some constant R+3 K (R). Because of (4.1), we can insert the weight p into this estimate to establish the existence of another constant K˜ (R), which depends only on R, such that
p |∇ f |2 ≤ K˜ (R) p (| f |2 + | f |2 ). (4.4) BR
B R+1
2 if f ∈ F 2 . In other words, p 1/2 ∇ f ∈ L loc w A second step consists in showing that p 1/2 ∇ f is in L 2 (Rn ) and satisfies (3.1). Let χ R ∈ C ∞ (Rn ) be a truncation function such that 0 ≤ χ R ≤ 1, with χ R (x) = 1 if |x| ≤ R, and χ R (x) = 0 if |x| ≥ R + 1. We can take χ R so that ∇χ R ∞ is independent of R. Let f ∈ Fw2 , and set f R = χ R f . Since ∇ f R = χ R ∇ f + f ∇χ R , we see that p 1/2 ∇ f R L 2 (B R ) = p 1/2 ∇ f L 2 (B R ) , and
lim p 1/2 ( f R − f ) L 2 (Rn ) → 0,
R→∞
by Lebesgue dominated convergence. By the same argument with f R = χ R f + f χ R + 2∇χ R ∇ f , lim p 1/2 ( f − (χ R f + f χ R )) L 2 (Rn ) = 0.
R→∞
We have the estimate p 1/2 ∇χ R ∇ f 2L 2 (B ) ≤ c2 B R+1 \B R p 1/2 |∇ f |2 , for some R+1 constant c2 , independent of R. We can cover the set B R+1 \ B R by a finite set of balls {B1 ( j)} j=1,...,N (R) , of radius 1, centered at points x j such that |x j | = R + 1/2. In each
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G. A. Hagedorn, A. Joye
of these balls B1 ( j), we can apply (4.4) (with a constant K˜ 1 , independent of R), to see that
N (R) p |∇ f |2 ≤ c2 K˜ 1 p (| f |2 + | f |2 ), B R+1 \B R
j=1
B2 ( j)
N (R)
where B2 ( j) has radius 2 instead of 1. Using ∪ j=1 B2 ( j) ⊂ B R+3 \ B R−3 , and taking into account that certain points are counted (uniformly) finitely many times in the integral, we eventually obtain
1/2 2 p ∇χ R ∇ f L 2 (Rn ) ≤ c3 p (| f |2 + | f |2 ), B R+1 \B R
where c3 is uniform in R. By the dominated convergence theorem again, this integral goes to zero as R goes to infinity. So, we finally obtain lim p 1/2 ( f − f R )) L 2 (Rn ) = 0.
R→∞
Since f R belongs to Hc2 (Rn ), the set of compactly supported functions in H 2 , we compute ∇ · p f¯R ∇ f R = p f¯R f R + |∇ f R |2 p + f¯R ∇ p ∇ f R . Since f R has compact support, Stokes Theorem and our hypotheses on ∇ p show that
2 ¯ ¯ fR fR p + f R ∇ p ∇ f R p | fR| =
≤
1/2
| f R |2 p
1/2
+2C
1/2
| f R |2 p
| fR| p 2
1/2 |∇ f R | p 2
,
or, in other words, ∇ f R 2p ≤ f R w f R w + 2 C ∇ f R w f R w . This estimate implies (3.1) for f R . The right hand side of that estimate has a finite limit as R → ∞ with f in place of f R on the right-hand side. Since
2 p |∇ f | ≤ p |∇ f R |2 , BR
we deduce that p |∇ f |2 ∈ L 1 (Rn ) and satisfies (3.1).
Proof of Proposition 3.6. We use the following Paley–Wiener theorem, Theorem IX.13 of [14]: Let f ∈ L 2 (Rn ). Then ea|x| f ∈ L 2 (Rn ) for all a < a if and only if f has an analytic continuation to the set { p : |Im p| < a } with the property that for each t ∈ Rn with |t| < a , f (· + it) ∈ L 2 (Rn ), and for any a < a , sup|t|≤a f (· + it)2 < ∞. We refer to the conditions on f in this theorem as “the Paley–Wiener conditions.”
Mathematical Theory for Vibrational Levels of Symmetric Hydrogen Bonds
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Since ea|x| ϕ ∈ L 2 (Rn ) is equivalent to ϕ ∈ D(eax ), Proposition 3.3 shows that ϕ is analytic everywhere and satisfies the Paley–Wiener conditions. The functions p → pj ϕ ( p) and p → j p 2j ϕ ( p) also satisfy these conditions. n As a preliminary remark, we note that for any fixed t 2∈ R , there exist K (t) > n K (t) > 0 and R(t) > 0, such that if p ∈ R satisfies j p j ≥ R(t), then (t) K
n j=1
n n 2 2 pj ≤ ( p j + it j ) ≤ K (t) p 2j . j=1 j=1
(4.5)
So, if B R is a ball of radius R with center at the origin, − ϕ satisfies ⎛
Rn \B R(t)
⎝
n
⎞2 p 2j ⎠ |ϕ( ˆ p + it)|2 dp < ∞,
(4.6)
j=1
uniformly for t in compact sets of Rn . We now start an induction on the length |α| of the multi-index α in D α ϕ. We first show that p → p j pk ϕ ( p) satisfies the Paley–Wiener conditions for any j, k ∈ {1, · · · , n}. Note that we only need to prove estimates for large values of the | p j |’s. Also, note that if nj=1 p 2j ≥ R(t) > 1, there exists a constant C(t) > 0 such that | ( p j + it j ) ( pk + itk ) | ≤ C(t)
n
p 2j .
(4.7)
j=0
Therefore, (4.6) implies that
Rn \B R(t)
| ( p j + it j ) ( pk + itk ) |2 | ϕ ( p + it)|2 dp ⎛
≤ C 2 (t)
Rn \B R(t)
⎝
n
⎞2 p 2j ⎠ | ϕ ( p + it)|2 dp
j=1
< ∞, uniformly for t in compact sets of Rn . Hence, ∂x j ∂xk ϕ ∈ D(eax ) for any a > 0. We next turn to third order derivatives. Consider the derivative of −ϕ + (V − E)ϕ = 0. For any j ∈ { 1, · · · , n}, ∂x j ϕ = (∂x j V )ϕ + (V − E) ∂x j ϕ. Since V is a polynomial, Proposition (3.3) shows that ∂x j ϕ ∈ D(eax ), for any a > 0. Thus, the function p → p j ( nj=0 p 2j ) ϕ ( p) satisfies the Paley–Wiener conditions. Consider now any triple of indices j, k, l. For nj=0 p 2j ≥ R(t), we have | ( p j + it j ) ( pk + itk ) ( pl + itl ) | ≤ C(t) | p j + it j |
n j=0
p 2j .
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Hence, using this estimate with (4.6), we deduce that
| ( p j + it j ) ( pk + itk ) ( pl + itl ) |2 | ϕ ( p + it)|2 dp Rn \B R(t)
⎛
≤ C 2 (t)
Rn \B R(t)
| p j + it j |2 ⎝
n
⎞2 p 2j ⎠ | ϕ ( p + it)|2 dp
j=1
< ∞, uniformly for t in compact sets of Rn . Therefore, the Paley–Wiener Theorem asserts that ∂x j ∂xk ∂xl ϕ ∈ D(eax ), for any a > 0. We now proceed by assuming D β ϕ ∈ D(eax ), for any a > 0 and any β, such that |β| ≤ m. Let α have |α| = m + 1. Let α˜ be any multi-index of length m − 1. Differentiating the eigenvalue equation again, Leibniz’s formula yields ˜ D α˜ ϕ = Cγα˜ D α−γ (V − E) D γ ϕ, (4.8) 0≤γ ≤α˜
where the Cγα˜ are multinomial coefficients. The induction hypothesis and the assumption ϕ ( p) that V is a polynomial show that D α˜ ϕ ∈ L 2 (Rn ). Therefore, p → p α˜ ( nj=1 p 2j ) satisfies the Paley–Wiener conditions. In α, there are two indices, α j and αk , not necessarily distinct, which are larger or equal to one, such that we can write ( p + it)α = ( p1 + it1 )α1 · · · ( p j + it j )α j −1 · · · ( pk + itk )αk −1 · · · ( pn + itn )αn × ( p j + it j ) ( pk + itk ).
(4.9)
Estimating the absolute value of the last two factors by (4.7) and using that α˜ = (α1 , · · · , α j − 1, · · · , αk − 1, · · · , αn ) has length m − 1, we see that p → p α ϕ ( p) satisfies the Paley–Wiener conditions. Hence, D α ϕ ∈ D(eax ) for any a > 0. Acknowledgements. The authors would like to thank Thierry Gallay for several helpful conversations, and Bernard Helffer and the referees for pointing out Proposition 3.2 to them. George Hagedorn would like to thank T. Daniel Crawford for teaching him to use the Gaussian software of computational chemistry.
References 1. Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edition. New York: Springer, 2001 2. Elghobashi, N., González, L.: A Theoretical Anharmonic Study of the Infrared Absorption Spectra of F H F − , F D F − , O H F − , and O D F − Ions. J. Chem. Phys. 124, article 174308 (2006) 3. Hagedorn, G.A.: High order corrections to the time-independent Born-Oppenheimer approximation I: smooth potentials. Ann. Inst. H. Poincaré Sect. A. 47, 1–16 (1987) 4. Hagedorn, G.A., Joye, A.: A mathematical theory for vibrational levels associated with hydrogen bonds II: The Non–Symmetric Case. In preparation 5. Hagedorn, G.A., Toloza, J.H.: Exponentially accurate quasimodes for the time–independent Born–Oppenheimer approximation on a one–dimensional molecular system. Int. J. Quantum Chem. 105, 463–477 (2005) 6. Hislop, P.: Exponential decay of two-body eigenfunctions: A review. In: Mathematical Physics and Quantum Field Theory, Electron. J. Diff. Eqns., Conf. 4, 265–288 (2000) 7. Hislop, P., Sigal, M.: Introduction to Spectral Theory with Applications to Schrodinger Operators. Applied Mathematics Sciences, Volume 113, New York: Springer, 1996
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8. Hörmander, L.: Linear Partial Differential Operators, Berlin-Göttingen-Heidelberg: Springer, 1964 9. Helffer, B., Nourrigat, J.: Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs. Boston: Birkhäuser, 1985 10. Kato, T.: Perturbation Theory for Linear Operators, New York: Springer, 1980 11. Kawaguchi, K., Hirota, E.: Diode Laser Spectroscopy of the ν3 and ν2 Bands of F H F − in 1300 cm−1 Region. J. Chem. Phys. 87, 6838–6841 (1987) 12. Mohamed, A., Nourrigat, J.: Encadrement du N (λ) pour un opérateur de Schrödinger avec un champ magnétique et un potentiel électrique. J. Math. Pures Appl. 70, 87–99 (1990) 13. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I, Functional Analysis, New YorkLondon: Academic Press, 1972 14. Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness, New York-London: Academic Press, 1975 15. Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of Operators, New YorkLondon: Academic Press, 1978 Communicated by H. Spohn
Commun. Math. Phys. 274, 717–735 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0288-1
Communications in
Mathematical Physics
Finite-Time Blow-up of Solutions of an Aggregation Equation in R n Andrea L. Bertozzi1,2 , Thomas Laurent2 1 Department of Mathematics, University of California, Los Angeles 90095, USA.
E-mail: [email protected]
2 Department of Mathematics, Duke University, Durham, NC 27708, USA.
E-mail: [email protected] Received: 2 August 2006 / Accepted: 31 December 2006 Published online: 3 July 2007 – © Springer-Verlag 2007
Abstract: We consider the aggregation equation u t + ∇ · (u∇ K ∗ u) = 0 in R n , n ≥ 2, where K is a rotationally symmetric, nonnegative decaying kernel with a Lipschitz point at the origin, e.g. K (x) = e−|x| . We prove finite-time blow-up of solutions from specific smooth initial data, for which the problem is known to have short time existence of smooth solutions. 1. Introduction The aggregation equation u t + ∇ · (u∇ K ∗ u) = 0
(1)
arises in a number of contexts of recent interest in the physics and biology literature. In this paper we are concerned with a class of potentials that have a Lipschitz point at the origin, e.g. K (x) = e−|x| . In biology, a swarming mechanism, in which individual organisms sense others at a distance, and move towards regions in which they sense the presence of others, involves a complex neurological process at the individual level. The exponentially decaying potential reflects the typical degradation of biological sensors with distance. The individual interactions lead to large scale patterns in nature for which it is desirable to have a tractable mathematical model. The model with this same potential e−|x| also arises in the context of self-assembly of nanoparticles (see e.g. Holm and Putkaradze [15, 16] and references therein). Equation (1) with different classes of potentials and with additional regularizing terms appears in a number of recent and older papers. Topaz and Bertozzi [27] derive the model as multi-dimensional generalization of one-dimensional aggregation behavior discussed in the biology literature [13, 23]. Bodnar and Velazquez [2] consider well-posedness on R for different types of kernels, including the one we consider here. Burger and collaborators [5, 4] consider well-posedness of the model with an additional ‘porous media’ type smoothing term. This class of models, with diffusion, is also derived by Topaz et al.
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[28] who cite some earlier rigorous work from the 1980s in one space dimension. We specifically consider the case where the kernel K has a Lipschitz point at the origin, as in the case of e−|x| which arises in both the biological and nanoscience applications. As a mathematics problem, Eq (1) is an active scalar problem [10] in which a quantity is transported by a vector field obtained by a nonlocal operator applied to the scalar field. Such problems commonly arise in fluid dynamics, for example, the two dimensional vorticity equation ∂ω + v · ∇ω = 0, v = ∇ ⊥ Ne ∗ ω, ∂t where Ne is the Newtonian potential in two space dimensions. These active scalar equations sometimes serve as model problems for the study of finite time blowup of solutions of the 3D Euler equations in which the vorticity vector satisfies ∂ω + v · ∇ ω = ω · ∇v, v = K 3 ∗ ω, ∂t
(2)
where K 3 is the 3D Biot-Savart kernel, homogeneous of degree −2 in 3D. Stretching of vorticity is accomplished by the right-hand side of (2) in which the nonlocal amplification of vorticity occurs. For this problem, finite time blowup of solutions is an open problem. The 2D-quasi-geostrophic equations also pose an interesting family of active scalar equations [9, 17]. An important distinction, between our problem and that of the Euler-related problems, is that our transport field v is a gradient field whereas the v in fluid dynamics is divergence free. Nevertheless it is interesting to note this analogy which also includes the well-known one dimensional model problem studied by Constantin, Lax, and Majda [8] u t = H (u)u, where H denotes the Hilbert transform. There is no transport in this problem and the equation is known to have solutions that blow up in finite time. By differentiation, our problem can be written as ∂u + v · ∇u = (−K ∗ u)u, ∂t
(3)
an advection-reaction equation in which the solution u is amplified by the nonlocal operator (−K ∗ u). As in the Euler examples, there is a conserved quantity, namely the L 1 norm of the solution. In one space dimension, (3) takes on a particularly simple form for which it is easy to show solutions become singular in finite time for pointy potentials such as e−|x| . That is, the K ∗ u operator splits into convolution with a Dirac mass plus convolution with a bounded kernel. Thus, for a smooth solution one can apply a maximum principle argument to obtain an estimate of the form (u m )t ≥ Cu 2m − Du m (where u m is the maximum value of u) which implies finite time blowup of the solution provided a suitable continuation theorem holds. This argument is described in a nonrigorous fashion in [15]. Bodnar and Velazquez consider the one-dimensional problem in [2] and prove finite time blowup by direct comparison with a Burgers-like dynamics. In higher space dimensions one does not obtain such a straightforward blowup result. This is because K , for n ≥ 2, in (3) does not have a Dirac mass, instead its singular part is of the form 1 |x| . As a convolution operator, K is increasingly less singular, as the dimension of space, n, increases. Note for example that the Newtonian potential, which has the form n of |x|c2−n in R n , n ≥ 3, introduces a gain of two derivatives, as a convolution operator.
Finite-Time Blow-up of Solutions of an Aggregation Equation in R n
719
Indeed, if K has bounded second derivative, there is no finite time blowup, so the type of kernel considered here poses an interesting problem for nonlocal active scalar equations. Another related problem of biological relevance is the chemotaxis model ρt + ∇ · (ρ · ∇c) = ρ, −c = ρ,
(4)
where ρ is a mass density of bacteria and c is concentration of a chemoattractant [3]. The model is related to the much-studied Keller-Segel model [18] and it also arises as an overdamped version of the Chandrasekhar equation for the gravitational equilibrium of isothermal stars [6]. In (4) the left-hand side has the same structure as (1) where the kernel K is now the Newtonian potential, which is significantly more singular than the aggregation kernels considered in this paper. Finite time singularities for (4) are known to occur even with the linear diffusion on the right-hand side. The paper [3] shows that the behavior of the blowup depends strongly on the dimension of space. See [3] for a comprehensive literature on this problem. Finally we mention a body of literature addressing both discrete and dynamic models for aggregation [7, 11, 12, 14, 21, 24, 26, 29]. In discrete models, individuals appear as points rather than as a continuum density. The analogue of (1) for the discrete problem is a kinematic gradient flow model for particles interacting via the pairwise potential K . As pointed out in [20], a pointy potential has a discontinuity in the flow field which can result in finite time aggregation of a finite number of particles. For smooth potentials, the aggregation occurs in infinite time since the velocity of individuals approaches zero as they amass. This manuscript considers the same issue for the continuum problem. In our paper we prove, in dimension two and higher, that Eq (1) with a pointy potential, such as e−|x| , has solutions that blowup in finite time from smooth initial data. The proof involves two important steps. The first step, outlined in the next section, is to obtain a continuation theorem for the problem, that smooth solutions can be continued in time provided a low norm (in our case we use L q ) is bounded. This step is, for example, the analogue of the Beale-Kato-Majda result for 3D Euler [1]. The second step is to prove that there exist initial data for which the solution can not be continued past some finite time. Putting these results together, we obtain Cauchy data for which the unique smooth solution blows up in finite time in the L q norm. Since we take our data to have compact support with radial symmetry, the support remains uniformly bounded on the time interval of existence, and thus we also obtain blowup in the L ∞ norm (and L p for all p ≥ q.) There remain a number of interesting and open problems regarding the details of the blowup. We discuss these in the last section of the paper. 2. Continuation of Smooth Solutions with H s Initial Data In [19, 20] one of the authors proves local existence of solutions with data in H s and global existence in the case of a sufficiently smooth kernel. In this section we show that the local existence has a continuation result with control in an L q norm. For simplicity we consider solutions with compact support as we are interested in localized blowup associated with aggregation. To prove the continuation result, we reconstruct the localin-time H s solution by a simple approximating problem involving smoothing the kernel K and smoothing the initial data. For the continuation result, we need only some very general properties for the potential K as described below. Definition 1. The potential K on R n , n ≥ 2 is acceptable if ∇ K ∈ L 2 (R n ) and K ∈ L p (R n ) for some p ∈ [ p ∗ , 2], where p1∗ = 21 + n1 .
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Note that the ‘pointy’ potentials of interest satisfy the above definition. Consider u 0 ∈ H0s (R n ) (i.e. H s ) with compactly supported initial data. Define the approximating problem as follows: u (x, t) is the smooth solution of the problem u t + ∇ · (u∇ K ∗ u) = 0, u (x, 0) = J u 0 .
(5)
Here J denotes convolution with a standard mollifier ρ = 1n ρ(x/), where ρ ∈ C0∞ (R n ) is a nonnegative radially symmetric function, with compact support and mass one. K = J ∗ K is a smoothed kernel. Then from [19, 20] we have the following result: Theorem 1. The approximating problem (5) above has a unique smooth solution for all time. The smoothness of the kernel K is the key to obtaining smooth solutions globally in time. We now show that in passing to the limit one obtains u ∈ C[0, T ; H s ], and thus one can continue the solution as long as the H s norm is controlled. Moreover, we derive an a priori bound for this norm which shows that it is controlled by L q for any q > 2. We also need the following lemma (2.4 from [20]) where we denote by T α the trilinear form α α T α (u, v, w) = (6) D u D ∇ · [v (w ∗ ∇ K )] d x. Rn
Lemma 1. Assume K is an acceptable potential and s ∈ Z + . (i) If α is a multiindex of order s ≥ 1, then α T (u, u, w) ≤ C u 2 s w H s ∀u ∈ H s+1 (R n ) and ∀w ∈ H s (R n ) . H The constant C > 0 depends only on α, K and n. (ii) If α = 0 then 0 (7) ∀u, w ∈ H 1 R n , T (u, u, w) ≤ C u 2L 2 w H 1 0 T (u, v, w) ≤ C u L 2 v H 1 w L 2 ∀u, w ∈ L 2 R n and ∀v ∈ H 1 R n . (8) The constant C > 0 depends only on K and n. The following argument is closely based on the arguments in Chapter 3 of [22] proving continuation of solutions of the 3D Euler equation C[0, T ; H s ]. The reader can review that material for some additional details. First we prove the following local in time existence theorem. The result is then used to prove Theorem 3 on continuity of the solution in time with values in the high norm. Theorem 2. Local in Time Existence of Solutions to the Aggregation Equation. Given K satisfying Definition 1, an initial condition u 0 ∈ H s , and s ≥ 2 is a positive integer, then i) There exists a time T with the rough upper bound T ≤
1 , cs u 0 H s
(9)
Finite-Time Blow-up of Solutions of an Aggregation Equation in R n
721
such that there exists the unique solution u ∈ C([0, T ]; C 2 (R n ))∩C 1 ([0, T ]; C(R n )) to the aggregation equation (1). The solution u is the limit of a subsequence of approximate solutions, u , of (5). ii) The approximate solutions u and the limit u satisfy the higher order energy estimate sup u H s ≤ 0≤t≤T
u 0 H s . 1 − cs T u 0 H s
(10)
iii) The approximate solutions and the limit u are uniformly bounded in the spaces L ∞ ([0, T ], H s (R n )), Li p([0, T ]; H s−1 (R n )), C W ([0, T ]; H s (R n )). Definition 2. The space C W ([0, T ]; H s (R n )) denotes continuity on the interval [0, T ] with values in the weak topology of H s , that is for any fixed φ ∈ H s , (φ, u(t))s is a continuous scalar function on [0, T ], where (u, v)s = D α u · D α vd x. α≤s
Rn
We note that most of the above theorem is already proved in [20]. In particular uniqueness of the H s solution is proved in Sect. 2.2 of [20] so we do not rederive that here. However we rederive the short term existence result with a different approximating problem (5) in order to concisely derive a continuation result in the high norm and with a necessary condition for blowup involving the L q norm. The strategy for the local existence proof, that we implement below, is to first prove the bound (10) in the high norm, then show that we actually have a contraction in the L 2 norm. We then apply an interpolation inequality to prove convergence as → 0. Following the local existence proof, we establish Theorem 3 below, that the solution u is actually continuous in time in the highest norm H s , and can be continued in time provided that its H s norm remains bounded. We need this fact in order to discuss the link between the existence of these solutions globally in time and blowup in a lower norm. To prove Theorem 2 we need a priori higher derivative estimates that are also independent of . Proposition 1. The H s Energy Estimate. Let u 0 ∈ H s with integer s ≥ 2. Then the unique regularized solution u ∈ C 1 ([0, ∞); H s ) satisfies d 1 2 u H s ≤ cs u H s−1 u 2H s . (11) dt 2 Proof. To derive this estimate we need the following slightly improved version of Lemma 1: if α is a multiindex of order s ≥ 2, then |T α (u, u, u)| ≤ Cα ( ∇ K L 2 + K L p ) u 2H s u H s−1
(12)
for all u ∈ H s+1 . Here p can be chosen to be 2 if we are in dimension N ≥ 3. If we are in dimension N = 2, p can be chosen to be 3/2. The constant Cα is a positive constant depending only on α. The proof of (12) is very similar to the one of Lemma 2.4 from [20]) and we refer the reader there for details. To prove (11), if α is a multiindex of order s then (D α u , D α u t ) = −T α (u , u , u ) ≤ Cα (∇ K 2 + K
2 ) u H s u H s−1 L Lp 2 ≤ Cα ( ∇ K L 2 + K L p ) u H s u H s−1 .
(13) (14) (15)
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A. L. Bertozzi, T. Laurent
The estimate of derivatives of lower order are simpler and left as an exercise for the reader. We just need to use Lemma 1 and proceed as above. We can now complete the Proof of Theorem 2. First we show that the family (u ) of regularized solutions is uniformly bounded in H s . The H s energy estimate implies the time derivative of u H s can be bounded by a quadratic function of u H s independent of , sup u H s ≤ 0≤t≤T
u 0 2H s cs T u 0 H s = u 0 H s + . 1 − cs T u 0 H s 1 − cs T u 0 H s
(16)
Thus the family (u ) is uniformly bounded in C([0, T ]; H s ), provided that T < (cs u 0 H s )−1 . Furthermore, the family of time derivatives ( du dt ) is uniformly bounded in H s−1 ,
du s−1 ≤ ∇ · (u ∇ K ∗ u ) H s−1 ≤ u ∇ K ∗ u H s dt H ≤ u H s ∇ K ∗ u W s,∞ ≤ u 2H s ∇ K L 2 .
We now show that the solutions u to the regularized equation (5) form a contraction in the low norm C([0, T ]; L 2 (R n )). Specifically we prove Lemma 2. The family u forms a Cauchy sequence in C([0, T ]; L 2 (R n )). In particular, there exists a constant C depending only on u 0 1 and the time T so that for all and ,
sup u − u L 2 ≤ C max(, ).
0
Proof. We have
d 1 u − u 2L 2 = u − u , ∇ · (u − u )∇ K ∗ u dt 2 + u − u , ∇ · u ∇ K ∗ (u − u ) + u − u , ∇ · u ∇ K ∗ (J − J )u ) = T1 + T2 + T3 .
Using Lemma 1 we have the following estimates: Applying (7) to T1 we have
|T1 | ≤ C u − u 2L 2 J u H 1 . Applying (8) to T2 and T3 we have
|T2 | ≤ C u − u 2L 2 u H 1 , and
|T3 | ≤ C u − u L 2 max(, ) u 2H 1 , where in the last inequality we use the fact that ||J w − w|| L 2 ≤ C|w| H 1 .
Finite-Time Blow-up of Solutions of an Aggregation Equation in R n
723
A proof of this property of mollifiers, using the Fourier transform, can be found at the end of Chapter 3 in [22]. Putting this all together gives d u − u L 2 ≤ C(M)(max(, ) + u − u L 2 ), dt where M is an upper bound, from (16) for the u H 1 on [0, T ]. Integrating this yields
sup u − u L 2 ≤ C(M, T ) max(, ),
(17)
0
where the final inequality is established by recalling that u 0 − u 0 is bounded in L 2 by max(, ) times the H 1 norm of u 0 . Thus u is a Cauchy sequence in C([0, T ]; L 2 (R n )) so that it converges strongly to a value u ∈ C([0, T ]; L 2 (R n )). We have just proved the existence of a u such that sup u − u L 2 ≤ C.
(18)
0≤t≤T
We now use the fact that the u are uniformly bounded in a high norm to show that we have strong convergence in all the intermediate norms. In order to do this, we need the following well-known interpolation lemma for the Sobolev spaces (see [22] and references therein): Lemma 3. Given s > 0, there exists a constant Cs so that for all v ∈ H s (R N ), and 0 < s < s, 1−s /s
v s ≤ Cs v L 2
s /s
v H s .
(19)
We now apply the interpolation lemma to the difference u − u. Using (16) and (18) gives
sup u − u H s ≤ C( u 0 H s , T ) 1−s /s .
0≤t≤T
Hence for all s < s we have strong convergence in C([0, T ]; H s (R n )). A bounded sequence u H s ≤ C in H s (R N ) has a subsequence that converges weakly to some limit in H s , u u. The preceding arguments show that sup u H s ≤ M
and
(20)
0≤t≤T
sup 0≤t≤T
∂u s−1 ≤ M1 . ∂t H
(21)
Hence, u is uniformly bounded in the Hilbert space L 2 ([0, T ]; H s (R n )) so there exists a subsequence that converges weakly to u ∈ L 2 ([0, T ]; H s (R n )).
(22)
Moreover, if we fix t ∈ [0, T ], the sequence u (·, t) is uniformly bounded in H s , so that it also has a subsequence that converges weakly to u(t) ∈ H s . Thus we see that for each t, u s is bounded. This combined with (22) implies that u ∈ L ∞ ([0, T ]; H s ). A similar argument, applied to the estimate (21) shows that u ∈ Li p([0, T ]; H m−1 (R n )).
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A. L. Bertozzi, T. Laurent
Now we conclude that u is continuous in the weak topology of H s . To prove that u ∈ C W ([0, T ]; H s (R n )), let [φ, u], φ ∈ H −s denote the dual pairing of H −s and H s through the L 2 inner product. Estimates (20) and (21), the uniform compact support of u e , and the Lions-Aubin Lemma 10.4 from [22] imply u → u in C([0, T ]; H s ); thus it follows that [φ, u (·, t)] → [φ, u(·, t)] uniformly on [0, T ] for any φ ∈ H −s . Using (20) and the fact that H −s is dense in H −s for s < s, via an /2 argument using (20), we have [φ, u (·, t)] → [φ, u(·, t)] uniformly on [0, T ] for any φ ∈ H −s . This fact implies that u ∈ C W ([0, T ]; H s ). We now show that the limit u is a distribution solution of the original equation. Recall that the sequence of approximate u satisfies u ∈ C([0, ∞); H s ) ∩ C 1 ([0, ∞); H s−1 ) for all s, u t + ∇ · (u ∇ K ∗ u ) = 0, u (0) = J u 0 .
(23) (24) (25)
Since u 0 ∈ H 2 , then we know that there exists a time T1 such that u is a Cauchy sequence in C([0, T1 ]; H 1 ). Therefore there exists a function u such that u → u
in C([0, T1 ]; H 1 ).
(26)
Lemma 4. The function u satisfies u ∈ C([0, T1 ]; H 1 ) ∩ C 1 ([0, T1 ]; L 2 ), u t + ∇ · (u∇ K ∗ u) = 0, u(0) = u 0 ,
(27) (28) (29)
with the dynamic equation in the sense of distributions. Proof. From (26) one can easily obtain ∇ · (u ∇ K ∗ u ) → ∇ · (u∇ K ∗ u),
in C([0, T1 ]; L 2 ).
(30)
In the above we use the fact that u ∇ K ∗ u H 1 ≤ C u H 1 ∇ K ∗ u W 1,∞ ≤ C u 2H 1 ∇ K L 2 . Let v = ∇ · (u∇ K ∗ u). The proof will be completed once we establish that v is the distributional derivative of u, i.e.: T1 T1 u(t, x)φ (t)dt = − v(t, x)φ(t)dt (31) 0
0
C0∞ (0, T1 ).
u
Since ∈ C 1 ([0, ∞); L 2 ) it is clear that for all test functions φ ∈ T1 T1 u (t, x)φ (t)dt = − u t (t, x)φ(t)dt. 0
(32)
0
Also note that (30) can be written u t → v
in C([0, T1 ]; L 2 ).
This convergence, together with (26), allows us to pass to the limit in (32), thus proving (31).
Finite-Time Blow-up of Solutions of an Aggregation Equation in R n
725
Now we show that u is continuous in time with values in the highest H s norm. Theorem 3. Continuity in the High Norm. Let u be the solution described in Theorem 2. Then u ∈ C([0, T ]; H s ) ∩ C 1 ([0, T ]; H s−1 ). Proof. The argument follows the same reasoning as in [22, 25] for the Euler equations. By virtue of the equation it is sufficient to show that u ∈ C([0, T ]; H s ). It is important to obtain this sharper result of continuity in time with values in H s in order to prove the blowup result of the following section. By virtue of the fact that u ∈ C W ([0, T ]; H s (R n )), it suffices to show that the norm u(t) s is a continuous function of time [22, 25]. Passing to the limit in this equation and using the fact that for fixed t, lim sup→0 u H s ≥ u H s we obtain sup u H s − u 0 H s ≤ 0≤t≤T
u 0 2H s cs T . 1 − cs T u 0 H s
(33)
From the fact that u ∈ C W ([0, T ]; H s (R n )), we have lim inf t→0+ |u(·, t)| H s ≥ u 0 H s . The estimate (33) gives lim supt→0+ ||u(·, t)|| H s ≤ u 0 H s . In particular, limt→0+ u(·, t) H s = u 0 H s . This gives us strong right continuity at t = 0. Since the analysis is time reversible, we could likewise show strong left continuity at t = 0. It remains to prove continuity of the · H s norm of the solution at times other than the initial time. Consider a time T0 ∈ [0, T ], and the solution u(·, T0 ). At this fixed time, u(·, T0 ) ≡ u 0T0 ∈ H s (R n ) and from (16), u 0T0 H s ≤ u 0 H s +
u 0 2H s cs T0 . 1 − cs T0 u 0 H s
(34)
So we can take u 0T0 as initial data and construct a forward and backward in time solution as above by solving the regularized equation (5). We obtain approximate solutions u T0 (·, t) that satisfy d u H s ≤ cs u T0 2H s , dt T0
(35)
we can pass to a limit in u T0 as before and find a solution u˜ to the equation on a time interval [T0 − T , T0 + T ] with initial data u T0 . Following the same estimates as above, we obtain that the time T satisfies the constraint 0 < T <
1 − T0 . cs u 0 H s
Furthermore, this solution u˜ must agree with u on [T0 − T , T0 + T ] ∩ [0, T ] by virtue of uniqueness of H s solutions and the fact that u and u˜ agree at t = T0 ∈ [0, T ]. Following the argument above used to show that u H s is continuous at t = 0, we conclude that |u| ˜ H s is continuous at T0 , hence |u| H s itself must be continuous at T0 . Since T0 ∈ [0, T ] was arbitrary, we have just showed that u H s is a continuous function on [0, T ] and hence by the fact that u ∈ C W ([0, T ]; H s (R n )), we obtain u ∈ C([0, T ]; H s (R n )). We obtain the following corollary.
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Corollary 1. Given K satisfying Definition 1 and an initial condition u 0 ∈ H s , s ≥ 2 is a positive integer, then there exists a maximal time of existence T ∗ ∈ (0, ∞] and a unique solution u ∈ C([0, T ∗ ); H s ) ∪ C 1 ([0, T ∗ ); H s−1 ) to (1). Moreover if T ∗ < ∞ then necessarily limt→T ∗ u(·, t) H s = ∞. The above result says that boundedness of the solution in the H s norm is sufficient to guarantee a global solution. We now show that we can weaken this condition to boundedness of the solution in a lower norm. In this sense we say that existence of the solution is ‘controlled’ by a certain norm. To continue solutions with control in a lower norm, we first show that the H s−1 control is inherited by the limit u and follow with an a priori bound in terms of the L q norm of the solution. Recall that u ∈ C 1 ([0, ∞); H s ) for every s and satisfies d u 2 s ≤ C u s−1 u 2 s . H H H dt
(36)
d u 2 s ≤ C u 3 s . H H dt
(37)
Of course this implies
Since u 0 ∈ H s , (37) implies that there exists a time T1 such that u L ∞ ([0,T1 ];H s ) ≤ C,
(38)
where C is independent of . Moreover, we know that on this time interval [0, T1 ] we have the following convergence: u → u u u
in C([0, T1 ]; H s−1 ) strong, in L ∞ (0, T1 ; H s ) weak-star.
(39) (40)
See the discussion after Lemma 3 for the proof of (39). Using the Grönwall inequality we deduce from (36) that |u | L ∞ ([0,T1 ];H s ) ≤ eC
T 0
u (τ ) H s−1 dτ
. u 0 H s .
(41)
Fix now a time T ≤ T1 . From (39) it is clear that T T u (τ ) s−1 dτ → u(τ ) H s−1 dτ, H 0
0
and therefore lim inf u L ∞ (0,T1 ;H s ) ≤ eC →0
T 0
u(τ ) H s−1 dτ
u 0 H s .
(42)
Finally, because of (40) we get u L ∞ (0,T1 ;H s ) ≤ eC
T 0
u(τ ) H s−1 dτ
u 0 H s .
(43)
Since this inequality is true for every T ≤ T1 , and since u ∈ C([0, T1 ]; H s ), we have just proven the following lemma:
Finite-Time Blow-up of Solutions of an Aggregation Equation in R n
727
Lemma 5. For every t ∈ [0, T1 ], u(t) H s ≤ eC
t
0 u(τ ) H s−1 dτ
u 0 H s .
(44)
We have proven existence of a continuous solution with value in H s up to time T1 . By iterating the argument, we can continue the solution on [T1 , T2 ], then [T2 , T3 ], . . . . We have also proven that u(t) H s ≤ eC
t
0 u(τ ) H s−1 dτ
u 0 H s
on [0, T1 ].
(45)
The same proof shows that u(t) H s ≤ e
C
t
T1 u(τ ) H s−1 dτ
u(T1 ) H s
on [T1 , T2 ].
(46)
Combining (45) and (46), we obtain that, for t ∈ [T1 , T2 ], u(t) H s ≤ e
C
≤ eC
t
T1
T1 u(τ ) H s−1 dτ
eC
0 u(τ ) H s−1 dτ
u 0 H s .
t
0
u(τ ) H s−1 dτ
u 0 H s
And therefore one can easily iterate the argument to find that, as long as the solution exists, it must satisfy the estimate u(t) H s ≤ eC
t
0 u(τ ) H s−1 dτ
u 0 H s .
(47)
To prove L q control of blowup we need the following version of the Young inequality. Lemma 6. Suppose 1 ≤ p ≤ 2, and q is the conjugate of p. If φ1 ∈ L 2 (R n ), φ2 ∈ L q (R n ), φ3 ∈ L 2 (R n ) and φ4 ∈ L p (R n ), then
R
φ1 φ2 (φ3 ∗ φ4 ) d x ≤ φ1 L 2 φ2 L q φ3 L 2 φ4 L p . n
Proof. Using Young’s inequality, we get φ3 ∗ φ4 L r ≤ φ3 L 2 φ4 L p for r defined by 1/r = 1/2+1/ p −1. Note that the condition p ≤ 2 ensures that 1/r is nonnegative. One can then check that 1/r + 1/q = 1/2. This allows us to use Hölder’s inequality to show that φ2 (φ3 ∗ φ4 ) L 2 ≤ φ2 L q φ3 ∗ φ4 L r . Then the Schwarz inequality concludes the proof. Proposition 2. Suppose n ≥ 3. Then, for every q ∈ [2, +∞], d u 2H 1 ≤ C D 2 K p u L q u 2H 1 , (48) L dt where p is the conjugate of q and D 2 K L p = supi, j K xi ,x j L p . The constant C depends only on n. If n = 2, then (2) holds for every q ∈ (2, +∞].
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Note that in dimension n = 2, D 2 K L 2 = ∞. This is why, when n = 2, we exclude the case p = q = 2. In dimension n = 3, D 2 K L 2 < ∞ and we do not need to exclude the case p = q = 2. Proof. Integrating by parts twice we easily obtain d u 2L 2 = ∇ K ∗ u · ∇(u 2 ) = − u 2 u ∗ K (49) dt ≤ u 2L 2 u ∗ K L ∞ ≤ u 2L 2 u L q K L p . (50) Differentiating the PDE we obtain u t,xi = −∇u xi · (u ∗ ∇ K ) − ∇u xi (u ∗ K ) −∇u · (u ∗ ∇ K xi ) − u(u xi ∗ K ), and then, after some integration by parts d u x 2 2 = − u 2 u ∗ K − 2 u x ∇u · u ∗ ∇ K x − 2 u x u · u ∗ K x xi i L i i i i dt = −A − 2B − 2C. It is clear that |A|, |B| ≤ D 2 K
Lp
u L q ∇u 2L 2 ,
and, using Lemma 6 we get |C| ≤ u xi L 2 u L q u xi L 2 K L p ≤ D 2 K p u L q ∇u 2L 2 . L
Note that in order to apply Lemma 6 it is necessary that p ≤ 2 and therefore q ≥ 2. From the estimate of |A|, |B| and |C| we easily obtain d u x 2 2 ≤ C D 2 K p u L q ∇u 2L 2 . i L L dt This together with (50) conclude the proof. d 2 Note that in order to estimate dt u H 1 it was necessary for u(t) to be in H 2 (if u(t) was just in H 1 then the integration by parts would not have been justified). The final result of this section is the continuation theorem: Theorem 4. Continuation Theorem. Given initial data u 0 ∈ H s (R n ), n ≥ 2, for positive integer s ≥ 2, there exists a unique solution u(x, t) of (1) and a maximal time interval of existence [0, T ∗ ) such that either T ∗ = ∞ or limt→T ∗ sup0≤τ ≤t u(·, τ ) L q = ∞. The result holds for all q ≥ 2 for n > 2 and q > 2 for n = 2. We use this result in the following section to prove that for some specific initial data, the L q norm blows up in finite time.
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729
3. An Energy Estimate Proving Finite Time Blowup In this section we consider the specific potential K = e−|x| and prove finite time blowup for specific smooth initial data. The argument extends to slightly more general potentials. The main feature is that ∇ K = −N (x) ≡ −x |x| at the origin. Specifically, we assume that ∇ K = −cN (x) + S(x), where S is Lipschitz continuous. They key idea is to use an energy, which was first derived in [28] for biological aggregation with additional ‘porous media’ dissipation. See [15, 16] for a discussion of energies as they related to nanoparticle applications. We consider E(u) = u K ∗ ud x, (51) Rn
for which integration by parts gives the following rate of increase of E(u): dE =2 u|∇ K ∗ u|2 d x. n dt R
(52)
We now show that E has an a priori uniform bound for all u ∈ L 1 (R n ) while at the same time ddtE has a positive lower bound for certain initial data. Thus the smooth solution can not be continued past some critical finite time. We have the following lemma: Lemma 7. For all u ∈ L 1 (R n ), we have E(u) ≤ u 2L 1 . Proof.
E(u) =
Rn
u K ∗ ud x ≤ K ∗ u L ∞ u L 1 ≤ K L ∞ u 2L 1 = u 2L 1 .
Before proceeding with the proof of blowup we need the following result from [20] Sect. 3. Theorem 5. Let u be the solution described in the preceding section. Assume that u 0 ∈ W 1,1 (R n ) and u 0 is nonnegative. Then for each t in the domain of existence, the solution u(x, t) is nonnegative and moreover u(t) L 1 = u 0 L 1 . Using the above facts, the main idea of this section is to show that, for a specific class of functions, the right hand side of (52) has a positive pointwise lower bound. To do this we consider special smooth initial data close to a delta-distribution. Consider a radially symmetric C0∞ function ρ ≥ 0 with compact support in B1 (0), R n ρd x = 1. Define u δ (x) =
1 x ρ δn δ
(53)
to be a rescaling of the smooth function ρ to approximate a delta mass at the origin. We now prove the following kinematic estimate associated with all functions that satisfy the above geometric constraint.
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Proposition 3. There exists a constant C > 0 such that for all δ sufficiently small, we have, for any radially symmetric L 1 function u δ with support inside a ball of radius δ, u δ |∇ K ∗ u δ |2 d x ≥ C. (54) Rn
The main idea is that for data with small support, ∇ K ∗ u δ is approximately −N ∗ u δ , x where N is the kernel, homogeneous of degree zero, |x| . Thus we first prove the bound for this kernel and then show that for data with small support, the support remains small and the correction to the kernel results in a small perturbation of the constant. The key lemma for N is described below. Lemma 8. Let u be a radially symmetric, nonnegative function, with compact support. Then v(|x|) := (N ∗ u(x)) · N (x) is a nonnegative, nondecreasing function of |x|. Proof. Since u is radially symmetric, it suffices to prove the result for u a delta ring concentrated on radius one. This is because v is a linear functional of u which can be decomposed into a linear superposition of delta rings. Any such delta ring can be rescaled to radius one with the nonnegativity and nondcreasing property preserved. For the delta ring (uniform distribution) of unit mass on ∂ B1 (0), we have 1 N (x) · N (x − y)d S, (55) vδ(1) (x) := ω(n) ∂ B(1) where ω(n) is the area of the unit ball in R n . Note that the integral is a sum of unit vectors from a point in space to the boundary of the unit ball, therefore the size of the convolution depends on the degree of cancellation for vectors in different directions. Let us first prove that vδ(1) (x) is nonnegative. We include a figure in 2D illustrating the notation and ideas. Without loss of generality, since u is radially symmetric, we take x on the x1 axis and positive: x = (x1 , . . . , xn ). If x1 ≥ 1 then the quantity N (x) · N (x − y) is clearly nonnegative for every y in the domain of integration and therefore vδ(1) (x) is nonnegative. If 0 ≤ x1 < 1 we divide the domain of integration into regions where the integrand N (x) · N (x − y) is positive and negative. It is negative for y1 > x1 , and positive for −1 ≤ y1 < x1 . Note that a larger mass of the delta-ring is on the positive part and moreover there is less cancellation of vectors for these values because y is farther away from x and thus y − x has a direction more in line with the x vector. More precisely, for y1 > x1 > 0, compare a point y := (y1 , y2 , . . . , yn ) on ∂ B1 (0) with its reflection y˜ := (−y1 , y2 , . . . , yn ) and note that |N (x) · N (x − y˜ )| is always greater than |N (x) · N (x − y)| with equality when x = 0. Thus vδ(1) (x) is still nonnegative when x is inside the ball. To show that vδ(1) is nondecreasing as a function of radius, note that for each y, the integrand in (55) increases monotonically as x1 increases. Thus the integral inherits this monotonicity in x1 . We have the following corollary. Corollary 2. For any decomposition u = u 1 + u 2 , where u, u 1 , and u 2 are nonnegative radially symmetric functions, let v, v1 , and v2 be the respective associated functionals as in (55). Then v, vi are all nonnegative and moreover v = v1 + v2 and thus vi ≤ v.
Finite-Time Blow-up of Solutions of an Aggregation Equation in R n
~
y
731
y
x1
x
B1(0)
reflective plane Fig. 1. Figure for Lemma 8
Lemma 9. Given a nonnegative radial function u with support inside B1 (0) and unit mass, for |x| ≥ 1, the associated v satisfies v(x) ≥ C1 , where 1 N (e1 − y)ds > 0, (56) C1 = e1 · ω(n) ∂ B1 (0) where e1 is the unit vector in the x1 direction. Proof. Since u has support inside the unit ball, following the arguments in Lemma 8, we see that v is nondecreasing for |x| increasing. Thus it suffices to compute an estimate for |x| = 1. Now we consider for what radially symmetric u does one minimize v(1). By the geometry of the problem, v(1) decreases by moving more of the mass of u to the edge of the support. Thus the minimum occurs for u as a delta ring concentrated on the unit ball, yielding formula (56). Proof of Proposition 3. First we prove the proposition for the kernel ∇ K = −N , where u is not required to have small support. Given a radially symmetric u of unit mass, decompose u into two nonnegative, radially symmetric parts, u 1 , u 2 , defined as (57) u 1 = χ B R (0) u, u 2 = [1 − χ B R (0) ]u, (58) where R is defined to be the radius such that B R (0) ud x = 1/2. Define v, v1 and v2 to be the respective functionals as in (55). Note that N (x − y)u 1 (y)dy = N (R xˆ − R yˆ )u 1 (R yˆ )R n d yˆ = Nˆ (xˆ − yˆ )u 1 ( yˆ )d yˆ , B R (0)
B1 (0)
B1 (0)
where yˆ = y/R, xˆ = x/R, uˆ 1 ( yˆ ) = R n u 1 (R yˆ ), and we use the fact that N is homogeneous of degree zero. Since uˆ 1 has total mass 1/2 when integrated in the yˆ variable,
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A. L. Bertozzi, T. Laurent
then v ≥ v1 and moreover v1 (x) ≥ C1 for |x| > R. Now we compute 1 u|N ∗ u|2 d x ≥ u|N ∗ u|2 d x ≥ C12 ud x = C12 . n 2 R |x|>R |x|>R In the above we use the pointwise lower bound for N ∗ u outside of B R (0). To finish the proof of the proposition, we consider the general kernel K and smooth initial data of unit mass and support inside Bδ (0). Note that such data can be written as u δ defined in (53). We write R n u δ |∇ K ∗u δ |2 d x and rescale space as xˆ = x/δ, yˆ = y/δ. Thus 1 y ∇ K (x − y) n ρ( )dy = ∇ K (x − δ yˆ )ρ( yˆ )d yˆ , ∇ K ∗ u δ (x) = δ δ Rn Rn and
Rn
u δ (x)|∇ K ∗ u δ (x)|2 d x =
where
Rn
ρ(x)|∇ ˆ K ∗ u δ (δ x)| ˆ 2 d x, ˆ
∇ K ∗ u δ (δ x) ˆ =
B1 (0)
∇ K (δ xˆ − δ yˆ )ρ( yˆ )d yˆ .
We now use the fact that ∇ K (x) = −N (x) + S(x), where the vector field S(x) is Lipschitz continuous. We use this decomposition to estimate the integral in (54) from below. We have |∇ K ∗ u δ |2 u δ ≥ u δ |N ∗ u δ |2 − 2 Bδ (0) Bδ (0) |N ∗ u δ ||S ∗ u δ |u δ d x ≥ C − 2 |N ∗ u δ ||S ∗ u δ |u δ d x. (59) Bδ (0)
Bδ (0)
In the above we use the pointwise lower bound for |Nδ | as described above. It remains to show the the last term in the above expression is small for δ small, |N ∗ u δ ||S ∗ u δ |u δ d x ≤ N ∗ u δ L ∞ S ∗ u δ L ∞ (Bδ (0)) , Bδ (0)
where we assume u δ has unit mass. Note that N ∗ u δ L ∞ ≤ N L ∞ = 1. Using the above change of variables, we have S(δ xˆ − δ yˆ )ρ( yˆ )d yˆ ≤ δ S Li p S ∗ uδ = B1 (0)
B1 (0)
|xˆ − yˆ |ρ( yˆ )d yˆ ≤ 2δ S Li p .
Thus S ∗ u δ L ∞ (Bδ (0)) ≤ 2δ S Li p . Here we use that sup B1 (0)×B1 (0) |xˆ − yˆ | = 2 and ρ has unit mass. Thus the last term in (59) is bounded by 2δ S Li p , where we use the fact that u δ has mass one. Thus we have finished the proof of Proposition 3.
Finite-Time Blow-up of Solutions of an Aggregation Equation in R n
733
Putting things together, we have the following theorem. Theorem 6. Blowup Theorem. Consider Eq. (1) on R n for n ≥ 2 with C0∞ initial data, radially symmetric, nonnegative, with compact support inside a ball of radius δ as defined by Proposition 3. Then there exists a finite time T ∗ and a unique solution u(x, t) ∈ C([0, T ∗ ); H s ) ∩ C 1 ([0, T ∗ ); H s−1 ) ∀s ≥ 2, such that for all q ≥ 2 (q > 2 for n = 2) sup0≤τ ≤t u(·, τ ) L q → ∞ as t → T ∗ . Proof. Note that since u 0 has compact support and since the vector field ∇ K ∗ u points inward, the support of the solution u, by method of characteristics, can only shrink over time. We take the initial data smooth enough so that u is a classical solution on its time interval of existence, thereby guaranteeing characteristics exist. Thus the solution, on its time interval of existence, has support contained inside the support of the initial data. The symmetry of the kernel and the equations (along with uniqueness of solutions) guarantees that the solution retains radial symmetry as long as it exists. Lastly, Theorem 5 guarantees that the L 1 norm is preserved and the solution remains nonnegative. Thus the solution will satisfy Proposition 3 as long as it remains in H s , which implies a lower bound on the rate of increase of the energy. Since the energy has a finite upper bound for all L 1 functions, the only choice is for the solution to cease to exist as an H s solution after some finite time. The continuation theorem of the previous section tells us that the L q norm must blowup as t approaches the blowup time. Note that for s > n/2 the solution is pointwise bounded up to the blowup time. Thus the initial blowup involves the solution going to infinity in finite time in such a way that the L q norm blows up for all q ≥ 2 (q > 2 for n = 2). 4. Discussion We have proved that (1) on R n , n ≥ 2, with H s initial data (s ≥ 2), with sufficiently small compact support, and nonnegative and radially symmetric, has a unique solution that blows up in finite time. The blowup result involves a singularity in the L q (R n ) norm for q ≥ 2, if n ≥ 3, and q > 2 for n = 2. The blowup proof uses an energy estimate for the solution that makes careful use of the fact that ∇ K has a discontinuity at the origin which approximates the homogeneous kernel (of degree zero) N (x) = x/|x|. Without this discontinuity, in particular when K is bounded, the finite time blowup result is false and one can obtain an a priori bound for the L ∞ norm of the solution. This bound was originally derived in [27] and global existence of smooth solutions with such kernels was established in [20]. This paper settles the question, in dimensions two and higher, of whether finite time blowup is possible for less regular kernels, in particular the commonly used biological kernel K (x) = e−|x| . However a number of open questions remain regarding the nature of the blowup. Bodnar and Velazquez [2] prove the shape of the blowup for the one-dimensional problem, by analogy to asymptotic theory for shock waves. In multi-dimensions, one very important question is whether the blowup results in a concentration of mass at the blowup time. The energy estimate is highly suggestive of this, but does not constitute a proof. In fact, many PDEs exist for which an energy argument proves blowup, yet the actual nature of the blowup is not as suggested by the energy, but rather involves a more subtle form of singularity at a time preceding the blowup time predicted by the a priori energy estimate. The rigorous results of this paper do not preclude the possibility of blowup happening at a point, with a singularity weaker than Dirac delta formation, but strongly enough to give L q blowup as described above. The precise nature of the blowup could be
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important for modelling if one wants to include diffusion effects on small lengthscales, which would desingularize the singularity. To our knowledge, there are no careful computational results in dimension n ≥ 2 that address blowup of these kinds of problems. We also mention that it would be interesting to know about local and global well-posedness of the problem for weaker classes of initial data, such as L ∞ ∩ L 1 , L p ∩ L 1 , and the space of nonnegative measures. Another point of further study is to link the discussed phenomena to the problem of dynamic models for aggregation (see e.g. [7, 11, 12, 14, 21, 26, 29]). Two recent papers [7, 11] show a connection between the scaling properties of dynamic discrete swarms and the notion of H-stability of the interaction potential (from classical statistical physics). Dynamic aggregation, defined by models in which the velocity of motion is not determined by a kinematic rule but rather by a separate momentum equation for the motion, exhibits much richer dynamics than its kinematic cousin. It would be interesting to know how the singular part of the potential affects solutions of dynamic swarms, in particular in regards to the scaling issues discussed in the recent literature. Acknowledgements. This research was supported by ARO grant W911NF-05-1-0112 and ONR grant N000140610059. The authors thank Peter Constantin, Jim Ralston, and the anonymous referee for helpful comments on the manuscript.
References 1. Beale, J.T., Kato, T., Majda, A.J.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984) 2. Bodnar, M., Velazquez, J.J.L.: An integro-differential equation arising as a limit of individual cell-based models. J. Differ. Eqs. 222(2), 341–380 (2006) 3. Brenner, M.P., Constantin, P., Kadanoff, L.P., Schenkel, A., Venkataramani, S.C.: Diffusion, attraction, and collapse. Nonlinearity 12, 1071–1098 (1999) 4. Burger, M., Capasso, V., Morale, D.: On an aggregation model with long and short range interactions. Nonlinear Analysis: Real World Applications 8(3), 939–958 (2007) 5. Burger, M., Di Francesco, M.: Large time behavior of nonlocal aggregation models with nonlinear diffusion. Preprint, 2006 6. Chandrasekhar, S.: An introduction to the study of stellar structure. Dover, New York, 1967 7. Chuang, Y.-L., D’Orsogna, M.R., Marthaler, D., Bertozzi, A.L., Chayes, L.: State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Preprint, 2006 8. Constantin, P., Lax, P.D., Majda, A.: A simple one-dimensional model for the three-dimensional vorticity equation. Commun. Pure. Appl. Math. 38, 715–724 (1985) 9. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 10. Constantin, P.: Geometric statistics in turbulence. SIAM Review 36(1), 73–98 (1994) 11. D’Orsogna, M.R., Chuang, Y.-L., Bertozzi, A.L., Chayes, L.: Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96, 104302 (2006) 12. Ebeling, W., Erdmann, U.: Nonequilibrium statistical mechanics of swarms of driven particles. Complexity 8, 23–30 (2003) 13. Edelstein-Keshet, L., Watmough, J., Grunbaum, D.: Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts. J. Math. Bio. 36, 515–549 (1998) 14. Flierl, G., Grünbaum, D., Levin, S., Olson, D.: From individuals to aggregations: the interplay between behavior and physics. J. Theor. Biol. 196, 397–454 (1999) 15. Holm, D.D., Putkaradze, V.: Aggregation of finite size particles with variable mobility. Phys. Rev. Lett. 95, 226106 (2005) 16. Holm, D.D., Putkaradze, V.: Clumps and patches in self-aggregation of finite size particles. Preprint, 2006 17. 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) 18. Keller, E.F., Segel, L.A.: J. Theor. Biol. 26, 399–415 (1970) 19. Laurent, T.: PhD thesis, Duke University, Department of Mathematics, 2006 20. Laurent, T.: Local and global existence for an aggregation equation. Preprint 2006. to appear in Communications in PDE
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21. Levine, H., Rappel, W.J., Cohen, I.: Self-organization in systems of self-propelled particles. Phys. Rev. E 63, 017101 (2000) 22. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. University Press, Cambridge, 2002 23. Mogilner, A., Edelstein-Keshet, L.: A non-local model for a swarm. J. Math. Bio. 38, 534–570 (1999) 24. Parrish, J., Edelstein-Keshet, L.: Complexity, pattern, and evolutionary trade-offs in animal aggregation. Science 294, 99–101 (1999) 25. Taylor, M.E.: Partial Differential Equations, Volume 3. Springer-Verlag, New York, 1996 26. Toner, J., Tu, Y.: Long-range order in a two-dimensional dynamical xy model: how birds fly together. Phys. Rev. Lett. 75, 4326–4329 (1995) 27. Topaz, C.M., Bertozzi, A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65(1), 152–174 (electronic) (2004) 28. Topaz, C.M., Bertozzi, A.L., Lewis, M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Bio. 68(7), 1601–1623 (2006) 29. Vicsek, T., Czirk, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995) Communicated by P. Constantin
Commun. Math. Phys. 274, 737–750 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0290-7
Communications in
Mathematical Physics
Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory Jürg Fröhlich1 , Enno Lenzmann2 1 Institute for Theoretical Physics, ETH Zürich–Hönggerberg, 8093 Zürich, Switzerland.
E-mail: [email protected]
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
E-mail: [email protected] Received: 9 August 2006 / Accepted: 19 March 2007 Published online: 17 July 2007 – © Springer-Verlag 2007
Abstract: We study finite-time blow-up for pseudo-relativistic Hartree- and HartreeFock equations, which are model equations for the dynamical evolution of white dwarfs. In particular, we prove that radially symmetric initial configurations with negative energy lead to finite-time blow-up of solutions. Furthermore, we derive a mass concentration estimate for radial blow-up solutions. Both results are mathematically rigorous and are in accordance with Chandrasekhar’s physical theory of white dwarfs, stating that stellar configurations beyond a certain limiting mass lead to “gravitational collapse” of these objects. Apart from studying blow-up, we also prove local well-posedness of the initial-value problem for the Hartree- and Hartree-Fock equations underlying our analysis, as well as global-in-time existence of solutions with sufficiently small initial data, corresponding to white dwarfs whose stellar mass is below the Chandrasekhar limit. 1. Introduction and Description of the Problem This paper is a contribution to the mathematical physics of white dwarfs and neutron stars. White dwarfs are dense stars composed of electrons and nuclei, which form a completely ionized plasma. The electrostatic Coulomb forces between these particles establish local electric neutrality to a high degree. For this reason, these forces are screened almost perfectly and make only a very modest contribution to the dynamical evolution and the energy of a white dwarf. Local electric neutrality implies that the spatial and momentum distributions of nuclei are approximately equal to those of the electrons. Since the masses of nuclei are much larger than the mass of an electron, the leading contribution to the total kinetic energy of a white dwarf comes from the electron gas, while the main contribution to its potential energy is due to the gravitational interaction among the nuclei. To simplify matters, we consider a single species of nuclei of electric charge Z e and mass m Z m, where −e is the electric charge of the electron and m its mass. Throughout this paper, we use units such that Planck’s constant = 1 and the velocity of light c = 1.
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1.1. Heuristic discussion. Let N denote the number of electrons in a white dwarf, and let R be its radius. Then the number of nuclei is N /Z , and the average momentum, p, of a nucleus or electron is given by p N 1/3 /R. A rough estimate for the groundstate energy, E(N ), of such a star is thus given by N 1 N 2 Gm 2Z 2 2 E(N ) = min N p + m + m Z − . (1.1) p R=N 1/3 p −1 Z 2 Z R The constant NZ m Z is the rest energy of the nuclei and will be subtracted from the groundstate energy henceforth; the constant G is Newton’s gravitational constant. The function of p within curly brackets on the right side of (1.1) has a minimum > −∞ only if N ≤ Ncr :=
2Z 2 3/2 , Gm 2Z
(1.2)
where Ncr is the “Chandrasekhar number.” Thus, a white dwarf of total mass M larger than the so-called “Chandrasekhar mass” Mcr :=
Ncr mZ, Z
N Z mZ
(1.3)
is energetically unstable and is expected to undergo gravitational collapse; see [2]. For N Ncr , the momentum p minimizing the right side of (1.1) is of the order of m; (more precisely, p m(N /Ncr )3/2 , with p → ∞, as N → Ncr −, see e. g. [14]). The use of relativistic kinematics for the electrons in a calculation of the groundstate energy of a white dwarf is therefore mandatory. Furthermore, for typical white dwarfs, the ratio between the Schwarzschild radius, 2G M, and the radius R of the star, as determined by (1.1), is of the order of 10−4 , so that effects of general relativity are unimportant, and gravity can be described by unretarded Newtonian two-body forces. The outline of a heuristic description of neutron stars is similar, except that effects of general relativity become more important. 1.2. Hartree- and Hartree-Fock equations. We now propose to describe a white dwarf, or a neutron star, quantum-mechanically, but within the approximation described above and assuming that the number N of electrons is conserved. This leads us to consider the Hamilton operator H
(N )
=
N k=1
pk2 + m 2 − κ
1≤k
1 , |xk − xl |
(1.4)
where pk = −i∇xk and κ = Gm 2Z /Z 2 , acting on the Hilbert space ∧N
. H(N ) = L 2 (R3 ) ⊗ C2
(1.5)
Here L 2 (R3 ) is the space of square-integrable one-electron wave functions on physical space R3 , C2 is the space of states of the spin of an electron, and “∧ N ” denotes an N -fold antisymmetric tensor product, in accordance with the fact that electrons (and neutrons) are fermions, i. e., they satisfy the Pauli principle. In the following, electron
Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory
739
spin plays a completely uninteresting rôle. To simplify our notation, we will therefore ignore it. Special state vectors in the Hilbert space H(N ) are given by Slater determinants, := ψ1 ∧ . . . ∧ ψ N ,
(1.6)
where ψ1 , . . . , ψ N are N orthonormal one-particle wave functions; i. e.,
ψk , ψl = δkl , for all k, l = 1, . . . , N ,
(1.7)
with ·, · the scalar product on L 2 (R3 ) (remember that we neglect electron spin). We note that the (dimensionless) coupling constant κ = Gm 2Z /Z 2 is tiny, κ ∼ O(10−38 ), while the number N of electrons in a star is huge: N O(1057 ). One expects that, in this regime, the groundstate of the Hamiltonian H (N ) is well approximated by a Slater determinant, 0 ; see [10, 11]. Moreover, the quantum-mechanical time evolution, as described by the one-parameter unitary group {exp(−it H (N ) )}t∈R , is expected to evolve a Slater determinant, φ1 ∧ . . . ∧ φ N , describing the state of the star at time t = 0, to another Slater determinant, ψ1 (t) ∧ . . . ∧ ψ N (t),
(1.8)
at time t > 0, up to an error term that tends to 0 in the “mean-field limit” κ → 0, with N ∼ O(κ −3/2 ). In (1.8), the one-particle wave functions ψ1 (t), . . . , ψ N (t) are solutions of the N coupled equations i∂t ψk =
−∆ + m 2 ψk −
N N
κ
κ ∗ |ψl |2 ψk + ∗ {ψ l ψk } , ψl |x| |x| l=1
(HF)
l=1
with initial conditions ψk (t = 0) = φk , k = 1, . . . , N , and time 0 ≤ t < T , where N . In (HF), the 0 < T ≤ ∞ is the maximal time of existence of the solution {ψk (t)}k=1 √ pseudo-differential operator −∆ + m 2 , which is defined by its symbol p 2 + m 2 in Fourier space, describes the kinetic energy, including the rest energy, of an electron, and the symbol ∗ denotes convolution of functions on R3 . The last term on the right side of (HF) is the so-called “exchange term,” which is a consequence of the Pauli principle. When compared to the second term on the right side of (HF), the “direct term,” it is subleading in the mean-field limit (N → ∞). It is therefore often neglected. Then Eq. (HF) is replaced by the N coupled equations i∂t ψk =
N
κ ∗ |ψl |2 ψk , −∆ + m 2 ψk − |x|
(H)
l=1
with ψk (t = 0) = φk and k = 1, . . . , N . Equations (HF) are called (dynamical) Hartree-Fock equations, while (H) are called Hartree equations. These systems of evolution equations are the main characters studied in this paper. The logics leading from the quantum-mechanical time evolution generated by the Hamiltonian H (N ) to the nonlinear evolution equations (HF) and (H) for N orthonormal one-particle wave functions, ψ1 (t), . . . , ψ N (t), in the mean-field limit, κ → 0 with N = O(κ −3/2 ), has been studied in [5].
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1.3. Hamiltonian structure and conserved quantities. Let H 1/2 (R3 ) denote the inhomogeneous Sobolev space of index 1/2. We define
×N Γ (N ) = H 1/2 (R3 ) .
(1.9)
This space can be interpreted as an affine “classical phase space” with complex coordinates Ψ = (ψ1 , . . . , ψ N ) and Ψ = (ψ 1 , . . . , ψ N ). The symplectic 2-form, ω, is given by i ω= dψk ∧ dψ k . 2 N
(1.10)
k=1
We define a Hamilton functional, H#(N ) , on Γ (N ) by setting H#(N ) (Ψ , Ψ ) = E# (Ψ ),
(1.11)
with # = HF or H, respectively, where the energy functionals EHF and EH are defined in Sect. 2, below. Then Eqs. (HF) and (H) turn out to be the Hamiltonian equations of motion (N ) (N ) corresponding to the Hamilton functionals HHF and HH , respectively. Formally, the quantities E# (Ψ (t)) and ψk (t), ψl (t), with k, l = 1, . . . , N , are conserved under the Hamiltonian flow determined by Eqs. (HF), (H), respectively. These conservation laws play an important rôle in our analysis.
1.4. Notation. Throughout this text, we make use of inhomogeneous and homogenous Sobolev spaces of order√s, denoted by H s (R3 ) and H˙ s (R3 ), which are equipped with norms u H s = (1 + −∆)s/2 u L 2 and u H˙ s = (−∆)s/2 u L 2 , respectively. The scalar product on L 2 (R3 ) is defined as u, v = R3 u v dx. With some abuse of notation, we sometimes identify collections of wave functions, N , with ordered tuples Ψ = (ψ , . . . , ψ ). For solutions of (HF) and (H), Ψ = {ψk }k=1 1 N this procedure is legitimate, thanks to their U (N )-gauge invariance; see Sect. 2, below. In what follows, we write X Y if X ≤ CY , where C is some universal constant. With regard to physical applications of (H) and (HF), we remind the reader that we use units such that = c = 1. 2. Main Results We begin by reviewing some aspects of Eqs. (H) and (HF). First, we recall from Sect. 1 that both sets of coupled equations exhibit (formally, at least) conservation of energy. That is, the Hartree energy, EH (Ψ ) =
ρΨ (x)ρΨ (y) κ dx dy,
ψk , −∆ + m 2 ψk − 2 R3 R3 |x − y| k=1
N
(2.1)
Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory
741
and the Hartree-Fock energy, N ρΨ (x)ρΨ (y) − |ρΨ (x, y)|2 κ 2 dx dy, EHF (Ψ ) =
ψk , −∆ + m ψk − 2 R3 R3 |x − y| k=1
(2.2) N are conserved for solutions Ψ = {ψk }k=1 of (H) and (HF), respectively. Here and in what follows, we make use of the density matrix,
ρΨ (x, y) =
N
ψk (x)ψ k (y),
(2.3)
k=1
and the particle density, ρΨ (x) = ρΨ (x, x).
(2.4)
In addition to conservation of energy, we also have conservation of the particle number (proportional to the stellar mass) given by N (Ψ ) = ρΨ (x) dx. (2.5) R3
As we will see below, the conservation of N (Ψ ) is a special consequence of the U (N )N Tkl ψl , with T ∈ U (N ), yields gauge symmetry; i. e., every transformation ψk → l=1 another solution of (H) and (HF), respectively. 2.1. Initial-value problem. Our first main result states local well-posedness of the initial-value problems for (H) and (HF), provided that the set of initial data belongs to H s (R3 ), for some s ≥ 1/2. Theorem 2.1 (Local Well-Posedness). Let (#) denote either (H) or (HF). Suppose that s ≥ 1/2 and let N ≥ 1 be an integer. Then the initial-value problem for (#) is locally well-posed in H s (R3 ). N By this we mean the following. For every collection of initial data, Φ = {φk }k=1 ⊂ N H s (R3 ), there exists a unique solution, Ψ (t) = {ψk (t)}k=1 ⊂ H s (R3 ), solving (#) such that
ψk (0) = φk and ψk ∈ C 0 [0, T ); H s (R3 ) ∩ C 1 [0, T ); H s−1 (R3 ) holds, for all k = 1, . . . , N . Here 0 < T ≤ ∞ denotes the maximal time of existence, N and T < ∞ implies that limt→T − k=1 ψk (t) H 1/2 = ∞ holds. In addition, the solution Ψ (t) depends continuously on Φ, and E# (Ψ (t)) = E# (Φ) and N (Ψ (t)) = N (Φ) hold for all times 0 ≤ t < T . Moreover, we have that
ψk (t), ψl (t) = φk , φl , for all 1 ≤ k, l ≤ N and all times 0 ≤ t < T .
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Remark. The proof of this theorem proceeds along the lines of [8] and will only be sketched in Subsect.√3.1, below. We remark that no use of Strichartz-type estimates for 2 the propagator, e−it −∆+m , is made throughout this proof. Note that when considering initial data below the energy norm, e. g., belonging to H s (R3 ), for some s < 1/2, one would have to resort to such estimates; see [3]. Since we are only interested in finite-energy solutions of (H) and (HF), we have no reason to pursue this issue here. The next theorem shows that sufficiently small initial data lead to global-in-time solutions. The smallness condition corresponds to a number of particles below the Chandrasekhar number, Ncr , mentioned in Sect. 1. Theorem 2.2 (Global solutions for N Ncr ). Every solution of either (H) or (HF), given by Theorem 2.1, exists for all times, 0 ≤ t < ∞, whenever the corresponding N initial data, Φ = {φk }k=1 ⊂ H s (R3 ), form a collection of L 2 -orthonormal functions whose number N = N (Φ) satisfies κ 3/2 cr . N< κ Here κcr > 0 is a universal constant of order 1. Remark. If, in addition, the initial data Φ satisfy E# (Φ) < N m (the rest energy of the electrons) then Ψ (t) L p does not tend to 0, as t tends to ∞, for any p > 2. The physical interpretation of this result is that Ψ (t) describes the evolution of a bound configuration of matter forming a star-like object. For details and a proof of a closely analogous result, see [7]. 2.2. Finite-time blow-Up. We now turn our attention to finite-time blow-up for the Hartree equation (H), which, by Theorem 2.2, is only encountered for sufficiently large initial data. N Theorem 2.3 (Radial blow-up for (H) with negative energy). Let Φ = {φk }k=1 ⊂ N ∞ 3 2 Cc (R ) be a collection of functions with the property that ρΦ (x) = k=1 |φk (x)| is radially symmetric. If the Hartree energy is strictly negative, i. e.,
EH (Φ) < 0, N , of (H) with initial data Φ blows up within finite then the solution, Ψ (t) = {ψk (t)}k=1 time. That is, we have that
lim
t→T −
N
ψk (t) H 1/2 = ∞,
k=1
for some 0 < T < ∞. Remarks. 1) It is not difficult to construct an initial configuration Φ with EH (Φ) < 0, as follows: We consider a ball in R3 of radius R > 0 centered at the origin. We then pick an integer N > 0 and let φ1 , . . . , φ N denote eigenfunctions of the Laplacian with Dirichlet boundary conditions at the boundary of the ball corresponding to the lowest N eigenvalues and spanning a rotation-invariant subspace of L 2 (R3 ). N . The relativistic In accordance with the Pauli Principle, we choose Φ := {φk }k=1
Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory
743
kinetic energy of this configuration is proportional to N 4/3 , while the gravitational potential energy is proportional to −N 2 . Thus, if N is sufficiently large then EH (Φ) is strictly negative. 2) We have also found an analogous blow-up result for (HF), but with the additional assumption that each function φk (x) has to be spherically symmetric. From the physical point of view such a hypothesis appears unnaturally strong, so that we refrain from formulating this blow-up result for (HF) as a theorem. 3) The requirement that φk ∈ Cc∞ (R3 ) can be relaxed to weaker conditions on regularity and spatial decay. For the sake of simplicity of our presentation, we will not pursue this issue here. N 4) By invariance of Φ = {φk }k=1 under spatial rotations we mean that, for every N R ∈ S O(3), φk (Rx) = l=1 Tkl (R)φl (x) holds, where T (R) ∈ U (N ) is some unitary matrix. This implies in particular that the density matrix obeys ρΦ (x, y) = ρΦ (Rx, Ry), for any R ∈ S O(3). Moreover, we remark that it is easy to see that the corresponding unique solution, Ψ (t), of (HF) or (H) is also invariant under spatial rotations, for all times, 0 ≤ t < T , provided that Ψ (0) = Φ has this property. Our last result shows that, when approaching the time of blow-up, radial blow-up solutions of (H) or (HF) exhibit a concentration of particles at the origin, whose number is at least of order of the Chandrasekhar number Ncr . Theorem 2.4 (Chandrasekhar mass concentration for radial blow-up). Let Φ = N ⊂ H 1/2 (R3 ) be a collection of L 2 -orthonormal functions, and suppose that Φ {φk }k=1 is invariant under spatial rotations in the sense defined above. Moreover, let Ψ (t) = N be the corresponding solution of either (H) or (HF). If Ψ (t) blows up at time {ψk (t)}k=1 T > 0, then, for every R > 0, we have that κ 3/2 cr lim inf ρΨ (t) (x) dx ≥ , t→T − |x| 0 is the same universal constant as in Theorem 2.2. Remarks. 1) The assumption that Φ consists of L 2 -orthonormal functions (representing the Pauli principle for fermions) is crucial when deriving a lower bound on mass concentration that is proportional to κ −3/2 in accordance to Chandrasekhar’s theory. 2) In view of the physical interpretation of (HF) and (H) discussed in Sect. 1, it would be of considerable interest to gain more insight into the properties of blow-up solutions for these equations and to arrive at a state of affairs comparable to what is known about blow-up for nonlinear Schrödinger equations (NLS) with L 2 -critical, focusing nonlinearities; see, e. g., the monograph [1] and references given there; (see, in particular, [12, 15] for mass concentration of blow-up solutions for NLS). 3. Proof of Main Results The proofs of Theorems 2.1–2.3 are extensions of arguments derived in [8, 6], showing local and global well-posedness, as well as finite-time blow-up for the pseudo-relativistic Hartree equation, i. e., the equation
i∂t ψ = −∆ + m 2 ψ − |x|−1 ∗ |ψ|2 )ψ, (3.1) where ψ : [0, T )×R3 → C. We therefore only sketch the proofs of Theorems 2.1–2.3 in Subsects. 3.1–3.3. In contrast, the proof of Theorem 2.4 is given in detail in Subsect. 3.4.
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3.1. Proof of Theorem 2.1. For definiteness we consider the initial-value problem for (HF) in H s (R3 ), and we observe that all arguments apply to (H), with almost no change. The initial-value problem for (HF) can be written as follows: √ i∂t Ψ = −∆ + m 2 Ψ + κ F(Ψ ), (IVP) Ψ (0) = Φ ∈ H s,N , 0 ≤ t < T. Here
×N H s,N := H s (R3 )
(3.2)
is the N -fold cartesian product of H s (R3 ), equipped with the norm Φ H s,N =
N
φk 2H s
1/2
.
(3.3)
k=1 N with the vector Ψ , With some abuse of notation, we sometimes identify Ψ = {ψk }k=1 and likewise for Φ. In (IVP) the nonlinearity, F = (F1 , . . . , FN ), is given by
(Fk (Ψ )) = −
N N
−1
|x| ∗ |ψl |2 ψk + ψl |x|−1 ∗ (ψ l ψk ) . l=1
(3.4)
l=1
Lemma 3.1. Suppose s ≥ 1/2, and let N ≥ 1 be an integer. Then F : H s,N → H s,N is locally Lipschitz such that
F(Ψ ) − F(Φ) H s,N Ψ 2H s,N + Φ2H s,N Ψ − Φ H s,N , (3.5) F(Ψ ) H s,N Ψ 2H r,N Ψ H s,N ,
(3.6)
for all Ψ , Φ ∈ H s,N , where r = max{s − 1, 1/2}. Remark. The proof of Lemma 3.1 for the first term in (3.4) follows from a straightforward extension of [8, Lemma 3], where an analogous result is shown for the nonlinearity J : H s (R3 ) → H s (R3 ) with J (u) = (|x|−1 ∗ |u|2 )u, corresponding to Hartree nonlinearities and N = 1. Also, the proof of the estimates (3.5), (3.6) for the second term in (3.4), i. e., the “exchange term”, can be shown in a similar fashion. By Lemma 3.1, local-in-time existence and uniqueness of Ψ (t) ∈ H s,N , as well as continuous dependence on Φ, now follow by standard methods for evolution equations with locally Lipschitz nonlinearities; see [8] and references given there. In addition, estimate (3.6) and Gronwall’s inequality allow us to deduce that, for any s > 1/2, sup Ψ (t) H s,N ≤ C(T∗ , Ψ (0) H s,N , sup Ψ (t) H 1/2,N ), for 0 ≤ T∗ < T .
0≤t≤T∗
0≤t≤T∗
In particular, this implies that the maximal time of existence of any H s -valued solution of (HF), with s > 1/2, coincides with its maximal time of existence when viewed as an H 1/2 -valued solution; see also [8]. Finally, we note that U (N )-charge conservation is due to dtd ψk (t), ψl (t) = 0, which follows from a direct calculation. Moreover, conservation of energy stems from the fact that dtd E(Ψ (t)) = 0 holds whenever the initial data satisfy Φ ⊂ H 1,N . To prove conservation of energy for H s (R3 )-valued solutions when 1/2 ≤ s < 1, one can proceed in a standard way, i. e., by using the continuous dependence on initial data in H s,N and by appealing to the density of H 1,N ⊂ H s,N . This completes our sketch of the proof of Theorem 2.1.
Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory
745
3.2. Proof of Theorem 2.2. Let (#) either stand for (H) or (HF). Then, by our hypothesis N on the initial data and Theorem 2.1, we have that the solution, Ψ (t) = {ψk (t)}k=1 ⊂ 1/2 3 2 H (R ), of (#) forms a collection of L -orthonormal functions. By Lemma A.1 (in Appendix A) and the estimate
2/3
ρ(x)ρ(y) dx dy ≤ C ρ(x) dx ρ(x)4/3 dx), (3.7) R3 R3 |x − y| R3 R3 for some constant C 1 (see [11] and references given there), we obtain that E# (Ψ ) ≥
N
≥ 1−
ρΨ (t) (x)ρΨ (t) (y) dx dy R3 R3 |x−y|
2/3 N κ 2 k=1 ψk (t) H˙ 1/2 , R3 ρΨ (t) (x) dx κcr
2 k=1 ψk (t) H˙ 1/2
−
κ 2
(3.8)
for some universal constant κcr of order 1. Using that E# (Ψ (t)) = E# (Φ) and R3 ρΨ (t) = N , we deduce the a-priori bound N
sup
0≤t
κ 2/3 −1 ψk (t)2H 1/2 N + 1 − N E# (Φ), κcr
(3.9)
provided that N < (κcr /κ)3/2 holds. By Theorem 2.1, this implies that the maximal time of existence is T = ∞. 3.3. Proof of Theorem 2.3. The proof of Theorem 2.3 follows [6], with some cosmetic changes only. We remark that smoothness and sufficient spatial decay of the initial data (e. g., that φk ∈ H 2 (R3 ) and | φk , |x|4 φk | < ∞) guarantee that all quantities involved in the following calculations are well defined. First, we notice that N
a(t) :=
ψk (t), Aψk (t), with A := − 2i (x · ∇ + ∇ · x),
(3.10)
k=1
is found to satisfy the differential inequality a(t) ˙ ≤ EH (Ψ (t)) = EH (Φ), for 0 ≤ t < T .
(3.11)
Further, a calculation similar to the one in [6] shows that m(t) :=
N
ψk (t), Mψk (t), with M := x ·
√
−∆ + m 2 x,
(3.12)
k=1
obeys m(t) ˙ ≤ 2a(t) + C1 , for 0 ≤ t < T ,
(3.13)
where C1 > 0 is some constant only depending on N (Φ). In our proof of (3.13), we make use of Newton’s theorem for radially symmetric densities ρΨ (t) (x), which forces us to assume that ρΦ (x) be radially symmetric; see also Remark 4) following Theorem 2.3.
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J. Fröhlich, E. Lenzmann
By combining (3.11) and (3.13) and integrating, we thus obtain m(t) ≤ EH (Φ)t 2 + C1 t + C2 , for 0 ≤ t < T .
(3.14)
Since m(t) is a nonnegative quantity, we conclude that if EH (Ψ (t)) = EH (Φ) < 0 then the maximal time of existence T must be finite. By Theorem 2.1, the solution, Ψ (t), of (H) thus blows up at time T , which is bounded from above by the positive root of the right-hand side of (3.14). 3.4. Proof of Theorem 2.4. We consider (HF); but the proof for (H) is almost identical to the one presented below. The proof makes use of variational arguments; see also [12, 15] for a proof of mass concentration for NLS. We argue by contradiction, as follows. Suppose the conclusion of Theorem 2.4 were not true. That is, there exists some R > 0 such that κ 3/2 cr lim ρΨ (tn ) (x) dx < (3.15) n→∞ |x| 0 is the same universal constant as in Theorem 2.2. To prove that assumption (3.15) cannot hold, we begin by introducing a collection k,n } N , where of N sequences, Ψn = {ψ k=1 k,n (x) := σ (tn )−3/2 ψk (tn , σ (tn )−1 x). ψ
(3.16)
Here the strictly positive function σ (t), defined on [0, T ), is given by σ (t) :=
N
ψk (t)2H˙ 1/2 .
(3.17)
k=1
Note that, since ψk (t) L 2 = ψk (0) L 2 , we have that σ (t) → ∞, as t → T −. Next, we define the functional E(Φ) :=
N k=1
φk 2H˙ 1/2
ρΦ (x)ρΦ (y) − |ρΦ (x, y)|2 κ dx dy, (3.18) − 2 R3 R3 |x − y|
N ⊂ H 1/2 (R3 ). A simple calculation then yields for any collection Φ = {φk }k=1
(tn )). E( Ψn ) = σ (tn )−1 E(Ψ
(3.19)
Moreover, by using conservation of mass and energy, it follows that (tn ))| ≤ |E(Ψ (0))| + mN (Ψ (0)), (3.20) |E(Ψ √ where we use that 0 ≤ φ, −∆ + m 2 φ − φ2H˙ 1/2 ≤ mφ2L 2 . Combining estimate (3.20) with the fact that σ (tn )−1 → 0, as n → ∞, we conclude that Ψn ) = 0. lim E(
n→∞
(3.21)
Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory
747
For later use, we record that this implies that N
2 ρΨ (x)ρΨ (y) − |ρΨ (x, y)| n
lim
n
n
|x − y|
n→∞ R3 R3
dx dy =
2 , κ
(3.22)
k,n . = 1 for all n ∈ N, by construction of ψ N As a next step, we notice that Ψn = {ψk,n }k=1 is a collection of bounded sequences k,n ψ k,∗ , weakly in H 1/2 (R3 ). Hence, by passing to a subsequence, we find that ψ k,n (x) → ψ k,∗ (x) pointwise, for almost every x ∈ R3 , as in H 1/2 (R3 ), as well as ψ k,∗ } N , and the sequence of density matrin → ∞. Correspondingly, we write Ψ∗ = {ψ k=1 3 ces obeys ρΨ n (x, y) → ρΨ ∗ (x, y) pointwise, for almost every x and y in R , as n → ∞. Next, we claim that
since
2 k=1 ψk,n H˙ 1/2
n→∞
p 3 ρΨ n (x) −→ ρΨ ∗ (x), strongly in L (R ), for 1 < p < 3/2,
(3.23)
after possibly passing to a subsequence. To prove (3.23), we define the functions N k,n (x)|2 . |ψ (3.24) θn (x) := ρΨ (x) = n
k=1
One can show (using [9, Theorem 7.13]) that N
θn 2H 1/2
k,n 2 1/2 . ψ H
(3.25)
k=1
Thus, {θn }n∈N forms a bounded sequence in H 1/2 (R3 ) and, by hypothesis of Theorem 2.4 1/2 and Remark 4) following Theorem 2.4, we actually have that {θn }n∈N ⊂ Hrad (R3 ), which denotes the subspace of spherically symmetric functions in H 1/2 (R3 ). Since the embedding 1/2
Hrad (R3 ) → L p (R3 )
(3.26)
is compact if and only if 2 < p < 3 (see [13]), we deduce (after passing to a subsequence) that θn → θ∗ , as n → ∞, strongly in L p (R3 ), for every 2 < p < 3. This proves our claim (3.23). Combining all the convergence properties shown above, we infer that
2 ρΨ (x)ρΨ (y) − |ρΨ (x, y)| n
lim
n→∞ R3 R3
=
n
n
|x − y| 2 ρΨ (x)ρΨ (y) − |ρΨ (x, y)| ∗
∗
R3 R3
|x − y|
∗
dx dy dx dy.
(3.27)
Here we have used (3.23) together with the Hardy-Littlewood-Sobolev inequality, as well as the dominated convergence theorem combined with the pointwise estimate 2 |ρΨ n (x, y)| ≤ ρΨ n (x)ρΨ n (y), which follows from the Cauchy-Schwarz inequality. We note, further, that Ψn ) ≥ E( 0 = lim E( Ψ∗ ), n→∞
(3.28)
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J. Fröhlich, E. Lenzmann
by using (3.27) and the weak lower semicontinuity of the first term in (3.18). In addition, k,∗ , ψ l,∗ ≤ δkl (in the sense of Hermitian N × N -matrices), we obtain that 0 ≤ ψ since ψk,n ψk,∗ weakly in L 2 (R3 ). Invoking Lemma A.1 (see Appendix A) for k,∗ } N we find (similar to the proof of Theorem 2.2) that Ψ∗ = {ψ k=1 2/3 κ
4/3 E( Ψ∗ ) ≥ K 1 − ρΨ (x) dx ρΨ dx, (3.29) ∗ (x) 3 κcr R3 ∗ R where K ≥ 1.63 is the constant from Lemma A.1, and κcr > 0 denotes the same universal constant as in Theorem 2.2. Moreover, we deduce from (3.27) and (3.22) that ρΨ ∗ (x) ≡ 0 must hold. Thus, we see that inequalities (3.29) and (3.28) imply κ 3/2 cr ρΨ (x) dx ≥ . (3.30) ∗ κ R3 By using (3.30), we find that assumption (3.15) leads to a contradiction as follows. 1 3 In view of (3.23), we deduce that ρΨ n (x) → ρΨ ∗ (x) strongly in L loc (R ), as n → ∞. For every A > 0, we thus obtain that ρΨ (x) dx = lim ρ (x) dx ∗ n n→∞ |x|
|x|
where we use that σ (tn )−1 → 0 as n → ∞. Since estimate (3.31) holds for every A > 0, we deduce from (3.30) that assumption (3.15) leads to a contradiction. This completes the proof of Theorem 2.4. A. Lower Bound for Kinetic Energy The following result is a slight extension of an estimate derived in [4]. N Lemma A.1. Suppose Φ = {φk }k=1 ⊂ H 1/2 (R3 ) satisfies 0 ≤ φk , φl ≤ δkl in the sense of Hermitian N × N -matrices. Then
N √
φk , −∆ φk ≥ K k=1
where ρΦ (x) =
N
k=1 |φk (x)|
2
R3
ρΦ (x)4/3 dx,
and K ≥ 1.63 is some constant.
Proof. First we remark that both sides of the inequality to be shown are invariant under N Akl φl , unitary transformations of the φ’s, i. e., under the transformations φk → l=1 for an arbitrary unitary matrix A ∈ U (N ). Therefore we can assume without loss of generality that
φk , φl = λk δkl , with 0 < λk ≤ 1,
(A.1)
where we have also discarded any possible zero vector, φk ≡ 0, corresponding to λk = 0.
Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory
Next, we consider the N × N -matrix, H = (h kl )1≤k,l≤N , with entries √ h kl = φk , −∆ φl − c φk , U φl .
749
(A.2)
Here U (x) := ρΦ (x)1/3 , and c > 0 is some constant to be chosen below. Since H is = B ∗ H B has entries Hermitian, there exists B ∈ U (N ) such that H h kl = k δkl ,
(A.3)
with eigenvalues 1 ≤ 2 ≤ . . . ≤ N . Let {E√j } j≥0 denote the set of negative eigenvalues of the relativistic Schrödinger operator −∆ − cU , acting on L 2 (R3 ), and we k : φ k = l Bkl φl , consider the set of orthogonal vectors given by {φ k < 0}. Noting that 0 < φk , φk ≤ 1 holds, we deduce that N
h kk =
k=1
N k=1
≥
j≥0
k ≥
N
k ≥
k=1 k <0
E j ≥ −Lc4
N
k=1 k <0
R3
k k , φ k
φ
U (x)4 dx.
(A.4)
Here the first inequality in the second line follows from the min-max principle applied √ k L 2 } and the operator −∆ − cU . Moreover, k /φ to the set of orthonormal vectors {φ the last inequality in (A.4) is a standard estimate, where L > 0 is some constant; see [4]. By choosing c = 2−2/3 L −1/3 , we complete the proof of Lemma A.1, where the lower bound K = 3/4 · 2−2/3 L −1/3 ≥ 1.63 follows from known bounds for L. Acknowledgement. We are grateful to the referee for corrections and useful suggestions.
References 1. Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, Vol. 10, New York: New York University Courant Institute of Mathematical Sciences, 2003 2. Chandrasekhar, S.: The maximum mass of ideal white dwarfs. Astrophys. J. 74, 81–82 (1931) 3. Cho, Y., Ozawa, T.: On the semi-relativistic Hartree type equation. SIAM J. Math. Anal. 38, 1060–1074 (2006) 4. Daubechies, I.: An uncertainty principle for fermions with generalized kinetic energy. Comm. Math. Phys. 90(4), 511–520 (1983) 5. Elgart, A., Erd˝os, L., Schlein, B., Yau, H.-T.: Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. (9) 83(10), 1241–1273 (2004) 6. Fröhlich, J., Lenzmann, E.: Blow-up for nonlinear wave equations describing Boson Stars, To appear in Comm. Pure Appl. Math. 2007, DOI: 10.1002/cpa.20186 7. Lenzmann, E.: Nonlinear dispersive equations describing Boson stars. ETH Dissertation No. 16572, (2006) 8. Lenzmann, E.: Well-posedness for semi-relativistic Hartree equations of critical type. To appear in Mathematical Physics, Analysis, and Geometry 2007, DOI: 10.1007/s11040-007-9020-9 9. Lieb, E.H., Loss, M.: Analysis. Second ed., Graduate Studies in Mathematics, Vol. 14, Providence, RI: Amer. Math. Soc. 2001 10. Lieb, E.H., Thirring, W.E.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Physics 155(2), 494–512 (1984) 11. Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112(1), 147–174 (1987) 12. Merle, F., Tsutsumi, Y.: L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. J. Differ. Eqs. 84(2), 205–214 (1990)
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13. Sickel, W., Skrzypczak, L.: Radial subspaces of Besov and Lizorkin-Triebel classes: extended Strauss lemma and compactness of embeddings. J. Fourier Anal. Appl. 6(6), 639–662 (2000) 14. Straumann, N.: General relativity. Texts and Monographs in Physics, Berlin: Springer-Verlag, 2004 15. Weinstein, M.I.: The nonlinear Schrödinger equation—singularity formation, stability and dispersion. The connection between infinite-dimensional and finite-dimensional dynamical systems (Boulder, CO, 1987), Contemp. Math., Vol. 99, Providence, RI: Amer. Math. Soc., 1989, pp. 213–232 Communicated by H.-T. Yau
Commun. Math. Phys. 274, 751–774 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0286-3
Communications in
Mathematical Physics
Cohomogeneity One Einstein-Sasaki 5-Manifolds Diego Conti Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi 53, 20125 Milano, Italy. E-mail: [email protected] Received: 22 August 2006 / Accepted: 13 March 2007 Published online: 13 July 2007 – © Springer-Verlag 2007
Abstract: We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one. Introduction From a Riemannian point of view, an Einstein-Sasaki manifold is a Riemannian manifold (M, g) such that the conical metric on M × R+ is Kähler and Ricci-flat. In particular, this implies that (M, g) is odd-dimensional, contact and Einstein with positive scalar curvature. The Einstein-Sasaki manifolds that are simplest to describe are the regular ones, which arise as circle bundles over Kähler-Einstein manifolds. In five dimensions, there is a classification of regular Einstein-Sasaki manifolds [12], in which precisely two homogeneous examples appear, namely the sphere S 5 and the Stiefel manifold (1) V2,4 = SO(4)/SO(2) ∼ = S2 × S3 . In fact these examples are unique (up to finite cover), as homogeneous contact manifolds are necessarily regular [3]. Among regular Einstein-Sasaki 5-manifolds, these two are the only ones for which the metric is known explicitly: indeed, the sphere is equipped with the standard metric, and the metric on V2,4 has been described in [20]. Notice however that both S 5 and S 2 × S 3 carry other, non-regular Einstein-Sasaki metrics [4, 5]. Only recently other explicit examples of Einstein-Sasaki manifolds have been found, as in [13] the authors constructed an infinite family Y p,q of Einstein-Sasaki metrics on S 2 × S 3 (see also [10] for a generalization). These metrics are non-regular, and thus they are not included in the above-mentioned classification. The isometry group of each Y p,q acts with cohomogeneity one, meaning that generic orbits are hypersurfaces.
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In this paper we give an alternative construction of the Y p,q , based on the language of cohomogeneity one manifolds. In fact we prove that, up to finite cover, they are the only Einstein-Sasaki 5-manifolds on which the group of isometries acts with cohomogeneity one. In particular, this result settles a question raised in [14], concerning the family of links L(2, 2, 2, k), k > 0 defined by the polynomial z 12 + z 22 + z 32 + z 4k . The homogeneous metrics mentioned earlier provide examples of Einstein-Sasaki metrics on L(2, 2, 2, k), k = 1, 2 such that the integral lines of the characteristic vector field are the orbits of the natural action of U(1) with weights (k, k, k, 2). The authors of [14] show that for k > 3 no such metric exists. Since L(2, 2, 2, 3) is diffeomorphic to S 5 , and a metric on L(2, 2, 2, 3) of the required type would necessarily be of cohomogeneity one, our classification extends this result to k ≥ 3. In five dimensions, another characterization of Einstein-Sasaki manifolds is by the existence of a real Killing spinor or, more precisely, a spinor satisfying 1 ∇ X ψ = − X · ψ, 2 where ∇ is the Levi-Civita connection and · denotes Clifford multiplication. Such a spinor exists on every Einstein-Sasaki manifold in any dimension. The converse only fails to be true to the extent that a complete simply-connected Riemannian manifold which admits a real Killing spinor but no Einstein-Sasaki structure is either a sphere S 2n , a 7-manifold with a nearly-parallel G2 structure or a nearly-Kähler 6-manifold [1]. In all of these cases the metric is Einstein with positive scalar curvature. Thus, Einstein-Sasaki manifolds can be viewed as part of a slightly more general class. We shall not make much use of the spinor formalism in this paper, because once one fixes the dimension, the spinor can be replaced with differential forms. However, the characterization in terms of spinors establishes an analogy which is suggestive of the fact that the methods of this paper can be adapted to the nearly-Kähler case, potentially leading to the construction of non-homogeneous nearly-Kähler 6-manifolds. This expectation is supported by the fact that the corresponding exterior differential system is involutive in the nearly-Kähler as in the Einstein-Sasaki case. As far as the author knows, the only known examples of nearly-Kähler 6-manifolds are homogeneous (see [7] for a classification). On the other hand, cohomogeneity one nearly-parallel G2 manifolds are classified, and none of them is complete [8]. Our method to classify cohomogeneity one Einstein-Sasaki 5-manifolds consists in considering a generic orbit, which by hypothesis is a hypersurface. We prove that it is tangent to the characteristic vector field; this fact enables us to define an induced global frame on the hypersurface, and write down certain equations that it must satisfy. Resuming the analogy with higher dimensions, these equations correspond to the nearly hypo and nearly half-flat equations of [11]. In general, a hypersurface in an Einstein-Sasaki 5-manifold (which we take to be tangent to the characteristic vector field) determines the Einstein-Sasaki structure locally. We express this fact in terms of evolution equations like in [16, 9, 11]. These evolution equations are of independent interest, as their solutions correspond to local Einstein-Sasaki metrics; the fact that a solution always exists for real analytic initial conditions is not obvious, but it follows from the exterior differential system being involutive. However, this is not essential to our classification: all we need to do is find all homogeneous solutions of our nearly-hypo-like equations, and solve the evolution equations explicitly in an interval (t− , t+ ), with these solutions as
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initial conditions. Having done that, we prove that for suitable choices of the parameters the resulting metric can somehow be extended to the boundary of (t− , t+ ), leading to cohomogeneity one manifolds with an invariant Einstein-Sasaki metric. Conversely, all such Einstein-Sasaki 5-manifolds are obtained this way. This construction is similar to the one of [8], where cohomogeneity one manifolds with (weak) holonomy G2 were classified. However, such manifolds are non-compact, in sharp contrast with our case. Indeed, the metrics we obtain appear to be the first compact examples obtained by an evolution in the sense of [16]. 1. Invariant Einstein-Sasaki SU(2)-Structures In the usual terminology, an Einstein-Sasaki structure on a 5-manifold is a type of U(2)structure. However, simply connected Einstein-Sasaki manifolds carry a real Killing spinor [12], which reduces the structure group to SU(2). The relevant representation of SU(2) can be described by the diagram SU(2)−
/ SU(2)+ × SU(2)− 2:1
SU(2)
/ SO(4)
∼ =
/ Spin(5) 2:1
/ SO(5)
Giving an SU(2)-structure on a 5-manifold M is the same as giving differential forms (α, ω1 , ω2 , ω3 ), such that locally there exists a basis of orthonormal forms e1 , . . . , e5 satisfying α = e5 ω1 = e12 + e34 . (2) ω2 = e13 + e42 ω3 = e14 + e23 Here and in the sequel, we abbreviate e1 ∧e2 as e12 , and so on. By [9], the Einstein-Sasaki condition can be written dα = −2 ω1 ,
dω2 = 3α ∧ ω3 ,
dω3 = −3α ∧ ω2 .
(3)
By this we mean that, up to passing to the universal cover, every Einstein-Sasaki U(2)structure on a 5-manifold has an SU(2)-reduction satisfying (3). Remark 1. The constant 3 appearing in Eq. (3) is in some sense not essential: one could replace it with an unspecified constant, obtaining a possible definition of an α-EinsteinSasaki SU(2)-structure. Most of our arguments generalize to this more general setting in a straightforward way. However, since this paper is mainly concerned with cohomogeneity one metrics, we shall focus on the Einstein case, as the generalization does not seem to produce any new example. We are interested in Einstein-Sasaki manifolds M of cohomogeneity one, namely those for which the principal orbits of the isometry group are hypersurfaces. Since we require M to be complete, and Einstein-Sasaki manifolds have positive Ricci, by Myers’ theorem M will be compact with finite fundamental group. We shall assume that M is simply connected, which amounts to passing to the universal cover. In this hypothesis, we now prove some facts that play an important rôle in the classification. Recall that the characteristic vector field is by definition the vector field dual to α.
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Lemma 1. Let a compact Lie group G act on a contact manifold M with cohomogeneity one, preserving the contact form. Then the characteristic vector field is tangent to each principal orbit. Proof. Let α be the contact form, and let M have dimension 2n + 1. On every principal orbit Gx we have (dα)n = d(α ∧ (dα)n−1 ) ; by Stokes’ Theorem,
0=
(dα)n . Gx
On the other hand, (dα)n is invariant under G, so it must vanish identically. Now observe that at each point x, the characteristic direction is the space {X ∈ Tx M | X (dα)n = 0} ; the statement follows immediately. Lemma 2. Let M be a compact, simply-connected, Einstein-Sasaki 5-manifold. Suppose that the group of isometries of M acts with cohomogeneity one. Then the action of its identity component I preserves the Einstein-Sasaki U(2)-structure of M, and one can define a homomorphism eiγ : I → U(1), L ∗g (ω2 + iω3 ) = eiγ (g)(ω2 + iω3 ) . If I has dimension greater than four, I = ker eiγ is a 4-dimensional Lie group that acts on M with cohomogeneity one, beside preserving the Einstein-Sasaki SU(2)-structure. Proof. The action of I preserves both metric and orientation on M, and therefore the spin structure. Thus, I acts on the space of spinors (), and the space K ⊂ () of Killing spinors with Killing constant −1/2 is preserved by I. By the assumption on the isometry group, M is not isometric to the sphere. Hence, K is a complex vector space of dimension one (see [12]), and it determines the U(2)-structure of M. The action of I on K ∼ = C determines the homomorphism eiγ . Suppose I has dimension greater than four, and let K be the stabilizer at a point of a principal orbit M; by dimension count, K is not discrete. We claim that k ⊕ i = i , i
I
(4)
where k, i, are the Lie algebras of K , I and respectively. We shall prove by contradiction that k is not contained in i , which implies (4) by a dimension count. The principal orbit M has trivial normal bundle, and the unit normal is invariant. Since the action of I preserves K, it also preserves the contact form, and so by Lemma 1 the characteristic vector field is tangent to M. Consequently, the unit normal is an invariant section of ker α| M . The structure group SU(2) acts freely on unit vectors in R4 = ker e5 , and so I preserves an {Id}-structure on M. Hence k ⊂ i acts trivially on i/k, implying that k is an ideal of i. Then the identity component of K is a normal subgroup of I, and so it acts trivially on M; since I acts effectively, this implies that K is discrete, which is absurd. Thus, (4) holds and I acts with cohomogeneity one. By the same token, the stabilizer I ∩ K has trivial isotropy representation and is therefore normal in I . But then it is also normal in I, which acts effectively, implying that I ∩ K is the trivial group and I acts freely on M. In particular, I has dimension equal to four.
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Remark 2. Part of the argument of Lemma 2 was exploited in [18] to prove that unit Killing fields do not exist on nearly-Kähler 6-manifolds other than S 3 × S 3 . The 6-dimensional case is simpler in this respect, because Killing spinors form a real vector space of dimension one. Thus, a Lie group acting isometrically on a nearly-Kähler 6-manifold automatically preserves the nearly-Kähler SU(3)-structure. It will follow from our classification that the isometry group of M has dimension five, but we cannot prove it directly. The best that we can do at the moment is the following: Proposition 1. If M is a simply-connected compact Einstein-Sasaki 5-manifold of cohomogeneity one, then the Lie group SU(2) × U(1) acts on M with cohomogeneity one preserving the Einstein-Sasaki U(2)-structure, and each principal stabilizer K is a finite subgroup of a maximal torus T2 ⊂ SU(2) × U(1). Proof. Since the isometry group acts with cohomogeneity one, its dimension is greater than or equal to four. If greater, by Lemma 2 there is a four-dimensional subgroup that acts with cohomogeneity one. Thus, there is a four-dimensional compact Lie group G that acts with cohomogeneity one. Up to finite cover, G is either a torus or SU(2)×U(1). Under the projection G → G/K ∼ = M, where M is a principal orbit, the invariant oneform α pulls back to a non-closed left-invariant one-form on G. No such form exists on a torus, and so we can assume that G is SU(2) × U(1). Now, the stabilizer K is discrete, because G acts with cohomogeneity one, and compact, because G is; hence, K is finite. The tangent space of a principal orbit splits as T M = G × K (su(2) ⊕ u(1)) = G × K su(2) ⊕ G × K u(1) into two integrable distributions. Since ker α is not integrable on M, ker α| M = G × K su(2) . Thus α has a non-zero component in G × K (su(2))∗ . On the other hand, the subgroup of G that fixes a non-zero element in su(2)∗ is a maximal torus T2 . Since α is fixed by the isotropy representation, it follows that K ⊂ T2 . 2. Cohomogeneity One In this section we recall some standard facts and notation concerning cohomogeneity one manifolds, referring to [6] for the details. Like in the statement of Proposition 1, in this section M is a compact simply-connected manifold on which a Lie group G acts with cohomogeneity one. By the general theory of cohomogeneity one manifolds, the orbit space M/G is a one-dimensional manifold, which by our topological assumptions is a closed interval. More precisely, we can fix an interval I = [t− , t+ ] and a geodesic c : I → M that intersects principal orbits orthogonally, so that the induced map I → M/G is a homeomorphism. The stabilizer of c(t) is fixed on the interior of I , and we can define H± = Stab c(t± ),
K = Stab c(t), t ∈ (t− , t+ ) .
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Then H± ⊃ K , and moreover there is a sphere-transitive orthogonal representation V± of H± with principal stabilizer K . By the simply-connectedness assumption, V± has dimension greater than one, so that H± /K is diffeomorphic to a sphere of positive dimension. The orbits Gc(t± ) ∼ = G/H± are called special. In a neighbourhood of each special orbit, M is G-equivariantly diffeomorphic to an associated bundle G × H± D± ⊂ G × H± V± ,
(5)
where D± is the closed unit disk in V± . One can reconstruct M by glueing together two disk bundles of the form (5) along the boundaries, namely along the sphere bundles G × H± ∂ D± . By hypothesis these boundaries consist of a single principal orbit; identifying both of them with G/K , the glueing map is defined by a G-equivariant automorphism of G/K . All such automorphisms have the form g K → ga K , a ∈ N (K ) . It is customary to represent any cohomogeneity one manifold obtained this way by the diagram (6) K ⊂ {H− , H+ } ⊂ G . Strictly speaking, this diagram does not determine M (up to G-equivariant diffeomorphism), because the glueing map is also involved. However in many cases, e.g. if N (K ) is connected, any glueing map can be extended to an equivariant diffeomorphism of at least one of the two disk bundles, and so all maps give the same manifold up to equivariant diffeomorphism (see [21] for details). More specifically to our case, by Proposition 1 K is a finite subgroup of T 2 ⊂ G. Since H± /K is diffeomorphic to a sphere, each H± can be either one-dimensional or three-dimensional. However, since M is simply-connected, H+ and H− cannot both be three-dimensional (see e.g. [15]). If H± is three-dimensional, then its identity component is (H± )0 = SU(2) × {1} and H± = K · (H± )0 . Now (H± )0 H± = , S3 ∼ = K (H± )0 ∩ K and thus K ∩ (SU(2) × {1}) is the trivial group. It follows that the tube about the special orbit G/H± satisfies (7) G × H± V± = G ×(H± )0 V± . Concerning the case that H± is one-dimensional, we remark that if one assumes that H± is contained in T 2 , then K is normal in H± . Then K acts trivially in the slice representation, and again (7) holds. These facts will be used in the proof of Lemma 4. 3. Evolution and Hypersurfaces We now introduce a convenient language to determine the cohomogeneity one metrics in terms of the invariant structure induced on a principal orbit. This section is motivated by Sect. 1, but otherwise independent, and should be read in the context of evolution equations in the sense of [16]. In this section, M is an oriented hypersurface in a 5-manifold M with an SU(2)structure; notice that we make no hypotheses of invariance. Let X be the unit normal to
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M, compatible with the orientations. One can measure the amount to which α fails to be tangent to M by the angle β : M → [−π/2, π/2] , sin β = ι∗ (α(X )) . The relevant case for our classification is β ≡ 0 (Lemma 1). Remark 3. If α is a contact form on M, the angle β cannot equal ±π/2 on an open subset of M, as that would mean that the distribution ker α is integrable. Given a hypersurface ι : M → M, one can look for deformations of ι that leave β unchanged. Under suitable integrability assumptions on M, there is a canonical deformation obtained by the exponential map with this property. We shall say that an SU(2)-structure is contact if the underlying almost contact metric structure is contact, i.e. dα = −2ω1 , and it is K-contact if in addition the characteristic vector field is Killing. There is a well-known characterization of K -contact structures [2], which in our language reads ∇ X α = −2X ω1 ∀X ∈ T M , where ∇ is the Levi-Civita connection. Lemma 3. Let M be a K-contact 5-manifold and let ι : M → M be an oriented, compact embedded hypersurface. Consider the one-parameter family of immersions ιt : M → M given by ιt (x) = expι(x) (t X x ) , where X is the unit normal. Then the angle βt : M → [−π/2, π/2] of the hypersurface ιt : M → M does not depend on t. Proof. Using the exponential map, we can extend X to a neighbourhood of M, in such a way that X is normal to all of the ιt (M) for small t. We must prove ι∗t (α(X )) = ι∗ (α(X )) ,
(8)
which holds trivially for t = 0. By the definition of the exponential map, ∇ X X = 0. So, L X (α(X )) = (∇ X α)(X ) = (−2X ω1 )(X ) = 0 , and (8) holds for all t. With notation from Lemma 3, the exponential map produces an inclusion M × (t− , t+ ) (x, t) → ιt (x) ∈ M ; with respect to which the Riemannian metric g on M pulls back to a metric in the “generalized cylinder form” dt 2 + ι∗t g. Using Lemma 3, we can do the same for the SU(2)-structure:
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Proposition 2. Let ι : M → M be a compact, oriented hypersurface with angle β = 0 in a K -contact 5-manifold (M, α, ωi ). Then there is a one-parameter family of {Id}structures (η0 (t), . . . , η3 (t)) on M satisfying η0 (t) = ι∗t α ,
η23 (t) = ι∗t ω1 ,
η31 (t) = ι∗t ω2 ,
η12 (t) = ι∗t ω3 ,
and M is locally given as the product M × (t− , t+ ), with SU(2)-structure determined by α = η0 (t) ,
ω1 = η23 (t) + η1 (t) ∧ dt ,
ω2 = η31 (t) + η2 (t) ∧ dt ,
ω3 = η12 (t) + η3 (t) ∧ dt .
Proof. Let x be a point of ιt (M). Choose a basis e1 , . . . , e5 of Tx∗ M such that Eqs. 2 hold. Using the metric on M, we can write Tx∗ M = Tx∗ ιt (M) ⊕ dt , where dt represents the unit normal 1-form compatible with the orientations. By Lemma 3, e5 lies in Tx∗ ιt (M). We can act on e1 , . . . , e5 by some element of SU(2) to obtain e4 = dt. Then η0 = e5 , η1 = e3 , η2 = −e2 , η3 = e1 , dt = e4 . It is well known that Einstein-Sasaki manifolds are K -contact. In the hypotheses of Proposition 2, assume that (α, ωi ) satisfy (3). Then it is clear that for all t, the following hold: dη0 = −2η23 ,
dη31 = 3η012 ,
dη12 = −3η031 .
(9)
Thus, every oriented hypersurface in a 5-dimensional Einstein-Sasaki manifold has a natural {Id}-structure satisfying (9). By Proposition 2, a one-parameter family of {Id}structures satisfying (9) is induced on the hypersurface. Conversely, we have the following: Proposition 3. Let (ηi (t)) be a 1-parameter family of {Id}-structures on M such that (9) holds for t = t0 . The induced SU(2)-structure on M = M × (t− , t+ ) is Einstein-Sasaki if and only if ∂t η0 = 2η1 ∂t η23 = −dη1 . (10) ∂t η31 = 3η03 − dη2 ∂t η12 = −3η02 − dη3 In this case, (9) holds for all t. Proof. Suppose Eqs. 10 define an Einstein-Sasaki structure on M; then (9) holds for all t. Equations 3 give dt ∧ ∂t η0 + dη0 = dα = −2ω1 = −2η23 − 2η1 ∧ dt ; hence ∂t η0 = 2η1 . Moreover dη31 + dt ∧ ∂t η31 + dη2 ∧ dt = dω2 = 3α ∧ ω3 = 3η012 + 3η03 ∧ dt , and therefore ∂t η31 = 3η03 − dη2 ;
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similarly, for ω3 we get ∂t η12 = −3η02 − dη3 . Since (3) implies that ω1 is closed, we also obtain dη1 = −∂t η23 . So the “only if” part is proved. To prove the “if” part, it suffices to show that (10) forces (9) to hold for all t; (3) will then follow from the calculations in the first part of the proof. Observe first that ∂t (dη0 + 2η23 ) = d∂t η0 + 2∂t η23 = 0 , so the condition dη0 = −2η23 holds for all t. Using this, we compute ∂t (dη12 + 3η031 ) = −3dη02 − 3η0 ∧ dη2 = −3dη0 ∧ η2 = 6η23 ∧ η2 = 0, and a similar argument shows that all of (9) are preserved in time. Remark 4. The evolution equations (10) are not in Cauchy-Kowalewsky form. Indeed, write ηi (t) = a ij (x, t)η j , and dηi = cijk η jk , where cijk are functions of x. Then (10) reduce to a system of 22 equations in 16 unknowns; so the system is overdetermined. However, it turns out that a solution always exists, at least in the real analytic case, as will be proved elsewhere. Remark 5. Given an {Id}-structure satisfying (9), we can define another {Id}-structure satisfying (9) by (η0 , η1 , η2 , η3 ) → (η0 , −η1 , −η2 , −η3 ) ; (11) this has the effect of reversing the orientation, and reflects the ambiguity in the reduction from Z2 to the trivial group. In terms of the evolution equations, this change in orientation corresponds to changing the sign of t. As an example, consider the Lie group SU(2) × U(1). Fix a basis e1 , e2 , e3 , e4 of left-invariant 1-forms such that de1 = −e23 ,
de2 = −e31 ,
de3 = −e12 ,
de4 = 0 .
(12)
One can then define a left-invariant {Id}-structure satisfying (9) by η0 =
1 1 e + e4 , 3
η1 = e4 ,
1 η2 = √ e2 , 6
1 η3 = √ e3 . 6
The solution of the evolution equations is: 1 1 1 1 e + cos t e4 , η1 = − sin t e4 , η2 = e2 , η3 = e3 , (13) 3 2 √ where we have set = 6. The resulting Einstein-Sasaki structure can be extended to a compact 5-manifold, which can be realized as a circle bundle over the Kähler-Einstein manifold S 2 × S 2 . Indeed, this is the homogeneous Kobayashi-Tanno metric (see [20]) on the space (1). To see this, let S 3 be another copy of SU(2), with global invariant forms e˜i satisfying relations analogous to (12). We can introduce coordinates θ, ψ, φ on S 3 such that e˜1 = −dψ + cos θ dφ e˜2 = − sin ψ sin θ dφ + cos ψdθ . (14) e˜3 = cos ψ sin θ dφ + sin ψdθ η0 =
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If we set t = 1 θ and identify e4 with 13 dφ, then (13) restricted to SU(2) × {ψ = 0} becomes 1 1 1 1 1 η0 = (e1 + e˜1 ) , η1 = − e˜3 , η2 = e2 , η3 = e3 , dt = e˜2 . (15) 3 On the other hand, if viewed as forms on SU(2) × S 3 , the forms (15) annihilate a vector field which we may denote by e1 − e˜1 , and their Lie derivative with respect to e1 − e˜1 is zero; so, they pass onto the quotient. Thus the forms (α, ωi ) defined as in Proposition 2 are invariant under the left action of SU(2) × SU(2), and they define a homogeneous Einstein-Sasaki structure on a homogeneous space equivalent to (1). Notice that this is not a symmetric space. 4. Solutions on SU(2) × U(1) In this section we classify solutions of (9) on G = SU(2) × U(1) which are invariant in a certain sense. In order to determine the relevant notion of invariance, let us go back to the hypotheses of Proposition 1. Thus, G acts transitively on a hypersurface M ⊂ M; let (η˜ i ) be the {Id}-structure induced on M by the invariant Einstein-Sasaki structure. Then (η˜ i ) pulls back to an {Id}-structure on G which we also denote by (η˜ i ). By Lemma 2, L ∗g acts on η˜ 2 + i η˜ 3 as multiplication by eiγ . So if we set η˜ 2 + i η˜ 3 = eiγ (η2 + iη3 ) , and η0 = η˜ 0 , η1 = η˜ 1 , then the ηi are left-invariant forms on G. The Lie algebra of G is g = su(2) ⊕ u(1), whose dual will be represented by the basis e1 , . . . , e4 satisfying (12). Since g has only one 3-dimensional subalgebra, the kernel of eiγ contains SU(2) × {1}. We can therefore write dγ = me4 for some integer m. In terms of the invariant basis (ηi ), Eqs. 9 become: dη0 = −2η23 ,
dη31 = 3η012 + me4 ∧ η12 ,
dη12 = −3η031 − me4 ∧ η31 .
(16)
Observe that the space of left-invariant solutions of (9) or (16) is closed under right translation. Left-invariant {Id}-structures on G can be identified with {Id}-structures on the Lie algebra g, and right translation on G correponds to the adjoint action of G on g. Observe also that replacing eiγ with ei(γ +A) does not affect (9). Hence solutions of (16) are closed under the transformation (η0 , η1 , η2 , η3 ) → η0 , η1 , cη2 + sη3 , −sη2 + cη3 , (17) where c and s are real constants with c2 + s 2 = 1. The action of Int g and transformations of type (17), (11) commute; they generate an equivalence relation on the space of solutions. Proposition 4. Every invariant solution of (16) on G can be written up to equivalence as either m η1 = a e1 , η2 = h e2 , η3 = c e2 + k e3 , (18) η0 = 2hk e1 − e4 , 3 where h > 0, k > 0, a > 0 and c is an arbitrary constant, or η0 = 2h 2 e1 + µ e4 ,
η 1 = a1 e 1 + a4 e 4 ,
η2 = h e2 ,
where h > 0, a4 = 0 and 3a1 µ = 6h 2 a4 − a4 − a1 m.
η3 = h e3 ,
(19)
Cohomogeneity One Einstein-Sasaki 5-Manifolds
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Proof. Up to an inner automorphism, we can assume that η0 is in e1 , e4 . Using (16), we deduce that η23 = − 21 dη0 is a multiple of e23 , meaning that η2 , η3 = e2 , e3 . Up to a transformation of type (17), we can assume η2 = h e2 for some positive constant h. Now suppose that η31 is closed, and therefore η1 ∈ e1 , e2 , e3 . Then (16) gives 3η012 + m e4 ∧ η12 = 0 . Hence η0 = − m3 e4 + qe1 and η1 lies in e1 , e2 . The same argument applied to 3η031 + m e4 ∧ η31 shows that η1 is a multiple of e1 , and so (18) holds. Up to changing the signs of both e1 and e3 , we can assume a > 0. Finally, we can fix the overall orientation to obtain k > 0. Suppose now that η31 is not closed, so that η1 = β + a4 e4 , a4 = 0, β ∈ e1 , e2 , e3 . By (16), dη31 ∧ e2 is zero, and therefore dη3 ∧ e42 = 0, i.e. η3 = ke3 for some constant k. Since we are allowed to change the signs of both e1 and e3 , we can assume that k is positive. On the other hand every exact 2-form, as is dη31 , gives zero on wedging with e4 ; so, by (16) β is in e1 , e2 . The same argument applied to the last of (16) shows that β = a1 e1 for some constant a1 . Now write η0 = µ1 e1 + µ4 e4 , where µ1 = 0 because η0 is not closed. Then (16) can be rewritten as hk =
1 µ1 , −a4 k = h(3a1 µ4 − 3µ1 a4 + a1 m), −a4 h = k(3a1 µ4 −3µ1 a4 +a1 m); 2
the solution is k = h , µ1 = 2h 2 , 3a1 µ4 = 6h 2 a4 − a4 − a1 m . 5. The Local Metrics In order to give a local classification, i.e. classify invariant Einstein-Sasaki structures on non-compact cohomogeneity one manifolds SU(2) × U(1) × (t− , t+ ), it is now sufficient to solve the evolution equations (10) using the solutions of Sect. 4 as initial data. Observe that taking into account the eiγ rotation of Lemma 2, we must replace the bottom row of (10) with ∂t η31 = 3η03 − dη2 + dγ ∧ η3 ,
∂t η12 = −3η02 − dγ ∧ η2 − dη3 .
Notice that eiγ does not depend on t. We shall distinguish among three cases. (i) The first case is given by (19) when a1 = 0. Then 6h 2 − 1 = 0, and η0 =
1 1 e + µe4 , 3
η 1 = a4 e 4 ,
1 η2 = √ e2 , 6
1 η3 = √ e3 . 6
762
D. Conti
This family is closed under evolution; explicitly, setting = evolution equations” are solved by η0 =
√ 6 for short, the “rotated
1 1 m 4 k 1 e , η1 = − sin t e4 , η2 = e2 , e + k cos t − 3 3 2
η3 =
1 3 e .
It is not hard to check that, regardless of k and m, the resulting metric is the KobayashiTanno metric on V2,4 described in Sect. 3. (ii) The second case is given by (19) when a1 = 0. We can set a4 = Ca1 , so that writing a for a1 : C +m 4 e , η0 = 2h 2 e1 + 2Ch 2 − 3
η1 = a(e1 + Ce4 ),
η2 = h e2 , η3 = h e3 .
This family is closed under evolution for every choice of C. Indeed, the evolution equations read d 2 d h =a, (ah) = h − 6h 3 , (20) dt dt which we can rewrite as (2h h 2 ) = −6h 3 + h. Fix initial conditions h(t0 ) = h 0 , a(t0 ) = a0 ; we assume that (ηi (t)) is a well-defined {Id}-structure at t = t0 , and therefore h 0 = 0 = a0 . Up to changing the orientation of the 4-manifold—or equivalently, changing the sign of t in (20)—we can assume that a0 h 0 is positive. By continuity and non-degeneracy, a(t)h(t) > 0 in (t− , t+ ). Thus, the first of (20) tells us that dh/dt > 0 in (t− , t+ ). In particular, we can use h as the new variable; writing x = dh dt , (20) gives 4x 2 + 2hx
dx = 1 − 6h 2 . dh
We can solve this differential equation explicitly about h 0 ; the solution is x(h) =
1 A + h 4 − 4h 6 , 2h 2
where the sign of the square root has been chosen consistently with h 0 > 0, a0 > 0, and A is a constant determined by the initial data. More precisely, by the first of (20), A = 4h 0 6 − h 0 4 + (a0 h 0 )2 . 1 Since the function h → 4h 6 − h 4 has − 108 as its minimum and a0 h 0 > 0, it follows that 1 . A>− 108
By the above argument, h(t) is a solution of h (t) =
1 A + h 4 − 4h 6 ; 2h 2
(21)
Cohomogeneity One Einstein-Sasaki 5-Manifolds
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conversely, it is clear that every solution of (21) gives rise to a solution of (20). However we do not need to solve (21) explicitly: it is sufficient to set ⎧ C +m 4 1 0 2 1 4 1 4 ⎪ 4 6 1 ⎪ ⎨ η = 2h (e + Ce ) − 3 e , η = h A + h − 4h (e + C e ) , (22) 2h 2 ⎪ ⎪ ⎩ dh . η2 = h e2 , η3 = h e3 , dt = √ A + h 4 − 4h 6 Whether this metric extends to a compact Einstein-Sasaki 5-manifold depends only on the initial conditions, in a way that we shall determine in Sect. 7. (iii) The structure (18) can be rewritten as η0 = 2(hk − bc)e1 −
m 4 e , η1 = a e1 , 3
η2 = h e2 + b e3 ,
η3 = c e2 + k e3 ,
where we can assume that a and hk − bc are positive. We shall also assume that at least one of h − k, b and c is non-zero, since otherwise we are reduced to case (ii). This family is closed under evolution: indeed, the evolution equations give ⎧ ∂t (ak) = −6k(hk − bc) + h, ∂t (ac) = −6c(hk − bc) − b, ⎨ ∂t (ab) = −6b(hk − bc) − c, ∂t (ah) = −6h(hk − bc) + k, (23) ⎩ ∂t (hk − bc) = a. Any solution of these equations defines a local Einstein-Sasaki metric, which cannot however be extended to a complete metric, as we shall prove without solving (23) explicitly. 6. Extending to Special Orbits In Sect. 5 we have constructed invariant Einstein-Sasaki metrics on G × (t− , t+ ). The problem remains of determining whether one can extend the resulting Einstein-Sasaki structure to a compact cohomogeneity one manifold with diagram (6). In this section we study this problem in slightly greater generality. Given a differential form (or, more generally, a tensor) defined away from the special orbits, we want to determine conditions for it to extend to all of M. By Sect. 2, this problem reduces to extending a form on G × H (V \ {0}) to the zero section. Fix a leftinvariant connection on G as a principal bundle over G/H , i.e. a H -invariant one-form ω : g → h extending the identity on h. Then the tangent bundle of G × H V can be identified with g ⊕V , (G × V ) × H h where the principal H action on G × V is given by (Rh , ρ(h −1 )). Explicitly, the identification is induced by the H -equivariant map
g T (G × V ) (g, v; A, v ) → Ag/h, v − ω(A) · v ∈ ⊕ V , h where we have used the trivialization of T (G × V ) given by left translation.
(24)
764
D. Conti
A differential form on G × H V can be viewed as a H -equivariant map ∗ g ⊕V ; τ : G × (V \ {0}) → h the form is invariant if the map τ is also G-invariant, i.e. τ (gh, v) = τ (h, v). So invariant forms are H -equivariant maps g g τ : V \ {0} → ∗ ⊕ V = ∗ ⊗ ∗ V . h h Now decompose the target space of τ into irreducible H -modules. It is clear that τ extends smoothly across 0 if and only if its components extend. Thus, we only need to determine when an equivariant map τ : V \ {0} → W extends, where W is an irreducible H -module. For the rest of this section we shall assume that H ∼ = U(1) and K = Zσ is the cyclic subgroup of order σ ; this is essentially the only situation we will need to consider in this paper. The representations of H are modeled on C, on which we shall use both polar and Euclidean coordinates, writing the generic element as x + i y = r eiθ . More precisely, the non-trivial irreducible representations of H are two-dimensional real vector spaces Vn ∼ = C, on which eiθ acts as multiplication by einθ . We have Vn ⊗ Vm = Vn−m ⊕ Vn+m . The U(1)-space Vσ ∼ = C has a canonical section [0, +∞) ⊂ C, and any equivariant map from Vσ to Vn is determined by its restriction to this section. Proposition 5 (Kazdan-Warner). A non-zero smooth map τ : [0, +∞) → Vn extends to a smooth equivariant map defined on all of Vσ if and only if σ divides n and the k th derivative of τ at zero vanishes for n n n k = 0, 1, . . . , − 1, + 1, + 3 . . . . σ σ σ Proof. Set m = n/σ . The equivariant extension of τ to R ⊂ Vσ must satisfy τ (−r ) = (−1)m τ (r ) , implying that m is integer. We need this extension to be smooth, which is equivalent to requiring that the odd derivatives at zero vanish if m is even, and the even derivatives vanish if m is odd. The equivariant extension of τ to Vσ is given in polar coordinates by f (r, θ ) = τ (r eiθ ) = eimθ τ (r ) . Set f˜(r cos θ, r sin θ ) = f (r, θ ). If f˜ is smooth at the origin, then k imθ
r e
k k dkτ ∂ f˜ k ∂ k h k−h (0) = r f (0, θ ) = r cos θ sin θ (0, 0) dr k ∂r k ∂ x h ∂ y k−h h=0
is a homogeneous polynomial of degree k in r cos θ , r sin θ . This condition is also sufficient for f˜ to be smooth [17]. On the other hand, it is easy to check that r k eimθ is not a homogeneous polynomial of degree k unless k − |m| is even and non-negative.
Cohomogeneity One Einstein-Sasaki 5-Manifolds
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We now illustrate the method in the case of G = SU(2) × U(1). On the Lie algebra of G, fix a basis e1 , . . . , e4 such that the dual basis satisfies (12). If we identify SU(2) with unit quaternions and its Lie algebra with imaginary quaternions, our choice can be expressed as e1 = i/2, e2 = j/2, e3 = k/2. In particular, we see from this that e1 has period 4π . Let H be the connected subgroup of G with Lie algebra spanned by ξ = pe1 + qe4 , where p/2 and q are coprime integers. Then ξ has period 2π , and we can identify H with U(1). The adjoint action of H on g∗ is trivial on e1 , e4 , and so its action is determined by eiθ · e2 = cos pθ e2 + sin pθ e3 , eiθ · e3 = − sin pθ e2 + cos pθ e3 . So, for instance, the space of G-invariant 2-forms is identified with the space of U(1)invariant maps from Vσ to 2 (V p ⊕ Vσ ⊕ R) ∼ = V0 ⊕ V p+σ ⊕ V p−σ ⊕ V p ⊕ Vσ . More explicitly, fix the connection form p2
1 ( pe1 + qe4 ) , + q2
where we have used ξ to identify h with R. This choice determines a horizontal space h⊥ = qe1 − pe4 , e2 , e3 , and realizes the identification map (24) as (Id, r ; A, v ) → (A, v ) ,
A ∈ h⊥ ,
(Id, r ; pe1 + qe4 , v ) → (0, v − σ r ∂/∂ y). By duality, we see that: – –
the 1-forms e2 , e3 and qe1 − pe4 are mapped to the corresponding elements of (g/h)∗ ; the connection form is mapped to − σ1r dy.
This is all one needs to know in order to translate a differential form (or a tensor) into an equivariant map τ , and apply the criterion of Proposition 5. 7. From Local to Global We can now apply the criteria from Sect. 6 to determine conditions for the local metrics of Sect. 5 to extend smoothly across the special orbits. Consider the one-parameter family of {Id}-structures ⎧ ⎨ η0 = 2∆(e1 + Ce4 ) − C + m e4 , η1 = ∆ (e1 + Ce4 ) , 3 (25) ⎩ 2 2 3 3 2 3 η = he +be , η = ce +ke ,
766
D. Conti
where we have set ∆ = hk − bc, and we assume the constant C + m and the functions ∆(t), ∆ (t) to be non-zero in the interval (t− , t+ ). In particular, solutions of the evolution equations (22) or (23) are of this form. Recall from Proposition 1 that the principal stabilizer K is contained in a torus T 2 ; by the expression of (25), it follows that the Lie algebra of T 2 is spanned by e1 and e4 . We introduce the variables r± = ±(t± − t) , so that the special orbit G/H± corresponds to r± = 0. For brevity’s sake, we shall drop the subscript sign and simply write H , r rather than H± , r± . We shall refer to the point r = 0 as the origin, and say that a function f (r ) on [0, ), > 0 is odd (resp. even) at the origin if it extends to a smooth odd (resp. even) function on (− , ). We can now determine conditions on the functions h, k, b, c, and ∆ in order that the metric extend smoothly across each special orbit. Lemma 4. The one-parameter family of {Id}-structures (25) defines an invariant U(2)structure on the cohomogeneity one manifold G ×(t− , t+ ). If this structure extends across the special orbit G/H , then – H0 = SU(2) × {1},
1 (hb r4
+ ck) is even, and
∆/r 2 ,
1 2 1 (h + c2 ), 2 (k 2 + b2 ) 2 r r
are even functions, all taking the value 41 at the origin; or – H0 ∼ = U(1), with Lie algebra h = pe1 +qe4 , where p/2 and q are coprime integers satisfying p + qC > 0; ∆ , h 2 + c2 + b2 + k 2 are even functions satisfying ∆ (0) =
σ q(C + m) , ∆(0) = = 0 , p + qC 6( p + qC)
where σ is the order of K ∩ H0 , and hk + bc, h 2 + c2 − b2 − k 2 are smooth functions that, if p = 0, vanish at the origin. The first condition is also sufficient, whereas the second one becomes sufficient if hk + bc and h 2 + c2 − b2 − k 2 are identically zero. Proof. We think of a U(2)-structure as defined by a 1-form α, a 2-form ω1 and a Riemannian metric g; the U(2)-structure extends across a special orbit if and only if these tensors extend smoothly, while remaining non-degenerate. Assume first that H has dimension three. By the final remarks of Sect. 2, the identity component of H is the subgroup SU(2) × {1}, and the normal bundle at G/H is given by (7). Thus, V is the 4-dimensional irreducible representation of SU(2) and G ×H V ∼ = V × U(1) .
Cohomogeneity One Einstein-Sasaki 5-Manifolds
767
In particular we can identify G/K × (t− , t+ ) with an open set in V × U(1). Explicitly, we define V to be the field of quaternions, identifying SU(2) with the unit sphere in V , acting on the left. We write the generic element of V as x 0 + x 1 i + x 2 j + x 3 k, and identify r with the radial coordinate r = (x 0 )2 + (x 1 )2 + (x 2 )2 + (x 3 )2 . Then
2 0 1 x d x − x 1d x 0 − x 2d x 3 + x 3d x 2 , 2 r 2 e2 = 2 x 0 d x 2 + x 1 d x 3 − x 2 d x 0 − x 3 d x 1 , r 2 e3 = 2 x 0 d x 3 − x 1 d x 2 + x 2 d x 1 − x 3 d x 0 , r 1 0 0 ∓dt = x d x + x 1d x 1 + x 2d x 2 + x 3d x 3 , r e1 =
where the sign of ∓dt depends on the special orbit G/H± we consider, and the coefficient 2 is a consequence of our choice (12). Rewriting in these terms the 2-form ω1 = ∆ (e1 + Ce4 ) ∧ dt + ∆e23 , we see that ω1 is smooth and non-degenerate at the origin only if ∆ ∼ ∓ 2∆ r , so that 2∆ . ω1 ∼ 2 2d x 2 ∧ d x 3 −2d x 0 ∧ d x 1 + Ce4 ∧ x 0 d x 0 + x 1 d x 1 + x 2 d x 2 + x 3 d x 3 r Hence, ∆/r 2 is even and non-zero at the origin. In particular α is smooth, and it does not vanish at G/H since C + m = 0. Moreover ∆(0) = 0 = ∆ (0). The rest of the statement now follows from the fact that the metric tensor dt ⊗ dt + i ηi ⊗ ηi has the form C +m 2 2 2 1 1 2 2C∆ − 4∆ + (∆ ) e ⊗ e + + (C∆ ) e4 ⊗ e4 3 C +m e1 e4 + (h 2 + c2 )e2 ⊗ e2 + C(∆ )2 + 2∆ 2C∆ − 3 + (k 2 + b2 )e3 ⊗ e3 + (hb + ck)e2 e3 + dt ⊗ dt and thus smoothness requires 1 4∆2 + (∆ )2 ∼ h 2 + c2 ∼ k 2 + b2 ∼ r 2 . 4 U(1). Then ∆ is non-zero at the Now suppose H is one-dimensional, so that H0 ∼ = origin. Indeed, assuming otherwise, we shall prove that α is not smooth. In general, we are not allowed to replace H with H0 , since a smooth invariant form on G × H0 V does not necessarily correspond to a smooth form on G × H V . However, the converse always holds. Thus, we can apply the language of Sect. 6 in order to prove that α is not smooth. Let the Lie algebra of H0 be spanned by ξ = pe1 + p e2 + p e3 + qe4 .
768
D. Conti
The connection form corresponding to H0 has the form pe1 + p e2 + p e3 + qe4 . p 2 + ( p )2 + ( p )2 + q 2 Since α(ξ ) = 2 p∆ + q(2C∆ − C+m 3 ), we can write C +m 1 2 p∆ + q 2C∆ − dy α≡− σr 3
(mod (g/h)∗ ).
So if ∆(0) were zero, then α could only be smooth if C + m = 0, which is absurd. Having shown that ∆(0) = 0, we can conclude that the isotropy representation of H fixes e1 , and so H is contained in T 2 . Recall from Sect. 2 that this condition ensures that (7) holds; so, it is now safe to replace H with H0 . Let the Lie algebra of H be generated by ξ = pe1 + qe4 , where 2p and q are coprime integers. Write 1 1 C +m C +m qe − pe4 α =− 2 p∆+q 2C∆− dy + 2q∆− p 2C∆ − . σr 3 3 p2 + q 2 By Proposition 5, α is smooth if and only if 2( p + qC)∆(0) −
q(C + m) = 0 , ∆ is even. 3
Write ω1 = ( p + qC)
∆ qe1 − pe4 d x ∧ dy + ∆ (q − pC) 2 ∧ d x + ∆e23 . σr p + q2
We already know that ∆(r ) is even; hence, ∆ (r ) =
d∆ d∆ (r ) = ∓ (r ) dt dr
is odd, implying that ω1 is smooth. On the other hand, requiring that ω1 be nondegenerate gives ∆ (0) = 0 ,
p + qC = 0 .
Replacing ξ with −ξ if necessary, we can assume that p + qC is positive. The metric tensor can be decomposed as 1 2 qe1 − pe4 qe − pe4 d x ⊗ d x + s(dy ⊗ dy) + f dy + g p2 + q 2 p2 + q 2 + (h 2 + c2 ) e2 ⊗ e2 + (b2 + k 2 ) e3 ⊗ e3 + (hb + ck)e2 e3 whence 1 s= 2 2 σ r
q(C + m) 2( p + qC)∆ − 3
2
2
+ ( p + qC) (∆ ) 2
Cohomogeneity One Einstein-Sasaki 5-Manifolds
769
must be even and s(0) = 1. The former condition is automatic, whereas the latter gives σ ∆ (0) = . p + qC Moreover f must be odd, g must be even, and g, h 2 + k 2 + c2 + b2 must be non-zero at the origin; all these conditions follow from those we have already obtained. The rest of the statement follows from the fact that e2 ⊗ e2 + e3 ⊗ e3 is H -invariant, whereas e2 e3 , e2 ⊗ e2 − e3 ⊗ e3 is isomorphic to V2 p . We can now prove the main result of this section. We shall use the identification iθ iψ T 2 = (exp 2θ e1 , exp ψe4 ) ∈ SU(2) × U(1) ∼ = (e , e ) ∈ U(1) × U(1) . Theorem 1. There is no solution of (23) that defines an Einstein-Sasaki metric on a compact manifold. The Einstein-Sasaki structure (22) extends to an invariant EinsteinSasaki structure on the compact cohomogeneity one manifold with diagram (6) if and only if: –
A = 0, K is a finite subgroup of T 2 intersecting (H+ )0 in a group of order σ , 1 H+ = e 2 (σ +mq)iθ , eqiθ ∈ T 2 , θ ∈ R · K , H− = (SU(2) × {1}) · K ,
where q and 21 (qm + σ ) are coprime integers, and σ = − 13 (C + m)q. The resulting Riemannian manifold is locally isometric to S 5 . – A < 0, K is a finite subgroup of T 2 intersecting (H± )0 in a group of order σ± ; 1 H± = e 2 (σ± +mq± )iθ , eq± iθ ∈ T 2 , θ ∈ R · K , where q± and 21 (q± m + σ± ) are coprime integers, and the equation A + ∆2 − 4∆3 = 0
(26)
has two distinct roots ∆− , ∆+ such that q± 6∆± 1 = . σ± C + m 1 − 6∆± Then the Einstein-Sasaki structure is quasi-regular if C is rational, and irregular otherwise. Proof. In (23), we can assume that at some t = t0 , h, k, ∆ > 0, b = 0, (k − h, c) = (0, 0) . Introduce the variables u =h+k , v =h−k , z =b+c , w =b−c ;
770
D. Conti
by above, either u, v = 0 at t = t0 or we can apply a transformation of type (17) to obtain u, v = 0. Then about t0 (23) reads ∆ + 6∆ = −1 − ∆
z w u v = 1 − ∆ = 1 − ∆ = −1 − ∆ . z w u v
(27)
Every solution of (27) satisfies w = λu and z = µv, where λ and µ are constants. Thus 1 + λ2 1 + µ2 1 u2 − v2 . ∆ = (u 2 − v 2 − z 2 + w 2 ) = 4 4 4 √ √ 2 1+µ2 u, V = v, transforming (27) into This motivates us to set U = 1+λ 2 2 ∆ + 6∆ = 1 − ∆
U V = −1 − ∆ , ∆ = U V
2 U U
−
V V
, ∆ = U2 − V 2 .
(28)
These equations only make sense at points where U, V = 0. However, U cannot vanish in (t− , t+ ) because ∆ is positive, and if V were zero at some t0 in (t− , t+ ) then it would be zero on all the interval, as one can see by applying a transformation of type (17) and reducing to case (ii) of Sect. 5. Suppose that H is one-dimensional. Since C = 0, Lemma 4 implies that p > 0, and hb + ck =
1 (λ + µ)uv , h 2 + c2 − b2 − k 2 = (1 − λµ)uv 2
must vanish at the origin. So, U (0)V (0) = 0. Since, again by Lemma 4, ∆ is a smooth function with ∆(0) > 0, it follows that V (0) = 0, U (0) = 0. Now observe that h 2 + k 2 + b2 + c2 = 4V 2 + 2∆ = 4U 2 − 2∆ ,
(29)
is even by Lemma 4, so that U 2 and V 2 are also even. This implies that U , V are smooth functions, and U /U vanishes at 0. By (28), it follows that V V 2 U = lim − − = lim − = −∞ . lim r →0 V r →0 r →0 ∆ U V On the other hand, since V (r ) is non-zero for all r > 0 we have lim
r →0
dV dr
V
≥0.
Since dr = ∓dt, we conclude that this possibility may only occur at t+ . Suppose now that H has dimension three. Then Lemma 4 and (29) imply that V 2 /r 2 , U 2 /r 2 are even. Dividing (29) by r 2 and evaluating at the origin, we find U 2 /r 2 = 41 , V 2 /r 2 = 0 at the origin. It follows from (28) and Lemma 4 that 1 1 V rV = −1 − lim ∆ = −1 ± lim , r →0 2 V 2 r →0 V and so we reach the contradiction
Cohomogeneity One Einstein-Sasaki 5-Manifolds
771
r ddrV rV = ∓ lim = −3 . r →0 V r →0 V This completes the proof of the first part of the theorem. Now consider a solution of (22). By Lemma 4, at the origin either h = 0 or (h 2 ) = 0. The first possibility can only occur if A = 0. In this case, (21) gives 1 h = − h2 , 4 0 ≤ lim
and thus ∆ = h 2 is a smooth even function such that ∆ (0) = 21 . Then, by Lemma 4, H0 = SU(2) × {1} and the structure extends across G/H . Notice that one cannot have h = 0 at both special orbits, because otherwise h would have to vanish at some point in between. On the other hand, h vanishes at both special orbits if and only if A is negative. Suppose that h = 0 at the origin. It follows from (21) that ∆ is automatically even, and Lemma 4 only requires that, at the origin, ∆=
q(C + m) , 6( p + qC)
σ = |1 − 6∆| , p + qC
where we have used a formula analogous to (27) to express ∆ in terms of ∆. Since σ is determined up to sign, we can write q=
σ 6∆(0) , C + m 1 − 6∆(0)
p = qm + σ .
(30)
The condition h = 0 means that ∆(0) is a root of (26). Thus, if A = 0, ∆(t+ ) = 41 and (30) gives 3σ ; C +m the remaining conditions on p, q and σ follow from Lemma 4. A straightforward computation shows that the Riemannian metric has constant sectional curvature, and therefore the Riemannian manifold we obtain is locally isometric to S 5 . If A is non-zero, both special orbits are three-dimensional and satisfy (30), which together with Lemma 4 determines the required conditions. Finally, the statement about regularity follows from the fact that the characteristic vector field is Ce1 − e4 , up to an invariant function. q=−
It was proved in [13] that there are countably infinite values of A in the interval 1 < A < 0 for which (26) has roots ∆+ , ∆− with ∆+ − ∆− ∈ Q. One can show, − 108 using (26), that for any such value of A and any choice of m there exists a value of C satisfying the hypotheses of Theorem 1. Moreover, there also exist infinite values of A for which both ∆+ and ∆− are rational, so that C is also rational and the metric is quasi-regular. Remark 6. It follows from (22) that the right action of T 2 on G preserves the U(2)structure. One can then replace G with a subgroup SU(2) × U(1) ⊂ G × T 2 that acts transitively on G preserving the Einstein-Sasaki SU(2)-structure. Since T 2 ⊃ K , H± is abelian, this action is well defined on the cohomogeneity one manifold (6). Thus, we discover that the cohomogeneity one action can be assumed to preserve the SU(2)-structure, which amounts to setting m = 0 in (22).
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By the above remark, there is no loss of generality in assuming m = 0. Thus, we recover the Einstein-Sasaki metrics of [13]. Indeed, introduce coordinates (θ, ψ, φ) on SU(2) = S 3 as in (14), a coordinate α on U(1) such that e4 = dα, and a coordinate y on (t− , t+ ) such that y = 1 − 6∆. Set 2(108A + 1 − 3y 2 + 2y 3 ) . β = −ψ + C α , wq = 1−y Then our family of local Einstein-Sasaki metrics can be written as 1−y 1 1 (dθ 2 + sin2 θ dφ 2 ) + dy 2 + wq(dβ + cos θ dφ)2 6 wq 36 1 + (dψ − cos θ dφ + y(dβ + cos θ dφ))2 9 which is the local form of a Y p,q metric if A < 0, and the standard metric on the sphere if A = 0. 8. Classification We can finally prove that the only cohomogeneity one Einstein-Sasaki 5-manifolds are the Y p,q . Theorem 2. Let M be a compact, simply-connected Einstein-Sasaki 5-manifold on which the group of isometries acts with cohomogeneity one. Then M can be represented by the diagram (6), where K = −1, e2πiq+ /σ+ , −1, e2πiq− /σ− ⊂ T 2 , 1 H± = e 2 σ± iθ , eq± iθ ∈ T 2 , θ ∈ R · K , the integers q± , σ± satisfy Theorem 1, and the metric is given by (22). Moreover, M is diffeomorphic to S 2 × S 3 . Proof. Since the group of isometries of S 5 acts in a homogeneous way, M is not isometric to S 5 . Thus, by Theorem 1 and the subsequent remark M can be represented by the diagram K˜ ⊂ H˜ ± ⊂ G, where K˜ is a finite subgroup of T 2 intersecting ( H˜ ± )0 in a group of order σ± , and 1 H˜ ± = e 2 σ± iθ , eq± iθ ∈ T 2 , θ ∈ R · K˜ . Now define K , H± as in the statement; then H˜ ± /H± ∼ = K˜ /K . Thus, up to an equivariant covering map M is represented by the diagram K ⊂ H± ⊂ G. Since M is simply connected, this covering map is actually a diffeomorphism. Recall from Sect. 2 that the diagram determines M only up to a glueing map. However, in our case K is contained in the center of G, and so the diagram does determine M up to equivariant diffeomorphism. To determine the topology of M, we make use of Smale’s classification theorem [19]; indeed, the existence of an Einstein-Sasaki metric implies that M is spin, whence it suffices to prove that H2 (M) = Z, π1 (M) = 0 .
Cohomogeneity One Einstein-Sasaki 5-Manifolds
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Let D± be the 2-dimensional disc, and write M = U+ ∪ U− , where U± = G × H± D± . Then G/H± is a retract of U± , whereas G/K is a retract of U+ ∩ U− . Define loops γ4 , γ± : S 1 → G by γ4 (eit ) = exp(te4 ), γ± (eit ) = exp(tσ± e1 + tq± e4 ) . It is not difficult to show that Zγ+ ⊕ Zγ− ⊕ Zγ4 , σ+ γ+ − q+ γ4 , σ− γ− − q− γ4 Zγ− ⊕ Zγ4 π1 (G/H± ) = , H2 (G/H± ) = 0 . σ∓ γ∓ − q∓ γ4 , q± γ4 π1 (G/K ) =
By the Seifert - van Kampen theorem, it follows that π1 (M) =
Zγ4 , q+ γ4 , q− γ4
and so M is simply connected if and only if q+ and q− are coprime. On the other hand the Mayer-Vietoris sequence gives H2 (U+ ) ⊕ H2 (U− )
/ H2 (M)
/ H1 (U+ ∩ U− )
j
/ H1 (U+ ) ⊕ H1 (U− ) .
By above, we conclude H2 (M) ∼ = ker j = Z(q+ q− γ4 ) ∼ =Z, where we have used the fact that q+ and q− are coprime. Acknowledgements. I would like to thank A. Ghigi and S. Salamon for helpful discussions. I am also in debt to K. Galicki and the referee for some useful comments on earlier versions of this paper.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Bär, C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154(3), 509–521 (1993) Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Basel-Boston: Birkhäuser, 2002 Boothby, W.M., Wang, H.C.: On contact manifolds. Ann. of Math. (2) 68, 721–734 (1958) Boyer, C.P., Galicki, K., Kollár, J.: Einstein metrics on spheres. Ann. of Math. (2) 162(1), 557–580 (2005) Boyer, C.P., Galicki, K., Nakamaye, M.: On the geometry of Sasakian-Einstein 5-manifolds. Math. Ann. 325(3), 485–524 (2003) Bredon, G.E.: Introduction to compact transformation groups. Number 46 in Pure and Applied Mathematics, London-New York: Academic Press, 1972 Butruille, J.-B.: Classification des variétés approximativement Kähleriennes homogènes. Ann. Global Anal. Geom. 27(3), 201–225 (2005) Cleyton, R., Swann, A.: Cohomogeneity-one G 2 -structures. J. Geom. Phys. 44, 202 (2002) Conti, D., Salamon, S.: Generalized Killing spinors in dimension 5. To appear in Trans. Amer. Math. Soc., DOI: S-0002-9947(07)04307-3 Cvetiˇc, M., Lü, H., Page, D.N., Pope, C.N.: New Einstein-Sasaki spaces in five and higher dimensions. Phys. Rev. Lett. 95, 071101 (2005) Fernández, M., Ivanov, S., Muñoz, V., Ugarte, L.: Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities, http://arxiv.org/list/math.DG/0602160, 2006 Friedrich, T., Kath, I.: Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator. J. Differ. Geom. 29, 263–279 (1989) Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Sasaki-Einstein metrics on S 2 × S 3 . Adv. Theor. Math. Phys. 8, 711 (2004)
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14. Gauntlett, J.P., Martelli, D., Sparks, J., Yau, S.-T.: Obstructions to the existence of Sasaki-Einstein metrics. Commun. Math. Phys. 273(3), 803–827 (2007) 15. Grove, K., Wilking, B., Ziller, W.: Positively curved cohomogeneity one manifolds and 3-Sasakian geometry. http://arxiv.org/list/math.DG/0511464, 2005 16. Hitchin, N.: Stable forms and special metrics. In: Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Volume 288 of Contemp. Math., Providence, RI: American Math. Soc., 2001, pp. 70–89 17. Kazdan, L., Warner, F.W.: Curvature functions for open 2-manifolds. Ann. Math. 99(2), 203–219 (1974) 18. Moroianu, A., Nagy, P.-A., Semmelmann, U.: Unit Killing vector fields on nearly Kähler manifolds. Internat. J. Math. 16(3), 281–301 (2005) 19. Smale, S.: On the structure of 5-manifolds. Ann. Math. 75, 38–46 (1962) 20. Tanno, S.: Geodesic flows on C L -manifolds and Einstein metrics on S 3 × S 2 . In: Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), Tokyo: Kaigai Publications, 1978, pp. 283–292 21. Uchida, F.: Classification of compact transformation groups on cohomology complex projective spaces with codimension one orbits. Japan. J. Math. (N.S.) 3(1), 141–189 (1977) Communicated by G.W. Gibbons
Commun. Math. Phys. 274, 775–794 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0262-y
Communications in
Mathematical Physics
Complex/Symplectic Mirrors Wu-yen Chuang1,2 , Shamit Kachru1,2 , Alessandro Tomasiello1 1 ITP, Stanford University, Stanford, CA 94305, USA.
E-mail: [email protected]
2 SLAC, Stanford University, Menlo Park, CA 94309, USA
Received: 25 August 2006 / Accepted: 8 January 2007 Published online: 10 July 2007 – © Springer-Verlag 2007
Abstract: We construct a class of symplectic non-Kähler and complex non-Kähler string theory vacua, extending and providing evidence for an earlier suggestion by Polchinski and Strominger. The class admits a mirror pairing by construction. Comparing hints from a variety of sources, including ten-dimensional supergravity and KK reduction on SU(3)-structure manifolds, suggests a picture in which string theory extends Reid’s fantasy to connect classes of both complex non-Kähler and symplectic non-Kähler manifolds. 1. Complex and Symplectic Vacua The study of string theory on Calabi–Yau manifolds has provided both the most popular vacua of the theory, and some of the best tests of theoretical ideas about its dynamics. Most manifolds, of course, are not Calabi–Yau. What is the next simplest class for theorists to explore? The answer, obviously, depends on what the definition of “simplest” is. However, many leads seem to be pointing to the same suspects. First of all, it has been suggested long ago [1] that type II vacua exist, preserving N = 2 supersymmetry (the same as for Calabi–Yau’s), on manifolds which are complex and non-Kähler (and enjoy vanishing c1 ). Calabi–Yau manifolds are simultaneously complex and symplectic, and mirror symmetry can be viewed as an exchange of these two properties [2]. The same logic seems to suggest that the proposal of [1] should also include symplectic non-Kähler manifolds as mirrors of the complex non-Kähler ones. Attempts at providing mirrors of this type (without using a physical interpretation) have indeed already been made in [3, 4]. In a different direction, complex non-Kähler manifolds have also featured in supersymmetry-preserving vacua of supergravity, already in [5]. More recently the general conditions for preserving N = 1 supersymmetry in supergravity have been reduced to geometrical conditions [6]; in particular, the manifold has to be generalized complex [7]. The most prominent examples of generalized complex manifolds are complex and
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symplectic manifolds, neither necessarily Kähler. It should also be noted that complex and symplectic manifolds seem to be natural in topological strings. In this paper we tie these ideas together. We find that vacua of the type described in [1] can be found for a large class of complex non-Kähler manifolds in type IIB and symplectic non-Kähler manifolds in type IIA, and observe that these vacua come in mirror pairs. Although these vacua are not fully amenable to ten-dimensional supergravity analysis for reasons that we will explain (this despite the fact that they preserve N = 2 rather than N = 1 supersymmetry), this is in agreement with the supergravity picture that all (RR) SU(3)-structure IIA vacua are symplectic [8], and all IIB vacua are complex [9, 10, 8], possibly suggesting a deeper structure. In Sect. 2, in an analysis formally identical to [1], we argue for the existence of the new vacua. In Sect. 3 we show that the corresponding internal manifolds are not Calabi–Yau but rather complex or symplectic. More specifically, in both theories, they are obtained from a transition that does not preserve the Calabi–Yau property. As evidence for this, we show that the expected physical spectrum agrees with the one obtained on the proposed manifolds. The part of this check that concerns the massless spectrum is straightforward; we can extend it to low-lying massive fields by combining results from geometry [11] and KK reduction on manifolds of SU(3) structure. We actually try in Sect. 4 to infer from our class of examples a few properties which should give more control over this kind of KK reduction. Specifically, we suggest that the lightest massive fields should be in correspondence with pseudo-holomorphic curves or pseudo-Special-Lagrangian three-cycles (a notion we will define at the appropriate juncture). Among the motivations for this paper were also a number of more grandiose questions about the effective potential of string theory. One of the motivations for mathematicians to study the generalized type of transition we consider in this paper is the hope that many moduli spaces actually happen to be submanifolds of a bigger moduli space, not unlike [12] the realization of the various 19-dimensional moduli spaces of algebraic K3’s as submanifolds of the 20-dimensional moduli space of abstract K3’s. It might be that string theory provides a natural candidate for such a space, at least for the N = 2 theories, whose points would be all SU(3)-structure manifolds (not necessarily complex or symplectic), very possibly augmented by non-geometrical points [13]. We would not call it a moduli space, but rather a configuration space: on it, a potential would be defined, whose zero locus would then be the moduli space of N = 2 supersymmetric string theory vacua, including in particular the complex and symplectic vacua described here. In this context, what this paper is studying is a small neighborhood where the moduli space of N = 2 non-Kähler compactifications meets up with the moduli space of Calabi–Yau compactifications with RR flux, inside this bigger configuration space of manifolds. 2. Four-Dimensional Description of the Vacua We will now adapt the ideas from [1] to our needs. The strategy is as follows. We begin by compactifying the IIB and IIA strings on Calabi–Yau threefolds, and we switch on internal RR fluxes, F3 in IIB and F4 in IIA (our eventual interest will be the case where the theories are compactified on mirror manifolds M and W, and the fluxes are mirror to one another). As also first noted in [1], this will make the four-dimensional N = 2 supergravity gauged; in particular, it will create a potential on the moduli space. This potential has supersymmetric vacua only at points where the Calabi–Yau is singular. However, on those loci of the moduli space new massless brane hypermultiplets have to be taken into account, which will then produce the new vacua.
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2.1. The singularities we consider. Let us first be more precise about the types of singularities we will consider. In IIB, as we will review shortly, if we switch on F3 with a non-zero integral along a cycle B3 of a Calabi–Yau M, a supersymmetric vacuum will exist on a point in moduli space in which only the cycle A3 conjugate to B3 under intersection pairing shrinks. It is often the case that several cycles shrink simultaneously, with effects that we will review in the next section, but there are definitely examples in which a single B cycle shrinks. These are the cases we will be interested in. (We will briefly explain in Sect. 3.2 how this condition could be relaxed.) In IIA, switching on F4 with a non-zero integral on a four-cycle A˜ 4 of W will generate a potential which will be zero only in points in which the quantum-corrected volume of the conjugate two-cycle B˜ 2 (the Poincaré dual to F4 ) vanishes. This will happen on a wall between two birationally equivalent Calabi–Yau’s, connected by a flop of B˜ 2 . These points will be mirror to the ones we described above for IIB. The converse is not always true: there can be shrinking three-cycles which are mirror to points in the IIA moduli space in which the quantum volume of the whole Calabi– Yau goes to zero. These walls separate geometrical and Landau–Ginzburg, or, hybrid, phases. One would obtain a vacuum at such a point by switching on F0 instead of F4 , for instance. The example discussed in [1] (the quintic) is precisely such a case. Since in the end we want to give geometrical interpretations to the vacua we will obtain, we will restrict our attention only to cases in which a curve shrinks in W - that is, when a flop happens. Although this is not strictly necessary for IIB, keeping mirror symmetry in mind we will restrict our attention to cases in which the stricter IIA condition is valid, not only the IIB one: in the mirror pairs of interest to us, the conifold singularity in M is mirror to a flop in W. It would be interesting, of course, to find the IIA mirrors to all the other complex non-Kähler manifolds in IIB. Looking for flops is not too difficult, as there is a general strategy. If the Calabi– Yau W is realized as hypersurface in a toric manifold V , the “enlarged Kähler moduli space” [14, 15] (or at least, the part of it which comes from pull-back of moduli of V ) is a toric manifold W K itself. The cones of the fan of W K are described by different triangulations of the cone over the toric polyhedron of V . Each of these cones will be a phase [16]; there will be many non-geometrical phases (Landau–Ginzburg or hybrid). Fortunately, the geometrical ones are characterized as the triangulations of the toric polyhedron of V itself (as opposed to triangulations of the cone over it). This subset of cones gives an open set in W K which is called the “partially enlarged” Kähler moduli space. This is not the end of the story, however. In many examples, it will happen that a flop between two geometrical phases will involve more than one curve at a time, an effect due to restriction from V to W. Worse still, these curves might have relations, and sometimes there is no quick way to determine this. Even so, we expect that there should be many cases in which a single curve shrinks (or many, but without relations). Such an example is readily found in the literature [17, 18]: taking W to be an elliptic fibration over F1 (a Calabi–Yau whose Hodge numbers are h 1,1 = 3 and h 2,1 = 243), there is a point in moduli space in which a single curve shrinks (see Appendix A for more details). By counting of multiplets and mirror symmetry, on the mirror M there will be a single three-cycle which will shrink. This implies that the mirror singularity will be a conifold singularity. Indeed, it is a hypersurface singularity, and as such the shrinking cycle is classified by the so-called Milnor number. This has to be one if there is a single shrinking cycle, and the only hypersurface singularity with Milnor number one is the conifold.
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2.2. Gauged supergravity analysis. After these generalities, we will now show how turning on fluxes drives the theory to a conifold point in the moduli space; more importantly, we will then show how including the new massless hypermultiplets generates new vacua. We will do this in detail in the IIB theory on M, as its IIA counterpart is then straightforward. The analysis is formally identical to the one in [1] (see also [19, 20]); the differences have been explained in the previous subsection. As usual, define the symplectic basis of three-cycles A I , B J and their Poincaré duals α I , β I such that I I A · BJ = δ J , αI = β J = δI J (2.1) AJ
BI
along with the periods = A I and FI = B I . Additionally, the basis is taken so that the cycle of interest described in Subect. 2.1 is A = A1 . When X 1 = 0, the cycle A1 degenerates to the zero size and M develops a conifold singularity. By the monodromy argument, the symplectic basis (X 1 , F1 ) will transform as follows when we circle the discriminant locus in the complex moduli space defined by X 1 = 0: X 1 → X 1 F1 → F1 + X 1 . (2.2) XI
From this we know F1 near the singularity: F1 = constant +
1 1 X lnX 1 + . . . . 2πi
(2.3)
The metric on the moduli space can be calculated from the formulae K V = − ln i( X¯ I FI − X I F¯ I ).
G I J¯ = ∂ I ∂ J¯ K V , Therefore we obtain
G11¯ ∼ ln(X 1 X¯1 ).
(2.5)
Now, the internal flux we want to switch on is F3 = n 1 F5 =
F2I
(2.4)
β 1.
The vectors come from
∧ α I − G 2,I ∧ β , I
(2.6)
where the F2I (G 2I ) is the electric (magnetic) field strength. The Chern–Simons coupling in the IIB supergravity action is then j i j F21 ∧ B2 , (2.7) F˜5 ∧ H3i ∧ B2 = n 1 M4 ×CY
M4
where M4 is the spacetime. By integration by parts, and since B2 dualizes to one of the (pseudo)scalars in the universal hypermultiplets, we see that the latter is gauged under the field A1 whose field strength is d A1 = F21 . The potential is now given by the “electric” formula V = h uv k uI k vJ X¯ I X J e K V + (U I J − 3 X¯ I X J e K V )P Iα P Jα , where
¯
U I J = Da X I g a b Db¯ X J ,
(2.8) (2.9)
Pα
are together the so-called Killing prepotential, or hypermomentum map. In and the our situation only the flux over B1 is turned on, and the Killing prepotential is given by ˜
P11 = P12 = 0; P13 = −e K H n 1 = −e2φ n 1 ,
(2.10)
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where φ is the dilaton. The potential will then only depend on the period of the dual A1 cycle, call it X 1 : (n 1 )2 V ∼ . (2.11) ln X 1 X¯ 1 The theory will thus be driven to the conifold point where X 1 = 0. This is not the end of the story: at the singular point, one has a new massless hypermultiplet B coming from a brane wrapping the shrinking cycle A1 . The world–volume coupling between the D3-brane and F5 gives then R×A1 A4 = R A1 , where R is the worldline of the resulting light particle in M4 . (The coincidence between the notation for the cycle A1 and the corresponding vector potential A1 is rather unfortunate, if standard.) This means that both the universal and the brane hypermultiplet are charged under the same vector; we can then say that they are all electrically charged and still use the electric formula for the potential (2.8), with the only change being that the Killing prepotential is modified to be P1α = P1α | B=0 + B + σ α B; (2.12) the black hole hypermultiplet is an SU (2) doublet with components (B1 , B2 ). These states arise by compactifying a D3 brane on the cycle A1 ; since their mass is proportional to the volume of A1 , they become massless in the limit in which A1 shrinks. Loci on which the P α ’s are zero are new vacua: it is easy to see that they are given by ˜ 1/2 (2.13) B = ((e K H n 1 )1/2 , 0) = (eφ n 1 , 0). The situation here is similar to [1]: the expectation value of the new brane hypermultiplet is of the order gs = eφ . So, as in that paper, the two requirements that gs is small and that B be small (the expression for the P α is a Taylor expansion and will be modified for large B) coincide, and with these choices we can trust these vacua. After Higgsing the flat direction of the potential, namely, the massless hypermultiplet B˜ 0 , would be a linear combination of the brane hypermultiplet and the universal hypermultiplet while the other combination would become a massive one B˜ m . 2.3. The field theory capturing the transition. It is useful to understand the physics of the transition from a 4d field theory perspective, in a region very close to the transition point on moduli space. While this analysis is in principle a simple limit of the gauged supergravity in the previous subsection, going through it will both provide more intuition and also allow us to infer some additional lessons. In fact, in the IIB theory with n 1 units of RR flux, the theory close to the transition point (focusing on the relevant degrees of freedom) is simply a U(1) gauge theory with two charged hypers, of charges 1 and n 1 . Let us focus on the case n 1 = 1 for concreteness. Let us call the N = 1 chiral ˜ In N = 1 language, this theory has a multiplets in the two hypers B, B˜ and C, C. superpotential ˜ B + CϕC, ˜ W ∼ Bϕ (2.14) where ϕ is the neutral chiral multiplet in the N = 2 U(1) vector multiplet. It also has a D-term potential ˜ 2 − |B|2 + |C| ˜ 2 − |C|2 )2 . (2.15) |D|2 ∼ (| B| There are two branches of the moduli space of vacua: a Coulomb branch where ϕ = 0 and the charged matter fields vanish, and a Higgs branch where ϕ = 0 and the hypers
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have non-vanishing vevs (consistent with F and D flatness). The first branch has complex dimension one, the second has quaternionic dimension one. These branches meet at the point where all fields have vanishing expectation value. At this point, the theory has an SU(2) global flavor symmetry. This implies that locally, the hypermultiplet moduli space will take the form C2 /Z2 [21]. In fact, the precise geometry of the hypermultiplet moduli space, including quantum corrections, can then be determined by a variety of arguments [21, 22] (another type of argument [23] implies the same result for the case where the hypermultiplets coming from shrinking three-cycles in IIB). The result is the following. Locally, the quaternionic space reduces to a hyperKähler manifold which is an elliptic fibration, with fiber coordinates t, x and a (complex) base coordinate z. Let us denote the Kähler class of the elliptic fiber by λ2 . Then, the metric takes the form (2.16) ds 2 = λ2 V −1 (dt − A · dy)2 + V (dy)2 , where y is the three-vector with components (x, λz , λz¯ ). Here, the function V and the vector of functions A are given by ⎛ ⎞ ∞ 1 1 1 ⎠ ⎝ V = − + constant (2.17) 2 2π n=−∞ |n| |z| (x − n)2 + λ2 and
∇ × A = ∇V.
(2.18)
This provides us with detailed knowledge of the metric on the hypermultiplet moduli space emanating from the singularity, though it is hard to explicitly map the flat direction to a combination of the universal hypermultiplet and the geometrical parameters of M or W . We shall discuss some qualitative aspects of this map in §3.3. For the reader who is confused by the existence of a Coulomb branch at all, given that e.g. in the IIB picture F3 = 0, we note that the Coulomb branch will clearly exist on a locus where gs → 0 (since the hypermultiplet vevs must vanish). This is consistent with supergravity intuition, since in the 4d Einstein frame, the energetic cost of the RR fluxes vanishes as gs → 0. 3. Geometry of the Vacua We will first of all show that the vacua obtained in the previous section cannot come from a transition to another Calabi–Yau. To this aim, in the next subsection we will review Calabi–Yau extremal transitions. We will then proceed in Subsect. 3.2 to review the less well-known non-Calabi–Yau extremal transitions, and then compare them to the vacua we previously found in Subsect. 3.3. 3.1. Calabi–Yau extremal transitions. Calabi–Yau extremal transitions sew together moduli spaces for Calabi–Yaus whose Hodge numbers differ; let us quickly review how. For more details on this physically well-studied case, the reader might want to consult [24–26, 14]. Consider IIB theory on a Calabi–Yau M. (Some of the explanations in this paper are given in the IIB case only, whenever the IIA case would be an obvious enough modification). Suppose that at a particular point in moduli space, M develops N nodes (conifold
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points) by shrinking as many three-cycles Aa , a = 1, . . . , N , and that these three-cycles satisfy R relations N ria Aa = 0, i = 1, · · · , R (3.1) a=1
in H3 . We are not using the same notation for the index on the cycles as in Sect. 2, as these Aa are not all elements of a basis (as they are linearly dependent). Notice that it is already evident that this case is precisely the one we excluded with the specifications in Sect. 2.1. To give a classic example [24], there is a known transition where M is the quintic, N = 16 and R = 1. Physically, there will be N brane hypermultiplets Ba becoming massless at this point in moduli space. Vectors come from h 2,1 ; since the Ba only span N − R directions in H 3 , they will be charged under N − R vectors X A only, A = 1, . . . N − R. Call the matrix of charges Q aA , A = 1, . . . , N − R, a = 1, . . . N . In this case, when looking for vacua, we will still be setting the Killing prepotential Pa (which is a simple extension of the one in (2.12)) to zero: the flux is now absent, and the B 2 term now reads PA = Q aA Ba+ σ α Ba . (3.2) a
Notice that we have switched no flux on in this case; crucially, P = 0 now will have an R-dimensional space of solutions, due to the relations. Let us suppose this new branch is actually the moduli space for a new Calabi–Yau. This new manifold would have h 2,1 − (N − R) vectors, because all the X A have been Higgsed; and h 1,1 + R hypers, because of the N Ba , only R flat directions have survived. This is exactly the same result one would get from a small resolution of all the N nodes. Indeed, let us call the Calabi–Yau resulting from such a procedure M , and let us compute its Betti numbers. It is actually simpler to first consider a case in which a single three-cycle undergoes surgery1 , which is the case without relations specified in Sect. 2.1; we will go back to the Calabi–Yau case, in which relations are necessary, momentarily. The result of this single surgery along a three-cycle is that H 3 → H 3 −2, H 2 → H 2 . This might be a bit surprising: one is used to think that an extremal transition replaces a three-cycle by a two-cycle. But this intuition comes from the noncompact case, in which indeed it holds. In the compact case, when we perform a surgery along a three-cycle, we really are also losing its conjugate under Poincaré pairing; and we gain no two-cycle. The difference is illustrated in a low-dimensional analogue in Fig. 1, in which H 2 and H 3 are replaced by H 0 and H 1 . Coming back to the Calabi–Yau case of interest in this subsection, let us now consider N shrinking three-cycles with R relations. First of all H 3 only changes by 2(N − R), because this is the number of independent cycles we are losing. But this is not the only effect on the homology. A relation can be viewed as a four-chain F whose boundary is Aa . After surgery, the boundary of F by definition shrinks to points; hence F becomes a four-cycle in its own right. This gives R new elements in H4 (or equivalently, in H 2 ). The change in homology is summarized in Table 1, along with the IIA case and, more importantly, in a more general context that we will explain. By comparing with the physical counting above, we find evidence that the new branches of the moduli space correspond to new Calabi–Yau manifolds obtained by extremal transitions. 1 This is a purely topological computation; in a topological context, an extremal transition is called a surgery, and we will use this term when we want to emphasize we are considering purely the topology of the manifolds involved.
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C A D B Fig. 1. Difference between compact and non-compact surgery: in the noncompact case (up), one loses an element in H 1 and one gains an element in H 0 (a connected component). In the compact case (down), one loses an element in H 1 again, but the would-be new element in H 0 is actually trivial, so H 0 remains the same. This figure is meant to help intuition about the conifold transition in dimension 6, where H 0 and H 1 are replaced by H 2 and H 3 . We also have depicted various chains on the result of the compact transition, for later use
To summarize, Calabi–Yau extremal transitions are possible without fluxes, but they require relations among the shrinking cycles. This is to be contrasted with the vacua in the previous section, where there are no relations among the shrinking cycles to provide flat directions. Instead, the flux (and resulting gauging) lifts the old Calabi–Yau moduli space (as long as gs = 0), but makes up for this by producing a new branch of moduli space (emanating from the conifold point or its mirror). 3.2. Non-Calabi–Yau extremal transitions. In this section we will waive the Calabi–Yau condition to reproduce the vacua of the previous section. This is, remember, a case in which cycles shrink without relations. However, we will start with a review of results in the more general case, to put in perspective both the case we will eventually consider and the usual Calabi–Yau case. All the manifolds we will consider from now on are compact. We will consider both usual conifold transitions, in which three-cycles are shrunk and replaced by curves, and so-called reverse conifold transitions, in which the converse happens.2 As a hopefully useful shorthand, we will call the first type a 3 → 2 transition and the second a 2 → 3. Though the manifolds will no longer be (necessarily) Calabi– Yau, we will still call the initial and final manifold M and M in the 3 → 2 case (which is relevant for our IIB picture), and W and W in the 2 → 3 case (which is relevant for our IIA picture). We will first ask whether a 3 → 2 transition takes a complex, or symplectic, M into a complex, or symplectic, M , and then turn to the same questions about W, W for 2 → 3 transitions. These questions have to be phrased a bit more precisely, and we will do so case by case. 2 Implicit in the use of the word “conifold” is the assumption that several cycles do not collapse together in a single point of the manifold M. More general cases are also interesting to consider, see for example [26] for the complex case and [4] for the symplectic case.
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It is also useful to recall at this point the definitions of symplectic and complex manifolds, which we will do by embedding them in a bigger framework. In both cases, we can start with a weaker concept called G-structure. By this we mean the possibility of taking the transition functions on the tangent bundle of M to be in a group G. This is typically accomplished by finding a geometrical object (a tensor, or a spinor) whose stabilizer is precisely G. If we find a two-form J such that J ∧ J ∧ J is nowhere zero, it gives an Sp(6, R) structure. In presence of a (1, 1) tensor (one index up and one down) j such that j 2 = −1 (an almost complex structure), we speak of a Gl(3, C) structure. For us the presence of both will be important; but we also impose a compatibility condition, which says that the tensor jm p J pn is symmetric and of positive signature. This tensor is then nothing but a Riemannian metric. The triple is an almost hermitian metric: this gives a structure Sp(6, R)∩ Sl(3, C) =U(3). By themselves, these reductions of structure do not give much of a restriction on the manifold. But in all these cases we can now consider an appropriate integrability condition, a differential equation which makes the manifold with the given structure more rigid. In the case of J , we can impose that d J = 0. In this case we say that the manifold is symplectic. For j, a more complicated condition (that we will detail later, when considering SU(3) structures) leads to complex manifolds. Let us now consider a complex manifold M (which we will also take to have trivial canonical class K = 0). First order complex deformations are parameterized by H 1 (M, T ) = H 2,1 . Suppose that for some value of the complex moduli N three-cycles shrink. Replace now these N nodes by small resolutions. The definition of small resolution, just like the one of blowup, can be given locally around the node and then patched without any problem with the rest of the manifold. So the new manifold M is still complex. Also, the canonical class K is not modified by the transition because a small resolution does not create a new divisor, only a new curve.3 Actually, the conjecture that all Calabi–Yau are connected was initially formulated by Reid [12] for all complex manifolds (and not only Calabi–Yaus) with K = 0, extending ideas by Hirzebruch [27]. If now we consider a symplectic M, the story is different. For one thing, now symplectic moduli are given by H 2 (M, R) [28], so it does not seem promising to look for a point in moduli space where three-cycles shrink. But 2.1 in [3] shows that we can nevertheless shrink a (Lagrangian) three-cycle symplectically, and replace it by a two-cycle. Whether the resulting M will also be symplectic is not automatic, however. This can be decided using Theorem 2.9 in [3]: the answer is yes precisely when there is at least one relation in homology among the three-cycles.4 The case of interest in this paper is actually a blending of the two questions considered so far, whether complex or symplectic properties are preserved. In IIB, we will take a Calabi–Yau M (which has both properties) and follow it in moduli space to a point at which it develops a conifold singularity. Now we perform a small resolution to obtain a manifold M and ask whether this new manifold is still Kähler; this question has been considered also by [31]. As we have seen, the complex property is kept, and the 3 The conditions for N = 1-preserving vacua in ten-dimensional type II supergravity actually only require c1 = 0. The role of this condition is less clear for example in the topological string: for the A model it would seem to be unnecessary, as there is no anomaly to cancel; for the B model, it would look like the stronger condition K = 0 is required, which means that the canonical bundle should be trivial even holomorphically. 4 We should add that the relations must involve all the three-cycles. If there is one three-cycle A which is not involved in any relation, it is possible to resolve symplectically all the other cycles but not A. Examples of this type are found in [30, 31] when M is Kähler, which is the case of interest to us and to which we will turn shortly. These examples would play in our favor, allowing us to find even more examples of non-Kähler M , but for simplicity of exposition we will mostly ignore them in the following.
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symplectic property is not (though the question in [3] regards more generally symplectic manifolds, disregarding the complex structure, and in particular being more interesting without such a path in complex structure moduli space). Let us see why M cannot be Kähler in our case. A first argument is not too different from an argument given after Fig. 1 to count four-cycles. If the manifold M after the transition is Kähler, there will be an element ω ∈ H4 dual to the Kähler form. This will have non-zero intersection ω · Ca = vol(Ca ) with all the curves Ca produced by the small resolutions. Before the transition, then, in M, ω will develop a boundary, since the Ca are replaced by three-cycles Aa ; more precisely, ∂ω = r a Aa for some coefficients r a . This proves there will have to be at least one relation between the collapsing three-cycles. We can rephrase this in yet another way. Let us consider the case in which only one nontrivial three-cycle A is shrinking. Since, as remarked earlier (see Fig. 1), in the compact case the curve C created by the transition is trivial in homology, there exists a three-chain B such that C = ∂ B; then we have, if J is the two-form of the SU(3) structure, 0 = J= d J. (3.3) C
B
Hence d J = 0: the manifold cannot be symplectic.5 Even if a symplectic J fails to exist, there is actually a non-degenerate J compatible with j (since the inclusion U(3) ⊂ Sp(6, R) is a homotopy equivalence, not unlike the way the homotopy equivalence O(n) ⊂ Gl(n) allows one to find a Riemannian metric on any manifold). In other words, the integrable complex structure j can be completed to a U(3) structure (and then to an SU(3) structure, as we will see), though not to a Kähler one. This is also a good point to make some remarks about the nature of the curve C that we will need later on. The concept of holomorphic curve makes sense even without an integrable complex structure; the definition is still that (δ + i j)m n ∂ X n = 0, where X is the embedding C in M. For j integrable this is the usual condition that the curve be holomorphic. But this condition makes sense even for an almost complex structure, a fact which is expressed by calling the curve pseudo-holomorphic [29]. We will often drop this prefix in the following. In many of the usual manipulations involving calibrated cycles, one never uses integrability properties for the almost complex or symplectic structures on M. For example, it is still true that the restriction of J to C is its volume form volC . Exactly in the same way, one can speak of Special Lagrangian submanifolds even without integrability (after having defined an SU(3) structure, which we will in the next section), and sometimes we will qualify them as “pseudo” to signify this. Let us now consider 2 → 3 transitions. It will turn out that the results are just mirror of the ones we gave for 3 → 2, but in this case it is probably helpful to review them separately. After all, mirror symmetry for complex-symplectic pairs is not as well established as for Calabi–Yaus, which is one of the motivations of the present work. (Evidence so far includes mathematical insight [2], and, in the slightly more general context of SU(3) structure manifolds, comparisons of four-dimensional theories [32, 33] and direct SYZ computation [34].) Suppose now we start (in the IIA theory) with a symplectic manifold W (whose moduli space is, as we said, modeled on H 2 (M, R)), and that for some value of the symplectic moduli some curves shrink. Then, it turns out that one can always replace the resulting singularities by some three-cycles, and still get a symplectic manifold 5 In the mirror picture, a similar argument shows immediately that d = 0 on W , and hence the manifold cannot be complex.
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Table 1. The conditions for a transition to send a complex or symplectic conifold to a complex or symplectic manifold
IIA IIB
transition
keeps symplectic
2→3 3→2
yes ([3], 2.7) if ria Ba = 0 ([3], 2.9)
keeps complex if ria Ca = 0 ([35, 36]) yes
b2
b3
N−R R
2R 2(N − R)
(Theorem 2.7, [3]). The trick is that T ∗ S 3 , the deformed conifold, is naturally symplectic, since it is a cotangent bundle. Then [3] proves that this holds even globally: there is no problem in patching together the modifications around each conifold point. One should compare this with the construction used by Hirzebruch and Reid cited above. It is not automatic that the resulting manifold W is complex, even if W is complex itself. The criterion is that there should be at least one relation in homology between the collapsing curves Ca [35, 36] (see also [37] for an interesting application).6 Let us collect the transitions considered so far in a table; we also anticipate in which string theory each transition will be relevant for us. The symmetry among these results is clear; we will not need all of them, though. 3.3. Vacua versus geometry. We can now apply the results reviewed in the previous subsection to our vacua. Remember that in IIB we have chosen a point in moduli space in which a single three-cycle shrinks, and in IIA one in which a single curve shrinks. From our assumptions, the singularities affect the manifold only locally (as opposed for example to the IIA case of [1], in which the quantum volume of the whole manifold is shrinking); it is hence natural to assume that the vacua of Sect. 2 are still geometrical. Given the experience with the Calabi–Yau case, it is also natural that the brane hypermultiplet B describes a surgery. But then we can use the results of the previous subsection. In IIB, where we have shrunk a three-cycle, we now know that the manifold obtained by replacing the node with a curve will be naturally complex, but will not be symplectic, since by assumption we do not have any relations. As we have explained, the reason for this is that on the manifold M after the transition, there will be a holomorphic curve C which is homologically trivial; and by Stokes, we conclude that the manifold cannot be symplectic. Summing up, we are proposing that in IIB the vacua we are finding are given by a complex non-symplectic (and hence non-Kähler7 ) manifold. This manifold M is defined by a small resolution on the singular point of M, and it has (see Table 1) Betti numbers b2 (M ) = b2 (M), b3 (M ) = b3 (M) − 2 . (3.4) In the example described in Sect. 2.1, when M is the mirror of an elliptic fibration over F1 , M has b2 = 243, b3 /2 = 3. 6 Actually, the criterion also assumes W to satisfy the ∂ ∂-lemma, ¯ to ensure that H 2,1 ⊂ H 3 , which is not always true on complex non-Kähler manifolds; this assumption is trivially valid in the cases we consider, where W is a Calabi–Yau. Moreover, a similar comment as in footnote 4 applies: the relation has to involve all the two-cycles. 7 There might actually be, theoretically speaking, a Kähler structure on the manifold which has nothing to do with the surgery. This question is natural mathematically [3], but irrelevant physically: such a Kähler structure would be in some other branch of moduli space, far from the one we are considering, which is connected and close to the original Calabi–Yau by construction.
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In IIA, a similar reasoning lets us conjecture that the new vacua correspond to having a symplectic non-complex (and hence non-Kähler) manifold W , obtained from the original Calabi–Yau W by replacing the node with a three-cycle. This manifold W has b2 (W ) = b2 (W) − 1, b3 (W ) = b3 (W).
(3.5)
In the example from Sect. 2.1, when W is an elliptic fibration over F1 , W has b2 = 2, b3 /2 = 244. Notice that these two sets of vacua are mirror by construction: we localize in IIA and in IIB to points which are mirror to each other, and in both cases we add the appropriate brane hypermultiplets to reveal new lines of vacua. What is conjectural is simply the interpretation of the vacua. We now proceed to give evidence for that conjecture. In the IIB case, the spectrum before the transition is clearly given by b3 (M)/2 − 1 vector multiplets and b2 (M) + 1 hypermultiplets (the “ + 1” is the universal hypermultiplet). We have seen that the potential generated by F3 gives mass to one of the vector multiplets, fixing it at a certain point in the complex moduli space. On the other side, the number of massless hypermultiplets remains the same. Indeed, we have added a brane hypermultiplet B; but this combines with the universal hypermultiplet to give only one massless direction, the one given in (2.13). This is to be compared with the Betti numbers of the proposed M from Table 1: indeed, b2 remains the same and b3 changes by 2. Since the manifold is now non-Kähler, we have to be careful in drawing conclusions: “Kähler moduli” a priori do not make sense any more, and though complex moduli are still given by H 2,1 (by Kodaira–Spencer and K = 0), a priori this number is = b3 /2 − 1, since the manifold is non-Kähler. However, two circumstances help us. The first is that, by construction, the moduli of the manifolds we have constructed are identified with the moduli of the singular Calabi– Yau on which the small resolution is performed. Then, indeed we can say that there should be b3 (M )/2 − 1 + b2 (M ) complex geometrical moduli in total (after complexifying the moduli from b2 with periods of the anti-symmetric tensor field appropriately, and neglecting the scalars arising from periods of RR gauge fields). A more insightful approach exists, and will also allow us to compare low-lying massive states. Reduction on a general manifold of SU(3) structure (along with a more general class which will not concern us here) has been performed recently in [33]. (Manifolds with SU (3) structures and various differential conditions were also considered from the perspective of supergravity vacua, starting with [38, 39]). We have introduced a U(3) structure in the previous section as the presence on the manifold of both a complex and a symplectic structure with a compatibility condition. The almost complex structure j allows us to define the bundle of (3, 0) forms, which is called the canonical bundle as in the integrable case. If this bundle is topologically trivial the structure reduces further to SU(3). The global section of the canonical bundle can actually be used to define the almost complex structure by ∗ ¯ = 0}. Thol = {v1 ∈ T ∗ |v1 ∧
(3.6)
The integrability of the almost complex structure is then defined by (d)2,2 = 0, something we will not always require. Let us now review the construction in [33] from our perspective. In general the results of [33] require one to know the spectrum of the Laplacian on the manifold, which is not always at hand; but in our case we have hints for the spectrum, as we will see shortly. We have seen that a U(3) structure, and hence also an SU(3) structure, defines a metric.
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Let us see it again: since J ∧ = 0, J is of type (1, 1), and then a metric can be defined as usual: gi j¯ = − i Ji j¯ . We can now consider the Laplacian associated to this metric. The suggestion in [33, 32] is to add some low-lying massive eigen-forms to the cohomology. Since [ , d] = 0 and [ , ∗] = 0, at a given mass level there will be eigen-forms of different degrees. Suppose for example ω2 = m 2 ω2 for a certain m. Then dω2 ≡ mβ3
(3.7)
will also satisfy β3 = m 2 β3 , and similarly for α3 ≡ ∗β3 and ω4 ≡ ∗ω2 . (The indices denote the degrees of the forms.) We can repeat this trick with several mass levels, even if coincident. After having added these massive forms to the cohomology, we can use the resulting combined basis to expand = X I α I + β I FI and J = ti ωi , formally as usual but with some of the α’s, β’s and ω’s now being massive. Finally, these expansions for and J can be plugged into certain “universal” expressions for the Kähler prepotential P α . Without fluxes (we will return to this point later) and with some dilaton factor suppressed, this looks like [33] 1 2 3 P + iP = d(B + i J ) ∧ , P = (dC2 − C0 d B) ∧ . (3.8) Since the reader may be confused about the interpretation of the expressions d(B + i J ) ∧ and (dC2 − C0 d B) ∧ which appear above (given the ability to integrate by parts), let us pause to give some explanation. Our IIB solutions indeed correspond to complex manifolds, equipped with a preferred closed 3-form which has d = 0. However, the 4d fields which are given a mass by the gauging actually include deformations of the geometry which yield d = 0, as we discussed above. Therefore, the potential which follows from (3.8) is a nontrivial function on our field space. Let us try to apply the KK construction just reviewed to the manifold M . First of all we need some information about its spectrum. We are arguing that M is obtained from surgery. In [11], it is found that the spectrum of the Dirac operator changes little, in an appropriate sense, under surgery. If we assume that this result goes through after twisting the Dirac operator, we can in particular consider the Dirac operator on bispinors, also known as the signature operator, which has the same spectrum as the Laplacian. All this suggests that for very small B and gs the spectrum on M will be very close to the one on M. Hence there will be an eigenform of the Laplacian ω with a relatively small eigenvalue m (and its partners discussed above), corresponding to the extra harmonic forms generating H 3 before the surgery. By the reasoning above, this will also give eigenforms α, β and ω. ˜ Expanding now = X 1 α + 0 , J = t 1 ω + J0 , B = b1 ω + B0 and C2 = c1 ω + C20 (where 0 , J0 , B0 and C20 represent the part of the expansion in cohomology) and using the relation M β3 ∧ α3 = 1, we get from (3.8): P 1 + iP 2 ∼ m(b1 + it 1 )X 1 ,
P 3 ∼ m(c1 − C0 b1 )X 1 .
(3.9)
The parameter m measures the non-Kählerness away from the Calabi–Yau manifold M, and should be proportional to the vev of the brane hypermultiplet B˜ 0 of §2.2. Clearly the formula is reminiscent of the quadratic dependence on the B hypermultiplet in (2.12). The size of the curve C is measured by t 1 . Of course B˜ 0 is really a function of the t 1 and universal hypermultiplets. Presumably, it and the massive hyper B˜ m in Sect. 2.2 are
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different linear combinations of the curve volume and gs . It is even tempting to map the M and M variables by mapping B directly to C J = t 1 , and (very reasonably) mapping the dilaton hypermultiplet on M directly into the one for M . Indeed, the size of C would then be proportional to gs (at least when both are small), which is consistent with both being zero at the transition point. Fixing this would require more detailed knowledge of the map between variables. However, since the formula for the Killing prepotentials has the universal hypermultiplet in it (which can be seen from (3.9), where C0 is mixed with other hypers and some dilaton factor is omitted in the front), it could have α corrections. Moreover, (3.8) is only valid in the supergravity regime where all the cycles are large compared with the string length. Hence an exact matching between the Killing prepotentials is lacking. We can now attempt the following comparison between the spectrum of the vacua and the KK spectrum on the conjectural M : • On M, one of the vectors, X 1 , is given a mass by the gauging F3 ∧ . On M , this vector becomes a deformation of which makes it not closed, → + α,
α = m 2 α. In both pictures, the vacuum is at the point X 1 = 0. On M, this is because we have fixed the complex modulus at the point in which A1 shrinks. On M , the manifold which is natural to propose from Table 1 is complex, and hence d = 0. • The remaining vectors are untouched by either gauging and remain massless. • Both for M and for M , there are b2 + 1 massless hypermultiplets. • From the perspective of the gauged supergravity analysis on M there is a massive hypermultiplet too: B and the universal hypermultiplet have mixed to give a massless direction, but another combination will be massive. On M , there is also a massive hypermultiplet: it is some combination of gs and t 1 , which multiplies the massive form ω (with ω = m 2 ω) in the expansion of J . To determine the precise combination one needs better knowledge of m(t 1 , gs ) in (3.9). Again, this comparison uses the fact that there is a positive eigenvalue of the Laplacian which is much smaller than the rest of the KK tower, and this fact is inspired by the work in [11]. This comparison cannot be made too precise for a number of reasons. One is, as we have already noticed, that it is hard to control the spectrum, and we had to inspire ourselves from work which seemed relevant. Another is that the KK reduction of tendimensional supergravity on the manifold M will not capture the full effective field theory precisely, as we are close (at small B vevs) to a point where a geometric transition has occurred. Hence, curvatures are large in localized parts of M , though the bulk of the space can be large and weakly curved. And indeed, we know that ten-dimensional type II supergravities do not allow N = 2 Minkowski vacua from non-Kähler compactification manifolds in a regime where all cycles are large enough to trust supergravity (though inclusion of further ingredients like orientifolds, which are present in string theory, can yield large radius N = 2 Minkowski vacua in this context [40]). The vacua of [1], and our own models, presumably evade this no-go theorem via stringy corrections arising in the region localized around the small resolution. Some of these corrections are captured by the local field theory analysis reviewed in §2.3, which gives us a reasonable knowledge of the hyper moduli space close to the singularity. It should be noted that the family of vacua we have found cannot simply disappear as one increases the expectation values of the B fields and eφ : the moduli space of N = 2 vacua is expected to be analytic even for the fully-fledged string theory. However, new terms in the expansion of the
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P α ’s in terms of the B hypermultiplet will deform the line; and large gs will make the perturbative type II description unreliable. An issue that deserves separate treatment is the following. Why have we assumed F3 = 0 in (3.8)? It would seem that the integral B F3 cannot simply go away. Usually, in conifold transitions (especially noncompact ones) a flux becomes a brane, as the cycle becomes contractible and surrounds a locus on which, by Gauss’ law, there must be a brane. This would be the case if, in Fig. 1, the flux were on A: this would really mean a brane on C. In our case, the flux is on B, on a chain which surrounds nothing. Without sources, and without being non-trivial in cohomology, F3 has no choice but disappear on M . To summarize this section, we have conjectured to which manifolds the vacua found in Sect. 2 correspond. In this way, we have also provided explicit symplectic-complex non-Kähler mirror pairs. 4. The Big Picture: A Space of Geometries There are a few remarks that can be made about the type of complex and symplectic manifolds that we have just analyzed, and that suggests a more general picture. This is a speculative section, and it should be taken as such. One of the questions which motivated us is the following. The KK reduction in [33] says that d J ∧ encodes the gauging of the four-dimensional effective supergravity on M . Hence in some appropriate sense (to be discussed below), d J must be integral - one would like d J ∧ to be expressed in terms of integral combinations of periods of . This is just because the allowed gauge charges in the full string theory form an integral lattice. But from existing discussions, the integral nature of d J is far from evident. Though one can normalize the massive forms appropriately in such a way that the expression does give an integer, this does not distinguish between several possible values for the gauging: it is just a renormalization, not a quantization. Without really answering this question, we want to suggest that there must be a natural modification of cohomology that somehow encodes some of the massive eigenvalues of the Laplacian, and that has integrality built in. It will be helpful to refer again to Fig. 1: on M (the manifold on the right in the lower line of Fig. 1), we have depicted a few relevant chains, obviously in a low-dimensional analogy. What used to be called the A cycle is now still a cycle, but trivial in homology, as it is bounded by a four-cycle D. The dual B cycle, from the other side, now is no longer a cycle at all, but merely a chain, its boundary being the curve C. This curve has already played a crucial role in showing that M cannot be symplectic. We want to suggest that a special role is played by relative cohomology groups H3 (M , C) and H4 (M , A). Remember that relative homology is the hypercohomoloιC gy of C• (C) −→ C• (M ), with Ck being chains and the map ιC being the inclusion. In plain English, chains in Ck (M , C) are pairs of chains (ck , c˜k−1 ) ∈ Ck (M )×Ck−1 (C), and homology is given by considering the differential ∂(ck , c˜k−1 ) = (∂ck + ιC (c˜k−1 ), −∂ c˜k−1 ).
(4.1)
So cycles in Hk (M , C), for example, are ordinary chains which have boundary on C. B is precisely such a chain. A long exact sequence can be used to show that, when C is a curve trivial in H2 (M ) as is our case, dim(H3 (M , C)) = dim(H3 (M )) + 1. So (B, C) and the usual cycles generate H3 (M , C). Similarly, dim(H4 (M , A)) = dim(H4 (M )) + 1, and the new generator is (D, A).
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Similar and dual statements are valid in cohomology. This is defined similarly as for homology: pairs (ωk , ω˜ k−1 ) ∈ k (M ) × k−1 (C), with a differential ∗ d(ωk , ω˜ k−1 ) = (dωk , ιC (ωk ) − d ω˜ k−1 ).
(4.2)
A non-trivial element of H 3 (M , C) is (0, volC ). Since C is a holomorphic curve, ∗ J and hence this representative is also equivalent to (d J, 0), using the volC = J|C ≡ ιC differential above. When we deform M with the scalar in the massive vector multiplet X 1 , the manifold becomes non-complex, as we have shown in the previous section; but one does not require the almost complex structure to be integrable to define an appropriate notion of holomorphic curve. In fact, one might expect then that, when d = 0, which corresponds to M being non-complex, one can also choose A to be SLag (as we remarked earlier, the definition will not really require that the almost symplectic structure be closed).8 Definitely, the logic would hold the other way around - if such a SLag A can be found, A = 0 and then, again by integration by parts, it follows that d = 0. In our example, we expect the number of units n 1 of F3 flux present before the transition in the IIB picture, to map to “n 1 units of dJ ” on M . The phrase in quotes has not been precisely defined, but it is reasonable to think that it is defined by some kind of intersection theory in relative homology. We will now try to make this more precise. As we have seen, the dimension of the relative H3 can be odd (and it is in our case), so we should not expect a pairing between A and B cycles within the same group. One might try nevertheless to define a pairing between chains in H3 (M , C) and H4 (M , A); it would be defined by (B, C) · (D, A) ≡ #(B ∩ A) = #(C ∩ D).
(4.3)
In fact, if we think of another lower-dimensional analogy, in which both A and C are one-dimensional in a three-dimensional manifold, it is easy to see that what we have just defined is a linking number between C and A. Indeed, dim(C)+dim(A) = dim(M )−1. This can also be rephrased in relative cohomology. Consider a bump-form δ A which is concentrated around A and has only components transverse to it, and similarly for C. These can be defined more precisely using tubular neighborhoods and the Thom isomorphism [41]. Since A and C are trivial in homology, we cannot quite say that these bump forms are the Poincaré duals of A and C. But we can say that (δ A , 0) ∈ H 3 (M, C) is the Poincaré dual to the cycle (D, A) ∈ H4 (M, A), with natural definitions for the pairing between homology and cohomology. δ A is non-trivial in relative cohomology but trivial in the ordinary cohomology H 3 (M), and hence there exists an FA such that d FA = δ A . Then we have FA ∧ δC = FA = d FA = #(C ∩ D) ≡ L(A, C). (4.4) M
C
B
In other words, in cohomology we have L(A, C) = d −1 (δ A ) ∧ δC . Suppose we have now another form δ˜ A which can represent the Poincaré dual (in relative cohomology) to (D, A). Then we can use this other form as well to compute the linking, with identical result. This is because (δ A , 0) ∼ (δ˜ A , 0) in H 3 (M , A) means 8 The reader should not confuse this potential SLag, which may exist off-shell in the IIB theory, with the pseudo-SLag manifold that exists on W where d = 0 even on the N = 2 supersymmetric solutions.
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that, by the definition of the differential above, δ A − δ˜ A = dω2 with ω2 satisfying ∗ ω = dω ιC ˜ 1 for some form ω˜ 1 on C. Then 2 M
d −1 (δ A − δ˜ A ) ∧ δC =
M
ω2 ∧ δC =
ω2 =
C
d ω˜ 1 = 0
(4.5)
C
so L(A, C) does not depend on the choice of the Poincaré dual. But now, remember that (d J, 0) is also a non-trivial element of H 3 (M , C); if we normalize the volume of C to claim to be called a Poincaré dual to (D, A). Indeed, 1, it then has an equally valid (d J, 0) ≡ d J = J = 1 = (D, A) · (B, C), and for all other cycles the (B,C) B C result is zero. Similar reasonings apply to d. Then we can apply the steps above and conclude that d J ∧ . (4.6) L(A, C) = M
In doing this we have normalized the volumes of C and A to one; if we reinstall those volumes, we get precisely that d J ∧ is a linear function of the vectors and hypers with an integral slope. Another point which seems to be suggesting itself is the relation between homologically trivial Special Lagrangians and holomorphic curves on one side, and massive terms in the expansion of and J on the other. The presence of a holomorphic but trivial curve, as we have already recalled, implies that d J = 0: in the previous section we have seen that one actually expects that such curves are in one-to-one correspondence with massive eigenforms of the Laplacian present in the expansion of J (whose coefficients represent massive fields, which vanish in vacuum). We have argued for this relation close to the transition point, and for the M that we have constructed, but it might be that this link persists in general. This would mean that inside an arbitrary SU(3) structure manifold, one would have massive fields which are naturally singled out, associated to homologically trivial holomorphic curves. Similarly, in the IIA on W , there is a 3-cycle which is (pseudo) Special Lagrangian but homologically trivial. Its presence implies that d = 0, in keeping with the fact that the IIA vacua are non-complex. Reid’s fantasy [12] involved the conjecture that by shrinking -1 curves, and then deforming, one may find a connected configuration space of complex threefolds with K = 0. Here, we see that it is natural to extend this fantasy to include a mirror conjecture: that the space of symplectic non-complex manifolds with SU(3) structure is similarly connected, perhaps via transitions involving the contraction of (pseudo) Special Lagrangian cycles, followed by small resolutions. The specialization to -1 curves in [12] is probably mirror to the requirement that the SLags be rigid, in the sense that b1 = 0. In either IIB or IIA, we have seen that (at least close to the transition) there is a natural set of massive fields to include in the low-energy theory, associated with the classes of cycles described above. Allowing these fields to take on expectation values may allow one to move off-shell, filling out a finite-dimensional (but large) configuration space, inside which complex and symplectic manifolds would be zeros of a stringy effective potential. While finding such an N = 2 configuration space together with an appropriate potential to reveal all N = 2 vacua is clearly an ambitious goal, it may also provide a fruitful warm-up problem for the more general question of characterizing the string theory “landscape” of N ≤ 1 vacua [42].
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In this bigger picture, this paper is a Taylor expansion of the master potential around a corner in which the moduli space of M meets the moduli space of compactifications on M with RR flux. Acknowledgements. We would like to thank P. Aspinwall, B. Florea and A. Kashani-Poor for useful discussions, and I. Smith and R. Thomas for some patient explanations of their work. The authors received support from the DOE under contract DE-AC03-76SF00515 and from the National Science Foundation under grant 0244728. SK was also supported by a David and Lucile Packard Foundation Fellowship for Science and Engineering.
A. Details about an Example We will detail here the transition for the example mentioned in Sect. 2.1. We will do so on the IIA side, which is the one which involves the strictest assumptions, as explained there. The Calabi–Yau W is an elliptic fibration over the Hirzebruch surface F1 . It is convenient to describe it as a hypersurface in a toric manifold V . The fan for the latter is described by the columns of the matrix ⎡
v1 v2 v3 v4 v5 v6 v7
0 0 0 0 1 ⎢ 0 ⎣ 0 −1 2 −1 0 3
⎤ 1 0 −1 0 1 −1 0 0 ⎥ . 2 2 2 2⎦ 3 3 3 3
The last five vectors lie in the same plane, determined by the last two coordinates; let us plot the first two coordinates, along with three different triangulations: v3
@ @
@
-
v6
v7 @ @
@ v5
v4
-
@ @ @
The vectors of the fan are indeed the right ones to describe the F1 base. The fan is further specified by the higher-dimensional cones in the picture, with the first triangulation really describing the elliptic fibration over F1 , the last describing a space related to the first by a flop, and the middle triangulation describing the singular case. (The points have been labeled in the singular case only to avoid cluttering the picture.) We associate as usual a homogeneous coordinate z i to each of the vi ’s, with charge matrix given by the (transposed) kernel of the matrix above: ⎡ ⎤ 0 0 3 −2 1 −2 0 ⎣6 4 1 0 1 0 0⎦. 3 2 0 0 0 0 1 From the picture we see that the flopped locus in V lies at z 3 = z 4 = z 6 = z 7 = 0. One has to check whether this locus intersects the Calabi–Yau only once. This is done
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by looking at the equation for W ⊂ V , which for a certain point in the complex moduli space reads z 12 + z 23 + z 312 z 418 z 76 + z 512 z 66 z 76 + z 312 z 618 z 76 + z 46 z 512 z 76 = 0; hence we get the singular locus z 12 + z 23 = 0 on W. Taking into account the C∗ actions, this corresponds to only one point p as desired. To verify that the normal bundle of the shrinking curve has charges (−1, −1), one can identify the combination of the charges that keeps p invariant; this action turns out to be (1, 1, λ, λ−1 , 1, λ−1 , λ), λ ∈ C∗ , which is the right one for a conifold point. References 1. Polchinski, J., Strominger, A.: New Vacua for Type II String Theory. Phys. Lett. B 388, 736 (1996) 2. Kontsevich, M.: Homological algebra of mirror symmetry. Proceedings of ICM (Zürich, 1994), Basel: Birkhäuser, 1995, pp. 120–139 3. Smith, I., Thomas, R.P., Yau, S.-T.: Symplectic conifold transitions. J. Diff. Geom. 62, 209 (2002) 4. Smith, I., Thomas, R.P.: Symplectic surgeries from singularities. Turkish J. Math. 27, 231 (2003) 5. Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274, 253 (1986) 6. Grana, M., Minasian, R., Petrini, M., Tomasiello, A.: Generalized structures of N = 1 vacua. http:// arxiv.org/list/hep-th/0505212, 2005 7. Hitchin, N.: Quart. J. Math. Oxford Ser. 54, 281 (2003) 8. Grana, M., Minasian, R., Petrini, M., Tomasiello, A.: Supersymmetric backgrounds from generalized Calabi–Yau manifolds. JHEP 0408, 046 (2004) 9. Dall’Agata, G.: On supersymmetric solutions to IIB supergravity with general fluxes. Nucl. Phys. B 695, 243 (2004) 10. Frey, A.: Notes on SU (3) structures in type IIB supergravity. JHEP 0406, 027 (2004) 11. Bär, Ch., Dahl, M.: Surgery and the Spectrum of the Dirac Operator. J. Reine Angew. Math. 552, 53– 76 (2002) 12. Reid, M.: The moduli space of 3-Folds with K = 0 may nevertheless be irreducible. Math. Ann. 278, 329– 334 (1987) 13. For recent discussions of supersymmetric non-geometric vacua, see: Shelton, J., Taylor, W., Wecht, B.: Nongeometric flux compactifications. JHEP 0510 085 (2005), and references therein 14. Greene, B.R.: String theory on Calabi–Yau manifolds. http://arxiv.org/list/hep-th/9702155, 1997 15. Aspinwall, P.S., Greene, B.R., Morrison, D.R.: Calabi–Yau moduli space, mirror manifolds and spacetime topology change in string theory. Nucl. Phys. B 416, 414 (1994) 16. Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B 403, 159 (1993) 17. Morrison, D.R., Vafa, C.: Compactifications of F-Theory on Calabi–Yau Threefolds - I. Nucl. Phys. B 473, 74 (1996) 18. Louis, J., Sonnenschein, J., Theisen, S., Yankielowicz, S.: Non-perturbative properties of heterotic string vacua compactified on K 3 × T 2 . Nucl. Phys. B 480, 185 (1996) 19. Michelson, J.: Compactifications of type IIB strings to four dimensions with non-trivial classical potential. Nucl. Phys. B 495, 127 (1997) 20. See for instance: Andrianopoli, L., Ferrara, S., Trigiante, M.: Fluxes, supersymmetry breaking, and gauged supergravity. Contribution to the proceedings of the “Sugrazo” Conf. (Northeastern Univ. Phys. 2003) http://arxiv.org/list/hep-th/0307139, 2003 and references therein 21. Seiberg, N.: IR dynamics on branes and space-time geometry. Phys. Lett. B 384, 81 (1996), Seiberg, N., Witten, E.: Gauge dynamics and compactification to three dimensions. In: Saclay 1996: The mathematical beauty of physics, Adv. Ser. Math. Phys. 29. River Edge, NJ: World Scientific, 1997 pp. 333–366 22. Seiberg, N., Shenker, S.: Hypermultiplet moduli space and string compactification to three dimensions. Phys. Lett. B 388, 521 (1996) 23. Greene, B., Morrison, D., Vafa, C.: A geometric realization of confinement. Nucl. Phys. B 481, 513 (1996); Ooguri, H., Vafa, C.: Summing up D instantons. Phys. Rev. Lett. 77, 3296 (1996) 24. Candelas, P., Green, P.S., Hübsch, T.: Rolling Among Calabi–Yau Vacua. Nucl. Phys. B 330, 49 (1990); Finite Distances Between Distinct Calabi–Yau Vacua: (Other Worlds Are Just Around The Corner). Phys. Rev. Lett. 62, 1956 (1989) 25. Greene, B., Morrison, D., Strominger, A.: Black hole condensation and the unification of string vacua. Nucl. Phys. B 451, 109 (1995) 26. Chiang, T.M., Greene, B.R., Gross, M., Kanter, Y.: Black hole condensation and the web of Calabi–Yau manifolds. Nucl. Phys. Proc. Suppl. 46, 82 (1996) 27. Hirzebruch, F.: Some examples of threefolds with trivial canonical bundle. Notes by Werner, J., Max Planck Inst. preprint no.85–58, Bonn, 1985
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28. 29. 30. 31. 32.
Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc. 120, 286–294 (1965) Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985) van Geemen, B., Werner, J.: New examples of threefolds with c1 = 0. Math. Z. 203, 211–225 (1990) Werner, J.: Kleine Auflösungen spezieller dreidimensionaler Varietäten. Ph. D. Thesis, Bonn Gurrieri, S., Louis, J., Micu, A., Waldram, D.: Mirror symmetry in generalized Calabi–Yau compactifications. Nucl. Phys. B 654, 61 (2003) Grana, M., Louis, J., Waldram, D.: Hitchin functionals in N = 2 supergravity. JHEP 01, 008 (2006) Fidanza, S., Minasian, R., Tomasiello, A.: Mirror symmetric SU(3)-structure manifolds with NS fluxes. Commun. Math. Phys. 254, 401 (2005) Friedman, R.: Simultaneous resolution of threefold double points. Math. Ann. 274, 671 (1986) Tian, G.: Smoothing 3–folds with trivial canonical bundle and ordinary double points. In: Essays on Mirror Manifolds S.-T. Yau, ed., Hong Kong: International Press, 1992, pp. 458–479 Lu, P., Tian, G.: The complex structures on connected sums of S 3 × S 3 . In: Manifolds and geometry, Cambridge: Cambridge University Press, 1993 Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G2 structures. Proc. Conf. Differential Geometry Valencia 2001, http://arxiv.org/list/math.DG/0202282, 2002 Gauntlett, J.P., Martelli, D., Pakis, S., Waldram, D.: G-structures and wrapped NS5-branes. Commun. Math. Phys. 247, 421 (2004) Kachru, S., Schulz, M., Tripathy, P., Trivedi, S.: New supersymmetric string compactifications. JHEP 0303, 061 (2003) See for instance Bott, R., Tu, L.: Differential Forms in Algebraic Topology. Berlin-Heidelberg, New York: Springer, 1982, pp. 65–69 Bousso, R., Polchinski, J.: Quantization of four-form fluxes and dynamical neutralization of the cosmological constant. JHEP 0006, 006 (2000); Giddings, S.B., Kachru, S., Polchinski, J.: Hierarchies from fluxes in string compactifications. Phys. Rev. D 66, 106006 (2002); Maloney, A., Silverstein, E., Strominger, A.: de Sitter space in noncritical string theory. http://arxiv.org/list/hep-th/0205316, 2002; Kachru, S., Kallosh, R., Linde, A., Trivedi, S.: de Sitter vacua in string theory. Phys. Rev. D 68, 046005 (2003); Susskind, L.: The anthropic landscape of string theory. http://arxiv.org/list/hep-th/0302219, 2003; Douglas, M.: The statistics of string/M theory vacua. JHEP 0305, 046 (2003)
33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
Communicated by M.R. Douglas
Commun. Math. Phys. 274, 795–819 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0298-z
Communications in
Mathematical Physics
One-Parameter Continuous Fields of Kirchberg Algebras Marius Dadarlat1, , George A. Elliott2, 1 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.
E-mail: [email protected]
2 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3.
E-mail: [email protected] Received: 2 September 2006 / Accepted: 28 February 2007 Published online: 17 July 2007 – © Springer-Verlag 2007
Abstract: We prove that all unital separable continuous fields of C*-algebras over [0, 1] with fibers isomorphic to the Cuntz algebra On (2 ≤ n ≤ ∞) are trivial. More generally, we show that if A is a separable, unital or stable, continuous field over [0, 1] of Kirchberg C*-algebras satisfying the UCT and having finitely generated K-theory groups, then A is isomorphic to a trivial field if and only if the associated K-theory presheaf is trivial. For fixed d ∈ {0, 1} we also show that, under the additional assumption that the fibers have torsion free K d -group and trivial K d+1 -group, the K d -sheaf is a complete invariant for separable stable continuous fields of Kirchberg algebras. 1. Introduction A separable nuclear purely infinite simple C*-algebra is called a Kirchberg algebra. In this paper we shall consider separable unital or stable continuous fields of Kirchberg algebras over the unit interval. We shall prove an approximation result, Theorem 6.1, and an inductive limit representation, Theorem 6.2, for these fields. These results lead us to Theorem 7.3 (a triviality result for unital On -fields), Theorem 7.5 (a triviality result for fields whose associated K -theory presheaf is isomorphic to the presheaf of a trivial field), and Theorem 8.2 (a classification result based on the K d -theory sheaf). The fields classified by Theorem 8.2 may have a rather complicated structure and can fail to be locally trivial at any point in [0, 1], as illustrated by Example 8.4. We shall rely heavily on the classification theorem (and related results) of Kirchberg and Phillips [24], and on the work on non-simple nuclear purely infinite C*-algebras of Blanchard and Kirchberg [6] and Rørdam and Kirchberg [15, 16]. The results of Blackadar [2] and Spielberg [28] on the semiprojectivity of Kirchberg algebras and the results of Spielberg [27] and Lin [19] on the weak semiprojectivity of Kirchberg algebras also play an important role. M.D. was supported in part by NSF Grant #DMS-0500693.
G.A.E. held a Discovery Grant from NSERC Canada.
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There are two key ideas on which we base our approach. The first is to approximate continuous fields by what we shall call elementary fields—i.e., fields which are locally trivial at all but finitely many points; see Sect. 6. The second is to introduce the notion of fibered morphism of fields, a natural blow-up construction based on the usual notion of morphism of fields; see Sect. 5. The triviality of O2 -stable continuous fields was announced by Kirchberg [14] in a vastly more general context. An isomorphism theorem for continuous fields of Kirchberg algebras over zero dimensional spaces was given in [10]. One word concerning the terminology: in many of our statements, we will refer to the C*-algebra of continuous sections associated to a continuous field of C*-algebras over a compact space X as a continuous C*-bundle over X . This is consistent with the terminology used in [4, 17 and 5]. 2. Semiprojective Algebras Recall that a separable C*-algebra D is weakly semiprojective if for any finite subset F ⊂ D and any ε > 0, any C*-algebra B, any increasing sequence (Jn ) of (closed, two-sided) ideals of B and any ∗-homomorphism ι : D → B/J (where J is the closure of n Jn ) there is a ∗-homomorphism ϕ : D → B/Jn (for some n) such that πn ϕ(c) − ι(c) < ε for all c ∈ F (where πn : B/Jn → B/J is the natural map). If we assume that there is ϕ such that πn ϕ = ι, then A is semiprojective. We shall use (weak) semiprojectivity in the following context (see Sect. 3 for terminology). Let B be a continuous C*-bundle over a compact metric space X , let x ∈ X and consider the sets Un = {y ∈ X : d(y, x) ≤ 1/n}. Then Jn = CUn (X )B is an increasing sequence of ideals of B such that B/Jn ∼ = B(x). Here CUn (X ) denotes the ideal = B(Un ) and B/J ∼ of C(X ) consisting of all continuous functions that vanish on Un . Let us recall that all unital Kirchberg algebras A with K ∗ (A) finitely generated and satisfying the UCT are weakly semiprojective by a result of Spielberg [27] and Lin [19]. If moreover K 1 (A) is torsion free, then A is semiprojective by a result of Spielberg [28]. Blackadar showed that a Kirchberg algebra A is semiprojective if and only if A ⊗ K is semiprojective [2]. Similarly, a Kirchberg algebra A is weakly semiprojective if and only if A ⊗ K is weakly semiprojective [8]. Let A be a C*-algebra, a ∈ A and F, G ⊆ A. If ε > 0, we write a ∈ε F if there is b ∈ F such that a − b < ε. Similarly, we write F ⊂ε G if a ∈ε G for every a ∈ F. Proposition 2.1 [12, Thms. 3.1, 4.6]. Let D be a separable weakly semiprojective C*algebra. For any finite subset F ⊂ D and any ε > 0 there exist a finite subset G ⊂ D and δ > 0 such that for any C*-algebra B ⊂ A and any ∗-homomorphism ϕ : D → A with ϕ(G) ⊂δ B, there is a ∗-homomorphism ψ : D → B such that ϕ(a) − ψ(a) < ε for all a ∈ F. The following two results appear in Loring’s book [20], except that he makes a certain assumption that is not necessary. Specifically, the assumption that the semiprojective algebras are finitely presented can be removed. While the proofs of the sharpened results do not really require new ideas, the consequences are quite useful. Although it is not clear whether or not O∞ or other semiprojective Kirchberg algebras are finitely presented in Loring’s sense, Propositions 2.2 and 2.3 now allows us to deal with arbitrary semiprojective Kirchberg algebras. Proposition 2.2. Let A be a separable semiprojective C*-algebra and let (xi ) be a sequence dense in the unit ball of A. For any n ≥ 1 and ε > 0 there are m ≥ 1 and
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δ > 0 such that for any diagram =B {{ { { π {{ {{ σ / B/J A ψ
of C*-algebras and ∗-homomorphisms (J an ideal of B and π the quotient map) such that π ψ(xi ) − σ (xi ) < δ for all 1 ≤ i ≤ m, there is a ∗-homomorphism ϕ : A → B such that π ϕ = σ and ϕ(xi ) − ψ(xi ) < ε for all 1 ≤ i ≤ n. Proposition 2.3. Let A be a separable semiprojective C*-algebra and let (xi ) be a sequence dense in the unit ball of A. For any n ≥ 1 and ε > 0 there are m ≥ 1 and δ > 0 such that for any C*-algebra B and any two ∗-homomorphisms ϕ, ψ : A → B such that ϕ(xi ) − ψ(xi ) < δ for all 1 ≤ i ≤ m, there is a homotopy χt : A → B, t ∈ [0, 1], of ∗-homomorphisms from ϕ to ψ that satisfies ϕ(xi ) − χt (xi ) < ε, for all 1 ≤ i ≤ n and all t ∈ [0, 1]. Propositions 2.2 and 2.3 remain valid if one requires that all C*-algebras and morphisms are unital. Proposition 2.4. Let A be a semiprojective Kirchberg algebra. Let B be a separable C*-algebra with a full projection such that B ∼ = B ⊗ O∞ ⊗ K, let J be a proper ideal of B and let π : B → B/J denote the quotient map. Let σ : A → B/J be a full ∗-homomorphism. Assume that there is α ∈ K K (A, B) such that [π ]α = [σ ] in K K (A, B/J ). Then there is a full ∗-homomorphism ϕ : A → B such that [ϕ] = α and πϕ = σ. Proof. Let (xi ) be a sequence dense in the unit ball of A. By Proposition 2.3 there are n ≥ 1 and ε > 0 such that any two ∗-homomorphisms ϕ, ψ1 : A → B satisfying ϕ(xi ) − ψ1 (xi ) < ε, for all 1 ≤ i ≤ n, are homotopic. Let m and δ be as in Proposition 2.2. By [24, Thm. 8.2.1] there is a full ∗-homomorphism ψ : A → B such that [ψ] = α. Since [π ψ] = [σ ] in K K (A, B/J ) and both maps are full, it follows from [24, Thm. 8.2.1] that there is a unitary v ∈ M(B/J ) such that π ψ(xi ) − v ∗ σ (xi )v < δ, for all 1 ≤ i ≤ m. Since B is stable, so is B/J [24]. Hence the unitary group of M(B/J ) is path connected. It follows that v lifts to a unitary u ∈ M(B). The map ψ1 = uψu ∗ is an approximate lifting of σ in the sense that π ψ1 (xi ) − σ (xi ) < δ for all 1 ≤ i ≤ m. By Proposition 2.2 there is a ∗-homomorphism ϕ : A → B lifting σ and such that ϕ(xi ) − ψ1 (xi ) < ε for all 1 ≤ i ≤ n. By Proposition 2.3, ϕ is homotopic to ψ1 and hence [ϕ] = [ψ1 ] = [ψ] = α. Moreover ϕ is full, since for every nonzero projection e ∈ A, ϕ(e) is homotopic to the full projection uψ(e)u ∗ .
The following result was proved by H. Lin [18] in the case that A is a separable nuclear unital C*-algebra. While the result extends to exact C*-algebras (as stated below; the proof is similar) the nuclear case is all we shall need in the present paper. Theorem 2.5. Let A be a separable exact unital C*-algebra and let B be a unital C*algebra. Let ϕ, ψ : A → B be two unital nuclear full ∗-monomorphisms with [ϕ] = [ψ] in K K nuc (A, B). Let j : B → B ⊗ O∞ be defined by j (b) = b ⊗ 1. Then j ◦ ϕ is approximately unitarily equivalent to j ◦ ψ.
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Let G ⊂ A and let δ > 0. A map ϕ : A → B is called (G, δ)-multiplicative if ϕ(a)ϕ(b) − ϕ(ab) < δ for all a, b ∈ G. Theorem 2.6. Let A be a unital Kirchberg C*-algebra. Suppose that K ∗ (A) is finitely generated and that A satisfies the UCT. Then for any finite subset F of A and any ε, ε > 0, there are a finite subset G of A and δ > 0 with the following property. For any unital Kirchberg C*-algebra B, any (G, δ)-multiplicative unital completely positive map ϕ : A → B and any unitary w ∈ B with spectrum δ-dense in T and satisfying [ϕ(a), w] < δ, for all a ∈ G, there is a unital full ∗-homomorphism φ : C(T) ⊗ A → B such that φ(1 ⊗ a) − ϕ(a) < ε for all a ∈ F, and φ(z ⊗ 1) − w < ε. If in addition K K (S A, B) = 0, then there is a continuous path (wt )t∈[0,1] of unitaries in B with w0 = 1, w1 = w and [ϕ(a), wt ] < ε , for all a ∈ F and all t ∈ [0, 1]. For the second part of the theorem the condition that the spectrum of w is δ-dense in T is not needed. Proof. Suppose, to obtain a contradiction, that there are a finite subset F of the unit ball of A and ε > 0 for which no G and δ can be found satisfying the conclusion of the theorem. Choose a sequence (an ) dense in the unit ball of A. In particular, for each n, the finite set Gn = {a1 , . . . , an } and the tolerance δn = 1/n will not do. In other words, for each n there exist a unital Kirchberg algebra Bn , a unital completely positive map ϕn : A → Bn which is (Gn , δn )-multiplicative, and a unitary wn with [wn , ϕn (ai )] < δn for 1 ≤ i ≤ n, with the spectrum of wn δn -dense in T, such that the pair (ϕn , wn ) cannot be approximated as in the statement. The sequence (ϕn ) defines a unital ∗-homomorphism ϕ∞ : A → ∞ (Bn )/c0 (Bn ). The sequence (wn ) defines a unitary w∞ ∈ ∞ (Bn )/c0 (Bn ) which commutes with the image of ϕ∞ . Thus we obtain a ∗-homomorphism ∞ : C(T) ⊗ A → ∞ (Bn )/c0 (Bn ). One verifies immediately that ∞ is a unital full ∗-monomorphism. Indeed, if I is a nonzero ideal of C(T) ⊗ A, let us show that ∞ (I ) is contained in no proper ideal of ∞ (Bn )/c0 (Bn ). Since A is simple, I = C F (T) ⊗ A for some proper closed subset F of T, where C F (T) = { f ∈ C(T) : f | F = 0}. Let g ∈ C F (T) with 0 ≤ g ≤ 1 be such that g(t) = 1 on some arc c disjoint from F. It suffices to show that ∞ (g ⊗ 1) is full in
∞ (Bn )/c0 (Bn ). Note that (g(wn )) is a lifting of ∞ (g ⊗ 1) to ∞ (Bn ). Since there is n 0 such that the spectrum of wn intersects the arc c for n ≥ n 0 , we have g(wn ) = 1 for n ≥ n 0 . Since Bn is unital simple and purely infinite, by [24, Lemma 4.1.7] we find bn ∈ Bn of norm at most two with bn g(wn )bn∗ = 1 Bn , for all n ≥ n 0 . Thus (g(wn )) is full in ∞ (Bn ) and hence so is its image in ∞ (Bn )/c0 (Bn ). Since the extension 0 → c0 (Bn ) → ∞ (Bn ) → ∞ (Bn )/c0 (Bn ) → 0 K( ∞ (Bn )/c0 (Bn ))
(1)
is quasidiagonal, the boundary map ∂ : → K(c0 (Bn )) vanishes by [7, Thm. 8]. By the UMCT of [9], if D is a separable C*-algebra that satisfies the UCT and such that K ∗ (D) is finitely generated, then there is a natural isomorphism K K (D, E) ∼ = Hom (K(D), K(E)),
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for any σ -unital C*-algebra E. Under this isomorphism, the boundary map K K (D, ∞ (Bn )/c0 (Bn )) → K K (S D, c0 (Bn )) corresponds to composition with ∂, so that it also vanishes. Therefore, by the six-term exact sequence in KK-theory, the map K K (D, ∞ (Bn )) → K K (D, ∞ (Bn )/c0 (Bn )) is surjective. Using Kirchberg’s theorem [24, Thm. 8.3.3], and thehypothesis that Bn are Kirchberg algebras, we verify immediately that the natural map n Hom(D, Bn ) → Hom(D, ∞ (Bn )) induces a surjection n K K (D, Bn ) → K K (D, ∞ (Bn )). In view of the above discussion, we obtain a surjective map K K (C(T) ⊗ A, Bn ) → K K (C(T) ⊗ A, ∞ (Bn )/c0 (Bn )). n
By applying [24, Thm. 8.3.3] again, we find a sequence of unital ∗-monomorphisms φn : C(T) ⊗ A → Bn which induces a unital full ∗-monomorphism ∞ : C(T) ⊗ A →
∞ (Bn )/c0 (Bn ) and which has the same KK-theory class as ∞ . Therefore, by Theorem 2.5, j ◦ ∞ , j ◦ ∞ : C(T) ⊗ A → ( ∞ (Bn )/c0 (Bn )) ⊗ O∞ are approximately unitarily equivalent. In particular, there is a unitary v ∈ ( ∞ (Bn )/c0 (Bn )) ⊗ O∞ such that v j ◦ ∞ (a) v ∗ − j ◦ ∞ (a) < ε, (2) for all a ∈ {1 ⊗ b : b ∈ F} ∪ {z ⊗ 1}, where z denotes the identity map of T. Since the extension (1) is quasidiagonal, we can lift v to a unitary in ∞ (Bn ) ⊗ O∞ given by a sequence of unitaries vn ∈ Bn ⊗O∞ . By a result of Kirchberg (see also [24, Thm. 8.4.1]), for each n, there is an isomorphism νn : Bn ⊗ O∞ →Bn , whose composition with the inclusion map jn : Bn → Bn ⊗ O∞ is approximately unitarily equivalent to id Bn . Using this in conjunction with (2), we find unitaries u n ∈ Bn such that lim sup u n φn (1 ⊗ a) u ∗n − ϕn (a) < ε, n→∞ lim sup u n φn (z ⊗ 1) u ∗n − wn < ε, n→∞
for all a in F. Letting φn (−) = u n φn (−) u ∗n , we have lim sup φn (1 ⊗ a) − ϕn (a) < ε, n→∞
lim sup φn (z ⊗ 1) − wn < ε, n→∞
which produces a contradiction. Consider now the second part of the theorem. Once again we will assume that the conclusion is false for some finite set F ⊂ A and ε > 0 and will seek a contradiction. Let α : [0, +∞) → [0, +∞) be a continuous function with α(0) = 0 which will be specified later. There is ε > 0 such that α([0, ε]) ⊂ [0, ε ). Let (ϕn ), Gn , δn and (wn ) be as in the first part of the proof, except that we make no assumptions on the spectrum K n of the unitary wn . After passing to a subsequence of (wn ) we may assume that the sequence of compact subsets (K n ) converges in the Gromov-Hausdorff distance to a nonempty compact subset K of T by [23, Prop. 7.2]. If K = T, then we apply the first
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part of the theorem to ϕ = ϕn and w = wn for some sufficiently large n. Thus, we obtain a unital ∗-monomorphism φ : C(T) ⊗ A → B such that φ(1 ⊗ a) − ϕ(a) < ε for all a ∈ F, and φ(z ⊗ 1) − w < ε. Observe that if K K (S A, B) = 0, then the canonical injection K K (A, B) → K K (C(T) ⊗ A, B) is bijective. Thus, if ν : B ⊗ O∞ → B is an isomorphism as above, then the class of [φ] in K K (C(T) ⊗ A, B) is equal to the class of some unital ∗-monomorphism φ of the form ν ◦ (θ ⊗ ψ ) for some ∗-homomorphisms ψ : A → B and θ : C(T) → O∞ . By Phillips’s theorem [24, 8.2.1], φ is approximately unitarily equivalent to φ . Without any loss of generality, we may assume that φ is unitarily equivalent to φ . Set φ(z ⊗ 1) = u 1 . Since the unitary group of O∞ is path connected, we find a continuous path of unitaries (u t )t∈[0,1] in O∞ joining u 1 with 1, satisfying [φ(1 ⊗ a), u t ] = 0 for all a ∈ A and t ∈ [0, 1]. It follows that [ϕ(a), u t ] < 2ε for all a ∈ A and t ∈ [0, 1]. If vt = (1 − t)w + tu 1 , then vt |vt |−1/2 is a path of unitaries in B joining w to u 1 . Then the juxtaposition of this path with u t gives a path wt from w to 1 with [ϕ(a), wt ] < α(ε), for all a ∈ F, and t ∈ [0, 1]. Here α is a universal non-negative continuous function with α(0) = 0 (recall that a ≤ 1 for all a ∈ F). Since α(ε) < ε we ran into a contradiction. It remains to argue the case when K is a proper subset of T. After dropping finitely many wn ’s we may assume that there is some fixed closed neighborhood V of K in T such that V = T and each K n is contained in V . By functional calculus we find a sequence of selfadjoint elements h n ∈ Bn such that limn→∞ [ϕn (a), h n ] = 0 for all a ∈ A, wn = exp(i h n ) and supn h n < ∞. Let us set vn (t) = exp(ith n ). One shows immediately that for some sufficiently large n, [ϕn (a), vn (t)] < ε , for all a ∈ F and all t ∈ [0, 1], which gives a contradiction, since vn (0) = 1 and vn (1) = wn .
Corollary 2.7. Let A be a unital Kirchberg C*-algebra. Suppose that K ∗ (A) is finitely generated and that A satisfies the UCT. Then for any finite subset F of A ⊗ K and any ε > 0 there are a finite subset G of A ⊗ K and δ > 0 with the following property. For any unital Kirchberg C*-algebra B with K K (S A, B) = 0, any full ∗-homomorphism ϕ : A ⊗ K → B ⊗ K and any unitary w ∈ M(B ⊗ K) satisfying [ϕ(a), w] < δ, for all a ∈ G, there is a continuous path (wt )t∈[0,1] of unitaries in M(B ⊗ K) with w0 = w, w1 = 1 and [ϕ(a), wt ] < ε, for all a ∈ F and all t ∈ [0, 1]. Proof. We may assume that F ⊂ A ⊗ eKe for some projection e ∈ K. Let δ > 0 and G ⊂ A ⊗ eKe (with 1 A ⊗ e ∈ G) be given by the second part of Theorem 2.6 applied to A ⊗ eKe, F and ε. Let us choose δ > 0 small enough such that if [ϕ(a), w] < δ for all a ∈ G, then there is a unitary v ∈ M(B ⊗ K) such that v commutes with f = ϕ(1 A ⊗ e) and v is sufficiently close to w so that [ϕ(a), v] < δ for all a ∈ G and there is a
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continuous path of unitaries (ωt ) from w to v such that [ϕ(a), ωt ] < ε for all a ∈ F. In particular [ϕ(a), f v f ] < δ for all a ∈ G. By Theorem 2.6, there is a continuous path (yt ) of unitaries in f (B ⊗K) f joining f v f to f such that [ϕ(a), yt ] < ε for all a ∈ F. Let us argue that there is a continuous path (z t ) of unitaries in (1 − f )M(B ⊗ K)(1 − f ) joining (1 − f )v(1 − f ) to 1 − f . Indeed, since f ∈ B ⊗ K, (1 − f )H B ∼ = H B by Kasparov’s absorption theorem [13] (where H B is the Hilbert B-module B ⊕ B ⊕ · · · ), (1 − f )M(B ⊗ K)(1 − f ) ∼ = L(H B ) ∼ = M(B ⊗ K). = L((1 − f )H B ) ∼ Since the unitary group of M(B ⊗ K) is path connected, we have verified the existence of the path (z t ). Finally let us observe that the juxtaposition of the paths (ωt ) and (yt + z t ) gives a continuous path of unitaries (wt ) in M(B ⊗ K) such that the path t → wt has the desired properties.
3. C*-Bundles Let X be a compact Hausdorff space. A C*-bundle A over X is a C*-algebra A endowed with a unital ∗-monomorphism from C(X ) to the center of the multiplier C*-algebra M(A) of A. If Y ⊆ X is a closed subset, we let CY (X ) denote the ideal of C(X ) consisting of functions vanishing on Y . Then CY (X )A is a closed two-sided ideal of A. The quotient of A by this ideal is a C*-bundle denoted by A(Y ) and called the restriction of A = A(X ) to Y . The quotient map is denoted by πY : A(X ) → A(Y ). If Z is a closed subset of Y we have a natural restriction map π ZY : A(Y ) → A(Z ) and π Z = π ZY ◦ πY . If Y reduces to a point x, we write A(x) for A({x}) and πx for π{x} . The C*-algebra A(x) is called the fiber of A at x. The image πx (a) ∈ A(x) of a ∈ A is denoted by a(x). For any a ∈ A, the map x → a(x) is upper semi-continuous. If the map x → a(x) is continuous for all a ∈ A, then A is called a continuous C*-bundle. If ϕ : A → B is a morphism of C(X )-bundles and Y is a closed subset of X , then the induced map A(Y ) → B(Y ) is denoted by ϕY . Lemma 3.1. Let X be a compact Hausdorff space and let A be a continuous C*-bundle over X . If Y , Z are closed subsets of X , then M(A(Y ∪ Z )) is isomorphic to the pullback of the two restriction maps M(A(Y )) → M(A(Y ∩ Z )) and M(A(Z )) → M(A(Y ∩ Z )). π
π
/ A(Y ∩ Z ) o Proof. The pullback of the diagram A(Y ) A(Z ) , is isomorphic to A(Y ∪ Z ) by [11, Prop. 10.1.13]. The statement follows now from the description of the multiplier algebra of a pullback given by [21, Prop. 7.2]. Alternatively, one can derive the statement from [1, Thm. 3.3] which identifies M(A) with the set of sections (s(x))x∈X , s(x) ∈ M(A(x)) which are bounded and strictly continuous.
Proposition 3.2. Let A be a stable Kirchberg algebra. Suppose that K ∗ (A) is finitely generated and that A satisfies the UCT. Then for any finite subset F of A and any ε > 0 there are a finite subset G of A and δ > 0 with the following property. Let B be a stable continuous C*-bundle of Kirchberg algebras over [α, β] and let φ : A → B be a full ∗-homomorphism. Let z ∈ [α, β] and let w ∈ M(B(z)) be a unitary such that [πz φ(a), w] < δ, for all a ∈ G. Assume that K K (S A, B(z)) = 0. Then there is a neighborhood [z 1 , z 2 ] of z and there is a unitary W ∈ M(B) such that [φ(a), W ] < ε, for all a ∈ F, W (z) = w, and W (x) = 1 for all x ∈ [α, β]\[z 1 , z 2 ].
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Moreover one may arrange that [z 1 , z 2 ] is contained in a given neighborhood of z. Proof. If G and δ are as in Corollary 2.7, then there is a continuous path (wt )t∈[0,1] of unitaries in M(B(z)) with w0 = 1, w1 = w and [πz φ(a), wt ] < ε, for all a ∈ F. Let [z 1 , z 2 ] be a neighborhood of z. Since w1 = 1, the path (wt )t∈[0,1] lifts to a continuous path of unitaries in M(B[z 1 , z 2 ]), denoted by (t )t∈[0,1] , such that 1 = 1. After passing to a smaller neighborhood of z if necessary, we may arrange that [π[z 1 ,z 2 ] φ(a), t ] < ε, for all a ∈ F and t ∈ [0, 1]. Let h : [z 1 , z 2 ] → [0, 1] be a continuous map such that h(z) = 1 and h vanishes on {z 1 , z 2 }\{z}. Then the map h(x) (x), if x ∈ [z 1 , z 2 ], x → 1, if x ∈ [α, β] \ [z 1 , z 2 ], defines a unitary W in M(B) (by Lemma 3.1) which satisfies the conclusion of the proposition.
Corollary 3.3. Let A be a stable Kirchberg algebra satisfying the UCT and such that K ∗ (A) is finitely generated. Let Z = [z 1 , z 2 ] be an interval with z 1 < z 2 . Then for any finite subset F of C(Z ) ⊗ A and any ε > 0 there exist a finite subset G of A and δ > 0 with the following property. Let B be a stable continuous C*-bundle of Kirchberg algebras over Z and let φ, ψ : C(Z ) ⊗ A → B be two injective ∗-homomorphisms which are C(Z )-linear. Let u i ∈ M(B(z i )), i = 1, 2, and let v ∈ M(B) be unitaries such that u i φzi (a)u i∗ − ψzi (a) < δ, for all a ∈ G and i = 1, 2, vφ(a)v ∗ − ψ(a) < δ, for all a ∈ G.
(3) (4)
Assume that K K (S A, B(z i )) = 0, i = 1, 2. Then there is a unitary u ∈ M(B) such that u(z i ) = u i , i = 1, 2, and uφ(a)u ∗ − ψ(a) < ε, for all a ∈ F. Proof. Since both φ and ψ are C(Z )-linear it suffices to prove the statement for ε > 0 and F ⊂ A. Choose G and δ as in Proposition 3.2 such that δ < ε. Assume that (3) and (4) are satisfied for δ/2. Then [v(z i )∗ u i , φzi (a)] < δ, for all a ∈ G.
(5)
By applying Proposition 3.2 to both ends of Z , we find a unitary w ∈ M(B) such that w(z i ) = v(z i )∗ u i , i = 1, 2, and [φ(a), w] < ε, for all a ∈ F. Consider the unitary u = vw ∈ M(B). We have u(z i ) = u i , i = 1, 2, and uφ(a)u ∗ − ψ(a) ≤ v wφ(a)w∗ − φ(a) v ∗ ] + vφ(a)v ∗ − ψ(a) < ε + δ < 2ε for all a ∈ F, as desired.
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4. Invariants Let X be the unit interval. Let U be the set of all closed subintervals of X of positive length. We regard U as a category with morphisms given by inclusions. A presheaf on U consists of the following data: (a) An assignment to each U ∈ U of a set S(U ). (b) A collection of mappings (called restriction homomorphisms) r VU : S(U ) → S(V ) for each pair U , V in U such that V ⊂ U , satisfying (1) rUU = idU (the identity map), U = r V rU . (2) for W ⊂ V ⊂ U , r W W V If S and S are presheaves over U, then a morphism of presheaves α ∈ Hom(S, S ) is a collection of maps αU : S(U ) → S (U ) for each U ∈ U such that for V ⊂ U the following diagram commutes: S(U )
αU
r VU
S(V )
/ S (U ) r VU
αV
/ S (V )
(6)
In other words, S is a contravariant functor from the category U to the category of sets and α is a natural transformation of functors. In our examples, S will take values in a category that has some algebraic structure (e.g., abelian semigroups, groups, ordered groups, etc.) Naturally, we shall require that the restriction maps and the morphisms preserve the algebraic structure. Our main example is the presheaf S = Kd (A) associated to a continuous C*-bundle A on X : S(U ) = Kd (A)(U ) = K d (A(U )), r VU = K d (πVU ), where d ∈ {0, 1} is fixed and πVU : A(U ) → A(V ) is the natural quotient map. A morphism of C*-bundles over X , ϕ : A → B induces a morphism of presheaves Kd (ϕ) ∈ Hom(Kd (A), Kd (B)). Let us observe that we can extend S to singletons V = {x} by setting S(V ) = K d (A(x)) and r VU = K d (πxU ). The stalk of S at x ∈ X is the direct limit Sx = lim ◦ S(U ) with respect to the restriction maps (r VU ), where U runs through − →x∈U those elements of U which contain x in their interior. Since A(x) = lim ◦ A(U ), we − →x∈U can identify S({x}) with Sx , by continuity of K-theory. Consequently, if B is a C*bundle over X and α ∈ Hom(Kd (A), Kd (B)), then for any V = {x} ⊂ U we have a commutative diagram as in (6), where α{x} : S({x}) → S ({x}) corresponds to the map Sx → Sx induced by α, and S = Kd (B). Since a map α{x} which makes the diagram (6) commutative for all U with x ∈ U and V = {x} is uniquely determined by a morphism of presheaves α : Kd (A) → Kd (B), we see that for any morphism of C*-bundles ϕ : A → B, Kd (ϕ){x} = K d (ϕx ). We are going to verify that the presheaf S = Kd (A) on U satisfies the following properties which are similar to those of a sheaf. For every U ∈ U and any collection U1 , . . . , Un of elements of U with U = i Ui , (i) If s, t ∈ S(U ) and rUUi (s) = rUUi (t) for all i, then s = t.
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(ii) If si ∈ S(Ui ) for all i and if for Ui ∩ U j = ∅, we have rUUii∩U j (si ) = rUij∩U j (s j ), then there exists s ∈ S(U ) such that rUUi (s) = si for all i. Every sheaf S is isomorphic to the sheaf of sections of an etale bundle π : E S → X [29]. This is also the case with S = Kd (A). There is a natural map r xU : S(U ) → Sx . If ◦
s ∈ S(U ) and x ∈U , then sx := r xU (s) is called the germ of s at x. Consider the disjoint union ES = Sx x∈X
and denote by π : E S → X the natural projection which maps Sx to x. For each ◦
s ∈ S(U ) consider the section sˆ : U → E S defined by sˆ (x) = sx . The family of sets ◦
{ˆs (U ) : U ∈ U, s ∈ S(U )} forms a basis for a topology on E S . This topology glues together all the stalks (Sx )x∈X forming the space E S . The bundle π : E S → X is called etale since the continuous surjection π is a local homeomorphism. For U ∈ U let (U, E S ) denote the set of continuous sections from U to E S . Using the conditions (i) and (ii) (that we verify next) and the continuity of the K-theory one shows (as in the proof of [29, Thm. 2.2, Chap. II]) that the natural map S(U ) → (U, E S ), s → sˆ , is an isomorphism. Thus we are justified in calling Kd (A) a sheaf. Proposition 4.1. Let A be a continuous C*-bundle over X = [0, 1]. Fix d ∈ {0, 1} and assume that K d+1 (A(x)) = 0 for all fibers of A. Then Kd (A)(U ) = K d (A(U )) is a sheaf and the natural map Kd (A)(U ) → (U, E Kd (A) ) is an isomorphism of sheaves. Proof. Let us give the proof in the case d = 0, the case d = 1 being similar. We need to verify the conditions (i) and (ii) from above. It is clear that, given (Ui ), it suffices to verify (i) and (ii) for some cover of U which refines the cover (Ui ). Therefore we may assume that Ui ∩ Ui+1 reduces to a point, and that Ui ∩ U j = ∅ if |i − j| > 1. Set X i = U1 ∪ · · · ∪ Ui and note that K 1 (A(X i ∩ Ui+1 )) = K 1 (A(Ui ∩ Ui+1 )) = 0 by assumption. If Y, Z are closed subsets of U , one has the Mayer-Vietoris exact sequence ([26]):
K 0 (A(Y )) ⊕O K 0 (A(Z ))
/ K 0 (A(Y ∩ Z ))
ρ
K 0 (A(Y ∪ Z )) o
K 1 (A(Y ∩ Z )) o
/ K 1 (A(Y ∪ Z )) K 1 (A(Y )) ⊕ K 1 (A(Z ))
This yields the exact sequence 0
/ K 0 (A(X i+1 ))
ρ
/ K 0 (A(X i ))⊕ K 0 (A(Ui+1 ))
/ K 0 (A(X i ∩Ui+1 )), (7)
where ρ is the natural restriction map and = K 0 (π XXii∩Ui+1 ) − K 0 (π XUii+1 ∩Ui+1 ). One easily verifies (i) and (ii) by induction using the exactness of the sequence (7), at its first term for (i) and at its second term for (ii).
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5. Fibered Morphisms of C*-Bundles Let C be a class of Kirchberg algebras and set X = [0, 1]. Let us use the term admissible cover of X to mean closed intervals Yi = [a2i , a2i+1 ] and Z j = [a2 j+1 , a2 j+2 ], where 0 = a0 < a1 < · · · < a2m+1 = 1. Let us set Y = [a0 , a1 ] ∪ [a2 , a3 ] ∪ . . ., Z = [a1 , a2 ] ∪ [a3 , a4 ] ∪ . . . .; thus Y ∩ Z = {a1 , a2 , . . . , a2m }. For the sake of brevity let us refer to the above cover by {Y, Z }. Consider diagrams D of the form π
D
η
/Fo
(8)
E
based on an admissible cover {Y, Z } of X . Let us assume that D, E, F are locally trivial C*-bundles over Y , Z and Y ∩ Z respectively with fibers in C. More precisely, m−1 m D = i=0 C(Yi , Di ) and E = j=0 C(Z j , E j ), where Di , E j ∈ C. Let us also assume that F is the restriction of D to Y ∩ Z , so that
F = D(Y ∩ Z ) =
C(Yi ∩ Z j , Di ) =
Yi ∩Z j =∅
m−1
(Di ⊕ Di+1 ).
i=0
The map π : D → F is the restriction πY ∩Z : D(Y ) → D(Y ∩ Z ) and the map η : E → F is obtained as the composition E(Z )
πY ∩Z
γ
/ E(Y ∩ Z )
/ D(Y ∩ Z ) = F,
where γ is a monomorphism of C*-bundles. It is useful to denote the components of γ by γi,i = γa2i+1 : E i → Di and γi,i+1 = γa2i+2 : E i → Di+1 , and the corresponding components of η by ηi,i : C(Z i , E i ) → Di and ηi,i+1 : C(Z i , E i ) → Di+1 . The pullback of D is the continuous C*-bundle over X , PD = {(d, e) ∈ D ⊕ E : π(d) = η(e)}. Its fibers are isomorphic to E j on Z j and to Di on Yi \Z . Let us call a diagram D satisfying all the conditions described above admissible. Let us call a continuous C*-bundle A elementary if there is an admissible diagram D such that A ∼ = PD . For a fixed isomorphism ι : A → PD there is a commutative diagram A(Y )
π
/ A(Y ∩ Z ) o
ιY
π
ιY ∩Z
D
π
/Fo
η
A(Z ) E
ιZ
(9)
with the vertical maps monomorphisms of C*-bundles, such that the induced ∗-homomorphism A → PD can be identified with ι. An admissible diagram D comes with a closed cover of X , namely {Y, Z }. If A is a continuous C*-bundle over X , it is convenient to denote by D A the (not necessarily admissible) diagram A(Y )
π
/ A(Y ∩ Z ) o
π
A(Z ),
whose pullback is isomorphic to A by [11, Prop. 10.1.13]. Moreover if ϕ : A → B is a morphism of C*-bundles we denote by Dϕ the corresponding morphism of diagrams D A → D B, with components ϕY , ϕY ∩Z , ϕ Z . Thus, a continuous C*-bundle A
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is elementary if there is an admissible diagram D and a unital morphism of diagrams ι : D A → D which induces a ∗-isomorphism A → PD . In that case let us say that ι : D A → D is a fibered presentation of A. Let A, B be continuous C*-bundles over X such that A is elementary. A fibered morphism φ from A to B consists of a fibered presentation of A, ι : D A → D, together with injective morphisms of C*-bundles φY , φY ∩Z , φ Z such that the following diagram is commutative: π
A(Y )
/ A(Y ∩ Z ) o
ιY
π
A(Z )
ιY ∩Z
D
/Fo
π
φY
B(Y )
π
E
η
φY ∩Z
/ B(Y ∩ Z ) o
ιZ
φZ
B(Z ).
π
Equivalently, a fibered morphism from A to B is given by a triple (ι, D, φ), DA
ι
φ
/D
/ D B,
where ι : D A → D is a fibered presentation of A and φ = (φY , φY ∩Z , φ Z ). To simplify notation we will write φ for (ι, D, φ). Note that a fibered morphism φ induces a mor : A → B. A morphism ϕ of C*-bundles is called elementary phism of C*-bundles φ
. The set of all fibered morphisms if it is induced by a fibered morphism φ, i.e. ϕ = φ from A to B, corresponding to a given fibered presentation ι : D A → D of A will be denoted by HomD (A, B). There is a natural composition of fibered morphisms HomD (A, B) × HomD (B, C) → HomD (A, C),
) ◦ φ. In other words the components of ψ ◦ φ are (ψ ◦ φ)Y = (φ, ψ) → ψ ◦ φ = D( ψ
Y ◦ φY , (ψ ◦ φ)Y ∩Z = ψ
Y ∩Z ◦ φY ∩Z and (ψ ◦ φ) Z = ψ
Z ◦ φ Z . ψ Two fibered morphisms φ, ψ ∈ HomD (A, B) are approximately unitarily equivalent if there is a sequence of unitaries (u n ) in M(B) such that the sequence (πY (u n )) induces an approximate unitary equivalence between φY and ψY and the sequence (π Z (u n )) induces an approximate unitary equivalence between φ Z and ψ Z . It follows that φY ∩Z and ψY ∩Z are also approximately unitarily equivalent since they are restrictions of φY and ψY , respectively. It is obvious that if φ is approximately unitarily equivalent to ψ,
is approximately unitarily equivalent to ψ
. then φ Fix d ∈ {0, 1}. By a fibered K d -morphism from A to B, corresponding to a given fibered presentation ι : D A → D of A, let us mean a triple of maps α = (αY , αY ∩Z , α Z ), where αY has components αi : K d (D(Yi )) → K d (B(Yi )), α Z has components α j : K d (E(Z j )) → K d (B(Z j )), and αY ∩Z has components αi, j : K d (D(Yi ∩ Z j )) → K d (B(Yi ∩ Z j )), such that the following diagram is commutative: K d (A(Y ))
π∗
ι∗
K d (D)
π∗
ι∗
π∗
αY
K d (B(Y ))
/ K d (A(Y ∩ Z )) o / K d (F) o
ι∗
η∗
αY ∩Z
π∗
/ K d (B(Y ∩ Z )) o
K d (A(Z )) K d (E) αZ
π∗
K d (B(Z ))
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Let us summarize the above diagram by the notation K d (D A)
K d (ι)
/ K d (D)
α
/ K d (D B).
(10)
Note that αY ∩Z (K d (D(Yi ∩ Z j ))) ⊆ K d (B(Yi ∩ Z j )) and αY (K d (D(Yi ))) ⊆ K d (B(Yi )), α Z (K d (E(Z j ))) ⊆ K d (B(Z j )), by definition. The set of fibered K d -morphisms from A to B corresponding to a given fibered presentation ι : D A → D of A is denoted by Hom(K d (D), K d (D B)). If B and H are C*-bundles over X and D is as above let us denote by Hom(K d (D B), K d (D H )) the families of group morphisms βY = (βYi ), βY ∩Z = (βYi ∩Z j ), β Z = (β Z j ) which make the following diagram commutative: K d (B(Y ))
π
/ K d (B(Y ∩ Z )) o
π
/ K d (H (Y ∩ Z )) o
βY
K d (H (Y ))
π
βY ∩Z
K d (B(Z )) βZ
π
K d (H (Z ))
Note that despite the conspicuous notation, the role of D here is limited to giving an admissible cover. One has an obvious composition (α, β) → β ◦ α, Hom(K d (D), K d (D B)) × Hom(K d (D B), K d (D H )) → Hom(K d (D), K d (D H )). There is a natural restriction map Hom(Kd (B), Kd (H )) → Hom(K d (D B), K d (D H )), β → Dβ which depends only on the admissible cover {Y, Z } of [0, 1] that appears in the definition of D. This map takes the family of maps β = (βU ) to the subfamily Dβ consisting of those βU with U of the form Yi , Z j or Yi ∩ Z j . If U = {x}, βU stands for βx : K d (B(x)) → K d (H (x)). If D is a C*-algebra and B is a C*-bundle over X , then any morphism of groups α : K d (D) → K d (B) induces a morphism of presheaves
α : Kd (C(X, D)) → Kd (B), where for a closed subinterval U ⊂ X ,
αU : K d (C(U, D)) ∼ = K d (D) → K d (B(U )) is defined by
αU = (πU )∗ α. This observation extends to elementary C*-bundles: Proposition 5.1. Let A, B and H be continuous C*-bundles over [0, 1] whose fibers have vanishing K d+1 -groups. Assume moreover that A is an elementary C*-bundle given by a fibered presentation ι : D A → D. Then, there is a map Hom(K d (D), K d (D B)) → Hom(Kd (A), Kd (B)), α →
α , such that
) for all φ ∈ HomD (A, B), (a) K d (φ) = Kd (φ (b) ((Dβ) ◦ α) = β ◦
α for all α ∈ Hom(K d (D),K d (D B)) and β ∈ Hom(Kd (B), Kd (H )). Proof. Let α ∈ Hom(K d (D), K d (D B)). For each closed subinterval U of [0, 1] we shall construct a morphism of groups
αU : K d (A(U )) → K d (B(U )) as follows. Let us consider the diagram DU , D(Y ∩ U )
π
/ F(Y ∩ Z ∩ U ) o
η
E(Z ∩ U ),
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M. Dadarlat, G. A. Elliott
and note that it enjoys all the essential properties of an admissible diagram. The only possible differences are that Yi ∩ U or Z j ∩ U can reduce to boundary points of U (for at most two indices) and that the cover {Y ∩ U, Z ∩ U } of U may begin or end by a subinterval of the form Z j ∩ U rather than Yi ∩ U . We also consider the diagram DU B, π
B(Y ∩ U )
/ B(Y ∩ Z ∩ U ) o
π
B(Z ∩ U ).
Let I (U ) and J (U ) consist of those indices i and j such that Yi ∩ U = ∅ and respectively Z j ∩ U = ∅. The proof relies on three observations. First, we note that A(U ) is a pullback of DU . Second, we note that since K d (D(Yi ∩ U )) ∼ = K d (D(Yi )) ∼ = K d (Di ) if i ∈ I (U ) and K d (E(Z j ∩ U )) ∼ = K d (E(Z j )) ∼ = K d (E j ) if j ∈ J (U ), the diagram K d (DU ) can be naturally identified with the following full subdiagram of K d (D):
i∈I (U )K d (D(Yi ))
π
/
i∈I (U ), j∈J (U )K d (F(Yi ∩ Z j ))
η
o
j∈J (U )
K d (E(Z j )),
denoted by K d (D)U . Third, we observe that K d (B(U )) is isomorphic to the pullback of the diagram K d (DU B), as a consequence of the Mayer-Vietoris exact sequence and the assumption that all the fibers of B have vanishing K d+1 -groups. Let us denote by K d (D B)U the following full diagram of K d (D B):
i∈I (U )K d (B(Yi ))
π
/
i∈I (U ), j∈J (U )K d (B(Yi ∩ Z j ))
o
π
j∈J (U ) K d (B(Z j )).
The first observation gives a morphism of groups from K d (A(U )) to K d (PDU ) and hence to PK d (DU ) , the pullback of the diagram K d (DU ). The second observation allows us to restrict α to a map of diagrams K d (DU ) ∼ = K d (D)U → K d (D B)U → K d (DU B) (as explained in more detail below), giving a map PK d (DU ) → PK d (DU B) . By the third observation we can identify PK d (DU B) with K d (B(U )). Then we define
αU : K d (A(U )) → K d (B(U )) as the composition of the three maps from above. Finally it is easy to verify that the family
α = (
αU )U ∈U is a morphism of sheaves and that the properties (a) and (b) are satisfied. Let us verify the properties (a) and (b) using an explicit calculation of
α : Kd (A) → Kd (B). While this calculation is not really needed, we think that it is useful to compute the basic invariant Kd (A) at least for elementary C*-bundles. Since the fibers of A have vanishing K d+1 -groups, we may identify K d (A(U )) with PK d (DU ) by the Mayer-Vietoris exact sequence. Thus K d (A(U )) ∼ K d (Di ) ⊕ K d (E j ) : K d (γ ji )(e j ) = di }, = {(di , e j ) ∈ i∈I (U )
j∈J (U )
where the condition K d (γ ji )(e j ) = di is required for j ∈ {i − 1, i}. To simplify notation we let (di , e j ) stand for ((di )i∈I (U ) , (e j ) j∈J (U ) ). One can rewrite the calculation of K d (A(U )) in the following equivalent form. If J (U ) = ∅ then I (U ) = {i} for some i and K d (A(U )) ∼ = K d (Di ). If J (U ) = {s, . . . , s + k}, then K d (A(U )) ∼ = K d (E s ) if k = 0 and K d (A(U )) ∼ K d (E j ) : K d (γ j−1, j )(e j−1 ) = K d (γ j, j )(e j ), = {(e j ) ∈ j∈J (U )
j = s + 1, . . . , s + k}
One-Parameter Continuous Fields of Kirchberg Algebras
809
if k ≥ 1. Let x ∈ K d (A(U )) be viewed as an element of the pullback of the diagram Z αU (x) = K d (πYYii∩U )αi (di ), K d (π Z jj∩U )α j (e j ) viewed K d (DU ), x = (di , e j ). Then
as an element of the pullback of the diagram K d (DU B), where i runs in I (U ) and j
U ): runs in J (U ). To verify (a) we need to show that K d (φ)U = K d (φ Zj Yi K d (φ)U (x) = K d (πYi ∩U )K d (φYi )(di ), K d (π Z j ∩U )K d (φ Z j )(e j ) = K d (φYi ∩U )(di ), K d (φ Z j ∩U )(e j )
U )(x). = K d (φ To verify (b) we need to show that ((Dβ) ◦ α) U = βU ◦
αU . With notation as above we have Z ((Dβ) ◦ α)) U (x) = K d (πYYii∩U )βYi αi (di ), K d (π Z jj∩U )β Z j α j (e j ) Z = βYi ∩U K d (πYYii∩U )αi (di ), β Z j ∩U K d (π Z jj∩U )α j (e j ) αU (x). = βU ◦
6. Approximation and Inductive Limit Results The results in this section are useful for describing the structure of continuous C*-bundles of Kirchberg algebras. The following result applies to C*-bundles whose fibers are Kirchberg algebras satisfying the UCT. Indeed, such Kirchberg algebras are isomorphic to inductive limits of weakly semiprojective Kirchberg algebras (see [24]). Theorem 6.1. Let C be a class of unital weakly semiprojective Kirchberg algebras. Let A be a unital continuous C*-bundle over the unit interval such that all of its fibers are inductive limits of sequences of algebras in C. For any finite subset F ⊂ A and any ε > 0 there are an elementary continuous C*-bundle A1 with fibers in C and an elementary
(A1 ). A similar result is
: A1 → A such that F ⊂ε unital morphism of C*-bundles valid if one assumes that all the C*-algebras in C are stable rather than unital. Proof. We give the proof of the unital case. The stable case is entirely similar. Let F and ε be as in the statement. We must find points 0 = a0 < a1 < · · · < a2m+1 = 1 and C*-algebras Di , E j ∈ C (0 ≤ i ≤ m, 0 ≤ j ≤ m − 1) suchthat if we set Yi = [a2i , a2i+1 ], Z j = [a2 j+1 , a2 j+2 ] and D = i C(Yi , Di ), E = j C(Z j , E j ), Y = i Yi , Z = j Z j , then there are C*-bundle monomorphisms ϕ : D → A(Y ), ψ : E → A(Z ) such that πYZ∩Z ψ(E) ⊂ πYY∩Z ϕ(D) , (11) πY (F) ⊂ε ϕ(D), π Z (F) ⊂ε ψ(E). (12) Let A1 be the pullback of the maps πYY∩Z ϕ and πYZ∩Z ψ, A1 = {(d, e) ∈ D ⊕ E : πYY∩Z ϕ(d) = πYZ∩Z ψ(e)}. If D is defined by D
π
/Fo
η
E,
(13)
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M. Dadarlat, G. A. Elliott
where F = D(Y ∩ Z ) and η is obtained as the composition E(Z )
πY ∩Z
γ
/ E(Y ∩ Z )
/ D(Y ∩ Z ) = F,
∼ PD . Letting ϕ F be the restriction of ϕ to F, where γ (e) = (ϕ −1 ψ)|Y ∩Z (e), then A1 = we obtain a commutative diagram D
π
ϕ
A(Y )
π
/Fo
η
ϕF
/ A(Y ∩ Z ) o
E ψ
π
A(Z )
and hence a unital fibered morphism ∈ HomD (A1 , A). Using a partition of unity
: A1 → A satisfies one verifies immediately that the induced ∗-homomorphism
(A1 ) as a consequence of (12). Therefore it remains to construct D, E, ϕ and ψ F ⊂ε with properties as above. By hypothesis, if x ∈ [0, 1], then A(x) is an inductive limit of a sequence of C*-algebras in C with unital connecting maps. Therefore there are E x ∈ C and a unital ∗-homomorphism ı x : E x → A(x) such that πx (F) ⊂ε/2 ı x (E x ). Let H be a finite subset of E x such that for each a ∈ F there is h a ∈ H satisfying ı x (h a )−πx (a) < ε/2. Since E x is weakly semiprojective, there are a closed neighborhood Ux of x and a ∗-homomorphism ηx : E x → A(Ux ) such that πx ηx (h) − ı x (h) < ε/2 for all h ∈ H. Therefore, πx (ηx (h a ) − πUx (a)) = πx ηx (h a ) − πx (a) < ε for all a ∈ F. Using the semicontinuity of the norm for C*-bundles, after passing to a smaller neighborhood if necessary, we may arrange that ηx is unital and ηx (h a ) − πUx (a) < ε for all a ∈ F. In particular, πUx (F) ⊂ε ηx (E x ). By compactness of [0, 1], there are points 0 = y0 < y1 < · · · < ym = 1, C*-algebras E j ∈ C (0 ≤ j ≤ m − 1), unital ∗-homomorphisms η j : E j → A(U j ), where U j = [y j , y j+1 ], and finite sets F j ⊂ E j such that πU j (F) ⊂ε/2 η j (F j )
(14)
for all 0 ≤ j ≤ m − 1. Let G j ⊂ E j and δ j be given by Proposition 2.1 applied to E j for the input data F j and ε/2. Choose δ > 0 such that δ < min{δ0 , . . . , δm−1 }. By repeating the reasoning from above for the fibers A(yi ) we obtain mutually disjoint closed intervals Yi = [a2i , a2i+1 ] (0 ≤ i ≤ m), such that Yi is a neighborhood of yi and there are C*-algebras Di ∈ C and unital ∗-homomorphisms ϕi : Di → A(Yi ) such that πYi (F) ⊂ε ϕi (Di ) and πU j ∩Yi (η j (G j )) ⊂δ ϕi (Di ) for all i, j for which U j ∩ Yi is nonempty, i.e., for which j ∈ {i − 1, i}. Consider the C*-subbundle of A, B = {a ∈ A : πx (a) ∈ πx (ϕi (Di )), whenever x ∈ Yi , 0 ≤ i ≤ m}. By construction we have η j (G j ) ⊂δ B(U j ) for all j. Since each E j is weakly semiprojective, by Proposition 2.1 we can perturb η j to a unital ∗-homomorphism ψ j : E j → B(U j ) such that ψ j (a) − η j (a) < ε/2
(15)
for all a ∈ F j . Hence if we set Z j = [a2 j+1 , a2 j+2 ] ⊂ U j , then the sets in the family (Z j ) are mutually disjoint and π Z j ∩Yi ψ j (E j ) ⊂ π Z j ∩Yi ϕi (Di ) (whenever Z j ∩Yi = ∅). Extend ϕi : Di → A(Yi ) and ψ j : E j → A(Z j ) to maps of continuous C*-bundles and define ϕ and ψ as above. Then ϕ and ψ satisfy (11). Moreover, ϕ satisfies (12) since πYi (F) ⊂ε ϕi (Di ), and ψ satisfies (12) as a consequence of (14) and (15).
One-Parameter Continuous Fields of Kirchberg Algebras
811
Let A be a separable C*-algebra. A sequence (Ak ) of C*-subalgebras of A is called exhaustive if for any finite subset F of A and any ε > 0 there is k such that F ⊂ε Ak . If we further assume that each Ak is weakly semiprojective, then A is isomorphic to the inductive limit of a subsequence (Ak(n) ) of (Ak ), where the connecting maps are perturbations of the inclusion maps Ak(n) → A; see [20, 15.2.2]. Let us now show that the elementary continuous C*-bundles with semiprojective fibers satisfy a similar weak semiprojectivity property in the category of fibered morphisms. The following result applies to C*-bundles whose fibers are Kirchberg algebras satisfying the UCT and having torsion free K 1 -groups. Indeed, such Kirchberg algebras are isomorphic to inductive limits of semiprojective Kirchberg algebras (see [24]). Theorem 6.2. Let C be a class of unital semiprojective Kirchberg algebras. Let A be a separable unital continuous C*-bundle over the unit interval such that all of its fibers are inductive limits of sequences of algebras in C. There exists an induc k,k+1 ) consisting of elementary continuous C*-bundles with fibers tive system (Ak ,
k,k+1 ∈ Hom(Ak , Ak+1 ) such that in C and elementary morphisms of C*-bundles
k,k+1 ) ∼ lim(Ak , A. A similar result is valid if one assumes that all the C*-algebras = − → in C are stable rather than unital. Proof. We give the proof of the unital case. The stable case is entirely similar. By Theorem 6.1 there is a sequence of admissible diagrams (Dk ) and fibered unital morphisms
k,∞ (Ak ), then k,∞ ∈ HomDk (Ak , A), where Ak = PDk , such that if we set Bk = the sequence (Bk ) is exhaustive for A. Arguing as in the proof of [20, 15.2.2], we see that it suffices to prove a natural weak semiprojectivity property for elementary bundles and fibered morphisms that we describe below. Using that property we then perturb k,∞ to a fibered morphism k,n ∈ HomDk (Ak , Bn ) for some large n that depends on k and
n,∞ )−1 ◦ k,n to conclude the proof. then we set k,k+1 = Dk ( Let D be an admissible diagram (with components Di , E j semiprojective Kirchberg algebras and based on a cover {Y, Z } of X as above) and let π
D
/Fo
ϕ
A(Y )
π
η
ϕF
E ψ
π
/ A(Y ∩ Z ) o
A(Z )
be a commutative diagram with the vertical maps unital morphisms of C*-bundles. Then, we assert that for any finite sets F D ⊂ D, F E ⊂ E and any ε > 0 there are finite sets G D ⊂ D, G E ⊂ E and δ > 0 such that for any C*-subbundle B ⊂ A with ϕ(G D ) ⊂δ B(Y ) and ψ(G E ) ⊂δ B(Z ), there is a commutative diagram π
D ϕ
B(Y )
π
/Fo
η
ϕ F
/ B(Y ∩ Z ) o
E ψ
π
B(Z )
(16)
with the vertical maps unital morphisms of C*-bundles, such that ϕ(d) − ϕ (d) < ε for all d ∈ F D and ψ(e) − ψ (e) < ε for all e ∈ F E . Let us outline the proof of the above assertion. Using the semiprojectivity of Di and E j , for given F D , F E and ε we can find G D , G E and δ such that ϕ and ψ perturb to ϕ and ψ , by Proposition 2.1.
812
M. Dadarlat, G. A. Elliott
By defining ϕ F to be the restriction of ϕ to F, we arrange that the left square of the diagram (16) is commutative, whereas we only have that the right square is approximately commutative. However, since the degree of approximate commutativity of the right square can be controlled by choosing G D , G E and δ appropriately, we may invoke Proposition 2.2 to perturb ψ further to a ∗-homomorphism which makes the right square of (16) commutative and which approximates ψ as desired.
7. Unital C*-Bundles of Cuntz Algebras Proposition 7.1. Let A be an elementary continuous C*-bundle given by an admissible diagram as in (9) such that all its maps are unital and Di = E j = B for all i and j where B is a unital Kirchberg algebra. Assume that all the components of γ are K K -equivalences. Then A is isomorphic to the trivial C*-bundle C(X ) ⊗ B. Proof. Since X is contractible it suffices to prove that A is locally trivial. Therefore we may assume that the spaces Y and Z are closed intervals with Y ∩ Z consisting of a single point and that D = C(Y ) ⊗ B and E = C(Z ) ⊗ B. Then γ is a unital ∗-homomorphism γ : B → B which induces a KK-equivalence. For the sake of simplicity, let us assume that Y = [0, 1] and Z = [1, 2]. Since [γ ] ∈ K K (B, B)−1 , by the Kirchberg-Phillips theorem [24, Thm. 8.4.1 and Cor. 8.4.10], there is an automorphism θ : B → B and a continuous unitary-valued map t → u(t) ∈ B, with t ∈ [0, 1), such that lim u(t)θ (b)u(t)∗ − γ (b) = 0,
t→1
for all b ∈ B. By hypothesis A is isomorphic to {(g, h) ∈ C[0, 1] ⊗ B ⊕ C[1, 2] ⊗ B : g(1) = γ (h(1))}. One verifies immediately that the equation ι( f ) = (g, h), where g(t) =
u(t)θ ( f (t))u(t)∗ , if 0 ≤ t < 1, γ ( f (1)), if t = 1,
and h(t) = f (t) for 1 ≤ t ≤ 2, defines an isomorphism ι : C[0, 2] ⊗ B → A.
Proposition 7.2. Let n ∈ {2, 3, . . . , ∞} be fixed. If A is an elementary unital continuous C*-bundle over [0, 1] with all fibers isomorphic to On , then A is ∗-isomorphic to C[0, 1] ⊗ On . Proof. Since K ∗ (On ) = K 0 (On ) is cyclic and generated by the class of the unit, any unital ∗-homomorphism γ : On → On induces an automorphism of K ∗ (On ) and therefore [γ ] ∈ K K (On , On )−1 , since On satisfies the UCT. Therefore A is trivial by Proposition 7.1.
Theorem 7.3. Let n ∈ {2, 3, . . . , ∞} be fixed. Any separable unital continuous C*-bundle over an interval, or over a circle, with all fibers isomorphic to On is trivial.
One-Parameter Continuous Fields of Kirchberg Algebras
813
Proof. Let A be as in the statement. It suffices to prove that A is locally trivial. Indeed, if that is the case, then A is given by a principal Aut(On )-bundle. On the other hand Aut(On ) is path connected [22, Thm. 4.1.4]. Since all principal G-bundles over the circle are trivial whenever the structure group G is path-connected, our assertion is justified. To prove that A is locally trivial it suffices to prove that A is trivial if its spectrum is an interval. By Theorem 6.2 (applied with C = {On }) and Proposition 7.1, A is isomorphic to the limit of an inductive system (Ai , φi ), where Ai = C[0, 1] ⊗ On and φi : Ai → Ai+1 are unital ∗-homomorphisms. We assert that φi is approximately unitarily equivalent to the identity map of C[0, 1] ⊗ On . Indeed, since φi is C[0, 1]linear, it suffices to verify that the restriction of φi to On is approximately unitarily equivalent to the unital ∗-homomorphism which maps On onto the constant functions in C[0, 1] ⊗ On . This holds by [24, Thm. 8.2.1]. We conclude that A ∼ = C[0, 1] ⊗ On by Elliott’s intertwining argument [24, Cor. 2.3.3].
We need the following results of Blanchard, Kirchberg and Rørdam: Theorem 7.4. Let A be a separable continuous C*-bundle over a finite dimensional compact space with fibers which are Kirchberg algebras. If A is either stable or unital, then A ⊗ O∞ ∼ = A. A is stable if and only if each fiber is stable. Proof. By [6, Cor. 5.11], A ⊗ O∞ ⊗ K ∼ = A ⊗ K, and hence A ⊗ K is strongly purely infinite by [16, Thm. 9.1]. It follows that A is strongly purely infinite by [15, Prop. 5.11]. Therefore A ⊗ O∞ ∼ = A by [16, Thm. 8.6]. The last part of the statement is a result from [25].
We give now a trivialization result for C*-bundles A in terms of the K-theory presheaf K∗ (A), where for a closed interval V of [0, 1], K∗ (A)(V ) is the graded group K ∗ (A(V )). For unital C*-bundles we require that the morphisms of presheaves preserve the class of unit. Theorem 7.5. Let A be a separable unital continuous C*-bundle over [0, 1] the fibers of which are Kirchberg algebras satisfying the UCT. Assume that the K -theory presheaf of A is (unitally) isomorphic to the K -theory presheaf of C[0, 1] ⊗ D, for some unital Kirchberg algebra D with finitely generated K -theory groups and satisfying the UCT. Then A ∼ = C[0, 1] ⊗ D. A similar result is valid if one assumes that all the fibers of A are stable rather than unital. Proof. We give only the proof of the unital case as the stable case is entirely similar. By assumption, for every closed nondegenerate subinterval U of X = [0, 1] there is an isomorphism αU : K ∗ (D) = K ∗ (C(U, D)) → K ∗ (A(U )) such that the family (αU ) defines a morphism of presheaves. We assert that for any finite set F ⊂ A and ε > 0 there is a unital morphism of C*-bundles φ : C(X, D) → A such that K ∗ (φx ) : K ∗ (D) → K ∗ (A(x)) is bijective for each x ∈ X and F ⊂ε φ C(X, D) . Let us show how this assertion implies the theorem. Since K ∗ (C(X, D)) = K ∗ (D) is finitely generated, there exists a finite set G ⊂ D ⊂ C(X, D) and a number δ > 0 such that if φ and ψ are two unital ∗-homomorphisms defined on C(X, D) and satisfying φ(a) − ψ(a) < δ for all a ∈ G, then K ∗ (φ) = K ∗ (ψ). Let (εk ) be a sequence of numbers such that 0 < εk < δ and k εk < ∞. Using the assertion, and the weak semiprojectivity of D, we construct inductively two sequences of unital morphisms of C*-bundles, φk : C(X, D) → A, θk : C(X, D) → C(X, D), k = 1, 2, . . . , and a sequence of finite sets Fk ⊂ C(X, D) that contain G, such that
814
(i) (ii) (iii) (iv)
M. Dadarlat, G. A. Elliott
K ∗ ((φk )x ) is bijective for all x ∈ X and k ≥ 1; φk+1 θk (a) − φk (a) < εk for all a ∈ Fk and all k ≥ 1; θk (Fk ) ⊂ Fk+1 for all k ≥ 1; −1 ∞ (F j+1 ) is dense in C(X, D) and ∞ j=k θ j ◦ · · · ◦ θk j=k φ j (F j ) is dense in A for all k.
Arguing as in the proof of [24, Prop. 2.3.2], one verifies that the sequence (φk ) induces an isomorphism of unital C*-bundles limk (C(X, D), θk ) → A. Let us show that each − → K ∗ (θk ) is bijective. It suffices to check that K ∗ ((θk )x ) is bijective for some x. Since G ⊂ Fk and εk < δ, we deduce from (ii) that (φk+1 θk )x (a) − (φk )x (a) < δ for all a ∈ G and hence K ∗ ((φk+1 )x )K ∗ ((θk )x ) = K ∗ ((φk )x ) for all x ∈ X . Consequently K ∗ ((θk )x ) is bijective since K ∗ ((φk )x ) is so for all k, by (i). Since D satisfies the UCT, θk is a KK-equivalence. By [24, Thm. 8.2.1], each θk is approximately unitarily equivalent to a C(X )-linear automorphism of C(X, D) of the form idC(X ) ⊗σk , where σk ∈ Aut(D). Hence A is isomorphic to C(X, D) by Elliott’s intertwining argument. Let us prove now the assertion made at the beginning of the proof. The first part of the argument is essentially a repetition of the proof of Theorem 6.1 together with the observation that one can arrange that the components of the fibered morphism ∈ HomD (A1 , A) have the property that K ∗ (φYi ) = αYi and K ∗ (φ Z j ) = α Z j . We have that A ⊗ O∞ ∼ = A by Theorem 7.4. Since A absorbs O∞ and D satisfies the UCT, we can apply Phillips’s Theorem [24, Thm. 8.2.1] to lift α X to a unital ∗-homomorphism η : D → A. If U is a closed subinterval of X , set ηU = πU η. Since the horizontal maps in the commutative diagram K ∗ (C(X ) ⊗ D) K ∗ (C(U ) ⊗ D)
αX
αU
/ K ∗ (A(X )) / K ∗ (A(U ))
are bijections by hypothesis, the restriction map K ∗ (A(X )) → K ∗ (A(U )) is also a bijection. Let x ∈ X and consider the sets Un = {y ∈ X : |y − x| ≤ 1/n}. We deduce that K ∗ (A(X )) → K ∗ (A(x)) = lim K ∗ (A(Un )) is also a bijection, and hence − → ηx : D → A(x) induces a bijection K ∗ (D) → K ∗ (A(x)) and K 0 (ηx )[1] = [1]. By the Kirchberg-Phillips classification theorem, ηx is approximately unitarily equivalent to an isomorphism. Therefore there is a unitary u x ∈ A(x) such that πx (F) ⊂ε/2 u x ηx (D)u ∗x . Using the continuity of the norm for sections of continuous C*-bundles, we find a closed neighborhood Ux of x and a unitary u Ux ∈ A(Ux ) such that πUx (F) ⊂ε/2 ∗ . We also have K (η ) = α u Ux ηUx (D)u U ∗ Ux Ux by construction. In this manner we x obtain a family of maps η j : D → A(U j ), 0 ≤ j ≤ m, as in the proof of Theorem 6.1, with the additional property that K ∗ (η j ) = αU j . Let ϕi : D → A(Yi ) be constructed as in the proof of Theorem 6.1, with the modifications described above, so that we can arrange to have K ∗ (ϕi ) = αYi . Arguing as there, we perturb η j to ψ j such that ψ j (a) − η j (a) < ε/2 for all a ∈ F j . By choosing F j appropriately this last condition also implies that K ∗ (ψ j ) = K ∗ (η j ) and hence after restricting both ψ j and η j to Z j ⊂ U j , we have K ∗ (ψ j ) = α Z j . Consider the maps φYi : C(Yi ) ⊗ D → A(Yi ) and φ Z j : C(Z j ) ⊗ D → A(Z j ) obtained by extending ϕi and ψ j to morphisms of C*-bundles. They satisfy the conclusion of Theorem 6.1 and moreover they have the property that K ∗ (φU ) = αU for any closed interval U ⊂ Yi or U ⊂ Z j . This implies immediately that the components of γ induce the identity map on K ∗ (D). Let A1 be the
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pullback of the first row of the commutative diagram C(Y ) ⊗ D
π
φX
A(Y )
π
/ C(Y ∩ Z ) ⊗ D o
η
φY ∩Z
/ A(Y ∩ Z ) o
C(Z ) ⊗ D φZ
π
A(Z )
This diagram defines a fibered morphism which induces a unital ∗-homomorphism
A1 . By
: A1 → A such that K ∗ (φ
x ) is bijective for all x ∈ X and F ⊂ε φ φ Proposition 7.1, A1 ∼
= C(X, D). 8. Classification Results For the remainder of the paper we fix d ∈ {0, 1}. When employing the terminology of Sects. 5 and 6 we shall restrict ourselves to the class C = Cd of stable Kirchberg algebras satisfying the UCT, with K d -group finitely generated and torsion free and K d+1 -group equal to zero. If B is a stable Kirchberg C*-algebra satisfying the UCT and such that K d (B) is torsion free and K d+1 (B) = 0, then B can be written as an inductive limit of a sequence of C*-algebras in Cd [24]. By the UCT, K K (S A, B) = 0 for all A ∈ Cd and B as above. The following theorem gives an existence and uniqueness result for fibered morphisms. Theorem 8.1. Let A and B be continuous C*-bundles over [0, 1] such that the fibers of A are in Cd and the fibers of B are nonzero inductive limits of sequences of C*-algebras in Cd . Suppose that A is elementary with fibered presentation ι : D A → D. For any K d -fibered morphism α ∈ Hom(K d (D), K d (D B)) there is a fibered monomorphism φ ∈ HomD (A, B) such that K d (φ) = α. If ψ ∈ HomD (A, B) is another fibered monomorphism satisfying K d (ψ) = α, then φ is approximately unitarily equivalent to ψ, and
is approximately unitarily equivalent to ψ
. hence φ Proof. We need to find φ ∈ Hom(D, D B) such that K d (φY ) = αY , K d (φ Z ) = α Z , and K d (φY ∩Z ) = αY ∩Z . By assumption we have αY (K d (D(Yi ))) ⊆ K d (B(Yi )), α Z (K d (E(Z j ))) ⊆ K d (B(Z j )) and αY ∩Z (K d (D(Yi ∩ Z j ))) ⊆ K d (B(Yi ∩ Z j )). Let αiD and α Ej denote the corresponding components of αY and α Z . Similarly let αi, j : K d (Di ) → K d (B(Yi ∩ Z j )) denote the components of αY ∩Z when Yi ∩ Z j = ∅. We have B ∼ = B ⊗ O∞ ⊗ K by Theorem 7.4 and B contains a properly infinite full projection by [6, Prop. 5.6]. By [24, Thm. 8.2.1] there are ∗-monomorphisms φi, j : Di → B(Yi ∩ Z j ), such that K d (φi, j ) = αi, j . By Proposition 2.4, for each i = 0, . . . , m there is a ∗-monomorphism φiD : Di → B(Yi ), which we then extend to a monomorphism of C*-bundles φiD : C(Yi ) ⊗ Di → B(Yi ), with K d (φiD ) = αiD and such that φiD lifts simultaneously the maps φi,i−1 ◦ πYi ∩Z i−1 and φi,i ◦ πYi ∩Z i . Similarly, by applying Proposition 2.4 again, for each j = 0, . . . , m − 1 there is a monomorphism of C*-bundles φ Ej : C(Z j ) ⊗ E j → B(Z j ) which lifts simultaneously the maps φ j, j ◦ η j, j and φ j+1, j ◦ η j, j+1 , such that K d (φ Ej ) = α Ej . Then φY = (φiD ), φ Z = (φ Ej ) and φY ∩Z = (φi, j ) is the desired lifting of α. We must now address the degree of uniqueness of a lifting. Let φ and ψ be as in the statement, with K d (φ) = K d (ψ) = α and components φiD , ψiD , φ Ej , ψ jE , φi, j , ψi, j . By [24, Thm. 8.2.1], φiD is approximately unitarily equivalent to ψiD and
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similarly, φ Ej is approximately unitarily equivalent to ψ jE . This yields unitaries u i ∈ M(B(Yi )) and v j ∈ M(B(Z j )) which approximately intertwine the corresponding pairs of ∗-monomorphisms. While the restrictions of these unitaries to Yi ∩ Z j are not necessarily equal, we may use Corollary 3.3 to replace each unitary v j by a unitary which agrees with u i on Yi ∩ Z j for i = j, j + 1. This procedure yields unitaries u (n) ∈ M(B) implementing an approximate unitary equivalence between φ and ψ. More explicitly, let us focus on a fixed component Z i = [a2i+1 , a2i+2 ]. Its neighbors are Yi and Yi+1 . If we let ≈u denote approximate unitary equivalence for ∗-homomorphisms, from the discussion above we have: φiD ≈u ψiD (implemented by a sequence (u i(n) )n ), D ≈ ψ D (implemented by (u (n) ) ), and φ E ≈ ψ E (implemented by (v (n) )). φi+1 u u i+1 n i i+1 i i After restricting to the endpoints of Z i and composing with ηi,i and ηi,i+1 (in the first two equivalences), we obtain (φiE )a2i+1 ≈u (ψiE )a2i+1 implemented by πa2i+1 (u i(n) ) n , (n) and (φiE )a2i+2 ≈u (ψiE )a2i+2 implemented by πa2i+2 (u i+1 ) n . This enables us to apply (n)
Corollary 3.3 to φiE , ψiE : C(Z i ) ⊗ E i → B(Z i ) and replace the sequence (vi ) by a (n) sequence of unitaries (wi ) in M(B(Z i )) which still implements φiE ≈u ψiE and such (n) (n) (n) (n) (n) that πa2i+1 (wi ) = πa2i+1 (u i ) and πa2i+2 (wi ) = πa2i+2 (u i+1 ). The unitaries u i and (n) wi then glue together to a unitary u (n) ∈ M(B), by Lemma 3.1, and the sequence (u (n) )n gives an approximate unitary equivalence between φ and ψ.
Recall that if A is a C*-bundle over X = [0, 1], we are working with the K d -theory sheaf Kd (A)(U ) = K d (A(U )), defined on the category of closed subintervals U of [0, 1]. We are now ready to prove the main isomorphism result of the paper. Theorem 8.2. Fix d ∈ {0, 1}. Let A, B be separable continuous C*-bundles over [0, 1] the fibers of which are stable Kirchberg algebras satisfying the UCT, with torsion free K d -groups and vanishing K d+1 -groups. Then any morphism α : Kd (A) → Kd (B) of K d -sheaves lifts to an injective morphism of C*-bundles : A → B, which is unique up to approximate unitary equivalence. If α is an isomorphism of sheaves, then we may arrange that is an isomorphism of C*-bundles. Proof. By Theorem 6.2 there exist a sequence (Ak ) of elementary C*-bundles with fibers in Cd such that Ak = PDk for admissible diagrams Dk and injective fibered morphisms
k,∞ = φ
k+1,∞ ◦ φ
k,k+1 . φk,k+1 ∈ HomDk (Ak , Ak+1 ), φk,∞ ∈ HomDk (Ak , A) such that φ ∼
Moreover, these morphisms induce an isomorphism A = lim (Ak , φk,k+1 ). In the first − → part of the proof we establish the following uniqueness result. If , : A → B are monomorphisms of C*-bundles such that Kd () = Kd () then ≈u . It suffices to
k,∞ ≈u ◦ φ
k,∞ for each k ≥ 1. This would follow provided that we show that ◦ φ show that Dk ◦ φk,∞ ≈u Dk ◦ φk,∞ . These maps are illustrated in the diagram Dk
φk,∞
/ Dk A
Dk Dk
// D B. k
By Theorem 8.1 it suffices to show that K d (Dk ◦ φk,∞ ) = K d (Dk ◦ φk,∞ ). But this is verified using the functoriality of K d and the assumption that Kd () = Kd (): K d (Dk ◦φk,∞ ) = Dk Kd () ◦ K d (φk,∞ ) = Dk Kd () ◦ K d (φk,∞ ) = K d (Dk ◦φk,∞ ).
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In the second part of the proof we shall lift a morphism of sheaves α : Kd (A) → Kd (B) to a monomorphism of C*-bundles A → B. To that purpose we are going to construct a sequence (ϕk ) of fibered monomorphisms ϕk ∈ HomDk (Ak , B) such
k,k+1 ≈u
k,∞ ). The conclusion of the thethat
ϕk+1 ◦ φ ϕk and Kd (
ϕk ) = α ◦ Kd (φ orem will then follow by applying Elliott’s intertwining argument [24, Sec. 2.3]. Let αk ∈ Hom(K d (Dk ), K d (Dk B)) be defined by the commutative diagram K8 d (DO k B) r r r rr Dk α r r rrr / K d (Dk A) K d (Dk ) αk
K d (φk,∞ )
By Theorem 8.1, for each k ≥ 1 there is a fibered monomorphism ϕk ∈ HomDk (Ak , B) which lifts αk , i.e. K d (ϕk ) = αk . By Proposition 5.1,
k,∞ ) = α ◦ K d (φ α ◦ Kd (φ (ϕk ) = Kd (
ϕk ), k,∞ ) = (Dk α ◦ K d (φk,∞ )) = K d
and hence the diagram Kd (B) t9 O
Kd (
ϕk )ttt
t tt tt Kd (Ak )
α
/ Kd (A)
k,∞ ) Kd ( φ
is commutative for each k ≥ 1. Therefore the left triangle of the diagram f2 Kd (B) O fffflflfll6 l l f l α ffff lll ϕk+1 ) ffffff lll Kd (
ffffff / Kd (Ak+1 ) / Kd (A) Kd (Ak ) Kd (
ϕk f ) fffffff
k,k+1 ) Kd ( φ
k+1,∞ ) Kd ( φ
is commutative. By the uniqueness result established in the first part of the proof we
k,k+1 ≈u
obtain that
ϕk+1 ◦ φ ϕk .
Corollary 8.3. Let A, B be separable unital continuous C*-bundles over [0, 1] the fibers of which are Kirchberg algebras satisfying the UCT, with torsion free K 0 -groups and vanishing K 1 -groups. Suppose that there is an isomorphism of K 0 -sheaves α : K0 (A) → K0 (B) such that α[1 A ] = [1 B ]. Then A ∼ = B. Proof. By the previous theorem there is an isomorphism ϕ : A ⊗ K → B ⊗ K such that K 0 (ϕ)[1 A ⊗ e] = [1 B ⊗ e] for a rank one projection e ∈ K. Since both ϕ(1 A ⊗ e) and 1 B ⊗ e are properly infinite and full projections in B ⊗ K, we may arrange that ϕ(1 A ⊗ e) = 1 B ⊗ e after conjugating ϕ by a suitable unitary in M(B ⊗ K), by [24, Lemma 4.1.4]. It follows that ϕ induces an isomorphism A ∼
= A⊗e → B ⊗e ∼ = B. Example 8.4. In order to illustrate the possible complexity of the continuous C*-bundles classified by Corollary 8.3, we construct now a continuous C*-bundle A satisfying all the assumptions of that theorem and such that for any closed nondegenerate subinterval
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I of [0, 1], A(I ) is not isomorphic to a trivial continuous C*-bundle, even though all the fibers of A are mutually isomorphic. Let D be a unital Kirchberg algebra satisfying the UCT and such that K 0 (D) = Z⊕Z, [1 A ] = (1, 0) and K 1 (D) = 0. Let γ : D → 0D be a unital ∗-homomorphism such that K 0 (γ )(0, 1) = (0, 0). Let (xn ) be a sequence of numbers dense in [0, 1]. For each n consider the continuous C*-bundle over [0, 1], Dn = { f ∈ C([0, 1], D) : f (xn ) ∈ γ (D)}. Let us define A1 = D1 , An+1 = An ⊗C[0,1] Dn+1 , and A = lim(An , θn ), where θ : − → An → An+1 is defined by θn (a) = a ⊗ 1. In other words A is the infinite tensor product over C[0, 1] of C[0, 1]-modules A = ∞ n=1 Dn . It is a continuous C*-bundle of Kirchberg algebras over [0, 1] (see [3]) which satisfies the assumptions of Theorem 8.2. It is
not hard to see that, while all its fibers are isomorphic to ∞ n=1 D, A is nowhere locally trivial. Let us verify the latter assertion. This is done by showing that for any closed nondegenerate subinterval I of [0, 1] and any x ∈ I , the evaluation map A(I ) → A(x) induces a non-injective map K 0 (A(I )) → K 0 (A(x)). Such a situation cannot occur for trivial continuous C*-bundles. Fix I as above and observe that Dn (I ) = C(I, D) if xn ∈ / I and that the map φn : D → Dn (I ), defined by φn (d)(x) = γ (d), x ∈ I , induces an isomorphism K 0 (D) → K 0 (Dn (I )) if xn ∈ I . Moreover, if xn ∈ I and if x ∈ I \{xn }, then the projection πx : Dn (I ) → Dn (x) ∼ = D induces a map K 0 (πx ) which can be identified with K 0 (γ ) and hence it is not injective. Using the Künneth Theorem, one verifies that the inclusion θn : An → An+1 ∼ = An ⊗C[0,1] Dn+1 induces an injective map K 0 (An (I )) → K 0 (An+1 (I )) for any I as above. Therefore the map η : K 0 (An (I )) → K 0 (A(I )) is injective. For x ∈ I , let us consider the commutative diagram induced by evaluating at x: K 0 (An (I )) (n)
K 0 (πx )
K 0 (An (x))
η
/ K 0 (A(I ))
. (∞)
K 0 (πx
/ K 0 (A(x))
)
Fix x ∈ I . Since the sequence (xn ) is dense in [0, 1], there is n such that xn ∈ I and (n) x = xn . Using the Künneth Theorem again, one verifies that the map K 0 (πx ) is not injective if x ∈ I \{xn }, since it can be identified with K 0 (πx(n−1) ) ⊗ K 0 (πx ) : K 0 (An−1 (I )) ⊗ K 0 (Dn (I )) → K 0 (An−1 (x)) ⊗ K 0 (Dn (x)), (∞)
and as seen earlier K 0 (πx ) is not injective. Since η is injective, it follows that K 0 (πx is not injective and hence A(I ) cannot be isomorphic to C(I, ∞ n=1 D).
)
Acknowledgement. We would like to thank E. Blanchard for a number of useful comments. We are also grateful to the referee whose extensive suggestions and criticism helped us to improve the exposition.
References 1. Akemann, C.A., Pedersen, G.K., Tomiyama, J.: Multipliers of C ∗ -algebras. J. Funct. Anal. 13, 277– 301 (1973) 2. Blackadar, B.: Semiprojectivity in simple C ∗ -algebras. In: Operator algebras and applications, Volume 38 of Adv. Stud. Pure Math., Tokyo: Math. Soc. Japan, 2004, pp. 1–17
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3. Blanchard, É.: Tensor products of C(X )-algebras over C(X ). In: Recent advances in operator algebras (Orléans, 1992), Astérisque 232, 81–92 (1995) 4. Blanchard, É.: Déformations de C ∗ -algèbres de Hopf. Bull. Soc. Math. France 124(1), 141–215 (1996) 5. Blanchard, É., Kirchberg, E.: Global Glimm halving for C*-bundles. J. Op. Theory 52, 385–420 (2004) 6. Blanchard, É., Kirchberg, E.: Non-simple purely infinite C*-algebras: the Hausdorff case. J. Funct. Anal. 207, 461–513 (2004) 7. Brown, L.G., Dadarlat, M.: Extensions of C ∗ -algebras and quasidiagonality. J. London Math. Soc. 53(3), 582–600 (1996) 8. Dadarlat, M.: Continuous fields of C*-algebras over finite dimensional spaces. http://arxiv.org/list/math. OA/0611405, 2006 9. Dadarlat, M., Loring, T.A.: A universal multicoefficient theorem for the Kasparov groups. Duke Math. J. 84(2), 355–377 (1996) 10. Dadarlat, M., Pasnicu, C.: Continuous fields of Kirchberg C*-algebras. J. Funct. Anal. 226, 429–451 (2005) 11. Dixmier, J.: C ∗ -algebras. Amsterdam: North Holland, 1982 12. Eilers, S., Loring, T.A.: Computing contingencies for stable relations. Internat. J. Math. 10(3), 301–326 (1999) 13. Kasparov, G.G.: Hilbert C ∗ -modules: Theorems of Stinespring and Voiculescu. J. Op. Theory 4, 133–150 (1980) 14. Kirchberg, E.: Das nicht-kommutative Michael-auswahlprinzip und die klassifikation nicht-einfacher algebren. In: C ∗ -algebras, (Münster, 1999), Berlin: Springer, 2000, pp. 92–141 15. Kirchberg, E., Rørdam, M.: Non-simple purely infinite C ∗ -algebras. Amer. J. Math. 122(3), 637–666 (2000) 16. Kirchberg, E., Rørdam, M.: Infinite non-simple C ∗ -algebras: absorbing the Cuntz algebra O∞ . Adv. in Math. 167(2), 195–264 (2002) 17. Kirchberg, E., Wassermann, S.: Operations on continuous bundles of C ∗ -algebras. Math. Ann. 303(4), 677–697 (1995) 18. Lin, H.: Full extensions and approximate unitary equivalence. http://arxiv.org/list/math.OA/0401242, 2004 19. Lin, H.: Weak semiprojectivity in purely infinite C*-algebras. Preprint 2002 20. Loring, T.: Lifting Solutions to Perturbing Problems in C ∗ -Algebras. Volume 8 of Fields Institute Monographs. Providence, RI: Amer. Math. Soc., 1997 21. Pedersen, G.K.: Pullback and pushout constructions in C ∗ -algebra theory. J. Funct. Anal. 167(2), 243–344 (1999) 22. Phillips, N.C.: A classification theorem for nuclear purely infinite simple C ∗ -algebras. Doc. Math. 5, 49–114 (2000) 23. Roe, J.: Lectures on coarse geometry. Volume 31 of University Lecture Series. Providence, RI: Amer. Math. Soc. 2003 24. Rørdam, M.: Classification of nuclear, simple C ∗ -algebras. Volume 126 of Encyclopaedia Math. Sci. Berlin: Springer, 2002 25. Rørdam, M.: Stable C ∗ -algebras. In: Operator algebras and applications, Volume 38 of Adv. Stud. Pure Math., Tokyo: Math. Soc. Japan, 2004, pp. 177–199 26. Schochet, C.: Topological methods for C ∗ -algebras. III. Axiomatic homology. Pacific J. Math. 114(2), 399–445 (1984) 27. Spielberg, J.S.: Weak semiprojectivity for purely infinite C*-algebras. http://arxiv.org/list/math.OA/ 0206185, 2002 28. Spielberg, J.S.: Semiprojectivity for certain purely infinite C ∗ -algebras. http://arxiv.org/list/math.OA/ 0102229, 2001 29. Wells, R.O. Jr.: Differential analysis on complex manifolds. Volume 65 of Graduate Texts in Mathematics Second edition. New York: Springer-Verlag, 1980 Communicated by Y. Kawahigashi
Commun. Math. Phys. 274, 821–839 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0263-x
Communications in
Mathematical Physics
Thermodynamics and Universality for Mean Field Quantum Spin Glasses Nicholas Crawford Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555, USA. E-mail: [email protected] Received: 13 October 2006 / Accepted: 16 January 2007 Published online: 16 May 2007 – © Springer-Verlag 2007
Abstract: We study aspects of the thermodynamics of quantum versions of spin glasses. By means of the Lie-Trotter formula for exponential sums of operators, we adapt methods used to analyze classical spin glass models to answer analogous questions about quantum models.
1. Introduction Classical spin glass models have seen a flurry of activity over the last few years, see [2, 6, 14–16, 18–20, 25–27] to name a few; in particular [2, 14, 15, 27] all consider aspects of the (generalized) Sherrington-Kirkpatrick model, a mean-field model in which the interactions between spins are mediated by an independent collection of Gaussian random variables. In contrast, though physicists have considered both short and long range quantum spin glass models for quite some time, see [3, 12, 23, 24], there are few rigorous mathematical results; we mention [7], which provides a proof that the quenched free energy of certain short ranged quantum spin glasses exists. Here we extend classical spin glass results to quantum models in two directions. First, using the ideas of [16], we demonstrate the existence of the quenched free energy of the Sherrington-Kirkpatrick spin glass with a transverse external field. Second, under conditions made precise below, we give a complementary result which shows that a large class of quantum spin glasses, including the transverse S-K model, satisfies universality. By this we mean that as the size of the system goes to infinity, the asymptotics of the free energy of the system do not depend on the type of disorder used to define model. This latter result is based on the work of [6]. One may view this paper as an attempt to adapt the methods of classical spin systems to the analysis of various quantum models. Guerra’s interpolation scheme [14–16] and the Gaussian integration-by-parts formula are ubiquitous tools in the classical setting. A major theme of the present work is that through the systematic use of the Lie-Trotter
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product formula (i.e. that A+B
e
= lim
k→∞
k
A
B
ek ek
(1.1)
j=1
for any pair of matrices A, B) one may extend the interpolation scheme and integration-by-parts formula to quantum systems in useful ways. We comment here that the use of the Lie-Trotter formula in the context of quantum spin systems has a long history, going back at least to the work [13]. Other references on the application of path integral representations to quantum spin systems include [1, 10], among many others. For concreteness, the remainder of the introduction, Sect. 2 and Sect. 4 all use the language of the spin-1/2 representation of su(2) to describe quantum spin systems (see [9] for more background), though a careful reading shows that our methods apply with minor modifications to spin-S representations as well. In this setting, one describes each particle using a two dimensional Hilbert space C2 along with a representation of su(2) generated by the triplet of Pauli operators S = (S (x) , S (y) , S (z) ). To represent an N -particle system, we introduce the tensor product V N = Nj=1 C2 , one factor for each particle, along with a sequence (S j ) Nj=1 of N copies of the Pauli vector S, where S j acts on the j th factor. The particles interact by means of the Hamiltonian H N acting on V N with weights of configurations described by the associated GibbsBoltzmann operator e−β H N . Here, H N is a self adjoint operator acting on V N , typically a polynomial in the N -tuple of spin operators (S j ) Nj=1 . For example, the Hamiltonian of the simplest non-trivial quantum spin system, the transverse Ising model, is described by −H N =
N N 1 (z) (z) (x) Si S j + λ Sj , N i, j=1
(1.2)
j=1
where λ > 0. Once we specify the Hamiltonian, statistical quantities of the system may be defined. The partition function and free energy of a quantum spin system are defined via the trace of the Gibbs-Boltzmann operator as Z N (β) =Tr e−β H N , f N (β) =
−1 log Z N . Nβ
Self adjoint operators on V N replace functions as observables of the system and the thermal average of an observable A is defined as Tr Ae−β H N . A = Tr e−β H N For future reference, we refer to the functional · on observables as the Gibbs state of the system corresponding to the Hamiltonian H N . With this formalism we present a few examples of spin glasses of particular interest. Traditionally the modeling of any spin glass necessitates the introduction of disordered
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interactions between spins. In general, as will be the case here, the interactions are i.i.d. The basic example, the transverse S-K model, has a Hamiltonian defined by N N 1 (z) (z) (x) Ji, j Si S j + λ Sj . −H N = √ 2 N i, j=1 j=1
A more complicated class of models, the quantum Heisenberg spin glasses, are described by one of the Hamiltonians N 1 (y) (y) (z) (z) (x) (x) −H N = √ Ji, j Si S j + αSi S j + γ Si S j , 2 N i, j=1
1 −H˜ N = √ 2 N
N
(1.3)
(ν) (ν) (ν)
Ji, j Si S j ,
ν∈{x,y,z} i, j=1
where α, γ ∈ R. There is a sharp and interesting contrast in the scope of application of our results. Unlike the quantum Heisenberg spin glasses, applying the Lie-Trotter expansion to the transverse S-K model has the added benefit of allowing a natural path integral representation, known as the Feynman-Kac representation, of statistical quantities like the free energy. More precisely, the Feynman-Kac representation allows the expression of the partition function as an integral over paths which are right continuous and have left limits (i.e. càdlàg paths) (see [4] for details on the use of such paths) with state space the classical Ising spin configurations on N sites, {−1, +1} N . For this and other technical reasons we can only resolve the existence of the free energy for the one quantum model. The remainder of the paper is structured as follows. Section 2 introduces notation and states the main theorems of the paper. Section 3 presents quantum generalizations of [6] which allow us to prove universality and control the fluctuations of the free energy for a class of quantum spin glasses which include the Spin-1/2 quantum Heisenberg spin glasses. In addition, we adapt the techniques of [16] to prove exponential decay of the fluctuations of models with Gaussian disorder. Finally, Sect. 4 details the existence of the pressure of the transverse S-K model. 2. Results We begin by giving a bit of notation to be used below. We consider a collection ξ I of random variables, with I ∈ I for some finite index set I, and denote E [·] and P (·) integration with respect to this collection. Examples of index sets that we have in mind include the collection of all r -tuples of sites in a system of N particles. In the simplest case the index set I consists of all pairs (i, j) (or more generally some subset of pairs) with i, j ≤ N . Unless otherwise specified, the variables ξ I are assumed to be i.i.d. according to some fixed random variable ξ satisfying the conditions E [ξ ] = 0, E ξ 2 = 1, E |ξ |3 < ∞. (2.1) When explicitly considering Gaussian environments we denote the random variables by gI .
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N. Crawford
Let S denote the N -tuple of Pauli vectors and I N index some collection of interactions between particles at the system size N . Consider the Hamiltonian H N (ξ ) = ξ I X I (S), I ∈I N
where each X I (S) is a self-adjoint polynomial in the components of S, i.e. X ∗I (S) = X I (S) and is regarded as an operator on V N via a trivial embedding. We define the associated partition function and quenched ‘pressure’ by
Z N (β, ξ ) = Tr eβ H N ; α N (β, ξ ) = E log Z N (β, ξ ) . Note that for convenience we have omitted the minus sign from the expression for the Gibbs-Boltzman operator. For any operator A on V N , we denote its operator norm by 1 2 A := supv∈V N (Av,Av) . With this notation we have the following result: (v,v)
Theorem 1. Let ξ be a random variable with mean 0, variance 1, and E |ξ |3 < ∞ and let g be a standard normal random variable. Consider a sequence of Hamilitonians H N defined as above so that X I (S) 3 = o(N ). I ∈I N
Then for any β ∈ R,
1 α N (β, ξ ) − 1 α N (β, g) = o(1) N N
as N tends to infinity. Moreover
3 √ 1 |I N |o(N ) 1 E log Z N (β, ξ ) − α N (β, ξ ) ≤ . N N N3
(2.2)
Remark 1. Choosing the operators X I and index sets I N appropriately allows application to a wide variety of systems. For example, in (1.3) the norm of each summand is bounded by √C and the number of pairs is of the order N 2 . As a result, N
1
X I (S) 3 = O(N 2 )
I ∈I N
and the theorem is immediately applicable. The concentration estimate (2.2) gets better if one assumes the random variable ξ has higher moments. At the extreme end, we consider the case of fluctuations for Gaussian environments: Proposition 1. Let g be a standard normal random variable. Then
P (|log Z N (β, g) − α N (β, g)| ≥ u) ≤ 2e
−
u2 2 2 I ∈I N β X I
.
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Next we take up the more subtle question of the existence of the free energy for mean field models. In light of Theorem 1, to prove convergence of free energies as the system size goes to infinity we may assume the interactions are Gaussian without loss of generality. As mentioned above, we specialize to the transverse S-K model: N N 1 (z) (z) (x) −H N (λ) = √ gi, j Si S j + λ Sj . 2 N i, j=1 j=1
(2.3)
Theorem 2. Let β, λ > 0 be fixed. Then −1 log Tr e−β H N (λ) N →∞ β N lim
exists, is finite and is constant a.s. Moreover, there exists a K > 0 so that the following concentration property holds: 2 1 −1 −N K u2 β . log Tr e−β H N (λ) + E log Tr e−β H N (λ) ≥ u ≤ 2e P N N Remark 2. For readability, we assume here that the disordered portion of the Hamiltonian in (2.3) is a two body interaction. However analogous arguments allow one to treat p-spin models where we replace the disordered portion by (z) −Hdis N (S ) =
∞ ar r =1
N
r −1 2
gi1 ...ir
i 1 ,...ir
r
(z)
Sik .
k=1
Here i 1 ...ir } is a collection of independent standard Gaussian random variables and ∞ {g 2 q r is even, convex and continuous as a function of q on [−1, 1]. See [2] and a 1 r [16].
3. Universality and Fluctuations Before proving Theorem 1, we adapt the methods of [6] so as to apply them to a wide variety of quantum spin systems. It turns out that the line of argument given there is robust enough to be followed in the quantum case, though the calculations must be adjusted to accommodate the non-commutative setting. d In the general setup, we consider a collection of self adjoint operators {X i }i=1 and H0 defined on some finite dimensional Hilbert space. Let d
1 H0 , β 1 Z (β, ξ ) = Tr eβ H(ξ ) ,
α(β, ξ ) = E log Z (β, ξ ) H(ξ ) =
ξi X i +
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N. Crawford
denote the Hamiltonian, partition function, and quenched ‘pressure’ of a system. We define the thermal average, Duhamel two point function, and the three point function for any bounded linear operators A, B, and C on V N as follows: Tr Aeβ H(ξ ) , A = Z (β, ξ ) 1 uβ H(ξ ) (1−u)β H(ξ ) du Tr Ae Be , (A, B) = 0 Z (β, ξ ) 11 suβ H(ξ ) (1−s)uβ H(ξ ) (1−u)β H(ξ ) ds du Be Ce 0 0 uTr Ae . (A, B, C) = Z (β, ξ ) For the convenience of the reader, we recall a generalized integration-by-parts lemma proved in [6]. The left-hand side of (3.1) is well known to be zero if the randomness is Gaussian. This fact will play a role in what follows.
Lemma 1. Let ξ be a real valued random variable such that E |ξ |3 < ∞ and [ξ ] = E 0. Let F : R → R be twice continuously differentiable with F ∞ = supx∈R F (x) < ∞. Then
3 (3.1) E [ξ F(ξ )] − E ξ 2 E F (ξ ) ≤ F ∞ E |ξ |3 . 2 We first express the derivative of the quenched pressure in terms of thermal averages and Duhamel two-point functions of the operators X i and a bound on the difference between the quantities: Lemma 2.
d d ∂α(β, ξ ) 2 2 3 3 = βE X i , (X i , X i ) − X i + 9O(β )E |ξ | ∂β i=1
i=1
where |O(β 2 )| ≤ β 2 . Remark 3. There is no correction in this formula when ξ is Gaussian since the integration-by-parts formula is exact in this case. Proof. To begin the derivation, we have ∂α(β, ξ ) = E [ξi X i ] . ∂β d
i=1
For z ∈ R, let us define Hi (z) = (z)
Gibbs state by ·i
j =i
ξ j X j + z X i + β1 H0 , denote the corresponding (z)
and define the function Fi (z) = X i i . Then E [ξi X i ] = E [ξi Fi (ξi )] .
(3.2)
To evaluate the right-hand side of Eq. (3.2), we calculate the first and second deriva A B tives of Fi . Applying the Lie-Trotter expansion, i.e. that e A+B = limk k1 e /k e /k , with
Mean Field Quantum Spin Glasses
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A = βz X i and B = βHi (z) − βz X i , we find that the first and second derivatives of Fi take the form: 2 (z) (z) Fi (z) =β (X i , X i )i − X i , 3 2 (z) (z) (z) (z) Fi (z) =β 2 (X i , X i , X i ) − 3 (X i , X i ) X i + 2 X i . In order to prove the lemma we must bound Fi . To this end we claim that for any a1 , . . . , an > 0 so that a1 + · · · + an = 1 and any self-adjoint matrices X, H, (3.3) Tr X ea1 H · · · X ean H ≤ X n Tr eH . k
Indeed, we may assume by continuity that a j = 2mj for some m > 0 and some sequence of positive integers (k j ) summing to 2m . For any sequence of matrices (B j )2k j=1 , ⎛ ⎞ 2k 2k k 2k1 ∗ Tr ⎝ B j ⎠ ≤ Tr B j B j . (3.4) j=1 j=1 This can be seen as a specific case of the general method of chessboard estimates, H H Theorem 4.1 of [11]. Applying (3.4) to the left-hand side of (3.3) with B j ∈ {X e 2m , e 2m } we have 2m−1 2nm 2m −n k1 H kn 2m H H Tr eH . (3.5) Tr X e 2m · · · X e 2m ≤ Tr X e 2m−1 X Another application of (3.4) implies 2m−1 1 m+1 1 H 2 2 Tr e2H . ≤ Tr X 2 Tr X e 2m−1 X Using this bound on the right-hand side of (3.5) and letting m pass to infinity proves the bound (3.3).
It follows that Fi ∞ ≤ 6β 2 X i 3 . Recalling that E ξ 2 = 1, Lemma 1 in conjunction with the previous calculations imply
d d ∂α(β, ξ ) 2 2 3 3 − βE X i . (X i , X i ) − X i ≤ 9β E |ξ | ∂β i=1
i=1
Next we use Lemma 2 to compare the quenched ‘pressure’ for ξ to that of a Gaussian environment g where g is a standard normal. For this, recall the assumptions (2.1) on ξ . Lemma 3. Let {ξi } and {gi } be collections of i.i.d random variables distributed according to ξ and g respectively. For any β ∈ R, d 3 3 3 |α(β, ξ ) − α(β, g)| ≤ 9|β| E |ξ | X i . i=1
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Proof. The similarity between the proof of this lemma and that of Proposition 7 [6] means that we will be extremely brief. Consider the interpolating partition function and corresponding quenched ‘pressure’ defined by Z (s, t − s) =e
√ √ d d s i=1 ξi X i + t−s i=1 gi X i +H0
α (t) (s) =E log Z (s, t − s) respectively. We have √ √ α (t) (t) = α( t, ξ ); α (t) (0) = α( t, g). Lemma 2 along with independence between the two environments implies that for all s ∈ [0, t] we have d ∂α (t) (s) √ 3 X i 3 . ≤ 9 tE |ξ | ∂s i=1
and integrate this inequality the result follows. For β < 0 For β ≥ 0, if we let t = we instead consider the environments −ξ, −g. β2
Next we attend to the fluctuations of the ‘pressure’ determined by the random environment (ξi ): Lemma 4. There exists some universal constant c > 0 so that d √ 3 3 3 3 E | log Z (β, ξ ) − α(β, ξ )| ≤ c E |ξ | β d X i . i=1
Proof. Consider the filtration Fk = σ {ξ1 . . . ξk }, k ≥ 1 determined by the sequence of independent random variables (ξk ). Let
i := E log Z (β, ξ ) Fi − E log Z (β, ξ ) Fi−1 . We have log Z (β, ξ ) − α(β, ξ ) =
d
i .
i=1
Burkholder’s martingale inequality (see [5] for a statement) implies the existence of a universal constant c so that d d 23 3 E i ≤ c E i2 . i=1
i=1
To bound the increment i , consider the partition function Z i (β, ξ ) = Tr eβψi , where ψi = ψi (ξ ) := H(ξ ) − ξi X i . Since Z i (β, ξ ) is independent of ξi , Z (β, ξ ) Z (β, ξ ) i = E log Fi − E log Fi−1 . Z i (β, ξ ) Z i (β, ξ )
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We use this identity to estimate i . We claim that Z (β, ξ ) ≤ eβ|ξi |· X i . Z i (β, ξ ) Indeed, the Lie-Trotter formula implies ⎛ k ⎛ k ⎞ ⎞ 2 2 βψi βξi X i βξi X i βψi βξi X i e 2k e 2k ⎠ = lim Tr ⎝ e 2k+1 e 2k e 2k+1 ⎠ . Z (β, ξ ) = lim Tr ⎝ k→∞
k→∞
j=1
j=1
βξ X β|ξi |· X i i i e 2k ≤ e 2k ,
Since
Inequality (3.3) applied to the right-hand side for k finite gives ⎞ ⎛ k 2 βψi βξi X i e 2k e 2k ⎠ ≤ eβ|ξi |· X i Tr eβψi Tr ⎝ j=1
from which our claim follows. From this we estimate: | i | ≤ β X i (|ξi | + E|ξi |). Therefore,
⎡ ⎤ 23 d d 3 2 ⎢ ⎥ E | log Z (β, ξ )−α(β, ξ )|3 ≤cE ⎣ i2 ⎦ ≤cβ 3 X i 2 E (|ξi | + E|ξi |)2 i=1
i=1
≤8cβ E |ξ | 3
3
√
d
d
X i
3
,
i=1
where the constant c is independent of d by Burkholder’s inequality and we have used Hölder’s inequality in the last bound. Proof (Proof of Theorem 1). Our first theorem now follows as an application of the above machinery: the first statement is an application of Lemma 3, while the second follows from Lemma 4. Finally, we consider fluctuations in Gaussian environments: Proof (Proof of Proposition 1). Let Z β (t) be defined as the auxiliary partition function given by two independent collections of Gaussian disorder g (1) and g (2) , √ d (1) √ d (2) Z β (t) = Tr eβ t i=1 gi X i +β 1−t i=1 gi X i +H0 with t an interpolation parameter varying between 0 and 1. Let E j denote the average with respect to the random variables g ( j) for j = 1, 2. Given any s ∈ R, let Y (t) = exp(sE2 log Z β (t)); φ(t) = log E1 [Y (t)] .
(3.6)
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N. Crawford
We note that
φ N (1) − φ N (0) = log E exp(s(log Z (β, g) − α(β, g))) .
(3.7)
In order to estimate this difference, consider # d # d
$ $ (2) 1 (1) sβ 1 E1 Y (t)E2 √ φ (t) = g Xi − √ g Xi , 2E1 [Y (t)] t i=1 i 1 − t i=1 i t
t
where the notation ·t represents the Gibbs state induced by the interpolating Hamiltonian. A calculation involving the Lie-Trotter expansion and the Gaussian version of the integration-by-parts formula implies
d s2β2 2 E1 E2 Y (t) φ (t) = X i t 2E1 [Y (t)]
i=1
so that
|φ (t)| ≤
s2β2
d
2 i=1 X i
2
.
(3.8)
Using Eq. (3.8) and the fact that e|x| ≤ e x + e−x we have d s 2 β 2 i=1 X i 2 exp(|s|| log Z (β, g) − α(β, g)|) ≤ 2 exp . 2 Finally, applying Markov’s inequality we have P (| log Z (β, g) − α(β, g)| ≥ u) ≤ 2 exp
s2β2
d
2 i=1 X i
2
− su
for any s ∈ R. Optimizing over s concludes the proof. 4. Existence of the Pressure Recalling the su(2) formalism from the introduction, via the trivial embedding of N one particle spaces into V N we choose as a preferred basis the collection of all simple (z) tensor products of eigenvectors for the operators {S j }. If we denote the eigenvector for S (z) which corresponds to the eigenvalue +1 by |+ and the eigenvector for which corresponds to the eigenvalue −1 by |−, we may identify this preferred basis with classical Ising spin configurations σ ∈ {−1, +1} N . For each σ , we denote the corresponding basis vector by |σ . The proof of Theorem 2 proceeds in two steps. The first step consists of a concentration estimate following essentially the same argument as that of Proposition 1. Widening the scope beyond the transverse Ising spin glass, for this first step we consider quantum Hamiltonians of the form (z) det H N := Hdis (4.1) N (S ) + H N .
Mean Field Quantum Spin Glasses
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We assume that only Hdis N involves Gaussian disorder and that the deterministic operator takes a sufficiently nice form so as to admit a Feynman-Kac representation in terms Hdet N of the basis of eigenvectors for the S (z) operators. By this we mean that σ | exp(−uHdet ˜ ≥ 0 N )|σ for all u ≥ 0 and all spin configurations σ, σ˜ . Assuming that all off-diagonal matrix elements of Hdet N are non-positive gives a necessary and sufficient condition which guarantees the existence of a Feynman-Kac representation. As mentioned in Sect. 2, we assume for our treatment that Hdis N takes the form N 1 (z) (z) (z) (S ) = gi j Si S j , −Hdis √ N 2 N i, j=1
where the collection {gi, j } is assumed to be i.i.d. standard normal, though more general (z) interactions involving the {S j } may be considered. To illustrate the significance of the Feynman-Kac representation, note that if we are willing to modify Hdet N by adding a uniquely determined matrix D N , diagonal with respect to the preferred basis, we can force the rows of the matrix representation of Hdet N to sum to zero. This implies that the modified Hamiltonian H¯ det N is the generator of a continuous time Markov chain with state space {−1, +1} N . Let N be the measurable space of {−1, +1} N valued càdlàg paths. Let σ denote the spin path {σ (u) : u ∈ [0, β]} and let us denote the induced Markov chain path measure started from initial configuration σ by dνσ (σ ). Now (z) ¯ det H N = Hdis N (S ) − D N + H N (z) and the term Hdis N (S ) − D N is diagonal with respect to the preferred basis. Consider any matrix element σ |e−u H N |σ˜ . Expanding the exponential for finite k using (z) ¯ det the Lie-Trotter formula (1.1) with A = −u(Hdis N (S ) − D N ) and B = −u H N and A
B
inserting the complete orthogonal set {|σ } in between each factor e k e k we have A B k σ | e k e k |σ˜ =
k−1
u u ¯ det dis (z) σ (i)|e− k H N (S −D N ) e− k H N |σ (i + 1), (4.2)
σ (0),...,σ (k) i=0 (z) where σ (0) = σ and σ (k) = σ˜ . Let µ(σ ) denote the eigenvalue for −Hdis N (S ) +
u dis (z) D N corresponding to the eigenvector |σ . Contracting the operator e− k H N (S −D N ) against σ (i)| in (4.2), we have
A B k σ | e k e k |σ˜ =
e
k−1
u i=0 k µ(σ (i))
σ (0),...,σ (k)
k−1
¯ det
σ (i)|e− k H N |σ (i + 1). u
(4.3)
i=0
Interpreting the last expression in terms of the path measure dνσ (σ ), σ (i) is the value of the process at time uik . Therefore % A B k k−1 u iu σ | e k e k |σ˜ = e i=0 k µ(σ ( k )) 1{σ (u)=σ˜ } dνσ (σ ). (4.4) N
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N. Crawford
Passing to the limit as k tends to infinity and using the bounded convergence theorem on the right-hand side, we have % u e 0 µ(σ (s))ds 1{σ (u)=σ˜ } dνσ (σ ). σ |e−u H N |σ˜ = N
Remark 4. In the case of most interest at present, the exact form of the matrix D N and the dynamics of the induced Markov chain are not difficult to determine: for the transverse S-K model, the diagonal matrix D N is λN · I, and the induced measure is defined by starting from an initial configuration and evolving in time via spin flips which occur independently at each site according to the arrivals of a Poisson process of rate λ. Thus for (4.1), β 1 N % √ g σ (s)σ j (s)ds Z N = Tr e−β H N = eλβ N e 0 2 N i, j=1 i j i 1{σ (β)=σ (0)} dνσ σ . σ ∈{−1,1} N
N
We use this representation throughout Sect. 4. Returning to the general set up let us introduce definitions and notation to be used below. In order to prove convergence of the quenched pressure we need to consider truncated partition functions Z NF defined via the Feynman-Kac transformation by ⎛ ⎞ % β 1 1 log Z NF = log ⎝ e 0 µ N (σ (s))ds 1{σ (β)=σ (0)} dνσ (σ )⎠, N N N F σ ∈{−1,1}
where F is some deterministic (i.e. not depending on Gaussian disorder) subset of N . For any pair of spin configurations σ, σ˜ ∈ {−1, 1} N , let R(σ, σ˜ ) =
N 1 σi σ˜ i . N i=1
Let
% ζ N (σ ) = 0
β
1 √
2 N
N
gi j σi (u)σ j (u)du
i, j=1
denote the Gaussian random variable defined by the classical Gibbs weight of the spin path σ . An easy calculation shows that % %
N β β 2 R (σ (u), σ˜ (s))duds, E ζ N (σ )ζ N (σ˜ ) = 4 0 0 where the function R is the classical overlap for Ising spin configurations. Finally, let γ N (σ ) denote the eigenvalue of D N corresponding to the eigenvector |σ Lemma 5. Let β > 0 be given. Fix any Feynman-Kac representable deterministic Hamiltonian Hdet N . Let F ⊂ N . Then 1 2u 2 1 F F (4.5) P log Z N − E log Z N ≥ u ≤ 2 exp − 2 N . N N β
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Proof. With the appropriate modifications, the method of proof of Proposition 1 may be followed here: Let Z NF (t) be defined as the auxiliary partition function given by two independent collections of Gaussian disorder g (1) and g (2) , Z NF (t) =
%
√
e
β √ (1) (2) tζ N (σ )+ 1−tζ N (σ )+ 0 γ N (σ (s))ds
F
σ ∈{−1,1} N
1{σ (β)=σ (0)} dνσ (σ )
with t an interpolation parameter varying between 0 and 1 and ζ (i) the Gaussian corresponding to g (i) . Let Y N (t) and φ N (t) be defined in terms of Z NF (t) as in (3.6). Replacing the corresponding quantities appearing in the proof of Proposition 1 we have, using the Feyman-Kac representation, φ N (t) =
s2 2E1 [Y N (t)] ×E1 Y N (t)
%
σ,σ˜ ∈{−1,1} N
F×F
N 4
%
β
%
0
β
R 2 (σ (u), σ˜ (s))duds
wN
0
×(t, σ )w N (t, σ˜ )dνσ (σ )dνσ˜ (σ˜ ) where
√
w N (t, σ ) =
e
β √ (1) (2) tζ N (σ )+ 1−tζ N (σ )+ 0 γ N (σ (s))ds
Z NF (t)
1{σ (β)=σ (0)}
is the truncated ‘Gibbs weight’ corresponding to the event F. Thus 2 2 φ (t) ≤ s β N . N 8
The bound now follows as in the proof of Proposition 1. Remark 5. More generally Lemma 4.5 applies to the p-spin models defined in Remark 2, though the bound stated in the lemma must be modified slightly. Unfortunately the use of the Feynman-Kac transformation alone does not allow our method to go through. In particular, we were unable to treat the quantum system with deterministic quadratic couplings in the x and y directions: the ferromagnetic version does permit a Feynman-Kac representation but the interaction is convex, which turns out to have exactly the wrong sign in the expression for the derivative of the interpolating ‘pressure’. The only natural example that we found amenable to our method is the transverse field Ising model. Notice that the deterministic portion of this particular Hamiltonian is linear, which simplifies the interpolation scheme that we employ to compare the thermodynamics at different system sizes. For inverse temperature β and transverse field strength λ > 0, we refer to the partition function of this model by Z N (β, λ) and denote p N (β, λ) = − N1 log Z N (β, λ). As mentioned in Sect. 2, in light of the preceding results the proof of Theorem 2 reduces to the following lemma:
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Lemma 6. Let β, λ > 0 be fixed. Then lim E [ p N (β, λ)] ≡ p(β, λ)
N →∞
exists. Proof. For notational convenience let dν σ =
1{σ (β)=σ (0)} dνσ σ .
σ ∈{−1,1} N
This proof, like that of the previous lemma, relies on the proof of an analogous statement in [16]. The main idea is to partition the space of paths into subsets on which we may control the time correlated self overlap R(σ (u), σ (s)). To this end, let , δ > 0 be fixed, where for convenience we assume βδ ∈ N. Let N be fixed. For any function g : [0, β] × [0, β] → [−1, 1] we define the event A(δ, , g) ⊂ N by β − 1, δ |g(u, s) − R(σ (u), σ (s))| ≤ 2 ∀ (u, s) ∈ [0, β) × [0, β)}.
A(δ, , g) = {σ : g(iδ, jδ) ≤ R(σ (iδ), σ ( jδ)) < g(iδ, jδ) + ∀ i, j ≤
When there is no possibility of confusion we suppress the dependence on δ, and denote this event by A g . Let Sδ () be the set of functions which are constant on the squares [ jδ, ( j + 1)δ) × [kδ, (k + 1)δ) for j, k ∈ {0, . . . βδ − 1} and take values in {i : i ∈ [− 1 , 1 ] ∩ N}. We define the event A = A(δ, ) = ∪g∈Sδ () A(δ, , g). Though A definitely does not cover the full sample space N , we claim it is enough to prove convergence of the truncated pressure p NA (β, λ) = −
1 log Z NA . N
More precisely suppose = N , δ = δ N . If − N log δ N is sufficiently large as N tends to infinity, then A(δ , ) lim E p N (β, λ) − p N N N (β, λ) = 0. (4.6) N →∞
Indeed, let Niδ (σ ) denote the total number of jumps made by the spin path σ in the time interval [iδ, (i +1)δ]. To determine how large to take − log δ with N , let A∗ ⊂ N be defined by N }. A∗ = {σ : max Niδ (σ ) ≤ β 4 i≤ δ −1 Then Ac ⊂ Ac∗ so that
Ac
c
Z NA (β, λ) Z NA (β, λ)
≤
Z N ∗ (β, λ) Z NA∗ (β, λ)
.
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By Jensen’s inequality,
% Ac exp ζ N (σ ) Z N ∗ (β, λ) E log 1 + A E ≤ log 1 + dν(σ ) . Ac∗ Z N ∗ (β, λ) Z NA∗ (β, λ)
(4.7)
By the Cauchy-Schwarz inequality,
−2 21 1
exp ζ N (σ ) A∗ 2 E Z N (β, λ) E ≤ E exp 2ζ N (σ ) Z NA∗ (β, λ) 1 % 2
1 2 2 ≤ E exp 2ζ N (σ ) E exp ζ N (σ ) dν(σ ) ν(A∗ )−1 . ν(A∗ ) A∗ The last line follows from Jensen’s inequality applied with respect to the path measure β β 2 1 A∗ dν ν(A∗ ) . Since 2ζ N (σ ) is a Gaussian random variable with variance N 0 0 R (σ (u), σ (s))duds, % β% β
N 2 E exp 2ζ N (σ ) = exp R (σ (u), σ (s))duds . 2 0 0 Similarly, after a short calculation we have
2 E exp ν(A∗ )
% ζ N (σ ) dν(σ ) A∗
exp
N 2
= %
% A∗ ×A∗
β 0
%
β
R 2 (σ (u), σ˜ (s))duds
0
dν(σ )dν(σ˜ ) . ν(A∗ )2
As a result of the preceding estimates
% Nβ 2 ν(A c ) exp ζ N (σ ) dν(σ ) ∗ E dν(σ ) ≤ e 2 . c ν(A exp ζ (σ ) dν(σ ) ∗) A∗ N A∗ Standard calculations imply that for all small enough and δ < 2 , 1 ν(Ac∗ ) ≤ eC N log δ ν(A∗ ) δ
(4.8)
for some constant C > 0 depending only on the transverse field strength λ. If we require δ and to be chosen so that C log 1δ ≥ β 2 + N1 log 1δ , then putting estimates together and taking the appropriate limits proves our claim. Let us note that these conditions can be arranged by letting N = N − /4 , δ N = e 1
1/ 2 2β 2
−N
(4.9)
and taking N large. to showing that the mean of the truncated quenched pressure Thus we are reduced E p NA(δ N , N ) (β, λ) converges. For convenience of exposition we first consider subsequences of the form Nk = N0 n k for some N0 , n ∈ N. For any k, we may view σ ∈ Nk
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N. Crawford
as an n-tuple of spin paths (σ (1) , . . . , σ (n) ) so that σ (l) ∈ Nk−1 . In order to compare thermodynamics at consecutive system sizes, let us define the interpolating Hamiltonian ζ Nk (t, σ ) =
√
tζ Nk (σ ) +
n √ (l) 1−t ζ Nk−1 (σ (l) ), l=1
(l) where ζ Nk−1 (σ (l) ) involve disorder couplings which are mutually independent and independent from the couplings in ζ Nk (σ ) . In addition, for any F ⊂ N , we introduce the
partition function
% Z NF (t)
exp ζ N (t, σ ) dν(σ ).
= F
Let , δ > 0 be fixed for the moment, not necessarily as N , δ N . Let A˜ g = A˜ Nk (δ, , g) = {σ = (σ (1) , . . . , σ (n) ) : σ (l) ∈ A Nk−1 (δ, , g)}. Obviously, for all k, A˜ g ⊂ A g . Therefore A˜
A
− log Z Nkg (t) ≤ − log Z Nkg (t).
(4.10)
For any g ∈ Sδ () consider (g)
φ Nk (t) = −
1 A˜ E log Z Nkg (t). Nk
After a bit of work employing the integration-by-parts formula we arrive at d (g) φ (t) = dt Nk ⎡# $ A˜ g ,(1)⎤ % % n % % 1 β β 2 (l) 1 ⎣ β β 2 ⎦ R (σ (u), σ (s))duds− R (σ (u), σ (l) (s))duds − E 8 n 0 0 0 0 1 t ⎡# ⎤ $ ˜ ,(2) A % β% β g n % β% β 1 ⎣ 1 ⎦, + E R 2 (σ (u), σ˜ (s))duds − R 2 (σ (l) (u), σ˜ (l) (s)) 8 n 0 0 0 0 1
t
(4.11) A˜ g ,(1)
where ·t
corresponds to the truncated Gibbs weight determined by the Hamiltonian A˜ ,(2)
ζ Nk (t, σ ) and ·t g corresponds to the product Gibbs weight determined by an independent pair of spin paths σ , σ˜ and Hamiltonian ζ Nk (t, σ ) + ζ Nk (t, σ˜ ). We stress that corresponding terms in this pair Hamiltonian involve the same realizations of disorder. As the function f (x) = x 2 is Lipschitz continuous on [−1, 1] with constant 2, from the definition of A˜ g the first term in (4.11) can be bounded by β 2 in absolute value. (g) Since f is convex, the second term is less than or equal to zero. Thus, by evaluating φ Nk at zero and one and applying (4.10) we have −1 −1 A Ag + β2 E log Z Nkg ≤ E log Z Nk−1 Nk Nk−1 for any g ∈ Sδ ().
Mean Field Quantum Spin Glasses
837
Next, by Lemma 4.5, we have 1 2u 2 1 Ag Ag P log Z Nk (β, λ) − E log Z Nk (β, λ) ≥ u ≤ 2 exp − 2 Nk . Nk Nk β Setting u = , there exists a set S so that
2 2 4 P(S) ≥ 1 − 2 exp − 2 Nk−1 δ β
and so that on S −1 −1 A Ag log Z Nkg (β, λ) ≤ log Z Nk−1 (β, λ) + (2 + β 2 ) Nk Nk−1 for all g ∈ Sδ ().
1
−1/4
N2 Now let us specialize so that δ = exp(− 2β 2 ) and = Nk Let
Fk = {
, i.e. δ Nk , Nk as in (4.9).
−1 A(δ N , N ,g) log Z Nk k k (β, λ) ≥ Nk −1 A(δ N , N ,g) −1/ −1/ log Z Nk−1k k (β, λ) + (2 + β 2 )Nk 4 for some g ∈ Sδ(Nk ) (Nk 4 )}. Nk−1
A standard application of the Borel-Cantelli lemma implies that P (Fk i.o.) = 0. Therefore, for Nk large enough and δ = δ Nk , = Nk we have A(δ,,g) A(δ,) Z Nk (β, λ) = Z Nk (β, λ) g∈Sδ ()
≥
e−(2+β
3 2 )N /4 k
g∈Sδ ()
≥ e−(2+β
3 2 )N /4 k
e−(n−1)β
A(δ,,g)
Z Nk−1
1 −2 N /2 k
(β, λ)
1−n
Nk 4
n
A(δ,)
Z Nk−1 (β, λ)
n
,
where we used the inequality k1 xin ≥ k 1−n ( k1 xi )n for n ≥ 1 and xi ≥ 0. By (4.6) we have A(δ N , N ) 1 A(δ N , N ) E log Z Nk−1k k (β, λ) − log Z Nk−1k−1 k−1 (β, λ) −→ 0. k→∞ Nk−1 A(δ Nk , Nk )
Therefore the sequence of truncated pressures − N1k log Z Nk decreasing and converges a.s., though the limit may be −∞. A(δ N , N )
(β, λ) is nearly
Let X k = − N1k log Z Nk k k (β, λ). Then (E [X k ])∞ k=1 is a uniformly bounded sequence, so the concentration inequality (4.5) along with the Borel-Cantelli lemma implies that (X k )∞ k=1 is a bounded sequence a.s. and the convergence is to a finite nonrandom constant f (β, λ). It is now a small matter to prove convergence of the truncated quenched averages along these subsequences: The bound (4.5) implies that X k are uniformly integrable, i.e.
838
N. Crawford
lim sup E |X k |1|X k |≥κ = 0,
κ→∞ k
N
1/ 4
thus E [X k ] → f (β, λ) as well. Finally, as − Nk log δ Nk = 2βk 2 , the arguments given
after (4.6) imply that limk E p Nk (β, λ) = f (β, λ). Another application of the concentration inequality then implies that p Nk (β, λ) → f (β, λ) a.s. It remains to prove uniqueness of this limit and then convergence along arbitrary subsequences. We shall briefly sketch a standard argument from which these two statements follow. Fix any (not nesc. geometric) sequence (Nk )∞ k=1 and some integer N ∈ N. For k large we compare the system at size N with the system at size Nk . Now Nk = Q k N + Rk , where Rk ≤ N . Dividing the spin path σ Nk into Q k subpaths of size N and one subpath of size Rk , we may proceed analogously to the above to show that there exists k0 ∈ N and a set S with 2 2 4 P(S ) ≥ 1 − 2 exp − 2 N δ β on which we have −1 Q k N −1 1 A(δ,,g) A(δ,,g) log Z N log Z Nk (β, λ) ≤ (β, λ) + O(1) + (2 + β 2 ) Nk Nk N Nk for all g ∈ Sδ () and for all k > k0 . Here, the term O(1) accounts for the system of size Rk (among other errors) and is maximized so that it depends on N , δ and but not on Nk . Thus on S , A(δ,,g) A(δ,) Z Nk (β, λ) Z Nk (β, λ) = g∈Sδ ()
≥
e−(2+β
2 ) N
g∈Sδ ()
≥ e−(2+β
2 ) N
k +O(1)
k +O(1)
A(δ,,g)
ZN
(β, λ)
Qk
Q k −1 Qk δ2 Z NA(δ,) (β, λ) .
If we choose −C log δ > β 2 then the argument in support of (4.6) along with (4.5) implies −1 1 A(δ,) lim log Z Nk (β, λ) + log Z Nk (β, λ) = 0 a.s. k→∞ Nk Nk Therefore, combining this observation with the previous paragraph lim sup k→∞
on
−1 1 −1 A(δ,) log Z N log Z Nk (β, λ) ≤ (2 + β 2 ) − log(δ 2 ) + (β, λ) Nk N N
S.
Choosing δ = δ N , = N as in (4.9) and applying the Borel-Cantelli lemma once more −1 −1 log Z N (β, λ) a.s. log Z Nk (β, λ) ≤ lim inf lim sup N →∞ N N k k→∞
As the sequence (Nk )∞ k=1 was arbitrary, we may conclude convergence of the pressures occurs a.s. To prove L 1 convergence, we proceed as above.
Mean Field Quantum Spin Glasses
839
Acknowledgement. The author would like to thank Marek Biskup for a number of suggestions (in particular with respect to the universality results), Shannon Starr for numerous useful discussions, Lincoln Chayes for his help in preparing this manuscript, and referees the for a number of useful comments. This research was partially supported by the NSF grants DMS-0306167 and DMS-0505356.
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