Coherent States in Geometric Quantization

Home , Heat kernel

arXiv:math/0502026v2 [math.SG] 15 Mar 2005 hyaentobt fafiuilvco ne h cino i gro Lie a of ex action for the – under coincide states vector coherent fiducial Perelomov a and of states coherent orbits not instance are specific they from them coh distinguish the to to order refer ma in will st that We states coherent demonstrates coherent considerations. general here corresponding from taken the follow approach states for quantiza abstract geometric form the from explicit hand, arising an space finding Hilbert available, quantum the aut for the to basis clear not simply direction). is this it in – case progress non-compact recent the in different are osqec,i sdffiutt oto h lblbhvo ftequ the of behavior global the control to h difficult that necessarily condition is not it does consequence, state a quantum a Kähler quantization almost ipe nteams ¨he ae ncnrs oteqatzto o quantization althoug the cases, to both contrast in In Kähler similar case. almost is non-Kähler ca the describe the in will simpler In we which cor simplicity. states metaplectic coherent for The quantizing omitted of be methods quantization. will geometric and of construction machinery the using hoyi h otx fgoercqatzto.Ti ae sa at an is paper This quantization. geometric of context literat the physics mathematical in the theory in ubiquitous are states Coherent Introduction 1 1 esol on u htsmlci oeetsae r o,i gen in not, are states coherent symplectic that out point should We semi-construc is make will we that states coherent of definition The ewl en oeetsae soitdt nabtayintegral arbitrary an to associated states coherent define will We -aladdress: E-mail ahmtc ujc lsicto 20) rmr 53D50, Primary (2000): Classification genera Subject state, Mathematics coherent quantization, geometric Keywords: formula explicit find also We surfaces. sp Riemann symplecti function states. coherent analytic of weighted symplectic properties of Bargmann’s the of ove to properties study sic dedicated we Specifically, is quantization. the paper Berezin-Toeplitz use the and spaces of Hilbert rest quantum the non- Kähler to and the associated both resu kernels in recent quantization some geometric collect goal of to tions main and systems Our such of quantization. construction geometric by manifolds plectic nti ae esuyoecmlt ytm fchrn stat coherent of systems overcomplete study we paper this In M [email protected] scmatbcmsesnil hsi o osyta h properties the that say to not is This essential. becomes compact is oeetSae nGoercQuantization Geometric in States Coherent M lotKalr[,2]adSpin and 24] Kähler [6, almost : thSaeUiest,Lgn T83230,USA 84322-3900, UT Logan, University, State Utah eateto ahmtc n Statistics and Mathematics of Department ila .Kirwin D. William Abstract 1 cmltns,smlci nlge fteba- the of analogues symplectic rcompleteness, c ie ega kernel Bergman lized ¨ he ae.W hnsuytereproducing the study then We Kähler cases. 1,2,2,2]qatzto.Tedfiiinof definition The quantization. 29] 25, 23, [13, t.W ei yrcligtebscconstruc- basic the recalling by begin We lts. cs n h mxmlycascl behavior classical’ ‘maximally the and aces, o ypetcchrn ttso compact on states coherent symplectic for s odfiesmlci oeetsae.The states. coherent symplectic define to m r ogv ytmtcteteto the of treatment systematic a give to are s eodr 32A36 Secondary ml nteqatztosof quantizations the in ample oeetsae n h corresponding the and states coherent c yo h rdtoa rpriso coherent of properties traditional the of ny 1 rn ttscntutdhr ssymplectic as here constructed states erent r.Ytteesest ealc fgeneral of lack a be to seems there Yet ure. sascae ocmatitga sym- integral compact to associated es s. ep oprilyfilti gap. this fill partially to tempt v iehlmrhclclfr.As form. local holomorphic nice a ave ¨he aiod nbt Spin both in Kähler manifold, a f in ic uhabssi o always not is basis a such Since tion. p nsm pcfi ae symplectic cases specific some In up. nu ttsad sw ilse the see, will we as and, states antum tsmyb icl.O h other the On difficult. be may ates o o opoed(e 2]frsome for [24] (see proceed to how hor opc ypetcmnfl ( manifold symplectic compact eto ilntpa oei this in role a play not will rection o ehia esn ssomewhat is reasons technical for h eteeaea es w common two least at are there se ie twl eedo hieof choice a on depend will It tive. rl fPrlmvtp 2] i.e. [27]; Perelomov-type of eral, ftechrn states coherent the of math.SG/0502026 C and S ,ω M, 2 c (see and ) 6), where the methods of Perelomov yield a reproducing system in the quantum Hilbert space that arises from§ geometric quantization. In fact, when this happens, the two construction must agree (Theorem 3.2). The coherent state map we will study has appeared in a different (but in some ways equivalent) form in [7] where it is used to prove a symplectic analogue of Kodaira’s embedding theorem. In [7], the coherent states lie in a different quantum Hilbert space and are associated to a circle bundle over M. In [24], Ma- Marinescu analyze the semiclassical properties of generalized Bergman kernels. This analysis leads to the notion of a peak section, which is related to symplectic coherent states ( 3.2). In [28], Rawnsley globalized the constructions of Perelomov [27] to make a general definition of a coherent§ state on Kähler manifolds. When applied to the specific case of a principal C×-bundle over M the coherent states constructed here and the Rawnsley type coherent states are related (see 3.3). The properties of Rawnsley-type coherent states, as well as the geometric interpretation of Berezin quantization§ which they provide, are studied in [9, 10]. Symplectic coherent states are associated to M itself, and can be associated to a non-Kähler symplectic manifold. Symplectic coherent states generalize many of the systems constructed in the literature. The most basic examples of coherent states are those associated to the complex plane – the simplest Kähler manifold. These states are known as Segal-Bargmann-Heisenberg-Weyl (or some permutation thereof) coherent states, or often more simply as canonical coherent states; see [12], for example, where they are developed using projective representations of the symplectic group on the quantum Hilbert spaces of 2.2. Canonical coherent states are briefly described in 6. [17] contains a survey on many traditional mathematical§ aspects of canonical coherent states, the introduction§ of which also includes a discussion of what, in general, should be called a coherent state. Following [17] (and to some extent popular opinion) we will define a system of coherent states to be a set x H x M of quantum states in some quantum Hilbert space H, parameterized by some set M, such{| that:i ∈ ∈ }

1. the map x x is smooth, and 7→ | i 2. the system is overcomplete; i.e.

x x dµ(x)= 1H. | ih | ZM Physicists usually call property (2) completeness. As we will see, the map x x is actually antiholomorphic in a sense appropriate to non-K¨ahler manifolds. The parameterizing set M7→is | generally,i and for us will be, a classical phase space; i.e. an integral symplectic manifold. We motivate our construction of coherent states by recalling some basic quantum mechanics. The following observations are well known ([17] and [20, Chap. 3]). Let us, for a moment, eschew deﬁnitions and rigor in order to see how to proceed. The position space wave function representing a state ψ is ψ(x)= x ψ , where x is a coherent state localized at x. The position space wave function of the coherent| i stateh x| iis then | i | i K(x, y) := K (y) := y x . x h | i We can use this to rewrite the equation x ψ = ψ(x) as h | i

K(x, y)ψ(y) dy = ψ(x). (1.1) Z A function K that satisfies (1.1) for some space of functions is called a reproducing kernel for that space. Reading the above discussion backwards, we see that a coherent state can be defined in terms of a reproducing kernel. We will use this approach and define symplectic coherent states in terms of reproducing kernels for the quantum Hilbert spaces arising from geometric quantization, sometimes called generalized Bergman kernels; properties (1) and (2) will then follow. This construction is well known for the Kähler quantization of C and yields the Bergman reproducing kernel, which in turn yields the canonical coherent states. The asymptotics of generalized Bergman kernels are studied in [6, 7, 11, 23, 24] and will be important when we consider the semiclassical limit. Example A reproducing kernel for the space S(R) of Schwartz functions on R is the Dirac distribution δ(x y). −

2 Symplectic coherent states associated to the Poincarédisc, and hence via Riemann uniformization to compact Riemann surfaces of genus g 2, were used in [18, 19] to study the semiclassical limit of the deformation quantizations of these surfaces.≥ We will give explicit formulas for these coherent states in 6. The rest of the paper is organized as follows: in 2 we review the geometric quantization of a Kähler§ manifold (M,ω) and two generalizations to the non-Kähler§ case known as the almost Kähler and Spinc quantizations of (M,ω). In 3 we define the reproducing kernel for the quantum Hilbert space associated to M by geometric quantization§ and use it to define symplectic coherent states. We also describe symplectic analogues of some analytic function space results of [2], and the relationship between Rawnsley-type and symplectic coherent states. In 4 we discuss the overcompleteness relation and the coherent state quantization induced by the symplectic coherent§ states. In 5 we show that symplectic coherent states are the most classical quantum states and consider the semiclassical§ limit. Finally, in 6 we apply the constructions of 3 to compact Riemann surfaces. § §

2 Background and Notation 2.1 Prequantization ω Throughout we assume that (M,ω) is an integral compact symplectic manifold; i.e. 2π is in the image 2 Z 2 of the map H (M; ) HDR(M). The basic object of geometric quantization is an Hermitian line bundle π : ℓ M with compatible→ connection with curvature iω, known as the prequantum line bundle. The → ∇ − k k ω existence of ℓ is guaranteed by the integrality of ω; in fact the Chern character of ℓ⊗ is ch(ℓ⊗ ) = exp(k 2π ). For detailed accounts of geometric quantization see [14, 31]. We denote by h : ℓx ℓx C the Hermitian structure on ℓ. We follow the physics convention that the ⊗ → 2 first term is conjugate linear. All tensor products will be taken over C. The norm of q Lx is q = h(q, q). h induces an Hermitian structure on Γ(ℓ) : for s ,s Γ(ℓ) ∈ | | 1 2 ∈ s s = h(s (x),s (x))ǫ (x), h 1| 2i 1 2 ω ZM where 1 n ω n ǫ = ∧ ω 2π n! is the Liouville volume form on M. We have included recurring factors of 2π in the Liouville form. This will simplify some formulas later on, but will also have the effect that our formulas differ slightly from some of those in the literature. The norm of s Γ(ℓ) is s 2 = s s . ∈ k k h | i k We will be occasionally interested in the semiclassical k = 1/~ limit of ℓ⊗ . The structures k → ∞ (h, , ) on ℓ induce corresponding structures on ℓ⊗ which we will denote by the same symbols. The h·|·i ∇ k curvature of ℓ⊗ is ikω. − The program of geometric quantization associates to (M,ω) a Hilbert space H and a map Q : C∞(M) H H0 2 → Op( ). To begin, we define the prequantum Hilbert space k to be the L completion, with respect to the k Liouville measure, of the set of square integrable sections of ℓ⊗ :

H0 = s Γ(ℓ k) s < . k { ∈ ⊗ k k ∞}

The Kostant-Souriau quantization of the Poisson-Lie algebra C∞(M) is the map

(k) i 0 f C∞(M) Q (f)= + f Op(H ) ∈ 7→ KS −k ∇Xf ∈ k

where Xf is the Hamiltonian vector ﬁeld deﬁned by

y Xf ω = df.

In 4 we will recall an alternate quantization of C∞(M). § H0 As is well known, k is too large for the purposes of quantization [31, Ch 9]. If (M,ω) is Kähler there H H0 is a standard method of choosing a subspace k k. In the non-Kähler case, there are (at least) two reasonable methods: almost Kähler quantization [6,⊂ 24] and Spinc quantization [13, 23, 25, 29]. We will review these three constructions in the next three sections.

3 2.2 Kähler Quantization If (M,ω,J) is a Kähler manifold with complex structure J, there is a natural method of reducing the prequantum Hilbert space H0. The complexified tangent bundle of M decomposes into i eigenspaces of J : k ± (1,0) (0,1) TMC = TM TM . (2.2) J ⊕ J k (1,0) A section s Γ(ℓ⊗ ) is said to be polarized if it is tangent to the Kähler polarization TMJ ; i.e. if X s =0 for each X ∈ Γ(TM (0,1)). ∇ ∈ 2 k The quantum Hilbert space is defined to be the L closure of the set of polarized sections of ℓ⊗ :

H = s Γ(ℓ k) s < ,s polarized . k { ∈ ⊗ k k ∞ }

This quantum Hilbert space can be described in terms of a Dirac-type operator [4, Chap. 3]. The decomposition (2.2) induces a decomposition

n n p,q p (1,0) q (0,1) Λ∗(T ∗M)= Λ (T ∗M)= Λ (T ∗MJ ) Λ (T ∗MJ ). (2.3) p,q⊕=0 p,q⊕=0 ⊗

p,q k p,q+1 k k Let ∂ :Ω (M,ℓ⊗ ) Ω (M,ℓ⊗ ) denote the Dolbeault operator twisted by ℓ⊗ . Hodge’s theorem says k → ∗ 2 k that ker(∂k +∂k) is isomorphic to the sheaf cohomology space H∗(M, (ℓ⊗ )). Kodaira’s vanishing theorem q k O H then tells us that for k suﬃciently large, H (M, (ℓ⊗ )) = 0 for q > 0, and hence that k = ker ∂k Γ(ℓ⊗k). H O The dimension of k can be computed with the Riemann-Roch-Hirzebruch theorem:

H k k (1,0) dk := dim k = RR(M,ℓ⊗ )= ch(ℓ⊗ )T d(TMJ ). (2.4) ZM In particular, since we assume M is compact, d < . k ∞ 2.3 Almost Kähler Quantization We suppose now that (M,ω) is an integral compact symplectic manifold, not necessarily Kähler. Every such manifold admits an ω-compatible almost complex structure J, and any two such choices are homotopic. The complexified tangent bundle again decomposes as in (2.2). If J is not integrable (i.e. M is not a complex k (1,0) manifold), then there may be no sections s Γ(ℓ⊗ ) which are tangent to TMJ . In this case, more work is required to define a quantum Hilbert space.∈ In this section, we describe one such method, known as almost Kähler quantization, which was introduced ∗ 2 in [6] based on results in [15] and further studied in [11, 23, 24]. The idea is to replace (∂k + ∂k) , which does not exist if J is not integrable, with the rescaled Laplacian ∆k := ∆ nk, where ∆ is the Laplacian for the metric g = ω( ,J ). In the Kähler case, these two quantities are equated− by the Bochner-Kodaira · · ∗ 2 formula: ∆k = 2(∂k + ∂k) . The main result of [15] is (in a slightly sharpened form due to [6, 24]):

Theorem 2.1 Given an integral symplectic manifold (M,ω) with ω-compatible almost complex structure, there exists a constant C and a positive constant a such that for k sufficiently large, 1. the first d eigenvalues of ∆ (in nondecreasing order) lie in the interval ( a,a), and k k − 2. the remaining eigenvalues lie to the right of nk + C, where dk is the Riemann-Roch number as in (2.4). Motivated by this, the quantum Hilbert space is defined to be

(k) (k) Hk := spanC θ ,...,θ { 1 dk } where θ(k) is an eigenfunction of ∆ with eigenvalue λ and λ λ λ . Note that just as in j k j 1 ≤ 2 ≤···≤ j ≤ · · · the K¨ahler case, the dimension is given by the Riemann-Roch number dk. This quantization has excellent semiclassical (~ = 1/k 0) properties [5, 6, 23, 24]. It also has the advantage that the quantum Hilbert →

4 k c space consists of sections of ℓ⊗ . Spin quantization, described in the next section, does not have either of these properties, although the Spinc quantum states are true zero-modes of a Dirac-type operator. In analogy with the K¨ahler case, we will call the elements of Hk polarized sections. Since M is compact and the polarized sections are smooth and the quantum Hilbert space is ﬁnite dimensional, Hk is a closed 2 k subspace of L (M,ℓ⊗ ).

2.4 Spinc Quantization

c The idea of Spin quantization is to find a suitable generalization of ∂k for the non-Kähler case. We will briefly review the relevant details here. See [22, App D] for a more complete account of the Spinc bundle, and [13, 23, 25, 29] for Spinc quantization. Let J be an ω-compatible almost complex structure on (M,ω) so that g = ω( ,J ) is a Riemannian metric c · · 0, on M. The Spin bundle associated to the data (M,ω,J) is defined as S(M) := Λ ∗(T ∗M) according to 0, k 0, k the decomposition (2.3). There is a Dirac-type operator ∂/k :Ω ∗(M,ℓ⊗ ) Ω ∗(M,ℓ⊗ ) that decomposes 0,even k 0,odd k 0,odd k → 0,even k into (∂/k)+ :Ω (M,ℓ⊗ ) Ω (M,ℓ⊗ ) and (∂/k) :Ω (M,ℓ⊗ ) Ω (M,ℓ⊗ ). The quantum Hilbert space→ associated to the data (M,ω,J− ) is the virtual→ vector space

Hk := ker(∂/k)+ ker(∂/k) . ⊖ − c The dimension is again given by the Riemann-Roch number dk (2.4). There is a Spin analogue of the Kodaira vanishing theorem [6, 23] which insures that Hk is an honest vector space for k sufficiently large. k The metric g on M and the Hermitian structure h on ℓ⊗ combine to give an Hermitian structure, also k denoted by h, on S(M). Although the zero-modes of ∂/k are not sections of ℓ⊗ since they have higher degree components, their norms are asymptotically concentrated on the zero degree part; i.e. there exists a constant 1/2 C > 0 such that for k sufficiently large s+ Ck− s0 , for each s Hk and where s = s0 + s+ denotes the decomposition of s into zero and higherk k≤ degree componentsk k [6, 23].∈ Just as in the Kähler and almost Kähler cases, we will refer to the elements of Hk as polarized sections. Also, since M is compact and polarized sections are again smooth, and the space of them is finite dimensional, 2 Hk is a closed subspace of L (M,S(M)). We will assume throughout that k is chosen sufficiently large to ensure the validity of the relevant vanishing/existence theorem.

3 Coherent States

In this section we will construct coherent states associated to an integral symplectic manifold (M,ω).

3.1 Reproducing Kernels For the Kähler, almost Kähler and Spinc quantizations of a compact symplectic manifold (M,ω) the quantum Hilbert space is finite dimensional with dimension given by the Riemann-Roch formula (2.4). We will see below that since dk < there exists a reproducing kernel for the quantum Hilbert space. In the Kähler ∞ k case, the existence of a reproducing kernel may also be established by trivializing ℓ⊗ – polarized sections are locally holomorphic and standard methods from complex analysis can be used. In the non-Kähler cases no such nice local form is known to the author, and the compactness assumption and resulting finite dimensionality become essential. For two vector bundles π : E M , j =1, 2 we define E ⊠E := π∗E π∗E M M . If a vector j j → j 1 2 1 1 ⊗ 2 2 → 1 × 2 bundle E has an Hermitian structure h, we will identify E E∗. For u, v Ex we define u v := h(u, v). Similarly, we identify u u = h(u,u). Moreover, if L is a line≃ bundle, we will∈ identify v w ·= h(v, w) for ⊗ ⊗ v Lx, w Lx so that L L C. Combining these definitions, we see that u v w = h(v, w)u. These conventions∈ ∈ agree since u ⊗v ≃w = h(v, w)u = h(v,u)w = u v w. Finally, we define⊗ · u v := u v. Many of the results in⊗ this⊗ section hold for all three methods⊗ · of quantization. To unify⊗ notation⊗ and avoid repetition we define

k k ℓ⊗ for Kahler and almost Kahler quantization L := 0, k c Λ ∗(T ∗M) ℓ⊗ for Spin quantization. ( ⊗

5 Let θ(k) dk be a unitary basis for H . { j }j=1 k Deﬁnition 3.1 The reproducing kernel K(k) Γ(Lk ⊠ Lk) is the section ∈

dk K(k)(x, y) := θ(k)(x) θ(k)(y). j ⊗ j j=1 X K(k) is also known as a generalized Bergman kernel. Note that K(k) does not depend on the choice of unitary basis. In the K¨ahler and Spinc quantization schemes, the quantum Hilbert space is the kernel of a Dirac-type operator, and the reproducing kernel is the large t limit of the associated heat kernel (see 4.2). Although it is not a function on M M, the reproducing kernel K(k) has many of the same properties§ enjoyed by reproducing kernels for analytic× function spaces.

Theorem 3.2 K(k) is the unique polarized section of Lk ⊠ Lk such that

K(k)(x, y) s(y) ǫ (y)= s(x) s H . · ω ∀ ∈ k ZM Proof: Suppose there are two reproducing kernels. Their difference evaluated against an arbitrary section s Hk must be zero. Hence this difference must be in Hk (Hk)⊥. But since both kernels are polarized, the∈ difference is in H H and is therefore zero. ⊗ k ⊗ k The restriction of K(k) to the diagonal is a smooth function.

(k) Deﬁnition 3.3 The coherent density is the smooth function ε C∞(M) deﬁned by ∈

dk ε(k)(x) := K(k)(x, x)= θ(k)(x) 2. | j | j=1 X (k) (k) (k) Since M is compact and ε is smooth and nonnegative, we may deﬁne a measure on M by µ = ε ǫω which we will call the coherent measure.

2 k 2 k Since Hk L (M,L ) is a closed subspace, there is a projection Πk : L (M,L ) Hk. To find the Schwartz kernel⊆ of this projection, we need the following observation (which follows from→ the facts that L2(R2n, C) is separable and that since M is compact it has a finite cover by open sets which are diffeomorphic to subsets of R2n).

Theorem 3.4 L2(M,Lk) is a separable Hilbert space.

(k) Theorem 3.5 The Schwartz kernel of Πk is the reproducing kernel K .

Proof: We need to show that

(Π s) (x)= K(k)(x, y) s(y)ǫ (y) s L2(M,Lk). k · ω ∀ ∈ ZM H 2 H Since Πk is a projection, it is uniquely characterized by im Πk = k and Πk = Πk = 1 k . (k) 2 k (k) dk Let θ ∞ be a unitary basis for L (M,L ) such that spanC θ = H . Then for each s { j }j=1 { j }j=1 k ∈ 2 k j N j (k) 2 L (M,L ) there exists s C ∞ such that s s θ 0 as N . We then have { ∈ }j=1 k − j=1 j k → → ∞

dk P ∞ K(k)(x, y) s(y)ǫ (y)= θ(k)(x) slh(θ(k)(y),θ(k)(y))ǫ (y). (3.5) · ω j j l ω M j=1 M Z X Z Xl=1

6 Using H¨older’s inequality we see that

∞ l 2 (k) (k) ∞ l 2 (k) 2 (k) 2 2 s h(θj (y),θl (y)) ǫω(y) s θj (y) θl (y) = s . | | M | | ≤ | | k k k k k k Xl=1 Z Xl=1 Hence, the integrand is absolutely integrable and we may interchange the integral and sum in (3.5) to obtain

dk K(k)(x, y) s(y)ǫ (y)= θ(k)(x) H · ω j ∈ k M j=1 Z X as desired. Moreover, for s = dk sj θ(k)(x) H , we easily obtain Π2 s = Π s in terms of K(k) since all of the j=1 j ∈ k k k relevant sums are ﬁnite. We conclude that K(k) is the Schwartz kernel of Π as desired. P k

3.2 Coherent States We now deﬁne the coherent states associated to an integral compact symplectic manifold (M,ω).

Definition 3.6 The coherent state localized at x M is ∈ dk Φ(k) := K(k)(x, )= θ(k)(x) θ(k). x · j ⊗ j j=1 X In order to distinguish these coherent states from others, we will sometimes refer to them as symplectic coherent states. Observe that Φ(k) depends smoothly and antiholomorphically on x in the generalized sense: a section is holomorphic on a symplectic manifold if it is polarized. (k) k H (k) Since Φx Lx k, it is necessary to investigate how Φx should be interpreted as a quantum state. ∈ ⊗ k (k) Consider the case of almost Kähler quantization. If we trivialize ℓx⊗ with a unit, Φx becomes a well-defined H (k) (k) k state in k via the identification 1 θj θj . The different unit trivializations of ℓx⊗ are parameterized by (k) ⊗ ≃ U(1) which means that Φx is a well-defined quantum state up to a phase – the usual situation in quantum mechanics. On the other hand, quantum states are most properly regarded as rays in the projective Hilbert (k) space PHk. It follows from the above discussion that the map x M C Φx PHk is well-defined and k ∈ 7→ · ∈ (k) smooth. If it is not possible to find a global unit section of ℓ⊗ then there is no smooth lift of Φ to Hk. In [24], this map is shown to be asymptotically symplectic (as k ), asymptotically isometric with respect to the metric g = ω J, and, for k sufficiently large, an embedding→ ∞ (see [7] for similar results). We now return our attention◦ to the general case of almost Kähler or Spinc quantization. It is sometimes convenient to work with normalized states. In terms of coherent states, Definition 3.3 reads

(k) 2 (k) Theorem 3.7 Φx = ε (x). k k This is the reason ε(k) is called the coherent density. (k) Let Mk := x M ε (x) =0 . By Corollary 3.10, Mk is the complement of the base locus of Hk. For x M we will{ denote∈ the normalized6 } coherent state localized at x by ∈ k

(k) (k) (k) Φx Φx := x := . | i ε(k)(x) e (k) (k) p For x M we deﬁne Φx := x := 0. We can now state the reproducing property concisely: 6∈ k | i Φ(k) s = s(x) e h x | i ε(k)(x) x(k) s = s(x) x M, s H . h | i ∀ ∈ ∀ ∈ k q For x M the above is justiﬁed by Corollary 3.10. 6∈ k

7 C (k) C (k) PH H H If x Mk then Φx = x k so that we may deﬁne the projection proj x(k) : k k by ∈ · · | i ∈ | i → (k) 1 (k) (k) (k) (k) s (ε (x))− Φ Φ s = x x s . (3.6) 7→ | x ih x | i | ih | i (k) (k) (k) 1 (k) (k) We will also write projC (k) = x x = (ε (x))− Φx Φx for this projection. Φx | ih | | ih | The following is a generalization· of a result in [2] and is the basic reason why the quantum Hilbert space behaves in many ways like a weighted analytic function space, even on non-K¨ahler manifolds. We will further develop this analogy in 4.1. § Theorem 3.8 For each polarized section s H , ∈ k s(x) s ε(k)(x). | |≤k k q Proof: Let s H . We use the reproducing property of the coherent states to write, for x M , ∈ k ∈ k s(x) 2 = h(s(x),s(x)) = h( Φ(k) s , Φ(k) s )= Φ(k) s 2. | | h x | i h x | i |h x | i | The result then follows from the Cauchy-Schwartz inequality and Theorem 3.7. (k) (k) (k) If x Mk then ε (x) = 0 implies θj (x) = for all j. Since s is a linear combination of θj , this implies s(x)=06∈. This Theorem has two useful corollaries. The proofs are simple and left to the reader.

Corollary 3.9 The evaluation map ev : s H s(x) Lk is continuous. x ∈ k 7→ ∈ x We can also prove Corollary 3.9 directly – it follows from the facts that dk < and that the sets (k) dk (k) dk ∞ θj (x) j=1 and θj j=1 are bounded, which in turn follow from our assumption that M is compact. {| In the|} Kähler case,{k thek} compactness assumption on M can be lifted. The existence of the reproducing kernel, as well as Theorem 3.8, can then be deduced from Jensen’s formula [21, p324]: if f is holomorphic on the closed disc of radius R and f(0) =0, and the zeroes of f in the open disc, ordered by increasing moduli 6 and repeated according to multiplicity, are z1, ..., zN , then f f(0) k kR z z . | |≤ RN | 1 ··· N | Unfortunately, the author is unaware of nice local forms of polarized sections in the Spinc and almost Kähler quantizations of a non-Kähler manifold. This makes the assumption that M is compact, or more precisely that dk < , essential. Of course, it is also not clear whether the spectrum of the rescaled Laplacian has the requisite structure∞ to define the almost Kähler quantization of M in the non-compact case. On the other hand, most of the results of this paper hold for any choice of finite dimensional subspace H0 c of the prequantum Hilbert space k. The primary advantages of almost Kähler and Spin quantization are that they provide canonical methods for choosing such a subspace and that they provide enough structure to ensure a meaningful semiclassical limit (c.f. (5.15)). The next corollary justifies our definition of the normalized coherent state x(k) in the case that ε(k)(x)=0. | i

Corollary 3.10 ε(k)(x) = 0 if and only if s(x) = 0 for each polarized section s. In 3.4, 5, and 6 we will be interested in the semiclassical limit of the symplectic coherent states. It § § § (k) will be useful to express Φx in terms of the peak sections of Ma-Marinescu [24], deﬁned as follows: The (k) PH H Kodaira map Ψ : Mk k∗, which sends x Mk to the hyperplane s k s(x)=0 of sections → ∈ { ∈ (k) } (k) (k) which vanish at x, is base point free for large enough k. Construct an unitary basis θ ,...,θ ,Sx 1 dk 1 (k) (k) { − } such that θ (x)=0for 1 j d 1. Then Sx , called a peak section, is a unit norm generator of the j ≤ ≤ k − orthogonal complement of Ψ(k)(x). Observe that

(k) (k) (k) (k) Φ (y)= K (x, y)= Sx (x) S (y) (3.7) x ⊗ x (k) (k) 2 and also that ε (x)= Sx (x) . Moreover, | | S(k)(y) 2ǫ (y)=1. (3.8) | x | ω ZM

8 3.3 Rawnsley-Type Coherent States The coherent states defined in [28] for compact Kähler manifolds are a generalization of Bargmann’s principal vectors [2] to spaces of holomorphic sections. We will describe their relation to symplectic coherent states in this section. Due to Corollary 3.9, we are able to construct Rawnsley-type coherent states on any compact integral symplectic manifold. In this section, we will consider only the almost Kähler quantization of M so k that the prequantum bundle is ℓ⊗ . By Corollary 3.9, for each q ℓx we get a continuous map δq : Hk C by composing the evaluation evx ∈ k → (k) with the trivialization s(x) = δq(s)q⊗ . By the Riesz representation theorem, there exists eq Hk such that ∈ (k) k k e s q⊗ = δ (s)q⊗ = s(π(q)) s H . h q | i q ∀ ∈ k Observe that (k) k (k) ecq = c− eq (k) for 0 = c C. We will refer to the section eq as a Rawnsley-type coherent state. We can express the 6 ∈ (k) reproducing kernel, and therefore the symplectic coherent states, in terms of eq :

Theorem 3.11 Let x M and q ℓ . Then ∈ ∈ x (k) k (k) K (x, y)= q⊗ e (y). ⊗ q (k) k (k) Equivalently, Φx = q⊗ eq . ⊗ Proof: In terms of the unitary basis θ(k) dk for H we have { j }j=1 k

dk dk (k) (k) (k) (k) q k (k) k s(x)= Φ s = θ s θ (x)= θ s θ (x)q⊗ = e s q⊗ h x | i h j | i j h j | i j h q | i j=1 j=1 X X e q (k) k where θj is the trivialization of θj determined by a local section with value q⊗ at x. Therefore

e (k) q (k) eq = θj (π(q))θj . (3.9) j X e Hence we obtain, k (k) (k) q⊗ e =Φ . ⊗ q π(q)

Let s : M ℓ×. In [28], Rawnsley deﬁnes a function 0 → (1) (1) 2 η(x) := e e s0(x) . h s0(x)| s0(x)i | |

It is easy to check that this function is independent of s0. This function was also studied for K¨ahler M in [9, 10] and in the almost K¨ahler case in [7]. A short calculation using the previous Theorem yields:

Corollary 3.12 η = ε(1).

3.4 Transition Amplitudes In this section we deﬁne the 2-point transition amplitude for symplectic coherent states and show that it can be interpreted as a probability density on M.

Deﬁnition 3.13 The 2-point function, or transition amplitude, is

(k) (k) (k) 2 ψ (x, y) := x y C∞(M M). |h | i | ∈ ×

9 In terms of the reproducing kernel, the 2-point function is

K(k)(y, x) K(k)(x, y) ψ(k)(x, y)= · . ε(k)(x)ε(k)(y)

As expected, ψ(k)(x, x)=1. The Cauchy-Schwartz inequality

Φ(k) Φ(k) 2 Φ(k) 2 Φ(k) 2 |h x | y i | ≤k x k k y k (k) (k) implies ψ (x, y) [0, 1]. Since the map x C Φx is an embedding for k suﬃciently large [24], x = y (k) (k∈) (k) 7→ · 6 implies Φx =Φy so that ψ (x, y) = 1 if and only if x = y. 6 (k) Theorem 3.14 For each x Mk, ψ (x, y) is a probability density on M with respect to the coherent measure µ(k)(y). ∈

Proof: For each x with ε(k)(x) =0, 6 1 ψ(k)(x, y) dµ(k)(y)= K(k)(y, x) K(k)(x, y)ǫ (y)=1. (3.10) ε(k)(x) · ω ZM ZM In [10], Cahen-Gutt-Rawnsley deﬁne a 2-point function for K¨ahler M in terms of Rawnsley-type coherent states: (1) (1) 2 eq eq′ ψ′(x, y)= |h | i | (1) 2 (1) 2 eq e ′ k k k q k (1) where x = π(q) and y = π(q′). By Theorem 3.11 we see that ψ′ = ψ . Moreover, if the quantization is regular (i.e. ε(k)(x)= const for all k) then it follows from [10, Prop. 2 and eq 1.7] that ψ(k) = (ψ(1))k. The transition amplitude can be expressed in terms of peak sections by a simple calculation using equation (3.7):

(k) (k) (k) 2 Theorem 3.15 ψ (x, y)µ (y)= Sx (y) | | Theorem 3.14 is therefore equivalent to (3.8). (k) (k) Finally, we note here that the association to each ~ = 1/k, k Z+ of Hk, C Φx and µ deﬁnes a pure state quantization of the integral symplectic manifold (M,ω)∈ (see [20, p113] for· the deﬁnitions).

4 Berezin-Toeplitz Quantization

In this section we study the Berezin quantization [3] induced by the coherent state map Φ(k). This method of quantization is studied in detail in the context of analytic function spaces in [20]. The extension to Hilbert spaces of sections of the prequantum bundle can be described in terms of symplectic coherent states.

4.1 Overcompleteness and Characteristic Sets In this section we consider the most important property of coherent states: overcompleteness.

Deﬁnition 4.1 A system of coherent states x Hk x M is overcomplete with respect to a measure µ if {| i ∈ ∈ }

1. x y = 0 for all x, y M with x , y =0, and h | i 6 ∈ | i | i 6 2. x x dµ(x)= 1H . M | ih | k R

10 Theorem 4.2 The system of symplectic coherent states x(k) x M deﬁned in 3 is overcomplete with {| i ∈ } § respect to the coherent measure µ(k). In particular,

(k) (k) (k) x x dµ (x)= 1H . (4.11) | ih | k ZM Proof: We compute, for every s ,s H , 1 2 ∈ k s x(k) x(k) dµ(k)(x) s = s Φ(k) Φ(k) s ǫ (x) h 1| | ih | | 2i h 1| x i h x | 2i ω ZM ZM = h(s (x),s (x)) ǫ (x)= s s . 1 2 ω h 1| 2i ZM so that x(k) x(k) dµ(k) = 1H as desired. M | ih | k R (k) (k) Corollary 4.3 There exist points x1,...,xd M such that x ,..., x is a basis for Hk. k ∈ {| 1 i | dk i} S (k) S H Proof: Let x1 Mk and set 1 = x1 . If dk = 1 then is a basis for k. Suppose dk > r 1 and let S (k) ∈ (k) {| i} H ≥ r = x1 ,..., xr be a set of linearly independent vectors in k. Since dk > r, there is some vector {| i | i} (k) ψ spanC S . Suppose for every x M that x spanC S . Then | i 6∈ r ∈ | i ∈ r

(k) (k) (k) x x ψ dµ (x) spanC S . | ih | i ∈ r ZM which implies x(k) x(k) ψ dµ(k)(x) = ψ . | ih | i 6 | i ZM (k) This contradicts Theorem 4.2. Hence, there is some x M such that x spanC S . Let x = x. Then ∈ | i 6∈ r r+1 Sr+1 is a linearly independent set in Hk. We continue inductively. Since dk < , the process must stop, and the resulting set is the required linearly independent set. ∞ This corollary motivates the following deﬁnition, which is a generalization of the characteristic point sets introduced in [2].

Deﬁnition 4.4 A set S M is characteristic if for every s H , ⊆ ∈ k

s S = 0 implies s =0.

Theorem 4.5 If S M is characteristic, then x(k) x S is complete. ⊆ {| i ∈ }

Proof: If s(x) = 0 for all x S implies s =0, then x (k) s = 0 for all x S implies s =0. Hence, the only vector orthogonal to x(k) ∈x S is 0, which meansh S is| i complete. ∈ {| i ∈ }

4.2 Quantization

In this section we study the quantizing map Q : C∞(M) Op(Hk) resulting from the overcompleteness relation (4.11). Applying Berezin’s method of quantization→ [3], we have

(k) Deﬁnition 4.6 The Berezin quantization Q (f) of f C∞(M) is the operator ∈

Q(k)(f) := f(x) x(k) x(k) dµ(k)(x). | ih | ZM

11 (k) In fact, Q (f) converges for f L∞(M). Some basic theorems about Berezin’s method of quantization of analytic function spaces apply in∈ this case; see for example [20, Thm 1.3.5]. There is another way to describe the Berezin quantization of f. For each s L2(M,Lk) we have ∈ (Π s) (x)= K(k)(x, y) s(y)ǫ (y)= Φ(k) s . k · ω h x | i ZM Therefore

(Q(k)(f)s)(x)= f(y) y(k) y(k) s dµ(k)(y) (x) | ih | i ZM = K(k)(x, y) (f(y)(Π s)(y))ǫ (y) = (Π M Π s) (x) · k ω k ◦ f ◦ k ZM (k) where Mf denotes the multiplication operator. In this form, Q (f) is known as the Toeplitz quantization of f. The Berezin-Toeplitz quantization and Kostant-Souriau quantizations of f are related by Tuynman’s formula [30]: 1 Π Q(k)(f) Π = Q(k)(f ∆f) k ◦ KS ◦ k − 2k where ∆ is the Laplacian associated to the metric g = ω( ,J ). See [8] for the theory of generalized Toeplitz operators, and [5] for an analysis of the semiclassical properties· · of Q(k). We can recast Berezin’s covariant symbol [3] in terms of the symplectic coherent states.

Definition 4.7 The covariant symbol A C∞(M) associated to the operator A Op(H ) is ∈ ∈ k A(x) := x(k) A x(k) , b h | | i d (k) (k) where A x(k) := k θ (x) Aθ . b | i j=1 j ⊗ j A consequenceP of Theorem 3.11 is that Definition 4.7 agrees with the covariant symbol defined in [9] for Kähler M using Rawnsley-type coherent states. A standard result involving Berezin’s covariant symbol is true in our case as well (see [10] for an analogous computation with Rawnsley-type coherent states):

(k) Theorem 4.8 Tr A = M A(x) dµ (x). We conclude this sectionR by pointing out the relationship, in the Spinc and K¨ahler cases, between sym- b plectic coherent states and the heat kernel of the appropriate Dirac-type operator. See [4, Chap. 3] for a detailed analysis of the heat kernel, some properties of which we will use below. (k) R k ⊠ k c The heat kernel Kt C∞( + M M, L L ) of the Laplacian associated to ∂k (or ∂/k in the Spin case) admits an expansion∈ × × ∞ (k) λj t (k) (k) K (x, y)= e− θ (x) θ (y) (4.12) t j ⊗ j j=0 X where 0 λ λ are the eigenvalues of the Laplacian with corresponding eigenmodes ≤ 1 ≤ 2 ≤ ··· → ∞ θ(k) Γ(Lk). Moreover, j ∈ ∞ tλj (k) (k) λd t e− θ θ Ce− k+1 j ⊗ j ≤ j=dk +1 X

for some constant C > 0 [4, Prop 2.37]. Hence, the large time limit of the heat kernel is a symplectic coherent state: lim K(k)(x, y)=Φ(k)(y). t t x →∞ In the almost Kähler case, although the heat kernel has an expansion of the form (4.12), the low lying eigenvalues of the polarized states are not necessarily zero, and so the large time limit of the heat kernel is not directly related to the symplectic coherent states. Finally, observe that µ(k) = lim Tr K(k) and so Theorem 4.8, applied to the identity operator, yields the t t →∞ (k) familiar index formula dk = lim Tr K . M t t →∞ R 12 5 Classical and Semiclassical Behavior 5.1 Classical Behavior for Finite k In this section we show that the coherent states defined in 3 are the quantum states that behave most classically: they are maximally peaked and evolve classically. § Consider for a moment the almost Kähler quantization of M so that the prequantum bundle Lk is the k (k) line bundle ℓ⊗ . In this case, the coherent state Φx is the projection of the Dirac distribution onto Hk. To k (k) see why, let x M and trivialize ℓ⊗ over an open set U containing x with a unit section s0. Let δ˜x denote the Dirac distribution∈ on U centered at x and define ˜(k) (k) δx (y)s0(x) s0(y) for y U δx (y)= ⊗ ∈ (0 otherwise

Then (k) 1 k δ s = s(x) s C (M,ℓ⊗ ). h x | i ∀ ∈ (k) (k) (k) δx does not depend on our choice of s0. We now see that Φx is the projection onto Hk of δx since

(k) (k) (k) (k) Πkδx (y)= K (y,z) δx (z)ǫω(z)=Φx (y). M · Z We next observe that symplectic coherent states are maximally peaked quantum states. The following result holds for coherent states arising from K¨ahler, almost K¨ahler and Spinc quantization.

(k) 2 k 2 (k) Theorem 5.1 Φx maximizes s(x) over all s ℓx⊗ H with s = ε (x). | | ∈ ⊗ k k k Proof: As in Theorem 3.8 we write s(x) 2 = Φ(k) s 2. | | |h x | i | This is minimized when we have equality in the Cauchy-Schwartz inequality, which occurs when s is propor- (k) tional to Φx . (k) In the almost Kähler case, we can say more: Φx evolves classically. Suppose f C∞(M) is such that ∈ the Hamiltonian vector field Xf is complete. The flow of Xf induces a Hamiltonian evolution of sections in the quantum Hilbert space as follows [31, 8.4]: § k We can lift the Hamiltonian vector field Xf to a vector field Vf on T (ℓ⊗ ); that is, there exists a unique k 1 y 1 y vector field on ℓ⊗ defined by π Vf = Xf and k Vf α = k Vf α = f π, where α is the connection 1-form ∗ k ◦ iφ on the complement of the zero section of ℓ⊗ . If the fiber coordinate is z = re in a local trivialization, then ∂ V = X + kL f f ∂φ

y k where L = Xf τ + f is the Lagrangian associated to f by a local symplectic potential τ. Locally, Tℓ⊗ TM C and we have identified X TM and ∂ T C with the corresponding vector fields in TM C≃. × f ∈ ∂φ ∈ × The flow ξt of Vf is fiber preserving and projects to the flow ρt of Xf . Moreover, ξt induces a linear pull-back k k action ρ : Γ(ℓ⊗ ) Γ(ℓ⊗ ) by t → ξt(ρts(x)) = s(ρtx). (5.13) In fact,b this action is infinitesimally generated by the Kostant-Souriau quantization of f : b d ρ = ikQ(k)(f)ρ . dt t KS t

(k) H0 This is one of the motivations for the Kostant-Souriaub quantizationb QKS of f. The restriction of ρt to k is a 1-parameter unitary group. k ⊠ k Extending the action of ξt to ℓ⊗ ℓ⊗ in the obvious way, we have: b

13 Theorem 5.2 For f C∞(M) such that X is complete, ∈ f (k) (k) ξtΦx (y)=Φρtx(ρty); (5.14) i.e. the symplectic coherent states evolve classically. Equivalently,

(k) (k) ρtΦx =Φx .

Proof: H0 Since ρt is a unitary endomorphism ofb k, we have

dk b ρ Φ(k)(y)= ρ θ(k)(x) ρ θ(k)(y) t x t j ⊗ t j j=1 X b dk b b = θ(k)(x) tρ ρ θ(k)(y) j t ⊗ t j j=1 X (k) =Φx (y). b b

5.2 The Semiclassical Limit The asymptotic analysis of generalized Bergman kernels by Ma-Marinescu reveals the semiclassical behavior of peak sections [24, eq 3.24]. Let rk be a sequence of real numbers with rk 0 and √k rk as k . Denote by B(x, r) the open{ geodesic} ball of radius r centered at x M. Then→ → ∞ → ∞ ∈ S(k)(y) 2ǫ (y)=1 O(1/k), k . (5.15) | x | ω − → ∞ ZB(x,rk) (k) Comparing this with (3.8) we see that the peak section Sx is asymptotically concentrated about x. In terms of the transition amplitude, (5.15) is

ψ(k)(x, y) µ(k)(y)=1 O(1/k), k . (5.16) − → ∞ ZB(x,rk) Combining this with Theorem 3.14, we have:

Theorem 5.3 If f C1(M) then ∈ lim f(y)ψ(k)(x, y)µ(k)(y)= f(x); k →∞ ZM (k) (k) that is, limk ψ (x, y)µ (y)= δx(y). →∞

Proof: Let rk be a sequence of positive real numbers with rk 0 and √k rk . By Theorem 3.15 we have, for each{ }x M , → → ∞ ∈ k (f(y) f(x))ψ(k)(x, y) dµ(k)(y) − ZM (5.17) (k) 2 (k) 2 f(y) f(x) Sx (y) ǫω(y)+ f(y) f(x) Sx (y) ǫω(y). ≤ B(x,rk) | − | | | M B(x,rk) | − | | | Z Z \ The ﬁrst integral on the right hand side of (5.17) goes to zero as k because of (5.15) and the fact that f is continuous. The second integral on the right hand side of (5.17)→ ∞ goes to zero since f(x) f(y) is bounded (speciﬁcally as a function of y) and equations (5.15) and (3.8) imply that the peak sections− go to zero outside the ball B(x, rk). Of course, there is a physical reason to expect this behavior. The coherent state localized at x should be the quantum state most concentrated about x. In the semiclassical limit, we expect to recover the classical picture – in particular the classical state most concentrated about x is the Dirac distribution at x.

14 6 Examples 6.1 The Complex Plane This example is well known [17, 31]. We will take z = 1 (x + iy), use the standard symplectic form √2 ω = idz dz,¯ and trivialize the prequantum line bundle (globally since C is contractible) with the symplectic ∧ i k z 2/2 k potential τ = (zdz¯ zdz¯ ). The space H = f(z)e− | | of polarized sections of ℓ⊗ relative to the 2 − k { } standard complex structure can be identiﬁed with a weighted Bargmann space [2]. A unitary basis for Hk k j k z 2/2 is z e− | | j N. The reproducing kernel is the usual Bergman kernel { j! } ∈ q j (k) ∞ (¯wz) k z 2/2 k w 2/2 wz¯ k z 2/2 k w 2/2 K (w,z)= k e− | | − | | = ke − | | − | | . j! j=0 X (k) k z w 2 The coherent density is ε = k and the 2-point function is e− | − | . In this case, it is easy to see that the semiclassical limit yields the expected results:

(k) (k) k k w z 2 lim ψ (w,z)µ (z) = lim e− | − | ω = δ(w z)ω. k k 2π − →∞ →∞

6.2 The 2-Sphere Coherent states on S2 are constructed by Perelomov in [27] using Lie group techniques. As we will see, the construction of 3 yields the same results without using any group structure. The correspondence between the two methods§ is due to Theorem 3.2 and the fact that Perelomov’s coherent states are reproducing. 2 1 We trivialize S CP = [z0,z1] /C over the open set U0 = [z0,z1] z0 =0 . Deﬁne a local coordinate z = z /z . The Fubini-Study≃ symplectic{ } form on U is { 6 } 1 0 0

idz dz¯ ω = ∧ . (1 + z 2)2 | | k 2 1 k Trivializing ℓ⊗ with the symplectic potential τ = (1+ z )− izdz,¯ the Hermitian form on ℓ⊗ is given by 2 k | | h(p, q)=(1+ z )− pq. We then have the following unitary basis for H : | | k k (k + 1) zj 0 j k . j ≤ ≤ (s )

The coherent state localized at w is therefore

k k Φ(k)(z) = (k + 1) (¯wz)j = (k +1)(1+wz ¯ )k. w j j=0 X The corresponding coherent density is ε(k) = k + 1; one must take care to include the extra factors arising (k) from the Hermitian structure when evaluating Φw (w). In this case, Theorem 5.3 becomes

k +1 (1+wz ¯ )(1 + wz¯) k idz dz¯ lim ψ(k)(w,z)µ(k)(z) = lim ∧ = δ(w z). (6.18) k k 2π (1 + z 2)(1 + w 2) (1 + z 2)2 − →∞ →∞ | | | | | |

6.3 The 2-Torus

2 For λ = λ1 + iλ2 C with Im λ> 0, let T (λ)= C/ m + nλ m,n Z . λ is known as the modulus of the ∈ 1 { C ∈ } 2 torus. The standard symplectic form ω =2πiλ2− dz dz¯ on , normalized to be integral on T (λ), descends 2 ∧ k to a symplectic form on T (λ). The prequantum line bundle ℓ⊗ can be lifted to a line bundle over C (since C

15 is the universal cover of T 2(λ)). The resulting line bundle can be globally trivialized. Hence we will identify k sections of ℓ⊗ with appropriately pseudoperiodic functions on C. 1 If we trivialize with the symplectic potential τ = iπλ2− (zdz¯ zdz¯ ) then the Hermitian form is given by h(p, q)= pq. A unitary basis for the quantum Hilbert space can− be given in terms of ϑ-functions. Let

iλπ(kn2 +2jn)+2πi√2(j+kn)z ϑj (λ; z)= e , n Z X∈ (k) kπz(z z¯)/λ2 ψj (z, z¯)= e − ϑj (λ; z), and

2 (k) 2 1 2πj λ2/k Nk,j = ψj = e . k k √2kλ2

1/2 (k) k 1 A unitary basis for H is N − ψ (z, z¯) − . From this we construct the coherent state localized at k { k,j j }j=0 w = 1 (w + iw ): √2 1 2

k 1 − 2 (k) 2πj λ2/k √2πik(zy ww¯ 2)/λ2 Φw (z)= 2kλ2 e− e − ϑj (λ; w)ϑj (λ; z). j=0 X p The coherent density is

k 1 − (k) 2πky2/λ 2πj2λ /k 2 ε (z)= 2kλ e− 2 e− 2 ϑ (λ; z) . 2 | j | j=0 p X The semiclassical limit (Theorem 5.3) yields the identity

1 k 1 − 2 − 2 (k) (k) 2πkw /λ2 2πj λ2/k 2 δ(w z) = lim ψ (w,z)µ (z) = lim 2kλ2 e− 2 e− ϑj (λ; z) − k k  | |  →∞ →∞ j=0 p X k 1   − 2 2 2π(j +l )λ2/k 2πi e− ϑj (λ; w)ϑl(λ, z)ϑj (λ; z)ϑl(λ; w) dz dz.¯ · λ2 ∧ j,lX=0 6.4 Higher Genus Riemann Surfaces

We will construct coherent states on a compact Riemann surface Σg of genus g 2 by uniformizing Σg as the quotient of the complex upper half plane H = z C Im z > 0 by a Fuschian≥ group Γ (see [1, 16] for details). The coherent states in this section correspond{ ∈ to those of Klimek-Lesniewski} [19, eq 4.5].

Let Γ < PSL(2, Z) be a Fuschian group. PSL(2, Z ), and hence Γ, acts on H by fractional linear transformations. The space Σg := Γ H is a compact manifold if and only if Γ is a hyperbolic group, which \ 2 we will henceforth assume. The Kähler form ω = i(Im z)− dz dz¯ descends to a symplectic form on Σg, as do the complex structure and Kähler metric. ∧ a b For γ = Γ and z H define j(γ,z) := cz + d. An automorphic form of weight k relative to c d ∈ ∈ k Γ is a function f : H C such that f(γz)= j(γ,z) f(z). The space k(H) of automorphic forms on H of weight k relative to Γ→ has an Hermitian product A ω f g = f(z)g(z)(Im z)k . h | i k 2π ZH (1,0) The bundle ℓ := T ∗Σg is a prequantum bundle for Σg since its curvature is iω. The quantum Hilbert (1,0) k − space Γ(Σ , (T ∗Σg )) is isomorphic to (Σ ). Sections of ℓ⊗ correspond to automorphic forms of weight g O A2 g 2k relative to Γ restricted to Σg. The Hermitian form descends to Σg and is known as the Weil-Petersson inner product.

16 The reproducing kernel for 2k(H) is known (see for example [26]) and descends to Σg via a Poincar´e series. The resulting coherent stateA localized at w is

k (k) k 1 2i k Φ (z)= − j(γ,z)− . w 2 γz w¯ γ Γ X∈ − The associated coherent density is

2i 2k ε(k)(z) = (k 1) Re . − (γz z¯)j(γ,z) γ,γ−1 Γ { X}⊂ − Finally, the semiclassical limit of Propostion 5.3:

δ(w z) = lim ψ(k)(w,z)µ(k)(z) − k →∞ 1 2k − k 1 k 1 k 2 k 2i = lim − 4 − (Im z) − (Im w) Re k 2π  (γz z¯)j(γ,z)  →∞ γ,γ−1 Γ { X}⊂ − 2k  ( γz w¯ j(γ,z) )− idz dz.¯ · | − | | | ∧ γ,γ′ Γ X∈ Acknowledgments. The author is grateful to Siye Wu for many valuable discussions and suggestions. He would also like to thank Xiaonan Ma for bringing to his attention the papers [11, 23, 24] as well as for several helpful comments.

References

[1] Alvarez-Gaumé, L. and Moore, G. and Vafa, C., Theta Functions, Modular Invariance and Strings, Communications in Mathematical Physics 106 (1986), 1–40. [2] V. Bargmann, On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Com- munications and Pure and Applied Mathematics 14 (1961), 187–214. [3] Berezin, F.A., Quantization, Mathematics of the USSR Izvestija 8 (1974), 1109–1165. [4] Berline, N. and Getzler, E. and Vergne, M., Heat Kernels and Dirac Operators, Springer, 2004. [5] Bordemann, M. and Meinrenken, E. and Schlichenmaier, M., Toeplitz Quantization of Kähler Manifolds and gl(N),N Limits, Communications in Mathematical Physics 165 (1994), 281–296. → ∞ [6] Borthwick, D. and Uribe, A., Almost Complex Structures and Geometric Quantization, Mathematical Research Letters 3 (1996), no. 6, 845–861. [7] , Nearly Kählerian Embeddings of Symplectic Manifolds, Asian Journal of Mathematics 4 (2000), no. 3, 599–620. [8] Boutet de Monvel, L. and Guillemin, V., The Spectral Theory of Toeplitz Operators, Annals of Mathe- matical Studies, vol. 99, Princeton University Press, 1981. [9] Cahen, M. and Gutt, S. and Rawnsley, J.H., Quantization on Kähler Manifolds I, Journal of Geometry and Physics 7 (1990), 45–67. [10] , Quantization on Kähler Manifolds II, Transactions of the American Mathematical Society 337 (1993), no. 1, 73–98. [11] Dai, X., Liu, K. and Ma, X., On the asymptotic expansion of bergman kernel, C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 193–198, Full version: math.DG/0404494.

17 [12] Daubechies, I., Coherent states and projective representation of the linear canonical transformations, Journal of Mathematical Physics 21 (1980), no. 6. [13] Duistermaat, J.J., The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Birkh¨auser, 1996. [14] Guillemin, V. and Sternberg, S., Geometric Asymptotics, Mathematical Surveys, vol. 14, American Mathematical Society, 1977. [15] Guillemin, V. and Uribe, A., The laplace operator on the n-th tensor power of a line bundle: Eigenvalues which are uniformly bounded in n, Asymptotic Analysis 1 (1988), 105–113. [16] Jost, J., Compact riemann surface, Springer-Verlag, Berlin, 2002. [17] Klauder, J. R. and Skagerstam, B.-S. (ed.), Coherent states, World Scientiﬁc Publishing, 1985. [18] Klimek, S. and Lesniewski, A., Quantum Riemann Surfaces: I. The Unit Disc, Communications in Mathematical Physics 146 (1992), 103–122. [19] , Quantum Riemann Surfaces: II. The Discrete Series, Letters in Mathematical Physics 24 (1992), 125–139. [20] Landsman, N.P., Mathematical topics between classical and quantum mechanics, Springer, 1998. [21] Lang, S., Complex Analysis, Springer-Verlag, 1993. [22] Lawson, H.B., Jr. and Michelsohn, M.-L., Spin geometry, Princeton University Press, 1989. [23] Ma, X. and Marinescu, G., The spinc dirac operator on high tensor powers of a line bundle, Mathema- tische Zeitschrift 240 (2002), no. 3, 651–664. [24] , Generalized bergman kernels on symplectic manifolds, C. R. Math. Acad. Sci. Paris 339 (2004), no. 7, 493–498, Full Version: math.DG/0411559. [25] Meinrenken, E., Symplectic Surgery and the Spinc-Dirac Operator, Advances in Mathematics 134 (1998), no. 2, 240–277. [26] Miyake, T., Modular Forms, Springer-Verlag, 1989. [27] Perelomov, A.M., Generalized Coherent States and Applications, Springer-Verlag, 1986. [28] Rawnsley,J.H., Coherent States and Kahler Manifolds, Quarterly Journal of Mathematics 28 (1977), 403–415. [29] Tian, Y. and Zhang, W., An analytic proof of the geometric quantization conjecture of Guillemin- Sternberg, Inventiones Mathematicae 132 (1998), 229–259. [30] Tuynman, G.M., Quantization: Towards a comparison between methods, Journal of Mathematical Physics 28 (1987), 2829–2840. [31] Woodhouse, N.M.J., Geometric Quantization, 2nd Edition, Oxford University Press, Inc., New York, 1991.