arXiv:1905.03027v2 [math.DG] 1 Mar 2020 connection onay noe ihaHriinln ude( bundle line Hermitian a with endowed boundary, h osrcino nascae ibr pc fqatms quantum of a space of Hilbert data associated an of construction The ie lsia hs space phase classical a Given Introduction 1 hs pc srpeetdb 2 a by represented is space phase H quantization iemrhss r apdi aua a oteassociate the to way natural a H in mapped are diffeomorphisms, h orsodn unu yaisaesmtmsmc easi much sometimes are dynamics quantum corresponding the hsclstaina ad npriua,tecascldy classical the particular, ( In hand. at situation physical ,ω X, 1 ecie s1prmtrfmle fuiayoeaos I operators. unitary of families 1-parameter as described , atal upre yteErpa eerhCuclStart Council Research European the by supported Partially fqatmsae,sc httecascldnmc over dynamics classical the that such states, quantum of emti uniaino Hamiltonian of quantization Geometric osadteGtwle rc formula trace Gutzwiller the and flows setrl eemndb a by determined entirely is ) opt xlctytelaigtr.W hndsrb pote a describe then We topology. term. contact Kostant leading the the for formula explicitly trace compute Gutzwiller a establish semi-classic in we various the lar, establish over we spaces and quantum structures, the complex in associa transport the parallel express c We via tors the Souriau. in and Kostant manifolds of symplectic quantization cla prequantized with closed associated over dynamics flows Hamiltonian quantum the study eueteter fBrznTelt prtr fM n Mar and Ma of operators Berezin–Toeplitz of theory the use We polarization ∇ nrdcdidpnetyb otn [ Kostant by independently introduced , L ihcurvature with n h etcoc fsc oaiainuulydpnso depends usually polarization a such of choice best the and , X R h olo uniaini opoueaHletspace Hilbert a produce to is quantization of goal the , L iedpnetHamiltonian time-dependent aifigtefollowing the satisfying n oi IOOS Louis ω dmninlsmlci aiod( manifold symplectic -dimensional = Abstract √ 2 1 − π 1 R L . 1 ,h L, 23 n ora [ Souriau and ] aiso ypetcmanifold symplectic a of namics n rn 757585 grant ing L oehrwt Hermitian a with together ) letmts nparticu- In estimates. al rqatzto condition prequantization Suiuoeao and operator –Souriau ae eed nteextra the on depends tates h otx of context the n ue aho almost of path duced rt ne o n specific one for infer to er e vlto opera- evolution ted X neto geometric of ontext F unu yaisof dynamics quantum d ta plcto to application ntial sclHamiltonian ssical ecie sflw of flows as described , ∈ C ∞ 34 ( R nsuto inescu ,ω X, ,teclassical the ], × X, without ) geometric R ,and ), the n (1.1) , choice of polarization. For the general theory as well as numerous examples, we refer to the classical book of Woodhouse [35, Chap. 5, Chap. 9]. In this paper, we focus our attention on compact symplectic manifolds, and consider the polarization induced by an almost complex structure J End(TX) compatible with ∈ ω, which always exists. The associated Hilbert space Hp of quantum states will depend of an integer p N, representing a quantum number, and the goal of this paper is to ∈ study the behaviour of the quantum dynamics associated with the classical Hamiltonian flow of F C ∞(R X, R) as p tends to infinity. This limit is called the semi-classical limit, when∈ the scale× gets so large that we recover the laws of classical mechanics as an approximation of the laws of quantum mechanics. We show in particular that the quantum dynamics approximate the corresponding classical dynamics as p + . The construction we present in this paper holds for any compact prequantized→ ∞ symplectic manifold, and coincides with the holomorphic quantization of Kostant and Souriau in the particular case when the almost complex structure J End(TX) is ∈ integrable, making (X,J,ω) into a Kähler manifold. Then (L, hL) admits a natural L holomorphic structure for which is its Chern connection, and the space Hp of quan- tum states coincides with the associated∇ space of holomorphic sections of the pth tensor p p 2 power L := L⊗ , for all p N big enough. The natural L -Hermitian product (2.3) on p ∈ p the space C ∞(X, L ) of smooth sections of L then gives Hp the structure of a Hilbert space. In the very restrictive case when the Hamiltonian flow acts by biholomorphisms on (X, L), the quantum dynamics is simply given by the induced action on the space of holomorphic sections H for all p N. In contrast, our results apply to the holomorphic p ∈ quantization of general Hamiltonian flows. In Section 2, we consider a general almost complex structure J End(TX) com- ∈ patible with ω, and for all p N, we define in (2.7) the space H of quantum states ∈ p as the direct sum of eigenspaces associated with the small eigenvalues of the following renormalized Bochner Laplacian
p ∆ := ∆L 2πnp , (1.2) p − p p Lp acting on the smooth sections C ∞(X, L ) of L , where ∆ is the usual Bochner Lapla- cian of (Lp, hLp ) associated with the Riemannian metric gT X := ω( ,J ). This follows an idea of Guillemin and Uribe [17], and we consider here a more general· · construction due to Ma and Marinescu [27], where Lp is replaced by E Lp for all p N, for any Hermitian vector bundle with connection (E, hE, E). Note⊗ that both constructions∈ admit an extension to the case of a general J-invariant∇ metric gT X, and the general construction of [27] also deals with the case when one adds a potential term to ∆p. In Theorems 2.5 and 2.6, we describe the results of [21] on the dependence of the quanti- zation to the choice of an almost complex structure J End(TX) at the semi-classical ∈ limit p + . Specifically, we introduce the parallel transport operator between → ∞ Tp,t different quantum spaces Hp along a path Jt End(TX) t R of almost complex struc- { ∈ } ∈ tures with respect to the L2-connection (2.17), and we describe in Theorem 2.5 how p,t behaves like a Toeplitz operator as p + , giving an explicit formula for the Thighest order coefficient. This is based on→ the∞ theory of Berezin–Toeplitz operators
2 for symplectic manifolds developed by Ma and Marinescu in [28], and extended to this context in [22]. In Section 3, we show how one can use this parallel transport to construct the quantum Hamiltonian dynamics associated with any F C ∞(R X, R). In fact, ∈ × the corresponding Hamiltonian flow ϕt : X X defined by (3.2) for all t R does not preserves any almost complex structure→ in general, and thus does not induce∈ an action on Hp for any p N. Instead, fix an almost complex structure J0 End(TX) ∈ ∈ 1 compatible with ω, and consider the almost complex structure Jt := dϕt J0 dϕt− , as well as the spaces H of quantum states associated with J , for all t R and p N. p,t t ∈ ∈ This flow, together with its lift to L defined by (3.4), induces a unitary isomorphism p ϕ∗ : H H by pullback on C ∞(X, L ), and considering the parallel transport t,p p,t → p,0 : H H along the path s J for s [0, t], the associated quantum evolution Tp,t p,0 → p,t 7→ s ∈ operator at time t R is given by the unitary operator ϕ∗ End(H ). ∈ t,pTp,t ∈ p,0 In the rest of the Introduction, we assume that the Hamiltonian F =: f C ∞(X, R) ∈ does not depend on time, and write ξf C ∞(X,TX) for the Hamiltonian vector field of f, as defined in (3.1). In that case, we∈ show in Lemma 3.1 that
ϕ∗ = exp( 2π√ 1tpQ (f)) , (1.3) t,pTp,t − − p for all t R and p N, where Qp(f) is the Kostant–Souriau operator associated with ∈ ∈ p f, defined as an operator acting on C ∞(X, L ) by the formula
√ 1 Lp Qp(f) := Pp f − ξ Pp , (1.4) − 2πp ∇ f ! p where f is the operator of pointwise multiplication by f and P : C ∞(X, L ) H is p → p,0 the L2-orthogonal projection. This is the holomorphic version of the Blattner–Kostant– Sternberg kernel, as described for example in [35, § 9.7]. Under this form, it was first noticed by Foth and Uribe in [16, § 3.2], who interpreted the trace of Qp(f) as a mo- ment map for the group of Hamiltonian diffeomorphisms acting on the space of almost complex structures compatible with ω. Using the results described in Section 2, we establish the following semi-classical estimate on its Schwartz kernel (2.26), where τt,p p denotes the parallel transport in L along s ϕs(x) for s [0, t]. Here and in all the k 7→ ∈ paper, we use the notation O(p− ) in the sense of the corresponding Hermitian norm k as p + , and O(p−∞) means O(p− ) for all k N. → ∞ ∈ Proposition 1.1. For any t R and x, y X such that ϕt(y) = x, we have the following estimate as p + ,∈ ∈ 6 → ∞ exp 2π√ 1tpQ (f) (x, y)= O(p−∞) . (1.5) − p Furthermore, there exist a (t, x) C (r N)smooth in x X and t R, such that for r ∈ ∈ ∈ ∈ any k N∗, as p + we have ∈ → ∞ exp 2π√ 1tpQ (f) (ϕ (x), x) − p t k 1 n 2π√ 1tpf(x) − r k = p e − p− ar(t, x)+ O(p− ) τt,p , (1.6) ! rX=0
3 with a (t, x) =0 for all t R and x X. 0 6 ∈ ∈ This follows from the more precise Proposition 3.3, which gives in particular a formula for the first coefficient a0(t, x). As explained there, this result shows that the quantum dynamics approximates the classical dynamics at the semi-classical limit p + in a precise sense. In Theorem 3.4, we also use the results described in Section→ ∞2 to give an associated semi-classical trace formula. Note that the estimate (1.5) is not uniform in t R, x, y X, and the estimate (1.6) shows that there is in ∈ ∈ fact a jump when ϕt(y) tends to x. In Section 4, we consider a time-independent Hamiltonian f C ∞(X, R), and we ∈ use the setting of Section 2 to study the operatorsg ˆ(pQp(f c)) End(Hp) defined for all p N by the formula − ∈ ∈ 2π√ 1tpc gˆ(pQp(f c)) := g(t)e − ϕ∗ p,t dt , (1.7) − R t,pT Z where g : R R is smooth with compact support, where c R is a regular value → ∈ of f and where ϕt,p∗ p,t End(Hp,0) is the quantum evolution operator associated with f. Via the interpretationT ∈ (1.3) in terms of quantum evolution operators, the Gutzwiller trace formula predicts a semi-classical estimate for the trace Tr[ˆg(pQp(f c))] as p + , in terms of the periodic orbits of the Hamiltonian flow of f included in− the → ∞1 level set f − (c). This formula was first worked out by Gutzwiller in [19, (36)] for usual Schrödinger operators over Rn as the Planck constant ~ tends to 0, using path integral methods. As explained in his book [20], this formula plays a fundamental role in the theory of quantum chaos, which studies the quantization of chaotic classical systems. Specifically, let Supp g R be the support of g, and suppose that for all t Supp g, ϕt ⊂ ∈ the fixed point set X X of ϕt : X X is non-degenerate over a neighborhood 1 ⊂ → 1 of f − (c) in the sense of Definition 2.7 and intersects f − (c) transversally such that ϕt 1 X f − (c) is non-empty only for a finite subset T Supp g. Let Y 6 6 be the ∩ ⊂ { j}1 j m set of connected components of
ϕt 1 X f − (c) , (1.8) t T ∩ a∈ ϕt 1 and for any 1 6 j 6 m, write tj T for the time such that Yj X j f − (c). In particular, these hypotheses are automatically∈ satisfied if Supp g ⊂R is a small∩ enough ⊂ neighborhood of 0, so that T = 0 and Xϕ0 = X. { } 1 Let λj R be the action of f over Yj as in Definition 3.5, and let Volω(f − (c)) > 0 ∈ 1 be the volume of f − (c) with respect to the natural Liouville measure (4.20) induced 1 by ω and f on f − (c). Theorem 1.2. Under the above assumptions, there exist b C (r N), depending j,r ∈ ∈ only on geometric data around Yj for all 1 6 j 6 m, such that for any k N∗ and as p + , we have ∈ → ∞ m k 1 − (dim Yj 1)/2 2π√ 1pλj r k Tr [ˆg(pQp(f c))] = p − g(tj)e− − p− bj,r + O(p− ) . (1.9) − ! jX=1 rX=0 4 Furthermore, there is an explicit geometric formula for the first coefficients bj,0 for all 1 6 j 6 m, and as p + we have → ∞ n 1 1 n 2 Tr [ˆg(pQ (f c))] = p − g(0) Vol (f − (c)) + O(p − ) . (1.10) p − ω Note that formula (1.9) does not follow from (1.3) and Proposition 1.1 by integrating over t R, due to the jump in the estimates (3.25) and (1.6) when ϕ (y) tends to x. ∈ t This is illustrated by Proposition 4.1, where it is shown howg ˆ(pQp(f c)) localizes 1 − around f − (c) after integrating in t R via stationary phase estimates, with a precise 1 ∈ control on the constant around f − (c). The general formula for the first coefficients bj,0 of the expansion (1.9) is given in Theorem 4.3, and reduces to the so-called Weyl term (1.10) of the trace formula in the case 0 Supp g. However, the relevance of this formula for quantum chaos mainly lies ∈ in the terms associated with isolated periodic orbits, and one would like to consider general situations where this formula exhibits natural geometric quantities associated to such orbits. To describe such situations, let us first consider the general case of a Hermitian vector bundle with connection (E, hE, E) over R X. Writing E for its restriction to X over ∇ × t t R, we take more generally the quantum spaces Hp,t to be the almost holomorphic ∈ p 2 sections of Et L with respect to Jt, for all p N, together with the L -Hermitian ⊗ p ∈ product induced by hEt and hL . Following Definition 2.4, we can again consider the 2 parallel transport p,t : Hp,0 Hp,t with respect to the associated L -connection, and T → E if ϕt : X X is a Hamiltonian flow lifting to a bundle map ϕt : E0 Et over X for R→ → all t , we again have a unitary evolution operator ϕt,p∗ p,t End(Hp,0). Then the right∈ hand side of formula (1.7) still makes sense, and TheoremT ∈ 4.3 gives the general version of Theorem 1.2 in this context. Consider now the canonical line bundle (K , hKX , KX ) over R X associated with X ∇ × Jt End(TX) t R defined in Section 2 by the formula (2.14), and assume that (X,ω) ∈ {admits∈ a metaplectic} structure, so that this canonical line bundle admits a square root 1/2 R KX over X with induced metric and connection, called the metaplectic correction. Then ϕ admits× a natural lift for all t R, and we call the associated unitary operator t ∈ ϕ∗ as above the metaplectic quantum evolution operator. t,pTp,t Theorem 1.3. Assume that (X,ω) admits a metaplectic structure, and consider the assumptions of Theorem 1.2. Let 1 6 j 6 m be such that dim Yj = 1 and such that [Jξf ,ξf ]=0 over Yj. Then the first coefficients of the analogous expansion (1.9) as p + for the metaplectic quantum evolution operator satisfy the following formula, → ∞
n−1 t(Yj) b =( 1) 2 , (1.11) j,0 − det (Id dϕ ) 1/2 | Nx N − tj |N | for a natural choice of square root and for any x Yj, where N is the normal bundle of 1 ∈ Yj inside T f − (c) and where t(Yj) > 0 is the primitive period of Yj as a periodic orbit 1 of the flow t ϕ inside f − (c). 7→ t
5 For (X,J ,ω) Kähler for all t R and endowed with a metaplectic structure, one can t ∈ show using [21, (5.2)] that the generator of the metaplectic quantum evolution operator considered above coincides with the metaplectic Kostant–Souriau operator considered by Charles in [12, Th. 1.5]. Theorem 1.3 follows from Theorem 4.4, which gives also a formula for general (E, hE, E) as an integral along the associated periodic orbit. In the case of usual ∇ Schrödinger operators over a compact Riemannian manifold, the Gutzwiller trace for- mula has been established by Guillemin and Uribe [18, Th. 2.8], Paul and Uribe [32, Th. 5.3] and Meinrenken [30, Th. 3], while in the case of Toeplitz operators over the smooth boundariy of a compact strictly pseudoconvex domain, it has been established by Boutet de Monvel and Guillemin [9, Th, 9, Th. 10]. In the case of Berezin–Toeplitz operators over a compact prequantized Kähler manifold with metaplectic structure, instead of Kostant–Souriau operators over a general compact prequantized symplectic manifold as in Theorem 1.2, it has been established by Borthwick, Paul and Uribe [8, Th. 4.2] using the theory of Boutet de Monvel and Guillemin [9]. In all the works cited above, the corresponding formulas for the first coefficients involve an undetermined subprincipal symbol term with no obvious geometric inter- pretation. In contrast, the general formula for the first coefficient given in (4.22) is completely explicit in terms of local geometric data. Furthermore, the formula for iso- lated periodic orbits given in Theorem 1.3 is the same as the corresponding formulas in all the cases mentioned above, but without the undetermined subprincipal symbol term. This makes it much simpler to use in practical applications. In fact, let the Hamiltonian f C ∞(X, R) and the almost complex structure J End(TX) be such that ∈ ∈ 1 dιJξf ω = ω over f − (I) , (1.12) for some interval I R of regular values of f containing c R. As explained at the end ⊂ 1 ∈ 1 of Section 4, this induces a contact form α Ω (Σ, R)onΣ:= f − (c), and ξ generates ∈ f the Reeb flow of (Σ, α). Then Proposition 4.5 shows how Theorem 1.2 can be used to detect the periods of the non-degenerate isolated periodic orbits of this flow, and Theorem 1.3 allows in principle to compute the associated action. This is of particular interest in contact topology, where the study of non-degenerate isolated periodic orbits of the Reeb flow is a major topic, usually tackled via methods of Floer homology. Note that to recover the action from the expansion (1.9) in practice, one needs a completely explicit formula for the first coefficient, and formula (1.11) is the best that one can hope for. In Theorem 4.2, we also establish semi-classical estimates on the Schwartz kernel of the operatorg ˆ(pQp(f c)) as p + , analogous to the corresponding estimates in [8, Th. 1.1] for Berezin–Toeplitz− operators→ ∞ over compact prequantized Kähler manifolds admitting a metaplectic structure. Once again, our formula for the first order term is completely explicit in terms of geometric data, without the undetermined subprincipal symbol term appearing in the corresponding formula in [8, Th. 2.7]. In the case of compact prequantized Kähler manifolds and when the Hamiltonian
6 flow ϕ : X X acts by biholomorphisms, so that one can define the quantization of t → ϕt simply by its induced action on holomorphic sections, the pointwise semi-classical estimates of Theorem 4.2 for E = C have been obtained by Paoletti in [31, Th. 1.2]. In this case, the general version of Theorem 1.2 is a direct consequence of the following Kirillov formula,
p Tr ϕ∗ = Td −1 (TX) ch −1 (L ) , (1.13) tj +t,p ϕt ϕ , tξf ϕ , tξf X j tj − tj − h i Z as described in [6, (2.38)] for any fixed 1 6 j 6 m and t > 0 small enough, using the stationary phase lemma as p + . Paoletti recovers this| | special case in [31, Th. 1.3] → ∞ without using formula (1.13). The theory of Berezin–Toeplitz operators over compact prequantized Kähler mani- folds with E = C was first developed by Bordemann, Meinreken and Schlichenmaier [7] and Schlichenmaier [33]. Their approach is based on the work of Boutet de Monvel and Sjöstrand on the Szegö kernel in [10], and the theory of Toeplitz structures developed by Boutet de Monvel and Guillemin in [9]. The present paper is based instead on the approach of Ma and Marinescu using Bergman kernels, and we refer to the book [26] for a detailed presentation of this method. The quantization of symplectic maps over compact prequantized Kähler manifolds has first been considered by Zelditch in [36], using a unitary version of the theory of Toeplitz structures of [9], and Zelditch and Zhou use it in [37, Th. 0.9] to establish the pointwise semi-classical estimates of Theorem 4.2 in the Kähler case for E = C. Note that Theorem 1.2 is not a consequence of these pointwise semi-classical estimates, as they are not uniform in x X. The applications of parallel transport to the quantum dynamics associated with∈ Hamiltonian flows have also been explored by Charles [13, Th. 5.3] in the case of compact prequantized Kähler manifolds with metaplectic struc- ture, where he establishes an analogue of Proposition 1.1 in the language of Fourier integral operators.
Acknowledgements. The author wants to thank Pr. Xiaonan Ma for his support and Pr. Leonid Polterovich for helpful discussions. The author also wants to thank the anonymous referees for useful comments and suggestions. This work was supported by the European Research Council Starting grant 757585.
2 Setting
Let (X,ω) be a compact symplectic manifold without boundary of dimension 2n, and let (L, hL) be a Hermitian line bundle over X, endowed with a Hermitian connection L satisfying the prequantization condition (1.1). Let J be an almost complex structure∇ on TX compatible with ω, and let gT X be the Riemmanian metric on X defined by
gT X ( , ) := ω( ,J ) . (2.1) · · · ·
7 T X We write for the associated Levi–Civita connection on TX, and dvX for the ∇ T X n Riemannian volume form of (X,g ). It satisfies the Liouville formula dvX = ω /n!. For any Hermitian vector bundle with connection (E, hE, E) over X, we write , and for the Hermitian product and norm induced by∇hE, and write RE for h· ·iE |·|E the curvature of E. We denote by C the trivial line bundle with trivial Hermitian metric and connection.∇ For any p N, we write Lp for the p-th tensor power of L, and ∈ for any Hermitian vector bundle with connection (E, hE, E), we set ∇ E := Lp E , (2.2) p ⊗ equipped with the Hermitian metric hEp and connection Ep induced by hL, hE and L E 2 ∇ , . The L -Hermitian product , on C ∞(X, E ) is given for any s ,s ∇ ∇ h· ·ip p 1 2 ∈ C ∞(X, Ep) by the formula
s1,s2 p := s1(x),s2(x) Ep dvX (x) . (2.3) h i ZX h i 2 Let L (X, E ) be the completion of C ∞(X, E ) with respect to , . p p h· ·ip Definition 2.1. For any p N, the Bochner Laplacian ∆Ep of (E , hEp , Ep ) is the ∈ p ∇ second order differential operator acting on C ∞(X, Ep) by the formula
2n Ep Ep 2 Ep ∆ := ( ej ) TX e , (2.4) − ∇ − ∇∇ej j jX=1 where e 2n is any local orthonormal frame of (TX,gT X). { j}j=1 2 This defines an unbounded self-adjoint elliptic operator on L (X, Ep), and by stan- dard elliptic theory, its spectrum Spec(∆Ep ) is discrete and contained in R.
Definition 2.2. For any p N, the renormalized Bochner Laplacian ∆ is the second ∈ p order differential operator acting on C ∞(X, Ep) by the formula
n ∆ := ∆Ep 2πnp RE(w , w¯ ) , (2.5) p − − j j jX=1
n (1,0) where wj j=1 is an orthonormal basis of T X for the Hermitian metric induced by gT X. { }
2 As above, ∆p is an unbounded self-adjoint elliptic operator on L (X, Ep), and has discrete spectrum Spec(∆p) contained in R. Furthermore, we have the following refine- ment of [17, Th.2.a].
Theorem 2.3. [25, Cor. 1.2] There exist constants C,C > 0 such that for all p N, ∈ Spec(∆ ) [ C, C ] ]4πnp e C, + [ , (2.6) p ⊂ − ∪ − ∞ e e 8 and the constants C, C> 0 are uniform in the choice of J End(TX) varying smoothly ∈ in a compact set of parameters. Furthermore, the direct sum e H := Ker(λ ∆ ) (2.7) p − p λ [ C,C ] ∈ M− is naturally included in C ∞(X, E ), and there is p N such that for any p > p , we p e e 0 ∈ 0 have (1,0) dim Hp = Td(T X)ch(E) exp(pω), (2.8) ZX where Td(T (1,0)X) represents the Todd class of T (1,0)X and ch(E) represents the Chern character of E. The integer p N is uniform in the choice of J End(TX) varying 0 ∈ ∈ smoothly in a compact set of parameters.
For any p N, the Hilbert space H L2(X, E ) defined by (2.7) is called the space ∈ p ⊂ p of almost holomorphic sections of Ep. In the special case when J is integrable, so that (X,J,ω,gT X) is a Kähler manifold and the Hermitian bundles (L, hL)and(E, hE) admit natural holomorphic structures such that L and E are their Chern connections, then p ∇ ∇ the subspace H C ∞(X, L ) coincides with the space of holomorphic sections of E p ⊂ p for all p > p0. In fact, as explained for example in [26, § 1.4.3, § 1.5], by the Bochner- Kodaira formula, the formula (2.5) is twice the Kodaira Laplacian of Ep, and there is a spectral gap, so that C = 0in(2.6). It is then a basic fact of Hodge theory that the kernel of the Kodaira Laplacian in C ∞(X, Ep) is precisely the space of holomorphic sections of E , for all p e N. p ∈ The goal of this section is to describe the results of [21] about the dependence of this quantization scheme on the choice of an almost complex structure J End(TX). To this end, we consider a smooth path ∈
t J End(TX), for all t R , (2.9) 7−→ t ∈ ∈ of almost complex structures over X compatible with ω. We will see Jt End(TX) t R ∈ as the endomorphism of the vertical tangent bundle TX over R X{ of∈ the tautological} fibration × π : R X R × −→ (2.10) (t, x) t , 7−→ restricting to Jt End(TX) over t R. We then have an induced vertical Riemannian metric on TX over∈ R X, defined∈ by its restriction on the fibre X over any t R via × ∈ the formula gT X( , ) := ω( ,J ) . (2.11) t · · · t · Following Bismut in [3, Def. 1.6] and in [4, (1.2)], we consider the induced vertical Levi– Civita connection T X on the subbundle TX of the tangent bundle of R X, defined by the formula ∇ × T X T X R T X T X := Π ⊕ Π , (2.12) ∇ ∇ 9 R T X where ⊕ is the Levi–Civita connection on the total space of R X for the Rie- ∇ × mannian metric defined on T (R X)= R TX as the orthogonal sum of the canonical R T X× R⊕ T X R metric of and the metric gt over t , with Π : TX TX the canonical ∈ n ⊕ → projection. Note that by the Liouville formula dvX = ω /n!, the Riemannian volume form dv of (X,gT X) does not depend on t R. X t ∈ Let TXC := TX R C be the complexification of the vertical tangent bundle TX ⊗ over R X. The family of complex structures Jt End(TX) t R induces a splitting × { ∈ } ∈ (1,0) (0,1) TXC = T X T X (2.13) ⊕ into the eigenspaces of Jt corresponding to the eigenvalues √ 1 and √ 1 over t X −T X − − T X { }× for all t R. We endow TXC with the Hermitian product h given by g ( ,¯) over ∈ t · · t X for all t R. The canonical line bundle associated with Jt End(TX) t R is { } × ∈ { ∈ } ∈ the line bundle (1,0) KX := det(T ∗X) (2.14) over R X equipped with the Hermitian metric hKX and connection KX induced by × ∇ the vertical Hermitian metric hT X and the vertical Levi–Civita connection (2.12) via the splitting (2.13). For any Hermitian vector bundle with connection (E, hE, E) over R X, we write Et Et ∇ × (Et, h , ) for the Hermitian vector bundle with connection induced on X by re- striction∇ to the fibre over t R. For all t R, we write ∈ ∈ τ E : E E (2.15) t 0 −→ t for the bundle isomorphism over X induced by parallel transport in E with respect to E along horizontal directions of π : R X R. We still write (L, hL, L) for the ∇Hermitian line bundle with connection over×R →X defined by pullback of (L, h∇L, L) over E×p Ep ∇ p X via the second projection, and write (Ep, h , ) for the tensor product Ep = E L over R X for any p N, with induced Hermitian∇ metric and connection. For any p⊗ N × ∈ ∈ and t R, we write ∆p,t for the renormalized Bochner Laplacian acting on C ∞(X, Ep,t) ∈ T X associated with the metric gt as in Definition 2.2, and write Hp,t C ∞(X, Ep,t) for the associated space of almost holomorphic sections defined in Theorem⊂ 2.3. Let us assume that there exists p N such that the two intervals in (2.6) are 0 ∈ disjoint and such that Hp,t satisfies the Riemann-Roch-Hirzebruch formula (2.8) for all p > p0 and t R. By Theorem 2.3, such a p0 N always exists if we ask Jt and Et Et ∈ ∈ (Et, h , ) to be independent of t R outside a compact set of R. On the other hand, this∇ assumption will be automatically∈ satisfied for all t R in the main case of interest considered in Section 3. In the sequel, we fix such a p ∈ N. 0 ∈ Following for instance [2, §9.2], for all t R and p > p , we define the orthog- ∈ 0 onal projection operator P : L2(X, E ) H with respect to the associated L2- p,t p,t → p,t Hermitian product (2.3) by the following contour integral in the complex plane,
1 Pp,t := (λ ∆p,t)− dλ , (2.16) ZΓ − 10 where Γ C is a circle of center 0 and radius a > 0 satisfying C 0 such that ∈ ∈ k k 1 − l k p,t p− Pp,tµl,tPp,0 6 Ckp− , (2.19) T − l=0 p,0,t X for all p N and t R. Furthermore, there is a natural function µ C ∞(X, C) such ∈ ∈ t ∈ that the first coefficient µ0,t satisfies E µ0,t = µtτt . (2.20) To describe the function µt C ∞(X, C)of (2.20), let us describe the local setting involved in the proof of the above∈ theorem in [21]. For any t R, using the fact that ∈ the almost complex structures J0 End(TX) and Jt End(TX) are both compatible with the same symplectic form ω,∈ we get a splitting ∈ (1,0) (0,1) TXC = T X T X (2.21) 0 ⊕ t (1,0) into the holomorphic subspace T X0 of TXC associated to J0 End(TX) and the (0,1) ∈ anti-holomorphic subspace T Xt of TXC associated with Jt End(TX)asin (2.13). We write ∈ t Π End(TXC) (2.22) 0 ∈ 11 (1,0) (0,1) for the projection operator onto T X0 with kernel T Xt. In a dual way, we write 0 (0,1) (1,0) Πt End(TXC) for the projection operator onto T Xt with kernel T X0. Consid- ∈ (0,1) (0,1) (1,0) ering its restriction to T X0 and via the isomorphism T Xt T ∗Xt induced by gT X for all t R, it induces an isomorphism ≃ t ∈ 0 det(Π ) : K K (2.23) t X,0 −→ X,t of the respective canonical line bundles over X. Recall the connection KX on the K ∇ canonical line bundle K over R X of (2.14), inducing τ X : K K by (2.15). X × t X,0 → X,t Then by [21, (5.4)], the function µ C ∞(X, C)of (2.20) satisfies t ∈ 0 2 1 KX µ¯t (x) = det(Πt )− τt , (2.24) R C for all t , via the canonical identification KX,0 KX,∗ 0 . The main∈ tool of the proof of Theorem 2.5 in [21⊗] is the≃ local study of the Schwartz kernel with respect to dvX of the parallel transport operator. For any linear operator L (H , H ), write ( , ) C ∞(X X, E ⊠ E∗ ) for the Schwartz kernel Kp,t ∈ p,0 p,t Kp,t · · ∈ × p,t p,0 with respect to dvX of the bounded operator := P P : L2(X, E ) L2(X, E ) , (2.25) Kp,t p,tKp,t p,0 p,0 −→ p,t defined for any s C ∞(X, E ) and x X by the formula ∈ p,0 ∈ p,ts (x)= p,t(x, y)s(y) dvX(y) . (2.26) K ZX K The existence of a smooth Schwartz kernel is an immediate consequence of the fact that the image of (2.25) is finite dimensional. In the case t = 0, so that End(H ) Kp,0 ∈ p,0 and (x, x) End(E ) for all x X, we have the following basic trace formula, Kp,0 ∈ p,0 x ∈ Tr[ p,0]= Tr [ p,0(x, x)] dvX (x) . (2.27) K ZX K Fix ε> 0 and consider a collection of diffeomorphisms T X B x0 (0,ε) ∼ V X , (2.28) −−−→ x0 ⊂ varying smoothly with x0 X, sending 0 to x0 X and with differential at 0 induc- ∈ ∈ R L L ing the identity of Tx0 X. For any x0 X and t , we pullback (L, h , ) and Et Et ∈ ∈ ∇ (Et, h , ) over Vx0 in this chart, and identify them with their central fibre Lx0 and ∇ Tx0 X Et,x0 by parallel transport along radial lines of B (0,ε). We then identify Lx0 with C by the choice of a unit vector. For any p,t L (Hp,0, Hp,t), we write p,t,x0 ( , ) for the image in this trivialization of its SchwartzK ∈ kernel over V V . ThenK · ·( , ) x0 × x0 Kp,t,x0 · · can be seen as the evaluation at x0 X of the pullback of Et E0∗ over the fibred T X T X ∈ ⊗ product B (0,ε) B (0,ε) over X, and for any m N, let m be a local ×X ∈ |·|C (X) C m-norm on this bundle induced by derivation with respect to x X. 0 ∈ 12 For any Z, Z′ T X and t R, we use the following local model ∈ x0 ∈ t T (Z,Z′) := exp π Π (Z Z′), (Z Z′) + √ 1ω(Z,Z′) , (2.29) t,x0 − 0 − − − hD E i where , is the scalar product on T X induced by the metric gT X defined by (2.11). h· ·i x0 0 For any F (Z,Z′) E E∗ polynomial in Z, Z′ T X, we write t,x0 ∈ t ⊗ 0 ∈ x0 F T (Z,Z′) := F (Z,Z′)T (Z,Z′) E E∗ . (2.30) t,x0 t,x0 t,x0 ∈ t ⊗ 0 For any function h C ∞(X, R), we will use repeatedly in the sequel the following form ∈ of Taylor expansion around any x0 X up to order k 1 N, as Z 0 in the chart (2.28) above, ∈ − ∈ | |→ k 1 r α − ∂ h Z k h(Z)= h(x0)+ α + O( Z ) r=1 α =r ∂Z α! | | X |X| (2.31) k 1 r α − r/2 ∂ h (√pZ) k k − − 2 = h(x0)+ p α + p O( √pZ ) . r=1 α =r ∂Z α! | | X |X| X T X Write d ( , ) for the Riemannian distance of (X,g ), and for any m N, let m · · 0 ∈ |·|C be the local C m-norm induced by derivation with respect to Ep over R X. Then ∇ × Theorem 2.5 is based on the following result. Theorem 2.6. [21, Th.4.3] Consider a collection of charts of the form (2.28), varying smoothly with x0 X, sending 0 to x0 and with differential at 0 inducing the identity ∈ N R of Tx0 X. Then for any m, k , θ ]0, 1[ and any compact subset K , there is C > 0 such that for all p N∈and t ∈K, we have ⊂ k ∈ ∈ k X θ (x, y) 6 C p− as soon as d (x, y) > εp− 2 . (2.32) |Tp,t |C m k Furthermore, there is a family Gr,t,x0 (Z,Z′) Et,x0 E0∗,x r N of polynomials in { ∈ ⊗ 0 } ∈ Z,Z′ T X of the same parity as r, depending smoothly on x X, such that for any ∈ x0 0 ∈ m , m′, l, k N, δ ]0, 1[ and any compact subset K R, there is C > 0 and θ ]0, 1[ ∈ ∈ ∈ θ/2 ∈ such that for any x X,p N and Z, Z′ T X with Z , Z′ < εp− , we have 0 ∈ ∈ ∈ x0 | | | | l α α′ ∂ ∂ ∂ n sup ′ p− p,t,x0 (Z,Z′) ′ l α α α + α =m ∂t ∂Z ∂Z′ T | | | | k 1 − − r/2 k m +δ 6 2 p− GrTt,x0 (√pZ, √pZ′) ′ Cp− , (2.33) m − r=0 C (X) X where G (Z,Z′) is constant in Z, Z′ T X, given by 0,t,x0 ∈ x0 1 E G0,t,x0 (Z,Z′)=¯µt− (x0)τt,x0 . (2.34) 13 Let us now consider a diffeomorphism ϕ : X X preserving the symplectic form ω, → together with a lift ϕL : L L to the total space of L preserving metric and connection, and assume that for some→t R, we have 0 ∈ 1 Jt0 = dϕJ0 dϕ− . (2.35) E Assume further that there is a lift ϕ : E0 Et0 of ϕ preserving metric and connection, and write ϕ : E E for the induced→ lift on E , for all p N. For any s p p,0 → p,t0 p ∈ ∈ C ∞(X, E ), we define the pullback ϕ∗s C ∞(X, E ) by the formula p,t0 p ∈ p,0 1 (ϕp∗s)(x) := ϕp− s(ϕ(x)) . (2.36) This induces by restriction a unitary isomorphism ϕ∗ : H ∼ H . (2.37) p p,t0 −−−→ p,0 Then Theorem 2.8 gives a semi-classical estimate for the trace of the endomorphism ϕp∗ p,t0 End(Hp,0) as p + , using the trace formula (2.27). To this end, we need theT following∈ assumption.→ ∞ Definition 2.7. The fixed point set Xϕ X of a diffeomorphism ϕ : X X is said ⊂ → to be non-degenerate over an open set U X if Xϕ U is a proper submanifold of U satisfying ⊂ ∩ T Xϕ = Ker(Id dϕ ) for all x Xϕ U. (2.38) x TxX − x ∈ ∩ As the lift ϕL preserves hL and L, its value β C via the canonical identification ∇ ϕ ∈ L L∗ C is locally constant over X , and satisfies β = 1. ⊗ ≃ | | Let Y X be a submanifold, and let N be a subbundle of TX over Y transverse ⊂ N T X to TY . Write g for the Euclidean metric on N induced by the metric g0 defined T X by (2.11), and write dv T X and dv N for the Riemannian densities of (TX,g0 ) and (N,gN ). We denote by| |dv |the| density over Y defined by the formula | |T X/N dv = dv dv . (2.39) | |T X | |T X/N | |N N T X We write P : TX N for the orthogonal projection with respect to g0 of vector bundles over Y . → Theorem 2.8. [21, Th.1.2] Assume that the fixed point set Xϕ of ϕ : X X is → non-degenerate over X, and write Xj 16j6m for the set of its connected components. { ϕ } Then there exist densities ν over X for any r N such that for any k N∗ and as r ∈ ∈ p + , → ∞ m k 1 − dim Xj /2 2π√ 1pλj r k Tr[ϕp∗ p,t0 ]= p e− − p− νr + O(p− ) , (2.40) T Xj ! jX=1 rX=0 Z 14 2π√ 1λ L where e − j is the constant value of ϕ over X , for some λ R. Furthermore, for j j ∈ any x Xϕ we have ∈ 1 ν 0 0 E, 1 E 1 KX − 2 (x) = TrEx [ϕ − τt0 ] det(Πt0 )− τt0 Tt0,x(dϕ.Z, Z)dZ dv x Nx T X/N Z | | 1 0 E, 1 E 1 KX − 2 = TrE [ϕ − τ ] det(Π )− τ x t0 t0 t0 x 1 2 N t0 1 0 N det − P (Π dϕ− Π )(Id dϕ)P , (2.41) Nx 0 − t0 T X − for some natural choices of square roots,h where N is any subbundle of TXi over Xϕ transverse to TXϕ. The first coefficient (2.41) acquires a geometric interpretation in the special case ϕ when the bundles TX and N are both preserved by ϕ and J0. In order to describe KX it, let ϕ : KX,0 KX,t0 be the natural action induced by ϕ, and recall that KX −→ K , 1 KX τ : K K has been defined in (2.15). Then ϕ X − τ C ∞(X, C) via the t0 X,0 −→ X,t0 t0 ∈ canonical identification K K∗ C, and one can compute the following. X,0 ⊗ X,0 ≃ Proposition 2.9. [21, Lemma 5.1] Assume that the fixed point set Xϕ of ϕ : X X is non-degenerate over X, and that there exists a subbundle N of TX over Xϕ transverse→ ϕ ϕ to TX such that TX and N are both preserved by ϕ and J0. Then we have the following formula, for all x Xϕ, ∈ 1 0 1 KX − 2 det(Π )− τ Tt ,x(dϕ.Z, Z)dZ t0 t0 x 0 ZNx 1 dim Nx 1 K , 1 K 2 =( 1) 4 (ϕ X − τ X )x− det (Id dϕ ) − 2 , (2.42) − t0 | Nx N − |N | for some natural choices of square roots. The previous result acquires an even cleaner formulation in the case when (E, hE, E) satisfies ∇ 2 E = KX , (2.43) as a line bundle over R X with induced metric and connection. Such a line bundle × 2 exists if and only if the first Chern class c1(TX) H (X, Z) of TX is even, and the choice of a complex line bundle E satisfying (2.43∈) is called a metaplectic structure 1/2 on X. We write E =: KX , and call it the metaplectic correction. We then get the following straightforward corollary of Proposition 2.9. Corollary 2.10. Consider the assumptions of Proposition 2.9, and assume further 1/2 that X admits a metaplectic structure. Then if E = KX is the associated metaplectic correction over R X, the first coefficient ν of (2.41) satisfies the formula × 0 dim N 1 ν =( 1) 4 det (Id dϕ ) − 2 dv , (2.44) 0 − | N N − |N | | |T X/N for some natural choices of square roots. Ep In the sequel, we will write p for the norm induced on Ep Ep∗ by h , for all p N. |·| ⊗ ∈ 15 3 Quantum evolution operators Let (X,ω) be a compact symplectic manifold without boundary endowed with (L, hL, L) ∇ satisfying the prequantization condition (1.1), and consider a smooth function f ∈ C ∞(X, R). The Hamiltonian vector field ξf C ∞(X,TX) associated to f is defined by the formula ∈ ιξf ω = df . (3.1) The Hamiltonian flow of f is the flow of diffeomorphisms ϕt : X X defined for all t R by → ∈ ∂ ∂t ϕt = ξf , (3.2) ϕ0 = IdX . By definition (3.1) of a Hamiltonian vector field and by Cartan formula, the Hamiltonian flow ϕt : X X preserves ω for all t R. Let ξf C ∞(L, T L) be the horizontal lift of → ∈ L ∈ ξf to the total space of L with respect to , and let t C ∞(L, T L) be the canonical ∇ e ∈ vector field on the total space of L defined by ∂ 2π√ 1t t = e − , (3.3) ∂t t=0 2π√ 1t for the action of e − by complex multiplication in the fibres. Then the flow (3.2) L R lifts to a flow ϕt : L L on the total space of L over X, defined for all t by the formula → ∈ ∂ L ∂t ϕt = ξf + ft , (3.4) L e ϕ0 = IdL . L Note that both ϕt and ϕt define 1-parameter groups, as both vector fields defining them do not depend on t R. From the definition (3.1) of the Hamiltonian vector ∈ field of f, we see that ϕL is the unique lift of ϕ to L preserving the connection L, t t ∇ for all t R. More specifically, for any t R, recall the pullback of s C ∞(X, L) by ∈ ∈ ∈ ϕt defined by formula (2.36). Then for any vector field v C ∞(X,TX), we get from (3.1) and (3.4) that ∈ L L ϕ∗ s = ϕ∗s . (3.5) t ∇v ∇dϕt.v t On the other hand, we also deduce from (3.4) the following Kostant formula, for all t R, ∈ ∂ L ϕ∗s = 2π√ 1f ϕ∗s . (3.6) ∂t t ∇ξf − − t N R L For any p and t , let us write ϕt,p for the flow induced by ϕt on the total space p ∈ ∈ p of L , and ϕt,p∗ for the associated pullback as in (2.36). Then for any s C ∞(X, L ), the Kostant formula (3.6) becomes ∈ ∂ Lp ϕ∗ s = 2π√ 1pf ϕ∗ s . (3.7) ∂t t,p ∇ξf − − t,p 16 This formula characterizes f C ∞(X, R) as the Kostant moment map for the action p p ∈ of R on (Lp, hL , L ) induced by ϕ for all t R. ∇ t ∈ Note that (3.1) implies that f(ϕt(x)) = f(x) for all t R and x X, and (3.2) implies that dϕ .ξ = ξ for all t R. For all x X, we write∈ ∈ t f f ∈ ∈ p p τt,p : L L (3.8) x −→ ϕt(x) for the parallel transport along the path s ϕ (x) for s [0, t]. We can then reformu- 7→ s ∈ late the Kostant formula (3.7) as 1 2π√ 1tpf ϕ− τ = e− − C ∞(X, C) , (3.9) t,p t,p ∈ via the canonical identification L L∗ C. ⊗ ≃ Let us now consider an almost complex structure J0 End(TX) over X compatible with ω. Then for any t R, the formula ∈ ∈ 1 J := dϕ J dϕ− End(TX) (3.10) t t 0 t ∈ defines a path Jt End(TX) t R of almost complex structures over X compatible { ∈ } ∈ with ω. For any t R, we write H for the space of almost holomorphic sections with ∈ p,t respect to Jt defined in Theorem 2.3. Then for any t0 R and p N, the pullback ?? induces by restriction a bijective linear map ∈ ∈ ϕ∗ : H H , (3.11) t0,p p,t+t0 −→ p,t for all t R. For any fixed p N, this implies in particular that the dimension dim Hp,t ∈ ∈ Hp Hp does not depend on t R, so that the quantum bundle (Hp, h , ) of Definition 2.4 is well defined over R∈for all p N. Recall the tautological fibration∇ π : R X X ∈ × → considered in (2.10) together with all the data induced by Jt End(TX) t R, and ∈ consider the flow over R X defined for all t R by { ∈ } × 0 ∈ Φ : R X R X t0 × −→ × (3.12) (t, x) (t + t ,ϕ (x)) . 7−→ 0 t0 N R p Lp Lp For any p and t0 , the lift ϕt0,p of ϕt0 to (L , h , ) over X induces ∈ ∈ p Lp Lp∇ R tautologically a lift Φt∗0,p of Φt0 to the pullback of (L , h , ) over X via the p ∇ × second projection. For any section s C ∞(R X, L ) over R X and any t R, write p ∈ × × ∈ st C ∞(X, L ) for the section over X defined by st(x) := s(t, x) for all x X. Then ∈ R N R p ∈ for any t0 and p , the pullback of s C ∞( X, L ) by Φt0 is given for any t R by the∈ formula ∈ ∈ × ∈ (Φt∗0,ps)t = ϕt∗0,pst+t0 . (3.13) R By (3.11), the pullback Φt∗0,p preserves the smooth sections C ∞( , Hp) of the quantum p bundle, seen as a subspace of C ∞(R X, L ) as in Definition 2.4. × We still write ξf for the pullback of the Hamiltonian vector field ξf C ∞(X,TX) to a vertical vector field over π : R X R via the second projection, and∈ write ∂ for × → t 17 the horizontal vector field over π : R X R induced by the canonical vector field of × L→ L p R. By definition of the pullback of (L, h , ) to R X, for any s C ∞(R X, L ) and t R we have ∇ × ∈ × ∈ Lp ∂ s = st . (3.14) ∇∂t t ∂t Recall Definition 2.4 for the connection Hp , and note that for all t, t R, we have ∇ 0 ∈ Φt∗0,pPp,t = Pp,t+t0 Φt∗0,p . (3.15) p Using (3.11) to (3.15), for any smooth section s C ∞(R, Hp) C ∞(R X, L ) ∈ p ⊂ × and seeing the orthogonal projection Pp : C ∞(R X, L ) C ∞(R, Hp) as a global endomorphism, we get the following quantized version× of the→Kostant formula (3.7), for all t R, 0 ∈ ∂ Lp Lp ∗ √ ∗ ∗ Φt,ps = ξf 2π 1pf Φt0,ps + ∂t Φt0,ps ∂t t=t ∇ − − ∇ 0 Lp = Pp 2π√ 1pf Φ∗ Pps ∇ξf +∂t − − t0,p Lp Lp (3.16) = Pp Pp + Pp( 2π√ 1pf)Pp Φ∗ s ∇∂t ∇ξf − − t0,p Hp √ 1 Lp = ∂ 2π√ 1pPp f + − ξ Pp Φt∗ ,ps . ∇ t − − 2πp ∇ f ! ! 0 Seeing the path Jt End(TX) t R as lying in the space Jω of almost complex { ∈ } ∈ structures compatible with ω and comparing with the usual Kostant formula (3.7), we can interpret the quantized Kostant formula (3.16) by stating that the Kostant– Souriau operator Q (f) End(H ) given by formula (1.4) induces a moment map on p ∈ p the quantum bundle Hp over Jω for the natural action of the group of Hamiltonian diffeomorphisms Ham(X,ω) on Jω defined by (3.10). The relevance of the Kostant–Souriau operator in Kähler geometry goes back to the work of Cahen, Gutt and Rawnsley [11] relating it to Berezin’s quantization of Kähler manifolds [1], and the moment map picture described above has been introduced by Donaldson in [14]. As explained for example in [16, §1], we can consider the line bundle det(Hp) over any compact submanifold of Jω for p N big enough, and the det Hp 2 ∈ curvature of the connection induced by the L -connection (2.17) on det(Hp) defines a natural symplectic∇ form via the prequantization formula (1.1). In the spinc Dirac operator case, which implies the Kähler case, it follows from the asymptotics of the curvature of Hp as p + established by Ma and Zhang in [29, Th. 2.1]. Then the quantized Kostant∇ formula→ (∞3.16) shows that the Hamiltonian flow associated with the function det(Qp(f)) on Jω is precisely the action of the Hamiltonian flow of f on Jω defined by (3.10). On the other hand, as described for example in [35, § 9.7], the quantum dynamics is given by the 1-parameter family of unitary operators generated by the quantum Hamiltonian operator acting on a fixed space of quantum states. For any p N, we ∈ thus consider the quantum Hamiltonian operator Qp(f) restricted to the space Hp,0 18 of almost holomorphic sections with respect to our initial almost complex structure J . This induces a one-parameter family exp 2π√ 1tpQ (f) End(H ) of unitary 0 − p ∈ p,0 operators defined for all t R by ∈ ∂ exp 2π√ 1tpQ (f) =2π√ 1pQ (f) exp 2π√ 1tpQ (f) , ∂t − p − p − p (3.17) exp 2π√ 1tpQp(f) = IdH . − t=0 p,0 Writing : H H for the parallel transport with respect to Hp over R as in Tp,t p,0 → p,t ∇ Section 2, the following Lemma establishes formula (1.3). Lemma 3.1. For any p N and all t R, we have the following equality ∈ ∈ exp 2π√ 1tpQ (f) = ϕ∗ End(H ) . (3.18) − − p t,pTp,t ∈ p,0 2 Hp Proof. By Definition 2.4 of the L -connection , using (3.14) and the fact that Φt∗0,p ∇ p commutes with P when acting on C ∞(R X, L ) for any t R and p N, we have p × 0 ∈ ∈ Hp Hp Φ∗ = Φ∗ . (3.19) t0,p∇∂t ∇∂t t0,p Lp p Furthermore, as ϕ∗ commutes with 2π√ 1pf when acting on C ∞(X, L ), t0,p ∇ξf − − by (1.4) and the pullback formula (3.13), we get Φt∗0,pQp(f)= Qp(f)Φt∗0,p . (3.20) This implies the following analogue of (3.9) for the quantized Kostant formula (3.16), for all t R, ∈ ∂ ϕ∗ = 2π√ 1pQ (f)ϕ∗ , (3.21) ∂t t,pTp,t − − p t,pTp,t which follows from the quantized Kostant formula (3.16) in the same way as (3.9) follows from the usual Kostant formula (3.7). This proves the lemma. Before giving the applications of the results described in Section 2 to the quan- tum evolution operator defined above, let us illustrates its behaviour via the following definition. Definition 3.2. For any x X and any unit vector ζ L , the associated coherent 0 ∈ ∈ x0 state is the sequence sx0,p Hp,0 p N defined for all x X by { ∈ } ∈ ∈ p sx0,p(x)= Pp,0(x, x0)ζ , (3.22) p p where Pp,0( , ) C ∞(X X, L ⊠ (L )∗) is the Schwartz kernel with respect to dvX of · · ∈ × p the orthogonal projection operator P : C ∞(X, L ) H . p,0 → p,0 19 Coherent states represent the quantization of a classical particle located at x X 0 ∈ in phase space. As one can readily check from the definition, it is characterized by the fact that its orthogonal in H consists of sections vanishing at x X. As shown p,0 0 ∈ in [27, Th. 0.1, § 1.1], the sequence sx0,p p N decreases rapidly as p + outside { } ∈ n → ∞ p any open set containing x0, while sx0,p(x0) L is of order p . Now using the following | | p tautological identity of operators acting on C ∞(X, L ), exp 2π√ 1tpQ (f) = exp 2π√ 1tpQ (f) P , (3.23) − p − p p,0 and by the classical formula for the Schwartz kernel of the composition of two operators, for any t R and x X, we get from Definition 3.2, ∈ ∈ exp 2π√ 1tpQ (f) s (x) − p x0,p p = exp 2π√ 1tpQp(f) (x, w)Pp,0(w, x0)ζ dvX (w) (3.24) X − Z = exp 2π√ 1tpQ (f) (x, x )ζp . − p 0 This shows that the last line of (3.24), seen as a section of Lp with respect to the variable x X, can be interpreted as the quantum evolution at time t R of the ∈ ∈ quantization of a classical particle at x0 X. In particular, we expect this section to decrease rapidly as p + outside any∈ open set containing the classical evolution ϕ (x ) X, while its value→ ∞ at ϕ (x ) should be of order pn. The following result shows t 0 ∈ t 0 that this is indeed the case. Proposition 3.3. For any ε> 0, k, m N, θ ]0, 1[ and any compact subset K R, ∈ ∈ ⊂ there exists Ck > 0 such that k X θ exp 2π√ 1tpQp(f) (x, y) 6 Ckp− as soon as d (x, ϕt(y)) > εp− 2 , (3.25) − C m for all t K. Furthermore, there exist a (t, x) C for any r N, depending smoothly ∈ r ∈ ∈ on x X and t R, such that for any k N∗, ∈ ∈ ∈ k 1 n 2π√ 1tpf(x) − r k exp 2π√ 1tpQp(f) (ϕt(x), x)= p e − p− ar(t, x)+ O(p− ) τt,p , − ! r=0 X (3.26) with first coefficient a0 satisfying the formula 0 1 2 1 KX − a0(t, x) = det(Π t)− τ t . (3.27) − − x In particular, it satisfies a (t, x) =0 for all t R and x X. 0 6 ∈ ∈ Proof. Recall that for any x, y X and t R, we get from Lemma 3.1 and formula (2.36) that ∈ ∈ 1 exp 2π√ 1tpQ (f) (x, y)= ϕ− (ϕ (x),y) . (3.28) − − p t,p Tp,t t 20 Then (3.25) is a consequence of the Kostant formula (3.6), together with the rapid decrease of p,t( , ) outside of the diagonal given by (2.32). Using theT exponentiation· · (3.9) of Kostant formula, rewrite (3.28) as exp 2π√ 1tpQp(f) (ϕt(x), x)= ϕt,p p, t(ϕ t(ϕt(x)), x) − T − − (3.29) 2√ 1πtpf(x) = e − τt,p p, t(x, x) . T − We can thus apply Theorem 2.6 with x0 = x for Z = Z′ = 0, and noting that J2q+1(0, 0) = 0 for all q N for parity reasons, we then get the expansion (3.26), ∈ 1 R with first coefficient satisfying a0(t, x)=¯µ−t (x), for all x X and t . This implies − ∈ ∈ formula (3.27) via the formula (2.24) for µ C ∞(X, C). ∈ Recall the non-degeneracy assumption of Definition 2.7. We also have the following semi-classical trace formula for the quantum evolution operator, where we use the notations of Theorem 2.8. Theorem 3.4. Let t R be such that the fixed point set Xϕt of ϕ : X X is non- ∈ t → degenerate, and write Xj 16j6m for the set of its connected components. Then there { ϕt} exist densities ν over X for any r N such that for any k N∗ and as p + , r ∈ ∈ → ∞ Tr exp 2π√ 1tpQ (f) − − p h iq k 1 − dim Xj /2 2π√ 1pλj r k = p e− − p− νr + O(p− ) , (3.30) Xj ! jX=1 rX=0 Z 2π√ 1λ L where e − j is the constant value of ϕ over X , for some λ R. Furthermore, we j j ∈ have 1 0 1 1 KX − 2 2 N t 1 0 N ν = det(Π )− τ det − P (Π dϕ− Π )(Id dϕ )P dv , (3.31) 0 t t N 0 − t t T X − t | |T X/N h i for some natural choices of square roots. Proof. Using Lemma 3.1, this is a straightforward consequence of Theorem 2.8. The coefficient λj R appearing in the expansion (3.30) has a natural geometric interpretation, which∈ fits in a more general context. In fact, note that the evolution equations (3.2)and(3.4) generalize to time-dependent Hamiltonians F C ∞(R X, R), ∈ × so that f C ∞(X, R) is replaced by f C ∞(X, R) depending on t R, with f (x) := ∈ t ∈ ∈ t F (t, x) for all x X. In that case, we also get a Hamiltonian flow ϕt : X X together L ∈ → with a lift ϕt to the total space of L preserving metric and connection. The main difference here is that the term replacing L 2π√ 1f in the Kostant formula ∇ξf − − (3.6) will depend on time, and the corresponding exponentiation as in (3.9) at x X ∈ reads t 1 2π√ 1p fs(ϕs(x))ds (ϕt,p− τt,p)x = e− − 0 . (3.32) R 21 Let now x X and t R be such that ϕt(x)= x, and via the canonical identification ∈C ∈ Lx Lx∗ , let us write ⊗ ≃ L 2√ 1πλt(x) ϕt,x =: e − . (3.33) Note that the path s ϕs(x) defines a loop γ inside X. Assuming that this loop bounds an immersed disk7→ D X, by the prequantizaton condition (1.1) and via (3.32) ⊂ above, we get t λt(x)= ω + fs(ϕs(x))ds . (3.34) ZD Z0 This is a familiar quantity in symplectic topology, called the action of F around the loop γ. Recall that as ϕL preserves the connection L, the quantity λ (x) R is ∇ t ∈ constant when x X varies continuously over a submanifold of fixed points of ϕt. This discussion motivates∈ the following definition. Definition 3.5. Let F C ∞(R X, R) be a time-dependent Hamiltonian, and let t R be such that its Hamiltonian∈ flow× ϕ : X X at time t R has non-degenerate fixed∈ t → ∈ point set Xϕ X in the sense of Definition 2.7. Then for any connected component Y of Xϕ, the associated⊂ real number λ R defined over Y by 0 ∈ L 2√ 1πλ0 ϕt =: e − (3.35) is called the action of F over Y . In the general case of a time-dependent Hamiltonian F C ∞(R X, R), we get ∈ × a quantum Hamiltonian as before from the corresponding quantized Kostant formula (3.16), and we can define the associated evolution operator by equation (3.17). Then the analogue of Lemma 3.1 holds in that case, and the analogues of Proposition 3.3 and Theorem 3.4 hold in the same way. Finally, the situation described above also generalizes to the case when (Lp, hLp , Lp ) p ∇ over R X is replaced by Ep = L E with induced metric and connection, where (E, hE,× E) is a Hermitian vector bundle⊗ with connection over R X. In that case, we ∇ × make the further assumption that the Hamiltonian flow ϕt : X X lifts to a bundle E → R map ϕt : E0 Et over X preserving metric and connection for all t , and we still write ϕ →for the corresponding action on E for any p N. Then∈ the analogue t,p p ∈ of Lemma 3.1 for the quantum evolution operator defined by equation (3.17) holds as before, and using the general setting of Section 2, we also get the corresponding 1/2 analogues of Proposition 3.3 and Theorem 3.4. The case of E = KX , with the lift KX R induced by ϕt : KX,0 KX,t for all t , will be of particular interest in the next section. → ∈ 4 Gutzwiller trace formula Consider the setting of the previous section, with Hamiltonian f C ∞(X, R) not depending on time, and a Hermitian vector bundle with connection∈ (E, hE, E) over ∇ 22 R X, so that Lp is replaced by E = Lp E for all p N. Let g : R R be a smooth × p ⊗ ∈ → function with compact support, and for all t R, set ∈ 2π√ 1t gˆ(t) := g(t)e− − dt . (4.1) R Z We define the family of operators gˆ(pQp(f)) End(Hp) p N by the formula { ∈ } ∈ gˆ(pQp(f)) := g(t) ϕ∗ p,t dt . (4.2) R t,pT Z In the particular case of E = C, so that Lemma 3.1 holds, we get gˆ(pQp(f)) = g(t) exp( 2π√ 1tpQp(f))dt , (4.3) R Z − − recovering the usual definition via functional calculus from (4.1). Note that formula (1.7) forg ˆ(pQ (f c)) with c R follows from (2.36), (3.2) and (3.4), as replacing p − ∈ f C ∞(X, R) by f c does not change the Hamiltonian flow ϕt : X X but multiplies ∈ 2π√− 1tc → its lift to L by e− − . In the context of semi-classical analysis, the Gutzwiller trace formula predicts a semi-classical estimate for the trace Tr[ˆg(pQ (f c))] as p + , p − → ∞ where c R is a regular value of f, showing that it localizes around the periodic orbits ∈ 1 of the Hamiltonian flow of f inside the level set f − (c). The following preliminary result shows that the Schwartz kernel ofg ˆ(pQp(f c)) 1 − decreases rapidly outside f − (c) as p + . → ∞ Proposition 4.1. Let c R be a regular value of f C ∞(X, R). For any k N, there ∈ 1 ∈ ∈ exists C > 0 such that for any y X, x X f − (c) and p N, we have k ∈ ∈ \ ∈ Ck n k gˆ(pQ (f c))(x, y) 6 p − 2 . (4.4) | p − |p f(x) c k | − | Proof. To simplify notations, we assume E = C, the case of general E being completely analogous. Recall the definition (3.8) of τ , and p N. For any x X and t R, t,p ∈ ∈ 0 ∈ consider a chart around x0 := ϕt0 (x) X asin(2.28), such that the radial line generated Tx0 X ∈ p by ξf,x0 in B (0,ε) is sent to the path s ϕs(x0) in Vx0 . Then L is identified with Lp along this path by the parallel transport7→ τ , for all p N and t < ε. Thus for x0 t,p ∈ | | Tx0 X R any Z B (0,ε) sent to y Vx0 in the chart (2.28) and for any t small enough, we have∈ ∈ ∈ 1 τ − (ϕ (x),y)= (tξ ,Z) . (4.5) t,p Tp,t0+t t0+t Tp,t0+t,x0 f,x0 Using τt0+t,p = τt,pτt0,p for all t0 T and t <ε, we can apply Theorem 2.6 in such charts for all t R, so that for any k ∈N we get| |C > 0 such that for any x, y X, t Supp g 0 ∈ ∈ k ∈ ∈ and all p N, we have ∈ k ∂ k 1 n+ 2 τt,p− p,t(ϕt(x),y) 6 Ckp . (4.6) ∂tk T p 23 Recall from (3.1) that the Hamiltonian flow of f is the same as the Hamiltonian flow of f c, for all c R. Then exponentiating Kostant formula as in (3.9) and as Supp g − ∈ is compact, we can integrate by parts to get from the definition (4.2)ofg ˆ(pQp(f)) that for any x, y X and k N, ∈ ∈ 1 2π√ 1tp(f(x) c) gˆ(pQp(f c))(x, y)= g(t)τ − p,t(ϕt(x),y)e− − − dt R t,p − Z T k 1 1 ∂ 2π√ 1tp(f(x) c) = g(t)τ − p,t(ϕt(x),y) e− − − dt k R t,p k ( 2π√ 1p(f(x) c)) Z T ∂t − − k − k k p− ∂ 1 2π√ 1tp(f(x) c) = (2π√ 1)− g(t)τ − p,t(ϕt(x),y) e− − − dt . (4.7) − (f(x) c)k R ∂tk t,p T − Z This proves the result by (4.6). Let us now estimate the Schwartz kernel ofg ˆ(pQp(f c)) as p + . Proposi- 1 − → ∞ tion 4.1 shows that it localizes around the level set f − (c), in contrast with Proposi- tion 3.3. In the following theorems and their proofs, we will use freely the notations of Sections 1 and 2. Theorem 4.2. Let c R be a regular value of f C ∞(X, R). If x, y X do not ∈ ∈ ∈ satisfy ϕt(x)= y for some t R or do not satisfy f(x)= f(y)= c, then for any k N, there exists C > 0 such that∈ for all p N, ∈ k ∈ k gˆ(pQ (f c))(x, y) 1 E, 1 gˆ(pQp(f c))(x, y)= g(t)τ − ϕt − p,t(ϕt(x),y)dt . (4.11) R t,p − Z T 24 As c R is a regular value of f, the Hamiltonian vector field ξf does not vanish over 1 ∈ f − (c). By (2.32), this implies that for any θ ]0, 1[ and ε > 0, we get the following estimate as p + , ∈ → ∞ − θ t0+εp 2 1 E, 1 gˆ(pQp(f c))(x, y)= − θ g(t)τt,p− ϕt − p,t(ϕt(x),y)dt + O(p−∞) , (4.12) − t0 εp 2 T t0 T Z X∈ − where all terms but a finite number vanish by compacity of Supp g. Consider a chart around y as in (2.28), sending the radial line generated by ξf,y in BTyX (0,ε) to the path s ϕ (y) in V . Then Lp is identified with Lp along this path 7→ s y y by the parallel transport τ , for all p N and t < ε. Using τ = τ τ for all t,p ∈ | | t0+t,p t,p t0,p t T and t < ε, we can then apply Theorem 2.6 to get a family G (Z,Z′) 0 ∈ | | { r,t,y ∈ Et,y E0∗,y r N of polynomials in Z,Z′ TyX of the same parity as r and smooth in ⊗ } ∈ ∈ t R, such that for any δ ]0, 1[ and k N∗, there is θ ]0, 1[ such that for all p N, ∈ ∈ ∈ ∈ ∈ − θ εp 2 n 1 E, 1 − gˆ(pQp(f c))(x, y)= p τp,t− 0 − θ g(t0 + t)ϕt0+t − εp 2 t0 T Z X∈ − k 1 − r θ k 2 n 2 2 +δ p− GrTt0+t,y(√ptξf,y, 0)dt + p − O(p− ) . (4.13) rX=0 Consider the right hand side of (4.13), and let us apply the Taylor expansion in t R described in (2.31) on all terms depending on t R respectively inside and outside∈ ∈ the exponential of the local model (2.29) for T . We then get F (t) E E∗, t0+t,y r,t0 ∈ t0 ⊗ 0 polynomial in t R and of the same parity as r for any r N, such that for all t0 T and as t 0, ∈ ∈ ∈ | |→ k 1 − r E, 1 2 − p− g(t0 + t)ϕt0+t GrTt0+t,y(√ptξf,y, 0) = rX=0 k 1 − k E, 1 r/2 t0 2 M g(t )ϕ − p− F (√pt) exp pπ Π ξ ,ξ t + p− 2 O( √pt k ) , (4.14) 0 t0 r,t0 − h 0 f,y f,yi | | rX=0 for some M N∗ depending on the degrees of the polynomials G for all t <ε, k ∈ r,t0+t,y | | t0 T Supp g and 1 6 r 6 k. Furthermore, from the formula (2.34) for the first ∈ ∩ 1 E coefficient G , we know that F (t)=µ ¯− (y)τ for all t R. Note that by the 0,t,y 0,t0 t0 t0,y ∈ definition (2.22) of Πt0 , the real part of Πt0 ξ ,ξ is strictly positive, so that the right 0 h 0 f f i hand side of (4.14) decreases exponentially in t R. Writing ∈ M (1 θ) δ = δ + k − , (4.15) k 2 and after a change of variable t t/√p, we then get 7→ 1 E, 1 E 1 n 1 gˆ(pQ (f c))(x, y)= g(t )¯µ− (y)ϕ − τ τ − p − 2 p − 0 t0 t0 t0 p,t0 t0 T X∈ k 1 − r 1 k t0 2 n +δk p− 2 Fr,t (t) exp π Π ξf,y,ξf,y t dt + p − 2 O(p− 2 ) . (4.16) R 0 − h 0 i rX=0 Z 25 As F (t) is odd as a function of t R for all q N, we get 2q+1,t0 ∈ ∈ t0 2 F2q+1,t (t) exp π Π ξf,y,ξf,y t dt =0 . (4.17) R 0 − h 0 i Z As δ δ when θ 1 by (4.15) and as δ can be chosen arbitrary small, this gives k → → the expansion (4.9), and we get the formula (4.10) for the first coefficient via the classical formula for Gaussian integrals, using the formula (2.24) and the fact that 1 E F (t)=¯µ− (y)τ for all t R. 0,t0 t0 t0 ∈ Consider now the hypotheses and notations of Theorem 1.2. Recall that c R is a 1 ∈ regular value of f, so that Σ := f − (c) is a smooth manifold. Then there exists ε > 0 and diffeomorphisms 1 Ψ : f − (]c ε,c + ε[) ∼ ]c ε,c + ε[ Σ (4.18) t0 − −−−→ − × for all t T , such that f(u, x) = u for all (u, x) ]c ε,c + ε[ Σ under this 0 ∈ ∈ − × identification, and such that ϕt 1 Ψ X 0 f − (]c ε,c + ε[) = ]c ε,c + ε[ Y , (4.19) t0 ∩ − − × j 16j6m tj[=t0 ϕt 1 where the connected components Yj of X j f − (c) are seen as submanifolds of Σ, for all ∩ T Σ T X 1 6 j 6 m. We endow Σ with the Riemannian metric g induced by g0 := ω(J0 , ) 1 · · via the inclusion Σ = f − (c) X. For any 1 6 j 6 m, let dv Yj be the Riemannian T⊂Σ | | density over Yj induced by g and let N be the normal bundle of Yj inside Σ. We write P N : T Σ N for the orthogonal projection with respect to gT Σ over Y for all → j 1 6 j 6 m. 1 Recall that the Liouville measure on the level set f − (c) is induced by the volume form n ιv ω 2n 1 1 Ω − (f − (c), R) , (4.20) (n 1)! ∈ − 1 for any v C ∞(f − (c),TX) satisfying ω(ξf , v) = 1, and does not depend on such a ∈ 1 1 choice. We write Volω(f − (c)) > 0 for the volume of f − (c) with respect to (4.20). Recalling Definition 3.5, the following theorem is a version of the Gutzwiller trace formula in geometric quantization of compact prequantized symplectic manifolds, and is the main result of this section. Theorem 4.3. Under the above assumptions, there exist b C for all r N and j,r ∈ ∈ 1 6 j 6 m, such that for any k N∗, we have as p + , ∈ → ∞ m k 1 − (dim Yj 1)/2 2π√ 1pλj r k Tr [ˆg(pQp(f c))] = p − g(tj)e− − p− bj,r + O(p− ) , (4.21) − ! jX=1 rX=0 26 where λ C is the action of f over Y . Furthermore, for all 1 6 j 6 m we have j ∈ j 1 E, 1 E 0 1 K 2 − X − bj,0 = TrE[ϕtj τt ] det(Πt )− τt Y j j j Z j 1 2 N tj 1 0 N dv Yj − − det N P (Π0 dϕtj Πtj )(IdT X dϕtj )P | | , (4.22) − − ξf gTX h i | | 0 for some natural choices of square roots. In particular, we have as p + , → ∞ n 1 1 n 2 Tr[ˆg(pQ (f c))] = p − g(0) rk(E) Vol (f − (c)) + O(p − ) . (4.23) p − ω Proof. First note that replacing f C ∞(X, R) by f c, we are reduced to the case ∈ − c = 0. Consider thus f C ∞(X, R) satisfying the hypotheses of Theorem 1.2 with ∈ c = 0. Using the trace formula (2.27) and the definition ofg ˆ(pQp(f)) in (4.2) for all p N, we know that ∈ Tr gˆ(pQp(f)) = Tr [ˆg(pQp(f))(x, x)] dvX (x) X h i Z (4.24) 1 = g(t) Tr ϕt,p− p,t(ϕt(x), x) dtdvX (x) . X R T Z Z h i For any ε> 0, write 1 U(ε) := f − (] ε,ε [) . (4.25) − Then by Proposition 4.1, for any θ ]0, 1[ and k N, we get Ck > 0 such that for all θ ∈ ∈ x X U(εp− 2 ), ∈ \ − Ck n k(1 θ) gˆ(pQ (f))(x, x) 6 p − 2 , (4.26) | p |p εk so that in particular, we have as p + , → ∞ Tr gˆ(pQp(f)) = Tr[ˆg(pQp(f))(x, x)] dvX (x)+ O(p−∞) . (4.27) U(εp−θ/2) h i Z 1 Recall that T := t Supp g x f − (0), ϕ (x)= x is finite, and let ε> 0 be such { ∈ | ∃ ∈ t } that all u ] ε,ε[ are regular values of f, so that the Hamiltonian vector field ξf does not vanish∈ on− the closure of U(ε). Then in the same way as in (4.12), we get from the rapid decrease (2.32) of ( , ) outside the diagonal that as p + , Tp,t · · → ∞ Tr gˆ(pQp(f)) = −θ/2 h i t0+εp 1 g(t) Tr ϕ− p,t(ϕt(x), x) dtdvX (x)+ O(p−∞) . (4.28) −θ/2 −θ/2 t,p U(εp ) t0 εp T t0 T Z Z X∈ − h i Take ε > 0 small enough so that the identification Ψ of (4.18) holds for any t T . t0 0 ∈ Recall from (3.1) that the Hamiltonian vector field ξf is tangent to the level sets of f. 27 Then for any t T , we get diffeomorphisms ϕ : Σ Σ, depending smoothly on 0 ∈ u,t → u ] ε,ε[ and t ]t ε, t + ε[, such that ∈ − ∈ 0 − 0 ϕ (u, x)=(u,ϕ (x)) and ϕ (x)= x for all x Y , (4.29) t u,t u,t0 ∈ j in the coordinates (u, x) Ψ (U(ε)) of (4.18) and for all 1 6 j 6 m such that t = t . ∈ t0 j 0 For any ε> 0 and 1 6 j 6 m, consider the normal geodesic neighbourhood Vj(ε) Σ T Σ ⊂ of Yj inside (Σ,g ). Then by the non-degeneracy assumption of Definition 2.7 and as all u ] ε,ε[ are regular values of f, the map Φ:(u, t, x) (u,t,ϕu,t(x)) is also non-degenerate∈ − around the fixed point set ] ε,ε[ t Y inside7→ ] ε,ε[ R Σ, so − ×{ j} × j − × × that working in local charts, we see that there exists ε′ > 0 such that for all θ ]0, 1[ ∈ and p N, ∈ Σ θ/2 d (x, ϕu,t(x)) >ε′p− as soon as θ/2 θ/2 θ/2 (t, x) / ]tj εp− , tj + εp− [ Vj(εp− ) , (4.30) ∈ 6 6 − × 1 [j m for all u ] ε,ε[. On the other hand, as f(x, u) = u in the coordinates (u, x) ∈ − ∈ Ψtj (U(ε)) of (4.18), by the definition (3.1) of the Hamiltonian vector field ξf and the T X R definition (2.11) of g0 , we get a function ̺ C ∞(U(ε), ) such that over U(ε), we have ∈ 1 dvX = ̺dudvΣ and ̺(0, x)= ξf,x g−TX for all x Σ . (4.31) | | 0 ∈ We can then rewrite (4.28) as Tr gˆ(pQp(f)) = h i − θ − θ m εp 2 εp 2 θ θ Tr [Ij,p(t, u, x)] dtdudvΣ(x)+ O(p−∞) , (4.32) −θ/2 − − Vj (εp ) εp 2 εp 2 jX=1 Z Z− Z− where in the coordinates (u, x) Ψ (U(ε)) and for any t ]t ε, t + ε[, we set ∈ tj ∈ j − j 1 I (t, u, x)= g(t + t)ϕ− ((u,ϕ (x)), (u, x))̺(u, x) . (4.33) j,p j tj +t,pTp,tj +t u,tj +t N Recall that N denotes the normal bundle of Yj inside Σ equipped with the metric g induced by gT Σ, and consider the natural identification N V (ε) ∼ B (ε) := w N w <ε . (4.34) j −−−→ { ∈ | | |N } As ϕ (x) = x implies ϕ (ϕ (x)) = ϕ (x) for all t R and x X by the 1-parameter tj tj t t ∈ ∈ group property of ϕt, we know that its flow is transverse to the fibres of the ball bundle N B (ε) via the identification (4.34) for ε > 0 small enough. We can then pick x0 Yj N ∈ and u ] ε,ε[, and consider the natural embedding defined from the fibre Bx0 (ε) of the ball∈ bundle− (4.34) over x and a neighbourhood I R of 0 by 0 ⊂ N I B (ε) ∼ W Σ × x0 −−−→ u ⊂ (4.35) (t, w) ϕ (w) . 7−→ u,tj +t 28 We identify in turn Wu with a subset of Tx0 Σ via the inclusion I BN (ε) T Σ × x0 −→ x0 ∂ (4.36) (t, w) w + tξ , where ξ := ϕ (x ) T Σ . 7−→ f,u f,u ∂t u,t 0 ∈ x0 N For any u ] ε,ε[, we identify L over Wu with the pullback of L over Bx0 (ε) via ∈ − L parallel transport with respect to along flow lines of t ϕu,t, then trivialize L over BN (ε) via parallel transport with respect∇ to L along radial7→ lines. Then Lp is trivialized x0 ∇ by the parallel transport τt,p along the flow lines of ϕt, and by the exponentiation of Kostant formula (3.9), we have in this trivialization, 1 2√ 1πptu ϕ− = e − for all t <ε and p N . (4.37) t,p | | ∈ 1 Now by the definition (3.1) of ξ and as u Σ corresponds to the level set f − (u) via f { } × the identification Ψtj of (4.18), we know that for any vector field v C ∞(Wu, T Σ), we have ∈ 2π RL (v,ξ )= ω (v,ξ )=0 . (4.38) f √ 1 f − This shows that the trivialization of L described above coincides with a trivialization along radial lines of W in (4.36), for all u ] ε,ε[, so that we are under the hypotheses u ∈ − of Theorem 2.6. As in Definition 3.5 and using (4.19) under the identification Ψtj of (4.18), define the action λ R by j ∈ 2π√ 1pλ ϕ =: e − j over ] ε,ε[ Y . (4.39) tj ,p − × j By a standard computation, which can be found for example in [26, (1.2.31)] and which holds in any trivialization of L along radial lines, the connection L at w BN (ε) ∇ ∈ x0 inside W T Σ as in (4.36) has the form u ⊂ x0 1 L = d + RL(w, .)+ O( w 2) . (4.40) ∇ 2 | | Using this formula together with the fact that ϕL preserves L, we get in our coordinates a smooth bounded function λ of u ] ε,ε[ and w BN∇(0,ε) such that x0 ∈ − ∈ x0 1 2π√ 1pλ 3 ϕ− = e− − j exp(p w λ (u,w)) over ] ε,ε[ V (ε) . (4.41) tj ,p | | x0 − × j We can then apply Theorem 2.6 in a chart of the form (2.28) around (u, x0) and con- taining W T Σ T X via the identifications (4.36) and (4.18), to get from u ⊂ x0 ⊂ (u,x0) (4.33) as p + , → ∞ n 2π√ 1pλ 3 2√ 1πptu I (t,u,w)= p e− − j g(t + t) exp(p w λ (u,w))e − ρ(u,w) j,p j | | x0 k 1 E, 1 − r n k +δ − 2 2 ϕtj +t p− GrTtj +t,(u,x0)(√pϕu,tj +t(w), √pw)+ p O(p− ) , (4.42) rX=0 29 θ/2 N θ/2 for all t, u R with t t , u < εp− and w B (εp− ). Following (4.14), ∈ | − j| | | ∈ x0 we apply the Taylor expansion described in (2.31) to the right hand side of (4.42) in t, u and w, inside and outside the exponential of the local model (2.29) for Ttj +t,(u,x0). We then get a family Fr,x0 (t,u,w) Etj ,x0 E0∗,x r N of polynomials in w Nx0 { ∈ ⊗ 0 } ∈ ∈ and t, u R, of the same parity as r, depending smoothly in x0 Yj and with ∈ 1 E ∈ F (t,u,w)=µ ¯− (x )τ for all t, u and w, such that for any k N, there is 0,x0 tj 0 tj ,x0 ∈ M N∗ such that as p + , k ∈ → ∞ k 1 n 2π√ 1pλ 1 E, 1 − r − − j − − − 2 Ij,p(t,u,w)= p e g(tj) ξf,x0 gTX ϕtj ,x0 p Fr,x0 (√pt, √pu, √pw) | | 0 rX=0 2√ 1πptu n k +δ M e − T (√ptξ + √pdϕ .w, √pw)+ p − 2 O( √pt k + 1) . (4.43) tj ,x0 f,x0 tj | | Using the non-degeneracy assumption of Definition 2.7 together with the explicit for- mula (2.29) for the local model, we see that (4.43) decreases exponentially in t R and w N , but not in u R. Now to get an exponential decrease in u R, we will∈ make ∈ x0 ∈ ∈ the change of variables t t/√p and u u/√p, integrate first with respect to t R and then with respect to7→u R. With this7→ in mind, from the local model (2.29),∈ we ∈ get for any t, u R and w N , ∈ ∈ x0 2√ 1πtu e − Ttj ,x0 (tξf,x0 + dϕtj .w,w)= Ttj ,x0 (dϕtj .w,w) 2 tj tj tj exp πt Π ξ ,ξ πt (Π + (Π )∗)ξ ,dϕ .w w +2√ 1πtu . (4.44) − h 0 f,x0 f,x0 i − h 0 0 f,x0 tj − i − h i Using the classical formula for the Fourier transform of Gaussian integrals, we compute 2 tj exp πt Π0 ξf,x0 ,ξf,x0 R R Z Z − h i tj tj πt (Π0 + (Π0 )∗)ξ,dϕtj .w w +2√ 1πtu dt du − h − i − tj 1 2 = Π0 ξf,x0 ,ξf,x0 − h i (4.45) tj tj 2 π 2u + √ 1 (Π0 + (Π0 )∗)ξf,x0 ,dϕtj .w w exp − h − i du R − 4 tj Z Π0 ξf,x0 ,ξf,x0 h i 2 tj 1 u 2 = Π0 ξf,x0 ,ξf,x0 − exp π t du =1 , h i R "− Π j ξ ,ξ # Z h 0 f,x0 f,x0 i for a suitable choice of square root in the middle terms. In particular, this computation shows that we get an exponential decrease in u R after integration with respect to ∈ t R. Using successive integration by parts, this computation readily generalizes to ∈ the case when the exponential is multiplied by a polynomial Fr,x0 (t,u,w) of the same N parity as r , to get as a result a polynomial Hr,x0 (w) Etj ,x0 E0∗,x0 in w Nx0 ∈ 1 E ∈ ⊗ ∈ N − of the same parity as r , with H0,x0 (w)=¯µtj (x0)τtj ,x0 for all w Nx0 . Thus from ∈ ∈ N θ/2 (4.43), for all k N we get δ ]0, 1[ as in (4.15) such that for all w B (εp− ) as ∈ k ∈ ∈ x0 30 p + , → ∞ − θ − θ εp 2 εp 2 − θ − θ Tr [Ij,p(t,u,w)] dtdu εp 2 εp 2 Z− Z− 1−θ 1−θ εp 2 εp 2 1 1/2 1/2 = p− 1−θ 1−θ Tr Ij,p(p− t, p− u,w) dtdu εp 2 εp 2 Z− Z− h i 1 1/2 1/2 = p− Tr Ij,p(p− t, p− u,w) dt du + O(p−∞) R R Z Z h k 1 i n 1 2π√ 1pλ − r E, 1 j 2 − = p − e− − g(tj) p− Tr ϕtj ,x0 Hr,x0 (√pw) r=0 h i X k n 1 2 +δk Ttj ,x0 (√pdϕtj .w, √pw)+ p − O(p− ) . (4.46) Working locally, we can suppose that Yj is orientable. Let dvYj be the Riemannian T Σ volume form on Yj induced by g , let dw be the Euclidean volume form on the fibres N N of (N,g ) and let ρ C ∞(B (ε), R) be such that via the identification (4.34) of Vj(ε) ∈ N with the ball bundle B (ε) over Yj, we have dv = ρdwdv and ρ(0, x)=1 forall x Y . (4.47) Σ Yj ∈ j Through the change of variable w w/√p, using the exponential decrease of (4.46) in 7→ w Nx0 and taking the Taylor expansion in w Nx0 of ρ(w, x0) as described in (2.31) ∈ ∈C N for all x0 Yj, we then get polynomials Kr,x0 [w] of the same parity as r , with ∈ 1 E, 1 E ∈ ∈ − − K0,x0 (w)=¯µtj (x0) Tr ϕtj ,x0 τtj ,x0 , such that using (4.46), we can rewrite (4.32) as h i (dim Yj 1)/2 2π√ 1pλj Tr gˆ(pQp(f)) = p − e− − g(tj) x Y BN (εp(1−θ)/2) h i Z ∈ j Z x k 1 − r k ′ 1/2 2 2 +δk ρ(p− w, x) p− Kr,x(w)Ttj ,x(dϕtj .w,w)dw dvYj (x)+ O(p− ) rX=0 k−1 ⌊ 2 ⌋ (dim Yj 1)/2 2π√ 1pλj q = p − e− − g(tj) p− qX=0 ′ dvYj k 2 +δk K2q,x(w)Ttj ,x(dϕ.w,w)dw (x)+ O(p− ) , (4.48) x Yj Nx ξf gTX Z ∈ Z | | 0 for some δ′ ]0, 1[ satisfying δ′ δ as θ 1, where all the terms with r odd vanished k ∈ k → → in the same way as in (4.17) by the odd parity of K2q+1,x for all q N. This shows the ex- pansion (4.21), and formula (4.22) follows from (4.48) and the second∈ equality of (2.41). Finally, formula (4.23) follows from the fact that function µt C ∞(X, C)of (2.20) is ∈ 1 constant equal to 1 for t = 0 and the fact that the vector field v C ∞(f − (c),TX) 1 ∈ 2 defining the volume form (4.20) over f − (c) can be chosen to be v = J0ξf / ξf gTX . | | 0 Under the assumptions of Theorem 1.2, consider now the case when Yj Σ satisfy ⊂ 1 dim Y = 1 for some 1 6 j 6 m, so that Y = ϕ (x) 6 , for some x f − (c) j j { s }0 s This shows that the quantity det Nx (IdN dϕtj N ) is actually independent of x Yj. 1/2 − | ∈ Considering the case E = KX and recalling that ξf is the tangent vector field of the curve Yj, we get K , 1 K 1 E, 1 E dv Yj dv Yj X − X 2 − (ϕtj τtj )− TrE[ϕtj τtj ] | | = | | = t(Yj) . (4.52) Yj ξf gTX Yj ξf gTX Z | | 0 Z | | 0 This gives formula (4.50). Let us now show how these results can be applied to contact topology. To that end, consider f C ∞(X, R) and an almost complex structure J End(TX) satisfying ∈ ∈ 1 dιJξf ω = ω over f − (I) , (4.53) over some interval I R of regular values of f. Then considering the Riemannian metric gT X = ω( ,J )as⊂ in (2.11), the form · · ι ω Jξf 1 R 2 Ω (X, ) (4.54) − ξ TX ∈ | f |g 32 1 1 restricts to a contact form α Ω (Σ, R) over Σ := f − (c) for any c I, meaning n 1 ∈ ∈ that α dα − is a volume form over Σ. The restriction of the Hamiltonian vector ∧ 1 field ξ C ∞(X,TX) over f − (c) induces the Reeb vector field of (Σ, α), which is the f ∈ unique vector field ξ C ∞(Σ, T Σ) satisfying ∈ ιξα =1 , (4.55) ιξdα =0 . The corresponding Reeb flow is the flow of diffeomorphisms ϕt : Σ Σ generated by ξ, and its periodic orbits are called the Reeb orbits of (Σ, α). An isolated→ Reeb orbit of period t R is said to be non-degenerate if it satisfies Definition 2.7 as a fixed point 0 ∈ set of ϕt0 inside Σ. Conversely, given a compact manifold Σ endowed with a contact form α Ω1(Σ, R), ∈ we define a symplectic form ωSΣ Ω2(SΣ, R) over SΣ := R Σ by the formula ∈ × SΣ u ω := d(e π∗α) , with π : SΣ := R Σ Σ . − × −→ (4.56) (u, x) x 7−→ The symplectic manifold (SΣ,ωSΣ) is called the symplectization of (Σ, α). Consider the function f C ∞(SΣ, R) defined by ∈ f(u, x) := eu for all (u, x) R Σ . (4.57) ∈ × Then its Hamiltonian vector field ξ C ∞(SΣ,TSΣ) restricts over any level set of f to f ∈ the Reeb vector field ξ C ∞(Σ, T Σ) of (Σ, α). On the other hand, consider an almost complex structure J End(∈ TSΣ) compatible with ωSΣ satisfying ∈ ∂ Jξ = , (4.58) f ∂u in the coordinates (u, x) R Σ = SΣ. Such an almost complex structure always ∈ × exists. Finally, consider the situation when an open set of the form I Σ SΣ can be symplectically embedded into a compact prequantized symplectic manifold× ⊂ (X,ω) without boundary. We can then extend the Hamiltonian and the almost complex struc- ture defined by (4.57) and (4.58) to the whole of (X,ω) using partitions of unity. The typical case when such an embedding exists is when Σ is the boundary of a star-shaped domain in R2n, with contact form α Ω1(Σ, R) given by the restriction of the stan- dard Liouville form λ Ω1(R2n, R). Then∈ using a Darboux chart, an open set of the form I Σ SΣ can∈ always be symplectically embedded in (X,ω). More generally, × ⊂ by the results of [15, Th. 1.3] and [24, Cor. 1.11], any fillable contact manifold satisfies this property. The following result then shows how to use the above picture to ex- tract informations on isolated non-degenerate Reeb orbits of (Σ, α) from the geometric quantization of (X,ω). 33 Proposition 4.5. Let Σ be a compact manifold without boundary endowed with a con- tact form α Ω1(Σ, R), and assume that an open set of the form I Σ in its symplec- tization (SΣ∈,ωSΣ) can be symplectically embedded in a compact prequantized× symplectic manifold (X,ω) without boundary. Consider the Hamiltonian f C ∞(X, R) and the ∈ almost complex structure J End(TX) defined via (4.57) and (4.58). Then for any c I, Theorem∈ 1.2 holds as soon as the fixed point set of the Reeb flow ∈ ϕt : Σ Σ is non-degenerate for all t Supp g, and Theorem 1.3 holds for the non- degenerate→ isolated Reeb orbits of (Σ, α)∈in case (X,ω) is endowed with a metaplectic structure. Proof. Recall that the Hamiltonian vector field ξ C ∞(SΣ,TSΣ) of f C ∞(SΣ, R) f ∈ ∈ defined by (4.57) restricts to the Reeb vector field ξ C ∞(Σ, T Σ) of (Σ, α) over u Σ, ∈ { }× for all u R. Then the Hamiltonian flow ϕ : X X of f over I Σ is of the ∈ t → × form ϕt(u, x)=(u,ϕt(x)) for all (u, x) I Σ. 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