Asymptotic representations of Hamiltonian diffeomorphisms and quantization
AAsymptotic representations of Hamiltoniandiffeomorphisms and quantization
Laurent Charles and Leonid Polterovich a October 14, 2020
Abstract
We show that for a special class of geometric quantizations with“small” quantum errors, the quantum classical correspondence givesrise to an asymptotic projective unitary representation of the group ofHamiltonian diffeomorphisms. As an application, we get an obstruc-tion to Hamiltonian actions of finitely presented groups.
Geometric quantization is a mathematical theory modeling the quantumclassical correspondence. The latter is a fundamental physical principlestating that the quantum mechanics contains the classical mechanics in thelimit when the Planck constant goes to zero. In the present paper we focus onthe correspondence between Hamiltonian diffeomorphisms modeling motionsof classical mechanics, and their quantum counterparts, unitary operatorscoming from the Schr¨odinger evolution. We show that for a special classof geometric quantizations with “small” quantum errors, which exist on acertain class of phase spaces (see Theorem 1.4), this correspondence gives riseto an asymptotic unitary representation of the universal cover of the group ofHamiltonian diffeomorphisms (Theorem 1.5). Interestingly enough, togetherwith recent results from group theory [9, 14], this yields an obstruction toHamiltonian actions of finitely presented groups (Corollary 1.7). Let us passto precise definitions. Partially supported by the Israel Science Foundation grant 1102/20 a r X i v : . [ m a t h - ph ] O c t .1 Hamiltonian diffeomorphisms Let ( M n , ω ) be a closed symplectic manifold. Here ω is a closed differential2-form, whose n -th power does not vanish at any point and, thus, gives rise toa volume form on M . For a function f ∈ C ∞ ( M ) introduce its Hamiltonianvector field sgrad f as the unique solution of the equation i sgrad f ω = − df .Given a smooth function f : M × [0 , → M , denote f t ( x ) := f ( x, t ), andconsider the time dependent vector field sgrad f t . Its evolution defines a pathof diffeomorphisms φ t on M with φ = . This path is called a Hamiltonianpath , and the diffeomorphisms f t are called Hamiltonian diffeomorphisms.The latter form a group denoted by Ham(
M, ω ) (see [16] for further details).Denote by (cid:93)
Ham(
M, ω ) the universal cover of Ham(
M, ω ). Its elements (cid:101) φ are Hamiltonian paths { φ t } , t ∈ [0 ,
1] with φ = , considered up to ahomotopy with fixed end points. We write φ = φ for the projection of (cid:101) φ toHam( M, ω ). Every path { φ t } is uniquely determined by a time-dependentgenerating Hamiltonian f t ∈ C ∞ ( M ), where the functions f t are assumedto have zero mean: (cid:82) M f t ω n = 0 for all t . We shall say that (cid:101) φ ∈ (cid:93) Ham(
M, ω )is generated by a Hamiltonian f ∈ C ∞ ( M × [0 , π (Ham( M, ω )) is an abeliangroup, and we have a central extension1 (cid:47) (cid:47) π (Ham( M, ω )) (cid:47) (cid:47) (cid:93) Ham(
M, ω ) τ (cid:47) (cid:47) Ham(
M, ω ) (cid:47) (cid:47) . (1) Define a fundamental operation on functions on a symplectic manifold called the Poisson bracket : { f, g } = L sgrad f g , where L stands for the Lie derivative.We write (cid:107) f (cid:107) = max | f | for the uniform norm of a function f .In what follows we denote by L ( H ) the space of Hermitian operatorsacting on a finite-dimensional complex Hilbert space H , and write U ( H ) forthe unitary group of H . Definition 1.1. A fine quantization of ( M, ω ) consists of a sequence ofpositive numbers (cid:126) k with lim k →∞ k (cid:126) k = 1, a family of finite-dimensionalcomplex Hilbert spaces H k such thatdim H k = (cid:16) k π (cid:17) n Vol(
M, ω ) + O ( k n − ) , (2)and a family of R -linear maps Q k : C ∞ ( M ) → L ( H k ) with Q k (1) = ,satisfying the following properties: 2P1) ( norm correspondence ) (cid:107) Q k ( f ) (cid:107) op = (cid:107) f (cid:107) + O (1 /k );(P2) ( bracket correspondence ) [ Q k ( f ) , Q k ( g )] = (cid:126) k i Q k ( { f, g } )+ O (1 /k ),where the remainder is understood in the operator norm (cid:107) · (cid:107) op .The wording “fine” is chosen in order to emphasize that the remainderin (P2) is O (1 /k ), as opposed to O (1 /k ), as it happens for a wide class ofgeometric quantizations. For K¨ahler quantizations (see Section 2 below), theorder of the remainder cannot be improved to O (1 /k ), see [4, p.470]. It isunknown whether the same holds true for “abstract” quantizations definedby axioms (P1) and (P2).Recall that ( M, ω ) is quantizable if the cohomology class [ ω ] / (2 π ) is in-tegral. The following conditions on the first Chern class c ( T M ) and thecohomology class of symplectic form [ ω ] of a quantizable symplectic manifoldare equivalent:(C1) the line c ( T M ) − R [ ω ] in H ( M, R ) intersects the lattice of integralclasses H ( M, Z ) / torsion;(C2) c takes even values on Ker([ ω ]), where both c and [ ω ] are consideredas morphisms H ( M, Z ) / torsion → R .Indeed, (C1) yields (C2) immediately. In the opposite direction, choose a ba-sis in Ker([ ω ]), say e , ..., e m − , and extend it to a basis in H ( M, Z ) / torsionby e . Then ω ( e ) = 2 πN , where the number N ∈ Z is defined as an integersuch that [ ω ] / (2 πN ) is a primitive vector. To get (C1) from (C2), we choose λ = ( c ( e ) + 2 p ) / (2 N ), with any integer p . Definition 1.2.
We say that (
M, ω ) satisfies condition (C) if it satisfies oneof the equivalent conditions (C1) or (C2).Condition (C) may be viewed as a generalisation of the existence of meta-plectic structure. It is more general: all complex projective spaces satisfycondition (C) because their second cohomology groups are one-dimensional.However, only the projective spaces with an odd complex dimension have ametaplectic structure.
Example 1.3.
Take M to be C P blown up at one point. Let L, E bethe basis in H ( M, Z ) with L being the class of a general line and E ofthe exceptional divisor, respectively. There exist a symplectic forms on M with ω ( L ) = 2 πm , ω ( E ) = 2 πn , for any integral m > n > c ( nL − mE ) = 3 n − m , and hence (C2) is satisfied iff m = n mod 2.3 heorem 1.4. Every quantizable closed symplectic manifold M satisfyingcondition (C)) admits a fine quantization. The proof is given in Section 2.
Let Q k be a fine quantization. For a Hamiltonian f t as above consider theunitary quantum equation U k ( t ) : H k → H k described by the Schroedingerequation ˙ U k ( t ) = − i (cid:126) k Q k ( f t ) U k ( t ) , U k (0) = . (3)One can view the time-one map U k = U k (1) as a quantization of the element (cid:101) φ represented by f t [13].Consider family of maps µ := { µ k } , µ k : (cid:93) Ham(
M, ω ) → U ( H k ) , (cid:101) φ (cid:55)→ U k . Let us emphasize that µ k ( (cid:101) φ ) depends on the specific choice of a Hamiltonianpath joining the identity with φ , in the class of paths homotopic with fixedendpoints. Theorem 1.5. (i) The unitaries µ k ( (cid:101) φ ) and µ (cid:48) k ( (cid:101) φ ) defined via two different choices of pathshomotopic with fixed endpoints representing φ ∈ (cid:93) Ham(
M, ω ) , satisfy (cid:107) µ k ( (cid:101) φ ) − µ (cid:48) k ( (cid:101) φ ) (cid:107) op = O (1 /k ) . (4) (ii) For every (cid:101) φ, (cid:101) ψ ∈ (cid:93) Ham(
M, ω ) (cid:107) µ k ( (cid:101) φ ) µ k ( (cid:101) ψ ) − µ k ( (cid:101) φ (cid:101) ψ ) (cid:107) op = O (1 /k ) . (5) (iii) If φ (cid:54) = , (cid:107) µ k ( (cid:101) φ ) − (cid:107) op ≥ / O (1 /k ) . (6)The proof is given in Section 3.2. 4 .4 First constraints on Hamiltonian group actions The collection of maps µ k gives rise to an interesting algebraic object. Inorder to describe it, we need some preliminaries from [9, 14]. For p ≥ A : H → H acting on a d -dimensional Hilbert space H denoteby (cid:107) A (cid:107) p its p -th Schatten norm given by (cid:107) A (cid:107) p = (cid:16) tr (cid:0)(cid:0) √ A ∗ A (cid:1) p (cid:1)(cid:17) /p . Recall that (cid:107) A (cid:107) op ≤ (cid:107) A (cid:107) p ≤ d /p (cid:107) A (cid:107) op . (7) Definition 1.6 ([14]) . A group Γ is called p -norm approximated if thereexists a family of maps ρ k : Γ → U ( H k ) , where H k is a sequence of Hilbert spaces of growing dimension, such thatlim (cid:107) ρ k ( x ) ρ k ( y ) − ρ k ( xy ) (cid:107) p = 0 , ∀ x, y ∈ Γ , (8)and lim inf (cid:107) ρ k ( x ) − (cid:107) p > , ∀ x ∈ Γ , x (cid:54) = 1 . (9)We call any sequence of maps ρ k satisfying (8) an asymptotic representation of Γ in the sequence of unitary groups equipped with the p -norms.Theorem 1.5 combined with estimate (7) and formula (2) immediatelyyields the following result. Corollary 1.7.
Assume that a n -dimensional closed symplectic manifold M admits a fine quantization. Let Γ ⊂ (cid:93) Ham(
M, ω ) be a finitely presentedsubgroup with Γ ∩ π (Ham( M, ω )) = { } . (10) Then Γ is p -norm approximated for every p > n . Denote by LO p the class of finitely presented groups with are not p -norm approximated. Existence of such groups for p > (cid:96) -adic Lie groups belong to this class.Corollary 1.7 yields obstructions to actions of groups from LO p on certainsymplectic manifolds. 5 xample 1.8. Let M be a closed oriented surface of genus ≥ ω . Then π (Ham( M, ω )) = 1 (e.g. see [16]). Furthermore, H ( M, Z ) = Z , and hence M satisfies condition (C) of Theorem 1.4. Thusno group of class LO p admits a faithful Hamiltonian action on ( M, ω ).Denote by K p ⊂ π (Ham( M, ω )) the subgroup formed by elements (cid:101) φ ∈ (cid:93) Ham(
M, ω ) with lim k →∞ (cid:107) µ k ( (cid:101) φ ) − (cid:107) p = 0. Assumption (10) in Corollary1.7 can be replaced to Γ ∩ K p = { } . (11)It would be interesting to explore the subgroup K p . What can we say about the restriction of the approximate representation µ k to the fundamental group π (Ham( M, ω )) ⊂ (cid:93) Ham(
M, ω ) ? The followingenhancement of Theorem 1.4 sheds light on this question.
Theorem 1.9.
Every K¨ahler closed symplectic manifold M satisfying con-dition (C) admits a fine quantization which satisfies µ k ( γ ) = e ir k ( γ ) + O (1 /k ) , (12) where r k : π (Ham( M, ω )) → R / (2 π Z ) is a sequence of homomorphisms. The proof is given in Section 4. The homomorphisms r k will be explicitlydescribed in terms of action and Maslov invariants. The result follows from[8], which is developed in the K¨ahler setting. But there is no serious reasonto think that the K¨ahler assumption is essential here.Denote by PU ( H k ) = U ( H k ) /S the projectivization of the unitary groupof the Hilbert space H k . We equip this group with the quotient metric δ p ([ A ] , [ B ]) = inf θ (cid:107) A − e iθ B (cid:107) p . Let us state an analogue of Definition 1.6for projective representations. Definition 1.10.
A group Γ is called p -norm projectively approximated ifthere exists a family of maps ρ k : Γ → PU ( H k ) , where H k is a sequence of Hilbert spaces of growing dimension, such thatlim δ p ( ρ k ( x ) ρ k ( y ) , ρ k ( xy )) = 0 , ∀ x, y ∈ Γ , (13)6nd lim inf δ p ( ρ k ( x ) , ) > , ∀ x ∈ Γ , x (cid:54) = 1 . (14)We call any sequence of maps ρ k satisfying (13) an asymptotic projectiverepresentation of Γ in the sequence of unitary groups equipped with the p -norms.With this language, the asymptotic unitary representation µ k from The-orem 1.9 descends to an asymptotic projective representation ν k : Ham( M, ω ) → PU ( H k ) , φ (cid:55)→ [ µ k ( (cid:101) φ )] , where (cid:101) φ is any lift of φ . Furthermore, every finitely presented subgroup ofHam( M, ω ) is p -norm projectively approximated. The proof is analogous tothe one of Theorem 1.5, with the only extra ingredient being explained inRemark 3.2 below.Write PLO p for the class of finitely presented groups which are not p -norm projectively approximated. We sum up the previous discussion in thefollowing theorem, which is the main application of our quantization-basedtechnique to group actions on symplectic manifolds. Theorem 1.11.
Let ( M, ω ) be a closed K¨ahler manifold of dimension n with [ ω ] / (2 π ) being an integral class and c ( T M ) taking even values onKer [ ω ] . Then every finitely presented subgroup of the group of Hamilto-nian diffeomorphisms Ham(
M, ω ) is p -norm projectively approximated withany p > n . In other words, groups from the class PLO p , p > n do not aadmit a faithful Hamiltonian action on ( M, ω ) . Question 1.12.
Can groups from the class
PLO p act by volume-preservingdiffeomorphisms on closed manifolds? PLO p ? (following [9, 14]) De Chiffre, Glebsky, Lubotzky, and Thom [9] and Lubotzky and Oppenheim[14] came up with a technique leading to examples of groups of the class LO p for p ∈ (1 , + ∞ ). It was explained to us by Lubotzky that the samemethod shows that these groups lie in PLO p . The argument from [14, 9]extends verbatim . For reader’s convenience we provide a brief outline of thisargument adjusted to projective case.Fix a non-principal ultrafilter U , and consider the ultra-product: V p := (cid:89) n →U (Mat( C , k n ) , || · || p ) . π p of Γ on V p by conjugation. The crux of the matter isthat the action by conjugation is well defined since for U = e iθ U , we have U AU ∗ = U AU ∗ .Given a class of groups P , we say that a group Γ is residually P if forevery element x ∈ Γ \ there exists a homomorphism from Γ to a groupfrom P whose kernel does not contain x . Interesting classes of groups include linear groups (those, admitting a faithful finite-dimensional representations)and finite groups. Proposition 1.13 ([9]) . Let Γ be a finitely presented group with the follow-ing properties:(a) H (Γ , π p ) = 0 ;(b) Γ is not residually linear.Then Γ / ∈ PLO p . Indeed, assumption (a) enables one to apply a Newton-type processwhich yields a genuine representation of Γ on Mat( C , k n ) for almost all n with respect to the ultra-filter. Moreover, every x (cid:54) = does not lie in itskernel for almost all n . But this contradicts assumption (b).The group Γ is constructed in two steps:(i) Take a cocompact lattice Γ in a simple Lie group G of rank ≥ (cid:96) -adic numbers with (cid:96) sufficiently large;(ii) Take a special finite central extension Γ of Γ which is not residuallyfinite (Deligne).The paper [9] proposes a specific example of the lattice Γ ,Γ = U (2 n ) ∩ Sp (2 n, Z [ √− , /p ])considered as a cocompact lattice in Sp (2 n, Q p ).The central extension Γ → Γ , based on technique of Deligne, is quitecomplicated, and we refer to [9] for details.In order to verify assumption (a) of Proposition 1.13, the following fea-tures are used: first, the Lie group G acts on a special simplicial complex(a Bruhat-Tits building); here one uses the (cid:96) -adic nature of the situation.Second, the representation π p is a particular case of an isometric representa-tion on Banach spaces from a special class: they are obtained from Pisier’s8 -Hilbertian spaces (where θ depends on p ) by using quotients, l -sums andultra-products.For verifying assumption (b) of Proposition 1.13, one uses the (immediateconsequence of) Malcev Theorem: any residually linear group is residuallyfinite. This completes our outline of the argument from [9, 14]. Another application of Theorem 1.5 deals with the following stability ques-tion: given a subgroup Γ ⊂ (cid:93) Ham(
M, ω ), is its quantization µ k | Γ : Γ → U ( H k )close to a genuine representation? It follows that the answer is affirmativefor the class of p -norm stable groups defined as follows [14, 9]. Here weinclude the case p = ∞ , i.e. of the operator norm. Let Γ be a finitelypresented group defined by finite collections of generators S and relations R , considered as subsets of the free group F S generated by S . The p -normstability means that for every (cid:15) > δ > H and every homomorphism t : F S → U ( H )with max r ∈ R (cid:107) t ( r ) − (cid:107) p ≤ δ , there exists a homomorphism ρ : Γ → U ( H ) whose lift ρ : F S → U ( H )satisfies max s ∈ S (cid:107) t ( s ) − ρ ( s ) (cid:107) p < (cid:15) . Let us mention that all finite groups are operator norm stable by [10, 12].
Corollary 1.14.
Assume that a n -dimensional closed symplectic manifold M admits a fine quantization. Let Γ = (cid:104) S | R (cid:105) ⊂ (cid:93) Ham(
M, ω ) be a finitelypresented p -norm stable subgroup, where p > n . There exists a family ofhomomorphisms ρ k : Γ → U ( H k ) such that max s ∈ S (cid:107) µ k ( s ) − ρ k ( s ) (cid:107) p → , k → ∞ . Remark 1.15.
Some examples of finite subgroups of (cid:93)
Ham(
M, ω ) come fromthe following construction. Let F ⊂ Ham(
M, ω ) be a finite group acting ina Hamiltonian way on a closed quantizable symplectic manifold (
M, ω ). Forinstance, any unitary representation of F on a finite-dimensional complexHilbert space V yields an action of F on the projectivization P ( V ). Denoteby (cid:101) F ⊂ (cid:93) Ham(
M, ω ) as the full lift of F . If F is perfect, there exists a finite9belian extension G of F , called the universal extension [18] such that thefollowing diagram commutes: G (cid:15) (cid:15) (cid:47) (cid:47) F (cid:15) (cid:15) (cid:101) F τ (cid:47) (cid:47) F This provides a monomorphism of G into (cid:93) Ham(
M, ω ).Let us note also that for any finite subgroup F ⊂ Ham(
M, ω ), the restric-tion of the asymptotic projective representation ν k , which we constructed forquantizable K¨ahler manifolds satisfying condition (C), the restriction ν k | F is close to a genuine projective representation, see [10]. A few bibliographical remarks are in order. For K¨ahler quantization withmetaplectic correction an asymptotic representation of the quantomorphismsgroup of a prequantum circle bundle over a closed symplectic manifold isconstructed by Charles in [4]. In the present paper we generalize this re-sult in two directions: first, we prove it for arbitrary fine quantizations, andsecond, for K¨ahler quantization, we impose Condition (C) instead of theassumption that the canonical bundle admits a square root.Charles showed that quantization enables one to reconstruct Shelukhin’squasi-morphism on (cid:93)
Ham(
M, ω ) [7]. Ioos, Kazhdan and Polterovich exploreda link between quantization and almost representations of Lie algebras [11].Constraints on smooth actions of finitely presented groups on closedmanifolds is a classical and still rapidly developing subject. Its highlightis Zimmer’s famous conjecture [19] which, roughly speaking, states thathigher rank lattices in semisimple Lie groups cannot act on manifolds ofsufficiently small dimension. This conjecture was recently resolved in abreakthrough work by Brown, Fisher, and Hurtado [2]. Some results onHamiltonian actions were obtained by Polterovich, Franks and Handel. Werefer to Fisher’s survey in [19] for a more detailed discussion. It would beinteresting to explore potential actions of the group constructed in [9, 14] anddescribed above, which is a finite extension of a higher rank (cid:96) -adic latticewith sufficiently high (cid:96) , along the lines of [2]. As we have learned fromDavid Fisher, this problem is at the moment open. Furthermore, Fisherconjectured existence of constraints on actions of such groups.10
Constructing fine quantizations
In this section we prove Theorem 1.4 by constructing a fine quantization,which will be denoted by Op k .In the usual Toeplitz-K¨ahler quantization, we consider a compact K¨ahlermanifold ( M, ω ) equipped with a holomorphic Hermitian line bundle L whose Chern connection has curvature i ω . The quantum space is definedas the space H k of holomorphic sections of L k ⊗ L (cid:48) , where L (cid:48) is an auxilliaryHermitian holomorphic line bundle. Here, the parameter k is a positive inte-ger. The large k limit is the semiclassical limit where in first approximationthe quantum mechanics reduces to the classical mechanics of M consideredas the classical phase space. In this context, a standard way to define aquantum observable from a classical one is the Berezin-Toeplitz quantiza-tion: for any f ∈ C ∞ ( M, R ), we let T k ( f ) be the endomorphism of H k suchthat (cid:104) T k ( f ) ψ, ψ (cid:48) (cid:105) = (cid:104) f ψ, ψ (cid:48) (cid:105) (15)for any ψ, ψ (cid:48) ∈ H k . Here the scalar product of C ∞ ( M, L k ⊗ L (cid:48) ) is given byintegrating the pointwise scalar product against the Liouville volume form.The basic properties of these operators are the following equalities whichholds for any f, g ∈ C ∞ ( M ) T k ( f g ) = T k ( f ) T k ( g ) + O ( k − )[ T k ( f ) , T k ( g )] = ( ik ) − T k ( { f, g } ) + O ( k − )tr( T k ( f )) = (cid:16) k π (cid:17) n (cid:90) M f µ + O ( k n − ) (16)see [3], [1]. Furthermore (cid:107) T k ( f ) (cid:107) op = (cid:107) f (cid:107) + O ( k − ). Observe that in thebracket correspondence (second line of (16)), the remainder is a O ( k − ), sowe miss the fine quantization condition given in Definition 1.1.The first order correction to (16) have been computed in [4]. Introducefor any f ∈ C ∞ ( M ), the operatorOp k ( f ) := T k ( f − (2 k ) − ∆ f ) (17)where ∆ is the holomorphic Laplacian of M (in complex coordinates ∆ f = (cid:80) G ij ∂ z i ∂ z j with ( G ij ) the inverse of ( G ij ) given by ω = i (cid:80) G ij dz i ∧ dz j ).Since Op k ( f ) = T k ( f ) + O ( k − ), the operators Op k ( f ) satisfy (16) as well.11he novelty is that we have now some explicit formulas for the first correc-tionsOp k ( f ) Op k ( g ) = Op k ( f g ) + i k Op k ( { f, g } ) + O ( k − )[Op k ( f ) , Op k ( g )] = ( ik ) − Op k ( { f, g } − k − ω ( X f , X g )) + O ( k − )tr(Op k ( f )) = (cid:16) k π (cid:17) n (cid:90) M f ( ω + k − ω ) n /n ! + O ( k − ) (18)see [4]. Here ω = i (Θ (cid:48) − Θ K ) where Θ (cid:48) and Θ K are the Chern curvatureof L (cid:48) and the canonical bundle K respectively. In complex coordinates asabove, Θ K = ∂∂ ln det( G ij )In the case where M has a metaplectic structure, one can choose for L (cid:48) a square root of the canonical bundle, so that ω = 0 and we get our finequantization. More generally, to prove the existence of fine quantizationsunder assumption (C), we construct a convenient auxiliary bundle L (cid:48) . Lemma 2.1.
Assume condition (C) . Then there exists a holomorphic Her-mitian line bundle L (cid:48) such that ω = λω with λ ∈ Q .Proof. The basic observation we need is that for any line bundle D andinteger m such that D m is equipped with a Hermitian and holomorphicstructures, D has natural holomorphic and Hermitian structures inducingthe ones of D m . Furthermore the Chern curvature of D is 1 /m times theChern curvature of D m .Now, the assumption that c R ( K )+ R [ ω ] intersects the lattice of integralclasses means that there exists a line bundle L (cid:48) such that c R ( L (cid:48) ) = c R ( K )+ λc R ( L ). Since c R ( L ) (cid:54) = 0, λ = p/q is rational. So ( L (cid:48) ) q = K q ⊗ L p ⊗ T where T is a torsion line bundle, i.e. T m = 1 for some m ∈ N . We endow T withthe holomorphic and Hermitian structures such that T m becomes the trivialHermitian and holomorphic line bundle, so that the Chern curvature of T is zero. Then we endow L (cid:48) with the Hermitian and holomorphic structurecompatible with the isomorphism ( L (cid:48) ) q = K q ⊗ L p ⊗ T . So the Cherncurvature Θ (cid:48) , Θ and Θ K of L (cid:48) , L and K satisfy Θ (cid:48) = Θ K + λ Θ. So ω = iλ Θ = λω .In the case where ω = λω , the second and third equations of (18) reads[Op k ( f ) , Op k ( g )] = ( i ( k + λ )) − Op k ( { f, g } ) + O ( k − )tr(Op k ( f )) = (cid:16) k + λ π (cid:17) n (cid:90) M f µ + O ( k n − ) (19)12hich proves Theorem 1.4 for a K¨ahler manifold with (cid:126) k = ( k + λ ) − .Let us generalize this to symplectic manifolds. So we start with a sym-plectic compact manifold ( M, ω ) such that π [ ω ] is integral. We introducea Hermitian line bundle L with Chern class π [ ω ] and a second Hermitianline bundle L (cid:48) . We denote by Ω ∈ H ( M, R ) the cohomology classΩ = π (cid:0) c R ( L (cid:48) ) − c R ( K ) (cid:1) . Here, the canonical bundle K is defined through any almost complex struc-ture compatible with ω . It is well known that the Chern class of K onlydepends on ω . If H k is a finite dimensional subspace of C ∞ ( M, L k ⊗ L (cid:48) ), wecan define as before the Toeplitz operators T k ( f ) by (15). Then we have thefollowing results:1. by [5], cf. also [3], [15], one can choose the family ( H k ) so that theoperators T k ( f ) satisfy (16).2. by [6], there exists a real differential operator P : C ∞ ( M ) → C ∞ ( M )such that Op k ( f ) = T k ( f ) + k − T k ( P f ) satisfies (18) with ω a rep-resentative of Ω . Furthermore, by adding to P a vector field, onemodifies ω by an exact form. Choosing conveniently this vector field,we can obtain any representative of Ω .If condition (C) holds, we can choose L (cid:48) so that Ω = λ [ ω ] for some λ ∈ Q .Choosing P so that ω = λω , we obtain equations (19). We start with the Egorov theorem for fine quantizations. Let f t be a clas-sical Hamiltonian generating the Hamiltonian flow φ t , and let U k ( t ) be thecorresponding quantum evolution. Theorem 3.1.
For every function g ∈ C ∞ ( M ) (cid:107) Q k ( g ◦ φ − ) − U k Q k ( g ) U − k (cid:107) op = O ( 1 k ) , (20) where the remainder depends on f and g . This formula readily follows from [13, Proposition 2.7.1]. Let us empha-size that the quantum map U k depends on the Hamiltonian f generatingthe diffeomorphism φ . This dependence will be analyzed later.13 roof of the Egorov theorem (20) : Recall that if φ t is the Hamiltonian flow generated by a time-dependentHamiltonian f t ( x ), the flow φ − t is generated by ¯ f t := − f t ◦ φ t . It followsthat for any function g ∈ C ∞ ( M ) ddt g ◦ φ − t = ( φ − t ) ∗ ( L sgrad ¯ f t g ) = ( φ − t ) ∗ { ¯ f t , g } = −{ f t , g ◦ φ − t } . (21)Next, turn to the analysis of the Schr¨odinger equation ˙ ξ = − i (cid:126) k Q k ( f t ) ξ .Introduce the family of unitary operators U ( s, t ) : H k → H k , ξ ( s ) (cid:55)→ ξ ( t )which sends the solution at time s to the solution at time t . Observe that U (0 , t ) = U k ( t ) is the Schr¨odinger evolution, U ( t, t ) = and U ( s, t ) = U ( t, s ) − = U ( t, s ) ∗ . The Schr¨odinger equation yields ∂∂s U ( t, s ) = − i (cid:126) k Q k ( f s ) U ( t, s ) , ∂∂s U ( s, t ) = − i (cid:126) k U ( s, t ) Q k ( f s ) . (22)Put now B ( s ) := U ( s, Q k ( g ◦ φ − s ) U (1 , s ), so that B (0) = U k Q k ( g ) U k = − B (1) = Q k ( g ◦ φ − ). From (21) and (22) we get that dBds = U ( s, (cid:18) i (cid:126) k [ Q k ( f s ) , Q k ( g ◦ φ − s )] − Q k ( { f s , g ◦ φ − s } (cid:19) U (1 , s ) . Observe that the functions f s and g ◦ φ − s , s ∈ [0; 1] form a compact familywith respect to C ∞ -topology, and hence by bracket correspondence (P2)max s (cid:107) dB/ds (cid:107) op = O (1 /k ). Thus (cid:107) Q k ( g ◦ φ − ) − U k Q k ( g ) U − k (cid:107) op = (cid:107) (cid:90) dB/ds ( s ) ds (cid:107) op = O (1 /k ) , as required. Suppose that we have two Hamiltonian paths γ = φ ,t and γ = φ ,t , t ∈ [0; 1] with φ , = φ , = φ , which are homotopic with fixed end pointsthrough a family φ t,s , s ∈ [0 , U k ( φ ,j ) the time one map of theSchroedinger evolution obtained by the quantization of γ j . We claim that (cid:107) U k ( φ , ) − U k ( φ , ) (cid:107) op = O (1 /k ) . (23)14o see this, look at the family φ t,s and denote by p t,s the generating Hamil-tonian when s is fixed, t varies, and by q t,s the Hamiltonian when t is fixed, s varies. All the Hamiltonians are assumed to have zero mean. Then ∂ s p = ∂ t q + { p, q } . (24)Put A = (cid:126) − k Q k ( p ), C = (cid:126) − k Q k ( q ). Let U ( t, s ) be the unitary evolutionof ∂ t U = − iAU with U (0 , s ) = . Note that U k ( φ , ) = U (1 , , U k ( φ , ) = U (1 , . Define B by ∂ s U = − iBU . (25)Then ∂ s ∂ t U = − iA∂ s U − i∂ s AU = − iABU − i∂ s AU ,∂ t ∂ s U = − iB∂ t U − i∂ t BU = − iBAU − i∂ t BU .
Subtracting and rearranging we get ∂ t B = ∂ s A − i [ A, B ] . Further, by (24) ∂ t C = (cid:126) − k Q k ( ∂ t q ) = (cid:126) − k Q k ( ∂ s p ) + (cid:126) − k Q k ( { p, q } ) = ∂ s A + (cid:126) − k Q k ( { p, q } ) . Thus ∂ t ( B − C ) = (cid:126) − k ( − i [ Q k ( p ) Q k ( q )] − (cid:126) k Q k ( { p, q } ) = O (1 /k ) , by bracket correspondence (P2). Observe that ∂ s U (0 , s ) = 0, so B (0 , s ) = 0.Further, q (0 , s ) = 0, so C (0 , s ) = 0. Thus (cid:107) B (1 , s ) − C (1 , s ) (cid:107) op = O (1 /k ) . But C (1 , s ) = 0 since q (1 , s ) = 0. Thus (cid:107) B (1 , s ) (cid:107) op = O (1 /k ) and hence by(25) (cid:107) U (1 , − U (1 , (cid:107) op = O (1 /k ) , and (23) follows. This proves item (i) of the theorem.Let’s analyze the quantization of the product of two Hamiltonian paths.Let φ t and ψ t be two paths generated by normalized Hamiltonians f t and g t θ t = φ t ψ t . Consider the corresponding Schroedingerevolutions ˙ U k = − i (cid:126) − k Q k ( f t ) U k , U k (0) = , ˙ V k = − i (cid:126) − k Q k ( g t ) V k , V k (0) = . Put S ( t ) = Q k ( f t ) + U k ( t ) Q k ( g t ) U k ( t ) − , W k ( t ) = U k ( t ) V k ( t ) . Observe that ˙ W k = − i (cid:126) − k S ( t ) W . (26)Since θ t is generated by h t := f t + g t ◦ φ − t , the Egorov theorem (Theorem3.1 ) yields Q k ( h t ) = S ( t ) + O (1 /k ) . Denote by Z k ( t ) the Schroedinger evolution of θ t , that is˙ Z k = − i (cid:126) − k Q k ( h t ) Z k = ( − i (cid:126) − k S ( t ) + O (1 /k )) Z k , Z k (0) = . Comparing this equation with (26) we conclude that (cid:107) U k (1) V k (1) − Z k (1) (cid:107) op = O (1 /k ) . Thus µ k is an almost-representation, which proves item (ii) of the theorem.Finally, assume that a Hamiltonian f t generates a Hamiltonian path φ t with φ (cid:54) = . Thus φ displaces an open set Y ⊂ M : φ ( Y ) ∩ Y = ∅ . Take anon-vanishing function g supported in φ ( Y ). Observe that (cid:107) g ◦ φ − − g (cid:107) = (cid:107) g (cid:107) . (27)Put A k := Q k ( g ). Let U k be the unitary operator quantizing φ . By theEgorov theorem, Q k ( g ◦ φ − ) = U k A k U − k + O (1 /k ). It follows from (27)and (P1) that (cid:107) U k A k U − k − A (cid:107) op = (cid:107) A (cid:107) op + O (1 /k ). Estimating (cid:107) A (cid:107) op + O (1 /k ) = (cid:107) U k A k U ∗ k − A (cid:107) op = (cid:107) U k AU ∗ k − U k A + U k A − A (cid:107) op ≤ (cid:107) A (cid:107) op · (cid:107) − U k (cid:107) op , we get that (cid:107) − U k (cid:107) op ≥ / O (1 /k ), which proves item (iii) of thetheorem. 16 emark 3.2. Replacing U k by e iθ U k in the proof of (iii), we get that (cid:107) U k − e iθ (cid:107) op ≥ / O (1 /k )for every phase θ . This implies that the approximate projective representa-tion ν k appearing right after Theorem 1.9 satisfies, for every φ ∈ Ham(
M, ω ), δ p ( ν k ( φ ) , ) ≥ const > , ∀ k ∈ N , provided φ (cid:54) = . In this section we prove Theorem 1.9 from the introduction. A more detailedformulation of this result appears in Theorem 4.1 below.
Let (
M, ω ) be a compact symplectic manifold equipped with a prequantumline bundle L and an auxiliary line bundle L (cid:48) such that c R ( L (cid:48) ) = λc R ( L ) + c R ( K ) (28)where K is the canonical line bundle.Since i ω is the curvature of L , the periods of ω are multiple of 2 π , so theaction of any contractible periodic trajectory γ ( t ), t ∈ [0 , T ] of a Hamiltonian( H t ) is well-defined modulo 2 π Z and given by the usual formula A ( γ ) = (cid:82) D ω − (cid:82) T H t ( γ ( t )) dt (29)where D is a disc with boundary γ . We can even define the action modulo2 π of any periodic trajectory, by using parallel transport in L instead of theintegral of ω .If ( H t ) generates a loop L = ( φ t , t ∈ [0 , L (cid:48) allows to define a mixed action-Maslovinvariant as follows [17]. By Floer theory, any trajectory φ t ( x ), t ∈ [0 ,
1] isthe boundary of a disc D . We set I ( L ) = λ (cid:0)(cid:82) D ω − (cid:82) H t ( φ t ( x )) dt (cid:1) + πm ( ψ ) (30)where ψ is the loop of Sp(2 n ) obtained by trivialising the symplectic bundle T M over D and defining ψ ( t ) := T x φ t , m ( ψ ) = 0 or 1 according to the classof ψ in π (Sp(2 n )) = Z is even or odd. One readily checks that I ( L ) is welldefined modulo 2 π Z . 17 .2 Quantization of a Hamiltonian loop Assume now that (
M, ω ) is K¨ahler, that L and L (cid:48) are holomorphic hermitianline bundles with Chern curvatures Θ and Θ (cid:48) satisfying Θ = i ω , Θ (cid:48) = λ Θ + Θ K . Consider the space H k of holomorphic sections of L k ⊗ L (cid:48) . Forany f ∈ C ∞ ( M, R ), we define the operator Op k ( f ) as in (17)Let ( H t ) be a Hamiltonian of M generating a loop L = ( φ t , t ∈ [0 , U t,k ,1 i ( k + λ ) ∂ t U k,t + Op k ( H t ) U k,t = 0 , U k, = We assume from now on that M is connected, so the periodic trajectories( φ t ( x ) , t ∈ [0 , A ( L ). Theorem 4.1.
We have U k, = e ikA ( L )+ iI ( L ) + O ( k − ) .Proof. We can rewrite the Schr¨odinger equation as ik ∂ t U k,t + (1 + λk ) Op k ( H t ) U k,t = 0Then, by [8, Theorem 4.2] the Schwartz kernel of U k,t is a Lagrangian stateassociated to the graph of φ t . We refer to [8] for the precise definitions.What is important to us here is that since φ is the identity, U k, = e ikθ T k ( σ ) + O ( k − ) (31)where θ is a real number, σ ∈ C ∞ ( M ) and T k ( σ ) is the Berezin-Toeplitzoperator with multiplicator σ defined as in section 2.Furthermore, we can compute θ and σ by introducing a half-form bundle(i.e., the square root of the canonical bundle) denoted by δ . It is possiblethat such a bundle does not exist on M but we only need it on the trajectory γ of a given point x . In this case we take a disk D with boundary γ andchoose the square root δ which extends to D .Then by [8, Theorem 1.1] U k,t ( φ t ( x ) , x ) ∼ (cid:16) k π (cid:17) n e i (cid:82) t H sub r ( φ r ( x )) dr (cid:104) φ Lt ( x ) (cid:105) ⊗ k ⊗T L t ( x ) ⊗ (cid:2) D t ( x ) (cid:3) / . Here φ Lt is the prequantum lift of φ t to L , and H sub r = λH t is the subprincipalsymbol of (1 + λk ) Op k ( H t ). The second term T L t ( x ) : L | x → L | φ t ( x ) is theparallel transport in the line bundle L = L (cid:48) ⊗ δ − . It is the multiplicationby exp( iλ (cid:82) D ω ) because the curvature of L is Θ (cid:48) − Θ K = λ Θ = λi ω . The18ast term is the square root of an isomorphism D t ( x ) : K x → K φ t ( x ) definedby D t ( x )( α )(( T x φ t ) , u ) = α ( u ) , ∀ α ∈ K x , u ∈ det T , x M .
Here the square root is chosen so as to be continuous and equal to 1 at t = 0.On the other hand, by (31), U k, ( x, x ) = (cid:0) k/ π (cid:1) n e ikθ ( σ ( x ) + O ( k − )).Now φ L ( x ) = e iA ( L ) implies that θ = A ( L ) and it remains to prove that e i (cid:82) H sub r ( φ r ( x )) dr T L ( x ) ⊗ (cid:2) D ( x ) (cid:3) / = e iI ( L ) (32)Since T x φ is the identity of T x M , D ( x ) is the identity of K x so( D t ( x )) / = ± δ x . To determine the sign, we trivialize
T M along γ with an symplectic frame,so that ( T x φ t ) becomes a loop α of symplectic matrices based at the identityand in the corresponding trivialisation of K , D t ( x ) is the multiplication by acomplex number. The sign we search depends only on the homotopy class of α . Since Sp(2 n ) deformation retracts to its subgroup U( n ), we can assumethat α is a loop of U( n ), in which case D t ( x ) is the complex determinantof α ( t ). Thus, our sign is positive or negative according to the class of α in π (Sp(2 n )) = Z is even or odd. We conclude that each factor in (32)corresponds to a summand in (30), which completes the proof. Acknowledgments.
We are grateful to Alex Lubotzky for his help withthe class of groups
PLO p , and for valuable comments on [9, 14]. We thankDavid Fisher for useful discussions. References [1] Bordemann, M., Meinrenken, E., and Schlichenmaier, M.,
Toeplitz quan-tization of K¨ahler manifolds and gl( N ) , N → ∞ limits, Comm. Math.Phys. (1994), 281–296.[2] Brown, A., Fisher, D., Hurtado, S.
Zimmer’s conjecture: Subexpo-nential growth, measure rigidity, and strong property (T),
PreprintarXiv:1608.04995, 2016.[3] Boutet de Monvel, L., and Guillemin, V.,
The spectral theory of Toeplitzoperators , vol. 99, Annals of Mathematics Studies, Princeton UniversityPress, Princeton, NJ, 1981. 194] Charles, L.,
Semi-classical properties of geometric quantization withmetaplectic correction,
Comm. Math. Phys. (2007),445–480.[5] Charles, L.,
On the quantization of compact symplectic manifold , JGeom. Anal. (2016), 2664–2710.[6] Charles, L., Subprincipal symbol for Toeplitz operators.
Lett. Math. Phys. (2016), no. 12, 1673-1694.[7] Charles, L.,
On a quasimorphism of Hamiltonian diffeomorphisms andquantization, preprint arXiv:1910.05073, 2019.[8] Charles, L., Le Floch, Y.
Quantum propagation for Berezin-Toeplitz op-erators , preprint hal-02935681, 2020.[9] De Chiffre, M., Glebsky, L., Lubotzky, A., Thom, A.,
Stability, coho-mology vanishing, and nonapproximable groups,
Forum of Mathematics,Sigma (2020), e18.[10] Grove, K., Karcher, H. and Ruh, E.A., Jacobi fields and Finsler met-rics on compact Lie groups with an application to differentiable pinchingproblems,
Math. Ann., (1974), 7–21.[11] Ioos, L., Kazhdan, D., Polterovich, L.,
Almost representations of Liealgebras and quantization, preprint arXiv:2005.11693, 2020[12] Kazhdan, D., On (cid:15) -representations, Israel J. Math. (1982), 315 –323.[13] Landsman, N.P., Mathematical Topics between Classical and QuantumMechanics,
Springer, 2012.[14] Lubotzky, A., Oppenheim, I.,
Non p -norm approximated groups, preprint arXiv:1807.06790, 2018.[15] Ma, X., and Marinescu, G., Holomorphic Morse inequalities andBergman kernels , volume 254, Progress in Mathematics, Birkh¨auser Ver-lag, Basel, 2007.[16] Polterovich, L.,
The geometry of the Group of Symplectic Diffeomor-phisms,
Birkh¨auser; 2012[17] Polterovich, L.,
Hamiltonian loops and Arnold’s principle,
Translationsof the American Mathematical Society-Series 2, (1997), 181–188.[18] Rosenberg, J.,
Algebraic K-theory and its Applications,
Springer, 1995.2019] Zimmer, R.J.,
Group Actions in Ergodic Theory, Geometry, and Topol-ogy: Selected Papers,
University of Chicago Press, 2019.
Laurent Charles Leonid Polterovich
Sorbonne Universit´e, Universit´e de Paris, CNRS Tel Aviv UniversityInstitut de Math´ematiques de Jussieu-Paris Rive Gauche School of Mathematical SciencesF-75005 Paris, France 69978, Tel Aviv, Israel
E-mail: [email protected] [email protected]@imj-prg.fr [email protected]