A geometric construction of representations of the Berezin-Toeplitz quantization
AA GEOMETRIC CONSTRUCTION OF REPRESENTATIONS OF THEBEREZIN-TOEPLITZ QUANTIZATION
KWOKWAI CHAN, NAICHUNG CONAN LEUNG, AND QIN LIA
BSTRACT . For a K¨ahler manifold X equipped with a prequantum line bundle L , we givea geometric construction of a family of representations of the Berezin-Toeplitz deformationquantization algebra ( C ∞ ( X )[[ ¯ h ]] , (cid:63) BT ) parametrized by points z ∈ X . The key idea is touse peak sections to suitably localize the Hilbert spaces H ( X , L ⊗ m ) around z .
1. I
NTRODUCTION
Quantization plays important roles in both physics and in mathematics. Two outstand-ing approaches are the deformation quantization ([1–3, 11, 20, 24]) and geometric quanti-zation [18, 21, 29, 30, 32]. This paper is an attempt to understand the intriguing relationshipbetween these two quantization schemes.To begin with, let us consider a symplectic vector space X = R n equipped with thestandard symplectic form ω = ∑ nj = dx j ∧ dy j . A complex polarization (i.e., complex struc-ture) identifies X with C n with coordinates z j = x j + iy j ’s. Then geometric quantizationgives the Bargmann-Fock space H L ( C n , µ ¯ h ) , elements of which are L integrable entireholomorphic functions with respect to the density µ ¯ h ( z ) : = ( π ¯ h ) − n e −| z | /¯ h . A smoothfunction f = f ( z , ¯ z ) ∈ C ∞ ( X ) acts on H L ( C n , µ ¯ h ) as a Toeplitz operator T f defined bysetting T z j = m z j (i.e., multiplication by z j ) and T ¯ z j = ¯ h ∂∂ z j . The composition of theseoperators defines a star product via the formula T f ◦ T g = T f (cid:63) g , which is given explicitlyby f (cid:63) g : = exp (cid:32) − ¯ h n ∑ i = ∂∂ z i ∂∂ ¯ w i (cid:33) ( f ( z , ¯ z ) g ( w , ¯ w )) | z = w .This endows C ∞ ( X )[[ ¯ h ]] with a noncommutative algebra structure, or a deformationquantization of ( X , ω ) , and H L ( C n , µ ¯ h ) is naturally its representation.Note that the space of polynomials C [ z , . . . , z n ] is preserved under this action, andwe have the decomposition C [ z , . . . , z n ] = ∑ m C [ z , . . . , z n ] m , where C [ z , . . . , z n ] m is thespace of degree m homogeneous polynomials on C n which can be regarded as the spaceof holomorphic sections of L ⊗ m with L being the trivial line bundle. Mathematics Subject Classification.
Key words and phrases.
Deformation quantization, geometric quantization, Berezin-Toeplitz star product,Toeplitz operator, peak section. a r X i v : . [ m a t h . QA ] J a n CHAN, LEUNG, AND LI
In general, we are interested in a compact K¨ahler manifold ( X , ω , J ) with integral [ ω ] ,so that there exists a prequantum line bundle L whose curvature F L satisfies √− π F L = ω . Geometric quantization of ( X , m ω ) gives the Hilbert space H ( X , L ⊗ m ) , the space ofholomorphic sections of L ⊗ m . To a smooth function f ∈ C ∞ ( X ) , we can similarly associatethe Toeplitz operator T f , m : = Π m ◦ m f : H ( X , L ⊗ m ) → H ( X , L ⊗ m ) ,where m f is multiplication by f and Π m denotes the orthogonal projection from the spaceof L sections L ( X , L ⊗ m ) to H ( X , L ⊗ m ) .An important result in the Berezin-Toeplitz quantization is that this gives rise to a starproduct (cid:63) BT , and hence the Berezin-Toeplitz deformation quantization algebra ( C ∞ ( X )[[ ¯ h ]] , (cid:63) BT ) [3, 17, 28]: f (cid:63) BT g : = ∑ i ≥ ¯ h i C i ( f , g ) ,where C i ( − , − ) are bi-differential operators, || · || is the operator norm, and K N ( f , g ) isindependent of m such that the following estimates hold:(1.1) || T f , m ◦ T g , m − N − ∑ i = (cid:18) m (cid:19) i T C i ( f , g ) , m || ≤ K N ( f , g ) (cid:18) m (cid:19) N .Unlike the flat case, however, the estimate (1.1) says that the difference T f , m ◦ T g , m − T f (cid:63) BT g , m is only asymptotically zero when m tends to infinity. So ( C ∞ ( X )[[ ¯ h ]] , (cid:63) BT ) doesnot quite act on H ( X , L ⊗ m ) ; we do not even expect a representation of ( C ∞ ( X )[[ ¯ h ]] , (cid:63) BT ) on the product ∏ m H ( X , L ⊗ m ) .On the other hand, as m → ∞ , physically speaking we are scaling X to the large volumelimit . We would expect the physical system to behave like that on a flat space aroundany given point z ∈ X . We are going to show that this is indeed the case. To be moreprecise, we will use peak sections S m , p , r of H ( X , L m ) to appropriately localize the Hilbertspaces around z and produce a representation H z of the Berezin-Toeplitz deformationquantization algebra ( C ∞ ( X )[[ ¯ h ]] , (cid:63) BT ) .In a suitably chosen coordinates (and frame of L ) around z , S m , p , r is equal to z p · · · z p n n up to order 2 r −
1. Because of the error terms, the peak sections in ∏ m H ( X , L m ) with a fixed r behave in a compatible way with the actions of T z j = m z j and T ¯ z j = ¯ h ddz j around z only up to order 2 r −
1, which is not enough to produce a representation of ( C ∞ ( X )[[ ¯ h ]] , (cid:63) BT ) . To construct our representation H z , we need to find a clever way toincrease the order r of Taylor expansions at z to infinity when the choices of peak sections S m , p , r ’s are changing in r .To achieve this, we consider ∑ m α m , r ∈ ∏ m H ( X , L ⊗ m ) , which represents a sum of peaksections of various tensor powers L m ’s and where we are actually only keeping track of itsTaylor expansion around z up to order r −
1. A key observation is that { ∑ m α m , r } ∞ r = , orsimply a double sequence { α m , r } , will define elements in ∏ m H ( X , L ⊗ m ) with more and EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 3 more terms of their Taylor expansions around z being identical if the following conditionholds: there exists a sequence of complex numbers { a p , k } p , k ≥ such that, for each fixed r >
0, we have the following estimates:(1.2) (cid:104) α m , r − ∑ k + | p |≤ r a p , k · m k · S m , p , r + , S m , q , r + (cid:105) m = O (cid:18) m r + (cid:19) ,for any multi-index q with | q | ≤ r .We call such { α m , r } an admissible sequence at z , and they span a linear subspace V z ⊂ ∏ r (cid:0) ∏ m H ( X , L ⊗ m ) (cid:1) . In fact, only the coefficients (cid:8) a p , k (cid:9) of { α m , r } would record thewhole Taylor expansions at z . Thus this defines an equivalence relation ∼ on V z andthe desired vector space can be constructed as H z : = V z / ∼ . Theorem 1.1 (=Theorem 3.16) . The vector space H z is a representation of the Berezin-Toeplitzdeformation quantization algebra ( C ∞ ( X )[[ ¯ h ]] , (cid:63) BT ) . We will prove that this representation possesses various nice properties, as expectedfrom the physical point of view. First of all, it is local , namely, for any smooth function f ∈ C ∞ ( X ) , the action of the Toeplitz operator T f on H z depends only on the infinitejets of f at z ; this will be proved in Theorem 3.18. Also, for every real-valued function f , the operator T f on H z is self-adjoint; see Proposition 3.14. Last but not the least, it isirreducible in a suitable sense, as we will see in Theorem 3.21. Remark . The space H z has the structure of a formal Hilbert space with inner productstaking values in C [[ ¯ h ]] , as in the work [5] of Bordemann-Waldmann, where they gavesome algebraic constructions of representations of deformation quantization algebras.On the other hand, our results are closely related to the work [27] of Reshetikhin-Takhtajan, where they related the star product to formal Feynman-Laplace expansionsof formal integrals. Acknowledgement.
We thank Si Li and Siye Wu for useful discussions. The first named author thanks Mar-tin Schlichenmaier and Siye Wu for inviting him to attend the conference GEOQUANT2019 held in September 2019 in Taiwan, in which he had stimulating and very helpfuldiscussions with both of them as well as Jrgen Ellegaard Andersen, Motohico Mulase,Georgiy Sharygin and Steve Zelditch.K. Chan was supported by grants of the Hong Kong Research Grants Council (ProjectNo. CUHK14302617 & CUHK14303019) and direct grants from CUHK. N. C. Leung wassupported by grants of the Hong Kong Research Grants Council (Project No. CUHK14301117& CUHK14303518) and direct grants from CUHK. Q. Li supported by a grant from Na-tional Natural Science Foundation of China for young scholars (Project No. 11501537).
CHAN, LEUNG, AND LI
2. T HE F EYNMAN -L APLACE T HEOREM AND PERTURBATIONS OF THE B ARGMANN -F OCK SPACE
In this section, we perform the local computations needed for proving Theorem 1.1.2.1.
The Bargmann-Fock space and Wick algebra.
Recall that in the flat case when X = C n (and prequantum line bundle L is trivial), theHilbert space on which the Toeplitz operators act is the well-known Bargmann-Fock space H L ( C n , µ ¯ h ) consisting of L integrable entire holomorphic functions with respect to thedensity µ ¯ h ( z ) = ( π ¯ h ) − n e −| z | /¯ h ; here ¯ h is regarded as a positive real number. It is easy tosee, by direct computations, that the holomorphic polynomials z I (cid:112) I !¯ h | I | ,where I runs over all multi-indices, form an orthonormal basis of H L ( C n , µ ¯ h ) .We will only need the Toeplitz operators associated to polynomials, i.e., multiplying bya polynomial f ∈ C [ z , ¯ z ] which is in general non-holomorphic and then projecting backto the holomorphic subspace. For example, when n =
1, we have T z = m z , T ¯ z = ¯ h ddz , T f ( z ) f ( ¯ z ) = f (cid:18) ¯ h ddz (cid:19) ◦ m f ( z ) ,Let f , g ∈ C [ z , ¯ z ] . Then there is a formula for the composition of Toeplitz operators: T f ◦ T g = T f (cid:63) g , where f (cid:63) g : = exp (cid:32) − ¯ h n ∑ i = ∂∂ z i ∂∂ ¯ w i (cid:33) ( f ( z , ¯ z ) g ( w , ¯ w )) | z = w .This product can be extended to formal power series, giving the definition of the Wickalgebra: Definition 2.1.
The
Wick algebra is W C n : = C [[ y , ¯ y ]][[ ¯ h ]] equipped with the multiplication:(2.1) f (cid:63) g : = exp (cid:32) − ¯ h n ∑ i = ∂∂ y i ∂∂ ¯ y (cid:48) i (cid:33) ( f ( y , ¯ y ) g ( y (cid:48) , ¯ y (cid:48) )) | y = y (cid:48) . Remark . Here we use y , ¯ y instead of z , ¯ z in order to distinguish functions on C n andWick algebra in later sections.There is an action of the Wick algebra on the Bargmann-Fock space F C n : = C [[ y , · · · , y n ]][[ ¯ h ]] , EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 5 which we call the
Bargmann-Fock action , defined as follows: let y i · · · y i k ¯ y j · · · ¯ y j l be anymonomial in the Wick algebra, then we turn it to a differential operator on F C n by(2.2) y i · · · y i k ¯ y j · · · ¯ y j l (cid:55)→ (cid:32) ¯ h · ∂∂ y j (cid:33) ◦ · · · ◦ (cid:32) ¯ h · ∂∂ y j l (cid:33) ◦ m y i ··· y ik ;in other words, holomorphic polynomials are mapped to creation operators and anti-holomorphic ones are mapped to annihilation operators. This assignment is also knownas the Wick normal ordering . Remark . There is another Fock representation on W C n where y i ’s act as annihilators and¯ y j ’s act as creators. Explicitly, F (cid:48) C n : = C [[ ¯ y , · · · , ¯ y n ]][[ ¯ h ]] , and the operators associated toa monomial is given by y i · · · y i k ¯ y j · · · ¯ y j l (cid:55)→ m ¯ y j ··· ¯ y jl ◦ (cid:18) − ¯ h · ∂∂ ¯ y i (cid:19) ◦ · · · ◦ (cid:18) − ¯ h · ∂∂ ¯ y i k (cid:19) This Fock representation is equivalent to the GNS representation on C n in [5]. Notation 2.4.
Throughout this paper, we will use the following notation for multi-indices:let I = ( i , · · · , i n ) and J = ( j , · · · , j m ) then we set y I : = y i · · · y i n n , ¯ y J : = ¯ y j · · · ¯ y j m n .We also use the notations: | I | : = i + · · · + i n and I ! : = i ! · · · i n !.Let I , J be as above. We assign a Z -grading on W C n by letting the monomial ¯ h k y I ¯ y J tohave degree 2 k + | I | + | J | . There is an associated decreasing filtration on W C n given bythe set ( W C n ) k of power series in W C whose terms are all of degree ≥ k . In a similar way,we can define a grading and filtration on both F C n and C [[ ¯ h ]] . Note that this grading ispreserved by both the Wick product and the Bargmann-Fock action.We will also need the following extension of the Wick algebra: Definition 2.5.
The extension W + C n of the Wick algebra W C n is defined as follows: • Elements of W + C n are given by power series, possibly with negative powers of ¯ h , • The degrees of terms of an element U ∈ W + C n have a uniform lower bound whichcould be negative; equivalently, there exists k ≥ h k · U hasnon-negative degree. • For an element U ∈ W + C n , there exists a finite number of terms for any given non-negative total degree. Remark . The definition of W + C n here is different from the one in [11, p. 224]; in thatdefinition, monomials in W + C n must have non-negative total degrees. Our extension willbe important later for proving the irreducibility of our representation.Note that W + C n is closed under the Wick product. One of the motivations of this exten-sion is to allow the following exponentials: CHAN, LEUNG, AND LI
Example 2.7.
Let H ∈ ( W C n ) , i.e., every term in H is of degree at least 3, then the follow-ing classical and quantum exponentials both live in W + C n :exp ( H /¯ h ) = + H ¯ h + H · H ¯ h + · · · ,exp (cid:63) ( H /¯ h ) = + H ¯ h + H (cid:63) H ¯ h + · · · Notation 2.8.
In this paper, we will use the notation e H /¯ h to denote the classical exponen-tial of H /¯ h .We can define F + C n in a similar way. It is clear that there is a naturally extendedBargmann-Fock action of W + C n on F + C n . The following lemma shows that the subspace F C n ⊂ F + C n is closed under the action of elements in W + C n of a special form. Lemma 2.9.
Suppose H = ∑ k , | I |≥ | J |≥ ¯ h k a k , I , J · y I ¯ y J ∈ ( W C n ) has no purely holomorphicterms, i.e., a k , I ,0 = . Then F C n is closed under the action of exp ( H /¯ h ) and exp (cid:63) ( H /¯ h ) .Proof. Each ¯ y j in every monomial of H /¯ h acts as ¯ h ∂∂ y j , and the ¯ h in this differential oper-ator will cancel with the ¯ h in the denominator. So the output can only have nonnegativepowers of ¯ h . (cid:3) Lemma 2.10.
The classical exponential exp ( H /¯ h ) , where H ∈ ( W C n ) , can also be written as aquantum exponential: exp ( H /¯ h ) = exp (cid:63) ( H (cid:48) /¯ h ) , with H (cid:48) ∈ W + C n . In particular, exp ( H /¯ h ) is invertible in W + C n and ( exp ( H /¯ h )) − = exp (cid:63) ( − H (cid:48) /¯ h ) . Proof.
Let A = exp ( H /¯ h ) − ∈ ( W C n ) . Then H (cid:48) is defined via the following formallogarithm with respect to the quantum product (cid:63) : H (cid:48) = ∞ ∑ k = ( − ) k + k A k ,where A k = A (cid:63) A (cid:63) · · · (cid:63) A denotes the k -th power with respect to the quantum product.The fact that A ∈ ( W C n ) implies that each term of H (cid:48) is of positive degree, and in eachdegree there are only finitely many terms in H (cid:48) , i.e., H (cid:48) ∈ W + C n . (cid:3) Formal Hilbert spaces.
In the K¨ahler geometry setting, we cannot in general reduce to the local model of theBargmann-Fock space, and will need to consider a more general situation. For this pur-pose, we need the following theorem:
EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 7
Theorem 2.11 (Feynman-Laplace) . Let X be a compact n-dimensional manifold (possibly withboundary), and let f be a smooth function attaining a unique minimum on X at an interior pointx ∈ X, and assume that the Hessian of f is non-degenerate at x ; also, let µ = α ( x ) · e g ( x ) d n xbe a top-degree form. Then the integralI ( ¯ h ) : = (cid:90) X µ e − h f ( x ) = (cid:90) X α ( x ) · e − f ( x )+ ¯ hg ( x ) ¯ h dx · · · dx n , has the following asymptotic expansion as ¯ h → + :I ( ¯ h ) ∼ ∑ k ≥ a k · ¯ h k , where each coefficient a k is a sum of Feynman weights which depends only on the infinite jets ofthe functions f , g at the point x . More explicitly, each a k is a sum over connected graphs of genus k . Recall that the genusof a graph γ is the sum of the genera of the vertices in γ (in our situation, each vertex hasgenus either 0 or 1, labeled by f ( x ) and g ( x ) respectively, since the integrand is e f ( x )+ ¯ hg ( x ) ¯ h ),and k = − χ ( γ ) where χ ( γ ) denotes the Euler characteristic of γ . The propagator in theFeynman weights is given by the inverse of the Hessian of f at x . The following pictureshows a Feynman graph:Here every vertex labeled by f must be at least trivalent, and every vertex labeled by¯ h · g must be univalent. For more details on the Feynman-Laplace Theorem, we referthe readers to Pavel Mnev’s excellent exposition in [25], and for a detailed exposition ofFeynman graph computations, we refer the readers to [7].We will mainly apply the Feynman-Laplace Theorem to a function f ( z , ¯ z ) on a closeddisk D n ⊂ C n such that the origin 0 ∈ D n is the unique minimum of f and f ( ) = z = ( z , · · · , z n ) centered at 0,the Taylor expansion of f at the origin is given by f ( z , ¯ z ) = | z | + O ( | z | ) .Theorem 2.11 gives an asymptotic expansion of the following integral: ( √− ) n ¯ h n (cid:90) D n h ( z , ¯ z ) e − f ( z ,¯ z )+ ¯ h · g ( z ,¯ z ) ¯ h dz d ¯ z · · · dz n d ¯ z n . Remark . The above integral clearly depends on the radius of D n , but its asymptoticexpansion is actually independent of the radius. CHAN, LEUNG, AND LI
Theorem 2.11 implies that the asymptotic expansion of the above integral depends onlyon the Taylor expansions of the functions f , g and h at the origin. We can thus replacethese functions by formal power series in W C n , and define a formal integral: Definition 2.13.
For φ ( y , ¯ y ) ∈ ( W C n ) and h ( y , ¯ y ) ∈ W C n , we define the following formalintegral: 1¯ h n (cid:90) h ( y , ¯ y ) · e −| y | + φ ( y , ¯ y ) ¯ h ∈ C [[ ¯ h ]] via the Feynman rule in Theorem 2.11. Remark . We omit the standard differential form ( √− ) n dz d ¯ z · · · dz n d ¯ z n in the nota-tion of formal integral. Lemma 2.15.
The formal integral preserves the decreasing filtration on W C n and C [[ ¯ h ]] ; moreprecisely, if h ( y , ¯ y ) ∈ ( W C n ) k , then the formal integral lies in ( C [[ ¯ h ]]) k .Proof. The leading term of the formal integral1¯ h n (cid:90) h ( y , ¯ y ) · e −| y | + φ ( y , ¯ y ) ¯ h ∈ C [[ ¯ h ]] is the same as that in the free case, i.e., when φ =
0. Thus the leading term of the integralhave the same degree as the leading degree of h ( y , ¯ y ) . (cid:3) Using this formal integral, we can define a
Hilbert space in the formal sense , namely, itsinner product takes values in the formal Laurent series C (( √ ¯ h )) : Definition 2.16.
On the C (( √ ¯ h )) -vector space W C n ⊗ C [[ ¯ h ]] C (( √ ¯ h )) , we define a complexconjugation by extending the complex conjugation on polynomials in C n : ( √ ¯ h ) k a I , J y I ¯ y J (cid:55)→ ( √ ¯ h ) k ¯ a I , J ¯ y I y J .Fix φ ( y , ¯ y ) ∈ ( W C n ) . Then for f , g ∈ W C n (( √ ¯ h )) , we define their formal inner product asthe following formal integral:(2.3) (cid:104) f , g (cid:105) : = h n · (cid:90) f ¯ g · e −| y | + φ ( y , ¯ y ) ¯ h ,which is in turn defined using Feynman graph expansions as in Definition 2.13 and takesvalue in C (( √ ¯ h )) .The following are some simple properties of this formal inner product. Lemma 2.17.
Suppose that φ is real , i.e., φ = ¯ φ . Then the formal inner product (2.3) is Hermit-ian, namely, (cid:104) f , g (cid:105) = (cid:104) g , f (cid:105) . Lemma 2.15 implies the following
Corollary 2.18.
The formal inner product of f , g ∈ W + C n is a formal power series in ¯ h, i.e., (cid:104) f , g (cid:105) ∈ C [[ ¯ h ]] . EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 9
Remark . We allow φ to have ¯ h -dependence. In particular, the fact that φ is of at leastdegree 3 guarantees that the graph expansion of (2.3) is valid. In the K¨ahler geometrysetting, f will be given by the logarithm of the norm of a local holomorphic frame of theprequantum line bundle, and g will be the logarithm of the volume form.The following lemma explains the reason for considering an extension of W C n by C (( ¯ h )) : Lemma 2.20.
The holomorphic polynomials y I (cid:112) I !¯ h | I | form a basis of the formal Hilbert space, which is orthonormal modulo ¯ h, i.e., (cid:104) y I (cid:112) I !¯ h | I | , y J (cid:113) J !¯ h | J | (cid:105) = δ I , J + O ( ¯ h ) . Proof.
The proof for the cases where I = J is obvious since the computation of the leadingterm is the same as that in the Bargmann-Fock space. For the cases where I (cid:54) = J , the terms y I ¯ y J cannot be fully contracted using the quadratic part −| y | /¯ h . The “interaction” part e φ /¯ h needs to come in so that we can get a full contraction which takes value in C [[ ¯ h ]] .Notice that this contraction preserves the filtration induced by the grading on W C n and C [[ ¯ h ]] . Now y I √ I !¯ h | I | , ¯ y J √ J !¯ h | J | have degree 0, and all terms in e φ /¯ h have degrees strictly greaterthan 0. The result follows. (cid:3) Corollary 2.21.
Given two different multi-indices I (cid:54) = J, we have the following asymptotics: ¯ h − n (cid:90) y I ¯ y J e −| y | + φ ( y , ¯ y ) ¯ h = O ( ¯ h max {| I | , | J |} ) . Let φ = ∑ k , I , J ¯ h k φ k , I , J y I ¯ y J , and suppose φ satisfies the property that φ I , J = if either | I | = or | J | = . Then we further have the refinement: ¯ h − n (cid:90) y I ¯ y J e −| y | + φ ( y , ¯ y ) ¯ h = o ( ¯ h max {| I | , | J |} ) . Proof.
Let K = ( k , · · · , k n ) be the multi-index given by k l = max { i l , j l } , 1 ≤ l ≤ n . Theworst scenario is when y I ¯ y J together with terms in e φ /¯ h form a multiple of y K ¯ y K so thatwe get full contraction. These terms coming from e φ /¯ h must be a multiple of¯ h − l · y K − I ¯ y K − J for some l . Thus the leading term of the integral in the statement is O ( ¯ h | K |− l ) . If l ≤ | K | − l ≥ | K | ≥ max {| I | , | J |} . Thus we assume that l >
0. Since every monomial in φ /¯ h contains at least one y i ’s, it follows that(2.4) l ≤ | K − I | . From Lemma 2.20, we know that¯ h − n (cid:90) y I ¯ y J · h l y K − I ¯ y K − J = ¯ h − n (cid:90) y K ¯ y K ¯ h l = O (cid:16) ¯ h | K |− l (cid:17) .The statement follows since | K | − l ≥ | K | − | K − I | = | I | , and also | K | − l ≥ | J | by asimilar argument. For the refinement under the additional condition on φ , we only needto notice that the inequality (2.4) can be refined to l ≤ | K − I | < | K − I | . (cid:3) We want to define the notion of orthogonal projection and formal Toeplitz operatorsusing this formal inner product. To do so, we need the following technical theorem (whichis also important in the sequel [6] to this paper):
Theorem 2.22.
Suppose φ ∈ ( W C n ) contains no purely holomorphic monomials. For anyf ∈ W C n , there exists a unique O f ∈ W C n such that (1) For any s ∈ F C n , the element T O f ( s ) ∈ F C n satisfies the following equalities: (cid:104) T O f ( s ) , y I (cid:105) = (cid:104) f · s , y I (cid:105) , for every multi-index I; here T O f denotes the Bargmann-Fock action by O f . (2) IF f is a monomial, then the leading term of O f is exactly f , i.e.,O f = f + higher order terms . Proof.
Given s ∈ F C n , suppose there exists s (cid:48) ∈ F C n such that(2.5) T f · e φ /¯ h ( s ) = T e φ /¯ h ( s (cid:48) ) .There is the following straightforward computation: (cid:90) (cid:16) e φ /¯ h · s (cid:48) (cid:17) · ¯ y I · e − | y | h = (cid:90) T e φ /¯ h ( s (cid:48) ) · ¯ y I · e − | y | h = (cid:90) T f · e φ /¯ h ( s ) · ¯ y I · e − | y | h = (cid:90) (cid:16) f · e φ /¯ h · s (cid:17) · ¯ y I · e − | y | h for any purely antiholomorphic monomial ¯ y I ; here the first equality follows from the factthat T e φ /¯ h ( s (cid:48) ) is the orthogonal projection of e φ /¯ h · s (cid:48) with respect to the standard Gaussianmeasure. So for every multi-index I , we have (cid:90) s (cid:48) · ¯ y I · e −| y | + φ ( y , ¯ y ) ¯ h = (cid:90) f s · ¯ y I · e −| y | + φ ( y , ¯ y ) ¯ h .Hence we only need to solve the equation (2.5) for s (cid:48) . EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 11
By Lemma 2.10, e φ /¯ h is invertible under the Wick product and its inverse is: (cid:16) e φ /¯ h (cid:17) − = exp (cid:63) (cid:32) − ∞ ∑ k = ( − ) k + k ( e φ /¯ h − ) k (cid:33) .From the following expansion − ∞ ∑ k = ( − ) k + k (cid:16) e φ /¯ h − (cid:17) k = − ∞ ∑ k = ( − ) k + k (cid:18) φ ¯ h + φ ¯ h + · · · (cid:19) k ,it is easy to see that each monomial in the expansion of (cid:0) e φ /¯ h (cid:1) − satisfies the followingproperty: in every ¯ h − k term, the antiholomorphic components must have degrees at least k . By a similar argument as in the proof of Lemma 2.9, we see that there is a well-definedaction of (cid:0) e φ /¯ h (cid:1) − on F C n .Therefore we get the following explicit description of s (cid:48) :(2.6) s (cid:48) = T ( e φ /¯ h ) − ◦ T f · e φ /¯ h ( s ) .The next step is to look at the term T f · e φ /¯ h ( s ) more closely. According to (2.1), we have f · e φ /¯ h = e φ /¯ h (cid:63) f − ∑ k ≥ ¯ h k C k (cid:16) e φ /¯ h , f (cid:17) .Since C k ( − , − ) is a bi-differential operator, the term C k (cid:0) e φ /¯ h , f (cid:1) is still of the form e φ /¯ h · g k ( y , ¯ y ) for some g k ∈ W C n and satisfies the condition thatleading degree of g k − leading degree of f ≥ k .Thus by an induction on the degree, this procedure can be iterated, and we can find O f ∈ W C n whose first terms are exactly f such that f · e φ /¯ h = e φ /¯ h (cid:63) O f . This implies that(2.7) T f · e φ /¯ h ( s ) = ( T e φ /¯ h ◦ T O f )( s ) .In particular, we see that s (cid:48) = T O f ( s ) . (cid:3) By the first statement of this theorem, we have the following:
Definition 2.23.
The orthogonal projection operator (2.8) π φ : W C n → F C n = C [[ y , · · · , y n ]][[ ¯ h ]] .is defined by requiring that (cid:104) f , y I (cid:105) = (cid:104) π φ ( f ) , y I (cid:105) for all multi-indices I ; here (cid:104)− , −(cid:105) is the inner product defined by equation (2.3).We can also define the formal Toeplitz operators: Definition 2.24.
The formal Toeplitz operator T φ , f associated to f ∈ W C n is defined as thecomposition of multiplication by f and the projection π φ : T φ , f : = π φ ◦ m f . Theorem 2.22 gives an explicit algorithm to compute T φ , f : we only need to find O f ∈W C n associated to f , and then T φ , f = T O f . A simple observation is that if f = f ( y ) is aholomorphic power series, then T f is simply the multiplication m f since then O f = f .Here we give a description of the adjoint operator of a formal Toeplitz operator: Lemma 2.25.
Suppose φ ∈ W C n is real, i.e., φ = ¯ φ . Then for any f ∈ W C n , the adjoint ofthe formal Toeplitz operator T φ , f is given by T φ , ¯ f . In particular, T φ , f is self-adjoint if and only iff ∈ W C n is real.Proof. According to the definition of formal Toeplitz operators, for any elements s , s ∈F C n , we have (cid:104) T φ , f ( s ) , s (cid:105) = (cid:90) T φ , f ( s ) · ¯ s · e −| y | + φ ( y , ¯ y ) ¯ h = (cid:90) f · s · ¯ s · e −| y | + φ ( y , ¯ y ) ¯ h = (cid:90) s · ¯ f · s · e −| y | + φ ( y , ¯ y ) ¯ h = (cid:104) s , T φ , ¯ f ( s ) (cid:105) . (cid:3) Local asymptotics via formal Hilbert space.
Let D n ⊂ C n be a ball centered at 0, with dvol D n : = ( √− ) n e ψ ( z , ¯ z ) dz d ¯ z · · · dz n d ¯ z n the volume form. For every smooth function f on D n , we will let J f ∈ W C n denote theTaylor expansion of f at the origin: J f : = ∑ I , J ≥ I ! J ! ∂ | I | + | J | f ∂ z I ∂ ¯ z J ( ) y I ¯ y J .The previous algebraic computations together with the Feynman-Laplace Theorem 2.11give the following asymptotics as ¯ h → + : Theorem 2.26.
Suppose ϕ ( z , ¯ z ) is a smooth function on D n which attains its unique minimumat the origin. Let f , ϕ , s be functions on D n such that ¯ ∂ s = , ϕ has a unique minimum at theorigin and satisfies (2.9) J ϕ = | y | + ∑ I , J ≥ I ! J ! ∂ | I | + | J | ϕ∂ z I ∂ ¯ z J ( ) y I ¯ y J . There exist complex numbers a k , I so that for every fixed multi-index J, we have the followingasymptotics as ¯ h → : (2.10) 1¯ h n (cid:90) D n (cid:18) f · s − ∑ k + | I |≤ r h k a k , I z I (cid:19) · ¯ z J e − ϕ ( z ,¯ z ) ¯ h dvol D n = O ( ¯ h r + ) . In particular, these a k , I ’s only depend on the Taylor expansions of f , s , ϕ and ψ at the origin. EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 13
Proof.
We define a function φ = | z | − ϕ ( z , ¯ z ) + ¯ h ψ ( z , ¯ z ) . Then we define a k , I ’s via thefollowing equation: ∑ k , | I |≥ a k , I ¯ h k · z I = T ( e J φ /¯ h ) − ◦ T J f · e J φ /¯ h ( J s ) .From Theorem 2.11, we have, for any ¯ z J , the following equality of asymptotic ¯ h -expansions1¯ h n · (cid:90) D n f · s · ¯ z J e − ϕ ( z ,¯ z ) ¯ h dvol D n = h n (cid:90) J f · J s · ¯ y J e −| y | + J φ ¯ h .On the other hand, there is the following identity by Theorem 2.22:1¯ h n (cid:90) J f · J s · ¯ y J e −| y | + J φ ¯ h = h n (cid:90) (cid:18) ∑ k , | I |≥ a k , I ¯ h k · y I (cid:19) · ¯ y J e −| y | + J φ ¯ h .Now equation (2.10) follows from Corollary 2.21 since the truncated higher order termswill only contribute to integrals of type o ( ¯ h r + ) . (cid:3)
3. G
EOMETRIC REPRESENTATIONS OF THE B EREZIN -T OEPLITZ QUANTIZATION
In this section, we construct a family of representations of the Berezin-Toeplitz defor-mation quantization parametrized by the points in the K ¨ahler manifold X , and describeits basic properties such as locality and irreducibility.The organization of this section is as follows: In Section 3.1, we study C n as the motivat-ing example to illustrate the idea behind the general definition of admissible sequences.In Section 3.2, we construct representations of the Berezin-Toeplitz deformation quantiza-tion using peak sections (whose properties are reviewed in Section A), which reduce theproof of Theorem 1.1 to the local computations done in Section 2. In Section 3.3, we provelocality and (modified) irreducibility of our representations.3.1. Admissible sequences on C n . Recall that the prequantum line bundle L on C n is trivialized by a global holomorphicframe . The Hermitian inner product of L ⊗ m under this trivialization is given by h m ( ⊗ m , ⊗ m ) = e − m ·| z | .We would like to define an action of C ∞ ( C n )[[ ¯ h ]] on F C n such that its restriction to polyno-mials is exactly the Bargmann-Fock action. To do this, we first apply asymptotic analysisto give an equivalent description of F C n .First of all, we consider the vector space V : = ∏ r ≥ (cid:32) ∏ m ≥ H ( C n , L ⊗ m ) (cid:33) ,an element of which is a double sequence α = ( α m , r ) with α m , r ∈ H ( C n , L ⊗ m ) . We also consider the map(3.1) F : F C n → V , a = ∑ k , I a k , I ¯ h k z I (cid:55)→ α = { α m , r } ,defined by setting α m , r : = ∑ k + | I |≤ r m − k · a k , I z I ⊗ m ∈ H ( C n , L ⊗ m ) .If the element in F C n is of the form ∑ I a I z I , i.e., it does not include ¯ h , then each α m , r is aholomorphic section of L ⊗ m which is a polynomial of degree ≤ r truncated from α underthe trivialization ⊗ m . For general elements, ¯ h k is mapped to 1/ m k in the correspondingcomponents of the double sequence.We now consider an action of smooth functions on the image of the map (3.1). Let f be any smooth function on C n . It is clear that the Toeplitz operator T f , m is in general notwell-defined since C n is noncompact. We apply the asymptotic analysis in Section 2.3 toobtain the following Proposition 3.1.
Let f be any smooth function on C n , and let F ( a ) = { α m , r } be defined asabove. Then there exists b = ∑ k , I ¯ h k b k , I z I ∈ F C n with F ( b ) = { β r , m } , satisfying the followingasymptotics as m → ∞ : (3.2) m n · (cid:90) C n ( f · α m , r − β m , r ) · ¯ z J · e − m ·| z | = O (cid:18) m r + (cid:19) for every fixed r ≥ and multi-index J. In particular, the formal power series b ∈ F C n is uniquelydetermined by α and the Taylor expansion of f at the origin ∈ C n .Proof. Explicitly, we need to prove the following: m n · (cid:90) C n ( f · α m , r − ∑ k + | I |≤ r m k · b k , I z I ) · ¯ z J · e − m ·| z | = O (cid:18) m r + (cid:19) .Since α m , r ’s are truncated from the same formal power series, we see that { f · α m , r } hasthe same property. Thus the result follows from Theorem 2.26. (cid:3) Example 3.2.
Let us consider the simplest example where n =
1, and let a = z . Then thedouble sequence { α m , r } is explicitly defined by α m , r : = z ⊗ m . Let f = ¯ z , then a simplecomputation of the Wick ordering gives ¯ h ∂∂ z ( z ) = ¯ h . It follows that b k , I = (cid:40) ( k , I ) = (
1, 0 ) ,0, otherwise .We can interpret Proposition 3.1 as follows. Let T f , m denote the Toeplitz operators on H ( C n , L ⊗ m ) associated to the function f , then the double sequence { T f , m ( α m , r ) } ∈ V canbe “approximated” by a vector in the image of F . To make this precise, we define thesubspace of admissible sequences V ⊂ V : EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 15
Definition 3.3.
We call α = { α m , r } ∈ V an admissible sequence if there exists b = ∑ k , I b k , I ¯ h k z I ∈F C n with F ( b ) = { β r , m } such that, for every fixed r >
0, we have m n · (cid:90) C n ( α m , r − β m , r ) · ¯ z J · e − m ·| z | = O (cid:18) m r + (cid:19) for any multi-index J .It is clear that there are inclusions F ( V ) ⊂ V ⊂ V . Furthermore, there is a naturalequivalence relation ∼ on V : If α i ∈ V , i =
1, 2 are two admissible sequences with b i their corresponding elements in F C n respectively, then we say α is equivalent to α if b = b ∈ F C n .3.2. Construction of the representation.
For a general compact K¨ahler manifold X equipped with a prequantum line bundle L ,we define V : = ∏ r ≥ (cid:32) ∏ m ≥ H ( X , L ⊗ m ) (cid:33) as before. The situation is more complicated than C n because L is non-trivial and thereare no obvious global holomorphic sections analoguous to z I ⊗ m on C n . The best re-placement (or approximation) for the polynomial sections on C n are given by so-called peak sections , with which we can define double sequences of holomorphic sections withasymptotic properties similar to equation (3.2): Definition 3.4.
For every point z ∈ X , we fix a set { S m , p , r } of normalized peak sec-tions centered at z , as introduced in Section A. A sequence of holomorphic sections α = { α m , r ∈ H ( X , L ⊗ m ) } , regarded as an element in V , is called an admissible sequence atz if it satisfies the following two conditions:(1) For every fixed r , the norm of the sequence { α m , r } m > has a uniform bound: || α m , r || m ≤ C r .(2) There is a sequence of complex numbers { a p , k } p , k ≥ such that, for each fixed r > (cid:104) α m , r − ∑ k + | p |≤ r a p , k · m k · S m , p , r + , S m , q , r + (cid:105) m = O (cid:18) m r + (cid:19) ,for any multi-index q with | q | ≤ r .We define the subspace V z ⊂ V as the C -linear span of admissible sequences at z .Equation (3.3) is the analogue of equation (3.2) in the flat case. According to LemmaA.9, the coefficients a p , k are uniquely determined. Remark . The index r in admissible sequences corresponds to the degree in the Wickalgebra and Bargmann-Fock space. The complex numbers { a p , k } are called the coefficients of the admissible sequence α .Note that they are independent of either the tensor power m and the weight index r . Thecoefficients define a natural equivalence relation ∼ on V z , namely, α is equivalent to β (denoted as α ∼ β ) if the coefficients of α − β are all 0. Remark . It follows from this definition that, for each fixed r , even if we change finitelymany terms of the double sequence { α m , r } , its equivalence class remains the same (cf.direct limits).The vector space we would like to construct is then simply the quotient by this equiva-lence relation: H z : = V z / ∼ .It follows from asymptotics of inner products of peak sections S m , p , r ’s as m → ∞ andequation (3.3) that H z is a formal Hilbert space. Remark . The vector space H z is defined as a sub-quotient, instead of just as a linearspan of peak sections. This is because in general the Toeplitz operators do not preservethe space of peak sections.We now give some examples of admissible sequences: Example 3.8.
Suppose we fix any multi-index q . Then we define an admissible sequence α as follows: we let α m , r : = m is too small and there is no (normalized) peak section of L ⊗ m corresponding to the index r , and let α m , r = S m , q , r + be simply the normalized peaksection. It is then easy to see that this is indeed an admissible sequence with coefficients a p , r = ( p , r ) (cid:54) = ( q , 0 ) and a q ,0 = Example 3.9.
We give an example of equivalent admissible sequences. Let α be the admis-sible sequence as in the previous example. We now construct a sequence also consistingof normalized peak sections similar to α but with a higher order error term. Namely, welet β = { β m , r } be the admissible sequence given by β m , r = S m , q , r + . It is easy to show that α ∼ β .It is not difficult to see that the admissible sequences in Example 3.8 form a basis of V z as a C -vector space, and thus every vector can be written as { a p , k } . More precisely, wehave the following lemma: Lemma 3.10.
We have the following isomorphism of C -vector spaces: (3.4) H z ∼ = C [[ y , · · · , y n ]][[ ¯ h ]] . Proof.
The above isomorphism is given by { a p , k } (cid:55)→ ∑ p , k a p , k · ¯ h k y p . (cid:3) Remark . The isomorphism in the above lemma depends on a choice of K -coordinatescentered at z , and thus is unique only up to a U ( n ) -transformation. EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 17
Now for every sequence of operators { A m } m ≥ , where A m ∈ End ( H ( X , L ⊗ m )) , wehave an obvious action on V : { α m , r } (cid:55)→ { A m ( α m , r ) } .We apply this to the sequence of Toeplitz operators { T f , m } m ≥ associated to any givensmooth function f ∈ C ∞ ( X ) . Lemma 3.12.
Suppose that α = { α m , r } is an admissible sequence. Then { T f , m ( α m , r ) } is also anadmissible sequence for any smooth function f .Proof. We consider the sequence T f ( α ) : = { T f , m ( α m , r ) } m , r ≥ . First of all, since the operatornorm of T f , m is bounded by || f || ∞ for any fixed r >
0, the sequence { T f , m ( α m , r ) } m > hasbounded norm.To show that T f ( α ) satisfies the second asymptotic property, we split the integral whichdefines that property into two parts: one inside the disk { ρ ( z ) < } and the other outside: m n (cid:90) X h m ( T f , m ( α m , r ) , S m , p , r + ) · dV g = m n (cid:90) X h m ( f · α m , r , S m , p , r + ) · dV g = m n (cid:90) X \{ ρ ( z ) < } h m ( f · α m , r , S m , p , r + ) · dV g + m n (cid:90) { ρ ( z ) < } h m ( f · α m , r , S m , p , r + ) · dV g ,where the first equality follows from the fact that T f , m ( α m , r ) is the orthogonal projection of f · α m , r to the space of holomorphic sections, and that S m , p , r + are all holomorphic sections.For the integral outside the disk, we have the following estimate: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m n (cid:90) X \{ ρ ( z ) < } h m ( f · α m , r , S m , p , r + ) · dV g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m n (cid:18) (cid:90) X \{ ρ ( z ) < } || f · α m , r || h m · dV g (cid:19) · (cid:18) (cid:90) X \{ ρ ( z ) < } || S m , p , r + || h m · dV g (cid:19) ≤|| f || ∞ · C r · O (cid:18) m ( r + + | p | ) /2 (cid:19) = O (cid:18) m r + (cid:19) .Here the multi-index p satisfies | p | ≤ r , and the constant C r is given by the upper boundof the sequence { α m , r } m > . For the second inequality, we have used boundedness of { α m , r } m > , and equation (A.9).Hence it remains to consider the integral inside the disk { ρ ( z ) < } . In this local setting,the computation is the same as the asymptotics of the Gaussian integral on D n , and alsothat in the formal Hilbert space. So the statement follows from Theorem 2.26. (cid:3) This lemma shows that for any smooth function f , the sequence of Toeplitz operators { T f , m } m > gives a well-defined linear operator T f : V z → V z , α (cid:55)→ T f ( α ) . Lemma 3.13.
Suppose that two admissible sequences are equivalent, i.e., α ∼ β . Then for anysmooth function f , we have T f ( α ) ∼ T f ( β ) .Proof. We only need to show that if the coefficients a k , I of α = { α m , r } vanish, then T f ( α ) has the same property. Notice that the condition a k , I = r , q , as m → ∞ : m n (cid:90) X h m ( α m , r , S m , q , r + ) · dV g = O (cid:18) m r + (cid:19) .Similar to the argument of Lemma 3.12, there is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m n (cid:90) X h m ( T f , m ( α m , r ) , S m , q , r + ) · dV g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m n (cid:90) X h m ( f · α m , r , S m , q , r + ) · dV g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m n · || f || ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) X h m ( α m , r , S m , q , r + ) · dV g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) m r + (cid:19) . (cid:3) Hence, for every smooth function f on X , the sequence of Toeplitz operators { T f , m } m > gives a well-defined linear operator T f on the vector space H z . We can further extend itto an action of C ∞ ( X )[[ ¯ h ]] on H z by letting ¯ h k · f act as T ¯ h k · f : { α m , r } (cid:55)→ (cid:26) m k · T f , m ( α m , r ) (cid:27) .Lemma 2.25 implies the following: Proposition 3.14.
Suppose f ∈ C ∞ ( X ) is a real function. Then for every z ∈ X, the operatorT f on H z is self-adjoint. Lemma 3.15.
Let A = { A m } m ≥ and B = { B m } m ≥ be two sequences of bounded operatorspreserving asymptotic sequences and satisfying the condition that (3.5) || A m − B m || = O (cid:18) m k + (cid:19) . Then for any admissible sequence α , the two admissible sequences A ( α ) and B ( α ) have the samecoefficients up to weight k.Proof. Equation (3.5) implies that the operators { A m − B m } will increase the weight of α by k +
1. The lemma follows. (cid:3)
Here is our main theorem:
EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 19
Theorem 3.16.
Let z ∈ X be any point. The action of C ∞ ( X )[[ ¯ h ]] on the vector space H z satisfies the following relation: (3.6) T f ◦ T g = T f g + ∑ k ≥ ¯ h k · T C k ( f , g ) , f , g ∈ C ∞ ( X ) , where C k ( − , − ) are the bi-differential operators which appear in the Berezin-Toeplitz quantiza-tion. Therefore, H z is a representation of the Berezin-Toeplitz deformation quantization algebra ( C ∞ ( X )[[ ¯ h ]] , (cid:63) BT ) .Proof. We first recall the property of Toeplitz operators: || T f , m ◦ T g , m − ( T f g , m + n ∑ k = m k · T C k ( f , g ) , m ) || = O (cid:18) m n + (cid:19) .We apply Lemma 3.15 by putting A m : = T f , m ◦ T g , m and B m : = T f g + ∑ nk = m k · T C k ( f , g ) , m .Then A ( α ) and B ( α ) have the same coefficients up to order n . The theorem follows byletting n → ∞ . (cid:3) Remark . The representation and also the isomorphism (3.4) are independent of thechoice of the set of peak sections because for every multi-index p and p (cid:48) > | p | , differentchoices of peak sections only differ by higher order terms.3.3. Locality and irreducibility of the representation.
Locality.
We will give an explicit formula of our representation under the isomor-phism (3.4). Given any K -coordinates ( z , · · · , z n ) centered at z , we define J f , z ∈ W C n by(3.7) J f , z : = ∑ | I | , | J |≥ I ! J ! ∂ | I | + | J | f ∂ z I ¯ z J ( z ) y I ¯ y J ,where the sum is over all multi-indices. Theorem 3.18.
Let f be any smooth function on X, and J f , z be defined as above. We defineO f , z ∈ W C n as the unique solution of the following equation:J f , z · e Φ /¯ h = e Φ /¯ h (cid:63) O f , z . Then the action of T f on α ∈ H z is given byT f ( α ) = O f , z (cid:63) α . In particular, this implies that the representation H z is local in f ∈ C ∞ ( X ) , i.e., it only dependson the infinite jets of f at z .Proof. The representation H z depends on an arbitrarily small neighborhood of z . So theresult follows from the computation of the formal Toeplitz operators on C n in Theorem2.22. (cid:3) As a straightforward corollary, we have:
Corollary 3.19.
Let f , g ∈ C ∞ ( X ) be smooth functions on X. ThenO f (cid:63) BT g , z = O f , z (cid:63) O g , z .This gives an algorithm for computing f (cid:63) BT g : for every z ∈ X , in order to find ( f (cid:63) BT g )( z ) , we only need to compute the Wick product O f , z (cid:63) O g , z and then collect all theconstant terms in the Wick algebra.3.3.2. Irreducibility.
We now consider irreducibility of our representation. The first obser-vation is that the Bargmann-Fock space is not an irreducible representation of W C n : forevery f ∈ W C n , we have the invariant subspaces T f (( F C n ) k ) ⊂ ( F C n ) k .But this is the only reason why the respresentation fails to be irreducible. So to have asuitable notion of irreducibility, we make use of the extended algebra W + C n , which allowsterms with negative degrees, and the corresponding extension F + C n . It is then quite easyto check that we indeed obtain an irreducible representation.We now define an extension of C ∞ ( X )[[ ¯ h ]] , which is the geometric analogue of W + C n . Definition 3.20.
For every smooth function f ∈ C ∞ ( X ) , let deg z ( f ) be the vanishingorder of f at z . Then let ( C ∞ ( X )[[ ¯ h ]]) + z be the extension of C ∞ ( X )[[ ¯ h ]] which consists offormal functions: ∑ i ∈ Z ¯ h i · f i ,where f i ∈ C ∞ ( X ) are smooth functions on X satisfying the conditions that • the sum deg z ( f i ) + i has a uniform lower bound for all i , and • for every degree k , the following expression is a finite sum: ∑ i + deg z ( f i )= k ¯ h i · f i .In the same way we can define the extension H z ⊂ H + z . It is easy to check that the ex-tension ( C ∞ ( X )[[ ¯ h ]]) + z is closed under the star product (cid:63) BT , and acts on H + z . Furthermore,the map f (cid:55)→ O f , z can be extended to(3.8) ( C ∞ ( X )[[ ¯ h ]]) + z → W + C n . Theorem 3.21.
For every z ∈ X, the representation H + z of ( C ∞ ( X )[[ ¯ h ]]) + z is irreducible.Proof. Let W be a sub-representation of H + z . We choose any non-zero a ∈ W , which canbe written as: a = ∑ i + | I |≥ k a i , I ¯ h i · y I . EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 21
Since for a local holomorphic function f , we have O f , z = J f , z ,which consists of only creators in ( H z ) + , we only need to find f ∈ ( C ∞ ( X )[[ ¯ h ]]) + z suchthat T f ( a ) = ¯ h l for some l , and the result will follow.We choose a non-zero term in a of leading order a i , I ¯ h i · z I , 2 i + I = k , such that i isthe least possible. Let f ∈ ( C ∞ ( X )[[ ¯ h ]]) + z be a formal function which is a − i , I ¯ z I near z .So the leading term of the image of f under the map (3.8) is a − i , I ¯ y I , and the degree ofthe function f is exactly | I | . We have T f (cid:18) ∑ i + | I |≥ k a i , I ¯ h i · y I (cid:19) = ¯ h i + | I | + ∑ i + | I | = ( i + | I | )+ b i , I ¯ h i y I + higher degree terms.The next step is to find a formal function f ∈ ( C ∞ ( X )[[ ¯ h ]]) + z , so that ( T f + T f ) (cid:18) ∑ i + | I |≥ k a i , I ¯ h i · y I (cid:19) = ¯ h i + | I | + higher degree terms.Let g be a formal function which equals − h i + | I | ∑ j + | J | = ( i + | I | )+ b j , J ¯ h j z J near z . It iseasy to see that the total degree of g is 1, and we have ( T f + T g ◦ T f ) (cid:18) ∑ i + | I |≥ k a i , I ¯ h i · y I (cid:19) = ¯ h i + | I | + ( terms of degree ≥ ( i + | I | ) + )) .Although T g ◦ T f = ∑ i ≥ ¯ h i T C i ( g , f ) is an infinite sum, but those high enough ¯ h will mapterms in a to terms of high degree. More precisely, we can simply let f = ∑ Ni = ¯ h i C i ( g , f ) ,where N = i + | I | − k /2 +
2. Then we have ( T f + T f ) (cid:18) ∑ i + | I |≥ k a i , I ¯ h i · y I (cid:19) = ¯ h i + | I | + ( terms of degree ≥ ( i + | I | ) + )) .From the formula of (cid:63) BT , it is easy to see that the total degree of ¯ h i C i ( g , f ) is no less thanthe sum of the degree of g and f which equals 1 + | I | . This implies that the degree ofterms in f is strictly greater than the degree of f . This procedure can be repeated andwe obtain the desired formal function f = ∑ i ≥ f i ∈ ( C ∞ ( X )[[ ¯ h ]]) + z . (cid:3) A PPENDIX
A. P
EAK SECTIONS IN
K ¨
AHLER GEOMETRY
In this appendix, we briefly review of the notion of peak sections, which was intro-duced in [31] and plays an important role in the study of asymptotic expansions of theBergmann kernel [33] with applications to balanced embeddings and constant scalar cur-vature metrics as well as in the theory of the geometric quantization. For our purpose, we need to introduce a normalized version of peak sections and describe their basic prop-erties. We first recall the notion of K -coordinates and K -frame of the prequantum linebundle L . Definition A.1.
Let e L , z be a holomorphic frame of the prequantum line bundle L in aneighborhood of a point z ∈ X , and let ( z , · · · , z n ) be a holomorphic coordinate systemcentered at z . Let ϕ ( z ) : = − log || e L , z || . We say that ( z , · · · , z n ) are K-coordinates withK-frame e L , z if the Taylor expansion of ϕ ( z ) at z is of the following form:(A.1) ϕ ( z ) ∼ | z | + ∑ a JK z J ¯ z K , | J | ≥ | K | ≥ K -coordinatesand K -frames was shown by Bochner. For K¨ahler manifolds with only smooth K ¨ahlerform, such a coordinate system and frame do not exist in general; then we may consider aweaker K -coordinates and K -frames of finite order. But to avoid further technical compli-cations, let us assume that the K¨ahler manifolds in this paper always admit K -coordinatesand K -frames.It is obvious that this local holomorphic frame e L , z is unique up to a multiplication bya complex number of modulus 1. In particular, the leading term of the Taylor expansionof ϕ with degree at least 3 is given by the curvature:(A.2) ϕ ( z , ¯ z ) = | z | + ∑ i , j , k , l R i ¯ jk ¯ l z i z k ¯ z j ¯ z l + O ( | z | ) .Also note that equation (2.9) is satisfied. Lemma A.2.
Suppose the volume form is ( √− ) n · e ψ ( z , ¯ z ) · dz · · · dz n d ¯ z · · · d ¯ z n = ω n . Thenthe purely (anti-)holomorphic derivatives of ψ ( z , ¯ z ) vanish at z under the K-coordinates, i.e., ∂ | I | ψ∂ z I ( z ) = ∂ | J | ψ∂ ¯ z I ( z ) = for all multi-indices with | I | , | J | > .Proof. We will only prove the vanishing of purely holomorphic derivatives at z ; the prooffor antiholomorphic ones is the same. It suffices to show that the statement is valid forfunctions ω i ¯ j · · · ω i n ¯ j n , where ω i ¯ j = ∂ ϕ∂ z i ∂ ¯ z j ,and φ is K¨ahler potential. But then equation (A.1) implies that ∂ | I | + ϕ∂ z I ∂ ¯ z j ( z ) = | I | ≥ (cid:3) EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 23
Equation (A.1) together with Lemma A.2 tell us that the Taylor expansions of ϕ ( z , ¯ z ) and ψ ( z , ¯ z ) satisfy the technical conditions in Theorem 2.26. This allows us to apply thealgebraic computations in the formal setting in Section 2.2.We now recall the following proposition in Tian’s paper [31] and, in particular, thedefinition of peak sections. Let ( z , · · · , z n ) be a K -coordinate with K -frame e L , z at z ∈ X ,and consider the function ρ ( z ) : = (cid:112) | z | + · · · + | z n | . Proposition A.3 (Lemma 1.2 in [31]) . For an n-tuple of integers p = ( p , · · · , p n ) ∈ Z n + andan integer r > | p | = p + · · · + p n , there exists an m > such that, for m > m , there is aholomorphic global section S, called a peak section , of the line bundle L ⊗ m , satisfying (A.3) (cid:90) X || S || h m dV g = (cid:90) X \{ ρ ( z ) ≤ log m √ m } || S || h m dV g = O (cid:18) m r (cid:19) , and locally at z , (A.4) S ( z ) = λ m , p · (cid:16) z p · · · z p n n + O ( | z | r ) (cid:17) e mL , z (cid:18) + O (cid:18) m r (cid:19)(cid:19) , where || · || h m is the norm on L ⊗ m given by h m , and O (cid:16) m r (cid:17) denotes a quantity dominated byC / m r with the constant C depending only on r and the geometry of X, moreover (A.5) λ − m , p = (cid:90) ρ ( z ) ≤ log m / √ m | z p · · · z p n n | · e − m · ϕ ( z ) dV g , where dV g = det ( g i ¯ j )( √− ( π )) n dz ∧ d ¯ z ∧ · · · ∧ dz n ∧ d ¯ z n is the volume form. Here we use the same notation as in the introduction of K -frame: h m ( e L , z , e L , z ) = e − m · ϕ ( z ) . Geometrically, a peak section is, roughly speaking, a global holomorphic sectionof a high enough tensor power of L whose norm is 1 and is concentrated around a givenpoint z on the K¨ahler manifold.We want to define a section S m , p , r of the line bundle L ⊗ m by normalizing the peaksection S ( z ) in Proposition A.3 so that its Taylor expansion at z under the K-frame e ⊗ mL , z is exactly equal to z p · · · z p n n up to order 2 r −
1. This forces S m , p , r to be of the form:(A.6) S m , p , r : = λ − m , p · (cid:18) + O (cid:18) m r (cid:19)(cid:19) · S ( z ) .We now give an estimate of λ − m , p . We have, for m >> λ − m , p = (cid:90) ρ ( z ) ≤ log m / √ m | z p · · · z p n n | · e − m · ρ ( z ) dV g ≤ (cid:90) ρ ( z ) ≤ | z p · · · z p n n | · e − m · ρ ( z ) dV g = O (cid:18) m | p | + n (cid:19) , where the estimate follows from Theorem 2.11. In particular, there is a constant C p , de-pending only on the point z and the multi-index p , such that λ − m , p · (cid:18) + O (cid:18) m r (cid:19)(cid:19) ≤ C p · (cid:32) m | p | + n (cid:33) .We define a normalized version of the inner product of sections of L ⊗ m : Definition A.4.
Let s , s be (smooth) sections of L ⊗ m . Their (normalized) inner productis defined as(A.7) (cid:104) s , s (cid:105) m : = m n · (cid:90) X h m ( s , s ) dV g ,where n = dim C X , and we let || s || m be the norm of a section s under this inner product. Remark
A.5 . An explanation of the normalization factor m n is the following: consider C n with the volume form (cid:32) √− π (cid:33) n e − m ·| z | dz ∧ d ¯ z ∧ · · · ∧ dz n ∧ d ¯ z n ,then the factor m n normalizes the volume to 1 under this volume form.We summarize the properties of S m , p , r as follows:(A.8) || S m , p , r || m ≤ m n · C · λ − m , p = O (cid:18) m | p | (cid:19) ;(A.9) m n · (cid:90) M \{ ρ ( z ) ≤ } || S m , p , r || h m dV g ≤ m n · (cid:90) M \{ ρ ( z ) ≤ log m / √ m } || S m , p , r || h m dV g ≤ m n · C · λ − m , p · O (cid:18) m r (cid:19) = O (cid:18) m r + | p | (cid:19) . Remark
A.6 . The constant C in the above estimates comes from the number 1 + O (cid:16) m r (cid:17) in equation (A.6).Locally around z , we have(A.10) S m , p , r ( z ) = (cid:16) z p · · · z p n n + O ( | z | r ) (cid:17) · e mL .The first property (A.8) implies that for fixed p , r , the sequence { S m , p , r } is bounded forall m . The second property (A.9) is saying that the sections S m , p , r are asymptotically “con-centrated” around the point z . The third property (A.10) is saying that asymptotically, S m , p , r has an assigned leading term of the Taylor expansion at the point z . Remark
A.7 . The third property of S m , p , r is the reason for calling it a normalized peak section :its Taylor expansion at z has leading term exactly exactly equal to the monomial z p · e mL corresponding to the multi-index p . EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 25
Remark
A.8 . According to [31], for every fixed p , r , peak sections exist only when m is bigenough. We will adopt the follows two conventions • S m , p , r : = m , • S m , p , r : = r ≤ | p | .There is the following estimate of the inner product between peak sections: Lemma A.9.
Given two normalized peak sections S m , p , r , S m , p , r , we have the following asymp-totic expansion of their inner product up to order r: (A.11) (cid:104) S m , p , r , S m , p , r (cid:105) m − p (cid:48) − ∑ k = a k · m k = O (cid:18) m p (cid:48) (cid:19) , where the coefficients a k ’s are the same as those in the expansion of the following formal integral: (cid:90) z p ¯ z p e −| z | + φ ( z ,¯ z ) ¯ h = ∑ k ≥ a k · ¯ h k . Thus, for fixed multi-indices p , p , the a i ’s are independent of r >> . In particular, the leadingterm of the asymptotic expansion of || S m , p , r || m is given by (cid:18) m (cid:19) | p | p !, which is non-zero.Proof. We split the integral defining the inner product to two parts:1 m n (cid:90) { ρ ( z ) < } h m ( S m , p , r , S m , p , r ) · dV g + m n (cid:90) X \{ ρ ( z ) < } h m ( S m , p , r , S m , p , r ) · dV g ,where the second part is O (cid:32) m r + | p | + | p | (cid:33) by using Cauchy-Schwarz inequality and equa-tion (A.9). Thus to show equation (A.11), the integral outside the disk { ρ ( z ) < } can beignored. For the integral over the disk, we can apply Theorem 2.11 to obtain the desiredasymptotic expansion. In particular, the coefficients a k ’s are the same as those comingfrom the formal integral. (cid:3) As an immediate corollary, we have the following:
Corollary A.10.
Let p , p be multi-indices, and let r > max {| p | , | p |} , then we have thefollowing estimate of the inner product between S m , p , r and S m , p , r : (cid:104) S m , p , r , S m , p , r (cid:105) m = O (cid:16) m | p | (cid:17) , p = p o (cid:16) m max {| p | , | p |} (cid:17) , p (cid:54) = p . Proof.
The case where p = p is given by equation (A.8). For p (cid:54) = p , we need toestimate an integral. For the integral inside the disk { ρ ( z ) < } , this estimate is givenby Corollary 2.21 where the technical condition on φ is implied by the existence of K -coordinates and K -frame. For the estimate of the integral outside the disk, we use Cauchy-Schwarz inequality:(A.12) m n · (cid:90) X \{ ρ ( z ) < } h m ( S m , p , r , S m , p , r ) · dV g = O (cid:32) m r + | p | + | p | (cid:33) = o (cid:18) m max {| p | , | p |} (cid:19) . (cid:3) R EFERENCES [1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer,
Deformation theory and quantiza-tion. I. Deformations of symplectic structures , Ann. Physics (1978), no. 1, 61–110.[2] ,
Deformation theory and quantization. II. Physical applications , Ann. Physics (1978), no. 1, 111–151.[3] M. Bordemann, E. Meinrenken, and M. Schlichenmaier,
Toeplitz quantization of K¨ahler manifolds and gl ( N ) , N → ∞ limits , Comm. Math. Phys. (1994), no. 2, 281–296.[4] M. Bordemann and S. Waldmann, A Fedosov star product of the Wick type for K¨ahler manifolds , Lett. Math.Phys. (1997), no. 3, 243–253.[5] , Formal GNS construction and states in deformation quantization , Comm. Math. Phys. (1998),no. 3, 549–583.[6] K. Chan, N. C. Leung, and Q. Li. in preparation.[7] K. Costello,
Renormalization and effective field theory , Mathematical Surveys and Monographs, vol. 170,American Mathematical Society, Providence, RI, 2011.[8] ,
A geometric construction of the Witten genus, II , available at arXiv:1111.4234[math.QA] .[9] ,
Renormalization and the Batalin-Vilkovisky formalism , available at arXiv:0706.1533[math.QA] .[10] S. K. Donaldson,
Planck’s constant in complex and almost-complex geometry , XIIIth International Congresson Mathematical Physics (London, 2000), 2001, pp. 63–72.[11] B. V. Fedosov,
A simple geometrical construction of deformation quantization , J. Differential Geom. (1994),no. 2, 213–238.[12] , Deformation quantization and index theory , Mathematical Topics, vol. 9, Akademie Verlag, Berlin,1996.[13] M. Kapranov,
Rozansky-Witten invariants via Atiyah classes , Compositio Math. (1999), no. 1, 71–113.[14] A.V. Karabegov,
Deformation quantizations with separation of variables on a K¨ahler manifold , Comm. Math.Phys. (1996), no. 3, 745–755.[15] ,
On Fedosov’s approach to deformation quantization with separation of variables , Conf´erence Mosh´eFlato 1999, Vol. II (Dijon), 2000, pp. 167–176.[16] ,
A formal model of Berezin-Toeplitz quantization , Comm. Math. Phys. (2007), no. 3, 659–689.[17] A.V. Karabegov and M. Schlichenmaier,
Identification of Berezin-Toeplitz deformation quantization , J. ReineAngew. Math. (2001), 49–76.[18] A. A. Kirillov,
Geometric quantization , Current problems in mathematics. Fundamental directions, Vol.4, 1985, pp. 141–178, 291.[19] M. Kontsevich,
Feynman diagrams and low-dimensional topology , First European Congress of Mathemat-ics, Vol. II (Paris, 1992), 1994, pp. 97–121.[20] ,
Deformation quantization of Poisson manifolds , Lett. Math. Phys. (2003), no. 3, 157–216. EOMETRIC REPRESENTATIONS OF THE BT QUANTIZATION 27 [21] B. Kostant,
Quantization and unitary representations. I. Prequantization , Lectures in modern analysis andapplications, III, 1970, pp. 87–208. Lecture Notes in Math., Vol. 170.[22] Z. Lu and B. Shiffman,
Asymptotic expansion of the off-diagonal Bergman kernel on compact K¨ahler manifolds ,J. Geom. Anal. (2015), no. 2, 761–782.[23] X. Ma and G. Marinescu, Toeplitz operators on symplectic manifolds , J. Geom. Anal. (2008), no. 2, 565–611.[24] , Berezin-Toeplitz quantization on K¨ahler manifolds , J. Reine Angew. Math. (2012), 1–56.[25] P. Mnev,
Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory ,available at arXiv:1707.08096[math-ph] .[26] N. Neumaier,
Universality of Fedosov’s construction for star products of Wick type on pseudo-K¨ahler mani-folds , Rep. Math. Phys. (2003), no. 1, 43–80.[27] N. Reshetikhin and L. A. Takhtajan, Deformation quantization of K¨ahler manifolds , L. D. Faddeev’s Semi-nar on Mathematical Physics, 2000, pp. 257–276.[28] M. Schlichenmaier,
Deformation quantization of compact K¨ahler manifolds by Berezin-Toeplitz quantization ,Conf´erence Mosh´e Flato 1999, Vol. II (Dijon), 2000, pp. 289306.[29] J. ´Sniatycki,
Geometric quantization and quantum mechanics , Applied Mathematical Sciences, vol. 30,Springer-Verlag, New York-Berlin, 1980.[30] J.-M. Souriau,
Structure des syst`emes dynamiques , Maˆıtrises de math´ematiques, Dunod, Paris, 1970.[31] G. Tian,
On a set of polarized K¨ahler metrics on algebraic manifolds , J. Differential Geom. (1990), no. 1,99–130.[32] N. M. J. Woodhouse, Geometric quantization , Second, Oxford Mathematical Monographs, The Claren-don Press, Oxford University Press, New York, 1992. Oxford Science Publications.[33] S. Zelditch,
Szego kernels and a theorem of Tian , Internat. Math. Res. Notices (1998), 317–331.D EPARTMENT OF M ATHEMATICS , T HE C HINESE U NIVERSITY OF H ONG K ONG , S
HATIN , H
ONG K ONG
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HATIN , H
ONG K ONG
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OUTHERN U NIVERSITY OF S CIENCE AND T ECHNOLOGY , S
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