Review Article Berezin-Toeplitz Quantization for Compact Kähler Manifolds. A Review of Results

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1 Advances in Mathematical Physics Volume 2010, Article ID , 38 pages doi: /2010/ Review Article Berezin-Toeplitz Quantization or Compact Kähler Maniolds. A Review o Results Martin Schlichenmaier Mathematics Research Unit, University o Luxembourg, FSTC, 6, rue Coudenhove-Kalergi 1359 Luxembourg-Kirchberg, Luxembourg Correspondence should be addressed to Martin Schlichenmaier, martin.schlichenmaier@uni.lu Received 30 December 2009; Accepted 5 March 2010 Academic Editor: S. T. Ali Copyright q 2010 Martin Schlichenmaier. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deormation quantization or compact quantizable Kähler maniolds. The basic objects, concepts, and results are given. This concerns the correct semiclassical limit behaviour o the operator quantization, the unique Berezin- Toeplitz deormation quantization star product, covariant and contravariant Berezin symbols, and Berezin transorm. Other related objects and constructions are also discussed. 1. Introduction For quantizable Kähler maniolds the Berezin-Toeplitz BT quantization scheme, both the operator quantization and the deormation quantization, supplies canonically deined quantizations. Some time ago, in joint work with Martin Bordemann and Eckhard Meinrenken, the author o this review showed that or compact Kähler maniolds it is a welldeined quantization scheme with correct semiclassical limit 1. What makes the Berezin-Toeplitz quantization scheme so attractive is that it does not depend on urther choices and that it does not only produce a ormal deormation quantization, but one which is deeply related to some operator calculus. From the point o view o classical mechanics, compact Kähler maniolds appear as phase space maniolds o restricted systems or o reduced systems. A typical example o its appearance is given by the spherical pendulum which ater reduction has as phase-space the complex projective space. Very recently, inspired by ruitul applications o the basic techniques o the Berezin- Toeplitz scheme beyond the quantization o classical systems, the interest in it revived considerably.

2 2 Advances in Mathematical Physics For example, these techniques show up in a noncommutative geometry. More precisely, they appear in the approach to noncommutative geometry using uzzy maniolds. The quantum spaces o the BT quantization o level m, deined in Section 3 urther down, are inite-dimensional, and the quantum operator o level m constitutes inite-dimensional noncommutative matrix algebras. This is the arena o noncommutative uzzy maniolds and gauge theories over them. The classical limit, the commutative maniold, is obtained as limit m. The name uzzy sphere was coined by Madore 2 or a certain quantized version o the Riemann sphere. It turned out to be a quite productive direction in the noncommutative geometry approach to quantum ield theory. It is impossible to give a rather complete list o people working in this approach. The ollowing is a rather erratic and random choice o Another appearance o Berezin-Toeplitz quantization techniques as basic ingredients is in the pioneering work o Jørgen Andersen on the mapping class group MCG o suraces in the context o Topological Quantum Field Theory TQFT. Beside other results, he was able to prove the asymptotic aithulness o the mapping group action on the space o covariantly constant sections o the Verlinde bundle with respect to the Axelrod-Witten-de la Pietra and Witten connection 11, 12 ;seealso 13. Furthermore, he showed that the MCG does not have Kazhdan s property T. Roughly speaking, a group which has property T says that the identity representation is isolated in the space o all unitary representations o the group 14. In these applications, the maniolds to be quantized are the moduli spaces o certain lat connections on Riemann suraces or, equivalently, the moduli space o stable algebraic vector bundles over smooth projective curves. Here urther exciting research is going on, in particular, in the realm o TQFT and the construction o modular unctors In general, quite oten moduli spaces come with a Kähler structure which is quantizable. Hence, it is not surprising that the Berezin-Toeplitz quantization scheme is o importance in moduli space problems. Noncommutative deormations and a quantization being a noncommutative deormation, yield also inormation about the commutative situation. These aspects clearly need urther investigations. There are a lot o other applications on which work has already been done, recently started, or can be expected. As the Berezin-Toeplitz scheme has become a basic tool, this seems the right time to collect the techniques and results in such a review. We deliberately concentrate on the case o compact Kähler maniolds. In particular, we stress the methods and results valid or all o them. Due to space-time limitations, we will not deal with the noncompact situation. In this situation, case by case studies o the examples or class o examples are needed. See Section 3.7 or reerences to some o them. Also we have to skip presenting recent attempts to deal with special singular situations, like orbiolds, but see at least O course, there are other reviews presenting similar quantization schemes. A very incomplete list is the ollowing This review is sel-contained in the ollowing sense. I try to explain all notions and concepts needed to understand the results and theorems only assuming some background in modern geometry and analysis. And as such it should be accessible or a newcomer to the ield both or mathematicians as or physicists and help him to enter these interesting research directions. It is not sel-contained in the strict sense as it does supply only those proos or sketches o proos which are either not available elsewhere or are essential or the understanding o the statements and concepts. The review does not require a background in quantum physics as only mathematical aspects o quantizations are touched on.

3 Advances in Mathematical Physics 3 2. The Set-Up o Geometric Quantization In the ollowing, I will recall the principal set-up o geometric quantization which is usually done or symplectic maniolds in the case when the maniold is a Kähler maniold Kähler Maniolds We will only consider phase-space maniolds which carry the structure o a Kähler maniold M, ω. Recall that in this case M is a complex maniold and ω, thekähler orm, is a nondegenerate closed positive 1, 1 -orm. I the complex dimension o M is n, then the Kähler orm ω can be written with respect to local holomorphic coordinates {z i } i 1,...,n as n ω i g ij z dz i dz j, i,j with local unctions g ij z such that the matrix g ij z i,j 1,...,n is hermitian and positive deinite. Later on we will assume that M is a compact Kähler maniold Poisson Algebra Denote by C M the algebra o complex-valued arbitrary oten dierentiable unctions with the point-wise multiplication as an associative product. A symplectic orm on a dierentiable maniold is a closed nondegenerate 2-orm. In particular, we can consider our Kähler orm ω as a symplectic orm. For symplectic maniolds, we can introduce on C M a Lie algebra structure, the Poisson bracket Poisson bracket {, }, in the ollowing way. First we a assign to every C M its Hamiltonian vector ield X, and then to every pair o unctions and g the Poisson bracket {, } via ω ( X, ) d, {, g } : ω ( X,X g ). 2.2 One veriies that this deines a Lie algebra structures and that urthermore, we have the Leibniz rule { g,h } { g,h } {, h } g,, g, h C M. 2.3 Such a compatible structure is called a Poisson algebra.

4 4 Advances in Mathematical Physics 2.3. Quantum Line Bundles A quantum line bundle or a given symplectic maniold M, ω is a triple L, h,, where L is a complex line bundle, h a Hermitian metric on L, and a connection compatible with the metric h such that the pre quantum condition curv L, X, Y : X Y Y X X,Y i ω X, Y, resp., curv L, i ω 2.4 is ulilled. A symplectic maniold is called quantizable i there exists a quantum line bundle. In the situation o Kähler maniolds, we require or a quantum line bundle to be holomorphic and that the connection is compatible both with the metric h and the complex structure o the bundle. In act, by this requirement will be uniquely ixed. I we choose local holomorphic coordinates on the maniold and a local holomorphic rame o the bundle, the metric h will be represented by a unction ĥ. In this case, the curvature in the bundle can be given by log ĥ and the quantum condition reads as i log ĥ ω Example: The Riemann Sphere The Riemann sphere is the complex projective line P 1 C C { } S 2. With respect to the quasiglobal coordinate z, the orm can be given as i ω dz dz zz For the Poisson bracket, one obtains { }, g i 1 zz 2( z g z ) g. 2.7 z z Recall that the points in P 1 C correspond to lines in C 2 passing through the origin. I we assign to every point in P 1 C the line it represents, we obtain a holomorphic line bundle, called the tautological line bundle. The hyper plane section bundle is dual to the tautological bundle. It turns out that it is a quantum line bundle. Hence P 1 C is quantizable.

5 Advances in Mathematical Physics Example: The Complex Projective Space Next we consider the n-dimensional complex projective space P n C. The example above can be extended to the projective space o any dimension. The Kähler orm is given by the Fubini- Study orm ω FS : i ( 1 w 2) n i 1 dw i dw i n i,j 1 w iw j dw i dw j (1 w 2) The coordinates w j, j 1,...,n, are aine coordinates w j z j /z 0 on the aine chart U 0 : { z 0 : z 1 : : z n z 0 / 0}. Again, P n C is quantizable with the hyper plane section bundle as a quantum line bundle Example: The Torus The complex- one-dimensional torus can be given as M C/Γ τ, where Γ τ : {n mτ n, m Z} is a lattice with Im τ>0. As Kähler orm, we take ω i π dz dz, Im τ 2.9 with respect to the coordinate z on the covering space C. Clearly this orm is invariant under the lattice Γ τ and hence well deined on M. For the Poisson bracket, one obtains ( { } Im τ, g i π z g z ) g z z The corresponding quantum line bundle is the theta line bundle o degree 1, that is, the bundle whose global sections are scalar multiples o the Riemann theta unction Example: The Unit Disc and Riemann Suraces The unit disc D : {z C z < 1} 2.11 is a noncompact Kähler maniold. The Kähler orm is given by 2i ω dz dz zz

6 6 Advances in Mathematical Physics Every compact Riemann surace M o genus g 2 can be given as the quotient o the unit disc under the ractional linear transormations o a Fuchsian subgroup o SU 1, 1. IR a/b b/a with a 2 b 2 1 as an element o SU 1, 1, then the action is z R z : az b bz a The Kähler orm 2.12 is invariant under the ractional linear transormations. Hence it deines a Kähler orm on M. The quantum bundle is the canonical bundle, that is, the bundle whose local sections are the holomorphic dierentials. Its global sections can be identiied with the automorphic orms o weight 2 with respect to the Fuchsian group Consequences o Quantizability The above examples might create the wrong impression that every Kähler maniold is quantizable. This is not the case. For example, only those higher-dimensional tori complex tori are quantizable which are abelian varieties, that is, which admit enough theta unctions. It is well known that or n 2 a generic torus will not be an abelian variety. Why this implies that they will not be quantizable, we will see in a moment. In the language o dierential geometry, a line bundle is called a positive line bundle i its curvature orm up to a actor o 1/i is a positive orm. As the Kähler orm is positive, the quantum condition 2.4 yields that a quantum line bundle L is a positive line bundle Embedding into Projective Space In the ollowing, we assume that M is a quantizable compact Kähler maniold with quantum line bundle L. Kodaira s embedding theorem says that L is ample, that is, that there exists a certain tensor power L m 0 o L such that the global holomorphic sections o L m 0 can be used to embed the phase space maniold M into the projective space o suitable dimension. The embedding is deined as ollows. Let Γ hol M, L m 0 be the vector space o global holomorphic sections o the bundle L m 0. Fix a basis s 0,s 1,...,s N. We choose local holomorphic coordinates z or M and a local holomorphic rame e z or the bundle L. Ater these choices, the basis elements can be uniquely described by local holomorphic unctions ŝ 0, ŝ 1,...,ŝ N deined via s j z ŝ j z e z. The embedding is given by the map M P N C, z φ z ŝ 0 z : ŝ 1 z : : ŝ N z Note that the point φ z in projective space neither depends on the choice o local coordinates nor on the choice o the local rame or the bundle L. Furthermore, a dierent choice o basis corresponds to a PGL N, C action on the embedding space and hence the embeddings are projectively equivalent. By this embedding, quantizable compact Kähler maniolds are complex submaniolds o projective spaces. By Chow s theorem 26, they can be given as zero sets o homogenous polynomials, that is, they are smooth projective varieties. The converse is also true. Given

7 Advances in Mathematical Physics 7 a smooth subvariety M o P n C, it will become a Kähler maniold by restricting the Fubini- Study orm. The restriction o the hyper plane section bundle will be an associated quantum line bundle. At this place a warning is necessary. The embedding is only an embedding as complex maniolds are not an isometric embedding as Kähler maniolds. This means that in general φ 1 ω FS / ω.seesection 7.6 or results on an asymptotic expansion o the pullback. A line bundle, whose global holomorphic sections will deine an embedding into projective space, is called a very ample line bundle. In the ollowing, we will assume that L is already very ample. I L is not very ample, we choose m 0 N such that the bundle L m 0 is very ample and take this bundle as quantum line bundle with respect to the rescaled Kähler orm m 0 ω on M. The underlying complex maniold structure will not change. 3. Berezin-Toeplitz Operators In this section, we will consider an operator quantization. This says that we will assign to each dierentiable dierentiable will always mean dierentiable to any order unction on our Kähler maniold M i.e., on our phase space the Berezin-Toeplitz BT quantum operator T. More precisely, we will consider a whole amily o operators T m. These operators are deined in a canonical way. As we know rom the Groenewold-van Howe theorem, we cannot expect that the Poisson bracket on M can be represented by the Lie algebra o operators i we require certain desirable conditions see 27 or urther details. The best we can expect is that we obtain it at least asymptotically. In act, this is true. In our context also the operator o geometric quantization exists. At the end o this section, we will discuss its relation to the BT quantum operator. It will turn out that i we take or the geometric quantization the Kähler polarization then they have the same asymptotic behaviour Tensor Powers o the Quantum Line Bundle Let M, ω be a compact quantizable Kähler maniold and L, h, a quantum line bundle. We assume that L is already very ample. We consider all its tensor powers ( L m,h m, m ). 3.1 Here L m : L m.iĥ corresponds to the metric h with respect to a local holomorphic rame e o the bundle L, then ĥm corresponds to the metric h m with respect to the rame e m or the bundle L m. The connection m will be the induced connection. We introduce a scalar product on the space o sections. In this review, we adopt the convention that a hermitian metric and a scalar product is antilinear in the irst argument and linear in the second argument. First we take the Liouville orm Ω 1/n! ω n as a volume orm on M and then set or the scalar product and the norm ϕ, ψ : M h m ( ϕ, ψ ) Ω, ϕ : ϕ, ϕ, 3.2

8 8 Advances in Mathematical Physics on the space Γ M, L m o global C -sections. Let L 2 M, L m be the L 2 -completion o Γ M, L m,andγ hol M, L m its due to the compactness o M inite-dimensional closed subspace o global holomorphic sections. Let Π m :L 2 M, L m Γ hol M, L m 3.3 be the projection onto this subspace. Deinition 3.1. For C M,theToeplitz operator T m (o level m) is deined by T m : Π m ( ) : Γ hol M, L m Γ hol M, L m. 3.4 In words, one takes a holomorphic section s and multiplies it with the dierentiable unction. The resulting section s will only be dierentiable. To obtain a holomorphic section, one has to project it back on the subspace o holomorphic sections. The linear map T m : C M End Γ hol M, L m, T m Π m ( ), m N is the Toeplitz or Berezin-Toeplitz quantization map (o level m). It will neither be a Lie algebra homomorphism nor an associative algebra homomorphism as in general T m T m g Π m ( )Π m ( g )Π m / Π m ( g )Π T m g. 3.6 Furthermore, on a ixed level m, it is a map rom the ininite-dimensional commutative algebra o unctions to a noncommutative inite-dimensional matrix algebra. The initedimensionality is due to the compactness o M. A lot o classical inormation will get lost. To recover this inormation, one has to consider not just a single level m but all levels together. Deinition 3.2. The Berezin-Toeplitz quantization is the map C M ( End (Γ hol M, L m )) (, m N 0 T m ) m N We obtain a amily o inite-dimensional matrix algebras and a amily o maps. This ininite amily should in some sense approximate the algebra C M Approximation Results Denote or C M by, the supnorm o on M and by T m : sup s Γ hol M,L m s / 0 s s, 3.8 T m

9 Advances in Mathematical Physics 9 the operator norm with respect to the norm 3.2 on Γ hol M, L m. The ollowing theorem was shown in Theorem 3.3 Bordemann et al. 1. a For every C M,thereexistsaC>0 such that C T m m. 3.9 In particular, lim m T m. b For every, g C M, [ m i T m,t m g ] ( T m 1 {,g} O m ) c For every, g C M, T m T m g ( T m 1 g O m ) These results are contained in Theorems 4.1, 4.2, and in Section 5 in 1. We will indicate the proo or b and c in Section 5. It will make reerence to the symbol calculus o generalised Toeplitz operators as developed by Boutet de Monvel and Guillemin 28. The original proo o a was quite involved and required Hermite distributions and related objects. On the basis o the asymptotic expansion o the Berezin transorm 29, a more direct proo can be given. I will discuss this in Section 7.3. Only on the basis o this theorem, we are allowed to call our scheme a quantizing scheme. The properties in the theorem might be rephrased as the BT operator quantization has the correct semiclassical limit Further Properties From Theorem 3.3 c, we have the ollowing proposition. Proposition 3.4. Let 1, 2,..., r C M ;then T m 1... r T m 1 T m r O (m 1) 3.12 ollows directly. Proposition 3.5. [ lim m T m,t m g ]

10 10 Advances in Mathematical Physics Proo. Using the let side o the triangle inequality, rom Theorem 3.3 b, it ollows that [ m T m,t m g ] T m {,g} [ m i T m,t m g ] ( T m 1 {,g} O m ) By part a o the theorem T m {,g} {, g} m, and it stays inite. Hence T be a zero sequence. Proposition 3.6. The Toeplitz map,t m g has to ( C M End (Γ hol M, L m )), T m, 3.15 is surjective. For a proo, see 1,Proposition4.2. This proposition says that or a ixed m every operator A End Γ hol M, L m is the Toeplitz operator o a unction m. In the language o Berezin s co- and contravariant symbols, m will be the contravariant symbol o A. We will discuss this in Section 6.2. Proposition 3.7. For all C M, T m m T In particular, or real valued unctions the associated Toeplitz operator is sel-adjoint. Proo. Take s, t Γ hol M, L m ; then s, T m t s, Π m ( t ) s, t s, t T m s, t The opposite o the last statement o the above proposition is also true in the ollowing sense. Proposition 3.8. Let A End Γ hol M, L m be a sel-adjoint operator; then there exists a real valued unction, such that A T m. Proo. By the surjectivity o the Toeplitz map A T m 0 i 1 with real unctions 0 and 1.AsT m 0. From this we conclude that A T m hence T m 1 A A T m with a complex-valued unction T m 1., it ollows that T 0and We like to stress the act that the Toeplitz map is never injective on a ixed level m. Only i T m 0orm 0, we can conclude that g. g

11 Advances in Mathematical Physics 11 Proposition 3.9. Let C M and n dim C M. Denote the trace on End Γ hol M, L m by Tr m,then Tr m ( ) ( T m m n 1 vol P n C M ( Ω O m 1)) See 1, respectively 30 or a detailed proo Strict Quantization The asymptotic results o Theorem 3.3 say that the BT operator quantization is a strict quantization in the sense o Rieel 31 as ormulated in the book by Landsman 32. We take as base space X {0} {1/m m N}, with its induced topology coming rom R. Note that {0} is an accumulation point o the set {1/m m N}. AsC algebras above the points {1/m}, we take the algebras End Γ hol M, L m and above {0} the algebra C M. For C M, we assign 0 and 1/m T m. Now the property a in Theorem 3.3 is called in 32 Rieel s condition, b Dirac s condition, and c von Neumann s condition. Completeness is true by Propositions 3.6 and 3.8. This deinition is closely related to the notion o continuous ields o C -algebras; see Relation to Geometric Quantization There exists another quantum operator in the geometric setting, the operator o geometric quantization introduced by Kostant and Souriau. In a irst step, the prequantum operator associated to the bundle L m or the unction C M is deined as P m : m i id. X m 3.19 Here m is the connection in L m,andx m the Hamiltonian vector ield o with respect to the Kähler orm ω m m ω, thatis,mω X m, d. This operator P m acts on the space o dierentiable global sections o the line bundle L m. The sections depend at every point on 2n local coordinates and one has to restrict the space to sections covariantly constant along the excessive dimensions. In technical terms, one chooses a polarization. In general such a polarization is not unique. But in our complex situation, there is a canonical one by only taking the holomorphic sections. This polarization is called Kähler polarization. The operator o geometric quantization is then deined by the ollowing proposition. Q m : Π m P m The Toeplitz operator and the operator o geometric quantization with respect to the Kähler polarization are related by the ollowing.

12 12 Advances in Mathematical Physics Proposition 3.10 Tuynman Lemma. Let M be a compact quantizable Kähler maniold; then Q m i T m 1/2m Δ, 3.21 where Δ is the Laplacian with respect to the Kähler metric given by ω. For the proo, see 33, 34 or a coordinate independent proo. In particular, the Q m and the T m have the same asymptotic behaviour. We obtain or Q m similar results as in Theorem 3.3. For details see 35. It should be noted that or 3.21 the compactness o M is essential L Approximation In 34 the notion o L, respectively, gl N, respectively, su N quasilimit were introduced. It was conjectured in 34 that or every compact quantizable Kähler maniold, the Poisson algebra o unctions is a gl N quasilimit. In act, the conjecture ollows rom Theorem 3.3; see 1, 35 or details The Noncompact Situation Berezin-Toeplitz operators can be introduced or noncompact Kähler maniolds. In this case the L 2 spaces are the space o bounded sections and or the subspaces o holomorphic sections one can only consider the bounded holomorphic sections. Unortunately, in this context the proos o Theorem 3.3 do not work. One has to study examples or classes o examples case by case in order to see whether the corresponding properties are correct. In the ollowing, we give a very incomplete list o reerences. Berezin himsel studied bounded complex-symmetric domains 36. In this case the maniold is an open domain in C n. Instead o sections one studies unctions which are integrable with respect to a suitable measure depending on ħ. Then 1/ħ corresponds to the tensor power o our bundle. Such Toeplitz operators were studied extensively by Upmeier in a series o works Seealso the book o Upmeier 41. For C n see Berger and Coburn 42, 43. Klimek and Lesniewski 44, 45 studied the Berezin-Toeplitz quantization on the unit disc. Using automorphic orms and the universal covering, they obtain results or Riemann suraces o genus g 2. The names o Borthwick, Klimek, Lesniewski, Rinaldi, and Upmeier should be mentioned in the context o BT quantization or Cartan domains and super Hermitian spaces. Aquite dierent approach to Berezin-Toeplitz quantization is based on the asymptotic expansion o the Bergman kernel outside the diagonal. This was also used by the author together with Karabegov 29 or the compact Kähler case. See Section 7 or some details. Engliš 46 showed similar results or bounded pseudoconvex domains in C N.Maand Marinescu 18, 19 developed a theory o Bergman kernels or the symplectic case, which yields also results on the Berezin-Toeplitz operators or certain noncompact Kähler maniolds and even orbiolds.

13 Advances in Mathematical Physics Berezin-Toeplitz Deormation Quantization There is another approach to quantization. Instead o assigning noncommutative operators to commuting unctions, one might think about deorming the pointwise commutative multiplication o unctions into a noncommutative product. It is required to remain associative, the commutator o two elements should relate to the Poisson bracket o the elements, and it should reduce in the classical limit to the commutative situation. It turns out that such a deormation which is valid or all dierentiable unctions cannot exist. A way out is to enlarge the algebra o unctions by considering ormal power series over them and to deorm the product inside this bigger algebra. A irst systematic treatment and applications in physics o this idea were given in 1978 by Bayen et al. 47, 48. There the notion o deormation quantization and star products were introduced. Earlier versions o these concepts were around due to Berezin 49, Moyal 50, and Weyl 51. For a presentation o the history, see 24. We will show that or compact Kähler maniolds M, there is a natural star product Deinition o Star Products We start with a Poisson maniold M, {, }, thatis,adierentiable maniold with a Poisson bracket or the unction such that C M,, {, } is a Poisson algebra. Let A C M ν be the algebra o ormal power series in the variable ν over the algebra C M. Deinition 4.1. A product on A is called a ormal star product or M or or C M i it is an associative C ν -linear product which is ν-adically continuous such that 1 A/νA C M,thatis, gmod ν g, 2 1/ν g g mod ν i{, g}, where, g C M. Alternatively, we can write ( ) g C j, g ν j, j with C j, g C M such that the C j are bilinear in the entries and g. The conditions 1 and 2 can be reormulated as C 0 (, g ) g, C1 (, g ) C1 ( g, ) i {, g }. 4.2 By the ν-adic continuity, 4.1 ixes on A. A(ormal) deormation quantization is given by a (ormal) star product. I will use both terms interchangeable. There are certain additional conditions or a star product which are sometimes useul. 1 We call it null on constants i 1 1, which is equivalent to the act that the constant unction 1 will remain the unit in A. In terms o the coeicients, it can be ormulated as C k, 1 C k 1, 0ork 1. In this review, we always assume this to be the case or star products.

14 14 Advances in Mathematical Physics 2 We call it sel-adjoint i g g, where we assume ν ν. 3 We call it local i supp C j (, g ) supp supp g,, g C M. 4.3 From the locality property, it ollows that the C j are bidierential operators and that the global star product deines or every open subset U o M a star product or the Poisson algebra C U. Such local star products are also called dierential star products Existence o Star Products In the usual setting o deormation theory, there always exists a trivial deormation. This is not the case here, as the trivial deormation o C M to A, which is nothing else as extending the point-wise product to the power series, is not allowed as it does not ulil Condition 2 in Deinition 4.1 at least not i the Poisson bracket is nontrivial. In act, the existence problem is highly nontrivial. In the symplectic case, dierent existence proos, rom dierent perspectives, were given by Marc De Wilde and Lecomte 52, Omori et al. 53, 54, and Fedosov 55, 56. The general Poisson case was settled by Kontsevich Equivalence and Classiication o Star Products Deinition 4.2. Given a Poisson maniold M, {, }. Two star products and associated to the Poisson structure {, } are called equivalent i and only i there exists a ormal series o linear operators B B i ν i, B i : C M C M, 4.4 i 0 with B 0 id such that B ( ) B ( g ) B ( g ). 4.5 For local star products in the general Poisson setting, there are complete classiication results. Here I will only consider the symplectic case. To each local star product,itsfedosov-deligne class cl 1 i ν ω H2 dr M ν 4.6 can be assigned. Here H 2 M denotes the 2nd derham cohomology class o closed 2- dr orms modulo exact orms and H 2 M ν the ormal power series with such classes as dr

15 Advances in Mathematical Physics 15 coeicients. Such ormal power series are called ormal derham classes. In general we will use or the cohomology class o a orm. This assignment gives a 1 : 1 correspondence between the ormal derham classes and the equivalence classes o star products. For contractible maniolds, we have H 2 M 0and hence there is up to equivalence dr exactly one local star product. This yields that locally all local star products o a maniold are equivalent to a certain ixed one, which is called the Moyal product. For these and related classiication results, see Star Products with Separation o Variables For our compact Kähler maniolds, we will have many dierent and even nonequivalent star products. The question is the ollowing: is there a star product which is given in a natural way? The answer will be yes: the Berezin-Toeplitz star product to be introduced below. First we consider star products respecting the complex structure in a certain sense. Deinition 4.3 Karabegov 63. A star product is called star product with separation o variables i and only i h h, h g h g, 4.7 or every locally deined holomorphic unction g, antiholomorphic unction, and arbitrary unction h. Recall that a local star product or M deines a star product or every open subset U o M. We have just to take the bidierential operators deining. Hence it makes sense to talk about -multiplying with local unctions. Proposition 4.4. A local product has the separation o variables property i and only i in the bidierential operators C k, or k 1 in the irst argument only derivatives in holomorphic and in the second argument only derivatives in antiholomorphic directions appear. In Karabegov s original notation the rôles o the holomorphic and antiholomorphic unctions are switched. Bordemann and Waldmann 64 called such star products star products o Wick type. Both Karabegov and Bordemann-Waldmann proved that there exist or every Kähler maniold star products o separation o variables type. In Section 4.8, we will give more details on Karabegov s construction. Bordemann and Waldmann modiied Fedosov s method 55, 56 to obtain such a star product. See also Reshetikhin and Takhtajan 65 or yet another construction. But I like to point out that in all these constructions the result is only a ormal star product without any relation to an operator calculus, which will be given by the Berezin-Toeplitz star product introduced in the next section. Another warning is in order. The property o being a star product o separation o variables type will not be kept by equivalence transormations.

16 16 Advances in Mathematical Physics 4.5. Berezin-Toeplitz Star Product Theorem 4.5. There exists a unique (ormal) star product BT or M BT g : ν j ( ) ( ) C j, g, Cj, g C M, 4.8 j 0 in such a way that or, g C M and or every N N we have with suitable constants K N, g or all m T m T m g 0 j<n ( ) 1 j T m m C j,g K ( ) ( ) 1 N N, g. 4.9 m The star product is null on constants and sel-adjoint. This theorem has been proven immediately ater 1 was inished. It has been announced in 66, 67 and the proo was written up in German in 35. A complete proo published in English can be ound in 30. For simplicity we might write T m T m g j 0 ( ) 1 j T m m C j m,,g 4.10 but we will always assume the strong and precise statement o 4.9. The same is assumed or other asymptotic ormulas appearing urther down in this review. Next we want to identiy this star product. Let K M be the canonical line bundle o M, that is, the nth exterior power o the holomorphic 1-dierentials. The canonical class δ is the irst Chern class o this line bundle, that is, δ : c 1 K M. I we take in K M theibremetric coming rom the Liouville orm Ω, then this deines a unique connection and urther a unique curvature 1, 1 -orm ω can. In our sign conventions, we have δ ω can. Together with Karabegov the author showed the ollowing theorem. Theorem 4.6 see 29. a The Berezin-Toeplitz star product is a local star product which is o separation o variable type. b Its classiying Deligne-Fedosov class is cl BT 1 i ( 1 ν ω δ ) or the characteristic class o the star product BT. c The classiying Karabegov orm associated to the Berezin-Toeplitz star product is 1 ν ω ω can. 4.12

17 Advances in Mathematical Physics 17 The Karabegov orm has not yet deined here. We will introduce it below in Section 4.8. Using K-theoretic methods, the ormula or cl BT was also given by Hawkins Star Product o Geometric Quantization Tuynman s result 3.21 relates the operators o geometric quantization with Kähler polarization and the BT operators. As the latter deine a star product, it can be used to give also a star product GQ associated to geometric quantization. Details can be ound in 30. This star product will be equivalent to the BT star product, but it is not o the separation o variables type. The equivalence is given by the C ν -linear map induced by B ( ) : ν Δ ( 2 id ν Δ ) We obtain B BT B g B GQ g. 4.7.TraceortheBTStarProduct From 3.18 the ollowing complete asymptotic expansion or m can be deduced 30, 69 : Tr m ( ) T m m n ( 1 m j 0 ) j ( ) ( ) τ j, with τ j C We deine the C ν -linear map Tr : C M ν ν n C ν, Tr : ν n ν j ( ) τ j, j where the τ j are given by the asymptotic expansion 4.14 or C M and or arbitrary elements by C ν -linear extension. Proposition 4.7 see 30. The map Tr is a trace, that is, we have Tr ( g ) Tr ( g ) Karabegov Quantization In 63, 70 Karabegov not only gave the notion o separation o variables type, but also a proo o existence o such ormal star products or any Kähler maniold, whether compact, noncompact, quantizable, or nonquantizable. Moreover, he classiied them completely as individual star product not only up to equivalence. He starts with M, ω 1 a pseudo-kähler maniold, that is, a complex maniold with a nondegenerate closed 1, 1 -orm not necessarily positive.

18 18 Advances in Mathematical Physics A ormal orm ω 1/ν ω 1 ω 0 νω 1 is called a ormal deormation o the orm 1/ν ω 1 i the orms ω r,r 0, are closed but not necessarily nondegenerate 1, 1 -orms on M. It was shown in 63 that all deormation quantizations with separation o variables on the pseudo-kähler maniold M, ω 1 are bijectively parameterized by the ormal deormations o the orm 1/ν ω 1. Assume that we have such a star product A : C M ν,. Then or, g A the operators o let and right multiplications L,R g are given by L g g R g.the associativity o the star-product is equivalent to the act that L commutes with R g or all, g A. I a star product is dierential, then L,R g are ormal dierential operators. Karabegov constructs his star product associated to the deormation ω in the ollowing way. First he chooses on every contractible coordinate chart U M with holomorphic coordinates {z k } its ormal potential Φ ( ) 1 Φ 1 Φ 0 νφ 1, ω i Φ ν Then construction is done in such a way that we have or the let right multiplication operators on U L Φ/ zk Φ, R Φ/ zl Φ. z k z k z l z l 4.18 The set L U o all let multiplication operators on U is completely described as the set o all ormal dierential operators commuting with the point-wise multiplication operators by antiholomorphic coordinates R zl z l and the operators R Φ/ zl. From the knowledge o L U, the star product on U can be reconstructed. The local star-products agree on the intersections o the charts and deine the global star-product on M. We have to mention that this original construction o Karabegov will yield a star product o separation o variable type but with the role o holomorphic and antiholomorphic variables switched. This says or any open subset U M and any holomorphic unction a and antiholomorphic unction b on U that the operators L a and R b are the operators o point-wise multiplication by a and b, respectively, that is, L a a and R b b Karabegov s Formal Berezin Transorm Given such a star products, Karabegov introduced the ormal Berezin transorm I as the unique ormal dierential operator on M such that or any open subset U M, holomorphic unctions a, and antiholomorphic unctions b on U, therelationi a b b aholds see 71. He shows that I 1 νδ, where Δ is the Laplace operator corresponding to the pseudo-kähler metric on M. Karabegov considered the ollowing associated star products. First the dual starproduct on M is deined or, g Aby the ormula g I 1( Ig I ) It is a star-product with separation o variables on the pseudo-kähler maniold M, ω 1. Its ormal Berezin transorm equals I 1, and thus the dual to is. Note that it is not a star

19 Advances in Mathematical Physics 19 product o the same pseudo-kähler maniold. Denote by ω 1/ν ω 1 ω 0 ν ω 1 the ormal orm parameterizing the star-product. Next, the opposite o the dual star-product, op, is given by the ormula g I 1( I Ig ) It deines a deormation quantization with separation o variables on M, but with the roles o holomorphic and antiholomorphic variables swapped with respect to. It could be described also as a deormation quantization with separation o variables on the pseudo- Kähler maniold M, ω 1, where M is the maniold M with the opposite complex structure. But now the pseudo-kähler orm will be the same. Indeed the ormal Berezin transorm I establishes an equivalence o deormation quantizations A, and A,. How is the relation to the Berezin-Toeplitz star product BT o Theorem 4.5? There exists a certain ormal deormation ω o the orm 1/ν ω which yields a star product in the Karabegov sense. The opposite o its dual will be equal to the Berezin-Toeplitz star product, that is, BT op The classiying Karabegov orm ω o will be the orm 4.12.Noteas and BT are equivalent via I, we have cl cl BT ; see the ormula We will identiy ω in Section The Disc Bundle and Global Operators In this section, we identiy the bundles L m over the Kähler maniold M as associated line bundles o one unique S 1 -bundle over M. The Toeplitz operator will appear as modes o a global Toeplitz operator. A detailed analysis o this global operator will yield a proo o Theorem 3.3 part b and part c. Moreover, we will need this set-up to discuss coherent states, Berezin symbols, and the Berezin transorm in the next sections. For a more detailed presentation, see The Disc Bundle We will assume that the quantum line bundle L is already very ample, that is, it has enough global holomorphic sections to embed M into projective space. From the bundle as the connection will not be needed anymore, I will drop it in the notation L, h, wepassto its dual U, k : L,h 1 with dual metric k. Inside o the total space U, we consider the circle bundle Q : {λ U k λ, λ 1}, 5.1

20 20 Advances in Mathematical Physics the open disc bundle, and closed disc bundle, respectively D : {λ U k λ, λ < 1}, D : {λ U k λ, λ 1}. 5.2 Let τ : U M be the projection maybe restricted to the subbundles. For the projective space P N C with the hyperplane section bundle H as quantum line bundle, the bundle U is just the tautological bundle. Its ibre over the point z P N C consists o the line in C N 1 which is represented by z. In particular, or the projective space the total space o U with the zero section removed can be identiied with C N 1 \{0}. The same picture remains true or the via the very ample quantum line bundle in projective space embedded maniold M. The quantum line bundle will be the pull-back o H i.e., its restriction to the embedded maniold and its dual is the pull-back o the tautological bundle. In the ollowing we use E \ 0 to denote the total space o a vector bundle E with the image o the zero section removed. Starting rom the real-valued unction k λ : k λ, λ on U, we deine ã : 1/2i log k on U \ 0 the derivation is taken with respect to the complex structure on U and denote by its restriction to Q. With the help o the quantization condition 2.4, weobtaind τ ω with the derham dierential d d Q and that in act μ 1/2π τ Ω is a volume orm on Q. Indeed is a contact orm or the contact maniold Q. As ar as the integration is concerned we get Q ( τ ) μ M Ω, C M. 5.3 Recall that Ω is the Liouville volume orm on M The Generalized Hardy Space With respect to μ, we take the L 2 -completion L 2 Q, μ o the space o unctions on Q. The generalized Hardy space H is the closure o the space o those unctions in L 2 Q, μ which can be extended to holomorphic unctions on the whole disc bundle D. The generalized Szegö projector is the projection Π :L 2( Q, μ ) H. 5.4 By the natural circle action, the bundle Q is an S 1 -bundle and the tensor powers o U can be viewed as associated line bundles. The space H is preserved by the S 1 -action. It can be decomposed into eigenspaces H m 0 H m, where c S 1 acts on H m as multiplication by c m. The Szegö projector is S 1 invariant and can be decomposed into its components, the Bergman projectors Π m :L 2( Q, μ ) H m. 5.5

21 Advances in Mathematical Physics 21 Sections o L m U m can be identiied with unctions ψ on Q which satisy the equivariance condition ψ cλ c m ψ λ, that is, which are homogeneous o degree m. This identiication is given via the map γ m :L 2 M, L m L 2( Q, μ ), s ψ s, where ψ s m s τ, 5.6 which turns out to be an isometry onto its image. On L 2 M, L m, we have the scalar product 3.2. Restricted to the holomorphic sections, we obtain the isometry γ m : Γ hol M, L m H m. 5.7 In the case o P N C, this correspondence is nothing else as the identiication o the global sections o the mth tensor powers o the hyper plane section bundle with the homogenous polynomial unctions o degree m on C N The Toeplitz Structure There is the notion o Toeplitz structure Π, Σ as developed by Boutet de Monvel and Guillemin in 28, 72. I do not want to present the general theory but only the specialization to our situation. Here Π is the Szegö projector 5.4 and Σ is the submaniold Σ : {t λ λ Q, t > 0} T Q \ o the tangent bundle o Q deined with the help o the 1-orm. It turns out that Σ is a symplectic submaniold, called a symplectic cone. A generalized Toeplitz operator o order k is an operator A : H H o the orm A Π R Π, where R is a pseudodierential operator ΨDO o order k on Q. The Toeplitz operators constitute a ring. The symbol o A is the restriction o the principal symbol o R which lives on T Q to Σ.NotethatR is not ixed by A, but Guillemin and Boutet de Monvel showed that the symbols are well deined and that they obey the same rules as the symbols o ΨDOs. In particular, the ollowing relations are valid: σ A 1 A 2 σ A 1 σ A 2, σ A 1,A 2 i{σ A 1,σ A 2 } Σ. 5.9 Here {, } Σ is the restriction o the canonical Poisson structure o T Q to Σ coming rom the canonical symplectic orm on T Q A Sketch o the Proo o Theorem 3.3 For this we need only to consider the ollowing two generalized Toeplitz operators. 1 The generator o the circle action gives the operator D ϕ 1/i / ϕ, where ϕ is the angular variable. It is an operator o order 1 with symbol t. It operates on H m as multiplication by m.

22 22 Advances in Mathematical Physics 2 For C M,letM be the operator on L 2 Q, μ corresponding to multiplication with τ.weset T Π M Π : H H As M is constant along the ibres o τ, the operator T commutes with the circle action. Hence we can decompose T T m m 0, 5.11 denotes the restriction o T to H m. Ater the identiication o H m where T m with Γ hol M, L m, we see that these T m are exactly the Toeplitz operators T m the introduced in Section 3. We call T the global Toeplitz operator and the T m local Toeplitz operators. The operator T is o order 0. Let us denote by τ Σ : Σ T Q Q M the composition, then we obtain or its symbol σ T τ Σ. Now we are able to prove First we introduce or a ixed t>0 Σ t : {t λ λ Q} Σ It turns out that ω Σ Σt tτ Σ ω. The commutator T,T g is a Toeplitz operator o order 1. From the above, we obtain with 5.9 that the symbol o the commutator equals σ ([ T,T g ]) t λ i { τ Σ, τ Σ g} Σ t λ i t 1{, g } M τ λ We consider the Toeplitz operator A : D 2 ϕ[ T,T g ] i Dϕ T {,g} Formally this is an operator o order 1. Using σ T {,g} τ Σ {, g} and σ D ϕ t, weseethat its principal symbol vanishes. Hence it is an operator o order 0. Now M and hence also Q are compact maniolds. This implies that A is a bounded operator ΨDOs o order 0 on compact maniolds are bounded. It is obviously S 1 -invariant and we can write A m 0 A m, where A m is the restriction o A on the space H m. For the norms we get A m A. But A m A H m m 2[ T m,t m g ] i mt m {,g} Taking the norm bound and dividing it by m, we get part b o Theorem 3.3.Using 5.7,the norms involved indeed coincide. Quite similar, one can prove part c o Theorem 3.3 and more general the existence o the coeicients C j, g or the Berezin-Toeplitz star product o Theorem 4.5. See 30, 35 or the details.

23 Advances in Mathematical Physics Coherent States and Berezin Symbols 6.1. Coherent States Let the situation be as in the previous section. In particular, L is assumed to be already very ample, U L is the dual o the quantum line bundle, Q U the unit circle bundle, and τ : Q M the projection. In particular, recall the correspondence 5.6 ψ s m sτ o m-homogeneous unctions ψ s on U with sections o L m. To obtain this correspondence, we ixed the section s and varied a. Now we do the opposite. We ix U \ 0 and vary the section s. Obviously, this yields a linear orm on Γ hol M, L m and hence with the help o the scalar product 3.2, we make the ollowing. Deinition 6.1. a The coherent vector (o level m) associated to the point U \ 0 is the unique element o Γ hol M, L m such that,s ψ s m s τ 6.1 or all s Γ hol M, L m. b The coherent state (o level m) associated to x M is the projective class x [ : ] P Γ hol M, L m, τ 1 x, / O course, we have to show that the object in b is well deined. Recall that, denotes the scalar product on the space o global sections Γ M, L m. In the convention o this review, it will be antilinear in the irst argument and linear in the second argument. The coherent vectors are antiholomorphic in and ulil c c m, c C : C \ {0}. 6.3 Note that 0 would imply that all sections will vanish at the point x τ. Hence, the sections o L cannot be used to embed M into a projective space, which is a contradiction to the very ampleness o L. Hence, / 0anddueto 6.3 the class [ ] { : s Γ hol M, L m c C : s c } 6.4 is a well-deined element o the projective space P Γ hol M, L m, only depending on x τ M. This kind o coherent states goes back to Berezin. A coordinate independent version and extensions to line bundles were given by Rawnsley 73. It plays an important role in the work o Cahen et al. on the quantization o Kähler maniolds 74 77, viaberezin s covariant symbols. I will return to this in Section 6.5. In these works, the coherent vectors are parameterized by the elements o L\0. The deinition here uses the points o the total space o the dual bundle U. It has the advantage that one can consider all tensor powers o L together on an equal ooting.

24 24 Advances in Mathematical Physics Deinition 6.2. The coherent state embedding is the antiholomorphic embedding M P Γ hol M, L m P N C, x [ ]. 6.5 τ 1 x Here N dim Γ hol M, L m 1. In this review, in abuse o notation, τ 1 x will always denote a non-zero element o the iber over x. The coherent state embedding is up to conjugation the embedding o Section 2.9 with respect to an orthonormal basis o the sections. In 78 urther results on the geometry o the coherent state embedding are given Covariant Berezin Symbols We start with the ollowing deinition. Deinition 6.3. The covariant Berezin symbol σ m A (o level m) o an operator A End Γ hol M, L m is deined as σ m A : M C, x σ m A x :,A,, τ 1 x. 6.6 As the actors appearing in 6.3 will cancel, it is a well-deined unction on M. Ithe level m is clear rom the context, I will sometimes drop it in the notation. We consider also the coherent projectors used by Rawnsley P m x,, τ 1 x. 6.7 Here we used the convenient bra-ket notation o the physicists. Recall, i s is a section, then P m x s,s,s,,. 6.8 Again the projector is well deined on M. With its help, the covariant symbol can be expressed as ( σ m A Tr AP m x ). 6.9 From the deinition o the symbol, it ollows that σ m A is real analytic and σ m A σ m A. 6.10

25 Advances in Mathematical Physics Rawnsley s ɛ Function Rawnsley 73 introduced a very helpul unction on the maniold M relating the local metric in the bundle with the scalar product on coherent states. In our dual description, we deine it in the ollowing way. Deinition 6.4. Rawnsley s epsilon unction is the unction M C M, h m ( x ɛ m x :,, ) x, τ 1 x With 6.3, it is clear that it is a well-deined unction on M. Furthermore, using /, m( ) τ, 6.12 it ollows that Hence, we can deine the modiied measure x / 0, or x τ, and ɛ m > Ω m ɛ x : ɛ m x Ω x 6.14 or the space o unctions on M and obtain a modiied scalar product, m ɛ Proposition 6.5. For s 1,s 2 Γ hol M, L m, we have or C M. h m s 1,s 2 x,s 1,,s 2 ɛ m x s 1,P m x s 2 ɛ m x Proo. Due to 6.13, we can represent every section s locally at x as s x ŝ x local unction ŝ. Now,s m ( ) ŝ x x ŝ x m ( ) x ŝ x, with a We rewrite h m s 1,s 2 x ŝ 1 s 2 h m, x,andobtain h m s 1,s 2 x,s 1,,s 2, h m (, ) x. 6.17

26 26 Advances in Mathematical Physics From the deinition 6.11, the irst relation ollows. Obviously, it can be rewritten with the coherent projector to obtain the second relation. There exists another useul description o the epsilon unction. Proposition 6.6. Let s 1,s 2,...,s k be an arbitrary orthonormal basis o Γ hol M, L m.then k ɛ m x h m ( ) s j,s j x. j Proo. For every vector ψ in a inite-dimensional hermitian vector space with orthonormal basis s j,j 1,...,k, the coeicient with respect to the basis element s j is given by ψ j s j,ψ. Furthermore, ψ, ψ ψ 2 j ψ jψ j. Using the relation 6.15 we can rewrite k h ( ) ɛ m x k s j,s j x,s j,s j j 1, j 1 Hence the claim ollows. In certain special cases, the unctions ɛ m will be constant as a unction o the points o the maniold. In this case, we can apply Proposition 6.11 below or A id, the identity operator, and obtain ɛ m dim Γ hol M, L m vol M Here vol M denotes the volume o the maniold with respect to the Liouville measure. Now the question arises when ɛ m will be constant, respectively, when the measure Ω m ɛ will be the standard measure up to a scalar. FromProposition 6.6, it ollows that i there is a transitive group action on the maniold and everything, or example, Kähler orm, bundle, metric, is homogenous with respect to the action this will be the case. An example is given by M P N C. By a result o Rawnsley 73, respectively, Cahen et al. 74, ɛ m const i and only i the quantization is projectively induced. This means that under the conjugate o the coherent state embedding, the Kähler orm ω o M coincides with the pull-back o the Fubini-Study orm. Note that in general this is not the case; see Section Contravariant Berezin Symbols Recall the modiied Liouville measure 6.14 and modiied scalar product or the unctions on M introduced in the last subsection.

27 Advances in Mathematical Physics 27 Deinition 6.7. Given an operator A End Γ hol M, L m, then a contravariant Berezin symbol ˇσ m A C M o A is deined by the representation o the operator A as integral i such a representation exists. A M ˇσ m A x P m x Ω m ɛ x 6.21 Proposition 6.8. The Toeplitz operator T m admits a representation 6.21 with ˇσ m ( ) T m, 6.22 that is, the unction is a contravariant symbol o the Toeplitz operator T m. Moreover, every operator A End Γ hol M, L m, has a contravariant symbol. Proo. Let C M and set A : M x P m x Ω m ɛ x, 6.23 then ˇσ m A. For arbitrary s 1,s 2 Γ hol M, L m, we calculate using 6.15 s 1,As 2 M M M x s 1,P m x s 2 Ω m ɛ x x h m s 1,s 2 x Ω x h m ( s 1,s 2 ) x Ω x s 1,s 2 s 1,T m s Hence T m A. As the Toeplitz map is surjective Proposition 3.6, every operator is a Toeplitz operator, hence has a contravariant symbol. Note that given an operator its contravariant symbol on a ixed level m is not uniquely deined. We introduce on End Γ hol M, L m the Hilbert-Schmidt norm A, C HS Tr A C Theorem 6.9. The Toeplitz map T m A, T m and the covariant symbol map A σ m A are adjoint: HS σ m A, m ɛ. 6.26

28 28 Advances in Mathematical Physics Proo. A, T m ( ) Tr A T m Tr (A M x P m x Ω m ɛ ) x M ( x Tr A P m x ) Ω m ɛ x Now applying Deinition 6.7 and 6.10 A, T m M x σ m A Ω m ɛ x M σ m A x x Ω m ɛ x σ m A, x m ɛ As every operator has a contravariant symbol, we can also conclude A, B HS σ m A, ˇσ m B m ɛ From Theorem 6.9 by using the surjectivity o the Toeplitz map, we get the ollowing proposition. Proposition The covariant symbol map σ m is injective. Another application is the ollowing. Proposition Tr A M σ m A Ω m ɛ Proo. We use Id T 1 and by 6.26 Tr A A, Id HS σ m A, 1 m ɛ Berezin Star Product Under certain very restrictive conditions, Berezin covariant symbols can be used to construct a star product, called the Berezin star product. Recall that Proposition 6.10 says that the linear symbol map ( σ m :End (Γ hol M, L m )) C M 6.31 is injective. Its image is a subspace A m o C M, called the subspace o covariant symbols o level m. Iσ m A and σ m B are elements o this subspace the operators, A and B will

29 Advances in Mathematical Physics 29 be uniquely ixed. Hence also σ m A B. Now one takes σ m A m σ m B : σ m A B 6.32 as a deinition or an associative and noncommutative product m on A m. It is even possible to give an analytic expression or the resulting symbol. For this we introduce the two-point unction ψ m ( x, y ), β,, β, β β 6.33 with τ 1 x x and β τ 1 y. This unction is well deined on M M. Furthermore, we have the two-point symbol σ m A ( x, y ),A β, β It is the analytic extension o the real-analytic covariant symbol. It is well deined on an open dense subset o M M containing the diagonal. Using 6.15, we express σ m,a B A B x A M M M,,B, h m ( A,A β β,b ) (y ) Ω ( y ),B β, β, ɛ m ( y ) Ω ( y ), σ m A ( x, y ) σ m B ( y, x ) ψ m ( x, y ) ɛ m ( y ) Ω ( y ) The crucial problem is how to relate dierent levels m to deine or all possible symbols a unique product not depending on m. In certain special situations like these studied by Berezin himsel 36 and Cahen et al. 74, the subspaces are nested into each other and the union A m N A m is a dense subalgebra o C M. Indeed, in the cases considered, the maniold is a homogenous maniold and the epsilon unction ɛ m is a constant. A detailed analysis shows that in this case a star product is given.

30 30 Advances in Mathematical Physics For urther examples, or which this method works not necessarily compact, see other articles by Cahen et al For related results, see also work o Moreno and Ortega- Navarro 79, 80. In particular, also the work o Engliš 46, Reshetikhin and Takhtajan 65 gave a construction o a ormal star product using ormal integrals in the spirit o the Berezin s covariant symbol construction. 7. Berezin Transorm 7.1. The Deinition Starting rom C M, we can assign to it its Toeplitz operator T m and then assign to T m Deinition 7.1. The map the covariant symbol σ m T m End Γ hol M, L m. It is again an element o C M. C M C M, I m ( ) : σ m ( ) T m 7.1 is called the Berezin transorm (o level m). From the point o view o Berezin s approach, the operator T m has as a contravariant symbol. Hence I m gives a correspondence between contravariant symbols and covariant symbols o operators. The Berezin transorm was introduced and studied by Berezin 36 or certain classical symmetric domains in C n. These results were extended by Unterberger and Upmeier 84 ; see also Engliš 46, 81, 82 and Engliš and Peetre 85. Obviously, the Berezin transorm makes perect sense in the compact Kähler case which we consider here The Asymptotic Expansion The results presented here are joint work with Karabegov 29.Seealso 86 or an overview. Theorem 7.2. Given x M, then the Berezin transorm I m evaluated at the point x has a complete asymptotic expansion in powers o 1/m as m I m ( ) x i 0 I i ( ) x 1 m i, 7.2 where I i : C M C M are maps with I 0 ( ), I1 ( ) Δ. 7.3 ω. Here the Δ is the usual Laplacian with respect to the metric given by the Kähler orm

31 Advances in Mathematical Physics 31 Complete asymptotic expansion means the ollowing. Given C M, x M, and an r N, then there exists a positive constant A such that r 1 I m x I i x 1 m A i m r. 7.4 i 0 In Section 7.4, I will give some remarks on the proo but beore I present you a nice application Norm Preservation o the BT Operators In 87 I conjectured 7.2 which is now a mathematical result and showed how such an asymptotic expansion supplies a dierent proo o Theorem 3.3,part a. For completeness, I reproduce the proo here. Proposition 7.3. I m ( ) σ m ( ) T m T m. 7.5 Proo. Using Cauchy-Schwarz inequality, we calculate x τ σ m ( ) T m x 2,T m, 2 2 T m,t m, T m Here the last inequality ollows rom the deinition o the operator norm. This shows the irst inequality in 7.5. For the second inequality, introduce the multiplication operator M m on Γ M, L m. Then T m Π m M m Π m M m and or ϕ Γ M, L m, ϕ / 0 M m ϕ 2 ϕ 2 M h m ( ϕ,ϕ ) Ω M h m ( ϕ, ϕ ) Ω M z z h m ( ϕ, ϕ ) Ω M h m ( ϕ, ϕ ) Ω Hence, T m M m sup ϕ / 0 ϕ. 7.8 ϕ M m Proo (Theorem 3.3 part (a)). Choose as x e M a point with x e. From the act that the Berezin transorm has as a leading term the identity, it ollows that I m x e x e C/m with a suitable constant C. Hence, x e I m x e C/m and C m xe C ( ) m I m x e I m. 7.9

32 32 Advances in Mathematical Physics Putting 7.5 and 7.9 together, we obtain C T m m Bergman Kernel To understand the Berezin transorm better, we have to study the Bergman kernel. Recall rom Section 5, the Szegö projectors Π :L 2 Q, μ H and its components Π m :L 2 Q, μ H m, the Bergman projectors. The Bergman projectors have smooth integral kernels, the Bergman kernels B m, β deined on Q Q, thatis, Π m ( ψ ) ( ) ( ) ( ) B m, β ψ β μ β The Bergman kernels can be expressed with the help o the coherent vectors. Proposition 7.4. B m (, β ) ψ β Q ψ e m ( β ), β For the proos o this and the ollowing propositions, see 29 or 86. Let x, y M and choose, β Q with τ x and τ β y, then the unctions u m x : B m, ( ) ( ) ( ) v m x, y : Bm, β Bm β,,, β, β, are well deined on M and on M M, respectively. The ollowing proposition gives an integral representation o the Berezin transorm. Proposition 7.5. ( I m ( )) 1 ( ) ( ) x B m, β Bm β, τ ( β ) μ ( β ) B m, Q 1 ( ) ( ) ( ) v m x, y y Ω y. u m x M 7.15 Typically, asymptotic expansions can be obtained using stationary phase integrals. But or such an asymptotic expansion o the integral representation o the Berezin transorm, we will not only need an asymptotic expansion o the Bergman kernel along the diagonal which is well known but in a neighbourhood o it. This is one o the key results obtained in 29. It is based on works o Boutet de Monvel and Sjöstrand 88 on the Szegö kernel and in generalization o a result o Zelditch 89 on the Bergman kernel on the diagonal. The integral

33 Advances in Mathematical Physics 33 representation is used then to prove the existence o the asymptotic expansion o the Berezin transorm. Having such an asymptotic expansion, it still remains to identiy its terms. As it was explaining in Section 4.8, Karabegov assigns to every ormal deormation quantizations with the separation o variables property a ormal Berezin transorm I.In 29 it is shown that there is an explicitly speciied star product see 29, Theorem 5.9 with associated ormal Berezin transorm such that i we replace 1/m by the ormal variable ν in the asymptotic expansion o the Berezin transorm I m x we obtain I x. This inally proves Theorem 7.2. We will exhibit the star product in the next section Identiication o the BT Star Product Moreover in 29 there is another object introduced, the twisted product R m (, g ) : σ m ( T m ) T m g Also or it the existence o a complete asymptotic expansion was shown. It was identiied with a twisted ormal product. This allows the identiication o the BT star product with a special star product within the classiication o Karabegov. From this identiication, the properties o Theorem 4.6 o locality, separation o variables type, and the calculation to the classiying orms and classes or the BT star product ollow. As already announced in Section 4.8, the BT star product BT is the opposite o the dual star product o a certain star product. To identiy we will give its classiying Karabegov orm ω. As already mentioned above, Zelditch 89 proved that the the unction u m 7.13 has a complete asymptotic expansion in powers o 1/m. In detail he showed u m x m n 1 m b k x, b k k 0 I we replace in the expansion 1/m by the ormal variable ν, we obtain a ormal unction s deined by e s x ν k b k x. k Now take as ormal potential 4.17 Φ 1 ν Φ 1 s, 7.19

34 34 Advances in Mathematical Physics where Φ 1 is the local Kähler potential o the Kähler orm ω ω 1. Then ω i Φ. Itmight be also written in the orm ω 1 ) (i ν ω F log B m, Here we denote the replacement o 1/m by the ormal variable ν by the symbol F Pullback o the Fubini-Study Form Starting rom the Kähler maniold M, ω and ater choosing an orthonormal basis o the space Γ hol M, L m,weobtainanembedding φ m : M P N m 7.21 o M into projective space o dimension N m.onp N m we have the standard Kähler orm, the Fubini-Study orm ω FS. The pull-back φ m ω FS will not depend on the orthogonal basis chosen or the embedding. But in general it will not coincide with a scalar multiple o the Kähler orm ω we started with see 78 or a thorough discussion o the situation. It was shown by Zelditch 89, by generalizing a result o Tian 90, that Φ m ω FS admits a complete asymptotic expansion in powers o 1/m as m. In act it is related to the asymptotic expansion o the Bergman kernel 7.13 along the diagonal. The pull-back can be given as 89, Proposition9 ( φ m ) ωfs mω i log u m x I we again replace 1/m by ν, weobtainvia 7.20 the Karabegov orm introduced in Section 4.8 ω F ((φ m ) ) ωfs Reerences 1 M. Bordemann, E. Meinrenken, and M. Schlichenmaier, Toeplitz quantization o Kähler maniolds and gl n, n limits, Communications in Mathematical Physics, vol. 165, no. 2, pp , J. Madore, The uzzy sphere, Classical and Quantum Gravity, vol. 9, no. 1, pp , A. P. Balachandran, B. P. Dolan, J. Lee, X. M., and D. O Connor, Fuzzy complex projective spaces and their star-products, Journal o Geometry and Physics, vol. 43, no. 2-3, pp , A. P. Balachandran, S. Kurkcuoglu, and S. Vaidya, Lectures on Fuzzy and Fuzzy SUSY Physics, World Scientiic, Singapore, C. I. Lazaroiu, D. McNamee, and Sämann, Generalized Berezin quantization, Bergman metric and uzzy laplacians, Journal o High Energy Physics. In press. 6 B. P. Dolan, I. Huet, S. Murray, and D. O Connor, Noncommutative vector bundles over uzzy CP N and their covariant derivatives, Journal o High Energy Physics, vol. 7, article 007, P. Aschieri, J. Madore, P. Manousselis, and G. Zoupanos, Dimensional reduction over uzzy coset spaces, Journal o High Energy Physics, vol. 2004, no. 4, article 034, H. Grosse, M. Maceda, J. Madore, and H. Steinacker, Fuzzy instantons, International Journal o Modern Physics A, vol. 17, no. 15, pp , 2002.

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37 Advances in Mathematical Physics R. Nest and B. Tsygan, Algebraic index theory or amilies, Advances in Mathematics, vol. 113, pp , A. V. Karabegov, Deormation quantizations with separation o variables on a Kähler maniold, Communications in Mathematical Physics, vol. 180, no. 3, pp , M. Bordemann and St. Waldmann, A Fedosov star product o the Wick type or Kähler maniolds, Letters in Mathematical Physics, vol. 41, no. 3, pp , N. Reshetikhin and L. A. Takhtajan, Deormation quantization o Kähler maniolds, in L. D. Faddeev s Seminar on Mathematical Physics, vol. 201 o American Mathematical Society Translations: Series 2, pp , American Mathematical Society, Providence, RI, USA, M. Schlichenmaier, Berezin-Toeplitz quantization o compact Kähler maniolds, in Quantization, Coherent States and Poisson Structures, A. Strasburger, S. T. Ali, J.-P. Antoine, J.-P. Gazeau, and A. Odzijewicz, Eds., pp , Polish Scientiic Publisher PWN, 1998, Proceedings o the 14th Workshop on Geometric Methods in Physics Białowieża, Poland, July M. Schlichenmaier, Deormation quantization o compact Kähler maniolds via Berezin-Toeplitz operators, in Proceedings o the 21st International Colloquium on Group Theoretical Methods in Physics, H.-D. Doebner, P. Nattermann, and W. Scherer, Eds., pp , World Scientiic, Goslar, Germany, E. Hawkins, The correspondence between geometric quantization and ormal deormation quantization, 69 D. Borthwick, T. Paul, and A. Uribe, Semiclassical spectral estimates or Toeplitz operators, Annales de l Institut Fourier, vol. 48, no. 4, pp , A. V. Karabegov, Cohomological classiication o deormation quantizations with separation o variables, Letters in Mathematical Physics, vol. 43, no. 4, pp , A. V. Karabegov, On the canonical normalization o a trace density o deormation quantization, Letters in Mathematical Physics, vol. 45, no. 3, pp , V. Guillemin, Some classical theorems in spectral theory revisited, in Seminar on Singularities o Solutions o Linear Partial Dierential Equations, L. Hörmander, Ed., vol. 91 o Annals o Mathematics Studies, pp , Princeton University Press, Princeton, NJ, USA, J. H. Rawnsley, Coherent states and Kähler maniolds, The Quarterly Journal o Mathematics. Oxord. Second Series, vol. 28, no. 112, pp , J. Rawnsley, M. Cahen, and S. Gutt, Quantization o Kähler maniolds. I. Geometric interpretation o Berezin s quantization, Journal o Geometry and Physics, vol. 7, no. 1, pp , M. Cahen, S. Gutt, and J. Rawnsley, Quantization o Kähler maniolds. II, Transactions o the American Mathematical Society, vol. 337, no. 1, pp , M. Cahen, S. Gutt, and J. Rawnsley, Quantization o Kähler maniolds. III, Letters in Mathematical Physics, vol. 30, no. 4, pp , M. Cahen, S. Gutt, and J. Rawnsley, Quantization o Kähler maniolds. IV, Letters in Mathematical Physics, vol. 34, no. 2, pp , S. Berceanu and M. Schlichenmaier, Coherent state embeddings, polar divisors and Cauchy ormulas, Journal o Geometry and Physics, vol. 34, no. 3-4, pp , C. Moreno and P. Ortega-Navarro, -products on D1 C, S 2 and related spectral analysis, Letters in Mathematical Physics, vol. 7, no. 3, pp , C. Moreno, -products on some Kähler maniolds, Letters in Mathematical Physics, vol. 11, no. 4, pp , M. Engliš, Berezin quantization and reproducing kernels on complex domains, Transactions o the American Mathematical Society, vol. 348, no. 2, pp , M. Engliš, Asymptotics o the Berezin transorm and quantization on planar domains, Duke Mathematical Journal, vol. 79, no. 1, pp , M. Engliš, The asymptotics o a Laplace integral on a Kähler maniold, Journal ür die Reine und Angewandte Mathematik, vol. 528, pp. 1 39, A. Unterberger and H. Upmeier, The Berezin transorm and invariant dierential operators, Communications in Mathematical Physics, vol. 164, no. 3, pp , M. Engliš and J. Peetre, On the correspondence principle or the quantized annulus, Mathematica Scandinavica, vol. 78, no. 2, pp , M. Schlichenmaier, Berezin-Toeplitz quantization and Berezin transorm, in Long Time Behaviour o Classical and Quantum Systems, vol. 1, pp , World Scientiic, River Edge, NJ, USA, 2001.

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