Quantization of Kähler manifolds. II

Abstract

Abstract.We use Berezin's dequantization procedure to define a formal *-product on a dense subalgebra of the algebra of smooth functions on a compact homogeneous Kahler manifold M. We prove that this formal »-product is convergent when M is a hermitian symmetric space. IntroductionIn part I of this paper [7] we showed how to quantize certain compact Kahler manifolds (M, co, J). This means the following: Let (L, V, h) be a quantization bundle over M (i.e., a holomorphic line bundle L with connection V admitting an invariant hermitian structure h, such that the curvature is curv(V) = -2inoe).Let %? be the Hilbert space of holomorphic sections of L. To any linear operator A on %? is associated a symbol  which is a real analytic function on M. Denote by Ê(L) the space of these symbols. For any positive integer k , (Lk = ®fc L, V(A:), A(/c)) is a quantization bundle for (M, kco, J). If Xfc is the Hilbert space of holomorphic sections of Lk , we denote by Ê(Lk) the space of symbols of linear operators on %?k . If, for every k, a certain characteristic function e(/c) (which depends on L and k and which is real analytic on M) is constant, the space Ê(L') is contained in the space Ê(Lk) for any k > I. Furthermore U/^i Ê(Ll) (denoted by fé¿) is a dense subspace of the space of continuous functions on M. Any function / in ^l belongs to a particular Ê{Ll) and is thus the symbol of an operator AÍ ' acting on ^k for k > I. One has thus constructed, for a given /, a family of quantum operators parametrized by an integer k .From the point of view of deformation theory [1], where quantization is realised at the level of the algebra of functions, one can say that one has constructed a family of associative products on Ê(L'), with values in 8¿, parametrized by an integer k ; indeed f*kg = AfÀf, f,g£Ê(Ll),k>l.

Fuzzy complex Grassmannians and quantization of line bundles

“…One easily verifies that π(g) P N π(g) −1 is the projector onto the "coherent state" associated to x = gK ∈ M (compare [7]). Thus the coherent state map used in the Berezin-Toeplitz quantization of Kähler manifolds (see [6]) is here equal to…”

“…First, in the Berezin-Toeplitz quantization of a compact Kähler manifold (M , ω) with integral Kähler form ω. Denoting the N -th power of the pre-quantization line bundle L (with first Chern form equal to ω) by L ⊗N and its holomorphic section module Γ hol (M , L ⊗N ) by H N , the algebra A N := End C (H N ) is of course isomorphic to a matrix algebra. The Toeplitz quantization map T N : C ∞ (M ) −→ A N is defined by associating to a function f multiplication of holomorphic sections of L ⊗N by f followed by projection on the space of holomorphic sections (compare [6,7] resp. [5] for geometric resp.…”

“…We obtain by purely representationtheoretic methods that all complex Grassmannians M = Gr n (C n+m ) allow for a sequence of linear subspaces E N N ≥1 in their function algebra C ∞ (M ) such that E N ⊂ E N +1 and N ≥1 E N is dense in C ∞ (M ), and such that E N is isomorphic to a matrix algebra A N . In Section 3, we recall very briefly the Berezin-Toepliz quantization procedure (à la [6,7] and [4]) and prove that E N is in fact naturally identified with End C (Γ hol (M , L ⊗N )) via the Toeplitz quantization map T N . Let us note that this result does not come alone from general properties of T N but needs extra arguments from representation theory.…”

Normalized Weyl-type *-product on Kähler manifolds

“…This approach of quantization is applicable only to a flat manifold. There are papers on Berezin-type star-product on Kähler manifolds [5] [6]. The Berezin star-product is defined on certain Kähler manifolds by…”

Asymptotics of reproducing kernels on a plane domain

“…The latter, in turn, provides the basic tool for certain quantization procedures on Q (construction of *-products) [4], [5].…”