Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

Abstract

ABSTRACT. I repeat my definition for quantization of a vector bundle. For the cases of the Toeplitz and geometric quantizations of a compact Kähler manifold, I give a construction for quantizing any smooth vector bundle which depends functorially on a choice of connection on the bundle.

Generating functions for effective hamiltonians. Symmetry, topology, combinatorics

“…To see that we can use the deformation quantization approach which gives the following expression for the number of quantum states [24,25] (the so-called general index formula):…”

Fuzzy complex Grassmannians and quantization of line bundles

“…The quantization of smooth vector bundles over compact Kähler manifolds is defined and studied by E. Hawkins in [11,12], aiming for a generalization of the Berezin-Toeplitz quantization procedure. Let L be, as before, the dual of the determinant line bundle over the Grassmannian M = Gr n (C n+m ) (n ≥ m ≥ 1).…”

“…Recall here that every complex line bundle over M is (up to isomorphism) of the form L ⊗k for some k ∈ Z. We then consider the "SU (n + m)-equivariant quantization" of the bundle L ⊗k −→ M in the sense of, e.g., E. Hawkins (see [11,12]). This quantization is given by a sequence of…”

Deformation quantization of surjective submersions and principal fibre bundles

“…However, a global description of the deformed frame bundle is still missing and structure groups beyond the general linear group do not seem to be accessible by this approach. Also in [29,30] the deformation theory of vector bundles in the context of strict deformation quantizations is discussed.…”