Categorical quantization on K\"ahler manifolds

TL;DR

This paper develops a new categorical framework for deformation quantization on K"ahler manifolds, generalizing previous notions and establishing an equivalence with categories of holomorphic vector bundles and differential operators.

Contribution

It constructs a sheaf-enriched category for quantization of Hermitian holomorphic vector bundles on K"ahler manifolds and relates it to holomorphic differential operator categories.

Findings

Constructed a category $\

Established an equivalence with holomorphic vector bundle categories when $M$ is prequantizable.

Abstract

Generalizing deformation quantizations with separation of variables of a K\"ahler manifold $M$, we adopt Fedosov's gluing argument to construct a category $\mathsf{DQ}$, enriched over sheaves of $\mathbb{C}[[\hbar]]$-modules on $M$, as a quantization of the category of Hermitian holomorphic vector bundles over $M$ with morphisms being smooth sections of hom-bundles. We then define quantizable morphisms among objects in $\mathsf{DQ}$, generalizing Chan-Leung-Li's notion [4] of quantizable functions. Upon evaluation of quantizable morphisms at $\hbar = \tfrac{\sqrt{-1}}{k}$, we obtain an enriched category $\mathsf{DQ}_{\operatorname{qu}, k}$. We show that, when $M$ is prequantizable, $\mathsf{DQ}_{\operatorname{qu}, k}$ is equivalent to the category $\mathsf{GQ}$ of holomorphic vector bundles over $M$ with morphisms being holomorphic differential operators, via a functor obtained from Bargmann-Fock actions.