Symmetry in Deformation quantization and Geometric quantization

TL;DR

This paper investigates the relationship between deformation and geometric quantization on Kähler manifolds, classifies certain formal functions in deformation quantization, and examines symmetry compatibility in the quantization process.

Contribution

It provides a classification of degree 1 formal quantizable functions and analyzes symmetry compatibility between deformation and geometric quantization.

Findings

Formal functions are of the form $f = f_0 - rac{ar}{4\u00pi}(\uar f_0 + c)$

Real-valued $f_0$ correspond to Hamiltonian Killing vector fields

Compatibility conditions for symmetry actions in quantization are established.

Abstract

In this paper, we explore the quantization of K\"ahler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form $f = f_0 - \frac{\hbar}{4\pi}(\Delta f_0 + c)$ for a certain smooth (non-formal) function $f_0$. If $f_0$ is real-valued then $f_0$ corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian $G$-symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.