Symmetry in Deformation quantization and Geometric quantization

Abstract

In this paper, we explore the quantization of Kähler 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 $f=f_0-\frac{\hslash }{4\ddot{I}€}(\hat{I}”f_0+c)$ 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.