Singular Poisson-K\"ahler geometry of stratified K\"ahler spaces and quantization

TL;DR

This paper explores the use of stratified K"ahler spaces for quantization in classical phase spaces with singularities, establishing a framework that aligns geometric reduction with quantum procedures.

Contribution

It introduces a method of holomorphic quantization on stratified K"ahler spaces, linking classical singularities with quantum structures and demonstrating the compatibility of reduction and quantization.

Findings

Holomorphic quantization yields a costratified Hilbert space reflecting classical singularities.

Reduction after quantization matches quantization after reduction for certain group actions.

The framework extends geometric quantization to singular phase spaces with stratified structures.

Abstract

In the presence of classical phase space singularities the standard methods are insufficient to attack the problem of quantization.In certain situations the difficulties can be overcome by means of K\"ahler quantization on stratified K\"ahler spaces. Such a space is a stratified symplectic space together with a complex analytic structure which is compatible with the stratified symplectic structure; in particular each stratum is a K\"ahler manifold in an obvious fashion. Holomorphic quantization on a stratified K\"ahler space then yields a costratified Hilbert space, a quantum object having the classical singularities as its shadow. Given a K\"ahler manifold with a hamiltonian action of a compact Lie group that also preserves the complex structure, reduction after quantization coincides with quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond but the invariant unreduced and reduced quantum observables as well