Stratified Pseudobundles and Quantization

TL;DR

This paper extends Geometric Quantization to singular spaces called Symplectic Stratified Spaces, developing stratified pseudobundles and demonstrating their use in quantizing singular quotients and examples beyond singular reduction.

Contribution

It introduces stratified pseudobundles as a new tool for quantization on singular spaces and applies this framework to specific classes of singular quotients.

Findings

Established [Q,R]=0 results for singular quotients of toric manifolds.

Developed the theory of stratified pseudobundles for singular geometric quantization.

Provided an example of singular quantization not arising from singular reduction.

Abstract

Geometric Quantization is a term used to describe a wide collection of techniques dating back to the 1960s in the work of Kirillov, Kostant, and Souriau, which take symplectic manifolds and produce complex vector spaces. The name comes from the natural interpretation of symplectic manifolds as the phase spaces of classical mechanical systems and complex vector spaces as the natural domains of wave functions in quantum mechanics. In this thesis, I extend the classical framework of Geometric Quantization to handle a class of singular spaces called Symplectic Stratified Spaces, which date back to the work of Sjamaar and Lerman in the 1990s. As part of this work, I develop the theory of stratified pseudobundles to serve as singular replacements for important auxiliary information in Geometric Quantization: prequantum line bundles and polarizations. I then use this formalism to provide [Q,R]=0 results for singular quotients of toric manifolds and cotangent bundles. I also provide an example of singular Geometric Quantization that does not arise from singular reduction.