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Stratified Pseudobundles and Quantization

Ethan Ross

TL;DR

The thesis extends geometric quantization to singular spaces by developing a framework of symplectic stratified spaces equipped with stratified prequantum data and polarizations. It introduces stratified pseudobundles as singular replacements for prequantum line bundles and polarizations, and proves that equivariant quantization data descend under singular reduction, yielding a canonical $[Q,R]$ map. The work establishes $[Q,R]=0$ for singular quotients of toric manifolds and for cotangent bundles, demonstrating the viability of quantization in singular settings and providing a concrete mechanism to relate quantized data before and after reduction. It also develops a non-singular groundwork (including mixed polarizations) and a systematic linear-to-singular reduction theory, with detailed treatments of toric Delzant constructions and cotangent bundle reductions. Together, these contributions extend geometric quantization to a broad class of singular spaces, enabling new connections between symplectic geometry, representation theory, and 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.

Stratified Pseudobundles and Quantization

TL;DR

The thesis extends geometric quantization to singular spaces by developing a framework of symplectic stratified spaces equipped with stratified prequantum data and polarizations. It introduces stratified pseudobundles as singular replacements for prequantum line bundles and polarizations, and proves that equivariant quantization data descend under singular reduction, yielding a canonical map. The work establishes for singular quotients of toric manifolds and for cotangent bundles, demonstrating the viability of quantization in singular settings and providing a concrete mechanism to relate quantized data before and after reduction. It also develops a non-singular groundwork (including mixed polarizations) and a systematic linear-to-singular reduction theory, with detailed treatments of toric Delzant constructions and cotangent bundle reductions. Together, these contributions extend geometric quantization to a broad class of singular spaces, enabling new connections between symplectic geometry, representation theory, and 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.
Paper Structure (84 sections, 228 theorems, 1000 equations, 4 figures)

This paper contains 84 sections, 228 theorems, 1000 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Non-Singular Preliminaries
  3. Background in Equivariant Geometry
  4. Proper Group Actions and Notation
  5. Local Normal Form for Proper Actions
  6. The Slice Theorem for Proper Group Actions
  7. Tangent Spaces of Manifolds of Symmetry and Orbit-Type Classes
  8. Equivariant Vector Bundles
  9. Symplectic Geometry
  10. Symplectic Vector Spaces
  11. Symplectic Complements and Distinguished Subspaces
  12. Linear Symplectic Reduction
  13. Symplectic and Poisson Manifolds
  14. Hamiltonian Spaces and Reduction
  15. Linear Symplectic Representations
  16. ...and 69 more sections

Key Result

Proposition 2.6

Let $M$ be a $G$-space. Then the following are equivalent.

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (625)

  • Definition 2.1: $G$-space
  • Example 2.2
  • Definition 2.3: Equivariant Map
  • Definition 2.4: fixed point Set, Manifold of Symmetry, Orbit-Type Class
  • Definition 2.5: Proper Action/ Proper $G$-Space
  • Proposition 2.6: lee_introduction_2013
  • Corollary 2.7
  • proof
  • Corollary 2.8
  • proof
  • ...and 615 more