Geometric quantization via cotangent models

TL;DR

This paper introduces a comprehensive geometric quantization model for integrable systems with singularities, extending previous cotangent models by including singular orbits and prequantum line bundles, applicable to various physical and mathematical systems.

Contribution

It develops a universal cotangent model for geometric quantization that accounts for singularities and aligns with classical methods, improving upon previous models.

Findings

Models handle elliptic, hyperbolic, and focus-focus singularities.

Corrects infinite contributions in previous approaches.

Applicable to physical systems like the spherical pendulum.

Abstract

In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the previous cotangent models by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler's equations of the rigid body on $T^*(SO(3))$ to a sphere. Our geometric quantization formulation coincides with the previous models away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.