Geometric quantization of integrable systems with hyperbolic singularities

TL;DR

This paper explores the geometric quantization of compact surfaces with integrable systems, focusing on how hyperbolic singularities influence the quantization process and its dependence on polarization.

Contribution

It provides a detailed analysis of the impact of hyperbolic singularities on geometric quantization, highlighting the polarization dependence in the process.

Findings

Hyperbolic singularities contribute infinite-dimensional components to the quantization.

Quantization depends strongly on the choice of polarization due to singularities.

The study advances understanding of quantization in the presence of nondegenerate singularities.

Abstract

We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we compute the effect of hyperbolic singularities, which make an infinite-dimensional contribution to the quantization, thus showing that this quantization depends strongly on polarization.