Locally toric manifolds and singular Bohr-Sommerfeld leaves

TL;DR

This paper investigates the geometric quantization of compact symplectic manifolds modeled locally on toric manifolds, deriving a counting rule for Bohr-Sommerfeld leaves that excludes singular leaves, contrasting with Kähler polarization methods.

Contribution

It extends Sniatycki's quantization results to locally toric manifolds with real polarization, explicitly excluding singular Bohr-Sommerfeld leaves from the count.

Findings

Derived a quantization formula similar to Sniatycki's for locally toric manifolds.

Established that singular Bohr-Sommerfeld leaves are not included in the quantization count.

Provided a comparison between real polarization and Kähler polarization quantizations.

Abstract

When geometric quantization is applied to a manifold using a real polarization which is "nice enough", a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld leaves. Subsequently, several authors have taken this as motivation for counting Bohr-Sommerfeld leaves when studying the quantization of manifolds which are less "nice". In this paper, we examine the quantization of compact symplectic manifolds that can locally be modelled by a toric manifold, using a real polarization modelled on fibres of the moment map. We compute the results directly, and obtain a theorem similar to Sniatycki's, which gives the quantization in terms of counting Bohr-Sommerfeld leaves. However, the count does not include the Bohr-Sommerfeld leaves which are singular. Thus the quantization obtained is different from the quantization obtained using a Kaehler polarization.