On geometric quantization of $b$-symplectic manifolds

TL;DR

This paper develops a framework for geometric quantization of $b$-symplectic manifolds with Hamiltonian torus actions, demonstrating that the resulting quantizations are finite-dimensional modules, extending quantization theory to this class of manifolds.

Contribution

It introduces a notion of pre-quantization for $b$-symplectic manifolds and constructs a formal geometric quantization for those with Hamiltonian torus actions, highlighting finite-dimensionality.

Findings

Quantizations form finite-dimensional $T$-modules.

Pre-quantization notion adapted to $b$-symplectic manifolds.

Extension of geometric quantization to $b$-symplectic setting.

Abstract

We study a notion of pre-quantization for $b$-symplectic manifolds. We use it to construct a formal geometric quantization of $b$-symplectic manifolds equipped with Hamiltonian torus actions with nonzero modular weight. We show that these quantizations are finite dimensional $T$-modules.