On geometric quantization of $b^m$-symplectic manifolds

TL;DR

This paper investigates the geometric quantization of $b^m$-symplectic manifolds with torus actions, revealing finite-dimensional modules for odd $m$ and asymptotic behavior for even $m$, advancing understanding of symplectic geometry and representation theory.

Contribution

It provides a detailed analysis of the geometric quantization of $b^m$-symplectic manifolds, including the behavior of resulting modules depending on the parity of $m$, which was not previously understood.

Findings

Finite-dimensional virtual $T$-modules for odd $m$

Asymptotic analysis of representations for even $m$

Extension of geometric quantization to $b^m$-symplectic manifolds

Abstract

We study the formal geometric quantization of $b^m$-symplectic manifolds equipped with Hamiltonian actions of a torus $T$ with nonzero leading modular weight. The resulting virtual $T$-modules are finite dimensional when $m$ is odd, as in [GMW2]; when $m$ is even, these virtual modules are not finite dimensional, and we compute the asymptotics of the representations for large weight.