Reduction theory for singular symplectic manifolds and singular forms on moduli spaces

TL;DR

This paper develops a general reduction theory for singular symplectic manifolds, extending existing frameworks to include broader classes of Hamiltonian functions and new constructions of quasi-Hamiltonian spaces, advancing geometric quantization efforts.

Contribution

It establishes a comprehensive Marsden-Weinstein reduction framework for general singular symplectic manifolds, including $b^m$-symplectic and folded symplectic types, with novel constructions of quasi-Hamiltonian spaces.

Findings

Extended reduction theory to $b^m$-symplectic manifolds

Broadened admissible Hamiltonian functions to include singularities

Constructed new singular quasi-Hamiltonian spaces via reduction

Abstract

The investigation of symmetries of b-symplectic manifolds and folded-symplectic manifolds is well-understood when the group under consideration is a torus (see, for instance, [GMPS,GLPR, GMW18a] for b-symplectic manifolds and [CGP, CM] for folded symplectic manifolds). However, reduction theory has not been set in this realm in full generality. This is fundamental, among other reasons, to advance in the "quantization commutes with reduction" programme for these manifolds initiated in [GMW18b, GMW21]. In this article, we fill in this gap and investigate the Marsden-Weinstein reduction theory under general symmetries for general $b^m$-symplectic manifolds and other singular symplectic manifolds, including certain folded symplectic manifolds. In this new framework, the set of admissible Hamiltonian functions is larger than the category of smooth functions as it takes the singularities of the differential forms into account. The quasi-Hamiltonian set-up is also considered and brand-new constructions of (singular) quasi-Hamiltonian spaces are obtained via a reduction procedure and the fusion product.