Groupoid equivariant prequantization

TL;DR

This paper extends the concept of prequantization for quasi-presymplectic Lie groupoids to include non-exact structures using Dixmier-Douady bundles, generalizing previous exact-structure definitions and exploring related invariance properties.

Contribution

It introduces a new prequantization framework for non-exact quasi-presymplectic Lie groupoids using Dixmier-Douady bundles, expanding prior exact-structure approaches.

Findings

Compatible with Laurent-Gengoux and Xu's definition when structures are exact

Establishes Morita invariance of the prequantization

Demonstrates properties related to symplectic reduction

Abstract

In their 2005 paper, C. Laurent-Gengoux and P. Xu define prequantization for pre-Hamiltonian actions of quasi-presymplectic Lie groupoids in terms of central extensions of Lie groupoids. The definition requires that the quasi-presymplectic structure be exact (i.e. the closed 3-form on the unit space of the Lie groupoid must be exact). In the present paper, we define prequantization for pre-Hamiltonian actions of (not necessarily exact) quasi-presymplectic Lie groupoids in terms of Dixmier-Douady bundles. The definition is a natural adaptation of E. Meinrenken's treatment of prequantization for quasi-Hamiltonian Lie group actions with group-valued moment map. The definition given in this paper is shown to be compatible with the definition of Laurent-Gengoux and Xu when the underlying quasi-presymplectic structure is exact. Properties related to Morita invariance and symplectic reduction are established.