Groupoid Equivariant Prequantization

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 S1-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-preysmplectic structure is exact. Properties related to Morita invariance and symplectic reduction are established.