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 S 1 -central extensions of Lie groupoids. The definition requires that the quasipresymplectic 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.

Differential cocycles and Dixmier–Douady bundles

“…1 (M) results in a 'strictification' of the bicategory of DD-bundles to the strict 2-category of differential cocycles, which can be useful in practice. (For example, they were used in [13] to verify the compatibility among certain definitions of prequantization in the context of Hamiltonian actions of quasi-symplectic/twisted presymplectic groupoids.) Second, the equivalence as 2-stacks provides an equivalence of equivariant objects as well.…”