Equivariant Dixmier-Douady Classes

TL;DR

This paper establishes a precise relationship between two models of equivariant cohomology in degree 3, linking the Dixmier-Douady classes of equivariant bundle gerbes and their stack counterparts.

Contribution

It proves that the natural morphism between Cartan and simplicial models of equivariant cohomology maps the Dixmier-Douady class appropriately.

Findings

The morphism relates the Dixmier-Douady classes in different models.

It clarifies the connection between equivariant bundle gerbes and $S^1$-gerbes over stacks.

The result enhances understanding of equivariant cohomology in degree 3.

Abstract

An equivariant bundle gerbe \`a la Meinrenken over a $G$-manifold $M$ is known to be a special type of $S^1$-gerbe over the differentiable stack $[M/G]$. We prove that the natural morphism relating the Cartan and simplicial models of equivariant cohomology in degree 3 maps the Dixmier-Douady class of an equivariant bundle gerbe \`a la Meinrenken to the Behrend-Xu-Dixmier-Douady class of the corresponding $S^1$-gerbe.