Differential Cocycles and Dixmier-Douady Bundles

TL;DR

This paper establishes equivalences between models of $ ext{S}^1$-gerbes and differential 3-cocycles, connecting Dixmier-Douady bundles with higher geometric structures and extending to equivariant cases over Lie groupoids.

Contribution

It provides a comprehensive framework linking Dixmier-Douady bundles to differential cocycles and gerbes, including equivariant extensions over Lie groupoids.

Findings

Equivalence between Dixmier-Douady bundles and differential 3-cocycles of height 1.

Height 2 and 3 cocycles correspond to gerbes with connection and curving.

Extension of these equivalences to the equivariant setting over Lie groupoids.

Abstract

This paper exhibits equivalences of 2-stacks between certain models of $\mathbb{S}^1$-gerbes and differential 3-cocycles. We focus primarily on the model of Dixmier-Douady bundles, and provide an equivalence between the 2-stack of Dixmier-Douady bundles and the 2-stack of differential 3-cocycles of height 1, where the 'height' is related to the presence of connective structure. Differential 3-cocycles of height 2 (resp. height 3) are shown to be equivalent to $\mathbb{S}^1$-bundle gerbes with connection (resp. with connection and curving). These equivalences extend to the equivariant setting of $\mathbb{S}^1$-gerbes over Lie groupoids.