G-gerbes, principal 2-group bundles and characteristic classes

TL;DR

This paper establishes a correspondence between principal 2-group bundles and G-gerbes over Lie groupoids, introduces universal characteristic classes, and generalizes Dixmier–Douady classes with integral properties.

Contribution

It provides an explicit classification of 2-group bundles over Lie groupoids and introduces universal characteristic classes for these structures.

Findings

Correspondence between 2-group bundles and G-gerbes over Lie groupoids

Introduction of universal characteristic classes for 2-group bundles

Dixmier–Douady classes coincide with universal characteristic classes and are integral

Abstract

Let $G$ be a Lie group and $G\to\Aut(G)$ be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group $[G\to\Aut(G)]$-bundles over Lie groupoids and, on the other hand, $G$-extensions of Lie groupoids (i.e.\ between principal $[G\to\Aut(G)]$-bundles over differentiable stacks and $G$-gerbes over differentiable stacks). This approach also allows us to identify $G$-bound gerbes and $[Z(G)\to 1]$-group bundles over differentiable stacks, where $Z(G)$ is the center of $G$. We also introduce universal characteristic classes for 2-group bundles. For groupoid central $G$-extensions, we introduce Dixmier--Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier--Douady classes are integral.