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

Let $G$ be a Lie group and $G \to \mathrm{Aut}(G)$ 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 $[G \to \mathrm{Aut}(G)]$-bundles over Lie groupoids and, on the other hand, $G$-extensions of Lie groupoids (i.e. between principal $[G \to \mathrm{Aut}(G)]$-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.

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Published 17 March 2016