Journal of Symplectic Geometry

Volume 13 (2015)

Number 4

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

Pages: 1001 – 1047



Grégory Ginot (Institut de Mathématiques de Jussieu, Paris Rive Gauche UPMC, Sorbonne Universités Paris 6, Paris, France)

Mathieu Stiénon (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)


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