2008/01/31 by Gregory Ginot, Mathieu Stienon · 1 citation
Mathematics · Physics and Astronomy · #math.AT #hep-th #math.CT #math.DG
paper · pdf · doi:10.4310/jsg.2015.v13.n4.a6
published as J. Symplectic Geom. 13 (2015), no. 4, 1001-1047 · Presentation improved, 38 pages
arxiv created 2014/11/02 · arxiv updated 2019/10/15
Let G be a Lie group and G→\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→\Aut(G)]-bundles over Lie groupoids and, on the other hand, G-extensions of Lie groupoids (i.e. between principal [G→\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)→ 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.