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

Grégory Ginot, Mathieu Stiénon

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

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 dierentiable stacks and G-gerbes over dierentiable stacks). This approach also allows us to identify G-bound gerbes and [Z(G) → 1]-group bundles over dierentiable 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.

Original languageEnglish (US)
Pages (from-to)1001-1048
Number of pages48
JournalJournal of Symplectic Geometry
Volume13
Issue number4
DOIs
StatePublished - Jan 1 2015

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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