Differentiable stacks and gerbes

Kai Behrend, Ping Xu

Research output: Contribution to journalArticle

46 Scopus citations

Abstract

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S1-bundles and S1-gerbes over differentiable stacks. In particular, we establish the relationship between S1-gerbes and groupoid S1-central extensions. We define connections and curvings for groupoid S1-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S1-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analog of connections and curvatures. We also describe a prequantization result for both S1-bundles and S1-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of S1-central extensions with prescribed curvature-like data.

Original languageEnglish (US)
Pages (from-to)285-341
Number of pages57
JournalJournal of Symplectic Geometry
Volume9
Issue number3
DOIs
StatePublished - Sep 2011

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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