Differentiable stacks and gerbes

By K. Behrend, P. Xu. 2006

199 citations recorded by the source corpus.

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connections and curvings for groupoid $S^1$-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for $S^1$-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 analogues of connections and curvatures. We also describe a prequantization result for both $S^1$-bundles and $S^1$-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of $S^1$-central extensions with prescribed curvature-like data.

Subfields: Geometry and Topology, Mathematical Physics, Pathology and Forensic Medicine

View Paper: https://www.semanticscholar.org/paper/2339fa2f8424f79cfef58f6116fb2ca579bcdcbe