Open Access
September 2011 Differentiable stacks and gerbes
Kai Behrend, Ping Xu
J. Symplectic Geom. 9(3): 285-341 (September 2011).


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 analog 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.


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Kai Behrend. Ping Xu. "Differentiable stacks and gerbes." J. Symplectic Geom. 9 (3) 285 - 341, September 2011.


Published: September 2011
First available in Project Euclid: 11 July 2011

zbMATH: 1039.58016
MathSciNet: MR2817778

Rights: Copyright © 2011 International Press of Boston

Vol.9 • No. 3 • September 2011
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