Journal of Symplectic Geometry

Differentiable stacks and gerbes

Kai Behrend and Ping Xu

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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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J. Symplectic Geom., Volume 9, Number 3 (2011), 285-341.

First available in Project Euclid: 11 July 2011

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Behrend, Kai; Xu, Ping. Differentiable stacks and gerbes. J. Symplectic Geom. 9 (2011), no. 3, 285--341.

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