Differentiable Stacks and Gerbes

Kai Behrend; Ping Xu

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Differentiable Stacks and Gerbes

Kai Behrend·Ping Xu

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Abstract

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.

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BibTeX RIS

Kai Behrend, Ping Xu. 2008-12-31. Differentiable Stacks and Gerbes. https://arxiv.org/abs/math/0605694

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