Differentiable Stacks and Gerbes
Kai Behrend, Ping Xu
TL;DR
The paper develops a geometric framework for differentiable stacks by bridging them with Lie groupoids, and uses this bridge to study $S^1$-bundles and $S^1$-gerbes on stacks. It establishes a precise correspondence between $S^1$-gerbes over a differentiable stack and Morita equivalence classes of groupoid $S^1$-central extensions, enabling a concrete realization of $H^2(\mathfrak X,S^1)$ and the Dixmier–Douady class in $H^3(\mathfrak X,\mathbb Z)$ via Chern–Weil theory and pseudo-connections. The work develops a de Rham/Čech double complex calculus for groupoids, proves prequantization theorems in this generalized setting, and investigates how Lie algebroid central extensions capture infinitesimal data and integration to groupoid central extensions. These results provide a robust geometric toolkit for gerbes, twisted cohomology, and prequantization on differentiable stacks with potential applications to twisted K-theory and quantization in broader contexts.
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.