Differentiable Stacks and Gerbes

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.

48 pages, minor revision, examples added, references added and updated

References in corpus (3)

• Foundations of Topological Stacks I
• Topological and Smooth Stacks
• Introduction to the language of stacks and gerbes

Cited by in corpus (2)

• The geometry of unitary 2-representations of finite groups and their 2-characters
• Anomalies in gauge theory and gerbes over quotient stacks