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 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.