Differentiable Stacks and Gerbes

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S-bundles and S-gerbes over differentiable stacks. In particular, we establish the relationship between S-gerbes and groupoid S-central extensions. We define connections and curvings for groupoid S-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S-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-bundles and S-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of S-central extensions with prescribed curvature-like data.