Differentiable stacks and gerbes

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹-bundles and S¹-gerbes over differentiable stacks. In particular, we establish the relationship between S1-gerbes and groupoid S1-central extensions. We define connections and curvings for groupoid S1-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S1-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 S¹-bundles and S¹-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of S¹-central extensions with prescribed curvature-like data.