Portal de Periódicos da CAPES

Differentiable stacks and gerbes

2011; Volume: 9; Issue: 3 Linguagem: Inglês

10.4310/jsg.2011.v9.n3.a2

1540-2347

Kai Behrend, Ping Xu,

Algebraic structures and combinatorial models

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.