Differentiable Stacks and Gerbes

TL;DR

This paper develops a comprehensive theory of differentiable stacks, exploring their relation to Lie groupoids, and extends classical concepts like connections, characteristic classes, and prequantization to the setting of $S^1$-gerbes and central extensions.

Contribution

It introduces the notion of differentiable stacks, relates $S^1$-gerbes to groupoid central extensions, and generalizes Chern-Weil theory and prequantization to this broader context.

Findings

Established the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions.

Developed a generalized Chern-Weil theory for characteristic classes on differentiable stacks.

Provided explicit constructions of $S^1$-central extensions with prescribed curvature data.

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.