Differentiable stacks and gerbes
Differentiable stacks and gerbes
复制标题
可微的堆栈和非洲菊
DOI：
10.4310/jsg.2011.v9.n3.a2
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发表时间：
2006
影响因子：
0.7
通讯作者：
P. Xu
中科院分区：
文献类型：
--
作者：
K. Behrend;P. Xu
我们介绍可微栈并解释与李群群的关系。然后我们在可微堆栈上研究 $S^1$-bundles 和 $S^1$-gerbes。特别是，我们建立了 $S^1$-gerbes 和 groupoid $S^1$-central 扩展之间的关系。我们定义群胚$S^1$-中心扩展的连接和弯曲，扩展了 Brylinski、Hitchin 和 Murray 的相应概念
流形上的 $S^1$-gerbes。我们通过根据连接和曲率的类似物提出陈氏类和迪克斯米尔-杜阿迪类的构造，在这种一般背景下发展了特征类的陈-韦尔理论。我们还描述了 $S^1$-bundles 和 $S^1$-gerbes 的预量化结果，扩展了 Weil 和 Kostant 的众所周知的结果。特别是，我们使用规定的类曲率数据给出了 $S^1$-中心扩展的显式构造。
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.