Higher gauge theory
Higher gauge theory
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DOI：
10.1090/conm/431/08264
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发表时间：
2005-11
期刊：
影响因子：
--
通讯作者：
J. Baez;U. Schreiber
中科院分区：
文献类型：
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作者：
J. Baez;U. Schreiber
就像规范理论描述了点粒子在束上的平行输运一样，更高规范理论描述了一维物体（例如弦）在2-束上的2-连接的平行输运。2-丛是丛的范畴化版本：也就是说，纤维不是流形而是具有适当光滑结构的范畴。在规范理论使用李群和李代数的地方，高规范理论使用它们的分类类似物：李2-群和李2-代数。我们描述了主2-丛上的2-联系理论，并解释了这与Breen和Messing关于非阿贝尔盖尔贝的联系理论的关系。我们的理论的显着特点是，2连接允许并行运输沿着路径和表面在参数化无关的方式。根据布林和梅辛的框架，这要求“假曲率”必须消失。在本文中，我们总结了我们的理论的主要结果没有证明。
Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2-groups and Lie 2-algebras. We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing's theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2-connection allows parallel transport along paths and surfaces in a parametrization-independent way. In terms of Breen and Messing's framework, this requires that the "fake curvature" must vanish. In this paper we summarize the main results of our theory without proofs.