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arxiv: math/0511710 · v2 · submitted 2005-11-29 · 🧮 math.DG · math.CT

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Higher Gauge Theory

John C. Baez , Urs Schreiber
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classification 🧮 math.DG math.CT
keywords theorygaugeconnectionsbundleshigherparalleltransportalgebras
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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.

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