Abstract
In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity.
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Acknowledgments
This paper is based on JH’s notes of lectures given by JB at the 2nd School and Workshop on Quantum Gravity and Quantum Geometry, held as part of the 2010 Corfu Summer Institute. We thank George Zoupanos, Harald Grosse, and everyone else involved with the Corfu Summer Institute for making our stay a pleasant and productive one.We thank Urs Schreiber for many discussions of higher gauge theory, and thank Tim van Beek and David Roberts for catching some errors. This research was partially supported by an FQXi grant.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Baez, J.C., Huerta, J. An invitation to higher gauge theory. Gen Relativ Gravit 43, 2335–2392 (2011). https://doi.org/10.1007/s10714-010-1070-9
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DOI: https://doi.org/10.1007/s10714-010-1070-9