An invitation to higher gauge theory |
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Authors: | John C Baez John Huerta |
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Institution: | (1) Department of Mathematics, University of California, Berkeley, CA, 94720-3840, U.S.A;(2) KdV Institute for Mathematics, University of Amsterdam, Science Park, 904 1098 Amsterdam, XH, The Netherlands;(3) Department of Mathematics, University of Aarhus, Aarhus, Denmark |
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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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