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Bundle gerbes
 J. London Math. Soc
, 1996
"... Abstract. An introduction to the theory of bundle gerbes and their relationship to HitchinChatterjee gerbes is presented. Topics covered are connective ..."
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Abstract. An introduction to the theory of bundle gerbes and their relationship to HitchinChatterjee gerbes is presented. Topics covered are connective
Higher gauge theory
"... Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1dimensional objects (e.g. strings) using 2connections on 2bundles. A 2bundle is a categorified version of a bundle: that is, one where t ..."
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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 1dimensional objects (e.g. strings) using 2connections on 2bundles. A 2bundle 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 2groups and Lie 2algebras. We describe a theory of 2connections on principal 2bundles 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 2connection allows parallel transport along paths and surfaces in a parametrizationindependent 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. 1
Categorified symplectic geometry and the classical string
, 2008
"... A Lie 2algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poi ..."
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Cited by 48 (10 self)
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A Lie 2algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an ndimensional field theory using a phase space that is an ‘nplectic manifold’: a finitedimensional manifold equipped with a closed nondegenerate (n + 1)form. Here we consider the case n = 2. For any 2plectic manifold, we construct a Lie 2algebra of observables. We then explain how this Lie 2algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2plectic structure for the string.
Gerbes, M5Brane Anomalies and E8 Gauge Theory
, 2004
"... Abelian gerbes and twisted bundles describe the topology of the NS 3form gauge field strength H. We review how they have been usefully applied to study and resolve global anomalies in open string theory. Abelian 2gerbes and twisted nonabelian gerbes describe the topology of the 4form field streng ..."
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Abelian gerbes and twisted bundles describe the topology of the NS 3form gauge field strength H. We review how they have been usefully applied to study and resolve global anomalies in open string theory. Abelian 2gerbes and twisted nonabelian gerbes describe the topology of the 4form field strength G of Mtheory. We show that twisted nonabelian gerbes are relevant in the study and resolution of global anomalies of multiple coinciding M5branes. Global anomalies for one M5brane have been studied by Witten and by Diaconescu, Freed and Moore. The structure and the differential geometry of twisted nonabelian gerbes (i.e. modules for 2gerbes) is defined and studied. The nonabelian 2form gauge potential living on multiple coinciding M5branes arises as curving (curvature) of twisted nonabelian gerbes. The nonabelian group is in general ˜ ΩE8, the central extension of the E8 loop group. The twist is in general necessary to cancel global anomalies due to the nontriviality of the 11dimensional 4form field strength G and due to the possible torsion present in the cycles the M5branes wrap. Our description of M5branes global anomalies leads to the D4branes one upon compactification of Mtheory to Type IIA theory. a
An Invitation to Higher Gauge Theory
, 2010
"... In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie gr ..."
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In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2group; 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 2group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2group, 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 2group’, which serves as a gauge 2group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2group’. We also touch upon higher structures such as the ‘gravity 3group’, and the Lie 3superalgebra that governs 11dimensional supergravity. 1
Central extensions of smooth 2groups and a finitedimensional string 2group, Geom. Topol. 15 (2011) 609–676
 Department of Mathematics, Massachusetts Institute of Technology
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Higher gauge theory
"... I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to ce ..."
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Cited by 16 (0 self)
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I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to certain 2categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2category of 2bundles over a given 2space under a given 2group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2space is the 2space corresponding to a given space and the 2group is the automorphism 2group of a given group, then this 2category is equivalent to the 2category of gerbes over that space under that group (being described by the same cohomological data).
Mbrane models from nonabelian gerbes
 JHEP
"... We make the observation that Mbrane models defined in terms of 3algebras can be interpreted as higher gauge theories involving Lie 2groups. Such gauge theories arise in particular in the description of nonabelian gerbes. This observation allows us to put M2 and M5brane models on equal footing ..."
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We make the observation that Mbrane models defined in terms of 3algebras can be interpreted as higher gauge theories involving Lie 2groups. Such gauge theories arise in particular in the description of nonabelian gerbes. This observation allows us to put M2 and M5brane models on equal footing, at least as far as the gauge structure is concerned. Furthermore, it provides a useful framework for various generalizations; in particular, it leads to a fully supersymmetric generalization of a previously proposed set of tensor multiplet equations. ar X iv
Crossed module bundle gerbes; classification, string group and differential geometry. Available as arXiv:math/0510078v2
"... We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module (H → D) there is a simplicial group NC(H→D), the nerve of the 1category defined by the crossed module and its geometric realization NC(H→D). Equivalence classes of princi ..."
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We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module (H → D) there is a simplicial group NC(H→D), the nerve of the 1category defined by the crossed module and its geometric realization NC(H→D). Equivalence classes of principal bundles with structure group NC(H→D)  are shown to be onetoone with stable equivalence classes of what we call crossed module bundle gerbes. We can also associate to a crossed module a 2category ˜ C(H→D). Then there are two equivalent ways how to view classifying spaces of NC(H→D)bundles and hence of NC(H→D)bundles and crossed module bundle gerbes. We can either apply the Wconstruction to NC(H→D) or take the nerve of the 2category ˜ C(H→D). We discuss the string group and string structures from this point of view. Also a simplicial principal bundle can be equipped with a simplicial connection and a Bfield. It is shown how in the case of a simplicial principal NC(H→D)bundle these simplicial objects give the bundle gerbe connection and the bundle gerbe Bfield.
AKSZ models of semistrict higher gauge theory
 JHEP 1303, 014 (2013) [arXiv:1112.2819 [hepth
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