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47
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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Cited by 41 (12 self)
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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
The basic gerbe over a compact simple Lie group
"... Abstract. Let G be a compact, simply connected simple Lie group. We give a construction of an equivariant gerbe with connection on G, with equivariant 3curvature representing a generator of H 3 G(G, Z). Technical tools developed in this context include a gluing construction for gerbes and a theory ..."
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Cited by 29 (1 self)
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Abstract. Let G be a compact, simply connected simple Lie group. We give a construction of an equivariant gerbe with connection on G, with equivariant 3curvature representing a generator of H 3 G(G, Z). Technical tools developed in this context include a gluing construction for gerbes and a theory of equivariant bundle gerbes. 1.
Combinatorics of nonabelian gerbes with connection and curvature
, 203
"... Abstract: We give a functorial definition of Ggerbes over a simplicial complex when the local symmetry group G is nonAbelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a fibered category equipped with a functorial connectio ..."
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Cited by 22 (2 self)
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Abstract: We give a functorial definition of Ggerbes over a simplicial complex when the local symmetry group G is nonAbelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a fibered category equipped with a functorial connection over the space of edgepaths. By computing the curvature of the latter on the faces of an infinitesimal 4simplex, we recover the cocycle identities satisfied by the curvature of this gerbe. The link with BFtheories suggests that gerbes provide a framework adapted to the geometric formulation of strongly coupled gauge theories.
A homotopy double groupoid of a Hausdorff space II: A van Kampen Theorem
 THEORY AND APPLICATIONS OF CATEGORIES
, 2005
"... This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2dimensional, localtoglobal ..."
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Cited by 20 (11 self)
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This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2dimensional, localtoglobal problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These results have recently been generalised to all dimensions by Philip Higgins.
Higher YangMills theory
"... Electromagnetism can be generalized to Yang–Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2form electromagnetism to a kind of ‘higherdimensional Yang–Mills theory’. It turns out that to do this, one should repla ..."
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Cited by 20 (1 self)
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Electromagnetism can be generalized to Yang–Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2form electromagnetism to a kind of ‘higherdimensional Yang–Mills theory’. It turns out that to do this, one should replace the Lie group by a ‘Lie 2group’, which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a ‘Lie crossed module’: a pair of Lie groups G, H with a homomorphism t: H → G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing’s ideas on the geometry of nonabelian gerbes, one can define ‘principal 2bundles ’ for any Lie 2group C and do gauge theory in this new context. Here we only consider trivial 2bundles, where a connection consists of a gvalued 1form together with an hvalued 2form, and its curvature consists of a gvalued 2form together with a hvalued 3form. We generalize the Yang–Mills action for this sort of connection, and use this to derive ‘higher Yang– Mills equations’. Finally, we show that in certain cases these equations admit selfdual solutions in five dimensions. 1
Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory
, 2003
"... Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge field ..."
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Cited by 18 (3 self)
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Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of YangMills theories with 2form gauge potential.
Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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Cited by 18 (7 self)
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
Homotopy quantum field theories and the homotopy cobordism category in dimension 1+1
"... Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor ..."
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Cited by 16 (0 self)
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Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor
Gerbes and homotopy quantum field theories
, 2002
"... For manifolds with freely generated first homology, we characterize gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1dimensional homotopy quantum field theories, and we show that flat gerbes on such spaces a ..."
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Cited by 15 (5 self)
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For manifolds with freely generated first homology, we characterize gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1dimensional homotopy quantum field theories, and we show that flat gerbes on such spaces as above are the same as a specific class of rank one homotopy quantum field theories.
Parallel transport and functors
"... Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. Th ..."
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Cited by 15 (5 self)
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Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to substitute categories of functors for categories of smooth fibre bundles with connection. We indicate that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport. Table of Contents