Holonomy and Parallel Transport for Abelian Gerbes

Mackaay, Marco; Picken, Roger

doi:10.1006/aima.2002.2085

Cited by 55 publications

(90 citation statements)

References 20 publications

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“…In both cases, our claim shows to be true. Similar results for abelian gerbes have been obtained in [MP02]. Connections on a certain class of (possibly) non-abelian gerbes have been introduced by Breen and Messing [BM05].…”

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confidence: 66%

Smooth functors vs. differential forms

Schreiber

Waldorf

2011

Homology, Homotopy and Applications

153

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We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as derivatives of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces.

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supporting

confidence: 66%

Smooth functors vs. differential forms

Schreiber

Waldorf

2011

Homology, Homotopy and Applications

153

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“…This work was deeply inspired by the ideas of Breen and Messing [28,29], who considered a special class of 2-groups, and omitted the equation t(B) = d A + A ∧ A, since their sort of connection did not assign holonomies to surfaces. One should also compare the closely related work of Mackaay, Martins, and Picken [65,67,68], and the work of Pfeiffer and Girelli [76,56].…”

Section: -Connectionsmentioning

confidence: 99%

An invitation to higher gauge theory

2010

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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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“…Interest in gerbes has been revived recently following a concrete approach due to Hitchin and Chatterjee [15]. Gerbes can be understood both in terms of local geometry, local functions and forms, and in terms of non-local geometry, holonomies and parallel transports, and these two viewpoints are equivalent, in a sense made precise by Mackaay and the author in [16], following on from work by Barrett [5] and Caetano and the author [11]. In [16] holonomies around spheres and parallel transports along cylinders were considered.…”

Section: Introductionmentioning

confidence: 99%

TQFT’s and gerbes

Picken¹

2004

Algebr. Geom. Topol.

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We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a one-to-one correspondence between their local description in terms of locally-defined functions and forms and their non-local description in terms of a suitable class of embedded TQFT's.

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Holonomy and Parallel Transport for Abelian Gerbes

Cited by 55 publications

References 20 publications

Smooth functors vs. differential forms

Smooth functors vs. differential forms

An invitation to higher gauge theory

TQFT’s and gerbes

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