Higher Holonomies, Geometric Loop Groups and Smooth Deligne Cohomology

Gajer, Paweł

doi:10.1007/978-1-4612-1770-1_10

Cited by 16 publications

(28 citation statements)

References 6 publications

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“…7.2), because (F y + dB y ) (v, w), according to our formula for H at the end of Sect. 5, is equal to −i ∂ 2 ∂k∂l log H(s y (k, l))| (k,l)=(0,0) = 0, following the notation in (15). The smooth 2-parameter family of 2-loops s y (k, l) starts at the trivial 2-loop at y after recentering, so we can indeed apply Lem.…”

Section: Theorem 81 Given An Arbitrary 2-holonomymentioning

confidence: 99%

“…Finally we note that there is also a notion of n-gerbe connection, and the parallel transport of an n-gerbe connection is defined along n + 1-dimensional paths. Some details of these notions remain to be worked out completely, but Gajer [14,15] has worked out a considerable part already. The truly categorical nature of gerbe-connections comes to light when we study the general setting of their parallel transport.…”

Section: Introductionmentioning

confidence: 99%

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Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹

2002

Advances in Mathematics

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In this paper we establish a one-to-one correspondence between U (1)-gerbes with connections, on the one hand, and their holonomies, for simply-connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with * presently working as a postdoc at the University of Nottingham, UK 1 group U (1) on a simply-connected manifold M is a group morphism from the thin second homotopy group to U (1), satisfying a smoothness condition, where a homotopy between maps from [0, 1] 2 to M is thin when its derivative is of rank ≤ 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply-connected case.

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Section: Theorem 81 Given An Arbitrary 2-holonomymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹

2002

Advances in Mathematics

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“…We should remark that, when finishing this paper, we found an article by Gajer [15] in which he already obtains the characterization of Abelian ngerbes with n-gerbe-connections in terms of holonomy maps. However, Gajer's approach is very different from ours.…”

Section: Introductionmentioning

confidence: 95%

“…Finally, we note that there is also a notion of n-gerbeconnection, and the parallel transport of an n-gerbe-connection is defined along ðn þ 1Þ-dimensional paths. Some details of these notions remain to be worked out completely, but Gajer [14,15] has worked out a considerable part already.…”

Section: Introductionmentioning

confidence: 98%

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Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹,

Picken²

2002

Advances in Mathematics

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In this paper, we establish a one-to-one correspondence between Uð1Þ-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group Uð1Þ on a simply connected manifold M is a group morphism from the thin second homotopy group to Uð1Þ; satisfying a smoothness condition, where a homotopy between maps from ½0; 1 2 to M is thin when its derivative is of rank 42: For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. # 2002 Elsevier Science (USA)

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Differential Characters and Geometric Chains

Bär

Becker

2014

Differential Characters

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ABSTRACT. We study Cheeger-Simons differential characters and provide geometric descriptions of the ring structure and of the fiber integration map. The uniqueness of differential cohomology (up to unique natural transformation) is proved by deriving an explicit formula for any natural transformation between a differential cohomology theory and the model given by differential characters. Fiber integration for fibers with boundary is treated in the context of relative differential characters. As applications we treat higher-dimensional holonomy, parallel transport, and transgression.

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Higher Holonomies, Geometric Loop Groups and Smooth Deligne Cohomology

Cited by 16 publications

References 6 publications

Holonomy and Parallel Transport for Abelian Gerbes

Holonomy and Parallel Transport for Abelian Gerbes

Holonomy and Parallel Transport for Abelian Gerbes

Differential Characters and Geometric Chains

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