# Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles

@article{Waldorf2011TransgressionTL, title={Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles}, author={Konrad Waldorf}, journal={arXiv: Differential Geometry}, year={2011} }

We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops. The equivalence is established by a functor called regression, and complements a similar equivalence for bundles and gerbes equipped with connections, derived previously in Part II of this series of papers. The two… Expand

#### 16 Citations

Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection

- Mathematics, Physics
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We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with… Expand

Spin structures on loop spaces that characterize string manifolds

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Classically, a spin structure on the loop space of a manifold is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension. Heuristically, the… Expand

String geometry vs. spin geometry on loop spaces

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Abstract We introduce various versions of spin structures on free loop spaces of smooth manifolds, based on a classical notion due to Killingback, and additionally coupled to two relations between… Expand

Connes fusion of spinors on loop space

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The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the… Expand

Loop-fusion cohomology and transgression

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- 2015

`Loop-fusion cohomology' is defined on the continuous loop space of a manifold in terms of \vCech cochains satisfying two multiplicative conditions with respect to the fusion and figure-of-eight… Expand

Transgressive loop group extensions

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A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group.… Expand

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We describe the quantization of 2-plectic manifolds as they arise in the context of the quantum geometry of M-branes and non-geometric flux compactifications of closed string theory. We review the… Expand

Equivalence of string and fusion loop-spin structures

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The importance of the fusion relation of loops was recognized in the context of spin structures on the loop space by Stolz and Teichner and further developed by Waldorf. On a spin manifold M the… Expand

The Canonical 2-Gerbe of a Holomorphic Vector Bundle

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For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a… Expand

Geometric quantization in representation theory

- 2018

The pioneering works on geometric quantization are due to J.M. Souriau [24], B. Kostant [12], although many of their ideas were based on previous works by A.A. Kirillov [13]. In this theory the… Expand

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