# String geometry vs. spin geometry on loop spaces

@article{Waldorf2015StringGV, title={String geometry vs. spin geometry on loop spaces}, author={Konrad Waldorf}, journal={Journal of Geometry and Physics}, year={2015}, volume={97}, pages={190-226} }

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 loops: thin homotopies and loop fusion. The central result of this article is an equivalence between these enhanced versions of spin structures on the loop space and string structures on the manifold itself. The equivalence exists in two settings: in a purely topological one and in a geometrical one… Expand

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#### References

SHOWING 1-10 OF 44 REFERENCES

Spin structures on loop spaces that characterize string manifolds

- Mathematics, Physics
- 2012

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 connections and Chern-Simons theory

- Mathematics
- 2009

We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is… Expand

Loop Spaces, Characteristic Classes and Geometric Quantization

- Mathematics
- 1994

This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical… Expand

A LOOP SPACE FORMULATION FOR GEOMETRIC LIFTING PROBLEMS

- Mathematics
- Journal of the Australian Mathematical Society
- 2011

Abstract We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and by Gomi. Lifting gerbes… Expand

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

- Mathematics
- 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… Expand

Equivalence of string and fusion loop-spin structures

- Mathematics, Physics
- 2013

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

ON THE GEOMETRY OF FREE LOOP SPACES

- Mathematics
- 2002

We verify the following three basic results on the free loop space LM. (1) We show that the set of all points, where the fundamental form on LM is nondegenerate, is an open subset. (2) The… Expand

Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps

- Mathematics
- 2009

We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles.… Expand

String structures on loop bundles

- Mathematics
- 1989

Differential geometry and topology of principal loop bundles (bundles of loop groups over loop spaces) are investigated. String structures, defined as bundle extensions corresponding to the central… Expand

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

- Mathematics, Physics
- 2010

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