Spin structures on loop spaces that characterize string manifolds

Waldorf, Konrad

doi:10.2140/agt.2016.16.675

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Spin structures on loop spaces that characterize string manifolds

Konrad Waldorf

Algebraic & Geometric Topology 16 (2016) 675–709

DOI: 10.2140/agt.2016.16.675

Abstract

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 loop space of a manifold is spin if and only if the manifold itself is a string manifold, against which it is well known that only the if part is true in general. In this article we develop a new version of spin structures on loop spaces that exists if and only if the manifold is string. This new version consists of a classical spin structure plus a certain fusion product related to loops of frames in the manifold. We use the lifting gerbe theory of Carey and Murray, recent results of Stolz and Teichner on loop spaces, and some of our own results about string geometry and Brylinski–McLaughlin transgression.

Keywords

string structures, loop group, transgression, fusion product

Mathematical Subject Classification 2010

Primary: 57R15

Secondary: 58B05, 53C08

References

Publication

Received: 1 December 2013

Revised: 10 April 2015

Accepted: 27 July 2015

Published: 26 April 2016

Authors

Konrad Waldorf
Institut für Mathematik und Informatik Ernst-Moritz-Arndt-Universität Greifswald Walther-Rathenau-Str. 47 D-17493 Greifswald Germany
http://waldorf.math-inf.uni-greifswald.de

Volume 16, issue 2 (2016)