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Spin structures on loop spaces that characterize string manifolds

Konrad Waldorf

Algebraic & Geometric Topology 16 (2016) 675–709

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.

string structures, loop group, transgression, fusion product
Mathematical Subject Classification 2010
Primary: 57R15
Secondary: 58B05, 53C08
Received: 1 December 2013
Revised: 10 April 2015
Accepted: 27 July 2015
Published: 26 April 2016
Konrad Waldorf
Institut für Mathematik und Informatik
Ernst-Moritz-Arndt-Universität Greifswald
Walther-Rathenau-Str. 47
D-17493 Greifswald