Spin structures on loop spaces that characterize string manifolds

Waldorf, Konrad¹

¹Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität Greifswald, Walther-Rathenau-Str. 47, D-17493 Greifswald, Germany

Algebraic and Geometric Topology, Tome 16 (2016) no. 2, pp. 675-709

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.675

Classification : 57R15, 58B05, 53C08
Keywords: string structures, loop group, transgression, fusion product

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