Spin structures on loop spaces that characterize string manifolds

TL;DR

This paper introduces a new type of spin structure on loop spaces that exists precisely when the underlying manifold is a string manifold, linking loop space geometry with string topology.

Contribution

It develops a novel spin structure on loop spaces characterized by a fusion product, aligning the existence of such structures with the string condition on manifolds.

Findings

New spin structures exist iff the manifold is string

Fusion product relates loops of frames to spin structures

Uses lifting gerbe theory and recent loop space results

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, as desired. 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-Murray, recent results of Stolz-Teichner on loop spaces, and some own results about string geometry and Brylinski-McLaughlin transgression.