From String structures to Spin structures on loop spaces

Abstract

Let X be a n-dimensional smooth manifold, with n 3. In a series of papers culminating in Spin structures on loop spaces that characterize string manifolds, arXiv:1209.1731, Konrad Waldorf recently gave the first rigorous proof that a String structure on X induces a Spin structure on its loop space. Here we give a closely related but independent proof of this result by working in the more general setting of smooth stacks. In particular, the crucial point in our proof is the existence of a natural morphism of smooth stacks BSpin ! B2(BU(1)conn) refining the first fractional Pontryagin class. Once this morphism is exhibited, we show how Waldorf’s result follows from general constructions in the setting of smooth stacks.