String geometry vs. spin geometry on loop spaces
Abstract
We introduce various versions of spin structures on free loop spaces of smooth manifolds, based on a classical notion due to Killingback, and additionally coupled to two relations between loops: thin homotopies and loop fusion. The central result of this article is an equivalence between these enhanced versions of spin structures on the loop space and string structures on the manifold itself. The equivalence exists in two settings: in a purely topological one and a in geometrical one that includes spin connections and string connections. Our results provide a consistent, functorial, one-to-one dictionary between string geometry and spin geometry on loop spaces.
Cite
@article{arxiv.1403.5656,
title = {String geometry vs. spin geometry on loop spaces},
author = {Konrad Waldorf},
journal= {arXiv preprint arXiv:1403.5656},
year = {2015}
}
Comments
54 pages. In v2 two errorneous lemmata (2.3.3 and 3.1.3) have been removed, with corresponding changes in Prop. 2.3.4 and Def. 3.1.4 (now Prop. 2.3.3 and Def. 3.1.3, respectively); otherwise minor changes. v3 comes with few minor changes and is the published version