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.

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

References in corpus (5)

• More Morphisms between Bundle Gerbes
• Spin structures on loop spaces that characterize string manifolds
• Transgressive loop group extensions
• T-duality of current algebras and their quantization
• Equivalence of string and fusion loop-spin structures

Cited by in corpus (7)

• Principal $\infty$-Bundles and Smooth String Group Models
• String structures associated to indefinite Lie groups
• Smooth Fock bundles, and spinor bundles on loop space
• Connes fusion of spinors on loop space
• $S^1$-equivariant Chern-Weil constructions on loop space
• Current Groups and the Hamiltonian Anomaly
• String${}^{\mathbf{c}}$ Structures and Modular Invariants