The co-Riemannian Structure of Smooth Loop Spaces

TL;DR

This paper introduces a co-Riemannian structure on smooth loop spaces of Riemannian manifolds, demonstrating that for string manifolds, this space admits a Dirac operator, linking geometry and analysis.

Contribution

It constructs a natural co-Riemannian structure on smooth loop spaces and shows that string manifolds' loop spaces are co-spin manifolds with Dirac operators.

Findings

Loop space of a Riemannian manifold has a natural co-Riemannian structure.

Loop space of a string manifold admits a Dirac operator.

The structure is per-Hilbert-Schmidt locally equivalent to a co-spin manifold.

Abstract

We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.