How to Construct a Dirac Operator in Infinite Dimensions

TL;DR

This paper introduces a method to define the Dirac operator on infinite-dimensional manifolds, specifically applying it to the loop space of string manifolds using co-Riemannian structures.

Contribution

It presents a novel framework for constructing Dirac operators in infinite dimensions via co-Riemannian structures, extending geometric analysis tools.

Findings

Successfully defines Dirac operator on loop spaces of string manifolds

Establishes the use of co-Riemannian structures for infinite-dimensional geometry

Provides a foundation for further analysis of infinite-dimensional Dirac operators

Abstract

We define the notion of a co-Riemannian structure and show how it can be used to define the Dirac operator on an appropriate infinite dimensional manifold. In particular, this approach works for the smooth loop space of a so-called string manifold.