An Index Theorem for Loop Spaces

TL;DR

This paper develops an index theorem for loop spaces of compact manifolds within $KK$-theory, proposing a noncommutative geometric approach to the Witten genus and establishing an equivariant index theorem for non-compact manifolds with $S^1$-actions.

Contribution

It introduces a new index theorem for loop spaces using $KK$-theory, linking noncommutative geometry with the Witten genus and equivariant index theory.

Findings

Formulated and proved an index theorem for loop spaces.

Established an equivariant index theorem for non-compact manifolds with $S^1$-actions.

Proposed a noncommutative geometric framework for the Witten genus.

Abstract

We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of $KK$-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order to find out an "appropriate form" of the index theorem to formulate a loop space version, we formulate and prove an equivariant index theorem for non-compact manifolds equipped with $S^1$-actions with compact fixed-point sets. In order to formulate it, we use a ring of formal power series.