An Analytical Approach to the equivariant index and Witten genus on spin manifolds

TL;DR

This paper develops an analytical framework for computing equivariant indices and the Witten genus on spin manifolds, considering fixed points of circle actions on manifolds and their loop spaces, linking analysis with topology.

Contribution

It introduces a novel analytical approach to relate indices on spin manifolds with topological invariants like the Witten genus, considering fixed points of circle actions.

Findings

Analytical index matches the equivariant Atiyah-Singer index in the first case.

Analytical index equals a topological expression involving the Witten genus in the second case.

Framework connects analysis, topology, and equivariant geometry on spin manifolds.

Abstract

This work is divide in two cases. In the first case, we consider a spin manifold $M$ as the set of fixed points of an $S^{1}$-action on a spin manifold $X$, and in the second case we consider the spin manifold $M$ as the set of fixed points of an $S^{1}$-action on the loop space of $M$. For each case, we build on $M$ a vector bundle, a connection and a set of bundle endomorphisms. These objects are used to build global operators on $M$ which define an analytical index in each case. In the first case, the analytical index is equal to the topological equivariant Atiyah Singer index, and in the second case the analytical index is equal to a topological expression where the Witten genus appears.