The (anti-)holomorphic sector in $\mathbb{C}/\Lambda$-equivariant cohomology, and the Witten class

TL;DR

This paper explores how equivariant localization techniques applied to the conformal double loop space of a rationally string manifold yield the Witten genus, extending Atiyah's classical work on equivariant cohomology.

Contribution

It introduces an antiholomorphic sector in $ ext{C}/ ext{Λ}$-equivariant cohomology that produces the Witten genus, linking localization methods to supersymmetric derivations.

Findings

Equivariant localization in the antiholomorphic sector yields the Witten genus.

Connection established between Atiyah's localization and supersymmetric derivations.

Provides a new perspective on the geometric origin of the Witten genus.

Abstract

Atiyah's classical work on circular symmetry and stationary phase shows how the $\hat{A}$-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable ''antiholomorphic sector'' in the $\mathbb{C}/\Lambda$-equivariant cohomology of the conformal double loop space $\mathrm{Maps}(\mathbb{C}/\Lambda,X)$ of a rationally string manifold $X$ produces the Witten genus of $X$. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation of the Witten genus.