Supersymmetric localization, modularity and the Witten genus

TL;DR

This paper explores the geometric and modular properties of the Witten genus using equivariant localization, linking double loop space geometry with elliptic cohomology and proposing a candidate for the elliptic Bismut-Chern character.

Contribution

It provides a rigorous geometric interpretation of the Witten genus's modularity and connects double loop space geometry with elliptic cohomology, proposing a new candidate for the elliptic Bismut-Chern character.

Findings

Witten genus interpreted as an integral over double loop space

Modularity properties explained through geometric localization

Identification of a candidate target for elliptic Bismut-Chern character

Abstract

Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut-Chern character.