A-Genus and the Sigma Model

TL;DR

This paper provides a detailed derivation linking the $ ext{ extasciitilde}A$-genus of spacetime's loop space to the genus one partition function of the $N=1/2$ sigma model, highlighting potential generalizations from a string theory perspective.

Contribution

It offers a pedantic derivation connecting the $ ext{ extasciitilde}A$-genus and the sigma model's partition function, and discusses possible new generalizations inspired by string theory.

Findings

Derived the connection between $ ext{ extasciitilde}A$-genus and sigma model partition function

Outlined potential generalizations of the $ ext{ extasciitilde}A$-genus from a stringy viewpoint

Targeted at a mathematical audience unfamiliar with the sigma model

Abstract

This set of lecture notes presents a pedantic derivation of the connection between the $ {\hat A} $-genus of spacetime's loop space and the genus one partition function of the $ N=1/2 $ sigma model. It concludes with some remarks on possible generalizations of the $ {\hat A} $-genus which follow naturally from the `stringy' point-of-view but have yet to be explored mathematically. This set of lecture notes is geared towards a mathematical audience unfamiliar with the $ N=1/2 $ sigma model.