Beyond the String Genus

TL;DR

This paper extends the elliptic genus to higher genus surfaces using path integral methods, revealing a modular invariant with a multiplier, and analyzes its behavior under the mapping class group.

Contribution

It provides a simplified formula for the higher genus elliptic genus and studies its modular properties and invariance under the mapping class group.

Findings

The invariant is a modular function with a multiplier.

The formula for the higher genus elliptic genus is simplified.

The invariant is a cobordism invariant parametrized by Teichmuller space.

Abstract

In an earlier work we used a path integral analysis to propose a higher genus generalization of the elliptic genus. We found a cobordism invariant parametrized by Teichmuller space. Here we simplify the formula and study the behavior of our invariant under the action of the mapping class group of the Riemann surface. We find that our invariant is a modular function with multiplier just as in genus one.