Genus four superstring measures

TL;DR

This paper explicitly constructs the genus four superstring measure using modular forms, extending previous results from genus two and three, and confirming the approach's validity in higher genus.

Contribution

It provides the explicit modular form for the genus four superstring measure, confirming the generalization of earlier methods to higher genus.

Findings

Explicit genus four superstring measure derived

Method successfully extended from genus two and three

Supports the generalization of superstring measure construction

Abstract

A main issue in superstring theory are the superstring measures. D'Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are determined by modular forms of weight eight and the bosonic measure. They also suggested a generalisation to higher genus. We showed that their approach works, with a minor modification, in genus three and we announced a positive result also in genus four. Here we give the modular form in genus four explicitly. Recently S. Grushevsky published this result as part of a more general approach.