Modular Forms and Three Loop Superstring Amplitudes

TL;DR

This paper refines the mathematical framework for three-loop superstring amplitudes by identifying a unique modular form solution, providing explicit formulas, and confirming key conjectures in genus 3 and 4.

Contribution

It introduces a modified set of constraints leading to a unique solution for the superstring measure at genus three, with explicit polynomial formulas and verification of conjectures.

Findings

A unique solution for the genus three superstring measure was found.

An explicit polynomial formula in theta constants was derived.

The results are consistent with the vanishing cosmological constant and Igusa's conjectures.

Abstract

We study a proposal of D'Hoker and Phong for the chiral superstring measure for genus three. A minor modification of the constraints they impose on certain Siegel modular forms leads to a unique solution. We reduce the problem of finding these modular forms, which depend on an even spin structure, to finding a modular form of weight 8 on a certain subgroup of the modular group. An explicit formula for this form, as a polynomial in the even theta constants, is given. We checked that our result is consistent with the vanishing of the cosmological constant. We also verified a conjecture of D'Hoker and Phong on modular forms in genus 3 and 4 using results of Igusa.