Superstring scattering amplitudes in higher genus

TL;DR

This paper advances the understanding of superstring scattering amplitudes by constructing new modular forms for higher genus cases, verifying their properties, and proposing a general conjecture for all genera.

Contribution

It provides new expressions for superstring measure ans"atze in genera 2 and 3, proposes a genus 4 ansatz satisfying key constraints, and conjectures a universal formula for all genera.

Findings

New expressions for genus 2 and 3 superstring measures

A genus 4 ansatz satisfying factorization and vanishing cosmological constant

A conjecture for superstring amplitudes in arbitrary genus

Abstract

In this paper we continue the program pioneered by D'Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ans\"atze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots.