The vanishing of two-point functions for three-loop superstring scattering amplitudes

TL;DR

This paper proves the two-point function vanishes for a three-loop superstring measure ansatz, using advanced mathematical tools, and revisits the two-loop case with a new proof, discussing higher genus cases.

Contribution

It demonstrates the vanishing of the two-point function for the three-loop superstring measure ansatz using theta functions and Jacobian theory, extending previous results to higher genus.

Findings

Two-point function vanishes at three-loop level.

Provides a new proof for two-loop vanishing.

Discusses approaches for higher genus cases.

Abstract

In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen vanishes. Our proof uses the reformulation of ansatz in terms of even cosets, theta functions, and specifically the theory of the $\Gamma_{00}$ linear system on Jacobians introduced by van Geemen and van der Geer. At the two-loop level, where the amplitudes were computed by D'Hoker and Phong, we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera.