Binary Forms and the Hyperelliptic Superstring Ansatz

TL;DR

This paper presents a hyperelliptic formulation of the D'Hoker-Phong superstring Ansatz, introducing a unique family of binary invariants satisfying the Ansatz and connecting it to Siegel modular forms via Thomae's map.

Contribution

It provides a hyperelliptic formulation of the superstring Ansatz, constructs a unique family of binary invariants, and links these to Siegel modular forms through Thomae's map.

Findings

Explicit family of binary invariants for each genus

Proof of uniqueness of the binary invariants satisfying the Ansatz

Connection between the invariants and Siegel modular forms via Thomae's map

Abstract

We give a hyperelliptic formulation of the Ansatz of D'Hoker and Phong. We give an explicit family of binary invariants, one for each genus, that satisfies this hyperelliptic Ansatz. We also prove that this is the unique family of weight eight binary forms over the theta group on the hyperelliptic locus that satisfies this Ansatz. Futhermore, we prove that this solution may also be obtained by applying Thomae's map to multivalued Siegel modular forms of Grushevsky and making certain choices of roots.