Binary Forms and the Hyperelliptic Superstring Ansatz

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.