On the splitting of genus two supermoduli

TL;DR

This paper explains why the supermoduli space of genus two curves splits, using deformation theory and superstring amplitude techniques, contrasting with the behavior of higher genus supermoduli spaces.

Contribution

It demonstrates that genus two supermoduli space splits due to the structure of odd deformations and supermoduli obstructions, providing a geometric explanation.

Findings

Genus two supermoduli space splits, unlike higher genus cases.

Odd deformations are generated by Schiffer variations at specific points.

The splitting is linked to superstring amplitude calculations by D'Hoker and Phong.

Abstract

This article investigates why the genus two, supermoduli space of curves will split in contrast to, potentially, almost all other supermoduli spaces. We use that the dimension of the odd, versal deformation space of a genus two, super Riemann surface is two dimensional. As a consequence, the odd versal deformations can be generated by Schiffer variations at the associated points of a Szego kernel. This idea is present in D'Hoker and Phong's two loop, superstring amplitude calculation. We show how this idea, combined with Donagi and Witten's characterization of supermoduli obstructions, will result in a splitting of supermoduli space in genus two.