On the splitting of genus two supermoduli

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.