Regularity of the superstring supermeasure and the superperiod map

TL;DR

This paper investigates the regularity of the superstring supermeasure on the moduli space of supercurves, demonstrating regularity for genus up to 11 and analyzing the superperiod map's properties and its relation to the Schottky ideal.

Contribution

It proves the supermeasure's regularity for genus up to 11 and studies the algebraic properties of the superperiod map and its connection to the Schottky ideal.

Findings

Supermeasure is regular for g ≤ 11.

The minimal power of the Schottky ideal matching the genus g.

Insights into the superperiod map's image and algebraic structure.

Abstract

The supermeasure whose integral is the genus $g$ vacuum amplitude of superstring theory is potentially singular on the locus in the moduli space of supercurves where the corresponding even theta-characteristic has nontrivial sections. We show that the supermeasure is actually regular for $g\leq 11$. The result relies on the study of the superperiod map. We also show that the minimal power of the classical Schottky ideal that annihilates the image of the superperiod map is equal to $g$ if $g$ is odd and is equal to $g$ or $g-1$ if $g$ is even.