Superperiods and superstring measure near the boundary of the moduli space of supercurves

TL;DR

This paper investigates the behavior of superperiod maps near the boundary of the moduli space of supercurves, revealing similarities to classical periods, and explores implications for the geometry of genus 2 supermoduli space and superstring measure expansions.

Contribution

It characterizes the boundary behavior of the superperiod map, proves the non-projectedness of the genus 2 supermoduli space, and analyzes superstring measure expansions near the boundary.

Findings

Superperiod map behaves similarly to classical periods near the boundary.

Genus 2 supermoduli space is not projected.

Superstring measure expansion near the boundary is characterized.

Abstract

We study the behavior of the superperiod map near the boundary of the moduli space of stable supercurves and prove that it is similar to the behavior of periods of classical curves. We consider two applications to the geometry of this moduli space in genus $2$, denoted as $\bar{\mathcal S}_2$. First, we characterize the canonical projection of $\bar{\mathcal S}_2$ in terms of its behavior near the boundary, proving in particular that $\bar{\mathcal S}_2$ is not projected. Secondly, we combine the information on superperiods with the explicit calculation of genus $2$ Mumford isomorphism, due to Witten, to study the expansion of the superstring measure for genus $2$ near the boundary. We also present the proof, due to Deligne, of regularity of the superstring measure on $\bar{\mathcal S}_g$ for any genus.