Complex Geometry and Supergeometry

TL;DR

This paper surveys recent advances in understanding the relationship between complex geometry and supergeometry, particularly in superstring theory, highlighting progress at genus 2 and future research directions.

Contribution

It provides an overview of recent solutions to longstanding obstacles in superstring perturbation theory related to supermoduli at genus 2.

Findings

Overcoming supermoduli obstacle at genus 2

Improved understanding of supergeometry and superholomorphicity

Framework for future research in superstring amplitudes

Abstract

Complex geometry and supergeometry are closely entertwined in superstring perturbation theory, since perturbative superstring amplitudes are formulated in terms of supergeometry, and yet should reduce to integrals of holomorphic forms on the moduli space of punctured Riemann surfaces. The presence of supermoduli has been a major obstacle for a long time in carrying out this program. Recently, this obstacle has been overcome at genus 2, which is the first loop order where it appears in all amplitudes. An important ingredient is a better understanding of the relation between geometry and supergeometry, and between holomorphicity and superholomorphicity. This talk provides a survey of these developments and a brief discussion of the directions for further investigation.