Super Riemann surfaces and fatgraphs

TL;DR

This paper develops a combinatorial approach to describe superconformal structures on super Riemann surfaces using fatgraphs, Strebel differentials, and supermoduli space analysis, including dual supermanifolds.

Contribution

It introduces a novel method to characterize superconformal structures on super Riemann surfaces via fatgraph data and explores their relation to supermoduli space and dual supermanifolds.

Findings

Characterization of superconformal structures using fatgraph data.

Description of moduli and deformations via Strebel differentials.

Analysis of superconformal structures for N=1 and N=2 super Riemann surfaces.

Abstract

Our goal is to describe superconformal structures on super Riemann surfaces (SRS), based on data assigned to a fatgraph. We start from the complex structures on punctured $(1|1)$-supermanifolds, characterizing the corresponding moduli and the deformations using Strebel differentials and certain \v{C}ech cocycles for a specific covering, which we reproduce from a fatgraph data, consisting of $U(1)$-graph connection and odd parameters at the vertices. Then we consider dual $(1|1)$-supermanifolds and related superconformal structures for $N=2$ super Riemann surfaces. The superconformal structures $N=1$ SRS are computed as the fixed points of involution on supermoduli space of $N=2$ SRS.