Super Riemann surfaces and fatgraphs

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 $(1\mathrm{\mid }1)$-supermanifolds, characterizing the corresponding moduli and the deformations using Strebel differentials and certain Čech 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 $(1\mathrm{\mid }1)$-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.