Super Riemann Surfaces and Fatgraphs

Our goal is to describe superconformal structures on super Riemann surfaces (SRSs) 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 Čech cocycles for a specific covering, which we reproduce from 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 the supermoduli space of N=2 SRS.