N=2 super-Teichmüller theory

Based on earlier work of the latter two named authors on the higher super-Teichmüller space with N=1, a component of the flat OSp(1|2) connections on a punctured surface, here we extend to the case N=2 of flat OSp(2|2) connections. Indeed, we construct here coordinates on the higher super-Teichmüller space of a surface F with at least one puncture associated to the supergroup OSp(2|2), which in particular specializes to give another treatment for N=1 which is simpler than the earlier work. The Minkowski space in the current case, where the corresponding super Fuchsian groups act, is replaced by the superspace R2,2|4, and the familiar lambda lengths are extended by odd invariants of triples of special isotropic vectors in R2,2|4 as well as extra bosonic parameters, which we call ratios, defining a flat R+-connection on F. As in the pure bosonic or N=1 cases, we derive the analogue of Ptolemy transformations for all these new variables.