Based on earlier work of the latter two named authors on the higher super-Teichmueller space with $\mathcal{ N}=1$, a component of the flat $OSp(1|2)$ connections on a punctured surface, here we extend to the case $\mathcal{ N}=2$ of flat $OSp(2|2)$ connections. Indeed, we construct here coordinates on the higher super-Teichmueller 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 $\mathcal{ N}=1$ simpler than the earlier work. The Minkowski space in the current case, where the corresponding super Fuchsian groups act, is replaced by the superspace $\mathbb{ R}^{ 2,2|4}$, and the familiar lambda lengths are extended by odd invariants of triples of special isotropic vectors in $\mathbb{ R}^{ 2,2|4}$ as well as extra bosonic parameters, which we call ratios, defining a flat $\mathbb{ R}_{ +}$-connection on $F$. As in the pure bosonic or $\mathcal{ N}=1$ cases, we derive the analogue of Ptolemy transformations for all these new variables.
Categories: math.GT hep-th math-ph math.DG math.MP