$\mathcal{N}=2$ Super-Teichm\"uller Theory

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.