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

TL;DR

This paper extends the super-Teichmüller theory to the $ ext{OSp}(2|2)$ case, constructing coordinates on the higher super-Teichmüller space that incorporate new odd and bosonic invariants, generalizing previous $ ext{OSp}(1|2)$ results.

Contribution

The authors develop a coordinate system for the $ ext{OSp}(2|2)$ super-Teichmüller space, generalizing earlier $ ext{OSp}(1|2)$ work and introducing new invariants and transformations.

Findings

Constructed coordinates on the $ ext{OSp}(2|2)$ super-Teichmüller space.

Extended lambda lengths with odd invariants and ratios.

Derived Ptolemy transformations for the new variables.

Abstract

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.