Decorated Super-Teichm\"uller Space

Abstract

We introduce coordinates for a principal bundle $S\tilde T(F)$ over the super Teichmueller space $ST(F)$ of a surface $F$ with $s\geq 1$ punctures that extend the lambda length coordinates on the decorated bundle $\tilde T(F)=T(F)\times {\mathbb R}_+^s$ over the usual Teichmueller space $T(F)$. In effect, the action of a Fuchsian subgroup of $PSL(2,{\mathbb R})$ on Minkowski space ${\mathbb R}^{2,1}$ is replaced by the action of a super Fuchsian subgroup of $OSp(1|2)$ on the super Minkowski space ${\mathbb R}^{2,1|2}$, where $OSp(1|2)$ denotes the orthosymplectic Lie supergroup, and the lambda lengths are extended by fermionic invariants of suitable triples of isotropic vectors in ${\mathbb R}^{2,1|2}$. As in the bosonic case, there is the analogue of the Ptolemy transformation now on both even and odd coordinates as well as an invariant even two-form on $S\tilde T(F)$ generalizing the Weil-Petersson Kaehler form. This finally solves a problem posed in Yuri Ivanovitch Manin's Moscow seminar some thirty years ago to find the super analogue of decorated Teichmueller theory and provides a natural geometric interpretation in ${\mathbb R}^{2,1|2}$ for the super moduli of $S\tilde T(F)$.