Decorated super-Teichmüller space

Abstract

We introduce coordinates for a principal bundle $S\tilde{T}(F)$ over the super Teichmüller 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 Teichmüller 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 $OSp (1\vert 2)$ on the super Minkowski space $\mathbb{R}^{2, 1 \vert 2}$, where $OSp (1\vert 2)$ denotes the orthosymplectic Lie supergroup, and the lambda lengths are extended by fermionic invariants of suitable triples of isotropic vectors in $\mathbb{R}^{2, 1 \vert 2}$. 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 Kähler 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 Teichmüller theory and provides a natural geometric interpretation in $\mathbb{R}^{2, 1 \vert 2}$ for the super moduli of $S\tilde{T}(F)$.