Decorated Super-Teichmüller 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 $s\ge 1$ 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 $S\tilde{T}(F)$.

39 pages, 10 figures, v4: minor changes, to appear in Journal of Differential Geometry

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