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March 2019 Decorated super-Teichmüller space
R. C. Penner, Anton M. Zeitlin
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J. Differential Geom. 111(3): 527-566 (March 2019). DOI: 10.4310/jdg/1552442609

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 $S\tilde{T}(F)$.

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R. C. Penner. Anton M. Zeitlin. "Decorated super-Teichmüller space." J. Differential Geom. 111 (3) 527 - 566, March 2019. https://doi.org/10.4310/jdg/1552442609

Information

Received: 25 January 2016; Published: March 2019
First available in Project Euclid: 13 March 2019

zbMATH: 07036515
MathSciNet: MR3934599
Digital Object Identifier: 10.4310/jdg/1552442609

Rights: Copyright © 2019 Lehigh University

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Vol.111 • No. 3 • March 2019
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