An Expansion Formula for Decorated Super-Teichm\"uller Spaces

TL;DR

This paper develops formulas for super $ ext{lambda}$-lengths in super-Teichmüller spaces, extending existing theories with combinatorial and algebraic insights, and making progress towards super cluster algebra structures.

Contribution

It introduces super Ptolemy relations for super $ ext{lambda}$-lengths and provides combinatorial expansion formulas for polygons, advancing the understanding of super cluster algebras.

Findings

Formulas for super $ ext{lambda}$-lengths in bordered surfaces.

Combinatorial expansion formulas for diagonals of polygons.

Partial progress towards super cluster algebra structures.

Abstract

Motivated by the definition of super-Teichm\"uller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichm\"uller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super $\lambda$-lengths associated to arcs in a bordered surface. In the special case of a disk, we are able to give combinatorial expansion formulas for the super $\lambda$-lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's $T$-path formulas for type $A$ cluster algebras. We further connect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type $A_n$. In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the $\mu$-invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.