An Expansion Formula for Decorated Super-Teichmüller Spaces

Abstract

Motivated by the definition of super-Teichmüller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichmüller spaces, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super λ-lengths associated to arcs in a bordered surface. In the special case of a disk, we can give combinatorial expansion formulas for the super λ-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 𝐴ₙ. In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the μ-invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.