An Expansion Formula for Decorated Super-Teichm\"uller Spaces
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 -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 -lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's -path formulas for type 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 . In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the -invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.
Cite
@article{arxiv.2102.09143,
title = {An Expansion Formula for Decorated Super-Teichm\"uller Spaces},
author = {Gregg Musiker and Nicholas Ovenhouse and Sylvester W. Zhang},
journal= {arXiv preprint arXiv:2102.09143},
year = {2021}
}