Matrix Formulae for Decorated Super Teichm\"uller Spaces

TL;DR

This paper develops matrix formulas for arcs in decorated super Teichmüller spaces, generalizing previous models, and applies these to derive new results on super Fibonacci numbers.

Contribution

It introduces new matrix formulas for super Teichmüller spaces that unify and extend prior models, with applications to super Fibonacci numbers.

Findings

Matrix formulas yield super λ-lengths in decorated super Teichmüller space.

Formulas agree with previous combinatorial approaches.

Application to annulus case produces new super Fibonacci results.

Abstract

For an arc on a bordered surface with marked points, we associate a holonomy matrix using a product of elements of the supergroup $\mathrm{OSp}(1|2)$, which defines a flat $\mathrm{OSp}(1|2)$-connection on the surface. We show that our matrix formulas of an arc yields its super $\lambda$-length in Penner-Zeitlin's decorated super Teichm\"uller space. This generalizes the matrix formulas of Fock-Goncharov and Musiker-Williams. We also prove that our matrix formulas agree with the combinatorial formulas given in the authors' previous works. As an application, we use our matrix formula in the case of an annulus to obtain new results on super Fibonacci numbers.