Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 080, 34 pages      arXiv:2102.09143      https://doi.org/10.3842/SIGMA.2021.080

An Expansion Formula for Decorated Super-Teichmüller Spaces

Gregg Musiker, Nicholas Ovenhouse and Sylvester W. Zhang
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received March 31, 2021, in final form August 27, 2021; Published online September 01, 2021

Abstract
Motivated by the definition of super-Teichmüller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichmüller 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.

Key words: cluster algebras; Laurent polynomials; decorated Teichmüller spaces; supersymmetry.

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