Combinatorial Expansion Formulas for Decorated Super-Teichmüller Spaces

Abstract: Motivated by the definition of super Teichmuller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super Teichmuller 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. 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. This is joint work with Nicholas Ovenhouse and Sylvester Zhang.

K-theory and homologyquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Paris algebra seminar

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