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 λ-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 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.
|Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
|Published - Sep 1 2021
Bibliographical noteFunding Information:
The authors would like to thank the support of the NSF grant DMS-1745638 and the University of Minnesota UROP program. We would also like to thank Misha Shapiro and Leonid Chekhov for inspiring conversations, as well as the anonymous referees for their helpful feedback.
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- Cluster algebras
- Decorated teichmüller spaces
- Laurent polynomials