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This article is cited in 1 scientific paper (total in 1 paper)
An Expansion Formula for Decorated Super-Teichmüller Spaces
Gregg Musiker, Nicholas Ovenhouse, Sylvester W. Zhang School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
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.
Keywords:
cluster algebras, Laurent polynomials, decorated Teichmüller spaces, supersymmetry.
Received: March 31, 2021; in final form August 27, 2021; Published online September 1, 2021
Citation:
Gregg Musiker, Nicholas Ovenhouse, Sylvester W. Zhang, “An Expansion Formula for Decorated Super-Teichmüller Spaces”, SIGMA, 17 (2021), 080, 34 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1762 https://www.mathnet.ru/eng/sigma/v17/p80
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