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An Expansion Formula for Decorated Super-Teichmüller Spaces
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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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, decorated Teichmüller spaces, supersymmetry.
Mots-clés : Laurent polynomials 
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     title = {An {Expansion} {Formula} for {Decorated} {Super-Teichm\"uller} {Spaces}},
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Gregg Musiker; Nicholas Ovenhouse; Sylvester W. Zhang. An Expansion Formula for Decorated Super-Teichmüller Spaces. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a79/

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