装饰超Teichmüller空间的一个展开公式
An Expansion Formula for Decorated Super-Teichmüller Spaces
摘要：
受super-Teichmüller空间定义的启发，以及Penner-Zeitlin最近将此定义扩展到装饰super-Teichmüller空间，作为超黎曼曲面的例子，我们使用超托勒密关系获得与有边曲面中的弧相关的超$λ$ -长度的公式。在圆盘的特殊情况下，我们能够在Ralf Schiffler的$A$型簇代数的$T$ -路公式的基础上，给出与多边形对角线相关的超$λ$ -长度的组合展开公式.我们进一步将我们的公式连接到Morier-Genoud，Ovsienko和Tabachnikov的超friezes，并在定义$A_n$型超簇代数方面取得了部分进展。特别是，以下彭纳-蔡特林，我们能够得到的公式（符号）的$μ$ -不变量相关的三角形在一个三角形多边形，并解释如何这些提供了一个步骤，了解奇变量的超级集群代数。
英文摘要：
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 $A_n$. 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.
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