Cluster algebras and Weil-Petersson forms

Abstract

In the previous paper [GSV] we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents. Our leading idea is that a relevant geometric object in this case is a certain closed 2-form compatible with the cluster algebra structure. The main example is provided by Penner coordinates on the decorated Teichmüller space, in which case the above form coincides with the classical Weil-Petersson symplectic form.

Cluster algebras: an introduction

“…In particular, we will associate a cluster algebra to any bordered surface with marked points, following work of Fock and Goncharov [8], Gekhtman, Shapiro, and Vainshtein [20], and Fomin, Shapiro, and Thurston [11]. This construction provides a natural generalization of the type A cluster algebra from Section 2.2, and realizes the lambda lengths (also called Penner coordinates) on the decorated Teichmüller space associated to a cusped surface, which Penner had defined in 1987 [39].…”

“…We start by associating a cluster algebra to any bordered surface with marked points, following work of Fock and Goncharov [8], Gekhtman, Shapiro, and Vainshtein [20], and Fomin, Shapiro, and Thurston [11]. This construction specializes to the type A example from Section 2 when the surface is a disk with marked points on the boundary.…”

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

“…Furthermore, the mutation of Stokes graphs corresponds to the mutation of triangulations called signed flips and signed pops. These facts provide a natural bridge between exact WKB analysis and cluster algebra theory with the help of the surface realization of cluster algebras developed by [GSV05,FG06,FST08,FT12]. Along the mutation of Stokes graphs, the Voros symbols also mutate (or jump) due to the Stokes phenomenon.…”

Positive basis for surface skein algebras

“…This algebra includes many of the cluster variables in the cluster algebra associated with a surface (8)(9)(10), namely the cluster variables without notched arcs. In particular, for a surface without punctures, this gives a positive, natural basis for an algebra between the cluster algebra and the upper cluster algebra of the surface.…”