Cluster Configuration Spaces of Finite Type

Abstract

For each Dynkin diagram D, we define a “cluster configuration space” MD and a partial compactification fMD. For D = An−3, we have MAn−3 =M0,n, the configuration space of n points on P1, and the partial compactification fMAn−3 was studied in this case by Brown. The space fMD is a smooth affine algebraic variety with a stratification in bijection with the faces of the Chapoton-Fomin-Zelevinsky generalized associahedron. The regular functions on fMD are generated by coordinates uγ, in bijection with the cluster variables of type D, and the relations are described completely in terms of the compatibility degree function of the cluster algebra. As an application, we define and study cluster algebra analogues of tree-level open string amplitudes.