Cluster Configuration Spaces of Finite Type

Abstract

For each Dynkin diagram 𝐷, we define a ''cluster configuration space'' ℳ𝐷 and a partial compactification ℳ˜𝐷. For 𝐷 = 𝐴ₙ₋₃, we have ℳ𝐴ₙ₋₃ = ℳ₀,ₙ, the configuration space of 𝑛 points on ℙ¹, and the partial compactification ℳ˜𝐴ₙ₋₃ was studied in this case by Brown. The space M˜𝐷 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 ℳ˜𝐷 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.