Symmetry, Integrability and Geometry: Methods and Applications | 2021
Cluster Configuration Spaces of Finite Type
Abstract
For each Dynkin diagram D, we define a cluster configuration space M_D and a partial compactification \\tM_D. For D = A_{n-3}, we have M_{A_{n-3}} = M_{0,n}, the configuration space of n-points on P^1, and the partial compactification \\tM_{A_{n-3}} was studied in this case by Brown. The space \\tM_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 \\tM_D are generated by coordinates u_\\gamma, 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.