Cluster Configuration Spaces of Finite Type

Abstract

For each Dynkin diagram $D$, we define a ''cluster configuration space'' ${\mathcal{M}}_D$ and a partial compactification ${\tilde{\mathcal{M}}}_D$. For $D=A_{n-3}$, we have ${\mathcal{M}}_{A_{n-3}}={\mathcal{M}}_{0,n}$, the configuration space of $n$ points on ${\mathbb{P}}^1$, and the partial compactification ${\tilde{\mathcal{M}}}_{A_{n-3}}$ was studied in this case by Brown. The space ${\tilde{\mathcal{M}}}_D$ 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 ${\tilde{\mathcal{M}}}_D$ 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.

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