Cluster Configuration Spaces of Finite Type

For each Dynkin diagram $D$, we define a ''cluster configuration space'' ${\mathcal{M}}_D$ and partial compactification ${\widetilde {\mathcal{M}}}_D$. $D = A_{n-3}$, have ${\mathcal{M}}_{A_{n-3}} {\mathcal{M}}_{0,n}$, the space of $n$ points on ${\mathbb P}^1$, {\mathcal{M}}}_{A_{n-3}}$ was studied in this case by Brown. The {\mathcal{M}}}_D$ is smooth affine algebraic variety with stratification bijection faces Chapoton-Fomin-Zelevinsky generalized associahedron. regular functions are generated coordinates $u_\gamma$, cluster variables type relations described completely terms compatibility degree function algebra. As an application, study algebra analogues tree-level open string amplitudes.