Cluster configuration spaces of finite type

DOI10.3842/SIGMA.2021.092zbMATH Open1484.13049arXiv2005.11419MaRDI QIDQ2665994

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Publication date: 22 November 2021

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Abstract: For each Dynkin diagram $D$, we define a cluster configuration space ${mathcalM}_D$ and a partial compactification ${widetildemathcalM}_D$. For $D=A_{n-3}$, we have ${mathcalM}_{A_{n-3}}={mathcalM}_{0,n}$, the configuration space of $n$ points on ${mathbbP}^1$, and the partial compactification ${widetildemathcalM}_{A_{n-3}}$ was studied in this case by Brown. The space ${widetildemathcalM}_D$ 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 ${widetildemathcalM}_D$ are generated by coordinates $u_{g}amma$, 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.

Full work available at URL: https://arxiv.org/abs/2005.11419

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