有限型的簇构型空间
Cluster Configuration Spaces of Finite Type
摘要：
对于每个Dynkin图$D$，我们定义了一个“簇配置空间”${\mathcal{M}}_D$和一个部分紧化${\widetilde {\mathcal{M}}}_D$。对于$D = A_{n-3}$，我们有${\mathcal{M}}_{A_{n-3}} = {\mathcal{M}}_{0,n}$，${\mathbb P}^1$上的$n$点的配置空间，Brown在这种情况下研究了部分紧化${\widetilde {\mathcal{M}}}_{A_{n-3}}$。空间${\widetilde {\mathcal{M}}}_D$是一个光滑的仿射代数簇，它具有Chapoton-Fomin-Zelevinsky广义结合面体的双射分层。${\widetilde {\mathcal{M}}}_D$上的正则函数是由坐标$u_γ$与$D$型簇变量的双射生成的，它们之间的关系完全用簇代数的相容度函数来描述.作为应用，我们定义和研究簇代数类似物的树水平开弦振幅。
英文摘要：
For each Dynkin diagram $D$, we define a ''cluster configuration space'' ${\mathcal{M}}_D$ and a partial compactification ${\widetilde {\mathcal{M}}}_D$. 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 ${\widetilde {\mathcal{M}}}_{A_{n-3}}$ was studied in this case by Brown. The space ${\widetilde {\mathcal{M}}}_D$ 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 ${\widetilde {\mathcal{M}}}_D$ are generated by coordinates $u_γ$, 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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