论文标题
有限类型的集群配置空间
Cluster Configuration Spaces of Finite Type
论文作者
论文摘要
对于每个Dynkin图$ D $，我们定义了'''''''''''''''''''''''''''''''''''''''''''$ {\ MATHCAL {M}} _ D $和部分紧凑型$ {\ widetilde {\ Mathcal {M}}}}} _ d $。对于$ d = a_ {n-3} $，我们有$ {\ Mathcal {m}} _ {a_ {a_ {n-3}} = {\ Mathcal {M}} _ {0，n} $，$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $的配置空间{\ Mathcal {M}}} _ {a_ {n-3}} $在这种情况下，Brown研究了。空间$ {\ widetilde {\ Mathcal {m}}} _ d $是一种平滑的仿射代数品种，分层进行了分层，并带有副骗子 - 芬斯基 - Zelevinsky pentryiziahedron。 $ {\ widetilde {\ Mathcal {m}}} _ d $上的常规函数是由坐标$u_γ$生成的，与$ d类型的群集变量进行了培养，并且关系由群集Algebra的兼容度功能完全描述。作为一种应用，我们定义和研究树级开放式弦振幅的群集代数类似物。
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.
