Cluster Configuration Spaces of Finite Type
Cluster Configuration Spaces of Finite Type
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DOI：
10.3842/sigma.2021.092
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发表时间：
2020-05
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影响因子：
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通讯作者：
N. Arkani-Hamed;Song He;T. Lam
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文献类型：
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作者：
N. Arkani-Hamed;Song He;T. Lam
对于每个动态图D，我们定义了一个“簇配置空间”M_D和一个部分紧化\TM_D。对于D=A_{n-3}，我们有M_{A_(n-3)}=M_{0，n}，P^1上n个点的配置空间，Brown在这种情况下研究了部分紧化\TM_{A_(n-3)}。空间TM_D是一种光滑仿射代数簇，具有与Chapoton-Fomin-Zlevinsky广义联合体的面的双射分层。TM_D上的正则函数由坐标u_γ与D型簇变量双射生成，并用簇代数的兼容度函数完全描述了它们之间的关系。作为应用，我们定义并研究了树级开弦振幅的簇代数类似物。
For each Dynkin diagram D, we define a "cluster configuration space" M_D and a partial compactification \tM_D. For D = A_{n-3}, we have M_{A_{n-3}} = M_{0,n}, the configuration space of n-points on P^1, and the partial compactification \tM_{A_{n-3}} was studied in this case by Brown. The space \tM_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 \tM_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.