We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type $\mathcal{A}$ and $\mathcal{X}$ over both the real and complex numbers. Extensions of these groupoids to the completions of the cluster varieties where cluster variables are allowed to vanish are also considered. In the real case, we construct source-simply-connected groupoids for the cluster charts via the Poisson spray technique of Crainic and Mărcuţ. These groupoid charts and their analogues for the symplectic double and blow-up groupoids are glued by lifting the cluster mutations to groupoid comorphisms whose formulas are motivated by the Hamiltonian perspective of cluster mutations introduced by Fock and Goncharov.

中文翻译：

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簇流形的辛群

我们在实数和复数上构建了在 $\mathcal{A}$ 和 $\mathcal{X}$ 类型的簇变体上集成对数规范泊松结构的辛群。还考虑将这些 groupoids 扩展到允许集群变量消失的集群变体的完成。在实际情况下，我们通过 Crainic 和 Mărcuţ 的泊松喷雾技术为聚类图构建源单连通群。这些群形图及其对辛双和膨胀群形的类似物通过将簇突变提升为群形同态而粘合在一起，其公式是由 Fock 和 Goncharov 引入的簇突变的哈密顿观点激发的。