Dimers和Beauville可积系统
Dimers and Beauville integrable systems
AI总结：
本文研究了标准三角形多边形对应的可积系统，证明谱变换是可积系统的birational同构，并展示Beauville系统具有簇代数结构。
摘要：
与平面中的凸积分多边形N相关的是两个可积系统：Goncharov和Kenyon构造的簇可积系统，由双色环面图上的dimer模型构造，以及与N的toric表面相关的Beauville可积系统。这两个系统通过一种称为谱变换的birational映射相关联。在本文中，我们研究当N是边长为d的标准三角形时的情况，即toric表面是¶²的情况，并证明谱变换是可积系统的birational同构。由于哈密顿量通过构造被识别，其本质内容是谱变换 intertwines 两个泊松结构。特别是，这表明Beauville可积系统具有簇代数结构。
英文摘要：
Associated to a convex integral polygon $N$ in the plane are two integrable systems: the cluster integrable system of Goncharov and Kenyon, constructed from the dimer model on bipartite torus graphs, and the Beauville integrable system associated with the toric surface of $N$. These two systems are related by a birational map called the spectral transform. In this paper we study the case when $N$ is the standard triangle of side length $d$, equivalently when the toric surface is $¶^2$, and prove that the spectral transform is a birational isomorphism of integrable systems. Since the Hamiltonians are identified by construction, the essential content is that the spectral transform intertwines the two Poisson structures. In particular, this shows that Beauville integrable systems admit cluster algebra structures.
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