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arXiv 1107.1244hep-thmath-phmath.MP
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二聚体模型与可积系统

Dimer Models and Integrable Systems

Richard Eager, Sebastian Franco, Kevin Schaeffer

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29 pages, 16 figures. v2: typos fixed
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摘要：

我们探讨二聚体模型和可积系统最近推出的Goncharov和Kenyon之间的对应关系的各个方面。二聚体模型产生的相对论性可积系统与涅克拉索夫研究的5d N=1规范理论相匹配。我们将对应应用于与Y^{p，0}几何相关的二聚体模型，表明它们引起了最初由Ruijsenaars研究的周期Toda链的相对论推广。的对应关系减少了所有保守电荷的计算，以一个简单的组合问题，枚举非相交路径的二聚体模型。我们展示了如何通常的周期户田链出现在非相对论性的限制，以及如何Lax运营商对应的Kasteleyn矩阵的二聚体模型。讨论了一般Y^{p，q}流形的二聚体模型如何产生其他相对论性可积系统，推广了周期Toda链，并明确地构造了一般Y^{p，p}的可积系统.在Y^{p，q}箭图结构中引入的杂质与扭曲sl（2）XXZ自旋链中的杂质相一致。最后，我们讨论了如何higgsing一个二聚体模型的物理概念提供了一个有效的方法，产生新的可积系统从已知的。我们通过构造Y^{4，0}的higgsings的可积系统来说明这一思想。

英文摘要：

We explore various aspects of the correspondence between dimer models and integrable systems recently introduced by Goncharov and Kenyon. Dimer models give rise to relativistic integrable systems that match those arising from 5d N=1 gauge theories studied by Nekrasov. We apply the correspondence to dimer models associated to the Y^{p,0} geometries, showing that they give rise to the relativistic generalization of the periodic Toda chain originally studied by Ruijsenaars. The correspondence reduces the calculation of all conserved charges to a straightforward combinatorial problem of enumerating non-intersecting paths in the dimer model. We show how the usual periodic Toda chain emerges in the non-relativistic limit and how the Lax operator corresponds to the Kasteleyn matrix of the dimer model. We discuss how the dimer models for general Y^{p,q} manifolds give rise to other relativistic integrable systems, generalizing the periodic Toda chain and construct the integrable systems for general Y^{p,p} explicitly. The impurities introduced in the construction of Y^{p,q} quivers are identified with impurities in twisted sl(2) XXZ spin chains. Finally we discuss how the physical concept of higgsing a dimer model provides an efficient method for producing new integrable systems starting from known ones. We illustrate this idea by constructing the integrable systems for higgsings of Y^{4,0}.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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