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DIMERS AND CLUSTER INTEGRABLE SYSTEMS

机译：裁切器和集群集成系统

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We show that the dimer model on a bipartite graph Γ. on a torus gives rise to a quantum integrable system of special type, which we call a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space ￡Γ of line bundles with connections on the graph Γ. The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ1 and Γ2 are equivalent if the Newton polygons of the corresponding partition functions coincide up to translation. We define elementary transformations of bipartite surface graphs, and show that two equivalent minimal bipartite graphs are related by a sequence of elementary transformations. For each elementary transformation we define a birational Poisson isomorphism L_(Γ1)→L_(Γ2) providing an equivalence of the integrable systems. We show that it is a cluster Poisson transformation, as defined in [10]. We show that for any convex integral polygon N there is a non-empty finite set of minimal graphs Γ for which N is the Newton polygon of the partition function related to Γ. Gluing the varieties LΓ for graphs Γ related by elementary transformations via the corresponding cluster Poisson transformations, we get a Poisson space χ_N. It is a natural phase space for the integrable system. The Hamiltonians are functions on χ_N, parametrized by the interior points of the Newton polygon N. We construct Casimir functions whose level sets are the symplectic leaves of χ_N. The space χ_N has a structure of a cluster Poisson variety. Therefore the algebra of regular functions on χ_N has a non-commutative q-deformation to a *-algebra O_q(χ_N). We show that the Hamiltonians give rise to a commuting family of quantum Hamiltonians. Together with the quantum Casimirs, they provide a quantum integrable system. Applying the general quantization scheme [11], we get a *-representation of the *-algebra O_q(χ_N) in a Hilbert space. The quantum Hamiltonians act by commuting unbounded selfadjoint operators. For square grid bipartite graphs on a torus we get discrete quantum integrable systems, where the evolution is a cluster automorphism of the *-algebra O_q(χ_N)commuting with the quantum Hamiltonians. We show that the octahedral recurrence, closely related to Hirota's bilinear difference equation.[20], appears this way. Any graph G on a torus T gives rise to a bipartite graph Γ_G on T. We show that the phase space χ related to the graph Γ_G has a Lagrangian subvariety R, defined in each coordinate system by a system of monomial equations. We identify it with the space parametrizing resistor networks on G. The pair (χ, R) has a large group of cluster automorphisms. In particular, for a hexagonal grid graph we get a discrete quantum integrable system on χ whose restriction to R is essentially given by the cube recurrence [33], [4]. The set of positive real points χ_N (R>o) of the phase space is well defined. It is isomorphic to the moduli space of simple Harnack curves with divisors studied in [26]. The Liouville tori of the real integrable system are given by the product of ovals of the simple Harnack curves. In the sequel [17] to this paper we show that the set of complex points χ_N(C) of the phase space is birationally isomorphic to a finite cover of the Beauville complex algebraic integrable system related to the toric surface assigned to the polygon N.
机译：我们证明了二分图Γ上的二聚体模型。在圆环上产生了一种特殊类型的量子可积系统，我们称其为簇可积系统。经典系统的相空间包含在图Γ上具有连接的线束的模空间￡Γ作为开放密集子集。哈密​​顿量的和本质上是二聚体模型的分配函数。我们说，如果相应分区函数的牛顿多边形符合平移，则两个这样的图Γ1和Γ2是等效的。我们定义了二分曲面图的基本变换，并显示了两个等效的最小二分图通过一系列基本变换相关。对于每个基本变换，我们定义了一个双等位泊松同构L_（Γ1）→L_（Γ2），提供了可积系统的等价性。我们证明这是一个聚类泊松变换，如[10]中所定义。我们表明，对于任何凸整数多边形N，都有一个非空的最小图Γ的有限集，其中N是与Γ相关的分配函数的牛顿多边形。通过相应的聚类泊松变换，将与基本变换相关的图Γ的变体LΓ进行胶合，得到泊松空间χ_N。它是可积系统的自然相空间。哈密​​顿量是χ_N上的函数，由牛顿多边形N的内点参数化。我们构造卡西米尔函数，其水平集是χ_N的辛叶。空间χ_N具有簇泊松变种的结构。因此，在χ_N上的正则函数代数具有到*代数O_q（χ_N）的非交换q变形。我们表明，哈密顿量产生了量子哈密顿量的通勤族。它们与量子卡西米尔一起提供了一个量子可积系统。应用一般的量化方案[11]，我们得到了希尔伯特空间中*-代数O_q（χ_N）的*-表示。量子哈密顿量是通过交换无界自伴算子来进行的。对于圆环上的正方形网格二部图，我们得到了离散的量子可积系统，其中演化是*-代数O_q（χ_N）与量子哈密顿量交换的簇自同构。我们证明，八面体递归与Hirota的双线性差分方程密切相关[20]。圆环T上的任何图G都会生成T上的二部图Γ_G。我们证明，与图Γ_G相关的相空间χ具有拉格朗日子变量R，该拉格朗日子变量R在每个坐标系中由一阶方程组定义。我们用G上的空间参数化电阻网络来识别它。线对（χ，R）具有大量的簇自同构。特别是，对于六边形网格图，我们在χ上获得了一个离散的量子可积系统，其对R的限制本质上是由立方递归给出的[33]，[4]。相空间的正实点χ_N（R> o）的集合定义良好。它与具有除数的简单Harnack曲线的模空间同构[26]。实际可积系统的Liouville托里由简单的Harnack曲线的椭圆形乘积给出。在本文的续集[17]中，我们证明了相空间的复点集χ_N（C）是与分配给多边形N的复曲面有关的Beauville复代数可积系统的有限覆盖的同构同构。

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