双q参数熔化晶体模型的可积结构
Integrable structure of melting crystal model with two q-parameters
摘要：
研究了具有两个$q$参数$q_1,q_2$的广义熔融晶体模型的可积结构.这个模型，像一个普通的一个单一的$q$参数，制定为一个随机的平面分区（或等效的，随机三维杨氏图）的模型。玻尔兹曼权重包含无限多个外部势，这些势取决于平面分区的对角切片的形状。因此配分函数是无限个耦合常数$t_1,t_2,...$和额外的一个$Q$的函数。这个配分函数在二维复自由费米子系统的语言中有一个紧凑的表达式，从中可以看出这个模型背后量子环面代数的存在。这个配分函数是一个同时包含两个可积结构的τ函数（乘以一个简单因子）。第一个可积结构是双分次Toda族，它决定了对$t_1,t_2,...$的依赖性。当$q$ -参数$q_1,q_2$取特殊值时，出现这种可积结构。第二个可积结构是一维Toda方程的$q$ -差分模拟。配分函数满足关于$Q$的该$q$ -差分方程。与双阶Toda族不同，这种可积结构对$q_1,q_2$的任何值都存在。
英文摘要：
This paper explores integrable structures of a generalized melting crystal model that has two $q$-parameters $q_1,q_2$. This model, like the ordinary one with a single $q$-parameter, is formulated as a model of random plane partitions (or, equivalently, random 3D Young diagrams). The Boltzmann weight contains an infinite number of external potentials that depend on the shape of the diagonal slice of plane partitions. The partition function is thereby a function of an infinite number of coupling constants $t_1,t_2,...$ and an extra one $Q$. There is a compact expression of this partition function in the language of a 2D complex free fermion system, from which one can see the presence of a quantum torus algebra behind this model. The partition function turns out to be a tau function (times a simple factor) of two integrable structures simultaneously. The first integrable structure is the bigraded Toda hierarchy, which determine the dependence on $t_1,t_2,...$. This integrable structure emerges when the $q$-parameters $q_1,q_2$ take special values. The second integrable structure is a $q$-difference analogue of the 1D Toda equation. The partition function satisfies this $q$-difference equation with respect to $Q$. Unlike the bigraded Toda hierarchy, this integrable structure exists for any values of $q_1,q_2$.
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。
