Toda hierarchies and their applications
Toda hierarchies and their applications
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DOI：
10.1088/1751-8121/aabc14
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发表时间：
2018-01
期刊：
影响因子：
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通讯作者：
K. Takasaki
中科院分区：
文献类型：
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作者：
K. Takasaki
二维Toda层次结构在Toda类型的可积层次结构家族中占据中心位置。一维Toda层次和Ablowitz-Ladik（又名相对论Toda）层次可以从二维Toda层次作为约化而得到。自20世纪90年代以来，这些可积层次已被应用于各种数学和数学物理问题。最近的一个例子是一系列关于统计力学模型的研究，称为熔融晶体模型。这项研究揭示了上述两种二维Toda层次结构的减少是两种不同的熔融晶体模型的基础。技术线索是量子环面代数的费米子实现，其中称为移位对称的特殊代数关系，以及矩阵分解问题。因此，这两种熔融晶体模型与厄米矩阵和酉矩阵模型具有显著的相似性，其中二维Toda层次的两种约简起着基本可积结构的作用。
The 2D Toda hierarchy occupies a central position in the family of integrable hierarchies of the Toda type. The 1D Toda hierarchy and the Ablowitz–Ladik (aka relativistic Toda) hierarchy can be derived from the 2D Toda hierarchy as reductions. These integrable hierarchies have been applied to various problems of mathematics and mathematical physics since 1990s. A recent example is a series of studies on models of statistical mechanics called the melting crystal model. This research has revealed that the aforementioned two reductions of the 2D Toda hierarchy underlie two different melting crystal models. Technical clues are a fermionic realization of the quantum torus algebra, special algebraic relations therein called shift symmetries, and a matrix factorization problem. The two melting crystal models thus exhibit remarkable similarity with the Hermitian and unitary matrix models for which the two reductions of the 2D Toda hierarchy play the role of fundamental integrable structures.