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Massive Integrable Models and Infinite Dimensional Symmetries

大规模可积模型和无限维对称性

基本信息

批准号：
12640261
负责人：
ODAKE Satoru
金额：
$ 1.34万
依托单位：
Shinshu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640261/
关键词：
deformed Virasoro algebra Lepowsky-Wilson's Z algebra deformed W algebra free field realization elliptic quantum group solvable lattice model Calogero-Moser model principal gradation 古典平衡点 Coxeter群ルート系変形Viraroso代数 Macdonald予想変

项目摘要

In order to answer the question "What symmetries ensure the integrability or infinitely many conserved quantities of massive integrable models (quantum field theory and lattice models)?", we have been studying infinite dimensional symmetries and solvable lattice models.Corresponding to the face type and the vertex type of solvable lattice models whose Boltzmann weights are elliptic solutions of the Yang-Baxter equation, elliptic quantum groups have the face type and the vertex type. One of their differences is the counting of the energy eigenvalues; the former is homogeneous gradation and the latter is principal gradation. Free filed realizations of solvable lattice models were obtained for some face type models but not for the vertex type models direct way. To get some hint for this problem, we studied (quantum) affine Lie algebra with principal gradation. We construct a free field realization of sl^^^^_2 with principal gradation. We point out that the Lepowsky-Wilson's Z algebra, whi … More ch is obtained from sl^^^^_2 with principal gradation by splitting the Cartan part, is certain limit of the deformed Virasoro algebra. In other word the deformed Virasoro algebra can be considered as a q-deformation of Lepowsky-Wilson's Z algebra. This observation may help us to study the vertex models. For higher rank case we establish the same relationship between the sl_N version of Z algebra and the deformed W_N algebra by calculating the defining relation of the deformed W_N algebra explicitly.Calogero-Moser models are very interesting integrable models as both classical and quantum mechanics, for example, they are related to the Seiberg-Witten theory of super Yang-Mills theory, the deformed Virasoro and W algebras, etc. Recently it is pointed out that various quantities at equilibrium positions are 'quantized in integer values' even in classical theory. Motivated by this observation, we calculate the classical equilibrium position of Calogero-Moser models with rational and trigonometric potential for all finite root systems, and define Coxeter (Weyl) invariant polynomials. Coefficients of these polynomials are also integer values. Less

为了回答“什么对称性保证了大质量可积模型（量子场论和格点模型）的可积性或无穷多守恒量？与Boltzmann权为Yang-Baxter方程椭圆解的可解格点模型的面型和顶点型相对应，椭圆量子群也有面型和顶点型。它们的区别之一是能量特征值的计算，前者是均匀级配，后者是主级配。对于某些面型模型，直接得到了可解格模型的自由场实现，而对于顶点型模型，直接得到的可解格模型的自由场实现是不可能的。为了给这个问题提供一些启示，我们研究了具有主阶化的（量子）仿射李代数。我们构造了sl ^^_2的自由场实现。本文指出，Lepowsky-Wilson的Z代数， ...更多信息 ch是由具有主阶的sl ^^_2分裂Cartan部分得到的，是变形Virasoro代数的一定极限.换句话说，变形的Virasoro代数可以被认为是Lepowsky-Wilson Z代数的q-变形。这一发现有助于我们对顶点模型的研究。Calogero-Moser模型是经典和量子力学中非常有趣的可积模型，它与超Yang-Mills理论中的Seiberg-Witten理论，变形Virasoro和W代数，最近有人指出，即使在经典理论中，平衡位置上的各种量也是“以整数值量子化的”。受此启发，我们计算了具有有理势和三角势的Calogero-Moser模型在所有有限根系下的经典平衡位置，并定义了Coxeter（Weyl）不变多项式.这些多项式的系数也是整数值。少