Massive Integrable Models and Infinite Dimensional Symmetries

大规模可积模型和无限维对称性

• 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

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

In order to answer the question "What symmetries ensure the integrability or infinitely many configured 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量子组具有面部类型和顶点类型。他们的差异之一是计算能量特征值。前者是均匀的等级，后者是主要等级。对于某些面部类型模型而言，获得了免费的可解决晶格模型的现实，但对于顶点类型模型的直接方式没有。为了获得这个问题的提示，我们研究了（量子）主体代数，并以主等级为单位。我们构建了sl ^^^^ _ 2的自由场实现，并具有主等级。我们指出，Lepowsky-Wilson的Z代数，whi…更多CH是从Sl ^^^^ _ 2获得主级别通过分裂cartan部分获得的，是变形的Virasoro代数的某些限制。换句话说，变形的virasoro代数可以被视为Lepowsky-Wilson's Z代数的Q信息。该观察结果可能有助于我们研究顶点模型。对于较高等级的情况，我们通过计算明显的变形W_n代数的定义关系是非常有趣的集成模型，因为它们是非常有趣的集成模型，因为它们是大小和量子机制相关，例如，它们与Seclos of Seply and ynoge ynoge ynoge yourgation nowder y，是相关的，我们建立了Z代数SL_N版本与变形的W_N代数之间相同的关系。代数等。最近指出，即使在经典理论中，等效位置处的各种数量也被“量化”。在这种观察过程中，我们计算了所有有限根系的合理和三角学潜力的Calogero-Moser模型的经典等效位置，并定义了Coxeter（Weyl）不变的多项式。这些多项式的系数也是整数值。较少的

项目成果

M. Jimbo, H. Konno, S. Odake, Y. Pugai and J. Shiraishi: ""Free Field Construction for the ABF Models in Regime II""Journal of Statistical Physics. 102. 883-991 (2001)

M. Jimbo、H. Konno、S. Odake、Y. Pugai 和 J. Shiraishi：““Regime II 中 ABF 模型的自由场构造””统计物理学杂志。

Y.Saint-Aubin(S.Odake): "Theoretical Physics at the End of the Twentieth Century"Springer. 636(307-449) (2002)

Y.Saint-Aubin（S.Odake）：《二十世纪末的理论物理学》施普林格。

S.Odake: "Comments on the Deformed W_N Algebra"International Journal of Modern Physics. B16. 2055-2064 (2002)

S.Odake：“变形 W_N 代数的评论”国际现代物理学杂志。

Y.Hara: "On Lepowsky-Wilson's Z-algebra"Proceedings of the Conference on Infinite-Dimensional Lie Theory and Conformal Field Theory,(CONM series AMS に掲載).

Y. Hara：“论 Lepowsky-Wilson 的 Z 代数”无限维李理论和共形场论会议论文集（发表于 CONM 系列 AMS）。

Y. Hara, M. Jimbo, H. Konno, S. Odake and J. Shiraishi: ""On Lepowsky-Wilson's Z-algebra""Contemporary Mathematics. 297. 143-149 (2002)

Y. Hara、M. Jimbo、H. Konno、S. Odake 和 J. Shiraishi：“论 Lepowsky-Wilson 的 Z 代数”当代数学。