Finite dimensional integrable structure in systems with infinite degree of freedom

无限自由度系统中的有限维可积结构

• 批准号: 10640165
• 负责人: TAKASAKI Kanehisa
• 金额: $ 2.11万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640165/
• 关键词: integrable system; Whitham deformation; isomonodromic deformation; supersymmetric field theory; topological field theory; solvable spin system; conformal field theory; Toda lattice; 共形場理論; モノドロミー保存変形; 有限次元可積分系; 超対称性; ゲージ理論; 位相不変量; 表現論; 複素力学

This research project aimed to search for various finite dimensional integrable systems in systems with infinite degrees of freedom, and to elucidate their mathematical structures. Related issues on geometry and non-integrable systems were also investigated.The head investigator obtained interesting results on Whitham deformation equations, isomonodromic deformations, systems arising in supersymmetric/topological gauge theories and Calogero-Moser systems, all of which are mutually related. Firsly, he could find an explicit form of the Whitham deformation equations for asymptoci description of isomonodromic systems on the Rie-mann sphere. Secondly, an extension, to a torus, of such isomonodromic systems was achieved on the basis of methods in solvable spin sysmtes and conformal field theories. Thirdly, he clarified the roles that the tau functions and the Whitham deformations play in four dimensional supersymmetric and topolotical gauge theories. Finally, he considered the elliptic Calogero-Moser systems, which are also closely related to four dimensional supersymmetric gauge theories, and discoveref that a non-autonomous analogues of those systems give an example of isomonodromic dermations on a torus.The achievement due to other members of the project group respectively ranges over low-dimensional topology, stochastic analysis of hypoelliptic operators, complex dynamical systems, transcendental number theory related to nonlinear dynamical systems, celular automata, intelligent agents, hypoelliptic and hyperbolic equations with infnite degeneracy,

本研究计画旨在于无限自由度系统中寻找各种有限维可积系统，并阐明其数学结构。此外，还对几何学和不可积系统的相关问题进行了研究，主要研究者在Whitham形变方程、等单点形变、超对称/拓扑规范理论中产生的系统以及Calogero-Moser系统等相互关联的问题上获得了有趣的结果。首先，他可以找到一个显式形式的Whitham变形方程的渐近描述的isomonodromic系统的黎曼球。其次，利用可解自旋系统和共形场论的方法，将这类等单轨道系统推广到环面上。第三，他澄清的作用，tau功能和惠瑟姆变形发挥四维超对称和拓扑规范理论。最后，他考虑了与四维超对称规范理论密切相关的椭圆Calogero-Moser系统，并发现这些系统的非自治类似物给出了环面上的等单轨道dermations的例子。项目组其他成员的成就分别涉及低维拓扑，亚椭圆算子的随机分析，复杂动力系统，与非线性动力系统、细胞自动机、智能代理、具有无限退化性的亚椭圆和双曲方程相关的超越数论，

项目成果

高崎 金久: "Integrable Hierachies and Contact Terms in u-plane Integrals of Topologically Twisted Supersymmetric Gauge Theories"Int.J.Mod.Phys.. 14巻7号. 1001-1013 (1999)

Kanehisa Takasaki：“拓扑扭曲超对称规范理论的 u 平面积分中的可积层次和接触项” Int.J.Mod.Phys.. 第 14 卷，第 7 期。1001-1013 (1999)

Kanehisa Takasaki: "Calogero-Moser Models IV : Limits to Toda theory"Prog. Theor. Phys. Vol.102 No.4. 749-776 (1999)

高崎兼久：“Calogero-Moser 模型 IV：户田理论的限制”Prog。

Kanehisa Takasaki: "Calogero-Moser Models II : Symmetries and Foldings"Prog. Thero. Phys. Vol. 101 No. 3. 487-518 (1999)

Kanehisa Takasaki：“Calogero-Moser 模型 II：对称性和折叠”Prog。

高崎金久: "Integrable Hierarchies and Contact Terms in u-plane Integrals of Topologically Twisted Supersymmetric Gauge Theories" Int.J.Mod.Phys.A.

Kanehisa Takasaki：“拓扑扭曲超对称规范理论的 u 平面积分中的可积层次和接触项”Int.J.Mod.Phys.A。

高崎 金久: "Calogero-Moser Models IV:Limits to Toda theory"Prog.Theor.Phys.. 102巻4号. 749-776 (1999)

Kanehisa Takasaki：“Calogero-Moser 模型 IV：Toda 理论的限制”Prog.Theor.Phys.. Vol. 102，No. 4. 749-776 (1999)