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Geometric structure and integrable systems in mathematical physics

数学物理中的几何结构和可积系统

基本信息

批准号：
16340040
负责人：
TAKASAKI Kanehisa
金额：
$ 5.63万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16340040/
关键词：
integrable hierarchy dispersionless integrable system solvable many body system gauge theory free fermion conformal mapping Grassmann manifold hyperbolic structure 量子可積分系無分散極限ランダム行列等モノドロミー変形ハミルトン構造同変コホモロジー Calogero系 Loew

项目摘要

1.We considered the instanton sum of four and five dimensional supersymmetric gauge theories as a model of random (plane) partitions, and applied the method of free fermions for integrable hierarchies to derive the Seiberg-Witten curve.2.We pointed out a relation between a special class of deformation process of conformal mapping and a kind of dispersionless integrable systems. A solution technique (hodograph method) of such integrable systems was also studied.3.We derived several new dispersionless integrable systems.as quasi-classical limit from integrable hierarchies. An example is related to a q-analogue of the modified KP (and Toda) hierarchy. Another example is obtained from the two-component BKP hierarchy. Moreover, we could identify the so called genus-zero universal Whitham hierarchy to be quasi-classical limit of a multi-component analogue of the KP hierarchy.4.We elucidated some new features of solvable many body systems (the Calogero-Moser system, the Sutherland systems, and their variants) such as : equilibrium configuration, shape invariance, creation-annihilation operator (as quantum mechanics), direct integration method (as classical mechanics), etc.5.We obtained several geometric results on Grassmann manifolds, noncommutative algebraic varieties, invariants of low dimensional manifolds, hypergeometric equations related to hyperbolic cones, etc.6.We did some other researches on random matrices, Seiberg-Witten integrable systems, isomonodromic deformations, integrable systems related to a moduli space of vector bundles, etc.

1.将四维和五维超对称规范理论的瞬子和看作是一个随机(平面)划分模型，并应用可积族的自由费米子方法导出了Seiberg-Witten曲线。2.指出了保角变换的一类特殊形变过程与一类无色散可积系统之间的关系。研究了这类可积系统的一种求解方法(速度图法)。3.从可积族出发，导出了几个新的无色散可积系统，作为准经典极限。一个例子与修正的KP(和Toda)层次的Q-模拟有关。另一个例子是从两分量BKP层次结构中得到的。4.阐明了可解多体系统(Calogero-Moser系统、Sutherland系统及其变种)的一些新特征，如：平衡位形、形状不变性、产生-湮灭算符(作为量子力学)、直接积分方法(作为经典力学)等。5.我们得到了Grassmann流形、非对易代数变元、低维流形的不变量、与双曲锥有关的超几何方程等几何结果Seiberg-Witten可积系统、等单形变形、与向量丛的模空间相关的可积系统等。