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.我们将四个维度超对称理论的intsanton总和作为随机（平面）分区的模型，并应用了自由费米子的方法来推导Seiberg-witten曲线。22。我们指出了在特殊的合格映射的特殊形象映射和无用集成系统的特殊错误类别之间的关系。还研究了此类可集成系统的解决方案技术（Hodograph方法）。3。我们得出了几种新的无散集成系统。作为准分层次结构的准经典限制。一个示例与修改后的KP（和TODA）层次结构的Q Analogue有关。另一个示例是从两个组件BKP层次结构中获得的。此外，我们可以确定所谓的零属通用层次结构是KP层次结构的多组分类似物的准经典限制。机械师），直接整合方法（作为经典的力学）等。5。我们在格拉斯曼流形，非交易性代数品种，低维歧管的不变式，与超圆锥体相关的超几何方程等方面获得了几种几何结果。6。我们对随机矩阵的其他相关系统进行了一些相关的整合效果。矢量束的模量空间等。