Comprehensive Studies on Hypergeometric and Painleve Systems

超几何和 Painleve 系统的综合研究

• 批准号: 11304007
• 负责人: NOUMI Masatoshi
• 金额: $ 13.43万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11304007/
• 关键词: Painleve equation; affine Weyl group symmetry; hypergeometric equation; configuration space; integrable system; ベックルント変換; 離散可積分系; 初期値空間; モジュライ空間; ミラー対称性; ハミルトン構造

Main achievements of this research are summarized as follows.1. Affine Weyl group symmetry of Painleve systems: Generalizing the structure of Backlund transformations for Painleve equations, Noumi and Yamada proposed a universal framework of Weyl group symmetry as groups of birational transformations. This class of groups of birational transformations is defined in terms of root systems, and has its origin in Kac-Moody groups. Also, they found a method for constructing a class of nonlinear (partial) differential equations of Painleve type by similarity reduction from the infinite integrable systems of Drinfeld-Sokolov. This work brought out a new unified point of view on the Lie theoretic background of the symmetry of Painleve equations.2. Geometric aspects of hypergeometric and Painleve systems: In the joint work with Yoshida, Sasaki constructed the uniformizing differential equation for the moduli space of cubic surfaces. This system of linear differential equations, with Weyl group symmetry of type E_6, arises as a subsystem of the hypergeometric system of type E(3, 6) associated with hyperplane configurations. On the basis of Sakai's geometric approach to Painleve equations, Saito and others found a method of constructing the Painleve equations and their defining manifolds from the algebraic- geometric theory of deformations of rational surfaces.

本研究的主要成果概括如下： 1． Painleve 系统的仿射 Weyl 群对称性：Noumi 和 Yamada 推广 Painleve 方程的 Backlund 变换结构，提出了作为双有理变换群的 Weyl 群对称性的通用框架。这类双有理变换群是根据根系统定义的，并且起源于 Kac-Moody 群。此外，他们还发现了一种通过 Drinfeld-Sokolov 无限可积系统的相似性约简来构造一类 Painleve 型非线性（偏）微分方程的方法。这项工作对Painleve方程对称性的Lie理论背景提出了新的统一观点。 2．超几何和 Painleve 系统的几何方面：在与 Yoshida 的合作中，Sasaki 构造了立方曲面模空间的均匀化微分方程。该线性微分方程组具有 E_6 类型的 Weyl 群对称性，是作为与超平面配置相关的 E(3, 6) 类型超几何系统的子系统而出现的。在 Sakai 对 Painleve 方程的几何方法的基础上，Saito 等人从有理曲面变形的代数几何理论中找到了构造 Painleve 方程及其定义流形的方法。

项目成果

Saito,Masa-Hiko: "Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry""The arithmetic and geometry of algebraic cycles" (Banff,AB,1998), Kluwer Acad, Publ.. 385-425 (2000)

Saito，Masa-Hiko：“某些 Calabi-Yau 3 重和镜像对称的 Yukawa 耦合的预势”“代数圈的算术和几何”（Banff，AB，1998），Kluwer Acad，Publ.. 385-425（

Saito,Mutsumi: "Hypergeometric polynomials and Integer Programming"Compositio Math.. 115. 185-204 (1999)

Saito Mutsumi：“超几何多项式和整数规划”复合数学.. 115. 185-204 (1999)

Yamada,Yasuhiko: "Determinant formulas for the γ-functions of the Painleve equations of type A"Nagoya Math.J.. 156. 123-134 (1999)

山田康彦：“A 型 Painleve 方程的 γ 函数的行列式”Nagoya Math.J.. 156. 123-134 (1999)

K.Kajiwara et al.: "A study on the fourth q-Painleve equation"J.Phys. A : Math.Gen.. 34. 8563-8581 (2001)

K.Kajiwara 等：“第四个 q-Painleve 方程的研究”J.Phys。

T.Sasaki, M.Yoshida: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of a marked cubic surface.II"J.Phys.A. 34. 2319-2328 (2001)

T.Sasaki，M.Yoshida：“标记立方表面模空间上复杂双曲结构的均匀化微分方程。II”J.Phys.A。