Applications of integrable systems in geometry and topology

可积系统在几何和拓扑中的应用

基本信息

批准号：
12640083
负责人：
GUEST Martin
金额：
$ 2.24万
依托单位：
Tokyo Metropolitan 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 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640083/
关键词：
Integrable system harmonic map holomorphic map quantum cohomology

项目摘要

Results were obtained on the geometry and topology of harmonic maps and spaces of harmonic maps, especially in the case where the domain is a Riemann surface and the target space is a compact Lie group or symmetric space. Guest used a generalization of the Weierstrass representation for minimal surfaces to study harmonic maps from the two-dimensional sphere (or, more generally, harmonic maps of finite uniton number, from any Riemann surface) to the unitary group. Earlier results of Uhlenbeck, Segal, Dorfmeister-Pedit-Wu, Burstall-Guest were developed into an effective tool for describing such maps. In particular, an explicit canonical form was obtained, and this was used to study the space of all such maps. The main application was a description of the connected components of the space of harmonic maps from the two-dimensional sphere to the unitary group. Ohnita used a different approach, based on earlier work of Hitchin in gauge theory, to obtain a framework for studying the geometry (in particular, the pre-symplectic geometry) of spaces of harmonic maps.The harmonic map equation can be regarded as an integrable system, and the above work sheds light on other integrable systems. Two other examples of integrable systems were studied from this point of view, and preliminary results obtained. The first example, studied by Guest, was the theory of quantum differential equations. Parallels with harmonic maps were established, forming the basis for future work in this direction. Results on quantum cohomology of symmetric spaces were obtained also by Ohnita and Nishimori, and on quantum cohomology of flag manifolds by Guest and Otofuji. The second example, studied by Burstall and Calderbank, was the integrable systems aspect of conformal and Mobius geometry, and a new approach was initiated.

获得了调和映射和调和映射空间的几何和拓扑的结果，特别是在域是黎曼曲面且目标空间是紧李群或对称空间的情况下。 Guest 使用最小曲面的 Weierstrass 表示的推广来研究从二维球体（或者更一般地说，从任何黎曼曲面的有限单位数的调和映射）到酉群的调和映射。 Uhlenbeck、Segal、Dorfmeister-Pedit-Wu、Burstall-Guest 的早期结果已发展成为描述此类地图的有效工具。特别是，获得了明确的规范形式，并将其用于研究所有此类地图的空间。主要应用是描述从二维球体到酉群的调和映射空间的连通分量。 Ohnita 基于希钦规范理论的早期工作，采用了不同的方法，获得了研究调和映射空间几何（特别是前辛几何）的框架。调和映射方程可以被视为可积系统，上述工作为其他可集成系统提供了启示。从这个角度研究了另外两个可积系统的例子，并得到了初步结果。盖斯特研究的第一个例子是量子微分方程理论。建立了与谐波图的平行关系，为未来这一方向的工作奠定了基础。 Ohnita 和 Nishimori 也获得了对称空间量子上同调的结果，Guest 和 Otofuji 也获得了旗形流形量子上同调的结果。 Burstall 和 Calderbank 研究的第二个例子是共形几何和莫比乌斯几何的可积系统方面，并提出了一种新方法。