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Geometry of the space of Yang-Mills connections and its dual.

杨-米尔斯连接空间的几何及其对偶。

基本信息

批准号：
19540104
负责人：
KORI Toshiaki
金额：
$ 2.25万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2007
资助国家：
日本
起止时间：
2007 至 2009
项目状态：
已结题

项目摘要

(1) The moduli space of flat connections on four manifolds is given a pre-symplectic structure. And a geometric pre-quantization of this space is constructed. When the 4-manifold has the boundary the gauge transformation group on the boundary acts on the moduli space infinitesimally symplectically. This actionlifts to the pre-quantization equivariantly. (2) The Lie group extension of SU(n)-current group was constructed by J. Mickelsson for n bigger than 3. The similar construction for the SU(2)-current group had not been solved. I have constructed two kind of Lie group extensions of SU(2)-current group. (There exist actually two types of extensions.) (1) and (2) were the subjects of former research, but here stated results are improved ones and given the final form. (3) One of the purpose of this research is to construct a general frame work of the dual spaces of the space of connections and the transformation on it so that one can see transparently the "Zakharov-Shabat method for integrable systems". For that we constructed the theory of residue and duality on the solution space of gauge-coupled Dirac operators that may describe the behaqvior of the solutions near their singular points. As an application we arranged in a clear form the ADHM construction of solitons.

(1)给出了四流形上平坦联络的模空间的准辛结构。并构造了该空间的几何预量子化。当四维流形有边界时，边界上的规范变换群无穷小辛地作用在模空间上。该作用等变地提升到预量化。(2)J. Mickelsson构造了SU（n）-流群的李群扩张，其中n大于3。SU（2）-流群的类似构造一直没有解决。构造了SU（2）-流群的两种李群扩张。（实际上有两种类型的扩展。(1)（2）是前人研究的对象，但本文所述结果是改进的结果，并给出了最终形式。(3)本文研究的目的之一是建立联络空间的对偶空间及其上的变换的一般框架，从而使“可积系统的Zakharov-Shabat方法”变得透明。为此，我们在规范耦合Dirac算子的解空间上建立了留数和对偶理论，可以描述其奇点附近解的性质。作为应用，我们以一种清晰的形式安排了孤子的自洽介质结构。