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

(1)给出了四个流形上的平联络的模空间具有预辛结构。并构造了该空间的几何预量化。当4-流形有边界时，边界上的规范变换群在模空间上无限辛地作用。该动作等量地提升到预量化。(2)J.Mickelsson对n大于3构造了SU(N)-流群的Lie群扩张，但SU(2)-流群的类似构造尚未解决。构造了SU(2)-当前群的两种李群扩张。(实际上存在两种类型的扩展。)(1)和(2)是以前研究的对象，但这里所述的结果是改进的结果，并给出了最终形式。(3)本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，使人们能够透明地看到“可积系统的Zakharov-Shabat方法”。为此，我们在规范耦合Dirac算子的解空间上构造了可描述解在奇点附近的行为的留数和对偶理论。作为应用，我们以清晰的形式安排了孤子的ADHM结构。

(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.