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

Research Project

Project/Area Number	19540104
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Geometry
Research Institution	Waseda University
Principal Investigator	KORI Toshiaki Waseda University, 理工学術院, 教授 (50063730)
Project Period (FY)	2007 – 2009
Project Status	Completed (Fiscal Year 2009)
Budget Amount *help	¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000) Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000) Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000) Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords	ヤング・ミルズ接続 / 幾何的量子化 / ディラック作用素 / ADHM構成 / スピノール解析 / 接続のモジュライ空間 / シンプレクティク構造
Research Abstract	(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.