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

2009 Fiscal Year Final Research Report

• 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
• Keywords: ヤング・ミルズ接続 / 幾何的量子化 / ディラック作用素

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.