Mathematical Sciences: Non-Self-Dual Yang-Mills Connections

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The principal investigator will continue his study of non- self-dual Yang-Mills connections. The existence of these connections on any bundle satisfying a topological condition was recently proved by Sadun and the principal investigator. He will now use algebraic and analytic methods to attempt to obtain additional topological and geometric information. Spectacular geometric results about four-dimensional manifolds were obtained from a study of the self-dual Yang-Mills moduli spaces and it is hoped that equally important breakthroughs will result from a study of the non-self-dual case. The Yang-Mills equations were first studied by physicists in an attempt to create a unified field theory. Deep mathematical techniques were needed to analyze these equations and it soon became apparent that this area of mathematical physics possessed a fascinating algebraic and geometric structure. The field has developed to become one which is important both to physics and mathematics and in which ideas from each discipline contribute to the other.