Analysis on analytic spaces and non-self-dual Yang-Mills fields

Abstract:

This paper gives a cohomological description of the Witten-Isenberg-Yasskin-Green generalization to the non-self-dual case of Ward’s twistor construction for self-dual Yang-Mills fields. The groundwork for this description is presented in Part I: with a brief introduction to analytic spaces and differential forms thereon, it contains an investigation of the exactness of the holomorphic relative de Rham complex on formal neighbourhoods of submanifolds, results giving sufficient conditions for the invertibility of pull-back functors on categories of analytic objects, and a discussion of the extension problem for analytic objects in the context of the formalism earlier introduced. Part II deals with non-self-dual Yang-Mills fields: the Yang-Mills field and current are identified in terms of the Griffiths obstructions to extension, including a proof of Manin’s result that "current = obstruction to third order". All higher order obstructions are identified, there being at most ${N^2}$ for a bundle of rank $N$. An ansatz for producing explicit examples of non-self-dual fields is obtained by using the correspondence. This ansatz generates ${\text {SL}}(2,\mathbb {C})$ solutions with topological charge $1$ on ${S^4}$.