Part 1 : Topological aspects of Yang-Mills fields in curved spaces. (Exact solutions)

Abstract

To explore in its full richness the topological possibilities of gauge fields one should allow for simultaneous presence of gravitational and Yang-Mills ones. Thus if the integral topological indices of the Yang-Mills field for a flat Euclidean base space is associated with the structure of the vacuum, one may ask among other questions of interest, how this spectrum might be modified when the base space itself has non trivial indices. Exact solutions of SU(2) Yang-Mills fields are presented for metrics corresponding to well-known gravitational instantons. Such selfdual solutions, with vanishing energy monien-tunl tensor Tμv for Euclidean signature of the base space, do not perturb the metric. Thus they provide solutions of the combined gravitational-Y.M. system. New topological possibilities, such as finite action SU(2) fields with fractional indices for many centre inetrics are displayed explicitly. As another type of possibility non selfdual, finite action solutions are constructed explicitly on Schwarzschild and de Sitter metrics, the solution being real in the first and complex in second case respectively. It is also shown how various meron type solutions in flat space can be derived systematically from a very simple static solution in de Sitter.