Yang-Mills theories in curved space-times

Multi-instanton solutions of four dimensional HP(^1) models are sought, and a singular two instant solution in flat Euclidean space time is constructed. Non-singular multi-instanton solution can be constructed if a gravitational field is introduced, as first pointed out by Gursey et al. Their method is developed, and in the process a formalism for the construction of an (anti) self-dual SU(2) Yang- Mills field tensor in curved space-times is exhibited. Demanding that a potential for the SU(2) field exists implies that, for a space of non-zero scalar curvature, Einstein's field equations must be satisfied, and conditions on the Weyl tensor are found. It is shown how the formalism relates to the work of Charap and Duff Finally the method is applied to the four dimensional complex projective space and the four dimensional manifold consisting of the outer product of two two spheres.