Spinor and gauge connections over oriented spheres

The author presents here some work done on a remarkable relation between spinor connections over oriented spheres S/sub n/ (n ≥ 2) and topologically non-trivial gauge configurations. The Levi-Civita connection of the standard Riemannian metric on S/sub n/ lifts to a spinor connection which gives a solution of the Yang-Mills equations on S/sub n/. The structure group is Spin (n) and there is invariance under the action of Spin (n+1). If one transforms the spinor connection into a connection on the Euclidean space R/sub n/ by means of a stereographic projection, then the resulting connection is a solution of the YM equations on R/sub n/ if and only if n=4. This is easily understood in view of the fact that the YM equations are conformally invariant only in four dimensions