Non-minimal Yang-Mills fields and dynamics (Q1187491)

From the variational viewpoint Yang-Mills fields on Riemannian 4- manifolds that are neither self-dual nor anti-self-dual correspond to non-minimal critical points of the Yang-Mills action function. The paper describes a dynamical systems approach to Yang-Mills fields, and uses it to construct non-minimal Yang-Mills fields. Theorem 3.1. Let \(P\to S^ 1_ L\times S^ 3\) be the trivial \(\text{SU}(2)\) bundle over the product of the circle of length \(L\) and the 3-sphere of unit radius. For each integer \(N>0\) there exists an \(L\) such that \(S^ 1_ L\times S^ 3\) admits at least \(N\) non-gauge- equivalent irreducible non-minimal Yang-Mills fields. Theorem 8.3. There exists a family of metrics on \(S^ 4\) each of which admits an irreducible non-minimal Yang-Mills field on the trivial SU(2) bundle. There are metrics in this family arbitrarily close to the standard metric. The key observation is that, when enough symmetry is imposed, Yang-Mills becomes a classical mechanics problem. This correspondence is described in the beginning and used further to construct non-minimal Yang-Mills fields.