Yang–Mills Equations of General Connections and a Certain Solutions in the Quaternionic Hopf Fibration Over Four-Sphere

Abstract

We investigate a version of Yang–Mills theory by means of general connections. In order to deduce a basic equation, which we regard as a version of Yang–Mills equation, we construct a self-action density using the curvature of general connections. The most different point from the usual theory is that the solutions are given in pairs of two general connections. This enables us to get nontrivial solutions as general connections. Especially, in the quaternionic Hopf fibration over four-sphere, we demonstrate that there certainly exist nontrivial solutions, which are made by twisting the well-known BPST anti-instanton.