Solutions to Yang-Mills field equations in eight dimensions and the last Hopf map

Abstract

We will show that the Hopf map\(S^{15} \xrightarrow{{S^7 }}S^8\) admits a sourceless, topologically non-trivial gauge field. This result will be cast in the form of a solution to eight dimensional Euclidean Yang-Mills field equations with topological chargeQ=1. This solution is Spin (9) symmetric and leads to a new generalized duality conditionF∧F=±(F∧F)*.