Solutions to Yang-Mills Equations that are not Self-Dual

The Yang-Mills functional for connections on principle SU(2) bundles over S⁴ is studied. Critical points of the functional satisfy a system of second-order partial differential equations, the Yang-Mills equations. If, in particular, the critical point is a minimum, it satisfies a first-order system, the self-dual or anti-self-dual equations. Here, we exhibit an infinite number of finite-action nonminimal unstable critical points. They are obtained by constructing a topologically nontrivial loop of connections to which min-max theory is applied. The construction exploits the fundamental relationship between certain invariant instantons on S⁴ and magnetic monopoles on H³. This result settles a question in gauge field theory that has been open for many years.