Solutions to Yang-Mills equations that are not self-dual. (English) Zbl 0731.53031

Sibner, L. M.; Sibner, R. J.; Uhlenbeck, K.

Proc. Natl. Acad. Sci. USA 86, No. 22, 8610-8613 (1989).

Summary: The Yang-Mills functional for connections on principal SU(2) bundles over \(S^ 4\) 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 \(S^ 4\) and magnetic monopoles on \(H^ 3\). This result settles a question in gauge field theory that has been open for many years.

MSC:

53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
37D99	Dynamical systems with hyperbolic behavior
53C80	Applications of global differential geometry to the sciences

Keywords:

Yang-Mills functional; connections; Yang-Mills equations; critical points; instantons; monopoles; gauge field