Algebraically special Yang–Mills solutions without sources

Abstract

We consider source-free Yang–Mills solutions for which the curvature is decomposable in the sense that the curvature 2-form is the product of a single Lie-algebra-valued function and a real 2-form. If the curvature is everywhere nonnull (or null with twisting rays), then the solution is a connection in a principal fiber bundle, which is reducible to a source-free Maxwell principal bundle. All such solutions are therefore readily obtained, locally or globally, from Maxwell solutions. Our analysis uses the Ambrose–Singer theorem to show that the holonomy group is one-dimensional. A principal bundle-with-connection is reducible to the holonomy subbundle of any point, and, in this case, since the holonomy group is one-dimensional, the reduced bundle has the structure of a Maxwell bundle. On the other hand, if the curvature is null and twist-free on a full neighborhood of some point, then the bundle need not be reducible. The holonomy group is generally the entire gauge Lie group. The solutions can still be constructed locally from Maxwell solutions, there being extra freedom in the construction over regions where the Maxwell field is algebraically null with twist-free rays. We extend all these results to the class of solutions for which the self-dual curvature is (complex) decomposable. These are all the solutions of type D and type N in the classification of Anandan and Tod.