Non-Abelian Generalization of Electric-Magnetic Duality — a Brief Review

A loop space formulation of Yang-Mills theory highlighting the significance of monopoles for the existence of gauge potentials is used to derive a generalization of electric-magnetic duality to the non-Abelian theory. The result implies that the gauge symmetry is doubled from SU(N) to {SU}(N) × widetilde {SU}(N), while the physical degrees of freedom remain the same, so that the theory can be described in terms of either the usual Yang-Mills potential A_μ(x) or its dual Ã_μ(x). Non-Abelian "electric" charges appear as sources of A_μ but as monopoles of Ã_μ, while their "magnetic" counterparts appear as monopoles of A_μ but sources of Ã_μ. Although these results have been derived only for classical fields, it is shown for the quantum theory that the Dirac phase factors (or Wilson loops) constructed out of A_μ and Ã_μ satisfy the 't Hooft commutation relations, so that his results on confinement apply. Hence one concludes, in particular, that since colour SU(3) is confined, dual colour widetilde{SU}(3)$ is broken. Such predictions can lead to many very interesting physical consequences, which are explored in a companion paper.