Nonabelian Generalization of Electric-Magnetic Duality - a Brief Review

Abstract

A loop space formulation of Yang-Mills theory high-lighting the significance of monopoles for the existence of gauge potentials is used to derive a generalization of electric-magnetic duality to the nonabelian theory. The result implies that the gauge symmetry is doubled from SU(N) to $SU(N) \times \widetilde{SU}(N)$, 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 $\tilde{A}_μ(x)$. Nonabelian `electric' charges appear as sources of $A_μ$ but as monopoles of $\tilde{A}_μ$, while their `magnetic' counterparts appear as monopoles of $A_μ$ but sources of $\tilde{A}_μ$. 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 $\tilde{A}_μ$ 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 then dual colour $\widetilde{SU}(3)$ is broken. Such predictions can lead to many very interesting physical consequences which are explored in a companion paper.