Nonabelian Generalization of Electric-Magnetic Duality - a Brief Review

TL;DR

This paper reviews a loop space formulation of nonabelian gauge theories, proposing a duality that doubles the gauge symmetry and relates electric and magnetic charges, with implications for confinement and dual symmetry breaking.

Contribution

It introduces a nonabelian generalization of electric-magnetic duality using loop space formalism, doubling gauge symmetry and linking electric and magnetic charges in Yang-Mills theory.

Findings

Gauge symmetry doubles from SU(N) to SU(N) × ˜SU(N)

Electric charges act as sources for Aμ and monopoles for ˜Aμ

Dual color symmetry ˜SU(3) is broken in QCD

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 $A_\mu(x)$ or its dual $\tilde{A}_\mu(x)$. Nonabelian `electric' charges appear as sources of $A_\mu$ but as monopoles of $\tilde{A}_\mu$, while their `magnetic' counterparts appear as monopoles of $A_\mu$ but sources of $\tilde{A}_\mu$. 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 $A_\mu$ and $\tilde{A}_\mu$ 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.