A Nonabelian Yang-Mills Analogue of Classical Electromagnetic Duality

The classic question of a nonabelian Yang-Mills analogue to electromagnetic duality is here examined in a minimalist fashion at the strictly 4-dimensional, classical field and point charge level. A generalisation of the abelian Hodge star duality is found which, though not yet known to give dual symmetry, reproduces analogues to many dual properties of the abelian theory. For example, there is a dual potential, but it is a 2-indexed tensor $T_{\mu\nu}$ of the Freedman-Townsend type. Though not itself functioning as such, $T_{\mu\nu}$ gives rise to a dual parallel transport, $\tilde{A}_\mu$, for the phase of the wave function of the colour magnetic charge, this last being a monopole of the Yang-Mills field but a source of the dual field. The standard colour (electric) charge itself is found to be a monopole of $\tilde{A}_\mu$. At the same time, the gauge symmetry is found doubled from say $SU(N)$ to $SU(N) \times SU(N)$. A novel feature is that all equations of motion, including the standard Yang-Mills and Wong equations, are here derived from a `universal' principle, namely the Wu-Yang (1976) criterion for monopoles, where interactions arise purely as a consequence of the topological definition of the monopole charge. The technique used is the loop space formulation of Polyakov (1980).Comment: We regret that, due to a technical hitch, parts of the reference list were mixed up. This is the corrected version. We apologize to the authors whose papers were misquote

On loop space formulation of gauge theories

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A Generalized duality symmetry for nonabelian Yang-Mills fields

It is shown that classical nonsupersymmetric Yang-Mills theory in 4 dimensions is symmetric under a generalized dual transform which reduces to the usual dual *-operation for electromagnetism. The parallel phase transport $\tilde{A}_\mu(x)$ constructed earlier for monopoles is seen to function also as a potential in giving full description of the gauge field, playing thus an entirely dual symmetric role to the usual potential $A_\mu(x)$. Sources of $A$ are monopoles of $\tilde{A}$ and vice versa, and the Wu-Yang criterion for monopoles is found to yield as equations of motion the standard Wong and Yang-Mills equations for respectively the classical and Dirac point charge; this applies whether the charge is electric or magnetic, the two cases being related just by a dual transform. The dual transformation itself is explicit, though somewhat complicated, being given in terms of loop space variables of the Polyakov type.Comment: Latex file, 26 pages, 3 figures and 2 charts not included but supplied on reques