A Nonabelian Yang-Mills Analogue of Classical Electromagnetic Duality

TL;DR

This paper explores a classical, four-dimensional nonabelian Yang-Mills analogue to electromagnetic duality, introducing a dual potential and a doubled gauge symmetry, with all equations derived from monopole topological principles.

Contribution

It generalizes abelian Hodge star duality to nonabelian Yang-Mills fields, revealing a dual potential, doubled gauge symmetry, and deriving equations from monopole topology using loop space methods.

Findings

Identifies a dual potential $T_{\mu\nu}$ of Freedman-Townsend type.

Reveals a doubled gauge symmetry from $SU(N)$ to $SU(N) \times SU(N)$.

Derives equations of motion from monopole topological principles.

Abstract

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).