Non-Abelian 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 Tμν of the Freedman-Townsend type. Though not itself functioning as such, Tμν gives rise to a dual parallel transport, Ãμ, 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 Ãμ. At the same time, the gauge symmetry is found doubled from say SU(N) to SU(N)× 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).