S-duality in SU(3) Yang-Mills Theory with Non-abelian Unbroken Gauge Group

TL;DR

This paper explores S-duality in SU(3) Yang-Mills theory with a non-abelian unbroken gauge group, revealing a hidden magnetic symmetry, classifying charge sectors, and demonstrating an SL(2,Z) duality mapping that preserves fusion rules.

Contribution

It provides a detailed analysis of magnetic charges, dual group actions, and the formulation of non-abelian electric-magnetic duality as an SL(2,Z) transformation in Yang-Mills theory.

Findings

Magnetic charges correspond to coherent states of a dual group.

Electric and magnetic symmetries are mutually obstructive.

S-duality acts as an SL(2,Z) map preserving fusion rules.

Abstract

It is observed that the magnetic charges of classical monopole solutions in Yang-Mills-Higgs theory with non-abelian unbroken gauge group $H$ are in one-to-one correspondence with coherent states of a dual or magnetic group $\tilde H$. In the spirit of the Goddard-Nuyts-Olive conjecture this observation is interpreted as evidence for a hidden magnetic symmetry of Yang-Mills theory. SU(3) Yang-Mills-Higgs theory with unbroken gauge group U(2) is studied in detail. The action of the magnetic group on semi-classical states is given explicitly. Investigations of dyonic excitations show that electric and magnetic symmetry are never manifest at the same time: Non-abelian magnetic charge obstructs the realisation of electric symmetry and vice-versa. On the basis of this fact the charge sectors in the theory are classified and their fusion rules are discussed. Non-abelian electric-magnetic duality is formulated as a map between charge sectors. Coherent states obey particularly simple fusion rules, and in the set of coherent states S-duality can be formulated as an SL(2,Z)-mapping between sectors which leaves the fusion rules invariant.