S-duality in Yang-Mills theory with non-abelian unbroken gauge group

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 H̃. In the spirit of the Goddard-Nuyts-Olive conjecture this observation is interpreted as evidence for a hidden magnetic symmetry of Yang-Mills theory. 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 Yang-Mills 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.