Asymptotic Symmetry Groups of Long-Ranged Gauge Configurations

We make some general remarks on long-ranged configurations in gauge or diffeomorphism invariant theories where the fields are allowed to assume some non vanishing values at spatial infinity. In this case the Gauss constraint only eliminates those gauge degrees of freedom which lie in the connected component of asymptotically trivial gauge transformations. This implies that proper physical symmetries arise either from gauge transformations that reach to infinity or those that are asymptotically trivial but do not lie in the connected component of transformations within that class. The latter transformations form a discrete subgroup of all symmetries whose position in the ambient group has proven to have interesting implications. We explain this for the dyon configuration in the $SO(3)$ Yang-Mills-Higgs theory, where we prove that the asymptotic symmetry group is $Z_{\mathrm{\mid }m\mathrm{\mid }}\times \mathrm{\Re }$ where $m$ is the monopole number. We also discuss the application of the general setting to general relativity and show that here the only implication of discrete symmetries for the continuous part is a possible extension of the rotation group $SO(3)$ to $SU(2)$.