On the Topology of Configurations of Gauge Fields and Higgs Fields

Starting with the topological classification of gauge fields on the sphere (Dirac monopoles) and of Higgs vacuum configurations (through the second homotopy group of the space of vacuum states), the features of minimum energy solutions to gauge theories with spontaneous symmetry breaking and in the presence of singularities (monopoles) are deduced. For each theory, this analysis reveals whether a monopole can exist isolated or must be attached to a massive string and how its flux must be distributed.