Charge quantization in theories with an adjoint representation Higgs mechanism

Gauge theories are considered in which an overall simple gauge symmetry group G is broken, by a Higgs field in the adjoint representation, to an exact symmetry with a structure locally of the form U(1)×K. Necessary and sufficient conditions on the Higgs field for K to be semisimple are given, together with an algorithm for determining K from the Dynkin diagram. The theory of weight lattices is used to show that both the electric and magnetic charges are multiples of certain basic units. These are calculated and related to each other by formulae more precise than quantization conditions given previously. It is shown that globally the exact symmetry group has the form $\text{H}\text{=(}\text{U}\text{(1)×}\text{K}\text{)/}\text{Z}$, where $\text{K}$ is simply connected (assuming G is) and Z is a cyclic group whose order can be calculated from the Dynkin diagram.