Topological charges in monopole theories

Let us consider a monopole theory with a compact, simply connected gauge group and the Higgs field in the adjoint representation. Using root theory we show that.(i) The homotopy class of the Higgs field is ap-tuple of integers wherep is the dimension of the centre of the residual symmetry group. These Higgs charges can be expressed as surface integrals of differential forms.(ii) To any invariant polynomial on the Lie algebra is associated a topological invariant which turns out to be a combination of the Higgs charges.(iii) Electric charge is quantized. The monopole's magnetic charge is a combination — with the Higgs charges as coefficients — ofp basic magnetic charges which satisfy generalized Dirac conditions.The example ofG=SU(N) is worked out in detail.