Exact Monopole Solutions inSU(n)Gauge Theory

Abstract

We construct exact monopole and dyon solutions in a renormalizable $\mathrm{SU}\mathrm{}(N+1\mathrm{})\mathrm{}$ gauge theory broken down to $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}\otimes \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ by one Higgs multiplet in the adjoint representation. The solutions saturate the Bogomolny lower bound and are spherically symmetric with respect to the angular momentum operator $\overrightarrow{\mathrm{J}}=-i\overrightarrow{\mathrm{r}}\times \nabla +\overrightarrow{\mathrm{T}}$, where $\overrightarrow{\mathrm{T}}$ spans the maximal SO(3) subalgebra of $\mathrm{SU}\mathrm{}(N+1\mathrm{})\mathrm{}$. The solutions are obtained after a remarkable factorization of the relevant coupled nonlinear second-order radial equations into a product of coupled linear first-order equations.