ExactSU(N)monopole solutions with spherical symmetry

Exact monopole solutions are constructed for an $\mathrm{SU}\mathrm{}(N+1\mathrm{})\mathrm{}$ gauge theory spontaneously broken by a single Higgs field in the adjoint representation. The solutions saturate the Bogomolny lower bound on the energy and are spherically symmetric with respect to the angular momentum operator $\overrightarrow{\mathrm{J}}=-i\overrightarrow{\mathrm{r}}\times \overrightarrow{\nabla }+\overrightarrow{\mathrm{T}}$, where $\overrightarrow{\mathrm{T}}$ generates the maximal SU(2) subalgebra of $\mathrm{SU}\mathrm{}(N+1\mathrm{})\mathrm{}$. Our solutions are the most general ones with the above symmetry and contain $N$ real parameters which may be thought of as specifying the nature of symmetry breaking. When this symmetry breaking is such that the scalar field matrix has repeated eigenvalues, it is found that only one of the possible point monopoles has a corresponding finite-energy solution saturating the Bogomolny bound.