More SO(3) monopoles

Spontaneously broken gauge theories describing gauge bosons coupled in the manner of the Yang-Mills prescription to a Lorentz scalar $\phi$ transforming as an arbitrary ($2n+1$)-dimensional irreducible representation of the gauge group SO(3) are considered. It is shown that given the topologically stable, static solution of 't Hooft and Polyakov for the isovector ($n=1\mathrm{}$) field there exists a recipe for constructing solutions to all higher-dimensional fields $\phi$. The case $n=2\mathrm{}$ is worked out in some detail. The same recipe is applicable to any other homotopy class where the isovector problem is solved, and the solutions so generated are seen to be the only possible stable ones. Since the above solutions exist only if the vacuum is U(1) symmetric, arguments supporting that contingency for a general rank-$n$ Lagrangian are given. In two space dimensions, the tower of solutions corresponding to the only stable homotopy class are outlined and the case $n=2\mathrm{}$ is described in detail. In all cases the electric potential that may be added in the manner of Julia and Zee is specified.