Topology, and (in)stability of non-Abelian monopoles

The stability problem of non-Abelian monopoles with respect to “Brandt-Neri-Coleman type” variations reduces to that of a pure gauge theory on the two-sphere. Each topological sector admits exactly one stable monopole charge, and each unstable monopole admits $2\sum (2\mathrm{\mid }q\mathrm{\mid }-1)$ negative modes, where the sum goes over the negative eigenvalues $q$ of an operator related to the non-Abelian charge $Q$ of Goddard, Nuyts and Olive. An explicit construction for the [up-to-conjugation] unique stable charge, as well as the negative modes of the Hessian at any other charge is given. The relation to loops in the residual group is explained. From the global point of view, the instability is associated with energy-reducing two-spheres, which, consistently with the Morse theory, generate the homology of the configurations space, and whose tangent vectors at a critical point are negative modes. Our spheres might indicate possible decay routes of an unstable monopole as a cascade into lower lying critical points.