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∑(2|q|−1)2∑(2|q|−1) negative modes, where the sum goes over the negative eigenvalues qq of an operator related to the non-Abelian charge QQ 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 configuration space. Our spheres are tangent to the negative modes at the considered critical point, and may indicate possible decay routes of an unstable monopole as a cascade into lower lying critical points.

► Topological classification of non-Abelian monopoles is carried out in Lie algebra terms.
► Monopole charges are classified, and the number of negative modes are determined. Each topological sector contains one stable monopole.
► Negative modes are explicitly constructed.
► Unstable solutions are associated with non-contractible two-spheres in configuration space.