Symetric Monopoles

We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry, and monopoles having the symmetry of a regular solid. We introduce the notion of a strongly centred monopole and show that the space of such monopoles is a geodesic submanifold of the monopole moduli space. By solving Nahm's equations we prove the existence of a tetrahedrally symmetric monopole of charge $3$ and an octahedrally symmetric monopole of charge $4$, and determine their spectral curves. Using the geodesic approximation to analyse the scattering of monopoles with cyclic symmetry, we discover a novel type of non-planar $k$-monopole scattering process.