The classification of monopoles for the classical groups (Q1812530)

The moduli spaces of monopoles with maximal symmetry breaking at infinity are computed for the groups \(SU(N)\), \(SO(N)\) and \(Sp(N)\) by using the construction of Nahm. The Nahm's equations are divided into two parts, one invariant under a real group of gauge transformations, the other under a large complex group \(\mathcal G\) of gauge transformations. It is shown that each \(\mathcal G\)-orbit contains an essentially unique solution to the real equations. Then, the solutions to the complex equations are classified in terms of rational maps. These maps are interpreted in terms of twistor construction of monopoles. It is concluded that the moduli spaces of monopoles are equivalent to spaces of holomorphic maps from \(\mathbb{P}_ 1\) into flag manifolds.