Euclidian monopoles and rational maps

Abstract

A famous theorem of Donaldson describes a correspondence between $\mbox{SU}(2)$ monopoles over three-dimensional Euclidean space and maps from $\Bbb{CP}^1$ to itself. This paper generalises this to monopoles with arbitrary gauge group $G,$ and new maps from $\Bbb{CP}^1$ to flag manifolds $G/H$ are produced. This is in line with a conjecture of Atiyah and Murray, following a similar result of Atiyah's on hyperbolic monopoles.

Donaldson's approach, also followed by Hurtubise and Murray in previous proofs of many important cases of the current result, depends on a description of monopoles in terms of a system of ordinary differential equations known as Nahm's equations. In contrast, our approach is more direct, and the bulk of the paper is concerned with describing the rational map associated to a particular framed monopole, via solutions to a scattering equation (first introduced by Hitchin) along parallel lines.

A subsidiary section of the paper analyses rational maps into flag manifolds, constructing canonical lifts into larger flag manifolds, and into the (complexified) Lie group. The main result is dependent upon additional work in a companion paper, ‘Construction of Euclidean monopoles’, to appear in the same journal, where a procedure is described for recovering a monopole from its rational map.

1991 Mathematics Subject Classification: 53C80, 58D27, 58E15, 58G11.