Construction of Euclidian monopoles

Abstract

This paper describes a procedure for the construction of monopoles on three-dimensional Euclidean space, starting from their rational maps. A companion paper, ‘Euclidean monopoles and rational maps’, to appear in the same journal, describes the assignment to a monopole of a rational map, from $\Bbb{CP}^1$ to a suitable flag manifold. In describing the reverse direction, this paper completes the proof of the main theorem therein.

A construction of monopoles from solutions to Nahm's equations (a system of ordinary differential equations) has been well-known for certain gauge groups for some time. These solutions are hard to construct however, and the equations themselves become increasingly unwieldy when the gauge group is not $\mbox{SU}(2).$

Here, in contrast, a rational map is the only initial data. But whereas one can be reasonably explicit in moving from Nahm data to a monopole, here the monopole is only obtained from the rational map after solving a partial differential equation.

A non-linear flow equation, essentially just the path of steepest descent down the Yang-Mills-Higgs functional, is set up. It is shown that, starting from an ‘approximate monopole’ - constructed explicitly from the rational map - a solution to the flow must exist, and converge to an exact monopole having the desired rational map.

1991 Mathematics Subject Classification: 53C07, 53C80, 58D27, 58E15, 58G11.