Algebraic construction of static axially symmetric self-dual fields

Abstract

In this short Note we will present a short introduction to the recent work by us 1),2) on the problem of solving self-dual gauge field equations for an arbitrary compact semisimple gauge group G. The discussion is mostly focused on the simple case of the static and axially symmetric configuration, in which the effective spacetime dimension is reduced to two and we can apply some of the solution generation techniques of two dimensional soliton theories. 3) The central result is an algebraic method for the construction of solutions starting from one particular solution and solutions of the associated linear scattering problem. Here we see an interesting interplay of soliton theoretic techniques and group theoretic concepts. A main application is the construction of axially symmetric monopole solutions in gauge theories with an arbitrary group G and a single Higgs field in the adjoint representation.