On the complete integrability of the static axially symmetric self-dual gauge field equations for an arbitrary group

Abstract

We present an analysis of static axially symmetric gauge fields for an arbitrary gauge group G. Two ansätze are considered. The full ansatz involves a total of 4 d( d = dim G), the reduced ansatz only 2 d functions of ( ϱ, z). Imposing self-duality is shown to reduce the problem to a sigma model in the curved two-dimensional ( ϱ, z) space over the coset spaces G̃/G for the full, and G ∗/K for the reduced ansatz. G̃ is the complexification of G. ∗ is a particular non-compact form of G, and K the local form-preserving symmetry group of the reduced ansatz. We give explicitly the Lax-pair type representations (linear scattering problem) of the sigma model, indicating that the standard methods available for certain non-linear two-dimensional problems can be used to generate solutions. Our procedure has the advantage that only real fields over a real manifold enter the analysis.