Applications of hypercomplex automorphic forms in Yang-Mills gauge theories

Abstract

In this paper we show how hypercomplex function theoretical objects can be used to construct explicitly self-dual SU(2)-Yang-Mills instanton solutions on certain classes of conformally flat 4-manifolds. We use a hypercomplex argument principle to establish a natural link between the fundamental solutions of $D \Delta f = 0$ and the second Chern class of the SU(2) principal bundles over these manifolds. The considered base manifolds of the bundles are not simply-connected, in general. Actually, this paper summarizes an extension of the corresponding results of G\"ursey and Tze on a hyper-complex analytical description of SU(2) instantons. Furthermore, it provides an application of the recently introduced new classes of hypercomplex-analytic automorphic forms.