Applications of Hypercomplex Automorphic Forms in Yang–Mills Gauge Theories

In this paper we show how hypercomplex function theoretical objects can be used to construct explicitly self-dual S U ( 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 Δ f = 0 and the second Chern class of the S U ( 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 Gürsey and Tze on a hyper-complex analytical description of S U ( 2 ) instantons. Furthermore, it provides an application of the recently introduced new classes of hypercomplex-analytic automorphic forms.