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Applications of hypercomplex automorphic forms in Yang-Mills gauge theories
Rolf Sören Kraußhar and Jürgen Tolksdorf
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 Gürsey 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.