Applications of Hypercomplex Automorphic Forms in Yang–Mills Gauge Theories
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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.
KeywordsYang–Mills gauge theory \(SU(2)\) instantons Quaternionic analyticity Conformally flat manifolds Hypercomplex argument principle Chern numbers Hypercomplex automorphic forms
Mathematics Subject Classification30G35 70S15
The authors are very thankful to Professor John Ryan from the University of Arkansas and to Professor Vladimir Soucek from Charles University of Prague for the very fruitful discussions which lead to a successful development of this paper. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 267087.
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