A constraint on the geometry of Yang-Mills theories
The aim of this paper is an explanation of the fact that ``attempts to generalize pure Yang-Mills theories by allowing the standard fibre to be a non-group manifold have been unsatisfactory'' [see for example \textit{W. Hann}, An octonionic generalization of Yang-Mills, CERN preprint TH-2489 (1978)]. It is known from physics that ``a connection form on fibre bundle can only be regarded as a gauge potential if the dependence of the fibre coordinate in the transformation rule for the connection can be eliminated.'' So a fibre bundle (\({\mathcal E},{\mathcal M},{\mathcal F},\pi)\) is considered and the transformation rule for the connection form is analysed in detail. This analysis shows that the physical constraint, mentioned above, can be fulfilled only if the fibre \({\mathcal F}\) admits a parallelism. Further investigation leads to the conclusion that the parallelism on \({\mathcal F}\) must be globally integrable and so \({\mathcal F}\) is a factor space \({\mathcal G}/{\mathcal D}\), where \({\mathcal G}\) is a Lie group and \({\mathcal D}\) is a discrete subgroup of \({\mathcal G}\) (the definition of the parallelism is as usual and for the definition of a globally integrable parallelism and corresponding results see \textit{P. Libermann} [J. Differ. Geom. 8, 511-539 (1973; Zbl 0284.53023)]. Thus, if the fibre \({\mathcal F}\) is assumed to be simply connected then \({\mathcal F}\) must be a Lie group and, what is more, the bundle transformations consist of either right or left multiplications (but not both) and (\({\mathcal E},{\mathcal M},{\mathcal F},\pi)\) is a principal bundle.
scientific article; zbMATH DE number 4062081