Generalized gauge theories and the Weinberg-Salam model with Dirac-Kähler fermions (Q2762020)

Ten years ago, \textit{N. Kawamoto} and \textit{Y. Watabiki} proposed a generalized gauge theory formulation for Chern-Simons type actions and topological Yang-Mills actions [cf. Commun. Math. Phys. 144, 641-648 (1992; Zbl 0749.53047) and 148, 169-176 (1992; Zbl 0762.53052)], and that in order to contribute towards the construction of the standard model with quantum gravity in a unified way. On the other hand, A. Connes has pointed out that the famous Weinberg-Salam model can be formulated as a particular case of his non-commutative geometry formulation of a gauge theory [cf. \textit{A. Connes}, Noncommutative Geometry, Academic Press, New York (1994; Zbl 0818.46076)]. This type of non-commutative geometry approach to the Weinberg-Salam model has been intensively investigated ever since, and the present paper provides another contribution in this direction. More precisely, the authors present an extension of the previously proposed gauge theory formulation by Kawamoto and Watabiki, the simplest model of which turns out to be just the Weinberg-Salam model in its non-commutative geometry formulation. This is done as follows.NEWLINENEWLINENEWLINEIn Section 2, after an enlightening introduction, the generalized gauge theory formulation à la Kawamoto-Watabiki is briefly summarized. Section 3 discusses an extension of this approach to the quantization of the generalized topological Yang-Mills action in the \(N=2\) supersymmetry case, and Section 4 provides a generalized theory of the Dirac-Kähler fermion formulation. This is used to formulate the Weinberg-Salam model by the generalized Yang-Mills action and Dirac-Kähler formulation with the \(SU(2|1)\)-graded Lie algebra in Section 5.NEWLINENEWLINENEWLINEIn this approach of reconstructing the Weinberg-Salam model, only gauge transformations with respect to the \(O\)-form gauge parameters are considered, while the authors' generalized gauge theory includes all degrees of the occuring differential forms. This might be an essential feature with regard to unifying the standard model, together with quantum gravity on the simplicial lattice.