HIGGS FIELDS AS YANG-MILLS FIELDS AND DISCRETE SYMMETRIES

We consider an extension of the formalism of gauge theories where the connection is no longer described by a Lie algebra valued one-form on space-time, but incorporates both the Yang-Mills fields and the Higgs fields. Higgs fields are associated here with the gauging of discrete directions. The formalism presented here is therefore not a Kaluza-Klein-like approach to Yang-Mills theory, but is, in a sense, a discrete analog of it. We first study a U(1) x U(1) model, then a more realistic SU(2) x U(1) model. Algebraic and geometric aspects of those models in relation with the old and new ideas of "Non-Commutative Geometry" are also discussed.