Noncommutative differential geometry and new models of gauge theory

The noncommutative differential geometry of the algebra C^∞(V)⊗M_n(C) of smooth M_n(C)‐valued functions on a manifold V is investigated. For n≥2, the analog of Maxwell’s theory is constructed and interpreted as a field theory on V. It describes a U(n)–Yang–Mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential. The Euclidean action, which is positive, vanishes on exactly two distinct gauge orbits, which are interpreted as two vacua of the theory. In one of the corresponding vacuum sectors, the SU(n) part of the Yang–Mills field is massive. For the case n=2, analogies with the standard model of electroweak theory are pointed out. Finally, a brief description is provided of what happens if one starts from the analog of a general Yang–Mills theory instead of Maxwell’s theory, which is a particular case.