The noncommutative differential geometry of the algebra Mn (C) of complex n×n matrices is investigated. The role of the algebra of differential forms is played by the graded differential algebra C(sl(n,C),Mn (C))=Mn (C)⊗Λsl(n,C)*,sl(n,C) acting by inner derivations on Mn (C). A canonical symplectic structure is exhibited for Mn (C) for which the Poisson bracket is, to within a factor i, the commutator. Also, a canonical Riemannian structure is described for Mn (C). Finally, the analog of the Maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge.
The noncommutative differential geometry ofthe algebra COO (V) ®M n (C) of smooth Mn (C)valued functions on a manifold Vis 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. 323
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