UNIFIED GAUGE-THEORIES IN NONCOMMUTATIVE GEOMETRY

We generalize the prescription of Connes in non-commutative geometry to construct unified gauge theories in particle physics. The physical space-time is taken to be a product of a riemannian four-manifold by a discrete set of points. The "algebra of functions" is defined over a space of matrices, and three models are constructed. The first one in which the discrete space consists of two points gives the SU(2) x U(1) gauge theory, broken spontaneously to U(1), and corresponds to the standard model. In the second model, the discrete space has three points, two of which are permutation-symmetrical. It is similar to the minimal SU(5) model of Georgi and Glashow, and all the fermions fit into one spinor representation. The third is the left-right symmetric model SU(2)L X SU(2)R X U(1)B-L model, where the discrete space consists of four points, with a permutation symmetry between the first two and the last two.