We propose a geometrized Higgs mechanism based on the gravitational sector in the Connes-Lott formulation of the standard model, which has been constructed by Chamseddine, Fröhlic and Grandjean. The point of our idea is that Higgs-like couplings depend on the local coordinates of the four-dimensional continuum, M 4 . The localized couplings can be calculated by the Wilson loops of the U (1) EM gauge field and the connection, which is defined on Z 2 × M 4 .

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The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus Tn with its algebra, C(Tn), of continuous complex-valued functions under pointwise multiplication. But C(Tn) is the universal C*-algebra generated by n commuting unitary operators. By definition, [15, 16, 50], a non-commutative n-torus is the universal C*-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [8, 10], these algebras have a natural differentiable structure, defined by a natural ergodic action of Tn as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [37] that the K-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK-equivalent to ordinary tori by Corollary 7.5 of [52].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [17].)

One can describe an n-dimensional noncommutative torus by means of an antisymmetric n × n matrix θ. We construct an action of the group SO(n, n|Z) on the space of n × n antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched. By definition [R5], an n-dimensional noncommutative torus is an associative algebra with involution having unitary generators U 1 , ..., U n obeying the relationswhere e(t) = e 2πit and θ is an antisymmetric matrix. The same name is used for different completions of this algebra. In particular, we can consider the noncommutative torus as a C ⋆ -algebra A θ (the universal C ⋆ -algebra generated by n unitary operators satisfying (1) ). Noncommutative tori are important in many problems of mathematics and physics. It was shown recently that they are essential in consideration of compactifications of M(atrix) theory ([CDS]; for further development see [T]). The results of the present paper also have application to physics. If two algebras A and are Morita equivalent (see the definition below), then for every A-module R one can construct anÂ-moduleR in such a way that the correspondence R →R is an equivalence of categories of A-modules andÂ-modules. * M. A. R.