A.H. Chamseddine and A. Connes
Examining the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space a new tensorial notation to present the various constructs of noncommutative geometry is developed. In particular, this notation is used to determine the spectral data of the standard model. The particle spectrum with all of its symmetries is derived, almost uniquely, under the assumption of irreducibility and of dimension 6 modulo 8 for the finite space.The leading order terms in the spectral action are re-derived. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. The following predictions emerge:
(i) The number of fermions per family is 16.
(ii) The symmetry group is U(1) 6times; SU(2) × SU(3).
(iii) There are quarks and leptons in the correct representations.
(iv) There is a doublet Higgs that breaks the electroweak symmetry to U(1).
(v) Top quark mass of 170–175 GeV.
(vi) There is a right-handed neutrino with a see-saw mechanism.
The influence of higher order corrections to the predicted mass of the Higgs field, the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations are discussed.