SUPERCONNECTIONS AND NONCOMMUTATIVE GEOMETRY AT THE SERVICE OF THE STANDARD MODEL

This talk presents some results of work in collaboration with R Coquereaux, F Scheck and G Vaillant. We show that the Higgs fields and the gauge bosons of electroweak interactions can be unified in a generalized Yang-Mills field valued in the Lie superalgebra SU(2\1), which plays the role of a connection on two parallel spacetimes. The bosonic sector of the Standard Model thus appears as a discrete analogue of Kaluza-Klein theories, where the fibre is replaced by a two-point set. The Z2-graded symmetry corresponding to this superalgebra does not exchange bosons with fermions as in usual SUSY theories, but p-forms with (p +/- 1)-forms in the bosonic sector, and left-handed with right-handed particles in the fermionic one. The smallest typical and non-typical irreducible representations of SU(2\1) allow one to describe respectively quarks and leptons, and reducible indecomposable representations lead naturally to flavour mixing. Besides providing a geometrical and aesthetic way to rewrite the Standard Model, this approach also reduces the number of free parameters of the theory.