SU(3) ⊗ SU(2) ⊗ U(1) from D = 11 supergravity

In this paper we show that D = 11 supergravity admits an infinite discrete class of solutions having the phenomenological group SU(3) ? SU(2) ? U(1) as a symmetry of the internal space M₇. These solutions lead, in dimensional reduction, to SU(3) ? SU(2) ? U(1) gauge fields.In general all these spaces produce a complete breaking of supersymmetry except in one case where N = 2 supersymmetry survives. The parameter which classifies the solutions is a rational number q/p which describes the embedding of the stability subgroup SU(2) ? U(1) ? U(1) of M₇ in SU(3) ? SU(2) ? U(1). For all q/p ≠ 1 the holonomy group is SO(7) and all supersymmetries are broken. For q/p = 1 the holonomy group is SU(3) and two supersymmetries survive. In this last case we can also find a solution with internal photon curl $\text{F}{}_{\mathrm{\alpha \beta \gamma \delta }}\text{≠ 0}$. It breaks all sypersymmetries.