Family unification in five and six dimensions

In family unification models, all three families of quarks and leptons are grouped together into an irreducible representation of a simple gauge group, thus unifying the standard model gauge symmetries and a gauged family symmetry. Large orthogonal groups, and the exceptional groups ${E}_{7}$ and ${E}_{8},$ have been much studied for family unification. The main theoretical difficulty of family unification is the existence of mirror families at the weak scale. It is shown here that family unification without mirror families can be realized in simple five-dimensional and six-dimensional orbifold models similar to those recently proposed for $\mathrm{SU}(5)$ and $\mathrm{SO}(10)$ grand unification. It is noted that a family unification group that survived to near the weak scale and whose coupling extrapolated to high scales unified with those of the standard model would be evidence, accessible in principle at low energy, of the existence of small (Planckian or GUT-scale) extra dimensions.