ON CERTAIN TOPOLOGICAL ASPECTS OF THE FAMILY PROBLEM

Certain topological aspects of potential significance to the problem of family formation are investigated. Specifically, it is shown that the conventional vacuum state for gauge theories with non-trivial topology and fermions in the fundamental representation of the universal covering group of the gauge group, is not, in general, gauge invariant. The true gauge invariant vacuum, as well as the corresponding generating functionals are constructed. As a result, for the particular case of an SO(10) gauge symmetry over R3×S1 which is explicitly considered here, gauge and Lorentz invariance requires the existence of four-fermion generations.