Grand unified string theories with non-Abelian gauge family symmetry in the free fermion formulaton

In the framework of four-dimensional heterotic superstring theory with free fermions we investigate the rank eight grand unified string theories (GUST's) which contain the SU(3)H-gauge family symmetry. GUST's of this type accommodate naturally three fermion families presently observed and, moreover, can describe a fermion mass spectrum without high-dimensional representations of conventional unification groups. We explicitly construct GUST's with the gauge symmetry G=SU(5)xU(1)x[SU(3)xU(1)]H contained-in SO(16) in the free complex fermion formulation. As the GUST's originating from Kac-Moody algebras (KMA's) contain only low-dimensional representations it is usually difficult to break the gauge symmetry. We solve this problem by taking for the observable gauge symmetry the diagonal subgroup Gsym of the rank 16 group GxG contained-in SO(16)xSO(16)contained-in E8xE8. Such a construction effectively corresponds to a level-two KMA, and therefore some higher-dimensional representations of the diagonal subgroup appear. This (because of GxG-tensor Higgs fields) allows one to break GUST symmetry down to SU(3c)xU(1)em. In this approach the observed electromagnetic charge Qem can be viewed as a sum of two QI and QII charges of each G group. In this case below the scale where GxG breaks down to Gsym the spectrum does not contain particles with exotic fractional charges. In these GUST's there has to exist ''superweak'' light chiral matter