Grand Unified Theories in Superstrings in the Free World-Sheet Fermion Formulation

In the framework of the four dimensional heterotic superstring with free world-sheet fermions we study the GUSTs which contain non-abelian horizontal symmetry. To solve the problem of breaking the G_GUT gauge symmetry of the level one current algebra we extend the gauge symmetry to G x G. Using the autoduality of the fermionic charge lattice we study the possibilities of constructing of some GUST gauge groups. We also study the string amplitudes and for one realistic model explicitly calculate the renormalizable and non-renormalizable (N=4,5,6) contributions to the superpotential. We observe that the existence of the mass matrix ansatz depends on the way of the breaking of the horizontal gauge symmetry.