Fermion spectra from superstrings

Some compactifications of the ten-dimensional anomaly-free E₈ × E₈ and SO(32) theories that correspond to superstrings are studied. Compactification is achieved by setting the classical gauge field equal to the spin connection. The resulting chiral fermion spectra are obtained for any six-dimensional manifold, under the condition Tr F² = 30 Tr R², plus a quantization condition for U(1) charges. For E₈ × E₈ these conditions lead to potentially realistic models for any irreducible six-dimensional manifold and any embedding of the holonomy group. Apart from a few more exotic examples, the four-dimensional models we obtain are more or less standard SU(5), SO(10), SU(4) × SU(2) × SU(2) or E₆ models.