On orbifolds and free fermion constructions

This work develops the correspondence between orbifolds and free fermion models. A complete classification is obtained for orbifolds X/GX/G with XX the product of three elliptic curves and GG an abelian extension of a group (Z2)2(Z2)2 of twists acting on XX. Each such quotient X/GX/G is shown to give a geometric interpretation to an appropriate free fermion model, including the geometric NAHE+ model. However, the semi-realistic NAHE free fermion model is proved to be non-geometric: its Hodge numbers are not reproduced by any orbifold X/GX/G. In particular cases it is shown that X/GX/G can agree with some Borcea–Voisin threefolds, an orbifold limit of the Schoen threefold, and several further orbifolds thereof. This yields free fermion models with geometric interpretations on such special threefolds.