In a project with Prof. Ron Donagi we investigate the geometry underlying semi-realistic string models. We are studying the "realistic free fermion models" constructed by Antoniadis, Ellis, Hagelin, Nanopoulos, Faraggi, Yuan, Cleaver and others, and we search for a mathematical understanding of the orbifolds involved.
develops the correspondence between orbifolds and free fermion models. A complete classification is obtained for orbifolds X/G with X the product of three elliptic curves and G an abelian extension of a group (Z2)2 of twists acting on X. These orbifolds exhaust all candidates for geometric interpretations of the realistic free fermion models mentioned above. Each such orbifold 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/G. In particular cases it is shown that X/G can agree with some Borcea-Voisin threefolds, an orbifold limit of the Schoen threefold, and several further orbifolds thereof. Thus we obtain free fermion models with geometric interpretations on such special threefolds. This may be of considerable interest in model building, since free fermion models are mathematically rather simple and hence could dramatically simplify constructions of models on these threefolds. Indeed, along with our classification scheme this result is repeatedly used by Vaudrevange and his coworkers in the construction of an MSSM with three generations. Our classification also fits neatly into the classification of so-called Calabi-Yau 3-folds of type K proposed by Hashimoto and Kanazawa.