On the Number of Chiral Generations in Z 2 × Z 2 Orbifolds

The data from collider experiments and cosmic observatories indicates the existence of three light matter generations. In some classes of string compactifications the number of generations is related to a topological quantity, the Euler characteristic. However, these do not explain the existence of three generations. In a class of free fermionic string models, related to the Z2 ×Z2 orbifold compactification, the existence of three generations is correlated with the existence of three twisted sectors in this class of compactifications. However, the three generation models are constructed in the free fermionic formulation and their geometrical correspondence is not readily available. In this paper we classify quotients of the Z2 ×Z2 orbifold by additional symmetric shifts on the three complex tori. We show that three generation vacua are not obtained in this manner, indicating that the geometrical structures underlying the free fermionic models are more esoteric. [email protected] [email protected]