  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 X 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 X 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.

High Energy Physics - Theory;High Energy Physics - Phenomenology;Algebraic Geometry