Naturalness of three generations in free fermionicZ2n⊗Z4string models

We study the construction of free fermionic spin structure models with $Z_2^n\otimes Z_4$ boundary-condition vectors. We argue that requiring chiral space-time fermions in the massless spectrum and the existence of a well-defined hidden gauge group severely constrain the allowed boundary-condition vectors. We show that the minimal way to obtain these requirements is given by a unique set of $Z_2^5$ boundary-condition vectors. We classify the possible extensions to this basic set. We argue that a result of this fundamental set is that obtaining three generations in this construction is correlated with projecting out all the enhanced gauge symmetries which arise from nonzero vacuum expectation values of background fields. We propose that this correlation and the properties of the fundamental $Z_2^5$ subset suggest that three generations is natural in this construction.