Number of Generations in Free Fermionic String Models,

TL;DR

This paper explores the relationship between the number of generations in free fermionic string models and the index of the supersymmetry generator, providing a formula sensitive to boundary conditions and verifying results through explicit spectrum construction.

Contribution

It derives a new formula linking the number of generations to the internal superconformal field theory's index, applicable to realistic string models.

Findings

Derived a formula for the number of generations based on the index of the supersymmetry generator.

Applied the formula to realistic models and confirmed results by explicit spectrum analysis.

Established a connection between boundary conditions, spin-structure contributions, and generation count.

Abstract

In string theory there seems to be an intimate connection between spacetime and world-sheet physics. Following this line of thought we investigate the family problem in a particular class of string solutions, namely the free fermionic string models. We find that the number of generations $N_g$ is related to the index of the supersymmetry generator of the underlying $N=2$ internal superconformal field theory which is always present in any $N=1$ spacetime supersymmetric string vacuum. We also derive a formula for the index and thus for the number of generations which is sensitive to the boundary condition assignments of the internal fermions and to certain coefficients which determine the weight with which each spin-structure of the model contributes to the one-loop partition function. Finally we apply our formula to several realistic string models in order to derive $N_g$ and we verify our results by constructing explicitly the massless spectrum of these string models.