Summary

Abstract

In this work, chiral four-dimensional covariant lattice models were revisited and a classification of all possible right-mover lattices was performed. Also, it was found that once a right-mover lattice is fixed, modular invariance requires that the set of possible left-mover lattices forms a genus. The result is that there are in total 99 right-mover lattices which may lead to chiral models, and only 19 of them lead to $\mathcal {N}=1$ spacetime supersymmetry. Then, using the Smith-Minkowski-Siegel mass formula, a lower bound of $O(10^{10})$ models with $\mathcal {N}=1$ supersymmetry was calculated. Furthermore, some of the relevant genera were enumerated completely. In particular, for two classes of covariant lattice models that correspond to $Z_3$ and $Z_6$ asymmetric orbifolds we performed an exhaustive enumeration of models, which resulted in exactly 2030 models in the $Z_3$ case, and in $O(10^7)$ models (including duplicates) for the $Z_6$ case. We also considered some generic phenomenological properties of these models, such as discrete flavor and R-symmetries, and also gave spectra of three-generation models. Finally, we studied how the equivalence between certain covariant lattice and twist-orbifold models fits into our picture, and found that there exist some covariant lattices which cannot be obtained as a twist-orbifold theory.