Chiral Abelian gauge theories with few fermions

We construct chiral theories with the smallest number $n_{\chi }$ of Weyl fermions that form an anomaly-free set under various Abelian gauge groups. For the $U(1)$ group, where $n_{\chi }=5$, we show that the general solution to the anomaly equations is a set of charges given by cubic polynomials in three integer parameters. For the $U(1)\times U(1)$ gauge group we find $n_\chi = 6$, and derive the general solution to the anomaly equations, in terms of 6 parameters. For $U(1)\times U(1)\times U(1)$ we show that $n_{\chi }=8$, and present some families of solutions. These chiral gauge theories have potential applications to dark matter models, right-handed neutrino interactions, and other extensions of the Standard Model. As an example, we present a simple dark sector with a natural mass hierarchy between three dark matter components.