Chiral Abelian gauge theories with few fermions

We construct chiral theories with the smallest number ${n}_{\ensuremath{\chi}}$ of Weyl fermions that form an anomaly-free set under various Abelian gauge groups. For the $U(1)$ group, where ${n}_{\ensuremath{\chi}}=5$, 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 $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ gauge group we find ${n}_{\ensuremath{\chi}}=6$, and derive the general solution to the anomaly equations, in terms of 6 parameters. For $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ we show that ${n}_{\ensuremath{\chi}}=8$, 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.