Abelian F-theory Models with Charge-3 and Charge-4 Matter

This paper analyzes U(1) F-theory models admitting matter with charges $q=3$ and $4$. First, we systematically derive a $q=3$ construction that generalizes the previous $q=3$ examples. We argue that U(1) symmetries can be tuned through a procedure reminiscent of the SU(N) and Sp(N) tuning process. For models with $q=3$ matter, the components of the generating section vanish to orders higher than 1 at the charge-3 matter loci. As a result, the Weierstrass models can contain non-UFD structure and thereby deviate from the standard Morrison-Park form. Techniques used to tune SU(N) models on singular divisors allow us to determine the non-UFD structures and derive the $q=3$ tuning from scratch. We also obtain a class of a $q=4$ models by deforming a prior U(1)$\times$U(1) construction. To the author's knowledge, this is the first published F-theory example with charge-4 matter. Finally, we discuss some conjectures regarding models with charges larger than 4.