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) x 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.