Orders of Vanishing and U(1) Charges in F-theory

TL;DR

This paper links the U(1) charge of matter in F-theory models to the orders of vanishing of section components in elliptic fibrations, providing formulas that could simplify charge determination and explore the F-theory landscape.

Contribution

It introduces a novel relationship between U(1) charges and orders of vanishing in elliptic fibrations, extending understanding of F-theory models and their charges.

Findings

Formulas relate U(1) charges to orders of vanishing.

Correctly describe properties of models in prior literature.

Predict geometric info for models with high U(1) charges.

Abstract

Many interesting questions about F-theory models, including several concerning the F-theory swampland, involve massless matter charged under U(1) gauge symmetries. It is therefore important to better understand the geometric properties of F-theory models realizing various U(1) charges. We propose that, for F-theory models described by elliptic fibrations in Weierstrass form, the U(1) charge of light matter is encoded in the orders of vanishing of the section components corresponding to the U(1) gauge symmetry. We give specific equations relating the U(1) charges to the orders of vanishing that seem to hold for both U(1)-charged singlets and for matter additionally charged under a simply-laced nonabelian gauge algebra. Our formulas correctly describe properties of F-theory models in the prior literature, and we give an argument that they should describe the orders of vanishing for arbitrarily high U(1) charges. They also resemble formulas for the $p$-adic valuations of elliptic divisibility sequences developed by Stange [arXiv:1108.3051]. These proposals could serve as a U(1) analogue of the Katz-Vafa method, allowing one to determine U(1) charges without resolution. Additionally, they predict geometric information about F-theory models with general U(1) charges, which may be useful for exploring the F-theory landscape and swampland.