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 [1]. 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.
Highlights
A major goal of the string theory program is understanding how to construct a compactification of string theory realizing a desired massless spectrum
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
As evidenced by eq (2.15) and (2.16), the formulas we develop here use T(ν) and τ(ν) for the group corresponding to the codimension-two singularity type
Summary
A major goal of the string theory program is understanding how to construct a compactification of string theory realizing a desired massless spectrum. One can determine matter representations with the Katz-Vafa method [10], in which one breaks the adjoint of the enhanced singularity type’s corresponding Lie algebra to representations of the nonabelian gauge algebra These techniques allow one to calculate the gauge group and charged matter content of a model by considering orders of vanishing, making the process of constructing and analyzing nonabelian F-theory models significantly easier. We only consider matter in generic [60] representations of the nonabelian gauge factors, and we focus on matter loci where the elliptic fiber singularity type undergoes a rank-one enhancement..
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