Orders of vanishing and U(1) charges in F-theory

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.

Automatic Enhancement in 6D Supergravity and F-theory Models

“…The gauge algebra is therefore su(18) ⊕ u(1), and one can confirm with the Katz-Vafa method [55] and the proposals in [56] that the charged hypermultiplet spectrum agrees with Eq. (4.7).…”

“…and the gauge algebra is su(2) ⊕ u(1). Because [γ] is trivial, the only u(1)-charged matter occurs at {σ = f 2 1 − γ 2 φ}; this locus supports 2 1/2 matter, as can be verified with the techniques in [56]. 23 This matter spectrum satisfies the anomaly cancellation conditions for b = −2K B −…”

“…implying that the gauge algebra is enhanced to su( 19)⊕u (1). And by the Katz-Vafa method [55] and the proposals in [56], one can again verify that the charged hypermultiplet spectrum agrees with Eq. (4.8).…”

Snowmass White Paper: String Theory and Particle Physics

“…The range of allowed representations for heterotic orbifold models was studied some time ago in [87]. More recently, analysis from the F-theory point of view seems to strongly limit the range of possible matter representations of non-Abelian groups that may arise from geometry [88][89][90], although there is more apparent flexibility for U(1) charges [91][92][93][94][95]. Furthermore, the global structure of the gauge group affects the set of allowed matter fields; the observed Standard Model chiral matter spectrum is much more natural if the global structure of the group is (SU (3) × SU (2) × U (1))/Z 6 (as occurs in most GUT models and many of the Ftheory constructions delineated above) than the group without the quotient; in the F-theory context these issues have been studied recently in [44,45,96,97] 1 .…”