$\frac{1}{2}$Calabi-Yau 4-folds and four-dimensional F-theory on Calabi-Yau 4-folds with U(1) factors

Kimura, Yusuke

doi:10.48550/arxiv.1911.03960

“…1 (0, 0, 0, 0) (2, 3, 1, 4) 2 (0, 0, 0, 0) (8, 12, 4, 16) 3 (0, 0, 0, 0) (18, 27, 9, 36) 4 (0, 0, 0, 0) (32, 48, 16, 64) 5 (0, 0, 0, 0) (50, 75, 25, 100) 6 (0, 0, 0, 0) (72, 108, 36, 144) 7 (0, 0, 0, 0) (98, 147, 49, 196) 8 (0, 0, 0, 0) (128,192,64,256) (a) ν = 0 for both 32q ⊕ 12q and 32 q ⊕ 12q matter |q| ord (1) ord…”

Section: Jhep03(2022)051mentioning

confidence: 99%

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

Raghuram

¹

,

Turner

²

2022

J. High Energ. Phys.

3

2

0

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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.

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“…1 (0, 0, 0, 0) (2, 3, 1, 4) 2 (0, 0, 0, 0) (8, 12, 4, 16) 3 (0, 0, 0, 0) (18, 27, 9, 36) 4 (0, 0, 0, 0) (32, 48, 16, 64) 5 (0, 0, 0, 0) (50, 75, 25, 100) 6 (0, 0, 0, 0) (72, 108, 36, 144) 7 (0, 0, 0, 0) (98, 147, 49, 196) 8 (0, 0, 0, 0) (128,192,64,256) (a) ν = 0 for both 32 q ⊕ 1 2q and 32 q ⊕ 1 2q matter |q| ord (1) ord…”

Section: Spinor Representations Of So(4k)mentioning

confidence: 99%

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

Raghuram,

Turner

2021

Preprint

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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.

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“…Very recently, the construction of a 1 2 K3 surface is generalized to complex-three and -four dimensions with interesting applications to F-theory compactifications[19,20].…”

mentioning

confidence: 99%

More on a dessin on the base: Kodaira exceptional fibers and mutually (non-)local branes

Fukuchi

¹

,

Kan

²

,

Kuramochi

³

et al. 2020

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A "dessin d'enfant" is a graph embedded on a two-dimensional oriented surface named by Grothendieck. Recently we have developed a new way to keep track of non-localness among 7branes in F-theory on an elliptic fibration over P 1 by drawing a triangulated "dessin" on the base.To further demonstrate the usefulness of this method, we provide three examples of its use. We first consider a deformation of the I * 0 Kodaira fiber. With a dessin, we can immediately find out which pairs of 7-branes are (non-)local and compute their monodromies. We next identify the paths of string(-junction)s on the dessin by solving the mass geodesic equation. By numerically computing their total masses, we find that the Hanany-Witten effect has not occurred in this example. Finally, we consider the orientifold limit in the spectral cover/Higgs bundle approach. We observe the characteristic configuration presenting the cluster sub-structure of an O-plane found previously. *

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$\frac{1}{2}$Calabi-Yau 4-folds and four-dimensional F-theory on Calabi-Yau 4-folds with U(1) factors

Cited by 3 publications

References 89 publications

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

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

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

More on a dessin on the base: Kodaira exceptional fibers and mutually (non-)local branes

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