Towards the Standard Model in F-theory

Abstract

This article explores possible embeddings of the Standard Model gauge group and its matter representations into F-theory. To this end we construct elliptic fibrations with gauge group SU (3) × SU (2) × U (1) × U (1) as suitable restrictions of a Bl 2 P 2 -fibration with rank-two MordellWeil group. We analyse the five inequivalent toric enhancements to gauge group SU (3) × SU (2) along two independent divisors W 3 and W 2 in the base. For each of the resulting smooth fibrations, the representation spectrum generically consists of a bifundamental (3, 2), three types of (1, 2) representations and five types of (3, 1) representations (plus conjugates), in addition to charged singlet states. The precise spectrum of zero-modes in these representations depends on the 3-form background. We analyse the geometrically realised Yukawa couplings among all these states and find complete agreement with field theoretic expectations based on their U (1) charges. We classify possible identifications of the found representations with the Standard Model field content extended by right-handed neutrinos and extra singlets. The linear combination of the two abelian gauge group factors orthogonal to hypercharge acts as a selection rule which, depending on the specific model, can forbid dangerous dimension-four and -five proton decay operators.

TASI Lectures on Abelian and Discrete Symmetries in F-theory

“…The three The non-abelian part of the Standard Model gauge algebra is engineered via toric methods (so-called "tops" [83,84]). In this case, we obtain five inequivalent tops that realize su(3)×su (2) in codimension one of the elliptic fibration (5.2) [85]. Furthermore, in each such top, we have the freedom of identifying the hypercharge u(1) with a linear combination of the two geometrically realized u(1)s; the orthogonal combination then serves as the selection rule.…”

“…Furthermore, in each such top, we have the freedom of identifying the hypercharge u(1) with a linear combination of the two geometrically realized u(1)s; the orthogonal combination then serves as the selection rule. All such identifications compatible with the geometric spectrum have been listed in [85], together with the possible dimension four and five operators of the Standard Model, which are and are not forbidden by the selection rule.…”

Yukawa hierarchies in global F-theory models

“…2 Specifically, when the rank of Φ reduces over a complex codimension one sub-variety Σ ⊂ S, we find localized fields that are trapped on Σ. 3 The reduction of the rank of Φ implies that a larger gauge algebra is preserved over Σ, a phenomenon that exactly mirrors what happens in the geometry, and the localized matter and its representation under the gauge group can be read off from the enhancement pattern following [34].…”

Tate’s algorithm for F-theory GUTs with two U(1)s

“…Our results are not restricted to F-theoretic GUT model building, and we hope that they are also useful in other areas of F-theory, for example in direct constructions of the Standard Model [51,52], in the determination of the network of resolutions of elliptic fibrations [53][54][55][56][57], or in the recent relationship drawn between elliptic fibrations with U(1)s and genus one fibrations with multisections [58][59][60].…”