Towards the Standard Model in F-theory

TL;DR

The paper constructs F-theory vacua with gauge group $SU(3)\times SU(2)\times U(1)_1\times U(1)_2$ as toric restrictions of a ${\rm Bl}_2\mathbb{P}^2$-fibration, exploiting rank-two Mordell-Weil geometry to realize two abelian factors and two independent non-abelian enhancements on separate base divisors. By analyzing three $SU(2)$-tops and three $SU(3)$-tops, the authors enumerate five inequivalent toric realizations of the full gauge group, compute the charged singlet spectrum and Yukawa couplings, and match the resulting states to MSSM-like embeddings (with right-handed neutrinos and a $\mu$-singlet) subject to a hypercharge remaining massless under fluxes and an orthogonal $U(1)$ acting as a proton-decay selection rule. They develop a detailed geometric framework for non-perturbative Yukawas and abelian selection rules, including a prime-ideal analysis of Yukawa points and explicit fibre enhancements (e.g., to affine $SU(5)$ and $SO(8)$ diagrams) at codimension-three loci. The work also discusses the role of vertical fluxes in generating a chiral spectrum, the constraints required to keep hypercharge massless, and outlines future directions toward a systematic flux search, moduli stabilisation, and engineering of vector-like matter. Overall, the paper demonstrates that toric F-theory constructions can realize Standard Model-like gauge groups with controlled abelian selection rules and consistent Yukawa structures beyond perturbative Type II setups, paving the way for explicit, globally consistent SM embeddings in string 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)xSU(2)xU(1)xU(1) as suitable restrictions of a ${\rm Bl}_2{\mathbb P}^2$-fibration with rank-two Mordell-Weil group. We analyse the five inequivalent toric enhancements to gauge group SU(3)xSU(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.

Figures (4)

• Figure 1: Non-standard fibre structure at the $(\mathbf{3},\mathbf{2}) - (\mathbf{3},\mathbf{2}) - \mathbf{3}$ Yukawa point.
• Figure 2: Polygon 5 (in the classification of Bouchard:2003bu) describing the fibre ambient space ${\rm Bl}_2 {\mathbb P}^2$; every lattice point of the dual polygon (right) gives an individual term of the hypersurface equation. The reflection symmetry along the dotted diagonal is manifest.
• Figure 3: Possible $SU(2)$ (upper) and $SU(3)$ (lower) tops. The coloured lines and vertices are the 'top layer' of the three-dimensional toric diagram, projected down onto the layer containing the base polygon representing the fibre ambient space.
• Figure 4: The combination of $SU(2)$ and $SU(3)$ tops gives rise to five inequivalent models. Their toric diagram lies in a four-dimensional lattice, where the 'top layers' corresponding to $SU(2)$ and $SU(3)$ resolution divisors extend into two linearly independent directions that do not lie in the plane spanned by the base polygon. For this figure we have projected the tops down into said plane. The four pairs of tops that are equivalent are related to each other by reflection along the diagonal in the plane of the base polygon.