Local Models in F-Theory and M-Theory with Three Generations

TL;DR

The paper develops a framework for geometrically engineering local, phenomenological models in F-theory and M-theory using ALE-fibrations, arguing that three generations arise generically from the $E_8$ adjoint branching and that triple-intersections (F-theory) correspond to supersymmetric three-cycles (M-theory). It provides an explicit $\widehat{E_7}$-fiber example realizing $SU_5$ with three generations and discusses how Yukawa couplings and chirality are controlled differently in F-theory via flux and in M-theory via instantons, including mechanisms to achieve on-diagonal Yukawas and realistic superpotentials. The work then constructs concrete local models—the Diamond Ring in F-theory and a local MSSM in M-theory—by unfolding $\widehat{E_8}$ through $E_6$ and $SO_{10}$ down to $SU_5$ or directly to the Standard Model gauge group, using flux choices and moduli constraints to manage exotics and coupling structure. Overall, the results show that purely local, highly structured geometries can produce realistic three-generation phenomenology, offering predictive, testable features while acknowledging global embedding challenges and future directions toward global compactifications and phenomenological constraints.

Abstract

We describe a general framework that can be used to geometrically engineer local, phenomenological models in F-theory and M-theory based on ALE-fibrations, and we present several concrete examples of such models that feature three generations of matter with semi-realistic phenomenology. We show that the geometric structures required for generating interactions--triple-intersections of matter-curves in F-theory and supersymmetric three-cycles supporting multiple conical singularities in M-theory--are generic in such ALE-fibred manifolds, and that they can be understood in correspondence with one another. The models we can construct in this way are strictly limited in complexity by the maximality of the E8-ALE space, but turn out to be just complex enough to accommodate some of the most realistic string models to date.

Figures (10)

• Figure 1: The lines of various enhanced singularities over the base of the fibration $\widehat{E_7}(a(t),b(t),c(t),0,0,0,0)$, where $t\in\mathbb{C}^2$. Every fibre over the plane has at least an $SU_5$-singularity. Thick, solid blue lines correspond to the locations where lattices of $\mathbf{10}$'s (or $\overline{\mathbf{10}}$'s) shrink to zero size; widely-dashed red lines are $\mathbf{5}$ or $\overline{\mathbf{5}}$'s, and thin, finely-dashed black lines are singlets (which generally lie off the plane). The green line, or 'slice,' defines the base space of an analogous model in M-theory, with dots along it indicating the locations of conical singularities supporting matter representations. The parenthetical section along the M-theory slice refers to Figure \ref{['three_cycle']}.
• Figure 2: The geometry in M-theory described by the section of the thick, solid green slice in Figure \ref{['e7_exempli']} enclosed in parentheses. The purple line through the fibres represents the location of the shrunk $SU_5$-lattice of two-cycles in each fibre. The blue dots on either side are the conical singularities giving rise to massless $\mathbf{10}$'s of $SU_5$ and the singularity indicated by the red dot between the two gives rise to a massless $\mathbf{5}$. Notice the foliated three-cycle that supports all three singularities.
• Figure 3: Matter curves along the $E_6$-base. $~$
• Figure 4: An illustration of the fibration (\ref{['e8toso10']}) which generates a canonical $SO_{10}$-model in F-theory with matter indicated in Table \ref{['fthso10model']}. Here, thick, solid blue lines represent $\mathbf{16}$'s and $\overline{\mathbf{16}}$'s, widely-dashed red lines are $\mathbf{10}$'s, and the finely-dashed black lines are $SO_{10}$-singlets. Black dots indicate triple-intersections that generate gauge-invariant Yukawa couplings based on the choice of fluxes generating the conjugations indicated in Table \ref{['fthso10model']}.
• Figure 5: The set of matter-curves listed on the left-hand side of Table \ref{['fthsu5modelall']} resulting from the fibration (\ref{['e8tosu5resolution']}). $\mathbf{10}$-dimensional representations are coloured in solid blue, $\mathbf{5}$'s and $\overline{\mathbf{5}}$'s in widely-dashed red, and $SU_5$-singlets are in finely-dashed black.