Non-Higgsable QCD and the Standard Model Spectrum in F-theory

TL;DR

This work investigates how non-Higgsable clusters in F-theory can realize the Standard Model gauge group, focusing on an unbroken $SU(3)$ (QCD) factor realized by a Kodaira type $IV$ singularity and exploring three distinct ways to realize the $SU(2)$ and $U(1)$ factors. The authors develop a geometric framework based on a base $B_3$, detect non-Higgsable sectors, and analyze matter content arising at codimension-two intersections via string junctions and anomaly considerations, including a detailed study of a $IV$-$III$ collision that yields SM-like representations. They present explicit 6D and 4D examples demonstrating non-Higgsable $SU(3)$ and $SU(3) imes SU(2)$ configurations, show how G-flux can generate chirality, and argue that a minimal chiral $SU(3) imes SU(2) imes U(1)$ spectrum consistent with SM generations is achievable in this framework. The results imply that non-Higgsable QCD could provide a natural mechanism for an unbroken QCD sector and a path toward realistic SM-like spectra within the F-theory landscape, with further work needed on abelian factors and flux-induced chirality.

Abstract

Many four-dimensional supersymmetric compactifications of F-theory contain gauge groups that cannot be spontaneously broken through geometric deformations. These "non-Higgsable clusters" include realizations of $SU(3)$, $SU(2)$, and $SU(3) \times SU(2)$, but no $SU(n)$ gauge groups or factors with $n> 3$. We study possible realizations of the standard model in F-theory that utilize non-Higgsable clusters containing $SU(3)$ factors and show that there are three distinct possibilities. In one, fields with the non-abelian gauge charges of the standard model matter fields are localized at a single locus where non-perturbative $SU(3)$ and $SU(2)$ seven-branes intersect; cancellation of gauge anomalies implies that the simplest four-dimensional chiral $SU(3)\times SU(2)\times U(1)$ model that may arise in this context exhibits standard model families. We identify specific geometries that realize non-Higgsable $SU(3)$ and $SU(3) \times SU(2)$ sectors. This kind of scenario provides a natural mechanism that could explain the existence of an unbroken QCD sector, or more generally the appearance of light particles and symmetries at low energy scales.

Figures (2)

• Figure 1: Displayed are the local geometries of an $I_3$-$I_2$ collision and a $IV$-$III$ collision, respectively. In both cases the $SU(2)$ gauge theory is on the vertical line, the $SU(3)$ gauge theory is on the horizontal line, and quark doublets are localized at the intersection of the solid lines; the integers denote the number of branes in each stack. Note the additional brane $\tilde{\Delta}$ participating in the intersection in the $IV$-$III$ case.
• Figure 2: In an elliptic surface near a $IV$-$III$(-$I_1$) collision the geometry exhibits an $E_6$ singularity, where the the $SU(3)$ and $SU(2)$ Dynkin diagram embed as displayed above. The crossed nodes are simple roots of $E_6$ whose associated states only become massless in codimension two. The left-most node corresponds to the $E_6$ simple root junction charged under the extra brane.