Non-Higgsable clusters for 4D F-theory models

TL;DR

This work develops a geometric framework for non-Higgsable clusters in 4D F-theory by deriving local divisor-based bounds on the vanishing orders of the Weierstrass coefficients $f$ and $g$ and cataloguing the resulting gauge algebras and matter content. The authors identify a finite set of possibilities: nine isolated simple gauge factors and five two-factor products, including the Standard Model-like $SU(3)\times SU(2)$. Unlike the 6D case, 4D clusters support rich quiver structures with branchings, long chains, and loops, arising from how divisors intersect within the threefold base. They discuss realizations on various surfaces (e.g., $\mathbb{P}^2$, $dP_k$, and $\mathbb{F}_n$-type bases) and the localization of matter on curves, as well as potential implications for phenomenology and dark sectors; they also highlight open questions about codimension-3 singularities, flux effects, and the global classification of Calabi–Yau fourfold bases.

Abstract

We analyze non-Higgsable clusters of gauge groups and matter that can arise at the level of geometry in 4D F-theory models. Non-Higgsable clusters seem to be generic features of F-theory compactifications, and give rise naturally to structures that include the nonabelian part of the standard model gauge group and certain specific types of potential dark matter candidates. In particular, there are nine distinct single nonabelian gauge group factors, and only five distinct products of two nonabelian gauge group factors with matter, including $SU(3) \times SU(2)$, that can be realized through 4D non-Higgsable clusters. There are also more complicated configurations involving more than two gauge factors; in particular, the collection of gauge group factors with jointly charged matter can exhibit branchings, loops, and long linear chains.

Figures (3)

• Figure 1: The quiver diagram associated with a non-Higgsable cluster having gauge algebra ${\mathfrak{su}}_2 \oplus {\mathfrak{su}}_2 \oplus {\mathfrak{g}}_2 \oplus {\mathfrak{g}}_2$, with bifundamental (geometrically non-chiral) matter connecting the first ${\mathfrak{su}}_2$ component with the other three gauge factors.
• Figure 2: A global model with a non-Higgsable geometry giving rise to a linear chain in the quiver structure of the gauge algebra, ${\mathfrak{su}}_2 \oplus {\mathfrak{su}}_3 \oplus {\mathfrak{su}}_3 \oplus {\mathfrak{su}}_3 \oplus {\mathfrak{su}}_2$. (A) the quiver diagram of the chain, (B) A schematic depiction of the triangulation of the 3D toric fan describing the global geometry, formed from a sequence of blowups at points $7 (E_1), \ldots,$ from an initial fan describing the threefold $\mathbb{P}^1 \times \mathbb{F}_4$. (Note that points associated with the toric rays $v_3$-$v_6$ and $v_{14}$ are not shown.) Large blue dots represent divisors supporting an ${\mathfrak{su}}_3$ gauge summand, smaller red dots are divisors supporting an ${\mathfrak{su}}_2$ gauge algebra. Open circles represent $(4, 6)$ curves that must be blown up to divisors after the blow-ups up to $v_{25}$. Note that before blowing up to $v_{25}$, the divisors associated with points $9, 10, 11$ are connected del Pezzo $dP_3$ surfaces (as can be seen from the structure of solid lines). Similar constructions are possible with up to (at least) 11 factors of ${\mathfrak{su}}_3$ in the linear chain.
• Figure 3: A global model with a non-Higgsable geometry giving rise to a loop in the quiver structure of the gauge algebra. (A) the quiver diagram of the loop, (B) A schematic depiction of the triangulation of the 3D toric fan describing the global geometry, formed from a sequence of blowups at points $7, \ldots,$ from an initial fan describing the threefold $\mathbb{P}^1 \times \mathbb{P}^1 \times\mathbb{P}^1$. (Note that the points on the left should be identified with the points on the right with the same labels.)