$\mathbb{P}^1$-bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models

TL;DR

This work analyzes a large, explicit class of 4D F-theory compactifications with base threefolds B that are P^1-bundles over toric surfaces, generating 109,158 distinct bases. It provides substantial evidence that geometrically non-Higgsable gauge groups and matter are ubiquitous in these 4D vacua, with 98.3% of bases exhibiting NHCs and only a small subset being weak-Fano. The study details how the base geometry constrains non-Higgsable content, including single and two-factor gauge factors, cluster structures, and potential phenomenological implications such as non-Higgsable QCD, dark matter sectors, and Higgs-sector fields, using Weierstrass models and toric methods. It also develops a framework to compute approximate Hodge numbers of the associated elliptic Calabi–Yau fourfolds and discusses minimal model perspectives within a Mori-theory-inspired context. Collectively, the results suggest non-Higgsable structure is a robust and generic feature of the 4D F-theory landscape, with important implications for phenomenology and the global structure of vacua.

Abstract

We explore a large class of F-theory compactifications to four dimensions. We find evidence that gauge groups that cannot be Higgsed without breaking supersymmetry, often accompanied by associated matter fields, are a ubiquitous feature in the landscape of ${\cal N} = 1$ 4D F-theory constructions. In particular, we study 4D F-theory models that arise from compactification on threefold bases that are $\mathbb{P}^1$ bundles over certain toric surfaces. These bases are one natural analogue to the minimal models for base surfaces for 6D F-theory compactifications. Of the roughly 100,000 bases that we study, only 80 are weak Fano bases in which there are no automatic singularities on the associated elliptic Calabi-Yau fourfolds, and 98.3% of the bases have geometrically non-Higgsable gauge factors. The $\mathbb{P}^1$-bundle threefold bases we analyze contain a wide range of distinct surface topologies that support geometrically non-Higgsable clusters. Many of the bases that we consider contain $SU(3)\times SU(2)$ seven-brane clusters for generic values of deformation moduli; we analyze the relative frequency of this combination relative to the other four possible two-factor non-Higgsable product groups, as well as various other features such as geometrically non-Higgsable candidates for dark matter structure and phenomenological ($SU(2)$-charged) Higgs fields.

Figures (18)

• Figure 1: Hodge numbers for generic elliptic Calabi-Yau manifolds fibered over the 61,539 toric base surfaces that support elliptically fibered Calabi-Yau's mt-toricHodge. Only seven Hodge numbers (on the left in orange) correspond to elliptic fibrations over weak Fano bases that do not give rise to non-Higgsable clusters.
• Figure 2: The average number of distinct $\mathbb{P}^1$ bundles $B$ over a toric base surface $S$ in the set found in mt-toric, as a function of $h^{1, 1}(S)$.
• Figure 3: Number of bases $S$ that support non-Higgsable groups on a section of the $\mathbb{P}^1$ bundle over $S$, as a function of $h^{1, 1}(S)$.
• Figure 4: Average number of different normal bundles compatible with a non-Higgsable group on a section divisor, as a function of $h^{1, 1}(S)$
• Figure 5: Plotted are the Hodge numbers for each example in our data set, as estimated from the geometry of each base.