ℙ 1 P¹ -bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models

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 N=1 4D F-theory constructions. In particular, we study 4D F-theory models that arise from compactification on threefold bases that are P¹ 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 P¹ -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) × 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.