Gauge groups and matter spectra in $F$-theory compactifications on genus-one fibered Calabi–Yau $4$-folds without section: Hypersurface and double cover constructions

Abstract

We investigate gauge theories and matter contents in F-theory compactifications on families of genus-one fibered Calabi-Yau 4-folds lacking a global section. To construct families of genus-one fibered Calabi-Yau 4-folds that lack a global section, we consider two constructions: hypersurfaces in a product of projective spaces, and double covers of a product of projective spaces. We consider specific forms of defining equations for these genus-one fibrations, so that genus-one fibers possess complex multiplications of specific orders. These symmetries enable a detailed analysis of gauge theories. $E_6$, $E_7$, and $SU(5)$ gauge groups arise in some models. Discriminant components intersect with one another in the constructed models, and therefore, discriminant components contain matter curves. We deduce potential matter spectra and Yukawa couplings.