Discrete gauge groups in certain F-theory models in six dimensions

Abstract

We construct six-dimensional (6D) F-theory models in which discrete Z 5 , Z 4 , Z 3 , and Z 2 gauge symmetries arise. We demonstrate that a special family of "Fano 3-folds" is a useful tool for constructing the aforementioned models. The geometry of Fano 3-folds in the constructions of models can be useful for understanding discrete gauge symmetries in 6D F-theory compactifications. We argue that the constructions of the aforementioned models are applicable to Calabi-Yau genus-one fibrations over any base space, except models with a discrete Z 5 gauge group. We construct 6D F-theory models with a discrete Z 5 gauge group over the del Pezzo surfaces, as well as over P 1 ×P 1 and P 2 . We also discuss some applications to four-dimensional F-theory models with discrete gauge symmetries.

F-theory models with U(1) × ℤ2, ℤ4 and transitions in discrete gauge groups

“…There are situations in which a genus-one fibration has a global section and in which it does not have a global section; when a genus-one fibration does not have a global section, a discrete gauge group forms in F-theory on this fibration, as mentioned. Recent discussions of F-theory on genus-one fibrations without a global section can be found, for example, in [20,19,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45] 2 . When a genus-one fibration has a global section (in which case, the fibration is often called an elliptic fibration 3 in the F-theory literature), the U (1) gauge group forms in F-theory if the fibration has two or more independent global sections.…”

“…However, there are various other manners in which a multisection splits into multisections of smaller degrees in the moduli of multisection geometry. When studying these, physically unnatural phenomena are identified [44,45]. Under certain conditions, a four-section splits into a pair of bisections [34,44].…”

“…Under certain conditions, a four-section splits into a pair of bisections [34,44]. When one considers the process [44] wherein a four-section splits into a pair of bisections, and these bisections further split into four global sections, 1 See, e.g., [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] for recent progress of discrete gauge groups. 2 [46,47] discussed F-theory on genus-one fibrations without a global section.…”

“…when seen from the physical viewpoint, U (1) 3 breaks into a discrete Z 2 gauge group, and this discrete Z 2 gauge group transitions further to a discrete Z 4 gauge group via Higgsing. It was pointed out in [44] that this process appears unnatural because a discrete Z 2 gauge group appears "enhanced" to a discrete Z 4 gauge group, rather than broken down into another discrete gauge group of a smaller degree. In contrast, another puzzling physical phenomenon was observed in [45], wherein an n-section (with n ≥ 3) splits into a global section and an (n − 1)-section.…”

“…The aim of this note is to examine whether the proposal in [45] to resolve the puzzle pointed out in [44] is consistent along bisection geometries loci in a four-section geometry. We confirm in Section 3.1 that a discrete Z 2 gauge group actually enlarges and U (1) also forms in F-theory along any bisection geometries locus in a four-section geometry realized as the complete intersections of two quadrics in P 3 fibered over any base space.…”