# F-theory models with $U(1)\times \mathbb{Z}_2,\, \mathbb{Z}_4$ and transitions in discrete gauge groups

## TL;DR

This paper investigates F-theory models with combined $U(1)$ and discrete gauge groups, confirming a proposal that relates geometric transitions to gauge group Higgsing, and explores models with $U(1) imes bZ_4$ symmetry.

## Contribution

It demonstrates how a four-section geometry in F-theory can realize gauge group transitions from $U(1) imes bZ_2$ to $bZ_4$, confirming the proposed geometric-Higgsing correspondence.

## Key findings

- Discrete $bZ_2$ gauge group enlarges along bisection loci.
- Vacuum expectation values break $U(1) imes bZ_2$ to $bZ_4$ via Higgsing.
- Constructed models with $U(1) imes bZ_4$ gauge symmetry.

## Abstract

We examine the proposal in the previous paper to resolve the puzzle in transitions in discrete gauge groups. We focus on a four-section geometry to test the proposal. We observed that a discrete $\mathbb{Z}_2$ gauge group enlarges and $U(1)$ also forms in F-theory along any bisection geometries locus in the four-section geometry built as the complete intersections of two quadrics in $\mathbb{P}^3$ fibered over any base. Furthermore, we demonstrate that giving vacuum expectation values to hypermultiplets breaks the enlarged $U(1)\times \mathbb{Z}_2$ gauge group down to a discrete $\mathbb{Z}_4$ gauge group via Higgsing. We thus confirmed that the proposal in the previous paper is consistent when a four-section splits into a pair of bisections in the four-section geometry. This analysis may be useful for understanding the Higgsing processes occurring in the transitions in discrete gauge groups in six-dimensional F-theory models. We also discuss the construction of a family of six-dimensional F-theory models in which $U(1)\times\mathbb{Z}_4$ forms.