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

## TL;DR

This paper constructs six-dimensional F-theory models with various discrete gauge groups using Fano 3-folds, providing new geometric tools and extending understanding of discrete symmetries in string compactifications.

## Contribution

It introduces a novel use of Fano 3-folds for constructing 6D F-theory models with discrete gauge groups, including the first models with a $ ext{Z}_5$ symmetry over specific bases.

## Key findings

- Constructed 6D F-theory models with $ ext{Z}_5$, $ ext{Z}_4$, $ ext{Z}_3$, and $ ext{Z}_2$ gauge groups.
- Demonstrated applicability of Fano 3-folds in model building.
- Discussed potential extensions to 4D models with discrete symmetries.

## Abstract

We construct six-dimensional (6D) F-theory models in which discrete $\mathbb{Z}_5, \mathbb{Z}_4, \mathbb{Z}_3,$ and $\mathbb{Z}_2$ 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 $\mathbb{Z}_5$ gauge group. We construct 6D F-theory models with a discrete $\mathbb{Z}_5$ gauge group over the del Pezzo surfaces, as well as over $\mathbb{P}^1\times\mathbb{P}^1$ and $\mathbb{P}^2$. We also discuss some applications to four-dimensional F-theory models with discrete gauge symmetries.