# Heterotic/$F$-theory Duality and Narasimhan-Seshadri Equivalence

## TL;DR

This paper introduces a novel approach to finding the $F$-theory dual of Heterotic models with Wilson-line symmetry breaking by leveraging a specific choice in Narasimhan-Seshadri equivalence involving real $E_8$-bundles and complex conjugation.

## Contribution

It proposes using a particular automorphism in Narasimhan-Seshadri equivalence to incorporate $E_8$ symmetry into the $bZ_2$-action, aiding duality construction.

## Key findings

- Restores $E_8$ and $SU(5)_{gauge}$ symmetry in the dual model.
- Provides a new method for constructing $F$-theory duals with Wilson-line breaking.
- Ensures the $bZ_2$-quotient remains Calabi-Yau with an Enriques GUT surface.

## Abstract

Finding the $F$-theory dual of a Heterotic model with Wilson-line symmetry breaking presents the challenge of achieving the dual $\mathbb{Z}_{2}$-action on the $F$-theory model in such a way that the $\mathbb{Z}_{2}$-quotient is Calabi-Yau with an Enriques $\mathrm{GUT}$ surface over which $SU\left(5\right)_{gauge}$ symmetry is maintained. We propose a new way to approach this problem by taking advantage of a little-noticed choice in the application of Narasimhan-Seshadri equivalence between real $E_{8}$-bundles with Yang-Mills connection and their associated complex holomorphic $E_{8}^{\mathbb{C}}$-bundles, namely the one given by the real outer automorphism of $E_{8}^{\mathbb{C}}$ by complex conjugation. The triviality of the restriction on the compact real form $E_{8}$ allows one to introduce it into the $\mathbb{Z}_{2}$-action, thereby restoring $E_{8}$- and hence $SU\left(5\right)_{gauge}$- symmetry on which the Wilson line can be wrapped.