# Higgs Bundles for M-theory on $G_2$-Manifolds

## TL;DR

This paper analyzes the gauge sector in M-theory compactifications on $G_2$-manifolds by studying Higgs bundles derived from 7d super Yang-Mills theory, using Morse theory to determine the massless spectrum and exploring chiral matter engineering.

## Contribution

It introduces a mathematical framework using Morse and Morse-Bott theory to determine the 4d spectrum from Higgs bundles on $G_2$-manifolds, including TCS examples.

## Key findings

- Derived BPS equations for Higgs bundles on $G_2$-manifolds.
- Determined massless spectra for abelian and non-abelian gauge groups.
- Provided a method to engineer chiral matter through singular transitions.

## Abstract

M-theory compactified on $G_2$-holonomy manifolds results in 4d $\mathcal{N}=1$ supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle $M_3$. We derive the BPS equations and find the massless spectrum for both abelian and non-abelian gauge groups in 4d. The mathematical tool that allows us to determine the spectrum is Morse theory, and more generally Morse-Bott theory. The latter generalization allows us to make contact with twisted connected sum (TCS) $G_2$-manifolds, which form the largest class of examples of compact $G_2$-manifolds. M-theory on TCS $G_2$-manifolds is known to result in a non-chiral 4d spectrum. We determine the Higgs bundle for this class of $G_2$-manifolds and provide a prescription for how to engineer singular transitions to models that have chiral matter in 4d.