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

Understand

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.