Hitchin's Equations and M-Theory Phenomenology

TL;DR

This paper explores the geometric and physical structures arising from M-theory compactifications on G_2 manifolds with singularities, using Higgs bundles and spectral covers to analyze the low-energy effective theories.

Contribution

It introduces a framework connecting Higgs bundles, spectral covers, and G_2 holonomy in M-theory compactifications, with new tools for analyzing instanton corrections and effective theories.

Findings

Spectral covers constructed as Lagrangian branes with flat connections.

Dictionary established between spectral covers and ALE fibrations.

Morse theoretic methods used to compute corrected massless spectra.

Abstract

Phenomenological compactifications of M-theory involve 7-manifolds with G_2 holonomy and various singularities. Here we study local geometries with such singularities, by thinking of them as compactifications of 7d supersymmetric Yang-Mills theory on a three-manifold Q_3. We give a general discussion of compactifications of 7d Yang-Mills theory in terms of Higgs bundles on Q_3. We show they can be constructed using spectral covers, which are Lagrangian branes with a flat connection in the cotangent bundle T^*Q_3. We explain the dictionary with ALE fibrations over Q_3 and conjecture that these configurations have G_2 holonomy. We further develop tools to study the low energy effective theory of such a model. We show that the naive massless spectrum is corrected by instanton effects. Taking the instanton effects into account, we find that the massless spectrum and many of the interactions can be computed with Morse theoretic methods.