Hitchin’s equations and MM-theory phenomenology

Phenomenological compactifications of MM-theory involve seven-manifolds with G2G2 holonomy and various singularities. Here we study local geometries with such singularities, by thinking of them as compactifications of 7d7d supersymmetric Yang–Mills theory on a three-manifold Q3Q3. We give a general discussion of compactifications of 7d7d Yang–Mills theory in terms of Higgs bundles on Q3Q3. We show that they can be constructed using spectral covers, which are Lagrangian branes with a flat connection in the cotangent bundle T∗Q3T∗Q3. We explain the dictionary with ALE fibrations over Q3Q3 and conjecture that these configurations have G2G2 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.