Hitchin's Equations and M-Theory Phenomenology

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.

Figures (8)

• Figure 1: The extended $E_8$ Dynkin diagram and Dynkin indices.
• Figure 2: Fermion zero mode localized on a defect.
• Figure 3: The extended $A_5$ Dynkin diagram and Dynkin indices.
• Figure 4: The extended $E_7$ Dynkin diagram and Dynkin indices.
• Figure 5: Positively charged (red) and negatively charged (black) boundaries of Morse functions on $S^3$. Case (a) yields an electro-static potential that we call $f_6$, case (c) yields the potential $f_1$, and case (b) corresponds to the linear combination $-f_1 - f_6$. Note that we took one of the pieces of the boundary common to $f_1$ and $f_6$, but with the charge density for $f_6$ dominant there.