Musings

G₂ Compactifications

There have been several papers recently on compactifications of M-Theory on “manifolds”, X, of G₂ holonomy. I put ‘manifolds’ in quotes because to get nonabelian gauge groups with chiral matter, we need certain types of singularities.

Friedmann and Witten have written about threshold corrections in such theories. At least in the limit that they consider, the threshold corrections are given by the Ray-Singer torsion of the associative 3-manifold, Q, where X can be thought of as a K3 fibration over a base isomorphic to Q, and where the generic K3 fiber has a A₄ singularity. In layman’s terms, the threshold corrections are entirely due to massive adjoints of SU(5); there is no contributions (in this limit) from massive fundamentals or other representations.

I’ve been meaning to post some comments on this paper; I guess I’ll get around to that eventually.

More recently, Bobby Acharya has written about the superpotential for models with G-flux. He finds that, in addition to the usual contribution linear in G, there’s also a term in the superpotential proportional to a complex Chern-Simons term on Q.

This ends up fixing the moduli of the compactification, in a manner reminiscent of what happens in some (doubtless related) Type IIA orientifold models. Of course, he ends up with a theory with unbroken N=1 supersymmetry in (3+1)-dimensional anti-de Sitter space. Perhaps (with considerable handwaving) after supersymmetry breaking we might end up with a vanishing or small positive cosmological constant. But understanding the resolution of that question seems as far away as ever.