Seven-disk manifold, α-attractors, and B modes

Cosmological alpha-attractor models in N = 1 supergravity are based on the hyperbolic geometry of a Poincare disk with the radius square R-2 = 3 alpha. The predictions for the B modes, r approximate to 3 alpha 4/N-2, depend on moduli space geometry and are robust for a rather general class of potentials. Here we notice that starting with M theory compactified on a 7-manifold with G(2) holonomy, with a special choice of Betti numbers, one can obtain d = 4, N = 1 supergravity with the rank 7 scalar coset [SL(2)/SO(2)](7). In a model where these seven unit size Poincare disks have identified moduli one finds that 3 alpha = 7. Assuming that the moduli space geometry of the phenomenological models is inherited from this version of M theory, one would predict r approximate to 10(-2) for N = 53 e-foldings. We also describe the related maximal supergravity and M/string theory models leading to preferred values 3 alpha = 1, 2, 3, 4, 5, 6, 7.