Seven-Disk Manifold, alpha-attractors and B-modes

Cosmological alpha-attractor models in \cN=1 supergravity are based on hyperbolic geometry of a Poincar\'e disk with the radius square {\cal R}^2=3\alpha. The predictions for the B-modes, r\approx 3\alpha {4\over 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 \cN=1 supergravity with rank 7 scalar coset \Big[{SL(2)\over SO(2)}\Big]^7. In a model where these 7 unit size Poincar\'e 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 \approx 10^{-2} for 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.Cosmological α-attractor models in N=1 supergravity are based on the hyperbolic geometry of a Poincaré disk with the radius square R2=3α. The predictions for the B modes, r≈3α4N2, 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 G2 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 Poincaré disks have identified moduli one finds that 3α=7. Assuming that the moduli space geometry of the phenomenological models is inherited from this version of M theory, one would predict r≈10-2 for N=53 e-foldings. We also describe the related maximal supergravity and M/string theory models leading to preferred values 3α=1, 2, 3, 4, 5, 6, 7.Cosmological alpha-attractor models in \cN=1 supergravity are based on hyperbolic geometry of a Poincar\'e disk with the radius square {\cal R}^2=3\alpha. The predictions for the B-modes, r\approx 3\alpha {4\over 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 \cN=1 supergravity with rank 7 scalar coset \Big[{SL(2)\over SO(2)}\Big]^7. In a model where these 7 unit size Poincar\'e 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 \approx 10^{-2} for 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