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

Sergio Ferrara,Renata Kallosh

doi:10.1103/physrevd.94.126015

Abstract

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.

Highlights

To compare the predictions of theoretical models with the observational data on inflationary cosmology [1] one has to use some form of the d 1⁄4 4 Einstein theory
Before looking at M theory on a 7-manifold with G2 holonomy we will explain the origin of the seven-disk geometry starting from D 1⁄4 4, N 1⁄4 8 supergravity
We may study the same cases as we did in the previous section: the conclusion is as in M theory compactified on the 7-manifold with G2 holonomy in Eq (4.16) which gives us seven possible values of r, according to (1.1), for example, for N 1⁄4 55, r ≈ f1.3; 2.6; 3.9; 5.2; 6.5; 7.8; 9.1g × 10−3: ð5:15Þ

Summary

INTRODUCTION

To compare the predictions of theoretical models with the observational data on inflationary cosmology [1] one has to use some form of the d 1⁄4 4 Einstein theory. One may try to motivate certain preferred values of the Poincaré disk radius square 3α as originating from a fundamental theory underlying N 1⁄4 1 supergravity. We assume that when the maximally supersymmetric theories are reduced to N 1⁄4 1 phenomenological α-attractor models, some mechanism of generating the required potentials will take place, but the moduli space geometry will be inherited from the more fundamental theories. In this setting we will find reasonably well motivated models of the Poincaré disk with radius square 3α taking values 1,2,3,4,5,6,7. SERGIO FERRARA and RENATA KALLOSH and the lowest one in the context of maximal N 1⁄4 4 superconformal theory is 1∶

POINCARÉ DISK WITH THE RADIUS SQUARE 3α

SEVEN-DISK GEOMETRY IN MAXIMAL SUPERGRAVITY

M THEORY ON A 7-MANIFOLD WITH G2 HOLONOMY

VALUES OF 3α IN STRING THEORY

CONCLUSION