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.


Introduction
To compare the predictions of theoretical models with the observational data on inflationary cosmology [1] one has to use some form of d=4 Einstein theory. In particular one can use N = 1 supergravity models making a choice of the Kähler potential and a superpotential to fit the data. Cosmological models called α-attractor models [2,3,4], based on hyperbolic geometry of a Poincaré disk with the radius square 3α, are in good agreement with the data. The tilt of the spectrum of fluctuations and the level of B-models depend on the number of e-foldings N and on the moduli space curvature R Kähler = − 2 3α : This prediction is valid for α-attractor models with α O(10) for rather general class of potentials described in [2,3,4]. The early versions of these models were derived in [2], the more advanced versions were presented in [3,4]. At the level of phenomenological N = 1 supergravity any value of 0 < 3α < ∞ is acceptable, so one can view the future detection of the B-modes, or a new bound on r, as an experimental information about the curvature of the moduli space in these phenomenological models.
However, one may try to motivate certain preferred values of the Poincaré disk radius square 3α as originating from a fundamental theory underlying N = 1 supergravity. It was already suggested in [3] that the lowest possible value 3α = 1, with one unit size Poincaré disk, is motivated by a maximal superconformal N = 4 theory [5] and N = 4 pure supergravity without matter [6].
In this note we will study the possible origin of the moduli space geometries in maximal N = 8 supergravity and M/string theory. We assume that when the maximally supersymmetric theories are reduced to N = 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 a reasonably well motivated models of the Poincaré disk with radius square 3α taking values 1, 2, 3, 4, 5, 6, 7. In particular, the case with the highest value of 3α = 7 suggests that r is only slightly below 10 −2 .
Joint analysis of the data from BICEP2/Keck and Planck experiments [1] yields an upper limit on B-modes, r ≤ 7×10 −2 . The new interesting target with preferred values of α originating in M/string theory, for the number of e-foldings 47 < N < 57, is now and the lowest one in the context of maximal N = 4 superconformal theory is 2 Poincaré disk with the radius square 3α Consider the Kähler potential It describes a Poincaré disk with the radius square 3α. The metric of the moduli space is The Kähler manifold curvature computed from this metric depends on α: The kinetic term for the complex scalar field is where Z = (x + iy)/ √ 3α. For the vanishing sinflaton 1 the kinetic term becomes in terms of In these models the potentials depend on a geometric variable Z =Z (2.5)

Half-plane variables
One can use an alternative description of the same physical system by making a choice 1+Z The kinetic term for the complex scalar field is (2.7) In this form the kinetic term has an SL(2, R) symmetry where a, b, c, d are real numbers and dτ dτ (2.9) When the sinflaton τ +τ vanishes at τ = −τ = ie (2.10)

Seven-disk geometry in maximal supergravity
Before looking at M-theory on a 7-manifold with G 2 holonomy we will explain the origin of the seven-disk geometry starting from D=4 N = 8 supergravity. M theory/d=11 supergravity can be compactified on a 7-torus, which leads to d=4 maximal N = 8 supergravity [7] upon dualization of the form fields. This model has 70 scalars in the coset space SU (8) and E 7(7) symmetry. Following [8], we consider truncation of N = 8 supergravity [7] to N = 4 supergravity interacting with six N = 4 vector multiplets. The E 7(7) symmetry is decomposed as follows The 70 scalars of N = 8 supergravity [7] are first truncated to In the next step one takes into account that This truncation has a Kähler structure supporting N = 1 supersymmetry. One can identify 7 Poincaré disks due to the decomposition The original kinetic term is reduced to a form with the Kähler potential of the form with 7 pairs of independent scalars and the [SL(2, R)] 7 symmetry, a seven-disk manifold. The fact that the disk of the SL(2) commuting with SO(6, 6) has the same Kähler curvature of the other six SL(2)/SO(2) (each separately corresponding to α = 1/3) can be understood by string triality arguments [9] and by the underlying special geometry of the N=2 truncation [10].

M theory on a 7-manifold with G 2 holonomy
Instead of a compactification on a 7-torus, one can compactify M theory on a 7-manifold with G 2 holonomy. The early investigation of G 2 holonomy in physics was performed in [11], with review of the first 20 years in [12]. One of the most recent application of this compactification can be found in [13] and, of course, many more studies of M theory on G 2 were performed over the years.
Here we will focus on the model studied in [14,15], it requires the following choice of the Betti numbers This theory is identified with the maximal rank reduction on the seven torus and leads directly to d=4 N = 1 'curious supergravity' where 7 complex scalars are coordinates of the coset space The corresponding Kähler potential describing the scalar sector of this theory is the one in eq. (3.7) with 7 pairs of independent scalars and the [SL(2, R)] 7 symmetry. This model is one of the 'Four curious supergravities' defined in [15]. The other 3 have N = 2, N = 4, N = 8 supersymmetries, we are interested only in N = 1 'curious supergravity'. It has the field content defined by Betti numbers : the d=4 fields originating from the d=11 metric g M N are The ones from d=11 gravitino ψ M are The ones from the 3-form A M N P are To summarize, the field content of the M theory compactified on a 7-manifold with G 2 holonomy and Betti numbers (4.1) is a metric, a gravitino and a 3-form (which has no degrees of freedom, but affects trace anomaly) g µν , ψ µ , A µνρ (4.6) and 7 scalars, 7 spin 1/2 fields and 7 pseudoscalars The corresponding Kähler geometry is the seven-disk manifold in (3.7). For generic Betti numbers (b 0 , b 1 , b 2 , b 3 ) these models are known to have a generalized mirror symmetry, which flips one set of Betti numbers into the other one, and ρ ≡ 7b 0 − 5b 1 + 3b 2 − b 3 changes the sign. One of the reason the model we describe here was given a name 'curious supergravity' is that it has ρ = 0, it is a self-mirror in the above sense. It also means that it has a vanishing Weyl anomaly g µν T µν = − ρ 24×32π 2 R * µνρσ R * µνρσ = 0, the presence of the 3-form A µνρ is important for this.
To connect this compactified M theory model to α-attractor geometry we can make a choice that all moduli in our 7 unit radius disks in (3.7) are identified, namely 3α = 7 : We are left with one Poincaré disk of the radius square 7 times larger than the unit radius square. (4.10) (4.11) The following interpretation of this identification can be suggested: the diagonal components of the internal space metric g ij are taken to be the same in all 7 directions, g ij ∼ δ ij , and the 3form A ijk , which leads to 7 pseudoscalars in d=4, since b 3 = 7, is also the same in all directions. An analogous identification was performed in [16], where an early dimensional reduction of superstring theories was studied. The resulting d=4 N = 1 supergravity, neglecting the matter fields C in [16], has the following Kähler manifold: We will show in the next section that using string theory compactification on a product of 3 tori T 2 × T 2 × T 2 ⊂ T 6 one can get the seven-disk geometry.

Values of 3α in string theory
Here we will show how to derive the 7-disk geometry (4.13) in string theory. We start with the derivation of non-compact symmetries in string theory following [18], [19]. The toroidal compactification to d=4 of the N = 1 supergravity/string theory in d=10 space-time leads to scalars in SO(6,6) SO(6)×SO(6) coset space 2 upon truncation of non-geometric moduli from the d=10 vector multiplets.
As the result of the dimensional reduction one finds a d=4 action for the scalars of the following form, Here and Here M is a symmetric O(6, 6) matrix where G αβ and B αβ are the internal space metric and a 2-form, α, β = 1, . . . , 6. Together they represent the 36 coordinates of the coset space SO(6,6) SO(6)×SO(6) , we recover the moduli space of the six-torus T 6 in string theory. We would like now to perform the truncation of the 6-torus to three T 2 so that This means that we keep the following 9 components of G αβ G (IJ) = (g 11 , g 22 , g 12 ; g 33 , g 44 , g 34 ; g 55 , g 66 , g 56 ) , (5.7) and 3 components of B αβ We also introduce notation Now we observe that the coset SO(2,2) SO(2)×SO(2) is isomorphic to SL(2,R) SO(2) × SL(2,R) SO (2) and so we can package the SO(2, 2) matrix into an SL(2, R) × SL(2, R) matrix. We will do this for all three 2 In general, in the case of the heterotic string theory one finds scalars in the SO(6,6+n) SO(6)×SO(6+n) coset space.
Here the scalars in the SO(6,6) SO(6)×SO (6) part of the coset space originate from the geometric moduli, whereas the additional ones with n = 0 originate from the matter vector multiplets in d=10. If we keep some of the vector multiplets, so that n > 0 we do not find models with Poincaré disk geometry. copies of SO (2,2) SO(2)×SO(2) cosets, following an example of one of them in [19]. We have 4 real scalars from g 11 , g 22 , g 12 , b 12 . We package them as follows: t 1 ≡ b 1 + i √ g 1 and u 1 ≡ g 12 g 22 + i √ g 1 g 22 . The inverse relation is for the 2x2 matrices G = g 11 g 12 g 21 g 22 = Im t 1 Im u 1 In the same way we can organize all 6 complex scalars, 3 of them are often called Kähler moduli 12) and the other 3 are called complex structure moduli . (5.13) This corresponds to reorganizing One more important step here is to switch from the string frame as in (5.1) to the Einstein frame in d=4, which is a well known procedure of rescaling the metric, see for example [16]. As the result, we find an action with N = 1 supersymmetry and 7 complex scalars. The axion, dual to H µνλ , and dilaton as shown in eq. (5.2) form a complex scalar s = a + ie φ (5.14) with the Kähler potential K = − ln(−i(s −s)). The complete Kähler potential of the string theory dimensionally reduced on T 2 × T 2 × T 2 ⊂ T 6 is now given by the expression in (4.13) in the previous section, as promised there. Thus here again we reproduced the 7 Poincaré disk geometry of the unit radius each. We may now study the same cases as we did in the previous section: the conclusion is as in Mtheory compactified on 7-manifold with G 2 holonomy in eq. (4.16) which gives us seven possible values of r, according to (1.1), for example for N = 55 r ≈ {1.3, 2.6, 3.9, 5.2, 6.5, 7.8, 9.1} × 10 −3 (5.15)

Conclusion
In conclusion, we made an assumption that the moduli space geometry of the phenomenological N = 1 α-attractor models in [2,3,4] is inherited from the M-theory compactified on 7-manifold with G 2 holonomy to a 'curious N = 1 supergravity' [15], or from truncated N = 8 maximal supergravity, or from toroidally compactified string theory. In such case we argued that the possible cosmological α-attractor models might come with the values of 3α = 1, 2, 3, 4, 5, 6, 7 when some of the higher dimensional fields are discarded, following the procedure employed in the past in [16] and presented in eq. (4.17). To make a step from preferred values for 3α to a realistic prediction we would need to find the origin of the suitable class of potentials in these theories.
The relevant preferred values of the ratio of the tensor to scalar fluctuations during inflation are shown in eq. (5.15). We illustrated the position of these models in n s − r plane in Fig. 1. The highest one r ≈ 10 −2 will be the first interesting target for the B-mode experiments as well as for the theoretical studies of realistic cosmological models based on the seven-disk geometry.