Kuperstein embedding

where in the last step we have used (4.21), that translates intor3/2 µhere and implies that the D3-brane is located further down in the throat than the D7-brane (which extends down tor3_D/₇2 =µ). As we will see in the next section (see also [68]), the effective potential

V(φ) that one obtains using this maximum (unstable in the angular directions) is exactly the same as the one for the Kuperstein embedding (4.52), where now the angular directions are at a minimum.

4.5.2 Kuperstein embedding

The simplest Kuperstein embedding [73] (that we will also consider in the chapter 5) is obtain from (4.5) with the choice ˜g(z23) = µ. To facilitate the comparison with the

Ouyang case, we factorize out from the Kuperstein embedding a factor µ and absorb it into the definition of A0. Explicitly, we use the embedding

g(z) = 1− z1

µ , (4.51)

where now we parameterize the conifold with alternative coordinates{zi}(the relation to

Two trajectories extremize the potential in the angular directions: z1 = ±r3/2/

√

2, but only the one with the negative sign is actually a minimum. The correction to the potential then becomes [68, 67]: ∆V = κ 2 4|A|2e −2aτ n2_R2 −2πRe z1 µ−z1 + r γ|µ−z1|2 1− |z1| 2 2r3 = κ 2 4|A|2e −2aτ n2_R2 2πr3/2 √ 2µ+r3/2 + r γ(√2µ+r3/2₎2 , (4.52)

which is exactly the same as in the Ouyang case after choosing the (in that case unstable) trajectory w1 =−r3/2. The fact that the minus sign corresponds to the stable trajectory

(z1 =−r3/2/

√

2) is crucial for the fine tuning ofη. Indeed it determines that the correction toηKKLT '2/3 comes with a minus and a cancellation is possible.

The potential we have written still depends on τ. To obtain the effective potential for the inflaton we have to extremize the potential with respect to τ, i.e. use (4.30). The minimization of the volume can straightforwardly be carried out numerically. For an analytical estimate we will use (see appendix B.1)

τc =τ0+ β a2_τ 0 + r 3/2 anµ +. . . , (4.53)

where the dots stand for terms suppressed by higher powers in r3/2_{/µ or 1/τ}

0. We use

this expression for the r-dependent critical value of τ to transform the potential V(τ, r) into a potential for a single field V(r) = V(τc(r), r). This implicitly assumes that the dynamics in theτ direction is much faster than in the r direction such that the evolution of the system is well approximated by the trajectory τc(r) in the (τ, r) space. Eventually, the effective potential has to be expressed in terms of the canonically normalized field φ.

4.6 Inflation

In the previous sections we calculated the potential for the radial position r of the D3- brane in the throat, once all other moduli have reached their minimum6. In this chapter we investigate if the potential we have obtained can provide phenomenologically viable inflation.

The first step is to rewrite the potential in terms of a canonically normalized field (to which we will refer in the following as the inflaton)

φ=pTD3r , (4.54)

6_{In section 4.5.2, we have not been specific about the moduli coming from the angular position of the} D3-brane. As we said at the end of section 4.4.4, we assume that we start in a configuration where these moduli are already at their repective minima, from which values they do not move anymore.

4.6 Inflation 55

where we notice thatrhas the dimension of a length whileφof a mass, as it should be for a canonically normalized scalar in 4-dimensions. We remember that τ0 is dimensionless

and measures the four cycle volume in units of l4

s = (α

0₎2_.

As we have seen in section 4.3, VKKLT,0 depends on the inflaton as7

VKKLT,0 = 3H2 36M6 P l (φ2 _−6M2 P l)2 '3H2M_{P l}2 +H2φ2+. . . (4.55)

for small φ. This prevents slow roll as

η =M_{P l}2 V 00

V &

2

3. (4.56)

If we want to have a flat potential, we need another term in the potential of the same size but opposite sign that we can fine tune to cancel with the 2/3. The new terms in the potential, coming from the dependence of the non-perturbative superpotential onφ as in (4.39), are proportional to|g|1/n_{or toφ|g|}1/n_{. The known supersymmetric embeddings all} depend on integer powers of wi ∝ zi ∝φ3/2. This, in particular, implies that there is no term, in the smallφ expansion, that can exactly cancel theφ2 _fromV

KKLT,0. The absence

of fractional power embeddings, i.e. g ∝ wp_i with p non-integer, might be traced back to the holomorphicity of g(wi) (see also [67, 68]); it seems therefore hard to circumvent this problem.

Also, all those embeddings for which g ∝ 1 +wp_i with p > 1 vanish much faster than

VKKLT,0 for φ → 0 and do not help to flatten the potential. From this observation, it

follows that embeddings of the ACR family [71] with p >1 are not helpful to cancel the

ηKKLM M T '2/3, at least for small φ. Further study is needed to see if there is a region where φ is large enough so that the effects of higher ACR embeddings become relevant and at the same time, where that region is still well described by the conifold geometry (i.e. before the cut of the conifold and the gluing to the Calabi-Yau manifold become relevant).

Two embeddings that produce corrections to the scalar potential proportional to φ and

φ3/2 (as opposed to φp with p > 2) are the Ouyang (which is in the ACR family as well but with p = 1) and the Kuperstein embedding. For the former, once the angular minimization is performed, the corrections to the scalar potential vanish [66]. For the latter this is not the case and the potential is indeed modified as in (4.52) [67, 68].

f

V

e

h

Figure 4.3: The plot shows the potential V(φ) (red) and the slow-roll parameters η(φ) (blue) and (φ) (black). The latter is so small that it can hardly be distinguished from the φ axis. Next to the tip of the throat the potential has generically a maximum and a minimum. For φ

large enough the potential grows like φ2 and η is of order one (or bigger). But for φ→ 0 the curvature of the potential changes at the inflection point and η switches sign (and eventually diverges at φ= 0).