Antecedentes

In this chapter we have proposed and studied a model of D3-brane inflation that takes place along the angular directions of a warped conifold and where no anti D3-branes are needed. A D3-brane moves on the tip of a warped deformed conifold [107] embedded in a compact Calabi-Yau manifold, which is an S3 _{(see figure 5.1). The potential comes}

from the F-term where the threshold corrections to the non-perturbative superpotential are taken into account. Inflation at the tip is an explicit example of how several issues of brane inflation can be addressed in a way different from the standard radial case that we studied in chapter 4.

We have looked for inflationary trajectories, considering two classes:

1. Slow-roll inflation: it is possible to achieve but only allowing for fine tuning. In a small region of the stringy parameter space (where Λ ' 2C) the inflaton potential becomes very flat on the top (see figure 5.4). This is a kind of hilltop model [105, 106], i.e. of the form

V 'Λ− 1

2m

2_φ2_−λφ4_{. (5.66)}

The negative squared mass of the inflaton can be made small byfine tuning, and we obtain a string theory model of slow-roll inflation in perfect agreement with CMB data (see chapter 3 and section 5.5.2). When the mass termm2φ2 is negligible with respect to theλφ4 _{term, the model predictsn}

s'0.94, negligible tensor modes and the scale of inflation (Λ)1/4 _∼d×10−3_{, whered was defined in (5.29).}

2. DBI inflation: as the inflaton corresponds to the position of a D3-brane, the kinetic term comes from the DBI action. This can allow inflation to take place even with a steep potential [50, 51]. We investigated this possibility and discovered that, although perturbations compatible with CMB data can be produced, the DBI regime can never lead to more than a few e-foldings. We argued that more general embeddings than the one we studied here (Kuperstein embedding [73] with constang ˜

g) can possess flat regions that would increase the number of e-foldings and lead to a successful model. We leave a more detailed study of this interesting possibility for future investigation.

During the study of both radial (chapter 4) and angular (present chapter) inflation, we have neglected perturbative quantum corrections. These can arise in different ways: there

5.7 A summary on angular brane inflation 97

are α0 corrections, suppressed by powers of the string scale Ms = (α0)−1/2, which are induced by the higher oscillation modes of the string that we have ignored by looking at the massless spectrum (see chapter 2). There are field theory and/or string loop corrections suppressed by the appropriated coupling. There is the tower of KK modes and the effects due to the large warping. For a completely successful string inflation model, all these effects have to be carefully taken into account. This is a formidable challenge and we are still far from that goal.

On the other hand it is very important to check the consistency of our effective description, to know if string theory and/or quantum corrections can change the qualitative features of a certain model. This is why we will dedicate the next chapter to this issue. We will consider an explicit model and carefully analyze the effects of several types of corrections. On the other hand, we will abandon the subject of string cosmology and switch to string particle phenomenology. Particle physics is the other wide field of research, where by general considerations (see chapter 1) string theory can be applied most usefully. We postpone a further discussion to the next chapter.

Finally, in the following we list some interesting directions for future research

• The case of a generic embedding (with a generic ˜g in (4.5) or those discussed in [72]) has to be studied in detail looking for an explicit working model. A criteria to keep in mind in this search follows from our no go result in section (5.6): a successful model needs to have a potential with at least one slow-roll flat region.

• The absence of an anti D3-brane makes the mechanism of graceful exit and reheating in the present model very different from the standard brane-anti brane inflation model. A mechanism naturally embedded in the model is the D-brane trapping (which was one of the motivations of the original proposal of brane inflation [49]). In [93] it was shown that in the collision of branes, the kinetic energy can be transferred to the gauge fields living on the newly created stack. This would create a thermal bath in the world volume of the branes that could evolve into our universe. It would be interesting to investigate quantitatively the phenomenological viability of this idea.

• It would be desirable to develop a gauge dual description of the brane inflation at the tip proposed in this paper, a work in this direction is [141] (see also [142]). Although there the origin of the potential is different, the authors provide a gauge theory description of the radial motion of a brane in a resolved warped deformed conifold.

6 On soft terms from large volume

compactifications

As we have seen in chapter 2, the KKLT strategy [8, 20] for producing stabilized string vacua of type IIB, can serve as a starting point for phenomenological constructions. In chapters 4 and 5 for example, we have studied two models of brane inflation in this setup. Here we will focus on a generalization of the KKLT setup, known as the “large vol- ume scenario” (LVS for short, we introduce it in section 6.2.3), where many phenomeno- logical issues have been addressed, such as for example soft supersymmetry breaking [143, 144, 145], the QCD axion [146, 147], neutrino masses [148], first attempts at LHC phenomenology [149], and where also some closed string inflationary models have been constructed [58, 150, 151, 152].

Although tantalizing, the models discussed in the aforementioned papers as well as the inflationary models of chapters 4 and 5 raise many questions. It remains an open prob- lem to construct complete KKLT models in string theory, as opposed to supergravity. Problems one faces include things like the description of RR fluxes in string theory, showing that the necessary nonperturbative effects actually can and do appear in a way consistent with other contributions to the potential (for progress in this direction, see [61, 153, 154, 23, 155, 156, 157, 158, 159, 160, 161, 162, 114, 163, 164, 165, 166]), and verifying that one can uplift to a Minkowski or deSitter vacuum without ruining stabiliza- tion [22, 167, 168, 161, 125]. These issues become possibly even more important in some extensions of KKLT and in particular in LVS [169, 143], where corrections to the tree-level supergravity effective action (computed in [170]) play a significant role, and where the compactification volume can be stabilized as large as 1015 in string units. In LVS, since string corrections play a crucial role, striving for actual string constructions seems quite important. In the end, the restrictiveness this entails may greatly improve predictivity, or kill the models completely as string compactifications.

In this chapter, we will not improve on the consistency of KKLT or LVS in general, but rather assume the existence of LVS models in string theory, and then perform self- consistency checks. This is a modest step on the way towards reconciling phenomeno- logically promising scenarios with underlying string models. We will see that although a priori the situation looks very bleak, and one might have hastily concluded that even our modest consistency check would put very strong constraints on LVS, things are more interesting. It turns out that LVS jumps through every hoop we present it with, and instead of broad qualitative changes, we find only small quantitative changes.

The structure of this chapter is as follows: we start in section 6.1 with some preliminaries; in section 6.2 we critically review the KKLT construction of section 2.6 and introduce the LVS; in section 6.3 we argue which kind of loop corrections we expect in LVS and compute their effects on the potential in an explicit example. In section 6.4 we compute the gaugino masses in LVS and comment on other susy-breaking terms. In section 6.5 we extend our discussion of string loop corrections to other classes of Calabi-Yau manifolds. We conclude in section 6.6 with a discussion of further corrections and a summary. A series of technical details are left to appendix D.

6.1 Preliminaries

The main difference between KKLT and LVS is that LVS includes a specific stringα0 cor- rection ∆Kα0 in the K¨ahler potentialK of the 4-dimensionalN = 1 effective supergravity.

Naturally, the 4-dimensional string effective action also contains other string corrections. Here, we will focus on gs corrections due to sources (D-branes and O-planes). For some

N = 1 andN = 2 toroidal orientifolds, these corrections were computed in [171] (see also [172]; for a comprehensive introduction to orientifolds, see [173]). Compared to the α0

correction ∆Kα0 considered in LVS, the g_s corrections to the K¨ahler potential ∆K_g

s will scale as ∆Kα0 : ∆K_g s ∼O(α03) :O(g_s2α02) (string frame) . (6.1) By naive dimensional analysis, one would expect that in a 1/V expansion, where V is the overall volume in the Einstein frame, (6.1) implies

∆Kα0 ∼ O(g−3/2

s V

−1_{) , ∆K}

gs ∼ O(gsV

−2/3_{) (Einstein frame). (6.2)}

If, contrary to what we assumed in chapter 4 and 5, there is more than one K¨ahler modulus various combinations of K¨ahler moduli may appear in ∆Kgs in (6.2), and a priori this

could lead to even weaker suppression in 1/V than that shown. However, we will argue that (6.2) is actually correct as far as the suppression factors in the 1/V expansion go. Nevertheless, even the suppression displayed in (6.2) seems to be a challenge for LVS, if indeedV ∼1015. ForV this large, ∆Kgs would dominate ∆Kα0, since we do not expect the

string couplinggsto be stabilized extremely small. On the other hand, if we are interested

in the effects gs corrections may have on the existence of the large volume minima, the

relevant quantity to look at is thescalar potential V, rather than the K¨ahler potentialK. It turns out that certain cancellations in the expression for the scalar potential leave us with leading correction terms to V that scale as

∆Vα0 ∼ O(g−1/2

s V

−3_{) , ∆V}

gs ∼ O(gsV

−3_{) . (6.3)}

This is already much better news for LVS. However, restoring numerical factors in (6.3), and with gs typically not stabilized extremely small, it would seem that ∆Kgs could still