Advances in High Energy Physics

Volume 2016 (2016), Article ID 4303752, 17 pages

http://dx.doi.org/10.1155/2016/4303752

## Aspects of Moduli Stabilization in Type IIB String Theory

Center for Fundamental Physics, Zewail City of Science and Technology, Giza 12588, Egypt

Received 1 October 2015; Revised 28 December 2015; Accepted 30 December 2015

Academic Editor: Elias C. Vagenas

Copyright © 2016 Shaaban Khalil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We review moduli stabilization in type IIB string theory compactification with fluxes. We focus on KKLT and Large Volume Scenario (LVS). We show that the predicted soft SUSY breaking terms in KKLT model are not phenomenological viable. In LVS, the following result for scalar mass, gaugino mass, and trilinear term is obtained: , which may account for Higgs mass limit if TeV. However, in this case, the relic abundance of the lightest neutralino cannot be consistent with the measured limits. We also study the cosmological consequences of moduli stabilization in both models. In particular, the associated inflation models such as racetrack inflation and Kähler inflation are analyzed. Finally, the problem of moduli destabilization and the effect of string moduli backreaction on the inflation models are discussed.

#### 1. Introduction

Ever since the invention of the Kaluza-Klein mechanism, it was realized that extradimensional models are plagued with massless scalar fields when compactified to 4D. In the original Kaluza-Klein construction, the radius of the -circle, is not fixed by the dynamics and appears as a scalar field with no potential in the effective 4D theory. This is a generic feature of most compactifications of higher-dimensional gravitational/Yang-Mills theories. The parameters which describe the shape and size of the compactification manifold give rise to massless scalar* moduli* with no potential at tree level in 4D (flat directions), that is, nonstabilized moduli fields. These moduli are gravitationally coupled and as such their existence would be in conflict with experiment (see [1] for a review).

String theory being a candidate for a unified theory of the forces of nature must be able to reproduce the physics of our real world which is 4-dimensional with a small positive cosmological constant and chiral gauge interactions. This amounts to finding de Sitter vacua of string theory with all moduli stabilized. The seminal work in [2] constructed the first models of Grand Unified Theories (GUTs) from Calabi-Yau compactification of the heterotic string. In these models, the 10D gauge group ( or ) is broken to Standard-Model-like gauge groups by turning on background gauge fields on the internal space. The chiral fermions are obtained from the dimensional reduction of the 10D gaugino and the number of the generations is half the Euler characteristic of the internal space [2]. These compactifications are purely geometric and give rise to an supersymmetric (SUSY) 4D theories with large number of moduli. Other compactifications which leads to SUSY in 4D are type II theories on Calabi-Yau orientifolds with D-branes and fluxes, M-theory on manifolds with holonomy, and F-theory on Calabi-Yau fourfold (see [3] for a review and references therein).

One essential fact about our universe is that it has a small positive cosmological constant, that is, a de Sitter space-time. The moduli scalar potential should be such that its minimum produces the observed value of the cosmological constant. This turns out to be a very difficult problem since de Sitter vacua are known to break supersymmetry. The first attempt at finding realistic vacua of string theory was mainly concerned with Minkowski and Anti-de Sitter vacua [2, 4–6]. It was shown that turning on magnetic fluxes in the internal manifold leads to a nontrivial warp factor and a non-Kähler geometry. The situation improved drastically since the introduction of D-branes as nonperturbative objects in string theory. This resulted in the celebrated KKLT scenario in which a moduli fixing mechanism was introduced [7]. The Large Volume Scenario followed after [8]. Therein, one can turn on a vacuum expectation value for the fluxes in the internal space without breaking the 4D Lorentz invariance.

In this paper, we review the flux compactifications and moduli stabilization in type IIB string theory. The Kachru-Kallosh-Linde-Trivedi (KKLT) [7] scenario was a major step in constructing dS vacua with all the moduli stabilized including the volume modulus. The Large Volume Scenario (LVS) by Quevedo et al. [9, 10] was proposed as an alternative to stabilize all the moduli with the volume moduli stabilized at extremely large volume. Realistic models of moduli stabilization must come as close as possible to the observed phenomenology at low energy and also to account for cosmological inflation at high energy scales. From this point of view, we analyze the low energy phenomenology of both KKLT and LVS. We emphasize that these models provide specific set of soft SUSY breaking terms, which are not phenomenologically viable. In addition, we study the impact of these scenarios on inflation. It turns out that inflation could destabilize the moduli again. This problem has been analyzed in details in [11, 12]. On the other hand, moduli stabilization may have backreaction effects on the inflationary potential, which could change the inflationary parameters.

This paper is organized as follows. In Section 2, we review the flux compactification and moduli stabilization in type IIB string theory. Section 3 is devoted to KKLT model and its variants, where we discussed several examples for vacuum uplifting. In Section 4, we present the LVS as an alternative scenario that overcomes some of the KKLT drawbacks. The phenomenological implications and SUSY breaking soft terms of these two models are studied in Section 5. In Section 6, we highlight the cosmological implications of these two models. Finally, we state our conclusions in Section 7.

#### 2. Flux Compactifications and Moduli Stabilization in String Theory

We start by reviewing the heterotic string compactification which has been considered to connect string theory to four-dimensional physics [2]. Then, we discuss type IIB flux compactifications, where the complex structure moduli (CSM) and the dilaton are stabilized by the RR and NS-NS 3-form fluxes.

##### 2.1. Heterotic String Compactification

To compactify string theory down to 4D, one looks for vacuum solutions of the form , where is assumed to have 4D Poincaré invariance and (or simply ) is a compact internal 6D Euclidean space. The most general metric compatible with these requirements can be written as [2] where are the coordinates on with the metric and the requirement of Poincaré symmetry of still allows for a warp factor which depends on only.

We also require an supersymmetry in 4D since this is phenomenologically appealing and gives more analytic control. For , one considers homogenous and isotropic maximally symmetric solutions which implies that the Riemann tensor takes the formwhere is fixed by contracting both sides with and turns out to be equal to . The constant scalar curvature could be (Minkowski), (AdS), or (dS).

When the radius of curvature of is large compared to the Planck scale, one can use the supergravity approximation of string theory. In order to have a supersymmetric background, the supergravity transformations of the fermions must vanish [2]For heterotic string theory, the supergravity variations of the fermions are given by [2]where , and are the gravitino, dilatino, and the gaugino, respectively. The -flux is and ; and is Yang-Mills field strength of the or gauge fields in 10D.

The Bianchi identity of is given bySince is exact, that is, zero in cohomology, then the cohomology classes of and are the same.

The vanishing of the above variations for a given spinor will put some restrictions on the background fields and in particular on the geometry and topology of . The compactification of heterotic string theory with vanishing -flux (or vanishing torsion) was first studied in [2, 5] and it leads to the following conditions on the string background:That is, the external space is Minkowski with a constant warp factor.

The dilatino variation givesThat is, the dilaton is constant over .

The gravitino variation givesThis equation says that admits a covariantly constant spinor. The integrability condition resulting from the above equation implies that is Ricci flatHence, the first Chern class of vanishesIt was conjectured by Calabi and proved by Yau that Ricci-flat compact Kähler manifolds with admit a metric with holonomy. These metrics come in families and are parameterized by continuous parameters which defines the shape and sizes of . The parameters appear as scalar fields (moduli) in 4D with no potential and a major goal in string theory is to generate a potential which stabilizes these moduli in a way that is consistent with observations.

One can describe the 4D models resulting from the heterotic string compactification in terms of an effective SUSY theory. This theory is characterized by a Kähler potential , a gauge kinetic function , and a superpotential . The tree-level superpotential does not fix the moduli. Due to nonrenormalization theorems, is not renormalized at any order in perturbation theory [14, 15]. This means that if supersymmetry is unbroken at tree level, it will remain unbroken at all orders of perturbation theory. Nonperturbative effects such as gaugino condensation [16] can correct the superpotential and fix some of the moduli.

It is worth mentioning that there is an alternative way to stabilize the moduli if the string model is nonsupersymmetric. In this case, perturbative corrections can generate a one-loop potential for the moduli, for example, a vacuum energy. For example, in the nonsupersymmetric tachyon-free heterotic string models studied in [17–19], the one-loop vacuum energy was shown to be finite and extremized at the symmetry-enhancement points in the moduli space. The one-loop potential can stabilize some of the moduli fields. Another way to generate masses for the moduli is by the breaking of supersymmetry through a Scherk-Schwartz mechanism [20].

##### 2.2. Type IIB Compactification

We now turn our discussion to type IIB string theory. The massless bosonic spectrum of type IIB consists of the metric , RR 0-form , and scalar dilaton which are combined into the axiodilaton , where the string coupling is given by . In addition, the spectrum contains RR 2-form and 4-form , as well as the NS 2-form . It is convenient to combine the RR and NS 3-forms and into . The classical action of type IIB supergravity is divided into a bulk action , the Chern-Simons action , and contributions from the D-brane sources [21] (more precisely, represents the action of the localized sources for the case of a D-brane wrapping a ()-cycle )In the string frame, , , and are given bywhere and are, respectively, the tension and charge of the -brane. The string tension is expressed in terms of string length asThe 5-form is defined aswhich is self dual and satisfies Bianchi identity

One would like to consider warped compactifications of type IIB on a compact manifold . The metric ansatz for a 4D warped compactification is given by [22]The 10D Einstein equation of motion iswhereis the total stress tensor of supergravity plus the localized objects; that is

In this regard, the space-time components of the latter action reduce towhere is a function on the compact space. Integrating both sides of this equation over the compact manifold , the left-hand side gives zero since it is a total derivative. If there are no localized sources, then the right-hand side is a sum of positive terms and vanishes only if and are constants and . This is the familiar no-go theorem of flux compactifications [23, 24]. However, the existence of localized sources in string theory like orientifold planes can balance the contribution coming from fluxes to give a nontrivial warp factor. This was realized in string theory in [22]. The setup in [22] allows for a stabilization of complex structure moduli by turning on RR and NS fluxes in the internal space [25, 26].

##### 2.3. Type IIB Fluxes and Moduli Stabilization

One way to see the problem in a simple setting is nicely reviewed in [25, 27] in a toy model and we review it here. One can generate a potential for the moduli by turning on fluxes in the internal space. The potential in 4D results from the Maxwell term of the fluxeswhere is the metric of the internal space. The metric will depend on the moduli of the internal space and, after doing the integral on , one gets a potential for the moduli . For example, consider a 6D Maxwell-Einstein theory compactified on a two-sphere with a nonzero flux of piercing This flux contributes a positive energy to the effective 4D potential which can then balance the negative contribution coming from the curvature of . More specifically, the contribution to the effective potential coming from the flux originates from the Maxwell term in 6Dwhere the determinant of the metric contributes a factor of and two metric contractions contribute a factor of while the transformation to the Einstein frame gives a factor . Therefore, total 4D potential takes the formwhich is minimized at , and if is large the curvature is small and the supergravity approximation is reliable [25, 27]. In string theory, additional ingredients beside the fluxes are needed to construct stable vacua. These ingredients are the D-branes and orientifold planes [28].

The main idea of flux compactification is that there are solutions of the string tree-level equations in which some of the -form fields are nonzero in the vacuum. In these constructions, one needs to make sure that the backreaction of the flux on the geometry does not take us outside the supergravity approximation. This turns out to be possible [5] with the introduction of a warp factor varying over the internal manifold and hence the new geometry is conformal to the nonflux case. The fluxes, which can be turned on, are the RR fluxes of type II and the flux. In this case, the quantization condition on the fluxes iswhere the integrality of the cohomology classes of is due to Dirac’s charge quantization. The nonvanishing of the cohomology classes of these -form fields leads to obstructions which lifts some of the flat directions of the compactification; that is, it leads to potential which freezes some of the moduli.

In the presence of sources, the modified Bianchi identity now reads [28]Integrating this equation over the compact internal manifold, one gets the tadpole cancelation condition [28]where

The type IIB string theory will be compactified on Calabi-Yau orientifolds in order to obtain a 4D model. It turned out that one needs to make an orientifold projection in order to have supersymmetric compactification [28]. The orientifold action projects out one of the two gravitinos and breaks the SUSY down to . The orientifold projection also introduces O-planes with a background charge and a negative energy density which balances the contribution of D-branes and leads to stable compactification. The RR and NS-NS 3-form fluxes are restricted via the integral cohomology which determines the quantized background fluxes as follows:where are 3 cycles of the Calabi-Yau manifold. In this case, tadpole condition (27) readswhere is the net number of () branes filling the noncompact dimensions and is the Euler characteristic of the elliptically fibred Calabi-Yau fourfold .

The compactification to 4D on a Calabi-Yau manifold with orientifold planes will give rise to an supergravity theory which is characterized by a Kähler potential , a superpotential , and a gauge kinetic function [29–31]. The tree-level Kähler potential is given by the Weil-Petersson metric using the Kaluza-Klein reduction of type IIB supergravityHere, represents the volume modulus and is one of the Kähler moduli. The conditions on the fluxes in type IIB areSince the hodge depends on the metric, then one expects the above conditions can fix the geometric moduli except for the overall scale of the metric since the hodge is conformally invariant in six dimensions. This leaves the overall volume of the compactification manifold undetermined. These conditions can be derived from a superpotential given by the Gukov-Vafa-Witten (GVW) form [32]This superpotential depends on the complex structure moduli through and is independent of the Kähler moduli.

The supergravity scalar potential is given bywhere run over all the moduli. Due to the no-scale structure of the Kähler potential (31), the sum over Kähler moduli cancels the term and the potential (34) reduces to the no-scale structure [33–35]where run over the dilaton and complex structure moduli. Accordingly, the dilaton and complex structure moduli can be stabilized in a supersymmetric minimum by solving the equation which may have a solution for generic choice of the flux. In this case, at the minimum.

The above discussion shows that the no-scale structure does not fix the value of the volume modulus ; that is, the modulus is a flat direction. The stabilization of is of uttermost importance in order for string theory to make contact with realistic models. This issue will be addressed in the upcoming sections.

#### 3. KKLT and Its Variants

In order to stabilize the volume modulus , a nonperturbative superpotential was considered by Kachru, Kallosh, Linde, and Trivedi (KKLT) [7]. The source of these nonperturbative terms could be either D3-brane instantons or gaugino condensation from the nonabelian gauge sector on the D7-branes. As advocated in the previous section, the dilaton and the complex structure moduli are stabilized at a high scale by the flux induced superpotential and hence their contribution to the superpotential is a constant . Thus, the total effective superpotential is given byThe coefficient or is the correction arising from instantons or gaugino condensation, and is constant of order . In addition, the Kähler potential is given byHere, , where is the volume modulus of the internal manifold and is the axionic part. A supersymmetric minimum is obtained by solving the equationwhere is the value of that minimizes the scalar potential.

Substituting this solution in the potentialone finds the following negative minimum:The scalar potential as a function of is given by (the imaginary part is frozen at zero)

It is important to uplift this AdS minimum to a Minkowski or a dS minimum, as shown in Figure 1, in order to have realistic models. The uplift of the above AdS vacuum to a dS one will break SUSY where one needs another contribution to the potential which usually has dependence like [36]In this case, a new minimum is obtained due to shifting to , where is given bySince the consistency of the KKLT requires that [7], the shift in the minima is much small and we can calculate physical quantities such as masses in terms of .