We obtain the vacuum solutions for M-theory compactified on eight-manifolds with non-vanishing four-form flux by analyzing the scalar potential appearing in the threedimensional theory. Many of these vacua are not supersymmetric and yet have a vanishing three-dimensional cosmological constant. We show that in the context of Type IIB compactifications on Calabi-Yau threefolds with fluxes and external brane sources α ′corrections generate a correction to the supergravity potential proportional to the Euler number of the internal manifold which spoils the no-scale structure appearing in the classical potential. This indicates that α ′ -corrections may indeed lead to a stabilization of the radial modulus appearing in these compactifications. April 2002the results of [17] by determining the minima of the potential calculated in [21]. This analysis will also show that the non-supersymmetric vacua found in [17] are classically stable. Our second aim is to investigate the fate of the no-scale structure of the potential if one considers the effect of higher derivative corrections of string theory and M-theory.We confirm the expectation of [5] that the no-scale structure in Type IIB compactifications does not survive in the quantum theory once higher order α ′ -corrections are taken into account. In particular, this implies that breaking supersymmetry via a (0,3)-flux induces a non-vanishing potential. Due to the relationship between Type IIB compactifications with three-form flux and M-theory compactifications with four-form flux [19], [25] a similar result should be valid for the non-supersymmetric fluxes in M-theory [17], which also lead to a vanishing cosmological constant at leading order.This paper is organized as follows. In section 2 we rederive the results of [17] from the scalar potential derived in [21] and show that the non-supersymmetric vacua are classically stable. In section 3 we calculate higher order α ′3 -corrections to the scalar potential computed in [5]. We show that these corrections generate a scalar potential that depends on the Calabi-Yau volume and is proportional to the Euler number of the internal manifold.This spoils the no-scale structure of the classical scalar potential for manifolds with nonvanishing Euler number and suggests that further α ′ -corrections may lead to a stabilization of the radial modulus. Some of the technical details of the computation are relegated to an appendix. (Non)-Supersymmetric Solutions in M-theoryIn this section we derive the non-supersymmetric vacuum solutions with vanishing cosmological constant computed in [17] from the superpotentials found in [19] and [21].We use the notation and conventions of [21]. The Scalar PotentialThe scalar potential of M-theory compactified on a fourfold Y 4 to three dimensions ground fluxes is performed in [22], [23] while the potential for M-theory on G 2 -holonomy manifolds with fluxes has been computed in [24].

We construct electrically charged AdS 5 black hole solutions whose charge, mass and boost-parameters vary slowly with the space-time coordinates. From the perspective of the dual theory, these are equivalent to hydrodynamic configurations with varying chemical potential, temperature and velocity fields. We compute the boundary theory transport coefficients associated with a derivative expansion of the energy momentum tensor and Rcharge current up to second order. In particular, for the current we find a first order transport coefficient associated with the vorticity of the fluid.

We determine one-loop string corrections to Kähler potentials in type IIB orientifold compactifications with either N = 1 or N = 2 supersymmetry, including D-brane moduli, by evaluating string scattering amplitudes.2 One way to do this is to relax momentum conservation [11,12,13,14,15]. We confirm the validity of this prescription in our case in appendix E by considering a 4-point function. In the case of [10], the result was confirmed by a 3-point function of gravitons in [16].3 Notice the twofold use of the term "Kähler" here, because both the compactification manifold and the moduli space are Kähler. 4The moduli dependence of our result for the N = 2 orientifold is in agreement with the one-loop Kähler metric given in [10] for the case of vanishing open string scalars, there inferred from the one-loop correction to the Einstein-Hilbert term. 4 As another check, we explicitly derive the one-loop correction to the prepotential from the corrected Kähler potential. This confirms that our result is consistent with supersymmetry.Let us next summarize the content and organization of this paper. We consider the same three orientifold models as in [20], i.e. the T 4 /Z 2 × T 2 model with N = 2 supersymmetry (in chapter 2), the T 6 /(Z 2 × Z 2 ) model (in chapter 3) and the T 6 /Z ′ 6 model (in chapter 4), both with N = 1 supersymmetry. In section 2.4 we verify for the T 4 /Z 2 × T 2 model that one can straightforwardly reproduce the tree level sigma-model metrics by calculating 2-point functions on the sphere and disk using the Kähler structure adapted vertex operators (doing so, we built on previous work on disk amplitudes, in particular [21,22,23,24,25,26]). We then continue by determining a particular 2-point function at one-loop that allows us to read off the one-loop Kähler potential. We perform several checks on the result. In section 2.6 we verify that the result is consistent with N = 2 supersymmetry by determining the corresponding prepotential. A second check is performed in appendix C, where we calculate five other 2-point functions for the (vector multiplet) scalars of this model and show that the result is consistent with the proposed Kähler potential. Given the fact that the prepotential in N = 2 theories only gets one-loop and non-perturbative corrections, we conclude that our result holds to all orders of perturbation theory in the T 4 /Z 2 × T 2 case. The main result of chapter 2 is given in formula (2.77).We then continue in chapters 3 and 4 to generalize this result to the N = 1 cases of the T 6 /(Z 2 × Z 2 ) and T 6 /Z ′ 6 models. The main results in these chapters can be found in formulas (3.30) and (4.3), respectively.In section 5 we draw some conclusions and, in particular, translate our results to the T-dual picture with D3-and D7-branes.Finally, we relegated some of the technical details to a series of appendices. 4 The case with D9-brane Wilson line moduli was also considered in [19]. Their integral representation for the Kähler metric (given in (2.4) of that paper) is less straightforward to ...

The recent progress in embedding inflation in string theory has made it clear that the problem of moduli stabilization cannot be ignored in this context. In many models a special role is played by the volume modulus, which is modified in the presence of mobile branes. The challenge is to stabilize this modified volume while keeping the inflaton mass small compared to the Hubble parameter. It is then crucial to know not only how the volume modulus is modified, but also to have control over the dependence of the potential on the inflaton field. We address these questions within a simple setting: toroidal N = 1 type IIB orientifolds. We calculate corrections to the superpotential and show how the holomorphic dependence on the properly modified volume modulus arises. The potential then explicitly involves the inflaton, leaving room for lowering the inflaton mass through moderate fine-tuning of flux quantum numbers. 7 We would like to emphasize that our results are important also for any other effective theory in which closed-string and open-string moduli appear simultaneously. Another example, D3/D7-brane inflation, will be mentioned later on. 8 The effective action of these orientifold models has for instance been discussed in [27,28,29,30,31]. Further generalizations with non-abelian gauge groups and chiral matter have also been proposed in [32]. 9 See [33] for explicit examples of this stabilization of complex structure moduli in Calabi-Yau compactifications with fluxes.

We subject the phenomenologically successful large volume scenario of hep-th/0502058 to a first consistency check in string theory. In particular, we consider whether the expansion of the string effective action is consistent in the presence of D-branes and O-planes. Due to the no-scale structure at tree-level, the scenario is surprisingly robust. We compute the modification of soft supersymmetry breaking terms, and find only subleading corrections. We also comment that for large-volume limits of toroidal orientifolds and fibered Calabi-Yau manifolds the corrections can be more important, and we discuss further checks that need to be performed.