We demonstrate that flux compactifications of type IIA string theory can classically stabilize all geometric moduli. For a particular orientifold background, we explicitly construct an infinite family of supersymmetric vacua with all moduli stabilized at arbitrarily large volume, weak coupling, and small negative cosmological constant. We obtain these solutions from both ten-dimensional and four-dimensional perspectives. For more general backgrounds, we study the equations for supersymmetric vacua coming from the effective superpotential and show that all geometric moduli can be stabilized by fluxes. We comment on the resulting picture of statistics on the landscape of vacua. * On leave from Steklov Mathematical Institute, Moscow, Russia
We study properties of flux vacua in type IIB string theory in several simple but illustrative models. We initiate the study of the relative frequencies of vacua with vanishing superpotential W = 0 and with certain discrete symmetries. For the models we investigate we also compute the overall rate of growth of the number of vacua as a function of the D3-brane charge associated to the fluxes, and the distribution of vacua on the moduli space. The latter two questions can also be addressed by the statistical theory developed by Ashok, Denef and Douglas, and our results are in good agreement with their predictions. Analysis of the first two questions requires methods which are more number-theoretic in nature. We develop some elementary techniques of this type, which are based on arithmetic properties of the periods of the compactification geometry at the points in moduli space where the flux vacua are located.
The presence of RR and NS three-form fluxes in type IIB string compactification on a Calabi-Yau orientifold gives rise to a nontrivial superpotential W for the dilaton and complex structure moduli. This superpotential is computable in terms of the period integrals of the Calabi-Yau manifold. In this paper, we present explicit examples of both supersymmetric and nonsupersymmetric solutions to the resulting 4d N = 1 supersymmetric no-scale supergravity, including some nonsupersymmetric solutions with relatively small values of W . Our examples arise on orientifolds of the hypersurfaces in ,6 . They serve as explicit illustrations of several of the ingredients which have played a role in the recent proposals for constructing de Sitter vacua of string theory. December 2003 * On leave from Steklov Mathematical Institute, Moscow, Russia 1 giryav@stanford.edu 2 skachru@stanford.edu 3 prasanta@theory.tifr.res.in 4 sandip@tifr.res.inspaces with reduced holonomy. Here, we present some explicit solutions of the IIB flux equations for orientifolds of "generic" Calabi-Yau threefolds, whose holonomy fills out SU (3). This is of more than academic interest: such examples are closely related to some proposals for constructing de Sitter vacua in string theory [18,19], and for more precisely estimating the number of metastable string vacua [20].We will find two surprises in our analysis. First, we will find that supersymmetric solutions of the flux equations do exist. Given an elementary counting argument which we will review below, this is by itself somewhat surprising. Perhaps more importantly, we will find that simple nonsupersymmetric solutions to the flux equations (still at vanishing potential V = 0 in the no-scale approximation, as described in [5]) with small values of W also exist. This is a bit surprising given the small numbers of fluxes we will be turning on.These examples provide support for the assertion in e.g. [18] that by discretely tuning the choice of fluxes in manifolds with large b 3 , one can attain small values of W . 1The organization of this paper is as follows. In §2, we describe the basic facts about the two models (which we call model A and model B) that we will be studying -the threefold geometries, the relevant orientifold actions, and the lift to an F-theory description. We also describe the special (small) subclass of fluxes we will be turning on, and the symmetries of the resulting potential which guarantee that we can consistently solve the equations with many of the CY moduli frozen at a special symmetric locus. This saves us from having to solve the Picard-Fuchs equations for hundreds of independent periods in the two models. In §3, we give a more precise formulation of the problems of interest, and we present details about the period integrals in the two models. In §4, we give examples of supersymmetric solutions in model B. In §5, we give examples of nonsupersymmetric solutions in both models, including some with small W . We close with a discussion in §6.In two appendices, we include more det...
We investigate several predictions about the properties of IIB flux vacua on Calabi-Yau orientifolds, by constructing and characterizing a very large set of vacua in a specific example, an orientifold of the Calabi-Yau hypersurface in W P 4 1,1,1,1,4 . We find support for the prediction of Ashok and Douglas that the density of vacua on moduli space is governed by det(−R − ω) where R and ω are curvature and Kähler forms on the moduli space. The conifold point ψ = 1 on moduli space therefore serves as an attractor, with a significant fraction of the flux vacua contained in a small neighborhood surrounding ψ = 1. We also study the functional dependence of the number of flux vacua on the D3 charge in the fluxes, finding simple power law growth.
We develop a theory of static Bogomol'nyi-Prasad-Sommerfield (BPS) domain walls in stringy landscape and present a large family of BPS walls interpolating between different supersymmetric vacua. Examples include Kachru, Kallosh, Linde, Trivedi models, STU models, type IIB multiple flux vacua, and models with several Minkowski and anti-de Sitter vacua. After the uplifting, some of the vacua become de Sitter (dS), whereas some others remain anti-de Sitter. The near-BPS walls separating these vacua may be seen as bubble walls in the theory of vacuum decay. As an outcome of our investigation of the BPS walls, we found that the decay rate of dS vacua to a collapsing space with a negative vacuum energy can be quite large. The parts of space that experience a decay to a collapsing space, or to a Minkowski vacuum, never return back to dS space. The channels of irreversible vacuum decay serve as sinks for the probability flow. The existence of such sinks is a distinguishing feature of the landscape. We show that it strongly affects the probability distributions in string cosmology.
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