Abstract
We construct supersymmetric AdS4 vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the α′ expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude < 10−123 in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol.
Highlights
Our goal in this work is to find supersymmetric AdS4 vacua of the type proposed by Kachru, Kallosh, Linde, and Trivedi (KKLT) [1]
22Note that as we can neglect Euclidean D3-brane corrections to the Kähler potential, nothing is lost by using the axion shift symmetries Ti → Ti + δTi with δTi ∈ iR to absorb the complex phases in h1,1 Pfaffian numbers, but in some examples we find h1,1 + 1, h1,1 + 2, or h1,1 + 3 leading contributions, and in such cases there are one, two, or three phases remaining, respectively
The triangulation depicted in figure 1 corresponds to a phase in which we have found a solution to the F-flatness conditions for the Kähler moduli
Summary
To understand the effects of the quantization of gravity in cosmology, one can search for cosmological solutions of string theory. A first step is to characterize isolated vacua in well-controlled settings, such as the four-dimensional N = 1 supergravity theories that arise in compactifications of type IIB string theory on Calabi-Yau orientifolds. Our goal in this work is to find supersymmetric AdS4 vacua of the type proposed by Kachru, Kallosh, Linde, and Trivedi (KKLT) [1]. Three conditions are necessary for such vacua to exist. The expectation value of the classical flux superpotential must be exponentially small. The nonperturbative superpotential must contain at least h1,1 independent terms for the h1,1 Kähler moduli. There must exist a point inside the Kähler cone at which the F-terms for all the Kähler moduli vanish, and the α and gs expansions are well-controlled. An important open problem is to determine how widely these requirements are fulfilled in ensembles of flux compactifications on Calabi-Yau orientifolds
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