New class of de Sitter vacua in string theory compactifications

String theory contains few known working examples of de Sitter vacua, four-dimensional universes with a positive cosmological constant. A notorious obstacle is the stabilization of a large number—sometimes hundreds—of moduli fields that characterize the compact dimensions. We study the stability of a class of supersymmetric moduli (the complex structure moduli and dilaton in type-IIB flux compactifications) in the regime where the volume of the compact space is large but not exponentially large. We show that, if the number of moduli is very large, random matrix theory provides a new stability condition, a lower bound on the volume. We find a new class of stable vacua where the mass spectrum of these supersymmetric moduli is gapped, without requiring a large mass hierarchy between moduli sectors or any fine-tuning of the superpotential. We provide the first explicit example of this class of vacua in the ${\mathbb{P}}_{[1,1,1,6,9]}^4$ model. A distinguishing feature is that all fermions in the supersymmetric sector are lighter than the gravitino.

Figure 1

Scalar mass spectrum, (3), of the complex structure sector at tree level with a large number of fields, $N\to \infty$. The spectrum is always tachyon-free, but when the heaviest fermion is heavier than the gravitino, $m_h>m_{3/2}$ (left), the spectral density diverges as $\rho ({\mu }^2)\sim 1/\mu$ near $\mu =0$. By contrast, if the heaviest fermion is lighter than the gravitino, $m_h<m_{3/2}$ (right), the stability of the configuration is protected by a gap in the mass spectrum of size ${\mu }_{\min }^2=(m_{3/2}-m_h{)}^2$.

Figure 2

Percentage of (real) scalars in the complex structure sector with tree-level masses smaller than the size of the leading quantum corrections, ${\mu }^2\le m_{3/2}^2·\widehat{\xi }/\mathcal{V}$. The horizontal axis represents the typical mass scale in this sector, $m_h$. The spectrum of perturbations of the LSF vacua ($m_h<m_{3/2}$), contains no light scalar modes at tree level. Stability is also ensured if there is a large hierarchy between the masses of the supersymmetric complex structure sector and the supersymmetry breaking scale, $m_h\gg m_{3/2}$ (KKLT regime), or an exponentially large volume, $\widehat{\xi }/\mathcal{V}\sim {10}^{-10}$ (LVS).