On the distribution of stable de Sitter vacua

TL;DR

The paper tackles the problem of locating stable de Sitter vacua in string theory by developing a systematic method to solve the equations of motion at the origin of moduli space, expressed in terms of SUSY-breaking parameters. It analyzes both geometric IIA setups (over-constrained) and non-geometric IIB flux backgrounds, using isotropic STU reductions and a six-parameter solution space to study stability. Its key finding is that isotropic, non-geometric fluxes can yield thin, rare regions where dS critical points are tachyon-free, but fully stable vacua across all moduli remain elusive; Random Matrix Theory expectations do not directly apply to these constrained models. The work highlights the essential role of non-geometric fluxes and provides a framework for assessing the distribution of stable dS vacua in flux parameter space, pointing to substantial fine-tuning and the need for broader parameter exploration in higher-dimensional settings.

Abstract

The possible existence of (meta-) stable de Sitter vacua in string theory is of fundamental importance. So far, there are no fully stable solutions where all effects are under perturbative control. In this paper we investigate the presence of stable de Sitter vacua in type II string theory with non-geometric fluxes. We introduce a systematic method for solving the equations of motion at the origin of moduli space, by expressing the fluxes in terms of the supersymmetry breaking parameters. As a particular example, we revisit the geometric type IIA compactifications, and argue that non-geometric fluxes are necessary to have (isotropically) stable de Sitter solutions. We also analyse a class of type II compactifications with non-geometric fluxes, and study the distribution of (isotropically) stable de Sitter points in the parameter space. We do this through a random scan as well as through a complementary analysis of two-dimensional slices of the parameter space. We find that the (isotropically) stable de Sitter vacua are surprisingly rare, and organise themselves into thin sheets at small values of the cosmological constant.

Figures (8)

• Figure 1: $T^{6} = T^{2}_{1} \times T_{2}^{2} \times T_{3}^{2}$ torus factorisation and the coordinate basis. We will use indices $a,b,c$ for horizontal $\,"-"$ directions $1,\,3,\,5$ and indices $i,j,k$ for vertical $\,"\,|\,"$ directions $2,\,4,\,6$ in the $2$-tori $\,T_{i}\,$ with $\,i=1,2,3$.
• Figure 2: The dS region as well as the tachyon-free region are subsets of positive measure in the $N$-dimensional parameter space of solutions. Generically, they will overlap in a region that may be small but still is expected to be of positive measure.
• Figure 3: Left: the one-parameter branch of dS solutions of geometric type IIA isotropic flux compactifications. The value of the energy $V_{0}$ is given as a function of the scan parameter ${\delta}=\exp(A_{1})$. The curve crosses the Minkowski line at ${\delta}\sim 5.127$. Right: the value of the $\eta$ parameter within the isotropic sector as a function of the scan parameter ${\delta}$ already reveals the presence of tachyons all along the dS branch.
• Figure 4: Comparison between the value of ${\gamma}$ along the dS line and ${\gamma}_{\textrm{critical}}=\frac{n_{\textrm{eff}}}{3}-1$, i.e. the maximal value of ${\gamma}$ for which the averaged sGoldstino mass is not negative.
• Figure 5: Distribution of the fraction of tachyon-free critical points of the isotropic scalar potential with $N=6$ fields as a function of the uplift parameter $\tilde{{\gamma}}\,\equiv\,\frac{|DW|^{2}}{3|W|^{2}}$. Data produced from the analysis of $\,10^{7}$ random points. The zoomed diagram in the upper-right corner shows the presence of a stable dS tail with $1<\tilde{{\gamma}} \lesssim 1.02$. This tail was analysed using a sample bigger by a factor of $\,10$.