A Global View on The Search for de-Sitter Vacua in (type IIA) String Theory

TL;DR

The paper addresses whether classically stable de-Sitter vacua can arise in Type IIA string theory and argues that their likelihood is Gaussianly suppressed as the moduli count grows. It uses random-matrix models to study the Hessian at extrema, comparing complete GOE matrices and hierarchically coupled ones, and corroborates the picture with explicit Type IIA constructions showing pervasive tachyonic directions. The main finding is that raising the cosmological constant increases off-diagonal couplings in the Hessian, driving instabilities and making high-CC dS vacua exponentially unlikely in the Type IIA landscape. This work provides a quantitative framework for understanding the scarcity of classical dS vacua in IIA and informs where to focus efforts, including potential checks in Type IIB regions with non-perturbative effects for viable dS vacua.

Abstract

The search for classically stable Type IIA de-Sitter vacua typically starts with an ansatz that gives Anti-de-Sitter supersymmetric vacua and then raises the cosmological constant by modifying the compactification. As one raises the cosmological constant, the couplings typically destabilize the classically stable vacuum, so the probability that this approach will lead to a classically stable de-Sitter vacuum is Gaussianly suppressed. This suggests that classically stable de-Sitter vacua in string theory (at least in the Type IIA region), especially those with relatively high cosmological constants, are very rare. The probability that a typical de-Sitter extremum is classically stable (i.e., tachyon-free) is argued to be Gaussianly suppressed as a function of the number of moduli.

Figures (4)

• Figure 1: The probabilities of positive definite real symmetric random matrices. Each diagonal entry is independently assigned a real number obeying the normal distribution with central value zero and variance one, while off-diagonal entries has variance $1/\sqrt{2}$, i.e., we assume a Gaussian orthogonal ensemble. The fitting function is $a e^{-b N^2 - c N}$ where $a \sim 0.938, b \sim 0.277, c\sim 0.382$.
• Figure 2: The probability estimation for variances $\sigma_A = 10-100$ in the positive definite diagonal matrix $A$ with $N=4-20$. The variance of real-symmetric matrix $B$ is fixed at $\sigma_B = 1$. We use the probability fitting function of the form $P= a\, e^{-b N^2 - c N}$ as before, while each coefficient depends on the relative ratio of variances $y=\sigma_B/\sigma_A$. We found that $a$, $b$, $c$ can be fitted by (\ref{['probability of suppressed off-diagonal comp']}) and $a$ is mostly one between $\sigma_A=10-100$.
• Figure 3: Stable points at positive extrema, given values for some coefficients. The horizontal plane is spanned by values of the determinant and the trace of the matrix. The determinant is reaching zero as we increase the cosmological constant, with a fixed trace.
• Figure 4: Probabilities fits with the different form of probability function ${\cal P} = a e^{-f N^g}$. The exponent $g$ interpolates between 1 and 2 as a function $g= 1.05 + 1.03 e^{-0.0105/y}$.