The Wasteland of Random Supergravities

TL;DR

This work shows that in general ${ m N}=1$ supergravity with many scalars, metastable de Sitter vacua are exponentially rare when $W$ and $K$ are treated as random functions. The Hessian at a critical point is modeled as a sum of a Wigner matrix and two Wishart matrices, and its spectrum is obtained via free convolution; typical spectra have many negative eigenvalues, implying tachyonic directions. Using Tracy–Widom fluctuations and large-deviation theory, the authors derive that the probability of all eigenvalues becoming positive scales as $P\propto e^{-cN^{p}}$ with $p\approx1.5$ for generic points and $p\approx1.3$ for approximately-supersymmetric points, meaning stability is exponentially suppressed but not impossible. The results hold broadly due to universality in random-matrix theory, with decoupling of sectors or special constructions (e.g., KKLT-like setups) potentially yielding larger, though still exponentially suppressed, numbers of metastable vacua. Overall, the paper provides a quantitative, statistically grounded view of the scarcity of metastable de Sitter vacua in large-$N$ supergravity landscapes and implications for string compactifications.

Abstract

We show that in a general \cal{N} = 1 supergravity with N \gg 1 scalar fields, an exponentially small fraction of the de Sitter critical points are metastable vacua. Taking the superpotential and Kahler potential to be random functions, we construct a random matrix model for the Hessian matrix, which is well-approximated by the sum of a Wigner matrix and two Wishart matrices. We compute the eigenvalue spectrum analytically from the free convolution of the constituent spectra and find that in typical configurations, a significant fraction of the eigenvalues are negative. Building on the Tracy-Widom law governing fluctuations of extreme eigenvalues, we determine the probability P of a large fluctuation in which all the eigenvalues become positive. Strong eigenvalue repulsion makes this extremely unlikely: we find P \propto exp(-c N^p), with c, p being constants. For generic critical points we find p \approx 1.5, while for approximately-supersymmetric critical points, p \approx 1.3. Our results have significant implications for the counting of de Sitter vacua in string theory, but the number of vacua remains vast.

Figures (7)

• Figure 1: The eigenvalue spectra for the Wigner ensemble (left panel), and the Wishart ensemble with $N=Q$ (right panel), from $10^3$ trials with $N=200$.
• Figure 2: The eigenvalue spectrum for the C$I$ ensemble, from $10^5$ trials with $N=200$. The full spectrum appears in the left panel, while the right panel shows the details of the cleft at $\lambda=0$. Notice that the boundary of the linear regime occurs for $\lambda \sim \frac{1}{N}$.
• Figure 3: The histogram shows the spectrum of eigenvalues of the full Hessian matrix ${\cal H}$ (\ref{['MasterHessian']}) for $N=200$ and $\omega=.1$, in units of $F^2$, while the curve gives the analytic result (\ref{['MasterSpectrum']}) from the Wigner $\boxplus$ Wishart $\boxplus$ Wishart model, with no adjustable parameters.
• Figure 4: The logarithm of the probability $P(\lambda_{\rm min}>0)$ that the smallest eigenvalue of ${\cal H}$ is positive, as a function of $N$, with $\omega=1$. Upper branch: simulations of the full Hessian matrix ${\cal H}$, with best-fit values $p=1.50 \pm 0.10$, $c=0.29 \pm 0.06$. Lower branch: simulations of the Wigner $\boxplus$ Wishart $\boxplus$ Wishart model, with best-fit values $p=1.90 \pm 0.04$, $c=0.21 \pm 0.02$. The error bars give the 2$\sigma$ statistical uncertainty.
• Figure 5: The ellipses show the 2$\sigma$ allowed regions of the $p-c$ plane, cf. equation (\ref{['eq:prob']}), for $\omega=0.1,0.2,\ldots, 1.0$, from left to right, with $2 \le N \le 23$. As $\omega$ increases (so that for fixed $F$ the cosmological constant decreases), $c$ increases substantially, while $p$ increases slightly.