The Wasteland of Random Supergravities

TL;DR

This paper analyzes the stability of de Sitter critical points in large N supergravity models using random matrix theory, revealing that metastable vacua are exponentially rare due to eigenvalue repulsion.

Contribution

It introduces a random matrix model for the Hessian in large N supergravity, analytically computes eigenvalue spectra, and quantifies the probability of metastability.

Findings

A significant fraction of eigenvalues are negative in typical configurations.

The probability of all eigenvalues being positive decreases exponentially with N.

Metastable vacua are exponentially suppressed, but still numerous.

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.