Supersymmetric Vacua in Random Supergravity

TL;DR

This paper analyzes the mass spectrum of scalar fields in random supersymmetric vacua of N=1 supergravity, revealing the prevalence of tachyons and their impact on vacuum stability and uplift prospects.

Contribution

It introduces a random matrix model for the Hessian in supergravity and derives exact probabilities for tachyon absence in large N limits.

Findings

Tachyons are generically present in supersymmetric vacua.

The probability of a tachyon-free vacuum is given by an explicit exponential formula.

Tachyonic instabilities are common when the superpotential magnitude exceeds a certain threshold.

Abstract

We determine the spectrum of scalar masses in a supersymmetric vacuum of a general N=1 supergravity theory, with the Kahler potential and superpotential taken to be random functions of N complex scalar fields. We derive a random matrix model for the Hessian matrix and compute the eigenvalue spectrum. Tachyons consistent with the Breitenlohner-Freedman bound are generically present, and although these tachyons cannot destabilize the supersymmetric vacuum, they do influence the likelihood of the existence of an `uplift' to a metastable vacuum with positive cosmological constant. We show that the probability that a supersymmetric AdS vacuum has no tachyons is formally equivalent to the probability of a large fluctuation of the smallest eigenvalue of a certain real Wishart matrix. For normally-distributed matrix entries and any N, this probability is given exactly by P = exp(-2N^2|W|^2/m_{susy}^2), with W denoting the superpotential and m_{susy} the supersymmetric mass scale; for more general distributions of the entries, our result is accurate when N >> 1. We conclude that for |W| \gtrsim m_{susy}/N, tachyonic instabilities are ubiquitous in configurations obtained by uplifting supersymmetric vacua.