Supersymmetric Vacua in Random Supergravity

TL;DR

This work analyzes the scalar mass spectrum in a general N=1 supergravity vacuum with random Kahler potentials and superpotentials across N complex fields, applying random-matrix theory to map the Hessian to Wishart and Altland-Zirnbauer CI ensembles. The spectrum arises from two branches, governed by the ratio $m_{susy}/|W|$, and tachyons allowed by the BF bound are common in AdS vacua, with the no-tachyon probability given by $P = \exp(-2 N^2 |W|^2 / m_{susy}^2)$ for Gaussian entries. By combining MP-law fluctuations with Edelman’s result for the smallest Wishart eigenvalue, the paper derives the full Hessian spectrum and the likelihood of tachyon-free vacua, and discusses how uplifting to de Sitter space is affected, finding metastability is unlikely unless the uplift scale is sufficiently small or the SUSY mass scale is large. The results provide quantitative constraints on uplifting in string-inspired landscapes and highlight the roles of D-terms and uplift models in stabilizing otherwise tachyonic directions, with implications for model-building and the search for metastable de Sitter vacua.

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.

Figures (6)

• Figure 1: The eigenvalue spectrum $\rho_{MP}(\mu)$ for an almost square Wishart matrix with $M=N+1=6$ and $\sigma=1/\sqrt{N}$.
• Figure 2: The eigenvalue spectrum of the Altland-Zirnbauer C$I$ ensemble for $N=100$.
• Figure 3: The mass spectra, in units of $m_{susy}^2$, of generic supersymmetric AdS vacua with $N=100$ complex fields. The purple (darker) regions correspond to the contribution from the positive branch of \ref{['eq:omega2']}, and the blue (lighter) regions correspond to the contribution from the negative branch. The black vertical line on the left side is at the BF bound $m^2_{\text{BF}} =-\tfrac{9}{4} |W|^2$ (not shown for plot \ref{['fig:last']}).
• Figure 4: The mass spectra, in units of $m_{susy}^2$, of supersymmetric AdS vacua with $N=100$ complex fields that have fluctuated to positivity. The purple (darker) regions correspond to the contribution from the positive branch of \ref{['eq:omega2']}, and the blue (lighter) regions correspond to the contribution from the negative branch. Blue curve: analytic result from equation \ref{['eq:fluctuatedH']}. Black line: equilibrium eigenvalue distribution from Metropolis simulation (see Appendix \ref{['theappendix']}).
• Figure 5: The probability of the absence of tachyons in $\mathcal{H}_{tot}=\mathcal{H}+\mathcal{H}_{up}$ for $|\widehat{W}|=c_{1}/N$, versus $c_{1}$ for $N$=10 and $c_{2}=2\sqrt{3}c_{1}, \, \sqrt{3}c_{1},\,c_1,\, 0$ (from left to right).