Supersymmetric vacua in random supergravity

Abstract

We determine the spectrum of scalar masses in a supersymmetric vacuum of a general N = 1 supergravity theory, with the Kähler 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 2 susy ), 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 | m susy /N , tachyonic instabilities are ubiquitous in configurations obtained by uplifting supersymmetric vacua.

Inflation and String Theory

“…An analysis of uplifting of supersymmetric AdS vacua of type IIA string theory, leading to the same conclusion, appears in [474]. 44 Unbroken supersymmetry in AdS does not guarantee the absence of tachyons allowed by the Breitenlohner-Freedman bound [475], but the techniques described above can be used to compute the probability that there are no tachyons [476]. See also [477].…”

“…largest and smallest) eigenvalues of a random matrix. For a large class of random matrix ensembles, including the W W W ensemble [469], one finds [470][471][472] 158) where the function Ψ(ζ), which depends on the particular ensemble and is computable in simple cases, is N -independent at leading order in large N . In summary, eigenvalue repulsion dictates that the probability that a Hessian matrix in the ensemble defined by (3.149) has only positive eigenvalues is given by…”

Supersymmetry breaking and the cosmological constant

“…If SUSY breaking is small in Planck units, then this fraction increases to e −c ′ N 1.3 . Even more interesting is the paper [27] which shows that the probability of obtaining a meta-stable dS minimum by "uplifting" a supersymmetric AdS point is small unless…”

“…The O' Raifeartaigh fields have to include standard model singlets, and might include the Higgs fields 27 . Other candidates among the standard model superfields are excluded by symmetries (see previous footnote).…”

“…vanishes, so that the super-partners of the Goldstino have to appear in the low energy Lagrangian. 27 One cannot include lepton superfields at the renormalizable level. As a consequence of the R symmetry, the model is lepton flavor conserving at the renormalizable level when Λ = 0, and the diagrams that violate R symmetry involve gravitino exchange with the horizon and conserve lepton flavor.…”