Stability in Einstein-Scalar Gravity with a Logarithmic Branch

TL;DR

This paper establishes a lower energy bound for Einstein-scalar gravity with logarithmic scalar field behavior at the BF bound, resolving previous issues and exploring implications for AdS/CFT and symmetry breaking.

Contribution

It derives a conserved energy lower bound for theories with logarithmic scalar branches at the BF bound, addressing a key stability challenge in designer gravity.

Findings

Derived a lower bound on conserved energy for logarithmic scalar fields.

Resolved previous divergence issues in spinor charge calculations.

Explored implications for AdS/CFT and spontaneous symmetry breaking.

Abstract

We investigate the non-perturbative stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass saturating the Breitenlohner-Freedman bound. Such "designer gravity" theories admit a large class of boundary conditions at asymptotic infinity. At this mass, the asymptotic behavior of the scalar field develops a logarithmic branch, and previous attempts at proving a minimum energy theorem failed due to a large radius divergence in the spinor charge. In this paper, we finally resolve this issue and derive a lower bound on the conserved energy. Just as for masses slightly above the BF bound, a given scalar potential can admit two possible branches of the corresponding superpotential, one analytic and one non-analytic. The key point again is that existence of the non-analytic branch is necessary for the energy bound to hold. We discuss several AdS/CFT applications of this result, including the use of double-trace deformations to induce spontaneous symmetry breaking.