A tale of two superpotentials: Stability and Instability in Designer Gravity

TL;DR

This paper examines the stability of anti-de Sitter gravity with tachyonic scalars in designer gravity theories, revealing that the existence of a specific superpotential branch is crucial for ensuring a lower energy bound.

Contribution

It clarifies the role of two superpotential branches in the energy bounds of designer gravity, resolving previous discrepancies and providing rigorous proofs and numerical confirmation.

Findings

Presence of P_- superpotential branch ensures energy is bounded below.

Numerical solutions confirm the theoretical lower energy bound.

Existence of P_- may be necessary for stability in designer gravity.

Abstract

We investigate the stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound. The boundary conditions in these ``designer gravity'' theories are defined in terms of an arbitrary function W. Previous work had suggested that the energy in designer gravity is bounded below if i) W has a global minimum and ii) the scalar potential admits a superpotential P. More recently, however, certain solutions were found (numerically) to violate the proposed energy bound. We resolve the discrepancy by observing that a given scalar potential can admit two possible branches of the corresponding superpotential, P_{\pm}. When there is a P_- branch, we rigorously prove a lower bound on the energy; the P_+ branch alone is not sufficient. Our numerical investigations i) confirm this picture, ii) confirm other critical aspects of the (complicated) proofs, and iii) suggest that the existence of P_- may in fact be necessary (as well as sufficient) for the energy of a designer gravity theory to be bounded below.