A tale of two superpotentials: Stability and instability in designer gravity

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_±. 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.