Stability in Holographic Theories with Irrelevant Deformations

We investigate the non-perturbative stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass near the Breitenlohner-Freedman bound. Such scalars are characterized by power-law radial decay near the AdS boundary, and typical boundary conditions are "Dirichlet" (fix the slower fall-off mode) or "Neumann" (fix the faster fall-off mode). More generally though, these "designer gravity" theories admit a large class of boundary conditions defined by a functional relation between the two modes. While previous stability proofs have considered boundary conditions that are deformations of the Neumann theory, the goal of this paper is to analyze stability in designer gravity with boundary conditions that are irrelevant deformations of the Dirichlet theory. We obtain a lower bound on the energy using spinor charge methods and show that for the most interesting class of such boundary conditions, the theory is always stable. We argue that the deformed theory flows to a new fixed point in the ultraviolet, which is just the Neumann theory. We also derive a corresponding "effective potential" that implies stability if it has a global minimum.