Bound States in the AdS/CFT Correspondence

TL;DR

The paper demonstrates that boundary terms in the AdS bulk action govern the existence of bound states for a scalar field and that unstable boundary double-trace perturbations correspond to bulk tachyonic modes. By carefully identifying the correct sources for operators of dimension $\Delta_{-}$ and employing a generalized Legendre transform, the authors connect boundary perturbations to bulk spectral properties across minimally and non-minimally coupled cases, including a Gibbons-Hawking term. They derive explicit criteria for bound-state existence for several boundary-term families and show that the Breitenlohner-Freedman bound alone is insufficient to guarantee bulk stability. This work deepens the AdS/CFT dictionary by linking boundary coefficient choices to bulk tachyonic behavior and highlights the essential role of boundary data in holographic stability analyses.

Abstract

We consider a massive scalar field theory in anti-de Sitter space, in both minimally and non-minimally coupled cases. We introduce a relevant double-trace perturbation at the boundary, by carefully identifying the correct source and generating functional for the corresponding conformal operator. We show that such relevant double-trace perturbation introduces changes in the coefficients in the boundary terms of the action, which in turn govern the existence of a bound state in the bulk. For instance, we show that the usual action, containing no additional boundary terms, gives rise to a bound state, which can be avoided only through the addition of a proper boundary term. Another notorious example is that of a conformally coupled scalar field, supplemented by a Gibbons-Hawking term, for which there is no associated bound state. In general, in both minimally and non-minimally coupled cases, we explicitly compute the boundary terms which give rise to a bound state, and which ones do not. In the non-minimally coupled case, and when the action is supplemented by a Gibbons-Hawking term, this also fixes allowed values of the coupling coefficient to the metric. We interpret our results as the fact that the requirement to satisfy the Breitenlohner-Freedman bound does not suffice to prevent tachyonic behavior from existing in the bulk, as it must be supplemented by additional conditions on the coefficients in the boundary terms of the action.