A note on causality in the bulk and stability on the boundary

TL;DR

The paper shows that an unstable double-trace deformation in the boundary CFT corresponds to a bound state in the radial spectrum of a bulk scalar in $AdS_{d+1}$. Using a rigorous Sturm–Liouville analysis of the radial equation, the authors identify a discrete eigenvalue $oldsymbol{\lambda = -c^{1/ u}}$ that appears when the boundary-condition parameter $oldsymbol{c}$ is positive, signaling instability via a pole in the bulk Green function. They map this bulk signal to the boundary theory through the relation between the boundary data $A$ and $c$, demonstrating that the bound state arises for boundary configurations with $oldsymbol{eta/oldsymbol{ ext{α}}}<0$ (the wrong sign for the double-trace perturbation). This work thus extends the holographic dictionary by showing how stability properties of boundary deformations are encoded in bulk spectral data, while clarifying the interplay between boundary conditions, causality, and the Breitenlohner–Freedman bound.

Abstract

By carefully analyzing the radial part of the wave-equation for a scalar field in AdS, we show that for a particular range of boundary conditions on the scalar field, the radial spectrum contains a bound state. Using the AdS/CFT correspondence, we interpret this peculiar phenomenon as being dual to an unstable double trace deformation of the boundary conformal field theory. We thus show how the bulk theory holographically detects whether a boundary perturbation is stable.