Multi-Trace Deformations in AdS/CFT: Exploring the Vacuum Structure of the Deformed CFT

We present a general and systematic treatment of multi-trace deformations in the AdS/CFT correspondence in the large N limit, pointing out and clarifying subtleties relating to the formulation of the boundary value problem on a conformal boundary. We then apply this method to study multi-trace deformations in the presence of a scalar VEV, which requires the coupling to gravity to be taken into account. We show that supergravity solutions subject to `mixed' boundary conditions are in one-to-one correspondence with critical points of the holographic effective action of the dual theory in the presence of a multi-trace deformation, and we find a number of new exact analytic solutions involving a minimally or conformally coupled scalar field satisfying `mixed' boundary conditions. These include the generalization to any dimension of the instanton solution recently found in hep-th/0611315. Finally, we provide a systematic method for computing the holographic effective action in the presence of a multi-trace deformation in a derivative expansion away from the conformal vacuum using Hamilton-Jacobi theory. Requiring that this effective action exists and is bounded from below reproduces recent results on the stability of the AdS vacuum in the presence of `mixed' boundary conditions.