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 inthe AdS/CFT correspondence in the large N limit, pointing out and clarifyingsubtleties relating to the formulation of the boundary value problem on aconformal boundary. We then apply this method to study multi-trace deformationsin the presence of a scalar VEV, which requires the coupling to gravity to betaken into account. We show that supergravity solutions subject to `mixed'boundary conditions are in one-to-one correspondence with critical points ofthe holographic effective action of the dual theory in the presence of amulti-trace deformation, and we find a number of new exact analytic solutionsinvolving a minimally or conformally coupled scalar field satisfying `mixed'boundary conditions. These include the generalization to any dimension of theinstanton solution recently found in hep-th/0611315. Finally, we provide asystematic method for computing the holographic effective action in thepresence of a multi-trace deformation in a derivative expansion away from theconformal vacuum using Hamilton-Jacobi theory. Requiring that this effectiveaction exists and is bounded from below reproduces recent results on thestability of the AdS vacuum in the presence of `mixed' boundary conditions.Comment: 41 pages, 1 figure; v2 typos corrected, references added; v3 few more typos corrected and minor comments added. Version to appear in JHE