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

Ioannis Papadimitriou

TL;DR

The paper develops a comprehensive, large-N framework for multi-trace deformations in AdS/CFT, clarifying how such deformations map to boundary conditions on bulk fields and how back-reaction with gravity arises when a scalar VEV is present. It introduces a systematic boundary-value formalism for AlAdS spaces, uses a dilatation-based expansion and Hamilton–Jacobi theory to compute the holographic effective action, and analyzes both minimally and conformally coupled scalars, including exact analytic solutions such as domain walls and instantons. A central result is the one-to-one correspondence between mixed boundary conditions and critical points of the holographic effective action under multi-trace deformations, with explicit constructions of vacua, stability conditions, and non-perturbative decay channels. The work also provides practical tools—the toy model, the Hamilton–Jacobi approach, and the minisuperspace reductions—to derive two-derivative effective actions and exact potentials in curved boundaries, offering insights into vacuum structure and stability in deformed CFTs with holographic duals.

Abstract

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.

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

TL;DR

The paper develops a comprehensive, large-N framework for multi-trace deformations in AdS/CFT, clarifying how such deformations map to boundary conditions on bulk fields and how back-reaction with gravity arises when a scalar VEV is present. It introduces a systematic boundary-value formalism for AlAdS spaces, uses a dilatation-based expansion and Hamilton–Jacobi theory to compute the holographic effective action, and analyzes both minimally and conformally coupled scalars, including exact analytic solutions such as domain walls and instantons. A central result is the one-to-one correspondence between mixed boundary conditions and critical points of the holographic effective action under multi-trace deformations, with explicit constructions of vacua, stability conditions, and non-perturbative decay channels. The work also provides practical tools—the toy model, the Hamilton–Jacobi approach, and the minisuperspace reductions—to derive two-derivative effective actions and exact potentials in curved boundaries, offering insights into vacuum structure and stability in deformed CFTs with holographic duals.

Abstract

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.

Paper Structure

This paper contains 20 sections, 2 theorems, 127 equations, 1 figure, 10 tables.

Table of Contents

  1. Introduction and summary of results
  2. Multi-trace deformations in QFTs with a large N limit
  3. The boundary value problem for the Klein-Gordon operator in AlAdS spaces
  4. The variational problem in the presence of a conformal boundary
  5. Boundary conditions
  6. Solution of the boundary value problem
  7. The on-shell action and the AdS/CFT dictionary
  8. Toy model
  9. General solution with linear boundary conditions
  10. Vacua with non-linear boundary conditions
  11. Effective action from Hamilton's characteristic function
  12. Minimal coupling
  13. Two-derivative effective action for conformal boundary conditions
  14. Minisuperspace approximation
  15. The '2/3' potential
  16. ...and 5 more sections

Key Result

Lemma 6.1

Provided the BF bound holds and $\Delta_->0$, any $W_-$ solution of equation (superpotential) lies on a continuous family of $W_-$ solutions while any $W_+$ solution is isolated, or corresponds to an end point of an one-parameter family of $W_-$ solutions, at an infinite distance in parameter space

Figures (1)

  • Figure 1: Plot of the effective potential (\ref{['exact_effective_potential_k']}), with $f(\phi_-)$ given by (\ref{['conformal_deformation']}), on $S^3$ for $\alpha<\sqrt{\lambda}$ (long dashes), $\alpha=\sqrt{\lambda}$ (short dashes), and $\alpha>\sqrt{\lambda}$.

Theorems & Definitions (2)

  • Lemma 6.1
  • Lemma 6.2