Multi-Trace Operators and the Generalized AdS/CFT Prescription

TL;DR

This work extends the AdS/CFT prescription to accommodate multi-trace perturbations by coupling them to generalized boundary conditions and a full Legendre transform, linking bulk quantization to boundary deformations via a canonical-energy framework. It derives a precise consistency relation between boundary couplings $\beta$ and $\tilde{\beta}$, shows how double-trace terms map to Dirichlet/Neumann/mixed boundary conditions, and introduces boundary surface terms that realize these deformations in both minimally and non-minimally coupled scalar theories. A key result is that irregular bulk modes can propagate only under specific coupling constraints, with the Legendre transform interpolating between conformal dimensions $\Delta_{+}$ and $\Delta_{-}$; in the non-minimal case, double-trace perturbations generate a natural extension of the Gibbons-Hawking term. Overall, the paper provides a robust, calculable framework to incorporate multi-trace deformations in holography and clarifies the bulk/boundary data mapping under generalized boundary conditions.

Abstract

We show that multi-trace interactions can be consistently incorporated into an extended AdS/CFT prescription involving the inclusion of generalized boundary conditions and a modified Legendre transform prescription. We find new and consistent results by considering a self-contained formulation which relates the quantization of the bulk theory to the AdS/CFT correspondence and the perturbation at the boundary by double-trace interactions. We show that there exist particular double-trace perturbations for which irregular modes are allowed to propagate as well as the regular ones. We perform a detailed analysis of many different possible situations, for both minimally and non-minimally coupled cases. In all situations, we make use of a new constraint which is found by requiring consistence. In the particular non-minimally coupled case, the natural extension of the Gibbons-Hawking surface term is generated.