AdS/CFT correspondence for n-point functions

TL;DR

The paper develops a non-perturbative AdS/CFT framework for general interacting scalar QFTs on the covering of ${AdS}_{d+1}$, establishing a limiting procedure that yields Luscher–Mack conformal field theories on the asymptotic cone and connecting bulk correlations to boundary CFTs via explicit maps between Wightman and Schwinger functions. It shows how Minkowskian boundary theories arise from horocyclic (Poincaré) foliations of AdS, producing Poincaré invariant QFTs on branes that satisfy the Wightman axioms and exhibit conformal covariance at the boundary. A complete analysis of two-point functions is provided, including the Klein–Gordon case with its BF bound, maximal analyticity, and explicit boundary limits that reproduce the expected $1/[-(x-x')^2]^{\Delta}$ behavior. The work also gives a detailed brane decomposition of AdS fields into a continuum of Minkowski modes, clarifying the spectrum and self-adjoint extensions, and offers a unified, non-perturbative route from bulk AdS dynamics to boundary conformal dynamics.

Abstract

We provide a new general setting for scalar interacting fields on the covering of a d+1-dimensional AdS spacetime. The formalism is used at first to construct a one-paramater family of field theories, each living on a corresponding spacetime submanifold of AdS, which is a cylinder $R\times S_{d-1}$. We then introduce a limiting procedure which directly produces Luescher-Mack CFT's on the covering of the AdS asymptotic cone. Our AdS/CFT correspondence is generally valid for interacting fields, and is illustrated by a complete treatment of two-point functions, the case of Klein-Gordon fields appearing as particularly simple in our context. We also show how the Minkowskian representation of these boundary CFT's can be directly generated by an alternative limiting procedure involving Minkowskian theories in horocyclic sections (nowadays called (d-1)-branes, 3-branes for AdS_5). These theories are restrictions to the brane of the ambient AdS field theory considered. This provides a more general correspondence between the AdS field theory and a Poincare' invariant QFT on the brane, satisfying all the Wightman axioms. The case of two-point functions is again studied in detail from this viewpoint as well as the CFT limit on the boundary.