From Fixed Points to the Fifth Dimension

TL;DR

This work presents a detailed blueprint for how a 4D Lorentzian CFT can be dual to a 5D AdS theory, with a large-N planar limit yielding a tree-level AdS bulk description of an infinite tower of primary operators. It shows how 4D locality and conformal symmetry translate into 5D locality in AdS, using a confining IR deformation to connect four-dimensional glueball dynamics to a five-dimensional effective field theory, and derives the AdS bulk fields corresponding to vector and tensor primaries, including gauge fields and gravity. The analysis clarifies the generation of Witten diagrams from CFT correlators, resolves Lorentzian versus Euclidean subtleties, and argues for emergent relativity and gravity under strong coupling and a large scaling-dimension gap. It also discusses the robustness and limitations of emergent spacetime physics, and outlines how EFTs on AdS5 can be UV completed by the dual CFT, with potential implications for holography in broader contexts.

Abstract

4D Lorentzian conformal field theory (CFT) is mapped into 5D anti-de Sitter spacetime (AdS), from the viewpoint of "geometrizing" conformal current algebra. A large-N expansion of the CFT is shown to lead to (infinitely many) weakly coupled AdS particles, in one-to-one correspondence with minimal-color-singlet CFT primary operators. If all but a finite number of "protected" primary operators have very large scaling dimensions, it is shown that there exists a low-AdS-curvature effective field theory regime for the corresponding finite set of AdS particles. Effective 5D gauge theory and General Relativity on AdS are derived in this way from the most robust examples of protected CFT primaries, Noether currents of global symmetries and the energy-momentum tensor. Witten's prescription for computing CFT local operator correlators within the AdS dual is derived. The main new contribution is the derivation of 5D locality of AdS couplings. This is accomplished by studying a confining IR-deformation of the CFT in the large-N "planar" approximation, where the discrete spectrum and existence of an S-matrix allow the constraints of unitarity and crossing symmetry to be solved (in standard fashion) by a tree-level expansion in terms of 4D local "glueball" couplings. When the deformation is carefully removed, this 4D locality (with plausible technical assumptions specifying its precise nature) combines with the restored conformal symmetry to yield 5D AdS locality. The sense in which AdS/CFT duality illustrates the possibility of emergent relativity, and the special role of strong coupling, are briefly discussed. Care is taken to conclude each step with well-defined mathematical expressions and convergent integrals.