AdS/CFT Equivalence Transformation

TL;DR

The paper demonstrates a classical, invertible holographic map between conformal field theories on $d$-dimensional Minkowski space with spontaneously broken conformal symmetry and AdS$_{(d+1)}$-brane worldvolume theories. It shows that a field-dependent change of variables converts Minkowski-space conformal actions (including the dilaton) into AdS-brane actions in static gauge, with the conformal group realized as AdS isometries and extrinsic-curvature couplings arising in the AdS image. The construction uses two nonlinear realizations of the $SO(2,d)$ symmetry and an explicit equivalence relation between the two bases, enabling the translation of invariants and dynamics between the conformal and AdS descriptions. For branes such as D3 on $AdS_5 imes S^5$, the conformal basis leads to nonpolynomial, higher-derivative actions while the AdS basis yields standard brane actions with transverse coordinates interpreted as AdS radial modes. The results offer a new lens on AdS/CFT, suggesting avenues to extend to supersymmetric cases and to study quantum effective actions through this holographic correspondence.

Abstract

We show that any conformal field theory in d-dimensional Minkowski space, in a phase with spontaneously broken conformal symmetry and with the dilaton among its fields, can be rewritten in terms of the static gauge (d-1)-brane on AdS_(d+1) by means of an invertible change of variables. This nonlinear holographic transformation maps the Minkowski space coordinates onto the brane worldvolume ones and the dilaton onto the transverse AdS brane coordinate. One of the consequences of the existence of this map is that any (d-1)-brane worldvolume action on AdS_(d+1)\times X^m (with X^m standing for the sphere S^m or more complicated curved manifold) admits an equivalent description in Minkowski space as a nonlinear and higher-derivative extension of some conventional conformal field theory action, with the conformal group being realized in a standard way. The holographic transformation explicitly relates the standard realization of the conformal group to its field-dependent nonlinear realization as the isometry group of the brane AdS_(d+1) background. Some possible implications of this transformation, in particular, for the study of the quantum effective action of N=4 super Yang-Mills theory in the context of AdS/CFT correspondence, are briefly discussed.