AdS/CFT equivalence transformation

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 $(d\ensuremath{-}1)$-brane on ${\mathrm{AdS}}_{(d+1)}$ 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 $(d\ensuremath{-}1+m)$-brane worldvolume action on ${\mathrm{AdS}}_{(d+1)}\ifmmode\times\else\texttimes\fi{}{X}^{m}$ (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 ${\mathrm{AdS}}_{(d+1)}$ background. Some possible implications of this transformation, in particular for the study of the quantum effective action of $\mathcal{N}=4$ super-Yang-Mills theory in the context of AdS/CFT correspondence, are briefly discussed.