`Conformal Theories - AdS Branes' Transform, or One More Face of AdS/CFT

TL;DR

This paper explores the classical AdS/CFT transformation relating conformal field theories in Minkowski space to scalar branes in AdS space, providing geometric insights and implications for conformal mechanics.

Contribution

It elucidates the origin and implications of the AdS/CFT coordinate map, including a geometric interpretation in extended conformal space, and discusses its effects in conformal and superconformal models.

Findings

The AdS/CFT transform maps field variables to brane coordinates.

It provides a geometric interpretation involving extended conformal space.

Implications for conformal mechanics models are discussed.

Abstract

The AdS/CFT transformation relates two nonlinear realizations of (super)conformal groups: their realization in the appropriate field theories in Minkowski space with a Goldstone dilaton field and their realization as (super)isometry groups of AdS (super)spaces. It exists already at the classical level and maps the field variables and space-time coordinates of the given (super)conformal field theory in $d$-dimensional Minkowski space ${\cal M}_d$ on the variables of a scalar codimension one (super)brane in AdS$_{d+1}$ in a static gauge, the dilaton being mapped on the transverse AdS brane coordinate. We explain the origin of this coordinate map and describe some its implications, in particular, in $d=1$ models of conformal and superconformal mechanics. We also give a suggestive geometric interpretation of this AdS/CFT transform in the pure bosonic case in the framework of an extended $2d+1$-dimensional conformal space involving extra coordinates associated with the generators of dilatations and conformal boosts.