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

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