Conformal Theories–A{d}S Branes Transform, or One More Face of A{d}S/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 already exists 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 ${\mathcal M}_d$ to the variables of a scalar codimension-one (super)brane in $AdS_{d+1}$ in a static gauge, the dilaton being mapped onto the transverse AdS brane coordinate. We explain the origin of this coordinate mapping 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 transformation in the purely 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.