No, scale invariant field theories are not always equivalent to field theories living on the conformal boundary of an Anti-de Sitter (AdS) spacetime. While there is a well-known correspondence called the AdS/CFT (Anti-de Sitter/Conformal Field Theory) correspondence, which relates certain conformal field theories to gravitational theories in AdS spacetime, not all scale invariant theories have an AdS dual.
The AdS/CFT correspondence, proposed by Juan Maldacena, establishes a duality between a conformal field theory (CFT) defined on the boundary of an AdS spacetime and a gravitational theory in the bulk of the AdS spacetime. The correspondence states that the physics of the bulk theory is equivalent to the physics of the boundary CFT. This correspondence has been extensively studied and has provided valuable insights into both gravitational and quantum field theories.
However, the AdS/CFT correspondence is limited to specific cases where there is a known duality. It typically applies to conformal field theories with a large number of degrees of freedom and certain properties, such as supersymmetric theories or certain strongly coupled gauge theories. Not all scale invariant field theories have a known AdS dual or can be described by the AdS/CFT correspondence.
In general, scale invariance is a property of quantum field theories that have a well-defined scale transformation symmetry. On the other hand, the AdS/CFT correspondence is a specific duality that relates conformal field theories to gravitational theories in AdS spacetime. While there may be connections between scale invariance and the AdS/CFT correspondence in certain cases, they are not always equivalent or directly related.