  Every renormalization group flow in $d$ spacetime dimensions can be
equivalently described as spectral deformations of a generalized free CFT in
$(d-1)$ spacetime dimensions. This can be achieved by studying the effective
action of the Nambu-Goldstone boson of broken conformal symmetry in anti-de
Sitter space and then taking the flat space limit. This approach is
particularly useful in even spacetime dimension where the change in the Euler
anomaly $ a_{UV}-a_{IR}$ can be related to anomalous dimensions of lowest twist
multi-trace operators in the dual CFT. As an application, we provide a simple
proof of the 4d $a$-theorem using the dual description. Furthermore, we
reinterpret the statement of the $a$-theorem in 6d as a conformal bootstrap
problem in 5d.
