Consistency conditions for holographic duality

We show that if the beta functions of a field theory are given by the gradi ent of a certain potential on the space of couplings, a gravitational backg round in one more dimension can express the renormalization group (RG) flow of the theory. The field theory beta functions and the gradient ow constra int together reconstruct the second-order spacetime equations of motion. Th e RG equation reduces to the conventional gravitational computation of the spacetime quasilocal stress tensor, and a c-theorem holds true as a consequ ence of the Raychaudhuri equation. Conversely, under certain conditions, if the RG evolution of a field theory possesses a monotonic c-function, the f low of couplings can be expressed in terms of a higher-dimensional gravitat ional background.