Holographic Renormalization Group

The holographic renormalization group (RG) is reviewed in a self-contained manner. The holographic RG is based on the idea that the radial coordinate of a space-time with asymptotically AdS geometry can be identified with the RG flow parameter of boundary field theory. After briefly discussing basic aspects of the AdS/CFT correspondence, we explain how the concept of the holographic RG emerges from this correspondence. We formulate the holographic RG on the basis of the Hamilton-Jacobi equations for bulk systems of gravity and scalar fields, as introduced by de Boer, Verlinde and Verlinde. We then show that the equations can be solved with a derivative expansion by carefully extracting local counterterms from the generating functional of the boundary field theory. The calculational methods used to obtain the Weyl anomaly and scaling dimensions are presented and applied to the RG flow from the N = 4 SYM to an N = 1 superconformal fixed point discovered by Leigh and Strassler. We further discuss the relation between the holographic RG and the noncritical string theory and show that the structure of the holographic RG should persist beyond the supergravity approximation as a consequence of the renormalizability of the nonlinear σ-model action of noncritical strings. As a check, we investigate the holographic RG structure of higher-derivative gravity systems. We show that such systems can also be analyzed based on the Hamilton-Jacobi equations and that the behavior of bulk fields are determined solely by their boundary values. We also point out that higher-derivative gravity systems give rise to new multicritical points in the parameter space of boundary field theories