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 the boundary field theory. After briefly discussing basic aspects
of the AdS/CFT correspondence, we explain how the notion of the holographic RG
comes out in the AdS/CFT correspondence. We formulate the holographic RG based
on the Hamilton-Jacobi equations for bulk systems of gravity and scalar fields,
as was 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 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 a 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 sigma
model action of noncritical strings. As a check, we investigate the holographic
RG structure of higher-derivative gravity systems, and show that such systems
can also be analyzed based on the Hamilton-Jacobi equations, and that the
behaviour 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 the boundary field theories.Comment: 95 pages, 6 figures. Typos are corrected. References and a discussion
about continuum limit are adde
