Holographic Renormalization Group

TL;DR

The paper reviews the holographic renormalization group (RG) in the context of AdS/CFT, detailing its formulation via Hamilton-Jacobi equations, applications to supersymmetric flows, and extensions to higher-derivative gravity systems.

Contribution

It provides a comprehensive, self-contained formulation of the holographic RG using Hamilton-Jacobi equations and explores its applications and extensions beyond supergravity.

Findings

Derivation of local counterterms from boundary generating functionals.

Application to RG flow from N=4 SYM to N=1 superconformal fixed point.

Analysis of higher-derivative gravity systems and their boundary behavior.

Abstract

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.