Holographic Renormalization Group

TL;DR

The paper develops a self-contained holographic renormalization group framework in asymptotically AdS spacetimes by recasting bulk gravity and scalar dynamics into a Hamilton-Jacobi flow for the on-shell action. Through a derivative expansion, local counterterms are extracted and the Callan-Symanzik equation, Weyl anomalies, and operator dimensions are derived, with explicit checks against AdS/CFT predictions such as the Leigh-Strassler flow and N=4 SYM. The authors extend the formalism to higher-derivative gravity and noncritical string theory, showing that holographic RG persists and yields consistent 1/N corrections to Weyl anomalies, as well as novel multicritical behavior in the boundary theories. The work provides a unified, extendable toolkit for computing RG data from gravity and demonstrates the depth and resilience of the AdS/CFT correspondence beyond the supergravity limit, including connections to noncritical strings and higher-curvature corrections.

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.

Figures (6)

• Figure 1: Superpotential $W(\phi)$. The fixed points are at $(\pm\ln 3,(2/\sqrt{6})\ln 2)$.
• Figure 2: Scalar potential $V(\phi)$. The fixed points $(\pm\ln 3,(2/\sqrt{6})\ln 2)$ are saddle points, so that one direction is relevant and the other irrelevant.
• Figure 3: The evolution of the classical solutions $\bigl(\overline{g}_{ij}(x,\tau),\bar{\phi}^a(x,\tau)\bigr)$ along the radial direction $\tau$. The region I is defined by $\tau\ge \tau_{\rm R}$, and the region II is defined by $\tau_0\le \tau < \tau_{\rm R}$.
• Figure 4: Classical solutions $Q(t)$ for $A>0$.
• Figure 5: Classical solutions $Q(t)$ for $A<0$.