A Note on the Weyl Anomaly in the Holographic Renormalization Group

TL;DR

The work develops a dimension-agnostic prescription for computing the holographic Weyl anomaly using the Hamilton-Jacobi flow equation in holographic RG. By decomposing the on-shell action into a local counterterm part and a nonlocal remainder and solving weight-by-weight, it yields the Weyl anomaly in arbitrary $d$ and reproduces known $d=4$ and $d=6$ results. It also analyzes continuum limits, showing that holographic RG tracks the renormalized trajectory by tuning bare sources along bulk classical paths, and relates the formalism to Henningson–Skenderis through the divergent structure. Overall, the work provides a practical, renormalization-group–aware framework for holographic anomalies with clear connections to established analyses.

Abstract

We give a prescription for calculating the holographic Weyl anomaly in arbitrary dimension within the framework based on the Hamilton-Jacobi equation proposed by de Boer, Verlinde and Verlinde. A few sample calculations are made and shown to reproduce the results that are obtained to this time with a different method. We further discuss continuum limits, and argue that the holographic renormalization group may describe the renormalized trajectory in the parameter space. We also clarify the relationship of the present formalism to the analysis carried out by Henningson and Skenderis.

Figures (2)

• Figure 1: The evolution of the classical solutions $\bar{\phi}^i$ along the radial direction. The region I is defined by $r\ge r_R$, and the region II is defined by $r_0\le r < r_R$.
• Figure 2: The classical solution $\overline{G}(x,\rho)$ with $\rho=\exp(2r)$ is chosen such that it passes through the point $\widehat{g}(x)/\epsilon$ at $\rho=\epsilon$ ( i.e., $r=r_0$).