Holographic Renormalisation and Anomalies

TL;DR

The paper analyzes Weyl anomalies within holographic renormalisation using the Hamilton–Jacobi framework and descent equations near a UV fixed point. It formalises the on-shell action as $S = [S]_0 + [S]_2 + [S]_4 + ⋯ + Γ$ and derives level-zero through level-four descent equations to extract anomaly terms, distinguishing bare from renormalised boundary data via a bulk–boundary propagator. Key contributions include explicit level-two and level-four anomaly structures for general dimensions and a single scalar, the translation to renormalised quantities via a Jacobian, and a discussion of a possible holographic c-function in $d=2$. The work clarifies how holographic RG encodes boundary Weyl anomalies and the role of non-local bulk effects and renormalisation schemes, while highlighting challenges in extending the descent interpretation non-perturbatively along the full RG flow.

Abstract

The Weyl anomaly in the Holographic Renormalisation Group as implemented using Hamilton-Jacobi language is studied in detail. We investigate the breakdown of the descent equations in order to isolate the Weyl anomaly of the dual field theory close to the (UV) fixed point. We use the freedom of adding finite terms to the renormalised effective action in order to bring the anomalies in the expected form. We comment on different ways of describing the bare and renormalised schemes, and on possible interpretations of the descent equations as describing the renormalisation group flow non-perturbatively. We find that under suitable assumptions these relations may lead to a class of c-functions.