The Holographic Weyl anomaly

TL;DR

This work computes the holographic Weyl anomaly for conformal field theories with AdS/CFT duals by a covariant regularization of the bulk gravitational action. The authors derive a general structure for the anomaly, decomposing it into Euler density, conformal invariants, and total derivatives, with odd boundary dimensions yielding no anomaly and even dimensions producing explicit expressions. In detail, they verify the $d=2$ result against Brown–Henneaux, reproduce the $d=4$ ${\\cal N}=4$ SYM anomaly, and announce new $d=6$ results for the $(0,2)$ theory showing $N^3$ scaling and vanishing on Ricci-flat backgrounds, hinting at tensionless string dynamics for coincident M5 branes. The findings reinforce the holographic connection between IR bulk divergences and UV boundary anomalies and provide concrete, testable expressions across dimensions. Overall, the paper extends holographic renormalization techniques to higher-dimensional conformal theories and offers new insights into the Weyl anomaly structure of the six-dimensional (0,2) theory.

Abstract

We calculate the Weyl anomaly for conformal field theories that can be described via the adS/CFT correspondence. This entails regularizing the gravitational part of the corresponding supergravity action in a manner consistent with general covariance. Up to a constant, the anomaly only depends on the dimension d of the manifold on which the conformal field theory is defined. We present concrete expressions for the anomaly in the physically relevant cases d = 2, 4 and 6. In d = 2 we find for the central charge c = 3 l/ 2 G_N in agreement with considerations based on the asymptotic symmetry algebra of adS_3. In d = 4 the anomaly agrees precisely with that of the corresponding N = 4 superconformal SU(N) gauge theory. The result in d = 6 provides new information for the (0, 2) theory, since its Weyl anomaly has not been computed previously. The anomaly in this case grows as N^3, where N is the number of coincident M5 branes, and it vanishes for a Ricci-flat background.