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The holographic Weyl anomaly. (English) Zbl 0958.81083
Summary: 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 $$\text{AdS}_3$$. 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.

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 83E30 String and superstring theories in gravitational theory 81T50 Anomalies in quantum field theory
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