More on counterterms in the gravitational action and anomalies

TL;DR

The work extends holographic renormalization to AdS gravity coupled to matter by deriving explicit boundary counterterms for p-form and scalar fields and clarifying how their logarithmic divergences map to Weyl anomalies in the dual CFT. It confirms agreement between the 4D vector-current–stress-tensor anomaly computed from supergravity and that of the $\mathcal{N}=4$ SU(N) gauge theory, and analyzes the structure of anomalies in higher dimensions, including a 6D case with a vector-current anomaly basis. Through explicit solutions (magnetic strings and branes) and a 7D gauged supergravity truncation, it shows that bulk dynamics can preclude certain boundary currents, thereby shaping which boundary anomalies can be realized. Overall, the paper refines the holographic renormalization program for matter-coupled AdS gravity and links bulk counterterms to dual CFT anomalies across dimensions.

Abstract

The addition of boundary counterterms to the gravitational action of asymptotically anti-de Sitter spacetimes permits us to define the partition function unambiguously without background subtraction. We show that the inclusion of p-form fields in the gravitational action requires the addition of further counterterms which we explicitly identify. We also relate logarithmic divergences in the action dependent on the matter fields to anomalies in the dual conformal field theories. In particular we find that the anomaly predicted for the correlator of the stress energy tensor and two vector currents in four dimensions agrees with that of the ${\cal{N}} = 4$ superconformal SU(N) gauge theory.