Counterterms, critical gravity and holography

TL;DR

The paper investigates AdS counterterms in odd boundary dimensions and shows there exist choices that render the CFT free energy on $S^d$ and $S^1×S^{d-1}$ cut-off independent, simultaneously yielding a cut-off independent Schwarzschild entropy. When these counterterm actions are treated as independent theories, their linearized gravity content matches that of critical gravity after a field redefinition, and in certain parameter regimes they realize a DBI-like, non-dynamical limit. The authors explicitly construct counterterm actions for $D=4,6,8$ (i.e., $d=3,5,7$) and discuss Weyl corrections, anomalies, and connections to logarithmic CFTs, including vanishing central charges and log-like graviton modes. These results bridge holographic renormalization with higher-derivative gravity theories and offer insights into holographic duals of non-unitary log-CFTs, with potential implications for c-theorems and entropy in AdS/CFT.

Abstract

We consider counterterms for odd dimensional holographic CFTs. These counterterms are derived by demanding cut-off independence of the CFT partition function on $S^d$ and $S^1 \times S^{d-1}$. The same choice of counterterms leads to a cut-off independent Schwarzschild black hole entropy. When treated as independent actions, these counterterm actions resemble critical theories of gravity, i.e., higher curvature gravity theories where the additional massive spin-2 modes become massless. Equivalently, in the context of AdS/CFT, these are theories where at least one of the central charges associated with the trace anomaly vanishes. Connections between these theories and logarithmic CFTs are discussed. For a specific choice of parameters, the theories arising from counterterms are non-dynamical and resemble a DBI generalization of gravity. For even dimensional CFTs, analogous counterterms cancel log-independent cut-off dependence.