Counterterms, critical gravity and holography

TL;DR

This paper derives counterterms for odd-dimensional holographic CFTs ensuring cut-off independence of physical quantities, revealing connections to critical gravity theories and logarithmic CFTs, with implications for black hole entropy and anomalies.

Contribution

It introduces a method to determine counterterms that render holographic CFT partition functions and black hole entropy cut-off independent, linking these to critical gravity and logarithmic CFTs.

Findings

Counterterms ensure cut-off independence of partition functions and entropy.

Connections established between counterterm actions and critical gravity theories.

Special parameter choices lead to non-dynamical, DBI-like gravity theories.

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.