Trace Anomaly and Counterterms in Designer Gravity

TL;DR

This paper develops counterterms for Einstein-scalar theories in AdS4 to ensure finite actions, analyzes the trace anomaly in the dual field theory, and applies the formalism to exact hairy black hole solutions related to supergravity.

Contribution

It introduces explicit counterterms for designer gravity boundary conditions in AdS4 and explores their implications for holographic mass, trace anomaly, and dual field theories.

Findings

Holographic mass matches Hamiltonian mass for scalar fields with conformal mass.

Trace anomaly vanishes for AdS-invariant boundary conditions.

Exact hairy black hole solutions correspond to triple trace deformations in dual theories.

Abstract

We construct concrete counterterms of the Balasubramanian-Kraus type for Einstein-scalar theories with designer gravity boundary conditions in AdS$_{4}$, so that the total action is finite on-shell and satisfy a well defined variational principle for an arbitrary scalar field potential. We focus on scalar fields with the conformal mass, $m^{2}=-2l^{-2}$, and show that the holographic mass matches the Hamiltonian mass for any boundary conditions. We compute the trace anomaly of the dual field theory in the generic case, as well as when there exist logarithmic branches of non-linear origin. As expected, the anomaly vanishes for the boundary conditions that are AdS invariant. When the anomaly does not vanish, the dual stress tensor describes a thermal gas with an equation of state related to the boundary conditions of the scalar field. When the anomaly vanishes, we recover the dual theory of a massless thermal gas. As an application of the formalism, we consider a general family of exact hairy black hole solutions that, for some particular values of the parameters in the moduli potential, contains solutions of four-dimensional gauged $\mathcal{N}=8$ supergravity and its $\omega$-deformation. Using the AdS/CFT duality dictionary, they correspond to triple trace deformations of the dual field theory.