Trace Anomaly and Counterterms in Designer Gravity

Understand

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.