Holographic Counterterm Actions and Anomalies for Asymptotic AdS and Flat Spaces

Counterterm actions are constructed along the ADM formalism. It is shown that the counterterm action can be intrinsically written in terms of intrinsic boundary geometry. Using the expression of counterterm action, we obtain a general form of the counterterm action available for any $d$-dimensional spherical boundary. In the description, we also derive {\it arbitrary} dimensional holographic conformal anomaly. It is also shown that counterterm actions for AF spaces can be obtained from the AdS description just as taking the limit of $\mathrm{\ell }\to \mathrm{\infty }$. An asymptotically flat spacetime with non-spherical boundary is speculated. In the example, additional counterterms to eliminate (leading) divergent terms due to deviation of boundary from round sphere are imagined by observing (4-dimensional) holographic anomaly proportional to $\mathrm{\square }R$. Argument of the deceptive-like anomaly is given by comparing with the holographic description of 5-dimensional Kerr-AdS spacetime.