Matching renormalization schemes in holography

In holography there is a one-to-one correspondence between physical observables in the bulk and boundary theories. To define physical observables, however, regularisation needs to be implemented in both sides of the correspondence. It is arguable whether the correspondence should extend to regularisation and renormalisation scheme which are not physical in the conventional sense. However, if we are to take the renormalisation group (RG) interpretation of holography seriously, its precise understanding appears to require the matching of regularisations and renormalisation schemes in the bulk and boundary theories. We address this question in the AdS_5/CFT_4 correspondence by considering a simplest physical quantity, the Casimir energy of the N=4 super Yang-Mills (SYM) theory on R x S^3, in a {\zeta}-function regularisation and show that there are choices of scheme which match the bulk AdS result including the radial cutoff dependent corrections when the cutoff is kept finite. We further discuss the implication of this result to the RG interpretation of holography.