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One-loop Test of Free SU(N) Adjoint Model Holography

We consider the holographic duality where the CFT side is given by $SU(N)$ adjoint free scalar field theory. Compared to the vector models, the set of single trace operators is immensely extended so that the corresponding AdS theory also contains infinitely many massive higher spin fields on top of the massless ones. We compute the one-loop vacuum energy of these AdS fields to test this duality at the subleading order in large $N$ expansion. The determination of the bulk vacuum energy requires a proper scheme to sum up the infinitely many contributions. For that, we develop a new method and apply it first to calculate the vacuum energies for the first few `Regge trajectories' in AdS$_4$ and AdS$_5$. In considering the full vacuum energy of AdS theory dual to a matrix model CFT, we find that there exist more than one available prescriptions for the one-loop vacuum energy. Taking a particular prescription, we determine the full vacuum energy of the AdS$_5$ theory, whereas the AdS$_4$ calculation still remains technically prohibitive. This result shows that the full vacuum energy of the AdS$_5$ theory coincides with minus of the free energy of a single scalar field on the boundary. This is analogous to the $O(N)$ vector model case, hence suggests an interpretation of the \emph{positive} shift of the bulk coupling constant, i.e. from $N^2-1$ to $N^2$.