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The bulk locality in the constructive holographic renormalization grouprequires miraculous cancellations among various local renormalization groupfunctions. The cancellation is not only from the properties of the spectrum butfrom more detailed aspects of operator product expansions in relation toconformal anomaly. It is remarkable that one-loop computation of the universallocal renormalization group functions in the weakly coupled limit of the$\mathcal{N}=4$ super Yang-Mills theory fulfils the necessary condition for thecancellation in the strongly coupled limit in its $SL(2,\mathbf{Z})$ dualityinvariant form. From the consistency between the quantum renormalization groupand the holographic renormalization group, we determine some unexplored localrenormalization group functions (e.g. diffusive term in the beta function forthe gauge coupling constant) in the strongly coupled limit of the planar$\mathcal{N}=4$ super Yang-Mills theory.

Local renormalization group functions from quantum renormalization group and holographic bulk locality