Wilsonian renormalisation and the exact cut-off scale from holographic duality

We propose a method for determining the exact correspondence between the Wilsonian cut-off scale on the boundary and its holographically dual bulk theory. We systematically construct the multi-trace Wilsonian effective action from holographic renormalisation and evolve it by integrating out the asymptotically Anti-de Sitter bulk geometry with scalar probes. The Wilsonian nature of the effective action is shown by proving that it must be either double-trace, closing in on itself under successive integrations, or have an infinite series of multi-trace terms. Focusing on composite scalar operator renormalisation, we relate the Callan-Symanzik equation, the flow of the scalar anomalous dimension and the multi-trace beta functions to their dual RG flows in the bulk. Establishing physical renormalisation conditions on the behaviour of the large-$N$ anomalous dimension then enables us to extract the energy scales. Examples of pure AdS, GPPZ flow, black brane in AdS, M2 and M5 branes are studied before we generalise our results to arbitrary numbers of mass and thermal deformations of an ultra-violet CFT. Relations between the undeformed Wilsonian cut-off, deformation scales and the deformed Wilsonian cut-off are discussed, as is phenomenology of each considered background. We see how a mass gap, the emergent infra-red CFT scaling, etc. arise in different effective theories. We also argue that these results can have alternative interpretations through the flow of the conformal anomaly or the Ricci scalar curvature of boundary branes. They show consistency with the c-theorem.