Finite Cutoff AdS$_{5}$ Holography and the Generalized Gradient Flow

TL;DR

This work analyzes finite-cutoff holography via double-trace deformations of four-dimensional large $N$ holographic CFTs, showing how the deformation flow can be mapped to the bulk ADM Hamiltonian of $AdS_{5}$. By matching boundary operator expectations to canonical momenta and employing a two-derivative ansatz for a counterterm functional $S[g,\phi]$, the authors derive a bulk gravity–scalar theory with a superpotential that enforces $V(\phi)$ and fixes the deformation scale to satisfy $\mu=\kappa$ and $a=c$. The results reveal a generalized gradient-flow structure for the induced boundary gravity, with the gradient-flow equations for $g_{\mu\nu}$ and $\phi$ reproducing the radial evolution via the Hamilton–Jacobi formalism, linking finite-radius holography to gradient-flow regularization. The findings illuminate how bulk diffeomorphism invariance emerges from boundary RG flows, identify a single controlling scale for the double-trace deformations, and point toward extensions to other even dimensions and holographic entropy studies.

Abstract

Recently proposed double trace deformations of large $N$ holographic CFTs in four dimensions define a one parameter family of quantum field theories, which are interpreted in the bulk dual as living on successive finite radius hypersurfaces. The transformation of variables that turns the equation defining the deformation of a four dimensional large $N$ CFT by such operators into the expression for the radial ADM Hamiltonian in the bulk is found. This prescription clarifies the role of various functions of background fields that appear in the flow equation defining the deformed holographic CFT, and also their relationship to the holographic anomaly. The effect of these deformations can also be seen as triggering a generalized gradient flow for the fields of the induced gravity theory obtained from integrating out the fundamental fields of the holographic CFT. The potential for this gradient flow is found to resemble the two derivative effective action previously derived using holographic renormalization.