$SO(1, d + 1)$ symmetry of the Exact RG equation

TL;DR

The paper proves that the Polchinski ERG equation for a boundary $d$-dimensional Euclidean field theory exhibits an $SO(1,d+1)$ symmetry for any UV cutoff, with a cutoff-dependent realization of the generators; a special cutoff yields the standard AdS isometry. Through both a functional (S[φ]) and a Hamiltonian formulation, the authors derive scale and conformal transformations, identify Noether charges, and demonstrate that the full Wilson action inherits the same symmetry via a field redefinition to a diffusion-type equation. They construct the explicit realization of the $SO(1,d+1)$ algebra in momentum space, show how the generators map onto bulk AdS isometries, and extend the symmetry to ERG for the full action. The results strengthen the holographic RG perspective by showing the symmetry persists beyond Polchinski’s equation and clarifying the role of the cutoff in shaping the representation. Overall, the work provides a robust, cutoff-agnostic account of conformal symmetry in ERG and its bulk AdS interpretation, with implications for holographic reconstructions and beyond.

Abstract

There is a method for constructing from first principles, a holographic bulk dual action in Euclidean $AdS_{d+1}$ space for a $d$-dimensional Euclidean CFT on the boundary, starting from the Polchinski's Exact Renormalization Group (ERG) equation that describes the RG evolution of the interaction part of the boundary Wilson action. The bulk action in $AdS_{d+1}$ has an $SO(1,d+1)$ symmetry and is obtained from the evolution operator of the Polchinski's ERG equation by a map that involves a field redefinition and requires a $\textit{special}$ form of the UV cutoff function in the ERG equation. In this paper, we show that for $\textit{any form}$ of the cutoff function, the ERG evolution operator has an $SO(1,d+1)$ symmetry. The generators of the special conformal transformation depend on the cutoff function. For the special cutoff function that maps to $AdS$ space, the transformations have the standard form of $AdS$ isometry. We also show that the ERG evolution operator for the $\textit{full}$ Wilson action can be put in the same form as the Polchinski's ERG equation by a field redefinition and consequently also has an $SO(1,d+1)$ symmetry for any cutoff function.