Dual holography as functional renormalization group

TL;DR

This work connects functional renormalization group (FRG) methods with a dual holography framework by recasting the FRG as a path-integral for the probability distribution $P_{\Lambda}[]$ governed by a Fokker-Planck–type equation. In the semiclassical limit, the RG flow maps to a Hamilton–Jacobi problem with a Hamiltonian derived from a bulk action, motivating a holographic interpretation where the RG scale is the radial coordinate in $AdS_{d+1}$/CFT$_d$. The authors extend this picture by incorporating RG beta-functions as gradient flows of an effective potential $\mathcal{V}_{eff}$, yielding a generalized bulk action that embeds the RG flow into the gravitational dynamics, and they introduce Weyl-anomaly terms to ensure consistency with holography. A BRST-topological construction underlies the path integral representation, and the framework accommodates nonperturbative renormalization via a gradient-flow structure tied to the Weyl anomaly. Overall, the paper provides a nonperturbative, RG-aware holographic formalism that unifies FRG and holography through a gradient-flow perspective and paves the way for incorporating information-theoretic and quantum-information concepts into holographic RG.

Abstract

We investigate the relationship between the functional renormalization group (RG) and the dual holography framework in the path integral formulation, highlighting how each can be understood as a manifestation of the other. Rather than employing the conventional functional RG formalism, we consider a functional RG equation for the probability distribution function, where the RG flow is governed by a Fokker-Planck-type equation. The central idea is to reformulate the solution of Fokker-Planck type functional RG equation in a path integral representation. Within the semiclassical approximation, this leads to a Hamilton-Jacobi equation for an effective renormalized on-shell action. We then examine our framework for an Einstein-Hilbert action coupled to a scalar field. Applying standard techniques, we derive a corresponding functional RG equation for the distribution function, where the dual holographic path integral serves as its formal solution. By synthesizing these two perspectives, we propose a generalized dual holography framework in which the RG flow is explicitly incorporated into the bulk effective action. This generalization naturally introduces RG $β$-functions and reveals that the RG flow of the distribution function is essentially identical to that of the functional RG equation.