Dual holography as functional renormalization group

TL;DR

This paper explores the deep connection between the functional renormalization group and dual holography, reformulating RG flow as a path integral and deriving a generalized holographic framework that incorporates RG dynamics.

Contribution

It introduces a novel reformulation of the functional RG as a path integral, establishing a direct link with dual holography and deriving a generalized framework including RG beta functions.

Findings

Reformulation of functional RG as a Fokker-Planck type equation.

Derivation of a Hamilton-Jacobi equation for an effective action.

Establishment of a dual holographic path integral as a solution to the RG equation.

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 $\beta$-functions and reveals that the RG flow of the distribution function is essentially identical to that of the functional RG equation.