# A Holographic form for Wilson's RG

## TL;DR

This paper formulates Wilson's renormalization group in a holographic framework by mapping the RG flow of a scalar field to a scalar field in AdS space, emphasizing the role of cutoff functions and invariance.

## Contribution

It provides a precise holographic representation of Wilson's RG for scalar fields, connecting RG flow with AdS geometry under specific cutoff conditions.

## Key findings

- Mapped Wilson's RG to scalar fields in AdS space
- Identified conditions for the cutoff function in the holographic mapping
- Discussed scale and conformal invariance with finite UV cutoff

## Abstract

An attempt is made to make precise the connection between Wilson's RG and "Holographic RG" by writing Wilson's RG in a holographic form. A functional formulation is given for the exact RG evolution of a scalar field in $d$ (flat) dimensions. It is shown that a change of variables maps the action to that for a scalar field in $AdS_{d+1}$. This provides a holographic form for Wilson's RG that can be called "Holographic RG". This mapping can only be done for a specific form of the cutoff function in the Exact Renormalization Group formalism. The notion of scale and conformal invariance in the presence of a {\em finite} UV cutoff is emphasized. The discussion is primarily about the two-point function and the Gaussian fixed point. Some remarks are made about nontrivial fixed points.