Gradient Flow and Holography from a Local Wilsonian Cutoff

TL;DR

This paper explores how a local Wilsonian cutoff in a 4d scalar quantum field theory leads to a gradient flow of beta functions and insights into holographic duality, emphasizing Weyl invariance and consistency conditions.

Contribution

It introduces a framework for incorporating a local cutoff into the vacuum partition function, revealing a gradient flow structure and partial holographic constraints in a curved background.

Findings

Local cutoff can be absorbed by metric and coupling rescaling.

Vacuum partition function obeys new consistency conditions.

Partial holographic Hamiltonian constraints are satisfied.

Abstract

We consider the vacuum partition function of a 4d scalar QFT in a curved background as function of bare marginal and relevant couplings. A local UV cutoff $\Lambda(x)$ transforming under Weyl rescalings allows to construct Weyl invariant kinetic terms including Wilsonian cutoff functions. The local cutoff can be absorbed completely by a rescaling of the metric and the bare couplings. The vacuum partition function satisfies consistency conditions which follow from the Abelian nature of local redefinitions of the cutoff, and which differ from Weyl rescalings. These imply a gradient flow for beta functions describing the cutoff dependence of rescaled bare couplings. The consistency conditions allow to satisfy all but one Hamiltonian constraints required for a holographic description of the flow of bare couplings with the cutoff.