Consistency Conditions for Holographic Duality

TL;DR

This paper establishes conditions under which the renormalization group flow of a quantum field theory can be represented by a gravitational background, linking beta functions, gradient flows, and holographic duality.

Contribution

It demonstrates that gradient flow conditions on beta functions lead to a gravitational dual description of RG flow, connecting field theory and gravity in a new way.

Findings

Gradient beta functions imply a gravitational dual for RG flow.

The spacetime equations of motion are reconstructed from RG equations.

A c-theorem is derived from the Raychaudhuri equation in this framework.

Abstract

We show that if the beta functions of a field theory are given by the gradient of a certain potential on the space of couplings, a gravitational background in one more dimension can express the renormalization group (RG) flow of the theory. The field theory beta functions and the gradient flow constraint together reconstruct the second order spacetime equations of motion. The RG equation reduces to the conventional gravitational computation of the spacetime quasilocal stress tensor, and a c-theorem holds true as a consequence of the Raychaudhuri equation. Conversely, under certain conditions, if the RG evolution of a field theory possesses a monotonic c-function, the flow of couplings can be expressed in terms of a higher dimensional gravitational background.