General Covariance from the Quantum Renormalization Group

TL;DR

This paper demonstrates that applying the quantum renormalization group to boundary field theories naturally leads to a bulk gravitational theory with general covariance, linking boundary consistency conditions to bulk constraint algebra closure.

Contribution

It establishes a connection between boundary Wess-Zumino consistency conditions and bulk general covariance through the QRG framework, revealing how gravitational features emerge from boundary dynamics.

Findings

Bulk covariance arises from boundary consistency conditions.

The scalar and vector constraints form a closed algebra similar to general relativity.

The metric beta function must be of gradient form.

Abstract

The Quantum renormalization group (QRG) is a realisation of holography through a coarse graining prescription that maps the beta functions of a quantum field theory thought to live on the `boundary' of some space to holographic actions in the `bulk' of this space. A consistency condition will be proposed that translates into general covariance of the gravitational theory in the $D + 1$ dimensional bulk. This emerges from the application of the QRG on a planar matrix field theory living on the $D$ dimensional boundary. This will be a particular form of the Wess--Zumino consistency condition that the generating functional of the boundary theory needs to satisfy. In the bulk, this condition forces the Poisson bracket algebra of the scalar and vector constraints of the dual gravitational theory to close in a very specific manner, namely, the manner in which the corresponding constraints of general relativity do. A number of features of the gravitational theory will be fixed as a consequence of this form of the Poisson bracket algebra. In particular, it will require the metric beta function to be of gradient form.