Large $N$ von Neumann algebras and the renormalization of Newton's constant

TL;DR

This paper develops a family of Ryu--Takayanagi formulae in the large N limit of holographic codes, linking bulk entropy, von Neumann algebras, and a renormalization group flow that balances area and entropy terms.

Contribution

It introduces a novel framework connecting von Neumann algebras, holographic entropy, and renormalization group flow, providing explicit proofs and realizations of theoretical conjectures.

Findings

Renormalization of area and entropy terms exactly cancel each other.

Established a concrete realization of the ER=EPR paradigm.

Provided an explicit proof of a conjecture by Susskind and Uglum.

Abstract

I derive a family of Ryu--Takayanagi formulae that are valid in the large $N$ limit of holographic quantum error-correcting codes, and parameterized by a choice of UV cutoff in the bulk. The bulk entropy terms are matched with a family of von Neumann factors nested inside the large $N$ von Neumann algebra describing the bulk effective field theory. These factors are mapped onto one another by a family of conditional expectations, which are interpreted as a renormalization group flow for the code subspace. Under this flow, I show that the renormalizations of the area term and the bulk entropy term exactly compensate each other. This result provides a concrete realization of the ER=EPR paradigm, as well as an explicit proof of a conjecture due to Susskind and Uglum.