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

Elliott Gesteau

TL;DR

The paper constructs a rigorous framework in which the large-N bulk von Neumann algebra can be regulated by nested type I factors connected by conditional expectations, enabling well-defined bulk entropies and a robust RT-type formula in holographic codes.A renormalization-group flow of code subspaces is introduced, implemented by conditional expectations, and shown to guarantee that the renormalization of the bulk entropy exactly cancels the renormalization of the area term, proving the Susskind–Uglum conjecture within this formalism.This work reframes entanglement and geometry as two faces of the same underlying structure (ER=EPR) in the large-N limit, clarifies the role of UV regulation in holographic entropy, and lays out generalizations to infinite-dimensional boundaries and larger codes, with pathways toward concrete realizations in holographic models.

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.

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

TL;DR

The paper constructs a rigorous framework in which the large-N bulk von Neumann algebra can be regulated by nested type I factors connected by conditional expectations, enabling well-defined bulk entropies and a robust RT-type formula in holographic codes.A renormalization-group flow of code subspaces is introduced, implemented by conditional expectations, and shown to guarantee that the renormalization of the bulk entropy exactly cancels the renormalization of the area term, proving the Susskind–Uglum conjecture within this formalism.This work reframes entanglement and geometry as two faces of the same underlying structure (ER=EPR) in the large-N limit, clarifies the role of UV regulation in holographic entropy, and lays out generalizations to infinite-dimensional boundaries and larger codes, with pathways toward concrete realizations in holographic models.

Abstract

I derive a family of Ryu--Takayanagi formulae that are valid in the large 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 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.
Paper Structure (18 sections, 13 theorems, 108 equations, 2 figures)

This paper contains 18 sections, 13 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Hilbert space of effective field theory
  3. Large $N$ algebras and the crossed product
  4. Embedding into UV-complete theories and quantum error correction
  5. Code subspace renormalization
  6. What should a consistent renormalization procedure be?
  7. Formal setup
  8. A proof of the Susskind--Uglum conjecture
  9. The bulk to boundary map
  10. A Ryu--Takayanagi formula
  11. Invariance under renormalization group flow
  12. Susskind--Uglum as ER=EPR
  13. Generalizations
  14. Infinite-dimensional boundary at finite $N$
  15. Type $I_\infty$ factors in the bulk
  16. ...and 3 more sections

Key Result

Proposition 3.5

Let $(\Lambda,(M_\lambda)_{\lambda\in\Lambda},(\mathcal{E}_\lambda)_{\lambda\in \Lambda},(E_{\lambda\mu})_{\lambda,\mu\in \Lambda,\;\lambda\geq \mu},\omega)$ be a code subspace renormalization scheme, and let $\lambda\geq\mu$. There exist decompositions of the form and where $M_\mu$ and $M_{\lambda\mu}$ are type $I$ factors. Moreover, the Hilbert spaces $\mathcal{H}_\lambda$ and $\mathcal{H}_\mu

Figures (2)

  • Figure 3.1: A commutative diagram summarizing the structure of code subspace renormalization. Here the full bulk von Neumann algebras $M$ and $M^\prime$, which are commutants of each other, are mapped to the subalgebras $M_\lambda$ and $M_\mu$ and their commutants, corresponding to different cutoff scales, through the conditional expectations $\mathcal{E}_\lambda$, $\mathcal{E}_\mu$ and $\mathcal{E}^\prime_\lambda$, $\mathcal{E}^\prime_\mu$. The prime on the horizontal arrows denotes the commutant structure implemented by modular conjugation. Given that the states in $\mathcal{H}_\lambda$ and $\mathcal{H}_\mu$ are invariant under the conditional expectations, Takesaki's theorem guarantees that the commutant structure is respected, and that the diagram commutes.
  • Figure 4.1: The code in the case of two entangled CFT's on a compact space. The large $N$ algebras $M^L$ and $M^R$ need to be regulated in order for the map to the finite $N$ algebras $\mathcal{B}(\mathcal{H}_N^L)$ and $\mathcal{B}(\mathcal{H}_N^R)$ to allow the derivation of an entropy formula.

Theorems & Definitions (34)

  • Conjecture 1.1: Susskind--Uglum, Susskind_1994
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • Definition 4.1
  • ...and 24 more