[Paper Review] Large N algebras and generalized entropy

Venkatesa Chandrasekaran, Geoff Penington+1 more|arXiv (Cornell University)|Sep 21, 2022

Advanced Algebra and Logic4 citations

TL;DR

This paper constructs a Type II∞ von Neumann algebra in the large N limit of AdS/CFT that describes single-trace operators in the microcanonical ensemble without 1/N corrections. Using the extrapolate dictionary, it derives the generalized entropy of black hole bifurcation surfaces as the entropy of this algebra, providing a holographic derivation of the quantum-corrected Bekenstein-Hawking formula and the QES prescription without Euclidean gravity or replicas, while also deriving the generalized second law from algebraic entropy monotonicity.

ABSTRACT

We construct a Type II$_\infty$ von Neumann algebra that describes the large $N$ physics of single-trace operators in AdS/CFT in the microcanonical ensemble, where there is no need to include perturbative $1/N$ corrections. Using only the extrapolate dictionary, we show that the entropy of semiclassical states on this algebra is holographically dual to the generalized entropy of the black hole bifurcation surface. From a boundary perspective, this constitutes a derivation of a special case of the QES prescription without any use of Euclidean gravity or replicas; from a purely bulk perspective, it is a derivation of the quantum-corrected Bekenstein-Hawking formula as the entropy of an explicit algebra in the $G o 0$ limit of Lorentzian effective field theory quantum gravity. In a limit where a black hole is first allowed to equilibrate and then is later potentially re-excited, we show that the generalized second law is a direct consequence of the monotonicity of the entropy of algebras under trace-preserving inclusions. Finally, by considering excitations that are separated by more than a scrambling time we construct a "free product" von Neumann algebra that describes the semiclassical physics of long wormholes supported by shocks. We compute Rényi entropies for this algebra and show that they are equal to a sum over saddles associated to quantum extremal surfaces in the wormhole. Surprisingly, however, the saddles associated to "bulge" quantum extremal surfaces contribute with a negative sign.

Motivation & Objective

• To derive the generalized entropy of black hole horizons in AdS/CFT without relying on Euclidean gravity or replica tricks.
• To construct a Type II∞ von Neumann algebra that captures the large N physics of single-trace operators in the microcanonical ensemble.
• To show that the entropy of this algebra reproduces the quantum-corrected Bekenstein-Hawking formula via the extrapolate dictionary.
• To derive the generalized second law from the monotonicity of algebraic entropy under trace-preserving inclusions.
• To construct a free product von Neumann algebra for long wormholes with shocks and compute Rényi entropies via quantum extremal surface saddles, including negative contributions from 'bulge' surfaces.

Proposed method

• Construct a Type II∞ von Neumann algebra from the large N limit of single-trace CFT operators in the microcanonical ensemble.
• Use the extrapolate dictionary to map boundary algebraic structures to bulk generalized entropy on the black hole bifurcation surface.
• Apply the crossed product construction with a compact non-Abelian group G to model time-translation symmetry and derive the algebraic structure dual to the bulk QFT.
• Prove trace-preservation of the isomorphism between boundary and bulk algebras using Peter-Weyl decomposition and modular theory.
• Derive the generalized second law from the monotonicity of algebraic entropy under trace-preserving inclusions of subalgebras.
• Construct a free product von Neumann algebra for long wormholes with shockwave excitations, computing Rényi entropies as sums over quantum extremal surface saddles.

Experimental results

Research questions

• RQ1 How can the generalized entropy of a black hole horizon be derived from a boundary von Neumann algebra without using Euclidean gravity or replicas?
• RQ2 What is the algebraic structure dual to the bulk quantum field theory in the large N limit of the microcanonical ensemble?
• RQ3 Why does the entropy of a boundary algebra reproduce the quantum-corrected Bekenstein-Hawking formula?
• RQ4 How does the generalized second law emerge from the monotonicity of algebraic entropy under inclusion of subalgebras?
• RQ5 Why do 'bulge' quantum extremal surface saddles contribute with a negative sign in Rényi entropy computations for long wormholes?

Key findings

• The generalized entropy of the black hole bifurcation surface is holographically dual to the entropy of a Type II∞ von Neumann algebra constructed from large N single-trace operators in the microcanonical ensemble.
• The quantum-corrected Bekenstein-Hawking formula emerges as the entropy of this algebra in the G→0 limit of Lorentzian effective field theory quantum gravity.
• The generalized second law is a direct consequence of the monotonicity of algebraic entropy under trace-preserving inclusions of subalgebras.
• For long wormholes with shockwave excitations, the Rényi entropies are computed as a sum over quantum extremal surface saddles, including negative contributions from 'bulge' surfaces.
• The isomorphism between boundary and bulk algebras is trace-preserving, confirmed via Peter-Weyl decomposition and modular theory.
• The construction provides a derivation of the QES prescription without relying on Euclidean path integrals or replica tricks.