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Gravity and the Crossed Product

Edward Witten

TL;DR

The paper analyzes $1/N$ corrections to the emergent Type III$_1$ operator algebra in ${\cal N}=4$ SYM at large $N$, showing how these corrections promote the right/left algebras to a Type II$_\infty$ crossed product by the modular automorphism group. This construction provides a well-defined trace and entropy on the extended algebra, with entropy defined only up to an additive constant, mirroring classical thermodynamic notions. The framework links boundary modular flow to bulk conserved charges, clarifying how black hole entropy can be treated within quantum statistical structure, and extends to extra conserved charges via symmetry-modulus considerations. The results offer a background-independent, algebraic route toward understanding black hole observables beyond the strict large-$N$ limit, with concrete implications for entropy and information in holographic gravity.

Abstract

Recently Leutheusser and Liu [1,2] identified an emergent algebra of Type III$_1$ in the operator algebra of ${\mathcal N}=4$ super Yang-Mills theory for large $N$. Here we describe some $1/N$ corrections to this picture and show that the emergent Type III$_1$ algebra becomes an algebra of Type II$_\infty$. The Type II$_\infty$ algebra is the crossed product of the Type III$_1$ algebra by its modular automorphism group. In the context of the emergent Type II$_\infty$ algebra, the entropy of a black hole state is well-defined up to an additive constant, independent of the state. This is somewhat analogous to entropy in classical physics.

Gravity and the Crossed Product

TL;DR

The paper analyzes corrections to the emergent Type III operator algebra in SYM at large , showing how these corrections promote the right/left algebras to a Type II crossed product by the modular automorphism group. This construction provides a well-defined trace and entropy on the extended algebra, with entropy defined only up to an additive constant, mirroring classical thermodynamic notions. The framework links boundary modular flow to bulk conserved charges, clarifying how black hole entropy can be treated within quantum statistical structure, and extends to extra conserved charges via symmetry-modulus considerations. The results offer a background-independent, algebraic route toward understanding black hole observables beyond the strict large- limit, with concrete implications for entropy and information in holographic gravity.

Abstract

Recently Leutheusser and Liu [1,2] identified an emergent algebra of Type III in the operator algebra of super Yang-Mills theory for large . Here we describe some corrections to this picture and show that the emergent Type III algebra becomes an algebra of Type II. The Type II algebra is the crossed product of the Type III algebra by its modular automorphism group. In the context of the emergent Type II algebra, the entropy of a black hole state is well-defined up to an additive constant, independent of the state. This is somewhat analogous to entropy in classical physics.
Paper Structure (9 sections, 67 equations, 1 figure)

This paper contains 9 sections, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Large $N$ Limit
  3. The $1/N$ Expansion and the Crossed Product
  4. Beyond the Large $N$ Limit
  5. The Crossed Product
  6. Classical-Quantum States And The Modular Operator
  7. Traces
  8. Density Matrices and Entropies
  9. Other Conserved Charges

Figures (1)

  • Figure 1: (a) A two-sided eternal black hole, with shaded regions to the future of the future horizon and the past of the past horizon. Operators on the right and left boundaries, labeled $R$ and $L$, describe physics in the unshaded regions to the right or left of the black hole, respectively. Also sketched is a Cauchy hypersurface $S$ that passes through the bifurcate horizon. The horizon divides $S$ in a natural way into left and right portions $S_\ell$ and $S_r$. (b) A time band $B=(t_0,\infty)$ of the right boundary. According to LLLL2, in the large $N$ limit, operators in $B$ describe physics in the causal wedge of $B$, which is sketched in black.