Algebra of operators in an AdS-Rindler wedge

TL;DR

This paper analyzes the operator algebra in the AdS-Rindler wedge within AdS/CFT, showing how renormalization at finite N transforms the algebra into a Type II$_{\infty}$, enabling entropy definition.

Contribution

It explicitly constructs the operator algebra at infinite N and introduces a novel renormalization method to incorporate 1/N corrections, changing the algebra type.

Findings

Algebra is Type III$_{1}$ at infinite N.

Renormalization renders the algebra Type II$_{\infty}$.

Allows associating a density matrix and entropy to states.

Abstract

We discuss the algebra of operators in AdS-Rinlder wedge, particularly in AdS$_{5}$/CFT$_{4}$. We explicitly construct the algebra at $N=\infty$ limit and discuss its Type III$_{1}$ nature. We will consider $1/N$ corrections to the theory and using a novel way of renormalizing the area of Ryu-Takayanagi surface, describe how several divergences can be renormalized and the algebra becomes Type II$_{\infty}$. This will make it possible to associate a density matrix to any state in the Hilbert space and thus a von Neumann entropy.