Cluster algebraic description of entanglement patterns for the BTZ black hole

TL;DR

This paper reveals that the entanglement structure of a thermal state dual to a BTZ black hole can be described using cluster algebras, geometric polytopes, and Y-systems, linking quantum entanglement with algebraic and geometric frameworks.

Contribution

It introduces a novel algebraic and geometric framework, based on cluster algebras and cyclohedra, to describe entanglement patterns in the BTZ black hole's dual conformal field theory.

Findings

Entanglement patterns form a $C_{N-1}$ cluster algebra.

The geometric representation is a cyclohedron ${ m C}_{N-1}$ polytope.

Entanglement can be modeled by a Zamolodchikov $Y$-system with BTZ entropy boundary conditions.

Abstract

We study the thermal state of a two dimensional conformal field theory which is dual to the static BTZ black hole in the high temperature limit. After partitioning the boundary of the static BTZ slice into $N$ subsystems we show that there is an underlying $C_{N-1}$ cluster algebra encoding entanglement patterns of the thermal state. We also demonstrate that the polytope encapsulating such patterns in a geometric manner for a fixed $N$ is the cyclohedron ${\mathcal C}_{N-1}$. Alternatively these patterns of entanglement can be represented in the space of geodesics (kinematic space) in terms of a Zamolodchikov $Y$-system of $C_{N-1}$ type. The boundary condition for such an $Y$-system is featuring the entropy of the BTZ black hole.