The centaur-algebra of observables

TL;DR

This paper investigates how the algebra of observables in asymptotically AdS spacetimes transitions between different types due to gravitational constraints, using centaur geometries and boundary protocols to reveal a type II$_1$ algebra structure.

Contribution

It introduces the concept of the centaur-algebra of observables and demonstrates how gravitational constraints lead to a type II$_1$ algebra in various AdS spacetime scenarios.

Findings

Transition from type II$_ ext{infty}$ to type II$_1$ algebra due to constraints

Use of centaur geometries to interpolate between AdS$_2$ and dS$_2$

Modification of algebra structure with boundary infalling observer protocol

Abstract

This letter explores a transition in the type of von Neumann algebra for asymptotically AdS spacetimes from the implementations of the different gravitational constraints. We denote it as the \emph{centaur-algebra} of observables. In the first part of the letter, we employ a class of flow geometries interpolating between AdS$_2$ and dS$_2$ spaces, the centaur geometries. We study the type II$_\infty$ crossed product algebra describing the semiclassical gravitational theory, and we explore the algebra of bounded sub-regions in the bulk theory following $T\overline{T}$ deformations of the geometry and study the gravitational constraints with respect to the quasi-local Brown-York energy of the system at a finite cutoff. In the second part, we study arbitrary asymptotically AdS spacetimes, where we implement the boundary protocol of an infalling observer modeled as a probe black hole proposed by arXiv:2211.16512 to study modifications in the algebra. In both situations, we show how incorporating the constraints requires a type II$_1$ description.