Spacetime topology from holographic entanglement

TL;DR

This paper develops a holographic method to determine the spacetime topology from entanglement structures in boundary CFT states, applicable to arbitrary dimensions and topologies, including handlebodies.

Contribution

It introduces a prescription to compute the boundary state expansion for spacetimes with general topology, linking Euclidean correlation functions to bulk geometry.

Findings

The expansion matches the Schmidt decomposition at large N.

States exhibit quantum coherence properties.

Method applied to characterize a genus one handlebody spacetime.

Abstract

An asymptotically AdS geometry connecting two or more boundaries is given by a entangled state, that can be expanded in the product basis of the Hilbert spaces of each CFT living on the boundaries. We derive a prescription to compute this expansion for states describing spacetimes with general spatial topology in arbitrary dimension. To large N, the expansion coincides with the Schmidt decomposition and the coefficients are given by $n$-point correlation functions on a particular Euclidean geometry. We show that this applies to all spacetime that admits a Hartle-Hawking type of wave functional, which via a standard hypothesis on the spatial topology, can be (one to one) mapped to CFT states defined on the asymptotic boundary. It is also observed that these states are endowed with quantum coherence properties. Applying this as holographic engineering, one can to construct an emergent space geometry with certain predetermined topology by preparing an entangled state of the dual quantum system. As an example, we apply the method to calculate the expansion and characterize a spacetime whose initial spatial topology is a (genus one) handlebody.