Entanglement entropy and $T\bar T$ deformations beyond antipodal points from holography

TL;DR

This paper investigates entanglement entropy in de Sitter sliced (A)dS spaces with a cutoff, revealing a formal similarity to antipodal points on a sphere via an effective radius, and confirms holographic results with field theory calculations including counterterms.

Contribution

It introduces a novel geometric interpretation of entanglement entropy in cutoff (A)dS spaces using an effective radius and extends the field theory calculation of entanglement entropy in DS/dS holography.

Findings

Entanglement entropy can be expressed using an effective radius related to the interval length.

Holographic and field theory calculations of entanglement entropy agree, including counterterm contributions.

Counterterms' effects match Wald entropy calculations on the gravity side.

Abstract

We consider the entanglement entropies in dS$_d$ sliced (A)dS$_{d+1}$ in the presence of a hard radial cutoff for $2\le d\le 6$. By considering a one parameter family of analytical solutions, parametrized by their turning point in the bulk $r^\star$, we are able to compute the entanglement entropy for generic intervals on the cutoff slice. It has been proposed that the field theory dual of this scenario is a strongly coupled CFT, deformed by a certain irrelevant deformation -- the so-called $T\bar T$ deformation. Surprisingly, we find that we may write the entanglement entropies formally in the same way as the entanglement entropy for antipodal points on the sphere by introducing an effective radius $R_\text{eff}=R\,\cos(\beta_\epsilon)$, where $R$ is the radius of the sphere and $\beta_\epsilon$ related to the length of the interval. Geometrically, this is equivalent to following the $T\bar T$ trajectory until the generic interval corresponds to antipodal points on the sphere. Finally, we check our results by comparing the asymptotic behavior (no Dirichlet wall present) with the results of Casini, Huerta and Myers. We then switch on counterterms on the cutoff slice which are important with regards to the field theory calculation. We explicitly compute the contributions of the counterterms to the entanglement entropy by considering the Wald entropy. In the second part of this work, we extend the field theory calculation of the entanglement entropy for antipodal points for a $d$-dimensional field theory in context of DS/dS holography. We find excellent agreement with the results from holography and show, in particular, that the effects of the counterterms in the field theory calculation match the Wald entropy associated with the counterterms on the gravity side.