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

TL;DR

This work analyzes entanglement entropy in dS$_d$ sliced (A)dS$_{d+1}$ with a hard radial cutoff, revealing that EE for generic intervals can be re-expressed as antipodal-point EE on a sphere with an effective radius $R_{\mathrm{eff}}=R\cos(\beta_{\epsilon})$, thereby connecting interval length to a $T\bar T$ trajectory. Using a one-parameter bulk solution characterized by the turning point $r^*$, the authors derive exact holographic EE for general intervals and show how counterterms on the cutoff, incorporated via Wald entropy, yield results consistent with renormalized field theory. They derive the d-dimensional $T\bar T$-like deformation operator $X_d$ from trace flow and compute sphere partition functions $Z_{S^d}$, obtaining antipodal EE expressions that agree with holography when counterterm effects are included. Extending to DS/dS holography in higher dimensions, they demonstrate precise matching of renormalized EE between field theory and gravity, including the role of counterterms, thereby clarifying previous apparent mismatches. Overall, the paper provides a unified holographic and field-theoretic treatment of EE under $T\bar T$ deformations in (A)dS, with broad implications for DS/dS holography and higher-dimensional entanglement studies.

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 $R_\text{eff}=R\,\cos(β_ε)$, where $R$ is the radius of the sphere and $β_ε$ 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.

Figures (2)

• Figure 1: The interval under consideration on the circle of radius $R$ is depicted in green. The effective radius $R_\text{eff}=R\,\cos(\beta_\epsilon)$ corresponds by the definition of the cosine (dashed blue line) to the radius, where the points of the interval are antipodal.
• Figure 2: Left: The entangling surface for $r^\star/L=\pi/3$ -- $(\theta,r)$ are the polar and azimuthal angles, respectively, in the static patch of Euclidean dS$_3$ in presence of a cutoff $\epsilon$ (magenta surface). The cutoff surface restricts the entangling surface to the bolder line. We can rotate this surface by $\theta_0=\pi/3$ to bring it to the top of the sphere. If we draw a line through the ending points, we see that this corresponds exactly to a cutoff surface with radius $R_\text{eff}=R\,\cos(\beta_\epsilon(r^\star))$, which is depicted in blue. By rotating the surface on the circle, we see that the entangling surface exactly corresponds to the half-circle, i.e. the interval consists of antipodal points. The field theory lives on the circle on the magenta surface. Right: The analogous picture for Euclidean AdS$_3$. Note that the transformation consists of a spacetime rotation and a special conformal transformation.