Entanglement entropy and $T \overline{T}$ deformation

TL;DR

The paper tests the conjecture that a $T \overline{T}$-deformed CFT is holographically dual to gravity in a finite region by focusing on entanglement entropy. Using the flow equation for the sphere partition function, the authors compute the entanglement entropy across two antipodal points on $S^2$, finding a UV-finite result $S=\frac{c}{3}\sinh^{-1}\left(\sqrt{\frac{24\pi}{c\mu}}\,r\right)$ that matches the Ryu-Takayanagi formula with a finite bulk cutoff. They also develop a conical entropy $\tilde{S}_n$ for $n<1$, expressing it in terms of the complete elliptic integral of the third kind and showing it remains finite; the holographic comparison via Dong’s proposal confirms $\tilde{S}_n$ corresponds to a brane/geometry length in AdS$_3$. The UV finiteness of entanglement entropy supports the view that the $T \overline{T}$ deformation implements a UV cutoff and renders the deformed CFT dual to a finite bulk region, with implications for entanglement structure and potential generalizations. Overall, the work reinforces the holographic interpretation of $T \overline{T}$ deformations and elucidates how UV and IR cutoffs emerge in entanglement measures.

Abstract

Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory deformed by the irrelevant operator $T \overline{T}$. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary while also probing the bulk geometry. We find that the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi formula applied to a finite region consistent with the conjecture of McGough, Mezei and Verlinde. We also consider a one-parameter family of conical entropies, which are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local, we conclude that the $T \overline{T}$ deformation acts as an ultraviolet cutoff on the entanglement entropy. Our results support the conjecture that the $T \overline{T}$-deformed CFT is the holographic dual of a finite region of spacetime.

Figures (3)

• Figure 1: Entanglement entropy in the $T \overline{T}$ theory agrees with the CFT result for $r \gg \sqrt{\mu c}$, but is UV finite.
• Figure 2: The entropy $\tilde{S}_n$ is a smoothly increasing function of $r$. At fixed $r$, $\tilde{S}_n$ is a decreasing function of $n$, with a kink at $n = 1$.
• Figure 3: Embedded surfaces with $r = \ell = 1$ in Euclidean AdS${}_3$. For this graph we have used Poincaré disk coordinates in which the metric is $ds^2 = \frac{4 d \vec{x}^2}{(1 - \vec{x}^2)^2}$ with the third coordinate suppressed. The AdS boundary is the black circle, and finite boundaries are shown in color. At $n = 1$, the embedding is a circle. As $n$ is decreased the embeddings have an increasingly elongated "football" shape. At $n = 0$, the embedding degenerates to a line.