Extremal surfaces in glue-on AdS/$T\bar T$ holography

TL;DR

This paper introduces a glue-on version of the Ryu-Takayanagi formula for $Tar T$-deformed CFTs, using extremal surfaces in glue-on AdS$_3$ geometries to compute entanglement entropy, revealing phase transitions and a minimum length scale.

Contribution

It proposes a new extremal surface prescription in glue-on AdS$_3$ geometries that reproduces entanglement entropy in $Tar T$-deformed CFTs and explores its properties across various spacetimes.

Findings

The signed area formula reproduces entanglement entropy for $Tar T$-deformed CFTs.

Extremal surfaces exhibit phase transitions with multiple intervals.

A minimum length scale related to the deformation parameter is identified.

Abstract

$T\bar T$ deformed CFTs with positive deformation parameter have been proposed to be holographically dual to Einstein gravity in a glue-on $\mathrm{AdS}_3$ spacetime. The latter is constructed from AdS$_3$ by gluing a patch of an auxiliary AdS$_3^*$ spacetime to its asymptotic boundary. In this work, we propose a glue-on version of the Ryu-Takayanagi formula, which is given by the signed area of an extremal surface. The extremal surface is anchored at the endpoints of an interval on a cutoff surface in the glue-on geometry. It consists of an RT surface lying in the AdS$_3$ part of the spacetime and its extension to the AdS$_3^*$ region. The signed area is the length of the RT surface minus the length of the segments in AdS$_3^*$. We find that the Ryu-Takayanagi formula with the signed area reproduces the entanglement entropy of a half interval for $T\bar T$-deformed CFTs on the sphere. We then study the properties of extremal surfaces on various glue-on geometries, including Poincar\'e $\mathrm{AdS}_3$, global $\mathrm{AdS}_3$, and the BTZ black hole. When anchored on multiple intervals at the boundary, the signed area of the minimal surfaces undergoes phase transitions with novel properties. In all of these examples, we find that the glue-on extremal surfaces exhibit a minimum length related to the deformation parameter of $T\bar T$-deformed CFTs.