Timelike entanglement entropy and $T\bar{T}$ deformation

TL;DR

This work demonstrates that a complete characterization of Tar{T}-deformed (2d) CFT entanglement requires both spacelike and timelike entanglement entropies, with a general pseudoentropy framework unifying them. Field-theoretic calculations using the replica trick reveal topology-dependent TTbar corrections: in finite temperature systems, spacelike EE receives a leading correction while timelike EE does not; in finite size systems, timelike EE is corrected while spacelike EE is not. Holographic checks using cutoff-AdS/RT confirm these results: BTZ (finite temperature) respects the spacelike EE correction only, while global AdS3 (finite size) shows timelike EE correction. The findings reinforce the necessity of a dual, complete entanglement measure and illuminate how TTbar deformations imprint on both boundary theories and their gravitational duals, with potential implications for spacetime emergence and black hole information.

Abstract

In a previous work arXiv:1811.07758 about the $T\bar{T}$ deformed CFT$_2$, from the consistency requirement of the entanglement entropy theory, we found that in addition to the usual spacelike entanglement entropy, a timelike entanglement entropy must be introduced and treated equally. Inspired by the recent explicit constructions of the timelike entanglement entropy and its bulk dual, we provide a comprehensive analysis of the timelike and spacelike entanglement entropies in the $T\bar{T}$ deformed finite size system and finite temperature system. The results confirm our prediction that in the finite size system only the timelike entanglement entropy receives a correction, while in the finite temperature system only the usual spacelike entanglement entropy gets a correction. These findings affirm the necessity of a complete measure including both spacelike and timelike entanglement entropies.

Figures (2)

• Figure 1: Geodesics connecting to $\partial A$ in the Poincare patch of $\text{AdS}_{3}$. The red line denotes one timelike geodesic and two green lines denote two spacelike geodesics. The blue line denotes a timelike interval $A$ on the conformal boundary.
• Figure 2: The figure illustrates a Euclidean cylindrical manifold with a black cylinder representing the original boundary and a violet cylinder representing the cut-off boundary. Two red lines denote an interval $\mathcal{A}$ in the compact direction and an interval $\mathcal{B}$ in the non-compact direction. Two orange lines indicate their corresponding intervals in the cut-off boundary.