Geometric Interpretation of Timelike Entanglement Entropy

Abstract

Analytic continuations of holographic entanglement entropy in which the boundary subregion extends along a timelike direction have brought a promise of a novel, time-centric probe of the emergence of spacetime. We propose that the bulk carriers of this holographic timelike entanglement entropy are boundary-anchored extremal surfaces probing analytic continuation of holographic spacetimes into complex coordinates. This proposal not only provides a geometric interpretation of all the known cases obtained by direct analytic continuation of closed-form expressions of holographic entanglement entropy of a strip subregion but crucially also opens a window to study holographic timelike entanglement entropy in full generality. We initialize the investigation of complex extremal surfaces anchored on a timelike strip at the boundary of anti-de Sitter black branes. We find multiple complex extremal surfaces and discuss possible principles singling out the physical contribution.

Figures (6)

• Figure 1: (a) Spatial single interval subregion in 1+1 dimensions and (c) its higher-dimensional generalization as a strip. (b,d): Analogous timelike regions can be obtained from (a,c) by making one of the spatial coordinates imaginary.
• Figure 2: Our HTEE proposal \ref{['proposal']} entails considering codimension-two complex (blue) extremal surfaces that are anchored on the asymptotic boundary on a desired real (red) timelike subregion, here the timelike strip from Fig. \ref{['fig:regions']}(d).
• Figure 3: $z_t$ for all the known complex extremal hypersurfaces in an AdS$_4$-Schwarzschild black brane. Blue (green) curves correspond to v.c. (v.d.) solutions. Horizons [roots of $f(z) = 0$] are represented as black stars, and critical extremal surfaces as red crosses.
• Figure 4: Regularized area density $\mathcal{A}_\textrm{reg}$ for the v.c. (blue curves) and v.d. (green curves) extremal surfaces. Real (imaginary) parts correspond to solid (dashed) curves.
• Figure 5: Left: comparison between the $\Delta t \to 0$ limit of the regularized area density of the v.c. extremal surfaces, $\mathcal{A}_\textrm{reg}^{v.c.}$, and the vacuum result \ref{['A_vac']}. Right: comparison between the $\Delta t \to 0$ limit of the regularized area density of the v.d. extremal surfaces, $\mathcal{A}_\textrm{reg}^{v.d.}$, and the prediction of the singularity probing solution, Eq. \ref{['A_0']}.