Holographic Timelike Entanglement Across Dimensions

TL;DR

This work develops a comprehensive holographic framework for timelike entanglement entropy (tEE), extending the Ryu–Takayanagi construction to Lorentzian settings and providing exact and approximate tEE expressions for slab, spherical, and hyperbolic regions across three classes of ten-dimensional backgrounds. By applying the formalism to infinite families of holographic SCFTs in d=2–6 (via rank-function data encoded in a potential V) and to gapped/confining models, the authors uncover universal scaling relations among timelike, slab, and holographic central charges, and elucidate how mass gaps modify tEE and its phase structure. The results unify tEE across conformal and nonconformal theories and establish tEE as a probe of causal structure, universal data, and nonperturbative dynamics in holography, while also highlighting caveats related to UV locality in phase-transition diagnostics. The paper outlines future directions, including extending to spherical entangling regions, incorporating quantum corrections, and exploring thermal or charged backgrounds to broaden the applicability of real-time holographic entanglement measures.

Abstract

We develop a holographic framework for computing timelike entanglement entropy (tEE) in quantum field theories, extending the Ryu-Takayanagi prescription into Lorentzian settings. Using three broad classes of supergravity backgrounds, we derive both exact and approximate tEE expressions for slab, spherical, and hyperbolic regions, and relate them to the central charges of the dual conformal field theories. The method is applied to infinite families of supersymmetric linear quivers in dimensions from d=2 to d=6, showing that Liu-Mezei and slab central charges scale universally like the holographic central charge. We then analyse gapped and confining models, including twisted compactifications and wrapped brane constructions, identifying how a mass gap modifies tEE and when approximate formulas remain accurate. In all cases, we uncover robust scaling with invariant separations and signature dependent phase behaviour, distinguishing spacelike from timelike embeddings. Our results unify the treatment of tEE in both conformal and nonconformal theories, clarifying its role as a probe of causal structure, universal data, and nonperturbative dynamics in holography.

Figures (9)

• Figure 1: Light cone structure that clearly distinguishes between spacelike and timelike separated events. The red line ($c_y>c_t$) corresponds to spacelike separated events (and hence a Type I or usual RT like extremal surface). On the other hand, the dotted line ($c_y<c_t$) corresponds to a timelike separated events that corresponds to a Type II (or complex) extremal surface. Clearly, $c_y=0$ corresponds to pure timelike separated events that are along the time ($T$) axis of the diagram.
• Figure 2: A linear quiver. The balancing condition implies $F_i = 2 N_i - N_{i-1}-N_{i+1}$
• Figure 3: Interval $\Delta$ and entanglement entropy vs. $u_0$, with $u_\Lambda = Q = 1$.
• Figure 4: Parametric plot of $S_{EE}$ vs. $\Delta$, with $u_\Lambda = Q = 1$. Here, we set $u_0=1.01$ which corresponds to a real turning point and hence a Type I extremal surface in the bulk. From \ref{['u0conf']} one could see that this corresponds to $c_y>c_t$ and hence a spacelike separated interval in the dual confining QFTs.
• Figure 5: Plot for tEE vs system size where we set $\epsilon=1$, $q=1$, $L=1$ and $\ell=0.001$. An almost identical plot can be obtained for the choice $\epsilon =-1$.