Timelike entanglement in 2D conformal field theory is defined by time‑ordered twist correlators and tied to complex geodesics in holography
Main idea: The authors define a notion of entanglement that lives along time, not space. They do this in two-dimensional conformal field theory (CFT) by analytically continuing the usual replica trick. The result is a complex-valued quantity called timelike entanglement entropy (TEE). The same idea extends to Rényi entropies, which are standard generalizations of entanglement entropy.
What they did: In the usual replica method one computes the Rényi entropies from correlation functions of special “twist” operators at two spatial points and then continues a replica index to one. The authors take those twist correlators and continue them from Euclidean to Lorentzian signature, placing the two twist operators at timelike-separated, time-ordered points. This gives a direct field-theory definition of timelike Rényi entropies and of TEE itself as the replica index approaches one.
How it works in holography: For CFTs that have a gravity dual in three bulk dimensions, the semiclassical limit of the time-ordered twist correlator picks out complex saddle points in the bulk. These saddles are boundary-anchored complex geodesics. The prescription is to choose the complex geodesic whose length has the smallest real part. For Rényi index n>1 the authors construct the dual object as a complex codimension-two “cosmic brane” and compute its complexified area in the vacuum. In this way the paper connects the field-theory construction directly to a bulk gravitational picture built from complex extremal surfaces.
Tests and examples: The paper works through several settings where explicit calculations can be done. These include the vacuum on the line and on the circle, locally and globally excited states, local operator quenches, and a time-dependent AdS–Vaidya spacetime. In all presented examples the boundary twist-correlator results agree with the complex-geodesic bulk computations. Notably, for AdS–Vaidya their method gives a different answer than earlier constructions that glued spacelike and timelike geodesic pieces, while reproducing the CFT result.