Modular Flow and Holographic Timelike Entanglement Entropy in AdS and Flat Holography
Timelike entanglement entropy provides a useful probe of holographic theoris, yet an explicit bulk realization of the modular flow underlying its timelike extremal curve construction has so far been lacking. We construct modular flows associated with spacelike and timelike intervals in AdS$_{3}$/CFT$_{2}$ and extend this framework to flat$_{3}$/CCFT$_{2}$. Beginning with the boundary symmetry generators, we explicitly derive the boundary modular flows and extend them into the bulk, providing a systematic procedure that relates bulk modular flows to the extremal curves and holographic (timelike) entanglement entropy. For spacelike intervals, the resulting bulk modular flows determine the extremal curves, whose integrated line elements reproduce the standard holographic entanglement entropy. We then generalize this approach to timelike intervals, explicitly constructing the modular flows whose fixed points reproduce the extremal spacelike and timelike curves which gives the holographic timelike entanglement entropy. To the best of our knowledge, this is the first explicit construction of the modular flow associated with a timelike extremal curve for a timelike interval. The same procedure is then carried over to flat$_{3}$/CCFT$_{2}$, which give rise to the holographic (timelike) entanglement entropy. Finally, we show that the flat space results obtained in this work can be recovered from their AdS$_3$ counterparts in the flat space limit.