Composite AdS geodesics for CFT correlators and timelike entanglement entropy

TL;DR

This work develops a concrete prescription to reconstruct timelike bulk worldlines from boundary CFT data by extremizing a composite geodesic length that concatenates timelike and spacelike segments between timelike-separated boundary points. The resulting complex length $\\ell=\\ell_s+i\\ell_t$ is fixed by an $i\\epsilon$ analytic continuation and shown to reproduce lengths extracted from CFT two-point functions and timelike entanglement entropy in AdS$_3$/CFT$_2$, with explicit checks in Poincaré AdS, global AdS, and BTZ. Across these geometries, the timelike segment length is universally $\\ell_t=\\pi$, while the spacelike legs encode boundary separations, yielding $\\ell_s$ that matches the appropriate correlator under the continuation. The approach refines prior complexified-geodesic methods by staying in the original geometry and using a precise $i\\epsilon$ prescription, offering a path toward a CFT-based derivation via geodesic Witten diagrams or worldline techniques and advancing holographic observer reconstruction and timelike entanglement in AdS/CFT.

Abstract

We study how to recover timelike worldlines in AdS from CFT data as a toy model for holographically reconstructing realistic observers. We give a bulk extremization procedure that determines composite timelike-spacelike geodesics that connect timelike-separated boundary points. The total geodesic length matches the length extracted from CFT correlators at the timelike-separated points. We show agreement in Poincaré AdS, for generic boundary points in global AdS, and also for the BTZ solution, in which the timelike segment probes behind the horizon. We refine related methods to compute timelike entanglement entropy in AdS$_3$/CFT$_2$ and recover known results.

Figures (1)

• Figure 1: For timelike-separated boundary points $x_1, x_2$, we extremize the lengths $\ell_s, \ell_t$ over bulk points $y_1, y_2$. We show this uniquely specifies a composite geodesic whose complex length $\ell$ is equal to the length extracted from $\braket{\mathcal{O}(x_1) \mathcal{O}(x_2)}$.