Timelike entanglement entropy with gravitational anomaly from complexified geometry
We investigate the timelike entanglement entropy (TEE) in a two-dimensional conformal field theory with a gravitational anomaly, $c_L\neq c_R$, dual to topologically massive gravity in AdS$_3$. Holographically, the TEE is determined by the on-shell action of a massive spinning particle on a single complex geodesic anchored at the endpoints of the timelike interval. This action consists of the complex geodesic length and the twist of a parallel-transported normal frame. We carry out the computation in Poincaré AdS$_3$, non-extremal and extremal rotating BTZ backgrounds, and provide an equivalent Chern-Simons derivation in which the bulk connection stays real while only the saddle of the Wilson-line probe becomes complex. In each case, the holographic result matches the corresponding TEE in dual field theory. The real part of the TEE is fixed unambiguously whereas its imaginary part depends on the boundary $iε$ prescription and the orientation of the interval. In the bulk, the dependence on orientation reflects an ambiguity in the choice of framing. Along the complex geodesic, the residual frame group becomes $SO(2,\mathbb C)\simeq\mathbb C^*$, so the endpoint conditions determine the frame twist only up to integer multiples of $2πi$. The difference between the two orientations, $\frac{c_L-c_R}{6}\,iπ$, is fixed entirely by the gravitational anomaly and persists for the Rényi pseudo-entropies.