Precision tests of bulk entanglement: $AdS_3$ vectors

We consider single-particle excitations of the massive Chern-Simons field of mass $M$ in $AdS_3$ and evaluate their contribution at the first sub-leading order in $G_N$ to the entanglement entropy across the Ryu-Takayanagi surface. Quantizing the Chern-Simons field in $AdS_3$, we evaluate the corrections to the holographic entanglement entropy using the Faulkner-Lewkowycz-Maldacena formula. The massive Chern-Simons field also obeys the equations of motion of a massive vector in $AdS_3$. The lowest-energy single-particle excitation of this field is dual to the primary operator of conformal dimension $M+1$ with spin one in the dual CFT; all other single-particle excitations are dual to its global descendants. We compare the entanglement entropy result from the FLM formula to the single-interval entanglement entropy in large-charge holographic CFT obtained using the replica trick for the primary and its tower of holomorphic descendants. The two results agree precisely in the leading and sub-leading terms of the short interval expansion. We evaluate the contribution of the edge mode to the vacuum-subtracted entanglement and show that it vanishes, which is crucial for the FLM formula to agree with the CFT result. On taking the massless limit, the result coincides with the contribution of a $U(1)$ current to the single interval entanglement entropy. This is surprising since an earlier calculation in the literature reproduced the CFT result entirely from the edge $U(1)$ degrees of freedom on the RT surface.