Ryu-Takayanagi area from Virasoro modular data

Abstract: We show that in holographic 2d CFTs, the entanglement entropies across several choices of global state and subregion can be written in a way that at once has a microscopic interpretation and matches the leading large$-c$ organization of the Ryu-Takayanagi formula. This representation is obtained by applying crossing symmetry to the replica manifolds. From the boundary point of view, each rewritten entropy looks like an algebraic entanglement entropy for the Virasoro algebra restricted to the region, with center labels obtained by coarse-graining heavy primaries of the BCFT on the regulated region into bins labeled by Liouville momenta. At large $c$, a resulting sum over bins is dominated by a saddle, and the $O(c)$ part of the entropy comes from the Cardy density of heavy primaries in the dominant bin. We identify this $O(c)$ part of the entropy with the Ryu-Takayanagi area. Physically, this suggests a concrete statistical origin for the Ryu-Takayanagi area as coming from coarse-grained Virasoro intertwiners across the entangling cut. The result also provides quantitative criteria for the amount of coarse-graining allowed for the consistency of the interpretation.