Toward a worldsheet theory of entanglement entropy
We propose a new action for entanglement entropy in the framework of the AdS$_{3}$/CFT$_{2}$ correspondence. This action is constructed directly from the entanglement entropy of the CFT$_{2}$, and we show that the Einstein equations of AdS$_{3}$ gravity can be derived from it. In the near-coincidence limit, using Riemann normal coordinates, the action reduces to a string worldsheet action in a curved background that naturally includes the symmetric spacetime metric, an antisymmetric Kalb-Ramond field, and a dilaton. The Kalb-Ramond field gives rise to a string charge density, from which we demonstrate that bit threads can be exactly reproduced. This correspondence provides a clear physical interpretation of bit threads. Exploiting this correspondence, we establish explicit relations between the emergent string worldsheet and the Ryu-Takayanagi (RT) surface, providing new insights into entanglement entropy. In particular, entanglement entropy can be computed from open string charge, while Bekenstein-Hawking entropy arises from closed string charge through open-closed string duality. These results suggest a unified picture in which the Susskind-Uglum conjecture, open-closed string duality, and the ER=EPR proposal emerge as equivalent manifestations of the same underlying principle. Finally, we propose a quantization of the RT surface, pointing to a possible connection with loop quantum gravity that refines Wall's conjecture.