Generalized entropy in topological string theory

The Ryu Takayanagi formula identifies the area of  extremal surfaces in AdS with the entanglement entropy of the boundary CFT. However the bulk microstate interpretation of the extremal area remains mysterious. Progress along this direction requires understanding how to define entanglement entropy in the bulk closed string theory. As a toy model for AdS/CFT, we study the entanglement entropy of closed strings in the topological A model in the context of Gopakumar Vafa duality. We give a self consistent factorization of the closed string Hilbert space which leads to string edge modes transforming under a q-deformed surface symmetry group. Compatibility with this symmetry requires a q-deformed definition of entanglement entropy. Using the topological vertex formalism, we define the Hartle Hawking state for the resolved conifold and compute its q-deformed entropy directly from the closed string reduced density matrix.  We show that this is the same as the generalized entropy, defined by prescribing a contractible replica manifold for the closed string theory on the resolved conifold.  We then apply the Gopakumar Vafa duality to reproduce the closed string entropy from Chern Simons dual  using the un-deformed definition of entanglement entropy. Finally we relate non local aspects of our factorization map to analogous phenomenon recently found in JT gravity.