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### Generalized entropy in topological string theory

## Abstract

The holographic entanglement entropy formula identifies the generalized entropy of the bulk AdS spacetime with the entanglement entropy of the boundary CFT. However the bulk microstate interpretation of the generalized entropy remains poorly understood. Progress along this direction requires understanding how to define Hilbert space factorization and 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, which enjoys a gauge-string duality. We define a notion of generalized entropy for closed strings on the resolved conifold using the replica trick. As in AdS/CFT, we find this is dual to (defect) entanglement entropy in the dual Chern Simons gauge theory. Our main result is a bulk microstate interpretation of generalized entropy in terms of open strings and their edge modes, which we identify as entanglement branes.

More precisely, we give a self consistent factorization of the closed string Hilbert space which introduces open 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 reduced density matrix. We show that this is the same as the generalized entropy. Finally, we relate non local aspects of our factorization map to analogous phenomenon recently found in JT gravity.