Abstract
Progress in identifying the bulk microstate interpretation of the Ryu-Takayanagi formula requires understanding how to define entanglement entropy in the bulk closed string theory. Unfortunately, entanglement and Hilbert space factorization remains poorly understood in 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 will present our results in two separate papers. In this work, we consider the bulk closed string theory on the resolved conifold and give a self-consistent factorization of the closed string Hilbert space using extended TQFT methods. We incorporate our factorization map into a Frobenius algebra describing the fusion and splitting of Calabi-Yau manifolds, and find string edge modes transforming under a q-deformed surface symmetry group. We define a string theory analogue of the Hartle-Hawking state and give a canonical calculation of its entanglement entropy from the reduced density matrix. Our result matches with the geometrical replica trick calculation on the resolved conifold, as well as a dual Chern-Simons theory calculation which will appear in our next paper [1]. We find a realization of the Susskind-Uglum proposal identifying the entanglement entropy of closed strings with the thermal entropy of open strings ending on entanglement branes. We also comment on the BPS microstate counting of the entanglement entropy. Finally we relate the nonlocal aspects of our factorization map to analogous phenomenon recently found in JT gravity.
Highlights
The holographic principle states that the number of degrees of freedom in a spacetime region scales with the area of its boundary, and is exemplified by the Bekenstein-Hawking (BH) entropy formula
We find a realization of the Susskind-Uglum proposal identifying the entanglement entropy of closed strings with the thermal entropy of open strings ending on entanglement branes
Using the topological quantum field theory (TQFT) description, we propose a factorization of the closed string Hilbert space that is consistent with the entanglement entropy as computed by the replica trick
Summary
The holographic principle states that the number of degrees of freedom in a spacetime region scales with the area of its boundary, and is exemplified by the Bekenstein-Hawking (BH) entropy formula. In the follow-up paper [1], we will give a dual Chern-Simons gauge theory description of the entanglement entropy and the corresponding edge modes, giving an independent check of our closed string calculations. WR is precisely the quantum dimension which captures the topological degeneracy associated with the fusion Hilbert space of an anyon By superposing such Wilson loops in all possible representations, we will reproduce the string theory Hartle-Hawking state as well as its entanglement entropy in an appropriate large-N limit of the quantum dimensions. This limit gives a precise relation between the closed string edge modes and the anyons of Chern-Simons theory. We give a summary of our paper, starting with an overview of the GV duality and a description of the closed string state whose factorization and entanglement entropy we will be studying
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