Abstract
This is the second in a two-part paper devoted to studying entanglement entropy and edge modes in the A model topological string theory. This theory enjoys a gauge-string (Gopakumar-Vafa) duality which is a topological analogue of AdS/CFT. In part 1, we defined a notion of generalized entropy for the topological closed string theory on the resolved conifold. We provided a canonical interpretation of the generalized entropy in terms of the q-deformed entanglement entropy of the Hartle-Hawking state. We found string edge modes transforming under a quantum group symmetry and interpreted them as entanglement branes. In this work, we provide the dual Chern-Simons gauge theory description. Using Gopakumar-Vafa duality, we map the closed string theory Hartle-Hawking state to a Chern-Simons theory state containing a superposition of Wilson loops. These Wilson loops are dual to closed string worldsheets that determine the partition function of the resolved conifold. We show that the undeformed entanglement entropy due to cutting these Wilson loops reproduces the bulk generalized entropy and therefore captures the entanglement underlying the bulk spacetime. Finally, we show that under the Gopakumar-Vafa duality, the bulk entanglement branes are mapped to a configuration of topological D-branes, and the non-local entanglement boundary condition in the bulk is mapped to a local boundary condition in the gauge theory dual. This suggests that the geometric transition underlying the gauge-string duality may also be responsible for the emergence of entanglement branes.
Highlights
In the context of the AdS/CFT correspondence [1,2,3], the HRRT/generalized entropy [4,5,6,7] formula provides the basis for our understanding of how spacetime emerges from quantum entanglement
We show that under the Gopakumar-Vafa duality, the bulk entanglement branes are mapped to a configuration of topological D-branes, and the non-local entanglement boundary condition in the bulk is mapped to a local boundary condition in the gauge theory dual
We find that the quantum group edge mode symmetry of the closed string theory is described in the dual gauge theory by the large N limit of Kac-Moody symmetry of the CFT edge modes
Summary
In the context of the AdS/CFT correspondence [1,2,3], the HRRT/generalized entropy [4,5,6,7] formula provides the basis for our understanding of how spacetime emerges from quantum entanglement. As first discussed in [9] and emphasized recently in [10], the leading area term in (1.1) only arises from saddles in which the circle shrinks smoothly in the interior, which gives the cigar geometry in the left of figure 1 This represents an apparent obstruction to the trace interpretation (1.4) from the viewpoint of effective field theory and obscures the bulk quantum mechanical origin of area term. We defined an analogue of generalized entropy for closed strings on the resolved conifold geometry (see left of figure 4) and showed that it has a canonical Hilbert space interpretation despite the presence of a non-local shrinkable boundary condition. Using a q-deformed version of the extended TQFT sewing relations, we determined the factorization of the closed string Hilbert space and showed that the generalized entropy has a quantum mechanical description as a q-deformed entanglement entropy:.
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