Entanglement entropy and edge modes in topological string theory II: The dual gauge theory story

TL;DR

This work embeds the entanglement structure of the A-model topological string on the resolved conifold into a dual Chern-Simons gauge theory via Gopakumar-Vafa duality. By mapping the closed-string Hartle-Hawking state to a superposition of Wilson loops in CS theory, the authors show that the undeformed entanglement entropy from cutting Wilson loops reproduces the bulk generalized entropy, with edge-mode degeneracies captured by a q-deformed measure tied to the Drinfeld element. The GV duality further relates bulk entanglement branes to a configuration of topological D-branes and maps the non-local shrinkable boundary condition to a local boundary condition in the gauge theory, clarifying the role of open-closed string duality in entanglement. Large-N analyses of Wilson loops and their worldsheet images demonstrate a precise correspondence between string-edge degrees of freedom and CS edge modes, reinforcing the interpretation that spacetime entanglement and geometric transitions are intimately linked through topological dualities. The results suggest a unified picture where entanglement branes emerge from geometric transitions and where quantum-group edge structures on the string side correspond to conventional CFT/Kac-Moody edge modes on the gauge side.

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.

Figures (28)

• Figure 1: The left figure shows the cigar geometry which is the saddle point that contributes the the area term in the generalized entropy. On the right we have removed a cap at the tip of the cigar and inserted a shrinkable boundary condition $e$.
• Figure 2: In this figure we have flattened out the cigar geometry into a disk. On the right figure, we can view the lower half of the annulus as a path integral preparation of a factorized state with a shrinkable boundary condition at the entangling surface. Quantizing $Z(\beta)$ with respect to the time variable around the origin shows that it can be viewed as the trace of a reduced density matrix on $V$.
• Figure 3: Susskind and Uglum considered the generalized entropy of perturbative closed strings in flat space, viewed as a limit of the cigar geometry. Using off shell arguments, they computed generalized entropy by inserting a conical singularity in the background, corresponding to the tip of the cigar geometry. In perturbative string theory, the area term comes from the sphere diagram which intersects the conical singularity. Viewed in the open string channel, this is a one-loop open string diagram. This interpretation amounts to an open-closed string duality which identifies Bekensten Hawking entropy as thermal entropy of open strings that end on the conical singularity. Figure borrowed from Ref. Donnelly:2016jet.
• Figure 4: Gopakumar-Vafa duality relates closed A-model string on the resolved conifold to the open A-model string on the deformed conifold
• Figure 5: The left figure shows worldsheet instantons ending on D-branes which cut the minimal volume $S^2$ of the resolved conifold along the equator. The branes extend into the non compact fiber directions and wrap a Lagrangian submanifold with the topology $\mathbb{C} \times S^1$