Entanglement entropy and edge modes in topological string theory: I

TL;DR

This work develops a complete framework for defining and computing entanglement entropy in a bulk string theory setting by embedding the closed A-model on the resolved conifold into an open–closed extended TQFT with a quantum-group structure. By constructing a self-consistent factorization of the closed-string Hilbert space through zipper/cozipper maps and introducing entanglement-brane boundary states, the authors produce edge modes transforming under a $q$-deformed surface symmetry, and derive a $q$-deformed von Neumann entropy that matches replica-trick calculations and an eventual Chern–Simons dual. The Hartle–Hawking state is interpreted as a coherent open-string boundary state whose entanglement entropy is governed by quantum dimensions and the Drinfeld element, realizing Susskind–Uglum-type edge-mode counting in the string context. The work also connects the formalism to JT gravity analogies and outlines how BPS microstate counting enters the entropy, highlighting a rich structure of edge-mode physics and nonlocal factorization tied to Calabi–Yau constraints. A companion paper will provide the dual CS computation and further elucidate the role of edge modes and quantum-group symmetry in the bulk/boundary correspondence.

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 \cite{secondpaper}. 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.

Figures (10)

• Figure 1: The partition function of the $A$-model on a line bundle over $S^2$ has two interpretations. In the closed string channel (left), it represents the overlap $\braket{HH^*|HH}$ between the Hartle-Hawking state and its orientation reversal. In the open string channel (right), it represents a trace in the Hilbert space of open strings. Figure borrowed from Ref. Donnelly:2016jet.
• Figure 2: Gopakumar duality relates closed A-model string on the resolved conifold to the open A-model string on the deformed conifold
• Figure 3: The left figure shows the codimension-1 slice $\Sigma$ of the resolved conifold where a QFT state would be defined. In the closed string theory, the analogue of a time slice is a set $\mathcal{F}_{\Sigma}$ of loops configurations associated with $\Sigma$. For the A-model string, we will restrict these loop configurations to lie in a Lagrangian submanifold $\mathcal{L} \subset \Sigma$. The string wavefunctional assigns an amplitude to each configuration of such loops.
• Figure 4: The left figure shows a D-brane on $\mathcal{L} \subset \Sigma$ which intersects the base $S^2$ along the equator and extends in to the fiber directions along a hyperbola. In the right figure, we show the string loops in the time slice $\mathcal{F}_{\Sigma}$ which lives in $\mathcal{L}$. The state $\ket{HH}$ state is defined via worldsheets which end on these loops and wrap the upper hemisphere, as shown in the left figure. Similarly $\bra{HH^*}$ describes worldsheets on the southern hemisphere which end on anti-branes.
• Figure 5: On the left, we show the splitting of the worldsheet boundary into $A$ and $B$. On the right, the brane $\mathcal{L}$ on which the closed string configurations $X(\sigma)$ live is split into subregions by the entanglement branes. We show an open string configuration $X_{ij}(\sigma) \in \mathcal{F}_{\Sigma_{A}}$. These end on the entanglement branes intersecting $\mathcal{L}$ along two open disks.