Entanglement branes in a two-dimensional string theory

TL;DR

The paper analyzes how entanglement is realized in a string-theoretic setting by studying spatial entanglement between intervals in the Gross–Taylor model, the string dual of 2D Yang–Mills at large N. It develops a canonical open-string framework for edge states, introduces the E-brane as the entangling-surface brane, and derives a modular Hamiltonian that governs the Hartle–Hawking state, showing that the entanglement entropy is the thermal entropy of open strings ending on the E-brane. The work unpacks how closed strings decompose into open strings at the entangling surface, how Omega-point singularities encode edge-mode counting, and how tubes connecting chiral sectors arise from unitarity constraints, providing a comprehensive picture of entanglement in a tractable string theory. These insights offer a concrete string-theoretic mechanism for edge modes and have potential implications for understanding entanglement in higher-dimensional string theories and holographic dualities.

Abstract

What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large $N$. The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the Modular hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an E-brane.

Figures (12)

• Figure 1: In string theory, $\log Z(\beta)$ is a sum of connected string diagrams embedded in a spacetime with a conical defect of angle $\beta$ at the entangling surface. Only closed string diagrams that intersect the entangling surface, such as the sphere diagram on the left, contribute to the entanglement entropy. Sliced transverse to the entangling surface, the middle diagram describes a closed string emitted from a point on the entangling surface and then reabsorbed. Sliced in angular time $\phi$, the sphere diagram is a one loop open string diagram with the endpoints fixed on the entangling surface as depicted on the right.
• Figure 2: The closed string configuration corresponding to the permutation $\sigma: 1\rightarrow 2,2\rightarrow 3,3\rightarrow1,4\rightarrow5,5\rightarrow4$. The cycle lengths $(3,2)$ correspond to the winding numbers of the closed strings.
• Figure 3: A 3-sheeted covering map of the torus $T^{2}$ with two interaction branch points $q_{1},q_{2}$. The cover map $\nu$ is defined here by vertical projection. A counterclockwise loop encircling the target space branch point $q_1$ lifts to a permutation $p_{1}:1\rightarrow 1, 2\rightarrow 3,3\rightarrow 2$. A vertical time slicing of $\Sigma$ shows an initial closed string breaking into two and then joining back together again.
• Figure 4: $Z^{+}_{S^2}$ can be expressed as a sum over worldsheets with holes corresponding to closed strings that wind around the $\Omega$-points on $S^2$.
• Figure 5: By closing up the boundaries of the worldsheet into branch points and introducing appropriate branch cuts, we can present the worldsheet as a covering space of the sphere, with covering map represented by vertical projection. The first is a single covering with no interaction and $\Omega$-point singularities. The second is a double cover of the sphere with two $\Omega$-points. The third term corresponds to the "pair of pants" diagram, now presented as a double cover with an interaction branch point inserted, which is connected to an $\Omega$-point via a branch cut.