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Entanglement branes, modular flow, and extended topological quantum field theory

William Donnelly, Gabriel Wong

TL;DR

This work shows that entanglement in two-dimensional gauge theories and their string duals can be naturally described within open-closed TQFTs augmented by an entanglement brane axiom. Edge modes on entangling surfaces are captured by the open sector, yet entanglement quantities such as entropy and negativity can be computed entirely via the closed sector thanks to the entanglement-brane sewing rules. The authors derive explicit formulas for entanglement entropy, modular Hamiltonians, and negativity in 2D Yang–Mills and connect these to worldsheet pictures involving Omega points in the Gross–Taylor string. The framework suggests deep links between edge-mode counting, Morse-theoretic saddle points of modular flow, and gravitational entropy, with potential applications to 2D gravity and beyond.

Abstract

Entanglement entropy is an important quantity in field theory, but its definition poses some challenges. The naive definition involves an extension of quantum field theory in which one assigns Hilbert spaces to spatial sub-regions. For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal. With the addition of one additional axiom characterizing the `entanglement brane' we show how entanglement calculations can be cast in this framework. We use this formalism to calculate modular Hamiltonians, entanglement entropy and negativity in two-dimensional Yang-Mills theory and relate these to singularities in the modular flow. As a byproduct we find that the negativity distinguishes between the `log dim R' edge term and the `Shannon' edge term. We comment on the possible application to understanding the Bekenstein-Hawking entropy in two-dimensional gravity.

Entanglement branes, modular flow, and extended topological quantum field theory

TL;DR

This work shows that entanglement in two-dimensional gauge theories and their string duals can be naturally described within open-closed TQFTs augmented by an entanglement brane axiom. Edge modes on entangling surfaces are captured by the open sector, yet entanglement quantities such as entropy and negativity can be computed entirely via the closed sector thanks to the entanglement-brane sewing rules. The authors derive explicit formulas for entanglement entropy, modular Hamiltonians, and negativity in 2D Yang–Mills and connect these to worldsheet pictures involving Omega points in the Gross–Taylor string. The framework suggests deep links between edge-mode counting, Morse-theoretic saddle points of modular flow, and gravitational entropy, with potential applications to 2D gravity and beyond.

Abstract

Entanglement entropy is an important quantity in field theory, but its definition poses some challenges. The naive definition involves an extension of quantum field theory in which one assigns Hilbert spaces to spatial sub-regions. For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal. With the addition of one additional axiom characterizing the `entanglement brane' we show how entanglement calculations can be cast in this framework. We use this formalism to calculate modular Hamiltonians, entanglement entropy and negativity in two-dimensional Yang-Mills theory and relate these to singularities in the modular flow. As a byproduct we find that the negativity distinguishes between the `log dim R' edge term and the `Shannon' edge term. We comment on the possible application to understanding the Bekenstein-Hawking entropy in two-dimensional gravity.

Paper Structure

This paper contains 27 sections, 113 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Open-Closed TQFT
  3. Closed TQFT
  4. Open TQFT
  5. Open-closed TQFT
  6. Branes
  7. The entanglement brane
  8. Two-dimensional Yang-Mills theory
  9. Two-dimensional Yang-Mills as an area-preserving QFT
  10. Entanglement
  11. Single interval
  12. Two-intervals
  13. Thermal state
  14. Reduced density matrix for general states and regions
  15. Negativity
  16. ...and 12 more sections

Figures (2)

  • Figure 1: These two cobordisms (along with the closed multiplication) describe saddle points of the modular flow. In these figures, modular time flows from top to bottom and can be interpreted as a height function, which is the standard example of a Morse function. A saddle point corresponds to a critical point of the Morse function where the Hessian has one positive and one negative eigenvalue. Figure adapted from Ref. Lauda:2005wn.
  • Figure 2: The cobordism associated with half of the modular flow of two intervals on $S^2$, and an associated Morse function with one singularity. Figure adapted from Ref. Lauda:2005wn.