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Topological subregions in Chern Simons theory and topological string theory

Gabriel Wong

TL;DR

This work tackles the challenge of defining subsystems in topological field theories by developing a purely topological subregion in Chern-Simons theory through combinatorial quantization. It builds an operator-algebraic framework with a lattice of quantum-group holonomies, yielding a finite, $q$-deformed entanglement entropy $S = \log \dim_{q}R$ arising from anyonic edge modes and governed by a $q$-tracial state. The authors introduce the balancing element $D$ as the shrinkable holonomy, establish a diagrammatic spacetime-ribbon calculus, and show how triangulation-independent Haar measures lead to a consistent $q$-deformed entropy across subregions, including disconnected cuts. In the large-$N$ limit, these structures provide a bridge to topological string theory and a path toward understanding spacetime emergence in gravity-like settings via edge-mode multiplicities and quantum-trace structures.

Abstract

The standard, gapped entanglement boundary condition in Chern Simons theory breaks the topological invariance of the theory by introducing a complex structure on the entangling surface. This produces an infinite dimensional subregion Hilbert space, a non-trivial modular Hamiltonion, and a UV-divergent entanglement entropy that is a universal feature of local quantum field theories. In this work, we appeal to the combinatorial quantization of Chern Simons theory to define a purely topological notion of a subregion. The subregion operator algebras are spaces of functions on a quantum group. We develop a diagrammatic calculus for the associated $q$-deformed entanglement entropy, which arise from the entanglement of anyonic edge modes. The $q$-deformation regulates the divergences of the QFT, producing a finite entanglement entropy associated to a $q$-tracial state. We explain how these ideas provide an operator algebraic framework for the entanglement entropy computations in topological string theory \cite{Donnelly:2020teo,Jiang:2020cqo, wongtopstring}, where a large- $N$ limit of the $q$-deformed subregion algebra plays a key role in the stringy description of spacetime.

Topological subregions in Chern Simons theory and topological string theory

TL;DR

This work tackles the challenge of defining subsystems in topological field theories by developing a purely topological subregion in Chern-Simons theory through combinatorial quantization. It builds an operator-algebraic framework with a lattice of quantum-group holonomies, yielding a finite, -deformed entanglement entropy arising from anyonic edge modes and governed by a -tracial state. The authors introduce the balancing element as the shrinkable holonomy, establish a diagrammatic spacetime-ribbon calculus, and show how triangulation-independent Haar measures lead to a consistent -deformed entropy across subregions, including disconnected cuts. In the large- limit, these structures provide a bridge to topological string theory and a path toward understanding spacetime emergence in gravity-like settings via edge-mode multiplicities and quantum-trace structures.

Abstract

The standard, gapped entanglement boundary condition in Chern Simons theory breaks the topological invariance of the theory by introducing a complex structure on the entangling surface. This produces an infinite dimensional subregion Hilbert space, a non-trivial modular Hamiltonion, and a UV-divergent entanglement entropy that is a universal feature of local quantum field theories. In this work, we appeal to the combinatorial quantization of Chern Simons theory to define a purely topological notion of a subregion. The subregion operator algebras are spaces of functions on a quantum group. We develop a diagrammatic calculus for the associated -deformed entanglement entropy, which arise from the entanglement of anyonic edge modes. The -deformation regulates the divergences of the QFT, producing a finite entanglement entropy associated to a -tracial state. We explain how these ideas provide an operator algebraic framework for the entanglement entropy computations in topological string theory \cite{Donnelly:2020teo,Jiang:2020cqo, wongtopstring}, where a large- limit of the -deformed subregion algebra plays a key role in the stringy description of spacetime.

Paper Structure

This paper contains 39 sections, 218 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Local edge modes in Chern Simons theory
  3. Shrinkable boundary condition for Chern Simons theory
  4. A non-local constraint on the shrinkable boundary
  5. Shrinkable holonomy in Chern Simons theory
  6. Shrinkable holonomy in q2DYM
  7. Combinatorial quantization and quantum group holonomies
  8. Algebra of observables for Chern Simons theory
  9. The lattice algebra $\mathcal{B}$ and Fun$_{q}(G)$
  10. The GNS state space
  11. Gauge transformations and the gauge invariant algebra
  12. Example: The Torus Observable Algebra.
  13. The triangulation independent Haar measure and relation to q2DYM
  14. Hilbert space factorization and q-deformed entropy
  15. Categorical traces and anyonic entanglement entropy
  16. ...and 24 more sections

Figures (3)

  • Figure 2.1: A state prepared with a Wilson loop insertion along the b cycle can also be specified by the holonomy along the dual a cycle
  • Figure 5.1: Factorizaing a Chern Simons state on a torus with a Wilson loop insertion produces a lattice simulation of the quantum state.
  • Figure 6.1: A closed string wavefunction is produced by the topological string amplitude for worldsheets that end on a stack of D branes with the topology of a solid torus. This is a state in the large $N$ Hilbert space of the worldvolume Chern Simons theory. Combinatorial quantization of the worldvolume theory leads to quantum holonomies which couple to the boundary of the string worldsheet