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Levin-Wen is a gauge theory: entanglement from topology

Kyle Kawagoe, Corey Jones, Sean Sanford, David Green, David Penneys

TL;DR

This work shows that the Levin-Wen string-net model for any unitary fusion category ${\\mathcal{C}}$ can be understood as a gauge theory with gauge symmetry given by the tube algebra ${\\mathrm{Tube}}({\\mathcal{C}})$. The authors construct a ${\\mathrm{Tube}}({\\mathcal{C}})$-symmetric (gapped) theory on a spine-coral skein module and define a gauging map that yields the Levin-Wen model whose anyons realize the Drinfeld center $Z({\\mathcal{C}})$. They develop the categorified trace $\\mathrm{Tr}:{\\mathcal{C}}\to Z({\\mathcal{C}})$ and Ocneanu’s tube algebra to organize the gauging, and demonstrate both untwisted and twisted gauging in group-categorical cases ${\\mathsf{Hilb}}(G)$ and ${\\mathsf{Hilb}}(G,\\omega)$, including an explicit Fibonacci example. The framework provides a diagrammatic, operator-algebraic route to 2+1D topological orders from categorical data, with potential extensions to nontrivial Tube$(\\mathcal{C})$ SPTs and practical ground-state preparation techniques.

Abstract

We show that the Levin-Wen model of a unitary fusion category $\mathcal{C}$ is a gauge theory with gauge symmetry given by the tube algebra $\operatorname{Tube}(\mathcal{C})$. In particular, we define a model corresponding to a $\operatorname{Tube}(\mathcal{C})$ symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case $\mathcal{C}=\mathsf{Hilb}(G,ω)$, we show how our procedure reduces to the twisted gauging of a trivial $G$-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter.

Levin-Wen is a gauge theory: entanglement from topology

TL;DR

This work shows that the Levin-Wen string-net model for any unitary fusion category can be understood as a gauge theory with gauge symmetry given by the tube algebra . The authors construct a -symmetric (gapped) theory on a spine-coral skein module and define a gauging map that yields the Levin-Wen model whose anyons realize the Drinfeld center . They develop the categorified trace and Ocneanu’s tube algebra to organize the gauging, and demonstrate both untwisted and twisted gauging in group-categorical cases and , including an explicit Fibonacci example. The framework provides a diagrammatic, operator-algebraic route to 2+1D topological orders from categorical data, with potential extensions to nontrivial Tube SPTs and practical ground-state preparation techniques.

Abstract

We show that the Levin-Wen model of a unitary fusion category is a gauge theory with gauge symmetry given by the tube algebra . In particular, we define a model corresponding to a symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case , we show how our procedure reduces to the twisted gauging of a trivial -SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter.
Paper Structure (17 sections, 75 equations, 2 figures)

This paper contains 17 sections, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Levin-Wen string net model
  3. Categorified trace and the tube algebra
  4. Categorified trace
  5. Tube algebra
  6. Ungauging the Levin-Wen model
  7. The local on-site Hilbert space
  8. The tensor product space of tube algebra modules
  9. The invariant subspace and the gauging map
  10. Local operator algebras of observables on either side of the gauging map
  11. Bounded spread isomorphism of quasi-local operator algebras of observables
  12. A modified model to see all excitations
  13. Group categories
  14. Trivial cocycles
  15. Twisted Gauging
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Left: The spine coral skein module produced by an iterated Day convolution product. Right: A sketch of Lophohelia prolifera, Pallas (sp.)CoralImage
  • Figure 2: $\mathop{\mathrm{Tube}}\nolimits({\mathcal{C}})$-invariant subspace and the gauging map. The Levin-Wen lattice is depicted by the dotted grid.

Theorems & Definitions (5)

  • Definition 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4