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Sector Theory of Levin-Wen Models II : Fusion and Braiding

Alex Bols, Boris Kjær

Abstract

This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category $\mathcal{C}$ on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided $\rm C^*$-tensor category of superselection sectors $\mathsf{SSS}$. By constructing explicit isomorphisms between the fusion spaces of $\mathsf{SSS}$ and those of the Drinfeld center $Z(\mathcal{C})$, we show that these two categories have isomorphic $F$- and $R$-symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center, $$\,\mathsf{SSS}_f \simeq Z(\mathcal{C}).$$ This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum dimensions.

Sector Theory of Levin-Wen Models II : Fusion and Braiding

Abstract

This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided -tensor category of superselection sectors . By constructing explicit isomorphisms between the fusion spaces of and those of the Drinfeld center , we show that these two categories have isomorphic - and -symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center, This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum dimensions.
Paper Structure (51 sections, 39 theorems, 182 equations, 7 figures)

This paper contains 51 sections, 39 theorems, 182 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Levin-Wen model and its category of superselection sectors
  3. The Levin-Wen model
  4. Local degrees of freedom
  5. The Levin-Wen Hamiltonian and its ground state
  6. The category of superselection sectors
  7. Cones and anyon representations
  8. Localized and transportable endomorphisms of the allowed algebra
  9. Tensor product and braiding
  10. Direct sums and subobjects
  11. Summary of Part I
  12. Skein modules
  13. Unitary fusion categories and the Drinfeld center
  14. Skein modules on extended surfaces
  15. Gluing
  16. ...and 36 more sections

Key Result

Theorem 1.1

[theorem]thm:main theorem There is a unitary braided monoidal equivalence

Figures (7)

  • Figure 1: An allowed cone $\Lambda$ together with allowed cones $\Delta$ and $\Delta_L$, both disjoint from $\Lambda^{+s}$ and lying respectively clockwise and counter clockwise from $\Lambda$ w.r.t. the forbidden direction $\hat{f}$.
  • Figure 2: A link $L$ together with the region $C^L$ indicated in light grey, and the associated surface $\Sigma_L$ in light yellow. The anchor $\mathop{\mathrm{ }}\nolimits_L$ is shown in blue.
  • Figure 3: Schematic representation of the region $D$ and its anchor $\mathop{\mathrm{ }}\nolimits_D$. The enumeration of the punctures of $D$ induced by $\mathop{\mathrm{ }}\nolimits_D$ corresponds to the ordering $(e_{n+1}, e_n, e_{N+2}, e_{N+1})$. The anchor $\mathop{\mathrm{ }}\nolimits_D$ extends the standard anchor $\mathop{\mathrm{ }}\nolimits_{L_{n+1}}$ and the standard anchor $\mathop{\mathrm{ }}\nolimits_{L_{N+2}}$ with offset $2$.
  • Figure 4: Isotopy of links $L$ and $L'$ supported by region $E$. The regions $\Sigma_L, \Sigma_{L'}$ and $\Sigma_E$ are shown in green, blue, and light yellow respectively. A choice of anchors $\mathop{\mathrm{ }}\nolimits_E = \mathop{\mathrm{ }}\nolimits_{L}$ and $\mathop{\mathrm{ }}\nolimits_{L'}$ is also shown, together with the enumeration of their strands.
  • Figure 5: Part of a bridge $( \{ Q_n \}, \{ E_n \} )$ between chains ${\mathscr C} = (L_n)$ and ${\mathscr C}' = (L'_n)$.
  • ...and 2 more figures

Theorems & Definitions (55)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • ...and 45 more