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Some universal properties of Levin-Wen models

Liang Kong

TL;DR

This work analyzes Levin-Wen lattice models enriched with boundaries and defects via unitary tensor category methods. It establishes boundary-bulk duality, relating boundary theories to bulk centers $Z(\EuScript{C})$, and duality-defect correspondence, connecting invertible domain walls to braided autoequivalences, forming a deep link between boundary data and bulk physics. A detailed boundary analysis shows excitations are classified by $\EuScript{C}_{\mathcal{M}}^\vee=\mathrm{Fun}_{\EuScript{C}}(\mathcal{M},\mathcal{M})^{\otimes^{op}}$, with bulk theories encoded by the monoidal center, while defects and defect junctions are described by bimodule functors and higher morphisms, all compatible under Morita equivalence. The paper also proposes a functorial viewpoint for $Z$, embedding the LW data into a potential tricategory and extending the framework toward fully extended TQFTs, RCFT analogies, and generalizations to multi-fusion categories and fiber-functor realizations.

Abstract

We review the key steps of the construction of Levin-Wen type of models on lattices with boundaries and defects of codimension 1,2,3 in a joint work with Alexei Kitaev. We emphasize some universal properties, such as boundary-bulk duality and duality-defect correspondence, shared by all these models. New results include a detailed analysis of the local properties of a boundary excitation and a conjecture on the functoriality of the monoidal center.

Some universal properties of Levin-Wen models

TL;DR

This work analyzes Levin-Wen lattice models enriched with boundaries and defects via unitary tensor category methods. It establishes boundary-bulk duality, relating boundary theories to bulk centers , and duality-defect correspondence, connecting invertible domain walls to braided autoequivalences, forming a deep link between boundary data and bulk physics. A detailed boundary analysis shows excitations are classified by , with bulk theories encoded by the monoidal center, while defects and defect junctions are described by bimodule functors and higher morphisms, all compatible under Morita equivalence. The paper also proposes a functorial viewpoint for , embedding the LW data into a potential tricategory and extending the framework toward fully extended TQFTs, RCFT analogies, and generalizations to multi-fusion categories and fiber-functor realizations.

Abstract

We review the key steps of the construction of Levin-Wen type of models on lattices with boundaries and defects of codimension 1,2,3 in a joint work with Alexei Kitaev. We emphasize some universal properties, such as boundary-bulk duality and duality-defect correspondence, shared by all these models. New results include a detailed analysis of the local properties of a boundary excitation and a conjecture on the functoriality of the monoidal center.

Paper Structure

This paper contains 5 sections, 3 theorems, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The original Levin-Wen models
  3. Boundary excitations
  4. Defects of codimension 1,2,3
  5. Summary and outlooks

Key Result

Lemma 2

The algebras $A^{(m,m)}$ and $A^{(n,n)}$ are Morita equivalent for any $m,n\in \mathbb{N}$. Moreover, the $A^{(m,m)}$-$A^{(n,n)}$-bimodule $A^{(m,n)}$ is invertible and defines the Morita equivalence.

Figures (3)

  • Figure 1: An upward-oriented planar graph with edge and vertex labels.
  • Figure 2: The action of the plaquette operator $B_{\mathbf{p}}^k$: a) the initial state of the plaquette; b) a symbolic representation of the action of $B_{\mathbf{p}}^k$; c) the loop is partially fused using Eq. (\ref{['eq:decomp_id']}) (some labels and the overall factor are not shown); d) the corner triangles have been evaluated to trivalent vertices (summation over $j_p'$, $\alpha_q'$ is assumed).
  • Figure 3: Boundary excitations (the unexcited part of the lattice is shown in gray): a) a boundary excitation is localized in the region $R$ and $R'=R \cup \partial R'$ where the lattice configuration at the boundary of $R'$ is $\partial R'=G^{(2,n)}$ (see equation (\ref{['eq:G']})); b) three $B_{\mathbf{p}}$operators act on the adjacent plaquette; c) the loop is partially fused in each plaquette so that it splits into two parts, one of which acts on $\EuScript{H}_R$, the other acts on $\EuScript{H}_{\text{ext}}$; d) the loop is completely fused.

Theorems & Definitions (6)

  • Definition 1
  • Lemma 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Conjecture 6