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Topological orders and factorization homology

Yinghua Ai, Liang Kong, Hao Zheng

TL;DR

This work develops a factorization-homology framework for 2d topological orders, showing that anomaly-free data on a closed (and stratified) surface determine global observables via a universal 0-disk algebra $(\mathbf{H},u_Σ)$, with $u_Σ$ exactly matching the Hilbert space assigned by Reshetikhin–Turaev 2+1D TQFT. By organizing local observables as UMTCs and their module/bimodule structures, the authors derive explicit factorization-homology computations for disks and stratified surfaces, derive dimensional-reduction correspondences with tensor-excision and pushforward properties, and connect these to ground-state degeneracy in lattice models. The paper extends to defects of codimensions 0–2, establishing that anomaly-free configurations yield well-defined global observables and that GSDs can be computed purely from categorical data via factorization-homology. Overall, the results provide a rigorous, computable bridge between tensor-categorical descriptions of 2d topological orders and their global, high-dimensional observables, including concrete lattice-model interpretations and dimensional reduction insights.

Abstract

In the study of 2d (the space dimension) topological orders, it is well-known that bulk excitations are classified by unitary modular tensor categories. But these categories only describe the local observables on an open 2-disk in the long wave length limit. For example, the notion of braiding only makes sense locally. It is natural to ask how to obtain global observables on a closed surface. The answer is provided by the theory of factorization homology. We compute the factorization homology of a closed surface $Σ$ with the coefficient given by a unitary modular tensor category, and show that the result is given by a pair $(\mathbf{H}, u_Σ)$, where $\mathbf{H}$ is the category of finite-dimensional Hilbert spaces and $u_Σ\in \mathbf{H}$ is a distinguished object that coincides precisely with the Hilbert space assigned to the surface $Σ$ in Reshetikhin-Turaev TQFT. We also generalize this result to a closed stratified surface decorated by anomaly-free topological defects of codimension 0,1,2. This amounts to compute the factorization homology of a stratified surface with a coefficient system satisfying an anomaly-free condition.

Topological orders and factorization homology

TL;DR

This work develops a factorization-homology framework for 2d topological orders, showing that anomaly-free data on a closed (and stratified) surface determine global observables via a universal 0-disk algebra , with exactly matching the Hilbert space assigned by Reshetikhin–Turaev 2+1D TQFT. By organizing local observables as UMTCs and their module/bimodule structures, the authors derive explicit factorization-homology computations for disks and stratified surfaces, derive dimensional-reduction correspondences with tensor-excision and pushforward properties, and connect these to ground-state degeneracy in lattice models. The paper extends to defects of codimensions 0–2, establishing that anomaly-free configurations yield well-defined global observables and that GSDs can be computed purely from categorical data via factorization-homology. Overall, the results provide a rigorous, computable bridge between tensor-categorical descriptions of 2d topological orders and their global, high-dimensional observables, including concrete lattice-model interpretations and dimensional reduction insights.

Abstract

In the study of 2d (the space dimension) topological orders, it is well-known that bulk excitations are classified by unitary modular tensor categories. But these categories only describe the local observables on an open 2-disk in the long wave length limit. For example, the notion of braiding only makes sense locally. It is natural to ask how to obtain global observables on a closed surface. The answer is provided by the theory of factorization homology. We compute the factorization homology of a closed surface with the coefficient given by a unitary modular tensor category, and show that the result is given by a pair , where is the category of finite-dimensional Hilbert spaces and is a distinguished object that coincides precisely with the Hilbert space assigned to the surface in Reshetikhin-Turaev TQFT. We also generalize this result to a closed stratified surface decorated by anomaly-free topological defects of codimension 0,1,2. This amounts to compute the factorization homology of a stratified surface with a coefficient system satisfying an anomaly-free condition.

Paper Structure

This paper contains 19 sections, 20 theorems, 52 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Elements of unitary modular tensor categories
  3. Unitary multi-fusion categories
  4. Tensor products of module categories
  5. Unitary modular tensor categories
  6. Factorization homology
  7. Factorization homology
  8. Factorization homology for stratified surfaces
  9. Computation of factorization homology
  10. Anomaly-free coefficient systems
  11. A few useful mathematical results
  12. Closed stratified surfaces without 1-stratum
  13. Closed stratified surfaces with 1-stratum
  14. Topological orders
  15. Topological orders and anomaly-free defects
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Sigma$ be a closed stratified surface with a coefficient system $A$ satisfying an anomaly-free condition (see Def. def:af-cond). We have When $\Sigma$ has no 1-stratum and $A$ is determined by a single unitary modular tensor category $\EuScript{C}$, $u_\Sigma$ is nothing but the Hilbert space that is assigned to the surface $\Sigma$ in Reshetikhin-Turaev 2+1D TQFT determined by $\EuScript{C

Figures (12)

  • Figure 1: For a given coefficient system $A$, the values of $A$ on three types of $L$-stratified 2-disks are shown in figures (a), (b) and (c) and discussed in details in Example \ref{['expl:coeff']}. In particular, in figure (c), 2-disk algebras $\EuScript{A}_0,...,\EuScript{A}_n=\EuScript{A}_0$ are assigned to 2-cells, 1-disk algebras $\EuScript{M}_1, \cdots, \EuScript{M}_n$ are assigned to 1-cells, and a 0-disk algebra $\EuScript{P}$ is assigned to the unique 0-cell.
  • Figure 2: A stratified 2-disk $K=(\mathbb{R}^2; \mathbb{R} \cup \mathbb{R})$ with a coefficient system $A_K$ determined by 2-disk algebras $\EuScript{C},\EuScript{D},\EuScript{E}$ for 2-cells and 1-disk algebras $\EuScript{M},\EuScript{N}$ for 1-cells.
  • Figure 5: This figure depicts a stratified 2-disk with an anomaly-free coefficient system $A$ determined by its target labels.
  • Figure 6: These two pictures depicts the two steps in computing the factorization homology of a sphere with the coefficient system determined by a single UMTC $\EuScript{C}$.
  • Figure 7: Figure (a) shows a stratified cylinder with a unique 2-cell labeled by a UMTC $\EuScript{C}$ and a unique 1-cell labeled by a closed multi-fusion $\EuScript{C}$-$\EuScript{C}$-bimodule $\EuScript{M}$. Figure (b) is a disjoint union of two open 2-disks with 2-cells labeled by $\EuScript{C}$, 1-cells labeled by $\EuScript{M}$ and $\EuScript{M}^\mathrm{rev}$, respectively, and two 0-cells simultaneously labeled by $(\EuScript{X}\boxtimes \EuScript{X}^\mathrm{op}, \oplus_{i\in \mathrm{O}(\EuScript{X})} i\boxtimes i)$ (see Rem. \ref{['rem:0-cell-label']}).
  • ...and 7 more figures

Theorems & Definitions (84)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: kong-zheng Thm. 3.3.6
  • Theorem 2.5: kong-zheng Thm. 3.3.7
  • Corollary 2.6
  • Corollary 2.7
  • Definition 3.1
  • Definition 3.2
  • ...and 74 more