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Defects in the 3-dimensional toric code model form a braided fusion 2-category

Liang Kong, Yin Tian, Zhi-Hao Zhang

TL;DR

The paper realizes and verifies the conjectured braided fusion 2-category structure of all codimension-2+ defects in the 3+1D toric code, showing it matches the center Z(2Vec_{Z_2}). Using an explicit lattice model, it identifies four simple string types (1, 1_c, m, m_c), catalogs their 0d defects, fusion rules, loop/link invariants, and dualities, and computes double braidings. It then ties bulk data to gapped boundary theories via the boundary-bulk relation, constructing smooth and rough boundaries whose half-braidings reproduce the known center data (KTZ20). The results provide a concrete bridge between higher category theory and lattice realizations of 3D topological order, confirming nondegeneracy and the predicted braided monoidal 2-category structure of 3+1D Z_2 gauge theory.

Abstract

It was well known that there are $e$-particles and $m$-strings in the 3-dimensional (spatial dimension) toric code model, which realizes the 3-dimensional $\mathbb{Z}_2$ topological order. Recent mathematical result, however, shows that there are additional string-like topological defects in the 3-dimensional $\mathbb{Z}_2$ topological order. In this work, we construct all topological defects of codimension 2 and higher, and show that they form a braided fusion 2-category satisfying a braiding non-degeneracy condition.

Defects in the 3-dimensional toric code model form a braided fusion 2-category

TL;DR

The paper realizes and verifies the conjectured braided fusion 2-category structure of all codimension-2+ defects in the 3+1D toric code, showing it matches the center Z(2Vec_{Z_2}). Using an explicit lattice model, it identifies four simple string types (1, 1_c, m, m_c), catalogs their 0d defects, fusion rules, loop/link invariants, and dualities, and computes double braidings. It then ties bulk data to gapped boundary theories via the boundary-bulk relation, constructing smooth and rough boundaries whose half-braidings reproduce the known center data (KTZ20). The results provide a concrete bridge between higher category theory and lattice realizations of 3D topological order, confirming nondegeneracy and the predicted braided monoidal 2-category structure of 3+1D Z_2 gauge theory.

Abstract

It was well known that there are -particles and -strings in the 3-dimensional (spatial dimension) toric code model, which realizes the 3-dimensional topological order. Recent mathematical result, however, shows that there are additional string-like topological defects in the 3-dimensional topological order. In this work, we construct all topological defects of codimension 2 and higher, and show that they form a braided fusion 2-category satisfying a braiding non-degeneracy condition.

Paper Structure

This paper contains 20 sections, 53 equations, 24 figures.

Table of Contents

  1. Introduction
  2. 3+1D topological orders and braided fusion 2-categories
  3. Categorical description of topological defects
  4. Braiding structure
  5. Dualities
  6. 3d toric code model
  7. Hamiltonian
  8. Ground state degeneracy
  9. Topological defects
  10. Four types of strings and 0d defects
  11. 2-category Toric
  12. Monoidal structure on Toric
  13. Loops and links
  14. Dualities in Toric
  15. Double braidings in Toric
  16. ...and 5 more sections

Figures (24)

  • Figure 2: (a): Moving a string $X$ around $Y$ in the 3d space is a braiding of $X$ and $Y$; (b): The trajectory of a double braiding is a cylinder; (c): The braiding is a 1+0D defect embedded in the 3+1D spacetime.
  • Figure 3: (a): The time slice of the world sheet $S_1$ of braiding two strings at time $t$ is $B_{1,t}$; (b): The section of $S_1$ at $z = z_0$ is a braid $B_1' \coloneqq S_1 \cap H_{z_0}$; (c): After a rotation in $zt$-plane, the braid $B_1'$ is isotopic to $B_2$.
  • Figure 4: (a) After a rotation, the braiding of two strings gives a 0+1D defect; (b): The braiding structure is a 0d domain wall between $X \otimes Y$ and $Y \otimes X$.
  • Figure 5: The right dual $X^*$ of a string $X$ is given by reversing the orientation. Both the evaluation and coevaluation 1-morphisms are 0d domain walls.
  • Figure 6: Straightening a string $X$ is an instanton.
  • ...and 19 more figures

Theorems & Definitions (18)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • ...and 8 more