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Finite 2-group gauge theory and its 3+1D lattice realization

Mo Huang

Abstract

In this work, we employ the Tannaka-Krein reconstruction to compute the quantum double $\mathcal D(\mathcal G)$ of a finite 2-group $\mathcal G$ as a Hopf monoidal category. We also construct a 3+1D lattice model from the Dijkgraaf-Witten TQFT functor for the 2-group $\mathcal G$, generalizing Kitaev's 2+1D quantum double model. Notably, the string-like local operators in this lattice model are shown to form $\mathcal D(\mathcal G)$. Specializing to $\mathcal G = \mathbb{Z}_2$, we demonstrate that the topological defects in the 3+1D toric code model are modules over $\mathcal D(\mathbb{Z}_2)$.

Finite 2-group gauge theory and its 3+1D lattice realization

Abstract

In this work, we employ the Tannaka-Krein reconstruction to compute the quantum double of a finite 2-group as a Hopf monoidal category. We also construct a 3+1D lattice model from the Dijkgraaf-Witten TQFT functor for the 2-group , generalizing Kitaev's 2+1D quantum double model. Notably, the string-like local operators in this lattice model are shown to form . Specializing to , we demonstrate that the topological defects in the 3+1D toric code model are modules over .

Paper Structure

This paper contains 32 sections, 7 theorems, 142 equations, 8 figures.

Table of Contents

  1. Preliminaries
  2. Introduction
  3. Review of finite group gauge theory
  4. Basic notions
  5. Dijkgraaf-Witten theory
  6. Finite 2-group gauge theory
  7. Flat G-connections on nD manifold
  8. Finite 2-group and classifying space
  9. Finite 2-group flat connections
  10. Gauge transformations of flat 2-group connections
  11. Equivalence of gauge transformations and the 2-groupoid CG(M)
  12. Finite 2-group gauge theory
  13. Partition function
  14. nD TQFT functor
  15. Quantum double model for finite 2-groups
  16. ...and 17 more sections

Key Result

Theorem 5.1

Let $\mathsf{C}$ be a fusion 2-category $\mathsf{C}$ and $\text{f} \colon \mathsf{C} \to 2\mathrm{Vec}$ be a fiber 2-functor. Then the category $\mathop{\mathrm{End}}\nolimits(\text{f})$ of endo-natural transformations and modifications admits a canonical finite semisimple Hopf monoidal category str

Figures (8)

  • Figure 1: The standard triangulation of $x \times I$ for a 2-simplex $x$. Here we write $x_j$ for $(x_j,0)$ and $x_j'$ for $(x_j,1)$.
  • Figure 2: The product CW structure of $x \times I$ for a 2-simplex $x$. Here we write $x_j$ for $(x_j,0)$ and $x_j'$ for $(x_j,1)$.
  • Figure 3: The composition of two cylinders $[x_0,x_1] \times I$. The 2-simplices $[x_0,x_0',x_0"]$ and $[x_1,x_1',x_1"]$ are labeled by the unit $1 \in A$.
  • Figure 4: The equivalence between two gauge transformations.
  • Figure 5: A ribbon consists of a string $s$ and a dual string $\tilde{s}$.
  • ...and 3 more figures

Theorems & Definitions (37)

  • Remark 1.1
  • Remark 2.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Remark 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Example 4.1
  • ...and 27 more