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(3+1)D topological orders with only a $\mathbb{Z}_2$-charged particle

Theo Johnson-Freyd

TL;DR

This work classifies (3+1)D topological orders with a single nontrivial particle by leveraging braided fusion 2-categories and their centers. It identifies three canonical orders $\mathcal{R},\mathcal{S},\mathcal{T}$, with $\mathcal{R}$ bosonic ($\Omega\mathcal{R}\cong\mathbf{Rep}(\mathds Z_2)$) and $\mathcal{S},\mathcal{T}$ fermionic ($\Omega\mathcal{B}\cong\mathbf{SVec}$), the latter split by a gravitational anomaly; $\mathcal{S}$ is nonanomalous while $\mathcal{T}$ carries a nontrivial 't Hooft anomaly. The paper provides three proofs—direct $\pi_0$ analysis, categorified Galois descent, and a long exact sequence via the Witt spectra—to establish this trichotomy and to characterize automorphisms and boundary data, thereby completing the (3+1)D classification in this setting. These results connect higher-categorical centers, boundary conditions, and cohomological invariants, with implications for understanding 3+1D topological phases and their anomalies.

Abstract

There is exactly one bosonic (3+1)-dimensional topological order whose only nontrivial particle is an emergent boson: pure $\mathbb{Z}_2$ gauge theory. There are exactly two (3+1)-dimensional topological orders whose only nontrivial particle is an emergent fermion: pure "spin-$\mathbb{Z}_2$" gauge theory, in which the dynamical field is a spin structure; and an anomalous version thereof. I give three proofs of this classification, varying from hands-on to abstract. Along the way, I provide a detailed study of the braided fusion $2$-category $\mathcal{Z}_{(1)}(Σ\mathbf{SVec})$ of string and particle operators in pure spin-$\mathbb{Z}_2$ gauge theory.

(3+1)D topological orders with only a $\mathbb{Z}_2$-charged particle

TL;DR

This work classifies (3+1)D topological orders with a single nontrivial particle by leveraging braided fusion 2-categories and their centers. It identifies three canonical orders , with bosonic () and fermionic (), the latter split by a gravitational anomaly; is nonanomalous while carries a nontrivial 't Hooft anomaly. The paper provides three proofs—direct analysis, categorified Galois descent, and a long exact sequence via the Witt spectra—to establish this trichotomy and to characterize automorphisms and boundary data, thereby completing the (3+1)D classification in this setting. These results connect higher-categorical centers, boundary conditions, and cohomological invariants, with implications for understanding 3+1D topological phases and their anomalies.

Abstract

There is exactly one bosonic (3+1)-dimensional topological order whose only nontrivial particle is an emergent boson: pure gauge theory. There are exactly two (3+1)-dimensional topological orders whose only nontrivial particle is an emergent fermion: pure "spin-" gauge theory, in which the dynamical field is a spin structure; and an anomalous version thereof. I give three proofs of this classification, varying from hands-on to abstract. Along the way, I provide a detailed study of the braided fusion -category of string and particle operators in pure spin- gauge theory.

Paper Structure

This paper contains 12 sections, 23 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Fusion 2-categories
  4. (3+1)D topological orders
  5. The three examples
  6. The unique topological order in which e is a boson
  7. The two topological orders in which e is a fermion
  8. Automorphisms of R, S, and T
  9. Three proofs of the Main Theorem
  10. First proof of the Main Theorem: direct analysis of pi0 B
  11. Second proof of the Main Theorem: categorified Galois descent
  12. Third proof of the Main Theorem: a long exact sequence

Key Result

Lemma 1.2.3

Let $\mathcal{A}$ be a finite semisimple $2$-category over $\mathds C$, and $X,Y \in \mathcal{A}$ a pair of indecomposable objects. Then any nonzero $1$-morphism $X \to Y$ extends to a condensation $X \mathrel{\,\space\joinrel\rhook\joinrel\space\joinrel\rightarrow} Y$. ∎

Figures (1)

  • Figure 1: Left: a graphical depiction of the definition (\ref{['eqn.slab']}) of the 2+1D topological order $\mathcal{A}$ as a "slab" built from the Neumann boundary condition $\mathcal{N} : \mathcal{S} \to 2\mathbf{Vec}$ and the automorphism $f \in \mathop{\mathrm{Aut}}\nolimits(\mathcal{S})$. Right: the same figure, rotated a bit, recovers the boundary condition $\mathcal{N} \circ \mathcal{S}_f$ from a Morita equivalence $\mathcal{F} : \mathcal{C} \otimes \mathbf{SVec} \simeq \mathbf{SVec}$.

Theorems & Definitions (63)

  • Remark 1.1.1
  • Remark 1.1.2
  • Definition 1.2.1
  • Definition 1.2.2
  • Lemma 1.2.3
  • Definition 1.2.4
  • Remark 1.2.5
  • Definition 1.3.1
  • Remark 1.3.2
  • Remark 2.1.1
  • ...and 53 more