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On the classification of topological orders

Theo Johnson-Freyd

TL;DR

This work provides a rigorous higher-categorical framework for topological orders by identifying them with centreless, fully dualizable, Karoubi-complete multifusion $n$-categories, whose Morita-invertibility captures anomalous fully extended TQFTs. It develops the higher condensation/isomorphism theory that generalizes idempotent splitting to $n$-categories, proving dualizability results and linking centres to invertibility. The authors justify the definition through correctness and completeness arguments, showing every centreless multifusion $n$-category yields an anomalous TQFT and that topological orders effectively classify a quotient of physical phases by invertible ones. In low dimensions, they recover and refine known classifications: central simple algebras in $(0+1)$D, equivariant cohomology for $(1+1)$D, Witt-group-based modular tensor categories for $(2+1)$D, and anomalous sigma-model descriptions with finite groupoids for $(3+1)$D, including fermionic and symmetry-enriched cases. Overall, the paper extends Wen’s conjectures by embedding topological orders in a precise higher-categorical Morita framework, offering a path toward systematic classification in higher dimensions via cohomology and Galois descent.

Abstract

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete $n$-categories which are mildly dualizable and have trivial centre. Dualizability encodes the word "topological," and we take it as the definition of "(separable) multifusion $n$-category"; triviality of the centre implements the physical principle of "remote detectability." We show that such $n$-categorical algebras are Morita-invertible (in the appropriate higher Morita category), thereby identifying topological orders with anomalous fully-extended TQFTs. We identify centreless fusion $n$-categories (i.e. multifusion $n$-categories with indecomposable unit) with centreless braided fusion $(n{-}1)$-categories. We then discuss the classification in low spacetime dimension, proving in particular that all $(1{+}1)$- and $(3{+}1)$-dimensional topological orders, with arbitrary symmetry enhancement, are suitably-generalized topological sigma models. These mathematical results confirm and extend a series of conjectures and proposals by X.G. Wen et al.

On the classification of topological orders

TL;DR

This work provides a rigorous higher-categorical framework for topological orders by identifying them with centreless, fully dualizable, Karoubi-complete multifusion -categories, whose Morita-invertibility captures anomalous fully extended TQFTs. It develops the higher condensation/isomorphism theory that generalizes idempotent splitting to -categories, proving dualizability results and linking centres to invertibility. The authors justify the definition through correctness and completeness arguments, showing every centreless multifusion -category yields an anomalous TQFT and that topological orders effectively classify a quotient of physical phases by invertible ones. In low dimensions, they recover and refine known classifications: central simple algebras in D, equivariant cohomology for D, Witt-group-based modular tensor categories for D, and anomalous sigma-model descriptions with finite groupoids for D, including fermionic and symmetry-enriched cases. Overall, the paper extends Wen’s conjectures by embedding topological orders in a precise higher-categorical Morita framework, offering a path toward systematic classification in higher dimensions via cohomology and Galois descent.

Abstract

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete -categories which are mildly dualizable and have trivial centre. Dualizability encodes the word "topological," and we take it as the definition of "(separable) multifusion -category"; triviality of the centre implements the physical principle of "remote detectability." We show that such -categorical algebras are Morita-invertible (in the appropriate higher Morita category), thereby identifying topological orders with anomalous fully-extended TQFTs. We identify centreless fusion -categories (i.e. multifusion -categories with indecomposable unit) with centreless braided fusion -categories. We then discuss the classification in low spacetime dimension, proving in particular that all - and -dimensional topological orders, with arbitrary symmetry enhancement, are suitably-generalized topological sigma models. These mathematical results confirm and extend a series of conjectures and proposals by X.G. Wen et al.

Paper Structure

This paper contains 18 sections, 16 theorems, 56 equations.

Table of Contents

  1. Introduction and summary of results
  2. Multifusion $n$-categories
  3. $n$-categorical condensation
  4. $\mathbf{Mor}_1(n\mathbf{KarCat}_\mathds C)$
  5. A dualizability result
  6. Centres and invertibility
  7. Justifying our definition of topological order
  8. Correctness
  9. Completeness
  10. Nondegenerate ground states and braided fusion $(n{-}1)$-categories
  11. From fusion to braided
  12. Braided centres
  13. Topological orders with no lines
  14. Classification in low spacetime dimension
  15. $(0{+}1)$-dimensional topological orders
  16. ...and 3 more sections

Key Result

Lemma II.5

Suppose that $\mathcal{S}$ is a Karoubi-complete, additive, and $\mathds C$-linear $n$-category, pointed by an object $\mathbf 1 \in \mathcal{S}$. Define $\Omega\mathcal{S} = \mathop{\mathrm{End}}\nolimits_\mathcal{S}(\mathbf 1)$, $\Omega^2\mathcal{S} = \mathop{\mathrm{End}}\nolimits_{\Omega\mathcal

Theorems & Definitions (51)

  • Definition I.1
  • Remark I.2
  • Remark I.3
  • Remark II.1
  • Remark II.2
  • Definition II.3
  • Remark II.4
  • Lemma II.5
  • proof
  • Remark II.6
  • ...and 41 more