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arxiv: 2605.28794 · v1 · pith:J6U2OTSPnew · submitted 2026-05-27 · ❄️ cond-mat.str-el · math-ph· math.CT· math.MP· math.QA· quant-ph

Non-invertible symmetry enriched string net topological orders

Luisa Eck , Peter Huston , Kyle Kawagoe , David Penneys This is my paper

Pith reviewed 2026-06-29 09:43 UTC · model grok-4.3

classification ❄️ cond-mat.str-el math-phmath.CTmath.MPmath.QAquant-ph
keywords non-invertible symmetrysymmetry enriched topological orderstring-net modelunitary fusion categoryrelative centeranyon condensationWalker-Wang modeltube algebra
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The pith

Non-invertible symmetry enriched topological orders are realized as relative centers of unitary fusion categories in string-net models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines non-invertible symmetry enriched topological orders (NI-SETOs) and constructs them explicitly inside string-net models. It does this in two equivalent ways: by taking full inclusions of unitary fusion categories and by performing anyon condensation. In both constructions the resulting order is the relative center of the input categories. The approach covers all NI-SETOs and supplies chiral examples by placing enriched categories on the boundary of a three-dimensional Walker-Wang model whose bulk encodes the anomaly. Tube-algebra calculations then determine how the symmetries act on anyons and defects.

Core claim

NI-SETOs are defined and realized in string-net models as the relative center of unitary fusion categories, obtained either from full inclusions or from anyon condensation; both routes produce every possible NI-SETO, and enriched categories placed on the boundary of a 3D Walker-Wang model give chiral realizations whose anomaly is carried by the bulk.

What carries the argument

The relative center of unitary fusion categories, which encodes the full data of the non-invertible symmetry enrichment inside the string-net model.

If this is right

  • Every NI-SETO admits a string-net realization via either construction.
  • Chiral NI-SETOs appear on the boundary of a 3D Walker-Wang model whose bulk encodes the anomaly.
  • Symmetry actions on anyons and defects are computable by tube-algebra techniques in both constructions.
  • The two routes (inclusions and condensation) yield equivalent realizations of the same NI-SETO.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The relative-center method may extend to other lattice models that support anyonic excitations.
  • It supplies a concrete route to classify anomalies associated with non-invertible symmetries in two-dimensional topological phases.
  • Tube-algebra computations of defect actions could be used to predict measurable signatures in candidate materials.

Load-bearing premise

The relative-center construction coming from full inclusions of unitary fusion categories or from anyon condensation captures every consistency condition of the symmetry enrichment and introduces no extra anomalies.

What would settle it

An explicit example in which a relative center obtained from a full inclusion or condensation produces an anomaly or consistency violation that is absent from the original non-invertible symmetry data.

Figures

Figures reproduced from arXiv: 2605.28794 by David Penneys, Kyle Kawagoe, Luisa Eck, Peter Huston.

Figure 1
Figure 1. Figure 1: c ∈ Irr(YA) xA ∼= c U(c) 1 0A 0 ⊕ 6 ψ (10)A 4 ⊕ 10 σ − 3 ⊕ 7 a 1A 1 ⊕ 5 ⊕ 7 x 2A 2 ⊕ 4 ⊕ 6 ⊕ 8 b 9A 3 ⊕ 5 ⊕ 9 [PITH_FULL_IMAGE:figures/full_fig_p036_1.png] view at source ↗
read the original abstract

We propose a definition of a non-invertible symmetry enriched topological order (NI-SETO), and we implement our definition for string net models. We do so in two ways, using full inclusions of unitary fusion categories (UFCs), as well as anyon condensation. In both cases, the NI-SETO is a relative center of UFCs. All NI-SETOs can be realized in either model, where we can use enriched UFCs to get chiral examples on the boundary of a 3D Walker-Wang model representing the anomaly. We describe several examples of NI-SETOs and compute the qualitative symmetry action on anyons and symmetry defects using tube algebra techniques.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes a definition of non-invertible symmetry enriched topological orders (NI-SETOs) and implements the definition in string-net models in two ways: via full inclusions of unitary fusion categories (UFCs) and via anyon condensation. In both cases the resulting NI-SETO is realized as a relative center of UFCs. The central claim is that every NI-SETO arises in this manner; the authors further assert that enriched UFCs yield chiral examples realized on the boundary of a 3D Walker-Wang model that encodes the anomaly, and they illustrate the construction with several explicit examples whose symmetry actions on anyons and defects are computed via tube-algebra techniques.

Significance. If the relative-center construction is shown to be complete and anomaly-free, the work supplies a systematic algebraic route to constructing and classifying NI-SETOs, extending the symmetry-enrichment program beyond invertible (group-like) symmetries. The explicit tube-algebra computations for concrete examples and the link to Walker-Wang boundaries constitute verifiable, falsifiable content that strengthens the contribution.

major comments (2)
  1. [Abstract] Abstract (paragraph on definition and implementation): the claim that 'All NI-SETOs can be realized in either model' rests on the assertion that the relative-center construction obtained from full UFC inclusions or anyon condensation automatically encodes the complete physical data, including consistent fusion of symmetry defects with anyons, higher associators, and anomaly cancellation. The manuscript supplies tube-algebra computations only for examples; a general argument that no extra anomalies or missing consistency conditions are introduced by the construction is required to support the universality statement.
  2. [Abstract] Abstract (paragraph on definition and implementation): it is not shown that the relative-center output reproduces the full set of braiding and fusion data required for a non-invertible symmetry enrichment; non-group-like fusion rules may impose additional higher-category constraints that are not guaranteed to be satisfied by the relative-center functor alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. Below we respond point-by-point to the major comments, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on definition and implementation): the claim that 'All NI-SETOs can be realized in either model' rests on the assertion that the relative-center construction obtained from full UFC inclusions or anyon condensation automatically encodes the complete physical data, including consistent fusion of symmetry defects with anyons, higher associators, and anomaly cancellation. The manuscript supplies tube-algebra computations only for examples; a general argument that no extra anomalies or missing consistency conditions are introduced by the construction is required to support the universality statement.

    Authors: We agree that an explicit general argument would strengthen the universality claim. By definition, a NI-SETO is realized as the relative center Z_C(D), which encodes the full module category structure, fusion rules, and associators via the coherence theorems of tensor categories; the string-net and Walker-Wang constructions realize this data without additional anomalies because the anomaly is captured by the 3D bulk. Tube-algebra computations in examples confirm consistency, but we will add a short subsection deriving the absence of extra constraints from the relative-center axioms. revision: yes

  2. Referee: [Abstract] Abstract (paragraph on definition and implementation): it is not shown that the relative-center output reproduces the full set of braiding and fusion data required for a non-invertible symmetry enrichment; non-group-like fusion rules may impose additional higher-category constraints that are not guaranteed to be satisfied by the relative-center functor alone.

    Authors: The relative center is constructed precisely to include the complete braiding and fusion data between anyons and defects, with non-invertible fusion rules inherited from the UFCs; higher associators and pentagon identities hold by the standard coherence results for fusion categories. The tube algebra extracts this data explicitly in the model. We will revise the definition section to state this more explicitly, though the categorical construction already guarantees it. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes an independent definition of NI-SETO, then implements that definition in two distinct models (full UFC inclusions and anyon condensation), both of which produce relative centers by construction of the models themselves. The statement that all NI-SETOs are realized in these models follows directly from the proposed definition and the explicit constructions rather than from any fitted parameter, self-citation chain, or renaming of prior results. No equations, self-referential definitions, or load-bearing self-citations appear in the provided text, and the derivation remains self-contained as a mathematical proposal with example computations via tube algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the relative-center construction is treated as a standard categorical object rather than a new postulate.

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Forward citations

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