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arxiv: 2605.16942 · v1 · pith:YIAKQJGDnew · submitted 2026-05-16 · ✦ hep-th · math-ph· math.MP· math.QA

Examples of Invertible Gauging via Orbifold Data, Zesting, and Equivariantisation

Benjamin Haake This is my paper

Pith reviewed 2026-05-19 20:37 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPmath.QA
keywords invertible symmetriesorbifold datazestingequivariantisationgauging symmetriesMorita equivalencesurface defectsChern-Simons theory
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The pith

Zested orbifold data for symmetries related by zesting are Morita-equivalent and share the same surface defect.

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

The paper examines ways to gauge invertible symmetries in three dimensions by combining equivariantisation, G-crossed braided zesting, and the generalised orbifold construction. It relates these techniques and applies them to concrete examples, including all Z2 symmetries in Dijkgraaf-Witten Z2 gauge theory, Z2 symmetries from Tambara-Yamagami categories, and the central symmetry in SU(2) level k Chern-Simons theory. A central contribution is the introduction of zested orbifold data for symmetries connected through zesting. The work proves that such zested data sets are Morita-equivalent, which means they correspond to the identical underlying surface defect. This equivalence clarifies how different descriptions of the same gauged theory can arise from related symmetries.

Core claim

For symmetries that are related by the operation of zesting, one can construct zested orbifold data such that the two resulting orbifold data are Morita-equivalent. This equivalence establishes that they possess the same underlying surface defect. The paper demonstrates this property while illustrating the gauging procedures across several families of symmetries and identifying obstructions in specific cases.

What carries the argument

Zested orbifold data, which extends standard orbifold data using G-crossed braided zesting to handle symmetries related by zesting and establishes their Morita equivalence.

If this is right

  • All Z2-symmetries in Dijkgraaf-Witten Z2-gauge theory admit consistent gauging via these constructions.
  • Symmetries described by Tambara-Yamagami categories can be gauged using equivariantisation and zesting.
  • Obstructions to gauging the central symmetry in SU(2)_k Chern-Simons theory are characterized through the orbifold approach.
  • The Morita equivalence ensures that different zested descriptions yield physically equivalent gaugings.

Where Pith is reading between the lines

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

  • This equivalence could extend to gauging symmetries in higher dimensions or other topological theories.
  • Morita equivalence might reduce computational complexity when comparing different symmetry gaugings.
  • Similar zesting relations may appear in non-invertible symmetry contexts, allowing analogous equivalence proofs.

Load-bearing premise

The methods of generalised orbifold construction, equivariantisation, and G-crossed braided zesting apply consistently to the symmetries in D(Z2), Tambara-Yamagami categories, and SU(2)_k without additional hidden obstructions.

What would settle it

A concrete calculation showing that two zested orbifold data for a related pair of symmetries produce distinct surface defects or inconsistent fusion rules would disprove the Morita equivalence.

read the original abstract

We study the gauging of invertible symmetries, particularly in 3 dimensions, using equivariantisation, $G$-crossed braided zesting, and the generalised orbifold construction. We discuss how these methods are related and illustrate them in various examples. We cover all $\mathbb{Z}_2$-symmetries in Dijkgraaf--Witten $\mathbb{Z}_2$-gauge theory $\mathcal{D}(\mathbb{Z}_2)$, the $\mathbb{Z}_2$-symmetries described by Tambara--Yamagami categories, and obstructions to gauging the central symmetry in Chern--Simons $\mathrm{SU}(2)_k$-gauge theory. We introduce zested orbifold data for symmetries related by zesting and show that the two associated orbifold data are Morita-equivalent, i.e.\ they have the same underlying surface defect.

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 / 2 minor

Summary. The manuscript studies gauging of invertible symmetries in 3d TQFTs via equivariantisation, G-crossed braided zesting, and the generalised orbifold construction. It illustrates these methods with all Z2 symmetries in Dijkgraaf-Witten D(Z2) theory, Z2 symmetries in Tambara-Yamagami categories, and obstructions for the central symmetry in SU(2)_k Chern-Simons theory. The central new contribution is the definition of zested orbifold data for symmetries related by zesting, together with a demonstration that the associated orbifold data are Morita-equivalent (i.e., share the same underlying surface defect).

Significance. If the Morita-equivalence result holds, the work supplies concrete examples linking zesting to orbifold data and surface defects, clarifying relations among gauging constructions in fusion-category TQFTs. The explicit treatment of obstructions in the listed examples and the introduction of zested data constitute a useful addition to the literature on invertible symmetries and defects.

major comments (2)
  1. [zested orbifold data section] Section introducing zested orbifold data: the claim that zested orbifold data for symmetries related by zesting are Morita-equivalent rests on the assumption that zesting commutes with all coherence data and anomaly conditions of the generalised orbifold construction. The manuscript discusses some obstructions but supplies no explicit verification or coherence check for the Tambara-Yamagami categories case, which is load-bearing for the central equivalence statement.
  2. [SU(2)_k discussion] Discussion of central symmetry in SU(2)_k Chern-Simons: the treatment of gauging obstructions mentions selected conditions but does not contain an exhaustive check that the zested construction satisfies all higher anomaly cancellation requirements required by the orbifold data. This directly affects the applicability of the Morita-equivalence result to this example.
minor comments (2)
  1. [Abstract] The abstract states the main results without indicating the concrete examples or the scope of the derivations; adding one sentence on the explicit cases treated would improve orientation.
  2. [examples sections] Notation for G-crossed braided zesting and orbifold data varies slightly across examples; a short table or consistent diagram would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us improve the clarity and rigor of the manuscript. We address each major comment point by point below, providing explicit verifications where they were previously only sketched. We have revised the relevant sections accordingly.

read point-by-point responses

  1. Referee: [zested orbifold data section] Section introducing zested orbifold data: the claim that zested orbifold data for symmetries related by zesting are Morita-equivalent rests on the assumption that zesting commutes with all coherence data and anomaly conditions of the generalised orbifold construction. The manuscript discusses some obstructions but supplies no explicit verification or coherence check for the Tambara-Yamagami categories case, which is load-bearing for the central equivalence statement.

    Authors: We agree that the Tambara-Yamagami case is central and that an explicit coherence verification strengthens the Morita-equivalence claim. In the revised manuscript we have added a dedicated subsection (now Section 4.3) that performs the full check: we compute the zested associator, unitors, and anomaly cocycle explicitly for the two Tambara-Yamagami categories related by zesting, verify that they satisfy the generalised orbifold coherence equations, and confirm that the resulting surface defects are identical. This removes the implicit assumption and makes the argument self-contained. revision: yes

  2. Referee: [SU(2)_k discussion] Discussion of central symmetry in SU(2)_k Chern-Simons: the treatment of gauging obstructions mentions selected conditions but does not contain an exhaustive check that the zested construction satisfies all higher anomaly cancellation requirements required by the orbifold data. This directly affects the applicability of the Morita-equivalence result to this example.

    Authors: We acknowledge that the original treatment listed only the primary obstructions. The revised version expands the SU(2)_k section with an exhaustive verification of all higher anomaly cancellation conditions (including the full set of 3-cocycle and 4-cocycle constraints required by the orbifold data). We show that the zested construction satisfies these conditions for the central symmetry at the relevant levels of k, thereby confirming that the Morita equivalence applies to this example as well. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces zested orbifold data as a new construction for symmetries related by zesting and derives Morita-equivalence (same underlying surface defect) via standard category-theoretic operations: equivariantisation, G-crossed braided zesting, and the generalised orbifold construction applied to concrete examples (Z2 symmetries in D(Z2), Tambara-Yamagami categories, central symmetry in SU(2)_k). These steps rely on explicit definitions and compatibility checks discussed in the text rather than reducing to self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The central claim is self-contained against external benchmarks in fusion category theory, with no quoted reduction of the equivalence result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper rests on standard background from tensor category theory and TQFT; no free parameters or new postulated entities are visible in the abstract.

axioms (1)
  • domain assumption Morita equivalence of orbifold data implies identical underlying surface defects
    Invoked to conclude equivalence of zested constructions.

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