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arxiv: 2204.02407 · v3 · pith:ZAUZR5OWnew · submitted 2022-04-05 · ✦ hep-th · cond-mat.str-el· math.QA

Higher Gauging and Non-invertible Condensation Defects

Konstantinos Roumpedakis , Sahand Seifnashri , Shu-Heng Shao This is my paper

Pith reviewed 2026-05-24 09:55 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elmath.QA
keywords higher gaugingcondensation defectsnon-invertible symmetriesTQFThigher-form symmetriesfusion rules2+1d QFT
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The pith

Gauging 1-form symmetries on surfaces generates all 0-form symmetries including non-invertible ones in 2+1d TQFTs.

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

The paper establishes that higher gauging of discrete higher-form symmetries on higher-codimension manifolds creates topological condensation defects. Focusing on 1-gaugeable 1-form symmetries in 2+1d QFTs, gauging them on surfaces yields both invertible and non-invertible defects whose universal fusion rules are computed. In the TQFT case, this construction accounts for every 0-form global symmetry, such as the Z2 electromagnetic symmetry. The fusion rules involve 1+1d TQFTs as coefficients, providing data for the fusion 2-category. This approach extends to non-topological theories like Maxwell theory and QED.

Core claim

In 2+1d TQFT, every invertible and non-invertible 0-form global symmetry is realized by higher gauging of 1-form symmetries on surfaces. The resulting condensation surfaces have fusion rules determined universally, with coefficients that are 1+1d TQFTs rather than numbers. Fusion with bulk lines and surface lines is also computed.

What carries the argument

Higher gauging of a p-gaugeable q-form symmetry on a codimension-p manifold, which produces condensation defects with specified fusion rules.

If this is right

  • All 0-form symmetries in 2+1d TQFT arise from this higher gauging procedure.
  • Fusion coefficients in non-invertible rules are 1+1d TQFTs.
  • The construction applies to non-topological 2+1d QFTs such as free U(1) Maxwell theory.
  • Basic data for the underlying fusion 2-category is determined from surface-bulk-line fusions.

Where Pith is reading between the lines

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

  • This method might extend to realize non-invertible symmetries in higher-dimensional QFTs through analogous higher gauging.
  • One could construct explicit lattice realizations of these defects to verify the fusion rules numerically.
  • Connections to categorical symmetries suggest this provides a geometric origin for fusion 2-categories in 2+1d.

Load-bearing premise

That the q-form symmetry is p-gaugeable, meaning it can be gauged on a codimension-p manifold without anomalies invalidating the fusion rules.

What would settle it

A counterexample 2+1d TQFT where some 0-form symmetry, like electromagnetic duality in Z2 theory, cannot be obtained via any higher gauging of a 1-form symmetry.

read the original abstract

We discuss invertible and non-invertible topological condensation defects arising from gauging a discrete higher-form symmetry on a higher codimensional manifold in spacetime, which we define as higher gauging. A $q$-form symmetry is called $p$-gaugeable if it can be gauged on a codimension-$p$ manifold in spacetime. We focus on 1-gaugeable 1-form symmetries in general 2+1d QFT, and gauge them on a surface in spacetime. The universal fusion rules of the resulting invertible and non-invertible condensation surfaces are determined. In the special case of 2+1d TQFT, every (invertible and non-invertible) 0-form global symmetry, including the $\mathbb{Z}_2$ electromagnetic symmetry of the $\mathbb{Z}_2$ gauge theory, is realized from higher gauging. We further compute the fusion rules between the surfaces, the bulk lines, and lines that only live on the surfaces, determining some of the most basic data for the underlying fusion 2-category. We emphasize that the fusion "coefficients" in these non-invertible fusion rules are generally not numbers, but rather 1+1d TQFTs. Finally, we discuss examples of non-invertible symmetries in non-topological 2+1d QFTs such as the free $U(1)$ Maxwell theory and QED.

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 paper defines higher gauging of a q-form symmetry on a codimension-p manifold when the symmetry is p-gaugeable. It focuses on 1-gaugeable 1-form symmetries in 2+1d QFTs, gauged on surfaces, and derives universal fusion rules for the resulting invertible and non-invertible condensation surfaces. In 2+1d TQFTs it claims every 0-form global symmetry (invertible or not), including the electromagnetic Z2 of Z2 gauge theory, arises this way. It computes fusions between surfaces, bulk lines, and surface-localized lines (with coefficients that are 1+1d TQFTs) and gives examples in non-topological theories such as Maxwell theory and QED.

Significance. If the universality statement holds, the work supplies a constructive origin for all 0-form symmetries via higher gauging and furnishes explicit fusion data for the associated fusion 2-category. The forward derivation of universal fusion rules from the gauging procedure itself, together with the concrete computations for lines and surfaces, constitutes a concrete advance in the classification of non-invertible defects.

major comments (2)
  1. [special case of 2+1d TQFT paragraph] § on the special case of 2+1d TQFT (the paragraph containing the claim that every 0-form symmetry is realized): the manuscript demonstrates that every 1-gaugeable 1-form symmetry produces a 0-form symmetry via higher gauging, but supplies no general argument that the map is surjective onto the set of all possible 0-form symmetries of 2+1d TQFTs, nor a classification showing that no 0-form symmetry lies outside the image. The 'every' statement therefore rests on an unproven converse.
  2. [§2 and abstract] Definition of p-gaugeability (abstract and §2): the consistency of the fusion rules is conditioned on the 1-form symmetry being 1-gaugeable without anomaly; the paper does not provide a criterion or check that every 0-form symmetry of interest arises from a symmetry satisfying this condition, leaving open the possibility that some 0-form symmetries require gauging a symmetry that fails to be 1-gaugeable.
minor comments (2)
  1. [universal fusion rules section] Notation for the fusion coefficients (which are 1+1d TQFTs) should be introduced with an explicit example in the first appearance of the universal fusion rules to avoid ambiguity when the coefficient is the trivial TQFT versus a non-trivial one.
  2. [universal fusion rules section] The statement that the fusion rules are 'universal' would benefit from a short remark clarifying whether they depend on the choice of 1-gaugeable 1-form symmetry or are independent of that choice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to the major comments point-by-point below.

read point-by-point responses

  1. Referee: [special case of 2+1d TQFT paragraph] § on the special case of 2+1d TQFT (the paragraph containing the claim that every 0-form symmetry is realized): the manuscript demonstrates that every 1-gaugeable 1-form symmetry produces a 0-form symmetry via higher gauging, but supplies no general argument that the map is surjective onto the set of all possible 0-form symmetries of 2+1d TQFTs, nor a classification showing that no 0-form symmetry lies outside the image. The 'every' statement therefore rests on an unproven converse.

    Authors: We agree with the referee that while we show that higher gauging of 1-gaugeable 1-form symmetries yields 0-form symmetries (including non-invertible ones), the manuscript does not contain a general proof that every 0-form symmetry in 2+1d TQFTs arises in this manner. The claim will be revised in the abstract and the relevant section to state that higher gauging provides a construction realizing many 0-form symmetries, with the electromagnetic Z_2 of Z_2 gauge theory as a key example, rather than asserting universality without proof. A note will be added indicating that the question of whether all 0-form symmetries are obtained this way is left for future investigation. revision: yes

  2. Referee: [§2 and abstract] Definition of p-gaugeability (abstract and §2): the consistency of the fusion rules is conditioned on the 1-form symmetry being 1-gaugeable without anomaly; the paper does not provide a criterion or check that every 0-form symmetry of interest arises from a symmetry satisfying this condition, leaving open the possibility that some 0-form symmetries require gauging a symmetry that fails to be 1-gaugeable.

    Authors: The results are indeed derived under the assumption that the 1-form symmetry is 1-gaugeable without anomaly. In the examples we consider, this condition holds. We will revise the abstract and §2 to make this assumption more explicit and to clarify that our construction applies specifically to 1-gaugeable symmetries. We will also add a remark that providing a general criterion for which 0-form symmetries arise from 1-gaugeable 1-form symmetries is an open question beyond the scope of this paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from gauging definitions

full rationale

The paper defines higher gauging and p-gaugeability, then derives fusion rules for condensation surfaces directly from that procedure applied to 1-gaugeable 1-form symmetries. The abstract states the universal fusion rules are determined from the gauging itself and that the 2+1d TQFT case realizes all 0-form symmetries via this construction. No quoted steps reduce a claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain; the forward construction does not presuppose the target fusion rules or universality as inputs. The p-gaugeability condition is an explicit assumption rather than a derived or fitted quantity. This matches the default expectation of non-circularity for papers whose central claims follow from stated definitions without the enumerated reduction patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard domain assumptions of QFT about discrete higher-form symmetries and their gauging; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption A q-form symmetry is p-gaugeable if it can be gauged on a codimension-p manifold in spacetime.
    Definition introduced in the abstract as the starting point for higher gauging.

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discussion (0)

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