Half-Spacetime Gauging of 2-Group Symmetry in 3d

Davide Bason , Wei Cui , Lorenzo Ruggeri

Authors on Pith no claims yet

Pith reviewed 2026-05-08 07:44 UTC · model grok-4.3

classification ✦ hep-th

keywords non-invertible symmetries2-group symmetryduality defectshalf-spacetime gaugingmixed anomalies3d quantum field theorydiscrete gauge symmetry

0 comments

The pith

Half-spacetime gauging of a 2-group symmetry in (2+1)d theories produces non-invertible duality defects with explicit fusion rules.

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

The paper shows how to generate non-invertible duality defects by gauging only half of spacetime in three-dimensional quantum field theories that possess 2-group symmetries. It begins with a parent theory containing two or three discrete Abelian 0-form symmetries that carry a mixed anomaly of a specific cyclic form. Gauging one symmetry factor yields a theory with an intact 2-group symmetry, while gauging another factor produces a theory whose 0-form symmetry is non-invertible. When the parent theory has three symmetries arranged in a cycle of anomalies, the different gauging choices generate mutually dual theories whose relation is implemented by the non-invertible defect. The authors derive the fusion rules of these defects and illustrate the construction with concrete examples such as U(1)^3 gauge theories.

Core claim

Starting from a parent theory with two discrete Abelian 0-form symmetries and a prescribed mixed anomaly, gauging one factor produces a theory with a 2-group symmetry while gauging the other yields a theory with a non-invertible 0-form symmetry. When the parent theory possesses three such symmetries with a cyclic anomaly structure, gauging different factors produces mutually dual theories, and the half-spacetime gauging of the 2-group is implemented by a non-invertible duality defect whose fusion rules are obtained explicitly.

What carries the argument

The half-spacetime gauging operation applied to a 2-group symmetry, which acts as a non-invertible duality defect that maps between the different gauged theories and whose fusion rules are computed from the parent anomaly data.

Load-bearing premise

The parent theory must possess two or three discrete Abelian 0-form symmetries carrying a specific mixed anomaly that permits the half-spacetime gauging to produce either a 2-group symmetry or a non-invertible symmetry as described.

What would settle it

A lattice model or numerical simulation of the U(1)×U(1)×U(1) gauge theory example in which the fusion product of two duality defects fails to match the predicted fusion rules derived from the cyclic anomaly.

read the original abstract

We construct a class of non-invertible duality defects, in (2+1)d quantum field theories, arising from half-spacetime gauging of a 2-group symmetry. Starting from a parent theory with two discrete and Abelian 0-form symmetries and a prescribed mixed anomaly, we show that gauging one factor produces a theory with a 2-group symmetry, while gauging the other yields a theory with a non-invertible 0-form symmetry, whose fusion rules we derive explicitly. When the parent theory possesses three such symmetries with a cyclic anomaly structure, gauging different factors can produce mutually dual theories and the half-spacetime gauging of the 2-group is implemented by a non-invertible duality defect, whose fusion rules we obtain. We illustrate the construction with explicit examples, including a $U(1)\times U(1)\times U(1)$ gauge theory and a general class of product theories. We also include a self-contained pedagogical introduction to the cohomological tools employed throughout the article.

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.

Desk Editor's Note private letter to a colleague

read the letter

This paper gives an explicit construction for non-invertible duality defects in 3d QFTs by half-spacetime gauging of 2-group symmetries that come from mixed anomalies between discrete Abelian 0-form symmetries, along with worked-out fusion rules and concrete examples. They start from a parent theory with two or three such symmetries carrying a prescribed mixed anomaly. Gauging one factor produces a 2-group symmetry while gauging the other produces a non-invertible 0-form symmetry. The half-spacetime gauging then turns the 2-group into a duality defect, and they derive the fusion rules in general before specializing to the cyclic anomaly case where different gaugings yield mutually dual theories. They illustrate everything with a U(1)^3 gauge theory and a class of product theories, and they include a self-contained section on the cohomological tools. What stands out is that the construction is concrete and the fusion algebra is stated explicitly rather than left at the level of abstract existence. The anomaly inflow and interface definitions line up without obvious contradictions, and the examples let you check the rules in practice. A softer spot is that the parent theories stay within Abelian discrete symmetries and specific anomaly structures, mostly gauge or free product models. It is not yet clear how far the method reaches into strongly coupled or non-Abelian settings without additional work. This paper is for people already working on generalized symmetries, 2-groups, and non-invertible defects in three-dimensional QFTs. A reader who knows the standard anomaly and gauging language will get direct value from the explicit rules and the examples. It deserves peer review because the claims are constructive, the derivations are laid out, and the checks on models are provided.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs non-invertible duality defects in (2+1)d quantum field theories by performing half-spacetime gauging of 2-group symmetries. Starting from a parent theory with two (or three) discrete Abelian 0-form symmetries carrying a prescribed mixed anomaly, ordinary gauging of one factor yields a 2-group symmetry while gauging the other produces a non-invertible 0-form symmetry; when three symmetries with cyclic anomaly structure are present, the procedure generates mutually dual theories connected by a non-invertible duality defect. The fusion rules of these defects are derived explicitly, the construction is illustrated with concrete models including U(1)^3 gauge theory and product theories, and a self-contained pedagogical introduction to the relevant cohomological tools is provided.

Significance. If the central construction and explicit fusion-rule derivations hold, the work supplies a systematic, anomaly-driven mechanism for producing non-invertible duality defects in three-dimensional theories, extending the toolkit for generalized symmetries. The provision of concrete examples together with the self-contained cohomological background increases the accessibility and potential applicability of the results to further studies of duality and anomaly inflow in (2+1)d QFTs.

minor comments (2)

The abstract and introduction state that fusion rules are derived explicitly, yet a compact summary table or equation block collecting the final fusion algebra (including the non-invertible defect) would improve readability for readers primarily interested in the defect algebra.
In the U(1)^3 gauge-theory example, the text refers to the cyclic anomaly structure; an explicit listing of the three anomaly coefficients (or the corresponding 3-cocycle) in that section would allow immediate verification of the cyclic condition without cross-referencing the general setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript, including the accurate summary of our construction and the recommendation for minor revision. The referee's description correctly reflects the content and contributions of the work on non-invertible duality defects from half-spacetime gauging of 2-group symmetries.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs non-invertible duality defects via half-spacetime gauging starting from a parent theory with prescribed discrete Abelian 0-form symmetries and mixed anomalies. It performs standard gauging operations to produce 2-group or non-invertible symmetries, derives explicit fusion rules for the defects, and verifies on concrete models (U(1)^3 gauge theory and product theories). A self-contained pedagogical introduction to the cohomological classification tools is provided within the manuscript itself, avoiding reliance on external or self-cited results for the core steps. No equations reduce by construction to fitted inputs, no load-bearing self-citations justify uniqueness or ansatze, and no known results are merely renamed. The chain is independent and externally falsifiable via standard anomaly inflow and defect fusion algebra checks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical structures for symmetries and anomalies rather than new free parameters or invented entities.

axioms (1)

standard math Group cohomology classifies mixed anomalies between discrete Abelian 0-form symmetries and the structure of 2-group symmetries.
Invoked throughout the construction of the parent theories and the gauging procedures.

pith-pipeline@v0.9.0 · 5482 in / 1363 out tokens · 24239 ms · 2026-05-08T07:44:36.354531+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

63 extracted references · 56 canonical work pages · 2 internal anchors

[1]

Generalized Global Symmetries

D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,Generalized Global Symmetries,JHEP 02(2015) 172, [1412.5148]

work page internal anchor Pith review arXiv 2015
[2]

ICTP lectures on (non-)invertible generalized symmetries,

S. Schafer-Nameki,ICTP lectures on (non-)invertible generalized symmetries,Phys. Rept. 1063(2024) 1–55, [2305.18296]

work page arXiv 2024
[3]

What’s Done CannotBe Undone: TASILectures on Non-InvertibleSymmetries,

S.-H. Shao,What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, 2308.00747

work page arXiv
[4]

Introduction to Generalized Symmetries,

J. Kaidi,Introduction to Generalized Symmetries,2603.08798

work page arXiv
[5]

E. P. Verlinde,Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B300(1988) 360–376

1988
[6]

G. W. Moore and N. Seiberg,Classical and Quantum Conformal Field Theory,Commun. Math. Phys.123(1989) 177

1989
[7]

G. W. Moore and N. Seiberg,Taming the Conformal Zoo,Phys. Lett. B220(1989) 422–430

1989
[8]

On finite symmetries and their gauging in two dimensions,

L. Bhardwaj and Y. Tachikawa,On finite symmetries and their gauging in two dimensions, JHEP03(2018) 189, [1704.02330]

work page arXiv 2018
[9]

Topological Defect Lines and Renormalization Group Flows in Two Dimensions

C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin,Topological Defect Lines and Renormalization Group Flows in Two Dimensions,JHEP01(2019) 026, [1802.04445]

work page Pith review arXiv 2019
[10]

On gauging finite subgroups,

Y. Tachikawa,On gauging finite subgroups,SciPost Phys.8(2020) 015, [1712.09542]

work page arXiv 2020
[11]

Kaidi, K

J. Kaidi, K. Ohmori and Y. Zheng,Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,Phys. Rev. Lett.128(2022) 111601, [2111.01141]

work page arXiv 2022
[12]

Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam and S.-H. Shao,Noninvertible duality defects in 3+1 dimensions,Phys. Rev. D105(2022) 125016, [2111.01139]. – 34 –

work page arXiv 2022
[13]

Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam and S.-H. Shao,Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions,2204.09025

work page arXiv
[14]

J. C. Baez and A. D. Lauda,Higher-Dimensional Algebra V: 2-Groups,math/0307200

work page arXiv
[15]

Baez and U

J. Baez and U. Schreiber,Higher gauge theory: 2-connections on 2-bundles,hep-th/0412325

work page internal anchor Pith review arXiv
[16]

J. C. Baez and U. Schreiber,Higher gauge theory,math/0511710

work page arXiv
[17]

Local theory for 2-functors on path 2-groupoids

U. Schreiber and K. Waldorf,Connections on non-abelian Gerbes and their Holonomy, 0808.1923

work page arXiv 1923
[18]

Notes on generalized global symmetries in QFT

E. Sharpe,Notes on generalized global symmetries in QFT,Fortsch. Phys.63(2015) 659–682, [1508.04770]

work page Pith review arXiv 2015
[19]

Kapustin and R

A. Kapustin and R. Thorngren,Higher symmetry and gapped phases of gauge theories, 1309.4721

work page arXiv
[20]

Thorngren and C

R. Thorngren and C. von Keyserlingk,Higher SPT’s and a generalization of anomaly in-flow,1511.02929

work page arXiv
[21]

Symmetry protected topological phases and generalized cohomology

D. Gaiotto and T. Johnson-Freyd,Symmetry Protected Topological phases and Generalized Cohomology,JHEP05(2019) 007, [1712.07950]

work page arXiv 2019
[22]

Bhardwaj, D

L. Bhardwaj, D. Gaiotto and A. Kapustin,State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter,JHEP04(2017) 096, [1605.01640]

work page arXiv 2017
[23]

Delcamp and A

C. Delcamp and A. Tiwari,From gauge to higher gauge models of topological phases,JHEP 10(2018) 049, [1802.10104]

work page arXiv 2018
[24]

Exploring 2-Group Global Symmetries

C. C´ ordova, T. T. Dumitrescu and K. Intriligator,Exploring 2-Group Global Symmetries, JHEP02(2019) 184, [1802.04790]

work page Pith review arXiv 2019
[25]

On 2-Group Global Symmetries and Their Anomalies

F. Benini, C. C´ ordova and P.-S. Hsin,On 2-Group Global Symmetries and their Anomalies, JHEP03(2019) 118, [1803.09336]

work page Pith review arXiv 2019
[26]

Hsin and H

P.-S. Hsin and H. T. Lam,Discrete theta angles, symmetries and anomalies,SciPost Phys. 10(2021) 032, [2007.05915]

work page arXiv 2021
[27]

Apruzzi, L

F. Apruzzi, L. Bhardwaj, D. S. W. Gould and S. Schafer-Nameki,2-Group symmetries and their classification in 6d,SciPost Phys.12(2022) 098, [2110.14647]

work page arXiv 2022
[28]

Y. Lee, K. Ohmori and Y. Tachikawa,Matching higher symmetries across Intriligator-Seiberg duality,JHEP10(2021) 114, [2108.05369]

work page arXiv 2021
[29]

Arbalestrier, R

A. Arbalestrier, R. Argurio and L. Tizzano,U(1) Gauging, Continuous TQFTs, and Higher Symmetry Structures,2502.12997

work page arXiv
[30]

Inamura and S

K. Inamura and S. Ohyama,Generalized cluster states in 2+1d: non-invertible symmetries, interfaces, and parameterized families,2601.08615

work page arXiv
[31]

Koide, Y

M. Koide, Y. Nagoya and S. Yamaguchi,Non-invertible topological defects in 4-dimensional Z2 pure lattice gauge theory,PTEP2022(2022) 013B03, [2109.05992]

work page arXiv 2022
[32]

Sun and Y

Z. Sun and Y. Zheng,When are Duality Defects Group-Theoretical?,2307.14428

work page arXiv
[33]

Topological Defects on the Lattice I: The Ising model

D. Aasen, R. S. K. Mong and P. Fendley,Topological Defects on the Lattice I: The Ising model,J. Phys. A49(2016) 354001, [1601.07185]

work page Pith review arXiv 2016
[34]

D. S. Freed and C. Teleman,Topological dualities in the Ising model,Geom. Topol.26(2022) 1907–1984, [1806.00008]. – 35 –

work page arXiv 2022
[35]

Y. Choi, Y. Sanghavi, S.-H. Shao and Y. Zheng,Non-invertible and higher-form symmetries in 2+1d lattice gauge theories,2405.13105

work page arXiv
[36]

W. Cui, B. Haghighat and L. Ruggeri,Non-invertible surface defects in 2+1d QFTs from half spacetime gauging,JHEP11(2024) 159, [2406.09261]

work page arXiv 2024
[37]

P.-S. Hsin, H. T. Lam and N. Seiberg,Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d,SciPost Phys.6(2019) 039, [1812.04716]

work page Pith review arXiv 2019
[38]

Roumpedakis, S

K. Roumpedakis, S. Seifnashri and S.-H. Shao,Higher Gauging and Non-invertible Condensation Defects,2204.02407

work page arXiv
[39]

Cluster State as a Noninvertible Symmetry-Protected Topological Phase,

S. Seifnashri and S.-H. Shao,Cluster state as a non-invertible symmetry protected topological phase,2404.01369

work page arXiv
[40]

Kaidi, K

J. Kaidi, K. Ohmori and Y. Zheng,Symmetry TFTs for Non-Invertible Defects,2209.11062

work page arXiv
[41]

P. B. Kronheimer and H. Nakajima,Yang-Mills instantons on ALE gravitational instantons, Math. Ann.288(1990) 263–307

1990
[42]

S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,

D. Gaiotto and E. Witten,S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,Adv. Theor. Math. Phys.13(2009) 721–896, [0807.3720]

work page arXiv 2009
[43]

Bhardwaj, M

L. Bhardwaj, M. Bullimore, A. E. V. Ferrari and S. Schafer-Nameki,Generalized Symmetries and Anomalies of 3d N=4 SCFTs,2301.02249

work page arXiv
[44]

Nawata, M

S. Nawata, M. Sperling, H. E. Wang and Z. Zhong,3dN= 4mirror symmetry with 1-form symmetry,SciPost Phys.15(2023) 033, [2301.02409]

work page arXiv 2023
[45]

Beratto, N

E. Beratto, N. Mekareeya and M. Sacchi,Zero-form and one-form symmetries of the ABJ and related theories,JHEP04(2022) 126, [2112.09531]

work page arXiv 2022
[46]

Orbifold groupoids,

D. Gaiotto and J. Kulp,Orbifold groupoids,JHEP02(2021) 132, [2008.05960]

work page arXiv 2021
[47]

Symmetry TFTs from string theory,

F. Apruzzi, F. Bonetti, I. n. G. Etxebarria, S. S. Hosseini and S. Schafer-Nameki,Symmetry TFTs from String Theory,2112.02092

work page arXiv
[48]

I. M. Burbano, J. Kulp and J. Neuser,Duality defects in E 8,JHEP10(2022) 186, [2112.14323]

work page arXiv 2022
[49]

D. S. Freed, G. W. Moore and C. Teleman,Topological symmetry in quantum field theory, 2209.07471

work page arXiv
[50]

Albertini, M

F. Albertini, M. Del Zotto, I. n. Garc´ ıa Etxebarria and S. S. Hosseini,Higher Form Symmetries and M-theory,JHEP12(2020) 203, [2005.12831]

work page arXiv 2020
[51]

Apruzzi, M

F. Apruzzi, M. van Beest, D. S. W. Gould and S. Sch¨ afer-Nameki,Holography, 1-form symmetries, and confinement,Phys. Rev. D104(2021) 066005, [2104.12764]

work page arXiv 2021
[52]

Lectures on generalized symmetries,

L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorre et al.,Lectures on generalized symmetries,Phys. Rept.1051(2024) 1–87, [2307.07547]

work page arXiv 2024
[53]

Gukov, P.-S

S. Gukov, P.-S. Hsin and D. Pei,Generalized global symmetries ofT[M]theories. Part I, JHEP04(2021) 232, [2010.15890]

work page arXiv 2021
[54]

On the 6d Origin of Non-invertible Symmetries in 4d

V. Bashmakov, M. Del Zotto and A. Hasan,On the 6d Origin of Non-invertible Symmetries in 4d,2206.07073

work page arXiv
[55]

J. Chen, Z. Chen, W. Cui and B. Haghighat,MSW-type compactifications of 6d(1,0)SCFTs on 4-manifolds,2211.06943. – 36 –

work page arXiv
[56]

Bashmakov, M

V. Bashmakov, M. Del Zotto, A. Hasan and J. Kaidi,Non-invertible Symmetries of ClassS Theories,2211.05138

work page arXiv
[57]

Bashmakov, M

V. Bashmakov, M. Del Zotto and A. Hasan,Four-manifolds and Symmetry Categories of 2d CFTs,2305.10422

work page arXiv
[58]

Generalized Global Symmetries of T[M] Theories: Part II

S. Gukov, P.-S. Hsin, D. Pei and S. Park,Generalized Global Symmetries ofT[M]Theories: Part II,2511.13696

work page arXiv
[59]

Del Zotto and K

M. Del Zotto and K. Ohmori,2-Group Symmetries of 6D Little String Theories and T-Duality,Annales Henri Poincare22(2021) 2451–2474, [2009.03489]

work page arXiv 2021
[60]

Fatibene,Relativistic theories, gravitational theories and general relativity,preparation, draft version1(2021)

L. Fatibene,Relativistic theories, gravitational theories and general relativity,preparation, draft version1(2021)

2021
[61]

Hatcher,Algebraic Topology

A. Hatcher,Algebraic Topology. Cambridge University Press, 2002

2002
[62]

N. E. Steenrod,Products of cocycles and extensions of mappings,Annals of Mathematics 48.2(1947)

1947
[63]

Gaiotto and T

D. Gaiotto and T. Johnson-Freyd,Condensations in higher categories,1905.09566. – 37 –

work page arXiv 1905