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arxiv: 2604.25820 · v1 · submitted 2026-04-28 · ✦ hep-th

Recognition: unknown

Candidate Gaugings of Categorical Continuous Symmetry

Qiang Jia , Cheng Ma , Jiahua Tian
Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:35 UTC · model grok-4.3

classification ✦ hep-th
keywords gauging continuous symmetriesSymTFTmodular S and T kernelsChern-Simons theoryBF theoryLagrangian algebraHopf-link correlatorsanomalies
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The pith

Candidate modular S- and T-kernels derived from Hopf-link and framing correlators in BF plus level-k Chern-Simons theory supply candidate gaugings for continuous global symmetries.

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

The paper constructs a semiclassical model for the symmetry topological field theory of a quantum field theory that has a continuous global symmetry G and a possible anomaly labeled by an integer k. It combines BF theory with level-k Chern-Simons theory and extracts candidate modular S- and T-kernels by evaluating Hopf-link and framing correlators on the three-sphere. These kernels are then applied to generate candidate modular invariants and gaugings. The formulas recover all previously known discrete cases and indicate how the same kernel approach might extend to compact Lie groups.

Core claim

Using the BF plus level-k Chern-Simons theory as a semiclassical model for the SymTFT of a QFT with continuous G symmetry and anomaly k, the authors derive candidate modular S- and T-kernels from Hopf-link and framing correlators in S^3. These kernels are then employed to construct candidate modular invariants and gaugings, recovering established results and suggesting an extension of the kernel-theoretic approach to compact Lie groups.

What carries the argument

The candidate modular S- and T-kernels obtained from semi-classical Hopf-link and framing correlators in the BF+kCS SymTFT model; their common +1 eigenspaces are proposed to encode candidate Lagrangian algebra data that determine the gaugings.

If this is right

  • All established gaugings for finite groups appear as special cases of the derived formulas.
  • The same procedure supplies explicit candidate gaugings for any compact Lie group G once the Hopf-link and framing correlators are evaluated.
  • The modular invariants classify possible phases of the original theory after gauging, consistent with the anomaly level k.
  • The kernel construction supplies a uniform computational route to Lagrangian data in the Drinfeld center without requiring a direct algebraic description of the center for continuous groups.

Where Pith is reading between the lines

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

  • If the kernels prove accurate, the same correlator method could be applied to mixed anomalies or higher-form symmetries where direct Lagrangian-algebra constructions are unavailable.
  • Testing the predicted gaugings in concrete models such as two-dimensional sigma models with target space a Lie group would provide a direct check of whether the +1 eigenspace prescription selects physically allowed phases.
  • The recovery of discrete cases suggests that existing classification results for finite-group gaugings can be viewed as the k=0 or finite-subgroup limit of a single continuous construction.

Load-bearing premise

The BF plus level-k Chern-Simons theory correctly models the relevant SymTFT and the common +1 eigenspaces of the resulting modular kernels correctly detect candidate Lagrangian algebra data for gaugings in the continuous case.

What would settle it

Explicitly compute the modular kernels and resulting gaugings for a concrete group such as SU(2) at small level k and compare them against independent determinations of allowed gaugings obtained from anomaly inflow or known dualities in a specific quantum field theory.

read the original abstract

Different gaugings of the global symmetry of a quantum field theory are closely related to its various phases. In this work, we study candidate gaugeable symmetries by analyzing candidate Lagrangian algebra data in the Drinfeld center of a symmetry category $\mathscr{C}^k(G)$ associated to a QFT with continuous global $G$-symmetry and possible 't Hooft anomaly labeled by an integer $k$. We use the combination of the $BF$ theory and the level-$k$ Chern-Simons theory with gauge group $G$ as a semiclassical kernel-theoretic model for the corresponding SymTFT. Under two explicit assumptions, namely that this $BF{+}k$CS theory provides the relevant SymTFT model and that the common $+1$ eigenspaces of the resulting modular kernels detect candidate Lagrangian algebra data in the continuous setting, we derive candidate modular $S$- and $T$-kernels from Hopf-link and framing correlators in $S^3$ semi-classically. We then use these kernels to obtain candidate modular invariants and candidate gaugings. The resulting formulas recover the established cases and suggest a possible extension of this kernel-theoretic picture to compact Lie groups.

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 proposes a kernel-theoretic approach to candidate gaugings of continuous global G-symmetries (with possible 't Hooft anomaly labeled by integer k) in QFTs. It models the symmetry category C^k(G) via a semi-classical BF + level-k Chern-Simons theory as SymTFT, computes candidate modular S- and T-kernels from Hopf-link and framing correlators in S^3, and uses the common +1 eigenspaces of these kernels to extract candidate modular invariants and gaugings. The resulting formulas recover known discrete cases and suggest an extension to compact Lie groups, under two explicit assumptions about the SymTFT model and the eigenspace detection of Lagrangian algebra data.

Significance. If the assumptions hold and the eigenspace correspondence extends, the work would provide a concrete semi-classical route to candidate gaugings for continuous symmetries, recovering established results for finite groups as a consistency check and offering a potential unification of discrete and continuous symmetry gauging via modular kernels. The explicit semi-classical computation of kernels from correlators is a methodological strength.

major comments (2)
  1. [Abstract and assumptions section] Abstract and § on assumptions: The derivation of candidate modular invariants and gaugings rests on the unverified extrapolation that common +1 eigenspaces of the semi-classically computed S- and T-kernels detect candidate Lagrangian algebra data. This identification is standard for finite semisimple fusion categories but the manuscript provides no justification for its validity in the non-semisimple continuous setting of C^k(G) for compact Lie G, where irreps form continuous families.
  2. [SymTFT model section] SymTFT model section: The choice of BF + level-k Chern-Simons theory as the relevant SymTFT for C^k(G) is introduced as an explicit assumption without derivation, comparison to alternative models, or verification that its Hopf-link and framing correlators correctly reproduce the expected symmetry category data in the continuous limit.
minor comments (2)
  1. [Introduction] Notation for the symmetry category C^k(G) and the kernels could be clarified with explicit definitions early in the text to aid readers unfamiliar with the continuous extension.
  2. [Discussion] The manuscript would benefit from a brief discussion of potential limitations or counterexamples where the eigenspace method might fail in the continuous case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comments point by point below, acknowledging the limitations of our assumptions while strengthening the presentation of their scope and motivation. Revisions have been made to clarify these points without altering the candidate nature of the results.

read point-by-point responses

  1. Referee: [Abstract and assumptions section] Abstract and § on assumptions: The derivation of candidate modular invariants and gaugings rests on the unverified extrapolation that common +1 eigenspaces of the semi-classically computed S- and T-kernels detect candidate Lagrangian algebra data. This identification is standard for finite semisimple fusion categories but the manuscript provides no justification for its validity in the non-semisimple continuous setting of C^k(G) for compact Lie G, where irreps form continuous families.

    Authors: We agree that the common +1 eigenspace identification is an extrapolation from the finite semisimple case and is presented explicitly as one of the two core assumptions in the manuscript. The work is exploratory, deriving candidate modular invariants and gaugings under these assumptions, with recovery of known discrete results serving as a consistency check. In the revised manuscript, we have expanded the assumptions section with additional discussion of the challenges posed by continuous families of irreps and the Drinfeld center structure, explaining why the eigenspace approach provides a natural semiclassical starting point. A rigorous proof of the correspondence for non-semisimple categories with continuous spectra lies beyond the present scope. revision: partial

  2. Referee: [SymTFT model section] SymTFT model section: The choice of BF + level-k Chern-Simons theory as the relevant SymTFT for C^k(G) is introduced as an explicit assumption without derivation, comparison to alternative models, or verification that its Hopf-link and framing correlators correctly reproduce the expected symmetry category data in the continuous limit.

    Authors: The BF + kCS theory is adopted as a semiclassical SymTFT model because its Hopf-link and framing correlators in S^3 yield the expected modular kernels for the anomaly-labeled symmetry category C^k(G), reducing correctly to established discrete cases. It is stated as an assumption since a complete first-principles derivation of the SymTFT for continuous G remains an open question. In the revision, we have added a dedicated paragraph comparing this choice to alternative topological models (e.g., those based on higher-form symmetries or other CS variants) and clarifying the semiclassical nature of the correlator computations. Full non-perturbative verification in the continuous limit is not feasible within the semiclassical framework employed here. revision: partial

Circularity Check

0 steps flagged

No significant circularity: kernels computed explicitly from model correlators under stated assumptions

full rationale

The paper explicitly introduces two modeling assumptions (BF+kCS as SymTFT and +1 eigenspaces detecting Lagrangian data) before computing S- and T-kernels from Hopf-link and framing correlators in S^3. These kernels are then used to propose candidate invariants and gaugings via the assumed eigenspace correspondence. Because the central computations are direct extractions from the semi-classical model rather than fits or self-referential definitions, and because the correspondence to gaugings is labeled 'candidate' without claiming a first-principles derivation that collapses to the inputs, the chain does not reduce by construction. Recovery of known discrete cases supplies an external consistency check, leaving the work self-contained against the stated premises.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The claim rests on two ad-hoc modeling assumptions introduced to bridge the discrete-to-continuous gap; no independent evidence or parameter-free derivation is supplied for the continuous case.

free parameters (2)
  • k
    Integer labeling the 't Hooft anomaly of the G-symmetry
  • G
    Continuous compact Lie group serving as the global symmetry
axioms (2)
  • ad hoc to paper BF + level-k Chern-Simons theory supplies the relevant SymTFT for the symmetry category C^k(G)
    Explicit assumption 1 stated in the abstract
  • ad hoc to paper Common +1 eigenspaces of the modular kernels detect candidate Lagrangian algebra data in the continuous setting
    Explicit assumption 2 stated in the abstract
invented entities (1)
  • candidate Lagrangian algebra data in the continuous setting no independent evidence
    purpose: To label consistent gaugings of the continuous symmetry
    Introduced via the +1 eigenspace construction; no independent falsifiable handle given

pith-pipeline@v0.9.0 · 5500 in / 1614 out tokens · 50293 ms · 2026-05-07T15:35:43.598369+00:00 · methodology

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Reference graph

Works this paper leans on

57 extracted references · 41 canonical work pages · 1 internal anchor

  1. [1]

    Landau et al.,On the theory of phase transitions,Zh

    L.D. Landau et al.,On the theory of phase transitions,Zh. eksp. teor. Fiz7(1937) 926

    1937

  2. [2]

    Tol´ edano and P

    J.C. Tol´ edano and P. Tol´ edano,The Landau Theory of Phase Transitions, WORLD SCIENTIFIC (1987), 10.1142/0215, [https://www.worldscientific.com/doi/pdf/10.1142/0215]

    work page doi:10.1142/0215 1987

  3. [3]

    Hohenberg and A

    P. Hohenberg and A. Krekhov,An introduction to the ginzburg–landau theory of phase transitions and nonequilibrium patterns,Physics Reports572(2015) 1

    2015

  4. [4]

    Moore and N

    G.W. Moore and N. Read,Nonabelions in the fractional quantum Hall effect,Nucl. Phys. B 360(1991) 362

    1991

  5. [5]

    Wen,Topological orders and edge excitations in FQH states,Adv

    X.-G. Wen,Topological orders and edge excitations in FQH states,Adv. Phys.44(1995) 405 [cond-mat/9506066]

    work page arXiv 1995

  6. [6]

    Wen,Topological Order in Rigid States,Int

    X.G. Wen,Topological Order in Rigid States,Int. J. Mod. Phys. B4(1990) 239

    1990

  7. [7]

    Wen,Theory of the edge states in fractional quantum Hall effects,Int

    X.-G. Wen,Theory of the edge states in fractional quantum Hall effects,Int. J. Mod. Phys. B 6(1992) 1711

    1992

  8. [8]

    Moore and N

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

    1989

  9. [9]

    Anyons in an exactly solved model and beyond

    A. Kitaev,Anyons in an exactly solved model and beyond,Annals Phys.321(2006) 2 [cond-mat/0506438]

    work page Pith review arXiv 2006

  10. [10]

    Witten,Quantum Field Theory and the Jones Polynomial,Commun

    E. Witten,Quantum Field Theory and the Jones Polynomial,Commun. Math. Phys.121 (1989) 351

    1989

  11. [11]

    Reshetikhin and V.G

    N. Reshetikhin and V.G. Turaev,Invariants of 3-manifolds via link polynomials and quantum groups,Inventiones mathematicae103(1991) 547

    1991

  12. [12]

    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

  13. [13]

    Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

    L. Kong and X.-G. Wen,Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions,1405.5858

    work page Pith review arXiv

  14. [14]

    Kong, X.-G

    L. Kong, X.-G. Wen and H. Zheng,Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers,1502.01690

    work page arXiv

  15. [15]

    Boundary-bulk relation in topological orders

    L. Kong, X.-G. Wen and H. Zheng,Boundary-bulk relation in topological orders,Nucl. Phys. B922(2017) 62 [1702.00673]

    work page Pith review arXiv 2017

  16. [16]

    Symmetry TFTs from String Theory,

    F. Apruzzi, F. Bonetti, I. Garc´ ıa Etxebarria, S.S. Hosseini and S. Schafer-Nameki,Symmetry TFTs from String Theory,Commun. Math. Phys.402(2023) 895 [2112.02092]

    work page arXiv 2023

  17. [17]

    Topological symmetry in quantum field theory,

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

    work page arXiv

  18. [18]

    ICTP lectures on (non-)invertible generalized symmetries,

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

    work page arXiv 2024

  19. [19]

    Bhardwaj, L

    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 [2307.07547]

    work page arXiv 2024

  20. [20]

    Q. Jia, R. Luo, J. Tian, Y.-N. Wang and Y. Zhang,Categorical Continuous Symmetry, 2509.13170. – 49 –

    work page arXiv

  21. [21]

    Stockall and M

    D. Stockall and M. Yu,Geometric Categories for Continuous Gauging,2511.08254

    work page arXiv

  22. [22]

    Freed, M.J

    D.S. Freed, M.J. Hopkins, J. Lurie and C. Teleman,Topological Quantum Field Theories from Compact Lie Groups, inA Celebration of Raoul Bott’s Legacy in Mathematics, 5, 2009 [0905.0731]

    work page arXiv 2009

  23. [23]

    SymTFTs for Continuous non-Abelian Symmetries,

    F. Bonetti, M. Del Zotto and R. Minasian,SymTFTs for continuous non-Abelian symmetries,Phys. Lett. B871(2025) 140010 [2402.12347]

    work page arXiv 2025

  24. [24]

    Q. Jia, R. Luo, J. Tian, Y.-N. Wang and Y. Zhang,Symmetry Topological Field Theory for Flavor Symmetry,2503.04546

    work page arXiv

  25. [25]

    Q. Jia, R. Luo, J. Tian, Y.-N. Wang and Y. Zhang,Anomaly of Continuous Symmetries from Topological Defect Network,2510.14722

    work page arXiv

  26. [26]

    Higher-dimensional Algebra and Topological Quantum Field Theory

    J.C. Baez and J. Dolan,Higher dimensional algebra and topological quantum field theory,J. Math. Phys.36(1995) 6073 [q-alg/9503002]

    work page Pith review arXiv 1995

  27. [27]

    On the Classification of Topological Field Theories

    J. Lurie,On the Classification of Topological Field Theories,0905.0465

    work page Pith review arXiv

  28. [28]

    Freed,The cobordism hypothesis,1210.5100

    D.S. Freed,The cobordism hypothesis,1210.5100

    work page arXiv

  29. [29]

    Henriques,What Chern-Simons theory assigns to a point,1503.06254

    A. Henriques,What Chern-Simons theory assigns to a point,1503.06254

    work page arXiv

  30. [30]

    Ji and X.-G

    W. Ji and X.-G. Wen,Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions,Phys. Rev. Research.1(2019) 033054 [1905.13279]

    work page arXiv 2019

  31. [31]

    Kaidi, Z

    J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri and S.-H. Shao,Higher central charges and topological boundaries in 2+1-dimensional TQFTs,SciPost Phys.13(2022) 067 [2107.13091]

    work page arXiv 2022

  32. [32]

    You,Gapped Boundaries of Fermionic Topological Orders and Higher Central Charges, Phys

    M. You,Gapped Boundaries of Fermionic Topological Orders and Higher Central Charges, Phys. Rev. Lett.133(2024) 086601 [2311.01096]

    work page arXiv 2024

  33. [33]

    Yetter,Measurable categories,Applied Categorical Structures13(2005) 469

    D.N. Yetter,Measurable categories,Applied Categorical Structures13(2005) 469

    2005

  34. [34]

    Waldorf,Transgressive loop group extensions,Mathematische Zeitschrift286(2017) 325

    K. Waldorf,Transgressive loop group extensions,Mathematische Zeitschrift286(2017) 325

    2017

  35. [35]

    Amabel, A

    A. Amabel, A. Debray and P.J. Haine,Differential cohomology: categories, characteristic classes, and connections,arXiv preprint arXiv:2109.12250(2021)

    work page arXiv 2021

  36. [36]

    Costello and O

    K. Costello and O. Gwilliam,Factorization Algebras in Quantum Field Theory, New Mathematical Monographs (41), Cambridge University Press (9, 2021), 10.1017/9781316678664

    work page doi:10.1017/9781316678664 2021

  37. [38]

    Ben-Zvi, J

    D. Ben-Zvi, J. Francis and D. Nadler,Morita equivalence for convolution categories: Appendix to arxiv: 0805.0157,arXiv preprint arXiv:1209.0193(2012)

    work page arXiv 2012

  38. [39]

    Q. Jia, R. Luo, J. Tian, Y.-N. Wang and Y. Zhang, To appear

  39. [40]

    Turaev,Quantum invariants of knots and 3-manifolds, vol

    V.G. Turaev,Quantum invariants of knots and 3-manifolds, vol. 18, Walter de Gruyter GmbH & Co KG (2016)

    2016

  40. [41]

    Bakalov, A.A

    B. Bakalov, A.A. Kirillov et al.,Lectures on tensor categories and modular functors, vol. 21, American Mathematical Society Providence, RI (2001)

    2001

  41. [42]

    Lyubashenko,Invariants of three manifolds and projective representations of mapping – 50 – class groups via quantum groups at roots of unity,Commun

    V.V. Lyubashenko,Invariants of three manifolds and projective representations of mapping – 50 – class groups via quantum groups at roots of unity,Commun. Math. Phys.172(1995) 467 [hep-th/9405167]

    work page arXiv 1995

  42. [43]

    Lyubashenko,Modular transformations for tensor categories,Journal of Pure and Applied Algebra98(1995) 279

    V. Lyubashenko,Modular transformations for tensor categories,Journal of Pure and Applied Algebra98(1995) 279

    1995

  43. [44]

    De Renzi, A.M

    M. De Renzi, A.M. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel,3-Dimensional TQFTs from non-semisimple modular categories,Selecta Math.28(2022) 42 [1912.02063]

    work page arXiv 2022

  44. [45]

    Hofer and I

    A. Hofer and I. Runkel,Modular functors from non-semisimple 3d TFTs,2405.18038

    work page arXiv

  45. [46]

    Coupling a QFT to a TQFT and Duality

    A. Kapustin and N. Seiberg,Coupling a QFT to a TQFT and Duality,JHEP04(2014) 001 [1401.0740]

    work page Pith review arXiv 2014

  46. [47]

    Dijkgraaf and E

    R. Dijkgraaf and E. Witten,Topological Gauge Theories and Group Cohomology,Commun. Math. Phys.129(1990) 393

    1990

  47. [48]

    Cattaneo and C.A

    A.S. Cattaneo and C.A. Rossi,Wilson surfaces and higher dimensional knot invariants, Commun. Math. Phys.256(2005) 513 [math-ph/0210037]

    work page arXiv 2005

  48. [49]

    Cordova, K

    C. Cordova, K. Ohmori and T. Rudelius,Generalized symmetry breaking scales and weak gravity conjectures,JHEP11(2022) 154 [2202.05866]

    work page arXiv 2022

  49. [50]

    A SymTFT for continuous symmetries,

    T.D. Brennan and Z. Sun,A SymTFT for continuous symmetries,JHEP12(2024) 100 [2401.06128]

    work page arXiv 2024

  50. [51]

    Anomalies and gauging of U(1) symmetries,

    A. Antinucci and F. Benini,Anomalies and gauging of U(1) symmetries,Phys. Rev. B111 (2025) 024110 [2401.10165]

    work page arXiv 2025

  51. [52]

    Lin and S.-H

    Y.-H. Lin and S.-H. Shao,Z N symmetries, anomalies, and the modular bootstrap,Phys. Rev. D103(2021) 125001 [2101.08343]

    work page arXiv 2021

  52. [53]

    Di Francesco, P

    P. Di Francesco, P. Mathieu and D. Senechal,Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997), 10.1007/978-1-4612-2256-9

    work page doi:10.1007/978-1-4612-2256-9 1997

  53. [54]

    Gepner and Z.-a

    D. Gepner and Z.-a. Qiu,Modular Invariant Partition Functions for Parafermionic Field Theories,Nucl. Phys. B285(1987) 423

    1987

  54. [55]

    Gannon,U(1)-m modular invariants, N=2 minimal models, and the quantum Hall effect, Nucl

    T. Gannon,U(1)-m modular invariants, N=2 minimal models, and the quantum Hall effect, Nucl. Phys. B491(1997) 659 [hep-th/9608063]

    work page arXiv 1997

  55. [56]

    Thorngren and Y

    R. Thorngren and Y. Wang,Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond,JHEP07(2024) 051 [2106.12577]

    work page arXiv 2024

  56. [57]

    Lynker and R

    M. Lynker and R. Schimmrigk,Geometric Kac-Moody modularity,J. Geom. Phys.56(2006) 843 [hep-th/0410189]

    work page arXiv 2006

  57. [58]

    Witten,Two-dimensional gauge theories revisited,J

    E. Witten,Two-dimensional gauge theories revisited,J. Geom. Phys.9(1992) 303 [hep-th/9204083]. – 51 –

    work page arXiv 1992