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arxiv: 2606.17708 · v2 · pith:2UHMPIIRnew · submitted 2026-06-16 · ✦ hep-th · cond-mat.str-el· hep-lat

Monopoles, Center Vortices, Confinement in (3+1)d, and the Lens-Space Twisted Partition Function

Yui Hayashi , Yuya Tanizaki This is my paper

Pith reviewed 2026-06-27 00:08 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elhep-lat
keywords center vorticesmonopolesconfinementtwisted partition functionslens spaceadjoint Higgs phasetopological field theorysymmetry fractionalization
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The pith

The gapped phase with center-vortex condensation necessarily exhibits monopole condensation as well.

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

The paper proposes gauge-invariant diagnostics for center-vortex condensation and monopole condensation based on twisted partition functions under a discrete one-form symmetry. The torus twisted partition function serves as the diagnostic for vortex condensation, while the lens-space version diagnoses monopole condensation. These criteria are justified by explicit calculations in the adjoint Higgs phase showing that the leading nontrivial contributions arise from vortices and monopoles, respectively. Using methods from topological field theory, the authors prove that any gapped phase with vortex condensation must also feature monopole condensation, and they illustrate the result with a model exhibiting symmetry fractionalization for higher-charge monopoles.

Core claim

The authors establish gauge-invariant criteria for center-vortex and monopole condensation via Z_N^{[1]}-symmetry twisted partition functions, with the torus version detecting vortex condensation and the lens-space version detecting monopole condensation. They prove that in the gapped phase, center-vortex condensation entails monopole condensation as well, and illustrate this with a center-vortex model where higher-charge monopoles exhibit symmetry fractionalization beyond the standard Wilson-'t Hooft classification.

What carries the argument

The Z_N^{[1]}-symmetry twisted partition functions on the torus (for center-vortex condensation) and on lens space (for monopole condensation), which act as order parameters derived from topological field theory.

If this is right

  • In the adjoint Higgs phase the leading contributions to the respective twisted partition functions arise from center vortices and monopoles.
  • Any gapped phase with center-vortex condensation must also exhibit monopole condensation.
  • Higher-charge monopole condensation in a center-vortex model realizes symmetry fractionalization outside the conventional Wilson-'t Hooft classification.

Where Pith is reading between the lines

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

  • Lattice simulations could test the criteria by measuring the twisted partition functions directly without additional gauge fixing.
  • The implication suggests that proposed confinement mechanisms based on vortex condensation in four dimensions automatically incorporate monopole effects.
  • The lens-space construction may extend to other discrete symmetries or spacetime topologies for diagnosing topological defects.

Load-bearing premise

The leading nontrivial contributions to the twisted partition functions in the adjoint Higgs phase come specifically from center vortices and monopoles respectively.

What would settle it

A concrete counterexample consisting of a gapped phase in which the torus twisted partition function indicates center-vortex condensation but the lens-space twisted partition function shows no monopole condensation would falsify the claimed implication.

read the original abstract

We propose the gauge-invariant criteria of center-vortex condensation and monopole condensation using the $\mathbb{Z}_N^{[1]}$-symmetry twisted partition functions: The torus twisted partition function characterizes the center-vortex condensation, and the lens-space twisted partition function characterizes the monopole condensation. To justify our proposal, we study how these twisted partition functions behave in the adjoint Higgs phase and show that their leading nontrivial contributions come from the center vortex and monopole, respectively. Using the techniques of topological field theories, we uncover the relation between the center-vortex and monopole condensations, and in particular, we prove that the gapped phase with the center-vortex condensation necessarily shows the monopole condensation, too. We then study a center-vortex model with monopoles as an illustrative example, and the higher-charge monopole condensation gives an example of the symmetry fractionalization, which goes beyond the conventional Wilson-'t Hooft classification.

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

1 major / 0 minor

Summary. The paper proposes gauge-invariant criteria for center-vortex condensation (via torus twisted partition functions) and monopole condensation (via lens-space twisted partition functions) using the Z_N^{[1]}-symmetry. It justifies the criteria by studying the adjoint Higgs phase, where leading nontrivial contributions are attributed to vortices and monopoles respectively. TFT techniques are then used to prove that a gapped phase with center-vortex condensation necessarily exhibits monopole condensation as well. An illustrative center-vortex model with monopoles is analyzed, with higher-charge monopole condensation providing an example of symmetry fractionalization beyond the Wilson-'t Hooft classification.

Significance. If the identification of leading contributions holds and the TFT relation is robust, the work supplies a concrete link between two condensation mechanisms relevant to confinement, with potential implications for (3+1)d gauge theories. The use of twisted partition functions as order parameters and the explicit example of symmetry fractionalization are strengths that could guide lattice or model-building studies.

major comments (1)
  1. [Abstract and adjoint Higgs phase justification] Abstract and the justification of the proposal in the adjoint Higgs phase: the assertion that the leading nontrivial contributions to the torus and lens-space twisted partition functions come specifically from center vortices and monopoles requires an explicit demonstration (e.g., via classification of sectors or exponential suppression estimates) that other configurations such as non-minimal windings, dyonic combinations, or non-topological fluctuations do not contribute at the same order. This identification is load-bearing for both the characterization of the condensations and the subsequent TFT implication that vortex condensation entails monopole condensation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract and the justification of the proposal in the adjoint Higgs phase: the assertion that the leading nontrivial contributions to the torus and lens-space twisted partition functions come specifically from center vortices and monopoles requires an explicit demonstration (e.g., via classification of sectors or exponential suppression estimates) that other configurations such as non-minimal windings, dyonic combinations, or non-topological fluctuations do not contribute at the same order. This identification is load-bearing for both the characterization of the condensations and the subsequent TFT implication that vortex condensation entails monopole condensation.

    Authors: We agree that strengthening the justification with an explicit demonstration would improve the manuscript. In the adjoint Higgs phase analysis, the twisted partition functions receive contributions from topological sectors labeled by winding numbers around the torus or lens space; the minimal-winding vortex and monopole configurations dominate because higher-winding or dyonic sectors carry larger action (proportional to the square of the winding number) and are exponentially suppressed at weak coupling. Non-topological fluctuations are further suppressed by the Higgs mass gap. We will revise the manuscript to include a dedicated paragraph with these suppression estimates and a brief classification of sectors, added to the discussion of the adjoint Higgs phase. This clarification supports but does not alter the subsequent TFT argument, which relies only on the existence of the condensates once identified. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via external TFT and explicit phase study

full rationale

The paper's central claim (vortex condensation implies monopole condensation) is derived using topological field theory techniques presented as an external framework, combined with direct study of twisted partition functions in the adjoint Higgs phase to identify leading contributions. No steps reduce by construction to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations whose content is unverified within the paper. The justification for leading contributions is asserted via explicit analysis rather than tautological mapping, and the implication proof does not rely on renaming or smuggling ansatze. This matches the default expectation of non-circularity for papers with independent external tools.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract relies on standard TFT techniques and Z_N symmetry properties without detailing new free parameters or invented entities; full text would be needed to audit specific axioms.

axioms (1)
  • standard math Standard properties of topological field theories for gauge theories with Z_N^{[1]} symmetry
    Invoked to relate vortex and monopole condensations.

pith-pipeline@v0.9.1-grok · 5698 in / 1100 out tokens · 30674 ms · 2026-06-27T00:08:24.539135+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

99 extracted references · 46 linked inside Pith

  1. [1]

    Strings, Monopoles and Gauge Fields,

    Y. Nambu, “Strings, Monopoles and Gauge Fields,”Phys. Rev.D10(1974) 4262

    1974

  2. [2]

    Vortices and Quark Confinement in Nonabelian Gauge Theories,

    S. Mandelstam, “Vortices and Quark Confinement in Nonabelian Gauge Theories,” inPhys. Rep. 23 (1976) 245-249, In *Gervais, J.L. (Ed.), Jacob, M. (Ed.): Non-linear and Collective Phenomena In Quantum Physics*, 12-16, vol. 23, pp. 245–249. 1976

    1976

  3. [3]

    Compact Gauge Fields and the Infrared Catastrophe,

    A. M. Polyakov, “Compact Gauge Fields and the Infrared Catastrophe,”Phys. Lett.B59 (1975) 82–84

    1975

  4. [4]

    On the phase transition towards permanent quark confinement,

    G. ’t Hooft, “On the phase transition towards permanent quark confinement,”Nucl.Phys.B 138(1978) 1–25

    1978

  5. [5]

    Quark confinement and vortices in massive gauge invariant QCD,

    J. M. Cornwall, “Quark confinement and vortices in massive gauge invariant QCD,”Nucl. Phys. B157no. UCLA/79/TEP/5, (1979) 392–412

    1979

  6. [6]

    A quantum liquid model for the QCD vacuum: Gauge and rotational invariance of domained and quantized homogeneous color fields,

    H. B. Nielsen and P. Olesen, “A quantum liquid model for the QCD vacuum: Gauge and rotational invariance of domained and quantized homogeneous color fields,”Nucl. Phys. B 160no. NBI-HE-79-17, (1979) 380–396

    1979

  7. [7]

    A color magnetic vortex condensate in QCD,

    J. Ambjorn and P. Olesen, “A color magnetic vortex condensate in QCD,”Nucl. Phys. B 170no. NBI-HE-80-14, (1980) 265–282

    1980

  8. [8]

    Abelian Dominance and Quark Confinement in Yang-Mills Theories,

    Z. F. Ezawa and A. Iwazaki, “Abelian Dominance and Quark Confinement in Yang-Mills Theories,”Phys. Rev. D25(1982) 2681

    1982

  9. [9]

    A Ginzburg-Landau Type Theory of Quark Confinement,

    T. Suzuki, “A Ginzburg-Landau Type Theory of Quark Confinement,”Prog. Theor. Phys. 80(1988) 929

    1988

  10. [10]

    Color confinement, quark pair creation and dynamical chiral symmetry breaking in the dual Ginzburg-Landau theory,

    H. Suganuma, S. Sasaki, and H. Toki, “Color confinement, quark pair creation and dynamical chiral symmetry breaking in the dual Ginzburg-Landau theory,”Nucl. Phys. B 435(1995) 207–240,arXiv:hep-ph/9312350

    Pith/arXiv arXiv 1995

  11. [11]

    Abelian projected effective gauge theory of QCD with asymptotic freedom and quark confinement,

    K.-I. Kondo, “Abelian projected effective gauge theory of QCD with asymptotic freedom and quark confinement,”Phys. Rev. D57(1998) 7467–7487,arXiv:hep-th/9709109

    Pith/arXiv arXiv 1998

  12. [12]

    Topology and Dynamics of the Confinement Mechanism,

    A. S. Kronfeld, G. Schierholz, and U. J. Wiese, “Topology and Dynamics of the Confinement Mechanism,”Nucl. Phys. B293(1987) 461–478

    1987

  13. [13]

    A possible evidence for Abelian dominance in quark confinement,

    T. Suzuki and I. Yotsuyanagi, “A possible evidence for Abelian dominance in quark confinement,”Phys. Rev. D42(1990) 4257–4260. – 31 –

    1990

  14. [14]

    Center dominance and Z(2) vortices in SU(2) lattice gauge theory,

    L. Del Debbio, M. Faber, J. Greensite, and S. Olejnik, “Center dominance and Z(2) vortices in SU(2) lattice gauge theory,”Phys. Rev. D55no. LBL-39424, LBNL-39424, SWAT-96-119, (1997) 2298–2306,arxiv:hep-lat/9610005

    Pith/arXiv arXiv 1997

  15. [15]

    Center vortices of Yang-Mills theory at finite temperatures,

    K. Langfeld, O. Tennert, M. Engelhardt, and H. Reinhardt, “Center vortices of Yang-Mills theory at finite temperatures,”Phys. Lett. B452no. UNITU-THEP-5-98, (1999) 301, arxiv:hep-lat/9805002

    Pith/arXiv arXiv 1999

  16. [16]

    Vortices and confinement at weak coupling,

    T. G. Kovacs and E. T. Tomboulis, “Vortices and confinement at weak coupling,”Phys. Rev. D57no. UCLA-97-TEP-22, COLO-HEP-392, (1998) 4054–4062,arxiv:hep-lat/9711009

    Pith/arXiv arXiv 1998

  17. [17]

    Deconfinement in SU(2) Yang-Mills theory as a center vortex percolation transition,

    M. Engelhardt, K. Langfeld, H. Reinhardt, and O. Tennert, “Deconfinement in SU(2) Yang-Mills theory as a center vortex percolation transition,”Phys. Rev. D61(2000) 054504, arxiv:hep-lat/9904004

    Pith/arXiv arXiv 2000

  18. [18]

    On the relevance of center vortices to QCD,

    P. de Forcrand and M. D’Elia, “On the relevance of center vortices to QCD,”Phys. Rev. Lett.82(1999) 4582–4585,arxiv:hep-lat/9901020

    Pith/arXiv arXiv 1999

  19. [19]

    Center dominance, center vortices, and confinement,

    L. Del Debbio, M. Faber, J. Greensite, and S. Olejnik, “Center dominance, center vortices, and confinement,” inNATO Advanced Research Workshop on Theoretical Physics: New Developments in Quantum Field Theory, pp. 47–64. 6, 1997.arXiv:hep-lat/9708023

    Pith/arXiv arXiv 1997

  20. [20]

    Vortex structure versus monopole dominance in Abelian projected gauge theory,

    J. Ambjorn, J. Giedt, and J. Greensite, “Vortex structure versus monopole dominance in Abelian projected gauge theory,”JHEP02(2000) 033,arxiv:hep-lat/9907021

    Pith/arXiv arXiv 2000

  21. [21]

    Center vortices and monopoles without lattice Gribov copies,

    P. de Forcrand and M. Pepe, “Center vortices and monopoles without lattice Gribov copies,” Nucl. Phys. B598(2001) 557–577,arXiv:hep-lat/0008016

    Pith/arXiv arXiv 2001

  22. [22]

    Center projection vortices in continuum Yang-Mills theory,

    M. Engelhardt and H. Reinhardt, “Center projection vortices in continuum Yang-Mills theory,”Nucl. Phys. B567(2000) 249,arXiv:hep-th/9907139

    Pith/arXiv arXiv 2000

  23. [23]

    Topology of center vortices,

    H. Reinhardt, “Topology of center vortices,”Nucl. Phys. B628(2002) 133–166, arXiv:hep-th/0112215

    Pith/arXiv arXiv 2002

  24. [24]

    Center vortices, nexuses, and fractional topological charge,

    J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,”Phys. Rev. D 61(2000) 085012,arXiv:hep-th/9911125

    Pith/arXiv arXiv 2000

  25. [25]

    Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism,

    Y. Hayashi and Y. Tanizaki, “Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism,”Phys. Rev. Lett.133no. 17, (2024) 171902,arXiv:2405.12402 [hep-th]

    arXiv 2024

  26. [26]

    Minimum Action Solutions for SU(2) Gauge Theory on the Torus With Nonorthogonal Twist,

    M. Garcia Perez, A. Gonzalez-Arroyo, and B. Soderberg, “Minimum Action Solutions for SU(2) Gauge Theory on the Torus With Nonorthogonal Twist,”Phys. Lett. B235(1990) 117–123

    1990

  27. [27]

    Numerical study of Yang-Mills classical solutions on the twisted torus,

    M. Garcia Perez and A. Gonzalez-Arroyo, “Numerical study of Yang-Mills classical solutions on the twisted torus,”J. Phys. A26no. FTUAM-92-08, (1993) 2667–2678, arxiv:hep-lat/9206016

    Pith/arXiv arXiv 1993

  28. [28]

    Constituents of doubly periodic instantons,

    C. Ford and J. M. Pawlowski, “Constituents of doubly periodic instantons,”Phys. Lett. B 540(2002) 153–158,arXiv:hep-th/0205116

    Pith/arXiv arXiv 2002

  29. [29]

    Doubly periodic instantons and their constituents,

    C. Ford and J. M. Pawlowski, “Doubly periodic instantons and their constituents,”Phys. Rev. D69(2004) 065006,arXiv:hep-th/0302117

    Pith/arXiv arXiv 2004

  30. [30]

    Fractional instanton of the SU(3) gauge theory in weak coupling regime,

    E. Itou, “Fractional instanton of the SU(3) gauge theory in weak coupling regime,”JHEP05 (2019) 093,arxiv:1811.05708 [hep-th]. – 32 –

    Pith/arXiv arXiv 2019

  31. [31]

    SU(N) fractional instantons and the Fibonacci sequence,

    J. Dasilva Gol´ an and M. Garc´ ıa P´ erez, “SU(N) fractional instantons and the Fibonacci sequence,”JHEP12(2022) 109,arXiv:2208.07133 [hep-th]

    arXiv 2022

  32. [32]

    Numerical fractional instantons in SU(2): center vortices, monopoles, and a sharp transition between them,

    F. D. Wandler, “Numerical fractional instantons in SU(2): center vortices, monopoles, and a sharp transition between them,”arXiv:2406.07636 [hep-lat]

    arXiv

  33. [33]

    Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications,

    Y. Tanizaki and M. ¨Unsal, “Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications,”PTEP2022(2022) 04A108,arXiv:2201.06166 [hep-th]

    arXiv 2022

  34. [34]

    Semiclassics with ’t Hooft flux background for QCD with 2-index quarks,

    Y. Tanizaki and M. ¨Unsal, “Semiclassics with ’t Hooft flux background for QCD with 2-index quarks,”JHEP08(2022) 038,arXiv:2205.11339 [hep-th]

    arXiv 2022

  35. [35]

    Semiclassical analysis of the bifundamental QCD onR 2 ×T 2 with ’t Hooft flux,

    Y. Hayashi, Y. Tanizaki, and H. Watanabe, “Semiclassical analysis of the bifundamental QCD onR 2 ×T 2 with ’t Hooft flux,”JHEP10(2023) 146,arXiv:2307.13954 [hep-th]

    arXiv 2023

  36. [36]

    Non-supersymmetric duality cascade of QCD(BF) via semiclassics onR 2 ×T 2 with the baryon-’t Hooft flux,

    Y. Hayashi, Y. Tanizaki, and H. Watanabe, “Non-supersymmetric duality cascade of QCD(BF) via semiclassics onR 2 ×T 2 with the baryon-’t Hooft flux,”arXiv:2404.16803 [hep-th]

    arXiv

  37. [37]

    Semiclassics for the QCD vacuum structure through T2-compactification with the baryon-’t Hooft flux,

    Y. Hayashi and Y. Tanizaki, “Semiclassics for the QCD vacuum structure through T2-compactification with the baryon-’t Hooft flux,”JHEP08(2024) 001, arXiv:2402.04320 [hep-th]

    arXiv 2024

  38. [38]

    Monopole-vortex continuity ofN= 1 super Yang-Mills theory onR 2 ×S 1 ×S 1 with ’t Hooft twist,

    Y. Hayashi, T. Misumi, and Y. Tanizaki, “Monopole-vortex continuity ofN= 1 super Yang-Mills theory onR 2 ×S 1 ×S 1 with ’t Hooft twist,”JHEP05(2025) 194, arXiv:2410.21392 [hep-th]

    arXiv 2025

  39. [39]

    The metamorphosis of semi-classical mechanisms of confinement: from monopoles onR 3 ×S 1 to center-vortices onR 2 ×T 2,

    C. G¨ uvendik, T. Schaefer, and M. ¨Unsal, “The metamorphosis of semi-classical mechanisms of confinement: from monopoles onR 3 ×S 1 to center-vortices onR 2 ×T 2,”JHEP11 (2024) 163,arXiv:2405.13696 [hep-th]

    arXiv 2024

  40. [40]

    Center-vortex semiclassics with non-minimal ’t Hooft fluxes onR 2 ×T 2 and center stabilization at large N,

    Y. Hayashi, Y. Tanizaki, and M. ¨Unsal, “Center-vortex semiclassics with non-minimal ’t Hooft fluxes onR 2 ×T 2 and center stabilization at large N,”JHEP02(2026) 126, arXiv:2505.07467 [hep-th]

    arXiv 2026

  41. [41]

    From 4d Yang-Mills to 2dCP N−1 model: IR problem and confinement at weak coupling,

    M. Yamazaki and K. Yonekura, “From 4d Yang-Mills to 2dCP N−1 model: IR problem and confinement at weak coupling,”JHEP07(2017) 088,arXiv:1704.05852 [hep-th]

    Pith/arXiv arXiv 2017

  42. [42]

    The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles,

    A. A. Cox, E. Poppitz, and F. D. Wandler, “The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles,”JHEP10(2021) 069,arXiv:2106.11442 [hep-th]

    arXiv 2021

  43. [43]

    Abelian duality, confinement, and chiral symmetry breaking in QCD(adj),

    M. Unsal, “Abelian duality, confinement, and chiral symmetry breaking in QCD(adj),”Phys. Rev. Lett.100(2008) 032005,arXiv:0708.1772 [hep-th]

    Pith/arXiv arXiv 2008

  44. [44]

    Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,

    M. Unsal, “Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,”Phys. Rev.D80(2009) 065001,arXiv:0709.3269 [hep-th]

    Pith/arXiv arXiv 2009

  45. [45]

    Center-stabilized Yang-Mills theory: Confinement and large N volume independence,

    M. Unsal and L. G. Yaffe, “Center-stabilized Yang-Mills theory: Confinement and large N volume independence,”Phys. Rev.D78(2008) 065035,arXiv:0803.0344 [hep-th]

    Pith/arXiv arXiv 2008

  46. [46]

    QCD-like Theories on R(3) x S(1): A Smooth Journey from Small to Large r(S(1)) with Double-Trace Deformations,

    M. Shifman and M. Unsal, “QCD-like Theories on R(3) x S(1): A Smooth Journey from Small to Large r(S(1)) with Double-Trace Deformations,”Phys. Rev.D78(2008) 065004, arXiv:0802.1232 [hep-th]

    Pith/arXiv arXiv 2008

  47. [47]

    Universal mechanism of (semi-classical) – 33 – deconfinement and theta-dependence for all simple groups,

    E. Poppitz, T. Sch¨ afer, and M. ¨Unsal, “Universal mechanism of (semi-classical) – 33 – deconfinement and theta-dependence for all simple groups,”JHEP03(2013) 087, arXiv:1212.1238 [hep-th]

    Pith/arXiv arXiv 2013

  48. [48]

    Continuity, Deconfinement, and (Super) Yang-Mills Theory,

    E. Poppitz, T. Sch¨ afer, and M. ¨Unsal, “Continuity, Deconfinement, and (Super) Yang-Mills Theory,”JHEP10(2012) 115,arXiv:1205.0290 [hep-th]

    Pith/arXiv arXiv 2012

  49. [49]

    Monopoles, affine algebras and the gluino condensate,

    N. M. Davies, T. J. Hollowood, and V. V. Khoze, “Monopoles, affine algebras and the gluino condensate,”J. Math. Phys.44(2003) 3640–3656,arXiv:hep-th/0006011 [hep-th]

    Pith/arXiv arXiv 2003

  50. [50]

    Gluino condensate and magnetic monopoles in supersymmetric gluodynamics,

    N. M. Davies, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, “Gluino condensate and magnetic monopoles in supersymmetric gluodynamics,”Nucl. Phys. B559(1999) 123–142, arXiv:hep-th/9905015

    Pith/arXiv arXiv 1999

  51. [51]

    Phases of Theories with ZN 1-Form Symmetry, and the Roles of Center Vortices and Magnetic Monopoles,

    M. Nguyen, T. Sulejmanpasic, and M. ¨Unsal, “Phases of Theories with ZN 1-Form Symmetry, and the Roles of Center Vortices and Magnetic Monopoles,”Phys. Rev. Lett.134 no. 14, (2025) 141902,arXiv:2401.04800 [hep-th]

    arXiv 2025

  52. [52]

    Emergent Photons and Confinement: A Numerical Study on ZN Lattice Gauge Theory,

    J. Giansiracusa, D. Lanners, and T. Sulejmanpasic, “Emergent Photons and Confinement: A Numerical Study on ZN Lattice Gauge Theory,”Phys. Rev. Lett.135no. 22, (2025) 221901, arXiv:2505.00079 [hep-lat]

    arXiv 2025

  53. [53]

    Generalized Global Symmetries,

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP02(2015) 172,arXiv:1412.5148 [hep-th]

    Pith/arXiv arXiv 2015

  54. [54]

    Coupling a QFT to a TQFT and Duality,

    A. Kapustin and N. Seiberg, “Coupling a QFT to a TQFT and Duality,”JHEP04(2014) 001,arXiv:1401.0740 [hep-th]

    Pith/arXiv arXiv 2014

  55. [55]

    Notes on generalized global symmetries in QFT,

    E. Sharpe, “Notes on generalized global symmetries in QFT,”Fortsch. Phys.63(2015) 659–682,arXiv:1508.04770 [hep-th]

    Pith/arXiv arXiv 2015

  56. [56]

    Quantum Field Theory and the Jones Polynomial,

    E. Witten, “Quantum Field Theory and the Jones Polynomial,”Commun.Math.Phys.121 (1989) 351–399

    1989

  57. [57]

    Topological quantum field theories,

    M. Atiyah, “Topological quantum field theories,”Inst. Hautes Etudes Sci. Publ. Math.68 (1989) 175–186

    1989

  58. [58]

    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) 6073–6105,arXiv:q-alg/9503002

    Pith/arXiv arXiv 1995

  59. [59]

    On the Classification of Topological Field Theories,

    J. Lurie, “On the Classification of Topological Field Theories,”arXiv:0905.0465 [math.CT]

    Pith/arXiv arXiv

  60. [60]

    Topological Field Theory, Higher Categories, and Their Applications,

    A. Kapustin, “Topological Field Theory, Higher Categories, and Their Applications,” arXiv:1004.2307 [math.QA]

    Pith/arXiv arXiv

  61. [61]

    Reflection positivity and invertible topological phases,

    D. S. Freed and M. J. Hopkins, “Reflection positivity and invertible topological phases,” Geom. Topol.25(2021) 1165–1330,arXiv:1604.06527 [hep-th]

    arXiv 2021

  62. [62]

    Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons,

    T. Lan, L. Kong, and X.-G. Wen, “Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons,”Phys. Rev. X8no. 2, (2018) 021074

    2018

  63. [63]

    Classification of 3+1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions,

    T. Lan and X.-G. Wen, “Classification of 3+1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions,”Phys. Rev. X9no. 2, (2019) 021005, arXiv:1801.08530 [cond-mat.str-el]

    Pith/arXiv arXiv 2019

  64. [64]

    Anyons in an exactly solved model and beyond,

    A. Kitaev, “Anyons in an exactly solved model and beyond,”Annals Phys.321no. 1, (2006) 2–111,arXiv:cond-mat/0506438 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2006

  65. [65]

    On the Classification of Topological Orders,

    T. Johnson-Freyd, “On the Classification of Topological Orders,”Commun. Math. Phys.393 no. 2, (2022) 989–1033,arXiv:2003.06663 [math.CT]. – 34 –

    arXiv 2022

  66. [66]

    Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order,

    Z.-C. Gu and X.-G. Wen, “Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order,”Phys. Rev. B80(2009) 155131,arXiv:0903.1069 [cond-mat.str-el]

    Pith/arXiv arXiv 2009

  67. [67]

    Twisted partition functions as order parameters,

    J. Maeda and Y. Tanizaki, “Twisted partition functions as order parameters,”JHEP08 (2025) 128,arXiv:2505.16546 [hep-th]

    arXiv 2025

  68. [68]

    A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories,

    G. ’t Hooft, “A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories,” Nucl. Phys.B153(1979) 141–160

    1979

  69. [69]

    Study of gapped phases of 4d gauge theories using temporal gauging of theZ N 1-form symmetry,

    M. Nguyen, Y. Tanizaki, and M. ¨Unsal, “Study of gapped phases of 4d gauge theories using temporal gauging of theZ N 1-form symmetry,”JHEP08(2023) 013,arXiv:2306.02485 [hep-th]

    arXiv 2023

  70. [70]

    Magnetic Monopoles in Unified Gauge Theories,

    G. ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,”Nucl. Phys.B79(1974) 276–284

    1974

  71. [71]

    Particle Spectrum in the Quantum Field Theory,

    A. M. Polyakov, “Particle Spectrum in the Quantum Field Theory,”JETP Lett.20(1974) 194–195

    1974

  72. [72]

    Higher symmetry and gapped phases of gauge theories,

    A. Kapustin and R. Thorngren, “Higher symmetry and gapped phases of gauge theories,” arXiv:1309.4721 [hep-th]

    Pith/arXiv arXiv

  73. [73]

    Topological quantum field theory, symmetry breaking, and finite gauge theory in 3+1D,

    R. Thorngren, “Topological quantum field theory, symmetry breaking, and finite gauge theory in 3+1D,”Phys. Rev. B101no. 24, (2020) 245160,arXiv:2001.11938 [cond-mat.str-el]

    arXiv 2020

  74. [74]

    Anomalous Symmetry Fractionalization and Surface Topological Order,

    X. Chen, F. J. Burnell, A. Vishwanath, and L. Fidkowski, “Anomalous Symmetry Fractionalization and Surface Topological Order,”Phys. Rev. X5no. 4, (2015) 041013, arXiv:1403.6491 [cond-mat.str-el]

    Pith/arXiv arXiv 2015

  75. [75]

    Symmetry Fractionalization, Defects, and Gauging of Topological Phases,

    M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, “Symmetry Fractionalization, Defects, and Gauging of Topological Phases,”Phys. Rev. B100no. 11, (2019) 115147, arXiv:1410.4540 [cond-mat.str-el]

    arXiv 2019

  76. [76]

    Symmetry-enriched quantum spin liquids in (3 + 1)d,

    P.-S. Hsin and A. Turzillo, “Symmetry-enriched quantum spin liquids in (3 + 1)d,”JHEP09 (2020) 022,arXiv:1904.11550 [cond-mat.str-el]

    arXiv 2020

  77. [77]

    Anomalies and symmetry fractionalization,

    D. G. Delmastro, J. Gomis, P.-S. Hsin, and Z. Komargodski, “Anomalies and symmetry fractionalization,”SciPost Phys.15no. 3, (2023) 079,arXiv:2206.15118 [hep-th]

    arXiv 2023

  78. [78]

    Fractionalization of coset non-invertible symmetry and exotic Hall conductance,

    P.-S. Hsin, R. Kobayashi, and C. Zhang, “Fractionalization of coset non-invertible symmetry and exotic Hall conductance,”SciPost Phys.17no. 3, (2024) 095,arXiv:2405.20401 [cond-mat.str-el]

    arXiv 2024

  79. [79]

    Consequences of symmetry fractionalization without 1-form global symmetries,

    T. D. Brennan, T. Jacobson, and K. Roumpedakis, “Consequences of symmetry fractionalization without 1-form global symmetries,”JHEP11(2025) 153, arXiv:2504.08036 [hep-th]

    arXiv 2025

  80. [80]

    Center vortex model for the infrared sector of Yang-Mills theory: Confinement and deconfinement,

    M. Engelhardt and H. Reinhardt, “Center vortex model for the infrared sector of Yang-Mills theory: Confinement and deconfinement,”Nucl. Phys. B585(2000) 591–613, arXiv:hep-lat/9912003

    Pith/arXiv arXiv 2000

Showing first 80 references.