pith. sign in

arxiv: 2606.13428 · v2 · pith:LU4TFXOVnew · submitted 2026-06-11 · ✦ hep-lat · cond-mat.str-el· hep-ph· hep-th

Numerical Hints for Dyon Condensation at θ=2π via Wilson-'t Hooft Loops in SU(2) Yang-Mills Theory

Hiromasa Watanabe , Issaku Kanamori , Okuto Morikawa , Yuki Nagai , Yuya Tanizaki , Akio Tomiya This is my paper

Pith reviewed 2026-06-27 04:59 UTC · model grok-4.3

classification ✦ hep-lat cond-mat.str-elhep-phhep-th
keywords SU(2) Yang-Millstheta termdyon condensationWilson-t Hooft loopslattice gauge theorygradient flowcenter symmetrySPT phase
0
0 comments X

The pith

Lattice measurements of Wilson-'t Hooft loops at θ=2π support dyon condensation in SU(2) Yang-Mills.

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

The paper measures the long-distance behavior of Wilson-'t Hooft loop operators in SU(2) lattice gauge theory at θ=2π. It applies the 1-form covariant DBW2 gradient flow to define topological charge in the presence of defects. The resulting data are consistent with the theoretical expectation of dyon condensation at θ=2π rather than the monopole condensation that occurs at θ=0. This distinction follows from the fact that the two confining vacua realize different symmetry-protected topological phases under the 1-form center symmetry. A reader would care because the result tests how the 2π periodicity of the θ term is realized nontrivially through a change in the condensing object.

Core claim

Numerical evidence from Wilson-'t Hooft loops at θ=2π, obtained using the 1-form covariant DBW2 gradient flow to identify topological charge with defects, is consistent with dyon condensation rather than monopole condensation, as predicted by generalized global symmetry arguments that place the θ=0 and θ=2π vacua in distinct SPT states.

What carries the argument

Wilson-'t Hooft loop operators whose long-distance decay is analyzed after the 1-form covariant DBW2 gradient flow is used to assign topological charge near the defects.

Load-bearing premise

The 1-form covariant DBW2 gradient flow correctly identifies the gauge topological charge in the presence of Wilson-'t Hooft defects and does not introduce artifacts that alter the long-distance loop behavior.

What would settle it

If the Wilson-'t Hooft loops at θ=2π displayed the same perimeter-law or area-law behavior as those at θ=0, the interpretation of dyon condensation would be ruled out.

read the original abstract

Yang-Mills theories at $\theta$ and $\theta+2\pi$ are unitarily equivalent, but their $2\pi$ periodicity has a nontrivial realization. Recent developments in generalized global symmetries show that confinement vacua at $\theta=0$ and $2\pi$ should belong to different symmetry-protected topological (SPT) states with the $1$-form center symmetry. For its examination, we measure the Wilson-'t Hooft loop operators at $\theta=2\pi$ for the $SU(2)$ Wilson lattice gauge action and discuss their long-distance behaviors. This requires us to identify the gauge topological charge in the presence of defects, and we employ the $1$-form covariant DBW2 gradient flow to smear lattice gauge fields. We then obtain numerical evidence consistent with dyon condensation at $\theta=2\pi$, rather than monopole condensation, as theoretically predicted.

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 / 0 minor

Summary. The manuscript reports lattice simulations of SU(2) Yang-Mills theory at θ=2π. It measures Wilson-'t Hooft loop operators after applying 1-form covariant DBW2 gradient flow to identify topological charge in the presence of defects, and obtains numerical evidence consistent with dyon condensation (rather than monopole condensation) at long distances, supporting the theoretical expectation that confinement vacua at θ=0 and θ=2π realize different SPT phases under the 1-form center symmetry.

Significance. If the evidence is robust, the work supplies a direct numerical test of recent generalized-global-symmetry predictions for the nontrivial 2π periodicity of the θ term. The approach relies on direct measurement of loop operators rather than parameter fits, which is a methodological strength. Confirmation of dyon condensation would sharpen understanding of confinement mechanisms in θ-dependent vacua.

major comments (2)
  1. [Abstract/Results] Abstract and results presentation: The claim of obtaining 'numerical evidence consistent with dyon condensation' is stated without accompanying quantitative diagnostics (fit parameters with uncertainties, χ² values, finite-volume extrapolations, or statistics on the loop correlators). This omission makes it impossible to evaluate the strength of support for the central claim and is load-bearing for any assertion of consistency with the theoretical prediction.
  2. [Methods] Methods: The 1-form covariant DBW2 gradient flow is introduced to handle topological charge identification with defects, yet no explicit validation (e.g., comparison against known topological sectors or checks for artifacts in the long-distance loop behavior) is referenced. Because this step is required to interpret the measured operators, additional controls are needed to substantiate the interpretation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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

read point-by-point responses

  1. Referee: [Abstract/Results] Abstract and results presentation: The claim of obtaining 'numerical evidence consistent with dyon condensation' is stated without accompanying quantitative diagnostics (fit parameters with uncertainties, χ² values, finite-volume extrapolations, or statistics on the loop correlators). This omission makes it impossible to evaluate the strength of support for the central claim and is load-bearing for any assertion of consistency with the theoretical prediction.

    Authors: We acknowledge that additional quantitative diagnostics would allow readers to better assess the strength of our numerical evidence. The manuscript presents the evidence via the observed long-distance scaling of the Wilson-'t Hooft loop operators (perimeter law consistent with dyon condensation rather than area law). In the revised manuscript we will add explicit fit parameters with uncertainties, χ² values, and a discussion of finite-volume effects drawn from the existing simulation data. revision: yes

  2. Referee: [Methods] Methods: The 1-form covariant DBW2 gradient flow is introduced to handle topological charge identification with defects, yet no explicit validation (e.g., comparison against known topological sectors or checks for artifacts in the long-distance loop behavior) is referenced. Because this step is required to interpret the measured operators, additional controls are needed to substantiate the interpretation.

    Authors: We agree that explicit validation of the 1-form covariant DBW2 gradient flow strengthens the methodological foundation. The manuscript describes the implementation, but we will revise to include direct comparisons against standard topological charge measurements on defect-free configurations and checks confirming the absence of artifacts in the long-distance loop operators. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper reports direct numerical measurements of Wilson-'t Hooft loop operators on the lattice at θ=2π after applying 1-form covariant DBW2 gradient flow to define topological charge. The claimed evidence for dyon condensation is an empirical observation extracted from these simulations rather than any derivation, ansatz, or fitted parameter that reduces to the input data by construction. No self-citation chain, self-definitional step, or renaming of known results is load-bearing for the central claim.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on standard lattice regularization of SU(2) Yang-Mills, the existence of a well-defined 1-form covariant gradient flow for topological charge in the presence of defects, and the assumption that long-distance loop behavior directly diagnoses condensation type. No new free parameters or invented entities are introduced beyond conventional lattice parameters such as beta and volume.

free parameters (1)
  • lattice coupling beta
    Standard bare coupling in the Wilson action; its value is chosen to reach the continuum limit but is not fitted to the condensation signal itself.
axioms (1)
  • domain assumption The 1-form covariant DBW2 gradient flow preserves the topological charge classification when Wilson-'t Hooft defects are inserted.
    Invoked to enable measurement of topological charge with defects; stated in the methods description.

pith-pipeline@v0.9.1-grok · 5723 in / 1322 out tokens · 18263 ms · 2026-06-27T04:59:11.454085+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

96 extracted references · 41 linked inside Pith

  1. [1]

    Confinement of Quarks,

    K. G. Wilson, “Confinement of Quarks,”Phys. Rev.D10(1974) 2445–2459

    1974

  2. [2]

    Strings, Monopoles and Gauge Fields,

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

    1974

  3. [3]

    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

  4. [4]

    Compact Gauge Fields and the Infrared Catastrophe,

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

    1975

  5. [5]

    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

  6. [6]

    Dyons of Chargeeθ/2π,

    E. Witten, “Dyons of Chargeeθ/2π,”Phys. Lett.B86(1979) 283–287

    1979

  7. [7]

    Large N Chiral Dynamics,

    E. Witten, “Large N Chiral Dynamics,”Annals Phys.128(1980) 363

    1980

  8. [8]

    Chiral Dynamics in the Large n Limit,

    P. Di Vecchia and G. Veneziano, “Chiral Dynamics in the Large n Limit,”Nucl. Phys. B171 (1980) 253–272

    1980

  9. [9]

    Theta dependence in the large N limit of four-dimensional gauge theories,

    E. Witten, “Theta dependence in the large N limit of four-dimensional gauge theories,” Phys. Rev. Lett.81(1998) 2862–2865,arXiv:hep-th/9807109 [hep-th]

    Pith/arXiv arXiv 1998

  10. [10]

    Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories,

    G. ’t Hooft, “Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories,”Nucl. Phys.B190(1981) 455–478

    1981

  11. [11]

    Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory,

    N. Seiberg and E. Witten, “Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory,”Nucl. Phys.B426(1994) 19–52, arXiv:hep-th/9407087 [hep-th]. [Erratum: Nucl. Phys.B430,485(1994)]

    Pith/arXiv arXiv 1994

  12. [12]

    Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD,

    N. Seiberg and E. Witten, “Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD,”Nucl. Phys.B431(1994) 484–550,arXiv:hep-th/9408099 [hep-th]

    Pith/arXiv arXiv 1994

  13. [13]

    Theta, Time Reversal, and Temperature,

    D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, “Theta, Time Reversal, and Temperature,”JHEP05(2017) 091,arXiv:1703.00501 [hep-th]

    Pith/arXiv arXiv 2017

  14. [14]

    Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking,

    Y. Tanizaki, “Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking,”JHEP08(2018) 171,arXiv:1807.07666 [hep-th]

    Pith/arXiv arXiv 2018

  15. [15]

    Time-reversal breaking in QCD 4, walls, and dualities in 2 + 1 dimensions,

    D. Gaiotto, Z. Komargodski, and N. Seiberg, “Time-reversal breaking in QCD 4, walls, and dualities in 2 + 1 dimensions,”JHEP01(2018) 110,arXiv:1708.06806 [hep-th]

    Pith/arXiv arXiv 2018

  16. [16]

    Anomaly matching for phase diagram of masslessZ N-QCD,

    Y. Tanizaki, Y. Kikuchi, T. Misumi, and N. Sakai, “Anomaly matching for phase diagram of masslessZ N-QCD,”Phys. Rev.D97(2018) 054012,arXiv:1711.10487 [hep-th]

    Pith/arXiv arXiv 2018

  17. [17]

    Anomaly matching in QCD thermal phase transition,

    K. Yonekura, “Anomaly matching in QCD thermal phase transition,”JHEP05(2019) 062, arXiv:1901.08188 [hep-th]

    Pith/arXiv arXiv 2019

  18. [18]

    On the baryon-color-flavor (BCF) anomaly in vector-like theories,

    M. M. Anber and E. Poppitz, “On the baryon-color-flavor (BCF) anomaly in vector-like theories,”JHEP11(2019) 063,arXiv:1909.09027 [hep-th]. – 23 –

    arXiv 2019

  19. [19]

    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

  20. [20]

    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

  21. [21]

    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

  22. [22]

    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,”JHEP07(2024) 033, arXiv:2404.16803 [hep-th]

    arXiv 2024

  23. [23]

    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

  24. [24]

    Baryons as Quantum Hall Droplets,

    Z. Komargodski, “Baryons as Quantum Hall Droplets,”arXiv:1812.09253 [hep-th]

    Pith/arXiv arXiv

  25. [25]

    Chiral symmetry and eta-x mixing,

    M. Kobayashi and T. Maskawa, “Chiral symmetry and eta-x mixing,”Prog. Theor. Phys.44 (1970) 1422–1424

    1970

  26. [26]

    Symmetry breaking of the chiral u(3) x u(3) and the quark model,

    M. Kobayashi, H. Kondo, and T. Maskawa, “Symmetry breaking of the chiral u(3) x u(3) and the quark model,”Prog. Theor. Phys.45(1971) 1955–1959

    1971

  27. [27]

    Spontaneous Symmetry Breaking in Vector-Gluon Model,

    T. Maskawa and H. Nakajima, “Spontaneous Symmetry Breaking in Vector-Gluon Model,” Prog. Theor. Phys.52(1974) 1326–1354

    1974

  28. [28]

    Symmetry Breaking Through Bell-Jackiw Anomalies,

    G. ’t Hooft, “Symmetry Breaking Through Bell-Jackiw Anomalies,”Phys. Rev. Lett.37 (1976) 8–11

    1976

  29. [29]

    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

  30. [30]

    Quantized Singularities in the Electromagnetic Field,

    P. A. M. Dirac, “Quantized Singularities in the Electromagnetic Field,”Proc. Roy. Soc. Lond.A133(1931) 60–72

    1931

  31. [31]

    Magnetic charge and quantum field theory,

    J. S. Schwinger, “Magnetic charge and quantum field theory,”Phys. Rev.144(1966) 1087–1093

    1966

  32. [32]

    Quantum field theory of particles with both electric and magnetic charges,

    D. Zwanziger, “Quantum field theory of particles with both electric and magnetic charges,” Phys. Rev.176(1968) 1489–1495

    1968

  33. [33]

    Gauge Theories and Magnetic Charge,

    P. Goddard, J. Nuyts, and D. I. Olive, “Gauge Theories and Magnetic Charge,”Nucl. Phys. B125(1977) 1–28

    1977

  34. [34]

    Reading between the lines of four-dimensional gauge theories,

    O. Aharony, N. Seiberg, and Y. Tachikawa, “Reading between the lines of four-dimensional gauge theories,”JHEP08(2013) 115,arXiv:1305.0318 [hep-th]

    arXiv 2013

  35. [35]

    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

  36. [36]

    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

  37. [37]

    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]. – 24 –

    arXiv 2023

  38. [38]

    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

  39. [39]

    Wilson-’t Hooft classification and the perimeter law for dyonic loops in 3d monopole semiclassics,

    Y. Hayashi and Y. Tanizaki, “Wilson-’t Hooft classification and the perimeter law for dyonic loops in 3d monopole semiclassics,”JHEP05(2026) 303,arXiv:2601.02058 [hep-th]

    arXiv 2026

  40. [40]

    Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories,

    S. Gukov and A. Kapustin, “Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories,”arXiv:1307.4793 [hep-th]

    Pith/arXiv arXiv

  41. [41]

    Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement,

    A. Kapustin and R. Thorngren, “Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement,”Adv. Theor. Math. Phys.18no. 5, (2014) 1233–1247, arXiv:1308.2926 [hep-th]

    Pith/arXiv arXiv 2014

  42. [42]

    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

  43. [43]

    A Study of the ’t Hooft loop in SU(2) Yang-Mills theory,

    P. de Forcrand, M. D’Elia, and M. Pepe, “A Study of the ’t Hooft loop in SU(2) Yang-Mills theory,”Phys. Rev. Lett.86(2001) 1438,arXiv:hep-lat/0007034

    Pith/arXiv arXiv 2001

  44. [44]

    θdependence of the deconfinement temperature in Yang-Mills theories,

    M. D’Elia and F. Negro, “θdependence of the deconfinement temperature in Yang-Mills theories,”Phys. Rev. Lett.109(2012) 072001,arXiv:1205.0538 [hep-lat]

    Pith/arXiv arXiv 2012

  45. [45]

    Phase diagram of Yang-Mills theories in the presence of aθterm,

    M. D’Elia and F. Negro, “Phase diagram of Yang-Mills theories in the presence of aθterm,” Phys. Rev.D88no. 3, (2013) 034503,arXiv:1306.2919 [hep-lat]

    Pith/arXiv arXiv 2013

  46. [46]

    θdependence inSU(3) Yang-Mills theory from analytic continuation,

    C. Bonati, M. D’Elia, and A. Scapellato, “θdependence inSU(3) Yang-Mills theory from analytic continuation,”Phys. Rev. D93no. 2, (2016) 025028,arXiv:1512.01544 [hep-lat]

    Pith/arXiv arXiv 2016

  47. [47]

    θdependence of 4DSU(N) gauge theories in the large-Nlimit,

    C. Bonati, M. D’Elia, P. Rossi, and E. Vicari, “θdependence of 4DSU(N) gauge theories in the large-Nlimit,”Phys. Rev.D94no. 8, (2016) 085017,arXiv:1607.06360 [hep-lat]

    Pith/arXiv arXiv 2016

  48. [48]

    θ-dependence and center symmetry in Yang-Mills theories,

    C. Bonati, M. Cardinali, M. D’Elia, and F. Mazziotti, “θ-dependence and center symmetry in Yang-Mills theories,”Phys. Rev. D101no. 3, (2020) 034508,arXiv:1912.02662 [hep-lat]

    arXiv 2020

  49. [49]

    Theθ-dependence of theSU(N) critical temperature at largeN,

    C. Bonanno, M. D’Elia, and L. Verzichelli, “Theθ-dependence of theSU(N) critical temperature at largeN,”JHEP02(2024) 156,arXiv:2312.12202 [hep-lat]

    arXiv 2024

  50. [50]

    IsN= 2 Large?,

    R. Kitano, N. Yamada, and M. Yamazaki, “IsN= 2 Large?,”JHEP02(2021) 073, arXiv:2010.08810 [hep-lat]

    arXiv 2021

  51. [51]

    Peeking into theθvacuum,

    R. Kitano, R. Matsudo, N. Yamada, and M. Yamazaki, “Peeking into theθvacuum,”Phys. Lett. B822(2021) 136657,arXiv:2102.08784 [hep-lat]

    arXiv 2021

  52. [52]

    Subvolume method for SU(2) Yang-Mills theory at finite temperature: topological charge distributions,

    N. Yamada, M. Yamazaki, and R. Kitano, “Subvolume method for SU(2) Yang-Mills theory at finite temperature: topological charge distributions,”JHEP07(2024) 198, arXiv:2403.10767 [hep-lat]

    arXiv 2024

  53. [53]

    θdependence of T c in SU(2) Yang-Mills theory,

    N. Yamada, M. Yamazaki, and R. Kitano, “θdependence of T c in SU(2) Yang-Mills theory,” JHEP02(2025) 211,arXiv:2411.00375 [hep-lat]

    arXiv 2025

  54. [54]

    Evidence of a CP broken deconfined phase in 4D SU(2) Yang-Mills theory atθ=πfrom imaginaryθ simulations,

    M. Hirasawa, M. Honda, A. Matsumoto, J. Nishimura, and A. Yosprakob, “Evidence of a CP broken deconfined phase in 4D SU(2) Yang-Mills theory atθ=πfrom imaginaryθ simulations,”JHEP05(2025) 009,arXiv:2412.03683 [hep-th]

    arXiv 2025

  55. [55]

    The Phase Structure of SU(N)/Z(N) Lattice Gauge Theories,

    I. G. Halliday and A. Schwimmer, “The Phase Structure of SU(N)/Z(N) Lattice Gauge Theories,”Phys. Lett. B101(1981) 327. – 25 –

    1981

  56. [56]

    Z(2) Monopoles in Lattice Gauge Theories,

    I. G. Halliday and A. Schwimmer, “Z(2) Monopoles in Lattice Gauge Theories,”Phys. Lett. B102(1981) 337–340

    1981

  57. [57]

    Topology of Lattice Gauge Fields,

    M. L¨ uscher, “Topology of Lattice Gauge Fields,”Commun. Math. Phys.85(1982) 39

    1982

  58. [58]

    Topology of SU(N) lattice gauge theories coupled withZ N 2-form gauge fields,

    M. Abe, O. Morikawa, S. Onoda, H. Suzuki, and Y. Tanizaki, “Topology of SU(N) lattice gauge theories coupled withZ N 2-form gauge fields,”JHEP08(2023) 118, arXiv:2303.10977 [hep-lat]

    arXiv 2023

  59. [59]

    Magnetic operators in 2D compact scalar field theories on the lattice,

    M. Abe, O. Morikawa, S. Onoda, H. Suzuki, and Y. Tanizaki, “Magnetic operators in 2D compact scalar field theories on the lattice,”PTEP2023no. 7, (2023) 073B01, arXiv:2304.14815 [hep-lat]

    arXiv 2023

  60. [60]

    Instanton density operator in lattice QCD from higher category theory,

    J.-Y. Chen, “Instanton density operator in lattice QCD from higher category theory,” SciPost Phys.19no. 6, (2025) 158,arXiv:2406.06673 [hep-lat]

    arXiv 2025

  61. [61]

    An explicit categorical construction of instanton density in lattice Yang-Mills theory,

    P. Zhang and J.-Y. Chen, “An explicit categorical construction of instanton density in lattice Yang-Mills theory,”JHEP06(2025) 085,arXiv:2411.07195 [hep-lat]

    arXiv 2025

  62. [62]

    Abelian gauge theories on the lattice:θ-terms and compact gauge theory with(out) monopoles,

    T. Sulejmanpasic and C. Gattringer, “Abelian gauge theories on the lattice:θ-terms and compact gauge theory with(out) monopoles,”Nucl. Phys.B943(2019) 114616, arXiv:1901.02637 [hep-lat]

    Pith/arXiv arXiv 2019

  63. [63]

    Self-dual U(1) lattice field theory with a θ-term,

    M. Anosova, C. Gattringer, and T. Sulejmanpasic, “Self-dual U(1) lattice field theory with a θ-term,”JHEP04(2022) 120,arXiv:2201.09468 [hep-lat]

    arXiv 2022

  64. [64]

    2d Cardy-Rabinovici model with the modified Villain lattice: exact dualities and symmetries,

    N. Katayama and Y. Tanizaki, “2d Cardy-Rabinovici model with the modified Villain lattice: exact dualities and symmetries,”JHEP11(2025) 004,arXiv:2505.19412 [hep-th]

    arXiv 2025

  65. [65]

    Some Results for SU(N) Gauge Fields on the Hypertorus,

    P. van Baal, “Some Results for SU(N) Gauge Fields on the Hypertorus,”Commun. Math. Phys.85(1982) 529

    1982

  66. [66]

    Vacuum structure of bifundamental gauge theories at finite topological angles,

    Y. Tanizaki and Y. Kikuchi, “Vacuum structure of bifundamental gauge theories at finite topological angles,”JHEP06(2017) 102,arXiv:1705.01949 [hep-th]

    Pith/arXiv arXiv 2017

  67. [67]

    Comments on Abelian Higgs Models and Persistent Order,

    Z. Komargodski, A. Sharon, R. Thorngren, and X. Zhou, “Comments on Abelian Higgs Models and Persistent Order,”SciPost Phys.6no. 1, (2019) 003,arXiv:1705.04786 [hep-th]

    Pith/arXiv arXiv 2019

  68. [68]

    Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics,

    Y. Kikuchi and Y. Tanizaki, “Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics,”Prog. Theor. Exp. Phys.2017(2017) 113B05,arXiv:1708.01962 [hep-th]

    Pith/arXiv arXiv 2017

  69. [69]

    Anomaly and global inconsistency matching:θ-angles, SU(3)/U(1) 2 nonlinear sigma model,SU(3) chains and its generalizations,

    Y. Tanizaki and T. Sulejmanpasic, “Anomaly and global inconsistency matching:θ-angles, SU(3)/U(1) 2 nonlinear sigma model,SU(3) chains and its generalizations,”Phys. Rev.B98 no. 11, (2018) 115126,arXiv:1805.11423 [cond-mat.str-el]

    Pith/arXiv arXiv 2018

  70. [70]

    Anomalies in the Space of Coupling Constants and Their Dynamical Applications I,

    C. C´ ordova, D. S. Freed, H. T. Lam, and N. Seiberg, “Anomalies in the Space of Coupling Constants and Their Dynamical Applications I,”SciPost Phys.8no. 1, (2020) 001, arXiv:1905.09315 [hep-th]

    arXiv 2020

  71. [71]

    Anomalies in the Space of Coupling Constants and Their Dynamical Applications II,

    C. C´ ordova, D. S. Freed, H. T. Lam, and N. Seiberg, “Anomalies in the Space of Coupling Constants and Their Dynamical Applications II,”SciPost Phys.8no. 1, (2020) 002, arXiv:1905.13361 [hep-th]

    arXiv 2020

  72. [72]

    Properties and uses of the Wilson flow in lattice QCD,

    M. L¨ uscher, “Properties and uses of the Wilson flow in lattice QCD,”JHEP08(2010) 071, arXiv:1006.4518 [hep-lat]. [Erratum: JHEP 03, 092 (2014)]. – 26 –

    Pith/arXiv arXiv 2010

  73. [73]

    Comparison of the gradient flow with cooling inSU(3) pure gauge theory,

    C. Bonati and M. D’Elia, “Comparison of the gradient flow with cooling inSU(3) pure gauge theory,”Phys. Rev. D89no. 10, (2014) 105005,arXiv:1401.2441 [hep-lat]

    Pith/arXiv arXiv 2014

  74. [74]

    Comparison of topological charge definitions in Lattice QCD,

    C. Alexandrou, A. Athenodorou, K. Cichy, A. Dromard, E. Garcia-Ramos, K. Jansen, U. Wenger, and F. Zimmermann, “Comparison of topological charge definitions in Lattice QCD,”Eur. Phys. J. C80no. 5, (2020) 424,arXiv:1708.00696 [hep-lat]

    arXiv 2020

  75. [75]

    On-shell improved lattice gauge theories,

    M. L¨ uscher and P. Weisz, “On-shell improved lattice gauge theories,”Commun. Math. Phys. 98no. 3, (1985) 433. [Erratum: Commun.Math.Phys. 98, 433 (1985)]

    1985

  76. [76]

    Instantons and Topological Charge in Lattice Gauge Theory,

    Y. Iwasaki and T. Yoshie, “Instantons and Topological Charge in Lattice Gauge Theory,” Phys. Lett. B131(1983) 159–164

    1983

  77. [77]

    Renormalization Group Analysis of Lattice Theories and Improved Lattice Action. II. Four-dimensional non-Abelian SU(N) gauge model,

    Y. Iwasaki, “Renormalization Group Analysis of Lattice Theories and Improved Lattice Action. II. Four-dimensional non-Abelian SU(N) gauge model,”arXiv:1111.7054 [hep-lat]

    Pith/arXiv arXiv

  78. [78]

    Instantons from over - improved cooling,

    M. Garcia Perez, A. Gonzalez-Arroyo, J. R. Snippe, and P. van Baal, “Instantons from over - improved cooling,”Nucl. Phys. B413(1994) 535–552,arXiv:hep-lat/9309009

    Pith/arXiv arXiv 1994

  79. [79]

    Topological susceptibility and instanton size distribution from over improved cooling,

    P. de Forcrand and S. Kim, “Topological susceptibility and instanton size distribution from over improved cooling,”Nucl. Phys. B Proc. Suppl.47(1996) 278–281, arXiv:hep-lat/9509081

    Pith/arXiv arXiv 1996

  80. [80]

    Lattice gradient flows (de-)stabilizing topological sectors,

    Y. Tanizaki, A. Tomiya, and H. Watanabe, “Lattice gradient flows (de-)stabilizing topological sectors,”JHEP04(2025) 123,arXiv:2411.14812 [hep-lat]. [81]QCD-TAROCollaboration, P. de Forcrand, M. Garcia Perez, T. Hashimoto, S. Hioki, H. Matsufuru, O. Miyamura, A. Nakamura, I. O. Stamatescu, T. Takaishi, and T. Umeda, “Renormalization group flow of SU(3) lat...

    arXiv 2025

Showing first 80 references.