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A bosonic lattice model yields chiral fermions with no doublers by making the reconstructed Dirac operator non-local while keeping the microscopic theory ultra-local.

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2026-07-14 14:28 UTC pith:Z5GPGUXZ

load-bearing objection Clean 2D analytic demo that ultra-local bosonization yields doubler-free chiral fermions whose reconstructed Dirac kernel is non-local, consistent with Nielsen-Ninomiya.

arxiv 2607.09935 v1 pith:Z5GPGUXZ submitted 2026-07-10 hep-th cond-mat.str-elhep-lat

Bosonization versus the Nielsen-Ninomiya theorem

Saif Ullah Baig , Shi Chen , Aleksey Cherman , Maria Neuzil This is my paper
classification hep-th cond-mat.str-elhep-lat
keywords bosonizationNielsen-Ninomiya theoremlattice chiral fermionsmodified Villain modelnon-local Dirac operatordoublers2D compact scalar
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Putting massless chiral fermions on a lattice without unwanted copies (doublers) is blocked by the Nielsen-Ninomiya theorem whenever the Dirac operator is local, translation-invariant, and has the usual chiral structure. This paper shows that bosonization offers a way around that obstruction. In the two-dimensional modified Villain scalar model, which is ultra-local in its bosonic variables and symmetries, the authors build composite lattice operators that behave as Weyl fermions. Exact two-point functions of these operators, after inversion, produce a lattice Dirac operator with a single zero and no doublers. Consistency with the theorem is restored because that reconstructed operator is non-local, decaying only as 1 over distance. The non-locality is an artifact of the derived fermionic description; the underlying bosonic model remains ultra-local, so its non-anomalous symmetries can still be gauged without obstruction. The result clarifies that the theorem constrains the form of a Dirac kernel, not the existence of chiral fermions in the long-distance spectrum of an ultra-local bosonic lattice theory.

Core claim

In the 2D modified Villain model at the self-dual radius, lattice operators built from the scalar and its dual, dressed by topological rays, produce exact two-point functions whose continuum asymptotics match free massless Weyl correlators. Inverting those correlators yields a hermitian lattice Dirac operator that anti-commutes with the usual chiral matrix, has a single Dirac zero at vanishing momentum, and therefore no doublers, yet falls off only as O(1/|x-y|) and is therefore non-local. The microscopic bosonic action and its symmetries remain ultra-local, so non-anomalous symmetries can be gauged directly.

What carries the argument

The reconstructed lattice Dirac operator obtained by inverting the exact two-point functions of the composite Weyl operators: it is doubler-free and chirally structured, yet non-local, thereby saturating the Nielsen-Ninomiya assumptions while the underlying modified Villain model stays ultra-local.

Load-bearing premise

That the composite lattice operators, once attached to topological rays and fixed by a constant renormalization, continue to match continuum Weyl fields in the continuum limit, verified only by matching two-point asymptotics and the leading factorization of four-point functions.

What would settle it

Compute the full spectrum of the reconstructed Dirac operator on a large finite lattice (or its Fourier transform) and check whether any extra zeros appear at non-zero lattice momenta once the contact term is fixed by the continuum asymptotics; an extra zero would restore doublers and falsify the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper constructs lattice Weyl operators (Eqs. 43a–b) as composite operators of the ultra-local 2D modified Villain scalar model at the self-dual radius, evaluates their exact two-point functions by path-integral reduction to the lattice Green function (App. C.1, Eq. 49), and shows that after a fixed renormalization (Eq. 52) the large-distance asymptotics reproduce the continuum Weyl propagator (App. C.2, Eq. 51). Inverting these correlators yields a reconstructed hermitian Dirac operator (Eqs. 59–60) that anti-commutes with γ3, has a single Dirac zero at p=0, and is non-local (kernel ~O(1/|x−y|)). The non-locality is forced by the Poincaré–Hopf theorem applied to the momentum-space two-point function (Sec. IV.D, App. C.3), so the construction remains consistent with Nielsen–Ninomiya while the microscopic bosonic theory and its non-anomalous symmetries stay ultra-local and gaugeable. Connected four-point functions are non-vanishing at finite spacing but irrelevant in the continuum limit (Sec. IV.E).

Significance. The work supplies an explicit, analytically controlled illustration of how bosonization evades the Nielsen–Ninomiya theorem: the microscopic theory is ultra-local and free of Grassmann fields, yet the reconstructed inverse propagator of the emergent chiral operators is non-local and doubler-free. The exact two-point evaluation, continuum asymptotics, and topological argument for poles are self-contained and reproducible from the appendices. This clarifies the status of propagator zeros in the symmetric-mass-generation literature and strengthens the conceptual case for gauging chiral symmetries directly on the bosonic side. The result is limited to 2D free fermions, but the distinction it draws between microscopic ultra-locality and reconstructed non-locality is of broader interest for lattice chiral gauge theory.

minor comments (4)
  1. The continuum operator dictionary (Eq. 27) is verified only through two-point asymptotics and continuum four-point factorization. A short remark in Sec. IV.B or the Outlook noting that a full lattice OPE or spectral reconstruction is left for future work would make the scope of the identification clearer.
  2. Figs. 6–7 show that the locations of zeros of eS+(p) depend on the contact term S+(0). Adding a sentence that the total Poincaré–Hopf index is contact-term independent (while individual zeros are not) would prevent misreading of the figures.
  3. Table I is a useful scorecard; a brief footnote clarifying that the “bosonization” row refers to the reconstructed Dirac operator rather than a microscopic Grassmann action would avoid confusion with SLAC/Stacey fermions.
  4. The spin θ-angle is deliberately omitted on the lattice (Sec. IV). A one-sentence reminder that its only effect on infinite-volume correlators is to select a single-valued branch would help readers unfamiliar with the continuum review in Sec. III.

Circularity Check

0 steps flagged

No significant circularity: reconstructed Dirac operator and its non-locality/no-doublers properties are derived outputs of explicit correlator inversion, not inputs.

full rationale

The derivation chain is self-contained and non-circular. Lattice Weyl operators are defined from the microscopic modified Villain fields plus topological rays (Eqs. 43a–b); their two-point functions S_pm are computed exactly from the path integral (App. C.1, Eq. 49) using only the lattice Green function and sum-by-parts identities; continuum asymptotics follow from the known large-|x| expansion of G (App. C.2, Eq. 51) after a constant renormalization Z fixed by matching the leading 1/|x| coefficient; the lattice Dirac kernel is then defined by inversion of those correlators (Eqs. 59–60); Fourier analysis plus the Poincaré–Hopf theorem applied to the single pole of eS (App. C.3) forces the absence of extra zeros of 1/eS while the slow decay of S implies poles of P and hence non-locality of /D ~ O(1/|x–y|). Contact-term dependence of zero locations is acknowledged and does not affect the topological index or the non-locality conclusion. Continuum bosonization (Coleman) is used only for operator identification and is an external standard result; multi-point connected correlators are computed independently to show residual interactions are irrelevant. No fitted parameters are re-labeled as predictions, no uniqueness theorem is imported from the authors’ prior work to force the spectrum, and the ultra-locality of the bosonic action is an independent microscopic fact. Self-citations (e.g. [30]) supply the model but are not load-bearing for the reconstructed-operator claims.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 1 invented entities

The paper rests on standard continuum bosonization, the Poincaré-Hopf theorem, and the known continuum limit of the modified Villain model. The only free parameters are overall operator normalizations fixed by matching continuum asymptotics; no data fitting occurs. The lattice Weyl operators with attached rays are new composite objects defined inside the paper, but they are constructed from already-present fields rather than postulated new degrees of freedom.

free parameters (1)
  • Z± Z̄± renormalization product = 2 e^{γ_E}(1±i)
    Overall multiplicative constants for the lattice Weyl operators, fixed in Eq. (52) to 2 e^{γ_E}(1±i) so that the large-|x| two-point function matches the continuum Weyl propagator. They are pure normalizations, not dynamical fits.
axioms (4)
  • standard math Poincaré-Hopf theorem applied to continuous sections of the trivial rank-2 real bundle over T^{2}
    Used in Sec. II and again in Sec. IV D to force extra zeros (or poles of the inverse) once a single zero of known index is present.
  • domain assumption Coleman’s continuum boson-fermion duality at R=1/√2, including the operator dictionary of Eq. (27)
    Taken as established continuum QFT; the lattice construction is designed so that its continuum limit reproduces this dictionary.
  • domain assumption The modified Villain model at R=1/√2 flows to the free compact boson (and hence, after spin θ-angle, to the free Dirac fermion)
    Assumed from prior lattice literature (Refs. [28–30]); the present paper does not re-derive the continuum limit of the bosonic theory itself.
  • standard math Contact-term redefinitions of the lattice fermion operators do not alter the topological index of the momentum-space pole at p=0
    Used to argue that the existence of compensating zeros (hence poles of the inverse) is robust.
invented entities (1)
  • Lattice Weyl operators ψ±(C_x), ψ̄±(C_x) attached to topological rays C_x no independent evidence
    purpose: Provide gauge-invariant composite operators that flow to continuum Weyl fields and whose correlators define the reconstructed Dirac kernel.
    Defined in Eqs. (43a–b) from the microscopic fields φ, n, θ and a lattice path; they are not new fundamental fields but new composite operators.

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read the original abstract

Thanks to bosonization, bosonic lattice models can offer a lattice regularization of chiral fermions. We construct chiral lattice fermion operators in the 2D modified Villain scalar model and evaluate their correlation functions. This microscopic bosonic model has an ultra-local action and an ultra-local symmetry that realizes the fermionic chiral symmetry under bosonization. The reconstructed lattice Dirac operator has no doublers, but is consistent with the Nielsen-Ninomiya theorem because it turns out to be non-local. The non-locality of this derived quantity at finite lattice spacing does not pose any obstructions to gauging the non-anomalous symmetries of the model, which is itself ultra-local.

Figures

Figures reproduced from arXiv: 2607.09935 by Aleksey Cherman, Maria Neuzil, Saif Ullah Baig, Shi Chen.

Figure 1
Figure 1. Figure 1: FIG. 1: Lattice sites, links and plaquettes are labeled [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Two equivalence classes of lattice paths in the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The site-plaquette nature of the Weyl operators in combination with the topological line gives rise to the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: When we drag [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Magnitude (top) and phase (bottom) of the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Magnitude (top) and argument (bottom) of the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: A reproduction of Fig [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The tetrahedron of topological manipulations [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The lattice differential ( [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The lattice codifferential ( [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The lattice path decomposition of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗

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Reference graph

Works this paper leans on

107 extracted references · 60 linked inside Pith

  1. [1]

    Fermionic description 5 IV

    Physical effect 4 B. Fermionic description 5 IV. Lattice chiral fermion in the 2D Modified Villain Model 6 A. 2D Modified Villain Model 6 B. Fermionic operator 7 C. Two-point correlation function 7 D. Non-local lattice Dirac operator 10 E. Multi-point correlation function 11 V. Outlook 12 A. Appendix: 2D boson-fermion transformation tetrahedron 12 B. Appe...

  2. [2]

    Lattice differential operators 13

  3. [3]

    Appendix: Calculation details in the 2D modified Villain model 15

    Lattice Green function 14 C. Appendix: Calculation details in the 2D modified Villain model 15

  4. [4]

    Two-point correlation function 15

  5. [5]

    Asymptotic expansion 17

  6. [6]

    spinθ-angle,

    Momentum-space discontinuity 18 References 19 ∗ sbaig.phys@gmail.com † s.chern.phys@gmail.com ‡ acherman@umn.edu § neuzi008@umn.edu I. Introduction It is notoriously challenging to put massless fermions on a Euclidean spacetime lattice while preserving chiral symmetries. This poses an obstruction to the lattice reg- ularization of chiral gauge theories, s...

    Pith/arXiv arXiv 2026

  7. [7]

    On 2D oriented and spin spacetime manifolds, theθ-angles are classified by the bordism groups (see e.g

    Spinθ-angle The topology of anS 1 scalar brings not only theU(1) W symmetry but also topologicalθ-angles. On 2D oriented and spin spacetime manifolds, theθ-angles are classified by the bordism groups (see e.g. Refs. [62–65]) Hom eΩSO 2 (S1), U(1) = 0,(8a) Hom eΩSpin 2 (S1), U(1) =Z 2 ,(8b) respectively. All 2D orientable manifolds are spinnable, but in ge...

  8. [8]

    protects

    Physical effect Since the spinθ-angle (9) is trivial onS 2, it does not affect local dynamics, and only affects the global struc- ture. Hence the fermionic theory (13) is still a conformal 2 See e.g. Refs. [58–61] for useful reviews about the Arf invariant. field theory with central chargesc= ¯c= 1. Evaluating the path integral (13) on a flat torus, T 2 =...

  9. [9]

    and ( 1 2 ,0) are two links em- anating from the site (0,0), while ( 1 2 , 1

  10. [10]

    is the plaquette attached to the links (0, 1

  11. [11]

    Y s∈Γ0 Z R dφ(s) 2π #

    and ( 1 2 ,0). We shall use various operators on lattice fields includ- ing the lattice differential d, the lattice codifferentialδ, the lattice laplacian ∆≡dδ+δd, and the lattice Hodge star⋆. They are discretizations of their continuum coun- terparts and satisfy similar properties. In Appendix B 1, we summarize their definitions and prove their propertie...

    2026

  12. [12]

    We use Γ 0, Γ1, and Γ 2 to de- note the collection of sites, links, and plaquettes, respec- tively

    Lattice differential operators In this subsection, we review the discrete differential operators on a 2D lattice. We use Γ 0, Γ1, and Γ 2 to de- note the collection of sites, links, and plaquettes, respec- tively. As we explained in Section IV A, it is convenient 14 to label the lattice elements with the 1 2-notation, Γ0 ∪Γ 1 ∪Γ 2 ≃ 1 2 Z⊕ 1 2 Z(B1) such ...

  13. [13]

    We fix this constant by requiring G(0) = 0.(B10) The Poisson equation (B9) and the convention (B10) de- termine a uniqueG(x)

    Lattice Green function In this subsection, we review the lattice Green function G:Z 2 7→Rthat solves the 2D lattice Poisson equation, −∆G(x) =δ x,0 .(B9) IfG(x) is a solution, so isG(x) +c. We fix this constant by requiring G(0) = 0.(B10) The Poisson equation (B9) and the convention (B10) de- termine a uniqueG(x). G(x) does not have a particularly illumin...

  14. [14]

    Based on our discussion in Sec- tion IV C before Eq

    Two-point correlation function In this subsection, we evaluate the two-point correla- tion functionS ±(x). Based on our discussion in Sec- tion IV C before Eq. (47), we have S±(x−y) = Z DφDnDθe −S(φ,n,θ) ψ±(Cx) ¯ψ±(Cy) Z DφDnDθe −S(φ,n,θ) (C1) with the equivalence class of pathsC x,y ≡C y −C x rep- resented by Fig. 2a (and Fig. 4). First let us integrate ...

  15. [15]

    ∓iπ X ℓ∈Γ1 Cx,0(ℓ)δσx(ℓ) # ,(C16a) eiΘ±(x) ≡exp

    Asymptotic expansion In this subsection, we derive the large-|x|asymptotic expansion of the correlation functionS ±(x). Let us rewrite Eq. (49) as S±(x) =Z ±Z ±e2πG(x)eiΦ±(x)eiΘ±(x) ,(C15) where the phase functions eiΦ±(x) ≡exp " ∓iπ X ℓ∈Γ1 Cx,0(ℓ)δσx(ℓ) # ,(C16a) eiΘ±(x) ≡exp " ±iπ X ℓ∈Γ1 Cx,0(ℓ)δσ0(ℓ) # ,(C16b) for the equivalence class of pathsC x,0 re...

  16. [16]

    Since the leadingO(|x| −1) term in the asymptotic expansion (51) ofS +(x) decays more slowly thanO(|x| −2), the Fourier transform eS+(p) will contain singularities

    Momentum-space discontinuity In this subsection, we discuss the singularities of the momentum-space correlation function eS+(p). Since the leadingO(|x| −1) term in the asymptotic expansion (51) ofS +(x) decays more slowly thanO(|x| −2), the Fourier transform eS+(p) will contain singularities. We first show that eS+(p) is continuous onT 2 except forp= (0,0...

  17. [17]

    H. B. Nielsen and M. Ninomiya, Absence of Neutrinos on a Lattice. 1. Proof by Homotopy Theory, Nucl. Phys. B185, 20 (1981)

    1981

  18. [18]

    H. B. Nielsen and M. Ninomiya, Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof, Nucl. Phys. B193, 173 (1981)

    1981

  19. [19]

    L. H. Karsten and J. Smit, Lattice Fermions: Species Doubling, Chiral Invariance, and the Triangle Anomaly, Nucl. Phys.B183, 103 (1981), [,495(1980)]

    1981

  20. [20]

    L. H. Karsten, Lattice Fermions in Euclidean Space-time, Phys. Lett. B104, 315 (1981)

    1981

  21. [21]

    H. B. Nielsen and M. Ninomiya, No Go Theorem for Reg- ularizing Chiral Fermions, Phys. Lett.105B, 219 (1981)

    1981

  22. [22]

    Friedan, A Proof of the Nielsen-Ninomiya Thorem, Commun

    D. Friedan, A Proof of the Nielsen-Ninomiya Thorem, Commun. Math. Phys.85, 481 (1982)

    1982

  23. [23]

    Luscher, Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation, Phys

    M. Luscher, Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation, Phys. Lett. B428, 342 (1998), arXiv:hep-lat/9802011

    Pith/arXiv arXiv 1998

  24. [24]

    P. H. Ginsparg and K. G. Wilson, A Remnant of Chiral Symmetry on the Lattice, Phys. Rev.D25, 2649 (1982)

    1982

  25. [25]

    Neuberger, Exactly massless quarks on the lattice, Phys

    H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B417, 141 (1998), arXiv:hep-lat/9707022

    Pith/arXiv arXiv 1998

  26. [26]

    Neuberger, More about exactly massless quarks on the lattice, Phys

    H. Neuberger, More about exactly massless quarks on the lattice, Phys. Lett.B427, 353 (1998), arXiv:hep- lat/9801031 [hep-lat]

    arXiv 1998

  27. [27]

    Hasenfratz, V

    P. Hasenfratz, V. Laliena, and F. Niedermayer, The Index theorem in QCD with a finite cutoff, Phys. Lett. B427, 125 (1998), arXiv:hep-lat/9801021

    Pith/arXiv arXiv 1998

  28. [28]

    D. B. Kaplan, A Method for simulating chiral fermions on the lattice, Phys. Lett.B288, 342 (1992), arXiv:hep- lat/9206013 [hep-lat]

    arXiv 1992

  29. [29]

    Shamir, Chiral fermions from lattice boundaries, Nucl

    Y. Shamir, Chiral fermions from lattice boundaries, Nucl. Phys.B406, 90 (1993), arXiv:hep-lat/9303005 [hep-lat]

    Pith/arXiv arXiv 1993

  30. [30]

    Wang and Y.-Z

    J. Wang and Y.-Z. You, Symmetric Mass Generation, Symmetry14, 1475 (2022), arXiv:2204.14271 [cond- mat.str-el]

    Pith/arXiv arXiv 2022

  31. [31]

    You, Y.-C

    Y.-Z. You, Y.-C. He, C. Xu, and A. Vishwanath, Symmetric Fermion Mass Generation as Deconfined Quantum Criticality, Phys. Rev. X8, 011026 (2018), arXiv:1705.09313 [cond-mat.str-el]

    Pith/arXiv arXiv 2018

  32. [32]

    You, Y.-C

    Y.-Z. You, Y.-C. He, A. Vishwanath, and C. Xu, From Bosonic Topological Transition to Symmetric Fermion Mass Generation, Phys. Rev. B97, 125112 (2018), arXiv:1711.00863 [cond-mat.str-el]

    Pith/arXiv arXiv 2018

  33. [33]

    Fidkowski and A

    L. Fidkowski and A. Kitaev, The effects of interactions on the topological classification of free fermion systems, Phys. Rev. B81, 134509 (2010), arXiv:0904.2197 [cond- mat.str-el]

    Pith/arXiv arXiv 2010

  34. [34]

    Tong, Comments on symmetric mass generation in 2d and 4d, JHEP2022(07), 001, arXiv:2104.03997 [hep-th]

    D. Tong, Comments on symmetric mass generation in 2d and 4d, JHEP2022(07), 001, arXiv:2104.03997 [hep-th]

    Pith/arXiv arXiv

  35. [35]

    Eichten and J

    E. Eichten and J. Preskill, Chiral Gauge Theories on the Lattice, Nucl. Phys.B268, 179 (1986)

    1986

  36. [36]

    M. Zeng, Z. Zhu, J. Wang, and Y.-Z. You, Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model, Phys. Rev. Lett.128, 185301 (2022), arXiv:2202.12355 [cond-mat.str-el]

    Pith/arXiv arXiv 2022

  37. [37]

    Golterman and Y

    M. Golterman and Y. Shamir, Propagator Zeros and Lat- tice Chiral Gauge Theories, Phys. Rev. Lett.132, 081903 (2024), arXiv:2311.12790 [hep-lat]

    Pith/arXiv arXiv 2024

  38. [38]

    Golterman and Y

    M. Golterman and Y. Shamir, Constraints on the symmetric mass generation paradigm for lattice chi- ral gauge theories, Phys. Rev. D113, 014503 (2026), arXiv:2505.20436 [hep-lat]

    arXiv 2026

  39. [39]

    Golterman and Y

    M. Golterman and Y. Shamir, Symmetric mass gener- ation and the Nielsen-Ninomiya theorem, in42th In- ternational Symposium on Lattice Field Theory(2026) arXiv:2603.15985 [hep-lat]

    arXiv 2026

  40. [40]

    T. D. Schultz, D. C. Mattis, and E. H. Lieb, Two- dimensional Ising model as a soluble problem of many fermions, Rev. Mod. Phys.36, 856 (1964)

    1964

  41. [41]

    S. R. Coleman, The Quantum Sine-Gordon Equation as the Massive Thirring Model, Phys. Rev. D11, 2088 (1975)

    2088

  42. [42]

    Witten, Nonabelian Bosonization in Two-Dimensions, Commun

    E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys.92, 455 (1984), [,201(1983)]

    1984

  43. [43]

    D. J. Gross and I. R. Klebanov, One-dimensional string theory on a circle, Nucl. Phys. B344, 475 (1990)

    1990

  44. [44]

    Sulejmanpasic and C

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

    Pith/arXiv arXiv 2019

  45. [45]

    Gorantla, H

    P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, A modified Villain formulation of fractons and other exotic theories, J. Math. Phys.62, 102301 (2021), arXiv:2103.01257 [cond-mat.str-el]

    Pith/arXiv arXiv 2021

  46. [46]

    Berkowitz, A

    E. Berkowitz, A. Cherman, and T. Jacobson, Exact lat- tice chiral symmetry in 2D gauge theory, Phys. Rev. D 110, 014510 (2024), arXiv:2310.17539 [hep-lat]

    Pith/arXiv arXiv 2024

  47. [47]

    Seifnashri, Lieb-Schultz-Mattis anomalies as obstruc- tions to gauging (non-on-site) symmetries, arxiv (2023), arXiv:2308.05151 [cond-mat.str-el]

    S. Seifnashri, Lieb-Schultz-Mattis anomalies as obstruc- tions to gauging (non-on-site) symmetries, arxiv (2023), arXiv:2308.05151 [cond-mat.str-el]

    Pith/arXiv arXiv 2023

  48. [48]

    DeMarco, E

    M. DeMarco, E. Lake, and X.-G. Wen, A Lattice Chiral Boson Theory in 1 + 1d, arxiv (2023), arXiv:2305.03024 [cond-mat.str-el]

    Pith/arXiv arXiv 2023

  49. [49]

    Cheng and N

    M. Cheng and N. Seiberg, Lieb-Schultz-Mattis, Lut- tinger, and ’t Hooft – anomaly matching in lattice sys- tems, SciPost Phys.15, 051 (2023), arXiv:2211.12543 [cond-mat.str-el]

    Pith/arXiv arXiv 2023

  50. [50]

    Fazza and T

    L. Fazza and T. Sulejmanpasic, Lattice quantum Vil- lain Hamiltonians: compact scalars, U(1) gauge theories, fracton models and quantum Ising model dualities, JHEP 05, 017, arXiv:2211.13047 [hep-th]

    Pith/arXiv arXiv

  51. [51]

    Seifnashri, Exactly Solvable 1+1d Chiral Lattice Gauge Theories, arxiv (2026), arXiv:2601.14359 [hep- th]

    S. Seifnashri, Exactly Solvable 1+1d Chiral Lattice Gauge Theories, arxiv (2026), arXiv:2601.14359 [hep- th]

    Pith/arXiv arXiv 2026

  52. [52]

    Shamir, Constraints on the existence of chiral fermions in interacting lattice theories, Phys

    Y. Shamir, Constraints on the existence of chiral fermions in interacting lattice theories, Phys. Rev. Lett.71, 2691 (1993), arXiv:hep-lat/9306023

    Pith/arXiv arXiv 1993

  53. [53]

    Gurarie, Single-particle Green’s functions and inter- acting topological insulators, Phys

    V. Gurarie, Single-particle Green’s functions and inter- acting topological insulators, Phys. Rev. B83, 085426 (2011)

    2011

  54. [54]

    Xu and C

    Y. Xu and C. Xu, Green’s function Zero and Symmetric Mass Generation, arXiv (2021), arXiv:2103.15865 [cond- mat.str-el]

    Pith/arXiv arXiv 2021

  55. [55]

    D.-C. Lu, M. Zeng, and Y.-Z. You, Green’s function zeros in Fermi surface symmetric mass generation, Phys. Rev. B108, 205117 (2023), arXiv:2307.12223 [cond-mat.str- el]

    Pith/arXiv arXiv 2023

  56. [56]

    S. D. Drell, M. Weinstein, and S. Yankielowicz, Strong Coupling Field Theories. 2. Fermions and Gauge Fields on a Lattice, Phys. Rev. D14, 1627 (1976)

    1976

  57. [57]

    J. M. Rabin, Perturbation Theory for SLAC Lattice 20 Fermions, Phys. Rev. D24, 3218 (1981)

    1981

  58. [58]

    Weinstein, Anomalies and the Lattice Schwinger Model: Paradigm Not Paradox, Phys

    M. Weinstein, Anomalies and the Lattice Schwinger Model: Paradigm Not Paradox, Phys. Rev. D26, 839 (1982)

    1982

  59. [59]

    Stacey, Eliminating lattice fermion doubling, Phys

    R. Stacey, Eliminating lattice fermion doubling, Phys. Rev. D26, 468 (1982)

    1982

  60. [60]

    Stacey, GAUGE INVARIANCE AND FERMION DOUBLING ON THE LATTICE, Z

    R. Stacey, GAUGE INVARIANCE AND FERMION DOUBLING ON THE LATTICE, Z. Phys. C19, 75 (1983)

    1983

  61. [61]

    Stacey, SIMPLE DOUBLING OF LATTICE FERMIONS, Phys

    R. Stacey, SIMPLE DOUBLING OF LATTICE FERMIONS, Phys. Lett. B127, 440 (1983)

    1983

  62. [62]

    Stacey, SPECTRUM DOUBLING AND GAMMA MATRICES AND LATTICE BASED REGULARIZA- TIONS, J

    R. Stacey, SPECTRUM DOUBLING AND GAMMA MATRICES AND LATTICE BASED REGULARIZA- TIONS, J. Phys. G11, 1097 (1985)

    1985

  63. [63]

    Susskind, Lattice Fermions, Phys

    L. Susskind, Lattice Fermions, Phys. Rev.D16, 3031 (1977)

    1977

  64. [64]

    W. Bock, J. Smit, and J. C. Vink, Fermion Higgs model with reduced staggered fermions, Phys. Lett. B291, 297 (1992), arXiv:hep-lat/9206008

    Pith/arXiv arXiv 1992

  65. [65]

    Catterall, Lattice Regularization of Reduced K¨ ahler- Dirac Fermions and Connections to Chiral Fermions, Sci- Post Phys.16, 108 (2024), arXiv:2311.02487 [hep-lat]

    S. Catterall, Lattice Regularization of Reduced K¨ ahler- Dirac Fermions and Connections to Chiral Fermions, Sci- Post Phys.16, 108 (2024), arXiv:2311.02487 [hep-lat]

    Pith/arXiv arXiv 2024

  66. [66]

    K. G. Wilson, Quarks and strings on a lattice, inNew Phenomena in Subnuclear Physics: Part A, edited by A. Zichichi (Springer US, Boston, MA, 1977) pp. 69–142

    1977

  67. [67]

    J. W. Milnor,Topology from the Differentiable Viewpoint (Princeton University Press, 1997) based on notes by David W. Weaver

    1997

  68. [68]

    Luscher, Abelian chiral gauge theories on the lat- tice with exact gauge invariance, Nucl

    M. Luscher, Abelian chiral gauge theories on the lat- tice with exact gauge invariance, Nucl. Phys. B549, 295 (1999), arXiv:hep-lat/9811032

    Pith/arXiv arXiv 1999

  69. [69]

    Luscher, Weyl fermions on the lattice and the non- Abelian gauge anomaly, Nucl

    M. Luscher, Weyl fermions on the lattice and the non- Abelian gauge anomaly, Nucl. Phys. B568, 162 (2000), arXiv:hep-lat/9904009

    Pith/arXiv arXiv 2000

  70. [70]

    Chatterjee, S

    A. Chatterjee, S. D. Pace, and S.-H. Shao, Quan- tized Axial Charge of Staggered Fermions and the Chi- ral Anomaly, Phys. Rev. Lett.134, 021601 (2025), arXiv:2409.12220 [hep-th]

    Pith/arXiv arXiv 2025

  71. [71]

    Clancy, D

    M. Clancy, D. B. Kaplan, and H. Singh, General- ized Ginsparg-Wilson equations, arXiv e-prints (2023), arXiv:2309.08542 [hep-lat]

    Pith/arXiv arXiv 2023

  72. [72]

    Singh, Generalized Ginsparg-Wilson relations: Fermionic anomalies on the lattice, PoSLAT- TICE2024, 375 (2025), arXiv:2503.05900 [hep-lat]

    H. Singh, Generalized Ginsparg-Wilson relations: Fermionic anomalies on the lattice, PoSLAT- TICE2024, 375 (2025), arXiv:2503.05900 [hep-lat]

    Pith/arXiv arXiv 2025

  73. [73]

    Singh, Ginsparg-Wilson Hamiltonians with Im- proved Chiral Symmetry, arXiv e-prints (2025), arXiv:2505.20419 [hep-lat]

    H. Singh, Ginsparg-Wilson Hamiltonians with Im- proved Chiral Symmetry, arXiv e-prints (2025), arXiv:2505.20419 [hep-lat]

    Pith/arXiv arXiv 2025

  74. [74]

    Tachikawa, Topological phases and relativistic QFTs (2018), lectures given in the CERN winter school, Febru- ary 2018

    Y. Tachikawa, Topological phases and relativistic QFTs (2018), lectures given in the CERN winter school, Febru- ary 2018

    2018

  75. [75]

    Thorngren, Anomalies and Bosonization, Commun

    R. Thorngren, Anomalies and Bosonization, Commun. Math. Phys.378, 1775 (2020), arXiv:1810.04414 [cond- mat.str-el]

    Pith/arXiv arXiv 2020

  76. [76]

    Karch, D

    A. Karch, D. Tong, and C. Turner, A Web of 2d Dualities: Z2 Gauge Fields and Arf Invariants, SciPost Phys.7, 007 (2019), arXiv:1902.05550 [hep-th]

    Pith/arXiv arXiv 2019

  77. [77]

    Ji, S.-H

    W. Ji, S.-H. Shao, and X.-G. Wen, Topological Transition on the Conformal Manifold, Phys. Rev. Res.2, 033317 (2020), arXiv:1909.01425 [cond-mat.str-el]

    Pith/arXiv arXiv 2020

  78. [78]

    D. S. Freed, Z. Komargodski, and N. Seiberg, The Sum Over Topological Sectors andθin the 2+1-Dimensional CP1 σ-Model, Commun. Math. Phys.362, 167 (2018), arXiv:1707.05448 [cond-mat.str-el]

    Pith/arXiv arXiv 2018

  79. [79]

    Y. Lee, K. Ohmori, and Y. Tachikawa, Revisiting Wess- Zumino-Witten terms, SciPost Phys.10, 061 (2021), arXiv:2009.00033 [hep-th]

    Pith/arXiv arXiv 2021

  80. [80]

    Chen and Y

    S. Chen and Y. Tanizaki, Solitonic Symmetry beyond Ho- motopy: Invertibility from Bordism and Noninvertibil- ity from Topological Quantum Field Theory, Phys. Rev. Lett.131, 011602 (2023), arXiv:2210.13780 [hep-th]

    Pith/arXiv arXiv 2023

Showing first 80 references.