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Chiral gauge theories revisited

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Martin Lüscher

High Energy Physics - Theory · Submitted 6 Feb 2001 · 90 references

Pith's verdict: Authoritative, compact lecture notes on chiral lattice gauge theory; the central theorem is quoted from the author's own published paper, and the only wrinkle is a printed index convention in item (2).

T0 · AI review

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  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDhep-th/0102028 (version 2)
Pith IDTS3YYXLB, Pith's permanent record of this paper and its review
CitationMartin L\"uscher. "Chiral gauge theories revisited." arXiv:hep-th/0102028 (2001). https://pith.science/paper/TS3YYXLB
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The paper

The paper claims

Anomaly-free chiral gauge theories can be regularized on a lattice without breaking gauge symmetry, to all orders in perturbation theory.

The reading

These lecture notes explain how to put an anomaly-free chiral gauge theory on a four-dimensional lattice while keeping the gauge symmetry exact. The obstacle is that the fermion measure—the integration measure over left-handed fields—has a gauge-dependent phase, and a naive lattice projection produces unwanted fermion copies. The author shows that a lattice Dirac operator obeying the Ginsparg–Wilson relation supplies an exact modified chiral symmetry, and that the measure phase can be fixed by a local current satisfying both the anomalous conservation law and an integrability condition. The resulting theorem states that, for any anomaly-free representation, this can be done to all orders of perturbation theory; the full recursive construction is cited to ref. [37], and non-perturbative global issues are explicitly left aside.

What carries the argument

The load-bearing object is the Ginsparg–Wilson relation, a lattice identity that replaces the continuum anticommutation $[6pt]

Extended reading notes

Core claim

The central claim is that anomaly-free chiral gauge theories admit a gauge-invariant lattice regularization in four dimensions, to all orders in the gauge coupling. The construction starts from a local lattice Dirac operator $D$ that satisfies the Ginsparg–Wilson relation $[6pt]

Load-bearing premise

The load-bearing premise is that, for every anomaly-free fermion representation, a local current exists that satisfies both the anomalous conservation law and the field-space integrability condition to all orders of the gauge coupling; the lectures state this as a theorem and cite the recursive construction rather than reproducing it.

Original abstract

Contents: 1. Introduction, 2. Chiral gauge theories & the gauge anomaly, 3. The regularization problem, 4. Weyl fermions from 4+1 dimensions, 5. The Ginsparg-Wilson relation, 6. Gauge-invariant lattice regularization of anomaly-free theories.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,594 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Contents: 1. Introduction, 2. Chiral gauge theories & the gauge anomaly, 3. The regularization problem, 4. Weyl fermions from 4+1 dimensions, 5. The Ginsparg-Wilson relation, 6. Gauge-invariant lattice regularization of anomaly-free theories.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Contents: 1. Introduction, 2. background 0.95 0.03 0.01 unclear 1.00
s1 Chiral gauge theories & the gauge anomaly, 3. background 0.92 0.06 0.03 unclear 1.00
s2 The regularization problem, 4. background 0.89 0.06 0.02 unclear 0.99
s3 Weyl fermions from 4+1 dimensions, 5. contribution 0.43 0.17 0.03 unclear 0.94
s4 The Ginsparg-Wilson relation, 6. background 0.62 0.10 0.04 unclear 0.99
s5 Gauge-invariant lattice regularization of anomaly-free theories. contribution 0.81 0.39 0.21 unclear 0.73

What kind of paper the classifier reports this is: review 0.93 · evidence from proof or derivation 0.46.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Feynman diagram contributing to the muon decay at two
Figure 2. The vertices V (n) (p1, . . . , pn) are equal to the sum of all (n − 1)! fermion one-loop diagrams with n amputated external gauge field lines. Since all entries in the functional integral respect the gauge symmetry, it seems obvious that the effective action must be a gauge-invariant expression in the gauge potential. However, the fermion one-loop diagrams with less than 5 external lines are ultra-violet-divergent and thus only incompletely defined. Now it can happen that any consistent way to make the diagrams finite necessarily breaks the gauge symmetry of the effective action. In
Figure 3. In lattice gauge theory the fermion fields
Figure 4. Qualitative shape of the background field
Figure 5. For negative chirality, the potential V (s) has a well at s = 0 in which the massless mode is trapped. The right-handed modes, on the other hand, are always heavy independently of whether they are bound to the domain wall or not. is a non-negative operator. Moreover, all non-zero mass values m must be of order M, since there is no other characteristic scale around. (c) Massless modes. In this case the Dirac equation reduces to {−γ5∂s + φ(s)}u(s) = 0, γµpµ u(s) = 0, (4.7) from which we infer that u(s) = exp  ± Z s 0 dt φ(t)  v, γµpµ v = 0, P±v = v. (4.8) While the positive chi
Figure 6. The projector Pˆ− maps the space of all lattice Dirac fields to the subspace of left-handed fields. Since the projector involves the lattice Dirac operator D, the sub￾space changes when the gauge field is varied. any such algebra is unique up to a complex proportionality factor. In particular, if we pass to a different orthonormal basis, v˜j (x) = X l vl(x)
Figure 7. Equation (6.8) describes a curve in the space of all la
Figure 8. At low energies, massive fermions in a five-dimension

Figures reproduced from arXiv:hep-th/0102028 by the authors.

References
How the literature uses it

35 arXiv papers cite this in 45 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Papers citing it · 25 sentences without a confident label

Cites it as background 18 papers
  • During the past two decades, since the work of Ginsparg and Wilson (GW) [ref] , there has been significant progress in understanding chiral symmetries on the lattice [ref] – [ref] ; further references are given in the reviews [ref] , while [ref] contain more recent work.

    Lattice chirality and the decoupling of mirror fermions · Motivation · 1.00

  • There have been numerous attempts to find a lattice regularization of a chiral gauge theory, using domain wall or overlap fermions as a starting point, or by other approaches [One approach has been to gauge the Lüscher symmetries, which has only been proved possible for Abelian gauge symmetries [ref] .

    Chiral gauge theory at the boundary between topological phases · Introduction · 0.84

  • A number of ingenious approaches have sidestepped this no-go theorem for anomalous or anomaly-free chiral models [ref] , including the overlap-fermion approach [ref] , which computes correlation functions as the overlaps of successive ground states [ref] , and the related domain wall approach [ref] .

    A Lattice Chiral Boson Theory in $1+1$d · Introduction · 0.81

Discussion

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