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arxiv: 2606.12358 · v1 · pith:H4JB4Q52new · submitted 2026-06-10 · ✦ hep-lat · hep-th

Lattice chiral non-Abelian gauge symmetry via bosonization

Soma Onoda This is my paper

Pith reviewed 2026-06-27 07:29 UTC · model grok-4.3

classification ✦ hep-lat hep-th
keywords lattice gauge theorychiral fermionsbosonizationanomaly cancellationWess-Zumino-Witten modelnon-Abelian gauge symmetrytwo-dimensional theories
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The pith

Bosonized lattice construction cancels non-Abelian gauge anomalies at finite spacing when quadratic indices match.

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

The paper develops a lattice formulation for two-dimensional non-Abelian chiral gauge theories by using non-Abelian bosonization of the fermions. It begins from the continuum structure in which the gauge anomaly appears as inflow from a three-dimensional bulk term inside the gauged Wess-Zumino-Witten model and translates that structure onto the lattice with the help of gauge-neutral spectator fermions. The resulting discrete model is defined so that the left and right bulk contributions cancel exactly inside the exponentiated action as soon as the left and right representations have identical quadratic indices. This cancellation is shown to occur at finite lattice spacing and does not require first taking the continuum limit. Readers may care because the approach supplies an explicit mechanism that places the continuum anomaly-cancellation condition directly on the lattice for these theories.

Core claim

In the bosonized description the gauge anomaly of chiral fermions is represented as anomaly inflow from a three-dimensional Chern-Simons-type bulk contribution contained in a gauged Wess-Zumino-Witten model. The lattice formulation constructs the counterpart of this model with a three-dimensional bulk extension under appropriate smoothness conditions. A salient feature is that the left and right bulk contributions cancel in the exponentiated action even before the continuum limit whenever the anomaly-free condition is satisfied, that is, when the left and right representations have identical quadratic indices. Thus the construction realizes the anomaly-cancellation mechanism at finite lattic

What carries the argument

The lattice counterpart of the gauged Wess-Zumino-Witten model with three-dimensional bulk extension, whose left and right contributions cancel in the exponentiated action under matching quadratic indices.

If this is right

  • Anomaly cancellation occurs at finite lattice spacing for any pair of representations whose quadratic indices are identical.
  • Gauge-neutral spectator fermions make the bosonized lattice description possible.
  • The cancellation appears in the exponentiated action without requiring a continuum limit.
  • The method supplies a concrete lattice realization of two-dimensional anomaly-free non-Abelian chiral gauge theories.

Where Pith is reading between the lines

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

  • The same bulk-cancellation idea could be examined for lattice formulations of chiral theories in dimensions higher than two.
  • Numerical checks on small lattices could test whether the required smoothness conditions preserve the exact cancellation.
  • The construction might be compared with other lattice bosonization techniques to see whether they also achieve finite-spacing cancellation for anomaly-free cases.

Load-bearing premise

The lattice gauged Wess-Zumino-Witten model with three-dimensional bulk extension reproduces the continuum anomaly inflow structure before the continuum limit is taken under appropriate smoothness conditions.

What would settle it

A direct evaluation on a small lattice showing that the left and right bulk contributions fail to cancel when the quadratic indices of the representations are equal.

Figures

Figures reproduced from arXiv: 2606.12358 by Soma Onoda.

Figure 1
Figure 1. Figure 1: Labeling of the vertices of a cube. The coordinate axes (x1, x2, x3) are also shown. We use 0 = (0, 0, 0), 1 = (1, 0, 0), 2 = (0, 1, 0), 3 = (0, 0, 1), 4 = (1, 1, 1), 5 = (1, 0, 1), 6 = (1, 1, 0), and 7 = (0, 1, 1). 3.2 Construction of the gauge-invariant lattice action We define the lattice counterpart of the continuum action (3.1) by S[g1, g2, U] = X n∈Λ2 1 4λ2 X µ Tr h DL g1(n, µ) † [PITH_FULL_IMAGE:f… view at source ↗
Figure 2
Figure 2. Figure 2: Unfolded net of the cube. Each face is labeled by i = 1, . . . , 6 together with the corre￾sponding coordinate condition. a b c d xα xβ [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Choice of the vertices (a, b, c, d) and the local coordinates (xα, xβ) on each face. The coordinates are chosen so that a = (0, 0), b = (1, 0), c = (0, 1), and d = (1, 1). 1, . . . , 6, we first define P L i (xα, xβ; g1) := h g1(a) †U L acg1(c) ixβ × " h g1(c) †U L cag1(a) ixβ g1(a) †RL [(UacUcdUdbUba) xβ ]U L abg1(b) h g1(b) †U L bdg1(d) ixβ #xα . (3.39) Here a, b, c, d denote the vertices of the face i a… view at source ↗
read the original abstract

A central issue in lattice formulations of chiral gauge theories is how the anomaly cancellation mechanism of the continuum theory can be realized at finite lattice spacing. In the present paper, based on non-Abelian bosonization, we propose a lattice formulation of the bosonic theory corresponding to a two-dimensional non-Abelian chiral gauge theory. In the continuum theory, the gauge anomaly of chiral fermions is represented, in the bosonized description, as anomaly inflow from a three-dimensional Chern--Simons-type bulk contribution contained in a gauged Wess--Zumino--Witten model. Motivated by this structure, we introduce gauge-neutral spectator fermions and use the resulting bosonized description. We then construct a lattice counterpart of the gauged Wess--Zumino--Witten model with a three-dimensional bulk extension under appropriate smoothness conditions. A salient feature of this lattice formulation is the cancellation of the left and right bulk contributions in the exponentiated action. This cancellation occurs even before taking the continuum limit when the anomaly-free condition is satisfied, namely when the left and right representations have identical quadratic indices. Thus, the present construction realizes the anomaly-cancellation mechanism at finite lattice spacing via the bosonized description of two-dimensional anomaly-free chiral gauge theories. Establishing the desired continuum limit remains an important open problem.

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 a lattice formulation of two-dimensional non-Abelian chiral gauge theories based on non-Abelian bosonization. Gauge-neutral spectator fermions are introduced to obtain a bosonized description, from which a lattice counterpart of the gauged Wess-Zumino-Witten model with three-dimensional bulk extension is constructed under appropriate smoothness conditions. The central claim is that left and right bulk contributions cancel exactly in the exponentiated action at finite lattice spacing whenever the anomaly-free condition holds (identical quadratic indices for left and right representations). The desired continuum limit is explicitly left as an open problem.

Significance. If the finite-spacing cancellation can be established rigorously, the construction would provide a concrete realization of the anomaly-cancellation mechanism at finite lattice spacing for 2D chiral gauge theories via bosonization. This is a potentially useful technical step in a field where exact anomaly cancellation on the lattice has been difficult; the paper correctly flags the continuum limit as remaining open, which appropriately bounds the current claim.

major comments (1)
  1. [Abstract] Abstract: the claim that cancellation of left and right bulk contributions occurs 'even before taking the continuum limit' when quadratic indices match is asserted for a 'lattice counterpart of the gauged Wess-Zumino-Witten model with a three-dimensional bulk extension under appropriate smoothness conditions.' No explicit lattice definition, measure, or discretization of the bulk term is supplied, so it is unclear whether the cancellation is an identity of the discrete model itself or an artifact that requires the smoothness conditions to reproduce continuum-like behavior at finite spacing. This is load-bearing for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to clarify the status of the lattice construction in the abstract. The manuscript defines a lattice counterpart under the stated smoothness conditions, and we address the concern point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that cancellation of left and right bulk contributions occurs 'even before taking the continuum limit' when quadratic indices match is asserted for a 'lattice counterpart of the gauged Wess-Zumino-Witten model with a three-dimensional bulk extension under appropriate smoothness conditions.' No explicit lattice definition, measure, or discretization of the bulk term is supplied, so it is unclear whether the cancellation is an identity of the discrete model itself or an artifact that requires the smoothness conditions to reproduce continuum-like behavior at finite spacing. This is load-bearing for the central claim.

    Authors: The lattice counterpart is constructed in the body of the paper by discretizing the gauged WZW model together with its three-dimensional bulk extension, subject to the smoothness conditions that render the discretization well-defined on the lattice. These conditions are part of the model definition and ensure that the bulk term can be evaluated at finite spacing without additional lattice artifacts. Under this definition the left and right bulk contributions cancel exactly in the exponentiated action as soon as the quadratic indices match; the cancellation is therefore an identity of the discrete model itself rather than a continuum artifact. The smoothness requirement is the minimal condition needed to make the bulk term a valid lattice object, analogous to how continuum anomaly inflow is realized only for sufficiently smooth fields. We will revise the abstract to state this more explicitly and to reference the relevant sections where the discretization is specified. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper proposes an original lattice formulation of the bosonized gauged WZW model with 3D bulk extension, defined under explicit smoothness conditions, and states that left/right bulk terms cancel in the exponentiated action precisely when the quadratic indices match. This cancellation is asserted as a direct property of the defined lattice model rather than a reduction to a fitted parameter, self-citation, or prior ansatz. The argument invokes standard continuum bosonization identities for motivation but does not derive the central finite-spacing result by re-labeling inputs or by load-bearing self-citation; the lattice construction itself supplies the claimed mechanism.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on established continuum bosonization and WZW model properties rather than introducing new fitted parameters or postulated entities; the lattice construction itself is the novel element.

axioms (2)
  • domain assumption Continuum non-Abelian bosonization maps the chiral fermion anomaly to inflow from a gauged WZW bulk term
    Invoked to motivate the lattice construction from the continuum theory.
  • standard math Anomaly cancellation occurs precisely when left and right quadratic indices are identical
    Standard result in gauge anomaly literature used to identify the cancellation condition.

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discussion (0)

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Reference graph

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