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arxiv: 2605.19363 · v1 · pith:7GUGMIWCnew · submitted 2026-05-19 · ✦ hep-th

Non-invertible Symmetries in Weyl Fermions, and Applications to Fermion-Boundary Scattering Problem

Pengcheng Wei , Yunqin Zheng This is my paper

Pith reviewed 2026-05-20 04:43 UTC · model grok-4.3

classification ✦ hep-th
keywords non-invertible symmetriesWeyl fermionstopological defectsconformal boundary conditionsduality defectsfermion scatteringgauging symmetries
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The pith

Non-invertible topological defects in two-dimensional Weyl fermion theories are constructed by unfolding G-symmetric conformal boundary conditions from Dirac fermions.

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

The paper shows how to build non-invertible topological defects for n Weyl fermions in two dimensions. It starts from the existence of G-symmetric conformal boundary conditions for n Dirac fermions and unfolds them into defects that mix different G-representations. For abelian symmetries like U(1)^n these defects correspond to duality transformations from gauging finite groups. The approach also clarifies fermion scattering off conformal boundaries using the duality picture. This offers a systematic way to find and classify such defects in fermionic systems.

Core claim

We construct a family of non-invertible topological defects in two-dimensional theories of n Weyl fermions. The construction relies on the existence of G-symmetric conformal boundary conditions for n Dirac fermions. Upon unfolding, these boundary conditions become topological defects D of n Weyl fermions that intertwine the two G-representations, and they are generically non-invertible. For G=U(1)^n, we show that D is a duality defect associated with gauging a finite Abelian group Γ, and we give an explicit algorithm for determining Γ and its action on the fermions. We also show that the same finite-Abelian gauging description applies in certain restricted examples with non-Abelian G. By the

What carries the argument

The non-invertible topological defect D obtained upon unfolding G-symmetric conformal boundary conditions for n Dirac fermions into n Weyl fermions, which intertwines G-representations.

If this is right

  • For U(1)^n symmetries, the defect D is realized as a duality defect from gauging a finite Abelian group Γ.
  • An explicit algorithm determines Γ and its action on the fermions.
  • The finite-Abelian gauging description extends to some restricted non-Abelian G cases.
  • For G=SU(2) in the 1-5-7-8-9 problem, D cannot be a duality defect from gauging any finite Abelian group.
  • The duality perspective streamlines the derivation of fermion scattering from a conformal boundary.

Where Pith is reading between the lines

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

  • This method could be extended to classify non-invertible symmetries in other two-dimensional conformal field theories with fermions.
  • Applications might include understanding boundary states in condensed matter systems with Weyl fermions.
  • The distinction between abelian and non-abelian cases suggests different physical realizations for these defects.

Load-bearing premise

That G-symmetric conformal boundary conditions exist for the n Dirac fermions and can be unfolded to produce the topological defects in the Weyl fermion theory.

What would settle it

An explicit example of a symmetry G for which no G-symmetric conformal boundary conditions exist for the Dirac fermions, or a calculation showing the unfolded defect is invertible rather than non-invertible.

Figures

Figures reproduced from arXiv: 2605.19363 by Pengcheng Wei, Yunqin Zheng.

Figure 1
Figure 1. Figure 1: The folding trick. On the left is the theory of Weyl fermions with a topological [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Topological defect D is G-intertwining. The defects Ug and Vg can terminate topologically on D at a common junction. D¯ D Ug Vg Ug A Ug Ug = [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Transverse intersection of Ug with D ⊗ D ¯ = A. denote by A [57]: D ⊗ D ¯ = 1 ⊕ · · · ≡ A. (2.2) In fact, we can assign a canonical (co)algebra structure to A via (co)evaluation maps, pro￾moting it to a separable Frobenius algebra. The defect D is then the duality defect obtained by gauging the algebra A. We remark that this statement applies to a general topological defect, and D is not necessarily a Tamb… view at source ↗
Figure 4
Figure 4. Figure 4: Transverse intersection of Vg with D ⊗ D¯ = A′ . D O(x) D Oe(x) L = [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Passing the topological defect D through O, the point operator will become at￾tached to a line operator L ⊂ D ⊗ D¯ = A′ . seeing [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A sequence of topological moves, showing the relation on torus partition function [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Scattering process require mj = ⌈ Pn u=1 Vuj qui Nu − 1 2 ⌉. In other words, we derive the scattering process, as in [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
read the original abstract

We construct a family of non-invertible topological defects in two-dimensional theories of $n$ Weyl fermions. The construction relies on the existence of $G$-symmetric conformal boundary conditions for $n$ Dirac fermions. Upon unfolding, these boundary conditions become topological defects $\mathcal D$ of $n$ Weyl fermions that intertwine the two $G$-representations, and they are generically non-invertible. For $G=U(1)^n$, we show that $\mathcal D$ is a duality defect associated with gauging a finite Abelian group $\Gamma$, and we give an explicit algorithm for determining $\Gamma$ and its action on the fermions. We also show that the same finite-Abelian gauging description applies in certain restricted examples with non-Abelian $G$. By contrast, for certain non-Abelian symmetry structures, including the $G=SU(2)$ symmetry appearing in the $1$-$5$-$7$-$8$-$9$ problem, we prove that $\mathcal D$ cannot be realized as a duality defect for gauging any finite Abelian group. Finally, we explain how the duality-defect perspective gives a streamlined derivation of fermion scattering from a conformal boundary.

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 / 2 minor

Summary. The paper constructs a family of non-invertible topological defects D in two-dimensional theories of n Weyl fermions. The construction starts from the existence of G-symmetric conformal boundary conditions on n Dirac fermions; upon unfolding, these become defects that intertwine the two G-representations and are generically non-invertible. For G = U(1)^n the defects are shown to be duality defects obtained by gauging a finite Abelian group Γ, with an explicit algorithm given to determine Γ and its action. The same description applies in certain restricted non-Abelian examples. For G = SU(2) in the 1-5-7-8-9 problem the paper proves that D cannot be realized as an Abelian duality defect. The duality-defect viewpoint is used to streamline the derivation of fermion scattering off a conformal boundary.

Significance. If the central constructions hold, the work supplies a systematic route to non-invertible defects in free-fermion theories and clarifies their relation to duality defects and boundary states. The explicit algorithm for determining Γ in the U(1)^n case and the proof that the SU(2) defect in the 1-5-7-8-9 problem is not an Abelian duality defect are concrete, falsifiable contributions that strengthen the paper. The streamlined scattering derivation is a useful application.

major comments (2)
  1. [§2] The central construction (abstract and §2) assumes the existence of G-symmetric conformal boundary conditions for n Dirac fermions without providing a general construction or classification for arbitrary G. This assumption is load-bearing: if such boundary conditions do not exist or fail to be conformal while preserving the full G action, the unfolding step does not produce the claimed topological defects D. The paper supplies an explicit algorithm only for G = U(1)^n and a counter-example for one non-Abelian case; a general existence proof or additional explicit constructions would be required to support the family of defects for general G.
  2. [§4.2] §4.2, the proof that the SU(2) defect in the 1-5-7-8-9 problem cannot be realized as an Abelian duality defect: the argument relies on representation-theoretic obstructions, but the precise matching between the unfolded defect and the would-be gauging data is not spelled out in sufficient detail to verify that all possible finite Abelian groups have been ruled out.
minor comments (2)
  1. [§3] Notation for the two G-representations that the defect intertwines is introduced without a dedicated table or diagram; adding one would improve readability when comparing Abelian and non-Abelian cases.
  2. [§5] The scattering derivation in §5 re-derives known results; a short comparison table with the original boundary-state calculation would make the streamlining benefit more transparent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§2] The central construction (abstract and §2) assumes the existence of G-symmetric conformal boundary conditions for n Dirac fermions without providing a general construction or classification for arbitrary G. This assumption is load-bearing: if such boundary conditions do not exist or fail to be conformal while preserving the full G action, the unfolding step does not produce the claimed topological defects D. The paper supplies an explicit algorithm only for G = U(1)^n and a counter-example for one non-Abelian case; a general existence proof or additional explicit constructions would be required to support the family of defects for general G.

    Authors: We agree that the general construction is conditional on the existence of G-symmetric conformal boundary conditions. The manuscript provides a complete, explicit algorithm for the case G = U(1)^n (the primary family) together with concrete non-Abelian illustrations and one counter-example. A full classification of such boundaries for arbitrary G is a separate, extensive problem that lies beyond the present scope. In the revision we will add an explicit statement that the defects D are constructed whenever G-symmetric conformal boundaries exist, and we will emphasize the concrete results already obtained for U(1)^n and the restricted non-Abelian examples. revision: partial

  2. Referee: [§4.2] §4.2, the proof that the SU(2) defect in the 1-5-7-8-9 problem cannot be realized as an Abelian duality defect: the argument relies on representation-theoretic obstructions, but the precise matching between the unfolded defect and the would-be gauging data is not spelled out in sufficient detail to verify that all possible finite Abelian groups have been ruled out.

    Authors: We thank the referee for this observation. In the revised manuscript we will expand §4.2 to spell out the matching in greater detail: we will explicitly compare the representation content and fusion rules of the unfolded SU(2) defect against the possible actions of any finite Abelian group Γ, showing why no such Γ can reproduce the required intertwining. This will make the representation-theoretic obstructions fully explicit and confirm that all candidate Abelian gaugings are excluded. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is one-way from assumed boundary conditions

full rationale

The paper's central construction explicitly takes the existence of G-symmetric conformal boundary conditions on n Dirac fermions as an external input, then unfolds them to produce the claimed non-invertible defects D on n Weyl fermions. This is a forward derivation rather than a self-referential loop. For G=U(1)^n an explicit algorithm determines the gauging group Gamma; for selected non-Abelian cases the paper proves non-realizability as Abelian duality defects. No quoted step reduces a prediction or first-principles result to a fitted parameter, self-citation, or definitional renaming of the output itself. The derivation therefore remains self-contained against the stated external benchmark of boundary-condition existence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of certain boundary conditions and the unfolding procedure; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Existence of G-symmetric conformal boundary conditions for n Dirac fermions
    Directly invoked as the starting point for constructing the defects via unfolding, per the abstract.

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