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

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 $G=U(1)<sup>n$, we show that $\mathcal D$ 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 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.

• The paper develops explicit criteria and an algorithm for constructing non-invertible duality defects in two-dimensional Weyl fermion theories.
• It demonstrates how finite abelian group gauging underpins fermion-boundary scattering and connects to monopole-electron interactions.
• The analysis rigorously establishes limitations in extending these methods to genuine non-abelian symmetries, refining symmetry classification.

Non-Invertible Symmetries in Weyl Fermions: Defects and Boundary Scattering

Overview

This manuscript presents a systematic construction and analysis of non-invertible topological defects in two-dimensional theories of $n$ Weyl fermions, grounded in recent advances in conformal boundary conditions of Dirac fermion theories. Key results establish explicit criteria and algorithms for generating non-invertible duality defects for abelian symmetry groups, discuss their realization via finite abelian group gauging, and probe the limitations of such constructions for non-abelian symmetry structures. Furthermore, the paper elucidates applications to fermion-boundary scattering phenomena, notably linking the presence of non-invertible symmetries in Weyl fermion systems to longstanding puzzles in four-dimensional monopole scattering problems via dimensional reduction.

Non-Invertible Symmetries and Defect Construction

Topological defects in QFT are recognized as the natural language for generalized symmetries, including non-invertible fusion rules. The paper targets $n$ free Weyl fermion CFTs, employing the "folding trick" to relate topological defects in Weyl fermion systems to conformal boundary conditions in Dirac fermion theories. By folding along $x=0$, the boundary in the Dirac system corresponds to a defect, $D$, in the Weyl system, which generically interwines two representations $V_g$ and $U_g$ of a symmetry group $G$.

When $G = U(1)^N$, the exploration hinges on anomaly-free charge assignments, leading to a duality defect $D$ associated with gauging a finite abelian group $\mathcal{T}$. The explicit algorithm leverages rational orthogonal matrices relating left and right charge assignments, their Smith normal decomposition, and the construction of $n$0 and its action on the fermions. The defect $n$1 is shown to be non-invertible for generic charge assignments, and invertible only for cases associated with permutation symmetry on the charge matrix.

For restricted non-abelian cases, such as in the Cartan sector of $n$2, finite abelian duality defects persist. However, for genuinely non-abelian symmetry structures (e.g., full $n$3 in the 1-5-7-8-9 model), the authors prove that no duality defect from finite abelian group gauging exists. This is achieved by analyzing commutation and intertwining relations and the structure of the charge matrices, revealing that no orthogonal rational matrix satisfies all consistency constraints.

Duality Defects and Quantum Symmetry

The duality defects constructed for abelian $n$4 exhibit explicit quantum dimension (e.g., $n$5 for cyclic group $n$6). The gauging procedure attaches a quantum symmetry line to the fermion operators, thus transforming local fermions into twisted sector states. The paper provides detailed partition function identities and prescriptions for extracting quantum symmetry lines and calculating their associated charges.

A key technical claim is the absence of mixed anomalies between $n$7 and $n$8—the global symmetry persists consistently across all sectors of the theory. The quantum symmetry of the gauged theory is explicitly derived via background field insertions in the partition function and tracked through the lattice condition on the charge assignments.

Fermion-Boundary Scattering via Defect Perspective

The manuscript advances a symmetry-based derivation of fermion-boundary scattering processes, linking the Callan-Rubakov effect and similar phenomena in four-dimensional gauge theories (notably monopole-electron scattering) to topological defects in 2D Weyl fermion theories. By folding, the boundary condition in Dirac fermion theory becomes a topological defect, enabling a systematic description of the scattering process where incoming fermions acquire quantum symmetry line attachments upon traversing the defect.

Strong numerical results are provided for specific models, including the 3-4-5-0 model and unit-charge monopole models, establishing the explicit cyclic group symmetry, quantum dimension, and charge assignments. Generalizations to higher-charge monopoles are discussed, proving that only uniform cyclic group gauging captures all symmetry constraints in these setups.

Crucially, the boundary scattering is mapped to the transformation of operators under the action of $n$9, and explicit expressions are given for charges and spectral properties of outgoing twisted sector states. In models where no non-invertible defect exists for non-abelian symmetry, the failure of boundary symmetry preservation is rigorously proven via the non-existence of orthogonal matrix solutions—contradicting previous proposals and establishing a clear boundary on the scope of abelian gauging constructions.

Implications and Future Directions

The investigation deepens the understanding of generalized symmetries in fermionic CFTs, particularly illuminating the classification and construction of non-invertible duality defects and their practical impact on observable boundary phenomena. On a theoretical level, the work demonstrates the utility of functional analytic tools (orthogonal matrices, Smith normal form, partition functions) in symmetry classification and defect construction.

Practically, the reformulation of boundary scattering in terms of duality defects suggests new perspectives on anomaly matching, defect-induced symmetry breaking, and the role of twisted sector excitations in low-dimensional quantum field theory. The systematic limitations for non-abelian symmetry preservation highlight open questions in the search for more general non-invertible symmetry structures, especially relevant for understanding boundary conditions in the Standard Model and strongly interacting field theories.

Future research avenues include:

• Extending the classification to non-invertible defects beyond duality defects (e.g., those not realized by finite group gauging).
• Exploring the implications for bulk-boundary entanglement and topological phases in higher-dimensional analogues.
• Investigating the emergence of generalized quantum symmetry lines in lattice and continuum models with strongly non-abelian charge sectors.

Conclusion

This comprehensive analysis provides a formal framework for understanding non-invertible symmetries in 2D Weyl fermion theories, their realization as duality defects via finite abelian group gauging, and their central role in boundary scattering problems. The explicit algorithms, strong technical results, and rigorous limitations discovered for non-abelian cases set the stage for further explorations of generalized symmetries and topological defects in both mathematical physics and quantum field theoretic settings.