Non-Invertible Symmetries and Boundaries for Two-Dimensional Fermions

Abstract: We study the relation between boundary conditions and categorical symmetries of two-dimensional fermionic conformal field theories. We determine all anomaly-free invertible global symmetries of two free complex Weyl fermions, which take the form $\mathbb{Z}_k$ for each primitive Pythagorean triple $a^2 + b^2 = k^2$. The theory is self-dual under gauging any of these symmetries, and so to each there is associated a non-invertible topological defect. We study the properties of these lines, and show that any conformal boundary condition of two Dirac fermions that preserves a $U(1)^2$ symmetry can be found by dressing a trivial Dirichlet boundary with one of them. We discuss two microscopic descriptions of these defects: fermions coupled to a quantum-mechanical rotor degree of freedom; and an abelian gauge theory that realises symmetric mass generation in a half-space.

• The paper reveals that the exhaustive classification of anomaly-free discrete symmetries in 2D fermion CFTs corresponds to cyclic groups determined by primitive Pythagorean triples.
• It demonstrates that gauging these symmetries realizes non-invertible topological defects with Tambara–Yamagami fusion rules and self-duality properties under discrete transformations.
• The work provides explicit constructions of symmetric conformal boundaries and UV models via symmetric mass generation, linking categorical symmetry to practical boundary phenomena.

Non-Invertible Symmetries and Boundaries in 2D Fermion CFTs: Structure, Classification, and Implications

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Introduction and Scope

This work rigorously investigates the correspondence between anomaly-free invertible and non-invertible symmetries and boundary conditions in two-dimensional (2D) fermion conformal field theories (CFTs), focusing specifically on the theory of two free complex Weyl (Dirac) fermions. The core of the study is a detailed classification of anomaly-free discrete invertible symmetries, their gauging, and the resultant non-invertible topological defects, as well as the implications for possible conformal boundaries preserving certain global symmetries. The analysis combines categorical symmetry, modular invariance, anomaly theory, fusion categories, and constructions of explicit UV models realizing the topological features.

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Classification of Anomaly-Free Invertible Symmetries

A main technical result is the exhaustive classification of all anomaly-free invertible discrete 0-form symmetries in the theory of two complex Weyl fermions. These are shown to be cyclic groups, with orders determined by arithmetic data equivalent to primitive Pythagorean triples $(a,b,k)$ such that $a^2 + b^2 = k^2$, where $a,b$ are coprime and $k > 0$. More precisely, for every such triple, an anomaly-free $\mathbb{Z}_k$ symmetry exists, generated by group actions with explicit charge assignments on the fundamental fermions. The group structure and equivalence relations among distinct parameterizations are analyzed using methods from number theory and the theory of Gaussian integers.

Non-abelian discrete subgroups are shown to always be anomalous in this context. The classification has strong implications: it asserts that only cyclic groups labeled by Pythagorean arithmetic data are realized as non-anomalous, and that the corresponding discrete gauging is unique (up to invertible lines) for each such group.

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Self-Duality Under Gauging and Categorical Structures

Gauging any of these anomaly-free discrete symmetries leaves the theory in the same universality class: the theory of two Weyl fermions is shown to be self-dual under such discrete gauging operations. The paper provides explicit modular-invariant partition functions in all spin structures, constructing the action of symmetry on Ramond and Neveu-Schwarz sectors. The action on twisted sectors is fixed by modular constraints and uniqueness arguments; only a unique choice (modulo stacking with invertible spin-TQFT phases) gives back a fermionic CFT with the correct gravitational anomaly.

Making the self-duality manifest, the authors provide explicit isomorphisms between the operator content (including local operators and extended operators/twist fields) before and after gauging. This involves careful accounting for shifts in fugacities and charge assignments under global $U(1)^2$ symmetries and the action of modular transformations.

Categorically, the non-invertible defects associated with self-dualities under discrete gauging are realized as topological lines with Tambara–Yamagami ($\mathrm{TY}(\mathbb{Z}_k)$) fusion rules when the symmetry and its quantum dual are appropriately identified. The additional gauge ambiguity (stacking with invertible lines or Arf phases) is classified and its physical meaning clarified.

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Construction and Properties of Non-Invertible Defects

The paper systematically constructs a family of non-invertible codimension-one topological defects ("self-duality defects") for each primitive Pythagorean triple. These are formed by "half-space gauging": gauging the discrete symmetry in one region and gluing the resulting theory to the ungauged theory via an isomorphism. Fusion rules, action on operators, scattering processes, and the relation to categorical quantum symmetry are analyzed. These defects, by construction, necessarily act non-invertibly on the local operator content.

A crucial physical effect is elucidated: the presence of such a non-invertible defect generically causes an excitation (say, a left-moving local field) to be reflected into a state attached to a topological line ending at the boundary, corresponding to the twisted sector operator. This is manifest in the scattering data for fermions in the presence of a boundary preserving the relevant symmetry.

Moreover, the detailed categorical structure (including the bicharacter data) of the non-invertible defects is explicitly determined and tied to the arithmetic of the associated Pythagorean triple.

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Classification and Construction of Symmetric Boundary Conditions

The theory realizes two distinct classes of simple conformal boundary conditions preserving a vector-like or chiral anomaly-free $U(1)^2$ symmetry, labeled as Class~$\mathcal{V}$ and Class~$\mathcal{A}$. The charge preserving reflection or rotation boundaries are categorized and their SPT classes identified via cobordism and anomaly inflow arguments.

A central result is that every conformal boundary condition for two Dirac (or four Majorana-Weyl) fermions preserving an anomaly-free $a^2 + b^2 = k^2$0 symmetry arises from dressing a trivial (vector-like) Dirichlet boundary with one of the constructed non-invertible lines and, for rotation-type boundaries, an additional invertible topological line. For Class $a^2 + b^2 = k^2$1 symmetries, the only possible "boundary" is actually an interface to the Arf topological phase; a true boundary preserving the symmetry requires adding an unpaired boundary Majorana zero mode (i.e., a non-simple boundary).

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UV Realizations: Rotor Model and Symmetric Mass Generation

Microscopic realizations of these topological defects and boundary conditions are discussed. The non-invertible lines are shown to be realized in the IR by coupling the fermions to a quantum-mechanical abelian rotor degree of freedom at a point—a scenario already encountered in discussions of monopole scattering and the Kondo problem. The symmetry and charge assignments required for anomaly-freedom and absence/presence of boundary Majorana modes are established.

In addition, the paper constructs gauge-theoretic models dynamically realizing the boundaries through symmetric mass generation (SMG), wherein the mass deformation and confluence of Higgs/confining phases correspond to the different boundary SPT classes. It is demonstrated that boundaries preserving vector-like (Class $a^2 + b^2 = k^2$2) symmetries flow to the trivial phase, while those preserving chiral (Class $a^2 + b^2 = k^2$3) symmetries flow to the Arf topological phase, in some cases necessitating the presence of additional localized boundary Majoranas.

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Implications and Outlook

This comprehensive analysis advances the structural understanding of anomaly-free discrete symmetries, their non-invertible avatars, and their manifestation in conformal boundaries in 2D fermion CFTs. Practically, the results give concrete recipes for constructing and classifying symmetric gapped and gapless boundaries (critical for both condensed matter and high-energy contexts such as the study of monopole-fermion dynamics, quantum wires, and edge states in topological phases). The explicit arithmetic classification enables generalization and provides diagnostic tools for determining possible categorical symmetries and non-invertible topological operators in similar theories.

Theoretically, the identification between categorical symmetries and boundary SCFT data adds to the landscape of dualities, topological defects, and modular bootstrap. It also offers templates for extending the results to higher $a^2 + b^2 = k^2$4 (numbers of fermions), interacting CFTs, and more complicated global symmetry assignments.

An open direction is the full generalization to cases with $a^2 + b^2 = k^2$5 free fermions, where non-invertibility and the associated fusion categories become more intricate and may be tied to the still-unresolved monopole scattering in certain gauge theories. The techniques of half-space gauging, rotor realization, and modular bootstrap developed here provide a robust framework for such generalizations.

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Conclusion

The paper establishes a complete and explicit classification of anomaly-free invertible and non-invertible symmetries in the theory of two 2D Weyl fermions and elucidates their deep connection to the structure of conformal boundary conditions. It provides comprehensive constructions, explicit UV models, and a rigorous categorical analysis, clarifying the interplay between boundary SPT classes, fusion rules, and the full global symmetry group. These results have broad implications for the understanding of topological defects, boundary phenomena, and symmetry-protected phases in two-dimensional quantum field theories.