Non-invertible Symmetries in Quantum Field Theory

Updated 12 March 2026

Non-invertible symmetries are generalized symmetry structures characterized by fusion categories of topological defects rather than by groups.
They manifest in quantum field theories by governing dualities, constraining renormalization group flows, and encoding complex anomaly structures.
Their categorical construction provides concrete operator frameworks in diverse systems, including lattice models, gauge theories, and holographic duals.

A non-invertible symmetry in quantum field theory is a generalized symmetry structure, where the algebra of symmetry operators is encoded by a fusion category or higher fusion category of topological defects. Unlike ordinary (group-like) symmetries, whose fusion is invertible and forms a group, non-invertible symmetries are realized by topological defects whose fusion closes only under a generalized fusion algebra—with non-negative integer coefficients and, generically, without inverses. Non-invertibility is manifest at the level of both operator product expansions and the action on boundary conditions, local operators, and extended observables. Non-invertible symmetries have broad implications: they underlie dualities, constrain renormalization group flows, encode generalized anomaly structures, and can obstruct the existence of symmetric gapped phases. Their mathematical underpinning is in the theory of fusion categories and module categories, and recent developments place non-invertible symmetries at the forefront of quantum field theory, condensed matter, and quantum information.

1. Algebraic and Categorical Foundations

Non-invertible symmetries are specified by unitary fusion categories (in 1+1d), fusion $n$ -categories in higher dimensions, or weak Hopf algebra data. The simple objects of the fusion category correspond to codimension-1 topological symmetry defects $L_i$ , whose fusion rules take the general form

$L_i \otimes L_j = \bigoplus_k N_{ij}^k L_k, \quad N_{ij}^k \in \mathbb{Z}_{\ge 0},$

rather than a group law (Choi et al., 2023, Ahmad et al., 22 Sep 2025). These fusion coefficients encode how non-invertible symmetries combine when placed on parallel defect lines or surfaces. Junction operators between fusions live in Hom spaces, implementing a rich associativity structure determined by $F$ -symbols, which satisfy the pentagon equation.

Non-invertible symmetry defects act on local and extended operators as quantum operations—specifically unital, completely positive (CP) maps (Okada et al., 2024). These maps generalize automorphisms: in the invertible case, the action is by unitary conjugation; for non-invertible defects, the action is by quantum channels with Kraus decompositions and a composition law directly reflecting the categorical fusion structure. In the most general framework, non-invertible actions arise as trace-preserving channels or isometries between collection of Hilbert spaces corresponding to twisted sectors, compatible with a unitarity structure on the symmetry category (Bartsch et al., 6 Feb 2026, Ortiz et al., 29 Sep 2025).

In higher dimensions (e.g., 2+1d and above), non-invertible symmetries are organized as higher categories—such as 2-categories in 3d QFTs (Bhardwaj et al., 2022) and fusion 5-categories in 6d (Apruzzi et al., 2024). The construction and analysis of such structures require categorical tools (module categories, condensation algebras, etc.), and their physical interpretation encompasses dualities, generalized gauging operations, and higher-form symmetry "cosets" and "condensates."

2. Physical Manifestations and Examples

Non-invertible symmetries appear in a wide range of quantum field theoretic and statistical mechanical contexts:

Dualities and Half-space Gauging: Kramers–Wannier duality in the Ising model, higher Kramers–Wannier or Tambara–Yamagami type symmetries in lattice spin systems, and T-duality in 2d nonlinear sigma models all correspond to non-invertible defects constructed via half-space gauging and fusion (Okada et al., 2024, Arias-Tamargo et al., 26 Mar 2025, Cao et al., 2024). The key operator identity is that the square of the duality defect produces a sum of invertible symmetries:

$D^2 = 1 \oplus g,$

where $g$ is a group-like symmetry (e.g., spin-flip), reflecting non-invertibility.

Gauge Theories with Discrete Gauging: In $d$ -dimensional gauge theories with disconnected gauge groups, such as $O(N)$ or $Pin(N)$ , gauging an outer automorphism (e.g., charge conjugation) produces non-invertible 1-form symmetries. The fusion algebra of associated Gukov–Witten defects exhibits explicit non-invertibility, as in (Arias-Tamargo et al., 2022):

$T_{[h_1]}^G \cdot T_{[h_2]}^G = T_{[h_1 h_2]}^G + T_{[h_1 h_2^{-1}]}^G,$

and none of the $T_{[h]}^G$ admits an inverse.

Condensation Defects in Topological Phases: In 2+1d TQFTs and Maxwell theory, condensation defects associated with gauging a subgroup (e.g., $\mathbb{Z}_N \subset U(1)$ ) realize non-invertible symmetries with fusion rules involving both non-invertible and invertible constituents:

$S_N \otimes S_{N'} = \mathcal{Z}_{\gcd(N,N')} \cdot S_{\mathrm{lcm}(N,N')},$

where $\mathcal{Z}_k$ denotes a $\mathbb{Z}_k$ gauge theory (Choi et al., 2023).

Non-invertible Dualities in 3+1d: Gauging a discrete one-form symmetry in 3+1d can produce non-invertible duality defects, whose existence is tied to precise arithmetic conditions (e.g., $N = k^2 \ell$ and $-1$ a quadratic residue mod $\ell$ ) for the possibility of a symmetric gapped phase (Apte et al., 2022).
Gaillard-Zumino and Continuous Dualities: In 4d abelian gauge theories coupled to neutral sectors (including supergravity models), the quantum theory can admit non-invertible defects implementing rational symplectic transformations, realizing a fusion category associated to the rational points of the duality group $Sp(2n,\mathbb{Q})$ (Apruzzi et al., 21 Oct 2025).
Stringy Non-invertible Symmetries: In the tensionless limit of string theory, worldsheet orbifolds realize non-invertible symmetrical structures in the target space (Heckman et al., 2024). These are realized by non-invertible worldsheet defect lines associated to the representation category of the orbifold group, and topological properties survive only in the strict $T \to 0$ or $g_s \to 0$ limit.
Holographic Realizations: Non-invertible symmetries are systematically realized in AdS/CFT via the Hamiltonian quantization of symmetry topological field theories (SymTFTs), manifest in D-brane constructions and the Myers effect (Apruzzi et al., 2022).
Finite Group Gauge Theories: Non-invertible 0-form symmetries in finite group gauge theories are classified by subgroups and cocycles in $G \times G$ and have explicit fusion rules, including examples such as Fibonacci fusion in non-abelian dihedral group gauge theories (Cordova et al., 2024).

3. Action on Local Operators and Physical Observables

Non-invertible symmetries act on operators via CP maps or families of partial isometries between (possibly extended) Hilbert spaces. Crucially, the basic structure differs from group symmetries:

The action on local operators can be invertible, especially in 3+1d in the absence of topological lines; in such cases, non-invertible symmetries act as group symmetries when restricted to local operator algebras (Putrov et al., 3 Mar 2026).
When topological lines are present (e.g., in 1+1d), the action is genuinely non-invertible, leading to operator-valued sum rules rather than simple group actions.
On density matrices, the action corresponds to trace-preserving quantum channels or partial isometries; Wigner's theorem is generalized to allow symmetry operators to be isometries between different twisted-sector Hilbert spaces, provided probability preservation is maintained (Bartsch et al., 6 Feb 2026, Ortiz et al., 29 Sep 2025).

For 1d or 2d many-body systems, these actions can be explicitly constructed via Kraus operators, e.g., in the Ising chain, the Kramers–Wannier duality defect acts as a non-invertible channel with fusion $D^2 = 1 \oplus g$ (Okada et al., 2024).

4. Boundary Conditions, Anomalies, and Gauging

Non-invertible symmetries give rise to refined notions of symmetry-preserving boundary conditions:

Weak vs. Strong Symmetry: In 1+1d, a boundary is weakly symmetric if symmetry defects can end on it, resulting in conserved operators on an interval. Strongly symmetric boundaries require the boundary state to be an eigenstate of all symmetry operators. For invertible symmetries, these notions coincide; for non-invertible symmetries, they generically differ (Choi et al., 2023).
The existence (or absence) of strongly symmetric boundaries is related to anomalies: a strongly anomaly-free fusion category symmetry permits strongly symmetric boundaries and fiber functors; otherwise, only weakly symmetric or no symmetric boundaries exist.
Generalized gauging procedures (e.g., gauging algebra objects within a fusion category) can be implemented when suitable algebra objects (Morita trivial or not) exist, and this can generate new, possibly anomalous, non-invertible symmetries.

Anomalies for non-invertible symmetries appear as obstructions to the existence of a trivially gapped, symmetry-preserving phase. For instance, in 3+1d, a non-invertible duality symmetry may forbid the possibility of a unique gapped vacuum unless the aforementioned arithmetic criteria are satisfied (Apte et al., 2022).

5. Fusion Rules, Higher Categories, and Lattice Constructions

Non-invertible symmetry operators obey generalized fusion algebras (fusion categories, 2-categories, etc.):

In $d$ -dim QFTs, gauging a finite 0-form symmetry produces a dual $(d-1)$ -category of topological defects; for 3d gauge theory, this is a fusion 2-category $2\mathrm{Rep}(G)$ (Bhardwaj et al., 2022).
Lattice models give explicit operator constructions with non-invertible fusion, as in the case of Tambara–Yamagami and dipole Kramers–Wannier symmetries (Cao et al., 2024). In such models, fusion can produce Fibonacci-type relations, i.e., $\mathcal N \times \mathcal N = 1 + \mathcal N$ .
Non-invertible chiral and reflection symmetries have been classified in boundary and defect theories via module category and Tannaka–Krein duality, further encoded by weak Hopf algebras (Ahmad et al., 22 Sep 2025).

The full fusion algebra, associators, and higher categorical data determine the structure of constraint equations (such as generalized Verlinde formulas), the possible operator algebras, and the spectrum of emergent symmetry selection rules.

6. Implications for Quantum Gravity, RG Flows, and Constraints

Non-invertible symmetries play a critical role in constraining quantum field theory and quantum gravity:

Quantum Gravity and the Swampland: Swampland considerations suggest that exact non-invertible symmetries cannot survive in quantum gravity, much as for invertible global symmetries. Discrete gauging and the production of "twist vortices" ensure that any attempt to rigidify a non-invertible symmetry in the presence of gravity is ultimately obstructed (Arias-Tamargo et al., 2022, Heckman et al., 2024).
Renormalization Group Flow: Non-invertible symmetries are RG invariants; they cannot be spontaneously broken along flows (except by explicit symmetry breaking or anomaly matching). Their presence can forbid symmetric gapped phases and restrict the set of allowed RG endpoints, e.g., in the case of self-dual points in Maxwell and $\mathbb{Z}_N$ lattice gauge theories (Apte et al., 2022, Cordova et al., 2024).
Exotic RG Flows: Non-invertible twisted compactifications—implementing RG flows by twisting with non-invertible defects—produce new strongly coupled fixed points (e.g., 3d $\mathcal{N}=6$ SCFTs not reachable by ordinary group actions) (Kaidi et al., 2022).
Selection Rules and Fractionalization: Non-invertible symmetry selection rules lead to fractional charge or matter quantum numbers, as in axion electrodynamics, and predict nontrivial correlations among extended operators associated to domain walls, flux tubes, and their intersections (Yokokura, 2022).

7. Quantum Information-Theoretic Perspective

Recent developments contextualize non-invertible symmetries in the language of quantum information:

Actions of non-invertible symmetries correspond to quantum channels (completely positive, trace-preserving maps) rather than automorphisms; relative entropy with respect to the induced conditional expectation (the Haar integral for a weak Hopf algebra) provides a robust order parameter for symmetry breaking (Ahmad et al., 22 Sep 2025).
Unlike group averaging, non-invertible "averaging" operators emerge naturally in the weak Hopf setting, with new entropic upper bounds reflecting the quantum dimension (or more generally the index) of the fusion category symmetry.

This formalism unifies statistical, algebraic, and categorical properties, providing new tools for diagnosing and classifying phases with non-invertible symmetry.

References (arXiv IDs):

(Choi et al., 2023, Arias-Tamargo et al., 2022, Okada et al., 2024, Ahmad et al., 22 Sep 2025, Bartsch et al., 6 Feb 2026, Ortiz et al., 29 Sep 2025, Apte et al., 2022, Cao et al., 2024, Arias-Tamargo et al., 26 Mar 2025, Bhardwaj et al., 2022, Niro et al., 2022, Apruzzi et al., 2024, Apruzzi et al., 21 Oct 2025, Putrov et al., 3 Mar 2026, Heckman et al., 2024, Kaidi et al., 2022, Cordova et al., 2024, Yokokura, 2022, Apruzzi et al., 2022)