Non-Invertible Duality Defects

• Non-invertible duality defects are topological entities that implement dualities rather than conventional symmetries, characterized by fusion rules that decompose into multiple sectors.
• They appear in diverse settings such as the critical Ising model, gauge theories, and string worldsheet models, with behavior captured by fusion categories and higher-categorical frameworks.
• Their construction via half-space gauging, lattice realizations, and entanglement-spectrum analysis elucidates connections to symmetry TFTs, anomaly constraints, and the emergence of irrational quantum dimensions.

Searching arXiv for recent and foundational papers on non-invertible duality defects.

Non-invertible duality defects are topological defects that implement dualities rather than ordinary symmetries, and whose fusion is not group-like. In the modern formulation, they are organized by fusion categories or higher-categorical analogues rather than by groups, and their characteristic data include nontrivial fusion rules, non-integer or irrational quantum dimensions, and twisted sectors that are not captured by conventional symmetry operators alone (Choi et al., 2021). They arise in settings ranging from the Kramers–Wannier defect of the critical Ising chain to 3+1-dimensional gauge theories, worldsheet theories of compact scalars, class $\mathcal{S}$ theories, lattice integrable models, the Conway module, and K3 non-linear sigma models (Liang, 1 Jul 2026).

1. Categorical definition and quantum dimension

A non-invertible defect is topological but does not admit an inverse under fusion. In the categorical language used throughout the literature, a symmetry may be generalized from a group $G$ of invertible operators to a fusion category $\mathcal{C}$ of topological defect lines, and an object $X\in\mathcal{C}$ is non-invertible when $X\times X$ decomposes into several simple defects rather than into a single inverse pair (Liang, 1 Jul 2026). The canonical example is the Kramers–Wannier defect $\mathcal{D}$ of the Ising model, whose fusion rule is

$\mathcal{D}\times \mathcal{D}=1+\eta,$

with $1$ the trivial defect and $\eta$ the Ising $\mathbb{Z}_2$ spin-flip defect (Liang, 1 Jul 2026).

The associated quantum dimension measures the effective “size” of the defect. In the Ising example, the abstract of Liang’s lattice study states $G$0, while the detailed exposition notes a lattice/SymTFT normalization with $G$1; this suggests a normalization dependence in how the same categorical datum is reported (Liang, 1 Jul 2026). More generally, Tambara–Yamagami categories furnish duality defects $G$2 obeying

$G$3

for a finite abelian group $G$4, with $G$5 (Angius et al., 22 Dec 2025). The paper on the Conway module and K3 sigma models also exhibits Fibonacci defects with

$G$6

and quantum dimension

$G$7

showing that irrational quantum dimensions occur naturally beyond Tambara–Yamagami constructions (Angius et al., 22 Dec 2025).

A recurrent misconception is that every duality defect should be reducible to an ordinary symmetry after a change of presentation. The group-theoretical analysis shows that this is false in general: some duality defects are “intrinsically non-invertible,” meaning that their Symmetry TFT is not a Dijkgraaf–Witten theory and no reformulation in terms of purely invertible symmetries exists (Sun et al., 2023).

2. Half-space gauging, self-duality, and symmetry TFT

A general mechanism for constructing non-invertible duality defects is to gauge an abelian $G$8-form symmetry only on one side of an interface. For a $G$9-dimensional theory $\mathcal{C}$0 with anomaly-free $\mathcal{C}$1, this produces a topological interface between $\mathcal{C}$2 and $\mathcal{C}$3; if the theory is self-dual under gauging, the interface becomes a topological defect of $\mathcal{C}$4 itself (Choi et al., 2021). Self-duality under gauging requires

$\mathcal{C}$5

so the basic cases are $\mathcal{C}$6 in $\mathcal{C}$7, $\mathcal{C}$8 in $\mathcal{C}$9, and $X\in\mathcal{C}$0 in $X\in\mathcal{C}$1 (Choi et al., 2021).

In 3+1 dimensions, the construction uses a $X\in\mathcal{C}$2 one-form symmetry. Gauging on half of spacetime yields a codimension-one defect $X\in\mathcal{C}$3 whose fusion with its orientation reverse on a closed 3-manifold $X\in\mathcal{C}$4 is

$X\in\mathcal{C}$5

where $X\in\mathcal{C}$6 are one-form symmetry surfaces (Choi et al., 2021). This is the higher-dimensional analogue of the Kramers–Wannier line: fusion produces a condensate of higher-form symmetry defects rather than the identity.

The same logic appears in worldsheet theories of compact scalars. There, topological codimension-one defects act on fields by elements of $X\in\mathcal{C}$7 and on momentum–winding charges by $X\in\mathcal{C}$8; when the charge action is rational, $X\in\mathcal{C}$9, the defect can be realized by combining gauging of anomaly-free discrete subgroups of momentum and winding symmetry with elements of $X\times X$0 that leave the couplings invariant (Bharadwaj et al., 2024). Generically, such defects map local operators into non-genuine operators attached to lines, which is precisely the hallmark of non-invertibility (Bharadwaj et al., 2024).

The Symmetry TFT viewpoint packages these constructions into a bulk topological theory in one higher dimension. In this formulation, gapped boundary conditions encode absolute theories, topological manipulations correspond to gauging operations, and duality defects are boundaries of higher-dimensional symmetry defects. This framework is central in the analyses of $X\times X$1 SYM, class $X\times X$2, and obstruction theory (Antinucci et al., 2022).

3. Gauge-theoretic and lattice realizations

In 4d $X\times X$3 $X\times X$4 SYM, non-invertible self-duality defects appear at fixed points of the $X\times X$5 coupling: $X\times X$6 and $X\times X$7. In the holographic description, type IIB string theory has a gauged $X\times X$8 duality symmetry, and at these points the axio-dilaton preserves finite subgroups $X\times X$9 or $\mathcal{D}$0, giving an emergent discrete gauge field in the 5d bulk. Reduction on the internal manifold yields a 5d topological theory

$\mathcal{D}$1

whose twisted sectors and gapped boundaries reproduce the categorical fusion rules of self-duality and triality defects in the boundary theory (Antinucci et al., 2022).

A broader “zoology” occurs in 4d class $\mathcal{D}$2 theories. Compactification of the 6d $\mathcal{D}$3 theory of type $\mathcal{D}$4 on a genus-$\mathcal{D}$5 Riemann surface produces a duality group acting through $\mathcal{D}$6, and the global variants are specified by Lagrangian sublattices $\mathcal{D}$7. A duality defect associated with $\mathcal{D}$8 is written as

$\mathcal{D}$9

and its rank is determined by the intersection $\mathcal{D}\times \mathcal{D}=1+\eta,$0. The fusion algebra takes the schematic form

$\mathcal{D}\times \mathcal{D}=1+\eta,$1

with a decoupled TQFT coefficient $\mathcal{D}\times \mathcal{D}=1+\eta,$2 and a condensation defect $\mathcal{D}\times \mathcal{D}=1+\eta,$3 (Antinucci et al., 2022).

The lattice realization in 4d $\mathcal{D}\times \mathcal{D}=1+\eta,$4 pure gauge theory is fully explicit. At the Kramers–Wannier–Wegner self-dual point, Koide, Nagoya, and Yamaguchi construct a codimension-one duality defect with local weight

$\mathcal{D}\times \mathcal{D}=1+\eta,$5

and find that the expectation value of the defect wrapped on $\mathcal{D}\times \mathcal{D}=1+\eta,$6 is

$\mathcal{D}\times \mathcal{D}=1+\eta,$7

so its quantum dimension is $\mathcal{D}\times \mathcal{D}=1+\eta,$8 and it is non-invertible (Koide et al., 2021). They also construct the 1-form $\mathcal{D}\times \mathcal{D}=1+\eta,$9 symmetry defects, the junctions between duality and symmetry defects, and the crossing relations needed to evaluate defect configurations (Koide et al., 2021).

Non-invertible duality defects also organize exact maps between lattice many-body systems. The XXZ chain, a three-state antiferromagnet, a Rydberg-blockade ladder, and a zigzag-coupled pair of Ising chains share a local algebra whose categorical content is $1$0. The corresponding lattice defects $1$1 satisfy fusion rules such as

$1$2

and generate non-invertible maps between models while preserving integrability (Eck et al., 2023).

4. Entanglement-spectrum fingerprints

A major recent development is the identification of non-invertible symmetry data directly in entanglement spectra. For the Kramers–Wannier defect in the critical Ising chain, Liang studies the duality-twisted Hamiltonian

$1$3

which realizes the non-local Kramers–Wannier twist on the lattice (Liang, 1 Jul 2026). In the Majorana representation, the duality-twisted chain has $1$4 Majorana modes, or effective length

$1$5

with a localized unpaired Majorana zero mode (Liang, 1 Jul 2026).

Using the correlation-matrix method, the single-particle entanglement levels are

$1$6

and the defect sector exhibits an exact entanglement zero mode,

$1$7

while the homogeneous chain has no such level (Liang, 1 Jul 2026). This maximally mixed Majorana contributes $1$8 to the entanglement entropy and is identified as the microscopic origin of the boundary entropy

$1$9

which in turn encodes the non-invertible quantum dimension (Liang, 1 Jul 2026).

The same duality-twisted ground state also determines the twist-field conformal weight in two independent ways. A modified translation operator yields momenta

$\eta$0

while finite-size scaling of the ground-state energy gives

$\eta$1

The defect Hilbert space organizes into a half-integer $\eta$2-twisted tower with many-body levels on

$\eta$3

providing a level-resolved spectral signature of non-invertibility (Liang, 1 Jul 2026).

This result is methodologically significant because it promotes boundary entropy from an integrated constant to a level-resolved diagnostic, and supplies an exactly solvable benchmark for tensor-network studies of non-invertible defects in interacting models (Liang, 1 Jul 2026).

5. Group-theoreticality, anomalies, and obstructions to gapped phases

A central structural question is when a duality defect is “group-theoretical,” meaning that it can be obtained from invertible symmetries together with gauging of a non-anomalous subgroup. The Symmetry TFT criterion is sharp: a duality defect is group-theoretical if and only if its Symmetry TFT is a Dijkgraaf–Witten theory (Sun et al., 2023). For $\eta$4 in 2d, this happens if and only if $\eta$5 is a perfect square; for $\eta$6 in 4d, under the stated assumptions, it happens if and only if

$\eta$7

(Sun et al., 2023).

The anomaly analysis refines this further. In both 2d and 4d, there are two obstructions to gauging non-invertible self-duality symmetries: first, the existence of a duality-invariant Lagrangian algebra in a Dijkgraaf–Witten theory in one dimension more; second, the vanishing of a pure anomaly for the invertible duality symmetry after accounting for symmetry fractionalization (Antinucci et al., 2023). A notable consequence is that intrinsically non-invertible duality symmetries are necessarily anomalous (Antinucci et al., 2023).

These anomalies translate into dynamical constraints. In 3+1 dimensions, the existence of a non-invertible duality defect associated with gauging a $\eta$8 one-form symmetry often forbids a symmetry-preserving gapped phase. A self-dual theory with $\eta$9 can flow to a TQFT with a unique local vacuum only if

$\mathbb{Z}_2$0

and $\mathbb{Z}_2$1 is a quadratic residue modulo $\mathbb{Z}_2$2; otherwise it must be gapless or spontaneously break the self-duality symmetry (Apte et al., 2022). The same paper extends the statement to more general gauging operations, including triality symmetries (Apte et al., 2022).

The Cardy–Rabinovici model provides an explicit 4d example. There, the $\mathbb{Z}_2$3 self-duality at

$\mathbb{Z}_2$4

is realized by half-space gauging of a $\mathbb{Z}_2$5 symmetry with a discrete topological term. The corresponding defect obeys

$\mathbb{Z}_2$6

and a mixed gravitational anomaly detected on K3 rules out a trivially gapped phase at the self-dual point (Hayashi et al., 2022).

6. Worldsheet and non-rational SCFT realizations

On the worldsheet of compact bosons, non-invertible duality defects provide a string-theoretic realization of the same general mechanism. Topological codimension-one defects act on the fields by $\mathbb{Z}_2$7, on momentum and winding charges by $\mathbb{Z}_2$8, and when the latter action is rational they can be realized by discrete gauging together with $\mathbb{Z}_2$9 dualities (Bharadwaj et al., 2024). In the simplest nontrivial $G$00 example at radius $G$01, the resulting T-duality defect acts as

$G$02

has quantum dimension $G$03, and maps many local operators to non-genuine operators attached to lines (Bharadwaj et al., 2024). The paper verifies the modified selection rules explicitly on higher-genus worldsheets (Bharadwaj et al., 2024).

Non-rational SCFTs supply further examples with explicit topological defect lines of irrational quantum dimension. In the Conway module $G$04, duality defects for $G$05, $G$06, $G$07, and Fibonacci fusion categories are constructed through Leech lattice endomorphisms $G$08, with $G$09 (Angius et al., 22 Dec 2025). In K3 non-linear sigma models, the same paper describes a duality defect of irrational quantum dimension $G$10 for $G$11 on a 16-dimensional slice of moduli space, and constructs Fibonacci and $G$12 defects in particular Gepner models (Angius et al., 22 Dec 2025).

A plausible implication of these results is that non-invertible duality defects are not confined to rational CFT or topological phases, but persist in holomorphic SCFTs, torus orbifolds, and K3 sigma models with full control over defect-twined elliptic genera (Angius et al., 22 Dec 2025). Across these settings, the common structure is the same: duality implemented by topological defects, fusion controlled by categorical rather than group-theoretic data, and physical observables—from partition functions to entanglement spectra—encoding the resulting non-invertibility.