---
title: Non-Invertible Duality Defects
url: https://www.emergentmind.com/topics/non-invertible-duality-defects
type: topic
---

# Non-Invertible Duality Defects

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 [2111.01139]. 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 [2607.01137].

## 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 [2607.01137]. 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 [2607.01137].

The associated quantum dimension measures the effective “size” of the defect. In the Ising example, the abstract of Liang’s lattice study states \(d_\sigma=\sqrt{2}\), while the detailed exposition notes a lattice/SymTFT normalization with \(d_\sigma=2\); this suggests a normalization dependence in how the same categorical datum is reported [2607.01137]. More generally, Tambara–Yamagami categories furnish duality defects \(N\) obeying
\[
N^2=\sum_{a\in A}\mathcal{L}_a,\qquad N\mathcal{L}_a=\mathcal{L}_aN=N,
\]
for a finite abelian group \(A\), with \(d(N)=\sqrt{|A|}\) [2512.19640]. The paper on the Conway module and K3 sigma models also exhibits Fibonacci defects with
\[
W\times W=\mathbf{1}+W,
\]
and quantum dimension
\[
d(W)=\frac{1+\sqrt{5}}{2},
\]
showing that irrational quantum dimensions occur naturally beyond Tambara–Yamagami constructions [2512.19640].

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 [2307.14428].

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

A general mechanism for constructing non-invertible duality defects is to gauge an abelian \(q\)-form symmetry only on one side of an interface. For a \(d\)-dimensional theory \(\mathcal{T}\) with anomaly-free \(G^{(q)}\), this produces a topological interface between \(\mathcal{T}\) and \(\mathcal{T}/G^{(q)}\); if the theory is self-dual under gauging, the interface becomes a topological defect of \(\mathcal{T}\) itself [2111.01139]. Self-duality under gauging requires
\[
q=\frac{d-2}{2},
\]
so the basic cases are \(q=0\) in \(d=2\), \(q=1\) in \(d=4\), and \(q=2\) in \(d=6\) [2111.01139].

In 3+1 dimensions, the construction uses a \(\mathbb{Z}_N^{(1)}\) one-form symmetry. Gauging on half of spacetime yields a codimension-one defect \(\mathcal{D}\) whose fusion with its orientation reverse on a closed 3-manifold \(M\) is
\[
\mathcal{D}\times \overline{\mathcal{D}}
= \frac{1}{N}\sum_{S\in H_2(M;\mathbb{Z}_N)} \eta(S),
\]
where \(\eta(S)\) are one-form symmetry surfaces [2111.01139]. 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 \(O(d;\mathbb{R})\times O(d;\mathbb{R})\) and on momentum–winding charges by \(O(d,d;\mathbb{R})\); when the charge action is rational, \(M_B\in O(d,d;\mathbb{Q})\), the defect can be realized by combining gauging of anomaly-free discrete subgroups of momentum and winding symmetry with elements of \(O(d,d;\mathbb{Z})\) that leave the couplings invariant [2408.14556]. Generically, such defects map local operators into non-genuine operators attached to lines, which is precisely the hallmark of non-invertibility [2408.14556].

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 \(\mathcal{N}=4\) SYM, class \(\mathcal{S}\), and obstruction theory [2210.09146].

## 3. Gauge-theoretic and lattice realizations

In 4d \(\mathcal{N}=4\) \(\mathfrak{su}(N)\) SYM, non-invertible self-duality defects appear at fixed points of the \(SL(2,\mathbb{Z})\) coupling: \(\tau=i\) and \(\tau=e^{2\pi i/3}\). In the holographic description, type IIB string theory has a gauged \(SL(2,\mathbb{Z})\) duality symmetry, and at these points the axio-dilaton preserves finite subgroups \(G=\mathbb{Z}_4\) or \(G=\mathbb{Z}_6\), giving an emergent discrete gauge field in the 5d bulk. Reduction on the internal manifold yields a 5d topological theory
\[
S_{\text{CS}}=\frac{N}{2\pi}\int b\,dc
=\frac{N}{4\pi}\int B^T\epsilon\,dB,
\]
whose twisted sectors and gapped boundaries reproduce the categorical fusion rules of self-duality and triality defects in the boundary theory [2210.09146].

A broader “zoology” occurs in 4d class \(\mathcal{S}\) theories. Compactification of the 6d \((2,0)\) theory of type \(A_{N-1}\) on a genus-\(g\) Riemann surface produces a duality group acting through \(Sp(2g,\mathbb{Z}_N)\), and the global variants are specified by Lagrangian sublattices \(L\subset (\mathbb{Z}_N)^{2g}\). A duality defect associated with \(M\in Sp(2g,\mathbb{Z}_N)\) is written as
\[
D_L^M=M\circ \Phi_L^M,
\]
and its rank is determined by the intersection \(K=L\cap ML\). The fusion algebra takes the schematic form
\[
D_L^{M_1}\times D_L^{M_2}=N^{(1,2)}\,C^{A_{1,2}}\,D_L^{M_1M_2},
\]
with a decoupled TQFT coefficient \(N^{(1,2)}\) and a condensation defect \(C^{A_{1,2}}\) [2212.09549].

The lattice realization in 4d \(\mathbb{Z}_2\) 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
\[
D(a,\tilde a)=(-1)^{a\tilde a},
\]
and find that the expectation value of the defect wrapped on \(S^3\) is
\[
\langle \text{duality defect on }S^3\rangle=\frac{1}{\sqrt{2}},
\]
so its quantum dimension is \(\sqrt{2}\) and it is non-invertible [2109.05992]. They also construct the 1-form \(\mathbb{Z}_2\) symmetry defects, the junctions between duality and symmetry defects, and the crossing relations needed to evaluate defect configurations [2109.05992].

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 \(so(3)_2\subset su(2)_4\). The corresponding lattice defects \(\mathcal{K},\mathcal{M},\mathcal{O},\mathcal{D}\) satisfy fusion rules such as
\[
\mathcal{K}^\dagger\mathcal{K}=1+F,\qquad
\mathcal{O}^\dagger\mathcal{O}=1+F_3,\qquad
\mathcal{D}\mathcal{D}^\dagger=1+\mathcal{S},
\]
and generate non-invertible maps between models while preserving integrability [2302.14081].

## 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
\[
H^{\rm d} = -\sum_{j=1}^{L-1} Z_j Z_{j+1} - \sum_{j=2}^{L} X_j - Z_L Y_1,
\]
which realizes the non-local Kramers–Wannier twist on the lattice [2607.01137]. In the Majorana representation, the duality-twisted chain has \(2L-1\) Majorana modes, or effective length
\[
L_{\rm eff}=L-\tfrac12,
\]
with a localized unpaired Majorana zero mode [2607.01137].

Using the correlation-matrix method, the single-particle entanglement levels are
\[
\xi_k=\ln\frac{1+\nu_k}{1-\nu_k},
\]
and the defect sector exhibits an exact entanglement zero mode,
\[
|\xi_{\min}|\approx 4.4\times 10^{-16}\approx 0,
\]
while the homogeneous chain has no such level [2607.01137]. This maximally mixed Majorana contributes \(\ln 2\) to the entanglement entropy and is identified as the microscopic origin of the boundary entropy
\[
\log g=\tfrac12\ln 2,
\]
which in turn encodes the non-invertible quantum dimension [2607.01137].

The same duality-twisted ground state also determines the twist-field conformal weight in two independent ways. A modified translation operator yields momenta
\[
p=\left(L-\tfrac12\right)\frac{\arg\lambda}{2\pi}\bmod 1
\in \left\{\pm \tfrac1{16},\pm \tfrac7{16}\right\},
\]
while finite-size scaling of the ground-state energy gives
\[
h^{\rm dual}=0.062501\approx \frac1{16}.
\]
The defect Hilbert space organizes into a half-integer \(\sigma\)-twisted tower with many-body levels on
\[
\{0,\tfrac12,1,\tfrac32,\dots\},
\]
providing a level-resolved spectral signature of non-invertibility [2607.01137].

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 [2607.01137].

## 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 [2307.14428]. For \(G=\mathbb{Z}_N^{(0)}\) in 2d, this happens if and only if \(N\) is a perfect square; for \(G=\mathbb{Z}_N^{(1)}\) in 4d, under the stated assumptions, it happens if and only if
\[
N=L^2 M
\quad\text{with}\quad
-1 \text{ a quadratic residue mod } M
\]
[2307.14428].

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 [2308.11707]. A notable consequence is that intrinsically non-invertible duality symmetries are necessarily anomalous [2308.11707].

These anomalies translate into dynamical constraints. In 3+1 dimensions, the existence of a non-invertible duality defect associated with gauging a \(\mathbb{Z}_N^{(1)}\) one-form symmetry often forbids a symmetry-preserving gapped phase. A self-dual theory with \(\mathbb{Z}_N^{(1)}\) can flow to a TQFT with a unique local vacuum only if
\[
N=k^2\ell
\]
and \(-1\) is a quadratic residue modulo \(\ell\); otherwise it must be gapless or spontaneously break the self-duality symmetry [2212.14605]. The same paper extends the statement to more general gauging operations, including triality symmetries [2212.14605].

The Cardy–Rabinovici model provides an explicit 4d example. There, the \(ST^{-1}\) self-duality at
\[
\tau_* = e^{\pi i/3}
\]
is realized by half-space gauging of a \(\mathbb{Z}_N^{[1]}\) symmetry with a discrete topological term. The corresponding defect obeys
\[
\mathscr{D}(M^{(3)}) \times \bar{\mathscr{D}}(M^{(3)})
= \frac{1}{N}\sum_{\Sigma\in H_2(M^{(3)};\mathbb{Z}_N)} (-1)^{Q(\Sigma)}\,\eta(\Sigma),
\]
and a mixed gravitational anomaly detected on K3 rules out a trivially gapped phase at the self-dual point [2204.07440].

## 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 \(O(d;\mathbb{R})\times O(d;\mathbb{R})\), on momentum and winding charges by \(O(d,d;\mathbb{R})\), and when the latter action is rational they can be realized by discrete gauging together with \(O(d,d;\mathbb{Z})\) dualities [2408.14556]. In the simplest nontrivial \(d=1\) example at radius \(R^2=N\), the resulting T-duality defect acts as
\[
\begin{pmatrix} w \\ m \end{pmatrix}
\to
\begin{pmatrix} 0&1/N\\ N&0 \end{pmatrix}
\begin{pmatrix} w \\ m \end{pmatrix},
\]
has quantum dimension \(\sqrt{N}\), and maps many local operators to non-genuine operators attached to lines [2408.14556]. The paper verifies the modified selection rules explicitly on higher-genus worldsheets [2408.14556].

Non-rational SCFTs supply further examples with explicit topological defect lines of irrational quantum dimension. In the Conway module \(V^{f\natural}\), duality defects for \(TY(\mathbb{Z}_2)\), \(TY(\mathbb{Z}_3)\), \(TY(\mathbb{Z}_2\times \mathbb{Z}_2)\), and Fibonacci fusion categories are constructed through Leech lattice endomorphisms \(\rho(\mathcal{L})\in \mathrm{End}(\Lambda)\), with \(\rho(\mathcal{L}_1\times \mathcal{L}_2)=\rho(\mathcal{L}_1)\rho(\mathcal{L}_2)\) [2512.19640]. In K3 non-linear sigma models, the same paper describes a duality defect of irrational quantum dimension \(\sqrt{2}\) for \(TY(\mathbb{Z}_2,-1)\) on a 16-dimensional slice of moduli space, and constructs Fibonacci and \(Rep(S_3)\) defects in particular Gepner models [2512.19640].

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 [2512.19640]. 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.

Source: https://www.emergentmind.com/topics/non-invertible-duality-defects