Non-Invertible Topological Defects

• Non-invertible topological defects are extended operators in QFT that lack an inverse, altering fusion rules to produce sums of other operators.
• They are constructed via methods like half-gauging discrete symmetries, with measurable effects such as a fractional quantum dimension (e.g. 1/√2 in 4d Z₂ lattice gauge theory).
• Their fusion rules and junctions form higher-categorical algebraic structures that enforce dualities, anomaly matching, and the classification of topological phases.

Non-invertible topological defects are codimension-one or higher extended operators in quantum field theory which cannot be assigned an inverse under fusion: their composition with their orientation-reversed counterpart does not yield the identity defect, but produces instead a sum or condensation of operators, often parameterized by lower-dimensional topological data or even full topological quantum field theories (TQFTs). These structures generalize symmetry operators, encompassing both conventional (invertible) symmetries and richer categorical symmetries. Non-invertible defects naturally arise in lattice gauge theories, continuum gauge theories with higher-form symmetries, and topological field theories, where they control dualities, condensation phenomena, and generalized anomaly matching.

1. Construction of Non-Invertible Topological Defects

The paradigmatic construction involves discrete gauge theories, often $\mathbb{Z}_N$ lattice gauge theories, or more general quantum field theories with $q$-form symmetries that can be gauged on submanifolds of various codimension. A prominent example is the construction of duality defects in four-dimensional $\mathbb{Z}_2$ pure lattice gauge theory (Koide et al., 2021). Here, the system is formulated on a hypercubic lattice with active degrees of freedom (binary gauge variables) on links. The Kramers-Wannier-Wegner (KWW) duality defect is constructed by "cutting" the lattice along a three-dimensional surface and locally exchanging active and dual (inactive) variables. Explicitly, the 4d lattice is decomposed into 16-cells (pairs of glued hypercubes), and the duality defect is inserted as a doubled 3-cell (tetrahedral prism), which swaps active/inactive assignments across the surface.

Similarly, in higher or lower dimensions, defects are defined by "half-gauging" – gauging a discrete symmetry in only half of spacetime (or along submanifolds) (Choi et al., 2021, Roumpedakis et al., 2022). In $(d+1)$d, gauging a $q$-form symmetry on a $p$-codimensional submanifold yields a $p$-codimensional defect. For example, duality defects generalizing the Kramers-Wannier defect in 2d Ising are constructed by gauging a $q=1$ form symmetry in 3+1d, resulting in a codimension-one duality interface.

In all these settings, the topological defect is defined so that it alters the structure of path integrals or partition functions by enforcing new edge-matching or duality conditions across its worldvolume, and is implemented using local "gluing" rules, additional auxiliary fields (often TQFTs), or categorical data.

2. Properties: Non-Invertibility, Quantum Dimension, and Fusion

Non-invertibility is witnessed at both algebraic and computational levels. The key diagnostic is the quantum dimension (or topological dimension) of the defect operator. For instance, in the $\mathbb{Z}_2$ 4d lattice gauge theory, the expectation value of a closed duality defect wrapped on $S^3$ is found to be $\langle \mathcal{D}(S^3) \rangle = 1/\sqrt{2}$, not $1$ as would be for an invertible symmetry operator (Koide et al., 2021). This fractional quantum dimension signals the inherent non-invertibility: fusion of the defect with its orientation reverse (or "inverse") yields a nontrivial sum (or condensation) of other operators.

More generally, the fusion rules of non-invertible duality defects are of Tambara-Yamagami type for discrete symmetries. In 3+1d with a one-form $\mathbb{Z}_N$ symmetry, one finds

$$\mathcal{D} \times \overline{\mathcal{D}} = \frac{1}{N} \sum_{S \in H_2(M; \mathbb{Z}_N)} \eta(S)$$

where $\eta(S)$ are symmetry surface operators labeled by two-cycles $S$ in the three-manifold $M$ supporting the defect (Choi et al., 2021, Kaidi et al., 2022). In yet higher dimensions, fusion involves categorified structures, such as fusion 5-categories in 6d $(2,0)$ SCFTs (Apruzzi et al., 14 Nov 2024).

The non-invertibility is also reflected in operator algebras: non-invertible defects obey non-group fusion rules, and their repeated composition may yield a sum over invertible symmetry operators, or in some constructions (such as higher gauging), coefficients in the fusion laws are partition functions of lower-dimensional TQFTs (Roumpedakis et al., 2022, Choi et al., 2022).

3. Crossing Relations, Junctions, and Algebraic Structure

Topological defects must be compatible with the set of allowed deformations (crossing relations) and the rules for local operator insertions at junctions where defects meet. In the $\mathbb{Z}_2$ lattice gauge theory, explicit crossing relations and commutation (or Reidemeister) moves are derived: defects may be deformed without changing the partition function up to known weights, and defects terminating on one another must be assigned precise junction weights.

For the duality defect and 1-form symmetry defect junction, weights such as $J(a) = (-1)^a$ and $\widetilde{J}(b, c) = \sigma^x_{b,c}$ (Pauli matrices) are used, following from fusion consistency constraints (Koide et al., 2021). In all known cases, such junction data matches the crossing structure studied in the tensor symbolic calculus for fusion categories and extended topological field theories (Huang et al., 2021).

At the algebraic level, non-invertible symmetry defects and their fusion, junctions, and crossing data organize into higher-categorical generalizations of groups (fusion categories, 2-categories, or higher). The full algebraic structure respects associativity, possibly up to multiplication by TQFTs (Roumpedakis et al., 2022, Apruzzi et al., 14 Nov 2024).

4. Higher Gauging and Categorification

A general mechanism for producing non-invertible defects is higher gauging: gauging a $q$-form symmetry on a submanifold of spacetime with codimension $p$, resulting in a $p$-codimensional condensation defect (Roumpedakis et al., 2022, Cordova et al., 21 Dec 2024). The fusion, action, and junction rules of these condensation defects exhibit categorified features: fusion coefficients are not numbers, but partition functions of 1+1d (or higher) TQFTs. These defects implement higher-category symmetries, with relationships to higher-group global symmetry structures, including nontrivial higher-group anomalies and relations between different symmetry types (Bhardwaj et al., 2022, Schafer-Nameki, 2023).

Such higher gauging provides a unifying framework where even non-invertible 0-form, 1-form, and higher-form symmetries emerge via condensation of higher-form symmetry generators, often described explicitly in lattice Hamiltonian or TQFT models (Cordova et al., 21 Dec 2024).

5. Expectation Values, Observable Signatures, and Physical Consequences

Explicit computation of expectation values of non-invertible defect networks, such as duality defects wrapping various 3-manifolds or configurations with junctions and crossings, reveals their characteristic quantum dimensions and algebraic properties. For example, for the duality defect in the 4d $\mathbb{Z}_2$ theory, wrapping the defect on $S^1 \times S^2$ yields an expectation value of $1$, while the value for connected sums of 3-manifolds $X \# Y$ obeys

$$\langle X \# Y \rangle = \sqrt{2} \, \langle X \rangle \langle Y \rangle$$

(Koide et al., 2021). In the continuum, similar analyses show that duality defects severely constrain the possible low-energy phases: the existence of non-invertible defects often implies an obstruction to flowing to a trivial gapped phase and gives rise to generalized anomaly matching conditions (Choi et al., 2021, Choi et al., 2022).

In topological and condensed matter systems, such as (2+1)d topological orders or quantum Hall states, non-invertible defects determine fractionalization patterns, modified selection rules, and novel edge dynamics. For instance, coset non-invertible symmetries obtained by gauging non-normal subgroups yield fractionalized quantum numbers for anyons and lead to physical consequences such as enforced gapless edge states (Hsin et al., 30 May 2024).

6. Mathematical Formulations: Lattice and Continuum, Explicit Expressions

The mathematical definition of non-invertible defects admits both lattice and continuum formulations:

• Lattice Gauge Theory: Defects are specified by local modifications of the lattice Boltzmann weights, e.g., the duality defect weight $D(a, \tilde{a}) = (-1)^{a\tilde{a}}$, site/link normalization factors $l = s = 1/\sqrt{2}$, and explicit construction of doubled cells (tetrahedral prisms) (Koide et al., 2021).
• Topological Quantum Field Theory (TQFT): Higher gauging is encoded via higher cocycles, relative cobordism invariants, auxiliary BF or Chern-Simons terms, or algebra objects in fusion categories (Roumpedakis et al., 2022, Huang et al., 2021). In field theory, the duality defect in continuum Maxwell theory is often represented by an interface with a Chern-Simons term coupling the fields on either side (Choi et al., 2021).

For example, in 4d $\mathbb{Z}_2$ lattice gauge theory, the local crossing relation encoding the quantum dimension of the duality defect on $S^3$ is

$$\sum_{a_1,a_2,a_3,a_4} W(a_1,a_2,a_3,a_4) s^8 l^8 \overline{s}^8 \overline{l}^8 \prod_{(m,\tilde{n})\in U} D(a_m,\tilde{a}_n) = \frac{1}{\sqrt{2}} W(\tilde{a}_1,\tilde{a}_2,\tilde{a}_3,\tilde{a}_4) s^4 l^4 \overline{s}^4 \overline{l}^4$$

which identifies the quantum dimension $1/\sqrt{2}$.

The general framework encompasses more elaborate categorical, cohomological, and representation-theoretic tools, including Poincaré duality, induced representations for condensation defects (Cordova et al., 21 Dec 2024), and the explicit construction of fusion 2- and higher-categories.

7. Outlook and Impact

Non-invertible topological defects constitute a fundamental generalization of symmetry in quantum and statistical systems, with implications for dualities, anomaly constraints, categorical symmetries, and the classification of topological phases. Their systematically non-group-like fusion, the appearance of higher-category structure, and their role in constraining RG flows and phase diagrams mark them as central to the emerging language of quantum field theory beyond the group-centric paradigm. Current research explores their connections to higher-form and higher-group symmetries, the design of new lattice models and topological phases, and their appearance in strongly coupled field theories, string theory, and quantum gravity (Huang et al., 2021, Kaidi et al., 2022, Schafer-Nameki, 2023, Apruzzi et al., 14 Nov 2024).