Non-Abelian Kramers–Wannier Dualities

• Non-Abelian Kramers–Wannier dualities are defined via fusion categories and TQFT, generalizing abelian duality by incorporating non-invertible duality defects and categorical symmetries.
• They are operationally constructed through gauging a fusion subcategory, applying a categorical Fourier transform, and using module equivalence to map between dual lattice models.
• The framework uncovers new order/disorder correspondences and fusion rules that are critical for understanding symmetry transitions in low-dimensional quantum and statistical systems.

Non-Abelian Kramers–Wannier duality refers to a categorical and algebraic generalization of the classic Kramers–Wannier (KW) duality in lattice models, replacing abelian symmetries (e.g., $\mathbb{Z}_2$ in the Ising model) by non-abelian structures such as non-abelian groups, fusion categories, or Hopf algebras. This results in duality defects whose algebra and fusion are governed by non-group-like (generically non-invertible) categorical data, leading to profound modifications of duality, order/disorder correspondences, and the global symmetry content in quantum and statistical lattice models in dimensions greater than one.

1. Framework of Non-Abelian Kramers–Wannier Dualities

The non-abelian generalization of KW duality is naturally formulated using the machinery of fusion categories, module categories, and topological quantum field theory (TQFT). In generalized Ising-type lattice models on a 2D surface $\Sigma$, local degrees of freedom (“spins”) can take values in a finite group $G$ (group case), or more generally labels from the simple objects of a spherical fusion category $\mathcal{C}$ (categorical case). Interactions are encoded by Boltzmann weight functions $\{\theta_e\}$, with the partition function expressed as a sum (or contraction) over configurations determined by the combinatorics of $\mathcal{C}$, with a natural topological line defect algebra embedded in correlation functions.

These lattice models admit a precise realization as boundary theories of a three-dimensional TQFT—the Turaev–Viro–Barrett–Westbury state-sum associated to $\mathcal{C}$—where the choice of indecomposable $\mathcal{C}$-module categories (“brane” boundary conditions) determines the physical boundary state (Delcamp et al., 2024). This boundary realization provides a categorical and TQFT-theoretic underpinning to non-abelian KW dualities.

2. Defects, Fusion Categories, and Non-Invertible Duality Maps

Physical global symmetries and dualities in these generalized lattice models are implemented by topological line defects labeled by simple objects of $\mathcal{C}$, whose algebra is governed by the fusion rules and $F$-moves of the category (Eck et al., 2023, Lootens et al., 2021). In particular, dualities are realized by explicitly constructed non-invertible defect operators—often as matrix product operators (MPOs)—which intertwine the bond algebra of dual presentations of the model.

The fusion rules of these duality defects go beyond group structure, typically obeying non-invertible fusion relations of the form

$\Sigma$0

where $\Sigma$1 is a non-trivial symmetry defect, and $\Sigma$2 is the identity sector (as in the Ising case), or more generally as determined by the fusion coefficients $\Sigma$3 of $\Sigma$4 (e.g., in so(3)$\Sigma$5, $\Sigma$6) (Eck et al., 2023). This non-invertibility reflects the categorical (rather than group) structure of the symmetries and dualities.

3. Operational Construction: Gauging, Fourier Transform, and Categorical Duality

Non-abelian KW duality is systematically realized through a sequence of exact operations:

• Gauging a Fusion Subcategory: Internal symmetry lines corresponding to a fusion subcategory $\Sigma$7 can be summed over, changing the boundary module category to $\Sigma$8.
• Categorical Fourier Transform: A non-abelian generalization of the Fourier transform replaces group-valued edge labels by labels of irreducible representations (moving from $\Sigma$9- to $G$0-type models).
• Duality via Module (Morita) Equivalence: After applying both gauging and Fourier transform (in either order), one arrives at a canonically isomorphic (dual) boundary state in the same 3D TQFT, characterized by a new realization of the fusion category (i.e., $G$1 for finite $G$2) (Delcamp et al., 2024, Freed et al., 2018).

Explicitly, for group symmetry: $G$3 where $G$4 is the Fourier transform of the original weights, and both the original and dual theories can be written as state sums with weights defined by the fusion data.

4. Examples and Algebraic Structures

A canonical example is provided by lattice models with $G$5 symmetry. The original model assigns spins $G$6 and Boltzmann weights $G$7, while the Fourier-dual uses representations $G$8 and weights $G$9. Gauging a subgroup (e.g., $\mathcal{C}$0) can be performed on either side, yielding the same physical content. Order parameters in one model map to disorder operators in the dual, reflecting the swap of Wilson and 't Hooft lines under duality (Delcamp et al., 2024).

In 1+1d, integrable spin chains provide systematic access to non-abelian KW duality. In the XXZ chain and related models, non-invertible duality defects arise from algebraic structures such as so(3)$\mathcal{C}$1, leading to families of models (including the three-state AFM and the Rydberg-blockade ladder) related by categorical duality transformations. Their dualities are explicitly realized as commuting MPOs whose fusion exactly matches the underlying fusion category (Eck et al., 2023, Zhu, 2 Sep 2025). The progression to non-abelian settings, as in the Hopf-Ising models with self-dual Hopf algebra symmetries, further enriches the duality structures with non-cocommutativity and new categorical invariants that go beyond the group-theoretic framework (Lu et al., 10 Feb 2026).

5. Symmetry Content, Superselection Sectors, and Categorical Symmetry Algebras

Gauging a non-abelian finite group $\mathcal{C}$2 in quantum spin chains realizes non-invertible KW duality and leads to a nontrivial decomposition of the Hilbert space. States organize into superselection sectors labeled by irreducible representations $\mathcal{C}$3 and conjugacy classes, with the dual model's global symmetry captured not by a group but by an algebraic ring of double-cosets together with a categorical $\mathcal{C}$4 factor (Cao et al., 21 Jan 2025). The duality operator thus transports symmetry from the original group to the dual categorical structure, and its non-invertible nature projects or embeds the Hilbert space onto the appropriate sectors.

On the categorical side, the construction and classification of these dualities rely critically on the Morita equivalence of module categories over the symmetry fusion category. Duality is defined whenever there exist two inequivalent but Morita-equivalent module categories, with the MPO intertwiner providing an exact mapping of spectra and operator content (Lootens et al., 2021).

6. Higher-Dimensional and Higher-Form Generalizations

These structural features extend to higher dimensions and higher-form symmetries. In $\mathcal{C}$5D gauge theories with non-abelian gauge groups (e.g., SO(3), SU(2)), non-invertible codimension-one duality defects are constructed via half-space gauging of a one-form symmetry, resulting in fusion rules that generalize the Ising defect algebra to fusion $\mathcal{C}$6-categories (Choi et al., 2021, Kaidi et al., 2021). Such defects enforce self-duality under composite gauging and counterterms (e.g., $\mathcal{C}$7 or $\mathcal{C}$8 operations), and their presence is intrinsic to non-invertible mixed anomaly structure, obstructing the existence of symmetric trivially gapped phases. The algebra of defects is controlled by higher analogs of Ising fusion categories and involves, in full generality, higher categorical data.

Orbifold and categorical TQFT techniques provide explicit partition functions and link the lattice realizations to continuum CFT modular invariants, confirming non-invertible symmetry actions and dual sector decompositions. The fusion algebras of the defects and modular data precisely match categorical expectations, with non-invertible duality defects playing the role of lattice incarnations of Verlinde lines or Ising-type fusion lines in the continuum.

7. Classification, Extensions, and Outlook

The non-abelian Kramers–Wannier framework is expansive: all models built from a chosen fusion category $\mathcal{C}$9 (e.g., $\{\theta_e\}$0, $\{\theta_e\}$1, $\{\theta_e\}$2, Haagerup, etc.) support a network of non-invertible duality defects exact at the lattice level (Eck et al., 2023, Lootens et al., 2021). Non-invertible KW duality thus generalizes both the symmetry and duality content of lattice and quantum models well beyond group symmetries, encoding intrinsic categorical and higher-form data. The formalism is sufficiently robust to encompass models in 2+1 and 3+1 dimensions and to integrate into the program of classifying topological phases and integrable models by non-invertible (categorical) duality structures.

In summary, non-abelian Kramers–Wannier duality is realized as an exact equivalence between boundary TQFT states, implemented through categorical Fourier transforms and module category swaps, and physically corresponds to non-invertible duality defects controlling criticality, phase structure, and the fusion algebra of order/disorder operators in models without abelian symmetry. This provides a unifying perspective on self-duality and non-invertible symmetries in low-dimensional quantum and statistical systems (Freed et al., 2018, Delcamp et al., 2024, Lu et al., 10 Feb 2026, Eck et al., 2023, Cao et al., 21 Jan 2025, Lootens et al., 2021, Choi et al., 2021, Kaidi et al., 2021, Zhu, 2 Sep 2025).