Kramers-Wannier Duality: Non-Invertible Symmetry

• Kramers-Wannier duality is a transformation in lattice models that maps order variables to disorder variables and interchanges coupling regimes.
• It features non-invertible fusion rules and acts as a categorical symmetry, linking to topological quantum field theory and defect structures.
• Extensions to non-abelian groups and higher dimensions connect the duality to gauging operations, brane redefinitions, and the classification of gapped phases.

Kramers-Wannier (KW) duality-symmetry is a paradigmatic transformation in lattice statistical mechanics and quantum spin systems, originally formulated for the two-dimensional Ising model. In modern understanding, KW duality is not merely a mapping between models at high and low temperatures, but realizes a non-invertible symmetry of quantum systems, generalizing to a categorical structure applicable to a broad class of models in arbitrary dimensions, for abelian and non-abelian symmetry groups, and linked to topological quantum field theory (TQFT) and topological defects.

1. Definition and Lattice Construction

Kramers-Wannier duality is most simply realized as a transformation on lattice spin models, mapping local “order” variables (such as Ising spins) to nonlocal “disorder” or “domain wall” variables and interchanging coupling regimes. For a general two-dimensional lattice model with symmetry group $G$, the configuration space consists of “spins” $\sigma_v \in G$ at each vertex $v$ and Boltzmann weights $W(\sigma_v\sigma_w^{-1})$ assigned to each edge $e=(v,w)$. The partition function is

$$Z[\sigma] = \sum_{\sigma \in G^{V(\Sigma)}} \prod_{e \in E(\Sigma)} W(\sigma_{v(e)}\sigma_{w(e)}^{-1}) .$$

The duality transformation proceeds by Fourier analysis on the group $G$, introducing dual variables $\tau_e \in \hat{G}$ (irreducible reps). The dual partition function, after Fourier transform, is

$$\hat{Z}[\tau] = \sum_{\tau: E(\Sigma) \to \hat{G}} \prod_{e} \hat{W}(\tau_e) \prod_{v} \delta\left(1, \prod_{e \ni v} \tau_e^{\epsilon(v,e)}\right) ,$$

where $\hat{W}(\tau_e)$ are Fourier-transformed weights and the delta enforces a “flatness” constraint that guarantees dual configurations correspond to consistent domain-wall configurations.

For the usual $\mathbb{Z}_2$ (Ising) case, the mapping specializes to a correspondence between local spin products $\sigma^z_j\sigma^z_{j+1}$ and dual “bond” (domain wall) variables, e.g. $\mu^x_{j+1/2}\equiv\sigma^z_j\sigma^z_{j+1}$, $\mu^z_{j+1/2} \equiv \prod_{k=1}^j \sigma^x_k$, with the duality explicitly exchanging the transverse magnetic field and the nearest-neighbor spin coupling.

2. Non-Invertibility and Fusion Rules

A fundamental feature of Kramers-Wannier duality as a symmetry operation is its non-invertibility. In the 1d Ising chain, the duality operator $D$ (or $N$) satisfies

$D^2 = 1 + g ,$

where $g$ is the generator for the global $\mathbb{Z}_2$ spin-flip symmetry; $D$ projects onto the even-parity sector. In language of topological defects,

$D \times D = 1 \oplus \eta ,$

a hallmark of non-invertible categorical symmetry (Tambara-Yamagami fusion). The physical implication is that duality lines or defects cannot be simply undone by an inverse operation—fusion generates a sum of sectors rather than a unique one.

This algebraic structure generalizes: for a $\mathbb{Z}_p$ duality, the defect $D$ satisfies

$D^p = \sum_{a=0}^{p-1} \eta^a ,$

with $\eta$ the generator of global symmetry associated to $G$.

3. Topological Field Theory and the Symmetry-Duality Correspondence

In modern terms, Kramers-Wannier duality is described as a topological defect encoded within a (d+1)-dimensional TQFT, specifically via the Turaev-Viro/Barrett-Westbury construction derived from a spherical fusion category $\mathcal{C}$. The 3d bulk TQFT $\mathcal{Z}_\mathcal{C}$ attaches to a boundary anyonic lattice model, whose space of “topological ground states” is a subspace $Z_\mathcal{C}(\Sigma)$ of the microscopic boundary Hilbert space $H_\mathcal{C}(\Sigma_\Upsilon)$.

Defects in the bulk correspond to objects in the Drinfeld center $\mathcal{Z}(\mathcal{C})$; defects that survive (i.e., do not condense) on the boundary align with the Morita dual category $\mathcal{C}^{\vee}_\mathcal{M}$ relative to the choice of module category $\mathcal{M}$ (brane boundary condition). Under compactification, these become topological line defects in the lattice boundary theory, exactly implementing duality operations such as Kramers-Wannier.

In this setting, KW duality is interpreted as a change in boundary condition (e.g., from Dirichlet to Neumann brane) effected via a defect interpolating between the corresponding module categories, and algebraically realized as a combination of Fourier transform and group-sub-symmetry gauging.

4. Non-Abelian and Higher Dimensional Generalizations

The KW construction extends to non-abelian spin models and higher dimensions. For non-abelian $G$, the Fourier transform leads to dual partition functions over edge labellings by irreducible representations, and the fusion category formalism ensures a systematic description of dualities and their algebraic structure. Gauging a subgroup $A \subset G$ manifests in the TQFT as choosing a different brane boundary, and the resulting gauged boundary partition function

$$Z_{\text{gauged}} = \frac{1}{|A|} \sum_{a \in A^E} \sum_{\sigma \in G^V} \prod_{e} W(a_e \sigma_v \sigma_w^{-1})$$

is encoded topologically in the module category $\mathcal{M}(A)$.

The combination of subgroup-gauging and Fourier transform—performed in either order—produces the non-abelian Kramers-Wannier dual, which is realized in the boundary theory as a model with $G$-domain wall weights replaced by $\hat{W}(\tau)$ on the dual domain, with brane boundary given by $\mathcal{M}(A)$; this boundary theory is Morita equivalent to the original, and the topological lines are traced through the duality via categorical equivalence.

In higher dimensions, duality defects generalize to codimension-1 (or higher) objects. For 3+1d gauge theories, half-space gauging construction produces non-invertible duality defects: the interface between theory $\mathcal{T}$ and its orbifold (gauged) $\mathcal{T}/G^{(q)}$ is topological, and when the theory is self-dual under this construction, the interface becomes a defect in a single theory. Fusion rules in 3+1d are given by condensation over topological surfaces, e.g.,

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

mirroring the 1+1d fusion but in higher-form symmetry language.

5. Renormalisation Group Fixed Points and Gapped Phases

Gapped symmetric phases correspond, in the categorical framework, to algebra objects $\mathcal{A}$ in the fusion category $\mathcal{C}$. The boundary partition function is specified by the insertion of an $\mathcal{A}$-network on the brane. KW duality acts at the level of these phases by mapping algebra objects $\mathcal{A}$ in $\mathcal{C}$ to their Morita duals $\mathcal{A}^\vee$ in $\mathcal{C}_\mathcal{M}^\vee$.

Gauging a subsymmetry $A$ corresponds to switching branes from $\mathcal{M} = \mathrm{Vect}_G$ to $\mathcal{M}(A)$ (or, in the Fourier dual, from $\mathrm{Rep}(G)$ to $\mathrm{Rep}(A)$), and the duality intertwines the respective fixed (gapped) phases. This explicit mapping at the RG level ensures that RG flows are constrained by duality, and that the correspondence is robust to perturbations as long as symmetry considerations are maintained. In particular, KW duality maps gapped symmetric phases into dual gapped phases within the categorical framework, generalized to non-abelian and even non-invertible contexts.

6. KW Duality as an Exact Symmetry: Categorical and Physical Manifestations

While traditionally viewed as a mapping between high- and low-temperature phases, in quantum spin chains and 2d field theories, KW duality acts at the operator level as a non-invertible symmetry. Physically, at criticality (self-dual point), it intertwines order/disorder parameters, exchanges phases, and enforces fusion rules on local and non-local operators. In categorical language, the KW line is an invertible (or non-invertible) topological defect in the 3d SymTFT, and its endpoint on the boundary implements the duality transform, thereby unifying the concepts of symmetry and duality lines.

For general groups and fusion categories, KW duality is a particular example of a non-invertible categorical symmetry arising from the state-sum construction and persists as a global, though not necessarily invertible, symmetry exchanging order and disorder variables.

In particular, for 2d lattice models with symmetry corresponding to a general spherical fusion category $\mathcal{C}$, the model is realized as the boundary of a 3d Turaev-Viro theory, and the combinations of gauging and Fourier transform operations (implementing KW duality) correspond to transparent changes in boundary conditions.

7. Physical and Mathematical Significance

The reconstruction of Kramers-Wannier duality as an example of categorical symmetry, as explored in both quantum spin models and TFT boundary frameworks, captures its non-invertible character, operator-level consequences, and its role in classifying gapped phases and constraining renormalization group flows. The explicit connection between gauging, brane redefinition in TQFT, and Fourier transform provides a powerful toolkit for extending duality constructions to non-abelian, higher-dimensional, and categorical settings. This generalizes the classic order-disorder correspondence and links dualities to topological, algebraic, and quantum-information-theoretic structures, as well as to the systematic classification of symmetry-protected phases and their duals (Delcamp et al., 12 Aug 2024).