Higher-Form Kramers–Wannier Duality

• Higher-form Kramers–Wannier duality extends the classic duality concept by linking gauged higher-form symmetries with non-invertible topological defects across various dimensions.
• Lattice constructions reveal that gauging discrete Z₂ symmetries produces dual (n–1)-form symmetries, uniting bosonic and fermionic systems via nontrivial unitary mappings.
• Fermionic bosonization and continuum reformulations demonstrate that gauging leads to order–disorder transformations and constraints on infrared phases in gauge theories.

Higher-form Kramers–Wannier duality is the extension of the ordinary Kramers–Wannier paradigm from a $0$-form symmetry in $1+1$ dimensions to settings in which gauging produces dual higher-form symmetries or, at self-dual points, non-invertible topological defects. In recent lattice constructions, gauging a $\mathbb Z_2^B$ spin-flip symmetry or a $\mathbb Z_2^F$ fermion parity produces $\mathbb Z_2$ gauge theories with dual $(n-1)$-form symmetries in $n$ spatial dimensions (Su, 23 Oct 2025). In continuum and lattice gauge theories, gauging a self-dual higher-form symmetry on only half of spacetime yields the higher-dimensional analogue of the Kramers–Wannier defect, with non-invertible fusion rules and order/disorder conversion for extended operators (Choi et al., 2021). Bosonization frameworks sharpen the same structure in arbitrary dimensions by relating parity-gauged fermions, flat gauge fields, modified Gauss laws, and projections onto fixed higher-form charge sectors (Su et al., 27 Aug 2025).

1. Duality defects beyond $1+1$ dimensions

The defect-theoretic formulation begins from the observation that the familiar $1+1$-dimensional Kramers–Wannier defect line can be generalized to even spacetime dimensions by gauging only half of spacetime. For a $q$-form symmetry in $1+1$0 dimensions, the self-dual case occurs when

$1+1$1

so that gauging returns a theory of the same type. In $1+1$2 dimensions this gives the central example $1+1$3: a $1+1$4 one-form symmetry and a codimension-one topological defect supported on a $1+1$5-manifold $1+1$6 (Choi et al., 2021).

The construction splits spacetime into regions $1+1$7 and $1+1$8 separated by an interface $1+1$9, gauges the higher-form symmetry only in $\mathbb Z_2^B$0, and imposes a topological Dirichlet boundary condition

$\mathbb Z_2^B$1

For a discrete $\mathbb Z_2^B$2 symmetry, the gauging sector is the topological gauge theory

$\mathbb Z_2^B$3

The full interface theory is therefore

$\mathbb Z_2^B$4

This defect is the direct higher-form analogue of the Kramers–Wannier line. Crossing it takes charged operators to operators attached to higher-dimensional symmetry operators. In $\mathbb Z_2^B$5 dimensions, a charged line operator crossing the defect becomes a line bounded by a topological surface operator. This is the higher-form version of order/disorder conversion in the Ising model. The defect is non-invertible because fusing it with itself does not return the identity defect; instead, the fusion produces a sum over higher-form symmetry defects (Choi et al., 2021).

2. Gauging on lattices and the emergence of dual $\mathbb Z_2^B$6-form symmetries

A lattice realization of higher-form Kramers–Wannier structure arises from gauging $\mathbb Z_2^B$7 symmetries in a way that treats bosonic and fermionic systems in parallel. One construction gauges the $\mathbb Z_2^B$8 spin-flip symmetry of the transverse-field Ising model by inserting $\mathbb Z_2^B$9 gauge qubits and imposing Gauss law plus flatness. A second construction gauges $\mathbb Z_2^F$0 fermion parity by inserting Majorana fermions and using a fermionic analogue of the disentangling unitary. These procedures produce, respectively, the usual Ising-dual $\mathbb Z_2^F$1 gauge theories and Majorana-dual $\mathbb Z_2^F$2 gauge theories (Su, 23 Oct 2025).

In two dimensions, the bosonic gauging of the transverse-field Ising model yields the standard $\mathbb Z_2^F$3 lattice gauge theory with dual $\mathbb Z_2^F$4-form symmetry generated by loop operators

$\mathbb Z_2^F$5

while Wilson loops

$\mathbb Z_2^F$6

are the charged operators. The fermionic version places Majoranas on edges, with local parity

$\mathbb Z_2^F$7

and the dual $\mathbb Z_2^F$8-form symmetry becomes a loop of fermion parity,

$\mathbb Z_2^F$9

At $\mathbb Z_2$0, the resulting fermionic dual Hamiltonian is described as a Majorana stabilizer code (Su, 23 Oct 2025).

The higher-dimensional statement is explicit: gauging a $\mathbb Z_2$1 $\mathbb Z_2$2-form symmetry yields a $\mathbb Z_2$3 gauge theory with a dual $\mathbb Z_2$4-form symmetry, and gauging $\mathbb Z_2$5 fermion parity yields a gauge theory whose dual $\mathbb Z_2$6-form symmetry is generated by loops of emergent fermions. In this sense, the original $\mathbb Z_2$7-form symmetry and the dual $\mathbb Z_2$8-form symmetry are exchanged under gauging. The same construction extends to general polyhedral decompositions of space, provided the relevant faces are even-edged in the fermionic case. In three dimensions, the product of the three Gauss-law operators around a cube is the identity, so only two are independent; this enforces that excitations form loops (Su, 23 Oct 2025).

A notable feature of this lattice framework is that the bosonic and fermionic gauge theories are unitarily equivalent, but not trivially so. The equivalence is implemented by a linear-depth local unitary circuit and induces a direction-dependent anyonic transmutation connecting bosonic and fermionic toric-code-like theories. In one direction a dressed flux-creation operator maps to a bare $\mathbb Z_2$9; in the other direction the map introduces a large Wilson-loop attachment. The result is a transmutation rather than a simple anyon permutation (Su, 23 Oct 2025).

3. Fermionic bosonization, spin structure, and translation-induced duality

A complementary arbitrary-dimensional formulation treats higher-form Kramers–Wannier duality as the bosonized image of a minimal Majorana translation. The construction starts with a complex fermion on each top-dimensional cell and a $(n-1)$0 gauge spin on each codimension-one cell. Gauging fermion parity imposes a Gauss law such as

$(n-1)$1

in three dimensions, or

$(n-1)$2

in two dimensions. A local disentangling unitary

$(n-1)$3

maps the gauge constraint to pure fermion parity, $(n-1)$4, so that the fermions and gauge spins disentangle (Su et al., 27 Aug 2025).

To recover the original ungauged fermion theory, the gauge field must be flat. In two dimensions this requires

$(n-1)$5

and in three dimensions

$(n-1)$6

After the disentangling unitary, flatness becomes a modified Gauss law in the spin system. The resulting bosonization duality is therefore

$(n-1)$7

The sign structure of that Gauss law depends on a Kasteleyn orientation, equivalently on a discrete spin structure. In two dimensions, inequivalent Kasteleyn orientations are in bijection with $(n-1)$8, and the obstruction is the second Stiefel–Whitney class $(n-1)$9 (Su et al., 27 Aug 2025).

Within this framework, a minimal translation of a Majorana lattice becomes a higher-dimensional Kramers–Wannier duality after bosonization. On a hypercubic lattice in arbitrary dimension, the canonical map is

$n$0

so the translation-induced duality takes the form

$n$1

This is the direct higher-dimensional analogue of the one-dimensional exchange between a spin operator and a domain-wall operator (Su et al., 27 Aug 2025).

Non-invertibility is built into the lattice operator. On periodic lattices, the bosonized theory carries higher-form symmetry operators on nontrivial cycles, and the Kramers–Wannier operator includes projection onto a fixed higher-form symmetry sector. In two dimensions, for example,

$n$2

Because the projector annihilates the $n$3 sector, the duality has no inverse on the full Hilbert space. The duality is therefore an isomorphism only after restricting to a sector with fixed higher-form charge (Su et al., 27 Aug 2025).

4. Four-dimensional gauge theories and non-invertible fusion

In four-dimensional gauge theory, higher-form Kramers–Wannier duality appears as a non-invertible topological defect tied to self-duality under gauging a discrete one-form symmetry. One route begins with a theory carrying an anomaly-free $n$4 one-form symmetry. Gauging that symmetry on half of spacetime yields a $n$5-dimensional defect $n$6, and for a connected orientable $n$7-manifold $n$8 its fusion rule is

$n$9

where $1+1$0 is the symmetry surface operator on the $1+1$1-cycle $1+1$2. On $1+1$3,

$1+1$4

which is a direct indicator of non-invertibility (Choi et al., 2021).

The same paper shows that such defects constrain infrared phases. If the theory is invariant under gauging the relevant one-form symmetry, then a trivially gapped infrared phase would have to be a one-form SPT phase compatible with the self-duality. For bosonic $1+1$5 symmetry, the required arithmetic condition includes

$1+1$6

As a result, bosonic theories invariant under gauging $1+1$7 can flow to a trivially gapped phase only if every prime factor of $1+1$8 is $1+1$9; for even $1+1$0, bosonic theories cannot flow to a trivially gapped phase. Fermionic theories obey modified constraints in which $1+1$1 must be even and every prime factor of $1+1$2 must be $1+1$3 (Choi et al., 2021).

A related $1+1$4-dimensional construction starts from a mixed anomaly between a $1+1$5 symmetry and a $1+1$6 one-form symmetry,

$1+1$7

After gauging the one-form symmetry, the original codimension-one symmetry wall is no longer gauge-invariant. Gauge invariance is restored by attaching the spin TQFT $1+1$8 on the defect worldvolume, producing a genuine non-invertible defect

$1+1$9

Its fusion is

$q$0

where $q$1 generates the one-form symmetry and $q$2 is the mod-$q$3 triple intersection number inside $q$4 (Kaidi et al., 2021).

This framework gives explicit self-duality defects in $q$5 Yang–Mills at $q$6, $q$7 $q$8 super Yang–Mills, and $q$9 $1+1$00 super Yang–Mills at $1+1$01. The self-duality transformation is $1+1$02 when the $1+1$03 symmetry is linear and $1+1$04 when it is anti-linear. The characteristic Kramers–Wannier feature remains unchanged: the square of the duality defect is not the identity, but a sum over higher-form symmetry operators (Kaidi et al., 2021).

5. Continuum, homological, and infrared reformulations

Higher-form Kramers–Wannier duality also admits a continuum Landau-type formulation in which the fundamental variable is not a point field but a functional field $1+1$05 defined on closed $1+1$06-surfaces. In this approach, the $1+1$07-dimensional Wilson-surface operator of a lattice higher gauge theory is promoted to the basic field, charged under the $1+1$08-form global symmetry. The continuum kinetic term is written using an area derivative rather than an ordinary derivative, and the field dimension is

$1+1$09

for the quartic coupling (Kawana, 9 Jul 2025).

The classical solutions reproduce the expected order/disorder behaviour of higher gauge theory. In the strong-coupling limit, the solution takes the form

$1+1$10

which is the area law and corresponds to the unbroken $1+1$11-form symmetry. In the weak-coupling limit, a nonzero expectation value develops,

$1+1$12

the low-energy mode is a phase modulation

$1+1$13

and the classical behaviour is perimeter-like. Compact $1+1$14 and finite $1+1$15 theories further admit topological defects that generalize vortices and domain walls (Kawana, 9 Jul 2025).

At the duality level, the lattice higher gauge theory with gauge group $1+1$16 is mapped to a gauged $1+1$17-form theory with dual group $1+1$18. Near criticality, this induces an infrared duality between the corresponding Landau theories, exchanging order and disorder operators and relating a $1+1$19-form theory to a gauged $1+1$20-form theory. The order/disorder dictionary is

$1+1$21

This is the higher-form counterpart of the standard Kramers–Wannier exchange between local order and disorder variables (Kawana, 9 Jul 2025).

A distinct but complementary topological reformulation uses normal factor graphs to encode chains, cochains, boundary maps, and homology. In that framework, the $1+1$22-torus decomposition underlying finite-size Ising duality appears as four homology sectors, while on the $1+1$23-torus one has $1+1$24, so the duality relation contains $1+1$25 topological sectors. The same language explains why nontrivial homology forces sums over topological sectors in dual partition functions and why higher-dimensional dual models need not remain pairwise-interaction models (Al-Bashabsheh et al., 2016).

6. Adjacent generalizations and conceptual delimitations

Several nearby constructions are often grouped with higher-form Kramers–Wannier duality but are technically distinct. In $1+1$26 dimensions with subsystem $1+1$27 symmetry, gauging line-like row and column symmetries produces a subsystem Kramers–Wannier transformation whose duality operator obeys

$1+1$28

a sum over a grid of subsystem symmetry operators. The corresponding defects are mobile in both spatial directions by local unitaries. The same work stresses that subsystem symmetry is not simply higher-form symmetry: the distinction is not only codimension, but also mobility and topologicalness (Cao et al., 2023).

A related operator-level program constructs Kramers–Wannier duality in subsystem-symmetric lattice models as a sequential circuit followed by projection onto the symmetric subspace. In two dimensions and higher-dimensional hypercubic models, the duality squares to translation times projection,

$1+1$29

and the authors explicitly present this as a non-invertible symmetry operator. The same paper remarks that the philosophy should extend to higher-form symmetries, but its main body is devoted to subsystem symmetry rather than to higher-form gauge symmetry proper (Mana et al., 2024).

In $1+1$30 dimensions, gauging finite Abelian modulated symmetries yields further Kramers–Wannier-like dualities in which the dual symmetry generators are reflected versions of the original ones,

$1+1$31

and the non-invertible symmetry operator is a reflection-dressed sequential circuit,

$1+1$32

These constructions generalize the Ising story to dipole, quadrupole, and other modulated symmetries, but they are not higher-form constructions in the strict sense (Pace et al., 2024).

A final conceptual distinction concerns fermionic higher-dimensional self-dualities obtained from minimal Majorana translations. In the Majorana-dual $1+1$33 gauge theories, the resulting Kramers–Wannier self-dualities are direction-dependent and inherited from the translation and folding structure of the Majorana lattice. They are not simply the same as gauging the $1+1$34-form symmetry of the folded Ising chain, nor are they identical to gauging along a space-covering path (Su, 23 Oct 2025).

Taken together, these distinctions delimit the higher-form case. The strict higher-form Kramers–Wannier framework concerns dualities generated by gauging higher-form or parity-related symmetries so that extended operators, higher-form charges, and non-invertible defect fusion become the primary structures. Subsystem, modulated, and translation-induced variants are closely related generalizations, but the cited works treat them as structurally different categories of duality.