Continuous Non-Invertible Symmetries

• Continuous non-invertible symmetries are continuously parameterized topological defects whose fusion decomposes into sums rather than following a standard group law.
• They are realized in frameworks like 1+1d CFT and lattice systems, where non-local conserved currents replace the usual local symmetry generators.
• Applications of these symmetries span ABJ anomalies, duality systems, and quantum gravity, offering fresh perspectives on gauging and quantum channel actions.

Searching arXiv for papers on continuous non-invertible symmetries and closely related foundational work. Continuous non-invertible symmetries are generalized symmetries implemented by continuously parameterized topological defects whose composition is not group-like. In place of a one-parameter group of automorphisms, the known constructions involve fusion laws that decompose products into sums of defects, defect-local or non-local conserved currents, flat gauging of continuous symmetries, and, in anomalous settings, continuously labeled topological charge defects. The subject is technically most developed in $1+1$d CFT and compact-boson/orbifold models, but closely related structures also appear in lattice Hamiltonian systems, ABJ-anomalous gauge theories, continuous $2$-groups in three dimensions, and as rational non-invertible remnants or approximate realizations of continuous duality groups in higher-dimensional theories and string theory (Delmastro et al., 30 Jul 2025, Etxebarria et al., 2022, Jia et al., 14 Jun 2026).

1. Definition and basic forms

Ordinary continuous internal symmetry is encoded by topological codimension-one operators $U_\alpha$ obeying a group law, typically $U_\alpha U_\beta = U_{\alpha+\beta}$. Non-invertible symmetry relaxes that requirement: topological defects $\mathcal N_i$ fuse according to a more general rule

$$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$

with $\mathcal T_{ij}^k$ c-number coefficients or, more generally, partition functions of decoupled TQFTs. A continuous non-invertible symmetry is then a continuously parameterized family of such topological operators whose composition is not a group law (Heckman et al., 2024).

The cleanest prototype is the compact-boson orbifold. Before orbifolding, a compact boson has invertible $U(1)$ momentum operators $\mathcal U_\theta$. After gauging $X\to -X$, the gauge-invariant operators are orbit sums

$2$0

with fusion

$2$1

This is continuous because $2$2 is continuous, and non-invertible because multiplication produces a sum rather than a single inverse element (Heckman et al., 2024).

A related lattice realization is the cosine family

$2$3

which satisfies

$2$4

and contains discrete fusion-category substructures at special angles such as $2$5 (Seifnashri et al., 4 Mar 2025).

Setting	Representative operators	Characteristic law
$2$6 orbifold CFT	$2$7	$2$8
1d lattice cosine symmetry	$2$9	$U_\alpha$0
ABJ-anomalous 4d theory	$U_\alpha$1	continuously $U_\alpha$2-labeled, but non-invertible in flux sectors

These examples already show that continuity and non-invertibility are logically independent. Continuous labeling does not force group structure, and non-invertibility does not require a finite set of simple defects.

2. Generalized Noether theory and local action

A central development is the generalized Noether picture for $U_\alpha$3d CFT. Continuous non-invertible symmetries are associated not to local conserved currents, but to non-local conserved currents: dimension-$U_\alpha$4 or $U_\alpha$5 point operators attached to topological defect lines. If $U_\alpha$6 is such a current at the end of a line $U_\alpha$7, and a baseline defect $U_\alpha$8 satisfies

$U_\alpha$9

then one can integrate $U_\alpha U_\beta = U_{\alpha+\beta}$0 along $U_\alpha U_\beta = U_{\alpha+\beta}$1 and build a continuous topological family

$U_\alpha U_\beta = U_{\alpha+\beta}$2

The resulting defects commute with the Virasoro generators and are therefore topological. This gives a generalized Noether theorem: $U_\alpha U_\beta = U_{\alpha+\beta}$3 (Delmastro et al., 30 Jul 2025).

A second foundational question concerns the action on local observables. Ordinary symmetry acts locally by automorphisms,

$U_\alpha U_\beta = U_{\alpha+\beta}$4

but non-invertible symmetry does not. A general defect $U_\alpha U_\beta = U_{\alpha+\beta}$5 with dual $U_\alpha U_\beta = U_{\alpha+\beta}$6 acts on a local operator by a Stinespring-type formula

$U_\alpha U_\beta = U_{\alpha+\beta}$7

hence as a completely positive map rather than an algebra automorphism. In the Ising/Kramers–Wannier example this becomes an explicit Kraus decomposition,

$U_\alpha U_\beta = U_{\alpha+\beta}$8

and the failure of invertibility is visible in relations such as

$U_\alpha U_\beta = U_{\alpha+\beta}$9

This places non-invertible symmetry in the same formal class as quantum channels, measurements, and ancilla-assisted operations (Okada et al., 2024).

The state-space version sharpens the relation to Wigner’s theorem. In a unitary fusion-category symmetry, a defect does not act on a single Hilbert space, but as an isometry

$\mathcal N_i$0

between twisted-sector Hilbert spaces, equivalently as a trace-preserving quantum channel. Transition probabilities are preserved only after one enlarges to all twisted sectors and defect-junction channels (Bartsch et al., 6 Feb 2026). A complementary operator-theoretic formulation shows that Wigner-compatible non-invertible symmetries are realized by partial isometries on an extended gauged Hilbert space, rather than by arbitrary non-unitary maps on the original physical Hilbert space (Ortiz et al., 29 Sep 2025).

3. Canonical $\mathcal N_i$1d constructions

The compact boson and its orbifolds remain the canonical arena. The broad picture is that gauging a discrete operation acting on a continuous symmetry converts a continuous invertible family into a continuous non-invertible one. In the $\mathcal N_i$2 orbifold, the action on orbifold-even and orbifold-odd operators is

$\mathcal N_i$3

so the defects act continuously on local operators even though their fusion is non-group-like (Heckman et al., 2024).

The same logic underlies the “cosine symmetry” emphasized in survey literature, often written as

$\mathcal N_i$4

This example established early that continuous non-invertible symmetry exists in $\mathcal N_i$5d and that it is naturally produced by orbifolding a theory with continuous $\mathcal N_i$6 symmetry (Shao, 2023).

Recent work systematizes this beyond the $\mathcal N_i$7 moduli space. In diagonal $\mathcal N_i$8 WZW models, continuous non-invertible symmetries arise when a Verlinde line $\mathcal N_i$9 supports a defect current of weight $$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$0. The allowed levels are

$$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$1

that is,

$$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$2

For these $$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$3, the defect currents form an $$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$4 multiplet and generate continuous non-invertible symmetries. Under the restriction that $$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$5 be preserved, the examples with

$$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$6

are intrinsic in the sense that the non-local currents cannot be made local by any gauging consistent with that global symmetry (Delmastro et al., 30 Jul 2025).

Products of minimal models furnish another infinite source. In $$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$7, a non-local $$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$8 current appears whenever

$$\mathcal N_i\otimes \mathcal N_j=\sum_k \mathcal T_{ij}^k\,\mathcal N_k,$$9

These defect currents generate continuous non-invertible symmetries and, after folding and unfolding, produce new defect conformal manifolds in a single minimal model (Delmastro et al., 30 Jul 2025).

4. Lattice realizations and non-invertible gauging

The lattice counterpart is particularly explicit in the qubit-chain realization of the $\mathcal T_{ij}^k$0 symmetry generated by the Kennedy–Tasaki transformation. The Hamiltonian

$\mathcal T_{ij}^k$1

respects the continuous cosine symmetry, implemented by MPO operators $\mathcal T_{ij}^k$2 with the same cosine fusion law as in continuum orbifold CFT. Special values recover discrete substructures: $\mathcal T_{ij}^k$3 Thus a finite fusion-category symmetry sits inside a larger continuous non-invertible family (Seifnashri et al., 4 Mar 2025).

This family is not merely decorative. Gauging the non-maximal algebra object

$\mathcal T_{ij}^k$4

is implemented by the specific cosine element

$\mathcal T_{ij}^k$5

The construction introduces two qubits around each link as gauge fields and imposes Gauss-law projectors

$\mathcal T_{ij}^k$6

These constraints are explicitly not ordinary gauge transformations in disguise; the authors stress that the local operators $\mathcal T_{ij}^k$7 do not act as ordinary gauge transformations preserving each Hamiltonian term separately (Seifnashri et al., 4 Mar 2025).

The locality question has an independent answer from the Ising chain. There, the Kramers–Wannier duality defect acts on local operators through ancilla insertion, local unitaries, and ancilla removal,

$\mathcal T_{ij}^k$8

which is manifestly of Stinespring form and therefore completely positive (Okada et al., 2024). The lattice and continuum pictures are therefore tightly aligned: continuous non-invertible symmetry is compatible with strict topological locality, but locality is implemented defect-theoretically rather than by algebra automorphisms.

5. Anomalies, higher dimensions, and quantum gravity

A distinct route to continuity comes from ABJ anomalies. In four-dimensional QED-like theories with

$\mathcal T_{ij}^k$9

ordinary axial $U(1)$0 is obstructed, but one can define a continuously labeled family of topological charge defects

$U(1)$1

with $U(1)$2. These defects act on charged local operators by

$U(1)$3

yet are non-invertible in magnetic-flux sectors. From this construction one obtains a Goldstone theorem: if a charged operator acquires an expectation value, then a gapless mode must exist. In axion-like effective theory the corresponding coupling is

$U(1)$4

(Etxebarria et al., 2022).

In three dimensions, gauging compact scalar backgrounds can convert anomalies in coupling space into continuous $U(1)$5-group structures. In the Goldstone model one finds transformation laws such as

$U(1)$6

which exhibit the $U(1)$7-form/$U(1)$8-form mixing characteristic of a continuous $U(1)$9-group. In the Goldstone–Maxwell model the same mechanism yields infinitely many TQFT-dressed non-invertible defects labeled by rational data rather than a straightforward continuous group (Damia et al., 2022).

Four-dimensional duality systems provide another, more arithmetic, variant. In Gaillard–Zumino models with classical continuous duality group $\mathcal U_\theta$0, the usual quantum statement is that only

$\mathcal U_\theta$1

survives invertibly. A stronger statement is that the much larger rational subgroup

$\mathcal U_\theta$2

survives through codimension-one non-invertible defects. These defects are topological, act classically on local operators, and act non-invertibly on line operators because rational symplectic transformations preserve only finite-index sublattices of the charge lattice (Apruzzi et al., 21 Oct 2025). This is not a full continuous quantum symmetry, but it is a precise non-invertible remnant of a continuous classical one.

In string theory and holography, exact continuity is more fragile. Worldsheet orbifolds provide exact continuous non-invertible lines at tree level, but the paper on stringy non-invertible symmetries argues that higher-genus worldsheets generally break such symmetries to their maximal invertible subsector; the $\mathcal U_\theta$3 example leaves only the invertible half-shift unbroken once loops are included (Heckman et al., 2024). In $\mathcal U_\theta$4 theories, abelian Maxwell defects can realize continuous bonus-$\mathcal U_\theta$5 rotations on local operators exactly, whereas in nonabelian $\mathcal U_\theta$6 SYM the corresponding broader family appears only approximately in a large-$\mathcal U_\theta$7, large-$\mathcal U_\theta$8 supergravity regime (Sela, 2024).

6. Constraints, misconceptions, and open directions

A common misconception is to identify dense rational families with genuinely continuous symmetry. The distinction is explicit in several places. QED/QCD examples labeled by $\mathcal U_\theta$9 are infinite and dense but are not continuous, because irrational angles are not supplied by the half-gauging construction (Shao, 2023). Likewise, the Gaillard–Zumino construction realizes $X\to -X$0, not the full classical $X\to -X$1, as exact quantum defects (Apruzzi et al., 21 Oct 2025). Dense arithmetic remnants and continuous families are therefore structurally different.

A second misconception is that non-invertibility on local operators is generic in higher dimensions. For finite $X\to -X$2d non-invertible symmetries described by fusion $X\to -X$3-categories, the action on local operators is severely constrained: if there are no nontrivial topological line operators,

$X\to -X$4

then codimension-one defects act invertibly on local operators. More generally, the local action can be decomposed into an invertible action in a related line-free theory followed by a gauging interface. This suggests that any continuous $X\to -X$5d generalization with genuinely non-invertible local action will need an analogue of the topological-line/gauging-interface mechanism (Putrov et al., 3 Mar 2026).

A third open issue concerns the correct definition of continuous non-invertible gauging. On the orbifold branch of the compact boson, direct integration over continuous non-invertible defects misses contributions supported on measure-zero fixed loci. The resulting ambiguity is visible in the prescription

$X\to -X$6

which leads to different answers depending on $X\to -X$7. The noncompact orbifold $X\to -X$8 corresponds to $X\to -X$9, while the limit of finite non-invertible gaugings gives $2$00. The paper explicitly leaves the intrinsic measure-theoretic definition of continuous non-invertible gauging as an open problem (Jia et al., 14 Jun 2026).

The operational and geometric side of the subject is also expanding. A distinguishability-based metric and generalized complexity geometry have been proposed for both continuous and discrete non-invertible symmetries, with continuous families treated as analogues of Lie-group manifolds and discrete families embedded via linear combinations of unitary operators (Heckman et al., 15 Apr 2026). This suggests that continuous non-invertible symmetry may ultimately require an overview of defect categories, quantum channels, flat-gauging constructions, and noncompact or operator-algebraic geometry rather than a direct generalization of Lie theory.

At present, the literature supports a stratified view. Exact continuous non-invertible symmetries are firmly established in $2$01d CFT, compact-boson/orbifold models, and ABJ-type anomalous constructions. In higher dimensions, the most robust results either produce continuous $2$02-group structures, dense rational non-invertible remnants, or approximate/singular-limit realizations. This suggests that “continuous non-invertible symmetry” is not a single uniform object, but a family of related structures whose common denominator is continuous topological data combined with fundamentally non-group-like composition.