---
title: Continuous Non-Invertible Symmetries
url: https://www.emergentmind.com/topics/continuous-non-invertible-symmetries
type: topic
---

# Continuous Non-Invertible Symmetries

Searching arXiv for recent 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 [2507.22976] [2211.09570] [2606.15732].

## 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 [2402.00118].

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
\[
\mathcal L_\theta=\mathcal U_\theta\oplus \mathcal U_{-\theta},\qquad \theta\in(0,\pi),
\]
with fusion
\[
\mathcal L_\theta\mathcal L_{\theta'}=\mathcal L_{\theta+\theta'}+\mathcal L_{\theta-\theta'}.
\]
This is continuous because \(\theta\) is continuous, and non-invertible because multiplication produces a sum rather than a single inverse element [2402.00118].

A related lattice realization is the cosine family
\[
\mathsf L_\theta=\frac{1+\eta}{2}\left(e^{i\theta Q}+e^{-i\theta Q}\right),
\]
which satisfies
\[
\mathsf L_\theta\mathsf L_{\theta'}=\mathsf L_{\theta+\theta'}+\mathsf L_{\theta-\theta'},
\qquad
\mathsf L_{\theta+2\pi}=\mathsf L_\theta,
\]
and contains discrete fusion-category substructures at special angles such as \(\mathsf L_{\pi/2}=D\) [2503.02925].

| Setting | Representative operators | Characteristic law |
|---|---|---|
| \(c=1\) orbifold CFT | \(\mathcal L_\theta=\mathcal U_\theta\oplus \mathcal U_{-\theta}\) | \(\mathcal L_\theta\mathcal L_{\theta'}=\mathcal L_{\theta+\theta'}+\mathcal L_{\theta-\theta'}\) |
| 1d lattice cosine symmetry | \(\mathsf L_\theta=\frac{1+\eta}{2}(e^{i\theta Q}+e^{-i\theta Q})\) | \(\mathsf L_\theta\mathsf L_{\theta'}=\mathsf L_{\theta+\theta'}+\mathsf L_{\theta-\theta'}\) |
| ABJ-anomalous 4d theory | \(U_\alpha(\Sigma_3)\) | continuously \(U(1)\)-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 \(1+1\)d CFT. Continuous non-invertible symmetries are associated not to local conserved currents, but to non-local conserved currents: dimension-\((1,0)\) or \((0,1)\) point operators attached to topological defect lines. If \(J\) is such a current at the end of a line \(\mathcal L\), and a baseline defect \(\mathcal B\) satisfies
\[
\mathcal B\times \mathcal L \supset \mathcal B,
\]
then one can integrate \(J\) along \(\mathcal B\) and build a continuous topological family
\[
\mathcal N_\alpha
=
\sum_{n=0}^{\infty}\frac{(i\alpha)^n}{n!}
\oint \prod_i \frac{dz_i}{2\pi i}\,\mathcal P[\cdots J(z_i)\cdots].
\]
The resulting defects commute with the Virasoro generators and are therefore topological. This gives a generalized Noether theorem:
\[
\text{continuous non-invertible symmetry}
\quad\longleftrightarrow\quad
\text{non-local conserved current}
\]
[2507.22976].

A second foundational question concerns the action on local observables. Ordinary symmetry acts locally by automorphisms,
\[
g(O)=A_g^\dagger O A_g,
\]
but non-invertible symmetry does not. A general defect \(X\) with dual \(\overline X\) acts on a local operator by a Stinespring-type formula
\[
X(O)=V^\dagger\,\pi_{X\overline X}(O)\,V,
\]
hence as a completely positive map rather than an algebra automorphism. In the Ising/Kramers–Wannier example this becomes an explicit Kraus decomposition,
\[
D(O)=\sum_t K_t O K_t^\dagger,
\]
and the failure of invertibility is visible in relations such as
\[
\mathcal R(D^2(O))=O+g(O),\qquad D(Z_i)=0.
\]
This places non-invertible symmetry in the same formal class as quantum channels, measurements, and ancilla-assisted operations [2403.20062].

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
\[
U(A)_X:\mathcal H_X\to \bigoplus_S D_{XS}^A\,\mathcal H_S,
\]
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 [2602.07110]. 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 [2509.25327].

## 3. Canonical \(1+1\)d 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 \(S^1/\mathbb Z_2\) orbifold, the action on orbifold-even and orbifold-odd operators is
\[
\begin{aligned}
\mathcal L_\theta:\;&\mathcal O_m^+\mapsto \cos(m\theta)\,\mathcal O_m^+ + \sin(m\theta)\,\mathcal O_m^-,\\
\mathcal L_\theta:\;&\mathcal O_m^-\mapsto -\sin(m\theta)\,\mathcal O_m^+ + \cos(m\theta)\,\mathcal O_m^-,
\end{aligned}
\]
so the defects act continuously on local operators even though their fusion is non-group-like [2402.00118].

The same logic underlies the “cosine symmetry” emphasized in survey literature, often written as
\[
\mathcal L_\theta = 2\cos(\theta Q),
\qquad
\mathcal L_\theta\times \mathcal L_{\theta'}
=
\mathcal L_{\theta+\theta'}+\mathcal L_{\theta-\theta'}.
\]
This example established early that continuous non-invertible symmetry exists in \(1+1\)d and that it is naturally produced by orbifolding a theory with continuous \(U(1)\) symmetry [2308.00747].

Recent work systematizes this beyond the \(c=1\) moduli space. In diagonal \(SU(2)_k\) WZW models, continuous non-invertible symmetries arise when a Verlinde line \(\mathcal L_\lambda\) supports a defect current of weight \((1,0)\). The allowed levels are
\[
k=(n+2)(n-1),\qquad \lambda=2n,
\]
that is,
\[
\mathbb K=\{4,10,18,28,40,\dots\}.
\]
For these \(k\), the defect currents form an \(SU(2)\) multiplet and generate continuous non-invertible symmetries. Under the restriction that \(SU(2)\times SU(2)\) be preserved, the examples with
\[
k\in \mathbb K\setminus\{4,10,28\}
\]
are intrinsic in the sense that the non-local currents cannot be made local by any gauging consistent with that global symmetry [2507.22976].

Products of minimal models furnish another infinite source. In \(\mathcal M_m\otimes \overline{\mathcal M_m}\), a non-local \((1,0)\) current appears whenever
\[
h_{r_1,s_1}+h_{r_2,s_2}=1,
\qquad
h_{r,s}=\frac{((m+1)r-ms)^2-1}{4m(m+1)}.
\]
These defect currents generate continuous non-invertible symmetries and, after folding and unfolding, produce new defect conformal manifolds in a single minimal model [2507.22976].

## 4. Lattice realizations and non-invertible gauging

The lattice counterpart is particularly explicit in the qubit-chain realization of the \(\mathrm{Rep}(D_8)\) symmetry generated by the Kennedy–Tasaki transformation. The Hamiltonian
\[
H = h_0 \sum_{j=1}^L X_j + h_1 \sum_{j=1}^L Z_{j-1}Z_{j+1}(1+X_j)
\]
respects the continuous cosine symmetry, implemented by MPO operators \(\mathsf L_\theta\) with the same cosine fusion law as in continuum orbifold CFT. Special values recover discrete substructures:
\[
\mathsf L_0=1+\eta,\qquad \mathsf L_{\pi/2}=D,\qquad \mathsf L_\pi=\eta^e+\eta^o.
\]
Thus a finite fusion-category symmetry sits inside a larger continuous non-invertible family [2503.02925].

This family is not merely decorative. Gauging the non-maximal algebra object
\[
\mathcal A=1\oplus \eta \oplus D
\]
is implemented by the specific cosine element
\[
\mathsf G=\mathsf L_{\pi/4},
\qquad
\mathsf L_{\pi/4}^\dagger\mathsf L_{\pi/4}=1+\eta+D.
\]
The construction introduces two qubits around each link as gauge fields and imposes Gauss-law projectors
\[
P_j
=
\left(\frac{1+\tau_{j-1}^z\sigma_{j-\frac12}^x\tau_j^z}{2}\right)
\left(\frac{1+\sigma_{j-\frac12}^z\,\tau_j^x X_j\,\sigma_{j+\frac12}^z}{2}\right)=1.
\]
These constraints are explicitly not ordinary gauge transformations in disguise; the authors stress that the local operators \(\mathcal G_{j-\frac12}\) do not act as ordinary gauge transformations preserving each Hamiltonian term separately [2503.02925].

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,
\[
D(O)=V_{D\overline D,k}^\dagger
\bigl(R_{D,k}\cdots R_{D,j}\bigr)\,
O\,
\bigl(R_{D,j}^\dagger\cdots R_{D,k}^\dagger\bigr)
V_{D\overline D,k},
\]
which is manifestly of Stinespring form and therefore completely positive [2403.20062]. 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
\[
d\star j^A=\frac{1}{4\pi^2}F\wedge F,
\]
ordinary axial \(U(1)\) is obstructed, but one can define a continuously labeled family of topological charge defects
\[
U_\alpha(\Sigma_3)=
\int[\mathcal D\theta]\,
\exp\left(
i\frac{\alpha}{2}
\int_{\Sigma_3}
\left(
\star j_A-\frac{1}{4\pi^2}(A-d\theta)\wedge dA
\right)
\right),
\]
with \(e^{i\alpha}\in U(1)\). These defects act on charged local operators by
\[
U_\alpha(S^3)\,\mathcal O(x)=e^{\frac{i\alpha q}{2}}\mathcal O(x),
\]
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
\[
S[A,\phi]=\int\left(\frac{1}{2f^2}d\phi\wedge *d\phi+\frac{1}{2e^2}F\wedge *F+ig\,\phi\,F\wedge F\right),
\qquad
g=\frac{1}{4\pi^2}
\]
[2211.09570].

In three dimensions, gauging compact scalar backgrounds can convert anomalies in coupling space into continuous \(2\)-group structures. In the Goldstone model one finds transformation laws such as
\[
B^{(2)}\to B^{(2)}+d\Lambda_B^{(1)}-\frac{1}{2\pi}\lambda_A^{(0)}F_A^{(2)},
\]
which exhibit the \(0\)-form/\(1\)-form mixing characteristic of a continuous \(2\)-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 [2206.14093].

Four-dimensional duality systems provide another, more arithmetic, variant. In Gaillard–Zumino models with classical continuous duality group \(\mathscr G\subset Sp(2n,\mathbb R)\), the usual quantum statement is that only
\[
\mathscr G_{\mathbb Z}=\mathscr G\cap Sp(2n,\mathbb Z)
\]
survives invertibly. A stronger statement is that the much larger rational subgroup
\[
\mathscr G_{\mathbb Q}=\mathscr G\cap Sp(2n,\mathbb Q)
\]
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 [2510.18997]. 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 \(S^1/\mathbb Z_2\) example leaves only the invertible half-shift unbroken once loops are included [2402.00118]. In \(\mathcal N=4\) theories, abelian Maxwell defects can realize continuous bonus-\(U(1)\) rotations on local operators exactly, whereas in nonabelian \(\mathcal N=4\) SYM the corresponding broader family appears only approximately in a large-\(N\), large-\(\lambda\) supergravity regime [2401.05032].

## 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 \(\mathbb Q/\mathbb Z\) are infinite and dense but are not continuous, because irrational angles are not supplied by the half-gauging construction [2308.00747]. Likewise, the Gaillard–Zumino construction realizes \(\mathscr G_{\mathbb Q}\), not the full classical \(\mathscr G\), as exact quantum defects [2510.18997]. 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 \(3+1\)d non-invertible symmetries described by fusion \(3\)-categories, the action on local operators is severely constrained: if there are no nontrivial topological line operators,
\[
\Omega^2 C\cong \mathrm{Vec},
\]
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 \(3+1\)d generalization with genuinely non-invertible local action will need an analogue of the topological-line/gauging-interface mechanism [2603.03438].

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
\[
\delta_{\widetilde\alpha_1,\widetilde\alpha_2}\to \kappa\,\delta(\widetilde\alpha_1-\widetilde\alpha_2),
\]
which leads to different answers depending on \(\kappa\). The noncompact orbifold \(\mathbb R/\mathbb Z_2\) corresponds to \(\kappa=\tfrac12\), while the limit of finite non-invertible gaugings gives \(\kappa=1\). The paper explicitly leaves the intrinsic measure-theoretic definition of continuous non-invertible gauging as an open problem [2606.15732].

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 [2604.14275]. This suggests that continuous non-invertible symmetry may ultimately require a synthesis 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 \(1+1\)d CFT, compact-boson/orbifold models, and ABJ-type anomalous constructions. In higher dimensions, the most robust results either produce continuous \(2\)-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.

Source: https://www.emergentmind.com/topics/continuous-non-invertible-symmetries