Non-Invertible Symmetries

A non-invertible symmetry is implemented by a topological defect network that has no two-sided fusion inverse. Branching is a useful witness in a finite semisimple setting, but it is not the definition by itself: one must first establish topological deformation, then specify protected fusion, junction and coherence data, and an action on operators or sectors. Gaugeability, anomaly, and infrared obstruction are later and independent verdicts.

Choose the fusion route when the question is whether a defect is genuinely non-invertible. Choose the action route when ordinary charge eigenvalues or group-character selection rules have stopped working. Choose the construction route when gauging, duality, or condensation is supposed to produce the defect, and continue to the anomaly/RG route only after the complete network and background family are fixed. A compact-Maxwell duality wall supplies one controlled four-dimensional thread through all four pages.

Helpful background. Fusion, Junctions, and Endpoints supplies regulated collisions, channel spaces, junctions, and associators. What Is an Anomaly? supplies the counterterm-quotiented obstruction test used at the end of the chapter. Neither is a hard prerequisite merely to choose a route here.

The specialist evidence supporting the operator, construction, anomaly, and infrared-obstruction claims summarized here was checked through 10 August 2026. The dated cutoff matters because the general higher-dimensional and nonsemisimple frameworks are still developing; it does not alter the bounded Maxwell and finite semisimple calculations stated below.

Parent volume. Symmetry and Gauge Structure

Jump to: separate the first verdicts · choose a route · follow the Maxwell wall · open the exact guide · review the chapter

Topologicality and two-sided fusion decide the first verdict

Let $\mathcal D_a(M^{d-1})$ be a codimension-one defect in an oriented Euclidean $d$-manifold. If $M_0$ and $M_1$ are related by a deformation that crosses no insertion, endpoint, junction, physical boundary, singular background, or framing obstruction, topologicality means

$$\left\langle\mathcal D_a(M_0)\mathcal X\right\rangle = \left\langle\mathcal D_a(M_1)\mathcal X\right\rangle .$$

This conservation property does not decide invertibility. In a controlled finite semisimple line sector, protected fusion takes the form

$$\mathcal D_a\otimes\mathcal D_b \simeq \bigoplus_c N_{ab}^{\ c}\mathcal D_c, \qquad N_{ab}^{\ c}\in\mathbb Z_{\geq0}.$$

Invertibility requires one defect $b$ and chosen equivalences in both orders,

$$\mathcal D_a\otimes\mathcal D_b\simeq\mathbf1, \qquad \mathcal D_b\otimes\mathcal D_a\simeq\mathbf1.$$

Orientation reversal and categorical duality are separate notions. Within a rigid simple-object control, an additional channel in

$$\mathcal D_a\otimes\mathcal D_{\bar a} \simeq \mathbf1\oplus(\text{additional channels})$$

decisively obstructs inversion. Outside this domain, higher-dimensional wall fusion can carry topology-, network-, or TQFT-valued coefficients rather than a finite direct sum. These distinctions and the conditional fusion formulas are reviewed in Schäfer-Nameki 2024, arXiv v2, printed pp. 5–11, eqs. (1.1)–(1.9) and figs. 2–4, PDF.

The shared diagram places deformation, inversion, branching, and the missing junction/coherence data side by side. Read the arrows as distinct operations, not as time evolution.

Topological deformability supplies conservation, while fusion decides invertibility. In the finite semisimple line control, a group-like defect has a two-sided identity channel and no other output; a noninvertible defect can fuse with its orientation dual into the identity plus an additional same-support channel. Multiplicities count junction spaces only under the stated hypotheses, associators compare resolved networks, and a scalar dimension is conditional. The diagram is an original monochrome schematic, not to scale; arrows denote deformation, fusion, or comparison rather than time evolution, and higher-dimensional wall fusion may require TQFT- or network-valued data instead of the displayed finite sum.

The same relationships remain available without the figure:

Operational tests for group-like and noninvertible defect fusion

Test	Required data	Group-like case	Noninvertible conclusion or stop
Deformation	Support, orientation, framing, backgrounds, and forbidden crossings	Correlators are unchanged in the allowed sweep	Topologicality is still required; branching alone is not a symmetry
Fusion output	A regulated collision and closed output sector	One group-law output	Several same-support channels, multiplicity, or more general wall data may appear
Two-sided inverse	Left and right fusion equivalences with the unit	An inverse exists in both orders	No such defect exists; this is the decisive definition
Orientation and dual	Reversal plus evaluation, coevaluation, and snake identities	The dual is also the inverse	A dual can exist while extra fusion channels obstruct inversion
Multiplicity and junction	Typed junction spaces and chosen basis operators	A selected one-dimensional fusion equivalence implements multiplication	An allowed coefficient does not construct or normalize a junction
Associativity	Comparison maps for two resolved fusion trees	Coherent comparison data remain required and may carry an anomaly	A fusion ring without coherent comparison maps is incomplete
Action	Crossing or surrounding maps on operators and sectors	Often an invertible representation or automorphism	May mix sectors, attach defects, or act as a quantum operation
Dimension	A finite semisimple rigid setting and a chosen dimension notion	A simple invertible object has dimension one	Dimension greater than one diagnoses noninvertibility only in that setting
Higher-dimensional ceiling	Wall topology and lower-dimensional worldvolume theories	A group-like wall still cancels to the identity wall	Coefficients can be surface networks or TQFT data, not integers

The multiplicity is the dimension of a typed junction space only under the finite semisimple hypotheses,

$$\mathcal V_{ab}^{\ c} = \operatorname{Hom}(\mathcal D_a\otimes\mathcal D_b,\mathcal D_c), \qquad N_{ab}^{\ c}=\dim\mathcal V_{ab}^{\ c}.$$

It does not choose or normalize a junction vector. Nor does the fusion ring determine the associator, action, anomaly, or physical phase.

Choose a route from the missing data

The route is determined by the first datum that has not yet been established. Chapter order is a guide, not a substitute for each leaf’s hard preparation.

Three routes from a defect question to a controlled noninvertible-symmetry verdict

Starting question	Route	Preparation	Capability at the end
How do I prove that a topological defect is non-invertible?	Topological defects and fusion	Fusion, Junctions, and Endpoints plus the group-like higher-form comparator	A deformation domain, protected fusion law, and two-sided inverse verdict with framework limits
What replaces charge eigenvalues and ordinary selection rules?	Fusion → actions, generalized charges, and selection rules	The fusion page; multiplets and linking are useful but not hard inputs	A network action on sectors, invariant-junction test, and projection-versus-attachment diagnosis
How is the defect constructed, and what can it constrain?	For construction: fusion + finite gauging → gauging, duality, and condensation. For anomaly/RG: actions + the obstruction test → anomalies and RG limits	An actual quotient/interface construction for the first branch; a complete exact network and background map for the second	A bounded construction or obstruction, with its dimensional and categorical assumptions explicit

A non-invertible action is a network operation

An ordinary group symmetry often acts on a simple sector by one character. A non-invertible defect can instead mix sectors, project components, or turn a genuine operator into a defect-attached one. In a finite semisimple line sector, let a protected finite-dimensional space be invariant under closed defect loops and fix their nesting order. The fusion algebra then constrains the corresponding loop operators,

$$\mathsf L_a\mathsf L_b = \sum_c N_{ab}^{\ c}\mathsf L_c,$$

but scalar eigenvalues exist only in suitable one-dimensional eigensectors. For compatible oriented charge objects $Q_i$ in a declared finite semisimple charge category $\mathcal C_{\mathrm{ch}}$, fix the cyclic or fusion order and replace every incoming or orientation-reversed label by its dual. The robust vacuum-channel multiplicity is then

$$m_{\mathrm{vac}} = \dim\operatorname{Hom}_{\mathcal C_{\mathrm{ch}}} \!\left(\mathbf1,Q_1\otimes\cdots\otimes Q_n\right).$$

If $m_{\mathrm{vac}}=0$, the protected channel vanishes; if it is positive, the channel is merely allowed and still needs a chosen junction vector and dynamics. In a local operator-algebra setting satisfying its own hypotheses, a non-invertible action can be a completely positive quantum operation rather than unitary conjugation Okada and Tachikawa 2024, version of record, pp. 191602-1–191602-4, especially eqs. (1)–(2), PDF. The broader network and generalized charge viewpoint is developed in Bhardwaj and Schäfer-Nameki 2025, §§ 3.2.2–3.2.3, version-of-record pp. 39–41, eqs. (123)–(126), and § 4.3, pp. 58–60, eqs. (202)–(212).

One compact-Maxwell wall threads the chapter

The shared four-dimensional model is deliberately narrow: Euclidean pure compact $U(1)$ gauge theory on an oriented spin manifold, with no dynamical electric charges or monopoles, exact electric $\mathbb Z_N^{(1)}$, and the magnetic spectator background held trivial during electric gauging. Write

$$S_E= \frac{1}{2e^2}\int f\wedge\star f +\frac{i\theta}{8\pi^2}\int f\wedge f, \qquad \tau=\frac{\theta}{2\pi}+\frac{2\pi i}{e^2}.$$

Gauging the electric $\mathbb Z_N^{(1)}$ changes the compact presentation by $\tau\mapsto\tau/N^2$. At the special point

$$\tau=iN, \qquad \theta=0, \qquad e^2=\frac{2\pi}{N},$$

gauging sends $iN$ to $i/N$ and electromagnetic $S$ returns it to $iN$. The fixed operation is therefore gauging followed by $S$, not $S$ alone. Half-gauging followed by this background-compatible equivalence produces an endodefect $\mathcal D$. A local wall representative is

$$S_{\mathcal D} = \frac{iN}{2\pi}\int_{M^3}a_L\wedge\mathrm d a_R,$$

but compact sectors, measures, global line and flux lattices, counterterms, and wall junctions remain part of its definition.

The opposite orientations fuse to the condensation wall

$$\bar{\mathcal D}\otimes\mathcal D \simeq \mathcal D\otimes\bar{\mathcal D} \simeq \mathcal C_0,$$

whose worldvolume sum is not the identity. On connected $M^3=S^2\times S^1$,

$$\mathcal C_0 = \frac{1}{N}\sum_{k=0}^{N-1}\eta_k(S^2).$$

The coefficient $1/N$ is a gauging normalization, not an integer fusion multiplicity. The transverse action on an unattached Wilson charge is the Fourier projector

$$\mathsf P_N(q) = \frac{1}{N}\sum_{k=0}^{N-1}e^{2\pi i kq/N} = \delta_{[q]_N,0}.$$

This is projection, not screening: the model contains no charged endpoint. A minimally charged Wilson line crossing the wall instead emerges with an attached symmetry surface. Same-orientation fusion also remembers charge conjugation,

$$\mathcal D\otimes\mathcal D \simeq \mathcal U_C\otimes\mathcal C_0.$$

The construction, global normalization, surface junctions, and orientation-sensitive fusion are derived in Choi, Córdova, Hsin, Lam, and Shao 2023, arXiv v2, printed pp. 9–10, 17–20, 23–24, and § 6.1, pp. 31–36, especially eqs. (2.5)–(2.9), (3.1)–(3.7), (4.1)–(4.3), and (6.1)–(6.9), PDF.

The thread supplies the chapter’s minimum evidence package: an explicitly topological fusion network, a projected and attached line action, an actual gauging-plus-duality construction, and a complete-network obstruction test. It does not make the Maxwell wall representative of every dimension or nonsemisimple symmetry.

Construction and anomaly are separate gates

A half-gauging interface

$$\mathcal I_A:\mathcal T\longrightarrow\mathcal T/A$$

becomes an internal symmetry defect only after an actual background-compatible equivalence

$$\Phi:\mathcal T/A\xrightarrow{\simeq}\mathcal T, \qquad \mathcal D_\Phi=\Phi\circ\mathcal I_A$$

has been supplied. Equality of one coupling or partition number does not prove equivalence of the absolute theories. Nor does an independently constructed nonidentity condensation wall automatically prove that the wall itself is non-invertible; it must face its own two-sided inverse test.

An anomaly is later still. For a complete defect mesh $\mathbb B$, a network move can leave an obstruction after permitted local counterterms and junction redefinitions. Matching that class under RG requires the entire network—not only $N_{ab}^{\ c}$—to be transported: associators, junctions, attachments, measures, global sectors, and the self-gauging equivalence all matter.

The final leaf tests these statements for $N=3$ on a closed smooth simply connected spin four-manifold. A candidate short-range-entangled response labelled by $r\in\mathbb Z_3$ would have to satisfy

$$4r^2\equiv-1\pmod 3,$$

which has no solution. Under the stated exact-network hypotheses, this rules out a unique symmetry-preserving short-range-entangled infrared phase. It does not prove a particular gapless phase, and the stronger exclusion of a unique-local-vacuum TQFT requires additional theorem hypotheses. The arithmetic obstruction and its limits are developed in Choi et al. 2022, arXiv v3, introduction and § 3 opening, printed pp. 4–6 and 14–17, especially eqs. (1.4), (3.1), (3.2), and (3.6), PDF and Apte, Córdova, and Lam 2023, § III.B, arXiv v1, printed pp. 8–9, Theorem 5 and proof sketch, PDF.

Check your preparation

Use these questions as repair cues, not as a score.

Repair the first missing capability before entering a technical route

Can you do this?	Ready	If unsure	Repair route
Separate a regulated fusion collision from a chosen junction and associator	Enter the fusion page	Review typed channel spaces and resolved fusion trees	Fusion, Junctions, and Endpoints
Distinguish an orientation reverse, a dual, and a two-sided fusion inverse	Use the branching test	Check both fusion orders and the evaluation/coevaluation data	Fusion without a two-sided inverse
Promote a finite background to a dynamical field with the correct global sectors and measure	Enter the construction page	Repair finite gauging before using a half-gauging interface	Gauging Continuous and Finite Symmetries
Test a network phase modulo admissible counterterms and junction redefinitions	Enter the anomaly/RG page	Repair the obstruction test before making an infrared claim	What Is an Anomaly?

Exact chapter guide

The four pages appear once, in manifest and sidebar order. Their hard preparation is explicit; a later page is not automatically required merely because it follows in the list.

Four pages from fusion diagnosis to construction and infrared obstruction

Page	Question and capability	Required background	Stop condition
Non-Invertible Topological Defects and Fusion	Establish topologicality, protected fusion, and the absence of a two-sided inverse; separate orientation duals, junctions, associators, and conditional dimensions	Fusion, Junctions, and Endpoints; Higher-Form Symmetry from Operators and Linking	Finite scalar fusion data do not classify a general higher-dimensional wall
Actions, Generalized Charges, and Selection Rules	Replace universal charge eigenvalues by crossing, mixing, projection, attachment, defect Hilbert spaces, and invariant-junction tests	Non-Invertible Topological Defects and Fusion	A scalar eigenvalue or nonzero vacuum multiplicity is not a universal action or a dynamical amplitude
Constructions from Gauging, Duality, and Condensation	Build a half-gauging interface, supply an equivalence back to the theory, and track condensation normalization and new sectors	Non-Invertible Topological Defects and Fusion; Gauging Continuous and Finite Symmetries	An interface is not an internal symmetry defect until the absolute-theory equivalence is proved
Anomalies, RG Constraints, and Framework Limits	Separate gaugeability from infrared exclusion, transport a complete defect network under RG, and state dimensional and categorical hypotheses	Actions, Generalized Charges, and Selection Rules; What Is an Anomaly?	Matching or a congruence obstruction does not construct a unique infrared phase

Keep the four obstruction questions separate

Nearby observations that answer different questions

Observed fact	What it does not imply	Additional test or scope
A defect has a branching fusion formula	It is topological or defines a symmetry action	Check the deformation domain, junctions, associators, and action
A defect has an orientation reverse or categorical dual	It has a two-sided fusion inverse	Fuse in both orders and inspect every output channel
A fusion multiplicity is nonzero	A canonical junction or nonzero physical amplitude exists	Choose and normalize the junction, then apply dynamics and other selection rules
An unattached component is projected out	The complete action vanishes or a particle screened the operator	Include twisted and attached sectors; distinguish group averaging from endpoints
Half-gauging produces an interface	The interface is an internal symmetry defect	Supply a background-compatible equivalence between the quotient and the original absolute theory
A condensation wall is not the identity	The condensation wall itself is non-invertible	Apply the wall's own two-sided inverse test
The complete network is anomalous	The theory is nonunitary, inconsistent as a relative system, or forced into one infrared phase	Separate standalone gauging, inflow, and RG matching verdicts
A self-duality congruence has or lacks a solution	A QFT has been constructed or all gapped phases have been classified	State the exact dimension, spin/global data, faithfulness, and phase class tested
A finite semisimple line formula is valid	The same scalar language applies to continuous, nonsemisimple, or higher-dimensional walls	Allow topology-, network-, or field-theory-valued coefficients and declare the target framework

What the chapter establishes—and where it stops

The chapter establishes a sequence of typed questions. First, topologicality makes a defect deformable on a declared domain. Second, protected fusion and the two-sided inverse test decide noninvertibility. Third, actions, junctions, and associators make the symmetry network operational. Fourth, a concrete gauging/duality/condensation construction can realize such a network. Finally, anomaly and RG arguments constrain complete realizations under explicitly identified backgrounds.

The scalar fusion coefficients, Hom-space dimensions, and positive quantum dimensions used as controls belong to finite semisimple settings. The local quantum-operation theorem has its own zero-form/local-operator hypotheses. Fiber-functor criteria are mature in finite semisimple $1+1$-dimensional fusion-category symmetry, not as a universal higher-dimensional theorem Thorngren and Wang 2019, § 2.3, printed pp. 11–16, Theorem 1 and eqs. (2.11)–(2.17), PDF. General continuous, nonsemisimple, approximate, and higher-dimensional cases have no universal scalar anomaly formula or complete classification in this chapter.

Review the chapter

The prompts below cover the chapter’s distinct capabilities. Each names the owner, a success invariant, and a repair route; they are not a separate formal assessment system.

Eight ways to demonstrate a controlled noninvertible-symmetry analysis

Mode and task	Owner pages	Successful response and invariant	Characteristic repair
Retrieval — state the definition of a non-invertible topological defect	Fusion diagnosis	Separates topologicality from the absence of any two-sided fusion inverse	Recheck both fusion orders if branching is used as the definition
Explanation — explain why an orientation reverse need not be an inverse	Fusion and duality	Names evaluation/coevaluation and shows that extra same-support channels prevent cancellation	Return to Fusion, Junctions, and Endpoints if duality data are omitted
Derivation check — recover the Maxwell-wall projector on Wilson charge q	Compact-Abelian action	Uses character orthogonality to obtain one for q=0 mod N and zero otherwise; calls this projection, not screening	Repair the finite sum at Gauging Continuous and Finite Symmetries
Representation change — translate from scalar charge language to a network action	Actions and generalized charges	Allows mixing, twisted sectors, projection, and attachments while preserving the fusion-composition relation	Revisit the network operation if every simple object is assigned a scalar eigenvalue
Comparison — distinguish full gauging, half-gauging, and condensation	Construction page	States the source and target of the interface, the required quotient-to-original equivalence, and the worldvolume normalization	Repair at the interface-to-endodefect gate
Transfer — test the Ising fusion rule 𝒩2 ≃ 1 ⊕ η	Conditional dimension control	Finds d𝒩=√2 only after declaring a finite semisimple positive dimension function; does not export it to a generic wall	Return to the framework ceiling if scalar dimension is treated as universal
Failure diagnosis — locate the error in “the N=3 congruence has no solution, so the infrared is uniquely gapless”	Anomalies and RG limits	Restricts the exclusion to the stated symmetry-preserving short-range-entangled class and lists other allowed realizations	Repair at What an infrared theory may do
Synthesis — decide whether a proposed duality wall defines an exact non-invertible symmetry with an RG obstruction	All four chapter pages	Checks topologicality, two-sided fusion, junction/coherence/action data, the gauging-plus-equivalence construction, counterterm quotient, and complete-network transport under RG	Return to the route matrix and repair the first absent datum rather than inferring later verdicts

Continue from non-invertible symmetry data

• To compare the result with an ordinary group-like higher-form symmetry, return to Higher-Form Symmetry from Operators and Linking.
• To repair the global sum behind a half-gauging construction, continue to Gauging Continuous and Finite Symmetries.
• To classify an obstruction before using it in RG flow, continue to What Is an Anomaly?.
• To place defect fusion, boundary choice, gauging, and anomaly in one one-higher-dimensional system, continue to Symmetry TFT Encoding of Symmetry, Anomaly, and Gauging.

References

• Apte, Anuj, Clay Córdova, and Ho Tat Lam. “Obstructions to Gapped Phases from Non-Invertible Symmetries.” Physical Review B 108, no. 4 (2023): 045134. DOI. Open PDF, arXiv v1.
• Bhardwaj, Lakshya, and Sakura Schäfer-Nameki. “Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT.” SciPost Physics 19, no. 4 (2025): 098. DOI. Open PDF, arXiv v3.
• Choi, Yichul, Clay Córdova, Po-Shen Hsin, Ho Tat Lam, and Shu-Heng Shao. “Non-Invertible Duality Defects in 3+1 Dimensions.” Physical Review D 105, no. 12 (2022): 125016. DOI. Open PDF, arXiv v3.
• Choi, Yichul, Clay Córdova, Po-Shen Hsin, Ho Tat Lam, and Shu-Heng Shao. “Non-Invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions.” Communications in Mathematical Physics 402, no. 1 (2023): 489–542. DOI. Open PDF, arXiv v2.
• Okada, Masaki, and Yuji Tachikawa. “Noninvertible Symmetries Act Locally by Quantum Operations.” Physical Review Letters 133, no. 19 (2024): 191602. DOI. Open PDF, arXiv v2.
• Schäfer-Nameki, Sakura. “ICTP Lectures on (Non-)Invertible Generalized Symmetries.” Physics Reports 1063 (2024): 1–55. DOI. Open PDF, arXiv v2.
• Thorngren, Ryan, and Yifan Wang. “Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases.” arXiv:1912.02817v1 [hep-th], 5 December 2019. Stable record. Open PDF.