Actions, Generalized Charges, and Selection Rules

An invertible symmetry can act by conjugation because its defect has an inverse. A noninvertible topological defect has no such operation. Its action is instead a network move: encircle or cross an operator, deform the defect through a protected region, and resolve every junction and attachment that the move creates. The output may be a linear combination, a projection, an operator in a twisted sector, or an extended operator with an attached defect.

Accordingly, a generalized charge is not usually one phase attached to one operator. It is the sector and response data needed to describe that network action: the multiplet of ordinary and twisted operators, the matrices or intertwiners by which defects act, and the junctions and attachments that make the action well-defined. A selection rule asks whether the full external network admits an invariant functional or compatible junction. In the ordinary Abelian case—and in specially chosen one-dimensional eigensectors more generally—this can reduce to scalar charge arithmetic.

The clean algebra below first uses a finite protected semisimple line sector, where fusion sums and finite matrices are available. The first QFT application then returns to the four-dimensional compact-Maxwell duality wall and its finite one-form surface network. Outside those domains—especially in nonsemisimple theories or for higher-dimensional defects carrying their own QFT—the network statement survives, while a finite matrix or scalar-charge description need not.

Required background. Non-Invertible Topological Defects and Fusion supplies topological deformation, regulated fusion, junction spaces, and the fact that an action is additional data rather than a consequence of a fusion table.

Helpful background. Multiplets, Invariants, and Selection Rules supplies the ordinary singlet-projector comparison. Linking, Braiding, and Framing fixes the orientation, linking, and incidence conventions used in the finite surface example.

A defect action is a network operation

Let $\mathcal D_a$ be a topological defect in a $d$-dimensional QFT. Its label $a$ specifies the defect type, not a number acting on every observable. For a $q$-dimensional operator $\mathcal O_q$, a protected encircling or crossing move defines schematically

$$\mathsf A_a: \mathcal O_q \longmapsto \mathcal O'_q.$$

The prime is consequential. The output can be another operator on the same support, but it can also end a $(q+1)$-dimensional defect or lie in a twisted sector. A closed line can become a line with a surface attachment; a local operator can be exchanged with a disorder operator at the end of a defect. The multiplet on which $\mathsf A_a$ closes must contain all such outputs. Not every abstract charge label need be realized by an operator in a given QFT.

This is the operational content of a generalized $q$-charge in the finite symmetry framework: it labels a multiplet of $q$-dimensional operators closed under the declared network moves. Bhardwaj and Schäfer-Nameki formulate this action and its possible passage into defect-attached sectors in Bhardwaj–Schäfer-Nameki 2025, Definition 3.1 and § 3.1, VOR pp. 30–34, eqs. (110)–(112).

Four pieces of local data must accompany every displayed action:

1. the support, orientation, and allowed deformation region of $\mathcal D_a$;
2. whether the move is an encircling, a crossing, or an endpoint move;
3. the junction morphism used to resolve the new network; and
4. every outgoing attachment or twisted-sector label.

Changing one of these can change the map. A topological action is invariant only while the deformation avoids charged insertions, endpoints, boundaries, and singular backgrounds and preserves any required framing. Topologicality does not make the defect transparent.

Encircling and crossing answer different questions

Encircling keeps the operator inside a closed defect and is the closest analogue of measuring a charge. In a finite protected space it may produce a matrix or, on a one-dimensional eigenspace, a scalar. Crossing transports an operator through a defect. It can change the operator type and can leave an attachment behind. An encircling eigenvalue therefore does not determine a crossing map.

Nor should either operation be written as $U\mathcal O U^{-1}$. There is no inverse defect with which to define that conjugation. For a locally supported operator acted on by a 0-form noninvertible wall, under the hypotheses of Okada and Tachikawa, the induced local action is instead a completely positive quantum operation with a Stinespring-type realization,

$$\mathsf X(\mathcal O) = V^\dagger\,\pi_{\mathsf X\overline{\mathsf X}}(\mathcal O)\,V,$$

rather than a unitary algebra automorphism. This is a precise corrective example, not a universal construction for every QFT defect; higher-form cases are outside that theorem. See Okada–Tachikawa 2024, VOR pp. 191602-1–191602-3, eqs. (1)–(2) and Fig. 2.

Fusion constrains composition without making a group action

Suppose, for this subsection, that the relevant topological lines form a finite semisimple unitary sector. Their regulated fusion is

$$\mathcal D_a\otimes\mathcal D_b \simeq \bigoplus_c N_{ab}^{\ c}\,\mathcal D_c.$$

Closed defect operators on a spatial slice obey the corresponding decategorified relation

$$\widehat{\mathcal D}_a\widehat{\mathcal D}_b = \sum_c N_{ab}^{\ c}\widehat{\mathcal D}_c.$$

Fix the nesting convention so that $b$ acts first and $a$ is the outer loop; this order realizes the written product $a\otimes b$. Reversing the nesting can realize the opposite algebra when fusion is noncommutative. If a finite protected space $V$ is invariant under all these closed-loop operators, then

$$\mathsf L_a := \widehat{\mathcal D}_a\big\rvert_V, \qquad \mathsf L_a\mathsf L_b = \sum_cN_{ab}^{\ c}\mathsf L_c.$$

For crossing or endpoint actions, a chosen fusion-junction basis gives the channel-aware compatibility

$$\mathsf A_a\circ\mathsf A_b \simeq \bigoplus_c\bigoplus_{\alpha=1}^{N_{ab}^{\ c}} \mathsf A_{c,\alpha}.$$

Here $\alpha$ labels the selected junction channel; changing the fusion tree relates these maps by the associator. Only after all outputs are included and the channel labels are consistently contracted does a finite protected space carry matrices for the decategorified fusion algebra. That representation is not automatically faithful, unitary, commutative, or diagonalizable.

The distinction between defect ends, fusion channels, and the induced action is made explicit in Bhardwaj–Schäfer-Nameki 2025, §§ 3.2.2–3.2.3, VOR pp. 39–41, eqs. (123)–(126), Figs. 15–17.

On a common one-dimensional eigenspace the eigenvalues must satisfy

$$\lambda_a\lambda_b = \sum_c N_{ab}^{\ c}\lambda_c.$$

These numbers need not be phases. They are fusion-algebra eigenvalues on a particular sector, not universal charges of the theory. If the fusion algebra is noncommutative, or if the action mixes twisted sectors, there may be no simultaneous scalar description at all.

The Ising line separates eigenvalues from complete action data

The minimal control has an invertible line $\eta$ and a self-dual noninvertible line $\mathcal N$ with

$$\eta^2\simeq\mathbf 1, \qquad \eta\mathcal N\simeq\mathcal N, \qquad \mathcal N^2\simeq\mathbf 1\oplus\eta.$$

This finite line control is reviewed in Schäfer-Nameki 2024, arXiv v2, printed p. 7.

On any common eigenvector, the last relation gives

$$\lambda_{\mathcal N}^{2}=1+\lambda_\eta.$$

Thus an $\eta$-even sector can have $\lambda_{\mathcal N}=\pm\sqrt2$, while an $\eta$-odd sector has $\lambda_{\mathcal N}=0$. The zero does not say that the entire symmetry action is absent: a nonzero map can still land in a twisted or attached sector. The explicit operator realization is deferred until after the first compact-Abelian application.

Generalized charges are sector and response data—not universal eigenvalues

For an ordinary Abelian symmetry, an irreducible charge is a character. For an ordinary non-Abelian symmetry, a charge is already a representation label and an operator belongs to a multiplet. Noninvertible symmetry extends this second pattern rather than the first. The physically usable charge data can include

• the ordinary, twisted, or defect-ending sector in which an operator lives;
• the matrices $\mathsf A_a$ for protected encircling moves;
• crossing and endpoint intertwiners, including their junction indices;
• the attachment left on an extended operator; and
• half-braiding or linking responses when those operations are defined.

None of these entries alone is a complete charge classification. Two defects can act identically on one selected set of operators but differently on other sectors or junctions. Quotienting by the kernel of one representation would therefore discard genuine global defect data.

Defect Hilbert spaces keep twisted sectors visible

In two dimensions, place a topological line $a$ along Euclidean time so that it pierces the spatial circle at one point, then quantize. The result is a defect or twisted Hilbert space $\mathcal H_a$. Under radial quantization, an operator at which that line ends creates a state in $\mathcal H_a$, not in the untwisted space $\mathcal H_{\mathbf 1}$. Fusion and endpoint junctions supply maps among the appropriate defect Hilbert spaces.

The defect-in-time construction and its modified spatial boundary condition are reviewed in Kaidi 2026, arXiv v2, printed p. 7, around eq. (2.22). Twisted-sector reorganization and direct-sum behavior under line fusion appear in Kaidi 2026, §§ 2.9 and 3.1, arXiv v2, printed pp. 26–32, especially eqs. (2.124)–(2.127) and (3.2)–(3.3).

This construction prevents an easy category error: $N_{ab}^{\ c}$ counts fusion-junction channels in the finite semisimple control, whereas $\dim\mathcal H_a$ counts states and is generally infinite in QFT. Neither is a scalar charge. In higher dimensions, the analogous defect state space depends on the transverse geometry and can carry a lower-dimensional QFT, so the notation $\mathcal H_a$ must always come with that geometry.

Selection rules test invariant junctions and allowed attachments

An ordinary selection rule says that a correlator is an invariant covector on the tensor product of its external representations. The network version says the same thing without assuming a group action.

In the finite semisimple two-dimensional control, let $\mathcal M$ denote a correlator or amplitude functional for a fixed set of external multiplets. Wrap a topological line $a$ around all insertions, then sweep it inward and resolve every crossing and junction. Call the resulting action on the external network $\Delta_a$, excluding the final empty outer bubble. Evaluate that separate contractible bubble as $d_a$. Topological deformation then gives

$$\mathcal M\circ\Delta_a=d_a\,\mathcal M.$$

This is an intertwining equation. It is not a universal coproduct formula: $\Delta_a$ depends on the chosen external sectors, junction bases, ordering, and associators. If no compatible functional exists, the amplitude vanishes. If the invariant space has dimension greater than one, symmetry allows that many independent dynamical structures. When $a=g$ is a group-like line, $d_g=1$ and the resolved network factorizes into the usual representation matrices, recovering ordinary singlet invariance.

This displayed equation is the page’s topological sweep derivation. The source ingredients—end and junction spaces, lasso actions, and their fusion and half-braiding resolutions—are developed in Bhardwaj–Schäfer-Nameki 2025, § 4.3, VOR pp. 58–60, eqs. (202)–(212), Figs. 28–34.

The same necessary test can be written without choosing a defect sweep. Give every external leg an orientation, replace a reversed or incoming label by its dual, and fix the cyclic and fusion order. For the resulting charge objects $Q_1,\ldots,Q_n$ in the declared finite semisimple charge category $\mathcal C_{\mathrm{ch}}$, the number of vacuum fusion channels is

$$m_{\mathrm{vac}} = \dim\operatorname{Hom}_{\mathcal C_{\mathrm{ch}}}\!\left( \mathbf 1, Q_1\otimes\cdots\otimes Q_n \right).$$

If $m_{\mathrm{vac}}=0$, the correlator has no topological channel to the vacuum and must vanish. A positive value only permits the correlator; it does not make its dynamical coefficient nonzero. When $m_{\mathrm{vac}}>1$, the allowed answer is a vector of junction structures, with basis changes controlled by the associator.

For extended operators, there is an additional support test. A sweep can leave a line or surface attachment. The corresponding correlator component is allowed only if that attachment ends on another declared operator, reaches an allowed boundary, or is absorbed by a compatible junction. An unmatched attachment is the geometric reason a channel can vanish.

The following table gives a bounded comparison in the declared finite setting. Its last column is part of the rule: seeing one response never licenses the missing action or coherence data.

What replaces scalar charge when the symmetry defect is noninvertible

Question	Ordinary Abelian	Ordinary non-Abelian	Noninvertible network	Required check or stop
Geometric action	Group defect sweeps past the operator	Group defect mixes a multiplet	Encircling, crossing, endpoint, and attachment move	State the support, orientation, and protected deformation domain
Composition	Multiply phases	Multiply representation matrices	Resolve fusion channels and junctions	Fusion coefficients alone omit associators and action data
Charge datum	A character	An irreducible representation and multiplet	A sector or multiplet with network-response data	A scalar exists only on a suitable one-dimensional eigenspace
Correlator test	Total character is trivial	An invariant tensor exists	A compatible invariant junction network exists	Count all allowed resolved channels before inferring a zero
Extended-operator output	The same type with a phase	The same support with a matrix action	May change type or acquire an attachment	Do not discard the attached line or surface
Defect Hilbert space	Optional twisted boundary condition	Twisted multiplet when present	Essential home for defect-ending operators	Do not confuse state dimension with fusion multiplicity
Nonfaithful response	Quotient only a globally trivial kernel	Check all representations	One sector can miss a globally distinct defect	Do not quotient using one action matrix alone

These exact network constraints should also be kept separate from a different perturbative mechanism sometimes called a selection rule without a group action. In the fusion-algebra and hypergroup examples of Kaidi, Tachikawa, and Zhang, the rule is exact at tree level, weakens with loop order, and eventually reduces to an ordinary group rule. That framework is useful but does not replace a topological-defect Ward network; see Kaidi–Tachikawa–Zhang 2024, §§ 2.3.1–2.3.3, VOR pp. 7–12.

First application: the compact-Abelian wall projects and attaches line sectors

Work in four-dimensional Euclidean pure compact $U(1)$ gauge theory on an oriented spin manifold $X$, with no dynamical electric charges or monopoles. Let $\mathfrak a$ be the compact connection, $f=\mathrm d\mathfrak a$ locally, and

$$\frac{1}{2\pi}\int_{\Sigma_2}f\in\mathbb Z$$

on every closed oriented two-cycle. Use

$$\begin{aligned} S_E[\mathfrak a] &= \frac{1}{2e^2}\int_X f\wedge\star f +\frac{i\theta}{8\pi^2}\int_X f\wedge f,\\ \tau&=\frac{\theta}{2\pi}+\frac{2\pi i}{e^2}. \end{aligned}$$

Fix $N\geq2$ and $\tau=iN$. This is the fixed point of gauging the electric $\mathbb Z_N^{(1)}$ subgroup followed by electromagnetic $S$-duality; it is not a fixed point of $S$ alone. Half-gauging on a separating region, or cutting along a general two-sided three-manifold and gluing with the duality kernel, produces the topological wall $\mathcal D$ used below. Its construction is taken as input here.

The finite network contains Wilson lines

$$W_q(C)=\exp\!\left(iq\oint_C\mathfrak a\right), \qquad q\in\mathbb Z,$$

topological electric surfaces $\eta_k(\Sigma)$ with $k\in\mathbb Z_N$, and selected two-in/one-out surface-junction operators

$$I_{k,\ell}^{\ m}(J): \eta_k\otimes\eta_\ell\longrightarrow\eta_m, \qquad k+\ell-m=0\pmod N.$$

In four dimensions $W_q$ and $J$ are one-dimensional lines, $\eta_k$ is a two-dimensional codimension-two surface, and $\mathcal D$ is a three-dimensional codimension-one wall. The congruence is a necessary signed incidence rule; it neither constructs nor normalizes the junction space.

A linking character becomes a projector after condensation

For closed, oriented, disjoint $C$ and $\Sigma$ in a linking ball, with all other insertions $\mathcal X$ outside the deformation sweep,

$$\begin{aligned} \big\langle\eta_k(\Sigma)W_q(C)\,\mathcal X\big\rangle &= \chi_q(k)^{\operatorname{Lk}(\Sigma,C)} \big\langle W_q(C)\,\mathcal X\big\rangle,\\ \chi_q(k)&=\exp\!\left(\frac{2\pi i}{N}kq\right). \end{aligned}$$

The invertible surface therefore measures the Wilson class $[q]_N$ by an ordinary character. The noninvertible operation appears when the reversed duality wall is composed with the wall:

$$\overline{\mathcal D}\otimes\mathcal D \simeq \mathcal C_0.$$

This fusion order is given in Choi et al. 2023, § 3.1, arXiv v2, printed p. 19, eq. (3.4), Open PDF.

On the connected transverse wall geometry $M^3=S^2\times S^1$, the relevant part of the condensation wall is

$$\mathcal C_0(M^3) = \frac1N\sum_{k=0}^{N-1}\eta_k(S^2).$$

If the $S^2$ links $W_q$ once and there is no compensating attachment, its action is the finite Fourier projector

$$\begin{aligned} \mathsf P_N(q) &= \frac1N\sum_{k=0}^{N-1} \exp\!\left(\frac{2\pi i}{N}kq\right)\\ &= \delta_{[q]_N,0}. \end{aligned}$$

Thus the unattached component survives precisely for $q=0\pmod N$. This is a projection by a topological network, not dynamical screening: the pure Maxwell theory has no charged endpoint on which $W_q$ can terminate. On a general closed wall support, $\mathcal C_0$ sums all $H_2(M^3;\mathbb Z_N)$ sectors with the finite-gauge normalization $1/\lvert H^0(M^3;\mathbb Z_N)\rvert$ and a local-counterterm convention. The one-line character sum is a controlled transverse action, not the full global definition of the condensation wall.

The general condensation-wall sum, its normalization, and the Euler- counterterm ambiguity are given in Choi et al. 2023, § 2.1, arXiv v2, printed p. 10, eq. (2.5), Open PDF.

Crossing the wall produces an attachment, not an eigenvalue

The same wall acts differently by crossing. The minimal Wilson line $W_1$ transported through $\mathcal D$ becomes an improperly quantized ’t Hooft line whose failure to be genuine is repaired by an attached $\eta$ surface. More generally, a Wilson line with $q\not\equiv0\pmod N$ has a nontrivial residue and requires the corresponding attachment; the quotient-neutral lines do not. The output therefore cannot be summarized by a number multiplying $W_q$. Its charge data include the magnetic line type, the attached surface, and the surface-to-wall junction at the crossing.

The charge dependence and compact-connection rescaling are displayed in Kaidi 2026, arXiv v2, printed pp. 74–75, eqs. (4.21)–(4.25).

The finite surface can itself be absorbed on the wall through a selected higher-codimension junction,

$$\eta_k\otimes\mathcal D\simeq\mathcal D, \qquad \mathcal D\otimes\eta_k\simeq\mathcal D.$$

These relations do not say $\eta_k\simeq\mathbf 1$ in the ambient four- dimensional theory. They state that its action on this wall has a chosen absorption channel. Together, the linking character, Fourier projection, line-to-attached-line crossing, and absorption junction give the four different meanings that the word “action” can hide.

Choi and collaborators define the Maxwell model and coupling convention in Choi et al. 2023, § 6.1, arXiv v2, printed pp. 31–32, eqs. (6.1)–(6.9), Open PDF. The combined gauging–duality fixed point, wall crossing, attachment, and reversed-wall composition are given in Choi et al. 2023, arXiv v2, printed pp. 35–36, eqs. (6.22)–(6.28), Open PDF. The surface-absorption morphism is developed in Choi et al. 2023, §§ 4–4.1, arXiv v2, printed pp. 23–24, eqs. (4.1)–(4.3), Open PDF. The example depends on pure Maxwell symmetries, the declared global data, and the combined fixed point; charged matter, monopoles, a physical boundary, or a different coupling requires a new action analysis.

An Ising control makes the twisted sector explicit

The earlier fusion polynomial has a concrete two-dimensional operator realization. In the Ising multiplet, the untwisted order operator $\sigma$ and the $\eta$-twisted disorder operator $\mu$ are paired. A closed $\mathcal N$ loop around $\sigma$ has a vanishing no-attachment channel. In the junction normalization of Kaidi, a resolved channel with an outgoing $\eta$ line instead gives

$$\mathcal N:\sigma\longmapsto\sqrt2\,\mu.$$

By contrast, the same-sector loop action on the energy operator is the scalar $-\sqrt2$. Thus the zero is an absent junction channel, not an ordinary zero-valued charge. The example displays a fusion-polynomial eigenvalue, a vanishing component, and a nonzero sector-changing map without identifying them.

The explicit networks and normalization are shown in Kaidi 2026, arXiv v2, printed pp. 65–67, Figs. 7–9. The order–disorder multiplet and its existence statement appear in Bhardwaj–Schäfer-Nameki 2025, Example 4.6 and Statement 4.4, VOR pp. 69–72, eqs. (276)–(278).

What the action establishes—and what it cannot classify

When the stated topology, sectors, and junctions are fixed, the analysis can establish

1. which protected operator space is closed under the defect network;
2. how fusion constrains the composition of action maps;
3. which correlator components admit invariant junctions; and
4. when an extended operator is projected, mixed, or forced to carry an attachment.

It does not classify every generalized charge from a fusion table, prove that all abstract charge labels are realized, or determine an anomaly, a phase, a renormalization-group endpoint, or whether the symmetry can be gauged. One matrix representation can have a kernel even when the global network is faithful, and an accidentally invertible matrix does not give the underlying defect a fusion inverse.

Evidence checked through 9 August 2026 supports the finite protected framework and the bounded four-dimensional examples used here. The current review literature emphasizes that two-dimensional noninvertible actions are far better understood than a general higher-dimensional classification; see Kaidi 2026, arXiv v2, printed pp. 68–69. The completely positive local-action result and the loop-order-dependent hypergroup examples are independent corrections to overly broad representation language. They do not imply that every noninvertible action is a quantum channel or that every network selection rule is perturbative. No correction, retraction, or withdrawal was located for the cited works through that cutoff.

Common pitfalls

Using conjugation without an inverse. Writing $\mathcal D\mathcal O\mathcal D^{-1}$ assumes the very fusion inverse that a noninvertible defect lacks. Specify the encircling or crossing network and its junction resolution instead.

Calling every eigenvalue a charge. A fusion eigenvalue is meaningful on a declared invariant eigenspace. Other operators can mix, enter twisted sectors, or require attachments, so that eigenvalue is not a complete label.

Confusing projection with screening. A finite defect sum can remove an unattached correlator component even when no dynamical endpoint exists. Screening is a statement about genuine line classes and endpoints, not about the value of a projector.

Dropping an attachment after a crossing. The attached line or surface is part of the output operator. Removing it can change a gauge-invariant operator into a nongenuine one and invalidate the selection rule.

Equating fusion multiplicity with a defect Hilbert-space dimension. A finite $N_{ab}^{\ c}$ counts chosen fusion-junction channels in the controlled semisimple setting. A QFT Hilbert space on a defect background is a different object and is usually infinite-dimensional.

Promoting a perturbative rule to an exact topological one. A tree-level fusion-algebra rule can weaken at loops. An exact topological selection rule instead requires an exact defect and the full protected network.

Check your understanding

1. Solve the Ising fusion polynomial

On a common eigenspace, let $\lambda_\eta=+1$. What values of $\lambda_{\mathcal N}$ are allowed by $\mathcal N^2\simeq\mathbf 1\oplus\eta$? What changes when $\lambda_\eta=-1$?

Checked answer

The relation gives $\lambda_{\mathcal N}^2=1+\lambda_\eta$. For $\lambda_\eta=+1$, the two scalar solutions are $\lambda_{\mathcal N}=\pm\sqrt2$. For $\lambda_\eta=-1$, the scalar response is zero. This conclusion applies only on a common eigenspace; it does not rule out a nonzero junction map into a twisted sector.

2. Recover the ordinary selection rule

Take a group-like defect $g$ in the network equation $\mathcal M\circ\Delta_g=d_g\mathcal M$. Show how the usual invariant-tensor condition follows.

Checked answer

A group-like defect has $d_g=1$. Its resolved action on external operators is the tensor product of their representation matrices, so $\Delta_g=R_1(g)\otimes\cdots\otimes R_n(g)$. The equation becomes

$$\mathcal M\circ \bigl(R_1(g)\otimes\cdots\otimes R_n(g)\bigr) = \mathcal M,$$

which is exactly the statement that $\mathcal M$ is an invariant covector.

3. Evaluate the finite projector

For $N=4$, evaluate $\mathsf P_4(q)$ for $q=0,1,2,3,4$.

Checked answer

The four roots of unity sum to zero unless $q$ is divisible by four. Hence

$$\mathsf P_4(0)=1, \quad \mathsf P_4(1)=\mathsf P_4(2)=\mathsf P_4(3)=0, \quad \mathsf P_4(4)=1.$$

The result projects the unattached character sector; it does not provide a dynamical endpoint for any Wilson line.

4. Distinguish projection from screening

In the pure Maxwell example, the condensation action removes the unattached $q=1$ component. Is $W_1$ screened?

Checked answer

No. Screening would require a dynamical charge-one endpoint or an equivalent relation in the genuine-line quotient. The model contains no dynamical electric matter. The zero instead comes from averaging the nontrivial $\mathbb Z_N$ character over the finite surface network.

5. Track the support after crossing the wall

List the supports involved when $W_1$ crosses $\mathcal D$ and becomes an improperly quantized ’t Hooft line with an attached $\eta$ surface.

Checked answer

The incoming and outgoing charged objects are one-dimensional lines. The repairing $\eta$ defect is a two-dimensional surface whose boundary or junction meets the outgoing line. The crossing occurs on the three-dimensional wall $\mathcal D$, and the surface can meet or be absorbed on that wall only through a selected higher-codimension junction. Omitting the surface changes the operator.

6. Diagnose a nonfaithful action

Two globally distinct defects have identical matrices on one finite operator subspace. May they be identified as the same symmetry defect?

Checked answer

Not from that subspace. They may differ on twisted sectors, extended operators, junctions, or other Hilbert spaces. One may quotient only a subgroup or ideal shown to act trivially on the complete QFT network, not the kernel of one selected representation.

Continue to anomalies, constructions, and mathematical structure

Continue next to Anomalies, RG Constraints, and Framework Limits to ask which complete actions can be gauged and which identified data survive a flow. Constructions from Gauging, Duality, and Condensation develops the half-gauging and condensation operations that were inputs to the Maxwell example.

For the categorical formulation of charge multiplets and module actions, continue to Categorical Symmetries, Higher Representations, and Charges and Generalized-Symmetry Sectors, Selection Rules, and Reconstruction. Interfaces, Folding, and Fusion specializes defect actions to two-dimensional conformal interfaces. Those frameworks add categorical or conformal data rather than turning the bounded matrix and Maxwell calculations here into universal classifications.

References

• 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 Condensation, Duality, and Triality Defects in 3+1 Dimensions.” Communications in Mathematical Physics 402, no. 1 (2023): 489–542. DOI. Open PDF, arXiv v2.
• Kaidi, Justin. “Introduction to Generalized Symmetries.” KYUSHU-HET-354; arXiv:2603.08798v2 [hep-th], revised 6 July 2026. DOI. Open PDF.
• Kaidi, Justin, Yuji Tachikawa, and Hao Y. Zhang. “On a Class of Selection Rules without Group Actions in Field Theory and String Theory.” SciPost Physics 17, no. 6 (2024): 169. DOI.
• 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.