Anomalies, RG Constraints, and Framework Limits

A non-invertible defect is not automatically anomalous. Four different questions must be answered separately: whether the defect has a two-sided fusion inverse, whether its complete background or defect network can be gauged consistently, whether that network has a counterterm-invariant anomaly, and whether preserved network data rule out a proposed infrared theory. Fusion coefficients alone answer none of the last three questions.

This page gives operational tests for finite protected defect networks and then applies them to a four-dimensional compact-Maxwell duality wall. The clean classification statements are deliberately restricted to declared finite semisimple or finite-Abelian settings. General higher-dimensional, nonsemisimple, continuous, and approximate non-invertible symmetries do not yet have one universal background-field or anomaly-classification framework.

Required background. Actions, Generalized Charges, and Selection Rules supplies the network action, attachment, and invariant-junction tests used below. What Is an Anomaly? supplies the counterterm quotient and keeps a global-symmetry anomaly distinct from a gauge inconsistency and explicit breaking.

Helpful background. Constructions from Gauging, Duality, and Condensation constructs the normalized Maxwell wall and its reverse fusion. ’t Hooft Anomaly Matching supplies the necessary-but-not-sufficient logic of infrared matching.

Scientific evidence cutoff. Source versions, published notices, and scope-refining results cited here were checked through 9 August 2026. The 2026 lecture notes used for a current overview are unrefereed and are not the sole authority for any theorem below; their warning that the higher-dimensional subject remains noncomprehensive is part of the scope of this page.

What obstruction data can—and cannot—decide

For an ordinary exact global symmetry, let $B$ denote the complete fixed background, including global form, tangential structure, boundary conditions, and any higher-form components. In a nonzero local trivialization of the partition function, a background transformation $g$ may act as

$$Z_X[B^g] = e^{i\alpha_X(B,g)}Z_X[B].$$

Adding an allowed local counterterm $C_X[B]$ changes the representative by

$$\alpha_X(B,g) \longmapsto \alpha_X(B,g)+C_X[B^g]-C_X[B].$$

The anomaly is the equivalence class under all such globally defined local counterterms. If $Z_X[B]=0$, the invariant object is parallel transport in the anomaly line rather than a ratio of numbers. Let $Y$ obey $\partial Y=X$, and let $\widetilde B$ extend $B$ into $Y$. With the inflow convention

$$e^{iS_{\mathrm{in}}[Y,\widetilde B^g]} = e^{iS_{\mathrm{in}}[Y,\widetilde B]}e^{-i\alpha_X(B,g)},$$

the boundary functional multiplied by the $(d+1)$-dimensional inflow theory is invariant. This familiar scalar formula is a model for the logic, not a universal formula for a non-invertible symmetry.

For a protected non-invertible defect mesh $\mathbb B$, the background data may include defect strata, their orientations, junction vectors, associators, attachments, and boundary completions. When a one-dimensional local trivialization exists, first use the prescribed associator and junction basis maps to identify the two resolved networks. A nontrivial $F$-symbol is structural recoupling data, not by itself an anomaly. Only the residual response after those identifications is denoted by $\mathcal A_X$ below. A permitted local move $m$ can then be written schematically as

$$Z_X[m\mathbb B] = \mathcal A_X(\mathbb B;m)Z_X[\mathbb B],$$

with the consistency condition

$$\mathcal A_X(\mathbb B;m_2\!\circ m_1) = \mathcal A_X(\mathbb B;m_1) \mathcal A_X(m_1\mathbb B;m_2).$$

Counterterms and junction redefinitions on every relevant stratum change the representative. In a more general theory, however, the move may act on a vector space, an anomaly line, or an invertible field theory rather than by a single $U(1)$ number. The formulas above then stand for coherent isomorphisms. Calling an inconsistency an anomaly is licensed only after the background/move system, its allowed redefinitions, and its gluing law have been specified.

This immediately gives a hierarchy of verdicts:

• In a finite rigid protected sector where $\overline D$ is the chosen categorical dual, $D\otimes\overline D\not\simeq\mathbf 1$ proves that $D$ is not invertible; it does not prove an anomaly.
• A failure to choose compatible measures, multiplication maps, junctions, or boundary data can obstruct a proposed gauging or condensation.
• A phase or line-valued failure that survives every allowed local counterterm is anomaly data for the full network.
• An anomaly or a self-gauging relation constrains an infrared proposal only if the corresponding backgrounds and network maps survive the flow.

Fusion multiplicities do not determine any of these higher verdicts by themselves. Associators, braiding, junction bases, operator attachments, global sectors, and spin or orientation data can change the answer while the fusion ring stays fixed. The defect, junction, and higher-categorical layers behind this network-first statement are reviewed in Schäfer-Nameki 2024, § 1.2, arXiv v2, printed pp. 9–11, especially eqs. (1.7)–(1.9), Open PDF. A concrete control is $\operatorname{Vec}_G^{\omega}$: the simple-object fusion $g\otimes h=gh$ is independent of $\omega$, but a monoidal fiber functor exists only when $[\omega]=0$. The associator can therefore change the gapped-phase obstruction without changing the fusion ring Thorngren and Wang 2019, § 2.3, arXiv v1, printed p. 16, eq. (2.22), Open PDF. Current higher-dimensional proposals explicitly retain this limitation Antinucci et al. 2023, Introduction, arXiv v1, printed pp. 2–4, Open PDF. The broader higher-dimensional status is summarized in Kaidi 2026, Introduction, arXiv v2, printed p. 2, Open PDF. A current framework analysis makes the ceiling sharper: mathematically precise tensor-functor statements are mature there only in the $1+1$-dimensional fusion-category setting. The Matching Equation equivalence is proved for fusion 1- and 2-categories, but the injective and surjective 2-functor notions needed for a parallel classification remain incomplete; still higher and continuous cases are treated at the level of a proposed definition Antinucci et al. 2025, §§ II and III.B and Appendices A.2 and A.6, arXiv v1, printed pp. 5, 11, 43, and 46–47, Open PDF.

Gauging obstruction and infrared obstruction are different tests

A proposed gauging is an operation on the complete background family. Before summing a finite defect mesh, one must establish all of the following:

1. the defects, junctions, and local moves close on the chosen family;
2. every anomaly that restricts to the gauged data is cancelled or trivialized;
3. the measure, topological weights, and normalization obey gluing;
4. the boundary, tangential, and global-sector data are complete; and
5. the resulting operation has the multiplication and coherence needed for the claimed quotient or condensation.

A failure at this gate is a gauging obstruction. It need not be encoded by an ordinary group-cohomology phase, and it is not established merely by the absence of a construction in one presentation. Conversely, successful gauging does not by itself construct or dynamically select a trivially gapped phase, and it does not imply a self-gauging fixed response.

An infrared obstruction asks a different question. Fix a class of candidate infrared theories—short-range-entangled phases, TQFTs with a unique local vacuum, or all gapped phases—and ask whether any member can realize the identified network, its anomaly, and its self-gauging maps. A negative answer rules out only that declared class. Enlarging the class can reopen the problem.

There are controlled cases where the distinction becomes sharp:

• For a finite semisimple fusion-category symmetry $\mathcal C$ in $1+1$ dimensions, a nondegenerate gapped symmetric phase is equivalent to a one-simple-object $\mathcal C$-module category, or fiber functor. Failure of this test obstructs that phase under those hypotheses Thorngren and Wang 2019, § 2.3, arXiv v1, printed pp. 11–16, especially Theorem 1 and eqs. (2.11)–(2.17), Open PDF.
• For finite-Abelian self-duality networks in $3+1$ dimensions, arithmetic and Lagrangian-algebra tests can obstruct specified gapped realizations. Their force depends on spin, faithfulness, semisimplicity, and the chosen global sectors.
• In a general higher-dimensional or nonsemisimple theory, no theorem makes “cannot gauge the chosen subgroup,” “cannot realize the full self-duality network in a trivial phase,” and “has a conventional ’t Hooft anomaly” interchangeable.

The last point is not a technicality. The finite subgroup used to build a noninvertible self-duality wall may be gaugeable while the full self-duality network is incompatible with a particular gapped response. The network can also fail a proposed higher gauging while admitting a nontrivial symmetric TQFT. Recent work explicitly distinguishes these possibilities and leaves parts of the full higher-equivariantization problem open Antinucci et al. 2023, Introduction, pp. 2–4, and Appendix E, pp. 65–66, Open PDF.

RG transport requires the whole defect network

Suppose an exact ultraviolet defect network is preserved by the deformation. Let $F$ identify its defects, junctions, attachments, and backgrounds with data in the infrared theory. The invariant matching statement is a pullback of the complete anomaly theory,

$$[\mathfrak A_{\mathrm{UV}}] = F^*[\mathfrak A_{\mathrm{IR}}],$$

not equality of bare fusion coefficients or microscopic defect labels. If only a subnetwork survives, restrict both sides to that subnetwork. If the IR has an accidental larger symmetry, matching constrains only the image of the UV data.

Finite gauging also commutes with an exact symmetry-preserving flow when the same measure, global sectors, and boundary completion are transported:

$$\operatorname{RG}(\mathcal T/A) \simeq \operatorname{RG}(\mathcal T)/A.$$

If the ultraviolet theory has a background-compatible self-gauging equivalence

$$\Phi:\mathcal T/A\xrightarrow{\ \simeq\ }\mathcal T,$$

then the infrared must admit the transported equivalence $F(\Phi)$ unless a hypothesis is lost. This is stronger than preserving a fusion rule: it keeps the Fourier kernel, genuine-operator lattice, junction maps, counterterms, and any spectator-background transformation.

The logic has three important exits. Explicit breaking removes the broken defects from the matching problem. Spontaneous breaking does not: the exact network must instead be represented by vacua, domain walls, Goldstone or Wess–Zumino data, or topological sectors. An emergent IR symmetry can enlarge the target network, but only its pullback along $F$ is fixed by the UV.

For the finite one-form self-dualities used below, the commutation of gauging with RG and the resulting self-duality constraint are stated in Choi et al. 2022, Introduction and § 3 opening, arXiv v3, printed pp. 4–5 and 14, especially eqs. (1.4) and (3.1), with the defect-flow picture in Fig. 6 on p. 16, Open PDF. The ordinary background-map logic and its necessary-not-sufficient interpretation come from the general matching statement.

First application: the compact-Abelian wall excludes a trivial infrared

Work on a closed smooth oriented simply connected spin four-manifold $X$. Take pure compact $U(1)$ Maxwell theory with no dynamical electric charges or monopoles and

$$\tau = \frac{\theta}{2\pi}+\frac{2\pi i}{e^2} = 3i.$$

Choose the representative $\theta=0$, so $e^2=2\pi/3$.

The electric and magnetic $U(1)$ one-form symmetries have a mixed response, but either factor can be gauged when the other background is held trivial. Select the exact electric $\mathbb Z_3^{(1)}$ subgroup and keep the magnetic spectator background zero. Gauging that subgroup sends $\tau\mapsto\tau/9=i/3$; electromagnetic $S$ then sends $\tau\mapsto-1/\tau=3i$. After matching the compact flux lattice, backgrounds, and $B_2$-independent counterterm convention, half-gauging followed by $S$ gives a codimension-one endowall $D$.

In the declared closed smooth simply connected convention, the normalized finite Fourier transform uses $B_2\in H^2(X;\mathbb Z_3)$ and is

$$(SZ)[X;B_2] = \frac{1}{\sqrt{\lvert H^2(X;\mathbb Z_3)\rvert}} \sum_{b\in H^2(X;\mathbb Z_3)} Z[X;b] \exp\!\left( \frac{2\pi i}{3}\int_X b\smile B_2 \right).$$

Here $B_2\in H^2(X;\mathbb Z_3)$ is the background for the emergent dual one-form symmetry after the electric field $b$ has been summed. The displayed normalization belongs to this closed-manifold self-duality convention; it is not a universal finite-gauging factor on disconnected manifolds or manifolds with boundary.

The finite network makes the RG hypothesis testable. A Wilson line and an electric symmetry surface are

$$W_q(C) = \exp\!\left(iq\oint_C a\right), \qquad \eta_k(\Sigma), \quad q\in\mathbb Z, \quad k\in\mathbb Z_3.$$

For closed disjoint supports in the declared orientation convention,

$$\langle\eta_k(\Sigma)W_q(C)\,\mathcal X\rangle = e^{\frac{2\pi i}{3}kq\,\operatorname{Lk}(\Sigma,C)} \langle W_q(C)\,\mathcal X\rangle.$$

The surfaces fuse as

$$\eta_k\otimes\eta_\ell \simeq \eta_{k+\ell\,\mathrm{mod}\,3}.$$

A chosen junction line

$$I_{k,\ell}^{m}(J): \eta_k\otimes\eta_\ell\longrightarrow\eta_m$$

obeys the necessary incidence condition $k+\ell-m=0\pmod 3$. This congruence does not construct or normalize the junction. On the wall, selected absorption junctions give

$$\eta_k(\Sigma\subset M)\otimes D(M) \simeq D(M) \simeq D(M)\otimes\eta_k(\Sigma\subset M).$$

Opposite wall orientations fuse to the condensation wall,

$$\overline D\otimes D \simeq D\otimes\overline D \simeq C_0.$$

On a connected wall $M=S^2\times S^1$, the normalized worldvolume sum is

$$C_0(M) = \frac{1}{3} \left[ \mathbf 1+\eta_1(S^2)+\eta_2(S^2) \right].$$

The same-orientation product is different:

$$D\otimes D \simeq U_C\otimes C_0,$$

where $U_C$ is the charge-conjugation wall. Orientation reversal also sends $\eta_k(-\Sigma)=\eta_{-k}(\Sigma)$ and $W_q(-C)=W_{-q}(C)$. These relations prevent the opposite-orientation projector from being mistaken for $D^2$.

The support dimensions are part of the check: in $X^4$, $W_q$ and $I_{k,\ell}^{m}$ are one-dimensional, $\eta_k$ is two-dimensional, and $D$ and $C_0$ are three-dimensional.

Thus “the symmetry survives” means more than retaining the label $D$: the line action, surface fusion, junction incidence, absorption maps, and the normalized reverse-fusion wall must all remain coherent. These network and normalization statements follow from Choi et al. 2023, §§ 2.1, 3.1, 4–4.1, and 6.1.2, arXiv v2, printed pp. 9–10, 19–20, 23–24, and 35–36, especially eqs. (2.5), (3.4)–(3.7), (4.1)–(4.3), and (6.22)–(6.28), Open PDF.

The short-range-entangled test

Assume that the full self-gauging network above survives the RG flow. If the IR were a unique short-range-entangled phase preserving the electric $\mathbb Z_3^{(1)}$, its response would lie among the odd-order one-form SPTs

$$Z_r[X;B_2] = \exp\!\left( \frac{2\pi i r}{3} \int_X B_2\smile B_2 \right), \qquad r\in\mathbb Z_3.$$

In this convention, self-gauging requires

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

But $4\equiv1\pmod3$, every nonzero square modulo $3$ equals $1$, and $-1\equiv2\pmod3$. No $r$ solves the condition. Therefore a unique short-range-entangled IR preserving the complete self-duality network is excluded.

This arithmetic and its odd-$N$ bosonic-or-fermionic domain are derived in Choi et al. 2022, § 3.1, arXiv v3, printed pp. 15–17, especially eqs. (3.2) and (3.6) and the theorem on p. 17, Open PDF. The same source’s even-$N$ spin analysis in § 3.2, printed pp. 17–18, especially eqs. (3.8)–(3.10), shows why $N=2$ is not a valid substitute: a self-dual invertible fermionic SPT response can exist there.

A stronger theorem needs stronger hypotheses

A separate theorem addresses a broader class of infrared TQFTs. On smooth simply connected spin four-manifolds, a unitary spin TQFT with a unique local vacuum, an exact $\mathbb Z_N^{(1)}$ symmetry—possibly unfaithful on lines—and invariance under its gauging up to a $B_2$-independent gravitational counterterm exists only if

$$N=k^2\ell \quad\text{and}\quad -1\ \text{is a quadratic residue modulo }\ell.$$

For $N=3$, necessarily $k=1$ and $\ell=3$, and $-1\equiv2$ is not a square modulo $3$. Under those additional hypotheses, the IR cannot be a gapped unique-local-vacuum TQFT. “Unique local vacuum” here means uniqueness on $\mathbb R^3$ or $S^3$, not one ground state on every spatial topology. This statement is stronger than the SPT calculation and must not be inferred from it alone. It includes topologically ordered unitary TQFTs with a unique local vacuum and it tracks an unfaithful one-form action in the proof. It relies on the theorem’s smooth, simply connected, spin, unitarity, exact-topological, absolute-theory, and normalization assumptions Apte, Córdova, and Lam 2023, § III.B, arXiv v1, printed p. 8, Theorem 5 and proof sketch on pp. 8–9; detailed proof in Appendix C.1, pp. 14–17, Open PDF.

The same arithmetic appears as the first duality-invariant Lagrangian-algebra obstruction in a scoped spin Symmetry-TFT proposal Antinucci et al. 2023, § 4.3, arXiv v1, printed pp. 39–41, especially eq. (4.28), Open PDF. Agreement in this finite setting is a useful cross-check, not a universal definition of a non-invertible anomaly.

The converse direction of that theorem also supplies a useful separation test. In its spin-response convention, $N=4$ has no invertible solution of $p^2\equiv-1\pmod4$, yet $4=2^2\cdot1$ passes the TQFT criterion and a $\mathbb Z_2$ gauge TQFT supplies a gapped realization. “No self-dual SPT” therefore does not mean “no self-dual topological order.”

The minimal conclusion remains deliberately narrower: the preserved self-duality network excludes a unique trivial symmetric infrared. It does not by itself prove a conventional ’t Hooft anomaly, select one phase, or show that every gapped phase is impossible without the stronger theorem. Once all of that theorem’s hypotheses are imposed, however, the $N=3$ endpoint is gapless or has multiple local vacua and spontaneously breaks the duality. Escapes require losing a stated hypothesis—for example exactness, unitarity, the spin or absolute-theory setting, or the full self-gauging map. In the microscopic Maxwell theory, the photon supplies the familiar gapless realization.

As an adversarial check, $N=5$ has no analogous SPT arithmetic obstruction: $r=\pm1$ obeys $4r^2\equiv-1\pmod5$. That congruence merely permits a self-dual response; it neither constructs an IR phase nor proves the wall anomaly-free. Noninvertibility alone therefore cannot be the source of the $N=3$ exclusion.

Framework limits by dimension and categorical hypothesis

The following table is a bounded comparison of the settings used in current practice. Each row has its own input category and stop condition; conclusions must not be moved from one row to another without a theorem.

Framework assumptions and licensed obstruction conclusions

Setting	Symmetry or background object	Required assumptions	Obstruction datum	Licensed conclusion	Failure ceiling
Ordinary finite or higher-form symmetry	Bundles, cocycles, or higher connections	Complete global and tangential data; local counterterm class fixed	Invertible anomaly theory or background-gauge variation	A nontrivial class obstructs standalone gauging; the class matches under preserved RG	A local polynomial can miss torsion, boundaries, or global form
1+1d finite semisimple fusion-category symmetry	Topological lines, junctions, associator, and module action	Finite semisimple unitary category and nondegenerate gapped phase	Fiber-functor or one-simple-object module-category test	Existence or obstruction of that symmetric gapped phase	Not a theorem for nonsemisimple theories or higher dimensions
3+1d finite-Abelian self-duality	One-form surfaces, duality walls, junctions, and finite Fourier kernel	Exact topological network; declared spin, action/faithfulness data, and global sectors	SPT arithmetic or a duality-invariant Lagrangian-algebra test	Exclusion of the precisely stated gapped candidate class	Does not classify all higher-categorical anomalies or phases
Symmetry-TFT diagnostic	Bulk topological defects and candidate boundary or condensation data	A controlled finite topological model and a complete set of local moves	Linking, braiding, or absence of a compatible condensable algebra	A case-specific sufficient obstruction under the model’s hypotheses	Need not be a complete anomaly invariant of the boundary QFT
General higher-dimensional, continuous, nonsemisimple, or approximate case	Possibly higher-categorical defects with incomplete background theory	Only the explicitly constructed network and deformation domain	Case-specific coherence failure or response	Only the bounded obstruction actually derived	No universal scalar anomaly formula or classification is presently licensed

The first two rows have mature mathematical formulations within their stated domains. The third and fourth contain powerful results but depend on the chosen finite topological model. The last row is not empty—it supports real operator and network calculations—but it forbids extrapolating a complete classification from a suggestive fusion algebra or one linking phase. The case-specific nature of current $3+1$-dimensional tests is emphasized in Antinucci et al. 2023, § 4.3, arXiv v1, printed pp. 39–41, especially eqs. (4.18)–(4.29), Open PDF. A refereed $3+1$-dimensional analysis likewise distinguishes obstructions to a symmetric gapped phase from the stronger obstruction to an invertible gapped phase Córdova, Hsin, and Zhang 2024, § 2.2, arXiv v2, printed pp. 11–15, Open PDF.

What an infrared theory may do

An obstruction prunes candidate infrared descriptions; it does not choose the remaining branch. A complete IR can match the preserved data through several mechanisms:

• Gapless modes. Massless fields or an interacting fixed point carry the network and its anomaly data.
• Spontaneous breaking. Multiple vacua, domain walls, and their junctions realize an exact symmetry that is not preserved by one vacuum.
• Topological order or extra sectors. In generic matching problems a nontrivial TQFT can supply missing lines, surfaces, and junctions. In the $N=3$ application, however, Theorem 5 already includes unitary TQFTs with a unique local vacuum and unfaithful one-form actions; such a phase escapes only by violating another theorem hypothesis or by having multiple local vacua.
• A relative realization. The QFT is consistently defined only together with a fixed bulk or boundary completion.
• A changed comparison problem. Explicit breaking, dynamical endpoints, monopoles, or loss of topologicality removes some of the original network; only the surviving subnetwork remains constrained.

Conversely, a successful match means only that a proposal passes this necessary test. It does not establish dynamics, stability, uniqueness, or the absence of another matching phase. A current formulation of the $1+1$-dimensional module-category classification and its ground-state-count interpretation appears in Thorngren and Wang 2019, § 2.4, arXiv v1, printed pp. 16–18, especially Theorem 3, Open PDF. Its hypotheses are not silently extended to the Maxwell wall.

Common pitfalls

Calling every non-invertible defect anomalous. Noninvertibility is the absence of a two-sided fusion inverse. An anomaly is a counterterm-invariant failure of the complete background or network transformations.

Equating failure to gauge with failure to gap. These tests use different inputs and candidate classes. State the theorem, dimension, categorical hypotheses, and whether the conclusion concerns a trivial phase, a unique local vacuum, or every gapped realization.

Matching only the fusion ring. RG transport must preserve or map the junctions, associators, attachments, measures, global sectors, and counterterms that enter the claimed obstruction. The same fusion coefficients can support inequivalent anomaly data.

Treating a self-duality congruence as a construction. Solving the arithmetic condition only removes one obstruction. It does not construct a unitary QFT, prove a normalized defect exists, or establish dynamical flow to that response.

Forgetting that spontaneous breaking still matches. Explicit breaking removes a symmetry constraint; spontaneous breaking retains the exact symmetry and realizes it through vacua and defects. The latter is an allowed matching mechanism, not an evasion.

Exporting a controlled categorical result. A fiber-functor theorem in $1+1$ dimensions or a Lagrangian-algebra test in a finite $3+1$-dimensional Symmetry TFT is not a universal theorem for continuous, nonsemisimple, or approximate defects.

Check your understanding

1. Separate the four verdicts

A defect obeys $D\overline D\simeq\mathbf 1\oplus X$. What follows without additional input?

Checked answer

Only noninvertibility follows: the reverse fusion has an extra channel. One still needs a complete background network and counterterm quotient to diagnose an anomaly, coherent sum and multiplication data to diagnose gaugeability, and a preserved background map plus a declared class of candidate IR theories to derive an RG obstruction.

2. Check counterterm dependence

If $Z[B^g]=e^{i\alpha(B,g)}Z[B]$ and $Z'[B]=e^{iC[B]}Z[B]$, find the transformed phase.

Checked answer

Direct substitution gives

$$Z'[B^g] = e^{i\{\alpha(B,g)+C[B^g]-C[B]\}}Z'[B].$$

Thus only the class of $\alpha$ modulo differences of allowed globally defined local counterterms is invariant.

3. Run the Z₃ self-duality test

Solve $4r^2\equiv-1\pmod3$ for $r\in\mathbb Z_3$.

Checked answer

The squares modulo $3$ are $0$ and $1$, and $4\equiv1$. Hence the left side is $0$ or $1$, whereas $-1\equiv2$. There is no solution, so the declared unique short-range-entangled self-dual response is excluded.

4. Use Z₅ as a failure control

Does the same congruence obstruct $N=5$?

Checked answer

No. For $r=\pm1$, $4r^2=4\equiv-1\pmod5$. This removes the arithmetic obstruction but does not construct a symmetric phase or prove that every other consistency condition holds.

5. Identify the preserved Maxwell data

Why is preserving the label $D$ insufficient for the $N=3$ RG argument?

Checked answer

The argument also uses the electric surfaces $\eta_k$, their fusion and linking with $W_q$, the junction incidence rule, the absorption junctions on $D$, the normalized condensation wall $C_0$, and the background-compatible self-gauging equivalence. Losing any of these can invalidate the transported constraint even if an operator still happens to be called $D$.

6. Distinguish explicit from spontaneous breaking

What happens to matching if an RG deformation explicitly breaks the network, and what changes if the symmetry is instead spontaneously broken?

Checked answer

After explicit breaking, only the surviving exact subnetwork is constrained. Under spontaneous breaking the full symmetry remains exact, so its data must still be realized by the vacuum structure, domain walls, junctions, Goldstone/Wess–Zumino terms, topological sectors, or a combination of them.

Continue to Symmetry TFT and categorical obstruction theory

Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging will package background transformations, anomaly inflow, and gauging in a bulk topological model. Non-Invertible Symmetries: Fusion and Junction Data will supply theorem-level categorical definitions and obstruction classifications. These are distinct continuations: the first is the next physical encoding, while the second owns the rigorous classification boundary.

References

• Antinucci, Andrea, Francesco Benini, Christian Copetti, Giovanni Galati, and Giovanni Rizi. “Anomalies of Non-Invertible Self-Duality Symmetries: Fractionalization and Gauging.” arXiv:2308.11707v1 [hep-th], 22 August 2023. DOI. Open PDF.
• Antinucci, Andrea, Christian Copetti, Yuhan Gai, and Sakura Schäfer-Nameki. “Categorical Anomaly Matching.” arXiv:2508.00982v1 [hep-th], 1 August 2025. DOI. Open PDF.
• 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.
• 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.
• Córdova, Clay, Po-Shen Hsin, and Carolyn Zhang. “Anomalies of Non-Invertible Symmetries in (3+1)d.” SciPost Physics 17, no. 5 (2024): 131. 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.
• 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.