Constructions from Gauging, Duality, and Condensation

Finite gauging can manufacture a noninvertible defect, but not in one step. Gauging an exact finite symmetry on only one side of a hypersurface first produces an interface from a theory $\mathcal T$ to its gauging $\mathcal T/A$. Only after a background-compatible equivalence $\Phi:\mathcal T/A\simeq\mathcal T$ is supplied does that interface become an endodefect of $\mathcal T$. Fusing it with its orientation reverse then collapses the gauged slab to a normalized condensation network. If that network is not the identity wall, the endodefect is noninvertible.

Thus full gauging, half-gauging, duality, and condensation are related operations, not synonyms. The construction requires a gaugeable symmetry, the complete global sector sum and measure, a boundary condition for the gauged region, and an actual equivalence of the resulting absolute theories. This page works in a finite protected setting, gives the compact-Maxwell wall as its first QFT application, and uses one finite categorical fixture to check projection, twisted sectors, simple objects, and the emergent dual symmetry. It does not classify condensation algebras or all higher-dimensional noninvertible defects.

Required background. Non-Invertible Topological Defects and Fusion supplies the regulated collision, orientation, junction, and coherence data needed to interpret the final fusion. Gauging Continuous and Finite Symmetries supplies the finite groupoid sum, automorphism weights, twisted sectors, and gaugeability test.

Helpful background. Boundaries, Interfaces, and Domain Walls supplies the interface-composition and folding conventions. Gauging a Higher-Form Symmetry supplies the higher background degree, attachment, and finite-dual network used below.

Evidence on half-space gauging, condensation defects, and the four-dimensional Maxwell construction was checked through 9 August 2026. The checked sources support the bounded method and its exact network data, while the current review literature emphasizes that higher-dimensional noninvertible constructions are not yet described by one exhaustive framework. In particular, anomaly dressing and half-space gauging can give related walls but are not interchangeable constructions.

Half-gauging produces an interface

Let $\mathcal T$ be a QFT on an oriented Euclidean $d$-manifold $X$, and let $A^{(p)}$ be a finite exact $p$-form symmetry, Abelian when $p>0$. Before gauging, specify its full family of flat $(p+1)$-form backgrounds, its action on operators and sectors, all allowed topological weights, and any tangential or boundary data. The restriction of every ‘t Hooft anomaly to the subgroup being gauged must vanish or be canceled. A mixed anomaly with a spectator symmetry need not vanish, but it must be transported consistently to the output.

On a closed manifold $Y$, finite gauging has the schematic form

$$Z_{\mathcal T/A}[Y] = \sum_{[b]} \mu_A(Y;[b])\, Z_{\mathcal T}[Y;b]\, \Theta[b].$$

Here $[b]$ runs over gauge-equivalence classes of the admissible higher backgrounds, $\mu_A$ is the gluing-compatible groupoid or higher-groupoid measure, and $\Theta$ is a chosen gauge-invariant topological weight. The formula is deliberately schematic: the cohomological degree, gauge-for-gauge automorphisms, boundary data, and normalization depend on $p$, $A$, and the spacetime topology. A universal factor such as $1/\lvert H^{p+1}(Y;A)\rvert$ is generally wrong.

Now let a closed two-sided hypersurface $M^{d-1}$ separate $X=X_L\cup_M X_R$. Gauge $A^{(p)}$ only in $X_R$ and impose the topological boundary condition on the dynamical finite gauge field that implements the ungauged-to-gauged gluing. The output is an oriented interface

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

This is not yet a symmetry defect: the theories on its two sides are different. For a general two-sided but nonseparating $M$, one instead cuts $X$ along $M$ and defines the interface by gluing the two new boundary copies with the same finite-gauging kernel. A collar gives local sides, but by itself does not choose a global half-space.

The finite-gauging sum projects gauge-charged states and operators while also adding twisted or flux sectors. Keeping only the invariant sector is not gauging. Likewise, calling $b$ flat does not remove its global holonomies or the automorphism factors in the sum. These are the same projection-plus- twisted-sector ingredients developed in the finite-gauging prerequisite.

Schäfer-Nameki formulates half-space gauging with its boundary condition and self-duality gate in Schäfer-Nameki 2024, § 4.4, arXiv v2, printed pp. 71–72, eqs. (4.68)–(4.72), Open PDF. Kaidi separates this construction from anomaly-dressing mechanisms in Kaidi 2026, §§ 4.1.1–4.1.2, arXiv v2, printed pp. 69–73, eqs. (4.1)–(4.16), Open PDF.

A duality turns the interface into an endodefect

To obtain an internal defect of $\mathcal T$, supply an actual equivalence

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

The equivalence must match more than local equations of motion. It must identify the global form, genuine-operator lattice, background couplings, counterterms, spin or other tangential structure, boundary conventions, and any discrete-theta choice. A coincident value of one continuous coupling is not enough.

Composing the two arrows gives the endodefect

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

The orientation-reversed wall uses the reverse interface and the inverse equivalence. Fix the comparison junction and local counterterms so that $\Phi^{-1}\circ\Phi\simeq\mathrm{id}$. In that controlled self-dual construction, the equivalences cancel when the walls are fused in opposite orientations, while the chosen finite gauging layer remains:

$$\overline{\mathcal D}_{\Phi} \otimes \mathcal D_{\Phi} \simeq \mathcal C_{A,\Theta}.$$

$\mathcal C_{A,\Theta}$ is the condensation wall for the declared topological weight and normalization. If $\mathcal C_{A,\Theta}\not\simeq\mathbf 1$, the reversed wall is not an inverse and $\mathcal D_{\Phi}$ is noninvertible. The conclusion must be checked in both fusion orders when the construction is not known to be symmetric. A half-gauging interface without $\Phi$ cannot be declared noninvertible as an internal symmetry, because it is not an endodefect in the first place.

There are useful near misses. Gauging may return a theory only after changing its polarization, global form, or spectator background; then the object is an interface between distinct absolute theories. An anomalous subgroup may define a relative interface after adding inflow, but it does not define the same standalone gauging. A duality that closes only in the infrared gives an infrared defect, not an exact microscopic one.

Condensation is a worldvolume sum

Condensation can be defined without a self-duality by gauging suitable topological operators on a submanifold. In the half-gauging construction, the same operation appears when a thin gauged slab is collapsed. These two descriptions agree only after their allowed sectors, topological weights, junction multiplication, boundary data, and normalization have been matched.

The result has the dimension of the submanifold on which condensation is performed. In particular, condensing two-dimensional symmetry surfaces on a three-dimensional wall produces another wall. It is not a formal direct sum of codimension-two bulk surfaces. The notation for a worldvolume sum must therefore retain the embedding and the junctions that let a surface end on or fuse into the wall.

Four pieces of information are indispensable:

1. the allowed worldvolume sectors and their gauge equivalences;
2. the gluing-compatible measure, including automorphisms;
3. a multiplication and unit realized by selected junction operators, with coherent associativity; and
4. any discrete-torsion, quadratic, spin, or boundary choice.

A formal object $\bigoplus_a U_a$ without those data is not yet a condensation defect. Nor is the coefficient in a higher-dimensional fusion law necessarily an integer multiplicity. It can depend on the wall topology or be the partition function of an entire lower-dimensional TQFT. Overall normalization can also shift by an allowed Euler counterterm.

Choi and collaborators give the finite worldvolume sum, its discrete-torsion family, orientation reversal, and normalization ambiguity in Choi et al. 2023, § 2.1, arXiv v2, printed pp. 9–10, eqs. (2.5)–(2.9), Open PDF. Their fusion coefficients become three-dimensional TQFT data in Choi et al. 2023, §§ 3–3.1, arXiv v2, printed pp. 17–20, eqs. (3.1)–(3.7), Open PDF.

The table summarizes two related routes. The main branch runs from finite gauging through half-gauging, duality, and reverse fusion. The condensation row can also be entered independently, but only after its own worldvolume data are supplied. Within either branch, passing a later gate never repairs an earlier failure.

Two construction routes and their separate inversion tests

Route	Stage	Required input	Operation	Licensed output	Stop condition
Endodefect	Finite gauging	Exact symmetry, complete backgrounds, anomaly cancellation, measure, and weights	Sum gauge-equivalence classes and twisted sectors	A new theory T/A	An anomaly or incomplete sector family blocks standalone gauging
Endodefect	Half-gauging	A separating region, or cut-and-glue data, plus a topological boundary condition	Gauge on one side	An oriented interface from T to T/A	A collar alone does not define a global half-space
Endodefect	Duality	A background-compatible equivalence between the two absolute theories	Compose the interface with the equivalence	An endodefect of T	Matching one coupling or local Lagrangian is insufficient
Endodefect	Reverse fusion	A regulated collision and all transverse junction data	Collapse the gauged slab	A candidate worldvolume condensation network	The fusion order and the network's algebra and normalization still need checking
Endodefect	Inversion test for D	A validated, normalized condensation wall from reverse fusion, including its algebra and junction data; the identity wall and both fusion orders	Compare the reverse composite with the identity	D is noninvertible when its reverse composite is a nonidentity wall	Do not infer noninvertibility before computing the composite
Independent condensation	Worldvolume gauging	Worldvolume sectors, measure, multiplication, unit, associator, and topological weight	Gauge the induced finite network on the wall	A well-defined same-support wall with projector- or TQFT-valued fusion	A formal sum without junction and normalization data is not a defect
Independent condensation	Inversion test for C	The orientation reverse and the complete wall-fusion data	Compute reverse or projector fusion for the condensation wall itself	C is noninvertible only if its own two-sided inverse test fails	A nonidentity wall can still be invertible; C not equal to the identity is insufficient

On the endodefect branch, $\overline{\mathcal D}\otimes\mathcal D=\mathcal C\not\simeq\mathbf1$ proves that $\mathcal D$ is noninvertible. For an independently constructed $\mathcal C$, nonidentity alone proves nothing about invertibility; one must compute $\overline{\mathcal C}\otimes\mathcal C$ and the opposite order, or an equivalent projector- or TQFT-valued fusion test.

First application: construct the compact-Abelian duality wall

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

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

for every closed oriented two-cycle $\Sigma_2$. 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 select the exact electric $\mathbb Z_N^{(1)}\subset U(1)_e^{(1)}$. Hold the magnetic spectator background trivial during the gauging. After performing the finite two-form sector sum, one can use a rescaled presentation in which

$$\mathfrak a \longmapsto \frac{\mathfrak a}{N}, \qquad \tau \longmapsto \frac{\tau}{N^2}.$$

This is not an equality between compact connections with the same quantization. The rescaling changes the genuine electric-line lattice and the allowed magnetic flux units; those global data must be carried through the duality.

At

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

the gauging sends $iN$ to $i/N$. Electromagnetic $S$-duality then sends $\tau\mapsto-1/\tau$ and returns the theory to $iN$. The fixed operation is therefore electric $\mathbb Z_N^{(1)}$ gauging followed by $S$, not ordinary $S$-duality alone.

The coupling round trip is only the first check. After electric $\mathbb Z_N$ gauging, the external two-form background $B_2$ in the finite Fourier kernel couples to the emergent dual magnetic symmetry. The normalized kernel is given in Choi et al. 2023, § 2 opening, arXiv v2, printed p. 8, eq. (2.1), Open PDF. Electromagnetic $S$ then identifies that symmetry and background with the original electric subgroup; the electric–magnetic line map is Choi et al. 2023, § 6.1, arXiv v2, printed p. 32, eq. (6.6), Open PDF. Thus the combined operation also identifies the compact flux and genuine-line lattices on the declared spin manifold, with the magnetic spectator background set to zero. Turning on electric and magnetic backgrounds simultaneously requires the mixed one-form-anomaly completion rather than two independent background transformations. The lattice rescaling, attached-line, and mixed-anomaly checks are worked out in Kaidi 2026, § 4.2, arXiv v2, printed pp. 73–76, eqs. (4.17)–(4.26), Open PDF.

Fix the $B_2$-independent gravitational or Euler counterterms and the overall normalization so that the self-duality equality is evaluated in one convention. Without this choice, the interface can differ by an invertible gravitational decoration even when its background-dependent response agrees. See Choi et al. 2023, § 2.2, arXiv v2, printed p. 14, eq. (2.15), Open PDF.

Assume first that a closed oriented $M^3\subset X$ separates $X$. Gauge the electric subgroup on one component and compose the resulting interface with the $S$ wall. The resulting endowall $\mathcal D$ has the local compact-connection representative

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

Reversing the orientation and exchanging the sides flips the sign. This Chern–Simons-like kernel is local shorthand, not the complete definition: the compact global sectors, finite gauging measure, and wall junctions remain part of $\mathcal D$. A nonseparating support must be treated by cutting and gluing, and a physical boundary needs its own completion.

The Maxwell conventions and duality groups are set out in Choi et al. 2023, § 6.1, arXiv v2, printed pp. 31–32, eqs. (6.1)–(6.9), Open PDF. The combined fixed point and local wall kernel appear in Choi et al. 2023, § 6.1.2, arXiv v2, printed pp. 35–36, eqs. (6.22)–(6.28), Open PDF.

The finite network supplies line, surface, and junction data

The line operators are the Wilson loops

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

Let $\eta_k(\Sigma)$, $k\in\mathbb Z_N$, be the closed topological electric symmetry surfaces. For closed, disjoint, oriented $C$ and $\Sigma$ in a controlled linking region, and for all other insertions $\mathcal X$ outside the sweep,

$$\begin{aligned} \big\langle \eta_k(\Sigma)W_q(C)\,\mathcal X \big\rangle &= \exp\!\left[ \frac{2\pi i}{N} kq\,\operatorname{Lk}(\Sigma,C) \right] \big\langle W_q(C)\,\mathcal X\big\rangle, \\ \eta_k\otimes\eta_\ell &\simeq \eta_{k+\ell\;\mathrm{mod}\;N}. \end{aligned}$$

A two-in/one-out surface junction is supported on an oriented line $J$ and requires

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

The incidence rule is necessary but does not construct or normalize the junction. The wall construction needs a further selected absorption junction,

$$\eta_k(\Sigma\subset M^3)\otimes\mathcal D(M^3) \simeq \mathcal D(M^3), \qquad \mathcal D(M^3)\otimes\eta_k(\Sigma\subset M^3) \simeq \mathcal D(M^3).$$

A minimally charged $W_1$ crossing $\mathcal D$ becomes an improperly quantized ‘t Hooft line accompanied by an $\eta$-surface attachment. More generally, by fusing $q$ copies, the required attachment depends on $[q]_N$; a line with $q\equiv0\pmod N$ has no nontrivial finite residue. The output is therefore a line–surface composite, not a bare fractional magnetic line and not a scalar eigenvalue of the wall.

The absorption junction and the need for separate transverse junction data are developed in Choi et al. 2023, §§ 4–4.1, arXiv v2, printed pp. 23–24, eqs. (4.1)–(4.3), Open PDF. The attached line crossing is given in Choi et al. 2023, § 6.1.2, arXiv v2, printed pp. 35–36, especially the discussion before eq. (6.26) and eqs. (6.24)–(6.26), Open PDF.

Reverse fusion produces the condensation wall

In the untwisted convention, opposite orientations fuse in either order to

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

Let $\mathcal U_C$ be the invertible charge-conjugation wall, $\mathfrak a\mapsto-\mathfrak a$. The same-orientation relations are

$$\overline{\mathcal D} \simeq \mathcal U_C\otimes\mathcal D \simeq \mathcal D\otimes\mathcal U_C, \qquad \mathcal D\otimes\mathcal D \simeq \mathcal U_C\otimes\mathcal C_0 \simeq \mathcal C_0\otimes\mathcal U_C.$$

Thus the same-orientation square must not be replaced by the opposite-orientation equation. Locally, an intermediate compact connection $\mathfrak a_I$ exhibits the opposite-orientation composition,

$$\begin{aligned} S_{\mathcal D}[\mathfrak a_L,\mathfrak a_I] + S_{\overline{\mathcal D}}[\mathfrak a_I,\mathfrak a_R] &= \frac{iN}{2\pi} \int_{M^3} \mathfrak a_I\wedge \mathrm d(\mathfrak a_L-\mathfrak a_R). \end{aligned}$$

On a closed oriented wall $M^3$, the global untwisted condensation sum is

$$\mathcal C_0(M^3) = \frac{1}{\lvert H^0(M^3;\mathbb Z_N)\rvert} \sum_{[\Sigma]\in H_2(M^3;\mathbb Z_N)} \eta([\Sigma]).$$

For connected $M^3=S^2\times S^1$ this reduces to

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

The factor is the chosen finite-gauging normalization, not a fusion multiplicity, and an Euler counterterm can change the overall convention. Every summand is a surface network condensed on the wall worldvolume, so the output remains codimension one. Since $\mathcal C_0$ is not the identity wall for $N\geq2$, $\mathcal D$ has no two-sided inverse. At $N=1$ the sum is trivial and the construction reduces to the ordinary invertible $S$ wall at $\tau=i$.

The global sum and normalization are Choi et al. 2023, § 2.1, arXiv v2, printed p. 10, eq. (2.5), Open PDF. Both opposite-orientation fusion orders and the topology-dependent higher fusion are stated in Choi et al. 2023, § 3.1, arXiv v2, printed pp. 19–20, eqs. (3.4)–(3.7), Open PDF.

A modular-category fixture tracks the new sectors

A finite $2+1$-dimensional fixture shows why projection alone is incomplete. Start from the toric-code unitary modular tensor category

$$\mathcal B=\{\mathbf 1,e,m,\psi\}, \qquad \psi=e\otimes m,$$

and let $G=\mathbb Z_2$ exchange $e\leftrightarrow m$ while fixing $\mathbf 1$ and $\psi$. The action on the neutral category is not all the gauging data. Choose the unobstructed $G$-crossed braided extension with Frobenius–Schur indicator $\kappa_\sigma=+1$. It contains two flux-defect types $\sigma_+$ and $\sigma_-$, each of quantum dimension $\sqrt2$.

Full gauging is the $G$-crossed extension followed by equivariantization. With $\mathcal B_G^\times$ denoting the chosen crossed extension, define

$$\mathcal B_{\mathrm{gauged}} := (\mathcal B_G^\times)^G.$$

For this choice, the gauged modular category has nine simple objects:

$$\begin{array}{c|c} \text{simple objects} & \text{quantum dimension} \\ \hline (\mathbf 1,+),(\mathbf 1,-),(\psi,+),(\psi,-) & 1 \\ (\sigma_+,+),(\sigma_+,-),(\sigma_-,+),(\sigma_-,-) & \sqrt2 \\ [e]=\{e,m\} & 2 \end{array}$$

The orbit $[e]$ has two elements and trivial stabilizer, which accounts for its dimension two. The total dimension check is

$$\operatorname{Dim}(\mathcal B_{\mathrm{gauged}}) = 4(1)^2+4(\sqrt2)^2+(2)^2 =16, \qquad D_{\mathrm{tot}}(\mathcal B_{\mathrm{gauged}})=4 =\lvert G\rvert\,D_{\mathrm{tot}}(\mathcal B).$$

For the chosen $G$-crossed extension, $\mathcal B_{\mathrm{gauged}} \simeq\mathrm{Ising}\boxtimes\overline{\mathrm{Ising}}$. The bosonic invertible line $(\mathbf 1,-)$ squares to $(\mathbf 1,+)$ and toggles the two equivariant-representation labels. It generates the emergent $\widehat{\mathbb Z}_2^{(1)}\simeq\mathbb Z_2^{(1)}$ symmetry. The finite gauging degree shift is stated in Gaiotto et al. 2015, § 3, arXiv v2, printed p. 14, Open PDF. Let

$$A_{\mathrm{reg}} = (\mathbf 1,+)\oplus(\mathbf 1,-)$$

be the regular commutative algebra in this $\operatorname{Rep}(\mathbb Z_2)$ subcategory. Condensing this gauge-charge sector reverses the gauging: the deconfined sector is the neutral toric-code category $\mathcal B$, while retaining the confined graded sectors recovers the chosen $\mathbb Z_2$-crossed defect theory $\mathcal B_G^\times$. Barkeshli and collaborators state this inverse operation in Barkeshli et al. 2019, § VIII opening, arXiv v4, printed p. 56, Open PDF. The exact module-category theorem is left to the mathematical continuation.

This calculation distinguishes projection from full gauging. The $+$ and $-$ labels come from equivariant projection data, the $\sigma$ objects come from twisted defect sectors, and the orbit object records the nontrivial permutation of $e$ and $m$. Naively equivariantizing the neutral category without first choosing the crossed extension would miss the flux sectors.

The choice $\kappa_\sigma=+1$ is essential. The same permutation action admits another allowed defectification choice with a different gauged category, so the action on simple objects does not uniquely determine the gauging. Barkeshli, Bonderson, Cheng, and Wang give the general orbit–stabilizer construction and dimension formulas in Barkeshli et al. 2019, § VIII, arXiv v4, printed pp. 56–58, especially eqs. (407)–(418), Open PDF, and work this electric–magnetic $\mathbb Z_2$ example in Barkeshli et al. 2019, § XI.I, arXiv v4, printed pp. 84–86, eqs. (611)–(650), Open PDF. The general algebra objects, separability, module categories, and equivariantization theorem are left to Gauging, Equivariantization, Orbifolds, and Condensation.

The construction has sharp stop rules

The method establishes a noninvertible endodefect only after all of the following checks pass:

• the finite symmetry is exact and gaugeable with the declared spectators;
• the global background family, automorphism measure, and topological weight are complete;
• half-gauging is defined by a separating region or an explicit cut-and-glue construction;
• an equivalence $\mathcal T/A\simeq\mathcal T$ matches absolute global and operator data;
• wall collision is controlled and its worldvolume junctions are specified; and
• the reverse composite is demonstrably not the identity.

Failure at any gate weakens the conclusion. Without gaugeability there is no standalone gauged theory. Without self-duality there is an interface, not an internal symmetry. Without the sector sum there is only a projection. Without junction multiplication a formal condensate is not an operator. Without the reverse fusion calculation noninvertibility has not been shown.

None of these kinematic steps determines confinement, spontaneous breaking, a phase transition, or the infrared endpoint. Nor are half-space gauging and anomaly-dressing constructions exhaustive. Kaidi’s current review explicitly describes its higher-dimensional treatment as noncomprehensive in Kaidi 2026, § 4 opening, arXiv v2, printed p. 69, Open PDF.

Common pitfalls

Calling every gauging interface a symmetry. An interface $\mathcal T\to\mathcal T/A$ becomes an endodefect only after an equivalence returns the gauged theory to the same absolute theory.

Using a local wall action as the global definition. A compact Chern–Simons or BF kernel does not encode all flux sectors, automorphism weights, tangential choices, or transverse junctions. Keep the global sum and operator network.

Replacing condensation by an untyped direct sum. The condensed operators live on the wall worldvolume and require multiplication, a unit, associativity, and normalization. A bulk surface does not become a wall merely by writing a sum sign.

Confusing projection with screening. Gauging projects unattached nonneutral operators and adds twisted sectors. Dynamical screening is a separate statement about endpoints; the pure-Maxwell Wilson lines above remain unscreened.

Ignoring orientation. The reverse fusion gives $\mathcal C_0$ in the Maxwell example, whereas the same-orientation square includes charge conjugation. A bar is not decorative.

Reading dynamics from the construction. The wall and its fusion are kinematic symmetry data. They do not by themselves locate a phase or prove confinement, deconfinement, or symmetry breaking.

Check your understanding

1. Why is half-gauging not yet a symmetry defect?

Checked answer

It separates $\mathcal T$ from $\mathcal T/A$, so its two sides are different theories. It becomes an endodefect only after a background-compatible equivalence $\Phi:\mathcal T/A\simeq\mathcal T$ is composed with it.

2. Check the Maxwell coupling round trip

Show that electric $\mathbb Z_N^{(1)}$ gauging followed by $S$ fixes $\tau=iN$.

Checked answer

The gauging sends $\tau\mapsto\tau/N^2$, so $iN\mapsto i/N$. Then $S$ sends $i/N\mapsto-1/(i/N)=iN$. Ordinary $S$ alone would send $iN$ to $i/N$ and is not the fixed operation for $N>1$. This is only the coupling check; the background transformation and compact line and flux lattices must also match before the interface is an endodefect.

3. Evaluate the connected condensation sum

Why does $M^3=S^2\times S^1$ give $N$ surface sectors?

Checked answer

$H_2(S^2\times S^1;\mathbb Z_N)\simeq\mathbb Z_N$, so its classes are represented by $k[S^2]$, $k=0,\ldots,N-1$. Since $M^3$ is connected, $\lvert H^0(M^3;\mathbb Z_N)\rvert=N$. Therefore

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

4. Keep the supports straight

List the support dimensions of $W_q$, $\eta_k$, its fusion junction, and $\mathcal D$ in four dimensions.

Checked answer

$W_q$ is a one-dimensional line, $\eta_k$ is a two-dimensional surface, the two-in/one-out surface junction is a one-dimensional line, and $\mathcal D$ is a three-dimensional wall. Condensing the surfaces on the wall produces a three-dimensional wall, not a direct sum of two-dimensional bulk supports.

5. Check the gauged toric-code dimension

Use the nine simple objects in the modular-category fixture to recover the total quantum dimension and identify the emergent dual generator.

Checked answer

There are four dimension-one objects, four dimension-$\sqrt2$ objects, and one dimension-two object. Therefore

$$\operatorname{Dim} =4+4(2)+4 =16, \qquad D_{\mathrm{tot}}=4.$$

The invertible bosonic line $(\mathbf 1,-)$ squares to the unit and generates the emergent $\widehat{\mathbb Z}_2^{(1)}$ symmetry.

6. Diagnose two near misses

What remains if the gauging is valid but no equivalence $\mathcal T/A\simeq\mathcal T$ exists? What remains if the equivalence exists but the reverse fusion has not been computed?

Checked answer

In the first case there is a valid oriented interface between different theories, not an internal symmetry defect. In the second there is an endodefect, but its invertibility has not been decided. Noninvertibility requires the reverse composite to differ from the identity.

Continue to anomalies, categorical structure, and duality webs

Non-Invertible Anomalies, RG Constraints, and Framework Limits is the next same-chapter step and will ask when the constructed network is gaugeable and what its anomaly can constrain under RG flow. Gauging, Equivariantization, Orbifolds, and Condensation will develop algebra objects, equivariantization, module categories, and condensation theorems. Duality Operations: Gauging, Quotients, and Orbifolds will track gauging across duality dictionaries, while Duality Defects, Walls, Interfaces, and Fusion will develop duality-wall actions and fusion in specific webs.

References

• Barkeshli, Maissam, Parsa Bonderson, Meng Cheng, and Zhenghan Wang. “Symmetry Fractionalization, Defects, and Gauging of Topological Phases.” Physical Review B 100, no. 11 (2019): 115147. DOI. Open PDF, arXiv v4.
• 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.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. 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.