Duality Defects, Walls, Interfaces, and Fusion

A duality interface places two descriptions on opposite sides of a codimension-one wall and couples them so that energy, conserved charges, and selected supercharges pass consistently through it. An invertible topological wall realizes an equivalence: moving an operator across implements the dictionary, and fusing with the inverse removes the wall. More general interfaces can project, sum over sectors, or carry their own degrees of freedom; their existence alone does not prove a duality.

Required background. Duality claims and dictionaries defines the map to be realized, electric–magnetic dualization fixes the charge and period conventions used below, and boundaries and interfaces supplies the variational problem. Helpful background. Generalized symmetries and anomalies, fusion, junctions, and endpoints, and noninvertible gauging constructions clarify composition and inflow.

Folding an interface into a boundary

Let theory $A$ occupy $x^\perp<0$ and theory $B$ occupy $x^\perp>0$. Reflect the second half-space. The interface becomes a boundary condition for

$$\mathcal T_A\otimes\overline{\mathcal T_B},$$

where the bar reverses orientation. This folding trick turns interface consistency into familiar boundary questions:

• Does the combined variation have a well-posed boundary term?
• Is the normal stress continuous, so the wall is topological?
• Which currents and supercharges are preserved?
• Are gauge and gravitational anomalies canceled by inflow or wall fields?
• Which bulk operators may end on the boundary condition?

An interface is topological when correlation functions are unchanged under smooth deformations of its position that avoid operator insertions. Infinitesimal normal motion is generated by the displacement operator, which includes the jump of normal stress plus wall contributions. A genuinely topological wall has vanishing displacement operator. In a deliberately restricted $Q$-cohomological sector, $Q$-exact displacement can suffice, but that proves only topological behavior in that sector. A conformal interface need not be topological; it can reflect part of the energy.

Operator transport defines the dictionary

If a topological wall $\mathcal D_{A\to B}$ separates the theories, move an operator $\mathcal O_A$ through it. The resulting insertion defines

$$D(\mathcal O_A)=\mathcal O_B$$

or, more generally, a sum or complex of target operators. For an invertible or group-like wall with a chosen transparent junction, topological movement preserves local OPEs:

$$D(\mathcal O_1\mathcal O_2) =D(\mathcal O_1)D(\mathcal O_2).$$

For a noninvertible interface, transport instead involves bimodule and junction data and need not define one algebra homomorphism. For lines and surfaces, it must also account for linking, fusion, spins, and allowed endpoints. A Wilson line crossing an electric–magnetic wall may become an ’t Hooft line. If periods transform by $M_\Pi$, charges transform contragrediently by $M_\gamma=M_\Pi^{-T}$ so the central charge remains invariant.

Junction operators between walls provide natural transformations between such maps. Their associativity data are part of the duality structure, especially when fusion is noninvertible.

An abelian S-duality wall

Let $W=\partial M_L=-\partial M_R$ be oriented by the left region. For four-dimensional abelian gauge fields $A_L$ and $A_R$, an $S$ wall can carry the Lorentzian coupling

$$S_{\mathrm{wall}} =\frac{1}{2\pi}\int_W A_L\wedge dA_R$$

Use the unit-flux convention

$$G=\frac{2\pi}{e^2}*F-\frac{\theta}{2\pi}F$$

on each side. Varying the bulk-plus-wall action with the stated orientation gives

$$F_R|_W=G_L|_W, \qquad G_R|_W=-F_L|_W.$$

The wall is topological only when the stress tensors also match, which occurs for $\tau_R=-1/\tau_L$ in this convention. The coupling is well-defined under large gauge transformations only with the displayed integral normalization. Kapustin and Tikhonov derive the corresponding Euclidean package in Kapustin and Tikhonov 2009, §2.2, eqs. (5)–(6), arXiv PDF; its Euclidean theta and wall signs must be translated together if the Wick-rotation convention differs.

Let $p$ be magnetic and $q$ electric charge. In the active convention used here, transport gives

$$\binom{p}{q} \longmapsto \begin{pmatrix}0&1\\-1&0\end{pmatrix} \binom{p}{q}.$$

Thus a positive Wilson line $(0,1)$ becomes the same-oriented ’t Hooft line $(1,0)$, while a positive ’t Hooft line $(1,0)$ becomes the oppositely oriented Wilson line $(0,-1)$. Fusing two $S$ walls implements $S^2=-1$, namely charge conjugation, rather than the identity. The inverse wall carries $S^{-1}$.

On a manifold with boundary or nontrivial topology, extra zero modes, contact terms, and choices of global form can modify this simple local picture. The wall realizes a duality only between complete theories whose line lattices and flux sectors are mapped by $S$.

The global-form, line, anomaly, and wall map places these wall equations beside the active charge map and background-anomaly transport, while keeping the finite $\mathfrak{su}(2)$ line orbit in a separately labeled panel.

Invertibility from fusion

Let $\overline{\mathcal D}$ denote the orientation reverse. An invertible duality wall satisfies

$$\overline{\mathcal D}\circ\mathcal D\simeq\mathbf1_A, \qquad \mathcal D\circ\overline{\mathcal D}\simeq\mathbf1_B,$$

including possible declared invertible counterterm theories. This is stronger than finding a wall with the desired classical boundary equations.

For a finite zero-form, group-like schematic, if instead

$$\overline{\mathcal D}\circ\mathcal D =\sum_{g\in G}\mathcal U_g,$$

the wall is noninvertible. Crossing it loses information, and on states the normalized average $|G|^{-1}\sum_g\widehat{\mathcal U}_g$ is the projector onto invariants. For a $3+1$-dimensional one-form half-gauging wall, reverse fusion is instead a topology-dependent condensation sum over symmetry surfaces, as explained on the generalized-symmetry page. Such a defect may encode a powerful relation between theories, but it is not an equivalence functor without restricting or enlarging the category.

Fusion coefficients can themselves be three-dimensional TQFTs; ignoring such a factor can let a claimed inverse pass local bulk checks while failing on junctions and ground-state sectors Choi et al. 2023, §3.1, arXiv PDF. More generally, coupling a QFT to a TQFT can preserve local operators while changing global and extended-operator data Kapustin and Seiberg 2014, §§2–3 and 7, arXiv PDF.

Supersymmetric interfaces

For a supersymmetric wall, left and right supercharges are related by a projector involving the wall normal and possibly an R-symmetry rotation. A half-BPS interface in a four-dimensional $\mathcal N=2$ theory preserves three-dimensional $\mathcal N=2$ supersymmetry; a half-BPS interface in four-dimensional $\mathcal N=4$ can preserve three-dimensional $\mathcal N=4$.

Wall degrees of freedom can cancel boundary variations, mediate the operator map, and contribute anomaly inflow. In $\mathcal N=4$ SYM, an $S$-duality wall is associated with a three-dimensional theory often denoted $T[G]$, coupled to the gauge fields on both sides. Its Higgs- and Coulomb-branch symmetries are generically $G$ and the Langlands dual $G^\vee$, with faithful adjoint-form refinements; they couple to the electric and magnetic bulk data Gaiotto and Witten 2009, §4.1, arXiv PDF.

Supersymmetric partition functions with an interface can test the induced kernel on boundary states. Equality of such kernels is strong protected evidence, but invertibility on the full Hilbert space still requires fusion and unprotected sectors to be controlled.

Anomalies and wall consistency

If the bulk background responses differ across a wall, their inflow leaves an anomaly on $W$. A consistent interface must supply wall degrees of freedom or a counterterm whose anomaly equals the difference:

$$\mathcal A_W =\mathcal A_A- f^*\mathcal A_B.$$

For a true duality wall between theories with matched anomalies, this difference is trivial in the appropriate anomaly group after allowed counterterms. A nontrivial remainder signals either a relative interface attached to a higher-dimensional bulk or an incomplete duality map.

Gauge anomalies are stricter: an uncanceled anomaly for a dynamical gauge transformation makes the interface inconsistent, not merely non-topological.

Interfaces as integral transforms

On a spatial slice cut by the wall, the interface defines a linear map

$$\widehat{\mathcal D}:\mathcal H_A\to\mathcal H_B.$$

In a field basis, it can be represented by a kernel $K[\phi_A,\phi_B]$. Composition of interfaces integrates over the intermediate boundary data:

$$K_{CA}[\phi_C,\phi_A] =\int\mathcal D\phi_B\, K_{CB}[\phi_C,\phi_B]K_{BA}[\phi_B,\phi_A].$$

For an invertible wall, composing with the inverse gives the identity kernel after gauge volumes, zero modes, and contact terms are included. This formulation turns fusion into a calculable completeness test.

Common pitfalls

Calling every transparent classical boundary condition a duality wall. Quantum measures, anomalies, global sectors, and fusion can obstruct invertibility.

Checking local operators but not lines crossing the wall. Global form is visible precisely in which extended operators can pass, end, or split.

Confusing a second $S$ wall with the inverse. $\mathcal D_S\circ\mathcal D_S=\mathcal D_C$ implements charge conjugation, whereas $\mathcal D_{S^{-1}}\circ\mathcal D_S=\mathbf1$. For a noninvertible wall, even orientation-reverse fusion can yield a TQFT, projector, or sum of symmetry defects.

Exercises

An interface $\mathcal D$ obeys

$$\overline{\mathcal D}\circ\mathcal D =\mathbf1+\mathcal U,$$

where $\mathcal U$ is a nontrivial invertible symmetry defect satisfying $\mathcal U^2=\mathbf1$.

1. Is $\mathcal D$ invertible on the full theory?
2. What operation does the fusion resemble?
3. What additional information is needed to decide whether it becomes invertible after restricting sectors?

Solution

It is not invertible on the full theory because fusion with the orientation reverse does not give a single identity defect. On states, $\tfrac12(\mathbf1+\widehat{\mathcal U})$ is the normalized projector onto the $\mathcal U$-invariant sector; the unnormalized fusion coefficient $\mathbf1+\mathcal U$ records the two defect summands. To study a restricted equivalence, one must know how $\mathcal U$ acts on objects and morphisms, which sector the projector selects, and whether the restricted wall map is fully faithful and essentially surjective.

References

• 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 (2023): 489–542. DOI. Open PDF.
• Gaiotto, Davide, and Edward Witten. “S-Duality of Boundary Conditions in $\mathcal N=4$ Super Yang–Mills Theory.” Advances in Theoretical and Mathematical Physics 13 (2009): 721–896. DOI. Open PDF.
• Kapustin, Anton, and Mikhail Tikhonov. “Abelian Duality, Walls and Boundary Conditions in Diverse Dimensions.” Journal of High Energy Physics 11 (2009): 006. DOI. Open PDF.
• Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 04 (2014): 001. DOI. Open PDF.