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, while boundaries and interfaces supplies the variational problem. Helpful background. Fusion, junctions, and endpoints and noninvertible gauging constructions clarify composition.

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. Equivalently, the discontinuity of the normal momentum flux is a wall-exact operator. 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 local operators, topological movement preserves OPEs:

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

For lines and surfaces, the transport also must preserve linking, fusion, spins, and allowed endpoints. A Wilson line crossing an electric–magnetic wall may become an ’t Hooft line; its charge changes by the same integral symplectic matrix as the bulk periods.

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

For four-dimensional abelian gauge fields $A_L$ and $A_R$ on the two sides, an $S$ wall can carry the three-dimensional coupling

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

in Lorentzian signature, with the corresponding factor of $i$ in the Euclidean path integral. Varying the bulk-plus-wall action gives boundary conditions that exchange electric and magnetic flux. The coupling is well-defined under large gauge transformations only with the correct integral normalization. The Abelian wall action and its boundary conditions are derived in Kapustin and Tikhonov 2009, §§2.1–2.3.

Transporting a charge column through the wall gives

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

Thus a Wilson line becomes an oppositely oriented ’t Hooft line in this convention. Fusing two $S$ walls implements $S^2=-1$, namely charge conjugation, rather than the identity. The inverse wall is orientation-reversed and 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 $S$.

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.

If instead

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

the wall is noninvertible. It can implement gauging or condensation: crossing it loses information and the reverse operation returns a projector onto $G$-invariants, not the original state. Such a defect may encode a powerful relation between theories, but it is not an equivalence functor without restricting or enlarging the category.

Fusion can also leave a decoupled three-dimensional topological theory on the wall. If that factor is ignored, a claimed inverse can pass local bulk checks while failing on junctions and ground-state degeneracy. Examples of precisely such QFT–TQFT couplings and their effects on line operators are analyzed in Kapustin and Seiberg 2014, §§2–3 and 7.

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 a three-dimensional $\mathcal N=2$ or $\mathcal N=4$ subalgebra depending on its construction.

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 two $G$ symmetries couple electrically and magnetically, providing a nonabelian refinement of the BF wall; see Gaiotto and Witten 2009, §3.5 and §4.

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.

Assuming orientation reversal is automatically an inverse. Fusion may yield charge conjugation, a topological factor, a projector, or a 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.

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. The sum resembles a gauging or condensation projector over a two-element symmetry. 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

• 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. arXiv:0807.3720.
• Kapustin, Anton, and Mikhail Tikhonov. “Abelian Duality, Walls and Boundary Conditions in Diverse Dimensions.” Journal of High Energy Physics 11 (2009): 006. arXiv:0904.0840.
• Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 04 (2014): 001. arXiv:1401.0740.