Boundary Conditions and Interfaces in Functorial Field Theory

In functorial field theory, a boundary condition extends a theory to bordisms with a labeled boundary, while an interface is a boundary condition for the folded product of one theory with the orientation reverse of another. In finite topological models, boundaries are module categories, interfaces are bimodule categories, and composition is a relative tensor product. A valid composite additionally requires matching module actions, anomalies, tangential structures, and adjoints.

Required background. Boundaries, Defects, and Extended Operators in TQFT supplies extended bordisms. Defects on Stratified Spacetimes supplies junction coherence. Fusion Categories, Module Categories, and Bimodule Defects supplies relative tensor products. Helpful background. Boundaries, Interfaces, and Domain Walls supplies the physical distinctions, while Interfaces, Folding, and Fusion gives the conformal folding comparison.

Boundary extensions and folding

Let $Z:\operatorname{Bord}_d^\xi\to\mathcal V$ be a symmetric monoidal field theory with tangential structure $\xi$. A boundary condition is not merely a state in one Hilbert space: it is a coherent extension of $Z$ to bordisms whose free boundary components carry that label. Gluing a collar to the boundary must act as the identity, and gluing corners must agree with categorical composition.

An interface from theory $Z_1$ to $Z_2$ can be folded. Reverse the orientation on the $Z_2$ side and regard the wall as a boundary condition for

$$Z_1\boxtimes Z_2^{\mathrm{op}}.$$

The operation reverses orientation and therefore conjugates or dualizes the relevant tangential and anomaly data; it is not a purely pictorial reflection.

In a finite semisimple setting, let $\mathcal C$ and $\mathcal D$ encode topological lines in the two phases. A wall is a $(\mathcal C,\mathcal D)$-bimodule category $\mathcal M$. For a second $(\mathcal D,\mathcal E)$ wall $\mathcal N$, fusion along the shared phase is

$$\mathcal M\boxtimes_{\mathcal D}\mathcal N.$$

The balancing relation identifies the right action of $d\in\mathcal D$ on $\mathcal M$ with its left action on $\mathcal N$. Carqueville and Runkel develop algebras, bimodules, tensor products, and their defect interpretation in Carqueville and Runkel 2016, §§2.2–2.3 and 3.2, printed pp. 11–19 (PDF).

The orientation-reversed wall is a candidate adjoint $\mathcal M^\vee$. Evaluation and coevaluation walls must satisfy the two triangle identities. If

$$\mathcal M\boxtimes_{\mathcal D}\mathcal M^\vee\simeq\mathcal C, \qquad \mathcal M^\vee\boxtimes_{\mathcal C}\mathcal M\simeq\mathcal D,$$

then $\mathcal M$ is invertible and implements a Morita equivalence. An arbitrary wall can be dualizable without being invertible, just as a finite-dimensional vector space has a dual without being one-dimensional.

A finite-gauge interface and its adjoint

Let $G$ and $H$ be isomorphic finite groups and choose $\varphi:G\to H$. The pointed fusion categories $\mathrm{Vec}_G$ and $\mathrm{Vec}_H$ describe graded topological lines. Define a bimodule category $\mathcal M_\varphi$ whose simple labels form the $(G,H)$-bitorsor $H$: $H$ acts by right multiplication, while $g\in G$ acts on the left through $\varphi(g)$.

The orientation reverse is $\mathcal M_{\varphi^{-1}}$. Balancing over $\mathrm{Vec}_H$ cancels the intermediate $H$ action, giving

$$\mathcal M_\varphi\boxtimes_{\mathrm{Vec}_H} \mathcal M_{\varphi^{-1}}simeq\mathrm{Vec}_G.$$

The opposite composition returns $\mathrm{Vec}_H$. The bitorsor freeness supplies the independent invertibility check: each group element acts freely and transitively, so no nontrivial residual wall sector remains after composition.

This is the exact mathematical application returned to Boundaries, Interfaces, and Domain Walls. There the wall may also carry localized degrees of freedom and have non-topological dynamics. The bitorsor calculation establishes the finite topological interface and its adjoint, not a universal claim about gauge-theory domain walls.

Matching conditions and failure

The shared theory must match on both sides of a composite. If the right $\mathcal D$-action on $\mathcal M$ and left $\mathcal D$-action on $\mathcal N$ differ by an uncancelled anomaly, the balancing is not a physical quotient. The same problem occurs if one wall requires spin bordisms and the other is defined only as an oriented theory, or if their pivotal structures disagree.

The adversarial case composes two interfaces because their ungraded fusion multiplicities fit, while their $\mathcal D$-actions have different associator cocycles. The relative tensor product may exist after forgetting the cocycles, but it does not define the proposed field-theory wall. The strongest surviving conclusion is a composition in a reduced algebraic category. A physical composite requires an explicit equivalence of the full shared actions and cancellation of the interface anomaly.

Transparent, invertible, and merely topological walls should be separated. A transparent wall is identified with the identity bimodule and acts trivially on every bulk and boundary datum. An invertible wall has a two-sided inverse but can implement a nontrivial autoequivalence on line operators. A noninvertible topological wall still composes coherently, yet its product with the reverse wall contains additional summands. Confusing these notions incorrectly turns duality walls into invisible interfaces.

For a quantitative composition, one should verify Frobenius–Perron dimensions. In the finite semisimple case an invertible bimodule preserves the global dimension of the Morita-equivalent centers. Extra simple summands after fusion signal noninvertibility. This numerical check cannot replace the balancing equivalence, but it efficiently detects many incorrect claims of an adjoint being an inverse.

Exercises

Why does replacing the bitorsor by a nontransitive $(G,H)$-biset prevent invertibility?

Solution

A nontransitive biset has more than one orbit. After composing with its reverse, orbit labels survive as additional summands, so the result is not equivalent to the transparent regular bimodule. The wall may still be composable, but it is not invertible.

References

• Carqueville, Nils, and Ingo Runkel. “Orbifold Completion of Defect Bicategories.” Quantum Topology 7 (2016): 203–279. DOI; Open PDF.
• Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. “Fusion Categories and Homotopy Theory.” Quantum Topology 1 (2010): 209–273. DOI; Open PDF.