Boundaries, Defects, and Extended Operators in TQFT

In an extended TQFT, a boundary condition is lower-categorical data compatible with the bulk point object, a domain wall is a morphism between bulk theories, and a junction is a higher morphism between walls. Composition is geometric gluing and becomes relative tensor product or composition in the target. In the elementary semisimple two-dimensional algebraic model, boundary conditions are modules, walls are bimodules, and boundary-changing point operators are module homomorphisms. An arbitrary label or vector space is not a boundary condition until the required action, adjoints, traces, and sewing relations are supplied.

Required background. Fully extended TQFTs type the values on lower-dimensional strata; Atiyah–Segal gluing supplies the sewing law; and operators, boundaries, and relative topological theories supply the physical topological-operator interpretation.

Helpful background. Boundaries, interfaces, and domain walls own the physical gauge-theory treatment, while interfaces, folding, and fusion supply the defect-CFT comparison.

Strata as morphisms in the target

Fix an $n$-dimensional fully extended theory with point value $X$ in a symmetric monoidal $(\infty,n)$-category $\mathcal C$. A boundary condition can be represented, with an orientation convention fixed, by a $1$-morphism

$$B:\mathbb 1\longrightarrow X.$$

A wall from a bulk theory $X$ to another theory $Y$ is a $1$-morphism $W:X\to Y$. A codimension-two junction between walls is a $2$-morphism, and the pattern continues with codimension. Gluing adjacent walls is their categorical composition. Dual walls arise by reversing their normal orientation when the required adjoints exist.

The folding construction turns a wall $X\to Y$ into a boundary condition for the folded product $X^\vee\otimes Y$. This is an equivalence of descriptions only after orientation reversal, tensor-product conventions, and the target equivalence are specified. It does not identify the two physical bulks.

In the Morita $2$-category of algebras, the dictionary is concrete:

$$\begin{array}{c|c} \text{topological stratum}&\text{Morita value}\\ \hline \text{bulk region}&\text{algebra }A\\ \text{wall }A\to B&{}_B M_A\text{ bimodule}\\ \text{point junction}&\text{bimodule homomorphism}. \end{array}$$

Fusion of walls is relative tensor product,

$${}_C N_B\circ{}_B M_A \longmapsto N\otimes_BM.$$

Associativity is canonical only up to the coherent isomorphism appropriate to the bicategory. Treating it as literal equality can erase the coherence data used by a fully extended functor.

Module boundaries in two dimensions

Let $A$ be a finite-dimensional semisimple Frobenius algebra. In the elementary open–closed model, a left $A$-module $M$ supplies a boundary condition: a bulk operator $a\in A$ approaching the boundary acts by

$$\rho_M(a):M\longrightarrow M.$$

For two boundary conditions $M$ and $N$, the topological boundary-changing operators form

$$\mathcal H_{M,N}=\operatorname{Hom}_A(M,N).$$

Composition of junctions is composition of module maps. Sewing a bulk pair of pants to a boundary is compatible with associativity exactly because

$$\rho_M(ab)=\rho_M(a)\rho_M(b), \qquad \rho_M(1)=\mathrm{id}_M.$$

A complete open–closed TQFT also requires nondegenerate open-sector traces and the Cardy sewing relation; module data alone describe the basic boundary action, not every open–closed correlator. Moore and Segal formulate the bordism sewing conditions and the resulting category of boundary conditions in Moore and Segal 2006, §§2.4–2.5 and Theorem 1, pp. 9–13.

For the exact first application, choose

$$A=\mathbb C p_1\oplus\mathbb C p_2, \qquad p_ip_j=\delta_{ij}p_i.$$

Simple modules $M_1$ and $M_2$ are defined by $p_iM_j=\delta_{ij}M_j$. Hence

$$\operatorname{Hom}_A(M_i,M_j) \cong \begin{cases} \mathbb C,&i=j,\\ 0,&i\ne j. \end{cases}$$

The same-boundary junction has one topological channel, whereas no topological junction changes between the two superselection supports in this minimal model. Direct sums generate more general finite semisimple boundaries. This calculation is returned to boundaries, interfaces, and domain walls as the exact module-and-junction skeleton; dynamical wall excitations and non-topological correlators remain there.

An independent gluing check fuses an $A$–$B$ wall $M$ with a $B$–$C$ wall $N$ and then with a $C$–$D$ wall $P$. The two parenthesizations are canonically isomorphic:

$$(P\otimes_CN)\otimes_BM \cong P\otimes_C(N\otimes_BM).$$

This is the target image of two diffeomorphic ways to cut the stratified bordism.

The missing-action failure

Take a vector space $V$ and label a boundary by $V$ without specifying $\rho_V:A\to\operatorname{End}(V)$. A bulk insertion can then be brought to the boundary along two pair-of-pants decompositions, but there is no equation relating the two results. Even an arbitrary linear assignment $a\mapsto T_a$ fails unless

$$T_{ab}=T_aT_b, \qquad T_1=\mathrm{id}_V.$$

Thus gluing the bulk pair of pants to the boundary becomes ill-defined. The strongest surviving datum is a vector-space label on a chosen decomposition, not a functorial boundary condition. If an action is supplied but the open trace or Cardy relation fails, the module boundary may survive while the claimed full open–closed TQFT does not.

This page treats the algebraic topological skeleton. Physical line and surface operators, defect correlators, renormalization, and generalized-symmetry completeness require their separate owners; an extended target does not prove that all physical defects have been found.

Exercises

Compute the junction space between $M_1\oplus M_2$ and $M_1$ in the example.

Solution

An $A$-linear map preserves the idempotent support. The $M_2$ summand therefore maps to zero, while the $M_1$ summand maps by an arbitrary scalar. Hence $\operatorname{Hom}_A(M_1\oplus M_2,M_1)\cong\mathbb C$.

Why is an $A$–$B$ bimodule more appropriate for a wall than an ordinary vector space?

Solution

Bulk operators from the two sides must approach and act on the wall compatibly. The left $B$-action and right $A$-action encode those operations, and their commutation lets insertions from opposite sides pass each other topologically. A bare vector space carries neither action.

References

• Douglas, Christopher L., Christopher Schommer-Pries, and Noah Snyder. “Dualizable Tensor Categories.” Memoirs of the American Mathematical Society 268, no. 1308 (2020). DOI; Open PDF.
• Moore, Gregory W., and Graeme Segal. “D-Branes and K-Theory in 2D Topological Field Theory.” arXiv (2006). Open PDF.
• Schommer-Pries, Christopher J. The Classification of Two-Dimensional Extended Topological Field Theories. PhD thesis, University of California, Berkeley, 2009; expanded version 2014. Open PDF.