Defects on Stratified Spacetimes and Higher-Categorical Composition

A defect field theory assigns data not only to a smooth spacetime but to a spacetime decomposed into compatible strata. In two dimensions, regions carry phases, oriented lines carry interfaces, and points carry junction operators. Gluing regions, lines, and points becomes composition at three categorical levels. The construction is coherent only when every junction label is compatible with the cyclically ordered incident defects and when reassociation obeys the pentagon identity; a list of strata and labels alone is not yet a field theory.

Required background. Boundaries, Defects, and Extended Operators in TQFT supplies the extended-bordism assignment. Corners, Stratification, and Higher-Codimension Data explains why new compatibility data appear at each codimension. Boundaries, Interfaces, and Domain Walls supplies the physical distinction among phases, interfaces, and boundary conditions. Helpful background. Conformal Boundaries and Defects gives a non-topological comparison in which only selected defect motions are allowed.

Strata, links, and admissible labels

Let $X$ be a $d$-manifold equipped with a filtration

$$X_0\subset X_1\subset\cdots\subset X_d=X,$$

where $X_k\setminus X_{k-1}$ is a smooth $k$-manifold and a neighborhood of each point is locally a product of a disk with the cone on a stratified link. The link records which higher-dimensional strata meet the point and in what order. A label on a codimension-$r$ stratum is therefore not free-standing: its source and target are determined by the labeled link surrounding it.

For a two-dimensional oriented defect TFT, this local geometry is encoded by a pivotal bicategory $\mathcal B$:

• objects $a,b,\ldots$ label two-dimensional phases;
• a one-morphism $X:a\to b$ labels an oriented line defect, with orientation reversal represented by an adjoint $X^\dagger:b\to a$;
• a two-morphism $\phi:X\Rightarrow Y$ labels a point junction between parallel defect composites.

Carqueville and Runkel formulate precisely this passage from defect bordisms to a pivotal bicategory in Carqueville and Runkel 2016, §§2.1 and 3.1–3.2, printed pp. 8–19 (PDF). Their planar string diagrams are not decoration: the regions, oriented lines, and vertices reproduce the link of each stratum and make source–target compatibility visible.

Suppose a point has incident defects $X_1:a_0\to a_1,\ldots,X_n:a_{n-1}\to a_n$ and an outgoing defect $Y:a_0\to a_n$. An admissible junction lies in

$$\operatorname{Hom}_{\mathcal B(a_0,a_n)} \bigl(X_n\otimes\cdots\otimes X_1,Y\bigr).$$

If the intermediate phase of $X_i$ does not equal the source phase of $X_{i+1}$, the composite is not an object of any hom-category. There is then no junction vector space to choose from. This is the categorical form of the link condition.

Gluing and higher coherence

Horizontal composition fuses adjacent defect lines; vertical composition brings successive junctions together. Neither is strictly associative in a general bicategory. Instead there is a natural isomorphism

$$\alpha_{X,Y,Z}:(Z\otimes Y)\otimes X \xrightarrow{\ \cong\ } Z\otimes(Y\otimes X).$$

The pentagon identity says that the two composites of associators relating fourfold parenthesizations agree. Geometrically, it identifies two isotopies that shrink four adjacent junction networks in different orders. Unit constraints encode insertion or removal of a transparent line. The triangle identity makes unit removal compatible with reassociation. Adjunction zig–zag identities encode cancellation of a small cup–cap pair. These are local gluing laws, not statements about the spectrum or dynamics of a non-topological defect.

The exact first application belongs to Fusion, Junctions, and Endpoints. Take phases $a,b,c$, walls $X:a\to b$ and $Y:b\to c$, and point junctions that replace short segments of a network. Composing first on the left gives $((Z\otimes Y)\otimes X)$, while composing first on the right gives $Z\otimes(Y\otimes X)$. Inserting $\alpha_{X,Y,Z}$ identifies the two isotopic networks. For a fourth wall, the pentagon is exactly the condition that all five reassociation paths give one amplitude.

This mechanism also explains the hierarchy of defects in higher dimensions. Codimension one defects are one-morphisms, codimension two junctions are two-morphisms, and further strata continue the pattern in an $(\infty,d)$-category. A fully extended relative theory can be regarded as boundary data for a bulk theory, but that conclusion requires a specified bordism category and target; Freed and Teleman 2014, §2, printed pp. 2–7 (PDF) give the relative-theory formulation and explain its boundary interpretation.

Failure at an incompatible junction

Consider $X:a\to b$ and $Y:c\to d$ with $b\ne c$. Drawing the two lines so that they meet does not define $Y\otimes X$: the missing equality $b=c$ is a typing failure, not a small anomaly that can be repaired by choosing an $F$-symbol. Likewise, a cyclic point junction whose incident phase labels do not close has no admissible link label. The strongest surviving statement is that one has two separately valid defects. A new interface $K:b\to c$, or a change of phase labeling, is required before composition can even be posed.

Topological isotopy invariance is another independent hypothesis. In a conformal defect theory, bending a defect may be allowed while translating it across local insertions is not. The bicategorical composition of topological junctions does not automatically classify conformal defect correlators or their Ward identities.

Exercises

Let $X:a\to b$, $Y:b\to c$, and $Z:c\to d$. Identify the hom-category containing both parenthesizations of their composite and state what relates them.

Solution

Both $(Z\otimes Y)\otimes X$ and $Z\otimes(Y\otimes X)$ are one-morphisms from $a$ to $d$, hence objects of $\mathcal B(a,d)$. They are related by the associator $\alpha_{X,Y,Z}$; they need not be literally equal.

References

• Carqueville, Nils, and Ingo Runkel. “Orbifold Completion of Defect Bicategories.” Quantum Topology 7 (2016): 203–279. DOI; Open PDF.
• Freed, Daniel S., and Constantin Teleman. “Relative Quantum Field Theory.” Communications in Mathematical Physics 326 (2014): 459–476. DOI; Open PDF.