Topological interface between specified models

Use the selected-model and defect-insertion data from the preceding defect definition: a model specifies its manifold, background and boundary data, fields with regularity conditions, unit conventions, complete admissible insertion inputs, correlator, and topology on its value space. Let ${\mathcal{T}}_{-}$ and ${\mathcal{T}}_{+}$ be two such models in the same spacetime dimension $d$ on the same smooth manifold $M$. Their metric or signature, background bundles, boundary conditions, field normalizations, and units must either agree or be related by an explicitly supplied identification. No such identification is inferred from names or equal dimensions.

A codimension-one interface from ${\mathcal{T}}_{-}$ to ${\mathcal{T}}_{+}$ is data $(\mathcal{W},\mathcal{I})$ where $\mathcal{W}\hookrightarrow M^{\circ }$ is a closed smooth embedded hypersurface with a chosen coorientation and a specified collar $\mathcal{W}\times (-ϵ,ϵ)$, $ϵ>0$. The negative side uses ${\mathcal{T}}_{-}$ and the positive side uses ${\mathcal{T}}_{+}$. The interface data $\mathcal{I}$ is a specified local transmission condition, operator map, or boundary condition relating the one-sided field data in that collar, together with any orientation, framing, or junction labels required by the models. The combined model declares its admissible complete inputs, including combined background and boundary data and all other finite insertion data on each side. Write ${\mathrm{\Xi }}_{-}$ and ${\mathrm{\Xi }}_{+}$ for those ordered lists; each entry is an ordinary observable insertion or other model-admitted decorated defect/network insertion. The combined model assigns each such input a correlator $C_{{\mathcal{T}}_{-},{\mathcal{T}}_{+}}(b_{-};b_{+};\mathcal{I}(\mathcal{W});{\mathrm{\Xi }}_{-};{\mathrm{\Xi }}_{+})$. The interface is topological when, for every such input, this correlator is unchanged under every allowed decorated isotopy of the interface that preserves the source and target side labels and incidence data and crosses no other insertion.

For interfaces ${\mathcal{I}}_{01}:{\mathcal{T}}_0\to {\mathcal{T}}_1$ and ${\mathcal{I}}_{12}:{\mathcal{T}}_1\to {\mathcal{T}}_2$, define their fusion only when the model supplies a product collar containing two parallel interfaces at separation $\delta >0$ and a common Hausdorff topological value space $\mathcal{V}$ for the compared correlators. If the selected models use different value spaces, the model must also specify comparison maps into $\mathcal{V}$; convergence below means convergence after those maps, in the topology of $\mathcal{V}$. For every declared complete input with the pair and all other finite insertion data, including ordinary observables and other decorated defects or networks, held fixed outside the shrinking collar, the model must specify $C_{\delta }\in \mathcal{V}$ and a correlator $C_0\in \mathcal{V}$ of a single interface such that $C_{\delta }\to C_0$ as $\delta \downarrow 0$. The model also specifies the limiting insertion's normalization and junction data; when all these requirements hold, it is the fused interface ${\mathcal{I}}_{12}\star {\mathcal{I}}_{01}:{\mathcal{T}}_0\to {\mathcal{T}}_2$. If the comparison topology or the required correlator limits are absent, no single fused interface is asserted.

An endodefect has ${\mathcal{T}}_{-}={\mathcal{T}}_{+}$ as the same complete model data, not merely models with similarly named fields. The transparent identity interface ${\mathbf{1}}_{\mathcal{T}}$ is the model-supplied transmission condition whose correlator equals the no-interface correlator for every complete admissible input. For an endodefect $\mathcal{I}$ to induce a symmetry, its data must specify a bijection ${\alpha }_{\mathcal{I}}$ on the model's typed local-observable and other-insertion data and a bijection ${\beta }_{\mathcal{I}}$ on its combined background and boundary data; these maps must preserve types, supports, algebraic relations, and admissibility. They preserve correlators precisely when, for every admissible input $(b;\mathbf{O},D)$,

$C_{\mathcal{T}}({\beta }_{\mathcal{I}}(b);{\alpha }_{\mathcal{I}}(\mathbf{O},D))=C_{\mathcal{T}}(b;\mathbf{O},D)\mathrm{.}$

An endodefect is an invertible symmetry interface only if a two-sided inverse exists under the stated fusion operation, both composites equal ${\mathbf{1}}_{\mathcal{T}}$ in all declared correlators, and the induced bijections above are automorphisms preserving correlators. A topological interface between distinct source and target models is an interface, not an internal symmetry of either model. No further property is implied by this definition; any additional claim must be formulated and verified for the specified model.

No dimensional quantity is added by this definition. Each model and interface must retain its declared unit system and normalization, and this definition does not assert that a continuum fusion limit exists.

Topological defect insertion in a specified model

Gaiotto, Kapustin, Seiberg, and Willett, §1, PDF pp. 1–3

Carqueville, §2.1, PDF pp. 5–8

The diagram records only the directed source-to-target convention: the label $\mathcal{I}$ has source ${\mathcal{T}}_{-}$ and target ${\mathcal{T}}_{+}$. It does not assert topologicality, existence of a fusion limit, or invertibility.