Topological defect insertion in a specified model

Fix an integer $d\ge 1$ and a smooth $d$-manifold $M$, possibly with boundary. Fix the metric and signature, orientation and any time orientation, the geometric background bundles or lifts required by the model, and the boundary conditions. A selected model $\mathcal{T}$ on $M$ consists of a specified configuration space ${\mathcal{C}}_{\mathcal{T}}(M)$ of fields, including each field bundle, regularity class, and admissible boundary behavior; a collection of local observables with their types and supports; a set ${\mathcal{B}}_{\mathcal{T}}$ of admissible combined background and boundary data; and a set ${\mathcal{A}}_{\mathcal{T}}$ of admissible complete insertion inputs. An element of ${\mathcal{A}}_{\mathcal{T}}$ records data $b\in {\mathcal{B}}_{\mathcal{T}}$ and a finite ordered list whose entries are local observable insertions or other model-declared insertion data, together with the model's admissibility conditions on supports and labels. The model supplies a value space ${\mathcal{V}}_{\mathcal{T}}$ with a specified topology and a correlator functional $C_{\mathcal{T}}:{\mathcal{A}}_{\mathcal{T}}\to {\mathcal{V}}_{\mathcal{T}}$ with an explicit normalization convention, which may be normalized or unnormalized; the notation $C_{\mathcal{T}}(b;D;X_1,\dots ,X_n)$ denotes its value on the complete input with combined background and boundary data $b$, defect data $D$, and other insertion data $X_i$. The model also fixes its unit system, action and charge normalizations, and any regulator, measure, or renormalization prescription used to define $C_{\mathcal{T}}$. These are model inputs: this definition does not construct them or assume a general interacting continuum theory exists.

A defect insertion in $\mathcal{T}$ is data $(j,\lambda )$ where $j:\mathrm{\Sigma }\hookrightarrow M^{\circ }$ is a smooth embedding of a manifold $\mathrm{\Sigma }$ without boundary whose image is closed in the interior $M^{\circ }=M\setminus \mathrm{\partial }M$, and $\lambda$ is the complete model-specified field, operator, or boundary-condition data supported on $j(\mathrm{\Sigma })$, including any orientation, coorientation, framing, or label-transport rule the model requires. Its codimension is $d-\dim \mathrm{\Sigma }$. It is an allowed entry of ${\mathcal{A}}_{\mathcal{T}}$ only when the model defines $C_{\mathcal{T}}(b;D_{j,\lambda };O_1,\dots ,O_n)$ for each stated admissible combined background and boundary datum $b$ and list of other insertions.

A defect network is data $D=(K,\{S_s,j_s,{\lambda }_s{\}}_{s\in S},\{J_t{\}}_{t\in J})$ in which $K\subset M^{\circ }$ is closed, $S$ is finite, $K={⨆}_{s\in S}{S}_s$ is a smooth stratification, and each $j_s:{\mathrm{\Sigma }}_s\to S_s$ is a diffeomorphism from a smooth manifold without boundary onto its locally closed embedded stratum. The closure of each stratum is a union of strata, with the frontier condition that if $S_t$ meets the closure of $S_s$ and $S_t\mathrm{\ne }{S}_s$, then $S_t$ lies in that closure and has smaller dimension; the incidence relation records exactly these pairs. Each stratum carries its complete model-specified field, operator, boundary, orientation, coorientation, framing, and other label data ${\lambda }_s$ as required; $J\subseteq S$ is the set of junction strata, and each $S_t$ with $t\in J$ carries a model-specified junction operator $J_t$ with an ordering of all incident labels. The model must declare which such networks and other insertions form admissible inputs of ${\mathcal{A}}_{\mathcal{T}}$ and define their correlators. A one-stratum network with no junction stratum is the preceding single defect, and a list of strata or a picture without its field and junction data is not a defect insertion.

An isotopy of a defect network is a smooth one-parameter family of its stratum embeddings $j_{s,t}:{\mathrm{\Sigma }}_s\hookrightarrow M^{\circ }$, $t\in [0,1]$, compatible on stratum closures and incidence data, whose stratification and incidence relation are unchanged and which creates no additional intersections. The model-specified transport maps carry every stratum label ${\lambda }_s$, framing or orientation, and junction operator along the family; the combined background and boundary data remain fixed, and the network stays disjoint from every other insertion throughout the deformation. A single-stratum defect is the one-element special case. The insertion is topological in $\mathcal{T}$ if, for every complete admissible input containing the selected network, every other insertion data $X_1,\dots ,X_n$ in that input (each of which may itself be an observable or another defect/network), and every such decorated isotopy from $D_0$ to $D_1$ disjoint from all the other insertion data, the model-defined correlators satisfy

$C_{\mathcal{T}}(b;D_0;X_1,\dots ,X_n)=C_{\mathcal{T}}(b;D_1;X_1,\dots ,X_n)\mathrm{.}$

This is a property to verify for the specified model and insertion, not an automatic consequence of assigning a defect label or drawing a crossing. No physical quantity with units is introduced here; all dimensional quantities and normalizations remain those declared by $\mathcal{T}$.

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

Carqueville, §2.1, PDF pp. 5–8