Categorified braid-line model with an explicit realization boundary

Fix an integer $n\ge 2$, let $R=\mathbb{Q}[x_1,\dots ,x_n]$ with $\mathrm{deg}{x}_j=2$, and let $S_n$ act on $R$ by permuting variables. Let ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ be the idempotent-complete additive monoidal category of graded $(R,R)$-Soergel bimodules generated by the unit bimodule $R$, the bimodules $B_i=R{\otimes }_{R^{s_i}}R(1)$ for the adjacent transpositions $s_i=(ii+1)$, internal grading shifts, finite direct sums, tensor products over $R$, and direct summands. Use the shift convention $M(1{)}_d=M_{d+1}$; on the split Grothendieck ring it is recorded by $v[M]=[M(1)]$. Let $K^b({\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n)$ denote the category of bounded cochain complexes in ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ with chain maps modulo chain homotopy.

For $1\le i<n$, let $F_i=[B_i\stackrel{{\epsilon }_i}{\to }R(1)]$ in cohomological degrees $0,1$ and $F_i^{-1}=[R(-1)\stackrel{{\eta }_i}{\to }{B}_i]$ in degrees $-1,0$, with the maps and grading conventions of the positive and negative Rouquier generator complexes. For a signed word $\sigma ={\sigma }_{i_1}^{{ϵ}_1}\cdots {\sigma }_{i_r}^{{ϵ}_r}$, ${ϵ}_k\in \{+1,-1\}$, let $F(\sigma )=F_{i_1}^{{ϵ}_1}{\otimes }_R\cdots {\otimes }_R{F}_{i_r}^{{ϵ}_r}$, using signed tensor totalization and left-to-right bracketing; the empty word gives $R$ in degree zero. If $\beta \in B_n$ is the braid represented by $\sigma$, the Rouquier categorified line operator $F(\beta )$ is this complex considered up to the canonical homotopy equivalence between words representing $\beta$. Tensor product is the categorical stacking operation. The comparison maps give the unit and coherent multiplication maps $F(\beta ){\otimes }_{R}F(\gamma )\simeq F(\beta \gamma )$; equivalently, left tensoring defines the coherent braid-group action on $K^b(R\text{-grmod})$.

A physical defect realization is additional data. A monoidal category $\mathcal{D}$ here means a category with a tensor bifunctor, a unit object, associator and unit isomorphisms satisfying the pentagon and unit triangles; its objects are declared line-defect types, its morphisms are declared junction maps, and its tensor product is the chosen stacking operation. The realization is a strong monoidal functor $\mathcal{R}:K^b({\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n)\to \mathcal{D}$, meaning a functor with natural product and unit isomorphisms satisfying those same coherence equations and assigning images to every morphism, including junction maps, in its domain. Calling $\mathcal{D}$ a category of defects in a particular quantum field theory also requires an independently specified physical model. The Soergel and Rouquier data alone supply neither $\mathcal{D}$ nor this functor.

Split Grothendieck rings of the type-A Soergel categories

The positive and negative Rouquier generator complexes

The Rouquier complex of a braid word

The Rouquier complex is well defined up to canonical homotopy equivalence

Rouquier complexes form a coherent braid group action

The diagram shows the positive Artin generator on adjacent strands, with time increasing upward and the right input strand passing over the left input strand. Other strands are omitted. It records the braid-group convention only; the categorical operator assigned by this item is the Rouquier complex $F_i$, not a claim that the drawn crossing is itself a chain map or a physical trajectory. No proof or decategorification argument relies on the diagram.

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  \draw[white,line width=5pt] (1.4,0) .. controls (1.4,1.1) and (0,1.9) .. (0,3);
  \draw (1.4,0) .. controls (1.4,1.1) and (0,1.9) .. (0,3);
  \node at (0,-0.35) {$i$};
  \node at (1.4,-0.35) {$i+1$};
  \node at (0,3.3) {$i+1$};
  \node at (1.4,3.3) {$i$};
  \node at (0.7,1.5) {$\sigma_i$};
  \draw[->] (2.1,0.2) -- (2.1,2.8) node[midway,right] {$\text{time}$};
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The variable $v$ is a formal grading variable. If a later model interprets it as a charge or protected index, its exact power must be retained; this definition assigns no physical units or empirical meaning to $v$. A realization as topological or supersymmetric QFT defects requires the separate target category and functor stated above.