Categorified braid lines, grading and decategorification

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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  \node at (0.7,1.5) {$\sigma_i$};
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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.

Fix $n\ge 2$ and the type-A Rouquier categorical braid-line model. Let $\chi (C)={\sum }_m(-1{)}^m[C^m]$ be the alternating class of a bounded cochain complex in the split Grothendieck ring $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n)$, and use the shift convention $v[X]=[X(1)]$. Let $\mathrm{\Phi }:K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n)\to H_{S_n}$ be the algebra isomorphism with $\mathrm{\Phi }([B_i])=H_i=v(T_i+1)$ and Hecke parameter $q=v^{-2}$. For $\beta \in B_n$, let $e(\beta )$ be its exponent sum, the number of positive letters minus the number of negative letters in any signed word for $\beta$, and let $T_{\beta }$ be the image of $\beta$ under the group homomorphism $B_n\to H_{S_n}^{\times }$ sending ${\sigma }_i\mapsto T_i$. Then

Consequently the normalized map $\beta \mapsto v^{-e(\beta )}\mathrm{\Phi }(\chi (F(\beta )))$ is the standard Hecke braid homomorphism, and its normalization is multiplicative under braid stacking. Homotopy-equivalent complexes have the same Euler class, but need not be isomorphic as graded complexes.

Categorified braid-line model with an explicit realization boundary

Decategorification of a Rouquier complex is the Hecke braid generator

Euler classes of Rouquier complexes are homotopy invariant and multiplicative

The Rouquier complex is well defined up to canonical homotopy equivalence

The braid group by Artin presentation

Complexes, homotopies and contractibility in an additive category

Fix a field $\mathbb{k}$, a finite color set $I$, a symmetrizable Cartan matrix $A=(a_{ij}{)}_{i,j\in I}$, and positive integers $d_i$ satisfying $d_i{a}_{ij}=d_j{a}_{ji}$. Here $a_{ii}=2$, $a_{ij}\le 0$ for $i\mathrm{\ne }j$, and $a_{ij}=0$ exactly when $a_{ji}=0$. Fix a weight lattice $X$ (a free abelian group), simple roots ${\alpha }_i\in X$, and integer-valued pairings $⟨\lambda ,{\alpha }_i^{\vee }⟩$ with $⟨{\alpha }_j,{\alpha }_i^{\vee }⟩=a_{ij}$. Choose polynomials $Q_{ij}(u,v)\in \mathbb{k}[u,v]$ with $Q_{ii}=0$, $Q_{ij}(u,v)=Q_{ji}(v,u)$, and $Q_{ij}\mathrm{\ne }0$ for $i\mathrm{\ne }j$; each $Q_{ij}$ is homogeneous for $\mathrm{deg}u=2d_i$, $\mathrm{deg}v=2d_j$, of weighted degree $-2d_i{a}_{ij}$.

For $\nu ={\sum }_{i\in I}{m}_{i}i\in \mathbb{N}[I]$ of height $r={\sum }_i{m}_i$, a quiver Hecke algebra $R_Q(\nu )$ is the graded $\mathbb{k}$-algebra generated by idempotents $e(\mathbf{i})$ for sequences $\mathbf{i}=(i_1,\dots ,i_r)$ with content $\nu$, dots $y_1,\dots ,y_r$, and crossings ${\psi }_1,\dots ,{\psi }_{r-1}$. Put $s_k\mathbf{i}=(i_1,\dots ,i_{k+1},i_k,\dots ,i_r)$. The defining relations are

The displayed quotient is a polynomial because its numerator vanishes when $y_{k+2}=y_k$. The degrees are $\mathrm{deg}e(\mathbf{i})=0$, $\mathrm{deg}(y_{k}e(\mathbf{i}))=2d_{i_k}$, and $\mathrm{deg}({\psi }_{k}e(\mathbf{i}))=-d_{i_k}{a}_{i_k{i}_{k+1}}$.

For this definition, a small additive $\mathbb{k}$-linear graded abelian category has a set of objects and morphisms, finite biproducts, kernels and cokernels with every monomorphism a kernel and every epimorphism a cokernel, $\mathbb{k}$-vector-space Hom groups with bilinear composition, and a decomposition ${\mathrm{Hom}}_{\mathcal{C}}(M,N)={⨁}_{s\in \mathbb{Z}}{\mathrm{Hom}}_{\mathcal{C}}^s(M,N)$ with identities in degree zero and composition adding internal degrees. This internal degree is separate from any cohomological degree. An exact functor is a $\mathbb{k}$-linear functor preserving short exact sequences in the underlying abelian categories. A functor $G:\mathcal{C}\to \mathcal{D}$ between such graded categories is degree-preserving (degree zero) if

for every $M,N$ and $s\in \mathbb{Z}$. An adjunction $F⊣E$ is specified by natural unit and counit maps satisfying the two triangle identities. A natural transformation is a family of morphisms commuting with every source morphism; homogeneous degree $s$ means each component belongs to the internal degree-$s$ Hom space. If a proposed action uses internal shifts in an adjunction, those shifts and the degrees of its unit and counit maps are part of the specified data. For an abelian category $\mathcal{C}$, its exact Grothendieck group $K_0(\mathcal{C})$ is generated by isomorphism classes $[M]$ with relations $[B]=[A]+[C]$ for each short exact sequence $0\to A\to B\to C\to 0$.

A KLR categorical action contract specifies, for each $\lambda \in X$, such a small additive $\mathbb{k}$-linear graded abelian category ${\mathcal{C}}_{\lambda }$, degree-preserving exact functors $F_i:{\mathcal{C}}_{\lambda }\to {\mathcal{C}}_{\lambda -{\alpha }_i}$ with specified degree-preserving exact right adjoints $E_i:{\mathcal{C}}_{\lambda -{\alpha }_i}\to {\mathcal{C}}_{\lambda }$, and homogeneous natural transformations $x_i:F_i\Rightarrow F_i$ of degree $2d_i$ and ${\tau }_{ij}:F_i{F}_j\Rightarrow F_j{F}_i$ of degree $-d_i{a}_{ij}$. If a proposed adjunction convention uses internal shifts, those shifts and the degrees of its unit and counit maps must be stated explicitly. For each $\nu$ and $\lambda$, the direct sum of composite functors $F_{i_1}\cdots F_{i_r}$ over sequences of content $\nu$ must carry an action of $R_Q(\nu )$: $e(\mathbf{i})$ is projection to the indicated sequence summand, $y_k$ acts by the dot on the $k$th factor, and ${\psi }_k$ acts by the crossing on adjacent factors, all with the relations displayed above. Functor-composition order and grading shifts must be fixed in the proposed action and used consistently.

This contract records required data and checks; it does not assert that any such categories, functors, or transformations exist. A claim of an integrable action additionally states and proves local nilpotence: for every object $M$ and color $i$, some powers $E_i^N(M)$ and $F_i^N(M)$ vanish. A claimed highest-weight action additionally specifies a weight $\lambda$, a chosen object in ${\mathcal{C}}_{\lambda }$ killed by every $E_i$, and the precise subcategory generated from it by the $F_i$, shifts, finite sums, and summands. To assert that the induced operators on $K_0(\mathcal{C}):={⨁}_{\lambda \in X}{K}_0({\mathcal{C}}_{\lambda })$ form a representation of $U_q(\mathfrak{g})$, the contract must identify the exact Grothendieck-group convention, establish every required quantum-group relation, and cite a decategorification theorem whose hypotheses match the specified categories and functors. Merely satisfying the positive KLR relations does not establish those extra conclusions.

A physical interpretation as line-changing defects requires a separately specified defect 2-category, whose objects are declared defect sectors, whose 1-morphisms are declared defects changing sectors, and whose 2-morphisms are declared junction maps. A 2-functor here assigns objects, functors, and natural transformations while preserving identities and both compositions; a weak target additionally requires its coherence isomorphisms and equations. No physical defect realization is part of this mathematical contract.

The diagram depicts the generating 2-morphism that exchanges adjacent functor factors. The lower labels are the source order $(i,j)$ and the upper labels the target order $(j,i)$; time/composition is read upward. The crossing is the formal natural transformation ${\tau }_{ij}$, not an invertible topological braid crossing. The overpass convention is only a drawing convention and carries no extra algebraic relation; the displayed KLR relations, not the picture, define the action.

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The Khovanov–Lauda and Rouquier presentations use convention choices for root data, grading shifts, and polynomial parameters. The data above fixes one symmetrizable Cartan convention explicitly. A proposed action in another convention must provide a checked translation to these formulas before it can be used as a supplier.