Arc and wide-edge Khovanov-Rozansky factorizations

In the setting of Bigraded matrix factorizations with a potential, and writing a two-term factorization $(p,q)$ for $S\stackrel{p}{\to }S\{n_1,n_2\}\stackrel{q}{\to }S$ as a Koszul row (the notation fixed for this construction in the sequel to this definition; the shifts $n_1,n_2$ are the shifts of the middle term):

(1) Oriented arc. To an oriented arc $c$ oriented from the endpoint labelled $x_2$ to the endpoint labelled $x_1$ assign the two-term factorization $C_c:=(a,x_1-x_2)=[S\stackrel{a}{\to }S\{-1,1\}\stackrel{x_1-x_2}{\to }S]$ over $S=\mathbb{Q}[a,x_1,x_2]$, with potential $w=a(x_1-x_2)$; the differentials are $a$ and $x_1-x_2$, the middle term carries the bigrading shift $\{-1,1\}$, and both maps have bidegree $(1,1)$. It is an object of ${\mathrm{h}\mathrm{m}\mathrm{f}}_w$: the square of the differential is $(x_1-x_2)\circ a=a(x_1-x_2)=w\cdot \mathrm{i}\mathrm{d}$.

(2) Wide edge. To a wide edge $t$ whose four adjacent edge labels are $x_1,x_2$ on its outgoing ends and $x_3,x_4$ on its incoming ends assign the tensor product, over $S=\mathbb{Q}[a,x_1,x_2,x_3,x_4]$, of the two Koszul rows $(a,x_1+x_2-x_3-x_4)=[S\stackrel{a}{\to }S\{-1,1\}\stackrel{x_1+x_2-x_3-x_4}{\to }S]$ and $(0,x_1{x}_2-x_3{x}_4)=[S\stackrel{0}{\to }S\{-1,3\}\stackrel{x_1{x}_2-x_3{x}_4}{\to }S],$ with potential $w=a(x_1+x_2-x_3-x_4)$. For a two-fold tensor product of rows $(a_1,b_1)\otimes (a_2,b_2)$ the total differential satisfies $d^2=(a_1{b}_1+a_2{b}_2)\cdot \mathrm{i}\mathrm{d}$ with the Koszul sign convention for the totalization, so here $d^2=a(x_1+x_2-x_3-x_4)+0\cdot (x_1{x}_2-x_3{x}_4)=w$, and the middle terms are $S\{-1,1\}\oplus S\{-1,3\}$. It is an object of ${\mathrm{h}\mathrm{m}\mathrm{f}}_w$.

Both assignments produce objects of ${\mathrm{h}\mathrm{m}\mathrm{f}}_w$ with the stated potentials. Caveats: the potential of a wide edge is linear in the $x_i$; the quadratic entry $x_1{x}_2-x_3{x}_4$ enters only through the row whose first differential is $0$; and the shifts $\{-1,1\}$ and $\{-1,3\}$ are part of the definition and must be propagated exactly (Khovanov-Rozansky II, section 1, formulas (2)-(4)). Khovanov-Rozansky I, introduction printed pp. 6-8, gives the fixed-$n$ analogue with potentials $x_i^{n+1}$ and different row entries and shifts; it is not the source of these parameter-$a$ formulas.