Bigraded matrix factorizations with a potential

Fix a finite set $I$, let $S=\mathbb{Q}[a,x_i:i\in I]$ be the polynomial ring of The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution in a variable $a$ and the variables $x_i$, bigraded by

$\mathrm{deg}a=(2,0),\mathrm{deg}{x}_i=(0,2),$

and for a bigraded $S$-module $M$ and $(n_1,n_2)\in {\mathbb{Z}}^2$ write $M\{n_1,n_2\}$ for the internal shift

$M\{n_1,n_2{\}}_{(k,l)}=M_{(k-n_1,l-n_2)}$

in the sense of the internal shift of Associative graded algebras, bimodules, and internal shifts (the two-parameter refinement of the one-parameter shift, applied to each bigrading separately).

Fix signs ${ϵ}_i\in \{1,-1\}$ for $i\in I$ and put $w=a{\sum }_{i\in I}{ϵ}_i{x}_i$, an element of $S$ of bidegree $(2,2)$. A bigraded matrix factorization with potential $w$ is a pair $M=(M^0,M^1,d)$ consisting of free bigraded $S$-modules $M^0,M^1$ (of arbitrary, possibly infinite, rank) together with $S$-linear maps

$d\text{\hspace{0.17em}⁣}:M^0\to M^1,d\text{\hspace{0.17em}⁣}:M^1\to M^0$

of bidegree $(1,1)$ such that

$d^2=w\cdot \mathrm{i}\mathrm{d},$

i.e. $d^2(m)=wm$ for every $m\in M^0\oplus M^1$. A morphism $f\text{\hspace{0.17em}⁣}:M\to N$ is a pair of $S$-linear maps of bidegree $(0,0)$ on the two components commuting with $d$; a homotopy between two morphisms is a pair of maps of bidegree $(-1,-1)$ satisfying the usual homotopy formula. Write ${\mathrm{m}\mathrm{f}}_w$ for the category of factorizations with potential $w$ and their bidegree-preserving morphisms, and ${\mathrm{h}\mathrm{m}\mathrm{f}}_w$ for its homotopy category, in which the morphisms are the bidegree-preserving morphisms modulo null-homotopic ones. Explicitly, $f-g=d_{N}h+h{d}_M$, where $h$ reverses inner parity. Internal shifts $M\{u,v\}$ shift both modules and retain their inner parity. Write $\mathrm{\Pi }M$ for the parity reversal: $(\mathrm{\Pi }M{)}^0=M^1$, $(\mathrm{\Pi }M{)}^1=M^0$, with the same differential. It preserves the potential, acts on morphisms by the same component maps, and satisfies ${\mathrm{\Pi }}^{2}M=M$. For $w=0$, the direct sum of the two parity cohomologies of $\mathrm{\Pi }M$ is the same bigraded vector space as that of $M$; only their parity labels change.

Caveats. Unless $w=0$, $d$ does not square to zero, so a factorization is not an ordinary complex. If $I=\mathrm{\varnothing }$ then $S=\mathbb{Q}[a]$ and a factorization with $w=0$ is a $2$-periodic complex $M^0\stackrel{d}{\to }{M}^1\stackrel{d}{\to }{M}^0$ of free bigraded $\mathbb{Q}[a]$-modules; for $w\mathrm{\ne }0$ the only failure of the complex axioms is the identity $d^2=w\cdot \mathrm{i}\mathrm{d}$. The sign vector $({ϵ}_i)$ and the bigrading are part of the data, and on this page the potential is always $a$ times a linear form with coefficients in $\{1,-1\}$. The conventions (two-variable bigrading with $\mathrm{deg}a=(2,0)$, $\mathrm{deg}{x}_i=(0,2)$; $d$ of bidegree $(1,1)$; homotopies of bidegree $(-1,-1)$; morphisms commuting with $d$) follow Khovanov-Rozansky, Matrix factorizations and link homology II, section 1, formulas (1)-(2) and the lattice picture of its Figure 3; the ungraded definitions of a duplex and a free-module factorization are Definitions 1-2 in section 2 (printed pp. 13-14) of Khovanov-Rozansky, Matrix factorizations and link homology.