This is an old revision of the document!
Table of Contents
Matrix Factorization
$\gdef\ccal{\mathcal C}$
[N]
section 2.1
By a dg category $\ccal$, we mean a stable differential $\Z$-graded category. The hom complex is a $\Z$-graded cochain complex. We have a shift functor $[n]$ for any $n \in \Z$.
By a 2-periodic dg category $\ccal$, we mean a $\Z$-graded category that the functor $[2]$ is equivalent to identity. Note that the hom complex is $\Z$-graded.
By a $\Z/2$-dg category, we mean a stable differential $\Z/2$-graded complex.
To any dg category, we can associate to it the $\Z/2$ folding, denoted as $\ccal_{\Z/2}$, which takes the same object, and only remembering the $\Z/2$-grading on the complex.
To any $\Z/2$-graded category, we can associate to it a 2-periodic $\Z$-graded category, by unfurling. (It is like $p^*p_!$ where $p: \Z \to \Z/2$ )
section 2.2
$\gdef\Spec{\text{Spec}}$
Suppose $A = \C[x_1, \cdots, x_n]$, $W \in A$ is a polynomial with only one singular value $0$. And $B = A/(W)$. $X = \Spec B$.
We want to say $Coh(X) / Perf(X)$ is a 2-periodic dg category.
Why? What is perf? What is perfect complex? bounded complex of finite projective B-modules. But, what is projective?
References
- [N] Wrapped microlocal sheaves on pairs of pants by David Nadler. Section 2 on LG B-model