Derived Category of the derived critical locus, is it the category of Matrix Factorizations?

Question

Let $W \in \mathbb{C}[x_1, \dots, x_n]=R$ be a polynomial with an isolated critical point at the origin. A Matrix Factorizations for $W$ consists a $\mathbb{Z}/2\mathbb{Z}$-graded finite free $R$-module $E$ with an odd differential $d:E \to E$ satisfying $d^2 = W \cdot Id$ instead of the usual square zero condition.

Chain maps between matrix factorizations and chain homotopies are defined in the same way as for ordinary complexes. There is a category of matrix factorizations with morphisms homotopy equivalence classes of chain maps.

This category is supposed to be a model for the "derived category of the singularity". For example this point of view is advocated by Orlov, see arxiv/math/0302304.

Another way to get to the derived category of the singularity is via derived algebraic geometry, say in the sense of Vezzosi's Derived Critical Loci I - Basics. In this approach we let $X= spec(R)$ and consider the derived intersection of the zero section $0:X \to T^*X$ and $dW: X \to T^*X$. This will be a derived scheme, i.e. a scheme with a sheaf non-positively graded dg-algebras satisfying a few regularity conditions.

Presumably we should be able to take the bounded derived category of coherent sheaves for such a derived scheme.

My Question: Are these two versions of the derived category of a singularity related in any way? Are they in some sense equivalent? What is known about their relationship, if anything.

Note that the first one is a $\mathbb{Z}/2\mathbb{Z}$-graded triangulated category (shift squares to the identity), while that is not true for the later. Perhaps they become equivalent if we use a 2-periodic version of the derided category?

Answer 1

These are indeed related. The first thing to know is that they both ``live'' (i.e., sheafify) over the critical locus (this is not saying much if you assume $W$ has isolated critical points, but interesting in more general cases). This is clear for $\operatorname{Coh}(\operatorname{Crit}(W))$. For matrix factorizations, we can identify $\operatorname{MF}(W)$ with the category of singularities $\operatorname{Coh}(W^{-1}(0))/\operatorname{Perf}(W^{-1}(0))$, which vanishes on the regular part of $W^{-1}(0)$.

In the case where the superpotential $W$ is Morse, or Morse-Bott, the category of matrix factorizations is close to being the $2$-periodization of $\operatorname{Coh}(\operatorname{Crit}(W))$. This is the subject of Teleman's Matrix Factorisation of Morse-Bott functions.

The basic case to understand is when the superpotential is $W = x^2$ defined on $\mathbb{A}^1$. The category of matrix factorizations is $\operatorname{Coh}(k[x]/x^2)/\operatorname{Perf}(k[x]/x^2)$. A Koszul duality computation tells us that this is the category of modules over the algebra $k[\beta, \beta^{-1}]\langle \eta | \eta^2 = \beta \rangle$. Here $\beta$ is a variable of cohomological degree $2$ giving the $\mathbb{Z}/2$-grading, and $\eta$ is a variable of degree $1$. We interpret this result as a $2$-periodic Clifford algebra on a generator $\eta$ of degree $1$. The derived critical locus in this case is just $\operatorname{Spec}(k)$. You can then think about $\operatorname{MF}(x^2)$ as the $2$-periodization of $\operatorname{Coh}(\operatorname{Spec}(k))$, with an extra Clifford factor.

From this you can build higher dimensional Morse functions: the rule is that $\operatorname{MF}(f \boxplus g) = \operatorname{MF}(f) \otimes_{k[\beta, \beta^{-1}]} \operatorname{MF}(g)$ (see Preygel's Thom Sebastiani and Duality for Matrix Factorizations). The tensor product of $k[\beta, \beta^{-1}]\langle \eta | \eta^2 = \beta \rangle$ with itself turns out to be $k[\beta, \beta^{-1}]$. With this one can conclude that when $W = x_1^1 + \ldots x_n^2$ the category of matrix factorizations is the $2$-periodization of $\operatorname{Coh}(\operatorname{Crit}(W))$, with an added Clifford factor in odd dimensions. In the Morse-Bott case one has extra corrections arising from the topology of the normal bundle to the critical locus.

When you move beyond the Morse-Bott case the two categories start diverging. Their relationship in general is roughly like the relationship between the algebra $D_X$ of differential operators on a smooth scheme $X$, thought of as the deformation quantization of $T^*X$, and $\mathcal{O}_X$. The derived critical locus of $W$ has a $(-1)$-shifted symplectic structure. If one works in the $2$-periodic context this turns into a $1$-shifted symplectic structure, which allows one to deform the symmetric monoidal category $\operatorname{QCoh}(\operatorname{Crit}(W))$ into a monoidal category. This is the monoidal category of modules over the $E_2$-algebra associated to the Gerstenhaber algebra $\mathcal{O}_{\operatorname{Crit}(W)} \otimes k[\beta, \beta^{-1}]$. This $E_2$-algebra turns out to agree with the Hoschild cochains of $\operatorname{MF}(W)$, which implies that the deformation of $\operatorname{QCoh}(\operatorname{Crit}(W))\otimes k[\beta,\beta^{-1}]$ acts on (the ind-completion of) $\operatorname{MF}(W)$.