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

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?

• Are you familiar with the paper "Motivic realizations of singularity categories and vanishing cycles" by Blanc, Robalo, Toën and Vezzosi? – Simon Pepin Lehalleur Nov 2 '19 at 13:52
• @SimonPepinLehalleur They do not prove this sadly. Instead they generalize Orlov’s equivalence to the case where the base is a regular ring instead of a field of characteristic zero. But they are still in the case where the derived scheme happens to be classical, just like Orlov’s result. See their Remark 2.51. – Riza Hawkeye Nov 2 '19 at 17:22

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) $$\operatorname{MF}(W)$$.