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](https://ncatlab.org/nlab/show/matrix+factorization) 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](https://arxiv.org/abs/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](https://arxiv.org/abs/1109.5213). 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? 



Derived Category of the derived critical locus, is it the category of Matrix Factorizations?
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) $\operatorname{MF}(W)$.


