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?