Twisted Equivariant Matrix Factorizations

We introduce and study categories of twisted equivariant matrix factorizations MFαG(R, w), which are categories of matrix factorizations of a potential w over the local ring R = C[x1, . . . , xn] together with an action by a finite group G that is twisted via a 2-cocycle α. These categories provide rich examples of Z/2Z-differentially graded categories in the context of non-commutative geometry and come up naturally in the study of boundary conditions of 3d Rozansky–Witten theories. We prove that under certain assumptions on (R, w) the categories of twisted matrix factorizations MFαG(R, w) are smooth, proper and triangulated. In particular we prove that they are generated by a compact generator. Following Dyckerhoff and Polishchuk–Vaintrob, we compute the Hochschild homology and cohomology of these categories and determine explicitly all ingredients of the categorical Hirzebruch–Riemann–Roch theorem proved in [Shk13].