Gluing invariants of Donaldson--Thomas type -- Part II: Matrix factorizations

Abstract

This paper is a follow-up to arXiv:2407.08471. Let $X$ be a a $(-1)$-shifted symplectic derived Deligne--Mumford stack. Thanks to the Darboux lemma of Brav--Bussi--Joyce, $X$ is locally modeled by derived critical loci of a function $f$ on a smooth scheme $U$. In this paper we study the gluing of the locally defined $2$-periodic (big) dg-categories of matrix factorizations $MF^\infty(U,f)$. We show that these come canonically equipped with a structure of a $2$-periodic crystal of categories (\ie an action of the dg-category of $2$-periodic $D$-modules on $X$) compatible with a relative Thom--Sebastiani theorem expressing the equivariance under the action of quadratic bundles. As our main theorem we show that the locally defined categories $MF^\infty(U,f)$ can be glued along $X$ as a sheaf of crystals of 2-periodic dg-categories ``up to isotopy'', under the prescription of orientation data controlled by three obstruction classes. This result generalizes the gluing of the Joyce's perverse sheaf of vanishing cycles and partially answers conjectures by Kontsevich--Soibelman and Toda in motivic Donaldson--Thomas theory.