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

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 $M{F}^{\mathrm{\infty }}(U,f)$. 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 $M{F}^{\mathrm{\infty }}(U,f)$ 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.