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

TL;DR

The paper tackles gluing of 2-periodic dg-categories of matrix factorizations on a $(-1)$-shifted symplectic derived DM stack by developing a global crystal-structure framework that glues local $MF^infty(U,f)$ up to isotopy. It extends matrix factorization theory to derived inf-schemes, establishes a relative Thom–Sebastiani theory, and analyzes the obstruction data $(eta_1,eta_2,eta_3)$ that control orientation choices, linking to Clifford algebras and spinorial structures. The main result shows that, with orientation data, the locally defined MF∞ pieces glue along $X$ to yield a global invariant (a twisted crystal of 2-periodic dg-categories), partially confirming conjectures in Kontsevich–Soibelman and Toda motivic DT theory and recovering Behrend-type information via Euler characteristics. The framework relies on a robust Ind-coherent & de Rham–stack calculus, 1-affineness, and descent results to manage base-change, dualizability, and convolution structures across derived inf-schemes, opening a path toward a noncommutative/homological DT glueing theory.

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.

Theorems & Definitions (56)

• Theorem B: see \ref{['gluingDT-partII:theoremgluingMF']}
• Remark 1.1.1
• Lemma 2.1.2
• proof
• Lemma 2.1.3
• proof
• Corollary 2.1.4
• proof
• Corollary 2.1.5
• proof