Gluing invariants of Donaldson--Thomas type -- Part I: the Darboux stack

TL;DR

The paper develops a unified, local-to-global framework for invariants of singularities arising from (-1)-shifted symplectic derived stacks by introducing the Darboux stack, a moduli space of local Darboux presentations as derived critical loci. It shows that redundancies in local presentations are generated by quadratic bundles and, after enforcing A^1-isotopy identifications, the quotient is contractible, which yields a principled gluing mechanism for local invariants such as Milnor numbers, vanishing-cycle perverse sheaves, and matrix-factorization categories. The core contributions are (i) the contractibility theorem for the Darboux stack modulo A^1-isotopies and quadratic-bundle actions, (ii) a detailed framework relating Darboux data to formal LG-pairs and Liouville structures, and (iii) the groundwork for gluing refined invariants in motivic DT theory, with Part II promised to realize gluing of motives and 2-periodic dg-categories under orientation data. This provides a robust geometric mechanism to pass from local to global invariants in Donaldson–Thomas type theories connected to Behrend’s function, BBDJS’s perverse sheaf framework, and Kontsevich–Soibelman/Toda-type questions on gluing motives and MF categories.

Abstract

Let $X$ be a (-1)-shifted symplectic derived Deligne--Mumford stack. In this paper we introduce the Darboux stack of $X$, parametrizing local presentations of $X$ as a derived critical locus of a function $f$ on a smooth formal scheme $U$. Local invariants such as the Milnor number $μ_f$, the perverse sheaf of vanishing cycles $\mathsf{P}_{U,f}$ and the category of matrix factorizations $\mathsf{MF}(U,f)$ are naturally defined on the Darboux stack, without ambiguity. The stack of non-degenerate flat quadratic bundles acts on the Darboux stack and our main theorem is the contractibility of the quotient stack when taking a further homotopy quotient identifying isotopic automorphisms. As a corollary we recover the gluing results for vanishing cycles by Brav--Bussi--Dupont--Joyce--Szendr\H oi. In a second part (to appear), we will apply this general mechanism to glue the motives of the locally defined categories of matrix factorizations $\mathsf{MF}(U,f)$ under the prescription of additional orientation data, thus answering positively conjectures by Kontsevich--Soibelman and Toda in motivic Donaldson--Thomas theory.

Theorems & Definitions (97)

• Example 1.1.1
• Example 1.1.2
• Theorem A: Contractibility -- see \ref{['contractibilitytheoremdetailedversion']}
• Remark 1.1.4
• Example 1.2.1
• Example 1.2.2
• Example 1.2.3
• Theorem 1.2.4: Behrend MR2600874
• Theorem 1.2.6: MR3353002
• Theorem B: hennionholsteinrobaloII