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

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 ${\mu }_f$, the perverse sheaf of vanishing cycles ${\mathsf{P}}_{U,f}$ and the category of matrix factorizations $\mathsf{M}\mathsf{F}(U,f)$ 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 $\mathsf{M}\mathsf{F}(U,f)$ under the prescription of additional orientation data, thus answering positively conjectures by Kontsevich–Soibelman and Toda in motivic Donaldson–Thomas theory.