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 f, the perverse sheaf of vanishing cycles Pₔ, ₅ and the category of matrix factorizations 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 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 MF (U, f) under the prescription of additional orientation data, thus answering positively conjectures by Kontsevich--Soibelman and Toda in motivic Donaldson--Thomas theory.