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

TL;DR

This paper introduces the Darboux stack for (-1)-shifted symplectic derived stacks, enabling natural definitions of local invariants and establishing a contractibility result that aids in gluing motivic Donaldson--Thomas invariants.

Contribution

It defines the Darboux stack for (-1)-shifted symplectic stacks, proves its contractibility under certain automorphisms, and sets the stage for gluing local invariants in motivic DT theory.

Findings

Darboux stack parametrizes local presentations of (-1)-shifted symplectic stacks.

Contractibility of the quotient stack under automorphisms.

Framework for gluing motivic categories of matrix factorizations.

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 $\mu_f$, 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.