Marco Robalo: Categorification of Donaldson–Thomas invariants

Abstract

Given a (-1)-shifted derived scheme X with a convenient orientation data (in the sense of Kontsevich–Soibelman) Brav–Bussi–Dupont–Joyce–Szendroi (BBDJS) constructed a perverse sheaf over X and whose Euler characteristic recovers Behrend's counting of Donaldson–Thomas invariants. The BBDJS construction uses a Darboux local form for (-1)-shifted symplectic schemes: locally they are all derived critical loci of a function $f$ on a smooth scheme $U$ and the DT-invariants are obtained from the Euler characteristic of the sheaf of vanishing cycles of $f$. In this talk I will describe an ongoing joint work with B. Hennion and J. Holstein where we propose a strategy based on Toën–Vezzosi derived foliations, to glue over X a sheaf of 2-periodic dg-categories locally modeled on the categories of matrix factorisations MF(U,f). In particular, our strategy allows us to recover the construction of BBDJS.