Donaldson-Thomas invariants for the Bridgeland-Smith correspondence

Abstract: Celebrated work of Bridgeland and Smith shows a correspondence between quadratic differentials on Riemann surfaces and stability conditions on certain 3-Calabi--Yau triangulated categories. Part of this correspondence is that finite-length trajectories of the quadratic differential correspond to categories of semistable objects of a fixed phase. Categories of semistable objects have an associated Donaldson--Thomas invariant which, in some sense, counts the objects in the category. Work of Iwaki and Kidwai predicts particular values for these Donaldson--Thomas invariants for different types of finite-length trajectories, based on the output of topological recursion. The Donaldson--Thomas invariants produced by the category of Bridgeland and Smith do not always match these predictions. However, we show that if one replaces this category by the category recently studied by Christ, Haiden, and Qiu, then one does obtain the Donaldson--Thomas invariants matching the predictions. This is joint work with Omar Kidwai.