Cohomological integrality for 2-Calabi-Yau categories

In this talk, I will explain how to define BPS invariants of a large class of Abelian 2-Calabi-Yau (2CY) categories and their sheaf refinements (as perverse sheaves or mixed Hodge modules). Examples of relevance in geometry and representation theory are sheaves on symplectic surfaces, length-zero coherent sheaves on any surface, preprojective algebras of quivers, and fundamental groups of Riemann surfaces. These BPS invariants give cohomological integrality for the categories considered, and their categorification enjoys the richer structure of a Lie algebra, which happens to be a generalised Kac-Moody Lie algebra. This Lie algebra governs the homology of the stack of objects in the 2CY category via a PBW-type isomorphism.