Structure of cohomological Hall algebras

Abstract: Cohomological Hall algebras (CoHAs) are algebra structures on the Borel—Moore homology of the stack of objects of certain categories: representations of quivers and coherent sheaves on curves (1D CoHAs), representations of preprojective algebras, twisted fundamental group algebras of Riemann surfaces and Higgs sheaves on curves (2D CoHAs), representations of the Jacobi algebra of a quiver with potential and coherent sheaves on 3-Calabi—Yau manifolds (3D-CoHAs). Based on joint work with Ben Davison and Sebastian Schlegel Mejia, I will explain how to describe explicitly the algebraic structure of 2D CoHAs under mild conditions on the category under consideration. This structural result has striking consequences, yielding some positivity results and a nonabelian Hodge isomorphism for stacks, which we will also explain.