Cohomological Hall algebras of 2-Calabi-Yau categories and applications

Abstract: In this series of four lectures, I will explain the interactions between cohomological Hall algebras (CoHAs) and several questions of interest in algebraic geometry (in particular enumerative geometry) and representation theory (Kac--Moody algebras and their representations). CoHAs are associative algebra structures on the Borel--Moore homology of the stack of objects in some Abelian categories. We consider the CoHAs of various categories: sheaves on surfaces, representations of quivers, and representations of fundamental groups, which are $2$-Calabi--Yau. CoHAs lead to a fine understanding of the cohomology of the stacks and moduli spaces involved. They provide tools to study various conjectures in the subject: cohomological integrality, positivity, and purity. In the first two lectures, I will detail how CoHAs give a geometric construction of generalised Kac--Moody algebras (in the sense of Borcherds). The last two lectures will develop applications of CoHAs to the study of the cohomology of quiver varieties (following the groundbreaking work of Nakajima from the 1990s) and to nonabelian Hodge theory (following questions of Simpson). The four lectures will, to a large extent, be independent from each other and are largely based on joint work with Ben Davison and Sebastian Schlegel Mejia.