Cohomological Hall algebras, their categorification, and their representations via torsion pairs

Abstract

In this paper we provide a systematic way of producing representations of cohomological, K-theoretical and categorified Hall algebras, and study the output of our construction in several cases. We thus recover and categorify in a unified framework the action of the COHA of a quiver on the cohomology of Nakajima quiver variety, the action of the COHA of zero-dimensional sheaves on the the cohomology of Hilbert schemes of points and moduli spaces of Gieseker-stable sheaves on smooth surfaces, recovering the constructions of Negu\c{t} and DeHority. We also obtain new examples, associated to Pandharipande-Thomas stable pairs. Along the way, we explain carefully under which conditions one can associate to a pair $(\mathscr{C},\tau)$ consisting of a stable $\infty$-category with a t-structure a COHA. This requires a careful analysis and extension of Khan's theory of motivic Borel-Moore homology to the non quasi-compact setting, and it allows to produce new examples of COHAs arising from Bridgeland's stability conditions. The representations that we construct take an extra categorical input: that of a torsion pair $(\mathscr{T},\mathscr{F})$ on the heart $\mathscr{C}^\heartsuit$ of $\tau$. Under favorable conditions, the homology of the moduli stack associated to $\mathscr{T}$ acquires a Hall multiplication, that acts both on the left and on the right on the homology of the moduli stack associated to $\mathscr{F}$. The left action generalizes and categorifies Nakajima's positive operators, while the right action corresponds to negative operators.