Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians

Abstract

This paper provides the first algebraic characterization of an algebra of cohomological Hecke operators associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z\subset X$ (possibly singular and reducible), establishing a direct connection with Yangians. It is based on the theory of equivariant nilpotent cohomological Hall algebras ${\mathbf{HA}}_{X,Z}^T$, developed by the same authors. More precisely, let $X$ be a resolution of a Kleinian singularity (for example, $X=T^{\ast }{\mathbb{P}}^1$) and let $Z$ be the exceptional divisor. One of the main results of this paper is an explicit isomorphism ${\mathbf{HA}}_{X,Z}^T\simeq {\mathbb{Y}}_{\infty }^{+}$, where ${\mathbb{Y}}_{\infty }^{+}$ is a completed, nonstandard, positive half of the affine Yangian $\mathbb{Y}(\mathfrak{g})$ of the corresponding affine ADE Lie algebra $\mathfrak{g}$. Furthermore, the generators of ${\mathbf{HA}}_{X,Z}^T$--given by fundamental classes of substacks of zero-dimensional sheaves and of pushforwards of line bundles on $Z$--are expressed explicitly in terms of Yangian generators. Our main tools, which may be of independent interest, are: (i) a `continuity' theorem describing the behavior of cohomological Hall algebras of objects in the heart of $t$-structures ${\tau }_n$ when the sequence $({\tau }_n{)}_n$ converges, in an appropriate sense, to a fixed $t$-structure ${\tau }_{\infty }$; (ii) the definition of a multi-parameter Yangian ${\mathbb{Y}}_Q$ for an arbitrary quiver $Q$, given by generators and relations; (iii) a theorem relating the algebraic action of the braid group $B_Q$ on the Yangian ${\mathbb{Y}}_Q$ to the action of $B_Q$ on the equivariant 2-dimensional cohomological Hall algebra ${\mathbf{HA}}_Q^T$ of $Q$, where the latter can be described in terms of derived reflection functors of the bounded derived category of modules over the preprojective algebra of $Q$.

v2: references updated, 152 pages. v1: 150 pages. This paper is a revised version of Parts II, III, and IV of arXiv:2502.19445v2