Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras
Anton Mellit, Alexandre Minets, Olivier Schiffmann, Eric Vasserot
TL;DR
This work determines the cohomological Hall algebra of zero-dimensional sheaves on a smooth pure surface, giving a concrete presentation by generators and relations. It then constructs a doubled COHA and the deformed $W$-algebras $W^{(\mathbf{c})}(S)$, establishing triangular decompositions, Heisenberg/Virasoro subalgebras, and rich Fock-space representations, including level-$r$ modules, for both projective and open surfaces. The paper further develops derived and open-geometry Hecke correspondences to describe actions on tautological classes and moduli stacks (notably Hilbert schemes and Higgs moduli), and proves that Nakajima’s Heisenberg algebra and Virasoro algebra sit inside the doubled COHA. The results unify enumerative geometry of moduli spaces on $S$ with infinite-dimensional algebras, offering explicit operators on tautological rings and a toolbox for further applications to moduli of sheaves and instantons.
Abstract
We compute the cohomological Hall algebra of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface $S$ with pure cohomology, deriving an explicit presentation by generators and relations. When $S$ has trivial canonical bundle, this COHA is isomorphic to the enveloping algebra of deformed trigonometric $W_{1+\infty}$-algebra associated to the ring $H^*(S,\mathbb{Q})$. We also define a double of this COHA, show that it acts on the homology of various moduli stacks of sheaves on $S$ and explicitly describe this action on the products of tautological classes. Examples include Hilbert schemes of points on surfaces, the moduli stack of Higgs bundles on a smooth projective curve and the moduli stack of $1$-dimensional sheaves on a $K3$ surface in an ample class. The double COHA is shown to contain Nakajima's Heisenberg algebra, as well as a copy of the Virasoro algebra.