Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras
arxiv(2023)
摘要
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.
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