Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces

Abstract

AbstractFor any free oriented Borel–Moore homology theory A, we construct an associative product on the A-theory of the stack of Higgs torsion sheaves over a projective curve C. We show that the resulting algebra $$A\mathbf{Ha}_C^0$$ A Ha C 0 admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose A-theory admits an $$A\mathbf{Ha}_C^0$$ A Ha C 0 -action. These triples can be interpreted as certain sheaves on $$\mathbb {P}_C(\omega _C\oplus \mathcal {O}_C)$$ P C ( ω C ⊕ O C ) . In particular, we obtain an action of $$A\mathbf{Ha}_C^0$$ A Ha C 0 on the cohomology of Hilbert schemes of points on $$T^*C$$ T ∗ C .