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

For 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 AHaC0 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 AHaC0-action. These triples can be interpreted as certain sheaves on PC(omega C circle plus OC). In particular, we obtain an action of AHaC0 on the cohomology of Hilbert schemes of points on T*C.