Cohomological Hall algebra of Higgs sheaves on a curve

We define the cohomological Hall algebra AHA(Higgs(X)) of the (two-dimensional) Calabi-Yau category of Higgs sheaves on a smooth projective curve X, as well as its nilpotent and semistable variants, in the context of an arbitrary oriented Borel-Moore homology theory. In the case of usual Borel-Moore homology, AHA(Higgs(X)) is a module over the (universal) cohomology ring H of the stacks of coherent sheaves on X. We characterize its annihilator as an H-module, and we provide an explicit collection of generators (the collection of fundamental classes [Coh(r,d)] of the zero sections of the maps Higgs(r,d) -> Col(r,) (d) for r> 0 and d is an element of Z and for r= 0 and d is an element of Z(>0)).