Cohomological Hall algebra of Higgs sheaves on a curve

We define the cohomological Hall algebra $AHA_Higgs(X)$ of the ($2$-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 $mathbbH$ of the stacks of coherent sheaves on $X$ . We show that it is a torsion-free $mathbbH$-module, and we provide an explicit collection of generators (the collection of fundamental classes $[Coh_r,d]$ of the zero-sections of the map $Higgs_r,d o Coh_r,d$, for $r geq 0, d in Z$).