Kleinian orbifolds, Cohomological Hall Algebras, and Yangians

We establish, for each orbifold crepantly resolving a Kleinian singularity, the existence of the cohomological Hall algebra (COHA) of coherent sheaves supported on the exceptional locus and explicitly compute this COHA as a completion of some positive half of the associated affine Yangian. Tracking these categories under derived autoequivalences and the McKay correspondence, we show that (1) every point in Bridgeland’s space of stability conditions on the resolution arises from a Kleinian orbifold, and (2) every positive half of the affine Yangian can be recovered from the COHA associated to some such stability condition. This provides the first example of a family of (pointwise) COHAs defined over the space of stability conditions.