Kleinian orbifolds, Cohomological Hall Algebras, and Yangians
Francesco Sala, Olivier Schiffmann, Parth Shimpi
TL;DR
This work builds a bridge between the geometry of Kleinian orbifolds and the algebra of cohomological Hall algebras (COHAs), showing that COHAs associated to coherence on the exceptional locus realize completed positive halves of affine Yangians, with all halves recoverable from suitable hearts in Bridgeland stability spaces. By tracking how these categories transform under derived autoequivalences and McKay-type correspondences, the authors prove that every stability condition in Bridgeland’s manifold for the Kleinian resolution emerges from a Kleinian orbifold, and conversely every positive half of the affine Yangian can be recovered from a COHA attached to some stability condition. The introduction of limiting COHAs and the heart-fan stability-arcs provides a unified, geometric mechanism to describe a continuum of COHAs over the stability manifold, linking stability data to explicit subquotients of affine Yangians. In the classical limit, these constructions yield twisted positive halves of elliptic Lie algebras and illuminate a deep, structural connection between derived category theory, representation theory, and noncommutative resolutions of singularities, including a geometric realization of a broad family of COHAs that are pointwise defined over stability conditions.
Abstract
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.