Kleinian orbifolds, Cohomological Hall Algebras, and Yangians

TL;DR

This paper constructs and explicitly computes the cohomological Hall algebra (COHA) for Kleinian orbifolds, linking it to affine Yangians and stability conditions, revealing new algebraic structures associated with these geometric objects.

Contribution

It establishes the existence and explicit form of COHAs for Kleinian orbifolds and connects them to affine Yangians and Bridgeland stability conditions, a novel integration of these concepts.

Findings

COHA for Kleinian orbifolds is explicitly computed.

Every stability condition point arises from a Kleinian orbifold.

Positive halves of affine Yangians are recoverable from the COHA.

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.