Affine BPS algebras, W algebras, and the cohomological Hall algebra of ${\mathbb{A}}^2$

Abstract

We introduce affinizations and deformations of the BPS Lie algebra associated to a tripled quiver with potential, and use them to precisely determine the $T$-equivariant cohomological Hall algebra ${\mathcal{H}}_{{\mathbb{A}}^2}^T$ of compactly supported coherent sheaves on ${\mathbb{A}}^2$, acted on by a torus $T$. In particular we show that this algebra is spherically generated for all $T$.

v6: final version, 37 pp. v5: many improvements, suggested by the referee. v4: fixed a lot of typos. v3: fixed some signs+minor changes. v2: minor changes. v1: Prepared for the festschrift honouring the 60th birthday of Ezra Getzler. Comments welcome!

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