Cohomological Hall algebras, vertex algebras and instantons

TL;DR

This paper constructs an algebraic action on instanton moduli space cohomology, linking it to affine Yangians and vertex algebras, with conjectures extending to broader Calabi-Yau categories.

Contribution

It introduces a novel action of the double Cohomological Hall algebra on instanton moduli spaces and connects it to affine Yangians and vertex algebras.

Findings

Identified the algebraic action with the affine Yangian of gl(1)

Derived the vertex algebra at the Gaiotto-Rapcak corner

Conjectured applicability to a wide class of Calabi-Yau categories

Abstract

We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner $\mathcal{W}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds.