CoHA of Cyclic Quivers and an Integral Form of Affine Yangians

TL;DR

This paper computes the cohomological Hall algebra of cyclic quivers, linking it to affine Yangians and providing new algebraic structures and conjectures in the context of Donaldson-Thomas theory and M-theory.

Contribution

It explicitly describes the CoHA of cyclic quivers, relates it to affine Yangians, and explores its algebraic properties and connections to other mathematical physics frameworks.

Findings

CoHA of cyclic quivers is the universal enveloping algebra of a certain extension of matrix differential operators.

Deformation yields an integral form of Guay's affine Yangian.

Results recover and relate to known Yangians and algebraic structures in geometric representation theory.

Abstract

We calculate the deformed and non-deformed cohomological Hall algebra (CoHA) of the preprojective algebra for the case of cyclic quivers by studying the Kontsevich-Soibelman CoHA and using tools from cohomological Donaldson-Thomas theory. We show that for the cyclic quiver of length $K$, this algebra is the universal enveloping algebra of the positive half of a certain extension of matrix differential operators on $\mathbb{C}^{*}$, while its deformation gives a positive half of an explicit integral form of Guay's Affine Yangian $\ddot{\mathcal{Y}}_{\hbar_1,\hbar_2}(\mathfrak{gl}(K))$. By the main theorem of Botta-Davison (2023) and Schiffmann-Vasserot (2023), we also determine the Maulik-Okounkov Yangian for the case of cyclic quivers. Furthermore, we explain the construction of factorization coproduct, provide evidence for the strong rationality conjecture, calculate the spherical subalgebra of the non-deformed CoHA for any quiver without loops, recover results about the CoHA of compactly supported semistable sheaves on the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}_{K}$ and identify a commutative algebra inside the additive shuffle algebra associated to the cyclic quiver. We end by conjecturally relating the obtained integral form with the algebra defined by Gaiotto-Rap\v{c}\'ak-Zhou, in the context of twisted M-theory.