CoHA of Cyclic Quivers and an Integral Form of Affine Yangians

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-Rapčák-Zhou, in the context of twisted M-theory.