CoHA of Cyclic Quivers and an Integral Form of Affine Yangians

TL;DR

The work develops an integral, representation-theoretic framework for cohomological Hall algebras of cyclic quivers via the tripled quiver with canonical potential, linking CoHA structures to integral forms of affine Yangians. It shows that the nondeformed CoHA is the universal enveloping algebra of the positive half of a Poisson-type Lie algebra arising from differential operators on $C^ imes$, while its deformation realizes an integral form of Guay’s affine Yangian; this connects to Maulik–Okounkov Yangians and to shuffle-algebra realizations. The paper also establishes spherical subalgebras for loopless quivers, proves a strong rationality-type for Kleinian surfaces, and develops a detailed geometric program tying Nakajima quiver varieties, Hilbert schemes, and equivariant K-theory to BPS Lie algebras, with conjectural links to twisted M-theory constructions. Collectively, these results give new integral, geometric models for affine Yangians and related infinite-dimensional algebras, along with multiple applications to Kleinian singularities and Nakajima geometry. The constructions provide a rich dictionary between CoHA, Yangians, and the geometry of quiver varieties, enabling explicit computations and identifications across algebraic and physical perspectives.

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.

Figures (2)

• Figure 1: $\widetilde{Q^{3}},\widetilde{W^{3}}= \sum_{i \in \mathbb{Z}/4\mathbb{Z}} \omega_{i}(a_{i,i+1}^{*}a_{i,i+1} - a_{i,i-1}a_{i,i-1}^{*})$
• Figure 2: $\mathfrak{B}_i(\delta)$ for $K=3,i=2$

Theorems & Definitions (187)

• Definition 2.1
• Theorem A: Theorem \ref{['Theorem1']}
• Theorem B: Theorem \ref{['Theorem2']}
• Definition 2.2: CoHA Affine Yangian
• Theorem C: Theorem \ref{['equivcoha']}
• Corollary A: Theorem \ref{['moyangianconjecture']}
• Theorem D: Theorem \ref{['sphericalsubalgebra']}
• Corollary B: Corollary \ref{['strongrationalityconjectureproof']}
• Corollary C
• Corollary D: Corollary \ref{['elementsLn']}