Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds

TL;DR

This paper explores the structure of quiver Yangians associated with generalized conifolds, establishing their coproducts, isomorphisms under Seiberg duality, and connections to $\\mathcal{W}$-algebras, revealing their representation theory via truncated crystals.

Contribution

It constructs coproducts for quiver Yangians, proves their invariance under Seiberg duality, and links them to $\\mathcal{W}$-algebras, extending previous studies to generalized conifolds.

Findings

Coproducts of quiver Yangians are constructed.

Quiver Yangians related by Seiberg duality are isomorphic.

Universal enveloping algebras of $\\mathcal{W}$-algebras are truncations of quiver Yangians.

Abstract

We focus on quiver Yangians for most generalized conifolds. We construct a coproduct of the quiver Yangian following the similar approach by Guay-Nakajima-Wendlandt. We also prove that the quiver Yangians related by Seiberg duality are indeed isomorphic. Then we discuss their connections to $\mathcal{W}$-algebras analogous to the study by Ueda. In particular, the universal enveloping algebras of the $\mathcal{W}$-algebras are truncations of the quiver Yangians, and therefore they naturally have truncated crystals as their representations.