Quiver W-algebras
Taro Kimura, Vasily Pestun
TL;DR
The paper builds a two-parameter quiver W-algebra $W_{q_1,q_2}(\,\Gamma\,)$ from equivariant K-theory on the moduli space of $\Gamma$-quiver sheaves on $\mathbb{C}^2_{q_1,q_2}$, unifying Nekrasov’s qq-characters with a free-field realization and identifying the algebra as the commutant of screening charges in a Heisenberg framework. It demonstrates a gauge-theory–algebra correspondence, showing pole cancellation and current regularity in the $A_1$ case and extending to arbitrary quivers, including loops and hyperbolic types, thereby producing a rich family of W-algebras associated to generalized Borcherds–Kac–Moody algebras. A central result is the quantum $q$-geometric Langlands duality realized by exchanging $q_1$ and $q_2$, connecting to geometric Langlands limits and AB dualities in 4d/5d gauge theories, with further links to Nakajima quiver varieties and quantum loop algebras. The framework yields explicit constructions of higher-weight currents, a range of concrete examples, and several applications to Toda scaling limits, affine-type theories, and Nahm transforms, positioning quiver W-algebras as a versatile bridge between gauge theory, representation theory, and integrable systems.
Abstract
For a quiver with weighted arrows we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al., and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.