Quiver $\text{W}_{\epsilon_1,\epsilon_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories

TL;DR

This paper introduces an epsilon-deformation of W algebras that bridges the gap between q-deformed and classical W algebras, providing a new algebraic framework for understanding 4d N=2 gauge theories and their instanton partition functions.

Contribution

It constructs the epsilon-deformed W algebras as spectral duals of traditional W algebras, connecting 4d gauge theories with conformal blocks through a novel algebraic approach.

Findings

Epsilon-deformed W algebras reproduce instanton partition functions.

They provide dual integral representations of W conformal blocks.

The algebras fill the gap between q-deformed and classical W algebras.

Abstract

We construct an $\epsilon$-deformation of W algebras, corresponding to the additive version of quiver $\text{W}_{q,t^{-1}}$ algebras which feature prominently in the 5d version of the BPS/CFT correspondence and refined topological strings on toric Calabi-Yau's. This new type of algebras fill in the missing intermediate level between $q$-deformed and ordinary W algebras. We show that $\epsilon$-deformed W algebras are spectral duals of conventional W algebras, in particular the $\epsilon$-deformed conformal blocks manifestly reproduce instanton partition functions of 4d $\mathcal{N}=2$ quiver gauge theories in the full $\Omega$-background and give dual integral representations of ordinary W conformal blocks.