The q-AGT-W relations via shuffle algebras

TL;DR

This paper geometrically constructs the action of the q-deformed W-algebra on its level r representation using moduli spaces and shuffle algebras, establishing a q-deformed AGT-W relation.

Contribution

It provides an explicit LDU decomposition of W-algebra actions and links the Carlsson-Okounkov Ext operator with deformed W-algebra intertwiners, revealing a q-deformed AGT-W relation.

Findings

Explicit LDU decomposition for W-algebra currents

Relation between Ext operator and W-algebra intertwiners

Interpretation of results as q-deformed AGT-W relations

Abstract

We construct the action of the q-deformed W-algebra on its level r representation geometrically, using the moduli space of U(r) instantons on the plane and the double shuffle algebra. We give an explicit LDU decomposition for the action of W-algebra currents in the fixed point basis of the level r representation, and prove a relation between the Carlsson-Okounkov Ext operator and intertwiners for the deformed W-algebra. We interpret this result as a q-deformed version of the AGT-W relations.