Toward AGT for parabolic sheaves

TL;DR

This paper constructs specific elements in a shifted quantum toroidal algebra and explores their action on the K-theory of parabolic sheaves, aiming to connect algebraic structures with the AGT correspondence.

Contribution

It explicitly constructs elements in the shifted quantum toroidal algebra and investigates their relation to $q$-deformed $W$-algebras and the AGT correspondence.

Findings

Constructed elements $W_{ij}^k$ in the algebra.

Showed these elements act trivially on K-theory of parabolic sheaves.

Proposed a link to $q$-deformed $W$-algebras and AGT correspondence.

Abstract

We construct explicit elements $W_{ij}^k$ in (a completion of) the shifted quantum toroidal algebra of type $A$, and show that these elements act by 0 on the $K$-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements $W_{ij}^k$ will be related to $q$-deformed $W$-algebras of type $A$ for arbitrary nilpotent, which would imply a $q$-deformed version of the AGT correspondence between gauge theory with surface operators and conformal field theory.