# Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks

## TL;DR

This paper constructs explicit bosonizations of certain quantum toroidal algebra representations, studies twisted $W$-algebras acting on them, and proves a conjectured relation on $q$-deformed conformal blocks related to $q$-deformed isomonodromy and CFT correspondence.

## Contribution

It provides explicit bosonization formulas for Fock modules with nontrivial slope and establishes a key relation on $q$-deformed conformal blocks, advancing understanding of quantum toroidal algebras and $q$-deformed CFT.

## Key findings

- Explicit bosonization of Fock modules with nontrivial slope
- Analysis of twisted $W$-algebras acting on these modules
- Proof of a conjectured relation on $q$-deformed conformal blocks

## Abstract

We study certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra for $q=t$. We construct explicit bosonization of the Fock modules $\mathcal{F}_u^{(n',n)}$ with a nontrivial slope $n'/n$. As a vector space, it is naturally identified with the basic level 1 representation of affine $\mathfrak{gl}_n$. We also study twisted $W$-algebras of $\mathfrak{sl}_n$ acting on these Fock modules. As an application, we prove the relation on $q$-deformed conformal blocks which was conjectured in the study of $q$-deformation of isomonodromy/CFT correspondence.