Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and 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 ${\mathcal{F}}_u^{(n^{\prime },n)}$ with a nontrivial slope $n^{\prime }/n$. 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.