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

Abstract

We study certain representations of the quantum toroidal 𝖌𝔩₁ algebra for 𝑞 = 𝘵. We construct explicit bosonization of the Fock modules 𝓕⁽ⁿ′'ⁿ⁾ᵤ with a nontrivial slope 𝑛′/𝑛. As a vector space, it is naturally identified with the basic level 1 representation of affine 𝖌𝔩ₙ. We also study twisted 𝑊-algebras of s𝔩n 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.