The integrals of motion for the elliptic deformation of the Virasoro and $W_N$ algebra

Abstract

We review the free field realization of the deformed Virasoro algebra $Vir_{q,t}$ and the deformed $W$ algebra $W_{q,t}(\hat{gl_N})$. We explicitly construct two classes of infinitly many commutative operators ${\cal I}_m$, ${\cal G}_m$, $(m \in {\mathbb N})$, in terms of these algebras. They can be regarded as the elliptic deformation of the local and nonlocal integrals of motion for the conformal field theory.