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 $Vi{r}_{q,t}$ and the deformed $W$ algebra $W_{q,t}(\widehat{g{l}_N})$. We explicitly construct two classes of infinitly many commutative operators ${\mathcal{I}}_m$, ${\mathcal{G}}_m$, $(m\in \mathbb{N})$, 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.

Proceedings for 5-th World Congress for Nonlinear Analysts 2008, Florida, USA