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

TL;DR

This paper reviews the free field realization of deformed Virasoro and W algebras, constructing infinitely many commuting operators that serve as elliptic deformations of integrals of motion in conformal field theory.

Contribution

It explicitly constructs two classes of commuting operators in the elliptic deformation of Virasoro and W algebras, advancing understanding of their integrals of motion.

Findings

Construction of infinitely many commuting operators

Elliptic deformation of local and nonlocal integrals of motion

Enhanced algebraic structure understanding

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.