The Integrals of Motion for the Deformed Virasoro Algebra

We explicitly construct two classes of infinitly many commutative operators in terms of the deformed Virasoro algebra. We call one of them local integrals and the other nonlocal one, since they can be regarded as elliptic deformations of the local and nonlocal integrals of motion obtained by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov.

💡 Research Summary

The paper presents a systematic construction of two infinite families of commuting operators associated with the deformed Virasoro algebra, often denoted Vir_{q,t}. The authors begin by recalling the free‑field realization of the algebra in terms of two bosonic oscillator families β_{1,m}, β_{2,m} (m∈ℤ{0}) together with zero‑mode operators P and Q. Their commutation relations involve the elliptic theta function