# Integrals of motion from quantum toroidal algebras

## TL;DR

This paper connects transfer matrices of quantum toroidal algebras with integrals of motion, proving conjectures and exploring dualities in quantum integrable models, with implications for the quantum KdV spectrum.

## Contribution

It identifies transfer matrix coefficients with elliptic integrals of motion, proving Litvinov's conjectures and discussing dualities in quantum integrable systems.

## Key findings

- Proved Litvinov conjectures on the Intermediate Long Wave model.
- Established a duality between (gl(m),gl(n)) in quantum toroidal models.
- Conjectured the spectrum of non-local integrals via Gaudin Bethe ansatz.

## Abstract

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the (gl(m),gl(n)) duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine sl(2).