Toda lattice field theories, discrete W algebras, Toda lattice hierarchies and quantum groups

TL;DR

This paper explores the lattice $sl_3$ Toda theory, deriving the discrete $W_3$ algebra, establishing an integrable system, and connecting it to Toda hierarchies and quantum groups, including continuum limits and quantum algebra formulations.

Contribution

It introduces a new quadratic algebra for lattice $sl_3$ Toda theory and links it to discrete $W_3$ algebra, integrable systems, and quantum groups.

Findings

Derived the quadratic algebra for lattice $sl_3$ Toda theory.

Reconstructed the discrete $W_3$ algebra from the quadratic algebra.

Established the relation between the integrable system and Toda lattice hierarchy.

Abstract

In analogy with the Liouville case we study the $sl_3$ Toda theory on the lattice and define the relevant quadratic algebra and out of it we recover the discrete $W_3$ algebra. We define an integrable system with respect to the latter and establish the relation with the Toda lattice hierarchy. We compute the the relevant continuum limits. Finally we find the quantum version of the quadratic algebra.