Nonlinear realizations of $W_3$ symmetry

TL;DR

This paper presents a geometric derivation of the $W_3$ symmetry realization using Toda equations, linking higher-spin symmetries to integrable systems and suggesting broader applicability to other nonlinear algebras.

Contribution

It introduces a geometric method to realize $W_3$ symmetry from higher-spin algebra, connecting Toda equations to coset space constraints.

Findings

Derived $sl_3$ Toda realization of $W_3$ symmetry

Identified Toda equations as constraints on a coset space

Proposed extension to other nonlinear algebras

Abstract

We deduce the $sl_{3}$ Toda realization of classical $W_3$ symmetry on two scalar fields in a geometric way, proceeding from a nonlinear realization of some associate higher-spin symmetry $W_{3}^{\infty}$. The Toda equations are recognized as the constraints singling out a two-dimensional fully geodesic subspace in the initial coset space of $W_{3}^{\infty}$. The proposed geometric approach can be extended to other nonlinear algebras and integrable systems.