Multi-Field Coset Space Realizations of $w_{1+\infty}$

TL;DR

This paper develops multi-field coset space realizations of the $w_{1+ ablafty}$ algebra, introducing new two- and three-field models, and explores their connections to string theories and Toda models.

Contribution

It extends the coset space formulation to include multiple fields, deriving new realizations of $w_{1+ ablafty}$ with potential applications in string theory and integrable models.

Findings

Derived two-field realizations involving a 2D dilaton.

Constructed a three-field realization with an invariant action.

Discussed parallels with $N=2$ strings and Toda theories.

Abstract

We extend the coset space formulation of the one-field realization of $w_{1+\infty}$ to include more fields as the coset parameters. This can be done either by choosing a smaller stability subalgebra in the nonlinear realization of $w_{1+\infty}$ symmetry, or by considering a nonlinear realization of some extended symmetry, or by combining both options. We show that all these possibilities give rise to the multi-field realizations of $w_{1+\infty}$. We deduce the two-field realization of $w_{1+\infty}$ proceeding from a coset space of the symmetry group $\tilde{G}$ which is an extension of $w_{1+\infty}$ by the second self-commuting set of higher spin currents. Next, starting with the unextended $w_{1+\infty}$ but placing all its spin 2 generators into the coset, we obtain a new two-field realization of $w_{1+\infty}$ which essentially involves a $2D$ dilaton. In order to construct the invariant action for this system we add one more field and so get a new three-field realization of $w_{1+\infty}$. We re-derive it within the coset space approach, by applying the latter to an extended symmetry group $\hat{G}$ which is a nonlinear deformation of $\tilde{G}$. Finally we present some multi-field generalizations of our three-field realization and discuss several intriguing parallels with $N=2$ strings and conformal affine Toda theories.