Many Boson Realizations of Universal Nonlinear $W_{\infty}$-Algebras

TL;DR

This paper constructs numerous free field realizations of universal nonlinear $W_{ fty$-algebras, generalizing the Miura transformation, and relates them to graded $SL(p,q)$ Lie algebras, providing insights into KP hierarchies.

Contribution

It introduces many boson realizations of universal nonlinear $W_{ fty$-algebras using scalars of different signatures, extending the Miura transformation and linking to graded $SL(p,q)$ structures.

Findings

Realizations are expressed in terms of $p$ plus $q$ scalars with $p-q=N.

These realizations generalize the Miura transformation.

They naturally lead to modified KP hierarchies.

Abstract

An infinite number of free field realizations of the universal nonlinear $\hat{W}_{\infty}^{(N)}$ ($\hat{W}_{1+\infty}^{(N)}$) algebras, which are identical to the KP Hamiltonian structures, are obtained in terms of $p$ plus $q$ scalars of different signatures with $p-q=N$. They are generalizations of the Miura transformation, and naturally give rise to the modified KP hierarchies via corresponding realizations of the latter. Their characteristic Lie-algebraic origin is shown to be the graded $SL(p,q)$.