Bosonisation of the Complex-boson realisation of $W_\infty$

TL;DR

This paper develops bosonisation techniques for complex-boson realizations of $W_$ algebras, leading to new scalar-based representations of these infinite-dimensional symmetries.

Contribution

It introduces novel bosonisation methods for $W_$ algebras, deriving scalar and fermionic realizations that extend previous algebraic representations.

Findings

Derived nonlinear realizations of $W_$ and $W_{1+}$ in terms of fermions and scalars.

Obtained scalar-only realizations of classical $w_$ from the bosonisation process.

Provided explicit constructions of algebra representations using bosonic and fermionic fields.

Abstract

We bosonise the complex-boson realisations of the $W_\infty$ and $W_{1+\infty}$ algebras. We obtain nonlinear realisations of $W_\infty$ and $W_{1+\infty}$ in terms of a pair of fermions and a real scalar. By further bosonising the fermions, we then obtain realisations of $W_\infty$ in terms of two scalars. Keeping the most non-linear terms in the scalars only, we arrive at two-scalar realisations of classical $w_\infty$.