On Realisations of W Algebras

TL;DR

This paper explores a broad class of realizations of W algebras derived from the Miura transformation, providing explicit formulas and discussing their applications in W-string theory.

Contribution

It introduces a comprehensive framework for realizing W algebras via the Miura transformation, expanding previous methods and applications.

Findings

Explicit formulas for all realizations of the considered class.

Connections between realizations and canonical subalgebras.

Applications in W-string theory.

Abstract

It has been known for some time that $W$ algebras can be realised in terms of an energy-momentum tensor together with additional free scalar fields. Some recent results have shown that more general realisations are also possible. In this paper, we consider a wide class of realisations that may be obtained from the Miura transformation, related to the existence of canonical subalgebras of the Lie algebras on which the $W$ algebras are based. We give explicit formulae for all realisations of this kind, and discuss their applications in $W$-string theory.