Realisations of $W_3$ Symmetry

TL;DR

This paper explores new free-scalar realizations of the $W_3$ algebra by including a null spin-four operator, expanding previous models and providing a comprehensive analysis for two-scalar cases.

Contribution

It introduces generalized free-scalar realizations of the $W_3$ algebra that incorporate a null spin-four operator, broadening the scope of algebra representations.

Findings

Multiple realizations of $W_3$ algebra with null spin-four operator

Complete analysis for two-scalar realizations

Extension of previous models to include non-zero spin-four operator

Abstract

We perform a systematic investigation of free-scalar realisations of the Za\-mo\-lod\-chi\-kov $W_3$ algebra in which the operator product of two spin-three generators contains a non-zero operator of spin four which has vanishing norm. This generalises earlier work where such an operator was required to be absent. By allowing this spin-four null operator we obtain several realisations of the $W_3$ algebra both in terms of two scalars as well as in terms of an arbitrary number $n$ of free scalars. Our analysis is complete for the case of two-scalar realisations.