Null vectors of the W_3 algebra

TL;DR

This paper explicitly constructs null vectors for the W_3 algebra, extending methods from the Virasoro algebra, and provides alternative formulas involving specific modes, advancing understanding of algebraic structures in conformal field theory.

Contribution

It introduces explicit constructions of W_3 null vectors and extends Virasoro null vector formulas to the W_3 algebra, offering new computational tools.

Findings

Explicit W_3 null vectors constructed in two forms

Extended Virasoro null vector formulas involving only L_{-1} and L_{-2}

Analogous results to Benoit and St.~Aubin for W_3 algebra

Abstract

We construct $W_3$ null vectors of a restricted class explicitly in two different forms. The method we use is an extension of that of Bauer et al.~in the Virasoro case. Our results are analogous to the formulae of Benoit and St.~Aubin for the Virasoro null vectors. We derive in the Virasoro case some alternative formulae for the same null vectors involving only the $L_{-1}$ and $L_{-2}$ modes of the Virasoro algebra. }