W3 Constructions on Affine Lie Algebras

TL;DR

This paper constructs $W_3$ algebra realizations on arbitrary affine Lie algebras using Virasoro constructions, with detailed analysis for the $ ext{su}(2)_ ext{level}$ case and potential applications to $W$-strings.

Contribution

It introduces a method to realize $W_3$ algebras on any affine Lie algebra, expanding the scope of algebraic constructions in conformal field theory.

Findings

Realizations for generic levels are provided.

Explicit solutions for specific levels with parameters are given.

Discussion on applications to $W$-string theories.

Abstract

We use an argument of Romans showing that every Virasoro construction leads to realizations of $W_3$, to construct $W_3$ realizations on arbitrary affine Lie algebras. Solutions are presented for generic values of the level as well as for specific values of the level but with arbitrary parameters. We give a detailed discussion of the $\aff{su}(2)_\ell$-case. Finally, we discuss possible applications of these realizations to the construction of $W$-strings.