Null Fields Realizations of $W_3$ from $W(sl(4),sl(3))$ and $W(sl(3|1),sl(3))$ Algebras

TL;DR

This paper constructs null field realizations of the $W_3$ algebra from specific nonlinear algebras, identifying new realizations at particular central charges using currents and affine algebras.

Contribution

It provides new null field realizations of the $W_3$ algebra from $W(sl(4),sl(3))$ and $W(sl(3|1),sl(3))$, including previously unknown central charge values.

Findings

Realizations exist for $c_W=-30,-40/7,-98/5,-2$.

New realizations for $c_W=-30$ and $-40/7$ are listed for the first time.

Realizations involve arbitrary stress tensors and $u(1) imes sl(2)$ affine currents.

Abstract

We consider the nonlinear algebras $W(sl(4),sl(3))$ and $W(sl(3|1),sl(3))$ and find their realizations in terms of currents spanning conformal linearizing algebras. The specific structure of these algebras, allows us to construct realizations modulo null fields of the $W_3$ algebra that lies in the cosets $W(sl(4),sl(3))/u(1)$ and $W(sl(3|1),sl(3))/u(1)$. Such realizations exist for the following values of the $W_3$ algebra central charge: $c_W=-30,-40/7,-98/5,-2$. The first two values are listed for the first time, whereas for the remaining values we get the new realizations in terms of an arbitrary stress tensor and $u(1)\times sl(2)$ affine currents.