Linearizing W-Algebras

TL;DR

This paper demonstrates that nonlinear $W_3$ and $W_3^{(2)}$ algebras can be embedded into linear algebras, enabling new realizations and revealing relationships, which may aid in developing $W$-string theories.

Contribution

The authors construct linear embeddings of $W_3$ and $W_3^{(2)}$ algebras, providing new realizations and insights into their relationships, with potential applications in string theory.

Findings

Linear embeddings of $W_3$ and $W_3^{(2)}$ algebras are constructed.

New field realizations of $W_3$ and $W_3^{(2)}$ are obtained.

Hidden relationships between $W_3$ and $W_3^{(2)}$ are revealed.

Abstract

We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras $W_3$ and $W_3^{(2)}$ can be embedded as subalgebras into some {\em linear} algebras with finite set of currents. Using these linear algebras we find new field realizations of $W_3^{(2)}$ and $W_3$ which could be a starting point for constructing new versions of $W$-string theories. We also reveal a number of hidden relationships between $W_3$ and $W_3^{(2)}$. We conjecture that similar linear algebras can exist for other $W$-algebras as well.