Embedding of the Bosonic String into the $W_3$ String

TL;DR

This paper explores new realizations of the $W_3$ algebra with arbitrary central charge, enabling embeddings of the Virasoro string into $W_3$ strings through BRST operator constructions.

Contribution

It introduces novel realizations of the $W_3$ algebra with specific central charges and demonstrates their use in embedding the Virasoro string into $W_3$ string frameworks.

Findings

New $W_3$ algebra realizations at $c=102$ and $c=100$.

Construction of non-critical and critical $W_3$ BRST operators.

Embedding of the Virasoro string into $W_3$ strings via canonical transformations.

Abstract

We investigate new realisations of the $W_3$ algebra with arbitrary central charge, making use of the fact that this algebra can be linearised by the inclusion of a spin-1 current. We use the new realisations with $c=102$ and $c=100$ to build non-critical and critical $W_3$ BRST operators. Both of these can be converted by local canonical transformations into a BRST operator for the Virasoro string with $c=28-2$, together with a Kugo-Ojima topological term. Consequently, these new realisations provide embeddings of the Virasoro string into non-critical and critical $W_3$ strings.