On realizing the bosonic string as a noncritical $W_3$-string

TL;DR

This paper presents a way to realize the bosonic string as a noncritical $W_3$-string, using a specific Liouville sector and a BRST operator split, revealing new insights into string theory structures.

Contribution

It introduces a novel realization of the bosonic string within a noncritical $W_3$-string framework, utilizing a specific minimal model and a split BRST operator approach.

Findings

The noncritical $W_3$-string is characterized by a $(3,2)$ minimal model with $c_l=-2$.

The BRST operator is decomposed into two parts, with matter fields only in $Q_0$.

The $Q_1$-cohomology corresponds to a $(3,2)$ Virasoro minimal model at $c=0$.

Abstract

We discuss a realization of the bosonic string as a noncritical $W_3$-string. The relevant noncritical $W_3$-string is characterized by a Liouville sector which is restricted to a (non-unitary) $(3,2)$ $W_3$ minimal model with central charge contribution $c_l = - 2$. Furthermore, the matter sector of this $W_3$-string contains $26$ free scalars which realize a critical bosonic string. The BRST operator for this $W_3$-string can be written as the sum of two, mutually anticommuting, nilpotent BRST operators: $Q = Q_0 + Q_1$ in such a way that the scalars which realize the bosonic string appear only in $Q_0$ while the central charge contribution of the fields present in $Q_1$ equals zero. We argue that, in the simplest case that the Liouville sector is given by the identity operator only, the $Q_1$-cohomology is given by a particular (non-unitary) $(3,2)$ Virasoro minimal model at $c=0$.