On the BRST Operator of $W$-Strings

TL;DR

This paper investigates the structure of the BRST operator in $W$-strings, demonstrating conditions for its decomposition and exploring implications for the spectra of non-critical $W_n$-strings.

Contribution

It provides a new framework for decomposing the BRST operator in $W$-strings and proposes a conjecture relating the spectra of different $W$-string theories.

Findings

Decomposition conditions for the BRST operator in $W$-strings

Application to non-critical $W_3$-string example

Conjecture relating spectra of $W_n$ and $W_{n-1}$-strings

Abstract

We discuss the conditions under which the BRST operator of a $W$-string can be written as the sum of two operators that are separately nilpotent and anticommute with each other. We illustrate our results with the example of the non-critical $W_3$-string. Furthermore, we apply our results to make a conjecture about a relationship between the spectrum of a non-critical $W_n$-string and a $W_{n-1}$-string.