Parafermions, $W$ Strings and their BRST Charges

TL;DR

This paper explores the relationship between parafermions, $W$ strings, and their BRST charges, providing methods to construct and relate these charges for various $W_N$ string theories.

Contribution

It introduces a novel connection between parafermions in $W_3$ strings and standard parafermion constructions, and generalizes this to $W_N$ strings, enabling explicit BRST charge constructions.

Findings

Established the link between parafermions and $W_3$ string BRST charges.

Constructed the full BRST charge for $W_{2,N}$ strings.

Proposed a method to extend calculations to full BRST charges for $W_N$ strings.

Abstract

We show how to relate the parafermions that occur in the $W_3$ string to the standard construction of parafermions. This result is then used to show that one of the screening charges that occurs in parafermionic theories is precisely the non-trivial part of the $W_3$ string BRST charge. A way of generalizing this pattern for a $W_N$ string is explained. This enables us to construct the full BRST charge for a $W_{2,N}$ string and to prove the relation of such a string to the algebra $W_{N-1}$ for arbitrary $N$, We also show how to calculate part of the BRST charge for a $W_N$ string, and we explain how our method might be extended to obtain the full BRST charge for such a string.