Physical States and BRST Operators for Higher-spin $W$ Strings

TL;DR

This paper explores the structure of higher-spin $W$ strings by analyzing physical states, constructing BRST operators, and deriving new BRST charges for specific algebraic cases at singular points.

Contribution

It introduces linear realizations of $W_{1,2,s}$ algebras and constructs new BRST charges for $W_{2,s}$ strings at special central charges.

Findings

Derived BRST operators for $W_{2,s}$ algebras.

Constructed new BRST charges for Liouville systems.

Analyzed physical states and algebra singularities.

Abstract

In this paper, we mainly investigate the $W_{2,s}^{M}\otimes W_{2,s}^{L}$ system, in which the matter and the Liouville subsystems generate $W_{2,s}^{M}$ and $W_{2,s}^L$ algebras respectively. We first give a brief discussion of the physical states for corresponding $W$ stings. The lower states are given by freezing the spin-2 and spin-$s$ currents. Then, introducing two pairs of ghost-like fields, we give the realizations of $W_{1,2,s}$ algebras. Based on these linear realizations, BRST operators for $W_{2,s}$ algebras are obtained. Finally, we construct new BRST charges of Liouville system for $W_{2,s}^{L}$ strings at the specific values of central charges $c$: $c=-{22/5}$ for $W_{2,3}^{L}$ algebra, $c=-24$ for $W_{2,4}^{L}$ algebra and $c=-2,-{286/3}$ for $W_{2,6}^{L}$ algebra, at which the corresponding $W_{2,s}^L$ algebras are singular.