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

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 $c$: $c=-22\mathrm{/}5$ for $W_{2,3}^L$ algebra, $c=-24$ for $W_{2,4}^L$ algebra and $c=-2,-286\mathrm{/}3$ for $W_{2,6}^L$ algebra, at which the corresponding $W_{2,s}^L$ algebras are singular.