Noncritical $W$-strings and Minimal Models

We perform a BRST analysis of the physical states described by a general noncritical $W$-string. A crucial feature of our analysis is that we introduce a special basis in the Hilbert space of physical states in which the BRST operator splits into a nested sum of nilpotent BRST operators. We argue that the cohomology of each nilpotent BRST operator occurring in the “nested” sum is closely related to a specific $W$ mimimal model. We discuss in detail the special case of the noncritical $W_3$-string.