BRST COHOMOLOGY OF THE CRITICAL W4 STRING

We study the cohomology of the critical W4 string using the W4 BRST ch arge in a special basis in which it contains three separately nilpoten t BRST charges. This allows us to obtain the physical operators in thr ee steps. In the first step we obtain the cohomology associated to a s pin-4 constraint only, and it contains operators of the c = 4/5 W3 min imal model. In the next step, where the spin-3 constraint is added, th ese operators get dressed to operators of the c = 7/10 Virasoro minima l model. Finally, the Virasoro constraint is added to obtain the cohom ology of the critical W4 string. We describe the structure of the comp lete cohomology and compare with other results.