On the cohomology of the noncritical W-string

We investigate the cohomology structure of a general noncritical WN string. We do this by introducing a new basis in the Hubert space in which the BRST operator splits into a “nested” sum of nilpotent BRST operators. We give explicit details for the case N = 3. In that case the BRST operator Q can be written as the sum of two, mutually anticommuting, nilpotent BRST operators: Q = Q0 + Q1. We argue that if one chooses for the Liouville sector a (p, q) W3 minimal model then the cohomology of the Q1 operator is closely related to a (p, q) Virasoro minimal model. In particular, the special case of a (4,3) unitary W3 minimal model with central charge c = 0 leads to a c = 1/2 Ising model in the Q1 cohomology. Despite all this, noncritical W3 strings are not identical to noncritical Virasoro strings.