On the Cohomology of the Noncritical $W$-string

Abstract

We investigate the cohomology structure of a general noncritical $W_N$-string. We do this by introducing a new basis in the Hilbert 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: $Q=Q_0+Q_1$. We argue that if one chooses for the Liouville sector a $(p,q)$ $W_3$ minimal model then the cohomology of the $Q_1$ operator is closely related to a $(p,q)$ Virasoro minimal model. In particular, the special case of a (4,3) unitary $W_3$ minimal model with central charge $c=0$ leads to a $c=1/2$ Ising model in the $Q_1$ cohomology. Despite all this, noncritical $W_3$ strings are not identical to noncritical Virasoro strings.

38 pages, UG-7/93, ITP-SB-93-79

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• On the Algebraic Structure of Gravitational Descendants in CP(n-1) Coset Models
• On realizing the bosonic string as a noncritical $W_3$-string
• Spinor Field Realizations of the half-integer $W_{2,s}$ Strings
• Physical States and BRST Operators for Higher-spin $W$ Strings
• Explicit field realizations of W algebras