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arXiv · hep-th/9312185

On the Cohomology of the Noncritical $W$-string

E. Bergshoeff·J. de Boer·M. de Roo·T. Tjin
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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.

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E. Bergshoeff, J. de Boer, M. de Roo, T. Tjin. 1993-12-23. On the Cohomology of the Noncritical $W$-string. https://doi.org/10.1016/0550-3213(94)90387-5

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