On the Cohomology of the Noncritical $W$-string

TL;DR

This paper explores the cohomology of noncritical $W_N$-strings by introducing a new basis that simplifies the BRST operator, with explicit analysis for $N=3$, revealing connections to minimal models and the Ising model.

Contribution

It introduces a novel basis in the Hilbert space that decomposes the BRST operator into nested nilpotent parts, providing new insights into the structure of noncritical $W_N$-strings.

Findings

BRST operator splits into two anticommuting nilpotent operators for N=3

Cohomology of one operator relates to Virasoro minimal models

Special case yields a connection to the Ising model at c=1/2

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.