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On the Cohomology of the Noncritical $W$-string
High Energy Physics - Theory · Submitted 23 Dec 1993
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Despite all this, noncritical $W_3$ strings are not identical to noncritical Virasoro strings.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the cohomology structure of a general noncritical $W_N$-string. | contribution 0.96 | 0.93 | 0.05 | asserted 0.86 |
| s1 | 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. | method 0.99 | 0.95 | 0.04 | asserted 1.00 |
| s2 | We give explicit details for the case $N=3$. | contribution 0.86 | 0.93 | 0.04 | asserted 0.99 |
| s3 | In that case the BRST operator $Q$ can be written as the sum of two, mutually anticommuting, nilpotent BRST operators: $Q=Q_0+Q_1$. | result 0.65 | 0.86 | 0.08 | asserted 0.99 |
| s4 | 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. | result 0.60 | 0.76 | 0.18 | hedged possibility 0.71 |
| s5 | 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. | result 0.99 | 0.81 | 0.14 | asserted 1.00 |
| s6 | Despite all this, noncritical $W_3$ strings are not identical to noncritical Virasoro strings. | result 0.38 | 0.69 | 0.41 | asserted 1.00 |
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Original 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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3 arXiv papers cite this in 4 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 3 papers
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Since $W_3$ is the simplest $W$ algebra, most of the efforts in constructing $W$ string theories have been concentrated on it [ref] .
Spinor Field Realizations of the half-integer $W_{2,s}$ Strings · Introduction · 0.85
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Since $W$ algebras [ref] received considerable attention and application, much work [ref] has been carried out on the classification of $W$ algebras and the study of $W$ gravities and $W$ strings.
Physical States and BRST Operators for Higher-spin $W$ Strings · Introduction · 1.00
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Many $W$ algebras are known (for review see [ref] ) and much work has been carried out on their classification [ref] . $W$ algebras have many applications and become the subject of great interest in many branches of physics and mathematics, e.g., in $W$ gravity theories [ref] , critical and non-critical $W$ string theories [ref] , Wess-Zumino-Novikov-Witten(WZNW) models [ref] , quantum Hall effect [ref] , and especially in black holes [ref] , where it was shown that the Hawking radiation can be explained as the fluxes of chiral currents forming a $W_∞$ algebra.
Explicit field realizations of W algebras · Introduction · 1.00
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