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The study of W algebras began in 1985 in the context of
two-dimensional conf- mal field theories, the aim being to explore
higher-spin extensions of the Virasoro algebra. Given the
simultaneous growth in the understanding of two-dimensional metric
gravity inspired by analyses of string models, it was inevitable
that these algebras would be applied to give analogues of putative
higher-spin gravity t- ories. This book is an exposition of the
past few years of our work on such an application for the algebra:
in particular, the BRST quantization of the n- critical 4D string.
We calculate the physical spectrum as a problem in BRST cohomology.
The corresponding operator cohomology forms a BV algebra, for which
we provide a geometrical model. The algebra has one further
generator, of spin three, in addition to the (spin two)
energy-momentum tensor which generates the Virasoro algebra. C-
trary to the Virasoro algebra, it is an algebra defined by
nonlinear relations. In deriving our understanding of the resulting
gravity theories we have had to - velop a number of results on the
representation theory of W algebras, to replace the standard
techniques that were so successful in treating linear algebras.
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