The W3 Algebra

1. Front Matter

   Pages I-XI

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2. Introduction and Preliminaries

   Pages 1-16

3. W Algebras and Their Modules

   Pages 17-52

4. BRST Cohomology of the 4D W₃ String

   Pages 53-77

5. Batalin-Vilkovisky Algebras

   Pages 79-123

6. The BV Algebra of the W₃ String

   Pages 125-148

7. Back Matter

   Pages 149-204

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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.

• Virasaro algebra
• W*-geometry
• algebra
• cohomology
• field
• geometry
• gravity
• lie algebra
• linear algebra
• quantum gravity
• representation theory

• Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, Australia

  Peter Bouwknegt, Jim McCarthy

• Department of Physics and Astronomy, University of Southern California, Los Angeles, USA

  Krzysztof Pilch

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