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The \(\mathcal W_ 3\) algebra. Modules, semi-infinite cohomology and BV algebras. (English) Zbl 0860.17042

Lecture Notes in Physics. New Series m: Monographs. m42. Berlin: Springer. xi, 204 p. (1996).
\(W\) algebras are nonlinear extensions of the Virasoro algebra, and the simplest of them – beyond the Virasoro algebra itself – is the \(W_3\) algebra. This book is an exposition of the authors’ work on the structure and semi-infinite cohomology of the \(W_3\) algebra and its representations, as well as the relation between \(W\) algebras and Batalin-Vilkovisky (BV) algebras. The contents of the book are as follows:
(1) Introduction and preliminaries: 1.1. General introduction, 1.2. Outline and summary of results, 1.3. The BV algebra of the 2D \(W_2\) string.
(2) \(W\) algebras and their modules: 2.1. \(W\) algebras, 2.2. \(W_3\) modules, 2.3. Verma modules and Fock modules at \(c=2\), 2.4. Resolutions.
(3) BRST cohomology of the 4D \(W_3\) string: 3.1. Complexes of semi-infinite cohomology of the \(W_3\) algebra, 3.2. The \(W_3\) cohomology problem for the 4D \(W_3\) string, 3.3. Preliminary results, 3.4. The cohomology in the “Fundamental Weyl chamber”, 3.5. The conjecture for \(H(W_3,\mathbb{C})\).
(4) Batalin-Vilkovisky algebras: 4.1. G algebras and BV algebras, 4.2. The BV algebra of polyderivations of the ground ring algebra \(R_N\), 4.3. \(N=3\): The BV algebra structure of \(P(R_3)\), 4.4. G modules and BV modules, 4.5. \(N=3\): Twisted modules of \(P(R_3)\), 4.6. BV algebras on the base affine space \(A(G)\).
(5) The BV algebra of the \(W_3\) string: 5.1. Introduction, 5.2. A preliminary survey of \({\mathfrak H}\), 5.3. The relation between \({\mathfrak H}\) and \({\mathfrak B}\), 5.4. The bulk structure of \({\mathfrak H}\), 5.5. Towards the complete structure of \({\mathfrak H}\), 5.6. The complete structure of \({\mathfrak H}\), 5.7. Concluding remarks and open problems.

MSC:

17B68 Virasoro and related algebras
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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