Frenkel, L. B.; Garland, H.; Zuckerman, G. J. Semi-infinite cohomology and string theory. (English) Zbl 0607.17007 Proc. Natl. Acad. Sci. USA 83, 8442-8446 (1986). The semi-infinite cohomology of graded Lie algebras has been introduced in the mathematical literature by B. L. Feĭgin [Russ. Math. Surv. 39, No. 2, 155–156 (1984); translation from Usp. Mat. Nauk 39, No. 2(236), 195–196 (1984; Zbl 0544.17009)]. In this paper, it is shown that the structures related to the semi-infinite cohomology of the Virasoro algebra for the Fock module coincide with the corresponding structures associated to free bosonic string theories. In Section 1 the definitions and basic results are collected on the semi-infinite cohomology. The structures related to this cohomology are similar to those of Kähler geometry. With the help of spectral sequences a vanishing theorem is proved for the cohomology. In Section 2 the semi-infinite cohomology of the Virasoro algebra is studied. In the case of Fock module the authors define the so called Beechi-Rouet-Stora-Tyutin operator as the differential of some subcomplex. The calculations in physics for some operators related to Kähler algebra are explicitly obtained, which allows to construct the minimal local gauge invariant string theory in a standard mathematical setting. Reviewer: Alice Fialowski (Budapest) Cited in 4 ReviewsCited in 61 Documents MSC: 17B56 Cohomology of Lie (super)algebras 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 53B35 Local differential geometry of Hermitian and Kählerian structures 17B68 Virasoro and related algebras 81T70 Quantization in field theory; cohomological methods Keywords:no-ghost theorem; Kähler structure; semi-infinite cohomology; Virasoro algebra; Fock module; vanishing theorem; Beechi-Rouet-Stora-Tyutin operator; string theory Citations:Zbl 0544.17009; Zbl 0574.17008 PDFBibTeX XMLCite \textit{L. B. Frenkel} et al., Proc. Natl. Acad. Sci. USA 83, 8442--8446 (1986; Zbl 0607.17007) Full Text: DOI