Feigin, Boris; Frenkel, Edward Semi-infinite Weil complex and the Virasoro algebra. (English) Zbl 0726.17035 Commun. Math. Phys. 137, No. 3, 617-639 (1991). The Weil complex \(\wedge^*({\mathfrak g}')\otimes S^*({\mathfrak g}')\), where \({\mathfrak g}'\) is the dual of a finite-dimensional Lie algebra \({\mathfrak g}\) and \(\wedge^*\), \(S^*\) are the exterior and symmetric algebras thereof, is an important tool for the calculation of the cohomology of \({\mathfrak g}\) (being a cross between the Koszul and the standard complexes of \({\mathfrak g})\). It is also convenient for calculation of characteristic classes of principal \(G\)-bundles, where \({\mathfrak g}=\text{Lie}(G).\) In the paper a semi-infinite analogue of the Weil complex is introduced for the Virasoro algebra and some cohomologies are computed; some interesting geometrical consequences (characteristic classes) are conjectured. References are often messed up, regrettably. Reviewer: D.Leites (Stockholm) Cited in 3 ReviewsCited in 24 Documents MSC: 17B56 Cohomology of Lie (super)algebras 17B68 Virasoro and related algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 81T70 Quantization in field theory; cohomological methods Keywords:BRST-cohomology; Fock representations; Weil complex; semi-infinite analogue; Virasoro algebra; characteristic classes PDFBibTeX XMLCite \textit{B. Feigin} and \textit{E. Frenkel}, Commun. Math. Phys. 137, No. 3, 617--639 (1991; Zbl 0726.17035) Full Text: DOI References: [1] Alvarez-Gaume, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys.112, 503–552 (1987) · Zbl 0647.14019 [2] Atiyah, M.: New invariants of three and four dimensional manifolds. Proc. Symp. Pure Math.48, 285–299 (1988) [3] Atiyah, M., Bott, R.: The moment map and equivariant cohomology. Topology23, 1–28 (1984) · Zbl 0521.58025 [4] Bauileu, L., Singer, I.: Topological Yang-Mills theory. Nucl. Phys. B (Proc. Suppl.)5 B, 12–19 (1988) · Zbl 0958.58500 [5] Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators (to appear) · Zbl 1037.58015 [6] Bouwknegt, P., Ceresole, A., McCarthy, J.G., van Nieuwenhuizen, P.: Extended sugawara construction for the superalgebrasSU(M=1|N+1). Phys. Rev. D39, 2971–2986 (1989) [7] Bouwknegt, P., McCarthy, J., Pilch, K.: On the free field resolutions for coset conformal field theories, Preprint CTP 1861, June 1990 · Zbl 0784.17041 [8] Becchi, C., Rouet, A., Stora, R.: Ann. Phys.98, 287 (1976) [9] Brylinski, J.-L.: Representations of loop groups, Dirac operators on loop space, and modular forms. Topology29, 461–480 (1990) · Zbl 0715.22023 [10] Cartan, H.: La transgression dans un groupe de Lie et dans un espace fibre principal. Colloque de Topologie (Espaces Fibres), 57–71. C.B.R.M. Bruxelles 1950 [11] Date, E., Jimbo, M., Miwa, T., Kashiwara, A.: In: Proc. of RIMS Symp. Jimbo, M., Miwa, T. (eds.) pp. 39–120. Singapore: World Scientific 1983 [12] Dijkgraaf, R., Witten, E.: Mean field theory, topological field theory and multi-matrix models. Nucl. Phys.B342, 486–552 (1990) [13] Distler, J.: 2D quantum gravity, topological field theory and the multi-critical matrix models. Nucl. Phys.B 342, 523–538 (1990) [14] Dotsenko, Vl., Fateev, V.: Conformal algebra and multi-point correlation functions in statistical models. Nucl. Phys. B240, 312–348 (1984) [15] Feigin, B.L.: Semi-infinite cohomology of Lie, Virasoro and Kac-Moody algebras. Russ. Math. Surv.39, 155–156 (1984) · Zbl 0574.17008 [16] Feigin, B.L., Frenkel, E.V.: Affine Mac-Moody algebras and bosonization. In: Physics and mathematics of strings. V. Knizhnik Memorial Volume Brink, L., Friedan, D., Polyakov, A. (eds.) pp. 271–316. Singapore: World Scientific 1990 [17] Feigin, B.L., Frenkel, E.V.: Affine Kac-Moody algebras and semi-infinite flag manifolds, Commun. Math. Phys.128, 161–189 (1990) · Zbl 0722.17019 [18] Feigin, B.L., Frenkel, E.V.: Bosonic ghost system and the Virasoro algebra. Phys. Lett.B 246, 71–74 (1990) · Zbl 1243.81086 [19] Feigin, B.L., Frenkel, E.V.: Quantization of the Drinfeld-Sokolov reduction. Phys. Lett.B 246, 75–81 (1990) · Zbl 1242.17023 [20] Feigin, B.L., Fuchs, D.B.: Representations of the Virasoro algebra. In: Representations of infinite-dimensional Lie algebras and Lie groups. London: Gordon and Breach 1990 [21] Feigin, B.L., Tsygan, B.L.: Cohomology of the Lie algebra of the generalized Jacobian matrices. Funct. Anal. Appl.17, 86–87 (1983) · Zbl 0544.17011 [22] Feigin, B.L., Tsygan, B.L.: Riemann-Roch theorem and Lie algebra cohomology. Moscow Preprint 1988 · Zbl 0686.14007 [23] Figueroa-O’Farrill, J., Kimura, T.: Commun. Math. Phys.124, 105–132 (1989) · Zbl 0726.17034 [24] Floer, A.: Couran Institute Preprint, 1987 [25] Frenkel, I.: Two constructions of affine Lie algebras and boson-fermion correspondence in quantum field theory. J. Funct. Anal.44, 259–327 (1981) · Zbl 0479.17003 [26] Frenkel, I., Garland, H., Zuckerman, G.: Semi-infinite cohomology of the Virasoro algebra and the no-ghost theorem. Proc. Natl. Acad. Sci. USA83, 8842–8846 (1986) · Zbl 0607.17007 [27] Friedan, D.: Notes on string theory and two-dimensional conformal field theory. In: Unified string theories. Green, M., Gross, D. (eds.) pp. 169–213, Singapore: World Scientific 1986 · Zbl 0648.53057 [28] Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys.B 271, 93–165 (1986) [29] Fuchs, D.B.: Cohomology of infinite-dimensional Lie algebras. New York: Plenum Press 1986 [30] Gelfand, I.M., Fuchs, D.B.: Cohomology of the Lie algebra of formal vector fields. Izv. Acad. Sci. USSR34, N2, 332–337 (1970) [31] Hosono, S., Tsuchiya, A.: Lie algebra cohomology andN=2 SCFT, based on GKO construction, Preprint UT-561, May 1990 · Zbl 0748.17020 [32] Kac, V., van de Leur, W.: Super boson-fermion correspondence. Ann. Inst. Fourier37, 99–137 (1987) · Zbl 0625.58041 [33] Kac, V., van de Leur, W.: Super boson-fermion correspondence of typeB. In: Infinitedimensional Lie algebras and groups. Kac, V. (ed.) pp. 369–416. Singapore: World Scientific 1989 · Zbl 0748.17021 [34] Kac, V., Peterson, D.: Infinite-dimensional Lie algebras, theta-functions and modular forms. Adv. Math.53, 125–164 (1984) · Zbl 0584.17007 [35] Kac, V., Peterson, D.: Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981) · Zbl 0469.22016 [36] Kanno, H.: Weil algebra structure and geometrical meaning of BRST transformation in topological quantum field theory. Z. Phys. C43, 477–483 (1989) [37] Kostant, B., Sternberg, S.: Symplectic reduction BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Phys.176, 49–113 (1987) · Zbl 0642.17003 [38] Kravchenko, O., Semikhatov, A.: Phys. Lett.B 231, 85–92 (1989) [39] Lerche, W., Vafa, C., Warner, N.: Chiral rings inN=2 superconformal theories. Nucl. Phys.B 324, 427 (1989) [40] Lian, B., Zuckerman, G.: BRST cohomology of the supper Virasoro algebras. Commun. Math. Phys.125, 301–335 (1989) · Zbl 0693.17014 [41] Lian, B., Zuckerman, G.: BRST cohomology and highest weight vector. I Yale University Preprint, April 1990 · Zbl 0719.17015 [42] Mathai, V., Quillen, D.: Superconnections, Thom classes and equivariant differential forms. Topology25, 85–110 (1986) · Zbl 0592.55015 [43] Segal, G.: Elliptic cohomology, Seminaire Bourbaki,695, 1–15 (1987) [44] Tannery, J., Molk, J.: Elements de la theorie des fonctions ellipitiques. Paris 1898 · JFM 29.0379.11 [45] Tate, J.: Residues of differential on curves. Ann. Sci. ENS, 4 Ser., t 1, 149–159 (1968) · Zbl 0159.22702 [46] Tsuchiya, A., Kanie, Y.: Fock space representations of the Virasoro algebra–Intertwining operators. Publ. RIMS, Kyoto Univ.22, 259–327 (1986) · Zbl 0604.17008 [47] Tyutin, I.V.: Unpublished [48] Verlinde, E., Verlinde, H.: Princeton Preprint, April 1990 [49] Witten, E.: Introduction to cohomological field theory. Princeton Preprint August 1990 [50] Zuckermann, G.: Semi-infinite homology of the Virasoro algebra, handwritten manuscript, 1986 [51] Zuckerman, G.: Modular forms, strings and ghosts. In: Proc. Symp. Pure Math.49, 273–384 (1989) · Zbl 0687.17007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.