Abstract
A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Calogero, F.: J. Math. Phys. 10 (1969), 2197–2200; Sutherland, B.: J. Math. Phys. 12 (1970), 246–250, 251–256.
Belavin, A. A., Polyakov, A. M. and Zamolodchikov, A. B.: Nuclear Phys. B 241 (1984), 333–380.
Awata, H., Matsuo, Y., Odake, S. and Shiraishi, J.: Collective field theory, Calogero-Sutherland model and generalized matrix models, Phys. Lett. B 347 (1995), 49–55; A note on the Calogero-Sutherland model, W nsingular vectors and generalized matrix models, Preprint, hep-th/9503028, to appear in Soryushiron Kenkyu (Kyoto); Excited state of the Calogero-Sutherland model and singular vectors of the W nalgebra, Preprint, hep-th/9503043, to appear in Nuclear Phys. B.
Awata, H., Odake, S. and Shiraishi, J.: Integral representations of the Macdonald symmetric functions and generalized matrix models, Preprint, q-alg/9506006.
Mimachi, K. and Yamada, Y.: Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials, Preprint (November 1994).
Stanley, R.: Adv. Math. 77 (1989), 76–115; Macdonald, I. G.: Lect. Notes in Math. 1271, Springer, New York, 1987, pp. 189–200.
Feigin, B. L. and Fuchs, D. B.: Skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra, Funktsional. Anal. i Prilozhen 16 (1982), 47; Verma modules over the Virasoro algebra, in L. D. Faddeev and A. A. Malcev (eds), Topology, Proceedings of Leningrad Conference, 1982, Lecture Notes in Math. 1060, Springer, New York, 1985.
Macdonald, I. G.: Symmetric Functions and Hall Functions, Oxford University Press, 1979.
Jevicki, A. and Sakita, B.: Nuclear Phys. B 165 (1980), 511–527.
Andrić, I., Jevicki, A. and Levin, H.: Nuclear Phys. B 215 [FS7] (1983), 307–315.
Felder, G.: Nuclear Phys. B 317 (1989), 215–236; Errata, Nuclear Phys. B 324 (1989), 548.
Frenkel, E. and Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W-algebra, q-alg/9505025, May 1995.
Awata, H., Kubo, H., Odake, S. and Shiraishi, J.: in preparation.
Drinfeld, V. G.: Quantum groups, ICM 86 report.
Jimbo, M.: A q-difference analogue of \(U(\mathfrak{g})\) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63–69.
Gasper, G. and Rahman, M.: Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1990.
Kuniba, A. and Suzuki, J.: Analytic Bethe ansatz for fundamental representations of Yangians, Preprint hep-th/9406180.
Bazhanov, V., Lukyanov, S. and Zamolodchikov, A.: Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Preprint hep-th/9412229.
Shiraishi, J.: Free Boson Representation of \(U_q (\widehat{sl}_2 )\), Phys. Lett. A 171 (1992), 243–248; Awata, H., Odake, S. and Shiraishi, J.: Free Boson Representation of \(U_q (\widehat{sl}_3 )\), Lett. Math. Phys. 30 (1994), 207–216; Free Boson Realization of \(U_q (\widehat{sl}_N )\), Comm. Math. Phys. 162 (1994), 61–83.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shiraishi, J., Kubo, H., Awata, H. et al. A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett Math Phys 38, 33–51 (1996). https://doi.org/10.1007/BF00398297
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00398297