Letters in Mathematical Physics

, Volume 38, Issue 1, pp 33–51 | Cite as

A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions

  • Jun'Ichi Shiraishi
  • Harunobu Kubo
  • Hidetoshi Awata
  • Satoru Odake
Article

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.

Mathematics Subject Classifications (1991)

17B37 17B65 81R50 

Key words

quantum deformation Virasoro algebra bosonic realization Macdonald symmetric functions 

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References

  1. 1.
    Calogero, F.: J. Math. Phys. 10 (1969), 2197–2200; Sutherland, B.: J. Math. Phys. 12 (1970), 246–250, 251–256.Google Scholar
  2. 2.
    Belavin, A. A., Polyakov, A. M. and Zamolodchikov, A. B.: Nuclear Phys. B 241 (1984), 333–380.Google Scholar
  3. 3.
    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.Google Scholar
  4. 4.
    Awata, H., Odake, S. and Shiraishi, J.: Integral representations of the Macdonald symmetric functions and generalized matrix models, Preprint, q-alg/9506006.Google Scholar
  5. 5.
    Mimachi, K. and Yamada, Y.: Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials, Preprint (November 1994).Google Scholar
  6. 6.
    Stanley, R.: Adv. Math. 77 (1989), 76–115; Macdonald, I. G.: Lect. Notes in Math. 1271, Springer, New York, 1987, pp. 189–200.Google Scholar
  7. 7.
    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.Google Scholar
  8. 8.
    Macdonald, I. G.: Symmetric Functions and Hall Functions, Oxford University Press, 1979.Google Scholar
  9. 9.
    Jevicki, A. and Sakita, B.: Nuclear Phys. B 165 (1980), 511–527.Google Scholar
  10. 10.
    Andrić, I., Jevicki, A. and Levin, H.: Nuclear Phys. B 215 [FS7] (1983), 307–315.Google Scholar
  11. 11.
    Felder, G.: Nuclear Phys. B 317 (1989), 215–236; Errata, Nuclear Phys. B 324 (1989), 548.Google Scholar
  12. 12.
    Frenkel, E. and Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W-algebra, q-alg/9505025, May 1995.Google Scholar
  13. 13.
    Awata, H., Kubo, H., Odake, S. and Shiraishi, J.: in preparation.Google Scholar
  14. 14.
    Drinfeld, V. G.: Quantum groups, ICM 86 report.Google Scholar
  15. 15.
    Jimbo, M.: A q-difference analogue of \(U(\mathfrak{g})\) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63–69.Google Scholar
  16. 16.
    Gasper, G. and Rahman, M.: Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1990.Google Scholar
  17. 17.
    Kuniba, A. and Suzuki, J.: Analytic Bethe ansatz for fundamental representations of Yangians, Preprint hep-th/9406180.Google Scholar
  18. 18.
    Bazhanov, V., Lukyanov, S. and Zamolodchikov, A.: Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Preprint hep-th/9412229.Google Scholar
  19. 19.
    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.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Jun'Ichi Shiraishi
    • 1
  • Harunobu Kubo
    • 2
  • Hidetoshi Awata
    • 3
  • Satoru Odake
    • 4
  1. 1.Institute for Solid State PhysicsUniversity of TokyoTokyoJapan
  2. 2.Department of Physics, Faculty of ScienceUniversity of TokyoTokyoJapan
  3. 3.Uji Research Center, Yukawa Institute for Theoretical PhysicsKyoto UniversityUjiJapan
  4. 4.Department of Physics, Faculty of ScienceShinshu UniversityMatsumotoJapan

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