Conformal Algebras and Non-Linear Differential Equations

  • I. Bakas
Part of the NATO ASI Series book series (NSSB, volume 245)


The method of Hamiltonian reduction used by Drinfeld and Sokolov in the theory of integrable non-linear differential equations, is applied to two dimensional field theories. We show that conformai symmetries can be obtained from Kac-Moody Lie algebras by introducing appropriate gauge choices. We also present R-matrix generalizations of the underlying (Gelfand-Dickey) algebraic structures and outline possible extensions of the main results to higher dimensions. Finally, we discuss the significance of these ideas in theories of quantum gravity.


Pseudodifferential Operator Conformal Symmetry Conformal Algebra Kill Vector Field Hamiltonian Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. D. Faddeev, L. A. Takhtajan: “Hamiltonian Methods in the Theory of Solitons” (Springer-Verlag, 1987).MATHCrossRefGoogle Scholar
  2. [2]
    V. G. Drinfeld, V. V. Sokolov: Journ. Sov. Math. 30 (1985) 1975.CrossRefGoogle Scholar
  3. [3]
    A. M. Polyakov: Mod. Phys. Lett. A2 (1987) 893.MathSciNetADSGoogle Scholar
  4. V. G. Knizhnik, A. M. Polyakov, A. B. Zamolodchikov: Mod. Phys. Lett. A3 (1988) 819.MathSciNetADSGoogle Scholar
  5. [4]
    V. A. Fateev, S. L. Lykyanov: Int. Jour. Mod. Phys. A3 (1988) 507.MathSciNetADSCrossRefGoogle Scholar
  6. T. G. Khovanova: Funct. Anal. Appl. 20 (1986) 332.MATHCrossRefGoogle Scholar
  7. P. Mathieu: Phys. Lett. B208 (1988) 101.MathSciNetADSGoogle Scholar
  8. [5]
    I. Bakas: Phys. Lett. B213 (1988) 313; Phys. Lett. B219 (1989) 283; Commun. Math. Phys. (in press).MathSciNetADSGoogle Scholar
  9. [5a]
    I. Bakas: Phys. Lett. B219 (1989) 283;MathSciNetADSGoogle Scholar
  10. D. Smit: Utrecht preprint (1988)Google Scholar
  11. A. A. Belavin: in the proceedings of the Taniguchi symposium, Kyoto 1988 (to appear).Google Scholar
  12. [6]
    M. Adler: Invent. Math. 50 (1979) 219.ADSMATHCrossRefGoogle Scholar
  13. Yu. I. Manin: Journ. Sov. Math. 11 (1979) 1.MATHCrossRefGoogle Scholar
  14. [7]
    A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov: Nucl. Phys. B241 (1984) 333.MathSciNetADSCrossRefGoogle Scholar
  15. [8]
    L. A. Dickey: Commun. Math. Phys. 87 (1982) 127.MathSciNetADSMATHCrossRefGoogle Scholar
  16. I. M. Gelfand, I. Dorfman: Funct. Anal. Appl. 15 (1981) 173.MathSciNetCrossRefGoogle Scholar
  17. [9]
    M. A. Semenov-Tian-Shansky: Funct. Anal. Appl. 17 (1983) 259; Publ. RIMS, Kyoto Univ. 21 (1985) 1237.CrossRefGoogle Scholar
  18. [10]
    C. J. Isham: in “Relativity, Groups and Topology”, ed. B. S. DeWitt, R. Stora (North-Holland, 1984). J. R. Klauder: Phys. Rev. D2 (1970) 272.Google Scholar
  19. [10a]
    J. R. Klauder: Phys. Rev. D2 (1970) 272.ADSGoogle Scholar
  20. [11]
    A. B. Zamolodchikov, V. A. Fateev: Nucl. Phys. B280[FS18] (1987) 644.MathSciNetADSGoogle Scholar
  21. [12]
    A. A. Belavin, V. E. Zakharov: Phys. Lett. B73 (1978) 53.MathSciNetADSGoogle Scholar
  22. [13]
    R. S. Ward: Phil. Trans. Roy. Soc. Lond. A315 (1985) 451.ADSGoogle Scholar
  23. [14]
    E. Witten: Princeton preprint (1988).Google Scholar
  24. [15]
    N. Yu. Reshetikhin, M. A. Semenov-Tian-Shansky: LOMI preprint (1989).Google Scholar
  25. [16]
    I. Bakas, M. J. Bowick: work in progress (to appear).Google Scholar
  26. [17]
    A. Alekseev, S. Shatashvili: LOMI preprint (1988). M. Bershadsky, H. Ooguri: Princeton preprint (1989). Y. Matsuo: Chicago preprint (1989).Google Scholar
  27. [17a]
    M. Bershadsky, H. Ooguri: Princeton preprint (1989).Google Scholar
  28. [17b]
    Y. Matsuo: Chicago preprint (1989).Google Scholar
  29. [18]
    A. Bilal, J.-L. Gervais: Phys. Lett. B206 (1988) 412.MathSciNetADSGoogle Scholar
  30. [19]
    V. A. Belinskii, V. E. Sakharov: Sov. Phys. JETP 48 (1978) 985;ADSGoogle Scholar
  31. [19a]
    V. A. Belinskii, V. E. Sakharov: Sov. Phys. JETP 50 (1979) 1.ADSGoogle Scholar
  32. [20]
    P. Breitenlohner, D. Maison: Ann. Inst. Henri Poinc. 46 (1987) 215.MathSciNetMATHGoogle Scholar
  33. [21]
    I. Bakas: in preparation.Google Scholar
  34. [22]
    M. Wodzicki: in “K-Theory, Arithmetic and Geometry”, Lect. notes in Math., Vol. 1289, ed. Yu. I. Manin (Springer Verlag, 1987). A. Connes: Commun. Math. Phys. 117 (1988) 673.Google Scholar
  35. [22a]
    A. Connes: Commun. Math. Phys. 117 (1988) 673.MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • I. Bakas
    • 1
  1. 1.Center for Theoretical Physics Department of Physics and AstronomyUniversity of MarylandCollege ParkUSA

Personalised recommendations