Letters in Mathematical Physics

, Volume 28, Issue 3, pp 165–176 | Cite as

Quasi-classical limit of Toda hierarchy andW-infinity symmetries

  • Kanehisa Takasaki
  • Takashi Takebe


Previous results on quasi-classical limit of the KP hierarchy and itsW-infinity symmetries are extended to the Toda hierarchy. The Planck constantħ now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined.W1 + ∞ symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted tow1 + ∞ symmetries of the dispersionless hierarchy through their action on the tau function.

Mathematics Subject Classifications (1991)

17B65 35Q58 58F07 


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  1. 1.
    Lebedev, D. and Manin, Yu., Conservation laws and Lax representation on Benny's long wave equations,Phys. Lett. 74A, 154–156; (1979); Kodama, Y., A method for solving the dispersionless KP equation and its exact solutions,Phys. Lett. 129A, 223-226 (1988); Solutions of the dispersionless Toda equation,Phys. Lett. 147A, 477-482; (1990); Kodama, Y. and Gibbons, J., A method for solving the dispersionless KP hierarchy and its exact solutions, II,Phys. Lett,135A, 167-170 (1989).Google Scholar
  2. 2.
    Bakas, I., The structure of theW algebra,Comm. Math. Phys. 134, 487–508 (1990); Saveliev, M. V. and Vershik, A. M., Continual analogues of contragredient Lie algebras,Comm. Math. Phys. 126, 367-378 (1989).Google Scholar
  3. 3.
    Bakas, I., Area preserving diffeomorphisms and higher spin fields in two dimensions, in M. Duff, C. Pope and E. Sezgin (eds),Supermembranes and Physics in 2 + 1 Dimensions, Trieste, 1989, World Scientific, Singapore, 1990; Park, Q-Han, Extended conformal symmetries in real heavens,Phys. Lett. 236B, 429-432 (1990).Google Scholar
  4. 4.
    Takasaki, K. and Takebe, T., SDiff(2) Toda equation - hierarchy, tau function and symmetries,Lett. Math. Phys. 23, 205–214 (1991).Google Scholar
  5. 5.
    Avan, J.,w -currents in 3-dimensional Toda theory, BROWN-HET-855, March 1992; Avan, J. and Jevicki, A., Interacting theory of collective and topological fields in 2 dimensions, BROWN-HET-869, August 1992.Google Scholar
  6. 6.
    Takasaki, K. and Takebe, T., Quasi-classical limit of KP hierarchy,W-symmetries and free fermions, Kyoto preprint KUCP-0050/92, July 1992.Google Scholar
  7. 7.
    Ueno, K. and Takasaki, K., Today lattice hierarchy, in K. Okamoto (ed.),Group Representations and Systems of Differential Equations, Advanced Studies in Pure Math. 4, North-Holland, Amsterdam, 1984; Takasaki, K., Initial value problem for the Toda lattice hierarchy,ibid. Google Scholar
  8. 8.
    Date, E., Kashiwara, M., Jimbo, M., and Miwa, T., Transformation groups for soliton equations, in M. Jimbo and T. Miwa (eds),Nonlinear Integrable Systems - Classical Theory and Quantum Theory, World Scientific, Singapore, 1983.Google Scholar
  9. 9.
    Jimbo, M. and Miwa, T., Solitons and infinite dimensional Lie algebras,Publ. RIMS, Kyoto Univ. 19, 943–1001 (1983); Takebe, T., Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy: I,Lett. Math. Phys. 21, 77-84 (1991); II,Publ. RIMS, Kyoto Univ. 27 491-503, (1991).Google Scholar
  10. 10.
    Dubrovin, B. A., Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models,Comm. Math. Phys. 145, 195–207 (1992); Krichever, I. M., Theτ-function of the universal Whitham hierarchy, matrix models and topological field theories, LPTENS-92/18 May 1992.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Kanehisa Takasaki
    • 1
  • Takashi Takebe
    • 2
  1. 1.Department of Fundamental SciencesFaculty of Integrated Human Studies, Kyoto UniversityKyotoJapan
  2. 2.Department of Mathematical SciencesUniversity of TokyoTokyoJapan

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