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Vertex Operators in Mathematics and Physics pp 275–290Cite as

Monodromy, Solitons and Infinite Dimensional Lie Algebras

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Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 3))

Abstract

The present contribution is a brief survey of works done in Kyoto for the past 7 years. As such, it contains no new results. Rather, it is intended to cover an outline of the development that might be of some interest to the people working in vertex operators and related areas. Some more details can be found in several review articles [1], [2] as well as in the original papers [3], [4].

Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute.

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Comments and Bibliography

  1. M. Jimbo, T. Miwa, M. Sato and Y. Môri, Springer Lecture Notes in Physics 116 (1980) 119.

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  2. M. Sato, T. Miwa and M. Jimbo, ibid. 126 (1980), 429.

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  3. M. Jimbo and T. Miwa, Publ. RIMS. Kyoto Univ. 19 (1983) 943.

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  4. M. Sato, T. Miwa and M. Jimbo, Publ. RIMS, Kyoto Univ.14 (1978) 223,

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  5. M. Sato, T. Miwa and M. Jimbo, Publ. RIMS, Kyoto Univ.15 (1979) 201, 577, 871,

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  6. M. Sato, T. Miwa and M. Jimbo, Publ. RIMS, Kyoto Univ.16 ( 1980) 531.

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  7. M. Jimbo, T. Miwa, Y. Môri and M. Sato, Physica 1 D (1980) 80.

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  8. M. Kashiwara and T. Miwa, Proc. Japan Acad. 57 A (1981) 342.

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  9. E. Date, M. Kashiwara and T. Miwa, ibid. 57A (1981) 387.

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  10. E. Date, M. Jimbo, M. Kashiwara and T. Miwa, J. Phys. Soc. Jap. 50 (1981) 3806, 3813;

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  11. E. Date, M. Jimbo, M. Kashiwara and T. Miwa Physica 4 D (1982) 343

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  12. E. Date, M. Jimbo, M. Kashiwara and T. Miwa Publ. RIMS, Kyoto Univ. 18 (1982) 1077, 1111.

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  13. To be more precise, one lets also T tend to the “critical temperature” Tc. In the sequel, this limit is taken from above Tc. For simplicity of presentation, we assume also E1=E2.

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  14. T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Phys. Rev. B 13 (1976) 316.

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  15. A description of Painléve’s work can be found in the book: E.L. Ince, Ordinary differential equations, Dover, New York, 1956.

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  16. L. Schlesinger, J. für Math. 141 (1972) 96.

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  17. M. Jimbo, T. Miwa and K. Ueno, Physica 2D (1981) 306.

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  18. L. Onsager, Phys. Rev. 65 (1944) 117.

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  19. \( {\psi_{\pm }}(x) \) are obtained as the limit of \( {p_{{ij}}} = {V^i}{p_j}{V^{{ - i}}} \), \( {q_{{ij}}} = {V^i}{q_j}{V^{{ - i}}} \). They satisfy the 2-dimensional Dirac equation \( ( \pm \frac{\partial }{{\partial {x^0}}} + \frac{\partial }{{\partial {x^1}}}){\psi_\pm }(x) = m{\psi_\pm }(x) \). where m > 0 is the mass parameter.

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  20. Note that w±(x) solve the Dirac equation, which is elliptic in the Euclidean region. We are0 thus dealing with an analogue of Riemann’s monodromy problem for solutions of the Euclidean Dirac equation instead of the Cauchy-Riemann equation. In the limit m → 0, the former reduces to the latter.

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  21. T. Miwa, Publ. RIMS. Kyoto Univ. 17 (1981) 665.

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  22. It was M. and Y. Sato who first established this picture. A short account of their theory is found in: M. Sato, RIMS Kokyuroku 439 (1981) 30.

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  23. K. Okamoto, Transformation groups for Painlevé equations, in preparation (Univ. of Tokyo).

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  24. For the self-dual Yang-Mills equation, the transformation group of solutions is clarified by: K. Ueno and Y. Nakamura, Phys. Lett. 109 B (1982) 273.

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  25. For the self-dual Yang-Mills equation, the transformation group of solutions is clarified by: K. Ueno and Y. Nakamura, Phys. Lett. 117 B (1982) 208.

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  26. K. Takasaki, RIMS preprint 459 (1983).

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Author information

Authors and Affiliations

  1. R.I.M.S., Kyoto University, Kyoto 606, Japan

    Michio Jimbo & Tetsuji Miwa

Authors
  1. Michio Jimbo
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  2. Tetsuji Miwa
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    J. Lepowsky

  2. Department of Physics, University of California, Berkeley, CA, 94720, USA

    S. Mandelstam

  3. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    I. M. Singer

  4. Mathematical Sciences Research Institute, 2223 Fulton Street, Room 603, Berkeley, CA, 94720, USA

    I. M. Singer

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© 1985 Springer-Verlag New York Inc.

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Jimbo, M., Miwa, T. (1985). Monodromy, Solitons and Infinite Dimensional Lie Algebras. In: Lepowsky, J., Mandelstam, S., Singer, I.M. (eds) Vertex Operators in Mathematics and Physics. Mathematical Sciences Research Institute Publications, vol 3. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9550-8_13

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  • DOI: https://doi.org/10.1007/978-1-4613-9550-8_13

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  • Print ISBN: 978-1-4613-9552-2

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