Group theory approach to the τ-function and its quantization

Theoretical and Mathematical Physics

Volumn 114, Issue 2, 1998, Pages 127-183

  Mironov, A D ^a,b  

^a LEBEDEV PHYSICAL INSTITUTE   (Russian Federation)

^b INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032373185     PISSN: 00405779     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02557115     Document Type: Article

References (113)

1. A. Marshakov, "Exact nonperturbative solutions of quantum theory and integrable systems," Doctoral thesis, P. N. Lebedev Physical Institute, Moscow (1997).
2. A. Mironov, "τ-Functions and matrix models," Doctoral thesis, P. N. Lebedev Physical Institute, Moscow (1997).
3. A. B. Zamolodchikov, JETP Lett., 46, 160 (1987).
4. B. A. Kuperschmidt and P. Mathieu, Phys. Lett. B, 227, 245 (1989).
5. T. Eguchi and S. K. Yang, Phys. Lett. B, 244, 373 (1989).
6. A. B. Zamolodchikov, Adv. Studies in Pure Math., 19, 641 (1989).
7. R. Sasaki and I. Yamanaka, Adv. Studies in Pure Math., 16, 271 (1988).
8. A. Morozov, Usp. Fiz. Nauk, 162, No. 8, 84 (1992).
9. V. Knizhnik, Usp. Fiz. Nauk, 159, No. 3, 401 (1989).
10. S. Saito, Phys. Rev. Lett., 59, 1798 (1987).
11. S. Saito, Phys. Rev. D, 36, 1819 (1987).
12. A. Morozov, Phys. Lett. B, 196, 325 (1987).
13. A. R. Its, A. G. Izergin, V. E. Korepin, and N. A. Slavnov, Int. J. Mod. Phys. B, 4, 1003 (1990).
14. M. Jimbo, T. Miwa, and M. Sato, "Holonomic Quantum Fields, I-V," Publ. RIMS Kyoto Univ., 14, 223 (1978); 15, 201, 577, 871, 1531 (1979).
15. M. Jimbo, T. Miwa, and M. Sato, "Holonomic Quantum Fields, I-V," Publ. RIMS Kyoto Univ., 14, 223 (1978); 15, 201, 577, 871, 1531 (1979).
16. D. Bernard and A. Le Clair, "Differential equations for sine-Gordon correlation functions at the free fermion point," Preprint CLNS 94/1276, SPhT-94-021; hep-th/9402144.
17. A. G. Izergin, D. A. Coker, and V. E. Korepin, J. Phys.: Math. Gen. A, 25, 4315 (1992).
18. A. Morozov, Usp. Fiz. Nauk, 164, No. 1, 3 (1994).
19. A. Marshakov, Int. J. Mod. Phys. A, 8, 3831 (1993).
20. A. Mironov, Int. J. Mod. Phys. A, 9, 4355 (1994).
21. N. Seiberg and E. Witten, Nucl. Phys. B, 426, 19 (1994).
22. N. Seiberg and E. Witten, Nucl. Phys. B, 431, 484 (1994).
23. A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 355, 466 (1995).
24. A. Mironov, "WDVV equations in Seiberg-Witten theory and associative algebras," hep-th/9704205.
25. A. Gorsky, S. Gukov, and A. Mironov, "Multiscale N = 2 SUSY field theories, integrable systems and their stringy/brane origin," hep-th/9707120.
26. L. A. Takhtajan (Takhtadzhyan) and L. D. Faddeev, Usp. Mat. Nauk, 34, 13 (1979).
27. N. Yu. Reshetikhin, L. A. Takhtajan (Takhtadzhyan), and L. D. Faddeev, Algebra Anal., 1, 178 (1989).
28. V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions [in Russian], Nauka, Moscow (1992); English translation of previous edition: Cambridge Univ., Cambridge (1989).
29. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, London-New York (1989).
30. L. D. Faddeev, "Integrable models in 1 + 1-dimensional quantum field theory," in: Les Houches Lectures 1982, Elsevier, Amsterdam (1984), pp. 563-608.
31. E. Sklyanin, "Quantum inverse scattering method. Selected topics," in: Quantum Group and Quantum Integrable Systems (Nankai Lectures in Mathematical Physics) (Mo-Lin Ge, ed.), World Scientific, Singapore (1992), p. 63-97.
32. S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Int. J. Mod. Phys. A, 10, 2015 (1995).
33. K. Wilson, Phys. Rev. D, 10, 2445 (1974).
34. G. Weight, "On operator constructions in bosonic Liouville quantum gravity," in: Proceedings of the 28th International Symposium Ahrenshoop, Wendisch-Rietz, Germany, September 1993, DESY, Hamburg (1994).
35. A. Mironov, A. Morozov, and L. Vinet, Theor. Math. Phys., 100, 890 (1994).
36. A. Gerasimov, S. Khoroshkin, D. Lebedev, A. Mironov, and A. Morozov, Int. J. Mod. Phys. A, 10, 2589 (1995).
37. A. Mironov, CRM Proc. Lect. Notes, 9, 219 (1996).
38. S. Kharchev, A. Mironov, and A. Morozov, "τ-Functions: evolution and determinant representations," in: Proceedings of the 28th International Symposium Ahrenshoop, Wendisch-Rietz, Germany, September 1993, DESY, Hamburg (1994), p. 80.
39. S. Kharchev, A. Mironov, and A. Morozov, Theor. Math. Phys., 104, 866 (1995).
40. A. Gerasimov, S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and M. Olshanetsky, "Liouville-type models in the group theory framework. I. Finite-dimensional algebras," Preprint FIAN/TD-18/95, ITEP M4/TH-7/95; hep-th/9601161.
41. A. Mironov, "Group theory structures underlying integrable systems," Preprint FIAN/TD-12/96, ITEP/TH-24/96; hep-th/9607123, to appear in Proc. 2nd International A. D. Sakharov Conference.
42. M. Olshanetsky and A. Perelomov, Invent. Math., 54, 261 (1979).
43. B. Kostant, Adv. Math., 34, 195 (1979).
44. M. A. Olshanetskii and A. M. Perelomov, Theor. Math. Phys., 45, 843 (1980).
45. N. Hurt, Geometric Quantization in Action, Riedel, Dordrecht-Boston-London (1983).
46. H. P. B. Jacquet, Bull. Soc. Math. France, 95, 243 (1967).
47. G. Schiffmann, Bull. Soc. Math. France, 99, 3 (1971).
48. B. Kostant, Invent. Math., 48, 101 (1978).
49. M. Hashizume, Hiroshima Math. J., 12, 259 (1982).
50. S. G. Gindikin and F. I. Karpelevich, Dokl. Akad. Nauk SSSR, 145, 252 (1962).
51. S. G. Gindikin and F. I. Karpelevich, Izv. Akad. Nauk SSSR, Ser. Mat., 30, 1147 (1966).
52. M. Semenov-Tian-Shansky, "Quantization of the open Toda chains," in: Modern Problems in Mathematics, Vol. 16, VINITI, Moscow (1987), p. 194.
53. A. Gerasimov, A. Marshakov, A. Morozov, M. Olshanetsky, and S. Shatashvili, Int. J. Mod. Phys. A, 5, 2495 (1990).
54. A. Zamolodchikov and Al. Zamolodchikov, "Structure constants and conformal bootstrap in Liouville field theory," hep-t,h/9506136.
55. H. Dorn and H.-.J. Otto, Phys. Lett. B, 291, 39 (1992).
56. H. Dorn and H.-J. Otto, Nucl. Phys. B, 429, 375 (1994).
57. M. Kashiwara and T. Miwa, Proc. Japan Acad. A, 57, 342 (1981).
58. E. Date, M. Kashiwara, and T. Miwa, Proc. Japan Acad. A, 57, 387 (1981).
59. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, J. Phys. Soc. Japan, 50, 3806 (1981).
60. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Physica D, 4, 343 (1982).
61. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Publ. RIMS Kyoto Univ., 18, 1111 (1983).
62. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, J. Phys. Soc. Japan, 50, 3813 (1981).
63. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, "Transformation groups for soliton equations," in: Proc. RIMS Symp. Nonlinear Integrable Systems-Classical Theory and Quantum Theory (M. Jimbo and T. Miwa, eds.), World Scientific, Singapore (1983), p. 39.
64. M. Jimbo and T. Miwa, Publ. RIMS Kyoto Univ., 19, 943 (1983).
65. K. Ueno and K. Takasaki, Adv. Studies in Pure Math., 4, 1 (1984).
66. S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Nucl. Phys. B, 397, 339 (1993).
67. M. Sato, RIMS Kokyuroku, 439, 30 (1981).
68. I. M. Krichever, Funct. Anal. Appl., 11, 12 (1977).
69. I. M. Krichever, Russ. Math. Surv., 32, 185 (1977).
70. G. Segal and G. Wilson, Publ. IHES, 61, 5 (1985).
71. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevsky, Theory of Solitons. The Inverse Scattering Method, Plenum, New York (1984).
72. V. Kač and A. S. Schwarz, Phys. Lett. B, 257, 329 (1991).
73. S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and A. Zabrodin, Nucl. Phys. B, 380, 181 (1992).
74. S. Kharchev, A. Marshakov, A. Mironov, A. Orlov, and A. Zabrodin, Nucl. Phys. B, 366, 569 (1991).
75. P. J. Hansen and D. J. Kaup, J. Phys. Soc. Japan, 54, 4126 (1985).
76. M. A. Olshanetsky and A. M. Perelomov, Phys. Rep., 71, 313 (1981).
77. M. Olshanetsky and A. Perelomov, Phys. Rep., 94, 313 (1983).
78. O. I. Bogoyavlensky, Commun. Math. Phys., 51, 201 (1976).
79. R. Hirota, J. Phys. Soc. Japan, 57, 4285 (1987).
80. R. Hirota, Y. Ohta, and J. Satsuma, Prog. Theor. Phys. Suppl., 94, 59 (1988).
81. A. Leznov and M. Saveliev, Physica D, 3, 62 (1981).
82. A. N. Leznov and M. V. Saveliev, Group Theory Methods for Integrating Nonlinear Dynamic Systems [in Russian], Nauka, Moscow (1985); A. N. Leznov and M. V. Savel'ev (Saveliev), Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems, Birkhäuser, Basel (1992).
83. A. N. Leznov and M. V. Saveliev, Group Theory Methods for Integrating Nonlinear Dynamic Systems [in Russian], Nauka, Moscow (1985); A. N. Leznov and M. V. Savel'ev (Saveliev), Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems, Birkhäuser, Basel (1992).
84. O. Babelon, Commun. Math. Phys., 139, 619 (1991).
85. O. Babelon, Phys. Lett. B, 215, 523 (1988).
86. J.-L. Gervais, Nucl. Phys. B, 391, 287 (1993).
87. J.-L. Gervais and J. Schnittger, Nucl. Phys. B, 431, 273 (1994).
88. J. Schnittger, Theor. Math. Phys., 104, 892 (1995).
89. I. Frenkel and N. Reshetikhin, Commun. Math. Phys., 146, 1 (1992).
90. M. Jimbo and T. Miwa, Preprint RIMS-981, Kyoto Univ., Kyoto (1994).
91. C. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ., Cambridge (1994).
92. C. Fronsdal and A. Galindo, Preprint UCLA/93/TEP/2 (1993).
93. A. Morozov and L. Vinet, "Free-field representation of a group element for simple quantum groups," Preprint ITEP-M3/94, CRM-2202; hep-th/9409093.
94. D. P. Zhelobenko, Compact Lie Groups and their Representations [in Russian], Nauka, Moscow (1970).
95. L. L. Waksman and Ya. S. Soibelman, Funkts. Anal. Prilozhen., 23, 1 (1988).
96. S. Levendorskii and Ya. Soibelman, Commun. Math. Phys., 139, 141 (1991).
97. Y. Soibelman, Int. Math. Research Notices, No. 6, 151 (1993).
98. K. Kajiwara and J. Satsuma, J. Phys. Soc. Japan, 60, 3986 (1991).
99. K. Kajiwara, Y. Ohta, and J. Satsuma, "q-Discrete Toda molecule equation," solv-int/9304001.
100. M. Rosso, Commun. Math. Phys., 117, 581 (1988).
101. A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birkhäuser, Basel-Boston-Berlin (1990).
102. L. D. Faddeev and L. A. Takhtadzhyan, The Hamiltonian Methods in the Theory of Solitons, Springer, Berlin-Heidelberg New York (1987).
103. M. Olshanetsky and V. Rogov, J. Phys.: Math. Gen., 27, 4669 (1994).
104. I. M. Gel'fand, M. I. Graev, and L. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Saunders, Philadelphia (1969).
105. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series, and Products [in Russian], Nauka, Moscow (1963); English transl.: Acad. Press, New York (1969).
106. Harish-Chandra, Amer. J. Math., 80, 241, 553 (1958).
107. D. Mumford, Tata Lectures on Theta, Vols. I, II, Birkhäuser, Boston-Basel-Stuttgart (1983, 1984).
108. H. Awata, M. Noumi, and S. Odake, "Heisenberg realization for Uq(SL(n)) on the flag manifold," Preprint YITP/K-1016; hep-th/9306010 (1993).
109. P. Griffits and F. Adams, Topics in Algebraic Geometry and Analytic Geometry, Princeton Univ., Princeton (1974).
110. V. G. Drinfeld, Zap. Nauchn. Sem. LOMI, 155, 19 (1986).
111. M. Jimbo, Lett. Math. Phys., 10, 63 (1985).
112. N. Wallach, Harmonic Analysis in Homogeneous Spaces, Dekker, New York (1973).
113. S. Helgason, Groups and Geometrical Analysis, Acad. Press, New York (1987).