Tau-functions generating the conservation laws for generalized integrable hierarchies of KdV and affine Toda type

Nuclear Physics B

Volumn 547, Issue 3, 1999, Pages 623-663

  Miramontes, J Luis ^a  

^a UNIVERSITY OF SANTIAGO DE COMPOSTELA   (Spain)

Author keywords

Affine kac moody algebras; Generalized KdV equations; Integrable hierarchies; Non abelian affine; Solitons; Tau functions; Toda equations

Indexed keywords

EID: 0033577652     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0550-3213(99)00066-8     Document Type: Article

References (85)

1. H. Aratyn, E. Nissimov and S. Pacheva, Phys. Lett. A 201 (1995) 293; A 228 (1997) 164.
2. H. Aratyn, E. Nissimov and S. Pacheva, Phys. Lett. A 201 (1995) 293; A 228 (1997) 164.
3. M.F. de Groot, T.J. Hollowood and J.L. Miramontes, Commun. Math. Phys. 145 (1992) 57.
4. N.J. Burroughs, M.F. de Groot, T.J. Hollowood and J.L. Miramontes, Commun. Math. Phys. 153 (1993) 187; Phys. Lett. B 277 (1992) 89.
5. N.J. Burroughs, M.F. de Groot, T.J. Hollowood and J.L. Miramontes, Commun. Math. Phys. 153 (1993)187; Phys. Lett. B 277 (1992) 89.
6. T.J. Hollowood and J.L. Miramontes, Commun. Math. Phys. 157 (1993) 99.
7. L.A. Dickey, Soliton equations and hamiltonian systems, Advanced Series in Math. Phys., Vol. 12 (World Scientific, Singapore, 1991).
8. R. Hirota, Direct methods in soliton theory, in Soliton, ed. R.K. Bullough and P.S. Caudrey (Springer, Berlin, 1980) p. 157.
9. M. Jimbo and T. Miwa, Publ. RIMS, Kyoto Univ. 19 (1983) 943; E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Publ. RIMS, Kyoto Univ. 18 (1982) 1077.
10. M. Jimbo and T. Miwa, Publ. RIMS, Kyoto Univ. 19 (1983) 943; E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Publ. RIMS, Kyoto Univ. 18 (1982) 1077.
11. G. Segal and G. Wilson, Loop groups and equations of KdV type, Publ. Math. IHES 63 (1985) 1.
12. G. Wilson, Phil. Trans. R. Soc. Lond. A 315 (1985) 383; Habillage et fonctions τ C.R. Acad. Sc.paris 299 (I) (1984) 587; The τ-functions of the gAKNS equations, in Verdier Memorial Conferenceon Integrable Systems, Ed. O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach (Birkhäuser, Basel, 1993)p. 131.
13. G. Wilson, Phil. Trans. R. Soc. Lond. A 315 (1985) 383; Habillage et fonctions τ C.R. Acad. Sc. Paris 299 (I) (1984) 587; The τ-functions of the gAKNS equations, in Verdier Memorial Conferenceon Integrable Systems, Ed. O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach (Birkhäuser, Basel, 1993)p. 131.
14. G. Wilson, Phil. Trans. R. Soc. Lond. A 315 (1985) 383; Habillage et fonctions τ C.R. Acad. Sc.paris 299 (I) (1984) 587; The τ-functions of the gAKNS equations, in Verdier Memorial Conference on Integrable Systems, Ed. O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach (Birkhäuser, Basel, 1993) p. 131.
15. V.G. Kac and M. Wakimoto, Exceptional hierarchies of soliton equations, in Proceedings of Symposia in Pure Mathematics, Vol. 49 (1989) 191.
16. L.A. Ferreira, J.L. Miramontes and J. Sánchez Guillén, J. Math. Phys. 38 (1997) 882.
17. T.J. Hollowood, Nucl. Phys. B 384 (1992) 523; N. McKay and W.A. McGhee, Int. J. Mod. Phys. A 8 (1993) 2791; A 8 (1993) 3830 (E); Z. Zhu and D.G. Caldi, Nucl. Phys. B 436 (1995) 659.
18. T.J. Hollowood, Nucl. Phys. B 384 (1992) 523; N. McKay and W.A. McGhee, Int. J. Mod. Phys. A 8 (1993) 2791; A 8 (1993) 3830 (E); Z. Zhu and D.G. Caldi, Nucl. Phys. B 436 (1995) 659.
19. T.J. Hollowood, Nucl. Phys. B 384 (1992) 523; N. McKay and W.A. McGhee, Int. J. Mod. Phys. A 8 (1993) 2791; A 8 (1993) 3830 (E); Z. Zhu and D.G. Caldi, Nucl. Phys. B 436 (1995) 659.
20. T.J. Hollowood, Nucl. Phys. B 384 (1992) 523; N. McKay and W.A. McGhee, Int. J. Mod. Phys. A 8 (1993) 2791; A 8 (1993) 3830 (E); Z. Zhu and D.G. Caldi, Nucl. Phys. B 436 (1995) 659.
21. H. Aratyn, C.P. Constantinidis, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, Nucl. Phys. B 406 (1993) 727.
22. D. Olive, M.V. Saveliev and J.W.R. Underwood, Phys. Lett. B 311 (1993) 117; D. Olive, N. Turok and J.W.R. Underwood, Nucl. Phys. B 409 (1993) 509.
23. D. Olive, M.V. Saveliev and J.W.R. Underwood, Phys. Lett. B 311 (1993) 117; D. Olive, N. Turok and J.W.R. Underwood, Nucl. Phys. B 409 (1993) 509.
24. D. Olive, N. Turok and J.W.R. Underwood, Nucl. Phys. B 401 (1993) 663.
25. n-1affine Toda field theories II, hep-th/9803248.
26. n-1 affine Toda field theories II, hep-th/9803248.
27. n-1 affine Toda field theories II, hep-th/9803248.
28. n-1 affine Toda field theories II, hep-th/9803248.
29. n-1 affine Toda field theories II, hep-th/9803248.
30. L.A. Ferreira, J.L. Miramontes and J. Sánchez Guillén, Nucl. Phys. B 449 (1995) 631.
31. S. Saito, Phys. Rev. D 36 (1987) 1819; Phys. Rev. Lett. 59 (1987) 1798; D. Bernard and A. Le Clair, Nucl. Phys. B 426 (1994) 534; B 498 (1997) 619 (E); O. Babelon, D. Bernard and F.A. Smirnov, Commun. Math. Phys. 182 (1996) 319; T. Kojima, V. Korepin and N. Slavnov, Commun. Math. Phys. 189 (1997) 709; A. Mironov, τ-function Within Group Theory Approach and Its Quantization, q-alg/9711006.
32. S. Saito, Phys. Rev. D 36 (1987) 1819; Phys. Rev. Lett. 59 (1987) 1798; D. Bernard and A. Le Clair, Nucl. Phys. B 426 (1994) 534; B 498 (1997) 619 (E); O. Babelon, D. Bernard and F.A. Smirnov, Commun. Math. Phys. 182 (1996) 319; T. Kojima, V. Korepin and N. Slavnov, Commun. Math. Phys. 189 (1997) 709; A. Mironov, τ-function Within Group Theory Approach and Its Quantization, q-alg/9711006.
33. S. Saito, Phys. Rev. D 36 (1987) 1819; Phys. Rev. Lett. 59 (1987) 1798; D. Bernard and A. Le Clair, Nucl. Phys. B 426 (1994) 534; B 498 (1997) 619 (E); O. Babelon, D. Bernard and F.A. Smirnov, Commun. Math. Phys. 182 (1996) 319; T. Kojima, V. Korepin and N. Slavnov, Commun. Math. Phys. 189 (1997) 709; A. Mironov, τ-function Within Group Theory Approach and Its Quantization, q-alg/9711006.
34. S. Saito, Phys. Rev. D 36 (1987) 1819; Phys. Rev. Lett. 59 (1987) 1798; D. Bernard and A. Le Clair, Nucl. Phys. B 426 (1994) 534; B 498 (1997) 619 (E); O. Babelon, D. Bernard and F.A. Smirnov, Commun. Math. Phys. 182 (1996) 319; T. Kojima, V. Korepin and N. Slavnov, Commun. Math. Phys. 189 (1997) 709; A. Mironov, τ-function Within Group Theory Approach and Its Quantization, q-alg/9711006.
35. S. Saito, Phys. Rev. D 36 (1987) 1819; Phys. Rev. Lett. 59 (1987) 1798; D. Bernard and A. Le Clair, Nucl. Phys. B 426 (1994) 534; B 498 (1997) 619 (E); O. Babelon, D. Bernard and F.A. Smirnov, Commun. Math. Phys. 182 (1996) 319; T. Kojima, V. Korepin and N. Slavnov, Commun. Math. Phys. 189 (1997) 709; A. Mironov, τ-function Within Group Theory Approach and Its Quantization, q-alg/9711006.
36. S. Saito, Phys. Rev. D 36 (1987) 1819; Phys. Rev. Lett. 59 (1987) 1798; D. Bernard and A. Le Clair, Nucl. Phys. B 426 (1994) 534; B 498 (1997) 619 (E); O. Babelon, D. Bernard and F.A. Smirnov, Commun. Math. Phys. 182 (1996) 319; T. Kojima, V. Korepin and N. Slavnov, Commun. Math. Phys. 189 (1997) 709; A. Mironov, τ-function Within Group Theory Approach and Its Quantization, q-alg/9711006.
37. S. Saito, Phys. Rev. D 36 (1987) 1819; Phys. Rev. Lett. 59 (1987) 1798; D. Bernard and A. Le Clair, Nucl. Phys. B 426 (1994) 534; B 498 (1997) 619 (E); O. Babelon, D. Bernard and F.A. Smirnov, Commun. Math. Phys. 182 (1996) 319; T. Kojima, V. Korepin and N. Slavnov, Commun. Math. Phys. 189 (1997) 709; A. Mironov, τ-function Within Group Theory Approach and Its Quantization, q-alg/9711006.
38. M.R. Douglas, Phys. Lett. B 238 (1990) 176; E. Witten, Diff. Geom. 1 (1991) 243; M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A 6 (1991) 1385; Commun. Math. Phys.143 (1992) 371; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 438 ( 1991) 435; V. Kac and A. Schwarz, Phys. Lett. B 257 (1991) 329; M. Kontsevich, Commun. Math. Phys. 147 (1992) 1.
39. M.R. Douglas, Phys. Lett. B 238 (1990) 176; E. Witten, Diff. Geom. 1 (1991) 243; M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A 6 (1991) 1385; Commun. Math. Phys.143 (1992) 371; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 438 ( 1991) 435; V. Kac and A. Schwarz, Phys. Lett. B 257 (1991) 329; M. Kontsevich, Commun. Math. Phys. 147 (1992) 1.
40. M.R. Douglas, Phys. Lett. B 238 (1990) 176; E. Witten, Diff. Geom. 1 (1991) 243; M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A 6 (1991) 1385; Commun. Math. Phys.143 (1992) 371; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 438 ( 1991) 435; V. Kac and A. Schwarz, Phys. Lett. B 257 (1991) 329; M. Kontsevich, Commun. Math. Phys. 147 (1992) 1.
41. M.R. Douglas, Phys. Lett. B 238 (1990) 176; E. Witten, Diff. Geom. 1 (1991) 243; M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A 6 (1991) 1385; Commun. Math. Phys. 143 (1992) 371; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 438 ( 1991) 435; V. Kac and A. Schwarz, Phys. Lett. B 257 (1991) 329; M. Kontsevich, Commun. Math. Phys. 147 (1992) 1.
42. M.R. Douglas, Phys. Lett. B 238 (1990) 176; E. Witten, Diff. Geom. 1 (1991) 243; M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A 6 (1991) 1385; Commun. Math. Phys.143 (1992) 371; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 438 ( 1991) 435; V. Kac and A. Schwarz, Phys. Lett. B 257 (1991) 329; M. Kontsevich, Commun. Math. Phys. 147 (1992) 1.
43. M.R. Douglas, Phys. Lett. B 238 (1990) 176; E. Witten, Diff. Geom. 1 (1991) 243; M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A 6 (1991) 1385; Commun. Math. Phys.143 (1992) 371; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 438 ( 1991) 435; V. Kac and A. Schwarz, Phys. Lett. B 257 (1991) 329; M. Kontsevich, Commun. Math. Phys. 147 (1992) 1.
44. M.R. Douglas, Phys. Lett. B 238 (1990) 176; E. Witten, Diff. Geom. 1 (1991) 243; M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A 6 (1991) 1385; Commun. Math. Phys.143 (1992) 371; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 438 ( 1991) 435; V. Kac and A. Schwarz, Phys. Lett. B 257 (1991) 329; M. Kontsevich, Commun. Math. Phys. 147 (1992) 1.
45. T. Nakatsu and K. Takasaki, Mod. Phys. Lett. A 11 (1996) 157; Integrable system and N = 2 supersymmetric Yang-Mills theory, hep-th/9603129; A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Phys. Lett. B 355 (1995) 466.
46. T. Nakatsu and K. Takasaki, Mod. Phys. Lett. A 11 (1996) 157; Integrable system and N = 2 supersymmetric Yang-Mills theory, hep-th/9603129; A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Phys. Lett. B 355 (1995) 466.
47. M. Mariño and G. Moore, The Donaldson-Witten Function for Gauge Groups of Rank Larger Than One, hep-th/9802185; K. Takasaki, Integrable Hierarchies and Contact Terms in U-plane Integrals of Topologically Twisted Supersymmetric Gauge Theories, hep-th/9803217.
48. M. Mariño and G. Moore, The Donaldson-Witten Function for Gauge Groups of Rank Larger Than One,hep-th/9802185; K. Takasaki, Integrable Hierarchies and Contact Terms in U-plane Integrals of Topologically Twisted Supersymmetric Gauge Theories, hep-th/9803217.
49. B. Feigin and E. Frenkel, Integrals of motion and quantum groups, in Lect. Notes in Math. 1620 (Springer, Berlin, 1995) p. 349; Invent. Math. 120 (1995) 379; B. Enriquez and E. Frenkel, Commun. Math. Phys. 185 (1997) 211; E. Frenkel, Five Lectures on Soliton Equations, q-alg/9712005.
50. B. Feigin and E. Frenkel, Integrals of motion and quantum groups, in Lect. Notes in Math. 1620 (Springer, Berlin, 1995) p. 349; Invent. Math. 120 (1995) 379; B. Enriquez and E. Frenkel, Commun. Math. Phys. 185 (1997) 211; E. Frenkel, Five Lectures on Soliton Equations, q-alg/9712005.
51. B. Feigin and E. Frenkel, Integrals of motion and quantum groups, in Lect. Notes in Math. 1620 (Springer, Berlin, 1995) p. 349; Invent. Math. 120 (1995) 379; B. Enriquez and E. Frenkel, Commun. Math. Phys. 185 (1997) 211; E. Frenkel, Five Lectures on Soliton Equations, q-alg/9712005.
52. B. Feigin and E. Frenkel, Integrals of motion and quantum groups, in Lect. Notes in Math. 1620 (Springer, Berlin, 1995) p. 349; Invent. Math. 120 (1995) 379; B. Enriquez and E. Frenkel, Commun. Math. Phys. 185 (1997) 211; E. Frenkel, Five Lectures on Soliton Equations, q-alg/9712005.
53. A. Chodos, Phys. Rev. D 21 (1980) 2818.
54. V.G. Drinfel'd and V.V. Sokolov, J. Sov. Math. 30 (1985) 1975.
55. L. Fehér, J. Harnad and I. Marshall, Commun. Math. Phys. 154 (1993) 181; L. Fehér and I. Marshall, Commun. Math. Phys. 183 (1997) 423.
56. L. Fehér, J. Harnad and I. Marshall, Commun. Math. Phys. 154 (1993) 181; L. Fehér and I. Marshall, Commun. Math. Phys. 183 (1997) 423.
57. H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, J. Math. Phys. 38 (1997) 1559.
58. F. Delduc, L. Fehér and L. Gallot, Non-standard Drinfeld-sokolov Reduction, solv-int/9708002.
59. N.J. Burroughs, Non-linearity 6 (1993) 583; Nucl. Phys. B 379 (1992) 340; L. Fehér, KdV Type Systems and W-algebras in the Drinfeld-sokolov Approach, hep-th/9510001.
60. N.J. Burroughs, Non-linearity 6 (1993) 583; Nucl. Phys. B 379 (1992) 340; L. Fehér, KdV Type Systems and W-algebras in the Drinfeld-sokolov Approach, hep-th/9510001.
61. N.J. Burroughs, Non-linearity 6 (1993) 583; Nucl. Phys. B 379 (1992) 340; L. Fehér, KdV Type Systems and W-algebras in the Drinfeld-sokolov Approach, hep-th/9510001.
62. G. Wilson, Ergod. Th. Dynam. Sys. 1 (1981) 361.
63. C.R. Fernández-Pousa, M.V. Gallas, J.L. Miramontes and J. Sánchez Guillén, Ann. Phys. (N.y.) 243 (1995) 372.
64. M.A. Olshanetsky and A.M. Perelomov, Invent. Math. 54 (1979) 261; A.V. Mikhailov, M.A. Olshanetsky and A.M. Perelomov, Commun. Math. Phys. 79 (1981) 473.
65. M.A. Olshanetsky and A.M. Perelomov, Invent. Math. 54 (1979) 261; A.V. Mikhailov, M.A. Olshanetsky and A.M. Perelomov, Commun. Math. Phys. 79 (1981) 473.
66. A.N. Leznov and M.V. Saveliev, Group Theoretical Methods for Integration of Non-linear Dynamical Systems, Progress in Physics Series, Vol. 15 (Birkhaüser, Basel, 1992).
67. L.A. Ferreira, J-L. Gervais, J. Sánchez Guillén and M.V. Saveliev, Nucl. Phys. B 470 (1996) 236.
68. C.R. Fernández-Pousa, M.V. Gallas, T.J. Hollowood and J.L. Miramontes, Nucl. Phys. B 484 (1997) 609.
69. C.R. Fernández-Pousa, M.V. Gallas, T.J. Hollowood and J.L. Miramontes, Nucl. Phys. B 499 (1997) 673.
70. C.R. Fernández-Pousa and J.L. Miramontes, Nucl. Phys. B 518 (1998) 745.
71. M. Freeman, Nucl. Phys. B 433 (1995) 657.
72. V.G. Kac, Infinite Dimensional Lie Algebras, third ed. (Cambridge Univ. Press, Cambridge, 1990).
73. P. Goddard and D. Olive, Int. J. Mod. Phys. A 1 (1986) 303; J. Fuchs, Affine Lie Algebras and Quantum Groups (Cambridge Univ. Press, Cambridge, 1992).
74. P. Goddard and D. Olive, Int. J. Mod. Phys. A 1 (1986) 303; J. Fuchs, Affine Lie Algebras and Quantum Groups (Cambridge Univ. Press, Cambridge, 1992).
75. 8, in Symposium on Anomalies, Geometry and Topology, ed. W.A. Bardeen and A.R. White (World Scientific, Singapore, 1985) p. 276.
76. D.H. Peterson and V.G. Kac, Proc. Natl. Acad. Sci. USA 80 (1983) 1778; V.G. Kac, Constructing groups associated to infinite-dimensional lie algebras, in Infinite-dimensional Groups with Applications, ed. V.G. Kac, Berkeley MSRI pub. (Springer, Berlin, 1985) p. 167.
77. D.H. Peterson and V.G. Kac, Proc. Natl. Acad. Sci. USA 80 (1983) 1778; V.G. Kac, Constructing groups associated to infinite-dimensional lie algebras, in Infinite-dimensional Groups with Applications, ed. V.G. Kac, Berkeley MSRI pub. (Springer, Berlin, 1985) p. 167.
78. K. Ueno and K. Takasaki, Adv. Stud, in Pure Maths. 4 (1984) 1; D. Olive and N. Turok, Nucl. Phys. B 257 (1985) 277; B 265 (1986) 469; T. Takebe, Commun. Math. Phys. 129 (1990) 281.
79. K. Ueno and K. Takasaki, Adv. Stud, in Pure Maths. 4 (1984) 1; D. Olive and N. Turok, Nucl. Phys. B 257 (1985) 277; B 265 (1986) 469; T. Takebe, Commun. Math. Phys. 129 (1990) 281.
80. K. Ueno and K. Takasaki, Adv. Stud, in Pure Maths. 4 (1984) 1; D. Olive and N. Turok, Nucl. Phys. B 257 (1985) 277; B 265 (1986) 469; T. Takebe, Commun. Math. Phys. 129 (1990) 281.
81. K. Ueno and K. Takasaki, Adv. Stud, in Pure Maths. 4 (1984) 1; D. Olive and N. Turok, Nucl. Phys. B 257 (1985) 277; B 265 (1986) 469; T. Takebe, Commun. Math. Phys. 129 (1990) 281.
82. H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimmerman, Solitons from Dressing in An Algebraic Approach to the Constrained KP Hierarchy, hep-th/9709004; Vertex operators and solitons of constrained KP hierarchies, in Proc. Integrable Models and Supersymmetry.
83. H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimmerman, Solitons from Dressing in An Algebraic Approach to the Constrained KP Hierarchy, hep-th/9709004; Vertex operators and solitons of constrained KP hierarchies, in Proc. Integrable Models and Supersymmetry.
84. M.R. Niedermaier, Commun. Math. Phys. 160 (1994) 391; Lett. Math. Phys. 30 (1994) 217.
85. M.R. Niedermaier, Commun. Math. Phys. 160 (1994) 391; Lett. Math. Phys. 30 (1994) 217.