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Solitons and generalized tau-functions for affine integrable hierarchies

  • Integrable Models
  • Conference paper
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Book cover Supersymmetry and Integrable Models

Part of the book series: Lecture Notes in Physics ((LNP,volume 502))

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Abstract

We consider a very large class of hierarchies of zero-curvature equations constructed from affine Kac-Moody algebras Ĵ. We argue that one of the basic ingredients for the appearance of soliton solutions in such theories is the existence of “vacuum solutions” corresponding to Lax operators lying in some abelian (up to central term) subalgebra of Ĵ. Using the dressing transformation procedure we construct the solutions in the orbit of those vacuum solutions, and conjecture that the soliton solutions correspond to some special points in those orbits. The generalized tau-function for those hierarchies are defined for integrable highest weight representations of Ĵ, and it applies for any level of the representation and it is independent of its realization. We illustrate our methods with the recently proposed non abelian Toda models coupled to matter fields. A very special class of such theories possess a U (1) Noether charge that, under a suitable gauge fixing of the conformal symmetry, is proportional to a topological charge. That leads to a mechanism that confines the matter fields inside the solitons.

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References

  • H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, Phys. Lett. 254B (1991) 372.

    ADS  MathSciNet  Google Scholar 

  • H. Aratyn, C.P. Constantinidis, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, Nucl. Phys. B406 (1993) 727.

    Article  ADS  MathSciNet  Google Scholar 

  • H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, Vertex Operators and Solitons of Constrained KP Hierarchies, contribution to this volume

    Google Scholar 

  • O. Babelon and L. Bonora, Phys. Lett. 244B (1990), 220.

    ADS  MathSciNet  Google Scholar 

  • O. Babelon and D. Bernard, Phys. Lett. B260 81; Commun. Math. Phys. 149 (1992) 279–306, hep-th/9111036.

    Google Scholar 

  • O. Babelon and D. Bernard, Int. J. Mod. Phys. A8 (1993) 507–543, hep-th/9206002.

    ADS  MathSciNet  Google Scholar 

  • H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Phys. Lett. 227B (1989) 411, Nucl. Phys. B338 (1990) 689

    ADS  MathSciNet  Google Scholar 

  • M.D. Freeman Phys. Lett. 261B (1991) 57

    ADS  MathSciNet  Google Scholar 

  • A. Fring, H.C. Liao and D.I. Olive, Phys. Lett. 266B (1991) 82–86.

    ADS  MathSciNet  Google Scholar 

  • S.A. Brazovskii, N. N. Kirova, Sov. Phys. Rev. A Phys. 5 (1994) 99.

    Google Scholar 

  • A. Chodos, E. Hadjimichael, C. Tze, Solitons in Nuclear and Elementary Particle Physics, Proceedings of the Lewes Workshop, 1984, World Scientific.

    Google Scholar 

  • E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Proc. Japan. Acad. 57A (1981) 3806; Physica D4 (1982) 343; Publ. RIMS Kyoto University 18, (1982) 1077.

    MathSciNet  Google Scholar 

  • V.G. Drinfel'd and V.V. Sokolov, J. Sov. Math. 30 (1985) 1975.

    Article  MATH  Google Scholar 

  • L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, 1987.

    Google Scholar 

  • L.A. Ferreira, J.L. Miramontes and J. Sánchez Guillén, Nucl. Phys. B449 (1995) 631–679, hep-th/9412127.

    Article  ADS  Google Scholar 

  • L.A. Ferreira, J-L. Gervais, J. Sánchez Guillén and M.V. Saveliev, Affine Toda Systems Coupled to Matter Fields, Nucl. Phys. B470 (1996) 236, hep-th/9512105.

    Article  ADS  Google Scholar 

  • L.A. Ferreira, J.L. Miramontes and J. Sánchez Guillén, J. Math. Phys. 38 (1997) 2, hep-th/9606066.

    Article  Google Scholar 

  • A.P. Fordy and J. Gibbons, Commun. Math. Phys. 77 (1980) 21.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • P.J. Caudrey, J.D. Gibbon, J.C. Eilbeck and R.K. Bullough, Phys. Rev. Lett. 30, (1973), 237.

    Article  ADS  Google Scholar 

  • G.L. Lamb Jr., Elements of Soliton Theory, Wiley, New York, 1980. G.L. Lamb, Backlund transformations at the turn of the century. In: Lecture Notes in Mathematics, Vol. 515, pp. 69–79, Springer, 1976.

    MATH  Google Scholar 

  • M.D. Freeman Nucl. Phys. B433 (1995) 657.

    Article  ADS  MathSciNet  Google Scholar 

  • M.F. de Groot, T.J. Hollowood, and J.L. Miramontes, Commun. Math. Phys. 145 (1992) 57

    Article  MATH  ADS  Google Scholar 

  • 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.

    Article  MATH  ADS  Google Scholar 

  • A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.-P. Wu, Rev. Mod. Phys. 60 (1988) 782.

    Article  ADS  Google Scholar 

  • R. Hirota, Direct methods in soliton theory, in “Soliton” (R.K. Bullough and P.S. Caudrey, eds.), (1980) 157.; J. Phys. Soc. Japan 33 (1972) 1459.

    Google Scholar 

  • T.J. Hollowood and J.L. Miramontes, Commun. Math. Phys. 157 (1993) 99.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • R. Jackiw, C. Rebbi, Phys. Rev. D 13 (1976) 3398

    Article  ADS  MathSciNet  Google Scholar 

  • J. Goldstone, F. Wilczek, Phys. Rev. Lett. 47 (1981) 986.

    Article  ADS  MathSciNet  Google Scholar 

  • V.G. Kac and D.H. Peterson, 112 constructions of the basic representation of the loop group of E 8. In: Symp. on Anomalies, Geometry and Topology, W.A. Bardeen and A.R. White (eds.), World Scientific, Singapore, 1985.

    Google Scholar 

  • V.G. Kac, Infinite dimensional Lie algebras, Third Edition, Cambridge University Press, Cambridge, 1990.

    Book  MATH  Google Scholar 

  • V.G. Kac and M. Wakimoto, Exceptional hierarchies of soliton equations, in “Proceedings of Symposia in Pure Mathematics”, Vol. 49 (1989) 191.

    MathSciNet  Google Scholar 

  • A.N. Leznov and M.V. Saveliev, Group Theoretical Methods for Integration of Non-Linear Dynamical Systems, Progress in Physics Series, v. 15, Birkhaüser-Verlag, Basel, 1992.

    Google Scholar 

  • H.C. Liao, D. Olive and N. Turok, Phys. Lett. 298B (1993) 95–102.

    ADS  MathSciNet  Google Scholar 

  • J.L.Miramontes, private communication and in preparation with C.R.Fernandez-Pousa.

    Google Scholar 

  • C. Montonen and D.I. Olive, Phys. Lett. 72B (1977) 117

    ADS  Google Scholar 

  • P. Goddard, J. Nuyts and D.I. Olive, Nucl. Phys. B125 (1977) 1

    Article  ADS  MathSciNet  Google Scholar 

  • D.I. Olive, Magnetic monopoles and electromagnetic duality conjectures in Monopoles in quantum field theory eds. N. S. Craigie, P. Goddard and W. Nahm (World Scientific, Singapore, 1982) p. 157; D. Olive, Exact electromagnetic duality, hep-th/9508089, to appear in the Proc. of ICTP Conf. on Recent Developments in Statistical Mechanics and Quantum Field Theory, Trieste, 10–12 Apr. 1995.

    Google Scholar 

  • N. Seiberg and E. Witten Nucl. Phys. B426 (1994) 19–52, hep-th/9407087

    Article  ADS  MathSciNet  Google Scholar 

  • C. Vafa and E. Witten Nucl. Phys. B431 (1994) 3–77, hep-th/9408074.

    Article  ADS  MathSciNet  Google Scholar 

  • D. Olive, N. Turok and J.W.R. Underwood, Nucl. Phys. B401 (1993) 663–697.

    Article  ADS  MathSciNet  Google Scholar 

  • D. Olive, N. Turok and J.W.R. Underwood, Nucl. Phys. B409 (1993) 509–546, hep-th/9305160.

    Article  ADS  MathSciNet  Google Scholar 

  • D. Olive, M.V. Saveliev and J.W.R. Underwood, Phys. Lett. B311 (1993) 117–122, hep-th/9212123.

    ADS  MathSciNet  Google Scholar 

  • M. Semenov-Tian-Shansky, Functional Analysis and Its Application 17 (1983) 259; Publ. RIMS Kyoto Univ. 21 (1985) 1237.

    MATH  Google Scholar 

  • J.W.R. Underwood, Aspects of Non Abelian Toda Theories, hep-th/9304156.

    Google Scholar 

  • G. Wilson, Phil. Trans. R. Soc. Lond. A 315 (1985) 383; Habillage et fonctions τ C. R. Acad. Sc. Paris 299 (I) (1984) 587.

    ADS  Google Scholar 

  • G. Wilson, The τ-Functions of the gAKNS Equations, in “Verdier memorial conference on integrable systems” (O. Babelon, P. Cartier, and Y. Kosmann-Schwarzbach, eds.), Birkhauser (1993) 131–145.

    Google Scholar 

  • V.E. Zakharov, A.B. Shabat, Functional Analysis and Its Application 13 (1979) 166.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituto de Física Teórica — IFT/UNESP, Rua Pamplona 145, 01405-900, São Paulo - SP, Brazil

    L. A. Ferreira

  2. Facultad de Física, Universidad de Santiago de Compostela, 15706 Santiago de, Compostela, Spain

    J. Sánchez Guillén

Authors
  1. L. A. Ferreira
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  2. J. Sánchez Guillén
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Henrik Aratyn Tom D. Imbo Wai-Yee Keung Uday Sukhatme

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© 1998 Springer-Verlag

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Ferreira, L.A., Guillén, J.S. (1998). Solitons and generalized tau-functions for affine integrable hierarchies. In: Aratyn, H., Imbo, T.D., Keung, WY., Sukhatme, U. (eds) Supersymmetry and Integrable Models. Lecture Notes in Physics, vol 502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105317

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  • DOI: https://doi.org/10.1007/BFb0105317

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