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Exact vortex solutions in a CP N Skyrme-Faddeev type model

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Abstract

We consider a four dimensional field theory with target space being CP N which constitutes a generalization of the usual Skyrme-Faddeev model defined on CP 1. We show that it possesses an integrable sector presenting an infinite number of local conservation laws, which are associated to the hidden symmetries of the zero curvature representation of the theory in loop space. We construct an infinite class of exact solutions for that integrable submodel where the fields are meromorphic functions of the combinations (x 1 + ix 2) and x 3 + x 0 of the Cartesian coordinates of four dimensional Minkowski space-time. Among those solutions we have static vortices and also vortices with waves traveling along them with the speed of light. The energy per unity of length of the vortices show an interesting and intricate interaction among the vortices and waves.

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References

  1. S.R. Coleman, Quantum sine-Gordon equation as the massive Thirring model, Phys. Rev. D 11 (1975) 2088 [SPIRES].

    ADS  Google Scholar 

  2. S. Mandelstam, Soliton operators for the quantized sine-Gordon equation, Phys. Rev. D 11 (1975) 3026 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  3. C. Montonen and D.I. Olive, Magnetic monopoles as gauge particles?, Phys. Lett. B 72 (1977) 117 [SPIRES].

    ADS  Google Scholar 

  4. C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  5. N. Seiberg and E. Witten, Monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  6. P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 21 (1968) 467 [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  7. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Zh. Eksp. Teor. Fiz. 61 (1971) 118 [Soviet Phys. JETP 34 (1972) 62].

    Google Scholar 

  8. O. Alvarez, L.A. Ferreira and J. Sanchez Guillen, A new approach to integrable theories in any dimension, Nucl. Phys. B 529 (1998) 689 [hep-th/9710147] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  9. O. Alvarez, L.A. Ferreira and J. Sanchez-Guillen, Integrable theories and loop spaces: fundamentals, applications and new developments, Int. J. Mod. Phys. A 24 (2009) 1825 [arXiv:0901.1654] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  10. L.D. Faddeev, Quantization of solitons, Princeton IAS Print-75-QS70 (1975).

  11. L.D. Faddeev, 40 years in mathematical physics, World Scientific, Singapore (1995).

    MATH  Google Scholar 

  12. L.D. Faddeev and A.J. Niemi, Knots and particles, Nature 387 (1997) 58 [hep-th/9610193] [SPIRES].

    Article  ADS  Google Scholar 

  13. P. Sutcliffe, Knots in the Skyrme-Faddeev model, Proc. Roy. Soc. Lond. A 463 (2007) 3001 [arXiv:0705.1468] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  14. J. Hietarinta and P. Salo, Faddeev-Hopf knots: dynamics of linked un-knots, Phys. Lett. B 451 (1999) 60 [hep-th/9811053] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  15. J. Hietarinta and P. Salo, Ground state in the Faddeev-Skyrme model, Phys. Rev. D 62 (2000) 081701 [SPIRES].

    ADS  Google Scholar 

  16. L.A. Ferreira, Exact vortex solutions in an extended Skyrme-Faddeev model, JHEP 05 (2009) 001 [arXiv:0809.4303] [SPIRES].

    Article  ADS  Google Scholar 

  17. L.D. Faddeev and A.J. Niemi, Partially dual variables in SU(2) Yang-Mills theory, Phys. Rev. Lett. 82 (1999) 1624 [hep-th/9807069] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. L.D. Faddeev, Knots as possible excitations of the quantum Yang-Mills fields, arXiv:0805.1624 [SPIRES].

  19. H. Gies, Wilsonian effective action for SU(2) Yang-Mills theory with Cho-Faddeev-Niemi-Shabanov decomposition, Phys. Rev. D 63 (2001) 125023 [hep-th/0102026] [SPIRES].

    ADS  Google Scholar 

  20. L.D. Faddeev and A.J. Niemi, Partial duality in SU(N) Yang-Mills theory, Phys. Lett. B 449 (1999) 214 [hep-th/9812090] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  21. L.D. Faddeev and A.J. Niemi, Decomposing the Yang-Mills field, Phys. Lett. B 464 (1999) 90 [hep-th/9907180] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  22. K.-I. Kondo, T. Shinohara and T. Murakami, Reformulating SU(N) Yang-Mills theory based on change of variables, Prog. Theor. Phys. 120 (2008) 1 [arXiv:0803.0176] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  23. W.J. Zakrzewski, Low dimensional sigma models, Hilger, Bristol U.K. (1989).

    MATH  Google Scholar 

  24. A.M. Din and W.J. Zakrzewski, General classical solutions in the CP n−1 MODEL, Nucl. Phys. B 174 (1980) 397 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  25. A.M. Grundland and W.J. Zakrzewski, On CP 1 and CP 2 maps and Weierstrass representations for surface immersed into multi-dimensional Euclidean spaces, J. Nonliner Math. Phy. 10 (2003) 110.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. J. Hietarinta, J. Jaykka and P. Salo, Dynamics of vortices and knots in Faddeev’s model, in Workshop on Integrable Theories, Solitons and Duality (2002), July 1–6, São Paulo, Brazil (2002), PoS(unesp2002)017.

  27. J. Hietarinta, J. Jaykka and P. Salo, Relaxation of twisted vortices in the Faddeev-Skyrme model, Phys. Lett. A 321 (2004) 324 [cond-mat/0309499] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  28. J. Jaykka and J. Hietarinta, Unwinding in Hopfion vortex bunches, Phys. Rev. D 79 (2009) 125027 [arXiv:0904.1305] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  29. M. Hirayama, C.-G. Shi and J. Yamashita, Elliptic solutions of the Skyrme model, Phys. Rev. D 67 (2003) 105009 [hep-th/0303092] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  30. M. Hirayama and C.-G. Shi, A class of exact solutions of the Faddeev model, Phys. Rev. D 69 (2004) 045001 [hep-th/0310042] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  31. C.-G. Shi and M. Hirayama, Approximate vortex solution of Faddeev model, Int. J. Mod. Phys. A 23 (2008) 1361 [arXiv:0712.4330] [SPIRES].

    ADS  Google Scholar 

  32. M. Eto, Y. Isozumi, M. Nitta and K. Ohashi, 1/2, 1/4 and 1/8 BPS equations in SUSY Yang-Mills-Higgs systems: field theoretical brane configurations, Nucl. Phys. B 752 (2006) 140 [hep-th/0506257] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  33. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Solitons in the Higgs phase: the moduli matrix approach, J. Phys. A 39 (2006) R315 [hep-th/0602170] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  34. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York USA (1978).

    MATH  Google Scholar 

  35. H. Eichenherr and M. Forger, More about nonlinear σ-models on symmetric spaces, Nucl. Phys. B 164 (1980) 528 [Erratum ibid. B 282 (1987) 745] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  36. L.A. Ferreira and D.I. Olive, Noncompact symmetric spaces and the Toda molecule equations, Commun. Math. Phys. 99 (1985) 365 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. L.A. Ferreira and E.E. Leite, Integrable theories in any dimension and homogeneous spaces, Nucl. Phys. B 547 (1999) 471 [hep-th/9810067] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  38. C. Adam, J. Sanchez-Guillen and A. Wereszczynski, New integrable sectors in Skyrme and 4-dimensional CP n model, J. Phys. A 40 (2007) 1907 [hep-th/0610024] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  39. A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n expandable series of nonlinear σ-models with instantons, Nucl. Phys. B 146 (1978) 63 [SPIRES].

    Article  ADS  Google Scholar 

  40. N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press, Cambridge U.K. (2004), pag. 493 [SPIRES].

    Book  MATH  Google Scholar 

  41. P. Goddard and D.I. Olive, New developments in the theory of magnetic monopoles, Rept. Prog. Phys. 41 (1978) 1357 [SPIRES].

    Article  ADS  Google Scholar 

  42. N.D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (1979) 591 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Instituto de Física de São Carlos; IFSC/USP, Universidade de São Paulo, Avenida Trabalhador São-carlense 400, Caixa Postal 369, CEP 13560-970, São Carlos-SP, Brazil

    L. A. Ferreira & P. Klimas

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  1. L. A. Ferreira
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  2. P. Klimas
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Correspondence to L. A. Ferreira.

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ArXiv ePrint: 1007.1667

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Ferreira, L.A., Klimas, P. Exact vortex solutions in a CP N Skyrme-Faddeev type model. J. High Energ. Phys. 2010, 8 (2010). https://doi.org/10.1007/JHEP10(2010)008

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