Communications in Mathematical Physics

, Volume 249, Issue 2, pp 273–303 | Cite as

Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

  • Fanghua Lin
  • Yisong Yang
Article

Abstract

In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3(S2)=Open image in new window. We prove in the full space situation that there exists an infinite subset Open image in new window of Open image in new window such that for any mOpen image in new window, the Faddeev energy, E, has a minimizer among the topological class Q=m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for Open image in new window=Open image in new window. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality EC|Q|3/4, where C>0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.

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References

  1. 1.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary of solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959)MATHGoogle Scholar
  2. 2.
    Alexander, J.W.: Topological invariants of knots and links. Trans. A. M. S. 30, 275–306 (1928)MATHGoogle Scholar
  3. 3.
    Aratyn, H., Ferreira, L.A., Zimerman, A.H.: Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbers. Phys. Rev. Lett. 83, 1723–1726 (1999)CrossRefGoogle Scholar
  4. 4.
    Atiyah, M.: The Geometry and Physics of Knots. Cambridge: Cambridge Univ. Press, 1990Google Scholar
  5. 5.
    Babaev, E.: Dual neutral variables and knotted solitons in triplet superconductors. Phys. Rev. Lett. 88, 177002 (2002)CrossRefGoogle Scholar
  6. 6.
    Battye, R.A., Sutcliffe, P.M.: Knots as stable solutions in a three-dimensional classical field theory. Phys. Rev. Lett. 81, 4798–4801 (1998)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Battye, R.A., Sutcliffe, P.M.: To be or knot to be? Phys. Rev. Lett. 81, 4798–4801 (1998)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Battye, R.A., Sutcliffe, P.M.: Solitons, links and knots. Proc. Roy. Soc. A 455, 4305–4331 (1999)CrossRefMATHGoogle Scholar
  9. 9.
    Belavin, A.A., Polyakov, A.M.: Metastable states of two-dimensional isotropic ferromagnets. JETP Lett. 22, 245–247 (1975)Google Scholar
  10. 10.
    Bethuel, F.: A characterization of maps in H1(B3,S2) which can be approximated by smooth maps. Ann. Inst. H. Poincaré – Anal. non linéaire 7, 269–286 (1990)MATHGoogle Scholar
  11. 11.
    Bethuel, F.: The approximation problem for Sobolev maps between two manifolds. Acta Math. 167, 153–206 (1991)MathSciNetMATHGoogle Scholar
  12. 12.
    Bethuel, F., Brezis, H., Helein, F.: Ginzburg–Landau Vortices. Boston: Birkhäuser, 1994Google Scholar
  13. 13.
    Bogomol’nyi, E.B.: The stability of classical solutions. Sov. J. Nucl. Phys. 24, 449–454 (1976)Google Scholar
  14. 14.
    Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Berlin-New York: Springer, 1982Google Scholar
  15. 15.
    Cho, Y.M.: Monopoles and knots in Skyrme theory. Phys. Rev. Lett. 87, 252001 (2001)CrossRefGoogle Scholar
  16. 16.
    Dunne, G.: Self-Dual Chern–Simons Theories. Lecture Notes in Phys., Vol. 36, Berlin: Springer, 1995Google Scholar
  17. 17.
    Esteban, M.: A direct variational approach to Skyrme’s model for meson fields. Commun. Math. Phys. 105, 571–591 (1986)MathSciNetMATHGoogle Scholar
  18. 18.
    Esteban, M.J.: A new setting for Skyrme’s problem. In: Variational Methods, Boston: Birkhäuser, 1988, pp. 77–93Google Scholar
  19. 19.
    Esteban, M.J., Müller, S.: Sobolev maps with integer degree and applications to Skyrme’s problem. Proc. Roy. Soc. A 436, 197–201 (1992)MathSciNetMATHGoogle Scholar
  20. 20.
    Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. Regional Conference Series in Math. No. 74, Providence, RI: A. M. S., 1990Google Scholar
  21. 21.
    Faddeev, L.: Einstein and several contemporary tendencies in the theory of elementary particles. In: Relativity, Quanta, and Cosmology, Vol. 1, eds. M. Pantaleo, F. de Finis, New York: Johnson Reprint Co., 1979, pp. 247–266Google Scholar
  22. 22.
    Faddeev, L.: Knotted solitons. Plenary Address, In: ICM2002, Beijing, August 2002, Beijing: Higher Education Press of China, 2003Google Scholar
  23. 23.
    Faddeev, L., Niemi, A.J.: Stable knot-like structures in classical field theory. Nature 387, 58– 61 (1997)CrossRefGoogle Scholar
  24. 24.
    Faddeev, L., Niemi, A.J.: Toroidal configurations as stable solitons. Preprint. http//:arxiv.org/abs/hep-th/9705176Google Scholar
  25. 25.
    Finkelstein, D., Rubinstein, J.: Connection between spin, statistics, and kinks. J. Math. Phys. 9, 1762–1779 (1968)CrossRefMATHGoogle Scholar
  26. 26.
    Hang, F.B., Lin, F.H.: Topology of Sobolev mappings. Math. Res. Lett. 8, 321–330 (2001)MathSciNetMATHGoogle Scholar
  27. 27.
    Hang, F.B., Lin, F.H.: A Remark on the Jacobians. Comm. Contemp. Math. 2, 35–46 (2000)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Hardt, R., Riviere, T.: Connecting topological Hopf singularities. Annali Sc. Norm. Sup. Pisa. 2, 287–344 (2002)Google Scholar
  29. 29.
    Hietarinta, J., Salo, P.: Faddeev–Hopf knots: Dynamics of linked unknots. Phys. Lett. B 451, 60–67 (1999)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Husemoller, D.: Fibre Bundles (2nd ed.). New York: Springer, 1975Google Scholar
  31. 31.
    Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Boston: Birkhäuser, 1980Google Scholar
  32. 32.
    Jehle, H.: Flux quantization and particle physics. Phys. Rev. D 6, 441–457 (1972)CrossRefGoogle Scholar
  33. 33.
    Jerrard, R., Soner, M.H.: Functions of bounded high variation. Indiana Univ. Math. J. 51, 645– 677 (2002)MathSciNetMATHGoogle Scholar
  34. 34.
    Jones, V.F.R.: A new knot polynomial and von Neumann algebras. Notices A. M. S. 33, 219– 225 (1986)MathSciNetGoogle Scholar
  35. 35.
    Jones, V.F.R.: Hecke algebra representations of braid group and link polynomials. Ann. Math. 126, 335–388 (1987)MathSciNetMATHGoogle Scholar
  36. 36.
    Kauffman, L.H.: Knots and Physics. River Ridge, NJ: World Scientific, 2000Google Scholar
  37. 37.
    Kibble, T.W.B.: Some implications of a cosmological phase transition. Phys. Rep. 69, 183–199 (1980)CrossRefGoogle Scholar
  38. 38.
    Kibble, T.W.B.: Cosmic strings – an overview. In: The Formation and Evolution of Cosmic Strings, ed. G. Gibbons, S. Hawking, and T. Vachaspati, Cambridge: Cambridge U. Press, 1990, pp. 3–34Google Scholar
  39. 39.
    Lieb, E.H.: Remarks on the Skyrme model. In: Proc. Sympos. Pure Math. 54, Part 2, Providence, RI: Am. Math. Soc., 1993, pp. 379–384Google Scholar
  40. 40.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. Part I. Ann. Inst. H. Poincaré – Anal. non linéaire 1, 109–145 (1984)MATHGoogle Scholar
  41. 41.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. Part II. Ann. Inst. H. Poincare – Anal. non linéaire 1, 223–283 (1984)MATHGoogle Scholar
  42. 42.
    MacArthur, A.: The entanglement structures of polymers. In: Knots and Applications, L. H. Kauffman (ed.), Singapore: World Scientific, 1995, pp. 395–426Google Scholar
  43. 43.
    Makhankov, V.G., Rybakov, Y.P., Sanyuk, V.I.: The Skyrme Model. Berlin-Heidelberg: Springer, 1993Google Scholar
  44. 44.
    Murasugi, K.: Jones polynomials and classical conjectures in knot theory. Topology 26, 187–194 (1987)CrossRefMathSciNetMATHGoogle Scholar
  45. 45.
    Murasugi, K.: Knot Theory and its Applications. Boston: Birkhäuser, 1996Google Scholar
  46. 46.
    Prasad, M.K., Sommerfield, C.M.: Exact classical solutions for the ‘t Hooft monopole and the Julia–Zee dyon. Phys. Rev. Lett. 35, 760–762 (1975)CrossRefGoogle Scholar
  47. 47.
    Rajaraman, R.: Solitons and Instantons. Amsterdam: North-Holland, 1982Google Scholar
  48. 48.
    Riviere, T.: Minimizing fibrations and p-harmonic maps in homotopy classes from S3 to S2. Comm. Anal. Geom. 6, 427–483 (1998)MathSciNetMATHGoogle Scholar
  49. 49.
    Riviere, T.: Towards Jaffe and Taubes conjectures in the strongly repulsive limit. Manuscripta Math. 108, 217–273 (2002)CrossRefMathSciNetMATHGoogle Scholar
  50. 50.
    Rybakov, Y.P., Sanyuk, V.I.: Methods for studying 3+1 localized structures: The Skyrmion as the absolute minimizer of energy. Internat. J. Mod. Phys. A 7, 3235–3264 (1992)MathSciNetGoogle Scholar
  51. 51.
    Skyrme, T.H.R.: A nonlinear field theory. Proc. Roy. Soc. A 260, 127–138 (1961)MATHGoogle Scholar
  52. 52.
    Skyrme, T.H.R.: Particle states of a quantized meson field. Proc. Roy. Soc. A 262, 237–245 (1961)MATHGoogle Scholar
  53. 53.
    Skyrme, T.H.R.: A unified field theory of mesons and baryons. Nucl. Phys. 31, 556–569 (1962)CrossRefGoogle Scholar
  54. 54.
    Skyrme, T.H.R.: The origins of Skyrmions. Internat. J. Mod. Phys. A 3, 2745–2751 (1988)Google Scholar
  55. 55.
    Sumners, D.W.: Lifting the curtain: using topology to probe the hidden action of enzymes. Notices A. M. S. 42, 528–537 (1995)MathSciNetMATHGoogle Scholar
  56. 56.
    Tait, P.G.: Scientific Papers, Cambridge: Cambridge Uni. Press, 1900Google Scholar
  57. 57.
    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. 3rd ed. Amsterdam: North-Holland, 1984Google Scholar
  58. 58.
    Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, R.J. Knops (ed.), London: Pitman, 1979, pp. 136–212Google Scholar
  59. 59.
    Taubes, C.H.: The existence of a non-minimal solution to the SU(2) Yang–Mills–Higgs equations on R3, Parts I, II. Commun. Math. Phys. 86, 257–320 (1982)MATHGoogle Scholar
  60. 60.
    Vakulenko, A.F., Kapitanski, L.V.: Stability of solitons in S2 nonlinear σ-model. Sov. Phys. Dokl. 24, 433–434 (1979)MATHGoogle Scholar
  61. 61.
    Vassiliev, V.A.: Invariants of knots and complements of discriminants. In: Developments in Mathematics: the Moscow School. London: Chapman & Hall, 1993, pp. 194–250Google Scholar
  62. 62.
    Vilenkin, A.: Cosmic strings and domain walls. Phys. Rep. 121, 263–315 (1985)CrossRefMathSciNetMATHGoogle Scholar
  63. 63.
    Vilenkin, A., Shellard, E.P.S.: Cosmic Strings and Other Topological Defects. Cambridge: Cambridge Uni. Press, 1994Google Scholar
  64. 64.
    Ward, R.S.: Hopf solitons on S3 and ℝ3. Nonlinearity 12, 241–246 (1999)CrossRefMathSciNetMATHGoogle Scholar
  65. 65.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351– 399 (1989)MATHGoogle Scholar
  66. 66.
    Yang, Y.: Solitons in Field Theory and Nonlinear Analysis. New York: Springer, 2001Google Scholar
  67. 67.
    Zahed, I., Brown, G.E.: The Skyrme model. Phys. Rep. 142, 1–102 (1986)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Fanghua Lin
    • 1
  • Yisong Yang
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsPolytechnic UniversityBrooklynUSA

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