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

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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)=. We prove in the full space situation that there exists an infinite subset of such that for any m∈, 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 =. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E≤C|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. 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)

    MATH  Google Scholar 

  2. Alexander, J.W.: Topological invariants of knots and links. Trans. A. M. S. 30, 275–306 (1928)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  4. Atiyah, M.: The Geometry and Physics of Knots. Cambridge: Cambridge Univ. Press, 1990

  5. Babaev, E.: Dual neutral variables and knotted solitons in triplet superconductors. Phys. Rev. Lett. 88, 177002 (2002)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Battye, R.A., Sutcliffe, P.M.: To be or knot to be? Phys. Rev. Lett. 81, 4798–4801 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Battye, R.A., Sutcliffe, P.M.: Solitons, links and knots. Proc. Roy. Soc. A 455, 4305–4331 (1999)

    Article  MATH  Google Scholar 

  9. Belavin, A.A., Polyakov, A.M.: Metastable states of two-dimensional isotropic ferromagnets. JETP Lett. 22, 245–247 (1975)

    Google Scholar 

  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)

    MATH  Google Scholar 

  11. Bethuel, F.: The approximation problem for Sobolev maps between two manifolds. Acta Math. 167, 153–206 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Bethuel, F., Brezis, H., Helein, F.: Ginzburg–Landau Vortices. Boston: Birkhäuser, 1994

  13. Bogomol’nyi, E.B.: The stability of classical solutions. Sov. J. Nucl. Phys. 24, 449–454 (1976)

    Google Scholar 

  14. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Berlin-New York: Springer, 1982

  15. Cho, Y.M.: Monopoles and knots in Skyrme theory. Phys. Rev. Lett. 87, 252001 (2001)

    Article  Google Scholar 

  16. Dunne, G.: Self-Dual Chern–Simons Theories. Lecture Notes in Phys., Vol. 36, Berlin: Springer, 1995

  17. Esteban, M.: A direct variational approach to Skyrme’s model for meson fields. Commun. Math. Phys. 105, 571–591 (1986)

    MathSciNet  MATH  Google Scholar 

  18. Esteban, M.J.: A new setting for Skyrme’s problem. In: Variational Methods, Boston: Birkhäuser, 1988, pp. 77–93

  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)

    MathSciNet  MATH  Google Scholar 

  20. Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. Regional Conference Series in Math. No. 74, Providence, RI: A. M. S., 1990

  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. 247–266

  22. Faddeev, L.: Knotted solitons. Plenary Address, In: ICM2002, Beijing, August 2002, Beijing: Higher Education Press of China, 2003

  23. Faddeev, L., Niemi, A.J.: Stable knot-like structures in classical field theory. Nature 387, 58– 61 (1997)

    Article  Google Scholar 

  24. Faddeev, L., Niemi, A.J.: Toroidal configurations as stable solitons. Preprint. http//:arxiv.org/abs/hep-th/9705176

  25. Finkelstein, D., Rubinstein, J.: Connection between spin, statistics, and kinks. J. Math. Phys. 9, 1762–1779 (1968)

    Article  MATH  Google Scholar 

  26. Hang, F.B., Lin, F.H.: Topology of Sobolev mappings. Math. Res. Lett. 8, 321–330 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Hang, F.B., Lin, F.H.: A Remark on the Jacobians. Comm. Contemp. Math. 2, 35–46 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hardt, R., Riviere, T.: Connecting topological Hopf singularities. Annali Sc. Norm. Sup. Pisa. 2, 287–344 (2002)

    Google Scholar 

  29. Hietarinta, J., Salo, P.: Faddeev–Hopf knots: Dynamics of linked unknots. Phys. Lett. B 451, 60–67 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Husemoller, D.: Fibre Bundles (2nd ed.). New York: Springer, 1975

  31. Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Boston: Birkhäuser, 1980

  32. Jehle, H.: Flux quantization and particle physics. Phys. Rev. D 6, 441–457 (1972)

    Article  Google Scholar 

  33. Jerrard, R., Soner, M.H.: Functions of bounded high variation. Indiana Univ. Math. J. 51, 645– 677 (2002)

    MathSciNet  MATH  Google Scholar 

  34. Jones, V.F.R.: A new knot polynomial and von Neumann algebras. Notices A. M. S. 33, 219– 225 (1986)

    MathSciNet  Google Scholar 

  35. Jones, V.F.R.: Hecke algebra representations of braid group and link polynomials. Ann. Math. 126, 335–388 (1987)

    MathSciNet  MATH  Google Scholar 

  36. Kauffman, L.H.: Knots and Physics. River Ridge, NJ: World Scientific, 2000

  37. Kibble, T.W.B.: Some implications of a cosmological phase transition. Phys. Rep. 69, 183–199 (1980)

    Article  Google Scholar 

  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. 3–34

  39. Lieb, E.H.: Remarks on the Skyrme model. In: Proc. Sympos. Pure Math. 54, Part 2, Providence, RI: Am. Math. Soc., 1993, pp. 379–384

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  42. MacArthur, A.: The entanglement structures of polymers. In: Knots and Applications, L. H. Kauffman (ed.), Singapore: World Scientific, 1995, pp. 395–426

  43. Makhankov, V.G., Rybakov, Y.P., Sanyuk, V.I.: The Skyrme Model. Berlin-Heidelberg: Springer, 1993

  44. Murasugi, K.: Jones polynomials and classical conjectures in knot theory. Topology 26, 187–194 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  45. Murasugi, K.: Knot Theory and its Applications. Boston: Birkhäuser, 1996

  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)

    Article  Google Scholar 

  47. Rajaraman, R.: Solitons and Instantons. Amsterdam: North-Holland, 1982

  48. Riviere, T.: Minimizing fibrations and p-harmonic maps in homotopy classes from S3 to S2. Comm. Anal. Geom. 6, 427–483 (1998)

    MathSciNet  MATH  Google Scholar 

  49. Riviere, T.: Towards Jaffe and Taubes conjectures in the strongly repulsive limit. Manuscripta Math. 108, 217–273 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  51. Skyrme, T.H.R.: A nonlinear field theory. Proc. Roy. Soc. A 260, 127–138 (1961)

    MATH  Google Scholar 

  52. Skyrme, T.H.R.: Particle states of a quantized meson field. Proc. Roy. Soc. A 262, 237–245 (1961)

    MATH  Google Scholar 

  53. Skyrme, T.H.R.: A unified field theory of mesons and baryons. Nucl. Phys. 31, 556–569 (1962)

    Article  Google Scholar 

  54. Skyrme, T.H.R.: The origins of Skyrmions. Internat. J. Mod. Phys. A 3, 2745–2751 (1988)

    Google Scholar 

  55. Sumners, D.W.: Lifting the curtain: using topology to probe the hidden action of enzymes. Notices A. M. S. 42, 528–537 (1995)

    MathSciNet  MATH  Google Scholar 

  56. Tait, P.G.: Scientific Papers, Cambridge: Cambridge Uni. Press, 1900

  57. Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. 3rd ed. Amsterdam: North-Holland, 1984

  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. 136–212

  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)

    MATH  Google Scholar 

  60. Vakulenko, A.F., Kapitanski, L.V.: Stability of solitons in S2 nonlinear σ-model. Sov. Phys. Dokl. 24, 433–434 (1979)

    MATH  Google Scholar 

  61. Vassiliev, V.A.: Invariants of knots and complements of discriminants. In: Developments in Mathematics: the Moscow School. London: Chapman & Hall, 1993, pp. 194–250

  62. Vilenkin, A.: Cosmic strings and domain walls. Phys. Rep. 121, 263–315 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  63. Vilenkin, A., Shellard, E.P.S.: Cosmic Strings and Other Topological Defects. Cambridge: Cambridge Uni. Press, 1994

  64. Ward, R.S.: Hopf solitons on S3 and ℝ3. Nonlinearity 12, 241–246 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  65. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351– 399 (1989)

    MATH  Google Scholar 

  66. Yang, Y.: Solitons in Field Theory and Nonlinear Analysis. New York: Springer, 2001

  67. Zahed, I., Brown, G.E.: The Skyrme model. Phys. Rep. 142, 1–102 (1986)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY 10021, USA

    Fanghua Lin

  2. Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA

    Yisong Yang

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  1. Fanghua Lin
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  2. Yisong Yang
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Communicated by H.-T. Yau

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Lin, F., Yang, Y. Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model. Commun. Math. Phys. 249, 273–303 (2004). https://doi.org/10.1007/s00220-004-1110-y

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