Communications in Mathematical Physics

, Volume 153, Issue 2, pp 391–422 | Cite as

Geometry and kinematics of two Skyrmions

  • M. F. Atiyah
  • N. S. Manton


In Skyrme's soliton model of baryons, a single Skyrmion has six degrees of freedom, so it is expected that two-Skyrmion dynamics at modest energies can be modelled by motion on a 12-dimensional space of Skyrme fields. A candiate for this space is generated by the gradient flow of the potential energy function, descending from the unstable, baryon number two, hedgehog solutions of the Skyrme field equation. An apparently very similar space is obtained by restricting the gradient flow to the Skyrme fields derived fromSU(2) Yang-Mills instantons of charge two. On both of these spaces, one may quotient out by the group of translations and isospin rotations. Hartshorne's geometrical description of charge two instantons leads us to a conjecture for the global structure of the 6-dimensional quotient space. The conjectured structure is that of complex projective 3-space, with complex conjugate points on one projective plane identified and the real points in this plane removed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bogomol'nyi, E.B.: The stability of classical solutions. Sov. J. Nucl. Phys.24, 449 (1976)Google Scholar
  2. 2.
    Weinberg, E.: Parameter counting for multimonopole solutions. Phys. Rev.D20, 936 (1979)Google Scholar
  3. 3.
    Taubes, C.H.: ArbitraryN-vortex solutions of the first order Ginzburg-Landau equations. Commun. Math. Phys.72, 277 (1980)CrossRefGoogle Scholar
  4. 4.
    Manton, N.S.: A remark on the scattering of BPS monopoles. Phys. Lett.110B, 54 (1982)Google Scholar
  5. 4a.
    Atiyah, M.F., Hitchin, N.J.: The geometry and dynamics of magnetic monopoles, Princeton, NJ: Princeton University Press 1988Google Scholar
  6. 4b.
    Bates, L., Montgometry, R.: Closed geodesics on the space of stable two-monopoles. Commun. Math. Phys.118, 635 (1988)CrossRefGoogle Scholar
  7. 4c.
    Temple-Raston, M.: Closed 2-dyon orbits. Nucl. Phys.B313, 447 (1989)CrossRefGoogle Scholar
  8. 4d.
    Leese, R.: Low energy scattering of solitons in the ℂP 1 model. Nucl. Phys.B344, 33 (1990)CrossRefGoogle Scholar
  9. 4e.
    Samols, T.M.: Vortex scattering. Commun. Math. Phys.145, 149 (1992)Google Scholar
  10. 5.
    Gibbons, G.W., Manton, N.S.: Classical and quantum dyanmics of BPS monopoles. Nucl. Phys.B274, 183 (1986)CrossRefGoogle Scholar
  11. 5a.
    Schroers, B.J.: Quantum scattering of BPS monopoles at low energy. Nucl. Phys.B367, 177 (1991)CrossRefGoogle Scholar
  12. 6.
    Manton, N.S.: Unstable manifolds and soltion dynamics. Phys. Rev. Lett.60, 1916 (1988)CrossRefGoogle Scholar
  13. 7.
    Skyrme, T.R.H.: A unified field theory of mesons and baryons. Nucl. Phys.31, 556 (1962)CrossRefGoogle Scholar
  14. 7a.
    Nyman, E.M., Riska, D.O.: Low energy properties of baryons in the Skyrme model. Reps. Prog. Phys.53, 1137 (1990)CrossRefGoogle Scholar
  15. 8.
    Adkins, G.S., Nappi, C.R., Witten, E.: Static properties of nucleons in the Skyrme model. Nucl. Phys.B228, 552 (1983)CrossRefGoogle Scholar
  16. 9.
    Kopeliovich, V.B., Stern, B.E.: Exotic Skyrmions. JETP Lett.45, 203 (1987)Google Scholar
  17. 9a.
    Verbaarschot, J.J.M.: Axial symmetry of bound baryon number-two solution of the Skyme model. Phys. Lett.195B, 235 (1987)Google Scholar
  18. 9b.
    Schramm, A.J., Dothan, Y., Biedenharn, L.C.: A calculation of the deuteron as a biskyrmion. Phys. Lett.205B, 151 (1988)Google Scholar
  19. 10.
    Jackson, A.D., Rho, M.: Baryons as chrial solitons. Phys. Rev. Lett.51, 751 (1983)CrossRefGoogle Scholar
  20. 11.
    Wirzba, A., Bang, H.: The mode spectrum and the stability analysis of Skyrmions on a 3-sphere. Nucl. Phys.A515, 571 (1990)CrossRefGoogle Scholar
  21. 11a.
    Zenkin, S.V., Kopeliovich, V.B., Stern, B.E.: The soliton interaction in the Skyrme model. Sov. J. Nucl. Phys.45, 106 (1987)Google Scholar
  22. 12.
    Braaten, E., Carson, L.: Deuteron as a toroidal Skyrmion. Phys. Rev.D38, 3525 (1988)CrossRefGoogle Scholar
  23. 13.
    Atiyah, M.F., Manton, N.S.: Skyrmions from instantons. Phys. Lett.222B, 438 (1989)Google Scholar
  24. 13a.
    Manton, N.S.: Skyrme fields and instantons. In: Geometry of low dimensional manifolds: 1 (LMS Lecture Notes 150), Donaldson, S.K., and Thomas, C.B., eds. Cambridge: Cambridge University Press 1990Google Scholar
  25. 14.
    Jackson, A., Jackson, A.D., Pasquier, V.: The Skyrmion-Skyrmion interaction. Nucl. Phys.A 432, 567 (1985)CrossRefGoogle Scholar
  26. 15.
    Vinh Mau, R., Lacombe, M., Loiseau, B., Cottingham, W.N., Lisboa, P.: The static baryon-baryon potential in the Skyrme model. Phys. Lett.150B, 259 (1985)Google Scholar
  27. 16.
    Braaten, E., Townsend, S., Carson, L.: Novel structure of static multisoliton solutions in the Skyrme model. Phys. Lett.235B, 147 (1990)Google Scholar
  28. 17.
    Kugler, M., Shtrikman, S.: A new Skyrmion crystal. Phys. Lett.208B, 491 (1988)Google Scholar
  29. 17a.
    Castillejo, L., Jones, P.S.J., Jackson, A.D., Verbaarschot, J.J.M., Jackson, A.: Dense Skyrmion systems. Nucl. Phys.A501, 801 (1989)CrossRefGoogle Scholar
  30. 18.
    Atiyah, M.F.: Geometry of Yang-Mills fields. Lezioni Fermiane. Pisa: Scuola Normale Superiore, 1979Google Scholar
  31. 19.
    Schwartz, A.: On regular solutions of euclidean Yang-Mills equations. Phys. Lett.67B, 172 (1977)Google Scholar
  32. 19a.
    Jackiw, R., Rebbi, C.: Degrees of freedom in pseudoparticle systems. Phys. Lett.67B, 189 (1977)Google Scholar
  33. 19b.
    Brown, L., Carlitz, R., Lee, C.: Massles excitations in pseudoparticle fields. Phys. Rev.D 16, 417 (1977)CrossRefGoogle Scholar
  34. 19c.
    Atiyah, M.F., Hitchin, N., Singer, I.: Deformations of instantons. Proc. Natl. Acad. Sci. USA74, 2662 (1977)Google Scholar
  35. 20.
    't Hooft, G.: UnpublishedGoogle Scholar
  36. 20a.
    Corrigan, E., Fairlie, D.B.: Scalar field theory and exact solutions to a classicalSU(2) gauge theory. Phys. Lett.67B, 69 (1977)Google Scholar
  37. 21.
    Jackiw, R., Nohl, C., Rebbi, C.: Conformal properties of pseudoparticle configurations. Phys. Rev.D15, 1642 (1977)Google Scholar
  38. 22.
    Hartshorne, R.: Stable vector bundles and instantons. Commun. Math. Phys.59, 1 (1978)CrossRefGoogle Scholar
  39. 23.
    Berger, M.: Geometry 2. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  40. 24.
    Verbaarschot, J.J.M., Walhout, T.S., Wambach, J., Wyld, H.W.: Symmetry and quantization of the two-Skyrmion system: the case of the deuteron. Nucl. Phys.A468, 520 (1987)CrossRefGoogle Scholar
  41. 25.
    Walhout, T.S.: Multiskyrmions as nuclei. Nucl. Phys.A531, 596 (1991)CrossRefGoogle Scholar
  42. 26.
    Hosaka, A., Griffies, S.M., Oka, M., Amado, R.D.: Two Skyrmion interaction for the Atiyah-Manton ansatz. Phys. Lett.251B, 1 (1990)Google Scholar
  43. 26a.
    Hosaka, A., Oka, M., Amado, R.D.: Skyrmions and their interactions using the Atiyah-Manton construction. Nucl. Phys.A530, 507 (1991)CrossRefMathSciNetGoogle Scholar
  44. 27.
    Crutchfield, W.Y., Snyderman, N., Brown, V.R.: The deuteron in the Skyrme model. Phys. Rev. Lett.68, 1660 (1992)CrossRefGoogle Scholar
  45. 28.
    Massey, W.S.: The quotient space of the complex projective plane under conjugation is a 4-sphere. Geom. Dedicata2, 371 (1973)CrossRefGoogle Scholar
  46. 28a.
    Kuiper, N.H.: The quotient space of ℂP 1 by complex conjugation is the 4-sphere. Math. Ann.208, 175 (1974)CrossRefGoogle Scholar
  47. 29.
    Sommerville, D.M.Y.: Non-Euclidean geometry (Chap. 9). London: Bell 1914Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • M. F. Atiyah
    • 1
  • N. S. Manton
    • 2
  1. 1.Trinity CollegeCambridgeUK
  2. 2.Department of Applied Mathematics and Theoreitcal PhysicsCambridgeUK

Personalised recommendations