Communications in Mathematical Physics

, Volume 150, Issue 3, pp 519–535 | Cite as

Chern-Simons solitons, Toda theories and the chiral model

  • Gerald Dunne
Article

Abstract

The two-dimensional self-dual Chern-Simons equations are equivalent to the conditions for static, zero-energy solutions of the (2+1)-dimensional gauged nonlinear Schrödinger equation with Chern-Simons matter-gauge dynamics. In this paper we classify all finite chargeSU(N) solutions by first transforming the self-dual Chern-Simons equations into the two-dimensional chiral model (or harmonic map) equations, and then using the Uhlenbeck-Wood classification of harmonic maps into the unitary groups. This construction also leads to a new relationship between theSU(N) Toda andSU(N) chiral model solutions.

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References

  1. 1.
    Bilal, A., Gervais, J.-L.: Extendedc=∞ Conformal Systems from Classical Toda Field Theories. Nucl. Phys.B314, 646–686 (1989)CrossRefGoogle Scholar
  2. 2.
    Din, A., Zakrzewski, W.: Properties of General ClassicalCP N−1 Solutions. Phys. Lett.95B, 419–422 (1980); General Classical Solutions in theCP N−1 Model. Nucl. Phys.B174, 397–406 (1980); Interpretation and Further Properties of General ClassicalCP N−1 Solutions. Nucl. Phys.B182, 151–157 (1981)Google Scholar
  3. 3.
    Donaldson, S.: Twisted Harmonic Maps and the Self-Duality Equations. Proc. Lond. Math. Soc.55, 127–131 (1987)Google Scholar
  4. 4.
    Dunne, G., Jackiw, R., Pi, S.-Y., Trugenberger, C.: Self-Dual Chern-Simons Solitons and Two-Dimensional Nonlinear Equations. Phys. Rev.D43, 1332–1345 (1991)CrossRefGoogle Scholar
  5. 5.
    Dunne, G.: Self-Duality and Chern-Simons Theories. In: Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics. Catto, S., Rocha, A. (eds.) New York: Baruch College (1991)Google Scholar
  6. 6.
    Forgács, P., Wipf, A., Balog, J., Fehér, F., O'Raifeartaigh, L.: Liouville and Toda Theories as Conformally Reduced WZNW Theories. Phys. Lett.B227, 214–220 (1989); Toda Theory andW Algebra from a Gauged WZNW Point of View. Ann. Phys. (NY)203, 76–136 (1990)CrossRefGoogle Scholar
  7. 7.
    Ganoulis, N., Goddard, P., Olive, D.: Self-Dual Monopoles and Toda Molecules. Nucl. Phys.B205, 601–636 (1982)CrossRefGoogle Scholar
  8. 8.
    Grossman, B.: Hierarchy of Soliton Solutions to the Gauged Nonlinear Schrödinger Equation on the Plane. Phys. Rev. Lett.65, 3230–3232 (1990)CrossRefGoogle Scholar
  9. 9.
    Hitchin, N.: The Self-Duality Equations on a Riemann Surface. Proc. Lond. Math. Soc.55, 59–126 (1987)Google Scholar
  10. 10.
    See e.g. Humphreys, J.: Introduction to Lie Algebras and Representation Theory. Berlin, Heidelberg, New York: Springer-Verlag 1990Google Scholar
  11. 11.
    Jackiw, R., Pi, S.-Y.: Soliton Solutions to the Gauged Non-Linear Schrödinger Equation on the Plane. Phys. Rev. Lett.64, 2969–2972 (1990); Classical and Quantum Non-Relativistic Chern-Simons Theory. Phys. Rev.D42, 3500–3513 (1990)CrossRefGoogle Scholar
  12. 12.
    Kostant, B.: The Solution to a Generalized Toda Lattice and Representation Theory. Adv. Math.34, 195–338 (1979)CrossRefGoogle Scholar
  13. 13.
    Leznov, A., Saveliev, M.: Representation of Zero Curvature for the System of Non-linear Partial Differential Equations\(x_{\alpha ,z\bar z} = exp(kx)_\alpha \) and its Integrability. Lett. Math. Phys.3, 389–494 (1979); Representation Theory and Integration of Nonlinear Spherically Symmetric Equations of Gauge Theories. Commun. Math. Phys.74, 111–118 (1980)CrossRefGoogle Scholar
  14. 14.
    Liouville, J.: Sur l'équation aux différences partielles\(\frac{{d^2 }}{{dudv}}log \lambda \pm \frac{\lambda }{{2a^2 }} = 0\). J. Math. Pures, Appl.18, 71–72 (1853)Google Scholar
  15. 15.
    Mansfield, P.: Solution of Toda Systems. Nucl. Phys.B208, 277–300 (1982)CrossRefGoogle Scholar
  16. 16.
    Mikhailov, A., Olshanetsky, M., Perelomov, A.: Two-Dimensional Generalized Toda Lattice. Commun. Math. Phys.79, 473–488 (1981)CrossRefGoogle Scholar
  17. 17.
    O'Raifeartaigh, L., Ruelle, P., Tsutsui, I., Wipf, A.:W-Algebras for Generalized Toda Theories. Commun. Math. Phys.143, 333–354 (1992)CrossRefGoogle Scholar
  18. 18.
    Piette, B., Zakrzewski, W.: General Solutions of theU(3) andU(4) Chiral Sigma Models in Two Dimensions. Nucl. Phys.B300, 207–222 (1988); Some Classes of General Solutions of theU(N) Chiral σ Models in Two Dimensions. J. Math. Phys.30, 2233–2237 (1989)CrossRefGoogle Scholar
  19. 19.
    Piette, B., Zait, R., Zakrzewski, W.: Solutions of theU(N) Sigma Models with the Wess-Zumino Term. Zeit. für Phys.C39, 359–364 (1988)CrossRefGoogle Scholar
  20. 20.
    Sasaki, R.: General Classical Solutions of the Complex Grassmannian andCP N−1 Sigma Models. Phys. Lett.130B, 69–72 (1983)Google Scholar
  21. 21.
    Toda, M.: Studies of a Nonlinear Lattice. Phys. Rep.8, 1–125 (1975)CrossRefGoogle Scholar
  22. 22.
    Uhlenbeck, K.: Harmonic Maps into Lie Groups (Classical Solutions of the Chiral Model). J. Diff. Geom.30, 1–50 (1989)Google Scholar
  23. 23.
    Valli, G.: On the Energy Spectrum of Harmonic Two-Spheres in Unitary Groups. Topology27, 129–136 (1988)CrossRefGoogle Scholar
  24. 24.
    Ward, R.S.: Integrable and Solvable Systems and Relations Among Them. Phil. Trans. Roy. Soc. Lond.A315, 451–457 (1985)Google Scholar
  25. 25.
    Ward, R.S.: Classical Solutions of the Chiral Model, Unitons and Holomorphic Vector Bundles. Commun. Math. Phys.128, 319–332 (1990)CrossRefGoogle Scholar
  26. 26.
    Wood, J.C.: Explicit Construction and Parametrization of Harmonic Two-Spheres in the Unitary Group. Proc. Lond. Math. Soc.58, 608–624 (1989)Google Scholar
  27. 27.
    Yoon, Y.: Zero Modes of the Non-Relativistic Self-Dual Chern-Simons Vortices on the Toda Backgrounds. Ann. Phys. (NY)211, 316–333 (1991)CrossRefGoogle Scholar
  28. 28.
    Zakrzewski, W.: Low Dimensional Sigma Models. London: Adam Hilger 1989Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Gerald Dunne
    • 1
  1. 1.Department of Mathematics and Center for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Physics DepartmentUniversity of ConnecticutStorrsUSA

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