Advertisement

Communications in Mathematical Physics

, Volume 168, Issue 2, pp 321–336 | Cite as

Vortex condensation in the Chern-Simons Higgs model: An existence theorem

  • Luis A. Caffarelli
  • Yisong Yang
Article

Abstract

It is shown that there is a critical value of the Chern-Simons coupling parameter so that, below the value, there exists self-dual doubly periodic vortex solutions, and, above the value, the vortices are absent. Solutions of such a nature indicate the existence of dyon condensates carrying quantized electric and magnetic charges.

Keywords

Vortex Neural Network Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ab] Abrikosov, A.A.: On the magnetic properties of superconductors of the second group, Sov. Phys. JETP5, 1174–1182 (1957)Google Scholar
  2. [A1] Aubin, T.: Nonlinear Analysis on Manifolds: Monge-Ampére Equations. Springer: Berlin Heidelberg, New York, 1982Google Scholar
  3. [A2] Aubin, T.: Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de courburne scalaire. J. Funct. Anal.32, 148–174 (1979)Google Scholar
  4. [Av] Aviles, P.: Conformal complete metrics with prescribed non-negative Gaussian curvature inR 2. Invent. Math.83, 519–544 (1986)Google Scholar
  5. [CH] Chakravarty, S., Hosotani, Y.: Anyon model on a cylinder Phys. Rev. D44, 441–451 (1991)Google Scholar
  6. [CY] Chang, S.Y.A., Yang, P.C.: Prescribing Gaussian curvature onS 2. Acta Math.159, 215–259 (1987)Google Scholar
  7. [CHMcY] Chen, X., Hastings, S., McLeod J.B., Yang, Y.: A nonlinear elliptic equation arising from gauge field theory and cosmology. Proc. R. Soc. London, Series A,446, 453–478 (1994)Google Scholar
  8. [FM1] Fröhlich, J., Marchetti, P.: Bosonization, topological solitons and fractional charges in two-dimensional quantum field theory. Commun. Math. Phys.116, 127–173 (1988)Google Scholar
  9. [FM2] Fröhlich, J., Marchetti, P.: Quantum field theories of vortices and anyons. Commun. Math. Phys.121, 177–223 (1989)Google Scholar
  10. [HKP] Hong, J., Kim, Y., Pac, P.Y.: Multivortex solutions of the Abelian Chern-Simons theory. Phys. Rev. Lett.64, 2230–2233 (1990)Google Scholar
  11. [H] Hosotani, Y.: Gauge invariance in Chern-Simons theory on a torus. Phys. Rev. Lett.62, 2785–2788 (1989)Google Scholar
  12. [IL] Iengo, R., Lechner, K.: Quantum mechanics of anyons on a torus. Nucl. Phys.B 346, 551–575 (1990)Google Scholar
  13. [JLW] Jackiw, R., Lee, K., Weinberg, E.J.: Self-dual Chern-Simons solitions. Phys. Rev.D 42, 3488–3499 (1990)Google Scholar
  14. [JW] Jackiw, R., Weinberg, E.J.: Self-dual Chern-Simons vortices. Phys. Rev. Lett.64, 2234–2237 (1990)Google Scholar
  15. [JT] Jaffe, A., Taubes, C.H.: Vortices and Monopoles Boston: Birkhäuser, 1980Google Scholar
  16. [KW1] Kazdan, J.L., Warner, F.W.: Integrability conditions for Δu=k−Ke 2u with applications to Riemannian geometry. Bull. Am. Math. Soc.77, 819–823 (1971)Google Scholar
  17. [KW2] Kazdan, J.L., Warner, F.W.: Curvature functions for compact 2-manifolds. Ann. Math.99, 14–47 (1974)Google Scholar
  18. [KW3] Kazdan, J.L., Warner, F.W.: Curvature functions for open 2-manifold. Ann. Math.99, 203–219 (1974)Google Scholar
  19. [Mc] McOwen, R.C.: Conformal metrics inR 2 with prescribed Gaussian curvature and positive total curvature. Indiana U. Math. J.34, 97–104 (1985)Google Scholar
  20. [Ni] Ni, W.-M.: On the elliptic equation Δu+K(x)e 2u=0 and conformal metrics with prescribed Gaussian curvature. Invent. Math.66, 343–352 (1982)Google Scholar
  21. [O] Olesen, P.: Solition condensation in some self-dual Chern-Simons theories. Phys. Lett.B 265, 361–365 (1991); Erratum,267, 541 (1991)Google Scholar
  22. [P] Polychronakos, A.: Abelian Chern-Simons theories in 2+1 dimensions. Ann. Phys.203, 231–254 (1990)Google Scholar
  23. [RSS] Randjbar, S., Salam, A., Strathdee, J.A.: Anyons and Chern-Simons theory on compact spaces of finite genus. Phys. Lett.B 240, 121–126 (1990)Google Scholar
  24. ['t H] Hooft, G. 't.: A property of electric and magnetic flux in nonabelian gauge theories. Nucl. Phys.B 153, 141–160 (1979)Google Scholar
  25. [SY1] Spruck, J., Yang, Y.: The existence of non-topological solitons in the self-dual Chern-Simons Theory. Commun. Math. Phys.149, 361–376 (1992)Google Scholar
  26. [SY2] Spruck, J., Yang, Y.: Topological solutions in the self-dual Chern-Simons theory: Existence and approximation. Ann. l'I. H. P.-Anal. non Linéaire, to appearGoogle Scholar
  27. [T1] Taubes, C.H.: ArbitraryN-vortex solutions to the first order Ginzburg-Landau equations. Commun. Math. Phys.72, 277–292 (1980)Google Scholar
  28. [T2] Taubes, C.H.: On the equivalence of the first and second order equations for gauge theories. Commun. Math. Phys.75, 207–227 (1980)Google Scholar
  29. [WY] Wang, S., Yang, Y.: Abrikosov's vortices in the critical coupling. SIAM J. Math. Anal.23, 1125–1140 (1992)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Luis A. Caffarelli
    • 1
  • Yisong Yang
    • 1
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA

Personalised recommendations