Skip to main content
SpringerLink
Log in

Geodesic Incompleteness in the ℂP 1 Model on a Compact Riemann Surface

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

It is proved that the moduli space of static solutions of the ℂP 1 model on spacetime Σ ×∝, where Σ is any compact Riemann surface, is geodesically incomplete with respect to the metric induced by the kinetic energy functional. The geodesic approximation predicts, therefore, that lumps can collapse and form singularities in finite time in these models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Manton, N. S.: A remark on the scattering of BPS monopoles, Phys. Lett. B 110 (1982), 54–56.

    Article  Google Scholar 

  2. Speight, J. M.: Low-energy dynamics of a ℂP 1 lump on the sphere, J. Math. Phys. 36 (1995), 796–813.

    Article  Google Scholar 

  3. Speight, J. M.: Lump dynamics in the ℂP 1 model on the torus, Preprint hep-th/9707101.

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

    Google Scholar 

  5. Ward, R. S.: Slowly moving lumps in the ℂP 1 model in (2 + 1) dimensions, Phys. Lett. B 158 (1985), 424–428.

    Article  Google Scholar 

  6. Gallot, S., Hulin, D. and Lafontaine, J.: Riemannian Geometry, Springer-Verlag, Berlin, 1987, pp. 87–88.

    Google Scholar 

  7. Ibid. p. 95.

    Google Scholar 

  8. Atiyah, M. F. and Hitchin, N. J.: The Geometry and Dynamics of Magnetic Monopoles, Princeton University Press, Princeton, 1988; Samols, T. M.: Vortex scattering, Comm. Math. Phys. 145 (1992), 149-179; Stuart, D.: Dynamics of abelian Higgs vortices in the near Bogomolny regime, Comm. Math. Phys. 159 (1994), 51-91.

    Google Scholar 

  9. Zakrzewski, W. J. and Piette, B.: Shrinking of solitons in the (2 + 1)-dimensional S 2sigma model, Nonlinearity 9 (1996), 897–910.

    Article  Google Scholar 

  10. Cova, R. J. and Zakrzewski, W. J.: Soliton scattering in the O(3) model on a torus, preprint hep-th/9706166.

  11. Beardon, A. F.: A Primer on Riemann Surfaces, Cambridge University Press, Cambridge, 1984, p. 81.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, U.S.A

    L. A. Sadun & J. M. Speight

Authors
  1. L. A. Sadun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. M. Speight
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sadun, L.A., Speight, J.M. Geodesic Incompleteness in the ℂP 1 Model on a Compact Riemann Surface. Letters in Mathematical Physics 43, 329–334 (1998). https://doi.org/10.1023/A:1007433724535

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007433724535

Search

Navigation