Skip to main content
SpringerLink
Log in

On the equivalence of the first and second order equations for gauge theories

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

Abstract

We prove that every solution to the SU(2) Yang-Mills equations, invariant under the lifting to the principle bundle of the action of the group, O(3), of rotations about a fixed line in ℝ4, with locally bounded and globally square integrable curvature is either self-dual or anti-self dual. In other words we prove, under the above assumptions, that every critical point of the Yang-Mills functional is a global minimum.

We prove also that every finite extremal of the Ginzburg-Landau action functional on ℝ2, with the coupling constant equal to one, is a solution to the first order Ginzburg-Landau equations. The relationship between the Ginzburg-Landau equations and the O(3) symmetric, SU(2) Yang-Mills equations on ℝ2 ×S 2 is established.

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. Atiyah, M. F., Drinfeld, V. A., Hatchin, N. J., Mannin, Yu. I.,: Phys. Lett.65A, 185 (1978) Drinfeld, V. A., Manin, Yu. I.: Commun. Math. Phys.63, 177 (1978)

    Google Scholar 

  2. A proof that local minima of the Yang-Mills action are either self or anti-self dual is given in: Bourguignon, J. P., Lawson, H. B., Simons, J.: Proc. Nat. Acad. Sci., USA,76, 1550 (1979). A proof that there are no non-self dual solutions in an open neighborhood of a self-dual solution is given above and in: Flume: Phys. Lett.76B, 593 (1978)

    Google Scholar 

  3. Ginzburg, V. L., Landau, L. D.: Zh. Eksp. Teor. Fiz.20, 1064 (1950)

    Google Scholar 

  4. Weinberg, E.: Phys. Rev.D19, 10, 3008 (1979)

    Google Scholar 

  5. Bogomol'nyi, E. B.: Sov. J. Nucl. Phys.24, 449 (1976)

    Google Scholar 

  6. Taubes, C. H.: Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations. Commun. Math. Phys. to appear

  7. Witten, E.: Phys. Rev. Lett.38, 121 (1977)

    Google Scholar 

  8. Forgacs, P., Manton, N.: Space time symmetries in gauge theories. Commun. Math. Phys., to appear

  9. Palais, R.: Foundations of global nonlinear analysis. New York: Benjamin 1968

    Google Scholar 

  10. Adams, R.: Sobolev spaces. New York: Academic Press 1975

    Google Scholar 

  11. Gilbarg, D., Trudinger, S.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  12. Morrey, C.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966

    Google Scholar 

  13. Uhlenbeck, K.: Removable singularities in Yang-Mills fields, Preprint,

Download references

Author information

Authors and Affiliations

  1. Lyman Laboratory of Physics, Harvard University, 02138, Cambridge, MA, USA

    Clifford Henry Taubes (Harvard University GSAS Fellow)

Authors
  1. Clifford Henry Taubes
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

This work supported in part through funds provided by the National Science Foundation under Grant PHY 79-16812.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Taubes, C.H. On the equivalence of the first and second order equations for gauge theories. Commun.Math. Phys. 75, 207–227 (1980). https://doi.org/10.1007/BF01212709

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01212709

Keywords

Search

Navigation