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Communications in Mathematical Physics

, Volume 163, Issue 2, pp 257–291 | Cite as

A symmetric family of Yang-Mills fields

  • Lorenzo Sadun
Article

Abstract

We examine a family of finite energySO(3) Yang-Mills connections overS4, indexed by two real parameters. This family includes both smooth connections (when both parameters are odd integers), and connections with a holonomy singularity around 1 or 2 copies ofRP2. These singular YM connections interpolate between the smooth solutions. Depending on the parameters, the curvature may be self-dual, anti-self-dual, or neither. For the (anti)self-dual connections, we compute the formal dimension of the moduli space. For the non-self-dual connections we examine the second variation of the Yang-Mills functional, and count the negative and zero eigenvalues. Each component of the non-self-dual moduli space appears to consist only of conformal copies of a single solution.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Modulus Space 
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.

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References

  1. [ADHM] Atiyah, M.F., Drinfeld, V.G., Hitchin, N.J., Manin, Y.I.: Construction of Instantons. Phys. Lett.65A, 185–187 (1978)Google Scholar
  2. [AJ] Atiyah, M.F., Jones, J.D.S.: Topological aspects of Yang-Mills theory. Commun. Math. Phys.61, 97 (1978)Google Scholar
  3. [BHMM1] Boyer, C.P., Hurtubise, J.C., Mann, B.M., Milgram, R.J.: The Atiyah-Jones conjecture. Bull. Am. Math. Soc.26, 317–321 (1992)Google Scholar
  4. [BHMM2] Boyer, C.P., Hurtubise, J.C., Mann, B.M., Milgram, R.J.: The topology of instanton moduli spaces I: The Atiyah-Jones conjecture. Ann. Math.137, 561–609 (1993)Google Scholar
  5. [BoMo] Bor, G., Montgomery, R.: SO(3) Invariant Yang-Mills Fields Which Are Not Self-Dual. In: Harnad, J., Marsden, J.E. (eds.): Hamiltonian Systems, Transformation Groups, and Spectral Transform Methods. Proceedings, Montreal, 1989, Montreal: Les publications CRM, 1990Google Scholar
  6. [Bor] Bor, G.: Yang-Mills fields which are not Self-Dual. Commun. Math. Phys.145, 393–410 (1992)Google Scholar
  7. [BoSe] Bor, G., Segert, J.: Rational solutions of the quadrupole self-duality equation. Preprint, 1993Google Scholar
  8. [DK] Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. Oxford: Oxford University Press, 1990Google Scholar
  9. [FHP1] Forgacs, P., Horvath, Z., Palla, L.: An exact fractionally charged self-dual solution. Phys. Rev. Lett.46, 392 (1981)Google Scholar
  10. [FHP2] Forgacs, P., Horvath, Z., Palla, L.: One Can Have Noninteger Topological Charge. Z. Phys. C-Particles and Fields12, 359–360 (1982)Google Scholar
  11. [K] Kronheimer, P.B.: Embedded surfaces in 4-manifolds. Proceedings of the International Congress of mathematicians (Kyoto 1990), Tokyo Berlin, 1991Google Scholar
  12. [KM] Kronheimer, P.B., Mrowka, T.S.: Gauge theory for embedded surfaces I. Topology32, 773–826 (1992)Google Scholar
  13. [Pa] Parker, T.: Non-minimal Yang-Mills Fields and Dynamics. Invent. Math.107, 397–420 (1992)Google Scholar
  14. [R1] Råde, J.: Singular Yang-Mills Fields. Local theory I. J. reine angew. Math. (in press)Google Scholar
  15. [R2] Råde, J.: Singular Yang-Mills Fields. Local theory II. J. reine angew. Math. (in press)Google Scholar
  16. [SS1] Sadun, L., Segert, J.: Non-Self-Dual Yang-Mills connections with nonzero Chern number. Bull. Am. Math. Soc.24, 163–170 (1991)Google Scholar
  17. [SS2] Sadun, L., Segert, J.: Non-Self-Dual Yang-Mills connections with Quadrupole Symmetry. Commun. Math. Phys.145, 363–391 (1992)Google Scholar
  18. [SS3] Sadun, L., Segert, J.: Stationary points of the Yang-Mills action. Commun. Pure Appl. Math.45, 461–484 (1992)Google Scholar
  19. [SiSi1] Sibner, L.M., Sibner, R.J.: Singular Soblev Connections with Holonomy. Bull. Am. Math. Soc.19, 471–473 (1988)Google Scholar
  20. [SiSi2] Sibner, L.M., Sibner, R.J.: Classification of Singular Sobolev Connections by their Holonomy. Commun. Math. Phys.144, 337–350 (1992)Google Scholar
  21. [SSU] Sibner, L.M., Sibner, R.J., Uhlenbeck, K.: Solutions to Yang-Mills Equations which are not Self-Dual. Proc. Natl. Acad. Sci. USA86, 8610–8613 (1989)Google Scholar
  22. [T1] Taubes, C.H.: Stability in Yang-Mills theories. Comm. Math. Phys.91, 235–263 (1983)Google Scholar
  23. [T2] Taubes, C.H.: A framework for Morse theory for the Yang-Mills functional. Invent. Math.94, 327–402 (1988)Google Scholar
  24. [Ur] Urakawa, H.: Equivariant Theory of Yang-Mills Connections over Riemannian Manifolds of Cohomogeneity One. Indiana Univ. Math. J.37, 753–788 (1988)Google Scholar
  25. [W] Hong-Yu Wang: The existence of non-minimal solutions to the Yang-Mills equation with groupSU(2) onS 2 ×S 2 andS 1 ×S 3. J. Diff. Geom.34, 701–767 (1991)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Lorenzo Sadun
    • 1
  1. 1.Department of MathematicsUniversity of TexasAustinUSA

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