Communications in Mathematical Physics

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

A symmetric family of Yang-Mills fields

  • Lorenzo Sadun


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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Modulus Space 


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  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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