Communications in Mathematical Physics

, Volume 158, Issue 2, pp 241–266 | Cite as

Monopoles, braid groups, and the dirac operator

  • Ralph L. Cohen
  • John D. S. Jones


Using the relation between the space of rational functions on ℂ, the space ofSU(2)-monopoles on ℝ3, and the classifying space of the braid group, see [10], we show how the index bundle of the family of real Dirac operators coupled toSU(2)-monopoles can be described using permutation representations of Artin's braid groups. We also show how this implies the existence of a pair consisting of a gauge fieldA and a Higgs field Φ on ℝ3 whose corresponding Dirac equation has an arbitrarily large dimensional space of solutions.


Neural Network Statistical Physic Complex System Rational Function 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, J.F.: Lectures on generalised cohomology. Category Theory, Homology Theory and their Applications, Lectures Notes in Mathematics, vol.99. Berlin, Heidelberg, New York: Springer 1969, pp. 1–138Google Scholar
  2. 2.
    Adams, J.F.: Infinite Loop Spaces. Annals of Mathematics Studies, vol.90, Princeton, New Jersey: Princeton Univ. Press 1978.Google Scholar
  3. 3.
    Adams, J.F., Walker, G.: On complex Stiefel manifolds. Math. Proc. Camb. Phil. Soc.61, 81–103 (1965).Google Scholar
  4. 4.
    Atiyah, M.F.:K-Theory. New York: Benjamin 1967Google Scholar
  5. 5.
    Atiyah, M.F.: Bott periodicity and the index of elliptic operators. Quart. J. Math.19, 113–140 (1968)Google Scholar
  6. 6.
    Atiyah, M.F., Hitchin, N.: The Geometry and Dynamics of Magnetic Monopoles. Princeton, NJ: Princeton Univ. Press 1988Google Scholar
  7. 7.
    Atiyah, M.F., Jones, J.D.S.: Topological aspects of Yang-Mills theory. Commun. Math. Phys.61, 97–118 (1978)Google Scholar
  8. 8.
    Boyer, C., Mann, B.: Monopoles, non-linear σ-models, and two fold loop spaces. Commun. Math. Phys.115, 571–594 (1988)Google Scholar
  9. 9.
    Brown, E.H., Peterson, F.P.: On the stable decomposition ofΩ 2 S r+2. Trans. A.M.S.243, 287–298 (1978)Google Scholar
  10. 10.
    Cohen, F.R., Cohen, R.L., Mann, B.M., Milgram, R.J.: The topology of rational functions and divisors of surfaces. Acta Math.166, 163–221 (1991)Google Scholar
  11. 11.
    Cohen, F.R., Cohen, R.L., Mann, B.M., Milgram, R.J.: The homotopy type of rational functions. To appearGoogle Scholar
  12. 12.
    Cohen, F.R., Mahowald, M., Milgram, R.J.: The stable decomposition of the double loop space of a sphere, Algebraic and geometric Topology, Proc. Symp. Pure Math., vol. XXXII (2), Providence, Rhode Island: A.M.S. 1978, pp. 225–228Google Scholar
  13. 13.
    Cohen, F.R., May, J.P., Taylor, L.R.: Splitting of certain spacesC(X). Math. Proc. Camb. Phil. Soc.84, 45–496 (1978)Google Scholar
  14. 14.
    Cohen, R.L.: Stable proofs of stable splittings. Math. Proc. Camb. Phil. Soc.88, 149–151 (1980)Google Scholar
  15. 15.
    Cohen, R.L., Jones, J.D.S.: Representations of braid groups and operators coupled to monopoles. Geometry of low-dimensional manifolds: 1, LMS Lecture Note Series150, Cambridge: Cambridge University Press 1990, pp. 191–205Google Scholar
  16. 16.
    Cohen, R.L., Jones, J.D.S., Mahowald, M.E.: The Kervaire invariant of immersions. Invent. Math.79, 95–123 (1985)Google Scholar
  17. 17.
    Cohen, R.L., Shimamoto, D.H.: Rational functions, labelled configurations, and Hilbert schemes. J. Lond. Math. Soc.43, 509–528 (1991)Google Scholar
  18. 18.
    Donaldson, S.K.: Nahm's equations and the classification of monopoles. Commun. Math. Phys.96, 387–407 (1984)Google Scholar
  19. 19.
    Jaffe, A., Taubes, C.H.: Vortices and monopoles. Boston: Birkhäuser 1980Google Scholar
  20. 20.
    May, J.P.: The geometry of iterated loop spaces. Lecture Notes in Mathematics, vol.271. Berlin, Heidelberg, New York: Springer 1972Google Scholar
  21. 21.
    Milgram, R.J.: Iterated loop spaces. Ann. of Math.84, 386–403 (1966)Google Scholar
  22. 22.
    Segal, G.B.: Configuration spaces and iterated loop spaces. Invent. Math.21, 213–221 (1973)Google Scholar
  23. 23.
    Segal, G.B.: The topology of spaces of rational functions. Acta. Math.143, 39–72 (1979)Google Scholar
  24. 24.
    Taubes, C.H.: Monopoles and maps fromS 2 toS 2; the topology of the configuration space. Commun. Math. Phys.95, 345–391 (1984)Google Scholar
  25. 25.
    Taubes, C.H.: Min-max theory for the Yang-Mills-Higgs equations. Commun. Math. Phys.97, 473–540 (1985)Google Scholar
  26. 26.
    Taubes, C.H.: A note and erratum concerning “Min-max theory for the Yang-Mills-Higgs equation”. Commun. Math. Phys.122, 609–613 (1989)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Ralph L. Cohen
    • 1
  • John D. S. Jones
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Mathematics InstituteUniversity of WarwickCoventryEngland

Personalised recommendations