Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 043, 9 pages      hep-th/0703108      https://doi.org/10.3842/SIGMA.2007.043
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

A Journey Between Two Curves

Sergey A. Cherkis a, b
a) School of Mathematics, Trinity College Dublin, Ireland
b) Hamilton Mathematics Institute, TCD, Dublin, Ireland

Received October 31, 2006, in final form February 25, 2007; Published online March 11, 2007

Abstract
A typical solution of an integrable system is described in terms of a holomorphic curve and a line bundle over it. The curve provides the action variables while the time evolution is a linear flow on the curve's Jacobian. Even though the system of Nahm equations is closely related to the Hitchin system, the curves appearing in these two cases have very different nature. The former can be described in terms of some classical scattering problem while the latter provides a solution to some Seiberg-Witten gauge theory. This note identifies the setup in which one can formulate the question of relating the two curves.

Key words: Hitchin system; Nahm equations; monopoles; Seiberg-Witten theory.

pdf (220 kb)   ps (145 kb)   tex (14 kb)

References

  1. Nahm W., A simple formalism for the BPS monopole, Phys. Lett. B 90 (1980), 413-414.
  2. Hitchin N., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  3. Hitchin N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  4. Donagi R., Witten E., Supersymmetric Yang-Mills theory and integrable systems, Nuclear Phys. B 460 (1996), 299-334, hep-th/9510101.
  5. Kapustin A., Sethi S., The Higgs branch of impurity theories, Adv. Theor. Math. Phys. 2 (1998), 571-591, hep-th/9804027.
  6. Seiberg N., Witten E., Gauge dynamics and compactification to three dimensions, hep-th/9607163.
  7. Klemm A., Lerche W., Mayr P., Vafa C., Warner N.P., Self-dual strings and N = 2 supersymmetric field theory, Nuclear Phys. B 477 (1996), 746-766, hep-th/9604034.
  8. Witten E., Solutions of four-dimensional field theories via M-theory, Nuclear Phys. B 500 (1997) 3-42, hep-th/9703166.
  9. Chalmers G., Hanany A., Three dimensional gauge theories and monopoles, Nuclear Phys. B 489 (1997), 223-244, hep-th/9608105.
  10. Hanany A., Witten E., Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nuclear Phys. B 492 (1997), 152-190, hep-th/9611230.
  11. Katz S.H., Klemm A., Vafa C., Geometric engineering of quantum field theories, Nuclear Phys. B 497 (1997), 173-195, hep-th/9609239.
  12. Hitchin N., Brackets, forms and invariant functionals, math.DG/0508618.
  13. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, hep-th/0604151.
  14. Hanany A., Tong D., On monopoles and domain walls, Comm. Math. Phys. 266 (2006), 647-663, hep-th/0507140.
  15. Gerasimov A.A., Shatashvili S.L., Higgs bundles, gauge theories and quantum groups, hep-th/0609024.
  16. Donaldson S.K., Kronheimer P.B., The geomerty of four-manifolds, Oxford University Press, 1990.
  17. Jardim M., A survey on Nahm transform, J. Geom. Phys. 52 (2004), 313-327, math.DG/0309305.
  18. Nahm W., Self-dual monopoles and calorons, presented at 12th Colloq. on Group Theoretical Methods in Physics (September 5-10, 1983, Trieste, Italy).
    Nahm W., Self-dual monopoles and calorons, Physics, Vol. 201, Springer, New York, 1984.
  19. Nahm W., All selfdual multi-monopoles for arbitrary gauge groups, presented at Int. Summer Inst. on Theoretical Physics (August 31 - September 11, 1981, Freiburg, West Germany).
  20. Jardim M., Construction of doubly-periodic instantons, Comm. Math. Phys. 216 (2001), 1-15, math.DG/9909069.
  21. Seiberg N., Witten E., Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nuclear Phys. B 426 (1994), 19-52, Erratum, Nuclear Phys. B 430 (1994), 485-486, hep-th/9407087.
  22. Seiberg N., Witten E., Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nuclear Phys. B 431 (1994), 484-550, hep-th/9408099.
  23. Atiyah M.F., Hitchin N., The geometry and dynamics of magnetic monopoles, Princeton University Press, 1988.
  24. Diaconescu D.E., D-branes, monopoles and Nahm equations, Nuclear Phys. B 503 (1997), 220-238, hep-th/9608163.
  25. Cherkis S.A., Kapustin A., Singular monopoles and supersymmetric gauge theories in three dimensions, Nuclear Phys. B 525 (1998), 215-234, hep-th/9711145.
  26. Cherkis S.A., Kapustin A., Singular monopoles and gravitational instantons, Comm. Math. Phys. 203 (1999), 713-728, hep-th/9803160.
  27. Cherkis S.A., Hitchin N.J., Gravitational instantons of type D(k), Comm. Math. Phys. 260 (2005), 299-317, hep-th/0310084.
  28. Cherkis S.A., Kapustin A., Periodic monopoles with singularities and N = 2 super-QCD, Comm. Math. Phys. 234 (2003), 1-35, hep-th/0011081.
  29. Cherkis S.A., Kapustin A., Nahm transform for periodic monopoles and N = 2 super Yang-Mills theory, Comm. Math. Phys. 218 (2001), 333-371, hep-th/0006050.
  30. Howe P.S., Lambert N.D., West P.C., Classical M-fivebrane dynamics and quantum N = 2 Yang-Mills, Phys. Lett. B 418 (1998), 85-90, hep-th/9710034.
  31. Hitchin N.J., Monopoles and geodesics, Comm. Math. Phys. 83 (1982), 579-602.
  32. Hitchin N.J., On the construction of monopoles, Comm. Math. Phys. 89 (1983), 145-190.
  33. Jardim M., Nahm transform and spectral curves for doubly-periodic instantons, Comm. Math. Phys. 225 (2002) 639-668.
  34. Donaldson S.K., Nahm's equations and the classification of monopoles, Comm. Math. Phys. 96 (1984), 387-407.
  35. Friedman R., Morgan J., Witten E., Vector bundles and F theory, Comm. Math. Phys. 187 (1997), 679-743, hep-th/9701162.
  36. Friedman R., Morgan J.W., Witten E., Vector bundles over elliptic fibrations, alg-geom/9709029.
  37. Simpson C.T., Higgs bundles and local systems, IHES Publ. Math. 75 (1992), 5-95.
    Simpson C.T., The Hodge filtration on nonabelian cohomology, in Algebraic Geometry (1995, Santa Cruz), Proc. Symp. Pure Math., Vol. 62, Part 2, Amer. Math. Soc., Providence, 1997, 217-281, alg-geom/9604005.

Previous article   Next article   Contents of Volume 3 (2007)