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

SIGMA 5 (2009), 065, 22 pages      arXiv:0811.3056      https://doi.org/10.3842/SIGMA.2009.065
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Monopoles and Modifications of Bundles over Elliptic Curves

Andrey M. Levin a, b, Mikhail A. Olshanetsky a, c and Andrei V. Zotov a, c
a) Max Planck Institute of Mathematics, Bonn, Germany
b) Institute of Oceanology, Moscow, Russia
c) Institute of Theoretical and Experimental Physics, Moscow, Russia

Received November 20, 2008, in final form June 10, 2009; Published online June 25, 2009

Abstract
Modifications of bundles over complex curves is an operation that allows one to construct a new bundle from a given one. Modifications can change a topological type of bundle. We describe the topological type in terms of the characteristic classes of the bundle. Being applied to the Higgs bundles modifications establish an equivalence between different classical integrable systems. Following Kapustin and Witten we define the modifications in terms of monopole solutions of the Bogomolny equation. We find the Dirac monopole solution in the case R × (elliptic curve). This solution is a three-dimensional generalization of the Kronecker series. We give two representations for this solution and derive a functional equation for it generalizing the Kronecker results. We use it to define Abelian modifications for bundles of arbitrary rank. We also describe non-Abelian modifications in terms of theta-functions with characteristic.

Key words: integrable systems; field theory; characteristic classes.

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References

  1. Levin A., Olshanetsky M., Zotov A., Hitchin systems - symplectic hecke correspondence and two-dimensional version, Comm. Math. Phys. 236 (2003), 93-133, nlin.SI/0110045.
  2. Hitchin N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  3. Levin A., Olshanetsky M., Zotov A., Painlevè VI, rigid tops and reflection equation, Comm. Math. Phys. 268 (2006), 67-103, math.QA/0508058.
  4. Krichever I., Vector bundles and Lax equations on algebraic curves, Comm. Math. Phys. 229 (2002), 229-269, hep-th/0108110.
  5. Baxter R., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I, Ann. Physics 76 (1973), 48-71.
  6. Date E., Jimbo M., Miwa T., Okado M., Fusion of the eight vertex SOS model, Lett. Math. Phys. 12 (1986), 209-215.
  7. Felder G., Conformal field theory and integrable systems associated to elliptic curves, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zurich, 1994), Birkhauser, Basel, 1995, 1247-1255, hep-th/9609153.
  8. Belavin A., Dynamical symmetry of integrable system, Nuclear Phys. B 180 (1981), 189-200.
  9. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236, hep-th/0604151.
  10. Levin A., Zotov A., Integrable systems of interacting elliptic tops, Teoret. Mat. Fiz. 146 (2006), 55-64 (English transl.: Theoret. and Math. Phys. 146 (2006), 45-52).
  11. Flaschka H., Newell A.C., Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), 65-116.
  12. Krichever I., The τ-function of the universal Whitham hierarchy, matrix models and topological field theories, Comm. Pure Appl. Math. 47 (1994), 437-475, hep-th/9205110.
  13. Levin A., Olshanetsky M., Hierarchies of isomonodromic deformations and Hitchin systems, in Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 191, Amer. Math. Soc., Providence, RI, 1999, 223-262.
  14. Arinkin D., On λ-connections on a curve where λ is a formal parameter, Math. Res. Lett. 12 (2005), 551-565.
  15. Tong D., Quantum vortex strings: a review, arXiv:0809.5060.
    Shifman M., Yung A., Supersymmetric solitons and how they help us understand non-Abelian gauge theories, hep-th/0703267.
    Gorsky A., Shifman M., Yung A., N=1 supersymmetric quantum chromodynamics: how confined non-Abelian monopoles emerge from quark condensation, Phys. Rev. D 75 (2007), 065032, 16 pages, hep-th/0701040.
  16. Popov A., Bounces/dyons in the plane wave matrix model and SU(N) Yang-Mills theory, arXiv:0804.3845.
  17. Weyl A., Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 88, Springer-Verlag, Berlin - New York, 1976.
  18. Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevsky L.P., Theory of solitons. The method of the inverse scattering problem, Nauka, Moscow, 1980 (in Russian).
  19. Ward R., Integrable systems and twistors, in Integrable Systems (Oxford, 1997), Oxf. Grad. Texts Math., Vol. 4, Oxford Univ. Press, New York, 1999, 121-134.
  20. Vinberg E.B., Onishchik A.L., Seminar on Lie groups and algebraic groups, Moscow, 1988 (English transl.: Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990).
  21. Bourbaki N., Lie groups and Lie algebras, Chapters 4-6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002.

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