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Communications in Mathematical Physics

, Volume 280, Issue 2, pp 315–349 | Cite as

Calorons, Nahm’s Equations on S 1 and Bundles over \({\mathbb{P}^{1} \times \mathbb{P}^{1}}\)

  • Benoit Charbonneau
  • Jacques Hurtubise
Article

Abstract

The moduli space of solutions to Nahm’s equations of rank (k, k + j) on the circle, and hence, of SU(2) calorons of charge (k, j), is shown to be equivalent to the moduli of holomorphic rank 2 bundles on \({\mathbb{P}^{1} \times \mathbb{P}^{1}}\) trivialized at infinity (\({\{\infty\} \times \mathbb{P}^{1} \cup \mathbb{P}^{1} \times \{\infty\}}\)) with c 2 = k and equipped with a flag of degree j along \({\mathbb{P}^1 \times \{0\}}\). An explicit matrix description of these spaces is given by a monad construction.

Keywords

Modulus Space Normal Form Gauge Group Exact Sequence Vector Bundle 
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. 1.
    Atiyah, M.F.: Instantons in two and four dimensions. Commun. Math. Phys. 93(4), 437–451 (1984)zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Atiyah, M.F., Drinfel’d, V.G., Hitchin, N.J., Manin, Yu.I.: Construction of instantons. Phys. Lett. A 65(3), 185–187 (1978)CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Bruckmann, F., Nógrádi, D., van Baal, P.: Constituent monopoles through the eyes of fermion zero-modes. Nucl. Phys. B 666(1–2), 197–229 (2003)zbMATHCrossRefADSGoogle Scholar
  4. 4.
    Bruckmann, F., Nógrádi, D., van Baal, P.: Higher charge calorons with non-trivial holonomy. Nucl. Phys. B 698(1–2), 233–254 (2004)zbMATHCrossRefADSGoogle Scholar
  5. 5.
    Bruckmann, F., van Baal, P.: Multi-caloron solutions. Nucl. Phys. B 645(1–2), 105–133 (2002)zbMATHCrossRefADSGoogle Scholar
  6. 6.
    Buchdahl, N.P.: Stable 2-bundles on Hirzebruch surfaces. Math. Z. 194(1), 143–152 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Charbonneau, B., Hurtubise, J.: The Nahm transform for calorons. to appear in Proceedings of Hitchin Birthday Conference, 25 pages, 2007. http://arxiv.org/abs/0705.2412, 2007
  8. 8.
    Corrigan, E., Goddard, P.: Construction of instanton and monopole solutions and reciprocity. Ann. Phys. 154(1), 253–279 (1984)zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Donaldson, S.K.: Instantons and geometric invariant theory. Commun. Math. Phys. 93(4), 453–460 (1984)zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Donaldson, S.K.: Nahm’s equations and the classification of monopoles. Commun. Math. Phys. 96(3), 387–407 (1984)zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Garland, H., Murray, M.K.: Kac–Moody monopoles and periodic instantons. Commun. Math. Phys. 120(2), 335–351 (1988)zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley-Interscience [John Wiley & Sons], 1978Google Scholar
  13. 13.
    Harnad, J., Shnider, S., Vinet, L.: The Yang–Mills system in compactified Minkowski space; invariance conditions and SU(2) invariant solutions. J. Math. Phys. 20(5), 931–942 (1979)zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Harnad, J., Vinet, L.: On the U(2) invariant solutions to Yang–Mills equations in compactified Minkowski space. Phys. Lett. B 76(5), 589–592 (1978)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Hartshorne, R.: Algebraic geometry. Springer-Verlag, New York (1977)zbMATHGoogle Scholar
  16. 16.
    Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55(1), 59–126 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hurtubise, J.: The classification of monopoles for the classical groups. Commun. Math. Phys. 120(4), 613–641 (1989)zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Jardim, M.: A survey on Nahm transform. J. Geom. Phys. 52(3), 313–327 (2004)zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Jarvis, S.: Euclidean monopoles and rational maps. Proc. London Math. Soc. 77(1), 170–192 (1998)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Kempf, G.R.: Algebraic varieties. Volume 172 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1993Google Scholar
  21. 21.
    Kraan, T.C.: Instantons, monopoles and toric hyperKähler manifolds. Commun. Math. Phys. 212(3), 503–533 (2000)zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Kraan, T.C., van Baal, P.: Exact T-duality between calorons and Taub-NUT spaces. Phys. Lett. B 428(3–4), 268–276 (1998)ADSMathSciNetGoogle Scholar
  23. 23.
    Kraan, T.C., van Baal, P.: Monopole constituents inside SU(n) calorons. Phys. Lett. B 435, 389–395 (1998)CrossRefADSGoogle Scholar
  24. 24.
    Kraan, T.C., van Baal, P.: Periodic instantons with non-trivial holonomy. Nucl. Phys. B 533(1–3), 627–659 (1998)zbMATHCrossRefADSGoogle Scholar
  25. 25.
    Lee, K.: Instantons and magnetic monopoles on R 3  ×  S 1 with arbitrary simple gauge groups. Phys. Lett. B 426(3–4), 323–328 (1998)zbMATHADSMathSciNetGoogle Scholar
  26. 26.
    Lee, K., Lu., C.: SU(2) calorons and magnetic monopoles. Phys. Rev. D 58, 025011 (1998)CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Lee, K., Yi, S.-H.: 1/4 BPS dyonic calorons. Phys. Rev. D 67(2), 025012 (2003)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Nahm, W.: Self-dual monopoles and calorons. In: Group theoretical methods in physics (Trieste, 1983), Volume 201 of Lecture Notes in Phys., Berlin: Springer, 1984, pp. 189–200Google Scholar
  29. 29.
    Nahm, W.: All self-dual multimonopoles for arbitrary gauge groups. In: Structural elements in particle physics and statistical mechanics (Freiburg, 1981), Volume 82 of NATO Adv. Study Inst. Ser. B: Physics, New York: Plenum 1983, pp. 301–310Google Scholar
  30. 30.
    Norbury, P.: Periodic instantons and the loop group. Commun. Math. Phys. 212(3), 557–569 (2000)zbMATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Nye, T.M.W.: The geometry of calorons. PhD thesis, University of Edinburgh, 2001, available at http://arxiv.org/abs/hep-th/0311215, 2003
  32. 32.
    Nye, T.M.W., Singer, M.A.: An L 2-index theorem for Dirac operators on S 1  ×  R 3. J. Funct. Anal. 177(1), 203–218 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces, Volume 3 of Progress in Mathematics. Boston, Mass: Birkhäuser, 1980Google Scholar
  34. 34.
    Ward, R.S.: A Yang–Mills–Higgs monopole of charge 2. Commun. Math. Phys. 79(3), 317–325 (1981)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentDuke UniversityDurhamUSA
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

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