Cyclic Monopoles, Affine Toda and Spectral Curves

Abstract

We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe’s ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha vector) and base point of the linearising flow in the Jacobian of the spectral curve associated to the Nahm equations arise as pull-backs of Toda data. A theorem of Accola and Fay then means that the theta-functions arising in the solution of the monopole problem reduce to the theta-functions of Toda.

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References

  1. Acc71

    Accola Robert, D.M.: Vanishing Properties of Theta Functions for Abelian Covers of Riemann Surfaces. In: Advances in the Theory of Riemann Surfaces: Proceedings of the 1969 Sony Brook Conference, edited by L.V. Ahlfors, L. Bers, H.M. Farkas, R.C. Gunning, I. Kra, H.E. Rauch, Princeton, NJ: Princeton University Press, 1971, pp. 7–18

  2. BDE

    Braden H.W., D’Avanzo A., Enolski V.Z.: On charge-3 cyclic monopoles. Nonlinearity 24, 643–675 (2011)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  3. BE06

    Braden H.W., Enolski V.Z.: Remarks on the complex geometry of 3-monopole, math-ph/0601040 Part I appears as “some remarks on the Ercolani-Sinha construction of monopoles”. Theor. Math. Phys. 165, 1567–1597 (2010)

    Article  Google Scholar 

  4. BE07

    Braden, H.W., Enolski, V.Z.: Monopoles, Curves and Ramanujan. Reported at Riemann Surfaces, Analytical and Numerical Methods, Max Planck Instititute (Leipzig), #2007. Matem. Sborniki. 201, 19–74 (2010)

  5. BE09

    Braden H.W., Enolski V.Z.: On the tetrahedrally symmetric monopole. Commun. Math. Phys. 299, 255–282 (2010)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  6. CG81

    Corrigan E., Goddard P.: An n monopole solution with 4n−1 degrees of freedom. Commun. Math. Phys. 80, 575–587 (1981)

    MathSciNet  ADS  Article  Google Scholar 

  7. ES89

    Ercolani N., Sinha A.: Monopoles and Baker Functions. Commun. Math. Phys. 125, 385–416 (1989)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  8. Fay73

    Fay, J.D.: Theta functions on Riemann surfaces. Lectures Notes in Mathematics, Vol. 352, Berlin: Springer, 1973

  9. Hit82

    Hitchin N.J.: Monopoles and Geodesics. Commun. Math. Phys. 83, 579–602 (1982)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  10. Hit83

    Hitchin N.J.: On the Construction of Monopoles. Commun. Math. Phys. 89, 145–190 (1983)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  11. HMM95

    Hitchin N.J., Manton N.S., Murray M.K.: Symmetric monopoles. Nonlinearity 8, 661–692 (1995)

    MathSciNet  ADS  Article  Google Scholar 

  12. HMR00

    Houghton C.J., Manton N.S., Romão N.M.: On the constraints defining BPS monopoles. Commun. Math. Phys. 212, 219–243 (2000)

    ADS  MATH  Article  Google Scholar 

  13. HS96a

    Houghton C.J., Sutcliffe P.M.: Octahedral and dodecahedral monopoles. Nonlinearity 9, 385–401 (1996)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  14. HS96b

    Houghton C.J., Sutcliffe P.M.: Tetrahedral and cubic monopoles. Commun. Math. Phys. 180, 343–361 (1996)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  15. HS97

    Houghton C.J., Sutcliffe P.M.: su(n) monopoles and Platonic symmetry. J. Math. Phys. 38, 5576–5589 (1997)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  16. K

    Kostant B.: The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group. Amer. J. Math. 81, 973–1032 (1959)

    MathSciNet  MATH  Article  Google Scholar 

  17. MS04

    Manton N., Sutcliffe P.: Topological Solitons. Cambridge University Press, Cambridge (2004)

    MATH  Book  Google Scholar 

  18. Nah82

    Nahm, W.: The construction of all self-dual multimonopoles by the ADHM method. In: Monopoles in Quantum Field Theory, edited by N.S. Craigie, P. Goddard, W. Nahm, Singapore: World Scientific, 1982

  19. OR82

    O’Raifeartaigh L., Rouhani S.: Rings of monopoles with discrete symmetry: explicit solution for n = 3. Phys. Lett. 112, 143 (1982)

    MathSciNet  Google Scholar 

  20. Sut96

    Sutcliffe P.M.: Seiberg-Witten theory, monopole spectral curves and affine Toda solitons. Phys. Lett. B 381, 129–136 (1996)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  21. V95

    Vanhaecke P.: Stratifications of hyperelliptic Jacobians and the Sato Grassmannian. Acta. Appl. Math. 40, 143–172 (1995)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to H. W. Braden.

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Communicated by N.A. Nekrasov

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Braden, H.W. Cyclic Monopoles, Affine Toda and Spectral Curves. Commun. Math. Phys. 308, 303 (2011). https://doi.org/10.1007/s00220-011-1347-1

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Keywords

  • Theta Function
  • Spectral Curve
  • Period Matrix
  • Toda Equation
  • Cyclic Symmetry