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.
Similar content being viewed by others
References
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
Braden H.W., D’Avanzo A., Enolski V.Z.: On charge-3 cyclic monopoles. Nonlinearity 24, 643–675 (2011)
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)
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)
Braden H.W., Enolski V.Z.: On the tetrahedrally symmetric monopole. Commun. Math. Phys. 299, 255–282 (2010)
Corrigan E., Goddard P.: An n monopole solution with 4n−1 degrees of freedom. Commun. Math. Phys. 80, 575–587 (1981)
Ercolani N., Sinha A.: Monopoles and Baker Functions. Commun. Math. Phys. 125, 385–416 (1989)
Fay, J.D.: Theta functions on Riemann surfaces. Lectures Notes in Mathematics, Vol. 352, Berlin: Springer, 1973
Hitchin N.J.: Monopoles and Geodesics. Commun. Math. Phys. 83, 579–602 (1982)
Hitchin N.J.: On the Construction of Monopoles. Commun. Math. Phys. 89, 145–190 (1983)
Hitchin N.J., Manton N.S., Murray M.K.: Symmetric monopoles. Nonlinearity 8, 661–692 (1995)
Houghton C.J., Manton N.S., Romão N.M.: On the constraints defining BPS monopoles. Commun. Math. Phys. 212, 219–243 (2000)
Houghton C.J., Sutcliffe P.M.: Octahedral and dodecahedral monopoles. Nonlinearity 9, 385–401 (1996)
Houghton C.J., Sutcliffe P.M.: Tetrahedral and cubic monopoles. Commun. Math. Phys. 180, 343–361 (1996)
Houghton C.J., Sutcliffe P.M.: su(n) monopoles and Platonic symmetry. J. Math. Phys. 38, 5576–5589 (1997)
Kostant B.: The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group. Amer. J. Math. 81, 973–1032 (1959)
Manton N., Sutcliffe P.: Topological Solitons. Cambridge University Press, Cambridge (2004)
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
O’Raifeartaigh L., Rouhani S.: Rings of monopoles with discrete symmetry: explicit solution for n = 3. Phys. Lett. 112, 143 (1982)
Sutcliffe P.M.: Seiberg-Witten theory, monopole spectral curves and affine Toda solitons. Phys. Lett. B 381, 129–136 (1996)
Vanhaecke P.: Stratifications of hyperelliptic Jacobians and the Sato Grassmannian. Acta. Appl. Math. 40, 143–172 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N.A. Nekrasov
Rights and permissions
About this article
Cite this article
Braden, H.W. Cyclic Monopoles, Affine Toda and Spectral Curves. Commun. Math. Phys. 308, 303–323 (2011). https://doi.org/10.1007/s00220-011-1347-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1347-1