Abstract
We study SU(2) BPS monopoles with spectral curves of the form η 3+χ(ζ 6+b ζ 3−1) = 0. Previous work has established a countable family of solutions to Hitchin’s constraint that L 2 was trivial on such a curve. Here we establish that the only curves of this family that yield BPS monopoles correspond to tetrahedrally symmetric monopoles. We introduce several new techniques making use of a factorization theorem of Fay and Accola for theta functions, together with properties of the Humbert variety. The geometry leads us to a formulation purely in terms of elliptic functions. A more general conjecture than needed for the stated result is given.
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Accola, R.D.M.: Vanishing properties of theta functions for Abelian covers of Riemann surfaces. p7-18 In: Advances in the Theory of Riemann Surfaces: Proceedings of the 1969 Stony 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
Bateman H., Erdelyi A.: Higher Transcendental Functions: Vol. 2. McGraw-Hill, New York (1955)
Belokolos E.D., Bobenko A.I., Enolskii V.Z., Its A.R., Matveev V.B.: Algebro Geometric Approach to Nonlinear Integrable Equations. Springer, Berlin (1994)
Berndt B.C., Bhargava S., Garvan F.G.: Rananujans’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347(11), 4163–4244 (1995)
Bolza O.: Ueber die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades. Math. Ann. 28, 447–456 (1886)
Bolza O.: On Binary Sextics with Linear Transformations into Themselves. Amer. J. Math. 10(1), 47–70 (1887)
Berndt B.C.: Ramanujan’s Notebooks Part V. Springer-Verlag, New York (1998)
Braden, H.W., D’Avanzo, A., Enolski, V.Z.: In progress
Braden, H.W., Enolski, V.Z.: Remarks on the complex geometry of 3-monopole. http://arxiv.org/abs/math-ph/0601040v1, 2006
Braden, H.W., Enolski, V.Z.: Monopoles, Curves and Ramanujan, Reported at Riemann Surfaces, Analytical and Numerical Methods, Max Planck Instititute (Leipzig), 2007. Submitted, http://arxiv.org/abs/0704.3939v1[math-ph], 2007
Braden H.W., Enolski V.Z.: Finite-gap integration of the SU(2) Bogomolny equation. Glasgow Math. J. 51, 25–41 (2009)
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. Springer, Berlin (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)
Igusa J.: Theta Functions. Grund. Math. Wiss. Vol. 194. Springer, Berlin (1972)
Krazer, A.: Lehrbuch der Thetafunktionen. Leipzig: Teubner, 1903, reprinted by AMS Chelsea Publishing, 1998
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 Craigie, N.S., Goddard, P., Nahm, W.: Singapore: World Scientific, 1982
Matsumoto K.: Theta constants associated with the cyclic triple coverings of the complex projective line branching at six points. Publ. Res. Inst. Math. Sci 37, 419–440 (2001)
Murabayashi N.: The moduli space of genus 2 covering elliptic curve. Manuscripta math. 84, 125–133 (1994)
Sutcliffe P.M.: Seiberg-Witten theory, monopole spectral curves and affine Toda solitons. Phys. Lett. B381, 129–136 (1996)
Wellstein J.: Zur Theorie der Functionenclasse s 3 = (z − α 1)(z − α 2) . . . (z − α 6). Math. Ann. 52, 440–448 (1899)
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Communicated by N.A. Nekrasov
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Braden, H.W., Enolski, V.Z. On the Tetrahedrally Symmetric Monopole. Commun. Math. Phys. 299, 255–282 (2010). https://doi.org/10.1007/s00220-010-1081-0
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DOI: https://doi.org/10.1007/s00220-010-1081-0