On the Tetrahedrally Symmetric Monopole

We study SU(2) BPS monopoles with spectral curves of the form η3+χ(ζ6+bζ3−1) = 0. Previous work has established a countable family of solutions to Hitchin’s constraint that L2 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.

[1]  Seiberg-Witten theory, monopole spectral curves and affine Toda solitons , 1996, hep-th/9605192.

[2]  Bruce C. Berndt,et al.  Ramanujan’s Theories of Elliptic Functions to Alternative Bases , 1995 .

[3]  Adolf Krazer,et al.  Lehrbuch der Thetafunktionen , 1905 .

[4]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[5]  John D. Fay.  Theta Functions on Riemann Surfaces , 1973 .

[6]  Remarks on the complex geometry of the 3-monopole , 2006, math-ph/0601040.

[7]  B. Berndt.  Ramanujan’s Notebooks: Part V , 1997 .

[8]  On the Constraints Defining BPS Monopoles , 1999, hep-th/9909168.

[9]  N. Hitchin.  Monopoles and geodesics , 1982 .

[10]  H. Braden,et al.  Monopoles, Curves and Ramanujan , 2007, 0704.3939.

[11]  Oskar Bolza,et al.  On Binary Sextics with Linear Transformations into Themselves , 1887 .

[12]  N. Hitchin.  On the construction of monopoles , 1983 .

[13]  Theta Constants Associated with the Cyclic Triple Coverings of the Complex Projective Line Branching at Six Points , 2000, math/0008025.

[14]  N. Ercolani,et al.  Monopoles and Baker functions , 1989 .

[15]  William P. Reinhardt,et al.  Theta functions , 2010, NIST Handbook of Mathematical Functions.

[16]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[17]  E. Belokolos,et al.  Algebro-geometric approach to nonlinear integrable equations , 1994 .

[18]  Naoki Murabayashi.  The moduli space of curves of genus two covering elliptic curves , 1994 .

[19]  R. D. Accola.  Vanishing Properties of Theta Functions for Abelian Covers of Riemann Surfaces , 1971 .

[20]  M. Murray.  On the Construction of Monopoles for , 1989 .