Communications in Mathematical Physics

, Volume 299, Issue 1, pp 255–282 | Cite as

On the Tetrahedrally Symmetric Monopole

  • H. W. BradenEmail author
  • V. Z. Enolski


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.


Line Bundle Elliptic Curve Elliptic Curf Elliptic Function Theta Function 
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Authors and Affiliations

  1. 1.School of MathematicsEdinburgh UniversityEdinburghU.K
  2. 2.Institute of MagnetismNational Academy of Sciences of UkraineKyivUkraine

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