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On the Tetrahedrally Symmetric Monopole

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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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Authors and Affiliations

  1. School of Mathematics, Edinburgh University, Edinburgh, U.K

    H. W. Braden

  2. Institute of Magnetism, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    V. Z. Enolski

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  1. H. W. Braden
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  2. V. Z. Enolski
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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., 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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