### Abstract

We study solutions of the Bogomolny equation on ℝ^{2} × S^{1} with prescribed singularities. We show that the Nahm transform establishes a one-to-one correspondence between such solutions and solutions of the Hitchin equations on a punctured cylinder with the eigenvalues of the Higgs field growing at infinity in a particular manner. The moduli spaces of solutions have natural hyperkähler metrics of a novel kind. We show that these metrics describe the quantum Coulomb branch of certain N = 2 d = 4 supersymmetric gauge theories on ℝ^{3} × S^{1}. The Coulomb branches of the corresponding uncompactified theories have been previously determined by E. Witten using the M-theory fivebrane. We show that the Seiberg-Witten curves of these theories are identical to the spectral curves associated to solutions of the Bogomolny equation on ℝ^{2} × S^{1}. In particular, this allows us to rederive Witten's results without recourse to the M-theory fivebrane.

Original language | English (US) |
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Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Communications in Mathematical Physics |

Volume | 234 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2003 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*234*(1), 1-35. https://doi.org/10.1007/s00220-002-0786-0