### 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) |
---|---|

Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Communications in Mathematical Physics |

Volume | 234 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2003 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- 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

**Periodic monopoles with singularities and N = 2 super-QCD.** / Cherkis, Sergey; Kapustin, Anton.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 234, no. 1, pp. 1-35. https://doi.org/10.1007/s00220-002-0786-0

}

TY - JOUR

T1 - Periodic monopoles with singularities and N = 2 super-QCD

AU - Cherkis, Sergey

AU - Kapustin, Anton

PY - 2003/3

Y1 - 2003/3

N2 - We study solutions of the Bogomolny equation on ℝ2 × S1 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 ℝ3 × S1. 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 ℝ2 × S1. In particular, this allows us to rederive Witten's results without recourse to the M-theory fivebrane.

AB - We study solutions of the Bogomolny equation on ℝ2 × S1 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 ℝ3 × S1. 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 ℝ2 × S1. In particular, this allows us to rederive Witten's results without recourse to the M-theory fivebrane.

UR - http://www.scopus.com/inward/record.url?scp=0037346860&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037346860&partnerID=8YFLogxK

U2 - 10.1007/s00220-002-0786-0

DO - 10.1007/s00220-002-0786-0

M3 - Article

VL - 234

SP - 1

EP - 35

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -