### Abstract

We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkähler metric of the moduli space of the theory on ℝ^{3} × S^{1}. The wall-crossing formula reduces to the statement that this metric is continuous. Our construction of the metric uses a four-dimensional analogue of the two-dimensional tt* equations.

Original language | English (US) |
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Pages (from-to) | 163-224 |

Number of pages | 62 |

Journal | Communications In Mathematical Physics |

Volume | 299 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2010 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Gaiotto, D., Moore, G. W., & Neitzke, A. (2010). Four-dimensional wall-crossing via three-dimensional field theory.

*Communications In Mathematical Physics*,*299*(1), 163-224. https://doi.org/10.1007/s00220-010-1071-2