## Topological SolitonsTopological solitons occur in many nonlinear classical field theories. They are stable, particle-like objects, with finite mass and a smooth structure. Examples are monopoles and Skyrmions, Ginzburg-Landau vortices and sigma-model lumps, and Yang-Mills instantons. This book is a comprehensive survey of static topological solitons and their dynamical interactions. Particular emphasis is placed on the solitons which satisfy first-order Bogomolny equations. For these, the soliton dynamics can be investigated by finding the geodesics on the moduli space of static multi-soliton solutions. Remarkable scattering processes can be understood this way. The book starts with an introduction to classical field theory, and a survey of several mathematical techniques useful for understanding many types of topological soliton. Subsequent chapters explore key examples of solitons in one, two, three and four dimensions. The final chapter discusses the unstable sphaleron solutions which exist in several field theories. |

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### Contents

1 | |

15 | |

3 Topology in field theory | 47 |

4 Solitons general theory | 75 |

5 Kinks | 109 |

6 Lumps and rational maps | 131 |

7 Vortices | 158 |

8 Monopoles | 241 |

9 Skyrmions | 349 |

10 Instantons | 416 |

11 Saddle points sphalerons | 441 |

467 | |

491 | |

### Common terms and phrases

2-sphere ansatz approximation asymptotic baryon baryon number Bogomolny equation boundary conditions centre Chern-Simons components constant coordinates corresponding defined denote derivative dimensions dynamics dyon electric charge energy density example ﬁeld field configuration field equation field theory finite gauge field gauge group gauge invariant gauge potential gauge theory gauge transformation geodesic motion global gradient flow Higgs field homotopy infinity instantons integral interaction kinetic energy kink Lagrangian linear loop Lorentz lumps manifold matrix metric minimal energy moduli space monopoles Nahm data Nahm equation non-trivial nonlinear obtained parameter particles Phys plane polynomial profile function quantization radial rational map representation rotation saddle point satisfying scalar field scattering separated sigma model Skyrme field Skyrme model Skyrmions SO(d spatial spectral curve sphaleron symmetry group tensor tetrahedral topological charge topological solitons vacuum vector vortex vortices winding number x3-axis Yang-Mills zero