Topological solitons.

*(English)*Zbl 1100.37044
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (ISBN 0-521-83836-3/hbk; 0-511-20783-2/ebook). xi, 493 p. (2004).

The authors of this very interesting book consider the main examples of topological solitons and survey in detail static and dynamical multi-soliton solutions. They discuss several topics of interest as kinks in one dimension, lumps and vortices in two dimensions, monopoles and skyrmions in three dimensions, instantons in four dimensions. In some field theories, there are no static forces between solitons, and there is a large class of static multi-soliton solutions satisfying equations of Bogomolny type. The manifold of solutions can be considered as a moduli space. Its dimension increases with the soliton number.

The authors survey a lot of results in this area. Of special interest here is the discussion on the skyrmion quantization as well as the unstable analogue of the solitons, known as sphalerons. A very interesting topic is the question for the geodesic dynamics on moduli space. It is an adiabatic theory of multi-soliton motion at modest speeds when the static forces vanish. Many numerical results concerning solitons and their properties are shown as well. The techniques of modern differential geometry and algebra, such as Lie groups and algebras, moduli space dynamics, homotopy theory, Chern-Simons forms are used wherever appropriately to illuminate the considered topics.

The book is self-contained and beautifully written. It should remain for a long period of time as a standard reference for anyone interested in soliton theory and its application in physics.

The authors survey a lot of results in this area. Of special interest here is the discussion on the skyrmion quantization as well as the unstable analogue of the solitons, known as sphalerons. A very interesting topic is the question for the geodesic dynamics on moduli space. It is an adiabatic theory of multi-soliton motion at modest speeds when the static forces vanish. Many numerical results concerning solitons and their properties are shown as well. The techniques of modern differential geometry and algebra, such as Lie groups and algebras, moduli space dynamics, homotopy theory, Chern-Simons forms are used wherever appropriately to illuminate the considered topics.

The book is self-contained and beautifully written. It should remain for a long period of time as a standard reference for anyone interested in soliton theory and its application in physics.

Reviewer: Dimitar A. Kolev (Sofia)

##### MSC:

37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

57R22 | Topology of vector bundles and fiber bundles |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

35Q51 | Soliton equations |