##
**Quantum field theory and topology.
(Kvantovaya teoriya polya i topologiya.)**
*(Russian.
English summary)*
Zbl 0682.58001

Moskva: Nauka. 400 p. R. 4.60 (1989).

This book is devoted to the fundamental results of quantum field theory (QFT) obtained by topological methods. It is structured in three principal chapters: (i) Main Lagrangians of QFT; (ii) Topological methods of QFT; (iii) Foundations of topology. The book contains also some specific applications of the topological methods to QFT.

First, the fundamental results of gauge theories are reviewed: quadratic Lagrangians, standard SU(3)\(\times SU(2)\times U(1)\)-model, grand unification theories, etc. Then, the topological methods of QFT are introduced in a natural way: topological stable particle and strings, magnetic monopoles, nonlinear fields, multivalued action functionals, elliptic operators and anomalies, instantons, etc. The topological part represents a textbook of topology oriented towards physicists. It contains the results regarding the main topological concepts: degree of mapping, fundamental group and coverings, manifolds, differential forms, homology and cohomology groups, homotopy groups, fiber spaces, homogeneous manifolds, gauge fields and characteristic classes.

The author of this book holds a professorship at the Moscow Physical Engineering Institute. His work has played an important role in the development of topological methods in physics and a significant part of the book is based on his papers.

First, the fundamental results of gauge theories are reviewed: quadratic Lagrangians, standard SU(3)\(\times SU(2)\times U(1)\)-model, grand unification theories, etc. Then, the topological methods of QFT are introduced in a natural way: topological stable particle and strings, magnetic monopoles, nonlinear fields, multivalued action functionals, elliptic operators and anomalies, instantons, etc. The topological part represents a textbook of topology oriented towards physicists. It contains the results regarding the main topological concepts: degree of mapping, fundamental group and coverings, manifolds, differential forms, homology and cohomology groups, homotopy groups, fiber spaces, homogeneous manifolds, gauge fields and characteristic classes.

The author of this book holds a professorship at the Moscow Physical Engineering Institute. His work has played an important role in the development of topological methods in physics and a significant part of the book is based on his papers.

Reviewer: Gh.Zet

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

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

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

81T99 | Quantum field theory; related classical field theories |