### Abstract

This review is devoted to the discussion of the parallel existing between four-dimensional gauge theories and two-dimensional sigma models. We use sigma models as a laboratory allowing us to investigate such issues as the operator product expansion beyond perturbation theory, vacuum condensates, low-energy theorems and other non-perturbative aspects. All these questions are intensively discussed in the current literature, and we give a critical analysis of the situation. In particular, it is explained that, contrary to recent claims, one can define the operator product expansion beyond perturbation theory in a perfectly consistent way, with no ambiguities. The second part of the review represents a detailed discussion of the supersymmetric O(3) sigma model. After a simple description of the model we concentrate on instantons. The instanton-based method for calculating the exact Gell-Mann-Low function and bifermionic condensates is described. An analogue of this method has been previously used by us in four-dimensional Yang-Mills theories. Here we try to elucidate all aspects of the method in simplified conditions. The basic points are: (i) the instanton measure from purely classical analysis; (ii) a non-renormalization theorem in self-dual external fields; (iii) existence of vacuum condensates and their compatibility with supersymmetry. Pursuing pedagogical purposes we use much space for technical details and computations.

Original language | English (US) |
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Pages (from-to) | 103-171 |

Number of pages | 69 |

Journal | Physics Reports |

Volume | 116 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1984 |

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## Cite this

*Physics Reports*,

*116*(3), 103-171. https://doi.org/10.1016/0370-1573(84)90021-8