##
**Quantum fluctuations of instantons.**
*(English)*
Zbl 0535.58026

This excellent and authoritative review of instanton methods in two- dimensional (2D) sigma models and four-dimensional (4D) gauge theories deals more with the underlying mathematics of the semi-classical approximation than with physical applications. The parallel between 2D sigma models (O(3) and \(\mathrm{CP}(N-1))\) and 4D gauge theories \((\mathrm{SU}(2)\) and \(\mathrm{SU}(N))\) is emphasized. First the sigma models are considered, along with their instanton solutions which are known in full generality. The calculation of the quantum fluctuations about an arbitrary instanton configuration, followed by summation over all such configurations, leads to infrared finite results. In fact, this mathematical problem can be reexpressed in terms of a fictitious neutral gas of “instanton quarks” from which the physical properties of the semiclassical approximation are deduced. When one attempts to repeat this calculation for 4D gauge theories, the absence of an explicit arbitrary multi-instanton solution is an immediate and severe obstacle. Fluctuations about a dilute gas of instantons have infrared divergences, which (hopefully) the inclusion of dense instanton configurations will remove, as is the case for 2D sigma models. General aspects of the quantum problem are discussed. Ideas concerning the reformulation of the quantum fluctuation problem in terms of a gas of Yang-Mills “instanton quarks” are presented.

Reviewer: A. A. Actor (Fogelsville)

### MSC:

53D50 | Geometric quantization |

81T08 | Constructive quantum field theory |

53C80 | Applications of global differential geometry to the sciences |

### Keywords:

semiclassical approximation; gauge theory; sigma models; instanton quarks; multi-instanton solution
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\textit{V. A. Fateev} et al., Sov. Sci. Rev., Sect. C 2, 1--51 (1981; Zbl 0535.58026)

### References:

[1] | Belavin, A. A.; Polyakov, A. M.: Zhetf pisma. 22, 503 (1975) |

[2] | Seeley, R.: Proc. symp. Pure math.. 10, 288 (1971) |

[3] | Schwarz, A. S.: Comm. math. Phys.. 64, 233 (1979) |

[4] | Coleman, S.: Phys. rev.. 11, 2088 (1975) |

[5] | Fröhlich, J.: Comm. math. Phys.. 47, 233 (1976) |

[6] | Berezinskii, V. L.: Zhetf (USSR). 61, 1144 (1971) |

[7] | Kosterlitz, I. M.; Thouless, D. I.: J. phys.. 6, 1181 (1973) |

[8] | Kosterlitz, I. M.: J. phys. C. 7, 1046 (1974) |

[9] | Luther, A.; Scalapino, D. J.: Phys. rev.. 16, 1153 (1977) |

[10] | Mandelstam, S.: Phys. rev.. 11, 3026 (1975) |

[11] | Zomolodchikov, A. B.: ITEP preprint 91. (1976) |

[12] | Erdelyi, A.: Bateman manuscript project. (1953) |

[13] | Frolov, I. V.; Schwarz, A. S.: Zhetf pisma. 28, 273 (1978) |

[14] | Fateev, V. A.; Frolov, I. V.; Schwarz, A. S.: Yad. phys.. 8 (1979) |

[15] | Förster, D.: Nucl. phys.. 130, 38 (1977) |

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