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**Deformation quantization: Genesis, developments and metamorphoses.**
*(English)*
Zbl 1014.53054

Halbout, Gilles (ed.), Deformation quantization. Proceedings of the meeting of theoretical physicists and mathematicians, Strasbourg, France, May 31-June 2, 2001. Berlin: de Gruyter. IRMA Lect. Math. Theor. Phys. 1, 9-54 (2002).

This paper is a survey of the developments, both in physics and in mathematics, of the deformation quantization theory. Deformation quantization is born about 22 years ago in the seminal paper of Bayen-Flato-Fronsdal-Lichnerowicz-Sternheimer. Numerous developments and metamorphoses have accompanied this birth.

In this paper, the authors first review some important aspects of the theory, including basic results about star products, covariance and star representations of Lie groups. Then, they describe B. V. Fedosov’s construction of star products [J. Differ. Geom. 40, 213-238 (1994; Zbl 0812.53034)].

About five years ago, Kontsevich proved that for every Poisson manifold \(M\), there exists a canonically defined gauge equivalence class of star products on \(M\). This result appears in fact as a consequence of a more profound and stronger result: the formality theorem. The authors present here this theorem, which is a major metamorphosis of deformation quantization.

Alternative proofs of the existence of star products using operads are also shortly discussed.

For the entire collection see [Zbl 0986.00057].

In this paper, the authors first review some important aspects of the theory, including basic results about star products, covariance and star representations of Lie groups. Then, they describe B. V. Fedosov’s construction of star products [J. Differ. Geom. 40, 213-238 (1994; Zbl 0812.53034)].

About five years ago, Kontsevich proved that for every Poisson manifold \(M\), there exists a canonically defined gauge equivalence class of star products on \(M\). This result appears in fact as a consequence of a more profound and stronger result: the formality theorem. The authors present here this theorem, which is a major metamorphosis of deformation quantization.

Alternative proofs of the existence of star products using operads are also shortly discussed.

For the entire collection see [Zbl 0986.00057].

Reviewer: Angela Gammella (Metz)

### MSC:

53D55 | Deformation quantization, star products |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

81S10 | Geometry and quantization, symplectic methods |

81T70 | Quantization in field theory; cohomological methods |