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Formality theory. From Poisson structures to deformation quantization. (English) Zbl 1301.81003
SpringerBriefs in Mathematical Physics 2. Cham: Springer (ISBN 978-3-319-09289-8/pbk; 978-3-319-09290-4/ebook). xii, 90 p. (2015).
This book gives an introduction to the theory of deformation quantization of Poisson manifolds in the formalism developed by M. Kontsevich.
First, the author introduces the description of a classical mechanical system in terms of Poisson manifolds and defines the notion of formal Poisson structures. Then the notion of star product is discussed and Kontsevich’s formality theorem is introduced as an extension of the Hochschild-Kostant- Rosenberg theorem. Finally, the author gives a sketchly exposition of the Kontsevich’s formula for \({\mathbb R}^d\) and discusses the globalization approaches of Cattaneo-Felder-Tomassini.

MSC:
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
53D55 Deformation quantization, star products
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
53D17 Poisson manifolds; Poisson groupoids and algebroids
81S10 Geometry and quantization, symplectic methods
22E70 Applications of Lie groups to the sciences; explicit representations
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
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