# Fedosov vs. Kontsevich deformation quantization : a beginner survey

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I'm a condensed matter physicist who tries to understand the details of deformation quantization.

In my self-made training, I've found two huge pieces of work, namely

Fedosov, B. V. (1994). "A simple geometrical construction of deformation quantization". Journal of Differential Geometry, 40 : 213–238.

Kontsevich, M. (2003). "Deformation Quantization of Poisson Manifolds". Letters in Mathematical Physics, 66 : 157-216.

[remarks : Fedosov's work seems to be also available with details in a book Deformation quantization and index theory. Are the two references overlapping ? -- I've found several documents from Kontsevich having similar titles, from 1997 to 2003, but I've no access to the reference of 2003, is the arXiv version the same as the final one ?]

My problem is that these works are really deep, long and difficult to me, so before continuing reading them, I'd like to understand whether these two works are equivalent or not, if they overlap somehow, and which kind of problem these works solved. If the answers could be without too much details for a physicist I'd really appreciate continuing learning this interesting topic.

This post imported from StackExchange MathOverflow at 2018-03-22 21:59 (UTC), posted by SE-user FraSchelle
asked Mar 13, 2018
retagged Mar 22, 2018
One quick remark is that Fedosov's procedure only works for symplectic manifolds, whereas Kontsevich's works for general Poisson manifolds. You might also be interested in arxiv.org/abs/hep-th/0102208 - a "review aimed at a physics audience" by Cattaneo and Felder which gives an interpretation of Kontsevich's formulas using Poisson sigma models. As far as I understand this Poisson sigma model simplifies considerably when the target is symplectic, and this yields Fedosov's formulas.

This post imported from StackExchange MathOverflow at 2018-03-22 21:59 (UTC), posted by SE-user Bertram Arnold

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