Volume 17 (2011)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
MSP Books and Monographs
About this Series
Editorial Board
Ethics Statement
Author Index
Submission Guidelines
Author Copyright Form
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
Other MSP Publications

Deformation quantisation of Poisson manifolds

Simone Gutt

Geometry & Topology Monographs 17 (2011) 171–220


This introduction to deformation quantisation will focus on the construction of star products on symplectic and Poisson manifolds. It corresponds to the first four lectures I gave at the 2005 Summer School on Poisson Geometry in Trieste.

The first two lectures introduced the general concept of formal deformation quantisation with examples, with Fedosov’s construction of a star product on a symplectic manifold and with the classification of star products on a symplectic manifold.

The next lectures introduced the notion of formality and its link with star products, gave a flavour of Kontsevich’s construction of a formality for d and a sketch of the globalisation of a star product on a Poisson manifold following the approach of Cattaneo, Felder and Tomassini.

The notes here are a brief summary of those lectures; I start with a further reading section which includes expository papers with details of what is presented.

In the last lectures I only briefly mentioned different aspects of the deformation quantisation programme such as action of a Lie group on a deformed product, reduction procedures in deformation quantisation, states and representations in deformed algebras, convergence of deformations, leaving out many interesting and deep aspects of the theory (such as traces and index theorems, extension to fields theory); these are not included in these notes and I include a bibliography with many references to those topics.


deformation quantization, formality, Deligne characteristic classes

Mathematical Subject Classification

Primary: 53D55


Received: 18 May 2010
Accepted: 19 May 2010
Published: 14 April 2011

Simone Gutt
Université Libre de Bruxelles
Campus Plaine
Boulevard du Triomphe
1050 Brussels
Université de Metz
Ile du Saulcy
57045 Metz Cedex 01