Geometry & Topology Monographs 17 (2011) 171–220

Abstract

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 ℝ^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.

Keywords

deformation quantization, formality, Deligne characteristic classes

Mathematical Subject Classification

Primary: 53D55

References

Publication

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

Authors