Deformation Quantisation via Kontsevich Formality Theorem

TL;DR

This paper explains Kontsevich's proof of the formality theorem, which classifies deformation quantisation on Poisson manifolds using advanced algebraic structures, and discusses the construction and globalization of star products.

Contribution

It provides an exposition of Kontsevich's proof of the formality theorem and details the construction of deformation quantisation on Poisson manifolds.

Findings

Kontsevich's proof of the formality theorem in ^d

Construction of the Kontsevich star product

Discussion on globalization of star products on Poisson manifolds

Abstract

This dissertation is an exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold. We begin with an account of the physical background and introduce the Weyl-Moyal product as the first example. Then we develop the deformation theory via differential graded Lie algebras and $\mathrm{L}_\infty$-algebras, which allows us to reformulate the classification of deformation quantisation as the existence of a $\mathrm{L}_\infty$-quasi-isomorphism between two differential graded Lie algebras, known as the formality theorem. Next we present Kontsevich's proof of the formality theorem in $\mathbb{R}^d$ and his construction of the star product. We conclude with a brief discussion of the globalisation of Kontsevich star product on Poisson manifolds.