Deformation Quantisation via Kontsevich Formality Theorem

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 ${\mathrm{L}}_{\mathrm{\infty }}$-algebras, which allows us to reformulate the classification of deformation quantisation as the existence of a ${\mathrm{L}}_{\mathrm{\infty }}$-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.