Topological Deformation Quantizations

Abstract

In this chapter, we give a very short description of Kontsevich’s deformation quantization theorem, and of his algebraic deformation quantization methods, based on the deformation theory described in Chap. 10. We start by explaining the deformation quantization problem for Poisson manifolds and its relation with the Poisson sigma model, described in Chap. 16. We then explain the relation of this quantization problem with deformation theory for associative algebras and dg-categories. We then explain how this can be generalized to a categorical quantization problem for Artin stacks with a shifted symplectic form.