Associative algebras, punctured disks and the quantization of Poisson manifolds

The aim of the note is to provide an introduction to the algebraic, geometric and quantum field theoretic ideas that lie behind the Kontsevich–Cattaneo–Felder formula for the quantization of Poisson structures. We show how the quantization formula itself naturally arises when one imposes the following two requirements to a Feynman integral: on the one side it has to reproduce the given Poisson structure as the first order term of its perturbative expansion; on the other side its three-point functions should describe an associative algebra. It is further shown how the Magri–Koszul brackets on 1-forms naturally fits into the theory of the Poisson sigma-model.