FORMALITY CONJECTURE by Maxim Kontsevich

This paper is devoted to a conjecture concerning the deformation quantization. This conjecture implies that arbitrary smooth Poisson manifold can be formally quantized, and the equivalence class of the resulting algebra is canonically defined. In other terms, it means that non-commutative geometry, in the formal approximation to the commutative geometry of smooth spaces, is described by the semi-classical approximation. Recently an article by A. Voronov (see [V]) with the exposition of the formality conjecture appeared on the net. The present paper can be seen as a companion to [V]. Here I present the conjecture in a slightly different form. In order to do it I include some preparational material on deformation theory and homotopy theory of differential graded Lie algebras. Brute force calculations confirm (locally) the conjecture up to the 6-th order in the perturbation theory. As a by-product I obtained a formula for a new flow on the space of germs of Poisson manifolds. Also I propose a reformulation of the conjecture and further evidences.