Operads and Motives in Deformation Quantization

Abstract

This paper is dedicated to the memory of Moshe Flato, and will appear in Lett. Math. Phys. 48 (1) It became clear during last 5-6 years that the algebraic world of associative algebras (abelian categories, triangulated categories, etc) has many deep connections with the geometric world of two-dimensional surfaces. One of manifestations of this is Deligne's conjecture (1993) which says that on the cohomological Hochschild complex of any associative algebra naturally acts the operad of singular chains in the little discs operad. Recently D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Deligne's conjecture follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois group acts on deformation quantizations, and speculate on possible relations of higher-dimensional algebras and of motives to quantum field theories.