Operads,configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, [`(C)]bar C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A <sub>∞</sub>-algebras and their homotopy morphisms, (ii) the 2-coloured operad of L <sub>∞</sub>-algebras and their homotopy morphisms, and (iii) the 4-coloured operad of openclosed homotopy algebras and their homotopy morphisms. Two gadgets — a (coloured) operad of Feynman graphs and a de Rham field theory on [`(C)]bar C — are introduced and used to construct quantized representations of the (fundamental) chain operad of [`(C)]bar C which are given by Feynman type sums over graphs and depend on choices of propagators.