A proof of the Tsygan formality conjecture for chains

We extend the Kontsevich formality L_∞-morphism to an L_∞-morphism of L_∞-modules over . The construction of the map is given in Kontsevich-type integrals. The conjecture that such an L_∞-morphism exists is due to Boris Tsygan (Formality Conjecture for Chains, math. QA/9904132). As an application, we obtain an explicit formula for isomorphism is the Kontsevich deformation quantization of the algebra A by a Poisson bivector field, and {,} is the Poisson bracket). We also formulate a conjecture extending the Kontsevich theorem on cup-products to this context. The conjecture implies a generalization of the Duflo formula, and many other things.