A proof of the Tsygan formality conjecture for chains

We extend the Kontsevich formality L-infinity-morphism U:T-poly(.)(R-d) --> D-poly(.)(R-d) to an L-infinity- morphism of L-infinity-modules over T-poly(.)(R-d), U:C-.(A, A)-->Omega(.)(R-d), A = C-infinity(R-d). The construction of the map U is given in Kontsevich-type integrals. The conjecture that such an L-infinity-morphism exists is due to Boris Tsygan (Formality Conjecture for Chains, math. QA/ 9904132). As an application, we obtain an explicit formula for isomorphism A(*)/[A(*), A(*)]-->A/{A,A} (A(*) 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. (C) 2003 Elsevier Science (USA). All rights reserved.