A proof of the Tsygan formality conjecture for chains

Abstract

We extend the Kontsevich formality L ∞-morphism U : T poly •( R d)→ D poly •( R d) to an L ∞-morphism of L ∞-modules over T poly •( R d), U ̂ : C •(A,A)→Ω •( R d), A=C ∞( R d) . The construction of the map U ̂ 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 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.