A Proof of the Tsygan Formality Conjecture for Chains

We extend the Kontsevich formality L∞-morphism U : T • poly(R ) → D poly(R ) to an L∞-morphism of an L∞-modules over T • poly(R ), Û : C•(A, A) → Ω(R), A = C(R). The construction of the map Û is given in Kontsevich-type integrals. The conjecture that such an L∞-morphism exists is due to Boris Tsygan [Ts]. 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 the cup-products to this context. The conjecture implies a generalization of the Duflo formula, and many other things. 1. L∞-algebras and L∞-modules Here we recall basic definitions from [Ts] and construct an L∞-module over T • poly(R ) structure on the chain Hochschild complex C•(A,A), A = C ∞(Rd). 1.1. For a manifold M , T • poly(M) denotes the graded Lie algebra of smooth polyvector fields on M , Ω•(M) denotes the graded space of smooth differential forms on M . The space T • poly(M) is graded as a Lie algebra, i.e. T i poly(M) = {(i + 1)-polyvector fields}. The bracket is the Schouten–Nijenhuis bracket, generalizing the usual Lie bracket of vector fields. It is defined as follows: (1) [ξ0 ∧ · · · ∧ ξk, η0 ∧ · · · ∧ ηl] =