A proof of the Tsygan formality conjecture for chains

Original · EN

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