A proof of the Tsygan formality conjecture for chains

Abstract

We extend the Kontsevich formality $L_\infty$-morphism $\U\colon T^\ndot_\poly(\R^d)\to\D^\ndot_\poly(\R^d)$ to an $L_\infty$-morphism of an $L_\infty$-modules over $T^\ndot_\poly(\R^d)$, $\hat \U\colon C_\ndot(A,A)\toΩ^\ndot(\R^d)$, $A=C^\infty(\R^d)$. The construction of the map $\hat \U$ is given in Kontsevich-type integrals. The conjecture that such an $L_\infty$-morphism exists is due to Boris Tsygan \cite{Ts}. As an application, we obtain an explicit formula for isomorphism $A_*/[A_*,A_*]\simto A/\{A,A\}$ ($A_*$ 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.