Formality theorems for Hochschild chains in the Lie algebroid setting

In this paper we prove Lie algebroid versions of Tsygan’s formality conjecture for Hochschild chains both in the smooth and holomorphic settings. In the holomorphic setting our result implies a version of Tsygan’s formality conjecture for Hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary Poisson Lie algebroid. The proofs are based on the use of Kontsevich’s quasi-isomorphism for Hochschild cochains of R[[y_1,…,y_d]], Shoikhet’s quasi-isomorphism for Hochschild chains of R[[y_1,…,y_d]], and Fedosov’s resolutions of the natural analogues of Hochschild (co)chain complexes associated with a Lie algebroid.