Abstract

In this paper we prove Lie algebroid versions of Tsygan's formality conjecture for Hochschild chains both in the smooth and holomorphic settings. Our result in the holomorphic setting implies a version of Tsygan's formality conjecture for Hochschild chains of the structure sheaf of any complex manifold. 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. In the smooth setting we discuss an application of our result to the description of quantum traces for a Poisson Lie algebroid.