Caldararu's conjecture and Tsygan's formality

In this paper we complete the proof of Caldararu's conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet (the existence of the latter was conjectured by Tsygan). We also rely on the fact - recently proved by the first two authors - that Shoikhet's quasi-isomorphism is compatible with cap product after twisting with a Maurer-Cartan element.