Caldararu's conjecture and Tsygan's formality

TL;DR

This paper proves Caldararu's conjecture by showing that twisting with the Todd class square root creates an isomorphism between module structures on differential forms and Hochschild homology, using formal geometry and Kontsevich's formality.

Contribution

It completes the proof of Caldararu's conjecture by establishing a global isomorphism of precalculi via Todd class twisting, extending Kontsevich and Shoikhet's formality results.

Findings

Established the isomorphism of precalculi via Todd class twisting.

Extended Kontsevich and Shoikhet's formality to global settings.

Confirmed compatibility of Shoikhet's quasi-isomorphism with cap product.

Abstract

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.