Căldăraru’s conjecture and Tsygan’s formality

Abstract

In this paper we complete the proof of Căldăraru’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 products after twisting with a Maurer-Cartan element.

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Authors

Damien Calaque

ETH Zürich
8092 Zürich
Switzerland

On leave from:

Institut Camille Jordan
CNRS and Université Lyon 1
France

Carlo A. Rossi

MPIM Bonn
Vivatsgasse 7
53111 Bonn
Germany

Michel Van den Bergh

Departement WNI
Hasselt University
Agoralaan
3590 Diepenbeek
Belgium