The character map in deformation quantization
Abstract
The third author recently proved that the Shoikhet-Dolgushev L-infinity-morphism from Hochschild chains of the algebra of smooth functions on manifold to differential forms extends to cyclic chains. Localization at a solution of the Maurer-Cartan equation gives an isomorphism, which we call character map, from the periodic cyclic homology of a formal associative deformation of the algebra of functions to de Rham cohomology. We prove that the character map is compatible with the Gauss-Manin connection, extending a result of Calaque and Rossi on the compatibility with the cap product. As a consequence, the image of the periodic cyclic cycle 1 is independent of the deformation parameter and we compute it to be the A-roof genus of the manifold. Our results also imply the Tamarkin-Tsygan index Theorem.
Cite
@article{arxiv.0906.3122,
title = {The character map in deformation quantization},
author = {Alberto S. Cattaneo and Giovanni Felder and Thomas Willwacher},
journal= {arXiv preprint arXiv:0906.3122},
year = {2014}
}