The character map in deformation quantization

Abstract

The third author recently proved that the Shoikhet–Dolgushev L∞-morphism from Hochschild chains of the algebra of smooth functions on a 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.