The character map in deformation quantization

TL;DR

This paper studies the character map in deformation quantization, showing its compatibility with connections and computing its value as the A-roof genus, thereby extending index theorems.

Contribution

It proves the compatibility of the character map with the Gauss-Manin connection and computes its value as the A-roof genus, extending previous results.

Findings

The character map is compatible with the Gauss-Manin connection.

The image of the cyclic cycle 1 equals the A-roof genus.

The results imply the Tamarkin-Tsygan index Theorem.

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.