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arXiv · 0906.3122

The character map in deformation quantization

Alberto S. Cattaneo·Giovanni Felder·Thomas Willwacher
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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.

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Alberto S. Cattaneo, Giovanni Felder, Thomas Willwacher. 2009-06-17. The character map in deformation quantization. https://arxiv.org/abs/0906.3122

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