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The character map in deformation quantization


Cattaneo, A S; Felder, G; Willwacher, T (2011). The character map in deformation quantization. Advances in Mathematics, 228(4):1966-1989.

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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2011
Deposited On:09 Jan 2012 14:22
Last Modified:05 Apr 2016 15:05
Publisher:Elsevier
ISSN:0001-8708
Publisher DOI:https://doi.org/10.1016/j.aim.2011.06.026
Related URLs:http://arxiv.org/abs/0906.3122

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