The Character Map in Deformation Quantization

نویسندگان

  • ALBERTO S. CATTANEO
  • GIOVANNI FELDER
چکیده

The third author recently proved that the Shoikhet–Dolgushev L∞-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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تاریخ انتشار 2009