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The third author recently proved that the Shoikhet-DolgushevL-infinity-morphism from Hochschild chains of the algebra of smooth functionson manifold to differential forms extends to cyclic chains. Localization at asolution of the Maurer-Cartan equation gives an isomorphism, which we callcharacter map, from the periodic cyclic homology of a formal associativedeformation of the algebra of functions to de Rham cohomology. We prove thatthe character map is compatible with the Gauss-Manin connection, extending aresult of Calaque and Rossi on the compatibility with the cap product. As aconsequence, the image of the periodic cyclic cycle 1 is independent of thedeformation parameter and we compute it to be the A-roof genus of the manifold.Our results also imply the Tamarkin-Tsygan index Theorem.

The character map in deformation quantization