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Sign choices for M. Kontsevich’s formality. (Choix des signes pour la formalité de M. Kontsevich.) (French) Zbl 1055.53066
The authors give a detailed proof of Kontsevich’s formality theorem in the flat case, with all the signs and orientations precised. Let $$M$$ be a manifold, $$g_1$$ the graded Lie algebra of polyvector fields on $$M$$ and $$g_2$$ the graded Lie algebra of polydifferential operators on $$M$$. The Kontsevich formality theorem states the existence of a quasi-isomorphism between $$g_1$$ and $$g_2$$, and as a consequence, the existence of a star product on any Poisson manifold. In this paper, the authors concentrate on the flat case. They first introduce a coherent choice of orientations and signs. Then they make all signs precise appearing in the formality equation. With all the signs, they prove that the Kontsevich’s formality quasi-isomorphism indeed satisfies the formality equation. Finally, they also give, in an independent part, the detailed proof for the ‘inverse’ theorem (see Kontsevich’s paper). The link with star products is also reprecised in the appendix.

##### MSC:
 53D55 Deformation quantization, star products
##### Keywords:
star product; Kontsevich formality; formality equation
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