2003/07/31 by Vasiliy Dolgushev
Mathematics · Physics and Astronomy · #math.QA #hep-th #math.DG #msc:16E45 #msc:53C15
published as Adv. Math., Vol. 191, 1 (2005) 147-177 · 26 pages, no figures
arxiv created 2003/09/26 · arxiv updated 2009/12/01
We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is that it is based on the use of covariant tensors unlike Kontsevich's original proof, which is based on ∞-jets of polydifferential operators and polyvector fields. Using our construction we prove that if a group G acts smoothly on a manifold M and M admits a G-invariant affine connection then there exists a G-equivariant quasi-isomorphism of formality. This result implies that if a manifold M is equipped with a smooth action of a finite or compact group G or equipped with a free action of a Lie group G then M admits a G-equivariant formality quasi-isomorphism. In particular, this gives a solution of the deformation quantization problem for an arbitrary Poisson orbifold.