Covariant and Equivariant Formality Theorems

Details Covariant and Equivariant Formality Theorems Covariant and Equivariant Formality Theorems

2003-07-16

arxiv.org/abs/math/0307212v2

Adv. Math., Vol. 191, 1 (2005) 147-177

Mathematics

Quantum Algebra

26 pages, no figures

Scientific paper

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 $\infty$-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.

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