formality theorem by Maxim Kontsevich (arXiv 1997) states that there is an L-∞-algebra quasi-isomorphism from the dg-Lie algebra of polyvector fields (with zero differential and Schouten-Nijenhuis bracket) to the ? dg-Lie algebra of the shifted Hochschild cochain complex (with Hochschild differential and Gerstenhaber bracket), whose first Taylor coefficient is the HKR quasi-isomorphism.
Tamarkin alternatively proves the formality of the
little disks operad (see also Kontsevich 1999) and proves that it implies the Kontsevich formality.
The Kontsevich formality theorem implies that every
Poisson manifold has a canonical deformation quantization. References
Maxim Kontsevich, Deformation quantization of Poisson manifolds, q-alg/9709040, Lett. Math. Phys. 66 (2003), no. 3, 157–216 doi; Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35–72. Dmitry E. Tamarkin,
Another proof of M. Kontsevich formality theorem, math.QA/9803025; Formality of chain operad of little discs, Lett. Math. Phys. 66 (1-2):65–72, 2003.
Bernhard Keller, Notes for an Introduction to Kontsevich’s quantization theorem, pdf Dan Petersen,
Minimal models, GT-action and formality of the little disk operad, arxiv/1303.1448 Damien Calaque, Carlo A. Rossi,
Lectures on Duflo isomorphisms in Lie algebra and complex geometry, European Math. Soc. 2011 MPI pdf
Vladimir Hinich, Tamarkin’s proof of Kontsevich’s formality theorem, math.QA/0003052
In agreement with Tsygan’s philosophy of noncommutative differential calculus and its relations to braces algebra, Willwacher extends the Kontsevich formality to a homotopy braces morphism and to a
-morphism in G ∞
A note on Br-infinity and KS-infinity formality, arxiv/1109.3520
Revised on March 9, 2013 00:33:23