nLab
Kontsevich formality

The 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 $G_{\mathrm{\infty }}$-morphism in

• Thomas Willwacher, A note on Br-infinity and KS-infinity formality, arxiv/1109.3520

category: algebra, geometry