nLab Kontsevich formality

Idea

Original version for associative algebras

The formality theorem (Kontsevich 97) 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 deformation quantization, unique up to an element in the freely-acting piece of the automorphism infinity-group of the E1-operad. This is the Grothendieck-Teichmüller group (see there for more).

General version for higher $E_n$-algebras

More generally, a result of Kontsevich and Tamarkin (…) says that over a field $\mathbb{F}$ of characteristic 0 the canonical functor

$$E_n Algebras_{\mathbb{F}} \stackrel{\simeq}{\longrightarrow} P_n Algebras_{\mathbb{F}}$$

from E-n algebras to Poisson n-algebras is an equivalence (since $P_n$ is the homology of $E_n$, this says that The $E_n$“ operad is formal over a field of characteristic zero, see at the little n-disk operad is formal)

But of course the automorphism infinity-group of both $E_n$ and $P_n$ acts on both sides and makes the space of all possible such equivalences a torsor over this group. A $P_n$ algebra may be thought of as encoding a prequantum field theory of higher dimension, of sorts, and so formality says that the deformation quantization of a prequantum field theory always exists and that the choices are being acted on by the corresponding higher analog of the Grothendieck-Teichmüller group.

Related entries

• graph complex

• deformation quantization, Grothendieck-Teichmüller group

• the little n-disk operad is formal

• Hochschild cohomology – Differential calculus.

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 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

• Vasily Dolgushev, All coefficients entering Kontsevich’s formality quasi-isomorphism can be replaced by rational numbers, arxiv/1306.6733

• Damien Calaque, Thomas Willwacher, Triviality of the higher Formality Theorem, arxiv/1310.4605

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 $G_\infty$-morphism in

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

Campos has extended Kontsevich formality to a $BV_\infty$ morphism using a cyclic analog of the minimal operad

• Ricardo Campos, BV Formality, arxiv/1506.07715

category: algebra, geometry