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Kontsevich formality

Contents

Contents

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 nE_n-algebras

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

E nAlgebras 𝔽P nAlgebras 𝔽 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 nP_n is the homology of E nE_n, this says that The E nE_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 nE_n and P nP_n acts on both sides and makes the space of all possible such equivalences a torsor over this group. A P nP_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.

References

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 G_\infty-morphism in

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

category: algebra, geometry

Last revised on November 7, 2018 at 14:13:50. See the history of this page for a list of all contributions to it.