Kontsevich formality theorem

Builds on Kontsevich quantization formula Hochschild homology Poisson manifold Deformation quantization

Statement

Let $M = \R^d$ . Let $T_{\mathrm{poly}}$ be the differential graded Lie algebra of polyvector fields on $M$ , with zero differential and the Schouten–Nijenhuis bracket. Let $D_{\mathrm{poly}}$ be the differential graded Lie algebra of polydifferential operators on $M$ , with the Hochschild differential and the Gerstenhaber bracket. Both carry the grading in which $k$ -vector fields, and polydifferential operators with $k$ arguments, have degree $k - 1$ .

There is an $L_\infty$ quasi-isomorphism $T_{\mathrm{poly}} \to D_{\mathrm{poly}}$ whose first component is the Hochschild–Kostant–Rosenberg map.
As a consequence, every Poisson manifold structure on $M$ admits a star product. Moreover, equivalence classes of star products on $M$ correspond bijectively to classes of formal Poisson structures $\hbar \pi_1 + \hbar^2 \pi_2 + \cdots$ under formal diffeomorphisms.

Remarks

The Hochschild–Kostant–Rosenberg map identifies the cohomology of $D_{\mathrm{poly}}$ with $T_{\mathrm{poly}}$ , but it does not respect the brackets; the theorem corrects it by higher terms. Kontsevich built these terms from integrals over configuration spaces of points in the upper half-plane, indexed by graphs. Star products correspond to Maurer–Cartan elements of $\hbar D_{\mathrm{poly}}[\![\hbar]\!]$ , formal Poisson structures to those of $\hbar T_{\mathrm{poly}}[\![\hbar]\!]$ , and an $L_\infty$ quasi-isomorphism matches their equivalence classes. For a constant Poisson bivector on $\R^{2n}$ , Kontsevich's quantization formula gives the Moyal product. The result was later extended from $\R^d$ to all smooth manifolds.

Going deeper

For a student

Deformation theory says that deformations of an algebraic structure are governed by a differential graded Lie algebra through its Maurer–Cartan equation. For the algebra of functions on $M$ , this algebra is $D_{\mathrm{poly}}$ , whose cohomology is $T_{\mathrm{poly}}$ by the Hochschild–Kostant–Rosenberg theorem. Formality means that $D_{\mathrm{poly}}$ is equivalent, as an $L_\infty$ algebra, to its cohomology with the induced bracket. Equivalent $L_\infty$ algebras have the same deformation theory, which gives the classification of star products. The same circle of ideas links to the Gerstenhaber structure on Hochschild cohomology and to the Deligne conjecture.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Deformation quantization concept
Formal power series concept
Hochschild homology concept
Poisson manifold concept
Kontsevich quantization formula statement
Kontsevich formality theorem statement

Tag	`XXA9D45E0N`
URI	`m2://XXA9D45E0N`
Kind	statement, theorem
Writing	W0 stub
MSC	53D55, 16S80, 16E40
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About	Deformation quantization
Uses	Kontsevich quantization formula, Hochschild homology, Poisson manifold, Deformation quantization