Comptes Rendus
Homological algebra/Differential geometry
Formality theorem for differential graded manifolds
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 27-43.

We establish a formality theorem for smooth dg manifolds. More precisely, we prove that, for any finite-dimensional dg manifold (M,Q), there exists an L quasi-isomorphism of dglas from (Tpoly(M),[Q,],[,]) to (Dpoly(M),m+Q,,,) whose first Taylor coefficient (1) is equal to the composition hkr(td(M,Q))12:Tpoly(M)Dpoly(M) of the action of (td(M,Q))12k0(Ωk(M))k on Tpoly(M) (by contraction) with the Hochschild–Kostant–Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we prove the Kontsevich–Shoikhet conjecture: a Kontsevich–Duflo-type theorem holds for all finite-dimensional smooth dg manifolds.

Nous prouvons un théorème de formalité pour les variétés lisses différentielles graduées. Plus précisément, nous prouvons qu'il existe, pour toute variété différentielle graduée (M,Q), un quasi-isomorphisme L de l'algèbre de Lie différentielle graduée (Tpoly(M),[Q,],[,]) dans l'algèbre de Lie différentielle graduée (Dpoly(M),m+Q,,,), dont le premier coefficient de Taylor (1) est égal à la composée hkr(td(M,Q))12:Tpoly(M)Dpoly(M) de l'action (par contraction) de (td(M,Q))12k0(Ωk(M))k sur Tpoly(M) avec l'application de Hochschild–Kostant–Rosenberg et (2) respecte les structures d'algèbres associatives en cohomologie. Comme application, nous prouvons la conjecture de Kontsevich–Shoikhet : il existe un théorème de type Kontsevich–Duflo valable pour toute variété différentielle graduée de dimension finie.

Published online:
DOI: 10.1016/j.crma.2017.11.017

Hsuan-Yi Liao 1; Mathieu Stiénon 1; Ping Xu 1

1 Department of Mathematics, Pennsylvania State University, USA
     author = {Hsuan-Yi Liao and Mathieu Sti\'enon and Ping Xu},
     title = {Formality theorem for differential graded manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {27--43},
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     doi = {10.1016/j.crma.2017.11.017},
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Hsuan-Yi Liao; Mathieu Stiénon; Ping Xu. Formality theorem for differential graded manifolds. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 27-43. doi : 10.1016/j.crma.2017.11.017.

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Research partially supported by NSF grants DMS-1406668 and DMS-1707545.

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