Formality theorem for differential graded manifolds

Hsuan-Yi Liao; Mathieu Stiénon; Ping Xu

doi:10.1016/j.crma.2017.11.017

Homological algebra/Differential geometry

Formality theorem for differential graded manifolds
[Théorème de formalité pour les variétés différentielles graduées]

Présenté par : the Editorial Board

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

¹ Department of Mathematics, Pennsylvania State University, USA

Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 27-43.

Résumé
Abstract

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 $L_{\infty}$ de l'algèbre de Lie différentielle graduée $({}_{\oplus}T_{poly}^{•} (M), [Q, -], [-, -])$ dans l'algèbre de Lie différentielle graduée $({}_{\oplus}D_{poly}^{•} (M), 〚 m + Q, - 〛, 〚 -, - 〛)$ , dont le premier coefficient de Taylor (1) est égal à la composée $hkr \circ {({td}_{(M, Q)}^{\nabla})}^{\frac{1}{2}} : {}_{\oplus}T_{poly}^{•} (M) \to {}_{\oplus}D_{poly}^{•} (M)$ de l'action (par contraction) de ${({td}_{(M, Q)}^{\nabla})}^{\frac{1}{2}} \in \prod_{k \geq 0} {(Ω^{k} (M))}^{k}$ sur ${}_{\oplus}T_{poly}^{•} (M)$ 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.

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 $L_{\infty}$ quasi-isomorphism of dglas from $({}_{\oplus}T_{poly}^{•} (M), [Q, -], [-, -])$ to $({}_{\oplus}D_{poly}^{•} (M), 〚 m + Q, - 〛, 〚 -, - 〛)$ whose first Taylor coefficient (1) is equal to the composition $hkr \circ {({td}_{(M, Q)}^{\nabla})}^{\frac{1}{2}} : {}_{\oplus}T_{poly}^{•} (M) \to {}_{\oplus}D_{poly}^{•} (M)$ of the action of ${({td}_{(M, Q)}^{\nabla})}^{\frac{1}{2}} \in \prod_{k \geq 0} {(Ω^{k} (M))}^{k}$ on ${}_{\oplus}T_{poly}^{•} (M)$ (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.

Reçu le : 2017-11-02
Accepté le : 2017-11-23
Publié le : 2017-12-06

DOI : 10.1016/j.crma.2017.11.017

Affiliations des auteurs :

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

¹ Department of Mathematics, Pennsylvania State University, USA

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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/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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