The Atiyah class of a dg-vector bundle

Mehta, Rajan Amit; Stiénon, Mathieu; Xu, Ping

doi:10.1016/j.crma.2015.01.019

Differential geometry

The Atiyah class of a dg-vector bundle
[Classe d'Atiyah d'un fibré vectoriel différentiel gradué]

Présenté par : Charles-Michel Marle

Mehta, Rajan Amit ¹ ; Stiénon, Mathieu ² ; Xu, Ping ²

¹ Department of Mathematics and Statistics, Smith College, 44 College Lane, Northampton, MA 01063, USA
² Department of Mathematics, Penn State University, 109 McAllister Building, University Park, PA 16801, USA

Comptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 357-362.

Résumé
Abstract

Nous introduisons les notions de classe d'Atiyah et de classe de Todd d'un fibré différentiel gradué relatives à un algébroïde de Lie différentiel gradué. Nous prouvons que l'espace des champs de vecteurs sur une variété différentielle graduée admet une structure d'algèbre $L_{\infty} [1]$ ayant la dérivée de Lie par rapport au champ de vecteur cohomologique pour crochet unaire et le cocycle d'Atiyah associé à une connexion affine sans torsion pour crochet binaire.

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields $X (M)$ on a dg-manifold $M$ with homological vector field Q admits a structure of $L_{\infty} [1]$ -algebra with the Lie derivative $L_{Q}$ as unary bracket $λ_{1}$ , and the Atiyah cocycle ${At}_{M}$ corresponding to a torsion-free affine connection as binary bracket $λ_{2}$ .

Reçu le : 2014-10-23
Accepté le : 2015-01-30
Publié le : 2015-02-23

DOI : 10.1016/j.crma.2015.01.019

Affiliations des auteurs :

Mehta, Rajan Amit ¹ ; Stiénon, Mathieu ² ; Xu, Ping ²

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Mehta, Rajan Amit; Stiénon, Mathieu; Xu, Ping. The Atiyah class of a dg-vector bundle. Comptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 357-362. doi : 10.1016/j.crma.2015.01.019. http://www.numdam.org/articles/10.1016/j.crma.2015.01.019/

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^☆ Research partially supported by NSF grant DMS1406668 and NSA grant H98230-14-1-0153.