The Atiyah class of a dg-vector bundle

Rajan Amit Mehta; Mathieu Stiénon; Ping Xu

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

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

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

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     title = {The {Atiyah} class of a dg-vector bundle},
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Rajan Amit Mehta; Mathieu Stiénon; Ping Xu. The Atiyah class of a dg-vector bundle. Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 357-362. doi : 10.1016/j.crma.2015.01.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.019/

Bibliographie
Cité par

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[4] A. Kotov; T. Strobl Characteristic classes associated to Q-bundles, 2007 | arXiv

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[9] Yuri I. Manin Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 289, Springer-Verlag, Berlin, 1997 translated from the 1984 Russian original by N. Koblitz and J.R. King, with an appendix by Sergei Merkulov, MR 1632008 (99e:32001)

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Cité par Sources :

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

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