The Atiyah class of a dg-vector bundle

Rajan Amit Mehta; Mathieu Stiénon; Ping Xu

doi:10.1016/j.crma.2015.01.019

The Atiyah class of a dg-vector bundle

Rajan Amit Mehta, Mathieu Stiénon, Ping Xu

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

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_∞[1]-algebra with the Lie derivative L_Q as unary bracket λ₁, and the Atiyah cocycle AtM corresponding to a torsion-free affine connection as binary bracket λ₂.

Original language	English (US)
Pages (from-to)	357-362
Number of pages	6
Journal	Comptes Rendus Mathematique
Volume	353
Issue number	4
DOIs	https://doi.org/10.1016/j.crma.2015.01.019
State	Published - Apr 1 2015

All Science Journal Classification (ASJC) codes

Mathematics(all)

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title = "The Atiyah class of a dg-vector bundle",

abstract = "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∞[1]-algebra with the Lie derivative LQ as unary bracket λ1, and the Atiyah cocycle AtM corresponding to a torsion-free affine connection as binary bracket λ2.",

author = "Mehta, {Rajan Amit} and Mathieu Sti{\'e}non and Ping Xu",

note = "Funding Information: Research partially supported by NSF grant DMS1406668 and NSA grant H98230-14-1-0153 . Publisher Copyright: {\textcopyright} 2015 Acad{\'e}mie des sciences. Copyright: Copyright 2015 Elsevier B.V., All rights reserved.",

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AU - Mehta, Rajan Amit

AU - Stiénon, Mathieu

AU - Xu, Ping

N1 - Funding Information: Research partially supported by NSF grant DMS1406668 and NSA grant H98230-14-1-0153 . Publisher Copyright: © 2015 Académie des sciences. Copyright: Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/4/1

Y1 - 2015/4/1

N2 - 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∞[1]-algebra with the Lie derivative LQ as unary bracket λ1, and the Atiyah cocycle AtM corresponding to a torsion-free affine connection as binary bracket λ2.

AB - 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∞[1]-algebra with the Lie derivative LQ as unary bracket λ1, and the Atiyah cocycle AtM corresponding to a torsion-free affine connection as binary bracket λ2.

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