The Atiyah class of a dg-vector bundle

TL;DR

This paper introduces Atiyah and Todd classes for dg-vector bundles relative to dg-Lie algebroids, and shows that the vector fields on a dg-manifold form an L-infinity algebra with a structure derived from these classes.

Contribution

It defines Atiyah and Todd classes in the dg setting and establishes an L-infinity algebra structure on vector fields of a dg-manifold.

Findings

Vector fields on a dg-manifold form an L-infinity algebra.

The Lie derivative acts as the unary bracket in this algebra.

The Atiyah cocycle from a torsion-free connection forms the binary bracket.

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 on a dg-manifold with homological vector field $Q$ admits a structure of L-infinity algebra with the Lie derivative $L_Q$ as unary bracket, and the Atiyah cocycle corresponding to a torsion-free affine connection as binary bracket.