Basic curvature and the Atiyah cocycle in gauge theory

TL;DR

This paper explores the geometric structures of connections on Lie and Courant algebroids, introducing a basic curvature tensor and linking it to the Atiyah cocycle within the framework of graded geometry and higher gauge theories.

Contribution

It defines a basic curvature tensor for Courant algebroids, relates it to the homological vector field in QP2 manifolds, and connects gauge transformations to Kapranov L$_{ ext{infinity}}$ brackets.

Findings

Introduces a basic curvature tensor for Courant algebroids.

Shows the Atiyah cocycle as the binary bracket in a Kapranov L$_{ ext{infinity}}$ algebra.

Relates gauge transformations to graded geometric brackets.

Abstract

Motivated from target space covariant formulations of topological sigma models and from a graded-geometric approach to higher gauge theory, we study connections on Lie and Courant algebroids and on their description as differential graded (dg) manifolds. We revisit the notion of basic curvature for a connection on a Lie algebroid, which measures its compatibility with the Lie bracket and it appears in the BV-BRST differential of 2D gauge theories such as (twisted) Poisson and Dirac sigma models. We define a basic curvature tensor for connections on Courant algebroids and we show that in the description of a Courant algebroid as a QP2 manifold it appears naturally as part of the homological vector field together with the Gualtieri torsion of an induced generalised connection. Furthermore, we consider connections on dg manifolds and revisit the structure of gauge transformations in the graded-geometric approach to higher gauge theories from a manifestly covariant perspective. We argue that it is governed by the brackets of a Kapranov L$_{\infty}[1]$ algebra, whose binary bracket is given by the Atiyah cocycle that measures the compatibility of the connection with the homological vector field. We also revisit some aspects of the derived bracket construction and show how to extend it to connections and tensors.